Mathematical modeling of diffusion during multiphase layer growth
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THE formation and growth of intermediate phase layers by solid-state interdiffusion during dissimilar metal contact is a commonly occurring phenomenon. Such behavior is often observed in fiber-matrix reactions in composites, solid-sate joining of dissimilar metals by diffusion and formation a n d / o r degradation of diffusional coatings. In many instances, the formation of brittle intermetallic phases via diffusion processes between dissimilar metals can lead to a reduction in desirable mechanical and physical properties. Often, the kinetics of phase-layer growth at a given temperature is also of importance in defining the processing conditions necessary for optimum performance or in predicting the duration of successful usage. For many situations, phase-layer growth kinetics can be predicted analytically using Fick's second law and invoking conditions of local equilibrium at each interphase boundary. L2Closed-form solutions to such problems are well known for planar geometries in systems of infinite or semi-infinite extent. 3-8 More complex situations which involve nonplanar geometries of finite extent typically are treated by numerical methods and computer techniques. 9-H Mathematical analyses formulated by either of these methods are useful in predicting the time dependence of concentration-distance profiles in the diffusion zone and thicknesses of the corresponding phase layers. To understand and model the phase-layer growth kinetics, it is necessary to know the equilibrium composition ranges and lattice interdiffusion coefficients in each of the phase layers. In many systems where layer-growth kinetics is important, the phase solubility or diffusivity data may be unknown or unavailable with sufficient accuracy to permit mathematical modeling. Typically, intermediate phases possessing narrow composition ranges are often the most troublesome due to the experimental difficulties associated with solubility and diffusivity measurements. Sometimes wide variability in literature data, the necessity of extrapolation over large temperature ranges or contributions to trans-
port by mechanisms other than lattice diffusion can lead to significant errors in mathematical modeling. For many of these troublesome situations, a limited set of experimental phase-layer growth measurements may be obtained with relatively little difficulty especially for a phase with an incomplete set of parameters. In principle, the layer growth measurements can be used as input to the mathematical model (along with the known solubilities and diffusivities) in lieu of the missing parameters, providing a basis for back calculating the unknown solubilities a n d / o r diffusivities. The calculated and known parameters provide a self-consistent set of data which form the basis for calculating the growth kinetics of phase layers for geometrical, initial, and boundary conditions other than those characteristic of the experimental situation used to obtain the layergrowth input data.
WILLIAM C. JOHNSON is PostdoctoralResearch Associate, National Burea
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