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Related Experiment Videos

Nonlinear growth of periodic patterns.

Simon Villain-Guillot1, Christophe Josserand

  • 1Centre de Physique Moléculaire Optique et Hertzienne, Université Bordeaux I, 33406 Talence Cedex, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 9, 2002
PubMed
Summary

We analyzed the Cahn-Hilliard equation for spinodal decomposition, finding that intermediate-stage pattern growth follows a simple ordinary differential equation. This model accurately predicts density profiles and dynamics, validated by numerical simulations.

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Area of Science:

  • Materials Science
  • Mathematical Physics
  • Chemical Engineering

Background:

  • Spinodal decomposition is crucial for phase separation in alloys and polymers.
  • The Cahn-Hilliard equation models this process but is complex in nonlinear regimes.
  • Understanding intermediate-stage growth is key to controlling microstructure evolution.

Purpose of the Study:

  • To analyze the nonlinear growth of periodic patterns in one-dimensional spinodal decomposition.
  • To simplify the dynamics of the Cahn-Hilliard equation in the intermediate regime.
  • To characterize the stationary density profile after nonlinear growth.

Main Methods:

  • Employing a solubility condition on quasistatic solutions.
  • Deriving an ordinary differential equation to describe dynamics.

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  • Performing numerical simulations using three distinct methods.

    • Main Results:

    • The dynamics of pattern growth were reduced to a simple ordinary differential equation.
    • A well-characterized density profile for the stationary regime was identified.
    • Numerical simulations confirmed the analytical results and recovered asymptotic dynamics.

    Conclusions:

    • The simplified model accurately captures intermediate-stage spinodal decomposition dynamics.
    • The study provides a robust analytical framework for the Cahn-Hilliard equation.
    • The findings are validated across multiple numerical methods and regimes.