Differential Scanning Calorimetry

Differential Scanning Calorimetry

Springer-Verlag Berlin Heidelberg GmbH

G. W. H. H6hne . W. F. Hemminger H.-J. Flammersheim

Differential Scanning Ca lori metry

2nd revised and enlarged edition

With 130 Figures and 19 Tables

" Springer

Dr. G.W.H. H6hne (Retired from Universitat Ulm) Morikeweg 30 88471 Laupheim, Germany e-mail: [email protected]

Dr. W. F. Hemminger Physikalisch-Technische Bundesanstalt Bundesallee 100 38116 Braunschweig, Germany e-mail: [email protected]

Dr. H.-J. Flammersheim Universitat Jena Institut fur Physikalische Chemie Lessingstraf5e 10 07743 Jena, Germany e-mail: [email protected]

ISBN 978-3-642-05593-5

Library of Congress Cataloging-in-Publication Data Hohne, G. (Glinther) Differential scanning calorimetry: an introduction for practitioners / G. W. H. Hohne, W. Hemminger, H.-J. Flammersheim. -- 2nd rev. and en!. ed. p. cm. Includes bibliographical references and index. ISBN 978-3-642-05593-5 ISBN 978-3-662-06710-9 (eBook) DOI 10.1007/978-3-662-06710-9

1. Calorimetry. 1. Hemminger, W., 1941-II. Flammersheim. H.-J., 1942-II1. Title. QC29I.H64 2003 536'.6--dc21 2003050472

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Preface

Differential Scanning Calorimetry (DSC) is a well established measuring method which is used on a large scale in different areas of research, development, and quality inspection and testing. Over a large temperature range, thermal effects can be quickly identified and the relevant temperature and the characteristic caloric values determined using substance quantities in the mg range. Measurement values obtained by DSC allow heat capacity, heat of transition, kinetic data, purity and glass transition to be determined. DSC curves serve to identify substances, to set up phase diagrams and to determine degrees of crystallinity.

This book provides, for the first time, an overall description of the most important DSC measuring systems and measuring programs, including the modulated temperature DSC. Furthermore many examples of typical and widely used applications of Differential Scanning Calorimetry are presented. Prerequisites for reliable measurement results, optimum evaluation of the measurement curves and estimation of the uncertainties of measurement are, however, the knowledge of the theoretical bases of DSC, a precise calibration of the calorimeter and the correct analysis of the measurement curve.

The largest part of this book deals with these basic aspects: The theory of DSC is discussed for both heat flux and power compensated instruments, and for the recently introduced modulated temperature mode of operation (TMDSC) as well. Temperature calibration and caloric calibration are described on the basis of thermodynamic principles. Desmearing of the measurement curve in different ways is presented as a method for evaluating the curves of fast transitions.

The instrumental data which are most important for the characterization of Differential Scanning Calorimeters are defined, and it is explained how they are determined experimentally. This enables every potential instrument buyer to ask the manufacturer for measured characteristic data which will allow him to compare the different instruments available. To make measurement results comparable, in addition to their traceability (via reference materials) a standardized evaluation of measurement uncertainty is indispensable. How this may be done is explained in some detail.

We are indebted to S.M. Sarge for a critical examination of the manuscript and valuable suggestions for improvement, to S. Rudtsch for contributions to the chapter on calibration and to the expression concerning uncertainties and in particular to M. J. Richardson for helping us with the translation.

Braunschweig, Jena and Ulm March 2003

G. W. H. Hahne w. F. Hemminger H.-f. Flammersheim

Contents

1

2

2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.3

2.4 2.4.1 2.4.2

3 3.1 3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4

4 4.1 4.2 4.3 4.3.1 4.3.2 4.4 4.4.1 4.4.2 4.4.3 4.4.4

Introduction ........................ .

Types of Differential Scanning Calorimeters and Modes of Operation .................... . Heat Flux DSC .................... . Heat Flux DSC with Disk-Type Measuring System . Heat Flux DSC with Turret-Type Measuring System Heat Flux DSC with Cylinder-Type Measuring System Power Compensation DSC . . . . . Function Principle . . . . . . . . . . . . . . . . . . . . Special Power Compensating DSC .......... . DSC with Combined Heat Flux and Power Compensation Measuring System .. Modes of Operation ................ . Constant Heating Rate . . . . . . . . . . . . . . . . Variable Heating Rate (Modulated Temperature) .

Theoretical Fundamentals of Differential Scanning Calorimeters . Heat Flux DSC ............. . Power Compensation DSC . . . . . . . . . Temperature-Modulated DSC (TMDSC) The Temperature-Modulated Method Influences of the Sample . . . Influences of Heat Transport Conclusions . . . . . . . . . .

Calibration of Differential Scanning Calorimeters Aspects of Quality Assurance Basic Aspects of Calibration . . . . . Temperature Calibration . . . . . . . Temperature Calibration on Heating Temperature Calibration on Cooling Caloric Calibration . . . . . . Heat Flow Rate Calibration ... Heat (Peak Area) Calibration .. Examples of Caloric Calibration Caloric Calibration in Cooling Mode

1

9 10 10 13 14 17 17 22

25 25 26 27

31 31 48 50 51 52 57 63

65 66 66 69 69 84 86 87 90 92 97

VIII

4.5 4.6 4.6.1 4.6.2 4.6.3 4.7

4.7.1 4.7.2

5 5.1 5.2 5.3 5.3.1 5.3.2

5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.5 5.5.1 5.5.2 5.5.3 5.6

6 6.1 6.1.1 6.1.2 6.1.3

6.1.4 6.1.5 6.1.6 6.2 6.3 6.3.1 6.3.2 6.3.3

6.3.4 6.3.5 6.3.6

Conclusions Regarding the Calibration of DSCs Reference Materials for DSC Calibration . . . . . Reference Materials for Temperature Calibration Reference Materials for Heat Flow Rate Calibration Reference Materials for Heat (Peak Area) Calibration Additional Calibration in Temperature-Modulated Mode of Operation ...... . Calibration of Magnitude Calibration of Phase . . .

DSC Curves and Further Evaluation . Characteristic Terms of DSC Curves . Parameters Influencing the DSC Curve Further Evaluation of DSC Curves ... Determination of the Real Sample Heat Flow Rate The Baseline and the Determination of Peak Areas (Enthalpy Differences) ............... . Desmearing of the DSC Curve .......... . Correction of the Temperature and Heat Flow Rate Indicated Subtraction of the Zeroline ............... . Calculation of the True Heat Flow Rate into the Sample Advanced Desmearing Further Calculations . . . . . . . . . . . . . . TMDSC Curves . . . . . . . . . . . . . . . . . Reversing and Non-Reversing Heat Capacity Complex Heat Capacity ......... .. Curves from Step-Scan Evaluation .... . Interpretation and Presentation of Results .

Applications of Differential Scanning Calorimetry Measurement of the Heat Capacity "Classical" Three-Step Procedure . . . . . . . . . . . The ''Absolute'' Dual Step Method ......... . General Precautions for the Minimization of Errors and their Estimation . . . . . . . . . . Procedure of Small Temperature Steps . . . . . . . . The Temperature-Modulated Method ....... . Typical Applications of Heat Capacity Measurements Determination of Heats of Reaction Kinetic Investigations ................. . Introduction and Definitions ............. . Experimental Prerequisites for a Reliable Kinetic Analysis Selection of the Measuring Conditions -Isothermal or Non-Isothermal Reaction Mode Activation of the Sample by UV Irradiation Different Strategies of Kinetic Evaluation . Selected Examples and Possible Predictions

Contents

97 98 99

101 107

107 113 113

115 116 118 119 119

121 126 128 129 129 133 140 140 141 143 145 145

147 147 148 153

154 155 159 160 162 168 168 173

175 180 183 189

Contents

6.4 6.4.1 6.4.2

6.4.3

6.4.4 6.4.5 6.5 6.5.1 6.5.2 6.5.3 6.5.4 6.6 6.7 6.8 6.8.1 6.8.2 6.8.3 6.9

7

7.1 7.2 7.3

7.3.1 7.3.2 7.4

The Glass Transition Process ............. . The Phenomenology of the Glass Transition .... . The Nature of the Glass Transition and Consequences for DSC Measurements ................ . Definition and Determination of the Glass Transition Temperature T g • • • • • • • • • • • • • • • • • •

Applications of Glass Transition Measurements . . . The Dynamic Glass Process, an Example ..... . Characterization of Substances, the Phase Behavior Applications in Biology and Food Science Applications in Pharmacy Other Applications . . . . . . . . . Porosity Measurements ..... . Determination of Phase Diagrams Safety Aspects and Characterization of High-Energetic Materials Characterization of Polymers . . . . . . . . . Effects of Origin and Thermal History . . . . . . . . . Determination of the Degree of Crystallinity ..... Advanced Characterization with the TMDSC Method Purity Determination of Chemicals ......... .

Evaluation of the Performance of a Differential Scanning Calorimeter Characterization of the Complete Instrument Characterization of the Measuring System Characterization of the Results of a Measurement: Uncertainty Determination Black Box Method . GUM Method .... Check List for DSCs

Appendix 1

Appendix 2

References .

Subject Index

IX

200 200

201

203 212 217 219 219 221 224 228 230 232 233 234 236 238 241

245 245 245

251 253 253 257

259

263

281

291

List of Symbols

A area, pre-exponential factor C heat capacity, electric capacity D diffusivity E energy H enthalpy K factor (calibration), coefficient L thermal conductance P electric power, transfer function Q heat R resistance (thermal, electric), gas constant S entropy T temperature U internal energy, voltage V volume W work, electric energy a coefficient, apparatus function c specific heat capacity d distance i electric current k proportionality factor, calibration factor, rate constant, conductivity 1 length, distance m mass n reaction order p pressure r rate of reaction, radius t time w weight fraction x mole fraction a degree of reaction f3 heating rate 6 phase angle y expansivity coefficient E emissivity A thermal conductivity v stoichiometric number tP heat flow rate p density a standard deviation

XII List of Symbols

r time constant w angular frequency X compressibility ~ extent of reaction, composition, conversion

Subscripts A activation, amplitude a amorphous bl baseline c extrapolated offset, crystalline cal calorimeter e extropolated onset el electric eq equilibrium exp experimental F furnace f final fix fixpoint fus fusion g glass h, i, k, n running numbers

initial iso isothermal I liquid lin linear lit literature M measurement point m measured mix mixing o onset p peak, constant pressure prop proportional 4> related to heat flow rate Q related to heat R reference sample r reaction Ref reference material used for calibration (e. g. Certified Reference Material) S sample s solid st steady state th thermal tot total tr true trs transition u underlying V constant volume o zero, zero line

1 Introduction

The objective of calorimetry is the measurement of heat. To measure heat means to exchange heat. The exchanged heat tends to effect a temperature change in a body that can be used as a measure of the heat exchanged, or the process of heat exchange creates a heat flow which leads to local temperature differences along its path which again serve as a measure of the .flowing heat.

As chemical reactions and many physical transitions are connected with the generation or consumption of heat, calorimetry is a universal method for investigating such processes. Measuring devices in which an exactly known amount of heat is input into a sample, or abstracted from it, and the temperature change in the sample is measured (determination of the heat capacity, for example), are also referred to as calorimeters.

Caloric measurements have been carried out since the middle of the 18th century. Although modern Differential Scanning Calorimeters (DSC) are widely used today, the "classic" calorimeters cannot be dispensed with in precision measurements and for special applications. The most important classic calorimeters will be described only briefly in Appendix 2 to give the reader a more comprehensive survey of the field of calorimetry (for a more general presentation of calorimetry, cf. Hemminger, Hohne, 1984; Oscarson, Izatt, 1992).

The topic of this book is Differential Scanning Calorimetry (DSC) using as a measuring instrument the Differential Scanning Calorimeter (DSC) available in various designs (see Chapter 2). In addition to the measurement of heat, DSCs are used to measure heat flow rates (power) and characteristic temperatures of a reaction or a transition as well. The precise measurement of heat capacities, which is of an increasing importance, has distinctly been improved with modern DSCs. The measurement of heat not only includes integral (total) heats of reaction or transition but also the determination of "partial heats" developed within a selected temperature interval. Such values are of importance for kinetic evaluations, determination of crystallinity and purity (see Chapter 6).

An accurate definition of Differential Scanning Calorimetry (DSC) is as follows:

Differential Scanning Calorimetry (DSC) means the measurement of the change of the difference in the heat flow rate to the sample and to a reference sample while they are subjected to a controlled temperature program.

2 1 Introduction

It is important to understand that DSC measures the change of a property -namely of a heat flow rate difference - which normally is released due to an alteration of the sample temperature. When there is no alteration of the sample temperature, no change of a heat flow rate difference can be measured (except for possible chemical reaction heat flow rates). That means, a distinct temperature program, or in general a "mode of operation", is always part of a DSC measurement (cf. Chapter 2). In the case of thermally activated reactions/transitions, e. g. by a controlled stepwise change from one constant temperature to another, the sample's temperature undergoes an alteration by internal processes and causes the DSC signal.

It should even be mentioned, that heat only flows if there is a temperature difference present. In other words, a non-zero heat flow rate difference implies always a temperature difference between the sample and the reference and their surroundings and a change of the heat flow rate difference implies a change of the temperature as well.

Differential Thermal Analysis (DTA)

To distinguish a heat flux DSC from an apparatus for Differential Thermal Analysis (DTA), the latter will be characterized in the following.

Differential Thermal Analysis is applied to measure the temperature difference between the sample to be investigated and a reference sample as a function of temperature (or time). This temperature difference indicates a heat exchange qualitatively. DTA is of more recent date than classic calorimetry. Its advantages as compared with conventional calorimetry are the dynamic mode of operation ("scanning") which allows reactions or processes to be investigated which can be thermally activated, and the high sensitivity to anomalies of the temperaturetime function. DTA allows characteristic temperatures to be determined and qualitative statements made on heat due to reaction. The further development of DTA has led to the construction of Differential Scanning Calorimeters (DSCs) with disk-type measuring systems.

The widely used term Differential Thermal Analysis (DTA) means a thermoanalytical method, which is more than a thermoanalytical (measurement) technique (see the following definition), because the method (analysis) includes in principle the thermoanalytical investigation procedure, i. e. the evaluation and interpretation of the measured values. An accurate definition for the thermoanalytical technique is:

Differential Thermometry (DT) means the measurement of the change of the difference in temperature between the sample and the reference sample while they are subjected to a controlled temperature program.

The accurate term for the affiliated thermo analytical method is then Differential Thermal Analysis (DTA). For a short description of the principle of DTA measuring systems in comparison with DSC, see Appendix 1.

1 Introduction 3

Calorimetry and OTA Today

Owing to the new materials and sensors used and the application of modern mechanical manufacturing (micro and nano technologies), advanced data processing by modern electronic systems, all calorimeters available today - including the so-called "classic" calorimeters - are instruments which allow precise measurements to be carried out with high sensitivity and repeatability (see Sect. 7.2), their operation being relatively simple or perhaps even automated. Caloric methods are used in many fields for quality assurance purposes (cf. series of standards ISO 9000). The standards to be applied in the field of quality assurance (e.g., the standard ISO/IEC 17025, 1999: General requirements for the competence of testing and calibration laboratories) demand a large variety of measures to ensure reliable measurement results, for example, the application of well-proven methods for the preparation of samples, reliable measurement methods, calibration methods and calibration materials traceable to standards (see also Chapter 4).

In the following, reference is made to some of the fields in which calorimeters are widely used today (see Hemminger, Hohne, 1984; Wunderlich, 1990; Oscarson, Izatt, 1992). A suggestion how to classify calorimeters is given in Appendix 2.

- With bomb calorimeters, combustion heats (gross calorific values) are measured on a large scale in industry (costs of fossil fuels) (Sunner, Mansson, 1979).

- Gas calorimeters (flow calorimeters) are used for the continuous or discontinuous measurement of the calorific value of fuel gases (e.g., natural gas), both in frontier-crossing commercial transactions (between supplier and buyer) and for the calculation of the costs for the individual consumer (Hyde, Jones, 1960; Hemminger, 1988; Sarge, 1997; Ulbig, Hoburg, 2002; Jaeschke, 2002; Dale et al., 2002; Alexandrov, 2002).

- Drop calorimeters (usually self-made) allow mean heat capacities or enthalpy differences to be quickly measured. Drop sample temperatures of up to 2000°C are realized (Chekhovskoi, 1984).

- Different types of isoperibol mixing calorimeters are used to investigate reactions between two fluids or between a fluid and a solid (reaction heats, heats of solution, heats of mixing, adsorption heats) (see, e.g., Parrish, 1986).

- Reaction calorimeters and safety calorimeters allow model tests of procedures applied in industrial chemistry to be carried out. These instruments provide valuable support in development and optimization tasks, as all test parameters (temperature, time, addition of substances etc.) are completely documented and can be varied automatically (Landau, 1996). Questions of process technology which are also important for the dimensioning of production facilities, may be well in the fore here (Regenass, 1985). Reaction heat released is connected with the degree of conversion and exploitation, for example, in biotechnology, where calorimetric studies are carried out on industrial production plants (large-scale calorimetry, mega-calorimetry, see, e.g., von Stockar, Marison, 1991 in Lamprecht et aI., 1991).

4 1 Introduction

Special calorimeters (commercially available) are also used to investigate aspects of safety technique, for example, to determine characteristic temperatures of decomposition or the kinetics of reactions (decompositions, runaway reactions) under certain boundary conditions (see, for example, Grewer, 1987; Schwanebeck, 1991; Singh, 1993; for a survey,cf. Grewer,Steinbach, 1993).

- In highly sensitive flow calorimeters (usually of the isoperibol type, i. e., with surroundings at constant temperature), the heat generation of biological systems and their change at varying conditions of life are investigated (for example, addition of pharmaceuticals to bacterial cultures). In different types of biocalorimeters, the metabolism of organisms and their change due to external influences (optical, acoustical, mechanical, thermal, chemical) are studied (see, e.g., Spink, Wadso, 1976; Lamprecht, Schaarschmidt, 1977; Beezer, 1980; Lamprecht et aI., 1991).

- Heat capacities and heats of transition are directly measured with high accuracy using adiabatic calorimeters. The characteristic data of materials determined with their aid are an indispensable basis for the calibration of the DSCs (see, e.g., Gr0nvold, 1967; Kagan, 1984; Jakobi et aI., 1993; Gr0nvold, 1993). Properly calibrated DSCs allow specific heat capacities to be measured with an accuracy of 1 to 2 % (see Sect. 6.1; cf. Richardson, 1992a).

New fields of application are constantly being opened up for the modern, highly automated DSCs. DSC is increasingly used in the field of quality assurance for many purposes: for the inspection of raw materials, as an accompanying measure in the manufacture and for the control of the finished products. The basic limitations to, and the problems of these instruments should not, however, be forgotten in view of the ease of operation and evaluation.

DSCs allow reaction heats and heats of transition, or heat flow rates and their changes at characteristic temperatures, to be quickly measured on small sample masses (milligram range; in the case of classic calorimeters: gram range), in wide temperature ranges and with an accuracy which is usually sufficiently high for the respective purpose. DSCs are applied in the following fields (among others):

- characterization of materials (in particular polymers), - comparison (relative) measurements (quality control, identification of sub-

stances or mixtures), - stability investigations, - evaluation of phase diagrams, - purity determinations, - kinetic investigations, - safety investigations, - determination of heat capacity and complex heat capacity (with TMDSC).

Now as before, instruments used for Differential Thermal Analysis (DTA) offer particular advantages when special problems are to be investigated:

- they can be used at very high temperatures (up to about 2400°C), - they are highly sensitive, - they are most flexible as regards the volume and form of the crucibles, - their reasonably-priced measuring system can be easily exchanged.

1 Introduction 5

Characteristic temperatures of transitions or reactions can be very well determined by DTA. Heats can still be estimated with an uncertainty between 20 and 50 %. DTA is applied in the following fields:

- comparison (relative) measurements (identification, control, comparison), - safety research (stability investigations, also long-time investigations), see,

e. g., Hentze, 1984; Hentze, Krien, 1986, - investigation of transitions, decompositions, reactions with gases, - evaluation of phase diagrams.

DSC and DTA are also used together with other methods of thermal analysis or other analytical techniques (simultaneous thermal analysis), most frequently in connection with thermogravimetry (TG), more rarely with Evolved Gas Thermoanalysis (EGA), Thermomicroscopy (TOA) or Thermosonimetry (TS).

The coupling of DSC or DTA with Thermogravimetry (TG) is of particular importance. In addition to information on changes in the heat flow rate (due, for example, to changes of Cp) and heats of transition, the TG signal provides information on whether volatile components are involved and which changes in mass are to be attributed to a transition.

The different methods of gas analysis used together with DSC or DTA (usually together with TG/DSC or TG/DTA systems) are of increasing importance. Via a carrier gas stream, gaseous reaction products from the calorimeter or DTA device are transported to an apparatus (e. g., IR or mass spectrometer) in which these products are analyzed. This allows a correlation between the characteristic temperature of a reaction, gravimetric and/or enthalpic information and the composition of the volatile reaction product to be made (see examples in Mathot,1994a).

It is difficult to transfer the volatile products without adulteration from the hot sample to the analysis system (condensation must be avoided). Various systems for coupling quadrupole mass spectrometers are commercially available (up to temperatures of 2400 0c). Examples of the investigation of volatile components applying methods of gas analysis can be found in Ohrbach et al., 1987; Matuschek et al., 1991; Matuschek et al., 1993; Kaisersberger, 1997.

The advantage of simultaneous measurements is that the same sample is investigated under identical conditions and that diverse information is obtained by one measurement run which is important for interpretation purposes. The following potential disadvantages of simultaneous instruments should be mentioned: lower sensitivity, higher susceptibility to failure, increased time and effort required for preparation and operation, higher instrument costs.

In Differential Scanning Calorimetry which is widely applied, there are still some actual problems of which one should be aware:

- The theoretically well-founded complete understanding of DSC is not yet totally accepted. The average user does not yet know the limits to this method and the sources of the systematic errors by which it is affected. This is in particular true for the temperature-modulated DSC, its theoretical background is still a matter of controversial discussion.

6 1 Introduction

- There are not yet any practicable and experimentally tested recommendations for temperature and heat calibration and for measurement procedures which are internationally accepted (see Chapter 4; and some examples in Della Gatta et al., 2000).

- There is no international agreement on a single set of substances for the temperature and heat calibration of DSC which have been measured with sufficient accuracy, including a metrologically sound traceability to national/international standards; instead there is a confusing variety of "certified reference materials" whose characteristic data are in part contradictory. Substances which are certified on a metrologically basis (i. e., traceable to the SI units) are offered by the German Metrology Institute (PTB) (cf. Sect. 4.6).

Due to the rapidly increasing use of DSCs in various fields of application, some negligence has gained ground - favored by the ease of operation and evaluation - which would be inconceivable in "classic" calorimetry. Remedial measures should be taken; for example, the national societies of the International Confederation for Thermal Analysis and Calorimetry (ICTAC) should offer special training courses, and precise specifications for instruments and programs should be drawn up.

The following can frequently be observed:

- A realistic estimate of the uncertainty of measurement is rarely made (cf. Sect. 7.3). The calibration capability (cf. Chapter 4), different influencing quantities (cf. Sect. 5.2) and known theoretical considerations (cf. Chapter 3) should be taken into account. In many cases, the repeatability of a DSC is, for example, simply, but wrongly, indicated as accuracy of the measured data (cf. Sect. 7.3).

- Interpretation of the DSC measurement results is often insufficient or faulty, when

- uncertainties of measurement are not taken into consideration; - systematic error sources are disregarded; - the measured curve is not "desmeared" (cf. Sect. 5.4); - the laws of thermodynamics, kinetics, are not taken into account; - uncritical confidence is placed in the evaluation programs provided by the

manufacturer; - the results are not confirmed by other measuring methods.

A great number of calorimetric methods has not been mentioned here as, in comparison with DSC, they are used only in specific fields. To name a few examples:

- low-temperature calorimetry for measuring heat capacities (see, e. g., Gmelin, 1987),

- more recent techniques of cp measurement (see, e.g., Lakshmikumar, Gopal, 1981; Maglic et aI., 1984),

- measurement of the energy of particle radiation (see, e. g., Domen, 1987), - measurements on biological systems (see, e. g., Lamprecht et al., 1991; Wadso,

1993 and Kemp, 1993; Kemp et aI., 1998; Hansen, 2000),

1 Introduction 7

- deformation calorimetry on polymers (see, e. g., Kilian, Hahne, 1983; Godovsky, Hahne, 1994),

- high-temperature calorimetry in material science (see, e.g., Bruzzone, 1985; Bros, 1989).

- (high)-pressure calorimetry (see, e. g., in Mathot, 1994 b; Hahne, 1999 b), - AC calorimetry (Sullivan, Seidel, 1968) and 3w calorimetry (Rosenthal, 1961).

Summary

During the last decades the development of Differential Scanning Calorimetry from a half-quantitative DTA method (producing "thermograms") toward a true calorimetric method (yielding quantitative caloric quantities) forged ahead. This becomes manifest from the following items which hopefully will be proved within this book.

- The theory of (heat-flux) DSCs is well-known and the function principle understood.

- Systematic (unavoidable) error sources and methods of correction are known.

- Modern data treatment and powerful computers make it possible for everyone to make the necessary evaluations without problems.

- Metrological founded calibration and measurement procedures improved the certainty of the results which nowadays are comparable with those obtained with classical calorimeters.

- The difference in quality of the results from heat-flux and power-compensated DSCs has diminished.

- Precise heat capacity measurements are possible and open the door for other well-defined thermodynamic quantities.

- Coming to terms with the problem of the thermal inertia (thermal lag) enables kinetic investigations which otherwise need much more experimental efforts.

- DSC is a fast and reliable method in a wide field of different applications.

2 Types of Differential Scanning Calorimeters and Modes of Operation

Two basic types of Differential Scanning Calorimeters (DSCs) must be distinguished:

- The heat flux DSC, - The power compensation DSC.

They differ in the design and measuring principle what will be dealt with in this chapter. The theoretical bases are presented later in Chapter 3. Common to all DSC's is a differential method of measurement which is defined as follows:

A method of measurement in which the measured quantity (the measurand) is compared with a quantity of the same kind, of known value only slightly different from the value of the measurand, and in which the difference between the two values is measured (International Vocabulary of Basic and General Terms in Metrology, 1984).

The characteristic feature of all DSC measuring systems is the twin-type design and the direct in-difference connection of the two measuring systems which are of the same kind. It is the decisive advantage of the difference principle that, in first approximation, disturbances such as temperature variations in the environment of the measuring system and the like, affect the two measuring systems in the same way and are compensated when the difference between the individual signals is formed.

Moreover, the difference signal - which is the measurement signal actually of interest - can be strongly amplified, as the high basic signal (signal of the individual measuring system) is almost compensated when the difference is formed.

An extension to form multiple measuring systems (three or four) connected back to back does not mean a fundamental change of the difference principle.

The differential signal is the essential characteristic of each Differential Scanning Calorimeter. Another characteristic - by which it is distinguished from most classic calorimeters - is the dynamic mode of operation. In other words, the DSC can be operated by applying various "modes of operation": The temperature can not only be held constant but also raised or lowered at a preset rate which might be superimposed with a temperature modulation (see below).

A characteristic common to both types of DSC is that the measured signal is proportional to a heat flow rate iP and not to a heat as is the case with most of the classic calorimeters. This allows time dependences of a transition to be observed on the basis of the iP(t) curve. This fact - directly measured heat flow rates - enables all DSCs to solve problems arising in many fields of application (see Chapter 6).

10 2 Types of Differential Scanning Calorimeters and Modes of Operation

2.1 Heat Flux OSC

The heat flux DSC belongs to the class of heat-exchanging calorimeters (for the classification, see Appendix 2). In heat flux DSCs a defined exchange of the heat to be measured with the environment takes place via a well-defined heat conduction path with given thermal resistance. The primary measurement signal is a temperature difference; it determines the intensity of the exchange and the resulting heat flow rate tP is proportional to it.

In commercial heat flux DSCs, the heat exchange path is realized in different ways, but always with the measuring system being sufficiently dominating. The most important fundamental types are:

- The disk-type measuring system, where the heat exchange takes place via a disk which serves as solid sample support. Features: Simple and easily realizable design with a high sensitivity, the sample volume is small, but the heat exchange between furnace and sample is limited which allows only medium heating and cooling rates.

- The turret-type measuring system, where the heat exchange takes place via small hollow cylinders which serve as elevated sample support. Features: More sophisticated design with high sensitivity and fast thermal response which allows large heating and cooling rates, the sample volume is small.

- The cylinder-type measuring system, where the heat exchange between the (big) cylindrical sample cavities and the furnace takes place via a path with low thermal conductivity (often a thermopile). Features: Very sensitive with a large sample volume but with a large time constant which allows only low heating rates, the sensitivity per unit volume is, however, very high.

2.1.1 Heat Flux OSC with Oisk-Type Measuring System

The characteristic feature of this measuring system is that the main heat flow from the furnace to the samples passes symmetrically through a disk of medium thermal conductivity (Fig. 2.1 a). The samples (or the sample containers) are positioned on this disk symmetrical to the center. The temperature sensors are integrated in the disk. Each temperature sensor covers more or less the area of support of the respective container (crucible, pan) so that calibration can be carried out independent of the sample position inside the container (cf. Sect. 4.3). To keep the uncertainties of measurement as small as possible, the arrangement of sample and reference sample (or of the containers) and temperature sensor in relation to one another and to the support must always be the same (center pin or the like).

Metals, quartz glass or ceramics are used as disk materials. Type (and number) of the temperature sensors (e.g., thermocouples, resistance thermometers) differ. The use of modern sensors on the basis of semi -conducting material leads to a significant increase in sensitivity.

2.1 Heat Flux DSC

3

rzzzzzzzz£l~zzzhzzZl S R

a 6T

b

5

T( f) 1---7>'---,------.

calibra ticn KIT)

time, temperature ~

11

Fig.2.1. a Heat flux DSC with disk-type measuring system. 1 disk, 2 furnace, 3 lid, 4 differential thermocouple(s), 5 programmer and controller, S crucible with sample substance, R crucible with reference sample substance, IPps heat flow rate from furnace to sample crucible, IPpR heat flow rate from furnace to reference sample crucible, <Pm measured heat flow rate, K calibration factor. b Measured heat flow rate <Pm (schematic curve) (according to Hemminger, 1994)

When the furnace is heated (in general linearly in time, more recently also in a modulated way), heat flows through the disk to the samples. When the arrangement is ideally symmetrical (with samples of the same kind), the same heat flows into sample and reference sample. The differential temperature signal6.T (normally in form of an electrical potential difference) is then zero. If this steady-state equilibrium is disturbed by a sample transition, a differential signal


is generated which is proportional to the difference between the heat flow rates to the sample and to the reference sample:

c;llFS- c;llFR - - AT with AT= Ts- TR

As neither ideal thermal symmetry of the measuring system at all operating temperatures nor thermal identity of the samples can be attained in practical application, not even outside the transition interval, there will always be a signal ATwhich depends on the temperature and the sample properties (cf. definitions of zero line in Sect. 5.1). Calculations in Chapter 3 are based on the assumption that this portion of the total signal is zero or has already been subtracted from the measurement signal proper.

In the DSC the measurement signal AT is always obtained as electric voltage. The heat flow rate c;llm (m: measured) is internally (in the software) assigned to this signal AT by factory-installed provisional calibration:

c;llm = -k'·AT

The measurement signal output by the DSC (Fig. 2.1 b) and accessible to the user is c;llm (in II W or m W). The calibration of the DSC must be checked by the user: to what extent does c;llm represent the true heat flow rate c;lltrue which is released or consumed by the sample?

Such a test (verification of calibration) can be carried out

- by measuring the steady-state heat flow rate into a sample of known heat capacity C by "charging" this heat capacity in the quasi-steady-state of scanning with a constant heating rate f3 = dT/dt:

(2.1)

- by comparing the integral over a transition peak with the expected (known) heat of transition Qr (energy balance):

(2.2)

(for definition of baseline 4\" see Sect. 5.1) (When resistance thermometers are used as temperature sensors, the ther

mometers can be used also as heaters which is advantageous in calibration procedures, see Wolf et aI., 1994).

Questions of calibration capability and calibration are dealt with in Sect. 4.4; problems of "desmearing" arising in connection with it are discussed in Sect. 5.4.

Heat flux DSCs with disk-type measuring systems are available for temperatures between -190°C and 1600°C. The maximum heating rates are about 100 K min-I. Typical time constants of the AT-sensor (empty systems, no samples) are between 3 and 10 seconds. The noise (definition see Sect. 7.2) of the measurement signal lies between 0.511W and 20 llW (it also depends on the temperature and the heating rate). The total uncertainty of the heat measurement amounts to

2.1 Heat Flux DSC 13

about 5 %, and it is to be expected that it cannot be reduced to less than 2 % even if more time and effort were spent (cf. Sect. 4.4).

Heat Flux DS( with Triple Measuring System

An extension of the disk-type measuring system described above was presented by TA Instruments (formerly Du Pont). It is a "Dual Sample DSC" with three locations on the metallic (constantan) disk to receive sample crucibles.

The three locations are equipped with temperature sensors (cf. Fig. 2.1 a for the normal DSC) which can measure the temperature differences between them. This configuration makes it possible to measure the heat capacity in a single run, using one measuring position for the sample, the second for an empty crucible, and the third for the calibration (reference) material with well known heat capacity (Jin, Wunderlich, 1993; see even Sect. 6.1). The temperature range of this Dual Sample DSC is from 125 to 1000 K with heating rates up to 100 K min-I.

The development of an adiabatic high temperature triple-cell DSC system (up to 1500 K) has been described by Takahashi, Asou, 1993.

Pressure Heat Flux DS(s

Special containers for heat flux DSCs with disk-type measuring system allow higher pressure, in general up to 7 MPa (70 bar), to be applied. This makes it possible to determine, for example, vapour pressures and heats of evaporation by means of DSC (Wiedemann, 1991; Perrenot et al., 1992, see also in Mathot, 1994b).

2.1.2 Heat Flux DSC with Turret-Type Measuring System

The characteristic feature of this recently developed measuring system is that the essential heat flow passes from the bottom of the furnace through the jacket of two thin-walled cylinders to the top of them which serve as sample and reference sample support (cf. Fig. 2.2). Because of the short time since introduction of this system there are only few experiences yet, but it seems to offer some advantages against the common disk-type DSC which are:

- a small system size combined with low mass, - a very short heat conducting path, - a strictly direct connection of the sample and reference sample to the furnace, - no interference (cross talk) between sample and reference sample events.

This leads to a smaller furnace as well and the thermal response time becomes much lower, which allows faster heating and cooling rates and higher frequencies in the temperature-modulated mode.

In addition to the temperatures of the sample and reference sample support the temperature in the middle of the bottom plate To (see Fig. 2.2) is measured in this measuring system. With these data it is possible to determine additional quantities which are essential for the performance of the DSC and correct for the influence of thermal inertia. The theoretical background of this so-called


R ~::::::::1

Fig.2.2. Heat flux DSC with turret-type measuring system (TA Instruments). 1 elevated constantan platform for sample and reference sample, 2 chromel area thermocouple, 3 constantan body, 4 chromel - constantan thermocouple, 5 silver furnace, S sample substance, R reference sample substance,~T platform temperature difference, To body (furnace) temperature

Tzero ™ DSC technology will be outlined in Chapter 3, it is much more sophisticated than the simple formulae presented in Sect. 2.1.1 for the simple disk-type DSC. Modern computer techniques offer the possibility of expensive online calculations and enable the output of the corrected heat flow rate and the calculated temperature of the sample pan even while the measurement runs.

The tiny design necessitates a very accurate production of the components with very narrow tolerances to ensure a good symmetry which is indispensable for the construction of differential calorimeters. However, modern technology enables this new type of heat flux DSC which seems to be a large step in the right direction to overcome the well-known draw-backs of heat flux DSC (see Chapter 3) in near future.

2.1.3 Heat Flux DSC with Cylinder-Type Measuring System

A block-type cylindrical furnace is provided with two cylindrical cavities, each containing a cylindrical, fixed sample container which is connected with the furnace or directly with the other container by means of several thermocouples (thermopiles), which are the characteristic features of this type of measuring system.

In the original cylinder-type measuring system (according to Calvet,1948, Fig. 2.3), the outer surfaces of each sample container are in contact with a great number of thermocouples connected in series between the container and the furnace. The thermocouple bands or wires are the dominating heat conduction path from the furnace to the samples. Heat conduction path and temperature difference sensors are identical. Both sample containers are thermally decoupled; heat exchange takes place only with parts of the massive furnace. The


4

2

2

3

'-----0 !1 T 0-------'

Fig.2.3. Heat flux DSC with cylinder-type measuring system (Calvet, 1948; thermally decoupled sample containers) (according to Hemminger, 1994). 1 containers to take up sample and reference sample, 2 thermopiles, 3 furnace (with programmable temperature controller), 4 lid, S sample substance, R reference sample substance, 1'1 T temperature difference between the containers

measurement signal proper is the temperature difference I1T of both sample containers averaged over the surfaces; it is generated by differential connection of both thermopiles:

By analogy with the disk-type DSC, the following is valid for the (steady-state) heat flow rates exchanged between furnace and sample !PFS, between furnace and reference sample !PFR, and for the measured heat (output) flow rate !Pm and the true heat flow rate !Ptrue into the sample:

!PFS - !PFR - -I1T

!Pm = -k' ·I1T

!Ptrue = K~· !Pm

An electric voltage proportional to I1T, or a heat flow rate signal internally calculated from I1T (as in the case of the DSCs with disk-type measuring system, cf. Sect. 2.1.1) is put out as a measurement signal. It must be checked in both cases to what extent the measurement signal corresponds to the (steady-state) heat flow rate actually exchanged or to what extent the integral over the measured peak corresponds to the known heat of transition (cf. Sects. 4.4 and 5.4).

In modified cylinder-type measuring systems (for example, according to Petit et al., 1961), only a small fraction of the heat usually flows from the furnace to the samples via thermocouples (Fig. 2.4). The greatest part of the heat exchange be-


4

3

5

Fig.2.4. Heat flux DSC with modified cylinder-type measuring system (thermally coupled sample containers) (according to Hemminger, 1994). 1 containers to take up sample and reference sample, 2 thermopile(s), 3 furnace (with programmable temperature controller), 4 lid, 5 support of containers, S sample substance, R reference sample substance, I1T temperature difference between the containers

tween furnace and samples takes place via the holders of the sample containers, via leads and gas layers between furnace and sample containers. Thermocouples are preferably used to directly measure the temperature difference between sample and reference sample containers. Both containers are no longer thermally decoupled, however, the temperature difference measured between them is again proportional to the differential heat flow rate from the furnace to the samples, part of which is now also exchanged between the two containers.

In all cylinder-type measuring systems, the calibration factor may depend on the position and height of the sample in the container, the reason being that the efficiency of heat conduction paths, such as holders, leads and gas layers, depends on the position of the heat source (sample) in the container. When the sample comes close to the upper edge of the container, the calibration factor can change by more than 50 %, depending on the calorimeter type (cf. Sect. 4.4).

Compared with disk-type measuring systems, the classical cylinder-type measuring system according to Calvet has the advantage of a much larger useful volume. In return, greater thermal inertia (time constants of up to 40 min) must usually be put up with. The large cavity volume allows different special containers (for example, for electric calibration, gas flow, mixture, high pressure etc.) to be used and makes intervention (electrical, mechanical, gas exchange, ... ) and direct observations (acoustical, optical) in the sample cavity possible. The great number of thermocouples (up to 1000) generates a high output signal at relatively low noise.

Heat flux DSCs with cylinder-type measuring systems with sample volumes of about 10 mL are available for a temperature range between -196°C and 1500 0c. Due to the relatively great time constants of the measuring systems, the

2.2 Power Compensation DSC 17

maximum heating rates - depending on the cavity volume - lie at about 30 K min-I. Special instruments with large volumes (up to 100 cm3) are used, for example, in biology for the investigation of small animals at constant temperature (cf. Moratzky et al., 1993; Schmolz et al., 1993).

High temperature systems can also be used in isoperibol (isoperibol: surroundings at constant temperature) mode of operation, as a drop calorimeter (cf. Appendix 2) to measure, for example, enthalpies of mixture up to 1500°C (see, e.g., Fan et al., 1993).

Other systems with smaller sample volume (around 850 }lL) and semiconductor temperature sensors are used for highly sensitive measurements (detection limit around 50 n W). They are applicable in a temperature range from -45 to 120°C and can be equipped with special vessels according to various investigations (batch, ampoule, fluid-circulation, mixing, electric calibration by means of a "Joule-heater", etc.). These DSCs can usually be heated maximally with 1 K min-I, isothermal operation is possible as well (noise ca. 0.2 }lW).

For heat flux DSCs with cylinder-type measuring systems according to Calvet (decoupled sample containers), a rather simple theory has been developed which quantitatively describes the functional correlation between the instantaneous measurement signal IPm and the original event IPr in the calorimeter (cf. Sect. 3.1). According to this theory, "desmearing" (cf. Sect. 5.4) of the measured curve - to obtain the IPr(t) function - can be performed rather easily, even online, for example, by means of a commercially available electronic devices or via proper software on the computer.

For disk- and turret-type heat flux DSCs the desmearing procedure is not that simple but can be performed as well. For the latter DSC type the advanced Tzero™ technology makes it possible to do the necessary calculations during the measurement run (see Sects. 2.1.2 and 5.4.3).

2.2 Power Compensation DSC

The power compensation DSC belongs to the class of heat-compensating calorimeters (see Appendix 2). The heat to be measured is (almost totally) compensated with electric energy, by increasing or decreasing an adjustable Joule's heat.

2.2.1 Function Principle

The commercial power compensation DSCI most frequently used is an instrument with an isoperibol operation. The measuring system (Fig. 2.5) consists of two identical microfurnaces which are mounted inside a thermostated aluminum block. The furnaces are made of a platinum-iridium alloy, each of which contains a temperature sensor (platinum resistance thermometer) and a heating

1 The description of a power compensation DSC is based on the widely used DSC of PerkinElmer Instruments.


Fig.2.S. Power compensation DSC (Perkin-Elmer Instruments). Set-up of the measuring system (according to Hemminger, 1994). S sample measuring system with sample crucible, microfurnace and lid, R reference sample system (analogous to S), 1 heating wire, 2 resistance thermometer. Both measuring systems -separated from each other - are positioned in a surrounding (block) at constant temperature

,--___ @ Calibration ~ cf> m

tJp tJp

T (I) )----'-------+1 R e (0 r d er <Pm ( f)

Fig.2.6. Power compensation DSC (Perkin-Elmer Instruments). Block diagram showing the function principle (according to Hemminger, 1994). Ts temperature of the sample furnace, TR temperature of the reference sample furnace, IlT = Ts - TR, Pav average heating power, IlP compensation heating power, cf>m measured heat flow rate (measurement signal)


resistor (made of platinum wire). The micro furnace is about 9 mm in diameter, approx. 6 mm in height and has a mass of approx. 2 g. The time constant of the DSC without sample pans is 1.5 s, and the isothermal noise is about 2 II W. The maximum heating power of a microfurnace is about 15 W, the maximum heating rate is 500 K min -I. Maximum cooling rate can reach 200 K min-I, it depends on the temperature difference between block and sample (Perkin-Elmer Pyris Diamond DSC). The measuring range extends from -175°C (block cooled with liquid nitrogen) to 725°C,

During heating-up, the same heating power is supplied to both microfurnaces via a control circuit (Fig. 2.6) in order to change their mean temperature in accordance with the preset heating rate (see Watson et aI., 19642). If there is ideal thermal symmetry, the temperature of both micro furnaces is always the same. When an asymmetry occurs, for example, as a result of a sample reaction, a temperature difference results between the microfurnace accommodating the sample and the micro furnace containing the reference sample. The temperature difference is both the measurement signal and the input signal of a second control circuit. This second circuit compensates most of the reaction heat flow rate by proportional control by increasing or decreasing an additional heating power of the sample furnace. The compensating heating power I1P is proportional to the remaining temperature difference I1T (because of the proportional controller). The time integral over the compensating heating power is proportional to the heat Qr which was consumed or released in the sample (Fig. 2.7).

Again, a heat flow rate <Pm is assigned to the real measurement signal I1T as a result of a factory-installed calibration, and fed in. The relations between I1T, <Pm and the compensating heating power I1P are as follows:

I1P=-k l ·I1T

<Pm=-k2 ·I1T

The factor kl is a factory-set fIxed quantity of the proportional controller, k2 can be changed at the instrument with the aid of a potentiometer or it is adjusted via the software (calibration). The factor k2 is almost independent of measurement parameters (e.g., temperature), as - via kl - a given compensating heating power always corresponds to a given 11 T, independent of the temperature; k2 can therefore in principle be determined by one calibration measurement at one temperature only.

As regards the formal aspects (cf. Sect. 2.1 for the heat flux DSC), the output signal is also given as a heat flow rate signal <Pm (e. g. in m W), and the relation between <Pm and the true heat flow rate exchanged with the sample, <Ptrue = K<[>· <Pm, must also be determined by caloric calibration (see Sect. 4.4).

At higher temperatures of the measuring system, the heat flows exchanged with the (isoperibol) surroundings (conduction, radiation, convection) are relatively

2 In the basic paper by Watson et al. the term Differential Scanning Calorimeter (DSC) is coined, but the title of the paper is: A Differential Scanning Calorimeter for Quantitative Differential Thermal Analysis, i. e., the inventors of this DSC recognized clearly the systematic relationship between this type of DSC and classic DTA.


time--l>

t

t tJp

t tJT

t

Fig.2.7. Power compensation DSC. Diagrammatic view of the signals in question (according to Hemminger, 1994). tPr heat flow rate released in the sample (exothermic, therefore negative), qr heat released in the sample, ~ T* temperature increase in the sample furnace which would build up due to the exothermic effect unless it was compensated (as in heat flux DSC), ~p compensation heating power (negative, to compensate the exothermic effect), ~T temperature increase actually occurring in the sample furnace (corresponds to the residual deviation from the theoretical value, which cannot be completely compensated by the proportional control), tPm output signal (proportional to the (negative) amplified ~T signal)


large (several Watt) compared with the quantity to be measured which is three orders of magnitude lower. High requirements must, therefore, be met as far as the uniformity of the heat exchange between the two microfurnaces and the surroundings is concerned in order to keep a good symmetry and the uncertainties of measurement small. Moreover, the shares of the various heat exchange mechanisms and their respective amounts must depend only on the temperature, and strict repeatability must be ensured. This is a rather hard demand if one considers the huge heat exchange of the furnaces with the thermostated surroundings via radiation and convection. Consequently the conditions for this heat exchange should be kept as constant as possible. Conclusion: The microfurnaces must be covered with lids of the same kind in order to "cover up" possible inhomogeneity of sample and reference sample; the thermophysical properties of the crucibles and lids must depend only on the temperature and never on sample properties.

As a result an asymmetry of heat exchange with the surroundings of, say, 10-4

causes a differential signal of about 1 mW, the zero line of the power compensated DSC is never zero and often strongly curved. The manufacturer made allowance for these problems and took care of an electronically compensation of thermal asymmetries of the measuring system, which become apparent as a curvature of the zero line. In this way, it is also possible to "straighten" the zero line and/or incline it as desired, and with it the measured curve outside the peaks.

Compared with heat flux DSCs, the power compensation DSC offers the following advantages:

- The short heat conduction path between samples and heater and the relatively small masses of the microfurnaces allow an almost instantaneous response to a sample reaction. Due to the small time constant, desmearing is required only in a few cases (cf. Sect. 5.4).

- Reaction heat flow rates are rapidly and to a large extent compensated by electrical heating power. As a result, only small temperature differences I1T remain between the micro furnaces of sample and reference sample (approx. 1110 of those occurring in heat flux DSCs using the same sample and reference sample). This means that the calibration factor Kcp is practically independent of the intensity and kinetics of the sample reaction.

- The total compensating energy (J I1Pdt) is equal to the reaction heat or heat of transition.

- The temperature dependence of the control circuit properties (above all of the temperature sensors and the heating elements) is known and strictly repeatable. It can be taken into account with the aid of special electronics or by the software. A single caloric calibration is then - in principle - sufficient to determine the correlation between CPm (or CPr) and I1P (cf. Sect. 3.2).

The user must, however, keep in mind that the temperature difference between the two micro furnaces is not compensated totally in commercial power compensation DSCs. During the peak there is still a temperature difference 11 T proportional to the reaction heat flow rate. That is to say, the power compensation DSC can be considered a kind of DTA instrument with a 11 T as the measurement signal, but with a I1T arising from a specific thermal event which is much smaller than that developed in a heat flux DSC measuring system.


In conclusion it may be said that all the attributes of heat flux DSC systems which depend on this temperature difference can also be found in the "real" power compensation DSC, but to a lesser degree. The calibration factor in particular is not a constant figure but depends in principle on temperature, heat flow rate, heating rate and peak area (cf. Sects. 3.2, 4.4). Though these effects are not very pronounced, they should be carefully tested and a thorough verification (with the relevant parameters varied) should be performed if the demands on the accuracy of the measurements are high (H6hne, G16ggler, 1989).

2.2.2 Special Power Compensating DSC

Because of the importance of this type of calorimeters several modifications have been done for essential special purposes:

High Pressure DSC

High pressure measurements are of great importance from the thermodynamic point of view. The change of pressure enables a better insight into the thermodynamic behavior of materials. High pressure is often used during production and processing of materials and the change of properties with pressure is of great interest (H6hne, 1999b). While high pressure DTA already exists for a longer time in several laboratories worldwide (see, e.g., Szabo et aI., 1969) at pressures up to 1 GPa, the number of high pressure DSCs is very limited (Schmidt et aI., 1994). To our knowledge there exists only one power compensated DSC (based on the Perkin Elmer DSC-7), which works up to a pressure of 500 MPa. Design, specifications as well as applications have been described by Blankenhorn, H6hne, 1991 and H6hne, Blankenhorn, 1994.

Photo-DSC

Another modification concerns the possibility to follow light induced reactions in a DSC.

Slight changes in the design of a DSC make it possible to irradiate the sample with light. A DSC modified in this way is referred to as Photo-DSC. Irradiation of the sample with light, sufficiently rich in energy, leads to reaction in the Photo-DSC. The heat generated during these reactions is recorded. First applications of this type of DSC were described by Wight, Hicks, 1978; Tryson, Shultz, 1979; Flammersheim, 1981. Since 1987, the DPA-7 of Perkin-Elmer is commercially available as a supplementary device of the DSC-7. (Heat flux DSCs are also available with suitable additional devices.) The principle is shown in Fig. 2.8.

Sample and reference substance are contained in the calorimeter at constant temperature. The sample crucibles are either open or covered by quartz disks. The lid of the microfurnace is provided with openings sealed with quartz disks. If light high in energy is incident on the sample, it is either absorbed directly or - as is usually the case - with the aid of a photo-initiator. A reaction usually takes


Fig. 2.S. Light transfer in a Photo-DSC, schematic representation

LU W

light source

optics

quartz cuvette with water as IR-absorber

monochroma tor neutral density fil ters

calorimeter

23

place only as long as activation energy is supplied by irradiation. High-pressure mercury vapor lamps with a large number of spectral lines are frequently used as a light source; however, hydrogen, deuterium or xenon lamps are also suitable. The undesired infrared portion of the spectrum can be quantitatively absorbed by quartz cells filled with water. The portion of the spectrum which is of interest is selected using monochromatic filters and the desired intensity is adjusted by means of neutral density filters or metal sieves with different mesh width. It is a particular advantage that the DSC is also suitable for the direct measurement of the incoming radiation flow (light intensity). For this purpose, a graphite disk of known emissivity is substituted for the sample. This disk absorbs the greater portion of the incident light and transfers it to the calorimeter in the form of a heat flow. For most applications, it is of secondary importance whether sample and reference cells are irradiated or the sample alone. For precise measurements electronic stabilization of the light intensity of the lamp is important. Otherwise the total noise of the observed curve will be substantially higher than the noise of the DSC measuring system.

Light-activated reactions under conditions closely related to practice are usually very fast. The greatest part of the reaction takes place within a few seconds (cf. Figs. 6.13 and 6.14). There are two consequences:

- There is a considerable change in the measurement signal in periods comparable with that of the DSC time constant. The measured and the true heat flow rates differ substantially. Desmearing is necessary prior to every evaluation in which the time variable is involved, for example, when calculating the conversion versus time curve (see. Sect. 5.4).

- Heat flow rates are so large (see Fig. 6.13) that the sample temperature deviates considerably from the temperature of the microfurnace, even in powercompensated DSCs. If a thermal resistance of 40 K W-1 between sample and temperature sensor is assumed, temperature differences of 5 to 10K would result even for samples with very good thermal conductivity. For organic samples, these differences are even greater. This means that the measurements are no longer isothermal, not even as an approximation. Kinetic analysis must allow for this.


Another Photo-DSC is described by Saruyama, 1999. In this case the light is used to transfer energy to the sample by absorption. A commercial DSC (Rigaku DSC 8230) has been modified to enable the light to reach the lids of the crucibles, which were covered with carbon black for better absorption. The light intensity was modulated in time by two polarizers, one fixed and one rotating with a certain frequency. This way a periodical changing (sinusoidal) heat flow rate is added to the sample which causes a temperature modulation. The light-modulated DSC is used for similar investigations as the temperaturemodulated DSC (see Sects. 3.3, 5.5) but allows a wider range of frequencies up to 1 Hz.

Special DSC for Fluids

For precise biochemical investigations or for the investigation of reactions in fluids, commercial power compensation DSCs are often used that are operated adiabatically or quasi-adiabatically (Privalov, Plotnikov, 1989). In a former version of such a calorimeter the sample cell is heated with constant power and the (heated) jacket follows the cell temperature as closely as possible (adiabatic or quasi-adiabatic conditions). Consequences of this operation mode are that the heating rate is not constant but depends on the behavior of the sample (temperature change due to release of heat of reaction, change in heat capacity) and that cooling requires a jacket temperature that is significantly below the cell temperature to remove heat from the samples, which leads to non-adiabatic conditions.

A power compensation calorimeter for liquids (MicroCal LLC, see Plotnikov et al., 1997) uses twin coin-shaped measurement cells (non-removable, volume 0.5 mL) in an adiabatically (for heating) operated jacket. The cells are connected to the outside via capillary tubes. Thermoelectric sensors measure the temperature difference between the cells (power compensation control loop ) and between reference cell and jacket (adiabatic control loop ). The time constant is approx.5 s. The temperature range is from -10 to 130°C (-45 to 150°C for an extended temperature range version) with scan rates up to 1.5 K min-I. Heating (adiabatic) and cooling (nonadiabatic) are performed by means of Peltier elements. To avoid boiling of the liquids, the system can be pressurized up to 3 bar. The repeatibility of the baseline is stated to be around ± 0.2 p.W, the short time noise (RMS, s. Sect. 7.2) around ± 70 nW.

New calorimeters offered by Calorimetry Sciences Corp. (CSC) use a temperature-leading thermostat which is heated or cooled by means of Peltier elements according to the preset scanning rate. The cells are placed non-removable inside a jacket and are thermally connected to the thermostat via capillary tubes which also serve to fill, to clean etc. the cells. The temperature difference sensors between sample cell and reference sample cell are semiconductor (bismuth telluride) thermopiles.

The calorimeter can be operated between -10 °C to 130°C (extended temperature range version to 160°C) with scan rates (heating or cooling) up to 2 K min-I and with a pressure up to 6 bar. Small sample masses of 10 p.g or 50 p.g are investigated in capillary type or cylindrical cells.

2.4 Modes of Operation 25

Power compensation heaters serve to minimize the temperature difference between sample and reference sample cells. The time constant is approx. 5 s, the baseline noise is stated to be ~ ± 15 nW and the repeatibility of the baseline -even after refIlling the cells - to be within 0.5 ]l W. These calorimeters have successfully been used in biochemistry to investigate proteines, in particular their phase behavior in solution.

2.3 DSC with Combined Heat Flux and Power Compensation Measuring System

The "pure" power compensation DSC measuring system uses two thermally decoupled microfurnaces with an integrated temperature sensor and heater in each furnace (cf. Sect. 2.2). So called "hybrid systems" show a pair of sensorheater combinations on a disk. The temperature difference between the positions of the sample and the reference sample is measured by the temperature sensors and (almost) compensated for by means of controlling the integrated heaters. A good thermal coupling between temperature sensor and related heater is a prerequisite for short time constants and a negligible cross heat flow between the two sensor-heater elements. This construction combines advantages of both the pure heat flux and the pure power compensation measuring systems: stable baseline, short time constants, high resolution, low noise and small temperature differences between environment (furnace) and measuring system.

Another version with a combined heat flux and power compensation system uses the disk with integrated thermopile and compensation heaters as bottom plate for two distinct cylindrical chambers which are machined in a small cylindrical metal block. The crucibles for sample and reference sample are put into these chambers and are positioned on the detector disk (temperature range -150 to 550°C, detection limit approx. 0.5 ]lW, crucibles with 30 or 100 ]lL).

In principle the addition of heaters plus a power compensation control loop is not restricted to disk-type DSCs. Whenever a heater is part of the measuring system - e. g., a calibration heater inside a cylinder type system - this heater may be used for compensation purposes.

2.4 Modes of Operation

DSCs are generally operated by a controlled program which changes the temperature in time. We distinguish between those modes of operation which leave the heating rate constant (i.e., the classical DSC operation mode) and those with variable (periodical or non-periodical) change of the heating rate.


2.4.1 Constant Heating Rate

In this mode of operation the controlled program follows the time law:

(2.3)

where To is the starting temperature of the run and Po the heating or cooling (negative) rate. In other words the temperature changes linearly in time. With common DSCs heating rates up to 150 K min-1 (power compensated DSC: 500 K min-I) can be reached. The real cooling rates are lower (< 150 K min-I), because the transport of heat out of the sample needs time (a compensation is not possible).

Isothermal Mode

In this mode the heating rate [Po in Eq. (2.3)] is zero, i.e., the temperature To is kept constant. Consequently, if no transitions or reactions take place in the sample, there is no heat exchanged with the sample and the heat flow rate should read zero. This is in practice not the case, as all DSCs are not perfectly symmetric and the heat exchange of the sample and reference sample with the surroundings is somewhat different. This results in a non-zero heat flow rate even in the isothermal case. One reason to perform isothermal measurements in a DSC is to measure and check this asymmetry. Isotherms are used for calculation of heat flow rate corrections due to asymmetries, they are included before and after scanning sections in the case of precise heat capacity measurements, for example (see Sect. 6.1). Another reason to perform isothermal measurements is to determine the latent heat of reactions or transitions taking place in the sample at a certain temperature (examples can be found in Chapter 6). Of course, the baseline (the isotherm obtained without processes of the sample) must be subtracted from the measured curve to get the true reaction heat flow rate.

Scanning Mode

In this mode the temperature changes linearly in time. Every material needs a certain heat to be warmed up and the heat flow rate is proportional to the heating rate:

dT tPcp= Cp'

dt (2.4)

with Cp as the proportionality factor. In a DSC the differential heat flow rate depends on the differential heat capacity and heating rate. Generally the measured heat flow rate in scanning mode is never zero and is made up of three parts:

(2.5)

The first term on right hand side is caused by the (unavoidable) asymmetry of the DSC, the second term is caused by the difference in heat capacity of sample


and reference sample and the third term is the heat flow contribution from a reaction or transition (latent heat) occurring in the sample. The first two parts define the "baseline" and the third part the "peak" of the measured curve. In case of total symmetry and absence of processes, only the sample heat capacity causes a signal (and can be determined from it). Scanning is the mostly used mode of operation with DSCs.

High-Speed Scanning Mode

Recently Pijpers et al., 2002, published a method to perform DSC measurements at very high, controlled (including constant) cooling and heating rates of hundreds of degrees per minute with sub-milligram amounts of substance. To overcome the thermal inertia (thermal lag) of the heat transfer to the sample special measures have to be taken: the mass of the sample has to be very low and the thermal contact between DSC and sample as good as possible. This is needed to reduce the time constant of the respective RC-element (cf. Sect. 3.3.3) which otherwise can be much larger than the time constant of the DSC and the controllers causing a huge smearing (cf. Sect. 5.4) of the measured heat flow rate curve.

The method works, however, well and allows, for example, one to investigate melting kinetics and to override recrystallization processes which are unavoidable when testing polymers and often indistinguishably superimposed at lower heating rates.

2.4.2 Variable Heating Rate (Modulated Temperature)

In this case a certain modulation term is added to the linear part of the temperature-time function. The simplest and most used modulation type is the periodic (harmonic, i. e., sinusoidal) one and Eq. (2.3) reads:

T(t) = To + Po· t + TA • sin (wt) (2.6)

with TA the amplitude and w the (angular) frequency of the modulation. Together with Po, the "underlying" heating rate, there are three parameters which can be chosen freely within certain limits. This influences the heating rate, which follows from Eq. (2.6) as:

dT - = Po + TA • W· cos (wt) dt

(2.7)

Depending on the relation of Po to TAW, we have to distinguish four cases (Fig. 2.9).

Quasi-Isothermal Mode

In this mode the underlying heating rate is zero and the temperature varies around a constant temperature (see Fig. 2.9 curve a). As the amplitude of the modulation is low (0.01 to 0.5 K normally) the temperature is almost constant


Fig.2.9. Temperature-mod- S ulated modes of operation:

I (a) quasi-isothermal, 4 (b) heating-cooling, (c) heat-ing-iso, (d) heating-only

3 ClJ '--::J

2 ~

d '--ClJ Cl. E ClJ

0

-1 0 10

time • (quasi-isothermal). The heating/cooling rate varies between ± TAW. The advantage of using this mode is the possibility to determine heat capacities even in the isothermal case (see Sect. 6.1.5) what otherwise is not possible.

Heating-Cooling Mode

In this mode the amplitude is so large, that the heating rate (Eq. 2.7) changes its sign periodically (see Fig. 2.9 curve b),i.e., the sample is periodically heated and cooled. This is the case if the product TAW is larger than the underlying heating rate Po.

The advantage of this mode is a rather large heating rate and therefore good signal-to-noise ratio which is in particular useful if we have small sample masses. The disadvantage on the other hand is that the sample is heated and cooled during the run, this may cause problems in some cases, e. g., for polymers where different processes (with different time constants) could occur within the heating and cooling period. This would complicate the evaluation of the processes.

Heating-Only Mode

In this mode the amplitude is so small that the heating rate (Eq. 2.7) is always positive (see Fig. 2.9 curve d), i.e. the sample is only heated and never cooled during a run. This is the case if the product TAW is smaller than the underlying heating rate Po.

The advantage of this mode is that all processes only occurring during cooling of the sample are suppressed and the remaining processes are easier to evaluate. The disadvantage is clear from the rather low amplitude which gives a bad signal-to-noise ratio for all quantities derived from it.

Heating-Iso mode

This is the limiting case between the two modes above where TAW = Po: the heating rate varies periodically between TAW + Po and zero. This is the mode


Fig.2.1O. Special tempera- 5 ture-modulated modes

t of operation: (a) step-scan, 4 (b) sawtooth

<lJ 3 L..

:::I -0 L.. 2 <lJ CI.. E (lJ

10 time ..

with maximum of amplitude (and signal-to-noise ratio) but without cooling the sample (see Fig. 2.9 curve c).

Another advantage is that, at heating rate zero, the (even periodical changing) heat flow rate has a minimum with no contribution from the sample heat capacity [which follows from Eq. (2.4) and (2.7)]. This way the lower contour of the fluctuating heat flow rate curve can be used to detect endo- or exothermic processes occurring in the sample. After subtracting of cPo there is only cPr remaining in the moment of minimum (i.e. zero) heat flow rate [ef. Eq. (2.5)].

Sawtooth Mode

In this mode of operation different linear scanning rates change periodically (Fig. 2.10 curve b). Again we can distinguish between heating-cooling, heating only, and heating-iso modes of operation, depending on the choice of the respective rates of temperature change.

Every DSC can be operated in this mode, as programming of several different linear scanning ramps in one run normally is possible without special additional electronics or software. This is one advantage of this mode of operation. Another advantage follows from the fact that every (non-sinusoidal) periodical function can be expanded as a Fourier series (see textbooks of mathematics). In the case of the sawtooth mode, the periodical function, superimposed on the linear temperature change, reads:

_ (2n 1 3 . 2n 1 5 . 2n ) T(t) = c sin--2"sin--+2"sin--- ...

tp 3 tp 5 tp

with c as a proportionality factor containing the amplitude and period tp of the periodic modulation. This function contains not only the basic frequency f = lItp but also (odd) higher harmonics. The same is true for the measured heat flow rate. By Fourier analysis the modulated heat flow rate at different frequencies can be determined from one measurement.


A disadvantage of this temperature modulation mode is the discontinuous (step-like) change of the heat flow rate, which may cause problems with the furnace controller (overshoot) and produce additional (non-linear) heat flow rate fluctuations.

Step-Scan Mode

The step-scan mode, recently introduced by Perkin-Elmer Instruments for power-compensated DSCs and by Mettler-Toledo for heat flux DSCs, is a special mode of operation: After changing the sample temperature, say, by 0.5 K at a high heating rate (step-like, see Fig. 2.10 curve a) the heat flow rate is allowed to equilibrate again. The respective response is evaluated. This can be done many times in a periodical or non-periodical manner. In the periodical case the stepscan is a special sawtooth mode, which is similar to the heating-iso mode in the sinusoidal case, but with an asymmetric triangle shaped form of the temperature-time curve. This results in more than only the odd harmonics which one can separate from the modulated part of the heat flow rate signal. Another advantage is that a series of temperature steps is equivalent to a series of heating rate pulses. The heat flow rate response on a pulse-like event enables one to determine the apparatus (Green's) function of the DSC with and without a sample at every moment of the run. This offers an easy possibility to come to correct ("desmeared", cf. Sect. 5.4.3) reaction heat flow rates as well as to measure the complex heat capacity Cp(w) from one run (see Sects. 4.7, 5.5.3).

3 Theoretical Fundamentals of Differential Scanning Calorimeters

In all DSCs, a temperature difference AT - given as a voltage - is the original measurement signal. In almost all instruments a heat flow rate tl>m (differential heat flow rate) is internally assigned to AT ( cf. Chapt. 2). Independent of whether the user obtains AT or tl>m from the respective DSC, knowledge of the functional relation between the measured signal (AT, tl>m) and the quantity searched for (the real heat flow rate tl>r consumed/produced by the sample) is important for

- the time-related assignment of tl>r to AT or tl>m (investigation into the kinetics of a reaction),

- the determination of partial heats of reaction, - the evaluation and assessment of the influences of operating parameters and

properties of the measuring system with regard to this relation, - the estimate of the overall uncertainty of measurement.

The relation between tl>r and AT or tl>m can be derived in varying degrees of approximation to real DSCs. Analytical solutions of the differential equations are possible only for simple boundary and initial conditions and for quasisteady states. Numerical procedures and solutions can approximate the actual conditions more exactly, however, without the clarity of the functional relations given by an analytical solution. Basic considerations in this field are given by Gray, 1968.

To ensure better differentiation from tl>n in the following section, AT - instead of tl>m - is assumed to be the measurement signal, i. e., we search for the relation tl>r(AT). These two quantities, AT and tl>m, are under normal circumstances strictly proportional, with the exception of the opposite sign.

3.1 Heat Flux DSC

Three steps of an analytical description of the functional principle of a heat flux DSC will be presented in the following. The results of a numerical calculation will be discussed afterwards.

32 3 Theoretical Fundamentals of Differential Scanning Calorimeters

Fig.3.1. Heat flux DSC / (disk-type), model for zeroth / tJ!

l approximation (linear model). S sample, R reference Tr S T, sample, F furnace, A cross section of the heat conductor between furnace and Sand R,M distance between tem-perature measurement point and furnace F F

Ts T. / /

Zeroth Approximation

First, the heat flux DSC is represented by a very simple linear model (Fig. 3.1). The following simplifications have been made in this approximation:

- steady-state conditions (i.e., constant heat flow rates), - only one thermal resistance, the apparent resistance between furnace and

sample, is taken into account with no interaction (cross-talk) between sample and reference sample,

- only the heat capacities of the sample and reference sample (Cs, CR) are taken into account, other heat capacities are neglected,

- sample temperature and measured temperature are assumed equal, - no heat exchange with the surroundings (i. e., no heat leak).

Figure 3.2 shows the equivalent electric circuit diagram for the zeroth approximation. This diagram serves to better understand the interrelations. From the physical point of view, electric charge transport and heat transport are equivalent processes, and many people find it easier to read electric circuit diagrams than to visualize heat flows in real equipment.

Fig. 3.2. Equivalent electric circuit for the linear model of the heat flux DSC (see Fig. 3.1). C capacitance, R resistance, i current, U voltage, Sub-scripts: S sample, R refer- U, ence, F furnace

'fS R,s

fjU

if. R'R

[. [s

/77J777 T T O----------------T---------~----~


The Biot-Fourier equation of the (steady-state) conduction of heat, together with the formulation in absolute values, reads as follows:

~ I~I - = - A . grad T or - = - A ·1 grad T 1 A A

The amount of the heat flux tP/A is proportional to the gradient of the temperature; the thermal conductivity .\ is the proportionality factor.

In the one-dimensional model referred to above, this equation is reduced as follows for the left-hand and right-hand subsystem shown in Fig.3.l (with Tp> Ts, TR in scanning mode):

In the case of thermal symmetry, tPps = tPPR is valid, with Ts = TR. If a constant (exothermic) heat flow rate (tPr < 0) is produced in the sample,

Ts increases by t:.Ts, the temperature difference Tp - Ts and thus the heat flow rate tPps decrease. When the steady-state (i.e., uniform heating rate) is reached again, for reasons of balance, the change of tPps, (t:. tPps), must be equal to tPr :

Nothing has changed on the side of the reference sample, hence:

Consequently

(Here, Ts > TR,hence tPr is negative: exothermic effect; t:.T = t:.TSR: measurement signal)

In this simple steady-state model, K is given completely by the properties of the heat conduction path between the furnace and the samples. This means that in the steady state, there is direct proportionality between the measurand tPr and the measurement signal t:. T. The conditions of constant heat consumption can be achieved when in scanning operation sample and reference sample have different, temperature-independent "heat capacities". A greater amount of heat will always flow into the sample whose heat capacity is higher, in order that the steady-state heating rate is maintained. With Cs > CR the following is then valid for the difference between the heat flow rates to sample and reference sample:


There is no steady state during sample transitions or reactions; the above approximation does not apply in these cases. Furthermore, Cs and CR (and thus I:! cPSR) change with temperature, but these changes are in many cases rather slow and do not affect the steady-state condition very much.

When there is such a quasi-steady state, the following is valid in approximation:

(3.1)

This relation describes the shape of the baseline and is the basic equation to determine the heat capacity Cs. For the empty reference crucible (CR = 0)

I:!T Cs = -K'·-

P is valid.

When the heat capacity CR is known (i. e., if CR = Cref "# 0, using a reference material), we get

I:!T Cs=CR-K' .-

P In practice, first a zero line I:!To will be recorded with both crucibles empty (to check the asymmetry of the apparatus) which is subtracted from the measured curves (for the measurement of heat capacities, see Sect. 6.1).

First Approximation

In the first approximation, non-steady-state processes in the sample are also permitted which manifest themselves as "peaks" of the measured curve. This means that I:!T is not constant in time. For the rest, the simplified arrangement as for the zeroth approximation is used here.

cPps is the heat flow rate from the furnace to the sample, cPr(t) the time dependent heat flow rate produced inside the sample (reaction, transition). The following balance for the heat flow rates is then valid for the sample of heat capacity Cs:

dTs Cs - = cPps - cPr

dt

(exothermic: cPr negative, endothermic: cPr positive)

Withl:!T= Ts - TR ,

dTR dl:!T Cs-- + Cs -- = cPps - cPr

dt dt

results.


Accordingly, the following holds for the reference sample (cPr = 0 by definition):

dTR CR--= cPFR

dt

When the difference between the two balance equations is calculated, the following is obtained:

The following is valid for the heat flow rates cPFS and cPFR :

and

where RFS and RpR are global heat resistances between the furnace and the samples. In the case of thermal symmetry, R FS = RFR = R, thus:

flT dTR dfiT cPr = --- (Cs - CR)-- Cs -

R dt dt (3.2)

This equation links the reaction heat flow rate cPr searched with the measured signal fl T. The second term takes the asymmetry of the measuring system into account as regards heat capacities of sample and reference sample. The third term considers the contribution of the thermal inertia of the system when a measured signal flT(t) appears. In analogy to the charging or discharging of a capacitor of capacity C, a time constant T can similarly be defined for the heat flow rates:

T=Cs·R

When fl T is changed, R is the effective thermal resistance to the "charging or discharging" of the "capacity" Cs. With this resistance and with dTR/dt = /3, the heating rate; as the reference sample is always in a steady-state heating mode, the following results from Eq. (3.2):

flT(t) T dfl T(t) cPr (t) = - -R - - (Cs - CR) • /3 - R . ~ (3.3)

The measured signal flT is not (I) proportional to the heat flow rate cPr at a given moment but delayed with time and thus also distorted ("smeared"). Furthermore, when Cs "# CR is valid, the measurement signal is not equal to zero - even if cPr is equal to zero and steady-state conditions prevail - but has the value


t DoT

" time --.. " Fig.3.3. Measured curve of a heat flux DSC (schematic) with heat released inside the sample (exothermic effect). t1T measured signal, R thermal resistance between furnace and sample, f3 heating rate, t1C = Cs - CR difference between the heat capacities of sample and reference sample (here negative, because CR > Cs), Q) peak area (exothermic effect); it is a measure of the heat released between t, and tz,@ area below the baseline; it is a measure of the heat required to heat the sample between t, and tz

-R . p. (Cs - CR) which is the initial deviation after the quasi-steady state has been reached in scanning operation [cf. Eq. (3.I)]. This contribution is the measured curve before/behind a peak which is parallel to the abscissa if R, l1C = Cs - CR and p are constant (Fig. 3.3).

In reality, the term R ( Cs - CR) reflects the temperature dependence of the thermal resistance R (in general: of the heat transfer conditions) and of the heat capacities Cs and CR causing a temperature dependence of the measured curve even without any thermal effect due to the sample. [NB: Eq. (3.3) is in principle the so called Tian equation, originally derived for cylinder-type calorimeters.]

Regarding Eq. (3.3), the conclusions are as follows:

1. When the signall1 T measured at a given moment is to be assigned to the heat flow rate IPr by which it is caused, the third term in Eq. (3.3) must be taken into account (cf. desmearing, Sect. S.4). R must be determined by calibration (cf. Sect. 4.4); the time constant T can also be obtained from calibration measurements (cf. below "Higher-order approximations" and Sect. 7.2). For t> to, the solution of the differential Eq. (3.3) for a heat pulse (/Jr at the moment to has the form (see textbooks of mathematics)

3.1 Heat Flux DSC

Fig. 3.4. Measured signal of a heat flux DSC for an exothermic heat pulse at the time to (model of the 1st approximation).

37

r time constant; symbols see Fig. 3.3

~~r-~-----------~-~-----

t t-T-j

IlT

t, time--....

with T = Cs ' R. Up to the moment t = to, the solution is the steady-state curve

For t ~ 00, the function ~T(t) returns to this curve (cf. Fig. 3.4).

2. For the total heat of reaction or transition Qr developed/consumed in the sample, the following balance equation is valid:

where t1, t2 are the beginning and end, respectively, of the peak. With Eq. (3.3) inserted, the following is obtained:

1 [t2 t2 ] t2 T d~T Qr=-- f~T(t)dt-f(-R·~C'f3)dt -f-·-dt

R tl tl tl R dt (3.4)

The content of the square brackets corresponds to area CD, i. e., the so-called peak area between the measured curve and the (interpolated) baseline (definition see Sect. S.1) in Fig. 3.3.

- When ~C and R are not temperature- (or time-) dependent, the measured curve outside the peak is a parallel to the abscissa. In this case, d~T/dt is zero before and behind the peak. The contribution of the 3rd term vanishes when integration is carried out over the whole peak.

- In the real case, the 2nd term of Eq. (3.4) is not constant, i. e., the curve before and behind the peak is not parallel to the abscissa. In this case, the 3rd term will not vanish but represents a correction of the values obtained by peak integration.


Fig. 3.5. Partial integration of a peak (heat flux DSC, exothermic effect). CR > Cs, CD partial peak area (between tl and t*),@ area below baseline (between tl and t*); symbols see Fig. 3.3

t 6.T

t, time -.

3. For the partial integration (Fig. 3.5) of the peak between t, and t*, the contribution of the 3rd term must be taken into acount at the point t*:

Qr(t*} = -~ [Y ilT(t} dt- f (-R· ilC·f3) dt] - f ~. dilT dt R tl tl tl R dt

\. y ) '-v-------l partial peak area correction term

Partial integrations of the peak are, for instance, necessary for kinetic investigations and to determine purity (see Sects. 6.3 and 6.9).

Higher-Order Approximations

The temperatures of sample and reference sample (assumed here to be homogeneous) are not measured directly. There is a certain distance and a thermal

Fig. 3.6. Heat flux DSC, (one half), model of the 2nd approximation. S sample, F furnace, T MS temperature at sample measurement point

/

/


s

4

Fig.3.7. Equivalent electric circuit for the 2nd approximation. S sample side, R reference side, F furnace, Ms measurement point on sample side, MR measurement point on reference side, ll.U corresponds to ll.T, UF corresponds to TF, i corresponds to tPr • RFM thermal resistance between furnace and measurement points Ms, MR , R MS thermal resistance between sample measurement point and sample, RMR thermal resistance between reference sample measurement point and reference sample, Cs heat capacity of the sample, CR

heat capacity of the reference sample, CFM apparent heat capacity between furnace and measurement points Ms, MR

resistance between the temperature measurement points and the respective sample (Fig. 3.6). Depending on the design of the measuring system, the resistance is made up of several parts differing in quantity and originating in the transition layer between sample and bottom of the crucible on the one hand and crucible and support on the other hand as well as further resistances between support and temperature sensor. When the sample temperature changes, the temperature measurement point reacts only after some delay. The analogue electric circuit diagram of a DSC in this "2nd approximation" is shown in Fig. 3.7. The following is valid for this so-called "thermometer problem":

dTMs T MS = Ts - r2 • -

dt

T MS temperature of the measurement point (e. g., junction of the thermocouple) for the sample

Ts sample temperature (assumed homogeneous) r2 characteristic time constant for the temperature relaxation between sample

and measurement point.

In analogy, the following is valid for the reference side:

dTMR TMR =TR - r2·--

dt

(r2 is assumed to be equal for both sides).


With these equations, the following results for the difference:

( dTMS dTMR) flTsR = Ts- TR= TMS - TMR + T2 -----

dt dt

fl T m = T MS - T MR is the measured temperature difference. In other words, with known T2 it is possible to calculate the temperature difference between sample and reference sample from the temperature difference measured at the position of the temperature probe. This was made use of in the so-called advanced Tzero ™ technique for the turret-type DSCs recently introduced by TA-Instruments (see Sect. 2.1.2). However, the following relation results in analogy to the mathematical procedure of the 1 st approximation if we use fl TSR instead of fl T:

1 [ dflTm d2flTm J iPr (t) = - Ii fl T m + R . (Cs - CR) • f3 + T 1 dt + TIT 2 ~ (3.5)

[Tl = T from the first approximation, see Eq. (3.3) 1

In addition to the 1st derivative (slope) of the measured curve flTm (t), the 2nd derivative (curvature) must be used in the 2nd approximation to get the heat flow rate converted in the sample. The two time constants must be determined experimentally by proper calibration procedures. The first time constant Tl is determined by the apparent thermal resistance and heat capacity between furnace and temperature sensor. The second time constant is determined by the apparent thermal resistance and the "effective" heat capacity between sample and sample temperature measurement point. Exact symmetry between the sample and the reference side is assumed in this calculation.

The above-described approximation can be refined as desired: Possible asymmetries can be taken into account by introducing different thermal resistors and capacities on the sample and reference side into the network. This results in more complicated equations which, however, can be solved. This enables one to correct not only the temperature but even the measured heat flow rate for possible influences of the asymmetries (e. g., a curved zero-line, which otherwise must be subtracted afterwards) during the measurement. Certain heat transfer problems and a temperature gradient inside the sample and its influence on the peak shape can be taken into account by introducing further thermal resistances and capacities in the network of Fig. 3.7. A thicker sample with poor conductivity can be considered as having been split up into different layers which are linked with one another by heat-conducting boundary layers. In the end an additional differential quotient in the differential equation and another time constant result for each additional thermal resistance in connection with a heat capacity. The solution of such a refined approximation is as follows:

1 [ dflT d2flT d3flT J iPr(t)=-- flT+ko+k1--+k2--2 +k3--3 + ...

Reff dt dt dt


In reality, the constants ki are terms into which the thermal resistances and capacities (and thus the time constants) of the arrangement enter. Calculation of the reaction heat flow rate r/Jr presupposes that all ki and the measurement signal AT and its time derivatives are known. It can be shown (Loblich, 1985) that, for practical application, the 2nd order differential equation is a sufficiently good approximation to calculate the true desmeared reaction heat flow rate; only the time constants f1 and f2 must be known for this purpose. For a detailed treatment how to determine the time constants experimentally, see Loblich, 1994.

So far we have only considered the case where it is assumed that no heat exchange takes place between sample and reference sample, which is true for the turret-type DSC as well as for some of the cylinder-type DSCs. This simplification is certainly not permissible for the widespread disk-type measuring systems. Figure 3.8 is a more realistic representation of the measuring system of such a calorimeter. When the differential equation is to be set up for this system, it may be convenient to use the analogue electric circuit represented in Fig. 3.9 for the 2nd approximation (Rohne, 1983). In order to formulate the desired differential equation, according to Kirchhoff's laws, the voltage balance and current balance are made up for each loop and each node of the analogue electric circuit. For the circuit of Fig. 3.9, five equations for both voltages and currents are then obtained. On the basis of the laws of electricity, these 10 equations are simplified and combined to form one differential equation which is retranslated into the language of heat transport. The result for a symmetric twin design (i.e., R FMS = RFMR = R, CFMS = CFMR = C) is as follows:

(3.6)

/ T,/

'--~==:='--~~---'% -- ¢'R

/JT

Fig.3.8. Disk-type measuring system of a heat flux DSC (to calculate the tIIr(~n dependence). Ms temperature measurement point on sample side, MR temperature measurement point on reference side, tIIr heat flow rate produced/consumed by the sample


s

UF

Fig.3.9. Equivalent electric circuit for the disk-type measuring system according to Fig. 3.S. S sample, R reference sample, M temperature measurement point, F furnace, U voltage, i current, R resistance, C capacitance (cf.legend to Fig. 3.7)

This equation is similar to that of the 2nd approximation [Eq. (3.5)]. However, as there is a thermal resistance RMM between sample system and reference sample system, a thermal effect in the sample will also affect the reference side and thus TR• Only in the steady-state case and at a sufficiently great distance from peaks is d T Rid t equal to the heating rate f3, and the second derivative of the reference sample temperature is equal to zero only there. For the steady-state (st) case,

dilT d2ilT with tPr = 0, --= 0 and --2- = 0 the following is then valid and characterizes

dt dt the baseline:

CR - Cs ilTst = . f3

1 2 -+--R RMM

When Eq. (3.6) is integrated to get the area of a transition peak, the following is valid with the approximation dTR/dt::::: f3:

12 ( 1 2) r 12 12 CR - Cs ] f tPr (t)dt = Qr = - - + - f ilT(t)dt- f . f3dt II R RMM II II 1 2

-+-R RMM

(1 2) 12 = - - + -. f (ilT(t) - ilT.t) dt

R RMM II

This integral describes the peak area between the measured curve and the (interpolated) baseline. The approximation is the better the smaller the last two summands in Eq. (3.6). From this follows the rule that, when heats of transition


are determined with heat flux DSCs, the sample and the reference sample should be as similar as possible (CR ::::: CS,RMR ::::: R MS)'

The factor (l/R + 2IRMM ) is decisive for the sensitivity of the calorimeter; the greater the thermal resistance of the disk, the higher the peak for a given heat of transition. However, as a result the time constant T increases as well, i. e., the system becomes more inert. In addition, this factor allows the conclusion to be drawn that the ratio of R to RMM plays an important role for the sensitivity. Depending on where sample and reference sample are arranged on the disk, RMM

(thermal resistance between sample and reference sample) change, meaning that high reproducibility for the location of sample and reference sample is of great importance for the ability of a DSC to be calibrated.

In the approximation Cs ::::: CR and RMS ::::: RMR , Eq. (3.6) changes into a 2nd order differential equation of the following form:

The solution of this equation, i. e., the curve fl.T (t) as measured by the calorimeter, for a pulse-like CPr at t = 0 is the sum of two exponential functions:

(3.7)

where there is a complex dependence of the time constants Tl, T2 on the coefficients K, K1, K2 • Here, T2 is determined in approximation by C . R, and Tl by Cs ' RMS (cf. Eq. 3.6). Such a peak, produced by a heat pulse, is shown in Fig. 3.10. The time constant Tl essentially determines the ascending slope, T2 the descending slope.

Fig.3.10. Measured signal /:"T of the heat flux DSC for a heat pulse generated in the sample at the time to (sum t of two exponential functions with the time constants 11 and 12 in 2nd approximation)

fl.T

1---+- - - - - - - -=-::-=-=-,...,-~------

to time


Numerical Simulation

As the equations become increasingly complex and as it is impossible to solve them analytically without introducing simplifications, it is no longer recommended to set up differential equations for calculations. This is in particular true if we want to cover heat exchange by convection and radiation and include non-linearities due to temperature-dependent thermal resistances and heat capacities. In this case, numerical simulation by the finite-element method could be applied. Appropriate software is commercially available and runs on modern PCs without problems. When the required time and effort are spent, the temperature and heat flow fields and the measured AT(t) curves can be simulated for any complex arrangement and any thermal process inside the sample.

The finite-element method consists in splitting the whole arrangement up into sufficiently small "cells" with given material properties for which the heat flow rates and temperatures are calculated. On the basis of considerations with regard to the energy balance, a large system of equations is obtained which is solved by conventional methods. To show the essential results, this method has been applied to a very simple one-dimensional model of a disk-type measuring system as well as a real commercial disk-type DSC in one-dimensional approximation (for details see Hohne, 1983).

The results for a simulated melting peak are represented in Figs. 3.11 to 3.13. As can be seen, the shape of the peak depends strongly on the given conditions and parameters. The position of the peak maximum, for example, changes with the heating rate, with the thermal conductance of the sample and with the mass (or heat of transition Qr) of the sample. The slope of the descending part is determined by the heating rate and the thermal conductance of the sample. Only the extrapolated peak onset temperature is relatively independent of experimental parameters. This is why, together with the area, this temperature (Te) is preferred to characterize the peak. In contrast to this, the peak temperature maximum (Tp) and the peak width are not values suited to characterize transitions (for definition of the characteristic temperatures, see Sect. 5.1). Figure 3.14

Fig.3.11. Dependence of the peak shape on the heating rate /1, calculated by numerical simulation for a heat flux DSC

1 bJ

3.1 Heat Flux DSC

Fig.3.12. Dependence of the peak shape on the heat of transition Q" calculated by numerical simulation for a heat flux DSC (heating rate: 2 Kmin-I)

1 6.[

Fig.3.13. Dependence of the peak shape on the thermal conductance L of the sample, calculated by numerical 1 simulation for a heat flux DSC (heating rate: 2 K min-I)

6.T

45

time ..

L=O.OOS W/K

shows the numerical simulation of a thermal event which is caused by the melting of a small sample of indium in a real commercial disk-type DSC. The result is very similar to the real measurement in such a DSC.

It has furthermore been possible to show that the calibration factors KQ = I Qrlpeak areal and Kw = 14'rf.6.TI clearly depend on measurement and sample parameters (e. g., Qr, A.ample, f3) when radiation and convection are included into such model calculations (Hahne, 1983). Table 3.1 gives the results for different parameters as an example. The dependence of the calibration factor on such parameters is a fact which is of importance in practical scanning calorimetry and which must by all means be taken into consideration during calibration (cf. Sect. 4.4). The obvious temperature dependence of the calibration factor is


0,25

!::,JIK

0,20

0,15

0,10

0,05

~ ° °

25 50 75 100 125

time .. Fig.3.14. Numerical simulation of the signal (absolute values) generated by the melting of an indium sample in a disk-type DSC (1.38 mg In, 10 K min-I)

Table 3.1. Numerical simulation of a commercial disk-type DSC: Changes of the two calibration factors K<[> and KQ depending on changes of the heat of reaction (Qr), emissivity (E), heating rate ({3), density «(J) and thermal conductivity (A) of the sample, respectively". The initial standard parameters (belonging to K<[> = 1 and KQ = 1) are given in parenthesis

Parameter which is varied

QrinJ

E

Numerical value of the parameter

(0.3965) 3.965 0.03965

0.5 (AI) 0.25 (Ag)

(10) 1 0.1 0.01

(7310) 73100 731

(157) 15.7 1570

K<[>

1.001 1.001

1 1.001 1.004 0.991

1 0.998 1.015

1 0.998 1.003

KQ

1 1.012 0.929

0.915 0.888

1 0.930 0.875 0.870

1 0.935 0.885

1 1.016 0.999

a The parameters have been changed in (unrealistic) steps of 10 only, to give a better impression of their influence.


another conspicuous feature although the temperature dependence of the heat capacities and of the heat resistance of the substances has not been included in these model calculations. The reason is to be found in the intensive heat exchange with the surroundings via radiation and convection which is strongly temperature-dependent (non-linear heat leak). The calibration factor is also strongly influenced by the surface quality (emissivity c) of the crucibles containing sample and reference sample, which therefore should be well defined and must not change during the measurement.

As far as steady-state reaction heat flow rates are concerned, the above calculations have shown that the calibration factor Kcp = - ([Jr/l1 Tst also depends on the quantity ([Jr itself (which is proportional to the difference between the heat capacities, Cs - CR) when radiation heat exchange is included.

In addition, the calibration factors used for peak integration (Le., for determination of Qr) and for heat flow rates (e. g., for determination of Cs) are generally not equal. The difference in the calibration factors Kq, and KQ can also be concluded from the following considerations. In a DSC there are always temperature differences between sample and reference sample during the measurement. As a result, the heat exchange between sample and reference sample and the respective environment varies. As the radiation exchange increases, non-linearly with temperature, the overall heat exchange is also non-linearly linked with the temperature difference between sample and reference sample. As a consequence, the measured temperature difference (and thus the measured differential heat flow rate ([Jm) is non-linearly linked with the true heat flow rate into the sample ([Jtrue' This results in the calibration factor Kef> in the (linearly formulated) Eq. (2.1) becoming a function of the differential heat flow rate ([Jm itself:

(3.8)

As the measured heat flow rate ([Jm depends on sample parameters (Cp, m) and on the heating rate, Kq, implicitly depends on these quantities as well. Furthermore, it follows from Eqs. (3.3) and (3.6) that Kq, is determined by the thermal conduction path and its properties. As these quantities are always a function of temperature, the calibration factor is always also temperature-dependent.

On the other hand, for the determination of the true heat from the peak area, Eq. (2.2) is valid, which is obtained from Eq. (2.1) by peak integration. If, however, Kq, depends on ([Jm, this factor is not constant and cannot be moved in front of the integral, the integration of Eq. (3.8) then yields:

From a comparison with Eq. (2.2) it follows that KQ is an integral mean value over the function Kq,( ([Jm) in the region of the peak. As ([Jm may considerably vary during a peak, the difference between Kq, and KQ is normally significant (see Sect. 4.4.3). In these cases, a quantitative peak evaluation may lead to results affected by systematic errors, as the calibration factor also undergoes substantial changes. Summarizing, on the basis of the results of the numerical simulations, the following can be stated for all types of heat flux DSCs:


- For steady-state conditions

is valid [ef. Eqs. (3.1) and (3.8)].

With this function, for a known heating rate fl, the unknown heat capacity Cs can be determined from the measured L1«Pst • It must be borne in mind that the calibration constant Kq, depends on the difference of the heat capacities, Cs - CR.

- For non-steady-state conditions (i.e., within a peak):

- The shape of the peak changes as a function of heating rate, thermal con-ductivity, heat capacity and shape of the sample, and of the amount of the heat of transition. As a result, the peak parameters (width, maximum, height) change as well; only the extrapolated peak onset temperature Te is to a certain degree independent of sample parameters.

- The calibration factor Kq, (true heat flow rate/measured heat flow rate) is not constant but a (weak) function of the heat flow rate itself.

- The calibration factor KQ (heat of transition/measured peak area) is not constant but changes more or less with temperature, heating rate, and thermal conductivity of the sample as well as with the heat of transition, surface quality (emissivity), and sample position.

- The calibration factor for peak areas (KQ) differs from the calibration fac-tor for steady-state heat flow rates (Kq,).

It is basically recommended that in DSC measurements the best symmetry possible between sample and reference sample be ensured (CR :::: Cs, RMR :::: R MS ,

identical sample containers). The results of the model calculations are confirmed by practical measurements (Sarge et aI., 1994).


In ideal power compensation DSCs, each L1T signal appearing between sample and reference sample would be immediately compensated by a corresponding change in the heating power. The differential heating power required for this purpose would be equal to the differential heat flow rate as the complete electric energy is converted into heat. If the differential heating power would be measured directly, this signal would directly stand for the heat flow rate into the sample searched for. The calibration factor would then be identical to unity and (neglecting "smearing" of the measured signal) «Pr = «Pm would be obtained. There exists however, no such ideal power compensation DSC. In real power compensation DSCs, the sample is always put into a container and then placed into the heater; at least one heat conduction path and, as a result, at least one time constant T between the controlled heater and the sample location must be taken into consideration. This leads to a measurement signal «Pm which is


"smeared" in comparison with the processes inside the sample, because of the thermal lag, and a differential equation of at least 1st order is therefore obtained to describe the behavior:

Without exception, commercial power compensation DSCs are instruments in which a temperature difference always occurs between sample and reference sample (as is the case in heat flux DSCs). This temperature difference serves, firstly, as the primary measurement signal and, secondly, it is used to electrically compensate the measurement effect by means of a proportional controller (cf. Sect. 2.2). However, the proportional control can never completely compensate the measurement effect so that, even in these calorimeters, a ATsR remains between the micro-furnaces of sample and reference sample, and this both in the steady state and during a peak.

The following is then valid:

with kprop set by the proportional control circuit. O'Neill, 1964, analyzed such a system with a proportional controller; the

theoretical analysis of two different systems of power compensation DSC, one with P (proportional) and one with PID (proportional, integral and differential) temperature control of the sample micro-furnace has been presented by Tanaka, 1992.

As different temperatures also result in a different exchange of radiation and convection with the surroundings, conclusions similar to those for heat flux DSCs would have to be drawn. This means that, in principle, what has been stated in the previous section is also valid for power compensation DSCs. However, it applies to a lesser extent as, due to the compensation, the temperature differences are much smaller than in heat flux DSCs. For power compensation DSCs at present commercially available, it is therefore basically to be expected that

- the shape of the peak depends on sample parameters, the time constant(s) being, however, smaller than those of heat flux DSCs, leading to a better resolution in time,

- the calibration factor is not exactly equal to 1 (i. e., calibration is necessary), - the calibration factors for steady-state (heat capacity measurement) and for

peak evaluation are not the same, - the calibration factor depends on the temperature, the heating rate, the thermal

conductivity of the sample, the amount of the heat of transition and on the sample location.

Practical measurements have confirmed these assumptions (Hahne, Glaggler, 1989). As had been expected, the effects are, however, substantially smaller than with heat flux DSCs. When power compensation DSCs are calibrated, this dependence on parameters must be taken into consideration.


When heat capacities are measured (see Sect. 6.1) the reference microfurnace very often does not contain a sample. Due to the complex heat transfer conditions in the non-symmetrical DSC measuring system, heat losses then arise which cannot be compensated. These heat losses are different in static and dynamic operation. An analysis shows that the dynamic measurement error can be determined by means of the losses measured in isothermal mode before and after the scan mode (for details, see PoeBnecker, 1993 and Sect. 6.1).

Another method to describe the behavior of an apparatus is deduced from the transfer theory, namely the theory of linear response which can be applied if the apparatus in question behaves linearly. This is the case if the DSC can be described with the aid of linear differential equations of any order. We have shown in Sect. 3.1 that heat flux DSCs can be described in this way if radiation and convection heat exchange are disregarded. The transfer theory has in particular been used successfully to describe the behavior of DSCs in temperaturemodulated mode (see Sects. 3.3 and 4.7). In the case of power compensated DSC the situation is more complicated because of the control electronics involved. For the most frequently used equipment from Perkin-Elmer it has been shown that this DSC can be considered as a linear apparatus in the first approximation (Tanaka, 1992; Hohne, Schawe, 1993; Schawe et al. 1993, 1994). However, any asymmetry between sample and reference side disturbs the linear behavior, and the theory of linear response is strictly speaking not valid. This has consequences for evaluation procedures which require linear response, such as deconvolution (Sect. 5.4) in particular in peak regions of a measured curve.

Note:

In contradiction to what has been presumed in this chapter, every real DSC is not strictly symmetric in its functionality. Even in the empty state, the temperatures of the sample cell and of the reference cell are not equal so that a residual temperature difference will arise, which may change with temperature during a scan run. As a consequence, the empty DSC will produce a measured curve, which is indeed neither zero nor constant with temperature but has an apparatusdependent curved shape 4'0(T). This so-called zeroline, which will mostly be measured with empty crucibles in the calorimeter, is a function, which is additive to the measurement signal which stems from the sample processes. In every real case, this function should first be subtracted from the measured curve before evaluations following the theoretical considerations can be made. We therefore must insert the term 4'm - 4'0 (or AT - ATo) instead of 4'm (or AT) in all formulas used so far, this has been omitted in this chapter for clarity.

3.3 Temperature-Modulated DSC (TMDSC)

Different temperature-modulated calorimetric techniques have been known since the beginning of the last century [for details see (Gmelin, 1997)]. Gobrecht et al., 1971, were the first who ran a DSC with periodical changing temperature to measure complex heat capacities. It is remarkable that the authors already in

3.3 Temperature-Modulated DSC (TMDSC) 51

this early paper point out that this method offers the possibility of "quasi-isothermal investigation of heat capacity changes during annealing procedures or chemical reactions", which nowadays belong to the most successful applications of TMDSC. They even mention: ''A combination of linear and periodic heating offers the advantage of the good temperature resolution of slow scan speeds and the higher output signal due to the faster oscillations".

Unfortunately this article was published in a physical journal and was not noticed by the manufacturers and thermoanalytical community. That's why another 20 years passed before M. Reading and coworkers came out with a temperaturemodulated DSC (Reading et al., 1993; Sauerbrunn et al., 1992) and the method became commercially available. During the past decade this method became widespread and it has proved its worth with numerous successful applications.

We want to present the fundamental ideas and the theoretical background of TMDSC in what follows, but we restrict ourselves to present only those facts which are absolutely essential for the practitioner. The complete theory of TMDSC would go far beyond the scope of this book, in particular as it is still a matter of development and discussion. The interested reader is referred to the original literature. Of course we are not able to give a complete list of the numerous authors which have contributed to the theory of TMDSC during the recent years. Many of them are named in review articles on this topic (see, e.g., Simon, 2001) and in special issues about temperature-modulated DSC of Thermochimica Acta (1997, 1999, 2001) and Journal of Thermal Analysis and Calorimetry (1998).

3.3.1 The Temperature-Modulated Method

At the beginning of TMDSC the common (linear) temperature program of the DSC was superimposed with a sinusoidal temperature fluctuation:

T(t) = To + (Jot + TA· sin(wt) (3.9)

with {Jo the underlying heating/cooling rate, TA the temperature fluctuation amplitude and w = 2 n f the angular frequency of modulation. But other periodical temperature fluctuations (sawtooth-like, triangular, rectangular, step-like, etc.) are possible and have been used as well. From mathematics it is known, that every periodic function can be written as a Fourier series and Eq. (3.9) can easily be generalized for the non-harmonic case e.g. for an odd function:

T(t) = To + (Jot + L TA,n sin(nwt) (3.10) n~l

i. e., every periodic temperature change can be considered as a sum of sinusoidal functions. However, there is not only one sinusoidal frequency but also higher harmonics at frequencies which are integer multiples of the base frequency but with (generally) decreasing amplitudes.

It is even possible to subject the sample to non-periodic temperature changes like a step or a steep ramp. In such cases the Fourier series in Eq. (3.10) must


be replaced by a Fourier integral and as a result we get in the heat flow rate function a continuous spectrum of frequencies instead of discrete harmonics. The theoretical background comes from linear response theory and Fourier transform mathematics. However, the measurement may yield more information if we use non-sinusoidal temperature modulations, but the mathematics and even the evaluation is more sophisticated although the background, as far as the theory of temperature-modulated method is concerned, remains generally the same. To simplify the formulae and focus the understanding to the essential points, we restrict ourselves here to the simple sinusoidal case without reservation of generality. If needed the results can be generalized to other periodic or non-periodic temperature fluctuations.

From Eq. (3.9) it follows that the heating rate is not constant as in the case of conventional DSC but reads:

dT - = /30 + TA • W· cos(wt) dt

(3.11)

The heating rate fluctuates between a maximum (/30 + TAW) and a minimum (/30 - TAW) value and, depending on the magnitude of the three measuring parameters /30, T A and w = 2 n f, we have to distinguish between different modes of TMDSC operation (see Sect. 2.4.2)

- quasi-isothermal mode: /30 = 0 - heating only mode: TAW < /30 - heating-iso mode: TAW = /30 - heating cooling mode: TAW> /30

All these modes are used in practice, often without consideration about possible consequences concerning the result of the measurements, which may be different depending on the sample and processes in question. However, there are different temperatures and different temperature fluctuations as well as different heating rates during the TMDSC run, and such processes which react either to temperature or to heating rate changes will give different signals.

As a consequence of the temperature modulation of the furnace in the DSC, the measured heat flow rate fluctuates as well. The DSC signal is, of course, influenced by the sample and possible processes occurring, but even by the apparatus and the heat transfer process to the sample. Although this in reality takes place together, we will look at the different influences separately to simplify matters.

3.3.2 Influences of the Sample

To see what the heat flow rate in a TMDSC looks like we start from the very general heat flow rate caused by any sample in a DSC [ef. Eqs. (2.4), (2.5), (4.2)]

dT 11> (T, t) = Cp (T) . - + l1>ex. (T, t)

dt (3.12)


which tells us that the heat flow into a sample has two components, one comes from the always non-zero heat capacity Cp and the other from additional endoor exothermic processes occurring at a certain temperature with a certain rate. To make the things more transparent we distinguish in the following between different cases. They may, in reality, often happen together in the sample, but this doesn't matter, because of the superposition principle - which is valid for heat -we then simply measure the sum of the different heat flow rates in the DSC.

Case 7: Heat Capacity Sample - No Processes

This is the simplest case, the sample has a certain (vibrational) heat capacity CiT), with a generally very weak temperature dependence, considered as constant during one period of the temperature fluctuation. In this case Eq. (3.12), with ~ex. = 0, together with Eq. (3.11) yields:

~(T, t) = CpfJo + Cp ' TA • W· cos(wt) (3.13)

the measured heat flow rate is the sum of two components: the first one, proportional to the underlying heating rate, is almost constant and the second one fluctuates cosinusoidal (i. e., sinusoidal, but shifted in phase). These two parts are referred to as the underlying ~u and the periodic ~ component of the heat flow rate. As Eq. (3.13) tells us, Cp can be determined from both components and here the question arises how to determine them from the total signal. The method is simple, we have to determine the average within one period by integration:

t+tpl2

I ~(T, t) dt = Cp • Po = ~u t-tp/2

If we shift this integral through the measured heat flow rate signal ("gliding integration") we get the underlying heat flow rate. This is almost that curve we would get from the conventional DSC (without modulation: TAW = 0). Now it becomes obvious why Cp must not change during one period, this would give faulty values.

Subtracting the underlying part from the measured heat flow rate yields the periodic part:

ci>(T, t) = ~(T, t) - ~u(T, t) = Cp ' TA • W· cos(wt) (3.14)

This is a function which fluctuates in time around zero with the same frequency as the temperature, but shifted n/2 in phase with the amplitude:

~A= Cpo TA • w

From the amplitude of the periodic part, which is normally determined via Fourier analysis or other suitable mathematical procedures, we get the heat capacity of the sample:

~A Cp=cp·m=-

TA·w (3.15)


The specific quantity cp - obtained by dividing Cp with the mass m - is often called "reversing heat capacity". Of course it should be the same as the specific heat capacity determined from the underlying curve or via a conventional measurement. If not, there may be a zeroline (see Sect. 5.3.1) or calibration problem (see Chapter 4) with the particular TMDSC.

Case 2: Heat Capacity Sample with Additional Processes

Again we assume that the sample has a certain (only vibrational) heat capacity CP(T) with a very weak temperature dependence, but additional processes with endo- or exothermic latent heat exchange are taking place as well. In this case Eq. (3.13) must be extended with the excess heat flow rate:

<f>(T, t) = Cpf30 + Cp • TA· W· cos(wt) + <f>ex'(T, t) (3.16)

The latter, a continuous function of temperature and time, can be expanded as a Taylor series around the mean temperature during the periodical fluctuations Tu = To + f3ot,

with generally very small temperature fluctuations and a slow change of Tu in time the excess heat flow rate can be substituted with sufficient accuracy by the first approximation of the Taylor series:

Inserting this into Eq. (3.16) yields:

a <f>ex. (T t) <f>(T, t) = Cpf30 + Cp • TA· W· cos(wt) + <f>ex'(Tu, t) + u, (T - Tu)

aT

from Eq. (3.9) we see that T - Tu = TA· sin(wt) and after substitution and rearrangement we get:

a <f>ex. (T t) <f>(T, t) = Cpf30 + <f>ex'(T, t) + Cp • TA· W· cos(wt) + aT u, • TA . sin (wt)

(3.17)

Again the heat flow rate is the sum of a non-periodic and a periodic part. The non-periodic part - got from the measured signal via gliding integration - now contains contributions from the heat capacity as well as from the processes occurring in the sample:

(3.18)


this is exactly the signal we would measure if we would switch the temperature modulation off [set TA = 0 in Eq. (3.17) and compare with Eqs. (3.12) and (4.2)]. Subtracting the underlying (non-periodic) part from the total signal yields the periodic part:

_ oq,eX.(Tu,t) . cP(T, t) = Cp • TA· W· cos(wt) + oT . TA· sm(wt) (3.19)

This is again a harmonically fluctuating function with the same frequency as the temperature modulation. There are two contributions, one from the heat capacity of the sample (the same as in case 1) and one from the temperature dependence (1st derivative) of the process involved. Both contributions are shifted nl2 in phase, i. e., they have to be added like orthogonal vectors. [Formally the resulting amplitude can be interpreted as a complex quantity with the real part Cp • TA· wand the imaginary part TA . 0 cPex•(Tu,t)loT]. In otherwords,Eq. (3.19) can be rewritten in the following form:

ci>(T, t) = cPA· cos(wt + 6)

This is a (co )sinusoidal heat flow rate with amplitude

and - compared to the pure Cp contribution - a phase shift of

ocPex.

ay-(Tu,t)

6 = arctan ----

(3.20)

(3.21)

Depending on the sign of the temperature derivative of the reaction heat o q,ex. (Tu,t)lo T, which is positive for endothermic and negative for exothermic events, the phase shift is positive or negative, respectively, whereas the amplitude becomes always greater than that of the pure Cp-sample, regardless of whether we have an endo- or exothermic event. The increase depends on the magnitude of the temperature derivative of the reaction heat flow (but not linearly). It should be emphasized that every reaction is somehow temperature dependent and contributes therefore to the measured heat flow rate amplitude. But the contribution is often small because 0 F(Tu, t)loT is generally a small quantity compared to the product Cp (T) . w. In addition the latter contribution can be made even more dominant by choosing a higher frequency (lower period) of modulation.

From the amplitude - as in case 1 determined from the modulated heat flow rate with proper evaluation methods - an "apparent" heat capacity can be calculated [see Eq. (3.15)]:

Gppp. = m· cppp· = (Cp (T»2 + (~ °o~x. (Tu, t) J (3.22)


This tells us that, in addition to the common vibrational heat capacity, there is always a contribution from processes occurring in the sample. This so-called excess heat capacity depends on the temperature derivative of the reaction heat flow rate as well as on the frequency. It causes always an increase of the apparent heat capacity regardless of whether the process is endo- or exothermic in character. To separate the excess heat capacity from the vibrational part, the latter must be known, or measurements at different frequencies, but with the same underlying heating rate, should be performed.

Formally the apparent heat capacity may be understood as the absolute value of a complex heat capacity which can be given either as real and imaginary parts or as magnitude (absolute value) and phase angle and the following relations apply (see textbooks of mathematics):

ct = C; + iC; = I ct I . eiD = I ct I . cos 6 + i . I ct I . sin 6

C" and tan6=L C;

(3.23)

by means of Eqs. (3.20), (3.21) and (3.15) it is easy to show that in our case the real and imaginary parts of the complex heat capacity read:

and 1 o~ex.

C; = --::1- (Tu, t) w uT

(3.24)

With other words, the real part is the common vibrational heat capacity and the imaginary part is marked by the change of the process heat flow rate with temperature divided by frequency.

Depending on the manufacturer of the TMDSC and the software used, these useful quantities are calculated from the modulated signal and can be further evaluated.

Case 3: Sample with Time-Dependent Heat Capacity

In this case we are free to allow time dependent changes of the degrees of freedom to occur in the sample, e. g., relaxation processes like vitrification or devitrification, but we consider the temperature dependence of the heat capacity to be very weak again. However, time dependent heat capacity means a nonequilibrium state of the system or certain subsystems of the sample. The time scales of the relaxation processes are comparable with the time scale of the experiment. Within the scope of linear response the total heat flow from the sample at any moment is the superposition of the heat flows from all subsystems at that moment. That is, the simple equation ~(T, t) = Cp(T) . a TIO t [see Eq. (3.12)] is not valid any more in this case and must be replaced with an integral equation:

d ~ ( OT(t')) ~(T, t) = - f Cp(T, t - t') . --::I dt' dt -~ u t

(3.25)


This defines the so-called convolution product of the time dependent heat capacity and the heating rate. The convolution theorem means that the convolution product in time domain transforms into a common product in the frequency domain via Fourier transform ~:

~ (If>(t» = ~ (Cp(T, t» . ~ (T(t»

this is a complex frequency dependent function, a product of two complex functions in Fourier space:

If>*(w) = Gp(w)· T*(w) (3.26)

In other words, a time-dependent heat capacity is equivalent to a complex frequency dependent heat capacity and the heat flow rate is frequency dependent and complex as well. All evaluation and calculation should be done in Fourier space to simplify matters. The resulting function [e. g., the heat capacity Gp (w) 1 can be inverse Fourier transformed to get the result in time domain again. It should be emphasized that an evaluation of the heat capacity with Eq. (3.15) instead of Eq. (3.26) will lead to incorrect results in the time-dependent case. The apparent heat capacity would depend on the measurement parameters like heating rate and temperature amplitude and the true time dependence would never be obtained. Fortunately the relaxation times of most of the degrees of freedom contributing to the common heat capacity are much smaller than the time window of TMDSC experiments and their contributions appear as time independent. One process, however, the glass transition, which plays an essential role in polymer science (see Sect. 6.4), necessitates to take a time dependent heat capacity into account and thus a careful evaluation.

The three cases discussed in this section are the most important ones from the practical point of view. There exist other possible cases which may be described mathematically in a similar way. In reality all these cases occur seldom as isolated events as presented here, but are combined jointly in the same sample under investigation. We refrain from presenting the rather complicated mathematics of such combined cases. The advanced reader may be able to set up the formulae himself by means of the guidelines presented here. However, we recommend to choose experimental conditions and the measurement parameters such that only one of the above cases is dominant in the TMDSC run and determines the measured results and the other influences can be neglected. Because of the complicated matter any interpretation of the measured results should be done very carefully and all possibilities should be thought of.

3.3.3 Influences of Heat Transport

Beside the sample and the processes occurring, there are at least two more physical effects which influence the measured heat flow signal: the transport of heat needs time and the limited conductivity together with the heat capacity of the DSC parts gives rise to a (damped) thermal wave with changing amplitude and


phase on moving through the DSC. As a result both the amplitude and the phase of the measured modulated heat flow rate depend on the DSC used, and the apparatus must be properly calibrated (see Sect. 4.7) to come to reliable values, and on the thermal diffusivity of the sample.

The calorimeter itself is constructed of several parts which the heat has to pass on its way from the furnace to the sample. All these parts have a certain thermal conductivity (i.e., a certain thermal resistance) and a certain heat capacity which influences the heat flow and the thermal wave which is propagating through the DSC in the case of modulation. Any contact area between different parts acts as an additional thermal resistance. In addition the DSC includes often some sophisticated electronics that amplifies the voltages from the sensors to the measured signal transferred to the computer. But every DSC, no matter how complicated it is, can be dissected into a network of simple mechanical elements and, hopefully, linear electronics. As mentioned in Sect. 3.1 the heat transport and the transport of electric charge are physically equivalent processes and the DSC may be described as an electrical network as well. The influence of such a network on AC signals is equivalent to the influence of a calorimeter on the heat flow rate in case of TMDSC.

Formally the total network describing a DSC (here called "box") has one "input" (the temperature-time program) and one "output" (the heat flow rate into the sample). To evaluate the behavior of an apparatus within the framework of linear response, it is often sufficient to look at the so-called "transfer function" P(w) of the "box" in question. This is a complex function in frequency domain (Hohne, Schawe, 1993; Schawe et al., 1993) defined as the quotient of the "output function" Out(w) and the "input function" In(w). The transfer function (in frequency domain) is mathematically connected with the "step response" or "pulse response" functions (in time domain) via Fourier transform. It will go beyond the scope of this book to derive all details of the features of these functions; the interested reader is referred to textbooks of transfer theory. However, the transfer theory has proved to be a useful tool for the theory of TMDSC and for the description of the apparatus influence. Actually the transfer function is closely linked to the "calibration function" needed to correct the measured heat flow rate amplitude and phase regarding apparatus influences.

From the transfer theory of linear systems it is known that the overall transfer function of a network can be calculated from the transfer functions of the individual components. In particular it holds that for transfer elements connected in series the total transfer function is the product of those from the elements. In other words, the magnitudes of the complex functions have to been multiplied, whereas the phases have to be added up. On the other hand, for transfer elements connected in parallel the total transfer function is the sum of those of the components. These facts enable one to breakdown the rather complicated heat transfer network of a DSC into simple components having rather simple transfer functions. These transfer functions can then be assembled properly and yield the transfer behavior of the total TMDSC (both with and without the sample included). Knowing the transfer function enables the determination of the correction function and thus a proper calibration.


Transfer Function of a Simple RC-Element

The simplest component of heat conducting networks is an object having a certain thermal resistance Rth and a certain heat capacity Cpo The respective electrical analogy element is the low pass fllter built of a resistor R and a capacitor C in series (see Fig. 3.2). The complex transfer function, taking AC-voltage (i. e., the respective temperatures) as input and output functions, can easily be calculated for this case (see textbooks of physics):

(3.27)

From this the magnitude (modulus or absolute value) and the phase (argument) of the complex function can easily be determined:

1 I P(w) I = V Re2 (p) + Im2 (p) = ---;=========~ V 1 + (WRtbCp)2

(1m (P»)

arg(P(w» = tan-1 -- = tan-1 (WRthCp) Re(P)

(3.28)

For a given Rtb and Cp these functions are plotted in Fig. 3.15 logarithmically (Bode plot). In the quasi-static case - for low frequencies w - the magnitude and phase are 1 and 0, respectively, i. e., the temperature on the output side is the same as on the input side of the RC-element. But in the case of periodic temperature changes like in Eq. (3.9) on the input side of a thermal path, the output temperature is lower, namely by a factor of P (w).

In other words the amplitude of the modulated part decreases and the phase shifts on the way from furnace to the sample. What do we learn from that fact?

Fig.3.15. Transfer function (magnitude and phase) of a simple RC-element in form of a Bode plot I

c en cs E

p

0.10

;; ;' ",'"

------". ...... :::;-

5

rad

, 1

<1J U1 cs

-'= CL

",'"

0.01 '""""'-=-==-:=-_-_--_-_--1-____ -'-____ ----' 0

001 01 1.0 rad S·l 10

w--


The temperature amplitude at the sample site, compared to the set value, is too low, resulting in a too low apparent heat capacity as well. The error is the larger the larger are the frequency and the thermal resistance and the heat capacity of the thermal path. In addition even the phase is shifted, in other words, the evaluation mathematics would result in a complex heat capacity with a non-zero imaginary part even in cases where the sample has no time dependent heat capacity at all. These artifacts are caused by the propagation of the thermal wave through the network of thermal resistors and capacities from the furnace to the sample, an unavoidable effect for which every TMDSC must be calibrated (see Sect. 4.7).

The RC-element may even serve as a simple model for the sample itself, which always has a certain heat capacity and of course even a thermal resistance. From this it follows that the effective temperature amplitude inside the sample depends strongly on frequency, sample size (heat capacity) and its thermal resistance. For larger sample sizes this influence must be corrected as well. In cases where the thermal resistance or the heat capacity changes during the measurement, the correction function (the reciprocal transfer function) changes as well. This is in particular the case during phase transitions (e. g., melting), where the apparent heat capacity and the thermal resistance changes dramatically.

Thermal Waves

Another approach to describe the influence of apparatus as well as sample properties on the modulated signal starts from the (one dimensional) thermal diffusion equation:

(3.29)

with Dtb the thermal diffusivity. Solving this equation with the boundary condition of an periodical temperature change on one side of an infinite rod yields:

-~ (X) ~ __ ~2~h T(x, t) = TA,o . e ~. sin rot - ~ with ~ LV (3.30)

This is obviously a damped temperature wave which propagates through the rod in time but with exponentially decreasing amplitude. From Eq. (3.30) follows: the local amplitude at the position x is determined by the thermal diffusivity of the heat path material as well as the frequency:

x

TA(x) = TA,o' e-~ = TA,o' e-vooso'poPoWlkth°x (3.31)

(Cp the specific heat capacity, p the density, kth the thermal conductivity, T A,O the temperature amplitude at the (input) boundary of the rod, the phase shift is the argument of the exponential).

From the known properties of common materials used in DSCs and the normal operation frequencies, we find 5 values of about 10 to 80 mm. As the di-


mensions of the thermal pathways usually are in the region of a few millimeters, the exponential in Eq. (3.31) can be approximated with a linear function which results in a formula similar to that found for the simple RC-element. The situation is, however, totally different if we have materials with low thermal diffusivity like polymers. In this case the ~ drops to values of 1 to 4 mm. This means that the temperature amplitude changes already within one millimeter, i. e., the amplitude cannot be considered as constant or linearly decreasing in such a sample anymore but decreases exponentially. From the practical point of view we have to consider an apparent temperature amplitude inside the sample which is lower than the amplitude at the input boundary. To calculate it for a sample of thickness d in the linear case, we have to integrate Eq. (3.31):

1 d _~ TA o' ~ ( _4) (TA(d»=d!TA,O·e~dx=-~- l-e~

= TA,o [1- ~ : + ~ (:r -2~ (:)3 + - ... J (3.32)

This formula contains the case of a simple RC-element as the first approximation, which can easily be seen if we introduce ~ from Eq. (3.30) and (3.31), and divide by TA,o to get the transfer function.

Network of Heat Conducting Elements

The basic elements presented so far can even be helpful to model the properties of the total equipment, i. e., the apparatus (DSC) plus sample what concerns the heat transport influence on the measurement. In a first approximation the heat flows along a one-dimensional pathway through the DSC and into the sample. This can be modeled by a series of RC-elements (or a series of "rods" from different material properties in the thermal wave approach). The simplest model contains three elements: one, for instance, for the part of the pathway from the oven to the thermometer, one from the thermometer to the sample (or reference) support (or pan) and one from there to the inner of the sample (or reference). All these RC-elements have a different (apparent) thermal resistance and different (apparent) heat capacities, which, of course, are unknown for the user but almost fixed concerning the apparatus part (but they differ for the sample part).

From transfer theory it is known that for transfer elements connected in series the total transfer function is the product of the transfer functions from the elements. In other words, the magnitudes (amplitudes) have to be multiplied, whereas the phases connect additively. This way the total transfer function can be deduced from Eq. (3.28) with different 1) = Rth,j CP,j for the different RC-elements in question:

1 IP(w)l = IT and arg(P(w» = ~ tan-l (WRth,jCp)

j VI + (WRth,jCpY J

The more RC-elements are included, the more precise is the approximation of the real transfer function of the thermal pathway of the DSC, but the more (ini-


Fig.3.16. Transfer function 5 (Bode plot) of a simple DSC rad model (solid lines) together

I p 4 with that of a more detailed

I model (dotted lines) 3

QJ "0 0.10 :::J

0:: 2 :.c ""

Vl

'" OJ E Vl

'" .<:: c.

0.01 0 0.01 0.1 10 rad 5-1 10

w----

tially unknown) parameters are included in the respective correction formula and have to be determined for proper correction.

To show one result, the transfer behavior of the simplest possible complete DSC network is as follows: one RC-element on the reference side (collecting the heat flow path and the pan together to one apparent Rth and Cp) and two RC-elements in series on the sample side (the apparatus part as on the reference side, and the sample separated from the latter). The transfer function for this model (which corresponds to a DSC model in between the 1 st and 2nd approximation of Sect. 3.1) is again defined as the quotient of the output function (i.e., the ATsignal of the DSC) over the input function (i.e., the temperature of the heater). It can be calculated straight forward like the single RC-element, with the network rules in mind (for details see Hohne, 1999a; Hohne et al., 2002). The respective transfer function is plotted in Fig. 3.16. If we compare this with that from the simple RC-element (Fig 3.15) we see a qualitatively similar behavior: the amplitude drops with increasing frequency and there is an increasing phase shift. However, there are quantitative differences, the frequency dependence is stronger, i. e., the influence is larger for this network than for a single RC-element. Of course even this result is only an approximation, but it fits already rather well to the real DSC. The network can be refined including more RC-elements, e. g., the crucibles and the heat transfer resistance to them, this results in a modified transfer function (see Fig. 3.16 dotted lines) which give an even better approximation to the reality, but is not very different from the simpler model.

Several correction formulae have been derived this way in the literature which describe the DSC influence more or less satisfactory. We refrain from presenting the formulae here, they impart no more information to the practitioners than the above derived simple models. Instead we recommend to determine the transfer function experimentally and correct the measured quantities as shown in Sect.4.7.

3.3 Temperature-Modulated DSC (TMDSC)

3.3.4 Conclusions

63

The TMDSC method starts from a temperature program which contains a nonperiodic and a periodic part. This causes a heat flow rate signal with two components too, a non-periodic underlying part and a periodic part. The former can be separated by gliding averaging. Subtraction of the underlying from the total signal yields the periodic part which can be treated in different ways to get the information it contains. One possibility is to evaluate the amplitude, this results in the "reversing" curve and after subtraction from the underlying curve in the "non-reversing" curve. Another approach evaluates amplitude and phase angle and results in a complex heat capacity with magnitude and phase or real and imaginary part. Different processes contribute to the different signals in a different way. This offers the possibility to separate the processes from one another by choosing the right measurement conditions and the proper evaluation method. In every case the measured quantities as well as the properties of the sample (e. g. Cp) must not change during one period.

But the fundamentals presented in this section show even that the effective temperature amplitude inside the sample is never that of the heater which we program in our TMDSC and use for the calculation. It is always lower and we have to correct it to get proper results from the measurements. The correction depends on the frequency and the thermal diffusivity (i. e., the heat capacity and the thermal resistance) of the materials of the heat conducting path in the calorimeter but unfortunately even on the properties of the sample itself. This necessitates careful calibration of the TMDSC as far as the influence of the thermal wave is concerned. To reduce the influence of the sample itself and its (changing) properties - which normally are unknown - we have to reduce the thickness of the sample to the lowest possible size, or we have to determine this influence by means of a special procedure (see Sect. 4.7).

In summary, it may be said that temperature-modulated DSC is a very powerful method to measure material properties and to get more insight into sample behavior, in particular as far as time dependent processes are concerned, but the theory is not simple and one has to be careful in choosing the right measurement parameters and the proper evaluation. But it should be emphasized that linearity and stationarity of the whole system are essential conditions for the temperature-modulated method to reveal meaningful results.

4 Calibration of Differential Scanning Calorimeters

"Calibration means the set of operations that establish, under specified conditions, the relationship between values of a quantity indicated by a measuring instrument or measuring system ... and the corresponding values realised by standards" (from International Vocabulary of Basic and General Terms in Metrology, 1994). "Standards" stands for Certified Reference Materials, or for otherwise calibrated measuring instruments like temperature sensors, standard resistors and the like, which are connected to international fixed quantities. Calibration needs we1l4efined and tested procedures suitable to be done by well trained users in the laboratory.

A recommendation for DSC calibration, given by the International Confederation for Thermal Analysis and Calorimetry (ICTAC), exists which describes procedures for temperature as well as caloric calibration. This recommendation is founded on a metrological basis.

However, a metrologically flawless, basic calibration of a DSC should be done very carefully and is very time-consuming. For many routine applications of DSCs, e.g., for (relative) comparisons of material properties, other written "standards" exist which recommend to use selected parameters for the measurements, e. g., certain sample masses, specific heating rates etc. The standards (the written ones and the recommended reference materials) follow a "consensus philosophy" (Archer, Kirklin, 2000) which is based on round robin test procedures and average values.

Before using one of the "standard" procedures common in practice, it's a good idea to understand the principles of a metrologically correct calibration. This would enable the user to adapt (simplify) the correct calibration procedure to their specific purposes instead of using the "standards" uncritically without considering their limitations. A collection of papers on the calibration of calorimeters (DSC and others) has been published by Della Gatta et al.,2000.

To what extent DSCs are amenable to reliable calibration depends on the quality of the DSCs, the measuring system, the measurement parameters and on the availability of calibration substances with precisely measured properties. In this point, the situation is still quite unsatisfactory. Moreover, no complete theory of the DSCs exists, which takes full account of the influences of the instrument on the measurement and inherent systematic error sources. It should be borne in mind that it is the overall uncertainty of the calibration which yields the smallest possible systematic uncertainty of the measurements.

66 4 Calibration of Differential Scanning Calorimeters

4.1 Aspects of Quality Assurance

Thermoanalytical instruments, including DSCs, are extensively used by industry in quality assurance systems to warrant trustworthy, reproducible properties of the products, even products in the sense of measurement results by which products are characterized. The ISO 9000 series and the ISO/lEC 17025,1999 are the basic standards in the field of quality management and quality assurance. The standard ISO/lEC 17025, in particular, specifies general requirements to be met by test laboratories. It will no longer be sufficient to have technical and scientific competence, this competence must be documented and made apparent to others. DSC methods are used for quality assurance in many ways: for example, for the inspection of incoming materials, as an accompanying measure in manufacture and for the control of the final products.

To enable DSC methods to be useful for quality assurance, they, too, must be subject to the requirements of the relevant standards. Calibration of the DSC is an important precondition to fulfil these requirements. Calibration involves the application of sensitive calibration procedures and the use of calibration substances which are - if possible - traceable 1 to national/international standards and - in the end - to the SI-units. This means: to create confidence in the quality of the measured values and to avoid repeat measurements, metrologically flawless calibration procedures and traceable calibration substances (Certified Reference Materials) must be developed on the basis of which reliable uncertainty values can finally be assigned to the measurement results.

In the following sections calibration procedures and materials will be presented which are mainly based on investigations carried out by the German Society for Thermal Analysis (GEFTA), which are the basis of the ICTAC recommendations mentioned above (temperature calibration: Hahne et aI., 1990; Cammenga et aI.1993; caloric calibration: Sarge et aI., 1994; calibration general: Gmelin, Sarge, 1995; Sarge et aI., 1997; Gmelin, Sarge, 2000).

4.2 Basic Aspects of Calibration

In the following, reference will be made to some important thermodynamic aspects of calibration. The same considerations are valid for the phase transitions of a pure substance and for chemical reactions.

The quantity assigned to the substance is the enthalpy difference f...H of the phase transition (with T = constant); it is the difference between two variables of state and, consequently, well defined. In a calorimeter, it is the heat of transi-

1 Traceability: Property of the result of a measurement or the value of a standard whereby it can be related to stated references, usually national or international standards, through an unbroken chain of comparisons all having stated uncertainties (from International Vocabulary of Basic and General Terms in Metrology, 1994).

4.2 Basic Aspects of Calibration 67

tion Q which is measured. The relation between heat and enthalpy is given by the 1st law of thermodynamics and the definition of the enthalpy function:

dU=dQ +dw+ IdE; ;

dH=dU+pdV+ Vdp

(1 st law of thermodynamics)

(fromH == U + pV)

From this it follows (with dW= -pdV):

dQ = dH - V dp - I dE; ;

(U internal energy, W work, E; different energy forms).

The enthalpy H of a system is a function of the variables of state, namely pressure p, temperature T and composition ~. The total differential then reads:

dH=(aH) dP+(aH) dT+(aaH) d~ ap T,s aT P,s ~ T,p

Calorimetrically, the measured quantity, the heat in differential form is:

dQrn = [( ~H) -vJ dp + (~;) dT + (~H) d~ -7 dE; (4.1) P r.s p,S ~ T,p

Under the conditions normally prevailing during DSC measurements, the first term can almost always be neglected, even if closed crucibles are used (maximum pressure changes: 1 to 2 bar, i.e., dp :::: 0). The second term is the heat capacityat constant pressure and constant composition of the (reacting) system:

The third term in Eq. (4.1) is the isothermal and isobaric enthalpy change due, for example, to a phase transition (AH = AtrsH), a mixing effect (AH = ArnixH) or a reaction (AH = ArH).

Only if p :::: const. and I dE; = 0 and if the Cp course is known (namely the cor-;

rect baseline during a thermal event, d. Sect. 5.3) the following equivalence can be assumed for the melting of a pure substance [obtained by integration of Eq. (4.1)]:

Afus H = Qfus

It should be emphasized that the heat of melting equals the enthalpy difference only if the above restrictions are valid. A change in the surface already takes place if many small single crystals fuse together and form a bigger bead of


the melting substance. As a result, surface heat is exchanged (dEsurface =j:. 0) and Eq. (4.1) becomes

Almost the same is valid when deformation energy is exchanged during melting or when there is an interaction between the melted substance and the surface of the sample pan (wetting energy). Though the difference between the heat of fusion Qfus and the enthalpy difference fiR is small, these contributions must not be basically ignored. It is furthermore pointed out that common thermodynamics starts from infinite phases and the fiR definition is based on this fact. For smaller phases (small crystals), fiR depends on the size of the sample grains because the surface energy of the crystallites must not be ignored. In other words, the specific heat of melting of a fine powder differs from that of a large crystal and, as a result, the temperature of melting changes as well.

For the heat flow rate under ideal conditions (p =: const., IdE i =: 0) the following is obtained by differentiation of Eq. (4.1)

6Qrn dT (OR) d5 -= cPrn= C ~(T)-+ - -

dt p. dt 05 p.T dt (4.2)

The heat flow rate has two parts, the first is caused by the heat capacity of the sample and the second is the reaction heat flow rate. The former is the baseline, which must be subtracted from the measured heat flow rate to get the latter which forms the basis for the kinetic evaluation of DSC curves (cf. Sect. 6.3).

DSC measurements are normally performed both in the heating and cooling mode. Generally the temperature distribution in the DSC measuring system is asymmetrical with respect to heating and cooling. The reason for this is that the heat transfer is not strictly linearly dependent on the temperature difference. Therefore heat flow rates are basically different during heating and cooling of the same sample with the same rate. This results in different calibration factors for heat and heat flow rate. In practice, the effect is very small, i. e., normally sufficient symmetry with respect to heating and cooling can be assumed for heat and heat flow rate calibration factors. In other words, the uncertainty of heat and heat flow rates is normally larger than the asymmetry effect. This must, however, be verified and the procedures are part of a careful calibration.

It is even obvious that the temperature of the sample deviates from the set value differently on heating and cooling because of the thermal lag. This deviation can even be asymmetric if we heat or cool the sample at a certain rate because of non-linearity effects. This necessitates a separate calibration in cooling mode as well, which creates another problem, because the conventionally used calibration substances show supercooling effects for the relevant phase transition. Consequently, temperature calibration in heating and in cooling mode has to follow different procedures with different substances (see Sect. 4.3).

After the calibration experiments have been completed, the electronics provided for this purpose will either be adjusted until the values indicated corre-

4.3 Temperature Calibration 69

sponds to the true ones, or adaptation will be ensured via the control software (in a way usually not apparent to the user).

Any calibration already carried out by the manufacturer must be checked (verification of calibration) for a newly purchased equipment and again after appropriate periods. Regular calibrations provide important information about the repeatability error and any long-term systematic variations (drift) of the measured quantities. (It should be noted that repeatability error is not the same as uncertainty of measurement, cf. Sects. 7.2 and 7.3).

4.3 Temperature Calibration

Temperature calibration means the unambiguous assignment of the temperature "indicated" by the instrument to the "true" temperature.

The "true" temperature is defined by fixed points with the aid of calibration substances. It is reasonable to choose as calibration substances, if possible, the substances used to realize the fixed points of the International Temperature Scale of 1990 (lTS-90, internationally legal by law). The temperature "indicated" by the instrument must be derived from the measured curves, which usually requires extrapolation to zero heating rate in order to eliminate/minimize the influences of instrument and sample parameters, if possible.

Static methods (thermodynamic equilibrium) are applied to realize the fixed points of the temperature scale. In a DSC, these can be achieved only approximately. As the point of temperature measurement is not the point where the sample is located, a systematic error will always occur in scanning operation which depends on instrument and experimental parameters. To establish a graph or table, showing the relation between the indicated and the true temperatures in dependence on experimental parameters, is normally very helpful in practice and for uncertainty estimation. In every case, a table or graph should be drawn up which shows the variation of the indicated temperature at different heating rates. The calibration procedures described in the following take these special features into account in a general way, independent of the respective DSC type (see Hahne et aI., 1990).

In the case of an endothermic event, the DSC (in heating mode) records the heat flow rate signal schematically shown in Fig. 4.1. The section between the initial peak temperature Ti and the final peak temperature Tf is defined as the peak (see Sect. 5.1). The baseline needed to get the different temperatures can be interpolated by various methods between Ti and Tf (cf. Sect. 5.3.2). The intersection between the auxiliary line and the baseline suffices to fix the characteristic temperature Te (extrapolated peak onset temperature) which is important here.

4.3.1 Temperature Calibration on Heating

- Selection of at least 3 calibration substances (cf. Sect. 4.6.1) which cover the desired temperature range as uniformly as possible (it should be at least 3 in order to detect possible non-linear temperature dependence).

70

Fig.4.1. Heat flow rate signal of a DSC during a transition in heating mode (schematic representation, according to Hemminger, 1994). CD baseline (interpolated), @ auxiliary lines, Ti initial peak temperature, Te extrapolated peak onset temperature, Tp peak maximum temperature, Tc extrapolated peak offset temperature, Tf final peak temperature

r o c: Ol

V)


I I \

r---~Y~---~-----~-~-~------1\ 'I I I \ ,I I I I I I CD I I I I I I

T; Tf

tempera ture ... - At least two calibration samples of each substance are prepared (for repeat

measurements). The sample mass should correspond to that commonly used in routine measurements.

- The transition is to be measured with each calibration sample at a minimum of 3 different heating rates in the range of interest, including the smallest possible one. The second calibration sample of the same substance is also measured at the different heating rates.

- It is to be checked whether there is a significant difference between the characteristic temperatures (Te) obtained at identical heating rates for the first and second calibration sample of the same substance. If necessary, it is to be checked whether the temperatures depend on other parameters (mass, location of the sample in the crucible etc.).

- If this is not the case, Te is represented as a function of the heating rate and the extrapolated value Te(f3 ~ 0) determined for zero heating rate (Fig. 4.2).

- The difference Ll Teorr (13 = 0) between the value Te (13 ~ 0) obtained in this way and the respective fixed-point value TflX or the value Tlit taken from the literature (cf., e.g., Marsh, 1987; Cammenga et aI., 1993; Gmelin, Sarge, 1995; Sarge et aI., 1997) is either used to change the instrument calibration according to the manufacturer's instructions or it enters into a calibration table or curve (cf. Fig. 4.3).

- If Te depends not only on the heating rate but also on other parameters, these dependences should be represented accordingly (location of the sample in the container, position of the sample container in the measuring system, open/ closed sample container, sample mass, sample shape (foil, bead), atmosphere, material of sample container etc.).

Temperature calibration has then been completed.


o

(3 ~

Fig. 4.2. The extrapolated peak onset temperature Te as a function of the heating rate f3, and construction of Te(f3 ~ 0). ATe/6.f3 variation of Te (f3) with f3; O),@ different calibration substances

t

Fig.4.3. Temperature correction ATcorr (f3 = 0) as a function of the extrapolated peak onset temperature Te(f3 ~ 0) for three calibration substances (O),@,®). ATearr (f3= 0) is the difference between Te(f3 ~ 0) (cf. Fig. 4.2) and the "true" value of the temperature of transition. The curve A Tcorr (Te) obtained by means of (at least) three calibration substances shows the corrections to be applied to the measured values Te(f3 ~ 0) at different temperatures

How are the correct temperatures assigned in practice?

1. When the accuracy requirements for temperature measurements are high (e.g., thermodynamic investigations), the substance to be investigated is measured at various heating rates. The desired characteristic temperature (e. g., Te) is determined by extrapolation to 13 = 0 (static case, quasi-equilibrium temperature). The correction ATeorr (13= 0) is applied using the proper calibration table or curve:

Ttrue = Te(f3 -7 0) + ATeorr (13 = 0)

2. When the accuracy required in the determination of Te is not so high and/or the process to be investigated depends strongly on the heating rate ("kinetic"


processes), the value Te{f3~O) may be calculated from the mean slope ATe/Af3 of the Te{f3) curve{s) of the calibration substance{s) (cf. Fig. 4.2 and example below):

A Teorr (f3 = 0) is again taken from the calibration table or curve (Fig. 4.3), so that Ttrue = Te (f3 ~ 0) + A Teorr (f3 = 0) . The best thing to do is to start by carrying out two measurements at clearly differing heating rates. Then it is checked whether ATe/Af3 corresponds with the (mean) slope of the Te{f3) curve{s) of the calibration substances. If so, extrapolation to Te{f3 ~ 0) can be carried out at once. If not, first the heating rate dependence Te (f3) must generally be determined as described above.

To simplify the described method, for each heating rate applied, the corresponding overall correction A Teorr (f3) = A Teorr (f3 = 0) - f3 . A Tel Af3 can be listed (cf. example below) so that the "true" temperature can be determined directly:

Here, a distinction by classes of substances (metals, organic substances) must possibly be made because of their very different thermal behavior.

It is to be expected that, for a certain instrument, ATeorr {f3 = 0) will vary with time which is why a recheck should be made after some time (every week to every other month, depending on the DSC in question) by verification or recalibration. The dependence of the extrapolated peak onset temperature A Tel Af3 on the heating rate is only related to the heat transport properties of the DSC and the sample substance; it remains unchanged for a certain DSC and substance and need not, therefore, be regularly checked.

In order to assign the correct temperature to the individual phases in the case of complex thermal events with non-zero heating rate, an auxiliary line must be drawn at the angle a to the extrapolated baseline. The angle a can be taken from the calibration experiment in which the same heating rate has been applied (Fig. 4.4 a, b). The scale division of the axes must be the same in both cases; otherwise the slope of the auxiliary line must be converted properly. (For more advanced methods of determining characteristic temperatures in the case of endothermic and exothermic events see Sarge, 1991; Schawe, 1993).

Uncertainties

Depending on the type of DSC, the minimum repeatability error of the determination of Te on pure metals amounts to approx. ± 0.02 K (sample exactly in the same place in the crucible or measuring system); in the other cases it varies between 0.1 and 0.8 K. An overall uncertainty of Te measurement between 0.3 and 1.0 K must be reckoned with. The overall uncertainty of the temperature calibration should in every case be carefully estimated (including uncertainty of temperature sensors, uncertainty of the determination of Te etc.).


CI c:: en

·Vi ...... _.-:;

b

tempera ture

T,1

73

TP2

temp ern ture Fig. 4.4 a, b. Assignment of the characteristic temperature Tp in complex thermal events. a Calibration measurement to determine the angie a at a specific heating rate. b Construction of characteristic peak maximum temperatures Tp of a complex thermal event with the aid of angle a (measurement with the same heating rate as for a). In heating operation, due to the thermal lag, each endothermic thermal event in the sample is "indicated" too late, i. e. at too high a temperature. To find the "true" temperature of a characteristic segment of the measured curve in good approximation, a lower temperature must always be assigned instead of the indicated temperature (read at an angle of 90°). This is done with the aid of the angle a. In the case of exothermic events the true temperature can be higher than the measured temperature (see T p3)


Notes:

1. It has to be checked whether Te depends on the location of the calibration sample in the sample container (in particular at high temperatures). In particular, for power compensation DSCs, due to the isoperibol operation, a temperature profile develops in the sample support (in bottom plate of the microfurnace, Fig. 4.5) which - depending on the sample position - results in an earlier (center) or later (boundary) melting of a calibration sample - and thus in a lower or higher temperature Te indicated (see Hahne, GlOggler, 1989). Such effects are also possible in heat flux DSCs (cf. Sect. 3.1), however, to a much lower extent due to the non-isoperibol operation. In cylinder-type measuring systems, above all the dependence on the (vertical) position of the sample in the cylindrical container must be checked.

2. In the case of exothermic events, the sample temperature can be higher than the measured temperature. The amount of this deviation depends strongly on the heat produced and on sample properties and cannot be precisely determined. A reliable assignment of a temperature is therefore only possible at the beginning of the exothermic event.

The increase in the sample temperature in the course of the reaction can be estimated on the basis of the following considerations:

Fig.4.5. Temperature profile in the microfurnace of a power compensation DSC (schematic). The isoperibol mode of operation (constant ambient temperature) results in a temperature profile in the sample support as shown schematically. During melting of a sample, this temperature profile leads to time-dependent zones of melting (Tfw is first reached in the centre) and consequently to different extrapolated peak onset temperatures Te when the radial position (r) of small samples differs (according to Hohne, GlOggler, 1989)


Fig. 4.6 a, b. Heat resistance Rth between temperature measurement point and sample. a Diagrammatic view of the sample system. b Curve measured during melting of a pure substance, q, heat flow rate into the sample, tany ascending slope (dq,/dt) of the q,(t) curve during melting (to calculate Rth)

75

time ..

If there is a thermal resistance Rth between sample (Ts) and sensor temperature (TM) (Fig.4.6a, b),

(4.3)

is valid for the difference between the indicated temperature T M - calibrated as described above - and the sample temperature Ts considered to be homogeneous. When a pure substance melts (as for calibration, for example), Rth can be determined approximately from the slope of the respective heat flow rate curve 4>r(t). From Eq. (4.3) the following results after differentiation (with Ts = constant during melting, dTM/dt:::: f3 and tan y= (d4>r/dt)melting):

f3 Rth ". -- (4.4)

tan y

(Experiments at various heating rates shows the validity of this relation.)

Arguments in Support of the Temperature Calibration Procedure

The existing recommendations for the temperature calibration of DSCs (e. g., ASTM E 967-97: Standard Practice for Temperature Calibration of Differential Scanning Calorimeters and Differential Thermal Analyzers, 1997) and the spec-

76

1 Te

m L


326,-_,,--_,,--_,,--_-,--_-,--_-,--_.J......._ o 10 20 30 K/min 40

{3 ~

Fig.4.7. Extrapolated peak onset temperature Te in DC as a function of the heating rate f3 (two measurement series of a heat flux DSC with 58 mg of lead) Q-{) measured values, .... curve of average values. The curves show a non-linear dependence Te (f3) only towards the highest heating rate and a dispersion of the Te-values, which depends on f3

ifications of most manufacturers take a standard heating rate (e.g., 10 K min-I) as a basis. This method cannot be recommended since

- different heating rates require different corrections; shifting of Te is not always linear due to the heating rate (cf. Fig. 4.7);

- the sample temperature is equal to the measured temperature only at zero heating rate; to ensure safe extrapolation to zero heating rate, measurements should be carried out at at least 3 different heating rates (starting with the lowest heating rate),

- the temperature fixed-points of the International Temperature Scale are defined for the reference substances in phase equilibrium (i.e., static case, zero heating rate).

Only Te can be used as a characteristic temperature of a peak to define the DSC's temperature scale (cf. Fig. 4.1), the reasons being the following:

- Ti cannot be determined with the required reliability because of the noise; the same applies to Tf (the repeatability error is between 2 and 15K depending on substance, kind of transition etc.)

- Tp and Tc strongly depend on the thermal conductivity, on mass and layer thickness (volume) of the sample substance, on the heating rate and on the heat transfer from sample to sample container (crucible), which may change on melting (cf. Fig. 4.8).

- Te depends least on heating rate and sample parameters (substance, thermal conductivity, mass, layer thickness); any possible effect of melting (heat transfer) should be checked by carrying out various similar experiments with the same calibration sample.


o

2

cJ>m

+ 4

6 300 320 340 360 380 °C 400

tempera ture ~

Fig.4.8. Measured curves showing the peak temperature maximum Tp changing with the heating rate p (heat flux DSC, lead, 58 mg, heating rate p from 5 to 50 K min-I). 4'm heat flow rate (arbitrary units). In addition to the shifting of Tp with p, the great changes of Tc and Trare obvious (for definitions see Fig. 4.1)

Examples of Temperature Calibration on Heating

The results of the temperature calibration of a power compensation DSC in heating mode are given as an example. Two samples each (between 2 and 6 mg) of highly pure indium, tin and lead were used (in hermetically closed aluminum crucibles). The results shown in Tables 4.1 to 4.3 were obtained for the non-calibrated device (cf. Figs. 4.9 a to c):

The correction curve shown in Fig. 4.10 has been obtained for the calibration of the device.

In practice, for the purpose of temperature calibration, the potentiometer adjustment would in this case be changed by the mean value of 0.83 K. If necessary, the continuation of the correction curve towards higher temperatures may be determined on the basis of another calibration (e.g., with Zn). Different corrections must possibly be applied in different temperature ranges. Nowadays with modern DSCs, the user normally enters the results of the calibration measurements (for example, with In and Pb) into the computer which internally calculates the correction curve and converts the temperature scale. Unfortunately this is in most cases only done in linear approximation (slope and position, 2-point calibration) which is not sufficient for precise measurements.

After the DSC has been calibrated with the aid of !lTeorr (f3= 0), various pure substances are measured at different heating rates and Te is determined respectively. Table 4.4 is obtained and includes !lTeorr(f3= 0) = Tlit - Te(f3 ~ 0), the corrections for the various heating rates calculated from !l Teorr (f3 = 0) and the slope of the respective linear fit curve:

!lTe !l Teorr (f3) = !l Teorr (f3 = 0) - f3-

!lf3

(The values for !lTeorr (f3) have in general been rounded to two decimals.)


Table 4.1 Temperature calibration with indium (cf. Fig. 4.9a)

Sample No.

1 2 2

2

2 1 2

2 2

2

Results:

Run No.

2

2

2

Te (f3 ~ 0) = 155.523 °C TflX = 156.5985 °C (ITS-90) f:.Tcorr (f3 = 0) = + 1.0755 K f:.Te/f:.f3 = 0.0682 K/(Kmin- 1)

Heating rate in K min- 1

10.0 10.0 10.0 10.0 5.0 5.0 2.5 2.5 1.0 1.0 0.5 0.5 0.5 0.1 0.1

Table 4.2 Temperature calibration with tin (cf. Fig. 4.9 b)

Sample No.

1 2

2 1 2 1 2

2

2

Results:

Run No.

2 1

Te (f3 ~ 0) = 231.184 °C TflX = 231.928 °C (ITS-90) f:.Tcorr (f3 = 0) = + 0.744 K f:.Te/f:.f3 = 0.103 K/(Kmin- 1)

Heating rate in K min-1

10.0 10.0 10.0 5.0 5.0 2.5 2.5 1.0 1.0 0.5 0.5 0.1 0.1

Te in °C

156.21 156.16 156.25 156.21 155.81 155.91 155.65 155.67 155.64 155.65 155.52 155.65 155.53 155.49 155.50

Te in °C

232.35 232.18 232.06 231.81 231.69 231.54 231.41 231.33 231.22 231.28 231.12 231.26 231.12


Fig.4.9a-c. Extrapolated 156.5 peak onset temperature Te 7;lp.O) ISS.5n t (in 0c) as a function of the

i t 7i. """4t~ heating rate f3 to determine M .. , IP= 0)= .1.111 K Te(f3 ~ 0). (Power compen-

156.0 sation DSC, two measure- 0

ment series each with two T.

~ In samples of different mass,

closed aluminum crucibles). a Sample material: indium, 155.5 To: 155.523 + 0.0682 . P b Sample material: tin, a c Sample material: lead. 5 K/min 10 Te(f3 ~ 0) extrapolated peak (3-onset temperature at zero heating rate, TflX fixed-point

232.5 temperature, i. e., true tem- T.lp.O) 231.1841"( 0 perature of melting (ITS-90, T'i, 231.9681 '(

0

in the case of lead converted '(

to the ITS-90), 6.Tcorr = LlTco"IP= 0)= .0.784 K

TflX - Te(f3 ~ 0) temperature

i m.o

correction

T. 231.5 Sn

€J 0.9--'

T.: 231.1841 • 0.103 .p 0

b 231.0 5 K/min 10

(3 -328.0

t 0

T,lp.O) : 326.793 '(

328.0 7ij, : 321.502 '( 0

LlT,o"IP= 0)= .0.709 K 0

! 321.5 9

T. 0

Pb

T.= 326.793 • 0.0997 . P

c 326.5 K/min 10


Table 4.3 Temperature calibration with lead (cf. Fig. 4.9c)

Sample No. Run No. Heating rate in K min~l Te in °C

1 10.0 328.22 2 10.0 327.50 3 10.0 327.52

2 1 10.0 327.95 2 2 10.0 327.87

5.0 327.10 1 2 5.0 327.09 2 5.0 327.42

2.5 326.94 2 2.5 327.23 1 1.0 326.76 2 1.0 326.97 1 0.5 326.78 2 0.5 326.95 1 0.1 326.75 2 0.1 326.93

Results: Te (fJ ~O) = 326.793°C Tnx = 327.462°C (converted from the IPTS~68 to the ITS~90) !:lTeorr (fJ = 0) = + 0.669 K !:lTe/!:lfJ = 0.0997 K/(Kmin~l)

1.4

K In

t 1.0

0.6 Sn f':.. Tcorr(P=O) Pb

0.2

150 200 250 300 ·C 350

Te (P """0) .. Fig.4.1O. Correction curve for the temperature calibration of a power compensation DSC. The individual corrections !:lTeorr(fJ = 0) have been taken from the results of the calibration measurements shown in Figs. 4.9 a to c. !:lTeorr(fJ = 0) difference between true temperature (fixed~point temperature) and extrapolated peak onset temperature Te(f3 ~ 0) for zero heating rate (for Pb see legend of Fig. 4.9), ---- mean value: +0.83 K


Table 4.4 Measurements with various substances to determine ~ Te/~{3

Diphenyl ether (phenoxybenzene)

Te in Heating rate in °C Kmin- 1

27.05 0.5 27.10 l.0 27.20 2.5 27.73 5.0

Gallium

Te in Heating rate in °C Kmin-1

29.92 0.5 29.97 l.0 30.11 2.5 30.35 5.0 30.94 lO.O

C33H68 (paraffin) 1st transition

Te in Heating rate in °C K min-1

67.35 0.5 67.36 l.0 67.47 2.5 67.66 5.0 68.06 10.0

C33H68 (paraffin) 2nd transition

Te in °C

70.98 70.98 71.10 7l.27 7l.70

Benzoic acid

Te in °C

122.28 122.23 122.34 122.46 122.72

Heating rate in Kmin- 1

0.5 l.0 2.5 5.0

10.0

Heatin~ rate in Kmin-

0.5 l.0 2.5 5.0

lO.O

~Ttot({3) in K

-0.19 -0.24 -0.34 -0.87

~ Ttot ((3) in K

-0.16 -0.21 -0.35 -0.59 -1.18

~ Ttot ((3) in K

0.05 0.04

-0.07 -0.26 -0.66

0.12 0.12 o

-0.17 -0.60

0.06 0.11 o

-0.12 -0.38

Tlit = 26.86°C Te({3 --+0) = 26.93°C ~ T corr({3 = 0) = - 0.07 K

Tlit = 29.7646°C = Tfix Te({3 --+ 0) = 29.851 °C ~Tcorr({3 = 0) = -0.09 K

Tlit = 67.4°C Te({3 --+0) = 67.3°C ~Tcorr({3 = 0) = +0.1 K

Tlit = 7l.1 °C Te ({3 --+ 0) = 70.91 °C ~Tcorr({3 =0) = +0.2K

Tlit = 122.34°C Te ({3 --+ 0) = 122.22°C ~Tcorr({3 =0) = +0.12 K


Table 4.4 (continued)

Indium

Te in Heating rate in tJ. Ttot (f3) in °C Kmin-1 K

156.63 0.5 - 0.03 Tlit = 156.5985 DC = Tfix 156.67 1.0 - 0.07 Te({3 ~O) = 156.53 DC 156.80 2.5 - 0.20 tJ. Teorr ({3 = 0) = + 0.07 K 156.96 5.0 - 0.36 157.32 10.0 - 0.72 158.01 20.0 - 1.41 159.34 40.0 - 2.74 162.17 80.0 - 5.57 168.60 160.0 -12.00

Tin

Te in Heating rate in tJ.Ttot ({3) in DC Kmin-1 K

232.18 0.5 -0.25 Tlit = 231.928 DC = Tfll{ 232.21 1.0 -0.28 Te({3 ~ 0) = 232.l2 DC 232.34 2.5 -0.41 tJ. Teorr({3 = 0) = - 0.19 K 232.52 5.0 -0.59 233.Ql 10.0 -1.08

Caffeine

Te in Heating rate in tJ.Ttot ({3) in DC Kmin-1 K

236.30 0.5 -0.20 Tlit = 236.1 DC 236.30 1.0 -0.20 Te ({3 ~ 0) = 236.24 DC 236.36 2.5 -0.26 tJ.Teorr({3 = 0) = -0.1 K 236.67 5.0 -0.57 236.96 10.0 -0.86

Lead

Te in Heating rate in tJ. Ttot ({3) in DC Kmin-1 K

237.99 0.5 -0.53 Tlit = 327.462 DC 238.00 1.0 -0.54 Te({3 ~ 0) = 327.93 DC 238.17 2.5 -0.71 tJ.Teorr({3 = 0) = -0.47 K 238.32 5.0 -0.86 238.81 10.0 -1.35


Table 4.5 The dependence of Te on the heating rate f3 for various classes of materials (results from Table 4.4)

Substance

Gallium Indium Tin Lead

Mean value metals

Diphenyl ether Paraffin C33H68 Benzoic acid Caffeine

Mean value organic substances

Overall mean value

a Standard deviation On-I.

0.11 0.07 0.09 0.09

0.09±0.02a

0.15 0.08 0.05 0.07

0.09 ± 0.04a

0.09±0.03a

ATtot = Tlit - Te (P) is defined as the difference between the "true" temperature Tlit and the extrapolated peak onset temperature at a heating rate p.

Obviously, for this calorimeter there is a linear relation between the shift of Te and the heating rate p. The gradient ATe/AP of the respective fit-line yields the values given in Table 4.5.

From this it follows that for this DSC, at a heating rate of 10 K min-I, a mean value of 0.9 ± 0.3 K results for the difference between Te(P) and the true temperature of the transition. But, depending on the heat transfer between substance and crucible bottom, the difference may vary between 0.5 and 1.5 K. This means an uncertainty of ± 0.5 K for the temperatures at this heating rate. Separate measurements at different heating rates must be carried out for a certain substance in order to determine the slope and thus the transition temperatures more accurately.

Note:

The values obtained for Te in the first measurement of a calibration sample (In, Sn, Pb) are systematically higher than those of the second and all subsequent measurements (at a given heating rate). The reason for this is the heat transfer between the sample and the bottom of the crucible, which has strongly improved after the first melting (larger and better contact). As a result, Te changes. If, for reasons of irreversibility of the process to be investigated, only the first measurement of a sample can be evaluated, this effect must also be taken into account when the uncertainties of Te (P ~ 0) are estimated. Another striking feature of the results of the calibration measurements is that the Te of individual samples of the same substance are very well situated on a straight line, but that some of the straight lines deviate strongly from one another (e.g., Figs. 4.9a to c). The


328.0

°C

327.8

1 327.6

Te 327.4

o 2 4 6 8 K/mln 10 {3 ~

Fig.4.11. Extrapolated peak onset temperatures T. of lead (0.2 mg) as a function of heating rate p with the samples in various positions (1 to 4) in the microfurnace of the power compensation DSC (cf. Fig. 4.5, according to Hohne, GlOggler, 1989)

reason is that, firstly, the sample containers and thus the heat transfer path to the microfurnace are not identical, i. e., each sample encounters different contact points and heat flux conditions, and that, secondly, the location of the sample in the microfurnace of the power compensation DSC clearly influences the Te of the measured curve. This is obvious from the results shown in Fig. 4.11 (cf. also Hahne, Glaggler, 1989).

4.3.2 Temperature Calibrati.on on Cooling

In measurements at negative heating rates (cooling), the sample temperature is higher than the indicated temperature. As a result, the correction !1 T (f3) from the calibration table or curve must be applied with the sign reversed as compared with heating (see Fig. 4.12).

The procedures to be applied comprise a check whether the calibration in heating mode, performed before, is also valid in the cooling mode (symmetry check). If an asymmetry is found, a separate calibration for this mode has to be done; suitable procedures and substances are given here which have been adopted from a paper of Sarge et al., 2000.

Generally, substances with two types of phase transitions are recommended. These are substances with first order phase transitions (liquid/solid) with only small as well as reproducible supercooling, and substances with phase transi-


PCII'

o .--- (Doling rate

D. T(P)

Pbf.at

heating rate --

85

Fig.4.12. Schematic representation of the temperature corrections in the heating and cooling mode

tions of higher order (e.g., between different liquid crystal states) with no or negligible supercooling.

Symmetry Check

- From the list of substances recommended (see Table 4.12), one substance is selected, whose phase transition lies within the temperature range of interest.

- One sample is weighed in, the mass should be in the range normally used. - With this sample the transition is to be measured at three different heating

and cooling rates, respectively. At least two runs each should be done. - For each peak obtained the extrapolated peak onset temperature Te or the

peak maximum temperature Tp (in accordance with Table 4.12) is determined.

- It is to be checked whether the results from the two experiments differ significantly.

- If this is not the case, Te is plotted as a function of the heating rate and Tp normally as a function of the square root of the product of heating rate and sample mass (cooling rates count negative). Regression lines are determined separately for heating and cooling sections and these are extrapolated to zero rate. The two temperatures obtained this way are compared with the respective true temperatures Ttrue , taking a possible supercooling to be expected into account.

- Significant discrepancies indicate an instrumental asymmetry and a separate calibration for the cooling mode is to be performed.

One example of such a symmetry test of three power compensated DSC, where good symmetry is established, is presented in Fig. 4.13.


340.6

K

r 340.2

340.0

.. 339.8 .... ::. 339.6

o -----:::=: _0--- _____ _0 0- c __ c

0- ~

~---- c __ - x

c- x----O~ x--c -~ c __ x---____ x

x 339.4

339.2 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 K/min 2.5 (3 ~

Fig.4.13. Test of the symmetry of three different power compensation DSCs with respect to the heating and cooling mode using a liquid crystal (according to Hohne et al., 1993)

Calibration procedure

The calibration procedure is similar to that in heating mode (see Sect. 4.3.1) except for those substances where the peak temperature is to be evaluated and Tp should be plotted as a function of the square root of the product of heating rate and mass of the sample. For details see Sarge et al., 2000, where also an example of a calibration in cooling mode can be found.

In many cases results of a calibration in the cooling mode are different from those obtained in heating mode, but, on the other hand, additional calibrations are time consuming and therefore steered clear of. In addition, a calibration in the cooling mode is not as accurate as a calibration in the heating mode. As a consequence, the temperatures measured in cooling mode are generally more uncertain than those determined in heating mode.

4.4 Caloric Calibration

By means of caloric calibration (for a review see Sarge et al., 1994), the proportionality factor between the measured heat flow rate cJ>rn and the true heat flow rate cJ>true on the one hand, and between the measured exchanged heat Qrn and the heat Qtrue really transformed, on the other hand, is to be determined:

cJ>true = Kif>' cJ>rn and Qtrue = KQ • Om

Strictly speaking cJ>rn in this equation should be considered the measured heat flow rate with the instrument zero line already subtracted, but as all correction calculations are usually done using the measured curve cJ>rn itself, only this is used in the following.

4.4 Caloric Calibration 87

The calibration is carried out either as "heat flow rate calibration" in the (quasi -) steady state

- by electrical heating applying the well-known power, - by "charging" the known heat capacity of the calibration sample

(cf. ASTM E 968-99: Standard Practice for Heat Flow Calibration of Differential Scanning Calorimeters, 1999),

or as "peak area calibration" by integration over a peak which represents a known heat

- by electrical heating applying the well-known energy, - by applying the known heat resulting from a phase transition (melting) of a

pure substance

(cf.ASTM E 793-01: Standard Test Method for Enthalpies of Fusion and Crystallization by Differential Scanning Calorimetry, 2001).

Since Qtrue = f IPtruedt and Qm = f (IPm - <!>t,l) dt, KcI> and KQ should be identical which is not, however, the case because in practice - throughout the duration of the peak - KcI> depends on the temperature T (and therefore also on the time t) and is in addition a function of IP (cf. Sects. 3.1 and 4.4.3). As a result, the equation IPtrue = KcI>' IPm can indeed be integrated but KcI> must not, however, be placed in front of the integral. As stated already, KQ is not equal to KcI>; KQ is rather a kind of integral mean value of KcI> over the area of the peak. In practice, the difference between the two calibration factors is between 0.5 and several per cent. Both types of calibration must therefore be carried out separately.

The advantages of the twin principle of DSCs become fully effective only in the case of perfect thermal symmetry of the measuring system. In this case, however, the measured signal is zero. In real measurements there will always be asymmetries in the temperature field. The effects of such asymmetries are dealt with in Sect. 4.4.3. Another problem leading to the same effect is the non-linear heat exchange via radiation and convection dealt with in Sect. 3.1. The conclusion to be drawn for heat calibration is as follows:

The thermophysical behavior of calibration sample and sample to be measured must be as similar as possible. As this is possible only approximately, systematic errors exist which must be estimated and included in the overall uncertainty of measurement (Sect. 7.3).

4.4.1 Heat Flow Rate Calibration

In almost all DSCs commercially available, a heat flow rate signal IPm is internally assigned as a measurement signal for the actual measurement signal flT. [When the measurement signal is put out as a voltage flU (for example in mY), the following applies analogously, IPm being replaced by flU.] The heat flow rate calibration defines the functional relation between IPm and the true heat flow rate IPtrue absorbed or emitted by the sample: IPtrue = KcI>' IPm (steady state, IPm with the zero line - i. e., empty crucibles, definitions see Sect. 5.1 - already subtracted).


The proportionality factor (calibration factor, calibration function) K~ usually depends on parameters such as temperature and - what is most important -heat flow rate. In some DSCs, K~ is made unity by electronic or software means. In these cases, too, the relation between cPm and cPt rue must be carefully checked.

The heat flow rate calibration can be performed in two ways:

1. Installation of an electric calibration heater in the place of the sample or inside the sample. This method offers the following advantages: - the electric power (heating power) can be measured easily and with high

accuracy, - heat flow rates of differing intensity can be generated without the calibra

tion set-up being modified, - the steady state can be adjusted for any period of time desired; the result

ing conditions are most similar to those of a Cp measurement, - the heater can be switched on or off at any temperature desired so that the

position of the baseline (cPtrue = 0) can also be checked in between (even with a sample placed in the crucible),

- by appropriately presetting the development of the heating power with time, measurement effects (peaks) can be "repeated" (simulated) so that heat flow rates leading to such a peak can be assigned without "desmearing" (see Sect. 5.4),

- the time constant of the measuring system can be easily determined (cf. Sect. 7.2).

The disadvantages of the electric heat flow rate calibration are the following:

- the heater can hardly be installed in disk-type measuring systems, - heaters permanently installed in measuring systems are not situated at the

sample location; this leads to systematic errors, - heat fluxes in the wires lead to systematic uncertainties, "heat leaks".

(Ensure thermal symmetry of sample and reference side!)

Figure 4.14a shows the electrical calibration in a DSC with cylinder-type measuring system. (Peak area calibration may also be carried out using the peak areas furnished by these measurements; see below.) The resulting calibration curve K~(T) = cPtrue(T)/( cPm (T) - cf>t,l (T» is represented in Fig. 4.14b; cf>t,l is the baseline heat flow rate (cPm-curve without electric power).

2. Heat flow rate calibration can also be carried out with a sample of known heat capacity (Fig. 4.15). The following is valid for the heat flow rate absorbed by the sample (without reference sample) in (quasi-) steady state heating mode:

K (T) _ Cp(T)· f3 cPtrue = Cp • f3 so that I/> - cPm (T) _ cPo (T)

(cPo - zero line value with empty crucibles).

The advantages of this calibration method are the following:

- applicable in all DSCs, - calibration heat flux at the sample location, - no leads (wires) required and thus no additional heat leak.


Fig. 4.14a, b. Electrical heat flow rate calibration of a heat flux DSC (cylindertype measuring system) by means of a built-in heater. a Calibration peaks generated electrically in isothermal operation or during heating (exo up). b Calibration factor K,p (calculated with the data of a), schematic) to determine the reaction (real) heat flow rate: K,p(n = ~truel (~m<n - 4>J,1(D) (with ~true as electrical heating power Pel), ~m measured heat flow rate, 4>J,1 baseline value of the heat flow rate

Fig.4.15. Heat flow rate calibration by means of a known heat capacity Cpo ~m measured heat flow rate, ~o zero line value calibration factor to determine the true heat flow rate: K,p= Cp<nf3I(~m<n - ~o<n)

r ¢m

a

r

89

¢m- ¢bl

4't,1

time, tempera ture ~

tempera ture

steady state region

.,.~

temperature ~


The disadvantages are:

- The calibration cannot be switched off in between, i. e., checking of the baseline (or zero line) is not possible during a run (this leads to uncertainties in cI>m - cI>o),

- there is a temperature profIle inside the sample (this leads to a "mean CP").

The calibration substance most frequently used is synthetic sapphire (a-AI20 3 ,

corundum), cf. Table 4.9.

4.4.2 Heat (Peak Area) Calibration

For peak area calibration, a known heat Qtrue, dissipated or consumed, is compared with the area of the resulting peak (Figs. 4.16 and 4.17) and the following is valid:

tf

Qtrue = KQ (T)f [ cI>m (t) - cI>t,\ (t )] dt or ti

tf

Qtrue = f [K~ . cI>m (t) - K~ . cI>t,\ (t)] dt ti

In principle, K~"# K;' is valid. Since K~ depends on cI>m and T - which is proportional to time in all heat flux DSCs and also in power compensation DSCs -, it cannot be placed in front of the integral. This can be done in a first approximation only if it is assumed that cI>m and cI>t,\ are of the same order of magnitude (no dependence on cI» and the peak width is small (no dependence on T), which is in general the case with phase transitions of organic or inorganic substances, but not, however, with melting and crystallization of many polymers.

Integration must be carried out over the whole peak in order that contributions of the 1st derivative of the measurement signal are insignificant.

Fig.4.16. Electrical peak area calibration. tPm measured heat flow rate, tio tf beginning and end of the calibration peak, 1 ~l baseline value of the heat flow rate (electric power switched off)

tf

f(¢m-¢"Idt= A

Ii

~ ~ time ...


Fig. 4.17. Peak area calibration by means of a known heat of fusion (schematic). 4im measured heat flow rate, ti, tf beginning, end of the t peak, 4it.l baseline value of the heat flow rate

time

Peak area calibration can be performed in two ways:

91

A

1. If electrical calibration is possible (Fig. 4.16), the advantages connected with electrical heat flow rate calibration (see above) apply analogously to peak area calibration: - Easy and accurate measurement of the electrically generated heat

Qtrue = - f Pel. dt = - WeI. = - U· i . 11 t ,

- peaks of differing size can be produced, - calibration is possible at any temperature, - measurement effects (peaks) can be "repeated" (simulated) byappropri-

ately presetting the development of the heating power with time, - measurement effects (peaks) can be "encompassed" during the run by

similar calibration peaks.

The disadvantages are the same as those stated in the case of electrical heat flow rate calibration:

- The electric heater can hardly be installed in disk-type measuring systems, - systematic errors result if heaters are permanently installed (different loca-

tions of heater and sample), - wires give rise to systematic uncertainties (heat leaks).

Care must be taken that electrical calibration is carried out only over a small temperature interval.

2. Peak area calibration of DSCs is usually carried out by means of well-known heats of melting of pure substances (Fig. 4.17). The advantages of this method are as follows: - applicable in all DSCs, - calibration heat at the sample location, - no wires required, - simultaneous temperature and heat calibration is possible with some (se-

lected) calibration materials.


The disadvantages are:

- calibration possible only at discrete temperatures, - no adaptation to the measured peak possible, - systematic uncertainties due to the special shape of the sample tempera-

ture curve Ts (t) during the calibration procedure (melting), - uncertainties resulting from the determination of the area (definition of

the integration limits and shape of the baseline, ef. Sect. 5.3).

Procedure

The peak area calibration procedure for heat flux DSCs is thus as follows:

- Selection of calibration substances which cover the desired temperature range and whose thermophysical characteristic data are similar to those of the sample,

- weighing-in of such masses which approximately generate a heat effect as is found in normal measurements,

- adjustment of customary heating rates. (Be careful in the case of calibration substances which melt close to the start temperature: the quasi-steady state must have been reached, the relaxation effects due to the start must have faded; apply lower heating rate if necessary),

- evaluation of peak (area and extrapolated peak onset temperature Te), - determination of KQ(Te), drawing of a calibration curve or (establishing of a

table) or inputting of measured data into the apparatus according to the manufacturer's specifications (turning of potentiometer, or by software),

- estimate of the uncertainty of the calibration (uncertainty of the weighing, of the baseline, of the integration limits, of the values for the heat of fusion taken from the literature; the estimated uncertainty from theoretical considerations must be taken into account too),

- measurement of the repeatability errors of the calibration factors (or calibration curves). This repeatability error must be clearly smaller than the estimated overall uncertainty of the calibration (see above). The repeatability error is the smallest possible uncertainty of measurement of caloric measurements.

The same calibration procedure is applied to power compensation DSCs,however, in this case, calibration with one calibration material may be sufficient (e.g., In, ef. Sects. 2.2, 3.2 and Fig. 4.20), as the calibration factor depends on temperature only to a smaller extent than the uncertainty of heat determination generally is.

4.4.3 Examples of Caloric Calibration

Calibration Curve

A DSC with cylinder-type measuring system was calibrated with the aid of electric heaters which could be inserted into the sample containers. Only the insert in the sample container was electrically heated, the second insert of the same type placed into the reference container served to establish thermal symmetry.

4.4 Caloric Calibration 93

t:.u

Itemperature), time ~

tempera ture ~

Fig.4.18. Electrical heat calibration of a heat flux DSC (cylinder-type measuring system with built-in heater, schematic curves). L'lU measured signal (voltage), KQ calibration factor for peak area evaluation (in J V-I S-I), A peak area (in V s), Wei electrical heating energy (belonging to one peak: i . L'lU . M). In contrast to Fig. 4.14, the peak areas of the electrical calibration peaks were evaluated here to obtain KQ• It is also possible to simultaneously determine K<[> from the electrical heating power

At regular intervals, heating pulses of defined duration and power were automatically generated over the whole temperature range, the measured signal being a voltage in ]l V. The peak areas were integrated applying evaluation software. Figure 4.18 shows schematically the measured curve and the resulting calibration curve.

A heat flux DSC with disk-type measuring system was calibrated with the aid of sapphire, the measured curves are shown in Fig. 4.19. From these curves the cp (1') function can be calculated (cf. Sects. 5.3.1 and 6.1). Comparison with literature values (Table 4.9) enables to calculate the calibration curve.

Differences Between K<p and KQ

When DSCs are calibrated with the aid of the heat of fusion of, for example, metals (peak area calibration) and via the specific heat capacity (heat flow rate calibration), significant differences occur in practice. This is shown in Fig. 4.20 by the example of a power compensation DSC.

The 1 % difference in the case of a power compensation DSC suggests that the difference will be substantially greater for heat flux DSCs (see Chapter 3). This is confirmed by measurements:


B

mW

4

1 0

-4

~ -B 0

~

3 ~

-12 0 OJ .c

-16

-20

00 4.0 B.O 120 16.0 20.0 24.0 2B.0 32.0 36.0 min 44.0

time ... Fig.4.19. Heat flow rate calibration of a disk-type DSC (endo down). Upper curve: Crucibles empty (zero line), lower curve: 129.6 mg of sapphire, 10 K min-I, (according to Sarge et aI., 1994)

300 500 K 600

tempera ture ..

Fig.4.20. Systematic difference between K.p and KQ in a power compensation DSC. Heat flow rate calibration was carried out with sapphire (solid line) and with copper (crosses); peak area calibration with In, Sn, Bi, Pb. (according to Hohne, Gloggler, 1989). KQ calibration factor for peak area evaluation, K.p calibration factor for heat flow rate measurement, I error bar of the sapphire measurement (heat flow rate calibration), I error bar of the heat-of-fusion calibration (peak area calibration)


10 !-LV mw

9

t a 7

11K

6

95

100 200 O( 300

temperature .. Fig.4.21. Differences between the heat-of-fusion calibration (peak area calibration) and the calibration by means of a known heat capacity (heat flow rate calibration) (according to Sarge, Cammenga,1985). K calibration factor, - calibration curve with sapphire (3 measurements at lO K min-I), .... average value curve of the calibration with melting samples

For a heat flux DSC with disk-type measuring system, Fig. 4.21 shows the differences between the average calibration curve from of a number of heat -of-melting (peak area) calibrations and the curves from three individual calibration runs carried out with the heat capacity of a sapphire sample (heat flow rate calibration).

Dependence on Other Parameters

The repeatability error by which the calibration with a known heat capacity is affected can additionally depend on the sample mass. To give an example: the repeatability error of the calibration of a heat flux DSC with disk-type measuring system amounted to 5 % when the sample mass was 7 mg; but for a sample mass of 27 mg, the scatter decreased to half this amount (Doelman et aI., 1977).

Figure 4.22 shows the (temperature-dependent) systematic difference between the calibration factors, which results when one calibration is carried out with the calibration heater permanently installed (under the bottom of the sample cavity) and the other with the electric heater installed inside the sample (heat flux DSC with cylinder-type measuring system).

The influences of certain parameters (sample mass, heating rate) on the measurement results described in the following of course appear analogously during calibration and must be taken into consideration as well.

For a heat flux DSC with disk-type measuring system a dependence on the heating rate resulted for the heat of transition of CsCI (Fig. 4.23).

Figure 4.24 shows the relative dependence of the calibration factor on the heating rate for two different sample masses for a power compensation DSC.


30 mJ

mV· s

t 25

20

KQ

15 .. '

0 100 200 300 "( 400

tempera ture ~

Fig.4.22. Differences in the electrical peak area calibration of a heat flux DSC with cylindertype measuring system (according to Hemminger, Schonborn, 1982). KQ calibration factor for peak area evaluation, - permanently installed calibration heater (below the sample container), .... miniature heater installed in a copper sample, inside the sample container

i q

3.2 kJ

mol

3.0

20 30 40 K/min SO

Fig. 4.23. Dependence of the molar heat of transition q of CsCl on the heating rate f3 for a disk-type DSC (according to Breuer, Eysel, 1982)

% 10 mg

r 2 x x

x x

x x IS 0 o 1 mg KQ x IS x 0 0 0 0 0 0 0

-1 0.31 0.64 1.25 2.5 5 10 20 40 80

(i/(Kmin-1 ) ~

Fig.4.24. Dependence of the calibration factor KQ change on the sample mass and the heating rate f3 for a power compensation DSC. KQ calibration factor for peak area evaluation (determined by means of the heat of fusion of In; according to Hohne, Gloggler, 1989)

4.5 Conclusions Regarding the Calibration of DSCs 97

4.4.4 Caloric Calibration in Cooling Mode

The heat flow rate calibration in the cooling mode can be performed by complete analogy to that in the heating mode. The substances recommended for heat flow rate calibrations in the heating mode (see Tables 4.9 and 4.10) can also be used for calibrations in the cooling mode. Heat capacities measured in the cooling mode can differ systematically from the results determined in the heating mode. This effect is more pronounced at measurements on low thermal diffusivity materials. The reason for that temperature- and! or instrument -dependent behavior is a non-symmetric heat flow in heating and cooling mode. It is recommended to examine this effect carefully. Unfortunately the recommended reference materials have a rather high thermal diffusivity and do not show this effect so much. Consequently the possible differences must be included into the uncertainty budget. However, a correction method based on the thermal lag determination is described briefly in Sect. 7.3.

The heat calibration (peak area calibration) in cooling mode can be performed also by complete analogy with that in the heating mode. The same substances (with first order phase transitions) can be used, but supercooling must be considered. That means the dependence of the measured enthalpy of freezing as a function of the freezing temperature must be included into the evaluation. In other words, if a substance freezes at a temperature Tfrez, lower than the temperature of fusion Tfus , and the specific heat capacity of the substance in the (supercooled) liquid state (Is) differs from that in the solid state (ss) a correction according to

is necessary. In Sarge et al., 2000, differences between the specific heat capacities of heat calibration substances in the supercooled and the solid state have been determined from literature data and approximated by a linear function. The resulting correction coefficients are given in Table 4.13 together with the substances recommended for the heat calibration in cooling mode.

4.5 Conclusions Regarding the Calibration of D5Cs

The following conclusions can be drawn from the explanations of Chapter 3 (Theoretical Fundamentals), Sect. 4.3 (Temperature Calibration) and Sect. 4.4 (Caloric Calibration).

Temperature calibration and temperature measurement

From among all possible characteristic temperatures which may be assigned to a peak, only the extrapolated peak onset temperature Te (cf. Fig. 4.1, definition in Sect. 5.1) is relatively independent of sample and test parameters (mass, layer thickness, heat transfer, thermal conductivity and heating rate). This is why only Te should be used to characterize phase transitions.


Te should always be corrected to zero heating rate (13 = 0) unless shifting of Te with the heating rate is the subject of the investigation (for example for kinetic problems). The location of the sample inside the crucible and the calorimetric measuring system is of importance; this should always be the same in precision measurements.

Caloric calibration and heat measurement

Different calibration factors are to be expected for measurements of heats of transition and heat capacities; they must always be determined separately.

It is to be expected that the calibration factor for the heat flow rate depends on the heat flow rate itself (i.e., on the heat capacity of the sample). This can be checked by means of two samples of as different a heat capacity as possible (which must, however, be well-known).

The uncertainty of heat capacity measurements lies generally between 5 % and 20% for disk-type and between 3 % and 5 % for power compensation DSCs.A higher accuracy requires considerable time and effort to be spent on a proper calibration.

Precise heat capacity measurements with heat flux DSCs are possible only if all potential systematic error sources are kept constant by ensuring as identical conditions as possible during calibration run and measurement run (equal mass, equal thermal conductivity, equal heating range, equal heat capacity).

Remarks: When a dual-sample heat flux DSC is used (triple cell DSC, see Sect. 2.1.1), the uncertainty of the heat capacity measurements is lower. A scatter (RMS, see Sect. 7.2) ofthe calibration factor K!I> of approx. 1 to 1.6% is stated in the literature (see Jin, Wunderlich, 1993).

For the recently introduced turret-type DSC (see Sect. 2.1.2) only insufficient experimental experience has so far been gained to state reliable figures for the total uncertainty of heat capacity measurements.

In the case of peak area calibration, dependence of the calibration factor on the mass of the sample, the heating rate, the peak size, the temperature and on sample properties must be taken into consideration. The respective correlations have to be clarified during calibration and taken into account when the reliability of the results is estimated.

To ensure precise measurements, calibrations prior to the measurements should in principle be as similar to the measurement proper as much as possible. The peak to be measured should be "repeated" (simulated) by electrical calibration if there is the possibility of doing this.

Due to the systematic error sources, a ± 5 % reliability can be attained when heats of transition are measured with heat flux DSCs. This limit is at ± 1 % for power compensation DSCs. Prerequisite for a higher reliability is a very elaborate calibration, for example by direct comparison with electric energy.

4.6 Reference Materials for DSC Calibration

The requirements to be met by substances to be used for the calibration of DSCs are as follows:

4.6 Reference Materials for DSC Calibration 99

- high (defined) purity (at least 99.999 %), - precisely known characteristic data of substances, - non-hygroscopic, insensitive to light, non-toxic, - no decomposition, chemically stable, - no reaction with the material of the crucible (see Table 4.8) or with atmosphere, - negligible vapor pressure, - grains not too small.

When there are any doubts as to the usability of a substance, i. e., when the risk of persons being injured or equipment damaged cannot be excluded, information should be obtained from experts prior to using such substances.

The following must be observed when calibration samples are prepared:

- clean preparation tools, - clean crucible, - no adsorption layers, - no oxide layers (at surface) in the case of metal samples, - no weighing errors (calibrated balance).

A survey of reference materials, including certified reference materials, for heat flow and heat calibration has been published by Sabbah, 1999. Several ISO Guides are related to reference materials in general, and their certification and usage:

- ISO Guide 30, 1992: Terms and definitions used in connection with reference materials,

- ISO Guide 31,2000: Reference materials - Contents of certificates and labels, - ISO Guide 32, 1997: Calibration in analytical chemistry and use of certified

reference materials, - ISO Guide 33,2000: Uses of certified reference materials (under revision), - ISO Guide 34, 2000: General requirements for the competition of reference

materials producers, - ISO Guide 35, 1989: Certification of reference materials - General and statis

tical principles (under revision).

4.6.1 Reference Materials for Temperature Calibration

For the temperature calibration of DSCs those substances should preferably be used with which the fixed points of the International Temperature Scale of 1990 (ITS-90) are realized 2•

Not all substances meet the above requirements. Cadmium, for example, has a high vapor pressure in the liquid state during melting, which may result in injuries to persons and damage to equipment.

2 The fixed-point temperatures of metals are associated with the flat part of the temperatureversus-time curve obtained during the slow solidification or melting (in the case of Ga) of very pure metals at a pressure of 101.325 kPa. To calibrate DSCs, it is, however, appropriate to apply the heating mode. (For other calibration substances, cf. Marsh, 1987.)


Table 4.6. Defining fixed points of the ITS-90 and some secondary reference points

TinK tin °C

Triple point of equilibrium hydrogen 13.8033 -259.3467 Triple point of neon (ITS-90) 24.5561 -248.5939 Triple point of oxygen (ITS-90) 54.3584 -218.7916 Triple point of argon (ITS-90) 83.8058 -189.3442 Triple point of mercury (ITS-90) 234.3156 - 38.8344 Water-ice point 273.15 0.00 Triple point of water (ITS-90) 273.16 0.01 Melting point of gallium (ITS-90) 302.9146 29.7646 Boiling point of water 373.124 99.974 Solidification point of indium (ITS-90) 429.7485 156.5985 Solidification point of tin (ITS-90) 505.078 23l.928 Solidification point of bismuth 544.552 27l.402 Solidification point of lead 600.612 327.462 Solidification point of zinc (ITS-90) 692.677 419.527 Solidification point of aluminium (ITS-90) 933.473 660.323 Solidification point of silver (ITS-90) 1234.93 96l.78 Solidification point of gold (ITS-90) 1337.33 1064.18 Solidification point of copper (ITS-90) 1357.77 1084.62 Solidification point of nickel 1728 1455 Solidification point of palladium 1828.0 1554.8 Solidification point of platinum 204l.3 1768.2 Solidification point of rhodium 2236 1963 Solidification point of iridium 2719 2446

The substances used to realize the fixed points of the ITS-90 are listed in Table 4.6 starting with the triple point of hydrogen. These substances have been marked by "ITS-90" (cf. ITS-90, Preston-Thomas 1990).

Substances which were used to realize the defining fixed points of the International Practical Temperature Scale of 1968 (cf. IPTS-68, 1976) have been changed. (The fixed-point temperatures indicated in the IPTS-68 were valid only until December 31,1989). In addition, a selected set of secondary reference points was included (see Beford et al., 1996).

To calibrate DSCs in the heating mode, the melting point at zero heating rate [Te(P= 0)] is set equal to the solidification point stated in Table 4.6.

Unfortunately, no suitable fixed-point materials are available for some temperature ranges which is why other substances, too, must be used for the temperature calibration of DSCs. Taking the above criteria into consideration, the German Society for Thermal Analysis (GEFTA) recommends that the substances listed in Table 4.7 are used for temperature calibration. These materials have proved their practical worth, they were carefully tested and cover the temperature range of customary DSCs (Cammenga et al., 1993).

For the compatibility between calibration substances and crucible materials see Table 4.8.

Only the uncertainty of the temperature measurement is decisive for the uncertainty of the calibration when these substances are used. It comprises the


Table 4.7. Materials recommended for temperature calibration of DSCs (cf. Cammenga et ai., 1993)

Substance Transition Uncertainty Transition a Remarks temperature

TinK TinoC dTinmK

Cyclopentane 122.38 -150.77 50 sIs Cyclopentane 138.06 -135.09 50 sIs Cyclopentane 179.72 - 93.43 50 sn 1 Water 273.15 0.00 10 sll 2 Gallium 302.9146 29.7646 sn 3 Indium 429.7485 156.5985 <1 sn Tin 505.078 231.928 <1 sn 4 Lead 600.61 327.46 10 sn 5 Zinc 692.677 419.527 sll 3 Lithium sulfate 851.43 578.28 250 sIs 6 Aluminum 933.473 660.323 sn 7 Silver 1234.93 961.78 sn 8 Gold 1337.33 1064.18 sn 8

a Solid/solid or solidlliquid transition. 1 Measure in a hermetically sealed crucible only. 2 Air-saturated, bidistilled water in hermetically closed crucible. 3 Ga and Zn very easily react with aluminum which is usually used as crucible material (also

when carefully oxidized). To avoid alloy formation with potential subsequent destruction of calorimeter components, it is therefore necessary to proceed very carefully and with great attention. It is advised to use only fresh samples, to examine the crucible bottom for cracks in the aluminum oxide layer, and to immediately cut short a series of experiments, as soon as during successive experiments an increase in the peak width or a decrease of the peak area is observed. The authors do not accept any responsibility for damages! Note high vapor pressure of Zn and the strong supercooling of Ga.

4 Melt reacts with Al and Pt. 5 Melt reacts with Pt. 6 Anhydrate is hygroscopic, dehydration takes place from 100°C, thus movement of particles

and high water-vapor pressure (do not use hermetically closed crucible). 7 Melt reacts strongly with Pt. 8 Reacts with Pt, melt dissolves oxygen.

pure uncertainty of measurement of the sensors, repeatability errors of the heat transfers in the measuring system and uncertainties in the processing of the analogue signal until it is recorded and evaluated.

Reference materials for the symmetry test and calibration on cooling are listed in Table 4.12.

4.6.2 Reference Materials for Heat Flow Rate Calibration

The substances Ah03 in form of a single crystal (sapphire) disc and pure copper (eu) are suitable for heat flow rate calibration purposes. They are chemically

Tab

le 4

.8 C

ompa

tibi

lity

bet

wee

n ca

libr

atio

n su

bsta

nces

an

d c

ruci

ble

mat

eria

ls (

acco

rdin

g to

Cam

men

ga e

t ai

., 19

93)

IS C

alib

rati

on

Q)

subs

tanc

e Q

) ....

.:: ..s

'" "3

a

1:: '"

::l

Q)

a a

a .::

0-

0 ....

::l

.8

::l

's ....

Q)

'"0

u

~ ~

'"0

Cru

cibl

e m

ater

ial

~ ~

::;;;

'"0

.:: '"

.:: ~

"0

.5

~

Q)

N

" ....

:I <Z

l

" C

oru

nd

um

, Aiz

03

0 0

+

+

+

+

+

+

+

+

+

Bor

on n

itri

de, B

N

0 0

+

+

+

+

+

+

+

Gra

phit

e,C

0

0 +

+

+

+

+

+

+

+

S

ilic

ate

glas

s +

+

+

+

+

+

+

x

x Q

uart

z gl

ass,

Si0

2 +

+

+

+

+

+

+

+

+

+

A

lum

inum

, Al

+

+

+

+

x x

x ....

Alu

min

um, o

xidi

zed

+

+

+

+

+

+

+

+

x x

x n ~

Sil

ver,

Ag

+

+

x x

g; .... G

old,

Au

+

+

+

x ~ o·

Nic

kel,

Ni

+

+

+

::l

0 Ir

on, F

e +

+

+

+

.....

t;

j S

tain

less

ste

el

+

+

+

+

+

~ P

lati

num

, Pt

+

+

+

.... t1>

Mol

ybde

num

, Mo

+

+

a.. T

anta

lum

, Ta

+

+

+

+

+

eo. C/)

Tun

gste

n,W

0

0 +

+

+

t"

\ ~

::l

::l

+ :

No

solu

bili

ty a

nd

infl

uenc

e o

n m

elti

ng te

mpe

ratu

re t

o b

e ex

pect

ed.

S·

C1CI

-:

Mel

t dis

solv

es c

ruci

ble

mat

eria

l, g

reat

er c

hang

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f m

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ng te

mpe

ratu

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n eo. :

Par

tial

sol

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n pr

oces

ses

poss

ible

wit

h ne

glig

ible

cha

nge

of

mel

ting

tem

pera

ture

. 0 ....

X

: C

ruci

ble

mel

ts.

S· ?

: C

ompa

tibi

lity

unk

now

n.

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Com

bina

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can

no

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real

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stable and the heat capacity is known in a wide temperature range with an uncertainty of about 0.1 % between 100 K and 300 K for Cu and between 100 K and 900 K for Ah03 (Tables 4.9 and 4.10).

Probably the most accurate measurements on Cu were carried out by Martin, 1987 and on Ah03 by Ditmars et al., 1982. The published results are based on the IPTS-68. There is an ongoing discussion about the correct method for the temperature scale adjustment of thermodynamic data from the IPTS-68 to the ITS-90 (cf. Goldberg, Weir, 1992, Weir, Goldberg, 1996, Archer, 1993, 1997).

Therefore, the use of the original IPTS-68 based data is recommended here. Because of the practical interest the uncertainty of the specific heat capacity of Cu and Ah03 and some results of the temperature scale adjustment are discussed here briefly.

The measurements of Martin on Cu have been carried out in the temperature range between 20 K and 320 K and he claimed an uncertainty of 0.1 % between 30 K and 300 K. Martin did not publish his measured values, only a polynomial representation of them (Table 4.lOa). The tabulated values (Table 4.lOb) have been calculated by means of Martin's polynomial representation. For DSC calibration purposes Cu is of interest in the temperature range between approx. 100 K and 300 K. Archer, 1997, has adjusted Martin's data to the ITS-90 and published modified polynomial coefficients. The difference between Martin's data

Table 4.9 a. Specific heat capacity of a-Ah03 (synthetic sapphire, corundum, NIST SRM 720) (according to Ditmars et al., 1982) Molar mass 101.9613 g mol-I

Temperature range Fitted function cp in J mol-I K-I To

45K:s; T< 125 K 40K

125 K :s; T < 273.15 K 125K

4

273.15 K < T:s; 2250 K L CiTi i=-3

ai bi Ci

-3 -1.32506 . 108

-2 4.54238.106

-1 -5.475599. 104

0 6.966.10-1 2.1993.101 2.574076. 102

1 5.9387 . 10-2 3.8853 . 10-1 -1.715032. 10-1

2 4.0357 . 10-3 1.3955 . 10-3 1.2897189 . 10-4

3 9.5173.10-5 - 8.3967 . 10-5 -4.60768. 10-8

4 -3.5910. 10-6 1.9133. 10-6 6.31755. 10-12

5 - 6.498 . 10-7 -3.1778.10-8

6 4.089.10-9 2.9562 . 10-10


Table 4.9 b. Specific heat capacity of a-Alz03 (synthetic sapphire, corundum, NIST SRM 720) Molar mass 101.9613 g mol-1 (according to Ditmars et al., 1982 )

TinK cp in J g-l K-1 cp in J mol-1 K-1

100 0.1261 12.858 150 0.3134 31.95 200 0.5014 51.12 250 0.6579 67.08 300 0.7788 79.41 350 0.8713 88.84 400 0.9423 96.08 450 0.9975 101.71 500 1.0409 106.13 550 1.0756 109.67 600 1.1039 112.55 650 1.1271 114.92 700 1.1467 116.92 750 1.1636 118.64 800 1.1783 120.14 850 1.1913 121.47 900 1.2030 122.66 950 1.2138 123.76

1000 1.2237 124.77 1050 1.2330 125.72 1100 1.2417 126.61 1200 1.2578 128.25 1250 1.2653 129.01 1300 1.2724 129.74 1350 1.2792 130.43 1400 1.2856 131.08 1450 1.2917 131.70 1500 1.2975 132.29 1550 1.3028 132.84 1600 1.3079 133.36 1650 1.3128 133.85 1700 1.3173 134.31 1750 1.3214 134.73 1800 1.3253 135.13 1850 1.3289 135.50 1900 1.3324 135.85 1950 1.3356 136.18 2000 1.3387 136.50 2050 1.3417 136.80 2100 1.3446 137.10 2150 1.3477 137.41 2200 1.3508 137.73 2250 1.3540 138.06

4.6 Reference Materials for DSC Calibration

Table 4.lOa. Specific heat capacity of copper (according to Martin, 1987) Molar mass 63.546 g mol-I

Temperature range Fitted function cp in J mol-I K- '

20K < T~ 320K

aj

0 - 0.8209550462989 0.1877774093791

2 -0.1572548380193. 10-1

3 0.5828318431167. 10-3

4 -0.1420296394933.10-5

5 -0.3466012703872 . 10-6

6 0.1030643882976.10-7

7 -0.1621415050746. 10-9

8 0.1678243825986. 10-11

9 -0.1225826347399.10-13

10 0.6497335630403 . 10-16

11 -0.2516918128676. 10-18

12 0.7065147741950. 10-21

13 -0.1400307720276.10-23

14 0.1858984964834.10-26

15 -0.1483657580275.10-29

16 0.5382629833814. 10-33

Table 4.10 b. Specific heat capacity of copper (according to Martin, 1987) Molar mass 63.546 g mol-I

TinK cp in J g-I K-1 cp in J mol-I K-1

100 0.2520 16.02 120 0.2871 18.25 140 0.3122 19.84 160 0.3306 21.01 180 0.3444 21.89 200 0.3550 22.56 220 0.3635 23.10 240 0.3703 23.53 260 0.3758 23.88 280 0.3806 24.19 300 0.3848 24.45 320 0.3884 24.68

105


0.008

-o. 008 L--~-'---~-'---'---'-~--'---'---'---'---' ______ L--.>--.J

250 750 1250 1750 K 2250

temperature --

Fig.4.25. Differences between tabulated and calculated (polynomial presentation) specific heat capacity data of sapphire (Ah03): I!1cp = cr -cryn. (cf. Ditmars et al., 1982) in the temperature range between 250 K and 2250 K

and the adjusted data by Archer is between -0.022% at 100 K and +0.054% at 320 K. Sabbah et al., 1999, used the Goldberg and Weir, 1992, method and came to the conclusion that the adjustment is smaller than ± 0.06 %. The adjustments of both authors are smaller than Martin's claimed uncertainty in the temperature range between 100 K and 300 K.

The specific heat capacity of Al20 3 has been determined in the temperature range between 10 K and 2250 K (Ditmars et al., 1982, Table 4.9a and 4.9b). Figure 4.25 shows the difference between the tabulated data and the data calculated by means of Ditmars polynomial representation.

Although Ditmars estimated an uncertainty of 0.1 % in the temperature range between 100 K and 350 K only, it is generally accepted that the uncertainty of the measurements is better than 0.1 % in the temperature range between 100 K and 900 K. At higher temperatures the uncertainty increases to values between 1 % and 2 % at 2250 K. The influence of the IPTS-68 to ITS-90 adjustment and the data treatment (fitting) has been determined by Archer, 1993, and amounts to approx. 0.12% at 100 K, up to 0.09% between 300 K and 430 K, up to 0.31 % between 900 K and 1700 K and 1.4% at 2250 K. The ITS-90 adjustment of Weir and Goldberg, 1996, amounts up to ± 0.18 % between 900 K and l300 K.

NB: eu should not be used at higher temperatures and only in an inert purge gas flow, to avoid oxidation which would produce an additional heat flow rate and falsify the measurement.

4.7 Additional Calibration in Temperature-Modulated Mode of Operation 107

4.6.3 Reference Materials for Heat (Peak Area) Calibration

The heats of transition of pure substances for peak area calibration should have been measured with precision calorimeters. Furthermore, the results should be traceable to basic SI units and an uncertainty should be stated. Such results are available for only a few substances consequently fulfilling the above requirements. When several independent precise measurements have been carried out on one substance, the ranges of uncertainty stated in some cases do not overlap so the estimate of the best value and of the overall uncertainty is problematic. Recommended substances are listed in Table 4.11 a. Table 4.11 b contains a list of further commercially available substances used for calibration. The latter are not recommended here because of a lack of information about the used certification procedure and the uncertainty of the results.

The analysis of the most reliable enthalpy of fusion measurements allows a statement of an uncertainty less than 0.3 % for Ga and In and 0.5 % for Sn and Bi.

For peak area calibration in cooling mode most of the substances used in heating mode are suitable as well, Table 4.13 list these substances together with the correction of the enthalpy of crystallization which takes care of the temperature dependence of the transition enthalpy. The correction must be applied in cases of large supercooling.

Heat flux DSCs have a number of systematic error sources (cf. Sect. 3.1). In our opinion, the reliability of the results obtained with them is therefore of the order of approx. ± 5 % in routine operation. The uncertainties of the enthalpies of transition indicated in Table 4.11 a are therefore sufficient to allow these substances to be used as calibration substances.

In special cases relative uncertainties less than 1 % (e. g., Archer, Carter, 2000) can be achieved. Calibration by means of electrically generated heat should be aimed for precision measurements. In addition, reference is made to the discussion of thermodynamic aspects with respect to the peak area calibration in Sect. 4.2.

The discussion about the suitability of various substances for caloric calibration is still going on at the international level (St01en, Gr0nwold, 1999). The following complexes of problems are concerned:

- backing-up and re-determination of characteristic data, including an estimate of the uncertainties,

- testing of new substances for their suitability for caloric calibration purposes, - search for substances suitable for special calibrations, for example polymers

(glass or other transitions), calibrations in the cooling mode, or substances suitable for high temperatures.

4.7 Additional Calibration in Temperature-Modulated Mode of Operation

Quantities determined from the periodic part of the heat flow rate in temperaturemodulated mode of operation (cf. Sect. 2.4.2) depend usually on the frequency of modulation. The reason is that heat transport needs time and the thermal inertia

Tab

le 4

.11

a.

Ref

eren

ce m

ater

ials

for

hea

t (p

eak

area

) ca

libr

atio

n

Sub

stan

ce

Cyc

lope

ntan

e C

yclo

pent

ane

Cyc

lope

ntan

e G

alli

um

Bip

heny

l In

dium

T

in

Bis

mut

h L

ithi

um s

ulfa

te

Alu

min

um

Silv

er

Gol

d

a s:

sol

id; 1

: liq

uid.

b

Arc

her,

200

2.

T trs

(IT

S-90

) in

°C

-150

.77

-135

.09

-93.

43

29.7

6 69

.26

156.

60

231.

93

271.

40

578.

28

660.

32

916.

78

1064

.18

Typ

e o

f Q

trs

tran

siti

on a

(St0

1en

et a

I., 1

999)

in

J g-

l

sis

sis

sll

sll

79.9

7 ±

0.2

7 sl

l sl

l 28

.58

± 0

,07

sll

60.4

2 ±

0.1

7 sl

l 53

.33

± 0

.26

sis

sll

399.

9 ±

1.3

sl

l 10

4.6

± 2

.1

sll

64.6

± 1

.5

C

NIS

T S

RM

222

2 (N

IST

: N

atio

nal I

nsti

tute

of

Sta

ndar

ds a

nd

Tec

hnol

ogy,

USA

). d

NIS

T S

RM

223

2.

e N

IST

SR

M 2

220.

Qtrs

(N

IST

) in

J g-

l

80.0

97 ±

0.0

32 b

120.

41

± 0

.57

c

28.5

1 ±

0.1

9d

60.2

2 ±

0.1

ge

f PT

B C

erti

fied

Ref

eren

ce M

ater

ials

(PT

B:

Phy

sika

lisc

h-T

echn

isch

e B

unde

sans

talt

, Ger

man

y).

Qtrs

Q

us

(PT

B)

(Sar

ge e

t ai.

, 199

7)

in J

g-l

in J

g-l

69.6

0 ±

0.

35

4.91

±

0.05

8.

63 ±

0.

09

80.1

4 ±

0.3

3f

79.8

8 ±

0.

72

28.6

4 ±

0.1

1 f

28.6

2 ±

0.

11

60.2

4 ±

0.2

7f

60.4

0 ±

0.

36

53.1

4 ±

0.2

2f

53.8

±

2.

1 22

8 ±

10

39

8 ±

9

......

o 00

... (") ~ .... ~ g" o .....,

CI ~

'" .... '" g. e. Vl

() § 5"

~

(") S- .... s" ~ '" .... en

4.7 Additional Calibration in Temperature-Modulated Mode of Operation

Table 4.11 b. Further Certified Reference Materials for temperature and heat calibration

Substance

Phenyl salicylate Biphenyl Naphthalene Benzil Acetanilide Benzoic acid Diphenylacetic acid Lead Zinc Aluminum

source: http://www.lgc.co.uk.

Ttrs

in °C

41.79 68.93 80.23 94.85

114.34 122.35 147.19 327.47 419.53 660.33

(LGC: Laboratory of the Government Chemist, UK).

Qtrs

inkJ mol-1

19.18 18.60 18.923 23.26 21.793 17.98 31.16 4.765 7.103

10.827

Table 4.12. Substances recommended for temperature calibration in the cooling mode

Substance Transition Evaluate Phase transition temperature

Ttrs in °C Ttr, in K

Adamantane s~s Te -64.49 208.66 M24 a SA~N Tp 66 339 HP-53 b SA~N Tp 116 389 BCH-52c N~I Tp 162 435 Indium s~l Te 156.6 429.8 NaN03 s~s Tp 276 549 Znd s~l Te 419.5 692.7

a 4-cyano-4' -octyloxybiphenyl. b 4-( 4-pentyl-cyclohexyl)-benzoic acid 4-propyl-phenyl ester. c 4' -ethyl-4-( 4-propyl-cyclohexyl)-biphenyl.

Supercooling

8TinK

<1 <0.5

<2 <0.5

109

d Zn very easily reacts with aluminum which is usually used as crucible material (also when carefully oxidized). To avoid alloy formation with potential subsequent destruction of calorimeter components, it is therefore necessary to proceed very carefully and with great attention. It is advised to use only fresh samples, to examine the crucible bottom for cracks in the aluminum oxide layer, and to immediately cut short a series of experiments as soon as during successive experiments an increase in the peak width or a decrease of the peak area is observed. The authors do not accept any responsibility for damages!

s solid. SA smectic A (liquid crystal phase). N nematic (liquid crystal phase). 1 liquid.


Table 4.13. Substances recommended for heat (peak area) calibration in the cooling mode

Substance Fusion Enthalpy of fusion Temperature dependence b

temperature 1'1fusH in J g-1 d1'1trs H/dT= C;(T) - Cp(T) Tfus in °C = 1'1trsCp (T) = a + b (Ttrs - Tfus)

a in Jg -IK-1 binJg-1K-2

Cydopentane -150.77 C 69.60 c +0.38 -5.8. 10-3

Cydopentane -135.09 c 4.91 c -0.058 0 Cydopentane -93.43 8.63 +0.16 +3.1 . 10-3

Gallium a 29.7646 see Table 4.11 a +0.031 -4.6. 10-4

Indium 156.5985 see Table 4.11 a -0.0026 -2.6. 10-4

Tin 231.928 see Table 4.11 a -O.oI8 -3.1 . 10-4

Zinc a 419.53 108.09 +0.012 -7.3 . 10-4

Lithium sulfate 578.28c 228.1 c -0.15 -7.7. 10-3

Aluminium 660.323 398.1 -0.19 +3.2. 10-4

a Ga and Zn very easily react with aluminum which is usually used as crucible material (also when carefully oxidized). To avoid alloy formation with potential subsequent destruction of calorimeter components, it is therefore necessary to proceed very carefully and with great attention. It is advised to use only fresh samples, to examine the crucible bottom for cracks in the aluminum oxide layer, and to immediately cut short a series of experiments, as soon as during successive experiments an increase in the peak width or a decrease of the peak area is observed. The authors do not accept any responsibility for damages!

b When the equation (Sarge et at, 2000)

1'1trsH(Ttrs) = -1'1fusH(Tfus) + (a - bTfus ) (Ttrs - Tfus ) + 1/2 b (nrs - Tfus)

is used for the determination of the enthalpy of solidification from the enthalpy of fusion, the signs of the quantities used are strictly to be taken into consideration: fusion processes are endothermic and, therefore, enthalpies of fusion are positive; solidification processes are exothermic and, therefore, enthalpies of solidification are negative. The signs of the given differences in heat capacity relate to the transition of the low-temperature phase to the hightemperature phase. Thus, for example, supercooling (Ttrs - Tfus < 0) leads to an increase in the enthalpy of solidification when the heat capacity during the solid/liquid transition in the temperature interval decreases (1'1ep < 0). The enthalpy of solidification being however negative, this means a reduction of the amount of the enthalpy of solidification. The procedure can be applied for solid/solid transitions in an analogous manner.

c Solid/ solid transition.

causes a loss in amplitude and a shifting of the phase of the heat flow rate modulation on the path from the furnace to and into the sample. Therefore the (apparent) heat capacity, determined from the modulated part of the heat flow rate signal depends strongly on frequency (or period of modulation) used. That's why the TMDSC must be calibrated relating to the frequency dependence of the heat capacity in addition to the normal calibration of temperature, heat flow rate and peak area mentioned above. The relevant equation in this special case reads:

Cp, true = k ( w) . Cp, meas. (4.5)

The normal (vibrational) heat capacity does not depend on frequency and this fact is used to determine the calibration factor for the frequency of the temper-


ature modulation used, or, in the case of investigations at different frequencies, as a function of frequency in the region of interest.

The temperature-modulated mode of operation enables to determine timedependent (complex) heat capacity (see Sects. 3.3.2 and 5.5.2) from the modulated heat flow rate. From the theory of TMDSC (see Sect. 3.3) the following equation is valid within the scope of linear response

T~eas.(W) = T~ue(w)' P*(w)

with pew) the complex transfer function of the device. Because of the proportionality of temperature gradient and the heat flow rate a similar equation can be formulated for the heat flow rate

q,~eas.(w) = q,~ue(w)· P*(w) (4.6)

and even for the heat capacity which is linearly connected with the heat flow rate

(4.7)

Generally the functions are complex (characterized by the star) but, of course, the equation is even valid for real functions. Comparing Eq. (4.7) with Eq. (4.5) yields:

_ 1 K(w)=-

P*(w) (4.8)

the calibration function sought after is the reciprocal transfer function of the DSC. The latter can be determined easily with the tools of transfer theory (Hahne et al., 2002). One elegant possibility is to measure the response of the heat flow rate on a sudden step-like temperature change of the DSC. This function, if normalized, is the apparatus (Green's) function of the DSC, which, after Fourier transform, results in the complex transfer function.

The procedure is simple, Fig. 4.26 shows an example of such a measurement. To correct for the unavoidable asymmetry influences on the signal, a measurement with empty pans (zeroline) was subtracted. Fourier transform results in a complex function, which after normalization is plotted as magnitude and phase in logarithmic form (Bode plot, Fig. 4.27). The reciprocal function is the calibration factor [see Eq. (4.8)], a complex function of was well. With that the calibration, in principle, is completed, as the correction of magnitude and phase at any frequency can be obtained according to the rules of complex numbers.

However, the normal evaluation software of temperature-modulated DSCs doesn't contain this calibration procedure and the user has to perform the calibration in a more usual way. Any complex quantity can be expressed as magnitude (absolute value) and phase angle, consequently calibration can be done for both quantities separately. Normally only the magnitude (absolute value) of the heat capacity is of interest and the phase information is disregarded. Therefore there is a demand for separate procedures of magnitude and phase calibration which are to be presented in what follows.

112

aJ

d

:3 o


/\ 141 I \ I \ I \

, ...... I ....... \ ........................... tg.r!~p.~r.gJ~xg.::::f.:......... 0 C ; I \ : I \ ; I \ ! I \

•...•......... , I \ 140 I \ ! ._ \,___ sample line

~: ", -.. -----=---... ----;;;~;;--"'='=--~=--

zero line subtraded OL---~---L __ ~==~====b===d 139 -10 0 10 20 30 s 50

time ----

I d <ill D..

E ill

Fig.4.26. Temperature-step (0.5 K, dotted) response signals (dashed) of a power compensation DSC containing a sapphire sample (25 mg) as well as empty pans (for zeroline). Subtraction results in the response function (solid line)

c 5

Plw) rad 4

0100

I 3 I 2 1i 0.010 ill

Vl d d

..c . D.. , , , 0.001 0

001 01 10 rad S-l 10 (;J --

Fig.4.27. Transfer function (Bode plot, solid lines), the Fourier transformed step response function from Fig. 4.26, together with the really measured values (symbols) at different discrete frequencies (dashed line: magnitude of the transfer function from non -zero line corrected step response)

Any complex quantity can be separated into magnitude and phase:

f* = 11* I· exp(i6)

from this follows, that for a product of complex quantities as in Eq. (4.5) or (4.7) the magnitudes have to be multiplied, whereas the phases add up. In other words the magnitu~e of the. complex heat capacity has to be corrected with a calibration factor IK (w) 1 = K(w), whereas the phase has to be shifted by a cer· tain 6K (w).

No reference substance with known complex (frequency dependent) heat capacity, however, exist. The normal vibrational heat capacity is a real-valued


quantity, in other words the magnitude of Cp is independent of frequency, in the range used with TMDSC, and the phase angle is zero, this makes it suitable for our purpose and the respective calibration procedure easy.

4.7.1 Calibration of Magnitude

A simple procedure for magnitude calibration starts from Eq. (4.5) and determines the calibration factor by means of a substance with known heat capacity:

The recommended reference materials for heat flow rate calibration (Sect. 4.6.2) can be used for this purpose as well, sapphire is inert and the vibration frequencies are many orders of magnitude faster than the modulation frequencies of TMDSC. As a result the heat capacity Cp,Ref (see Table 4.9) is the same for all frequencies (or periods) used in the temperature-modulation experiment. This has been done for several frequencies and the (reciprocal) calibration factors are added to Fig. 4.27. As expected the results of this magnitude calibration coincide with the magnitude of the step-response evaluation performed with the same sample in the same DSC, but needs much more time. Of course the simple procedure is sufficient and faster, if only one frequency (period) is used for the TMDSC experiments. In this case there is a certain calibration factor instead of a calibration function. It should, however, be born in mind that this factor changes if the frequency (period) is changed. For every DSC there exists a frequency range where the TMDSC calibration factor is close to one (proper heat flow rate calibration for granted).

4.7.2 Calibration of Phase

The phase signal is strongly influenced by heat transfer properties of the DSC as well as the sample itself. From Eq. (3.28) (see Sect. 3.3.3) follows that the phase shift of a simple RC-element depends not only on frequency but even on the thermal resistance and heat capacity:

i. e., the phase signal changes not only with temperature (as Cp and Rtb depend on temperature) but even when the heat transfer to the sample and its heat capacity changes. This makes the determination of an exact phase angle not so easy.

A possible calibration procedure starts from the fact that a pure vibrational heat capacity is real-valued, i. e., the phase shift is zero. The measured phase shift must be assigned to the heat transfer from the furnace to the sample, this seems to be the correction c5K at the frequency in question we were looking for. Unfortunately this is not true, as the heat capacity of the sample is different from the


heat capacity of the reference material (sapphire), even the phase shift is different. The correction depends strongly on the sample and its properties Rth and Cp• Consequently a phase calibration can only be done if the total arrangement DSC plus pan plus sample is unchanged. The only way out is to perform the above mentioned step response measurement and determine the phase shift from the transfer function. A temperature step can be done easily at every temperature with the experimental setup of the TMDSC. Of course, one should choose a temperature well outside of any transition or reaction, to be sure that only the vibrational heat capacity contributes to the signal.

However, there is another problem arising from the fact that the heat capacity of a sample (normally) changes during a transition or reaction. This implies that even the phase signal changes. In other words, the correction 15K changes during transitions and reactions and is not a constant value. This is in particular true for the glass transition region, where the heat capacity becomes complex, and the magnitude changes as well. Weyer et aI., 1997, suggested a special correction for this case where the correction 15K is assumed to change in a sigmoidal way proportional to the magnitude of the complex heat capacity. This assumption is true for RC-elements for small phase shifts, where the tangens can be approximated by the argument. The authors could show that this method is successful for the glass transition and yields the true imaginary part of the complex heat capacity.

To sum up one may state that the phase correction 15K has to be determined at different temperatures and, in particular, on both sides of a transition or reaction to take care of possible heat capacity changes. The arrangement of sample and DSC must be the same as for the TMDSC measurement, i. e., the sample pan should not be touched. It should be ensured that only vibrational heat capacity contributes to the signal at the temperatures in question. A special reference substance is not needed, as the vibrational heat capacity of any sample is realvalued and therefore results in a zero phase shift. The phase shift in dependence on frequency at a certain temperature can easily be determined by measuring the temperature-step response and Fourier transform the respective (normalized) heat flow rate function.

It should be emphasized that the phase correction 15K changes during transitions, the problem to get the right correction function for this region is not solved yet in a satisfactory manner. The knowledge of the correction function is, however, an indispensable condition to determine the proper complex heat capacity. This is not an easy task and the reader interested in these questions is referred to special literature (see Merzlyakov et aI., 2002).

5 OSC Curves and Further Evaluations

A Scanning Calorimeter measures heat flow rates in dependence of temperature or time. Modern DSCs are nowadays always connected with a data acquisition system and a powerful computer (PC). This allows one to present the measured data online on a monitor in form of a curve. Normally the heat flow rate versus the program temperature (or time) is plotted, but it is also possible to calculate other quantities from the originally measured values and draw the respective graphs on the screen as well. Modern computer techniques make it possible to do even complicated evaluations of the just measured values in the background while the measurement runs.

It should be remembered that the original quantities measured in a DSC are a temperature difference (in form of a voltage) and the set value of the temperature of the furnace (or sample support) which is electronically controlled to follow a certain temperature program (other quantities are in certain cases

t

T, T,

temperature ..

Fig.S.1. Curve measured by a DSC with step of the baseline (Cp change) and endothermic peak (Ist order transition). IlCp change of the sample's heat capacity, Ti> Tf initial peak temperature, final peak temperature, CD initial segment of the measured curve, @ step of the measured curve due to IlCp ,

@ measured curve, ® interpolated baseline (between Tj and Tf in the peak region), ® final segment of the measured curve, ® peak (measured curve)

116 5 DSC Curves and Further Evaluations

measured additionally). The temperature difference is internally transformed into a differential heat flow rate (see Chapter 3) which is gathered at regular intervals together with the temperature (or time) in question and stored internally. This raw data set establishes the DSC curve, it forms the basis for all further evaluation.

The DSC curve (the measured curve, see Fig. 5.1) offers quick information on the total measuring process. In addition to the usual measuring effects (Cp changes, transitions, reactions) it can be seen

- whether the predetermined temperature range has been completely covered, - whether disturbances of the apparatus (mechanical, electrical) occurred, - whether there were irregularities or unusual shapes of the baseline, - whether the characteristic temperatures and peak areas lie within the expect-

ed range.

5.1 Characteristic Terms of DSC Curves

Some characteristic terms are used to describe a measured curve. The respective definitions are given in the following (see Fig. 5.2).

- The zero line is the curve measured with the instrument empty (i. e., without samples and without sample containers, crucibles), or with empty sample containers (crucibles without samples). It shows the thermal behavior of the

f 1 ---I I

1 1 1

/

I 1 \

peak (endothermic)

-- I '\ I .-!-\-Q-

I 1 : baseline (interpolated) 1 I I

\ 1 zero line

T; T. T,

temperature ~

Fig.5.2. Definition of zeroline, baseline, peak and the characteristic temperatures (definitions see text). Ti initial peak temperature, T. extrapolated peak onset temperature, Tp peak maximum temperature, Tc extrapolated peak offset temperature, Tc final peak temperature

5.1 Characteristic Terms of DSC Curves 117

measuring system itself and the degree and influence of unavoidable asymmetries. The smaller its range of variation with temperature or in time (repeatability, see Sect. 7.2), the better the instrument.

- The baseline is the part of the heat flow rate curve produced of the DSC during steady state conditions (no reactions or transitions in the sample). The baseline is the sum of the zeroline and the heat flow rate caused by the difference of heat capacities on the sample and reference side in scanning mode.

- A peak in the measured curve appears when the steady state is disturbed by thermally activated heat production or consumption in the sample (e. g., from transitions or reactions). Peaks in heat flow rate curves, which are assigned to endothermic processes, are normally plotted "upwards" (positive direction), as heat added to a system is defined as positive in thermodynamics by international convention. A peak begins at Ti (first deviation from the baseline, see below), ascends/descends to the peak maximum/minimum, Tp (see below), and merges into the baseline again at Tf • Only processes associated with a heat (e.g., melting or crystallization) lead to peaks (except for changes in the heat transfer between the sample and the AT-sensor). Other transitions (e.g., glass transition) only lead to changes in the shape of the measured curve, for example step-like changes (see Sect. 6.4).

- The interpolated baseline is the line which in the range of a peak is constructed in such a way that it connects the measured curve before and behind the peak as if no reaction heat had been exchanged, i. e., as if no heat (peak) had developed (for interpolation see Sect. 5.3.2).

- The characteristic temperatures are defined as follows: - Ti Initial peak temperature

Here the curve of measured values begins to deviate from the baseline, the peak begins,

- Te Extrapolated peak onset temperature Here the auxiliary line through the ascending peak slope intersects the baseline. [The auxiliary line is drawn through the (almost) linear section of the ascending peak slope, either as inflectional tangent or as fitted line. The distinction between the two methods is of no significance in practice, as the resulting difference is smaller than the repeatability error of the measurement results].

- Tp Peak maximum temperature This temperature designates the maximum value of the difference between the curve of measured values and the baseline (not necessarily the absolute maximum of the curve of measured values).

- T, Extrapolated peak offset temperature Here the auxiliary line (see above) through the descending peak slope intersects the baseline.

- Tr Final peak temperature Here the curve of measured values reaches again the baseline, the peak is completed.


5.2 Parameters Influencing the DSC Curve

In Sects. 4.3 and 4.4, reference has already been made to the influence of some parameters in connection with the calibration. These statements apply by analogy to each measurement.

Note: For highly precise measurements, the zeroline must be determined prior to and after every measurement. Temperature and heat calibration must then also be checked (verification) at regular intervals (depending on the specific DSC) at least with one calibration substance, e. g., indium. If the DSC shows a distinct tendency towards drifting, a daily test is important. In this way, information is obtained about drift processes or scatter which cannot be assigned to an exactly known parameter. This information enters into the estimate of the overall uncertainty.

The influencing parameters are listed in the following:

1. The shape of the zeroline (without crucibles) is influenced by the heating rate, the kind of purge gas and its flow rate, likewise by the temperature of the surroundings and by surface properties of the measuring system. If the zeroline is measured with the crucibles empty added to this are influences due to unequal masses of the crucibles, differences in the heat transfer between the crucibles and the furnace and from differences between the emissivities of the two crucibles (lids), and influences due to type and material of the crucible used.

2. Point 1. is also applicable to the shape outside a peak of the measured curve with sample and reference sample placed in the crucibles. In addition, the properties of the sample and reference sample (heat capacity and its temperature dependence) are of importance (differential measurement). In the case of pure Cp changes of the sample (e. g., glass point, Curie point), these changes determine the course of the measured curve which then contains the desired information. Unfortunately, changes of the measured curve can also take place if the conditions of heat transfer to the sample change. If this occurs abruptly this causes a step-like or peak-like change as well, but this usually appears statistically and can thus be distinguished from real Cp-changes or transitions of uniform samples (with the same thermal history) which always occur at the same temperature.

3. Point 2. is also applicable to the shape of the measured curve with peak. The peak itself is additionally influenced by - the heating rate (cf. Figs. 3.11,4.8), - the thermal conductivity of the sample (cf. Fig. 3.13), - the mass and heat capacity of the sample (cf. Fig. 4.24), - the structure of the sample (powder, granulates, foil, ... ), - the thermal resistance between sample and temperature sensor (cf.

Reichelt, Hemminger, 1983), - the location of the sample in the crucible or measuring system (cf. Fig 4.11), - the kind of purge gas in the measuring system, which influences the sepa-

ration (resolution) of closely adjacent peaks (and the calibration).

5.3 Further Evaluation of DSC Curves

4. In addition, attention must be paid to: - the sample purity, - the thermal history of the sample

119

- the thermal history of the measuring system (cf. Suzuki, Wunderlich, 1984).

Conclusion

Ensure that all the parameters for the measurement are as similar as possible to those for calibration. For direct comparison of measurements all parameters should be as similar as possible. Use samples of defined state, shape and purity.

5.3 Further Evaluation of DSC Curves

The original curve, measured with a DSC, contains not only the heat flow rate of the sample, but even other parts caused by unavoidable asymmetries. In addition the heat flow rate is falsified (smeared) by the thermal inertia of the measuring system (thermal lag}. To get the real sample heat flow rate and - if possible - sample temperature and eliminate the disturbances, conversion of the original measured curve must be performed. This is done in two steps, firstly, the differential heat flow rate into the sample is calculated and, secondly, the influence of thermal lag is eliminated (desmearing) if needed.

5.3.1 Determination of the Real Sample Heat Flow Rate

To eliminate all influences from asymmetries of the DSC in question, the zeroline must be subtracted from the measured curve. For precision measurements the proper zeroline should be determined prior to and after a sample run, to check whether there are significant differences, and subtracted. Of course all parameters of the measurement must be kept the same, only the sample pan is exchanged for an empty pan of the same type and with the same mass. Depending on the DSC and the stability of the surrounding conditions the zeroline may be rather stable during a day (should be tested). If that is the case, and for less precise demands, it could be sufficient to measure the zeroline only once a day (after sufficient equilibration) and use it for all measurements of the same kind. However, the zeroline must be subtracted from the measured heat flow rate to get the true sample heat flow rate

The advantages of this somewhat time-consuming procedure are, however, striking:

- the influences from apparatus and reference sample are eliminated, - the remaining heat flow rate curve is almost straight and horizontal (outside

thermal events), as the heat capacity of the sample only changes slowly with temperature,


- determination of the baseline is easy (no curvature), - step-like changes of the heat capacity are easily detected and cannot be mixed

up with a peak, - weak exothermic events (e. g., chemical reactions) can easily be detected when

the remaining heat flow rate deviates from the straight line and curves downward (exothermic heat flow rate counts negative),

- changes of the heat capacity during transitions (melting, crystallization) can easily be detected,

- specific heat capacity and thermodynamic quantities can easily be calculated.

This is why we recommend to subtract the zeroline prior to all further evaluation. Note: The recently introduced turret-type DSC (TA Instruments) takes the

possible asymmetry of heat capacity and thermal conductivity (i.e., the zeroline) into account and corrects for it internally with the so-called Tzero™ technique. As a result the measured curve is automatically zeroline-corrected and the DSC produces a measured curve, which is almost zero if an empty sample pan is used.

Heat Capacity and Other Thermodynamic Functions

The specific heat capacity is easily calculated from the measured heat flow rate with the zeroline subtracted. From the definition dQ = CpdT follows:

dQs

Cp dt cPs cPm - cPo cp=-=---=--=

m dT m . f3 m . f3 m·-

dt

In other words the specific heat capacity in steady state (no transitions and reactions) is the sample heat flow rate divided by sample mass and heating rate. As the latter quantities are given and constant for a DSC run, the cp curve and the measured curve with zeroline subtracted are identical beside a constant factor. Therefore the evaluation programs of modern DSCs calculate the measured curve (after zeroline subtraction) in units of specific heat capacity (J g-l K-1). It should be mentioned that this quantity equals the real (vibrational) heat capacity only in the absence of transitions or reactions (peaks). In the case of thermal events in the sample (in the region of peaks) it is an "apparent heat capacity", different from the static (vibrational) heat capacity and, in addition, "smeared" (see Sect. 5.4). However, there are great advantages of such a presentation; the advantages mentioned above are still all valid and there are even more:

- the heat capacity curve (outside peaks) can be compared with heat capacity values from the literature, and serve as an calibration check,

- the heat capacity is a thermodynamically well defined quantity and opens the way to determine thermodynamic potential functions.

Because of these advantages we recommend to calculate the specific heat capacity curves generally from the original measured curve.

5.3 Further Evaluation of DSC Curves 121

Normally the DSC runs at constant pressure, this yields the heat capacity at constant pressure Cp (T). The enthalpy follows per definition by integration:

T

H(T) - H(To) = I Cp(T)dT To

This function offers the possibility to determine enthalpy changes connected with transitions or reactions in a thermodynamically exact way (cf. Sect. 6.1.6). Other thermodynamic functions can be determined similarly, the entropy reads:

S(T) - S(To) = J Cp(T) dT To T

and we may even calculate the free (Gibbs) energy: G(T) - G(To) = H(T) - H(To) - T(S(T) - S (To».

5.3.2 The Baseline and the Determination of Peak Areas (Enthalpy Differences)

The baseline is produced for steady state conditions (see Sect. 5.1 and Figs. 5.1,5.2). Within a peak, i. e., during a transition or reaction, the baseline is defined as that curve between Tj and Tf in the region of a peak, which would have been recorded if all cp changes (and changes of heat transfer and other influences) had occurred but no heat of transition had been released. This is a virtual line which never is recorded and therefore must be determined otherwise. It is, however, an essential curve, because the area between the baseline and measured curve is a measure for the latent heat of the respective process (heat of transition or reaction).

Construction of the Baseline

The baseline has to be constructed in the region of peaks of all kinds to be able to determine peak areas. For 1st order transitions, with or without coupled Cp change (and/or change of the heat transport mechanism), the baseline can be constructed in different ways (cf. Hemminger, Sarge, 1991). A definite baseline can be constructed only if a pure Cp change occurs; changes of heat transport conditions in the range of a peak give rise to uncertainties in the shape of the baseline. For certain evaluations (for example kinetics), the measured curve and the respective baseline must possibly be "desmeared" as well (cf. Sect. 5.4). The common baseline construction methods are as follows.

1. For irreversible transformations without measurable cp change, the baseline can basically be determined by repeating the measurement with the same sample whose reaction process has come to an end (example: annealing of lattice defects during recrystallization of a plastically deformed metal). In these cases, the baseline is interpolated in the course of the measured curve of the 2nd run. The baseline uncertainty corresponds to the repeatability error of the DSC. If the heat capacity of the reacted product is different from that of the original sample, this method cannot be used.


2. For transformations with continuous Cp changes (without changes of the heat transfer conditions), the transformed mass fraction is in principle known for any particular time of the transformation from the measured curve. Thus, the Cp change coupled to it can be calculated.

3. Expressed more generally and in a purely formal manner (for example, for a non-horizontal measured curve outside the peak), the following is valid: When the degree of reaction a (t) is known (possibly only by approximation), the baseline can be constructed according to van der Plaats, 1984. For the change of the slope of the baseline between Ti and Tf> the following is valid in good approximation (see Fig. 5.3).

( d<P) = (1- a)(d<P) + a(d<P) dT bl dT Ti dT Tr

That is to say, aCt) determines how quickly the slope of the baseline changes from the slope of the measured curve at Ti (with a = 0 for T < Ti) in the interval Ti ~ T ~ Td with a = a (t) 1 to the slope of the measured curve in T[ (with a = 1 for T> T[). In the interval between Ti and Tf> the following results by approximation for the functional values ~I:

~I = (1 - a) <Pi,ex + a· <p[,ex

where <Pi, ex and <p[, ex are the segments for the measured curve extrapolated into the peak range from the left-hand and right-hand side, respectively. That

T,

tempera ture .. Fig.5.3. Construction of the baseline taking the degree of reaction a (t) into account (according to van der Plaats, 1984). Tj initial peak temperature, Tr final peak temperature, <1\ex> <Pr,ex measured curve extrapolated from Tj , Tf into the peak region


is to say, q,i,ex and q,f,ex are to be calculated as polynomials and extrapolated into the peak range in order that the baseline can be calculated according to the above equation. (When deriving this relation it has been assumed that the difference between the slopes of the measured curves at Ti and Tf is not extremely great; otherwise, this could be taken into account as well.) Integration of the peak area can then be carried out using the difference between the (desmeared) measured curve and the baseline q,(T) - ~1(T).

4. If the heat capacity of the sample changes suddenly by L'lCs at the transition temperature Ttrs with a constant sample temperature prevailing during transition (first-order phase transitions), the change in the heat capacity is described by a step function at T = Ttrs • By the "RC-elements" (cf. Sect. 3.3.3) of the measuring system (cf. Figs. 3.7, 3.9), this step function is transformed into an exponential function (of time) (see Hemminger, Sarge, 1991).

5. For transformations showing a jump of the baseline the cause of which is not known and which is not necessarily coupled to the fraction transformed during the reaction (for example spontaneous change of the heat transfer between sample and container), there is no method for finding the "correct" baseline which can be backed up theoretically. Figure 5.4 shows several possible constructions. The peak area very much depends on the selected baseline. For this reason, several possibilities should always be tried out (see Hemminger, Sarge, 1991). The differences in the peak areas (or heat flow rates) appear as systematic uncertainties of the measurements and must be taken into account when the overall uncertainty of measurement is estimated.

Fig. 5.4. Possible baselines when the reasons for the baseline change between Ti and Tf are not known. Ti initial peak temperature, r Tf final peak temperature, - linear extrapolation into the peak region and jump somewhere between Ti and Tf - - - constructed by means rp of an "apparent" transformation function, .... straight line between Ti and Tf

T, 0 temperature ..


The baseline always undergoes changes when Cp changes; additional changes may result when:

- the thermal conductivity of the sample changes substantially during transition,

- the thermal resistance between sample and temperature sensor (heat exchange conditions) changes during transition (e.g., during melting),

- the conditions for the heat transfer between sample and surroundings change, for example when the emissivity of the sample changes due to a reaction and the sample is positioned in an open crucible, or when the crucible is deformed.

When such changes coincide with Cp changes, the uncertainty how the baseline should be determined generally increases. There is then no method for constructing a "true" baseline.

As emphasized before, all the theoretical discussions presume a zeroline of the DSC which is at least an absolute straight line. If not, the zeroline must first be subtracted from the measured curve before the described procedures can be carried out correctly. It is highly recommended to do this for every measurement. The advantage to transform the measured curve into heat capacity units (see above) is obvious in this context; taking possibly known heat capacities of the sample before and after transition as a reference the Cp change during transition (i. e., the baseline) can easily be calculated.

Determination of Latent Heat Connected With a Peak Area

Integration of the measured curve with the baseline subtracted yields the heat of transition or reaction. This heat equals the enthalpy of reaction or transition, however, only, if the heat capacity is constant during the transformation. Otherwise the enthalpy of transition is a function of temperature (cf. Sect. 4.2 and 6.1.6). As a consequence the peak area is something like an integral mean value of the enthalpy change within the temperature regime of the peak. To get the total H(T) curve and from that the enthalpy change, the Cp curve should be calculated from the measured curve and then integrated (see Sects. 4.2 and 6.1.6).

As stated above, the baseline for its part is proportional to the pure heat capacity of the sample in question. Bearing this in mind, a method can be specified, which allows the latent heat (enthalpy of a transition or reaction) to be determined from a Cp curve without a baseline being constructed. The way of proceeding will be demonstrated by the example of melting, which is a 1st order thermodynamic transition (Richardson, 1993). Figure 5.5 shows schematically the Cp curve prior to, during and after transition. The heat capacities Cp,s (T) for the solid prior to melting and Cp,\ (T) for the liquid after transition differ in absolute values (at equal temperature) and also in their temperature dependence. The kind of functional dependence of the heat capacity on the temperature (linear dependence, polynomials of higher order) is unimportant. Extrapolations into temperature ranges that are not directly measurable must, however, be reliable. If this is guaranteed, the heat absorbed over a wide temperature interval


f >. ~

'-' 0 0-0

(P .• (T) '-'

~

0 Q)

.c

tempera ture ..

Fig. 5.5. The determination of thermodynamically valid values for the enthalpies of transition (according to Richardson, 1993). The limiting temperatures Tl and T2 must be chosen well below or above the transition interval. fJ.H(T) = Q - A - B with

Q: ~::::: :1 (horizontal dashed) A: ~ (right dashed) B: ~ (left dashed)

can be divided into the Cp portions and the isothermal enthalpy of fusion that is of interest. The following is valid (see even Fig. 6.8).

11 H (T) = H[ (T) - Hs (T)

= (H[(T2 ) - Hs(Tj» - (Hs(T) - Hs(Tj» - (H[(T2 ) - H[(T)

T T2

= (H[(T2 ) - Hs (Td) - f Cp,s (T) dT - f Cp,[ (T) dT Tj T

=Q-A-B

The three quantities Q, A and B are accessible by experiment, these are the areas hatched differently in Fig. 5.5.

Enthalpies of reaction uniquely defined from the thermodynamic viewpoint can be determined analogously; however, this determination presents some special features (cf. Sect. 6.2) and Figs. 6.6 to 6.9.

From Cp curves, got at low heating and cooling rates to avoid irreversibility (if necessary after previous desmearing of the measured curve), the entropy change in temperature ranges in which no transitions take place can be determined with sufficient accuracy according to the following equation:

T C (T) S(T) - S(To) = f -p- dT

To T

This is not allowed in the region of phase transitions because the peaks are always smeared and the plotted temperature is never the true equilibrium tem-


perature Teq of the sample. However if, in compliance with the above prescription, Teq and t:.H(Teq) are known, the reversibly exchanged entropy of transition can easily be calculated from the quotient t:.H(Teq)/Teq. It is then possible to calculate changes in the entropy and the free enthalpy for arbitrary temperature intervals. If a certain substance exists in several forms, information on its thermodynamic stability as a function of temperature can be obtained.

5.4 Desmearing of the DSC Curve

The user of a DSC is usually interested in the heat flow rate into the sample to be investigated or in the related thermodynamic potential function (depending on the temperature), which is either the specific heat capacity cp(Tsample) or the enthalpy change t:.H(Tsample)' The measured curve produced by the calorimeter as a function of time, or a temperature proportional to it, is, even in the form of a cp curve, only a more or less "smeared" representation of the function searched. Degree and quality of this "smearing" differ depending on the type of DSC, and they, moreover, depend on measurement parameters (heating rate, temperature, sample size). The theoretical relations which lead to this smearing have been quantitatively described in Chapter 3 in various approximations.

In many cases the curve measured by the DSC is a sufficiently exact representation of the functions searched for, and the desired evaluations can be directly made. When the accuracy requirements are higher, or kinetic evaluation must be performed, the measured curve (and the interpolated baseline) must be corrected and converted prior to their being used to determine the thermodynamic potential function (t:.H) of the sample or its derivative (Cp)' This procedure is referred to as "desmearing" and will be described in detail in the following. It should therefore be made quite clear when individual corrections are required, when they can be dispensed with and what will be the consequences for the uncertainty of the results.

Figure 5.6 schematically shows the relation between the heat flow rate curve recorded by the DSC and the desired potential function for the case of a 1st order endothermic phase transition. Accordingly, desmearing takes place in several steps:

a) Linear transformation of the abscissa (indicated temperature) in compliance with the results of the calorimeter's temperature calibration, and transformation of the ordinate (indicated heat flow rate) in compliance with the results of the calorimeter's heat flow rate calibration.

b) Determination and subtraction of the zeroline (sample crucibles empty). c) Mathematical elimination (deconvolution) of the influence which heat trans

port phenomena inside the DSC (heat flow rate relaxation) exert on the measured curve.

d) Proper conversion of the heat flow rate function into the real specific heat capacity of the sample, i.e., (non-linear) variable transformation of the temperature measured by the DSC into the true sample temperature.

e) Integration leading to the thermodynamic potential function (enthalpy).

The required steps will be described in closer detail in the sections below.

5.4 Desmearing of the DSC Curve 127

Q

b

e

-Fig.S.6a-e. Curve measured by a DSC for a 1st order endothermic transition and its evaluation. a Scaling of temperature and heat flow rate axes on the basis of the calibration, b Zeroline (crucibles empty) subtracted from the measured curve, c Desmearing due to influences from the apparatus (thermal lag), d Conversion into the heat capacity function, e Calculation of the enthalpy function. <Pm heat flow rate (measured curve), <Ps heat flow rate (sample run), <Po zeroline heat flow rate, <Ptrue heat flow rate after calibration, 6. <Ptrue sample heat flow rate ( <Ps minus <Po), <Pr reaction heat flow rate into the sample, T m temperature (measured), Ttrue temperature after calibration, Ttr, transition temperature, Ts sample temperature, Cp heat capacity of the sample, 6.H enthalpy change


5.4.1 Correction of the Temperature and Heat Flow Rate Indicated

This procedure has already been described in Sect. 4.3 and 4.4, however, for the sake of completeness, reference to it will be made again in the following.

For the correction of the temperature indicated at the heating or cooling rate {3, the following is valid:

Ttrue = T m + I1Teorr{fJ) (S.1) I1Teorr {f3) = I1Teorr{{3= 0) -I1T{f3)

11 Teorr ({3 = 0) results from the calibration measurements, and 11 T (f3) must either be determined experimentally as it may be different from one sample to the other, or it is determined by approximation from the results of the calibration measurements (cf. Sect. 4.3.1). The above equations describe a shift of the temperature axis scaling of the output of the measured function, a shift which is different for each heating rate. A simple correction is concerned here which must of course be carried out only if it is significant for the accuracy required for the measurements. The correction of the displayed temperature according to Eq. (S.l) takes only influences from the instrument into account. Samples of great thickness possibly require an additional correction which also considers the temperature proflle inside the sample; this correction is described below (Sect. 5.4.4).

Note: The recently introduced turret -type DSC (TA Instruments, see Sect. 2.1.2) takes the heat capacity and thermal conductivity of the measuring system into account and corrects for the thermal lag with the so-called Tzero™ technique. As a result either the temperature of the sample sensor or the temperature of the sample crucible is calculated and used as temperature axis for the measured curve. These corrections are normally sufficient for common measurements and additional corrections of the temperature scale are not necessary if careful calibration has been done.

The following equations are valid for the correction of the displayed heat flow rate (cf. Chapter 3 and Sect. 4.4.1):

IPtrue = K~· IPm or IPtrue = - K . 11 T m

The factors K~ (or K) are determined by suitable calibration measurements (cf. Sect. 4.4.1). The negative sign in the above equation is not always used consistently; it results from the differing definitions of the sign of I1T m (measured temperature difference) and IPm. A linear transformation of the IP-axis of the graph of the measured curve, with the form of the function maintained, is, however, only possible if K~ (or K) is not temperature-dependent in the range of interest. This case is often given only for relatively small temperature ranges (e.g., for a transition peak). In all other cases (above all when the heat capacity is measured over temperature ranges of 100 K or more), the form of the function must be recalculated according to the following equation, the scaling of the IP-axis being retained:


5.4.2 Subtraction of the Zeroline

It is not possible to produce a perfectly symmetrical DSC in which the differential signal between empty sample system and empty reference sample system is exactly zero over the entire temperature range. This applies also to the measured curve with the sample crucibles empty (zeroline).

To obtain the true heat flow rate into the sample (for example, to calculate the sample's heat capacity), the zeroline must be separately determined by a second measurement and subtracted from the curve actually measured. For the correct determination of the zeroline see Sects. 5.3.1 and 6.1.1.

The correction of the instrument asymmetry presents the problem that the two measurements can be carried out only one after the other; it must therefore be ensured that the DSC's symmetry properties have not changed in the meantime. This is a strict condition which means that high requirements must be met by the calorimeter (cf. Sect. 7.2). In addition, it is indispensable that the DSC is opened between the two measurements and the sample crucible changed or moved. This alone can change the symmetry properties, which results in the zeroline being shifted.

The design of the evaluation software of the DSC computers is normally such that the zero line (empty crucibles) is first subtracted and then the corrections of the temperature and heat flow rate scales (see above) are made. This order is justified only if the scale corrections for the measurement and the baseline are the same. In all other cases, first the correction and then the subtraction must be made. The same is true of all advanced desmearing procedures presented below.

5.4.3 Calculation of the True Heat Flow Rate into the Sample

Owing to the design of the DSC's measuring system, the heat flow rate into the sample cannot be measured directly; the measurement always yields only a heat flow rate at a certain distance from the sample, outside the sample crucible. Due to the finite thermal conductivity of the material between this point of measurement and the sample, the measurement signal is always a smeared representation of the sample reaction heat flow rate. The kind and degree of smearing depends basically on the thermal inertia of the system, in other words it depends only on the DSC construction.

Good DSCs retain linearity and allow the theory of linear response to be applied to this problem. Using this tool, ways have been found to calculate the true reaction heat flow rate from the measured heat flow rate curve. These methods are generally referred to as deconvolution or desmearing; they will be described in the following.

Solution of the Differential Equation

For simple cases, the behavior of a DSC can be described by normal differential equations. The mathematical interrelation has already been dealt with in Chap-


ter 3. The desired function cPr(t) can therefore be calculated from the measured function cPm (or AT(t) which is linked with cPm by the equation cPm = -k'AT(t», using Eq. (3.6) in the simplified form:

dcPm(t) d2cPm(t) cPr(t)=cPm(t)+al +a2 2

dt dt (5.2)

Accordingly, the desired function is a simple sum of terms which include the measured function and its derivatives. The coefficients al and a2 include the time constants Tl and T2 of the instrument and thus thermal resistances and capacities of the measuring system. Equation (5.2) can easily be realized by an electronic circuit with operational amplifiers. The "desmeared" signal can then be determined "online" (simultaneously) from the measured signal cPm (t). It is, of course, also possible to calculate it from the (stored) function cPm (t) after the measurement has been concluded.

Although the problem is easily solved (mathematically or by electronic means), the determination of the proper coefficients al and a2 is not so easy in practice. The time constants Tl and T2 [Eq. (3.7)] can be determined from the measured function of a pulse- or step-like event (cf. Fig. 3.10), but the actual relation to al and a2 may be more complex than the approximate calculations in Sect. 3.1 possibly predict. There is no other way than to adjust these two coefficients (by trial and error or by suitable fit mathematics) until the results of the desmearing procedure (online or calculated) coincide with the original heat flow rate cPr inside the sample. The best method to simulate an event is to switch on a constant heat flow rate (for a certain time) with the aid of a built-in electric heater.

Note: The recently introduced turret-type DSC (TA Instruments, Sect. 2.1.2) takes the heat capacity and thermal conductivity of the measuring system into account and corrects for it internally with the so-called advanced Tzero™ technique. As a result the measured curve is automatically desmeared and the software produces a curve of the heat flow rate into the sample crucible as a function of sample crucible temperature. This correction includes the three steps of desmearing described so far. This method takes a careful calibration of the DSC and unchanged measuring conditions as granted to give reliable results.

Numerical Methods

If the behavior of the DSC cannot be described by simple differential equations within the framework of the required accuracy (e. g., in the case of power compensation DSCs), another method must be applied which has its roots in the theory of linear response. It is valid for all measuring systems which work linearly, that is to say the measured signal for two distinct pulse-like events in the sample must be the sum (superposition) of the two single functions from each individual event (Fig. 5.7). Another condition is that all measured functions (curves) of various pulse-like events should have the same shape, in other words all these measured functions divided by their peak area (normalizing) must yield the same function, the so-called apparatus function a (t)


T,

r

b

T2

temperature

tempera ture ~

Fig. 5.7 a, b. Linear response of a DSC.

131

a Pulse-like heat events inside the sample at temperatures TJ and T2 in the scanning mode, b Measured DSC curve, - - - -: hypothetic (measured) curves of the individual events at TJ , T2 ,

-: recorded (superposed) curve measured for two successive events, tPr heat flow rate developed in the sample, tPm measured heat flow rate (measured signal), Q heat of the event, a(t) apparatus function of the DSC

(or even called Green's function). If these conditions are fulfilled, the following is valid:

tl>m (t) = c f tl>r (t') . a (t - t') dt' == tl>r (t) * a (t ) (5.3)

This defines the so-called convolution product of two functions, a integral equation which often is abbreviated as "convolution product"with a star as operator. The equation is valid for all DSCs which work in the above-described linear manner, irrespective of whether a certain approximate formula [such as Eq. (3.6)] is explicitly known. Only the "apparatus function" a (t) must be known which can easily be derived from a pulse-like event, usually produced with a built-in electric heater, or by the sudden solidification of a strongly supercooled melt of a pure metal (e.g., Sn).


The seamy side of this desmearing method, also called deconvolution 1 , is the rather ambitious mathematics required to solve the integral equation (5.3) for the function of interest 4>r(t). There are essentially two methods, the Fourier transform and the recursion method. Both require numerical calculations with a computer.

The Fourier transform represents an integral operation:

{S (f(x» = nl2 . f fey) . exp(ixy)dy

Applied to the convolution integral [Eq. (5.3)], it results in the convolution theorem (see textbooks of mathematics):

Thus the convolution product turns into an ordinary product which can be solved for

The desired function is obtained by an inverse Fourier transform:

This method can be applied in all cases. Today, the Fourier transform is commonly included in many mathematical software packages and therefore easily available. The drawbacks of this procedure lie in its laborious course and abstract nature, since the calculations are performed in Fourier space. Those who lack experience in numerical Fourier transforms are advised to study some "pitfalls" such as the "break-off effect" and the "sampling theorem", both obtained by numerical treatment. Under specific conditions this simulates periodicities and fluctuations which do not reflect any actual processes in the sample (refer to the literature for further details, e.g., Bracewell, 1965).

The recursion method for solving the convolution integral, Eq. (5.3), starts from the following recursion formula:

4>r,O(t) = 4>m(t)

4>r,n(t) = 4>r,(n-I)(t) + (4)m(t) - aCt) * 4>r,(n-I)(t» (5.4)

The deviation between the "reconvoluted", still inaccurate synthetic function 4>m,n (t) = a (t) * 4>r,(n-I) (t) and the measured function 4>m (t) is used additively for the recursive correction of the approximation.

1 Deconvolution means to solve the convolution integral equation, it is a special case of desmearing.


The recursion formula (5.4) does not converge for all event functions. Abrupt changes and steps (on-off effects and similar phenomena) generate oscillations of the approximation function which diverge rapidly. However, in the case of the "smooth" curves commonly encountered in calorimetry, the procedure converges quickly and without problems.

It should be noted that every desmearing procedure increases the noise. The better the resolution in time the higher the noise. Desmearing should therefore be carried out only if it is necessary, i. e., for events whose width is in the same order of magnitude as the time constant of the DSC and if the exact time course of the heat flow rate is needed (e. g., for kinetic investigations).

Both numerical desmearing procedures can be applied only after the measurements have been concluded (offline). For the case of a discrete (sampled) function, the convolution integral [Eq. (5.3)] can, however, be represented as a linear system of equations.

If>m(ti) = c· M· L (If>r(tk)' a(ti-k+l)) (i = 1 ... n) k=l

which can be resolved for If>r (tj):

(5.5)

As can be seen, each value of the desired function can be calculated from the measured points If>m (tk) (k ~ i) measured before this moment and the (known) discrete apparatus function a (tJ (i = 1... n). Even with this numerical deconvolution it is then principally possible to calculate the desired function online during the measurement and to put it out.

The problem is that the measured values (and the apparatus function) are not quite exact but noisy, i. e., uncertain, and that the errors enter into the calculation progressively.

In addition, the measured values are small at the beginning of a transformation and the noise fraction is, therefore, relatively high; the initial values of the apparatus function are usually very small which - according to Eq. (5.5) - leads to a very noisy signallf>r (tj), which in turn results in numerical instabilities which may lead to a "run-time error" of the computer. For these reasons, online deconvolution according to Eq. (5.5) is usually not suitable without special precautions concerning the initial peak region in question. Of course, such calculations can be performed offline as well, i. e., after the experiment has been finished.

5.4.4 Advanced Desmearing

The desmearing procedures described so far have essentially taken into account the influences from the DSC measuring system and the interaction between sample and instrument. The events inside the sample have not yet been discussed.


Peak Shape During Melting

For a 1st order endothermic phase transition of a pure sample, desmearing as it has been carried out so far furnishes, for example, a deconvoluted measured curve as shown in Fig. 5.8a. When recorded as a cp function, the 1st order phase transition has, however, the shape according to Fig. 5.8 b. Both functions are obviously not alike. To be able to calculate the real cp function from the measured function, the reasons must be known quantitatively which have led to the particular function according to Fig. 5.8 a. In the case of an endothermic phase transition, the sample temperature is constant for the duration of the phase transition; as the sample crucible is heated linearly, the heat flow rate into the sample also increases linearly:

tPr = L . fl T = L . (T m - Ts)

dtPr = L . (dTm _ dTs) dt dt dt

'----,-' '----,-'

"" f3 = 0

dtPr dTm -=L-dt dt

d tPr dt d tPr -·-=-=L dt dTm dTm

(L apparent thermal conductance to the sample, f3heating rate).

(5.6)

Accordingly, the increase in the heat flow rate depends on the heat transport conditions to the sample and on the heating rate.

In this case, advanced desmearing of the measured curve of a 1st order phase transition consists in simply compressing the abscissa in the range Tl to T2 (Fig. 5.8) and dividing the ordinate by the mass and the heating rate. As a result, the triangular peak transforms to a 6-peak with the weighting factor of the phase transition heat fltrsH, and the temperature (abscissa) becomes the sample temperature.

For endothermic transitions of, for instance, non-pure substances one may start from the fact that the rather low heat flow rate at the beginning of the transition is scarcely influenced by the limited thermal conductivity. But if the heat flow rate increases and approaches the magnitude of the phase transition peak, the shape of the measured peak will be smeared more and more. At a certain moment (TJimit in Fig. 5.9a) it reaches the heat flow rate limit of the apparatus and the sample temperature falls behind the measured temperature. The limiting factor is the same as for the phase transition of a pure substance, namely the thermal conductance L of the path from the temperature measuring point to the sample. The measured end temperature is not the end temperature of the transition of the sample (Fig. 5.9 b). There is a non-linear connection between these two temperatures which generally cannot be specified in detail.

In the case of impure substances which can be described with the aid of thermodynamics of eutectic multi-component systems, the theoretical shape of the


Fig. 5.8 a, b. Advanced desmearing in the case of the melting of a pure substance. t a Deconvoluted measured curve, b Corresponding cp(T) function. ~r true heat flow rate into the sample, T m temperature cj> r measured, Ts sample temperature, Ttr• transition temperature, L apparent thermal conductance be-tween sample crucible and sample, cp specific heat capacity of the sample

a

t (p

b

135

Tl T2 Tm ..

l~

function shown in Fig. 5.9b is known (see Sect. 6.9). This allows the true melting curve to be determined from the measured one by calculating along the theoretical curve and comparing the areas which must be equal according to the law of the conservation of energy. This calculation may be called desmearing as well, but it is far away from deconvolution due to the theory of linear response, which in the melting region, of course, is not valid any more.

Thermal Lag Inside the Sample

Another problem is the smearing of the measured curve due to the temperature profile developing in the sample itself. It is evident that the peak assigned to a transition in a thin sample differs substantially from that in a thick sample (cf. Sect. 3.1}.1t can be shown by the method of Laplace transformation that the temperature profile in the sample has a parabolic shape and that the mean temperature (T) of the sample can be calculated according to the following formula (Hoff, 1991):

d2

(T) = llb - Cp • P . P . -3..\

(5.7)

136

Fig. 5.9a, b. Advanced desmearing in the case of a melting of an impure substance. a Deconvoluted measured curve, dashed line: hypotenuse of a right-angled triangle with the slope L, b Corresponding cp(T) function (the temperature scale is not correlated to that of a). q,r true heat flow rate into the sample, T m temperature measured, Ts sample temperature, Ttrs transition temperature, Te extrapolated peak onset temperature, L apparent thermal conductance, cp specific heat capacity of the sample

a

5 DSC Curves and Further Evaluations

• 1 /1

/ 1 / 1

/ 1 r--------=------f-----+------~------

I 1 1 1 I 1 1 1

Tend 11,,1

Tlb being the temperature at the lower (heated) boundary of the sample (Cp heat capacity, p density, A thermal conductivity, f3 heating rate, d sample thickness). The second term on the right-hand side of Eq. (5.7) is an additional correction of the temperature scale.

In reality both the finite thermal conductivity of the heat path to the sample and the temperature profile of the sample influence the shape of the measured peak. In the case of linearity, the total apparatus function (often called Green's function as well) is the convolution product [Eq. (5.3)] of both parts, as these events are connected in series (Hahne, Schawe, 1993). This seems to complicate the desmearing procedure, but fortunately linear response theory offers the possibility to determine the total apparatus function (including sample properties) from the "step-response". This can be done by analyzing the switch-on and switch-off behavior of the scanning mode.

Step Response Analysis

The starting and ending of a DSC run always implies switching the heating rate from zero to a constant value and vice versa. As a result the real heat flow rate


f 2~

ClJ ~

C L..

:J: 0 -~

c ClJ

.<=

time ~

Fig.5.10. Curve measured with the heating rate switched from zero to f3 at to. 1 measured curve, 2 theoretical heat flow rate into the sample

O.B

S·l

I 0.6

0.4

C;

0.2

0.0 0 10 20 30 40

time ~

Fig.5.11. Apparatus function a (t) from step response arising from 1 switch-on, 2 switch-off

s

137

60

into the sample should change in a step-like manner, whereas the measured heat flow rate rises with a certain delay (Fig. 5.1O). This step response function covers both the heat transport and the sample behavior. By differentiation the pulse response function can be derived from this function (Fig. 5.11), which is the apparatus function (Green's function) of the total system. Using this apparatus function desmearing can be carried out as described. Figure 5.12 shows that such a desmearing gives rise to a distinct change in the measured curve in the case of samples with poor thermal conductivity. This may be used as a proof of the influence which the thermal lag and the temperature profile inside the sample


2.2

J glf

f 1.8

1.6 Co

LJ

1.4

1.2 350 360 370 380 390 K 410

temperature .. Fig.5.12. Heat capacity measurement showing a glass transition. 1 measured curve, 2 desmeared curve (10 mg polystyrene, heating rate 20 K min-I)

exert on the measured curve. An advantage of this method is the easy measurement of the correct apparatus function by only switching from the isothermal to scanning mode and vice versa which is always done in the case of heat capacity measurements (cf. Sect. 6.1). Thus the apparatus function in question is always on hand.

Note: The newly introduced step-scan mode of operation (cf. Sect. 2.4.2) makes it possible to get the step-response function and thus the Green's function of the total system at every moment of the measurement - even during transitions occurring in the sample - this offers the possibility to determine pure sample properties with all apparatus influences corrected for. Such a correction is in particular fruitful for determination of the complex heat capacity (see Sect. 5.5.2) during transitions.

Simple Thermal Lag Correction

The lag between sample and measured temperature due to the thermal resistance between sensor and sample causes an error not only because the shape of a peak is smeared, but also the baseline and the measured curve when steady state cp

measurements are performed. Figure 5.13 may serve to explain this"thermallag" effect: There is in principle a difference between the measured temperature of the sensor and the true temperature of the sample during a scanning run (Sect. S.4.1).1t increases with increasing heat flow rate into the sample due to the proportionality of heat flow rate and associated temperature gradient. The heat flow rate for its part depends on the heat capacity of the sample and on the heating


t ,...:::-

-e. cu

..2:!. ~ c -e. '-

~ 0

~

..... c OJ

.<=

heating rate {3 ----~----~--O>

~/ /~

~/ CPtr" //

(cpm- CPtru.) = tl(3) ~/

j ~/ ............................................... :/.'< ......... . :::::::::· .. ::::::::::::::::::::::·.::::>Z<:::::.;····: :

/' . . ................. ~~~~»~... ... . ... \ i

( (T5 -·Tm)~t({3) : .. . 0> I

~ag

T

139

t "

Fig.5.13. Difference between true (dashed line) and measured (solid lines) heat flow rate (or heat capacity, respectively) curves. T program temperature, Tst starting temperature, Ts sample temperature, Tm measured temperature, 4'>true true sample heat flow rate, 4'>m measured heat flow rate

rate. As a consequence, the measured curves shift more and more to the right side in Fig. 5.13 with increasing heating rate and heat capacity of the sample.

Due to the proportional relation between heat flow rate and heat capacity of the sample, an analogous relation is valid for the difference between the measured apparent heat capacity (calculated from the measured heat flow rate) and the true heat capacity of the sample. As the sample temperature is lower than the temperature displayed, the heat flow rate (and the heat capacity) displayed are also too low - compared with the true one - due to the increase in the sample heat capacity with temperature (see Fig. 5.13).

As the temperature dependence of the heat capacity is normally not very strong, the effect discussed (the thermal lag) is not very important. For highly precise measurements ( < 1 %), it is, however, to be taken into account and, if necessary, it should be corrected by applying the desmearing procedures referred to above. The thermal lag 6T itself can be approximately determined (Richardson, Savill, 1975b and Vallebona, 1979) from the heat 6Q, which is proportional to the area A between the step function at varying scanning rate and the measured function which is the step response function (i. e., the area between the two functions in Fig. 5.10, cf. also Fig. 6.2):

6T= 6Q Cp,s

(6T: thermal lag in K; A: area in J; Cp,s: heat capacity of sample in J K-1)


With this value, the temperature scale can be corrected for the lag of the sample temperature caused of the finite thermal conductivity and the influences of the sample.

5.4.5 Further Calculations

The desmearing procedures described above furnish the true heat capacitytemperature function of the sample, Cp (Ts). An integration must be carried out to obtain the thermodynamic potential function change:

T2

t:.H(T) = f Cp(T) dT TJ

Usually, the pressure p is constant in DSC experiments and the enthalpy change t:.H(T) is obtained from Cp; (see Sect. 5.3.1).

If the measurement would be carried out at constant volume, in analogy to Cp , firstly Cv ( Ts) is obtained and from this, by integration, the (internal) energy t:.U(T) as the respective thermodynamic potential function. (This is not very important regarding DSC measurements as the realization of a constant volume is very difficult.)

The time and effort to be spent on desmearing seems to be large and the procedures complex. It should, however, be noted that the individual steps of the desmearing procedure described in this section must be carried out only if it is necessary because of the accuracy requirement. In other words, the increase of accuracy after desmearing should be larger than the uncertainty of the measurement and the increased uncertainty caused by the desmearing procedure itself. For most applications, both deconvolution of instrument influences and the advanced desmearing procedures can be dispensed with; in peak area evaluations they are absolutely unnecessary. They are, however, required when, for example, kinetic evaluations or a precise phase transition analysis have to be done.

5.5 TMDSC Curves

The temperature-modulated mode of operation (cf. Sect. 2.4.2) offers the possibility to separate different processes from one another. A special evaluation of the (modulated) heat flow rate curve results in different additional functions (curves) which, depending on the sample and the processes of investigation manifest themselves in a different way. Of course, a proper calibration and, if needed, zero line correction is taken for granted before further evaluation is performed.

With a sinusoidal temperature modulation [Eq. (3.9) 1 the heat flow rate curve has normally the following form [see Sect. 3.3.2 and Eq. (3.17)1:

5.5 TMDSC Curves 141

This is the sum of two parts: the first two terms on the right side describe the normal DSC curve got at a heating rate f30 [ef. Eq. (3.12)], the third and fourth term form the periodic part of the measured curve. The first part is obtained by "gliding integration" (over a period tp = 2n/w, see Sect. 3.3.2) which lets the periodic part get always zero and leaves the first two terms unchanged [apart from that this function is averaged (smoothed) within the integration interval]. The curve obtained in this way is called the underlying heat flow rate lPu, it is almost the same as the curve got in the DSC with the temperature modulation switched off (common mode of operation).

Subtracting the underlying curve from the measured curve yields the periodic part of the measured heat flow rate:

_ alPexo(Tu,t) IP(T, t) = IP(Tu, t) - lPu(T, t) = Cp· TA· W· cos (wt) + aT· TA . sin (wt)

= IPA • cos(wt + c5) (5.9)

This is in reality a periodical (sinusoidal) fluctuating function with an amplitude IPA which on the one hand is influenced by the heat capacity of the sample (lst term on the right side) and on the other hand by the reaction or transition, if the respective latent heat flow rate depends on temperature (2nd term on the right side). The latter causes even a phase shift 6. The periodic part of the heat flow rate function forms the basis for further evaluations of TMDSC. By Fourier analysis, which normally is included in the software of TMDSCs, the amplitude and phase angle of the modulated part of the heat flow rate function can be determined. Further evaluation results in the determination of an "apparent heat capacity", which in the case of processes occuring in the sample can be different from the common (vibrational) heat capacity. Depending on the evaluation (and mathematical) method several heat capacity curves, containing different information of the sample, can be obtained.

5.5.1 Reversing and Non-Reversing Heat Capacity

These curves are the result of the mostly used evaluation procedure. The basis is the following: if there are no transitions or reactions in the sample, the amplitude of the modulated heat flow rate is proportional to the heat capacity of the sample (see Sect. 3.3.2, case 1). Formally, even in the case of processes occurring in the sample, an apparent heat capacity can be defined:

IPA (T) cp,revo (T) = T

m· A·W (5.10)

This function (curve) is traditionally called the reversing heat capacity. Outside the region of transitions or reactions it corresponds to the normal (vibrational) heat capacity of the sample, within the region of processes it contains components of the process in question (see Eq. 5.9).


From the reversing heat capacity the reversing heat flow rate can be calculated:

IPrev.(T) = Cp,rev.(T)· m· Po

Subtracting that from the underlying heat flow rate, got via gliding integration of the measured heat flow rate function, yields the so-called non-reversing heat flow rate:

If the temperature dependence of IPex. [the second term on the right side of Eq. (5.9)] can be neglected we get [with Eq. (5.8)]:

(5.11)

the heat flow rate curve of the process (with a certain latent heat). But this is only true in the regions where Cp from the underlying curve and the reversing Cp really coincide. It must be emphasized that this is only the case in regions of pure (real-valued) vibrational heat capacity, in other words without contributions from possible complex heat capacity components. Normally this is true for chemical reactions and the crystallization of polymers, but not in the region of glass transition or melting. Outside of these critical regions the non-reversing heat flow rate curve reproduces the heat flow rate curve of a common DSC, but with the baseline (see Sect. 5.3.2) already subtracted.

The heat flow rate curves can easily be normalized and presented as heat capacity curves if needed:

IPnon-rev. (T) C (T)=---p,non-rev. a

m· /"0

and the following is valid:

Cp,non-rev. (T) = Cp,ll (T) - Cp,rev. (T) (5.12)

This equation is formally right, but it must be emphasized again, that these quantities are apparent heat capacities which are not really defined in a thermodynamic sense in the region of transitions and reactions. All three quantities differ from the respective heat flow rates only by a constant factor. In Fig. 5.14 the underlying (even called "total"), the reversing and the non-reversing heat capacity curves calculated from a TMDSC measurement of polyethylene terephthalate (PET) are presented as an example.

Within the scope of this evaluation approach the phase signal is totally ignored, all these curves are real-valued and describe either the real vibrational heat capacity, or, for reactions or transitions, the magnitude (absolute value) of an apparent complex heat capacity. To obtain the complex heat capacity we have to pursue a different path. .

5.5 TMDSC Curves

6

f: ""

2

1

o ~ -1

~ -2 "C;

~ -3 V)

60

143

100 140 180 220 O( 260

temperature ~

Fig.5.14. Reversing (upper), underlying (medium) and non-reversing (lower) heat capacity curves from a TMDSC measurement of PET (normalized to specific heat capacity units; m: 5 mg, Po: 2 K min-1,f: 28 mHz., T A: 1 K; according to Schawe, 1995 a)

5.5.2 Complex Heat Capacity

Another approach, which is valid within the limits of linear response, calculates an apparent complex heat capacity from the measured heat flow rate curve. Again this evaluation is only valid if the DSC is properly calibrated, and if the calibration of magnitude and phase signals is carefully done as well. The latter is not easy but, however, necessary to come to reliable results. If this is successful, the knowledge of the apparent complex heat capacity will be very helpful in understanding of time-dependent processes taking place in the sample.

To get the complex heat capacity we start from the same evaluation procedure as before: by gliding integration we determine the underlying heat flow rate curve, subtract that from the measured heat flow rate and get the periodic part. By Fourier analysis or another mathematical treatment (see, e.g., Hohne, 1997) the amplitude and phase angle are determined. From the heat flow rate amplitude the magnitude (absolute value) of the apparent complex heat capacity is determined [see Eqs. (3.15) and (5.1O)]:

Icp(T) I = 4iA (T} m·TA·w

(5.13)

This function is identical to the reversing heat capacity (see Sect. 5.5.1), it contains possible contributions from reactions or transitions (see Sect 3.3.2). Together with the properly corrected phase signal 8 (see Sect. 4.7.2), the apparent complex heat capacity can be defined:

cp = I cp I . exp (i . 8) = I cp I . cos (8) + i . I cp I . sin ( 8) (5.14)


From this follows the real part and the imaginary part of the complex heat capacity (the phase shift is assumed to be negative!):

Re(cp) = Cp = ICpl' cos (6) and Im(cp) = cp = ICpl' sin (6) (5.15)

As magnitude and phase always change with temperature, even the complex heat capacity and its real and imaginary part are functions of temperature. In Fig. 5.15 the result from such an evaluation of a TMDSC measurement of PET is presented. From comparing these curves with those of Fig. 5.14, it follows that there are huge differences, in particular in the region where transitions occur. Only the underlying curve coincides with the magnitude curve as expected, because they are calculated in a identical way.

In the region of transitions or reactions the complex heat capacity depends on the modulation frequency too (see Fig. 6,43). To gain information on time dependent processes, different measurements at different frequencies (periods) must be done. Performing non-sinusoidal (e.g., sawtooth) modulation enables one to get the apparent complex heat capacity not only at the frequency of the modulation but also at the frequencies of the higher harmonics from the same run (for details see Merzlyakov, Schick, 2001 a).

The most elegant method, however, is to analyze the heat flow rate response on a temperature step. After Fourier transform and proper correction using the transfer function (see Sect. 4.7), we obtain in this way the complex heat capacity cft (w) as a continuous function of frequency at the temperature where the step was performed. This function contains all information concerning the time dependent processes, after an inverse Fourier transform one gets the time dependent heat capacity function at the temperature in question. However, this elegant method is not available with the commercial evaluation software and therefore

6

}/lg:1 >--= w

'" Cl..

8 2 ..... '" OJ

.r;

w ~ 0 w OJ Cl.. Vl

-2 50 100 150 200 250 ·C 300

temperature _

Fig.5.15. Real part (dashed), magnitude (solid) and imaginary part (dotted) of complex specific heat capacity calculated from a TMDSC measurement of PET (m: 5 mg, /30: 2 K min-I, f: 28 mHz., T A: 1 K; according to Schawe, Hohne 1996)

5.6 Interpretation and Presentation of Results 145

not accessible to the normal user and we refrain from presenting the details of this special evaluation here. The advanced reader is referred to the original literature (Merzlyakov, Schick, 2001 b).

5.5.3 Curves from Step-Scan Evaluation

The recently, for power-compensated DSCs, introduced step-scan mode of operation (see Sect. 2.4.2; Cassel, 2000) orginates originally two curves called the specific heat capacity and the isokinetic baseline. The former is simply the interpolated curve through the discrete heat capacity values got from the areas A of the peaks caused by the discrete temperature steps t:..T (see Sect. 6.1.4):

A cp=--

m·t:..T

It contains the vibrational heat capacity as well as contributions of reactions and transitions (as with sinusoidal modulation, Sect. 5.5.1), in other words, it is an apparent heat capacity like the reversing Cp• Using the extrapolated isokinetic line (see below) as baseline makes a subtraction of a zeroline normally unnecessary. A careful peak area calibration is, of course, needed anyway.

The isokinetic baseline, on the other hand, is the interpolated line through the end point of the isotherms after each temperature step. This equals the (isothermal) zeroline of the DSC (caused by asymmetry) but contains all the reaction and transition heat flow rates produced by the sample even in isothermal mode. Within the region of processes, the shape of this contours curve is highly influenced by the period of time chosen for the equilibration after the temperature step. The longer the isothermal period, the lower is the end value of the heat flow rate caused by reactions or transitions which usually follow a time law. This curve must be evaluated with due care and attention to get reliable results concerning the reaction or transition heat.

The advanced step-response analysis presented in Sect. 5.5.2 can be carried out as well. As a result we get a set of complex heat capacities cp (w), one for every temperature step, which contains all necessary information about time dependent processes occurring in the sample during the run. An indisputable evaluation and interpretation is, however, not an easy task and the investigations on this topic are still not completed.

5.6 Interpretation and Presentation of Results

For certain tasks (e. g., quality control of goods and materials received) comparison of the measured curve with reference curves enables an identification of the sample substance on the basis of a yes/no decision. This generally concludes interpretation in these cases.

In the normal case, however, interpretation is preceded by an evaluation of the curve in order that data are obtained which are characteristic of the sample


substance and/or the transition investigated. The steps of a reliable evaluation are the following:

- desmearing, if necessary (cf. Sect. 5.4), - determination of the peak area or partial areas after construction of the base-

line (cf. Sect. 5.3.2 and 6.3), - determination of Cp changes (cf. Sect. 6.4), - determination of the uncertainties (cf. Sect. 7.3).

The data obtained from the evaluation form the basis of the interpretation. The presentation of non-interpreted measured curves is poorly informative for the non-specialist. A reliable interpretation can possibly be made only after several experiments with, for instance,

- variation of the sample parameters (particle size, mass, sample shape, ... ), - variation of operating parameters (heating rate, kind of atmosphere, ... ).

The interpretation must correctly describe the effects of the parameters changed. Sometimes, for example with complex reactions, a final interpretation by means of the DSC experiments alone is not possible. Other methods must then be applied in addition. Only if the results obtained by all test methods can be interpreted in the same way is it probable that the interpretation is correct. For example, in the case of kinetic investigations in inhomogeneous phases, one should always be aware of the fact that the calculated values are affected by considerable uncertainties.

When DSC results are presented, the following should be stated:

- sample characteristics (mass, purity, structure, ... ), - instrument characteristics (type of DSC), - test conditions (heating rate, purge gas), - the curve originally measured (and, if used, the desmeared curve), - details of the calibration procedures (materials and their characteristic data), - details of the evaluation of the measured curve (specification of characteris-

tic temperatures, construction of baseline, peak integration), - data obtained from the measured curve and the uncertainties by which they

are affected, - basic formulas and calculation procedures which are used (e. g., for glass

temperature, or kinetics), - interpretation on the basis of the DSC results (taking the uncertainties into

account), - whether the interpretation could be confirmed by variation of the parameters

or by other measuring methods, - comparable results from literature.

6 Applications of Differential Scanning Calorimetry

The output signal from a DSC, the heat flow rate as a function of temperature, and any derived quantity, such as the heat of transformation or reaction or any change of the heat capacity of the sample, may be used to solve many different problems. The work required to evaluate the measured curve may differ greatly from one case to another. This will become clear from the following text. Sometimes the required information can be obtained from only a qualitative evaluation of the DSC curve. But most of the examples described in this section demand precise measurements and critical, very often special, evaluation procedures of the measured curve. In every case the basis of reliable results is a careful calibration of the DSC (see Chapter 4). As a rule the separately measured zeroline (see Sect. 5.1) has to be subtracted from the measured curve before evaluation. In every case, the relationship between uncertainties in the measurements and the quantities to be determined must always be borne in mind.

6.1 Measurement of the Heat Capacity

The heat capacity is one of the most important material properties. There is no other method which supplies the temperature dependent heat capacity as quickly and over such a large temperature range with - for most purposes -sufficient accuracy. A knowledge of the heat capacity of a material as a function of temperature is the basis for determination of any thermodynamic quantity (cf. Sect. 5.3.1).

The use of normal, not hermetically sealed, DSC crucibles (with a lid which rests on the sample and may be lightly closed by crimping), always gives the heat capacity Cp at constant pressure. The situation is somewhat more complicated if one uses hermetically sealed crucibles or the special crucibles which are available for pressures up to the order of a hundred bar. In addition to the condensed phases, the heat capacity of which is required, sealed crucibles always contain a gaseous phase. In this case it makes no difference whether this phase is composed of air or of gaseous reaction products. Strictly speaking neither Cp nor Cv are obtained because the thermal expansion of the sample cannot be prevented and the pressure of the gas changes. However, the pressure dependence of the heat capacity of condensed phases is very small and as the change of pressure in the sealed crucibles is generally small, the measured heat capacity is nearly the same as that at normal pressure.

148 6 Applications of Differential Scanning Calorimetry

In the following of this section the suffIx "p" will be omitted so that C is the heat capacity at constant pressure Cp and c the corresponding specific (per mass unit) quantity cp for the sample (subscript S) or reference (subscript R).

According to Sects. 3.1 and 3.2 the basic equation for heat capacity determination (under steady-state conditions)

is valid both for heat flux calorimeters and for power compensating DSCs. As, normally, the true heating rates of the sample and the reference material are not accessible by experiment, they must be replaced by the average heating rate p. If the heat capacity CR is known, Cs can be determined easily and quickly from the measured differential heat flow rate d 4>SR' Several variants of the experimental procedure are known, four widely used techniques will be discussed.

6.1.1 uClassicalu Three-Step Procedure

The procedure is illustrated in Fig. 6.1. The temperature-time curve during an experiment is outlined in the lower figure and the response of the calorimeter is shown above. The three steps are:

1. Determination of the heat flow rate of the zeroline 4>0(1'), using empty crucibles (of equal weight) in the sample and the reference sides. The temperature program should only be started when the isothermal heat flow rate at the starting temperature Tst has been constant for at least one minute. If the DSC is computer controlled, this can easily be automated by checking the differences between the current average value of the heat flow rate and that one minute before with allowance for a predetermined noise level. The scanning region between Tst and Tend can be 50 to 150 K in modern calorimeters. At the isothermal end temperature Tend the above computer check must be repeated. For the evaluation procedure all three regions of the curve are needed. The zero line reflects the (inevitable) asymmetry of the DSC.

2. A calibration substance (Ref) of known heat capacity CRef is placed into the sample crucible (or into a crucible of same type and mass), whereas nothing is changed on the reference side. Using the same experimental procedure as for the zeroline, the following is valid:

K<t>(T) is a temperature dependent calibration factor (cf. Sect. 4.4.1). 3. The calibration substance in crucible S is replaced by the sample. In analogy

to the equation above we get:

6.1 Measurements of the Heat Capacity

:a o

co ... .c

a

1 ~ ::J

~ ... C1. E 2!

b

12

0

iso- • Iher~

mal al Tst

~

0 1st

T,t

lsi

sample

(ribralion subslance

scanning mode

'" .e-'" Ie; .e-I ~ ..

.e-emply pan

4

lime ..

time ..

149

isolhermal 01 Tend

"-:1

12 min 16

Fig.6.1. The conventional three-step technique for the determination of the heat capacity. a schematic course of measurement, b the temperature change during the run. Ts! start temperature at time tst> Tend end temperature at time tend, t1>s, t1>Ref, t1>o heat flow rates into sample, calibration substance and empty crucible, respectively,~ t1>SR differential heat flow rate between sample and reference crucible

The specific heat capacity Cs (at a given temperature) can be calculated by a simple comparison of the heat flow rates into the sample and into the calibration substance as illustrated in Fig. 6.1:

The calibration factor K~(T) need not, therefore, be known explicitly. If the condition mscs::::: mRefCRef holds, the experimental conditions are very similar to


those of the second step. Many of the possible sources of error for DSC measurements then tend to have at least partial compensation.

For the previous and the following considerations it is always assumed that the same crucible has been used on the sample side. If during the second and third step different crucibles must be used, crucibles of the same kind with nearly the same mass (mer) should be used. It is possible to make routine measurements using crucibles of different masses if allowance is made for the different thermal responses according to:

The specific heat capacity of the crucible material (second term on right side) serves as a correction only. The values for common crucible materials (e.g., AI} are known with sufficient accuracy. From an error estimation (see Sect. 7.3) one can find that omitting this pan correction would result in an error < 1 0/0, if the masses of all (AI) crucibles are selected to differ by less than 0.03 mg (at a sample mass> 10 mg and a specific heat capacity >0.5 J g-l K-1).

Sources of error: Ideal and real conditions during the recording of the zeroline and measured curve of the sample are compared in Fig. 6.2. Three differences are obvious:

1. The quasi-steady-state conditions in the scanning and final isothermal regions are not reached immediately after changes in the scanning program but with a certain delay.

2. The measured heat flow rate (with zeroline subtracted) may be smaller than the ideal (theoretical) one.

3. The isothermal levels at tst and tend differ from each other and from run to run (and may often have non-zero values).

These discrepancies result from the finite thermal conductivity of the path between temperature sensor and sample and from the limited thermal conductivity of the sample itself (cf. Sect. 5.4). The sample operates both as a heat capacity and as a heat resistance with respect to the thermal surroundings (PoeBnecker, 1990). The signal is therefore a summation of the heat flow stored in the sample and that which passes through it and to the surroundings (heat leak). To be precise, it always appears, of course, as the differential heat flow rate between sample and reference sides.

In the following the causes of the three above-mentioned deviations from ideal behavior are considered in detail and possibilities for their correction are given.

1. The smearing (caused by thermal inertia) of the measured heat flow rate curve during the beginning of the scanning region, before steady state is reached, reduces the temperature range over which calculations are valid. The initial unusable temperature range can be estimated by !1T= 5 to 10 times f3. reff· The effective time constant reff results from a coupling of the time con-

6.1 Measurements of the Heat Capacity 151

A

f r---I I

~ I c I ... ~ I 0

I ;:;::

c I C1J I oJ::

I

c B

I L ______ _

time .. Fig.6.2. Idealized (dashed line) and real conditions (solid line) during a heat capacity measurement. Curve section AC: delay function of the sample due to the restricted heat transfer between sample and sensor, hatched area ABC: the product of thermal lag r5T and heat capacity of the sample

stants for sample and apparatus and that of the heat transfer between support, crucible and sample. As a rule the influence of the apparatus is predominant. The time constants of modern DSCs may vary from 2 to lOs. For thicker samples with poor thermal conductivity (e. g., polymers) the influence of the sample may dominate reff.

2. As discussed in Sect. 5.4.4, the sample temperature is always lower (higher) than the program temperature during heating (cooling) and the measured heat flow rate <Pm differs always from the true value <Plr • Assuming the worst conditions (large samples, high heating or cooling rates, large heat capacity, poor thermal contact between crucible and sample holder), the difference between both temperatures may be more than 10 K. This temperature error cST (the thermal lag) can be estimated (see. Sect. 5.4.4) from the (delayed) heat cSQ, which is proportional to the area ABC in Fig. 6.2. This procedure gives a reasonable approximation even for thick samples and/or those with poor thermal conductivity. Although there is still a rather large temperature gradient in such samples, there is a marked reduction in the overall temperature error after correction for thermal lag (Hanitzsch, 1991). As an example the Curie temperatures of Ni (sample mass ca. 250 mg) differed by 10 K for the original heating and cooling runs, whereas the difference was reduced to 3.6 K after using this temperature correction method. For a particular sample


mass and heating (or cooling) rate the differences between true and measured heat flow rates are influenced by the thermal conductivity of the sample and by the heat transfer resistance between sample and sample holder. The heat transfer resistance, and thus the thermal lag, can be minimized by proper sample preparation and by correct positioning of the sample in the DSC. It is essential to ensure completely flat bases for the crucibles, uniform sample thickness, size and position. Thermal conductivity effects can be partially compensated if the calibration substance has a heat capacity and a thermal conductivity similar to that of the sample. But thermal conductivities of common calibration substances fall in the following order (values in W cm-1 K- 1): Cu (4.01) > sapphire disk (0.34) > organic materials (0.05), which limits the use. The best general method for the correction of all effects due to finite thermal conductivities is to use the special desmearing procedure described by Schawe, Schick, 1991 (cf. Sect. 5.4.4).

3. The isothermal levels at TSI and Tend (resp. tSI and tend, cf. Fig. 6.1b) for zeroline, calibration run and measurement differ from each other by amounts which depend on the type of calorimeter, TSI and Tend and the temperature interval in between as well as laboratory conditions. The offset of the isothermal levels must be corrected to a common level (normally zero) before the heat capacities are calculated. The correction is only meaningful, if almost comparable conditions for the total heat conduction path can be assumed for the three successive runs (zeroline, calibration substance, sample). However, PoeBnecker, 1990 has shown by a detailed theoretical treatment of the heat transfer in a power compensated DSC that measurements with large differences in the offsets of the isothermals should always be rejected. For precise measurements, as a rule, the heat flow rates of the isothermals at TSI and Tend

should not differ more than 5 % of the difference between the heat flow rates in the isothermal and the scanning region. If it is assumed that the change of the isothermal heat flow rates with temperature can be approximated by a straight line cPiso (T) (but see Sect. 6.1.3 for the limits, temperature intervals have to be sufficiently small), the offset correction is very simple. Figure 6.3 demonstrates the procedure. With cPiso,sl and cPiso, end the heat flow rates of the initial isothermal and the final isothermal, the following is valid:

The corrected experimental heat flow rates cPcorr(t) are then obtained by subtracting the above correction term:

which should be used for the calculation of the heat capacity then.


12

w

t 8

6 .2! 0 ... 3: 4 0

;;::

0

'" 2 .c-

O

-2 0

I

~ 2

t,t

expo curve <Pup ( TJ

............................ .......................... .

corr. curve <Peorr( TJ

3 4 5

time ... min

Fig.6.3. Correction of the experimental heat flow rate curve 4'exp for differing isothermals

6.1.2 The "Absolute" Dual Step Method

8

The second step (the calibration run) in the above outlined Cp measurement procedure is not necessary, if the temperature dependent calibration factor K<t>(T) is truly constant in time and has been determined carefully (see Sect. 4.4.1). The heat capacity of the sample can then be calculated as follows:

K<t> (T) • (I/>s - 1/>0) Cs =

f3. ms

This method obviously saves time compared with the procedure in Sect. 6.1.1. A disadvantage is that conditions of thermal symmetry between sample and reference sample are lost and with them the partial compensation of related errors, and that the conditions of the calibration may have changed for the measurement (and even the calibration factor) which increases the uncertainty of the results.

The dual step method is, however, always a good technique for the investigation of samples with high thermal conductivities. The comments of Sect. 6.1.1 regarding errors and their corrections remain valid also for this simpler procedure.


An interesting possibility to get better symmetry is to attempt to set up experimental conditions for the sample run which are very similar to those for the zeroline run. For this purpose the sample is put into the correct crucible Sand a reference substance CRef is placed in the crucible R on the reference side. The (differential) signal L1 tP will approach that of the zero line tPo when Cs ::::: CRef • The sample specific heat capacity is then given by:

mRef K</>(T) . (L1tP - tPo) Cs =-- CRef +

ms f3. ms

At first sight this seems to be an advantageous method, because the uncertainty of the heat capacity measurements should come close to that of the calibration substance. However, the quality of the measurements is markedly influenced by possible differences in the thermal conductivities and the heat transfer conditions of both substances (or runs). Unfortunately it is very difficult to recognize these influencing factors and to correct them because:

- Conditions for heat transfer may be poor, although rather small offsets of the isothermals may conceal this and even suggest high-quality measurements.

- Thermal lag cannot be corrected by the simple procedure outlined in Sect. 6.1.1. In addition, desmearing has not so far been carried out under these conditions.

6.1.3 General Precautions for the Minimization of Errors and their Estimation

The determination of reliable heat capacities with the procedures mentioned in Sects. 6.1.1 and 6.1.2 requires the knowledge of the true heat flow rate into the sample (i.e., the difference between recorded signal and the zeroline). The zeroline cannot be obtained simultaneously from the same run, it is even impossible to get it unless the sample crucible has been replaced once (or several times). This implies that the original setup of sample, crucible and temperature sensors can never be reproduced perfectly. The unavoidable changes in the conditions for the heat transfer, which normally are concealed, are responsible for the limited certainty of the obtained heat capacities. For minimum uncertainties during routine operation, the following precautions are advisable:

- very careful sample preparation taking care of optimized thermal contacts at all interfaces,

- sample and reference substance should have similar heat capacities and, as good as possible, similar thermal conductivities,

- temperature intervals must not exceed 100 to 200 K for a single run. If larger temperature ranges are needed, they should be divided into proper sub-intervals. For isoperibol DSCs the initial and end isotherms are in principle nonlinearly connected (due to asymmetries of radiation and convection losses) and this tendency increases at higher temperatures. This must be borne in mind when selecting appropriate temperature intervals. Overlapping temperature intervals are helpful to detect possible differences in the observed heat capacities reflecting the uncertainty of the experiments,


- heating and cooling runs should be performed and compared, - the measurement procedure should include calibration and zeroline runs

before and after the sample runs at the same conditions (possibly repeated several times),

- constant purge gas flow, - constant line voltage, - preheating of the calorimeter (including the sample, if possible) to at least

10 to 15 K above the maximum temperature of the measurement in question, to stabilize the system, optimize the heat transfer conditions, and remove adsorbed moisture or retained solvent,

- for low temperature operation, prevention of any condensation of water (ice) from the atmosphere or of volatile components of the sample on the cold surroundings of the measuring system,

- constant room temperature, - in the case of isoperibol DSCs, a very good temperature control of the ther-

mostated block, which influences the long-time stability of the recorded signal to a great extent. If, for instance, the DSC has been heated to a high final temperature during the zeroline run and the stabilization time before the sample run was not long enough, the following sample run would result in a different isotherm. The same happens if the scan is started before the isothermal baseline has reached a steady state (cf. Sect. 6.1.1).

Detailed discussions can be found in papers of Richardson, 1989; Suzuki, Wunderlich, 1984; Poe:Bnecker, 1990; Flynn, 1993 and Rudtsch, 2002.

For cp measurements performed with the mentioned methods, typical uncertainties of 1-5% are frequently reported in the literature. The comparison of the results of different authors is difficult, because the complete evaluation procedure is rarely published. Using the "Guide to the Expression of Uncertainty in Measurement" (GUM, see Sect. 7.3.2), Rudtsch, 2002, presents a detailed estimation of error for cp measurements according to the "three-step procedure" (Sect. 6.1.1) in a power compensated DSC. If the complete uncertainty budget is considered, the author obtains temperature-independent uncertainties of U (cpJ/cp = 1.5 % for samples of sapphire and glass ceramics between O°C and 600°C (at a level of confidence of approximately 95%). Rudtsch suggests the use of two semi-quantitative criteria to check whether a single run fulfils the repeatability requirements: the temperature-dependent function of isothermal heat loss (Poe:Bnecker, 1990, 1993; cf. Sect. 6.1.1) and the temperature-dependent function of thermal lag. At first, these functions must be determined using well defined reference samples. The single runs of unknown samples (within the respective temperature intervals) can be accepted if the deviations from the values of these functions fall within a preset confidence interval (e.g., ± 10%).

6.1.4 Procedure of Small Temperature Steps

The techniques described so far allow a large temperature range to be covered in one run. This gives a continuous heat capacity-temperature curve. An alternative procedure is to measure the exchanged heat (area) which is connected with

156

6

mW

t cu 3 <:> .... ~ o -<:> cu

.z::.

a o

12

r:

b o

01

o 2 3

Q 1

f--:

o 2 3


4 5 6 7 8 9 10 11 min 13

time ..

t:..T--_

r 4 5 6 7 8 9 10 11 mm 13

time ..

Fig. 6.4a, b. Discontinuous method for the determination of the heat capacity (upper curves: sample run, lower curves: zeroline). a Small temperature intervals f:.T, the heat flow rate does not reach the steady-state in the heating phase, b larger temperature intervals f:. T, the heat flow rate comes to steady-state


a small temperature step. Thus the total temperature range must be divided into narrow intervals (Fig. 6.4) which are successively scanned with isothermal periods in between (Flynn, 1974; 1993). Depending on the temperature interval the procedure corresponds either to Fig. 6.4 a or to Fig. 6.4 b. The same operation should then be repeated with empty crucibles (of same mass) to see whether there are remaining peak areas (caused by asymmetries) which have to be subtracted to get the correct areas.

If Ti and I1T represent the initial temperature and the temperature interval respectively, the average specific heat capacity for the j-th temperature interval between Tj-l = Ti + (j - 1) 11 T and Tj = Ti + j 11 T can be calculated from the heat Qj, which is proportional to the area enclosed by the sample and zero runs:

(cs (Tj» = KQ(Tj) . Qj I1T· ms

(6.1)

It is very important that the isothermal periods after a temperature step are long enough to attain steady-state conditions. These times differ according to the type of calorimeter and should be determined experimentally. The older powercompensated Perkin-Elmer DSC's (DSC 2, DSC 7), for example, need 1 to 3 minutes, longer times are needed for samples that are large and/or have poor thermal conductivity.

By analogy with Sect. 6.1.2, the empty reference crucible may be replaced by a crucible containing a calibration sample. The "11" in the symbol 11 (Qj) of the following equation is used to label this. If all crucibles, both on the sample and reference side, are assumed to have the same mass, we obtain:

KQ(Tj) ·11 (Q;) mRef' CRef (cs(Tj» = +---

I1T· ms ms

This discontinuous technique replaces the disadvantage of the determination of absolute values of the heat flow rates from the continuous method with the advantage of the integration method for heat determination. A further advantage is that no correction for thermal lag is required but the temperature calibration must be very precise and, of course, performed for a heating rate of zero. If the I1T-intervals are in the range of 1 to 2 K the calculated average value (cs (Tj» reproduces the searched value Cs (Tj ) sufficiently. The interval may be increased to 5 or 10 K when there is a low temperature dependence of the heat capacity.

Early applications of the discontinuous technique concern the purity determination of organic substances (Staub, Perron, 1974; Zynger, 1975), without really convincing success compared to the usual slow linear heating of the samples. All in all, the discontinuous procedure has been little used so far. Possible reasons for this are:

- Long measurement times are needed due to the longer response times, also in the case of the older power compensated DSC's. To determine the heat capacityover 100 K with the continuous procedures of Sects. 6.1.1 and 6.1.2 requires 30 to 60 min at a heating rate of 10 K min-I. In contrast the discontinuous technique takes 400 to 600 min for I1T-intervals of 1 K. This time can,


however, be reduced to 150 to 75 min if a 1 K interval is measured every 5 or 10 K only. This is perfectly acceptable when heat capacities are only slowly changing functions of temperature. The measurement time then becomes comparable with that of the other techniques.

- Commercial software is normally not available and a major effort must be put into the proper evaluation of the raw data.

- The accuracy of heat capacities measured in this way was not higher than for those determined using the continuous technique. For normal sample masses (l0 to 20 mg), temperature intervals of 1 K and specific heat capacities of 0.5 J g-I K-l, the heats to be determined are only in the order of 5 to 10 mJ. This is comparable with the area of the fusion peak of indium samples with masses of 0.15 to 0.3 mg. Unfortunately, minor errors in determination of the isothermal heat flow rates, which are unavoidable, affect the results of the cp determination much more than is the case for In melting peaks.

Step-Scan Procedure

The discontinuous heating or cooling technique made a comeback with the recently introduced method of StepScan-DSC™ (SSDSC) of Perkin-Elmer (Cassel, 2000). The method utilizes the very low masses of sample and reference furnaces and the rapid response times of the power-compensated DSC to perform a rather fast repetitive sequence of short heat-hold segments over a large temperature range (step-scan mode of operation, see Sect. 2.4.2). A zero line run and! or a run with a reference substance (sapphire) as needed in the three step method (Sect. 6.1.1) is in generally not necessary due to certainty and repeatability of the heat flow rate and peak area calibration valid over wide temperature ranges. However, an additional zeroline run (cf. Sect. 6.1.1 and 6.1.2) and its subtraction from the sample run before peak area evaluation is highly recommended for precision measurements of cp• If the DSC is well balanced (equal masses for sample and reference crucibles) a zeroline correction is not necessary.

The step-scan mode of operation is explained in Sect. 2.4.2. Typical temperature intervals (heating rates 10 K min-I) for each step are 1-2 K, typical isothermal increments are 20-30 s. The result is a series of peaks (cf. Fig. 6.4a). A cp-curve is generated from the peak areas as explained in Sect. 5.5.3.

Due to the very short times needed for each step, the measurement of the thermodynamic (vibrational) heat capacity with SSDSC is not sensitive to base line drifts, even at the highest accessible temperatures. Measurement of the heat capacity with the step-scan method has some advantages:

- High heating rates can be used for the temperature step and the equilibration time is short.

- The measuring time is distinctly lower (by a factor of 2-3) compared to the common short step method.

- The data treatment is straightforward, Fourier analysis of the raw data is not needed.

Moreover, for advanced applications it was shown (Merzlyakov, Schick, 2001 b) that this SSDSC technique can be considered as time domain TMDSC. If desired


the frequency dependent (complex) heat capacity Ceff(W) can be obtained from single experiments for nearly three decades of W (cf. Sect. 5.5).

6.1.5 The Temperature-Modulated Method

With the temperature-modulated DSC (TMDSC, see Sect. 2.4.2 and 3.3) the heat capacity of the sample is determined from the amplitude of the modulated part of the heat flow rate [see Eqs. (3.15), (5.13)]:

CPA Cp=cp·m=-

TA·w

with CPA the heat flow rate amplitude, T A the temperature modulation amplitude, w = 2n/tp the angular frequency (tp the period), and m the mass of the sample. If the TMDSC is carefully calibrated (see Sect. 4.7) the method offers the possibility to measure the heat capacity with some percent uncertainty in only one run, the subtraction of a zeroline is normally not required (but recommended for precision measurements). This method has the one big advantage of being practicable even in isothermal mode of operation, where the DSC methods presented so far fail.

Outside of the temperature region of transitions or reactions the TMDSC method results in the static (vibrational) heat capacity. The temperature dependent values are, of course, independent of the frequency or period used, provided the calibration has been done properly (see Sect. 4.7). That's why one may take one suitable value, say a period of 60 s, for all such measurements, which simplifies the calibration procedure markedly. The same is true for the temperature amplitude, which again should not influence the measured result if we move within the limits of linearity (see Sect. 3.3).A proper value for TA is 0.5 K at a heating rate of 2 K min-I. Figure 6.5 shows an example of such a measurement, the correspondence between the measured and literature values is conspicuous.

The situation is, however, quite different if transitions or reactions occur in the sample. Within the temperature region of such events the evaluation of the heat flow rate amplitude results in an "apparent heat capacity" which contains not only the vibrational, but even an "excess heat capacity" which contains contributions from the latent heat of reaction or transition as well as possibly time dependent changes of the heat capacity from relaxation processes (e.g., the glass transition, see Sect. 6.4). In such cases the measured apparent heat capacity may very well depend on frequency and it is just the frequency dependence which enables one to get detailed knowledge of the processes involved, one example is given in Sect. 6.8.3. The deconvolution of an apparent heat capacity made up of several components into its different contributions is not an easy task and needs in every case frequency dependent measurements and therefore an extensive calibration, in particular if a complex heat capacity (i.e., a time dependent heat capacity) has to be determined which necessitates a precise phase signal as well. Such a deconvolution needs special experiences and is still a matter of scientific discussion and normally outside the scope of common heat capacity measurements. However, for


Fig. 6.5. TMDSC heat capac- 3 ity measurement for poly-styrene (PS); solid line: J K- 1 g-1

reversing cp , dashed line: underlying cp , symbols: liter- 2 ature values for the solid and t liquid state (ATHAS, 2002) I (m:4mg,po: 2 Kmin-I, tp: 60 s, TA: 0.5 K) [p

OL-----~----~ ______ L-____ _

-50 o 50 100 O( 150 temperature ..

most cases it is true that the higher the frequency, the lower is the contribution of the excess heat capacity to the total apparent heat capacity. This offers the possibility to get the static vibrational heat capacity via extrapolation of the measured apparent heat capacity to infinite frequency where time dependent processes don't contribute anymore (Schick et al., 2003). To extend the normal frequency range of a common TMDSC, the evaluation of higher harmonics is essential, this requires a non-sinusoidal temperature modulation (see Sect. 5.5).

6.1.6 Typical Applications of Heat Capacity Measurements

The heat capacity is a basic quantity for determination of other thermodynamic quantities as well as material properties. Typical applications are listed in the following:

- Very important from the theoretical point of view is, on the one hand, the comparison of theoretical and measured heat capacities and, on the other, the calculation of the thermodynamic potential functions S(T), H(T), F(T) and G (T). For polymers numerous, reliable experimental heat capacity results are available and these have been collected in a data bank (Wunderlich et al.1990, ATHAS, 2002). The majority of these are DSC measurements and generally refer to relatively high temperatures (> 200 K). Values at lower temperatures (as close as possible to 0 K) must be measured using high-performance adiabatic calorimeters. The main microscopic cause of the measured, macroscopic heat capacity of solids is the vibrational motion. It is thus possible to calculate the heat capacity of solid polymers if the vibrational spectra are known. According to the procedure of Wunderlich and co-workers (Lau et aI., 1984; Wunderlich et al. 1990; Roles, Wunderlich 1993) the spectra may be approximately separated into group and skeletal contributions. The former are obtained from infrared and Raman spectra. The latter can be approximated by Tarasov functions, the characteristic parameters for which are obtained by fitting the experimental heat capacities at low temperatures, when the group


vibrations are not yet excited. The result of the calculation is Cv, the molar heat capacity at constant volume. To compare with the experimental Cp a relation between both quantities is needed. The thermodynamically exact conversion

T. V.y2 Cp-Cv=--

X

cannot often be used because neither the thermal expansivity coefficient y nor the compressibility X are known. A good approximation is to use a modified Nernst-Lindemann equation (Roles, Wunderlich, 1993).

3R· Ao' Ci- T Cp - Cv = 0

Cv ' Tfus

nus equilibrium fusion temperature; Ao empirical constant (for many polymers: Ao = 3.9 10-3 K mol rl; R universal gas constant).

When the heat capacity functions Cp(T) are known from as near as possible to a K on, the thermodynamic functions H (T), 5 (T), F (T) and G (T) can be calculated as normal:

T

H(T) = Ho + f Cp(T) dT o

F(T) = U(T) - T· 5(T)

T C (T) 5 (T) = 50 + f -p- dT

o T

G(T) = H(T) - T· 5(T)

Low temperature DSC measurements of Cp are not possible and for this technique only changes in these functions can be calculated.

- Calculation of the reaction enthalpies from the measured heat capacities of reactants and products using Kirchhoff's law:

T2

L1rH(T2) = L1rH(Tl) + f I (Vi' Cm,i(T)) dT Tl i

L1rH(Td, L1rH(T2) molar reaction enthalpies at Tl and T2 ; Cm,i(T) molar heat capacity of the i-th component; Vi stoichiometric factor of the i-th component (positive for products, negative for reactants).

- Refined models for the description of the melting of polymers over broad (> 100 K) ranges of temperatures require heat capacities of the amorphous and crystalline fraction in this region.

- Characterization of the glass transition process (cf. Sect. 6.4) requires heat capacity functions for the non-equilibrium glassy amorphous and the equilibrium liquid state. Most such applications need only the change of the heat capacity with temperature. Thus for commonly used polynomials of the type C(T) = ao + al T + a2 T2 + ... or C(T) = ao + al T + a2 T-2 , ao is not needed. It is generally found that a linear approximation (all ai > a 1 are zero) is sufficient.


6.2 Determination of Heats of Reaction

The aim is to determine a thermodynamically well defined (temperature dependent) reaction enthalpy. If a subsequent kinetic analysis is planned, the calculation of a similarly well defined conversion-time curve is of equal importance. The interpretation assumes that both the initial and the final state of the reaction are unambiguous and well-defined. Is this not the case, the results will be, at best, semi-quantitative. Because of the limited precision of DSCs minor effects due, for example, to changes in pressure, stress or surface contributions may be neglected. The following, simplified equation is then obtained for the measured heat flow rates in a DSC:

( dQ) = <Pm = Cp,5(T). dT + (OH) . d~ dt p dt o~ T,p dt

Cp,{;(T) is the heat capacity of the system at constant pressure and at constant extent of reaction. The partial molar reaction enthalpy flrH = (oHlo~h;p is nearly always replaced by the average reaction enthalpy (flrH), which is independent of the instantaneous composition of the system. The Kirchhoff equation

describes the relation between the change of the heat capacity of the reacting system and the temperature-dependent reaction enthalpy. Here the Vi are the stoichiometric numbers and Cp, i the partial, molar heat capacities of reactants and products.

Reactions may be carried out isothermally or non-isothermally (in scanning mode or following a special temperature program). Advantages and disadvantages of the two methods are discussed in detail in Sect 6.3.

Reactions in the Isothermal Mode

Here, the sample is heated as rapidly as possible from a temperature at which the system is inert (often room temperature) to the reaction temperature. The "drop in" technique (cf. Appendix 2) may also be used - the sample is dropped into the preheated calorimeter. The end of the reaction is indicated by a constant heat flow rate <Pend' Simple extrapolation of this <Pend to the start time yields the baseline of the reaction peak. As there is no heat capacity contribution to the heat flow rate in the isothermal mode, the relation between the heat produced or consumed and the average reaction enthalpy is very simple:

t (;

Qm(t) = f (<Pm - <Pend) dt = (flrH) f d~ o 0

The only uncertainty is due to possible drift in the signal. The caloric error for a typical sample mass of 10 mg and a heat production of 360 J g-l is less than 1 %,

6.2 Determination of Heats of Reaction 163

if drift during a one hour reaction is less than 10 ]l W. For long term reactions drift must therefore be comparable with the short time noise (cf. Sect. 7.2).

If a reaction is investigated at several temperatures (which is essential for a kinetic analysis), the temperature dependence of the reaction heat is also obtained. Alternatively, if the Cp(1') functions of reactants and products are known, .1.r H (1') may be calculated by means of the Kirchhoff equation.

The main problem with isothermal measurements results from the ill-defined behavior of the signal following the initial introduction of the sample. Even for DSCs with very short time constants this time amounts to at least 10 to 15 s. Most of this error can, however, be eliminated by subtracting the isothermal heat flow rate curve of a second run with the same (now totally reacted) sample and exactly the same conditions. Nevertheless, minor differences may remain because of the different thermal conductivities of products and reactants and because heat transfer conditions may have changed during the reaction.

In order to check the completeness of the reaction, the sample is often heated to higher temperatures. However this is only useful if side reactions or shifts of chemical equilibrium at higher temperatures can be excluded. If additional reaction has been detected, the clear assignment of a reaction heat to a definite temperature may be lost.

Reactions in the Scanning Mode

In principle the correct evaluation corresponds to that already discussed in Sect. 5.3.2. However the situation is more difficult, because it cannot be assumed that reactants and products have almost the same and temperature-independent heat capacities owing to the large temperature interval of the reaction (100 K and more). In contrast to phase transitions, which follow a zeroth order reaction law, chemical reactions do not terminate at, or shortly beyond, the reaction peak maximum. To complicate matters, the reaction products may have a glass transition within the reaction region. In addition, the glass temperature of the original reactants may shift into this temperature range. As a typical example, the curing of an epoxide resin is shown in Fig. 6.6 (Richardson, 1989). The reaction starts immediately after the glass transition of the reactants at A and is finished at B. In the following it is always assumed that the zeroline has no curvature or has been subtracted and that changes in heat transfer conditions can be neglected. Reaction enthalpies may be determined in the following ways (in the first case only the heat flow rate curve has to be measured, all other methods require Cp-measurements ).

Method 1: A linear baseline is drawn between the points A and B (Fig. 6.6). This simple and widely used procedure is only an approximation. It is better the smaller the heat capacity difference between reactants and products. The validity of this assumption can be decided very easily by comparing the original run with a rerun on the reacted material. Both curves should then fit at A and B. As can be seen from Fig. 6.7b, this is not true for this system, even after allowance for the glass transition.

164

Fig.6.6. Curing of an epoxide resin (according to Richardson, 1989) and the procedure mostly used to determine the heat of reaction. The linear baseline between start (A) and end (B) of the reaction ignores all possible changes in heat capacity

Method 2:


2.5

J K-g

t 10

>~

c

8" 1.S

c '" .c

~

~ 1.0 '"

300

B A

350 400 450 K 500

tempera ture ..

If the temperature dependent change of the heat capacity Cp,~(T) of the reacting system cannot be neglected, but the heat capacities of reactants and products are known over the relevant temperature range (and there are no glass transitions), then the baseline can be calculated by iterations similar to those of case 3 in Sect. 5.3.2. Method 1 above gives the Oth approximation to the extent of reaction function ~(T). The heat capacities of the products can easily be obtained from the rerun on the fully reacted sample. If the first run of the reaction mixture can be started at least 30 K before the reaction starts, the heat capacity of the reactants may be determined from this interval. Extrapolation into the reaction temperature range is generally accurate enough. Otherwise, the heat capacities of all components of the reaction mixture must be measured separately and the Cp function calculated using simple mixing rules.

After subtraction of the baseline constructed in this way from the measured curve, an average heat of reaction is obtained (the term reaction enthalpy should be avoided, the quantity has no thermodynamic significance) for the temperature range between TJ and T2 • Improvements relative to method 1 are only found if the heat capacity functions are known with sufficient precision (uncertainties < 1 %). Obvious improvements in the definition of the baseline should result from the use of temperature-modulated DSC (cf. Sect 6.1.5). By this method it is possible to get both the heat flow rate curve (which is proportional to the actual reaction rate) from the non-reversing curve, and the heat capacity function of the reacting mixture from the reversing curve, from the same run (see Sect. 6.3.6)

The degree of reaction function a(T) can be obtained from the ratio between partial heats (up to a certain temperature) and the overall heat of the reaction.

6.2 Determination of Heats of Reaction 165

Method 3: Many of the above problems and difficulties can be avoided by integration of the Cp curves (Richardson, 1989; 1992b; Flynn, 1993). The specific reaction enthalpy Arh {Td at the starting temperature TI (e.g., 350 K) may be determined in a thermodynamically correct way even if a glass transition occurs within the temperature range of the reaction. From Fig. 6.7 it follows that

The reaction enthalpy at temperature TI (Arh (TI» is obtained as difference between the areas X and Y. The baseline is not necessary in this case, because only differences are needed:

d r h (TI ) = X - Y = {hI (P, 490 K) - hI (R, 350 K» - (hI (P, 490 K) - hI (P, 350 K»

The enthalpy subscript l represents the liquid state of the reactants Rand the liquid-like or rubbery state of the products P. The difference hI (P, 490 K) -hI (R, 350 K) corresponds to the experimental quantity, the area X, defined by the lines at Tl> T2 , cp = 0 and the reaction curve. The horizontal hatched area Y follows from the rerun on the reacted sample.

490K

Y = ht (P, 490 K) - ht (P, 350 K) = I Cp,t (P) dT 350K

If the glass transition of the products is above TI , as is the case in Fig. 6.7, Cp,l

must be extrapolated as shown. Without this extrapolation another result would be obtained:

The procedure can also be extended to the determination of reaction enthalpies at any temperature Tbetween TI and T2 (Richardson, 1992b; cf. also Sect. 5.3.2). At first a possible glass transition is disregarded. One obtains:

Arh (T) = h (P, T) - h (R, T)

= {h (P, T2) - h (R, Td) - {h (P, 12) - h (P, T» - {h (R, T) - h (R, TI» or:

T2 T

Arh{T)=X- ICp{P)dT- ICp{R)dT=X-Z T TJ

This procedure is schematically shown in Fig. 6.8. The areas X and Z are hatched horizontally and vertically respectively. A simple rearrangement of the last equation gives:

T2 T

Arh (T) =X - I cp{P) dT+ I (cp{P) - cp{R» dT TJ TJ

T

= Arh (TI) + I {cp (P) - Cp (R» dT TJ


Fig. 6.7 a, b. The thermody-2.5 namically correct determi-

nation of the specific heat J/IKgl of reaction (according to

Richardson, 1989); for details

r see text. a 1 st run curve,

2.0 b 1st and 2nd run curves B Hatched areas extended to cp = 0; ~ h (350 K) = >-X - Y = hproducts (350 K) - ~

hreactants (350 K) = - 44.7 J g-l '" c. 1.5 '" ~ '" C1I

..r:::. ~

~

C1I 10 c. VI

a

300 350 400 450 K 500

tempera ture .. 2.5

J/IKgl

r 2.0 B

>-

~

'" c. 15 '" ~ '" OJ

..r:::. ~ -~ C1I 10 c. VI

b

300 350 400 450 K 500

tempera ture .. It is clear from Fig. 6.8 that this procedure is a direct application of the Kirchhoff equation. An enthalpy-temperature diagram (Fig. 6.9) is especially clear and instructive. Here, for the sake of simplicity, temperature independent heat capacities are assumed. The enthalpies are then linear functions of the temperature. The diagram also recognizes that both reactants and products may be in the glassy [curves Hg (R) and Hg (P)] or liquid [curves HI (R) and HI (P)] state. A reaction usually proceeds at a measurable rate only when the reactants are in the liquid state [above glass transition Tg(R)]. Further, the glass transition of the

6.2 Determination of Heats of Reaction

Fig.6.S. The thermodynamically correct determination of a reaction enthalpy (according to Richardson, 1992 b); for details see text

Fig. 6.9. The determination of reaction enthalpies from an enthalpy-temperature diagram (according to Flynn 1993); A: integral of Cp(n for the reactants, B: integral of Cp(n for the reaction mixture, C: integral of Cp(n for the products

1 >C1.

CI .c C Q.I

1 -'" ... .c ... -" .. ... a.. II>

167

~rHITI = X-Z

'p (R) 'p(PI

--- ~r"' I--!\

r Z

X

r, T temperature ..

_-K[ I I __ - I I ...l- I I I I I

Tg (R) T Tg (P)

temperature ..

products [Tg (P) 1 is very often somewhere between Tl and T2 • A formal (Flynn, 1993) and thermodynamically correct procedure yields four different reaction enthalpies and four different a values at each temperature (see Fig. 6.9):

glassy reactants -7 glassy products ArH = bd, ~ = bc/bd glassy reactants -7 liquid (rubbery) products ArH = be, ~ = be/be liquid reactants -7 glassy products ArH = ad, ~ = ac/ad liquid reactants -7 liquid (rubbery) products ArH = ae, ~ = ac/ae

Only the final case has any meaning for kinetic studies.


In exceptional cases, e. g., when there is a possibility of side or decomposition reactions, the heating run should be stopped at a temperature at which such disturbing reactions don't contribute to the heat flow rate. The reaction should then be completed isothermally at this temperature. Baselines are needed for both the scanning and isothermal parts of the reaction (Richardson, 1989). However, it may be more convenient to carry out the reaction totally in the isothermal mode.

6.3 Kinetic Investigations

6.3.1 Introduction and Definitions

Every chemical reaction is associated with a certain heat of reaction. From this it follows that the heat flow rate is proportional to the rate of reaction. The assignment of the time dependent heat flow rates to defined reactions leads to kinetic data. Every quantitative kinetic analysis starts with the determination of a continuous sequence of concentration-time data. DSC methods are widely used to solve kinetic problems because of the simple and fast sample preparation and the wide range of experimental conditions - much information is produced in a short time. The technique, by contrast with many other methods, immediately gives a series of "reaction rates" as function of the extent of reaction 5, dependent on time and temperature. The aim of kinetic investigations is then to find quantitatively this functional relation. The methodology is quite general and it is immaterial whether the reactions investigated come from inorganic, organic or macromolecular chemistry. Continuing advances in instrumentation and data treatment facilitate refined calculations.

The framework of the kinetic description of reactions was originally developed in physical chemistry for reactions in homogeneous phases. A generalized reaction with the educts A, B, ... and the products C, D, ... is described by the following stoichiometric equation:

aA + bB + ... ~ cC + dD + ...

According to the definition (ni = ni,O + Vi· 5),5 can be related to the consumption of an educt (Vi < 0) or to the formation of a product (Vi> 0). The rate of reaction is then quite generally given by:

1 d5 r=-·-

V dt

Considering a reaction in homogeneous phase and at constant volume, the rate law for the overall or the elementary reaction may then be formulated as usual:

1 d5 1 dCi r=-·-=-·-=!(CA,CB, ... T,p, ... )

V dt Vi dt

6.3 Kinetic Investigations 169

or quite general

r= f(n ·f(e}

If other possible variables such as concentrations of catalysts or inhibitors are kept constant, one obtains at constant temperature and constant pressure frequently:

1 dS n n r=-·-= k(T)· eAA • eBB ••••

V dt

k(n is the rate constant; the exponents nA, nB, ... are the partial orders of the respective reactants in the rate law. They are generally not identical with the stoichiometric coefficients. Their sum is equal to the overall reaction order n. Of course, a DSC curve does not provide the concentrations of the respective components themselves, but they may be obtained unambiguously from the measured heat flow rate if

- the heat of reaction is independent of the extent of reaction, - the overall reaction is an elementary reaction, i.e., only one heat-producing

reaction exists, - the initial and final states of the reaction are known.

It is clear from these restrictions that, in general, a kinetic evaluation of DSC measurements is only meaningful if supplemented by results from other analytical methods (e. g., from IR, UV, NMR, MS, GC, HPLC etc.).

Using thermo analytical methods for the investigation of kinetic problems, the concentrations of the reactants are usually replaced by the degrees of reaction a = S/Smax> in particular if reactions in heterogeneous systems are investigated.As a rule, a is set as the ratio of partial area and total area of the DSC peak:

Qt a(t} =

Q~ and

da If>(t} ----dt Q~

But this is only valid if the above restrictions are fulfilled! In every case the measured reaction heat Q~ must be checked and, if necessary, corrected with respect to the final degree of reaction actually reached. This correction must be made whatever the reason for the cessation of reaction - perhaps caused by a great increase in viscosity (vitrification) or coming to an equilibrium state. All subsequent kinetic analyses are incorrect if this modification is omitted. This is because the calculated a(t} is not related to the thermodynamic degree of reaction but to an apparent final state that is specific to the individual experiment.

A general rate law is written as

da -= k(n ·f(a} dt


The functionf(a) is very simple for the smallest steps (elementary reactions) of reactions in homogeneous systems. In the case of equimolar mixtures one obtains then:

da - = k(T) . c8- 1 • (1- a)n dt

da/dt is measured in units of reciprocal time, a ranges from 0 to 1. Using the DSC in the scanning mode at a constant rate (fJ = dT/dt = const), the term da/dt will be replaced by the term fJ· da/dT (minor self-heating of the sample during the exothermic reaction is neglected).

Using thermoanalytical methods, c8- 1 in k(T)· c8- 1 is usually neglected. Elementary reactions have always constant and integer reaction orders n within the whole range of reaction degrees, broken values suggest a more complicated mechanism. Solvent-free systems are normally investigated, because of the higher reaction heat per unit volume of sample. The measured heat flow rate curves are a reliable basis for subsequent kinetic analyses. To some extent the above rate equation also holds when reactants and products are completely or partially immiscible, the reaction is virtually irreversible and the products have no autocatalytic influences.

Many reactions in homogeneous systems cannot be described, at least within the total range of conversion, by the above n-th order rate law. A frequently used overall rate law, which in addition allows for catalytic and autocatalytic effects, is (Sourour, Kamal, 1976):

r = k(T) . am. (1 - a)n

The most general equation was suggested by Sestak, Berggren, 1971:

r= k(T)· am. (1- a)n. [-In(1- a)]P

The formal reaction orders n, m and p are only parameters of variation, to obtain a better fit to the experimental curves, an interpretation in the physicochemical meaning is difficult or impossible.

The situation is much more complicated for reactions in heterogeneous systems. During a chemical reaction of this type the reactants may always exist in different phases (liquid-solid, solid-solid) or the system may change due to the development of new phases and/or the vitrification of polymerizing components. Further, the results may be influenced by the energetic heterogeneity of the solid surface. It is typical of heterogeneous reactions that very different events such as nucleation, growth and diffusion can occur simultaneously and successively. A large number of special functions f(a) in the rate laws for these cases were derived in the past. Above all, the Avrami-Erofeev equation is of considerable importance for describing the kinetics of both decomposition reactions of solids and the isothermal or non-isothermal crystallization. The kinetic analysis is additionally complicated because we have numerous additional sources of error connected with sample preparation and measuring conditions.


Heat flow rate curves are frequently difficult to analyze without help from other analytical methods (especially temperature programmed X-ray diffraction, hot stage microscopy, thermogravimetry and possibly mass spectroscopy, atomic absorption spectrometry and gas chromatography}. A review and a critical judgement of the most frequently used rate laws is given by Brown et aI., 1980, and by Galwey and Brown, 1998.

The temperature dependence of the rate constant k (T) is usually described by the empirical Arrhenius equation or rarer by the Eyring equation, which follows from the activated complex theory:

Arrhenius: k(T} = A· exp (- ;~ )

Eyring: kB • T (~S") (AH" ) k(T} = -h- exp R . exp - RT

kB is the Boltzmann constant, h the Planck constant

Both equations were first developed for gas phase reactions, but they are valid also for reactions in homogeneous liquid phase. The Arrhenius parameters (A preexponential factor, EA activation energy) and the Eyring parameters (AS" activation entropy, AH" activation enthalpy) are for condensed systems related to one another by:

A·h AS"=R·ln--

e· kB • T

The Eyring activation parameters are more suitable for the understanding of relations between the structure of the reactants and the reactivity (Heublein et aI., 1984).

Although open to criticism (Flynn, 1990), as a rule the traditional concept of activation is also used for reactions in heterogeneous systems. The main theoretical difficulty results from the fact that the energy distribution of the immobilized constituents of a solid is not represented by the Maxwell-Boltzmann equation, respectively, that we find a number of surface sites with different energies and energy distributions. For instance, Hunger et al., 1999, have determined the activation energy distributions for the desorption of water from different zeolites and they have definitely proved the existence of several different adsorption places. But meanwhile two reasonable approaches supply also for this type of reaction a theoretical foundation for the exponential k- T relationship. Galwey and Brown, 2002, start from the band structure of a solid and consider discrete interface energy levels in the locally modified environment between the band structures of reactant and product phases. The occupancy of the interface levels is directed by energy distribution functions based on FermiDirac statistics for electrons and Bose-Einstein statistics for phonons. At higher energy differences between the interface levels and the highest occupied electron bands (compared with kB T) both distributions approximate to the expo-


nential term of the Maxwell-Boltzmann statistics. The second and promising physical approach to the interpretation of the observed exponential k-T relationship was developed by L'vov, 2001, 2002, for some types of solid state decomposition reactions as an alternative to the traditional chemical approach based on the Arrhenius concept. The reaction scheme (using the Hertz-Langmuir prediction of the dependence of reactant evaporation on the equilibrium vapor pressure) is based on the congruent dissociative evaporation of the reactants with simultaneous condensation of the low-volatile reaction products. In the interval between evaporation and condensation, the gaseous species can diffuse for some distance from the primary sites. L'vov has shown for different substance classes that this model also accounts for such features as: the mechanism of nucleation and growth, the autocatalytic development of some decomposition reactions and the influence of gaseous products on the reaction rate.

To summarize we can formulate three main objectives of kinetic investigations:

1. determination of the rate equation or the rate law and the corresponding activation parameters,

2. determination of the reaction mechanism, 3. prediction of the reaction behavior for any complex time-temperature pro-

files.

First of all, kinetic analysis is of course an efficient tool for data reduction. A large body of experimental data from a series of measurements with many data points has then been reduced to a model with few parameters. Further, reactions may be judged using different criteria. For example, the influence of variations of the activation parameters can be discussed together with changes in the composition of the reacting system (stoichiometry, catalysts, solvents and fillers) or with modifications of one reactant (number and position of substituents, homologous series).

But the true aim of a kinetic analysis is an evaluation of the correct reaction mechanism: this is fully defined, if the sequence of all elementary steps is known as are the activation parameters for each of these. The inseparable combination of rate law, activation energy and preexponential factor is sometimes named the "kinetic triplet" (Maciejewski, 2000). Frequently, the form of the overall rate law already permits conclusions according to a possible mechanism. The rate law can be simple or complicated. A complicated rate law always signals a complex mechanism with a number of steps. On the other hand, the opposite conclusion is not correct. A simple rate law can also be caused by the existence of a very slow reaction step - the rate-determining step - within a number of much faster steps. The investigation of the reaction mechanism is an extremely demanding task. Much time and effort is required to reach it and supplementary investigations using other analytical techniques are essential. Even then, the result is often only a most probable mechanism or the knowledge of the most important steps.

A third aspect of kinetic investigations becomes more and more important. The practitioner in industry needs a useful and quickly accessible reaction model, which on the one hand adequately describes the course of the reaction and on the other allows reliable predictions for any temperature-time conditions. The form of this reaction model (overall rate law, combination of formal steps in


a formal-kinetic model, use of the "true" mechanism as sum of all important elementary reactions) is unimportant.

6.3.2 Experimental Prerequisites for a Reliable Kinetic Analysis

Kinetic investigations are only meaningful after consideration of specific aspects due to sample preparation, some peculiarities of the DSC method, data sampling and the processing of the raw data.

1. Sample Requirements - After the introduction of the sample into the DSC the amount of reaction

should be negligible prior to the attainment of a stable steady state. If this is not the case it must be determined separately and allowance made for this effect.

- The reaction mixture must not react with the material of the crucibles nor should there be any catalytic influence.

- Samples with an appreciable vapor pressure must be loaded into special sealed crucibles. Errors due to the effect of pressure on the measured reaction rates are usually insignificant. Much larger errors can occur because of changes in the concentration of a volatile component, for example a catalyst, used in very low concentrations, could be partially in the gaseous phase outside the reaction mixture. The change in the concentration in the reaction mixture can be allowed for to a sufficient approximation, if the vapor pressure of the volatile component and the volume of the gaseous phase are known.

- Multiple measurements should always be made (with the same operation parameters) to check on the experimental repeatability. In addition some spot checks on other independently prepared reaction mixtures of the same composition should be carried out in order to exclude accidental errors during sample preparation. If the mixture is reactive at room temperature, loaded crucibles should be stored in liquid nitrogen.

2. Features Peculiar to DSC Methods - The reaction mixture in the small and sealed crucibles of a DSC cannot be

stirred. The results can, therefore, only be partially interpreted if concentration gradients occur during the reaction, or if there are transport processes to the gaseous phase (e.g., the decomposition of hydrates).

- It is not possible to add solid or liquid components to the reaction mixture after closing the crucibles.

- To apply these results to technical processes (e.g., to predict behaviour in a reactor) the very different conditions must be recognized and, in particular, allowance made for heat transfer effects.

3. Data Acquisition and Processing - Some varieties of one-point-evaluations, e.g., In(f3IT~ax) versus lITmax

(Kissinger, 1956,1957), lnf3 versus lITmax (Ozawa, 1965, 1970) or lnf3 versus lITa=o.s (van Dooren, Muller, 1983), are fast and easy to realize but no longer


state-of-the-art. Recommendable kinetic evaluations need the complete a(t, T) function and often also that of the rate of conversion daldt(t,T). Therefore, the correct baseline must be determined first (according to the methods 1-3 in Sect. 6.2) and then subtracted from the measured curve. Commercial evaluation programs offer usually the following possibilities after the reaction range was selected: - a straight line between start- and endpoint, - a sigmoidal baseline whose differences from the simple linear case are al-

ways proportional to the area of the peak, - horizontal lines starting either from the left or from the right. The left

starting base line should be used if for any reasons it is not possible to determine the end of reaction, whereas the reaction start is well defined. Using the right -hand horizontal line from the assumed end of the reaction is the standard baseline method for isothermal measurements.

The reliability of kinetic results strongly depends on the correctness of the baseline! Therefore, it would be ideal if the baseline could measured simultaneously and independently. This is nowadays possible by using the modern temperature-modulated or step-scan techniques (see Sect. 5.5 and Reading et al., 1994; van Assche et aI., 1995, 1996, 1997; Hu, Wunderlich, 2001; Flammersheim, Opfer mann, 1999a, b, 2001).

- In general, however, several reactions contribute to the measured heat flow rate. Apart from so-called model free evaluations (see Sect. 6.3.5), an evaluation is then possible only with respect to a certain model. If the gross reaction is made up of, say, n reactions, all contribute to the measured heat flow rate and their reaction heats can be treated as parameters and determined as such by variation procedures. It is obviously much better to determine as much supplementary information as possible (for the optimum case of n-1) for the individual heats of reaction by changes in the DSC conditions or other measurements, e. g., temperature dependent spectroscopic investigations of equilibrium states etc.

- The experimental curve has to be desmeared (Sect. 5.4) before any evaluation if there are significant changes in the heat flow rates at time periods, which are comparable with the time constants of the calorimeter.

- The true sample temperature may differ from that measured by the thermometer which is outside the sample, because of the limited thermal conductivity of the sample and its surroundings. This is especially true for larger sample weights and higher reaction rates. Strictly speaking, even "isothermal" reactions proceed non-isothermally. The true sample temperature can be calculated if the instantaneous heat generated by the reaction is related to the heat exchanged with the calorimeter (this is limited by the overall thermal relaxation time constant of sample and calorimeter). For example, with very fast light-activated reactions there are sometimes heat generation rates of 100 mW and more leading to temperature corrections greater than 10 Keven for calorimeters with small time constants.

- Assuming that absolute and random errors for the measured signal are independent of time, the relative errors of the measured reaction rates increase at the end of the reaction. Therefore, efficient software should weight the data appropriately.


- Correct calibration of the DSC is essential - the temperature scale, the heat flow and the scanning rates.

- Changes of the sample volume during the reaction (e. g., up to 8 % decrease in volume during epoxide amine curing reactions are possible) have to be considered, irrespective of whether subsequent calculations are made as functions of concentration or degree of reaction.

In much of the literature it is unfortunately not clear whether the above aspects have been considered and the corresponding results must be treated with reservation.

6.3.3 Selection of the Measuring Conditions - Isothermal or Non-Isothermal Reaction Mode

Classical techniques of chemical kinetics normally operate in the (quasi -) isothermal mode. By contrast, it is very convenient to make kinetic studies by DSC using the scanning mode. Both modes are possible in DSC and have particular advantages and disadvantages (Flammersheim, 1988, 1999a; b, 2000, 2001). They complement each other and should not be regarded as alternatives. Modern, highly sensitive DSC devices, powerful computerization and sophisticated evaluation methods together allow and favor the combined realization and evaluation of isothermal and non-isothermal experiments. Further, it does not matter whether model-free or model-based evaluations are made. Apparently different conclusions are always obtained if non-comparable reaction conditions were selected (Flammersheim, Opfermann, 2001). There is a permanent risk to overlook consecutive and concurrent reaction steps with higher activation energies in isothermal measurements. When planning measurements for a kinetic evaluation, the selection of the proper window in the global temperature-time-conversion (concentration) space is extremely important. This condition is easy to fulfil at dynamic experiments. They should be carried out using very different heating rates. At least three or better five measurements should be made, a practicable and reasonable range of heating rates is between 0.5 and 10 K min-I. This should be sufficient for scanning an adequate range within the global reaction field (Opfermann, 2000). Single-heating rate scans are useless for a kinetic analysis (Opfermann, 2000; Burnham, 2000). Burnham formulates in his paper that judges the results of the [CrAC kinetics project 2000: " ... kinetic analysis using single heating rate methods should no longer be considered acceptable in the thermal analysis community." On the other hand is it often difficult or impossible to fulfil the above condition for isothermal measurements. This is due to the necessary restriction to a relatively narrow temperature range, in practice rarely more than so K. All comments made up to now are just as valid if temperature-modulated or step-scan modes of operation are applied. As already mentioned the main advantage of such techniques is a reliable determination of the baseline, but at the expense of much longer times for the experiments.

In the following some typical advantages and disadvantages of the two main operating modes (isothermal and scanning) are particularized. But it should be


mentioned that several other temperature control modes have been used. The advantages and drawbacks were reviewed by Ozawa, 2001.

Isothermal mode: - A simple and immediate interpretation of the measured curve is possible be

cause of the complete decoupling of the two variables, temperature and time, if the minor changes of the sample temperature during the reaction can be neglected. This method is especially useful when searching for the most probable reaction mechanism. Two examples will illustrate this:

1 '" -c ... :3: 0 --c

'" .r:::.

.~ -'v '" C1. II>

During the polyaddition reaction of aniline and the diglycidyl ether of bisphenol-A (DGEBA) the maximum conversion rate is found at non-zero degrees of reaction (Fig. 6.10). This can only be due to autocatalytic and/or consecutive reactions. Furthermore it is obvious that not only the ratio of the concentrations of both reactants has substantial consequences but also that equal excess concentrations of amine or epoxide differ greatly in their effects. This is a very important pointer to the possible reaction mechanism. The second example (Fig. 6.11) is the polyaddition reaction of a mixture of the two isomers 2,2,4-trimethylhexamethylene diisocyanate and 2,4,4-trimethylhexamethylene diisocyanate (TMDI) with 3,6-dioxaoctane-l,8-dithiol (TGDT) in presence of varying concentrations of a catalyst. As expected, the initial conversion rate is proportional to the catalyst concentration but the

100

Wig

2.25

T : 405 K all sample masses about 15 mg

1.50 ra tio 0 f OGEBA to aniline

1: 1:1 2: 1:2 3: 1:3

0.75 4: 1:4 5: 2:1

0.00 o 17 34 51 68 min 85

time ~

Fig.6.10. Isothermal reaction curves for the polyaddition of aniline and bisphenol-A diglycidyl ether (DGEBA) in mixtures with different stoichiometry

Fig.6.11. Isothermal reaction curves for a stoichiometric system of trimethylhexamethylene diisocyanate and 3,6-dioxaoctane-l,8-dithiol with pyridine as a catalyst. The change in the reaction rate on the leading edge of the peaks shows direct coupling of catalysis and autocatalysis

increase in the slope of the leading edge of the reaction peak in direct proportion to the concentration of catalyst is unusual. This can only be explained if one assumes a direct coupling of catalysis and autocatalysis.

- The baseline follows unambiguously by extrapolating the measured heat flow rate after the complete reaction, assuming that all parameters, which can influence the measurement directly or indirectly (e. g., the temperature of the surroundings) are sufficiently constant. Demands on the apparatus can be considerable, depending on the reaction time in question. Above all the longtime drift of the signal should not be greater than the short-time noise.

- The reaction can be allowed to take place at low temperatures such that decomposition or side reactions can be avoided.

- Isothermal measurements need more time than scanning runs. - The main disadvantage, already mentioned in Sect. 6.2, is that the initial phase

of the reaction (30-60 s) cannot be measured precisely, as the steady state conditions are disturbed on introduction of the sample. The solution given, using the difference between the reaction curve and the fully reacted rerun, for kinetic analysis, is only partially successful. A better solution is to always choose sufficiently low reaction temperatures so that unavoidable errors are minimized to values which do not influence model calculations. In addition,


lime ------ lime ------100 10 min 30

0 Wg-' Wg-'

1 -0005 -005

1 30 '( SO '(

-0.10 -0010

¢ -0.15 ¢

0

1 Wg-' Wg-' -05

1 -10

¢ -10 70 '( ¢

-15 -20

a 0 2 min 50 65 °C 80

lime ------ temperature ------

0 os K min

-, Wg 1

1 -0.2 1 ¢ -10 ¢

10 K min -1 - 04

b 0 5 10 15 min -50 0 50 °C 100

time ----- temperature -----Fig.6.12. Calculated DSC curves for isothermal (left) and scanning (right) mode of operation. Reliable experimental values for isothermal mode are only accessible for t > 45 s (right of the vertical broken lines). a autocatalytic 2nd order reaction: A + B -7 C + 2B with 19 (A/s- I L mol-I) = l3, EA = 100 kJ mol-I, Qr = -50 kJ mol-I, b simple second order reaction: A + B -7 C with 19 (A/s- I L mol-I) = 5, EA = 50 kJ mol-I, Qr = - 35 kJ mol-I

smearing of the heat flow rate data and deviations of the sample temperature from that of the isothermal surroundings are so small that the experimental data need not be corrected.

- The useable range of temperatures is, however, often strongly restricted and too small for reliable kinetic evaluations. Two examples will demonstrate this. First, let us assume an autocatalytic 2nd order reaction, as frequently found in epoxy curing reactions (Sourour, Kamal, 1976; Klee et al., 1998; Flammersheim, 1998).


Such reactions are easy to investigate isothermally, as they often start with a low reaction rate. Three isothermal and three non-isothermal DSC curves (Fig. 6.12 a) were calculated for the assumed reaction path A + B ~ C + 2B, with 19 (Als- I L mol-I) = 13, EA = 100 kJ mol-I and Qr = -50 kJ mol-I. The calculated initial rates at 30 DC and 50 DC are so small that the distortion of measured DSC heat flow rates can be neglected or reliably corrected (Sect. 5,4, Flammersheim, 2000). But the time (> 6 h) taken to complete the reaction at 30 DC already requires a device with very good long-term stability. On the other hand the reaction at 70 DC is already so fast that a considerable part of this reaction occurs within the first 45 s (on the left of the broken line in Fig 6.12a), then each attempt to correct the initial heat flow rate must fail. In other words, the experimentally accessible temperature window for absolutely reliable isothermal measurements is relatively narrow (about 30-40 K). The situation becomes dramatic if an elementary step with an nth order reaction (n > 1) dominates. This is shown in Fig. 6.12b for second order kinetics (A + B ~ C, 19 [AI(s-1 L mol-I) 1 = 5, EA = 50 kJ mol-I, Qr = - 35 kJ mol-I). It is no longer possible to find a temperature range in which error-free measurements can be made. At temperatures higher than 25 DC the heat flow rate is strongly disturbed and falsified at the beginning, while at temperatures lower than 25 DC, the end of the reaction cannot be detected because signal and noise will have the same magnitude. Again, non-isothermal runs are clearly favorable.

- If isothermal experiments are planned, a rule-of-thumb may be helpful. Appropriate temperatures should cover the first half of the leading edge of a temperature programmed scan with low heating rates (0.5-2 K min-I)

Scanning mode: - Scanning measurements need less time than isothermal experiments but

construction of the proper baseline may be a problem. But as already mentioned this can now be bypassed using the temperature-modulated or the step-scan mode of operation. The measurement can be started at temperatures well below that of the beginning of the reaction, so steady-state conditions of the DSC are ensured. Unfortunately, reactions may go to completion, even at low heating rates, at such high temperatures that secondary reactions cannot be neglected (this is especially relevant for organic reactions). It is not clear from the shape of the curve what type of reaction is involved and this is a drawback when searching for the most probable reaction mechanism.

- The screening of the complete global reaction field, by carrying out nonisothermal measurements with strongly different heating-rates, is needed to avoid misinterpretations. This needs time but is always possible without problems (compare Figs. 6.12a and 6.12b).


6.3.4 Activation of the Sample by UV Irradiation

Reactions studied in the DSC are normally thermally activated. The wide range of experimental conditions (scanning rate, temperature range, even complicated procedures with intermediate annealing phases) is responsible for the extensive use of this technique.

Many reactions, however, can also be started by irradiation with sufficiently high energy. The investigation of such reactions has considerable value for the optimization of process control parameters and this is especially true for those polyaddition and polymerization reactions for materials with widespread use. Slight changes in the design of a conventional DSC yield a so-called Photo-DSC (see Sect. 2.2.2), normally used in the isothermal mode of operation. First applications of this type of DSC were described by Wight, Hicks, 1978; Tryson, Shultz, 1979; Flammersheim, 1981. The light (normally UV), having sufficient energy to start a chemical reaction, is brought to the sample by use of classical lens and mirror systems or by glass fiber optics. The portion of the spectrum which is of interest is selected using monochromatic filters and the desired intensity is adjusted by means of neutral density filters or metal sieves with different mesh width. For precise measurements electronic stabilization of the radiation intensity of the light source is extremely important. Light-activated reactions under conditions closely related to practice are usually very fast. The greatest part of

:. o

c cu

..c

300 Jase line of the reaction mixture with the absence of UV radia tion

mW 2nd run

.-.----.-~.-.---

150

15

base line of the reacted sample with the presence of UV radiation

o ~------~--------~--------~--------~--------~ o 0.7 1.4 2.1 2.8 min 3.5

time ~

Fig.6.13. 1st and 2nd run for the photopolymerization of a multifunctional acrylate at room temperature. (photoinitiator: 1 % 2,2-dimethoxy-2-phenylacetophenone (benzyl dimethyl ketal), light intensity: 5 m W cm-2 at 365 nm)


the reaction takes place within a few seconds (cf. Figs. 6.13 and 6.14). There are two consequences:

- There is a considerable change in the measured signal in periods comparable with that of the DSC time constant. The measured and true heat flow rates differ substantially. Desmearing is necessary prior to every evaluation in which the time variable is involved.

- Heat flow rates (cf. Fig. 6.13) are so great that the sample temperature deviates considerably from the temperature of the microfurnace, even in power compensated DSCs. If a thermal resistance of 40 K W-1 is assumed, temperature differences of 5 to 10 K would result even for samples with very good thermal conductivity. For organic samples, these differences are even greater. This means that the measurements are never isothermal, not even approximatively! Kinetic analysis must allow for this.

A typical example of a light-activated measurement is shown in Fig. 6.13. It is often found, even in the case of absolutely symmetric radiation conditions on both the sample and the reference side, that the baselines change during the dark and the illumination phase, as in the figure. Simple subtraction of a second curve measured with the same procedure but with the reacted substance removes this effect. Light-initiated reactions, using conditions in the DSC which come close to those used in practice, are always extremely fast. The apparatus (or Green's) function, which is needed for the desmearing (deconvolution, see Sect. 5.4)

500

-------mW

r 375

~ 250 I

c I ~ I ~ , 0 , ;:;:: , c , QJ

..c 125 " " "

" " r

0 0 0.5 10 15 min 2.0

time • Fig.6.14. Baseline corrected experimental (solid line) and desmeared (dashed line) curves for the photoinduced reaction of Fig. 6.13


of the very fast reaction, is obtained easily and is sufficiently precise if, after the completed reaction, the response of the sample to a light flash is recorded (Flammersheim et al., 1991). If it were not for the fact that the heat produced during the chemical reaction is evolved inside the sample, but during the light flash it is mostly a surface effect, this response would be an ideal apparatus function for the sample properties in question. The solid line in Fig 6.14 corresponds to the desmeared curve. The dashed curve is the original measured curve after subtraction of the baseline from Fig. 6.13.

The kinetic evaluation of light-activated reactions has so far only been partially successful because the true reaction course is overshadowed by additional influences:

- Irradiation of the initiator causes a time dependence of the concentration of the initiating radicals but this effect can be easily corrected if the kinetics of the decomposition are known.

- Samples have finite thickness (0.1 to 0.4 mm). It follows from the Beer-Lambert law that there is a varying light absorption and hence a gradient of the reaction rates within the sample. However, the calorimeter always records an overall heat flow rate. As result of this rate gradient there is a corresponding concentration gradient of the reactive component. An exact solution of the subsequent system of differential equations is only possible for drastically simplified boundary and initial conditions (Tryson, Shultz, 1979; Shultz, 1984). If the measurements have been desmeared, the expected correlation between average reaction rate on the one hand and sample thickness and light intensity on the other hand is found (Flammersheim, Kunze, 1998).

- The situation may be made more difficult by the increasing viscosity of the mixture during the reaction, which nearly always is carried out at room temperature. As a result, the reaction may be incomplete (Klemm et al., 1985). For example, at room temperature phenyl glycidyl ether (PGE) reacts completely during cationic polymerization induced by irradiation of diaryliodonium salts, whereas the reaction of DGEBA stops at about 50 to 55 % conversion (Klemm et al., 1985). In this case, a kinetic analysis is impossible and even the estimation of the final degree of reaction is difficult because of missing or insufficiently reliable experimental information from other analytical methods. To solve the problem, it was supposed in this special case, that the same heat is produced during the reaction of the oxirane ring both for PGE and DGEBA.

The experimental situation can be improved if extremely thin « 0.1 mm) samples are used with the lowest possible concentrations of photo-initiator. If diffusion limitations are small during the whole reaction, as in the case of the reaction of PGE, one then finds a simple first order rate law with respect to the monomer.

The advantages that lead to the application of DSCs for investigation of lightactivated reactions are essentially the fast, reliable and quantitative collection of data that reflect the effects of those parameters which influence the conversion rate:

- type of monomer, - type and concentration of solvents, photo-initiators and inhibitors, - wavelength and intensity of the radiation used.

6.3 Kinetic Investigations

6.3.5 Different Strategies of Kinetic Evaluation

The Traditional Way - Search of a Rate Law and its Activation Parameters for the Overall Reaction

183

In the past, many evaluations were directed to find a rate law and its activation parameters which adequately describes the overall reaction within a range of conversion as well as possible (cf. Sect. 6.3.1). For technological purposes a description of an even complex reaction by a mathematical model is often sufficient. Of course, reliable predictions are then only possible within the often very narrow limits of the data sets included into the evaluation. The physical and chemical use of such rate laws is severely restricted and - if at all of any value - it must be done very carefully. An overall rate law may give a good description over a certain range of reaction, because the rate reacts very sensitively to those parameters which are present in the rate law. It may be insensitive with respect to those parameters which do not appear explicitly (e. g., stoichiometry of the reaction, solvents, catalysts, packing efficiency for heterogeneous samples, flow rate and type of the purge gas). A rate law can often be found by systematic trial and error, this is a result of possible simplifications that may sometimes be made to the rather complicated differential equations that describe the true reaction mechanism. In spite of this, a determination of the gross rate law may be a useful first step in finding the reaction mechanism.

The risk of using gross rate laws for the description of the reaction is, that this possibly may hinder further advances towards the true mechanism. For instance, the validity within a certain range of conversion may be taken as confirmation of a postulated, but often too simple, model. Deviations in other ranges of conversion are then interpreted too narrowly within the framework of that model only. A typical example is the Sourour-Kamal equation (cf. Sect. 6.3.1), which is often used in the polymer chemistry literature to describe the epoxide amine polyaddition reaction. For stoichiometric reactions it reads:

da - = kJ . C • (1 - a)n + k2 . am . (1 - a)n dt

C is the concentration of a catalyst, nand m are formal reaction orders and kJ and k2 are the rate constants of the catalyzed or auto catalyzed reaction. The chemistry of the reaction is thus reduced to a simple rate law containing catalytic and autocatalytic steps.

With n = 2 and m = 1 this equation was originally derived by Smith, 1961, as the most probable reaction model for the epoxide amine system on the basis of all experimental results available at that time. However, solvent-free reactions can only be described by this equation up to conversion degrees of about 0.6, which is equivalent to a degree of polymerization of only 2. There have been numerous attempts to retain this simple rate law but any connection with the chemistry of the process is completely lost (n and m as fitting parameters alone, sometimes even as temperature dependent ones: Ryan, Dutta, 1979; Chung, 1984; Keenan, 1987; non-verified assumption of diffusion control at higher conversions: Barton, 1980, 1985; Huguenin, Klein, 1985). If the detailed chemistry of the complex


process is unimportant, the value of such corrections to this equation simply reflects the better fit that results from any increase in the number of parameters. It is hardly surprising, therefore, that the Sourour-Kamal equation is frequently used to describe successfully quite different reactions in polymer chemistry.

Three general methods have been developed for the determination of kinetic parameters, although nearly innumerable varieties were developed:

a. The direct method uses the procedure of Borchardt, Daniels, 1957, which needs both the concentrations (in DSC, the degrees of reaction) and the corresponding reaction rates:

~; = k(T) ·f(a) = A· exp (- :;) ·f(a)

To evaluate simple rate laws of the form

f(a) = (1- a)n or more exactly: f(a) = CS- 1 • (1- a)n

it is especially useful to proceed from the logarithmic form:

(da) EA In - = In A + n . In (1 - a) - - + (n - 1) . In Co

dt RT

Today multiple non-linear regression software from different calorimeter manufacturers allows to estimate the best values for n, InA and EA by evaluation of a single data set from one scanning experiment. The risk of misinterpretations is minimized if several data sets with different measuring conditions are simultaneously evaluated. Originally developed for n-th order reactions, at present various choices are offered for f(a) (simple rate laws, rate laws including autocatalytic steps, typical rate laws for the kinetics of reactions in heterogeneous systems). The evaluation presupposes isokinetic behavior (no changes of the mechanism within the examined temperature range). Traditionally this was proved using somehow created "reduced curves" for isothermal scans and subsequent comparison with master curves. For instance, all experimental time values for certain conversion degrees can be scaled by the reduced time factor, tred = t/ ta (ref) , where ta (ref) is the time for a reference conversion degree, frequent -ly a = 0.5. If we have isokinetic behavior, all experimental plots of a against tred

should fallon one single curve. The comparison with theoretical master curves for all known rate laws enables conclusions concerning the probable rate law. But even in the case of single step reactions the differentiation between various reaction models is sometimes difficult on the basis of a graphical comparison, especially using erroneous real measurements. Therefore, using the present possibilities of computerization and evaluation methods, the determination of the most probable rate law should be supported by statistical methods. The F-test yields the information which reaction model has the highest fit quality and whether the best reaction model is significantly better than other models. Modern variations of the Borchardt-Daniels procedure also allow the evaluation of nonisothermal measurements and avoid therefore the main disadvantage of isothermal runs, the not exactly known initial phase of the reaction (see Sect. 6.3.3).


b. If the known rate law is used in its integrated form (the integral method)

a 1 t

f-da =g(a) = k(T)· f dt o f(a) 0

only a few conversion/time data points are needed to calculate k(T) for isothermal measurements leading immediately to the activation parameters for a single temperature scan. Evaluation of isothermal experiments gives no problems when the rate law is known. For scanning experiments using

a 1 A T (EA ) f-da =g(a) =- f exp - - dT o f(a) f3 0 RT

there are difficulties because an exponential integral cannot be solved analytically. A solution is to use integral tables as calculated, for example, by Doyle, 1961. A review of numerous varieties using integral methods can be found in the literature (e.g., Hemminger, Cammenga, 1989; Galwey, Brown, 1998). The importance of integral procedures has decreased considerably with developments in computer technology.

c. Likewise, many varieties of differential methods are known. For instance, Freeman, Carroll, 1958, used the logarithmic equation for n-th order reactions in a form based on differences between neighboring data points

(da) EA ( 1 ) ~ In dT = n . ~ In (1 - a) - R' ~ T

A graph of ~ In (da/dt) versus ~ In (1- a) (for constant increments of liT) yields the reaction order and activation energy. This analysis is very sensitive to experimental errors and this is particularly true for the procedure of Ellerstein, 1968, in which the last equation is differentiated once more. It must repeatedly be remembered that the analysis of a single curve is practically useless. Moreover, preconceived ideas for a certain reaction mechanism can often be confirmed by data fitting of single heating-rate data over a particular range of the degree of reaction.

Isoconversional Methods

As already outlined, activation parameters and rate law are inseparable coupled as "kinetic triplet". Vyazovkin, 2001a, has shown for single step reactions that the activation parameters dramatically depend on the chosen model. Otherwise, the majority of chemical reactions proceed as multi-step reactions. If a series of non-isothermal measurements at different heating rates is carried out, the activation energy can be obtained without having to specify a certain model. Therefore, these methods are also called as model-free or model-independent methods, although this description is probably somewhat misleading. To obtain the preexponential factor, the rate law must be known. The only requirement for isoconversional methods is the independence of the model f (a) from the degree


of reaction a. The mathematics of isoconversional evaluations is derived from the basic types of differential and integral methods.

Using the differential isoconversional method according to Friedman, 1964,

( da) EA 1 In - = In (A . f (ai)) - - . -dT a=ai R Ti

In (daldt)a= ai is plotted versus lITi at different heating rates fJi' If the investigated reaction is a single-step reaction, the lines for the different degrees of reaction are parallel and have slopes (-EAIR) and intercepts In [A ·f(a)]. Otherwise, the slopes are different. The preexponential factor can only be obtained if f(a) is known.

The second frequently used variety, the integral isoconversional method, was independent from one another developed by Ozawa, 1965 and Flynn, Wall, 1966. In the equation

( A. EA) lng(a) =In -R- -lnfJ+lnp(z)

is p (z) (with z = EAIRT) an approximation for the so-called exponential integral. By rearranging and using an approximation for p (z) (Doyle, 1962) one obtains:

( A. EA) EA 1 In fJi = In -- -lng(a) - 5.3305 + 1.052· -.-

R R ~

Again, the plot In fJi versus lITi for constant a/s yields parallel lines for a singlestep reaction. EA results from the slope, A from the intercept if f(a) is known. The performance of modern computers allows to calculate much better approximations for all values of EAIR then that used by Doyle.

If the evaluation shows a significant dependence of the activation parameters on the degree of reaction, this is an unmistakable sign that the reaction mechanism has at least two elementary steps. Such behavior is very frequently found for condensed phase reactions. The relative contributions of these reactions to the overall reaction rate vary with temperature and the effective activation energy varies with the extent of conversion. Vyazovkin et aI., 1996 a to 2001 b, has therefore developed a variety of model-free evaluations, which no longer presupposes the independence of the modelf(a) from the degree of reaction a but rather the opposite, a conversion-dependent overall activation energy. He calculates for a set of n experiments carried out at different heating rates the minimum of the function

i i I(EA,a, Ti(ta))

i=1 jn I(EA,a, Tj(ta))

with

I(EA, Ti(ta)) == f exp (-~) dt o RTi(t)


Assuming the simple additive superposition of the individual reactions for a possible multiple-step mechanism, he obtains conversion-dependent formal activation energies. Then it is also possible to obtain predictions for isothermal conditions without the knowledge of both reaction model and pre-exponential factor, no doubt an important advantage for the chemical engineer. Vyazovkin, 2000 b, also tries to interpret the conversion -dependent activation energies from a mechanistic point of view but he modifies himself this aspect when he formulates (Vyazovkin, 2000a): "However, one should not forget that the mechanistic clues are not yet the reaction mechanism, but rather a path to it that can further be followed only by using species-specific experimental techniques". Sewry and Brown, 2002, emphasize this aspect on formulating: "Model-free methods of kinetic analysis postpone the problem of identifying a suitable kinetic model until an estimate of the activation energy has been made". However, in the meantime a number of problems show that Vyazovkin's procedure has fundamental limits in spite of all advantages of the model-free analysis (Opfermann et al., 2002). Vyazovkin and Lesnikovich, 1990, have also suggested a scheme that, using the function EA = f(a), allows some conclusions regarding complex processes. Another model-free evaluation method, the so-called non-parametric kinetics method was recently suggested by Serra et aI., 1998 a, b.

Kinetic Evaluation Using a Formal Reaction Model

The description of the reaction behavior by a formal-kinetic model (Opfermann et aI., 1995; Flammersheim, 2000; Flammersheim, Opfermann, 2001, 2002; Opfermann et aI. 2002) is, at present, the most flexible, simple and fastest solution concerning the fitting of reaction curves. Using the vocabulary and basic ideas of usual reaction kinetics, the overall reaction is described by the combination of formal reaction steps (independent, parallel, competitive or consecutive) with constant activation parameters. The single steps can be described by all known rate laws for homogeneous and heterogeneous reactions. Up to 16 curves obtained at different heating rates and/or temperatures are included simultaneously in the non-linear regression. Using statistical tests the formal model is searched which describes the experiment as well as possible. An interpretation of the single steps and their parameters should be made very cautiously, if at all. As a rule, the formal steps of the model do not correspond to real elementary reactions. In other words, a high value of the fitting quality for the best model does not mean that the kinetic description (Roduit, 2000) is correct in the physicochemical meaning! But the industrial practitioner is usually more interested in predictions of the reaction behavior for arbitrary temperature-time profiles, than knowing or understanding the real mechanism. As with the modelfree analysis, the knowledge of the composition of the reacting system is not necessary. The obtained results allow the prediction of reaction degrees, heat flow rates and relative concentrations of the formal reactants versus temperature or time for any complicated temperature/time profiles.

Like all other evaluations discussed so far, disadvantages result from the fact that the obtained process parameters and the derived conclusions are valid only for the tested composition of the reaction mixture. To be more exact, there are


two disadvantages: technically important reaction mixtures could have different amounts of catalysts, non-reactive fillers or solvents; and, in addition, if the stoichiometric composition of the reaction mixture cannot be varied, then important additional information concerning the verification, acceptance or rejection of an assumed reaction model is not available. This is true for such cases, in which one or several elementary steps of the mechanism are influenced only by a specific reactant.

Search and Use of the Real Mechanism

A new evaluation program (Flammersheim, Opfermann, 2002) uses the same basic strategies and mathematical routines as during the kinetic analysis with a formal-kinetic model. The crucial difference is that the individual steps of the assumed mechanism are now treated and formulated as real elementary steps. Even if the reaction mechanism is not too complex, such evaluations can be done only if the experimental information is adequate. Therefore, besides having a sufficient reaction window (a sufficient number of isothermal reaction temperatures and/or heating rates) the initial composition of the reaction mixture should also be varied as widely as possible. The new program is somewhat more time-consuming (it was not developed to replace faster formal-kinetic evaluations) but it is a valuable supplementary tool for kinetic evaluations. As a rule, results from other analytical methods must be considered in defining an adequate kinetic model, even if only the relevant elementary steps of the assumed mechanism are considered.

Important features are:

- Unrestricted combination of elementary steps (consecutive, parallel, equilibrium).

- Simultaneous evaluation of both isothermal and non-isothermal measurements.

- Simultaneous evaluation of curves for mixtures with variable compositions and differing amounts of reactive (catalysts) and non -reactive (fillers, solvents) substances.

- Consideration of changes in volume during the reaction. - Test of significance regarding the fitting quality of different models and re-

garding an assumed number of reaction steps. - Predictions for signal, fractional reaction and reactants for any temperature

time program and for any composition of the reaction mixture.

The important steps of the new program are:

- Input of up to 16 DSC curves, selection of the base line and optional desmearing and smoothing.

- Input of the assumed mechanism with all (important) elementary steps (maximum 12 steps). Using symbolic letters or short names for the reactants, the reaction mechanism can be formulated as usual in chemistry. Reversible reactions with very high rate constants for both forward and backward reactions can be formulated as equilibrium reactions.


- Input of the molar masses, specific densities, molar concentrations or initial weights of all reactants, including those of a possible solvent. Molar masses and densities are valid for all curves of the chosen model, while the stoichiometric composition of the reaction mixture can be different for each curve. The program checks the mass balance.

- Input of useful initial values for all variables of the kinetic scheme: for elementary reactions log A, E A and Qr, for equilibrium reactions L1r Hand L1r s.

- Use of the differential equation solver, which is a combination of embedded Prince-Dormand procedure of 4/5 degree with a Gear4 procedure (see EngelnM tillges, Reuter, 1996). In the Prince-Dormand procedure two checks regarding stiffness have been integrated. The calculation always starts with the PrinceDormand procedure. If both checks regarding stiffness are positive, then the calculation continues using the Gear4 procedure. There are three criteria to terminate the iterations. The first criterion is an inherent part of the algorithm and causes the iterations to finish automatically, if the relative change of all parameter values is lower than a defined precision. User-defined criteria are the number of iterations or a maximum value of the correlation coefficient.

- Output of all optimized parameters and their errors. - Predictions of the reaction behavior can be made for any complex tempera-

ture-time profile using the favored model and its parameters. These predictions concern the concentration behavior of all reactants and products, the DSC signal and the degree of reaction.

6.3.6 Selected Examples and Possible Predictions

1. Example: Simple Test Reactions

Photochemically produced molten cis-azobenzene converts into the trans form above room temperature in a well-defined 1st order reaction. The activation parameters have been found to be In (A/S-i) = 27.7 and EA = lO3.6 kJ mol- i

(Eligehausen et al., 1989). By contrast with the situation for temperature and caloric calibration, there are so far no internationally recommended test reactions for kinetic investigations. The use of this well-known reaction could be a first step in this direction, to test individual calorimeters, the sample preparation technique and the evaluation procedure (Eckardt et al., 1998).

The conversion of cis-azobenzene is only a 1st order reaction in the liquid phase. Before each measurement the solid substance (Tfus = 71.6 0c) must be rapidly melted and quenched to the starting temperature (35°C). Minor reaction at this stage has no influence on the subsequent calculation of the kinetic parameters. The true baseline of this non-isothermal measurement should be constructed using the heat capacities of cis- and trans-azobenzene, according to method 2 of Sect. 6.2. However, this is not possible because the liquid cis-azobenzene cannot be supercooled sufficiently to measure cp (T) down to temperatures below the start of the isomerization reaction and a straight baseline must be used. This is a good approximation because the differences between a calculated sigmoidal baseline (from literature cp-values) and the straight line are very

190

100

Wig

t 2.25

<11 ~

0 ~

~ 1.50 ~

0 <11

.c;

....

.... 0.75 <11 0.. 11"1

o a

1.00

Wig

1 '.75

..::!. c

~ 0.50 o

c '" .c::

~ 0.25 '" 0-Il>

b o

50

o


1 K min-1

2 K min-1

4 K min-1

8 K min-1

15 K min-1

71 92 113 134 O( 155

temperature

370 K

30 60 90 min 120 time ~

Fig. 6.1Sa, b. Measured (solid) and calculated (dashed) curves for the cis- trans isomerization of azobenzene in the liquid phase. a in scanning mode at different rates, b in isothermal mode at different temperatures


small. An approximate (because of different thermal conductivities) but adequate desmearing is possible using the crystallization peak of molten trans-azobenzene as the apparatus function. The correction is small because the isomerization is rate-determining even at the highest used heating rates. A precise temperature correction to zero heating rate values for all heating runs is very important.

Figure 6.15 shows the excellent agreement between experiments and calculations using multiple-curve analysis by non-linear regression. All curves from runs at five heating rates and at four isothermal temperatures have been included in the calculations (that for the lowest temperature is not shown in the figure). If minor self-heating of the sample because of the limited heat exchange with the calorimeter is allowed for, no systematic differences are found between the single-curve and the multi-curve analysis. In addition, the same activation parameters are found even at higher heating rates (40 K min-I). Assuming a fixed reaction order of n = 1 the procedure gives activation parameters of In (Als- I ) = 27.6 ± 0.2 and EA = (l02.5 ± 0.8) kJ mol-I, in good agreement with Eligehausen et al., 1989.

Another example is the dimerization of cyclopentadiene (Flammersheim, Opfermann, 1999b). The monomer (CPD) is easy to obtain from the commercial dimer (DCPD) by cracking distillation at 150-170 DC and repeated distillation at 40 DC in an argon stream. The mechanism according to the following scheme is slightly more complicated (Turnbull, Hull, 1968):

2CPD ~ DCPD

2 CPD + DCPD ~ 2 DCPD

DCPD ~ 2CPD

(l)

(2)

(3)

The second order Diels-Alder dimerization is auto catalyzed and the back reaction cannot be neglected above 140 DC. But this reaction should be suitable to test the efficiency of the used kinetic software. Using the program "Component kinetics" and taking into account the volume contraction during the reaction (~CPD = 0.7970 g cm-3, (JDCPD = 0.9770 g cm-3), the best activation parameters are shown in the Table 6.l.

Table 6.1

Reaction

1: 2 CPD ~ DCPD

2: 2 CPD + DCPD ~ 2 DCPD

3: DCPD ~ 2 CPD

Parameter

19 (Ails-II mol-I) EA.I/(kJ mol-I)

19 (A 2/s- 1 L2 mol-2)

EA•2 /(kJ mol-I)

Ig (A3/s-l) EA• 3 /(kJ mol-I)

correlation coefficient

Optimized value

5.74 ± 0.15 69.43 ± 1.00

S.SO ± 0.30 73.79 ± 2.40

13.04 ± 0.25 142.0 ± 2.2

0.9997


Fig.6.I6. Concentration 12 profile for the dimeriza-

mol l-1 tion of CPD (heating rate: 10 K min-I). 1: CPD, ) 2 and 3: DCPD formed 8 according to the simple

6 2nd order reaction (1) and c

O[

120

the autocatalytic reaction (2) ~ - 4 of the reaction scheme 0 <-

80

(see text) -'C <lJ 2 u c

40 0 u

0 o 0 4 8 12 16 min 20

time ---

Using these results, the heat flow rates, conversion degrees and concentrations of the reactants can be predicted for any temperature-time reaction profiles. As an example, Fig. 6.16 shows the concentration profile for a heating rate of 10 K min-I. The program allows to differentiate between CPD formed during the two reaction paths (1) and (2). With increasing temperature, in the case of f3 = 10 K min-I above 185°C, dominates the back reaction (3) for the overall reaction. But the two forward reactions show a different behavior. While the amount of CPD formed as a result of reaction (1) already decreases above 160°C, the percentage of the autocatalytic reaction (2) is still increasing. Another result is important. Every nth-order reaction starts with maximal rate. Using the parameters of the table we can calculate for instance the storage periods for which a predefined degree of conversion is not exceeded. In our example and for a = 0.02 these times are nearly 4 d at - 20°C, 9.5 h at O°C and only 72.3 min at 20°C. That means, the dimerization of CPD should always be investigated with freshly distilled monomer immediately after its preparation.

2. Example: Radical Polymerization of (2,2-Dimethyl-l,3-dioxolan-4-y/)methyl Methacrylate (DOMA)

The radical polymerization of a monofunctional monomer is a simple 1st order reaction, if a number of ideal conditions can be assumed (reactivity of the macro-radicals is independent of the chain length, approximately constant radical concentration during the whole reaction, bimolecular chain termination reactions). Under these circumstances, the rate of concentration change is proportional to the concentration of the monomer (CM) as follows:

dCM (r. kini)"2 1/2 ---=kpro • -- ·CI 'CM dt kter

The quantity r is the yield of radicals due to the thermal homolysis of the initiator (I) [e. g., 2,2' -azodiisobutyronitrile (AcBN) or dibenzoyl peroxide 1, kpro ,

kter and kini are the rate constants of the chain propagation, chain termination


and initiation respectively. If the thermal initiator is replaced by a photoinitiator, like benzoin methyl ether or 2,2' -dimethoxy-2-phenylacetophenone (benzyl dimethyl ketal), the analogous overall rate law reads:

dCM ( r· labS) 112 ---= kpro ' -- 'CM dt kter

Here r is the quantum yield of the initiator, labs is the radiant power absorbed by the sample. In contrast to polyfunctional acrylates, for which cross-linking reaction starts immediately (Fig. 6.13), monofunctional methacrylates (methyl, ethyl, propyl, butyl) follow the simple 1st order law up to degrees of reaction of 0.3 to 0.6 (dependent on the chemical nature during both thermal and lightinduced polymerization). After this stage the so-called gel or TrommsdorffNorrish effect dominates, this is recognizable by increasing reaction rates despite decreasing monomer concentrations (Malavasic et al., 1986). This is also valid (Fig.6.17) for the polymerization of {2,2-dimethyl-l,3-dioxolan-4-yl)methyl methacrylate (DOMA) (Flammersheim,Klemm, 1985). There are no qualitative differences between curves 1 and 2. In both cases the change of reaction rate with time is small enough during the entire reaction that desmearing is not necessary. Spectroscopic investigations on the soluble reaction product verify the complete reaction of the acrylate double bonds, whereas the dioxolane ring remains intact under these experimental conditions. Up to a degree of reaction of 0.3 the rate law is 1st order. Knowing the activation term {rkini CI )1/2 or {rlabs )112, the overall rate constant gives the ratio kpro / k~~;. During the gel effect this ratio is formally dependent on the degree of reaction reached. In contrast to the nearly unhindered chain propagation reactions of the small monomer molecules, the chain termination reactions of the growing macroradicals are in-

6.0

mW

t 4.5

~ 3.0

0 ..... ~ 0 .... 1.5 0 cu

.J:

0 0 10 20 30 min 40

time ~

Fig.6.17. Light-initiated (1) and thermally activated (2) polymerization of DOMA. (curve 1: A = 365 nm, 10 = 0.35 mW cm-2, T = 298 K, 1 % 2,2-dimethoxy-2-phenylacetophenone as photoinitiator; curve 2: T = 343 K, 1 % 2,2-azodiisobutyronitrile as initiator)


20

mW

1 15

5

illumination period dark reaction

o~------~--------~------~------~--------~ o 0.6 1.2 1.8 min

time ..

Fig.6.18. Photopolymerization of DOMA with interruption of the irradiation at reaction degrees of 0.35 and 0.80 (arrows). Both curves are consecutive runs of the same sample, but the second curve is shifted to the origin of the time scale. Solid lines: measured curves, dashed lines: desmeared curves (A= 365 nm,Io = 3.5 mW cm-2,

T = 298 K, 1 % 2,2-dimethoxy-2-phenylacetophenone as photoinitiator)

creasingly hindered in the more and more viscous matrix. The result is a pronounced dark reaction after switching off the light during the polymerization (Fig. 6.1B). The first run was interrupted at a = 0.35. After the dark reaction was complete the same sample was once more radiated and the light switched off again at a overall degree of reaction of O.B (curve 2). To get a better comparison the second curve is shifted to the origin of the time scale in Fig. 6.1B.As a result, the relative extent of the dark reaction is proportional to the degree of reaction at that moment at which the light was interrupted. The rate of the dark reaction (missing activation by Flab.) is described by:

CM kter CMo ---=-·t----

dcM/dt kpro dCM/dt

If the ratio kpro Ik~~; is determined from the measured curve immediately before the light is interrupted and, following this, the ratio kterl kpro from the dark reaction (last equation) the rate constants kter and kpro result from the combination of these. Though desmearing is not necessary during the illumination period with normal light intensities, the evaluation of the dark period signal is meaningless without it, especially if the dark reaction is performed at low conversions (because of the very rapid decrease of the heat flow rate due to a large kterl kpro ratio). The picture shows this effect very clearly. The linear relation between -cM/(dcMldt) and t (see the


last equation) is only found from the desmeared signal (kterlkpro is about 310 at a = 0.35 and about 11 at a = 0.80). Measurement of this reaction in DSCs is only possible if their time constants are lower than 2 to 4 s.

3. Example: Formal-Kinetic Evaluation of a Polymerization Reaction with Partial Diffusion Control

The curing reaction of a solid mixture of a commercial epoxy (RUETAPOX VE 3579 or 2,2',6,6' -tetrabromobisphenol-A diglycidyl ether, Tg: 25 DC, used without purification) with 5 wt.-% of a latent accelerator [Zn(OCNh(iMeImidh, crystalline powder, self-prepared] is initiated by the release of the 1-methylimidazole above 70 DC (Flammersheim, Opfermann, 1999a, 2001). Although the mechanism is complex, the measured curves can be fitted by a relatively simple, formal model. But the kinetic evaluation is complicated because, with increasing reaction degree, a partial or complete vitrification of the reaction mixture - dependent on the heating rate - is observed. The resultant, complex shape of the baseline is directly available from TMDSC measurements. Figure 6.19 presents the course of the heat capacity during the reaction for three heating rates; curve 1: 0.1 K min-I, curve 2: 0.2 Kmin-I and curve 3: 2.0 K min-I. Curve 4 corresponds to the averaged heat capacity of the reaction product, calculated from all second runs. The picture clearly shows the devitrification of the initial epoxide at 25 DC, the heating rate dependent vitrification (decrease of cp) and the second devitrification (increase of cp) at temperatures higher than the Tg, max of this system (163 DC). In other words, at low heating rates the glass temperature of the reaction product increases faster than the program temperature. Comparable results were obtained by van Assche et al., 1995, 1996, 1997, for other vitrifying systems. After partial or complete vitrification of the reaction mixture the reaction is no longer controlled by the kinetics of the chemical reaction, but by diffusion processes. Finally, if the program temperature exceeds the maximum glass transition temperature, one again enters the realm of chemical reaction control. Figure 6.20 shows the isothermal reaction at 95 DC. The range of the isothermal vitrification is clearly displayed by the decrease of the heat capacity (between

Fig.6.19. Specific heat capacity change during curing of RUETAPOX VE 3579 at a heating rate of 1: 0.1, 2: 0.2 and 3: 2.0 K min-I; 4: heat capacity of the reaction product

1.8

t JK-' g-' >..

-- 1.4 '-' a Cl-a u

~ 1.2 '" OJ

.s:::

'-' OJ CI-Vl

0 50 100 150 O( 200 temperature ----

196

Fig. 6.20. Specific heat capacity and reaction heat flow rate for isothermal curing of RUETAPOX VE 3579 at 95°C; the vitrification process is clearly visible by the decrease of the heat capacity between the two vertical lines

CL <:>

'" CL en

14

1.3


Cp

3 co

-04 -;; QJ

..c:

QJ CL

1 2 '----------'---'---------'----'-----'-------' -0 6 en

o 50 100 150 min 200 tim e -------

the both vertical lines}. But this is not coupled to a noticeable sudden increase of the reaction rate! On the other hand, the picture indicates definitely a feature of the reaction mechanism. One or more reaction steps are autocatalyticaI.

The kinetic evaluation must be capable of taking into account the more or less complete change from chemically- to diffusion-controlled reaction steps. The most frequently used possibility goes back to a formulation given by Rabinowitch, 1937:

1 1 1 -=-+-k kdiff kchem

As usual, the temperature dependence of kchem is described by the Arrhenius equation. Several approaches for kdiff (Huguenin, Klein, 1985; Deng, Martin, 1994; Karkanas et aI., 1996; Wise et aI., 1997; Liu et al., 2001) are discussed in the literature. Van Assche et aI., 1995, 1996, 1997,2001, were able to show that for epoxide curing reactions the molecular motions frozen out at vitrification correspond to the molecular motions needed for the chemical reaction. They define for each run a mobility factor (heating rate and temperature dependent) between 1 and 0 which is directly accessible from the heat capacity signal of TMDSC measurements. This factor parallels the diffusion factor, defined as ratio between the theoretical reaction rate without diffusion hindrances and the actual reaction rate. The calculation of theoretical rate and diffusion factor presupposes the knowledge of the correct reaction model, in general surely a non trivial problem. If the correspondence between main glass transformation and mobility of the reactive species can be assumed (this is not always observed, see van Assche et aI., 1997,2001), it should be favorable to use an equation which describes the diffusion hindrances independent of the chemistry of the process. This is also advantageous for another reason, predictions of the reaction behavior for any conditions require a model for both kchem and k diff • A usable approximation is - despite of all criticism regarding the region of validity - a so-called modified WLF equation as proposed by Wise et aI., 1997:


kdiff (1j is related by this equation to Tg(a). A simple and convenient regression function for a number of measured Tg and a values obtained from partially cured samples was proposed by Hesekamp, 1998:

( g!. a) Tg(a) = Tg(O) . exp --

g2- a

For the investigated example were found: Tg(O) = 296.3 K, g! = 4.48 and g2 = 12.26.

The general limitation of applying any types of WLF equation is based on the fact that they should be used only for the liquid state above Tg• Below Tg, one obtains values for kdiff(T) which are increasingly too low. But it is reasonable (Flammersheim, Opfermann, 2001) to assume:

- for T ~ Tg the WLF-equation is valid in the given form - for T < Tg an Arrhenius type equation is used:

Both equations fulill the boundary conditions that kdiff and its first derivative are continuously at T = Tg• The C-parameters of the modified WLF equation should not be discussed, as long as the true reaction mechanism is not known. Using the formal-kinetic evaluation it is not clear how certain reaction steps are influenced by the changing segmental length, van Assche, 1995, 1997,2001, during the freezing of the reactive species.

The most simple formal-kinetic model that produces a practically perfect fit for all heating rates (correlation coefficients> 0.9995) corresponds to a process with two consecutive partial steps:

A~B~C

The corresponding rate law for the formal component B is:

dCB ( EA'!) (EA'2) 2 dt = A! . exp - RT . CA - A2 . exp - RT . CB • (1 + Kcat,2 • cd

The decrease of the formal component A and the increase of the formal component C correspond to the negative first and second term of this equation respectively. Two data sets were available for the kinetic evaluation, each with 8 different heating rates. The fit to the measured curves is perfect. The second reaction step is auto catalyzed by the formal reactant "c" and this is taken into account by the parameter Kcat,2 in the above equation. Only this reaction step is influenced by the diffusion. If, as a trial, the diffusion control is not used for this step, no acceptable fit is reached. The model parameters for both data sets are summarized in Table 6.2.


Table 6.2

Parameter

19 (A2 /s- 1)

19 (A1/s-1) EA,l/(kJ mol-I) EA,2/(kJ mol-I) 19 Kcat,2

19 (Kdiffusion,2/mol-1 m3 S-l)

CdK C2 /K Contribution of consecutive reaction Correlation coefficient

Data set 1

7.94 ± 1.05 6.48 ± 0.70 70.0 ± 5.2 79.0 ± 8.2

0.826 ± 0.170 -4.77 ± 1.05

9.9 ± 3.1 25.9 ± 21.8 0.25 ± 0.06

0.9997

Data set 2

8.90 ± 1.12 6.20 ± 0.86 67.5 ± 6.6 86.7 ± 8.7

0.788 ± 0.170 -4.39 ± 0.78

8.1 ± 2.1 28.6 ± 19.4 0.21 ± 0.06

0.9996

Using these values, the course of the isothermal curing was correctly predicted. Further, at low temperatures, the conversion degrees never reach "1" after an extended period due to partial freezing. The corresponding values are correctly predicted for different temperatures (110 °C: calculated 0.58, found 0.57 and 0.61; 140°C: calculated 0.80, found 0.79 and 0.79; 170°C: calculated 0.94, found 0.93 and 0.94). Finally, a convincing argument for the validity of the selected model is given in Fig. 6.21 for an experiment with a very slow heating rate, 0.25 K min-I. The heat capacity function that is directly accessed by experiment (cf. Fig. 6.19) runs completely parallel to the increase of the glass temperature and the degree of reaction calculated by means of the model parameters. This is valid both for the vitrification at about 100°C and the slow devitrification starting at about 130°C. This temperature range becomes larger at even lower heating rates. At heating rates faster than 1 K min-I Tg permanently lags behind the program temperature, as the diffusion hindrances due to the partial freezing become more and more unimportant.

18 200 10

t JK-1 g-l O[

t 08 t

>--c

~ 06 0

d CL 14 100

OJ ~

d :::> d

/ ~ 0.4 :=':' -;:; ~

"0 OJ OJ

£ 12 Il 50 CL E OJ

~ ,,' T'!,./i OJ a 2 <lJ ~

OJ en CL :.....---- . OJ Vl ._cx./ 'D

a 200 400 600 min 800

lime ----Fig.6.21. Heat capacity, glass transition temperature and degree of reaction as function of temperature and time for the curing of RUETAPOX VE 3579, heating rate: 0.25 K min-1


Fig. 6.22. Calculated tem-

t 100

perature profile for the rate controlled reaction of

% RUETAPOX VE 3579 at a constant conversion rate of c:

!:! 0.1 % min-1 (i.e., a is propor- '" 60 <-

OJ

tional to time; Tg parallels > c:

the reaction temperature 0 40 '-'

within a large temperature -0

range) OJ 20 OJ

DC

t 150 <U '-::> ~

T 9 100 e OJ c.. E OJ

<-

"" <U 50 -

"0

0

time ---

Another kind of prediction is the temperature respectively the heating rate profile for a so-called rate controlled reaction. This is very important if in certain temperature ranges the change of any physical properties coupled with the increasing degree of reaction is of primary technological interest. This could be for example the thermal shrinkage during the production of ceramics. In Fig. 6.22 it is assumed that for the epoxy curing a constant rate of conversion of 0.1 % min-1 is desired. The result of such calculations for the chemical engineer is a table of temperature intervals and their incident heating rates, possibly including isothermal reaction steps. An interesting but surely not surprising result (cf. Fig. 6.21) is that over large temperature intervals the program temperature parallels the glass transition temperature.

Finally, a useful framework for understanding the changes which occur during the cure of a thermosetting system is a Time-Temperature-Transformation (TTT) diagram. Some examples including possible varieties were presented by Wisanrakkit, Gillham, 1990, van Hemelrijck, van Mele, 1997 and Theriault at al., 1999. The diagram in Fig. 6.23 is calculated for the isothermal cure as function of time and curing temperature. The vitrification curve (Tg = Teure , the solid, thick line in Fig. 6.23) separates the liquid range without diffusion hindrances (left) from the glassy range (right) with increasing diffusion hindrances for the chemical reaction. Sometimes, the practitioner needs the reaction behavior for a number of constant differences between Tg and Teure , as shown by the solid, thin lines in Fig. 6.23 for Tg - Teure = 10 and 20 K. The dashed curves show for a number of curing temperatures the evaluation of Tg against 19 (time). Eventually, also isoconversional curves (the dotted curves in Fig. 6.23) can be calculated for a number of conversion degrees, in the example for 0.3,0.6,0.8,0.9,0.95 and 0.99. It is also possible to construct a so-called Continuous-Heating-Transformation (CHT) diagram (van Hemelrijck, van Mele, 1997), at which 19 (t) on the x-axis is replaced by 19 (P>. Both types of diagrams enable a fast and rough assessment of the reaction behavior of the system within a wide range of experimental conditions. The common disadvantage is sometimes the limited practicability for real quantitative predictions, because true isothermal or true heating processes with constant heating rate are difficult to realize in technical processes. In such cases,

200

180 O(

140

w 100 ':::J

o .... ~ 60 E w


19 (100%)

10~

time

Fig.6.23. Time-Temperature-Transformation (TTT) diagram for the isothermal cure of RUETAPOX VE 3579 showing the increase of Tg at different curing temperatures (dashed), constant conversion lines (dotted) and two curves with Tg - Teure = constant (thin solid lines), the thick solid line corresponds to the vitrification line Tg - Teure = 0

the calculation of the reaction behavior (degree of reaction, Tg as function of a, heat production) for any concrete preset and technically feasible temperaturetime profiles (cf. Fig. 6.21) is the better and more reliable way.

6.4 The Glass Transition Process

6.4.1 The Phenomenology of the Glass Transition

Qualitative Cp measurements intended to provide information regarding the glass temperature are much commoner than direct determinations of the temperature dependent heat capacity (cf. Sect. 6.1). To find the best experimental procedure and an unambiguous interpretation of the results is, however, not possible without basic information about the nature of the glassy state and the transition process between glassy (amorphous) and liquid states.

Most materials can be obtained in the glassy state by suitable treatment. Inorganic systems have been known since ancient times, they are responsible for the name of this state of matter. However, from the DSC point of view, organic glasses and, especially, macromolecular glasses are of special importance.

Typical values for the temperature ranges of the transition are 10 K for lowmolar mass glasses, 20 to 50 K for most of the organic polymeric glasses and 100 K or more for silicate network glasses. The glass transition is characterized by an appropriately defined glass transition temperature Tg• Within the transition region many macroscopic properties, which may have great practical importance, change their values (viscosity, dielectric and especially mechanical

6.4 The Glass Transition Process 201

3.0 ( 0

B

r mW hea ting run

~ 1.5 " ..... ~ ~

" '" .c

cooling run

o ~ ______ ~ ______ ~ ________ L-______ ~ ______ ~

40 55 70 85 'C 115

temperature ..

Fig.6.24. Typical DSC curves of amorphous polymers in the glass transition region. A: the glassy non-equilibrium state, B: glass transition region, C: "enthalpy relaxation" peak, D: the equilibrium liquid state (sample: linear epoxide-amine polyadduct, heating and cooling rate: 5 K min-I)

properties). Each of these could form the basis of an experimental determination of Tg • With DSCs the glass transition is detectable by a step change of the heat capacity /}.cp on heating or cooling. Both the temperature and magnitude of this event are important. Observed values of /}.cp range from 0.1 to 2 J g-l K- 1•

Figure 6.24 shows a typical DSC curve for a linear, high-molar epoxide-amine polyadduct which was first cooled at the rate shown and then immediately reheated.

For polymeric glasses the temperatures of the transition range from - 50 to 300°C whereas the corresponding range for silicate glasses is from 500 to 1000 °C. Polymeric glasses are thus more suitable for DSC measurements than inorganic glasses. For polymer chemists and materials scientists a knowledge of the glass transition temperature is as least as important as is the melting temperature.

6.4.2 The Nature of the Glass Transition and Consequences for DSC Measurements

Problems result from the fact that the glassy (or vitreous) solid is, thermodynamically, far from equilibrium. The formation and behavior of a glass are exclusively kinetic events. There are only formal similarities between the Cp-change at an "ideal" glass transition and at a thermodynamically well defined second order transition. Only the liquid (or rubbery) state at the high tem-


perature end of the glass transition is at equilibrium and thermodynamics is valid without any restrictions in this region. Here any external (experimental) influence is slow compared with the mobility of the internal degrees of freedom which thus always are in equilibrium. Equilibrium thermodynamics can also be applied if the internal degrees of freedom react very slowly, in this case the system is in a frozen (vitrified) state with respect to external influences. This is true for temperatures well below the glass transition. The heat capacity function can, therefore, be determined (see Sect. 6.1) for both ranges without any restrictions. During the glass transition, changes of both the intrinsic and the measurement variables occur on the same time scale, the measured quantities become time-dependent, and classical thermodynamics is no longer valid. The system passes through a sequence of non-equilibrium states during heating or cooling. The typical asymmetric shape of the glass transition curve (an extensive tail on the low temperature side and a fairly abrupt end at high temperatures) is due to a distribution of the intrinsic variables which vitrify (or devitrify) over a wide range.

A special problem when investigating the glass transition (key word: relaxation phenomena) is caused by the effect of the previous history of the glass on the thermal behavior. The system has, so to speak, a memory of its thermal history. This is important not only for theoretical investigations but also for practical DSC measurements of the glass transition.

Conclusions: - In contrast to the measurement of equilibrium transitions, it is not possible

to get "equilibrium" values of the characteristic quantities Tg and /:"cp byextrapolation to zero heating or cooling rates: these quantities are determined by the thermal history (scanning rates and annealing times). If, for instance, the cooling rate is changed by an order of magnitude Tg will change by 3 to 20 K depending on the material in question. For flexible polymers the magnitude is generally 3 to 5 K, whereas it is 15 to 18 K for the considerably stiffer borosilicate network glasses. By contrast /:"cp shows much less dependence on thermal history.

- Characterization of glasses by DSC measurements (Fig. 6.24) is mostly carried out in the heating mode because this is practicable even for those heat flux DSCs having relatively sluggish furnaces. A heating run is also advantageous when it is important to characterize the "as received" glassy state (e. g., resulting from particular cooling or annealing procedures or chemical reaction). It must be remembered that, on heating, the original glassy state can change at temperatures as much as 50 K below the transition region (for instance, if the heating rate is slower than the previous cooling rate). Tg then depends on the heating rate. To sum up, values for Tg (and to a lesser extent /:"cp)

are only meaningful with respect to the chosen experimental conditions and - as will be pointed out later - in the context of the particular definition of the "glass transition temperature".

- If the glass sample to be investigated is formed only by cooling from the liquid, a cooling run is indeed better for its characterization. Problems due to the coupling of vitrification and devitrification processes (occurring during heating) are avoided. Cooling runs, therefore, immediately reflect - possibly after


desmearing of the experimental curve - the kinetics of the vitrification. The cooling rates must not be lower than 5 K/min for a reliable determination of the rather small changes of the heat capacity. This may sometimes be difficult because of the limited cooling rate of many DSC furnaces.

- In addition the measured curve of the glass process is falsified by the limited heat transfer between sample and temperature sensor (cf. Sect. 5.4.4). Correction of these influences is important, because larger sample masses (10-20 mg) and larger scanning rates (10 or 20 K min-I) are used in practice. For this purpose the advanced desmearing with the aid of the step response function (cf. Sect. 5.4.4) is very advantageous.

- Thermal lag can be determined using the following methods: 1. Determination as explained in the Sects. 5.4.4 and 6.1.1 (Figs. 5.10 and 6.2). 2. Placing a small piece of indium both on the bottom and in the middle of

the sample and determining the difference between the extrapolated onset temperatures from the In melting peaks which corresponds to the thermal lag. Attention must be focused on a good thermal contact between the indium and the sample.

3. A temperature correction suggested by Hutchinson et al., 1988. This method appears to be problematical, however, as it already presupposes the validity of one of the models which describes the glass process (Kovacs et al., 1979).

A disadvantage of all of these corrections is that only an average lag is obtained and thus can be corrected.

6.4.3 Definition and Determination of the Glass Transition Temperature 19

Conventional Glass Transition Temperature

The main parameters used to characterize the glass transition are shown in Fig. 6.25. For many applications it is important to know the temperature range Tg,i to Tg,f over which the substance vitrifies on cooling, or devitrifies on heating. Unfortunately, the practical determination of these temperatures is problematic, there are large errors in their definition and it is difficult to give clear instructions for their measurement. The situation is improved if clearer, more characteristic temperatures from the transition region are used. These are the extrapolated onset-temperature Tg,e (analogous to the peak onset temperature) and the half-step temperature Tg,1I2 related to the Cp change (the temperature at which Cp is midway between the extrapolated heat capacity functions of the glassy and liquid state). The use of the latter is more meaningful as this temperature is better related to the second characteristic quantity of the glass transition, the Cp change. A third definition, the temperature of the inflection point of the glass transition curve is seldom used.

Under certain circumstances glass transition temperatures defined in this way may be used without restriction to compare experimental data:

1. All investigations are made in cooling mode at the same rate. 2. If evaluation must be done on heating runs, the sample must be previously

cooled from the liquid at a fixed rate.

Fig. 6.25. Definition of the most frequently used conventional quantities for characterization of the glass transition. Tg, e: extrapolated onset temperature, Tg, 1/2: half-step temperature, !:J.Cp: Cp-change at the halfstep temperature, Tg,i and Tg,f: initial and final temperatures of the glass transition, Tg,f - Tg,i: temperature interval of the glass transition

Tg,e and Tg,1I2 can both easily be obtained from routine measurements, this is the reason for the nearly exclusive use of these pragmatically defined glass transition temperatures up to now. A repeatability error of ± 1 K is acceptable in practice. For heating runs the following practical and frequently used procedure is recommended:

- The sample is heated to a temperature at least 15 to 30 K above Tg•

- Short (5-10 min) annealing at this temperature in order to establish thermo-dynamic equilibrium and erase the "memory" (with respect to its thermal history) of the system.

- Rapid programmed cooling (or quenching) to a temperature at least 50 K below the glass transition.

- Immediate reheating at constant rate (10 or 20 K min-I). These rates lead to relatively high temperature errors (3 to 10K) and, in addition, a broadening of the transition, on the other hand problems caused by relaxation processes during the transition are avoided. The thermal history can be obtained quantitatively by comparison of the original run with the rerun under the same conditions.

Neither Tg,e nor Tg,l12 make any allowance for the non-equilibrium nature of the glass transition. This is especially striking (Fig. 6.26) if the glass transition is accompanied by "enthalpy relaxation peaks". These appear on heating curves as endothermic events at the high temperature end of the glass transition range.


4

mW

t 3

~ 2 c> ..... ~ 0

c>

"" ~

o

60 70

annealed glass

~

Tg •• lqul ~.l\lqU'Z: fg .• la' g.lIla'

80 90 temperature ..

205

100 O( 110

Fig.6.26. DSC heating curves for an annealed (or slowly cooled) and a quenched glass showing the paradoxical result that the conventional glass temperature(s) of the quenched (qu) glass seem to be lower than those of the annealed (a) glass, for details see text. Tg,e: extrapolated onset temperature, Tg,lf2: half-step temperature

Their height may be comparable with those of the melting peaks of crystalline materials (Petrie, 1972). For the example shown in Fig. 6.26 the glass annealed 70 h at 68°e has Tg,e = 89.1 °e, Tg,1/2 = 87.3°e whereas the quenched glass has Tg,e = 81.6°e, Tg,1I2 = 85.1 °e. This use of Tg,e and Tg,ll2 to characterize the glass process gives the paradoxical result that a slowly cooled or annealed glass seems to have a higher glass transition temperature than a rapidly cooled one. In addition Tg,e and Tg,1I2 react to the thermal history in a different manner.

Tg,e and Tg,1I2 are not useful for theoretical treatments of the kinetics of the glass process, this is also true for certain relationships between Tg and other properties (e. g., Tg as function of the molar mass or as a function of the degree of conversion in a reacting system).

The Thermodynamically Defined Glass Transition Temperature

The "glass transition", although a kinetic phenomenon, can be unequivocally defined thermodynamically. This uses the so-called fictive temperature Tg,fic=Tf

first introduced by Tool, 1946. Tf is a well defined quantity that reflects the current structural state of the glass to be characterized.

The concept of the fictive temperature was considerably extended by Narayanaswamy, 1971, Moynihan et al., 1976 and de Bolt et al., 1976. An analo-


1 H 9 (-P d _______

Hg(all~ >-C1. c; :E

H9 (-P21/ c: OJ

Hg(a21~

temperature .. Fig.6.27. Schematic enthalpy-temperature curves for the glass transition of an amorphous sample at different cooling rates (-Pi) and subsequent heating after isotherm annealing at T ai,

for details see text. Hg : enthalpy of the different non-equilibrium glassy states, HI: enthalpy function of the equilibrium liquid state, Tg (- Pi): glass transition temperatures on different cooling (11311 > 1132 D, Tg (ai) glass transition temperatures on heating after different annealing schedules

gously defined "thermodynamic" or "enthalpic" Tg-temperature, based on the specific information of a DSC, was introduced by Flynn, 1974 and Richardson, Savill, 1975a and Richardson, 1976. In the following section this thermodynamically defined temperature Tf is presented as Tg without the additional sufflx.

To understand the definition of Tg, the enthalpy versus temperature diagram (Fig. 6.27) will be discussed. Heat capacity functions for the glass and liquid can be described approximately by straight lines within a temperature range of 50 to 100 K. The enthalpy functions are then slightly parabolic curves. For simplicity, curvature of the enthalpy functions is neglected in this figure, in other words Cp

is assumed to be temperature independent. Depending on the cooling rate, the sample vitrifies (changes from HI to Hg) at different temperatures. The lower the cooling rates (negative {3) during this process the lower are the vitrification temperatures. The functions Hg (-{31) and Hg (-{32) characterize the enthalpy of the respective glasses. For simplicity it is further assumed that all enthalpy functions are parallel, in other words that the heat capacities of the glassy state are assumed to be independent of the conditions during vitrification [although very precise measurements (Gilmour, Hay, 1977) show slight differences]. If the glass is annealed at temperatures down to, at most, 50 K below the vitrification tem-


perature, the mobility of the frozen states is still so large that internal degrees of freedom are not totally frozen, and can relax towards equilibrium. During this process the Hg-function approaches the (extrapolated) HI-function (Petrie, 1972; Peyser, 1983; Cowie, Ferguson, 1986; Agrawal, 1989). The figure shows this for two annealing temperatures Tal and Ta2 of a glass, which was obtained at a cooling rate - Pl. At Tal the annealing time was sufficient to reach the equilibrium enthalpy value of the liquid at that temperature, whereas this was not the case at Ta2 • Reheating of the annealed glass then proceeds along the enthalpy lines Hg(al) or Hg(a2). From the theoretical point of view, the glass should devitrify exactly when the enthalpy line of the glass crosses that of the liquid but the transition from Hg to HI is not sharp. To determine the intersection of enthalpy curves for the glassy and liquid states therefore requires the extrapolation of these curves from temperatures, which are clearly above or below that of the transition region. The point of intersection, obtained in this way, defines the thermodynamic glass transition temperature Tg• Tool, 1946 called this temperature the "fictive temperature", because during heating nothing happens at that point. Hence Tg cannot be located directly on the measured curve, instead, on heating, the system progresses further along the Hg-curve (superheating effect). This is more pronounced the better the glass has been annealed (i. e., after annealing at T a2 it is far more intensive than after annealing at Tal). This is the reason for the paradoxical values, mentioned earlier, for the pragmatically defined Tg,e or Tg,1/2 temperatures, when comparing slowly cooled (or annealed) and quenched glasses. Superheating ends only at temperatures well above Tg, the return to the equilibrium curve is now rapid and produces the so-called relaxation peak. The reason for the superheating effect is the drastically decreased mobility in the glassy state, which parallels the slow enthalpy decrease during annealing.

If the enthalpy definition (Flynn, 1974; Richardson, Savill, 1975a; Richardson, 1976; Moynihan et al., 1976) is used, Tg can easily be calculated from DSC measurements.

The procedure in question (Richardson, Savill, 1975a; Richardson, 1976) is explained in Fig. 6.28. We start with the definition, the equality of hg(T) and hI (T) at Tg. The enthalpies h (T) for the glass and liquid are obtained by integration of the corresponding cp-functions, which can always be approximated by linear equations:

glass: Cp,g= a + b· T

liquid: Cp,l =A +B· T

and

and

1 hg(T) = a· T + - b· T2 + P

2

1 hl(T) = A . T + - B . T2 + Q

2

The integration constants P and Q are not known but Q-P maybe obtained from the difference hl (T2) - hg (Tl ), the hatched area in the Fig. 6.28, a directly accessible experimental quantity:

1 2 1 2 hl(T2)-hg(Tl ) =A· T2 - a· Tl +-B· T2 --b· Tl + (Q-P)

2 2


4.5

l/(Kgl

r 3.5

3.0 >-'u CI 2.5 0-CI ~

-;; C P.l -------------

cu 2.0 ..c: ~

~

cu 0- 1.5 '"

1.0 60 90 100 T2 'C 11

temperature ..

Fig.6.28. Determination of the thermodynamically glass temperature from DSC heating curves with "relaxation peak" (Richardson, Savill, 1975a; Richardson, 1976). Cp,g and cp,l are the (extrapolated) specific heat capacity functions of the glass and the liquid, respectively, the hatched area (extended to cp = O) corresponds to the difference hi (T2 ) -

hg (TI ), for details see text

TJ and T2 are convenient arbitrary temperatures that must be chosen to be below and above the glass transition region (i.e., in the glassy and in the liquid state, respectively). Using Q - P from the above equation the desired glass temperature is obtained by solving the quadratic equation:

1 - (B - b) . T~ + (A - a) Tg + (Q - P) = 0 2

Figure 6.29 shows an equivalent, graphical procedure (Moynihan et aI., 1976) for determining Tg • From the enthalpy definition it follows that:

T2 T2

f (Cp,I(T) - cp,g(T» dT = f (cp(T) - cp,g(T» dT Tg TJ

where cp(T) is the experimentally determined curve and Cp,g(T) and Cp,t(T) are the (linearily extrapolated) specific heat capacities of the glass and liquid respectively, the integration limits TJ and T2 have the same meaning as those in the previous figure. It must be guaranteed, however, that TJ (on heating) and T2 (on cooling) are definitely in the steady state region of the DSC. The lower limit of the left hand side integral, the thermodynamic glass temperature Tg, must be determined so that the integrals on both sides are equal. In other words, the area


4.5

J/!Kgl

1 3.5

3.0 >-

~

Cp.1 2.5 0 0..

------ ~~ =---=- =--=--0 ~

0

'" 2.0 ..c ~

'u / '" 1.5 0.. II>

cp•g

1.0 60 T, 70 90 100 Tz °C 11

I empera lure ..

Fig.6.29. Construction to determine the thermodynamic glass temperature Tg, which is defined by the equality of the different hatched areas (for other quantities see Fig. 6.28)

between Tl and Tz and the (extrapolated) cp-curves of glass and liquid (the left side integral) must be equal to that between the experimental curve and the (extrapolated) cp-curve of the glass (the right side integral). The two areas are hatched differently in Fig. 6.29.

It is clear from the equation above that absolute values of heat capacities are not required for calculating Tg• It is sufficient to know cp differences and this is also the case for the Richardson and Savill procedure. Nevertheless, their changes with temperature must be determined very precisely. This demand can only be fulfilled if the repeatability of the experimental curve is very good (for the same thermal history) and if a sufficiently large temperature range (more than 50 K on each side of Tg) is available for extrapolation.

An error estimation was done by Richardson, Savill, 1975a: for a typical /).cp of 0.3 J g-l K- 1 an uncertainty of 0.3 J g-l for the enthalpy change would result in a temperature uncertainty of ± 1 K. For total enthalpy changes of about 100 J g-l and caloric errors of ± 1 %, the determined Tg would be uncertain to ± 3K and this is not acceptable in practice. Fortunately, some errors tend to compensate each other in both procedures. For instance, an incorrectly extrapolated cp, I-curve (Fig. 6.29) has the same influence on both hatched areas, but the same only holds to a limited extend for the extrapolation of Cp,g' Tg cannot, therefore, be determined to better than ± 0.5 to ± 1 K with this method during routine measurements. However, this is not the reason why the thermodynamic Tg is so rarely used in practice - ± 1 K is adequate for many investigations of common Tg relations.

In fact, the scarcity of thermodynamic Tg data has mainly been due to a lack of suitable programs in manufacturers' software. The situation is now changing


rapidly and "fictive temperatures" (Tg,fic == Tg) can be calculated for most instruments. This is fortunate because for theoretical investigations of the kinetics of the glass processes the situation is clear: only the thermodynamically defined glass transition temperature reflects unambiguously the thermal history and all the other conditions during the formation of the investigated material. Tg is thus the central quantity for all kinds of relaxation studies. The aim of such investigations is to reproduce the behavior of the glass in the transition region, i. e., in the case of DSC measurements to reproduce the course of the function cp = cp (T) precisely. As the changes in Tg , caused by different thermal histories, may only be of the order of a few tenths K, Tg has to be determined at least with that precision.

To minimize the errors in determining Tg ,

- all error sources, associated with the determination of cp (cf. Sect. 6.1) must be borne in mind and carefully excluded,

- the sample should remain untouched in the apparatus during all experimental manipulations even for (often time consuming) annealing experiments. Annealing the sample outside the apparatus almost always yields unsatisfactory results because heat transfer conditions are not exactly reproducible after replacing the sample in the DSC.

The values for the glass transition temperatures obtained from DSC measurements need not necessarily agree with those of other methods. Discrepancies are caused by the different influences of the particular technique on the relaxation of the intrinsic variables (Duncan et aI., 1991). A formal conversion, taking into account the various experimental influences, can be made using the (modified) equation of Williams, Landel and Ferry (WLF) (Williams et aI., 1955). Principal differences must be attributed to different interactions between the method in question and the relaxation time spectrum of the intrinsic variables.

In studies of this kind it is indeed important to ensure that experimental errors have really been minimized and that any direct influence of the apparatus on the results is at least understood, if not avoidable. To illustrate this problem, in Fig. 6.30 the Tg values (open symbols) of polystyrene (determined as explained in this section) for different cooling (circles) and subsequent heating (triangles) runs at different rates are shown (Schawe, 1996).As can be seen, there is a significant difference between the results of heating and cooling obtained at a particular rate. In addition, this difference increases with the heating (or cooling) rate in question. From the theoretical point of view, there should not be any difference between the thermodynamic glass transition temperature measured in the heating and cooling mode if the sample has been cooled with the same rate before the heating run and if relaxation effects during the cooling run can be neglected. The glass transition should only depend on the procedure according to which the sample has been transformed from the liquid state to the glassy state. The differences measured result from the smearing effect due to the heat transfer path and the temperature profile inside the sample (cf. Sect. 5.4) which causes a lag of the sample temperature relative to the measured one. Thus the experimental glass temperatures are not fictive (or thermodynamic) values,


Fig. 6.30. Thermodynamic glass transition temperature Tg of polystyrene from DSC measurements as a function of heating rate. Circles: cooling mode; triangles: heating mode; open symbols: as measured, solid symbols: desmeared; according to Schawe, 1996

'-'

1,5

1.0

.9) 0,5

0,0

370 375

Tg In K

211

380

though determined as such. If the step response desmearing procedure described in Sect. 5.4.4 is applied and the glass transition temperature determined accordingly, the result is quite different (solid symbols in Fig. 6.30). Now the corrected values from the cooling and heating experiments almost superimpose. Furthermore, there is clearly a systematic change in Tg with the cooling rate (l.9 K per decade). This figure is comparable with the corresponding value from the activation plot of mechanical or dielectrical measurements carried out on the same sample. Nevertheless, absolute values are shifted half a decade with respect to dielectric results if one calculates an effective frequency from the cooling rate in question and this agrees with modulated temperature DSC measurements (Schawe,1996).

Otherwise, Fig. 6.30 also offers a simple and rapid procedure for the practitioner to obtain transition temperatures from non-desmeared curves. The mean values of the conventional midpoint temperatures obtained during subsequent heating and cooling runs with the same rate are practically independent of smearing effects and representative for the chosen rate. Analogous to Fig. 6.30 some measurements with different rates allow the extrapolation to zero heating rate. The temperature differences of such values between 10 participants of a GEFTA round robin test (Schick, 1999) were smaller than 1 K and hence absolutely sufficient for all practical purposes. Therefore this method is highly recommended to come to a well-defined and reproducible glass transition temperature.

Although the DSC method is very convenient for the characterization of the glass transition, it is not very sensitive. If the cp changes are small and take place over a broad temperature interval (as, for instance, is the case for lightly cross-linked polymers), the evaluation of the DSC curve is difficult and


uncertain. Dynamic mechanical or dielectric measurements are then more suitable.

6.4.4 Applications of Glass Transition Measurements

Many important applications of DSC measurement in the glass transition region are connected with polymer research, these are:

- Theoretical investigations concerning the thermokinetics of the glass transition. The quantitative description and modeling of relaxation phenomena implies either the determination of enthalpy changes (Petrie, 1972; Cowie, Ferguson, 1986; Agrawal, 1989; Montserrat, 1992; Hay, 1992) or of the thermodynamic glass transition temperature (Moynihan et al., 1976; Stevens, Richardson, 1985). An understanding of the relaxation processes is not only of theoretical interest for physicists but also allows a better access to physical aging phenomena in glassy, polymer materials (Struik, 1978; Cowie, Ferguson, 1986; Perez et al., 1991). Quantitative descriptions of the phenomena are often based on scanning experiments alone (Kovacs, 1963; Kovacs, Hutchinson, 1979; Moynihan et al., 1976; Ramos et al., 1984; Hutchinson, Ruddy, 1988; Hutchinson, 1990, 1992; Chang, 1988). However, some aspects of glassy behavior can be better studied following various isothermal annealing schedules (Petrie, 1972; Cowie, Ferguson, 1986; Agrawal, 1989; Montserrat, 1992; Hay, 1992). In this method the enthalpy difference between the annealed glass and the quenched glass is determined and evaluated using the empirical Williams-Watts function. It is well known that the approach of the structure toward equilibrium when held at a constant temperature (up to 50 K below the glass transition) is non exponential. This is caused by a broad distribution of different relaxation times. The non-exponentiality can be considered by assuming a "stretched" exponential expression (Moynihan et al., 1976).

T is a characteristic time, the exponent f3 is inversely proportional to the width of the corresponding distribution of the relaxation times (0 < f3 < 1). This model treats the relaxation behavior as thermorheologically simple, assuming that the relaxation function is independent of temperature. In addition, the relaxation rate according to the last equation does not depend linearly on the initial temperature jump. But linearity can be restored by the concept of the reduced-time integral (Narayanaswamy, 1971) for a temperature jump from To (at which the material was initially equilibrated) to T:

T dt ~=f--

To T(T, Tf )


Then, the dependence of the fictive temperature Tf can be expressed according to the Boltzmann superposition principle:

T

Tf(t) = To + f (1- exp (-sP» dT To

The most frequently used expression for T (T, Tf ) although purely empirical is the Moynihan equation (Moynihan et aI., 1976):

( i1h* i1h*) T (T, Tf ) = A . exp x· - + (1 - x) . -

RT RTf

x is the non-linearity parameter (O < x <1). A is the pre-exponential factor and i1h* is an effective activation energy. In other words, T is dependent on both temperature and structure. This model successfully describes practically all experimentally observed relaxation phenomena of inorganic and organic amorphous materials. Another model, based on the Adam-Gibbs equation (Adam, Gibbs, 1965) and advanced by Hodge, 1987, gives more meaningful model parameters but it is more rarely used and it does not provide for a better fit of the relaxation data.

- Tg as one of the most important properties of amorphous materials. For production of new materials with defined properties, the prediction of the glass transition temperature is a natural necessity. Predictions can be based on group contributions (Becker, 1976, 1977, 1978). Possible discrepancies between calculated and experimental Tg-values may then contribute to a better understanding of the relations between structure and properties of a specific material. Similar investigations concerning the correlation between structure and the caloric information (i1cp of the glass transition) are still only rarely found.

- Finding of relations between Tg and caloric quantities of the glass or the glass process (Becker, 1976; Batzer, Kreibich, 1982; Wunderlich, 1990).

- Relations between molar mass and glass transition temperatures of polymers. The most widely used equations are those of Fox, Flory, 1950:

and Ueberreiter, Kanig, 1952:

1 1 A' -=--+Tg Tg,= Mn

where Tg,= is the glass transition temperature of a polymer with infinite number average molar mass and A and A' are constants for certain broad classes of materials. Both equations can be used for high-molar mass polymers but that of Ueberreiter and Kanig is much better for oligomeric glasses. If, in such

214

Fig.6.31. DSC measurements of an epoxide cured at 160°C for the times shown: the glass transition temperature increases with extent of reaction (according to Wisanrakkit, Gillham, 1990)

1

'" QJ ..c


o 100 200 tempera ture

lb 11 4 h 3 h 2 h

15 h 1 h min

min

300 O( 400

investigations, the glass transition temperature must be determined from heating runs, the thermodynamic Tg should always be used. All other (empirical) Tg values are unsuitable because of the inevitable relaxation processes on heating (Agrawal, 1989; Aras, Richardson, 1989).

- Investigation of polymerization reactions. Tg is often more sensitive to the progress of a polymerization than is the heat production. This is especially true towards the end of the reaction where the heat production is low and barely detectable, whereas a distinct change of Tg may be observed (Mijovic, Lee, 1989; Wisanrakkit, Gillham, 1990; Hale et aI., 1991), for instance, if the cross-link density increases. As typical example, DSC curves for an epoxide-amine system, which has reacted for different times at 160 DC, are shown in Fig. 6.31. It is generally not possible, and in any case it would be far too time-consuming, to determine the degree of reaction only from the changes of Tg during polymerization. In addition the difficulties mentioned in determining exact Tg values would result in very uncertain results for any kinetic analysis. The situation is even more complicated because the glass transition for reacting systems may often be overlapped by the reaction heat flow and thus not precisely determinable. A "calibration curve" must be determined connecting the degree of reaction (obtained by another method such as an IR technique or, for soluble polymers, determination of the molar mass) with the Tg values obtained by DSC measurements, nearly identical experimental conditions should be chosen


for the different methods. The empirical function Tg = tea), obtained this way, is not related to a specific reaction model but it can be calculated if some assumptions are fulfilled: for instance, the Fox-Flory model can be used to describe the linear (L) polyaddition reaction:

Tg,Q and (Tg,=h are the limiting values of the glass transition temperatures for the monomer mixture and the linear polymer with infinite molar mass respectively. As early as 1970 Horie et al. tried to extend these considerations to reactions which give cross-linked polymers. In this case Tg shows a more pronounced rise, due to the increasing cross-link density, than is the case for linear polyaddition:

The relation between L".Tg,v and the cross-link density v is complex and the separation of L".Tg into L".Tg,L and L".Tg, v may not be possible but Min et aI., 1993 were successful. They found by NIR measurements that in the system bisphenol-A diglycidyl ether (DGEBA) and 4,4' -diaminodiphenyl sulfone (DDS) the reaction of the primary amine hydrogens only increases the molar mass of the linear polymer whereas that of the secondary amine hydrogen is responsible for cross-linking. Hence the total degree of reaction a can be separated into one part aL for the linear polymerization and another part a v for the cross-linking reaction. Both parts can be individually obtained from the experiment. If Tg,= represents the glass transition temperature of the completely reacted and cross-linked polymer, the following is valid:

- Investigation of enthalpy relaxation as a function of the degree of cure. Polymers heated for the first time after their production often show very large "relaxation peaks" (Sect. 6.4.3) which cannot be reproduced by any subsequent thermal manipulation (Montserrat Ribas, 1992). The reason for this exceptional relaxation behavior is not yet known. The shape and intensity of the relaxation peak reflect somehow the specific conditions of the polymer forming process and so far it has not been possible to decode this kind of "memory".

- The glass transition temperature of copolymers, blends and polymers containing plasticizers. The influence of plasticizers on Tg of polymers or the glass transition behavior of blends have often been investigated (Fox, Flory, 1950; Couchman, Karasz, 1978; Chee, 1985; Kalachandra, Turner, 1987; Braun et al., 1988; Brekner et aI., 1988; Tsitsilianis, Staikos, 1992; Pomposo et aI., 1993). Homogeneous, one-phase blends (random copolymers and miscible polymers) have only one Tg transition, whereas heterogeneous, two-phase blends (block copolymers and incompatible polymers) show the separate glass transitions of the components with L".cp values correlated with their relative


masses. As a rule, the glass transition range of homogeneous copolymers is distinctly enlarged compared with the pure components. The glass transition temperature of an ideal homogeneous system of two components can be calculated from their Tg values and mass fractions Wi by the Fox, Flory, 1950 equation:

1 WI Wz -=--+--Tg Tg•I Tg.z

or by the Karasz equation (Couchman, Karasz, 1978):

WI • ~cp I • In Tg I + Wz • ~cp z • Tg z In Tg = ' , ,. WI • ~Cp.I + Wz • ~cp,z

Possible interactions between the components may have very different effects depending on the composition of the system in question. Several equations have been suggested to describe the influence of molecular interactions on Tg with the aid of up to three additional parameters (Podesva, Prochazka, 1979; Braun et aI., 1988; Schneider, 1999). Relaxation experiments are helpful for deciding the sometimes difficult problem of whether a polymer blend is homogeneous or heterogeneous (Jorda, Wilkes, 1988; Tsitsilianis, Staikos, 1992). Following their proposals, a clearly detectable relaxation peak formed during the annealing of the sample is used as an "amplifier" for glass transitions, which otherwise only can be detected with difficulty. In this way glass transitions can even be detected which are only a few degrees apart from one another and which would normally overlap to give one broad transition and thus imply a homogeneous product. As an example Fig. 6.32 shows the DSC curves of one homogeneous and two heterogeneous glasses before and after annealing. The two heterogeneous blends differ in the amount of both copolymer components. After annealing the homogeneous sample shows only one non-structured relaxation peak, whereas the heterogeneous ones possess two separated relaxation peaks, though the DSC-curves of the unannealed glasses show no signs of heterogeneity.

- The glass transition of semi crystalline polymers. The inverse correlation between the glass transition and melting process has been investigated (Schick et aI., 1985, 1988). It is often found that the sum of the amounts of amorphous material (determined from ~cp) and of crystalline material (determined from the heat of fusion) is lower than expected for a two phase model. In these cases there must be a third phase (called "rigid amorphous"), which shows neither amorphous (glass transition) nor crystalline (melting) behavior. In addition, correlations have been found (Okui, 1990) between the glass transition temperature and other characteristic temperatures, e. g., the melting temperature or the temperature of the maximum crystallization rate.

- Testing of pharmaceuticals. Nowadays amorphous forms of pharmaceuticals are in widespread commercial usage. Substances in amorphous state show often different physical properties than the corresponding crystalline forms, in particular, an improved bioavailability and ease of processing due to


Fig. 6.32. DSC curves of a homogeneous (A) and two heterogeneous (B, C) blends which have been quenched from the melt or annealed a few K below Tg for 14 days. The relaxation peak( s) of the annealed samples clearly show that blends Band C are phase separated (according to Jorda, Wilkes, 1988)

f

)0 o --o

'" J:

30

217

A

quenched

annealed

B

quenched

annealed (

quenched

50 70°C 90

temperature ..

changed mechanical properties. On the other hand, there is a risk for an undesirable crystallization during storage, often connected with a change of the efficacy of the drug. Therefore, the knowledge of the molecular mobility in amorphous pharmaceuticals is of decisive importance for their practical use. The amorphous pharmaceuticals must be kinetically stable much longer than the expected lifetime of a drug. Both a possible tendency and the rate of crystallization can be estimated from the storage temperature in comparison with Tg and from the evaluation of corresponding relaxation experiments (Hancock, Shamblin, 2001; Crowley, Zografi, 2001).

6.4.5 The Dynamic Glass Process, an Example

As mentioned above the glass process is a relaxation (time dependent) process. Below the glass transition the system is in a non-equilibrium state which -


strongly depending on the temperature - slowly moves toward the equilibrium state. The vitrification or devitrification is coupled with a change of mobility (degrees of freedom) manifested to a change of the heat capacity, which thus becomes a function of time (at least in the range of, and below the glass temperature). This implies, that both the glass transition temperature as well as the heat capacity becomes a (complex) function of frequency (cf. Sect. 3.3 and 5.5.2). Therefore the temperature-modulated DSC method suggests itself (among other methods such as dielectric or dynamic mechanical measurements) to investigate the dynamics of the glass process.

As an example to show the usability of the TMDSC method, the "activation diagram" of the glass transition of glucose is shown in Fig. 6.33. The step-like change of the real part of the complex heat capacity cp (co) in the glass transition region is always coupled with a maximum in the imaginary part COmax • This maximum shifts in a characteristic way to another frequency if we change the temperature. The same is true if we, in a TMDSC experiment, change the temperature at a given frequency, again the imaginary part has a maximum at the glass transition temperature (see Fig. 5.15). It can be shown that both descriptions are equivalent. However, in Fig. 6.33 the maximum temperatures of dielectric experiments (DRS) - done with changing frequencies at fixed temperatures - are plotted together with the glass temperature - measured with different temperature-modulated calorimetric methods at fixed frequencies (Schick et al., 2002). The result is convincing, it is obvious that TMDSC is a suitable method to measure the frequency dependence (dynamics) of the glass transition. The experimental findings (symbols) fit to the theoretical prediction (solid line) of the Vogel Fulcher Tammann approach for the calorimetric as well as for the dielectric results. The systematic shift is due to the different stimulation of the glass process in the electric and thermal experiment.

It should be mentioned and emphasized, that the dynamic glass process investigated with TMDSC and the thermal glass process (vitrification or devitrifi-

Fig.6.33. Activation diagram of glucose from temperature-modulated calorimetric measurements (solid symbols) and dielectric results (open symbols) together with fit functions from the Vogel Fulcher Tammann approach (according to Schick et al., 2002)

t 'VJ

-.:::J d '---"-

"2l'§

= 0

6

5 4 3

2

1

0

-1

-2

-3

-4 29 3.0 3.1 3 2 K-1 33

1000/ T--

6.5 Characterization of Substances, the Phase Behavior 219

cation) visible in common DSC curves are different processes, which result in different glass transition temperatures. The glass temperature from the thermal process depends on the thermal history (cooling rate, annealing time etc) whereas the glass temperature from the dynamic process depends on the frequency of the measurements. In a TMDSC experiment the former manifests itself in the underlying, and the latter in the reversing cp curve. From this follows that the non-reversing curve (see Sect. 5.5.1) contains the difference of two in principle different cp curves. Every, however done, evaluation of the non-reversing cp curve in this region gives only accidental results.

6.S Characterization of Substances, the Phase Behavior

Such investigations are more meaningful only when they are coupled with other structure-sensitive analytical methods such as hot stage microscopy, infrared spectroscopy and X-ray diffraction. The main advantage of the DSC method lies in the ease and simplicity of operation - in particular both the choice and the rapid change of the required temperatures. This is very important for the investigation of substances with metastable phases. On the other hand there is the disadvantage that it is not possible to carry out experimental manipulations (e. g., nucleation of a supercooled liquid) inside the small and practically always closed crucibles.

6.S.1 Applications in Biology and Food Science

The investigation of polymorphism - defined as the ability of a chemical compound to crystallize into different states (modifications) - is increasingly important in food science.

A well-known example is cacao butter. Chocolate products with cacao butter have the desirable consumer properties only if the p-form with a melting point at 35°C is applied. DSC measurements are an effective tool for both optimizing the production method and the quality control.

Another example from food science is common butter. The objectives of investigations are an improvement of the cold consistency (spreading property) of butter and the concurrent reduction of softening when the temperature is increased. From DSC curves the influences of additives (triglycerides) with lowmelting point, and differences in the cream-ripening method can be directly concluded (Schaffer et al., 2001).

A frequently used sugar substitute is isomalt, a nearly equimolar mixture of two isomers containing crystal water. The crystalline material melts around 142-150°C, and after cooling results a glassy state with a glass transition temperature at about 60°C. Like other carbohydrates used in food industry, isomalt is very often used in its amorphous form. The final quality and the storage stability of isomalt containing products depends on the aging behavior of the non-equilibrium glassy state. Therefore, the properties of both crystalline and


0.03

1 WIg

0.02

!:! a ..... 0.01 =-~

a '" .&:.

u 3 u 0 C1J 0-III

-0.01 L...-____ --"-_____ -'--_____ '---____ --'

18 21 24 27 °C 30

temperature .-

Fig.6.34. DSC heating curves of the gel-liquid crystalline phase transition of a liposome. 1: 1.75% of dimyristoylphosphatidylcholine (DMPC) in water, 2: after addition of 3%, and 3: after addition of 5 % of vitamin D3 (mass: 15 mg, heating rate: 2 K min-I)

glassy isomalt were intensively studied (Cammenga et al., 1996; Borde, Cesaro, 2001).

The denaturation temperature is a measure for the thermal stability of proteins, essential in food science. The determination of the denaturation heat allows conclusions regarding a possible already existing thermal impairment of the proteins. Therefore Schubring, 1999, has studied the thermal behavior of herring muscle, skin and pyloric caeca during salting and ripening.

As a rule, applications in biochemistry or food industry require highly sensitive DSC instruments. As an example, Fig. 6.34 shows the transition from gel to liquid crystalline state of an appropriately prepared liposome of 1.75% of dimyristoyl phosphatidylcholine (DMPC) in water. The reversible transition (curve 1, Q = 0.47 J per g total sample mass, Tp = 24.6°C) corresponds to the change of the hydrocarbon chains from an ordered (crystal-like) to a liquid-like structure. The structural transition of the lipid can be perturbed by molecules which intercalate among the lipid chains and hence hinder the regular packing of the phospholipid molecules into gel-phase structures. This can be realized for instance by addition of the lipophilic vitamin D3. With increasing concentration of vitamin D3 in the liposomes, the peak becomes broader and smaller and the peak maximum shifts to lower temperatures (curve 2, 3 % of vitamin D3, Q = 0.15 J g-I, Tp = 22.7°C). Addition of more than 5% of vitamin D3 to the DMPC-water lipid (curve 3) causes the disappearance of this peak.

6.S Characterization of Substances, the Phase Behavior 221

Another point of view concerns the interaction of common environmental contaminants like phthalic esters and biomembranes (Bonora et aI., 2000). Dipalmitoyl phosphatidylcholine was used as a model of cellular membranes because lecithins are the major components in mammalian membranes. The authors have found noticeable effects on temperature, heat, shape and width of the main transition even in the presence of small amounts of phthalates.

6.5.2 Applications in Pharmacy

Without doubt most investigations on polymorphism concern compounds used as pharmaceuticals. The majority of drugs show one or the other form of polymorphism (Grunenberg et aI., 1996).

General Remarks

Phase transitions between different modifications may be reversible (enantiotropic) or irreversible (monotropic). This behavior can be understood best using energy-temperature diagrams (Fig. 6.35, Burger, Ramberger, 1979; Kuhnert-Brandstatter, 1996; Grunenberg et aI., 1996). Two modifications A and B are called enantiotropic or monotropic forms, if the free (Gibbs) energy curve of the liquid G1 crosses the respective curves of the solid modifications A and B above (Fig. 6.35 a) or below (Fig. 6.35b) the intersection point of GA and GB• Burger has formulated two important rules which immediately follow from Fig. 6.35.

The Heat-of Transition rule: If an endothermic (exothermic) transition enthalpy is found on heating, the two polymorphs are enantiotrops (monotrops). Because solid-state transitions are more or less kinetically hindered, apparent deviations from this rule are frequently observed and the actual behavior depends stronglyon the heating rates (compare the next two examples). Generally, on heating, the measured temperature of the DSC peak T;rs (AB) is higher than the thermodynamic transition point Ttrs(AB) (Fig. 6.35a).

The Heat-of Fusion rule: If the form with the higher melting point shows a lower heat of fusion the two forms are usually enantiotropic, otherwise they are monotropic. This rule is a good help if the melting-point difference of the two forms is small and heat capacity differences between supercooled liquid and crystalline B can be neglected.

In principle, the detailed thermodynamic analysis of DSC data enables the estimation of stability domains for possible modifications. This is possible if the cp functions, differing only very slightly, can be measured with the necessary precision. Of course, the quantitative determination of the complete energytemperature diagram according Fig. 6.35 is difficult and very time-consuming, often even impossible. Otherwise, to find the thermodynamically stable form of two modifications A and B within a given temperature range, the absolute values of their thermodynamic functions of state (G, Hand S) are not needed

222

Fig. 6.35 a, b. Energy-temperature diagram of a dimorphic system. a enantiotropic system, b monotropic system. G: Gibbs free energy, H: enthalpy, Ttrs (AB): thermodynamic transition temperature from A to B, T;rs (AB): measured transition temperature, Ttrs (BA): measured temperature for the monotropic transition B to A (A, B: solid phases, 1: liquid phase)


t H,

>--

C"l ~B~~~~-ill At-- - __

a

>-. en

--~ ,~-

. ~---G A

GB

~ __________ ~ __ ~ ____ ~ ____ ~ ____ ~G,

o T'rs(AB) T;rs(AB) Tfus(A) Tfus(B)

temperature/K -

H,

llf"HIAI

::;; I ~--==-~CHffi-:;;-- ___ lll"H<BAl HB c: ill B

A

b

o

-___ - HA

--~ --- "'" ; ~-_GB

temperature/K -----

~-GA ; G,

but only the differences against a reference state. In this case the liquid state of that modification with the highest melting temperature is used as reference state.

With (aH) = Cp and aT p

(as) Cp aT p = T one obtains:

Tfus,B T T

HB(T)-HA(T)=t1fusHA-t1fusHB+ f Cp,ldT+ f Cp,BdT- f Cp,AdT Tfus, A Tfus,B Tfus, A

and

t1f H t1f H Tfus,B C l T C T C SB(T)-SA(T)=~-~+ f ~dT+ f ~dT- f ~dT

Tfus,A Tfus,B Tfus,A T Tfus,B T Tfus,A T

The difference of the Gibbs free energies is then obtained as usual:


This way, Sacchetti, 2001, could verify that the transition temperature of two modifications of acetaminophen is clearly lower than previously thought (- 30 DC). This result is consistent with the recent observation that form B spontaneously converts to form A even at - 70 DC. Furthermore, Sacchetti could show that the greater stability of form A is not caused by a lower packing energy but by its higher entropy.

PharmaceuilcalAspecB

Investigations regarding the polymorphism of pharmaceutical substances are especially important, as different modifications have because of different efficacy, different solubilities and dissolution rates (bioavailability), and frequently also different processability (Burger, Ramberger, 1979; Kuhnert-Brandstatter, 1996; Giron, 1999). In general, the modification used should be the thermodynamic stable one. Metastable or glassy products showing better dissolution rates, can be used, if the stability is guaranteed for storage.

DSCs are also very useful for investigations of stability and physical and chemical interactions between drugs and excipients. The knowledge of the polymorphism of a excipient including its aging behavior is extremely important for the development of preparation forms such as suppositories, micro emulsions and solutions in soft gelatin capsules. Formulations with amorphous substances tend to form crystalline modifications with reduced bioavailability upon storage, strongly dependent on climatic conditions (temperature, moisture content). The determination of the amorphous part in semi-crystalline drugs is usually performed by X-ray diffraction. The detection limit is 5-10 %. Measuring the crystallization enthalpy, Giron, 1999, was able to detect amorphous parts less than 5%.

Examples

The first example of polymorphism concerns the melting of acetamide (Fig. 6.36). During the first heating the thermodynamically stable modification (the as received state) melts at about 80°C. On cooling from the melt and in the absence of crystal nuclei of the stable modification the metastable modification crystallizes first in accordance with Ostwald's step rule. The second run thus shows the melting peak of this metastable modification at about 65 DC. In the absence of crystal nuclei neither a possible monotropic transition below 65 DC nor the recrystallization of the supercooled melt between 65 and 80°C is found.

Remark: The formation of such metastable phases is one of the main problems encountered in the determination of phase diagrams, and this must always be borne in mind to avoid misinterpretation of the observed curves.

Phenylbutazone is another well-known material that shows polymorphism. The stable modification (curve 1 in Fig. 6.37a) melts at about 103°C. Rapid cooling of the melt yields a glassy phase. On reheating the glass (curve 2) this first recrystallizes to a metastable form at about 35°C. Subsequent behavior depends on the heating rates, either melting of the metastable modification at about 93 °C or of the stable structure at 103°C will be observed. This is because the rate of transformation from the metastable to the stable form is low in the solid phase (i. e., < 93°C), only at low heating rates « 1 K min-I) will there be enough


50.0

mW

1 37.5

"\ /

~ 25.0 /

:: =- / 0

/ 2

c 12.5 C1I -'=

/ ,/

.-._" - --.-. 0

50 58 66 74 82 O( 90 temperature ~

Fig.6.36. DSC melting curves of the thermodynamically stable (1) and of the metastable (2) modification of acetamide

time during the experiment for this to occur. At high heating rates (> 20 K min -I) the formation of the stable modification does not happen, not even in the liquid phase after the melting of the metastable modification. The corresponding DSC curves are shown in Fig. 6.37b. For better comparison of both curves, curve 2 was also run with a heating rate of 40 K min-I following a preliminary heating to 98°C at 1 K min-I and recooling to room temperature. At moderate heating rates (ca. 5 K min-I), however, the sample reaches some intermediate state. Any remaining metastable phase that has not transformed during the slow solid state reaction, melts at 93°C, it then transforms to the stable modification as an exothermic reaction and finally melts at 103 °C (curve 2 in Fig. 6.37a). This can be proved by quenching the sample immediately after completion of the exothermic reaction (at 100°C) and reheating it again, when only the melting peak of the stable phase is seen (curve 3).

6.5.3 Other Applications

Liquid Crystals

DSCs are used very frequently for the investigation of polymorphism of substances with liquid-crystalline mesophases. The reliable evaluation of mesophase transitions with small or very small transition energies (sometimes <0.1 J g-I) makes great requirements on apparatus and working conditions. An example is the behavior of cholesteryl myristate during heating and cooling (Fig. 6.38). The first peak (Q = 77.3 J g-I, T = 69.0°C) corresponds to the formation of a smectic

6.5 Characterization of Substances, the Phase Behavior

20

mW

15

f .!! 10 CI ... :;J: 0 -CI 5 2 ...

..c:

3

a 0 25 40 55 70 85 100 °e

tempera ture ... 70.0

mW

t 52.5 2

.!! 30.0 "" ....

:;J: 0 .... ~

"" .... .c;

17.5

b oL===~==~====~~~~~ 55 70 85

tempera ture

Fig.6.37. Polymorphism of phenylbutazone.

100 115 o( 130

225

115

a 1: first heating run, 2: heating run after quenching from the melt, 3: heating run after reheating the quenched melt up to the end of the exothermic peak at 100DC (mass: 5.1 mg, heating rate: 5 K min-I), b two limiting cases in heating the quenched melt, 1: complete melting of the metastable phase during fast heating, 2: melting of the stable phase after previous heating to 98 DC with 1 K min-I (heating rate: 40 K min-I)


20

mW

f 15

~ 10 E =-

-= c

• '" ..c: 5

o~------~------~------~------~------~~----~ 60 65 70 75 80 85 °C 90

tempera ture •

Fig. 6.38. Polymorphism of cholesteryl myristate. 1: first heating, 2: cooling to 60°C, 3: subsequent heating, 4: heating after cooling from the melt to room temperature (mass: 4 mg, heating and cooling rates: 5 K min-I)

mesophase. The following two peaks (Q = 2.5 J g-I, T= 77.7°C and Q = 1.9 J g-l, T = 83.0°C) correspond to the formation of the cholesteric mesophase or the transition into the isotropic melt. All phase transitions are reversible (compare curves 1 and 4). The reason for the narrower peak shape of the first peak of curve 4 (Q = 76.9 J g-l, T = 68.5 °C), compared with that of curve 1, is the better heat transfer to the sample after melting. If the cooling run (curve 2) is stopped before the crystalline phase appears, only the two liquid-crystalline phase transitions are observed during the subsequent reheating (curve 3). The heat of transition from the crystalline to the liquid-crystalline mesophase is usually - as in this example - much higher than those of the following mesophase transitions. Further, it can be demonstrated that there are practically no supercooling phenomena for mesophase transitions. This is due to the fact that mesophase transitions are probably second order transitions. The onset temperatures of the heating and cooling runs do not differ from each other after proper correction of the extrapolated onset temperatures to a heating or cooling rate of zero (onset differences of AT = 0.2 K and AT = 0.0 K for the two transitions). The crystalline phase, however, shows considerable supercooling. It crystallizes at about 30°C, equivalent to a AT of about 40 K.

Extreme Melting Ranges

Sometimes melting processes extend over very broad ranges of temperature. For low molar mass substances this may be due to large amounts of impurities


(cf. Sect. 6.9). For macromolecules this may be caused by the broad distribution of the molar masses or by a distribution of the thickness of the crystal-lamellae. The determination of the appropriate baseline and the choice of physically reasonable integration limits for the enthalpy calculation is then very problematic. Mathot, Pijpers, 1983; Mathot, 1984; Mathot, Pijpers, 1989, recommend the use of the heat capacity (cp) function rather than the normal heat flow rate curve of the sample for evaluation to overcome these difficulties.

Processes with Gas Exchange

Additional problems are found when DSC is used to investigate processes which lead to the production or absorption of gases, e. g., dehydrations or oxidations. Here both qualitative and quantitative results are influenced by the experimental conditions, the type of purge gas, its flow rate, the geometry of the sample holder, sample preparation and so on. A typical example is the thermal decomposition reaction of calcium oxalate monohydrate. The temperature of the first dehydration peak and the peak width are radically influenced both by the heating rate and the effectiveness of the water vapor transport by the purge gas. Thus every attempt at a kinetic evaluation risks interpreting the transport conditions rather than the kinetics of the decomposition process. The second step in the decomposition of calcium oxalate, the elimination of carbon monoxide, to yield CaC03 , is strongly influenced by traces of oxygen in the purge gas because in

t .!!. Q .... :J: 0

Q C1.I ~

50.0

mW

37.5

25.0

12.5

crystalline -+ smectic

/ mesophase

nematic -+

liquid pha se

\ smectic -+ nema tic

\

o L-______ ~ ______ ~ ______ ~ ________ ~ ____ ~

o 40 80 120 160 ·c 200

tempera ture ..

Fig.6.39. DSC curve of the phase behavior of calcium stearate monohydrate (mass: 3 mg, heating rate: 10 K min-I)


the presence of O2 the CO is completely or partially oxidized. The simultaneous coupling of the DSC with thermogravimetry or mass spectrometry is advantageous in these cases.

Another example, the phase behavior of calcium stearate monohydrate is shown in Fig. 6.39. Shape and temperature of the first peak (dehydration) and the separation of it from the second peak are strongly influenced by the effective removal of the evolved water vapor and thus by the experimental conditions (open, partially or completely closed sample crucibles; flow rate and type of purge gas). The three other peaks correspond to the transitions from the crystalline to the smectic phase, from smectic to nematic and finally from nematic to isotropic liquid. Within the homologous series of fatty acids the phase behavior changes both quantitatively (for rather small differences in the chain length) and qualitatively (for larger differences). The DSC method may therefore also be used to solve analytical problems.

6.5.4 Porosity Measurement

Porous materials have an increasing technological importance. Due to the surface contribution to the free energy of liquids within the confined geometry of a porous solid, their melting and freezing properties are different from those of the bulk phase. Frequently used experimental techniques are NMR, N2-adsorption, small and wide angle X-ray scattering (SAXS and WAXS), mercury porosimetry and thermoporosimetry. The latter method was developed by Brun et al., 1977, and permits the determination of both the mean pore size and the pore size distribution from the freezing and melting DSC-curves of a suitable liquid (e. g., water, cyclohexane, alkanes) confined in the pores. The steps of a typical DSC experiment are:

1. gasing-out of the porous solid by annealing under reduced pressure, 2. saturation of the pores with the desired liquid, 3. sealing of the sample in vapor-tight sample pans, together with a small excess

droplet of the liquid, 4. first cooling run, starting well above the freezing temperature of the bulk

liquid, 5. subsequent heating run, possibly followed by further thermal cycles, 6. determination of the total pore and bulk liquid by weighing before and after

piercing the pan and annealing the sample under reduced pressure.

Generally it is found that both the position and the shape of the peaks from repeated cooling/heating cycles are extremely reproducible. An alternative but very time-consuming technique replacing the steps 3 and 4 was suggested by Neffati et al., 1998 ("fractionation" of the crystallization or melting peak). It is applicable in all cases of a distinct hysteresis between freezing and melting temperatures.

A typical experiment is shown in Fig. 6.40 for a porous silica gel (f3 = -1 and + 1 K min-I) ftlled with water. The large peaks correspond to the excess bulk water, the broader and less intense ones to the "confined fluid". Contrary to the shape of the peaks for the bulk liquid, the width of the peaks for the pore liquid

6.5 Characterization of Substances, the Phase Behavior

1-I Wg-1

(onfined \later

OJ 01.----+---cJ

cJ OJ

-L

-1 1--- bulk water

-2~----~----~~------~----~ -30 -20 -10 o 10

temperQture ..

229

Fig.6.40. DSC curves on heating (upper) and cooling (lower) of water confined in a porous silica gel (heating and cooling rate: 1 K min-I)

doesn't become smaller for lower heating rates but is determined by the pore size distribution. The highest usable heating and cooling rates must be determined by preliminary experiments, in order to maintain thermodynamic "equilibrium" conditions andto avoid the "smearing" of the peak shape due to the thermal inertia of sample and DSC. Typical values are 0.25 to 2 K min-I, something dependent on the concrete system (Faivre et al., 1999; Neffati, Rault, 2001; Ishikiriyama et al., 1996). The pronounced but more or less statistical supercooling of the freezing peak for the bulk water due to the delay of heterogeneous nucleation is uninteresting but often very disturbing if overlapping with the freezing peak of the pore water. Other liquids, e.g., cyclohexane, show much smaller supercooling. Probably, the freezing in the nano-sized pores occurs through the slow penetration of the freezing front, dependent on the pore size aperture. The solid-liquid meniscus formed at the pore aperture should be spherical both for cylindrical and for spherical pores. Contrary to this, the melting of the confined material is initiated at the pore wall surface, the interface has the form of the pore surface. The result is the distinct hysteresis between freezing and melting.

The profile of the DSC curve and the hysteresis between freezing and melting allow conclusions on the shape of the mesopores. The shift of the transition temperature is given by a generalized form of the Gibbs-Thomson equation (Faivre et al., 1999):

o Ysl V. I1T= - a· T fus '-'--r I1fusH

where 11 T = (T?us - T) is the melting point temperature depression of the confined liquid, T?us is the melting temperature of a crystal of infinite dimension, Ysl is the average interfacial tension of the crystal, V. is the molar volume of the crystal, I1fusH is the molar enthalpy of melting and r is the radius of the pores. The factor a depends on the shape of the pores (Faivre et al., 1999). Apart from


uncertain values for Ysl the reliability of the results is limited by using the correct value for t:..fusH. If the total peak area is divided by the total amount of the used liquid and subsequently the part of the bulk phase obtained from the bulk liquid peak is subtracted, one always obtains values for t:..fusH in the pores which are distinctly smaller than those for the bulk phase. A distinct temperature dependence of t:..fusH suggested by Neffati and Raul, 2001, is rather improbable. The most probable explanation is the existence of anon-freezing layer on the surface of the pores. This can be verified if the above equation is modified and rewritten (Schmidt et aI., 1995) as

K t:..T=-

(r - t)

with o V. K= -a' Tf • VI'-us 's A H ilfus

That means the effective pore radius is reduced from r to (r - t). If the experimental data for defined and systematically varied pore sizes are fitted to this equation, t is in the order of one to three monolayers. The accessible range of r is limited for large pores because of overlapping and missing separation of the peaks for bulk and pores, and for very small pores because of missing filling with the test liquid (dependent on its required space).

The same technique can also be used to obtain the distances of the cross-link points of swollen polymers after equilibration in a appropriate swelling agent, e. g., cyclohexane.

6.6 Determination of Phase Diagrams

The determination of phase diagrams in the fields of metallurgy and mineralogy was one of the earliest applications of thermal methods. It was one of the decisive reasons for the introduction of Differential Thermal Analysis (DTA, see Appendix 1) at the beginning of the twentieth century.

As an example, the phase diagram of the binary system benzil!acetanilide is shown schematically in Fig. 6.41 together with the DSC curves of the heating runs of the pure components (curves 1 and 7) and of five mixtures including the eutectic (curves 2 to 6). In accordance with the phase diagram benzil and acetanilide are completely miscible in the liquid and immiscible in the solid phase. Pure acetanilide melts at 115°C, pure benzil at 95°C. The eutectic, with a mole fraction of about XBen = 0.58, melts at 78°C. In Fig. 6.41 a two different melting pathways (mole fractions Xl = 0.1 and X2 = 0.9 of benzil) are sketched. The measured DSC curve of every mixture shows two events. The first is the peak due to the melting of the eutectic, the subsequent broad endothermic effect is caused by the solution of the remaining solid component in the equilibrium melt (following the pathway along the liquidus curve in Fig. 6.41 a). The final point of this peak corresponds to the liquidus temperature according to that mole fraction in the phase diagram. Both the pure components and the eutectic

6.6 Determination of Phase Diagrams 231

115 6C

t 95°C

78°C

a

x ••• __

8

WIg

1 6 1--_2=--__

4

5

6

b 7

OL-____ ~~c=~====~====~~=_~ 60 73 86 99 112 °C 125

tempera lure •

Fig. 6.41 a, b. The phase behavior of the system acetanilide (Ac)-benzil (Ben). a Schematic phase diagram of the (eutectic) system, the arrows mark the course of the melting process for two mixtures with the mole fractions Xl and X2, b DSC curves of the pure components and of five mixtures, 1: acetanilide (XBen = 0),2: XBen = 0.1,3: XBen = 0.4,4: XBen = 0.578 (eutectic), 5: XBen = 0.75,6: XBen = 0.9,7: benzil (XBen = 1)

melt at well defined temperatures. These can be determined from the extrapolated peak onset temperatures in the usual way, whereas the determination of the liquidus temperature is not easy. The peak maximum temperature (if necessary corrected for thermal lag, see Sect. 5.4.4) of the second broad peak is a good approximation. In every case slow heating rates « 2 K min-I) and small sample masses should be used, to ensure approximate thermodynamic equilibrium at


every moment during the experiment and to avoid "smearing" (see Sect. 5.4) of the DSC curve and thus incorrect temperatures. The heating mode is more suitable for the determination of phase diagrams than the cooling mode, because many organic substances have a tendency to supercool considerably.

If the heats of fusion of the pure components are known and the binary system behaves ideally in the liquid phase, the theoretical melting course may be predicted, using the Schroder-Van Laar equation (Brezesinski, Dorfler, 1983; see also Sect. 6.9).

The molar heat of fusion of the eutectic ~fus Heu can be calculated for ideal eutectic systems (for which the eutectic is a simple blend of microcrystals) using the normal rule of mixtures:

XA,XB: mole fractions of the components A and B'~fusHA'~fusHB molar heats of fusion of the pure components.

For precise calculations ~fus HA and ~fus HB should be converted to the eutectic temperature using Kirchhoff's law. If this simple additivity holds, the exact composition of the eutectic can be determined by plotting the heats of fusion of the eutectic peak of different mixtures against the composition (Ding et aI., 2000). This gives two different straight lines to the right and to the left of the eutectic and the intersection gives the required composition. This procedure is generally also a good approximation for non-ideal mixtures.

For real systems, discrepancies between calculated and experimental eutectic heats of fusion can lead to conclusions regarding the nature of interactions between components. From these differences, the excess functions of entropy, enthalpy and Gibbs energy can be calculated (Rai, Shekhar, 1993, Rai, Rai, 1998). For positive (negative) excess free energy, the attractive interaction between molecules of the same kind is stronger (weaker) than that one between molecules of different components in the mixture.

Using available data from the literature Sangster, 1999, has published computer calculated and thermodynamically consistent phase diagrams for 60 binary eutectic systems of drugs. Among the results are also the excess Gibbs energy of liquid and solid solution phases and the thermodynamic properties of intermediate compounds. Due to the very long times for the diffusion-controlled establishment of the solid-liquid equilibrium, the determination of phase diagrams with mixed crystals is difficult and should be supplemented by adiabatic calorimetric measurements. Some examples are the systems trans-stilbeneltrans-azobenzene and p-dichlorobenzenelp-dibromobenzene (Stosch et al., 1996) and d-carvonel l-carvone (Gallis et aI., 1996a-c, 1999).

6.7 Safety Aspects and Characterization of High-Energetic Materials

Specific devices for this type of investigation are the reaction or safety calorimeters mentioned in Chapter 1. They provide detailed information for the optimization of process parameters, reducing production costs and avoiding safety

6.8 Characterization of Polymers 233

risks. However, valuable results may be obtained using conventional DTA or DSC (Hentze, 1984). Safety risks in the chemical industry occur when materials, known for their exothermic decomposition reactions, have to be stored for long periods at temperatures, such that the heat evolved is greater than that transferred to the surroundings. Under worst case conditions this may lead to a runaway and subsequent fire or explosion. The direct detection at ambient temperature of the extremely small heat flow rates in question is, in general, not possible with DTA/DSC (exceptions are Calvet calorimeters, cf. Sect. 2.1.3).

Hentze, 1984, avoided this problem by annealing the substance

- at constant temperature and various times, - constant times and various temperatures,

and comparing the subsequently measured melting and decomposition peaks with those of a fresh, unannealed sample. In this way quantitative results were obtained for the decomposition kinetics of, for example, p-nitroaniline. Using this procedure it is possible to determine decomposition heat flow rates down to 10 mW kg-' at storage temperatures.

Decomposition reactions often react sensitively to traces of foreign materials. With the annealing technique described above, the first detectable decomposition of azidophenylacetic acid was determined. In addition the effects of many kinds of "impurities" were investigated. In a similar way the autocatalytic influence of the decomposition products was definitely proved.

DSC measurements supply valuable results regarding the thermal decomposition of newer high-energetic materials as ammonium dinitramide (ADN) and 2,4,6,8,1O,12-hexanitrohexaazaisowurtzitane (HNIW). To avoid safety risks, small sample weights « 1 mg) should be investigated in open sample pans or in special high-pressure capsules. The decomposition kinetics of ADN is strongly influenced by the reaction conditions (Tompa,2000) or impurities and stabilizers (LObbecke et al., 1997). HNIW exists in several polymorphic forms with different thermodynamic stability and shelf life at room temperature (Foltz et al., 1994).

6.8 Characterization of Polymers

Quality control of a material is extremely important both for inspection on delivery and to insure the integrity of manufactured products, this is especially relevant in the plastics industry. Until recently the prime reason for inspection was to ensure continuity of properties if, for instance, the manufacturer, the batch number, or even the manufacturing process, were changed. Now, with the need for recycling used plastics many more questions must be answered, and thermoanalytical methods (DSC, TG, TMA, DMA) have proved very powerful in this respect. The results are easily comprehensible and readily interpreted.


6.8.1 Effects of Origin and Thermal History

Plastics are either amorphous or partial crystalline materials, i. e., the glass transition temperature and/or the melting region as well as the degree of crystallinity are essential quantities for the assessment of material properties and working conditions. They depend distinctly on manufacturing (synthesis) and storage conditions (thermal history) and a reliable determination of these quantities is indispensable in polymer technology. DSC offers the possibility to characterize plastics in a fast and satisfactory way. In fact the rapid development of a precise equipment for quantitative thermal analysis (DSC) was bound up with the increasing importance of plastics. Polymer science is still a major area of application for DSC (Mathot, 1994 a, b) and a polymer laboratory without a DSC is unthinkable.

To determine the properties and the thermal history of a polymer requires at least a cycle of three measurements: Heating the original sample to a temperature 20 to 50 K above the melting region, cooling the sample to get it into a defined state and reheat it again. The three DSC-curves are normally analyzed with respect to:

- The glass transition region: the temperature(s), the width(s) and step height(s) of the glass transition(s) (see Sect. 6.4) characterize amorphous polymers, blends of them, and even the amorphous part of partial crystalline samples. The glass transition temperature reflects the composition of the components, the conditions during production, and sometimes also the aging of the plastic product during its use (caused, for instance, by decreasing molar masses and a change of their distribution due to cracking processes by partial oxidation). In heterogeneous or micro-heterogeneous amorphous blends two or more glass transitions are found, which are characteristic of the individual components. The relative amounts can be estimated by evaluating the respective step heights. In partial crystalline polymers the degree of amorphous state can be determined from the step height as well, whereas the width of the glass transition region in blends is influenced by the miscibility of the components. Further important information can be taken from the possible existence and intensity of a relaxation peak in the region of the glass transition on heating. Glass transition evaluation is generally a very important application field in DSC and is discussed in detail in Sect. 6.4.

- The melting region: the melting peak(s) temperature(s), the temperature region of melting (peak width) and the peak area(s) (heats of fusion or transition and crystallization) characterize the crystalline polymer. As a rule it is not difficult to identify the components of a plastic mixture, using the characteristic temperatures of the melting peaks. Often even a quite accurate quantitative analysis is possible. Overlapping peaks can be separated by using appropriate evaluation methods (see Opfermann, 1992), since the necessary condition for using such programs - the additive superposition of the respective peaks because of immiscibility in the solid phase - is nearly always fulfilled for plastic mixtures. Some polymers appear in different polymorphous forms, for such plastics solid-solid phase transitions can be found. For semi-crystalline products the degree of crystallinity is an essential quantity


and the determination of crystallinity is an important application of DSC therefore we address this topic a special section (see Sect. 6.8.2). A relative (within the type of plastic) degree of crystallinity can be obtained very easily (ignoring all precautions described in Sect. 6.8.2) if the area of melting peak is compared with that of the pure material of the same type.

- The crystallization behavior: crystallization is visible in a DSC because of the exothermic heat connected with it. The exothermic peak is easily detected in scanning mode of operation, its temperature and width characterize the crystallization. Normally crystallization occurs on cooling from the melt, the crystallization rate depends strongly on temperature and can be followed in the DSC in isothermal mode of operation. If crystallization takes place during a heating run (prior to the melting peak), the sample was definitely in a non-equilibrium state caused by a special treatment such as quenching the sample from the melt. This way the thermal history of a polymer sample can be followed with DSC. In extreme cases, it is found that for bulk thermoplastic moulded parts the degree of crystallinity varies within the parts, due to uneven cooling rates during moulding, this can be quantified by determination of the crystallization heat from the peak area of the different samples.

- Chemical reactions and thermal and oxidative decomposition: the beginning of chemical reactions, in this case a polymerization or decomposition, is normally characterized by the appearance of an exothermic (or rarely endothermic) signal in the DSC. To determine this, the plastic is either annealed isothermally in the DSC until such a signal indicates the beginning of reaction or decomposition, or the sample is heated in scanning mode up to this event. Polymerization reactions can even be followed quantitatively and kinetic investigations can be performed (see Sect. 6.3). The curing of resins and lacquers depends on both the chemistry (including stoichiometry, filler and catalyst content) and on the specific heating procedures. As a result the cured products may differ considerably in their thermal and mechanical properties. This can be demonstrated either by repetition of the experiment under different conditions or by coupling with other methods (IR, GC, NMR, TMA, DMA). Even decomposition can be followed quantitatively. By choosing an appropriate gas atmosphere (inert or reactive, e.g., oxygen), information is available concerning the thermal or oxidative stability of the material. For these measurements there is always a risk of contaminating and destroying the measurement system, so the run should be interrupted immediately after detecting the start of decomposition. However, thermogravimetry is in principle much more suitable for this type of investigation.

From such measurements and the evaluation procedures mentioned, valuable information can be gained which is used to characterize the plastic material, to identify it and describe the properties. Such information is even useful for quality control and, of course, in case of complaints and material failure.

For products containing recycled plastics, most of the above mentioned thermal properties may have changed in a characteristic manner. The observed deviations from the original behavior provide reliable information whether the plastic in question is suitable for the intended application or not.


6.8.2 Determination of the Degree of Crystallinity

DSC is widely used for determination of the degree of crystallinity We (weight fraction crystallinity) of partially crystalline polymers. The most simple and obvious possibility is to relate the estimated heat of fusion (~fusH) to that of the same polymer but with known crystallinity, provided that this value was determined by an independent method such as X-ray diffraction, Raman scattering, density measurements etc. The determination of the required peak area is problematic, because it is not easy to fix the lower integration limit. The melting of polymers often starts more than 50 K below the maximum temperature of the peak and it is very difficult and rather arbitrary to determine the beginning of this, at first, very slow process on the DSC curve. The situation is made more difficult by the fact that both the degree of the crystallinity and the heat of fusion are dependent on temperature. The use of the heat of fusion measured at higher temperatures yields a value for the crystallinity which is almost certainly too small compared with the value that is actually required - that at room temperature. Nevertheless, this simple procedure is sufficient for determining relative (within the same type of plastic) crystallinities (e.g., for quality control, cf. Sect. 6.8), if one uses fixed integration limits and a fixed reference value for the heat of fusion of the totally crystallized sample.

Mathot, Pijpers, 1983, 1989, suggest a convenient procedure (Fig. 6.42), which avoids the difficulties mentioned. The heat flow rate after the melting peak, which is proportional to the heat capacity function Cp,a of the amorphous (liquid) polymer, is extrapolated to that temperature T, at which the degree of crystallinity We (T) is to be calculated. If, in doing so, the experimental curve is not intersected (T > T*, e.g. T2 in Fig. 6.42), only the area Al has to be evaluated. If the extrapolated line intersects the measured curve (T < T*, e.g. Tl in Fig. 6.42), the area A2 has to be subtracted from A l' The heat, corresponding to the areas Al (or to A 1-A2), is then divided by the reference value ~fush (T) of a completely crystalline sample at this temperature. For linear polyethylene Mathot, Pijpers, 1983, suggest the following equation for the latter function:

~fush(T) = 293 - 0.3092 ,10-5 • (414.6 - TIK)2. (414.6 + 2TIK) J g-1

The degree of crystallinity We (T) in dependence on temperature is then obtained according to:

Meanwhile the functions ha (T) for the completely amorphous and he (T) for the crystalline form are known for numerous linear macromolecules with sufficient accuracy (Wunderlich et al., 1990).

If the specific heat capacity curve is evaluated from the DSC measurement (see Sect. 6.1), a simple integration from To to T of the cp curve - and the respective functions of the amorphous and crystalline cp function (from literature) - yields


A,

f ~

A, ~ :a 0

;;:: ~

c:J GI

.c.

temperature ..

Fig. 6.42. The determination of the degree of crystallinity according to the procedure of Mathot, Pijpers, 1983, at two different temperatures T. < T* and T2 > T*, where T* is the temperature of intersection of the extrapolated liquid phase line and the measured curve (for details see text)

the enthalpy difference function h (T) [or respectively, ha (T) and he (T) ]. This opens a new thermodynamic based calculation of We (T), without the necessity of knowing the course of the, often non-linear, baseline in the melting range:

Of course, the baseline in the melting interval can be calculated too, if a suitable model and the functions ha (1'), he (1') and We (1') are known for the semicrystalline polymer. The simplest assumption is that the properties of a semicrystalline polymer can be adequately described by a two phase model with only crystalline and amorphous parts. The measuring curve is then calculated according to:

The first two terms correspond to the baseline, the third one describes the temperature dependent change of the degree of crystallinity (the peak). Unfortunately the two-phase model fails with some types of polymers, e.g., polyethyleneterephthalate. The baseline can be calculated in such cases by use of im-


proved models, which include a third phase called the "rigid amorphous phase" {Suzuki et al., 1984, Schick et al., 1985}. This phase is thought to exist on the lateral surface of the crystal lamellae and should have a very limited mobility. It follows from this that the rigid amorphous phase does not contribute either to the thermal glass transition or to the melting heat of the crystalline part. Nevertheless, the heat capacity of this non-crystallized material can be approximated by that of the crystalline phase.

For some polymers the rigid amorphous phase is absent and good results for We may be obtained for these {polydimethylsiloxane, polybutadiene}, if the measured step change of the heat capacity at the glass transition temperature Tg {cf. Sect. 6.4} is related to that of the completely amorphous polymer yielding the amorphous amount Wa = LiCp{Tg}/LiCp,a{Tg}. In this case the degree of crystallinity is We = 1 - Wa' The completely amorphous state can often be obtained by quenching or ultraquenching the molten polymer.

The temperature-modulated mode of operation {cf. Sect. 2.4.2} is a very suitable method to determine the {vibrational} heat capacity even in the regions of crystallization {see Sect. 5.5}. This way it is possible to precisely determine the heat capacity from below the glass transition up to the beginning of the melting region and even further if higher frequencies are used where the contributions from melting dynamics are suppressed {Schick et al., 2003}. Together with literature values of the liquid and solid heat capacities of the sample in question {ATHAS, 2002}, the degree of crystallinity {solid} can be determined precisely, even its change in dependence on temperature in the case of hidden crystallization. This way it is even possible to determine the rigid amorphous part and its change with temperature. Schick et al., 2001, showed with this method that the rigid amorphous part is closely connected with the crystallites and disappears together with their melting.

6.8.3 Advanced Characterization with the TMDSC Method

Recently TMDSC {see Sect. 2.4.2} has become relevant to polymer science. This method offers some more advantages and possibilities because of the additional modulated {reversing} signal, which contains information about the dynamics of the investigated process. The glass transition {see Sect. 6.4} is one example of a relaxation process which yields different signals depending on the frequency of modulation. The glass transition temperature depends on frequency in a characteristic manner and the resulting "activation diagram" (see Sect. 6.4.5) makes it possible to get a better insight into the glass process of polymers. Of course, this is mainly of interest for theoretical and model investigations and has hardly any relevance for more practical applications.

However, the situation is quite different for other time dependent processes which may happen in polymers. In particular melting and crystallization are generally time dependent and TMDSC offers the possibility to follow the change of, say, the apparent heat capacity in time. The crystallization is normally not seen in the modulated heat flow rate signal, since the crystallization process is not influenced by the rather small temperature changes of the modulation,


but with crystallization the heat capacity changes as well and the latter can precisely be calculated from the heat flow rate amplitude (see Sect. 6.1.5) which, in turn, enables the calculation of the degree of crystallinity and its change in time. With TMDSC it is even possible to measure the change of the heat capacity in a quasi-isothermal mode and follow the crystallization this way which opens the possibility to investigate crystallization kinetics (cf. Sect. 6.3).

Chemical reactions (e. g., polymerization or curing) are generally also not visible in the modulated signal of TMDSC, whereas any change of the heat capacity, such as the vitrification or devitrification (see Sect 6.3.6), can easily be determined from that signal, even in a quasi-isothermal mode. This enables valuable insight into the reaction conditions and possible changes from concentration control to diffusion control and the setting up of a TTT diagram for the reaction in question (see Sect. 6.3.6). The modulated (reversing) signal may even serve as baseline (see Sect. 5.1) to determine the enthalpy of reaction from the underlying (conventional) DSC curve (see Sect.6.2). This is in particular of great importance if the heat capacity changes during the reaction.

In the melting region of polymers the situation is more complicated, as the melting process contributes to the modulated signal in a complex manner. On the one hand the temperature modulation, though small in general, influences the melting and thus the heat flow rate quite well and on the other hand melting of polymers needs time which makes the modulated amplitude strongly frequency dependent (see Sect. 3.3.2). Both effects together with the reversible melting, which is confirmed for a large number of polymers (Wunderlich, 2003), affect the modulated signal and gives rise to the so-called excess heat capacity (see Sect. 3.3.2). In other words, the heat capacity determined from the modulated heat flow rate signal, the apparent heat capacity, is larger than the normal vibrational cp in the melting region and depends on frequency (see Fig. 6.43). From the

20

JK" g" , S ~H, 1 11.5 ~H, ) IS mH,

10 ~ .is mH7

Icpl OS( 10 Kim,"

5

0 90 100 110 120 130 O( 150

tempera t ure ~

Fig.6.43. The magnitude of the apparent heat capacity of nascent ultra-high-molar-mass polyethylene (UHMMPE) from TMDSC runs at different frequencies ({30: 0.1 K min-I, m: ca. 4 mg, WTA: 3.6 mK rad S-I). A common DSC run (dotted; fJo: 10 K min-I, m: 3 mg) and literature values of the amorphous (upper dashed line) and crystalline (lower dashed line) polyethylene (ATHAS, 2002) are added for comparison

240

Fig.6.44. Excess heat capacities (solid curves) calculated with Eq. (3.22) from the measured apparent heat capacity (dotted curves) from quasi-isothermal TMDSC measurements at different temperatures (UHMMPE; mass: 2-3 mg, frequency: 12.5 mHz, tp : 80 s, TA : 53 mK)

Fig.6.45. Activation dia-gram with the time constants of three different processes evaluated from the excess heat capacity curves of Fig. 6.44 via a multi-exponential fit. From the slopes activation energies can be calculated


10

0 0 50 100 150 min 250

time ----1000 0 1<>1

000 (>

[]

00

100 (>

c: E (>(>(> c ....... !-- 10 (><>

[] c:

[]

rP [][]

1 24 25 26 K-' 27

liT 10J

frequency dependence of the excess heat capacity conclusions concerning the time law of the delayed melting [causing the well known "superheating" of polymers (Wunderlich, Czorny, 1977)], can be drawn (Toda, Saruyama, 2001).

If we perform quasi-isothermal temperature-modulated measurements in the melting region it is possible to separate irreversible contributions from reversible excess cp and evaluate the time dependence quantitatively. Figure 6.44 shows as an example the result of such measurements in the case of a certain ultra-high-molar-mass polyethylene. The calculation of excess cp and further evaluation with proper fit functions (e. g., exponential functions) make it possible to separate different irreversible processes with different time constants in the melting region of this polymer (Fig. 6.45). Additional evaluation of the phase angle signal yields information about the exo- or endothermic character of the respective process (see Sect. 3.3.2 case 2).

In summary, it may be stated that DSC and in particular TMDSC offer very powerful tools in polymer science and technology. The method is indispensable

6.9 Purity Determination of Chemicals 241

for characterization of polymers and for quality control. Modern equipment yields reliable results in short time and is often superior to other thermoanalytical methods.

6.9 Purity Determination of Chemicals

The theory of purity determination using DSC is based on the thermodynamics of two-component systems. The simplest and most widely used theory presupposes an eutectic mixture of ideal behavior. For this case we find in textbooks of chemical thermodynamics that the mole fraction x of one component (defined here as the pure one) can be calculated as follows:

1 T i1H(T) lnx=- f --dT

R Ttrs T2

(R, gas constant: 8.31441 J mol-1 K- 1; i1H(T): phase transition enthalpy at temperature T of the pure component in question, Ttrs : phase transition temperature of this completely pure component}.

In the case of a negligible temperature dependence of the phase transition enthalpy, the integral can be solved and we get

lnx = i1H (_1 __ ~) R Ttrs T

or x = exp (i1H (_1 _ ~)) R Ttrs T

This equation serves often to calculate ideal eutectic phase diagrams and the melting behavior of slightly impure materials. If a change is made to the mole fraction of the (eutectic) impurity Ximp = 1 - x, and In (1- Ximp) is approximated (for small values of Ximp) by -Ximp as well as the product Ttrs • Tby Trrs> we obtain:

Ttrs - T Ximp (T) = i1H --2-

R· Ttrs (6.2)

This is the well-known van't Hoff equation which relates the decrease in the melting temperature of the impure component to the amount of impurity involved. In principle, any melting of an eutectic mixture starts at the eutectic temperature and ends at the temperature T which is related to the composition of the mixture via Eq. (6.2) (see Fig. 6.46) and may serve to determine the amount of impurities.

The theoretical heat flow rate curve of eutectic mixtures can be calculated from the phase diagram: an example is shown in Fig. 6.46 for different concentrations. With increasing impurity, the melting peak becomes lower and lower and less sharp. In reality, for small amounts of impurity, the eutectic melting peak and the starting of the main peak are hardly visible on the DSC curve. The real curves measured in a power compensation DSC (Fig. 6.47) differ consider-

242

1 c ...


5

mW

3

2

oL-________ ~==~~~~~~~L_~ 340 345 350 K 355

temperature ..

Fig.6.46. Calculated melting peaks of an ideal eutectic system, normalized to unit area (1: 0.20%; 2: 0.55%; 3: 1.05%; 4: 2.03%; 5: 4.67%; 6: 6.65% impurity)

ably from the theoretical shape because of the smearing due to the finite thermal conductivity of the DSC (cf. Sect. 5.4). As a result the end temperature of the peak cannot be determined with the required accuracy and the concentration of the impurity must therefore be determined in a different way.

The software in commercial DSCs usually starts from the assumption that, during the melting process, the instantaneous mole fraction of the mixture at the temperature T (along the eutectic curve, cf. Fig. 6.41 a) relative to the initial composition is the same as the relation between the heat of fusion used up to that temperature and the total heat:

Ximp (T) = Qr == F (T)

Ximp Qtot

(6.3)

The right-hand side of this equation is the quotient of the partial peak area up to the temperature T and the total peak area if smearing effects are disregarded. For this quotient, the abbreviated form F (T) (relative partial area) is often used. Inserting Eq. (6.3) into Eq. (6.2) furnishes an equation according to which the impurity Ximp can be determined from the slope of a plot of the temperature versus lIF(T) which should be a straight line. In practice this is not the case because of the difficulty in deciding when melting actually starts and the limited heat transfer to the sample. Commercial software generally corrects the measured values in such a way that a straight line is obtained in the plot. This is of course only an approximation to correct for deviations from experimental and

6.9 Purity Determination of Chemicals

f C1J ~

c .... ~ 0

;;:: ~

c C1J

..r:::.

0.6

mW

0.4

OJ

0.2

0.1

0.0 320 330

tempera ture ~

243

2

3

K 360

Fig.6.47. Measured melting peaks of an eutectic system, normalized to unit area (impurities the same as in Fig. 6.46)

theoretical shortcomings (e.g., the smearing effects), nevertheless it is widely used. To avoid larger errors, small sample masses and low heating rates should be used for purity determination. However, the signal will then be noisy and the partial areas cannot therefore be detected very precisely. These problems restrict the method and limit the certainty of the results to an extent which normally is not accepted in certification procedures which are an obligation in pharmacy and food industries.

Attempts have been made to improve the method and to bypass the smearing of the DSC results. Bader et al. (1993) presented a method which determines the impurity from the shape of the measured curve at the start of the melting process, just behind the eutectic point where the heat flow rates are still small and falsification due to smearing low. In principle, the shape of the melting peak is determined from the phase diagram of a mixture of two components and can be calculated (see Fig. 6.46). If the (normalized) heat flow rate equation in question is solved for the purity x as a function of temperature, a straight horizontal line should be obtained which intersects the ordinate at the value of the purity of the sample and drops to 1.0 at the maximum temperature of the peak. The same type of plot can be calculated if we insert the really measured heat flow rates (normalized and with the baseline subtracted, see Fig. 6.47) instead of the


theoretical heat flow rate curve. This should yield a horizontal line at the right purity value as well, but the result does not come up to what had been expected because of the difficulty of precisely determining the baseline and thus the beginning of the peak. It has, however, been shown (Bader et al., 1993) that some minor parameter variations (well below the uncertainty of the measurement) correct this problem satisfactorily. However, the problem, that even this method starts from an ideal eutectic mixture is not solved this way.

Another method to bypass both the thermodynamic and the thermal lag problem of the DSC was suggested by Sarge et al., 1988 and Stosch et al., 1998. They simulate the expected melting curve, using an improved thermodynamic approach (which includes even non-ideal mixing behavior) and a certain mathematical model for the influence for the unavoidable thermal lag of the DSC, and fit this to the measured melting curve with the purity as running parameter. However, even this simulation is not free from thermodynamic model assumptions and the obtained results are thus uncertain and not better than those from the other methods.

In summary, it can be stated that the accuracy of all purity evaluations is largely influenced by:

- The thermodynamic model used (ideal or real mixing, eutectic or non-eutectic behavior). At least the activity coefficients of the impurities are never known and cannot be used for exact calculation of the phase diagram and thus the expected melting curve.

- The smearing of the measured curve, which falsifies the calculated curve in one way or the other from the shape of the peak.

Bearing in mind that most theories of purity determination start from the assumption of eutectic mixtures, which is actually only one special case, the accuracy of every determination of purity by DSC should be considered to be a rather limited approximation to the truth. Therefore it can only be used for relative quality control, e. g., to compare different charges of the same chemical relative to its purity, but never for absolute measurements. The latter is nowadays much more precisely done with HPLC and other modern analytical methods. That's why the importance of the purity determination with DSC recently dropped distinctly. Nevertheless the method has still some importance because it is fast and easy and yields at least approximate values of possible impurities.

7 Evaluation of the Performance of a Differential Scanning Calorimeter

DSC furnishes information on temperature and heat flow rates (or respectively, heat). Whether it is suited to solve the respective problem depends on the efficiency of the instrument. The characteristic data of the DSC which describe the instrument unambiguously must therefore be known. They allow a decision to be taken as to whether the DSC will be suitable for the intended use, and they also make a comparison with other DSCs possible.

A distinction can be made between:

1. the characterization of the complete instrument, 2. the characterization of the measuring system, 3. the characterization of the results of a DSC measurement.

7.1 Characterization of the Complete Instrument

The following serves to characterize the DSC instrument as a whole:

- measuring principle (heat flux or power compensation DSC), - temperature range, - potential heating rates and temperature-time programs, - usable sample volume, - atmosphere (gases which may be used, vacuum, pressure).

7.2 Characterization of the Measuring System

In this section characteristic terms are presented which may be used to describe the efficiency of a DSC measuring system. Instructions how to determine the numerical values are suggested. The characteristic terms in question are the following:

- noise, - repeatability, - linearity, - time constant, - sensitivity, - resolution.

246 7 Evaluation of the Performance of a Differential Scanning Calorimeter

1 W} ~WNfol: ~WMvlRMS m

time

Fig.7.1. The various definitions of noise (according to Hemminger, 1994). pp peak-to-peak noise, p peak noise, RMS root-mean-square noise, ([Jrn measured heat flow rate

- The noise of the measured signal (given, for example, in ]l W) is indicated in different ways (see Fig. 7.1): - as peak-to-peak noise (pp): maximum variation of the measured signal in

relation to the mean signal value, - as peak noise (p): maximum deviation of the measured signal from the

mean signal value, - as root-mean-square noise (RMS): root of the mean value of the squared in-

stantaneous deviations of the measured signal from the mean signal value.

These three definitions of noise are statistically dependent on one another; in general, the following is valid: p = 0.5 pp; for a sine-shaped measurement signal, the RMS noise is equal to 0.35 pp noise; for a statistical random signal, the RMS noise is about 0.25 pp noise.

The noise of the DSC and DTA instrument depends on the heating rate, the temperature and on other parameters (e. g., purge gas). The signal-to-noise ratio is decisive for the smallest heat flow rate detectable ("heat flow rate resolution" or "detection limit" of the DSC). This threshold for the heat flow rate determination amounts to about 2 to 5 times the noise (cf. Wies et aI., 1992).

It is often expedient to indicate the sample volume- or sample mass-related noise (for example in ]lW/cm3 or ]lW/g) which allows the smallest detectable heat flow rate to be estimated for a given sample.

The noise ("short-time noise") can be measured as follows:

- In the desired operating mode (isothermal or scanning mode at a specified heating rate) and at the temperature of interest, the measured signal is amplified to such a degree (most sensitive measuring range) that the noise is clearly recognizable.

The mean variation (pp, p or RMS) of the signal over a period of about 1 min furnishes the respective (short-time) noise (for example in ]lW) (with the amplification factor taken into account, if necessary).

The isothermal noise should be the smallest noise possible (compare the scanning noise of the zeroline). It gives an impression of the disturbances from the environment to which the sensors are subject. It determines the maximum possible signal-to-noise ratio.

7.2 Characterization of the Measuring System 247

As regards the signal to noise ratio (and the resolution of DSCs, see below) a comparative test of 22 different models of DSCs (from 8 manufacturers) using 4,4'-azoxyanisole (which shows a reversible liquid crystal to isotropic liquid transition at ca. 134°C with a small heat of transition of ca. 2 Jg-1) was performed under defined experimental conditions (van Ekeren et aI., 1997). The signal to noise (pp) ratio was determined, the reported values showed a large spreading, even for one distinct DSC model. A real comparison with an understanding of the observed differences between classes of DSCs or even within one class (e. g., power compensated) was not possible. The repeatability for a particular DSC was reported to be good.

- The repeatability indicates the closeness of the agreement between the results of successive measurements of the same measurand carried out under the same conditions of measurement (according to the International Vocabulary of Basic and General Terms in Metrology, 1994; cf. Sect. 7.3).

DSCs can be characterized with the aid of repeatability by measuring a significant DSC quality (e. g., extrapolated peak onset temperature, peak area, course of the zeroline etc.) several times under the same conditions on samples of the same kind. Examples:

- The repeatability of the zeroline is determined by measuring 4 or 5 zero lines over the whole temperature range at medium scanning rate and superimposing the curves. The temperature-dependent range of the deviation from the «mean" zeroline (absolute or in percent) gives the repeatability.

This ± range of scatter gives an impression of the uncertainty of the heat flow rate calibration and is the systematic part of the uncertainty when an absolute heat flow rate must be determined that is related to the zeroline (for example when cp is measured).

- The repeatability of the peak area and of the baseline is determined by measuring for example, the peak area caused by the melting of a pure metal several times under the same conditions. (Either only repeat measurements without moving the sample crucible or with replacing the crucible after each run). The curves with the peaks and the baselines constructed for each peak are superimposed. The ± range related to the mean area shows the repeatability of the peak area (heat of fusion) determination. A separate baseline repeatability can be determined (according to the procedure with the zeroline, see above). The repeatability of the peak area (taking the baseline repeatability into account) furnishes the calibration uncertainty of the peak area (heat) calibration which is the smallest uncertainty for heat measurements.

When the repeatability is given as a ± range (scatter), it must be stated which measure is used: the standard deviation, the maximum deviation or another characteristic value.

Note:

The reproducibility describes the closeness of the agreement between the results of measurements carried out on a sample using different instruments in


---------

t

CD An '-J CD Q) C0 mn --P-t-+~

m ---<>

temperature .. Fig.7.2. Linearity (according to Hemminger, 1994). <Pm measured signal (heat flow rate), Al to A4 peak areas, belonging to sample masses ml to m4 (ml < m2 < m3 < m4). In the partial figure: ratio Anlmn with the range of uncertainty of the area determination, - - - -deviation from ideal linearity (here only for very small (1) and very big (4) sample masses)

different laboratories (round robin, inter-laboratory comparison; cf. Sect. 7.3). The absolute or percentage deviation of the mean value of the results of an instrument from the total mean value is a measure of the reproducibility. Low reproducibility may point to systematic errors of measurement or bias (see below) of certain instruments.

- The linearity of a DSC describes the relation between the measured heat flow rate IPm (signal) and the true heat flow rate IPtrue or between the measured heat Qm (peak area A) and the true heat Qtrue. Linear relations are: IPtrue = K</J . IPm or Qtrue = KQ • Qm.

To determine the linearity means to show the validity of these relations. If the proportionality factor K depends on parameters, e. g., the temperature, the mass of the sample, the heating rate etc the linearity is only given for the parameters in question. In such cases the dependence of K on parameters must be measured and represented graphically. Only with ideal linearity K is a constant. Examples, as the temperature dependence K = K (1') or the dependence of the factor KQon the mass of the sample (KQ= KQ (m)),have been shown in the calibration chapter. Figure 7.2 presents such a linearity test.

Note:

With decreasing mass, the minimum heat of fusion can be assessed which can be distinguished with reasonable accuracy from the noise and the baseline uncertainty.

7.2 Characterization of the Measuring System

The linearity can be tested with regard to, for example,

- the scanning rate, - the sample position in the crucible, - the surface-to-volume ratio of the sample.

249

- The time constant r of the DSC measuring system is the time interval in which - after a sudden (step-like) change of a constant heat flow rate produced at the place of the sample - the measurement signal has reached the e-th part of its final value (i. e., changed 63.2 %). In other words: if a constant heat flow rate, that causes a signal 4im,o, is switched off at the moment to, one time interval dater the signal has decreased to 4im,o/e (i. e. to 26.8 % of its original value or by 63.2 %) (Fig. 7.3). The time constant determines to what degree two successive thermal events (peaks) are recorded clearly separated from each other (time resolution, see "resolution" below). Knowledge of the time constant is important for the desmearing of DSC curves (see Sect. 5.4.).

The time constant of a DSC can be determined as follows (cf. Ulbrich, Cammenga, 1993 and L6blich, 1994): - When an electric calibration heater can be installed in the sample crucible,

4im,o, a constant heat flow rate is switched on until a constant signal is obtained; after switching-off of the heater, the descending curve is evaluated graphically (ace. to Fig. 7.3) or analytically:

4i (t) = 4im,o . exp (- tlr)

d4im (t) 4im (t) =---

dt

so for any time t*, r is determined by the quotient of the measured heat flow rate and its derivative

r= -d4im (t*)/dt

Fig.7.3. Determination of the time constant T (according to Hemminger, 1994). After a constant measurement signal 4>m,o has been reached, the constant heating current is switched off at the time to' After the time interval T, the measurement signal 4>m,o has dropped to 4>m,o/e

., ., . ., .. ., ., . r------,

t

hme


Fig.7.4. Determination of the time constant rwith the melting peak of a pure substance (according to Hemminger, 1994)

t

time

t* should not be too close to the switching-off time to because further, smaller time constants ( T2, T3 etc.) may there be superimposed on the maximum time constant Tmax (see below).

- When no calibration heater can be installed in the crucible, the descending section of a transition peak (when a pure substance melts or crystallizes) is evaluated by the "tangent method" (cf. Fig. 7.4). Several tangents are plotted to the descending section and the time constants T) ••• Tmax are graphically determined from the intersections with the linearly interpolated baseline. Towards the end of the descending curve, a constant value T

(= M between t* and the intersecting tangent on the baseline) results: this is the (greatest) time constant Tmax which describes the thermal inertia of the measuring system in good approximation.

The (usually sudden) solidification of the supercooled melt of a pure substance (e. g., pure tin) causes a pulse-like heat production in the sample. The resulting heat flow rate curve is the so-called apparatus (or Green's) function (see Sects. 3.3.3 and 4.7), this is ideally suitable to determine the apparent time constant of the system with the tangent method.

- The sensitivity of a measuring system is defined as the ratio between the change of the measurement signal and the change of the measured quantity that creates the signal. In DSC systems the measured quantity is the heat flow rate, and the signal output is usually an electric voltage, thus the ratio of f,U

(for example in }l V) and f, rP (for example in m W) yields the sensitivity given in }lV/mW.

- The resolution of a DSC measuring system describes its ability to clearly identify overlapping thermal events (peaks) as separate ones.

Van Ekeren et aI., 1997, suggested to use 4,4'-azoxyanisole to measure the resolution of DSCs. This substance shows two transitions close to one another: a solid

7.3 Characterization of the Results of a Measurement: Uncertainty Determination 251

Fig.7.5. Definition of "resolution" as ratio alb, for details see text (according to van Ekeren et ai., 1997)

1 \ I \ I \ I \ I

temperQ ture ..

to liquid crystal transition at about 117 °C (heat of transition approx. 120 Jg-1)

and a liquid crystal to isotropic liquid transition at approx.134 °C (heat of transition approx. 2 Jg-1). The two transitions are only 17 K apart and differ by a factor of 60 as regards their heats of transition. The degree of separation of the two transition peaks was used to quantify the "resolution" defined as the ratio between the distances baseline - shoulder (a) and baseline - peak maximum (b) (Fig. 7.5). From 22 DSCs of 8 different manufacturers tested with this method, most of the DSCs gave values between 0.10 and 0.35, i. e., a peak that was between 10 and approx. 3 times larger than the distance baseline-shoulder (a). But a few DSCs gave - under the experimental conditions of the test runs - values from 0.70 to 0.90 which makes a separation - and even a detection - of such narrow peaks almost impossible.

7.3 Characterization of the Results of a Measurement: Uncertainty Determination

The quality of measurement results must be evaluated prior to their being interpreted and published or used for further calculations (crystallinity, kinetics etc.). This evaluation is made on the basis of the data characterizing the efficiency of the measuring system (cf. Sect. 7.2).

To make measurements comparable, stating the measurement uncertainty is a must. This is in particular true when the laboratory in question is "certified" internally or by an administrative body. To come to an uncertainty measure different possibilities exist. Methods which use statistical tools applied to interlaboratory comparisons with terms like "random" and "systematic" measurement errors, which are used to describe deviations from the "true value" that is never known. A more recent, metrologically based, access to the assessment of the accuracy of a measurement is described by the "Guide to the Expression of Uncertainty in Measurement" (referred to as GUM, 1995; see Kessel, 2002). GUM specific terms are "standard measurement uncertainty", "uncertainty budget", "probability distribution" and others (cf. Sect. 7.3.2).


The description of the variability of results of measurements is usually separated into three parts. According to the "International Vocabulary of Basic and General Terms in Metrology", 1994, these parts are:

Repeatability characterizes the closeness of the agreement between the results of successive measurements of the same measurand carried out under the same conditions of measurement (cf. Sect. 7.2). Usually this means a multiple use of the same DSC, the same measurement procedure, and the same reference material for calibration by the same observer at the same conditions over a short period of time. Furthermore, it is known that the results of DSC measurements can depend on contact resistances between sample and furnace and the positioning of sample and reference sample inside of the furnace. A realistic determination of the repeatability of DSC-measurements should therefore include a suitable procedure to consider these effects. In most cases the specification of the repeatability is intended as a measure of the variability of DSC measurements within the specific laboratory.

Reproducibility characterizes the closeness of the agreement between the results of measurements of the same measurand carried out under changed conditions of measurement (cf. Sect. 7.2). For DSC measurements these changed conditions may include the measuring instrument, the observer, the method of measurement, the evaluation procedure or software, the calibration standard, conditions of use, or the time. In most cases the specification of a reproducibility is intended as a measure of the variability of DSC measurements between different laboratories.

Accuracy is an often used qualitative term which describes the closeness of the agreement between the result of a measurement and the true value of the measurand. But in all cases the true value of the measurand is not known.

Error is the difference between the result of a measurement and the true value of the measurand. A systematic error is a mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions minus the true value of the measurand.

Bias means a systematic error of the indication (reading) of a measuring instrument. This means in our case a systematic difference of the results of DSC measurements on well known (and stable) materials in comparison to the corresponding reference (literature) values.

Correction is a value added to or subtracted from the uncorrected result of a measurement to compensate for the systematic error. In some cases a correction is carried out by multiplication or division of the uncorrected results of a measurement with a correction factor to compensate for the systematic error.

A further source of confusion is the different meaning of "estimation" in common sense and in statistics. From the statistical point of view the word "estimation" means the application of a well defined mathematical procedure (e.g., the calculation of the standard deviation of the mean of a population) and not the making of a rough guess. Although in many cases experience based uncertainty estimations give reasonable results, a fair comparison of methods and instruments and a credible specification of the uncertainty should be based on a comprehensible (experimental and mathematical) basis.


Table 7.1. Results of an inter-laboratory study (according to ASTM EI269-01)

Material

Diphenyl ether

Linear polyethylene

Indium

Repeatability Reproducibility Bias (95% confidence limit) (95% confidence limit)

}~% +0.95%

-1.1%

+0.8% resp. + 1.8 %a

a Depending on the literature value used.

7.3.1 Black Box Method

The black box method starts from the assumption that the single apparatus (e. g., a DSC) is unknown, as far as the uncertainty of the measurements is concerned. The experimental basis for the specification of uncertainty are inter-laboratory studies on different materials. These inter-laboratory studies are evaluated by means of (classical) statistics including the comparison with reliable literature data. The uncertainty determination with the black box method yields mainly three values: the repeatability, the reproducibility and the bias. The method is rather easy but time consuming and needs a lot of cooperating laboratories to give reliable results, it is applied, e.g., by ASTM.

As an example, the specific heat capacities of diphenyl ether, linear polyethylene and indium at 67 DC were determined within an inter-laboratory study (ASTM EI269-0l) resulting in the relative uncertainty contributions given in Table 7.1.

7.3.2 GUM Method

With the "Guide to the Expression of Uncertainty in Measurement" (GUM, 1995) internationally accepted guidelines for evaluating and expressing the uncertainty of measurement results were published. This new approach for the unification of different uncertainty concepts is based on the principle of maximum (information) entropy and the Bayesian-statistics. It is somewhat complicated, but really helpful to determine the total uncertainty in a well defined and comparable way. In times where international comparability and certification of laboratories becomes more and more common, it is likely the method of the future. The main difference to the above method, used by ASTM, is that the instrument (DSC) is not considered as a black box. Therefore, a basic part of the uncertainty determination according to GUM is the mathematical model used (the evaluation equation with the corrections). It must include all relevant parts which contribute to the uncertainty of the measurement. The procedure to create an uncertainty budget consists of four steps:

1. Development of the models of evaluation for all relevant physical quantities and their relation to each other in mathematical terms. All significant corrections should be identified and applied.


2. Preparation of the input data for the evaluation, including a list of all sources of uncertainty and their contributions.

3. Calculation of the measurement result by means of the model in question and calculation of the uncertainty propagation.

4. Averaging of the results of the measurements and corresponding uncertainties.

As an example the procedure for the determination of the uncertainty of specific heat capacity measurements of a glassy ceramic by means of DSC according to GUM is outlined in the following (the detailed description can be found in Rudtsch,2002).

Considering a typical DSC measurement consisting of the empty (O), calibration sample (Ref) and sample (S) measurement. The specific heat capacity of the sample Cs (at a given temperature) can be calculated by the comparison of the heat flow rates into the sample <Ps and into the calibration sample <PRef according to (cf. Sect. 6.1):

(7.1)

According to this simple model the main sources of uncertainty are the masses of sample ms and calibration sample mRef> the heat flow rates <P of the three experiments and the specific heat capacity of the calibration material CRef.

In the next step additional quantities of influence and the corresponding corrections or uncertainty contributions must be identified and applied. One item is the consideration of the influence of the temperature measurement. Because of the fact that the heat capacities of sample, reference sample and empty system are functions of temperature and that their temperature dependencies differ from each other, a corresponding uncertainty contribution must be considered. One possibility for the inclusion of this effect is the introduction of different heat flow rate corrections L\ <P i for empty, sample and calibration sample measurement. In doing this, the slope of a measurement run and a temperature difference (i. e., the uncertainty of temperature measurement) are used for the determination of L\ <Pi.

In the following discussion it is assumed that the DSC was calibrated (according to Chapter 4) by measuring the phase transition temperatures of different calibration materials as a function of the scanning rate and subsequent extrapolation to zero scanning rate. The resulting uncertainty consists of three parts, the repeatability of the calibration procedure L\Tcalib , the uncertainty of the phase transition temperature of the calibration material L\ T mat and the uncertainty caused by the interpolation between the different phase transition temperatures of the calibration materials L\ Iiin .

Even if a careful temperature calibration of the instrument has been carried out, the reading of the temperature sensor doesn't agree with the sample temperature except for the limit of zero scanning rate. Therefore it is necessary to find a method to determine and correct for the corresponding bias in the temperature measurement during the scanning operation of the instrument. This temperature difference depends on the scanning rate, the thermal resistances between sample and temperature sensor and the thickness and thermophysical


properties of the sample itself (see Sect. 3.1 and 3.3.3). For some types of instruments a determination of the temperature difference between the temperature sensor of the DSC and the mean sample temperature is possible by the measurement of the thermal lag 61iag (Richardson, 1997). The results of the thermal lag analysis have shown that for power compensation DSCs the corresponding temperature difference is typically between 0.5 K and 3 K. Thus the resulting model equation for the sample temperature Ts reads

Ts = T + ~Tcalib + 6Tmat + ~Tlin - 6T1ag (7.2)

In the next step the standard uncertainties u of all input quantities Xi of Eq. (7.2) (e. g., 6 T mat or ~ Tcalib) must be determined. As an example, the standard uncertainty of the temperature calibration u (~Tcalib) can be determined by repeated measurements of the melting temperature (extrapolated peak onset temperature at zero scanning rate). If normal (Gaussian) distribution can be assumed for the measurand the standard deviation of the experiments is a measure of the standard uncertainty u (~Tcalib) and corresponds to a coverage probability of approximately 68 % (coverage factor k = 1). In most cases a standard coverage factor k = 2 should be used. The assigned expanded uncertainty U of a measurement corresponds to a coverage probability of approximately 95%. Table 7.2 shows the resulting uncertainty budget of the temperature measurement at 500°C. As a result, the expanded uncertainty (k = 2) of the temperature measurement is U (Ts) = 1.2 K.

A second additional uncertainty contribution comes from the heat capacity of the calibration sample. If NIST (National Institute of Standards and Technology, Gaithersburg, USA) SRM-720 sapphire as calibration material is used, the uncertainty can be taken from the certification report. But in many cases (noncertified) sapphire from other sources is used. In this case an additional contribution to uncertainty 6cRef is included which takes care of a possible specific

Table 7.2. Uncertainty budget for the temperature determination example

Input quantity a Estimate a Standard Probability Sensitivity Uncertainty Xi Xi uncertainty distribution coefficient contribution

U(Xi) in K oTs Ui(Y) in K

oXi

T 500°C 6 Tma, O.OK 0.1 normal 1.0 0.1 ll.Tcalib 1.9 K 0.3 normal 1.0 0.3 ll.Tjin O.OK 0.5 normal 1.0 0.5 6Tjag 1.0 K 0.2 normal 1.0 -0.2

Ts 500.9°C 0.6

a Input quantities Xi are measurable quantities like the temperature T or the thermal lag 6Tjag. The corresponding quantity in statistics is the estimator. The value of the estimator is the estimate or best estimate of the unknown quantity and corresponds to the result of a measurement (e. g., arithmetic mean).


heat capacity difference between the used calibration material and the NIST SRM -720 sapphire. The modified model equation for the calculation of the specific heat capacity of the sample is then given by

(IPs + ~IPs) - (IPo + ~IPo) (IPRef + ~ IPRef) - (IPo + ~ IPo)

(7.3)

In the next step the standard uncertainties U (Xi) of all input quantities Xi (e. g., mRef> ms, IPs, etc) in Eq. (7.3) must be determined. The corresponding contribution to the standard uncertainty Ui (cs) associated with the output estimate Cs

can be calculated according to

oCs Ui (cs) = oX

i • U (Xi) (7.4)

For uncorrelated input quantities the standard uncertainty of the specific heat capacity (combined uncertainty) is given by

(7.5)

Special investigations have shown that the relative uncertainty of the specific heat measurement between 0 °C and 600°C doesn't depend on temperature. The uncertainty budget for the specific heat capacity measurement has therefore as an example been calculated at a temperature of 250°C. Table 7.3 gives a summary of the results of this analysis.

The results of Table 7.3 show that the major source of uncertainty of DSC measurements are the heat flow rate measurements of sample and calibration sample. As a result, the final relative expanded uncertainty (k = 2) of the specific heat capacity determination is U (cs)/ Cs = 1.5 % with a reported result at 250°C of Cs = (1.015 ± 0.016) J . g-l. K-1 •

Table 7.3. Uncertainty budget for the specific heat capacity determination example

Input Estimate Standard Probability Uncertainty quantity Xj uncertainty distribution contribution Xj u(Xj) Uj (y)/J . g-l . K-1

mRef 55.41 mg 0.006mg rectangular 0.1. 10-3

CRef 1.060 J . g-l . K-1 0.001 J . g-l . K-1 rectangular 1.2 . 10-3

DcRef 0.0 J . g-l . K-1 0.003 J . g-l . K-1 rectangular 2.9.10-3

4>Ref 11.09 mW 0.05mW rectangular 4.6. 10-3

l1~ef -0.009mW 0.003mW rectangular 0.3 . 10-3

ms 52.29 mg 0.006mg rectangular 0.1. 10-3

4>s 1O.13mW 0.05mW rectangular 5.3. 10-3

l14>s -0.01 mW 0.003 mW rectangular 0.3. 10-3

4>0 1.2mW O.lmW rectangular 1.5. 10-3

l14>0 -0.002mW 0.0004mW rectangular 0.004.10-3

7.4 Check List for DSCs 257

7.4 Check List for OSCs

The following check list serves to collect the essential characteristic data of a DSC.1t may be used to

- ask the manufacturers for the values, - establish a guideline when characteristic data are measured in one's own lab-

oratory or in the manufacturer's laboratory, - compare different types of DSCs, - compare the DSC data with the values required to investigate a problem.

Manufacturer:

Type of measuring system:

Special features:

Sample volume (standard crucible):

Atmosphere (vacuum?, which gases? pressure?):

Temperature range:

Scanning rates:

Zeroline repeatability:

Peak area repeatability:

Total uncertainty for heat:

Extrapolated peak onset temperature repeatability:

Total uncertainty for temperature:

Scanning noise (pp) at ..... K min-I:

Isothermal noise (pp):

Time constant with sample:

Additional facilities:

o heat flux disk-type o heat flux turret-type o heat flux cylinder-type o power compensation

..... mm3

from ..... to ..... °C or K

from ... to ... K/min steps: ... K min-I

from ± ..... flW (at ..... °C) to ± ..... flW (at ..... 0c)

± ..... % (at ..... 0c)

± ..... % (at ..... oC)

....... K (at ..... 0c)

± ..... K (at ..... 0c)

from ± ..... flW (at ..... 0c) to ± ..... flW (at ..... 0c)

from ± ..... flW (at ..... 0c) to ± ..... flW (at ..... 0c)

..... s

Appendix 1

Comparison of Heat Flux Differential Scanning Calorimeters and Differential Thermal Analysis Instruments

The measurement signal generated in DTA instruments is the temperature difference AT between sample and reference sample. The measuring system comprises temperature sensors and a suitable holder or support to mechanically fix the temperature sensors and the sample containers. A defined, constant and good thermal contact must be established between temperature sensors and sample containers. Strict repeatability of the measuring system over the whole temperature range must be ensured; the system's properties are reflected in the shape of the zeroline (i. e., curve measured with the measuring system empty or with empty crucibles).

Depending on the desired measurement temperature, sample volume and sensitivity, a choice can be made from among various DTA measuring systems. In the following, the two contrasting basic types - the block-type measuring system and the measuring system with free-standing crucibles - will be described.

DTA: The block-type measuring system

The following is characteristic of this type of measuring system (cf. Fig. ALl):

- The temperature difference AT is measured directly in the sample substances. As a result, the temperature sensors are subject to the attack of the sample and reference sample substance.

- The duration (resolution with time) and the height (sensitivity) of the measurement signal are determined by the thermal resistance between sample substance and block, sample substance and temperature sensor and by the properties of the block material.

- The duration and height of the measurement signal and its shape also depend on the location of the temperature sensor in the sample substance and on the (changing) thermophysical properties of the sample substance. A quantitative evaluation of the heat in question is therefore impossible.

The generally high sensitivity and the rapid, almost instantaneous "response" to sample reactions are advantages of this measuring system.

260 Appendix 1

I1T

s R

2

4

a

r I1T

b

time ...

Fig. A 1.1. a Instrument for Differential Thermal Analysis (DTA) with block-type measuring system, b Schematic measured curve L'1 T (t) (exo up). 1 "block" with cavities to take up sample and reference sample substance, 2 furnace, 3 differential thermocouple, 4 programmer and controller, S sample substance, R reference sample substance, Tp furnace temperature, t time

Comparison of Heat Flux Differential Scanning Calorimeters 261

OTA: Measuring system with free-standing crucibles

This measuring system (Fig. A1.2) is more frequently used in DTA instruments than the block-type measuring system. A characteristic feature is that freestanding sample crucibles are put over the protective tubes of the thermocouples for the purpose of AT measurements. The thermocouple junction is in thermal contact with the bottom of the crucible.

The following is characteristic of this type of measuring system:

- The thermophysical properties of the sample substance affect the height and shape of the measured curve only slightly. The peak shape is to a large extent

2 4

a

t time

6.T

b

Fig. A 1.2. a Instrument for Differential Thermal Analysis (DTA), measuring system with freestanding crucibles, b Schematic measured curve ,iT(t) (endo down). 1 crucible,2 furnace, 3 differential thermocouple, 4 programmer and controller, S sample substance, R reference sample substance, Tp furnace temperature, t time

262 Appendix 1

determined by the containers, this allows a semi-quantitative evaluation of the heat in question from the peak area.

- Depending on the problems to be solved and the mass and structure of the samples, a choice can be made from among a great number of container types (crucibles), among them "poly-plate" crucibles, micro-crucibles worked out from the thermocouple bead, crucibles for "self-generated atmosphere" and others.

- The high flexibility as far as the usability of different furnaces is concerned (temperature ranges), the adjustment of different purge gases (including vacuum) and the usability in instruments for simultaneous thermal analysis.

In other types of DTA instruments, sleeves containing the sample substance are placed on the thermocouple junctions which are either unprotected or covered by thin tubes. With this the attempt is made to have the advantages offered by direct, undelayed temperature measurement, without having to put up with the unfavourable effects of the block material (resolution, sensitivity) or the (isolated) individual crucibles (delayed, "smeared" temperature measurement).

Heat flux OSC

In contrast to the majority of the DTA instruments of rather simple design, DSCs should be capable of being calibrated for heat measurement. An unambiguous and repeatable assignment of the measurement signal proper, ~ T, to the measurand d «P must be ensured (~ «P = «PFS - «PFR , differential heat flow rate from the furnace to the samples). The DSC should be suitable for the investigation of different substances and transitions under different test conditions. This is why the properties of the measuring system may depend as little as possible on the sample properties, the type of transition and on test parameters (e. g., heating rate).

Within certain limits, these requirements can be fulfilled in good approximation when the temperature field of the measuring system does not primarily depend on sample properties but on the properties of the measuring system itself which are given by its design.

This dominance of the measuring system, which is essential for DSCs, follows from the fact that - compared with the sample - the measuring system is made up of solid parts and has defined heat conduction paths. The sample reaction can then be regarded as a small disturbance of the steady-state temperature field established during heating. As this disturbance must be small, the measured signal itself, dT, will also be small. As a consequence, the noise of the whole measuring set-up must be low to allow the sensitive measurement of heat flow rates.

In all DSCs the calibration factor depends on instrument and sample parameters. This leads to systematic error sources, and the uncertainty of the result is, therefore, much larger than the repeatability error of the measurements. Very careful calibration, with the sample parameters adjusted, is, therefore, necessary.

Appendix 2

Calorimetry - a Synopsis

A great variety of calorimeters serve to measure heats and heat capacities in various fields of application. In the following, a classification system for calorimeters and a couple of examples of different types of calorimeters will be presented. The aim is to give a structured survey of the whole field of calorimetry which may help to better recognize and evaluate the advantages of and limitations to the DSCs which result from their mode of operation.

Classification

In a classification system, calorimeters are arranged in groups according to particular characteristics. Various classification systems are reasonable and practicable (cf. Hemminger, H6hne, 1984; Rouquerol, Zielenkiewicz, 1986). It may sometimes be difficult to arrange calorimeters in proper order in a relatively simple system, and such a classification may even be impossible. Classification systems covering the entire field of calorimetry tend to become very sophisticated, end in itself and are rather useless for practical applications. When thermodynamic principles of heat exchange or the aspect of how the caloric signal is formed are taken as a basis for classification, the result will always be that a certain number of calorimeters are characterized satisfactorily whereas additional explanations or auxiliary definitions are required for others. Nevertheless, a classification system is useful to show basic principles of calorimetry. It should be based on existing instruments and open to future developments. In the following, a simple system for practical use will be developed which will suffice to discuss characteristic features and error sources by groups.

Criteria for the classification are the following:

1. The Principle of Measurement 1.1 Measurement of the energy required for compensating the heat to be

measured (heat-compensating calorimeters). 1.2 Measurement of the temperature change of a substance due to the heat to

be measured (heat-accumulating calorimeters). 1.3 Measurement of the heat flow rate between sample and surroundings due

to the heat to be measured (heat-exchanging calorimeters).

264

2. Mode of Operation 2.1 Isothermal 2.2 Isoperibol 2.3 Adiabatic

2.4 Scanning of surroundings 2.5 Isoperibol scanning 2.6 Adiabatic scanning 2.7 Temperature-modulated

3. Construction Principle 3.1 Single calorimeter

}

}

3.2 Twin or differential calorimeter

Appendix 2

static modes

dynamic modes

Most of the calorimeters can be classified by the above-mentioned criteria. All combinations of 1,2 and 3 are, of course, not possible as some of the criteria are incompatible.

Calorimeters which are used today will be presented in the following according to this classification. As the classic calorimeters will be dealt with only briefly, reference will be made to the literature for further information on construction details, error sources, methods for the evaluation of the measured values, and ranges of application of these instruments. The DSCs are treated in closer detail in several sections in this book (see Chapters 2 and 3).

Examples of Colorimeters

The aim of this chapter is to give the reader an idea of the variety of calorimeters offered in addition to DSCs. This will help to better classify the range and potentialities of the DSC methods in comparison with different calorimetric methods.

Heat-Compensating Calorimeters

In the case of this calorimetric method, the effect of the heat to be measured is "suppressed", i.e., temperature changes of the sample or of the calorimeter's measuring system, or temperature differences in the measuring system due to the caloric effect are compensated. For this purpose, an equally high, wellknown amount of energy, with the sign reversed, is added.

Possibilities: Compensation of the heat to be measured with the aid of the "latent heat" of a phase transition (e.g., ice calorimeter) or with electric energy (Joule's heating or Peltier's cooling). The compensation by means of reversible expansion or compression of an ideal gas was described by Ter Minassian, Milliou, 1983.

It is an advantage of all compensating methods that the measurements are carried out under quasi-isothermal conditions and that heat leaks do not, therefore, represent important error sources. Moreover, in the case of electric compensation, no calibrated temperature sensor is required for the measurement but only a sensitive thermometer which controls the compensation power of a controller so that the temperature remains constant.

Appendix 2 265

1st example: "Ice calorimeter"

1. Compensation of the heat to be measured by latent heat 2. Isothermal 3. Single calorimeter

A warm sample placed in an ice calorimeter (Fig. A 2.1) transfers its heat to a O°C ice jacket. As a result, a certain mass of ice (to be determined) melts. In the case of the ice calorimeter according to Lavoisier, de Laplace, 1784, the melted water was weighed, whereas Bunsen, 1870, determined the change of the ice/water ratio on the basis of the change in the volume of the whole mixture (density difference of ice and water). The measurements which Ginnings and Corruccini carried out at the National Bureau of Standards (NBS) during the late forties of the last century were counted among the important applications of an ice calorimeter. They measured heat with an uncertainty of about 0.02 %, the temperature of the sample at the moment of its being dropped into the calorimeter lying between 100°C and 600 0c,

The liquid phase to gaseous phase transition was also made use of ("boil-off" calorimeter), in particular since in this case the difference between the density of both phases - and thus the sensitivity - is by two or three orders of magnitude higher than in solid to liquid transitions.

Phase transition calorimeters are relatively simple to construct and allow precise measurements to be performed. A disadvantage is that measurements can be carried out only at one temperature, i. e., at the temperature of transition of the respective calorimeter substance.

The quantity to be determined is the transformed mass of calorimeter substance (ice, liquid); the heat of transition must be known. (In general, the calorimeter is calibrated electrically.)

2nd example: "Dissolution calorimeter"

1. Compensation of the heat to be measured with the aid of electric energy 2. Isothermal 3. Single calorimeter

This calorimeter serves to measure heats of solution. A container fIlled with water is equipped with a stirrer, a controllable electric heater and a sensitive thermometer (Fig. A2.2). At constant temperature, an endothermically solving salt is added. The heater is adjusted so that the temperature of the liquid remains constant. The supplied electric energy is then equal to the heat of solution of the salt (Bronsted, 1906).

Expressed more generally, the following is valid: In calorimeters of this type, the heat to be measured is compensated by Joule's heat or with the aid of the Peltier effect. This is done in that a sensitive thermometer activates the compensation control circuit so that, if possible, no tem-

266

a

t x

b

o o o o o o

Appendix 2

1--..... [> X

7

time ... Fig. A2.1. a Ice calorimeter (Bunsen, 1870), b Measured curve (according to Hemminger, 1994).1 sample, 2 sample container (receiver), 3 ice, 4 water,S ice-water mixture, 6 mercury, 7 capillary tube, x position of the mercury meniscus. The displacement Lix* of the meniscus is proportional to the heat Q exchanged with the sample (positive for an endothermic effect). t.x* is determined taking the "pre-period" and the "post-period" into account

Appendix 2 267

3

v 5

2

T

a

1 ~l

b

time

Fig. A2.2. a "Compensation calorimeter" (Bronsted, 1906), b Measured curve (according to Hemminger, 1994). 1 sample (salt), 2 water, 3 stirrer, 4 electric heater, 5 temperature sensor, 6 controller, Pel electric heating power. The area below the measured curve Pel (t) corresponds to the compensation heat and (at constant temperature) also to the endothermic heat of solution Q of the sample substance, which has been searched:

t2

Q = J Pel (t) dt t,

268 Appendix 2

perature change due to reaction heat takes place. As the Peltier power for the compensation of exothermic effects cannot be measured with sufficient accuracy, the calorimeter substance (e.g., water) is generally cooled with constant Peltier power and at the same time heated with equally high Joule's heat (controlled). To compensate an endothermic effect, the heating power is increased in order to keep the temperature of the calorimeter substance constant. When an exothermic effect is to be compensated, the heating power is decreased with the Peltier power remaining unchanged (see, for example, Christensen et al., 1968). These calorimeters do not attain the strictly isothermal state of phase transition calorimeters, as the difference between actual and set temperature value must be non-zero in order that the electrical power control is activated. In addition, temperature control is delayed by the heat transfer processes, and it is very difficult to obtain spatially homogeneous temperature fields by electric heating.

The advantages of these quasi-isothermal calorimeters consist in the simple and very precise measurement of the electric compensation energy and the possibility of using highly sensitive sensors to measure temperature changes; these sensors must not, however, be calibrated.

This is done in that a sensitive thermometer activates the compensation control circuit so that, if possible, no temperature change due to reaction heat takes plase.

3rd example: "Adiabatic scanning calorimeter"

1. Compensation of the heat to be measured with the aid of electric energy 2. Adiabatic scanning 3. Single calorimeter

In these calorimeters (Fig. A2.3), the temperature program is preset. So much electrical heating power is supplied to the sample as is necessary to comply with the given temperature program. (In practical application, the electrical heating power required for sample heating is often preset and the resulting heating rate measured.) Heat losses are minimized by adapting the temperature of the surroundings as well as possible to the temperature of the sample (or sample container) (adiabacy). Calorimeters of this type allow the heat capacity to be measured with high accuracy (uncertainty :S0.1 %) (cf. 4th example of heat-accumulating calorimeters).

The following is valid for the heat capacity C(T):

or

C (T) = Pel, 1 - Pel,2

dT

dt

C(T) = P" (( ~~f (~:)J

(dT/dt: preset)

Appendix 2 269

5 T

2

3

4

6. T = 0 6

t f!1

~ ____ ---CD ---time ...

Fig. A2.3. a" Adiabatic scanning calorimeter". 1 sample, 2 sample furnace, 3 heatable (adiabatic) shield, 4 temperature difference sensor,S temperature sensor, 6 programmer and controller, Pel electrical heating power. When there is a variation of the temperature with time T(t), Pel (t) is controlled so that (dT/dth = (dT/dth (outside the peak). b Measured curve (according to Hemminger, 1994). CD curve measured with the calorimeter empty, @ curve measured with the sample placed in the calorimeter, sample transition between tJ and t2 (peak)

270 Appendix 2

(1: run with the calorimeter empty, 2: run with the sample placed in the calorimeter, Pel = i· U electrical heating power, dTldt heating rate at the respective moment, C(T) comprises the heat capacities of sample and container. The heat capacity of the empty container is determined by separate measurement).

When in a 1st order transition (e.g., melting of a pure metal) the sample temperature remains constant during transition in spite of the fact that heating power is continued to be supplied, the heat of transition is directly determined from the integral of the heating power over the transition time (Qtrs = Ii (t) U (t) dt).

The adiabatic shield guarantees quasi-isothermal conditions while the heat to be determined is compensated with the aid of electric energy.

Calorimeters of this type are used for the accurate, absolute and direct measurement of heat capacities and heats of transition (cf., for example, Nolting, 1985; Kagan, 1984; an extension to a low-temperature system has been described by Rahm, Gmelin, 1992).

4th example: "Power compensation DSC"

1. Compensation of the heat to be measured with the aid of electric energy 2. Isoperibol scanning 3. Twin calorimeter

The temperature of the sample surroundings remains constant (isoperibol). The calorimeter (see Figs. 2.5, 2.6) comprises two identical measuring systems (twin principle), one containing the sample, the other the reference sample. The temperature difference between the two systems is measured. In a 1st approximation, disturbances from the surroundings have the same effect on both measuring systems and therefore cancel out with respect to the temperature difference. The individual sample supports (microfurnaces) are heated separately so that they comply with the given temperature-time program. When there is ideal thermal symmetry between the two measuring systems, the same heating power is required for sample and reference sample. When additional heat is released or consumed during sample transition (exothermic or endothermic process), the sample's heating power is regulated by means of a proportional controller so that the electric heat supplied is decreased or increased by just the amount as has been generated or consumed during the exothermic or endothermic transition process. The measured signal is the temperature difference I1T (deviation from the set value) to which the compensation heating power I1P is proportional: I1P = k . 11 T. Calorimeters of this type ("power compensation DSC" or "differential power compensation scanning calorimeter, DPSC" or "dynamic power difference calorimeter") are widely used (cf., for example, Watson et al., 1964; Hemminger, Hohne, 1984). They are discussed at full length in Sects. 2.2 and 3.2 of this book.

Appendix 2 271

Heat-Accumulating Calorimeters

In the case of this calorimetric method, the effect of the heat to be measured is not "suppressed" by compensation but leads to a temperature change in the sample substance and a "calorimeter substance" with which the heat to be determined is exchanged. This temperature change is measured. When the change is not too large, it is proportional to the amount of heat exchanged. The proportionality factor must be determined by calibration with a known amount of heat.

5th example: "Drop calorimeter"

1. Measurement of the temperature change of a substance due to the heat to be measured

2. Isoperibol 3. Single calorimeter

The temperature of the surroundings is kept constant with the aid of a thermostat ("isoperibol": uniform surroundings). The heat Q to be measured is exchanged with the "calorimeter substance" and the temperature change !:.T(t) is measured (Fig. A 2.4).

The following is valid: Q = Ccal • !:.T* (cf. Fig. A2.4c). The proportionality factor Ccal is the heat capacity of the calorimeter substance

(the liquid in Fig. A2.4a) plus that of the other calorimeter components (stirrer, thermometer, etc.), which cannot be exactly defined. This factor is determined by calibration with electric energy (Joule's heat). As soon as there is a temperature difference between calorimeter substance and surroundings, heat is exchanged. This exchange must depend only on this temperature difference and must be reduced as far as possible by appropriate measures (Dewar vessel, radiation shields ... ). Otherwise, it cannot be determined by calibration and represents an error source. In order to guarantee strict repeatability of the heat exchange,!:. T must amount to only a few Kelvin. Since the temperature change of the calorimeter substance and the unavoidable heat exchange with the surroundings take place simultaneously, the temperature difference!:. T* used to calculate Q must be determined according to defined rules [e.g., International Standard ISO 1928-1976 (E); Rossini, 1956; Gunn, 1971; Oetting, 1970; Sunner, Mansson, 1979] from the shape of the !:.T(t) curve measured before and after the sample has been placed into the calorimeter (cf. also 2nd example). Simple drop calorimeters serve to measure mean heat capacities (temperature of the sample at the moment of its being placed into the calorimeter: up to about 500°C; sample mass between 10 and 100 g).

In "aneroid" drop calorimeters (Fig. A2.4b), the calorimeter substance is a solid body of good thermal conductivity. The advantage over calorimeters filled with liquid consists in that samples at high temperature (up to about 2000°C) can be dropped without the risk of evaporation or splashing.

Other examples of calorimeters of this class are instruments in which the sample placed into it (solid body, liquid, gas) reacts with the calorimeter substance; this results in heat of reaction being released.

272 Appendix 2

-3

T '~ 4

T

.;------;-Ir----. ~ c::=J I'--_~

__ L _

a b

I

1 (I

L1 r" I r I

c

time ~

Fig. A 2.4. "Drop calorimeter" in isoperibol mode of operation a with liquid, b with aneroid calorimeter substance and c measured curve (according to Hemminger, 1994). 1 sample, 2 calorimeter substance: (a) liquid, (b) solid, 3 temperature sensor, 4 stirrer, 5 radiation shields. The temperature change L'lT* of the calorimeter substance has to be determined from the measured curve according to defined rules, taking the "pre-period" and the "post-period" into account

Q = Cca1 . L'l T* is valid.

Ccal calibration factor ("heat capacity" of the calorimeter substance and of other calorimeter components)

Appendix 2 273

1 T

b

time ~

Fig. A2.S. a Adiabatic bomb calorimeter, b Measured curve (according to Hemminger, 1994). 1 sample in combustion pan, 2 vessel, 3 ignition device (electrodes with heating filament), 4 calorimeter substance (water), 5 adiabatic jacket, 6 stirrer, 7 temperature difference sensor, S temperature sensor, 9 controller

274 Appendix 2

6th example: ''Adiabatic bomb calorimeter"


2. Adiabatic 3. Single calorimeter

In the case of the adiabatic bomb calorimeter (Fig. A2.5), the "combustion bomb"which is (usually) filled with oxygen at high pressure immerses in water (calorimeter substance). The water temperature is continually measured before and after the electric ignition. With the aid of a controller, the temperature of the surroundings is always adapted to this temperature (adiabatic jacket). The temperature change with time in the surroundings serves as measurement signal; it is determined from the drift of the measured curve before the ignition (pre-period) and after the ignition (post-period) (as with the "drop calorimeter", see 5th example).

Bomb calorimeters of this type (usually automated) are widely used to measure the calorific value of solids or liquids under standardized conditions. In general, they are calibrated with benzoic acid. Their uncertainty of measurement lies in the per mil range. (There are also "dry" bomb calorimeters, in which the temperature change of the combustion vessel itself (of the bomb) is measured; bomb calorimeters with isoperibol surroundings are also used.) For a detailed representation of bomb calorimetry, see Rossini, 1956; Skinner, 1962; Sunner, Mansson, 1979.

7th example: "Flow calorimeter"


2. Isoperibol 3. Single calorimeter (also designed as "twin")

In the case of the so-called "gas calorimeters", the heat to be measured is transferred, if possible completely, to a flowing medium (Fig. A2.6). The temperature difference between the medium flowing in and the medium flowing out is proportional to the heat transferred. Calorimeters of this type are used to determine the calorific value of fuel gases; they are calibrated with gases of known calorific value (e. g., methane) so that the specific heat capacity of the heat -conveying medium must not be known. If an electric heater is used instead of the burner, the specific heat capacity of the heat-conveying medium can in principle be measured as well (cf. Hemminger, 1988).

In biology, biochemistry and chemistry, flow calorimeters with liquids serve to measure the heat development of microorganisms in certain nutrient solutions, or they are used to measure reaction heats. Two reacting solutions are, for example, mixed in a reaction tube (Fig. A2.7). The uniform temperature of the

Appendix 2 275

T = const.

2 "--~-4

3 5

5 T,

3 I

----~@J~ ------

Fig. A2.6. "Gas calorimeter". 1 burner in which the gas to be measured is burnt, 2 heat exchanger, 3 "heat conveying medium" (e.g., air or water), 4 combustion gases, 5 temperature sensor. Inside the heat exchanger, the hot combustion gases convey their heat to the heat conveying medium whose temperature increase T2 - TJ is measured. The "calorific value" (the combustion heat) is proportional to T2 - TJ

reacting agents is measured before they are mixed and then at a point at which the reaction in the flowing liquid has come to an end. The temperature difference is a measure of the reaction heat. The calorimeter must be calibrated, either with the aid of known reaction heats of liquids or with an electric heater installed in the reaction tube. A sophisticated example of a flow calorimeter with stimulated cardiac muscle developing heat in the perfused tube has been described by Daut et aI., 1991. [Flow calorimeters are also designed as twin calorimeters to cancel out the influence of the isoperibol surroundings (heat leaks) ].

8th example: "Adiabatic calorimeter"

1. Measurement of the temperature change of a substance due to the supply of a known amount of heat

2. Adiabatic 3. Single calorimeter

This type of calorimeter (Fig. A2.8) is not designed to measure an unknown heat; instead, a well-known, electrically generated heat Q = Wei serves to change the sample temperature by ~ T. The temperature of the surroundings (adiabatic jacket) is adapted to the measurement temperature with high accuracy in order to avoid any heat exchange with the surroundings. Calorimeters of this type are used to determine the phenomenological coefficient of the heat supplied, Q, and the temperature change, ~T, of a substance (at constant pressure): the heat capacity Cp (n. The following is valid: Cp = Q/ ~ T.

276

4

a

1

j

b

I

1 ...... --I

time

enter of readand

Appendix 2

Fig.A2.7. a "Flow-mix calorimeter", b Measured curve. 1,2 reactants, 3 reaction product, 4 temperature sensor. As in the case of the "drop calorimeter", the temperature change llT* required to determine the mixing or reaction heat is obtained from the pre- and post-period (affected by a drift) of the measured curve (T2 - T1) (t)

Appendix 2 277

t T

.--.---- switch on P"

b

time

Fig. A2.8. a Adiabatic calorimeter, b Measured curve (according to Hemminger, 1994). 1 sampie, 2 heatable sample container, 3 adiabatic jacket, 4 controller,S programmer and controller, Pol electric heating power. When the constant measurement temperature T is adjusted, the electric heating energy WeI is supplied to the sample, increasing the sample temperature by 6 T*

278 Appendix 2

Calorimeters of this type allow the specific heat capacity and latent heats to be measured with the greatest possible accuracy (Gmelin, Rodhammer, 1981; Jakobi et al., 1993).

Knowledge of the heat capacity and its temperature dependence is of utmost importance in solid state physics and thermodynamics. In practical application, I1T is kept as small as possible in order to determine the temperature dependence of Cp as precisely as possible and to avoid errors due to inhomogeneous temperature fields. The limits are determined by the noise and the uncertainty of measurement of the temperature sensors (cf. Kagan, 1984; Zhiying, 1986).

Heat-Exchanging Calorimeters

In calorimeters which measure the temperature change of the calorimeter substance, i. e., heat -accumulating calorimeters (e. g., drop calorimeters, bomb calorimeters), the heat exchange with the isoperibol surroundings is kept low to make the measured signal 11 T as great as possible. In the calorimeters referred to in this section, a defined exchange of the heat to be measured with the surroundings is deliberately aimed at, the reason for this being that the measured signal which describes the intensity of the exchange is then proportional to a heat flow rate ~ and not to a heat. This allows time dependences of a transition to be observed on the basis of the ~ (t) curve (the power compensation DSC also offers this possibility).

The twin design allows disturbances from the surroundings, which affect both systems in the same way, to be eliminated by taking only the difference between the individual measurement signals into account (Differential Scanning Calorimeter: DSC).

9th Example: "Heat flux differential scanning calorimeter"

1. Measurement of the exchange of the heat to be measured between sample and surroundings via a heat flow rate

2. Scanning of surroundings 3. Twin calorimeter

DSC with Disk-Type Measuring System (see Sect. 2. 7. 7)

A metal, ceramic or quartz glass disk with the sample and the reference sample (or the pans) positioned on it symmetrical to the center, is placed into a furnace (Fig. 2.1a). Heat exchange between furnace ("surroundings") and samples takes place by heat conduction, radiation and convection. Strict repeatability of this heat exchange as a function of the temperature (with the atmosphere remaining unchanged) must be ensured. This is why a solid heat-conducting disk is used which guarantees that the properties of the measuring system dominate. As a result, the different characteristics of the individual samples contribute less strongly to the kind of heat exchange than in the case of the DTA (the measurement signal itself must, of course, reflect the sample properties). The signal I1T is measured on the solid heat conductor (disk) between the supports for sample and reference sample (I1T = Ts - TR).

Appendix 2 279

Note: The disk-type DSC is not really a differential (twin) calorimeter, as there is an influence of sample temperature changes on the reference temperature via the center part of the solid disk (cross-talk).

DSC with Turret-Type Measuring System (see Sect. 2.1.2)

In this variant of the heat flux calorimeters the turret like sample and reference supports are soldered on the bottom plate of a silver furnace. This way the disadvantage of thermal coupling of sample and reference sample is get rid off and we have a real differential calorimeter (twin). The function principle is otherwise similar to the disk-type DSC (see above).

DSC with Cylinder-Type Measuring System (see Sect. 2.1.3)

In another type of heat flux DSC, the two cylindrical containers for sample and reference sample are connected with the furnace ("surroundings") by one thermopile each (Fig. 2.3). The heat from the furnace to the samples preferably flows through the thermocouple wires which are at the same time the dominant heat conduction paths and the temperature difference sensors. When a differential connection is provided between the outputs of both thermopiles, the measured signal (ilT) is proportional to the difference between the heat flow rates from the furnace to the sample (ilTps - <Pps) and from the furnace to the reference sample (ilTpR - <PPR).

Survey of the Classification of Calorimeters

PRINCIPLE Of MEASUREMENT

Measurement of the energy required to compensate the heat to be measured (heat-compensating calorimeters)

Measurement of the temperature change of the calorimeter substance measured (heataccumulating calorimeters)

Measurement of the heat flow rate between calorimeter substance and surroundings due to the heat to be measured (heatexchanging calorimeters)

MODE OF OPERATION

Static: isothermal isoperibol adiabatic

Dynamic: scanning of surroundings isoperibol scanning adiabatic scanning temperature-modulated

CONSTRUCTION

Single calorimeters

Twin (differential) calorimeters

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Subject Index

A AC calorimetry 7 accuracy 6,252 acetamide 223 acetaminophen 223 acetanilide 230 acrylate 192 activation energy 171,184-187 activation entropy 171 activation by radiation 180,192-195 adiabatic - bomb calorimeter 274 - calorimeter 4, 268 - DSC 13,24 - scanning calorimeter 268 amplitude (TMDSC) - temperature 51,60 - heat flow rate 53 - thermal wave 58 - transfer function 59 analogue electric circuit (see network) 32,

39,42,58 analytical equipment 5 aneroid drop calorimeter 271 aniline 176 annealing 202,217,228,233,235 apparent heat capacity (see heat capacity)

141 apparatus function 131,132,181,191,250 applications, DSC 147-244 Arrhenius equation 171,196 ASTM 75,87 autocatalysis, (autocatalytic) 170,172,176,

178,183,192,196 Avrami (Erofeev) equation 170 azobenzene 189

B baseline 42,116,117,121-124,162,227,

237 - changes 124 - construction 121-124

- reactions 162,174,177,179,181,189,195 - uncertainty 121,123 benzil 230 benzyl dimethyl ketal 180 bias 252 bioavailability 223 biochemistry 219,220 biological systems 6 Biot-Fourier equation 33 bisphenol-A-diglycidylether 176 Bode plot 59,62 bomb calorimeter 3,274 Borchardt & Daniels method 184 butter 219

C cacao butter 219 calcium oxalate monohydrate 227 calcium stearate monohydrate 228 calibrant 3 calibration 12,45,65-114 - caloric 86-92, - cooling 84-86, 97 - electrical 88, 91 - factor 12,19,45-47,86,93-97 - function 58,76,80,89-96 - heat 90-92 - heat flow rate 87-90 - influences 74,95,96 - magnitude 113 - peak area 90-92,107,108 - phase 113 - procedure 69,70,86,88,92 - substances 6, 98-107 - symmetry check 85 - temperature 69-75,84-86 - thermodynamic aspects 66 - TMDSC mode 107,110-114 caloric calibration 86-92, calorimeter 1,263 - classification 263 - examples 264-279

292

calorimeter - heat accumulating 271 - heat compensating 264 - heat exchanging 278 - types 3,263,264 calorimetry 1,3,263 - applications 3 - classification 263 - synopsis 263-279 Calvet type DSC 14-17 calcium - oxalate monohydrate 227 - stearate monohydrate 228 catalyst (catalytic) 169,170,173,176,183,

188 certified reference material 65 characteristic temperatures 70,76,117 characteristic terms 115-117 characterization - complete instrument 245 - materials 3,219-228 - measuring system 245 - results of a measurement 251 checklist DSC 257 cholesteryl myristate 224 classification of calorimeters 263 combined DSC 25 combustion calorimeter 274 comparison - manufacturers 257 - measurements 245 - types of DSC 259 compatibility of substance and crucible

102 compensating heating power 17-20 compensation calorimeter 264-270 complex heat capacity (see heat capacity)

56,60 conductance (conductivity), thermal 33,

45,57,134,152 construction principle 264 controller of power compensation 18 convection heat transfer 45 convolution integral 56, 132 convolution product 57,132 cooling mode 84, 97 copper 105 correction 71,252 - of errors 252 coupled apparatus 5 cp (see heat capacity) criteria of classification 263 cross-talk, thermal 42 crucible 10, 38, 102, 150

cryostate 6 crystal size 227 crystallinity, degree of 236 curve (see DSC curve) cyclopentadiene 191 cylinder-type DSC 14-17 - advantages 16 - noise 17

D

Subject Index

decomposition 5,168,227,235 deconvolution 126,132,181 deformation calorimetry 7 deformation energy 68 degree of reaction 122,164,169,175,

182,185,189,194,198,214 degree of crystallinity 236-238 dehydration 227,228 desmearing 126-140 - advanced 133-138 - cp-measurements 152,154 - heat flow rate 128 - measured curve 126 - numerical methods 130-133 - reaction curve 181,188,194 - temperature 127,128 - Tg-measurements 203,210 detection limit 246 devitrification 56,195,198,202,239 differential measurement (method) 9 Differential Scanning Calorimetry

(see DSC) 1 differential temperature 12 Differential Thermal Analysis (DTA)

2,259 Differential Thermometry (DT) 2 diffusivity, thermal 60 2,2-dimethyl-l ,3-dioxolan-4-yl)methyl

methacrylate 192 dimyristoylphosphatidylcholine 220 3,6-dioxaoctane-l,8-dithiol 176 disk-type DSC 10-13,278 - noise 12 - uncertainty 12, 13 drop calorimeter 3,271 DSC 1,7 - actual problems 5,6 - applications 4,147-244 - calibration 65-114 - characterization 245 - checklist 257 - combined types 25 - fluids 24 - heat flux 10-13,262,278

Subject Index

- light-modulated 24 - linearity 50, 130,248 - modulated (see TMDSC) 27 - noise 12,246 - oscillating (see TMDSC) - performance 245 - power compensation 17-25,48-50,

270 - pressure 13 - temperature-modulated 50-63,159,218,

238 - theory 31-63 - triple system 13 - types of 9-25 - uncertainty 12,13 DSC curve 115-146 - calculated 115 - check 116 - measured 115-119 - characteristic terms 116, 117 - desmearing 126-140 - evaluation 119-121 - influences 118, 119 - kinetic evaluation 168 - shape 118 DTA 2,259 - applications 4, 5 dual sample DSC 13 dynamic mode (see scanning mode) 26

E electric calibration 88, 91 electric network 32,39,42 Ellerstein method 185 endothermic event 55 energy (see internal energy) enthalpy 66,67,121,162,160-162 - determination 124-126 - reaction 124, 125, 162 - transition 67, 124, 125,221,241 enthalpy relaxation 202,212,216 enthalpy-temperature diagram 166,206 entropy 121,125,223,232 epoxide 163,175,176,183,195 equivalent electric circuit 32,39,42 errors 252 - systematic 252 eutectic system 134, 230,241 events, puIs-like 130 examples of calibration 77,77,86,92 examples of calorimeters 263 - heat accumulating 271 - heat compensating 264 - heat exchanging 278

293

excess - heat flow rate 54 - quantities 232 exothermic events 20,36,55,74 extent of reaction 162,164,168 extrapolated peak completion temperature

70,117 extrapolated peak onset temperature

70-72,117 Eyring equation 171

F fictive temperature 205 final peak temperature 70, 117 finite-element method 44 fixed point 69,100 flow calorimeter 4,274 flow-mix calorimeter 274 food 219 formal kinetic 173,187,195 Fourier - analysis 53 - integral 51,132 - series 51 - space 57 Fourier transform 57,58,132 - inverse 57,132 free energy (see Gibbs free energy) Freeman & Carrol method 185 frequency of modulation 51 fundamentals of DSC 31

G gas calorimeter 3,274 Gibbs free energy 121 glass transition 57,161,195,200-219,

234 - characterization of glasses 213 - during polymerization 163,195,214 - nature 201 - modeling of relaxation phenomena

212 - phenomenology 200 - relaxation peaks 204, 216 glass transition temperature 200 - applications 212 - definition 203 - influence of molar masses 213 - of copolymers 215,216 - of crosslinked polymers 215 - of semicrystalline polymers 216 - thermodynamic (fictive) 206,207 Green's function 131,136,138,181,250 GUM method 253

294

H harmonics 51 heat 1 - latent 54 - measured 86,91 - true 90 heat accumulating calorimeters 271 heat capacity (Cp) 34,53,67,68,88, 120,

147-160 - apparent 55,57,141,142,145 - change 123,200,201 - complex 56,60,138,143-145 - copper 105 - imaginary part 56, 144 - magnitude 56, 143 - non-reversing 142 - phase 56,143 - real part 56, 144 - reversing 141 - sapphire 103, 104 - time-dependent 56,57 - vibrational 53,56 heat capacity (Cp) measurement 6, 147-160 - classic procedure 148 - dual step method 153 - small step method 155 - sources of error 150 - step scan method 158 - with TMDSC 159 heat compensating calorimeters 264 heat convection 45 heat-exchanging calorimeters 278 heat flow 1 heat flow rate 12-20,19,52,31,34 - amplitude 53 - calibration 87-90,87-90 - excess 54 - measured 86-90 - non-reversing 142 - periodic part 53,55 - phase 53,55 - resolution (see noise) 246 - reversing 142 - true 12,15,19,47,87-90,119,129-133 - underlying part 53, 55 heat flux DSC 9-17, 31-48, 278 - cylinder-type measuring system 14-17 - disk-type measuring system 9,10-13,

31-48 - dual sample 13, - equivalent electric circuit 32,39,42 - models 31-48 - modulated heating mode 27-30 - numerical simulation 44-48

Subject Index

- turret-type measuring system 13,14 - theory 31-48 - triple measuring system 13 heating-cooling mode 28,52 heating-iso mode 28,52 heating-only mode 28, 52 heating rate 12,52 heat of crystallization 234 heat of fusion 234 heat of reaction 162-168 heat of dissolution 265 heat of transition 44,67,107,219 heat pulse 36,43,130,131 heat radiation 45 heat transfer 57 heterogeneous system 168, 170, 171, 215,

217 high-energetic materials 233 high-pressure calorimeter 7,22 high-speed mode 27 high-temperature calorimeter 7 homogeneous system 168,170,171,215,

217 hybrid system 25

ice calorimeter 265 ICTAC 6,65 impurity 241-244 indium 77 -80 initial peak temperature 117 internal energy 67 International Temperature Scale (ITS) 69,

99,100 interpolated baseline 117

interpretation of results 6, 57, 145 ISO 3,66,99 isoconversional method 185 isokinetic baseline 145 isomalt 219 isoperibol (constant surrounding

temperature) 264 isothermal - mode 26,159,162,175,180 - reaction curves 175 isothermals 148 ITS-90 (see International Temperature

Scale)

K kinetic analysis 162, 168 kinetic evaluation 168 - requirements 173 kinetic investigations 168

Subject Index

- calculation methods 184-188 - data sampling 173 - examples 189-200 - formal kinetic model 187,195 - heterogeneous phase 169,170,171,184 - homogeneous phase 168,170,171 - prediction 172, 183, 189 - rate laws 168-173 - reaction mechanism 172,176,183,186,

188 - sample preparation 173 - strategies 183-189 kinetics 168 kinetic triplet 172 Kirchhoff-equation 161,162,166,232

L lag, thermal (see thermal lag) lamellae crystal 227,238 latent heat 124-126 lead 77-80 light 23 light -activated reactions 23, 180, 192 light-modulated DSC 24 linear response 50, 56, 58, 130 linearity 50,63, 130, 248 liposome 220 liquid-crystalline mesophases 224 liquid crystal 86,109,224 - temperature calibration 109 low-temperature calorimetry 6

M measured curve (see DSC curve) measurement 251 - interpretation 57,145 - principle 263 measuring system 259 mega-calorimetry 3 mesophase transitions 224 metallography 230 metastable phases 219 methacrylates 193 metrology 9 mineralogy 230 mixing calorimeter 3 mode of operation 25, 264 - heating-cooling 28,52 - heating-iso 28,52 - heating-only 28,52 - high speed scanning 27 - isothermal 26,162,175,189 - quasi-isothermal 27,52 - sawtooth 29

295

- scanning 26, - step-scan 30 - temperature-modulated 27-30,50-63 modification (see polymorphism) 219 modulated DSC (see temperature-

modulated DSC) modulated temperature (see temperature-

modulated) modulation of temperature - frequency 51 - amplitude 51 - period 53 - phase 53,

N nematic phase 228 network model 32,39,42,58,59,61 nitroaniline 233 noise 12,17,19,246 non-equilibrium state 56 non-reversing curves 142 numerical methods 44, 130 numerical simulation 44

o 3-omega (3w) calorimetry 7 operation mode 25-30,264 order ofreaction 169 oscillating DSC (see TMDSC) overall rate laws 170,172,183,193 Ozawa-Flynn-Wall method 186

P partial peak area 38 particle energy 6 peak 34,116,117 - area 42,90,124-126 - definition 116 - characteristic temperatures 70,117 - heating rate dependence 44 - maximum 44,117 - overlapping 234 - partial integration 38 - repeatability 247 - shape 44-46,48,134 - temperature maximum 117 peak area calibration 90-92, 108 performance of DSC 245 period (time) 29,53 periodic - part of signal 53,55 - temperature change 51 PET 142,144 pharmaceutical applications 221

296

phase (TMDSC) 53,55 - thermal wave 58 - transfer function 59 phase behavior 219,241 - acetamide 223 - calcium oxalate monohydrate 227 - calcium stearate monohydrate 228 - cholesteryl myristate 224 - dimyristoylphosphatidylcholine 220 - liquid crystalline mesophases 224 - metastable phases 219 phase diagrams 5,230-232,241 - acetanilide/benzil 230 - eutectic 230,232,241 - ideal/real systems 232 phase transition 67,219-228 phenylbutazone 223 phenylglycidyl ether 182 photo-DSC 22-24,180 - base line 180 photo reactions 180-183,192 - dark reaction 194 plastics see polymers polybutadiene 238 polydimethylsiloxane 238 polyethylen 236,239,240 polyethylenterephtalate (see PET) 237 polymerisation 180,182,183,192,214,215 polymers 160,192,200,212,233-241 - amorphous 234,236 - crystalline 234,237 - crystallization 235 - glass transition 200-219 - melting 234 - recycling 235 - semi -crystalline (partial crystalline)

234,236,237 polymorphism 219-228 polystyrene (PS) 210 power compensation DSC 17-25,48-50,

270 - advantage 21 - controler 18,49 - fluids 24 - special types 22-25 pre-exponential factor 17l, 172, 186, 187,

213 presentation of results 145 pressure DSC 13, 22 principle of measurement 263 process 53,142 - endothermic 55 - exothermic 55 pulse 36,43,130,131

Subject Index

pulse response function 37, 43, 58, 131, 137 purity determination 241-244

Q quality assurance 4, 66 quality control 233 - aging of plastic 235 - degree of crystallinity 234,236 - oxidative decomposition 235 - plastic blends 234 - reactive resins 235 - semicrystalline polymers 234 - thermal decomposition 235 quasi-isothermal mode 27,52 quenching 206,217

R radiation activation 180 radiation heat transfer 44 radical polymerization 192 radioactivity 6 random errors 251 rate laws 168,182,183,189 RC-element 59,61 reaction 68, 162, 168 - autocatalytic 170,172,179,181,189,195 - consecutive 175,176,187,188,197 - curves 165,187 - heat 162-168 - isothermal 175 - mechanism 172,176,183,186,188 - non-isothermal 163,175 - order 169,183,185,189 reaction activation - by radiation 180,192 - thermal 180 reaction calorimeter 3 reaction heat flow rate 34 recycling 233 reference material - certified 65 - calibration 98-107 - recommended 101,101-105,

108-110 reference sample 32 relaxation process 56 repeatability 247,252 reproducibility 248,252 resistance (see thermal resistance) resolution 250 result 145, 251 reversing - heat capacity 54,141 - heat flow rate 142

Subject Index

S safety aspects 5, 232 - substances 233 safety calorimeter 3,232 safety risk 233 sample heat flow rate 119 sample model 34,60 sample preparation - cpmeasurement 150,154 - reaction 173 - Tg-measurements 204 sample properties 53, 58 sample temperature 38, 135, 136 sapphire 103, 104 sawtooth mode 29,51 semi-crystalline polymers 234,236,237 sensitivity 43,250 Sestak-Berggren equation 170 scanning mode 26 SI units 66 side reaction 163, 177 signal-to-noise ratio 247 silicate glass 201 simulation, numerical 44 simultaneous techniques 5 smearing 35,178,211,229 smectic phase 224, 228 Sourour-Kamal equation 170,183 stability investigations 235 standard 65 stationarity 63 steady state 32,37,42,48 step-scan mode 30,138,145,158 step response function 58, 136, 145 subscripts (table) XII substance identification 219 superposition principle 53,130,131 surface energy 68 symbols (table) XI symmetry 33,41,48,50,68,87 - check 85 systematic errors 251

T temperature - calibration 69-75,99 - correction 128, 135 - fixed points 100 - true 7l,72 - profIle in sample 136 temperature scale 69, 100 temperature-modulated mode 27-30,

50-63,159,195,238 - calibration 107,110-114

temperatures, characteristic 70, 117, 2,2',6,6' -tetrabromo-bisphenol-A

diglycidylether 195 theory - DSC 31-63 - heat flux DSC 31-48 - power compensation DSC 48-50 - TMDSC 50-63 theory of linear response 50, 56, 58,

130 thermal activation 180 thermal conductivity 33,45,57,58,136,

150,152,174,242 thermal diffusion equation 60 thermal history 202,234 thermal lag 135-140, - cp-measurements 151,154,155,157 - Tg-measurements 203 thermal resistance 32,35,59,60,75, 123 thermal wave 57,60 thermocouples 10,14 thermodynamic functions 67,120,121,

140,161 thermodynamics 67,121,161 thermometer problem 39 thermopile 15 Tian equation 36 time-temperature-transformation (TTT)

diagram 199,293

297

TMDSC (see temperature-modulated mode) 27-30,50-63,159,218,238

- calibration 107-114 - curves 140-145 - influence of sample 52-57 time constant 13, 16, 19,35,39,43,48, 130,

150,151,163,174,181,195,249,250 time dependent - heat capacity 57 - processes 54,238 tin 78-80 transfer function 58,59,61,62 - DSC model 62 transfer theory 58 transition enthalpy 67,125 trimethylene diisocyanate 176 triple measuring system 13 true heat 47,90 true heat flow rate 47,87,88,129-140 true temperature 7l,72 turret-type DSC 13,14, types - of calorimeters 3,263-279 - ofDSC 9-25 Tzero™ technology 14

298

U uncertainty 6,65,68,72,98,251-256 - black box method 253 - budget 253-256 - determination 251 - GUM method 253 - measurement 72,251 - temperature measurement 69, 72 - total 251 underlying signal 53, 55

V van't Hoff equation 241 variable heating rate 27 verification of calibration 69

Subject Index

vitrification 56,169,170,195,198,199,202, 239

Vyazovkin method 186

W wave (see thermal wave) WLF equation 196,197,210 work 67

Z zeroline 50,116,119,120,124,126,129,

147,163 - repeatability 247 zinc 77

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Differential Scanning Calorimetry

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