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Alkane Adsorption at the Water-Vapor Interface Henry S. Ashbaugh* Los Alamos National Laboratory, Theoretical Division, MS:B268, Los Alamos, New Mexico 87545
Brian A. Pethica Princeton University, Department of Chemical Engineering, The Engineering Quadrangle, Olden Street, Princeton, New Jersey 08544 Received April 1, 2003. In Final Form: June 18, 2003 The adsorption of methane and ethane at the water liquid-vapor interface is investigated using a combination of explicit molecular simulations and scaled-particle theory. Calculated adsorption coefficients are found to be approximately half the experimental values for both hydrocarbons. Simulation enthalpies of adsorption are in excellent agreement with the experimental values, while the heats predicted by scaledparticle theory are just below the lower limit of the experimental error. The simulation underprediction of the experimental adsorption coefficients corresponds to a ∼k/2 increase in the entropic adsorption penalty. Potential sources for the discrepancy between simulation and experiment are explored using the scaled-particle model. Based on this analysis, it is unlikely that the differences arise from either longwavelength capillary fluctuations or bulk water density differences between simple point charge and real water. Contributions from surface field polarization of these two adsorbates appear to be negligible.
Introduction The adsorption of surfactants, amphiphilic polymers, and proteins at aqueous interfaces impacts diverse phenomena from dish washing detergency1 and the spreading of liquids on surfaces2 to the function of lung alveoli during respiration3 and the integrity of cellular membranes.4,5 The interpretation of experiments on the aqueous surface activity of such species frequently appeals to the competition between hydrophilic groups, which favor dissolution in water, and hydrophobic groups, which resist mixing. It can be difficult to discriminate between these two opposing forces for a given adsorbate, however, since the constituent groups do not interact with water in isolation. The intermolecular forces between alkanes and water at aqueous interfaces are basic for understanding hydrophobic contributions to adsorption and thereby provide invaluable information for refining theoretical descriptions of these fundamental interactions. The available experimental information on the adsorption of alkanes and other hydrocarbons from the vapor phase to the surface of water up to 1996 has been critically reviewed.6 Further data on the adsorption of primarily aromatic hydrocarbons have since been made available.7 The heats of adsorption of pentane, hexane, heptane, and benzene on water were calculated by Vidal-Madjar et al.8 using a lattice model for the water and summing potential energies for the hydrocarbons with a semiempirical relation similar to the * To whom correspondence should be addressed. E-mail:
[email protected]. (1) McCoy, M. Chem. Eng. News 2003, 81 (3), 15. (2) Hill, R. M. Curr. Opin. Colloid Surf. Sci. 1998, 3, 247. (3) Lipp, M. M.; Lee, K. Y. C.; Zasadzinski, J. A.; Waring, A. J. Science 1996, 273, 1196. (4) Groot, R. D.; Rabone, K. L. Biophys. J. 2001, 81, 725. (5) Antonietti, M.; Thunemann, A. Curr. Opin. Colloid Surf. Sci. 1996, 1, 667. (6) Pethica, B. A. Langmuir 1996, 12, 5851. (7) Bruant, R. G.; Conklin, M. H. J. Phys. Chem. B 2002, 106, 2232. (8) Vidal-Madjar, C.; Gulochon, G.; Karger, B. L. J. Phys. Chem. 1976, 80, 394.
Lennard-Jones plus a term for the polarization of the adsorbate by the surface field of the water, itself calculated from the lattice model. The calculated heats, of which the surface field polarization term comprised 19%, were in good agreement with the experimental data. Molecular simulations provide detailed molecular-level information on surface adsorption inaccessible to experimental thermodynamic measurements.9-15 In particular, the potentials of mean force constraining the adsorbate at the surface can be directly calculated, from which the surface coverage can be determined under appropriate assumptions. Wilson and Pohorille calculated a surface excess for ethanol of 3 × 10-10 mol/cm2 at a bulk aqueous concentration of 1.2 M,13 compared to the experimental surface excess of (3-5) × 10-10 mol/cm2 at the same concentration determined by neutron reflectivity.16 In addition to adsorption potentials, calculated potentials of mean force for nonspherical species can provide information on the orientation of the adsorbate in the interface.9,10,14,15 It has been demonstrated that benzene, for example, lies coplanar with the air-water interface,9 while the aromatic ring of phenol is perpendicular to the aqueous interface with the OH group immersed in the liquid.14 In addition to equilibrium adsorption properties, recent simulations have begun to examine adsorption/desorption kinetics of gases and surface active species at the interface,10-13 important in understanding phenomena from the uptake of atmospheric pollutants by water droplets to the dynamic interfacial tension of newly created surfaces. (9) Dang, L. X.; Feller, D. J. Phys. Chem. B 2000, 104, 4403. (10) Shin, J. Y.; Abbott, N. L. Langmuir 2001, 17, 8434. (11) Somasundaram, T.; Lynden-Bell, R. M.; Patterson, C. H. Phys. Chem. Chem. Phys. 1999, 1, 143. (12) Somasundaram, T.; in het Panhuis, M.; Lynden-Bell, R. M.; Patterson, C. H. J. Chem. Phys. 1999, 111, 2190. (13) Wilson, M. A.; Pohorille, A. J. Phys. Chem. B 1997, 101, 3130. (14) Pohorille, A.; Benjamin, I. J. Chem. Phys. 1991, 94, 5599. (15) Pohorille, A.; Benjamin, I. J. Phys. Chem. 1993, 97, 2664. (16) Li, Z. X.; Lu, J. R.; Styrkas, D. A.; Thomas, R. K.; Rennie, A. R.; Penfold, J. Mol. Phys. 1993, 80, 925.
10.1021/la034559z CCC: $25.00 © 2003 American Chemical Society Published on Web 07/24/2003
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In this study, we present a combined molecular simulation and scaled-particle theory (SPT) analysis of methane and ethane adsorption at the aqueous liquid-vapor interface, for which reliable experimental data are available and offer the simplest illustration of the relevant factors determining adsorption. The work associated with pulling the alkanes through the interface is directly evaluated, providing insights into the relative contributions of excluded volume and longer-ranged dispersion interactions on the dissolution and adsorption process. From the calculated free energy profiles, the low-coverage surface affinities, quantified by the adsorption coefficient, are determined and compared directly with experiment. The simulations are in excellent agreement, within the statistical uncertainty, with the experimental standard integral enthalpy of adsorption for both alkanes. The simulations, on the other hand, ascribe a systematically greater standard entropy penalty of adsorption of ∼k/2, resulting in a predicted surface affinity of about half that of experiment. The SPT model readily permits us to tune the excluded volume and attractive interactions affecting adsorption and is used to examine potential sources for the discrepancy between the simulation and experimental results. Theoretical Considerations Adsorption Thermodynamics. The aqueous surface pressure, Π ) γ0 - γ (the difference between the surface tension of a pure solvent interface and that of a mixed interface), is moderated by alkane adsorption and desorption at an interface by the Gibbs adsorption equation,17
|
dΠ ) Γa dµa T
(1)
where µa is the alkane chemical potential, Γa is the alkane surface excess, and T is the temperature. The solute surface excess is determined by the expression
Γa )
∫-∞∞[Fa(z) - Faqa Θ(-z) - Fvap a Θ(z)] dz
(2)
where Fa(z) is the alkane concentration as a function of z perpendicular to the interface, Faq a is the alkane concenis the tration in the bulk aqueous phase (z f -∞), Fvap a solute concentration in the vapor phase (z f ∞), and Θ(z) is the Heaviside step function. The Gibbs dividing surface (assumed to be z ) 0 in eq 2) is chosen such that the solvent surface excess is zero by convention. In the lowpressure limit, Π is proportional to the solute partial pressure. The proportionality constant, R (the adsorption coefficient), is6
Γa dΠ ) limF vapf0 vap ) R ) limPaf0 dPa a F a
∫-∞∞[g(z) -
KΘ(-z) - Θ(z)] dz (3) where we have assumed the vapor phase behaves as an is the ratio ideal gas. In this expression, g(z) ) Fa(z)/Fvap a of the local solute concentration to the bulk vapor vap is the Ostwald partition concentration, and K ) Faq a /Fa coefficient. From statistical mechanical considerations, the chemical potential of an alkane positioned at z perpen(17) Hunter, R. J. Introduction to Modern Colloid Science; Oxford University Press: Oxford, U.K., 1993.
dicular to an interface can be expressed as18
µa(z) ) kT ln[Fa(z)qa-1] + µ/a(z)
(4)
where kT is the product of Boltzmann’s constant and the temperature, qa is the partition function of a single alkane in a vacuum, and µ/a(z) is the nonideal interaction (i.e., excess) contribution to the chemical potential at z. At adsorption equilibrium, the alkane chemical potential is independent of its position with respect to the interface, so that
) µa(z) µa ) µvap a
(5)
Assuming the vapor phase is ideal, eq 5 can be rewritten as
kT ln Fvap ) kT ln Fa(z) + µ/a(z) a
(6)
Rearrangement of this expression yields
) exp[-µ/a(z)/kT] g(z) ) Fa(z)/Fvap a
(7)
The Ostwald partition coefficient is governed by the excess chemical potential in the bulk aqueous phase,
K ) exp[-µ/a(-∞)/kT]
(8)
Combining eqs 3, 7, and 8, the adsorption coefficient is determined by the alkane’s excess chemical potential as a function of z through the aqueous interface. Adsorption Simulations. The solute excess chemical potential can be evaluated from Widom’s potential distribution theorem19
µ/a(z) ) -kT ln 〈exp[-Uaw(z)/kT]〉0
(9)
where the brackets, 〈...〉0, indicate averaging over an ensemble of water molecular configurations in the absence of the adsorbate, and Uaw(z) is the alkane-water interaction energy when a test alkane is randomly inserted in a plane at z parallel to the interface in a given conformation. To evaluate the chemical potential, we performed computer simulations of a lamellar slab of water in equilibrium with its vapor and calculated the appropriate test particle averages using eq 9. Canonical ensemble Monte Carlo simulations20 of a slab of liquid water in equilibrium with its vapor were carried out at 2, 25, and 50 °C. The simulations consisted of 300 simple point charge (SPC) water molecules21 in an 18 Å × 18 Å × 68 Å rectangular box under periodic boundary conditions, with the major axis normal to the liquid-vapor interface. This corresponds to a liquid water layer thickness of 28 Å with the two liquid interfaces buffeted by a 40 Å vapor layer. Simulations were equilibrated for 1 × 105 Monte Carlo cycles (one cycle corresponds to one attempted move per water molecule with 50% move acceptance) followed by 2 × 106 production Monte Carlo cycles. Lennard-Jones (LJ) and electrostatic interactions were truncated at half the minor simulation box length, Rtrunc ) 9 Å. While potential (18) McQuarrie, D. Statistical Mechanics; Harper Collins: New York, 1976. (19) Widom, B. J. Phys. Chem. 1982, 86, 869. (20) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1987. (21) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces: Proceedings of the Fourteenth Jerusalem Symposium on Quantum Chemistry and Biochemistry; Pullman, B., Ed.; Reidel: Dordrecht, The Netherlands, 1981; p 331.
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truncation introduces orientational changes in the interfacial water structure which affect the electrostatic potential drop across the interface,22 we do not expect these perturbations to affect the interactions between alkanes and water since the alkanes do not interact directly with the SPC water hydrogens. Water configurations were saved every 200 cycles for analysis of thermodynamic averages. Methane and ethane interactions were modeled using the OPLS united atom LJ parameters.23 Cross solute-water interactions were evaluated using Lorenz-Berthelot combining rules. The truncation of solute-water interactions at Rtrunc reduces the adsorption affinity of the alkane for the interface by a factor of ∼35%. Moreover, neglect of these interactions can affect barriers for dissolution, particularly for longer chain species, and influence the determination of mass accommodation coefficients. This truncation error can be corrected as / ∆µcorrect (z) ≈
∫|r-r |>R a
trunc
Faq(r)ΦLJ(|r - ra|) dr (10)
where solute-water pair correlations are assumed to be negligible beyond the truncation radius, Faq(r) is the water density as a function of position, and ΦLJ(r) is the alkanewater LJ interaction. To understand the contributions of repulsive excluded volume and attractive dispersion contributions to the free energy, we consider the solute as interacting with full LJ interactions and as a purely repulsive solute employing the Weeks-Chandler-Andersen division of interactions,24
ΦLJ(r) ) Φrepulsive(r) + Φattractive(r) Φrepulsive(r) )
{
(11a)
ΦLJ(r) + aw r < 21/6σaw r > 21/6σaw 0
Φattractive(r) )
{
(11b)
-aw
r < 21/6σaw ΦLJ(r) r > 21/6σaw
(11c)
where 21/6σaw is the location of the minimum in the LJ adsorbate site-water interaction. Scaled-Particle Adsorption Model. Historically the interpretation of hydrophobic hydration free energies has been divided between excluded volume, or cavity, contributions and longer range attractive interactions, / / (z) + µattract (z) µ/a(z) ) µcavity
(12)
SPT assumes the work for creating a solute sized cavity in water has been approximated by the work for creating the cavity in a hard sphere reference fluid.25,26 As such, the cavity hydration free energy is given as / µcavity (z)/kT ) -ln(1 - η3) +
( ) [
]( )
6η2 daa 12η1 daa 18η22 + + 2 1 - η3 2 1 - η3 (1 - η ) 2 3
2
(13a)
(22) Ashbaugh, H. S. Mol. Phys. 1999, 97, 433. (23) For methane, the LJ diameter and well depth are σMeMe ) 3.73 Å and MeMe ) 1.23 kJ/mol, respectively. For ethane, the individual methyl group LJ parameters are σEtEt ) 3.775 Å and EtEt ) 0.866 kJ/ mol with a carbon-carbon bond length of 1.54 Å. (24) Chandler, D.; Weeks, J. D.; Andersen, H. C. Science 1983, 220, 787. (25) Reiss, H. Adv. Chem. Phys. 1965, 9, 1. (26) Pierotti, R. A. Chem. Rev. 1976, 76, 717.
where
ηi )
π〈Faq〉d iww 6
(13b)
assuming that external pressure contributions, that is, terms dependent on the solute volume (va ∝ daa3), are negligible for molecular species in water at ambient conditions. In the above, expression 〈Faq〉 is the average density of water, and expressions dww and daa are the effective water and solute hard sphere diameters, respectively. This expression assumes the solute cavity is spherical and cannot describe orientational effects which are expected to be important for longer chain alkanes. In a homogeneous liquid or vapor phase, 〈Faq〉 is simply the bulk water density. At an interface, however, the solvent density varies with position. In this case, we assume the solvent density is averaged over the alkane’s excluded volume,
〈Faq〉 )
∫|r-r |d a
Φaw(|r - ra|)Faq(r) dr
aw
(16)
In a bulk phase, this integral reduces to the van der Waals mean-field expression for the attractive energy, that is, / µattract ) -(4πdaw3/3)〈Faq〉. At an interface, however, this integral systematically interpolates interactions between the bulk liquid and vapor phases. While the aqueous density at the interface determined from molecular simulation can be employed in the excluded volume (eq 13) and attractive interaction (eq 15) calculation, we choose to represent the aqueous interface using a Fermi function, (27) Hummer, G.; Garde, S.; Garcı´a, A. E.; Pohorille, A.; Pratt, L. R. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 8951. (28) Hummer, G.; Garde, S.; Garcı´a, A. E.; Paulaitis, M. E.; Pratt, L. R. J. Phys. Chem. B 1998, 102, 10469.
Alkane Adsorption at the Water-Vapor Interface
Faq(z) )
Fbulk aq 1 + exp(z/λ)
Langmuir, Vol. 19, No. 18, 2003 7641
(17)
where Fbulk aq is the bulk density of liquid water and λ is the interfacial thickness. This expression provides a reasonably accurate description of the interface.22 Moreover, it allows us to easily examine the sensitivity of the model predictions to perturbations in the interfacial properties. No significant differences were observed between model predictions made using the simulation density profile or that fitted to eq 17. Results and Discussion The water oxygen density at the liquid-vapor interface as a function of temperature is displayed in Figure 1. The density profiles are sharp and sigmoidal between the liquid (z < 0 Å) and vapor (z > 0 Å) phases. The interface becomes slightly more diffuse with increasing temperature due to increasing thermal fluctuations at the interface. Similarly, the interfacial structure is diminished with increasing temperature, as evidenced by damped density oscillations on the liquid side of the interface. While the appearance of these oscillations is reminiscent of freezing, the waterwater correlations in this region are liquidlike. Moreover, these oscillations disappear with increasing temperature and reappear with decreasing temperature, suggesting they are equilibrium structures. Finally, the bulk density of water (z , 0 Å) decreases with increasing temperature as a result of the expansion of water along the saturation line at the temperatures considered. Methane interfacial adsorption profiles as a function of temperature are shown in Figure 2. Enhanced adsorption of methane at the interface relative to the bulk phases is observed in the vicinity of the Gibbs dividing plane (z ) 0 Å). Notably the peak adsorption occurs on the vapor side of the interface at ∼1.8 Å, a result of attractive dispersion interactions between methane and the bulk liquid. Methane does not significantly penetrate the liquid side of the interface, however, as a result of the meager solubility of hydrophobic species in water. The magnitude of the adsorption peak decreases with temperature indicative of an enthalpically controlled adsorption. Methane adsorption coefficients determined by integration of the adsorption profiles are compared with the corresponding experimental values in Table 1. The simulation values of R are approximately half the experimentally observed values. This suggests that the simulation calculations neglect interactions between the methane and water important in determining the adsorption behavior. A reasonable candidate for this missing interaction is polarization interactions. The potential models used here do not explicitly include electronic polarizability and are only included implicitly in the optimization of the potentials to bulk liquid properties. We note that Wilson and Pohorille13 found excess surface coverages varying between 2.5 and 1-1.7 times less than the experimental values for methanol and ethanol, respectively, comparable to the underprediction found here. This suggests similar origins for the discrepancy with experiment in both simulation studies. Below we examine thermodynamic sources for this difference. The adsorption thermodynamic properties can provide insight into the differences between the experimental and simulation adsorption results. The adsorption coefficient can be expressed as
R ) exp(-∆Gadsorb/kT)
(18a)
Figure 1. Water density profile at the liquid-vapor interface as a function of temperature. z ) 0 Å corresponds to the Gibbs dividing surface (defined as the point at which the solvent excess adsorption is zero). The symbols denote simulation results at 2 °C (filled circles), 25 °C (open circles), and 50 °C (filled triangles).
Figure 2. Methane adsorption profile at the water liquidvapor interface as a function of temperature. z ) 0 Å corresponds to the Gibbs dividing surface. The adsorption profiles are normalized by the density of methane gas on the vapor side of the interface (z > 0 Å). The symbols denote simulation results at 2 °C (filled circles), 25 °C (open circles), and 50 °C (filled triangles). The dashed lines correspond to predictions from the SPT model developed here using the parameters given in Table 3.
where ∆Gadsorb is the adsorption free energy, defined here relative to a standard state of 1 mN/(m atm). Rearranging this expression yields the standard enthalpy and entropy of adsorption,
-ln R )
∆Hadsorb ∆Sadsorb kT k
(18b)
Assuming ∆Hadsorb and ∆Sadsorb are temperature independent, a plot of -ln R versus 1/T yields a straight line with a slope of ∆Hadsorb/k. Indeed, both the experimental and simulation adsorption coefficients plotted in this manner lie on essentially parallel lines (Figure 3a). Most interestingly, the experimental and simulation values of ∆Hadsorb differ by ∼0.1kT and are within the experimental uncertainty (Table 2). Thus the hypothesis that the differ-
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Table 1. Predicted versus Experimental Values of the Adsorption Coefficients
Table 2. Thermodynamics of Alkane Adsorption at the Water Liquid-Vapor Interface
R (mN m-1 atm-1) T (°C) 2 25 50 2 8.2 25 48 50 a
simulation
SPT model
Methane 0.106 ( 0.001 0.120 0.0822 ( 0.0003 0.0897 0.0649 ( 0.0002 0.0678 0.305 ( 0.005
Ethane 0.395
0.219 ( 0.002
0.276
0.162 ( 0.001
0.197
∆Hadsorb (kJ/mol) experimenta 0.194 ( 0.004 0.153 ( 0.003 0.116 ( 0.002
0.550 ( 0.011 0.440 ( 0.009 0.323 ( 0.06
References 6 and 29.
simulation SPT model experimenta
Methane -7.46 ( 0.11 -8.24 -7.60 ( 0.45
∆Sadsorb (k) -5.51 ( 0.04 -5.72 -5.00 ( 0.15
Ethane simulation orientationally averaged perpendicular (θ ) 0) parallel (θ ) π/2) SPT model experimenta a
-9.75 ( 0.22 -8.96 ( 0.23 -10.2 ( 0.2 -10.7 -10.0 ( 0.5
-5.45 ( 0.09 -5.39 ( 0.08 -5.50 ( 0.10 -5.61 -4.88 ( 0.20
References 6 and 29.
Figure 4. Ethane adsorption profiles at the water liquidvapor interface at 25 °C as a function of the orientation of ethane with respect to the dividing place. z ) 0 Å corresponds to the Gibbs dividing surface. The inset figure defines the orientation of ethane with respect to the interface where the angle θ is that between the normal, n, to the liquid-vapor interface and the ethane carbon-carbon bond vector, b. The symbols denote simulation results for methane oriented perpendicular (θ ) 0) (filled circles), parallel (θ ) π/2) (open circles), and randomly (filled triangles) with respect to the interface. The additional results denoted by the open triangles correspond to averaging of the perpendicular and parallel results using eq 20.
Figure 3. -ln R plotted against 1/T. The symbols denote experimental (filled circles), simulation (open circles), and SPT model (filled triangles) results for the adsorption coefficient. The lines are linear fits to the adsorption data using eq 18b. The enthalpy and entropy determined from these fits are reported in Table 2. Results for methane and orientationally averaged ethane are depicted in (a) and (b), respectively.
ences in adsorption are a result of differences in attractive polarization interactions at the interface appears to be erroneous. Rather, the differences arise from the adsorption entropy, with the simulations ascribing an entropic penalty for adsorption approximately k/2 (∆∆Sadsorb ) experiment simulation - ∆S adsorb ) greater than the experimental ∆S adsorb value (Table 2).
Ethane adsorption profiles at 25 °C are shown in Figure 4. In contrast to methane, the orientation of ethane and the longer chain alkanes relative to the interface plays a role in adsorption. Regardless of its orientation, ethane has a stronger affinity for the interface compared to methane as indicated by the larger magnitude of its adsorption peak. The adsorption peak has a significant orientation dependence, with stronger adsorption for ethane lying parallel to the interface. We anticipate this effect will be exacerbated for even longer alkanes, such that configurations in which the solute lies down in the interface dominate the adsorption as a result of the additive effect of attractive interactions between subsequent methylene groups and the aqueous surface.10 The net adsorption, however, results from averaging over all orientations in the interface. The result of inserting a randomly oriented ethane gives an adsorption peak lying between the perpendicular and parallel extremes (Figure 4). It is useful to consider how the net adsorption is related to the adsorption at the extreme parallel and perpendicular configurations. The distribution of ethane orientations
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can be expanded in powers of cos θ, where θ is the angle between the ethane carbon-carbon bond angle and the surface normal (Figure 4). As a result of the symmetry of ethane orientations about θ ) π/2, the full orientationally dependent distribution, expanded up to the third power of cos θ, can be expressed as
g(z,θ) ≈ g(z,π/2) + [g(z,0) - g(z,π/2)]cos2 θ (19) Averaging over orientations yields the average adsorption profile
〈g(z)〉θ ≈
2g(z,π/2) + g(z,0) 3
(20)
Results for this averaging are in excellent agreement with the explicit simulation averaging and validate the accuracy of this orientational expansion (Figure 4). Adsorption coefficients for ethane determined by integration of the orientationally averaged adsorption profiles are reported in Table 2. Similar to methane, the simulations underpredict the experimentally observed coefficients by a factor of approximately 2. Plotting -ln REt versus 1/T yields straight lines for ethane adsorption (Figure 3b). The corresponding ∆Hadsorb and ∆Sadsorb for ethane are reported in Table 2. As for methane, the agreement between the experimentally determined enthalpy of adsorption and that determined from simulation is excellent, lying within the experimental error. Thus we again conclude the discrepancy between the predicted and observed adsorption coefficients is entropic in origin, resulting from an entropic penalty for adsorption approximately 0.6k greater than the experimentally determined value (Table 2). Despite the fact that ethane loses some orientational freedom upon adsorption, this has a negligible impact on ∆Sadsorb. Indeed, the adsorption entropy of ethane does not appear to depend on the orientation at which it adsorbs and is the same as that for methane within the simulation error (Table 2). Furthermore, the difference between the simulation and experimental entropies is essentially the same for both alkanes, suggesting the conclusion that ethane orientations do not impact the entropy. The potentials of mean force for methane and ethane through the aqueous interface at 25 °C are shown in Figure 5. We note that the simulation excess chemical potentials of methane and ethane in the bulk liquid found here (10.0 ( 0.5 kJ/mol and 7.5 ( 1.2 kJ/mol, respectively) are in reasonable agreement with those previously determined by slow growth methods in bulk SPC water (9.8 ( 0.5 kJ/mol and 8.7 ( 0.5 kJ/mol).30 Since much of the computational effort in the present simulations focuses on determining surface effects, the temperature coefficient (e.g., the hydration entropy) and the bulk liquid excess chemical potentials for methane and ethane are not well estimated. Efforts aimed at understanding bulk hydration at numerous temperatures are described in a forthcoming publication.31 The SPT model fit to the simulation results is shown as well in this figure. The fitted model parameters are reported in Table 3. Clearly expulsion of these hydrophobic solutes from the aqueous phase is dominated by excluded volume interactions and less strongly favored by attractive interactions in the bulk aqueous phase. The expulsion from bulk solution is not monotonic, however, (29) Pethica, B. A.; Glasser, M. L.; Cong, E. G. Langmuir, in press. (30) Ashbaugh, H. S.; Paulaitis, M. E. J. Am. Chem. Soc. 1999, 121, 9243. (31) Ashbaugh, H. S.; Pratt, L. R. Los Alamos Technical Report No. LA-UR-03-2144.
Figure 5. Potential of mean force between the alkanes and the water liquid-vapor interface. The results shown are at 25 °C. The points are simulation values of the potential of mean force, and the lines are SPT model predictions. The symbols denote repulsive excluded volume contributions to the free energy (filled circles), attractive dispersion contributions (filled triangles), and the total potential of mean force (open circles). The dashed line indicates the zero reference state in the gas phase. Results for methane and orientationally averaged ethane are depicted in (a) and (b), respectively. Table 3. SPT Model Parameters for Alkane Adsorption dww dMeMe wMe dEtEt wEt
2.80 Å 4.08 Å 0.736 kJ/mol 4.74 Å 0.855 kJ/mol aqueous interface (eq 17)
-3 Fbulk aq (Å ) λ (Å)
2 °C
25 °C
50 °C
0.0336 0.623
0.0329 0.678
0.0323 0.762
and the total potential of mean force profile shows a weak minimum on the vapor side of the interface. This minimum is manifested as the maximum in the adsorption density profiles (Figures 2 and 4). The model does an excellent job at reproducing the simulation results capturing the salient adsorption features. It can be concluded that while dissolution is dominated by the excluded volume contributions, this interaction acts only over the size scale of the solute itself. Attractive interactions, on the other hand,
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are longer-ranged, acting over a few molecular diameters. Surface adsorption, therefore, is a result of the longer length scale of the dispersion interactions between the solute and interface on the vapor side of the interface where excluded volume interactions are negligible. While the difference in length scales for excluded volume and attractive interactions leads to a minimum in the potential of mean force on the vapor side of the interface, the same mismatch in length scales results in a weak maximum in the potential of mean force on the liquid side of the interface. While the simulation evidence suggests the existence of this maximum, the noise in the simulations obscures this feature. This maximum can play a role in the kinetics of adsorption of species like surfactants from the bulk liquid and depends sensitively on the modeling of attractive interaction at the interface. Indeed, we may expect this feature to be enhanced for longer chain adsorbates as a result of the additive effect of lost attractions near the aqueous side of the interface. Boltzmann weighting of the SPT potentials of mean force gives the adsorption density profiles, from which the adsorption coefficients can be calculated. The predicted mean-field density profiles for methane are shown in Figure 2. Despite the excellent agreement between the simulation and SPT model for the potential of mean force, slight differences in the model predict greater adsorption than observed from simulation. Nevertheless, the model predicts a decreasing adsorption peak with increasing temperature. Adsorption coefficients for methane and ethane determined by integration of the adsorption density profile are compared with the simulations and experiments in Table 1. The model predicts adsorption coefficients between those from simulation and experiment. This should not be taken as improved agreement with experiment compared to simulation, however, since the model is parametrized with respect to the simulations. The model does a reasonable job at predicting the adsorption thermodynamics (Figure 3 and Table 2), though it predicts more negative enthalpies, just outside the limit of the experimental uncertainty, and greater entropic penalties for adsorption compared to either experiment or the simulations. Despite its shortcomings, the SPT model allows us to test possible hypotheses regarding the differences in adsorption between the experiments and simulations. As noted above, these differences do not appear to arise from interactions not included in the simulations, such as polarizability, and result from entropic contributions to the adsorption free energy. Two possible sources of this discrepancy could be due to simulation constraints on capillary fluctuations and/or differences between the density of real and SPC water. In our simulations, long-wavelength capillary fluctuations are suppressed by the necessity of incorporating periodic boundary conditions to mimic an infinite slab of water.32 Long-wavelength fluctuations will tend to smear out and thicken the density profile of water at the interface.33,34 This can be mimicked within the mean-field model by increasing the thickness of the interface in eq 17 by increasing λ. The sensitivity of the adsorption coefficient for methane at 25 °C to λ is shown in Figure 6. Increasing λ is found to decrease the extent of methane adsorption, in disagreement with the adsorption coefficient differences. Moreover, adsorption is only weakly depend(32) Mitrinovic, D. M.; Tikhonov, A. M.; Li, M.; Li, Z. Q.; Schlossman, M. L. Phys. Rev. Lett. 2000, 85, 582. (33) Sides, S. W.; Grest, G. S.; Lacasse, M. D. Phys. Rev. E 1999, 60, 6708. (34) Senapati, S.; Berkowitz, M. L. Phys. Rev. Lett. 2001, 87, 1761011.
Ashbaugh and Pethica
Figure 6. Model predictions for the sensitivity of the methane adsorption coefficient at 25 °C on the thickness of the liquidvapor interface and the bulk density of water. The solid line is the effect of increasing λ, and the dashed line is the effect of increasing Fbulk aq . The point indicates the model adsorption prediction at the simulation conditions at 25 °C (Table 3). The arrows indicate the x-axis appropriate for each curve.
ent on this variable, however, such that a 50% increase in λ results in only a 3% decrease in RMe. We may anticipate then that the simulation results would not be significantly diminished if capillary fluctuations were accurately incorporated in the simulation calculations. The bulk simulation density of saturated liquid SPC water at 25 °C was found to be 0.985 g/cm3 ()0.0329 Å-3, Table 3), approximately 1% less than the experimental density of 0.997 g/cm3. Since both the attractive and excluded volume contributions to the adsorbate chemical potential depend on the bulk density (eqs 13 and 16), the discrepancy between the simulation and experimental densities could account for part of the differences in the adsorption coefficient. Indeed, the SPT model predicts that methane adsorption increases with increasing liquid density (Figure 6), in agreement with the differences in the adsorption coefficient. A 1% increase in the bulk liquid density, however, results only in a 1% predicted increase in methane interfacial adsorption. This increase is not sufficient to overcome the factor of 2 difference between simulations and experiment. Moreover, the increased adsorption largely results from increased attractive interactions between methane and water, inconsistent with the entropic origin of the adsorption differences. Summary In the paper, we have reported simulation results for the adsorption of methane and ethane at the water-vapor interface and compared these results with a SPT model for surface adsorption and experimental results for the effect of these gases on the surface pressure. Monte Carlo simulations were used to determine the potentials of mean force for pulling the adsorbates through the interface, which in turn were used to calculate the surface excess. The simulation adsorption coefficients were systematically less than the experimentally observed values. Thermodynamic analysis of the adsorption coefficients showed excellent agreement between the simulation and experimental adsorption enthalpies. Furthermore, a higher entropic penalty for adsorption of approximately k/2 for simulation versus experiment was implicated as the root of this discrepancy. This implies that the simulations
Alkane Adsorption at the Water-Vapor Interface
reasonably capture the energetic interactions at the interface, despite the fact they neglect important multibody interactions, such as surface polarization interactions, which could potentially be significant for an inhomogeneous system. We postulated that possible sources for these discrepancies could arise from the suppression of long-wavelength capillary fluctuations due to the implementation of periodic boundary conditions or differences in the bulk water densities for SPC versus real water. While it is not straightforward to address these issues using simulations, a model based on scaled-particle theory was presented to analyze the sensitivity of solute adsorption on these effects. After the model was validated by comparison with the simulation potential of mean force and adsorption results, the model demonstrated that neither long-wavelength
Langmuir, Vol. 19, No. 18, 2003 7645
surface fluctuations or equation of state differences appear to be significant enough to explain the simulation and experimental discrepancies. Indeed, capillary fluctuations were found to further reduce adsorption, though not enough to significantly change the simulation predictions. Nevertheless, the SPT model was found to be useful for interpreting the adsorption results for simple adsorbates and provides insights into the origin of interfacial barriers for adsorption from bulk solution. Acknowledgment. This work was supported by the U.S. Department of Energy, Contract W-7405-ENG-36, and a Los Alamos National Laboratory Director’s Fellowship (H.S.A.) (LA-UR-03-2141). LA034559Z
