Kinetic theory of coupled binary-fluid-surfactant systems

TL;DR

This work addresses the challenge of integrating microscopic surfactant physics with macroscopic hydrodynamics in binary-fluid systems. Using Rayleigh's minimum energy dissipation principle, the authors derive overdamped dynamics for surfactants modeled as dumbbells and systematically coarse-grain to a continuum theory for the binary fluid order parameter $\phi$, surfactant concentration $c$, polarization $\mathbf{p}$, and fluid velocity $\mathbf{v}$, all from a single mesoscopic free energy $F[\phi,c,\mathbf{p}]$. The resulting framework yields natural Marangoni-like flows and captures key surfactant phenomena, including surface-tension reduction and emulsion stabilization, without ad hoc stabilizing terms. The model is validated by perturbative analysis of a planar interface and direct numerical simulations, establishing thermodynamic consistency with Gibbs adsorption and Henry’s law. The formalism provides a versatile platform for exploring interfacial phenomena and can be extended to active interfacial systems and self-propelled particles at fluid interfaces.

Abstract

We derive a self-consistent hydrodynamic theory of coupled binary-fluid-surfactant systems from the underlying microscopic physics using Rayleigh's variational principle. At the microscopic level, surfactant molecules are modelled as dumbbells that exert forces and torques on the fluid and interface while undergoing Brownian motion. We obtain the overdamped stochastic dynamics of these particles from a Rayleighian dissipation functional, which we then coarse-grain to derive a set of continuum equations governing the surfactant concentration, orientation, and the fluid density and velocity. This approach introduces a polarization field, representing the average orientation of surfactants, and yields a mesoscopic free energy functional from which all governing equations are consistently derived. The resulting model accurately captures key surfactant phenomena, including surface tension reduction and droplet stabilization, as confirmed by both perturbation theory and numerical simulations.

Figures (4)

• Figure 1: Left: Schematic diagram illustrating how surfactants (black) are absorbed perpendicularly at the interface between two phases, e.g., water and oil phase (green and yellow respectively). Right: Diagram showing the surfactant molecule modelled as a dumbbell, adsorbed into a diffuse water-oil interface, with 'head' H, 'tail' T, rod of length $\ell$, centre of mass $\bm{r}_i$, and orientation vector $\hat{\bm{e}}_i$, directed from 'tail' to 'head'. The fluid exerts a force on each mass point, $\bm{F}_\mathrm{H}$ and $\bm{F}_\mathrm{T}$, due to the hydrophilic/hydrophobic attraction between said mass points and the corresponding fluid phases. The binary fluid order parameter $\phi(\bm{r},t)$ has values between $1$ and $-1$ to represent the two fluid phases.
• Figure 2: (a) A graph showing the analytical (line) and numerical (symbols) solutions for the fluid field $\phi(x)$ with the leading order $\phi_0(x) = \tanh{x}$ removed, at equilibrium for a variety of $\varepsilon$ and $C_0$ values. (b) A graph showing the analytical (line) and numerical (symbols) solutions for concentration, $c(x)$ with the leading order $c_0(x)=C_0$ removed, at equilibrium for a range of $\varepsilon$ and $C_0$ values. (c) A graph showing the analytical (line) and numerical (symbols) solutions for the polarization field $p_x(x,t)$, at equilibrium for a range of $\varepsilon$ and $C_0$ values. Parameters used: $\beta=2.0$, $B=0.5$, $M=3$, $\gamma_t=\gamma_r=0.01$.
• Figure 3: (a) Effective surface tension divided by the bare surface tension $\sigma_{\text{eff}}/\sigma{\text{I}}$ as a function of bulk surfactant concentration $C_0$ for different values of coupling strength $\varepsilon$ and fixed $\beta=1$. Dashed lines indicate the leading-order prediction from the Gibbs isotherm. (b) Equilibrium configuration of a planar interface located at $x=0$. Black arrows show the polarization field $\boldsymbol{p}$ which aligns perpendicular to the interface. (c) Under strong shear flow, the polarization $\boldsymbol{p}$ field becomes tilted and is no longer perpendicular to the interface. Blue arrows indicate the fluid velocity $\boldsymbol{v}$. Parameters used: $\beta=1.0$, $B=1.0$, $M=1.0$, $\gamma_t=\gamma_r=0.1$, $\eta=1.0$, and $\varepsilon=0.5$.
• Figure 4: Plots of binary fluid volume fraction $\phi(\bm{r},t)$ for bare emulsion (left column) and surfactant-containing emulsion (right column) at different time steps (rows) with values $t = 10, 110$ and $600$ being the first, second and third rows respectively. The black arrows on the right column indicate the polarization or average orientation of the surfactant molecules, $\bm{p}(\bm{r},t)$. The graphs show that the presence of the surfactants suppresses droplet coalescence and full phase separation. Parameters used: $(\varepsilon=0,C_0=0)$ (left column) and $(\varepsilon=1.5,C_0=0.244)$ (right column). Other parameters: $\beta=2$, $B=0.5$, $M=3$, and $\gamma_t=\gamma_r=0.01$.