Multicomponent Solids in Chemical Potential Gradients

We begin with the simplest case. A vacancy flux j° (driven, for example, by inhomogeneous particle radiation) flows across a multicomponent crystal (k = 1,2. n) and the component fluxes are restricted to one sublattice. We assume no other coupling between the fluxes except the lattice site conservation, which means that we neglect cross terms in the formulation of SE fluxes. (An example of coupling by cross terms is analyzed in Section 8.4.) The steady state condition requires then that the velocities of all the components are the same, independent of which frame of reference has been chosen, that is, 

There are ( -l) equations of type (8.3). Along with the Gibbs-Duhem equation, they can be solved for the unknown chemical potential gradients V. In combination with the (n-1) mass conservation equations 

A system which can easily be treated in this way is a single phase binary alloy. For preparation, however, let us consider an A crystal with a vacancy flux driven across it. In view of the fact that jA +y v = 0 in the steady state lattice system, the vacancy flux induces a counterflux of A, which shifts the whole crystal in the direction of the surface where the vacancy source is located. The shift velocity vb is yV 

For the slab of a binary alloy (A, B) across which the vacancy flux jw = j° flows, we derive from Eqn. (8.3) 

6) describes the steady state concentration profile of an (A, B) alloy which has been exposed to the stationary vacancy flux j°. The result is particularly simple if the mobilities, b are independent of composition, that is, if P = constant. From Eqn. (8.6), we infer that, depending on the ratio of the mobilities P, demixing can occur in two directions (either A or B can concentrate at the surface acting as the vacancy source). The demixing strength is proportional toy°-(l-p)/RT, and thus directly proportional to the vacancy flux density j°, and to the reciprocal of the absolute temperature, 1/71 For p = 1, there is no demixing. 

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