Lattice - Boltzmann Model of Amphiphilic Systems

– A lattice-Boltzmann model for the study of the dynamics of oil-water-surfactant mixtures is constructed. The model, which is based on a Ginzburg-Landau theory of amphiphilic systems with a single, scalar order parameter, is then used to calculate the spectrum of undulation modes of an oil-water interface and the spontaneous emulsification of oil and water after a quench from two-phase coexistence into the lamellar phase. A comparison with some analytical results shows that the model provides an accurate description of the static and dynamic behavior of amphiphilic systems. Introduction. – Self-assembling amphiphilic system are characterized by a large variety of phases [1, 2]. Many of these are structured on mesoscopic length scales. Examples include the swollen lamellar phase, the microemulsion phase in ternary amphiphilic mixtures, and the sponge phase in aqueous amphiphile solutions. In all of these cases, the amphiphile assembles into two-dimensional sheets which separate space into two networks of oiland/or water-channels, both of which span the whole system. The mesoscopic structure leads to dynamical behavior which differs qualitatively from that of simple fluid mixtures. In addition to the diffusive dynamics of all components, hydrodynamic flow plays an important role. Three classes of theoretical models which include hydrodynamic effects have been employed so far. The first are time-dependent Ginzburg-Landau models [3, 4, 5] which are coupled to a linearized Navier-Stokes equation (i.e. generalizations of model H of Hohenberg and Halperin [6] to amphiphilic systems). The second are membrane models, in which the hydrodynamics of the solvent is taken into account [7, 8]. Finally, a lattice-gas model [9, 10] — similar in spirit to vector lattice models employed to study equilibrium properties of microemulsions [1] — has been formulated and applied to study the dynamics of these systems. An interesting alternative to these descriptions of the hydrodynamics of complex fluids is the Lattice-Boltzmann (LB) approach [11, 12, 13, 14]. It has recently been shown how LB methods can be generalized to study the dynamics of non-ideal fluids [15, 16] and fluid mixtures Typeset using EURO-TEX 2 EUROPHYSICS LETTERS [17, 16, 18] with a given free-energy functional. In this paper we show how this approach can be extended to study the dynamical behavior of amphiphilic systems. Lattice-Boltzmann Approach. – The basic variables of a LB model for two-component fluid mixtures are the “microscopic” distribution functions fi(r) and gi(r), which are related to the density ρ, density difference ∆ρ, and velocity u by ρ(r) = ∑