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Lateral Intermolecular Forces in the Physisorbed State: Surface Field Polarization of Benzene and n-Hexane at the Water/ and Mercury/Vapor Interfaces Brian A. Pethica*,† and M. Lawrence Glasser‡ Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544, and Department of Physics, Clarkson University, Potsdam, New York 13699 Received June 29, 2004. In Final Form: October 6, 2004 The available experimental data on the dependence of the surface tensions of water and mercury on the adsorption of benzene and hexane from the vapor phase are critically analyzed and interpreted to obtain the two-dimensional second virial coefficients [B2(T)] for these adsorbed nonpolar molecules. Calculations based on the unperturbed Lennard-Jones (L-J) 12-6 formalism for benzene and the related 12-5 Salem formalism for long chains in two dimensions for hexane require that B2(T) should be negative for both adsorbates. On water, the experimental data indicate that B2(T) for both molecules is less negative than expected from the unperturbed L-J and Salem estimates, and on mercury the B2(T) values from experiment are positive. These findings are analyzed first in terms of a possible reduction in the attractive component of the potential of mean force between physisorbed molecules arising from their frequencydependent interaction with their electrostatic images in the bulk phases, as described by McLachlan. It is concluded that the McLachlan corrections are small for these molecules and surfaces. A second analysis considers the effect of an extra repulsion between the adsorbed molecules arising from the induction of dipoles normal to the interface by the surface electric field. Surface field polarization (SFP) accounts reasonably well for the experimental results, leading to estimates of the surface fields at the mercury and water surfaces which are consistent with estimates from contact potentials for mercury and computation from modeling the water surface. SFP may have a wide impact in determining the form of physisorption isotherms.
Introduction The understanding of lateral intermolecular forces between adsorbed molecules underlies the interpretation of many surface properties and processes. These forces can be approached via experimental estimates of the twodimensional (2D) second virial coefficients [B2(T)] for molecules spread or adsorbed at gas/liquid or liquid/liquid interfaces at moderate to low surface pressures, corresponding to unlocalized 2D fluids. B2(T) values have been estimated for lipids,1,2 alkanes,3,4 cyclic amides,5 and proprionic acid2 at the air/water interface and have been used to quantify the intermolecular potentials of mean force [Φ(r)] between pairs of identical molecules. Pair potentials for adsorbed gases such as methane have also been estimated from second virial coefficients at gas/metal interfaces.6,7 Intermolecular potentials for physisorbed molecules will differ in principle from those acting in the * Corresponding author. † Princeton University. ‡ Clarkson University. (1) Pethica, B. A.; Glasser, M. L.; Mingins, J. J. Colloid Interface Sci. 1981, 81, 41. (2) Hasmonay, D.; Billoudet, F.; Badiali, J. P.; Dupeyrat, M. J. Colloid Interface Sci. 1994, 165, 480. (3) Pethica, B. A. Langmuir 1996, 12, 5851. (4) Pethica, B. A.; Glasser, M. L.; Cong, E. H. Langmuir 2004, 19, 6820. (5) Lou, A.; Pethica, B. A.; Somasundaran, P. Langmuir 1996, 12, 5845. Pethica, B. A.; Senak, L.; Zhu, Z.; Lou, A. Colloids Surf., A 2001, 186, 113. (6) Kevan, S. D.; Skelton, D. C.; Wei, D.-H. Crit. Rev. Surf. Chem. 1994, 3, 77. Elliot, G. S.; Wei, D.-H.; Wu, K. J.; Kevan, S. D. J. Chem. Phys. 1990, 93, 4152. Elliot, G. S.; Wu, K. J.; Kevan, S. D. Phys. Rev. Lett. 1991, 66, 433. (7) Rauber, S.; Cole, M. W.; Bruch, L. W. Surf. Sci. 1982, 123, 173. Bruch, L. W. Surf. Sci. 1993, 125, 194. Bruch, L. W.; Cole, M. W.; Zaremba, E. Physical Adsorption: Forces and Phenomena; Clarendon Press: Oxford, 1997.
gas phase by modification of conformations on passing from the gas to the adsorbed state and by effects of the adsorbent on the adsorbate pair potentials. These include the interactions described by McLachlan,8 who included in the lateral interactions between adsorbed molecules the images of the frequency-dependent electrical fluctuations of the adsorbed molecules in the adsorbent phase. This effect has been shown to be relevant to gases adsorbed on metal crystals6,7 but has been little considered for other interfaces. Another aspect of the adsorbent-adsorbate interaction possibly contributing to repulsion in the adsorbate pair potentials is polarization of the adsorbed molecules in the electric field at the phase boundary (surface field polarization, SFP). A lattice model for the adsorption of the alkanes pentane to heptane and benzene9 on water indicates that SFP contributes significantly to the heat of adsorption. This is supported for benzene adsorption by a molecular dynamics study.10 Estimates of B2(T) for insoluble monolayers of phospholipids in the heptane/water interface (at which screening of the lipid chains by the paraffin solvent greatly reduces the van der Waals force component of the lipid pair potentials) are also available. They have been interpreted largely in terms of the electrostatics of the zwitterion headgroups and their zero-frequency images in the water phase11 but not including the dispersion terms and images of the McLachlan treatment or SFP. (8) McLachlan, A. D. Mol. Phys. 1964, 7, 381. (9) Vidal-Madjar, C.; Gulochon, G.; Karger, B. L. J. Phys. Chem. 1976, 80, 394. (10) Dang, L. X.; Feller, D. J. Phys. Chem. B 2000, 104, 4403. (11) Taylor, J. A. G.; Mingins, J.; Pethica, B. A. J. Chem. Soc., Faraday Trans. I 1976, 72, 2694. Mingins, J.; Stigter, D.; Dill, K. A. Biophys. J. 1992, 61, 1603. Stigter, D.; Mingins, J.; Dill, K. A. Biophys. J. 1992, 61, 1616.
10.1021/la040090b CCC: $30.25 © 2005 American Chemical Society Published on Web 12/30/2004
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The early estimates1 of B2(T) for uncharged lipid molecules at the air/water interface were based on experiments on insoluble spread air/water monolayers of long-chain compounds such as fatty acids, giving the surface pressure (Π) as a function of the surface density (Γ) or surface area/molecule (A). The results were interpreted in terms of the Lennard-Jones (L-J) vacuum pair potentials for extended chains lying flat in the interface, the probable configuration in dilute long-chain lipid air/ water monolayers. B2(T) for extended long chains in the surface plane is dominated by the large van der Waals forces when the chains are close and parallel. Summation of the pair potentials for long chains of CH2 groups obeying the L-J 12-6 dependence on separation gives a 12-5 form for the whole molecule, as shown by Salem.12 This 12-5 relationship gave a reasonable account of the chain length dependence of the 2D B2(T) values as estimated from the then available lipid monolayer data.1 Graphical representation of B2(T) on a scale of σ2 as a function of /k was given for both the 12-5 and 12-6 cases1 from the relation
Data Sources
∫
B2(T) ) π [1 - exp -Φ(r)/kT]r dr
(1)
where k is the Boltzman constant and σ, , and r have their customary significance in the L-J relationship for the potential of mean force Φ(r) ) 4{(σ/r)12 - (σ/r)6} and its 5th power analogue. The available experimental data on the dependence of the surface pressure on the alkane gas pressure and temperature for short-chain gaseous n-alkanes (methane to butane) adsorbed at the water/gas interface were analyzed to obtain estimates for the alkane 2D second virial coefficients at the water/vapor surface.3 These estimates were later improved by inclusion of the threedimensional (3D) gas-phase fugacities in the analysis of the experimental data and by upgrading the earlier computations using the 12-6 and 12-5 relations for the pair potentials.4,13 It was shown that for the adsorbed n-alkanes methane to propane the experimental estimates of B2(T) agree with values calculated from the 2D 12-6 L-J relation for Φ(r) using the available σ and /k values for the 3D gases substituted into the basic relation (eq 1) for a 2D gas of spherical molecules.4,14,15 The absence of McLachlan/SFP effects correlates with the small heats and entropies of adsorption for these molecules, which indicate a weak vibrational mode of adsorption with a correspondingly large separation from the water surface. The discrepancy between the corresponding calculated and experimental results for butane, which is more strongly adsorbed, may indicate that the McLachlan and SFP effects become significant components of the pair potential for butane, but use of the gas-phase L-J parameters may be invalidated by a change in conformation of butane on adsorption. In this paper, we extend the analysis for adsorbed hydrocarbon vapors on both water and mercury with emphasis on benzene and n-hexane. Benzene exemplifies nondipolar aromatic hydrocarbons without configurational complexities. Hexane is also nondipolar and extends the analysis to intermediate length n-alkane homologues. (12) Salem, L. J. Chem. Phys. 1962, 37, 2100; Nature 1962, 193, 476. (13) Glasser, M. L. Phys. Lett. A 2002, 300, 381. (14) Hirschfelder, J. O.; Curtis, C. F.; Byron Bird, R. Molecular Theory of Gases and Liquids; John Wiley and Sons: New York:, 1964. (15) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Liquids and Gases; McGraw-Hill: New York, 1987,
n-Alkanes on Water. Of the early published surface tension data only the results of King and colleagues16 using the capillary rise method over a range of temperatures with the homologous series from methane to butane and of Sachs and Meyn17 using the drop-shape method with methane at 25 °C were judged reliable for analysis for second virial coefficients at several temperatures.3,4 The extensive published data on the adsorption of higher alkanes using a vapor-flow system with Wilhelmy plates were shown to be unreliable and unable to give valid second virial coefficients and other thermodynamic properties of these alkanes on water.3 A restricted recent set of measurements using the drop-volume method for water exposed to static controlled partial vapor pressures of hexane at one temperature is available.18 Benzene on Water. The experimental surface tension data are taken from the recent extensive results on the adsorption of numerous aromatic molecules singly and in mixtures over a range of temperatures published by Bruant and Conklin,19,20 who also give full references to earlier work on aromatics adsorbed on water. The experimental methods and results for benzene are given in detail.19 Bruant and Conklin used the drop-shape method in a flow system in which the composition of the vapor was changed dynamically, measuring surface tension and gas composition simultaneously in both ascending and descending total vapor pressures. In the case of benzene, a number of “static” measurements were made in the flow at constant benzene vapor pressures at 25 °C to observe time effects and validate the assumption of rapid adsorption employed in interpreting the data from the dynamic runs. They analyze their data using the equation of state Π(A - a0) ) kT where k is Boltzman’s constant and a0 is negative and is described as an empirical constant. This equation is coupled to the Gibbs adsorption isotherm to obtain a relation between the vapor and the surface pressures and fitted to the data by a nonlinear regression method, which is said to give good correlations for a0. Benzene and Alkanes on Mercury. The classical work of Kemball and Rideal21 and Roberts22 measured vapor adsorption at the mercury surface by a sessile dropshape method at static controlled vapor pressures. The data are tabulated, and the virial coefficients are readily obtained. Where the two sets of data address the same vapors, the experimental differences are minor. The authors give extensive interpretation for the heats and entropies of adsorption and for co-areas obtained from coupling the Gibbs adsorption isotherm with the 2D Volmer equation Π(A - b) ) kT, where b is the co-area, following a graphical procedure for b. The data given by Smith23 for alkanes on mercury are presented as line graphs and agree with the data of Roberts where both workers take the same adsorbate. Smith includes data on (16) Jho, C.; Nealon, D.; Shogbola, S.; King, A. D. J. Colloid Interface Sci. 1978, 65, 141. Massoudi, R.; King, A. D. J. Phys. Chem. 1974, 78, 2262. Massoudi, R.; King, A. D. In Colloid and Interface Science; Kerker, M., Ed.; Academic Press: 1976; pp 3, 331. (17) Sachs, W.; Meyn, V. Colloids Surf., A 1995, 94, 291. (18) Lou, A.; Pethica, B. A. Langmuir 1996, 13, 4933. (19) Bruant, R. G., Jr.; Conklin, M. H. J. Phys. Chem. B 2000, 104, 11146. (20) Bruant, R. G., Jr.; Conklin, M. H. J. Phys. Chem. B 2002, 106, 2232. (21) Kemball, C.; Rideal, E. K. Proc. R. Soc. London, Ser. A 1946, 187, 53. Kemball, C. Proc. R. Soc. London, Ser. A 1946, 187, 117. (22) Roberts, N. K. J. Chem. Soc. 1964, 1907. (23) Smith, T. J. Colloid Interface Sci. 1968, 26, 509. Smith, T. In Hydrophobic Surfaces. Kendall Award Symposium Series; Fowkes, F., Ed.; Academic Press: New York, 1968; pp 5, 189.
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the change in the contact potential of mercury caused by both spread and adsorbed hydrocarbons. Methods For a pseudo-one-component surface phase, the virial expansion in the surface density is
Π/kTΓ ) 1 + B2(T)Γ + B3(T)Γ2 + ...
(2)
where B2(T), B3(T), and so forth are the second, third, and so forth temperature-dependent 2D virial coefficients. By taking the surface density from the Gibbs adsorption isotherm for an ideal vapor at pressure p, the virial expansion can be written3 as
Π/p ) kT
∑n Bn(T)pn-1(1/kT dΠ/dp)n
(3)
where Bn(T) is the nth virial coefficient, with B1(T) ) 1. B2(T) will be given by
B2(T)/kT ) -β/R2
(4)
where R ) lim(dΠ/dp) and β ) lim[d(Π/p)/dp] as p and Π go to 0. If the assumption that the alkane vapors can be treated to a good approximation as ideal gases is inapplicable, the equations above can be retained with substitution of the gas fugacity (p*) in place of the pressure and R* and β* in place of R and β.4 The deviations from ideality for hydrocarbon vapors at the temperatures and pressures considered here can be neglected. An alternative method of analyzing the data is via the 2D virial expansion as a power series in Π,
ΠA/kT ) 1 + (BΠ + C′Π2 + ...)/kT
(5)
where B, C′, and so forth are the second, third, and so forth temperature-dependent virial coefficients in the pressure expansion. B and C′ are related to the second and third virial coefficients in the density expansion by B ) B2(T)24 and
B3(T) ) [B2(T)]2 + kTC′
(6)
The surface fugacity (Π*) can be expressed for an ideal adsorbing vapor as4,24
Π* ) Rp ) Π exp(BΠ + C′Π2/2 + ...)/kT
(7)
The term R is accessible graphically or by regression methods from Π-p, Π/p-p, and ln(Π/p)-Π plots at low p, as discussed previously.4 For nonideal gases, p is replaced by the fugacity p*. In practice, the virial expansions are truncated at the highest coefficient which gives sufficient statistical consistency with the experimental data. Recalling that B ) B2(T), we note that truncation at the second virial coefficient leads from eq 5 to
Π(A - B) ) kT
(8)
With substitution of b (the co-area) for B, eq 8 becomes the 2D Volmer equation of state. Where this form is used and B corresponds with a realistic exclusion area for the molecules under study, it is well-recognized that the co(24) Guggenheim, E. A. Thermodynamics. An Advanced Treatment for Chemists and Physicists, 2nd ed.; N. Holland Publishing Co.: Amsterdam, 1950; p 99.
area is also the second virial coefficient, both positive. But in general, if eq 8 covers the experimental data, B gives the second virial coefficient, negative or positive. Plainly, if B is negative or of magnitude less than the molecular cross-section it cannot be simply a co-area and must be interpreted as a function of attractive, area exclusion and other contributions to the lateral potentials of mean force. Results Hexane on Water. As stated above, the bulk of the available data for alkanes on water is unreliable. Unfortunately, the surface pressure data for hexane on water obtained in equilibrium experiments are restricted to a range up to 2 mN m-1 at 23 °C only.18 These results were obtained using hexadecane or squalane as diluents to control the hexane vapor pressure because thermodynamic data on the activities of these mixtures are available.25,26 The results for the squalane diluent are preferred because hexadecane has a low but significant vapor pressure and may be a coadsorbent with the hexane. The calculation of the hexane vapor pressures over hexane/squalane mixtures involves an extrapolation for the hexane activity coefficient beyond the experimental range of the published data, with some consequent uncertainty. Using eqs 4 and 7, we find B2(T) for hexane on water of -0.9 ( 0.25 nm2/ molecule. We note that despite the uncertainties of much of the published data on alkane adsorption on water, all the longer alkanes at near-ambient temperatures show qualitatively that the B2(T) values are negative. Benzene on Water. The results of Bruant and Conklin19 agree fairly well with earlier published results also obtained by flow methods.27 The adsorption data are interpreted in terms of the equation of state Π(A - a0) ) kT. The parameter a0 is obtained via an unspecified statistical procedure for the nonlinear correlation and tabulated. The a0 are negative and, as explained above, are directly equivalent to negative B2(T) values from eq 8. Inspection of the tables19 shows that for the dynamic measurements a0 goes through a minimum at 25 °C in one set. The second set also shows a minimum at 25 °C but with a large decrease at 42 °C to a value almost 1 order of magnitude larger than that given at the same temperature in the first set. These are unrealistic findings and suggest experimental artifacts, possibly related to the solubility of benzene in water which would strictly require presaturation at each chosen vapor pressure. The results given in the “static” verification experiments at 25 °C show a spread of 15% around a mean a0 equivalent to B2(T) ) -0.29 nm2/molecule, in fair agreement with the results of the dynamic measurements at 25 °C. This result will be taken for discussion here, but further experiments are clearly desirable. Benzene and Hexane on Mercury. The data give excellent straight line plots (according to eq 7) of ln(Π/p) against Π over a good range of lower pressures, yielding physically acceptable values of the “co-area” of the adsorbed molecules with close agreement between the Kemball and Roberts data sets where they overlap for the same adsorbate. These convert directly to B2(T) values (nm2/molecule) at 25 °C of 0.337 for benzene and 0.322 for hexane, both positive. The experiments with benzene on Hg extend to 75 °C, with an increase in B2(T) to 0.378 (25) McGlashan, M. L.; Williamson, A. G. Trans. Faraday Soc. 1961, 57, 588. (26) Ashworth, A. J.; Everett, D. H. Trans. Faraday Soc. 1960, 56, 1609. (27) Blank, M.; Ottewill, R. H. J. Phys. Chem. 1964, 68, 2206.
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that the proximity of the aqueous phase also contributes an additional lateral repulsion. We now interpret these findings in terms of the possible McLachlan and SPF corrections to the potentials of mean force. McLachlan Corrections. In 1960 Sinanoglu and Pitzer28 proposed that the lateral interaction of molecules absorbed on a metal substrate was affected by image effects. McLachlan worked this out using the Dzyaloshinski-Lifshitz many-body formalism and obtained the correction in terms of the dielectric functions of the absorbate and substrate.8 Vidali and Cole29 subsequently put McLachlan’s expression into a form convenient for numerical calculation by adopting a Lorentzian model for the dielectric function; the necessary parameters for a variety of systems were given by Rauber et al.30 By explicitly evaluating the integrals in Vidali and Cole’s theory we obtain the following analytic expression for the correction to the lateral interaction potential, to be added to Φ(r) from the 2D L-J formalism.
[
EaEs F2 - 3z2 a02g0 (2E + E ) a s (Ea + Es)2r3 F5
]
2 2 3 Ea + 3EaEs + Es r3 g0 (9) 4 Ea + Es F6
Figure 1. B2(T)/πσ2 as a function of 4/kT for a 2D L-J gas from eq 9, with and without the surface field effect. The curves from bottom up refer to η ) 0 (uncorrected 12-6 L-J) and η ) 0.05, 0.08, 0.1, 0.12, 0.15, and 0.2. Computations for (a) low and (b) high range.
nm2/molecule. The data for hexane on Hg are only from 16.5 to 25 °C, showing a small positive increase in B2(T) with increasing temperature. Discussion We first compare the experimental results with the 2D second virial coefficients to be expected from the unperturbed L-J potentals using the parameters applicable to the 3D vapors of benzene and hexane.14,15 Benzene. For benzene at 25 °C we find the 2D B2(T) ) -0.731 nm2/molecule from eq 1 using the L-J 12-6 parameters derived from the 3D gas data (/k ) 412.3 K and σ ) 0.535 nm).15 This calculated B2(T) is to be compared with the experimental estimates of -0.29 for benzene on water and +0.34 on mercury. For benzene on mercury, the difference in B2(T) between the L-J calculation and the experiment is both definite and dramatic. If the customary two-parameter L-J 12-6 formalism for two dimensions is retained, the values of B2(T)/πσ2 (0.378 at 25 °C and 0.42 at 75 °C) for benzene on mercury using the experimental values of B2(T) are greater than its permitted L-J maximum of 0.35 if σ takes the 3D value (see Figure 1a). Furthermore, because B2(T) is positive /k would necessarily be below 186 K, less than 1/2 of the 3D value. The entropy of adsorption of benzene to mercury indicates that the molecule is mobile, rotating in the plane of the surface only.21 From the molecular dimensions of benzene, it is unlikely that this would increase the L-J σ sufficiently to bring B2(T)/πσ2 below 0.35. This qualitative conclusion requires further consideration, but the experimental results indicate a possible reduction in the net attractive component of the pair potential and the very probable intervention of an additional repulsive component to the potential of mean force between benzene molecule pairs in the adsorbed layer as compared with their interaction in the 3D gas phase. Given the experimental uncertainty for benzene on water, we may conclude to a lesser precision
where F ) (r2 + z2)1/2. The effect of the correction is small, raising the minimum in the interaction potential by 15% at the most when z approaches 0. A realistic estimate for benzene takes the parameter values as
Es ) 16.3 eV (ionization potential for Hg) Ea ) 8.5 eV (ionization potential for benzene) a0 ) 10.2 Å3 (polarizability of benzene) g0 ) 0.812 (dielectric function of Hg) z ) 1.0 Å (absorbate-reflection plane distance) and the L-J potential parameters for benzene as ) 0.0355 eV and σ ) 5.35 Å. This estimate gives only a 7% reduction in the minimum in Φ(r) and a positive shift in B2(T) of about 1%. If the value of Es is alternatively taken as the work function of liquid mercury, the McLachlan correction is approximately halved, and possible alternative values for the polarizability of benzene to reflect its orientation on the surface may further reduce the correction. We conclude that the McLachlan correction is insufficient to explain the large shift and reversal of sign of B2(T) for benzene on mercury as compared to the value to be expected from the 3D gasphase L-J parameters alone. SFP. If a nonpolar adsorbed molecule experiences an electric field at the surface, a dipole directed normal to the interface will be induced. These dipoles will contribute an additional repulsion to the pair potential. We take as a first step the simple point-dipole repulsion term as an addition to the L-J 6-12 relation to give
Φ(r) ) 4[(σ/r)12 - (σ/r)6 + η(σ/r)3]
(10)
where r is the intermolecular separation and and σ retain the values of the 3D gas. Because the dipole repulsive energy for the pair of induced dipoles is given by 2R´ 2F2/r3 (28) Sinanoglu, O.; Pitzer, K. S. J. Chem. Phys. 1960, 32, 1279. (29) Vidali, G.; Cole, M. N. Surf. Sci. 1981, 110, 10. (30) Rauber S.; et al. Surf. Sci. 1982, 123, 173.
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where R´ is the static polarizablity and F is the surface field, the coefficient η will be
η ) R´ 2F2/2σ3
(11)
Values of B2(T)/Πσ2 were computed as a function of 4/kT from eqs 1 and 10 over a range of η values. Examples of these calculations are shown in Figure 1. Taking the 3D gas-phase polarizability R´ ) 0.01 nm3/molecule for benzene,31 the presumed value of η is found by interpolating for the experimental result, leading to the surface field. For benzene on mercury at 25 °C we find η ) 0.15, corresponding to a surface field for mercury of 4.8 × 109 V m-1. Smith estimated the field at the mercury surface from his study of the change in the contact potential between the mercury and an ionizing probe electrode which is observed on spreading nonvolatile monolayers of nonpolar hydrocarbons or on adsorbing smaller molecularmass alkane vapors.23 Smith gives the field in the absence of adsorbed molecules as 1.37 × 109 V m-1 at a distance of approximately the diameter of an adsorbed atom from the surface dipole array, applying the electrostatic model of MacDonald and Barlow.32 The proximity of the estimates of the surface field from the two methods is encouraging and will be discussed below. A further test of these arguments can be made for benzene on Hg using the available experimental values of B2(T) at higher temperatures. If the values of R´ , σ, and η applied at 25 °C are used at 75 °C, the computed value of B2(T) from eqs 1 and 9 is 0.390 nm2/molecule, which compares reasonably with the experimental value of 0.378 nm2/molecule. For benzene on water, the experimental estimate of B2(T) is less reliable, as discussed above. Taking its value as -0.29 nm2/molecule at 25 °C and applying the same procedure as for mercury, we find η ) 0.145. This is equivalent to a field of 3.0 × 109 V m-1 at the surface of water. This estimate compares with the fields calculated from the model of Vidal-Madjar et al.9 for the surface water dipoles and multipoles. The field depends on the position relative to the oxygen and protons of the surface water molecules and is calculated as from 4.41 to 5.52 × 109 V m-1 at a distance 0.34 nm above the surface dipole layer of water, varying laterally to lower values at positions between the discrete water molecules and falling off rapidly as the distance from the surface increases. The agreement is encouraging at this stage of the analysis. Hexane. For hexane in two dimensions, we find B2(T) ) -1.10 nm2/molecule at 25 °C if the unperturbed L-J parameters for hexane vapor are used (/k ) 399.3 K; σ ) 0.5949 nm).15 However, the thermodynamic evidence for hexane on both water and mercury clearly indicate that hexane is extended in the surface at the lower pressures relevant to estimating B2(T), and it will be more appropriate to apply the Salem 12-5 form of the L-J relation for long-chain alkanes as a better approximation for hexane. Following the arguments given previously1 for extended long chains interacting in two dimensions, we take the effective values of /k as 432.1 K and σ as 0.465 nm and find the corresponding B2(T) ) -1.307 nm2/ molecule at 25 °C. This is to be compared with the experimental estimates of -0.9 on water and +0.322 on mercury. The B2(T) found for hexane on mercury is the more secure of these two estimates. These results again indicate that the bulk water and mercury phases modify the potentials of mean force between these physisorbed (31) Handbook of Chemistry and Physics, 76th ed.; CRC Press: New York, 1995-1996; Vol. 10, p 198. (32) MacDonald, J. R.; Barlow, C. A. Surf. Sci. 1966, 4, 381.
Table 1. Surface Field Estimates (Scale of 109 V m-1) for the Mercury/ and the Water/Vapor Interfaces from B2(T) for benzene from B2(T) for n-hexane electrostatic model9 from contact potentials23
water
mercury
3.0 2.0 4.41-5.52a
4.8 4.2 1.4
a
At a distance 3.4 Å above a surface water molecule in several assumed orientations.
molecules. In the case of hexane on mercury, the positive value of B2(T)/πσ2 using the experimental virial coefficient and σ from the Salem form of the L-J formalism is, as for benzene, much larger than permitted for the uncorrected formalism, again strongly indicating an additional repulsive component to the lateral potential of mean force. Because the McLachlan correction for benzene on mercury was found to be minor, the corresponding correction for hexane is unlikely to be more important than that from SFP, which we now discuss. Smith gave estimates23 of the induced dipole moments of several n-alkanes on mercury, from which the SFP correction according to eqs 1, 9, and 10 may be calculated, noting that the induced dipole moments (µ) are given by µ ) R´ F. A simple interpolation of Smith’s estimates of µ gives 0.4 D for the induced hexane dipole. Using the values of and σ in eq 11, we find that η takes a value of 0.013. From the computations for B2(T) from the Salem 12-5+3 form for Φ(r) as a function of η given previously,1 this result will not give a positive B2(T) for hexane, suggesting that Smith’s estimate of the mercury surface field is low. If we follow the procedure applied above for benzene, taking for hexane the experimental B2(T) at 25 °C and the polarizability R´ ) 0.0119 nm3/molecule31 to deduce η, the result is 0.25 and the corresponding surface field is 4.2 × 109 V m-1, essentially the same as estimated from the benzene data and a factor of 3 larger than Smith’s estimate. Repeating this procedure for hexane on water, using the tentative experimental estimate for B2(T), we find η ) 0.055 and the field at the surface of water as 2.0 × 109 V m-1, somewhat smaller than for mercury but close to the estimates from the data for benzene on water and from modeling.10 These surface field estimates are collected in Table 1. The approximations in our calculations of B2(T) for hexane should be noted. The use of the Salem potential function for hexane is an obvious approximation for a chain of this intermediate length, and the dependence of the potential of mean force on the rotation and flexing of the chains in the surface should be taken into account. Treating the induced dipole as a point dipole with R´ for hexane in the gas phase is a coarse approximation for a long chain flat on the surface, which will be polarized in each group along the chain, requiring a Salem-like correction to the r-3 relation for point dipoles. As for benzene, image contributions are omitted. The proximity of our surface field estimates to those of Smith for mercury is fair, but the comparison is probably incomplete. The MacDonald and Barlow model32 for the contact potentials used by Smith does not apply when R´ /z3 (where z is the distance from the center of the adsorbed molecule to the imaging plane) is more than 4, which is marginally possible for both benzene and hexane flat in the interface. Smith assumed the surface charge on the mercury to be 0, which would require that measurement of the surface potential be carried out in the compensation
Physisorbed State Lateral Intermolecular Forces
state.33 It would appear from the description of the experimental procedure that this was not the case, but given the small capacities in the experimental arrangements, any additional field due to the noncompensated charge surface charge density will be trivial compared with the estimates given in Table 1. A more probable reason for the lower field estimated from Smith’s contact potential results is that the reference electrode may also be adsorbing alkane vapors, even with the higher homologues spread as liquids directly to the surface, thereby reducing the measured contact potential. As for the estimated surface field of water, it appears too short-range to influence the 2D B2(T) for the more weakly adsorbed small alkane homologues. It was established4 that the experimental B2(T) for methane, ethane, and propane on water agree well with values derived from the L-J potentials for the 3D gases. This finding is partly a function of the smaller polarizabilities of the lower homologues,31 but it is also associated with the small heats and entropies of adsorption of these molecules. The small entropies correspond to the loss of less than one degree of translational freedom, indicating a weak vibrational mode normal to the surface in the adsorbed state and a correspondingly large average distance from the surface. Butane appears to be the first alkane homologue sufficiently closely bound to the water surface to show deviation from the L-J estimate using the 3D parameters for the 2D B2(T). These considerations imply that the field falls off sharply at the water surface, as would be expected for a dipole array of oriented water molecules. We note that, on mercury, the less reliable data for propane23 (the Π-p plot merges with the isotherm for butane at low Π) nonetheless show clearly that B2(T) is positive, in contrast with the negative value for the water surface. Smith does not give the heats and entropies of adsorption for propane to mercury, but comparison of the heats of adsorption for the n-alkanes in the range C5-C8 on mercury22 and water9 show that the heats on mercury are almost double those on water, indicating that propane is bound closer to the surface on mercury than on water. In general, if the Π-p isotherms for nonpolar physisorbed molecules at a liquid interface are concave to the gas pressure axis or if the Γ-p isotherms for adsorption on a homogeneous solid are likewise concave, B2(T) will be positive and SFP may be a contributing cause. The (33) Hall, D. G.; Pethica, B. A. Proc. R. Soc. London, Ser. A 1977, 354, 425.
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adsorption of benzene to a characterized annealed silica may be such an example.34 In addition to the critique of the assumptions of the theories applied to analysis of the experimental data in this report, we should observe that we have assumed simple additivity in the account of polarization (as in eq 10), which is known to be at best an approximation.35,36 Higher-order interactions of the induced dipoles, to include the static image terms, will also be relevant, and the estimated small effect of the McLachlan corrections will repay further consideration. Alternatives to the L-J representation of the potentials of mean force will also repay study. Conclusions Adsorbed hydrocarbons at the water and mercury liquid/ vapor interfaces, exemplified by benzene and n-hexane, are polarized by the surface electric field to give induced dipoles normal to the surface. This adds a repulsive component to the lateral potentials of mean force, leading to a positive increase in the 2D second virial coefficients. The signs of the virial coefficients are reversed in the case of mercury, for which the (negative) adsorption entropies are large and the adsorbed molecules are close the surface and, thus, more sensitive to the field, which decays rapidly with distance from the surface. Adsorption at the water surface is weaker, such that the SFP is not large enough to influence the lateral potentials for the smaller alkanes up to propane. The experimental second virial coefficients indicate that the surface fields sensed by adsorbed benzene and n-hexane are larger at the mercury surface than for water, in both cases on the order of 109 V m-1. These estimates are in fair agreement with surface fields estimated from surface potentials for mercury and from a surface dipole model for the water surface. The contribution of the McLachlan corrections to the lateral potentials of mean force appears to be minor. Further experiments are desirable to improve the estimates of B2(T) for benzene and for alkanes larger than butane adsorbed at the surface of water. Acknowledgment. M.L.G. thanks the NSF for support under Grant DMR-0121146. LA040090B (34) Hockey, J. A.; Pethica, B. A. Trans. Faraday Soc. 1962, 58, 2017. (35) Kunz, W; Lo Nostro, P.; Ninham, B. W. Curr. Opin. Colloid Interface Sci. 2004, in press. (36) Hall, D. G.; Pethica, B. A. Proc. R. Soc. London, Ser. A 1978, 364, 457.
