  Particle coarsening and grain growth take place to minimize the total
interfacial energy. The classical mean-field treatments by Lifshitz, Slyozov,
[1] Wagner [2] and Hillert [3] predicted cubic growth law under bulk-diffusion
controlled precipitation coarsening and parabolic growth law under interface
controlled grain growth, as well as their steady-state size distribution. When
the size distribution is the steady-state one, the average grain size satisfies
the following dependence: under bulk-diffusion control and under interface
control, with correct slopes definitely given by the theory. However, when the
size distribution does not satisfy the steady-state one, the growth kinetics
would deviate from theoretical prediction. As is shown by numerical simulations
in Ref. 4 and 5, the deviation is less obvious in the growth law, but more
evident in the slope of the growth curve. Specifically, starting from a
non-steady-state size distribution, the parabolic/cubic growth law would be
recovered quickly, yet the convergence of the slope in the or curve is
extremely slow. Therefore, it would be interesting to know how to calculate the
instantaneous slope of the growth kinetics from an arbitrarily given size
distribution. As will be shown below, assuming a known growth law, the
analytical solution can be easily obtained and the exact solution can be
obtained by numerical integration.
