Langmuir 2002, 18, 3549-3554
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Wetting Kinetics of Fluorinated Surfaces M. R. Sharpe,† I. R. Peterson,*,‡ and J. P. Tatum† Xaar plc, Cambridge Science Park, Cambridge CB4 4FD, U.K., and Centre for Molecular and Biomolecular Electronics, Coventry University SE, Priory Street, Coventry CV1 5FB, U.K. Received September 30, 2001. In Final Form: February 19, 2002 A comparison has been made of the wetting properties of surfaces formed from bulk perfluorinated thermoplastics, from perfluorinated thermoplastic coatings, and from sol-gel coatings containing perfluoroalkyl chains. All surfaces showed hysteretic behavior, i.e., different advancing and receding contact angles, that was well-described by the Blake-Haynes meniscus-fluctuation theory of wetting kinetics over 2 orders of magnitude of advancing and receding meniscus velocities. The behavior at high wetting speed, and particularly the maximum three-phase contact line velocity, was better described by the VoinovCox-de Gennes theory involving viscosity. Best agreement with experiment over the whole range of meniscus velocities was found with the theory of Petrov, which takes both types of effects into account.
Introduction The present investigation was carried out as part of a program for the development of a range of new highperformance inkjet printheads. In inkjet printing, it is of interest to achieve a faceplate that actively repels liquid. Xaar plc has patented a range of formulations of sol-gel coatings containing perfluoroalkyl chains,1 whose excellent static dewetting properties have been known since the work of Zisman.2 Zisman demonstrated an empirical correlation for any solid between the equilibrium contact angle θ0 of static wetting and the surface tension γL of the liquid. At a critical surface tension γC, the surface is completely wetted (θ0 ) 0), and this is also true for all liquids with surface tension less than γC. The lower the value of γC, the better the solid’s static dewetting properties. Zisman found the lowest values of critical surface tension for materials with perfluorinated chains. The value of γC of a material is correlated with its bulk cohesive energy. In part, the low cohesion and strong dewetting tendency of perfluorinated chains is related to their lack of polar groups, but compared to the similarly nonpolar hydrocarbons, the cohesion is anomalously low.3,4 Spontaneously adsorbed monolayers of long-chain perfluorinated derivatives can have values of γC under 10 mN/m. In mixtures with water, such compounds exhibit lamellar phases with a large range of stability.5 However, the lamellae are highly defective, perhaps as a result of the helical conformation of the perfluoro chains,6,7 and the phases have been called “discotic lamellar”.8 * Corresponding author. Tel.: +44 24 76888376. Fax: +44 24 76888702. E-mail:
[email protected]. † Xaar plc. ‡ Coventry University SE. (1) Griffin, M. C. A.; Howarth, L. G.; Tatum, J. P. (Xaar plc). International Patent Application PCT/GB95/02041; Publication WO 96/06895. (2) Hare, E. F.; Shafrin, E. G.; Zisman, W. A. J. Phys. Chem. 1954, 58, 236-239. (3) Newcomb, M. M.; Cady, G. H. J. Am. Chem. Soc. 1956, 78, 8, 5217. (4) Rotariu, G. J.; Hanrahan, R. J.; Fruin, R. E. J. Am. Chem. Soc. 1954, 76, 3752. (5) Smith, A. M.; Holmes, M. C.; Pitt, A.; Harrison, W.; Tiddy, G. J. T. Langmuir 1995, 11, 4202-4204. (6) Bunn, C. W.; Howell, E. R. Nature 1954, 174, 549. (7) Strobl, G. J. Chem. Phys. 1991, 95, 2800-2817.
The speed at which a substrate can be wetted or dewetted is important for the proposed application. In different experimental configurations, three different factors have been shown to take the dominant role in determining the wetting kinetics: (1) viscous forces in the liquid,9 (2) specific molecular-scale interactions at the three-phase contact line,10 and (3) viscoelastic deformation of the substrate.11-13 In addition to these dynamic aspects of wetting in which the meniscus moves at a constant speed, many surfaces display hysteresis, a phenomenon in which the contact angle depends on the previous history of the meniscus position.14 In this case, there appears to be no well-defined value of the equilibrium contact angle θ0, an important parameter in most theories of wetting. A common technique that is often claimed to yield a reproducible value of θ0 in hysteretic systems is to vibrate the system.15,16 The result must clearly lie between the values for the advancing and receding contact angles, but in our experience, it is not reproducible, and its theoretical basis is not clear. Voinov17 and Cox18 proposed a theory for the maximum horizontal velocity of the three-phase contact line when the viscosity of the fluid, η, is the dominant limiting factor, and this theory was considered in more detail by de Gennes and co-workers.19-21 The velocity V is related to the contact angle by (8) Boden, N.; Jolley, K. W.; Smith, M. H. J. Phys. Chem. 1993, 97, 7678-7690. (9) Savelski, M. J.; Shetty, S. A.; Kolb, W. B.; Cerro, R. L. J. Colloid Interface Sci. 1995, 176, 117. (10) Peterson, I. R.; Russell, G. J.; Roberts, G. G. Thin Solid Films 1983, 109, 371-378. (11) Long, D.; Adjari, A.; Leibler, L. Langmuir 1996, 12, 5221. (12) Extrand, C. W.; Kumagai, Y. J. Colloid Interface Sci. 1996, 184, 191-200. (13) Sedev, R. V.; Petrov, J. G.; Neumann, A. W. J. Colloid Interface Sci. 1996, 180, 36-42. (14) Extrand, C. W.; Kumagai, Y. J. Colloid Interface Sci. 1997, 191, 378-383. (15) Decker, E. L.; Garoff, S. Langmuir 1996, 12, 2, 2100-2110. (16) Della Volpe, C.; Maniglio, D.; Siboni, S.; Morra, M. Rev. Inst. Fr. Pe´ t. 2001, 56, 1. (17) Voinov, O. V. Fluid Dyn. 1976, 11, 714-721. (18) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (19) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (20) Brochard-Wyart, F.; de Gennes, P. G. Adv. Colloid Interface Sci. 1992, 39, 1-11. (21) Redon, C.; Brochard-Wyart, F.; Rondelez, F. Phys. Rev. Lett. 1991, 66, 715-718.
10.1021/la015602q CCC: $22.00 © 2002 American Chemical Society Published on Web 04/05/2002
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Table 1. Solvent and Substrate Combinations Investigated surface liquid
F
γL
E
K
H I O N A Y J X
hexane 2-propanol octane decane X2A Black Ink X2 Atlantic Ink X2A Pacific Ink Exxsol D140
19 21.8 22 24 28.5 28.5 28.6 28.6
L B
decanol X2 Black Ink
29 29.3
T
tripropylene glycol monomethyl ester triethyl citrate dimethylglycol propylene carbonate
31
Figure 7 Figure 5 Figure 7 Figure 7 Table 3 Table 3 Table 3 Figure 5 Table 3 Figure 7 Table 2 Table 3 Figure 7
35 36 41.8
Figure 7 Figure 7 Table 2
R M P
Table 3 Table 3 Table 3
Table 2 Table 3
Table 2
Table 2
Table 2
()
L ηV ln θ3 ) θ03 + 9 γL Ls
(1)
[
λ2γL (cos θ0 - cos θ) 2kBT
]
(2)
with parameters λ, an average barrier spacing, and Kw, the spontaneous rate of barrier crossing. On a simple molecular model, Kw can be derived from a parameter Ew that characterizes the inhomogeneities of interfacial tension
Kw )
( )
kBT λ2Ew exp h kBT
(3)
For the present purposes, it is more convenient to recast eq 2 with the meniscus velocity as an independent variable25
( )
cos θ ) cos θ0 - c sinh-1
D
U
G
Table 3
Table 3 Table 3 Table 3
Table 3 Table 3 Table 3
Table 3 Table 3 Table 3
Table 2
Table 2 Table 3
Table 2 Table 3
Table 3
V0 ) 2Kwλ
(5)
γLλ2c ) 2kBT
where L and Ls, the capillary length and a slip length, respectively, are parameters of the theory. In contrast, the Blake-Haynes theory of wetting dynamics focuses on molecular-scale fluctuations of the meniscus that overcome pinning by interfacial inhomogeneities. The inhomogeneities set up a series of energy barriers that the meniscus must cross.22-24 A discrepancy between the actual contact angle θ and the equilibrium value θ0 gives rise to a driving force γL(cos θ0 - cos θ) that promotes barrier crossing in one direction and reduces its likelihood in the opposite direction. The result is an overall meniscus velocity V. The initial equation proposed by Blake and Haynes had the form
V ) 2Kwλ sinh
C
V V0
(4)
The parameters V0 and c are related to the better-known parameters Kw and λ as follows (22) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421-423. (23) Blake, T. D. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993; Chapter 5, pp 251-309. (24) Blake, T. D.; Clarke, A.; De Coninck, J.; de Ruijter, M. J. Langmuir 1997, 13, 2164-2166. (25) Blake, T. D.; Bracke, M.; Shikhmurzaev, Y. D. Phys. Fluids 1999, 11, 1995-2007.
The form of eq 4 shows a very close similarity to the Butler-Volmer equation for electrode current density,26,27 with the parameter V0 playing the same role as the exchange current density J0. V0 can therefore be called the “exchange wetting velocity”. It should be noted that the standard Blake-Haynes parameters Kw, Ew, and λ have an interpretation in terms of a one-dimensional molecular model of the meniscus fluctuations and the nanoscale surface structure. In contrast, V0 and c are phenomenological parameters that do not presume a detailed model. In the present work, we employ two experimental techniques to investigate these questions. The first is an experimentally convenient configuration for measuring the maximum three-phase contact line velocity Vm. The second is a dynamic surface tensiometer, which, at low wetting speeds V , γL/η, provides experimental access to the dynamic contact angle.28 Experimental Section The present study involved a total of seven different surfaces and 11 different liquids, as shown in Table 1. The first surface, FEP 500L, symbolized F, was a 500-µmthick fluorohydrocarbon thermoplastic (DuPont). The next two, E and K, had a polyimide base with a fluorohydrocarbon coating. C, D, U, and G were each in-house proprietary films consisting of a polyimide substrate coated with a sol-gel formula containing fluoroalkyl chain derivatives.1 The films of bulk thermoplastic forming the first three surfaces F, E, and K had a rear surface that was nominally identical to the front. It was therefore straightforward to deduce the contact angle from the measurement in a model DST9005 dynamic surface tensiometer (Nima, Coventry, U.K.) of force on a Wilhelmy plate cut out of the film. For the asymmetrically coated films C, D, U, and G, the force on the rear surface had to be estimated by separately measuring the force on a symmetrical plate cut from uncoated Kapton. These values were considered to be less reliable. The liquids are ranked in order of their surface tensions γL. Nine of them were pure solvents and were obtained in the highest available grade. Exxsol D140 is a commercially available proprietary mixture of high-surface-tension hydrocarbons. X2 (26) Atkins, P. W. Physical Chemistry. Oxford UP 1998. (27) Bockris, J. O.; Reddy, A. K. N. Modern Electrochemistry; Plenum Press: New York, 1973; Vol. 2. (28) Rame´, E. J. Colloid Interface Sci. 1997, 185, 245.
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Figure 3. Data points and best fit to the Blake-Haynes theory as in Figure 2 but with abscissa reversed and split to show positive (advancing) and negative (receding) meniscus values on a logarithmic scale. Figure 1. Configuration for measurement of maximum threephase contact line velocity
Figure 4. Split-logarithmic-abscissa plot of data points and best-fit curve for propylene carbonate (liquid P, γL ) 41.8 mN/ m) on all three thermoplastic surfaces. Figure 2. Variation of contact angle cosine as a function of meniscus velocity for liquid B (X2 Black Ink) on surface K (FEPcoated Kapton). The circles are the measurement points, and the curve shows the best fit to Blake-Haynes theory. Black Ink and its variants contain, in addition, particulates and polymeric stabilizers. The maximum three-phase contact line velocity Vm was measured using the configuration shown in Figure 1. A droplet of liquid sitting on the surface was held captive by a strongly wetting probe fixed to an arm rotating about a vertical axis. The droplet was entrained across the surface in a circular path at a constant velocity. The velocity was increased slowly until the liquid droplet lost its integrity.
Results Figures 2 and 3 show the variation of contact angle with meniscus velocity plotted in two different ways. The data were obtained for liquid B (X2 Black Ink) wetting surface K (FEP-coated Kapton). In Figure 2, the abscissa is linear. The circles are the data points, and the continuous curve is the best fit to the phenomenological BlakeHaynes equation (eq 4) with the parameters given in the inset. For the purposes of the least-squares fitting routine, it is clear that the rather large scatter of the data points is due to errors in the measurement of cos θ and not of V. In some cases, this scatter and nonmonotonic behavior was clearly associated with pronounced meniscus instability, often called stick-slip behavior.
Figure 3 displays the same data as Figure 2, but the abscissa has been inverted and split into two regions corresponding to advancing and receding meniscus, each of which has been plotted on a logarithmic scale. It can be seen that the logarithmic plot essentially linearizes the theoretical curve for speeds above the exchange wetting velocity because, for large x
sinh-1(x) ≡ ln(x + x1 + x2)
(6)
≈ ln(2x) When the abscissa is logarithmic, the asymptote for speeds much less than V0 is a horizontal line, as clearly seen in Figure 3. The measurements shown in Figure 2 deviate significantly from the best-fit theoretical curve, but nevertheless display its systematic trends. The best-fit exchange wetting velocity V0 of 97 nm/s is over 2 orders of magnitude lower than the minimum speed available on the Nima DST9005 instrument. It was not possible to take measurements at or near V0 for any of the surface/liquid combinations investigated. Figure 4 shows a split-logarithmic-abscissa plot of the sort introduced in Figure 3 for the three thermoplastic surfaces wetted by liquid P (propylene carbonate). It can be seen that, despite the similarity of chemistry and fabrication of the surfaces and the very large measurement error, the contact angles of the three surfaces are
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Table 2. Best-Fit Parameters for the Data Shown in Figures 4-6 liquid/surface code
γL (mN m-1)
V0 (m s-1)
c
cos θ0
Kw (s-1)
λ (nm)
Ew (mN m-1)
BC BD BE BF BK BU IK PE PF PK XK
29.3 29.3 29.3 29.3 29.3 29.3 21.8 41.8 41.8 41.8 28.6
1.3817 × 10-5 1.5887 × 10-11 2.6398 × 10-6 4.1338 × 10-8 9.6771 × 10-8 2.7097 × 10-5 1.3124 × 10-10 1.3449 × 10-7 4.9047 × 10-6 3.9665 × 10-10 1.7635 × 10-8
4.4165 × 10-2 3.8349 × 10-3 1.8152 × 10-2 1.1925 × 10-2 1.7127 × 10-2 6.1649 × 10-3 4.6424 × 10-3 1.3219 × 10-2 8.0230 × 10-3 9.3715 × 10-3 7.5826 × 10-3
0.70114 0.75001 0.50900 0.74064 0.75157 0.85800 0.82617 0.24264 0.41057 0.32937 0.62978
2.76 × 103 9.36 × 10-4 3.38 × 102 4.30 × 100 1.21 × 101 2.02 × 103 7.34 × 10-3 1.76 × 101 4.99 × 102 4.36 × 10-2 1.44 × 100
2.49 8.44 3.88 4.79 3.99 6.66 8.89 3.81 4.88 4.52 6.07
13.91 2.05 6.28 4.89 6.76 1.97 1.74 7.34 3.89 6.38 3.15
Figure 5. Split-logarithmic-abscissa plot of data points and best-fit curve for four test liquids on FEP-coated Kapton (surface K).
Figure 6. Split-logarithmic-abscissa plot of data points and best-fit curve for X2 Black Ink (liquid B, γL ) 29.3 mN/m) on six surfaces.
significantly different. The gradients of the lines are, however, similar. As in Figure 2, there are significant differences between the advancing and receding contact angles, even at the slowest speeds achievable with the dynamic tensiometer. Figures 5 and 6 display split-logarithmic-abscissa plots showing similar characteristics for four liquids on surface K and for liquid B (X2 Black Ink) on all six surfaces, respectively. It can be seen that the fit to the BlakeHaynes theory is not always as good as in Figure 3, with erratic and nonmonotonic variations of cos θ with V. As mentioned above, this behavior was sometimes clearly associated with stick-slip instabilities of the meniscus. Table 2 shows the least-squares best-fit phenomenological parameters V0, θ0, and c and the Blake-Haynes model
Figure 7. Plot of contact angle cosines on surface K (FEPcoated Kapton) as a function of the liquid surface tension γL, and best fits to both a Zisman plot (straight line) and Fowkes’ model (curve).
parameters Kw, λ, and Ew for the data sets presented in Figures 4-6. In Figure 7, values of the cosine of the equilibrium contact angle (cos θ0) with substrate K (FEP-coated Kapton) are plotted against surface tension for a range of solvents. Unlike the values in Table 2 obtained from the data sets of Figures 2-4, these values were deduced from measurements at only two meniscus velocities of (83 µm s-1. The curve is the best fit to a + bγL-1/2 (Fowkes law29) and is a significantly better fit than the straight line a bγL (Zisman plot30). The Zisman theory is seen to give a systematically lower value for the surface energy. At velocities near the maximum three-phase contact line velocity Vm, the droplet developed a hump at its trailing edge, as reported by Redon et al.21 After a measurement of this value for the inks, the surface was always found, upon inspection under a microscope, to be covered with a high density of small droplets. The value of Vm for a given surface was found to decrease with exposure to ink. The longer the surface was maintained in contact with the ink, the slower the receding meniscus velocity. However, if the surface was then washed and dried, Vm was usually found to recover. Table 3 shows the receding meniscus velocities Vm of various combinations of surfaces and liquids (the latter all inks). cos θ0 is the cosine of the equilibrium contact angle, and cos θr is that of the receding contact angle at the maximum three-phase contact line velocity, both deduced using the Blake-Haynes formula by extrapolation from measured data (not listed here). The most (29) Fowkes, F. M. Ind. Eng. Chem. 1964, 56, 41-52. (30) Bernett, M. K.; Zisman, W. A. J. Phys. Chem. 1959, 63, 1241, 1911.
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Table 3. Receding Meniscus Velocities and Contact Angles for Various Surface/Liquid Combinations liquid/surface code
Vm (mm s-1)
cos θ0
cos θr
AC AC AD AF AG AK AU BD BF BG BK BU JD JF JG JK JU YD YF YG YK YU
12 12 6 8 3 14 4 3 5 1 14 2 12 9 5 15 12 8 6 7 15 9
0.597 0.61 0.704 0.649 0.713 0.614 0.694 0.752 0.722 0.837 0.605 0.774 0.678 0.637 0.679 0.604 0.664 0.677 0.676 0.711 0.616 0.66
0.691 0.711 0.791 0.757 0.873 0.703 0.797 0.898 0.842 1 0.681 0.953 0.765 0.725 0.745 0.677 0.741 0.766 0.785 0.788 0.702 0.738
Figure 8. Plot of contact angle cosines versus two-thirds power of the maximum three-phase contact line velocity as given in Table 3: (1) equilibrium contact angle (O) receding angle extrapolated using Blake-Haynes formula. Table 4. Comparison of Blake-Haynes-Extrapolated and Measured Values of the Maximum Three-Phase Contact Line Velocity Vm
common form of the Voinov-Cox-de Gennes formula has Vm ∼ θ3,21 but it is more convenient to plot the data as cos θ. To take account of the offset square-law behavior of the cosine at small angles, the two choices of cosine are plotted versus Vm2/3 in Figure 8. Neither set of points plotted in Figure 8 gives a perfect straight line, but the various measures of fit are better when the extrapolated receding contact angle rather than the equilibrium contact angle is used. This is, of course, consistent with de Gennes’ treatment,19 which does not actually make any assumptions about the constancy of the contact angle. Because the two values of cos θ are simply related by an additive term in the Blake-Haynes formula, the formula that best predicts the maximum three-phase contact line velocity is seen to be
cos θ0 + c sinh-1
( )
Vm + bVm2/3 ) 1 V0
(7)
maximum three-phase contact line velocity Vm (m s-1) liquid/surface code
extrapolated from Table 2
measured (Table 3)
BD BF BK BU
1.62 × 1017 5.77 × 101 9.64 × 10-2 1.37 × 105
5 × 10-3 5 × 10-3 1.4 × 10-2 2 × 10-3
Equation 4 together with the data in Table 2 can be used to estimate the time for the liquid to establish its equilibrium contact angle with its substrate. Simple calculations show that the kinetics is asymptotically exponential with the time constant
τ)
c dx V0 d cos θ
(9)
Although there is considerable measurement error apparent in all the graphs of Figures 4-6, the overall trend of linear variation with the logarithm of the velocity is well-described by eq 4. None of the graphs extends to speeds below the exchange wetting velocity, beyond which the contact angle would approach the thermodynamic equilibrium value. In some systems, this velocity was less than 1 nm/s.
From the Rayleigh expression relating meniscus height and contact angle, the derivative can be approximated by xγL/FL. For the combinations IK and PK, the time constant is on the order of magnitude of 1 week. Clearly, equilibrium is difficult to achieve experimentally, and this behavior is commonly described as hysteresis. However, the Blake-Haynes equation gives access to the equilibrium value. The best-fit values of the Blake-Haynes parameters λ and Ew shown in Table 2 all seem reasonable. However, unrealistic fits have been reported previously32 and indicate that the detailed molecular picture proposed for the Blake-Haynes model is not always appropriate. In fact, the model clearly makes no attempt to take account of the two-dimensionality of the interface. In addition, it is not implausible that the interfacial inhomogeneities can have a fractal character, described by a spectrum with no characteristic length scale. Objections of this nature do not affect the phenomenological eq 4, which can be justified on much more general considerations of Einsteinian stochastic transport. A number of liquid/surface combinations are common to Tables 2 and 3. Table 4 shows a comparison between the measured values of Vm and the values extrapolated
(31) Petrov, J. G.; Radoev, B. P. Colloid Polym. Sci. 1981, 259, 753760,
(32) Schneemilch, M.; Hayes, R. A.; Petrov, J. G. Langmuir 1998, 14, 7047-7051.
This equation can be derived from the formula of Petrov and Radoev31
( )
V F ) cos θ0 - c sinh-1 - bV2/3 γL V0
(8)
describing the combined effects of viscous and meniscus drag, in which the meniscus velocity V is set equal to -Vm and F/γL, the normalized overall driving force per unit meniscus length, is set equal to the maximum value of unity that can be exerted by a fluid surface. Discussion
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from the Blake-Haynes formula by setting cos θ ) 1.33 The extrapolated value is always higher, and often considerably higher, than the measurement. Clearly, the good fit of the Blake-Haynes formula at low speeds is not a reliable guide to the behavior at the highest speeds, and some other limiting factor must come into play. This is obviously the viscous drag, which varies as a power law of meniscus velocity, whereas the meniscus drag varies logarithmically. Hence, at some velocity, there must be a transition to a high-speed regime in which the viscous drag dominates. This is consistent with the satisfactory fits to the variation of Vm measured using the expansion of a hole induced in a liquid film of less-than-critical depth34 obtained despite the omission of the meniscus drag term. A theory of wetting kinetics taking both meniscus and viscous drag into consideration was put forward by Petrov and co-workers31,35,36 in connection with the deposition of Langmuir-Blodgett (LB) films. When this formula is used to calculate the maximum receding three-phase contact line velocity V ) -Vm, it corresponds to the Voinov-Coxde Gennes formula, in which the receding contact angle is set equal to the value predicted by the Blake-Haynes equation. This special case of the Petrov-Radoev formula is confirmed in Figure 8 for monolayer-free systems involving both simple and complex fluids. It should be noted that Petrov has more recently refined eq 8,37,38 obtaining a rather more complicated implicit relationship between driving force and velocity. Stick-slip behavior of the meniscus has been reported by a number of authors in a range of systems.39-42 As a result of the nonlinearity of the Blake-Haynes formula, the magnitude of the average force for a given average velocity is expected to decrease upon the onset of oscillations. Given the near-exponential variation of velocity with driving force, the relative meniscus-substrate velocity is expected to be small over much of the cycle of instability, consistent with the appearance of “sticking”. Instability of this sort, which might have an amplitude too small to be observed with the naked eye, can explain the large scatter and nonmonotonic behavior of many of the data sets. It should be noted that the Blake-Haynes theory cannot explain oscillatory behavior as it stands, because its forcevelocity curve always shows a positive gradient. However, the observation of the presence of small droplets remaining on the surface after measurement of Vm is consistent with surface inhomogeneities on a scale greater than molecular. In this scenario, the receding meniscus moves much faster (33) Blake, T. D.; Ruschak, K. J. Nature 1979, 282, 489-491. (34) Andrieu, C.; Sykes, C.; Brochard, F. J. Adhes. 1996, 58, 15-24. (35) Petrov, J. G. Z. Phys. Chem. Leipzig 1985, 266, 706-712. (36) Petrov, J. G. Colloids Surf. 1986, 17, 286-294. (37) Petrov, J. G.; Petrov, P. G. Langmuir 1992, 8, 1762. (38) Petrov, J. G.; Petrov, P. G. Langmuir 1998, 14, 2490-2496. (39) Peterson, I. R.; Russell, G. J. Philos. Mag. A 1984, 49, 463-473. (40) Merle, H. J.; Alberti, B.; Schwendler, M.; Peterson, I. R. J. Phys. D 1992, 25, 1556-1558. (41) de Gennes, P. G. Europhys. Lett. 1997, 39, 407-412. (42) Darot, M.; Reuschle´, T. Pure Appl. Geophys. 1999, 155, 119129.
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at some points than others, leading to the development of “fingers” of fluid that can then neck off as isolated droplets. It is possible that such inhomogeneities also lead to an avalanche-type response, which is known to lead to instabilities in formally similar systems.43 Conclusions Consistent with its superior theoretical grounding, the Fowkes theory for equilibrium contact angles has been found to give a better fit than the Zisman equation to experimental data for the variation of the equilibrium contact angle with solvent surface tension. The present demonstration is believed to be the first to use the equilibrium value cos θ0 interpolated from the BlakeHaynes theory, rather than advancing or receding angles. The systems investigated here show behavior that would usually be called hysteresis. We find that it is welldescribed by the Blake-Haynes theory over a range of meniscus velocities. It is possible that even better agreement will be achieved by taking account of stick-slip meniscus behavior.39 The fits to the Blake-Haynes theory suggest that the stick part of this cyclic behavior might involve not absolute cessation of movement of the meniscus relative to the substrate, but rather creep at a speed too low to be distinguished from zero. For commercial reasons, many of the liquids investigated here are quite complex, containing dissolved polymers and particulates. The Blake-Haynes theory nevertheless provides a good description of their wetting kinetics at low speeds. It provides theoretically based experimental access to the equilibrium contact angle for hysteretic systems in which equilibrium is not attained within an experimentally reasonable time. The meniscus drag described by the Blake-Haynes theory varies logarithmically with meniscus velocity. The theory of de Gennes only explicitly considers viscous drag, which varies as a power law. The former dominates at velocities satisfying the Rame´ condition,28 and the latter dominates at the much higher maximum three-phase contact line velocity. Best agreement with experimental measurements is obtained over the whole velocity range using the Petrov-Radoev equation, which is the de Gennes equation substituted with the dynamic contact angle given by the Blake-Haynes formula. Hence, the Petrov-Radoev equation explicitly takes both types of drag into account and has a range of applicability far wider than just Langmuir-Blodgett deposition. Acknowledgment. We thank Dr. C. Sykes and Prof. J. G. Petrov for critical examination of the manuscript. This work was partly funded by the DTI and the EPSRC LINK Surface Engineering Program (Grant GR/K87562). LA015602Q (43) Pal, B. B.; Khan, R. U.; Chakrabarti, P. J. Inst. Electron. Telecommun. Eng. 1994, 40, 35-42.
