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Forced Wetting Dynamics: A Molecular Dynamics Study F. Gentner, G. Ogonowski, and J. De Coninck* Centre for Research in Molecular Modeling, Materia Nova, Universite´ de Mons-Hainaut, Avenue Copernic, 1, 7000 Mons, Belgium Received August 15, 2002 We consider molecular dynamics simulations to study a polymerlike liquid meniscus between two parallel plates moving at constant opposite velocities. We investigate contact line motion versus the speed of the solid and the shapes of the liquid interface for several liquid/solid interaction amplitudes. The associated wetting dynamics are studied in detail.
1. Introduction Functional characteristics of nanodevices are to a large degree determined by the properties of material surfaces and their interactions with ambient media. One of the challenges is to understand how the interfacial characteristics, statically and dynamically, affect the performance of nanomechanical devices and thereby determine their technological value. This type of research is specifically concerned with understanding the relevant time scale(s) and mechanism(s) of kinetic energy flow at nanoscale interfaces under motion. Given the likely breakthrough of nanotechnologies soon, it seems clear that studies of liquid-solid interfaces at the nanoscale will be important. With this objective in mind, let us consider a liquid meniscus between two moving solid plates. How will the liquid meniscus behave versus the relative speed between the two plates? Dynamic wetting, or the displacement of one fluid, often air, in contact with a solid by another liquid, is central to this problem. From a macroscopic point of view, this phenomenon is commonly characterized by the macroscopic liquid-solid contact angle, that is, the angle at which the tangent to the liquid-air interface intersects the solid. If this angle is between 0 and π, it is said that the solid is partially wetted by the liquid. The dynamics of wetting is then determined by the variation of this angle with time or wetting velocity. In many published studies (ref 1 and references therein), it has been shown that, to understand dynamic wetting, several types of dissipation must be considered. Indeed, dissipation necessarily occurs due to the velocity gradients of flow of the liquid associated with the wetting process. Very recently,2-7 it has been shown that molecular dynamics techniques can usefully be applied to study the associated dynamics of liquid/solid interfaces. In particular, the spreading of sessile drops on a completely flat surface has been examined in great detail.8-11 Here, we (1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (2) Yang, J. X.; Koplik, J.; Banavar, J. Phys. Rev. Lett. 1991, 67, 3539. (3) Yang, J. X.; Koplik, J.; Banavar, J. Phys. Rev. A 1992, 46, 7738. (4) Thompson, P. A.; Robbins, M. O. Phys. Rev. Lett. 1989, 63, 766. (5) Thompson, P. A.; Brinckerhoff, W. B.; Robbins, M. O. J. Adhesion Sci. Technol. 1993, 7, 535. (6) De Coninck, J.; D’Ortona, U.; Koplik, J.; Banavar, J. Phys. Rev. Lett. 1995, 74, 928. (7) D’Ortona, U.; De Coninck, J.; Koplik, J.; Banavar, J. Phys. Rev. E 1996, 53, 562. (8) de Ruijter, M. J.; De Coninck, J.; Blake, T. D.; Clarke, A.; Rankin, A. Langmuir 1997, 13, 7293.
extend these techniques to study the forced wetting phenomena at the microscopic scale, reconsidering thus the work of Thompson and co-workers,4,5 but here for different liquid and solid interactions and speed regimes. This technique has the great advantage of allowing us to change a single parameter such as the liquid/solid affinity, which is never possible in a real experiment since changing the solid will also change other factors. Typically, using our cluster of Athlonxp 1600+, each simulation takes 1 week of calculation. The paper is organized as follows. Section 2 is devoted to the definition of our model system. The dynamics of wetting is then developed in Section 3. The results are summarized in Section 4. Comparisons with existing models are developed in Section 5. Concluding remarks are given in Section 6. 2. Model System In our simulations, we model a meniscus of liquid between two moving solid plates. Periodic boundary conditions are imposed in the direction parallel to the plates (y-direction). First, let us mention that all potentials between atoms, solid as well as liquid, are described by the standard pairwise Lennard-Jones 12-6 interactions: Vij(r) ) 4ij
[( ) ( ) ] σij r
12
-
σij r
6
(1)
where r is the distance between any pair of atoms i and j. The parameters ij and σij are related in the usual manner to the depth of the potential well and the effective molecular diameter, respectively. Translated into reduced (dimensionless) units (ru), eq 1 becomes
(
Vij*(r*) ) 4
Cij
r*
12
-
Dij
)
r*6
(2)
where the asterisk stands for reduced units. For simplicity, Cij and Dij are chosen the same for each type of atom. Following previous publications for such liquids and solids6-11and to speed up the calculations, we choose here Cff ) Dff ) 1.0, Css ) Dss ) 1.0, and Csf ) Dsf ) 0.5 or Csf ) Dsf ) 0.8, where the subscripts stand for fluid/fluid (9) de Ruijter, M. J. A Microscopic Approach to Partial Wetting: Statics and Dynamics. Ph.D. Thesis, University of Mons-Hainaut, Belgium, September 1998. (10) Blake, T. D.; Clarke, A.; De Coninck, J.; de Ruijter, M. J. Langmuir 1997, 13, 2164. (11) de Ruijter, M. J.; Blake, T. D.; De Coninck, J. Langmuir 1999, 15, 7836.
10.1021/la020724j CCC: $25.00 © 2003 American Chemical Society Published on Web 03/25/2003
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Figure 1. Side view of the liquid meniscus between both plates in the earliest stage of simulation.
Figure 3. Typical layer density profile versus x position.
Figure 2. Perspective view of the liquid meniscus between both plates in the earliest stage of simulation.
(ff), solid/solid (ss), and solid/fluid (sf) interactions. The intrafluid coefficients are standard and the solid-solid coefficients are chosen to produce a stable lattice structure at the temperature of interest. The solid/fluid interactions are characteristic of partial wetting.11 The theoretical range of the Lennard-Jones 12-6 interactions extends to infinity. Strictly, one should therefore evaluate the interactions between all possible pairs in the system. Fortunately, the interaction potentials decrease rapidly as the distance becomes large. We therefore apply a spherical cutoff at 2.5σij. As a result, we only consider short-range interactions in these simulations. We simulate a molecular structure for the liquid by including a strong elastic bond between adjacent atoms within a molecule, of the form Vconf ) Dconf r6 with Dconf ) 1.0. The liquid molecules are always 16 atoms long. This extra interaction forces the atoms within one molecule to stay together and reduces evaporation considerably. We apply a harmonic potential on the solid atoms, so that they are strongly pinned on their reference fcc lattice configuration, to give a realistic atomic representation of the solid surface. Typically, we use 12 160-atom molecules to represent the walls. Another set of 1720 16-atom chains is used to model the liquid meniscus, which will be spherical far away from the solid surface. To simulate a solid moving at constant speed, iterative shifts are applied to the reference fcc lattice configuration at each time step with appropriate periodic boundary conditions. Moreover, the high thickness of the liquid (profile view) gives good statistics for the density profiles. To summarize, we consider a very simple chainlike liquid system, made of 16-atom molecules, between two plates made of a fcc solid lattice. Typical perspective views of the system are given in Figure 1 and in Figure 2. For each atom, we use a computer time step of 5 × 10-3 in reduced units during our simulations, which is more than sufficient to resolve the behavior of the liquid. For carbon-like atoms, that corresponds to 5 fs/time step. Although the model described above is very simplistic, it nevertheless contains the basic ingredients to reproduce the experimental results within realistic computation times.
Figure 4. Typical instantaneous side view of a snapshot. The dots represent the (x, z) coordinates of the liquid atoms, and the points, the computed side profile of the liquid based on the layer density analysis. The static solid layers are located at heights z ) 1 and z ) 46 in reduced units.
3. The Dynamics of Wetting As an initial configuration, we consider a liquid meniscus between two parallel plates, with couplings Csf ) Dsf ) 0.5 or Csf ) Dsf ) 0.8, so that the liquid does not spread too much on the plates. First, we let the system equilibrate until the contact angles converge and the shape of the liquid became steady. Once the liquid and solid systems are equilibrated, we start to move the plates and observe the positions of the atoms as a function of time. Initially, we maintain the temperature of the complete system constant, but once the plates begin to move, we keep only the temperature of the solid constant, so as to mimic the dissipation of energy between the solid and the liquid. This technique is indeed reminiscent of real experiments and has already led to interesting results for the spontaneous spreading case.6,9 To compute the associated contact angles, we must first establish the shape of the liquid/vapor interface. We therefore subdivide the liquid meniscus into several horizontal layers of arbitrary thickness. The constraint on the number of layers is provided by the need to maximize the number of layers, while ensuring that each layer contains enough atoms to give a uniform density. For each layer as shown in Figure 3, we locate its center by symmetry and compute the density of particles as a function of the distance to the center. We then locate the extremities of each layer where the density falls below some cutoff value, usually 0.5, as shown in Figure 4. To check the consistency of the method,
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Figure 6. Side view of the system with points describing the right profiles of the stationary meniscus: each profile corresponds to a different plate velocity and has been translated by a constant in the x direction for the sake of clarity.
Figure 5. Contact angles measured for the static liquid profile.
different layer thickness and cutoff values were considered; these gave almost identical results for the interface shape. The above method enables us to construct the complete profile of the drop and to determine how it evolves with time. Before the plates move, i.e., when we let the system equilibrate, we reach a configuration corresponding to a partial wetting regime with a static contact angle. Since there is no gravity in the simulation, the associated equilibrium shape has to be cylindrical at the macroscopic scale, as shown in Figure 5, except very close to the solid surface. Here, we expect the profile to be perturbed by the solid for energetic and entropic reasons.12 To avoid this problem, we investigated the profile as a function of the number of layers removed near the solid. Evidently, to reproduce the macroscopic thermodynamics of the meniscus, we need to consider enough layers in the center of the interface and to stay sufficiently far from the substrate. The circular fit using all the experimental points except the last four above the substrate leads to stable results for menisci with more then 20 000 atoms as seen in Figure 5. The associated contact angles are then given by the tangents to the circle extrapolated to the solid surface. We have of course checked that the four contact angles are statistically equal at stationary solid walls. After this equilibration procedure, which takes around 200 000 time steps, we apply a constant velocity to both plates in opposite directions for two different affinities between the liquid and the plates: Csf ) Dsf ) 0.5 and Csf ) Dsf ) 0.8. To study the dynamic evolution of the profiles, we proceed in the following way. First, we locate the extremities of each layer as the points where the density falls below 0.5. We then fit each right and left profile with two circles, one for the top half and one for the bottom half. The associated tangents at contact with the substrate allow us to measure a contact angle θ as a function of the number of time steps during our simulations. We fit two circles instead of one as we did in the static case because we cannot rely on the fact that the flow fields are sufficiently weak at the center of the liquid meniscus for its profile for to be staticlike. We choose the two circles because they are simple yet give a robust result for the contact angle found by extrapolation to the solid surface. Moreover, we find that this procedure allows us to recover consistent results when compared with the case of a drop spreading spontaneously on a flat substrate. We (12) De Coninck, J.; Dunlop, F.; Menu, F. Phys. Rev. E 1993, 47, 3.
Figure 7. Typical side view of the meniscus versus time. Each profile is an average over five consecutive snapshots and has been translated by a constant in the x direction. The profiles represented here are separated by 125 000 time steps and are fitted by two pieces of circle as described in the text.
consider the angles found by this procedure to be the contact angles. While these angles are clearly defined, we could have tried to define a microscopic contact angle, which would measure the slope of the interface at the solid surface. However, such an angle would need to be fitted to very few interface points and would then be very susceptible to the fluctuations inherent to our small system, and thus not clearly representative of the fluid motions and interface configurations of the simulated fluid body. In Figure 6, we have plotted all the steady right-side profiles for the Csf ) Dsf ) 0.5 case, each of them corresponding to a different plate velocity. We observe that the liquid shape, initially with a contact angle of 110°, is modified by the displacement of the solid plates and that the global shape becomes inclined. As defined before, we have measured the contact angles by determining the tangent to the two circular fits from the middle of the meniscus for symmetry reasons. For the sake of clarity, we have reproduced in Figure 7 the associated right profiles generated every 125 000 time steps for one typical case [v ) 0.000 17 in reduced units (ru) per time step]. As can be seen, the fitting procedure is good. Of course, with this procedure, we have to ensure the continuity of the meniscus fits at the midpoints with respect to the plates. To ensure that the circular fits are consistent, we decided to take into account an angle measurement if the difference between the lower fit and
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Figure 8. Contact angle in degrees versus time in nanoseconds, when a velocity v is applied to the plates, for Csf ) Dsf ) 0.5 and v ) 1.2 × 10-4 Å/fs (left panel) and for Csf ) Dsf ) 0.8 and v ) 1.4 × 10-5 Å/fs (right panel). The continuous lines are given to guide the eye.
the upper fit at the midpoint was small enough, i.e., less than 1 in reduced units. More than 95% of the fits satisfied this condition. This procedure allows us to reduce the effect of profile fluctuations on the contact angle data. 4. Results Simulations have been launched for a range of velocities (1-16) × 10-4 Å by time step of 5 fs, or equivalently 2-32 m/s. As we consider left and right profiles, advancing angles are obtained by averaging both appropriate extremity profiles. The same procedure is also effective for the receding angle. Typical time evolution of the contact angle for a given solid speed is plotted in Figure 8. If we arbitrarily choose the unit distance to be 3.5 Å, the unit mass to be 10 times the mass of hydrogen, and the temperature of the solid to be 300 K, then the time unit in Figure 8 is 5 fs. The values chosen for these parameters do not affect the spreading behavior, but they are necessary to compare the measured contact angles with true experimental results. After some time and for each plate velocity on each side of the meniscus, we can determine steady advancing (where the solid comes to slide under the meniscus) and steady receding contact angles. Those are given in Figure 9 for Csf ) Dsf ) 0.5 and Csf ) Dsf ) 0.8. The associated error bars correspond to the statistical errors over the last 100 profiles recorded every 2500 time steps. For the Lennard-Jones coupling parameters Csf ) Dsf ) 0.5, the cosine of the advancing contact angle is clearly not linear with velocity and we can see an inflection point at zero velocity. For higher couplings Csf ) Dsf ) 0.8, the curve of cos θ becomes linear versus speed. However, this is over a smaller range of velocities than in the previous case. Beyond this velocity range the meniscus becomes unstable and ruptures, as it also does with the lower coupling at somewhat higher speeds. To check the consistency of the method, we have compared our contact angle measurements with those obtained for larger systems, with 117 600 liquid atoms instead of 27 520. For a velocity of 8 × 10-5 Å/fs and Csf ) Dsf ) 0.5, we get the dynamic contact angles given in Figure 10. Averaging the values over the last 100 frames, we get very compatible results, as reproduced in Table 1. Let us now study in more detail the flux lines in the meniscus. This will allow us to analyze the convection mechanisms between the two plates.
Table 1. Mean Value and the Associated Statistical Error of Both Advancing and Receding Contact Angles for the Last 100 Frames of the Dynamics for the Two Different System Sizes Described in the Text no. of liquid atoms advancing angle (deg) receding angle (deg) 27 520 117 600
98.64 ( 2.63 106.81 ( 2.68
131.39 ( 5.47 135.41 ( 1.50
Since the liquid meniscus relaxes independently of its thickness in the y direction, we project the volume of the meniscus onto the x-z plane. We then subdivide this plane into small square units, so as to make a grid. Every unit stands for the volume of a square-sectioned parallelepiped in the y direction. The size of the units in the grid is constrained by the need to maximize the number of bins, while ensuring that every bin contains enough atoms to give reproducible results. For each bin we compute the center of mass of the atoms, which are part of that bin. A short time later (typically 25 000 computer time steps), we determine the net displacement of its center of mass. This allows us to measure the velocity field of the meniscus in some coarse-grained sense. Assuming the density is uniform in the meniscus, the velocity field could also be considered as lines of flux (streamlines). In Figure 11, velocity fields are shown for the menisci with Csf ) Dsf ) 0.5 and with Csf ) Dsf ) 0.8 at different time steps. In Figure 11a,d, the velocity field of the meniscus during the initial stages of spreading is shown. Every arrow represents the net displacement of a bin over 25 000 computer time steps. The length of the arrow is a relative measure of the local velocity. The advancing contact angle is then lower than 90° for Csf ) Dsf ) 0.8 and higher for Csf ) Dsf ) 0.5. The middle of the meniscus is at x ) 220 Å. The velocity field is basically circular in the meniscus and strongly driven by the moving walls. Close to the solid wall and away from the three-phase line, the velocity is identical to the substrate velocity, indicative of a noslip boundary condition during the inclination of the meniscus and when it becomes steady, as indicated in Figure 11c,f. Regular and constant liquid recirculation is clearly shown in the whole meniscus. The first layer of liquid seems to be continuously entrained by the plate, as can be seen in Figure 12, so that the average velocity field versus time for the highest layer remains similar from the beginning of the simulation to the stabilization of the meniscus. More details about this slip phenomenon will be published elsewhere.
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Figure 9. Steady contact angles and cosine of contact angles versus velocity plates for Csf ) Dsf ) 0.5 and Csf ) Dsf ) 0.8.
Figure 10. Advancing and receding contact angles versus time for Csf ) Dsf ) 0.5 and v ) 8 × 10-5 Å/fs for the considered system with 27 520 atoms (2) and a larger system with 117 600 liquid atoms (b). The continuous lines are given to guide the eye.
At intermediate height, the velocity field becomes more vertical. Close to the middle of the meniscus this shift is even more pronounced. Except near the middle, the arrows are of about the same length. This means that the velocities
inside the meniscus are more or less uniform and that their amplitude changes only in the vicinity of the wall. The average direction of the flow field is toward the contact line. In Figure 11c,f, the final stages of wetting are shown. Away from the solid the flow is still vertical, while in the vicinity of the solid the flow is radial. The gradient along the solid is less pronounced. A ribbon of flow can still be observed near the liquid-vapor interface. In summary, the flow of the meniscus is basically circular with not much variation in the magnitude of the velocity. Closer to the solid wall, the velocity field shifts its direction toward the plate displacement. The center of the meniscus, closer to the axis of symmetry, seems to be relatively stagnant. This results in a ribbon of flow, which follows the liquid-vapor interface. There is a gradient of velocity, almost zero in the middle of the meniscus and maximal close to the contact line. It seems that the liquid is free to flow toward the three-phase zone and that the main dissipation occurs right there. For each coupling parameter a speed limit is observed, beyond which the meniscus ruptures, leaving two liquid drops, which spread on the plates. The typical systematic shape of the meniscus before it breaks is shown in Figure 13.
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Figure 11. Velocity fields for a meniscus (side view) with Csf ) Dsf ) 0.5 (left panels) and Csf ) Dsf ) 0.8 (right panels), respectively, with plate velocities of 1.4 × 10-4 Å/fs and 1.4 × 10-5 Å/fs for three different times, from top to bottom: beginning of the simulation, during wetting dynamics, and steady state at the end of the simulation.
5. Do the Results Agree with Existing Theories? Let us now try to compare our contact angle measurements to existing models. We know that dissipation necessarily occurs due to the confined flow of the liquid associated with the wetting process. One of the dissipation channels can be expressed in terms of the viscosity and can be calculated by locally solving the Navier-Stokes equations for wedge flow. This has been done by Cox.13 He showed that, upon certain simplifications, the contact angle depends on the speed of the liquid front in the following way: (13) Cox, R. G. J. Fluid Mech. 1986, 168, 169.
g(θ) ) g(ω0) +
R ηv ln γ s
()
(3)
with θ the macroscopic contact angle, ω0 a microscopic contact angle, η the viscosity, v the velocity of the liquid front, γ the liquid-vapor surface tension, R a characteristic macroscopic length, and s the slip length. The functions g(θ) and f′(θ) are given by
g(θ) ) f′(θ) )
∫0θ f′(θ˜ ) dθ˜
(θ - sin θ cos θ) 2 sin θ
(4)
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Figure 12. Liquid velocity field in the vicinity of the solid plate at the earliest stages of the simulation. The arrows given on top of the figure represent the displacement of the solid atoms over the considered time window.
Figure 13. Typical snapshot of the meniscus just before it breaks into two pieces at a speed limit v ) 3.2 × 10-4 Å/fs with Csf ) Dsf ) 0.5.
For droplet spreading with small stationary contact angles, this equation leads to the well-known Tanner’s law,14,15 which predicts that
v ∼ θ3
Figure 14. Cube difference for stationary advancing contact angles θ3 - θ03 for Csf ) Dsf ) 0.8 versus wall velocity. The straight line is the linear fit of the curve as predicted by Tanner.
(5)
It is clear that these derivations are essentially macroscopic. However, the hydrodynamic equations predict an infinite dissipation near the contact line. Cox (and many others) solved this problem by imposing some slip model, introducing the slip length s into the equations. The slip model between the liquid and the solid, or any other mechanism, is essentially nonhydrodynamic in nature and acts at the microscopic scale. It should depend on the surface characteristics of the solid and the specific solidliquid interactions. The value of s is a priori unknown. Very carefully obtained experimental results by Garoff and co-workers16,17 show that ω0, the microscopic contact angle, is velocity-dependent for some solid-liquid systems. These results indicate that the mechanism of dissipation near the solid is rather complex. However, if we now plot our results from Figure 9 for Csf ) Dsf ) 0.8 with the difference θ3 - θ03 versus plate velocity, we do indeed observe the validity of Tanner’s law as represented in Figure 14. This result is in perfect agreement with the results of Thompson and co-workers.4,5 The slope of the curve in Figure 14 equals to (9η/γLV) ln (R/s), according to eqs 3 and 5. The fitted value for this
slope is (7.38 × 10-1) ( (2.73 × 10-2) s/m, leading to ln (R/s) ) 53.96, where we have used the estimates for viscosity η ) 0.25 ru (0.04 mPa‚s) and liquid/vacuum surface tension γLV ) 0.47 ru (12.7 mN/m) from ref 11. The capillary and Reynolds numbers can also be estimated: Ca ) ηv/γLV, which varies from 10-4 to 2 × 10-3, and Re ) DvF/η, which varies from 4 × 10-2 to 8 × 10-1, with the height of the slot D ) 46 ru and the density F ) 0.73 ru from ref 11. Very recently,18 Shikhmurzaev has suggested not only that the macroscopic contact angle is velocity-dependent but also that it is dependent upon the whole flow field near the wetting line. This is indeed what we observe in our simulations, since by changing only the liquid/solid interaction, we observe not only a different contact angle but also a different flow field near the wetting line. In reality, a liquid is never allowed to slip freely across a solid surface as can be seen in Figure 12. Adsorption of liquid molecules, immobilizing them to some extent, is unavoidable. The mobility of liquid molecules in the vicinity of a solid has been considered by Blake and Haynes.19,20 They adopted the molecular-kinetic theory (MKT) of liquids, developed by Eyring and co-workers,
(14) Hoffman, R. L. J. Colloid Interface Sci. 1975, 50 (2), 228. (15) Tanner, L. J. Phys. D 1979, 12, 1473. (16) Marsh, J. A.; Garoff, S.; Dussan, E. B. Phys. Rev. Lett. 1993, 70, 2778. (17) Stoev, K.; Rame, E.; Garoff, S. Phys. Fluids 1999, 11, 3209.
(18) Shikhmurzaev, Y. D. Int. J. Multiphase Flow 1993, 19, 589. (19) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (20) Blake, T. D. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993.
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atoms. Let us here remark that a linear expansion of this function, corresponding to the small speed regime, would also allow us to fit the data given above for Csf ) Dsf ) 0.8 given above. In addition, we have verified that the data of Figure 15 cannot be fitted consistently with Hoffman’s law, taking into account the leading correction corresponding to the moderate regime. We thus observe that the wetting dynamics of the system with two different wettability properties can be fairly well described by the hydrodynamic theory at low speed and by the molecular-kinetic theory over all speeds, since in these simulations the associated steady contact angles θe are rather large. This result is in perfect agreement with the results of Brochard-Wyart and de Gennes.21
Figure 15. Cosine of the advancing contact angle versus the plate speed. The line represents the fit described in eq 6 with R2 ) 0.983.
which introduces an equilibrium frequency of molecular displacement K0. The velocity of the contact line is then
v ) 2K0λ sinh
[
]
γLV(cos θ0 - cos θ) 2nkBT
(6)
where T is the absolute temperature, λ is the characteristic length of displacement, n is the number of sites per unit area of solid, γLV is the liquid-vapor surface tension, and θ0 is the equilibrium contact angle; the dynamic contact angle θ used here is defined by thermodynamic considerations and is necessarily macroscopic. Inverting the relation, we get that the cosine of advancing contact angle versus the velocity of solid is given by
cos θ ) a - b sinh-1(cv)
(7)
with a ) cos θ0, b ) 2nkBT/γLV, and c ) 1/(2K0λ). Applying this functional dependence to our data given in Figure 9, we obtain for Csf ) Dsf ) 0.5 the result shown in Figure 15. This shows a good fit with a ) -0.333 ( 0.025, b ) 0.131 ( 0.011, and c ) 17 435.464 ( 6428.952, leading to K0λ = 698 m/s, or equivalently K0 = 2 × 1011 s-1, since λ can be approximated by the distance between two solid
6. Concluding Remarks We have used molecular dynamics to simulate contact line motion of a liquid meniscus between two parallel plates moving at constant but opposite velocity. For two liquid-solid interactions, we have studied the shape of the meniscus versus time for a range of velocities. The associated contact angles have been estimated locally by circular fits of the liquid profiles. The corresponding flux lines describing the displacement of the liquid molecules inside the meniscus have also been computed. It appears that, by changing only the amplitude of the interaction between the liquid and the solid, we observe not only a change in the steady contact angle (as could have been easily guessed) but also a complete change in the flux lines in the vicinity of the solid surface as predicted by Shikhmurzaev. The analysis of the contact angle data reveals apparently two regimes in perfect agreement with the observations of Brochard-Wyart and de Gennes: the hydrodynamic regime, which describes well the data at low speed, and the molecular-kinetic regime, which works well here since the associated steady contact angles θe are rather large. Acknowledgment. We thank T. Blake, A. Clarke, S. Garoff, M. Robbins, and P. G. de Gennes for very useful discussions. This research has been partially supported by the Re´gion Wallonne in the program Feder - Objective I. LA020724J (21) Brochard-Wyart, F.; de Gennes, P. G. Adv. Colloid Interface Sci. 1992, 39, 1.
