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A New Measure of Coarseness for Solutions to Cahn--Hilliard Equations

Peter Howard, Adam Larios, Quyuan Lin

TL;DR

The study addresses how to quantify coarsening in Cahn–Hilliard dynamics with a measure that remains meaningful from spinodal initiation through late-stage separation. It introduces an energy–period mapping based on exact periodic stationary solutions and uses a pseudoinverse to define a single coarseness parameter that evolves consistently with the system energy, independent of periodic structure. The authors compare three coarsening-rate frameworks—direct CH simulations, Langer's late-stage model, and H11's eigenvalue-based approach—against computational results, showing broad agreement and highlighting domain-size effects. This coarseness measure enables robust, cross-method comparisons and paves the way for studying coarsening in CH–NS couplings and higher dimensions, with potential impact on modeling phase separation kinetics in materials science. The work thus provides a practical, theory-grounded tool to quantify and analyze coarsening dynamics across the full evolution spectrum.

Abstract

We introduce a new measure of coarseness for characterizing phase separation processes such as those described by Cahn--Hilliard equations. An advantage of our measure is that it remains consistent throughout the evolution, including for solutions with no periodic structure. We use our measure to compare two previous models of coarsening dynamics with numerically generated dynamics, providing the first direct check that we are aware of for the efficacy of these methods.

A New Measure of Coarseness for Solutions to Cahn--Hilliard Equations

TL;DR

The study addresses how to quantify coarsening in Cahn–Hilliard dynamics with a measure that remains meaningful from spinodal initiation through late-stage separation. It introduces an energy–period mapping based on exact periodic stationary solutions and uses a pseudoinverse to define a single coarseness parameter that evolves consistently with the system energy, independent of periodic structure. The authors compare three coarsening-rate frameworks—direct CH simulations, Langer's late-stage model, and H11's eigenvalue-based approach—against computational results, showing broad agreement and highlighting domain-size effects. This coarseness measure enables robust, cross-method comparisons and paves the way for studying coarsening in CH–NS couplings and higher dimensions, with potential impact on modeling phase separation kinetics in materials science. The work thus provides a practical, theory-grounded tool to quantify and analyze coarsening dynamics across the full evolution spectrum.

Abstract

We introduce a new measure of coarseness for characterizing phase separation processes such as those described by Cahn--Hilliard equations. An advantage of our measure is that it remains consistent throughout the evolution, including for solutions with no periodic structure. We use our measure to compare two previous models of coarsening dynamics with numerically generated dynamics, providing the first direct check that we are aware of for the efficacy of these methods.
Paper Structure (12 sections, 4 theorems, 109 equations, 10 figures)

This paper contains 12 sections, 4 theorems, 109 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Cahn--Hilliard Equations
  3. The Cahn--Hilliard energy
  4. Periodic solutions of Cahn--Hilliard equations
  5. A New Measure of Coarseness
  6. Coarsening Rates Models
  7. Varying
  8. Computational Results
  9. Overview of the numerical methods
  10. Simulations for the Cahn--Hilliard Equation
  11. Conclusion
  12. Appendix: Proofs of Propositions \ref{['period-proposition']} and \ref{['specific-F-proposition']}

Key Result

Proposition 2.3

Assume $F$ is as described in Remark F-remark, and also that $F$ is an even function. Then for all $a \in (0, \phi_2)$ the period $p(a)$ specified in a-to-p satisfies In addition, if $F"'(\phi) > 0$ for all $\phi \in (0, \phi_2)$ then $p'(a) > 0$ for all $a \in (0, \phi_2)$.

Figures (10)

  • Figure 1: The bulk free energy $F$ along with its supporting line.
  • Figure 2: Periodic solutions for the Cahn--Hilliard equation. Note that the domain $[-1,1]$ is depicted for visual clarity, but by Proposition \ref{['specific-F-proposition']} (ii, the periods $p_s$ are approximately $0.2810$, $0.5620$, and $1.4050$.
  • Figure 3: Plot of $\mathcal{E} (p)$, computed with $\alpha = \beta = 1$, $\kappa = 0.001$, and $L = 1$.
  • Figure 4: Plot of $\mathcal{E} (p)$, with zoom-ins at $p \in [0.29, 0.32]$ and $p \in [0.3068, 0.3084]$ inset. Computed with $\alpha = \beta = 1$, $\kappa = 0.001$, and $L = 1$. We see that, as discussed in Section \ref{['MeasureofCoursening']}, $\mathcal{E} (p)$ can briefly lose monotonicity.
  • Figure 5: Plot of leading eigenvalues versus amplitude, computed with $\alpha = \beta = 1$, and $\kappa = 0.001$ and $\kappa = 0.0001$.
  • ...and 5 more figures

Theorems & Definitions (19)

  • Definition 1.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • proof
  • Definition 3.2
  • ...and 9 more