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Lipid domains are found in systems such as multi-component bilayer membranesand single component monolayers at the air-water interface. It was shown byAndelman et al. (Comptes Rendus 301, 675 (1985)) and McConnell et al. (Phys.Chem. {\bf 91}, 6417 (1987)) that in monolayers, the size of the domainsresults from balancing the line tension, which favors the formation of a largesingle circular domain, against the electrostatic cost of assembling thedipolar moments of the lipids. In this paper, we present an exact analyticalexpression for the electric potential, ion distribution and electrostatic freeenergy for different problems consisting of three different slabs withdifferent dielectric constants and Debye lengths, with a circular homogeneousdipolar density in the middle slab. From these solutions, we extend thecalculation of domain sizes for monolayers to include the effects of finiteionic strength, dielectric discontinuities (or image charges) and thepolarizability of the dipoles and further generalize the calculations toaccount for domains in lipid bilayers. In monolayers, the size of the domainsis dependent on the different dielectric constants but independent of ionicstrength. In asymmetric bilayers, where the inner and outer leaflets havedifferent dipolar densities, domains show a strong size dependence with ionicstrength, with molecular-sized domains that grow to macroscopic phaseseparation with increasing ionic strength. We discuss the implications of theresults for experiments and briefly consider their relation to other twodimensional systems such as Wigner crystals or heteroepitaxial growth.Comment: 13 pages, 5 figues in eps Replaced with new version, one citation  added and a few statements corrected. The results of the paper are unchange

Effect of dipolar moments in domain sizes of lipid bilayers and monolayers