Evolution of compositional variations

The study of stability of a spatially uniform composition in a solid solution under periodic small-amplitude fluctuations pursued in Section 9.6.2 was based on equilibrium considerations. A composition was considered to be stable or unstable, depending on whether the system free energy tended to increase or decrease as a result of small fluctuations. The approach leads to no conclusion on the rate of increase in amplitude of an unstable fluctuation or on the rate of decay of a stable fluctuation. The question of time evolution of spatial fluctuations in composition is pursued here, following the ideas introduced by Hillert (1961) and Cahn (1961). 

The local chemical potential field corresponding to a spatial distribution of component B in an substitutional solid solution was 

This is a nonlinear partial differential equation for as a function of time and of position throughout the region R. Analytical solutions of the equation are not available, in general. 

For the particular situation in which represents only a small departure from the spatially uniform composition, the equation (9.106) can be linearized and solutions are readily obtained. In this situation, mR( ) mR( ) = m, Kch(C) ch( ) = ch approximated 

Furthermore, if it is assumed that the composition varies periodically about the mean value with spatial period A as in (9.93), and that the corresponding mean normal stress is given by (9.94), the partial differential equation is reduced to the ordinary differential equation 

---

As was noted in the study of evolution of compositional variations in Section 9.6.4, the local rate of mass accumulation due to a mass flux vector is the negative of the flux divergence. The local addition of material is represented in continuum modeling as a stress-free volumetric strain. At fixed overall strain, the rate of material addition induces a proportional rate of increase in mean normal compressive stress. In the present onedimensional configuration, this connection is expressed as... 

---