Equilibrium thermodynamics of the perfect solid

The purpose of the chapter dealing with equilibrium thermodynamics of the perfect solid (Chapter 4) is to elaborate, on the one hand, simple expressions for the thermodynamic functions of the chemical ground state and, on the other hand, to make the reader familiar with questions of internal and external equilibria, not least with the intention to provide the equipment to deal with the thermodynamics of defect formation. (The major portion of the free enthalpy at absolute zero consists of bonding energy, while the temperature dependence is largely determined by the vibration properties.)... 

According to Fig. 1.2 we decompose thermodynamic functions into contributions that arise from (chemically) perfect solids and contributions that are brought in by defects. At this point we are now interested in the equilibrium thermodynamics of the (chemically) perfect state. Our aim is to sketch the fi ee enthalpy of the perfect solid with the aid of the previous chapters on chemical bonding and phonons, as well as to consider relevant aspects of the thermodynamic formalism and its apphcation to solids, in particular in view of interactions with the chemical environment. 

Systems involving an interface are often metastable, that is, essentially in equilibrium in some aspects although in principle evolving slowly to a final state of global equilibrium. The solid-vapor interface is a good example of this. We can have adsorption equilibrium and calculate various thermodynamic quantities for the adsorption process yet the particles of a solid are unstable toward a drift to the final equilibrium condition of a single, perfect crystal. Much of Chapters IX and XVII are thus thermodynamic in content. 

The shock-compression pulse carries a solid into a state of homogeneous, isotropic compression whose properties can be described in terms of perfect-crystal lattices in thermodynamic equilibrium. Influences of anisotropic stress on solid materials behaviors can be treated as a perturbation to the isotropic equilibrium state. ... 

Diffusion in general, not only in the case of thin films, is a thermodynamically irreversible self-driven process. It is best defined in simple terms, such as the tendency of two gases to mix when separated by a porous partition. It drives toward an equilibrium maximum-entropy state of a system. It does so by eliminating concentration gradients of, for example, impurity atoms or vacancies in a solid or between physically connected thin films. In the case of two gases separated by a porous partition, it leads eventually to perfect mixing of the two. 

The interesting point is that thermodynamically we do not expect a crystalline solid to be perfect, contrary, perhaps to our commonsense expectation of symmetry and order At any particular temperature there will be an equilibrium population of defects in the crystal. 

Wc have so far studied only perfect gases and have not taken up imperfect gases, liquids, and solids. Before we treat them, it is really necessary to understand what happens when two or more phases are in equilibrium with each other, and the familiar phenomena of melting, boiling, and the critical point and the continuity of the liquid and gaseous states. We shall now proceed to find the thermodynamic condition for the coexistence of two phases and shall apply it to a general discussion of the forms of the various thermodynamic functions for matter in all three states. 

In 1902, T. W. Richards found experimentally that the free-energy increment of a reaction approached the enthalpy change asymptotically as the temperature was decreased. From a study of Richards data, Nernst suggested that at absolute zero the entropy increment of reversible reactions among perfect crystalline solids is zero. This heat theorem was restated by Planck in 1912 in the form The entropy of all perfect crystalline solids is zero at absolute zero.f This postulate is the third law of thermodynamics. A perfect crystal is one in true thermodynamic equilibrium. Apparent deviations from the third law are attributed to the fact that measurements have been made on nonequilibrium systems. 

Some transitions that are only known for macromolecules, however, will not be mentioned at all since they are covered elsewhere in this Encyclopedia (see, eg. Gel Point). Also we shall not be concerned here with the transformations from the molten state to the solid state of polymeric materials, since this is the subject of separate treatments (see Crystallization Kinetics Glass Transition Viscoelasticity). Unlike other materials, polymers in the solid state rarely reach full thermal equilibrium. Of course, all amorphous materials can be considered as frozen fluids (see Glass Transition) Rather perfect crystals exist for metals, oxides, semiconductors etc, whereas polymers typically are semicrystalline, where amorphous regions alternate with crystalline lamellae, and the detailed structure and properties are history-dependent (see Semicrystalline Polymers). Such out-of-equilibrium aspects are out of the scope of the present article, which rather emphasizes general facts of the statistical thermodynamics (qv) of phase transitions and their applications to polymers in fluid phases. 

Liquid/solid equilibria also offer access to thermodynamic information. In this case, it is the differential interaction parameter of the polymer that is obtained according to (38) from the known molar mass of the polymer, its melting temperature in the pure state, and the corresponding heat of melting plus the polymer concentration in the solution that is in equilibrium with the pure polymer crystals. Because of the well-known problems in obtaining perfect crystals in the case of macromolecules, special care must be taken with the evaluation of such data. 

Experience indicates that the Third Law of Thermodynamics not only predicts that So — 0, but produces a potential to drive a substance to zero entropy at 0 Kelvin. Cooling a gas causes it to successively become more ordered. Phase changes to liquid and solid increase the order. Cooling through equilibrium solid phase transitions invariably results in evolution of heat and a decrease in entropy. A number of solids are disordered at higher temperatures, but the disorder decreases with cooling until perfect order is obtained. Exceptions are... 

---