Non-linear diffusion controlled particle growth problems, in which the diffusion coefficient in the matrix phase varies with composition, are examined. The growth of an expanding ellipsoid, initially of zero size is considered, limiting cases of which include planar, cylindrical and spherical growth. The theory for dendritic growth is also given. Variational principles are applied to generate numerical solutions to the full non-linear problems. Explicit formulae, relating to fast and slow 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, for situations in which a constant diffusion analysis is used as a substitute to the non-linear problem (by assuming the diffusion coefficient to be some weighted average of the variable one), the asymptotic expressions indicate which is the best average to adopt. It is found that for fast ellipsoidal growth (including planar, cylindrical and spherical growth) and for dendritic growth, to a first approximation, the most appropriate average to use for the constant diffusion analysis is when 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 and at infinity respectively. A similar conclusion obtains for slow spherical growth, although no such simple formulae exist for planar and cylindrical growth in this limit. The theory is also applied to a planar growth problem in which the diffusion profile is discontinuous, and the exact solution is used to judge the accuracy associated with the variational theory.
