Simulations of impinging droplets with surfactant-dependent dynamic contact angle

Abstract

An arbitrary Lagrangian-Eulerian (ALE) finite element scheme for computations of soluble surfactant droplet impingement on a horizontal surface is presented. The numerical scheme solves the time-dependent Navier-Stokes equations for the fluid flow, scalar convection-diffusion equation for the surfactant transport in the bulk phase, and simultaneously, surface evolution equations for the surfactants on the free surface and on the liquid-solid interface. The effects of surfactants on the flow dynamics are included into the model through the surfactant-dependent surface tension and dynamic contact angle. In particular, the dynamic contact angle (θ d ) of the droplet is defined in terms of the surfactant concentration at the contact line and the equilibrium contact angle (θ 0 e ) of the clean surface using the nonlinear equation of state for surface tension. Further, the surface forces are included in the model as the surface divergence of the surface stress tensor that allows to incorporate the Marangoni effects without calculating the surface gradient of the surfactant concentration on the free surface. In addition to a mesh convergence study and validation of the numerical results with experiments, the effects of adsorption and desorption surfactant coefficients on the flow dynamics in wetting, partially wetting and non-wetting droplets are studied in detail. It is observed that the effect of surfactants are more in wetting droplets than in the non-wetting droplets. Further, the presence of surfactants at the contact line reduces the equilibrium contact angle further when θ 0 e is less than 90• , and increases it further when θ 0 e is greater than 90• . The presence of surfactants has no effect on the contact angle, when θ 0 e = 90• . The numerical study clearly demonstrates that the surfactant-dependent contact angle has to be considered, in addition to the Marangoni effect, in order to study the flow dynamics and the equilibrium states of surfactant droplet impingement accurately. The proposed numerical scheme guarantees the conservation of fluid mass and of the surfactant mass well.

Numerical Study of Droplet Dynamics on a Solid Surface with Insoluble Surfactants

“…In addition, an arbitrary Lagrangian−Eulerian finite element scheme for computations of soluble surfactant droplet impingement on a horizontal surface was presented by Ganesan. 16 In his work, the dynamic contact angle was expressed as a function of the static contact angle and the local surfactant concentration. Clearly, these studies are all based on traditional computational fluid dynamics (CFD) methods, which solve the conservation equations of macroscopic properties (e.g., mass and momentum) numerically.…”

“…Their numerical results revealed that the overall behavior of the droplet is determined by the magnitudes of the capillary number and the viscosity ratio, and also by the sensitivity of the interfacial tension to variations in surfactant concentration. In addition, an arbitrary Lagrangian–Eulerian finite element scheme for computations of soluble surfactant droplet impingement on a horizontal surface was presented by Ganesan . In his work, the dynamic contact angle was expressed as a function of the static contact angle and the local surfactant concentration.…”

On the Navier-Slip Boundary Condition for Computations of Impinging Droplets

“…where P ν F = I − ν F ⊗ ν F is the tangential projection. The surface stress tensor, S Γ F [27] can be obtained as…”

Surfactant-dependent contact line dynamics and droplet spreading on textured substrates: derivations and computations

“…There are many recent theoretical and numerical studies on this subject, including insoluble or soluble surfactant; c.f. [13,19,18,16,30,27,2,9,29,17,10,26,8] and the references therein.…”