The wetting problem of fluids on solid surfaces. Part 1: the dynamics of contact lines

The understanding of the spreading of liquids on solid surfaces is an important challenge for contemporary physics. Today, the motion of the contact line formed at the intersection of two immiscible fluids and a solid is still subject to dispute. In this paper, a new picture of the dynamics of wetting is offered through an example of non-Newtonian slow liquid movements. The kinematics of liquids at the contact line and equations of motion are revisited. Adherence conditions are required except at the contact line. Consequently, for each fluid, the velocity field is multivalued at the contact line and generates an equivalent concept of line friction but stresses and viscous dissipation remain bounded. A Young-Dupré equation for the apparent dynamic contact angle between the interface and solid surface depending on the movements of the fluid near the contact line is proposed.Received: 5 June 2001, Accepted: 24 May 2003, Published online: 29 July 2003PACS: 47.17., 47.50, 66.20, 68.08, 83.50     

Recent progress in the moving contact line problem: a review

As pointed out long ago by Laplace, viscosity may become a large perturbation to capillary phenomena, especially close to solid surfaces where molecules may stick. A spectacular consequence of this is the impossibility for a triple line to move on a solid if the liquid/vapor interface is considered as a material surface and if the usual no slip boundary condition is enforced. As shown recently this specific phenomenon of contact line motion can be described by coupled van der Waals and fluid equations, yielding a rational theory that is divergence free and consistent with the equilibrium results. Far from the triple line, the equations of fluid mechanics are recovered in their usual form. In this approach, the contact line move close to the solid by evaporation or condensation, which requires (for evaporation) the molecules to jump above a high potential barrier on their way from the liquid to the vapor. An Arrhenius factor makes this process intrinsically slow, compared to molecular speeds. For (realistic) very small Arrhenius factors, the motion of the triple line induces a dynamical change of the functions in the van der Waals equations. This may lead to dynamical wetting and dewetting transitions, that is, to a change of the contact angle from a finite to a zero value or conversely. The dynamical wetting transition has been observed in liquids flowing down a plate (see Blake and Ruschak, Nature 282 (1979) 489–491) cusps on the contact line appear when it recedes faster than the speed of transition. Similar ideas account well also for the known sensitivity of contact line mobility to vapor pressure. To cite this article: Y. Pomeau, C. R. Mecanique 330 (2002) 207–222.     

A height function based momentum balance model to simulate contact angle dynamics with hysteresis

We report a combined method to deal with the contact angle dynamics with hysteresis. The momentum balance model is applied to obtain the transient contact angle by balancing the inertia and the capillary force where the curvatures are estimated by the height function at the contact line. This integrated approach provides a great convenience that no need for a prior knowledge of the contact angle, and possesses the good property of the sharp interface approximation. In order to facilitate the wider use of the present method, we incorporate a dynamic contact line model to estimate the contact angle when the contact line starts to move. This combination plus the hysteresis region will take the ability to solve most of problems related to wetting with more physical sense. This proposed model is finally validated by the droplet equilibrium, spreading, and sliding tests.     

Numerical simulation of dynamic contact angle using a force based formulation

A method for the numerical simulation of the dynamic response of the contact angle is presented and its development discussed. The proposed method was developed within a level-set framework by modelling forced capillary flows and it is based on the introduction of a force function to capture the balance of forces in the contact region between solid boundaries and a diffuse free-surface fluid interface. The proposed approach allows the system to define its own dynamic contact angle and its own contact line dynamics, without introducing numerical discontinuities such as locally prescribed angles or slip-length. The method was developed through numerical testing and comparisons with experimental and empirical models reported in the literature. These showed the validity of the proposed approach, which was able to reproduce the experimental correlation between the capillary number and the dynamic contact angle reported by [R.L. Hoffman, Study of advancing interface. 1. Interface shape in liquid-gas systems, J. Colloid Interf. Sci. 50 (1975) 228-241]. By using a single constitutive model for the force function, the simulation results of the dynamic contact angle showed an excellent agreement with the values predicted by Jiang’s empirical equation [T.S. Jiang, O.H. Soo-Gun, J.C. Slattery, Correlation for dynamic contact angle, J. Colloid Interf. Sci. 69 (1979) 74-77] through different material properties and flow speeds. The proposed approach also demonstrated the ability to work with meshes of low resolution.     

Some generic capillary-driven flows

This paper deals with numerical simulations of some capillary-driven flows. The focus is on the wetting phenomenon in sintering-like flows and in the imbibition of liquids into a porous medium. The wetting phenomenon is modeled using the coupled Cahn–Hilliard/Navier–Stokes system. The Cahn–Hilliard equation is treated as a system where the chemical potential is solved first followed by the composition. The equations are discretised in space using piecewise linear functions. Adaptive finite element method is implemented with an ad hoc error criterion that ensures mesh resolution along the vicinity of the interface. In the 3D case we use parallel adaptive finite element method. First, a basic wetting of a liquid drop on a solid surface is shown and is established the independence of the dynamic contact angle on the interface width. In addition, the dependence of the dynamic contact angle on the Capillary number is matched with experimental data. Next, some generic sintering-like flows with a fixed matrix is presented. Different geometries in 2D and 3D are considered. We observed rapid wetting, precursor films, coalescence, breakup of melt drops as well as pore migration and elimination that are all microstructural characteristics of a liquid phase sintering. Finally, the effect of equilibrium contact angles on imbibition of liquid into a porous medium is studied.     

梯度润湿表面上液滴定向迁移及合并行为的格子Boltzmann模拟

采用改进的格子Boltzmann方法，对梯度润湿性表面上液滴的定向迁移及合并行为进行了数值模拟，该模型在精度和稳定性上都有很大改善，同时，研究了梯度润湿性表面上液滴定向迁移和合并的动力学特性，并对液滴尺寸及润湿梯度对液滴动力学特性的影响规律进行了分析。数值结果表明，液滴在梯度润湿性表面运动时会发生形变，且动态接触角逐渐减小。润湿梯度对液滴定向迁移行为有显著影响，润湿梯度越大，液滴左右侧接触线位移越大，润湿长度增加越快。但是液滴尺寸对接触线位移影响较小。润湿梯度对液桥宽度基本无影响，但对液滴初始合并时间有显著影响。     

Motion of a three-phase contact line along a wet surface

The motion of the contact line in gas-liquid-solid systems is theoretically investigated for small values of the capillary number and Reynolds number. The possible existence on the solid substrate of a residual microscopic film formed by adsorbed liquid molecules is taken into account and the spreading characteristics of the liquid on dry and wet substrates are compared. It is shown that, in accordance with the experimental data, in the model employed the motion of the liquid during wetting is rolling motion, and that the increase in the dynamic contact angle is slower for a wet than for a dry substrate. The maximum dynamic contact angle is much less than 180°. The flow structure in the neighborhood of the moving contact line is analyzed and it is shown that under certain conditions regions with closed streamlines may be formed. The reason for this is the self-induced Marangoni effect — the reaction of the surface tension gradient on the liquid-solid boundary caused by the liquid flow on the flow that caused it.Based on a paper read at the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991. Presented by R. I. Nigmatulin.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 81–89, November–December, 1992.     

Flow patterns and interfacial velocities near a moving contact line

Experimental data on velocity fields and flow patterns near a moving contact line is shown to be at variance with existing hydrodynamic theories. The discrepancy points to a new hydrodynamic paradox and suggests that the hydrodynamic approach may be incomplete and further parameters or forces affecting the surfaces may have to be included. A contact line is the line of intersection of three phases: (1) a solid, (2) a liquid, and (3) a fluid (liquid or gas) phase. A moving contact line develops when the contact line moves along the solid surface. A flat plate moved up and down, inside and out of a liquid pool defines a simple, reliable experimental model to characterize dynamic contact lines. Highlighted are three important conclusions from the experimental results that should be prominent in the development of new theoretical models for this flow. First, the velocity along the streamline configuring the liquid–fluid interface is remarkably constant within a distance of a couple of millimeters from the contact line. Second, the relative velocity of the liquid–fluid interface, defined as the ratio of the velocity along the interface to the velocity of the solid surface, is independent of the solid surface velocity. Third, the relative interface velocity is a function of the dynamic contact angle.     

Comparative study of two lattice Boltzmann multiphase models for simulating wetting phenomena: implementing static contact angles based on the geometric formulation

Wetting phenomena are widespread in nature and industrial applications. In general, systems concerning wetting phenomena are typical multicomponent/multiphase complex fluid systems. Simulating the behavior of such systems is important to both scientific research and practical applications. It is challenging due to the complexity of the phenomena and difficulties in choosing an appropriate numerical method. To provide some detailed guidelines for selecting a suitable multiphase lattice Boltzmann model, two kinds of lattice Boltzmann multiphase models, the modified S-C model and the H-C-Z model, are used in this paper to investigate the static contact angle on solid surfaces with different wettability combined with the geometric formulation (Ding, H. and Spelt, P. D. M. Wetting condition in diffuse interface simulations of contact line motion. Physical Review E, 75(4), 046708 (2007)). The specific characteristics and computational performance of these two lattice Boltzmann method (LBM) multiphase models are analyzed including relationship between surface tension and the control parameters, the achievable range of the static contact angle, the maximum magnitude of the spurious currents (MMSC), and most importantly, the convergence rate of the two models on simulating the static contact angle. The results show that a wide range of static contact angles from wetting to non-wetting can be realized for both models. MMSC mainly depends on the surface tension. With the numerical parameters used in this work, the maximum magnitudes of the spurious currents of the two models are on the same order of magnitude. MMSC of the S-C model is universally larger than that of the H-C-Z model. The convergence rate of the S-C model is much faster than that of the H-C-Z model. The major foci in this work are the frequently-omitted important details in simulating wetting phenomena. Thus, the major findings in this work can provide suggestions for simulating wetting phenomena with LBM multiphase models along with the geometric formulation.     

Direct simulation of multiphase flows with modeling of dynamic interface contact angle

We describe a modeling technique for dynamic contact angle between a phase interface and a solid wall using a generalized Navier boundary condition in the context of a front-tracking-based multiphase method. The contact line motion is determined by the generalized Navier slip boundary condition in order to eliminate the infinite shear stress at the contact line. Applying this slip boundary condition only to the interface movement with various slip ratios shows good agreement with experimental results compared to allowing full fluid slip along the solid surface. The interface slip model performs well on grid convergence tests using both the slip ratio and slip length models. A detailed energy analysis was performed to identify changes in kinetic, surface, and potential energies as well as viscous and contact line dissipation with time. A friction coefficient for contact line dissipation was obtained based on the other computed energy terms. Each energy term and the friction coefficient were compared for different grid resolutions. The effect of varying the slip ratio as well as the contact angle distribution versus contact line speed was analyzed. The behavior of drop impact on a solid wall with different advancing and receding angles was investigated. Finally, the proposed dynamic contact model was extended to three dimensions for large-scale parallel calculations. The impact of a droplet on a solid cylinder was simulated to demonstrate the capabilities of the proposing formulation on general solid structures. Widely different contact angles were tested and showed distinctive characteristic behavior clearly.     

Spontaneous Inertial Imbibition in Porous Media Using a Fractal Representation of Pore Wall Rugosity

Considering the separable phenomena of imbibition in complex fine porous media as a function of timescale, it is noted that there are two discrete imbibition rate regimes when expressed in the Lucas–Washburn (L–W) equation. Commonly, to account for this deviation from the single equivalent hydraulic capillary, experimentalists propose an effective contact angle change. In this work, we consider rather the general term of the Wilhelmy wetting force regarding the wetting line length, and apply a proposed increase in the liquid–solid contact line and wetting force provided by the introduction of surface meso/nanoscale structure to the pore wall roughness. An experimental surface pore wall feature size regarding the rugosity area is determined by means of capillary condensation during nitrogen gas sorption in a ground calcium carbonate tablet compact. On this nano size scale, a fractal structure of pore wall is proposed to characterize for the internal rugosity of the porous medium. Comparative models based on the Lucas–Washburn and Bosanquet inertial absorption equations, respectively, for the short timescale imbibition are constructed by applying the extended wetting line length and wetting force to the equivalent hydraulic capillary observed at the long timescale imbibition. The results comparing the models adopting the fractal structure with experimental imbibition rate suggest that the L–W equation at the short timescale cannot match experiment, but that the inertial plug flow in the Bosanquet equation matches the experimental results very well. If the fractal structure can be supported in nature, then this stresses the role of the inertial term in the initial stage of imbibition. Relaxation to a smooth-walled capillary then takes place over the longer timescale as the surface rugosity wetting is overwhelmed by the pore condensation and film flow of the liquid ahead of the bulk wetting front, and thus to a smooth walled capillary undergoing permeation viscosity-controlled flow.     

Static wetting on flexible substrates: a finite element formulation

In static wetting on an elastic substrate, force exerted by the liquid–vapour surface tension on a solid surface deforms the substrate, producing a capillary ridge along the contact line. This paper presents a finite element formulation for predicting elastic deformation, close to the static wetting line (with angle of contact=90o and σSV=σSL).The substrate deformation is modelled with the Mooney–Rivlin constitutive law for incompressible rubber‐like solids. At the contact line, a stress singularity is known to arise, due to the surface tension acting on a line of infinitesimal thickness. To relive the stress singularity, either (i) the surface tension is applied over a finite contact region (of macroscopic thickness), or (ii) the solid crease angle is fixed. These two options suggest that normal component of Neumann's triangle law of forces, for the three surface tensions, is not applicable for elastic substrates (as for rigid ones). The vertical displacement of the contact line is a strong function of liquid/vapour surface tension and shear modulus of the solid. Copyright 2004 John Wiley & Sons, Ltd.     

Points singuliers d'une ligne de contact mobile

It is proposed to represent the dynamics of a moving contact line by an Onsager like mobility relation between the contact angle and the speed of the moving line, including an Arrhenius factor small enough in many physical situations to be the limiting factor for the motion. The liquid-vapor interface is then in quasiequilibrium, which allows one to analyse a dynamical wetting transition. This approach predicts well the formation of angular points on the rear edge of droplets sliding on a tilted plane.     

Two-phase boundary dynamics near a moving contact line over a solid

Consideration is given to the nonlinear problem on a shape of boundary between two viscous liquids, of which one displaces the other one from a solid surface, the Reynolds number being rather low. An asymptotic theory of wetting dynamics is developed that is of the second order with respect to small capillary numbers and valid for any ratio of viscosity coefficients of the media. A formula describing the dynamic contact angle (i.e. the inclination angle of the tangent to the interface) as a function of a distance to the solid is derived. Limitations on the angles for which the second-order theory is valid are shown. If the phase 2 viscosity is zero, the asymptotic second-order theory is valid for angles below 128.7°. A theory applicability domain depends on the ratio of viscosity coefficients. The applicability domain is not limited if the viscosity coefficients differ by a factor of less than four.     

Three-Phase Effective Contact Angle in a Model Pore

We consider a model pore (2D) in which a sharp interface between two fluids contact a third fluid which wets the solid boundary. If the configuration is capillary dominated, the geometry can be determined analytically in terms of the effective contact angle. This angle depends not only on the interfacial tensions, but also on the capillary pressures. However, if the height of the cusp formed by the wetting fluid is much smaller than the pore width, the effective contact angle is a simple function of the interfacial tensions. It turns out to be the same function as in the case of an undeformed wetting layer of molecular thickness. The analytical expression for the effective contact angle has been confirmed by a numerical technique, known as the lattice-Boltzmann method. This method, in turn, has been validated with Neumann's law for the three-phase contact angles.     

Issue Information ‐ TOC

Numerical modeling of multiphase flow generally requires a special procedure at the solid wall in order to be consistent with Young's law for static contact angles. The standard approach in the lattice Boltzmann method, which consists of imposing fictive densities at the solid lattice sites, is shown to be deficient for this task. Indeed, fictive mass transfer along the boundary could happen and potentially spoil the numerical results. In particular, when the contact angle is less than 90 degrees, the deficiencies of the standard model are major. Various videos that demonstrate this behavior are provided (Supporting Information). A new approach is proposed and consists of directly imposing the contact angle at the boundaries in much the same way as Dirichlet boundary conditions are generally imposed. The proposed method is able to retrieve analytical solutions for static contact angles in the case of straight and curved boundaries even when variable density and viscosity ratios between the phases are considered. Although the proposed wetting boundary condition is shown to significantly improve the numerical results for one particular class of lattice Boltzmann model, it is believed that other lattice Boltzmann multiphase schemes could also benefit from the underlying ideas of the proposed method. The proposed algorithm is two‐dimensional, and the D2Q9 lattice is used. Copyright © 2016 John Wiley & Sons, Ltd.     

Substrate elastic deformation due to vertical component of liquid-vapor interfacial tension

Young’s equation is a fundamental equation in capillarity and wetting,which reflects the balance of the horizontal components of the three interfacial tensions with the contact angle(CA).However,it does not consider the vertical component of the liquid-vapor interfacial tension(VCLVIT).It is now well understood that the VCLVIT causes the elastic deformation of the solid substrate,which plays a significant role in the fabrication of the microfluidic devices because of the wide use of the soft materials.In this paper,the theoretical,experimental,and numerical aspects of the problem are reviewed.The effects of the VCLVIT-induced surface deformation on the wetting and spreading,the deflection of the microcantilever,and the elasto-capillarity and electroelasto-capillarity are discussed.Besides a brief review on the historical development and the recent advances,some suggestions on the future research are also provided.