Two Species/Nonideal Solution Model for Amorphous/Amorphous Phase Transitions
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The model condidered here, in its simplest form, has been termed variously the "two species model" [9], "two state model" [10,11], "two level model" [12,13], "mixture model" [14] and "bond-lattice model" [15,16]. One assumes in a one component condensed phase (crystal or liquid) the existence of two "species", A and B, which can be interconverted via a physicochemical process or "chemical reaction" of the form: A
-B
(1)
The equilibrium concentrations of species A and B are dependent on temperature T and pressure P, so that the "chemical reaction" contributes to the T and P dependence of the thermodynamic properties. A simple crystal state example of such a process is vacancy formation in a metal. Of importance for the present discussion is the fact that processes of this sort are manifested in all liquids and are commonly referred to as "structural relaxation", whose kinetically induced disequilibration at low temperatures is responsible for the glass transition [13,15-18]. In the simplest versions of this model the two species, A and B, are assumed to form an ideal solution. This assumption leads to a continuous, monotonic variation in the concentrations of the two species as a function of T and P. If, however, the two species are 411 Mat. Res. Soc. Symp. Proc. Vol. 455 01997 Materials Research Society
assumed to form a nonideal solution, then, under certain circumstances, the concentrations of A and B may exhibit a discontinuous change as T or P are varied, i.e., a first order phase transition. This was first explicated in detail by Aptekar and Ponyatovskii [12], who used this treatment to model an unusual fcc-*fcc phase transition in crystalline Ce metal, where the two species were taken to be Ce+4 and Ce+ 3 ions. This approach, which we will *henceforth term the "two species/nonideal solution model", was also developed by Rapoport [9] to account for maxima in the melting point vs. P curves of metals. Although Rapoport mentioned that the model could lead to "separation (of a melt) into two phases", he did not explore this aspect in detail, and one cannot discern from his papers whether he thought formation of the second liquid phase was a true first order phase transition or was similar to liquid-liquid immiscibility in a two component melt. Thermodynamics of the Two Species/Nonideal Solution Model At a given T and P the Gibbs free energy per mole of species for a one component liquid or crystal consisting of interconvertible species A and B is given by G = (1- X)GA+
XGB
(2)
where X is the mole fraction of species B, and GA and GB are the molar free energies or chemical potentials of A and B. If we assume that A and B form a regular solution with interaction energy parameter W, then GA and GB can be expressed as [19]: GA = Go + VX(P-PO) + RTln(1 - X) + WX 2
(3a)
X) 2
(3b)
GB
= Go + V;(P-PO) + RTlnX + W(1-
where R is the ideal gas constant, Go and Go are respectively the standard molar free energies of A and B at temperature T (pure A and B at standard pressure P0 = 1 atm), and similarly for the sta
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