A hybrid lattice Boltzmann and finite difference method for droplet dynamics with insoluble surfactants

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.

Interfacial dynamics with soluble surfactants: A phase-field two-phase flow model with variable densities

“…Surfactants play a crucial role in everyday life and many industrial processes (Liu and Zhang, 2010), such as the cleanser essence, the crude oil recovery and pharmaceutical materials, thus having an understanding of their behavior is a necessity. Numerical simulation is taking an increasingly significant position in investigating interfacial phenomena, as it can provide easier access to some quantities such as surfactant concentration, pressure and velocity, which are difficult to measure experimentally (Liu et al, 2018;Yang et al, 2019). However, the computational modeling of interfacial dynamics with surfactants remains a challenging task.…”

Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants

“…Numerical modelling is taking an increasingly significant position in investigating the droplet dynamics on a solid surface with surfactants, as it can provide easier access to some quantities such as surfactant concentration, pressure and velocity, which are difficult to measure experimentally (Liu et al 2018). However, the efficient and accurate computational modelling of contact line dynamics with surfactants remains a challenging task.…”

“…The first challenge comes from the modelling of interfacial dynamics with surfactants. The presence of surfactants brings some difficulties to simulations (Booty & Siegel 2010;Zhang et al 2014;Liu et al 2018), including (1) a suitable equation of state is needed to account for the effect of surfactants in reducing the interfacial tension;…”

“…To address these difficulties, numerous numerical methods have been developed, such as the level set method (Xu et al 2012;Titta et al 2018), the volume of fluid method (Alke & Bothe 2009;James & Lowengrub 2004), the front tracking method (Zhang et al 2006;Muradoglu & Tryggvason 2008), the immersed boundary method (Lai et al 2010) and the lattice Boltzmann method (Van der Sman & Van der Graaf 2006;Zhang et al 2015;Zhao et al 2018;Wei et al 2019). Although these methods have made great progresses in simulating interfacial flows with surfactants, they still suffer from several drawbacks (Liu et al 2018), including (1) the level set and VOF methods require either unphysical re-initialization processes or complex interface reconstruction algorithms to represent the interface and surfactant concentration; (2) the front tracking and immersed boundary methods have difficulty in dealing with large topological changes, for example, the droplet breakup and coalescence; (3) most lattice Boltzmann models do not consider the Marangoni stress (Liu et al 2018), which is unphysical and could have an important impact on the droplet dynamics. The second challenge is the MCL problem.…”

A versatile lattice Boltzmann model for immiscible ternary fluid flows

“…Among these approaches, the front-tracking [16], volume-of-fluid (VOF) [17], level-set [18,19,20] and phase-field [21,22,23,24,25,26] methods are commonly used. However, the front-tracking method is not suitable for simulating interface breakup and coalescence; the VOF and level-set methods require either sophisticated interface reconstruction algorithms or unphysical re-initialization processes to represent the interfaces; and the phase-field method yields an interface thickness far greater than its actual value, which may lead to unphysical dissolution of small droplets and mobility-dependent numerical results [27]. It still remains an open question for the phase-field method to choose an optimal mobility, even for a two-phase flow problem [28].…”