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Langmuir 1987, 3, 648-654
Structural Forces as a Result of Nonlocal Water Polarizability? M. L. Belaya, M. V. Feigel'man,* and V. G. Levadny Landau Institute for Theoretical Physics, Moscow 117940,USSR Received December 12, 1986 A macroscopic theory of the structural (hydration) interaction between lipid membranes is proposed. The theory is based on nonlocal water electrostatics and generalizes the Gouy-Chapman theory of the electric double layer for the case of dipole-covered surfaces placed into a nonlocal polarizable electrolyte. It is shown that the structure of the electric double layer in that system differs substantially from the usual one. The behavior of hydration forces in pure water as well as in electrolyte solution is calculated and compared with experimental data.
I. Introduction It is now well-known that the properties of water in thin layers differ substantially from properties of bulk water.1-3 These properties of water are very important for various biological processes which are connected with the interaction and fusion of cell membranes. Intermembrane interactions have been extensively investigated in various model systems. One of them is multilamellar phospholipid dispersion, i.e., a pile of planar phospholipid bilayers, separated by water. DLVO theory was found to be valid for such systems if the thickness h of the water layer between two bilayers is larger than approximately 30 A. Strong exponentially decreasing repulsive forces have been shown to be dominant at distances h 6 30 A.67 The decay length of this force is about 2.5-3 A and the preexponential factor is of the order of (3 X 109)-(3 X 1O'O) dyn/cm2, both values being weakly dependent on the membrane charge and electrolyte concentration. These forces are in some sense similar (up to quantitative differences) to structural forces which were discovered in colloid ~ystems.~BIn biophysics these forces are known under the name "hydration forces". In the first theory of the structural forcesgit is assumed that the hydrophilous bilayer surface affects nearest water layers. This modified water can be described by some order parameter that decays exponentially with the increasing distance from the bilayer. It was then proposed to identify this order parameter with the normal component of polarizability.'OJ1 Gruen and Marcelje" have based their theory on the expression for the free energy density which was deduced from the statistical icelike water model with Bjerrum's defects. Jonsson and Wennerstrom12 have proposed a theory based on the interaction of discrete dipoles of lipid molecules with their electrostatical images "reflected" at the opposite bilayer. Sornette and Ostrowsky15have developed Helfriech's approach16and have connected structural forces with thermal fluctuations of the elastica1 bilayer in the presence of the other bilayer. Israelashvili and Sornettel' have demonstrated the existence of quantitative relation between the effective size of the lipid head and the magnitude of the hydration forces. Recently the molecular structure and properties of water have been investigated intensively. It seems reasonable to suppose that water has a local molecular structure with a typical size larger than the intermolecular distance.I8 In this connection there arises the problem of the dielectric response of water to the static strongly inhomogeneous electrical field. It was assumed that this response post Presented at the "VIIIth Conference on Surface Forces", Dec 3-5, 1985, Moscow; Professor B. V. Derjaguin, Chairman.
sessed the property of essential spatial dispersion in the case of a short characteristic length of the applied electric field's v a r i a t i ~ n . ' ~ In ? ~this ~ connection one can wonder about the applicability of usual local electrostatics to the water and water-containing systems. In this paper we shall analyze the effects of nonlocal polarizability in waterllipid systems. We shall examine the characteristics of the single bilayer placed into the water solution, as well as the interaction between two bilayers. We shall propose the macroscopic phenomenological theory of structural (hydration) forces based on the conventional electrostatics equation, the only modification being nonlocality of the water dielectric response. In the framework of this approach, it will be shown that surface membrane dipoles produce a rather strong rapidly decaying electric field. That is the field that leads to the appearance of structural (hydration) forces. Note that the neutral (covered with dipoles) surfaces being placed into the usual local polarizable media would not interact at all. We consider intermembrane interactions in pure water as well as in an electrolyte solution. In the latter case we obtain the repulsive force as a sum of two terms with different decay lengths. We discuss also the possibility of (1) Derjaguin, B. V.; Kusakov, M. M. Izv. Akad. Nauk USSR (Ser. Chem.) 1937,5, 1119; Acta Phys. Chem. USSR 1939,10, 153. (2) Derjaguin, B. V.; Landau, L. D. Zh. Eksp. Teor. Fiz. 1941,11,802; 1945,15,663. Derjaguin, B. V. Kolloid 2. 1955,17,207; Chem. Scr. 1976, 9, 97. (3) Derjaguin, B. V.; Churaev, N. V. Dokl. Akad. Nauk USSR 1972, 207, 512. (4) Derjaguin, B. V.; Churaev, N. V. J. Colloid Interface Sci. 1974,49, 249. (5) Le Neveu, D. M.; Rand, R. P.; Parsegian, V. A. Nature (London) 1976,259, 601. (6) Rand, R. P. Annu. Rev. Biophys. Bioeng. 1981, 10, 277. (7) Rand, R. P.; Das, S.; Parsegian, V. A. Chem. Scripta 1985,25,15. (8) Israelachvili,I. N.; Adams, C. E. Nature (London) 1976,262,774; J . Chem. SOC.,Faraday Trans. 2 1978, 75, 975. (9) Marcelja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (10)Cevc, G.; Podgornik, K. R.; ieH, B. Chem. Phys. Lett. 1982,82, 315. (11) Gruen, D. W. R.; Marcelja, S. J. Chem. Soc., Faraday Trans. 2 1983, 79, 225. (12) Jonsson, B.; Wennerstrom, H. J. Chem. SOC.,Faraday Trans. 2 1983, 29, 19. (13) Shiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (14) Cevc, G.; Marsh, D. Biophys. J. 1985, 47, 21. (15) Ostrowsky, N.; Sornette, D. Chemica Scripta 1985, 25, 108. (16) Helfriech, W. Z. Nuturforsch., C. 1975, 30, 841. (17) Israelachvili, I. N.; Sornette, D. J. Phys. 1985, 46, 2125. (18) Stillinger, F. H. Science (Washington,D.C.)1980, 209, 451. (19) Dogonadze,R. R.; Kornyshev, A. A.; Kuznetsov, A. M. Teor. Mat. Fit. 1973, 15, 121; Theor. Math. Phys. USSR (Engl. Transl.) 1975, 15, 407. Dogonadze, R. R.; Kornyshev, A. A. J. Chem. SOC.,Faraday Trans. 2 1974, 70, 1121. (20) Kornyshev, A. A. Electrochim. Acta 1982,26, 1. Kornyshev, A. A. In The Chemical Physics of Soluation. Part A; Dogonadze, R. R., Kalman, E., Kornyshev, A. A., Ulstrup, I., Eds.; Elsevier: Amsterdam, 1985; p 77.
0743-7463/87/2403-0648~0~.50/0 0 1987 American Chemical Society
Structural Forces due to Nonlocal Water Polarizability
experimental determination of the parameters characterizing the water nonlocal response by a careful measurement of the intermembrane interaction. Taking into account nonlocal polarizability in such systems we will demonstrate essential modification of the double electrical layer structure (nonlocal polarizability in the case of systems containing electrical charges leads to certain amendments). Practically the “true” values of double electrical layer parameters such as the surface potential and surface charge can differ from their “apparent” values obtained from experimental data by usual methods. With the nonlocal polarizability effects being neglected, our results are reduced to the results obtained in GouyChapman theory. It will be shown that the nonlocal polarizability leads to the increasing of the observed screening length of electrolyte (in comparison with its Debye’s value). This difference is of the order of unity in the case of electrolyte concentrations relevant for biological systems. An analogous approach was used by CevcZ1in order to describe hydration forces in pure water. The main difference between Cevc’s method and ours consists in a different choice of the electric field’s sources and also in the model of the lipid/water interface. The outline of the paper is as follows. In section 2 we shall discuss basic equations of the nonlocal water electrostatics. In section 3 we shall deduce the expression for the electrical field in water/lipid system (in cases of pure water and electrolyte solution). In section 4 we shall analyze the influence of nonlocal polarizability on the double electrical layer structure near the hydrophile surface. In section 5 we shall calculate the intermembrane interaction. It will be shown that the observed hydration forces can be explained as forces between bilayer-covered dipole layers. In conclusion we shall discuss our results. A short version of this paper has been published in ref 22. 2. Nonlocal Water Polarizability Let us discuss the physical origin of the nonlocality of water dielectric response (see also ref 20). Generally the relation between the electric polarization P ( r ) and the electric field E ( r ) is an integral one: 47r@(r3 = xK(F;T?@(r‘? d3i‘
(1)
For homogeneous medium K(r,r‘) = K(r - r? and in case of sufficiently smooth variations of E ( r ) the kernel K ( r r? can be replaced by (e0 - 1)6(r - r? so that eq 1 reduces to a local relation 47rP = (eo - l)E (with eo N 80 for water). It is hardly probable that this simple approximation is qualitatively correct in the case of a rapidly decaying electric field near the phospholipid membrane. In this case the dielectric constant eo has to be replaced by a wavevector-dependent dielectric function ~ ( q =) 1 + 47rK(q). The physical reason for the strong ‘ ( 4 ) dependence is as follows: the water is a strong polar liquid whose molecular dipoles possess very active rotational modes leading to high macroscopic polarizability. It is known from the radiofrequency measurement23that the frequency dispersion of dielectric permittivity is of Debye form, €(W)
=
€1
‘0 - €1 +1 + iwr
(21) Cevc, G. Chem. Scr. 1985,25,96. (22) Belaya, M. L.; Feigel’man, M. V.; Levadny, V. G. Chem. Phys. Lett. 1986, 126, 361. (23) Saxton, I. A. Proc. R. SOC.London, A 1952,213,473.
Langmuir, Vol. 3, No. 5, 1987 649
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with the characteristic time r = s (for w r >> 1 e(w) €1 N 5 ) . It is reasonable to assume that the value el is determined by fast modes (electronic and oscillatory) with the rotational mode frozen. This phenomenon leads to the dramatic fall of the dielectric “constant” value from eo to el = 5. It is probable that similar “freezing” occurs in the case of static but strongly space-dependent fields. Note, that the value eO/el is rather large, so the marked variations of the static dielectric response e ( q ) from eo can be observed on the spatial scale of q-l much more than lo.For the oscillatory mode analogous scale 3; must be sufficiently less. Consequently, there is some wave-vector domain >> q >> to-1 where 49) N el. These speculations give us an opportunity to describe l / e ( q ) in following form:
-
-
(3)
where a(q) = e> q >> lcland a(q) const for q 0. The asymptote of the Green function G(r) corresponding to eq 3 is (for r >> lo):
The simplest interpolation e(q) between co (for q = 0) and el (for q >> has a form that is similar to the real part of eq 2 and corresponds to a(q) = const = el-l - eo-1.
r’)
The nonlocal electrostatic approach to solvation phenomena was first proposed by Dogonadze and K0rny~hev.l~ It was later elaborated in a number of publications by Kornyshev and co-workers and others (for review see ref 20), including the dielectric function analysis and various calculation schemes for homogeneous and contact systems. It happened, however, that this activity was limited to charged field sources. Dealing with dipolar sources, one arrives at a new family of nonlocal effects. For example, it is interesting to compare the electric field produced by the single electrical pointlike dipole pp
(we retain the principal terms at r / l o >> l ) , with the electrical field produced by the single electrical charge e: (7)
The first terms of (6) and (7) are the usual classical parts of the electric fields. Note that in the case of the dipole this term decays faster than in the case of the charge. The second terms of (6) and (7) are caused by spatial dispersion of ‘(4). It is essential that these terms decay in the same way. The second term of (6) does not vanish (contrary to the first one) after the averaging of E,(r) over the ii orientations. Just this property of the dipole-induced field in nonlocal media leads to the existence of a strong electric field near a neutral lipid membrane. 3. Electric Field of a Planar Phospholipid Membrane Let us consider a single planar lipid membrane placed into an electrolyte solution. The sources of the applied
650 Langmuir, Vol. 3, No. 5, 1987
Belaya et al.
normal components of surface dipole moments. The value of the phospholipid molecular dipole moment is d = 30-40 D.27 The correct value of the normal component d, is not known now. It is reasonable to estimate it as d, N 10-15 D, which results in p = d,/So N 2 X lo3 cgs (here So = 50-70 A2 for the lateral area of a lipid molecule in the bilayer). It is interesting to compare (9) with the expression for the electric field of a charged membrane placed into the usual Gouy-Chapman solution with a screening length x - l :
$ I I
I
Figure 1. Scheme of the lipid/water interface.
electric field in this system are charges and dipoles of hydrophile head groups of the lipid molecules. The membrane thickness 6, N 50 A considerably exceeds the typical length of our problem X N 3 A, so we can consider the limit 6, m. Let axes y and z of the Cartesian coordinate system lie in the membrane plane and the axis x be the normal. The lipid/water interface is at x = 0 ( x > 0 corresponds to water). Actually water molecules closely stick around the polar head of each lipid molecule.24 Therefore, it is reasonable to consider molecular dipoles as continuously distributed over the plane at some distance 1 from the waterllipid interface (Figure 1). The value of 1 depends on the structure of the membrane surface and can probably vary in the range 0.5 + 3 A for different membranes. In the following we shall neglect the possible influence of the interface on the form of the kernel K ( r - r? (i.e., we shall adopt "dielectric approximation"%). For the case of membranes separated by a pure solvent (with no electrolyte) general and alternative K(r,r? models were considered in ref 26, 40. Let us first analyze the case of pure water. Then the basic electrostatic equation div D = 47rp for our system becomes d - [ E ( x ) + J m K ( x - x ? E ( x ? dx'] = -4sp6'(2 - 1 ) (8) dx 0 Here p is the normal component of the membrane dipole surface density. The boundary conditions for eq 4 are E ( x = m ) = 0 and E(x x-l ("Debye" term). At x > ql-'.
-
-
4. Structure of an Electric Double Layer Near the Hydrophilic Surface The electric field in the water electrolyte near the lipid membrane, which has surface dipoles as well as surface charges, can be obtained from eq 14-19. Therefore the distribution of electric potential in this system is &)
= (oloe-qlx
+ eoe-qZx
(20)
here Figure 2. Electric potential &) as a function of distance from the membrane for different values of electrolyte concentration (the corresponding Debye lengths x-l are indicated).
This result differs significantly from the well-known Gouy-Chapman result 4ra cp(x) = -e-xx COX
There are two main differences. First, in our case ~ ( x ) contains a sum of two exponential terms with different exponents and opposite signs of preexponential factors; therefore, the p(x) function is nonmonotonic. The second difference is that preexponential factors are determined by the values of the surface dipole density as well as surface charge density. Therefore the electric double layer has to exist near the neutral membrane as well as near the charged one. First account of the phenomenon was made ref 41 for the uncharged metal/electrolyte interface, where the surface polarization is due the surface density profile of s, p electrons. Similar effects were met at the lipid/ electrolyte system where the dipole potential drop is assumed ad hoc." The behavior of the p(x) function at different concentrations of electrolyte and different surface charge densities is shown on Figures 2 and 3, respectively. Now it is instructive to compare the main parameters of the electric double layer (such as surface potential and surface charge) obtained from experimental data by usual means from eq 23 with those values obtained with our eq 20. Here we restrict ourselves to the case of dilute elec(it will be shown below that qo-' trolyte x lo(we retain here the principal terms at q22/q12< x212 5 0.1) is
In the case of a charged membrane placed into dilute electrolyte ( P x > 2(t,/~&6)~/~. This restriction is a rather weak one due to the large value of the membrane thickness 6 50 A and the small value of the membrane dielectric susceptibility t, N 2. At last let us analyze the influence of membrane hydration (which is described by the parameter I ) on the structural (hydration) force. For this purpose it is necessary to obtain the solution of (32) for 1 # 0. Then we find P(h) (here we consider the case of pure water only)
-
(33)
The coefficients E , and E, are y
= [chl/lo + (l/P’/2)(shl/lo)12
Note, that y(l=lo)/y(l=O)
3.6.
6. Conclusion
Q = Q(h) = Yiqi2/ai - ~ 2 ~ 2 2 / a 2 here ai = 1 - e-qgh and yi = (qi - qo/31/2)-1 + (qi + qor1/2)-1.exp(-qih). Then we substitute eq 33, 34 in (31) and obtain the following expression for the free energy: F=
91- -- q2ff1
ff2
ff2
ff1
(35)
The corresponding expression for P(h) is rather cumbersome and we shall analyze here some limiting cases only. The exact form of P(h) for a neutral membrane (a = 0) in pure water is P(h) = 7=
8 w 2 v exp(-h/ld
do2[1 + 7 exp(-h/l~)l
(P1l2 - l)/(B1/’
+ 1)
N
0.6
(36)
We have analyzed the influence of nonlocal water polarizability on the electric field distribution near the hydrophilous surface of the lipid bilayer. As follows from (20)-(22) and Figures 2 and 3, the nonlocality of water polarizability alters substantially the double electric layer structure. Now the distribution of the electric potential becomes nonmonotonic and is described by the sum of two exponential terms. At short distance [x < [2/(q, - q2) In q l / q 2 ] ]the surface dipoles give the main contribution to the electric field, which is rather large due to the high density of surface dipoles. At large distances the usual Gouy-Chapman component is the main one. However, the nonlocality of the water dielectric response leads to deviations in the screening length as well as the preexponential factor values from the classical ones. These deviations are rather strong for the solution concentrations corresponding to biological conditions (Le., for x-l N 10 A). It is shown that nonlocality of the water dielectric response leads to a rather complicated structure of the electric field distribution near the membrane, so that “true” values of surface charge density and surface potential differ substantially from their “apparent” values obtained from experimental data by common methods. In particular, a neutral membrane in the nonlocal electrolyte looks like a charged one. This conclusion has some indirect confirmation in experiments (see ref 29, 30). In accordance with our theory the
654
Langmuir 1987, 3, 654-659
“additional” surface potential is caused by the contribution of the surface dipoles. Thus, it was shown that significant electric field created by surface dipoles exists near the surface of the phospholipid membrane. Just this field leads to strong “hydration” repulsion between two membranes at small separations. At present we are not aware of the value of the parameter lo entering the water dielectric function t ( q ) . Therefore we must use the experimental value of the decay length X N 2.5-3 A.7 Then from comparison of eq 36 with the experimental data it follows that lo E X = 2.5-3 A. The preexponential factor which we estimate for the pure water case is (see eq 36) P1 (5 X 1010)(d,/lo)2dyn/cm2 with ed, being the normal component of the head group dipole moment (we use S N 60 A2 for the lateral area of a lipid molecule). A precise value of d, is not known now but the estimation d, N X seems to be reasonable. Thus, we obtain P, in the range 101o-lO1l dyn/cm2, which is slightly above the measured interval. This overestimation can be attributed to the use of the simple linearized (with respect to the electric field) theory. Note that the increase of the intermembrane repulsion with the increase of parameter 1 (see eq 39) enables us to understand the measured decrease of P, in the course of a bilayer phase transition from the ‘‘liquid’’ to “solid”state? The point is that this transition leads to the pushing out of the water from the head group region and so to a decrease in l. (It is possible that thermal fluctuations of bilayer considered in ref 15 are also responsible for this effect.)
Thus, we have shown that structural (hydration) forces can be accounted for as a result of the dipole-dipole interaction of polar surfaces in media with nonlocal polarization. Nevertheless we cannot exclude the possibility of another (nonelectrostatic) origin of structural forces. For example, the general order parameter introduced in ref 9 could be connected with the orientational order parameter which was considered in some papers (see, e.g., ref 38,39) dealing with the problem of a melting phase transition. Indirect confirmation for our theory could be produced by careful measurement of the dependencies of the “polarization” (q1-l) and “screening” (q2-I)lengths on the electrolyte concentration [see (18) (19)]. If our theory of hydration forces were confirmed, it would be possible to determine both parameters (e1 and lo)of water dielectric response by comparison of measured values of q l , q2, and Powith our eq 18, 19, and 36. Acknowledgment. We are grateful to I. E. Dzyaloshinsky, A. Z. Patashinsky, B. V. Derjaguin, S. A. Leikin, I. L. Fabelinsky, Ya. A. Chizmadzev, N. B. Churaev, I. G. Abidor, and especially A. A. Kornyshev for many illuminating discussions. Registry No. Water, 7732-18-5. (38)Mitus, A. S.;Patashinsky, A. Z. Zh. Eksp. Teor. Fiz. 1981,80, 1554. (39)Bruinsma, R.;Nelson, D. R. Phys. Reu. B 1981,23,402.Nelson, D. R.;Sachdev, S. Phys. Reo. B 1985,32,4592. (40)Dzhavakhidze, P. G.;Kornyshev, A. A.; Levadny, V. G. Phys. Lett. A 1986,118A, 203. (41)Kornyshev, A. A.;Vorotyntsev, M. A. Can. J. Chem. 1981,59, 2031.
Temperature Dependence of the Stability of Natural Diamond Dispersions in Electrolyte Solutionst Yu. M. Chernoberezhskii,* V. I. Kuchuk, 0. V. Klochkova, and E. V. Golikova Leningrad Technological Institute of Cellulose-Paper Industry, 198092 Leningrad, USSR Received January 15, 1987 The flow ultramicroscopy method was used to investigate the aggregative stability and coagulation of diamond hydrosols having a particle size of about 0.5 pm at different pH values and electrolyte concentrations (LiC1and AlClJ. The determining role of structural forces in hydrosol stability has been demonstrated. An increase in temperature causes destruction of the boundary layers of water and a decrease in structural repulsion. It has been shown that the change of the structure of boundary layers with temperature is reversible. The existence of structural changes in boundary layers (BL) of a liquid near the surface of a solid body is beyond all doubt at present. When considering disperse particle interactions it was necessary to take into account structural and modified BL, which added a lot to the physical foundation of the stability theory. Derjaguin and Churaev introduced the concept of structural components of interaction forces between converging surfaces with BL overla~ped.’-~ The concept was based on experimental studies proving the existence of bound water layer^.^ Recently there appeared a great number of theoretical and experimental studies devoted to the investigation of ‘Presented at the ”VIIIth Conference on Surface Forces”, Dec 3-5, 1985,Moscow; Professor B.V. Derjaguin, Chairman.
structural forces. Some progress has been achieved in understanding their nature and regularities in their changes. Thus, based on studies6-16the law of structural (1)Derjaguin, B. V.; Churaev, N. V. J.Colloid Interface Sci. 1974,49, 49. (2)Derjaguin, B. V. Chem. Scr. 1976,9,95. (3)Derjaguin, B.V.; Churaev, N. V. Croat. Chem. Acta 1977,50,187. (4)Churaev, N. V. Kolloidn. Zh. 1984,46,301.Churaev, N. V.;Derjaguin, B. V. J. Colloid Interface Sci. 1985,103, 542. (5) Derjaguin, B. V. In Works of Allunion Conference of Colloid Chemistry; Acad. Nauk USSR: Kiev, 1952;p 26; Wear, 1958,1, 277; Discuss. Faraday SOC. 1966,42,109. (6) Marcelja, S.; RadiB, N. Chem. Phys. Lett. 1976,42,129. (7)Peschel, G.;Belouschek, P. 2. Phys. Chem. N.F. 1977,108, 145. (8)Peschel, G.;Belouschek, P.; Muller, M. M.; Muller, M. R.; Konig, R. Colloid Polym. Sci. 1982,260,441. (9)Israelachvili, J. N.J. Chem. Soc., Faraday Trans. I 1978,74,975.
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