Ternary and quaternary solutions

Similar models can be developed for more complex systems such as ternary and quaternary alloys. As an example, we will consider a modified regular solution model for quaternary alloys proposed by K. Onabe [8]. This model does not include some of the detailed treatment in the Ichimura calculation [6] described in Section 6.2.1. Specifically, it does not include strain and assumes a random alloy entropy. However, the calculation is simple, illustrative of the behavior of multinary alloys, and can easily be extended to include the strain and entropy terms. 

For a quaternary Ai xBxCyDi y alloy, the stability criterion is determined by the second partial derivative of the free energy change, AG, for mixing  

The regular solution theory energy in Equation 6.2 can be generalized to multinary solutions. Suppose that Wab is the bond energy of an AB compound, analogous to Eaa and Ebb in Equation 6.2. Further, suppose that aAc-Bc = Wac-Wbc being the 

The generalized regular solution entropy for a quaternary alloy is  

As with any spinodal decomposition, the curves shown are for an equilibrium condition. In a real crystal one almost never observes decomposition at the spinodal boundary, as the driving force for decomposition there is zero. Furthermore, there may be a substantial nucleation barrier preventing decomposition into small domains of different composition. (This reflects the contribution due to strain to some extent, which is ignored in the above treatment.) 

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In Sections 3.3.7 and 3.3.8 with the cases of enantiomers and the reciprocal conversion, examples of ternary and quaternary solution equilibria of industrial importance will be discussed. 

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