Spin-1 model of a microemulsion

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Abstract
We formulate a spin-1 model of a microemulsion in which the spin values +1, 0, and -1 correspond to water, surfactant, and oil, respectively. Physical considerations dictate our choice of interactions. In addition to the usual attractive interactions at short range between molecules of the same kind, the surfactant induces a competing attractive interaction between molecules of different kinds at a distance equal to the size of the surfactant. For sufficient surfactant concentrations, this induces a phase in which numerous walls of surfactant separate distinct regions of oil and water. In a continuum model, these walls would be rough, producing a critical phase without either the long-range order of a solid or the exponential decay of correlations of a disordered liquid. We identify this phase with the ‘‘bicontinuous’’ phase of Scriven. The phase diagram of the system within mean-field theory is calculated, as is the surface tension between oil and water as a function of surfactant concentration. Depending upon temperature, the value of the latter at the triple point can be reduced by a factor of a thousand or more from that in the absence of surfactant.

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