Free energy of a homogeneous solution

Consider a statistically homogeneous solid solution, with homogeneous and chemically distinct materials A and B as its endpoint compositions, at some fixed temperature T. Any intermediate composition is characterized by the parameter which is the fraction of lattice site atoms distinguishing 

A from B occupied by the atoms of material B as such, this parameter is in the range 0 1. In the SiGe system, the nnmber of lattice sites involved in this comparison is eight per unit cell, which represents all possible sites. On the other hand, in the InGaAs system, only fonr atom sites per unit cell are occupied by either In or Ga atoms. The free energy per nnit volume of the homogeneous stress-free solution at temperature T is commonly expressed in terms of the free energy densities of the pure substances under the same conditions of stress and temperature, say and ifB, and the volume fraction as 

The rate of change of free energy of a homogeneous unit volume of material, which is understood to include a fixed number of lattice sites, is 

The general issue of stability of composition of a solid solution is pursued further in the next subsection. Two potentially important physical effects are not taken into account in the discussion of energy variations with composition above. One of these effects arises from the possibility of atomic misfit of one species in the solution with respect to the other. The average unit cell dimension of a solid solution may depend on the composition, so that there is a stress-free volume change (or a more complex stress-free strain, perhaps) with change in concentration. For a spatially nonuniform composition, the associated stress-free strain field will be incompatible, in general, giving rise to a residual stress distribution. 

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