University of Cambridge, Materials Science & Metallurgy H. K. D. H. Bhadeshia

Thermal Analysis Techniques

Thermal analysis comprises a group of techniques in which a physical property of a substanceis measured as a function of temperature, while the substance is subjected to a controlledtemperature programme. In differential thermal analysis, the temperature difference that de-velops between a sample and an inert reference material is measured, when both are subjectedto identical heat–treatments. The related technique of differential scanning calorimetry re-lies on differences in energy required to maintain the sample and reference at an identicaltemperature.

Length or volume changes that occur on subjecting materials to heat treatment are detectedin dilatometry; X–ray or neutron diffraction can also be used to measure dimensional changes.

Both thermogravimetry and evolved–gas analysis are techniques which rely on samples whichdecompose at elevated temperatures. The former monitors changes in the mass of the specimenon heating, whereas the latter is based on the gases evolved on heating the sample. Electricalconductivity measurements can be related to changes in the defect density of materials or tostudy phase transitions.

Differential Thermal Analysis (DTA)

Introduction

DTA involves heating or cooling a test sample and an inert reference under identical conditions,while recording any temperature difference between the sample and reference. This differentialtemperature is then plotted against time, or against temperature. Changes in the sample whichlead to the absorption or evolution of heat can be detected relative to the inert reference.

Differential temperatures can also arise between two inert samples when their response to theapplied heat–treatment is not identical. DTA can therefore be used to study thermal propertiesand phase changes which do not lead to a change in enthalpy. The baseline of the DTA curveshould then exhibit discontinuities at the transition temperatures and the slope of the curveat any point will depend on the microstructural constitution at that temperature.

A DTA curve can be used as a finger print for identification purposes, for example, in thestudy of clays where the structural similarity of different forms renders diffraction experimentsdifficult to interpret.

The area under a DTA peak can be to the enthalpy change and is not affected by the heatcapacity of the sample.

DTA may be defined formally as a technique for recording the difference in temperature be-tween a substance and a reference material against either time or temperature as the twospecimens are subjected to identical temperature regimes in an environment heated or cooledat a controlled rate.

Apparatus

The key features of a differential thermal analysis kit are as follows (Fig. 1):

1. Sample holder comprising thermocouples, sample containers and a ceramic or metal-lic block.

2. Furnace.

3. Temperature programmer.

4. Recording system.

The last three items come in a variety of commercially available forms and are not be discussedin any detail. The essential requirements of the furnace are that it should provide a stableand sufficiently large hot–zone and must be able to respond rapidly to commands from thetemperature programmer. A temperature programmer is essential in order to obtain constantheating rates. The recording system must have a low inertia to faithfully reproduce variationsin the experimental set–up.

Fig. 1: Schematic illustration of a DTA cell.

The sample holder assembly consists of a thermocouple each for the sample and reference,surrounded by a block to ensure an even heat distribution. The sample is contained in a smallcrucible designed with an indentation on the base to ensure a snug fit over the thermocouplebead. The crucible may be made of materials such as Pyrex, silica, nickel or platinum, de-pending on the temperature and nature of the tests involved. The thermocouples should notbe placed in direct contact with the sample to avoid contamination and degradation, althoughsensitivity may be compromised.

Metallic blocks are less prone to base–line drift when compared with ceramics which containporosity. On the other hand, their high thermal conductivity leads to smaller DTA peaks.

The sample assembly is isolated against electrical interference from the furnace wiring with anearthed sheath, often made of platinum–coated ceramic material. The sheath can also be usedto contain the sample region within a controlled atmosphere or a vacuum.

During experiments at temperatures in the range –200 to 500 ◦C, problems are encounteredin transferring heat uniformly away from the specimen. These may be mitigated by usingthermocouples in the form of flat discs to ensure optimum thermal contact with the now flat–bottomed sample container, made of aluminium or platinum foil. To ensure reproducibility, itis then necessary to ensure that the thermocouple and container are consistently located withrespect to each other.

Experimental Factors

Care is necessary in selecting the experimental parameters. For example, the effects of specimenenvironment, composition, size and surface–to–volume ratio all affect powder decompositionreactions, whereas these particular variables may not affect solid–state phase changes. Experi-ments are frequently performed on powders so the resulting data may not be representative ofbulk samples, where transformations may be controlled by the build up of strain energy. Thepacking state of any powder sample becomes important in decomposition reactions and canlead to large variations between apparently identical samples.

In some circumstances, the rate of heat evolution may be high enough to saturate the responsecapability of the measuring system; it is better then to dilute the test sample with inertmaterial.

For the measurement of phase transformation temperatures, it is advisable to ensure that thepeak temperature does not vary with sample size.

The shape of a DTA peak does depend on sample weight and the heating rate used. Loweringthe heating rate is roughly equivalent to reducing the sample weight; both lead to sharperpeaks with improved resolution, although this is only useful if the signal to noise ratio is notcompromised. The influence of heating rate on the peak shape and disposition can be used toadvantage in the study of decomposition reactions, but for kinetic analysis it is important tominimise thermal gradients by reducing specimen size or heating rate.

Interpretation and Presentation of Data

A simple DTA curve may consist of linear portions displaced from the abscissa because the heatcapacities and thermal conductivities of the test and reference samples are not identical, andof peaks corresponding to the evolution or absorption of heat following physical or chemicalchanges in the test sample.

There are difficulties with the measurement of transition temperatures using DTA curves. Theonset of the DTA peak in principle gives the start–temperature, but there may be temperaturelags depending on the location of the thermocouple with respect to the reference and testsamples or the DTA block. It is wise to calibrate the apparatus with materials of preciselyknown melting points. The peak area (A), which is related to enthalpy changes in the testsample, is that enclosed between the peak and the interpolated baseline. When the differentialthermocouples are in thermal, but not in physical contact with the test and reference materials,

it can be shown that A is given by

A =mq

gK

wherem is the sample mass, q is the enthalpy change per unit mass, g is a measured shape factorand K is the thermal conductivity of sample. With porous, compacted or heaped samples, thegas filling the pores can alter the thermal conductivity of the atmosphere surrounding the DTAcontainer and lead to large errors in the peak area. The situation is made worse when gasesare evolved from the sample, making the thermal conductivity of the DTA–cell environmentdifferent from that used in calibration experiments.

The DTA apparatus is calibrated for enthalpy by measuring peak areas on standard samplesover specified temperature ranges. The calibration should be based upon at least two differentsamples, conducting both heating and cooling experiments.

It is possible to measure the heat capacity CP at constant pressure using DTA:

CP = K ′T2 − T1

mH

where T1 and T2 are the differential temperatures generated when the apparatus is first runwithout any sample at all and then with the test sample in position. H is the heating rate andthe constant K ′ is determined by calibration against standard substances.

University of Cambridge, Materials Science & Metallurgy H. K. D. H. Bhadeshia

Differential Scanning Calorimetry

Introduction

Differential scanning calorimetry (DSC) is a technique for measuring the energy necessary toestablish a nearly zero temperature difference between a substance and an inert reference ma-terial, as the two specimens are subjected to identical temperature regimes in an environmentheated or cooled at a controlled rate.

There are two types of DSC systems in common use (Fig. 1). In power–compensation DSC thetemperatures of the sample and reference are controlled independently using separate, identicalfurnaces. The temperatures of the sample and reference are made identical by varying thepower input to the two furnaces; the energy required to do this is a measure of the enthalpyor heat capacity changes in the sample relative to the reference.

In heat–flux DSC, the sample and reference are connected by a low–resistance heat–flow path(a metal disc). The assembly is enclosed in a single furnace. Enthalpy or heat capacity changesin the sample cause a difference in its temperature relative to the reference; the resulting heatflow is small compared with that in differential thermal analysis (DTA) because the sampleand reference are in good thermal contact. The temperature difference is recorded and relatedto enthalpy change in the sample using calibration experiments.

Fig. 1: (a) Heat flux DSC; (b) power–compensation DSC

Heat–flux DSC

This section is based largely on a description of the Dupont DSC system by Baxter and Greer.The system is a subtle modification of DTA, differing only by the fact that the sample and

reference crucibles are linked by good heat–flow path. The sample and reference are enclosedin the same furnace. The difference in energy required to maintain them at a nearly identicaltemperature is provided by the heat changes in the sample. Any excess energy is conductedbetween the sample and reference through the connecting metallic disc, a feature absent inDTA. As in modern DTA equipment, the thermocouples are not embedded in either of thespecimens; the small temperature difference that may develop between the sample and theinert reference (usually an empty sample pan and lid) is proportional to the heat flow betweenthe two. The fact that the temperature difference is small is important to ensure that bothcontainers are exposed to essentially the same temperature programme.

The main assembly of the DSC cell is enclosed in a cylindrical, silver heating black, whichdissipates heat to the specimens via a constantan disc which is attached to the silver block.The disc has two raised platforms on which the sample and reference pans are placed. Achromel disc and connecting wire are attached to the underside of each platform, and the re-sulting chromel–constantan thermocouples are used to determine the differential temperaturesof interest. Alumel wires attached to the chromel discs provide the chromel–alumel junctionsfor independently measuring the sample and reference temperature. A separate thermocoupleembedded in the silver block serves a temperature controller for the programmed heating cycle.An inert gas is passed through the cell at a constant flow rate of about 40 ml min−1).

The thermal resistances of the system vary with temperature, but the instruments can be usedin the ‘calibrated’ mode, where the amplification is automatically varied with temperature togive a nearly constant calorimetric sensitivity.

Heat Flow in Heat–Flux DSC Systems

A variety of temperature lags develop between the specimens and thermocouples, since thelatter are not in direct contact with the samples. The measured ∆T is not equal to TS − TRwhere TS and TR are the sample and reference temperatures respectively. TS − TR may bededuced by considering the heat flow paths in the system.

The following additional notation (due to Greer and Baxter) is relevant (Fig. 2):

TSP , TRP = Temperature of the sample and reference platforms, respectively, asmeasured by the thermocouples. TSP is normally plotted as the abscissa of a DSCcurve.

TF = Temperature of the silver heating block.

RD =Thermal resistance between the furnace wall and the sample or reference plat-forms (units C min J−1).

RS , RR = Thermal resistances between the sample (or reference) platform and thesample (or reference).

CS , CR = Heat capacity of the sample (or reference) and its container.

H = Imposed heating rate.

∆TR = Temperature lag of the reference platform relative to furnace.

∆TS = Temperature lag of the sample platform relative to furnace.

∆TL = Temperature lag of the sample relative to the sample thermocouple.

The following equations then hold:

∆TR = HRDCR (1)

∆TS = HRDCS (2)

∆T = HRD(CS − CR) (3)

∆TL = HRSCS (4)

∆TS = ∆TR + ∆T (5)

∆TL = RS/RD∆TS (6)

Fig. 2: Thermal resistance diagram representing a heat–flux DSC

Calibration: The Temperature Lag ∆TL

∆TL is non–zero because the thermocouple is not in direct contact with the sample. When thetransition temperature T ′ does not vary with heating rate, equation 4 indicates that a plot ofthe apparent T ′ versus H keeping the other quantities fixed, would at zero H extrapolate tothe true value of T ′; the apparent T ′ is the true value plus the lag.

A plot of the apparent T ′ versus CS would also extrapolate to the true T ′ at CS = 0, when Hand RS are kept constant.

Alternatively, the sample may be allowed to reach the temperature of the sample–platformby holding at a temperature just beyond T ′, and recording a DSC curve corresponding to theequilibration event. The area of this curve can then be used to deduce the temperature lag;this kind of an analysis requires more sophisticated equipment than is normally available.

Another method, due to Greer, is based on equation 6, and involves the evaluation of RS/RD.∆TR is measured for a particular reference, usually just an empty pan and lid. A heatingrun is first performed with an empty pan on both the sample and reference platforms. Thisprovides a baseline, from which measurements of ∆T can be carried out. A second run is thenperformed, with two pans on the sample side, and one on the reference side. The difference

between the first and second DSC curves is a measure of ∆TR, as a function of temperature.This becomes evident from equations 1 and 3; for the first run, CS and CR are identical andhence ∆T = 0, while for the second run CS = 2CR, so that ∆T = ∆TR.

By repeating this procedure, ∆TR can be obtained as a function of heating rate. To obtainthe temperature lag ∆TL, more tests are performed, bearing in mind that

∆TR + ∆T = ∆TS

Tests are conducted at a variety of heating rates, using a sample with a known transitiontemperature which is independent of heating rate, placed in the sample pan, with an emptypan on the reference side. These experiments give values of ∆T , and hence ∆TS , as a functionof heating rate; the gradient g1 of the graph of ∆TS versus heating rate is equal to RDCS ,equation 2. Another set of experiments, based on equation 4 then gives a plot of the apparenttransition temperature as a function of heating rate, and extrapolation to zero H yields thetrue transition temperature – hence a graph of (TL versus H can be plotted, whose gradient g2

is equal to RSCS , equation 4. Hence, g1/g2 = RD/RS . The temperature lag may be calculated(since RS/RD and ∆TR are known) for a given reference and at any heating rate or CS , usingequation 6.

Temperature Calibration

The temperature plotted on the abscissa of a DSC record is related to the emf generated at thethermocouple located under the sample. For standard thermocouple conditions, the emf maybe reliably converted to temperature units using established calibration charts, but a variety ofeffects can cause the thermocouple to age and shift calibration. It is advisable to calibrate theabscissa using substances with precisely known melting points; most DSC instruments havefacilities which allow calibration over limited temperature ranges. In changing the abscissascale to a true temperature reading, allowances have to be made for the thermal lag effect(∆TL), but this can be avoided by using very low heating rates for the purposes of calibration.

Calorimetric Calibration

Calibration is carried out by measuring the changes in specific heat or in enthalpy content ofsamples for which these quantities are known. When the DuPont instrument is used in thecalibration mode, the procedure related to equation 2 may be used to measure specific heatchanges. The heat balance equation for the heat–flux DSC system can be shown to be asfollows:

dH ′

dt=TSP − TRP

RD+ (CS − CR)H + CS

RD +RSRD

d(TSP − TRP )

dt(7)

dH ′/dt refers to the heat evolution of an exothermic transition; the first term on the right handside is the area under the DSC peak, after correcting for the baseline. The second term on theright refers to the actual baseline, and it is this which is used in specific heat determinations.The last term takes account of the fact that some of the evolved heat will be consumed bythe specimen to heat itself, and does not affect the are under the DSC peak, but may distortthe peak shape. From equation 7 it is clear that when dH ′/dt can be arranged to be zero, thesecond term can be used to determine specific heat. The method is involves a comparison ofthe thermal lag between the sample and reference; the system is first calibrated with a sapphirespecimen, so that

Csapphire = EqY/HM

where M is the mass of the specimen, E is a calibration constant, Csapphire, the specific heat

capacity of the sapphire, q Y –axis range (J s mm−1) and Y the difference in Y –axis deflectionbetween sample (or sapphire) and blank curves at the temperature of interest.

Enthalpy changes can be determined by measuring the areas under peaks on the DSC curve,when the latter is a plot of ∆T versus time. A relationship of the form indicated in equation 1then applies, again when the instrument is in the calibrated mode.

The Baseline and the Transformation Curve

In DTA or DSC, it is expedient to conduct experiments either isothermally or with the tem-perature changing at a constant rate. In the former case, the ordinate value would be plottedagainst time at isothermal temperature, whereas in the latter case it could be plotted againsttime or temperature. The following discussion is based on the abscissa being a time axis; theheight referred to is that beyond the baseline.

For DTA the height of the curve at any particular time t is a measure of the difference intemperature, ∆T , between the sample and the reference. For power compensated DSC, theheight of the curve at some particular time t is a measure of the heat evolving from the sampleper unit time, dH ′/dt (this also applies to heat flux DSC, after suitable calibration). Foreither DTA or DSC, one can assume that ∆T is proportional to dx/dt or dH ′/dt to dx/dt,respectively. Here, x refers to the volume fraction of transformation, t to the time measuredfrom the point where the appropriate curve departs from the baseline, and H ′ to the enthalpychange. The constants of proportionality follow from the condition that the total area underthe DTA or DSC corresponds to either x = 1, or to x equal to some constant value if thetransformation terminates prematurely.

This assumes that a reliable baseline can be obtained from the experimental information. Thebaseline can be visually estimated for sharp peaks without entailing large errors; for broadpeaks it is difficult to qualitatively establish the baseline. The problem is complicated by thefact that the DSC instrumental baseline on either side of the peak is not a no–signal line. Evenin the absence of a transition, the instrument measures the effect of the heat capacity of thesample, which may vary with temperature. This variation is usually nearly linear, but thecurvature becomes noticeable over wide temperature ranges.

One approximation to the baseline is a straight line connecting the start and finish of thetransformation. Other methods involve the use of stepped baselines; the parent and productparts of the experimental curve are linearly extrapolated towards the centre of the experimentalprofile, and are connected by a vertical step at the position of the peak. Again, this methodhas no fundamental basis. The most reliable way of constructing the baseline is an iterativetechnique due to Scott and Ramachandrarao. The fractions transformed are first calculatedapproximately, using a linear baseline between the initial and final points of the reaction.The baselines of the parent and product are then extrapolated under the peak; this gives twoseparate baselines, since the heat capacities of the parent and pure product differ. The truebaseline at any t is taken to be at a position between the extrapolated baselines. The exactpositioning of the new baseline between the extrapolated parent and product baselines dependson an estimated value of the amount of product at any time t, using a lever rule type of acalculation. The new baseline generated in this manner can then be used as the starting pointof another iteration and the process can be repeated to the desired accuracy. One iterationseems good enough for most purposes.

A subtle correction which has to be taken into account when constructing transformation

curves from DSC curves is that the peak shape (rather than peak area) can be expected tobe distorted, because some of any energy evolved may serve to the heat sample itself. Incontinuous heating experiments, the magnitude of this effect can be shown to be proportionalto the heat capacity of the sample and to the rate of change of the differential temperaturewith time.

Autocatalysis and Recalescence

Calorimetric experiments can be adiabatic or isothermal. The temperature is maintainedconstant in an isothermal experiment, whereas heat is neither added nor removed from thesystem during an adiabatic experiment. In practice, experiments fall somewhere between theideal isothermal and adiabatic conditions.

In an experiment where the rate of heat evolution is large relative to the capacity of thecalorimeter to maintain isothermal conditions, the specimen temperature rises beyond the de-sired level, until a steady state is reached. This adiabatic rise in temperature will affect the rateof reaction, which may in term exaggerate the evolution of heat. This effect is known as auto-catalysis. Recalescence describes the case where the release of heat reduces the transformationrate.

Kinetics of Glass Crystallisation

Both DSC and DTA have been used to study of the crystallisation of glasses. With fewexceptions, the results have been analysed using Johnson–Mehl–Avrami equations with littleattention to the mechanism of crystallisation. The general form of the equations is:

x = 1− exp{−ktn} (8)

where x is the volume fraction of transformation at time t, k is a function of transforma-tion temperature, and n is a parameter which can in special cases give an indication of themechanisms involved. The equation applies to isothermal transformations with the followingassumptions:

1. It is assumed that the growth rate is constant, i.e. there is no composition changeduring transformation.

2. Modern calorimetric experiments use small quantities of samples; it is assumed thefree surfaces of these samples do not affect the kinetics of transformation.

3. The extended volume concept on which the Avrami equation is based relies onrandom nucleation.

Activation Energy

The term k is temperature dependent since it is a function of the nucleation and growth rates ofthe transformation product; for most solid–state transformations both of these processes can beexpected to be thermally activated. Consider a transformation in which nucleation is random,the nucleation and growth rates are constant and where growth is isotropic. Equation 8becomes:

x = 1− exp{−Y 3It4/3} (9)

where Y is the growth rate and Iis the nucleation rate per unit volume. Hence,

k = Y 3I/3

= C1(C2 exp{−GY /RT})3(C3(exp{−GI/RT})= C4(exp((−3GY −GI)/RT})

(10)

where GY and GI are the activation free energies for growth and nucleation, respectively,and both are assumed to be independent of temperature (R is the gas constant). A furtherassumption is that the growth and nucleation events are both singly activated processes. Theactivation energies of equation 10 may be lumped together into a single effective activationenergy given by G′, which is the term really obtained from an analysis using equation 9. G′

cannot be isolated using this analysis since x depends on more than just the growth rate.

For isothermal transformation experiments, G′ can be obtained plotting the time taken toachieve a fixed amount of transformation (i.e. tx ) versus 1/T , a plot based on equation 11below, which is derived from equation 10:

tx = C5 exp{G′/nRT} (11)

It is difficult to determine the activation energy from anisothermal experiments. For anythermally activated process, the DTA or DSC peaks will shift with heating rate; Kissingerderived a relationship between the peak shift and the effective activation energy, assuminghomogeneous transformation:

dx/dt = C6(1− x)m exp{−G′/RT} (12)

where m is the order of the reaction, and the other terms have their usual meanings. Kissingershowed that

d(ln{H/T 2p })

d(1/Tp)= −G

′

R(13)

where H is the heating rate used and Tp is the sample temperature at which the maximumdeflection in the DTA or DSC curve is recorded. The equation requires that Tp equals thetemperature at which the maximum reaction rate occurs.

Most solid–state reactions are not homogeneous, but proceed by nucleation and growth events.Hence the G′ value obtained through equation 13 must not be compared with that obtainedfrom isothermal experiments which obey the Johnson–Mehl–Avrami equation. Henderson hasshown that for reactions that obey equation 8, a plot of ln{H/T 2

p } versus 1/Tp should have aslope of −G′/nR rather than the −G′/R of equation 13.

Marseglia has suggested that the activation energy G′ for anisothermal experiments can bededuced from a plot of ln{H/Tp} versus 1/Tp. The difference between Marseglia and Hendersonarises because the former takes account of the variation of k with time, whereas the latter doesnot. However, the manner in which the dependence of k on time is taken into account is notrigourous:

dk

dt=dk

dT

dT

dt=dk

dTH

Thus, the variation in growth rate with time is not fully accounted for.

Phase Transitions

Thermal analysis techniques have the advantage that only a small amount of material is nec-essary. This ensures uniform temperature distribution and high resolution. The sample can be

encapsulated in an inert atmosphere to prevent oxidation, and low heating rates lead to higheraccuracies. The reproducibility of the transition temperature can be checked by heating andcooling through the critical temperature range.

During a first order transformation, a latent heat is evolved, and the transformation obeys theclassical Clausius–Clapeyron equation. Second order transitions do not have accompanyinglatent heats, but like first order changes, can be detected by abrupt variations in compressibility,heat capacity, thermal expansion coefficients and the like. It is these variations that revealphase transformations using thermal analysis techniques.

Because of the sensitivity of liquid–vapour transitions to pressure, additional precautions arecalled for when testing for boiling points or enthalpy changes. The ambient pressure is required;the peak area no longer corresponds to the latent heat of vaporisation in any simple way. Thetransition temperature T ′ is related to the pressure P by the Clausius–Clapeyron equation

ln{P} = L/RT ′ + C

where L is the molar heat of vaporisation and C is an integration constant. L can be obtainedusing the Clausius-Clapeyron equation and a set of measured P, T ′ values, assuming L isindependent of temperature, that the volume of the vapour phase far exceeds that of theliquid, and that the vapour behaves as an ideal gas.

Greater care is needed when studying solid–solid transitions where the enthalpy changes aremuch smaller than those associated with vaporisation. Stored energy in the form of elasticstrains and defects can contribute to the energy balance, so that the physical state of theinitial solid, and the final state of the product, become important. This stored energy reducesthe observed enthalpy change.

Polymer Crystallinity

It is assumed that a volume fraction V of the polymer consists of perfectly crystalline materialwhich melts i.e. becomes amorphous, over the course of the experiment. The matrix which isnot crystalline is assumed to be perfectly amorphous. The transition from the crystalline tothe amorphous state is accompanied by a heat of “fusion”, written HFO when it occurs at thepure crystal “melting” point T0. The fraction V of crystalline phase can be determined for apartially crystalline specimen by comparing the measured heat of fusion with HFO. Imaginea DSC experiment in which a partially crystalline polymer is heated from a temperature T1 toT2 where the polymer becomes completely amorphous (T1 < T0 < T2). The enthalpy changescan be analysed in the following phenomenological sequence (Fig. 3):

a) Both the crystalline and amorphous phases are first heated, without transformationto T0. The enthalpy change for this process is

Ha = V (HC,1−0) + (1− V )(HA,1−0)

where the last two terms simply represent the change in heat content of the crys-talline and amorphous components, respectively on heating form T1 to T0. Ha canbe deduced from the DSC curve by measuring the area between the section of theDSC curve obtained before any change in V , linearly extrapolated over the range T1

to T0, and the instrumental baseline (i.e. the no-sample baseline).

b) At T0 the crystalline component is allowed to become amorphous. The enthalpy offusion for this is

Hb = V HF0

c) The now completely amorphous material is permitted to rise in temperature fromT0 to T2, so that the enthalpy change is

Hc = HA,0−2

Hc thus corresponds to the area between the DSC curve and the instrumental baseline, betweenthe temperatures T0 to T2.

If the total enthalpy change calculated from the separation of the DSC curve from the instru-mental baseline is given by H1−2, then

H1−2 = Ha +Hb +Hc and V = (H1−2 −Ha −Hc)/HF0

Fig. 3: Analysis of a DSC peak.

University of Cambridge, Materials Science & Metallurgy H. K. D. H. Bhadeshia

Dilatometry

The dilatometric method utilises either transformation strains or thermal strains; the basicdata generated are in the form of curves of dimension against time and temperature.

Thermal Expansion

The coefficient of linear expansion, also known as expansivity, is the ratio of the change in lengthper ◦C to the length at 0◦C. The coefficient of volume expansion for solids is approximatelythree times the corresponding linear coefficient. The coefficient of volume expansion of a liquidis the ratio of the change in volume per degree, to the volume at 0◦C. The value of the coefficientvaries with temperature. The coefficient of volume expansion for a gas under constant pressureis nearly the same for all gases and temperatures, and is equal to 0.00367.

If l0 is the length at 0◦C, then the length at a temperature T can be written:

lT = l0(1 + e1T + 22T2 + . . .) (1)

The expression is usually terminated after the second term; the same form of equation can beused to represent volume expansion.

The thermal expansion is a consequence of the nature of interatomic forces, and solid–statetheory predicts a simple relationship between specific heat capacity at constant volume (CV )and the coefficient of linear expansion. Thus, Gruneisen has demonstrated that e1 is propor-tional to CV , and this relationship can be exploited in dilatometry; Curie transitions in metalsare associated with an anomaly in the specific heat capacity, and can therefore be detected bydilatometry through the accompanying change in e1.

The sample should be free to move, i.e. without any mechanical constraints which would limitaccuracy. The length changes should not be transmitted through a second material upon whoseexpansion coefficient depends the evaluation of the test material. These conditions rule out theuse of most push–rod type dilatometers. A dilatometer capable of meeting the requirementsis described in the Journal of Scientific Instruments, 2, 515 (1969).

Study of Vacancies

Balluffi and co–workers first suggested that dilatometry, combined with precision X–ray latticeparameter measurements may be used to determine the concentrations of point defects inmetals. They showed that the the ratio of vacancies to lattice sites, cv, is given by

cv = 3

(∆l

l− ∆a

a

)(2)

where l is the length and a the lattice parameter of a material with a cubic lattice. Thepresence of point defects must alter the volume of the sample, but the lattice parametershould not be affected, apart from certain changes due to relaxation, to be discussed later.In comparing lengths at different temperatures, subtracting ∆a/a removes changes due tothermal expansion, leaving only the effect of point defects.

This subtraction is not necessary in isothermal experiments. The lattice parameter measuredis a mean value which includes the effect of local relaxations around a point defect. Sub-tracting ∆a/a from the length change therefore also removes the influence of local relaxations,leaving only the contribution to ∆l of new lattice sites formed or lost in creating the defects.During isothermal annealing of defects, lattice parameter measurements provide a measure ofthe lattice relaxation due to the defects and permit the determination of the absolute defectconcentration.

Vacancy clusters can lead to a misinterpretation of results. Fortunately, the concentration ofsuch clusters is expected to be relatively small. Note that it is the excess concentration ofvacancies that is revealed because interstitials can cancel some of the effect of the vacancies.

Push–Rod Dilatometer

The push–rod dilatometer can only be used for studies of length changes in solid materials.The sample rests between the tips of a fixed quartz rod and a similar frictionless sensing rodin the centre of a high–frequency induction furnace. Length changes are transmitted throughthe frictionless rod to an electronic transducer which in turn drives the recording system. Thethermocouple is spot welded to the sample, and referenced at 40◦C by means of a constanttemperature bath. Quenching gas enters the chamber, passes through the hollow cylindricalsample and escapes through the openings.

It may not be possible in conventional dilatometers to conduct experiments in which thespecimens require very fast heating or cooling rates, because of the high thermal inertia of thefurnaces involved. In a high–speed dilatometer, the specimen is positioned along the axis ofa cylindrical heating coil, which is connected to a radio–frequency power generator. Duringoperation, the magnetic field around the coil induces currents in the specimen, causing it towarm up. The induction coil itself is only mildly heated through resistive effects, but is in anycase water cooled, Hence, the response of the system depends on the thermal characteristics ofthe specimen rather than those of the furnace. Fast quenching rates can therefore be achievedby directing high–pressure jets of gas through the centre of a hollow specimen; quench ratesof up to 5000◦C/s can be obtained in favourable circumstances.

Helium is used as the quenching gas in high cooling rate experiments because its thermalconductivity is about six times that of nitrogen. Large thermal gradients can develop in thespecimen during the quench, which may cause difficulties in the interpretation of the observedchanges in specimen dimensions. The system should ideally be arranged to stabilise at theisothermal temperature concerned before the onset of any transformation.

The specimen can be enclosed in a vacuum, but it is more usual to use an inert atmosphere;in the case of steels, an inert atmosphere results in a smaller degree of decarburisation. Thespecimen chamber should be evacuated before triggering the quench gas; otherwise, the buildup of pressure in the chamber retards the quench, and may cause the lid of the dilatometerchamber to be blown off! The pumping system must be isolated before triggering the quench.

Specimen Design

Specimens for use in high–speed dilatometry are usually in the form of hollow rods, withinternal and external diameters of 1.5 and 3.0 mm respectively; the length is limited by theextent of the furnace to a maximum of about 3 cm. These dimensions are somewhat arbitrary,but experience suggests that they satisfy the following requirements:

1. The specimen should be sufficiently thick to prevent free surface effects from altering

transformation kinetics. The tests should reflect what happens in equivalent bulksamples. Nickel plating the specimens (to a thickness of about 0.08 mm) helps toreduce surface nucleation. Contrary to popular belief, this is not an effective way ofpreventing decarburisation in steels. To reduce decarburisation, the specimen shouldbe copper plated; carbon is only sparingly soluble in copper. The plating materialshould be chosen so as not to interact with the sample, for example by penetrationinto the grain boundaries of the substrate.

2. The specimen dimensions must be small enough to allow rapid changes in tempera-ture.

The specimen should obviously be representative of bulk material. Its ends should be groundflat and parallel to give a true cylindrical shape. Otherwise, slip at the specimen–quartzinterface can lead to erroneous interpretation especially under the influence of the high–pressurequench–gas jets.

It is normal to isolate the specimen from the RF coil with a length of quartz tubing, not only toavoid contaminating the coil, but also to guard against the potentially disastrous consequencesof accidental specimen melting.

High–speed dilatometers generally do not have long term electronic or mechanical stability.Equipment like this cannot be used for tests lasting more than a few hours. Prolonged holdingat high temperatures can also lead to slag–forming reactions between the specimen and quartzretaining–rods. A certain amount of pressure is always necessary to hold the specimen betweenthe quartz retaining rods and to remove backlash, so care should be taken to ensure that anyresulting creep effects are negligible.

Calibration

The temperature calibration is similar to that of differential scanning calorimeter and is notdiscussed further. However, for the calculation of thermal expansion coefficients, and for thepurposes of absolute dilatometry, it may be necessary to calibrate the magnification of thedisplacement transducer.

There are two ways of doing this:

1. A pure platinum specimen with known expansion characteristics is heated at a suf-ficiently slow rate over the temperature range of interest. The magnification M isthen given by

M =∆l

∆T lPtePt(3)

where ∆l is the deflection of the length recording pen, ∆T is the difference betweenthe initial and final temperatures T1 and T2 respectively of the test, lPt is thelength of the Pt specimen at T1, ePt the linear expansively of Pt (obtainable fromstandard handbooks). This method can be accurate, but does not take account ofthe expansion of the part of the quartz rods within the furnace assembly.

2. A micrometer attachment on the dilatometer allows the transducer to be stim-ulated independently of specimen movement. The magnification is then simply∆l/micrometer movement. Having calculated the magnification the additional ex-pansion to be expected from the quartz rods, lQeQ can be obtained from :

∆l = M(lPtePt + lQeQ) (4)

Interpretation of Transformation Curves

Dilatometric data are plotted as graphs of relative length change versus time. Theseplots are usually sigmoidal in shape, with the length change being dependent on theextent of transformation. If the length of the specimen before the beginning of trans-formation is l1, and that at the termination of transformation at a time t = t2 is l2,then at time t,

VtV2

= f

{lt − l1l2 − l1

}(5)

where t = 0 at the beginning of transformation and lt is the specimen at any time t. Vt isthe volume fraction of transformation corresponding to t, and V2 is that when reactionhas stopped. The latter quantity may either be deduced by using an independenttechnique, or, if sufficient data are available, it can be calculated from the magnitudeof ∆l2, where

∆l2 = l2 − l1

For an austenite (γ) to ferrite (α) transformation in a plain carbon steel, in whichthe formation of ferrite enriches the residual austenite with carbon,

∆l

l=

2V a3α + (1− V )a3

γ0− a3

γ

3a3γ

(6)

where ∆l/l is the length change per unit length; V is the volume fraction of ferrite;aα is the lattice parameter of ferrite at the transformation temperature; aγ0

is thelattice parameter of austenite at the transformation temperature when the austenitehas the mean composition of the steel (x); aγ is the corresponding lattice parameterof carbon–enriched austenite. The extent of enrichment can be estimated from massbalance:

xγ = x+ V (x− xα)/(1− V )

where xα is the carbon concentration in the ferrite.

Texture and Anisotropy of Thermal Expansion

Most polycrystalline materials are crystallographically textured. This becomes im-portant when testing materials with low crystalline–symmetry and anisotropic thermalexpansion characteristics. The assumption that the measured length change correspondsis about a third of the volume change is no longer valid and it becomes necessary tospecify the crystallographic directions along which measurements are made.

It turns out that such effects may be advantageously exploited. Under the influenceof neutron irradiation Uranium has a pronounced tendency to swell in the < 0 1 0 >direction and contact in the < 1 0 0 > direction. In polycrystalline specimens, theswelling tendency must depend on the texture, which can be determined using tediousX–ray techniques. Alternatively, an indirect measure of preferred orientation can beobtained by measuring thermal expansion and electrical resistivity along a variety ofdirections of the specimen shape. The expansion coefficient and thermal resistivitycharacteristics of uranium are both anisotropic. High resistivity in a particular directionindicates an excess of < 1 0 0 > directions over random, while a low expansion coefficientindicates an excess of < 0 1 0 > directions. Hence these two measurements made in the

same direction on the specimen give an estimate of the tendency to irradiation growth.(Journal of Nuclear Materials, 4, 109, 1969).

Miscellaneous

It is known that cycling a specimen through a phase transformation can lead topermanent length change, so that the apparent length change observed may alter withthe number of cycles, even though there may be no real changes in transformationbehaviour.

When dealing with specimens of low thermal conductivity, care should be taken toensure that the imposed heat–treatments allow the specimen to attain thermal equilib-rium within the time period of the test.

University of Cambridge, Materials Science & Metallurgy H. K. D. H. Bhadeshia

Other Techniques

Thermogravimetry

Thermogravimetry (TG) is a technique by which the weight of a substance, in an environmentheated or a cooled controlled rate is recorded as a function of time or temperature. Derivativethermogravimetry exploits the first derivative of the TG curve.

Thermogravimetry seems to have been used first in 1912 in the study of the efflorescence ofhydrated salts and has since been an aid in the quantitative investigation of decompositionreactions; it is at its best when used in conjunction with calorimetric analysis.

The equipment consists of a precision balance, a furnace with a programming facility, a reactionchamber and a suitable recording system. The assembly has to be capable of continuouslyregistering any weight changes in the test sample whilst the latter is being heat–treated. Thereare essentially two types of apparatus – the null–point balance and the deflection balance. Inthe former, any movement of the balance beam caused by weight changes is counteracted bya restoring force to bring the beam back to its original position; this force is then taken tobe proportional to the weight change concerned. Such instruments are readily adaptable tooperation in a vacuum. The deflection instrument can be more robust and reliable because itis based a conventional analytical balance.

The thermocouple is usually placed in direct contact with the sample. However, any connec-tions to the support must afford an extremely small torque, or in the case of null balanceinstruments, a small or highly reproducible torque. TG is a quantitative and dynamic tech-nique, and a number of factors can effect the shape of the TG curve. Temperature gradients,air buoyancy, convection currents within the furnace tube and other factors contribute to theso called buoyancy effect† in which the weight of an inert, empty crucible changes with tem-perature. A correction curve can be determined to compensate for this effect, making certainthat the conditions of such a calibration correspond to those of the actual experiment.

TG gives absolute changes in sample weight so that the calculated extent of reaction is notaffected by the heating rate used, although the start and finish temperatures are a function ofheating rate because of kinetic barriers.

The main applications of TG include the measurement of thermal stability, ageing character-istics, decomposition, reactivity and the structures of compounds. It has obvious uses in thedetermination of the moisture content of powders, water of hydration and of carbon monoxideand carbon dioxide evolution from carbonates etc. Decomposition reactions can be studiedin a variety of imposed environments to yield information on the reduction of metal ores. Inorganic chemistry the technique has been widely used to study the degradation of polymersand to investigate the pyrolysis of coals, peats and bitumens. TG can also be used to recordisothermal and isobaric weight changes.

† When a specimen is heated in ambient air or any gas of comparable density, the apparentweight changes with temperature due to the change in weight of the displaced gas; the specimenthen appears to gain weight on heating.

Evolved Gas Analysis

Useful information can often be gained by studying the gases which arise through the thermaldecomposition of compounds. Evolved Gas Analysis (EGA) apparatus can be of two kinds:

a) That in which the gas is evolved inside the gas detector or analysing apparatus,enabling the detection of transient species evolved during the course of the decom-position reaction. Difficulties can arise in the control of the temperature, pressureand degree of dilution of the sample environment.

b) That in which the evolved gas is led into a separate detection system, enabling moreflexible apparatus design, the possibility of pretreating the gas before analysis, andgenerally better control of the specimen environment. The possibility of secondaryreactions occurring during the passage of the gases from the source to the detectorcannot be discounted.

Many types of detectors have been used for evolved gas analysis; those which sense the thermalconductivity of gasses are popular, but not very discriminating; gas–density and ionisationdetectors are also used. Gases can be absorbed in suitable solvents for subsequent chemicalanalysis, and the use of mass spectrometers for high–resolution analysis is common.

References to Thermal Analysis Techniques

Books and Reviews

1 M. I. Pope and M.J. Judd: ‘Differential Thermal Analysis’, London, Heyden, 1977.Analysis’, 1,2, London, Academic Press.

2 J. Sestak, V. Satava and W.W. Wendlandt: Thermochimica Acta, 7, 333 (1973).

3 A. L. Greer: Ph.D. Thesis, University of Cambridge (1979).

4 V. Konryushin and L.N. Larikov: J. Mat. Sci., 13, 1 (1978).

Specific References

5 A. L. Greer: Thermochimica Acta, 42, 193 (1980). Curie point measurement in theDuPont DSC system.

6 R. A. Baxter:‘Thermal Analysis’, 1, 65 (1969) - eds. R.F. Sswenker and P.D. Garn,New York, Academic Press. This is the original paper on the Dupont DSC system.

7 H. K. Yuen, C.J. Yosel: Thermochimica Acta, 33, 281 (1979). Specific heat mea-surement in the DuPont DSC system.

8 L. M. Clareborough, M. E. Hargreaves, D. Mitchell and G. West: Proc. Roy. Soc.,A215, 507 (1952). Description of the original DSC concept, and measurement of thestored energy of deformation.

9 M. G. Scott and P. Ramachandrarao: Mat. Sci. & Eng. 29, 137 (1977). Deals withthe establishment of the baseline and some kinetic investigations.

10 M. G. Scott: J. Mat. Sci., 13, 291 (1978). Kinetics of glass crystallisation. Anal.Chem., 29, 1702 (1957). Deals with the Kinetics of non-isothermal, homogeneoustransformations.

11 D. W. Henderson: J. Non-Cryst. Sol., 30 301 (1979). Kinetics of non-isothermal,nucleation and growth transformations.

12 E. A. Marseglia: J. Non-Cryst. Sol., 4, 31 (1980). Kinetics of non-isothermalnucleation and growth transformations.

13 J. W. Christian: ‘Theory of Transformations in Metals and Alloys’. 2nd ed., (1975),Part 1, Pergamon Press, Oxford. Elegant and comprehensive review on the formaltheory of transformation kinetics.

14 K. Matusita and S. Sakha: Non-Cryst. Sol., 38-39, 741 (1980). Kinetics of glasscrystallisation.

15 A. P. Gray: Thermochimica Acta, 1, 563 (1970). Polymer crystallisation.

16 R. O. Simmons and R.W. Balluffi: Phys. Rev., 117, 52 (1960) & 125, 862 (1962).Dilatometric measurement of vacancy concentrations.

17 M. Perakh: Surface Technology, 4, 527 (1976) & 4, 538 (1976). Dilatometric deter-mination of stress.

18 T. Inoue and B. Raniecki: J. Mech. Phys. Sol., 26, 187 (1978). Use of dilatometryin the prediction of thermal hardening and transformation stresses.

19 J. J. Stobo and B. Pawelski: J. Nuc. Mat., 4, 109 (1961). The influence of textureon the anisotropy of thermal expansion.

20 J. Valentich: J. Mat. Sci., 14, 371 (1979). General dilatometry.

21 W. G. Hall and T.N. Baker: ‘Phase Transformations’, York Conference, (1979), No.11, Series 3, 2, 11-90. Resistively changes during the ageing of steels.

22 J. Bass: Advances in Physics, 21, 431 (1972). Resistance measurements and thedeviations from Mathiesson’s rule.