NONLINEAR DIFFUSION-CONTROLLED PARTICLE GROWTH - A VARIATIONAL AND ASYMPTOTIC APPROACH

Non-linear diffusion controlled particle growth problems, in which the diffusion coefficient in the matrix phase varies with composition, ar e examined. The growth of an expanding ellipsoid, initially of zero si ze is considered, limiting cases of which include planar, cylindrical and spherical growth. The theory for dendritic growth is also given. V ariational principles are applied to generate numerical solutions to t he full non-linear problems. Explicit formulae, relating to fast and s low growth are derived using perturbation techniques, and obtained for a general class of diffusion coefficient, dependent upon composition. The latter serve as a check on the numerical results. In addition, fo r situations in which a constant diffusion analysis is used as a subst itute to the non-linear problem (by assuming the diffusion coefficient to be some weighted average of the variable one), the asymptotic expr essions indicate which is the best average to adopt. It is found that for fast ellipsoidal growth (including planar, cylindrical and spheric al growth) and for dendritic growth, to a first approximation, the mos t appropriate average to use for the constant diffusion analysis is wh en D(C) is replaced by D-AV = integral (CM)(Cx) D(C)dC/(C-M-C-x) where C-M, C-x denote the concentrations at the particle-matrix interface a nd at infinity respectively. A similar conclusion obtains for slow sph erical growth, although no such simple formulae exist for planar and c ylindrical growth in this limit. The theory is also applied to a plana r growth problem in which the diffusion profile is discontinuous, and the exact solution is used to judge the accuracy associated with the v ariational theory.