Computational studies of coarsening rates for the Cahn–Hilliard equation with phase-dependent diffusion mobility

Abstract

We study computationally coarsening rates of the Cahn-Hilliard equation with a smooth double-well potential, and with phase-dependent diffusion mobilities. The latter is a feature of many materials systems and makes accurate numerical simulations challenging. Our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis. We demonstrate that the numerical solutions are consistent with the physical Gibbs-Thomson effect, even if the mobility is degenerate in one or both phases. For the two-sided degenerate mobility, we report computational results showing that the coarsening rate is on the order of l ∼ ct 1/4 , independent of the volume fraction of each phase. For the one-sided degenerate mobility, that is non-degenerate in the positive phase but degenerate in the negative phase, we illustrate that the coarsening rate depends on the volume fraction of the positive phase. For large positive volume fractions, the coarsening rate is on the order of l ∼ ct 1/3 and for small positive volume fractions, the coarsening rate becomes l ∼ ct 1/4 .

Non-integer temporal exponents in trans-interface diffusion-controlled coarsening

“…0. They did not report the temporal exponent for this model but affirmed the findings of Sheng et al Dai and Du [55] subsequently performed detailed mathematical and computational studies of the impact of three different dimensionless mobility expressions, M* = 1, M* = |1-u 2 | and M* = |1 ? u|/2, on latestage coarsening behavior.…”

“…In this context, it should be noted that concentration-dependent atomic mobilities have been incorporated into theories of late-stage spinodal decomposition (well into the coarsening regime) [47][48][49][50][51][52][53], and that phase field simulations of c 0 precipitate coarsening have used similar assumptions [54,55]. These works show that with suitably chosen functions for the mobility of solute, particularly functions that represent extremely slow diffusion in the precipitate phase, the temporal exponent n actually increases from 3 to as much as 4, with the irony that the kinetics in these papers is attributed to interface diffusion control.…”

Влияние Границ Зерен На Распределение Компонентов В Бинарных Сплавах

“…Данный метод решения уравнения Кана-Хилларда достаточно хорошо развит для случая посто-янной подвижности [33] и подвижности, зависящей от локального состава сплава [34] (см. также, [25,26]). Что-бы применить данный метод к разработанной модели, его необходимо адаптировать для случая, когда параметр квазихимического взаимодействия , а вместе с ним и параметр κ, зависят от координат.…”

“…В настоящей работе пред-полагается рассмотреть влияние границ зерен на рас-пределение компонентов сплава (включая формирование фаз) в бинарных сплавах на основе метода функционала плотности свободной энергии [19][20][21]. Данный подход наиболее часто используется для описания фазовых переходов в системах, не имеющих структурных де-фектов, как в области метастабильных [22][23][24], так и нестабильных [25,26] состояний. Этот метод обычно применяется в предположении безграничности рассмат-риваемой среды, при этом влияние эффектов, связанных с наличием границ зерен, как правило, не рассмат-ривается.…”

Three-dimensional coarsening dynamics of a conserved, nematic liquid crystal-isotropic fluid mixture

“…Unfortunately, this is computationally prohibitive for our current three dimensional system. We opt here to estimate the asymptotic coarsening rate by using the rate of decay of the mixing energy [23,15]. The argument is that, in the zero capillary length limit ε → 0,…”