Phase Separation Patterns from Directional Quenching

Rafael Monteiro, Arnd Scheel

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


We study the effect of directional quenching on patterns formed in simple bistable systems such as the Allen–Cahn and the Cahn–Hilliard equation on the plane. We model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across a trigger line. We are then interested in patterns forming in the bistable region, in particular as the trigger progresses with small speed and increases this bistable region. We find existence and nonexistence results of single interfaces and striped patterns. For zero speed, we find stripes parallel or perpendicular to the trigger line and exclude stripes with an oblique orientation. Single interfaces are always perpendicular to the trigger line. For small positive speed, striped patterns can align perpendicularly. Other orientations are excluded in Allen–Cahn for all nonnegative speeds. Single interfaces for positive trigger speeds are excluded for Cahn–Hilliard and align perpendicularly in Allen–Cahn.

Original languageEnglish (US)
Pages (from-to)1339-1378
Number of pages40
JournalJournal of Nonlinear Science
Issue number5
StatePublished - Oct 1 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017, Springer Science+Business Media New York.


  • Allen–Cahn
  • Cahn–Hilliard
  • Directional quenching
  • Phase separation


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