Curvature driven flow of bi-layer interfaces

Nir Gavish, Gurgen Hayrapetyan, Keith Promislow, Li Yang

Research output: Contribution to journalArticlepeer-review


We introduce the Functionalized Cahn-Hilliard (FCH) energy, a negative multiple of the Cahn-Hilliard energy balanced against the square of its own variational derivative, as a finite width regularization of the sharp-interface CanhamHelfrich energy. Mass-preserving gradient flows associated to the FCH energy are higher-order phase field models which develop not only single-layer, or front-type interfaces, but also bi-layer, or homoclinic interfaces with associated endcap and multi-junction structures. The single-layer interfaces manifest a fingering instability which grows into endcapped bi-layers. The meandering growth of the bi-layer interfaces and the subsequent merging lead to a multi-junction dominated network that bears a striking similarity to the phase separated domains of both perfluorosulfonic membranes and amphiphilic di-block co-polymer solutions. The bi-layers generated by the gradient flows of the FCH energy have an interfacial width which scales with ε≪1, however for fixed ε, there is a class of bi-layers parameterized by width and background state. Our primary result is the asymptotic derivation of the normal velocity of a closed bi-layer hypersurface in Rd (d<2) coupled to the evolution for the surface width, curvature, and background state. We also show the convergence of the FCH energy to a scaled CanhamHelfrich type energy for both single and bi-layer interfaces, with the surface area coefficient of the limiting CanhamHelfrich energy coupling to the bi-layer width. Thus the bi-layer networks grow to maximize surface area while minimizing the square of curvature, up to the point that the increase in surface area stretches the bi-layers too thin.

Original languageEnglish
Pages (from-to)675-693
Number of pages19
JournalPhysica D: Nonlinear Phenomena
Issue number7
StatePublished - 15 Mar 2011
Externally publishedYes


  • CanhamHelfrich energy
  • Functionalized Cahn-Hilliard energy
  • Geometric evolution
  • Network formation

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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