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Spreading Drop Dynamics on Porous Surfaces David Seveno, Vale´rie Ledauphin, Gre´gory Martic, Michel Voue´, and Joe¨l De Coninck* Universite´ de Mons-Hainaut, Centre for Research in Molecular Modeling, Place du Parc 20, B-7000 Mons, Belgium Received January 11, 2002. In Final Form: June 17, 2002 We study the dynamics of spontaneously spreading drops of viscous liquids on top of porous medium made by non-interconnected pores. A model for the single pore case is proposed on a theoretical basis and validated by molecular dynamics simulations. The generalization to the multiple non-interconnected pore cases allows unique characterization of the dynamics of spreading on a porous surface by essentially two parameters referring to the size of the considered pores and the friction between the liquid and the considered surfaces.
Introduction The dynamic of wetting, or the displacement of one fluid in contact with a solid by another fluid, is central in many industrial applications, such as coating deposition,1 printing,2 and enhanced oil recovery3 just to quote a few. It has frequently been characterized by the spontaneous spreading of droplets.4-17 Within this case of spontaneous wetting, the variation in time of the contact angle θ, base radius R, and/or contact line velocities ν is in general measured by means of optical inspection. The advantages of this method are that only small samples of flat solid substrates are required, that a large range of contact angles is covered by a single experiment, and that the geometry of the drops is well-known. One disadvantage exists in the fact that with spontaneous spreading, the dynamics are governed by the physics of the system, not by the user. From the experimental observations, it is clear that this type of dynamics refers to the details of the considered surface. Several models have been proposed in the literature to describe the dynamics of the associated contact angle or base radius. Theses theories differ essentially by the way they describe the dissipation of energy during the drop shape transformation. Among them, we have the molecular kinetic model due to Blake and Haynes,18,19 the linear (1) Blake, T. D.; Clarke, A.; Ruschak, K. J. AIChE J. 1994, 40, 229. (2) Karnik, A. R. J. Photogr. Sci. 1977, 25, 197. (3) Paterson, A.; Robin, M.; Fermigier, M.; Jenffer, P.; Hulin, J. P. JPSE 1998, 20, 133. (4) Strella, S. J. Appl. Phys. 1970, 41, 4242. (5) Dussan, V. E. B.; Davis, S. H. J. Fluid Mech. 1974, 65, 71. (6) Ogarev, V. A.; Timonina, T. N.; Arslanov, V. V.; Trapeznikov, A. A. J. Adhes. 1974, 6, 337. (7) Voinov, O. V. Fluid Dyn. 1976, 11, 714. (8) Tanner, L. H. J. Phys. D 1979, 12, 1473. (9) Chen, Q.; Rame´, E.; Garoff, S. J. Fluid Mech. 1997, 337, 49. (10) Dodge, F. T. J. Colloid Interface Sci. 1988, 121, 154. (11) Extrand, C. W. J. Colloid Interface Sci. 1993, 157, 72. (12) Zosel, A. Colloid Polym. Sci. 1993, 271, 680. (13) Brochard-Wyart, F.; de Gennes, P. G. Langmuir 1994, 10, 2440. (14) Seaver, A. S.; Berg, J. C. J. Appl. Polym. Sci. 1994, 52, 431. (15) Cazabat, A. M.; Gerdes, S.; Valignat, M. P.; Vilette, S. Interface Sci. 1997, 5, 129. (16) de Ruijter, M. J.; De Coninck, J.; Blake, T. D.; Clarke, A.; Rankin, A. Langmuir 1997, 13, 7293. (17) de Ruijter, M. J.; De Coninck, J.; Oshanin, G. Langmuir 1999, 15, 2209-2216. (18) Blake, T. D. The Contact Angle and Two-phase Flow. Ph.D. thesis, University Bristol, 1968. (19) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421.
version of it also known as the friction dissipation theory, and the hydrodynamic dissipation theory due to Voinov7 and many others (for a review, see ref 20). Recently, de Ruijter et al. presented17 a microscopic analysis validating the molecular kinetic theory (MKT) dynamics to describe effectively the spreading of drops. In that paper, we have shown that large-scale molecular dynamics simulations can be used to study droplet spreading. The dynamics of droplet spreading were followed for several drops, from an initial contact angle of 180° toward the equilibrium values. At equilibrium, we measured the density, the viscosity, the surface tension, and the contact angles. We also described a method to calculate directly the microscopic parameters appearing in the MKT. During spreading on the other hand, we measured the dynamic contact angle relaxation. Using all these measurements, we showed that the linear version of the molecular-kinetic model is valid in partial wetting. Indeed, we were able to predict the dynamic contact angle relaxation independently of its measurement. These results directly support the importance of the friction dissipation at the microscopic scale for spreading. Of course, this theory refers to the constant volume case since it deals with pure flat surfaces. On top of a porous surface however such as paper, one now has to take into account the fact that some part of the drop will enter the pores of the medium. It is precisely the aim of this paper to study the dynamics of this problem complementing interesting results obtained by Clarke et al.,21 Marmur,22 and Aradian et al.23 in this direction. Theoretical Considerations Consider a nonvolatile, sessile liquid drop of volume V, which is first placed on an ideal, horizontal solid substrate, exposed to a neutral gas phase, for example, air. At time t ) 0 the initial shape of the drop is characterized by the base radius R and the contact angle θ. In accordance with the experiments, we will suppose in what follows that θ is less than π/2. We are only concerned with the case of partial wetting. Consequently, the liquid drop will wet (20) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (21) Clarke, A.; Blake, T. D.; Carruthers, K.; Woodward, A. Spreading and imbibition of liquid droplets on porous surfaces. Preprint. (22) Marmur, A. J. Colloid Interface Sci. 1988, 122, 209. (23) Aradian, A.; Raphae¨l, E.; de Gennes, P. G. Eur. Phys. J. E 2000, 2, 367-376.
10.1021/la025520h CCC: $22.00 © 2002 American Chemical Society Published on Web 08/14/2002
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the solid substrate until it reaches its equilibrium shape with base radius R0 and contact angle θ0 (θ0 > 0), which obeys Young’s equation. It is also supposed that the drop, which is placed initially in a nonequilibrium configuration, retains the form of an ideal spherical cap at any moment in time. This implies that the instantaneous configuration of the drop can be totally described by a single parameter: the time-dependent base radius R(t) or the dynamic contact angle θ(t). By measurement of, for instance, the dynamic contact angle, the corresponding values of the instantaneous base radius can be calculated by virtue of the conservation of volume condition, which gives
R3(t) )
(3Vπ )Φ[θ(t)]
Figure 1. The characterization of a sessile drop.
(1)
where
Φ[θ(t)] )
sin3 θ(t) 2 - 3 cos θ(t) + cos3 θ(t)
(2)
Dissipation in the vicinity of the contact line results from various physicochemical processes, which lead to the attachment of liquid molecules to the solid. Such a dissipation channel has been considered already in refs 17 and 19 and subsequently scrutinized by several authors (see, e.g., refs 24 and 25). It is in general referenced to as the molecular-kinetic theory of wetting. For small velocities, this theory predicts that the wetting line moves with velocity, v, and the liquid exhibits a dynamic advancing contact angle θ ) θ(v) such that θ > θ0, where θ0 is the equilibrium contact angle, according to
v)
[
]
-γLV(1 + cos θ0) λ3 exp [γLV(cos θ0 - cos θ)] ηLυL nkBT (3)
where ηL is the viscosity of the liquid and where, strictly, υL is the volume of the “unit of flow” (however, for many simple liquids, the unit of flow is a single molecule, so that υL is equal to the molecular volume), γLV is the surface tension of the liquid, kB is the Boltzmann constant, T is the absolute temperature, λ is the length of the individual molecular displacements that occur along its length. In the simplest model, these displacements occur to and from adsorption sites on the solid surface, so that while λ will be influenced by the size of the liquid molecules, it will depend more strongly on the spacing of the putative adsorption sites which exist with a density n per unit of area, with for the simplest model we may imagine, λ ) 1/n1/2. In terms of friction, we then get
v ) γLV(cos θ0 - cos θ)/ζ
(4)
where ζ is considered to be a friction coefficient per unit length of the contact line on the solid surface.17 Let us now introduce the adsorption of the liquid within the substrate. Physically, this will be due to the presence of pores at the surface.For a single pore, the amount inside the pore can be described by the Lucas-Washburn equation which, without gravity, can be written as
( ) ((
( ))
2γLV 8ηx dx dx 2 d2x cos θt* ) 2 +F +x 2 r dt r dt dt
)
(5)
(24) De Coninck, J.; Voue´, M.; de Ruijter, M. Curr. Opin. Colloid Interface Sci. 2001, 6, 49-53. (25) Blake, T.; De Coninck, J. Adv. Colloid Interface Sci. To appear.
Figure 2. Early stages of liquid penetration into a capillary showing successive position of the meniscus.
where r denotes the radius of the pore, F is the density of the liquid, and θt* is the contact angle of the liquid inside the pore at time t. This contact angle θt* has also been shown to be associated to the friction between the liquid and the pore ζp,26 we thus get from (4)
cos θt* ) cos θ0* -
ζp dx γLV dt
(6)
where x˘ denotes the velocity of the liquid. Combining (5) and (6), we get the corrected LucasWashburn equation in which we now take into account the dissipation between the liquid and the surface of the pore.25 Now, from the sessile part of the drop, we will have an additional pressure due to the curvature of the reservoir.22 The next result is that the height x of the liquid within the pore can be described by the equation
() ( ))
2γLV sin θt 2γLV 2 dx 8ηx dx cos θ0* ) ζp + 2 + + Rt r r dt r dt F
((
dx 2 d2x +x 2 dt dt
)
(7)
Within this single pore case, and provided there is no hysteresis effect, we will then completely characterize the dynamics of drop spreading in terms of the three following equations: on top of the solid surface, from (1), (2), and (4)
( )
π dθ (t) ) dt 3V(t)
1/3
γLV (cos θ0 - cos θ(t))(cos(θ(t)) + ζsurf
2)((cos(θ(t)) + 2)(cos(θ(t)) - 1)2)1/3 (8) inside the pore, from (7)
() (( ) ( ))
2ζp dx 8ηx dx 2γLV sin θt 2γLV cos θ0* ) + 2 + + Rt r r dt r dt F
dx 2 d2x +x 2 dt dt
(9)
(26) Martic, G.; Gentner, F.; Seveno, D.; Coulon, D.; De Coninck, J.; Blake, T. To appear.
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Figure 3. The geometrical variables appearing in the problem.
the volume of the drop on top of the solid surface
Vt ) V0 - πr2x +
[
]
πδ3 πr2δ + 6 2
(10)
Figure 4. A typical time behavior for the radius of the sessile droplet (solving eqs 8-10 with the parameters given in the text).
where x denotes the height of the liquid meniscus in contact with the pore surface and
δ)
((
1 r -r cos θt* cos θt*
) ) 2
-1
1/2
(11)
This height x is in fact related to the normal current J23 describing the aspiration within the pore
J ) x/t
(12)
which asymptotically reduces to
J ∼ 1/t1/2
(13)
Our dynamical model differs from ref 23, where the case of a strongly pinned drop contact line has been considered due to hysteresis effects. Numerical Predictions This set of three differential equations can be solved numerically using a Runge-Kutta algorithm.27 For a liquid with a surface tension of 72 mN/m, let us consider a drop of typical volume 0.1 µL. Its viscosity, ηL, is chosen as 1 cP, υL is typically of the order of (3 Å),3 and we have chosen for simplicity λ equal to 3 Å. For the pore, we have considered a radius of 50 µm. To simplify our presentation, we have taken the same equilibrium contact angle for the surface and inside the pore, θ0 ) θ0* ) 0. We then get for the radius of the contact area R the prediction reproduced in Figure 4. In the same way, we can measure the remaining volume on the surface versus time. We get the results presented in Figure 5. Numerical Simulations To check the validity of these predictions, we have performed a detailed numerical simulation of a drop spreading on top of a capillary tube. Very recently, it has been shown that molecular dynamics techniques can usefully be applied to study the dynamics of spreading of sessile drops in great detail. Here our aim is to extend these techniques to study, at the microscopic scale, the dynamics of spreading of drops (27) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, 1992.
Figure 5. Typical behavior of the remaining volume of the sessile droplet (solving eqs 8-10 with the parameters given in the text).
on porous substrates. 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
(14)
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.28 Translated into reduced (dimensionless) units (r.u.), eq 14 becomes
(
Vij*(d*) ) 4
Cij 12
d*
-
Dij
)
d*6
(15)
where the asterisk stands for reduced units. For simplicity, Cij and Dij are chosen constant 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 intrafluid coefficients are standard, and the solid-solid coefficients are chosen to produce a stable lattice structure at the (28) Allen, M. P.; Tildesley, D. J. Computer Simulations of Liquids; Clarendon Press: Oxford, 1987.
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Figure 6. A typical snapshot of the system with 76000 atoms.
temperature of interest. The choice of the solid/fluid interactions ensures that the liquid wets the solid.29 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 ) 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 initial face centered cubic (fcc) lattice configuration, to give a realistic atomic representation of the solid surface. The cubic box containing all the atoms is set to be large enough so that the liquid atoms never reach the far end of the surface and the pore during the simulation. To summarize, we consider a very simple chain-like liquid system, made by 16 monomers with spherical symmetry, on a porous substrate made from a fcc solid lattice. A typical side view of the system is given in Figure 6. We always apply a reduced computer time step of 0.005 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 porous substrate. Using these numerical simulations, we can easily measure the covered surface versus time, the radius base, the contact angle, and the remaining liquid volume versus time t, as well as, into the pore, the dynamic contact angle and the liquid height penetration. To compute the associated contact angle for the sessile part of the drop, we proceed as already described in ref 16. Let us summarize it here for completeness. First, we subdivide the liquid droplet 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 molecules to give a uniform density. For each layer, 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 extremity of the layer as the distance where the density falls below some cutoff value, usually 0.5. To check the consistency of the method, different layer thickness and cutoff values were considered; these gave almost identical results. This method enable us to construct the complete profile of the drop and to determine (29) De Coninck, J.; D’Ortona, U.; Koplik, J.; Banavar, J. Phys. Rev. Lett. 1995, 74, 928-932.
Figure 7. The height inside the pore versus time t for carbonlike atoms.
how it evolves with time. The best circular fits through the profiles were always situated within the region where the density dropped from 0.75 to 0.25, except in the first few layers above the substrate. This indicates that the simulated drops 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 energetic and entropic reasons.16 To avoid this problem, we investigated the profile as a function of the number of layers used, from top to bottom. Evidently, to reproduce the macroscopic thermodynamics of the drop, we need to consider enough layers and to stay sufficiently far from the substrate. The circular fit using all the experimental points except the last five above the substrate leads to stable results. Thus, we are able to measure the contact angle θ (t) as a function of the number of time steps during our simulations. For the contact angle inside the pore, with diameter 75 Å, 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. 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. More details about this procedure can be found in ref 26. A typical snapshot of the observed profile is given in Figure 6. This method enables us to construct the complete profile of the meniscus and to determine how it evolves with time. The best circular fits inside the pore through the profiles 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 capillary wall. This indicates again that the simulated menisci always retain their spherical form during spreading, except very close to the solid surface. To avoid this problem, we investigated the profile as a function of the number of molecular layers used, from the center of the pore. We find that the circular fit using all the experimental points except the last few in contact with the substrate leads to stable results when we have a large number of liquid atoms, which is the case in this simulation with our 25600 liquid atoms. From the tangent at the wall, we are able to measure the contact angle θ and the height X as a function of the number of time steps.
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Figure 8. The contact angle of the sessile drop versus time t for carbon-like atoms.
Figure 9. The remaining volume of the sessile drop versus time t for carbon-like atoms.
Now, the question is how can we be sure that the set of equations (8-10) really describe the physics of this porous adsorption? This is the great advantage of the numerical experiment; we can determine simultaneously all the pertinent variables of the problem: the capillary height x; the volume of the remaining drop V; the contact angle of the remaining sessile drop θ; and the contact radius R versus the time t. The associated data are reproduced in the Figures 7-10: It now remains to solve eq 8-10 for this particular porous medium to check the validity of these theoretical predictions. Since we are here dealing with a very small pore, we can simplify our equations to get to the leading order in r:
1+
ζp 4ηx r sin θt dx dx ) + Rt cos θ0* γ cos θ0* dt rγ cos θ0* dt
( )
π ∂θ ) ∂t 3V(t)
1/3
(16)
Figure 10. The base radius of the sessile drop versus time for carbon-like atoms.
γLV (cos θ0 - cos θ(t))(cos(θ(t)) + ζsurf 2)((cos(θ(t)) + 2)(cos(θ(t)) - 1)2)1/3 (17)
Vt ) V0 - πr2x + δ)
((
[
]
πδ3 πr2δ + 6 2
(18)
) )
(19)
1 r -r cos θt* cos θt*
2
-1
1/2
In ref 26, we have computed the parameter ζp/γ cos θ0* for an equivalent pore imbibition problem; we thus known that
ζp/γ cos θ0* ) 0.0454 ( 0.0040
(20)
We have measured x, θ, R, and V versus time t, and we have only one free parameter, which is 4η/rγ cos θ0*, since for such a small pore, one may easily understand that the viscosity may well be different than the corresponding bulk one. Using a classical simplex algorithm, we get
4η/rγ cos θ0* ) 0.0013
Figure 11. A schematic top view of a sessile drop on top of a set of pores.
The Porous Surface Case To describe the spreading of a drop on a porous surface, it remains to take into account the multiple pores that the droplet will experience during its spreading. Obviously, at a certain time t, the number of pores is proportional to the contact area between the drop and the solid divided by the surface of one pore. If we neglect the interconnections between the pores, we can rigorously generalize the previous considerations. The total adsorbed amount becomes now equal to
(21)
To visualize the quality of the fitting procedure, we have reproduced in the Figures 7-10 the measured data and the associated fits. The agreement is quite remarkable. This result strongly supports the microscopic validity of the molecular-kinetic model of spreading and imbibition, at least within its linear form.
n
Vt ) V0 -
∑ i)1
{
δ3
πr2xi - π
}
δ - πr2 6 2
(22)
where the sum refers to the pores covered by the liquid surface at time t.As already pointed out in ref 29, we now have to deal with a heterogeneous liquid/solid surface.
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Figure 12. The base radius R versus time. From top to bottom the solid equilibrium angle is 0, 10, 20, ..., 80°.
Figure 13. The maximal value of the base radius R versus the equilibrium contact angle θ0.
Let us denote by L the liquid and by S the solid. Using Cassie’s approximation for the equilibrium contact angle, one has
cos θLS ) F cos θL + (1 - F) cos θS ≈ cos θS + F(1 - cos θS) (23) For the dynamics, it has been shown30 that to the leading order
ζsurf ) ζLS ) FζL + (1 - F)ζS ≈ (1 - F)ζS
(24)
One thus gets for each pore
() ( ))
2γLV sin θt 2γLV 2 dx 8ηx dx cos θ0* ) ζp + 2 + + Rt r r dt r dt F
((
dx d2x +x 2 dt dt 2
)
(25)
Figure 14. The volume Vt versus time. From left to right, the equilibrium solid contact angle is 0, 10, 20, ..., 80°.
and for the total volume n
Vt ) V0 -
∑ i)1
{
πr2xi - π
δ3 6
- πr2
}
δ 2
(26)
where
Rt ) Φ(θt)
(27)
γLV dR ) (cos θ0,LS - cos θt) dt (1 - F)ζS
(28)
and
The following numerical results are given for a liquid with γLV ) 72 mN/m, η ) 1 cP, V0 ) 0.1 µL, F ) 0.1, pores with radius of 10 µm, and various equilibrium contact angles θS ) 0, 10, 20, 30, 40, 50, and 80°. The behavior of the base radius R is represented in Figure 12. As can be seen, the drop initially spreads and then reaches its maximal base radius, and then, due to imbibition, the drop radius falls down. The maximal value reached by the base radius versus the equilibrium contact angle is given in Figure 13. (30) Cassie, A. B. D. Discuss. Faraday Soc. 1952, 57, 5041.
Figure 15. Time required to adsorb completely the liquid drop in the porous surface versus the equilibrium contact angle.
The volume of the sessile part of the drop is given in Figure 14. Contrary to the single pore case, we now have a concave shape. The time required for the drop to be completely adsorbed within the porous medium is thus also a function of the wettability of the solid surface. The results are given in Figure 15. Clearly the relationship is not linear.
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Discussion Combining the dynamics of spreading for sessile drops and the dynamics of penetration into a pore, we have been able to describe the dynamics of spreading of a drop on a porous surface by a closed set of three equations. We have then shown that large-scale molecular dynamics simulations can be used to study this droplet spreading and imbibition on porous surface. We adopted a very simple model of linear liquid molecules and an atomic solid: atoms within a molecule were connected with a strong harmonic potential, and intramolecular interactions were of the Lennard-Jones type. The dynamics of droplet spreading and imbibition was followed from an initial contact angle of 180° toward the equilibrium value. We measured the dynamic contact angle relaxation on the surface and in the pore, the height of penetration inside the pore, the relaxation of the base radius, and the volume of the drop versus time. Using all these measurements, we showed that the friction model, which corresponds to the linear version of the molecular-kinetic model, is valid in partial
Seveno et al.
wetting. Indeed, we were able to reproduce the dynamics of the four observed variables using a single fitted parameter. These results directly support the importance of the friction mechanism at the microscopic scale for spreading and imbibition. Generalizing these considerations to the case of multiple noninterconnected pores, we were able to derive interesting predictions concerning the critical time of complete adsorption within the porous surface, the maximal base radius reached during spreading, etc. The practical case where we also take into account the effect of hysteresis via molecular modeling is still an interesting open problem under consideration. Acknowledgment. It is a pleasure to thank T. Blake, A. Clarke, and E. Raphae¨l for stimulating discussions. We also thank the reviewers for pointing out interesting references on the subject. This research has been partially supported by the Structural European Funds, the Re´gion Wallonne, and by Kodak. LA025520H
