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Dilute Limit Coarsening with an Anisotropic Surface Tension

Arjun R. Anand, Melinda M. Andrews, Benjamin P. Vollmayr-Lee

TL;DR

The paper investigates how a weakly anisotropic surface tension affects late-stage coarsening in the dilute Lifshitz-Slyozov framework with a conserved order parameter. It develops a perturbative approach expanding both the surface tension and drop shape in spherical harmonics, derives an anisotropic Gibbs-Thomson condition, and solves for a scaling family of drop shapes a_{\ell m}(x) that depend on the scaled size x=R_0/R_c; notably, the first-order corrections yield nonequilibrium shapes that deviate from the equilibrium Wulff form, while the isotropic $t^{1/3}$ growth law remains valid. The analysis also shows that the drop-size distribution is modified at second order in the anisotropy, signaling a breakdown of universal LS morphology under anisotropy. Overall, the work reveals that even weak anisotropy can qualitatively alter coarsening morphology and its statistics, providing a framework extendable to other weakly anisotropic, diffusive systems and to non-dilute regimes in future studies.

Abstract

We investigate the impact of an anisotropic surface tension on the late-stage dilute phase separation dynamics, revisiting the seminal Lifshitz-Slyozov (LS) theory, which traditionally relies on the assumption of isotropic surface tension. Using a perturbative treatment for weak anisotropy, we demonstrate that although the characteristic $t^{1/3}$ drop growth law remains unchanged, the anisotropy causes a significant breakdown of morphological universality. Specifically, we calculate explicitly a one-parameter family of nonequilibrium drop shapes that depend on the scaled drop size. These shapes are close to the equilibrium Wulff shape, but the smaller drops are more spherical and the larger drops have an enhanced anisotropy in comparison to the Wulff shape. We also demonstrate that the the drop size distribution is modified from the isotropic LS distribution at second order in the anisotropy strength.

Dilute Limit Coarsening with an Anisotropic Surface Tension

TL;DR

The paper investigates how a weakly anisotropic surface tension affects late-stage coarsening in the dilute Lifshitz-Slyozov framework with a conserved order parameter. It develops a perturbative approach expanding both the surface tension and drop shape in spherical harmonics, derives an anisotropic Gibbs-Thomson condition, and solves for a scaling family of drop shapes a_{\ell m}(x) that depend on the scaled size x=R_0/R_c; notably, the first-order corrections yield nonequilibrium shapes that deviate from the equilibrium Wulff form, while the isotropic growth law remains valid. The analysis also shows that the drop-size distribution is modified at second order in the anisotropy, signaling a breakdown of universal LS morphology under anisotropy. Overall, the work reveals that even weak anisotropy can qualitatively alter coarsening morphology and its statistics, providing a framework extendable to other weakly anisotropic, diffusive systems and to non-dilute regimes in future studies.

Abstract

We investigate the impact of an anisotropic surface tension on the late-stage dilute phase separation dynamics, revisiting the seminal Lifshitz-Slyozov (LS) theory, which traditionally relies on the assumption of isotropic surface tension. Using a perturbative treatment for weak anisotropy, we demonstrate that although the characteristic drop growth law remains unchanged, the anisotropy causes a significant breakdown of morphological universality. Specifically, we calculate explicitly a one-parameter family of nonequilibrium drop shapes that depend on the scaled drop size. These shapes are close to the equilibrium Wulff shape, but the smaller drops are more spherical and the larger drops have an enhanced anisotropy in comparison to the Wulff shape. We also demonstrate that the the drop size distribution is modified from the isotropic LS distribution at second order in the anisotropy strength.

Paper Structure

This paper contains 14 sections, 96 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coarsening Phenomenology
  4. Isotropic Lifshitz-Slyozov Theory
  5. Anisotropic Gibbs-Thomson Relation
  6. Equilibruim Drop Shape
  7. The Wulff shape to first order
  8. Gibbs-Thomson relation for the Wulff shape
  9. Interface Dynamics
  10. Drop Center Motion
  11. Scaling Solution for the Drop Shape
  12. Drop Size Distribution
  13. Summary
  14. Anisotropic Gibbs-Thomson Evaluation

Figures (3)

  • Figure 1: (a) The differential area of the anisotropic interface is simply related to the differential area of sphere of radius $R$ by a cosine factor given by $\hat{\bf n}\cdot\hat{\bf r}$. (b) A displacement of the interface by an amount $\delta R$ in the radial direction results in a normal displacement of $\hat{\bf n}\cdot\hat{\bf r}\delta R$.
  • Figure 2: Update in the drop parametrization $R(\theta,\phi,t)$ due to combined effects of interface motion and drop center motion. This diagram is drawn in the plane defined by ${\bf v}_c$ and $\hat{\bf r}$.
  • Figure 3: The scaling solution for $a_{\ell m}(x)$ as compared to the Wulff shape. These represent a one-parameter family of nonequilibrium shapes.