Theory of DTA: historical basis SIDNEY SPEIL Application of thermal analysis to clays and alumino minerals Report of Investigation, Bureau of Mines, R.I. 3764 (1944) RIARJORIE J. VOLD Differential tharmal analysis Anal chem 21 (1949) 683 HAROLD L. SMYTH Temperature Distribution During Mineral Inversion and Its Significance in DTA J Amer Ceram Society 34 (1951) 222 S. L. BOERSMA A Theory of DTA and New Methods of Measurement and Interpretation J Amer Ceram Society 38 (1955) 282 IVO PROKS Rate of heating and its impact on the evolution of DTA curves Silikaty 5 (1961) 114 JAROSLAV SESTAK, PAVEL HOLBA Theory and practice of thermoanalytical methods based on the indication of enthalpy changes Silikaty 29 (1976) 83
Transcript
Theory of DTA: historical basis SIDNEY SPEIL Application of thermal analysis to clays and alumino minerals Report of Investigation, Bureau of Mines, R.I. 3764 (1944) RIARJORIE J. VOLD Differential tharmal analysis Anal chem 21 (1949) 683 HAROLD L. SMYTH Temperature Distribution During Mineral Inversion and Its Significance in DTA J Amer Ceram Society 34 (1951) 222 S. L. BOERSMA A Theory of DTA and New Methods of Measurement and Interpretation J Amer Ceram Society 38 (1955) 282 IVO PROKS Rate of heating and its impact on the evolution of DTA curves Silikaty 5 (1961) 114
JAROSLAV SESTAK, PAVEL HOLBA Theory and practice of thermoanalytical methods based on the indication of enthalpy changes Silikaty 29 (1976) 83
R. I. 3764
July 1944
united states
Department of the Interior
Harold L. Ickes, Secretary
BUREAU OF MINES
R. R. Sayers, Director
REPORT OF INVESTIGATIONS
APPLICATIONS OF THERMAL ANALYSIS TO
CLAYS AND ALUMINOUS MINERALS
The work upon which this report was based was done by the Bureau of Mines, United
States Department of the Interior, in cooperation with the Tennessee Valley Authority
BY
Sidney Speil
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R.I. 3764,
July 1944.
REPORT OF INVESTIGATIONS
UNITED STATES DEPARTMENT OF THE'INTERIOR - BUREAU OF MINES
APPLICATIONS OF- THERMAL ANALYSIS TO CLAYS AND
' ALUMINOUS- MINERALS!/
By Sidney Speilfj/
CONTENTS
Page
Introduction 3
Acknowledgments 4
Experimental technique 5
Apparatus 5
Procedure 5
Thermal curve 6
Experimental results 9
Rate of temperature rise 9'
Standard thermal curves 10
1. Kaolin minerals 11
2. Montmorillonite and related minerals 12
3. Accessory minerals 15
a. High-alumina minerals 15
b. Miscellaneous minerals 15
Quantitative analysis of mixtures 16
Georgia kaolin-quartz 17
Georgia kaolin-diaspore 17
Kaolin-bent onite , 18
Characteristic thermal peaks 18
Particle size 21
1. Differential thermal curve 21
2. Hydr othermal curves 22
1/ The Bureau of Mines will welcome reprinting of this paper, provided
the following footnote acknowledgment is used: "Reprinted from
Bureau of Mines Report of Investigations 3764."
2/ Associate nonmetals engineer, Bureau of Mines Electrotechnical
Laboratory, Norris, Tenn.
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R.I. 3734
CONTENTS Cont'd.
Pa2*e
Commercial clays .p>##> "~~24~~
1. Kaolins â– ^^^ * 24
2. Ball clays 25
Fractionation of Georgia kaolins !!!!!!!!!!!!!!!!!!!! 27
13. Calibration'curves'ior'Edgar (Fla.) plastic kaolin and Texas
montmorillom'te' mixture's. t 18
14. Effect of particle* size oh thermal analysis curves of ground
Pioneer (Ga.) kaolin, scale C. 20
15. Effect of particle size on the thermal decomposition of ground
Pioneer (Ga.) kaolin. 22
16. Thermal analysis curves of typical kaolins, scale C. 22
17. Thermal analysis curves of typical ball-clays, scale C. 24
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R.L 3764
ILLUSTRATIONS- C ont'd
Fig. Page
18. Thermal analysis curves of fractionated Pioneer (Ga.)
kaolin, scale C. 26
19. Thermal analysis curves of high-alumina Pacific Northwest
clays, scale C. 28
20. Thermal analysis curves of several North Pacific Coast
clays, scale C. 28
21. Thermal analysis curves of bauxite flotation products,
scale C. 32
22. Miscellaneous thermal analysis curves, scale A. 34
INTRODUCTION
In the study of clays and related minerals, the analytical techniques
commonly employed in mineral analysis are not always applicable. The
X-ray examination of a clay sample usually will show the major constitu-
ents, but often minorxlay impurities will not be detected because of line
broadening and the similarity of the main lines of clay: mineral diffraction
patterns. The petrographic analysis of clay minerals with clay particles
as small as a few microns is usually difficult. With still smaller particles,
a gross effect often is noted in which a mixture of several constituents
gives the appearance of a single constituent with the mean optical charac-
teristics of the mixture.^/ The electron microscope has recently been
used to study clay particle shapes of various aluminosilicates.4/ However,
this method has not yet been sufficiently explored. The chemical method
in which a mineralogical analysis is calculated from the oxide chemical
analysis has been discussed by Wilson^/ and others, and many of the short-
comings have been pointed out. It should also be recognized that the vari-
able alkali and alkaline earth content of bentonites makes it. inaccurate to
assign these basic elements arbitrarily to feldspar or mica in any clay that
contains appreciable bentonite.
3/ Grim, R. E,; and Rowland, R. A., Differential Thermal Analysis of
Clay Minerals and Other Hydrous Materials: Am. Mineral., vol.
27, 1942, pp. 746-761. '
4/ (a) Humbert, R. P., Particle Shape and the Behaviour of Clay as
Revealed by the Electron Microscope: Bull. Am. Ceram. Soc,
vol. 21, 1942, pp. 260-263.
(b) Humbert, R. P., and Shaw, B., Studies of Clay Particles with the
Electron Microscope: Soil Science, vol. 52, 1941, pp. 481-483.
5/ Wilson, Hewitt, Ceramics, Clay Technology: McGraw Hill Book Co.,
New York City, 1927, pp. 39-46.
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R.I. 3764*
As an alternatives process, the thermal analysis method was
selected. Starting with the original work of Le Chatelier,6/ this was
developed into a semiquantitative method by Granger,J/ Orcel,§/ Jourdain,^/
NortonlS 11/and Orim.12/ Hendricks!3/has used the differential thermal
method in studying the hydration of montmorillonite. Ins ley and EwelUi/
have studied the thermal decomposition of the kaolin minerals. The essen-
tial step in the process is the determination of the temperature at which
any thermal reactions take place in the sample being studied and the mag-
nitude of these thermal effects. The determination is most simply carried
out by heating two specimens at a constant and equal rate, one being the
clay sample and the other a thermally inert substance, and noting any tem-
perature differences between the two by means of a differential thermocouple,
The applications and limitations of this method to the study of various clays,,
bauxites, and aluminous minerals will be discussed in this paper.
ACKNOWLEDGMENTS
*
This investigation was made under the general supervision of
O. C. Ralston, principal chemical engineer, Bureau of Mines, Washington,
D. C:, and Hewitt Wilson, supervising engineer, Electrotechnical Lab-
oratory, Norris, Tenn. Acknowledgment is made to H. R. Shell, assistant
chemist, for chemical analyses and to the following-for supplying samples
of pure minerals:
6/ LeChatelier, H., (The Action of Heat on Clays): Bull. Soc. Franc. Min.,
vol. 10, 1887, pp. 204-211.
7/ Granger, A., (Thermal Analysis of Clay): Ceramique, vol. 37, 1934,
p. 58.
8/ Orcel, J., (Differential Thermal Analysis in the Determination of Clays,
Laterites, and Bauxites): Cong. Internat. Mines, Met. Geol. Appl.,
7th sess., Paris, 1935, pp. 359-373.
9/ Jourdain, M. A., (Studies of the Constituents of Refractory Clays by
. , Means of Thermal Analysis): Ceramique, vol. 40, 1937, pp. 135-141.
10/ Norton, F. H., Critical Study of Differential Thermal Method for the
Identification of the Clay Minerals: Jour. Am. Ceram. Soc.,- vol. 22,
1939, pp. 54-63.
11/ Norton, F. H., Analysis of High Alumina Clays by the Thermal Method:
Jour. Am. Ceram. Soc, vol. 23, 1940, pp. 281r-282.
12/ Grim, R. E., and Rowland, R. A., Work cited in footnote 3. '.
13/ (a) Hendricks, S. B., Nelson, R. A., and Alexander, L. T., Hydration
Mechanism of the Clay Mineral Montmorillonite Saturated with
Various Cations: Jour. Am. Chem. Sec, vol.. 62, 1940, pp. 1457-
1464. * â–
(b) Hendricks, S. B., Base Exchange of the Clay Mineral Montmorillon-
ite for Organic Cations and Its Dependence upon Adsorption Due
to Van de Waal's Forces: Jour. Phys. Chem., vol.' 45,1941, pp. 65-81.
\Aj Insley, H., andEwell, R. H., Thermal Behavior of Kaolin Minerals: Nat.
Bureau of Standards, Jour, of Research, vol. 14, 1935, pp. 615-627. ;:
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n Diff.
c
T.C.
T.C.
"I Rl
r—•-♦v'www-i—|
-r \ +â–
H
Motor
^0
Vanac
• • * 4 ii
R? -k
To galvanometer
^k
c T.C. Dry
cell
Automatic-
50°C. signaler
o °E
♦ 4 t ♦
m*
ii l in
3-position
temperature
controller
with cam
program
unit
Wiring diagram for thermal analysis apparatus
wywwww,wwAmTOm
E
b.'?tr>r>t;rrf»?/s/ss///w///ttl///t/A
B Vertical line source
C Horizontal line source
D Galvanometer
E Horizontal slit
F 1-R.P.H. motor
G Aluminum drum
H Sensitized paper
SECTION A-A
Recording system of thermal analysis apparatus
Figure 2.- Wiring diagram and recording system
of thermal analysis apparatus.
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RI. 3764
Herbert Insley, National Bureau of Standards
W. F. Foshag, U.S. National Museum
G. P. Nutting, Geological Survey
F. H. Norton, Massachusetts Institute of Technology
L. H. Berkelhamer, Bureau of Mines, Tuscaloosa, Ala.
R. E. Grim, Illinois State Division of Geology.
EXPERIMENTAL TECHNIQUE
Apparatus
The test apparatus used was similar to that described by Norton.-Пу
The samples are put into laterally positioned holes in a nickel block so
that each will receive equal heat treatment. This block is supported within
a small inside -wound crucible furnace equipped with replaceable heating
units (Arthur Thomas 5768, type 82). By insulating the furnace with alum-
inum foil, the required rate of temperature rise can be easily attained from
a 115-volt power supply controlled by a Variac voltage regulator. This type
of furnace reduces the thermal lag between the heating elements and the
nickel block. The measuring thermocouple, placed in the nickel block, also
acts as the controlling thermocouple for a three-position, program-con-
troller unit. The two junctions of a platinum-to-platinum, 10-percent -
rhodium differential thermocouple were placed centrally in the two speci-
men cavities, one of which contained calcined alumina, which is thermally
inactive up to 1,00C° C. The temperature difference between the two junc-
tions was then recorded photographically by reflecting a vertical line
source of light from an L. & N. type-P galvanometer onto a sheet of bromide
paper rotating behind a horizontal slit. At every 50° C. interval, as indicated
by the control thermocouple in the nickel block, a signal light was flashed
automatically, giving a direct temperature scale on the record. The galvano-
meter suspension and reflection distance from the galvanometer mirror to
the photographic sheet were chosen to give appreciable deflections for the
small thermal effects of bentonites.
In addition, tv/o series resistances were used to change the sensitivity
of the galvanometer system depending on the type of sample being investi- , .
gated. The furnace is shown in figure 1 and the control and recording system
in figure 2a and b.
-;- Procedure
Samples were ground to minus 100-mesh and placed in a desiccator v
over anhydrous magnesium Perchlorate for 24 hours before being placed /;..
in the test cavity in the nickel block. They were not dried in an oven,
15/ Norton, F. H., Work cited in footnote 10.
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B.L 3764
since the higher temperature would very markedly affect the initial
portion of the peak for many of the fine-grained minerals containing
absorbed water. By initially packing material on each side of the
thermocouple, forcing it under the couple wire, and then packing
the balance of the sample over the wire with a tightly fitting plunger '
it was, almost invariably, possible to pack the 0.35-gram specimen '
into the cavity without moving the thermocouple junction from its
central position.
The specimen and standard cavities were covered with a nickel
shield and a thin sheet of insulating brick to eliminate direct radiation
effects through the top of the sample. The furnace was warmed slightly
before the start of the test to minimize the initial thermal lag and then
slid over the stationary nickel block in its refractory holder. The ex-
ternal resistance of the thermocouple circuit was set at the value ap-
propriate for the type of specimen being studied. The sensitivity was
also varied during some runs to magnify certain sections of the curve
more than others. The furnace was brought to 1,000° C. or slightly higher
in some cases (for example, with talcs) at the rate of 12° C. per minute,
in all cases except for five special runs made to study the effect of '' :i
temperature rise.
Thermal curve
If a record were made of water loss vs. temperature for pure
kaolinite under equilibrium conditions, that is, by allowing the clay
to remain at a constant temperature and attain equilibrium before each
water loss determination was made, we would obtain a curve similar to
A'of figure 3,16/.which indicates that the reaction is not instantaneous.
On the other hand, the thermal analysis curve is a dynamic and not a
static or equilibrium record of thermal reactions occurring within the
sample being studied. On the thermal analysis record, this dehydration an
and decomposition reaction would appear to extend over a longer temper-
ature range because the actual temperature of the sample continues to â– ';
rise during the reaction. A downward deflection ofrthe curve indicates
an endothermic reaction while an upward deflection is caused by evolu-
tion of heat by the specimen.
In describing thermal analysis curves, the following conventions,
will be used: (1) Any deflection of the record from a straight line will
be called a peak whether a downward or upward deflection. The modi-
fying words "endothermic" or "exothermic" will describe the type of
peak. (2) The words "actual temperature" or "temperature" of the
16/ Nutting, P. G., The Bleaching Clays: Geol. Survey Cir. 3, 1933, 51 pp.
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H12
o 10
6'
â– : 8
6
CO
CO
o
*
'
0
+
T
Hydrothermal curve
Differential thermal curve
1
0 100 *0 300 t0 500 600 700 800
TEMPERATURE, °C.
Figure 3.- Hydrothermal and thermal analysis curves of kaolin,
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R.I. 3764
sample will be reserved to describe horizontal displacement along the
thermal curve, that is, actual temperature of the inert standard alumina.
(3) The vertical extent of the peak will be described by its relative height,
or if actual numerical data are used the terms "differential temperature"
or "<4T" will be employed. (4) The area of a peak will be that area in-
cluded by the actual curve and a line joining the two points where the de-
flection starts and ends (points a and c of fig. 3, B).
As the thermal curve is a differential function, we need consider only
those thermal effects that do not occur simultaneously and equally in the
specimen and in the standard alumina. Assuming that the two samples are
heated equally, normal heat input from the furnace may be neglected in ana-
lyzing the thermal analysis curve. Therefore, two opposite thermal effects
must be considered - the heat of the thermal reaction and the differential
heat inflow from the nickel block to the specimen which, in the case of an
end other mic reaction, is at a lower temperature than the standard. An
exothermic reaction will lead to a higher temperature in the sample than in
the standard alumina and will be indicated by an upward deflection of the
curve. The decomposition of kaolin, indicated by the valley abc of figure
3, B, may be used as an example of an endothermic reaction. At point b
the slope of the thermal curve is equal to that in the normal part of the
curve. The temperature differential between the two thermocouples is not
changing, and therefore the resultant heat input into each cavity must be
the same. Thus at this point the rate of heat absorption by the chemical
reaction must equal the rate of differential heat conductivity into the clay
specimen. The rate of heat absorption then continues to decrease more
rapidly than the equalizing heat input falls, and at some point d between
b and c the reaction ceases. However, this point cannot be established
exactly, and measurements are therefore made using the more definitely
established points a and c as limits.
Under static conditions, the heat effect would cause a rise in
temperature of a specimen consisting of kaolin and any thermally inert
substance given by:
T = M (zm),
M0.C
wherein: M = mass of reacting kaolin,
H = specific heat of reaction,
MQ = total mass of specimen,
C = mean specific heat of specimen.
However, we must also take into account the heat flow from the
nickel block toward the centers of the two sample cavities.
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R.I..3764
For any point between a and c (fig. 3,B) the simplified equation for
the thermally active constituent is:
rx
M I dH
s>
x
J ^- dt + g.kx | (T^T^dt = (T0-T1)M0.C1
and for the inert sample is:
' „x
g'k2| (To-T2)dt = (To"T2)Mo-C2 -
a â– â– .. ' ; .
wherein: t—time,
g = geometrical shape constant,
k2 = conductivity of the specimen,
kg = conductivity of the standard alumina,
TQ = temperature of the control couple in the nickel block,
T-, = temperature at the center of the sample,
To = temperature at the center of the standard alumina;
letting C2 = C1 +AC,
..and Kn = k +A.k.
2 1
gives the combined equation:
.x
x
X
M
§- dt H- g.k2
(T2-Tx) dt -g.&k (T0-T2)dt=
a
a
Mo(T2-Tl)Cl-Mo(To-T2)AC* ' '
As To-T-, = [\T = temperature indicated by the differential thermo-
couple, the equation can be considerably simplified by assuming that the
terms containing A C and./jk are small in comparison with the other terms,
By using "a" and "a" and "c" as integration limits, we get:
M
dH
dt
dt - g.k-L
A Tdt Z/\TQ • M0-C1 ;
o
a
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12° C/min.
16° C./min.
20° C./min.
600 °C.
Figure Y.- Effect of rate of temperature rise on the thermal curve of
Pioneer (Ga.) kaolin.
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R.I..3764
but/j T = 0,by definition of point c,
c
ra
dH
and 2±L dt = A H
j
c
so we finally get the relation:
M (Z\H)..
ac
g.kx
At dt.
u
a
The last expression is proportional to the area enclosed by the
straight line ac and the curve, abc if the scale deflection is a linear
function of the temperature. This area can be measured easily by
using a planimeter. In the limits of practical application, the linearity
holds within t 3 percent. The total heat effect equals the product of the
specific heat of reaction and the mass of reacting material. It is pro-
portional, therefore, to the percentage of reacting material in a given
weight of sample. Although the derivation of this relation does not take
into account the temperature gradient in the sample itself and also
neglects the differential terms,it does serve as a basis for using the
thermal area as a measure of mineral percentage. Using this approxi-
mate equation, the area under the curve is a measure of the total heat
effect, and if the conductivity is constant, the area under a peak should
be proportional to the amount of thermally active material present. In- '
the non-rigourous derivation, the area is considered to be independent
of the specific heat. However, this factor will actually affect the shape
of the peak and may also affect the area slightly.
EXPERIMENTAL RESULTS
Rate of Temperature Rise
The mathematical derivation is not confined to any specific rate of
temperature rise; therefore, the area should be independent of the rate
of temperature rise, except for the effect of variation of second order '
differential quantities with rate of temperature rise. A series of ex- -
periments was conducted on a Georgia kaolin to check this assumption.
The thermal-analysis curves obtained are! shown in figure,,. 4 and the data
from them in table 1. On this figure the heavy vertical line is at 600° C.,
and each line represents a 50° c. interval.
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R.I. 3764
TABLE 1. - Effect of rate of temperature rise on
the thermal analysis curve
Rate of rise,
Area
mm
Position of
Return to zero
°C./min.
peak °C
deflection, °C.
5
6030
575
620
8
6030
605 .
700
12
6100
625
700
16
6160
630
740
20
6200
650
750
Table 1 shows that the areas are equal within 3 percent although
there seems to be a slight tendency for smaller areas with low heating
rates. This variation is, however, within the experimental error of the
method. The slower the run, the lower and broader the peak when plotted
on the time scale used in making the thermal analysis record, because the
differential heat inflow into the specimen tends to keep the temperature
differential low. The apex of the peak and the return to zero deflection
occur at a lower actual temperature on the record with slow heating. r":ie.
These results are due primarily to the dynamic character of the test.
The actual peak temperature is the point where the differential heat
input equals the rate of heat absorption^ and therefore
.t
dH\
M
L\ T (peak) =\dt /(peak) g,k
Akigh rate of heating will causeiS_ to increase because more of the' re-
, '' dt
action will take place in the same interval of time, and therefore the
height at the apex or the differential temperature (AT peak) will be
greater. Asthe return to the zero line is a time function as well as a
temperature-difference function, the return to the zero line will occur
at a higher actual temperature with more rapid heating.
Standard Thermal Curves
Norton 17/ and Grimi^/ give thermal curves of many pure mineral
types. Most of these have also been analyzed in this laboratory, but only
the curves of those minerals essential to the study of common clays and
kaolins.will be discussed here. The sensitivity scale of each curve in
17/ Norton, F. H., Work cited in footnotes 10 and 11.
18/ Grim, R. E. and Rowland, R. A. Work cited in footnote 3.
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V O L U M E 21, NO. 6, J U N E 1949 683
( 3 ) Carson, F. T., Natl. Bur. Standards, Circ. C 445 (1944). (4) Clerc, L. P., “Photography,” New York, Pitman Publishing
(9) Lingane, J. J., IND. ESG. CHEM., ANAL. ED., 15,583-90 (19431. (10) Lingane, J. J., and Meites, L., Jr., I b i d . , 19, 159-61 (1947). (11) Mueller, 0. H., I b i d . , 14, 99-104 (1942). (12) ~ ~ ~ l ~ ~ , J, K., A ~ ~ ~ , cHEM., 19, 368-72 (1947).
(14) Zlotowski, I., and Kolthoff, I. M., J . Am. Chem. Soc., 66, 1431-5
Corp., 1944. (5) Institute Paper Chemistry, Paper Trade J . , TAPPI Section,
(6) Jablonski, V. F., and Moritz, H., Aluminium, 26, 97-9, 245-7 (13) Ib id . , 19, 478-80 (1947). 209-212 (1937).
(1944). (7) Koithoff, I. M., and Lingane, J. J., Chem. Revs., 24, 1-94 (1939). (8) Kolthoff, I. M., and Lingane, J. J., “Polarography,” New
(1944).
RECEIVED July 22, 1948. York, Interscience Publishers, 1941.
DIFFERENTIAL THERMAL ANALYSIS RIARJORIE J. VOLD
University of Southern California, Los Angeles, Calv.
Equations are derived which make possible the calculation of heats of trans- formation from differential heating curves, independent of external calibra- tions. The rate of restoration of a thermal steady state after a transformation is employed to establish the relation between the differential temperature and the heat absorption producing it. Valid results are obtained for such widely di- vergent processes as the melting of stearic acid and the vaporization of water, using exactly the same rapid and convenient experimental procedure. A fully automatic self-recording differential calorimeter is described, and the factors influencing the design finally adopted are discussed.
IFFERENTIAL thermal analysis is essentially a refinement D of the classical procedure of studying phase transformations by means of time-temperature records during uniform heating or cooling of the system. It appears to have been f i s t employed by Le Chatelier (6) in 1887 but has become popular again only re- cently. Examples of its employment are given by Kracek ( 6 ) for sodium sulfate, Partridge, Hicks, and Smith (8) for sodium poly- phosphates, Hendricks, Nelson, and Alexander ( 4 ) , Grim and Rowland (S), Speil et al. (10) and Norton (Y) for clays, and Vold (11-15) for soaps.
The experimental procedure consists of heating or cooling the sample side by side with an inert reference material in the same furnace, and measuring both the sample temperature and the temperature difference between sample and I eference material as a function of time. When a phase change occurs involving ab- sorption or evolution of heat, the temperature difference between reference and sample begins to increase; after the transformation is complete the temperature difference declines again, Thus each transformation produces a peak in the curve of temperature difference against time, from which it should be possible to derive information about the transformation temperature, heat of trans- formation, and rate of transformation.
This paper describes the construction and operation of a fully automatic differential calorimeter and presents an analysis of the course of the curve of differential temperature against time from which heats of transformation may be calculated.
DIFFERENTIAL HEATING CURVES
In order to analyze the differential heating curve, i t is conveni- ent to write down a formal expression for the rate a t which heat is transferred into and out of the sample or reference cell.
= K,(T, - T,) + u ( ~ , - T.) + L Y , ( ~ , - T,) ( l a ) dt
Here dq/dt is the rate a t which heat is received by the refer- ence material (subscript T ) and sample material (subscript s), respectively. K, and K. are heat transfer coefficients between the materials and the furnace wall. They are made as nearly identical as possible by choice of reference material and design of
cell and furnace. Sigma ( u ) is the heat transfer coefficient be- tween the cells, and alpha (ar and a.) is the heat loss (chiefly along the thermocouple wires) to the outside environment. T,, T,, T,, and To are the temperatures of the furnace wall, reference and sample materials, and external environment, respectively.
For experimental arrangements in which the furnace is a metal block with the materials contained in wells drilled in the block, a linear dependence of the rate of heat transfer on the temperature difference as given in the equations is certainly justified. When the materials are contained in metal capsules or cells suspended in air, and a portion of the transfer is by convection currents in the air, the validity of this assumption is open to question. White (16) has shown that heat transfer by convection, when the air flow in cylindrical space (up the walls and down the center of the cylinder) is not turbulent, is proportional to the square of the temperature difference between the walls and the center of the cylinder, but when the diameter of the cylinder is not small com- pared to its length he shows experimentally that the rate of heat transfer increases less rapidly with temperature difference, and approaches a first-power law.
Next use can be made of the identity
& = - - dH dH dT dt dt - ;it
In the case of the reference cell, d H / d t is simple C,, the heat capacity of the cell plus that of its contents. For the sample it is convenient to segregate the portion of the increased heat content arising from phase change, writing
(3)
Here C, is the heat capacity of the cell plus its contents, while AH is the heat of the transformation and d f / d t is its time rate of occurrence under the conditions of the experiment, f being the fraction of the sample transformed at any time t.
Every effort is made to have the two materials located sym- metrically within the furnace, so that one can write
( l a )
(4b)
K, = K, - 6K
cy* = LYr - 6cY
684 A N A L Y T I C A L C H E M I S T R Y
with the assurance that 6K and 6a are small. The various equations can then be combined to yield an expression for the rate of change of the differential temperature ( T , - T,) with time which is
At times when d f / d t = 0-Le., when the sample is not undergoing a transformation- Equation 5 can be integrated directly, subject to the assumptions that d T r / d t , ( T , - T 8 ) , ( T , - !Po), and C, and C, are independent of time and temperature. In the apparatus here de- scribed dT,/dt is controlled at a constant value. ( T , - T,) is a t least slowly varying. ( T , - T o ) appears with the coefficient 6a (small), so that its rise with increasing time can be safely neglected.
It is convenient a t this point to introduce a more compact nomenclature-Le., y = T, - T,; A = ( K , + a, + 2u)/C,; VI a given value of y a t t = t l , serving as a boundary condition, and
dT, d t . y8 = [(C. - C,) - - 6K(Tw - T,) + 6a(T, - T o )
( K , + a. + 2u) (6)
With these changes the integrated form of Equation 5 for d f / d t = Ois
y = y,( l - e-A(t--ti)) + y le -A( t - t i ) (7 )
I t is apparent that ya is a steady state value of the differential temperature achieved at a sufficiently long time after the initial condition y = yl at t = t l . At the outset of an experiment y, = 0 at t = tl = 0. The differential temperature rises to a value y,, dependent primarily on the difference in heat capacity of the sample and reference materials, the heating rate, and the heat transfer coefficients. After a transformation is complete, the differential temperature again approaches y8 according to the same equation. The Constant, A , can thus be evaluated from a plot of log ( y - y.) versus t . In the new nomenclature, Equation 5 may be written
Graphical or numerical integration of Equation 8 over the period of time during which the transformation is occurring then yields a value for the heat of the transformation.
Any time interval wide enough to include the whole transfor- mation may be taken. When d f / d t = 0 before the transforma- tion has begun d y / d t = 0 and y = ya. When d f / d t = 0 after the transformation, d y / d t is equal in value and opposite in sign to A ( y - y,). The integral of the second term on the right-hand side of Equation 8 is simply the area under the peak, while that of the first term is (yz - y l ) where y , and y~ are the values of y a t the beginning and end of the time period chosen.
In practice, in analyzing a differential heating curve, it is con- venient to plot (y - y,) against time, beginning a t the top of a peak. The points lie on a curve which becomes linear at the end of the transformation and thus yields a value of the time at which the transformation is over. The temperature of the sample at
L
T" 2t
t
T 6"
i t
A B Figure 1. Differential Calorimeter Furnaces - Metal (mild steel for A , brasa for B ) S&SSSSS Transite W////H& Asbestos magnesia packing C, C'. D . 0. Heating element
Glass tubes for entry of thermocouple leads Glass cell support for sample and reference cells
this time is of phase significance as the "liquidus" point in a binary system if it can be shown by experiment to be independent of the heating rate.
DIFFERENTIAL COOLING CURVES
If the rate of cooling is controlled, the analysis of differential cooling curves is identical to that given for differential heating curves. If the calorimeter is allowed to cool at its natural rate which is measured experimentally, an entirely similar, though numerically much more complex, analysis can be carried through. The complevity arises from the fact that y8 is now time-depend- ent. In many cases the rate of transformation on cooling is very small, so that the differential cooling curve exhibits a wide shallow dip whose area beneath the varying base line, us, is hard to determine with any accuracy. For these reasons, differential heating curves are much more convenient in the study of phase transformations in most cases.
RATE OF TRANSFORMATIOiY
Equation 8 can be used to calculate d f / d t at any time (and hence any T,) after AH has been obtained. -4t low heating rates, the values probably have little significance, as they are deter- mined primarily by the rate at which heat is made available to the sample. From differential cooling curves, or a t high heating rates, values of d f / d t may be an interesting property of the sys- tem under study.
INHERENT LIMITATIONS OF DIFFERENTIAL THERMAL ANALYSIS
Although sensitive and Convenient, this method contains two factors militating against its development into a technique of high precision. One is the assumption of a constant value of the heat capacity of the sample. The second is the assumption that the sample temperature is uniform throughout a t each time instant.
The heat capacity of the sample is that of the cell plus that of the transformed portion of the sample plus that of the untrans- formed portion. The relative proportions of transformed and untransformed sample change during the heating. In practice, if the heat capacity of the cell is deliberately made large, this fluctuation is minor, but sensitivity is sacrificed, because a given
0 25"
LO" WALL THICKNESS 0.0 05"
2.0"
and a high heating rate to achieve the same sensitivity. The most desirable type of furnace is believed therefore to consist of an air oven, suitably insulated. The designs given in Figure 1 have proved very satisfactory.
Furnace A consists of two concentric steel tubes 12 inches tall of 0,125-inch wall thickness, the outer 8 inches in diameter, the inner 2 inches. The inner is n rapped with a single layer of 0.125- inch asbestos cloth upon which are wound 20 feet of chrome1 A resistance wire (No. 22). The coil is more closely wound (4.75 turns per inch) around the upper and lower portions than around
v the central space (3.125 turns per inch). The
175"
A N A L Y T I C A L CHEMISTRY 686
15
10
v’ # R
c” I
c - 5
0
- 5 Y I I
40 40 60 80 TIME, MINUTES
Figure 4. Effect of Fluctuating Heating Rate on Differen- tial Heating Curves
1A. Fluctuating difference between actual and desired tempera- ture during heating (T - Tc)
1B. Cormaponding curve of temperature difference between ref- erence and #ample cella (Tr - TI) versus time. In both caees average
heating rate is 1.5’/min.
of aluminum alloy 24ST, but later stainless steel was employed because the threads machined in the aluminum alloy wore out very rapidly.
The control and measuring assembly is shown in Figure 3. The heating rate is measured and controlled by means of the
Brown Instrument Co. pneumatic time-temperature pattern controller. The desired time-temperature attern is plotted on an aluminum disk in polar coordinates a n z t h e disk is then cut evenly along the curve to give a control cam. This is mounted on the shaft of an electrically driven clock motor. A wheel rolling along the curved surface of the cam controls the position of a
ointer on the scale of the circular chart temperature recorder. &he furnace temperature is measured by means of a single-junc- tion iron-constantan thermocouple. When this temperature differs from the control temperature as indicated by the pointer, an e.m.f. proportional to the difference is a plied to open a valve by which air pressure is transmitted to a eonoflow air-operated motor, which drives the control spindle of an autotransformer (Variac model VMT5,O to 130 volts) governing the current pass- ing through the heating elements of the furnace. The drive is adjusted so that 30 volts are applied to the heating element when the applied air pressure is zero, and 130 volts when the applied air pressure reaches its maximum (14 pounds). The range of thi.; model is from 0 O to 400 O C.
A control knob enables the experimenter to apply a given frac- tion of the total available air pressure for a given percentage de- viation between recording pen and control pointer, based on the full scale. This is known as “per cent throttling.” Thus 20% throttling applies full air pressure for an 80” deviation, 10% for 8”, etc. Owing to lag in the furnace, low throttling gives con- siderable fluctuation of the temperature about its controlled value. With high throttling, the applied current is generally not enough to maintain the desired rate of heating, so that a further control is necessary. This is designated “automatic reset,” and provides a continuous, independent increase in the applied air pressure, again proportional to the deviation between actual and control temperature but applied a t a controllable rate to overcome the fluctuations that would result from a sudden increase in heat- ing current. The operation of these instruments has been de- scribed in greater detail (14).
Selection of the proper control settings for the given furnace is essential. Figure 4 shows two sets of curves for the deviation of
SLIDE WIRE UNIT
I - - - - - - - - - A U J U I I
(OR STANDARD CELL)
AMPLIFICATION VIA
ClRCul T BAT.
Figure 5. Wiring Diagram of Strip Chart Potentiometer
RA (3), 253.4 ohms; RC (4). 509.5 ohms; R E , 5.00 ohms; slide wire unit, 20 ohms. Ro (1) was originally 2.474 ohms but was made vari- able by placing a 20- to 150-ohm resistance in parallel with it. 2231 (2) was originally 1.474 ohms but was made variable at from 0.3
to 1.5 ohms, the lower figure giving maximum sensitivity.
actual and control temperatures, and for fluctuation of the dif- ferential temperature between sample and reference cells (in the absence of phase changes) for different values of throttling range and reset control. In set 1 the heating rate, dT,/dt, oscillates, and the differential temperature, y, oscillates also (as i t should according to Equation 7). The corresponding peaks in the curve of differential temperature against time simulate peaks due to phase changes. The better choice of control variable gives rise to the curves in set 2, where the y - t curve is almost a straight line, after time enough has elapsed for the temperature difference to reach its steady state value.
The temperature difference between sample and reference cells is measured by means of a three-junction iron-constantan thermo- pile, and recorded continuously on the Brown stripchart potenti- ometer. The thermocouple junctions are formed of No. 30 wire, spot welded, and insulated from each other by fiber glass sleeving further impregnated with Dow Corning silicone varnish 996. They have been found durable over periods of 2 to 3 months’ almost continuous use, and fail ultimately because of breakdown in the insulation.
The potentiometer as supplied has a pen travel of 12 inches (50 scale divisions) per 5 millivolts. As the rated sensitivity is such that full voltage is applied to the balancing motor for an e.m.f. across the thermocouples of 20 microvolts, i t is practicable to increase the sensitivity about fivefold by altering the resistance ratio in the arms of the bridge. The wiring diagram for the in- strument is given in Figure 5, showing this change and the addi- tion of a further auxiliary variable resistance to vary the position of zero e.m.f. on the scale. With this arrangement both positive and negative values of the temperature difference can be recorded; high values of the sensitivity are used for transformations having small heat effect and lower values for transformations in the same sample having larger heat effects.
Experimental Results. To test both the instrument and the equations described above, five runs were made on a specially pure sample of stearic acid (9) and one on Baker’s C.P. analyzed benzoic acid.
The powdered sample was tamped firmly into its calorimeter cell and hung in osition from the plug of the colorimeter furnace. The reference cef; was then filled with sufficient white mineral oil (Nujol) to give nearly the same calculated heat capacity, and assembly of the calorimeter was completed. After about 20 minutes the initial temperature difference between the cells (due to handling) had disappeared: and the automatic heating and recording had begun. For these relatively large heats of fusion the instrument was operated at a sensitivity of 0.131 O per scale
V O L U M E 21, NO. 6, J U N E 1 9 4 9 687
0
cg I
e 3 t P w Y
Y,
m
U
a
2 3
w
P
3 c
1
22
TIME
Benzoic Acids Figure 6. Differential Heating Curves of Stearic and
Pantograph reductions of automatic records. Arrows show time at which melting begins. Vertical full linea show time at which calculation indicates that transformation is complete. Vertical dotted lines correspond to breaks in automatic record produced
by deliberate test changes in sensitivity of recorder.
division for the measurenient of temperature difference as com- pared with its maximum of 0.089 O per scale division. However, in two cases the highest sensitivity was employed a t the beginning of the run, and decreased sensitivity later as required to keep the recording pen on the scale.
Pantograph reductions of some of the automatically recorded curves of temperature difference us. time are given in Figure 6.
Temperatures of Transition. The differential temperature, y, begins to rise as soon as the outside of the sample reaches a transition point, (Tm). The inside of the sample at the same instant, T,, hon-ever, is below T , by an amount dependent on the rate of heating and the thermal conductivity of the sample. The difference has been measured for the reference cell filled with Nujol and found to be of the order of 5" for a heating rate of 1.5' per minute. The thermal conductivity of the powdered sample, however, is not necessarily even approximately equal to that of Nujol nor reproducible from run to run, so no reliable esti- mate of the difference (T . - Tm) can be formed to serve as a systematic correction. By the time that the differential tem- perature is rising rapidly the sample temperature must be some- what above T , in order to keep up a steady flow of heat to the transforming portion. The best estimate of T , is therefore ob- tained by extrapolating the steeply rising portion of the curve backward to its intersection with the initial base line. As there is a degree of arbitrary choice involved in such an extrapolation, the values obtained are uncertain to about 3" to 4', with a tend- ency to be low rather than high. The arrows on Figure 6 show the times a t which the sample temperature was taken to be equal to Tm. The values obtained average 4" lower than the known melting points of the samples, as seen in Table I.
Heats of Transition. Heats of transformation were calculated from Equation 8.
Table I. Temperatures and Heats of Fusion of Stearic and Benzoic Acids5
Sample Tm Weight, $ T ~ / d t , AHc.lod.,
Run G. /Min. Obsd., Cal./G.
a M.P. of stearic acid, determined directly, was 69'.
b Run made using model B furnace and cells of Figures 1 and 2.
AH1 (8) is 47.6
All cal./g. For benzoic acid accepted m.p. is 122O and AH1 is 33.9 cal./g. ( 1 ) .
others with model A furnace and cells.
The value of A was first determined by lotting log (y - y.) against time. Two such plots are shown in gigure 7. The pointe fall on good straight lines after the transformation is over, a t the times indicated on the figure. At the end of the transformation, so determined, the base line y,, was assumed to have its h a 1 value-Le., c, equal to the heat capacity of the cell plus that of the melted sam le. At the beginning of the transformation-Le., the point segcted as T,,,-y, was assumed to have its initial value (c , equal to that of the cell plus that of the unmelted sample). The base line under the peak was taken as a straight line between these two points.
Figure 8 shows a typical run replotted to show the actual mag- nitude of TI, T,, and y. The base line and area taken in evaluat- ing AH are shown. The results, given in Table I, are obviously not of high precision, but when i t is remembered that they are absolutely independent of any empirical calibrations, they give an assurance of validity to the theoretical analysis.
In one experiment, employing technical stearic acid which meIted visually over a range from 56" to 60' C., two runs were made at heating rates of 0.5" and 1.5" per minute. Heat effects of 48.5 and 47.4 calories per gram were obtained, respectively.
To guard against the remote possibility that the agreement be- tween observed and theoretical values might be fortuitous, the same method was applied to an experiment in which powdered,
1 .P
1 .o
n
2 I 5 0.5 (3 s
0
-0.4
' \
50 60 70
Plot for Determining Constant A TIME, MINUTES
Figure 7. A governs rate at which a thermal steady state is re-established after e transformation occurs. Arrow. mark time et which transformation i s
finished; subsequent pointa lie on straight linen
688 A N A L Y T I C A L C H E M I S T R Y
125
J U ‘ I O O
4
0
K 3 I-
K E 3 I-
1 3
50
23
u
ui
4 13 $ I E
a ?
a t
B
F
L* 3 2
- 7 .. 0 PO 40 60
TIME, MINUTES
Figure 8. Acid and Nujol
I. Heating curve for Nujol cell. 11. Heating curve for stearic acid cell. 111. Differential heating curve. At 68‘ melting begins. a is time a t which calculation shows that melting is complete. Shaded area is that taken
into account in calculating AH.
Total and Differential Heating Curves for Stearic
of the cell as i t was heated at a rate of 1.5’ per minute. A value of 500 calories per gram for the heat of vaporization of n,ater was obtained. This type of experiment is of potential value in studying the thermal dehydration of clay minerals and similar problems.
LITERATURE CITED
(1) andrews, D. H., Lynn, G., and Johnston, J., J . Am.
(2) Garner, W. D., Madden, F. C., and Rushbrooke, J. E., J . Chem. SOC., 48, 1274 (1926).
Chem. SOC., 1926, 2941. (3) Grim, R. E., and Rowland, R. A , , Am. Mineral. , 27, 746-
(4) Hendricks, S. B., Nelson, R. A , , and Alexander, L. T.,
(5) Kracek, F. C., J . Phys . Chem., 33, 1281 (1929). (6) Le Chatelier, H., Z . physik. Chem., 1, 396 (1887). (7) Norton, F. H., J . Am. Ceram. SOC., 22, 54 (1939). (8) Partridge, E. P., Hicks, V., and Smith, G. IT., J . Am.
Chem. SOC., 63, 454 (1941). (9) Philipson, J. M., Heldman, M. J., Lyon, L. L., and Vold
R. D., Oil ck Soap, 21, 315 (1944). (10) Speil, S., Berkenhamer, L. H., Pask, J. A., and Davies,
B., U S. Bur. Mines, Tech. Paper 664 (1945). (11) Vold, R. D., J. Am. Chem. SOC., 63, 2915 (1941). (12) Vold, R. D., Grandine, J. D., 2nd, and Vold, M. J., J.
Colloid Sci., 3, 339 (1948). (13) Vold, R. D., and Vold, M. J., J . Phys. Chem., 49, 32
(1945). (14) Werey, R. B., “Instrumentation and Control in the Oil
Refining Industry,” Philadelphia, Pa., Brown Instru-
61, 801-18 (1942).
J . Am. Chem. Soc., 62, 1457 (1940).
ment Co., 1941.
York, Reinhold Publishing Corp., 1928. (15) White, W. P., “Modern Caloriheter,” pp. 74-6, New
oven-dried (1500) silica was used as reference material and the sample cell was packed with powdered glass and filled with water, which was allowed to vaporize through a hole drilled in the side
RECEIVED June 30, 1948. Presented before the Division of Physical and Inorganic Chemistry a t the 113th Meeting of the AMERICAN CHEMICAL S ~ ~ I E T ~ , Chicago, 111. Work supported in part by a grant from the Office of Naval Research, Contract NO. N6-onr-238-TO2; NR057057.
Analysis of Recycle Styrene P
AUGUSTUS R. GLASGOW, JR., NED C. KROUSKOP, VINCENT A. SEDLAK, CHARLES B. WILLINGHAM, AND FREDERICK D. ROSSINI
Sational Bureau of Standards, Washington, D . C .
Samples of “recycle” styrene, selected to be representative of stock from com- mercial processing, were analyzed with respect to their major components. The material w-as analyzed by a combination of procedure involving separation by distillation of the Cc portion of this material, high-efficiency azeotropic dis- tillation of the CS portion, and precision measurements of freezing points on the original material and appropriate portions of the distillates. The amounts of the various components in the recycle styrene samples, Rubber Reserve stand- ard blends 1, 2, and 4, respectively, were found to be as follows, in percentage by weight: 1,3-butadiene, plus other Ca hydrocarbons, 1.93, 4.50, 1.77; 4-vinyl-l- cyclohexene, 1.78, 3.84, 4.89; ethylbenzene, 1.36, 2.21, 3.20; styrene, 94.52,88.79, 88.63; Cg and higher hydrocarbons, 0.41, 0.66,1.51.
N COXXECTIOS with the government synthetic rubber pro- I gram, in the copolymer plants, the Office of Rubber Reserve requested the National Bureau of Standards to make analyses of selected samples of “recycle” styrene. The present report gives the method and other pertinent details of these analyses of re- cycle styrene.
SAMPLES ANALYZED
The samples of recycle styrene analyzed are identified as follows:
Rubber Reserve reference blend S o . 1, from the B. F. Goodrich Company, Agent for the Office of Rubber Reserve a t Port
Neches, Tex., was received on January 9, 1946, in a 2-gallon screw-cap can. The analysis was reported to the Office of Rub- ber Reserve on September 27,1946.
Rubber Reserve reference blend No. 2, from the B. F. Goodrich Company, Agent for the Office of Rubber Reserve a t Port Neches, Tex., was received on January 4, 1946, in a 2-gallon screw-cap can. The analysis was reported to the Office of Rub- ber Reserve on September 26,1946.
Rubber Reserve standard blend S o . 4, from the Copolymer Corporation, Agent for the Office of Rubber Reserve a t Baton Rouge, La., was received on June 12, 1946, in three 28-ounce bottles sealed with bottling caps. The analysis was reported on August 26,1946.
July 1951 Temperature Distribution During Mineral Inversion 22 1 Bcruard12 has shown that FeO in solid solutioii with COO
rcacts accordingly, 3Fe0 + COO @ Co + Fe30n. This reaction starts at approximately 1076 O F . , but the maximum temperature for the rcaction is 1652°F. I t was reported that cobalt needles aiid dendrites form at trmpcraturcs as high as 1800°F. The authors conductcd mother tcst, which was not reported in the papcrs. Iron oxide was spriukled on glass containing 0.5 and 8.0% cobalt oxide and fircd a t temperatures from 1200 to 1800 O F .
A t approximately 1700°F. all of the iron oxide bad bceii dissolved in the glass, but no metallic cobalt was visible. Also, iron oxide was smelted with the glasscs and no metallic cobalt was found in thc resultant glass. Thcsc tests further emphasize the fact that iron ions and cobalt ions do not react spontaneously in the glass structure to oxidize the iron and reduce the cobalt; however, data have indicated that hydrogen can be oxidized and cobalt rcduccd without iron ions being present.
Firing cnamcls in a vacuum should reduce thc amount of hydrogcii gcneratcd by the iron-water rcactions, sincc glasses
6.
R. M. Icing, “Mechanics of Enamcl Adherence: XIV (1) Role of Cobalt Oxide in Mctal and Oxide Precipitation During Grouud-Coat Firing Cycle and (2) Deteriniliation of Temperature and Time Intervals of Precipitation,” J . Am. Ceram. SOC., 26 [2] 43 (1943).
l 3 R D. Cooke, “Effect of Furnacc Attnosphcres on the Firing of Enamel,” ibid., 7 [4] 277-81 (1924).
l4 E. E. Geisinger and K. Berlinghof, “Effect of Furnace Gases Upon Glass Enamels,” ibid., 13 [2] 126-42 (1930).
l6 R. M. King, “Mechanics of Enamel Adhercnce: VIII, (A) Apparatus for Firing Enamels Under Accurate Control of Tempcrature, Pressure, and Atmosphere, (B) Studies in Firing Bnainels Undcr Reduced Pressurcs,” ibid., 16 [5] 232-38 (1933).
Temperature Distribution
fired in vacuum are dewatered. As a result of the reduction in the amount of hydrogen generated, a smaller number of cobalt- metal particles would be formed. C o ~ k e , ’ ~ Geisinger and Berlinghof, l4 and King15 all found normally processed enamels on iron fired in a vacuum, either “balled up,” foamed, or blistered badly. The degree of the defects depends principally on the inches of vacuum maintained, the composition of glass, which affects the amount of water in the glass, and the rate of heating. These factors affect the amount of iron oxide formed on the steel, and the time and amount of water vapor in contact with the iron. Completely dewatered glass should produce no reactions with iron, hence no hydrogen, no cobalt metal particles, and no ad- herence if an iron oxide layer cannot adhere to iron.
7. Hydrogen will be produced from glasses containing no metal oxides as long as the water in the glass has a chance of coming into contact with a metal (iron in the enameler’s case) which is capable of reducing the water. The degree of hydrogen produced is determined by the rate of rcaction and the avail- ability of the metal to the water vapor.
It is eiitirely possible to reduce the cobalt ion to metallic cobalt during the smelting operation in the absence of iron because of the dissociation of water at high temperatures; however, the amount of hydrogen available is not sufficient to maintain reduc- ing or neutral conditions. The cobalt metal would be instan- taneously reoxidized and redissolved in the glass. Strong oxi- dizing conditions are maintained.
The authors would like to emphasize the fact that as a result of the hydrogen generation and the weak adherence of iron oxide to iron, cobalt ions or other reducible and polarizable metal ions have to be added to the glass to consume the hydrogen in the enameling of iron.
During Mineral Inversion and Its Significance in Differential
Thermal Analysis by HAROLD 1. SMYTH
New Jersey Ceramic Research Station, Rutgers University, New Brunswick, New Jersey
Temperature distributions have been calculated in a thin slab of material for which the tempera- ture of the outside faces was raised at a uniform rate. Calculations cover the entire period dur- ing which the material is transformed from one form to another with an endothermic reaction. Application of these calculations to the inter- pretation of thermal analysis data is discussed.
1. Introduction N A usual form of differential thermal analysis apparatus‘ a metal block is placed in an electrically hcatcd furnace. The block has in it two cavities, equal in size and shape,
and, as far as is possible, symnietrically situated with respect to the block and the furnace. One cavity contains a material
I Received May 22, 1950; revised copy received September 15,
1950. The author is research professor a t Rutgers University.
Ch. V, pp. 76-93. Mc- Graw-Hill Book Co., Inc., New York, 1949. 782 pp.; Ceram. Abstracts, 1950, July, 145a.
F . H. Norton, Refractories, 3d ed.
whose thermal properties are being investigated; and the other, a reference material, such as alumina, whose thermal properties are reasonably well known and which does not itself show any polymorphic inversions or chemical reactions within the temperature range covered.
Located at the center of each cavity is a thermocouple. Proper electrical connection of these thermocouples enables the differential temperature between them to be recorded by a sensitive potentiometer. As the temperature of the furnace is raised a t a uniform rate, this differential temperature is plotted against time. On the same chart is plotted another temperature which may be the temperature of the furnace atmosphere, the temperature of the metal block, or, more usually, the temperature of the center of the reference ma- terial.
If at some temperature a transformation, such as a poly- morphic inversion, a loss of combined water, or any other chemical reaction, occurs in the material being investigated, and if this transformation requires heat or evolves heat, there will be a “bump” in the differential temperature recording. The bump will show on one or the other side of zero, depend- ing on whether the transformation is endothermic or exo- thermic.
222 Journal of The American Ceramic Society-Smyth Vol. 34, No. 7 For reasons which will become clear in the I A B M C D
analysis to follow, these bumps or peaks are not always instantaneous, but may spread over quite a range of temperature. I t was for the purpose of helping to interpret the exact mean- ing of the shape and location of these peaks that the calculations were undertaken. Only endothermic reactions were considered. The shape of exothermic peaks may be quite differ- ent since enough heat may be generated to trigger a more or less instantaneous reaction.
For reasons of mathematical simplicity the shape of the cavity was taken as the space be- tween two infinite parallel planes with the ther- mocouple located midway between the planes. The temperatures Of two bounding planes were equal and were assumed to be increasing at a uniform ratc. At each instant the boundary surfaces of the test specimen and reference specimen were equal. Although these shapes are physically unrealizable, the qualitative conclusions arrived at will ap- ply to cylindrical or other more practical shapes.
II. Temperature Distribution in Reference Specimen
Center of
e e AX-
Distance Distance from surface Fig. 1. Temperature gradients close to Fig 2. Diagram used in computing the
movement of the phase boundary as in- version proceeds.
boundary between low- and high-temper- ature forms.
distant planes separated from each other by a distance of AX cm., and the distribution is calculated at successive time intervals, each of which is called At seconds. If A x and At are so chosen that they satisfy the relation
(3) ( A.XY 4 t =
In both samples the heat flow is one dimensional, being always perpendicular to the boundary planes. If distance in thc direction 01 flow is denoted by x, then the temperature distribution in the reference sample and in the test sample, until the temperature of transformation is reached, is governed by the well-known heat-flow equation
where T is the temperature, t is the time, and a is the dif- fusivity of the material. If the zero of x is taken midway between the boundary walls, and if the walls are brought up at a uniform rate of Q: degrees per second, then the material will sooner or later arrive at a state where the temperature distribution is given by
where T, is a constant of the dimensions of a temperature. It is easy to verily that the distribution (2) actually satisfies the heat-flow equation (I), and that it represents a state in which the boundary walls (or, indeed, any point in the sample) are increasing uniformly in temperature.
It is assumed in the calculations that heating in both the reference and test samples has been going on for a time suffi- ciently long to allow the steady-state condition of equation (2) to be established in both samples.
Using equation (2), it is very simple to calculate the tem- perature distribution in the reference sample at any instant
111. Temperature Distribution in the Test Specimen
I t is assumed that at a certain temperature To a reaction in the test specimen takes place which requires, a t that tem- perature, L cal. of heat to convert 1 gm. of the specimen from the original state to the completely reacted state. This reaction may be a polymorphic inversion or any chemical reaction which takes place at a definite temperature.
Solutions of equation (1) which would satisfy the boundary conditions a t the surface of the sample and the conditions a t the interface of the unreacted and the reacted material could not be found. It was therefore necessary to use an approxi- mate method based on that given by Schack2 and attributed to Schmidt. In this method the slab is divided by equi-
the temperature at any one of these dividing planes at any instant will be the mean of the temperatures of its two im- mediate neighbors a t an instant At seconds earlier. Thus, if the distribution a t any instant in a uniform material and the boundary conditions are given, the distribution at any succeeding instant can be calculated.
The procedure used in calculating the activity a t the inter- face between the reacted and unreacted material is given. For simplification i t was assumed that the density p, specific heat C, and thermal conductivity k of the unreacted material were the same as those of the reacted material. As already stated L cal. of heat is required to convert 1 gm. of the un- reacted material to the completely reacted state at tempera- ture TO, the temperature at which the reaction takes place.
In Fig. 1 there is shown the temperature distribution close to the boundary between the unreacted and the reacted ma- terial if a chemical reaction is involved, or between the high and low forms of the material if a polymorphic inversion is being considered.
The heat flowing into the boundary layer from the left side is
- k - cal./sq. cm./second (3 and the heat flowing out to the right side is
- k r$)2 eal./sq. cm./second
The net gain in time At seconds is therefore
[- k (g)I -I- k r$)J At cal./sq. cm.
This is enough to convert a layer of thickness 6% from the low to the high form where
6x = "[- k (g:)l + k r 8 ) I . t P L x 1
Now At has been chosen such that ( Ax12
2a At = -
(4)
(3)
2 A. Schack, Industrial Heat Transfer, translated by Hans Goldschrnidt and Everett P. Partridge. John Wiley & Sons, Inc., New York, 1933. 371 pp.; Cerum. Abstracts, 13 [3] 64 (1934).
July 1951 Temperature Distribution During Mineral Inversion 223 I I I I I
a, E k- .-
.6 6 0 .66 Distance from center (cm.)
.66 0 06 .30 6 6 Distance from center (cm.)
Equation (8) gives the movement of the inter- face in time At expressed ,as a fraction A x which, in the practical calculations, is the most con- venient way to exmess it. It is necessary. how-
ever, to modify this to express ("'> a n d - ( g ) a x 1
as ratios of differences, rather than in their dif- ferential form.
For gj;, is used.
Therefore equation (8) can be rewritten in the form
In the 'calculations it is more convenient to exaress B M and MC as fractions of Ax.
B M MC Therefore - = f and ax = g is used, the
A x Fig. 3. Temperature distributions in the Fig. 4. Temperature distributions in the test reference material at intervals of 18.75 material ot intervals of 18.75 seconds.
quantities f and g being known. fore write for eauation (9)
One can there-
seconds. . ,
Hence This shows how, if the position of M at any instant of time and the temperatures, TZ and Tar are known, the position of M a t At seconds later can be calculated.
The temperature of B at the succeeding instant of time was taken to be the mean between TI and the temperature of
* = I [ - k rg), f k (:)A2 C if the slope is extrapolated a t B on to C as shown at T,' Ax PL (') on the drawing. Similarly the temperature a t C was taken
as the mean between Tz' and T,. The temperatures at the other planes were calculated as in the regular Schmidt2 method, the outside boundaries being brought up at the
2 P L [ - k ( g ) 1 + k r s ) ] @ ? x 2 2a (5)
or
But
a = - k ( 7 ) PC chosen uniform rate.
Thercfore
'This gives the distance the boundary moves in time At expressed as a fraction of Ax, the distance between neighbor- ing planes.
The actual method of evaluating the right side during the calculations is given. In Fig. 2 are shown four of the equally spaced planes, A , B, C, and D , by means of which the sample is divided for calculation purposes. These have been so chosen that the reaction interface M is between B and C. It is assumed that, by previous calculations, at a certain in- stant of time the temperatures TI , T2, T,, and Td at A , B , C, and D, respectively, are known, as well as the position of M between B and C. That is, the distances BM and MC are known. A method for calculating the temperatures and the position of M at the succeeding instant of time At seconds later is then needed.
IV. Numerical Values Chosen
The numerical values chosen were as follows:
Ax = 0.06cm. k = 0.003 cal./sq. crn./second/"C./crn. p = 2.0gm./cu.cm. C = 0.25 cal./gm.!'C.
At = 0 . 3 second L = 15 cal./gm. To = 50°C.
Rate of rise of outside = 8"C./iiiinute Total thickness of slab = 22 Ax = 1.32 cm.
The reference specimen was taken as identical with the test specimen except that it showed no inversion.
The physical properties of the test specimen above the in- version were taken as the same as those below the inversion.
6 Temp. of center of Temp of si reference specimen ("C.)
Fig. 5. Differential temperature plotted Fig. 6. Differential temperature plotted Fig. 7. Differential temperature plotted against the temperature at the center of the against the temperature at the center of the against the surface temperature of the test
reference material. or the reference sample. test specimen.
224 Journal of The American Ceramic Society-Smyth Vol. 34, No. 7
c
0 C c
p 20
n * !k 50 60 70 00 90
Temp. of center of reference specimen (“C.)
Fig. 9. Effect of having thermocouple 0.06 cm. from center of test specimen.
50 60 70 00 90 Temp. of center of
reference mecirnen ( “C.) Fig. 10. Effect of having thermocouple
0.30 cm. from center of test specimen. I I I
50 60 70 80 90 Temp. of surface of reference
and test specimens (“C.) Fig. 8. Temperature a t the center of the reference and the test specimens plotted against the surface temperature of either.
V. Results Temperatures were calculated for the reference specimen
and the test specimen a t intervals of 0.3 second, starting a t a time when the test specimen was completely below the in- version temperature and continuing until it had, within the accuracy of the calculations, recovered from the effects of the inversion and resumed its original parabolic temperature distribution. Selected distribution curves are shown for the reference specimen in Fig. 3 and for the test specimen in Fig. 4.
The distribution curves for the reference sample are at all instants of time identical parabolas, as one can see from equa- tion (2) . The test specimen curve starts out as a parabola; but as the outer layers reach the inversion temperature, so much heat is required to change them from the low to the high form that the heat supply to the interior is seriously interrupted. The rate of heating at the center, therefore, slows up quite a while before the material a t the center has reached the inversion temperature.
As soon as the rate of heating at the center starts slowing up, the pen tracing the differential temperature will start to deviate from its zero line. At this point neither the center of the test specimen nor the center of the reference specimen is at the inversion temperature, and one must be rather care- ful not to attach too much importance to this point of initial deviation.
It will be seen in Fig. 4 that when the inversion is complete the distribution curve comes to a sharp point a t the center. Since such a sharp point corresponds to a very high value of the second derivative (corresponding roughly to the curva- ture), one would correctly deduce from equation (1) that there would be a rapid rise in temperature a t the center. This rapid rise a t the center gradually slows down until, after more than 300 seconds, the test specimen has regained its original parabolic distribution and has caught up with the reference specimen bringing the differential temperature back to zero.
I n order to illustrate more clearly how these calculations relate to actual differential thermal analysis curves several curves were plotted.
In Fig. 5 the differential temperature is plotted against the temperature a t the center of the reference specimen. The curve departs €rom zero some distance below the inversion temperature (50°C.) and reaches its peak some 2OOC. above the inversion temperature.
In Fig. 6 the differential temperature is plotted against the temperature of the surface of the test (or reference) speci- men. This latter would correspond to the temperature of the metal block in which the sample cavities are located. This curve actually starts its deviation from the zero line a t
3 W. P. White, “Melting Point Determination,” Am. J . Sci., 28, 453 (1909).
the inversion temperature; and if this point of initial devia- tion could be accurately determined, i t would have useful significance on such a curve.
In Fig. 7 the differential temperature is plotted against the temperature a t the center of the test specimen itself. If by proper instrumental ingenuity this arrangement were used, the location of the peak of the curve would be a good indica- tion of the inversion temperature.
As an additional illustration of how heating progresses a t the center of the specimens, Fig. 8 was plotted showing the temperatures at the center of the reference and test samples plotted against the surface temperature of either. The center of the reference sample heats uniformly, always lag- ging 4.84OC. below the temperature of the surface. The in- crease in the temperature of the center of the test specimen slows down and finally effectively stops at the inversion tem- perature until the inversion is complete, at which time i t heats very rapidly finally catching up with the reference sample. The vertical distance between the two curves a t any point gives the differential temperature.
VI. Effect of lack of Symmetry
If the thermocouple in the test sample had been 0.06 cm. from the center, instead of at the center, the differential thermal analysis curve would be that shown in Fig. 9. If, in a rather extreme case, it had been 0.30 cm. from the center, the curve would be that shown in Fig. 10. The inversion occurs now, not at the peak of the curve, but at the first break. From there on the differential rises slowly until the material is completely inverted, after which it drops fast. White3 presented somewhat similar considerations semi- quantitatively in connection with discussions of the precision of melting-point determinations.
VII. Conclusions
It is shown that some rather simple mathematical calcula- tions on the mechanism can be of help in the interpretation of differential thermal analysis curves where an endothermic reaction or polymorphic inversion is involved.
If the differential temperature is plotted against the surface temperature of the sample, the point of initial departure from the straight line corresponds to the inversion temperature.
If the differential temperature is plotted against the temper- ature of the center of the sample while the outside is being heated uniformly, the peak of the curve corresponds to the inversion temperature.
If the differential temperature is plotted against the temperature of the center of the reference specimen, neither the point of initial departure from the straight line nor the peak of the curve corresponds to the inversion temperature.
Acknowledgmeni This work was carried out in connection with a project for the
investigation of high-temperature thermal and other physical properties of ceramic materials sponsored by the Office of Naval Research. The author would like to express his thanks to Rob- ert B. Sosman, Myril C. Shaw, and Harriet R. Wisely for reading the manuscript and offering useful suggestions.
A Theory of Differential Thermal Analysis and New Methods of Measurement and Interpretation
by S. 1. BOERSMA
Delft, Holland
A new-type nickel-block sample holder for use in m e r e n t i d thermal analysis is described. A simple machine to aid in the interpretation of
complex curves is suggested.
1. Introduction WELL-KNOWN method of mineral analysis is to heat a
sample in a furnace at a certain rate of temperature A rise and to determine the temperature lag or lead that results from the heat of transformation at certain tempera- tures specific for the mineral. A discussion of this method is given by Kronig and Snoodijk.'
An apparatus for differential thermal analysis built at the Laboratorium voor Grondmechanica has been described by de Bruyn and Marel.2 Since the completion of this appara- tus several thousand samples have been investigated. The results of these measurements have been analyzed by Marel, who showed that differential thermal analysis cannot be used for precise quantitative analysis. In this paper a theory of differential thermal analysis is given that accounts for the shortcomings found by Marel, and methods are outlined for improving measurement and interpretation.
11. Theory of Differential Thermal Analysis
( I ) Nickel Sample Holder The following symbols are used:
T = temperature. 0 = differential temperature. r = radius. a t = time. c = specific heat of sample material. X = thermal conductivity of sample material. p q
In a block of material with an infinitely high thermal conduc- tivity (e.g. nickel) there are two identical cylindrical or spheri- cal cavities. Cavity No. 1 is filled with an inert material having the same thermal properties as the sample. When the nickel block is heated (in practice at a uniform rate),* the surface temperature of both cavities is the same during the entire process because of the high conductivity of nickel. Cavity No. 2 contains the sample.
= radius of cavity filled with sample.
= specific density of sample material. = heat of transformation per unit volume.
The temperature TI in cavity No. 1 is given by pc (bT,/bt) - X div grad TI = 0
Received November 12, 1954. The work described in this paper was undertaken at the re-
quest of E. C. W. A. Geuze, director of the Laboratorium voor Grondmechanica (Soil Mechanics Laboratory) at Delft.
The author is a consulting engimyr for this laboratory. 1 R. Kronig and F. Snoodijk, Determination of Heats of
Transformation in Ceramic Materials," Appl . Sci . Research, A3 [I] 27-30 (1951); Ceram. Abstr., 1953, January, p. 17a.
2 C. M. A. de Bruyn and v. d. Marel, Geol. en Mijnbouw, pp. 69-83, March 1954.
* It can be shown that a uniform heating rate becomes neces- sary when there is a difference between the thermal properties of sample and reference material.
The temperature T2 in cavity No. 2 is given by
pc (dTz/bt) - X div grad TZ = dq/dt aq/dt = heat of reaction eventually produced (per unit time).
of the two linear equations The differential temperature B is found from the difference
a bq pc - - X div grad B = - at at with the boundary condition 0 = 0 for 7 = a and B = 0 for t = tl . It is assumed that t = tl is before the beginning of a reaction and that t = tt is far enough after its completion for the temperature difference B to become zero again.
The integration of equation (1) with respect to time gives
(2) pc(& - el) - XJlf' div grad 0 dl = p
q = entire heat of reaction per unit volume.
died away at the moment t 2 , so the first term vanishes.
tions in'the second term, equation (2) becomes
The temperature difference caused by the reaction will have
After interchanging the independent time and place opera-
(3)
Integrating equation (3) over the volume Ti within a sphere
-1 div grad J,f' e dt = q
of radius r gives
- - X ~ y d V d i v g r a d ~ ~ d t = J v q d V
or
-Ass dS grad ,fl e dt = q V (4)
where the left part is transformed by application of the Gauss theorem.
Because of radial symmetry, grad Jlf' B dt is everywhere perpendicular to the surface S and of the same magnitude; therefore
-SX grad Jl 0 dt = qV
(5) -SX - $, B df = q V
According to the boundary condition B = 0 for Y = a and The temperature integral in the center follows
d dr
so 0 dt = 0. from equation ( 5 ) :
For a cylindrical sample V / S = 7/2, so the peak area in the cylindrical case
For a sphere V/S = r /3 , so for a spherical sample
Jtf' Oo dt = qa2/6X
In the one-dimensional case of a flat plate
Jtf' Oo dt = qa2/2X
(7)
281
282
(2) For the sample holder some investigators use a ceramic
material instead of nickel. The thermal conductivity, A,, here is of the same order of magnitude as the conductivity, A,, of the sample itself. The boundary condition is 0 = 0 =
,f e dt for r = a. For an infinitely large ceramic sample holder, equation (6) becomes
Journal of The American Ceramic Society-Boersma Vol. 38, No. 8
Peak Area in a Ceramic Sample Holder
V8 = entire sample volume. qV, = all the heat produced that is carried away by the ceramic
For the case of an infinitely large ceramic block, equation (6a) has no finite solution in the one- and two-dimensional case; there is a solution only for a spherical sample (radius a, V = (49r/3)r3, V , = (47r/3)a3, S = 4rr2):
sample holder.
Peak area fi dt = (% + k) If A, becomes infinite, equation (8) reduces to equation
(7a) for the nickel block. In a ceramic block, A, is of the same order of magnitude as A, of the sample material. Thus, ac- cording to equation (8), the peak area becomes three times as large as with a nickel block. Although a ceramic block gives the greater peak area, the method is not attractive be- cause the temperature fields of several samples in the same block can penetrate into each other, thus causing mutual in- terference.
(3) Thermocouple Interference The temperature in the sample center is measured by means
of a thermocouple. Part of the heat produced in the sample is carried away by the thermocouple wires; too low a tempera- ture therefore is measured. The error from this effect is quite large. Figure 1 shows the sample-filled cavity in the nickel block. The thermojunction consists of a sphere of radius ro. During a reaction there is a temperature gradient in the leads over the lengths 1. It is assumed that at the dis- tance I (slightly larger than a ) the wires have attained the temperature of the nickel block. The area of the cross sec- tion of the leads will be A . If eo is the junction temperature, the thermocouple leads will carry away an amount of heat
X, = heat conductivity of wire material (platinum).
This heat no longer passes through the surface of integra- tion in equations (4) and (5); Q therefore must be sub- tracted there:
-s x J e at = qv - Q d Y
The correct peak area is now
(9)
For spherical samples ( S = 47rr2 and J' = (47r/3)(r3 - y o 3 ) )
the peak area is
a = 1 - ( Y o 2 / a ' ) [ 3 - z(ru/a)I A = X p ( Y o / l ) ( A / 4 ? ~ 7 ; ) [ 1 - ( Y u / a ) l
Equation (1 1) differs from the elementary expression (equation (7a)) by the factor a/1 + (A/A). Here a (very nearly unity) comes from the altered geometry and A from the heat leakage through the thermocouple leads.
I t follows from equation (1 1) that with low-conductivity samples [@/A) << 11 the peak area Will become independent
Fig. 1. Heat leakage through thermocouple wires of lengths 1.
of the sample conductivity A , whereas a high sample conduc- tivity will cause an inverse proportional relationship between peak area and thermal conductivity.
For cylindrical samples an expression very similar to equa- tion (11) exists:
The differential thermal analysis apparatus in use at the Laboratorium voor Grondmechanica has the following con- stants: I = a = 0.4 cm.; ro = 0.12 cm.; A = 4 x 10-3 cm. ; A, = 7 2 joules perm. sec. "C.
Most sample materials have thermal conductivities of about 0.3 joule perm. sec. "C., which, with the foregoing con- stants, makes A/A about unity (equation (1 1)). According, therefore, to equation (1 l), the heat leakage through the thermocouple leads reduces the peak area to less than .50y0 of its theoretical value. The peak area has become sensitive for the thermocouple geometry; in practice, changes in the calibration factor of about 20% have been observed when ex- changing the thermocouple.
111. Analysis
When used to obtain quantitative results, differential thermal analysis has several defects. The peak area is related not only to the amount of unknown material but is influenced by the following:
(1) Sample volume. The same sample will, when pressed into a smaller volume, give other readings because of the geometrical factor u2 in equations (7) and (11).
(2) Sample conductivity. According to equations (7), (8) , and (11) the peak area depends on the thermal conductiv- ity of the sample material. This conductivity varies with the composition and density and is mostly unknown.
Shortcomings of Existing Differential Thermal
August 1955 New Apparatus for Diferential Thermal Analysis 283
F
d I
EBlNickel. blElCeramic plate.
Fig. 2. A new sample holder for differential thermal analysis. S, sample; R, reference material; F, furnace.
(3) Thermocouple interference. As shown by equation (11) the thermocouple has (by its heat conduction) a very large effect on the measured peak area (approximately halves it). (4) Sample shrinkage. Some samples sinter at the higher
temperatures and shrink. An air gap between sample and nickel block is produced that completely upsets the tempera- ture distribution, resulting in a measured peak area that is much too high.
(5) Loss of material. In some reactions volatile products (e.g. water) are formed that evaporate. The sample mass is no longer constant. The escaping gas can alter the porosity and thermal conductivity, so the linear theory of this paper is no longer valid. When mass or conductivity is altered, the temperature may not return to the original value after the completion of the reaction; there is no longer a definite peak area. These shortcomings can be overcome by a new meas-
Fig. 3. Curve synthesis for differential thermal analysis. 1, lamp; S, slit; A, B, C, and x, moving templates; F, photocells; C, integrating .condensers; P, calibrated potentiometers; Ss, selector switch; and OSC,
cathode-ray oscillograph.
uring technique (new sample holder) and a new method of diagram interpretation (curve synthesis).
IV. A New Sample Holder In normal differential thermal analysis the heat of a reac-
tion is measured by the temperature difference it produces in the sample material itself. Thus the sample is used for two entirely different purposes: (a) a producer of heat and (b ) heat measuring resistance in which the flow of heat develops a temperature difference to be measured.
This can be done by leading the heat of reaction outside the sample through a spccial piece of material on which the temperature difference can be measured. The peak area then is solely de- pendent on the produced heat of reaction and on the calibra- tion factor of the instrument which no longer contains any sample properties (e.g., volume and conductivity). An ex- ample of such an apparatus is shown in Fig. 2. The sample S and reference material R are held in small nickel containers placed on a ceramic base plate. When a reaction takes place, the heat of reaction is led from the container through the ceramic plate, and the container takes on a temperature that is solely dependent on the heat of reaction and on the specific heat conduction between the container and the surrounding nickel dome. The heat passes for the greater part through the ceramic plate ; the remainder is transferred by radiation and convection. These properties are a calibration constant for the apparatus, independent of the kind or even the amount of sample material. The temperature peak area easily can be shown to be
A much better solution is to separate these functions.
O d t = G m = mass of sample. q = heat of reaction per unit mass. G = heat-transfer coefficient between small nickel container and
surrounding nickel.
The measurement has now become purely caloric: mp, the total heat of reaction, is measured. There is no need to dilute samples with A1203 or even to use samples of standardized weight or volume. Measurements made by this method con- firmed equation (12) quantitatively. Samples of CuSO4 of widely different packing density produced the same (pre- dicted) peak area.
V. Curve Synthesis The shortcomings of differential thermal analysis mentioned
in section 111 have been overcome with the new apparatus. A number of substances, however, do not produce clear, well- defined temperature peaks (e.g. illite), perhaps because there is no sharp reaction temperature or the properties of the sample vary considerably with temperature. These curves cannot be characterized by a mere number (peak area), which is a great difficulty in quantitative analysis. It seems much better to make use of all the data the curve can give. This can be done in the following manner:
From a number of pure minerals ( A , B , C, D ) are taken the differential thermal analysis curves (a , b, c, d). To analyze an unknown sample its curve x is taken and compared with a synthetic curve x’ = aa + pb + yc + 6d made up out of known fractions of the pure mineral curves a , b , c, d. The coef- ficients 01, p , y , and 6 that give the best match between the curves x and I’ represent the relative concentrations of the minerals A , B , C, and D in the sample.
A synthesis of a composite curve made up out of known fractions of pure mineral curves can easily be made with the apparatus shown in Fig. 3 (not yet completed when this paper was written). The curves obtained with the differen- tial thermal analysis recorder are cut out of opaque sheet ma- terial (e.g. black paper) to form templates. The templates move simultaneously in front of an illuminated slit, S; they
284 Journal of The American Ceramic Society-Fischer Vol. 38, No. 8
can be bent around a glass cylinder that rotates coaxially with respect to the lamp, L. The slit passes an amount of light that is partially intercepted by the templates. Behind each template is a photoelectric cell, F , that generates a current of the same wave form as the template to which it belongs. Known and adjustable fractions (a, 0, etc. of the generated signals) are taken from the calibrated potentiometers, P, and fed into a cathode-ray oscillograph, OSC. The oscillat- ing selector switch, Ss, puts alternatively the unknown sample curve x and the synthetic curve x' = au + 0b + yc + 6d on the screen. By manipulating the a, 0, 7 , and 6 potentiome- ters the two curves are made to cover each other as much as possible. When this is done, the composition of the sample can be read quantitatively from the potentiometer dials a, 0, y, and 6 . Because with differential thermal
analysis the temperature-time integral curve rather than the temperature itself is significant, i t is better to manipulate some form of integral curve on the screen. This is done readily by means of the electrically integrating condensers, C.
VI. Summary
A theory dealing with the calorimetric shortcomings of the differential thermal analysis of minerals has been given. These shortcomings can be overcome by the use of a new type of sample holder. To improve the interpretation of complex curves, a simple machine is suggested that performs the con- tinuous synthesis of differential thermal analysis curves of widely varying compositions.
Calcination of Calcite: 11, Size and Growth Rate of Calcium Oxide Crystallites
by H. C. FISCHER Massachusetts Jnstitute of Technology, Cambridge, Massachusetts
The size and the growth rate of CaO crystallites were determined by X-ray diffraction. The crystallites exhibited an extremely rapid rate of growth near 1650'F. Activation energies were obtained from the growth rates by means of
the Arrhenius relationship.
I. Introduction
(I) General The phenomenon of sintering or densification, for all prac-
tical purposes, amounts to bulk shrinkage and crystal growth by means of thermal-energy input. Most recent theories of sintering agree that the tendency of fine particles to sinter together originates in the energy of their surface area and that the forces involved are capillary forces associated with the curvature of their surfaces. Considerable controversy still exists as to the nature of the process by which material actually is transported during sintering. In general, either of two basic mechanisms seems to be possible: (1) viscous or plastic flow and (2) diffusion.
Received September 29, 1954; revised copy received February 10, 1955.
This paper is a portion of a thesis submitted in partial ful- fillment of the requirements for the degree of Doctor of Science at the Massachusetts Institute of Technology.
A t the time this work was done, the author was assistant pro- fessor of materials at the Massachusetts Institute of Technology and was associated with the Building Materials Research Labo- ratory. He is now assistant director of research and development, Wasco Chemical Company, Incorporated, Sanford, Maine.
For Part I, see J . A m . Cernrn. Soc., 38 [7] 245-51 (1955) ("Calcination of Calcite: I, Effect of Heating Rate and Tem- perature on Bulk Density of Calcium Oxide").
Inasmuch as the actual mechanism of sintering was not investigated in the present work, which is confined to a study of the size and growth of crystallites, only the basic funda- mentals of crystal growth are considered briefly.
Surface energy, the energy inherent in or at a surface or interface between phases, is commonly defined as the free energy of formation of a new surface. A large surface area consequently has large total surface energy.
The increase in crystal size with increasing temperature is a surface phenomenon, and it can be shown that the size at any particular temperature is a function of the size of the initial crystal. This original size is unique for the particular tem- perature. The final size also is governed somewhat by the rate of nucleation, since small units which are attracted to smaller units are continuously formed at high rates of nu- cleation. As the temperature is increased, causing increased growth and recrystaIlization, more material grows into the available pore space, and the bulk density increases.
Chemical reactions in general proceed at more rapid rates with an increase in temperature. The relation between the rate and the temperature often may be expressed by the Arrhenius equation, as follows:
k = A . e - E / R T
or E
4.576T log k = log A - ~
k = specific reaction rate. e = base of natural logarithms. E = activation energy (cal./mole). R = gasconstant (1.98cal./mole/"K.). T = temperature (OK.).
A = constant, containing such factors as (1) frequency of ther- mal vibration of the atoms, ( 2 ) the probability that an acti- vated unit will react, and (3) the effect of surface texture, par- ticle size, and conditions of contact.
