3261
Langmuir 1996,11,3261-3268
Extrapolated Dynamic Contact Angle and Viscous Deformation of a Steady Moving Meniscus at a Vertical Flat Wall Peter G. Petrovt and Jordan G. Petrov*vS Bulgarian Academy of Sciences, Institute of Biophysics, Acad. Georgi Bonchev Str., Block 21, 1113 Sofia, Bulgaria, and Max Planck Institute of Colloids and Interfaces, Rudower Chaussee 5, 12489 Berlin, Germany Received February 6, 1995. In Final Form: May 22, 1995@ The profile of a steady meniscus at a flat wall vertically withdrawn from a liquid is considered. The extrapolated dynamic contact angle, Oed, serving as a boundary condition of Laplace equation for the quasi-static part of the fluid interface is introduced and its dependence on contact line velocity, viscosity, surface tension,density ofthe liquid, and static wettabilityof the solid is obtained and numerically analyzed. The thickness of the hydrodynamic deformation of the meniscus, hqs,is also related to these properties and a nonmonotonous dependence ofthis thickness on contact line velocity is found. Two differentdynamic behaviors of the contact line, constant and velocity-dependentactual dynamic contact angle, are considered, and a strong difference between the corresponding dependences of Oe,t and h, on contact line velocity is established. If the other properties of the system remain the same, the velocity dependence of the actual dynamic angle makes the extrapolated dynamic angle decrease from the static value to 0” much faster and strongly reduces the hydrodynamic deformation of the moving meniscus.
Introduction The numerous studies of wetting dynamics appearing in the last decade pay relatively little attention to the correspondence between the theoretical definition and the experimental determination of the dynamic contact angle. The actual dynamic contact angle, e,, that is a boundary condition of the differential equation describing the profile of a moving meniscus, is a well-definedbut experimentally inaccessible quantity. For this reason and because of the difficulties arising from the application of the “no slip”boundary condition close to the moving contact line, hydrodynamic theory usually operates with local slopes of the fluid interface at some distance from it, supposing that they can be determined in the experiment. Another approach’ introduces an effective quantity, usually referred as “apparent dynamic contact angle”,which can be easily measured optically or mechanically. For a narrow capillary, a slot, or a small drop where the negligible effect of gravity does not change the spherical meniscus shape, flap, can be determined from the tube radius, R (halfwidth of the slot or radius of the drop basis), and the distance, H, between the apex of the dynamic meniscus (drop) and the plane of the contact line:
eapp= .,c,s(
)
2HIR 1+ (HlR)2
(1)
For a flat wall vertically withdrawn through a liquid-gas interface, Oapp can be obtained by introducing the dynamic capillary height, 2, in the corresponding static e q ~ a t i o n ~ - ~
* Author to whom correspondence may be addressed. Permanent address: Bulgarian Academy of Sciences, Institute of Biophysics, Acad. Georgi Bonchev Str., Block 21, 1113 Sofia, Bulgaria. Bulgarian Academy of Sciences. Max Planck Institute of Colloids and Interfaces. Abstract published inAdvance ACSAbstracts, August 1,1995. (1)Hansen, R.J.;Toong, T. Y. J.Colloid Interface Sci. 1971,37,196. ( 2 )Petrov, J. G.; Radoev, B. P. Colloid Polym. Sci. 1081,259,753. (3)Cain, J. B.; Francis, D. W.; Venter, R. D.; Neumann, A. W. J. Colloid Interface Sci. 1983,94,123. (4) Sedev, R. V.; Budziak, C.; Petrov, J. G.; Neumann, A. W. J.Colloid Interface Sci. 1093,159,392. ( 5 ) Petrov, J. G.; Sedev, R. V. Colloids Surf. 1993,74,233.
*
@
eapp= arcsin( 1 - Q$) where y is surface tension, Q is density of the liquid, and g is acceleration of gravity. For dynamic menisci around vertical fibers or cylinders, Oapp can be evaluated from 2 using Derjaguin’s formula6 or the numerical solution of Huh and Scriven,’ respectively. As realised by many investigators,1s8-12viscous deformation might strongly affect the meniscus profile close to the contact line, thus causing a significant difference between the real fluid interface and the imaginary one, defining e,,; the two interfaces should coincide at the contact line and far away from it, but “with no assurance of agreement along the remainder of the fluid interface”.12 Some hydrodynamic considerations suggest a comparison between the theoretically predicted and experimentally determined local slope of the fluid interface at a particular distance from the contact line.11-13 However, this distance is model dependent and the local slopevaries significantly when approaching the contact line.11J2 This introduces some arbitrariness and makes such a dynamic contact angle a delicate quantity.13 The existence of a quasi-static part of the moving meniscus far away from the moving contact line enables utilization of another easily measurable quantity here referred as “extrapolated dynamic angle”, Oext. Its definition14-16is illustrated in Figure l a for a dynamic mensicus not affected by gravity (in a narrow capillary or (6)Dejaguin, B. V. Dokl. Akad. Nauk SSSR 1946, 51, 517 (in Russian). (7) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1969,30,323. (8)Huh, C.; Mason, S. G. J. Fluid Mech. 1977,81, 401. (9)Kafka, F. Y.;Dussan V, E. B. J.Fluid Mech. 1979,95,539. (10) Greenspan, H. P. J. Fluid Mech. 1989,84,191. (11)Ngan, C. C.; Dussan V., E. B. J. Fluid Mech. 1989,209,191. (12)DussanV., E.B.;Rame, E.; Garoff, S. J.FluidMech. 1091,230,
97. (13)de Gennes, P. G.; Hua, X.;Levinson, P. J. Fluid Mech. 1990, 212, 55. (14)Starov, V. M.; Churaev, N. V.; Khvorostyanov, A. G. Colloid J. USSR (Engl. Transl.) 1977,39, 176. (15) Churaev, N. V. Rev. Phys. Appl. 1988,23,975. (16)Teletzke, G. F.; Davis, H. T.; Scriven, L. E. Rev. Phys. Appl.
1988,23,989.
0743-7463/95/2411-3261$09.00/00 1995 American Chemical Society
3262 Langmuir, Vol. 11, No. 8,1995
Petrov and Petrov
Ut
Ut
G
a
b
Figure 1. Extrapolated dynamic contact angle in a narrow capillary or a slot (case a) and at a vertical flat wall (case b) for
liquid receding against air.
a slot) and in Figure l b for a case where gravity plays an important role (a meniscus at a vertical flat wall or cylinder). The extrapolated dynamic contact angle serves as a boundary condition of the Laplace differential equation applied for the quasi-static part of the dynamic meniscus. The angle can be experimentally determined easily by fitting the solution of this equation to the profile of the liquid-gas interface far enough away from the contact line and theoretically extrapolating the fitted quasi-static profile up to its intersection with the solid surface. This extrapolation is not empirical, but relies on a rigorous theoretical equation describing the shape of the real fluid interface far away from the moving contact line. At zero velocity this definition is identical to the thermodynamic definition of the static contact angle, 80; the correct experimental determination of the latter requires the same procedure-a theoretical extrapolation of the static meniscus profile toward the solid wall from a zone unaffected by the intermolecular forces acting near the contact line. Thus both 80 and Oext are defined as boundary conditions of the Laplace differential equation, the first one for a static meniscus profile and the second one for the quasi-static part of a dynamic fluid interface. Dependences of Oext on contact line velocity, liquid viscosity, surface tension, and solid surface wettability have been reported mainly for systems with negligible gravity effects. Huh and Mason8investigated the steady movement of a liquid meniscus in a capillary tube with static and dynamic angles close to 90”. Voinov17reported such dependences for small (spherical)drops and narrow capillaries and slots. Dussan, Rame, and GarofF2 presented a solution for a system with gravity effects, a large cylinder entering a bath of liquid, but preferred to work with the local slope of the interface, OR, at a distance R = 10pm from the contact line, expecting that this quantity would depend only on the material properties of the system and not on its geometry. The aim of the present paper is to find similar theoretical dependence of OeXt on contact line velocity and properties of the liquid and solid for a flat wall vertically withdraw from a liquid for which Laplace equation has an exact analytical solution. Two different behaviors of the actual dynamic contact angle are compared-the one usually assumed, 8, = 80= const,18 and the case when 8, depends on contact line velocity through a nonhydrodynamic mechanism.lg
Theoretical Model The dynamic meniscus is divided in three regions as illustrated in Figure 2-a molecular region (l), with dimensions of several molecular diameters, a hydrody(17)Voinov, 0.V. Fluid Dyn. (Engl. Transl.) 1976, 11, 714. (18)Cox, R. G.J . Fluid Mech. 1986,168, 169. (19)Petrov, P. G.;Petrov, J. G. Langmuir 1992,8, 1762.
Figure 2. Theoretical model of a dynamic meniscus utilized in this paper: 1,molecular region; 2, hydrodynamic region; 3, quasi-static region.
namic region (2), where viscous and capillary forces are dominant, and a quasi-static region (31, where only gravitational and capillary forces play a role. The slope of the fluid interface at the boundary between the molecular and hydrodynamic region is defined as “actual dynamic angle”, e,, and the slope at the matching position of the hydrodynamic and quasi-static region is referred as a “quasi-staticangle”, e?,. The corresponding meniscus thicknesses at which regons 1 and 2 and 2 and 3 are matched are h, and hqs,respectively. A simple expression for the profile of the hydrodynamic region relating the local slope, 8, a t a given meniscus thickness, h, to contact line velocity, U,liquid viscosity, p, surface tension, y , and actual contact angle, e,, was given by Voinov17
(3) The profile of the quasi-static region can be found integrating the equation (4)
with Oex+,as a boundary condition at the solid wall (K is the local curvature of the meniscus). This yields20
where
At the particular meniscus thickness h,, where the hydrodynamic and quasi-static regions match, eqs 3 and 5 give, respectively:
(7) Thus the dependence of Oext on U,p, y , e, and 8, can be found if the relationship between h,, and these parameters is known. ~~
~~
(20) Princen, H. M. Suraf. Colloid Sci. 1969,2, 1.
Profile of a Steady Meniscus
Langmuir, Vol. 11, No. 8, 1995 3263
Following Levich’s21and Derjaguin’s22solutions to the problem of entrainment of a film by a tape withdrawn from a liquid, the thickness h,, can be determined by setting the local curvatures of the hydrodynamic and quasi-static regions equal at Itqs. Approximating the hydrodynamic region with a thin liquid film where lubrication theory holds, one can write the differential equation for its profile in dimensionless quantities only
d3H+ - 1= o dZ3 H2 Then the with H = h/h,, and Z = (~/h,,)(3pU/y)~~. dimensionless curvature of region 2 at h,, is a numerical constant2’
(B) = B = const dz2 H=l
Figure 3. Schematic representationof the heights Ze&,2,and Z,,, which can be used for experimental determination of &t, Oapp, and 8, according to eq 2.
2 a (9)
The dimensionless curvature of region 3 at h,, expressed from eq 4 for small interfacial slopes (K * d2h/dz2),is
80
60
d 40
20
Setting the right-hand sides of eqs 9 and 10 equal yields for h,
with p = 1.47B. For the present we avoid the exact determination of B and take p * 1having in mind the value /3 = 0.93 obtained for the case of liquid film entrainment.21*22 The system of eqs 6,7, and 11represents the dependence of the extrapolated dynamic contact angle, Oed, on contact line velocity, U,properties of the liquid, p, y, and e, and the actual dynamic angle, 8,. Two different dynamic behaviors of the molecular region are considered-the commonly accepted 0ne,l7J8where 8,is taken as constant and equal to the static contact angle 8 0 and the case when 8, depends on the contact line velocity via the nonhydrodynamic relationship of Blake.23924According to the latter the dependence of Oed on U and systems parameters is given by eqs 6, 7, 11, and 12
8, = arccos(cos 8,
+arsh 1 ) (12) 2K;1 Y
Equation 12 implies that the motion of the contact line occurs via adsorption/desorption of the fluid molecules onto/from the solid surface within the three-phase contact zone. The density of the adsorption centers, n, is related to the distance between them, il (n = and K is the equilibrium oscillation frequency of an adsorbed molecule at the contact line. It should be mentioned that although the system of eqs 6 and 7 works for angles up to about 3d4, eq 11is restricted to small interfacial slopes. This restriction follows from the lubrication approximation used to write eq 8 and (21) Levich, V. G.Physicochemical Hydrodynamics Prentice Hall: Engelwoods Cliffs, 1962; Chapter XI. (22) Dejaguin, B. V.;Levi, S. M. Film Coating Theory;Focal Press: London, 1964. (23) Blake, T. D. PhD Thesis, University of Bristol, Bristol, 1968. (24) Blake, T.D.;Haynes, J. M. J. Colloid Interface Sci. 1969,30, 421.
0 0
50
100
150
200
u, d
250
s
Figure 4. Velocity dependences of the extrapolated (curve 1) and the quasi-static (curve 2) receding dynamic contact angle for a constantactual dynamic angle equal to the static one. The curves are plotted according to eqs 6,7, and 11 at the following parameters values: 8 0 = 80°, y = 72.7 dyn/cm, p = 0.01 P, = 1 g/cm3, h, = low6cm.
because the local meniscus curvature Khas been expressed as d2h/dz2(eqs 8 and 10).
Numerical Results The values of Oext, Oapp, and 8, could be experimentally determined introducing Z e x t , 2, and Z,, in eq 2. The meaning of these heights is illustrated in Figure 3. (ed is directly obtained as a boundary condition &=O) = of the Laplace equation when it was fitted to the quasistatic part of the dynamic meniscus profile.) Z is the real dynamic height of the contact line above the horizontal liquid level. The interception of the theoretically extrapolated quasi-static profile with the solid wall defines Z e x t . The point at which the real and the Laplacian profile diverge determines Zqs and hqs. The difference Z - Zqs and h, represent the maximum length and thickness of the hydrodynamically deformed part of the meniscus, respectively. In this section we present a numerical simulation of the dependences of Oed and e,, on contact line velocity and analyze the effect of viscosity and static wettability upon them. The variation of the thickness of the hydrodynamic deformation, hqs, with U is also investigated. All dependences are obtained for both a constant and avelocitydependent actual dynamic contact angle in order to check the influence of the dynamic behaviors of the molecular region on 8,,t(U) and h,(U). Velocity Dependence of the Experimental and Quasi-Static Contact Angle. Dependences of Oed and 8,, on U, obtained from eqs 6, 7, and 11 for 8, = = constant, are plotted in Figure 4. The values of the parameters chosen, 8 0 = 80°, y = 72.7 dyn/cm, e = 1g/cm3, P, and h, = cm, are typical for the system p = polyethylene there~htalate-water-air.~~~~~ It can be seen
3264 Langmuir, Vol. 11,No.8, 1995
Petrov and Petrov
b
a
60
40
20
0 '
0
I
I
I
I
I
1
1
2
3
4
6
6
0
7
u,
0
8 cds
Figure 5. Velocity dependences of the extrapolated (curve 1) and the quasi-static(curve 2) receding dynamic contact angles for a velocity-dependentactualdynamic angle, plotted according to eqs 6, 7, 11, and 12. The parameters values used are as follows: Bo = 80", y = 72.7 dyn/cm, p = 0.01 P, e = 1g/cm3,h, = cm, K = lo5 s-l, 1 = 1 nm.
that both contact angles decrease monotonously and their difference, e,, - Oext, indirectly characterizing the viscous deformation, is significant at all velocities. Figure 5 shows Oext and e,, versus U for 8, = O,(U), plotted according to eqs 6,7,11, and 12 with the same parameter values as in Figure 4. Values of K = lo5s-l and 1 = 1 nm were used in eq 12 because they are often found for a polar liquid and a low energy so1id.19,23-28These dependences significantly differ in shape from those in Figure 4; they show strong initial and final decrease of eextand Oq8 and a shallow region with an inflection point at intermediate velocities. Their difference is comparable with the typical experimental scatter of the dynamic contact angle (up to U RZ 7.0 c d s , Oqs - eextI3") and therefore the hydrodynamic deformation of the meniscus should be small in this case. However, the main effect of the nonhydrodynamic dependence 6, = &(U)on Oext(U) and O,,.(U) becomes evident when one compares the abscissa of Figures 4 and 5. For 0, = eo the total variation of Oea and e,, from 80" to 0" requires U to be increased up to approximately 200 c d s (Figure 4), while for 8, = &(U)and the same values of 80, p , y , e, and h,, this change is complete by about 7.5 c d s (Figure 5). Therefore,the velocity dependence ofthe actual contact angle makes the dependences Oext( U ) and e,& U) considerably stronger. Effect of Viscosity of the Liquid. Figures 6 and 7 show the dependences of Oext and e,, versus U for two higher viscosities, p = 0.5 and 1.5 P, the values of the other parameters being the same as in Figures 4 and 5. For the sake of simplicity the surface tension was taken to be constant and equal to that of water, but the introduction of the exact values would not change the results significantly. At p = 0.5 P (curves 1) the dependences for @, = Bo (Figure 6) and & = &(U)(Figure 7) show the same characteristic trends as above, but OqS - Bed and the hydrodynamic deformation are now significant in both cases. At the higher viscosity the trend becomes similar and e,, - Oext and the hydrodynamic deformation becomes comparable (cf. curves 2 and the difference between 2a and 2b in Figures 6 and 7). The ratio of the velocities at which Bext 0" is this time about 3 compared to 27 for Figures 4 and 5. Effect of the Static Wettability. The effect of the static wettability on the velocity dependences of Oea and
-
(25)Blake, T.D.MChE International Symposium on the Mechanics of Thin Film Coating New Orleans, 1988,Paper la. (26)Petrov, J. G.;Petrov, P. G. Colloids Surf. 1992,64, 143. (27)Sedev, R. V.;Petrov, J. G. Colloids Surf. 1992,62, 141. (28)Hayes, R. A.;Ralston, J. J . Colloid Znterfuce Sci. 1993,159,429.
2
1
3
4
u, c d s
5
Figure 6. Effect ofviscosity ofthe liquid. Extrapolated (curves a) and quasi-static(curves b) receding dynamic contact angles versus contact line velocity at constant actual dynamic angle. 1, p = 0.5 P; 2, p = 1.5 P. All other parameters are the same as in Figure 4.
h
l
4
\bl
0.0
I
I
I
I
0.2
0.4
0.6
0.8
I
1.0
u, c d s
Figure 7. Effect of viscosity ofthe liquid. Extrapolated (curves a) and quasi-static(curvesb) receding dynamic con-tact angles versus contact line velocity for 8, = &(U:1,p = 0.5 P, 2, p = 1.5 P. All other parameters are the same as in Figure 5.
0 ' 0
5
1
I
10
15
u. c d s
Figure 8. Effect of the static wettability on 8,.,(v) and 8,,(U) for 8, = 80 = constant: 1,extrapolated; 2, quasi-staticreceding dynamic angles. The curves are plotted with the same parameters as in Figure 4, but for 80 = 30",instead of 80 = 80". Oqs for a constant actual contact angle is demonstrated by comparing Figure 8 plotted at 8, = 8 0 = 30" and Figure 4 in which 8, = O0 = 80". The values of the other P, y = 72.7 dydcm, e = 1 g/cma, and parameters,p = h, = cm, are the same. One can see that a better static wettability strongly reduces the velocity at which Oext 0";from about 200 c d s for Bo = 80" (Figure 41, it becomes approximately 13 c d s at 80 = 30" (Figure 8). Figure 9 shows the eea(U) and 8,,(U) for 0, = O,(v), 00 = 30" and the same values of p, y , e, h,, K, and A as in Figure 5. The comparison with Figure 5 ( 0 0 = 80") shows that the curves in Figure 9 do not possess the S-shape typical for velocity-dependent actual dynamic angle although the only difference between Figures 5 and 9 is the lower value of the static contact angle. At the same
-
Profile of a Steady Meniscus
Langmuir, Vol. 11, No. 8, 1995 3265
0.02
0.001
I
I
I
0.01
0.00
Yl u, cuds
0.000
eo = 800.
0.01
0.00
y '
0
I
I
I
I
2
4
6
0
u, c d s
Figure 11. Velocity dependence of the maximum thickness, hqs, of the hydrodynamically defomred part of the moving meniscus, estimated according to eqs 11and 6 and 12 for 8, = 8,(U). The parameters used are the same as in Figure 5.
deformation are in accord with our observations of dynamic meniscus profiles at velocities close to the velocityof liquid film entrainment. Values of the Parameters and Their Correspondence to Real Systems. Velocity dependences of receding dynamic contact angles for polar liquids on low-energy solids with p, y , Q, and 80 being similar to those used in the numerical simulations were reported in several publications. Fitting the equations of the hydrodynamic,l'J8 mole~ular-kinetic,~~ or molecular-hydrodynamicl9 theory to them yielded the values of h,, K, and il presented in Table 1. As one can see, h, varies between and cm, Kremains in the range 103-106 s-l, and 1 x 1nm is almost constant. We introduced these limitingvalues in eqs 6,7,11, and 12 in order to check the validity of the conclusions of the numerical simulations. Replacing h, = cm with h, = cm and maintaining the same values of K = lo5 s-l and il = lo-' cm stretched the abscissa of the dependences shown in Figures 4 and 5 and increased the difference between them resulting from the two dynamic boundary conditions (the ratio of the velocities at which Oe* 0" became 34 compared to 27 found for h, = cm). Taking the lowest, lo3 s-l, and the highest, lo6 s-l, values of K and keeping the values of il = lo-' cm and h, = cm constant, changed the highest steady velocity from 7.5 c d s (Figure 5) to 0.16 c d s and 25 c d s , respectively. However, all these values, obtained for 8, = 8,(U, are much lower than the maximum steady velocities found for 8, = 8, = constant (200 c d s and 315 c d s at h, = cm and cm, respectively). When h, = cm was used instead of h, = cm in the h,(U dependences in Figures 10 and 11 moved their maxima from 80 to 120 c d s and from 7.5 to 9.3 c d s , respectively, and both maximum values of h,, increased slightly. However, the relative position of these maxima with respect to the velocity at which Elea 0" remained unaltered. For K = lo3 s-l, il = lo-' cm, and h, = cm a very small hydrodynamic deformation (h, = 2pm) was found at the velocity at which Oea 0".When the highest value ofK = lo6s-l was used, a maximum h,, x 75 pm was obtained at 22 c d s that is slightly lower than the maximum steady velocity of 25 c d s . Both values are considerably smaller than the values of h, = 500 pm and 620 pm obtained for 8, = 80 = constant at h, = cm and cm, respectively. Summarizing the above results, one can conclude that variation of the values of the parameters h,, K, and il within the limits found for different experimental systems does not affect the main conclusions of the numerical simulations presented in the former sections. 19326-28
I
1
I
I
I
50
100
150
200
250
u,cm/s Figure 10. Velocity dependence of the maximum thickness, h,., of the hydrodynamically deformed part of the moving meniscus, estimated according to eqs 11 and 6 for 8, = 80 = constant. The parameters used are the same as in Figure 4.
time a complete coincidence between O e and ~ OqS, and therefore an absence of viscous deformation, is observed in Figure 9. Comparing Figures 8 and 9, one finds again significant differences in the values of e,, - e,*. The reduction of the velocity at which Oea 0" for 8, = 8,(U is even stronger in this case; the ratio of these velocities for Figures 8 and 9 is about 430. Hydrodynamic Deformation of the Meniscus. Figure 10 shows the thickness of the hydrodynamic deformation, h,,, versus contact line velocity U ,evaluated from eqs 11and 6 for ,d = 1and the same values of 8, = eo = SO",p = P, y = 72.7 dydcm, Q = 1 g/cms, and h, = cm as in Figure 4. This dependence passes through a maximum of 500pm at U sz 80 c d s that is much before the critical velocity U,, w 200 d s at which Be* 0". Such a large deformation at a small capillary number and a large dynamic contact angle is rather unlikely (at U x 80 c d s , Ca x 1 x and x 65"; cf. Figure 4). Figure 11represents the dependence h,(U calculated for 8, = 8,(U from eqs 11,6, and 12 with the same values of the parameters used to plot OeXt(V, and 8,.(V, in Figure 5. One can see that the maximum value of h,. = 30 pm is much smaller than the corresponding one of 500 pm in Figure 10. Therefore, the viscous deformation is considerably smaller for a velocity-dependent actual dynamic contact angle compared to the one when 8, = 80 = const. It is also worth noting that the maximum in Figure 11 appears very close to the velocity at which Oext abruptly decreases toward 0" (compare Figures 5 and 11). This location of the maximum and the magnitude of the
-
-
'
0
0.03
Figure 9. Effect of the static wettability on 8,(U) and 8,(U) for 8, = 8,(U): 1,extrapolated; 2, quasi-staticreceding dynamic angles. The dependences shown are plotted with the same parameters values as in Figure 5 but for 8 0 = 30",instead of
1:;:
I/
/
-
-
-
3266 Langmuir, Vol. 11, No. 8, 1995
Petrov and Petrov
Table 1. Literature Data for the Values of the Parameters h,, K, and 1 for Polar Liquids Receding against Air on a Low-Energy Solid h, (cm)
Ca = pU/y
system PET/water/air PET/glycerol-water/air
0-2 10-7 2 10-6-4 0-7 x
10-5
2
10-4
10-6-5
1 x 10-5-1 x 10-3 0-5 x 10-4 0-2 10-3 0-8 x
silanated glass/glycerol-waterlair
2 x 10-3-1 x
2.4 10-5 1.3 x loW6
10-2
0-8 x 10-3
2.6 10-4 3.3 x 10-5
K (8-l) 1.2 x 103 2.0 106 3.0 103 4.0 103 2.8 104 1.4x 105 1.8 104 5.3 103 6.0 x lo2 3.0 103 4.0 103 6.3 103 1.9 x 104 7.6 105 1.2 106
I (cm) 1.6 10-7 1.2 10-7 1.5 10-7 1.5 10-7 1.7 10-7 1.0 10-7 1.3 10-7 1.9 10-7 2.6 x 1.2 10-7 1.1 10-7 1.1 10-7 1.1 x 10-7 0.8 10-7 0.7 10-7
1.0
1.5 x 10-7
x
104
theoretical modela
ref
MK MK MK MK MK MK MK CMH CMH MK MK MK MK MK MK HD CMH
26 26 19 19 28 26 26 26 26 27 27 27 27 27 27 27 19
a MK,molecular kinetic theory,24 giving K and I ; HO, hydrodynamic theory,17J8 giving he; CMH,combined molecular hydrodynamic theory,l9 giving K, I , and h,.
Table 2. Comparison of Experimental Values of Extrapolated and Apparent Dynamic Contact Angles at a Tape, a Cylinder, and a Fiber, Vertically Withdrawn from or Plunging into a Glycerol-Water Mixture (Our Published Results and Data from Reference 6 ) polyester tape, receding polar liquid" siliconized glass-cylinder, receding polar liquid (R = 1.505cm) siliconized glass fiber, receding polar liquid (R = 0.013 cm) polyester tape, advancing polar, liquid polyester tape, advancing polar liquid a
1.4 1.4 3.2 x 0.32 1.2
5 x 10-4 1.2 x 1.1 x 2.7 x loT2 0.15
16.7 14.5 15.0 124.0 159.0
17.9 17.7 20.0 123.0 155.0
-1.2 -3.2 -5.0 1.0 4.0
Figure 12.
Discussion Huh and Masons theoretically studied the extrapolated contact angle for a liquid steady advancing against air in a narrow capillary with both 80 and Bed 90". In this case of a negligible effect of gravity they found that
eext= e, - r-(ln 4 u nY
L - 1.188)
eext= Oi - @(? In Ri - + 0.573)
(14)
eappx ei - LLu(4 - l n Ri - + 2.10)
(15)
Y n
R
R
Assuming a particular slip boundary condition they related the local fluid interface slope, 8i,at a distance Ri = 5 x lo-' cm from the contact line to the actual dynamic angle 8, and the slip length L,
(16)
Substitution of eq 16 in eq 14 yields a relationship similar
to eq 13.
From eqs 14 and 15 one finds that Oext > eapp for a liquid advancing against air '
(13)
Assuming that the slip length L, = 1nm and 8, = 80 they compared the difference OeA - e,, calculated from eq 13 with experimental data for Oapp - eofrom the studies of Hoffman,29Hansen and T~ong,~O and Rose and Heinsa31 The good coincidence found in the first case was considered to be an indication that 0, = 60 = constant and that Hoffman measured an apparent dynamic angle which is close to Oext. For the latter two references a considerable difference between 8, and eo was found and a velocity dependence of 8, was assumed, although the small static contact angles of about 20" observed for these systems made the application of eq 13 questionable. A thorough theoretical comparison of different dynamic contact angle definitions in a narrow capillary of radius R was performed by Kafka and D u s ~ a n . For ~ liquid advancing against air their expressions give
Y n
ei = e, + *V( l + l n 2 )
eext- eapp
1.527 @ >
Y
o
(17)
For a receding liquid being displaced by air, the contact line velocity has an opposite sign and one obtains that OeA e
eapp
eext- eapp
-1.527 @ < o
Y
(18)
These relationships were qualitatively confirmed in our former experimental investigation5 of the profiles of dynamic menisci at a vertical fiber, a large cylinder, or a tape, representing cases of negligible and significant effect of gravity and different solid surface curvature. For all systems studied we found that in the receding mode eext e eapp, while for an advancing liquid OeXt > OaPp. Some particular values of OeXt and Oapp are shown in Table 2 together with the correspondingBond numbers, Bo = egD/ y , and capillary numbers, Ca = pUIy. Each pair of Oext and eapp was obtained from the same profile picture so that their error of f0.2" originates from the accuracy of the determination ofthe meniscus coordinatesz andx but avoids the irreproducibility of the solid surface. Thus, all - Oapp, presented in Table 2 are significant differences, and confirm inequalities 17 and 18. Since the absolute value of the real capillary height (or depression), 121, is by definition always greater than the (29)Hoffman, R. L.J. Colloid Interface Sci. 1976,50,228. (30)Hansen, R. J.;Toong, T.V.J. Colloid Interface Sci. 1971,37, 410.
(31)Rose, W.;Heins, R. W. J. Colloid Sci. 1962,17,39.
Langmuir, Vol. 11, No. 8, 1995 3267
Profile of a Steady Meniscus 0.30 I
z, cm 0.20
LA'
0.05 0.00 0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.04
I 1 t
0.00~ 0.00
"
0.04
'
x, cm
Figure 12. Experimental steady dynamic meniscus at a tape of polyethylene terephthalate,verticallywithdrawn at U = 0.070 c d s from a glycerol-water mixture withp = 0.462P, y = 64.9 dyn/cm, and = 1.2116 g/cm3. The solid line represents the solution of the Laplace equation for a flat wall. Deviation of the point at the wall from the quasi-staticprofile can be seen. The values of dynamic contact angles obtained are Oca = 16.7", eapp= 17.9",and Os, = 19.8".
absolute value of the maximum height (depression) of the quasi-static part ofthe dynamic meniscus, lZqs1,one finds that in the receding mode e,, > Oapp (cf. Figure 3) and in the advancing mode e,, < Oapp. From these inequalities and eqs 17 and 18, one obtains that at receding 8, > Oapp > Oed while at advancing e,, < OaPp < 8,t. The first relationship agrees with our numencal results for the receding mode presented in Figures 4-9, which show that e,, 2 OeXt at all steady state velocities. A direct experimental confirmation that Os, > eapp > Oea was extracted from the dynamic meniscus profile at a polyester tape vertically withdrawn at U = 0.070 c d s from glycerol-water mixture withp = 0.462 P, 4 = 1.2016 g/cm3,and y = 64.9 dyn/cm, Figure 12. (In order to avoid the edge effects by photographing the meniscus profile, the 3.5 cm wide tape was slightly bent along its 10 cm length to form a cylindrical surface of large radius of curvature.) This profile was fitted by the solution of Laplace equation for a flat wall (the solid line) and values of e,, = 19.8",eapp = 17.9",and Oext = 16.7"were obtained with an experimental error of f 0 . 2 " . A similar relationship e,, = 23.5" =. Oapp = 17.7" > Oea = 14.5"was found in ref 5 for a siliconized glass cylinder of large radius from a glycerolvertically withdrawn at Ca = 1.2 x water mixture. The above relationships between Oed, 88pp,and e,,, predicted by theory and confirmed by expenment, yield an important practical conclusion-the dependence Oapp(U)is always located between eext(U)and 8,,(U) and all conclusions about the trend and peculiarities of OU(,) and e,,(U) relationships should hold at least qualitatively also for OaPp(U). Therefore, the main result of the numerical simulations that the dynamics of the inner (molecular) region strongly affects the dependences OeXt(v)and 8,,(U) which characterize the outer (quasi-static) region should be in force also for the dependence eaPp(U). The role of molecular region dynamics becomes smaller when the viscosity of the liquid increases by 2 orders of magnitude; under such conditions the maximum steady velocities obtained for 8, = constant and 8, = O,(U) become closer. However, the decrease of the static contact angle from 80" to 30" does not influence OeXt(U), €J,.(U), and Oapp-
"
0.08
0.12
'
'
'
0.16
0.20
0.24
x, cm
Figure 13. A static meniscus profile for the same system as in Figure 12. Comparison of the static contact angles determined by fitting the fluid interface profile, eext= 48.2",and introducing the capillary height 2 in eq 2, Oapp = 48.2". (U)in the same way; at 8 0 = 30"the two dynamic boundary
-
conditions produced much bigger difference of the values of U at which Oea 0". A similarity of the two effects might be expected from de G e n n e and ~ ~ Brochard ~ and de G e n n e predictions ~~~ that at small dynamic contact angles the viscous friction in the thin edge of the moving meniscus (region 2) should dominate the nonhydrodynamic friction in the molecular region 1 and the different dynamic behavior of region 1 should play a secondary role. The thickness of the hydrodynamic deformation also depends on the inner region dynamics; the maximum values of h, differ by more than an order of magnitude for 8, =constant and 8, = 8,(U) and the velocities at which they are achieved are far away from each other. Although the exact values ofh,, must be considered with precaution because of the somewhat arbitrary choice of B = 1, the ratio of the thicknesses obtained for the two boundary conditions should be representative. The dimensions of the hydrodynamic deformation for a real receding meniscus were estimated from Figure 12 and values of h,. x 30 pm and 2 - 2., x 60 pm were found at Ca = 5 x The profile of the dynamic meniscus at a siliconized large cylinder recorded in ref 5 at Ca = 1.2 x gave h,, x 100 pm and 2 - Z,, = 185 pm. All values are significant because of being greater than the standard regression error of the fit (7 pm). Substituting the data for h,, and 8, in eq 11 yields p = 1.7 and 0.7, for the two profiles, respectively. These are not so far away from the value B = 1 used in the numerical simulations but differ from each other in contrast to the prediction of the theory. Under static conditions such deformation is not present and all contact angle definitions should give the same values. This statement, although obvious, is illustrated in Figure 13 for a static meniscus profile of the system of Figure 12. All points of the fluid interface profile fit very well the flat wall solution of the Laplace equation (the solid line) which shows that the slight solid surface curvature does not play any role. This fit gives Oed = 48.2" and the same value Oapp = 48.2" is obtained when introducing the dynamic capillary height, 2,in eq 2. Figure 14 shows another extreme case-a dynamic profile at U U,,, where a strong instability (pulsing) of (32) de Gennes, P.G. Colloid Polym. Sci. 1986,264, 463. (33) Brochard, F.;de Gennes, P.G.Adv. Colloid Interface Sci. 1992, 39, 1.
Petrov and Petrov
3268 Langmuir, Vol. 11, No. 8, 1995 0.5
,
0.0 L 0.00
J
.
0.04
,
0.08
1
1
0.12
8
1
0.16
1
1
0.20
0.24
x, cm
Figure 14. Dynamic meniscus profile for the same system but at U = 0.077 c d s (veryclose to/or at the critical velocity of film entrainment, see the text). A fit of the experimental points to the solution of the Laplace equation is possible only if the theoretical curve is shifted away from the solid surface at a position represented by the vertical dashed line. The fit gives B,,t = 0" and a shift distance of 45 pm.
the contact line was observed. The determination of Oapp from eq 2 becomes impossible here (O, loses its physical meaning) because the real capillary height exceeds the maximum static one, Z,, = ( 2 y / ~ g ) At ~ . the same time Oext remains a well-defined quantity which has reached its limitingvalue of 0". Under these conditions the quasistatic profile can be fitted by the solution of Laplace equation only if it is shifted away from the solid wall at a distance shown by the vertical doted line. The same fit is usually performed for profiles of a static menisci being in contact with thick equilibrium wetting films ( 0 0 = 0"). In such a way the "contact thickness", thermodynamically characterizing the three-phase transition zone is defined15 and measured.34 Thus, Oea is a quantity that maintains its meaning under static conditions, at a steady wetting or dewetting and when a fluid film entrainment occurs. Some other fitting procedures for experimental determination of Bed were proposed in the literature. Damania and B o ~ fitted e ~ ~a parabolic function to the dynamic meniscus profiles at a flat wall and a rod close to the contact line but did not justify the use ofthis function theoretically. Another difference to our determination of Oea is that their fit was performed in a meniscus region where the hydrodynamic distortion must be most pronounced. We fit the Laplace equation to the meniscus profile enough far away from the contact line where its validity is rigorous. Dussan, Rame, and GarofP also fitted the dynamic profile of an advancing meniscus at a tilted large cylinder close to the contact line. However, they used a theoretical solution for the fluid interface shape of both the hydrodynamic and the quasi-static region and determined the extrapolated dynamic angle (in their motation W O ) as a free parameter of the fit. In our experiments the solution of Laplace equation was fitted far away from the contact line, i.e., only in the quasi-state meniscus region. The theoretical solution of Dussan, Rame, and GarofP is more general but in principle similar to ours. However, their strategy formulated in this and in some previous investigations of Dubsan et al.9J1is different. It replaces the actual dynamic contact angle as a boundary condition of the dynamic meniscus with the local slope, OR, at a definite distance from the contact line, and studies the dynamic behaviors of OR. When measured close enough
to the wetting perimeter, this angle is expected to be a material property of the system. This important advantage would enable one to transfer data between systems containing the same solid and fluid phases but having different geometries. Unfortunately, in spite ofthe skillful experiment performed in ref 12, the closest approach to the contact line achieved (18-25 pm) was not sufficient to ensure such an independence. The use of O,, as a boundary condition of the quasistatic part of the moving meniscus enables one to work at small magnifications and to use a simple but rigorous equation to fit the profile of the fluid interface. Since the capillary length which is a characteristic dimension of the meniscus at a flat wall is present in eqs 7 and 11,one might expect that the system geometry affects the extrapolated dynamic contact angle. The expressions for Oe* found in other studies for small drops17 or narrow ~ a p i l l a r i e s ~also , ~ Jcontain ~ the corresponding characteristic dimensions ofthe system h,, thus showing the geometry dependence explicitly. This dependence seems rather impractical but the expression obtained by Voinov17for OeXt of advancing liquid shows that a variation of h, does not strongly affect e,, because h, is subject to the logarithm
(19) The difference in the values of the numerical constant C for drops (C= LO), capillaries (C= 1.8)and slots (C= 1.5) is much smaller than the usual values of ldhdh,) which depending of the choice of h, have characteristic values between 10 and 12. On the other hand the extrapolated dynamic contact angle discussed here possessessome useful characteristics, which support its application in studies of wetting dynamics: At zero contact line velocity ita definition is identical with the rigorous thermodynamic definition of the static contact angle. The extrapolation .procedure is not empirical but relies on an exact theoretical equation describing the real fluid interface far away from the moving contact line. The quasi-static part of the dynamic meniscus is easily accessible and the experimental determination of flea possess no problems. The fitting procedure gives simultaneously an estimate of the dimensions of the hydrodynamically deformed region of the meniscus. Oea is sensitive to the dynamic behaviors of the molecular region. This sensitivity is comparable to the one of Oapp, but when studying fled(U),one operates with a quantity that has a clear physical meaning both for a quasi-static and a hydrodynamically deformed fluid interface.
Acknowledgment. This paper was prepared for publication during the academic visit of J. G. Petrov at Max Planck Institute of Colloids and Interfaces in Berlin. This author thanks Professor H. Mohwald for his kind hospitality and the Max Planck stipend which enabled this visit. The other author (P. G. Petrov) thanks the National Science Foundation of Bulgaria for the financial support under Project No. X7. LA950083A (34)Kolarov, T.; Zorin,Z.; Platikanov, D. Colloiak Surf.1990,51,37. (35) Damania, B. S.;Bose,A. J . ColloidZnterfmeSci.1988,113,321.
