The interdependence of intra-aggregate and inter-aggregate forces

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The interdependence of intra-aggregate and inter-aggregate forces

J.N. Israelachvili, D. Sornette

To cite this version:

J.N. Israelachvili, D. Sornette. The interdependence of intra-aggregate and inter-aggregate forces.

Journal de Physique, 1985, 46 (12), pp.2125-2136. �10.1051/jphys:0198500460120212500�. �jpa-

00210161�

The interdependence ^of intra-aggregate ^and inter-aggregate ^forces

J. N. Israelachvili

Department ^of Applied Mathematics, Institute of Advanced Studies, Research School of Physical Sciences, GPO Box 4, Canberra Act 2601, Australia

and D. Sornette

Laboratoire de Physique ^{de la Matière Condensée} (*), Université des Sciences, Parc Valrose, 06034 Nice Cedex, France

(Reçu ^{le 6 mai 1985,} accepté le 20 août 1985)

Résumé. 2014 Nous examinons l’interdépendance des forces responsables ^de l’association d’amphiphiles pour former des agrégats tels que micelles, microémulsions et bicouches et des forces s’exerçant ^{entre de telles} structures à courte distance. Nous présentons ^{d’abord une} analyse ^de l’origine ^{et de la nature} des forces d’hydratation ^{entre têtes} polaires qui prédominent ^{à très courte} portée. ^Un développement Landau-Ginzburg permet de relier naturellement la force d’hydratation

sein d’un même agrégat à ^celle s’exerçant ^entre deux bicouches. Le potentiel d’interaction unique ^obtenu permet ^{de démontrer la} corrélation entre l’aire _{par tête} polaire d’une bicouche isolée et la force de

répulsion ^entre bicouches. L’application ^{de cette} approche à d’autres systèmes ^{est discutée.}

Abstract

We examine the interdependence between forces responsible for the self-association of amphiphiles ⁱⁿ

aggregates ^{such as} micelles, microemulsions and bilayers ^and the forces occurring between such structures when

they

^close together. ^{We first} present

theoretical analysis ^{of the} origin ^{and nature of the} hydration ^{force which}

appears ^as the most important interaction occurring ^between head-groups ^at very short distances. A Landau-

Ginzburg approach ^{allows us to link} naturally ^the hydration ^force between the head-groups ^{within an} aggregate with that between two opposing bilayers. ^The single interaction potential ^{obtained between} amphiphilic ^head-

groups allows demonstration of

quantitative correlation between the measured head-groups

of isolated

bilayers ^{and the} repulsive pressures between bilayers. The relevance of this interdependence ^to various other

phenomena is discussed.

Classification

Physics ^Abstracts

82. 70K - 82.65D

1. Introduction

The forces acting ^between amphiphilic ^structures (aggregates) ^{such as micelles,} bilayers, vesicles and microemulsion droplets ^{in water determine} many

important phenomena, ^for example, ^their stability ^to aggregation. ^{in dilute} solution, ^their phase ^behaviour

in concentrated solution, the thickness of _soap films,

etc. By contrast, the forces acting between the same

amphiphilic molecules within these structures deter- mine the type ^{of structure} they self-assemble into, their shape, size, polydispersity ^and phase state, and in particular ^{how these} parameters ^{are sensitive to} changes in the solution conditions. For unlike solid

(i.e. rigid) ^colloidal particles, ^{the forces} holding amphiphiles together ^within aggregates ^{are weak}

(*) Laboratoire LA 190 Associe

C.N.R.S.

(viz. ^{van der} Waals, ^screened electrostatic, ^{and sol-}

vation forces), ^which usually results in fluid-like struc- tures that change their size and shape when the solu- tion conditions, temperature, ^or surfactant concen-

tration is changed. ^These factors, however, ^{are the same}

as those that affect the forces between these structures

(or any colloidal particles ⁱⁿ general).

The _purpose of this _paper is to investigate ^{how the}

intermolecular forces within amphiphilic aggregates

are correlated with the forces betweel aggregates, how this interdependence manifests itself in practical terms, and how it _may be quantified.

While such correlations have been noticed before [1],

no attempts appear ^to have been made to consider their generality ^or their theoretical basis. The reasons

for this are twofold. First, ^the short-range ^electro- static, hydration ^and hydrophobic forces involved

are still not sufficiently well understood. Second,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460120212500

theories of interparticle and intersurface forces are

often formulated in terms of some smeared-out surface property (rather than between the molecules consti-

tuting ^the surfaces). ^For example, ^a uniform surface

charge density ^{in the case of} double-layer ^forces [2],

or a uniform surface polarization density ^{in the case}

of hydration (solvation) ^forces [3, 4].

Further, the interface itself is usually ^{modelled as a}

discontinuous dielectric boundary. By ^so treating

surfaces as homogeneous planar discontinuities rather than made _up of discrete molecular _groups (amphi- philic head-groups, ^for example) ^{it is not} readily

apparent ^{how the} forces .between these _groups can be related to the forces between the surfaces.

Note however, that these formulations are the result of mean field approximations ^{and are not due}

to an inherent assumption ^{where one} disregards ^the

direct interaction between the ionic head _groups at the surface (1).

In this _paper we adopt ^another approach ^where right ^{from the start we assume} that the forces between individual head-groups ^within aggregates ^{are the same}

as those that give ^{rise to the net} force between the aggregates. ^{In other} words, ^a single interaction

potential ^is postulated ^{to describe both.}

This work can be considered as an attempt ^to generalize ^{to neutral} systems what has been established for charged systems.

In section 2 we proceed ^{with a} theoretical discussion of the problem, ^{and in} particular ^{we derive a} simple expression ^{for the} repulsive hydration potential

between two head-groups. ^{In section 3 we} apply ^this potential ^{in a detailed} quantitative analysis of lecithin

bilayers, while in the final section 4 we discuss the broader implications of the results and show that there are many other systems ^{to which the «} principle

of interdependence » applies.

2. Nature of the interactions within and between aggregates.

2.1 THEORETICAL BACKGROUND. - First ^we consi- der the interaction forces between amphiphiles (sur- factants, lipids) ^within aggregates ^{that determine the} type ^{of structure that} they self-assemble into. This has been well described and reviewed in the literature

[ 1, 5] ^and only the essential features will be given ^here.

For amphiphiles ^whose hydrocarbon ^{chains are in}

the fluid state (i.e. surfactants above their Krafft point)

their optimum head-.group ^area ao is determined by ^a

balance of two « opposing ^{forces »} (6) acting mainly

in the interfacial head-group region. The attractive

(hydrophobic) ^energy contribution _per molecule _may be expressed ^as ya where a is the head-group ^{area and}

where _y, the hydrocarbon-water interfacial (hydro- phobic) ^free energy, is in the _range 20-50 mJ/m2

[1, 7-9]. ^The repulsive energy contribution _per head-

(’) ^As pointed ^out by

^referee.

group may be expressed approximately ^as cla, ^{where c}

is a constant. The total energy per molecule is therefore ya ⁺ cla. ^{This has a} minimum value at a = ao

ffi,

which gives ^{the «} optimal ^{area »} per head-group [1, 9].

For fluid hydrocarbon chains the optimal ^{area should}

be independent of the number of chains or the chain

length, ^{as is} indeed found experimentally [10, 12]

since the van der Waals attraction between chains

gives ^a very small contribution compared ^{to these}

two terms.

The optimal head-group ^{area is a} very important

parameter ^since (when ^{it is taken} together ^{with the}

chain volume and chain length) ^it determines the types ^of aggregates ^{that are} formed and many of their

physical ^and thermodynamic properties [1, 5, 9].

From the preceding paragraph it is also apparent ^that the head-group ^area ao is mainly determined by ^the repulsive ^{forces between} head-groups, ^{since the attrac-}

tive component y is more or less constant [7-9] ^while

the repulsive component ^{c can} vary greatly depending

on the nature of the head-group and the solution conditions. Let us therefore consider these repulsive

forces in more detail.

The repulsive forces between adjacent head-groups

within a fluid micelle, bilayer ^or monolayer, ^which

determine their head-group area, include electrostatic,

steric and solvation (hydration) interactions. Since all these interactions are occurring ^over very short distances (a ^few Angstroms, corresponding ^{to the mean} spacing ^between head-groups) ^their theoretical model-

ling ^{is still at a} rudimentary stage. Simple equations ^for

the lateral electrostatic (double-layer) pressure within the interfacial region ^{of a} single monolayer ^or bilayer

have been derived by Payens [ 13] ^and Forsyth et ^al. [ 14]

on the assumption ^{of a smeared out surface} charge,

and their interplay with the attractive interfacial force has been analysed by Parsegian [7] ^and by ^Wenner-

str6m and Jonsson [8]. ^{The steric} repulsion, ^which depends primarily ^on the sizes of molecules or mole- cular _groups which _may be modelled by ^hard spheres,

discs or cylinders [ 15] ^was first considered by Lang-

muir [16]. However, ^the important ^{role of water of} hydration ⁱⁿ determining the effective head-group size, and/or ^its contribution to the inter head-group pair potential, ^{does not} appear ^to have been considered in the literature and is still unresolved.

Turning ^{our attention now to} inter-aggregate ^forces

we notice that the repulsive ^{forces are} fundamentally

the same as those between adjacent head-groups

discussed above, ^viz. repulsive electrostatic, ^{steric and} hydration ^forces. Consequently, ^{one should} expect ^a

correlation between intrabilayer ^and interbilayer ^for-

ces. For example, ^we might expect larger head-group

areas to be accompanied by larger repulsions ^between bilayers. This is borne out by experiments : ^the large hydration of the lecithin head-group results in a large

surface area of N 70 A2 as well as a large swelling ⁱⁿ

fully hydrated ^lecithin multibilayers. By ^{contrast the}

head-group repulsion ⁱⁿ phosphatidylethanolamines

is much less, which leads ^to a smaller head-group ^area

of 45-55 A2 and a much reduced swelling, ^{viz. 10 A} compared ^{to about 25 A} for lecithin [ 17,18]. ^{We shall}

consider other systems where such correlations occur

later. Here we proceed ^{with a} consideration of the three major types ^of repulsive inter-aggregate ^forces.

For simplicity ^{we shall} only ^talk about the forces between planar bilayers (while recognizing ^{that the}

same forces occur between micelles, vesicles, etc.).

The repulsive electrostatic double-layer ^{force bet-}

ween colloidal surfaces, lipid bilayers ^{and across} soap films is now well understood theoretically ^{and it has}

been measured in all three cases [19]. ^The importance

of steric repulsion between fluid bilayers ^{is still}

controversial [20], ^Huh [21] applied ^the theory ^of

Dolan and Edwards [22] ^to the interactions between oil in water microemulsion droplets ^{where the} pro-

truding hydrophilic head-groups ^{on each surface}

were considered to act as short polymer ^chains interacting ^{via a} repulsive ^osmotic (entropic) ^steric

interaction. Other steric contributions to the net

repulsion ^between bilayers would arise from local thermal fluctuations in bilayer ^{thickness and head-}

group ^area [23], ^and long wavelength elastic undu- lations of bilayers [24, 25]. Again, the effect of water of hydration ⁱⁿ increasing ^the effective excluded volume of head-groups ^{and in} modulating ^{the undu-}

lation forces is to enhance their magnitude ^and

range [25].

Finally, ^we briefly mention the various theories that have been proposed to account for the strong exponentially repulsive ^« hydration >> force measured between lecithin bilayers [26]. ^{The mean field} theory

of Marcelja ^{and Radic} [3] ^{later extended} by ^{Gruen and} Marcelja [4] ^considers hydration ^{forces in} general ^as

due to the decaying polarization ^{of water molecules}

away from a surface, although ^{a more} realistic mole- cular dynamics simulation of this system [27] ^shows

that an oscillatory ^{force is} expected when the discrete molecular nature of water is taken into ^account Jonsson and Wennerstrom [28] proposed ^a highly specific theory (for ^lecithin bilayers ^{alone) based on}

the interactions of the zwitterionic (dipolar) ^head-

groups ^{on one} surface with their electrostatic images

« reflected >> by the other. Somette and Ostrowsky [24, 25] ^{have extended a} previous calculation by

Helfrich [29] and have derived general expressions

for the steric repulsion contribution arising ^{from the}

thermal undulation of elastic bilayers ^{in the} presence of other interactions. Note that these three _very different theories each predict ^an exponential, ^{or near-} exponential short-range repulsion ^of magnitude ^and

range close to that measured, ^{and we} may also mention that the theory of Dolan and Edwards [22] ^also predicts

a short-range exponential ^steric repulsion ^between bilayers ^with freely ^mobile protruding head-groups.

(The ^one consolation for this state of affairs is that it is quite ^{common in the} history of science to find

completely different theories ^« explaining >> ^{one obser-}

vation

a phenomenon that is known as the « inva- riance of current theories to current experimental

results ».)

However, ^{we can make the} qualitative remark that this uniformity ^of exponential behaviour in all theories

only reflects the existence of a well defined correlation

length (not necessarily ^{the same for all} theories) leading unambiguously ^to exponential decaying ^cor-

relation functions.

We shall return to consider the likelihood of these interactions again ^at the end of this _paper. Here,

we shall continue with a more detailed analysis ^{of the} origin ^{and nature of the} hydration force since (i) ^{it is}

the least understood and yet appears ^to be the most

important (at ^{least in} contributing ^{to the} short-range repulsion ^between hydrophilic head-groups), ^and (ii) ^we require ^some formulation that naturally ^links

up the hydration force between head-groups (within

a bilayer) with that between two opposing bilayers.

2.2 A SIMPLE MODEL FOR THE HYDRATION INTERACTION BETWEEN HEAD-GROUPS.

The physical origin ^of

this so-called hydration ^or structural force is not yet fully understood. It has long ^been recognized ^that

water is an associated liquid which has rather unique

solvent properties [30].

A qualitative concept, developed ^{from the} beginning

of colloid and surface science, underlines the impor-

tance of the modification of solvent structure near a

solute molecule. This perturbation propagates ^over

some distance from the solute molecule. When a

second solute molecule approaches, the total _per- turbation of the solvent structure is changed, leading

to an indirect solvent mediated solute-solute inter- action.

A complete statistical mechanical treatment of

homogeneous ^{water is} extremely ^difficult (and ^all

the more so for inhomogeneous water) : indeed, ^the

structure of water is not determined by ^{hard core}

contacts and smooth attractive forces. Instead, ^water

constitutes an open ^structure akin to an imperfect

random tetrahedral network with oxygen ^{atoms at} the vertices and hydrogen ^atoms placed along ^{the 0-0}

bonds in configurations satisfying ^{the ice} rule, forming

the so-called hydrogen ^{bonds. As a result, water forms}

a structure where both position ^and orientation of

nearby ^{water molecules are} correlated. Theoretically,

one has thus to take into account both the transla- tional and orientational degrees of freedom and

analyse ^the way they ^are coupled

Some attempts have been made to derive general Landau-Ginzburg formalisms by expanding ^the

Ornstein-Zernike equation ^{in the weak} coupling ^limit

with the additional hypothesis that the interaction

potential ^{between water} molecules is short-ranged [31].

Marcelja et ^al. [31 have ^{obtained an} expression ^{for the}

free-energy density ^{in terms of the} density ^order

parameter. Realizing ^the incompleteness ^{of this}

approach, they ^{have also} proposed ^a generalized

Pople ^{model for water} focusing ^{on an} orientational order parameter, namely ^the average of the cosine of the angle measuring the deviation of either the lone

pair ^{direction or} of the OH bond from 0-0 line.

They also obtained in this case a Landau-Ginzburg expansion very similar to the density ^order parameter.

A similar formalism had been proposed previously by Marcelja ^{and Radic} [3] ^who considered the orientation effects of a planar membrane which polarizes ^the surrounding medium. This case may be simplified ^if

one assumes that the dominant effect of the pertur- bation of water structure is orientational ordering

of molecular dipoles and that the density ^variation

and dipolar polarization ^are decoupled.

The free _energy expansion describing ^{the solvent} perturbation ^{then reads}

where Eo ^{is the} applied ^electric field, ^P

XE, ^D

^sE and ~ ^{is the} corresponding correlation length.

This first approach ^{has been} subsequently ^{put on a}

firmer basis [4, 32] by showing ^that equation (1),

which implies ^a fixed local linear relation between the electric field and the dielectric polarization P, may be derived from a statistical ice-model of water with

Bjerrum ^defects.

The advantages ^{of this} type ^of approach ^{are :} a) ^The very difficult problem ^of determining ^the

translational and orientational correlation functions is replaced by the search for a continuous order parameter profile subjected ^to the condition that it minimizes the total free energy expressed ^{as a Landau} expansion. ^{This LG} expansion ^is extremely general

and has a wider validity ^than expected naively ^from

its derivation. It is essentially equivalent ^{to the} assumption ^{of the} Ornstein-Zernike form for the order parameter correlation function i.e.

where is the order parameter correlation length.

b) The unknown region where the interaction is strong (near ^{a solute} particle ^for example) ^which corresponds ^{to a breakdown of the} validity ^{of the}

weak coupling limit is subsumed in the boundary

conditions that one has to implement ^{to the Euler-} Lagrange equations derived from the minimization of the total free energy. Provided ^{one can} experi- mentally determine the boundary conditions (this

may be the difficult point) ^the problem ^is well-defined.

The continuous limit of the free _energy density ⁱⁿ

the LG expansion neglects the discrete nature of the solvent and should be ideally ^used only when relevant distances are larger than the molecular size. In this

limit, the variation of, say, ^an exponential ^sinusoidal

correlation function is averaged ^{to an} exponential.

However, ^{even if} the correlation length and molecular

separation ^are comparable ^{we can still} expect ^a qualitative description [31]. ^{In the case of} light ^solute particles ^or fluctuating ^fluid membranes, ^one expects

a smearing ^out of the molecular discrete structure of the solvent when averaging ^{over time} (or alternatively

over statistical configurations) with the result that the continuous description then becomes more rele- vant and appropriate.

We may then summarize the possible options ^for describing the solvation interaction between solute

particles ^{as follows :}

Conceptually, the notion of a disruption ^{of water}

structure by ^the presence of a solute molecule (or macroscopic body ^or surface) ^seems well established.

The difficulty remains in determining ^{what is} (are)

the relevant order parameter(s). ^{The most} likely

candidates _may be the solvent density, ^the hydrogen

bonds density ^{and the} polarization density. ^{The two}

first order parameters ^are scalar whereas the last one

is a vector. This recognition already ^{leads to an} interesting conclusion : when applied ^to solute-solute

interactions, the LG formalism with a single ^scalar

order parameter ^{can be shown to be} unable to dis-

tinguish clearly between the case of non-polar ^and polar solutes, ^and always predicts ^an attraction [31].

One could _argue that this results from the restriction to quadratic expansions of the free energy ^{as a} function of the order parameter ^{as in} equation (1). Indeed,

for large ^solute particles, it has been recognized ^that

the next quartic ^{term, as well as a surface} energy

density replacing ^{the fixed} boundary ^conditions usually imposed, ^{are essential.}

This surface free-energy which describes both the

change ⁱⁿ coupling of the order parameter ^induced

by ^the presence of the surface and a possible ^surface

chemical potential, together with the bulk free energy

density, ^{leads to the} qualitatively ^correct description

of the _very rich behaviour such as prewetting, ^first-

order and second-order wetting transitions [33].

However, it has been shown [34] ^{that one} always

expects ^an attractive contribution for the solvation interaction between like-solute particles ^{in the case}

of symmetric boundary conditions (i.e. ^{identical on}

the two solute particles). This result holds as long ^as

the scalar order parameters (say, ^molecular density

and hydrogen ^bonds density) ^are decoupled. ^This assumption ^is clearly questionable considering ^the highly correlated local orientational and translational order of liquid ^{water. Indeed, it can} be shown for the

case of two coupled, scalar, ^order parameters, ^that there exits suitable choices of the free energy density

and (symmetric) boundary conditions which lead to a

repulsive interaction.

Thus, within the hypothesis ^{that the} interaction between solute particles results from the disruption

of the density ^and hydrogen bonding ^{of water mole-} cules, ^we might conclude that the observed repulsion

can only emerge when fully taking ^{into account of the}

coupling between the two scalar order parameters.

The alternative description ^{in terms of the vector} polarization ^order parameter is richer since it embo- dies in a sense the density ^and hydrogen bonding description (to ^be precise, ^{a full} description ^{of the}

solvent is only attainable by defining ^all multipole

densities and not only ^the polarization profile) ^and

allows a simple ^{treatment in terms of a} single ^three-

dimensional order parameter.

In the following ^{to be} specific ^{we use this} repre- sentation. Let us consider a film of water bounded by

two infinite planes from which polar ^heads protrude

and _occupy on the _average the equilibrium ^area ao.

We assume that the planes ^are placed ^{at the} hydro- phobic-water interfaces and that the water molecules do not penetrate ^{in the} hydrocarbon ^core [35]. ^The

water disturbance can then be decomposed ^{into two} contributions, ^the hydrocarbon-water ^{and the} polar-

heads-water interactions. We assume for simplicity

and as a first approximation, ^{that the structure of}

water is mainly ^{due to} its interaction with the polar

heads. This hypothesis ^is clearly wrong when no polar

or ionic species ^{are fixed on} the surface as is the case

for hydrophobic ^surfaces immersed in water [36].

In this context, ^{a new} type ^of inter-aggregate ^force

has been measured, namely ^{a rather} long-range

attractive hydrophobic interaction which may be partly rationalized within the Marcelja’s type ^of models with a scalar order parameter ^and symmetric boundary conditions.

In the presence of polar ^{or ionic} species ^bounded

to the surfaces, ^we expect the electrostatic interaction to dominate.

We may ^now write the form of the generic ^total

Landau free-energy ^{of the water solvent as :}

where Vd ^{is the water} volume bounded by ^{the two} planes ^{and the} polar ^heads. Pj(r) ^{is the j th component}

of the polarization ^{at r and} f (P) ^{is the} free-energy density ^{of a} bulk solvent having ^{a uniform} polarization

P. The polarization profile actually ^carried by ^the

solvent minimizes Fd ~ P } and thus satisfies the Euler-

Lagrange equations :

This equation ^{must be} implemented by ^{either a}

surface _energy term or surface boundary conditions.

Since we only ^{want here to} convey the main ideas,

we assume for simplicity ^{that the} polar-heads ^are spherical with radius R The more general ^{case is}

treated in the Appendix ^{but the main} qualitative

results are not changed by ^this hypothesis.

We further assume that the polar-head group- water interaction is modelled by ^a spherically sym- metric radial boundary ^condition (see Fig. 1). ^The polarization ^{of a water} molecule in contact with a

head-group is thus taken radial and of amplitude,

say Po (2).

Finally, ^as is often done in many-body problems,

we simplify ^{the treatment} by considering only ^two- body interactions and assume pairwise additivity ^of

forces. This last hypothesis ^is usually ^{valid for} large

distances or weak interactions but has been shown to

yield incorrect results for short distances, ^{in a number}

of cases [37].

Now, the interaction between two polar-heads ^is computed using ^a differential method. If the distance d between the two head-groups is increased by ^bd by introducing ^a fluid slice of thickness bd in the vicinity

of the medium plane between the head-groups, ^the

total free-energy of the solvent will change by ^the

Fig. ^1.

Configuration ^{of two} spherical polar heads sepa- rated by

distance d + bd. The azimuthal angle T, ^not

represented,

^the deviation of the plane Sl rS2

with respect ^{to the} reference xOy.

(2) ^This approximation ^{is valid}

long

^{the « extra=}

polation

length A is shorter than the correlation length ~.

(The extrapolation length is defined

the ratio between the coefficient of the gradient ^{term and the} coefficient of the

quadratic ^term of the surface free energy [44].) ^The physical

status of the extrapolation length

^{be clarified} by relating

it to the range and anisotropy ^{of the} microscopic interactions in the bulk and

the surfaces [44].

When A is not small compared ^to ~, ^the physical ^behaviour

at surfaces is governed by ^the interplay ^{of these two charac-}

teristic lengths. ^{One can} then show that expression (14) ^for

the interaction energy between spherical head groups

(derived ^{in the limit h -} 0), is still valid if ~ ^is replaced by j

function of ~ and ,1,. For d - ~, ~ ^takes

simple ^{form :}

amount

which can be expressed ^{as :}

The notation Vd, Pd ... illustrates the fact that the volume of integration ^{and the} polarization profile

of the solvent are functions of

As shown on figure 1, ^we divide the volume into three parts, Vj, V2 ^{and the} slab of thickness 03B4d at x = 0

Then equation (6) ^reads

where x

0 denotes the position of the central plane. ^{We have used} Vd

V 1 ⁺ V2. ^{The first} integral ^{can be}

transformed using ^{the fact that} Pd +,5d(r) ^is only slightly different from Pd(r) :

The first term in the r.h.s. of equation (8) ^is nothing ^more than the functional derivative of Fd { P } ^with respect

to P at fixed volume Vd, ^{and is zero as seen on} inspection ^{of the} Euler-Lagrange equation (4).

The second term arises from the integration by part ^{and the} sign derives from the choice of the sense of the x-axis and recalling that the slab (3) ^{is not in} Yd. Equation (7) ^{can then be written as :}

Developing

allows to express 6F to first order in bd as :

The first term in the r.h.s. of equation (10) ^{does not}

contribute to equation (11) since it cancels out with the two contributions from x

± 6dl2.

The _pressure resulting from the interaction between the head-groups ^is simply

and F is computed up ^{to an} additive constant by integrating equation (11).

The calculation now requires the form of the polari-

zation profile P(x, r 1-) and of its first derivative. This is done in the appendix taking ^a specific model for the

free _energy density [31 ] :

and using ^a superposition approximation ^which

amounts to saying ^that P(x, r.1) and its first derivatives

are given by ^{the sum} of their values computed ^for

isolated head-groups. ^We finally ^obtain

where d is the distance between the two head-groups

and where A is given by

R is the radius of a polar ^head (Fig.1 ).

We can now use this formulation equation (14), ^for

the hydration force, ^{to link} up the solvation force between head-groups (within ^a bilayer) ^{with that}

between two opposing bilayers.

3. Lecithin multilayer interactions.

Let us consider the hydration interaction between two

parallel bilayers. ^{Within our} model, ^an important contribution comes from the interaction between

pairs ^of head-groups, ^{one sited in one} bilayer ^the

second on the other one. Using ^the assumption ^of pairwise additivity, the total interaction _{energy per} unit surface can be estimated by summing ^all pair

interactions between a given head-group ^{on one}

membrane and all head-groups ^{on the other one :}

ao is the equilibrium head-group ^{area and} p and r are

defined in figure ^2. Using d 2 ⁺ p2

r2, ^we obtain,

for the hydration interaction free energy per unit area

between two bilayers.

Now, ^{we can estimate} consistently ao by generalizing

the thermodynamic approach briefly ^{recalled in § 2. l.}

Arguing ^{that the} optimal head-group ^area ao is

determined by ^the balance between an attractive

hydrophobic energy and a repulsive energy which

Fig. ^2.

^Schematic drawing ^{of two} opposing bilayers

distance d apart ^whose head-groups occupy

^mean (optimall

area

ao. Each head-group ^{is assumed to} interact with all

other head-groups ^{via the}

potential ^function of equa- tion (14) regardless of whether these head-groups

^{in the}

same

bilayer ^{or the} opposite bilayer.

origin ^is mainly ^of hydration nature, ^we write the free energy per head-group ^{as :}

where a is the hard-core diameter of a head-group.

The integral ⁱⁿ equation (18) gives ^{the sum of all} pair hydration interactions between all head-groups ^{in a} bilayer ^{and the} head-group ^{at the} origin. ^Its expression

in equation (18) corresponds ^to the zeroth order term in a development of the free energy in powers of the

fluctuating part ^{of the} density [31]. ao is then given by

which yields

we can now replace expression (20) ^for ao in equation (17) ^{and obtain :}

where

The corresponding pressure is

with

Equations (22) ^and (23) constitute our main result Note that the repulsion ^between bilayers ^{does not} depend explicitely ^{on the} amplitude ^{A of the} hydration

forces between head-groups, ^but implicitely through

the dependence ^{of the} range ~ ^{on A} (or conversely through ^the dependence ^{of A} on ~ ^{as seen for} example

in Eq. (15)). ^{Indeed, since} G ~ A (Eq. (17)) ^and ao

ao - (Eq. (20)), it is clear that G should not

depend explicitely ^{on A. In other words,} this result

qualitatively says that when A increases (due ^{to ion} binding ^for example), ao increases so that the surface

density ^of head-groups ^decreases leading ^{to an} unchanged inter-bilayer interaction _energy, if we

neglect ^the change ^of ~. ^Note however that for our

approach ^{to be} consistent, ^we should have added the energy given by equation ^{(17) to the free-}

energy per polar heads. This gives ^a correction of

d-(1

amplitude (2 ^e ~ ) ^{times the} hydration ^force comput- ed in equation (18) ^{which is} negligeable ^as long ^as

-r ^is larger ^{than a few units.}

However for smaller separation, ^the coupling

between bilayers ^{as well as} the correction terms involv-

ing ^the higher ^order contributions ⁱⁿ _power of the

fluctuating part ^{of the} density ^should become relevant and equation (22) ^{should no more be valid. Let us now}

examine quantitatively equations (22)-(23). ^{We can}

extract an order of magnitude ^{for the} pressure ampli-

tude no by taking ^y

⁵⁰

which yields ¹⁰¹¹ dynes/cm’.

This is well within the experimental range [18].

Moreover, ^{our model} predicts ^{that the} pressure

amplitude is related to the hydration range ~ ^{as shown}

in equation (23). ^{To test this feature, we have} plotted

in figure 3, Log Ho ^{as a} function of 1 /~ which should

obey ^the following scaling ^{form :}

The grouping ^{of the} experimental points ^taken

from reference [18] ^{on a} straight line is rather remar-

kable considering the crudeness and level of approxi-

mation of the model.

The prediction (23) ^verified quite accurately ⁱⁿ figure ^{3 thus} presents strong evidence for the quanti-

tative correlation between the forces determining ^the

structure within a bilayer ^{and those} occurring ^between adjacent bilayers. ^{This can be stated} alternatively ^as

follows :

The decay length ~ varies from system ^to system ^{in a} way that correlates with the area per polar group.

However, ^{in the} Landau-Ginzburg theory presented

Fig. 3. - Plot of log 77o against I/j (where 1Io ^{is in}

dynes/cm2 ^{and ~ in} A) ^for eight ^different lecithins, egg PE, and lecithin-cholesterol mixtures, each in the fluid-state,

at 25 °C. The experimental data is from reference [18],

table I, ^which gives ^the repulsive hydration pressures

J7(J)

no ^{e-"/4 for these} lipid bilayer systems (as ^well

for three other lipids, ^two in the frozen state and one at 50 °C - not included in this plot). ^The slope ^{of the} straight line yields

estimate of the hard-core radius

8 A.

in § 2.2, ~ ^is identified as the correlation length ^and

should be constant for a given ^{solvent at a} given temperature (at ^constant pressure). ^This apparent

inconsistency ^{is clarified in note} (2) ^where the ~ ^{used in} equation (14) ^is argued ^to be related to the true correla- tion length ^{and also to another} characteristic ^« extra-

polation » length, function of surface quantities.

However, explaining ⁱⁿ details the precise origin

of the change ^{of the} decay length ~ ^{with each} lipid system, ^{would take us} outside the _scope of the _paper and does not seem feasible without a careful analysis

of the direct water correlation functions in the bulk and near the disturbing ^surfaces.

An idea, analogous ^{to the one discussed in this}

paper, has been put ^forward very recently ^{in the}

context of phase separation of micellar solutions [38].

Regarding intermicellar interactions as resulting ^from pairwise interactions between amphiphiles of different micelles the authors of reference [38] have been able to demonstrate the existence of a link between self- association and phase separation ^{via low-order mo-}

ments of the micelle distribution using ^{a classical} thermodynamic approach.

Let us finally mention that our development ⁱⁿ

section 3 ^can be applied also in the _presence of other types of interactions (and ^not only hydration) ^as long

as they ^are short-ranged (i.e. showing exponential decay) and that the pairwise additivity approxima-

tion holds.

4. Discussion.

In the preceding ^{sections we have} argued that there are

good theoretical reasons for expecting ^the properties

of individual amphiphilic aggregates ^to be correlated with the forces between the aggregates. ^{We have also} shown that for lecithin and mixed lecithin bilayers

this manifests itself as a correlation between the head- group ^areas of isolated bilayers ^{and the} repulsive

pressure between two bilayers, and further that a single

interaction potential ^between amphiphilic ^head-

groups ^{can account} for both. In this case the correlation

was surprisingly good (Fig. 3) and it is unlikely ^that

such quantitative agreement ^{would occur for other} systems. ^{Since the} possible presence of other relevant interactions of different and longer range (double- layers forces...) may break down the simplicity ^{of the} previous analysis. However, ^{we should} expect ^{at least} qualitative correlations between various aggregate

properties ^and inter-aggregate ^{forces and we now}

consider some of these.

(i) ^{We have} already ^noted (Sect. 2) the correlation between the head-group ^{area and amount of} swelling

of lecithins (PC) ^and phosphatidylethanolamines (PE). ^We should also expect ^{that the adhesion} of two PE bilayers ^is greater than that of PC bilayers. ^This

indeed is the case [23].

(ii) ^{All the} examples ^so far have involved zwitte-

rionic head-groups. ^{There are} many other examples

involving ^both cationic, anionic and non-ionic head- groups where a reduction in head-group ^{area results in} increased inter-aggregate ^adhesion (due ^{to a reduction}

in inter-aggregate repulsion). ^{Thus the} head-group

area of the nonionic lipid ^{DGDG is} greater ^{than that of} MGDG, which correlates with the shorter-range repulsion ^and stronger adhesion of MGDG bilayers [39]. Analogous correlations occur in anionic lipid bilayers ^{where in} general ^a decrease in pH ^{or addition}

of divalent cations reduces the electrostatic head- group repulsion and hence the surface area per lipid,

and also leads to reduced bilayers swelling ⁱⁿ multilayer phases ^{and to increased} adhesion of vesicles.

(iii) ^{Note that} any general correlation between

head-group ^{area and} inter-aggregate adhesion is

particularly interesting for self-assembly considerations since smaller head-group ^areas usually ^{lead to} larger

vesicles or inverted micellar structures (for ^doubled-

chained lipids), ^or larger ^micelles (for single-chained surfactants) [1]. Thus, aliphatic alcohols which are

often used as co-surfactants in microemulsion systems have a very small head-area (ao ²⁰ A2). ^{This causes}

oil in water microemulsion droplets ^to increase in size when alcohol is added. Recently, Pashley ^{and Mc} Guiggan (private communication) ^{have measured the}

adhesion forces between adsorbed surfactant bilayers

and found that the adhesion increases significantly ^as

more pentanol ^is incorporated ^{in the} bilayers.

(iv) ^{As a final} example, ^the aggregation ^{number of}

anionic alkyl sulphate micelles increases as the elec-

trolyte ^is changed from LiCl -+ NaCl -> CsCI [40].

This implies ^{a reduced} head-group ^{area which} probably

arises from the reduced hydration repulsion ^{of the}

bound counterions as we go from Li + to Cs + . This correlates with the increasing adhesion between oil-in- water emulsion droplets, stabilized by alkyl sulphate monolayers, ^{as monitored} by ^{the contact} angles

between them [41]. Similar correlations occur for CTABr and CTACI micelles.

We conclude that _many apparently ^unrelated properties ^of amphiphilic systems ^{are correlated} (interdependent), ^and that these correlations have

some sound theoretical basis. In the absence of a more

rigorous general theory ^to describe these correlations

quantitatively ^we propose that this phenomenon ^be

called the Principle of Interdependence ^of amphiphilic

systems.

The Principle ^of Interdependence ^should help ⁱⁿ testing ^the validity of theories of hydration ^forces

between surfaces, ^since any such theory ^{should be}

at least consistent with interdependence. ^We may ^note in passing ^{that the} theory ^of double-layer ^{forces is} inherently consistent in this regard [13]. ^{However, it}

is not yet clear whether the theory of Jonsson and Wennerstrom [28] ^or that of Somette and Ostrowsky [25] satisfy ^the Principle ^of Interdependence ^since

neither tell us anything about the lateral interactions between amphiphiles. On the other hand _any theory

of the entropic (steric) interaction between head-groups

must obviously satisfy ^the Principle, ^{as does the}

extension of Marcelja’s theory discussed in section 2 and applied in section 3.

It is also possible ^{that the} hydration force between

head-groups ^{involves a} combination of some of these effects. Thus, ^{it is} very likely ^{that the intrinsic} hydration

force is oscillatory, possibly superimposed ^{on a}

monotonic repulsion [42], and that the thermal fluctuations of head-groups ^and bilayers ^{smear out the} oscillatory ^nature of the force-law leaving ^a purely

monotonic repulsion that is enhanced by the thermal fluctuations (i.e. ^a combination of hydration ^{and steric}

effects [20, 23, 25]. However, ^these interesting develop-

ments take us outside the _scope of this _paper.

Appendix ^A.

In this appendix, ^we give ^{the main} steps leading ^to equation (14) ^{in the text. As seen from} equation (11),

in order to compute 6F/64 ^{one has to} estimate first the

polarization profile P(x, rl) and its first derivatives for the problem ^{of two} spherical head-groups ^distant

from d imposing ^a radial uniform water polarization Po ^{on their} boundary.

We first solve the case of a single sphere ^{for the} particular choice f(p) = 1 P 1. This form for the free energy density yields ^a linear differential equation

which allows in a second step ^{to solve the} two-spheres problem using ^a superposition approximation.

The Euler-Lagrange equation (4) then reads :

which must be solved with the boundary ^conditions

where R is the radius of the head-group.

The problem being spherically symmetric, ^{it is}

convenient to use spherical coordinates.

In this basis, equation (A .1) ^{reduces to :}

where Pr is the radial component ^of P, ^{the other two} angular component being ^zero.

Integration ^of (A. 2) ^is straightforward ^and gives :

Using ^the superposition principle, ^{the two} sphere

problem ^{is solved} immediately and within this approxi-

mation the polarization profile ^is given by :

The notations ^are defined in figure ^1.

We are now interested ⁱⁿ computing surface inte-

grals ^{of P2} and its derivatives in the mid-plane ^x

^0.

For ^x

0, 81 = 02 and ri = r2, expression (A. 4)

reduces to

Let us first estimate the integral

Using

and

we obtain

I I I I

where we have kept only ^the leading ^term in power of

j /d

Evaluation of equation (11) ^also requires computing

surface integrals of derivatives of P.

As an illustration, ^{let us} _compute ^for example

Using ^the equalities ^valid at constant p

we have

Equation (A. 9) ^{allows to} express in terms of _{r, p.}

Proceeding ^as before, ^{we end} up with the result

Comparison of the results given by (A. 10) ^and (A. 9)

shows that a large ^distances (~ld 1) ^the contribution of the terms involving derivations are dominant. This is reminiscent of the planar problem [3] where indeed

only ^the gradient ^term contribute to the repulsive

interaction. Examination of equation (11) ^{shows that}

the third negative ^{term under the} integral in the r.h.s.

dominates ^over the first and gives ^{an overall} negative contribution which can be easely ^{checked to} yield amplitude ^{of the form} given by (A. 10).

Therefore, ^as expected ^{from the} continuation of the

planar ^{case we obtain a} repulsive interaction which

strenght ^{scales as}

For large