Institute of Metals Division - Approximate Method for Calculations Using concentration-Dependent Diffusion Coefficients

IN the course of a research on steady-state diffusion it became necessary to make diffusion calculations for a finite solid. This problem was found to be sufficiently different from the corresponding problem for the semi-infinite solid for an entirely new approach to be necessary in obtaining a useful solution. The assumption of an exponential variation of the diffusion coefficient with concentration' was no longer a practical necessity, so that the new solution offered the possibility of handling diffusion in semi-infinite solids for arbitrary variations of the diffusion coefficient with concentration. In discussing the principle on which this method of calculation is based, it is convenient to consider the relatively simple problem of one dimensional diffusion into a matrix that initially contains none of the diffusing substance. The solution of the set- ond Fick Law,?C/?t=D0?2C/?x2, for diffusion into a semi-infinite solid is3 C/C=1-F(x/2vD0t [1] where C is the concentration of the diffusing substance at a depth x cm below the surface after dif- A. G. GUY, Member AIME, and M. GOLOMB are Professor of Metollurgicol Engineering and Professor of Mathematics, respectively, Purdue University, Lafayette, Ind., A. S. YUE, formerly with Purdue University, is now with the Research Laboratory, Dow Chemical Co., Midland, Mich. TP 4505E. Manuscript, July 18,1956. fusion has occurred for t seconds; C, is the concentration of the diffusing substance at the surface of the matrix; Do is the diffusion constant, expressed in sq cm per sec; and + represents the Gauss error function. The corresponding solution for diffusion into a finite solid of thickness, 1, is conveniently written for the change in concentration, C+, at a point midway through the matrix and for times, t, great enough for the following equation to be used4 c1/2/C1=(1/2-2/?e-?2D0t/l2) [2] In Eqs. 1 and 2 the diffusion coefficient is represented by the symbol Do (rather than by D) to emphasize that these equations are valid only for constant values of D. To understand how these equations can be modified to take account of actual variations of D, such as that shown in Fig. 1, first consider the fact that for a given time of diffusion theconcentration at a given point is determined, not by the D value alone, but by the ratio x/vD0 in Eq. 1 and by a similar ratio, l2/D0 in Eq. 2. Even though in fact D varies, it is possible to keep this ratio constant by using a special coordinate system, x,, that varies with D in the appropriate manner. The necessary character of this special coordinate system can be easily visualized by imagining that