Langmuir 1993,9,3618-3624
3618
Hydrophobic Force: Lateral Enhancement of Subcritical Fluctuations Vasili V. Yaminsky’ and Barry W. Ninham Department of Applied Mathematics, The Australian National University, Canberra, ACT 0200, Australia Received June 11,1993.I n Final Form: September 17,1999 The approach of two surfaces in a poorly wetting liquid (finite contact angles) is shown to lead to a decrease of the density of the liquid in the gap due to enhancement of thermal fluctuations in the lateral direction. A simple treatment based on classical nucleation theory shows that the decompression effect determined by the boundary conditions and the molecular discontinuity of the medium is essentially long range. The phenomenon accounts for the experimentallyobserved magnitude and range of the hydrophobic attraction and other deviations from Lifshitz theory, as well as for the much shorter range of the repulsive hydrophilichydration force. The triggering of the basic mechanismsof phase transition via critical cavitation either at the interfaces of thick liquid films or across the gap at small molecular distances between solid surfaces is substantially dependent on the wetting hysteresis. The boundary decompression-compression transition caused by surfactant adsorption is important in preventing rupture of thin soap films. The hydrophobic effects of nonpolar solutes are briefly discussed. Introduction
The dispersion force1 caused by electromagnetic fluctuations is usually considered as the universal source of long-distance intermolecular interactions in vacuum and condensed media. The general theoretical account of the effect in terms of electromagnetic properties of the interacting phases was given by L i f ~ h i t z .The ~ ~ ~macroscopic theory based on the continuous medium approximation is expected to be rigorous provided the distances are large compared to atomic (molecular) dimensions. Even at shorter distances the theory is applicable if finite molecular sizes are taken into account. It gives reasonable results by extrapolating down to the shortest intermolecular separations. A well-known example is the consistency of the values of the surface energy of nonpolar molecular phases with the values of the Hamaker-Lifshitz constants. When shorter range nondispersion attractive forces contribute to the surface energy of polar or metallic liquids and solids, the theory gives a reasonable estimate of the dispersion ~omponent.~ The validity of the theory in predicting the interaction at large distances was hardly disputed for decades. A perplexing challenge to the theory came from measurementa of surface forces between hydrophobized solids in aqueous media.”ll In several cases the long-range component of the attraction exceeded the predictions of Lifshitz theory by 1or even 2 orders of magnitude. Even for typical dispersion systems the theory fails to predict a Abstract published in Advance ACS Abstracts, November 1, 1993. (1) Mahantv, J.: Ninham, B. W. Dispersion Forces: Academic Press: New York, 19i6. (2) Lifshitz,E. M. Sou. Phys.-JETP (Engl. Transl.) 1966,2,73. (3) Devaloshinskii.. I. E.:. Lifshitz. E. M.: Pitaevskii. L. P. Adu. Phvs. 1961; 10,k65-209. (4) Fowkes, F. M. Ind. Eng. Chem. 1964,56,40. (6)Ieraelachvili, J. N.; Pashley, R. M. Nature 1982,300, 341-342. (6) Israelachvili,J. H.; Pashley, R. M. J. Colloid Interface Sci. 1984, 98,600. (7) Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Evans, D. F. Science 1986,229,10~38-1089. (8) Claeseon, P. M.; Blom, C. E; Herder, P. C.; Ninham, B. W. J. Colloid Interface Sci. 1986, 114, 234. (9) Christenson, H. K.; Claesson, P. M. Science 1988,239, 390. (10) Rabinovich, Ya. I.; Derjaguin, B. V. Colloids Surf. 1988,30,243. (11) Parker,J. L.; Cho, D. L.; Claesson,P. M. J. Phys. Chem. 1989,93, 6121.
even the sign of interaction in the range of large film thicknesses when it is supposed to be rigorous: this is indicated by finite contact angles of high-boiling-point nonpolar liquids in equilibrium with condensed multilayers at high-energy substrates.12 The assumption that the molecular granularity of the medium can be ignored at comparatively large distances is generally not obvious. In particular, fluctuations in molecular density which occur at finite temperatures may give rise to long-rangeeffects similar to the electromagnetic effects associated with electric charge density fluctuations. The fluctuations of mass distribution become especially pronounced near critical conditions for phase transitions. A typical phase transition caused by capillary effects is the nucleation of a macroscopic vapor cavity due to the contact between two surfaces in a nonwetting liq~id.l3-~6 The application of classical nucleation theory16 to a hydrophobic gap shows that subcritical cavitation can give rise to a long-range attractive force. The formalism developed here is based on general principles, and using known macroscopic parameters links together and accounts for a number of separate observations in a simple way. Description and Discussion 1. Preliminary Considerations. In a condensed phase at nonzerotemperature, fluctuations of density occur in the form of expansion and contraction of intermolecular voids that reflect thermal motion of the molecules. In the bulk of an isotropic liquid the averaged geometry of a void is spherical corresponding (in the macroscopic limit) to an equilibrium bubble shape. The voids obey the Boltzmann distribution and can be characterized by the mean parameters, where, eg., ( r ) is the mean radius of the effective cavity. (12) Beaglehole, D.; Christeneon, H. K. J. Phys. Chem. 1992,96,3396. (13) ScientificPapers by Lord Rayleigh;Dover: New York, 19s4; Vol. IV, p 430. (14) Yaminsky, V. V.; Yushchenko, V. S.; Amelina, E. A.; Shchukin, E. D. J. Colloid Interface Sci. 1983,96, 301. (16) Yushchenko, V. S.;Yaminsky, V. V.; Shchukin, E. D. J. Colloid Interface Sei. 1983,96,307. (16)Landau, L. D.; Lifshitz, E. M. Statistical Physics, 2nd ed.; Pergamon Press: London, 1968.
0743-7463/93/2409-3618$04.00/0 0 1993 American Chemical Society
Langmuir, Vol. 9, No. 12, 1993 3619
Hydrophobic Attraction and Phase Transition
..... :: ............
to%
::.i 0.
0.0
8.
0::.
p-[
Figure 1. Because of the boundary conditions, the energy of stretching of a poorly wetting liquid in the gap is less than in the bulk. As a result, thermal expansion of the liquid is enhanced in the lateral direction; the actual expansion is small (intermolecular distances t y p i d y increased by less than 1%at D > 1 nm) and structurally sensitive. A long-rangemolecular decompression predicted by classical theory of two-dimensional nucleation gives rise to an attractive density fluctuation force (see section 2).
. ........... . =.r-.,' .........:: ::I.
A . . 0..
Figure 3. In lyophilic systems the contact density exceeds the
bulk value. However, the propagation of the molecular compression into the f i b is limited to the thickness of adsorbed vapor or solute layers. The hydration repulsion is of shorter range than the hydrophobic attraction (see section 2). 0
L0o 0;
0.0
Figure 2. At 8 > 90° the film is metastable over the whole range of dietancee. However, the probabilityof the critical fluctuation resultingin the macroscopic cavitationbecomes physically finite when the layer is not more than several molecules thick. The presenceof an inert solute increasesthe criticalrupture thickness as well aa the range and magnitude of subcritical cavitation (see sections 3 and 5). In the macroscopic limit the work of the formation of a spherical cavity is
w = 4sr2y (2) where y is the surfacetension (surfaceenergy) of the liquid. A corresponding function, y = y ( r ) ,can be introduced to extend the formalism to the molecular level. An additional contribution due to a vapor pressure term proportional to 9 is unimportant for small cavities (see, however, section 5 for effects of dissolved gases). Substitution of eq 2 with y = const into eq 1 gives (r)= [kT/(4~~y)I~/~ (3) For water at ambient temperatures, take y = 70 dyn/cm and T = 300 K. Then the value of ( r ) predicted is 0.4 A, several times smaller than the molecular size, ~ m 3~ A / (u, is the molecular volume). Absurd as it may at first sight seem, this is a reasonable result. It is consistent with the magnitude of thermal expansion in a condensed phase. Intermolecular distances are typically increased by several percent on going from 0 K to room temperature for ordinary liquids far from the critical point. Note that the number of intermolecular voids is close to the number of molecules per unit volume, l/Um. Extension of this simplest dimensional argument can be made by considering correlation effects or by using different cohesion energy forms, y ( r ) ,to fit thermodynamic properties of the phase. With such extensions experimentalthermal expansion and compressibility can be fitted quantitatively by the model. But such extensions are not more revealing of the mechanism in the subsequent analysis, concerned with the immediate consequences of the basic formalism. At an isolated surface the equilibrium cavity geometry is a lens prescribed by the contact angle, 0. This is determined for an ideal (hysteresis-free)solid @)-liquid (L)-vapor (V)system by the Young equation Ysv - YSL = YLV cos ~ the equilibrium wetting tension, (ysv - y s is
7).
(4) The
Figure 4. The locally unbalanced surface tension under the conditions of essentially critical interfacial cavitation causes instantaneousstress which may lead to the propagation of a crack and film rupture at highthickneeaea exceedingthe range of surface forces. The boundary decompression-compression transition caused by surfactant adsorption is crucial in maintaining the stability of thin liquid films (see section 4). fluctuation modes at the molecular level should be changed by the surface accordingly. Any deviations from the macroscopic behavior arising at the molecular level can be taken into account in terms of line tension and/or y ( r )and T ( r ) corrections.17 A well-known example of the transition from homogeneous (in the bulk) to heterogeneous (at the surface) nucleation is an enhancement of air bubble nucleation by surface hydrophobicity. A straightforward consequence is that the density of a poorly wetting (finite contact angle) liquid is decreased in the immediate vicinity of the wall. For example, when the contact angle equals 90°, w = 27rr2y and the mean cavity radius at the surface is roughly 2ll2 = 1.41 times higher than in the bulk. The effect does not extend far beyond a few molecular layers into the liquid. Indeed, with the r2 form for w in the exponent the population of larger cavities given by the Boltzmann distribution is extremely scarce. The conclusion is consistent with the results of Monte Carlo simulations.18 ~ 2. Subcritical Attraction. For two surfaces situated at a finite distance another equilibrium cavity geometry can arise which corresponds to their bridging by a bubble. The bridge is exactly cylindrical at 90°, concave at larger angles, and convex at B below 90°. On a molecular scale, cylindrical bridging geometry corresponds to intermolecular channels normal to the surfaces. The work of formation of a cylindrical cavity of radiua r between surfaces a distance D apart is
w g = 2*rDyLV+ 2sr2(ySv- ySL) In the particular case where B = 90' (ysv substitution of eq 5 into eq 1leads to
YSL
(5) = 0)
( r ) = kT/2sDy (6) where y is ~ L V .From the comparison between eqs 6 and 5 w(r= ( r ) )= k T in that particular case. This is generally a reasonable result. For D larger than the molecular size, ~ ~ 1 1 about 3 , 3 A for water, the mean equivalent cavity (17)Frenkel, J. Kinetic Theory ofliquids; Clnrendon Press: Oxford, 1946. (18)Lee,C. Y.;McCammon, J. A.; Roaaky, P.J. J. Chem. Phys. 1984, BO, 4448.
Yaminsky and Ninham
3620 Langmuir, Vol. 9, No. 12, 1993
diameter is in the submolecular range, implying that such a fluctuation mode would be responsible for a slight increase of the intermolecular distances, e.g., r = 0.1 A at D = 10 A. It is to be stressed again that the "cylindrical cavity" is merely a mathematical presentation of the normal components of the fluctuating intermolecular void geometriesgiving rise to lateral expansion (decompression) determined by the boundary conditions. Alternative simple models relating elastic, thermal, and surface energies may be introduced at this point. In this paper the analysis will be restricted to the adopted thermodynamic formalism of the theory of nucleation and phase transition. To form a geometrically equivalent cavity in the bulk liquid, more energy has to be spent since (as long as 8 > 0) a smaller value of the wetting tension, 7 = ysv - YSL, in eq 5 is substituted for the larger one, y = y ~ v .So, similar fluctuation modes in the bulk should have a smaller average amplitude, ( r ) = rb, as compared with ( r ) = rg in the gap. Direct analytical solution of eqs 1 and 5 for fi is not so simple. However, both modes are very close in energy and geometry, and the relation between rgand fi can be obtained from the assumption wg(rg)= W b ( f i ) : 27rrp~y= 2 n r ~ + y 27rr:y
or rg- rb = r:/D
(7)
for 8 = 90°. In general, we have P g - rb r:(1 - COS 8)/D (8) Since rb c rg
