Langmuir 2004, 20, 737-742
737
Possibility of Different Time Scales in the Capillary Rise around a Fiber David Seveno and Joe¨l De Coninck* Centre for Research in Molecular Modeling, Materia Nova, University of Mons-Hainaut, Av. Copernic, 1, B-7000 Mons, Belgium Received June 30, 2003. In Final Form: October 10, 2003 By molecular modeling simulations, we study the dynamics of the rise of a meniscus on the outside of a fiber. We develop methods to measure simultaneously the height of the liquid interface and the contact angle versus time. We observe that in the complete wetting case (with an equilibrium contact angle equal to zero), the dynamic contact angle θt behaves asymptotically as t-1 contrary to some experimental results where one observes t-1/2 instead. Using the combined model describing the dynamics of wetting, we predict that there are two different time scale behaviors within this process related to the two dissipation channels: friction between the liquid and the solid, leading to t-1, and hydrodynamics, leading to t-1/2. Using this approach, we find that the maximal speed of spreading on a fiber is a nonmonotonic function of the equilibrium contact angle.
1. Introduction The coating of a flat surface has been the subject of many research works over the past decade. In particular, it has been established1 that the wetting properties of the considered liquid/solid pair play a crucial role in this mechanism. We should here stress that not only the static properties of wetting are relevant but also the dynamic ones. The static properties of wetting are characterized by the contact angle that a sessile drop of the considered liquid will make with the solid. The dynamics of wetting can be described by the dynamics of the contact angle θ or equivalently by its evolution to reach equilibrium. To understand the mechanisms controlling the dynamics of wetting, let us first recall here the simplest case: the spreading of a liquid drop on top of a flat solid substrate. When such a liquid drop is placed in contact with the solid, capillary forces drive the interface spontaneously toward equilibrium. As the drop spreads, the contact angle θ relaxes from its initial maximum of 180° at the moment of contact to its equilibrium angle, θ0 in the case of partial wetting or 0° if the liquid wets the solid completely. Since the shape of the drop is changing versus time, the liquid has to dissipate some energy. Several approaches devoted to the dynamics of a liquid interface have been considered in the literature and differ from each other mostly in the consideration of the effective dissipation channel.2-11 However, it should be clearly understood that all types of dissipation do exist simultaneously, and several at(1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (2) Brochard-Wyart, F.; de Gennes, P. G. Adv. Colloid Interface Sci. 1992, 1, 39. (3) Dussan, E. B.; Ann, V. Rev. Fluid Mech. 1979, 11, 371. (4) Voinov, O. V. Fluid Dyn. 1976, 11, 714. (5) Tanner, L. H. J. Phys. D 1979, 2, 1473. (6) Hoffmann, R J. Colloid Interface Sci. 1975, 50, 228. (7) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (8) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (9) Blake, T. D. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993. (10) Petrov, P. G.; Petrov, J. G. Langmuir 1992, 8, 1762. (11) de Ruijter, M.; De Coninck, J.; Oshanin, G. Langmuir 1999, 15, 2209.
tempts4,10 to work out a combined theory have been made. Following in particular de Gennes1 in his analysis of the energy dissipation in the complete wetting regime case, it was suggested that the unbalanced capillary force should be compensated by the total energy dissipation occurring during the spreading process: namely, the viscous dissipation in the edge of the drop, dissipation at the advancing contact line, and that in the precursor film. In his analysis, however, the emphasis was put on the latter dissipation channel and the energy dissipation due to frictional processes in the vicinity of the liquid/solid interface has been intentionally neglected. This point has been developed recently to describe the dynamics of a sessile liquid drop spreading on a solid surface.11 In this approach, the authors have taken into account these two different dissipation channels, that is, the dissipation due to viscous flows and due to frictional processes in the vicinity of the contact line. Closed-form equations describing the time evolution of the drop base radius have been derived, and several possible kinetic regimes associated with the different dissipation channels have been discussed. The interesting prediction within this combined model was that an hydrodynamic regime is preceded by a molecular kinetic regime where physicochemical aspects of the liquid and the solid are dominating. The identification of these two regimes is very important to assess the influence of the different parameters (viscosity, friction, etc.) on the dynamics of wetting. The time at which the crossover occurs between the two regimes was also precisely estimated. Recent experimental observations have confirmed the validity of this approach.12 Here we are willing to consider the dynamics of wetting on another type of substrate where the curvature may enhance the difference between these different channels: the case of the fiber. Que´re´ and Di Meglio13 have already shown that the capillary rise along a thin vertical fiber can be characterized by a dynamic contact angle which behaves, at least for some systems, according to θt ∼ t-1/2 in agreement with the hydrodynamic approach. Clanet (12) de Ruijter, M. J.; Charlot, M.; Voue´, M.; De Coninck, J. Langmuir 2000, 16, 2363. (13) Que´re´, D.; Di Meglio, J. M. Adv. Colloid Interface Sci. 1994, 48, 141.
10.1021/la030263h CCC: $27.50 © 2004 American Chemical Society Published on Web 01/07/2004
738
Langmuir, Vol. 20, No. 3, 2004
Seveno and De Coninck
and Que´re´14 also scrutinized the dynamics of the capillary rise around either large rods or small fibers for silicone oils with a wide range of viscosities. Furthermore, the dynamics of forced wetting of a fiber has been interestingly analyzed using the molecular kinetic theory by Schneemilch et al.15 Very recently,16-18 it has been shown that large-scale molecular dynamics techniques can usefully be applied to study the dynamics of spreading for sessile drops in great detail. It is thus here our aim to extend these techniques to study, at the nanoscopic scale, the dynamics of spreading of liquids around fibers. This approach has, in particular, 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 the other factors. The paper is organized as follows. Section 2 is devoted to the molecular modeling of the capillary rise around a fiber. Some associated predictions are given in section 3. We end the paper with some concluding remarks. 2. Molecular Modeling 2.A. The Model System. In our simulations, all potentials between atoms, solid as well as liquid, are described by the standard pairwise Lennard-Jones 12-6 interactions:
(( ) ( ) ) σij d
Vij(d) ) 4ij
12
-
6
σij d
(1)
where d is the distance between any pair of atoms i and j. The parameters ij and σij are in the usual manner related to the depth of the potential well and the effective molecular diameter, respectively. Translated into reduced (dimensionless) units (ru), eq 1 becomes
(
Vij*(d*) ) 4
Cij 12
d*
-
Dij
)
d*6
(2)
where the asterisk stands for reduced units. For simplicity, Cij and Dij are chosen to be constant for each type of atom. We choose Cff ) Dff )1.0, Css ) Dss ) 1.0, and Csf ) Dsf ) 0.9, 1.0, 1.05, 1.25, and 1.5 where the subscripts stand for fluid/fluid (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 choice of the solid/fluid interactions ensures that the liquid wets the solid.16-18 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 and shift the potential so that the energy and force are continuous at d* ) 2.5. 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 ) Dconfd6 with Dconf ) (14) Clanet, C.; Que´re´, D. J. Fluid Mech. 2002, 460, 131. (15) Schneemilch, M.; Hayes, R.; Petrov, J.; Ralston, J. Langmuir 1998, 14, 7047. (16) Blake, T. D.; Clarke, A.; De Coninck, J.; de Ruijter, M. Langmuir 1997, 13, 2164. (17) de Ruijter, M. J.; De Coninck, J.; Blake, T. D.; Clarke, A.; Rankin, A. Langmuir 1997, 13, 7293. (18) de Ruijter, M. J.; Blake, T. D.; De Coninck, J. Langmuir 1999, 15, 7836.
Figure 1. Top and side view of the fiber and the liquid annulus at the end of the equilibration. Black, gray, and light gray atoms represent frozen, solid, and liquid atoms, respectively.
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 an harmonic potential on the solid atoms, so that they are strongly pinned on their initial face-centered cubic (fcc) lattice configuration, to give a realistic atomic representation of the solid surface. To summarize, we consider a very simple chainlike liquid system, made by 16 monomers with spherical symmetry, in contact with a solid fiber made from a fcc solid lattice. We always apply a time step of 5 fs19 during our simulations with ij ) 0.276 × 103 J mol-1 and σij ) 3.5 Å, typical of carbon atoms. The simulation consists of a first part where the solid and the liquid are placed far from each other, so that they can equilibrate independently. During this interval, the liquid and the solid are kept at the same temperature. Then, the solid is brought close enough to the liquid to be wetted. From that moment on, the temperature of the liquid is free to evolve whereas the temperature of the solid is kept constant to mimic a real experiment. This leads to a slight decrease (a small percentage) of the liquid temperature over the time needed to reach equilibrium. Although the system is rather simple, it contains all the basic ingredients to model the wetting of a fiber. 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. Herewith, we consider a fiber with radius 20 Å and a meniscus of liquid molecules that we equilibrate inside an annulus of liquid frozen atoms as illustrated in Figure 1. Typical radii of the meniscus are of the order of 60 Å. Once all the elements are equilibrated, which takes around 1 000 000 time steps, we brought into contact the fiber and the meniscus. To compute the associated contact angle for the liquid around the fiber, we first 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 to the fiber. 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. To check the consistency of the method, different shell thickness and cutoff values were considered and these gave almost identical results. (19) Allen, M. P.; Tildesley, D. J. In Computer Simulation of Liquids; Oxford University Press: New York, 1987.
Capillary Rise around a Fiber
Langmuir, Vol. 20, No. 3, 2004 739
Figure 2. Equilibrium profile of a liquid meniscus in contact with a vertical fiber; the associated contact angle is θ0. Figure 4. Capillary rise for three different liquid/solid interactions versus time for carbon-like atoms; from top to bottom, Csf ) 1.05, 1.0, and 0.9. The full lines are given to guide the eye.
Figure 3. Typical profile of the interface. The squares correspond to the measured location of the interface, while the line corresponds to the best fit given by eq 4.
Without gravity, the equilibrium profile y0(x) obeys the James equation:20
(
y0(x) ) r0 cos(θ0) ln(4) - E ln
( (( ) x + r0
x r0
2
- cos2(θ0)
) )) 1/2
(3)
where y0 is the height of the meniscus, r0 is the fiber radius, θ0 is the equilibrium contact angle, E is Euler’s constant, and x is the abscissa of the considered point (Figure 2). Let us, as usual, assume that the liquid during its capillary rise takes an equilibrium meniscus shape or, in other words, that we do have local equilibrium. That is to say that this equation holds for any time t, substituting the dynamic contact angle θt for θ0:
(
y0,t(xt) ) r0 cos(θt) ln(4) - E ln
( (( ) xt + r0
xt r0
2
) ))
- cos2(θt)
1/2
(4)
Using our numerical simulations and eq 4, we can then measure, around the fiber, the dynamic contact angle and the liquid height. Typically, using our cluster of Athlonxp 1600+, each simulation takes 1 week of calculation. A typical snapshot of the profile is given in Figure 3. The line presented in Figure 3 corresponds to the best fit of the data using eq 4. The scattering of the data is (20) James, D. F. J. Fluid Mech. 1974, 63, 657.
Figure 5. Contact angle dynamics for three different liquid/ solid interactions; from top to bottom, Csf ) 0.9, 1.0, and 1.05. The full lines are given to guide the eye.
rather small. This is due to the large number of atoms that we are able to consider here. This method enables us to construct the complete profile of the meniscus and to determine how it evolves with time. The best fits through the profiles described by eq 4 were always located within the region where the density dropped from 0.75 to 0.25, except in the first few molecular layers in contact with the fiber wall. This indicates again that the simulated menisci always retain their equilibrium form during spreading, except very close to the solid surface where the fluctuations can be large. To avoid this problem, we investigated the averaged profile over 10 realizations every 2000 time steps as a function of the number of shells as we would do in a real experiment since the measurements are always averaged over a certain time window. We find that the fit using all the experimental points leads again to stable results when we have a large number of liquid atoms which is the case in this simulation with our 43 760 liquid atoms. By fitting the averaged profiles with eq 4, we are thus able to measure the contact angle θt and the height y0,t as a function of the number of time steps. 2.B. The Results. Versus the time t, we can measure the capillary rise for different solid-liquid couplings. The associated results are given in Figure 4. For the contact angle, we obtain the results presented in Figure 5. To check the consistency of the method, we compare our contact angle measurements with those obtained for a larger system, with 146 912 liquid atoms instead of 43 760. For Csf ) Dsf ) 1.0, we measure for the former
740
Langmuir, Vol. 20, No. 3, 2004
Seveno and De Coninck
Figure 6. Contact angle versus time for two different liquid/ solid interactions, Csf ) 1.5 (2) and 1.25 (9).
Figure 8. Velocity field between 3 ns and 3.5 ns for Csf ) 1.5.
Figure 7. Contact angle versus time for, from left to right, Csf ) 1.5 (2) and 1.25 (9) with a fitted slope of -0.96 ( 0.06 and -1.05 ( 0.09, respectively.
Figure 9. Velocity field between 5 ns and 5.5 ns for Csf ) 1.5.
system an equilibrium contact angle of 33.02° ( 3.17° and 33.11° ( 3.29° for the latter one. This is a good indication that our system is large enough. Increasing the coupling above 1.05, we may thus expect to reach the case where the equilibrium contact angle θ0 is zero. Indeed, by considering Csf ) 1.25 and 1.5, we obtain the contact angle dynamics which are described in Figure 6. Plotting these data in a logarithmic scale, we get the results presented in Figure 7 where we observe a t-1 behavior. This result seems in contradiction with the measurements of Que´re´ et al.,13 where a t-1/2 behavior was observed. 3. Time Behaviors Let us study in more detail the flux lines in the meniscus. Since our system is rotationally invariant, we project the volume of the meniscus onto the x-y plane. We then subdivide this plane into small square units, so as to make a grid. 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 every bin, we compute the center of mass of the atoms which are part of that bin. A short time later (typically 100 000 computer time steps, i.e., 0.5 ns), we determine the net displacement of the 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).
Figure 10. Velocity profiles in the meniscus during the capillary rise. The arrows represent the displacement of the liquid molecules over 0.5 ns (between 6.5 ns and 7.0 ns), i.e., the velocity field (fitted by eq 5, full line). The dotted lines represent the liquid/gas interface at these two different times.
As shown in Figures 8 and 9, regular and constant liquid recirculation is clearly shown in the reservoir. The flow of the reservoir is basically circular with not much variation in the magnitude of the velocity. This results in a ribbon of flow, which follows the liquid-vapor interface. Using these data, we are also able to scrutinize the flow lines in the meniscus. Figure 10 shows that the flow lines are nearly vertical. The lubrication approximation is then valid, and the velocity profile v(x) has the following form:1
v(x) )
3U (-(x - r0)2 + 2x0(x - r0)) 2 2x0
(5)
where U is the velocity of the contact line and x0 is the thickness of the meniscus.
Capillary Rise around a Fiber
Langmuir, Vol. 20, No. 3, 2004 741
A fit of our data given in Figure 10 by eq 5 gives U ) 4.71 ( 0.52 Å ns-1 and x0 ) 11.04 ( 1.49 Å. Following de Gennes,1 the dissipation during the capillary rise can be represented as the sum of two components.
D ) Df + DF
(6)
where Df is the flow dissipation and DF is the dissipation at the triple line. Within the lubrication approximation, the flow dissipation is then simply21
Df )
∫yy
max
min
dy
∫rx
0
0
(dv dx)
η
2
dx
(7)
For small contact angle θt, x0,t ) θty0,t, so that we get
Df )
( )
xmax 3ηU2 ln θt xmin
(8)
xmax is of millimetric order, whereas xmin is of molecular size. The logarithm factor can then be approximated to 12.21 The dissipation due to the attachment of the liquid molecules to the substrate is, at low velocities,
DF )
ζ0 2 U 2
Figure 11. The contact angle θt (degrees) versus time (seconds) obtained by solving eq 14 to the leading order in θt, for a viscosity of 10-3 (Pa.s), a surface tension of 72.4 mN/m, a friction ζ0 of 53.78 (Pa.s) (this particular value has been chosen to enhance the difference between the two regimes), a fiber radius of 1 mm, ln(xmax/xmin) ) 1221, and an equilibrium contact angle of 0°. The two straight full lines are the best fits with the following respective slopes: -0.915 ( 0.0007 and -0.554 ( 0.0004.
This leads for small values of θt and θ0 ) 0° to
dθt ≈ dt
(9)
Now, the relation between the driving force of spreading γ[cos θ0 - cos θt] and the dissipation is provided by the standard mechanical description of dissipative systems dynamics.22
γ[cos θ0 - cos θt] )
∂D ∂U
( ) ( )
θt2 γ 2
6η ln ζ0 +
xmax xmin
(15)
θt
Whenever there is no friction, we thus obtain
(10)
dθt ≈ θt3 dt
(16)
θt ≈ t-1/2
(17)
which leads to
( )
xmax ηU ln + ζ0U γ[cos θ0 - cos θt] ) 6 θt xmin
or
(11)
or in other words
and for the other case with a small viscosity,
U)
γ[cos θ0 - cos θt] xmax 6η ζ0 + ln θt xmin
( )
(12)
dθt ≈ θt2 dt
(18)
θt ≈ t-1
(19)
or
Besides,
U)
∂y0,t(r0) ∂y0,t(r0) dθt ) dt ∂θt dt
(13)
We then easily get
dθt γ(cos θ0 - cos θt) 1 ) dt x ∂y 6η max 0,t(r0) ζ0 + ln θt xmin ∂θt
( )
(14)
(21) de Gennes, P. G. Colloid Polym. Sci. 1986, 264, 463. (22) Landau, L. D.; Lifschitz, E. M., Me´ canique, 3rd ed.; Edition MIR: Moscow, 1969.
Very nice experimental evidence of the first regime has been obtained by Que´re´ and Di Meglio.13 If we solve numerically the differential eq 14 using some RungeKutta procedure for instance, we find that the hydrodynamic regime will be preceded by the friction regime as illustrated in Figure 11 for water on some polymeric fiber. The two straight lines which are represented correspond to the best fits with the powers -0.915 ( 0.0007 and -0.554 ( 0.0004, in agreement thus with the predicted powers. Equating the contact angle behaviors in the two regimes, we get the following estimate for the crossover
742
Langmuir, Vol. 20, No. 3, 2004
Seveno and De Coninck
Figure 12. Speed versus dynamic contact angle for the hydrodynamic (9), the friction model (2), and the combined model (f) for a viscosity of 10-3 (Pa.s), a surface tension of 72.4 mN/m, a friction ζ0 of 53.78 (Pa.s), and an equilibrium contact angle θ0 of 30°.
Figure 13. Plot of -ln(ζ0/γ) versus (1 + cos θ0) for Csf ) 0.9, 1.0, 1.05, and 1.25. The full line is the linear regression of the data.
Table 1. Results of the Fits Applying Equation 14 to the Data Given in Figures 5 and 6 Csf θ0 (deg) γ/ζ0 (Å ns-1 )
0.9
1.0
1.05
1.25
55.0 ( 3.0 25.0 ( 1.2
30.0 ( 3.6 23.0 ( 1.1
15.0 ( 4.1 22.0 ( 1.1
0.0 ( 5.2 21.7 ( 3.1
time T* between the friction regime and the hydrodynamic one.
T* ≈
2ζ02r0 xmax 3γη ln xmin
( )
(20)
Let us stress here that the second regime, the hydrodynamic one, will not be observable if the meniscus reaches its equilibrium height too rapidly. Among the possible predictions that can be derived from eq 14, we will concentrate on the existence of a maximum speed vmax for fiber coating. Indeed, plotting the dynamic contact angle θt versus the triple line velocity U, for a given value of the equilibrium contact angle θ0, we can observe that the maximal speed is obtained for θt ) 180°, as illustrated in Figure 12. Fitting the different contact angle dynamics for Csf ) 0.9, 1.0, 1.05, and 1.25 (Csf ) 1.5 is not shown as it has the same equilibrium contact angle as for Csf ) 1.25) with a purely friction model, we find that the friction ζ0 is a decreasing function of the equilibrium contact angle θ0, as shown in Table 1. Using the relation between the friction and the energy of adhesion as derived in ref 23, we plot -ln(ζ0/γ) versus (1 + cos θ0). This yields a straight line of negative slope (see Figure 13). This property is indicative of a nonmonotonic behavior of the maximal speed versus the equilibrium contact angle. Plotting the maximal speed of wetting as a function of the equilibrium contact angle for the pure hydrodynamic, the pure friction, and the combined models gives the results shown in Figure 14. (23) Blake, T. D.; De Coninck, J. Adv. Colloid Interface Sci. 2002, 96, 21.
Figure 14. Maximal speed versus equilibrium contact angle for the hydrodynamic model (9), the friction model (2), and the combined model (f) for a viscosity of 10-3 (Pa.s) and a surface tension of 72.4 mN/m.
As expected, we get a nonmonotonic behavior of this maximal speed versus the equilibrium contact angle for both the friction and the combined models. This is similar to that reported for flat surfaces23 and will obviously be a function of the different parameters appearing in the problem: viscosity, surface tension, radius of the fiber, and so forth. An experimental investigation of this result would be very interesting. 4. Concluding Remarks Using molecular modeling to mimic real experiments describing the rise of a low surface tension meniscus on the outside of a fiber, we have been able to show the existence of a new dynamic regime, which in terms of contact angles behaves as θt ≈ t-1. We have related this result to the two channels of dissipation during the process: friction between the liquid and the solid and hydrodynamics. The pure friction and the combined models predict a nonmonotonic behavior for the maximal speed of the liquid interface during the rise. Acknowledgment. It is a pleasure to acknowledge here T. D. Blake, A. Clarke, and D. Que´re´ for very stimulating discussions on the subject. This research has been partially supported by the Structural European Funds and by the Re´gion Wallonne. LA030263H
