Boundary Conditions for the Moving Contact Line Problem. Abstract

Transcription

1 Boundary Conditions for the Moving Contact Line Problem Weiqing Ren Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA Weinan E Department of Mathematics and PACM, Princeton University, Princeton, NJ 08540, USA and School of Mathematical Sciences, Peking University, Beijing, China (Dated: August 22, 2006) Abstract The physical processes near a moving contact line are investigated systematically using molecular dynamics and continuum mechanics. Constitutive relations for the friction force in the contact line region, the fluid-fluid interfacial force and the stresses in the fluid-solid interfacial region are studied. Verification of force balance demonstrates the importance of the normal stress difference across the contact line region. Effective boundary conditions are derived using force balance. The effective continuum model is solved numerically and the behavior of the apparent contact angle and the wall contact angle is studied. It is found that the fluid-fluid interface near the wall exhibits a universal behavior. The onset of the nonlinear response for the contact line motion is studied within the framework of Blake s molecular kinetic theory. 1

2 Contents I. Introduction 3 II. Equilibrium configuration and the Young s relation 5 III. Moving contact line: The Huh-Scriven singularity 6 IV. Slip models and the Cox relation 7 V. Earlier results on slip for single and two-phase systems 9 VI. The behavior near the contact line region 11 A. Friction force outside the contact line region 15 B. Friction force inside the contact line region 15 C. Fluid-fluid interaction outside the interfacial region 16 D. Fluid-fluid interaction inside the interfacial region 16 E. Normal stress inside the fluid-solid interfacial region 17 F. Force balance in the contact line region 17 G. Effective boundary conditions 18 VII. Effective continuum model and numerical results 20 VIII. The mechanism of energy dissipation 22 IX. The onset of nonlinear response and Blake s molecular kinetic theory 23 X. Summary 26 References 27 2

3 I. INTRODUCTION When two immiscible fluids are placed in a container or on a substrate, the line where the interface of the two fluid phases intersects the solid container or the substrate is called the contact line. The two fluid phases can be both liquids, such as water, oil, paint or lubricant, or a liquid and a gas (vapor) phase. The equilibrium configuration of the static contact line was the topic of classical work of Young, Laplace and Gauss, 1 and is described by Young s equation which relates the three coefficients of interfacial tension to the contact angle formed by the fluid-fluid interface with the solid surface. The moving contact line problem, however, has for many years remained an issue of controversy and debate. The difficulty stems partly from the fact that classical hydrodynamic equations, the Navier-Stokes equations, coupled with the conventional no-slip boundary condition predicts a singularity for the stress that results in a non-physical divergence for the energy dissipation rate. 2 5 Many models have been proposed in order to remove this singularity. These models either introduce modification to the linear constitutive relation for the bulk fluid, or modification to the no-slip boundary condition. While they succeed in regularizing the problem, they are all phenomenological in nature, and their validity is far from being obvious. Experimental verification is hindered by the difficulty of making direct measurements at the fluid-solid interface. A related but perhaps even more important issue is the underlying mechanism for the large slip that occurs at the contact line: In steady Couette flows, the fluid exhibits complete slip at the contact line, i.e. the relative slip velocity between the contact line and the solid wall is equal to the driving velocity of the solid wall. In recent years, molecular dynamics (MD) has been developed as a tool to address these problems, 6 11 and has shed considerable light on the nature of the physical processes near the moving contact line. The present paper continues on that path and presents a systematic study of the physical processes near the contact line, by combining molecular dynamics with continuum mechanics viewpoints. We will see that these efforts have led to a fairly complete understanding of the fundamental structure and the dynamic processes near a moving contact line including the physical mechanism that causes the large slip at the contact line. Simplified continuum models can be formulated as a result, which can be used effectively to describe the dynamics of the spreading of droplets on solid surfaces and immiscible two-phase flows in a capillary tube. This is also a very good example which 3

4 demonstrates that fundamental progresses can be made on classical problems of physics, by combining the atomistic and continuum viewpoints. Very briefly, let us summarize the main issues and our current understanding of these issues. The questions that we need to address include: 1. What is the underlying physical mechanism that regularizes the Huh-Scriven singularity found from continuum theory? 2. What drives the contact line motion and the associated anomalously large slip? Note that in a steady Couette geometry, the fluids experience complete slip at the contact line. 3. Such a large slip must give rise to a large strain rate. What governs the constitutive laws at such high strain rate? Based on extensive studies using molecular dynamics and continuum modeling, starting from the earlier work of Koplik, Robbins et al., 6 9 together with the recent work of Qian, Wang and Sheng, 10,11 as well as the present work, we have arrived at the following conclusions: 1. The Huh-Scriven singularity is regularized by the existence of a slip region. Inside the slip region, the no-slip boundary condition is replaced by the Navier boundary condition away from the contact line, and relaxational dynamics for the Young s stress at the contact line. 2. Small deviation of the contact angle from its equilibrium value can cause large stress which drives contact line motion and slip. 3. Even though the strain rate can be very large near the contact line (on the order of 10 9 s 1 ), for the examples of simple fluids considered here, it is still far from the values that would invalidate the linear constitutive relation. In addition, we also find that continuum models can be used to describe the dynamics in inhomogeneous systems up to the molecular scale, as long as the values of viscosity, surface tension, etc are adjusted near the fluid-solid interface, and the fluid-solid boundary conditions are suitably modified. This paper is organized as follows. We begin in section II with a quick review of the static problem and Young s relation. We then discuss the moving contact line problem and 4

5 the related stress singularity in section III. In section IV, we review the existing regularized continuum models and the results of the asymptotic analysis of these models. In section V, we review the results of earlier MD studies. In section VI, we present our MD results and formulate the effective continuum boundary condition. Numerical results for the effective continuum model are presented in section VII, and the energy dissipation relation is discussed in section VIII. These results demonstrate quite clearly that despite of the large slip at the contact line, linear constitutive relations provide an accurate description for the underlying physical process in most regimes of interest. To better understand the regime of validity of these linear constitutive relations, we discuss in section IX the onset of nonlinear response. The main conclusions are summarized in section X. The present paper will focus on the simplest situation when the solid surface is a perfectly clean and flat crystal surface, neglecting all chemical or geometric defects or heterogeneities. We will consider flows that are microscopically three dimensional but macroscopically two dimensional. II. EQUILIBRIUM CONFIGURATION AND THE YOUNG S RELATION To begin with, let us review the situation of the static contact line which has been understood for a long time. 1,5 The set-up is depicted in Fig. 1. Denote by γ 1 and γ 2 the surface tension coefficients between the solid and the two fluids, and γ the interfacial tension between the two fluids. By minimizing the total surface energy, one obtains the well-known Young s relation: γ 2 γ 1 = γ cos θ s. (1) The angle θ s in Eqn. (1) is called the static contact angle, which is the angle between the fluid-fluid interface and the solid wall measured from the side of fluid I, as shown in Fig. 1. Eqn. (1) can also be derived by tangential force balance. Note, however, that vertical force balance is not implied by Eqn. (1). In fact, to satisfy vertical force balance, we have to take into account the elastic forces from the solid. 5

6 III. MOVING CONTACT LINE: THE HUH-SCRIVEN SINGULARITY When the contact line moves, we have to model both the dynamics of the two-phase fluid flow and the dynamics of the contact line. The dynamics of the two-phase fluid over a solid wall is modeled by the incompressible Navier-Stokes equations: ρ i ( t u + (u )u) = p + µ i 2 u + f u = 0 (2) Here, ρ i and µ i (i = 1, 2) are the density and the viscosity of the fluids respectively, u = (u, v) is the velocity field, p is the pressure, and f denotes the external force. The fluid-fluid interface x Γ is advected by the velocity field. In addition, the following conditions are also imposed on x Γ : ˆn [τ] ˆn = γκ ˆt [τ] ˆn = 0 (3) where τ = pi + µ( u + u T ) is the stress tensor; [τ] denotes the jump of the stress tensor across the interface; κ is the curvature of the interface; ˆn and ˆt are the unit normal and tangent vectors to the interface respectively. Equations in (2) should be supplemented by boundary conditions at the solid surface. The standard boundary condition is the no-slip boundary condition: u = u b (4) where u b is the velocity of the solid wall. Huh and Scriven studied the problem of a planar fluid interface moving steadily over a flat solid surface. 2 The system is modeled by the two dimensional Stokes flow with no-slip boundary condition at the solid surface. The stream function for the flow field satisfies the biharmonic equation whose solution is given by ψ(r, θ) = r (a sin θ + b cos θ + cθ sin θ + dθ cos θ) (5) where r and θ are the polar coordinates with the origin at the contact line and θ = 0, π at the solid surface. The constants a, b, c and d are determined by the no-slip condition at the solid surface and conditions at the fluid interface. Although the velocity field corresponding to the stream function (5) appears to be realistic, the stresses and viscous dissipation increase 6

7 without bound when the contact line is approached. For example, the shear stress is given by τ rθ = 2µ (c cos θ d sin θ) (6) r and the pressure is given by p p 0 = 2µ r (c sin θ + d cos θ). (7) It is easy to check that this divergence of stress leads to a logarithmic divergence in the energy dissipation rate, which is non-physical. This divergence still remains when the full Navier-Stokes equation is used instead of the Stokes equation. See also the work of Dussan and Davis. 3 IV. SLIP MODELS AND THE COX RELATION Many attempts have been made to remove this non-physical singularity. These attempts fall into two categories: The first is to modify the governing equations, for example, by replacing the linear constitutive relation for the bulk fluids by some non-newtonian constitutive models, or by modifying the kinematic condition at the fluid-fluid interface and thus allowing non-zero mass flux across the interface near the contact line (diffuse interface models), or by including the effect of long-range interactions in the lubrication approximation, or by taking into account the gradient of interfacial energy, etc. The second is to replace the no-slip boundary condition by some slip model near the contact line. Some examples of the slip model include: 1. The Navier boundary condition: βu s = ˆt τ ˆn (8) where β is the friction coefficient, u s is the (tangential) slip velocity of the fluid at the solid surface, ˆt and ˆn are the unit tangent and outward normal vectors of the solid surface respectively, and τ is the stress tensor. 2. Stress-free boundary condition: ˆn u = 0 (9) in the vicinity of the contact line, where u is the tangential component of the velocity. 7

8 3. Prescribed slip profile, e.g. in Couette geometry u s = U (1 exp ( r ln (2/S))) (10) where U is the speed of the wall, r is the distance to the contact line, and S is the slip length Blake s model: ( ) γ (cos θs cos θ) u CL = 2κ 0 λ sinh 2mk B T where u CL denotes the contact line speed, κ 0 is the equilibrium frequency of molecular displacement between adsorption sites at the solid surface, λ is the average distance between the adsorption sites, m is the number of adsorption sites per unit area, and k B T is the thermal energy. The deviation of the contact angle θ from its static value θ s provides the driving force for contact line motion. 12,13 One common feature of these slip models is that they all introduce a new length scale S, which can be thought of as being the length of slip region around the contact line. Most of these models are phenomenological in nature. (11) They all serve the purpose of removing the singularity, but their physical origin and validity is far from being clear. Thus it is natural to ask which model better describes the actual physical processes near the contact line. This is one of the main issues that we will address in the present paper. Before we focus on the region near the contact line, let us discuss the implications of these slip models for the far field, away from the contact line region. This problem has been studied in the work of Hocking and his co-workers, 14,15 Huh and Mason, 16 Dussan, 4 and Cox, 17 using matched asymptotic analysis. These analyses revealed a fundamental fact, namely, what matters for the far field behavior is only the size of S and the contact angle at the wall θ w, not the details of the slip model. The main result, obtained by Cox, is the following: 17 To leading order in the capillary number Ca = µu γ where U is a typical scale for the velocity field, the apparent contact angle θ m, the wall angle θ w, Ca and the slip length S are connected by the following relation: g(θ m ) = g(θ w ) + Ca ( log(s 1 ) + Q i f(θ w ) ) Q o f(θ m ) where Q i and Q o are two constants that depend on the details of the flow in the inner region around the contact line and the overall geometry of the outer region respectively; g and f 8 (12)

9 are given by g(θ) = θ 0 dθ f(θ ), (13) f(θ) = 2 sin θ { λ 2 (θ 2 sin 2 θ) + 2λ ( θ(π θ) + sin 2 θ ) + ( (π θ) 2 sin 2 θ )} λ(θ 2 sin 2 θ) ((π θ) + sin θ cos θ) + ( (π θ) 2 sin 2 θ ) (θ sin θ cos θ) where λ is the ratio of the viscosities of the two fluids. Cox s relation was anticipated both by the experimental work of Hoffman 18 (14) and the analytical work of Dussan. 4 It gave a very satisfactory explanation of Hoffman s experimental results, and it has since been confirmed by various numerical results. 8,9,19,20 The fact that the details of the slip model do not affect the macroscopic flow behavior in a large system is certainly good news. However, if we would like to probe the physical process near the contact line region, we will not be able to do so by making macroscopic measurements: Direct measurement near the contact line region is important and this is very difficult to do. This is why direct simulation using molecular dynamics has been the main tool in probing the structure and dynamics near the contact line region. V. EARLIER RESULTS ON SLIP FOR SINGLE AND TWO-PHASE SYSTEMS Slip between fluids and solids has been studied by a number of researchers using molecular dynamics. The set-up is always quite similar to that presented below in section VI, except that some researchers (e.g. Troian et al. 21 ) also studied models of polymer melts using bead-and-spring models. For one-phase systems (e.g. single fluid phase), the representative work is that of Thompson and Troian. 22 Their careful MD study confirmed the validity of the Navier boundary condition, with a subtle twist that the slip length might be a function of shear rate γ: l s = l s ( γ). (15) In fact, a main result of Thompson and Troian is that l s has a universal behavior: l s ( γ) = l 0 s (1 γ/ γ c) 1/2 (16) where l 0 s is the slip length at zero shear rate and γ c is the critical shear rate. For the kind of systems we are interested in here, the nonlinear response represented by Eqn. (16) sets in only at shear rates close to atomic frequencies: γ s 1. Hence for all 9

10 practical purposes, the linear Navier boundary condition is a good model for describing the friction process at the single-phase fluid and solid interface. Slip at the solid surface for two-phase systems (two fluid phases) has been studied in the pioneering work of Koplik, Robbins and their co-workers. 6 9 Their MD studies provided clear evidence that there is a region near the contact line in which the no-slip boundary condition breaks down. This is not unexpected. After all, in the Couette flow setting, if the flow attends steady state at all, the contact line must experience complete slip, i.e. the slip velocity at the contact line should be equal to the imposed velocity of the wall. In Ref. 8, the authors also examined the validity of the linear constitutive relations in the contact line region. In particular, they calculated the viscous shear stress from the average velocity field using the linear constitutive relation, and compared the results with the stress computed directly from MD. A large discrepancy was found in the slip region which extends 2 3σ on each side from the contact line (σ is the unit of length in Lennard- Jones potential). The authors thus concluded that the linear response relation breaks down in this slip region, and the fluids exhibit non-newtonian behavior. Note that in their work, the authors assumed that the stress measured from MD only contained viscous component near the solid surface - the contribution of the Young stress at the fluid-fluid interface was neglected. As was found later by Qian et al. 10 and confirmed in our work, in the contact line region, the Young stress is dominant even in cases when the dynamic contact angle only deviates slightly from the static contact angle. In addition, the conventional concepts used in bulk fluids such as viscosity, surface tension coefficient and etc have to be generalized, for example, when applying the linear constitutive relation to calculate the stress: One should allow the viscosity to vary as the fluid-fluid interface is approached. The next milestone came with the recent work of Qian et al. 10,11 Through careful MD studies, they provided convincing evidence that the interfacial Young stress is dominant inside the contact line region. Once the Young stress is subtracted from the total stress, the remaining viscous contribution obeys linear response outside the contact line region. Based on a force balance argument, the authors proposed the following generalized Navier boundary condition in the framework of phase-field theory which takes into account the uncompensated (or unbalanced) Young stress in the contact line region: βu s = µ ˆn u + (K ˆn ϕ + ϕ γ wf ) x ϕ. (17) 10

11 Here ϕ is the composition (phase) field, K is a parameter in the Cahn-Hilliard free energy functional: ( ) 1 F (ϕ) = 2 K ( ϕ)2 + f(ϕ) dx (18) where f(ϕ) is a double-well potential. The left-hand side of Eqn. (17) is the friction force, the first term on the right-hand side is the viscous shear stress, and the last term is the uncompensated Young stress. A similar boundary condition as in (17) but with the viscous term left out was formulated in Ref. 23. In our MD study presented below, we shall distinguish the region of the bulk fluids and the interfacial regions. The latter includes the fluid-fluid interfacial region, the fluid-solid interfacial region as well as the contact line region where the three phases interact (see Fig. 5). While in the bulk we will examine the linear constitutive relations in the usual sense, in the interfacial regions we shall calculate integrated quantities. As we will show below, our MD studies suggest quite strongly that linear response holds for both the friction force (fluid-solid interaction) and the viscous force (fluid-fluid interaction) outside the contact line region. Thus away from the contact line, the conventional Navier boundary condition holds. This confirms the conclusions made by Qian et al. Inside the contact line region, the integrated friction force depends linearly on the contact line velocity. In addition, we find that the integrated normal stress across the fluid-solid interfacial region plays an important role in the force balance at the contact line. These studies suggest that at the contact line the Navier boundary condition has to be modified in order to take into account the interfacial Young stress, and the normal stress effect. VI. THE BEHAVIOR NEAR THE CONTACT LINE REGION In the following, we use molecular dynamics to study the details of the process near the contact line region and near the solid wall. Setup of the MD system. Our MD calculation follows the setup in Refs. 6 and 8, which has now become the standard practice in this area. Specifically, two species of fluid particles are placed between two solid plates which are parallel to the xy plane. The fluid particles interact via the Lennard-Jones (LJ) potential in a slightly modified form: ( (σ ) 12 ( σ ) ) 6 V LJ (r) = 4ε ζ (19) r r 11

12 where r is the distance between particles, ζ = 1 for particles of the same species and ζ = 1 for particles of different species. In the following we will express all the quantities in terms of the basic units ε, σ and m (the mass of the fluid particles). The fluid and solid particles interact via the same potential but with slightly different parameters: ε 1w = ε 2w = 1.16, σ 1w = σ 2w = 1.04 and ζ 1w = ζ 2w = 1 (The subscripts denote fluid 1, fluid 2 and the solid respectively). Each of the two solid walls is modeled by two layers of atoms in the facecentered cubic structure in the [001] direction with number density We have worked with different system sizes: and For the results shown below, the particle interactions are cut off at a finite range r c. In our calculation the cut-off radius is chosen to be r c = 2.5σ. The density and temperature of the fluids are 0.81 and 1.1 respectively. Correspondingly, the viscosity and interfacial tension in the bulk fluid are µ = 2.2 and γ = 3.7 respectively. The two walls are sheared with constant speed U in opposite directions along the x axis, generating a two-dimensional flow in the xz plane. Periodic boundary condition is used in both the x and y directions. The dynamics of the particles obey Newton s equation, which is solved by the velocity Verlet algorithm. The temperature of the system is controlled by adding a damping and a noise term to the dynamical equation of the particles in the y direction. Fig. 2 shows a snapshot of the instantaneous positions of all particles for U = Measurement of the stress and the friction force. To measure the stress, we divide the xz plane into bins of size 1σ 1σ. The fluid near the wall exhibits layered structure. The location of the bins are chosen to coincide with the fluid layers. The instantaneous stress and friction force are calculated using the Irving-Kirkwood formula: 24 τ(x, t) = i 1 m i (p i p i )δ(q i x) j i 1 ((q i q j ) F ij ) δ (λq i + (1 λ)q j x) dλ 0 (20) where q i s and p i s are the positions and momenta of the particles at time t, F ij is the force between the i-th and the j-th particle. The force includes the fluid-fluid interaction and fluid-solid interactions. The above formula defines the current of momentum flux. The 12

13 momentum flux in each bin is calculated by averaging (20) over space and time: 1 T2 τ = τ(x, t)dxdt (21) (T 2 T 1 ) Ω where T 1 is some transient time and T 2 is the total simulation time, Ω denotes the bin, and Ω is the volume of the bin. Finally, the stress tensor is obtained after subtracting the convection term: T 1 Ω τ = τ ρu u (22) where ρ is the mass density and u is the averaging velocity field. The friction force is measured in the void between the solid surface and the first fluid layer, using Eqn. (20). The height of the void is 0.85σ for the interactions noted above. The xz component of the stress tensor τ measured in this void gives the friction force. Fluid-solid and fluid-fluid interfacial regions. Due to the finite range of the molecular interaction, the fluid-solid and fluid-fluid interfaces have finite width. Let ξ and δ denote the width of the fluid-solid and fluid-fluid interfacial regions respectively. Inside the interfacial regions, most of the physical quantities, such as the pressure, particle density, and viscosity take different values from their bulk values. To see this, we next compare the pressure and the viscosity inside and outside the interfacial regions. First, let us look at the pressure. The hydrostatic pressure is calculated using the Irving- Kirkwood formula. Fig. 3 shows the xx and zz components of the stress tensor respectively. Only the stress in the lower half of the channel is shown here. From these results, we see that the stress inside the interfacial region exhibits different behavior from that of the bulk fluid. Furthermore the stress is anisotropic inside the interfacial regions. As a matter of fact, the difference of these two diagonal components defines the surface (or interfacial) tension. Specifically, Let τ and τ denote the stresses parallel and perpendicular to the interface respectively, then we have (τ τ ) = γ (23) I where the integral is over the interfacial region. For the system studied here, the integral of the difference between τ zz and τ xx over the fluid-solid interfacial region defines the surface tension γ wf, and the integral over the fluid-fluid interfacial region defines the interfacial tension γ ff. The last observation from Fig. 3 is that, for flat interfaces as in the static case of our system, the normal stress in the direction perpendicular to the interface is a constant 13

14 across the interface. On the other hand, the normal stress in the direction parallel to the interface is peaked inside the interfacial region. For example, τ xx is a constant function of x across the fluid-fluid interface, and τ zz is a constant function of z across the fluid-solid interface. The viscosity inside the interfacial regions also take a different value from its bulk value. The solid curves in Fig. 4 show the viscous part in the xx component of the stress as a function of z. Here the wall speed is U = The two panels show the cross sections of the normal stress at two different values of x. The dashed curves show the values calculated using the linear relation 2µ x u with µ = 2.2, the bulk viscosity and u being the average velocity in the x direction. The solid and dashed curves agree with each other within the error bars outside the fluid-solid interfacial region, but disagree near the wall. We have evidence that the fluid is still Newtonian inside the interfacial region, but with a larger viscosity than its bulk value. Main results of our MD study. We will study the following forces: 1. The friction force F at the bottom of the fluid-solid interfacial region; 2. The averaging friction force F CL across the contact line region, which is defined as the region of three phase co-existence; 3. The shear stress τ xz at the top of the fluid-solid interfacial region; 4. The integrated shear stress τ xz over the fluid-fluid interfacial region; 5. The normal stress τ xx inside the fluid-solid interfacial region. A schematic of these forces is shown in Fig. 5. The main results we will establish in the following are: 1. The friction force F satisfies a linear friction law. This linear relation defines a friction coefficient β for each fluid species; 2. The averaging friction force F CL also satisfies a linear relation with the contact line velocity. This linear relation defines the friction coefficient (of the contact line) β, which depends on the interactions between all three phases; 14

15 3. The shear stress τ xz obeys the standard linear constitutive relation for viscous fluids; 4. The integrated shear stress across the fluid-fluid interfacial region is the Young stress; 5. The normal stress in the fluid-solid interfacial region has a large jump across the contact line region. This jump, denoted by [τxx ], is linearly related to the contact line velocity. This term is important for the force balance near the contact line region. One of our aims is to arrive at a sharp-interface continuum model for the moving contact line problem. The details of the interfacial region are lumped into a few parameters such as the interfacial width ξ and δ, the surface or interfacial tensions, the contact line friction coefficient, the interfacial viscosity µ, etc. A. Friction force outside the contact line region Fig. 6 shows the slip velocity u s (upper panel) and friction force F (lower panel) for shear speed U = 0.02, 0.03, 0.04 and 0.05 (from the lower curve to the upper curve) respectively. The contact line is located at the position where the velocity profile attains the maximum value. It is evident from these curves that significant slip occurs in a region which extends about 20σ in the advancing fluid and 30σ in the receding fluid. Outside this region, slip is negligible. In Fig. 7 we plot the friction force (solid curve) for U = Also shown in the same figure is βu s (dashed curve) with β = 6. Outside the contact line region, the two curves agree very well indicating that the linear friction law holds with the same friction coefficient as its single-phase value. B. Friction force inside the contact line region In the previous section, we presented evidence suggesting quite strongly that the fluidsolid interaction force outside the contact line region obeys a linear friction law. Next we turn our attention to the contact line region. For the contact line region, it makes more sense to discuss integrated quantity, such as the integrated force and velocity. Fig. 8 displays F CL, which is the friction force averaged over the contact line region, versus the averaged slip velocity u CL, corresponding to the curves in Fig. 6. Note that the averaged friction force 15

16 vanishes in the static situation, and obeys a linear relation with the averaged slip velocity in the regime studied here, i.e. F CL = β u CL. The coefficient β, which takes a value of about 4.4, can be regarded as being the friction coefficient for the contact line. Note that β depends on interactions between all three phases. For the system considered here, β is smaller than the single phase friction coefficient. This is mainly due to the depletion of particles in the fluid-fluid interfacial region. At large slip velocity, this linear relation breaks down. The onset of the nonlinearity is discussed in section IX. C. Fluid-fluid interaction outside the interfacial region Here we consider the shear stress τ xz at the top of the fluid-solid interfacial region, as shown in Fig. 5. The two walls in our system are sheared at the speed U = 0.05 in opposite directions. We calculated the shear stress at a distance 2.5σ above the solid surface, and the result is plotted in Fig. 9 (the solid curve). Here the shear stress only contains the fluid-fluid interactions since the wall particles are out of the cutoff radius. The contact line is located at x = 40. The stress peaks inside the interfacial region. In Fig. 9 we also plotted the viscous stress (dashed curve) calculated using the linear constitutive relation: τ d = µ z u, where z u is the shear rate at the location 2.5σ above the solid surface, u is the averaging velocity in the x direction, and µ = 2.4 is the viscosity. Note that the viscosity is slightly larger than its bulk value. This is due to the nonlocal character of the microscopic stress and the layered structure of the fluid near the wall (which increases the particle density inside the fluid-solid interfacial region). The two curves agree well outside the contact line region, indicating that the fluids are Newtonian and we have a linear relation between the stress and the strain rate. D. Fluid-fluid interaction inside the interfacial region Here we present evidences that the shear stress integrated over the interfacial region is Young stress. The shear stress is integrated in the x direction over the fluid-fluid interfacial region which has a width 8σ. The interfacial region is centered at the location where the shear stress peaks. In Fig. 10 we plot the integrated shear stress as a function of z, the transversal 16

17 direction across the channel (solid curve). In the same figure we also plotted Young stress, γ cos θ, where θ(z) is the angle between the interface and the wall, and γ = 3.7 is the interfacial tension. These two curves agree well outside the fluid-solid interfacial region, indicating that the integrated shear stress over the interfacial region is Young stress. Inside the fluid-solid interfacial region, however, one needs to increase the value of γ to γ = 4.9 in order to get a good agreement. E. Normal stress inside the fluid-solid interfacial region Next we study the behavior of the normal stress τ xx inside the fluid-solid interfacial region. Fig. 11 shows the normal stress density τxx, which is obtained by averaging the normal stress τ xx in the z direction over the fluid-solid interfacial region. The two curves correspond to U = 0.04 (solid curve) and U = 0 (dashed curve) respectively. From these results we see that the normal stress has a large jump across the contact line region when the contact line moves. This jump, denoted by [τxx ], plays a similar role as the friction force: It retards the contact line motion. The magnitude of the normal stress jump increases as contact line speed increases, as shown in Fig. 12. The figure also shows a nice linear relation between the normal stress jump and the contact line speed: [τ xx] = ηu CL, where η = 6. F. Force balance in the contact line region We now look at the force balance in the contact line region along the x direction. We consider a pillbox ω which has a dimension 8σ 2.5σ and covers the contact line region. The total force acting on the pillbox in the x direction contains four contributions: F x = ( x τ xx + z τ xz )dx = τxx R τ xx L + τ xz T τ xz B (24) ω where τxx R and τ xx L are the integrated normal stresses on the right and left sides of the pillbox respectively, τxz T is the integrated shear stress at the top of the pillbox and τxz B is the integrated friction force at the surface of the wall (which is also the bottom of the pillbox). At steady state, the total force vanishes. This is verified by our MD calculation, as shows in table I, where the different rows are results for different driving speed U. We see that the difference between the shear stress and the friction force is balanced by the difference 17

18 between the normal stresses up to statistical errors. We also observe that the normal stress contribution is comparable to the friction force, and cannot be neglected. Some visual evidence is presented in Fig. 13, where the solid curve shows the difference between the shear stress at the top of the fluid-solid interfacial region (z = 2.5σ) and the friction force at the solid surface (z = 0), and the dashed curve shows the x derivative of the integrated normal stress over the fluid-solid interfacial region. i.e. x ξ 0 τ xxdz, where ξ = 2.5σ. It can be seen that the normal stress contribution is important and it balances the shear stress difference, especially inside the slip region. G. Effective boundary conditions The results from MD suggest the following set of boundary conditions: Boundary condition at the wall, away from the contact line: β i u s = µ i z u (25) where β i, i = 1, 2 is the single phase friction coefficient at the wall. Boundary condition at the contact line: δβ u CL = γ cos θ ξηu CL (26) where the left-hand side is the friction force, the first term in the right-hand side is the Young stress, and the second term represents the normal stress contribution. When the two fluids have different interactions with the solid, the normal stress jump contains an additional term which accounts for the surface tension difference across the contact line region. In this more general situation, we have δβ u CL = γ cos θ ξηu CL + [γ wf ]. (27) Both the friction force and the normal stress jump retard the motion of the contact line, therefore their combination can be thought of as being an effective friction force. In the following equation, we rewrite the above boundary condition by combining these two terms and also using Young s relation: β CL u CL = γ (cos θ s cos θ) (28) 18

19 where β CL is the effective friction coefficient and is given by β CL = δβ + ξη. (29) Eqn. (25) is the conventional Navier boundary condition. Eqn. (28) bears some similarities with the GNBC in Eqn. (17). Both models are based on the force balance argument. In both models, the main driving force for the slip is the unbalanced or uncompensated Young stress, which results from the deviation of the contact angle from its static value. However, the way in which force balance is examined in the two studies is different. GNBC is a result of the force balance at the solid surface (outside the fluid-solid interfacial region). Our model results from the force balance in a pillbox which covers the contact line region. Consequently, the two models are different in the following sense: 1. In GNBC, the contact angle is defined at the solid surface which is also the lower boundary of the fluid-solid interfacial region. In contrast, in our model the contact angle is defined at the upper boundary of the fluid-solid interfacial region; 2. The friction force in these two models has different meanings. In GNBC the friction force only contains the fluid-solid interaction. In our model, the friction force is an effective friction force: It not only contains the friction force from the solid, it also contains the normal stress jump inside the fluid-solid interfacial region (see the definition of the effective friction coefficient in Eqn. (29)). This latter hydrodynamic contribution plays an equally important role as the fluid-solid interaction force in the force balance; 3. Due to the large normal stress jump across the contact line region, the fluid interface has a large curvature inside the fluid-solid interfacial region. The model of Qian et al. requires resolving the details of this interfacial region. In contrast, our model is a sharp interface model and the normal stress contribution from the fluid-solid interfacial region is lumped into the parameter β CL and is already taken into account in the boundary condition. 19

20 VII. EFFECTIVE CONTINUUM MODEL AND NUMERICAL RESULTS Our continuum model contains the incompressible Navier-Stokes equations: ρ i ( t u + (u )u) = p + µ i 2 u + f u = 0 (30) The fluid interface x Γ is advected by the velocity field. In addition, the following stress conditions are imposed on x Γ : ˆn [τ] ˆn = γκ, ˆt [τ] ˆn = 0. On the boundary and away from the contact line we have: and at the contact line we have: (31) β i u s = µ i z u (32) β CL u CL = γ (cos θ s cos θ). (33) In the following, we present some numerical results for this continuum model. In the examples below, we take two immiscible fluids that are confined in a channel between two parallel solid walls at z = 0 and z = H respectively. The system is driven by an external force f in the x direction resulting in a two-dimensional flow field u = (u, v) in the xz plane (see Fig. 14). The equations in (30) are solved in a finite domain [ L, L] [0, H] in the xz plane. Far from the fluid-fluid interface the flow field has the Poiseuille flow profile. This suggests using the Neumann boundary condition at x = ±L: u x = 0, v = 0. (34) In our calculation the fluid-fluid interface is re-centered from time to time to keep it away from the two boundaries at x = ±L. At the solid wall z = 0 and z = H, we use the boundary conditions (32) and (33) for u and no-penetration condition for v: The static configuration is used as the initial condition. v = 0. (35) We assume that the two fluids interact identically with the wall, so the static contact angle is θ s = 90. The parameters are chosen as follows: L = 200, H = 200, ρ = 0.81, 20

21 µ = 2.2, γ = 3.7. The friction coefficients β 1, β 2 of the two fluids at the wall are the same. This value is denoted by β and is varied from 2 to 12. The effective friction coefficient at the contact line β CL is determined by Eqn. (29), where δ = 5, β = 0.6β, and ξη is taken as being equal to δβ. The system is symmetric about z = H/2 so the equations are solved on the lower half domain [ L, L] [0, H/2]. We used the central difference schemes to discretize the spatial derivatives on a staggered grid and forward Euler method to discretize the time derivative. The version of the projection method used in Ref. 25 was adopted here to keep the velocity field divergence-free. Interfacial forces were spread out to neighboring grid points as is done in the immersed boundary method. 25,26 Velocity field and structure of the interface. The upper panel in Fig. 14 shows the velocity field and the interface at steady state in the frame of the moving boundary for the parameters f = and β = 4. The lower panel in Fig. 14 shows the angle between the interface and the lower solid wall. Two observations are made from this curve: First there is a large deviation (about 25 ) in the dynamic contact angle θ d from its static value. Secondly, the interface exhibits a boundary layer behavior near the solid wall where the angle increases rapidly from θ d to θ m. We define θ m, the maximum of θ along the interface, as the apparent contact angle. Outside the boundary layer, the curve for the angle is close to a straight line. Thus the interface has a nearly constant curvature and can be well-fitted by a spherical cap. Dependence of θ d on u CL and β. We next vary the parameters f and β and study the behavior of θ d. In the upper panel of Fig. 15, we fix β at 6, and let f vary from to and plot the deviation of θ d from its static value against the contact line velocity. In the lower panel of Fig. 15, we fix f at , let β vary from 2 to 12, and plot the deviation of θ d from its static value against β. These results show that θ d increases linearly with the contact line velocity and the friction coefficient in the regime we studied. The large deviation of the contact angle θ d from its static value is also confirmed by our MD calculations. As shown in table I, the integrated shear stress across the fluid interface, i.e. the Young stress, is 0.613, and for u CL = 0.02, 0.04 and 0.06 respectively. These values correspond to contact angles of 100, 110 and 118 for γ = 3.7. These results are plotted as stars in the upper panel of Fig. 15. The MD results agree well with the results of our continuum calculation. 21

22 Dependence of θ m on u CL and β. We also show briefly that our numerical results are consistent with the analytical result in Eqn. (12). The results shown in Fig. 16 fit very well with the function g(θ) given in Eqn. (13) with λ = 1. Specifically, the solid curve in the figure is θ versus g(θ); the squares are the values of θ m plotted against g(θ d ) u CL for various values of u CL ; the diamonds are the values of θ m plotted against g(θ d )+0.003(log β+4.67) for various values of β. Here θ d and u CL are the corresponding contact angle at the boundary and the contact line velocity respectively. We found that the data from the numerical calculations fit very well with the solid curve from Cox s relation. Interface structure inside the boundary layer. As discussed earlier, near the boundary the fluid-fluid interface exhibits a boundary layer behavior in which the angle θ increases from θ d to θ m as z increases from 0 to z. A few examples are plotted in the upper panel of Fig. 17. Starting from the bottom, the solid curves show the results for β = 6, u CL = 0.015, 0.031, and 0.061; the dashed curves show the results for f = , β = 2, 6 and 12. We calculated (g(θ(z)) g(θ d ))/(g(θ m ) g(θ d )) along each curve for z z, and plotted them as a function of z in the lower panel of Fig. 17. Clearly these curves show universal behavior that can be described by a single function which we denote by h(z). The function h(z) increases from 0 to 1 monotonically when z is increased from 0 to 1. This suggests the following relation for the boundary layer structure: g(θ(z)) = g(θ d ) + (g(θ m ) g(θ d ))h(z/z ). (36) VIII. THE MECHANISM OF ENERGY DISSIPATION Now we calculate the rate of energy dissipation for the effective continuum model. Let Ω i, i = 1, 2 denote the advancing and receding fluids respectively; Ω i denotes the boundary of Ω i ; S ff and S wfi denote the fluid-fluid and fluid-solid interfaces respectively. Then the 22

23 dissipation of the kinetic energy is d dt i Ω i 1 2 ρ i u 2 dx = i Ω i u (µ i u p) dx = i Ω i (u τ d ˆn (u ˆn)p) ds i Ω i µ i u 2 dx = S ff [u τ d ˆn (u ˆn)p] ds i S wfi uµ i z u ds i Ω i µ i u 2 dx = S ff γκ(u ˆn) ds i S wfi β i u 2 s ds i Ω i µ i u 2 dx (37) where u = (u, v) is the velocity field, τ d = µ i ( u + u T ) is the viscous stress tensor, and the bracket [ ] denotes the jump of the quantity across the fluid-fluid interface. In the last step, we have used the conditions in (31) and (32). Denote by Γ ff and Γ wfi the energy of the fluid-fluid and fluid-solid interfaces respectively, then the rate of dissipation of the interfacial energies is d dt (Γ ff + Γ wf1 + Γ wf2 ) = S ff γκ (u ˆn) ds + u CL γ cos θ d u CL [γ wf ] (38) = S ff γκ (u ˆn) ds β CL u 2 CL where in the last step we have used the boundary condition (33) at the contact line. Combining (37) and (38) we obtain the following result for the rate of the total energy dissipation: ( d ) 1 dt i ρ Ω i 2 i u 2 dx + Γ ff + Γ wf1 + Γ wf2 = i Ω i µ i u 2 dx i S wfi β i u 2 s ds β (39) CLu 2 CL where on the right-hand side the first term corresponds to the energy dissipation in the bulk fluids, the second term corresponds to the energy dissipation on the solid surface, and the last term corresponds to the energy dissipation at the contact line. IX. THE ONSET OF NONLINEAR RESPONSE AND BLAKE S MOLECULAR KINETIC THEORY So far our main conclusion is that in the regime of shear rates that we have studied, the responses of the fluid, the interfaces and the contact line obey linear relations. To better appreciate the generality or limitations of this conclusion, let us ask how nonlinear responses set in. There are three major constitutive relations that we should consider: the constitutive 23

24 relation for the bulk fluid, the friction law at the fluid-solid surface and the friction law in the contact line region. We will focus on the onset of nonlinear response in the contact line region. The onset of nonlinearity for the first two constitutive laws have been very well studied in the literature. These nonlinearities set in much later than the nonlinear behavior that we considered here. The linear friction law at the contact line region breaks down at large contact line velocity. This can be seen from the MD result shown in Fig. 18. In this calculation, in order to achieve large slip velocity while maintaining a stable fluid-fluid interface, we weakened the fluid-solid interaction by taking ε 1w = ε 2w = 0.3 in the potential (19), and at the same time we increased the wall density to ρ w = 4. Fig. 18 shows the integrated friction force in the contact line region against the contact line velocity. It is seen that at large contact line velocity, the friction force and the slip velocity deviates substantially from the linear relation. We compared the above nonlinear behavior with the prediction of Blake s molecular kinetic theory. 12,13 In Blake s picture, the contact line motion is an activated process on the adsorption sites of the solid substrate. In the static situation, the contact line is confined at the adsorption sites for most of time, and hops back and forth between different sites occasionally at some rate κ 0. The forward and backward hopping rates are the same. However in a dynamic situation, the stresses near the wall (including the unbalanced Young stress and the normal stress jump across the contact line region) act as an external force on the contact line. This external force lowers the energy barrier for hopping in the forward direction and raises the barrier for hopping in the backward direction. Consequently, from Eyring s rate theory the hopping rates of the contact line in the forward and backward directions become κ + = κ 0 e w/2mk BT, κ = κ 0 e w/2mk BT (40) where w is the work done by the external force per unit displacement length of the contact line, m is the number of adsorption sites per unit area on the solid surface, and k B T is the thermal energy. As a net result of the forward and backward hopping, the contact line moves. Let λ be the lattice constant, then the contact line velocity is given by ( ) w u CL = (κ + κ )λ = 2κ 0 λ sinh. (41) 2mk B T Note that at steady state, the external force from the fluids on the contact line is balanced by the friction force (the fluid-solid interaction). Therefore the above equation gives the 24

25 following relation between the friction force at the contact line and the contact line velocity: δf CL = 2mk B T sinh 1 (u CL /2κ 0 λ). (42) where δ is the width of the contact line region. At small velocity the above equation reduces to the linear friction law F CL = β u CL, where the friction coefficient β = mk B T/κ 0 λδ. In Fig. 18 we compared the above theory with our MD calculations. We plotted the function (42) with parameters k B T = 1.1, m = 0.42, λ = m 1/2 = 1.5 and κ 0 = 0.32 (solid curve). The thermal energy k B T is the same as that in the MD calculation, m and κ 0 are chosen to optimize the fitting. Note that there is a large discrepancy between the parameter m used in the fitting and the actual number of adsorption sites on the surface. The latter is 2 for the fcc lattice with density ρ w = 4, which is five times larger than the value m = 0.42 used in the fitting. This discrepancy indicates that the contact line motion in the current system (with weak coupling between the fluid and solid, and high density of solid particles) is not a pure activated process. Here the contact line motion might be closer to a diffusive motion especially at large capillary numbers. Indeed, whether the contact line motion is an activated process or not depends on the height of the energy barrier that confines the motion of the fluid particles in the adsorption sites on the solid surface. When this energy barrier is comparable to or smaller than the thermal energy, the contact line motion will no longer be an activated process and the above molecular kinetic theory breaks down. In that situation, the contact line motion will become diffusive. The energy barrier can be lowered in different ways, for example, by weakening the fluid-solid interaction, or by increasing the density of the solid, or by applying large external forces (including large shear stress near the wall), etc. From this discussion we see that the contact line motion is an activated process at low capillary number (low driving force). The width of this regime depends on the material parameters of the system. The contact line motion becomes diffusive at large capillary number. From this viewpoint, the linear regime we discussed earlier is in the activated regime with small driving force. At large driving force, one expects either transition to nonlinear behavior in the activated regime or transition to the diffusive regime. Both transitions give rise to nonlinear response. More work is needed to fully clarify this picture. In Fig. 19 we studied the relaxation dynamics of a contact line. In this system, the 25