Shape Deformation and Circle Instability in Two-Dimensional Lipid Domains by Dipolar Force: A Shape- and Size-Dependent Line Tension Model

Abstract

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.

Nanoparticle shape selection by repulsive interactions: Metal islands on few-layer graphene

“…7,8 In our treatment this is regularized more naturally by representing the interaction potential using a nonsingular near field form controlled by the finite size of the relevant microscopic dipoles. More generally our expression for the interactionrenormalized line energy now can be applied to any repulsive two-point potential in the droplet interior.…”

“…They derived an explicit formula for the line energy of a twodimensional droplet in terms of a double line integral over its perimeter, a result that, in principle, can be used as input to an algorithm to compute the shape of a growing droplet. This approach was subsequently refined by Iwamoto et al 8,9 who replace the double line integral by an expression for the line energy parameterized by the amplitudes of its modes of deformation, and they applied this method to study effects of dipolar interactions both for the perpendicular geometry and for dipoles tilted with a nonzero component parallel to the tangent plane of the droplet. Our work is similar in spirit to that of Iwamoto et al, 8 though we present it here in a form that is applicable to a general repulsive potential.…”

“…This approach was subsequently refined by Iwamoto et al 8,9 who replace the double line integral by an expression for the line energy parameterized by the amplitudes of its modes of deformation, and they applied this method to study effects of dipolar interactions both for the perpendicular geometry and for dipoles tilted with a nonzero component parallel to the tangent plane of the droplet. Our work is similar in spirit to that of Iwamoto et al, 8 though we present it here in a form that is applicable to a general repulsive potential. For the electrostatic dipolar repulsive potential appropriate to the Yb/ graphene problem the model provides a good account of the experimental observations.…”

Probing motion of electric dipoles and carriers in organic monolayers by Maxwell Displacement Current and optical second harmonic generation

“…Hence domain shape of monolayer at the air-water interface is dependent on the electrostatic energy contribution [16,17]. Minimizing free energy of monolayers [18], the shape equation for the domain shape is derived and it has the Maxwell-stress term (2nd and 3rd terms) [19]:…”

Nano-interface Surface Dielectric Polarization Phenomena: Detection and Application

“…In a manner similar to the derivation of shape energy of a 3D vesicle by Helfrich and Ou-Yang [30,31] , the shape energy of a 2D domain with permanent dipoles pointing toward air normal to the water surface, corresponding to the state S1=1, is formalized [14] , and the domain energy is derived naturally taking into account the contribution of elongation by dipolar interaction [25, 261 and Such shapes with a micrometer size were found as domains in monolayers experimentally by BAM, fluorescence spectroscopy, etc [21][22][23][24] . Our experimental BAM system attached to the MDC-SHG experimental system have also visualized such textures in monolayers [29] .…”