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Langmuir 1996, 12, 1693-1695
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Comments Bridging Bubbles between Hydrophobic Surfaces
The intimate connection between bubbles and hydrophobic surfaces is due to the fact that it is unfavorable for water to be in contact with the latter. Water droplets on hydrophobic surfaces, by definition, subtend a large contact angle at the three-phase line, and one expects air bubbles in water to preferentially adhere to hydrophobic surfaces. One also expects capillary evaporation of the water between two closely spaced hydrophobic surfaces. The relationship between bubbles and hydrophobic surfaces is of particular interest in the light of the recent suggestion by Parker, Claesson, and me1 that submicroscopic air bubbles bridging between two macroscopic hydrophobic surfaces could be responsible for the long range attractions that have been measured. The experimental evidence for such a claim was the observation of steps or discontinuities in the measured forces at separations on the order of 100-300 nm.1 The viability of the putative mechanism relies upon the fact that such vapor bridges are stable (or metastable) at submicroscopic dimensions, and that these do not spread along the surfaces to microscopic (i.e. observable) sizes. Furthermore, for bridging bubbles (as distinct from spontaneously induced bridging cavities), it is necessary that submicroscopic bubbles pre-exist (i.e. are metastable and long-lived) on an isolated hydrophobic surface. This comment is concerned with the mechanical and thermodynamic stability of bubbles on and between hydrophobic surfaces. The stability of such bubbles has been questioned by Eriksson and Ljunggren,2 who in commenting upon the mechanism proposed in ref 1 for the long-ranged attraction between hydrophobic surfaces state that (i) a bubble on an isolated hydrophobic surface has a free energy maximum at the bulk contact angle and (ii) submicroscopic bridging cavities are mechanically unstable because they don’t have the bulk contact angle. Although the mathematical analysis of ref 2 is valuable since it confirms and clarifies the earlier numerical results, these two conclusions differ from those of ref 1. The stability of bubbles in contact with surfaces is most easily understood in terms of the thermodynamic stability of bubbles in bulk water. At the heart of the matter is the appropriate ensemble to use, and for fixed temperature one has a choice of two: the grand canonical ensemble (fixed chemical potential and volume) and the isobaric ensemble (fixed number of gas molecules and external pressure). (The ensemble defined only be the intensive variables, chemical potential, pressure, and temperature, is in general of indeterminant size unless surface terms are present; in this case the surface free energy would cause the bubble to shrink to zero.) The free energy is a function of the radius of the spherical bubble, R, and at the level of macroscopic thermodynamics one has for the grand potential
Ω[R] ) -∆pV + γlvA
(1)
where γlv is the liquid-vapor surface tension, and where the pressure drop ∆p ≡ pin - pout is fixed by the chemical potential. Setting the derivative to zero, one obtains the (1) Parker, J. L.; Claesson, P. M.; Attard, P. J. Phys. Chem. 1994, 98, 8468. (2) Eriksson, J. C.; Ljunggren S. Langmuir 1995, 11, 3325.
familiar Laplace-Young equation
R0 )
2γlv ∆p
(2)
However the second derivative, Ω′′[R0] ) -8πR0∆p + 8πγlv < 0, indicates that the extremum is a maximum. One concludes that bubbles comprised of gas in diffusive equilibrium with the atmosphere or other reservoir are thermodynamically unstable. The Gibbs free energy of the isobaric ensemble may be obtained by a Legendre transform from the canonical ensemble
G[R] ) poutV + γlvA + NkBT ln NΛ3/V
(3)
where the last term is the Helmholtz free energy of an ideal gas. Optimizing with respect to R, and using the ideal gas equation of state, pin ) NkBT/V(R), one again obtains the Laplace-Young equation. The second derivative yields G′′[R0] ) 8 R0pout + 8πγlv + 3NkBT/R20 > 0, which indicates that the extremum is a minimum. Hence bubbles are stable in the isobaric enzemble, which was the point of appendix B of ref 1. The fact that one actually observes bubbles in water indicates that the isobaric ensemble must be the appropriate one and that the encapsulated air cannot be in diffusive equilibrium with the atmosphere over macroscopic, measurable timescales. It is possible that for small, high-pressure bubbles the rate of diffusion is greater than that for microscopic bubbles, but a quantitative estimate of their lifetimes appears difficult. I shall return to this point below. The analysis of the stability of a bubble in contact with a hydrophobic surface differs only marginally from that of the bulk. In this case the free energy contains an additional contribution from the surface free energy of the solid. The radius is again given by the Laplace-Young equation, and the complement of the angle subtended by the bubble equals the bulk contact angle, which is given in terms of the surface energies by Young’s equation. (The bulk contact angle is measured through the liquid phase and is that at a straight three-phase line due to the mutual intersection of three planes of two phase coexistence. In this case Young’s equation can be obtained by simple force balance considerations, but this is not so straightforward for a curved interface, see below.) Young’s equation is
cos θlsv )
γsv - γsl γlv
(4)
where the numerator is the difference between the surface energies of the solid in contact with the vapor and liquid, which is negative for a very hydrophobic surface. One finds that it is the Gibbs free energy that is minimized when the bubble radius and contact angle satisfy the Laplace-Young and the Young equation, respectively.1 In contrast to this result, Figure 4 of ref 2 shows a local maximum in the energy when the bubble contact angle equals the bulk contact angle. The main reason for this discrepancy is that Eriksson and Ljunggren2 have used the grand potential rather than the Gibbs free energy. As discussed above and in ref 1, the isobaric ensemble is the appropriate ensemble for bubbles. Young’s equation corresponds to a minimum in the Gibbs free energy, but
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Figure 1. The relative change in the radius (upper) and the change in the Gibbs free energy (lower) compared to a free spherical bubble when it adsorbs to a surface, as a function of the surface’s energy, cos θslv ) (γsv - γsl)/γlv. Here γlv ) 72 mN/m, NkBT ) 2.7 × 10-4 J, and pout ) 1 × 105 Nm-2, which for a spherical bubble in the bulk correspond to R0 ) 200 nm and G0/NkBT ) 46.3.
just as for a bulk bubble, the grand potential here is a maximum and the adsorbed bubble is unstable in the grand canonical ensemble. What is also evident in Figure 4 of ref 2 is the local minimum in the grand potential at a contact angle of π/2. This can be easily understood as the angle that gives the maximum area of vapor-solid contact (for a fixed radius), which is favorable for a hydrophobic surface. Similar behavior occurs for submicroscopic bridging cavities, as discussed below. In Figure 1, I have plotted the change in the Gibbs free energy when a particular bubble is adsorbed onto surfaces with energies corresponding to contact angles up to 120°. It can be seen that the energy decreases and that the radius increases as the surface becomes increasingly hydrophobic. The adsorbed bubble is clearly stable with respect to being in bulk water. (Evidently, in this model, bubbles also stably adhere to low contact angle hydrophilic surfaces, but in practice the surface charge generally associated with such surfaces, which is not accounted for in the present calculations, would likely prevent adhesion from occurring.) As stated in ref 1: “Given that submicroscopic bubbles are present in the bulk water, then they preferentially segregate to the surfaces and adhere to them; the energy of a bubble on the surface is much lower than a free spherical bubble because the costly liquidvapor and solid-liquid interfaces are replaced by the less unfavorable vapor-solid contact.” This conclusion contrasts with that of Eriksson and Ljunggren2 who state that “no stable bubbles can form and adhere to hydrophobic surfaces”. The above discussion was concerned with air bubbles, and now I turn to bridging cavities. It turns out that for surfaces with high contact angles the grand potential of a bridging water-vapor cavity is negative, which implies that over a range of separations such cavities can be spontaneously induced in the water interlayer and are stable (capillary drying). (Since the gas in these induced cavities is just water vapor, the grand potential and not the Gibbs free energy is in this case appropriate.) Cavities of microscopic dimensions have been observed (see ref 1 and references therein), and the force between the surfaces
Comments
due to microscopic bridging cavities has been analyzed.1,3,4 (Ducker et al.5 have also given a rudimentary analysis of these microscopic cavities and their possible relation to the attractive forces between hydrophobic surfaces; their analysis has been critically discussed by Eriksson and Ljunggren.2) What I did with Parker and Claesson1 was to show that for a range of separations there was a local minimum in the grand potential for cavities of submicroscopic size. In the context of the long-ranged hydrophobic attraction this was significant because it accounts for the fact that cavities have never been observed prior to contact (they surely would have been seen if they were able to grow to microscopic size). As pointed out in ref 1, these submicroscopic cavities do not contact the solid surfaces with the bulk contact angle, but rather with an angle that locally minimizes the area of the vapor-solid interface (equivalent to the minimal convexity of the liquid-vapor interface) as a function of the radius of the cavity waist. (The situation is analogous to that discussed above, where a hemispherical bubble of fixed radius on an isolated hydrophobic surface has a local minimum in the grand potential for θ ) π/2.) Eriksson and Ljunggren2 confirmed the possibility of these metastable submicroscopic cavities, and they also confirmed that the contact angle did not equal the bulk contact angle and that the contact area was a minimum as a function of the radius of the cavity waist at this point, all of which was clearly stated in the final paragraph of appendix A of ref 1. However Eriksson and Ljunggren2 conclude that these submicroscopic cavities are mechanically unstable, because the surface tension forces are not in balance at the surface. However, the only criterion of mechanical stability is that the energy should be a minimum. The identification of forces with derivatives of particular contributions to the total energy, and their balance at equilibrium, is only straightforward in simple cases, in particular when there are no constraints. In the case of a bridging cavity, the geometry of the cavity is constrained by the Laplace-Young equation. Hence, the variation in the total energy due to a change in the interfaces at the solid surface is not equal to the surface tensions times the change in the interfacial areas. In other words, one cannot identify the minimum in the total energy with a local balance of surface tensions because it is not possible to make solely a local change in the geometry if one invokes the Laplace-Young equation for the liquidgas interface. (This comment obviously does not apply to a bubble or drop on a surface, since here one can minimize the energy by independent variations in the radius and the contact angle.) In conclusion, the present analysis and that of ref 1 indicates that (i) bubbles preferentially adhere to surfaces (compared to being free in the bulk) and that this is a thermodynamically stable state provided that the gas is not in diffusive equilibrium with some reservoir and (ii) it is possible for sub-microscopic metastable cavities to bridge between hydrophobic surfaces, and these are mechanically stable. As a mechanism that accounts for the long-ranged forces that are measured between hydrophobic surfaces, bridging bubbles or cavities are attractive, and they appear to be the only possibility that can account for the steps or discontinuities in the measured (3) Yaminsky, V. V.; Yuschenko, V. S.; Amelin, E. A.; Shchukin, E. D. J. Colloid Interface Sci. 1983, 96, 301. (4) Yuschenko, V. S.; Yaminsky, V. V.; Shchukin, E. D. J. Colloid Interface Sci. 1983, 96, 307. (5) Ducker, W. A.; Xu, Z.; Israelachvili, J. N. Langmuir 1994, 10, 3279. Note that when the penetration of the silica particle into the bubble due to the finite contact angle is taken into account, the range of the attractions measured in this paper differs little from that predicted by DLVO theory.
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Comments
forces of ref 1. By providing a mechanical link between the surfaces, the range of the measured force is related to the size of the bubbles and not to molecular interactions, and one avoids having to postulate polystructural effects extending thousands of water molecules away from a surface,6 which is the mechanism favored by Eriksson and Ljunggren.2 In comparing the two possibilities (bubbles or cavities), an argument against the latter is that the energy of forming submicroscopic cavities from bulk water is positive,1 which means that they will not form spontaneously. Microscopic bubbles have lifetimes comparable to those of the measurements, and because the diffusion times are so long, submicroscopic bubbles would probably not be removed by conventional deaeration treatments. Submicroscopic air bubbles that are formed in water by stirring or pouring preferentially segregate to hydrophobic surfaces and give an attractive force with a range comparable to their diameter. Whereas cavities are specific to hydrophobic surfaces, as are the measured forces, one needs to (6) Eriksson, J. C.; Ljunggren, S.; Claesson, P. M. J. Chem. Soc., Faraday Trans. 2 1989, 85, 163.
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augment the present thermodynamic analysis with an argument against bubbles adhering to hydrophilic surfaces, which would produce a force of similar range unless they were forced from the contact region as the two surfaces came together. In my opinion, of the various possibilities that have been proposed,1,6-8 bridging bubbles or cavities represent the most viable mechanism for the long-ranged attractions measured between macroscopic hydrophobic surfaces in ref 1. Phil Attard
Department of Physics, Faculty of Science, Australian National University, Canberra, ACT, 0200, Australia Received October 13, 1995 In Final Form: December 1, 1995 LA950866W (7) Attard, P. J. Phys. Chem. 1989, 93, 6441. (8) Be´rard, D. R.; Attard, P.; Patey, G. N. J. Chem. Phys. 1993, 98, 7236.
