Oil-Water Interface in the Widom Model: Is It Smooth, Rough or Crumpled?

Abstract

Trie interface in trie two-dimensional Ising model with only nearest-neighbour ferromagnetic interactions is known ta be rougir at ail temperatures T > o. Trie Widom model for temary microemulsions is, effectively, a spin-1/2 Ising model with, Dot Orly nearest-neighbour ferromagnetic interactions, but aise specific farther-neighbour antiferromagnetic interactions. We investigate trie effects of these farther-neighbour interactions m the Widom model on the roughening properties of the interface by carrymg eut Monte Carlo simulations. Temary microemulsions are three component mixtures where amphiphilic molecules act as surfactants between oil and water iii. Several lattice models of these systems bave been pro- posed and are similar in spirit to trie lattice gas model for trie liquid-gas transition (2). To our knowledge, trie Widom model (3) is trie simplest among trie lattice models of ternary microemulsions. This is, effectively, a spin-1/2 Ising model with not only nearest-neighbour ferromagnetic interactions, but also with farther neighbour antiferromagnetic interactions of specific type descnbed below. Trie farther-neighbour interactions arise from Widom's prescrip- tion for incorporating trie bending rigidity of trie amphiphilic membrane (a monolayer) formed by trie surfactants at the oil-water interface (4). Trie interface between trie up-spm and down- spin domains in trie two-dimensional Ising model with only nearest-neighbour interactions is known to be rougir at ail non-zero temperatures. Trie atm of this communication is to study trie effects of trie farther-neighbour antiferromagnetic interactions m trie Widom model on trie roughening properties of trie interface. In contrast to trie earlier series expansion study for trie three dimensional Widom model by Kahng et ai. (5) we carry out Monte Carlo Simulation of trie two dimensional Widom model. In trie Widom model trie amphiphilic molecules, as well as trie molecules of oit and water, are assumed to be located on trie nearest-neighbour bonds of a simple cubic lattice. Classical Ising spins are put at each of trie lattice sites and one uses trie convention that each bond between nearest-neighbour (+ +) spin pairs is occupied by a molecule of oil, that between (--) spin pairs is occupied by a molecule of water, and that between anti-parallel nearest-neighbour spin pairs is occupied by a surfactant. Trie Hamiltonian of trie system m ternis of trie Ising spin