The intermolecular interactions in the pseudo-potential lattice Boltzmann (PPLB) method can readily be extended to more than two components. We report about a three-component PPLB approach to explore whether the effect of a surfactant could be included in describing droplet behaviour in (liquid–liquid) emulsions. The two main liquid components are taken to follow the Carnahan-Starling equation of state (EoS), while the surfactant obeys an ideal EoS. We investigate the nature of the phases present at equilibrium and the dependence of the interfacial tension between the two liquid phases on the amount of surfactant. The response of a droplet subjected to simple shear is investigated in the absence and the presence of a surfactant. Our exploratory simulations show how during droplet deformation the surfactant re-distributes itself due to the action of the shear and flows towards the far ends of the deformed droplet, up to the moment the droplet breaks up. This inhomogeneous surfactant distribution along the interface increases the shear rate that is needed for droplet breakup such that the critical capillary number for breakup increases and the breakup process is delayed. The simulations also reveal the detailed flow fields inside and outside the deforming droplet.

We perform direct numerical simulations (DNS) of emulsions in homogeneous isotropic turbulence using a pseudopotential lattice-Boltzmann (PP-LB) method. Improving on previous literature by minimizing droplet dissolution and spurious currents, we show that the PP-LB technique is capable of long stable simulations in certain parameter regions. Varying the dispersed-phase volume fraction , we demonstrate that droplet breakup extracts kinetic energy from the larger scales while injecting energy into the smaller scales, increasingly with higher , with approximately the Hinze scale (Hinze, AIChE J., vol. 1 (3), 1955, pp. 289-295) separating the two effects. A generalization of the Hinze scale is proposed, which applies both to dense and dilute suspensions, including cases where there is a deviation from the inertial range scaling and where coalescence becomes dominant. This is done using the Weber number spectrum , constructed from the multiphase kinetic energy spectrum , which indicates the critical droplet scale at which . This scale roughly separates coalescence and breakup dynamics as it closely corresponds to the transition of the droplet size distribution into a scaling (Garrett et al., J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163-2171; Deane & Stokes, Nature, vol. 418 (6900), 2002, p. 839). We show the need to maintain a separation of the turbulence forcing scale and domain size to prevent the formation of large connected regions of the dispersed phase. For the first time, we show that turbulent emulsions evolve into a quasi-equilibrium cycle of alternating coalescence and breakup dominated processes. Studying the system in its state-space comprising kinetic energy , enstrophy and the droplet number density , we find that their dynamics resemble limit cycles with a time delay. Extreme values in the evolution of are manifested in the evolution of and with a delay of and respectively (with the large eddy timescale). Lastly, we also show that flow topology of turbulence in an emulsion is significantly more different from single-phase turbulence than previously thought. In particular, vortex compression and axial straining mechanisms increase in the droplet phase.