Entropic forces between amphiphilic surfaces in liquids - The Journal

Jan 1, 1992 - Stephen H. Donaldson , Jr. , Anja Røyne , Kai Kristiansen , Michael V. Rapp , Saurabh Das , Matthew A. Gebbie , Dong Woog Lee , Philipp...
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J. Phys. Chem. 1992, 96, 520-531

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FEATURE ARTICLE Entropic Forces between Amphiphliic Surfaces In Liquids Jacob N. Israelachvili* Department of Chemical and Nuclear Engineering, and Materials Department, University of California, Sanfa Barbara, California 93106

and Hhkan Wennerstrom Division of Physical Chemistry 1. Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden (Received: June 27, 1991; In Final Form: September 16. 1991) The forces between fluid amphiphilic surfaces such as surfactant micelles, lipid bilayers, and microemulsion droplets in liquids include the expected attractive van der Waals and repulsive electric double-layer forces (the two DLVO forces). However, because of the dynamic (fluid-like) nature of these interfaces, additional entropic fluctuation forces are also present which arise from the overlap of thermally excited surface modes. Three of these forces are the undulation,peristaltic, and protrusion forces. The fiist two arise from collective motions of bilayers or membranes and can be described in terms of their continuum elastic moduli, respectively. The last arises from molecular-scale fluctuations of hydrocarbon chains and other parts of the molecules protruding out of the surfaces (“molecular protrusion” force), and a similar osmotic repulsion between overlapping mobile headgroup (“headgroup overlap” force). It is shown that both these two forces are expected to decay roughly exponentially with distance with characteristic decay lengths in water of about 0.2 nm. These forces have long been believed to be due to water structure (and are commonly called the “hydration” force). By a review of recent experimental and theoretical progress in this area, it is concluded that this force is not primarily due to water structure (it occurs in other liquids than water) and that it is more akin to the steric repulsion between polymer-covered surfaces. Genuine hydration or solvation effects probably play only an indirect role in the interactions between amphiphilic surfaces, mainly in determining the hydrated sizes (excluded volumes) of the protruding groups and the positions of the planes of origin of other interaction potentials. A quantitative assessment is made of the relative contributions of DLVO forces, entropic forces, and genuine hydration forces between uncharged amphiphilic surfaces in water. It is concluded that between free bilayers the undulation repulsion dominates at large separations (>3 nm), the van der Waals attraction at intermediate separations (1.5-3 nm), and the protrusion and overlap repulsions at smaller separations (”showed that at distances D below about 1-3 nm they decay approximately exponentially with distance. The repulsive force per unit area, or pressure P, between two surfaces is, therefore, given by

p = +Ce-D/“a (1) where the “characteristic”decay length, A,, was found to be close to 0.2 nm. The observation that X,is about the size of a water molecule was given as further support for the notion that this force is due to water structure, and theoretical tended to confirm the possible existence of an exponentially repulsive force arising from the decaying “polarization” of water molecules by surfaces. However, the decay length was not easy to derive theoretically and had to be assumed or fitted, though it seemed conceivable that it could be close to the size of a water molecule. Subsequent molecular dynamics simulations4failed to predict the expected monotonically decaying force. Instead, with surfaces modeled on lecithin and mica, only decaying oscillatory profiles were obtained. In spite of these difficulties some authors considered the agreement between theory and experiment such that it was established that the hydration force was caused by water structure,I4 a view that seems to have been assimilated by a majority of scientists in the field. In any objective analysis of the force between two amphiphilic surfaces across water at short range, one has to recognize that the situation at the molecular level is extremely complex. The interface between the apolar hydrocarbon region and the polar solvent consists of headgroups with both polar and apolar parts. (15) Marra, J. Biophys. J. 1986, 50, 815; J . Phys. Chem. 1986, 90, 2145; J. Phys. Chem. 1986, 90, 815. Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Brady, J.; Evans, D. F.J. Phys. Chem. 1986, 90, 1637. (16) McIntosh, T. J.; Magid, A. D.; Simon, S.A. Biophys. J . 1990, 57, 1187. (17) Israelachvili, J. N.; WennerstrBm, H. Langmuir 1990, 6, 873. (18) Langmuir, I. J. Chem. Phys. 1938, 6, 873. Jordine, E. St. A. J . Colloid Interface Sei. 1973, 45, 435. (19) (a) Marplja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (b) Gruen, D. W. R.; Marcelja, S.J. Chem. SOC.,Faraday Trans. 2 1983, 79, 225. (20) Schiby, D.; Ruckenstein, E. Chem. Phys. Letr. 1983,95,435. Attard, P.; Batchelor, M. T. Chem. Phys. Lerr. 1988, 149, 206.

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Figure 1. Measured repulsive forces (or pressure) between egg phosphatidylcholine (egg lecithin or egg-PC) bilayers in three different solvents: water, formamide and 1 3 propanediol (PDO).22 The forces decay roughly exponentially according to eq 1 with preexponential factors, C, and decay lengths, &, given in Table I. Various properties of these solvents and of the hydrocarbon-lvent interface are also given in Table I. Note that X, increases as C decreases in the order propanediol > formamide > water, which is also the order of decreasing interfacial tension yi (decreasing solvophobicity) of the hydrocarbon-solvent interface.

The headgroup can be compact, as for ordinary soaps, or more extended, as for certain nonionic ethylene oxide surfactants, glycolipids, and phospholipids. In the highly anisotropic environment at the interface a number of intrinsically different molecular forces-electrostatic, dipolar, and solvation-balance one another. Between uncharged surfaces, the “hydration” force is obtained as the deviation from the van der Waals attraction which is rigorously derived for continuous media. The deviation is thus, in one way or another, caused by the molecularity of the system. In the conventional explanation based on decaying water structure the force is explained totally in terms of the molecularity of the soluent, while the interface is considered as a smooth boundary whose only influence is in terms of fixing a boundary condition for the solvent structure. It is appropriate to question whether or not the molecularity of the surface is in fact more important than that of the solvent. After all, the headgroups are substantially larger than the solvent molecules, and the interface is further “roughened” by collective motions and individual molecular protrusions of the liquid-like amphiphilic molecules.21 Experimentally, too, the picture is becoming more complex and increasingly less consistent with a hydration origin for these forces. Three key examples will now be mentioned; these also provide a convenient background for the ensuing theoretical analysis of thermal fluctuation forces. (1) Effect of Solvent. Recent measurementszzshow that the same forces also exist between bilayers in a variety of nonaqueous solvents. This is illustrated in Figure 1 for egg lecithin (egg-PC) in three different solvents, which shows that these forces are not unique to water. Moreover, the range increases with decreasing interfacial energy of the hydrocarbon-solvent interfaces (Table I), which is precisely what would be expected for thermal fluc(21) Pfeiffer, W.; Henkel, Th.; Sackmann, E.;Knoll, W.; Richter, D. Europhys. Letf. 1989,8, 201. (22) McIntosh, T. J.; Magid, A. D.; Simon, S.A. Biochemistry 1989,28, 7904. Persson, P. K. T.; Bergenstihl, B. A. Biophys. J. 1985, 47, 743.

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522 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992

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Figure 2. Solid curve: measured force between fluid state dipalmitoylphosphatidylcholine (DPPC) bilayers in water.”.22,28*58At larger separations the force decays roughly exponentially according to eq 1 with C = 5 X lo8 N/m2 and X, = 0.14 nm. A very similar force was measured between egg-PC for which C = 4 X l@ N/mZ and X, = 0.17 nm. Dashed curve: earlier results of Le Neveau et a].’’ and Lis et al.” for egg-PC, who reported C = 4 X 10” N/mZand X, = 0.22 nm for the same system.

tuation interactions (section 5 ) . (2) Range of Repulsive Forces. In the earlier force measurements by LeNeveu et a1.I0 and Lis et al.” the distances between bilayer surfaces were inferred from the repeat spacings of the multibilayers as measured using X-rays. More recently, McIntosh and Simon13 showed how the aqueous gap separation could be determined directly from the measured electron density profiles across the multibilayers. They also reevaluated the older data and concluded that the earlier experiments had significantly overestimated both the range and decay lengths of the repulsive forces, as illustrated in Figure 2. Values for X,obtained with different surfactant and lipid systems in water now range from below 0.1 nm to above 0.6 nm.11J4,39+46 With such a large range, Xo no longer appears to correlate with some unique correlation length in the liquid. (3) Efkct of Temperature. As shown in Figure 3, the repulsion between bilayers in water usually increases with temperature,8,12 and many surfactant bilayers do not swell at all when in the solid-crystalline state but do swell as soon as the temperature is raised to the gel state, and even more in the liquid-crystalline state where the chains are melted.23 Within the hydration model this trend would suggest that the water structure increases with increasing temperature. This is very unlikely: with increasing temperature, as the amphiphilic molecules become less ordered, one would expect the same to occur for the adjoining water molecules. Since deviations from the long-range van der Waals plus double-layer force are operationally identified as “hydration” forces, it is no surprise that such forces have also been invoked in a number of chemically very different systems such as mica, clays, silica, and DNA. Toward the end of this article we will consider the question whether or not the molecular mechanism behind these forces are the same as for amphiphilic surfaces. However, the next few sections concentrate on amphiphilic systems. We start by reviewing those entropic repulsive forces whose basic mechanism has been firmly established. This is followed by focusing more closely on the different kinds of thermal excitations that can occur at an amphiphile-solvent interface. The implications of these excitations for the overall force between two such surfaces are then discussed using several different models. A quantitative comparison between theory and experiment will then be made, leading to a final discussion of the roles of different forces in both amphiphilic and nonamphiphilic systems. 2. Repulsions Arising from Configurational Confmements (Pure Entropic Effects) 1. Historical Background. Langmuir was one of the first to consider the forces between amphiphilic molecules, both across an aqueous gapla as well as laterally within a m o n ~ l a y e r .He ~~ (23) Tiddy, G.J. T. Phys. Rep. 1980, 57, 1.

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Figure 3. Forces between various PC bilayers in water as measured using the osmotic stress technique by Lis et al.” (main figure), and the surface forces apparatus technique by Marra and Israelachvili12(inset). In their reevaluation of the data for lecithins McIntosh et aLz2obtained a range of 1.17 nm for DPPC and 1.54 nm for egg-PC, both in the fluid states. The range of the forces generally increases with increasing temperature, and especially during transitions from the solid crystalline to the gel state, and from the gel to the liquid-crystalline state (at T = T,).

identified three types of repulsive forces between them: First, there is a purely entropic (osmotic) repulsion P = pkT that, by analogy to Boyle’s law PV = RT for a gas, takes the following two-dimensional form for a monolayer: IIA = RT. When excluded volume and headgroup area effects are taken into account, these two equations become the well-known equations of state: p = - RT for a gas V-b (2) n = - RT for a monolayer A - A0 Equation 2 was proposed by LangmuirZ9as a suitable equation of state for monolayers and, like the van der Waals equation of state, is still used today. It is well to recall the very large osmotic pressure that a collection of noninteracting particles have in solution. If we define the mean interparticle or intersolute separation by D,eq 2 for a gas may be expressed as RT kT p=m(3) V-b 0 3 Figure 4 shows a plot of P = kT/D3over the range D = 0.3-1.6 nm. If a straight line is drawn through the points, one obtains the quasi-exponential force law P = Poe-D/bof decay length X, = 0.27 nm and magnitude Po 2 X lo9 dyn/cm2-both of which are typical of fitted values for many different amphiphilic systems! The m e in Figure 4 shows the maximum repulsive force (osmotic limit) that may be expected for any osmotic pressure measurement where the dissolved solute molecules or particles do not interact with each other, Le., where there is no real force between the particles. Thus, the mere measurement of such a repulsion is no indication of the existence of an interparticle force, whether attractive or repulsive. In more complex fluid systems there will always be an osmotic pressure between the fluid-like structures or aggregates, but this will be modified from that given by eq 3. One of the questions we attempt to answer here is, what is this modified force for amphiphilic surfaces? The second of the repulsive forces considered by Langmuir is the now well-known electric “double-layer” force that arises whenever the headgroups are charged. In one of his classic papers on the subject, Langmuir18 criticized “the past use of [potential] energy diagrams” because they ignored ‘the effect of the thermal agitation” of ions in the solution. He proposed that “instead of potential energy...the osmotic pressure be used, which includes these previously neglected factors”. Langmuir then demonstrated

The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 523

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that it is the osmotic pressure between ions that was responsible for the “double-layer” repulsion between similarly charged colloids and “coacervates”in solution-the purely electrostatic contribution to the interaction being attractive. Third, there are repulsive “hydration” forces, which Langmuirl* considered to arise from “the intense electric field [which] draws the dipole water molecules into the spaces between the particles and so holds them apart”. Langmuir felt that there was a need to invoke such forces to explain the observed short-range repulsion between oppositely charged “coacervates” (giant micelles, microemulsion droplets, etc.) and, later, of clay surfaces in water. It is not clear why Langmuir did not consider the possibility that here, too, the net electrostatic interaction involving charges and dipoles is also bound to be attractive (so long as the system is overall electroneutral). Nor is it clear why Langmuir only considered the osmotic repulsion between the quasi-free ions at surfactant-water interfaces and not that between the mobile surfactant molecules themselves. We now proceed with a more formal analysis of the various entropic forces between surfaces. 2. Entropic Nature of the Electric Double-Layer Force. The force between surfaces in a liquid is typically measured either by the surface force apparatus or by the osmotic stress technique.2J0 In analyzing these experiments, one normally assumes that the systems are bulk incompressible so that as the surfaces approach each other, the solvent leaves the gap between them in a way proportional to the change in gap size. Similarly, virtually all theories of surface forces in liquids use the incompressibility as a constraint. In thinking about forces it is natural to have a mechanical view of molecules pushing or pulling on surfaces. In a statistical mechanical formulation it is possible to maintain such a force balance point of view, but as soon as one either assumes incompressibility or effective medium approximations the applicability of this mechanical view is lost simply because one does not explicitly include all entities in the description. The general way to solve the problem is to express the force per unit area as a derivative of the free energy, which for incompressible systems can equally well be either the Helmholtz free energy A or the Gibbs free energy G. Thus, taking repulsive forces as positive

and one can identify energetic (U) and entropic (S)contributions to the force. If we consider the force between two similar (identical) surfaces across a liquid gap, it is possible to make some generalizations regarding the forces we presently understand. First, the energy term (aU/aD), is usually attractive. The typical example being the van der Waals (dispersion) force, while for two interacting double layers this contribution is also generally attractive (though not always the dominant term24). The entropic contribution is usually repulsive. For example, in the limit of small separations, and independent of the salt concentration, the double-layer re-

A. Undulation forces

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Figure 6. Four types of thermal fluctuation forces between amphiphilic surfaces such as surfactant and lipid bilayers. Note that from the geometry of a circle (the “cord theorem”) the following relation holds: x2 = (2R - D)D c 2RD. pulsion between two planar surfaces of fmed surface charge density CT always goes over to “osmotic limit”: 2akT P(D-4) = eD + 2kTPbulk where Pbulk is the solution concentration of ions. Note that the first term is the same as the pressure of an ideal gas of ions (or any noninteracting particles) of volume density 2a/eD confined uniformly within a gap of thickness D, as given by eq 3. The increasingly dominant entropic nature of the electric double-layer repulsion at small distances is illustrated in Figure 5, which shows the theoretically expected repulsion based on the full Poisson-Boltzmann equation, together with the measured repulsion between two bilayers of the charged lipid PG. The agreement between the two is remarkable, and yet this short-range force was considered to be due entirely to “hydration” effectss0 We may also note that at small separations the force in Figure 5 is quite similar to the purely osmotic pressure curve of Figure (24) Guldbrand, L.; Jiinsson, B.; Wennerstriim, H.; Linse, P. J . Chem. Phys. 1984,80,2221.

524 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992

4. Clearly, regardless of the system, so long as there are mobile species (whether ionic or molecular) trapped between two surfaces, we must always expect there to be a steep short-range repulsion of similar decay and magnitude (at least on a log scale). We now proceed to consider other entropic forces between amphiphilic surfaces of which there are at least four different types. The molecular origin of these forces is illustrated in Figure 6A to D. Unlike the double-layer force, none of these forces should exist between hard surfaces (e.g., solid colloidal particles), but between amphiphilic bilayers we shall see that both their magnitude and distance dependence quantitatively account for the measured repulsions. 3. Undulation Force. All elastic sheets, including fluid bilayers, have thermal undulations whose amplitude increases with increasing temperature, T, and decreasing bilayer bending modulus, K,,. Helfrich and co-workers showedzsthat for two bilayers, under no external tension, at a mean distance D apart (Figure 6A) the force per unit area between them is repulsive and given by 3~~(kT)~ p=64K,,D3 The undulation force is essentially an entropic force arising from the confinement of thermally excited modes (undulation waves) into a smaller region of space as two membranes approach each other. The undulation force can be easily derived from the ‘contact value theoremnz7which gives the entropic force per unit area between two surfaces as

P(D) = k T b , ( D ) - p,(=)l (7) where p,(D) is the volume density of molecules or molecular groups in contact with the surfaces when the distance between them is D. In the case of undulation forces these contacts can be associated with bilayer waves of amplitude D as shown in Figure 6A. Ignoring numerical factors, the density of surface contacts (or mods) for D = D and D = are

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p,(D) = l/(volume occupied per mode) = 1/*xzD and p , ( - ) = 0

(8)

Using the “cord theorem” (Figure 6 ) the pressure is p = - kT zs- kT (9) xx2D 2 u R S Now by definition, the elastic bending (curvature) energy per unit area of a membrane with radius R is ‘/&(2/R)’, and at temperature T we expect each mode, which occupies an area rxZ, to have energy -kT. Thus kT = 2?rXZKb/R:= 4ADKbIR. Substituting this into eq 9 and ignoring numerical factors, we obtain P = ( k q 2 / K @ , which is Helfrich’s undulation force, eq 6. The undulation force has been measured and the inverse third distance dependence verified experimentally by Safinya et al., McIntosh et al., and Abillon and Perez.28 4. Peristaltic Force. In addition to bending fluctuations, bilayers or membranes also undergo peristaltic (or squeezing) fluctuations wherein the thickness of the membrane fluctuates about the mean thickness (Figure 6B). The same approximate analysis as was used above to derive the undulation force can now be used to derive the peristaltic force associated with the entropic confinement of the peristaltic waves as two membranes approach each other. Referring to Figure 6B, consider a region, or mode, of area a = r x z where the local membrane thickness is greater (25) Helfrich, W. 2.Naturforsch. 1978, H a , 305. Servuss, R. M.; Helfrich, W. J . Phys. (France) 1989, 50, 809. (26) Marra, J. J . Colloid Interface Sci. 1985, 107, 446; 1986, 109, 11. (27) Henderson, D.;Blum, L.; Lebowitz, J. L. J . Electroanal. Chem. 1979, 102,315. Wennerstram, H.; Jansson, B.; Linse, P. J. Chem. Phys. 1982, 76,

4665. (28) Safinya, C. R.;Roux, D.;Smith, G.S.;Sinha, S. K.;Dimon, P.; Clark, N. A.; Bellocq, A. M. Phys. Reu. Lett. 1986,57, 2718. McIntosh, T. J.; Magid, A. D.; Simon, S.A. Biochemistry 1989, 28, 7904. Abillon, 0.; Perez, E. J . Phys. (France) 1990, 51, 2543. ( 2 9 ) Langmuir, I. J . Chem. Phys. 1933, I , 756. (30) Homola, A.; Robertson,A. A. J . Colloid Interface Sci. 1976.54.286.

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Figure 7. (A) Measured forces between adsorbed monolayers of the poly(ethy1ene oxide) surfactant C18EOnin water.30 The solid lines are fits to the Alexander-de Gennes theory for overlapping headgroups modeled as polymer ‘brush” layers, eq 12, using L = 7.8 nm for n = 20, L = 10.5 nm for n = 40, and L = 12 nm for n = 50 (all fitted for s = 13 nm). (B) Measured forces between bilayers of CI2EO,in water‘g and corresponding brush theory curve (solid line) using L = 1.6 nm (0.4 nm/EO group), s = 0.93 nm, and T = 25 O C .

than the mean thickness (adjacent regions will have thicknesses smaller than the mean, the total volume of the membrane being assumed constant). Using straightforward geometry, it can be shown that the surface area of the membrane exceeds the mean area, a, by ha = ?r@ per mode. Now, the elastic energy of a membrane whose equilibrium area a is stretched by Pa is

where, by definition, K, is the area expansion or compressibility modulus. Equating this energy per mode with kT as before, and again using the *contact value theorem”, eq 7, we obtain

which gives an estimate for the peristaltic pressure between two membranes. Note that in contrast to the undulation force, which is in terms of a membrane’s bending modulus Kb, the peristaltic force depends on the expansion modulus K,. These two elastic properties are quite different and have different dimensions, though they are not necessarily totally independent of each other. 5. Headgroup Overlap Force. A common method to prevent the coagulation of colloidal particles in solution is to cover the particle surfaces with long polymer molecules that repel each other in the solvent. This is usually referred to as “steric stabilization” and results in a ‘stable” colloidal suspension. The headgroups of many surfactants and lipids are longer than the chains. They also protrude into the aqueous phase where they repel each other. In a first approximation we may therefore consider a flexible headgroup as behaving like an end-grafted polymer whose exposed segments do not interact with each other. Given that the mean separation between headgroups is typically 0.6-0.9 nm and that this is also about how much they extend into the solution, we may expect to use the theory of polymer “brushes” to describe headgroup-headgroup interactions, especially for longer headgroups. Polymer brushes are formed when the molecules are ‘grafted” at one end to the surfaces either by covalent bonds or by using a block copolymer with one strongly adsorbing end. If the grafting density is high enough and if the nonadsorbing part is miscible with the solvent, the adsorbed layer takes on the character of a brush for which the theory is well-developed and t e ~ t e d . ~ ’ The Alexander-de Gennes theory gives for the repulsion between two brush layers3’

where L is the thickness of each brush layer and where s is the mean distance between grafting sites (corresponding to the mean (31) de Gennes, P. G. Adu. Colloid Interface Sci. 1987. 27, 189.

The Journal of Physical Chemistry, Vol. 96, No. 2, I992 525

Feature Article distance between headgroups). Equation 12 assumes that L >

s. There is an unknown numerical factor a t the front of eq 12 so that the exact magnitude of P is unknown even though the shape of the force curve is. Figure 7A shows how eq 12 nicely accounts for the experimentally measured forces between poly(ethy1ene oxide) (ClzEO,,) surfactants having n = 20, 40, and 50 segments. Even shorter ranged forces appear to be well described by eq 12 using reasonable values for L and s, as shown in Figure 7B for the forces between bilayers of CI2EO4. It appears, therefore, that headgroups longer than about 1 nm may be treated as quasi-brush layers. While there is no general theory that can currently describe the interactions between short headgroups (however, see section 3.3. for some simple models), one may show that the steric force between two end-grafted chains must always have the same shape as the Alexander-de Gennes equation, regardless of the details of the system. 6. General Features of Repulsive Entropic Forces. Although superficially very different the doublelayer, undulation, peristaltic, and polymer steric forces have some basic common features. In all cases the single surface has considerable thermal excitations that results in a density that decays out from the (average) location of the surface. This density distribution is typically a result of a compromise between an entropic factor which tends to make the interface as diffuse as possible and a restoring force that tends to make it as sharp as possible. What differs between the different types of forces is the nature of the density-ion density, surface wave density or flexible segment density-and consequently the nature of the restoring force, but the basic mechanism is always the same. It is pertinent to ask whether the above list contains all the possible types of density fluctuations that can occur at a liquidlike surface, and in particular a t an amphiphilic surface. Clearly the answer is no. There are in addition to the collective undulation and peristaltic modes local molecular-scale motions which involve either whole molecules or parts of them, and the confinement of these motions as two surfaces approach each other will also give rise to a repulsive force. On the basis of the same principles as used to analyze the other established repulsive forces, we now set out to estimate the range and magnitude of these molecular "protrusion" forces.

3. Molecular Protrusion Effects 1. Protnrsions at a Single Wace..An aggregate of amphiphilic molecules is in a dynamic equilibrium with monomers in the bulk phase. Particularly for micellar systems the dynamics of this process have been analyzed. The molecula entering and leaving a micelle are essentially diffusion controlled, with corrections for any possible long range electrostatic force. Motivated by his studies of micelle kinetics Aniansson was lead to an analysis of molecular protrusion effects.32 By assuming a potential energy V ( z ) of the form V ( z ) = Lyz

z

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proportional to the distance z out from the surface, Aniansson concluded that the density, p, decreases exponentially according to p ( z ) = p(zo)e-az/kr= p(zo)e-z/x

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where X = k T / a is the characteristic protrusion density decay length. Equation 14 was first used to analyze the protrusion dynamics of surfactant molecules in micelles and their exchange rates with the monomers in the bulk.32 For z equal to the maximum length of the chain, the density should reach a value approximately equal to the cmc. This was used by A n i a n s ~ o nto~ ~ estimate a value for ~y of -3 X lo-" J/m for single-chained surfactants, which corresponds to a decay length of X = k T / a = 0.14 nm at 25 O C . Additional experimental evidence for the (32) Aniansson, G. A. E. J . Phys. Chem. 1978,82, 2805. Aniansson, G. A. E.; Wall, S.N.; Almgren, M.; Hoffman, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J . Phys. Chem. 1976,80,905.

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Figure 8. (A) Theoreticallycomputed density profiles for lecithin (DPPC) bilayers in the fluid state. Mean-field lattice theory of Scheutjens to an isolated and Fleer, applied by Leermakers and Scheutjen~~~,''~ DPPC bilayer at T = 325 K. The total lipid concentration (heads and chains) well away from the bilayer reaches a constant value that has been fitted to the cmc, as required. Over a distance regime of 2 nm away from the surface the headgroup density decays roughly exponentially with a decay length of about X = 0.13 nm (this would correspond to a decay length in the protrusion interaction of X, = 1.15X = 0.15nm, which is very similar to the measured decay length of X, = 0.14 nm measured by McIntosh, Simon, and co-workers, as shown in Figure 2). (B) Shaded profiles: Same headgroup density as shown in Figure 8A, but plotted on a linear scale. Unshaded profiles: Monte Carlo computer ~imulation'~ of choline CH3group density profiles in DPPC bilayers in the fluid state in a multilamellar stack with 25% water. The computed distance between the choline peaks agrees with the measured value of 43.5 A determined from neutron diffraction experiments3' on DPPC bilayers in the L, liquid-crystalline phase with 25% water. Note how the headgroup distributions in a lamellar stack are compressed into the bilayers compared to those in a free bilayer (shaded profiles). This arises from the repulsive interaction between bilayers when confined within a lamellar structure with only 25% water. The headgroup distribution of a second bilayer surface is shown dashed on the left side only (note the overlap region across the water gap).

idea that the surfaces of amphiphilic aggregates are molecularly rough has been obtained from incoherent quasi-elastic neutron scattering studies of liquid crystalline DPPC bilayers by Pfeiffer et who revealed large out of plane molecular motions on a time-scale of s. We return to consider other experimental data relevant to both single-chained and double-chained amphiphiles in section 5.2. Theoretical support for the dynamic roughness of amphiphilic surfaces has come from computer simulations using various interatomic potentials. Simulations of both micelles33 and bilaye r ~show ~ a+polarapolar ~ ~ interface that is molecularly very rough (or diffuse). Additionally, in a theoretical study of fluid PC bilayers using the Scheutjens-Fleer mean-field lattice theory, Leermakers and S c h e u t j e n ~obtained ~~ a diffuse interface between the polar and apolar regions. This is shown in Figure 8A,B

-

~~

(33) JBnsson, B.; Edholm, 0.;Teleman, 0.J . Chem. Phys. 198685,2259. (34) Egberts, E. Thesis, Chapter IV, University of Groningen, 1988. (35) Egberts, E.; Berendsen, H. J. C. J. Chem. Phys. 1988, 89, 3718. (36) Leermakers, F. A. M. Thesis, University of Wageningen, 1988. Leermakers, F. A. M.;Scheutjens, J. M. H. M. J . Chem. Phys. 1988,89, 3264,6912.

526 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992

(shaded profiles), together with the simulation results of Egbert~’~ in Figure 8B (unshaded profiles). Both theories predict positions and distributions of different parts of the lecithin molecule in good agreement with experimental N M R data obtained by Billdt and co-workers3’ for lecithin bilayers in the liquid-crystalline state. 2. Protrusion Force-Mean-Field Theory. The undulation and peristaltic forces, given by eqs 6 and 11, are valid for large wavelength fluctuations where bilayers can be treated in terms of their continuum elastic properties. These equations should not apply to thermal excitations occurring at the molecular level, Le., when the lateral dimensions of the “modes”, x, approach molecular dimension, 6, as shown in Figure 6C. These molecular-scale protrusions and the additional thermal fluctuation force they give rise to must be treated in terms of a molecular rather than a continuum framework. On this scale the harmonic approximation of small deviations from equilibrium is no longer valid and the equipartion theorem can no longer be used. In the previous section we argued that at a single surface there are protrusions of molecules or groups out into the solvent. As two such surfaces approach there will be a configurational confinement. We can obtain a quantitative estimate of the resulting repulsive force by adopting the simple model of Aniansson, eqs 13 and 14, and neglect correlations between protrusions. Referring to Figure 6C, let each surface have molecular protrusions of lateral dimension 6, extending a distance zi into the solution, and let there be n protrusion sites per unit area (n = 1/62). For two surfaces facing each other, whose protrusions are not allowed to overlap, the configurational integral Z of the 2n degrees of freedom factorizes into n identical “two-particle” configurational integrals z, so that Z = z”, where

+

z = ~ D d z 2 ~ ~ z k x p [ - ~z ,() z/ klT ] dzl =

( k T / a ) 2 [ 1- (1

+ Da/kT)daDik7 (15)

which gives the force per unit area (the pressure) as1’ (ncw2D/kT)e-aDlkT P = aG/aD = kT a In Z / a D = [ l - (1 crD/kT)e-aD/kr)

+

or

P=

na(D / X)e-DiA [ l - (1

+ D/X)e-DIA]

where X = k T / a is the protrusion decay length as before. At large separations ( D >> X) the repulsion decays roughly exponentially, whereas at small separations ( D < A) it diverges according to P(D+O) = 2nkT/D (18) This is essentially the same as eq 5 and is the expected osmotic limit for an ideal gas of protrusions of density 2n/D confined within a gap of thickness D. Note that, by definition, the unfavorable energy of a fingerlike protrusion (Figure 7C) is equal to the excess area it exposes, d z , multiplied by the effective interfacial energy, yi. Thus, a and X may also be defined by (Y = ~ 6 y i A = kT/a = kT/dyi (19) 3. Effects of Headgroup Flexibility. Equation 17 was derived for the case of protrusions of whole molecules, but a diffuse polar/apolar interface can also be generated by the conformational freedom of flexible headgroups. For polymers this provides the dominant contribution to the force as discussed in section 2.5. Also for nonionic surfactants of the C,EO, type this is clearly an

(37) Biildt, G.; Gally, H. U.; Seelig, J. J. Mol. Biol. 1979, 134,673. Biildt, G.; Gally, H . U.; Seelig, J.; Zaccai, G. Nature 1978, 271, 182. (38) Claesson, P. M.; Kjellander, R.; Stenius, P.; Christensson, H. K. J . Chem. Soc., Faraday Trans. I 1986,82, 2735. (39) Lyle, I. G.; Tiddy, G. J . T. Chem. Phys. Lett. 1986, 124, 432. (40) Cevc, G. J . Chem. Soc., Faraday Trans., in press. (41) Rau, D. C.; Lee, B.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1984,81, 2621. Podgornik, R.; Rau, D. C.; Parsegian, V. A. Macromolecules 1989, 22, 1780.

Israelachvili and Wennerstrom

Figure 9. Computer-simulated repulsive pressures between surfaces modeled as follows (in order of increasing strength). (0)zwitterionic dipolar groups on each surface of length L = 0.5 nm and radius per charged group of R = 0.2 nm. Negative center fixed by a stiff harmonic potential!* (V) Same as previous calculation, but with protrusions also allowed. (A) Catanionic system with only protrusion^!^ (0)Each zwitterionic group characterized by two harmonic potentials (adapted from ref 44).

important mechanism for the force (as discussed earlier and shown in Figure 7). Even phospholipids like phosphatidylcholines have a substantial (10-15) number of degrees of freedom in the headgroup region so that the effects of headgroup flexibility cannot be neglected. Computer simulations are particu)arly well suited for exploring headgroups with only a few degrees of freedom, and recently a few papers have appeared focusing explicitly on the hydration force problem from the point of view of protrusion effects. Nilsson et a1.42,43 carried out a Monte Carlo simulation of zwitterionic headgroups which they modeled as charged hard spheres connected by a string. The anionic group was anchored at the surface by either a stiff harmonic potential or by a protrusion potential as in eq 13. To further test the role of protrusions of whole molecules, a so-called catanionic system was investigated. This consists of single-chain ionic surfactants with alternating positive and negative charges, making the surface electrically neutral. In this case there is no headgroup flexibility. In all the systems studied one finds a strong short range repulsion, as seen in Figure 9. A similar Monte Carlo study of DPPC bilayers (both isolated and interacting) was recently performed by Granfeldt and Mikl a ~ i c , 4who ~ used harmonic potentials to connect the negative centers to the surface and the negative and positive centers to each other. They also find a strongly repulsive entropic force between two bilayers of exponential decay length -0.2 nm having a range of 2-3 nm. In Figure 9 we have also included a curve from this study. A different approach was taken by Leermakers and Scheutjens (Figure 8), who used the Scheutjens-Fleer theory of inhomogeneous polymer systems. In this study the amphiphile is modelled as a short block copolymer. This allows for both protrusions of molecules and for conformational changes in the headgroups. An interesting prediction for double-chain amphiphiles is that conformations exist where a single chain protrudes out from the surface further than the headgroup. One could speculate that if this chain could penetrate through the polar groups of the opposing surface an attraction could arise. However, Leermakers basically finds4s an exponentially repulsive force with a decay length of about 0.3 nm. Additional work is in progress to obtain a more detailed picture of the different contributions to the force. 4. “Hydration” Forces: Experiment and Theory 1. Experimental Studies of ‘Hydration” Forces. Although of considerable age,’* the hydration force concept has attracted renewed interest through the experimental studies of Rand, (42) Nilsson, U.; Jonsson, B.; Wennerstrom, H. Faraday Discuss. 1990, 90, 107. (43) Nilsson, U.; Jonsson, B.; Wennerstrom, H., to be published. (44) Granfeldt, M.; Miklavic, S. J . Phys. Chem. 1991, 95, 6351. (45) Leermakers, F. A. M., private communication (to be published).

Feature Article

The Journal of Physical Chemistry, Vol. 96, No. 2, I992 527

Parsegian, and co-workersl0J 1 ~ 1 4 , 5 0on different phospholipid systems using the osmotic stress technique (see Figures 3 and 5). These studies have been followed up by similar studies in other laboratories (see Figures 1,2, and 7B and refs 13,22, 39,46, and 58) including work on other lipid and surfactant systems and employing other solvents than water (Figure 1). Measurements have also been performed using the surface force apparatus2with mica surfaces covered with phosph01ipids'~J~or other amphi~ h i l e s . ~ ~ J ~ *In~Figure ' @ 3 (inset) we show results for PC bilayers in the gel and liquid states. There is an overall qualitative agreement between data obtained on various phospholipid systems from different groups, as well as for data obtained by the surface forces apparatus and the osmotic stress techniques. The forces are near exponential, at least over a sizable part of their range, and they generally increase with temperature (Figure 3). However, the agreement between different researchers and techniques is not entirely quantitative, partly due to differences in the processing of the data to get the force profiles and partly due to actual physical differences in the preparation of the experiments. For example, when analyzed in terms of an exponential force, the decay length for the repulsion between egg-PC bilayers varies between 0.17 and 0.25 nm, and quoted experimental values for the preexponential factor C in eq 1 for egg-PC differ by 2 orders of magnitude.13J1 The strongest repulsions obtained for membrane lipids are found with phosphatidylcholines, which exhibit forces that are 1-2 orders of magnitude stronger at a given separation than those between the chemically very similar phosphatidylethanolamines.61 For other neutral surfactants such as monoglycerides, alkylamine oxides, alkylphosphine oxides, and alkylethylene oxides a strong shortrange repulsion is also found, but the measured forces have been of very short range (C1 nm) and an exponential distance dependence has not been established. The existence of a short-range repulsion can also be inferred indirectly from the phase behavior of many surfactant systems. For example, "catanionic" surfactants-a 50/50 mixture of an anionic and a cationic surfactant-form lamellar liquid crystals above the Krafft temperature. These lamellae swell much like the phospholipids, and the degree of swelling correlates with the size of the head group^.^^ For charged systems it is more difficult to establish the existence of any additional short-range repulsion because the double-layer force itself is often very strong at small separations. A short-ranged repulsion between supported CTAB bilayers in water was measured by Pashley and Israelachvili.*@ For ionic PG- bilayers, the measured forces appear to be satisfactorily accounted for by double-layer theory down to bilayer separations well below 1.0 nm (see Figure 5, and ref 16). Similar measurements on mica surfaces coated with bilayers of PG-, DHDAA+, and D H P (at low pH)I5sS1have also failed to demonstrate the existence of any extra repulsion between these charged bilayers at separations above 0.04.4 nm. We return to consider why charged systems appear to have little or no additional repulsive forces. 2. Theoretical Models of the "Hydration"Force. It was suggested at an early stage that the 'hydration" force, or-as it was called by the Russian School--'the structural component of disjoining pressure", could be explained in terms of water structure. However, it was not until 1977 when MarGelja and Radiclga published their theory of a decaying polarization profile that a quantitative theory was established. The basic ingredient of the theory is a Landau expansion of the solvent free energy density

using the solvent polarization as order parameter. One predicts an essentially exponentially decaying force where the amplitude is determined by a boundary condition at the assumed smooth interface. In later studies Gruen and Marqe1jalgbwere able to develop a microscopic model that gave a free energy consistent with the original Landau expansion. An explicit expression for the decay length emerged naturally. However, to account for the experimental decay length one had to invoke a picture of water as ice with 2% defects. In subsequent studies different variations of the Marplja-Radic theory have emerged.20J3 They all focus on water structural properties and have the common feature that the distance dependence of the force is a unique property of the solvent, with the repulsion caused by some structural order parameter decaying out from the surface generating a structural mismatch in the center of the gap. In terms of the thermodynamic terminology of section 2.2 the repulsion has an energetic source. A different approach to explaining the hydration force was adopted by Jonsson and W e n n e r ~ t r o m ,who ~ ~ focused on the solvation of highly polar surface groups. They developed an explicit electrostatic model based on the image charge concept, with the basic idea that an approaching colloidal particle displaces water from the outer hydration shell of surface groups generating a repulsive force. Although the focus in this case is also on the energetic contributions to the free energy, it was recognized that a repulsion could arise only for a disordered surface and that &he entropy was essential. However, the analysis stops short of allowing for an overlap of surface groups.44 Recently Kornyshev and Leikids developed the image charge idea by also allowing for a wave vector dependence of the solvent dielectric permittivity and thus combined the main features of the image charge and solvent polarization models of the hydration force. 3. Intermolecular Interactionsversus Configurational Freedom. At the surface of an amphiphilic aggregate there is an intricate interplay between a number of different intermolecular interactions: solventsolvent, solvent-headgroup, solvent-chain, chainheadgroup, and chain-chain. The net result of these is the creation of an interface between the polar solvent and the apolar chains with the headgroups located in the transition region. As a second similar surface approaches, the complexity of the interactions is even larger. There is a dispersion dominated van der Waals attraction between the apolar chains, largely compensated by a similar decrease of the dispersion interaction in the solvent. The polar headgroups also have a van der Waals-type electrostatic attraction. The net result is an enhanced van der Waals attraction between the surfaces. Closer in the surface groups interact strongly enough to cause deviations from the asymptotic (second-order perlurbation) form of the force. As these shorter range interactions increase in strength there are substantial effects on the configurational entropies of the surface groups. Above we mentioned how both analytical theories and computer simulations have demonstrated that such entropic effects are large and that they can dominate over the basic attractive interaction. Although the models used represent considerable simplifications of a highly complex reality, we feel that the qualitative conclusions remain valid. The extent of the fluctuations at an interface and the ability of a second surface to confine these fluctuations clearly depend on a delicate balance between attractive and repulsive intermolecular forces as mediated by the solvent. Hydration or, more generally, solvation effects are indeed crucial for modulating the interactions between headgroups at short range. Thus, in a good solvent the direct attractive headgroupheadgroup interaction is much reduced (or vanishes) so that there is no strong potential energy minimum at contact between the surfaces or headgroups. However, a repulsion will still be measured between these groups due to their osmotic pressure. How, then should one distinguish

(46) Marsh, D. Biophys. J . 1989, 55, 1093. (47) Herder, C. E.; Claesson, P. M.; Herder, P. C. J. Chem. Soc.,Faraday Trans. 1 1989, 85, 1933. (48) Pezron, I.; Pezron, E.; Bergenstihl, B.; Claesson, P. M. J . Phys. Chem. 1990, 94, 8255. (49) Jokela, P.; Jonsson, B. J . Phys. Chem. 1988, 92, 1923. (50) Cowley, A. C.; Fuller, N. L.; Rand, R. P.; Parsegian, V. A. Biochemistry 1978, 17, 3163. (51) Claesson, P.; Carmona-Ribeiro, A. M.; Kurihara, K. J . Phys. Chem. 1989, 93, 917. (52) Hartley, G . S. In Aqueous solutions of paraffin-chain salts; Hermann: Paris, 1936; p 60.

(53) Cevc, G. Chem. Scr. 1985, 25, 96. (54) JBnsson, B.; Wenncrstrom, H. J. Chem. SOC.,Faraday Trans. 2 1983, 79, 19; Chem. Scr. 1985, 25, 117. (55) Kornyshev, A. A.; Leikin, S. Phys. Rev. 1989, A40, 643 1. See also: Attard, P.; Patey, G. N. Phys. Rev. 1991, A43, 2953.

Israelachvili and WennerstrBm

528 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992

between the effects of a good solvent for a surface or group and the measured forces between them? One simple way to describe short-range solvation effects is to assign an intact solvation shell to the headgroups which simply increases their (hydrated) radius or (excluded) volume. This is often done in theoretical descriptions of electrolyte solutions and ion-ion interactions. By analogy with a van der Waals-type equation of state, eq 3, the effect of an excluded volume on the osmotic pressure may be expressed as =- 2nkT kT where Do = 2nb = 8nv (20) P= ( D / 2 n - b) ( D - Do) and where u is the excluded volume of a protruding group. Note that the above result suggests that, in a first approximation, the effect of finite headgroup size and hydration effects on the forces could simply be treated by shifting the origin from D = 0 out to D = D,,. The same conclusion was arrived at by Anians~on’~ regarding the effects of protrusions on shifting out the plane of origin of repulsive electrostatic double layer forces. This procedure would be equivalent to adding an effective “hydrated” layer of thickness 1/2D0= 2nu per surface. For example, assuming 10 ‘bound” water molecules per headgroup, each of volume 30 X m3, and n = 2 X lo1*m-,, we obtain a hydrated layer thickness of 0.24 nm/surface, i.e., the effective plane of origin of the forces or interaction potential is shifted out to Do = 0.5 nm. To summarize, the solvent (water) has a crucial role in establishing a repulsive force between fluid-like surfaces. But the force arises not because the solvent has difficulties in adjusting to the conditions at the surface due to structural constraints, but rather because it solvates the headgroups so well that it drastically reduces (screens) the intrinsically strong attractive interaction between them though without necessarily replacing it by another, e.g., repulsive, force.

5. Forces at Short Range. Quantitative Comparisons 1. van der Waals,Undulation, Peristaltic, and Headgroup Overlap Forces. Since we are mainly interested in the forces at a relatively short range, say below 4 nm, we now turn to a quantitative comparison of these different forces at small separations. Figure 10 shows theoretical plots of the undulation, peristaltic, protrusion, and headgroup overlap forces between two amphiphilic surfaces such as two lecithin bilayers in water at 25 OC. These have been computed as follows. van der Waals Force. The attractive van der Waals force has been included for comparing with all the other (repulsive) forces. This force is given by PvdW = -A/6rD3

(21)

where A is the Hamaker constant., For amphiphilic systems A = ( 5 f 2) X J should represent a reasonable value for the strength of the van der Waals force.26 Equation 21 is expected to remain valid out to bilayer separations of about 2.5-3.0 nm, above which the force decays faster than shown due to the finite thickness of bilayers and retardation effects. Thus at the larger separations the plotted van der Waals force is an upper bound. Undulation Force. According to eq 6, the strength of the undulation force is determined by the bending modulus Kb. Typical experimental values for lecithin bilayers are in the range 9X to 2 X J.61 At small separations the continuum, mean-field interaction is expected to become replaced by the protrusion and headgroup overlap force. Note that since the (56) Parsegian, V. A.; Rand, R. P. Lungmuir 1991, 7, 1299. (57) Pincus, P.; Joanny, J.-F.; Andelman, D. Europhys. Lett. 1990, I I , 763. Higgs, P.G.;Joanny, J.-F. J . Phys. (France) 1990, 51, 2307. (58) McIntosh, T. J.; Magid, A. D.; Simon, S.A. Biochemistry 1987,26, 7325. (59) Kwok, R.; Evans, E. Biophys. J. 1981,35, 637. (60)Tanford, C.In The Hydrophobic Effecr, 2nd 4.; Wiley: New York, 1980. (61) Cevc, G.;Marsh, D. In Phospholipid Bilayers; Wiley: New York, 1987;Chapter 2. Marsh, D. CRC Handbook of Lipid Bilayers; CRC Press: Boca Raton, FL, 1990.

I

I

I

I

I

I

I

I

I

I

I

I

I

DISTANCE, D (nm)

Figure 10. Theoretical plots of the various forces arising between two fluidlike amphiphilic surfaces such as two lecithin bilayers in water at 25 OC. The shaded region encompasses the experimentally measured range of forces between lecithin bilayers in the fluid state. The following parameters were used in the theoretical computations. For the van der Waals force: Hamaker constant, A = 5 X J. For the undulation J. For the peristaltic force: area force: bending modulus, Kb = expansion modulus, K, = 150 mJ/m2. For the protrusion force: interaction energy parameter, a = 2.5 X lo-” J/m, surface density of protruding groups: n = 2 X lo1*m-2. Note that all the forces plotted are repulsive except for the van der Waals force, which is attractive. Note too that between two rigid surfaces only two of the six forces shown can arise: the van der Waals force and, possibly, a hydration or solvation

force.

undulation and van der Waals forces have similar magnitude and decay characteristics (but of opposite sign), the short-range part of the van der Waals force can be simply considered as being effectively reduced by the undulation force, or vice versa. Peristaltic Force. The peristaltic force depends on the area expansion modulus of bilayers K,,which has been found to lie in the range 100-200 mJ/m2 for a variety of bilayer^.^^,^^ The range over which eq 11 is expected to apply is much smaller than the other forces. First, as a continuum, mean-field interaction it is not expected to apply at very small separations, but neither should it apply at separations greater than some fraction of the bilayer thickness (seeFigure 6C). The maximum range over which the peristaltic force, as given by eq 1 1 , is estimated to apply is between 0.2 and 2.0 nm. Headgroup Overlap Force. Figure 10 also shows the overlap force as determined from eq 12 using L = s = 0.8 nm. These values correspond to a headgroup area of 0.64 nm2 (64 A,) and to a headgroup layer thickness of 0.8 nm-both being typical values expected for lecithins and other surfactant/lipid molecules in bilayers. Longer headgroups should have longer-ranged forces, as illustrated in Figure 7 for the ethylene oxide surfactants. 2. Protrusion Force. Two parameters define the protrusion force of eq 17: the interaction parameter, a (in J m-l), and the density of protrusion sites, n (in m-2). For amphiphilic molecules in a bilayer in water, a would normally be the ‘hydrophobic” energy per unit length needed to extend the headgroup or a hydrocarbon chaii into the aqueous phase.’, The hydrophobic energy of a pure hydrocarbon chain in water is obtained from the solubility of alkanes in water60 and is about 6.3 X J/mol per CH2 group (900 cal/(mol CH,)) at 25 OC. Since the length of a CHI group along a hydrocarbon chain is 0.127 nm, this value corresponds to a = 5.0 X 10-” J/m and a protrusion decay length of A = k T / a = 0.082 nm. In the case of surfactant molecules in micelles, the hydrophobic energy per CH2 group or per unit length is normally measured from the ‘critical micelle concentration” (cmc, ref 60, Chapter 7 ) . The hydrophobicity of an amphiphilic chain is less than that of a pure alkane chain due to the proximity of the hydrophilic

The Journal of Physical Chemistry, Vol. 96, No. 2, I992 529

Feature Article Protrusion Energy per Unit Length for Single-and Doublechained Surfactants in Micelles As Determiwd from Cmc’s

TABLE II:

surfactant (lipid) alkyl sulfates@ alkyldimethylammonium chlorides62,63 phosphatidylcholines (lecithins)61

protrusion energy a, lo-” J m-I singledoublechained chained 2.3 2.1 3.5

3.7

4.1 5.4

head group and the higher chain ordering within micelles.61 Typical values for a are -3 X lo-” J/m (X = 0.14 nm) for single-chained surfactants and -5 X lo-” J/m (A = 0.10 nm) for double-chained surfactants (Table 11). The reason a for double-chained amphiphiles is less than twice that for singlechained is because the two monomer chains associate in water, hence reducing their area of hydrophobic contact with water.60$1,63.64 Turning now to the values of a and X for bilayers, Anianssons2 concluded that the protrusion energy a of a surfactant in a bilayer would be less, and X more, than in a micelle. Note that in bilayers composed of double-chained surfactants the protrusion of a headgroup now involves pulling out two chains for which the value of a is higher than for a single chain (Table 11). However, both chains do not always have to protrude together. Isolated chains can still protrude on their own. Indeed, since the protrusion energy of isolated, single chains remains unchanged or decreases, these are now energetically more favorable and therefore more probable than pair protrusions. This suggests that in bilayers composed of double-chained lipids, individual chains are likely to protrude farther than the headgroups (the second chain remaining in the bilayer). This is consistent with the theoretical analysis of lecithin bilayers by Leermakers and Scheutjens, who also concluded that the density profiles of the headgroups and individual chains, or “tails”, fall off more or less exponentially over a distance comparable with the fully extended length of the molecules (Figure 8A). The computed decay lengths were X 0.2 nm for the headgroups and X 0.3 nm for the tails, which corresponds to a = (1.0-2.0 X lo-” J/m. Further theoretical work by Leermakers4s based on the same theoretical model indicates that the repulsive force between two bilayers should be roughly exponential with a decay length of about 0.3 nm and that it arises from both entropic headgroup-headgroup and headgrouptail protrusion repulsions. These findings are consistent with the recent Monte Carlo simulations of Granfeldt and Miklavic,“ who computed a decay length for lecithin of about 0.2 nm. Recently, F‘feiffer et al.21used incoherent quasi-elastic neutron scattering (IQENS) to measure the various motions, such as molecular protrusions, tilting, rotation, diffusion, and collective bilayer undulations, in fully hydrated dipalmitoylphosphatidylcholine (DPPC) multibilayers in the fluid state ( T > 42 “C). They found that the lipid molecules exhibit out-of-plane vibrational motions (molecular protrusions) of amplitudes z = 0.2 and 0.6 and 4.3 X s, respectively. nm on time scales of T = 1 X Putting327 0: dA, these values correspond to a protrusion density decay length of X 2 0.27 nm (corresponding to a = 1.5 X J/m). Finally, a may also be estimated from the diameter and interfacial energy of the protruding groups via eq 19. For a surfactant chain of diameter 6 = 0.4 nm and using ri 20 mJ/m2 as determined for bilayer lamellar phases in ~ a t e r , 6we ~ *obtain ~ a = ?r6yi= 2.5 X lo-” J/m. From the preceding analysis of the current literature on protrusion it appears that realistic a values for bilayers in water should

-

-

-

(62) Mukerje, P.; Mysels, K. J. In Criricol Micelle Concentrations of Aqueous Systems; Nat. Bur. Stand. (US.), Nor. Srond. Ref.Dora. Ser. 1970, 36. (63) Shinoda, K.; Nakagawa, T.; Tamamushi, B.;Isemura, T. In Colloidal Surfocranrs; Academic Press: New York, 1963. (64) Pashley, R. M.;Israelachvili, J. N. Colloids Surf. 1981, 2, 169. (65) Jonsson, B.; Wennerstrom, H. J. Colloid Inrerfoce Sci. 1981,80,482. (66) Parsegian, V. A. Trans. Foraday SOC.1966, 62, 848.

range from above 5 X lo-” J/m to below 1.5 X lo-” J/m. This corresponds to X values in the range 0.08 to above 0.3 n m - a range that nicely spans the majority of force measurements (below we shall see that the decay length of the protrusion force A,,, is exw e d to be slightly larger than the decay length of the protrusion density A). In view of the above considerations, a value of a 2.5 X lo-” J/m (corresponding to X = 0.16 nm) was used in the calculation of the protrusion force shown in Figure 10. [This estimate of the protrusion energy was recently criticized by Parsegian and Rand,% who claimed that it would predict a cmc for dilauroyllecithin (DLPC) of 10-7-10” M, which according to the authors “is at least 4 orders of magnitude too high”. Actually, the literature value for the cmc of DLPC is (2-5) X M (Cevc and Marsh, ref 61, p 39), which is fully in agreement with Parsegian and Rand’s calculation based on the protrusion model.] Concerning the values for the density of protrusion sites, n, we may infer these from the known headgroup areas of amphiphilic structures. These range from 0.4 to 0.7 nm2 for single- and doublechained surfactants in micelles and bilayers, corresponding to n between 2.5 and 1.4 X 10l8m-2, and so a value of n = 2 X 10l8 m-2 was used (this corresponds to 1 protrusion/50 A2). Since we expect the protrusion force to dominate over the undulation and peristaltic forces at small distances, the protrusion force has been plotted down to the smallest separation. It is not, however, expected to extend beyond about 3-4 nm, corresponding to two fully extended hydrocarbon chain lengths (recall that the protrusion force precludes chains from overlapping at the center of the aqueous gap). The sharp drop in the repulsion beyond some distance has been observed experimentally (see Figure 3 and ref 11). 3. Comparison of Theoretical and Experimental Forces. From Figure 10, it appears that protrusion forces are expected to dominate the repulsive interaction between bilayers a t small separations (less than 1.5 nm), beyond which the undulation repulsion and van der Waals attraction take over. However, as mentioned above, due to retardation and finite bilayer thickness effects the undulation repulsion is expected to dominate at even larger separations (probably beyond 3-5 nm). We may also anticipate that for very long headgroups the overlap force will dominate the interaction at all separations (cf. Figure 7). The peristaltic force appears to be always small compared to the other four forces. Since the protrusion force appears to dominate a t small separations where most interbilayer force measurements have been made, it is worth characterizing this force further. From an analysis of eq 17, in the distance regime between 1 and 10 decay lengths (bilayer separations between 0.2 and 2.0 nm) the protrusion force varies roughly exponentially. Over this range it is adequately given by p = Ce-DIh, where C = 2.7na X, = 1.15X = 1.15kT/a (22)

-

The decay length of the protrusion force is therefore about 15% higher than the decay length of the protrusion density; the difference arising from the denominator in eq 17. For the typical values used in Figure 10, we obtain a preexponential factor of C = 1.4 X lo8 N/m2 and a decay length of A,, = 0.19 nm for the protrusion force in water. This is exactly within the range of values normally measured for the “hydration” forces between bilayers in water.11~’3J4~22*39*46*58 For example, for the much studied egg lecithin, the experimental values for Xo range from 0.17 to 0.26 nm, while the experimental values for C range over 2 orders of magnitude, from 4 X lo7 to 4 X lo9 N/m2, depending on how the experimental data are interpreted13J4,22 (these two limits are shown by the upper and lower shaded boundaries in Figure 10). For egg phosphatidylethanolamine (egg-PE) the measured forces are very similar to those of lecithins as long as the bilayers are in the fluid state,” as expected from the protrusion model, though-as already mentioned-for PES (and other lipids) in the solid crystalline state the protrusion repulsion is expected to be of much shorter range.

530 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992

Israelachvili and Wennerstrom

between mica surfaces.2*8.68While these forces, too, are generally As will now be shown, protrusion and overlap forces can account referred to as “hydration forces”$’ it is worth noting that there for many hitherto unexplained phenomena, either quantitatively are few similarities in the forces measured in these two types of or qualitatively. They also rationalize a number of trends that systems. First, with the smooth solid surfaces of clays and mica were always difficult to reconcile with the hydration model, there is always a short-range oscillatory force connected with the particularly the effects of nonaqueous solvents, chain number, and packing of the solvent molecules between the surfaces (e.g., of temperature on these forces: periodicity -0.25 nm in water). In the mica system the magnitude (1) The theory predicts that these forces should occur in noof the additional monotonic repulsion depends critically on the naqueous solvents. In addition, eq 19 predicts that A,, should pH and the binding affinity of the electrolyte ions. The decay increase as the interfacial energy (n)of the hydrocarbonsolvent length of the repulsion also depends on the counterion species and decreases. This trend has been found in the forces measured particularly on its valency.68 For monovalent electrolytes such between lecithin bilayers in water, formamide, and 1,3-propanediol as NaCl typical decay lengths are in the range 0.6-1.2 nm, and (Figure l), where the measured decay lengths parallel the dethus very different from the 0.1-0.3 nm found in+mostamphiphilic creasing interfacial energies of these liquids (Table I). systems. We may note that the surfaces of clays and mica are (2) Equation 19 suggests that longer decay lengths should arise solid crystalline lattices, where dynamic protrusions clearly cannot between singlechained bilayers than doublechained bilayers. This occur as they do at amphiphilic surfaces. It is our conclusion that is precisely what is observed: the three highest decay lengths so the molecular mechanism for the additional solvation force befar measured are all for lysdipids, viz. >3.0 8, for C12E03,39 4.5 tween these crystalline surfaces are different from the ones dis8, for lyso-pc46 (the data for lyso-PC may apply to the hexagonal cussed for liquid-like surfaces, although we do not exclude an phase where the aggregates are rodlike), and 6.4 8, for lyso-PS,46 entropic origin. In fact, one appealing feature of the mechanism while double-chained lipids such as PE, DLPC, DMPC, DPPC, proposed here for the amphiphiles is that it resolves a previous and egg-PC (all in the fluid state) generally have about half these dilemma that the water structure explanation should have been decay lengths. equally valid for mica and lipid surfaces, and yet the experimental (3) Equations 17-19 predict an increased repulsion with inobservations were generally quite different. creasing temperature, as is generally observed. Such a trend would The repulsive forces between bilayers have also been linked to also be expected for other entropic forces but not for forces arising apparently similar short-range forces measured between oriented from solvent structuring effects (unless one accepts the possibility DNA molecules.“’ Since DNA molecules are charged in water, of increased structure/ordering at higher temperatures). the net repulsive force between them is a combination of at least (4) For most of the measured force profiles low decay lengths are generally associated with high preexponential f a ~ t o r s ~ ~ * three ~ ~ *separate ~ ~ components: a repulsive electric doublelayer force, a thermal fluctuation force due to molecular motions, and any (see also Table I). Equation 22 readily accounts for this trend genuine hydration force. At small separations the magnitude of following the reciprocal relation between a and Xo. the doublelayer force between any two surfaces of constant charge ( 5 ) Equations 12 and 17 predict that repulsive entropic forces is expected to be determined by the osmotic pressure between the should not be simple exponential functions of distance. In partrapped ions which, as already discussed in section 2, is independent ticular, both predict a steep upturn in the repulsion at very small of salt concentration and very similar to the entropic forces arising D, which has been observed in many systems where forces have from molecular motions in bilayers. This, and the difficulty of These same been measured down to separations below 3-5 defining D = 0 for the DNA double helix, makes it difficult to two equations also predict that the forces decay more rapidly to say anything definitive about the separate contributions of these zero beyond some finite distance corresponding to the fully exthree forces in DNA systems. In our view, whether or not a tended chain and headgroup lengths. This probably explains the “hydration force” exists in these systems remains to be demonsharp cut-off in measured forces just prior to the potential well strated. (ref 11 and Figure 3). These sharp breaks in the curves cannot be explained in terms of a purely exponential repulsion plus a van 6. Concluding Remarks der Waals attraction, which would predict a much smoother From the good agreement obtained between experiment and transition toward the attractive minimum. If the protrusion force theory without any need to invoke any additional structuraldoes indeed decay faster beyond some finite separation, this may hydration interactions, it is concluded that the short-range reaccount for why very different and often unrealistically large pulsion between amphiphilic surfaces is mainly of entropic origin. Hamaker constants have been inferred by assuming that the The role of the solvent comes in solely for reducing the long-range exponential repulsion continues indefinitely until it is balanced attraction between the surface groups and for determining the by the van der Waals attraction (see Table I in ref 11). interfacial energy (rior a),after which the forces are determined (6) Since the solubilities of unsaturated hydrocarbons in water by thermal fluctuations and not by any decaying structural effects are less than those of the saturated homologues, larger decay of the solvent. Whether water structure provides an additional lengths should be seen with unsaturated and biological lipids, as long-range “hydration” force law between mobile amphiphilic has been observed. For example, Marsh4 measured X, = 2.2-2.3 interfaces as it does between rigid, crystalline surfaces such as 8, for the three saturated lipids di-C(18:O)PC, di-C( 16:0)PC, and silicate surfaces remains to be demonstrated. di-C(14:O)PC but obtained X, = 3.1-3.4 8, for their unsaturated It is clear that more careful thought must be given before using homologues di-C(18:l)PC and di-C(18:2)PC and X, = 2.8 8, for terms such as “structured water”, “water of hydration”, “hydration egg-PC, all at 22 OC. forces”, etc. For example, it is no longer obvious that the amount (7) Figure 10 shows that headgroup overlap forces should of water uptake during the swelling of a lyotropic lamellar phase become dominant at small separations. We may also expect can be unambiguously associated with the “structured water” or headgroup overlap forces to be suppressed by altering the solution “hydration” of the headgroups or surfaces. Likewise., it is probable conditions, for example, on going from “good” to “poor” solvent that in many cases where the “water structure”in a system is being conditions for the headgroup segments. Most interesting is the “disrupted” or ‘modified”, this does no more than simply change apparent absence of any additional repulsion between highly the flexibility of molecular groups and in this way alter their charged b i l a y e r ~ , ’ ~ Jan ~ +observation ~’ that appears to confirm entropic interaction. recent theoretical predictions5’ that electrostatic charges suppress undulation and other fluctuation interactions. By contrast, J. F. Acknowledgment. We thank Frans Leermakers, Ulrik Nilsson, Joanny, B. Simons, and M. Cates (private communication) have Bengt Jonsson, Jean-Frangois Joanny, Ben Simons, Mike Cates, recently concluded that headgroup dipoles should enhance fluctuations-a finding that may account for why the strongest steric repulsions have so far been seen with zwitterionic lipids. (67) van Olphen, H. In An Introduction to Clay Colloid Chemistry, 2nd 4. Nonamphiphilic Systems. Unexpected forces are also found ed.;Wiley: New York, 1977; Chapter 10. ( 6 8 ) Pashley, R. M. Adu. Colloid Interface Sci. 1982, 16, 57. in claylike systems6’ and have been measured in great detail

J. Phys. Chem. 1992,96, 531-537 and Stan Miklavic for making data available to us prior to publication. We also thank Frans Leermakers, Gregor Cevc, Christiane Helm, Wolfgang Knoll, Clay Radke, and Joe Zasadzinski for helpful comments and the NSF for supporting J.N.I. under Grant CTS-9015537.

Glossary force law

decay length

variation of force For pressure P with distance D between two surfaces. Since many force laws have a near expoor P = P$-D/A, nential distance dependence, F = these are usually plotted on a log linear (semi-log) graph where the data falls on a straight line whose inverse slope equals the decay length, A. for any exponential function of the type F = Foe-D/Athe decay length X is a measure of the rate of decrease of F with distance D F falls by a factor of 2.7 as D changes by A. The smaller the value of X the more rapid is the

a X

x, P e,

6 Yi

Q

531

decay and thus the shorter is the range of the force. area (m-’), Hamaker constant (J) distance (m) force (N), pressure (N bilayer compressibility modulus (J m-’), bilayer bending modulus (J) thickness of brush layer or headgroup layer thickness (m) density of molecular groups per unit area (m-’) mean distance between surface binding sites or between headgroups (m) protrusion energy per unit length (J m-I) decay length of the density of molecular groups protruding out from a surface decay length of the force number density (m3) electronic charge (C), surface charge density (C m-’) molecular diameter or width (m) interfacial energy (J m-’)

ARTICLES Global Potential Energy Surfaces from Limited ab Initio Data Alan D. Isaacson Department of Chemistry, Miami University, Oxford, Ohio 45056 (Received: May 1 , 1991)

-

Two new potential energy surfaces for modeling the dynamics of the reaction OH + H2 H 2 0 + H are presented. While the form of these surfaces is relatively simple, varying the surface parameters along the reaction path allows these surfaces to reproduce the ab initio information of Walch and Dunning as well as either the bamer shape computed by Dunning, Harding, and Kraka or the experimental rate constants of Ravishankara et al. By comparing the results obtained from these surfaces and an earlier one by Schatz and Elgersma, we are able to study the sensitivity of the predicted dynamics on both the form of the surface and the data to which the surface is fit. In particular, the results indicate that both the barrier shape and the degree of reaction path curvature can strongly influence the predicted rate constants at lower temperatures.

1. Introduction Any detailed theoretical study of reaction dynamics relies on a knowledge of the potential energy surface (PES) for the reacting system. However, since the PES for an N-atom system is a function of 3 N - 6 variables, the accurate calculation and representation of a PES poses a major challenge in dynamics studies even for simple systems. As a result, a common method for obtaining a PES of a reacting system involves the inversion of a limited amount of theoretical and/or experimental data on reactant, product, and transition-state properties as well as experimental rate constants and equilibrium constants. Such an inversion is always based on an assumed functional form for the PES, and one important question in this area concerns the sensitivity of the predicted results for the reaction dynamics on the choice of the functional form used for modeling the PES. Indeed, we present below a case in which different functional forms fit to the same set of data lead to profoundly different predictions for the rate constant. A related question involves the identification of the key properties of the PES that should be considered most carefully in the choice of a functional form. One such property is the barrier height, which strongly affects the reaction rate by determining the number of collisions that have sufficient energy to react. Another is the barrier shape or width, which controls the degree of reaction tunneling. As discussed below, the reaction path curvature arising from a particular functional form markedly

affects the results in some cases. Furthermore, we show below that requiring the PES have a particular bamer shape can strongly affect the degree of reaction path curvature obtained from a given functional form. A third important but more practical question deals with determining functional forms and fitting procedures that can provide efficient and accurate fits to a given set of property data. For example, we show below that the variation of surface parameters along the reaction path’-5 allows for an accurate and relatively systematic approach to the inversion of property data to yield a PES. In this paper we address the above questions within the context of the PES for the reaction OH + Hz HzO H (R1) +

+

which is important in combu~tion,~~’ and we present a new PES which better fits the available data than do previous (1) Blais, N. C.; Truhlar, D. G. J . Chem. Phys. 1974,61, 4186;1976,65, 3803E. (2)Truhlar, D.G.; Garrett, B. C.; Blais, N. C. J . Chem. Phys. q984,80, 232. (3)Steckler, R.;Truhlar, D. G.; Garrett, B. C. J . Chem. Phys. 1985.83, 2870. (4)Blais, N. C.;Truhlar, D. G. J. Chem. Phys. 1985,83, 5546. (5) Joseph, T.; Steckler, R.; Truhlar, D. G. J . Chem. Phys. 1987,87,7036. (6)Glassman, I. Combustion; Academic: New York, 1977. (7)Creighton, J. R.J . Phys. Chem. 1977,81, 2520.

0022-3654/92/2096-53 1%03.00/0 0 1992 American Chemical Society