Lattice model of microemulsions: The effect of fluctuations in one and two dimensions

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Abstract
A simple model of oil, water, and amphiphile, in which the third is favored energetically to sit between the other two, is studied in one and two dimensions by transfer-matrix methods, Monte Carlo simulations, and the Müller-Hartman-Zittartz approximation. Phase diagrams, correlation functions, scattering intensities, and the surface tension of the oil-water interface have been calculated. The results obtained here confirm many of the conclusions drawn from the analysis of the model in three dimensions via mean-field theory. In particular, we find that in the microemulsion, the water-water correlation function shows damped oscillatory behavior at large distances, whereas the amphiphile-amphiphile correlation function decays exponentially. With increasing amphiphile concentration, the phase diagram of the two-dimensional system, like that of the three-dimensional one, shows the commonly observed progression from three-phase coexistence through a microemulsion to two-phase coexistence with a lamellar phase. In contrast to the behavior in three dimensions, the microemulsion in two dimensions is stabilized even in the limit of very low temperatures. The comparison of results obtained here with those of mean-field theory elucidates the role that fluctuations play in this system.

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