Stability and the Reaction Path

In Section 4.3.3, it was explained how to construct the reaction (diffusion) path for ternary and higher solid solution systems. In practice, one plots, for example, in a ternary system, the composition variables (measured along the pertinent space coordinate of the reacting solid) into a Gibbs phase triangle, noting that the spatial information is thereby lost. For certain boundary conditions, such a reaction path is independent of reaction time and therefore characterizes the diffusion process. For a one dimensional ternary system with stable interfaces, these boundary conditions are c,-( = oo,f) = c°( oo) q( 0,0) = c (-oo) c,(f 0,0) = c (+oo). 

In the context of the morphological evolution of non-equilibrium systems, let us then ask whether the reaction path, when constructed for a system with stable interfaces, can tell us something about the instability of moving boundaries. For this we 

Normally, it is not possible to obtain analytical solutions for this transport problem and so we cannot a priori calculate the reaction path. Kirkaldy [J. S. Kirkaldy, D. J. Young (1985)] did pioneering work on metal systems, based on investigations by C. Wagner and the later work of Mullins and Sekerka. They used the diffusion path concept to formulate a number of stability rules. These rules can explain the facts and are predictive within certain limits if applied properly. One of Kirkaldy s results is this. The moving interface in a ternary system is morphologically stable if 

Although this equation is reminiscent of the rules given earlier in this chapter, there are differences. In Eqn. (11.21), the two independent concentration gradients of the ternary system are introduced instead of real driving forces, which are the chemical potential gradients. Also, other simplifying assumptions have been made in order to arrive at Eqn. (11.21), assumptions which hardly pertain to real systems. 

To conclude, we present an application of the reaction path concept and investigate the evolution of the phase boundaries in the ternary oxide system Fe-Mn-0 [Y. Ueshima, et al. (1989)]. Let us start with the a-crystal (Fe, Mn)0 which then is 

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