Moving contact line problem: Advances and perspectives

The solid-liquid interface, which is ubiquitous in nature and our daily life, plays fundamental roles in a variety of physical-chemical-biological- mechanical phenomena, for example in lubrication, crystal growth, and many biological reactions that govern the building of human body and the functioning of brain. A surge of interests in the moving contact line （MCL） problem, which is still going on today, can be traced back to 1970s primarily because of the exis- tence of the ＂Huh-Scriven paradox＂. This paper, mainly from a solid mechanics perspective, describes very briefly the multidisciplinary nature of the MCL problem, then summarizes some major advances in this exciting research area, and some future directions are presented.

Simulation of flows with moving contact lines on a dual-resolution Cartesian grid using a diffuse-interface immersed-boundary method

In this paper, we investigate flows with moving contact lines on curved solid walls on a dual-resolution grid using a diffuse-interface immersed-boundary（DIIB） method. The dual-resolution grid, on which the flows are solved on a coarse mesh while the interface is resolved on a fine mesh, was shown to significantly improve the computational efficiency when simulating multiphase flows. On the other hand, the DIIB method is able to resolve dynamic wetting on curved substrates on a Cartesian grid, but it usually requires a mesh of high resolution in the vicinity of a moving contact line to resolve the local flow. In the present study, we couple the DIIB method with the dual-resolution grid, to improve the interface resolution for flows with moving contact lines on curved solid walls at an affordable cost. The dynamic behavior of moving contact lines is validated by studying drop spreading, and the numerical results suggest that the effective slip length λ_n can be approximated by 1.9Cn, where Cn is a dimensionless measure of the thickness of the diffuse interface. We also apply the method to drop impact onto a convex substrate, and the results on the dual-resolution grid are in good agreement with those on a single-resolution grid. It shows that the axisymmetric simulations using the DIIB method on the dual-resolution grid saves nearly 60% of the computational time compared with that on a single-resolution grid.

A Level Set Method for the Simulation of Moving Contact Lines in Three Dimensions

We propose an efficient numerical method for the simulation of the twophase flows with moving contact lines in three dimensions.The mathematical model consists of the incompressible Navier-Stokes equations for the two immiscible fluids with the standard interface conditions,the Navier slip condition along the solid wall,and a contact angle condition(Ren et al.(2010)[28]).In the numerical method,the governing equations for the fluid dynamics are coupledwith an advection equation for a level-set function.The latter models the dynamics of the fluid interface.Following the standard practice,the interface conditions are taken into account by introducing a singular force on the interface in themomentum equation.This results in a single set of governing equations in the whole fluid domain.Similarly,the contact angle condition is imposed by introducing a singular force,which acts in the normal direction of the contact line,into theNavier slip condition.The newboundary condition,which unifies the Navier slip condition and the contact angle condition,is imposed along the solid wall.The model is solved using the finite difference method.Numerical results are presented for the spreading of a droplet on both homogeneous and inhomogeneous solid walls,as well as the dynamics of a droplet on an inclined plate under gravity.

Numerical Simulation of Moving Contact Lines with Surfactant by Immersed Boundary Method

In this paper,we present an immersed boundary method for simulating moving contact lines with surfactant.The governing equations are the incompressible Navier-Stokes equations with the usual mixture of Eulerian fluid variables and Lagrangian interfacial markers.The immersed boundary force has two components:one from the nonhomogeneous surface tension determined by the distribution of surfactant along the fluid interface,and the other from unbalanced Young’s force at the moving contact lines.An artificial tangential velocity has been added to the Lagrangian markers to ensure that the markers are uniformly distributed at all times.The corresponding modified surfactant equation is solved in a way such that the total surfactant mass is conserved.Numerical experiments including convergence analysis are carefully conducted.The effect of the surfactant on the motion of hydrophilic and hydrophobic drops are investigated in detail.

Molecular Hydrodynamics of the Moving Contact Line in Two-Phase Immiscible Flows

The no-slip boundary condition,i.e.,zero fluid velocity relative to the solid at the fluid-solid interface,has been very successful in describing many macroscopic flows.A problem of principle arises when the no-slip boundary condition is used to model the hydrodynamics of immiscible-fluid displacement in the vicinity of the moving contact line,where the interface separating two immiscible fluids intersects the solid wall.Decades ago it was already known that the moving contact line is incompatible with the no-slip boundary condition,since the latter would imply infinite dissipation due to a non-integrable singularity in the stress near the contact line.In this paper we first present an introductory review of the problem.We then present a detailed review of our recent results on the contact-line motion in immiscible two-phase flow,from molecular dynamics(MD)simulations to continuum hydrodynamics calculations.Through extensive MD studies and detailed analysis,we have uncovered the slip boundary condition governing the moving contact line,denoted the generalized Navier boundary condition.We have used this discovery to formulate a continuum hydrodynamic model whose predictions are in remarkable quantitative agreement with the MD simulation results down to the molecular scale.These results serve to affirm the validity of the generalized Navier boundary condition,as well as to open up the possibility of continuum hydrodynamic calculations of immiscible flows that are physically meaningful at the molecular level.

Numerical Simulation for Moving Contact Line with Continuous Finite Element Schemes

In this paper,we compute a phase field(diffuse interface)model of CahnHilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids.The generalized Navier boundary condition proposed by Qian et al.[1]is adopted here.We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time.We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable.Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid.We also derive the discrete energy law for the droplet displacement case,which is slightly different due to the boundary conditions.The accuracy and stability of the scheme are validated by examples,results and estimate order.

Stick-Slip Motion of Moving Contact Line on Chemically Patterned Surfaces

Based on our continuum hydrodynamic model for immiscible two-phaseflows at solid surfaces, the stick-slip motion has been predicted for moving contactline at chemically patterned surfaces [Wang et al., J. Fluid Mech., 605 (2008), pp. 59-78].In this paper we show that the continuum predictions can be quantitatively verifiedby molecular dynamics (MD) simulations. Our MD simulations are carried out fortwo immiscible Lennard-Jones fluids confined by two planar solid walls in Poiseuilleflow geometry. In particular, one solid surface is chemically patterned with alternating stripes. For comparison, the continuum model is numerically solved using material parameters directly measured in MD simulations. From oscillatory fluid-fluidinterface to intermittent stick-slip motion of moving contact line, we have quantitativeagreement between the continuum and MD results. This agreement is attributed tothe accurate description down to molecular scale by the generalized Navier boundary condition in our continuum model. Numerical results are also presented for therelaxational dynamics of fluid-fluid interface, in agreement with a theoretical analysisbased on the Onsager principle of minimum energy dissipation.

DECOUPLED, ENERGY STABLE SCHEME FOR HYDRODYNAMIC ALLEN-CAHN PHASE FIELD MOVING CONTACT LINE MODEL

In this paper, we present an efficient energy stable scheme to solve a phase field model incorporating contact line condition. Instead of the usually used Cahn-Hilliard type phase equation, we adopt the Allen-Cahn type phase field model with the static contact line boundary condition that coupled with incompressible Navier-Stokes equations with Navier boundary condition. The projection method is used to deal with the Navier-Stokes equa- tions and an auxiliary function is introduced for the non-convex Ginzburg-Landau bulk potential. We show that the scheme is linear, decoupled and energy stable. Moreover, we prove that fully discrete scheme is also energy stable. An efficient finite element spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed scheme is very efficient and accurate.

A θ-L APPROACH FOR SOLVING SOLID-STATE DEWETTING PROBLEMS

We propose a θ-L approach for solving a sharp-interface model about simulating solid-state dewetting of thin films with isotropic/weakly anisotropic surface energies.The sharp-interface model is governed by surface diffusion and contact line migration.For solving the model,traditional numerical methods usually suffer from the severe stability constraint and/or the mesh distribution trouble.In the θ-L approach,we introduce a useful tangential velocity along the evolving interface and utilize a new set of variables(i.e.,the tangential angle 6 and the total length L of the interface curve),so that it not only could reduce the stiffness resulted from the surface tension,but also could ensure the mesh equidistri-bution property during the evolution.Furthermore,it can achieve second-order accuracy when implemented by a semi-implicit linear finite element method.Numerical results are reported to demonstrate that the proposed θ-L approach is efficient and accurate.

A Variational Model for Two-Phase Immiscible Electroosmotic Flow at Solid Surfaces

We develop a continuum hydrodynamic model for two-phase immiscible flows that involve electroosmotic effect in an electrolyte and moving contact line at solid surfaces.The model is derived through a variational approach based on the Onsager principle of minimum energy dissipation.This approach was first presented in the derivation of a continuum hydrodynamic model for moving contact line in neutral two-phase immiscible flows(Qian,Wang,and Sheng,J.Fluid Mech.564,333-360(2006)).Physically,the electroosmotic effect can be formulated by the Onsager principle as well in the linear response regime.Therefore,the same variational approach is applied here to the derivation of the continuum hydrodynamic model for charged two-phase immiscible flows where one fluid component is an electrolyte exhibiting electroosmotic effect on a charged surface.A phase field is employed to model the diffuse interface between two immiscible fluid components,one being the electrolyte and the other a nonconductive fluid,both allowed to slip at solid surfaces.Our model consists of the incompressible Navier-Stokes equation for momentum transport,the Nernst-Planck equation for ion transport,the Cahn-Hilliard phase-field equation for interface motion,and the Poisson equation for electric potential,along with all the necessary boundary conditions.In particular,all the dynamic boundary conditions at solid surfaces,including the generalized Navier boundary condition for slip,are derived together with the equations of motion in the bulk region.Numerical examples in two-dimensional space,which involve overlapped electric double layer fields,have been presented to demonstrate the validity and applicability of the model,and a few salient features of the two-phase immiscible electroosmotic flows at solid surface.The wall slip in the vicinity ofmoving contact line and the Smoluchowski slip in the electric double layer are both investigated.