Thermal Analysis of Polymeric Materials
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Macromolecules behave always as real solutions because of the size dependence of the mixing parameters which is approximated by the Flory-Huggins equation, described in Chap. 7. A correction for the mixing of small molecules in the presence of ions is given by the Debye-Hückel theory, not discussed in this book.
The ideal mixing process can be visualized by the expansion of two pure gaseous components of volumes, V1 and V2, at an initial pressure p and temperature T, into the volume, Vt. Figure 2.24 represents the process schematically. Before mixing, two separate ideal gas equations describe the system (independent of the chemical nature of the molecules pV1 = n1 RT and pV2 = n2RT, see Fig. 2.23). The pressures one would find if components 1 and 2 alone would occupy volume Vt are called the partial pressures p1 and p2, respectively, so that one can write the overall gas equation as pVt = nt RT = (p1 + p2) RT = (n1 + n2)RT. For ideal gases the mixing process occurs without change in internal energy as long as the number of moles of molecules n and the temperature T stay constant (dU = 0). The change in enthalpy is also zero since dH = d[U + (pV = nRT)] = 0, so that the change in entropy dS = dG/T. Under these conditions only volume work can be done, which leads to dG = pdV and:
( S/ V)T,n = p/T = nR/V,
(see Fig. 2.19). Figure 2.24 illustrates the results on expansion of the given initial volumes V1 and V2 to Vt, and lists the final mixing equations that describe the change on mixing [Eqs. (1) and (2)]. The concentrations are expressed in terms of the mole fractions x1 and x2 which derive by insertion of the appropriate expressions for n1 and n2. Since mole fractions are always smaller than one, the entropy on mixing is always positive and G, negative, i.e., the mixing process must always be spontaneous and can only be reversed by a negative interaction term H, not present in ideal gases.
Fig. 2.24
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Because of the simplicity of the functions of state of the ideal gas, they serve well as models for other mixing experiments. Dilute solutions, for example, can be modeled as ideal gases with the empty space between the gas atoms being filled with a second component, the solvent. In this case, the ideal condition can be maintained as long as the overall interaction between solvent and solute is negligible. Deviations from the ideal mixing are treated by evaluation of the partial molar quantities, as illustrated on the example of volume, V, in Fig. 2.25. The first row of equations gives the definitions of the partial molar volumes VA and VB and shows the addition of their differentials. For ideal solutions the partial molar volumes are equal to the volumes in the pure states (VAo and VBo). The first equation in the second row shows the obvious integration of dV to V without change in concentration. The total differential of V, derived after this integration can be compared, in turn, with the starting equation for dV in the first row. Both can only hold simultaneously if the boxed equation, the Gibbs-Duham equation, is true. After measurement of V over the full concentration range, the construction and equations in Fig. 2.25 permits the full evaluation of the volume of the system. In the shown case there is an expansion on mixing, indicative of repulsion between the two types of molecules. Similar constructions can be made for other partial molar quantities of interest. More details, especially for solutions of polymers, are given in Sect. 7.1.
Fig. 2.25
2.2.6 Multi-phase Systems
Phases are described in Sect. 2.5. An equilibrium between phases involves the exchange of molecules (components) of different chemical nature or physical state. Two components of small molecules and two phases are treated in this section. Single and multi-phase systems of macromolecules are treated in Chaps. 6 and 7.
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The description of phase equilibria makes use of the partial molar free enthalpies,, called also chemical potentials. For one-component phase equilibria the same formalism is used, just that the enthalpies, G, can be used directly. The first case treated is the freezing point lowering of component 1 (solvent) due to the presence of a component 2 (solute). It is assumed that there is complete solubility in the liquid phase (solution, s) and no solubility in the crystalline phase (c). The chemical potentials of the solvent in solution, crystals, and in the pure liquid (o) are shown in Fig. 2.26. At equilibrium, of component 1 must be equal in both phases as shown by Eq. (1). A similar set of equations can be written for component 2. By subtracting1o from both sides of Eq. (1), the more easily discussed mixing (left-hand side, LHS) and crystallization (right-hand side, RHS) are equated as Eq. (2).
The mixing expression for the LHS of Eq. (2) is derived from Eq. (2) of Fig. 2.24. For dilute solutions ln x1 is close to zero (x1 1), so that the series representing the logarithm of x1 in terms of x2 (x1 = 1 x2) can be represented by the first term alone, giving a very simple LHS of Eq. (2) written in Fig. 2.26. At larger concentrations, the full logarithm must be used.
Fig. 2.26
The RHS of Eq. (2) of Fig. 2.26 applies to the crystallization of pure 1 (one component, two-phase system). It is written as the reverse of fusion (f), which accounts for the minus sign in the expression for the RHS. Assuming that the change of entropy and heat of fusion with temperature is negligible over the range of freezing point lowering, a simple expression for the RHS is derived. The final equation linking x2 to the entropy of fusion of the solvent ( Hf/Tmo) and the lowering of the melting temperature, T, is obtained by equating LHS with RHS.
Experimental use of the melting-point lowering can be made to determine molar concentrations, of importance for the molar-mass determinations (see Sect. 1.4), and
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the heats of fusion (diluent method, see Sect. 7.1.4). Figure 2.27 illustrates the use of the freezing point lowering for the description of the left side of a eutectic phase diagram. Analogous expressions can be derived for the right side by replacing 1 with 2 in the equations. The two curved boundaries of the area of the melt are the liquidus lines. At the intersection of the two lines one finds the eutectic temperature where both crystals have the same freezing point lowering.
A similar expression as for the freezing point lowering can be derived for the boiling point elevation. The second phase is now the pure vapor phase ( 1v). This case of a pure vapor in contact with a liquid solution occurs for two major groups of components, macromolecules and ions. Polymeric components have negligible vapor pressures due to the large size of the molecule. Ions without solvent are rigid macromolecules (see Sect. 1.1), and also have negligible vapor pressure. In solution they are solvated to large aggregates, still without significant vapor pressure. On subtraction of 1o, the LHS of Eq. (2) of Fig. 2.26 is again the change on dissolution, while the RHS represents the evaporation of the pure solvent. Since the heat of crystallization is exothermic and the heat of evaporation ( Hv) endothermic, the changes in the phase transition temperatures are going in opposite directions. The experimental use of boiling point elevation for the molar mass determination of macromolecules is discussed in Sect. 1.4, along with other methods making use of colligative properties (Fig. 1.52).
Fig. 2.27
A third case arises when a mixed gas is in equilibrium with immiscible pure liquid or solid phases. The vapor pressure of each condensed phase is then unaffected by the existence of the mixed vapor phase. Each pure condensed phase communicates only with its own partial pressure. This results in a total pressure that is the sum of the two vapor pressures, as illustrated above for ideal gas mixing: p1 = n1RT/Vt, and
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p2 = n2RT/Vt. The effect is a higher total pressure above the two pure phases than without the second component. This effect is made use of, for example, in steam distillations. Boiling water carries, on distillation, the low vapor-pressure material to the condensate in the molar ratio of the vapor pressures of the two condensed phases.
2.3 Heat Capacity
Heat capacity is the basic quantity derived from calorimetric measurements, as presented in Sects. 4.2 4.4, and is used in the description of thermodynamics, as shown in Sects. 2.1 and 2.2. For a full description of a system, heat capacity information is combined with heats of transition, reaction, etc. In the present section the measurement and the theory of heat capacity are discussed, leading to a description of the Advanced THermal Analysis System, ATHAS. This system was developed over the last 30 years to increase the precision of thermal analysis of linear macromolecules.
2.3.1 Measurement of Heat Capacity
In a measurement of heat capacity, one determines the heat, dQ, required to increase the temperature of the sample by dT, as is shown in Fig. 2.28. Classically this is done with an adiabatic calorimeter (see Sect. 4.2) which allows no heat transfer to the surroundings of the calorimeter (Gk. - , not able to go through). Even today, adiabatic calorimetry is the most precise method of measurement in the temperature range from 10 to 150 K. In an adiabatic calorimeter [9] (see Fig. 4.33),
Fig. 2.28
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an attempt is made to follow step-wise temperature changes of an internally heated calorimeter in well-controlled, adiabatic surroundings. Due to heat loss caused by deviations from the adiabatic condition, corrections must be made to the heat added to the calorimeter, Q. Similarly, the measured temperature increase, T, must be corrected for the temperature drifts of the calorimeter. The calculations are summarized in Fig. 2.28 (see also Fig. 4.34). The specific heat capacity, written as cp (in J K 1 g 1), is calculated by subtracting the heat capacity of the empty calorimeter, its water value C (or, in Fig. 4.34, Co), and dividing by the mass of the
sample, m (or Wunknown). The evaluation of these corrections is time consuming, but at the heart of good calorimetry.
Modern control and technology of measurement permitted to miniaturize the calorimeter some 30 40 years ago and to measure down to milligram quantities in a differential scanning mode (DSC, see Sect. 4.3 [10]). In a symmetrical 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 shown in Fig. 2.28, where K is the Newton’s-law constant; T, the temperature difference between reference and sample (Tr Ts), and q , the constant heating rate. The left equation is exact, the right approximation neglects the difference in heating-rate between reference and sample due to a slowly changing heat capacity of the sample. This limits the precision to ±3%.
In the more recently developed temperature-modulated DSC (TMDSC, see Sect. 4.4 [11]), a sinusoidal or other periodic change in temperature is superimposed on the underlying heating rate. The heat capacity is now given by the bottom equation in Fig. 2.28, where A and A are the maximum amplitudes of the modulation found in temperature difference and sample temperature, respectively, and is the modulation frequency 2 /period. 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 which are slower than the addition of the heat or which cannot be modulated at all (such as irreversible crystallization and reorganization, or heat losses).
Figure 2.29 illustrates a measurement by DSC. Central is the evaluation of the calibration constant at every temperature. Three consecutive runs must be made. The sample run (S) is made with the unknown, enclosed in the customary aluminum pan and an empty, closely-matched reference pan. The width of the temperature interval is dictated by the change of the isothermal baseline with temperature. In the chosen example, the initial isotherm is at 500 K and the final, at 515 K. Typical modern instrumentation may permit intervals up to 100 K. Measurements can also be made on cooling instead of heating. When measurements are made on cooling, all calibrations must necessarily also be made on cooling, using liquid-crystal transitions which do not supercool on ordering. As the heating is started, the DSC recording changes from the initial isotherm at 500 K in an exponential fashion to the steadystate amplitude on heating at constant rate q. After completion of the run, at tf, there is an exponential approach to the final isotherm at 515 K. The area between baseline and the DSC run is the integral over the amplitude, indicated as As in Fig. 2.30. A reference run (R) with a matched, second, empty aluminum pan instead of the sample
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Fig. 2.29
yields the small correction area AR which may be positive or negative and corrects for asymmetry. This correction can also be done online, as discussed in Appendix 11, below. Finally, a calibration run (C) with sapphire (Al2O3) 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, as indicated in Eq. (1). For a small temperature interval and constant heating rate q, it is sufficient to define an average
Fig. 2.30
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heat capacity to simplify the integral, as indicated. The uncalibrated heat capacity of Al2O3 is similarly evaluated, as shown by Eq. (2). The proportionality constant, K’, can now be evaluated between Eqs. (1) and (2), as shown by Eq. (3). The specific heat capacity of sapphire, cp(Al2O3), is well known, and the sapphire mass used, W(C) = W(Al2O3), 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 directly from the amplitudes aS(T), aR(T), and aC(T) at the chosen temperatures in the region of steady state. Equations
(4)and (5) of Fig. 2.30 are outlines of the computations.
In a typical DSC 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 should thus be measurable to a precision of ±0.25%, a very respectable value for such a fast-measuring instrument. For TMDSC the sample mass and modulation parameters must be chosen, so that the whole sample is modulated with the same amplitude (check with varying parameters).
2.3.2 Thermodynamic Theory
The importance of heat capacity becomes obvious when one looks at Eqs. (1–3) of Fig. 2.22. The quantities of enthalpy or energy, entropy, and Gibbs energy or free energy, are all connected to the heat capacity. Naturally, in cases where there are transitions in the temperature range of interest, the latent heats and entropies of these transitions need to be added to the integration, as shown in Fig. 2.23. The enthalpy or energy of a system can be linked to the total amount of thermal motion and interaction on a microscopic scale. The entropy of a system can be connected to the degree of order, and finally, the Gibbs energy (free enthalpy) or the Helmholtz free energy is related to the stability of the chosen system.
All experimental techniques lead to heat capacities at constant pressure, Cp. In terms of microscopic quantities, however, heat capacity at constant volume, Cv, is the more accessible quantity. The relationship between Cp and Cv is listed as Eq. (4) in Fig. 2.31 in continuation of Fig. 2.22. It involves several correlations, easily (but tediously) derivable from the first and second law expressions as will be shown next. To simplify the derivation, one starts assuming constant composition for the to be derived equation (no latent heats, dn = 0). From the first law, as given by Eq. (3) of Fig. 2.10, one differentiates dQ at constant pressure. Since ( Q/ T)p = Cp and ( U/ T)V = Cv, this differentiation gives Eq. (12) of Fig. 2.10:
Next, from the second law given in Fig. 2.17, one can write:
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Fig. 2.31
The expressions ( F/ V)T = p and ( S/ V)T = ( p/ T)V are derived in Fig. A.5.2. Combining the two equations via the common ( U/ V)T terms, gives:
The final step involves coefficient comparison for dT between the two expressions for dp that can be derived from the differential dV = ( V/ T)pdT +( V/ p)Tdp [Eq. (6) of Fig. 2.10] and dp = ( p/ T)VdT + ( p/ V)TdV. The comparison yields:
Insertion of the expressions for the expansivity, = ( V/ T)p/V, discussed in Sect. 4.1, and the compressibility, which is = ( V/ p)T/V, as discussed in Sect. 4.5, leads first to ( p/ T)V = / and then to the sought-after relationship of Eq. (4) of Fig. 2.31 which was to be proven. To establish Cp Cv, one needs, thus, both the expansivity and the compressibility for all temperatures of interest.
Frequently, however, experimental data for expansivity and compressibility are not available over the whole temperature range, so that one knows Cp, but has difficulties evaluating Cv. At moderate temperatures, as usually encountered below the melting point of linear macromolecules, one can assume that the expansivity is proportional to Cp. In addition, it was found that volume divided by compressibility does not change much with temperature. With these two empirical observations one can derive the approximate Eq. (5) of Fig. 2.31, where A is a constant that needs to be evaluated only at one temperature. If even this is not available, there exists the observation made by Nernst and Lindemann for elements and ionic solids that Eq. (5) holds with a universal constant Ao (= 5.1×10 3, in K mol J 1; Tm = melting temperature). In the survey of Nernst and Lindemann Ao varied from 2.4 to 9.6×10 3 K mol J 1, showing that Eq. (6) is only a rough approximation and should be avoided,
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if possible [12]. Furthermore, one must be careful to use the proper units, Ao refers to one mole of atoms or ions in the sample, so Cp and Cv must be expressed for the same reference amount. But, the difference between Cp and Cv remains small up the melting temperature, as seen in Fig. 2.51, below, for the polyethylene example. A rather large error in Eq. (6), thus, has only a small effect on Cv. For polyethylene, the difference becomes even negligible below about 250 K.
For organic molecules and macromolecules, the equivalent of the atoms or ions must be found in order to use Eq. (6). Since the equation is based on the assumption of classically excited vibrators, which requires three vibrators per atom (degrees of freedom), one can apply the same equation to more complicated molecules when one divides Ao by the number of atoms per molecule or repeating unit. Since very light atoms have, however, very high vibration frequencies, as will be discussed below, they have to be omitted in the counting at low temperature. For polypropylene, for example, with a repeating unit [CH2 CH(CH3) ], there are only three vibrating units of heavy atoms and Ao is 5.1×10 3/3 = 1.7×10 3 K mol J 1. Equation (7) of Fig. 2.31 offers a further simplification. It has been derived from Eq. (6) by estimating the number of excited vibrators from the heat capacity itself, assuming that each fully excited atom contributes 3R to the heat capacity as suggested by the Dulong–Petit rule [13]. The new Ao' for Eq. (7) is 3.9×10 3 K mol J 1 and allows a good estimate of CV even if no expansivity and compressibility information is available [14].
2.3.3 Quantum Mechanical Description
In this section, the link of Cv to the microscopic properties will be derived. The system in question must, of necessity, be treated as a quantum-mechanical system. Every microscopic system is assumed to be able to take on only certain states as summarized in Fig. 2.32. The labels attached to these different states are 1, 2, 3, ...
and their potential energies are 1, 2, 3 ... , respectively. Any given energy may, however, refer to more than one state so that the number of states that correspond to the same energy 1 is designated g1 and is called the degeneracy of the energy level. Similarly, degeneracy g2 refers to 2, and g3 to 3. It is then assumed that many such microscopic systems make up the overall matter, the macroscopic system. At least initially, one can assume that all of the quantum-mechanical systems are equivalent. Furthermore, they should all be in thermal contact, but otherwise be independent. The number of microscopic systems that are occupying their energy level 1 is n1, the number in their energy level 2 is n2, the number in their level 3 is n3, ... .
The number of microscopic systems is, for simplicity, assumed to be the number of molecules, N. It is given by the sum over all ni, as shown in Eq. (1) of Fig. 2.32. The value of N is directly known from the macroscopic description of the material through the chemical composition, mass and Avogadro’s number. Another easily evaluated macroscopic quantity is the total energy U. It must be the sum of the energies of all the microscopic, quantum-mechanical systems, making the Eq. (2) obvious.
For complete evaluation of N and U, one, however, needs to know the distribution of the molecules over the different energy levels, something that is rarely available. To solve this problem, more assumptions must be made. The most important one is