Thermal Analysis of Polymeric Materials
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Fig. 2.66
Fig. 2.67
MBPE-9. Macromolecules that contain aliphatic and aromatic sequences in their backbone structure have special possibilities to exist as liquid crystals or as condis crystals (see Sect. 2.5). The thermal analysis of a polyether based on the semiflexible mesogen 1-(4-hydroxyphenyl)-2-(2-methyl-4-hydroxyphenyl)ethane is copied in Fig. 2.68. The computed vibration-only heat capacity is reached somewhat
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Fig. 2.68
below the glass transition, but above the glass transition the heat capacity does not reach the computed value expected from the crystallinity derived from the heat of fusion (wc = 0.43). One assumes that there is some rigid-amorphous fraction present. Furthermore, the entropy of fusion indicates considerable conformational disorder in the crystal that freezes in its own glass transition, indicated by the cross-hatched area. The rigid-amorphous phase is discussed in Sects. 6.1.3 and 6.3.4.
Liquid selenium. The heat capacity of liquid selenium, which is also a linear macromolecule, illustrates the effect of a chemical equilibrium. The heat capacity for many liquid, linear macromolecules increases linearly with temperature, despite the fact that one expects an exponential increase in Cv from the vibrational contributions of the C H-bending and -stretching vibrations. To understand this observation, one has to remember that a decreasing heat capacity contribution with temperature results from the changes of the heat capacity from vibrational motion to rotational motion. Also the hole-theory of liquids, discussed in Sect. 6.1.3, includes a decrease in the hole contribution to the heat capacity since it is calculated similar to the case of two energy states, (see Fig. 2.33). The maximum in heat capacity occurs close to the glass transition temperature. The subsequent decrease seems to be able to compensate most of the vibrational increases in heat capacity for a considerable temperature range. Looking now at the heat capacity of the monatomic, liquid selenium, one finds that there are no group vibrations to produce the exponential increase in Cp and, indeed, the heat capacity of the liquid Se decreases with temperature from the glass-transition temperature. The decrease in Cp is nonlinear. At high temperature, however, Cp increases again. These changes in heat capacity with temperature are specific to Se, and an explanation is given in Fig. 2.69. Curve 1 represents the experimental data, curve 2, the vibrational contributions as derived from the crystalline vibrational spectrum. The difference between curves 1 and 3
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Fig. 2.69
represents the special contribution to the heat capacity that arises from the ring–chain equilibrium, i.e., from a chemical reaction. The melt of selenium consists of very long macromolecules (Se )x and rings (mainly Se8). The two components are in a temperature-dependent chemical equilibrium, and the difference between curves 1 and 3 is an estimate of the heat of reaction per kelvin of a temperature increase for this process. Finally, the difference between curves 3 and 4 is an estimate of the heat capacity due to the hole equilibrium. It decreases with temperature, as expected. The agreement between curves 4 and 2 is, finally, a measure of the quality of the model that was chosen for the interpretation of the change of heat capacity with temperature. Selenium represents thus a case where a chemical heat of reaction (latent heat) is included in the measurement of heat capacity.
From the addition scheme of heat capacities, it is possible to deduce the heat capacity of another, hypothetical, monatomic polymeric chain, namely (O )x. Its heat capacity is estimated by subtracting the (CH2 )C contribution to the heat capacity from the total heat capacities of the polyoxides which are shown in Fig. 2.59. As in the Se heat capacities, the heat capacities of (O )x decrease with temperature.
Poly(styrene-co-1,4-butadiene). To describe heat capacities of copolymers, it is naturally impossible to measure each and every composition. But since the heat capacities in the solid state at temperatures above, perhaps 100 K, are close-to additive with respect to composition, and the same is true for the liquid states, estimates can be made of the heat capacities of all copolymers. Figure 2.70 shows the additivity of the heat capacities of poly(styrene-co-1,4-butadiene). These measurements were done many years ago by adiabatic calorimetry, and it was possible later to derive the heat capacity of the copolymers from their molar composition and the heat capacities of the constituent homopolymers. As indicated in the equation, it is also possible to add more than two components. A conspicuous
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Fig. 2.70
feature of the 1,4-polybutadiene and copolymer heat capacities is the increase in temperature of the glass transition with styrene composition. The glass transition of pure polystyrene occurs at 373 K. The changes of glass transition temperature with concentration are discussed in Chap. 7. Comparing now the heat capacity measured by calorimetry with the heat capacity calculated from the addition scheme, one finds an error between 0 and 4% from 10 K to the beginning of the glass-transition region. But even above the glass-transition temperature, the additivity works well. The errors are less than 5%. Note that for the addition scheme contributions of both components, the butadiene as well as styrene, must be taken as their liquid heat capacities as soon as the copolymer glass-transition temperature has been exceeded. Position and breadth of the glass transition has to be predicted from other considerations.
Hypothetical entropy of the liquid at absolute zero of temperature. This topic was raised many years ago when extrapolating the heat capacity of the liquids in Figs. 2.59–61 to absolute zero and estimating the entropy, as outlined in Fig. 2.23. Since the heat capacity of the (metastable) liquid below the melting temperature is higher than that of the crystal, the step in entropy seen in Fig. 2.23 becomes less and finally reaches zero at the so-called Kauzmann temperature, 145 K for polyethylene. The third law of thermodynamics in Sect. 2.2.4, however, is hard to be reconciled with an amorphous structure of an entropy smaller than zero. The glass transition which intervenes below 237 K slows the structure rearrangements required for the liquid, so that the measured heat capacity drops and leaves a positive entropy of 2.59 J K 1 mol 1 at zero kelvin. Experimentally, thus, the entropy of the liquid at zero kelvin cannot be reached, but what if one could make sufficiently slow experiments to follow the liquid to low temperatures would the entropy be positive? Figure 2.71 suggests that these linear extrapolations of the liquid heat capacities of polymers shown in Figs. 2.46 and 2.59–61 are not correct at such low temperatures. The heavy
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Fig. 2.71
black line was calculated accounting for the conformational energy difference, cooperativity along the chain, and the external contribution which can be approximated by Cp CV. All three contributions are indicated in the figure. At high temperature, the fit to the experimental data of Fig. 2.46 is quite perfect, but the extrapolation below the glass transition temperature is far from linear. Figure 2.72 shows the corresponding entropy calculation. Instead of a negative entropy at zero
Fig. 2.72
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kelvin of 85 J K 1 mol 1 which a linear extrapolation of Cp would have given, one reaches now 6.8 J K 1 mol 1. A minor correction for the external contribution which is based on experiments at high temperature to account for the decrease in the conformational motion yields the indicated residual entropy at zero kelvin of +0.41 J K 1 mol 1. This refutes, at least for polyethylene, what is called the Kauzmann paradox, a negative entropy at zero K. Note, that the free enthalpy G has no such problem. The free enthalpy of the liquid is identical to that of the crystal at the equilibrium melting temperature as shown in Fig. 2.23. Relative to the crystal, it increases when approaching zero kelvin, where it reaches +2.1 kJ mol 1, indicating a substantial metastability of the liquid. The heat of fusion at the melting temperature is 4.11 kJ mol 1, and it decreases at zero kelvin to half of this value.
Starch and water. The final example of the analysis of heat capacities of linear macromolecules concerns the biological carbohydrates. Figure 2.73 illustrates as a first step the standard ATHAS analysis of the small molecule glucose, a key building block of the carbohydrates. Although glucose is not a linear polymer, the fitting to a vibrational skeletal and group-vibration spectrum is possible for the solid (glassy) state, and the melt yields a straight-line temperature dependence of the heat capacity as seen for the linear macromolecules in Figs 2.46 and 59–61. Using in a first step the heat capacity of liquid glucose as an estimate for the heat capacity of liquid starch, a polymer made out of glucose monomers with repeating 1,4-glucopyranose links, a similar heat capacity analysis is possible as displayed in Fig. 2.74. Besides the basic thermodynamic information available from the heat capacity information on amorphous, glassy starch, one can see that there is an increase of the heat capacity above room temperature. From the experience gathered in this section, one might interpret this increase in heat capacity as the beginning of the glass transition when compared to the polyethylene displayed in Fig. 2.46. The starch starts decomposing
Fig. 2.73
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Fig. 2.74
on heating long before the glass transition is completed and the heat capacity has reached the level expected for starch from the liquid glucose. Adding water to the starch, an important process in the food industry, the results of Fig. 2.75 have been obtained. The beginning of the glass transition of the starch is moved to lower temperature and in addition to the transition already seen in the amorphous, glassy starch, there is a second step in the glass transition, also not reaching the expected
Fig. 2.75
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heat capacity as derived from the glucose. Above 450 K, thus, there must be a third step in the devitrification of starch and water system. The next step in this analysis is the identification of the molecular motion responsible for the stepwise increase in large-amplitude motion, this research is not completed yet. Furthermore, the phase diagram of the partial molar quantities needs to be evaluated for all thermodynamic functions, as outlined in Fig. 2.25.
Conclusions. The heat capacity is one of the few macroscopic quantities with a well-understood link to the microscopic molecular motion. The Advanced Thermal Analysis System displayed in Fig. 2.76 allows to establish empirical addition schemes
Fig. 2.76
to compare similar structures and permit estimation of heat capacities of unknown materials. Figure 2.77 summarizes such addition schemes and their limits at low temperature were shown above in Sect. 2.3.7 to be caused by the nonlinear sum of the skeletal vibrations. A second step in ATHAS is the evaluation of the baseline of vibration-only heat capacity from the analysis of low-temperature solids of linear macromolecules, as well as related monomers. The key to this analysis is the separation of the frequency spectrum into group vibrations whose frequencies are experimentally available from Raman and IR data without calorimetric input, and skeletal vibrations, which can at present only be fitted to experimental data using two or three fitting parameters, the theta temperatures of Fig. 2.50. Figure 2.78 gives a schematic of the handling of the experimental data and the generation of the parameters for the calculations needed for the extrapolation into temperature ranges not available experimentally. The changes in the types of motion from the expected data can be found with the help of a fast, quantitative thermal analysis described in Chap. 4. Based on the accumulation of calorimetric data and evaluation together with kinetic information discussed in Chap. 3 and structural information discussed in
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Fig. 2.77
Fig. 2.78
Chap. 5, it is possible to interpret the behavior of polymeric materials as is described in more detail on-hand of a larger series of examples in Chaps. 6 and 7. The understanding of polymeric materials at this level has the goal to predict properties before samples become available, a result of great importance because of the unlimited number of macromolecules possible.
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2.4 Nonequilibrium Thermodynamics
For the understanding of thermal analysis, it is necessary to recognize that most encountered polymeric systems are not in equilibrium, may be microor nanophase separated, and could even be not homogeneous. In order to describe such situations, one has to turn to nonequilibrium, irreversible thermodynamics. Functions of state such as entropy and free enthalpy are defined for systems in global equilibrium. Some others, such as mass, volume, and total energy, on the other hand, are largely indifferent to equilibrium. They need special treatment to be used for the description of nonequilibrium situations which arise when deviating from the free enthalpy minimum characteristic for equilibrium, as shown in Fig. 2.79. The four possible functional forms of the free enthalpy lead to stable, metastable, neutral, and labile equilibria.
Fig. 2.79
The next step of importance is the description of inhomogeneous systems in terms of local equilibria. This process involves the division of the system to be described into subsystems, as discussed in Sect. 2.2.1, and shown in the upper half of Fig. 2.80. This is to be followed by the description of the system as a function of time as an additional variable. Following the system as a function of time allows the study of the kinetics of the processes seen when approaching equilibrium. Finally, it may be necessary to identify additional, internal variables to describe the nonequilibrium states as a function of time. The kinetics of polymers may be sufficiently slow to decouple consecutive steps. For very slow responses, it may lead to arrested equilibria, such as seen in materials below their glass transitions as described in Sect. 2.4.4. The following Section makes use of the just summarized concepts in an attempt to achieve a depiction of the nonequilibrium state of macromolecules.