Langmuir 1992,8,1762-1767
1762
A Combined Molecular-Hydrodynamic Approach to Wetting Kinetics Peter G. Petrov and Jordan G. Petrov' Central Laboratory of Mineral Processing, Bulgarian Academy of Sciences, Sofia 1126, P.O.Box 32, 1 James Bourcier Avenue, Bulgaria Received November 12,1991.In Final Form: February 11, 1992 A model describing wetting kinetics on the basis of the hydrodynamic equation of Cox [J.Fluid Mech. 1986,168,1693and the molecular-kinetic equation of Blake and Haynes [J.Colloid Interface Sei. 1969, 30,4211is proposed. It takes into account both viscous dissipation of energy in the bulk of the liquid and dissipation in the immediate proximity to the three-phase contact line. Our experimental results on the velocity dependences of receding dynamic angles and literature data for advancing of aqueous glycerol solutions on hydrophobic solid surfaces are used to compare theory and experiment. Good coincidence is observed in all cases of receding mode. At advancing the agreement is qualitative in systems with high Reynolds numbers, Re, and improves when Re decreases, as required by Cox theory.
Introduction Upon ita motion on a solid substrate, a gadliquid interface should overcome a resistance due to friction. A part of the latter is attributed to a viscous drag in the bulk of the liquid; i.e., it has a hydrodynamic origin. In the most general case, additional friction should be ascribed also to the moving contact line. Two approaches are established in the literature on wetting kinetics depending on what dissipative force is considered as predominant. By completelyneglecting the viscous drag, a theoretical dependence of the dynamic contact angle B on the contact line velocity Vis determined from the balance of the driving (nonequilibrium Young) force and the friction force in the three-phase contact zone. Such an approach, based on Eyring's theory for transport phenomena,l has been proposed by Blake and Haynes.2 Comparison of the equation obtained with different experimental data suggests that such a behavior is characteristic of various systems-siliconized glass/ benzene/water,2 polymer/glycerol-water/air (poly(ethy1ene terephthalate) and "Mylar" polyester),3i4etc. Good coincidence has been found also upon deposition of Langmuir-Blodgett multilayers of methyl ara~hidate.~ According to the hydrodynamic approach, the viscous friction is assumed to be the only significant dissipative force in the dynamic meniscus. Such a consideration (based on the creeping flow approximation), assuming a slippage of the liquid with respect to the solid in the contact line vicinity, is reported by COX.^ The applicability of the theory is illustrated by a comparison with the experimental resulta of Hoffman? obeying fairly well the equation obtained. Recently published data8 show that the Cox theory describes satisfactorily the wetting kinetics at advancing for the system glass/poly(dimethylsiloxane)/ ~~~~
~
~~
* To whom correspondence should be addressed.
(1) Glasstone, S.;Laidler, K. J.; Eying, H. J. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (2) Blake, T. D.;Haynea, J. M. J. Colloid Interface Sei. 1969,30,421. ( 3 ) Blake, T. D. WettingKinetics-HowDo WettingLinesMoue? 1988 AIChE Intamational Symposiumon the Mechanicsof Thin-FilmCoating, New Orleans, 1988, Paper la. (4) Petrov, J. G.; Petrov, P. G.Colloids Surf., in press. (5) Petrov, J. G.; Radoev, B. P. Colloid Polym. Sci. 1981, 259,753. (6) Cox, R. G.J. Fluid Mech. 1986, 168, 169. (7) Hoffman, R. L.J. Colloid Interface Sci. 1976,50, 228. (8)Fermigier, M.; Jenffer, P. Ann. Phys. (Paris) 1988, 13, 37.
air, while for some solid/liquid/liquid systems the coincidence between theory and experiment is only qualitative.819 Petrov and Radoev5 have considered both types of friction on the basis of Blake's equation and the hydrodynamic lubrication approximation, applied to the deformed part of the dynamic meniscus. The validity of the equation obtained was checked by studying the steady contact line motion upon deposition of Langmuir-Blodgett multilayers,5J0 as well as upon expansion of the wetting perimeter after the rupture of the thin liquid film between a gas bubble and a quartz surface." de Gennes12 and Brochard-Wyart and de Gennes13also pointed out that both viscous and molecular dissipation could prove commensurable in some cases and that both types of friction should be taken into account in a complete description of the wetting dynamics. An indirect indication of the necessity of such a general approach was found by Sedev and Petrov;14the experimental data on receding of a glycerolwater mixture on siliconized cylinders with different radii showed that the velocity dependences of the dynamic contact angles, BJV, cannot be explained by any of the above-mentioned theoretical equations. The present paper proposes a combined description of wetting kinetics based on a simultaneous use of BlakeHaynes2and Coxs approaches. The model used accounts for the complete energy dissipation upon the contact line motion. Comparison with experimental data for some solid/liquid/air systems is performed. The liquids considered (aqueous glycerol solutions) do not contain surfactants. Thus, the effect of a surface tension gradient, changing the force balance at the liquid/gas interface, is avoided.
Theoretical Model A model avoiding the difficulties of the hydrodynamic description of a moving contact line was proposed first by Hansen and T0ong.1~The flow in a dynamic meniscus is divided into two regions: an inner one, in immediate (9) Foister, R. T.J. Colloid Interjace Sci. 1990,136, 266. (10)Petrov, J. G.Z . Phys. Chem. (Leipzig) 1985,266, 706. (11) Hopf, W.; Stechemeaeer, H. Colloids Surf. lSS8,33,25. (12) de Gennes, P. G.Reo. Mod. Phys. 1985,57,827. (13) Brochard-Wyart, F.;de Gennea, P. G.To be published. (14) Sedev, R. V.;Petrov, J. G.Colloids Surf. 1992, 62, 141. (15) Hansen,R.J.; Toong,T. Y. J . Colloid Interface Sci. 1971,37,196.
0743-7463/92/2408-1762$03.00/00 1992 American Chemical Society
MoleculapHydrodynamic Approach to Wetting Kinetics
Langmuir, Vol. 8, No.7,1992 1763
proximity to the contact line (withcharacteristic dimension
L,)and an outer region (with dimension L >> L8).The
introduction of the inner region is imposed by the singularity of the viscous stresses appearing at the contact line.16 Either the inner region is excluded from the hydrodynamic c ~ n s i d e r a t i o n ~or~ Janother ~ mechanism removing the singularity is assumed-slippage of the liquid with respect to the solid18 or existence of long-range attraction forces in the three-phase contact zone.12 Recentlyde Gennes et al.lehave proposed a more precise consideration of the flow near the moving contact line. In their model four regions are distinguished: a molecular (nonhydrodynamic),a proximal (controlled by long-range van der Waals and viscous forces), a central (shear stress and capillary forces), and a distal one (shear stress and gravitational forces). This consideration gives the moat general scheme by taking into account all driving and dissipative forces, but its quantitative realization is rather complicated and should be a matter of future investigations. The solution of the hydrodynamic problem proposed by Coxs assumes slippage of the fluid in the inner region. The dependence of the dynamic contact angle B on the contact line velocity V is expressed by the equation g(B,c) = g(Bo,4 f Ca ln(L/L,) (1) B refers to the intermediate region between the inner and outer ones,6*20and the dimension L, of the inner region takes the meaning of a slip length. g(8,e) is defined as
((8’ - sin’ ~ ) l (-n8) + sin 8 cos 8) + ((?r - 8)’ - sin2@)(@ - sin 8 cos 8)
At advancing (plus sign), e = p R / p is~ the viscosity ratio of the receding and advancing fluid, and the capillary number should be written as Ca = p ~ V / y .At receding ~ Ca = p ~ V / (yy is the fluid/ (minus sign), e = p d p and fluid interfacial tension). The contact angle Bo is the “microscopic contact angle”, defined in the inner region. In most hydrodynamic considerations it is assumed to be constant and velocity independent.15*20121However, Cox: Voinov,17 and Dussan22admit the possibility of a nonhydrodynamic velocity dependence of Bo. Blake and Haynes2 consider the wetting kinetics as a process of desorption of the molecules of the receding fluid and adsorption of those of the advancing one onto adsorption centers of the solid surface. Their equation could be regarded as a balance of the driving force ylcos By - cos 81 and the friction force located in the three-phase contact zone (both taken per unit length of the contact line): cos 8 = cos 8, f (2nkTly) arsh (V/2KA) (2) k and T are the Boltzman constant and temperature, K (16) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971,35, 85. (17) Voinov, 0. V. Izu. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 1976, 5 7 6 ; Fluid Dun. (Engl. Tranal.) 1976,11,714. (18) Dusean V., E. B. Annu. Rev. Fluid Mech. 1979, 11, 371. (19) de Gennes, P. G.; Hua, X.; Levinson, P. J. Fluid Mech. 1990,212, 55. (20) Hocking, L. M.; Rivers, A. D.J. Fluid Mech. 1982, 121, 425. (21)Lowndes, J. J. Fluid Mech. 1980,101, 631. (22) Dusean V., E. B. International Symposium on Contact Angles and Wetting Phenomena, Invited Lecture, Toronto, Canada,June 21-23, 1990,to be published.
Table I. Prowrties of the Systems Studied
~ _ _ _ _
systemo
I I1 I11
hdeg
P,P
39 48 71
0.462 0.589 1.677
~ , d y n l c m p,g/cm3
64.9 65.0 65.7
1.2016 1.2050 1.2289
c
3.91 X lo-‘ 3.07 X l(r 1.08 X lo-‘
a The systems are as follows: I, PET/glycerol-water/air; 11, PET/ glycerol-water/air; 111, glass/glycerol-water/air.
is the equilibrium frequency of molecular oscillation between two adjacent adsorption centers, situated at a distance A, and n is the average two-dimensionalconcentration of these centers (n = A-2). The minus sign in eq 2 refers to advancing, while the plus sign denotes receding. Oy is the equilibrium Young contact angle. Voinov,17presenting the energy dissipation in the inner region as FV, where F is the nonhydrodynamic friction force in the inner region, obtains cos 00 = [YSG - YSL F(V)l/? (3) where ~ S and G y s are ~ the solid/gas and solid/liquid specific interfacial free energies, respectively. If the advancement or retraction of the liquid in the inner region takes place according to Blake’s molecularkinetic model, this presentation could be written explicitly. In this case F(V)should be given by the expression F(V)= 2nkTarsh (Vl2KA) (4) Taking into account that (YSG - y s ~ ) / y= cos Oy, an equation for a nonhydrodynamic velocity dependence of the angle Bo is obtained:
B o ( V ) = arccos [cos 8, f (2nkTly) arsh (V/2KA)l (5) This equation can be inserted in the equation of Cox (eq 11,thus giving a more general expression with a velocitydependent first term: g(O,e) = gW,(V),el f Ca In (LIL,) (6) Consideringthe system of eqs 5 and 6, it should be noted that both types of friction, the bulk viscous friction (accounted for by the term Ca In LIL,)and the nonhydrodynamic one (rationalized by the term g[B,(V),eI), contribute to the velocity dependence of the dynamic contact angle Wj. Thus, the dynamicbehavior of a threephase system depends on two sets of physical quantities: a microscopic one (K, A, L,)and a macroscopic set ( p , y, By, L). The three parameters remaining unspecified, K, A, and L,,are regarded as fitting parameters between theory and experiment. Comparison with Experiments RecedingDynamic ContactAngles. In order to check the validity of the model proposed, dependences of the receding dynamic contact angle, e,, on contact line velocity, V, were obtained for three solid/liquid/gas systems. Strips of poly(ethy1ene terephthalate) (PET), a siliconized glass cylinder (diameter of 3.01 cm), and aqueous glycerol solutions of different concentrations were used. The characteristics of the systems studied are presented in Table I. The difference of the static receding contact angles OR of systems I and 11at practically the same surface tension and liquid density values is due to the treatment of the solid substrate of system I with chromic acid, probably leading to oxidation and hydrophilization of the poly(ethy1ene terephthalate) surface. The solid substrates were withdrawn vertically through the liquid/gas interface at velocities increasing to the critical velocity of liquid film entrainment V$ The
Petrov and Petrov
1764 Langmuir, Vol. 8, No. 7, 1992
LO 1
3
I
5
9
v IO'kmhl Figure 1. Receding dynamic angle, Or, vs contact line velocity, V, for PET/glycerol-water/air (system I). The solid line represents the trend of eqs 7 and 8, the dashed line eq 1,the dotted line eq 2,and the solid circle the static receding contact angle.
20
-
J
0'
01
0.2
O3
vccm,sl
a4
Figure 3. Receding dynamic angle, Or,vs contact line velocity, V, for glass/glycerol-water/air (system 111). The solid line represents the trend of eqs 7 and 8 and the solid circle the static receding contact angle.
2ol 0
,
0.1
\ 02
Table 11. Values of the Parameters K,A, and L, for the Systems Studied at the Receding Mode of the Contact Line Motion.
K,s-'
systemb
I
~
I1 I11
03 Vkm&
Figure 2. Receding dynamic angle, Or, vs contact line velocity, V, for PET/glycerol-water/air (system 11). The solid line represents the trend of eqs 7 and 8 and the solid circle the static receding contact angle.
meniscus was photographed close to the three-phase contact line with a 16-mm Bolex H16 RX-5 camera, equipped with a Macro-Switar 75-mm lens and extension tubes. The contact angles were determined by tracing a tangent as close as possible to the contact line on the magnified images of the dynamic meniscus. The analysis of the experimental results according to eqs 5 and 6 can be simplified for the systems considered. When the viscosity of the advancing phase is negligibly low (e 0) and the dynamic contact angle is less than 3u/4, the Cox function g(B,e=O) can be approximated by 83/9.14J7 In our experimenta p d p < ~ 4X (Table I) and 8, < 71°, and the maximal error of such an approximation does not exceed 2 9%. Under these conditions the following system of equations is obtained:
-
e; = [e,(v)i3 - ( ~ P v / Yh) (LIL,) e,( v) = BrCCOS [COS 8R + (2nkT/y)arsh (v/%A)]
(7) (8)
5.3 x 103 6.0x lo2 1.0 x 104
A, cm 1.86 x 10-7 2.60 x 1.49 x
LS,cm 2.4 x 10-5 1.3 X lo* 3.3 X 10"
Lo,cm 0.332 0.332 0.330
Lo = ( 2 y / p g Y 2 ,and g is the acceleration due to gravity. * See footnote a in Table I. Table 111. Properties of the Systems from Reference 3 Analyzed at Advancing. systemb BA,deg 82 IV V 72 VI 67 VI1 64
P,P 0.0100 0.0152 0.0424 0.1008
r,dyn/cm 72.7 69.6 66.0 64.6
p,glcm3 0.9982 1.0350 1.1070 1.1500
c
1.81 X 1W2 1.19X 4.26X lW3 1.79 X
The values of 6A and y were indirectlyobtained from the graphical data in ref 3. The systems are as follows: IV, PET/water/air; V, PET/glycerol-water/air; VI, PET/glycerol-water/air; VII, PET/ glycerol-water/air.
*
data of Blake3 for several systems of PET/glycerol-water mixture/air were used, studied within a broad velocity range. The results for liquids of increasing viscosity and decreasing surface tension (Table 111)are shown in Figures 4-7. Since the advancing dynamic angles Be attain 180' and the velocity varies by several orders of magnitude, the data are presented in g(O,)/lgVscale. The termg[B,(v),el in eq 6 is approximated to 8,3/9, since 8, never exceeds 90'. The maximal error of this approximation is 8%. Consequently, the system of equations used to describe the experimental data at advancing is
Figures 1-3 represent the dynamic receding angles 0, versus contact line velocity for the three systems studied. The trend of eqs 7 and 8 is presented by the solid lines. The values of K, A, and L,, varied as free parameters, are given in Table 11. The capillary length, Lo = ( 2 ~ / p g ) ~ / ~ , is assumed to be the characteristic dimension of the outer region.23 The values of K and A characterizing the molecularkinetic behavior of the systems are physically reasonable The trend of eqs 9 and 10 is presented by the solid lines and close to those obtained for similar ~ y s t e m s . ~The ?~ in Figures 4-7. With the rise of the glycerol concentration, slip length L, is also quite acceptable; the literature data the coincidence between theory and experiment improves. are within the range 10-7-10-5cm depending on the While for the first and the second systems (Figures 4 and hydrodynamic models adopted.6*9*2°*21p24 5 ) the theoretical trend follows the experimental data only Advancing Dynamic Contact Angles. In order to qualitatively, the agreement for the last two systems check the validity of the model a t advancing, the literature (Figures 6 and 7) is fairly good. The obtained values of K, A, and L, in the last two cases (Table IV) are physically (23)de Gennes, P.G. Colloid Polym. Sci. 1986,264,463. reasonable, while the values of L, for systems IV and V (24)Huh,C.;Mason, S. G. J. Fluid Mech. 1977,81, 401.
MoleculapHydrodynamic Approach to Wetting Kinetics
I
2t
1
Langmuir, Vol. 8, No. 7,1992 1765
21 I
i
0'
1s v
Figure 4. g(8,) as a function of 1gV for PET/water/air (system IV). The data are from ref 3. The solid line represents the trend of eqs 9 and 10, the dashed line eq 1,and the dotted line eq 2.
d
-i
-i
i
i
i
l
1s v
4
Figure 7. g(0.) as a function of IgV for PET/glycerol-water/air (system VII). The data are from ref 3. The solid line represents the trend of eqs 9 and 10. Table IV. Values of the Parameters K,A, and L.for the Systems Studied at the Advancing Mode of the Contact Line Motion. systemb K,s-l A, cm LI,cm Lo,cm IV 2.0 X 10' 2.50 X 9.6 X 1 t 5 0.385 V VI VI1
4.0 X lo2 1.0 x 103 1.0 x 103
2.50 X 2.10 x 10-7 2.10 x 10-7
3.7 X 1 W 8.7 x 10-5 8.7 x i t 5
0.370 0.349 0.338
Lo = ( 2 y l ~ g ) ' / ~ See . footnote b in Table 111.
- 2 - 1
1
0
2
I
3
1s v Figure 5. g(8,) as a function of 1gV for PET/glycerol-water/air (system V). The data are from ref 3. The solid line represents the trend of eqs 9 and 10.
o
-
2
-
1
0
1
2
3
IgV
Figure 6. g(0,) as a function of 1gV for PET/glycerol-water/air (system VI). The data are from ref 3. The solid line represents the trend of eqs 9 and 10. where the coincidence between theory and experiment is only qualitative should be considered only as rough estimates.
Discussion The velocity dependences of the receding angle cannot be described within the entire velocity range solely on the basis of the moleculekinetic approach. This fact is illustrated for system I in Figure 1. The dotted line represents the best fit of eq 2 to the experimental data. The obtained values of K = 7.6 X lo3 s-l and X = 1.68 X 10-7 cm, are quite satisfactory. However, a t velocities cm/s the dynamic contact angles higher than 6 X predicted by the equation of Blake and Haynes differ substantially from those obtained experimentally.
The attempt to interpret the same data solely on the basis of the hydrodynamic equation (eq 1)is also unsuccessful (see the dashed line). The Cox theory describes well the results at V > 1.5 X cm/s, but below this value the predicted receding dynamic angles are lower than those actually observed. Moreover, the curve crosses the ordinate a t an angle e,* that is loo smaller than the static value. Comparison of these two theoretical curves with the trend predicted by eqs 7 and 8 (the solid line) demonstrates the advantage of the combined approach in the case of receding meniscus. It is seen that the combined equation closely follows the S-shaped trend of the experimental data in the velocity range studied. Similar analysis was performed also for the two branches of the experimental dependence in Figure 4 concerning system IV (PET/water/air). The dotted line represents the trend of eq 2 with K = 2.45 X lo6 s-l and X = 1.05 X cm, values reported by Blakee3 The dashed line is the theoretical trend of 8, according to eq 1. The description is quite good for velocities V < 1cm/s in the first case and V > lo2cm/s for the Cox equation provided that a "pseudostatic" angle do* = 73O (lowerby 9 O than the static angle) is used. The slip length obtained, L, = 5.3 X lV cm, seems quite reasonable. The plateau in the velocity range 1-100 cm/s appears as a transition between the two mechanisms. However, for reasons remaining so far unclear, this transition region cannot be described by eqs 9 and 10 and the coincidence between the combined theory (the solid line) and the experiment is only qualitative. A thorough examination of the experimental data in Figures 4-7 shows that such transition regions exist in all cases, becoming shorter and smoother with the increase of the liquid viscosity and the decrease of the Reynolds number Re. Since the Cox equation (eq 1)has been derived for small Reynolds numbers, the deviation of eqs 9 and 10 from the experimental data at advancing could be due to the presence of considerable inertial forces. While the maximum Reynolds numbed1 a t the receding mode are rather small (lO-LlO-l), their values at advancing, i.e., for the
Petrov and Petrov
1766 Langmuir, Vol. 8, No.7, 1992 Table V. Experimentally Obtained (V,,,.,) and Theoretically Estimated ( Vcr,tb,) Values of the Critical Velocities of Film Entrainment*
systemb
I I11
Vm.exn,cm/s Ver.thsor, cm/s Receding 0.086 0.377
6, %
0.077 0.337
10 11
Advancing
VI VI1 a
326.0 170.5
15 13
376.3 193.0
6 is the corresponding difference between these two magnitudes 6=
Ivcr,exp
- Vcr,theorl
Vcr,exp b
See footnote a of Table I and footnote b of Table 111.
systems analyzed in Figures 4-7, are quite high (10L102). This should give rise to an additional inertial deformation of the liquid meniscus and a greater disagreementbetween eqs 9 and 10 and the experimental data. Probably this also leads to the rather large values of L,(0.9-3.7 pm). The latter seem unrealistic for simple molecules, although Blake2s has obtained L, = 3 pm from the experiments of Schnel126on the Poiseuille flow of water inside silanated glass capillaries. An independent criterion for the applicability of eqs 5 and 6 may be found in the coincidence between the measured and theoretically predicted critical velocities of f i i entrainment at receding and advancing V;. Combining the critical condition for film entrainment2'
E
ea
-
180° at V-
Vi
with eqs 7 and 8 and eqs 9 and 10, one obtains
The expressions 11 and 12 represent transcendental equations which explicitly relate Vcr to the microscopic (K,A, L,)and the macroscopic (8R(A), y, p ) characteristics of the system. The experimental values of the critical velocities and those calculated by a numerical procedure show a coincidence within 10-15%, Table V. It should be mentioned that Vz is strongly overestimated when calculated from eq 2; see the dotted line in Figure 1. Equation 1gives correct values of but they are not directly related to the static contact angle; see the dashed line in Figure 1. The same holds for the advancing mode if only one pair of constants in the Blake's equation is used. As Blake3 has shown this makes the molecularkinetic description of the whole 8J V dependence impossible. Blake3 has interpreted his experimental data (Figures 4-7) only on the basis of eq 2 assuming that two different pairs of values of K and X are to be used for the low- and high-velocity range. In his opinion, such a behavior is related to the presence of two types of functional groups
Cr,
(25)Blake, T. D. Colloids Surf. 1990,47,135. (26)Schnell, E. J. Appl. Phys. 1956,27, 1149. (27)Sedev, R. V.; Petrov, J. G. Colloids Surf. 1991,53, 147.
on the polymer surface (polar ester groups and nonpolar CHz groups), situated at different distances and having different energies of interaction with the molecules of the liquid. This description of the wetting kinetics at advancing seems as good as the one based on the combined eqs 9 and 10. However, the reciprocal value of K obtained for the high-velocity branch in ref 3, 1/K = 10-10 s, is on the same order of magnitude as the escape time T of a water molecule in a bulk phase (T = 10-lo-lO-ll ~ 3 , ~ Moreover, the value of A = 3.19 A is close to the diameter of the water molecule. Therefore, one can doubt that at high contact line velocities the water molecules adsorb onto or desorb from the solid surface. The molecular transitions from a given potential well to the adjacent one more probably correspond to Eyring's model of bulk viscous friction. The existence of horizontal (plateau) regions in the e,! ZgV dependences and the stick-slip motion of the wetting perimeter might reflect the transition between two principally different dissipation mechanisms-a molecularkinetic and a hydrodynamic one. The alternative interpretation in ref 3 suggests an abrupt change of the molecular parameters K and A although both types of functional groups are always present on the solid surface and the change in the strength of the interaction with molecules of the liquid and the distance between the adsorption centers leading from one to the other pair of K and X in eq 2 should occur steadily, i.e., without stickslip effects. On the basis of molecular dynamic simulations of the contact line motion, Thompson and rob bin^^^ have found that 8, remains constant and equal to the static contact angle while the velocity dependence of the apparent dynamic angle follows the Cox equation. Although this statement is not obvious from the snapshot projections of the particle positions in their Figure 2b,c, it puts inquestion the adequacy of the effect considered here. However, its generality should be tested in further investigations. A publication of Koplik et al.,30treating the same problem but with another solid wall model, yields the opposite conclusion-that the motion of two slugs of fluids A and B in a channel occurs "with advancing and receding contact angles which differ from the static angle and from each other". Our results indicate that the dependence of Bo on Vis specific for any particular system. Thus, for the data shown in Figures 4-7 Bo increases only by about 15-36 5% above the corresponding static value whereas Be changes by 220-280%. The combined model proposed here employs the zeroth approximation of the Cox solution in orders of Ca, which is the only one available in an analytical form. The next step could be made using the higher order approximation O(Cal) g(8,t) = g(e,,t) f C a b (LIL,) + Qi,,/f(eo)- Q,,,/f(@l (13) where Qin depends on e, e,, and the particular slip model in the inner zone and Qout on c, 8, d8/dt, and the geometry of the outer hydrodynamic flow. Due to the general form of eq 13 the relative contribution of the terms in the brackets could hardly be estimated. The approximate (28)Elliott, G.E.P.; Riddiford, A. C. J. Colloid Interface Sci. 1967, 23,389.
(29)Thompson, P. A.;Robbins,M.0.Phys. Rev. Lett. 1989,63,766. (30)Koplik, J.; Banavar, J. R.; Willemsen,J. F. Phys. Reu. Lett. 1988, 60,1282. (31)Re, = pLVCr/r,where V, is the contact line velocity at which a liquid (air) film is entrained.
) .
Langmuir, Vol. 8, No. 7, 1992 1767
Moleculal-Hydrodynamic Approach to Wetting Kinetics
solution of Voinov17 for several flow geometries in solid/ liquid/gas systems shows that the third term is about 10% of the first one. Unfortunately nothing can be stated about the term Qh/f(O,,)at present. Further development of the molecular-hydrodynamic model should combine eq 13 with the nonlinear slip condition V, = A sinh (Ea) where V , is the slip velocity induced by the shear stress u and A and E are constants. The above expression has
been proposed by Blake3 on the basis of his eq 2. This combination will give the natural relationship between the slip length L, and the molecular-kinetic parameters K and X which is lacking in the present consideration. Acknowledgment. We thank Mr. R. V. Sedev who participated in a part of the experiments. Registry No.
PET,25038-59-9; glycerol, 56-81-5.
