Shape deformation and circle instability in two-dimensional lipid domains by dipolar force: a shape- and size-dependent line tension model

The dipolar energy of a solid monolayer domain surrounded by a fluid phase at an air-water interface is derived approximately as a sum of an additionally negative line tension and a curvature-elastic energy at the boundary. Variation of the domain energy yields an equilibrium domain shape equation. The obvious solutions of the domain shape equation clearly predict a circle, torus, D-form, S-form, and serpentine manner shape found experimentally, depending on the difference in the Gibbs free energy between the solid and fluid phases and the total line tension. Analysis of linear instability for a circle with a fixed area shows that, above a threshold size, the circle can be deformed into an m-sided quasipolygon. The good agreement with the observation and numerical calculation reported by Lee and McConnell J. Phys. Chem. 91, 9532 (1993)]] shows the quantitative validity of the present theory.