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A Molecular Dynamics Simulation of Capillary Imbibition G. Martic, F. Gentner, D. Seveno, D. Coulon, and J. De Coninck Center for Research in Molecular Modelling, Materia NovasUniversity of Mons-Hainaut, Parc Initialis, Av. Copernic, 7000 Mons, Belgium
T. D. Blake* Research and Development, Kodak Limited, Harrow HA1 4TY, U.K. Received January 22, 2002. In Final Form: July 31, 2002 To characterize porous media, use is often made of the Lucas-Washburn equation, which relates the rate of capillary penetration of a given liquid to an effective cylindrical pore radius and the contact angle between the liquid and the medium. Here, we extend previous large-scale molecular dynamics simulations developed for flat substrates to show how this tool can be used to study capillary imbibition in some detail. In particular, we demonstrate that the contact angle depends on the rate of wetting, especially during the early stages of pore filling, and that this leads to a reduction in the rate of penetration. The observed behavior can be modeled by a modified form of the Lucas-Washburn equation which takes specific account of these effects.
Introduction The wetting of porous media is a subject of primary importance in many applications such as oil recovery, printing, building conservation, dissolution of washing powders, etc. Although more sophisticated methods are becoming available,1 one practical approach, still used to characterize such systems, is to measure the rate of penetration of the liquid into the porous structure and then to model that structure as a bundle of uniform capillaries.2-6 Within each representative capillary, the driving force for penetration is the pressure drop across the liquid/vapor interface. Under steady conditions in a horizontal capillary, this pressure difference is balanced by the viscous drag of the liquid. Simple analysis of these effects leads to the Lucas-Washburn equation,7,8 which relates the distance of penetration x at time t to the wetting properties of the capillary and the viscosity of the liquid:
x)
x
γLVR cos θt xt 2η
(1)
where γLV is the surface tension of the liquid having viscosity η, θ is the contact angle between the liquid and the capillary wall, and R is the tube radius. Recent reviews have been given by Marmur9 and Zhmud et al.10 In general, the overall balance of forces on the liquid in the capillary may be expressed as
[
( )]
8 d2x dx dx 2 γLV cos θ ) 2 ηx +F x 2+ R dt dt R dt
2
(2)
* To whom correspondence should be addressed: Tel +44 (0)20 8424 4743; Fax +44 (0)20 8424 3750; e-mail
[email protected]. (1) Ridgway, C. J.; Schoelkopf, F.; Matthews, G. P.; Gane, P. A. C.; James, P. W. J. Colloid Interface Sci. 2001, 239, 417. (2) Eley, D. D.; Pepper, D. C. Trans. Faraday Soc. 1946, 42, 697. (3) Chwastiac, S. J. Colloid Interface Sci. 1973, 42, 298. (4) Bruil, H. G.; van Aartsen, J. J. Colloid Polym. Sci. 1974, 252, 32. (5) Lago, M.; Araujo, M. J. Colloid Interface Sci. 2001, 234, 35. (6) Marmur, A.; Cohen, R. D. J. Colloid Interface Sci. 1997, 189, 299. (7) Lucas, R. Kolloid Z. 1918, 23, 15. (8) Washburn, E. W. Phys. Rev. 1921, 17, 273.
where F is the density of the liquid. The left-hand side of eq 1 is the capillary driving force; the first term on the right-hand side gives the viscous resistance of the liquid in the capillary, and the remaining terms describe the inertial resistance. The effects of inertia were first considered by Rideal11 and Bosanquet12 and are usually significant only in the early stages of penetration or when R is large and/or η small. For very small radii, viscous forces are dominant, the inertial terms can be neglected, and one has only to solve the following differential equation:
2 8 dx γ cos θ ) 2 ηx R LV dt R
(3)
with the initial condition x ) 0, t ) 0. Integration leads to eq 1. The precise form of this equation is crucial for the effective characterization of porous media by capillary penetration experiments. Evidently, the rate of penetration is proportional to cos θ. In most treatments, the contact angle is assumed not to deviate significantly from its equilibrium value θ0, but while this may be expedient, it is not necessarily correct. The contact angle is associated with a moving wetting line, and its value may therefore depend on wetting-line velocity13 and hence on the time of penetration. The natural modification of the LucasWashburn equation is to assume that the balance of forces still holds at all times, but now with a velocity-dependent dynamic contact angle, θt. Indeed, in some studies (e.g., ref 14) this approach has been exploited as a way of determining the velocity dependence of the contact angle during capillary rise experiments. Empirically this is straightforward, the change in contact angle being ob(9) Marmur, A. In Modern Approach to Wettability: Theory and Applications; Schrader, M. E., Loeb, E. B., Eds.; Plenum: New York, 1992; pp 327-358. (10) Zhmud, B. V.; Tiberg, F.; Hallstensson, K. J. Colloid Interface Sci. 2000, 228, 263. (11) Rideal, R. K. Philos. Mag. 1922, 44, 1152. (12) Bosanquet, C. H. Philos. Mag. 1923, 45, 521. (13) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (14) Gribanova, E. V.; Mulchanova, L. Kolloidn. Zh. 1978, 40, 217.
10.1021/la020068n CCC: $22.00 © 2002 American Chemical Society Published on Web 09/17/2002
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tained from the variation in the rate of capillary rise with vertical position in the capillary. However, if we wish to model the effect a priori, we need to know the form of the relationship between θt and wetting velocity v ) dx/dt.
linear function of dx/dt, introducing the concept of “wettingline friction”:19
v)
Modified Lucas-Washburn Equation In their 1990 paper, Joos et al.15 combined the LucasWashburn equation with an empirical relationship between θt and the capillary number Ca ) ηv/γLV.16 Numerical integration of the resulting formula showed better agreement with experimental data describing capillary rise as a function of time than the standard LucasWashburn equation, especially at short times. Here, we make use of the relationship between θt and v predicted by the molecular-kinetic theory of wetting due to Blake and Haynes.17,18 This has proved effective in modeling dynamic wetting for a wide range of systems in various geometries, including capillary flow. According to this theory, the macroscopic behavior of the wetting line depends on the overall statistics of individual molecular displacements that occur within the three-phase zone where the fluid/fluid interface meets the solid surface. The velocity of the wetting line is characterized by K, the frequency of molecular displacements, and λ, their average length. In simple cases, λ is the distance between neighboring adsorption sites on the solid surface.17 If K+ is the frequency of molecular displacements in the direction of wetting and K- that in the reverse direction, then the net frequency is Knet ) K+ - K-. The velocity of the wetting line is therefore v ) λKnet. At equilibrium, Knet is zero and K+ ) K- ) K0. Using Eyring’s activated rate theory for transport in liquids and assuming the driving force at the wetting line is the nonequilibrium surface tension force γLV(cos θ0 - cos θt), then the final relationship between θt and v is
[
]
γLV(cos θ0 - cos θt) v ) 2K λ sinh 2nkBT 0
(4)
where n is the number of adsorption sites per unit area, kB Boltzmann’s constant, and T the temperature. Thus, by combining eqs 2 and 4, we can obtain a modified form of the Lucas-Washburn equation in which the contact angle is a function of the rate of liquid penetration, as required:
[
( )]
dx 2 γ cos θ0 - R arcsinh β ) R LV dt
[
( ) ] (5)
dx d2x dx 8 +F x 2+ ηx 2 dt dt R dt
γLV(cos θ0 - cos θt) ζ
(6)
where ζ ) kBT/K0λ3 is the coefficient of friction. Other mechanisms leading to such a linear relationship might be invoked; nevertheless, previous molecular dynamics simulations of droplet spreading19 suggest that the molecular-kinetic approach has an underlying validity. Given these assumptions, eq 5 can be written
1 - P2
dx dx - P3x )0 dt dt
(7)
with the initial condition x(0) ) P1. Here, P2 ) kBT/(γLVK0λ3 cos θ0) ) ζ/γLV cos θ0, P3 ) 4η/(RγLV cos θ0), and we assume n ) λ-2. Equation 7 has the exact solution
x(t) ) -
P2 + P3
x (
)
P2 2t + P1 + P3 P3
2
(8)
This yields the familiar Lucas-Washburn formula when P2 ) 0. Any deviation due to dynamic contact effects depends on the value of P2 or, more particularly, the ratio P2/P3 ) Rζ/4η. Within the context of the model used here, this ratio defines the relative importance of energy dissipation occurring at the wetting line and in the bulk. Thus, the significance of dynamic contact angle effects will be system specific but should tend to be smaller for high-viscosity liquids in narrow pores. Experimentally, it is quite difficult to establish the validity of eq 5 or 7 by measuring the two quantities x and θt vs the time t. New methods are being developed,20 but the experiments remain challenging, especially at short times. However, recent work has shown that molecular dynamics techniques can be a very effective tool with which to study the dynamics of solid/liquid interfaces and wetting processes during Couette and Poiseuille flow.21-24 Macroscopic behaviors and governing laws (e.g., the Laplace equation24) have been reproduced despite the microscopic scale of the simulations; in addition, much has been learned about the underlying mechanisms. Molecular dynamics simulations have also enabled the spreading of sessile drops to be reproduced in some detail.19,25-31 Nevertheless, the problem of capillary imbibition has not previously been tackled in this way. In this paper, we
2
where R ) (2nkBT)/γLV and β ) 1/(2K0λ). This equation can be solved numerically and compared with experiment. However, simplifications leading to an analytical solution are possible in certain circumstances. For small pores, inertial dissipation may be neglected as discussed. Moreover, for small arguments of the sinh function eq 4 reduces to v ) (K0λ/nkBT)(γLV(cos θ0 cos θt)). This is always true in the low-velocity limit when θt is not too far from θ0. We therefore treat cos θt as a (15) Joos, P.; van Remoortere, P.; Bracke, M. J. Colloid Interface Sci. 1990, 136, 189. (16) Bracke, M.; De Voeght, F.; Joos, P. Prog. Colloid Polym. Sci. 1989, 79, 142. (17) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (18) Blake, T. D. In Wettability; Berg, J. C., Ed.; Dekker: New York, 1993; pp 251-309.
(19) de Ruijter, M. J.; Blake, T. D.; De Coninck, J. Langmuir 1999, 15, 7836. (20) Gomez, F.; Denoyel, R.; Rouquerol, J. Langmuir 2000, 16, 4374. (21) Koplik, J.; Banavar, J. R.; Willemsen, J. F. Phys. Rev. Lett. 1988, 60, 1282. (22) Koplik, J.; Banavar, J. R.; Willemsen, J. F. Phys. Fluids A 1989, 1, 781. (23) Thompson, P. A.; Robbins, M. O. Phys. Rev. Lett. 1989, 63, 766. (24) Thompson, P. A.; Brinckerhoff, W. B.; Robbins, M. O. J. Adhes. Sci. Technol. 1993, 7, 535. (25) Yang, J. X.; Koplik, J.; Banavar, J. R. Phys. Rev. Lett. 1991, 67, 3539. (26) De Coninck, J.; D’Ortona, U.; Koplik, J.; Banavar, J. R. Phys. Rev. Lett. 1995, 74, 928. (27) D’Ortona, U.; De Coninck, J.; Koplik, J.; Banavar, J. R. Phys. Rev. E 1996, 53, 562. (28) De Coninck, J.; Voue´, M. Interface Sci. 1997, 5, 141. (29) Cazabat, A. M.; Valignat, M. P.; Villette, S.; De Coninck, J.; Louche, F. Langmuir 1997, 13, 7293. (30) de Ruijter, M. J. A Microscopic Approach to Partial Wetting: Statics and Dynamics. Ph.D. Thesis, University of Mons-Hainaut, Belgium, 1998. (31) Blake, T. D.; Clarke, A.; De Coninck, J.; de Ruijter, M. J. Langmuir 1997, 13, 2164.
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therefore extend these techniques to investigate the validity of the Lucas-Washburn equation with the above modification, specifically a velocity-dependent dynamic contact angle. Molecular Dynamics Simulations In our simulations, we consider liquid penetration into a single, cylindrical pore. All potentials between atoms, solid as well as liquid, are described by the standard pairwise Lennard-Jones 12-6 interaction:
[( ) ( ) ]
Vij(r) ) 4ij
σij r
12
-
σij r
6
(9)
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.32 Translated into reduced (dimensionless) units (ru), eq 9 becomes
Vij*(r*) ) 4
(
Cij 12
r*
-
Dij
)
r*6
(10)
For simplicity, Cij and Dij are the same for each type of atom. We choose Cff ) Dff ) 1.0, Css ) Dss ) 1.0, and Csf ) Dsf ) 1.0, where the subscripts stand for fluid/fluid (ff), solid/solid (ss), and solid/fluid (sf) interactions. The fluid/ fluid coefficients are standard, and the solid/solid coefficients produce a stable lattice structure at the temperature of interest. The choice of the solid/fluid interactions ensures that the liquid wets the solid.26 The theoretical range of the Lennard-Jones 12-6 interactions extends to infinity. Strictly, one should evaluate the interactions between all possible pairs in the system. However, the interaction potentials decrease rapidly as the distance becomes large. We therefore apply a spherical cutoff at 2.5σij, so that the pair potential is set to zero if r* g 2.5. As a result, we consider only shortrange interactions. We simulate a molecular structure for the liquid by including a strong elastic bond between adjacent atoms within a molecule, of the form Vconf ) Dconfr6 with Dconf ) 1.0. The liquid molecules are always 16 atoms long. This extra interaction forces the atoms of one molecule to stay together and reduces evaporation considerably. To give a realistic atomic representation of the solid surface, we apply a harmonic potential to the solid atoms, so that they are strongly pinned in their initial fcc lattice configuration. To avoid edge effects, we apply periodic boundary conditions in the lateral y and z directions, perpendicular to the x direction introduced before. The cubic box containing all the atoms is set to be large enough, such that the liquid atoms never reach the far end of the pore during the simulation. To summarize, we consider a very simple, chainlike liquid system, made of 16 monomers with spherical symmetry, in a pore of radius R made from a fcc solid lattice. A typical side view of the cylindrical system is given in Figure 1. We always apply a computer time step of 0.005 ps during our simulations, which is more than sufficient to resolve the behavior of the liquid. Although the system is rather simple, it contains all the basic ingredients to model the wetting of a microscopic pore. As the initial configuration, we consider a liquid droplet on top of the solid, which we let equilibrate with small (32) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford Science Publications: Oxford, 1989.
Figure 1. Typical snapshot of the system with 140 000 atoms.
couplings Csf and Dsf, in such a way that the droplet does not enter the pore. This procedure allows us to focus on the capillary dynamics by subsequently increasing the values of Csf and Dsf to 1 and so allowing the system to evolve within the pore. In this way we can measure the penetration distance x vs time t. Note that the increase in the coupling constants also causes the liquid to spread on the solid, yielding an effectively infinite reservoir with periodic boundaries and a flat surface as shown in Figure 1. Concerning the contact angle, we proceed in the following way, mimicking a real experiment. First, we subdivide the liquid index into several concentric cylindrical shells of arbitrary thickness. The constraint on the number of shells is provided by the need to maximize their number, while ensuring that each shell contains enough molecules to give a uniform density. For each shell, we compute the density of particles as a function of the distance x into the pore. We then locate the extremity of the shell at the distance where the density falls below a cutoff value of 0.5 times the liquid density. A circular fit to these extremities gives us the meniscus curvature and the contact angle at the pore wall. To check the consistency of the method, different shell thickness and cutoff values were considered and were found to give almost identical results. These methods enable us to construct the complete profile of the meniscus and to determine how it evolves with time. A typical profile is shown in Figure 2. The best circular fits through the profiles were always located within the region where the density dropped from 0.75 to 0.25, except over the first few molecular layers in contact with the capillary wall. This indicates that the simulated menisci always retain their spherical form during spreading, except very close to the solid surface. Indeed, we expect the profile to be perturbed in the vicinity of the solid for both energetic and entropic reasons.33 To avoid this problem, we investigated the profile as a function of the number of molecular layers used, from the center of the pore outward. Evidently, to reproduce the macroscopic thermodynamics of the meniscus, we need to consider enough layers yet stay sufficiently far from the substrate. We found that a circular fit using all the experimental points except the last few in contact with (33) De Coninck, J.; Dunlop, F.; Menu, F. Phys. Rev. E 1993, 47, 3.
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Figure 2. Typical meniscus profile. The line shows the circular fit and the contact angle.
Figure 4. Contact angle θt vs time t for R ) 70 Å (bottom) and R ) 50 Å (top). The curves were obtained using eq 4 and the parameters from Table 1. Table 1
Figure 3. Penetration distance x vs the time t for capillary radius R ) 70 Å (top) and R ) 50 Å (bottom). The smooth lines correspond to the fits given by eq 7, and the corresponding parameters are given in Table 1.
the substrate led to stable results, provided we had more than 46 000 liquid atoms in the simulation. From the tangent of the circular fit at the wall, we were able to measure the contact angle θt as a function of the number of time steps and combine this with the information on penetration distance x and rate dx/dt. The penetration distance was measured at a position corresponding to one-third of the radius from the wall to avoid fluctuations at the center of the meniscus. Separate simulations were performed for R ) 50 and 70 Å. Results and Discussion Several conclusions can be drawn from the results of our simulations. First, we are able to confirm that the liquid flow inside our pore obeys Poiseuille’s law, which is not necessarily expected at this scale. The corresponding parabolic velocity profile also confirms the validity of the molecular dynamics approach as a means to check the consistency of the Lucas-Washburn equation. In Figure 3 we have plotted penetration distance vs time for both capillary radii. After some short time (which we estimate equal to ≈1 ns), during which the influence of the initial conditions are felt, we find that the dynamics of x(t) can indeed usefully be described by eq 7. An adaptive step size Runge-Kunta algorithm34 combined with a downhill Simplex method was used to fit the data. The results are given in Table 1. To estimate the error bars, we used the bootstrap method.34 As can be seen from Table (34) Press, W.; Teukolsky, S.; Vetterling, W.; Flannery, B. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, 1992.
R (Å)
P1
P2
P3
70 50
10.5000 ( 1.0600 8.3963 ( 1.3319
0.0407 ( 0.0040 0.0436 ( 0.0054
0.00183 ( 0.0001 0.0025 ( 0.0001
1, the values of P2 do not depend on the tube radius R. This is as predicted by the molecular-kinetic theory, since P2 ) kBT/(γLVK0λ3 cos θ0) depends only on molecular parameters and the thermodynamic contact angle. We also find, as predicted, that P3(R ) 70) ) (5/7)P3(R ) 50). A more quantitative way of estimating the accuracy of the fit given by eq 7 is to compute the quantity χ2 ) (1/ n)∑t(x(t) - x*(t))2, where x*(t) is obtained from the fit. The resulting values are 1.23 for R ) 70 Å and 1.70 for R ) 50 Å, which confirm the effectiveness of our approach. In terms of the proposed wetting-line friction, the results indicate that ζ(R ) 70)/ζ(R ) 50) ∼ 1, meaning that ζ is independent of the radius of the pore, as anticipated. To confirm this, we have performed an independent simulation of spreading for a droplet of the same liquid on top of an equivalent flat substrate. From the resulting dynamics of the contact angle, we obtained ζ(R ) 70)/ζ(R ) +∞) ∼ 1. This shows that ζ is a local quantity, independent of the large-scale geometry of the substrate.18 Another way to test the utility of our approach is to take the parameters obtained by fitting eq 7 and use eq 4 to compare the predicted values of θt to those found in the simulations. As illustrated in Figure 4, the results are in very good agreement. The behavior of the dynamic contact angle revealed in Figure 4 also hints at a potentially important effect, clearly identified by Joos et al.15 but more usually overlooked. Because of velocity dependence of the contact angle and high initial velocities, the dynamic contact angle during the early stages of capillary penetration may approach 90°, thus reducing the capillary driving force, before decreasing toward its equilibrium value as the liquid penetrates further into the capillary and decelerates.
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Figure 5. Capillary rise as a function of time predicted by eq 11 for ethanol in a glass capillary of radius R ) 689 µm: (a) ζ ) 80 mPa‚s; (b) ζ ) 0. Open circles are data of Que´re´.36
Figure 6. Contact angle as a function of time predicted by eq 11 for ethanol in a glass capillary of radius R ) 689 µm, with ζ ) 80 mPa‚s.
On the basis of general principles,35 we can expect the imbibition velocity and contact angle to self-regulate in such a way as to minimize dissipation. Irrespective of any inertial or entrance effects,10,12 liquid penetration will therefore be slower than predicted by the conventional Lucas-Washburn equation. In our simulations, comparing the situation to the case for which P2 ) 0(θt ) θ0), we find that the difference [x(t) - xP2)0(t)] may reach 20%, which is quite significant and would, for example, have repercussions on any estimate of R. It might be argued that our results have been obtained only for pores of microscopic dimensions and that with pores of larger radius inertial effects (neglected here) are likely to be of greater significance than effects due to changes in the dynamic contact angle. Evidently, much depends on the relative magnitude of the relevant terms in eq 5. To assess this, we have numerically solved the full equation for a realistic system, specifically ethanol (η ) 1.17 mPa‚s, γLV ) 21.6 mN m-1, and density F ) 780 kg m-3) rising vertically against gravity up a glass capillary (R ) 689 µm). This system has been investigated experimentally by Que´re´,36 who found a final capillary rise height of 8.1 mm, implying an equilibrium contact angle of 8.7°. For a vertical capillary, eq 5 may be rewritten as
the equilibrium contact angle is 27°, indicative of weaker solid/liquid interactions than for ethanol on glass, so a smaller value of ζ is to be expected.38 Interestingly, if ζ is set equal to zero, we get the inflected curve (b) in Figure 5. This shows the oscillatory behavior characteristic of inertially dominated systems and reported by Que´re´ for diethyl ether in the same capillary. Apparently, for ethanol on glass, dynamic contact angle effects are sufficient to dominate the effects of inertia, but for diethyl ether the influence of the dynamic contact angle is much weaker. On the basis of his analysis of the initial rate of capillary rise for ethanol, Que´re´36 in fact postulated the existence of a dynamic contact angle due to viscous dissipation near the wetting line (rather than wetting-line friction) and suggested a value of order 55°. He also noted that the initial rate of rise was significantly increased if the capillary was prewetted with the liquid. Figure 6 shows the predicted behavior of the dynamic contact angle as a function of time obtained using eq 10 with ζ ) 80 mPa‚s (the value obtained by fitting the imbibition data). This reveals a contact angle which falls from an initially high value of about 70° toward its equilibrium value as the rate of capillary rise slows and the meniscus approaches its final, static position. A dynamic contact angle of 55° is attained after about 23 ms. In summary, on the basis of our simulations and the analysis of experimental systems investigated by Que´re´,36 it seems reasonable to conclude that dynamic contact angle effects can be significant, and their omission could lead to erroneous modeling of capillary imbibition experiments. This result should not be too surprising. Unless the effects were significant, it would not be possible to determine realistic data on the velocity dependence of the dynamic contact angle from the rate of capillary rise, as described in the literature.14 Nevertheless, this dependence may not have received sufficient attention in the past.
[
( ) ]
8 2 ζ dx dx γ cos θ0 + ) Fgx + 2 ηx R LV γLV dt dt R
[
F x
( ) ] (11)
d2x dx + 2 dt dt
2
where g is the acceleration due to gravity, and we assume, as before, that the system is in the linear dynamic contact angle regime, so that eq 6 applies. The monotonically increasing curve (a) in Figure 5 shows the result of fitting eq 11 to Que´re´’s data using the coefficient of wetting line friction ζ ) nkBT/K0λ as an adjustable parameter. Excellent agreement is found with ζ ) 80 mPa‚s. Unfortunately, there are no published dynamic contact angle data for ethanol on glass, so an independent check on the value of ζ is not possible. However, Bower37 has reported values of λ ) 0.76 nm, and K0 ) 2.27 × 108 s-1 for ethanol on poly(ethylene terephthalate), yielding ζ ) 41 mPa‚s. Here, (35) Batchelor, G. K. An Introduction to Fluid Dynamics; Cambridge University Press: Cambridge, 1967. (36) Que´re´, D. Europhys. Lett. 1997, 39, 533. (37) Bower, C. L. Dynamic Wetting in Solvent Systems, 4th European Coating Symposium: Brussels, 2001.
Conclusions We have been able to model the early stages of liquid penetration into a cylindrical capillary using molecular dynamics. Our simulations have shown that the familiar Lucas-Washburn equation should be corrected to take into account the velocity dependence of the dynamic contact angle. Our results are compatible with the modified form of the equation given by eqs 7 and 8. The effects we (38) Blake, T. D.; De Coninck, J. Adv. Colloid Interface Sci. 2002, 96, 21.
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observe are significant, especially during the initial stages of capillary penetration when the wetting velocities are highest. While inertial effects may also important, the influence of the dynamic contact angle should not be ignored.
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Acknowledgment. We acknowledge partial support from the Region Wallonne, the program Feder-Objective I, and Kodak Limited. LA020068N
