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solid systems
The Kinetics of Displacement Wetting in Liquid/Liquid/Solid ROBERT Polymers
Department,
General
Motors
Systems
T. FOISTER
Research
Laboratories,
Warren,
Michigan
48090-9055
Received May 24, 1989; accepted September 14, 1989 Spontaneous displacement wetting, where a drop of one liquid displaces a second immiscible liquid from a smooth solid surface, has been investigated experimentally. Relationships between dynamic advancing contact angle and displacement velocity were established for systems of widely ranging viscosity ratio. The experiments were interpreted by extending a previously established correlation for liquid/ vapor systems, as well as via a comparison with the hydrodynamic theory of Cox (J. Fluid Mech. 168, 169 ( 1986)). To compare experimental data with the hydrodynamic theory, a parameter which was unknown a priori-the ratio of slip length to macroscopic length scale-had to be estimated. Consequently, the relationships between displacement kinetics and material properties of the liquids (viscosities, interfacial tension) and the solid (equilibrium contact angle) were shown to depend on the microscopic dynamics, which were in turn characterized by a degree of uncertainty. Nonetheless, by estimating the ratios of slip lengths to macroscopic length scales, it was possible to obtain a reasonable theoretical description for experimental systems of widely ranging viscosity ratio. 0 1990 Academic PKSS.1~.
(designated “ 1”) and the solid expands.due to a decrease in total free energy, expressed as a Displacement wetting plays an important positive spreading coefficient. It was found (4) role in such diverse technologies as tertiary oil that a thin film of the displaced liquid (desrecovery, solid contaminant flotation in filtra- ignated “2”) was carried along ahead of the tion, lubrication, and adhesive bonding to contact line between the two liquids and the contaminated substrates. In spite of its rele- solid, and was probably responsible for the vance to many applied problems, however, measured dependence of the viscous retarding relatively little fundamental work has been force for TPB motion on the geometric mean done on the kinetics of displacement wetting. of the two liquid viscosities. This type of deEfforts thus far (see ( 1,2), for example) have pendence is not expected from simple hydrobeen largely empirical, and have concentrated dynamic considerations, from which one primarily on liquid/vapor/solid systems, since
266 0021-9797190 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal ofCo/loid and Inlerjzce Science, Vol. 136, No. 1, April 1990
DISPLACEMENT
arising from fluid motion in the two bulk liquid phases and at the liquid 1 /liquid 2 phase boundary which is of interest. Model liquid/liquid/solid systems have been used in a series of experiments designed to systematically characterize the process of spontaneous displacement wetting. The displacement velocity of liquid 1 has been related to the viscosities of the two liquids, their interfacial tension, and both the equilibrium, as well as non-equilibrium (dynamic advancing), contact angle. These experiments were confined to the simple geometry of a spreading drop (see Fig. 1) of liquid 1, surrounded by liquid 2. The solid in all cases was a “smooth”
A -
2% 2
1
-fsn 3
-B
261
WETTING
glass plate. Consequently, the role of surface roughness, as well as the surface energy of the solid itself, have not been investigated. Two different theoretical approaches, one based in large part on fundamental concepts of surface science and termed “microscopic”, the other based on classical hydrodynamics, have been used to interpret these experiments. While the hydrodynamic theory provides the most comprehensive description, it still lacks definition in terms of the proper model of the liquid in the immediate vicinity of the contact line. On the other hand, the microscopic theory provides a more detailed picture of this region, but only for liquid/vapor/solid systems. Finally, since drop spreading is a spontaneous process, an additional issue addressed implicitly in analysis of the data is whether the observed dependence on material properties in liquid/liquid/solid systems holds for forced spreading in other geometries, as has recently been demonstrated for liquid/vapor / solid systems (7 ) . If the results can be cast in a form which is independent of both the driving force and the geometry, then the conclusions drawn on the basis of this work are fundamental to the problem of how one liquid displaces another from a solid surface, whether the motion is forced or spontaneous. THEORETICAL
BACKGROUND
Spreading Dynamics in Liquid/ Vapor/Solid Systems: Microscopic Theory The spreading of a drop under capillary action is a non-steady process, with the instantaneous TPB velocity decreasing as a function of time. At equilibrium, the drop configuration corresponds to a minimum in free energy. This state is described by the Young-Dupre equation (8), 723
Frc . 1. Drop spreading sequence: (A) drop at rest (with radius &) prior to film rupture; (B) spreading underway with velocity U = dr/dt, dynamic advancing contact angle &; (C) equilibrium with static contact angle 0..
-
‘713
=
-flZcos
‘%
[II
where 8, is the equilibrium contact angle measured through the spreading liquid (phase 1)) the yii is the interfacial free energy per unit area between phases i and j ( i, j = 1, 2, 3 ) . Journal ofcolloid and Interface Science, Vol. 136, No. 1, April 1990
268
ROBERT
T. FOISTER
The nonsteady motion of a drop as its shape relaxes toward equilibrium can be viewed (9) as motion induced by an “unbalanced” Young-Dupre force per unit length F defined by
of fundamental arguments, de Gennes has shown that liquid motion in this “wedge” region (whose slope is the macroscopic contact angle) is driven by the angular term (the difference in cosines) in Eq. [2], and that the free energy term given by S, the spreading F = S + y,z( 1 - cos 0,) coefficient, is expended entirely in motion of [21 the precursor film. The Tanner-Hoffman law + 712(cos 0, - ’33s ed), therefore emerges in de Gennes’ theory as a where S = (yz3 - yi3 - -yr2) is the spreading consequence of this partition of energy dissicoefficient, and t& is the advancing dynamic pation between the film and wedge regions. contact angle. In terms of Eq. [ 21, the timeWhile fundamental advances have been dependent force arises due to the difference made in understanding spreading dynamics between 8, and &. Obviously, at equilibrium in liquid/vapor systems from the microscopic ee= ed, and F = 0. point of view, a parallel approach applicable De Gennes [ 91 has used Eq. [ 21 as a starting to liquid/liquid/solid systems has not been point for a “microscopic” approach to deduce fully developed, although Joanny and Andelthe empirical “Tanner-Hoffman Law,” which man ( 10) have reported progress in this area. essentially states that for 8, = 0, the TPB ve- There are, however, additional empirical relocity U varies with ed as sults which suggest an alternative approach.
u-e:
[31
during the latter stages of spreading, i.e., ed < 1. It is important to note two points concerning the Tanner-Hoffman law. First, Hoffman’s original experiments involved forced spreading at constant velocities in circular tubes. It has generally been assumed that the “law” (Eq. [3]) is both independent of geometry, and equally applicable to spontaneous, and therefore nonsteady, spreading. Concerning this latter point, Chen (7) has recently shown in a series of experiments with spreading drops that the Tanner-Hoffman Law does hold for spontaneous spreading in liquid/ vapor systems as well. Second, the dynamic contact angle measured experimentally is characteristic of the limiting slope of the macroscopic phase boundary, not necessarily of the true microscopic contact angle. In fact, for liquid/vapor spreading, the limiting slope of the liquid/vapor phase boundary changes gradually into a thin precursor film, typically within a few hundred angstroms of the three-phase boundary (9) _The Tanner-Hoffman law relates only to the macroscopic contact angle. Qn the basis Journal of Colioid and Interface
Science, Vol. 136, No. 1, April 1990
Spreading Dynamics in Liquid/Liquid/ Solid Systems: Extension of the Microscopic Theory Hoffman’s original data for spreading in liquid/vapor systems included several cases of non-zero equilibrium contact angle (2). Jiang et al. ( 11) have shown that all of Hoffman’s data could be fit to the equation H=
8, - COS ed I + cos ee
COS
= tanh( 4.96 Ca0.702),
[41
where Ca is the dimensionless capillary number defined by
Ca=!!L!.!!.
[51
712
A rationale for the form of Eq. [4] can be deduced by referring to the expression for the unbalanced Young-Dupre force (Eq. [ 21) . As explained above, since the free energy S is consumed in motion of the precursor film, the remaining energy, represented by the difference between cosines of the equilibrium and dynamic contact angles, is responsible for the
DISPLACEMENT
WETTING
269
source of this problem was a kinetic incompatibility of the no-slip condition with TPB motion, coupled with various approximations used to solve the hydrodynamic equations ( 3 ) . However, Dussan V. and Davis ( 13 ) presented convincing arguments to the contrary: the no-slip condition is in fact compatible with a general class of rolling, or “tank-tread” type motions. Velocity fields which describe these (02 - Ii?:) - Ca0,‘02 if31 motions are multivalued at the contact line, as the Tanner-Hoffman law for nonzero equi- however, and are therefore responsible for inlibrium contact angles. The microscopic the- finite velocity gradients and the nonintegrable ory of de Gennes leads to an analogous but stress singularity. Consequently, the real functionally somewhat different relationship problem with the hydrodynamic description for the case of nonzero equilibrium contact is that the normal continuum model (Newangles (12). tonian liquid) breaks down in the vicinity of As pointed out above, a relationship be- the three-phase boundary. Moreover, the cortween dynamic contact angle and spreading rect model and its associated constitutive velocity for liquid/liquid / solid systems, com- equation are not known a priori ( 14). parable to the empirical result given by Eq. Subsequent treatments (see (3, 14)) of [ 41, has not been developed. However, it may three-phase boundary motion have therefore be expected that the form of this equation utilized slippage at the TPB in an ad hoc fashwould hold for systems where the viscosity of ion, so that the stress singularity is removed, the spreading phase is much greater than that but at the expense of introducing an unknown of the displaced phase, and that a transition slip length. Once slip is introduced, the techto Eq. [4] would be observed as the ratio of niques of singular perturbation theory ( 15 ) can displaced to spreading phase viscosity ap- be used to obtain solutions of the hydrodyproached zero. For nonzero values of this ratio, namic equations. To date the most general we expect the two viscosities to enter explicitly treatment of the hydrodynamics of TPB mointo the relationship between dynamic contact tion using this approach is the theory of Cox angle and spreading velocity. It is possible to (5 ) . The elements of this theory needed for a develop an empirical approach based on an comparison with experiment are described extension of the liquid/vapor/solid treatment, below. and this is done below. However, a more rigTheory. There are two basic ideas which orous approach, based on hydrodynamics, can form the foundation of the hydrodynamic apbe utilized as well. proach to displacement wetting. The first is that a degree of slippage must be allowed at the contact line. The second is the definition Spreading Dynamics in Liquid/Liquid/ of appropriate length scales (i.e., physical reSolid Systems: Hydrodynamic Theory gions) for application of the method of Original attempts to apply hydrodynamic matched asymptotic expansions. theory to displacement wetting led to the conIntroduction of a slip boundary condition clusion that the classical no-slip boundary amounts to removing the constraint that the condition, which specifies that the tangential fluid tangential velocity at the three-phase velocity of the fluid at the solid surface be equal boundary be equal to the velocity of the solid. to the velocity of the solid, gives rise to a non- In the simplest case, for example, the differintegrable stress singularity at the three-phase ence between fluid and solid velocities is asboundary ( 3 ) . It was initially believed that the sumed to be proportional to the tangential
velocity dependence of the macroscopically observed contact angle (9). Equation [ 41 is consistent with the TannerHoffman law, as can be seen by expanding the left hand side for 8, = 0, Bd4 1, and the right hand side for Ca 4 1. This gives Bi - Ca0.‘02, which is essentially the relationship given by Eq. [ 31 above. Equation [ 41 also gives
Journal of Colloid and Inferface Science, Vol. 136, No. 1, April 1990
270
ROBERT
stress. Although the particular model of slip is undetermined a priori, to proceed theoretically all one needs is the concept of a slip length S, which is generally of microscopic dimension (
T. FOISTER
mediate region of dimension much greater than S, but still much less than a macroscopic length scale R0 (e.g., the drop radius), and (3) an outer region of dimension comparable to R0 . Using this approach for systems with negligible contact angle hysteresis Cox ( 5 ) derived, at the lowest order in the capillary number, the relationship between the dynamic contact angle, &, and the Capillary number g(dd)
-
g(k)
=
Ca
ln(e-‘),
[71
where 6 = s/Ro, Ca is the Capillary number defined above, 8, is the equilibrium contact angle, and the function g( (3) is given by g(o) = s,,‘f(P, X)dP>
[81
with f(p
>
x) = UP - sin2P) (( a - /3) + sin P cos p} + { (7r - p)2 - sin2P) { ,8 - sin fl cos /3} * 2 sin p[X2(,B2 - sin2P) + 2X{p(7r - ,B) + sin2/3} + {(x - p)2 - sin2P}] [91
Equation [7] is the hydrodynamic analogue of the empirical relationship given by Eq. [ 41. However, in contrast to Eq. [4], which was developed for negligible external phase viscosities (liquid/vapor systems with X N 0)) Eq. [ 71 is applicable to systems of arbitrary viscosity ratio. It can therefore be compared with experimental data for systems covering a broad range of viscosities for both fluid phases. However, the parameter 6 above is undetermined, by virtue of the fact that a specific model of slip has not been invoked at this stage of approximation. Considerably more complicated expressions, valid for higher orders in the capillary number, have been formally derived ( 5 ) , but they require specification of the slip model as well as the particular geometry (e.g., spreading drop, meniscus in a tube, etc.). The utility of Eq. [ 71 lies in its relative simplicity, and therefore it is desirable to make a self-consistent estimate of c to enable direct comparison of theory with experiment. A Journal of C&id
and Inteifiie
Science, Vol. 136, No. I, April 1990
means of estimating E for each of the experimental systems is described in a later section. It is also important to note that the dynamic contact angle in Eq. [ 71 is, as a consequence of the degree of approximation invoked in the derivation, equal to the contact angle calculated from measurements assuming a static interface shape ( 5). This is equivalent to the assumption made below in calculating dynamic contact angles from measurements of instantaneous drop shapes. Comparison of experiments and theory. The experimental data, which are cast in the form of contact angle as a function of instantaneous velocity, are interpreted in terms of the theoretical treatments described above in two ways. First, we have taken a semiempirical approach by simply extending Hoffman’s correlation to nonzero values of the viscosity ratio. This leads to a representation of the data in terms of a “master” curve. As a second approach, we
DISPLACEMENT
have compared the data with predictions based on the hydrodynamic theory. These two approaches were then analyzed and contrasted so that a unified physical interpretation could be made. EXPERIMENTAL
Materials The model systems are listed, together with their sources and other relevant information, in Table I. All of these materials were used as received, with no further modification. As model displacing liquids, we used a standard epoxy resin (diglycidyl ether of bisphenol A, EPON 825@ from Shell Chemical) and a chemically similar compound (2-phenoxyoxirane, “phenyl glycidyl ether,” from Aldrich Chemical). Liquids used to model the displaced phase were various viscosity grades of polydimethylsiloxane oils (DOW 200@ Series, from Dow Corning). Table I also gives experimental system designations, each of which consists of the displacing liquid (“liquid 1”) and the displaced liquid (“liquid 2”). Each system has a characteristic vicosity ratio X ( =y2 / pl ) , where y is the bulk viscosity at a shear rate of 1 s-l, and a characteristic interfacial tension y r2. These properties are given in Table II.
TABLE Experimental Phase I
EPON EPON EPON EPON PGE’ PGE PGE PGE
825” 825 825 825
Systems
20 cs Dow Si Fluidb 100 cs Dow Si Fluid 1000 cs Dow Si Fluid 12,500 cs Dow Si Fluid 20 cs Dow Si Fluid 100 cs Dow Si Fluid 1000 cs Dow Si Fluid 12,500 cs Dow Si Fluid
Determination
of Viscosities, Densities, and Interfacial Tension
Viscosities as a function of shear rate ( 1O-3 to lo2 s-l ) at room temperature were determined with a modified Brookfield viscometer, using a spring relaxation technique ( 17). Densities were measured using standard pycnometry, and interfacial tensions were determined by the pendant drop technique ( 18), using pendant drop profiles obtained from the video microscopy setup described below. These properties are listed for each experimental system in Table II. In all but one case (phenyl glycidyl ether), the liquids showed various degrees of shear thinning for shear rates less than 1 s-l. (This is somewhat different from the shear thinning of polymeric fluids which, for example, typically have a Newtonian plateau at lower shear rates, with a decreasing viscosity at higher shear rates.) It is possible that shear rates of less than 1 s-l could be characteristic of the final stages of TPB motion as the spreading velocity approaches zero, although the appropriate length scale for this regime of viscous flow is unclear. In any case, to take shear thinning into account in a systematic fashion when comparing the data with theory is beyond the scope of the current investigation. It was therefore assumed that the appropriate values of the viscosity were the constant values for shear rates > 1 sl.
Observation of Three-Phase Boundary Motion
I and Designations
Phase 2
271
WETTING
Designation
825/D20 825/DlOO 825/DlOOO 825/D12,500 PGE/D20 PGE/DlOO PGE/DlOOO PGE/D12,500
’ Diglycidyl ether of bisphenol A: Shell Chemical. b Polydimethylsiloxane fluid: Dow Corning. ’ Phenyl glycidyl ether (2-phenoxyoxirane): Aldrich.
Drop spreading experiments were carried out in a glass cell, using the video microscopy system and sample preparation techniques described previously (4). Generally, to minimize the effects of gravity on drop shape and motion, drop radii were of order 1 mm (see also the discussion of gravity effects below). Spreading dynamics were recorded for subsequent analysis using a Questar@ M 1 lens (35X magnification) attached to the video camera. To minimize the effects of ambient vibrations, the glass cell, camera, and lens were Journal oJColloid and Inferfoce Science. Vol. 136, No. I, April 1990
272
ROBERT
System system
825/D20 825/DlOO 825/DlOOO PGE/D20 PGE/DlOO PGE/DlOOO PGE/D12,500 a Viscosity
PI (Pa SY
BZ(Pa s)
4.00 4.00 4.00 X X x X
0.0236 0.0957 0.871 0.0236 0.0957 0.871 14.6
4.64 4.64 4.64 4.64 value
1O-3 1O-3 1O-3 1O-3
T. FOISTER TABLE
II
Material
Properties
0.207 0.196 0.185 0.182 0.140 0.130 0.125
7.34 8.60 11.2 4.23 5.53 5.06 4.76
5.90 x 1o-3 2.39 x lo-* 2.18 X 10-l 5.09 20.9 209 3180
line, similar to viscous fingering in Hele-Shaw cells ( 19), were observed. RESULTS
of TPB Velocity and Dynamic
Contact Angles Individual drops were dispensed from a microsyringe, and were allowed to approach the liquid 2/solid boundary under the influence of gravity. After rupture of the intervening liquid film, measurement of TPB motion began when the diameter of the contact area between the drop and the solid increased in an axisymmetric fashion (4 ) . Generally, this precluded measurement of dynamic contact angles close to 180”, so that the maximum in the calculated function H (Eq. [ 41) did not exceed approximately 0.93. Various stages in drop spreading are shown schematically in Fig. 1. TPB velocities were determined by graphical differentiation of TPB radius versus time plots, while dynamic contact angles were calculated from drop height (h) and instantaneous radius (Y) using the following “spherical cap” relation ( 3 ) :
coso = 1 - we* d
1 + (h/r)2 .
[lOI
Finally, it should be noted that even for extremely viscous displaced liquids, spontaneous spreading velocities were apparently low enough so that no corrugations of the contact Journal ofcoiloid
YIZ (mN/m)
is for shear rates 3 1.0 s-‘.
mounted on a vibration isolation table. Temperature was maintained at 23°C (-tl”C) during the experiments. Measurement
Ap (kg/m’ X IO-‘)
and Interface Science, Vol. 136, No. 1, April 1990
AND
DISCUSSION
Efects of Gravity and Drop Size on TPB Motion To minimize the effect of gravity on drop shape and TPB motion, drop radii were generally of order 1 mm or less. A quantitative assessment of the relative effects of gravity and capillary forces can be made on the basis of the dimensionless Bond number B ( 3 ), where B = ApgRi Y12
Here Ap is the density difference between the two liquids, g is the acceleration due to gravity, and RO is the “rest radius” of the drop (see Fig. 1) . Calculated values of the Bond number for the seven systems used are given in Table III. Obviously, capillary effects dominate gravity, since for all systems B < 1, although for one system B is as high as 0.49. Likewise, the relative importance of inertial and capillary effects can be characterized by the value of the Weber number W ( 3 ) , where w=
ApU*R; 712
.
The velocity U can be any characteristic velocity; the maximum measured velocity was used to calculate the Weber numbers listed in
DISPLACEMENT TABLE Drop
Rest Radii, Various
System
825/D20 825/DlOO 825/DlOOO PGE/D20 PGE/DlOO PGE/D 1000 PGE/D12,500 a Weber number b Bond number
III
Equilibrium Dimensionless
Ro (cm)
8, Wd
0.0929 0.0918 0.0964 0.108 0.112 0.104 0.101
12 71 70 76 73 82 90
Contact Angles, Groups
2.36 9.67 4.78 4.39 1.23 1.18 5.83
x X X x X x x
1O-6 LO-’ 10m6 1O-3 1O-4 1O-6 10m8
and
0.239 0.188 0.151 0.492 0.311 0.273 0.279
= (A~U’R&YIZ. = (ApgR$/yIz.
273
WETTING
very small values of X, the calculated values of H as a function of Ca were very close to the liquid/vapor (X N 0) correlation. This general behavior is reasonable and would have been expected on physical grounds. Since the data correlating dynamic contact angle and velocity (i.e., H vs Ca) retained the general shape of the liquid/vapor correlation, but were shifted to lower values of Ca with increasing X, a simple horizontal shift factor LYwas determined for each system such that the data could be superimposed on the correlation for X = 0. Thus a “new” value of the capillary number, Ca’, was defined as Ca’ = oL(X)Ca.
Table III. Since for all the seven experimental systems W < 1Op3, inertial effects did not significantly affect TPB motion. Finally, in previous work (4) it was shown that the early stages of the displacement process proceeded at rates which decreased with drop size. Although drop size effects were not investigated in detail for all of the systems considered here, for one particular system (825/DlOOO, X = 0.218) it was found that at a given dynamic contact angle, velocities generally decreased with drop size. The effect of drop size on the relationship between dynamic contact angle and capillary number is discussed in more detail below. However, all other experiments were carried out with drops of similar size (mean radius = 0.10 1 + 0.0077 cm, see Table III).
Using the values of the horizontal shift factors given in Table IV, the dependence of 01on h was found to be of the form a(X) = a( 1 + X)‘,
[I21
where a = 2.1 and b = 1.O, with a correlation coefficient of 0.98. The dependence of LXon X is particularly simple, and is physically reasonable in that the viscous dissipation arising from TPB motion enters simply as a sum of terms: piU + y2U. Also this dependence has a reasonable asymptotic form, so that in the limit of either h % 1 or X < 1, the dissipation is dominated by the liquid of largest viscosity. Obviously, if Eq. [ 121 holds, the master curve correlation would lead to the extension of the Tanner-Hoffman law, for nonzero values of X, 0: - (1 + X)Ca,
Empirical Extension of the Tanner/Hofman Law
1111
[I31
with an appropriate analog for nonzero values of 8, ( 12). It should be noted, however, that For all of the systems investigated the equi- the data shown in Fig. 3 do not superimpose librium contact angle, &, was nonzero (see well in the region which Eq. [ 131 is supposed Table III). When the function H in Eq. [4] to describe. Consequently, without further was plotted against capillary number, each justification, this result should be viewed as system gave a curve with the general shape of entirely empirical, and characterized, moreover, by a considerable uncertainty, which is the liquid/vapor correlation ( 10). Furthermore, as shown in Fig. 2, each curve was due primarily to inherent difficulties in meashifted to smaller values of Ca as the ratio of suring slowly varying velocities and contact angles. In the region where H decreases rapidly the external phase (2) viscosity to the internal ( 1) spreading phase viscosity increased. For with the capillary number, there appears to be
274
ROBERT
T. FOISTER
1 I 0.8
x 10-z
V
0 h = 2.39
x 10.;
V v
0 h = 0.218
Le
x
v
0.8
X A=209
H 0.4
0
O
: 0 ;
A A
n
d
;,
A
X
h-0
t
0
A
x
V
v A = 3180
a
k A A
x X
20.9
v
0
0
an
V xx
h -
0
A
0 A = 5.09 A
/
0
h - 5.00
0
a
J
r
0.2
0 10-e
10'6
10'4
10-'
100
Ca FIG. 2. Experimental data plotted vapor correlation (X N 0).
as Hoffman’s
correlation:
somewhat better superposition of the data on the liquid/vapor correlation. However, the reason for this and its significance are not clear. Comparison of Hydrodynamic Theory with Experimental Data E&et of viscosity ratio. The starting point for comparing predictions based on hydroTABLE Shift Factors System
825/D20 825/DlOO 825/DlOOO PGE/D20 PGE/D 100 PGE/D 1000 PGE/D12,500 Journal ofcolloid
IV
for Hoffman’s
Correlation
x (1(2/1(1) 5.90 x 1o-3 2.39 x lo-’ 2.18 x 10-l 5.09 20.9 209 3180
(Y(shift factors)
1.045 1.21 2.95 17.1 116 635 3.6 X lo3
and Interfare Science, Vol. 136, No. I, April I990
H vs Ca. Solid curve
is Eq. [4]
for liquid/
dynamic theory with the experimental data is Eq. [ 71. Although the viscosity ratio is measured independently, the other parameter which must be known to compare Eq. [ 71 with experiment is t, the ratio of slip length to the macroscopic length scale. As explained above, E is not uniquely determined, since at lowest order in Ca, neither the slip model nor the experimental geometry have been specified. However, it is possible to estimate E in a selfconsistent fashion for each experimental systern and this is discussed in detail below. From Eq. [ 71 and [ 91, once X and tie are specified, the dependence of the dynamic contact angle on Ca ln( c-l) can be determined by calculating g( 0,) - g( 6,). Thus a plot of ed vs. g(Bd) - g(0,) is equivalent to plotting 19~vs. Ca ln( 6-l). Figure 4 shows theoretical curves determined by numerical integration of the function f( p, X) given by Eq. [ 91 for
DISPLACEMENT
275
WETTING
1 m A = 5.90 x IO-3 3 A = 2.39 x IO-2
0.8
l
A = 0.218
3 A = 5.09 h A = 20.9
0.6
x A = 209
H
CJ A = 3180 0.4
0.2
FIG. 3. Master vs Ca’ (=&a).
curve
representation
of experimental
each of the experimental values of X and associated 0,‘s together with a calculated curve for X = 0. (An arbitrary & value of 70” was taken for X = 0). It is immediately obvious that the curves are shifted to smaller values of Ca ln( c -i ) as X increases, and that the y-axis intercept is 8,. Furthermore, as pointed out by Cox (5 ), for every ~lo~lze~ovalue of X, there is a maximum value of the capillary number for which ed = 7r. Although the theory may have limited applicability in the neighborhood of & = 7r (5), physically this corresponds to the phenomenon of entrainment. When X is rigorously equal to zero, i.e., when the external phase is a vacuum, then Eq. [ 71 predicts that there is no maximum value of Ca ln( e-l), and therefore no entrainment. Interestingly, this is also the only case for which the no-slip con-
data shifted
to liquid/vapor
(X N 0) correlation:
H
dition does not result in a nonintegrable stress singularity (20). Nonetheless, even for small values of X, there is still a maximum value of Ca for which tid = r. Using the computed value of g( r) - g( 0,)) a self-consistent value of t for each viscosity ratio can be estimated in the following way. From Fig. 2, the maximum value, Ca,, of the experimental capillary number can be obtained by extrapolation of each curve to H = 1, which corresponds to Bd = s. Thus, from Eq. [71,
c = exp{-[g(r) - g(kJllCa,}.
[141
Since this expression for E is very sensitive to small variations in the estimated value of Ca, , the computed value of 6 will have a rather large inherent uncertainty. Journal of
Colloid and Intefice
Science, Vol. 136, No. 1, April 1990
276
ROBERT
---
T. FOSTER
---_
-------______--------------_______________ __-----
-------
---_
--\
-\
.-.
-\ \
____---_.--._------.-_-___-_ _____ _ --------~. “l..
*\ \\ \\.
\
‘\ __----------____
\: --Y.
\ .‘\ ‘1% A \
Journal of Coiioid and Intefme Science, Vol. 136, No. 1, April 1990
DISPLACEMENT TABLE Estimated System
825/D20 825/DlOO 825/DlOOO PGE/D20 PGE/D 100 PGE/D 1000 PGE/D12,500
Values
V
of c and Slip Length
x (=PdPl) 5.90 X 1O-3 2.39 x lo-’ 2.18 x 10-l 5.09 20.9 209 3180
(s)
e 2.6 8.1 4.5 3.1 1.5 8.6 8.1
X x x x x x X
s (cm) 1O-4 10m5 1o-5 1om5 1om5 lo-’ 1O-6
2.4 7.4 4.4 3.4 1.7 8.9 8.2
X
x x x x x x
10m5 1O-6 1om6 1o-6 1o-6 lo-’ lo-’
Values of t calculated from Equation [ 141 are given in Table V, and they range from 2.6 X 10d4 to 8.1 X 10d6, as X ranges from 5.9 X 10e3 to 3.2 X 103. These values were then used with measured values of Ca to cast the experimental data in the form of Ba vs Ca ln( e-’ ) . The experimental curves are plotted in Fig. 5, together with the corresponding theoretical curves. Inspection shows that the overall trend of the data (curves are shifted to lower Ca with increasing X) is predicted by Eq. [ 7 1, as are the shapes of the individual curves. Overall, since Eq. [ 71 represents the relationship between dynamic contact angle and interface velocity (capillary number) to lowest order in Ca only, the agreement between experiment and theory can be judged quite satisfactory. Correspondence with experiment could in principle be improved by computing the theoretical solution to higher orders in Ca. Physically, this means that the instantaneous shape of the interface becomes an explicit function of the fluid velocities, therefore of the capillary number and X. Mathematically, however, one has to specify both the geometry (which is straightforward) and also a model for slippage (5 ). Unfortunately, this latter task is complicated by the fact that macroscopic measurements do not readily distinguish between models of slip on the microscale ( 3, 14). Two additional factors may have influenced the correspondence of theory and experiment depicted in Fig. 5. First, with one exception
277
WETTING
(PGE), the viscosities of all of the liquids used are characterized by shear thinning for shear rates below 1 set-‘. Thus, in the latter stages of displacement where velocities are very small, the characteristic viscosities could be different from the constant value used for shear rates > 1 set -l, but this would depend on the choice of length scale as well as the velocity. In fact, taking the slip length as characteristic of the latter stages of spreading would result in shear rates well into the Newtonian regime. In any case, shear rate-dependent viscosities are not taken into account in the hydrodynamic treatment of Cox ( 5 ) , and there is no clear indication of how such behavior might modify the relationship between the dynamic contact angle and the Capillary number. Second, the assumption that the drop shape is adequately described by the “spherical cap” approximation (Eq. [lo] ) could be called into question, particularly for Bond numbers greater than 0.1 (see Table III). Concerning this latter point, selected dynamic contact angles were measured directly from the magnified drop image, and were found to correspond closely ( *3” ) with values calculated from Eq. [ 3 1. This approximation was therefore judged satisfactory, particularly for characterizing trends in displacement with X. E&ct ofdro~ size. Even at the lowest order in Ca, it is clear that the external length scale RO (which corresponds approximately to the drop rest radius in the present case) can influence the relationship between Bd and Ca through the parameter c. In contrast to gravitational effects, which would tend to increase the displacement rate with drop size (2 1)) Eq. [ 7 ] predicts that for a given system, simply increasing this characteristic length will shift the theoretical curve of tid vs Ca ( = (g( 0,) - g( 0,) } /ln( t-l)) to smaller values of Ca. As shown in Fig. 6, this is in fact observed experimentally. As RO increases from 0.08 11 to 0.0964 to 0.148 cm, at constant X, the data for dd vs Ca is shifted toward smaller Ca. A similar trend has been observed by Ngan and Dussan V. (22), but for forced liquid/vapor Journal
of Co/bid
and Interfar
Scmce, Vol. 136, No. I, April I990
ROBERT
278
T. FOSTER
l
onma01XI
\
--. -
i
Journal ofcolioid and Interface Science, Vol. 136, No. 1, Apd 1990
0
DISPLACEMENT
279
WETTING
Ca Ro=0.0811
l FIG.
6. Dynamic
A contact
Ro=0.0964
angle vs ( :apillary
spreading between flat plates, and was discussed in terms of the hydrodynamic theory by Cox (5).
Variation of slip length with viscosity ratio. From the definition oft, a slip length s can be calculated from the estimated values of Egiven in Table III, keeping in mind, of course, that there is a large inherent uncertainty in these values of E. Assuming that the macroscopic length scale can be identified with the rest radius &,, slip lengths for the various experimental systems were calculated (Table III) and are plotted against X in Fig. 7. Previous estimates of s from experimental data were for systems where X < 1 ( 5, 15), and were of order 10 -5 to 10 -6 cm. This range is consistent with the present data. Figure 7 also indicates that in general s decreases with X. Little can be said definitively concerning the variation of the slip length with the viscosity ratio, simply because, as has been stated above, no firm a priori theoretical foundation exists upon which to distinguish between different models of slippage. Thus, the experiments themselves are unable to provide this
number:
l
effect
of drop
Ro=0.148
c
size.
distinction. However, it is interesting to note that Hocking (23) has developed a microscopic model of slippage where very small quantities of the displaced fluid are trapped due to surface roughness. He derived expressions for the slip length as a function of surface geometry (asperity height, wavelength, etc.) and the viscosities of displaced and displacing fluids. In this model s varies roughly as X-‘, whereas a linear regression on the data plotted in Fig. 7 gives s - X-O.*.
Connection Between Microscopic and Hydrodynamic Theories Tanner/Hoflman law for arbitrary viscosity ratios. As was demonstrated above, plotting the experimental data according to Hoffman’s correlation (Eq. [ 41) led eventually to a superposition of the data on a master curve. Calculation of the shift factor OL(Eq. [ 12]), together with Hoffman’s correlation, resulted in the following empirical extension of the Tanner-Hoffman law: 192- (1 + X)Ca. Journal ofcoiloid
[I51
and Interface Science, Vol. 136, No. I, April 1990
ROBERT
T. FOISTER
Viscosity
Ratio
FIG. 7. Estimated slip length (s) vs viscosity ratio ( A).
Although this result is particularly simple and therefore attractive, there was, as pointed out above, poor superposition of the experimental data as Bdapproached 0,. There is, therefore, a corresponding uncertainty in the relationship given by Eq. [ 15 1. In fact Joanny and Andelman have argued that the Tanner-Hoffman law should remain unchanged (except possibly for numerical constants), even when the external phase is of nonnegligible viscosity ( 10). The microscopic theory” has not been rigorously extended for all viscosity ratios; however, from Eqs. [ 7 ] and [ 91 it is possible to obtain an explicit prediction from the hydrodynamic theory for the variation of 19~with Ca as Bd approaches zero. Expanding Eq. [ 91 for p + 1 we find that
which, upon integration,
gives
[T* + 8(h - 4)]0i, + 0 1207r2
(a;).
[17]
Therefore, the hydrodynamic theory predicts that, irregardless of the viscosity ratio, the Tanner-Hoffman law is essentially unchanged. Clearly, however, the first term in Eq. [ 171 will dominate the second only when 6d 4 (8~/9X). In other words, the primary influence of a nonnegligible external phase viscosity will be to prolong the asymptotic approach to the Tanner-Hoffman law.
Synthesis of microscopic and hydrodynamic approaches. It is interesting from a fundamental point of view to compare the asymptotic form of Eq. [ 71 and the corresponding explicit expression derived by de Gennes (9 ). From the hydrodynamic theory,
-
[r2
+
8(X
-
4)lfi4
+
o
(>P5),
24~~ Journal of Colioid and Interface Science, Vol. 136, No. I, April I990
[l(j]
e,j - Ca
ln(Rols),
[I81
DISPLACEMENT
and de Gennes finds that 8; -
Ca
ln(&-d-%id,
[I91
where x,,, is, by definition, a macroscopic length scale, essentially equivalent to R0 in the present case, and Xmi, is the length of the precursor film ( 9 ) . In the hydrodynamic theory, the slip length s is not known a priori, but the same is true for the length of the precursor film in the microscopic theory. Ultimately, the slip model must be consistent with an appropriate constitutive equation for the contact line region, and, as pointed out by Dussan V. ( 13), no such equation is currently known. On the other hand, de Gennes has derived expressions for the length of the precursor film for at least two cases: one where slip predominates, and one where the precursor film is due to dominant van der Waals forces (the “disjoining pressure”) (9). In the former case, xmi” is proportional to the slip length divided by the dynamic contact angle, while in the latter case, Xmi” is proportional to a characteristic interaction distance (which is defined in terms of the Hamaker constant and the interfacial tension between the two liquids) divided by the square of the dynamic contact angle (9). Thus, except for the rather weak logarithmic dependence on Bd, in the limit ed < 1, both the hydrodynamic and microscopic approaches give essentially equivalent results. While these latter results offer some insight into the physical character of the contact line region, they do not, in their present form, serve to define the constitutive law which is needed to formulate the hydrodynamic theory in its most general form. In fact, whether precursor films exist as such in systems where X is not negligible is still an open question. Likewise, it should be kept in mind that “slippage” has not been established experimentally, but has been introduced formally to remove the stress singularity in the classical approach. Although progress has recently been made in elucidating possible mechanisms of slip from a phenomenological point of view ( 24, 25 ), a truly fundamental theory of contact line motion, ap-
281
WETTING
plicable to systems of arbitrary viscosity, will ultimately have to be based on a synthesis of the microscopic and hydrodynamic approaches. In all likelihood this synthesis will be the result of using microscopic models to derive constitutive equations for the contact line region. FINAL
REMARKS
The experimental data described above have been used to establish relationships between the material properties of liquid/liquid/solid systems and the rate of displacement wetting. It was found that a theoretical treatment based on hydrodynamics provides an adequate foundation for correlating the dynamic advancing and equilibrium contact angles with the velocity of the three-phase boundary. In general, as the viscosity of the external (displaced) phase increases relative to the viscosity of the displacing phase, the velocity at a given dynamic contact angle decreases. This has been placed on a quantitative basis by plotting dynamic advancing contact angle versus the dimensionless capillary number. It was shown that, in order to compare the experimental data with the hydrodynamic theory, a parameter which is unknown a priori-the ratio of slip length to macroscopic length scale-had to be estimated. Furthermore, using drop rest radius as the macroscopic length scale, it was found that slip lengths calculated from the estimated values of this parameter decreased with increasing viscosity ratio. The observed influence of drop size on displacement rate was also rationalized on the basis of this parameter. Consequently, the relationship between displacement kinetics and the material properties of the liquids (viscosities, interfacial tension) and the solid (equilibrium contact angle) was shown to depend on the microscopic dynamics; which were in turn characterized by a degree of uncertainty. This uncertainty arises from the well-documented shortcomings of classical hydrodynamics in describing liquid/liquid displacement, as well Journal of Co/hid and Inlerface Science, Vol. 136, No. I, April 1990
282
ROBERT
as the fact that a microscopic theory applicable to systems with nonnegligible external phase viscosity has yet to be developed. Nonetheless, by taking a self-consistent empirical approach to estimating the ratio of slip length to macroscopic length scale for a given system, we have shown that it is possible to obtain a reasonable theoretical description for experimental systems of widely ranging viscosity ratio. ACKNOWLEDGMENTS The author acknowledges Professor A. N. Gent (University of Akron) and Professor D. R. Paul (University of Texas at Austin) for very helpful discussions; Dr. J. C. Ulicny (General Motors Research Laboratories) for assistance in the numerical work; and Dr. H. H. Kuo, Mr. T. B. Pietrzyk, and Mr. K. S. Snavely (General Motors Research Laboratories) for viscosity measurements.
REFERENCES 1. Tanner, L. H., J. Phys. D 12, 1473 (1979). 2. Hoffman, R. L., J. Colloid Interface Sci. 50, 228 (1975); 94,470 (1983). 3. Dussan V., E. B., Ann. Rev. Fluid Mech. 11, 311 (1979). 4. Foister, R. T., J. Colloid Inter&e Sci. 116, 109 (1987). 5. Cox, R. G., J. FluidMech. 168, 169 (1986).
Journal ofColloid and Inte&ce Science, Vol. 136, No. 1, April 1990
T. FOISTER 6. Hocking, L. M., .I Fluid Mech. 79, 209 ( 1977). 7. Chen, J. D., J. Colloid Interface Sci. 122,60 ( 1988). 8. Adamson, A. W., “Physical Chemistry of Surfaces” (4th ed.), p. 338, Wiley, New York, 1982. 9. De Gennes, P. G., Rev. Mod. Phys. 57, 827 (1985). 10. Joanny, J.-F., and Andelman, D., J. Colloid Interface Sci. 119,451 (1987). 11. Jiang, T.-S., Oh, S.-G., and Slattery, J. C., J. Colloid
Interface Sci. 69, 74 ( 1979 ) 12. De Gennes, P. G., C. R. Acad. Sci. Paris 298, 111 (1984). 13. Dussan V., E. B., and Davis, S. H., .I. Fluid Mech. 65, 71 (1974). 14. Dussan V., E. B., in “Waves on Fluid Interfaces” (R. E. Meyer, Ed.), p. 303, Academic Press, New York, 1983. 15. Van Dyke, M., “Perturbation Methods in Fluid Mechanics,” Parabolic Press, Stanford, 1975. 16. Hocking, L. M., and Rivers, A. D., J. Fluid Mech. 121,425 (1982). 17. Kuo, H. H., J. Coat. Technol. 57, 57 (1985). 18. Ambwami, D. S., and Fort, T., Jr., “Surface and Colloid Science,” (E. Matijevic, Ed.), Vol. 11, p. 93, Plenum, New York, 1979. 19. Saffman, P. G., and Taylor, G. I., Proc. R. Sot. Lond. Ser. A 245,312 (1958). 20. Pismen, L. M., and Nir, A., Phys. Fluids 25,3 ( 1982). 2 1. Lopez, J., Miller, C. A., and Ruckenstein, E., J. Colloid Interface Sci. 56,460 ( 1976 ) . 22. Ngan, C. G., and Dussan V., E. B., J. Fluid Mech. 118,27 (1982). 23. Hocking, L. M., J. Fluid Mech. 76, 5 1 ( 1976). 24. Jansons, K. M., J. Fluid Mech. 167, 393 (1986). 25. Durbin, P. A., J. Fluid Mech. 197, 157 (1988).
Department,
General
Motors
Systems
T. FOISTER
Research
Laboratories,
Warren,
Michigan
48090-9055
Received May 24, 1989; accepted September 14, 1989 Spontaneous displacement wetting, where a drop of one liquid displaces a second immiscible liquid from a smooth solid surface, has been investigated experimentally. Relationships between dynamic advancing contact angle and displacement velocity were established for systems of widely ranging viscosity ratio. The experiments were interpreted by extending a previously established correlation for liquid/ vapor systems, as well as via a comparison with the hydrodynamic theory of Cox (J. Fluid Mech. 168, 169 ( 1986)). To compare experimental data with the hydrodynamic theory, a parameter which was unknown a priori-the ratio of slip length to macroscopic length scale-had to be estimated. Consequently, the relationships between displacement kinetics and material properties of the liquids (viscosities, interfacial tension) and the solid (equilibrium contact angle) were shown to depend on the microscopic dynamics, which were in turn characterized by a degree of uncertainty. Nonetheless, by estimating the ratios of slip lengths to macroscopic length scales, it was possible to obtain a reasonable theoretical description for experimental systems of widely ranging viscosity ratio. 0 1990 Academic PKSS.1~.
(designated “ 1”) and the solid expands.due to a decrease in total free energy, expressed as a Displacement wetting plays an important positive spreading coefficient. It was found (4) role in such diverse technologies as tertiary oil that a thin film of the displaced liquid (desrecovery, solid contaminant flotation in filtra- ignated “2”) was carried along ahead of the tion, lubrication, and adhesive bonding to contact line between the two liquids and the contaminated substrates. In spite of its rele- solid, and was probably responsible for the vance to many applied problems, however, measured dependence of the viscous retarding relatively little fundamental work has been force for TPB motion on the geometric mean done on the kinetics of displacement wetting. of the two liquid viscosities. This type of deEfforts thus far (see ( 1,2), for example) have pendence is not expected from simple hydrobeen largely empirical, and have concentrated dynamic considerations, from which one primarily on liquid/vapor/solid systems, since
266 0021-9797190 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal ofCo/loid and Inlerjzce Science, Vol. 136, No. 1, April 1990
DISPLACEMENT
arising from fluid motion in the two bulk liquid phases and at the liquid 1 /liquid 2 phase boundary which is of interest. Model liquid/liquid/solid systems have been used in a series of experiments designed to systematically characterize the process of spontaneous displacement wetting. The displacement velocity of liquid 1 has been related to the viscosities of the two liquids, their interfacial tension, and both the equilibrium, as well as non-equilibrium (dynamic advancing), contact angle. These experiments were confined to the simple geometry of a spreading drop (see Fig. 1) of liquid 1, surrounded by liquid 2. The solid in all cases was a “smooth”
A -
2% 2
1
-fsn 3
-B
261
WETTING
glass plate. Consequently, the role of surface roughness, as well as the surface energy of the solid itself, have not been investigated. Two different theoretical approaches, one based in large part on fundamental concepts of surface science and termed “microscopic”, the other based on classical hydrodynamics, have been used to interpret these experiments. While the hydrodynamic theory provides the most comprehensive description, it still lacks definition in terms of the proper model of the liquid in the immediate vicinity of the contact line. On the other hand, the microscopic theory provides a more detailed picture of this region, but only for liquid/vapor/solid systems. Finally, since drop spreading is a spontaneous process, an additional issue addressed implicitly in analysis of the data is whether the observed dependence on material properties in liquid/liquid/solid systems holds for forced spreading in other geometries, as has recently been demonstrated for liquid/vapor / solid systems (7 ) . If the results can be cast in a form which is independent of both the driving force and the geometry, then the conclusions drawn on the basis of this work are fundamental to the problem of how one liquid displaces another from a solid surface, whether the motion is forced or spontaneous. THEORETICAL
BACKGROUND
Spreading Dynamics in Liquid/ Vapor/Solid Systems: Microscopic Theory The spreading of a drop under capillary action is a non-steady process, with the instantaneous TPB velocity decreasing as a function of time. At equilibrium, the drop configuration corresponds to a minimum in free energy. This state is described by the Young-Dupre equation (8), 723
Frc . 1. Drop spreading sequence: (A) drop at rest (with radius &) prior to film rupture; (B) spreading underway with velocity U = dr/dt, dynamic advancing contact angle &; (C) equilibrium with static contact angle 0..
-
‘713
=
-flZcos
‘%
[II
where 8, is the equilibrium contact angle measured through the spreading liquid (phase 1)) the yii is the interfacial free energy per unit area between phases i and j ( i, j = 1, 2, 3 ) . Journal ofcolloid and Interface Science, Vol. 136, No. 1, April 1990
268
ROBERT
T. FOISTER
The nonsteady motion of a drop as its shape relaxes toward equilibrium can be viewed (9) as motion induced by an “unbalanced” Young-Dupre force per unit length F defined by
of fundamental arguments, de Gennes has shown that liquid motion in this “wedge” region (whose slope is the macroscopic contact angle) is driven by the angular term (the difference in cosines) in Eq. [2], and that the free energy term given by S, the spreading F = S + y,z( 1 - cos 0,) coefficient, is expended entirely in motion of [21 the precursor film. The Tanner-Hoffman law + 712(cos 0, - ’33s ed), therefore emerges in de Gennes’ theory as a where S = (yz3 - yi3 - -yr2) is the spreading consequence of this partition of energy dissicoefficient, and t& is the advancing dynamic pation between the film and wedge regions. contact angle. In terms of Eq. [ 21, the timeWhile fundamental advances have been dependent force arises due to the difference made in understanding spreading dynamics between 8, and &. Obviously, at equilibrium in liquid/vapor systems from the microscopic ee= ed, and F = 0. point of view, a parallel approach applicable De Gennes [ 91 has used Eq. [ 21 as a starting to liquid/liquid/solid systems has not been point for a “microscopic” approach to deduce fully developed, although Joanny and Andelthe empirical “Tanner-Hoffman Law,” which man ( 10) have reported progress in this area. essentially states that for 8, = 0, the TPB ve- There are, however, additional empirical relocity U varies with ed as sults which suggest an alternative approach.
u-e:
[31
during the latter stages of spreading, i.e., ed < 1. It is important to note two points concerning the Tanner-Hoffman law. First, Hoffman’s original experiments involved forced spreading at constant velocities in circular tubes. It has generally been assumed that the “law” (Eq. [3]) is both independent of geometry, and equally applicable to spontaneous, and therefore nonsteady, spreading. Concerning this latter point, Chen (7) has recently shown in a series of experiments with spreading drops that the Tanner-Hoffman Law does hold for spontaneous spreading in liquid/ vapor systems as well. Second, the dynamic contact angle measured experimentally is characteristic of the limiting slope of the macroscopic phase boundary, not necessarily of the true microscopic contact angle. In fact, for liquid/vapor spreading, the limiting slope of the liquid/vapor phase boundary changes gradually into a thin precursor film, typically within a few hundred angstroms of the three-phase boundary (9) _The Tanner-Hoffman law relates only to the macroscopic contact angle. Qn the basis Journal of Colioid and Interface
Science, Vol. 136, No. 1, April 1990
Spreading Dynamics in Liquid/Liquid/ Solid Systems: Extension of the Microscopic Theory Hoffman’s original data for spreading in liquid/vapor systems included several cases of non-zero equilibrium contact angle (2). Jiang et al. ( 11) have shown that all of Hoffman’s data could be fit to the equation H=
8, - COS ed I + cos ee
COS
= tanh( 4.96 Ca0.702),
[41
where Ca is the dimensionless capillary number defined by
Ca=!!L!.!!.
[51
712
A rationale for the form of Eq. [4] can be deduced by referring to the expression for the unbalanced Young-Dupre force (Eq. [ 21) . As explained above, since the free energy S is consumed in motion of the precursor film, the remaining energy, represented by the difference between cosines of the equilibrium and dynamic contact angles, is responsible for the
DISPLACEMENT
WETTING
269
source of this problem was a kinetic incompatibility of the no-slip condition with TPB motion, coupled with various approximations used to solve the hydrodynamic equations ( 3 ) . However, Dussan V. and Davis ( 13 ) presented convincing arguments to the contrary: the no-slip condition is in fact compatible with a general class of rolling, or “tank-tread” type motions. Velocity fields which describe these (02 - Ii?:) - Ca0,‘02 if31 motions are multivalued at the contact line, as the Tanner-Hoffman law for nonzero equi- however, and are therefore responsible for inlibrium contact angles. The microscopic the- finite velocity gradients and the nonintegrable ory of de Gennes leads to an analogous but stress singularity. Consequently, the real functionally somewhat different relationship problem with the hydrodynamic description for the case of nonzero equilibrium contact is that the normal continuum model (Newangles (12). tonian liquid) breaks down in the vicinity of As pointed out above, a relationship be- the three-phase boundary. Moreover, the cortween dynamic contact angle and spreading rect model and its associated constitutive velocity for liquid/liquid / solid systems, com- equation are not known a priori ( 14). parable to the empirical result given by Eq. Subsequent treatments (see (3, 14)) of [ 41, has not been developed. However, it may three-phase boundary motion have therefore be expected that the form of this equation utilized slippage at the TPB in an ad hoc fashwould hold for systems where the viscosity of ion, so that the stress singularity is removed, the spreading phase is much greater than that but at the expense of introducing an unknown of the displaced phase, and that a transition slip length. Once slip is introduced, the techto Eq. [4] would be observed as the ratio of niques of singular perturbation theory ( 15 ) can displaced to spreading phase viscosity ap- be used to obtain solutions of the hydrodyproached zero. For nonzero values of this ratio, namic equations. To date the most general we expect the two viscosities to enter explicitly treatment of the hydrodynamics of TPB mointo the relationship between dynamic contact tion using this approach is the theory of Cox angle and spreading velocity. It is possible to (5 ) . The elements of this theory needed for a develop an empirical approach based on an comparison with experiment are described extension of the liquid/vapor/solid treatment, below. and this is done below. However, a more rigTheory. There are two basic ideas which orous approach, based on hydrodynamics, can form the foundation of the hydrodynamic apbe utilized as well. proach to displacement wetting. The first is that a degree of slippage must be allowed at the contact line. The second is the definition Spreading Dynamics in Liquid/Liquid/ of appropriate length scales (i.e., physical reSolid Systems: Hydrodynamic Theory gions) for application of the method of Original attempts to apply hydrodynamic matched asymptotic expansions. theory to displacement wetting led to the conIntroduction of a slip boundary condition clusion that the classical no-slip boundary amounts to removing the constraint that the condition, which specifies that the tangential fluid tangential velocity at the three-phase velocity of the fluid at the solid surface be equal boundary be equal to the velocity of the solid. to the velocity of the solid, gives rise to a non- In the simplest case, for example, the differintegrable stress singularity at the three-phase ence between fluid and solid velocities is asboundary ( 3 ) . It was initially believed that the sumed to be proportional to the tangential
velocity dependence of the macroscopically observed contact angle (9). Equation [ 41 is consistent with the TannerHoffman law, as can be seen by expanding the left hand side for 8, = 0, Bd4 1, and the right hand side for Ca 4 1. This gives Bi - Ca0.‘02, which is essentially the relationship given by Eq. [ 31 above. Equation [ 41 also gives
Journal of Colloid and Inferface Science, Vol. 136, No. 1, April 1990
270
ROBERT
stress. Although the particular model of slip is undetermined a priori, to proceed theoretically all one needs is the concept of a slip length S, which is generally of microscopic dimension (
T. FOISTER
mediate region of dimension much greater than S, but still much less than a macroscopic length scale R0 (e.g., the drop radius), and (3) an outer region of dimension comparable to R0 . Using this approach for systems with negligible contact angle hysteresis Cox ( 5 ) derived, at the lowest order in the capillary number, the relationship between the dynamic contact angle, &, and the Capillary number g(dd)
-
g(k)
=
Ca
ln(e-‘),
[71
where 6 = s/Ro, Ca is the Capillary number defined above, 8, is the equilibrium contact angle, and the function g( (3) is given by g(o) = s,,‘f(P, X)dP>
[81
with f(p
>
x) = UP - sin2P) (( a - /3) + sin P cos p} + { (7r - p)2 - sin2P) { ,8 - sin fl cos /3} * 2 sin p[X2(,B2 - sin2P) + 2X{p(7r - ,B) + sin2/3} + {(x - p)2 - sin2P}] [91
Equation [7] is the hydrodynamic analogue of the empirical relationship given by Eq. [ 41. However, in contrast to Eq. [4], which was developed for negligible external phase viscosities (liquid/vapor systems with X N 0)) Eq. [ 71 is applicable to systems of arbitrary viscosity ratio. It can therefore be compared with experimental data for systems covering a broad range of viscosities for both fluid phases. However, the parameter 6 above is undetermined, by virtue of the fact that a specific model of slip has not been invoked at this stage of approximation. Considerably more complicated expressions, valid for higher orders in the capillary number, have been formally derived ( 5 ) , but they require specification of the slip model as well as the particular geometry (e.g., spreading drop, meniscus in a tube, etc.). The utility of Eq. [ 71 lies in its relative simplicity, and therefore it is desirable to make a self-consistent estimate of c to enable direct comparison of theory with experiment. A Journal of C&id
and Inteifiie
Science, Vol. 136, No. I, April 1990
means of estimating E for each of the experimental systems is described in a later section. It is also important to note that the dynamic contact angle in Eq. [ 71 is, as a consequence of the degree of approximation invoked in the derivation, equal to the contact angle calculated from measurements assuming a static interface shape ( 5). This is equivalent to the assumption made below in calculating dynamic contact angles from measurements of instantaneous drop shapes. Comparison of experiments and theory. The experimental data, which are cast in the form of contact angle as a function of instantaneous velocity, are interpreted in terms of the theoretical treatments described above in two ways. First, we have taken a semiempirical approach by simply extending Hoffman’s correlation to nonzero values of the viscosity ratio. This leads to a representation of the data in terms of a “master” curve. As a second approach, we
DISPLACEMENT
have compared the data with predictions based on the hydrodynamic theory. These two approaches were then analyzed and contrasted so that a unified physical interpretation could be made. EXPERIMENTAL
Materials The model systems are listed, together with their sources and other relevant information, in Table I. All of these materials were used as received, with no further modification. As model displacing liquids, we used a standard epoxy resin (diglycidyl ether of bisphenol A, EPON 825@ from Shell Chemical) and a chemically similar compound (2-phenoxyoxirane, “phenyl glycidyl ether,” from Aldrich Chemical). Liquids used to model the displaced phase were various viscosity grades of polydimethylsiloxane oils (DOW 200@ Series, from Dow Corning). Table I also gives experimental system designations, each of which consists of the displacing liquid (“liquid 1”) and the displaced liquid (“liquid 2”). Each system has a characteristic vicosity ratio X ( =y2 / pl ) , where y is the bulk viscosity at a shear rate of 1 s-l, and a characteristic interfacial tension y r2. These properties are given in Table II.
TABLE Experimental Phase I
EPON EPON EPON EPON PGE’ PGE PGE PGE
825” 825 825 825
Systems
20 cs Dow Si Fluidb 100 cs Dow Si Fluid 1000 cs Dow Si Fluid 12,500 cs Dow Si Fluid 20 cs Dow Si Fluid 100 cs Dow Si Fluid 1000 cs Dow Si Fluid 12,500 cs Dow Si Fluid
Determination
of Viscosities, Densities, and Interfacial Tension
Viscosities as a function of shear rate ( 1O-3 to lo2 s-l ) at room temperature were determined with a modified Brookfield viscometer, using a spring relaxation technique ( 17). Densities were measured using standard pycnometry, and interfacial tensions were determined by the pendant drop technique ( 18), using pendant drop profiles obtained from the video microscopy setup described below. These properties are listed for each experimental system in Table II. In all but one case (phenyl glycidyl ether), the liquids showed various degrees of shear thinning for shear rates less than 1 s-l. (This is somewhat different from the shear thinning of polymeric fluids which, for example, typically have a Newtonian plateau at lower shear rates, with a decreasing viscosity at higher shear rates.) It is possible that shear rates of less than 1 s-l could be characteristic of the final stages of TPB motion as the spreading velocity approaches zero, although the appropriate length scale for this regime of viscous flow is unclear. In any case, to take shear thinning into account in a systematic fashion when comparing the data with theory is beyond the scope of the current investigation. It was therefore assumed that the appropriate values of the viscosity were the constant values for shear rates > 1 sl.
Observation of Three-Phase Boundary Motion
I and Designations
Phase 2
271
WETTING
Designation
825/D20 825/DlOO 825/DlOOO 825/D12,500 PGE/D20 PGE/DlOO PGE/DlOOO PGE/D12,500
’ Diglycidyl ether of bisphenol A: Shell Chemical. b Polydimethylsiloxane fluid: Dow Corning. ’ Phenyl glycidyl ether (2-phenoxyoxirane): Aldrich.
Drop spreading experiments were carried out in a glass cell, using the video microscopy system and sample preparation techniques described previously (4). Generally, to minimize the effects of gravity on drop shape and motion, drop radii were of order 1 mm (see also the discussion of gravity effects below). Spreading dynamics were recorded for subsequent analysis using a Questar@ M 1 lens (35X magnification) attached to the video camera. To minimize the effects of ambient vibrations, the glass cell, camera, and lens were Journal oJColloid and Inferfoce Science. Vol. 136, No. I, April 1990
272
ROBERT
System system
825/D20 825/DlOO 825/DlOOO PGE/D20 PGE/DlOO PGE/DlOOO PGE/D12,500 a Viscosity
PI (Pa SY
BZ(Pa s)
4.00 4.00 4.00 X X x X
0.0236 0.0957 0.871 0.0236 0.0957 0.871 14.6
4.64 4.64 4.64 4.64 value
1O-3 1O-3 1O-3 1O-3
T. FOISTER TABLE
II
Material
Properties
0.207 0.196 0.185 0.182 0.140 0.130 0.125
7.34 8.60 11.2 4.23 5.53 5.06 4.76
5.90 x 1o-3 2.39 x lo-* 2.18 X 10-l 5.09 20.9 209 3180
line, similar to viscous fingering in Hele-Shaw cells ( 19), were observed. RESULTS
of TPB Velocity and Dynamic
Contact Angles Individual drops were dispensed from a microsyringe, and were allowed to approach the liquid 2/solid boundary under the influence of gravity. After rupture of the intervening liquid film, measurement of TPB motion began when the diameter of the contact area between the drop and the solid increased in an axisymmetric fashion (4 ) . Generally, this precluded measurement of dynamic contact angles close to 180”, so that the maximum in the calculated function H (Eq. [ 41) did not exceed approximately 0.93. Various stages in drop spreading are shown schematically in Fig. 1. TPB velocities were determined by graphical differentiation of TPB radius versus time plots, while dynamic contact angles were calculated from drop height (h) and instantaneous radius (Y) using the following “spherical cap” relation ( 3 ) :
coso = 1 - we* d
1 + (h/r)2 .
[lOI
Finally, it should be noted that even for extremely viscous displaced liquids, spontaneous spreading velocities were apparently low enough so that no corrugations of the contact Journal ofcoiloid
YIZ (mN/m)
is for shear rates 3 1.0 s-‘.
mounted on a vibration isolation table. Temperature was maintained at 23°C (-tl”C) during the experiments. Measurement
Ap (kg/m’ X IO-‘)
and Interface Science, Vol. 136, No. 1, April 1990
AND
DISCUSSION
Efects of Gravity and Drop Size on TPB Motion To minimize the effect of gravity on drop shape and TPB motion, drop radii were generally of order 1 mm or less. A quantitative assessment of the relative effects of gravity and capillary forces can be made on the basis of the dimensionless Bond number B ( 3 ), where B = ApgRi Y12
Here Ap is the density difference between the two liquids, g is the acceleration due to gravity, and RO is the “rest radius” of the drop (see Fig. 1) . Calculated values of the Bond number for the seven systems used are given in Table III. Obviously, capillary effects dominate gravity, since for all systems B < 1, although for one system B is as high as 0.49. Likewise, the relative importance of inertial and capillary effects can be characterized by the value of the Weber number W ( 3 ) , where w=
ApU*R; 712
.
The velocity U can be any characteristic velocity; the maximum measured velocity was used to calculate the Weber numbers listed in
DISPLACEMENT TABLE Drop
Rest Radii, Various
System
825/D20 825/DlOO 825/DlOOO PGE/D20 PGE/DlOO PGE/D 1000 PGE/D12,500 a Weber number b Bond number
III
Equilibrium Dimensionless
Ro (cm)
8, Wd
0.0929 0.0918 0.0964 0.108 0.112 0.104 0.101
12 71 70 76 73 82 90
Contact Angles, Groups
2.36 9.67 4.78 4.39 1.23 1.18 5.83
x X X x X x x
1O-6 LO-’ 10m6 1O-3 1O-4 1O-6 10m8
and
0.239 0.188 0.151 0.492 0.311 0.273 0.279
= (A~U’R&YIZ. = (ApgR$/yIz.
273
WETTING
very small values of X, the calculated values of H as a function of Ca were very close to the liquid/vapor (X N 0) correlation. This general behavior is reasonable and would have been expected on physical grounds. Since the data correlating dynamic contact angle and velocity (i.e., H vs Ca) retained the general shape of the liquid/vapor correlation, but were shifted to lower values of Ca with increasing X, a simple horizontal shift factor LYwas determined for each system such that the data could be superimposed on the correlation for X = 0. Thus a “new” value of the capillary number, Ca’, was defined as Ca’ = oL(X)Ca.
Table III. Since for all the seven experimental systems W < 1Op3, inertial effects did not significantly affect TPB motion. Finally, in previous work (4) it was shown that the early stages of the displacement process proceeded at rates which decreased with drop size. Although drop size effects were not investigated in detail for all of the systems considered here, for one particular system (825/DlOOO, X = 0.218) it was found that at a given dynamic contact angle, velocities generally decreased with drop size. The effect of drop size on the relationship between dynamic contact angle and capillary number is discussed in more detail below. However, all other experiments were carried out with drops of similar size (mean radius = 0.10 1 + 0.0077 cm, see Table III).
Using the values of the horizontal shift factors given in Table IV, the dependence of 01on h was found to be of the form a(X) = a( 1 + X)‘,
[I21
where a = 2.1 and b = 1.O, with a correlation coefficient of 0.98. The dependence of LXon X is particularly simple, and is physically reasonable in that the viscous dissipation arising from TPB motion enters simply as a sum of terms: piU + y2U. Also this dependence has a reasonable asymptotic form, so that in the limit of either h % 1 or X < 1, the dissipation is dominated by the liquid of largest viscosity. Obviously, if Eq. [ 121 holds, the master curve correlation would lead to the extension of the Tanner-Hoffman law, for nonzero values of X, 0: - (1 + X)Ca,
Empirical Extension of the Tanner/Hofman Law
1111
[I31
with an appropriate analog for nonzero values of 8, ( 12). It should be noted, however, that For all of the systems investigated the equi- the data shown in Fig. 3 do not superimpose librium contact angle, &, was nonzero (see well in the region which Eq. [ 131 is supposed Table III). When the function H in Eq. [4] to describe. Consequently, without further was plotted against capillary number, each justification, this result should be viewed as system gave a curve with the general shape of entirely empirical, and characterized, moreover, by a considerable uncertainty, which is the liquid/vapor correlation ( 10). Furthermore, as shown in Fig. 2, each curve was due primarily to inherent difficulties in meashifted to smaller values of Ca as the ratio of suring slowly varying velocities and contact angles. In the region where H decreases rapidly the external phase (2) viscosity to the internal ( 1) spreading phase viscosity increased. For with the capillary number, there appears to be
274
ROBERT
T. FOISTER
1 I 0.8
x 10-z
V
0 h = 2.39
x 10.;
V v
0 h = 0.218
Le
x
v
0.8
X A=209
H 0.4
0
O
: 0 ;
A A
n
d
;,
A
X
h-0
t
0
A
x
V
v A = 3180
a
k A A
x X
20.9
v
0
0
an
V xx
h -
0
A
0 A = 5.09 A
/
0
h - 5.00
0
a
J
r
0.2
0 10-e
10'6
10'4
10-'
100
Ca FIG. 2. Experimental data plotted vapor correlation (X N 0).
as Hoffman’s
correlation:
somewhat better superposition of the data on the liquid/vapor correlation. However, the reason for this and its significance are not clear. Comparison of Hydrodynamic Theory with Experimental Data E&et of viscosity ratio. The starting point for comparing predictions based on hydroTABLE Shift Factors System
825/D20 825/DlOO 825/DlOOO PGE/D20 PGE/D 100 PGE/D 1000 PGE/D12,500 Journal ofcolloid
IV
for Hoffman’s
Correlation
x (1(2/1(1) 5.90 x 1o-3 2.39 x lo-’ 2.18 x 10-l 5.09 20.9 209 3180
(Y(shift factors)
1.045 1.21 2.95 17.1 116 635 3.6 X lo3
and Interfare Science, Vol. 136, No. I, April I990
H vs Ca. Solid curve
is Eq. [4]
for liquid/
dynamic theory with the experimental data is Eq. [ 71. Although the viscosity ratio is measured independently, the other parameter which must be known to compare Eq. [ 71 with experiment is t, the ratio of slip length to the macroscopic length scale. As explained above, E is not uniquely determined, since at lowest order in Ca, neither the slip model nor the experimental geometry have been specified. However, it is possible to estimate E in a selfconsistent fashion for each experimental systern and this is discussed in detail below. From Eq. [ 71 and [ 91, once X and tie are specified, the dependence of the dynamic contact angle on Ca ln( c-l) can be determined by calculating g( 0,) - g( 6,). Thus a plot of ed vs. g(Bd) - g(0,) is equivalent to plotting 19~vs. Ca ln( 6-l). Figure 4 shows theoretical curves determined by numerical integration of the function f( p, X) given by Eq. [ 91 for
DISPLACEMENT
275
WETTING
1 m A = 5.90 x IO-3 3 A = 2.39 x IO-2
0.8
l
A = 0.218
3 A = 5.09 h A = 20.9
0.6
x A = 209
H
CJ A = 3180 0.4
0.2
FIG. 3. Master vs Ca’ (=&a).
curve
representation
of experimental
each of the experimental values of X and associated 0,‘s together with a calculated curve for X = 0. (An arbitrary & value of 70” was taken for X = 0). It is immediately obvious that the curves are shifted to smaller values of Ca ln( c -i ) as X increases, and that the y-axis intercept is 8,. Furthermore, as pointed out by Cox (5 ), for every ~lo~lze~ovalue of X, there is a maximum value of the capillary number for which ed = 7r. Although the theory may have limited applicability in the neighborhood of & = 7r (5), physically this corresponds to the phenomenon of entrainment. When X is rigorously equal to zero, i.e., when the external phase is a vacuum, then Eq. [ 71 predicts that there is no maximum value of Ca ln( e-l), and therefore no entrainment. Interestingly, this is also the only case for which the no-slip con-
data shifted
to liquid/vapor
(X N 0) correlation:
H
dition does not result in a nonintegrable stress singularity (20). Nonetheless, even for small values of X, there is still a maximum value of Ca for which tid = r. Using the computed value of g( r) - g( 0,)) a self-consistent value of t for each viscosity ratio can be estimated in the following way. From Fig. 2, the maximum value, Ca,, of the experimental capillary number can be obtained by extrapolation of each curve to H = 1, which corresponds to Bd = s. Thus, from Eq. [71,
c = exp{-[g(r) - g(kJllCa,}.
[141
Since this expression for E is very sensitive to small variations in the estimated value of Ca, , the computed value of 6 will have a rather large inherent uncertainty. Journal of
Colloid and Intefice
Science, Vol. 136, No. 1, April 1990
276
ROBERT
---
T. FOSTER
---_
-------______--------------_______________ __-----
-------
---_
--\
-\
.-.
-\ \
____---_.--._------.-_-___-_ _____ _ --------~. “l..
*\ \\ \\.
\
‘\ __----------____
\: --Y.
\ .‘\ ‘1% A \
Journal of Coiioid and Intefme Science, Vol. 136, No. 1, April 1990
DISPLACEMENT TABLE Estimated System
825/D20 825/DlOO 825/DlOOO PGE/D20 PGE/D 100 PGE/D 1000 PGE/D12,500
Values
V
of c and Slip Length
x (=PdPl) 5.90 X 1O-3 2.39 x lo-’ 2.18 x 10-l 5.09 20.9 209 3180
(s)
e 2.6 8.1 4.5 3.1 1.5 8.6 8.1
X x x x x x X
s (cm) 1O-4 10m5 1o-5 1om5 1om5 lo-’ 1O-6
2.4 7.4 4.4 3.4 1.7 8.9 8.2
X
x x x x x x
10m5 1O-6 1om6 1o-6 1o-6 lo-’ lo-’
Values of t calculated from Equation [ 141 are given in Table V, and they range from 2.6 X 10d4 to 8.1 X 10d6, as X ranges from 5.9 X 10e3 to 3.2 X 103. These values were then used with measured values of Ca to cast the experimental data in the form of Ba vs Ca ln( e-’ ) . The experimental curves are plotted in Fig. 5, together with the corresponding theoretical curves. Inspection shows that the overall trend of the data (curves are shifted to lower Ca with increasing X) is predicted by Eq. [ 7 1, as are the shapes of the individual curves. Overall, since Eq. [ 71 represents the relationship between dynamic contact angle and interface velocity (capillary number) to lowest order in Ca only, the agreement between experiment and theory can be judged quite satisfactory. Correspondence with experiment could in principle be improved by computing the theoretical solution to higher orders in Ca. Physically, this means that the instantaneous shape of the interface becomes an explicit function of the fluid velocities, therefore of the capillary number and X. Mathematically, however, one has to specify both the geometry (which is straightforward) and also a model for slippage (5 ). Unfortunately, this latter task is complicated by the fact that macroscopic measurements do not readily distinguish between models of slip on the microscale ( 3, 14). Two additional factors may have influenced the correspondence of theory and experiment depicted in Fig. 5. First, with one exception
277
WETTING
(PGE), the viscosities of all of the liquids used are characterized by shear thinning for shear rates below 1 set-‘. Thus, in the latter stages of displacement where velocities are very small, the characteristic viscosities could be different from the constant value used for shear rates > 1 set -l, but this would depend on the choice of length scale as well as the velocity. In fact, taking the slip length as characteristic of the latter stages of spreading would result in shear rates well into the Newtonian regime. In any case, shear rate-dependent viscosities are not taken into account in the hydrodynamic treatment of Cox ( 5 ) , and there is no clear indication of how such behavior might modify the relationship between the dynamic contact angle and the Capillary number. Second, the assumption that the drop shape is adequately described by the “spherical cap” approximation (Eq. [lo] ) could be called into question, particularly for Bond numbers greater than 0.1 (see Table III). Concerning this latter point, selected dynamic contact angles were measured directly from the magnified drop image, and were found to correspond closely ( *3” ) with values calculated from Eq. [ 3 1. This approximation was therefore judged satisfactory, particularly for characterizing trends in displacement with X. E&ct ofdro~ size. Even at the lowest order in Ca, it is clear that the external length scale RO (which corresponds approximately to the drop rest radius in the present case) can influence the relationship between Bd and Ca through the parameter c. In contrast to gravitational effects, which would tend to increase the displacement rate with drop size (2 1)) Eq. [ 7 ] predicts that for a given system, simply increasing this characteristic length will shift the theoretical curve of tid vs Ca ( = (g( 0,) - g( 0,) } /ln( t-l)) to smaller values of Ca. As shown in Fig. 6, this is in fact observed experimentally. As RO increases from 0.08 11 to 0.0964 to 0.148 cm, at constant X, the data for dd vs Ca is shifted toward smaller Ca. A similar trend has been observed by Ngan and Dussan V. (22), but for forced liquid/vapor Journal
of Co/bid
and Interfar
Scmce, Vol. 136, No. I, April I990
ROBERT
278
T. FOSTER
l
onma01XI
\
--. -
i
Journal ofcolioid and Interface Science, Vol. 136, No. 1, Apd 1990
0
DISPLACEMENT
279
WETTING
Ca Ro=0.0811
l FIG.
6. Dynamic
A contact
Ro=0.0964
angle vs ( :apillary
spreading between flat plates, and was discussed in terms of the hydrodynamic theory by Cox (5).
Variation of slip length with viscosity ratio. From the definition oft, a slip length s can be calculated from the estimated values of Egiven in Table III, keeping in mind, of course, that there is a large inherent uncertainty in these values of E. Assuming that the macroscopic length scale can be identified with the rest radius &,, slip lengths for the various experimental systems were calculated (Table III) and are plotted against X in Fig. 7. Previous estimates of s from experimental data were for systems where X < 1 ( 5, 15), and were of order 10 -5 to 10 -6 cm. This range is consistent with the present data. Figure 7 also indicates that in general s decreases with X. Little can be said definitively concerning the variation of the slip length with the viscosity ratio, simply because, as has been stated above, no firm a priori theoretical foundation exists upon which to distinguish between different models of slippage. Thus, the experiments themselves are unable to provide this
number:
l
effect
of drop
Ro=0.148
c
size.
distinction. However, it is interesting to note that Hocking (23) has developed a microscopic model of slippage where very small quantities of the displaced fluid are trapped due to surface roughness. He derived expressions for the slip length as a function of surface geometry (asperity height, wavelength, etc.) and the viscosities of displaced and displacing fluids. In this model s varies roughly as X-‘, whereas a linear regression on the data plotted in Fig. 7 gives s - X-O.*.
Connection Between Microscopic and Hydrodynamic Theories Tanner/Hoflman law for arbitrary viscosity ratios. As was demonstrated above, plotting the experimental data according to Hoffman’s correlation (Eq. [ 41) led eventually to a superposition of the data on a master curve. Calculation of the shift factor OL(Eq. [ 12]), together with Hoffman’s correlation, resulted in the following empirical extension of the Tanner-Hoffman law: 192- (1 + X)Ca. Journal ofcoiloid
[I51
and Interface Science, Vol. 136, No. I, April 1990
ROBERT
T. FOISTER
Viscosity
Ratio
FIG. 7. Estimated slip length (s) vs viscosity ratio ( A).
Although this result is particularly simple and therefore attractive, there was, as pointed out above, poor superposition of the experimental data as Bdapproached 0,. There is, therefore, a corresponding uncertainty in the relationship given by Eq. [ 15 1. In fact Joanny and Andelman have argued that the Tanner-Hoffman law should remain unchanged (except possibly for numerical constants), even when the external phase is of nonnegligible viscosity ( 10). The microscopic theory” has not been rigorously extended for all viscosity ratios; however, from Eqs. [ 7 ] and [ 91 it is possible to obtain an explicit prediction from the hydrodynamic theory for the variation of 19~with Ca as Bd approaches zero. Expanding Eq. [ 91 for p + 1 we find that
which, upon integration,
gives
[T* + 8(h - 4)]0i, + 0 1207r2
(a;).
[17]
Therefore, the hydrodynamic theory predicts that, irregardless of the viscosity ratio, the Tanner-Hoffman law is essentially unchanged. Clearly, however, the first term in Eq. [ 171 will dominate the second only when 6d 4 (8~/9X). In other words, the primary influence of a nonnegligible external phase viscosity will be to prolong the asymptotic approach to the Tanner-Hoffman law.
Synthesis of microscopic and hydrodynamic approaches. It is interesting from a fundamental point of view to compare the asymptotic form of Eq. [ 71 and the corresponding explicit expression derived by de Gennes (9 ). From the hydrodynamic theory,
-
[r2
+
8(X
-
4)lfi4
+
o
(>P5),
24~~ Journal of Colioid and Interface Science, Vol. 136, No. I, April I990
[l(j]
e,j - Ca
ln(Rols),
[I81
DISPLACEMENT
and de Gennes finds that 8; -
Ca
ln(&-d-%id,
[I91
where x,,, is, by definition, a macroscopic length scale, essentially equivalent to R0 in the present case, and Xmi, is the length of the precursor film ( 9 ) . In the hydrodynamic theory, the slip length s is not known a priori, but the same is true for the length of the precursor film in the microscopic theory. Ultimately, the slip model must be consistent with an appropriate constitutive equation for the contact line region, and, as pointed out by Dussan V. ( 13), no such equation is currently known. On the other hand, de Gennes has derived expressions for the length of the precursor film for at least two cases: one where slip predominates, and one where the precursor film is due to dominant van der Waals forces (the “disjoining pressure”) (9). In the former case, xmi” is proportional to the slip length divided by the dynamic contact angle, while in the latter case, Xmi” is proportional to a characteristic interaction distance (which is defined in terms of the Hamaker constant and the interfacial tension between the two liquids) divided by the square of the dynamic contact angle (9). Thus, except for the rather weak logarithmic dependence on Bd, in the limit ed < 1, both the hydrodynamic and microscopic approaches give essentially equivalent results. While these latter results offer some insight into the physical character of the contact line region, they do not, in their present form, serve to define the constitutive law which is needed to formulate the hydrodynamic theory in its most general form. In fact, whether precursor films exist as such in systems where X is not negligible is still an open question. Likewise, it should be kept in mind that “slippage” has not been established experimentally, but has been introduced formally to remove the stress singularity in the classical approach. Although progress has recently been made in elucidating possible mechanisms of slip from a phenomenological point of view ( 24, 25 ), a truly fundamental theory of contact line motion, ap-
281
WETTING
plicable to systems of arbitrary viscosity, will ultimately have to be based on a synthesis of the microscopic and hydrodynamic approaches. In all likelihood this synthesis will be the result of using microscopic models to derive constitutive equations for the contact line region. FINAL
REMARKS
The experimental data described above have been used to establish relationships between the material properties of liquid/liquid/solid systems and the rate of displacement wetting. It was found that a theoretical treatment based on hydrodynamics provides an adequate foundation for correlating the dynamic advancing and equilibrium contact angles with the velocity of the three-phase boundary. In general, as the viscosity of the external (displaced) phase increases relative to the viscosity of the displacing phase, the velocity at a given dynamic contact angle decreases. This has been placed on a quantitative basis by plotting dynamic advancing contact angle versus the dimensionless capillary number. It was shown that, in order to compare the experimental data with the hydrodynamic theory, a parameter which is unknown a priori-the ratio of slip length to macroscopic length scale-had to be estimated. Furthermore, using drop rest radius as the macroscopic length scale, it was found that slip lengths calculated from the estimated values of this parameter decreased with increasing viscosity ratio. The observed influence of drop size on displacement rate was also rationalized on the basis of this parameter. Consequently, the relationship between displacement kinetics and the material properties of the liquids (viscosities, interfacial tension) and the solid (equilibrium contact angle) was shown to depend on the microscopic dynamics; which were in turn characterized by a degree of uncertainty. This uncertainty arises from the well-documented shortcomings of classical hydrodynamics in describing liquid/liquid displacement, as well Journal of Co/hid and Inlerface Science, Vol. 136, No. I, April 1990
282
ROBERT
as the fact that a microscopic theory applicable to systems with nonnegligible external phase viscosity has yet to be developed. Nonetheless, by taking a self-consistent empirical approach to estimating the ratio of slip length to macroscopic length scale for a given system, we have shown that it is possible to obtain a reasonable theoretical description for experimental systems of widely ranging viscosity ratio. ACKNOWLEDGMENTS The author acknowledges Professor A. N. Gent (University of Akron) and Professor D. R. Paul (University of Texas at Austin) for very helpful discussions; Dr. J. C. Ulicny (General Motors Research Laboratories) for assistance in the numerical work; and Dr. H. H. Kuo, Mr. T. B. Pietrzyk, and Mr. K. S. Snavely (General Motors Research Laboratories) for viscosity measurements.
REFERENCES 1. Tanner, L. H., J. Phys. D 12, 1473 (1979). 2. Hoffman, R. L., J. Colloid Interface Sci. 50, 228 (1975); 94,470 (1983). 3. Dussan V., E. B., Ann. Rev. Fluid Mech. 11, 311 (1979). 4. Foister, R. T., J. Colloid Inter&e Sci. 116, 109 (1987). 5. Cox, R. G., J. FluidMech. 168, 169 (1986).
Journal ofColloid and Inte&ce Science, Vol. 136, No. 1, April 1990
T. FOISTER 6. Hocking, L. M., .I Fluid Mech. 79, 209 ( 1977). 7. Chen, J. D., J. Colloid Interface Sci. 122,60 ( 1988). 8. Adamson, A. W., “Physical Chemistry of Surfaces” (4th ed.), p. 338, Wiley, New York, 1982. 9. De Gennes, P. G., Rev. Mod. Phys. 57, 827 (1985). 10. Joanny, J.-F., and Andelman, D., J. Colloid Interface Sci. 119,451 (1987). 11. Jiang, T.-S., Oh, S.-G., and Slattery, J. C., J. Colloid
Interface Sci. 69, 74 ( 1979 ) 12. De Gennes, P. G., C. R. Acad. Sci. Paris 298, 111 (1984). 13. Dussan V., E. B., and Davis, S. H., .I. Fluid Mech. 65, 71 (1974). 14. Dussan V., E. B., in “Waves on Fluid Interfaces” (R. E. Meyer, Ed.), p. 303, Academic Press, New York, 1983. 15. Van Dyke, M., “Perturbation Methods in Fluid Mechanics,” Parabolic Press, Stanford, 1975. 16. Hocking, L. M., and Rivers, A. D., J. Fluid Mech. 121,425 (1982). 17. Kuo, H. H., J. Coat. Technol. 57, 57 (1985). 18. Ambwami, D. S., and Fort, T., Jr., “Surface and Colloid Science,” (E. Matijevic, Ed.), Vol. 11, p. 93, Plenum, New York, 1979. 19. Saffman, P. G., and Taylor, G. I., Proc. R. Sot. Lond. Ser. A 245,312 (1958). 20. Pismen, L. M., and Nir, A., Phys. Fluids 25,3 ( 1982). 2 1. Lopez, J., Miller, C. A., and Ruckenstein, E., J. Colloid Interface Sci. 56,460 ( 1976 ) . 22. Ngan, C. G., and Dussan V., E. B., J. Fluid Mech. 118,27 (1982). 23. Hocking, L. M., J. Fluid Mech. 76, 5 1 ( 1976). 24. Jansons, K. M., J. Fluid Mech. 167, 393 (1986). 25. Durbin, P. A., J. Fluid Mech. 197, 157 (1988).