Diffusiophoresis of a non-polar fluid droplet laden with soluble ionic surfactants

Abstract

We investigate the diffusiophoresis of a non-polarizable droplet laden with soluble ionic surfactant, for which the surface charge arises from adsorption of surfactant at the fluid-fluid interface. Unlike previous studies that assume either a fixed surface charge or instantaneous equilibrium between the interface and the adjacent electrolyte, we formulate the interfacial transport based on the mass-balance framework incorporating Langmuir adsorption-desorption kinetics and finite surface diffusivity. The coupled electrokinetic problem is solved using a perturbation approach. Analytical expressions for the droplet mobility and interfacial velocity are derived for insoluble surfactants. We demonstrate that assuming uniform, immobile surface charge leads to unphysical predictions, including negative chemiphoresis and singular mobility, whereas allowing the surface charge to evolve through interfacial surfactant redistribution yields continuous and physically consistent droplet diffusiophoresis. Interfacial kinetic exchange is found to play a central role. Increasing the desorption rate enhances surfactant redistribution and Marangoni stress, weakens the negative mobility, reverses the direction of motion through competition between electrophoretic and chemiphoretic contributions, and subsequently leads to a strong enhancement of positive mobility before eventual saturation in the transport-limited regime. The dependence of mobility on viscosity ratio and electrolyte composition of different salts further reveals how mixed electrolytes provides a robust means of tuning droplet motion. This study highlights the critical role of finite-rate surfactant dynamics and interfacial transport in determining the diffusiophoresis of fluid particles, with implications for manipulating droplets in microfluidic and varying-salinity environments.

Figures (12)

• Figure 1: Problem description and spherical coordinate system for the diffusiophoresis of a surfactant-laden liquid droplet in an electrolyte solution subjected to an ionic concentration gradient.
• Figure 2: Comparison of diffusiophoretic mobility $\mu_{D}$ with (a) fan2022diffusiophoresis for a non-polarizable droplet with uniform surfactant distribution at the interface for different viscosity ratio $\mu_{r}=0.01,0.1,0.5,1,10,100$ with $\Gamma^{0}=0.024~(\sigma^{0}=-10.53)$; (b) the analytical solution (\ref{['eq:mobex_1']}) for different $\Gamma^{0}=0.01,0.02,0.03,0.05$ at $\kappa a=10$ and (c) with the analytical solution (\ref{['eq:tla10_2']}) for different $\kappa a=30,50,100,150$ and $\mu_r=0.1$. Here, NaCl is considered as the electrolyte, $\Gamma^{\infty}=1~\rm nm^{-2}~(Ma=438.3)$ and the diffusion coefficient $D_{s}=3.94\times10^{-10}~\rm m^{2}/s$. In (a), square symbols, fan2022diffusiophoresis and solid lines, present results with uniform surfactant concentration. In (b,c), solid lines, analytical solution and dashed lines, numerical simulations.
• Figure 3: Variation of the diffusiophoretic mobility $\mu_{D}$ as a function of $\kappa a$ at (a) $\Gamma^{0}=0.01~(\sigma^{0}=-4.383)$, (b) $\Gamma^{0}=0.025~(\sigma^{0}=-10.958)$ and (c) $\Gamma^{0}=0.05~(\sigma^{0}=-21.915)$ for different values of $\mu_{r}=0.01,0.1,1,10,100$ when $\Gamma^{\infty}=1\rm ~nm^{-2}~(Ma=438.3)$ and $k_d=0$ in NaCl electrolyte solution. Here, dashed lines, uniform surfactant ($\delta\Gamma=0$); solid lines, non-uniform surfactant ($\delta\Gamma\neq0$). Dashed line with symbols, D-H based analytic solution (\ref{['eq:mobex_1_uniform']}) for uniform surfactant; solid lines with symbols, D-H based analytic solution (\ref{['eq:mobex_1']}) for non-uniform surfactant. Arrows indicate increasing values of $\mu_r$ and $D_{s}=10^{-9}~\rm m^{2}/s$.
• Figure 4: Variation of (a) the interfacial velocity at $\kappa a=3$ and $\Gamma^{0}=0.01$ for different $\mu_{r}=0.01,1,10$; (b) $\mu_{D}$ as a function of $\Gamma^{0}$ for different $\kappa a=50,100,150$ at $\mu_{r}=0.1$ (inset figure shows the complete range for uniform surfactant) and (c) $\mu_{D}$ as a function of $\Gamma^{0}$ for different salt (NaCl, KCl and HCl) at $\kappa a=1$ and $\mu_{r}=0.1$. Here, $\Gamma^{\infty}=1\rm ~nm^{-2}~(Ma=438.3)$, $k_d=0$, $D_{s}=10^{-9}~\rm m^{2}/s$, and dashed lines, uniform surfactant ($\delta\Gamma=0$); solid lines, non-uniform surfactant ($\delta\Gamma\neq0$). In (a,b), the electrolyte is NaCl.
• Figure 5: (a) Variation of mobility $\mu_{D}$ as a function of $\Gamma^{0}$ for different salts NaCl, KCl and HCl at $\kappa a=80$ when $\mu_{r}=0.1$. Variation of mobility $\mu_{D}$ and interfacial velocity $u_{s}$ as a function of higher values of $\Gamma^{0}$ at $\kappa a=50$ when $\mu_{r}=0.01$ in NaCl electrolyte solution for (b) uniform surfactant ($\delta\Gamma=0$) and (c) non-uniform surfactant ($\delta\Gamma\neq0$). Here, $\Gamma^{\infty}=1\rm ~nm^{-2}~(Ma=438.3)$, $k_d=0$, and $D_{s}=10^{-9}~\rm m^{2}/s$. In (a), dashed lines, uniform surfactant; solid lines, non-uniform surfactant. In (b,c), blue lines, interfacial velocity ($u_{s}$); red lines, mobility ($\mu_{D}$).