Heat capacity of polymers*

B. Wunderlich    Department of Chemistry, The University of Tennessee, Knoxville, TN 37996-1600, USA; and the Chemical and Analytical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6197, USA

Heat capacity is the basic quantity derived from calorimetric measurements and is used in the description of its thermodynamics. For a full description of a system, heat capacity information is combined with heats of transition, reaction, etc. In the present Chapter the measurement and theory of heat capacity are discussed which leads to a description of the Advanced THermal Analysis System, ATHAS [1]. This system was developed over the last 30 years to increase the precision of thermal analysis of linear macromolecules and related small molecules.

1 MEASUREMENT OF HEAT CAPACITY

In a measurement of heat capacity one measures the heat, dQ, required to increase the temperature of the sample by dT:

p≡dQdT=∂H∂Tp,n,

where H represents the enthalpy and p and n, are pressure and composition, which are kept constant. Classically, the measurement is done with an adiabatic calorimeter [2]. Even today, adiabatic calorimetry is the most precise method of measurement in the temperature range from 10 to 150 K.

In an adiabatic calorimeter an attempt is made to follow step-wise temperature changes of an internally heated calorimeter in a well-controlled, adiabatic enclosure. Due to loss of heat caused by deviations from the adiabatic condition, corrections must be made to the heat added to the calorimeter, ΔQ. Similarly, the measured increase of temperature, ΔT, must be corrected for the temperature drifts of the calorimeter. The calculations are then given by:

p=ΔQcorrected−C′ΔTcorrectedmΔTcorrected,

where the specific heat capacity cp (in J K− 1 g− 1) is calculated by subtracting the heat capacity of the empty calorimeter, its “water value” C′, and dividing by the mass of the sample, m. The evaluation of the corrections is time consuming, but at the heart of good calorimetry.

Modern control and measurement technology permitted some 30–40 years ago to miniaturize the calorimeter to measure down to milligram quantities in a differential scanning mode (DSC) [3]. In a symmetric setup, the difference in extraneous heat flux can be minimized and the remaining imbalance corrected for. Under the usual condition where sample and reference calorimeters (often aluminum pans) are identical and the reference pan is empty, one finds the heat capacity as:

cp=KΔTβ+KΔTβ+CpldΔT/dTr1−dΔT/dTr≈KΔTβ,

where K is the Newton’s law constant, ΔT is the temperature difference between reference and sample (Tr - Ts), and β is the constant heating rate (in K min− 1)’: The left equation is exact, the right approximation neglects the difference in heating rates between reference and sample due to a slowly changing heat capacity of the sample. This limits the precision to ± 3% or less.

In the more recently developed temperature-modulated DSC (TMDSC) [4], a sinusoidal or other periodic change in temperature is superimposed on the underlying heating rate. The heat capacity is now given by:

cp=AΔAKω2+Cp′2,

where AΔ and A are the maximum amplitudes of the modulation found in the temperature difference and sample temperature, respectively, and ω is the modulation frequency 2π/p (p = modulation period in seconds). The equation represents the reversing heat capacity. In case there is a difference between the result of the last two equations, this is called a nonreversing heat capacity, and is connected to processes within the sample that cannot be modulated properly, such as the change of temperature in the glass transition region, irreversible crystallization or reorganization, or heat losses.

Fig. 1 illustrates a heat-capacity measurement with a DSC. Central is the evaluation of the calibration constant at each temperature of measurement. Three consecutive runs must be made. The sample run (S) is made on the unknown, enclosed in the customary aluminum pan and an empty, closely-matched aluminum reference pan. The width of the temperature interval is dictated by the quality of the isothermal base line. In the example of Fig. 1 the initial isotherm is at 500 K and the final isotherm, at 515 K. Typical modern instrumentation may permit intervals as wide as 100 K. Measurements can also be made on cooling instead of heating. The cooling mode is particularly advantageous in the glass transition region since it avoids hysteresis. When measurements are made on cooling, all calibrations must necessarily also be made on cooling, using liquid-crystal transitions which do not supercool, for example, for temperature calibration [5]. As the heating is started, the DSC recording changes from the initial isotherm at 500 K in an exponential fashion to the steady-state amplitude on heating at constant rate β. After completion of the run, at time tf, there is an exponential approach to the final isotherm at 515 K. The area between base line and the DSC run, called AS, is the integral over the amplitude, as. A reference run (R) with a matched, second, empty aluminum-pan, instead of the sample, yields the small correction area AR which may be positive or negative. Finally, a calibration run (C) with sapphire crystals (A12O3) completes the experiment (area AC). The difference between the two areas, AS – AR, is a measure of the uncalibrated integral of the sample heat capacity:

′/β=WCc¯pCΔTAC−AR.

For a small temperature interval at constant heating rate β in the region of steady state in Fig. 1, it is sufficient to define an average heat capacity over the temperature range of interest to simplify the integral, as indicated. The uncalibrated heat capacity of A12O3 is similarly evaluated using Ac – AR. The proportionality constant, K′,can now be evaluated as:

0tfCpSβdt=C¯pSΔT=K′AS−AR/β.

The specific heat capacity of sapphire, cp(Al2O3), is well known, and the weight of sapphire used, W(C) = W(Al203), must be determined with sufficient precision (± 0.1%) so as not to affect the accuracy of the measurement (± 1% or better).

The initial and final exponential changes to steady state are sufficiently similar in area so that one may compute the heat capacities as a function of temperature directly from the amplitudes as(T), aR(T), and aC(T) in the region of steady state. Eq. (7) and Eq. (8) outline the computations involved in the calibration:

0tfCpSβdt=CpSΔT=K′aS−aR/βand

′/β=WCcpCΔTaC−aR.

In a typical DSC experiment, a sample may weigh 20 mg and show a heat capacity of about 50 mJ K 1. For a heating rate of 10 K min 1 there would be, under steady-state conditions, a lag between the measured temperature and the actual temperature of about 0.4 K. This is an acceptable value for heat capacity that changes slowly with temperature. If necessary, lag corrections can be included in the computation. The typical instrument precision is reported to be ± 0.04 mJ s− 1 at 700 K. Heat capacities can thus be measured to a precision of ± 0.25%, respectable for such fast measurements. For TMDSC, sample mass and modulation parameters are chosen to modulate the whole sample with the same amplitude, checked by measuring with varying sample masses [6], or calibration as a function of frequency is necessary.

2 THERMODYNAMIC THEORY

The importance of heat capacity becomes obvious when one realizes that by integration it is linked to the three basic quantities of thermodynamics, namely the enthalpy or energy (H, U), the Gibbs energy or free energy (G, F), and the entropy (S). Naturally, in cases where there are transitions in the temperature range of interest, their heats and entropies need to be added to the integration. The enthalpy or energy of a system can be linked to the total amount of thermal motion and interaction on a microscopic...