Abstract
Droplet dynamics in microfluidic applications is significantly influenced by surfactants. It remains a research challenge to model and simulate droplet behaviour including deformation, breakup and coalescence, especially in the confined microfluidic environment. Here, we propose a hybrid method to simulate interfacial flows with insoluble surfactants. The immiscible two-phase flow is solved by an improved lattice Boltzmann colour-gradient model which incorporates a Marangoni stress resulting from non-uniform interfacial tension, while the convection–diffusion equation which describes the evolution of surfactant concentration in the entire fluid domain is solved by a finite difference method. The lattice Boltzmann and finite difference simulations are coupled through an equation of state, which describes how surfactant concentration influences interfacial tension. Our method is first validated for the surfactant-laden droplet deformation in a three-dimensional (3D) extensional flow and a 2D shear flow, and then applied to investigate the effect of surfactants on droplet dynamics in a 3D shear flow. Numerical results show that, at low capillary numbers, surfactants increase droplet deformation, due to reduced interfacial tension by the average surfactant concentration, and non-uniform effects from non-uniform capillary pressure and Marangoni stresses. The role of surfactants on the critical capillary number ($Ca_{cr}$) of droplet breakup is investigated for various confinements (defined as the ratio of droplet diameter to wall separation) and Reynolds numbers. For clean droplets,$Ca_{cr}$first decreases and then increases with confinement, and the minimum value of$Ca_{cr}$is reached at a confinement of 0.5; for surfactant-laden droplets,$Ca_{cr}$exhibits the same variation in trend for confinements lower than 0.7, but, for higher confinements,$Ca_{cr}$is almost a constant. The presence of surfactants decreases$Ca_{cr}$for each confinement, and the decrease is also attributed to the reduction in average interfacial tension and non-uniform effects, which are found to prevent droplet breakup at low confinements but promote breakup at high confinements. In either clean or surfactant-laden cases,$Ca_{cr}$first remains almost unchanged and then decreases with increasing Reynolds number, and a higher confinement or Reynolds number favours ternary breakup. Finally, we study the collision of two equal-sized droplets in a shear flow in both surfactant-free and surfactant-contaminated systems with the same effective capillary numbers. It is identified that the non-uniform effects in the near-contact interfacial region immobilize the interfaces when two droplets are approaching each other and thus inhibit their coalescence.
Highlights
Emulsions, which consist of droplets of one fluid dispersed in another fluid, are common in our daily lives and can be found in the production processes of food, personal care products and pharmaceuticals (Sjöblom 2005; Seemann et al 2011)
The increased deformation, i.e. Df2 − Df0, consists of two parts: (1) δ1 = Df1 − Df0, which is attributed to a reduction of the interfacial tension caused by the average surfactant concentration ψ0, and (2) δ2 = Df2 − Df1, which is attributed to the non-uniform effects from non-uniform capillary pressure on the droplet surface and Marangoni stresses along the interface
The hybrid method solves the immiscible two-phase flows by using the lattice Boltzmann (LB) colour-gradient model, in which the Marangoni stress is incorporated into the perturbation term to account for the tangential stress created by the non-uniform interfacial tension, and describes the transport of surfactant concentration through a convection–diffusion equation of diffuse-interface form, which is solved in the entire fluid domain by the finite difference (FD) method
Summary
Emulsions, which consist of droplets of one fluid dispersed in another fluid, are common in our daily lives and can be found in the production processes of food, personal care products and pharmaceuticals (Sjöblom 2005; Seemann et al 2011). The surfactant evolution equation of diffuse-interface form does not need additional procedures to extend the surfactant concentration to a neighbourhood of the interface, greatly simplifying the modelling of surfactant dynamics Despite their great success in simulating droplet breakup and coalescence, the volume-of-fluid and level-set methods require either sophisticated interface reconstruction algorithms or unphysical re-initialization processes to represent the interface and surfactant concentration, and the phase-field method yields an interface thickness far greater than its actual value, which may lead to unphysical dissolution of small droplets (Zhang & Wang 2010) and mobility-dependent numerical results (van der Sman & van der Graaf 2008; Magaletti et al 2013). We will study the collision of two equal-sized droplets in a shear flow
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