Differential scanning calorimetry (DSC) of semicrystalline polymers

REVIEW

Differential scanning calorimetry (DSC)of semicrystalline polymers

C. Schick

Received: 4 May 2009 /Revised: 6 September 2009 /Accepted: 18 September 2009 /Published online: 14 October 2009# Springer-Verlag 2009

Abstract Differential scanning calorimetry (DSC) is aneffective analytical tool to characterize the physical propertiesof a polymer. DSC enables determination of melting,crystallization, and mesomorphic transition temperatures,and the corresponding enthalpy and entropy changes, andcharacterization of glass transition and other effects that showeither changes in heat capacity or a latent heat. Calorimetrytakes a special place among other methods. In addition to itssimplicity and universality, the energy characteristics (heatcapacity CP and its integral over temperature T—enthalpyH ), measured via calorimetry, have a clear physical meaningeven though sometimes interpretation may be difficult. Withintroduction of differential scanning calorimeters (DSC) inthe early 1960s calorimetry became a standard tool inpolymer science. The advantage of DSC compared withother calorimetric techniques lies in the broad dynamic rangeregarding heating and cooling rates, including isothermal andtemperature-modulated operation. Today 12 orders ofmagnitude in scanning rate can be covered by combiningdifferent types of DSCs. Rates as low as 1 μK s−1 arepossible and at the other extreme heating and cooling at1 MK s−1 and higher is possible. The broad dynamic range isespecially of interest for semicrystalline polymers becausethey are commonly far from equilibrium and phasetransitions are strongly time (rate) dependent. Nevertheless,there are still several unsolved problems regarding calorim-etry of polymers. I try to address a few of these, for example

determination of baseline heat capacity, which is related tothe problem of crystallinity determination by DSC, or theoccurrence of multiple melting peaks. Possible solutions byusing advanced calorimetric techniques, for example fastscanning and high frequency AC (temperature-modulated)calorimetry are discussed.

Keywords Differential scanning calorimetry (DSC) .

Fast scanning . Temperature modulation (AC) .

Semicrystalline polymers . Crystallization .Melting

Introduction

Differential scanning calorimetry (DSC) is an effectiveanalytical tool for characterizing the physical propertiesof a polymer. DSC enables determination of melting,crystallization, and mesomorphic transition temperatures,and the corresponding enthalpy and entropy changes, andcharacterization of the glass transition and other effectswhich show either changes in heat capacity or a latentheat. Calorimetry takes a special place among othermethods. In addition to its simplicity and universality,the energy characteristics (heat capacity CP and its in-tegral over temperature T—enthalpy H ), measured viacalorimetry, have a clear physical meaning even thoughsometimes interpretation may be difficult.

From thermodynamics it is well known that knowledgeabout heat capacity from zero Kelvin up to the temperatureof interest enables description of important materialproperties.

Enthalpy H ¼ZT

0

CpdT ð1Þ

C. Schick (*)Institute of Physics, University of Rostock,Wismarsche Str. 43-45,18051 Rostock, Germanye-mail: [email protected]

Anal Bioanal Chem (2009) 395:1589–1611DOI 10.1007/s00216-009-3169-y

Energy U ¼ZT

0

CVdT ð2Þ

Entropy S ¼ZT

0

Cp

TdT or S ¼

ZT

0

CV

TdT ð3Þ

Gibbs Energy G ¼ H � TS ð4Þ

Free Energy F ¼ U � TS ð5Þ

With a completely known heat capacity (and known heatof transitions) the thermodynamic properties of a materialare known, see, e.g., Ref. [1]: U, H are total thermalmotion and energy, S is disorder, and G, F are measuresof stability.

Therefore detailed knowledge of heat capacity in a broadtemperature range is of general interest. Precise measure-ments of heat capacity at low temperatures are commonlyperformed by use of adiabatic calorimeters. Uncertaintiesbelow 1% are reached at temperatures below roomtemperature. At higher temperatures or for low thermalconductivity materials, for example polymers, or if heatcapacity becomes time dependent some additional problemsarise, which significantly increase the uncertainty of heatcapacity measurements.

DSC has proven to be a very reliable technique to obtainheat capacity at elevated temperatures in a reasonable shorttime. DSC also enables study of the kinetics of transitionsin a wide dynamic range. Because of its simplicity and easeof use DSC is widely applied in polymer science.

Calorimetry is generally based on the followingrelationship:

dQ ¼ C � $T ¼ c � m � $T ð6Þor, in differential form, assuming time independent samplemass and specific heat capacity:

dQdt

¼ 6 ¼ C � dTdt

¼ c � m � b ð7Þ

where δQ is the heat exchanged, ΔT is the temperaturechange caused by the exchanged heat, Φ is the heat flowrate, C is the heat capacity, c=C/m is the specific heatcapacity, m is the sample mass, and β is the scan rate(heating or cooling).

Here we will mainly discuss applications of DSC tosemicrystalline polymers. Depending on the mode ofoperation (scan at constant rate, isothermal, temperature-

modulated) effects like melting, crystallization, glasstransition, reversing melting, etc. can be studied. After ashort introduction to the different techniques someapplications and limitations of the techniques will bediscussed.

Basic considerations, modes of operation

Differential scanning calorimeters commonly have twosample positions—one for the sample under investigationand the other for a reference sample, which is often anempty crucible or one filled with an inert material. Theprinciple of operation of a DSC, including temperature-modulated DSC (TMDSC), data treatment, and calibrationare described in much detail in several monographs [1–14]and will not be repeated here. Classification of the differentdevices, as shown in Fig. 1, can be based on differentcriteria. Regarding precision, an important question iswhether all heat absorbed or released by the sample ismeasured by the device. Therefore we will use this criterionfor classification of DSC equipment.

In two-dimensional measurement systems, as used inmost heat flow DSCs (shown in Fig. 1a) and ultra fastscanning calorimeters (see Fig. 14, below), the sample mayexchange heat directly with the surrounding oven which isnot measured by the heat flow sensor. The heat flow ratenot measured by the sensor may reach up to 50% of thetotal heat flow rate exchanged [15]. Assuming the samelosses for sample and reference side and more importantthat they are not changing between calibration runs and themeasurement, these losses can be theoretically included inthe calibration function. Different approaches are available,for example the sophisticated Tzero technology with T4Pcalibration [16]. But the general drawback of the two-dimensional systems—the heat exchanged other thanthrough the sensor—is still present and limits accuracy ofthe measurement. A truly three-dimensional measurementsystem avoids this problem by not allowing heat exchangewith the surrounding other than through the sensingelement. To a very good approximation, 94% of theexchanged heat is measured [15], such three-dimensionalmeasurement systems are realized by the Tian-Calvet typeheat flux DSCs (Fig. 1b; reviewed on Ref. [17]).Tian-Calvet calorimeters yield an accuracy of about 1%for heat capacity and latent heats [18]. But there is oneessential disadvantage of these calorimeters—the large timeconstant of the measurement system. Therefore thesecalorimeters are not frequently applied in studies ofsemicrystalline polymers. Another quasi three-dimensionalmeasurement system with very short response time wasintroduced by Watson and O’Neill [19, 20] and is shown inFig. 1c. Here sample and reference are placed in separately

1590 C. Schick

temperature controlled ovens made from highly thermallyconducting metals. Active temperature control (powercompensation) is needed in order to ensure reproducibleheat losses from the ovens to the heat sink, which must beindependent on any heat released or absorbed by the

sample. As shown by Wunderlich [21], Richardson [22],and Hoehne [23, 24] the accuracy of heat capacity of suchpower compensating DSC’s may reach 0.5% under perfectexperimental conditions. In everyday use uncertainties of2% can be achieved depending on the temperature rangeand the external conditions (ambient–sub-ambient operation,temperature range, etc.) [25]. In temperature-modulatedmode these instruments reach accuracy better 1% asshown in Ref. [26].

Scan

The most common mode of operation of a DSC is heatingor cooling at constant rates. The primary outcome of suchan experiment is heat flow rate as a function of time. Iftemperature of the sample position is known, data can berepresented as heat flow rate versus temperature also1.Figure 2 shows a typical example.

From the heat flow rate curves, as shown in Fig. 2, heatcapacity, Cp, and with known sample mass specific heatcapacity, cp, can be obtained according to Eq. 8:

cpðTÞ ¼ cp saphireðTÞ msaphire�bmsample�b �

6sampleðTÞ�6emptyðTÞ6sapphireðTÞ�6emptyðTÞ

¼ KðTÞ* 6sampleðTÞ�6emptyðTÞmsample�b

KðTÞ ¼ cp saphireðTÞ msaphire�b6sapphireðTÞ�6emptyðTÞ

ð8Þ

where K(T ) is a temperature dependent calibration factor,which can be stored for future use. Here all measurements arecollected at the same scanning rate. The isotherms at thebeginning and the end of the scan are commonly used tocorrect for small changes in heat losses between empty,sapphire, and sample measurements by aligning these parts ofthe curves. Small changes in losses are unavoidable becausethe thermal properties, like thermal conductivity, of thesamples are changing. On the other hand inspection of theheat flow rate at the isotherms allows for a check of correctplacement and thermal contacts of all the parts of themeasurement system moved during sample changes. Thehigh-temperature isotherm, especially, should not vary toomuch between successive measurements.

Specific heat capacity is the most useful quantityavailable from DSC, because it is directly related to sampleproperties and, according Eqs. 1–5, directly linked tostability and order. Nevertheless, often heat flow rate as

1 One should be always be aware that a temperature near to the sampleis measured and not sample temperature itself.

Furnace

Thermocouples

Sample ReferencePlatinum Alloy

PRT Sensor

Platinum

Heat Sink

SampleSample ReferenceReferencePlatinum Alloy

PRT Sensor

Platinum

Resistance Heater

Heat Sink

a

b

c

Fig. 1 Different types of differential scanning calorimeters (DSC). (a)two-dimensional plate like; (b) three dimensional cylindrical (Tian–Calvet); (c) three dimensional with power compensation

Differential scanning calorimetry (DSC) of semicrystalline polymers 1591

obtained from a single sample measurement only ispresented. There are several reasons why this should notbe presented:

1. Each heat flow rate graph needs indication of endothermicor exothermic direction because plot direction is notstandardized.

2. Curves measured at different scanning rates are noteasy to compare.

3. If not divided by sample mass curves for differentsamples cannot be compared.

4. If empty pan measurements are not subtracted, tracesmay be curved and baseline construction for peakintegration may be difficult.

5. If heat flow rate calibration factor K(T) is temperature-dependent the heats of fusion, etc., obtained may beerroneous.

Performing corrections (3)–(5) yields specific heatcapacity as given by Eq. 8. Because most DSC softwarepackages include determination of specific heat capacityaccording Eq. 8 determination of specific heat capacity, andnot presentation of heat flow rate curves, is stronglyrecommended. Although I am very much in favor ofpresenting specific heat capacity data there may be goodreasons not to do this. Then normalization of the heat flowrate curve by scan rate and sample mass may result in“pseudo cp measurements”, which can be used to determinetemperature-dependent crystallinity and other quantities asshown by Mathot [6]. But there is another very strongargument in favor of presenting specific heat capacity ratherthan “pseudo cp” or heat flow rate. For more than 200polymers specific heat capacity data from 0–1000 K areavailable from the ATHAS Data Bank (ATHAS-DB) [27].The data can be used for comparison of measured data inthe glassy or liquid state with the recommended values.

This allows an easy check of the quality of the measureddata although one should have in mind that accuracy of therecommended data bank data is about 6% only. Figure 3shows specific heat capacity according to Eq. 8 calculatedfrom the data shown in Fig. 2.

What information is available from such curves?

1. Below glass transition, below 140°C—where the entirepolymer is in the solid state, and above the meltingpeak, above 350°C—where all the material is in theliquid state, specific heat capacity from the measurementis in good agreement with the data from ATHAS-DB.This indicates that the quality of the data is reasonable andfurther evaluation is possible.

2. The specific heat capacity above the glass transition,150–170°C, which nearly coincides with the liquid heatcapacity from ATHAS-DB, shows that the sample isamorphous up to these temperatures.

3. From the relaxation strength (step height) at glasstransition, information about the mobile amorphousfraction is available [28].

4. From excess heat capacity, information about thedevelopment of the crystalline fraction can be obtained,see the section “Determination of the degree ofcrystallinity—the influence of melting–recrystallization–remelting and a possible rigid amorphous fraction”,below.

5. Comparison of measured specific heat capacity with thereference data from the ATHAS-DB provides moreinformation about crystallization and melting rangethan available from the peaks alone. Most of the cold

150 200 250 300 350-3

-2

-1

0

1

2

3

4

5

cp amorphous

from ATHAS-DB

cp crystalline

from ATHAS-DB

melting, Tm

cold crystallization, Tcc

glass transition, Tg

Hea

t cap

acity

in J

/ g K

Temperature in C

Fig. 3 Specific heat capacity versus temperature for an initiallyamorphous PEEK sample, data from Fig. 2. Reference data (straightlines) for the fully amorphous (liquid) and crystalline (solid) PEEKfrom ATHAS-DB [27]. Tg, glass transition; Tcc, cold crystallization;Tm, final melting

0 2 4 6 8 10 12 14 16

-20

0

20

40

60

HF

in m

W

t in min

PEEK sapphire empty

100

200

300

400

ϑ in C

endo

˚

Fig. 2 Temperature profile and measured heat flow rate for emptypans, sapphire calibration standard (31.3 mg), and initially amorphouspolyetheretherketone (PEEK) (29 mg). Heating rate β=20 Kmin−1

(Perkin–Elmer Pyris Diamond DSC)

1592 C. Schick

crystallization takes place between 165°C and 200°C.But up to about 230°C specific heat capacity is very closeto the crystalline reference line because an overallexothermic crystallization process is superimposed onbaseline heat capacity of the semicrystalline material.Without such an exothermic contribution specific heatcapacity should be somewhere in between the liquid andthe solid reference lines [6]. Only at temperatures above250°C the measured signal becomes larger than theliquid reference line, indicating an overall endothermicmelting process. As discussed in the section “Fastscanning calorimetry”melting–recrystallization–remeltingis a continuous process in this temperature region and thesign of the net latent heat depends on the balance of thetwo processes. For detailed analysis baseline heatcapacity Ref. [6] is needed, which is not easy to obtain.Obviously separate integration of the cold crystallizationand the melting peak is impossible because of theunknown baseline needed for peak integration.

Beside scan measurements on heating DSC enablescooling at a wide range of cooling rate also. Dependingon the instrument and temperature range of interest, coolingrates up to 500 Kmin−1 may be reached (HyperDSC)[29, 30]. But generally the temperature range for controlledcooling at the highest rates is limited. Measurements in a widerange of heating or cooling rates require optimization of theexperimental conditions. Sample mass should scale inverselywith scanning rate. At low rates, when thermal lag is not anissue, sample mass should be high to have a good signal-to-noise ratio. At high rates when signals are large sample massshould be small to minimize the heat flow to the sample,which is proportional to rate, and causes the thermal lag.Problems related to thermal lag, temperature calibration, andreproducibility in fast-scanning DSC experiments wereintensively studied and adequate recommendations weregiven by Mathot et al. [30, 31]. Figure 4 shows coolingcurves in the crystallization range of low-density polyethylene(LDPE). At rates higher than 200 Kmin−1 controlled coolingdown to 100°C was not possible, because of the limitedcooling power of the mechanical intercooler used. If highercooling rates are needed, liquid nitrogen has to be used. Forthe lower scanning rates shown in Fig. 4 sample mass mustbe large enough to ensure a good signal-to-noise ratio. Forhigher rates the large sample (4 mg) causes some thermallag, as discussed in text books or in Refs. [30–32], which isalso seen in the broadening of the crystallization peak at20 Kmin−1 compared with the 0.4 mg sample at the samecooling rate. Data as shown in Fig. 4 provide informationabout crystallization kinetics and can be analyzed usingdifferent kinetic models [33–42] to name a few.

As shown in Fig. 4 DSC has a broad dynamic range,which can be extended at least for one order of magnitude

towards lower rates, this way covering three orders ofmagnitude. In the section “Fast scanning calorimetry” anextension for several orders of magnitude to higher rateswill be discussed. The possibility of cooling a samplereasonably fast allows studying structure formation in farfrom equilibrium situations, for example “quasi” isothermalcrystallization at deep undercooling.

Isothermal heat flow rate measurements

A typical “quasi” isothermal crystallization experiment isshown in Fig. 5. In a first step the previous crystallization

100 105 110 115 120 125

100

80

60

40

20

0

Spe

cific

hea

t cap

acity

in J

/gK

Temperature in C

-1 K/min -2 K/min -5 K/min -10 K/min) -20 K/min -20 K/min -50 K/min -100 K/min -200 K/min

4 mg

0.4 mg

time

Fig. 4 Cooling curves in the crystallization range of LDPE. Sample ofmass 4 mg in a 25 mg aluminium pan for cooling rates up to -20 Kmin−1

and of 0.4 mg in 2 mg aluminium foil for higher rates. Heat capacity isplotted downwards (Perkin–Elmer Pyris 1 DSC)

0 500 1000 1500

-11.4

-11.2

-11.0

-10.8

-10.6

-10.4

Hea

t flo

w r

ate

in m

W

Time in s

exo

Fig. 5 Isothermal crystallization experiment on 5 mg isotactic polypro-pylene (iPP) at 122°C after cooling from 180°C at 20 Kmin−1. Thedotted line is the baseline for peak integration and the dashed line is a fitof Eq. 12 to the data


history is erased by keeping at a temperature above themelting temperature of the most stable polymer crystals.Next, the sample is cooled to the crystallization temperature.Cooling must be fast enough to avoid crystallization oncooling. If all nuclei were removed an induction time isobserved and after that crystallization occurs, yielding anexothermic heat flow. In a similar way the effect of “selfnucleation” is studied by keeping some crystal nuclei afterpartial melting [43–45].

After switching from fast cooling to isothermal con-ditions at time zero the measured heat flow rate exponen-tially approaches a constant value (−10.3 mW) with a timeconstant of about 3 s for the power-compensated Pyris 1DSC employed. The observed crystallization peak is oftensymmetric and then the time of the peak maximum(minimum) is a measure of crystallization half-time [46].Integration of the peak results in the correspondingenthalpy change as a function of time (Fig. 6) which canbe transformed into relative crystallinity (mass fraction) bydividing by the limiting value at infinite time. To obtainabsolute crystallinity (mass fraction) the curve has to bedivided by the enthalpy difference between crystal andliquid at the crystallization temperature, which is availablefrom ATHAS-DB. For iPP at 122°C it is 195 Jg−1, which issignificantly smaller than the 207 Jg−1 at the equilibriummelting temperature of 188°C [27].

The commonly applied Kolmogorov–Johnson–Mehl–Avrami (KJMA) model for the kinetic analysis of isothermalcrystallization data is based on volume fractions. Therefore

the mass fraction crystallinity, Wc, as always obtained fromDSC, is transformed into volume crystallinity, Vc, by:

VcðtÞ ¼ W ðtÞcWcðtÞ þ rc=rað Þ 1�WcðtÞð Þ ð9Þ

where ρc and ρa are, respectively, the densities of the 100%crystalline and amorphous polymers. The KJMA model canbe expressed as:

1� VcðtÞ ¼ exp �kntnð Þ ð10Þ

or, after taking the logarithm twice, as:

log � ln 1� VcðtÞð Þ½ � ¼ n logðkÞ þ n logðtÞ ð11Þ

Plotting the left hand side of Eq. 11 versus log (t) resultsin partially straight lines as shown in the inset of Fig. 6.From the slope of the line the Avrami exponent n and fromthe intersect the rate constant nlog(k) can be obtained(recently reviewed elsewhere [46, 47]). Another way todetermine the Avrami parameters is based on a fit of thetime derivative of Eq. 10 to the measured heat flow rate, Φ,[48]. Here, if Vc≈Wc, the exponential approach to isother-mal conditions can also be taken in to account:

dVc

dt≈ Φ ¼ Φ1 þ Φ0 � Φ1ð Þ exp � t � t0ð Þ=tð Þ

þ Ankn t � t0ð Þn�1 exp �kn t � t0ð Þn½ � ð12Þwhere Φ1 is heat flow rate at infinite time2, Φ0 is heat flowrate at time zero, t0 is the start time of the isotherm, t istime constant of the equipment’s approach towards isother-mal conditions, n and k are the Avrami parameters, and A isthe peak area. In Fig. 5 the fit curve is included and theparameters are: Φ1=−10.3 mW; Φ0=−16.9 mW; t0=14 s;t =3.4 s; n=3.1 (3 from the linear fit in Fig. 6); k=0.0013(the same as from the linear fit in Fig. 6); and A=−400 mJ.

Knowing the Avrami parameters allows calculation ofcrystallization half-time, t1/2, according to:

t1=2 ¼ lnð2Þ½ �1=nk

ð13Þ

In Fig. 6 the value calculated according to Eq. 13 and thevalues from the measured data are indicated. They coincidereasonable well. Values for t1/2 in a wide temperature rangefor iPP are shown in Fig. 17, below.

0 500 1000 1500-400

-300

-200

-100

0

t1/2

[ ]1/

1/ 2

ln(2)n

tk

=

Pea

k ar

ea in

mJ

Time in s

tmax

0 500 1000 1500

0.0

0.5

1.0

Nor

m. a

rea

Time in s

2.0 2.5 3.0 3.5

-4

-2

0

log(

Nor

m. a

rea)

log(time/s)

Fig. 6 Integral of the curve shown in Fig. 5. The normalized area isshown in the upper inset and the same in double logarithmicrepresentation in the bottom inset. The three vertical lines showcrystallization half-life determined, by use of Eq. 13, as the peakmaximum in Fig. 5 and as read from the curve in the main panel (fromleft to right)

2 At infinite time heat flow rate from the sample equals zero but,because of asymmetries of the instrument, a non-zero value ismeasured.

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Besides the exothermal heat flow rate caused by thecrystallization process a decreasing heat capacity can beobserved for most polymers during crystallization near theglass transition. Unfortunately, heat capacity cannot bemeasured under isothermal conditions. But applying a smallheating rate (temperature) perturbation during a quasi-isothermal measurement allows determination of heatcapacity as a function of time as discussed next.

Temperature modulation

Oscillating power (heating rate, temperature) has been used incalorimetry for a long time (reviewed by Kraftmakher [49]).Periodic perturbations have been in use in calorimetry since1910 when Corbino [50, 51] used the so called 3ω-method[52, 53] to determine the heat capacity of electricallyconducting wires. In the 1960s AC calorimetry wasproposed by Kraftmakher [54] and Sullivan and Seidel[55]. These authors considered heat capacity as a real valuedquantity though it was also known from ultrasoundpropagation in gases that, in general, heat capacity shouldbe considered as a frequency-dependent complex quantity[56]. The first direct measurement of the frequency-dependent complex heat capacity was performed in 1971by Gobrecht et al. [57] at the glass transition of an inorganicpolymer. Interestingly, they used for their experiments adifferential scanning calorimeter. Therefore, they not onlyperformed the first direct measurements of complex heatcapacity but also used, for the first time, a temperature-modulated DSC (TMDSC). This idea, the combination ofDSC and periodic temperature perturbations was reconsid-ered in 1992 by Reading et al. [58–60] and Salvetti et al.[61]. At that time it was possible to overcome the limitationsof the setup proposed by Gobrecht et al. [57] because ofimprovements in computer technology. ConsequentlyTMDSC became available as a standard tool in thermalanalysis and since then has been widely used in polymercharacterization [28, 60, 62].

Assuming an ideally symmetric and calibrated DSCEq. 8 simplifies to:

cp ¼ Φm � b and ΦðtÞ ¼ m � cp � dTdt respectively ð14Þ

For any temperature perturbation, dT/dt≠0, a heat flowrate Φ(t) occurs. According to Eq. 14 for any heating rateprofile the measured heat flow rate contains informationabout heat capacity Cp=mcp. Here we will discusssinusoidal heating rate profiles only, even though the wholeformalism can be applied to other signal shapes [63–67]and non-periodic perturbations [26, 62, 68–70]. This kindof DSC measurement is called temperature-modulated

(TMDSC) or modulated-temperature (MTDSC). But oneshould have in mind that scanning rate (heating; cooling)and not temperature is the perturbation in scanningcalorimetry. Only if temperature changes (scan rate≠0) acalorimetric signal can be obtained. Any heating rateperturbation (periodic, stochastic, harmonic, inharmonic)may be added on top of any temperature–time profile, forexample scan or isothermal. In the simplest case linearunderlying heating (cooling) is superimposed with aharmonic perturbation:

TðtÞ ¼ T0 þ b0t þ AT sin wtð Þ ð15Þwhere T0 is start temperature, β0 is underlying heating(cooling) rate, AT is amplitude of temperature perturbation,and w,=2π/tp, is angular frequency with tp the modulationperiod. The measured heat flow rate allows calculation ofan effective reversing or complex heat capacity accordingto:

Cp effectiv wð Þ ¼ AΦ wð ÞAb wð Þ ¼ AΦ wð Þ

wATð16Þ

where AΦ is the (complex) heat flow rate amplitude andAβ=wAT is heating rate amplitude. Under conditions oflinearity and stationarity [71, 72] this quantity equals the socalled reversing or complex heat capacity, details of whichare given elsewhere [62]. The (complex) amplitudes, asneeded for heat capacity determination, can be obtainedwith high accuracy and sensitivity by frequency-selectivetechniques, for example Fourier analysis or lock-in ampli-fiers. This way small changes in heat capacity, e.g. due tocrystallization, can be detected. For β0=0 the meantemperature is constant and only the oscillating partcontributes to the heat flow rate. This condition is called“quasi isotherm” and enables determination of heat capacityas a function of time (Fig. 7).

Tg Tm

cp solid

cp liquid

TC

Fig. 7 Schematic representation of liquid and solid (glass and crystal)heat capacities of a polymer. On isothermal crystallization at Tc adecrease of Cp with time is expected


If baseline heat capacity, heat capacity without anycontribution from latent heats, is measured by TMDSC3 theprogress in crystallinity is monitored by the changes in heatcapacity. Figure 8 shows a typical quasi-isothermalTMDSC measurement during crystallization of a low-molecular-mass liquid crystal.

The expected and measured specific heat capacitiescoincide very nicely. This indicates that the TMDSCreversing heat capacity for the low-molecular-mass sampleequals baseline heat capacity. For polymers this is notalways the case as shown below.

In Fig. 9 the change in specific heat capacity and theexothermic heat flow rate during quasi-isothermal crystal-lization of polyhydroxybutyrate (PHB) are shown.

Baseline heat capacity was measured as function of timeand compared with the predictions for a two and three-phase model (details are given in Ref. [74]). Such measure-ments not only allow the study of crystallization kinetics onvery long time-scales they provide information about thedevelopment of a rigid amorphous fraction (RAF) also.Details regarding RAF are available elsewhere (Ref. [28]and references therein). The RAF in PHB is establishedduring quasi-isothermal crystallization, as can be seen fromthe agreement of line e with the measured heat capacity atthe end of the crystallization process. For PHB it waspossible to measure the exothermic effect because of thecrystallization process simultaneously. The Pyris1 DSCenables quantitative measurement over 17 h even thoughthe peak maximum of the heat flow rate was less than40 μW. From the integral we obtain the enthalpy change,h(t), and the crystalline fraction as a function of time asdescribed in the section “Isothermal heat flow ratemeasurements”. The time dependence of baseline heatcapacity can then be determined from:

cp bðtÞ ¼ cp liquid � # CRFðtÞ# CRF 1ð Þ cp liquid � cp b 1ð Þ� � ð17Þ

The calculation can be performed for two cases:

1. the RAF is formed during the crystallization processfrom the very beginning, or

2. first the crystalline morphology is built up during maincrystallization and, in a second step, e.g. duringsecondary crystallization at longer times, the RAF isformed. Then during main crystallization no or only alittle RAF should be present.

Situation 2 should be described by Eq. 17 when cpb(∞)equals the value from a two-phase model, line d, taking intoaccount liquid and crystalline material only. Curve g inFig. 9 shows the result. Although the behavior at longer

times (>10,000 s) is not known, the result during themain crystallization is not in agreement with the measuredcurve. To calculate cp(t) according assumption 1, cpb(∞)equals the value from a three-phase model, line e. Here it isassumed that the rigid amorphous fraction is formed duringor just after the formation of each single lamella. Curve h inFig. 9 shows the result. The agreement is perfect withinthe scatter of the experimental points. This example isgiven here because it demonstrates the power of heatcapacity measurements and shows the possibility ofstudying morphologically induced (isothermal) vitrifica-tion of the RAF during crystallization by TMDSC. Herethe heat flow rate is a superposition of that caused by theheating rate perturbation and by a latent heat. Underthese particular conditions TMDSC is able to separateboth contributions and enables determination of baselineheat capacity [71, 75–78]. Furthermore, because of thequantitative heat capacity data, we are able to compare themeasured data with predictions from model calculations.This way we are able to make more substantial contribu-tions to polymer science compared with only qualitativeresults from DSC and TMDSC heat flow rate curves[74, 78–80].

For some polymers, for example isotactic polystyrene(iPS) [78], polycarbonate (PC) [80], and PHB [80],reversing heat capacity from TMDSC equals baseline heatcapacity for temperatures near the glass transition. For otherpolymers also in isothermal TMDSC measurements at thecommon low frequencies available by DSC, latent heatsmay contribute to the measured reversing heat capacity[81–85]. Figure 10 shows the development of measuredreversing heat capacity during isothermal crystallization ofpolyamide 12. Similar observations were made for poly-ethylene (PE) [86], poly(ε-caprolactone) (PCL) [75], andPEEK [87] to name a few.

After an induction time of about 100 min crystallizationstarts. Parallel to the increase in crystallinity (not shownhere) reversing heat capacity unexpectedly starts to increasealso. Specific reversing heat capacity reaches valuessignificantly above the liquid specific heat capacity. Thisobservation, which is also made on melting [81–85], cannotbe described by any mixing rule taking in to accountspecific heat capacities of the liquid and the crystallinepolymer only. To describe specific heat capacities largerthan the specific heat capacity of the liquid needs excesscontributions. Such excess heat capacities may originatefrom reversing melting–crystallization processes, e.g. at thegrowth front of the crystals [73, 88, 89] or by reversinglamellae thinning and thickening [90–92]. So far twoextreme cases have been discussed:

1. when no reversing melting is observed and baselineheat capacity is measured; and

3 Unfortunately, this is possible only close to glass transition but inmost cases not close to melting and crystallization transitions.

1596 C. Schick

2. when reversing heat capacity increases instead ofdecreasing, as is expected on crystallization.

All intermediate situations are feasible, and also observed,e.g. for PCL [75], for which reversing heat capacity decreasesbut not as much as expected from the increase in

crystallinity. Furthermore, it has been shown, and is seen inFig. 10, that the excess heat capacity due to reversingmelting is frequency-dependent. For infinite high frequen-cies, when the melting–crystallization process is muchslower than the temperature oscillation, baseline heatcapacity is measured for any polymer. In common TMDSCmeasurements frequency is in the mHz range and, therefore,contributions because of reversible melting can often not beavoided. In Fig. 11 the frequency dependence of the specificreversing heat capacity for polyamide 12 is presented. Thevalues at the maximum of the curves in Fig. 10 at about1,000 min are shown.

By combining different calorimeters and using sharpspikes in heating rate (Fig. 10a), a multi-frequency eval-uation [63, 69] and an appropriate calibration procedure[71, 76, 93–95] more than six orders of magnitude infrequency are covered by this very time-consumingexperiment. A reasonable fit to the data can be obtainedby using the so called Cool–Davidson-function:

cp wð Þ ¼ cpb þ $cp1þ i2p f tð Þð Þg ð18Þ

where cp(w) is the measured reversing heat capacity, cpb thebaseline heat capacity, Δcp the excess reversing heatcapacity at frequency zero, f the modulation frequency,and t the relaxation time. Reversing melting can beconsidered as an asymmetrically broadened relaxationprocess with a characteristic time constant t and a re-laxation strength Δcp. At high frequencies the excesscontribution vanishes and baseline heat capacity ismeasured. For polyamide 12 modulation frequencies above100 Hz are needed to reach this goal. So far TMDSCcannot reach such frequencies and special chip-based thin

1000 10000 100000Time in s

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p b(χ

CRF= 0.64)

e - cp b

(χsolid

(Tg)) = 0.88

c - cp liquid

b - cp solid

c p in J

/gK

Fig. 9 Time evolution of heat capacity during quasi-isothermalcrystallization of PHB at 296 K, AT=0.4 K, tp=100 s, curve (a).Curves (b) and (c) correspond to solid and liquid heat capacities,respectively, from ATHAS-DB. Curve (d) was estimated from a two-phase model and curve (e) from a three-phase model, using χsolid(Tg)from Δcp (details are given in Ref. [74]). Subscript solid means thesolid fraction of the polymer consisting of crystalline and glassyfractions. Subscript CRF means crystalline fraction alone. The squaresrepresent measurements for modulation periods ranging from 240 s to1,200 s. Curve (f) shows the exothermal effect in the total heat flowand curves (g) and (h) are the expected values from modelcalculations (see text and Ref. [74]) (Perkin–Elmer Instruments Pyris1DSC)

342.8343.0343.2

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A B

Fig. 8 (a) Temperatureprofile, measured heat flowrate, averaged heat flowrate, and heat flow rateamplitude (top to bottom) for2,5-bis-(2-propyloxycarbonyl-phenylsulfonyl)terephthalic aciddipropyl ester [73]. (b) Specificheat capacity for liquid andcrystal (horizontal lines). Theexpected baseline heat capacityis calculated from the integral ofthe averaged heat flow rate (theenthalpy change). The measuredspecific heat capacity resultsfrom the heating rate and heatflow rate amplitudes accordingEq. 16. T0=343 K, TA=0.2 K,tp=50 s, ms=14.47 mg


film calorimeters in AC-mode are needed [96, 97]. Only atsuch frequencies can crystallization be followed in time.

By heating a semicrystalline polymer through themelting range even larger contributions due to reversingmelting can be observed [81–85]. Therefore any discussionof baseline heat capacity measurements by TMDSC isquestionable as long as it is not shown that the highfrequency limit is reached. For poly(ethylene terephthalate)(PET), to give one example, 480 Hz is still not enough toavoid reversible melting at heating and cooling [98].

From the beginning of TMDSC [58, 59, 99, 100] it wasclaimed that TMDSC enables the separation of heat capacityand excess heat capacities. In that case recrystallizationprocesses appear as an exothermic peak in the non-reversingheat flow rate or heat capacity curves. This approach is

obviously questionable as soon as reversing melting occurs,as discussed above for quasi-isothermal experiments. As arule we can say that all processes, which are reversing on thetime scale of the temperature modulation, contribute to themeasured signals. Therefore the concept of separatingreversing and non-reversing events by subtracting thereversing heat flow rate from the total heat flow rate can beapplied generally neither to crystallization nor to melting andrecrystallization of polymers. An example demonstrating thiswill be given below.

Fast-scanning calorimetry

Another promising addition to conventional DSC is fast-scanning DSC. By going to the limits of conventionalDSCs heating and, more important, cooling rates up to500 Kmin−1 (8 Ks−1) can be reached [29, 30, 101]. At suchhigh rates care must be taken to avoid smearing of the DSCcurves by thermal lag effects. At increasing scanning ratesthe heat flow rate transferred to or from the sampleincreases in parallel to rate (Eq. 7), and several heatcapacities and thermal resistors contribute to smearing ofthe measured signals. Sophisticated models can be used tocorrect for these effects [2, 16, 20, 93–95, 102–119]. Ingeneral, a DSC is built from different parts, each of which,in principal, has characteristic heat conductivity and heatcapacity. Any contact area between different parts acts as anadditional heat resistance.

In Fig. 12 different elements affecting heat transfer fromthe DSC measurement system to the sample are shownschematically. The heat flow is limited by a thermalresistance between the cup (base-plate in case of a heat

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Setaram DSC 121

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AC-chip calorimetry

Fig. 11 Specific reversing heat capacity of PA 12 at the maximum ofthe curves in Fig. 10 at about 1,000 min for quasi-isothermalcrystallization at 173°C. The curve is a fit of Eq. 18, the so-calledCool–Davidson function

20 30 40 50 6057

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Spe

cific

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acity

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/gk

cp amcp c

30

200

173

t = 48h

ca. 2 days

A B

Fig. 10 Quasi-isothermal crystallization of polyamide 12 (PA 12) atT0=173°C, tP=600 s, AT=0.5 K. (a) Temperature profile consisting ofan asymmetric saw-tooth. The resulting heating rate and heat flow rateshow sharp spikes containing a broad spectrum of higher harmonics[63]. (b) Specific reversing heat capacity as a function of time for

different frequencies as indicated in the graph. The lines labeled cp am

and cp c indicate the data for liquid and crystalline polyamide 12 at173°C, respectively, from ATHAS-DB. The temperature–time profileused for sample preparation and crystallization is shown in the inset

1598 C. Schick

flow DSC) and the sample pan. The effective time constantis determined not only by the sample heat capacity but alsoby the sample pan heat capacity, which must also beconsidered [30, 95]. Another thermal contact, which, unfor-tunately, may change during the measurement [102, 103],exists between the sample pan and the sample. This changingthermal contacts seriously limits the applicability of desmear-ing algorithms, which are based on previously determinedproperties of the system as used in Refs. [16, 120]. And, lastbut not least, the thermal conductivity of the sample itselflimits the heat transfer and will affect the dynamic behaviorof the DSC–sample arrangement. This dynamic behavior israther complex because of the different elements contribut-ing. The model of choice to evaluate such a heat-transfernetwork is often that of the electrical analogy, which hasproved its worth in DTA and DSC analysis for decades(Ref. [121], or Ref. [93] for a recent paper) and will not bediscussed here.

For fast-scanning calorimetry it is therefore important toconsider that in any scanning calorimeter a certain thermalresistance exists between sample and measurement system.As a consequence a temperature difference appears acrossthis effective thermal resistance, ρth, because heat flow rateΦ has to be transferred across this resistance.

$T ¼ rth Φ ð19ÞBecause Φ increases with increasing rate, smearing

becomes a serious problem at higher scanning rates. Toavoid smearing, the thermal resistance should be small and,even more effective, the heat flow rate Φ transferred acrossthe resistance should also be small. For a sample with givenspecific heat capacity, c, the only way to achieve this atincreasing scanning rate is to reduce sample mass, m, in thesame way as the rate increases (Eq. 7). Additionally, thereis a thermal resistance between the measurement systemand the sample pan (Fig. 12), causing similar problems.Therefore sample pan heat capacity (sample pan mass)should be as small as possible also. In Ref. [30] it was

shown that wrapping the sample in thin aluminium foil isa good solution to this problem in the Hyper DSC.Similarly good is use of sample pans of a few mg mass.In general these sample pans or wrapped samples do nothave highly reproducible shapes, so heat exchange withthe oven may be on different heat paths for each sample.Only if all heat flow rates are recognized by the measure-ment system are the measurements reliable. This is thecase for Perkin–Elmer power-compensated DSCs but notfor two-dimensional heat-flow DSCs as discussed at thebeginning of the section “Basic considerations, modes ofoperation”. Even for small samples (below mg) and smallsample pans (few mg aluminium) temperature gradientsmay become serious at higher scanning rates and tem-perature calibration becomes very important [29–32, 122].Very often some smearing cannot be avoided as shown inthe examples below.

In melting curves of polymers multiple melting peaksare often observed. The reason for this observation isstill under debate and several papers dealing with theeffect can be found. Either melting–recrystallization–remelting mechanisms [123] or the occurrence of at leasttwo distinct crystal populations [124] are being considered.Scanning at different heating rates after isothermalcrystallization is a powerful tool for distinguishing bet-ween those two explanations. In Fig. 13 heating scans atdifferent heating rates are shown for isothermally crystal-lized iPS.

The high temperature melting peak moves to lowertemperatures and decreases in size with increasing heatingrate. This behavior is not a consequence of any thermal lag(one would expect an upward shift) and provides evidencethat this peak originates from a melting–recrystallization–remelting process on heating. The other peaks move tohigher temperatures and increase in size. As long as two ormore peaks are distinguishable it is not possible to assignthese peaks to melting–recrystallization–remelting orcrystal populations of different stability. In the example

1

2

3

2d

x

d

0

K ps

K op

Φp(t)

Φo(t)

S, cp, ρ, κs

To (t)

Tp (t)

Ts (x,t)

a bFig. 12 Schematic view ofthe single DSC cup in power-compensation Perkin–ElmerInstruments DSC (a) and itsblock diagram (b). 1, sample;2, pan; 3, oven [95]


shown in Fig. 13 the situation becomes definite at very highrates only. At a heating rate of 500 Kmin−1 and morepronounced at 30,000 Kmin−1 only one melting peakremains. Obviously there is only one population of crystalswhich melts between 200 and 240°C. All other meltingpeaks seen at lower heating rates are because of melting–recrystallization–remelting. At heating rate 10 Kmin−1

melting starts already at about 175°C, only 5 K abovecrystallization temperature. As soon as some crystals aremolten, having chain segments that still show localizedorder in the melt, the melt recrystallizes immediately,

forming slightly more stable (thicker) lamellae. Thisprocess yields only a very small excess heat capacitybecause melting and recrystallization nearly cancel eachother. This process continues until there is no possibility ofgaining further stability on recrystallization within theexisting lamellae stack. Then the whole stack has to meltto allow further recrystallization. This explains the secondmelting peak seen at 10 Kmin−1 above 200°C. The smallexothermic effect after this melting peak supports the viewof immediate recrystallization. The continuous melting–recrystallization–remelting process then proceeds until, at atime determined by the heating rate, recrystallization is nolonger possible and the crystals finally melt. Final meltingmoves therefore to lower temperatures at higher heatingrates, and at intermediate heating rates additional peaksmay appear depending on recrystallization rate and possiblegain in stability for the lamellae stacks actually present.

The shift of the first peak, often called the “annealingpeak”, to higher temperatures is because of superheating of themelting transition in polymers [125–138]. So far, for allinvestigated and isothermally crystallized polymers, only onemelting peak has been found at sufficiently high heatingrates suggesting the melting–recrystallization–remeltingprocess as the common behavior of isothermally crystallizedpolymers. As shown in Refs. [125, 135, 136, 139–143], andother unpublished data, recrystallization is a very fastprocess. For PET, as an example, it takes only millisecondsto recrystallize while crystallization from the isotropic meltis about two orders of magnitude slower [143, 144]. Inorder to avoid recrystallization fully, very high heating ratesare commonly needed. Recently developed chip-based thin-film fast-scanning calorimeters enable use of heating andcooling rates up to MK s−1 [140, 145, 146]. Next, the basicsof this technique will be described using, as an example,non-adiabatic fast-scanning calorimetry as used in Rostock[140, 147]; other groups have conducted similar experimentswith semicrystalline polymers [148, 149].

If scan rates larger than that of the Hyper DSC [30] areneeded, further reduction of sample mass is required. Inconventional DSC, reduction of sample mass is notunlimited. For polymers, sample masses of approximately0.1 mg are needed to achieve a good signal-to-noise ratio.The reason is the large addenda heat capacity of themeasurement system. For a Perkin–Elmer Pyris DiamondDSC, the heat capacity of the 1.5-g oven is approximately0.2 JK−1 whereas heat capacity for a 0.1-mg polymer sampleis approximately 10−4J K−1. Consequently, improvementis needed in the ratio of sample to addenda heat capacity.For even higher rates and smaller samples new calorimet-ric devices are therefore required. Chip-based thin-filmcalorimeters with heater and thermometer on a sub-μm-thick membrane are good candidates for such devices.MEMS technology enables the production of sensors

100 120 140 160 180 200 220 240

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Fig. 13 Temperature dependence of the specific heat capacity of 400 ngto 4 mg iPS samples at the heating rates: 1, 10 Kmin−1 (0.16 Ks−1) 4 mg;2, 50 Kmin−1 (0.83 Ks−1) 4 mg; 3, 100 Kmin−1 (1.6 Ks−1) 0.5 mg; 4,200 Kmin−1 (3.3 Ks−1) 0.5 mg; 5, 400 Kmin−1 (6.6 Ks−1) 0.5 mg; 6,500 Kmin−1 (8.3 Ks−1) 0.5 mg; 7, 6,000 Kmin−1 (100 Ks−1) 400 ng; 8,15,000 Kmin−1 (250 Ks−1) 400 ng; 9, 30,000 Kmin−1 (500 Ks−1)400 ng. The samples were crystallized at 170°C (a) and 140°C (b) for12 and 4 h, respectively, cooled to -50°C, and heated at the ratementioned. The heat-capacity scale corresponds to curve 9. The othercurves are successively vertically shifted by 1 Jg−1K and the straightlines are guides for the eyes, only [125]

1600 C. Schick

which can be used under quasi-adiabatic [145, 150] or non-adiabatic conditions [140, 146]. Non-adiabatic operation ofsuch thin-film calorimeters with well defined losses enablesfast, controlled cooling also [140, 147, 151] and is thereforepreferable for the study of crystallizing polymers.

Sensitive and ultrafast calorimeters for programmablethermal processing and heat capacity measurements withsub-millisecond resolution have been constructed usingsilicon nitride membrane technology. Commercially avail-able gauges from Xensor Integration [152] can be used forthis purpose. The cross sectional view of the calorimetercell with a sample is shown schematically in Fig. 14. TheXEN-39240 gauge consists of a submicron amorphoussilicon nitride membrane with a thermopile (of six thermo-couples) and a resistive heater (~1 kOhm) located in a 60×60 μm2 area at the center of the membrane. The central partof the XEN-39273 gauge with an even smaller heated regionand only one thermocouple is shown in the same figure. Theheater strips and all the electrical leaders (includingthermopile) are formed by p and n-doped polysilicon trackswith specified thermoelectric properties and resistivity. Thethermopile hot junctions are arranged in the central part ofthe heated area between the heater stripes. The cold junctionsare placed at the silicon frame, which fixes the membrane atthe TO-5 housing at a distance of ~1 mm from the center.Thus the cold-junction temperature equals the temperature ofthe holder which is close to the temperature of the thermostat

Tt. The silicon frame with the membrane is bonded on to astandard chip carrier. The whole assembly can be taken outof the thermostat and positioned under a microscope forsample handling. A gauge installed in a thermostat withcontrolled temperature and gas pressure can be used as acalorimeter and a thermal processing system for samples inthe nanogram range. To avoid adsorbed water on the thinsamples the thermostat is commonly first pumped to 2 mbarand then refilled with dry nitrogen or helium.

The XEN-39273 gauge allows heating and cooling ofpolymer samples up to MK s−1. Cooling and heating curvesobtained from ultra-high-molecular-weight PE (UHMWPE)at rates around 1 MK s−1 are shown in Fig. 15.

At a cooling rate of 1.6 MK s−1 no crystallization peakfor the UHMWPE is seen. This is in good agreement withexperiments by Geil [153], in which amorphous PE wasobtained by rapid quenching. Somehow unexpected is theobservation that on heating at the same rate no coldcrystallization is seen (left inset Fig. 15). To checkreliability of the data obtained at such high rates the samplewas isothermally crystallized for one minute at 106°C andthen scanned on heating from low temperature. Even at thehighest rate, a well pronounced melting peak is seen (rightinset). This shows the ability to acquire reliable data even atrates as high as 1.6 MK s−1.

Being able to quench at rates as high as 1 MK s−1

enables amorphous samples of nearly all polymers to beobtained. Consequently isothermal crystallization can bestudied at any temperature between melting temperatureand glass transition. Because of the short time constant ofthe chip calorimeter such isothermal crystallization experi-

200

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acity

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cooling

heating after isothermalcryst. @ 106 Cfor 1 min

heating after cooling

Fig. 15 Cooling curves of a 0.06 ng UHMWPE sample at the ratesindicated. The left inset shows the heating curves at the same ratesimmediately after cooling. The capability of the device to measure atsuch high rates is demonstrated in the right inset by the melting curvesafter isothermal crystallization of the same small sample

Fig. 14 Photograph of the gauge used as a measurement cell. (a) Therectangular 2×3 mm2 silicon frame supporting the submicron siliconnitride membrane, which is bonded to a standard chip holder. (b)Photograph of the central part of the XEN-39240 with the 60×60 μm2

heated region in the middle and six hot junctions of the thermopilelocated between two parallel heater stripes. (c) XEN-39273 with 14×26 μm2 heated region and one thermocouple. (d) Schematic cross-sectional view of the gauge with the sample (not to scale)


ments can be performed at time resolution of a fewmilliseconds.

The data shown in Fig. 16 were obtained by use of adifferential fast scanning device as described elsewhere[151]. As for conventional DSC, the heat flow rate signalcan be described by an exponential decay, because of theinstrument and an Avrami function (Eq. 15). The fit isreasonable, but at short times before the crystallization peakappears some differences between measured curve and fitare seen. The reason for this is not yet known. CombiningDSC and chip-based calorimeters yields a broad dynamicrange for both scanning and isothermal experiments. InFig. 17 crystallization half-time of iPP is shown in the timerange from a few tens of milliseconds up to 20,000 s.

The data in Fig. 17 cover more than five orders ofmagnitude in time. At high crystallization temperatures(above 40°C) acceleration of crystallization due to additionof a nucleating agent is seen in DSC and chip calorimeterdata. Below 40°C additional nuclei do not speed upcrystallization compared with the pure iPP. At suchtemperatures the mesophase is observed in iPP; this seemsto be formed by homogeneous nucleation [48]. Similarcurves with a local minimum in overall crystallization rateat about 130°Cwere obtained for poly(butylene terephthalate)(PBT) [142]. In Fig. 17 DSC and chip sensor datacoincide. Indicating similar crystallization kinetics forthe milligram-sized DSC and the nanogram-sized chipcalorimeter samples. For PBT [142] this is not the case.The nanogram-sized sample on the chip sensor crystallizesabout two orders of magnitude faster than the DSCsample. The larger surface-to-volume ratio for the smallerchip sensor sample results in significant deviations frombulk behavior; this must generally to be taken into accountfor small samples.

Besides fast scanning, the thin film chip sensors canalso be used as AC calorimeters in temperature-modulatedmode [97, 154, 155]. Because of the very small addendaheat capacity of the sensors, measurements on very smallsamples, e.g. films a few nanometers thick, can be per-formed [96, 97, 156–160]. Because the frequency oftemperature oscillation ranges from Hz to kHz there is agood chance of measuring baseline heat capacity duringcrystallization and melting of polymers as shown above inthe section “Temperature modulation”. This is especiallyimportant for determination of crystallinity, as discussednext.

Determination of the degree of crystallinity—the effectof melting–recrystallization–remelting and a possiblerigid amorphous fraction

DSC in scan mode is frequently used to determine heat offusion and from that the degree of crystallinity of semi-crystalline materials. In order to do so the melting peak hasto be integrated and the heat of fusion obtained has to becompared with the heat of fusion of a perfect crystal (100%crystallinity). For polymers, unfortunately, both tasks arenot easy to solve. The melting region of polymers is oftenvery broad (>200 K in some cases [161]) and multiplemelting peaks may appear. Then construction of the “peakbaseline” as needed for the integration is not a simple task.The problem becomes even more complex if not onlycrystalline and liquid amorphous fractions coexist. For mostsemicrystalline polymers a rigid amorphous fraction(reviewed recently [28]) coexists with the liquid fractionand its devitrification (vitrification on cooling) must also betaken into account in construction of the peak baseline.Furthermore, because of the width of the transition and thefact that the heat of fusion of the 100% crystalline material

0 20 40 60 80 100 120 140

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Fig. 17 Crystallization half-lives (time of peak maxima) of iPP andnucleated iPP as functions of crystallization temperature (data fromchip calorimeter and Perkin–Elmer Pyris Diamond DSC)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

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Fig. 16 Isothermal crystallization of ca 40 ng iPP at 80°C afterquenching from the melt at 1,000 Ks−1. The inset shows thetemperature profile and the right photograph shows the sample onthe sensor

1602 C. Schick

is commonly given for the equilibrium melting temperature,which may be significantly higher than the temperature ofthe observed melting or crystallization peak, the tempera-ture dependence of the heat of fusion of the 100%crystalline material has to be taken into account. For mostpolymers heat capacity of the liquid is larger than heatcapacity of the crystal in the temperature region of interest.Consequently, the enthalpy curves are not parallel and thedistance changes with temperature. This is illustrated forpolystyrene in Fig. 18. For more than 200 polymers thesedata are available from the ATHAS databank [27].

Heating scans of semicrystalline iPS are shown inFig. 19. If, for example, and neglecting all problems withfinding the correct peak baseline, the change in the degreeof crystallinity related to the small peak at approximately160°C has to be determined, the peak area must bedivided by approximately 80 Jg−1—the heat of fusion ofthe 100% crystalline iPS at that temperature. This is onlyapproximately 80% of the often used value of 96 Jg−1 atthe equilibrium melting temperature of 243°C. Using thevalue at the equilibrium melting temperature would resultin approximately 20% underestimation of the relatedchange in the degree of crystallinity. The temperaturedependence of the heat of fusion of the 100% crystallinematerial can be taken into account by using the enthalpyfunctions provided for more than 200 polymers in theATHAS data bank [27] or from the heat capacitydifference between crystal and liquid [162].

Finding the right baseline for integration is a moreserious problem in determination of the heat of fusion anddegree of crystallinity of polymers. Assuming a simple two-phase system consisting of a crystalline phase and an

amorphous phase which is always in the liquid state aboveits glass transition temperature, an enthalpy-based proce-dure has been suggested [6, 14, 161, 163, 164]. Theenthalpy of the sample, h(T), is the superposition ofthe enthalpy of the crystalline part, hC(T), and that of theamorphous part, hA(T). The degree of crystallinity, wC(T) asfunction of temperature is then obtained from:

wC ¼ hAðTÞ � hðTÞhAðTÞ � hCðTÞ ð20Þ

Whereas the denominator is available for most polymersfrom ATHAS data bank [27] the numerator can be obtainedby integrating the measured heat capacity (heat flow rate[161]), by using the heat capacity (heat flow rate) of theliquid polymer as baseline. The latter is also available fromthe ATHAS data bank, or can be extrapolated downwardsfrom the measured curve from temperatures above the endof melting [161, 163]. Integrating up to a temperature TE inthe liquid state of the sample, above the end of the meltingpeaks, and assuming h(TE)=hA(TE) yields the followingrelationship for the numerator of Eq. 20 for any temperatureTX below TE and above the glass transition temperature Tg:

hA TXð Þ ¼ hA TEð Þ � RTETX

cp;AðTÞdT

h TXð Þ ¼ hA TEð Þ � RTETX

cpðTÞdT

hA TXð Þ � h TXð Þ ¼ RTETX

cpðTÞ � cp;AðTÞ� �

dT

ð21Þ

This procedure can yield good results if good data areavailable (Fig. 20) [161, 163, 165, 166] but will, of course, fail

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Fig. 19 Temperature dependences of the specific heat capacity of a4-mg iPS sample at a heating rate 10 Kmin−1 [125]. The sample wascrystallized at TC=140°C, tC=12 h (solid line) and TC=170°C, tC=4 h(dashed line), and quenched below the glass transition. The curves foramorphous and crystalline iPS are also shown [27]

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Fig. 18 Enthalpy of amorphous and crystalline polystyrene. Thedifference between both equals the temperature-dependent heat offusion (crystallization). Equilibrium melting temperature and thecorresponding heat of fusion are indicated by the vertical andhorizontal thin lines, respectively (data from ATHAS-DB [27])


if—being based on the two-phase model—it is applied to athree-phase system, as shown in Fig. 21, see discussion below.

Comparison of the degree of crystallinity and the enthalpychange for the sample crystallized at 140°C shows that thetemperature dependent denominator of Eq. 20 yields abasically constant degree of crystallinity between 160°C and200°C even if the enthalpy change is continuously increasingwith temperature. This is the temperature range wherecontinuous melting–recrystallization–remelting is expectedfrom fast scanning calorimetry (Fig. 13b) [125]. Below

140°C, the temperature of isothermal crystallization,however, the degree of crystallinity decreases remarkably.This has been predicted (Figs. 5.25 and 5.29 in Ref. [6]),and experimental observations are reported for PE [166],PET [167], iPS [78, 168], and cis-1,4-polybutadiene[169], to give a few examples. The reason for this is thedeviation of the real polymer morphology from the so farassumed two-phase morphology. From the relaxation strengthof the glass transition it is known that not all of the non-crystalline fraction of a semicrystalline polymer contributes tothe glass transition. This was shown for dielectric relaxationdata by Ishida et al. [170] and for the step height of thespecific heat capacity by Wunderlich’s group [28, 171]. Adetailed description of the problem is given elsewhere [28].

Figure 22 is a magnified view of Fig. 19 enabling moredetailed discussion of the glass transition. From the degreeof crystallinity of approximately 0.3 a thirty percentreduction of the step in the specific heat capacity at theglass transition compared with the fully amorphous iPS isexpected. The expected value for the specific heat capacityabove the glass transition is indicted as cp two phase model. Itis calculated from the data shown in Fig. 21 using thedegree of crystallinity of 0.27 at 140°C according to:

cp;two phase modelðTÞ ¼ wAðTÞð Þ � cp;AðTÞ þ wCðTÞ� cp;CðTÞ � hAðTÞ � hCðTÞð Þ

� dwC

dTð22Þ

with wA þ wC ¼ 1 ð23Þ

100 120 140 160 180 200 220 240

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Tc 140 C

Tc 170 CC

ryst

allin

ity

Temperature in C

0

2

4

6

8

10

12

14

16

18

20

22

24

26

h(T) - h

a (T) in J/g

Fig. 21 The top curve, right axis, shows the numerator of Eq. 20determined by use of Eq. 21 from the data shown in Fig. 20. Thebottom curves show the degree of crystallinity from Eq. 20, using datafrom Fig. 18 for the denominator

100 120 140 160 180 200 220 240-1

0

1

2

3

4

5

Spe

cific

hea

t cap

acity

in J

/gK

Temperature in C

Fig. 20 The upper curves show the temperature dependences of thespecific heat capacity of semicrystalline iPS (4 mg), crystallized at140°C for 12 h, at heating rate 10 Kmin−1 [125]. The straight line isthe specific heat capacity for amorphous iPS from the ATHAS-DB[27]. The lower curve shows the difference between sample heatcapacity and heat capacity of the amorphous PS, as needed forintegration by use of Eq. 21

80 100 120 140 160 180 200 220 2401.4

1.6

1.8

2.0

2.2

2.4

c p two phase model

c p three phase model

c p crystal

Tc = 140 C; t

c = 12 h

Spe

cific

hea

t cap

acity

in J

/g*K

Temperature in C

cp liquid

Fig. 22 Specific heat capacity of iPS crystallized at 140°C for 12 h ata heating rate of 10 Kmin−1. Expected heat capacities [27] for theliquid iPS and for the crystalline and semicrystalline iPS, inaccordance with two and three-phase models, from the data at theglass transition (see text) are also shown

1604 C. Schick

Here the first two terms represent baseline heat capacityand the third term the excess heat capacity due to fusion. Justabove the glass transition the measured specific heat capacityis significantly smaller than expected. To explain thisobservation beside the mobile amorphous fraction (MAF,wMA) a rigid amorphous fraction (RAF, wRA) was introduced[28, 170, 171] (see also the discussion following Fig. 9,above). The RAF is not crystalline but does also not con-tribute to the common glass transition (at about 100°C foriPS). Above the common glass transition the RAF stays inthe glassy state and devitrification occurs at higher temper-atures. The temperature at which the RAF devitrifies isgenerally not known, because it is often superimposed bymelting.

From Eqs. 20 and 21 it follows that the too small heatcapacity above the glass transition yields an additionalexothermic contribution and consequently the decrease ofthe degree of crystallinity in Fig. 21 below 140°C. Thisbehavior causes serious problems if the degree of crystal-linity at room temperature is of interest. In order to take theRAF into account, the heat capacity for a three-phase modelcan be calculated by use of the equation:

cp;three phase modelðTÞ ¼ wMAðTÞð Þ � cp;AðTÞ þ wCðTÞ� cp;CðTÞ þ wRAðTÞ � cp;CðTÞ

� hAðTÞ � hCðTÞð Þ � dwC

dTð24Þ

with wMA þ wRA þ wC ¼ 1 ð25ÞHere the first three terms represent baseline heat capacity

and the fourth term the excess heat capacity due to fusion.If heat capacity and enthalpy functions are known, e.g.from the ATHAS-DB [27], Eqs. 22 and 23 can be solved.Because partition between wMA and wRA is not known,Eqs. 24 and 25 cannot be solved without further assump-tions. If at the lateral surfaces of each lamellae an RAFlayer with the same thickness (often ca. 2 nm) exists, theproblem can be solved iteratively [165, 166]. Righetti et al.developed a method based on an assumption regarding theavailability of baseline heat capacity in certain temperatureregions from TMDSC measurements [167, 168].

From TMDSC measurements during isothermal crystal-lization of iPS it is known that the rigid amorphous fractionis formed during isothermal crystallization [78]. On theother hand fast scanning data (Fig. 13) indicate the be-ginning of a continuous melting–recrystallization–remeltingprocess in the vicinity of the annealing peak [125]. Such acontinuous melting–recrystallization–remelting process cer-tainly requires mobility in the neighborhood of the moltencrystals and, consequently, devitrification of the rigid

amorphous fraction. This is further supported by theobserved step in heat capacity at the annealing peak. Bymaking use of these arguments construction of baselineheat capacity also becomes possible. Above glass transition(110–140°C) baseline heat capacity equals measured heatcapacity and takes into account the existence of a rigidamorphous fraction. At the annealing peak a step-likeincrease in heat capacity towards the expected value from atwo phase model is assumed to describe devitrification ofthe rigid amorphous fraction. Such baseline heat capacity isindicated in Fig. 23.

Using baseline heat capacity from Fig. 23 and Eqs. 24and 25 allows determination of all three fractions of asemicrystalline polymer as a function of temperature. Theresults for the mobile amorphous, rigid amorphous, andcrystalline fractions are shown in Fig. 24.

Now crystallinity stays constant up to the annealingpeak. There crystallinity decreases slightly while the rigidamorphous fraction disappears and the mobile amorphousfraction increases, as is assumed for construction of base-line heat capacity. Above the annealing peak crystallinity,and because of the assumption of the validity of a two-phase model, the mobile amorphous fraction stays nearlyconstant because of the continuous melting–recrystallization–remelting process. At about 200°C crystallinity increasesa few percent before final melting. Because we know a lotabout the particular behavior of isothermally crystallized iPSwe can construct baseline heat capacity as described and endup with reasonable data in Fig. 24. For non-isothermallycrystallized iPS this procedure does not work, because we donot know at what temperature the rigid amorphous fractiondevitrifies. Obviously it would be best to measure baselineheat capacity directly. Temperature-modulated calorimetry

80 100 120 140 160 180 200 220 2401.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

Tc = 140 C; t

c = 12 h

cp cryst ATHAS

cp amorph ATHAS

cp base line

Spe

cific

hea

t cap

acity

in J

/g*K

Temperature in C

Fig. 23 Measured and baseline heat capacity for iPS isothermallycrystallized at 140°C for 12 h as a function of temperature for heatingat 10 Kmin−1. For details see text


may provide a tool for such measurements if reversingmelting can be excluded. As discussed in the section“Temperature modulation”, temperature modulation atsufficiently high frequencies offers the possibility of directmeasurement of baseline heat capacity. A first attempt atsuch measurements is shown in Fig. 25 for PCL.

Neither on heating nor on cooling are peaks seen in thereversing heat capacity. The measured specific heat capacityat 80 Hz most probably corresponds to baseline heat capacity.In the example of PCL no separate steps as an indication ofdevitrification or vitrification of a RAF are observed. For amore detailed analysis of such “high” frequency heat capacity

data the frequency range must be further extended and theprecision of the data increased.

Availability of baseline heat capacity from indepen-dent measurements would enable more correct determi-nation of heat of fusion and degree of crystallinity. Buteven more important is the possibility of understandsolidification, a combination of crystallization and vitri-fication of the RAF, from a measurement of baseline heatcapacity in more detail. The same is true for softening ofsemicrystalline polymers on heating, which can beregarded as superposition of melting and devitrificationof the RAF. Until now there is no generally acceptedrelationship between the melting (crystallization) anddevitrification (vitrification) of the RAF. PET and iPSare typical examples of that, although there is disagree-ment on this [78, 125, 135, 143, 172–175].

Summary

Although the first calorimetry on polymers was performedin the first half of the twentieth century (reviewedelsewhere [176]), only since the introduction of differentialscanning calorimeters (DSC) in the early 1960s [19, 20] hascalorimetry became a standard tool in polymer science. Theadvantage of DSC compared with other calorimetric tech-niques lies in the broad dynamic range of heating and coolingrates, including isothermal and temperature-modulated oper-ation. Today 12 orders of magnitude in scanning rate can becovered by combining different types of DSCs. Rates as lowas 1 μK s−1 are possible [177] and at the other extremeheating [150] and cooling [140] at 1 MK s−1 and higher arepossible. The broad dynamic range is, especially, of interestfor semicrystalline polymers, because they are commonly farfrom equilibrium and phase transitions are strongly time(rate)-dependent.

Nevertheless, there are still several unsolved problemsregarding calorimetry of polymers. In this personal review Itried to address a few, for example determination of baselineheat capacity, which is related to the problem of crystallinitydetermination by DSC, or the occurrence of multiple meltingpeaks. I tried to indicate possible solutions by usingadvanced calorimetric techniques, for example fast scanningand high-frequency AC (temperature-modulated) calorime-try. It is my hope that further miniaturization of calorimeterswill enable even higher rates and frequencies and moredetailed study of surface and interfacial effects.

Acknowledgments I acknowledge the valuable contributions of allmy coworkers, especially A. Minakov, M. Merzlyakov, A. Wurm,H. Huth, S. Adamovsky, and E. Zhuravlev, and financial support fromthe German Science Foundation (DFG) and the European Unionwhich made this paper possible.

180 240 300 360

1.0

1.5

2.0

2.5

310 320 330 3401.8

2.0

cp,amorphous

Spe

cific

hea

t cap

acity

in J

/gK

Temperature in K

cp,crystalline

end of melting

glass transition

start ofcrystallization

Fig. 25 Specific heat capacity of PCL from DSC scans (dotted lines,sample mass ca. 10 mg; Mettler Toledo DSC 822) and temperature-modulated chip calorimeter (thick lines, sample mass ca. 200 ng,temperature-modulation amplitude ca. 0.4 K at modulation frequency80 Hz). Scanning rate for both measurements 1 Kmin−1

120 140 160 180 200 220 240

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0F

ract

ion

Temperature in C

mobile amorphous

crystalline

rigid amorphous

Fig. 24 Mobile amorphous, rigid amorphous, and crystalline fractionsfor iPS isothermally crystallized at 140°C for 12 h, as a function oftemperature for heating at 10 Kmin−1. For details see text

1606 C. Schick

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Differential scanning calorimetry (DSC) of semicrystalline polymers

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