Diffusion-controlled kink motion
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I.
the growth of kinks by volume diffusion is discussed, and singular valid for supersaturations much less than one, are used to derive coupled the motion of trains of (well-spaced) kinks. Numerical results are preof two- and three-kink trains.
INTRODUCTION
IN a
previous paper, m we have applied the method of Atkinson and Wilmott [2] to the transient motion of trains of two and three steps. Where possible, comparisons were made with the computer modeling (by numerical solution of the diffusion equation) of the growth kinetics of ledged interphase boundaries [3'41 and with the steadystate analyses of Atkinson. t5'61 Recently, the techniques of high-resolution transmission electron microscopy have been used by Howe et al. [7'sl and others to suggest that both the lengthening and thickening of Y' plates in an A1-4.2 at. pct Ag alloy occur by a terrace-ledge-kink mechanism where the limiting step in the growth process appears to be the substitutional diffusion of Ag atoms across kinks in the Shockley partial dislocation ledges which terminate in the Ag-rich, A planes. Thus, a picture like Figure 1 is suggested. Motivated by these experimental results, we consider here the diffusioncontrolled motion of a kink such as shown schematically in Figure 1. For the problem of volume diffusion-controlled kink motion, we apply asymptotic methods such as used by Atkinson and Wilmott.12] We therefore consider the limit of supersaturations tending to zero so that the methods of matched asymptotic expansion may be used. In the neighborhood of the kink, we use a somewhat simplified boundary condition to determine a solution; the method will work for a more exact representation, though at the possible expense of some computation. We begin with the diffusion equation in stationary rectangular coordinates (x, y, z), which are referred to a fixed origin in the solid, as
Of --
at
[02C =
D
L ox2
02C +
- -
oy 2
02C] +
072 J
[1.1]
where C, D, and t are the solute concentration in the matrix, the interdiffusion coefficient in the matrix, and the reaction time, respectively. Under the assumption that volume diffusion of solute
in the matrix to the ledge riser controls the growth rate, the flux equation at each riser can be written as [1.21
(Cp - CDVN = + D riser N
where Cp and Cm are the solute concentrations in the growing precipitate and in the matrix in contact with the riser, respectively, and VN is the velocity of the N-th step or kink. Further, it is assumed that the transfer of atoms to the precipitate is permitted only at the riser so there is no flux normal to the terraces; i.e., OC
ony=hN
- - = 0
and between risers
[1.3]
Oy
Corresponding conditions hold between kinks. Here, hN is the height above y = 0 of the N-th step (Figure 1) so that (hN-1 -- hN)/h = eN = 0(1), where h is a typical step or kink height with which we shall scale lengths. Nondimensionalizing the above by writing, we obtain r = (c -
c~)/(c
~ -
c~)
x.= hX = ~2Dt/h2
VN ~
OliN Oz
hVN
-- - -
Df~
where C ~ and C~ are the solute concentrat
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