Coalescence kinetics in surfactant stabilized emulsions: Evolution equations from direct numerical simulations

Abstract

Lattice Boltzmann simulations were used to study the coalescence kinetics in emulsions with amphiphilic surfactant, under neutrally buoyant conditions, and with a significant kinematic viscosity contrast between the phases (emulating water in oil emulsions). The 3D simulation domain was large enough (256(3) ~ 10(7) grid points) to obtain good statistics with droplet numbers ranging from a few thousand at early times to a few hundred near equilibrium. Increased surfactant contents slowed down the coalescence rate between droplets due to the Gibbs-Marangoni effect, and the coalescence was driven by a quasi-turbulent velocity field. The kinetic energy decayed at a relatively slow rate at early times, due to conversion of interfacial energy to kinetic energy in the flow during coalescence. Phenomenological, coupled differential equations for the mean droplet diameter D(t) and the number density n(d)(t) were obtained from the simulation data and from film draining theories. Local (in time) power law exponents for the growth of the mean diameter (and for the concomitant decrease of n(d)) were established in terms of the instantaneous values of the kinetic energy, coalescence probability, Gibbs elasticity, and interfacial area. The model studies indicated that true power laws for the growth of the droplet size and decrease of the number of droplets with time may not be justified, since the exponents derived using the phenomenological model were time dependent. In contrast to earlier simulation results for symmetric blends with surfactant, we found no evidence for stretched logarithmic scaling of the form D ~ [ln (ct)](α) for the morphology length, or exponential scalings associated with arrested growth, on the basis of the phenomenological model.

Multiphase Flow Research with a Surfactant Lattice Boltzmann Model

“…We have found the more standard LBM approach to be suitable for highly dynamic emulsion systems [4][5][6] where the Marangoni effect significantly reduces the coalescence rate so that the contact time between droplets is smaller than the film draining time. This corresponds to a pure hydrodynamic interaction between the droplets with a partial film draining phase, and with a subsequent separation of the droplets by the external flow.…”

“…The average coalescence rate is then controlled by the Marangoni effect, rather than the shorter range molecular forces. Therefore, in highly dynamic or turbulent cases, the surfactant lattice Boltzmann model proved to be adequate [4,6] .…”

“…The many side effects are well documented and discussed elsewhere in the literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] . We found that compressibility and mass transfer are effects have only a minor impact, if the simulations are run with moderate external forcing (in terms of applied pressure gradients, bodyforces, or moving walls in "rheometer" settings), and with moderate inter-phase forces (repulsion forces to separate the fluid components such as oil and water).…”

“…A clear advantage with the diffuse interface LBM is that the computational time does not increase with specific interfacial area (per unit volume). Realistic simulations of dense emulsions that contain a large number of droplets (about 100-3000 droplets in a simulation domain of typically 10 million grid points), is then feasible on affordable multi-processor workstations [4,5,6] . This property, together with numerical algorithms that are ideal for parallel computing, has been the main reason for choosing LBM in our work.…”

“…For example, we developed approximate evolution equations for the mean droplet size in non-sheared emulsions in decaying turbulent flows, where the Marangoni effect was a controlling factor [4] . We also studied the droplet size distribution and breakup criteria in forced homogeneous turbulence [4,6] , where we were able to systematize the different breakup criteria found in the literature, and analyze them in the context turbulence length scales.…”