DOMAIN SHAPE RELAXATION AND THE SPECTRUM OF THERMAL FLUCTUATIONS IN LANGMUIR MONOLAYERS

Isolated domains in Langmuir monolayers exhibit shape instabilities leading to branched structures as a consequence of the competing effects of Line tension and dipolar interactions. A theory for interfacial pattern formation in the presence of these forces, developed to study shape instabilities of magnetic fluids, is reformulated to treat electric dipolar systems and specialized to the case of ''ultrathin'' domains, for which the aspect ratio 2R/h much greater than 1, where R is the domain radius and h its thickness or a microscopic cutoff. Two phenomena studied in recent experiments are considered on the basis of this model: the spectrum of thermal fluctuations of the domain boundary and the dynamics of shape relaxation near the branching instability. The experimental spectrum of thermal fluctuations for monolayers of dimyristoylphosphatidylcholine and cholesterol (Seul, M. Physica A 1990, 168, 198) deviates in a small but measurable way from that expected in the presence of line tension alone and is described quantitatively by the theory. This analysis yields estimates for the line tension and dipole moment density which are in accord with previous determinations by other methods. The relaxation of branched shapes to a circular ground state is found, as in experiment (Seul, M. J. Phys. Chem. 1993, 97, 2941), to deviate in a characteristic way from the ''curve-shortening'' law which governs the motion in the absence of dipolar interactions. A heuristic argument explaining this phenomenon, in which the notion of a scale-dependent surface tension is introduced, is formulated on the basis of a ''localized induction approximation''. This approximation, familiar from the study of vortex motion in inviscid hydrodynamics, is applied here to the Biot-Savart integrals which represent the dipolar pressure at the boundary of the domain. A relationship between the energetics of fingering instabilities and domain fission is proposed on the basis of analytical results obtained in the ultrathin limit. Details of a numerical method for the study of this shape evolution are provided, with particular attention paid to a consistent treatment of cutoff effects.