A model for transport of soluble surfactants in two-phase flows

TL;DR

This work develops a robust, diffuse-interface model for transport of soluble surfactants in two-phase flows by coupling a three-equation transport system with an Allen-Cahn-based phase-field (ACDI) interface. The model includes interfacial and bulk surfactant concentrations, adsorption/desorption exchange, sharpening flux terms to prevent artificial leakage, and a Langmuir-type two-way coupling to update surface tension via a linearized EOS. Key contributions are the preservation of positivity and discrete mass conservation, avoidance of division by the volume fraction, and compatibility with multiple interface-capturing methods. Numerical demonstrations span 1D adsorption/desorption, selective adsorption, and multi-D scenarios including droplet oscillation and turbulence, showing accurate surfactant confinement, reduced oscillation frequency, and enhanced interfacial area due to inhibited coalescence. The approach provides a practical, non-dissipative framework for simulating soluble surfactants in complex two-phase flows with potential impact on engineering and environmental applications.

Abstract

In this work, we propose a novel transport model for soluble surfactants in two-phase flows. In a two-phase flow, the soluble surfactants can adsorb/desorb from/into the bulk of any of the phases to the interface and can modify the interface properties. This results in sharp gradients in the surfactant concentration on the interface and also between the two phases in the bulk when there is selective adsorption/desorption, presenting a serious challenge for the numerical simulations. To overcome this challenge, we propose a computational model for the transport of soluble surfactants that can model the adsorption and desorption processes accurately. The model is discretized using a central-difference scheme, which leads to a non-dissipative implementation that is crucial for the simulation of turbulent flows. The model is used with the ACDI diffuse-interface method (Jain, 2022), but can also be used with other algebraic-based interface-capturing methods. Furthermore, the provable strengths of the proposed model are: (a) the model maintains the positivity property of the surfactant concentration field, a physical realizability requirement for the simulation of surfactants, when the proposed criterion is satisfied, (b) the proposed model maintains discrete confinement of the interfacial and bulk surfactants and prevents artificial numerical diffusion of the surfactant between the interface and the bulk and between the two phases in the bulk. Finally, we present numerical simulations using the proposed model for both one-dimensional and multi-dimensional cases and assess: the accuracy and robustness of the model, the validity of the positivity property of the scalar concentration field, and the confinement of the surfactant at the interface. We also study the effect of surfactants on an oscillating droplet and on a complex droplet/bubble-laden turbulent flow.

Figures (14)

• Figure 1: A schematic showing a soluble surfactant in molecular and continuum (sharp and diffuse) representations. Here, $\gamma$ represents the two-dimensional interface embedded in a three-dimensional domain $\Omega$, where the dashed line represents the interface. In the molecular picture (not drawn to scale), the dispersed phase molecules are shown along with the surfactant molecules that are adsorbed on the interface and in the bulk of the dispersed phase. In the continuum representations, the colored solid line represents surfactant concentration on the interface, and the gradient filled color represents surfactant concentration in the bulk.
• Figure 2: Schematic representing the interfacial surfactant concentration, $c_i$, the bulk surfactant concentration, $c_b$, and the exchange between these two due to adsorption and desorption of the surfactant. The effect of the sharpening flux terms $f_i$ and $f_b$ in the model are shown. Here, the surfactant is only dissolvable in the bulk of the phase represented by $\phi=1$.
• Figure 3: One-dimensional simulation of adsorption of surfactant onto a stationary droplet interface, showing (a) initial bulk and interfacial concentrations at $t=0$, (b) final bulk concentrations at $t=1$, and (c) final interfacial concentration at $t=1$.
• Figure 4: One-dimensional simulation of desorption of surfactant from a stationary droplet interface into the bulk, showing (a) initial bulk and interfacial concentrations, (b) final bulk concentrations at $t=1$, and (c) final interfacial concentration at $t=1$.
• Figure 5: One-dimensional simulation of selective adsorption of surfactant from the bulk to a stationary droplet interface, showing the final time bulk concentrations at $t=1$, for the (a) symmetric case with $r_{a,l} = 1$, (b) non-symmetric case with $r_{a,1} = 0$, $r_{a,2} = 1$ (case 1), and (c) non-symmetric case with $r_{a,1} = 2$, $r_{a,2} = 1$ (case 2).