Linear Relationship Between dV/dt and Grain Volume During Grain Growth

The volumetric growth rate of individual grains has been found empirically to be directly proportional to their individual volume, dV/dt = β(V0 − V). This simple result extends from the relationship $${\mathrm{d}}V/{\mathrm{d}}t=-k{M}_{\mathrm{S}},$$ where MS is integral mean curvature of the boundaries of individual grains and k the grain growth rate constant, and the experimentally observed linear relationship between $${M}_{\mathrm{S}}$$ and individual grain volume, $${M}_{\mathrm{S}}$$ = α(V0 − V). Here, α is a scaling parameter that collapses the relationships for separate times of growth into a single trend. It has been shown that α = $$\gamma {\bar{V}}^{-2/3}$$ where γ is an experimentally determined constant ≃ − 2.7 and $${\bar{V}}$$ is the mean grain volume. Thus, β = −kα = 2.7k $${\bar{V}}^{-2/3}$$ , simply proportional to the scale of the overall grain structure. These relationships have been tested successfully in numerous 3D grain growth simulations and experiments. This paper describes the relationships among these kinetic and geometric grain characteristics that provide this surprisingly simple description of 3D grain growth, i.e., a linear relationship between the growth rate of an individual grain and its volume.