Return radius and volume of recrystallized material in Ostwald ripening

Within the framework of the Lifshitz-Slyozov-Wagner theory of Ostwald ripening, the amount of volume of the second (solid) phase in a liquid solution that is newly formed by recrystallization is investigated. It is shown that in the late stage, the portion of the newly generated volume formed within an interval from time $t_0$ to $t$ is a certain function of $t/t_0$ and an explicit expression of this volume is given. To achieve this, we introduce the notion of the return radius $r(t,t_0)$, which is the unique radius of a particle at time $t_0$ such that this particle has—after growing and shrinking—the same radius at time $t$. We derive a formula for the return radius, which later on is used to obtain the newly formed volume. Moreover, formulas for the growth rate of the return radius and the recrystallized material at time $t_0$ are derived.

Figure 1

Examples of the time evolution of the radius $R\left(t\right)$ of a spherical particle in the LSW theory. For example, the solid blue line indicates that $R\left(2t_0\right)=R\left(t_0\right)$, i.e., the return time is $t=2t_0$ and the return radius for $t=2t_0$ is given by $R\left(t_0\right)\approx 8.42$.

Figure 2

(a) The function $\alpha \left(z\right)=\ln z+\tau \left(z\right)$ in (15) has a unique maximum at $z=1$. Choosing $z_0>1$, there is a unique $z:=\rho \left(z_0\right)<1$ with $\alpha \left(z\right)=\alpha \left(z_0\right)$. (b) The function $\rho$ has been evaluated numerically using bisection.

Figure 3

Specific volume $\phi (t,t_0)$ of recrystallized material from time $t_0$ to time $t$ over normalized time $t/t_0$, as defined in (23).