(678b) Direct Numerical Simulation of Surfactant-Laden Two Phase Flows with Moving Contact Lines Above the Critical Micelle Concentration

Surfactant-laden interfacial flows are a branch of transport phenomena that couples mass transport (of surfactants) with the two-phase hydrodynamics (Slattery et al., 2007; Mavrovouniotis et al., 1989). Such studies are of central relevance to several applications, e.g., detergency, coatings, surfactant replacement therapy, and manufacturing (Miller and Raney, 1993; Jaensson et al., 2021). In the present work, we consider a comprehensive model that accounts for Marangoni stresses (arising from interfacial tension gradients), sorption kinetics (including adsorption/desorption associated with the deformable and liquid-solid interfaces), surface viscosity effects, interfacial and bulk diffusion, and moving contact lines. This model also accounts for situations wherein the surfactant bulk concentration exceeds the critical micelle concentration above which micellar aggregates are expected to form. We use a robust, highly parallelised, hybrid interface-tracking/level-set approach (Shin et al., 2013) to carry out the computations. As an exemplar problem, we use shear-driven flow past a sessile drop with surfactant to showcase the rich physics of surfactant-laden transport phenomena.

Figure 1(a) provides a schematic representation of the surfactant dynamics. The surfactants are differentiated based on their presence in the two-phase flow system: monomers adsorbed on the oil-water and water-solid interfaces, and monomers and micelles present in the bulk. The surfactant species interact via adsorption and desoprtion at and from the interfaces, and via micelle breakup to release monomers and re-formation via monomer aggregation. Importantly, adsorption and desorption at the contact line is also taken into consideration. Such a highly coupled transport process involves a large parametric space: 4 hydrodynamic parameters: Reynolds and Capillary numbers, density and viscosity ratio; 4 contact line dynamics parameters: initial, advancing and receding contact angle, and slip length; 2 interfacial rheological parameters: surface shear and dilatational Boussinesq numbers; 3 interfacial and bulk Peclet numbers for the monomers, and a bulk Peclet number for the micelles; a Fourier number for diffusion of surfactant on the solid surface; 3 Biot numbers describing the desorption timescales of the surfactant from the interface to the bulk, solid to the bulk and solid to the interface by the contact line; a parameter describing the breakup kinetics of the micelles; 3 adsorption ratios for the adsorption timescales from bulk to the interface and the solid, and solid to the interface by the moving contact line; 2 Damkohler numbers describing the adsorption depth at the interface and the solid surface; a surfactant-laden slip constant; an effective micelle size; 5 surfactant chemistry-related parameters: an elasticity parameter, the critical micelle concentration, the initial concentration of the adsorbed surfactant to the surface, solid, in the bulk as monomers and micelles. In total, there are 30 independent parameters which must be specified in order to investigate the coupled transport processes.

The surfactant dynamics in the bulk and the substrate are solved via GMRES. Robust surface differential discretisation on an Lagrangian mesh is used for the surfactant transport on the interface. Its numerical implementation is presented elsewhere (Shin et al., 2018a). A Generalised Navier Boundary condition involving the Navier slip and Youngs’ capillary stresses is implemented for modelling the moving contact line motion (Shin et al., 2018b). The interfacial viscous stresses are solved in a divide-and conquer rule, where we solve the dilatation surface forces via continuum surface force (CSF) and the shear via continuum surface stress method (CSS) (Panda et al., 2024).

The 30 parameters that are discussed are analysed in a heuristic way by setting the hydrodynamic and contact line parameters, and critical micelle concentration constant. Then, we set the initial state to be in equilibrium, thus, setting the initial concentration of surfactants in different phases. Two major groups are categorised as the sticky soil (pinned contact line) and the moving soil (unpinned contact line). The results show that the effect of surfactants delays the breakup of a sticky soil in a pinned that the unpinned contact line due to the Marangoni stresses along the contact line and the interface. We highlight the importance of the of ad/desorption of surfactants at the moving contact line that acc/deccelerates the contact line motion. A snapshot from our direct numerical simulations is shown in figure 1(b) for the interaction of monomers adsorbed on the interface and solid at the moving contact line. Moreover, we show that the micelles act as an additional reservoir for the adsorption of surfactant to the interface, which reduces the interfacial tension and induces rapid deformation of the soil. We also demonstrate the role of the various physical mechanisms outlined above via a systematic parametric study.

References

C. Slattery, L. Sagis, and E.-S. Oh, Interfacial transport phenomena. Springer Science
& Business Media, 2007.

M. Mavrovouniotis, H. Brenner, L. Ting, and D. T. Wasan, “Interfacial fluid mechanics
and transport processes,” Massachusetts Inst. of Tech., Cambridge (USA). Dept. of
Chemical Engineering, Tech. Rep., 1989.

A. Miller and K. H. Raney, “Solubilization—emulsification mechanisms of detergency,”
Colloids and Surfaces A: Physicochemical and Engineering Aspects, vol. 74, no. 2-3, pp.
169–215, 1993.

O. Jaensson, P. D. Anderson, and J. Vermant, “Computational interfacial rheology,”
Journal of Non-Newtonian Fluid Mechanics, vol. 290, p. 104507, 2021.

Shin, J. Chergui, D. Juric, A. Farhaoui, L. Kahouadji, L. S. Tuckerman, and N. P ́erinet,
“Parallel direct numerical simulation of three-dimensional two-phase flows,” in 8th Int.
Conf. on Multiphase Flow, Jeju, Korea, 2013.

Shin, J. Chergui, D. Juric, L. Kahouadji, O. K. Matar, and R. V. Craster, “A hy-
brid interface tracking–level set technique for multiphase flow with soluble surfactant,”
Journal of Computational Physics, vol. 359, pp. 409–435, 2018.

Shin, J. Chergui, and D. Juric, “Direct simulation of multiphase flows with modeling
of dynamic interface contact angle,” Theoretical and Computational Fluid Dynamics,
vol. 32, pp. 655–687, 2018.

Panda, S. Shin, L. Kahouadji, P. Pico, J. Chergui, D. Juric, and O. Matar, “A hybrid
interface-tracking method for surface rheology in multiphase flows,” in preparation, 2024.