Diffusion limited lengthening of plates/needles

Consider, for example, the growth of a platelike precipitate with coherent facetted faces and incoherent edges. In such a case, it is possible that the thickness of the plate would remain constant while the cylindrically curved edge would grow. In Fig. 26 we show the growth of such a precipitate along with the composition profile. One important change in the composition profile as compared to the one in Fig. 22 is that the composition on the matrix side is not the equilibrium composition given by the phase diagram but is elevated due to Gibbs-Thomson effect.

Figure 26: Diffusion limited lengthening of a plate of constant thickness; note the Gibbs-Thomson effect on composition at the curved interface on the $\alpha$ side.

[scale=0.4]Figures/PlatePptGro.pdf

In this case also, an equation similar to Eq. 51 holds (with the composition on the $\alpha$ side of the interface corrected for Gibbs-Thomson):

$$v = \frac{dx}{dt} = \frac{D}{c_e^{\beta} - c_r} \frac{dc}{dx}$$ (10)

As earlier, assuming equal molar volumes for both the phases, it can be shown that

$$\frac{dc}{dx} \approx \frac{\Delta X_0}{kr} \left(1 - \frac{r^{\star}}{r} \right)$$ (11)

where, $k$ is a cosntant of order unity and $r^{\star}$ is the radius of critical nucleus, that is, the radius for which $\Delta X$ is reduced to zero. Thus, we get, for the plate/needle lengthening velocity as

$$v = \frac{D}{X_e^{\beta} - X_r} \frac{\Delta X_0}{kr} \left(1 - \frac{r^{\star}}{r} \right)$$ (12)

An important consequence of such an expression as above is that the diffusion limited plate lengthening rate, for a given thickness of the plate, is a constant; that is, $x \propto t$; in other words, the rate of growth is linear.