Thermal Analysis of Polymeric Materials

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Fig. 2.80

2.4.1 Flux and Production

To describe the overall change of entropy in a system or subsystem with time, dS/dt, one makes a division into a term due to entropy flux, deS/dt, which gives the flow across the boundary, and to entropy production, diS/dt, which gives the internal entropy changes. The sketch in the lower part of Fig. 2.80 illustrates this statement. Equation (1) in Fig. 2.81 expresses the time dependence of the total rate of change of entropy of the system in terms of the flux and production. Calorimetry can give information on the heat-flow rate, dQ/dt, at constant temperature and pressure as it changes the entropy of a closed system described in Fig. 2.16. Since the temperature of the closed system is constant within the boundaries during dt, the heat flux can be considered reversible. The overall irreversibility caused by a higher temperature outside the boundaries, needed to bring about the heat flux, is pushed outside of the boundary and, thus, is of no importance to the description of the system.

For the open system of Fig. 2.16, one must express the entropy-flow rate as written in Eq. (2) of Fig. 2.81. Besides the entropy change due to the heat flux, there is an entropy change due to the flux of matter. The matter flux of component i is expressed by its flow-rate term, deni/dt. The matter flux can be measured, for example, by thermogravimetry as listed in Fig. 2.16. Each flux of matter across the boundary of the system, deni, needs to be multiplied with the partial molar change in the corresponding entropy, Si* [= ( S/ ni)T,p, see Fig. 2.19], to obtain the total contribution to deS/dt. The change in partial molar entropy must be known, or be measured independently by heat capacity measurements, for example, from zero kelvin to T, as outlined in Figs. 2.22 and 2.23.

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Fig. 2.81

More difficult is the evaluation of the rate of entropy production diS/dt. Entropy production cannot be measured directly, but one can make a number of general statements about it using the second law of thermodynamics treated in Figs. 2.17–2.20. Note that by separating the flux terms, one can treat the remaining production as being the changes in a hypothetical isolated system without flux. This hypothetical isolated system behaves like a small universe, meaning that the overall entropy statements of the second law apply: diS must increase for a spontaneous process as written in Eq. (3) of Fig. 2.81. A constant iS indicates equilibrium (diS= 0), and a negative diS has not been observed. In irreversible processes there is always an entropy production.

Other production terms are also restricted. The change diU, for example, must always be zero, since energy is conserved. All changes of U must come from flux across the boundary. Similarly, the change in mass, diM, must be zero. One cannot produce new mass. Equations (4) and (5) are expressing these conservation laws for energy and mass.

With this description of open and closed systems, melting and reorganization of lamellar crystals is treated next as an example of the irreversible thermodynamics of defect polymer crystals and their melting [24]. The added variable for the description of lamellae is the shape of the crystal. A more general treatment of internal variables is given in Sect. 2.4.4, and the detailed time-dependence of the processes is discussed in Sects. 2.4.5 and 2.4.6.

2.4.2 Melting of Lamellar Crystals

Most macromolecular, lamellar crystals are only about 10 nm thick (see Sect. 5.2), i.e., their surface area is about 200 m2 g 1. For any reasonable specific surface free energy, such a large surface will lead to a sizeable metastability, overshadowing most

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other defect contributions. The melting kinetics of such thin, lamellar crystals, is discussed further in Sect. 3.6, and a larger set of experimental data on melting of polymer crystals is given in Sect. 6.2. The sketch of Fig. 2.82 shows schematically a diagram for the flux of heat into a closed calorimeter as used for thermal analysis,

Fig. 2.82

and the production term caused by changing the crystal-to-melt ratio by flux across an open, internal boundary between crystal lamellae and melt. The lamellar crystal is the system with a total free enthalpy of Gc. As long as the lamellae in the sample are of the same thickness, have large lateral dimensions (so that the surfaces on the sides can be neglected), and are free of temperature and pressure gradients, all crystals can be described together as a single system or subsystem. If this were not so, one would have to sum over different subsystems, by either making measurements for every subsystem, a perhaps impossible task, or by deconvolution of the observed heat of fusion (see, Chap. 6, Fig. 6.3).

The amorphous part of the usually semicrystalline polymer and the melt produced on fusion are assumed to be coalesced to a single, amorphous system or subsystem with the free enthalpy Ga. If this were not so, additional amorphous subsystems would have to be introduced, and, if these subsystems are small, their surface effects would have to be considered via their area and the specific surface free energies. This leads, again, to experimental problems that are difficult to resolve.

The whole sample, consisting of crystals and melt, is enclosed in the sample holder of the calorimeter (see Sects. 4.2–4.4), making it the overall closed system, isolated from the surroundings. As melting progresses, mass is transferred in Fig. 2.82 from the left to the right. On crystallization, the mass transport goes in the opposite direction. The heat transport across the open boundary is minimal since it is assumed that the calorimeter remains at constant temperature.

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The heat exchange into the crystalline system from the outside is given by Qc, the heat exchange to the amorphous system by Qa. The lamellar crystals are described by the overall free enthalpy of Eq. (1) in Fig. 2.82, with mc representing the mass of the lamellar crystal; gc, the specific free enthalpy per gram of bulk crystal; , the lamellar thickness; , the density; and , the specific surface free energy. The side surface areas are, as suggested above, negligible. The amorphous system has a free enthalpy Ga, simply described by maga, as shown by Eq. (2). The free enthalpy of the amorphous subsystem is, thus, assumed to be described by an extrapolation of Ga from above the melt.

On melting, the change in free enthalpy of the bulk crystal is given in Fig. 2.82 by the boxed expression gf = ga gc, where gf represents the specific free enthalpy of fusion of a large crystal. One more simplifying fact arises from the schematic drawing. The change in mass of the amorphous system, dma, has to be equal to the negative of the change in mass of the crystalline system, dmc. There is conservation of mass within the closed system. The infinitesimal entropy flux deS during melting is given next in Fig. 2.83. It is derived directly from the heat flow, measured by thermal analysis.

Fig. 2.83

Making use of Eqs. (1) and (2) of Fig. 2.82, it is possible to derive the overall entropy production diS as shown by the boxed equation in the center of Fig. 2.83. It is caused by the flux between the two subsystems of Fig. 2.82. For the derivation one assumes that the entropy of melting, sf, does not change with temperature from its equilibrium value ( hf/Tmo) and writes that gf is simply hf T/Tmo as shown in Fig. 2.83 with T = Tmo T. The entropy production diS is, in contrast to deS, not directly measurable, but to keep the temperature constant, all enthalpy changes due to melting or crystallization were compensated by flux Qa and Qc.

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The first two terms of entropy production in Fig. 2.83 describe the melting (dmc < 0) or crystallization (dmc > 0) as a flux across the internal boundary. Termgfdmc/T is the bulk-entropy change. Since on melting the surface of the lamella decreases, one has to introduce the loss in surface area, expressed by the second term [ 2 /(T )]dmc. Overall, this term is positive on melting (dmc < 0) and negative on crystallization (dmc > 0).

The last term in the equation for diS of Fig. 2.83 is the entropy production term for the crystalline subsystem due to a change in thickness of the lamella by d . For the present discussion this is the only change assumed to be possible on annealing. This simplification is based on the experience that annealing to larger fold length is often the most important type of reorganization. If there are other changes, such as a decrease in concentration of nonequilibrium defects, one would simply expand the equation by the appropriate terms.

The three terms of diS must obey the restrictions of the second law of thermodynamics: The entropy production can be zero or positive, but never negative. Crystals with a large are treated first (list 1 in Fig. 2.83). In this case the last two terms of diS are negligible because they contain in the denominator [i.e., diS = ( gf/T)dmc]. This equation applies to perfect crystals with negligible surface areas. On melting and crystallization of such crystals under equilibrium conditions diS is zero since gf is zero at the equilibrium melting temperature (T = Tmo). Equilibrium melting and crystallization is, as expected, always a zero-entropy-production process. The total entropy of transition is provided by the measurable entropy flux (dQc + dQa)/Tmo (upper box of Fig. 2.83). The equilibrium melting can be visualized by the intersection of Gc and Ga in the free-enthalpy diagram of Fig. 2.84.

Whenever one crystallizes with supercooling, i.e., below Tmo, gf is positive sincehf and T are positive and dmc is positive, resulting in a positive diS. There is an entropy production, an indication that the process is permitted by the second law and can proceed irreversibly by a downward transition between corresponding free enthalpy curves shown in Figure 2.85. A quite similar heat flux dQc + dQa as in the equilibrium melting leads in this case to an entropy production because T< Tmo. Similarly, if one melts with superheating, i.e., melting occurs above Tmo, gf is negative, but dmc is also negative so that diS is positive again. Melting under conditions of superheating is thus also permitted by the second law. Melting with supercooling or crystallization with superheating are, in contrast, both forbidden because of the resulting negative diS, in good accord with experience.

For the description of thin lamellae, as listed in group 2 of Fig. 2.83, all three terms for the entropy production, diS, must be considered. First, one assumes that there is no overall crystallization or melting, i.e., dmc = 0. A process that leads to changes in crystal thickness, d , is reorganization or annealing. Only if d is positive, is diS positive and allows spontaneous reorganization with entropy production. The reorganization can also be recognized in a free enthalpy diagram as given in Fig. 2.86. Lamellar crystals can only increase in thickness or stay the same, a fact commonly born out by experience. A few observed cases of thinning of lamellae are caused by a tilting of the chains in the crystal with a decreasing specific surface free energy or a change in defect concentration, overcompensating the increase in surface area. The second case is described for polyethylene in Sect. 6.2.1 with Fig. 6.34.

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Fig. 2.84

Fig. 2.85

Crystallization and melting of thin lamellae without reorganization (d = 0) show important differences from equilibrium melting and crystallization. Crystallization

and melting is governed by the first two terms on the right-hand side of diS of Fig. 2.83 [( hf T/Tmo) {2 /(T )}]. The second term, {2 /(T )} is always

positive, i.e., crystallization with a positive dmc can only occur at a temperature

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equilibrium process. The metastable lamellae melt directly into a melt of equal metastability. The metastability of the melt is caused by being below the equilibrium melting temperature, at Tm of a thin crystalline lamella with a large surface free energy. Such melting looks superficially like equilibrium melting. The difference between the two zero-entropy-production cases is easily seen in the free enthalpy diagram of Fig. 2.4.88. Since heat capacities of crystals of different stability are usually similar (see Sect. 2.3), metastable free enthalpy curves are often parallel to equilibrium curves and can be superimposed by shifting from one Tm for zero- entropy-production melting to another.

Fig. 2.88

2.4.3 Experimental Data

Experimental use of the zero-entropy production melting temperatures allows a study of metastable crystals. Crystallization, in contrast, always requires a supercooling due to crystal and molecule nucleation , as is discussed in Sects. 3.5 and 3.6. It is a simple matter to derive the equation for the zero-entropy-production melting temperature from the equation for diS, derived in the previous section (Fig. 2.83) and shown once more in Fig. 2.89. The measurable lowering in melting temperature from equilibrium, T, is given by the easily derived Gibbs-Thomson equation which contains the surface free energy and the lamellar thickness . If one of the two latter quantities is known, the other can be calculated.

Thermal analysis data on lamellar crystals of polyethylene over a wide range of thicknesses are plotted in Fig. 2.90. The Gibbs-Thomson equation is a good mathematical description of the observed straight line and can be used to calculate the equilibrium melting temperature by setting = (Tmo = 414.2 K). Also, the ratio of the surface free energy to the heat of fusion can be obtained from the equation.

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Fig. 2.89

Fig. 2.90

2.4.4 Internal Variables

The following treatment applies to homogeneous systems with constant mass. The dissipative, time-dependent effects are caused in such homogeneous systems by disturbance of the equilibrium of the internal, molecular degrees of freedom. The

the system because of existing relations to the fixed internal equilibrium, to be shown as Eq. (15), below. To better understand the processes, however, it is useful to first consider the variables i explicitly. In the case of a glass transition, a single can be linked to the concentration of the microscopic vacancies in a liquid (number of holes, Nh, see Sects. 5.6 and 6.3). Similarly, the surface area of metastable crystals, as described in the previous two sections, could be expressed by a single . The actual time-dependence of the change of will be discussed in Sect. 2.4.6, the present discussion will deal with the equilibrium of a single internal variable .

The starting point for the description of the system considered is the Gibbs fundamental equation, Eq. (1), shown in Fig. 2.91 in terms of the specific quantities:

Fig. 2.91

free enthalpy, g, enthalpy, h, and entropy, s, per unit mass, written in lower-case letters, compare to Fig. 2.19 and also [19]. Temperature, T, pressure, p, and the internal variable are the mutually independent variables. Also listed in the figure are the expressions for s, Eq. (2), and h, Eq. (3), and the total differentials with respect to the independent variables, Eqs. (4) (6). The partial molar quantities of g, s, and h with respect to at constant T and p are shown as Eqs. (7) (9) and named: a, , and , respectively. The partial molar free enthalpy “a” is commonly called the activity and has the significance of a driving force (see Sect. 2.4.6).

Equation (10) of Fig. 2.92 is the Gibbs equation for the corresponding partial quantities at constant T and p. At internal equilibrium (designated by the superscript “e”) ae = 0, which leads to Eq. (11) and fixes one of the independent variables. At equilibrium, one can write with help of Eq. (10) and Eq. (12) the Eqs. (13) and (14) since da = 0. Finally, one can derive the change of the internal variable with temperature at equilibrium and constant pressure, as shown in Eq. (15).