Thermodynamics of Crystals

Since the state of a crystal in equilibrium is uniquely defined, the kind and number of its SE s are fully determined. It is therefore the aim of crystal thermodynamics, and particularly of point defect thermodynamics, to calculate the kind and number of all SE s as a function of the chosen independent thermodynamic variables. Several questions arise. Since SE s are not equivalent to the chemical components of a crystalline system, is it expedient to introduce virtual chemical potentials, and how are they related to the component potentials If immobile SE s exist (e.g., the oxygen ions in dense packed oxides), can their virtual chemical potentials be defined only on the basis of local equilibration of the other mobile SE s Since mobile SE s can move in a crystal, what are the internal forces that act upon them to make them drift if thermodynamic potential differences are applied externally Can one use the gradients of the virtual chemical potentials of the SE s for this purpose  

It has long been known that defect thermodynamics provides correct answers if the (local) equilibrium conditions between SE and chemical components of the crystal are correctly formulated, that is, if in addition to the conservation of chemical species the balances of sites and charges are properly taken into account. The correct use of these balances, however, is equivalent to the introduction of so-called building elements ( Bauelemente ) [W. Schottky (1958)]. These are properly defined in the next section and are the main content of it. It will be shown that these building units possess real thermodynamic potentials since they can be added to or removed from the crystal without violating structural and electroneutrality constraints, that is, without violating the site or charge balance of the crystal [see, for example, M. Martin et al. (1988)]. 

In a book on kinetics, the purpose of understanding the thermodynamics of point defects (= irregular SE s) is the elucidation of their role as carriers in the elementary steps of mass transport. For any given values of P, T, and component chemical potentials, their equilibrium concentrations can be calculated if the magnitudes of 

In this section, we will outline point defect thermodynamics and quantify the considerations of the introduction. 

Consider a crystal which is in equilibrium having n chemical components (k = 1,2. ). We can define (at any given P and T) a Gibbs function, G, as a homogeneous function that is first order in the amount of components 

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Thermodynamics of Crystallization of Flexible-Chain Polymers Under Conditions of Molecular Orientation. 217... 

Flory, P. J. (1949). Thermodynamics of crystallization in high polymers. IV. A theory of crystalline states and fusion in polymers, copolymers, and their mixtures with diluents. 

In the planetary geochemical literature, there are also extensive discussions about the thermodynamics of crystallizing melts. Sophisticated programs have been developed to model the compositions of the minerals that crystallize from a cooling liquid as well as changes in the residual melt. 

The results of the discussion on the phenomenological thermodynamics of crystals can be summarized as follows. One can define chemical potentials, /jk, for components k (Eqn. (2.4)), for building units (Eqn. (2.11)), and for structure elements (Eqn, (2.31)). The lattice construction requires the introduction of structural units , which are the vacancies V,. Electroneutrality in a crystal composed of charged SE s requires the introduction of the electrical unit, e. The composition of an n component crystal is fixed by n- 1) independent mole fractions, Nk, of chemical components. (n-1) is also the number of conditions for the definition of the component potentials juk, as seen from Eqn. (2.4). For building units, we have (n — 1) independent composition variables and n-(K- 1) equilibria between sublattices x, so that the number of conditions is n-K-1, as required by the definition of the building element potential uk(Xy For structure elements, the actual number of constraints is larger than the number of constraints required by Eqn. (2.18), which defines nk(x.y This circumstance is responsible for the introduction of the concept of virtual chemical potentials of SE s. 

Flory, P.J. (1947) Thermodynamics of crystallization in high polymers. I. Crystallization induced by stretching, J. Chem. Phys. 15(6), 397-408... 

Thermodynamics of Crystals by D. C. Wallace, Dover Publications, Inc., Mineola New York, 1998. Wallace s chaps. 2-A give a thorough discussion of the material that has been sketched in the present chapter. 

Wallace DC (1972) Thermodynamics of Crystals. John Wiley and Sons, New York. 

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