Comparison of analytical grain size distributions with three-dimensional computer simulations and experimental data

Ultra-large-scale phase-field simulation study of ideal grain growth

“…(2)) can be derived as K = 2ρ 2 α/γ from the mean-field theory. 18,24,25 Substituting the ρ and γ values obtained above into this formula yields K = 0.473, which is very close to the simulated value of 0.479 (Fig. 2c).…”

“…To our knowledge, the only analytical approach describing ideal grain growth without any fitting parameters is the mean-field theory, which was originally developed by Hillert 18 and recently reformulated to a more general form by Rios et al 24 and Darvishi Kamachali et al 25 The basic equation of this theory is the meanfield approximation for the growth rate of individual grains:…”

“…Using Eq. (3) and the Lifshitz-Slyozov theory 26 of particle ripening, recent studies 24,25 have derived a generalized form of Hillert's grain size distribution function, f (R/<R>), that depends on the mean-field parameter, γ, involved in the Lifshitz-Slyozov stability condition. The value of γ can be calculated using ρ = <R> 2 /<R 2 > and the relation 25 0.8557 + 0.0107γ − 0.5509 exp(-γ) ≈ ρ.…”

“…For most of the data points, the error bars are smaller than the symbols subsequent steady-state regime, the above parameters take almost constant values of ρ ≈ 0.866 and γ ≈ 3.17. For γ < 4, the specific form of f (R/<R>) is given by 24,25 :…”

Self-similar grain size distribution in three dimensions: A stochastic treatment

“…Essentially the modified LSW treatment gives a family of curves, which, similar to our case, all depend on a single parameter. Rios et al use this method, for example, in both 2-D [29,30] and 3-D [31] grain growth studies; this approach requires one adjustable parameter and this free parameter can be used to improve Hillert's mean-field approach. It gives a distribution without a theoretical cut-off.…”