A Lattice Boltzmann Model of Ternary Fluid Mixtures

– A lattice Boltzmann model is introduced which simulates oil–water–surfactant mixtures. The model is based on a Ginzburg-Landau free energy with two scalar order parameters. Diffusive and hydrodynamic transport is included. Results are presented showing how the surfactant diffuses to the oil–water interfaces thus lowering the surface tension and leading to spontaneous emulsification. The rate of emulsification depends on the viscosity of the ternary fluid. Introduction. – The addition of surfactant to a binary mixture of oil and water can produce many different complex structures on a mesoscopic length scale. The surfactant molecules move to the interface and lower the oil–water interfacial tension. This can result in, for example, lamellar, micellar, microemulsion or hexagonal arrangements of the oil and water domains[1, 2]. The equilibrium behaviour of such amphiphilic systems is well understood. However the dynamics of the self-assembly of the mesoscale phases and their rheology are less well investigated. This is a difficult problem because of the interplay between several relevant transport mechanisms, the diffusion of the constituent components and their hydrodynamic flow. To date models of amphiphilic rheology which treat hydrodynamic effects include time-dependent Ginzburg-Landau approaches[3, 4], molecular dynamics[5] and a lattice gas cellular automaton scheme based on microscopic interactions[6, 7]. The aim of this Letter is to introduce an alternative numerical scheme that can model the dynamics of amphiphilic systems in such a way that diffusive and hydrodynamic mechanisms are included. The numerical approach that we use is lattice Boltzmann simulations which have emerged as a useful tool to study the dynamics of complex fluids[8]. We base our approach on that described by Orlandini et. al. [9, 10] where the correct equilibrium of the fluid is imposed by choosing an appropriate free energy and including it in such a way that the fluid spontaneously reaches the equilibrium described by its minimum. Previous lattice Boltzmann models of amphiphilic systems have been based on a single order Typeset using EURO-TEX 2 EUROPHYSICS LETTERS parameter, that of the phase separating binary fluid[11, 12]. The effect of the amphiphilic molecules has been mimiced by varying the surface tension in the input free energy. Although this approach proved successful it has the disadvantage of not including the surfactant dynamics explicitly. We first give a description of the method and then present results showing how the surface tension of the binary fluid interface is lowered by surfactant at a rate which depends on the surfactant diffusion constant. As the surface tension becomes negative, this leads to the break-up of the interface and to spontaneous emulsification[13] to a lamellar phase. The Lattice-Boltzmann Scheme. – We consider a Ginzburg-Landau model defined by the free energy functional [3, 14] which depends on two scalar order parameters φ(r) and ρ(r) F [φ, ρ] = ∫ dr [a 2 φ + b 4 φ + κ 2 (∇φ) + c 2 (∇φ) + α 2 ρ + λ 2 (∇ρ) + γ 2 (∇ρ) +β1φρ 2 + β2φ (∇ρ) + β3ρφ(∇2φ) ] . (1) φ(r) and ρ(r) can be identified, respectively, with the local density difference of oil and water and with the difference of local surfactant concentration from its average ρ̄. ρ̄ enters the model via the parameter κ. κ is positive for small surfactant concentration. As κ decreases and eventually becomes negative, ρ̄ increases. The thermodynamic variables that we will need are the chemical potential difference between oil and water ∆μ, the chemical potential Λ of the surfactant and the pressure tensor Pαβ . The chemical potentials follow from the free energy as [15] ∆μ = δF δφ = aφ+ bφ − κ∇φ+ c(∇φ) + 2β1ρφ+ 2β2φ(∇2ρ) + β3ρ(∇2φ) + β3∇2(ρφ), (2) Λ = δF δρ = αρ− λ∇ρ+ γ(∇ρ) + β1φ2 + β2∇2φ2 + β3φ(∇2φ). (3) The derivation of the pressure tensor is slightly more complicated. Considering a linear combination of all symmetric tensors having two or four gradient operators, we find that a suitable choice, which allows the pressure tensor to obey the equilibrium condition ∂αPαβ = 0 is Pαβ = { pL + c [ (∇φ) + ∂σφ∂σ∇2φ ] + γ [ (∇ρ) + γ∂σρ∂σ∇2ρ ]