Langmuir 2008, 24, 1247-1253
1247
Attractive Surface Force in the Presence of Dissolved Gas: A Molecular Approach† Dusan Bratko*,‡,§ and Alenka Luzar*,‡ Department of Chemistry, Virginia Commonwealth UniVersity, Richmond, Virginia 23284, and College of Chemistry, UniVersity of California, Berkeley, California 94720 ReceiVed July 31, 2007. In Final Form: September 1, 2007 Despite widespread evidence of the influence of dissolved air on hydrophobic interaction, the mechanisms of observed effects are still unknown. Although some experiments indicate that adsorbed gases can modify the structure of water next to hydrophobic surfaces, gas effects on measured forces have been observed only at large surface separations. Gas-specific depletion of water at a hydrophobic surface has been detected but was not reproduced in subsequent measurements. We use computer simulations to study short-ranged hydrophobic attraction in the absence and presence of dissolved gas and monitor gas adsorption at molecular resolution inaccessible in experiments. Although we observe a significant accumulation of dissolved gases at hydrophobic surfaces, even in supersaturated gas solutions surface concentrations remain too low to induce any significant change in the local structure of water and short-range surface forces. We present direct calculations of the hydrophobic force between model hydrocarbon plates at separations between 1.5 and 4 nm. Although stronger, the calculated solvation force has a similar decay rate as deduced from recent surface force apparatus measurements at a somewhat lower contact angle. Within the statistical uncertainty, short-range attraction is not affected by the presence of dissolved nitrogen, even in supersaturated solution with a gas fugacity as high as 30 atm. Comparisons of the adsorption behavior of N2, O2, CO2, and Ar reveal similar features in contrast to the peculiar suppression of water depletion reported for an Ar solution in a neutron reflectivity experiment. Our calculations reveal a notable difference between pathways to the capillary evaporation of pure water and gas-phase nucleation in confined supersaturated gas solutions.
I. Introduction The nature of the boundary between water and a hydrophobic surface is crucial to the self-assembly processes, where solventinduced interaction is among the most important nonspecific interactions in biological systems. Water in apolar confinement thus continues to be the focus of intense experimental1,2 and theoretical research.3,4 The thermodynamics of confined hydrogenbonding liquids such as water is profoundly affected by reduced opportunities for interfacial molecules to form hydrogen bonds.5,6 The development of mean-field models for aqueous interfaces provided the first theoretical observation that interfacial water molecules preserve about three-fourths of the bulk average number of hydrogen bonds;7,8 this estimate was subsequently confirmed by molecular simulations9 and nonlinear optical experiments.10 The depletion of hydrogen bonds next to an extended hydrophobic surface is responsible for the reduced water density profile compared to the one at hydrophilic surfaces.11 So far, computational studies of the phenomenon focused on pure liquid whereas the role of dissolved gases attracted only †
Part of the Molecular and Surface Forces special issue. * E-mail:
[email protected] and
[email protected]. ‡ Virginia Commonwealth University. § University of California. (1) Christenson, H. K.; Claesson, P. M. AdV. Colloid Interface Sci. 2001, 91, 391. (2) Meyer, E. E.; Rosenberg, K. J.; Israelachvili, J. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 15739. (3) Weeks, J. D. Annu. ReV. Phys. Chem. 2002, 53, 533. (4) Hummer, G.; Garde, S.; Garcia, A. E.; Pratt, L. R. Chem. Phys. 2000, 258, 349. (5) Stillinger, F. H. J. Solution Chem. 1973, 2, 141. (6) Pratt, L. R.; Pohorille, A. Chem. ReV. 2002, 102, 2671. (7) Luzar, A.; Svetina, S.; Zeks, B. Chem. Phys. Lett. 1983, 96, 485. (8) Luzar, A.; Svetina, S.; Zeks, B. J. Chem. Phys. 1985, 82, 5146. (9) Lee, C. Y.; McCammon, J. A.; Rossky, P. J. J. Chem. Phys. 1984, 80, 4448. (10) Du, Q.; Superfine, R.; Freysz, E.; Shen, Y. R. Phys. ReV. Lett. 1993, 70, 2313. (11) Grigera, J. R.; Kalko, S. G.; Fischbarg, J. Langmuir. 1996, 12, 154.
limited attention.12,13 Experimental evidence (from optical cavitation, sonochemisty, and direct force measurements1) is growing in support of the idea that the presence of gas is crucial in determining the range and strength of hydrophobic interactions. The dissolved N2 and O2 molecules accumulated at the surface14,15 might initiate evaporation in a similar way as do hydrophobic surface areas (voids), which are shown to nucleate a vapor bubble.16 In addition to the potential influence of gases on the hydrophobic effect, their relevance is present in several other contexts as diverse as decompression sickness and particle flotation. Dissolved gas also plays an important role in intrusion/ extrusion processes of water in a nanopore.17 Colloidal suspensions and emulsions18,19 are other examples of the interplay between hydrophobic interfaces and dissolved gas. In such systems, the suspension stability is influenced considerably by the presence of dissolved gas.20 The above examples clearly demonstrate the importance of dissolved gases in a variety of systems controlled by surface interactions. A molecular-level understanding of their role, however, is lacking. Several questions remain to be answered: How significant is the influence of dissolved gases on the depletion of water next to an extended hydrophobic surface under ambient conditions? Recent experiments have yielded different results.21-31 (12) Dammer, S. M.; Lohse, D. Phys. ReV. Lett. 2006, 96, 206101. (13) Luzar, A.; Bratko, D. J. Phys. Chem. B 2005, 109, 22545. (14) Craig, V. S. J.; Ninham, B. W.; Pashley, R. M. Langmuir. 1999, 15, 1562. (15) Beaglehole, D. J. Phys. Chem. 1987, 91, 5091. (16) Luzar, A.; Leung, K. J. Chem. Phys. 2000, 113, 5836. (17) Qiao, Y.; Cao, G. X.; Chen, X. J. Am. Chem. Soc. 2007, 129, 2355. (18) Maeda, N.; Rosenberg, K. J.; Israelachvili, J. N.; Pashley, R. M. Langmuir 2004, 20, 3129. (19) Alfridsson, M.; Ninham, B.; Wall, S. Langmuir 2000, 16, 10087. (20) Gong, W. Q.; Stearnes, J.; Fornasiero, D.; Hayes, R. A.; Ralston, J. Phys. Chem. Chem. Phys. 1999, 1, 2799. (21) Ge, Z. B.; Cahill, D. G.; Braun, P. V. Phys. ReV. Lett. 2006, 96, 186101. (22) Takata, Y.; Cho, J. H. J.; Law, B. M.; Aratono, M. Langmuir 2006, 22, 1715.
10.1021/la702328w CCC: $40.75 © 2008 American Chemical Society Published on Web 11/03/2007
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Experiments32 also reveal long-wavelength capillary-like fluctuations of water next to a hydrophobic surface (contact angle of 110°), although several theoretical studies16,33-38 predict water in contact with the weakly attractive wall, in contrast to hardcore situation.5,39-41 Does the presence of air facilitate the formation of a vapor film near weakly attractive hydrophobic surfaces? In some measurements, the rather large density drop30 may also be interpreted as an averaged effect of larger density fluctuations occurring sporadically next to surfaces.42 These issues are important not only for understanding surface thermodynamics but also for resolving questions of where the slipping plane at a hydrophobic surface in water is located.43,44 The other unresolved issues are gas-specific effects on hydrophobic surfaces. Experiments, all done with octadecyltrichlorosilane (OTS) hydrophobic surfaces, give conflicting messages. X-ray reflectivity measurements do not reveal any significant gas-specific effects,29 in accord with related experiments on octadecyltriethoxysiloxane (OTE).24 Neutron reflectivity measurements, however, showed considerable gas-specific depletion of water at the hydrophobic(OTS)/water interface.30 These measurements also show counterintuitive behavior for argon. Although Ar is expected to contribute the least among measured gases to interfacial solvent depletion, it is surprising that the depletion of Ar-saturated D2O is reported to be 2 to 3 times smaller than for degassed D2O.30 If such gas-specific effects originate at hydrophobic/water interfaces, then the interfacial tension would be expected to vary with deaeration. Dynamic surface force apparatus (DSFA) measurements on OTE surfaces indicate the opposite.31 Molecular simulations offer a unique tool to clarify the apparently diverging conclusions coming from a range of recent measurements.24,29-31 We were the first to report on open ensemble simulations of thin aqueous confinements in equilibrium with atmospheric gases, specifically, N2 and CO2 at varied pore widths and gas fugacities.13 To minimize the interference with water hydrogen bonds, gas molecules adsorb at paraffin-like surfaces, increasing the pore gas concentration by over an order of magnitude relative to that of the bulk solution, with adsorption (23) Seo, Y. S.; Satija, S. Langmuir 2006, 22, 7113. (24) Poynor, A.; Hong, L.; Robinson, I. K.; Granick, S.; Zhang, Z.; Fenter, P. A. Phys. ReV. Lett. 2006, 97, 266101. (25) Mao, M.; Zhang, J. H.; Yoon, R. H.; Ducker, W. A. Langmuir 2004, 20, 1843. (26) McKee, C. T.; Ducker, W. A. Langmuir 2005, 21, 12153. (27) Zhang, X. H.; Khan, A.; Ducker, W. A. Phys. ReV. Lett. 2007, 98, 136101. (28) Cottin-Bizonne, C.; Cross, B.; Steinberger, A.; Charlaix, E. Phys. ReV. Lett. 2005, 94, 056102. (29) Mezger, M.; Reichert, H.; Schoder, S.; Okasinski, J.; Schroder, H.; Dosch, H.; Palms, D.; Ralston, J.; Honkimaki, V. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 18401. (30) Doshi, D. A.; Watkins, E. B.; Israelachvili, J. N.; Majewski, J. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 9458. (31) Meyer, E. E.; Lin, Q.; Israelachvili, J. N. Langmuir 2005, 21, 256. (32) Zhang, X. Y.; Zhu, Y. X.; Granick, S. Science 2002, 295, 663. (33) Bratko, D.; Curtis, R. A.; Blanch, H. W.; Prausnitz, J. M. J. Chem. Phys. 2001, 115, 3873. (34) Rudich, Y.; Benjamin, I.; Naaman, R.; Thomas, E.; Trakhtenberg, S.; Ussyshkin, R. J. Phys. Chem. A 2000, 104, 5238. (35) Huang, D. M.; Chandler, D. J. Phys. Chem. B 2002, 106, 2047. (36) Ashbaugh, H. S.; Pratt, L. R.; Paulaitis, M. E.; Clohecy, J.; Beck, T. L. J. Am. Chem. Soc. 2005, 127, 2808. (37) Ashbaugh, H. S.; Paulaitis, M. E. J. Am. Chem. Soc. 2001, 123, 10721. (38) Wallqvist, A.; Gallicchio, E.; Levy, R. M. J. Phys. Chem. B 2001, 105, 6745. (39) Katsov, K.; Weeks, J. D. J. Phys. Chem. B 2001, 105, 6738. (40) Weeks, J. D.; Selinger, R. L. B.; Broughton, J. Q. Phys. ReV. Lett. 1995, 75, 2694. (41) Lum, K.; Chandler, D.; Weeks, J. D. J. Phys. Chem. B 1999, 103, 4570. (42) Leung, K.; Luzar, A.; Bratko, D. Phys. ReV. Lett. 2003, 90, 065502. (43) Truesdell, R.; Mammoli, A.; Vorobieff, P.; van Swol, F.; Brinker, C. J. Phys. ReV. Lett. 2006, 97, 044504. (44) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H. J.; Craig, V. S. J. Rep. Prog. Phys. 2005, 68, 2859.
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restricted to the first monolayer.13 Because of a low concentration of atmospheric N2 and CO2 in air-saturated solutions at ambient conditions, the mole fraction of gas molecules at the solidliquid interface remains below a fraction of a percent, which is too low to produce noticeable changes in the water density structure or to visibly influence the competition between the liquid and vapor phases of water in hydrophobic confinement.45 The results of our previous work13 agreed with dynamic surface force apparatus (SFA) measurements that indicated no shortrange gas effect on the hydrophobic force (below 10 nm)31 but were in severe conflict with neutron reflectivity measurements at similar conditions indicating notable gas-specific effects on the surface depletion of water on hydrophobized surfaces.30 Because new experimental results became available,29 as well as their interpretation,2 we felt that more systematic modeling is warranted to shed light on the most current controversial experimental findings.29-31 To provide a molecular-level understanding of eventual gas-specific effects as indicated by neutron reflectivity,30 in this work we study the adsorption of Ar and O2 at atmospheric and elevated fugacities and compare the results with those for N2 and CO2 from our previous work.13 Our simulations should detect any significant differences between the adsorption behavior of Ar and that of other atmospheric gases. Direct measurements of forces between hydrophobic surfaces consistently show that the removal of dissolved gases decreases the range as well as the strength of surface attraction, but only at long range, leaving the short-range force (below 10 nm) unchanged.2,14,31,46-49 A rationale of the possible connection between the solvation force33 and the experimentally observed solvent density depression at the hydrophobic surface30 was discussed previously.50 Thus, to gain a molecular-level understanding of the influence of gases on surface forces at short range, if any, we explicitly calculate the solvation force (computationally limited to short-range) and study its distance dependence in the absence and presence of dissolved nitrogen at ambient and high gas fugacities (i.e., in supersaturated solutions (dissolved gas at a concentration significantly exceeding its solubility at atmospheric pressure)). Whereas such situations are of direct relevance to studies of decompression sickness and water extrusion from porous materials in energy-absorption devices,17 in the present study, our primary interest is to magnify the eventual gas effects on surface forces to enable the unambiguous detection of such effects in a computer experiment. We find the decay of the attractive surface force with distance to compare favorably with recent experiments that measure the solvent-induced interaction between hydrophobic surfaces in both aerated and dearted water.2 As in our previous work,13 we find a notable increase in gas concentration, in this case Ar and CO2, in the confinement, relative to that in the bulk solution. For typical atmospheric conditions, we show that this adsorption is still insufficient to produce water depletion even in the presence of strongly hydrophobic surfaces or in supersaturated gas solutions. We also give evidence for the lack of gas-specific effects and demonstrate that Ar is not an exception. These results are in conflict with the conclusions drawn from other theoretical work, based on simplified models,12 and (45) Luzar, A. J. Phys. Chem. B 2004, 108, 19859. (46) Stevens, H.; Considine, R. F.; Drummond, C. J.; Hayes, R. A.; Attard, P. Langmuir 2005, 21, 6399. (47) Meagher, L.; Craig, V. S. J. Langmuir 1994, 10, 2736. (48) Considine, R. F.; Hayes, R. A.; Horn, R. G. Langmuir 1999, 15, 1657. (49) Mahnke, J.; Stearnes, J.; Hayes, R. A.; Fornasiero, D.; Ralston, J. Phys. Chem. Chem. Phys. 1999, 1, 2793. (50) Forsman, J.; Jonsson, B.; Woodward, C. E.; Wennerstrom, H. J. Phys. Chem. B 1997, 101, 4253.
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Table 1. Lennard-Jones Potential Parameters of Different Atomic Species r (Oxygen, Hydrogen, Carbon, Nitrogen, and Argon Atoms in Water (w), Hydrocarbon (h), N2, O2, CO2, Ar) R
R/J mol-1
qR/eo
σR/A
Ow Hw Ch NN2 COMN2 OCO2 CCO2 OO2 Ar
650.2
-0.82 0.41 0.0 -0.4731 0.9462 -0.3086 0.6172 0.0 0.0
3.1656
648.8 392.2 776.6 405.8 401.6 995.0
a
3.754 3.3984 2.836 3.358 3.03 3.41
a COM denotes the midpoint between nitrogen atoms in the N2 molecule. qR is the atomic charge.
those from neutron reflectivity studies30 but agree completely with the most recent high-resolution X-ray studies.24,29 Finally, we discuss two distinct pathways of liquid-to-vapor transition in confined metastable water that we monitor in simulations (capillary evaporation of deareated water16,33,42,45 vs gas-phase nucleation from a supersaturated gas solution) and show how the latter phenomenon depends much less on the intersurface separation.
II. Model and Method Water is modeled using the simple point-charge model51 (SPC), where the intermolecular potential is represented by a pairwise sum of Lennard-Jones and Coulombic interatom potentials:
uRβ(r) )
[( ) ( ) ]
qRqβ σRβ + 4Rβ 4πor r
12
-
σRβ r
6
(1)
Subscripts R and β denote distinct interacting sites in the molecule, qR represents the atomic charges, and and σ are Lennard-Jones energy and size parameters, respectively. An analogous representation is used for dissolved gases (N2, CO2, O2, and Ar). o is the permittivity of vacuum. Interacting sites coincide with atom positions. An additional, positively charged site is introduced into the center of the model N2 molecule to capture the known quadrupolar moment of N2.53 All Lennard-Jones parameters and atomic charges are collected in Table 1. The values for water, N2, CO2, and hydrocarbon are based on refs 33,52-54 and our earlier work.13 Additional parameters for O2 and Ar are obtained from ref 55 and 56, respectively. For pairs of different atoms or interacting sites, interaction parameters are determined by the usual combining rules, σRβ ) (σR + σβ)/2 and Rβ ) (Rβ)1/2. Interatomic O-H distances in the SPC model equal 0.1 nm, and the bond angle is ∼109.5°. The distance between nitrogen atoms in the N2 molecule is taken to be 0.11 nm. The C-O distances in the linear CO2 molecule is 0.1162 nm, and the interatomic distance in the O2 molecule is 0.121 nm. By analogy to other reported studies in bulk CO2 solutions, the reaction of CO2 with water is not considered, a simplification that is not believed to have any serious consequences because it affects less than 0.5% of CO2 molecules.54 (51) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, The Netherlands, 1981. (52) Panhuis, M. I. H.; Patterson, C. H.; Lynden-Bell, R. M. Mol. Phys. 1998, 94, 963. (53) Somasundaram, T.; Lynden-Bell, R. M.; Patterson, C. H. Phys. Chem. Chem. Phys. 1999, 1, 143. (54) Somasundaram, T.; Panhuis, M. I. H.; Lynden-Bell, R. M.; Patterson, C. H. J. Chem. Phys. 1999, 111, 2190. (55) Swope, W. C.; Andersen, H. C. J. Phys. Chem. 1984, 88, 6548. (56) Shah, J. K.; Maginn, E. J. J. Phys. Chem. B 2005, 109, 10395.
In view of comparison with our previous work on N2 and CO2,13 we keep the generic confinement effect, ignoring the molecular details of the walls. We model the hydrophobic confinement by two parallel smooth walls with weakly attractive wall-water interaction. This interaction follows from integration of the Lennard-Jones (12-6) potential between oxygen atoms of water, Ow, and CHn groups of the hydrocarbon.33,57,58 Our simulations for these potentials yield a contact angle of 135 (5°, which is greater than the experimental value (∼110°) but in agreement with existing theoretical estimates.59 The system is of infinite lateral size simulated by periodic boundary conditions along lateral directions. Presuming a uniform number density of wall hydrocarbon groups FCh, at all distances |z| > D2, and FCh ) 0 for |z| < D2, where D ) 2D2 is the separation between the two surfaces, the potential energy of a water molecule or an atom of specified type R in a dissolved gas molecule in the field of the walls is33
uR(z) ) 4πFChChRσChR3
{[ ( ) ( ) ] [ ( ) ( ) ]}
9 3 1 σChR 1 σChR + 45 D2 - z 6 D2 - z 9 3 1 σChR 1 σChR (2) 45 D2 + z 6 D2 + z
The resulting water-wall interaction is the same as that used in several studies.33,57,58 The water-wall and Ar-wall potentials are effectively single-particle potentials independent of molecular orientation. However, the interactions of N2, CO2, and O2 molecules with the walls are orientation-dependent. The energy of the system is calculated as the sum of atom-wall interactions and a pairwise sum of interactions between interacting sites on distinct molecules, including interactions among the sites within the simulation box and their images introduced through periodic boundary conditions. In view of the prohibitive computational cost of 2D Ewald summations, we follow the procedure of Shelley and Patey57 with all intermolecular interactions subject to smooth spherical truncation. Comparisons33,60 with the results obtained upon inclusion of the 2D Ewald summation61-63 have revealed no significant differences between the two methods under conditions similar to those of our present study. To maintain conditions corresponding to the equilibrium between the confined liquid and the adjacent bulk solution, we used open ensemble simulations with fixed temperature, volume, and chemical potentials of water and gas components. The exchange of water molecules between the confinement and the bulk phase was implemented according to the grand canonical Monte Carlo (GCMC) algorithm as described previously.33,64 Using the formalism of Adams,65 the acceptance probabilities for attempted additions and deletions are given by
{
PNfN+1 ) min 1,
} {
〈N〉 β(µex - ∆U) e (N + 1)
PNfN - 1 ) min 1,
N β(-µex - ∆U) e 〈N〉
}
(3)
(57) Shelley, J. C.; Patey, G. N. Mol. Phys. 1996, 88, 385. (58) Lee, S. H.; Rossky, P. J. J. Chem. Phys. 1994, 100, 3334. (59) Hautman, J.; Klein, M. L. Phys. ReV. Lett. 1991, 67, 1763. (60) Bratko, D.; Daub, C. D.; Leung, K.; Luzar, A. J. Am. Chem. Soc. 2007, 129, 2504. (61) Bratko, D.; Henderson, D. Phys. ReV. E 1994, 49, 4140. (62) Bratko, D.; Henderson, D. J.; Blum, L. Phys. ReV. A 1991, 44, 8235. (63) Heyes, D. M.; Berber, M.; Clarke, J. H. R. J. Chem. Soc., Faraday Trans 2 1977, 73, 1485. (64) Luzar, A.; Bratko, D.; Blum, L. J. Chem. Phys. 1987, 86, 2955. (65) Adams, D. J. Mol. Phys. 1974, 28, 1241; ibid. 1975, 29, 307.
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where µex is the excess chemical potential of water, ∆U is the energy change upon addition or deletion of a molecule, and 〈N〉 is the average number of molecules occupying the simulation box of given thickness D under bulk-phase conditions. To improve exchange acceptances for somewhat bulkier gas molecules, the latter were added or deleted according to semigrand (SGCMC) or reaction ensemble (REMC)66-68 sampling. The details of this method, which are based on simulated changes in the chemical identity of particles, are given in our previous study.13 The acceptance probability of an attempted molecular exchange R f β (water and gas molecules) is determined from
{
PRfβ ) min 1,
NR 〈Nβ〉 β(µexβ - µexR - ∆U) e 〈NR〉 (Nβ + 1)
}
(4)
As shown in ref 13, eq 4 is equivalent to a more general equation (eq 18) of the reaction ensemble approach in ref 67. The density of water molecules in the bulk phase, 〈Nw〉/V, is known, and the ex product, 〈Ngas〉/V)eβµgas is determined from the desired bulk fugacity of the dissolved gas. We paid particular attention to the determination of the excess chemical potential for SPC water under the conditions of our simulation at T ) 298 K: βµex w ) -10.694. This value, which is slightly lower than the value of -10.596 determined in our former work,33 was selected to match the bulk-liquid water pressure, Pb ) 1 atm (calculated from the virial equation), within the simulation uncertainty of (20 atm. For the bulk phase, the above procedure reproduced experimental solubilities of N2 and CO2 within a statistical uncertainty of 10%. It underestimated those of O2 and Ar by about 35%, which is satisfactory agreement considering that, unlike N2 and CO2, the potential parameters for the latter two gases have not been parametrized for use in aqueous solutions. The average hydration pressure, Ph ) (F/S - Pb), where F/S is the average force per unit area of confinement walls and Pb is the bulk pressure, was calculated in two ways: by direct force calculation described in ref 33 and by a 2D virial pressure equation,69 with the two methods giving identical results. Open-ensemble Monte Carlo simulations of the confinement were performed in a rectangular box of lateral size Lxy fixed at 1.8 nm when the confinement width D did not exceed 1.8 nm. At larger D, the lateral size was increased to maintain approximately cubic geometry of the box. Trial moves (comprising simultaneous random translations and rotations) of particles were attempted in 20% of the attempted configuration changes, and the remaining attempts were used with equal probabilities for GCMC additions or deletions of water molecules and SGCMC exchanges between molecules of gas and water. To secure an acceptable speed of equilibration of distribution profiles and pressures on the two opposite walls, in addition to molecular moves, it was essential to include simultaneous moves of both confinement walls (in the perpendicular direction) at fixed wallwall separation. The inclusion of a small fraction (below 1%) of wall moves (equivalent to collective moves of all confined molecules relative to the walls) accelerated the equilibration of the water distribution by nearly two orders of magnitude. Because equilibration with respect to the gas content is very slow, (1-5) × 109 attempted moves were required in a typical run to obtain an average (5% statistical accuracy in gas molality (estimated from subaverage statistics) and (25 atm in the solvation force calculation. (66) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: San Diego, CA, 1996. (67) Smith, W. R.; Triska, B. J. Chem. Phys. 1994, 100, 3019. (68) Lisal, M.; Nezbeda, I.; Smith, W. R. J. Chem. Phys. 1999, 110, 8597. (69) Bratko, D.; Jonsson, B.; Wennerstrom, H. Chem. Phys. Lett. 1986, 128, 449.
III. Results and Discussion Thermodynamics. SolVation Force. By performing GCMC calculations, we determined the magnitude of attractive force mediated by metastable water between extended hydrophobic surfaces in absence and presence of dissolved nitrogen and compared the results with most recent experimental data.2 We choose to use N2 in force calculations because it represents the dominant component of air and therefore is most abundant in water exposed to the atmosphere. Furthermore, in laboratory studies, CO2 but not N2 is routinely removed from the solution. The solvent-mediated force per unit area, Fh/S, or hydration pressure, Ph, between parallel plates at separation D is defined as
Ph ) -
∂Ws(D) ∂D
(5)
Ws(D) is the solvent contribution to the free energy per unit area of the plates. If direct (van der Waals) interaction between the plates is included in W(D), then the latter can be identified as twice the surface tension between the walls and the liquid phase, and Ph ) -2[∂γ(D)/∂D]. In a typical SFA experimental setup, interacting surfaces are curved, and the measured quantity is the ratio F/R, where R is the radius of curvature. According to the Derjaguin approximation,70 F(D)/R can be related to the surface free energy of the corresponding planar system by F(D)/R ≈ 2πW(D) or (Fplates/S) ≈ (1/2π)∂(Fcylinders(D)/R)/∂D. If the ratio F(D)/R features an exponential decay with distance D (i.e., d ln[F(D)/R]/dD is approximately constant within the Derjaguin approximation), then the same holds true for hydration pressure Ph. Within the range of 2 to 4 nm, the hydration pressure between our model hydrocarbon walls remains on the order of -102 atm. This can be compared with recent experimental measurements for octadecyltriethoxysilane (OTE) surfaces, the most stable hydrophobic surfaces that can currently be produced in a laboratory.2 Using the Derjaguin approximation, the measured ratio F/R suggests the hydration pressure between planar OTE walls within the same separation interval to be about -2 atm. However, the reported OTE contact angle of θ ≈ 101° (average of advancing and receding angles) is well below the estimated60 angle of ∼135°, which is characteristic of the model hydrocarbon walls that we use. This suggests that the average force between modeled hydrocarbon walls should be about 4 times stronger than in the case of OTE. The simulated attraction is therefore about an order of magnitude stronger than that deduced from the measurement on the OTE sample characterized by about a 3040% lower contact angle. The same study also reports a roughly exponential decay of F/R with separation D, with a characteristic decay constant of d ln[F(D)/R]/dr ≈ -0.024 A-1 (Figure 6 in ref 2). This compares well with the average slope d ln[Fplates(D)/S] dr ≈ -0.016 A-1 that we observe over the range of simulated separations. We further note that the experiment performed in the crosscylinder geometry of SFA spans distances as small as 1 nm without the observation of water evaporation. In our simulation, capillary evaporation is invariably observed below D ) 1.4 nm for runs of O(109) simulation steps. The difference can be attributed to different hydrophobicities of the wall material in the two scenarios. This influences the kinetic threshold distance, Dkc .45 Taking into account the scaling of the kinetic barrier to capillary evaporation with contact angle, ∆G* ∝ 1/cos θ,45 the (70) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992.
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Figure 1. Hydration pressure as a function of slit width D in pure water and in a supersaturated solution of N2 (fugacity fN2 ) 30 atm). The dashed line is a fit to pure water data. The discontinuity at D ) 1.4 nm reflects the spontaneous expulsion of water from the slit at the kinetic threshold surface separation, Dkc .42,45
less hydrophobic OTE surface suggests that the kinetic threshold distance is smaller than in the case of our simulation study. Because ∆G* scales as ∼D2,45 for an equal rate of evaporation the kinetic threshold distance Dkc for θ ≈ 101.5° should be about one-half of that at θ ≈ 130° (i.e., 0.7 nm). This is the separation regime that is still not accessible to the current force measuring technique.2 Although all simulated states are metastable with respect to capillary (surface-induced) evaporation, we also present simulation data obtained in supersaturated gas solutions. We consider these solutions in order to monitor gas effects under extreme conditions where any effects on surface interactions would be magnified, facilitating detection in the simulations. The highest supersaturated concentrations of N2 we were able to simulate over prolonged simulation times corresponded to gas fugacity of about 30 atm, meaning a 30-fold supersaturation at an ambient pressure of 1 atm. In very long simulation runs, we often observed water expulsion in supersaturated samples; however, the rate of expulsion was low enough to accommodate a run length of O(109) simulation steps, which was sufficient to secure statistically meaningful sampling within the metastable supersaturated regime. Within the statistical uncertainty of the simulation, the magnitude of the force and the rate of decay remain unchanged in the supersaturated regime (Figure 1). These results can be favorably compared with experimental findings showing no effect of dissolved gas at ambient conditions on the measured force below 10 nm.2 Specifically, the two experimental force curves (one in aerated and the other in dearated water) follow an almost identical path in the final 10 nm before contact (Figure 4 in ref 2). The simulation results presented in Figure 1 indicate that the trend seen experimentally will be unchanged even in supersaturated gas solution, at least for separations that we can probe computationally. Clearly, the same applies at lower gas concentrations. Even though in principle some difference between the two force curves could remain undetected because of statistical uncertainty, even under supersaturated conditions the difference would be limited to a meager (20 atm or at most 20% of the total solvent-mediated interaction. Gas adsorption on a Single Surface. If there is significant gas adsorption, then the wall/liquid surface tension will change according to the Gibbs equation
∫
∆γ ) - Γ dµ
(6)
where Γ is the surface excess
Γ)c
∫0∞ [g(z) - 1] dz = Ac
(7)
and g(z) is the density distribution function of gas molecules in the gap.
Figure 2. Distribution functions for gas and water molecules in a hydrocarbon-like confinement of width D ) 1.6 nm, normalized by concentrations in the bulk phase. The data for water (dashed line) are multiplied by 10 for easier comparison.
For our system, the integral A ) ∫∞0 [g(z) - 1] dz is essentially constant for all realistic bulk concentrations, c, of dissolved gas.13 This behavior corresponds to the Henry’s law regime, hence dµ = RT d ln c and
∆γ = RTA
∫0cc′ d ln c′ ) AcRT
(8)
(i.e. the surface tension change, ∆γ, is directly proportional to the bulk concentration of the gas). For all gases that we considered, the change in the solid/liquid surface tension due to the presence of gas is less than 1 dyn/cm. Specifically, at a gas fugacity of f ) 1 atm, ∆γN2 ) -0.043 dyn/cm, ∆γ CO2 ) -0.59 dyn/cm, ∆γO2 ) -0.021 dyn/cm, and ∆γAr ) -0.019 dyn/cm. Therefore, gas fugacities that can be reached experimentally will not produce significant (>2 dyn/cm) changes in γsolid/liquid. For example, in the case of nitrogen, to reduce ∆γ by 1 dyn/cm, a pressure of 23 atm would be needed. The conclusion that we can draw from these calculations is that gas adsorption is still insufficient to produce water depletion (a nanovapor layer) at hydrophobic surfaces, even at contact angles as high as ∼130° and in supersaturated solutions. In summary, the surface tension on individual walls does not change appreciably by the adsorption of any of the gases considered in this work. Gas Adsorption in Confinement. For wall separations exceeding the kinetic threshold distance of capillary evaporation,33,42,45 adsorption on opposite plates happens independently. We substantiate this fact by direct calculation of density distribution profiles in the gap (Figure 2), the confirmed validity of Henry’s law in the confinement (Figure 3), and the ∼1/D dependence of scaled molality in the confinement with intersurface separation (Figure 4). Figure 2 compares density profiles of the four gases in aqueous confinements of width D ) 1.6 nm at gas fugacity f ) 1 atm. The density profile for water molecules (dashed curve) is included for comparison. According to our calculation, gas concentrations corresponding to atmospheric conditions result in no visible change in the structure of the water film in the confinement. All density profiles are normalized by the bulk value for each component. Calculations performed at varied f show that the gas distribution functions depend very weakly on the gas fugacity (and concentration) for all fugacities fCO2 e 2 to 3 atm and fN2,Ar,O2 e ∼40 atm. At higher fugacities, saturation with gas triggers the expulsion of metastable liquid water from the confinement. The invariance of the g(z) observation and the existence of a relatively broad interval around the pore midplane with gas density equal to the bulk value show that the two adsorption layers on opposite confinement walls build up essentially independently of each other and are, within the range of practically relevant fugacities,
1252 Langmuir, Vol. 24, No. 4, 2008
Figure 3. Average mole fractions of gas molecules in a hydrophobic confinement of width D ) 1.6 nm as functions of gas fugacities f for N2, O2, Ar (bottom axis) and CO2 (top axis).
Figure 4. Increase in the average molalities of dissolved gases N2, O2, Ar, and CO2 as a function of the width, D, of the hydrocarbonlike confinement. To make separate viewing possible, the labeled curves have been shifted as follows: N2 + 15, O2 + 10, and Ar + 5. Solid lines: theory predictions for uncoupled surfaces (see text and ref 13).
very similar to the case of adsorption on isolated walls. For all of the gases that we considered, we find the density profile of the water film in the confinement to be essentially unchanged upon saturation with dissolved gas at atmospheric gas fugacities. Saturated mole fractions of the four gases in a 1.6 nm slit are presented in Figure 3 as functions of gas fugacity. We include the data at fugacities exceeding the kinetic threshold for water expulsion under supersaturated conditions, when expulsion was delayed sufficiently to obtain reproducible samples for these metastable states. We find that Henry’s law for the confined solution remains valid throughout a wide range of fugacities for all gases studied. The validity of Henry’s law in the gap conforms with nonoverlapping g(z) values in Figure 2. In the case of N2, because of its low solubility, at ambient pressure the dissolved gas showed no effect on the longevity of the metastable liquid film. At confinement widths of ∼1.5-2.5 nm and typical simulation run lengths of O(109) attempted moves, capillary evaporation in the presence of N2, for example, was often observed at N2 fugacities of about 20 atm or higher. However, the system was usually stable in a quasi-equilibrium state long enough to obtain meaningful statistics for gas concentration and solvation pressure in this strongly metastable regime. Figure 4 shows the average confinement molalities of Ar and O2 (relative to those of the saturated bulk phase) as functions of the interwall separation, in comparison with N2 and CO2 reported previously.13 We use molality, rather than concentration, as the measure of gas solubility because of the somewhat ambiguous
Bratko and Luzar
definition of the solvent-accessible volume in a soft confinement. The average molality of the dissolved gas is shown to increase steadily upon decreasing the width of the confinement, reaching up to ∼30 and 15 times that of the saturated bulk phase for N2 and CO2, respectively.13 The new gases that we study here fall between these two numbers: an ∼ 25-fold increase for O2 and an ∼20-fold increase for Ar. The difference in the relative adsorptions of the four gases is primarily due to different interactions with water. The increase in the gas concentration at surfaces is strongest with the least soluble gas, N2, and weakest for the most soluble one, CO2. Although the four gases also differ in their direct attraction to the walls (Table 1), these differences play a secondary role. As demonstrated in our previous work13 and confirmed for Ar and O2 in this article (Figure 4), the overall molality in the confinement of the gas is successfully approximated by
〈mg(z)〉 A ≈1+ mb D
(9)
with constant A calculated using density profiles gg(z) for N2, Ar, O2, and CO2 from Figure 2. The agreement of eq 9 with the results of open ensemble simulations for all four gases confirms the absence of any significant coupling between adsorption processes on the opposing confinement walls for all relevant pore widths D exceeding the kinetic threshold distance for evaporation, Dk45 c . Gas Specificity. All of the results discussed above confirm the lack of significant gas specificity in our molecular simulations. This is in contrast to recent molecular dynamics simulations of Lennard-Jones models of confined liquids with dissolved gases by Dammer and Loshe12 showing greatly enhanced gas concentrations inside the hydrophobic pore (contact angle 115°) relative to the bulk. The authors ask, “What causes the dramatic gas enrichment?” and gave the answer, “The enrichment increases with increasing gas/liquid and σgas.” However, the argument applies only when interaction parameters of hypothetical Lennard-Jones model gases include a range of unrealistic interaction strengths that cannot be attributed to water solutions of gases found in the atmosphere. We show that detailed interaction site models for water and common gases with established Lennard-Jones and Coulombic interaction parameters produce significantly weaker effects. If anything, with our surfaces characterized by contact angle of ∼ 135°, the enhancement of gas concentrations would likely be stronger compared to ∼115° used in ref 12. We therefore conclude that a huge gas enhancement and gas specific effects seen in ref 12 might be a consequence of several factors: too crude a model; too high a gas concentration in the bulk (1:1000 gas/waters), specifically, 2 orders of magnitude larger in comparison to known gas solubility data; and simulations not performed at the required constant chemical potential. While offering interesting qualitative insights, generalizations of the results of this simplified model12 to real gases in aqueous confinements can be misleading and are not supported by experimental observations.29 Kinetics. Although the kinetics of capillary evaporation depends critically on the capillary width D,45 this dependence is much weaker in case of gas-phase nucleation from a supersaturated gas solution. In the former scenario, a critical vapor cavity spans the gap, and both walls must be engaged.42,71 Gas nucleation in a supersaturated solution, however, can occur within the liquid phase (homogeneous nucleation) or, preferably, (71) Yushchenko, V. S.; Yaminsky, V. V.; Shchukin, E. D. J. Colloid Interface Sci. 1983, 96, 307.
AttractiVe Surface Force
Langmuir, Vol. 24, No. 4, 2008 1253
whereas this dependence is much weaker for nucleation from a confined supersaturated gas solution.
IV. Concluding Remarks
Figure 5. Number of water molecules in the left (blue, red) and right (green, black) halves of the simulated planar pores filled by pure water (red, black) or water supersaturated with N2 at gas fugacity fN2 ) 30 atm (blue, green). The water-containing pore (red and black curves) was 1.4 nm wide, and the pore with solution was 1.8 nm wide. The expulsion of water requires the simultaneous emptying of both half-spaces (critical nucleus has to engage both walls) whereas gas-phase nucleation from the supersaturated gas solution represents a single wall event. The simulation time is measured by the number of attempted Monte Carlo (MC) moves: (a) full plot and (b) magnified time interval before evaporation.
at either of the (hydrophobic) confinement walls (heterogeneous nucleation). Whereas the kinetics is accelerated further by the proximity of the opposite wall, nucleation at a single wall is feasible, and the width of the pore is comparatively less important. Any dependence on the separation D of the nucleation rate of the gaseous phase in the supersaturated solution is expected to vanish when D appreciably exceeds the critical size of a nucleating bubble on a single wall. The two distinct pathways of gas-phase nucleation for the two scenarios are illustrated in Figure 5. The Figure shows time (number of MC passes) dependences of the number of water molecules in simulated pores before and during the liquid-to-gas transition. The pairs of curves comprising red and black or blue and green curves represent the numbers of water molecules in the left and right halves of the same simulation box separated by an imaginary midplane surface. The blue and green pair of curves illustrates an evaporation event from a capillary of 1.8 nm width initially filled by water supersaturated by N2 at gas fugacity 30 atm. The red and black curves correspond to pure water in a 1.4-nm-wide pore. In the case of a confined supersaturated gas solution, large fluctuations in the number of molecules can occur separately at either of the walls, and complete water expulsion is preceded by a gradual emptying of one of the slit halves (green curve), followed with a considerable delay by emptying the remainder of the pore (blue curve). For pure water, however, a large fluctuation is possible only when both walls are involved as reflected in abrupt and simultaneous emptying of both halves of the box (red and black curves). The above distinction between the pathways of confinement and supersaturated gas-induced liquid-to-vapor transitions was observed consistently in simulation runs involving the evaporation events of the two types. In view of the different mechanisms of nucleation, the kinetics of spontaneous water expulsion will depend strongly on the wall-wall separation D in pure water,
This work demonstrates one important point: modeling should contain enough realism to serve as a reasonable caricature of the real and truly complicated system. Simulated Lennard-Jones (LJ) spheres with varying LJ parameters give a “dramatic” effect of more than 2 orders of magnitude gas enrichment at hydrophobic surfaces.12 We show that by using more realistic and carefully parametrized interaction site models to capture molecular structure, matching correct charge distribution and quadrupole momenta,13,52,54-56 the big surprise disappears. Apart from some differences due to essentially stronger solubility in the case of CO2, our calculations reveal qualitatively identical behaviors for all four gases that we study. We observed 30-fold enrichment for N2 and 15-fold enrichment for CO2 compared to bulk values.13 In the present work, we find O2 and Ar between these two numbers. Our results are in accord with the most recent highresolution X-ray experiment29 that confirm insignificant gas adsorption, confined to the first molecular layer, and no dependence of the width of the hydrophobic surface water interface on the amount21 and type of dissolved gas in the adsorption layer.13 These results offer a molecular-level interpretation for the lack of influence of dissolved gas on the short-range hydrophobic force, a result that is consistently reported in experiments.2,14,31,46-49 Interestingly, within computational precision, our simulation results reveal an insignificant increase in short-ranged hydrophobic attraction even at supersaturated N2 concentrations corresponding to a 30-fold excess over the equilibrium value under ambient conditions. The new results presented here also rule out any dramatic effects of dissolved air in excess of saturation concentration,72 a situation that can occur upon release of gases from hydrophobic surface coatings where it may remain trapped since their preparation.73 However, a nanoscale gas state or surface microbubbles recently identified via infrared spectroscopy27 are still possible. Because of their mesoscopic size, an adequate coarse-grained description would be required to help unveil the physics of such microbubble formation and its role in shortrange solvent-induced interaction. Alternative explanations for air effects on long-range forces refer to the pH increase upon deaeration as a source of repulsive surface charge interactions.74 Finally, in view of the fact that polarizability has been found to play an important role in the local concentration of solutes near aqueous liquid/vapor and liquid/liquid interfaces,75,76 the possible effects of polarizability on gas solubility and adsorption are definitely worth exploring. We plan to do so in our future studies. Acknowledgment. We thank the National Science Foundation for financial support through awards CHE-0718724 (to A.L.) and BES-0432625 (D.B.) and the NSF (CHE-060047) and Hewlett-Packard for computational resources. LA702328W (72) Zhang, X. H.; Zhang, X. D.; Sun, J. L.; Zhang, Z. X.; Li, G.; Fang, H. P.; Xiao, X. D.; Zeng, X. C.; Hu, J. Langmuir 2007, 23, 1778. (73) Ishida, N.; Sakamoto, M.; Miyahara, M.; Higashitani, K. Langmuir 2000, 16, 5681. (74) Zhang, J. H.; Yoon, R. H.; Mao, M.; Ducker, W. A. Langmuir 2005, 21, 5831. (75) Jungwirth, P.; Tobias, D. J. Chem. ReV. 2006, 106, 1259. (76) Chang, T. M.; Dang, L. X. Chem. ReV. 2006, 106, 1305.
