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Physics of Hydrophobic Cavities Vassili Yaminsky* and Satomi Ohnishi Department of Applied Mathematics, Research School of Physical Sciences & Engineering, Australian National University, Canberra, ACT, 0200, Australia, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, 305-8565, Japan, and Advanced Technologies Centre, Moscow, Russia Received June 25, 2002. In Final Form: January 5, 2003 It has been long believed that modern physics fails in explaining the forces measured between hydrophobic surfaces. We show that the dispersion attraction and cavitation between two uncharged strongly hydrophobic surfaces agree with the classical capillarity and Lifshitz theories. The fluorinated macroscopic glass spheres, with contact angles over 90°, show several times weaker van der Waals attraction across water compared to measurements in air. The aqueous film, stable down to distances shorter than 3 nm, breaks on contact. A bridging vacuum cavity rapidly fills with diffusing air. The long-range capillary force is affected by contact angle hysteresis and pressure regulation effects. The dispersion attraction and cavitation are similar in air-supersaturated and undersaturated water; the stability of the bubbles forming on breaking the meniscus critically depends on the allocation of the system with respect to the coexistence boundary.
Introduction Hydrophobic interactions are a subject of special interest in the area of physical chemistry of aqueous systems, with distinctive biological flavors. The areas of interest include the hydrophobic effect of poor solubility of hydrocarbons in water in relation to cohesion and water structure entropy1 and interfacial phenomena related to large contact angles between the hydrogen-bonded liquid and nonpolar substances.2 The long-range attraction observed between some hydrophobic surfaces3 is beyond the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of stability of hydrophobic colloids4 based on the Lifshitz theory of van der Waals forces5 and the PoissonBoltzmann theory of osmotic interaction of diffuse double layers of ions.6 Originally taken as evidence of a new surface force not accounted for by the DLVO theory,7-11 the attraction was later attributed to an involvement of air bubbles whose origin also was unexplained.12-15 * To whom correspondence should be addressed. E-mail: afm110@ rsphysse.edu.au. (1) Yaminsky, V. V.; Vogler, E. A. Curr. Opin. Colloid Interface Sci. 2001, 6, 342. (2) Fowkes, F. M. Ind. Eng. Chem. 1964, 56 (12), 40. (3) Christenson, H. K.; Claesson, P. M. Adv. Colloid Interface Sci. 2001, 91, 391. (4) Verwey; E. G. W.; Overbeek, J. Th. G. Theory of Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (5) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. Adv. Phys. 1961, 10, 165. (6) Derjaguin, B. V. Acta Physicochim. URSS 1939, 10, 333. Derjaguin, B. V.; Landau, L. D. Acta Physicochim. URSS 1941, 14, 633. (7) Israelachvili, J. N.; Pashley, R. M. Nature 1982, 300, 341. Israelachvili, J. N.; Pashley, R. M. J. Colloid Interface Sci. 1984, 98, 500. (8) Rabinovich, Ya. I.; Derjaguin, B. V.; Churaev, N. V. Adv. Colloid Interface Sci. 1982, 16, 63. Rabinovich, Ya. I.; Derjaguin, B. V. Colloids Surf. 1988, 30, 243. (9) Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Evans, D. F. Science 1985, 229, 1088. (10) Christenson, H. K.; Claesson, P. M. Science 1988, 239, 390. (11) Christenson, H. K. In Modern Approaches to Wettability: Theory and Applications; Schrader, M. E., Loeb, G., Eds.; Plenum Press: New York, 1992; p 29. (12) Carambassis, A.; Jonker, L. C.; Attard, P.; Rutland, M. W. Phys. Rev. Lett. 1998, 80, 5357. (13) Ishida, N.; Kinoshita, N.; Miyahara, M.; Higashitani, K. J. Colloid Interface Sci. 1999, 216, 387. (14) Attard, P. Langmuir 2000, 16, 4455. (15) Considine, R. F.; Drummond, C. J. Langmuir 2000, 16, 631.
From the general thermodynamic standpoint, the formation of vacuum or air cavities in contact between two surfaces under a nonwetting liquid is explained on the same lines as the well-known effect of capillary condensation. Such cavities were originally observed in mercury16,17 and later in water.9,10 The capillary vaporization leads to bridging the surfaces by a concave (vacuum) or zero-curvature (air) meniscus. However, the assumption of the persistence of bubbles on single separated surfaces invoked in explaining the long-range attraction by coalescence on approach created a new difficulty.3 As explained by the Kelvin equation through surface tension (γ) and curvature effects, bubbles in liquids are thermodynamically unstable. Under excess pressure because of the convex shape of the surface, the air dissolves in the water, leading to complete bubble annihilation. The dissolution proceeds with acceleration because the Laplace overpressure (2γ/r) increases with decreasing radius (r). Diffusion theory predicts that microscopic bubbles formed by snapping the meniscus should vanish in seconds in air-saturated water.18 The bubble interpretation of the surface force effects thus encounters difficulties when it is reconciled with basic physical notions of capillary thermodynamics and chemical kinetics. The classical concepts might require reconsideration through the results of the surface force measurements. The bubble presence postulate could be substituted for the hypothesis of large distances of cavitation onset. However, this alternative explanation of the bridging attraction is equally difficult to reconcile with fundamental principles of the kinetic theory of matter. Indeed, the phenomenological theory of nucleation and phase transitions19,20 predicts that critical distances of cavity nucleation should be shorter than 3 nm,21,22 while experiments showed that bridging is onset from distances of tens of nanom(16) Scientific Papers by Lord Rayleigh; Dover: New York, 1964; Vol. IV, p 430. (17) Yaminsky, V. V.; Yushchenko, V. S.; Amelina, E. A.; Shchukin, E. D. J. Colloid Interface Sci. 1983, 96, 301. (18) Epstein, P. S.; Plesset, S. M. J. Chem. Phys. 1950, 18, 1505. (19) The Scientific Papers of J. Willard Gibbs; Dover: New York, 1961. (20) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; Wiley: New York, 1997.
10.1021/la026122h CCC: $25.00 © 2003 American Chemical Society Published on Web 01/29/2003
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eters23 and even larger.3 By taking into account the quadratic, according to the surface area, dependence of the activation energy on the distance, the capillary phase transition clearly shown by such experiments cannot be explained by thermal cavity nucleation. Even with decoration,24 the theory has to be stretched far beyond physical sense in order to account for the discrepancy. We earlier have shown that the long-range forces and the large bridging distances observed in these experiments are not related to bubbles. Rather, the effects are due to imperfections of the hydrophobic coatings resulting in condensed material bridging.25,26 Ordinary capillary condensation of simple27 and complex28 liquids is associated with similar force effects. The attraction by constant volume meniscus25 explains the “double exponential” surface forces. The confusing observations have been rationalized by considering the bridging by the soft and mobile hydrophobic layers.26 It indeed has been acknowledged that “robust”29 hydrophobic surfaces do not show the ultra-long-range interaction.30 However, also for the surfaces that apparently are more stable and hard the attraction range far exceeds expectations of the Lifshitz theory. As suggested by these results, surface defects, which are progressively important in measurements at shorter distances and are of different origin according to the structural and chemical complexity of real hydrophobic surfaces, explain the data divertissement. Direct comparison with the forces between the same surfaces in air could shed more light on the origin of these effects. Our older paper shows that the long-range patterns of hydrophobic attraction are observed in dry conditions.31 The Casimir force, the strongest dispersion attraction that can be measured between metal surfaces in a vacuum, is shorter-range than the assumed hydrophobic forces.32 The latter are measured in low Hamaker constant systems for which dispersion forces are even weaker. In this report, we experimentally confirm that the interaction between strongly hydrophobic surfaces that are smooth and stable in water agrees with current theories of molecular and capillary interactions. The attraction operating before the cavity forms is several times shorter-range than hitherto measured in hydrophobic systems. It is indeed shorter-range in water than in air as it should be according to the Lifshitz theory. As required by the theory of capillarity for systems with obtuse contact angles, water is spontaneously expelled between the hydrophobic surfaces when they contact. The experimental results show that the distances of cavitation onset are shorter than 3 nm, as predicted by the classical nucleation theory. The nucleated cavity expands and by filling with air grows to submacroscopic sizes according to the high contact angles and significant solubility of air (21) Yushchenko, V. S.; Yaminsky, V. V.; Shchukin, E. D. J. Colloid Interface Sci. 1983, 96, 307. (22) Yaminsky, V. V.; Ninham, B. W. Langmuir 1993, 9, 3618. (23) Parker, J. L.; Claesson, P. M.; Attard, P. J. Phys. Chem. 1994, 98, 8468. (24) Lum, K.; Chandler, D.; Weeks, J. D. J. Phys. Chem. B 1999, 103, 4570. Huang, D. M.; Chandler, D. Phys. Rev. E 2000, 61, 1501. (25) Yaminsky, V. V. Colloids Surf., A 1999, 159, 181. (26) Yaminsky, V.; Ohnishi, S.; Ninham, B. Long-range hydrophobic forces due to capillary bridging. Handbook of Surfaces and Interfaces of Materials. Volume 4. Solid Thin Films and Layers; Nalwa, H. S., Ed.; Academic Press: New York, 2001; pp 131-227. (27) Ohnishi, S.; Yaminsky, V. V. Langmuir 2002, 18, 5644. (28) Yaminsky, V. V. Langmuir 1997, 13, 2. (29) Wood, J.; Sharma, R. J. Adhes. Sci. Technol. 1995, 9, 1075. (30) Yaminsky, V. V.; Ninham, B. W.; Stewart, A. M. Langmuir 1996, 12, 836. (31) Yaminsky, V. V. Colloids Surf. 1997, 129-130, 415. (32) Ederth, T. Phys. Rev. A 2000, 6206 (6), 2104.
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Figure 1. The interaction between two hydrophobic surfaces on the first approach in water (filled symbols) is shorter range than in air (open symbols). The zero-force contact occurs at a negative displacement because of the elastic squashing due to the adhesion. The contact deformation in water is slightly larger than in air because of the added capillary force. The Young modulus of fused quartz and the measured pull-off forces were used in the calculations using the JKR and Hertz equations (ref 36). The plane of origin of the van der Waals force is taken for the zero distance. It is shifted to the right from the zero force contact by the magnitude of the contact deformation. The estimated Hamaker constants are about 7 × 10-20 J in air and 2 × 10-20 J for the interaction across water. The straight lines whose slope equals the spring constant follow the tracks of the jump into contact. While adhesion is enhanced in the nonwetting liquid, the long-range van der Waals attraction is several times weaker in water than in air. The curve in water is corrected for the viscous drag that becomes noticeable at short distances just before the jump; the viscosity of water (1 cP), the mean sphere radius (1.55 mm), and the loading speed (100 nm/s over the free path) were used for this calculation.
in water. The bubbles left on breaking the bridging meniscus are unstable, showing the expected lifetimes. Depending on the relative air concentration with respect to the saturation at the given pressure, they vanish in a minute or grow, in accordance with the predictions based on the Kelvin equation and diffusion theory. Materials and Methods The surfaces that we use were prepared by Langmuir-Blodgett monolayer deposition and lateral polymerization with grafting through annealing the silicone groups of a long-chain alkyl fluorinated surfactant (heptadecafluoro-1,1,2,2-tetrahydrodecyltriethoxysilane) to a fused silica substrate, as described in ref 33. The surfaces were then soaked in ethanol to remove traces of the unreacted material. The advancing contact angle of water, measured in that work via the Wilhelmy method in the process of retraction of a similarly treated glass slide, for sessile droplets directly on the spheres used in these surface force measurements, is about 110°. The receding contact angle is slightly above 90°. Forces between the spheres (several millimeters in diameter) were measured with the Interfacial Gauge,30 in the modes of magnetic loading of the piezoelectric spring sensor and using a macroscopic range motor drive. The spring stiffness equals the slope of the linear acceleration track over the range of short distances of the jump into contact as shown in Figure 1. The resolution, according to the scatter data points due to the mechanical noise, is about 10-1 µN in force (corresponding to F/R of about 10-1 mN/m) and about 0.5 nm in distance. The accuracy is several times higher after averaging. Systematic calibration errors are within 5%. The force is shown in the plots without scaling with the radius. Such scaling generally is valid under quasi-static and chemical equilibrium interaction conditions that (33) Ohnishi, S.; Ishida, T.; Yaminsky, V. V.; Christenson, H. K. Langmuir 2000, 16, 2722.
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justify the Derjaguin approximation that, under such conditions only, does apply to capillary forces of small bridging menisci.34 To avoid hydrodynamic effects, the loading rate was adjusted to the level when the measured interaction across water stops changing significantly on further reducing the speed. Small theoretical corrections for the drag repulsion were then introduced, practically not changing the appearance of the curves. After the cavity was formed, measurements were repeated at higher and lower speeds to test dynamic response as discussed in connection with the results. The effective sphere radius for calculations of surface interaction energy, adhesion and contact deformation, contact angle, and hydrodynamic force was calculated from the main curvature radii of the surfaces measured with greater than 1% accuracy using the general form of the Derjaguin formula.34 The pull-off forces varied by no more than several percent in measurements at different contact positions. The measured adhesion, in agreement with expectations following from surface energy and contact angle considerations, proves that the surfaces are uniform down to the atomic level. We have measured forces between the surfaces in dry and moist air, in distilled freshly collected Millipore water that is air supersaturated, and after boiling it and cooling to room temperature. The surfaces usually were immersed slowly while separated, and the internal volume was sealed by slowly rising up the liquid-filled cell. The mode of immersion is not essential as long as the distance is sufficiently high to avoid trapping air between the surfaces. The interaction in water as described in this work showed no change over time following the immersion. After the surfaces were taken out of water, the interaction in air was measured again to ensure reproducibility indicative of preserved stability of the hydrophobic layer. Data in the figures were obtained at 22 °C; changing the temperature by a few degrees in both directions had no significant influence on the results.
Results Forces measured between the surfaces in air and on first approach in water are presented in Figure 1. The experimental curves show a weaker van der Waals attraction in water than in air. The curves readily can be fitted to the Lifshitz theory. Because attraction gradients in water are several times lower, the critical distance from which the surfaces jump into contact in water is shorter than in air. Equating the force gradient at the jump-in point to the spring stiffness is an independent, complementary way to measure the attraction. It should be taken into consideration that the distance measured in these experiments corresponds to the central displacement. Unlike the “surface-to-surface” distances measured by surface forces apparatus (SFA) interferometry, the distance measured from the point of zero force includes the normal component of contact deformation of the surfaces by the adhesion. The elastic compliance can be calculated from the pull-off force and directly measured on unload up to the point of separation; it already has been shown35 that the results agree with the JohnsonKendall-Roberts (JKR) theory.36 The contact squashing, by 1-2 nm, in water is by about 0.5 nm larger than in air because of an additional compression by the capillary force. The result shows that the actual distance between the surfaces at the jump-in point of mechanical instability where the gradient of the force exceeds the stiffness of the spring (256 N/m) is just 3 nm. The measurement results, including the jump-in distances and the preceding attraction, by taking into account the shift of the curves due to the contact deformation in relation to the measured (34) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155. (35) Ohnishi, S.; Yaminsky, V. V.; Christenson, H. K. Langmuir 2000, 16, 8360. (36) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301.
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adhesion, agree self-consistently with the Lifshitz and JKR theories. The effective Hamaker constant that follows from these measurements in water is several times lower than in air. The numerical values as given in the legend are reasonable. It might be premature at this stage to go into speculation, with explicit consideration for the layered structure of the surfaces by ascribing different Hamaker constants to the fluorocarbon monolayer and silica substrate, and a possible involvement of retardation effects. At distances comparable to the layer thickness, just below the retardation range, both corrections might be of theoretical interest. Our recent measurements of retarded van der Waals forces between mica surfaces in air show resolutions allowing such detailed studies.37 Intricacies of theoretical curve fitting might not be possible to resolve using the current set of data. This is not the object of our study. The result as it is shows that explaining the attraction would not require new theoretical assumptions other than those already provided by the Lifshitz theory. Without invoking an additional assumption regarding interaction retardation through instability of bridging nodes above subcritical Rayleigh distances, the nucleation theory allows for the critical cavitation to be preceded by a density fluctuation attraction of the same form as the nonretarded dispersion force, albeit at a higher value of the Hamaker constant, close to that for the interaction across air.22 However, the experimental result shows that the strength of the interaction in water remains weaker than in air. The measured curves for the interaction across water show no coalescence discontinuities down to the short distance of van der Waals jump into contact. It is clear from the results that the jump itself is not caused by cavitation. The mechanical instability is a consequence of the observed force law. The same way as for the measurement in air, the attraction gradient reaches the critical value of the spring stiffness at the point of the jump, in continuation of the preceding part of the curve explicitly measured on the side of larger distances. The force profile in water is just an ordinary van der Waals form. This means that even at such short distances the surfaces remain separated by water, that is, embedded in a condensed continuum. The film of pure water between the two strongly hydrophobic surfaces does not break, even though the gap width at the jump-in point is just 10 times the diameter of water molecule. The cavity forms after the jump, at distances shorter than a few nanometers as the nucleation theory predicts. During the contact period, a fraction of a minute in these experiments, it grows and can be observed in a microscope. In the following cycles of separation and approach over a micron range, the interaction shows a hysteresis loop (Figure 2). The attraction, stronger on separation and weaker on approach, changes between the two levels determined by the advancing and the receding contact angles on changing the direction of the motion. In a few subsequent cycles of a period of half a minute following the first contact, the interaction continues to change slightly. The noticeable trend in systematic evolution of the loop can be explained by continued diffusion of air into the cavity. In a couple of minutes the force-displacement track stabilizes and hysteresis loops become reproducible, repeated between the cycles down to fine details. The result shows that a steady state of cyclically modulated air saturation is reached by that time. Speed dependencies resulting from pressure regulation (37) Stewart, A. M.; Yaminsky, V. V.; Ohnishi, S. Langmuir 2002, 18, 1453.
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Figure 2. The capillary vaporization transition is followed by cavitation leading to the capillary interaction. The figure shows the first approach of the surfaces across water and the hysteresis loop of the following cycle of the outward and inward motion with the jumps in and out of contact. On the first approach, the force remains zero down to the short distances that are not resolved on the distance scale of this figure. The first approach is enlarged in Figure 1, which shows details of entering the contact. The van der Waals attraction is onset before the cavity forms after the jump-in or on contact. The constant rate of magnetic load and unload corresponds to the same approach and separation speed. The force changes between the advancing and receding levels on changing the direction of motion because of the contact angle changes. After the transformation between the advancing and the receding contact angles has occurred on the steep parts of the curves, the force continues to change slightly with distance. It decreases by air decompression on the separation and increases by compression on the approach, allowing the upper curve to enter the repulsive range slightly above the zero level before entering contact. This pressure regulation modulates the static curves as can be noticed by changing the speed. The incurred small negative slope on both plateaux of the loop enhances the similarity of the surface force diagram with ordinary wetting tension vs depth of immersion charts measured with the Wilhelmy plate; in the latter case, qualitatively similar slopes of the receding and advancing traces are due to the buoyancy.
and dynamic contact angle effects are weak in measurements at this frequency. When the surfaces are separated with the motor, the force decreases many times on the long way out of contact (Figure 3). At a distance of 49 µm, it reduces to zero in a step. The meniscus breaks at this point, leaving two bubbles of roughly 20 µm in diameter adhered to the surfaces. In measurements using the preboiled water, the bubbles remained visible in the microscope for half of a minute. They were reducing their size in time with acceleration, at the beginning slowly, so that the process is not immediately obvious, and by the end more rapidly, disappearing suddenly out of view. Similar measurements were repeated with water that was not boiled, just stirred for a while before immersing the surfaces to reduce the existing supersaturation to a level that avoided air nucleation on the open immersed areas facing the solution, out of their mutual contact. The dispersion attraction on the first approach, contact cavitation, and subsequent capillary attraction show practically the same patterns as in the air-undersaturated water. The meniscus breaks at roughly the same distance of about 50 µm. The bubbles left are of almost the same size as in the air-undersaturated water. However, once the meniscus breaks, the bubbles do not perish but slowly
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Figure 3. On separation of the surfaces with the motor at a speed of 0.4 µm/s, the meniscus breaks by the Rayleigh instability from a distance of 49 µm. Here the force reduces to zero, as shown also on the enlargement. The attraction decreases more rapidly with increasing distance than theoretically predicted for the interaction at constant Laplace pressure (ref 21) because of the air decompression that occurs by increasing the meniscus volume on starting the separation. The effect is more pronounced in this case because of the higher measurement speed. If the surfaces are stopped halfway, relaxation of the force to the chemical equilibrium value is observed for about a minute.
grow, reaching in about half an hour a millimeter size. When the surfaces are immersed in freshly collected Millipore water, the entire surface becomes covered in minutes with air bubbles obstructing force measurements. The supersaturation, if not relieved in advance by intense stirring and prolonged storage, is higher in this case. Discussion While the results are almost self-explanatory, some features of these observations might be worth further comment. We set first of all the condition of chemical and mechanical equilibrium in the form γ ) KC - P0. Here γ is the surface tension; ) 1/r1 + 1/r2, where r1 and r2 are the main radii of curvature; C is concentration of the dissolved gas; and KC is the equilibrium pressure in the void, according to the distribution coefficient (K ) P/C). For a bubble (r1 ) r2 ) r), chemical equilibrium is unstable. On the other hand, a bridging cavity in coexistence with the surrounding water containing dissolved air is in the state of stable equilibrium. The surface of the meniscus retains nonzero, net negative or positive curvature, depending on whether the water is undersaturated (C < P0/K) or supersaturated (C > P0/K) at the given atmospheric pressure (P0). In a system free of contact angle hysteresis (θ ) const), with concentration changing around solubility (P0/K) or, equivalently, with external pressure changing at fixed concentration, the equilibrium meniscus volume is smaller or larger than that of the zero-curvature meniscus coexisting with air-saturated water. In systems with hysteresis, if the acting contact angle occurs in the interval between the advancing and receding values, the meniscus flexes on changing the pressure difference without changing the three-phase line position. Once the advancing or receding value is reached, the wetting front starts moving. Dependence of the contact angle on the three-phase line speed can be worked out experimentally. Pressure changes due to mass transfer across the meniscus surface can be calculated from diffusion coefficients in the same way as for a single bubble,18 by taking into account the confine-
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ment geometry. The self-consistent equation of motion constructed this way, with known parameters and boundary conditions, describes the cavity evolution in response to changing separation and/or external pressure. Further, by taking into account water viscosity and inertia, the description can be extended toward high-speed effects, such as the initial stage of vacuum cavity expansion. In systems with obtuse receding contact angles, the negative wetting tension (τ) applied at the perimeter of the cavity forming on contact by spontaneous nucleation is unbalanced. The capillary pressure (2τ/h) is initially very high, according to the small meniscus height, on the order of a nanometer for the critical cavity. The vacuum cavity expands rapidly in inertia/viscosity-controlled mode. The liquid shoots out of the gap in milliseconds, like in the similarly rapid process of capillary rise. Mechanical equilibrium (τ ) γ cos θ) is reached when the capillary pressure reduces to 1 atm. In the subsequent steady growth, the cavity fills with air in about a minute, according to the diffusion rates of air in the water. The state of chemical equilibrium is reached at ∆P ) KC - P0. In systems with acute receding contact angles, cavitation does not proceed spontaneously on contact because of the effectively receding conditions in the process of surface approach. The closing contact wedge remains filled with water until the beginning of unload when a vacuum starts forming in the opening slit between the separating surfaces. The acting pressure difference that forces water to advance changes from -1 atm to around zero in the course of air saturation. Whatever the situation, the kinetics of cavity evolution follows from the self-consistent condition of balance of the capillary pressure and the air pressure difference. In the measurements with the magnetic load (Figure 2), relative elongation of the meniscus, by moving the surfaces in and out over a micron range, is just a few percent of the full stretching range of the meniscus to the point where it breaks (Figure 3). With the relative volume changes remaining small, the pressure modulation amplitude is small, even in the constant air mass highfrequency regime. Because the measurement period is on the order of characteristic times of air diffusion, about a minute as is experimentally observed in accordance with theoretical results,18 the pressure oscillation is further damped. Wetting equilibration is also quite rapid. Relaxation times over which the contact angle changes from dynamic to the static advancing or receding values, measured with a Wilhelmy balance, are on the order of a minute. These measurements, not complicated by pressure regulation, show the three-phase line dynamics in pure form. It should be emphasized that the three-phase line translation paths associated with the same contact angle changes are much shorter for the confined meniscus. According to the microscopic cavity height (h), the distance ranges over which the contact angle changes between the receding and the advancing values are just a fraction of a micron, as shown in Figure 2, compared to a fraction of a millimeter for the macroscopic meniscus on the plate. In the D , h limit of much smaller distances (D), the force is close to the contact (D ) 0) value 2πRγ cos θ. The expression, exact in the h , R limit, remains valid for interaction in chemically nonequilibrium conditions.21,34 Over the short distance interval of the measurement in Figure 2, the force, once contact angle stabilized, changes by just several percent. The D , h condition justifies an approximate 2πR scaling of the force with the radius (R) of the surfaces. The contact angle (θ) estimated from the data in Figure 2 using the known surface tension of water
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(γ ) 72 mN/m) and the sphere radii (R ) 1.55 mm) alternates between the advancing value of 110° on separation and the receding value of 90° on approach. The result agrees with the previous Wilhelmy balance measurements.33 Unlike for the low-gradient capillary force, of submacroscopic range according to the meniscus size, the van der Waals attraction of the surfaces across the cavity, similar to that for measurements in air, steeply changes at short molecular-scale distances. The almost hard-wall contact repulsion is even steeper, much stiffer than the measuring spring. The residual compliance is due to the combined effect of adhesion and elasticity. As in the experiments in air, when the applied negative load reaches the critical value of the pull-off force the surfaces jump apart in inertia mode. The linear track at the slope of the spring constant follows to the distance at which the meniscus force, of lower differential stiffness, balances bending of the spring. In the provided condition of mechanical balance, the subsequent separation proceeds in the steady regime of maintained mechanical equilibrium. In the fast separation shown in Figure 3, the distance increases in about a minute on the order of the meniscus height. Under such conditions, the decrease of pressure is quite substantial, especially at the beginning of separation. On the small distance side, the force changes faster with separation compared to theoretical predictions for interaction in the ∆P ≈ 0 mode.21 The diffusion equilibration speeds up following the confinement relief on opening the gap and reducing the volume on narrowing the meniscus. The interaction proceeds closer to equilibrium conditions, leading to the expected more gradual decrease of the force with increasing D up to the rupture point. The critical distance, about 50 µm, at which the meniscus breaks, is indeed on the order expected for the ∆P ≈ 0 cavity, according to the given surface radius and the advancing contact angle. By taking into account that the instability occurs when the width of the meniscus occurs on the order of its height, the measured critical elongation agrees as well with the observed sizes of the bubbles forming after the collapse. Originally about 20 µm in diameter, the bubbles contain the same amount of air as the breaking meniscus of which they form, albeit at a slightly increased (by about 2γ/r) pressure, with the curvature changing to net positive. Provided the air concentration in water is below the point of Kelvin coexistence, P ) P0 + 2γ/r > C/K, the bubbles undergo Ostwald dissolution, even if P0 < C/K, that is, water remains supersaturated with respect to the atmosphere. On the other hand, if the supersaturation exceeds the critical level for the given radius, the bubbles grow macroscopic. The observed criticality of the bubble behavior is a consequence of the fact that the capillary equilibrium is in this case chemically unstable. Unlike for bubbles, for the bridging meniscus chemical equilibrium is stable. The meniscus volume and accordingly that of the bubbles into which it breaks increase with increasing absolute concentration. With relative changes remaining small while concentration varies around the solubility, the postcollapse bubbles have roughly the same original volume. Whether CK/P0 is slightly smaller or greater than unity, even if P > KC, allocation of the system with respect to the coexistence boundary is not critical for the cavity behavior. Similarly, unless in periods many times shorter than a minute, changing of dynamic conditions of the measurement has a relatively small effect on the volume. The experiment shows that the critical cavitation condition is fulfilled, as the theory predicts, at molecular
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distances. As follows from surface energy considerations, the critical nucleus energy for the bridging cavity is about γh2/cos θ.22 The result showing no criticality of the solute effect, if any effect at all, is anticipated. Indeed, the surface tension of water does not change upon evacuation. The two major air components, nitrogen and oxygen, nonpolar gases well above their critical temperatures, do not adsorb at the water-vapor interface. Also the contact angle does not change because for all the three coexisting interfaces air adsorption pressures are insignificant. By considering mechanically reversible water-oil (e.g., pure liquid hydrocarbon) systems, for which each of the three tensions can be directly measured, this result is easy to prove with very high accuracy. The physical meaning of this wellknown experimental fact can be readily illustrated theoretically. A simple, self-consistent consideration can be based on the mass action law. The thermodynamic transfer functions required for this calculation, that is, the free energies of adsorption and dissolution, are experimentally known from solubility measurements and can be theoretically estimated, for example in the form of molecular surface energies (γvm2/3). The values of adsorption for the zero cohesion supercritical gas solute in the low-pressure limit practically equal zero. The same basic principle of statistical thermodynamics is extended in the kinetic aspect of the nucleation theory. In the Boltzmann-Arrhenius exponent that describes the rate of nucleation, the prefactor is roughly the number of confined molecules (h/vm ≈ 1016 per unit area of the film) times the vibration frequency (1012 s-1). The nucleation rate is expected to become physically significant (e.g., 1 s-1) on reducing -γh2/cos θ to about 30 kT.22 Even in the extreme θ ) 180° case, which corresponds to zero adhesion (water film surrounded by vapor), this condition for water at room temperatures (γ ) 72 mN/m, kT ) 4 × 10-21 J) is fulfilled at distances of a few nanometers. Given the small critical volume (h3 on the order of a cubic nanometer), the PV contribution is insignificant on the background of the surface energy term. Changing the external pressure and/or air dissolution in the vacuum cavity practically do not influence the nucleation condition. By comparison, for a bubble nucleus in bulk water the free energy can be similarly reduced only at the expense of the volume PV term. Taking γs/3 on the order of tens of kT units requires reducing the critical Kelvin radius on the order of a nanometer. This demands manyfold supersaturation or, in the boiling context, overheating water well above the boiling point, with rising temperature close to the critical. The values of supersaturation involved in our experiments, according to typical values of the acting Kelvin radius on the order of 10 µm, are just several percent. For a capped sphere segment in sessile bubble geometry, with contact angles around 100°, the surface energy is several times lower compared to a full sphere of the same radius. According to this result, triggering surface nucleation requires lower supersaturation as is experimentally observed. However, at further lower supersaturation nucleation is not possible because the Kelvin radius diverges at saturation while the condition of complete nonwetting physically is not possible. Because of water adhesion, contact angles are well below 180°, even for most hydrophobic surfaces. For air, both cohesion and adhesion at room temperatures are effectively zero; that is why there is no air adsorption. On the other hand, cavitation on contact is met with no energy barrier. The cavity nucleates on contact even in water that does not contain dissolved air. The role of microscopic cracks, scratches, and other surface defects
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of the container walls in triggering boiling is well-known from daily observation. Our experiment shows the phenomenon in the pure, model form allowing quantitative study. The fact that the bubbles do not nucleate at open areas at mild supersaturation provides one more independent proof that the surfaces used in our experiments are free of defects. Interestingly, like the cavitation itself, the preceding van der Waals attraction does not depend on air saturation. The van der Waals form experimentally is resolved over a short distance interval before the jump into contact. It is the same, as we have shown, in the measurement on the first approach of the surfaces after the submersion in the undersaturated water and after breaking the meniscus and allowing about 1 min for the bubble dissolution before starting the new approach. In the experiments repeated using air-supersaturated water, the interaction on the first approach displays the same pattern as observed in the air-undersaturated (preboiled) water. This observation shows that whether the concentration of air dissolved in the water is above or below the saturation value P0/K, this does not influence the interaction. Of course, the result showing no criticality of the solute effect, if any effect at all, could be anticipated. The van der Waals interaction, according to the low solubility of air in water and practically zero adsorption, is not changed significantly. We confirm this theoretically expected result by the experiments. Cavitation from molecular distances or on contact proceeds spontaneously even for water that does not contain dissolved air. In the context of capillary cavitation, distance plays essentially the same role as air supersaturation or overheating in bubble nucleation in the bulk liquid and at single surfaces. For water films surrounded by fluid (vapor or oil), cavitation is substituted for coalescence. In consideration of asymmetric water films on solid and liquid substrates, essentially the same wellknown phenomenon takes the form of dewetting. Conclusions Experimental observations of the interaction of hydrophobic surfaces in water closely follow theoretical expectations. This concerns all major, thermodynamic and kinetic aspects of surface force and phase transition behavior. We summarize here the main points: (1) The results show that the interaction of the hydrophobic surfaces across water is explained by the Lifshitz theory of van der Waals forces. The attraction, several times weaker than in air, is typical of low Hamaker constant systems. (2) On approach, before the surfaces contact, continuity of the water layer is preserved down to distances of a few nanometers. The thin film is stable in the hydrophobic confinement in accordance with the classical nucleation theory. (3) As expected for systems with obtuse contact angles, a vapor meniscus surrounds the surfaces around contact. The nucleated cavity fills with air in a minute, in agreement with the diffusion theory. (4) The Young and Laplace equations, through the imposed contact angle and pressure regulation conditions, define the meniscus geometry and the force depending on separation, time, speed, and acting air concentration. (5) In cyclic measurements with preservation of the meniscus continuity, the capillary force shows a hysteresis loop. The force is stronger on separation and weaker on approach, according to the higher advancing (110°) and
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the lower receding (∼90°) contact angles. In short-period measurements, the loop widens following the dynamic contact angle and pressure changes. (6) The long-range capillary force abruptly reduces to zero on separating the surfaces 50 µm apart. At this point of mechanical instability, the meniscus breaks into two bubbles. Originally of about 10 µm radius, they vanish in a minute through Ostwald dissolution in air-saturated water, while in water supersaturated above the point of Kelvin coexistence, they grow toward macroscopic sizes. (7) Surface forces measured on the first approach and nucleation conditions are similar in under- and supersaturated water, according to the low solubility of air in water and practically zero adsorption. Hydrophobic phenomena can be comprehensively understood and rationally explained by the Lifshitz theory and the theory of capillarity, with time- and historydependent results extended in the standard context of chemical kinetics and thermodynamics. Examples of more complex systems studied over the past two decades, showing variable ranges of capillary attraction associated with condensed material bridging, can be found in the literature.3 We have considered and explained these observations in a comprehensive review.26 Of future developments suggested by this study, a
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wider variation of measurements conditions is clearly of interest. Among physical, chemical, and mechanical factors that can be varied are temperature, pressure and the extent of gas saturation, rates and ranges of approach and separation, and contact and out-of-contact duration. Thorough control of air concentration is particularly of interest in measurements with water and other liquids. Experimenting with mercury already invokes different limiting regimes. On the basis of the experimental results and advanced computational techniques, one should be able to obtain nonstationary and stationary, time-resolved solutions of capillary equations for the pressure regulation and contact angle hysteresis in dynamic hydrophobic cavities. The classical models and phenomenological theories of basic physical chemistry account in rigorous terms for the phenomenon of capillary vaporization. The first-order phase transition is at the core of the theory of boiling. Without casting doubts regarding the fundamental physics behind the hydrophobic effects, the problem is technologically and biologically important. LA026122H
