Langmuir 1995,11, 2213-2220
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Drainage of a Thin Liquid Film Confined between Hydrophobic Surfaces Olga I. Vinogradova Laboratory of Physical Chemistry of Modified Surfaces, Institute of Physical Chemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 117 915 Moscow, Russia Received January 17, 1995. I n Final Form: March 29, 1995@ We investigate theoretically the drainage of a thin liquid film between two undeformed hydrophobic spheres. The role of hydrophobicityis revealed in the apparent slippage of liquid over the solid. The origin of the slippage effect is probably linked with a decrease in viscosity in the very thin near-to-wall layer. The solution is obtained for arbitrary values of slip lengths (from zero to infinity) as well as for arbitrary radii of curvature of approaching surfaces. The main result consists in that the pressure and the drag force yield the product of corresponding expressions for similar hydrophilic spheres and some corrections for slippage. These corrections depend only on the relationships between the gap and the slip lengths. As a result, at distances that are much greater than both slip lengths of approaching surfaces, the liquid flow is the same as that for hydrophilic surfaces. If the gap width exceeds considerably only one of the slip lengths then the pressure and the resistance will be equal to those experienced by hydrophilic sphere moving toward the free bubble surface. If the gap is much smaller than both slip lengths, the flow will be like that which arises when two bubbles approach each other. In the latter case, the hydrodynamic drag is not inversely dependent on the gap but is inversely proportional to the slip lengths and only logarithmicallydependent on the gap. The correction for slippage plays a dramatic role in the coagulation processes. The main result for coagulation consists in the possibility for collision to occur at a finite time. Also, this correction needs to be taken into account when the various properties of confined liquids (first of all the hydrophobic attractive force) are investigated with the drainage technique.
I. Introduction Hydrophobization of a solid surface plays a crucial role in such phenomena as adhesion, wetting, film stability, cavitation, and coagulation. As a result, the process of hydrophobization is widely used in many industrial technologies, especially in those that take advantage of the destabilization of aqueous colloidal suspensions. For example, in mineral separation techniques the degree of hydrophobicity of mineral particles is the key to their floatability because the particle-bubble attachment is one of the decisive processes in floatation. All this makes the solid hydrophobic surface one of the main objects of modern colloid chemistry. In the last decade, it has been intensively studied. Most works reported so far concern the equilibrium physical parameters associated with the presence of a hydrophobic surface. The surface phenomena are usually related to surface forces between macroscopicbodies. That is why most experimental investigations have focused on the direct measurement of the force-distance profiles between a variety of hydrophobed surfaces immersed in different solutions (for a recent review see ref 1).However, even the equilibrium properties of hydrophobic surfaces are not yet understood. Thus, the very strong and longrange attraction which arises between macroscopic hydrophobicbodies occupies a unique place in surface science. There is currently no theory that can explain the results in terms of a physically reasonable model. This is surprising because electrical double-layerforces,attractive van der Waals forces, oscillatory solvation forces, etc., were all the subjects of considerable theoretical works long before accurate experimental data were available. Moreover, many experimental data seem to be unusual or contradictory. In fact, the conventional criteria of surface hydrophobicity such as the advancing contact angle or the adhesion between two surfaces (related to the interAbstract published in Advance A C S Abstracts, J u n e 1, 1995. (1)Christenson, H. K. In Modern approaches to wettability: theory and applications; Schrader, M. E., Loeb, G., Eds.; Plenum Press: New York, 1992;Chapter 2. @
0743-7463/95/2411-2213$09.00/0
facial energy y ) cannot give reliable predictions of the range or the strength of the hydrophobic interaction. At the same time, the force-distance profile appears to be very sensitive to the surface preparation procedure. The reports on the influence of solution conditions (temperature, electrolyte concentration, polarity of the solvent, etc.) remain controversial. Whether such confusion is the result of complexity due to the very nature of hydrophobic surfaces or whether it is a consequence of insufficient accuracy of the experimental techniques involved or incorrect interpretation of experimental data remains a n open question. Clearly, resolution of this question requires further investigations of such systems. Information about the behavior of hydrophobic bodies in dynamic conditions is rather scarce. It is limited, in general, to studies of liquid flow through hydrophobic capillaries. Now, it appears to be very timely to study the dynamic behavior of a hydrophobic surface more thoroughly. In this paper we examine the drainage of a thin liquid film confined between two solid hydrophobic surfaces. Apart from obvious engineering applications, such a n investigation seems to be very important in view of, a t least, two reasons. On the other hand, as we have mentioned above, the coagulation of colloids is especially effective when the particles are rendered hydrophobic, while the drainage of thin films of liquid between solid surfaces plays a key role in the coagulation p r o ~ e s s . ~ On -~ the other hand, the drainage method itself may be used to obtain information on various physical properties of the surfaces approaching each other. For example, even information on the equilibrium surface forces is very often obtained from the measurements of the drainage rate. To explain the rate of drainage, the system under investigation must be adequately described. Such a ~~~
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(2)Hocking, L. M. J.Eng. Math. 1973, 7, 207. (3) Vinogradova, 0.I. J. Colloid Interface Sci. 1995, 169, 306. (4)Vinogradova, 0.I. Colloids Surf., A 1994, 82, 247. (5) Potanin, A.A.; Ur’ev, N. B.; Muller, V. M. Kolloidn. Zh. 1988,50, 493.
0 1995 American Chemical Society
Vinogradova
2214 Langmuir, Vol. 11, No. 6, 1995 description comprises the exact solution of continuum hydrodynamics equations by exploiting the special geometry of film and boundary conditions. It is usually assumed that the hydrodynamic behavior of a thin macroscopic film confined between two moving solid surfaces conforms to the Reynolds theory. The latter predicts that the resistance to approach is inversely proportional to the gap between the surfaces, when this gap is small compared to their radii of curvature. A modification of the Reynolds theory for the case when the surface are hydrophobic is obtained in this paper. The paper is organized as follows. In section 2 we show how the hydrophobization of the solid surface can be revealed under dynamic conditions. We demonstrate that the viscosity of a thin layer adjacent to the hydrophobic wall decreases. As a result, a deviation of the flow of liquid from its theoretical value occurs. The simplest way of estimating this change in the flow is to use the slipflow approximation. The continuum equations with bulk viscosity terms are retained, but the boundary condition of no-slip is replaced in them by the condition that the relative velocity a t the boundary is proportional to the tangential stress. The constant of proportionality is not exactly defined, but it is linked with the size of the layer with modified viscosity and with the relative change in viscosity. The effect of slip on the motion of a hydrophobic sphere toward another sphere with, in a general case, different hydrophobicity degree, is considered in section 3. The gap is supposed to be small compared with the radius of the spheres, so that the lubrication approximation can be used to determine the flow. The solution is obtained for arbitrary values of slip coefficients (from zero to infinity) as well as for arbitrary radii of the spheres. Section 4 discusses the results obtained. The main result consists in that the pressure and the drag force yield the product of corresponding expressions for the no-slip boundary condition and some functions. These functions may be considered a s the corrections for slippage and depend only on the ratio of gap to the slip lengths of the approaching bodies. The corrections for slippage play a dramatic role in the coagulation processes. Also, they need to be taken into account when the properties of confined liquids (including the hydrophobic attraction) are investigated with the drainage technique. Our conclusions are summarized in section 5.
11. Boundary Conditions on the Hydrophobic Surface In classical fluid mechanics it is usually assumed that there is no slip a t the boundary between a liquid and a solid.6 This convention, which is in contrast to that adopted for a dilute gas and a solid,’ is largely a matter of practical experience. In most cases, the no-slip boundary condition yields the correct, i.e., the experimentally verified result for the flow of simple liquids. Nevertheless, both experimental and theoretical reasons exist for sometimes relaxing the no-slip hypothesis. Considerable increase in liquid flow compared with that expected when the no-slip condition is valid was observed experimentally in flow through capillaries of small diameters with walls made repellent to the liquid (hydrophobed). The simplest way of estimating this change in the flow is to use the slip-flow approximation. This means that hydrodynamic equations are retained, but the no-slip boundary condition is replaced by a some slippage (6)Stokes, G.C. Trans. Cambridge Philos. SOC.1851,9,8. (7) Maxwell, J. C. Philos. Trans. R. SOC.London, Ser. A 1879,231.
law. In this sense, more than a century ago, Helmholtzs found experimental evidenceof slippage for a liquid flowing over a solid surface. Schnellg detected the slip of water in a glass capillary pretreated with the vapor of dimethyldichlorosilane. TolstoilO and Somovll had measured a slip velocity for the flow of mercury in glass capillaries. Churaev et aZ.12also reported on the slippage for mercury in very narrow, fused quartz capillaries, as well as for water in the same capillaries but treated with trimethylchlorosilane. Thus, it is possible to conclude that most of the experimental data for the validity of the no-slip hypothesis come from systems that exhibit complete wetting, i.e. hydrophilic systems. In contrast, there is an increasing body of evidence to suggest that slippage may occur in systems where the walls are only poorly wetted, or hydrophobic systems. The slippage effect is not yet fully understood. However, there have been some theoretical attempts to explain it. These argued, first of all, from the fact that the strength of attraction of the liquid molecules is higher than that between the liquid and the solid. The first approach is based on the supposition of the existence of molecular slip, i.e. it is considered that molecules of liquid slip directly over the solid surface. The earlier molecular theory of slip was proposed by Tolstoi.lo It links the mobility of liquid molecules with the interfacial energy (or contact angle). This theory was later analyzed and reconsidered by Blake.13 Blake illustrated the Tolstoi theory by using it to predict the effect of slip on the rate of flow through a capillary. These predictions were compared with experiments. It was found that sometimes there is good agreement with theory. However, Ruckenstein and Ragora14 have shown that experimental results imply surface diffusion coefficients that are several orders of magnitude greater than those normally observed even for gases a t a solid surface. This inconsistency of molecular theories leads to the supposition of “apparent”slippage of liquid over the solid wall. To account for the relatively free motion of liquid molecules, Ruckenstein and Rajora14suggested that there may be a gas “gap”a t the interface between the solid and the liquid caused by the different nature of the two materials. A similar idea was discussed by Ruckenstein and Churaev.15 The importance of this supposition on the role of dissolved gas in the slippage phenomenon needs be stressed. However, the model of (‘gap”is a very strong idealization. In fact, in such a case, we may expect the boundary condition of absence of friction which is usually observed a t the liquidgas interface. But this is not experimentally confirmed (see below). Another model of “apparent” slippage is quite similar to that adopted in polymer physics.16J7 It links the slippage effect with a decrease in the viscosity of a boundary layer close to a hydrophobic s ~ r f a c e . ~This J~ fact follows from numerical simulation data, as well as from the direct experimental results obtained by the blow(8) Helmholtz, H.;Pictowsky, G. Sitzungsber. Kais.Akad. Wiss. Wein, Math.-Natunoiss. KI.,Abt. 2 1868,40, 607. (9) Schnell, E. J . Appl. Phys. 1956,27,1149. (10)Tolstoi, D. M.Dokl. Acad. Nauk SSSR 1952,85, 1329. (11) Somov. A.N.Kolloidn. Zh. 1982.44. 160. (12)Churaev, N.V.;Sobolev, V. D.; Somov’,A. N. J. Collozd Interface Sci. 1984.97. , 574. (13)Blake, T. D. Colloids Surf. 1990,47, 135. (14)Ruckenstein, E.; Rajora, P. J . Colloid Interface Sci. 1983,96, 488. (15) Ruckenstein, E.;Churaev, N. V. J . Colloid Interface Sci. 1991, 147,535. (16)Brochard, F.;de Gennes, P.-G. Langmuir 1992,8,3033. (17)de Gennes, P.-G. C.R.Acad. Sci. l979,288B,219. (18)Dejaguin, B. V.;Churaev, N. V. Langmuir 1987,3,607. ~~
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Drainage of a Liquid between Hydrophobic Surfaces
Langmuir, Vol. 11, No. 6, 1995 2215
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Figure 1. Slip length b characterizing the amount of slippage in viscous flow near a hydrophobic surface. off method.19 Macroscopically, the effect of reduced viscosity may be considered to be like the flow with slippage. In such a model, the slip velocity us on the wall is proportional to the bulk shear stress U, = b-
ofthe boundary layer, 6, with modifiedviscosity,and attain a very large value. We think so because the decrease in viscosity near the hydrophobic wall may be the result not only of a change in the structure of solvent itself but also of an increase in the concentration of gas-filled submicrocavities close to it as compared with the bulk case. The direct experimental evidence for such a n increase has been obtained recently by Vinogradova et aLZ1using the optical cavitation technique. Some observations made during the measurements of hydrophobic attractive f o r ~ e ~and ,~~n~~ slippagelZ indirectly support this point of view (about dissolved gas, see also ref 24). Thus, it is likely that the very high experimental value of b (about cm or even more) for water a t the hydrophobed quartz surface (contact angle 9O0)l2is due to the dramatic role of dissolved gas. To finish this section, let us briefly comment on the situation close to the hydrophilic surface. The numerical calculations unambiguously predict that the liquid layer adjacent to the hydrophilic surface has a n increased density. This should be accompaniedby a reduced mobility of molecules in the tangential direction, i.e. enhanced viscosity of the near-to-wall layer (see ref 3 and references therein). Experimental results obtained in thin capillariesZ6and by the blow-off methodlg have shown that there are discernible deviations from the bulk viscosity value for strongly wetting systems in the layer whose thickness considerably exceeds several molecular diameters. Again, endeavors were made to measure the viscosity near hydrophilic surfaces using a surface force a p p a r a t u ~ . ~ Interpreting ~,~’ the results of these experiments is not straightforward because the bulk liquid far outside the point of closest separation may contribute mightily to the total force measured (this has been discussed by van Alsten et a1.28). However, experiment unambiguously suggests that a narrow region of liquid near the hydrophilic surface is effectively immobili~ed.~’ Note that all method^'^^^^-^' give rather small deviations from the bulk viscosity and, as a result, it is possible to assume that b (according to its modulus) is less than 0.16. This is, of course, a negligibly small value in the scale of the drainage problem. Therefore, we suppose that for hydrophilic surface b = 0.
az
(2.1)
where u b is the liquid bulk flow rate, z is the axis perpendicular to the wall, and b is the so-called slip length, i.e. the product of the slippage coefficient and the bulk viscosity (Figure 1). If the viscosity of the near-to-wall layer is characterized by average value p8, the order of magnitude b can be estimated as
b=dk-l)
(2.2)
-
where 6 is the thickness of the boundary layer and pb is the bulk viscosity. At b 0, us = 0, which corresponds to the conventional no-slip condition a t the liquidsolid interface. In this case there is no change in viscosity close to the solid surface. At b 00, aut,& = 0, which corresponds to the conventional condition for a liquidgas interface. This means the complete slippage due to a negligibly small viscosity of the boundary layer (this is equivalent to the gas “gap” mentioned above). Such a n idealized case, of course, can hardly be effected when we consider the solid hydrophobic surface. It corresponds to a complete drying transition (contact angle 180°), which has never been observed a t the hydrophobic surface. However, it seems to be quite logical that the slippage law for the hydrophobic solidlliquid interface can be, in principle, transformed into the condition for a bubble. In fact, other physical properties of bubbles allow one to consider it as a very hydrophobic surface. We mean, first of all, a very high value of interfacial energy for the bubble surface ( y = 72 mJ/m2). Moreover, the first indirect reference to such a property of hydrophobic surface as long-range attraction was given by Blake and KitchenerZ0by observing just the rupture of air bubbles against hydrophobed silica surfaces. Of course, the slip length is not exactly defined in this model because a t present the information about both the thickness, 6, and the value ofp, is not sufficient. However, it is possible to expect that the value of b for the solid hydrophobic surface can several times exceed the thickness
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(19) Dejaguin, B. V.; Karasev, V. V. Surface Colloid Sci. l99S,15, 221. (20) Blake, T. D.; Kitchener, J. A. J . Chem. SOC.,Faraday Trans. 1 1972,68, 1435.
111. Solution of the Drainage Problem Consider two spherical particles Wl and Wz having radii R I and RZ (lR~l< IRzI), respectively (Figure 2). The particles move in a Newtonian liquid. We shall assume that the spheres are sufficiently rigid so that any deformation of their surfaces due to hydrodynamic stresses are negligible. The distance between particles is supposed to be small compared with the radius of curvature of surfaces. Contact can be realized only at a single point. Surface roughness effects are neglected. The particles move along the line connecting their centers a t velocities u1 and uz, respectively. We consider here only the inner region. This region must be matched to the outer one, but the leading term in the resistance to normal motion comes entirely from the inner region. Following are definitions (21) Vinogradova, 0. I.; Bunkin, N. F.; Churaev, N. V.; Kiseleva, 0. A,; Lobeyev, A. V.; Ninham, B. W. J . Colloid Interface Sei. 1996,in press. (22) Christenson, H. K.; Claesson, P. M. Science 1988,239,390. (23) Parker, J. L.; Claesson, P. M.; Attard, P. J . Phys. Chem. 1994, 98. 8468. (24) Craig, V. S. J.;Ninham, B. W.; Pashley, R. M. J . Phys. Chem. 1993.97. 10192. -___ ----(25) Kiseleva, 0. A.; Sobolev, V. D.; Starov, V. N.; Churaev, N. V. Kolloidn. Zh. 1979,41,192. (26) Israelachvili, J. N. J . Colloid Interface Sci. 1986,110, 263. (27)Chan, D. Y. C.; Horn, R. G. J . Chem. Phys. 1986,83,5311. (28) van Alsten,J.;Granick, S.;Israelachvili,J. N. J . Colloid Interface Sci. 1988,125,739. ~~
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2216 Langmuir, Vol. 11, No. 6, 1995
(2)The equation of the quantity of motion in the direction
I
r
h
r
w2
/
where is the density of liquid, p is the pressure, and p is the dynamic viscosity. (3) The expression for the quantity of motion in the direction z
R2
Figure 2. Schematic representation of a sphere approaching another one.
of cylindrical system (2, r ) of coordinates: (1)the axis 2 coincides with the line connecting the centers of two spheres and oriented in the direction of W1;(2) the origin of coordinates coincides with W2;( 3 )the plane 2 = 0 is the plane tangent to W2. In the region which is close to the origin of coordinates, the surface W2 of the sphere may be described locally as a paraboloid of revolution:
The boundary conditions express the slippage of liquid over the hydrophobic surface. Assume that for the surface W2 the value of slip length is equal to b2 = b, while for the surface W1 it is bl = b(1 k). Therefore, the boundary conditions will have the form
+
avr
At z = O(r4), u, = 0 and ur = baz At z = h
+ -1 r2 + O(r4), u, - " ' - -u 2 Re
(3.1)
Re
U, = -b(k
(3.6)
and
+ 1%av
(3.7)
In a similar way, the surface W1 of the second sphere may be expressed as follows:
r2 Z = h + -1 + O(r4)
(3.2)
2Rl
where h
