J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1941-1951 Hydration of Polar Interfaces A Generalised Mean-field Model Stephan Kirchner and Gregor Cevc* Medizinische Biophysik, Technische Universitat Munchen , Klinikum r.d.1. , lsmaningerstr. 22, D-81675, Munchen 80, Germany A generalized non-local electrostatic model has been proposed for the description of the hydration of arbitrary polar surfaces, including interfaces with a finite thickness. This model is suitable for the description of complex biological surfaces, such as the surfaces of lipid bilayer membranes. It is designed so as to resemble as closely as possible the Gouy-Chapman diffuse double-layer theory, for the sake of the simplicity of model use. The molecular meaning of the model parameters is discussed and the chief determinants of the surface hydration are identified. The concept of simple solvent polarization is shown to be less suitable for the description of the interfacial hydration than the local excess-charge density approach, which corresponds to a generalized, sub- molecular polarization. A general scheme for the calculation of the hydration between two thick, structured interfaces has been developed.The effects of surface structure on the hydration-dependent interfacial repulsion have been investigated. The magnitude and the range of the hydration pressure are shown to increase dramat- ically as a consequence of the water penetration and binding into the interfacial region. Interfacial swelling and dynamics, consequently, may affect the properties of, and the interactions between, structured surfaces.In most current descriptions of hydration phenomena it is customary to treat the interface between the aqueous sub- phase and the hydrophilic surface as a sharp water-perturbing boundary. In their seminal theory of hydration MarEelja and Radii:' have assumed, for example, that the reason for the occurrence of various hydration effects is the inability of the mutually coupled water (layers) near an infi- nitely thin interface to respond locally to the surface-induced structural perturbations. The concept of spatially varying water-order parameter has emerged from this. Early extensions of such a hydration model have retained essentially the same phenomenological picture. Correspond- ingly, their emphasis has been on the elucidation of the most relevant water-waterz or water-surface3 coupling mecha- nisms.Later, the picture of perturbed water structure was reformulated in terms of the distribution of '~ater-defects'.~,~ Various solvent-polarization concepts have also been devel- ~ped.~.~ is the treatment ofA more recent de~elopment~.~ surface polar residues and their associated local-excess charges as the chief source of interfacial hydration. This latter approach permits studies of interfacial hydration as a func- tion of the surface polarity and its spatial distribution.'-'' There is still confusion, however, about the most convenient and reliable formulation of surface-hydration theory. Even the precise significance of various model parameters is as yet unclear.The principal ideas of this paper are similar to those out- lined in ref. 10; here they are extended and formalized in more detail. Moreover, a rationale is given for the practical use of an electrostatic mean-field hydration model. We also propose the application of such a model for the extraction of interfacial polarity profile data from measured hydration force data. Details of this latter application and the corre- sponding computing protocols are described separately,' as are the effects of lateral surface structure.12 Here it suffices to say that, to a good approximation, for the thick interfaces such effects can be neglected. Standard Landau Theory of Hydration One of the most commonly used hydration models was pro- posed by MarEelja and Radik.' This model is based on the Landau postulate for the Helmholtz energy F = (A/C) (q(x)* + A2[dq(x)/dxI2) dx (1)r where A and C are two adjustable model parameters. An early modification of this model3 was aimed at facilitating the identification of the basic hydration-force sources and to permit molecular interpretation of the model predictions.(A more general but related ansatz is given in ref. 13.) d, in eqn. (1) is the separation between two interacting surfaces (the 'interlamellar water layer thickness'), each with an area A; A is the decay length of order parameter q. Boundary condi- tions cause the order parameter value at all polar surfaces to attain a non-zero value (qbulk= 0).The optimum value of order parameter is obtained from eqn. (1) by minimizing the Helmholtz energy. The result permits the hydration (disjoining) pressure between two hydrated surfaces to be derived from the simple thermodyna- mic relation, ph(d,) = (l/NAA)(dF/dd) [or for unit area: ph = (dF/dd)].This yields approximately the magnitude of po being determined by the boundary con- ditions. The 'Landau solution' to eqn. (1)is rather general. In spite of this, its agreement with experimental results is excellent. Owing to its entirely phenomenological origin, however, it is also dificult to interpret at the molecular level. However, an even more severe deficiency of the simplistic hydration-model approach is the assumed constancy of the hydration decay length A.All simple hydration models pos- tulate that this model parameter is determined only by the intrinsic properties of the medium and that it is independent of the interfacial properties. In contrast to this, experiments show that the measured decay lengths are highly variable. For example, the decay length of so-called hydration force between the fluid lamellar phosphatidylcholine multibilayers (L,-phase) is nearly twice as large as the corresponding decay length of this lipid in gel LBr-phase.14 A new mechanism is, therefore, needed to account for this variability. This can be based on the allowance for lateral' 5, l6 or transversal*-" interfacial structure. With this latter idea in mind we have introduced an exter- nal, hydration-inducing perturbation, h, which is spread over a region of finite thickness.(This represents a generalisation of the existing mean-field models of hydration for the case of spatially smeared interfaces.) An example of such a system is phospholipid bilayers in water. The boundary conditions are now no longer specified by the external field itself, but can be determined from the electroneutrality condition. Molecular Basis of the Hydration and Discreteness of Charge Effects All polar molecules have a non-uniform electric charge dis- tribution, even if the whole molecule is uncharged. ‘Local excess charges’ are, therefore, always present in such mol- ecules. This pertains to essentially all molecules which form interfaces as well as to the water molecules.The sites and the strength of water binding to a hydro- philic surface are determined largely by the quantum-mechanical fields that originate from the surface polar residues. Macroscopic fields which stem from the net surface charges or from the surface dipoles and higher multi-poles are weaker by one to two orders of magnitude than the very short-range, local fields that arise from the atomic local excess charges. Consequently, the former are much less important for the hydration of polar surfaces than the latter.? This is seen from the detailed computer simulations of surface hydration. These show that the whole water molecules as well as individual water OH-bonds follow closely the direction of local, atomic electric fields rather than the direction of Cou- lombic or dipolar surface fields.” On the other hand, the distributions of water molecules around the polar residues in protein crystals clearly show that the primary reason for water binding is hydrogen bonding to such residues.’* The principal trends in surface hydration, consequently, can be learned from the detailed electrostatic calculations on the atomic scale.Corresponding fields, or physical quantities related to these fields, as well as the corresponding inter- action potentials are thus suitable order parameters for the electrostatic description of surface hydration. This is tantamount to saying that the proximity of a polar surface alone may perturb the interfacial water structure.This causes the distribution of local excess charges on the bound water molecules to differ from the corresponding bulk values. This results in the accumulation of water-associated local-excess charges which compensates the oppositely charged local-excess charges on the polar residues. This is true for all polar residues in the contact with water and causes the electrical fields near such residues, and near the polar surfaces, to be extremely short-ranged. Computer simu- lations of the water binding to an isolated polar group clearly show this.Ig It is therefore reasonable to describe the polarity of a hydrophilic surface in terms of the surface electrostatics. The corresponding hydration can then be treated as a screen- ing process.Before extending these considerations to the real systems it is worthwhile to remember the basics of standard Gouy- Chapman theory of the charged surfaces in an electrolyte solution. Such theory can be derived from a Helmholtz energy expression that is very similar to (l),“*’if the decay length A is replaced by the Debye screening length A,,. The Gouy-Chapman theory is therefore a useful model for the construction and interpretation of the electrostatic theories of 7 This is to say that the contributions of standard macroscopic electrostatic polarization to the interfacial hydration is, in the first approximation, negligible. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 +-3 ---1 Lid-+* -I -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 distance Fig.1 (a) The distribution of surface-associated net charges (-, +) and the corresponding counterion clouds (curves) near an array of the anionic phosphatidylglycerol, A, or zwitterionic phospha-tidylacholine headgroups, B, in contact with a 0.1 molar monovalent electrolyte solution. The decay length value is 1 nm. (b) The distribu- tion of the water-associated local excess charges near a charged, A and non-ionic, B but hydrophilic, surface in water; the decay-length value of 0.1 nm gives the order of magnitude of the molecular-order decay length in water. hydration. By action of the net surface charges a region of non-zero local ion charge density, the so-called ionic diffuse double-layer, is created near a charged surface.’ The total charge density in such a surface-induced ionic diffuse double layer thus is always identical in magnitude, but opposite in sign, to the total structural surface electric charge.If the separation between two individual net surface charges is smaller than the average range of Coulombic inter- action [Debye screening length A, @ J(area per charge)] it is normally sufficient to consider only surface averages. A uniform surface charge density model then adequately replaces the more realistic discrete-charge approach [Fig. l(a)]. Surfaces with an equal density of positive and negative charges in such a situation behave as if they were elec-troneutral, owing to the mutual compensation of the oppositely charged groups on each surface.No interfacial repulsion is therefore observed between two electroneutral surfaces.? If the average intercharge separation far exceeds the Debye screening length, however, each individual surface charge tends to interact with its proximal ions independently, or nearly so [Fig. l(b)]. Clouds of the positive and negative ion charges then accumulate near the anionic and cationic surface sites, respectively.$ In such a situation even the sur- t Counterion distribution is always relatively uniform; the counterion concentration near a layer of charges is higher, however, than near a layer of zwitterions. In the ‘unscreened’ case pertaining to pure water with a very short characteristic decay length the inter- action of the individual charged segments with their associated counter charges is nearly spatially independent.This is the reason why lipid headgroup dipoles do more than just polarize the water molecules macroscopically. Indeed, the significance of the surface- induced water-dipole reorientation in such systems is rather limited. Hydration of the surface polar groups, consequently, is mirrored in the intra-molecular rather than multi-molecular polarization of the surface-perturbed water molecules. $ If one still wants to work with surface averages, and wishes to neglect the effects of interfacial correlations, the averaging procedure must be done separately for the positive and negative surface charges. J. CHEM. SOC.FARADAY TRANS., 1994, VOL. 90 -repulsion w attraction +-+-+ +-+-+ -+-+-+-+-+ Fig. 2 Schematic representation of the ionic charge distribution near two surfaces with a low density of two equally frequent, but different, types of net surface charge. faces with no net charge may repel or attract each other elec- trostatically, depending on the mutual positions of surface-charge lattices. This is schematically illustrated in Fig. 2. It is clear, however, that these interactions are weakened by the smearing of the charge lattices. Analogously, even in the absence of ions, non-zero local density of the local excess charges exists on the water mol- ecules that are contained in a small, surface-perturbed water volume fraction. This is principally a consequence of the interactions between the surface local excess charges and the water-associated local excess charges.Such interactions effec- tively mimic the screening of the former by the latter type of charges. Modified hydrogen bonding between the surface polar residues, surface-bound, and other water molecules are all manifestations of such local charge interaction pheno-mena. Indeed, any water binding to a polar residue that involves a charge transfer also tends to displace or shift the charges on the surrounding water molecules. Consequently, any such binding leads to at least partial screening of the surface local excess charges. Water alone, therefore, may give rise to the effects that resemble closely the screening of surface charges in an electrolyte solution. This is true on the appropriate scale, at least.The water-associated local excess charges that screen the local excess charges on the polar residues of a non-charged (i.e. zwitterionic) hydrophilic surface are always arranged in a number of distinct, but mutually nearly independent local maxima [Fig. lB(b)]. The size of such maxima is much smaller than the size of the individual phospholipid head- groups. It is also smaller than, or at best comparable to, the thickness of surface adsorbed water layers. The hydration of non-ionic polar surfaces, consequently, is not explicable in terms of simple macroscopic dipolar polar- ization or in terms of the water-dipole ordering alone. Such hydration is imaginable, however, in terms of the redistri- bution of local excess charges on the water molecules located near the hydrophilic surface.This may, but need not, give rise to some dipolar polarization as well. The distribution of water-associated local excess charges in the MarEelja-Gruen-Cevc approximation thus plays the same rde as the distribution of net ionic charges in the surface electrostatic models. Therefore, standard equations of the diffuse double-layer theory can also be adapted for the description of surface hydration. The similarity between Gouy-Chapman theory and the mean-field theory of hydra- tion is shown explicitly in Table 1. It is noteworthy that any approach based on the electro- static mean-field theory is macroscopic.This means that it is crucial to make a careful distinction between the molecular mechanism and theory. Quantities that appear in such a theory are not directly related to the atomic properties of the investigated system. In particular, if one uses the polarization vector (P)approach, the value of P is insensitive to the size of the solvent molecules. Consideration of the finite-size molecu- lar dipole orientation is then impermissible and misleading. Generalized Non-local Electrostatic Landau Theory of Hydra tion Our present formulation of the generalized theory of hydra- tion involves the following assumptions: (1) The Helmholtz Table 1 generalized Gouy-Chapman theory distribution of net surface- [pel(x)] and ion-charges [pi(x)] P(X) = Pedx) + Pi(x) electrostatic potential and electric field dE(x)/dx = P(X)/EE~ d$eI(x)/dx =z -E(x) d2$e1(x)/dx2= -P(X)/EEO linear Poisson-Boltzmann approximation for a planar, infinitely narrow, charged surface def dz$el(X)/dx2= $el(x)/'i d2E(x)/dx2= E(X)/'; $el(X) = (Gel 'd&~o)exP(-x/'d self-consistent spatial distribution of the ionic charges near a single charged surface def -rpi(x) dx = gel= E(x = 0) = EE0$:](X = 0) Helmholtz energy of a charged surface in ionic solution Fel = -3A(EEo/& $,",(x)+ ';C$:,(x)12lW Comparison of Gouy-Chapman and mean-field theories of hydration generalized theory of hydration distribution of surface polarity- [pp(x)] and water-associated local excess charges [p,(x)] PAXI = P,(X) + PJX) hydration potential and hydration (polarization) field dEh(x)/dx = Ph(X)/[EOEm(l-d$h(x)/dx = -Eh(X) d2$h(x)/dx2 = -Ph(X)/[EmEO(l-linear 'Poisson-Boltzmann ' approximation for a planar, infinitely narrow polar surface def d2$h(x)/dx2 = $h(x)/A2 d2Eh(x)/dx2= Eh(x)/A2 $h(X) = (ap A/&m EOk'xP(-x/A) self-consistent spatial distribution of the water-associated local excess charges near a single polar interface def -rp,(x) dx = oP= E,E~(~-E,/E)E,,(x = 0) = E,EO(l -E,,/&)$;(X = 0) Helmholtz energy of a polar surface in water Fh= -$A[E~E,/(~-E,/E)A~]p(x) + A2c$;(X)l2 -2('i/EEO)$el(X)P(X) dx -2[A2/E, -E,/E)l$(X)Ph(X)dx linear Poisson-Boltzmann equation for a charged interfacial linear Poisson-Boltzmann equation for the interfacial hydration zone with finite width potential of an interface with a finite width d211/e,(x)/dx2= $eLX)/'i -PeI(X)/&&O d2$h(X)/dX2= I(lh(X)/AZ-PP(~)/&O -Em/E) The meaning of various symbols is explained in the text.energy of a hydrated system can be described by a Landau- type order parameter expansion ; the coupling between water molecules is given by the fluctuation term -~4~(q’)~,which is proportional to the (medium specific) decay length value A. (2) The Helmholtz energy functional contains a linear contri- bution from the external interfacial field, h(x)= Eh(x) F = C c{q(~)~+ A2[q’(x)I2-q(x)h(x))dx (3) d, being the separation between the interacting interfaces. (3) The physical interpretation of model parameters is based on the assumption that hydration involves an electrostatic screening process.The Euler-Lagrange equation correspond- ing to eqn. (3) is therefore similar to the Poisson-Boltzmann equation. There are, however, some modifications of the con- stants. In eqn. (3) an arbitrary interface of finite thickness is con- sidered. This introduces a second length scale into the model. This allows for the different values of hydration decay length and, in our opinion, is essential for the understanding of experimentally determined hydration force. The generalized mean-field theory of hydration then pro- vides a means for studying of the interaction between two (thermally) smeared interfaces in a polar (associated) fluid medium which transmits the interaction.Steric contributions are neglected in this model but could easily be included.? Lateral surface inhomogeneity is also not considered here but is the centre of our interest in another paper.12 In the following, we thus confine our interest to the quan- tities which are averaged parallel to the plane of the interface. This introduces surface area, A, as a factor in some of our model results. The key to the understanding of the theory used in this paper is the concept of hydration as a screening process. This corresponds to using the following Poisson equation for hydration potential, $,, : (4) This states that the sources of hydration potential are given by the distribution of the external local excess charges, pp, and that they are screened by the internal local excess charges on the solvent molecules; the distribution of the latter is given by p,.[This distribution should be pro- portional to the Boltzmann factor exp(-q$,.JkT).] E~ is the permittivity of vacuum, E the static relative permittivity, and E, the corresponding high-frequency part. The somewhat strange factor on the right-hand side of eqn. (4) stems from the theory of non-local electrostatic^.^^^ In the field picture it includes the quantum modes arising from fluctuations of the polarization field in the so1vent.20n~b In the linear approximation, similar to Gouy-Chapman theory, the screening charge, p,, is proportional to the hydration potential. Comparing the corresponding form of eqn.(4): A2(d2$,/dx2) -$h = -Kpp, with the Euler-Lagrange equation, of eqn. (3), which is: A2(d2q/ dx2)-q = -h/2, then yields W,)= W/2W F{$:(x) + A2C$h(X)I2 -2Kh(x)Pp(x)dx (5) t It is possible to include such effects by introducing a polarity profile which explicitly depends on the thickness d, of the inter- lamellar water layer: p,(x) = pp(x; dw).This is without any influence on the formal development of our model. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 where and A is the total surface area investigated. As is shown in Appendix A, the minus sign pertains to all interfaces with a finite thickness. The plus sign is valid only when ppis a &function. The actual hydration potential is given by the solution to the differential equation A2(d2$Jdx2) -$h = -Kpp (7) The appropriate boundary conditions are discussed further in the text.The expression for the Helmholtz energy can be simplified by using eqn. (7) and a partial integration: F(d,) = --;[‘pp(x)$h(x)dx + surface terms (8) By the virtue of our boundary conditions the contribution from surface terms is zero. This term is thus omitted in the following discussion. The Helmholtz energy written in the form of eqn. (8) can be derived directly from the theory of non-local electrostatics. Appendix C illustrates how this is done. Field vs. Potential Approach For many years, the solvent polarization vector was con- sidered to be the natural order parameter of the mean-field theory of hydration.As is shown in Appendix A, identical model resuls are obtained, however, if the scalar quantities pp and $h are used instead. If the interfacial hydration is viewed in terms of water polarization, the former, field approach is appropriate. In the concept advocated in this work, which deals with the distorted local charge distribution as the basic hydration-associated variable, the hydration potential is to be given preference over the hydration field. Such a choice brings the advantage of using a scalar, direction-independent rather than a vectorial basic quantity. Consideration of the hydration potential, $h , rather than the hydration field also allows a straightforward derivation of the boundary conditions for t,bh, by the use of the condition of electroneutrality. Moreover, as is shown in Appendix C, it facilitates the formal comparison between the mean-field theory of hydration and the mean-field theory of surface elec- trostatics.[In particular, the approximation inherent in the use of eqn. (5) is better perceivable in such a derivation.] Note the following formal differences between the vectorial (polarization) and scalar (potential) approach to the surface hydration theory : the corresponding expressions differ by a factor +A2, which appears in the potential equations, the signs + and -being valid for pp 3 0 and pp # 0, respec-tively.t Interfacial Structure Effects Hydrophilic residues are never located all in one plane [cf: Fig. 3(a)]. Structural and/or dynamic smearing of the inter- facial region is thus often quite appreciable.The conse-quences of this smearing for the surface hydration can best be assessed by comparing the effective thickness of the polar t This is a direct consequence of: ( 1) the fact that the spatial varia- tion of any order parameter occurs on the length scale of A and (2) the fact that potential and field are related through a derivative. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Cs 0 -0.4 0.8 1 I 0.4 I-0.41 "I Id\ I h9 Pfl") / 0 -1.2 -0.8 -0.4 0 0.4 d/nm Fig. 3 Crystal structure of (a) the phosphatidylethanolamine layers at room temperature (from ref. 27) and (b) of the phos-phorylethanolamine crystals at 123 K (from ref.25). (c) The calcu- lated surface local-excess charge density, pp, and (6) the surface hydration potential, t+hb, of the latter. [The illustrated profiles were determined by means of eqn. (11) and (13.) To align panels (a) and (b),the phosphorus atoms were positioned at (0,O).The polarity peak that stems from the polar groups on the diacylglycerol is not shown, owing to the lack of firm information on the corresponding local excess charge density. Inclusion of this peak would give rise to two maxima in the interfacial polarity profile. Decay length used for potential calculations was A = 0.075 nm; higher A-values would cause the interfacial thickness to become even greater. surface region, d,, with the range of the surface-induced ord- ering in water, A ('solvent-order decay length ').We have recently measured the range of the water-mediated interactions between hydrated polar groups directly. This was achieved by so-called 'molecular rulers '. The pH-titration of such bifunctional water-soluble sub-stances of precisely known composition is studied potentiometrically2' or calorimetrically.22 By means of molecular modelling programs the separation between the identical terminal titratable polar groups is determined with an accuracy of better than 0.05 nm.$ From this length and the corresponding separation-dependence of the measured (de)protonation constant the solvent-dependent contribution to the Helmholtz energy was determined. This was again done as a function of the separation between the titrated polar groups.This yielded directly the range of the water- $ Previous attempts to use similar methods for the determination of A have given erroneous and too large values. This was chiefly owing to the ad hoc estimates of the molecular ruler length and to the fact that experimental data from different sources were used. 1945 mediated interactions, and thus a tentative value for the water-order decay length. For a series (n = 24) of different hydrophilic solutes we have found the hydration range of molecular rulers consist- ently to be smaller than 0.1 nm: A = 0.086 f0.003 nm, when E, = 2 was used. Even for the highest reasonable choice of the high-frequency relative permittivity, E, = 5, the water-order decay length value was still quite small: A = 0.144 f0.002 nm between 4 and 85 "C.? Although it may seem strange that the measured hydration decay length of water is smaller than the size of an individual solvent molecule (ca.0.25 nm) this result is in perfect agree- ment with the most recent theoretical results of Matyu~hov.~~ These show that the actual solvation decay length in polar liquids should be of the order of one-half of the solvent size, A z 0.5~.Some of the problems that arise in the non-local electrostatic theories of hydration thus might be due to the use of too large order decay values for water (A 2 a). At this point the ideal 'macroscopic' character of the cur- rently used simplified hydration theories is immediately clear.Many such theories invoke the propagation of an external perturbation of the solvent-order parameter throughout the bulk solvent on the length scale of A. The molecular graini- ness is typically neglected in such theories. The many-body effect, which is called 'hydration', is thus described by a smooth mean field which is also allowed to vary on a sub- molecular length scale. One must be therefore alert to avoid overly simplistic arguments and misleading conclusions based on such models. The explanation of the interfacial hydration solely in terms of simple 'dipole orientation' of water mol- ecules near the polar surface is one example of this. Indeed, the interfacial water layer thickness in phospho- lipid multilamellae, for example, is quite large; it may appre- ciably exceed the bulk value of A.To account for the resulting interfacial structure effects a fixed value of the surface local excess charge density, o,, should be replaced by the corresponding surface polarity profile, p,(r). In the sim- plest approximation this is achieved by introducing its one- dimensional kin, p,(x). In order to find such a spatial polarity profile, the magni- tude and the distribution of the individual local excess charge densities on all polar surface residues has to be known. These densities can be evaluated for each polar group or atom, r, from the corresponding entity charge, Q,, and volume, V,. The standard relation pp,, = Q,/K is used for this purpose. Individual local excess charge values can stem from suitable computer simulation^^^ or, even better, from direct meas~rements~~[cf.Fig. 3(b)].S The required volume values can deduced from Pauling radii (corrected for the effects of thermal smearing). Finally, the integral surface polarity profile is determined by summing up all contributions from the polar residues and by the subsequent surface averaging over molecular area, A, pp(r) = A, ' 1A, ~p,r(r) (9) r where A, = v,2/3 and the prime signifies that only polar groups that are not compensated directly by mutual intra- surface charge-transfer processes are included in the sum. The t Since all these values pertain to the dissolved substances it is impossible that the smallness of A could be due to the lateral- structure effects, which have repeatedly been invoked to explain the small A-values measured with the multilamellar lipid lamellae in the gel phase or for some DNA systems.$ If required and known, the differential physical accessibility to water, 0 d a, < 1, of the individual polar groups, moreover, can also be accounted for: pp,,= x, Q,/K. latter may arise from intra-surface hydrogen bonds or from other direct intra-surface interactions. (This means that only polar residues free to bind water are important for the surface hydration.) According to our data analysis, the relevant water-order decay length A is much smaller than the typical thickness of the phospholipid/water interface: d, z4 to 11 x A.The value of A thus being much smaller than the total length of the common phospholipid polar headgroups, it would seem that every realistic hydration model should allow for the dis- creteness of surface polar residues. The Poisson equation should, consequently, be solved in three dimensions. In this contribution we are primarily concerned with the energetics of surface hydration, however. We therefore neglect the effects of lateral surface inhomogeneity. This is justified in the first approximation at least.? With appropriate precautions the area integration then can be done in advance. This allows working with suitable surface averages without restricting the reliability of the qual- itative model conclusions.” Eqn. (9) then becomes r rx+ax A complex three-dimensional problem has thus been mapped into one dimension solely by means of eqn.(10) and a solu- tion to eqn. (7).$ The result of such a procedure is shown in Fig. 3(c). It is based on the local excess charge information measured for the model phosphatidylethanolamine ‘membrane’: phos-phorylethanolamine crystals at low temperature (123 K, ref. 25). To get the polarity profile shown in this figure each local excess charge value, Qi, was assumed to occupy a volume of qi= 4nRi3/3, where Ri is the van der Waals radius, and to be spread thermally in accordance with the experimentally determined value2 of the corresponding Gaussian Debye- Waller factor. The corresponding charge density values were then normalized with regard to the lipid area.The latter was taken to be constant, A, = 0.56 nm2, for the sake of simpli- city. The surface profile of a lipid layer was thus approx- imated by x exp[ -(7)I] dy where ri is atomic position, Rithe relevant van der Waals radius and ai the measure of the isotropic thermal mean-displacement of an atom i. Since the value of A is small com- pared to the intermolecular distances the absolute value of the local excess-charges were used. Boundary-value Formulation For practical applications it is more suitable to reformulate the theory of surface hydration directly as a boundary value problem. It is therefore convenient to introduce the Green’s function corresponding to eqn. (7). This function is given by t This is less true for a vector order parameter which is also sensi- tive to the effects of changing direction.Hydration models based on the concept of water polarization (vector) are thus more sensitive to lateral inhomogeneity than our present model which uses water binding energy or potential (scalars) as the basic variables. 1For a more realistic study of intersurface interactions it is advan- tageous to know and consider the positions of all polar residues on the interacting surfaces. The discreteness of water binding sites as well as interfacial correlation effects then may become impor-tant.12,15,16,26 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 the solution to d2G(x, y)/dx2 -G(x,y),1A2= 6(x -,v) (12) and relates the spatial profiles of interfacial charge and hydration potential by $h(X) = r.I.3 Y)[l-Kpp(Y)l dy (13) The actual form of Green’s function is determined by the boundary conditions.These must be specified in order to make the solution to eqn. (7) unique. The appropriate boundary conditions are easily derived from the usual condition of system electroneutrality. In the linear, one-dimensional approximation this means that one should have: j p, + p, x 1$h -KPP= 0. With eqn. (7) one then gets 1&(0) = t&(dw) for the first derivatives of the hydra- tion potential. For the identical surfaces this implies 9N)= 4ww) = 0 (14) owing to the system symmetry. By extending this argument one can claim the same boundary condition for any two sur- faces in the one-dimensional approximation.Eqn. (12) together with eqn. (14) determines the Green’s function of our model system. In the general case of two interacting interfaces separated by a solvent layer of thickness d, one then has (15) This expression is suitable for the calculation of all ther- modynamic quantities of our current interest, such as the Helmholtz energy of hydration or the hydration pressure. The Helmholtz energy, for example, can be written as a ‘matrix element’ (16) Likewise, the hydration pressure between both surfaces of interest is calculated from Eqn. (16) explicitly invokes parameters d, and A. It also depends on the separation dependence of p,. This clearly shows that the generalized mean-field theory of hydration contains two length scales: the solvent-order decay length A and the ‘typical interfacial thickness’ d, which depends on the actual form of the surface polarity profile p, .In principle, it should thus be possible to use a set of experimental hydration pressure data in order to deduce the spatial profile of surface hydration potential; from the latter, the surface polarity profile could ultimately be obtained. The fact that surface hydration potential and surface polarity profile are connected through a second derivative or double integration, however, restricts the intrinsic resolution of such an approach [cf. eqn. (7)]. The general Green’s function eqn. (15) is of little practical use. The reason for this is that the hydration pressure equa- tions are much too complicated if they are expressed in terms of this function.More suitable for the evaluation of the surface polarity profile is a conventional ‘linear superposition’ approx-imation. In this approach, the hydration pressure between J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 two (interacting) interfaces at a distance d, is calculated from the change of the Helmholtz energy which results from bring- ing two single interfaces from infinity to a separation dW/2. Fig. 3 illustrates the results of such a calculation. A more detailed description is given in ref. 11, where we derive the surface polarity profile, pp, of a phosphatidylcholine bilayer from the measured hydration force data.? In any case, the problem of a suitable starting interfacial polarity profile must first be solved.In this context it is con- venient to realize" that the application of the Green's func- tion to a polarity profile can be considered as the action of a self-adjoint operator. Therefore, every continuous function which fulfils appropriate boundary conditions can be written as a series expansion in terms of the eigenfunctions C,,(x)of Green's operator a, KPp(4 = c A"Cfl(4 (18)n=O With eqn. (8) and (18) it then follows owing to the orthonormality of eigenfunctions. Eigen-functions that correspond to eqn. (15) and pertain to our model are simple cosine-functions : n = 0, 1, 2, . . . (20) and have eigenvalues 12 UW = 0, 1, 2, . .. (21)A,,= -dt + (n7~A)~ Although the eigenvalue expansion seems to be quite simple one has to realize that both the Fourier coefficients A,, as well as the eigenvalues A,, depend on the interlamellar water layer thickness d, .Practical applications of the expansion (18) are discussed in detail in ref. 11. Discussion We have previously shown that many features of atomic and molecular hydration can be described in terms of a 'mean- field quantum mechanical model of hydration'.6 This was obtained by generalizing and modifying the non-local electro- static model of an ice-like water that was initially introduced by Onsager and Dupuis2* and later extended by Gruen and MarEelja.4 For the sake of convenience, all such models were devised so as to resemble as closely as possible the Gouy- Chapman diffuse double-layer theory.In this paper we discuss in detail the model generalizations which also allow for the interfacial structure effects. The simplifying assumptions of our current model are : First, a continuum solvent model with just one characteristic correlation (or order-decay) length is used. Secondly, the solvent parameters are taken to be independent of separation. (This is probably true for the very short correlation lengths and intramolecular modes considered in this work : librations and vibrations of the water molecules are unlikely to change t In ref. 11, the wrong boundary conditions are used. This affects just the Green's function, however, and has no influence on the resulting p,-profiles.The reason for this is that at all experimentally accessible distances such profiles are much greater than zero. There- fore, no reconstruction of the p,-profile near a boundary is possible. Beyond the boundaries, however, the profile is independent of the boundary conditions. much with the system hydration. This is true for the solvent- order decay length A as well as for the high-frequency rela- tive permittivity E, .) Thirdly, only a one-dimensional surface polarity profile is considered. No account is thus made for the lateral surface structure effects. This assumption is clearly an oversimplification which is relaxed in our next paper.I2 Fortunately, it does not affect our qualitative conclusions, however, owing to the scalar nature of the order parameter used.The formal parallelisms between the generalized Gouy- Chapman29 and the generalized mean-field hydration model8 are given in Table 1.It shows that all crucial equations of the non-local electrostatic hydration model closely resemble the corresponding equations of surface electrostatics. This simi- larity indicates that in both cases a similar physical mecha- nism is at work: the electrical screening of the interfacial charges. The only exceptions are expressions which depend on the distribution of the water-associated excess charge density, which contain a factor (1 -E,/E) instead of a simple relative permittivity E. This difference is a result of an 'ad hoc' electrostatic representation of the interactions between water molecules. This factor is the smallest possible tribute to the quantum- mechanical nature of molecular hydration.20 It ensures that the surface-induced accumulation of the water-associated local excess charges, p,,,(r),and the hydration-dependent field Eh(r) both disappear if the solvent structure is neglected. Water susceptibility for the interaction with the surface polar residues then, of course, also attains a zero value.In such situations, the aqueous sub-phase acts as a simple dielectric with E, -+ E. However, normally the situation is different. Polar surfaces immersed in water start to bind the solvent molecules. While doing so, they swell in all directions. In the majority of cases, the swelling proceeds chiefly, but not exclusively, in the direc- tion perpendicular to the hydrophilic surface.14 In the initial stages of water uptake by the phospholipid bilayers, for example, the molecular area' and the thickness of the hydrated headgroup region3' both increase strongly. For the phospholipids with two acyl chains the hydration-induced area expansion is up to 0.2 nm2 in the gel phase and up to 0.3 nm2 in the fluid lamellar phase, precise values depending on the lipid type and experimental conditions.The accompany- ing change in the interfacial thickness is estimated to be between 0.2 nm and 0.8 nm.I2 These values compare nicely with the thickness of the polar region that was deduced from the lipid crystal data (cf:Fig. 3).An assumed interfacial thickness of ca. 1 nm is thus cer- tainly not an exaggeration. This implies that between 30 and 75% of the 'interbilayer water layer thickness' can be reached, and possibly swept, by the hydrated lipid heads. These heads, consequently, must contribute to the total inter- facial repulsion at least at short and intermediate separations. It is important to realize, however, that this is not a standard, hard-core, steric repulsion discussed previously by Simon and co-worker~.~'It is also not identical to the protrusion force studied by Israelachvili and Wenner~trom.~~ The repulsion described in this work is driven and mediated by the polar headgroup hydration. In the absence of water our model reveals no net interfacial repulsion.If such a repulsion did exist dry lipid crystals would be unstable. Fig. 3 suggests that the interfacial fine-structure effects are masked in the surface potential and interfacial force profiles. The latter profiles are always smoother than the underlying polarity distribution profile. This is due to the integration which must be performed in order to obtain the surface hydration potential from the known surface-charge distribu- tion. The irregularities in damping are also a consequence of the fact that our present Landau form of the expression for the Helmholtz energy contains a gradient term which sup- presses the effects of all small-scale variations. In previous sections we have argued that Gouy-Chapman diffuse double- layer thesry is based on a very similar Helmholtz energy functional. From this theory and numerous computer simula- tions of the ionic double layers it is known that the surface potential profile is very insensitive to the fine details of the charge distribution in the interfacial region.33 Our previous suggestion” that the interfacial polarity profile is mirrored in the interfacial ‘hydration force’ thus should be interpreted properly. Rather than being a simple mirror, the interfacial force is an integrator of the overall surface hydrophilicity.However, this ‘integrator’ is relatively insensitive to the fine interfacial structural details. To illustrate the influence of a surface polarity profile and this thickness on the hydration phenomena, we have calcu- lated the hydration pressure for several model profiles: a box (with a variable width b), an exponentially decaying profile (with several decay lengths dp), and a Gaussian profile (with varying thickness ndJ2).All these models, their correspond- ing analytical expressions, and the results of our hydration force calculations are given in Fig. 4-6, respectively. They were all obtained by assuming that A = 0.1 nm. Comparison of the results from Fig. 4-6 shows that the ‘thickness’ of surface polarity profile may influence the final hydration pressure in a number of ways. For a box-like inter- face it only leads to an increase in the pre-exponential factor po. This is due just to the renormalisation of the effective 0.6 .-0.3 0.0 I I I II I I (b110 I I I I I I I I 1 0 1 2 3 4 d,/nm Fig.4 Model calcu lati ons for a box prof ile giv en by p,(x) = Q(@b - - -x)O(x) + 8[x (d, b)]8(dW-x)}, where x = 0.. d,, 8 is the usual step-function and b is the box width. (a)Polarity profile, p,(x) and (b) calculated hydration pressure as a function of the inter- lamellar water layer thickness, d,. Lines (from top to bottom) given forb = 0.25,0.15, 1.0, 0.08 and 0.04 nm. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 0.0 n h X vs’ Fig. 5 Model calculation for the exponentially decaying surface polarity profile given by p,(x) = N(d,)p,(?), with p,(?) = exp(-d,/2dp)cosh[(x -dW/2)d]-exp(-d,/d,) with the normal-ization constant N(d,) = Q/I$p,(?) dx.(a)Polarity profile pp(x) and (b)calculated hydration pressure as a function of interlamellar water layer thickness and exponential decay constant d, . Both were evalu- ated by means of the commercially available software package MAPLE V. (-) Total polarity profile and (---) polarity profile of a single interface. distance between the interfaces. However, a box-like polarity profile does not affect the apparent decay length of hydration pressure. On the contrary, for the exponential as well as for the Gaussian surface polarity profile, the decay length of hydration force is modified by the thickness of the interfacial region. In the case of an exponential surface polarity profile the result is essentially the same as that reported in ref.10: the calculated hydration pressure contains three terms. One depends on the water-order decay length A, one on the decay length of the surface polarity profile, d,, and the third on a mixture of both. The hydration pressure as a function of the interlamellar water layer thickness at large distances is always determined by the larger of these two characteristic length parameters. The results for the Gaussian surface polarity profiles are more complicated by also reveal the existence of two different decay regimes. At short distances the spatial decay of hydra- tion pressure is nearly exponential, even for a Gaussian polarity profile. The characteristic decay length of hydration force then increases with the interfacial thickness. Even an interfacial thickness which is only one-half of the water-order decay length increases the effective range of hydration pheno- mena appreciably. At large distances all profiles are parallel.Over small intervals the corresponding curves can be approx- imated by exponentials. The point at which this becomes pos- sible gets larger with increasing thickness of the surface polarity profile. In Fig. 5 the hydration pressure profiles for surfaces with an exponential polarity profile are shown. The pressure value J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.2 , 1 X 1o-2 d,/nm Fig. 6 Model calculation for a Gaussian surface polarity profile: PPH= exp{ -Cx -(~~p/2)12/(ndp/2)2}+ exp( -{x -Cd, -(.4/2)21}/[nd,/2I2).A, Polarity profile pp(x)and B, hydration pressure between two interacting surfaces with such polarity profile as a function of the interlamellar water layer thickness d,.(ndJ2) = (a)0.4, (b)0.3,(c)0.2, (d)0.1, (e) 0.05 and cf)0.02. (9)exp(d,jA) where A = 0.1 nm. diverges at short separations. However, if the area under the polarity profile decreases with d, +0 this divergence disap- pears (not shown). The latter situation is illustrated in Fig. 6 for a Gaussian surface polarity profile of variable height. For such a profile the hydration pressure at low distances goes to zero. We believe that the reality lies somewhere between these two limiting cases but closer to the latter. At short distances, which correspond to a low total water content, one does not expect the surface polarity profile to have a constant height.Indeed, if the total surface charge was a constant non-zero value, lipid drying would be nearly impossible. On the other hand, at large distances (i.e. at high water content) the surface charge distribution and accessibility is probably independent of the water content between the interfaces. This question of normalization is only of minor importance for practical pur- poses since differences in the calculated hydration pressures occur only at small distances. Realistic model calculations should allow for interfacial swelling during water uptake. The neglect of this phenome- non may provoke quantitatively false conclusions such as an underestimation of the effective decay length of the hydration force.However, the principal trends are always the same. We believe that our conclusions are relevant for most, if not all, hydrophilic systems. Interfacially bound water in such systems is known either to increase the average separation between the individual polar headgroups and the mobility of polar residues on such headgroups or else to shift the polar- apolar interface away from the solvent and into the ‘surface depth’. A more detailed account of these effects, as well as a discussion of the corresponding temperature variations, will be published elsewhere. Solvation force measurements in non-aqueous solvents34 imply an apparent correlation between the ‘solvation force decay length’ and the size of the solvent molecule.Our con- clusions are not necessarily borne out by such results. First, the interfacial swelling, and thus the interfacial thickness, is also likely to depend on solvent size; solvent partitioning and binding in the interfacial region are the reasons for this. Sec- ondly, from the three experimental points measured to date no general conclusions can be drawn. We believe that the validity of our conclusion is hardly affected by the neglect of the lateral surface structure. This is not surprising since the results of all models that are based on the vectorial order parameters are far more sensitive to the lateral surface structure than the models based on the use of scalar order parameters.Indeed, the corresponding insensi- tivity of the main conclusions of this study has been directly confirmed in an investigation in which the variations of the surface polarity parallel and perpendicular to the surface plane have been allowed for simultaneously.’2 In summary, we have shown that it is possible to describe and understand the hydration of polar surfaces either in terms of the surface hydration potential or else, but less pref- erably, in terms of its associated hydration field. We have derived several equations which relate these two basic quan- tities and have shown that they are formally identical to the results of standard electrostatic diffuse double-layer theory. Based on our model calculations, as well as on the experimen- tal data of other authors, we have concluded that the thick- ness of the polar membrane region (d, > 1 nm) is much greater than the typical water-order decay length (A % 0.1 nm).We suggest that any change of the interfacial thickness should modify appreciably the surface polarity profile and the hydration-dependent intermembrane repulsion. This pro- vides a rationale for the explanation of the measured hydra- tion force data. Financial support by the Deutsche Forschungsgemeinschaft (under grants Ce 2/1, Ce 19 5/1-1 and SFB 266/C8) is grate- fully appreciated. We would also like to thank A. A. Korny-shev for very stimulating discussions. Appendix A: Hydration in the Field and Potential Approaches In the field approach the Helmholtz energy of the hydration is described by the general expression’ F = s {El(x) + A2[VEh(x)l2-2Eh(x)Ep(x))dx (A1) Eh is the vectorial order parameter and E, the external per- turbing field.Auxiliary constant r is defined as r = -+A[E~E,(~ + _+A/2K,the use of positive and -E,/E)]/~A* negative signs being described in the main text. By minimizing eqn. (Al) in three dimensions the following Euler-Lagrange equation is obtained A2AEh-Eh = -Ep -vr\(VAEh) (A21 where A indicates the vector product. Since Eh = -vrl/h E, = -v$ ho 043) it also follows from eqn. (A2) that V(A2Arl/, -rl/h + $ho) = 0. This is equivalent to A2A+hh-rl/h = -lCIho + c where c is a constant. After the introduction of $co = $ho -c eqn. (A3) can be recast in the following form? A2A$h -$h = -$to (A4) which corresponds to the Euler-Lagrange equation of func- tional The Helmholtz energy of the investigated system can thus be described in two, almost equivalent, ways.First, the vecto- rial surface hydration field can be used as an independent system variable and a basic order parameter; secondly, this r61e is taken by the scalar surface hydration potential. To obtain two consistent sets of differential equations that describe the thermodynamic optimum of the system one should, furthermore, require that the boundary conditions in both cases are consistent. This implies that Vn $h lav = Eplav (A61 where n is the outer normal direction at the boundary dV. This relation connects the boundary conditions in the field and potential approaches.From the definition (A3) it is clear that the potential $h is a mean-field potential which includes contributions from the external as well as internal fields. What remains to be done is to find a connection between the pre-integral factors in the Landau expansion for the Helmholtz energy, when the latter is expressed in terms of the surface hydration field and hydration potential, respectively. To achieve this goal, a series of formal manipulations on the functionals F and I;* is made. The results of these manip- ulations are then combined with Euler-Lagrange equation for the surface hydration potential$ to get Starting with eqn. (Al) in three dimensions, setting again -Eh= -v$h, Ep = -Vt,bh0, r = ~~e,(l E,/E)/~, and per- forming partial integrations with regard to v$h and v$ho, one gets F = r bA$h(A2A$h -$h -k 2$h0) d3x + l,(1.-2$ho)v$h dS (A8) Combination of eqn.(A7) and (A8) then yields I-/. The same procedure applied to eqn. (A5) gives /. /. Now, F as well as F* describe the thermodynamic behav- iour of the same physical system. These two quantities, there- fore, must be identical, up to a constant term. This also pertains to their dimensions. Expressions (A9) and (A10) consist of sums only. The previous requirement is thus also for all individual terms. In order to find the connection between r and r*it is, consequently, enough to compare all t,$ho is a short notation of KPp. $ Owing to the fact that the constant difference between $h and t,$z depends only on the choice of the zero point of potential, and has no physical significance, the potentials $,, and t,$z from now on are taken to be identical.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table A1 Prefactors r and r*in 1 and 3 dimensions n=l n=3 equivalent terms. In the following, we use the volume inte- grals as an example. Substituting $hoA$h = $ho($h -$ho)/h2 [from eqn. (AT)] into eqn. (A9) gives + l>$h -2$ho)V$h dS (A11) If the potential $ho is non-zero, one gets, by comparison of eqn. (A10) and (All), r*= -T/A2; however, in the case of vanishing potential, $ho = 0,r*= +r/A2. Explicit expressions for the constant r are given for some cases in Table Al. The absolute values of all constants in the non-local electrostatic model are given in this table for the case of non-zero surface potential or external field (hf 0).In the one-dimensional case, the surface area A becomes a multiplicational factor owing to the fact that the area integra- tion is done in advance. The derivation given in this appendix is based on the use of one, standard thermodynamic minimization requirement and on the simple electrostatic relations between the local fields and potentials. No reference is made to the dimensions of the investigated system or to its boundary conditions. Final model results, consequently, are applicable to the isolated surfaces in water as well as to the interacting surfaces. Appendix B: External Hydration Potential and the Surface Polarity Profile In previous publication^^.^.^ the one-dimensional field approach was used.The external field was then assumed to be confined to the polar surfaces only, i.e. E, =f16(x + d/2) +f2qX-42).3 In the generalized model discussed in this work such an assumption would lead to rather unphysical situations, such as an infinitely high electrostatic potential in all space. To avoid this difficulty eqn. (A3) is useful. In the present case it gives E, = -KVp. In the one-dimensional approx- imation discussed in3 it corresponds to E2 + A2(E’)2+ ~EKAcT, (W if the density p is assumed to be purely a surface term. From the relation f(x)d’(x) = -f’(x)d(x) one gets: h2E” -E = E’KAo,[G(x + d/2) + 6(x -d/2)].However, the delta func- tions contribute only at the boundaries; integration across each surface thus affords the appropriate boundary condi- tions for the field derivative, E’. Within a sign, these are iden- tical to those given in ref. 3. In such a special case the distinction between the surface hydration potential or the surface charge density is somewhat arbitrary. After all, Poisson’s equation implies that a step- function-like potential is related to the charge density through 6’. With properly adjusted constants both interfacial descriptions are thus possible and consistent with each other. However, as soon as the hydrated interface becomes an interphase and attains a finite thickness, the interpretation of J. CHEM. SOC.FARADAY TRANS., 1994, VOL. 90 i+!ie0 as an electrostatic potential is no longer possible. For the thick interfaces (‘interphases’) only the terminology based on the charge distribution is appropriate. Appendix C: Generalized Theory of Hydration and Non-local Electrostatics The simplest method of obtaining eqn. (8) for the Helmholtz energy of hydration from the theory of non-local electro- statics is to begin with which is the general starting point of any theory of screening. Eqn. (Cl) relates the Fourier transforms of the potential, $(q), of the charge distribution p(q), and the (spatially varying) relative permittivity c(q). This relation is derived in nearly every textbook on solid-state physics (cf.ref. 35). Although the actual form of E(q) is not known this function has some very general features based on its physical proper- ties.(See ref. 36 for a thorough discussion of this aspect.) The simplest approximation for the relative permittivity of a polar fluid is given by so-called ‘single pole approximation’, which, together with eqn. (C2),results in Since q2$(q) is the Fourier transform of -A$(x), eqn. (C3) after the transformation into real space gives the integro- differential equation -A2A$(x) + $(x) = -A2 p(x) + -1 17exp(-iqx) dq ‘Q EEO (C4) This result is very similar to eqn. (7),except for the existence of the integral term. Neglect of this term together with the general electrostatic relation F x $(x)p(x) d3x then yields eqn. (8) for the Helmholtz energy, In the light of this deriva- tion the main simplification of the mean-field hydration theory is thus the use of a simplified relation between the charge distribution and the electrostatic potential.The differences in constants that appear in eqn. (7) and in the approximate equation eqn. (C3) are not important from the practical point of view. The reason for this is that E z 80 and E, z 2, so that (1 -E,/E) x 1. Furthermore, the given ‘derivation’ totally neglects the anisotropy of the system, (C2) pertaining to a homogeneous medium. Note that (C2) is only one special possible form of E(q). This means that even the most general or exact hydration model that is based on the concepts of non-local electro- statics relies on several ad hoc assumptions.? We are not t There is another possibility to derive the free energy of hydration within the framework of the non-local electrostatic model.This approach (ref. 12) works in q-space and uses a more general expres- sion for E. The hydration potential in such an approach is formally calculated for a very general Green’s function. For the numerical cal- culations, however, it is necessary to use a truncated Green’s function which is nearly identical to ours. aware of any first-principle arguments in favour or against a given plausible choice of the required model constants. It was, therefore, solely for the sake of convenience to keep the same definitions of constants as in our earlier papers, such as ref. 6. References 1 S. MarEelja and N.RadiE, Chem. Phys. Lett., 1976, 42, 129. 2 S. MarEelja, Croat Chem. Acta, 1977,49, 347. 3 G. Cevc, R. Podgornik and B. ZekS, Chem. Phys. Lett., 1982,91, 193. 4 D. W. R. Gruen and S. MarEelja, J. Chem. Soc., Faraday Trans. 2, 1983, 79, 225. 5 G. Cevc and D. Marsh, Biophys. J., 1985,47,21-31. 6 G. Cevc, Chem. Scr., 1985,2597. 7 G. Cevc and J. M. Seddon, in Surfactants in Solution, ed. K. L. Mittal and P. Bothorel, Plenum Press, New York, 1986, vol. 4, pp. 243-255. 8 G. Cevc and D. Marsh, Phospholipid Bilayers, Wiley, New York, 1987. 9 G. Cevc, Biochim. Biophys. Acta, 1990,1031-3, 3 11-382. 10 G. Cevc, J. Chem. SOC.,Faraday Trans., 1991,87,2733. 11 S. Kirchner and G.Cevc, Langmuir, submitted for publication. 12 M.Hauser, A. A. Kornyshev and G. Cevc, 1994, submitted for publication. 13 R. Podgornik, Chem. Phys. Lett., 1989, 163, 531. 14 R. P. Rand and V. A. Parsegian, Biochim. Biophys. Acta, 1990, 988,351. 15 A. A. Kornyshev and S. Leikin, Phys. Rev. A, 1989,40,6431. 16 A. A. Kornyshev and S. Leikin, Phys. Rev. A, 1991,44, 1156. 17 K. Zakrzewska, in Physical Chemistry of Transmembrane Ion Movements, ed. G. Spach, Elsevier, Amsterdam, 1983, pp. 45-66. 18 N. Thanki, J. M. Thornton and J. M. Goodfellow, J. Mol. Biol., 1988,202,637. 19 (a)A. Pullman, B. Pullman and H. Berthod, Theor. Chim. Acta, 1978, 47, 175; (b) A. A. Kornyshev, in The Chemical Physics of Solvation, ed. R. R. Dodonadze, E. Kalman and A. A. Korny- shev, Elsevier, Amsterdam, pp. 77-1 18. 20 (a) A. A. Kornyshev, J. Chem. Soc., Faraday Trans. 2, 1983, 79, 651; (b) R. R. Dogonadze, A. A. Kornyshev and A. M. Kuznet- sov, Theor. Mat. Fiz., 1973, 15, 127. 21 G. Schwarzenbach, Pure Appl. Chem., 1970,24, 307. 22 M. Hirth and G. Cevc, to be published. 23 D. V. Matyushov, Chem. Phys., 1993,173, 199. 24 G. Peinel, Chem. Phys. Lipids, 1975, 14, 268. 25 S. Swaminathan and B. M. Craven, Acta Crystallogr. B, 1984, 40, 511. 26 S. Leikin and A. A. Kornyshev, J. Chem. Phys., 1990,92,6980. 27 M. Elder, P. Hitchcock, R. Mason and G. G. Shipley, Proc. Royal SOC. London, A, 1977,354, 157. 28 L. Onsager and M. Dupuis, in Electrolytes, ed. B. Pesce, 1962, p. 27-46. 29 G. Cevc, S. Svetina and B. ZekS, J. Phys. Chem., 1981,85, 1762. 30 C. A. Helm, P. Tippman-Krayer, H. Mohwald, J. Als-Nielsen and C. Kjaer, Biophys. J., 1991,60, 1457. 31 T. J. McIntosh, A. D. Magid and S. A. Simon, Biochemistry, 1989, 28, 17. 32 J. N. Israelachvili and H. Wennerstroem, Langmuir, 1990,6, 873. 33 S. L. Carnie and G. M. Torrie, Adu. Chem. Phys., 1984,56, 141. 34 T. J. McIntosh, A. D. Magid and S. A. Simon, Biochemistry, 1989,28,7904. 35 N. W. Ashcroft and N. D. Mermin, Solid State Physics. Holt-Saunders, Philadelphia, International edn., 1976. 36 A. E. Blaurock, and T. J. McIntosh, Biochemistry, 1986,25,299. Paper 3/05964B; Received 5th October, 1993