Surfactant-Induced Pinning Effects

Updated 4 January 2026

Surfactant-induced pinning phenomena occur when surfactant monolayers accumulate and arrest interface motion, deviating from classical inertio-capillary thinning laws.
The interplay of capillary pressure, Marangoni stresses, and finite surface viscosities governs these effects, resulting in altered scaling exponents and singular stress fields.
Numerical simulations and ultra-high-speed experiments confirm that surfactant pinning critically influences droplet breakup and confined contact line behaviors.

Surfactant-induced pinning phenomena encompass a diverse set of interfacial effects wherein the localized accumulation or rheology of surfactant monolayers arrests or retards the motion of fluid interfaces, such as in the thinning neck of a droplet during pinch-off or at stationary contact lines in confined geometries. These phenomena are governed by the interplay between bulk hydrodynamics, Marangoni stresses due to surfactant concentration gradients, and the finite surface viscosities of the surfactant monolayer, with outcomes that often manifest as altered scaling laws and singular behaviors in stress and pressure fields.

1. Governing Frameworks for Surfactant-Interface Dynamics

Surfactant-induced pinning is typically described by the full three-dimensional Navier–Stokes equations for bulk fluid motion, coupled with boundary conditions at the free surface that account for both Marangoni and surface viscous stresses. For insoluble surfactants such as sodium dodecyl sulfate (SDS), the concentration field on the surface $\Gamma$ evolves according to an advection–diffusion equation. The interfacial stress balance incorporates three principal contributions:

Capillary pressure governed by the local interfacial curvature and surface tension $\sigma(\Gamma)$ ,
Marangoni stresses resulting from gradients in surfactant concentration,
Surface viscous stresses due to nonzero shear $\mu_s$ and dilatational $\kappa_s$ viscosities, quantified by dimensionless surface Ohnesorge numbers $\Oh_s^{(1,2)} = \mu_s,\kappa_s / \sqrt{\rho \sigma_0 R_0^3}$ for a characteristic length $R_0$ .

In confined, quasi-two-dimensional geometries, such as a fluid-filled cavity with pinned contact lines, the lubrication or Stokes approximation becomes applicable, suppressing inertial terms and prescribing the interface shape as nearly flat. Here, the surfactant transport and stress balances lead to spatial patterns of shear and pressure determined by eigenmode expansions and localized singularities near the pinned lines.

2. Pinning at Drop Neck During Pinch-Off: Scaling Laws and Regimes

For a clean interface free of surfactant, inertio-capillary dynamics drive the thinning of a droplet’s neck according to the classical similarity solution $R_{\min}(\tau) \sim \tau^{2/3}$ , where $R_{\min}$ is the minimum neck radius and $\tau = t_\text{pinch} - t$ the time to breakup. Inclusion of surface viscous effects for surfactant-laden interfaces alters this collapse:

Balancing capillary pressure $\sim \sigma_0/R$ against surface viscous stress $\sim \mu_s(V_s/R^2)$ , with stretch rate $V_s \sim R/\tau$ , yields new self-similar length and time scales $R_s \sim \mu_s^{2/3}$ , $\tau_s \sim \mu_s^1$ .
The resulting thinning law transitions from $R_{\min} \sim \tau^{2/3}$ to a steeper surface-viscous regime $R_{\min} \propto \tau^{\gamma} \mu_s^{2/3-\gamma}$ , with $\gamma \approx 1.4$ –$1.5$.
At the onset of this regime, surface viscous stresses balance the capillary pressure, and the thinning slows sharply compared to the inertio-capillary law, constituting a “pinning” of the interface (Ponce-Torres et al., 2020).

In cavity-confined spreading, surfactant transport and hydrodynamics are decomposed into a family of orthogonal modes by eigenfunction expansion. Near each stationary, pinned contact line, the governing biharmonic equation solutions reveal singular structures:

Shear stress at the free surface oscillates logarithmically and spatially in patterns $\tau_w(x) \approx -4A_2 + \operatorname{Re}\{A_{a_1} x^{a_1-2} f''_{a_1}(0)\}$ , with complex modal exponents $a_1 \approx 3.74 + 1.12i$ .
Pressure diverges logarithmically as $p(r, \theta) \sim -8A_2 \pi \ln r$ near the contact line, with $r$ the distance from the corner.
These structures originate from Marangoni compression of the monolayer against the fixed wall, imposing dynamic resistance and pinning at the contact line (Mcnair et al., 2021).

4. Numerical Methods and Experimental Evidence

Numerical integration of the coupled Navier–Stokes and surfactant transport system employs axisymmetric moving domain mapping, Chebyshev spectral collocation for spatial derivatives, and second-order implicit time-stepping on an adaptively remeshed grid. Ultra-high-speed imaging experiments—reaching $5 \times 10^6$ fps and 100 nm pixel resolution—have confirmed the following observations:

DI water droplets (no surfactant) closely follow the $R_{\min} \sim \tau^{2/3}$ law down to micron scales.
SDS-laden droplets at $0.8$ or $2$ cmc deviate systematically from this law once $R_{\min} \lesssim 5\,\mu$ m, evidencing retardation beyond Marangoni resistance alone.
Optimal agreement between theory and experiment is obtained by invoking either surface shear viscosity $\mu_s^* \approx 5 \times 10^{-10}\,\text{Pa}\cdot \text{s}\cdot\text{m}$ or dilatational viscosity $\kappa_s^* \approx 3.5 \times 10^{-9}\,\text{Pa}\cdot \text{s}\cdot\text{m}$ ; these values match upper bounds from microrheology (Ponce-Torres et al., 2020).
In confined cavity models, the predicted singular stress and pressure structures have been numerically resolved and shown to be regularized when a small, finite surface diffusivity is included, which “smears” the singularity on a length scale $O(D)$ .

5. Mechanistic Interpretation of Surfactant-Induced Pinning

In pinch-off scenarios, pinning arises when diverging surface velocity gradients elevate the contributions of surface‐viscous terms $\Oh_s^{(1)}\nabla^S(\nabla^S{\bf v}^S)$ and $\Oh_s^{(2)}\nabla^S(\nabla^S\!\cdot{\bf v}^S)$ to parity with the governing capillary pressure. Marangoni stresses, limited by finite slopes in the equation of state $\sigma(\Gamma)$ , cannot provide sufficient opposition to flow. The surfactant monolayer thus behaves as a viscous 2D sheet resisting the extensional flow at the neck, manifesting as a “creeping collar” that supports the capillary pressure until the thinning rate transitions to the surface-viscous regime.

In cavity spreading, the accumulation of surfactant near a pinned contact line creates strong, localized Marangoni stresses that arrest the advance of the interface, producing pressure and shear singularities indicative of pinning. Regularization by surface diffusion mitigates these singularities, but the effective resistance remains governed by the magnitudes of Marangoni and viscous stresses and the suppressed diffusion.

Key dimensionless groups governing the pinning effect include the Marangoni number $\mathrm{Ma} = E L/(\mu U)$ , comparing interfacial to viscous stresses, the surface Péclet number $\mathrm{Pe} = UL/D_s$ for advection/diffusion ratio, and the aspect ratio $H_{\ast} = H/L$ controlling decay rates and modal structure.

6. Significance and Implications in Interfacial Hydrodynamics

The emergence of surfactant‐induced pinning fundamentally alters the late-stage dynamics of interfacial breakup and spreading, with experimentally measurable consequences even for “nearly-inviscid” surfactants such as SDS. The transition to steeper thinning exponents, the tight quantitative bounds on surface viscosities, and the collapse of measured and computed $R_{\min}(\tau)$ demonstrate that surface rheology is integral at micrometer and submicron scales. In confined flows, the singular and oscillatory stress fields at pinned contact lines signify that interfacial transport and flow in microfluidic systems require consideration of surfactant-induced pinning effects and their regularization mechanisms.

A plausible implication is that in technological applications where control of droplet breakup, contact line mobility, or surfactant spreading is required, accounting for the full spectrum of surfactant-induced pinning phenomena—encompassing both Marangoni and surface viscous contributions—is essential for predictive modeling and design (Ponce-Torres et al., 2020 Mcnair et al., 2021).