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

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.

💡 Research Summary

This paper presents a comprehensive theoretical study of the diffusiophoretic motion of a non‑polarizable liquid droplet that carries soluble ionic surfactants. The droplet’s surface charge originates from the adsorption of surfactant ions at the fluid–fluid interface, and the authors explicitly model the interfacial transport using a mass‑balance framework that incorporates Langmuir adsorption–desorption kinetics together with finite surface diffusivity. The bulk electrolyte is described by the Stokes equations coupled to the Nernst–Planck ion‑transport equations and Poisson’s equation for the electrostatic potential.

To make the problem tractable, the authors adopt the thin‑double‑layer limit (κa≫1) and the Debye–Hückel linearisation, and treat the imposed external ionic concentration gradient ∇n∞ as a small perturbation parameter ε. A first‑order perturbation expansion yields analytical expressions for the droplet’s diffusiophoretic mobility U_D and the interfacial slip velocity. The mobility consists of two additive contributions: (i) an electrophoretic term proportional to the tangential Maxwell stress σ∂θψ, where σ = z_s Ma Γ is the surface charge density generated by the adsorbed surfactant, and (ii) a chemiphoretic‑Marangoni term proportional to the surface‑tension gradient (Ma∇_sγ) that arises from non‑uniform surfactant coverage.

A key finding is that the traditional assumption of a uniform, immobile surface charge leads to unphysical predictions: negative chemiphoresis, singular (infinite) mobility, and discontinuities as the surface charge density increases. By allowing the surface charge to evolve through surfactant redistribution, the model predicts a smooth, physically consistent dependence of mobility on Γ.

The analysis of parameter space reveals several controlling mechanisms:

1. Desorption rate (k_d) – Increasing k_d enhances surfactant exchange with the bulk, promotes surface charge redistribution, and strengthens Marangoni stresses. This weakens the negative electrophoretic contribution, eventually reversing the direction of motion from opposite to the concentration gradient (negative mobility) to the same direction (positive mobility). At high k_d the mobility saturates, indicating a transport‑limited regime.

2. Surface diffusivity (D_s) – Finite D_s smooths out concentration gradients along the interface, further regularising the mobility and preventing singular behaviour.

3. Viscosity ratio (μ_r = μ_drop/μ_bulk) – Low μ_r (a more fluid droplet) amplifies the chemiphoretic component, favouring negative mobility, whereas high μ_r enhances the electrophoretic component, leading to stronger positive mobility.

4. Electrolyte composition – Different salts (e.g., NaCl vs KCl) have distinct ionic diffusivities, which modify the dimensionless parameter λ governing the electrophoretic term. Mixing salts allows fine‑tuning of λ, thereby adjusting both the magnitude and sign of the mobility.

The analytical results agree closely with full numerical solutions for κa≥10, confirming the validity of the thin‑layer and linear‑potential approximations. The study thus bridges a gap in the literature: while previous works on droplet electrophoresis incorporated finite‑rate interfacial kinetics, no comparable framework existed for diffusiophoresis.

Implications are broad. In microfluidic platforms, one can manipulate droplets by designing appropriate surfactant chemistries, adjusting desorption kinetics (e.g., via temperature or pH), and selecting mixed electrolytes to achieve desired speeds and directions. In environmental or biomedical contexts, the findings suggest strategies for steering oil droplets in salinity‑gradient reservoirs or delivering drug‑laden droplets through tissues with spatially varying ionic strength.

Overall, the paper demonstrates that finite‑rate surfactant dynamics and Marangoni stresses are essential for accurately predicting and controlling diffusiophoretic motion of fluid particles, providing a robust theoretical foundation for future experimental and application‑driven work.