We introduce a "Kawasaki-type" spin-exchange dynamics for Widom's model of microemulsions, where the numbers of the molecules of oil, water and surfactants are conserved. We investigate the relaxation of concentration fluctuations in two dimensions, using this dynamics, and find diffusion for small-amplitude deviations from equilibrium. In addition we determine equilibrium concentrations, also in three dimensions, as a function of "field".
The molecules of a ternary microemulsion, consisting of oil, water and surfactant, are represented in the Widom model [1] by nearest-neighbor bonds of a spin-Β½ Ising model with non-zero interactions between nearest-neighbor as well as farther-neighbor spin pairs. The surfactants are amphiphilic molecules with a hydrophilic head and a hydrophobic tail; thus their head sticks to water and their tail prefers oil. The bond between an up-up nearest-neighbor spin pair represents an oil molecule, that between a down-down pair corresponds to water, while those between antiparallel pairs represent surfactant molecules ("soap"). To our knowledge, single-spin-flip Glauber dynamics has been used in all the earlier Monte Carlo simulations [2] (see ref. [2] where earlier literature on 3D simulations is cited) of the Widom model; the numbers of the three types of molecules were thus not conserved in these simulations.
For a realistic description of the dynamic properties of this lattice model of microemulsions, the numbers of the molecules must be conserved. In order to implement this conservation law in our simulations, we introduce here a Kawasaki-type spin-exchange dynamics. However, in contrast to the usual Kawasaki dynamics, in this new dynamics two neighboring spins exchange their position (i.e. a down-up pair is reversed into an up-down pair) in such a way