LETTER pubs.acs.org/JPCL
Effect of Hydrogen Bonding between Water Molecules on Their Density Distribution near a Hydrophobic Surface Eli Ruckenstein and Yuri S. Djikaev* Department of Chemical and Biological Engineering, SUNY at Buffalo, Buffalo, New York 14260, United States ABSTRACT: The number and energy of hydrogen bonds that a water molecule forms in the vicinity of a hydrophobic surface differ from their bulk values. Such an alteration gives rise to a hydrogen bond contribution to the external potential field whereto a water molecule is subjected in the vicinity of a hydrophobic surface. This contribution is repulsive and can be determined by using a recently developed probabilistic approach to waterwater hydrogen bonding. Combining the probabilistic approach with the density functional theory, we have found that the hydrogen bond contribution to the external potential plays a crucial role in the formation of a thin depletion layer (of thickness of a molecular diameter and of very low density) between liquid water and hydrophobic surface. SECTION: Statistical Mechanics, Thermodynamics, Medium Effects
H
ydrogen bonding constitutes a key element of many physical, chemical, and biological phenomena of utmost scientific interest.17 It is believed to play a crucial role in the hydration of hydrophobic particles and their solvent-mediated interactions in aqueous solutions (hydrophobic interactions). Various mechanisms have been suggested to understand hydrophobicity at a fundamental level812 and develop a general theory thereof. Although many controversies remain,13,14 there seems to be a convergence of views on its dependence on the length scales of hydrophobic particles involved. The hydration of small hydrophobic molecules (such as argon or methane) is considered to be entropically driven. Such molecules can fit into the water hydrogen-bond network without destroying any bonds,15 resulting in a negligible enthalpy of hydration. However, the presence of the solute constrains some degrees of freedom of the neighboring water molecules, which results in a negative entropy of hydration proportional to the solute excluded volume. Consequently, the hydration free energy is positive and increases with temperature and solute excluded volume. In contact, two hydrophobic molecules affect fewer solvent molecules than when they are far apart. Bringing them sufficiently close to each other should result in a positive change of the system entropy and should thereby lower the free energy of the solution. (Small enthalpy changes are ignored.) Thus, the solvent-mediated interaction of small hydrophobic molecules is also entropically driven. Note that such a mechanism of smallscale hydrophobic effects is now recognized to be overly simplistic. The hydrogen-bond structure is sufficiently flexible so that there remain considerable fluctuations in the adjacent solvent shells.13 Simulations16,17 and theory8 showed that two inert molecules in liquid water would more likely form a solventseparated pair rather than a contact pair (dimer). r 2011 American Chemical Society
Large hydrophobic particles behave differently15,18 upon hydration. When inserted into water, such a particle breaks some hydrogen bonds at the interface. The missing interfacial hydrogen bonds give rise to a large positive enthalpy of hydration and hence to a free energy change proportional to the solute surface area (as opposed to being proportional to the solute volume for small hydrophobes). Because the hydration of large hydrophobic particles is expected to be enthalpically driven, so is their hydrophobic interaction. Fewer water hydrogen bonds are broken when two large hydrophobes are “in contact” than when they are far apart, so there is a negative enthalpy change when such particles approach each other from larger separations. The free energy change (dominated by enthalpy) will be hence negative and will constitute a thermodynamic driving force toward aggregation. Furthermore, it was argued15 that if the solutewater attraction is sufficiently weak, then there may exist a microscopically thin film of water vapor in the immediate vicinity of a large smooth hydrophobic solute. This suggestion has generated much controversy.13,14,1820 There does seem to be an emerging consensus that the depletion layer, when it exists, is only of molecular size.13,14 To shed more light on this issue, we examine the distribution of water molecules near a macroscopic hydrophobic particle by combining the methods of the density functional theory21 (DFT) with the recently developed probabilistic hydrogen bond (PHB) model.22,23The DFT formalism is widely used for studying the fluid density profiles near rigid surfaces.24,25 It usually treats the interaction of fluid molecules with a foreign surface in the mean-field approximation whereby every fluid Received: April 14, 2011 Accepted: May 13, 2011 Published: May 24, 2011 1382
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molecule is considered to be subjected to an external potential because of its pairwise interactions with the molecules of an impenetrable substrate. This external potential gives rise to a specific contribution to the free energy functional. The minimization of the latter with respect to the number density of fluid molecules (as a function of the spatial coordinate r) provides their equilibrium spatial distribution. The effect of a hydrophobic surface on the ability of fluid (water) molecules to form hydrogen bonds had been so far ignored in the conventional DFT. This problem can now be solved by using the PHB approach to the hydrogen bonding ability of water molecules near the surface. Consider a “boundary” water molecule (BWM) in the vicinity of a hydrophobic substrate S (immersed in liquid water). Such a molecule forms a smaller number of hydrogen bonds (“boundary hydrogen bonds”) than in bulk water because the hydrophobic surface restricts the configurational space available to other water molecules necessary for a BWM to form hydrogen bonds. (See Figure 1.) The PHB model allows one to obtain an analytic expression for the average number of bonds that a BWM can form as a function of its distance to the surface. The probabilistic approach considers a water molecule, whereof the location is determined by its center, to have four hydrogenbonding (hb) arms (each capable of forming a single hydrogen bond) of rigid and symmetric (tetrahedral) configuration with the interarm angles R = 109.47. Each hb-arm can adopt a continuum of orientations. For a water molecule to form a hydrogen bond with another molecule, it is necessary that the tip of any of its hbarms coincide with the second molecule. The length of an hb-arm thus equals the hydrogen bond length η, which is assumed to be independent of whether the molecules are in the bulk or near a hydrophobic surface. The characteristic length of pairwise interactions between water and substrate molecules plays a trivial role in the hydrogen bond contribution to hydration or hydrophobic interaction22,23 and hence can be set equal to η. The energy of a bulk (waterwater) hydrogen bond is denoted by ɛb < 0, whereas the energy of a boundary hydrogen bond (BHB) is denoted by ɛs < 0, with no restriction on the latter so that the model proposed is valid independent of whether ɛb < ɛs, ɛb = ɛs, or ɛb > ɛs. (BHBs are mostly believed to be slightly energetically enhanced1113 compared with the bulk ones, but the opposite effect, that is, the weakening of boundary hydrogen bonds, is also suggested to take place.3) Denote the number of hydrogen bonds per bulk water molecule by nb and the average number of hydrogen bonds per BWM by ns (averaged with respect to all possible orientations of the water molecule). The latter is a function of distance x between the water molecule and the hydrophobic surface (assumed to be smooth on a molecular scale): ns ns(x). If x is larger than 2η, then the number of hydrogen bonds that the molecule can form is not affected by the presence of the surface. Therefore, ns(x) = nb for x g 2η. The function ns(x) attains its minimum at the minimal distance between the water molecule and the plate, that is, at x = η. The layer of thickness η from x = η to 2η is referred to as the surface hydration layer (SHL). The function ns = ns(x) can be shown to have the form (for details, see refs 22 and23) ns ¼ k1 b1 þ k2 b21 þ k3 b31 þ k4 b41
ð1Þ
where b1 is the probability that one of the hb-arms (of a bulk water molecule) can form a hydrogen bond and k1 k1(x), k2 k2(x), k3 k3(x), and k4 k4(x) are the coefficient functions
Figure 1. Water molecule (shown as a disk) at distance x from the hydrophobic surface S. The left boundary of the surface hydration layer (SHL) is marked as the plane Sl at distance η from the surface S, whereas the right (closer to the bulk water) boundary is shown as the plane Sr at distance 2η from the surface. Hydrogen bonding arms 1 and 2 are in the Figure plane, whereas arms 3 and 4 are off that plane (one under, the other above). The tips of hb-arms are shown as empty circles. The angle between any two hb-arms is R. The origin O of the Cartesian coordinate system (with the x axis normal to the plate) lies on the plate surface.
that can be evaluated by using geometric considerations, with their dependence on the BWM orientations being averaged. Note that b 1 can be determined22,23 from readily available experimental and simulational data on nb . Expression 1 takes into account the constraint that some orientations of the hbarms of a BWM cannot lead to the formation of hydrogen bonds because of the proximity to the hydrophobic particle. The severity of this constraint depends on the distance of the BWM to the surface and hence the x dependence of k1 , k2 , k3 , and k 4 . It assumes that the intrinsic hydrogen-bonding ability of a BWM (i.e., the tetrahedral configuration of its hb-arms and their lengths and energies) is unaffected by its proximity to the hydrophobic surface so that the latter only restricts the configurational space available to other water molecules necessary for this BWM to form hydrogen bonds. The deviation of ns from nb and the deviation of ɛs from ɛb give rise to a hydrogen bond contribution to the external potential field whereto a water molecule is subjected in the vicinity of a hydrophobic surface. This contribution, Vhb ext, can be determined as hb hb Vext ðxÞ ¼ Es ðxÞns ðxÞ Eb nb Vext
ðη e x < ¥Þ
ð2Þ
The first term on the RHS of this equation represents the total energy of hydrogen bonds of a water molecule at a distance x from the surface, whereas the second term is the energy of its hydrogen bonds in bulk (i.e., at x f ¥). Note that the dependence of Vhb ext on x may be due to not only the function ns(x) but also the x dependence of the hydrogen bond energy in the vicinity of the hydrophobic surface, ɛs ɛs(x). In the PHB model, ns(x) = nb for x g 2η, and hence it is reasonable to assume that ɛs(x) = ɛb for x g 2η as well. Thus, Vhb ext(x) is a very short-ranged function of x such that Vhb ext(x) = 0 for x g 2η. The effect of waterwater hydrogen bonding on the density profile of (liquid) water molecules in the vicinity of 1383
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a hydrophobic surface (assumed to be flat and smooth on a molecular scale) can be now examined by using DFT. In this formalism, the grand thermodynamic potential Ω of a nonuniform single component fluid, subjected to an external potential Vext, can be represented as a functional of the number density F(r) of fluid molecules Z Ω½FðrÞ ¼ dr fh ðFðrÞÞ V Z Z 1 dr dr 0 FðrÞFðr0 Þφat ðjr r0 jÞ þ 2 Z Z þ drVext ðrÞFðrÞ μ dr FðrÞ ð3Þ where V is the volume of the system, μ is the equilibrium chemical potential, and φat(|r r0 |) is the attractive part of the interaction potential between two fluid molecules located at r and r0 . In this expression, the contribution to the free energy due to the shortrange repulsive interactions (the first term on the RHS of the equation) is modeled by the hard-sphere free energy in a local density approximation (LDA), with fh at r being the Helmholtz free energy density of a hard sphere fluid of uniform density equal to F(r). The longer ranged attractive interactions are treated in a mean-field (van der Waals) approximation and represented by the second term on the RHS of eq 3. In an open system (grand canonical ensemble of constant μ, V, and T (temperature)), the equilibrium density profile is obtained by minimizing the functional Ω[F(r)] with respect to F(r), that is, by solving the EulerLagrange equation δΩ/δF = 0, which takes the form Z μ ¼ μh ðFðrÞÞ þ dr0 Fðr0 Þφat ðjr r0 jÞ þ Vext ðrÞ ð4Þ V
where μh(F) dfh(F)/dF is the chemical potential of the uniform reference (hard sphere) fluid of density, F. In the particular case of (fluid) water near a flat hydrophobic surface, one can use the planar symmetry of the system and choose the Cartesian coordinates so that the surface is located in the yz plane at x = 0 with the molecules of the fluid occupying the “half-space” x > 0. The equilibrium density profile obtained from eq 4 is then a function of a single variable x, that is, F(r) = F(x). The substitution of F(x) into eq 3 provides the grand thermodynamic potential Ω of the fluid. As already mentioned, the term Vext on the RHS of eq 4 (and the corresponding term on the RHS of eq 3) had been conventionally meant to represent the external potential exerted by all molecules constituting the hard wall (hydrophobic surface) on a fluid molecule. Various models for the external potential were designed to take into account pairwise interactions of a fluid molecule with molecules of the substrate24,25 as well as the effect of the latter on the pairwise interactions between fluid molecules themselves.26 The contribution of all of these effects (having pairwise nature) to Vext will be denoted by Vpw ext to distinguish it from the waterwater hydrogen bond contribution, Vhb ext. Thus, the overall external potential whereto a water molecule is subjected in the vicinity of a hydrophobic surface can be represented as pw
hb ðxÞ Vext ðxÞ ¼ Vext ðxÞ þ Vext
ð5Þ
Figure 2 presents the typical behavior of Vext and its components. The thick dashed curve presents Vext, where-
Figure 2. Typical behavior of the overall external potential Vext (exerted by a hydrophobic surface on water molecules in its vicinity) shown as a thick dashed-dotted curve and its components, Vpw ext (lower thin solid curve) and Vhb ext (upper thin solid curve). hb as the lower and upper thin solid curves present Vpw ext and V ext , pw respectively. The function Vext(x) is modeled as suggested in refs 24 and 25 ( ¥ ðx < ηÞ pw ð6Þ Vext ðxÞ ¼ Esw exp½ λsw ðx ηÞ ðx > ηÞ
with the parameters ɛsw and λsw characterizing its strength and range. (The latter parameter was set equal to 1/η for simplicity, while ɛsw was taken to be 7.959 1014 erg.) In Vhb ext (x) (see eq 2), the x dependence of ɛs was approximated by a linear function increasing from its minimum value of khɛb at x = η (with kh = 1.1 corresponding to a slightly enhanced hydrogen bonds11,13 for molecules closest to the hydrophobic surface) to its maximum (bulk) value ɛb for x g 2η. This results hb (x) = ɛbnb[(kh (kh 1)(x/η 1))ns(x)/nb 1] for in Vext 13 η e x < 2η and Vhb ext(x) = 0 for x g 2η (with ɛb = 3.3212 10 erg and nb = 3.63 at T = 293.15 K). As clear from Figure 2, the hydrogen bonding contribution Vhb ext(x) to the external potential has a repulsive character unlike the conventional pairwise contribution that has an attractive character (note also that the former dominates the latter in the most part of the range η < x < 2η). The repulsive character of Vhb ext is due to the fact that the total energy of hydrogen bonds per molecule near the hydrophobic surface is smaller (in absolute value) than that in the bulk, which in turn is due to ns(x) e nb and ɛs(x) ≈ ɛb for any x. To find the equilibrium density profile of (model) water molecules in the vicinity of the hydrophobic surface, it is necessary to solve eq 4 using, for example, an iterative procedure outlined in ref 25. Namely, the density profile Fi(x)at the ith iteration is found from the previous one, Fi1(x) via Z μh ðFi ðxÞÞ ¼ μ dr0 Fi1 ðx0 Þφat ðjr r0 jÞ Vext ðxÞ ð7Þ V
For the chemical potential μh(F), one can adopt the wellknown CarnahanStarling approximation24,25,27 ! 8 9ξ þ 3ξ2 3 μh ðFÞ ¼ kB T lnðΛ FÞ þ ξ ð1 ξÞ3 1384
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Lennard-Jones potential 8 > ðr12 < 21=6 dÞ < Eww" # φat ðr12 Þ ¼ d 12 d 6 > ðr12 g 21=6 dÞ : 4Eww r12 r12
Figure 3. Density profiles of a model water fluid near an infinitely large flat hydrophobic surface at T = 293.15 K shown as F(x)η3 versus (x η)/η. (a) Solid curve represents the case where the overall external potential includes both pairwise and hydrophobic contributions (i.e., hb Vext(x) = Vpw ext(x) þ Vext(x)), whereas the dashed-dotted curve is for the case where only the pairwise component is included in the external potential (i.e., Vext(x) = Vpw ext(x)). Both curves are for kp = 2.1. (b) Dashed-dotted, dashed, and dotted curves represent three different values of kp (2.1, 1.0, and 0.5, respectively), with the hydrogen bond contribution excluded from the external potential: Vext(x) = Vpw ext (x).
where ξ = (πd 3 /6)F, d is the diameter of a model molecule of mass m, and Λ = (h2 /2πmk BT)1/2 is its thermal de Broglie wavelength (with h and kB being Planck’s and Boltzmann’s constants, respectively). Because μ h is a single-valued (monotonically increasing) function of F, one can extract F i (x) from the LHS of eq 7 and continue iterations. For a numerical illustration, we obtained the density profiles of the model (fluid) water in contact with an infinitely large flat hydrophobic surface at T = 293.15 K and μ = 11.5989 kBT corresponding to two-phase equilibrium. The liquid state of the bulk water was ensured by imposing the appropriate boundary condition onto eq 7, F(x) f Fl as x f ¥, with Fl being the density of bulk liquid. The densities Fv and Fl of coexisting vapor and liquid, respectively, are determined by solving the equations μ(F,T)|F=Fv = μ(F,T)|F=Fl and p(F,T)|F=Fv = p(F,T)|F=Fl , expressing the requirement that the chemical potential μ μ(F,T) as well as the pressure p p(F,T) must be the same throughout both coexisting phases. The pairwise interactions of water molecules were modeled by using the Lennard-Jones potential. The attractive part φat of the pairwise waterwater interactions was modeled via a well-known perturbation scheme28 for the
where r12 is the distance between molecules 1 and 2 with the energy parameter ɛww = 3.79 1014 erg. (The diameter d of a model molecule was set to be equal to the length of a hydrogen bond.) The hydrogen bonding ability of water molecules was not included in φat directly, but its effect on substratewater interactions is explicitly implemented in eq 7 via the contribution Vhb ext(x) into Vext(x). The density profiles thus obtained from eq 7 are shown in Figure 3. The solid curve in Figure 3a represents the solution of eq 7 with the overall external potential including both pairwise and hydrohb phobic contributions, that is, Vext(x) = Vpw ext(x) þ Vext(x). (Note that the slope of the density profile has a slight discontinuity at x = η, which is the result of a sharp discontinuity in the slope of Vhb ext(x) at the same x.) For comparison, the dashed-dotted curve in Figure 3a shows the density profile obtained by solving eq 7 with only the pairwise component included in the external potential (i.e., with Vext(x) = Vpw ext(x)). Both curves in Figure 3a are for the same coefficient kp ɛsw/ɛww (characterizing the energy of water molecule attraction to the surface at distance η between them relative to ɛww), namely, kp = 2.1. As clear from the comparison of these two curves, hydrogen bonding plays a crucial role in the formation of a thin layer (of thickness of a molecular diameter and of density of vapor) between liquid water and hydrophobic surface. To illustrate further this finding, for the case where the hydrogen bond contribution was not included in the external potential, that is, for Vext(x) = Vpw ext(x), eq 7 was solved for three different coefficients kp equal to 0.5, 1.0, and 2.1. The corresponding density profiles are shown in Figure 3b as the dotted, dashed, and dashed-dotted curves, respectively. As clear, even a strongly hydrophobic surface (kp = 0.5) could not create a depletion layer between itself and liquid water if there were no hydrogen bond contribution to the overall surfacewater interaction potential. In conclusion, it is worth emphasizing that the combination of our previously developed probabilistic approach to waterwater hydrogen bonding with the classical DFT has allowed us to shed some light on the long-standing issue of whether of there is a microscopically thin film of water vapor in the immediate vicinity of a large smooth hydrophobic solute. We have shown that the alteration of the waterwater hydrogen bonding near a hydrophobic surface gives rise to an additional contribution to the external potential exerted by the substrate on a water molecule that plays a crucial role in the formation of a thin, “strong depletion” layer (of density much lower than liquid and of thickness of a molecular diameter, in agreement with previous suggestions13,14) between liquid water and hydrophobic surface, no matter how “weak” the hydrophobicity of the surface is. On the other hand, even for a relatively strong hydrophobic surface the conventional contribution to the external potential (due to pairwise interactions between a water molecule and those of the substrate) cannot cause the formation of a “strong depletion” layer near the surface.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: idjikaev@buffalo.edu. 1385
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’ ACKNOWLEDGMENT We are grateful to Dr. G. Berim for helpful discussions. ’ REFERENCES (1) Pimental, G. C.; McCellan, A. L. The Hydrogen Bond; W.H. Freeman: San Francisco, 1960. (2) The Hydrogen Bond: Recent Developments in Theory and Experiments; Schuster, P., Zindel, G., Sandorfy, C., Eds.; North-Holland: Amsterdam, 1976. (3) Chaplin, M. F. Water’s Hydrogen Bond Strength. In Water of Life: Counterfactual Chemistry and Fine-Tuning in Biochemistry; LyndenBell, R. M., Morris, S. C., Barrow, J. D., Finney, J. L., Harper, C. L., Jr., Eds. [Book in preparation. arXiv:0706.1355 (2007)]. (4) Anfinsen, C. B. Principles That Govern the Folding of Protein Chains. Science 1973, 181, 223–230. (5) Ghelis,C; Yan, J. Protein Folding; Academic Press: New York, 1982. (6) Kauzmann, W. Some Factors in the Interpretation of Protein Denaturation. Adv. Protein Chem. 1959, 14, 1–63. (7) Privalov, P. L. Cold Denaturation of Proteins. Crit. Rev. Biochem. Mol. Biol. 1990, 25, 281–305. (8) Pratt, L. R.; Chandler, D. Theory of the Hydrophobic Effect J. Chem. Phys. 1977, 67, 3683–3704. (9) Widom, B.; Bhimulaparam, P.; Koga, K. the Hydrophobic Effect. Phys. Chem. Chem. Phys. 2003, 5, 3085–3093. (10) Meng, E. C.; Kollman, P. A. Molecular Dynamics Studies of the Properties of Water around Simple Organic Solutes. J. Phys. Chem. 1996, 110, 11460–11470. (11) Silverstein, K. A. T.; Haymet, A. D. J.; Dill, K. A. Molecular Model of Hydrophobic Solvation. J. Chem. Phys. 1999, 111, 8000–8009. (12) Lee, B.; Graziano, G. a Two-State Model of Hydrophobic Hydration That Produces Compensating Enthalpy and Entropy Changes. J. Am. Chem. Soc. 1996, 118, 5163–5168. (13) Ball, P. Water As an Active Constituent in Cell Biology. Chem. Rev. 2008, 108, 74–108. (14) Berne, B. J.; Weeks, J. D.; Zhou, R. Dewetting and Hydrophobic Interaction in Physical and Biological Systems. Annu. Rev. Phys. Chem. 2009, 60, 85–103. (15) Stillinger, F. H. Structure in Aqueous Solutions of Nonpolar Solutes from the Standpoint of Scaled Particle theory. J. Solution Chem. 1973, 2, 141–158. (16) Pangali, C.; Rao, M.; Berne, B. J. Hydrophobic Hydration around a Pair of Apolar Species in Water. J. Chem. Phys. 1979, 71, 2982–2990. (17) Watanabe, K.; Andersen, H. C. Molecular Dynamics Study of the Hydrophobic Interaction in an Aqueous Solution of Krypton. J. Phys. Chem. 1986, 90, 795–802. (18) Lee, C. Y.; McCammon, J. A.; Rossky, P. J. The Structure of Liquid Water at an Extended Hydrophobic Surface. J. Chem. Phys. 1984, 80, 4448–4455. (19) Pratt, L. R. Molecular Theory of Hydrophobic Effects: She Is Too Mean to Have Her Name Repeated. Annu. Rev. Phys. Chem. 2002, 53, 409–436. (20) Chandler, D. Interfaces and the Driving Force of Hydrophobic Assembly. Nature. 2005, 437, 640–647. (21) Evans, R. Density Functional in the Theory of Nonuniform Fluids. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992. (22) Djikaev, Y. S.; Ruckenstein, E. Dependence of the Number of Hydrogen Bonds Per Water Molecule on Its Distance to a Hydrophobic Surface and a Thereupon-Based Model for Hydrophobic Attraction. J. Chem. Phys. 2010, 133, 194105. (23) Djikaev Y. S.; Ruckenstein, E. The Variation of the Number of Hydrogen Bonds Per Water Molecule in the Vicinity of a Hydrophobic Surface and Its Effect on Hydrophobic Interactions. Curr. Opin. Colloid Interface Sci. 2010, doi:10.1016/j.cocis.2010.10.002.
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