Consequences of Eq. 56 and 57

Let us consider Eq. 56:

$$x = \frac{\Delta X_0}{( X_e^{\beta}-X_e^{\alpha})} (\sqrt{Dt}).$$ (8)

This equation shows that the distance to which the interface moves is proportional to $\sqrt{Dt}$. In other words, the thickeninng of the precipitate is parabolic. As we have seen earlier such a parabolic growth is a common feature of all diffusion limited growth. Although this conclusion is derived under simplifying assumptions, more thorough treatments and allowing for curved boundaries do not change this conclusion: the linear dimensions of a spheroidal precipitate increases as $\sqrt{Dt}$ as long as the interface migration is diffusion controlled.

Let us consider Eq. 57:

$$v = \frac{\Delta X_0}{2 ( X_e^{\beta}-X_e^{\alpha})} \left( \sqrt{\frac{D}{t}} \right).$$ (9)

This equation shows that the velocity of the interface is proportional to $\sqrt{\frac{D}{t}}$ as well as to $\Delta X_0$ (the supersaturation at the beginning of the precipitation).

From the phase diagram, it is clear that $\Delta X_0$ increases with increasing superaturation. Thus, as undercooling increases, the velocity of the precipitate-matrix interface should increase. However, the diffusivity is high at high temperatures and low at lower temperatures. In other words, the diffusivity gives lower velocities at higher undercoolings. Hence, the velocity of the precipitate-matrix interface is high at intermediate undercooling as shown in Fig. 24.

Figure 24: The velocity change with undercooling; the velocity is maximum at intermediate undercooling because at high undercoolings diffusivity is high but supersaturation is low, whereas at low undercooling diffusivity is low but supersaturation is high.

[scale=0.4]Figures/VelocityTemperature.pdf