Journal of Colloid and Interface Science 270 (2004) 171–179 www.elsevier.com/locate/jcis The possibility of different time scales in the dynamics of pore imbibition G. Martic, a F. Gentner, a D. Seveno, a J. De Coninck, a,∗ and T.D. Blake b a Center for Research in Molecular Modeling, Materia Nova/University of Mons-Hainaut, ParcInitialis Av. Copernic, 1, 7000 Mons, Belgium b Kodak Limited Research and Development, Harrow, Middlesex HA1 4TY, UK Received 31 March 2003; accepted 29 August 2003 Abstract To model the imbibition of liquids into porous solids, use is often made of the Lucas–Washburn equation, which relates the distance of penetration of a liquid at a given time to the pore radius, the viscosity and surface tension of the liquid, and the effective contact angle between the liquid and the solid. In this paper, we extend previous large-scale molecular dynamics simulations to show how this tool can be used to study the details of liquid imbibition, including the impact of the contact angle on the dynamics of penetration and the evolution of the internal flow field. In particular, we show that the asymptotic behavior of the contact angle versus time for a completely wetting liquid is given by ∼t −1/4 . 2003 Elsevier Inc. All rights reserved. 1. Introduction The wetting of porous media is a subject of primary im- portance in many practical applications such as oil recovery, ceramics, and powder dissolution. Porous systems are usu- ally complex and difficult to model in detail. To characterize them, one approach is to measure the rate of penetration of a liquid into the given medium, which is then modeled as a bundle of uniform capillaries. Under steady conditions in a horizontal capillary, the capillary imbibition pressure is balanced by the viscous drag of the liquid. Simple analysis of this process leads to the Lucas–Washburn equation [1,2], which relates the distance of penetration x at time t to the capillary radius, R, the viscosity and surface tension of the liquid, η and γ LV , respectively, and the effective contact an- gle between the liquid and the capillary wall, θ : (1) x = γ LV R cos θ 2η √ t. Recent reviews have been given by Marmur [3] and Zhmud et al. [4]. * Corresponding author. E-mail address: [email protected] (J. De Coninck). In general, the overall balance of forces on the liquid in the capillary may be expressed as (2) 2 R γ LV cos θ = 8 R 2 ηx dx dt + ρ x d 2 x dt 2 + dx dt 2 , where ρ is the density of the liquid. The LHS of this equation is the capillary driving force, the first term on the RHS 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 Rideal [5] and Bosan- quet [6] and are usually significant only in the early stages of penetration, or when R is large. For small radii, viscous forces are dominant, the inertial terms can be neglected, and one has only to solve the differ- ential equation (3) 2 R γ LV cos θ = 8 R 2 ηx dx dt with the initial condition x = 0, t = 0. However, this equa- tion, like Eq. (2), holds only asymptotically, since the contact angle is associated with a moving wetting line and there is plenty of evidence to show that such dynamic contact angles are velocity-dependent, i.e., they depend on both speed and direction of wetting line displacement. Observation shows that advancing angles increase and receding angles decrease with increasing rates of displacement. As a result, the con- tact angle in the capillary will decrease with time as liquid 0021-9797/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.08.049
Transcript
stance ofact angleool can belution ofliquid is
Journal of Colloid and Interface Science 270 (2004) 171–179www.elsevier.com/locate/jcis
The possibility of different time scales in the dynamics ofpore imbibition
G. Martic,a F. Gentner,a D. Seveno,a J. De Coninck,a,∗ and T.D. Blakeb
a Center for Research in Molecular Modeling, Materia Nova/University of Mons-Hainaut, Parc Initialis Av. Copernic, 1, 7000 Mons, Belgiumb Kodak Limited Research and Development, Harrow, Middlesex HA1 4TY, UK
Received 31 March 2003; accepted 29 August 2003
Abstract
To model the imbibition of liquids into porous solids, use is often made of the Lucas–Washburn equation, which relates the dipenetration of a liquid at a given time to the pore radius, the viscosity and surface tension of the liquid, and the effective contbetween the liquid and the solid. In this paper, we extend previous large-scale molecular dynamics simulations to show how this tused to study the details of liquid imbibition, including the impact of the contact angle on the dynamics of penetration and the evothe internal flow field. In particular, we show that the asymptotic behavior of the contact angle versus time for a completely wettinggiven by∼t−1/4. 2003 Elsevier Inc. All rights reserved.
m-ery,usuzeationedonsis
ysis1,2]
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ud
).
in
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1. Introduction
The wetting of porous media is a subject of primary iportance in many practical applications such as oil recovceramics, and powder dissolution. Porous systems areally complex and difficult to model in detail. To characterithem, one approach is to measure the rate of penetrof a liquid into the given medium, which is then modelas a bundle of uniform capillaries. Under steady conditiin a horizontal capillary, the capillary imbibition pressurebalanced by the viscous drag of the liquid. Simple analof this process leads to the Lucas–Washburn equation [which relates the distance of penetrationx at time t to thecapillary radius,R, the viscosity and surface tension of tliquid, η andγLV , respectively, and the effective contact agle between the liquid and the capillary wall,θ :
(1)x =√γLVR cosθ
2η
√t .
Recent reviews have been given by Marmur [3] and Zhmet al. [4].
* Corresponding author.E-mail address:[email protected] (J. De Coninck
0021-9797/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2003.08.049
-
,
In general, the overall balance of forces on the liquidthe capillary may be expressed as
(2)2
RγLV cosθ = 8
R2ηx
dx
dt+ ρ
[xd2x
dt2+
(dx
dt
)2],
whereρ is the density of the liquid. The LHS of this equatiois the capillary driving force, the first term on the RHS givthe viscous resistance of the liquid in the capillary, andremaining terms describe the inertial resistance. The effof inertia were first considered by Rideal [5] and Bosquet [6] and are usually significant only in the early staof penetration, or whenR is large.
For small radii, viscous forces are dominant, the inerterms can be neglected, and one has only to solve the dential equation
(3)2
RγLV cosθ = 8
R2ηx
dx
dt
with the initial conditionx = 0, t = 0. However, this equation, like Eq. (2), holds only asymptotically, since the contangle is associated with a moving wetting line and therplenty of evidence to show that such dynamic contact anare velocity-dependent, i.e., they depend on both speeddirection of wetting line displacement. Observation shothat advancing angles increase and receding angles decwith increasing rates of displacement. As a result, the ctact angle in the capillary will decrease with time as liqu
172 G. Martic et al. / Journal of Colloid and Interface Science 270 (2004) 171–179
turabal-
ting,9],-eingde-that
the-ef-alion-
isstaousthe
thean-seav-of
hinhereloc-l
h-thece
on-
stheForthem-
f-olidainthare
bypar-ndnddy-
s ady-
n 4amicur
f an-d by],
d,ef-
oose
s),tsn to
nter-thattherin-eentionlarge.
us ater-
penetration proceeds and the meniscus slows. The namodification of Eqs. (2) and (3) is to assume that theance of forces still holds at every timet , but withθ replacedby its instantaneous, dynamic valueθt . Formally, we maymake use of any of the various theories of moving wetlines, based on hydrodynamics [7], molecular kinetics [8or phenomenology [10], in whichθt is some function of wetting line velocitydx/dt [11]. One result of this is that thdynamic contact angle becomes a function of the drivforce for wetting as well as being one of the factors thattermines it. What then is the correct form of the equationproperly describes the dynamics of capillary imbibition?
Bico and Quéré have already used the hydrodynamicory to confirm the importance of dynamic contact anglefects on the rate of fall of liquid indices or “slugs” in verticcapillaries [12]. Essentially, they made use of the relatship
(4)cosθt = cosθ0 − 3η
γLV
dx
dt
1
θtln(R/a),
wherea is some molecular cutoff length. According to ththeory, the change in the apparent contact angle from itstic value to a dynamic one arises as the result of viscdissipation very close to the wetting line, which causesliquid interface to bend.
We have used an alternative approach based onmolecular-kinetic theory [8,9] to model dynamic contactgle behavior during capillary imbibition and capillary ri[13,14]. According to this theory, the macroscopic behior of the wetting line depends on the overall statisticsthe individual molecular displacements that occur witthe three-phase zone, i.e., the microscopic region wthe fluid/fluid interface meets the solid surface. The veity of the wetting line is characterized byK0, the naturafrequency of molecular displacements, andλ, their averagelength. In simple cases,λ is the distance between two neigboring adsorption sites on the solid surface. Assumingdriving force for the wetting line to be the out-of-balansurface tension forceγLV (cosθ0 − cosθt ) and using Eyring’sactivated-rate theory for transport in liquids gives a relatiship betweenθt and the velocity of the wetting linedx/dt :
(5)dx
dt= 2K0λsinh
[γLV (cosθ0 − cosθt )
2nkBT
],
wheren is the number of adsorption sites per unit area,kB isBoltzmann’s constant, andT is the temperature. Within thimodel, the energy dissipation leading to the change incontact angle is localized within the three-phase zone.small arguments of the sinh function, which occur whendriving force is small (e.g., near equilibrium or at high teperatures), this simplifies to
(6)dx
dt= K0λ
nkBT
[γLV (cosθ0 − cosθt )
]or
(7)dx = 1[
γLV (cosθ0 − cosθt )],
dt ζ
l
-
where ζ = nkBT/K0λ is effectively the coefficient o
wetting-line friction and is a function of both liquid viscosity and the interactions between the liquid and the ssurface [15]. By combining Eqs. (7) and (3), we obtthe desired equation for capillary imbibition in which bothe capillary driving force and the viscous resistancevelocity-dependent:
(8)2
RγLV
[cosθ0 − ζ
γLV
(dx
dt
)]= 8
R2ηx
dx
dt.
The utility of this equation has been verified recentlylarge-scale molecular dynamics simulations and by comison with experiment [13]. Our objective here is to extethis analysis to study the details of capillary imbibition ato compare our results with those predicted by the hydronamic theory represented by Eq. (4).
Our paper is organized as follows. Section 2 containdescription of the model system. Results concerning thenamics of imbibition are detailed in Section 3. In Sectiowe consider the consequences of a combined hydrodynand molecular-kinetic treatment of dynamic wetting. Oconclusions are summarized in Section 5.
2. The model
We use molecular dynamics to model the imbibition oliquid into a cylindrical pore. In our simulations, all potetials between atoms, solid as well as liquid, are describestandard pairwise Lennard–Jones 12–6 interactions [16
(9)Vij (d)= 4εij
[Cij
(σij
d
)12
−Dij
(σij
d
)6],
whered is the distance between any pair of atomsi andj .The parametersεij andσij are in the usual manner relaterespectively, to the depth of the potential well and thefective molecular radius. For simplicity,Cij and Dij arechosen to be the same for each type of atom. We chCff =Dff = 1.0, Css = Dss = 1.0 and Csf = Dsf = 1.0,where the subscripts indicate fluid/fluid (ff), solid/solid (sand solid/fluid (sf) interactions. The intrafluid coefficienare standard, and the solid/solid coefficients are choseproduce a stable lattice structure at the temperature of iest. The choice of the solid/fluid interactions ensuresthe liquid wets the solid. Theoretically, the range ofLennard–Jones 12–6 interactions extends to infinity. In pciple, one should therefore evaluate interactions betwall possible pairs in the system. Fortunately, the interacpotentials decrease rapidly as the distance becomesWe therefore apply a spherical cutoff at 2.5σij and shiftthe potential so that the energy and force are continuod = 2.5σij . As a result, we consider only short-range intactions in these simulations.
G. Martic et al. / Journal of Colloid and Interface Science 270 (2004) 171–179 173
,000on-
in-eenrm
ex-taypplyare
olidboxtha
ringr-
ated
ris-, on
u-
im-tingrs dory tol re
piths,
t a1,
ore.
ircu-
nce
ow-ides.theon-achof
etoffen
, as
hellere
wecusof
lo-.75actal-pt
udyood
tate,uffi-t ingedd to
largee to
Fig. 1. Side view of typical snapshots of the system containing 145atoms: (left) the initial configuration after equilibration and (right) the cfiguration during capillary imbibition.
We simulate a molecular structure for the liquid bycluding a confining potential, a strong elastic bond betwthe adjacent atoms within a given molecule, of the foVconf = Dconf(d/d0)
6, with d0 = 21/6σss andDconf = 1.0.The liquid molecules are always 16 atoms long. Thistra interaction forces the atoms within one molecule to stogether and reduces evaporation considerably. We aa harmonic potential to the solid atoms, so that theystrongly pinned on their initialfcc lattice configuration, inorder to give a realistic atomic representation of the ssurface. To avoid undesirable edge effects, the cubiccontaining all the atoms is made large enough to ensurethe liquid atoms never reach the extremity of the pore duthe simulation. During imbibition, we control the tempeature of the solid by rescaling the speed of the associatoms to mimic a real experiment.
Thus, we consider a very simple chainlike liquid comping 16 monomers with spherical symmetry per moleculetop of a pore of radiusR within anfccsolid lattice. A typicalside view of the system is shown in Fig. 1.
We always apply a time step of 5 fs [17] during our simlations withεij = 0.066 kcal/mol andσij = 3.5 Å, which istypical for carbon atoms. Although the system is rather sple, it contains all the basic ingredients to model the wetof the pore. The values chosen for the above parametenot affect the spreading behavior, but they are necessacompare the measured contact angles with experimentasults.
As the initial configuration, we consider a liquid droon top of the pore and let the system equilibrate wa small affinity between the solid and the liquid atomCsf = Dsf = 0.5. Due to the boundary conditions we gelayer of liquid, which protrudes slightly into the pore (Fig.left). We then increase the couplingsCsf, Dsf between theliquid and the solid, causing the liquid to penetrate the p
t
-
Fig. 2. Typical snapshot of a meniscus profile showing the associated clar fit.
In this way, we can easily measure the penetration distax versus timet (Fig. 1, right).
To determine the contact angle, we proceed in the folling way, mimicking a real experiment. First, we subdivthe liquid index into cylindrical shells of arbitrary thicknesThe constraint on the number of shells is provided byneed to maximize their number while ensuring that each ctains enough molecules to give a uniform density. For eshell, we compute the density of particles as a functionthe distancex into the pore. We locate the extremity of thshell as the position at which the density falls below a cuvalue of half the liquid density. The contact angle can thbe obtained from the tangent to a circular fit to the profileillustrated in Fig. 2.
To check the consistency of the method, different sthickness and cutoff values were considered. These wfound to give almost identical results. Using this methodwere able to construct the complete profile of the menisand to determine how it evolved with time. A successionsuch profiles is reproduced in Fig. 3.
The best circular fits through the profiles were alwayscated within the region where the density dropped from 0to 0.25, except for the first few molecular layers in contwith the solid. This indicates that the simulated menisciways retain their spherical form during imbibition, excevery close to the solid surface. Note that in order to stthe associated dynamics of the meniscus profiles with gprecision, from the earliest stages until the stationary sit was necessary to consider the height of the profile sciently close to the wall. To reduce the fluctuations presenthis part of the profile, we considered the interface averaover 50 successive profiles, each of which correspondea snapshot of the system every 1000 time steps. Usingsystems over very long simulations, we were thus abldetermine the detailed behavior of the contact angleθt as afunction of time.
174 G. Martic et al. / Journal of Colloid and Interface Science 270 (2004) 171–179
tion-ps.
er of
wec-lot-iven
on-
d
d
ingig. 5
ofhewofor
ibi-c-m.on-laryoodat
Fig. 3. A succession of half-meniscus profiles showing their evoluwithin the pore forR = 70 Å andCsf = Dsf = 1.0. The profiles were averaged over 50 successive snapshots, each separated by 1000 time ste
Fig. 4. The average velocity profile versus the distance to the centthe pore. The continuous line corresponds to the parabolic fit 164.5 −A(R − r)(R + r), whereA= 0.00349± 0.00016 andR = 69.99± 1.135.
3. Results
3.1. Penetration distance and contact angle
Separate simulations were performed forR = 50 Å andR = 70 Å. Our first observation was that the liquid floinside our pore obeyed Poiseuille’s law, which was not nessarily expected at this small scale. In Fig. 4 we have pted the average displacement of atoms belonging to a gshell (betweenx = 170 Å andx = 160 Å) over a total of1,100,000 time steps. This parabolic velocity profile c
Fig. 5. Penetration distancex(t) versus timet for a pore of radiusR = 70 Åand a range of couplingsCsf = 0.8, 1.0, and 1.5. The solid lines were fitteusing Eq. (8).
Fig. 6. Penetration distancex(t) versus timet for a pore of radiusR = 50 Åand a range of couplingsCsf = 0.8, 1.0, and 1.5. The solid lines were fitteusing Eq. (8).
firms the suitability of our molecular approach for checkthe consistency of the Lucas–Washburn equation. In Fwe show the time behavior ofx(t) for three different liq-uid/solid interactions (Csf = 0.8, 1.0, and 1.5) and a poreradius 70 Å. To minimize the effects of fluctuations in tmeniscus profile,x(t) was measured at a position about tthirds of R from the pore axis. The analogous resultsthe smaller pore of radius 50 Å (also withCsf = 0.8, 1.0,and 1.5) are given in Fig. 6.
From these results, it is clear that the rate of pore imbtion is a nonmonotonic function of the liquid/solid interation. The caseCsf = 1.0 seems to correspond to a maximuFrom a practical point of view, this may have interesting csequences. For example, in order to fill a tube by capilrise, it would be reasonable to use a liquid that had a gaffinity for the wall of the tube. Our simulations indicate ththere is an optimum liquid to achieve this most rapidly.
G. Martic et al. / Journal of Colloid and Interface Science 270 (2004) 171–179 175
ty
e-ed aes-e
e
per-
rmssim-ofon-
m-
is
ay
e
be-
e
are
eenat-8).an
par-avior
be
The dynamics ofx(t) for small velocitiesdx/dt can bedescribed by Eq. (8), written as
(10)1− P2
(dx
dt
)− P3x
(dx
dt
)= 0
with an adjustable parameterx(0) = P1. Here, P2 =ζ/γLV cosθ0 andP3 = 4η/RγLV cosθ0. Equation (10) hasthe exact solution
(11)x(t) = −P2
P3+
√2t
P3+
(P1 + P2
P3
)2
.
Experimentally, it is quite difficult to establish the validiof these equations by measuring the two quantities,x andθt versus the timet . Here, however, we can perform a dtailed analysis of our numerical results. We therefore usLevenberg–Marquard method to fit our data, yielding thetimates ofP1, P2, andP3 given in Table 1. In all cases, thregression coefficients were above 0.95.
As can be seen from these results, the value ofP2 doesnot depend on the radiusR. This is in agreement with thmolecular-kinetic theory, which predicts thatP2 dependsonly on the material properties of the system and its temature:P2 = ζ/γLV cosθ0 = nkBT /K0λγLV cosθ0. We alsofind, as expected, thatP3(R = 70) ≈ (5/7)P3(R = 50).
In terms of the wetting-line friction, these results confithat ζ(R = 70)/ζ(R = 50) ∼ 1, meaning that the friction iindependent of the radius. We have performed a parallelulation of spreading for a droplet of the same liquid on topan equivalent flat substrate. From the dynamics of the ctact angle, we found thatζ(R = 70)/ζ(R = +∞) ∼ 1. Inother words,ζ is a local quantity, independent of the geoetry of the substrate, as anticipated in [8].
Another way to test the validity of Eqs. (8) and (10)to compare the predicted values ofθt with those found fromthe simulations. We proceed as follows. First, Eq. (10) mbe rearranged to give
(12)dx
dt= 1
P3(x + (P2/P3)).
Then, substituting forζ in Eq. (7) using the identityζ =P2γLV cosθ0, we obtain
(13)cosθt = cosθ0
[1− P2
(dx
dt
)].
Table 2Values of cosθ0 and θ0 obtained by fitting Eq. (14) to the data from thsimulations
Fig. 7. Contact angle in degrees versus time forR = 70 Å and the threecouplingsCsf = Dsf. The continuous lines correspond to the predictedhavior based on the molecular-kinetic theory.
Finally, eliminatingdx/dt from these two equations yields
(14)θt = arccos
[x (t)cosθ0
x(t)+ P2/P3
].
Since we knowP2 andP3 (Table 1), this equation can bfitted to our data to determine cosθ0. The resulting valuesfor the two capillary radii and the three couplings usedlisted in Table 2. As would be expected, couplingsCsf = 1.0and 1.5 lead to complete wetting, whileCsf = 0.8 inducespartial wetting. In the latter case, the discrepancies betwthe angles obtained for the two capillary radii can betributed to the noise in our simulations (see Figs. 7 and
Once the limiting, equilibrium angles are known, we cuse Eq. (14) to calculateθt as a function of the imbibitiontime for each case. In Figs. 7 and 8 we show a direct comison between our simulated data and the predicted behbased on the molecular-kinetic theory (Eq. (7)). As can
Table 1Values of fitted parameters for the equation 1− P2(dx/dt)− P3x(dx/dt) = 0, with x(0) = P1, for R = 50 and 70 Å and couplingsCsf = 0.8, 1.0, and 1.5
176 G. Martic et al. / Journal of Colloid and Interface Science 270 (2004) 171–179
be-
sed
inincethe
laneethisrep
oundtheined
ringle rescus
omsercenfluxve-meormflux
es-itho-a binthere,nter
ines
heiseichaenosepro-gth,areonlyw
on-usowro.ithing
Fig. 8. Contact angle in degrees versus time forR = 50 Å and the threecouplingsCsf = Dsf. The continuous lines correspond to the predictedhavior based on the molecular-kinetic theory.
seen, the agreement is quite good given the limits impoby the noise in the data.
3.2. Liquid flux within the pore
Here we consider the details of the flow within the poreorder to analyze the convective mechanisms at work. Sthe meniscus is symmetrical about the central axis ofpore, we project the volume of the meniscus onto a half pwith coordinatesr andx, respectively the distance from thaxis and the distance of penetration. We then divideplane into small squares to make a grid. Each squareresents the volume of a bin, a square-sectioned ring arthe pore axis. The larger the radial position of the bin,greater the volume represented. Grid spacing is determby the need to maximize the number of bins, whilst ensuthat each one contains enough atoms to give reproducibsults. Fortunately, the more interesting parts of the meniare away from the axis of symmetry.
For each bin we compute the center of mass of the atwithin it. A short time later (typically 100,000 computtime steps), we determine the net displacement of theter of mass by measuring the difference between theinto and out of the bin. This allows us to measure thelocity field associated with the moving meniscus in socoarse-grained sense. Assuming the density to be unifthe velocity field could also be considered as lines of(streamlines).
Figs. 9, 10, and 11 show velocity fields at succsive times for menisci in the 50-Å and 70-Å pores, wCsf =Dsf = 1.0 and 145,000 atoms (liquid and solid tgether). Each arrow represents the net displacement ofover a given period of time. The length and direction ofarrow is a measure of the local velocity relative to the powhich provides the stationary frame of reference. The ceof the meniscus is atr = 0.
-
-
-
,
Fig. 9. Velocity fields 5.0 ns after the start of the simulation. Dashed lindicate the positions of the solid surfaces.
In Fig. 9, the velocity field of the meniscus during tinitial stages of imbibition is shown. The contact anglenearly 60◦. The velocity field is essentially downward in thupper part of the meniscus. Close to the solid wall (whis atr = 50 andr = 70), the velocity is zero, indicative ofno-slip boundary condition. At intermediate radii (betwe10 and 40), the velocity field becomes more vertical. Clto the center of the meniscus this shift is even morenounced. Most of the arrows are of about the same lenwhich means that the velocities within the meniscusmore or less uniform, and that their magnitude changesin the vicinity of the wall. The average direction of the flofield is toward the contact line.
Fig. 10 shows the velocity at intermediate times. The ctact angle is close to 50◦. The basic features of the previoflow field are still observed: around the inner zone the flis vertical and downward; close to the wall it is almost zeAt the pore wall there seems to be a velocity gradient, wthe velocity near zero along most of the wall but increasnear the contact line.
G. Martic et al. / Journal of Colloid and Interface Science 270 (2004) 171–179 177
he
ss
si-hep-
osedi-eset of
el ofblenceboththeible
els.18],
ll
Fig. 10. Velocity fields 7.5 ns after the start of the simulation.
The final stage of the simulation is shown in Fig. 11. Tcontact angle is now smaller than 50◦. Away from the solid,the flow is still vertical. The gradient along the wall is lepronounced.
In summary, the flow in the top of the meniscus is bacally vertical and downward, with not much variation in tvelocity. Close to the wall, the velocity field almost disapears, indicative of a no-slip boundary condition, but clto the contact line there is a steep velocity gradient, incating significant dissipation processes in this region. Thbehaviors are typical of all our simulations, independenthe pore radius and coupling constants.
4. Combined approach
From the above results, it seems clear that the moddynamic wetting based on wetting-line friction is compatiwith our data. However, let us now consider the relevaof the hydrodynamic model. Suppose we assume thatwetting-line friction and viscous bending can influencedynamic contact angle, i.e., that the dissipation respons
Fig. 11. Velocity fields 10.0 ns after the start of the simulation.
for a nonequilibrium angle can occur through two channWe further assume that the two channels are additive [so that Eqs. (4) and (7) can be combined to give
(15)dx
dt= γLV (cosθ0 − cosθt )
ζ + 3η 1θt
ln(R/a),
i.e.,
(16)cosθt = cosθ0 −[ζ + 3η 1
θtln(R/a)
]dxdt
γLV.
Settingθ0 = 0 for simplicity, we get asymptotically for smaθt that when the viscous dissipation is negligible,
(17)θ2t ∼ ζ
γLV
dx
dt,
and when the wetting-line friction is negligible,
(18)θ3t ∼ 3η ln(R/a) dx
.
γLV dt
178 G. Martic et al. / Journal of Colloid and Interface Science 270 (2004) 171–179
Fig. 12. Contact angle versus time (left) and logarithm of contact angle versus logarithm of time (right) forR = 50 and 70 Å and coupling constantsCsf = Dsf = 1.0 and 1.5. The lines represent the best fits to the expressionaxb , omitting the first few points.
G. Martic et al. / Journal of Colloid and Interface Science 270 (2004) 171–179 179
In our simulations, we find thatθ0 = 0 for Csf = Dsf =1.0 and 1.5. In Fig. 12 we have therefore plotted logθt ver-sus logt for these systems in order to reveal the exponand so distinguish the dominant regime. The lines throthe data were fitted to the expressionaxb, and the result-ing values for the coefficients are listed in Table 3. In ecase we obtainb ∼ −0.25, which confirms that our data fain the molecular-kinetic regime. On the other hand, Band Quéré [12] have demonstrated that the hydrodynatreatment can successfully account for the variation ofdynamic contact angle in their experiments with falling luid indices in capillary tubes. In previous studies ofspreading of liquid drops on solid surfaces [19], we hfound clear evidence of different time scales pertainingthe molecular-kinetic and hydrodynamic regimes, withlatter dominant at long times and small angles. Evidenmore work needs to be done to investigate these effectsing capillary imbibition.
5. Summary and conclusions
In this paper we have shown how large-scale molelar dynamics simulations can be used to study the deof liquid imbibition into a cylindrical pore. We adoptedvery simple model comprising linear liquid molecules aan atomistic solid. Atoms within each molecule were cnected with a strong harmonic potential, and intramolecinteractions were of the Lennard–Jones type. By chanthe strength of the solid–liquid interactions, we were ablvary the wettability of the solid. The dynamics of liquid imbibition were followed for two pores of different radii: 5and 70 Å. During imbibition, we measured the distancemeniscus penetration and the relaxation of the dynamictact angle toward equilibrium. We were also able to anal
-
the flow field inside the pore. Our results are in agreemwith the Lucas–Washburn equation corrected to includeeffects of the dynamic contact angle. The latter’s impon the dynamics of capillary imbibition has been discusin detail. In particular, we have shown that the asymptbehavior of the contact angle versus time may follow dferent regimes according to whether the molecular-kinor hydrodynamic mechanism is dominant. The resultsthe simulations are compatible with a contact angle thalaxes asymptotically ast−1/4, and are thus consistent withe molecular-kinetic description.
Acknowledgments
We acknowledge partial support from the Region Wlonne, Kodak Limited, and the program Feder—Objectiv
References
[1] R. Lucas, Kolloid Z. 23 (1918) 15.[2] E.W. Washburn, Phys. Rev. 17 (1921) 273.[3] A. Marmur, in: M.E. Schader, E.B. Loeb (Eds.), Modern Approa
to Wettability: Theory and Applications, Plenum, New York, 199pp. 327–358.
[4] B.V. Zhmud, F. Tiberg, K. Hallstensson, J. Colloid Interface Sci. 2(2000) 263.
[5] R.K. Rideal, Philos. Mag. 44 (1922) 1152.[6] C.H. Bosanquet, Philos. Mag. 45 (1923) 521.[7] R.G. Cox, J. Fluid Mech. 168 (1986) 169.[8] T.D. Blake, J.M. Haynes, J. Colloid Interface Sci. 30 (1969) 421.[9] T.D. Blake, in: J.C. Berg (Ed.), Wettability, Dekker, New York, 199
pp. 251–309.[10] P. Joos, P. Van Remoortere, M. Bracke, J. Colloid Interface Sci.
(1990) 189.[11] For simplicity here we are ignoring any non-local hydrodynamic in
ences on the dynamic contact angle (T.D. Blake, Y.D. ShikhmurzPhys. Fluids 11 (1999) 1995), such influences mean that the dyncontact angle is not a simple function of wetting line velocity, but ifunctional, dependent on the precise details of the flow.
[12] J. Bico, D. Quéré, J. Colloid Interface Sci. 243 (2001) 262.[13] G. Martic, F. Gentner, D. Seveno, D. Coulon, J. De Coninck, T
Blake, Langmuir 18 (2002) 7971.[14] G. Martic, J. De Coninck, T.D. Blake, J. Colloid Interface Sc
in press.[15] T.D. Blake, J. De Coninck, Adv. Colloid Interface Sci. 96 (2002) 2[16] J. De Coninck, U. D’Ortona, J. Koplik, J.R. Banavar, Phys. R
Lett. 74 (1995) 928.[17] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxfo
Science Publications, Oxford, 1989.[18] F. Brochard-Wyart, P.G. de Gennes, Adv. Colloid Interface Sci.
(1992) 1.[19] M.J. de Ruijter, M. Charlot, M. Voué, J. De Coninck, Langmuir
