Phase behaviour of an ensemble of nonintersecting random fluid films

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Phase behaviour of an ensemble of nonintersecting random fluid films

David A. Huse, Stanislas Leibler

To cite this version:

David A. Huse, Stanislas Leibler. Phase behaviour of an ensemble of nonintersecting random fluid

films. Journal de Physique, 1988, 49 (4), pp.605-621. �10.1051/jphys:01988004904060500�. �jpa-

00210735�

Phase behaviour of an ensemble of nonintersecting ^random fluid films

David A. Huse (1) ^{and Stanislas Leibler} (2, *)

(1) AT & T Bell Laboratories, Murray Hill, NJ.07974, ^U.S.A.

(2) ^Baker Laboratory, ^Cornell University, Ithaca, NY.14853, ^{U. S. A.}

(Requ ^{le 17} septembre ^1987, accepte le 22 decembre 1987)

Résumé.

Nous considérons un modèle continu simple ^{de films fluides} aléatoires, ^dans lequel ^on néglige

toutes les interactions autres que celles du volume exclu, ^{et on} permet ^une topologie arbitraire de l’ensemble des films. L’hamiltonien phénoménologique ^{des films} (ou des surfaces aléatoires), qui ^est à la base de la

description statistique ^à l’équilibre ^{de notre} modèle, inclut les deux termes de courbure : celui de la courbure moyenne (extrinsèque) ainsi que celui de la courbure gaussienne (intrinsèque) ; ^nous considérons ici seulement le cas où les deux côtés du film sont symétriques ^{en mettant} les valeurs de la courbure spontanée ^{et de la}

différence des potentiels chimiques du volume à zéro. Nous examinons le modèle entièrement dans l’ensemble

grand canonique sans aucune contrainte sur la forme ou l’aire totale des films. En nous fondant sur les observations expérimentales ^{dans des} systèmes amphiphiliques, ^{nous soutenons} que le comportement thermodynamique ^{de ce} modèle très simple peut ^{en fait se} révéler fort riche. Nous trouvons sept phases

distinctes et étudions leur nature et les transitions entre elles. Nous discutons aussi des liens possibles ^{avec des}

modèles décrivant le polymorphisme ^de systèmes amphiphiliques ^ainsi qu’avec les modèles de surfaces aléatoires étudiés dans d’autres contextes.

Abstract.

A simple continuum model of random fluid films is considered in which one neglects any interactions other than simple self-avoidence, and allows arbitrary topology of the film assembly. ^The phenomenological ^{film or} random surface Hamiltonian, ^on which the equilibrium statistical mechanical

description of the model is based, includes both the usual area term and two curvature terms (with the extrinsic

(mean) ^{as well as} the intrinsic (Gaussian) curvatures) ; ^the spontaneous ^curvature and the bulk chemical

potential difference terms are set to zero so we are considering only the balanced case of symmetry ^between the two sides of the film. Our treatment is fully ^{in the} grand ^canonical ensemble, without any constraints ^on the

shape ^{and total area of film} present. We argue, inspired by ^the experimental observations in amphiphilic

systems, ^{that the} phase ^behaviour of this very simple ^{model can} in fact be quite rich. We find seven distinct

phases ^and study ^{their nature} and the transitions between them. We also discuss some possible connections to models describisng ^the polymorphism ^of amphiphilic systems, ^{as well as to} models of random surfaces studied in other contexts.

Classification

Physics ^Abstracts

05.20

64.70

82.00

1. Introduction

The problem of random surfaces has recently ^at-

tracted a lot of attention both in the field of

elementary particle physics [1], and that of con-

densed matter [1, 2]. Connections to experiment

appear promising in the latter field, ^where many

(quasi) two-dimensional objects ^{are known and} intensively studied. In fact, ^one hopes ^{that this}

(*) ^{On leave} from : Service de Physique Theorique, C.E.N.-Saclay, 91191 Gif-sur-Yvette Cedex, ^France (ad-

dress after September 1st).

bidimensional generalization ^{of the} random-walk

problem ^can apply ^{to such} systems ^as polymer

networks [3], bilayer ^membranes [4], ^{or microemul-}

sions [5]. ^{From a} theoretical point of view it seems

useful, therefore, ^to describe these materials in terms of random surfaces, ^and explore ^{the advan-} tages ^and shortcomings ^{of such a} formulation.

A microemulsion is an equilibrium phase ^of oil,

water and surfactants [6]. Surfactant molecules are

preferentially ^{adsorbed on} the oil/water interface due to their amphiphilic ^nature [7]. Microemulsions have the two following properties ^which, among others, have attracted the attention of chemists,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004904060500

physicists ^and engineers ^over the years : i) ^the

existence of very low surface tensions between different macroscopic phases [8], ii) ^{rich phase dia-}

grams [9] ^with particularly fascinating microstructure

[10]. A substantial amount of experimental ^and

theoretical work has been performed ^to study ^and explain ^these properties. Although microemulsions appear already ⁱⁿ ternary (or quasi-ternary) systems, these mixtures have quite complicated ^molecular

structures and molecular interactions. It is therefore

a challenge ^to catch the essential features of these

phases ^{in a} simple statistical-mechanical model

[2, 11].

Here we would like to explore ^a random film

(surface) model which is closely ^{related to the}

continuum description of microemulsions developed by Talmon and Prager [12], de Gennes et al. [13],

Widom [14], ^Safran and others [15]. ^{It is not our} aim, however, ^to explain the detailed properties ^of

microemulsions through ^{such a model.} Rather, ^in- spired by ^the experimental observations o f microemul- sions, ^we would like to explore ^a simple generalization of ^random surface models, ^{and show that} (despite ^its simplicity) ^{it can exhibit a} surprisingly ^rich phase

behaviour. Let us stress that throughout this paper

we discuss only two-dimensional films in a three- dimensional space. Generalizations to other dimen- sionalities are certainly interesting ^{but are not} pur- sued here.

We choose therefore to think about a microemul- sion as an ensemble of surfactant films ^{and to}

construct a statistical-mechanical model based on a

« film Hamiltonian ». This Hamiltonian can be writ- ten as a sum of three parts :

i) Hext , ^which depends ^{on the «} external » (as opposed ^{to « internal », see} below) degrees ^{of free-} dom, i.e. the size and the shape of the film. This part contains three important ^{terms :}

The first term here is proportional ^{to the area of the} film, ^{the constant} ro being ^a microscopic (bare)

surface tension. (We ^shall call ro the ^area coefficient

to avoid a possible ^{confusion with the} macroscopic

surface tension.) The second ^term is the curvature energy ^term [4], ^which originates ^{in the} splay ^{and the} saddle-splay of the film. Here H and K are, respect- ively, ^{the mean} and the Gaussian curvature at a

given point ^{of the film, while K0 and} Ro ^{are the} corresponding ^{bare elastic} constants. If cl and C2 ^are the two principle curvatures (inverse ^{radii of} curvature) ^{at a} point, then H = cl + c2 and K

cl C2. The spontaneous ^curvature Ho ^{reflects an}

asymmetry of the film : for Ho :0 0 the film prefers ^to

curve more towards one side. The integration ^{in the}

first two terms is performed ^{over the} entire random surface or film ; ^{we assume that this surface is non-} intersecting (self-avoiding), ^thus implicity taking

into account excluded volume effects. ^{We consider} only films with no edges, for which one can dis-

tinguish ^{two sides : we will denote one side « water »}

and the other « oil _», as suggested by ^microemul-

sions. The third term is due to the chemical potential

of the ^« water » (side) ^{relativer to the « oil »} (side),

tL 0, and the integral ^{in this term is over the entire} volume, Vw, ^{on the « water » side ;}

ii) Jein, which takes into account the ^« internal »

degrees ^{of freedom of the} film, e.g. positions,

orientations and conformations of the hydrocarbon

chains of molecules, ^{etc. It} also includes all the

couplings ^between the internal and external degrees

of freedom, ^{which can} give ^{rise to new} macroscopic

effects such as curvature instabilities, and the appear-

ance of ordered phases ^{within the film} [16] ; iii) Hinter, which includes all molecular interactions

not taken into account by previous ^{terms. The main}

contribution to this term is the interaction between different parts ^{of the} film, ^{such as van der Waals} forces, hydration repulsion and others [17]. ^{If the} description ^{of a} microemulsion as a self-interacting

ensemble of films is adequate, then the Hamilto- nian :

should lead to the experimentally observed ther-

modynamic behaviour of the oil/water/surfactant mixtures. The Boltzmann probability ^{of a} given configuration ^is exp(-/3H) ; ^where f3

1/kB T, kB is the Boltzmann constant and T is the tempera-

ture. We work without any explicit constraints on

the total amount of film, ^{« water » or « oil »} present ; this is the grand ^canonical ensemble. The partition

function is the integral ^of exp (- /3 H) ^{over all}

allowed configurations. ^In the continuum theory ^we

are considering, short-distance cutoffs on the fluctua- tions are necessary ^to make the partition ^function

well-defined. Precisely ^{how to} best introduce such cutoffs is at present ^{unclear to us. We assume that a} sensible cutoff procedure ^{can be} formulated. Note that in this continuum model the molecular nature of the system is reflected in the short-distance cut-offs present in the surface integrals, ^{as well as in the} microscopic coupling ^constants ro, K o, ILo, ^etc.

The purpose of this paper is ^to tentatively explore

a model based on the film Hamiltonian (2). ^{As a first}

and the most drastic simplification ^{we shall} neglect

the « internal », :rein’ ^{and the} interaction, 3center,

terms. A similar approach ^{has been taken} by ^the

authors of several phenomenological ^{models of}

microemulsions [14, 15]. ^{In these} models, however,

the system ^{is not} entirely ^{described as a random}

surface, ^a phenomenological ^{bulk «} entropy ^{of mix-} ing » ^term being ^used [14]. ^{Here we consider an}

idealized version of the pure surface ^{model in} which,

for simplicity, ^{we also set to zero the} only ^volume

term present ⁱⁿ (1), ^thus assuming that the relative chemical potential uo vanishes. Finally, ^{we also set}

the spontaneous curvature, Ho, ^{to zero.} This assump-

tion, together with JLo

0, ^{makes the two sides} of

the film equivalent. ^{All of these} assumptions ^{are, of}

course, oversimplifications ^{of real} microemulsions.

Note, however, ^{that we do not} make one other

common and drastic assumption, namely ^{we do}

allow the topology ^{of our film to} vary ; it ^can thus consist of many disconnected « components ^{», each} of them having ^an arbitrary ^{number of « handles »} [18]. ^{We can therefore} explore various microstruc- tures of the macroscopic phases, ^{and we} shall argue that even such a considerably simplified ^model

exhibits many of the different phases ^{which one}

encounters in real physical systems. Our results for this simplified film Hamiltonian are briefly ^sum-

marized in section 2 below. Note our reason for

simplifying JC (Eq. (2)) ^{is not} because the terms

dropped, namely Ho, tkO, acin ^and Winter? ^{are incom-} patible ^{with the} type of random surface treatment

we present. ^{These terms are} only dropped ^{to reduce}

the parameter space down ^{to a} size that we can fairly thoroughly explore. ^There are numerous interesting

effects associated with these terms that can be studied within this formulation, ^some of which have

already ^been investigated ^{in a} closely related fashion

(e.g. ^{see Refs} [2] ^and [12-15]).

Although ^the general understanding ^{of the} prob-

lem of random surfaces has progressed ^{a lot in recent}

years, many fundamental issues are still poorly

understood [1, 19]. ^The important question ^{of uni-} versality ^{is not at the moment} fully ^elucidaded [19, 20]. It appears that the random surfaces ^con- sidered in the context of high temperature expan- sions of lattice gauge theories [21], ^{which do not}

have a fixed internal metric [19, 22], behave diffe-

rently ^{from «} fixed-connectivity ^» surfaces, ^{such as}

the recently ^studied « tethered surfaces » [23], ^or triangulated ^{random surfaces} [24] (which ^{can be}

viewed as a discretization of the Polyakov string

model [25]). ^{In fact, recent} simulations of this last class of models [19], ^if confirmed, put in doubt the very concept ^of universality : ^the scaling properties

appear ^to depend ^on the short distance properties ^of

the triangulations [19]. ^In this paper ^{we are} mainly

concerned with fluid films. ^{If we were to discretize}

our continuum model (1) ^we would therefore

assume that the internal metric is not fixed, ^{as in} plaquette ^models [21] ^or triangulated ^{random sur-}

faces with a variable coordination number [26].

Plaquette models of random surfaces with fluctuat-

ing topology ^have recently been studied numerically [27, 28]. Even if the energy of ^a configuration

consists only ^{of the area term, two} distinguishable phases of different topologies, separated by ^{a con-}

tinuous transition (in ^three dimensions) [27, 28], ^are

found. By including ^{also a} topological ^{term, which in}

fact is similar to our Gaussian curvature energy in

(1), ^another phase of different microstructure [27] ^is

obtained. These results show that even very simple

« fluid ^» random surfaces can have quite ^{a rich} phase

behaviour if one does not restrain them to a planar topology ^{and to have} only ^{the area} energy ^term. In

fact, in section 4 we shall argue that the phase

behaviour found in references [27] ^and [28] ^is implicitly included in the phase diagram ^{of our}

model.

Before doing ^so, however, ^we shall summarize our

model and the main results of this paper in section 2.

Then (in ^section 3) ^we analyse ^{in more} detail the two cases of a vanishing ^{and of} a nonzero Gaussian curvature term. Finally, the last section includes the summary and ^some perspectives ^{on further} develop-

ments and applications.

2. The model and _summary of results.

We thus consider the simplified model film Hamilto- nian :

where the integral, ^as before, ^{is over a} surface of unconstrained topology ^{and total area.}

Experimentally, ^{the area} coefficient ro may be

adjusted by changing the chemical potential, JLs’ of

the surfactant molecules in the film. An increase in JL s will yield ^a decrease in ro, and vice ^versa. The

(bare) ^{curvature elastic constant,} K o, might ^be adjusted by adding cosurfactants or varying ^the hydrocarbon length of the surfactant [29]. ^There

exist now several experimental methods which can measure in principle ^the bending rigidity ^{K in} amphiphilic systems. ^{These are, for} example light scattering studies of thermal fluctuations of vesicles

[30, 31] ^or monolayer ^films [32], ^X-ray scattering

from multilayer crystals [33], ^the study ^{of the}

interactions between defects [34], ^{NMR measure-}

ments of local curvatures in lamellar systems [35],

and others. Much less is understood about the Gaussian curvature term [36] ^{and no} good ^measure-

ments of K _{are, to} our knowledge, available. Also it is not clear how one could independently vary the Gaussian rigidity coefficient, K0, ^{in real} systems.

Note that _{K o} and K0 ^have dimensions of energy, ^so the ^« reduced ^» parameters KolkB T ^and RolkB T

are dimensionless.

Let us for the moment forget ^{about the Gaussian} rigidity ^{term, thus} setting KO

^0. (We ^{return to the}

case K o ^{=1= 0} later.) ^A possible phase diagram ^{of this}

model as a function of ro and 1/K0 ^is presented schematically ⁱⁿ figure 1. We first briefly ^describe

the various possible phases ^before addressing ^{in the}

next section more detailed issues, ^{such as the nature} and locations of various phase transitions. For large positive 13 ro the system ^does not want to have much film present. ^It achieves this goal by breaking ^the symmetry between the two sides of the film, ^and forming ^{an « oil »} (« ^water ») ^rich phase ^with finite,

unconnected domains of the minority bulk compo- nent [37], ^{« water »} (« ^oil »), present. This is the so-

called droplet phase. ^The macroscopic ^{surface ten-} sion, a, of a film separating « oil »-rich and

« water »-rich phases ^{is nonzero} here, ^{so we denote}

the state of the film as tense in this phase [38].

Viewed in a larger parameter space including go :A 0 ^and Ho =1= ^{0 this « tense »} phase ^{of the film}

represents ^{a manifold where two bulk} phases (« oil »- and « water »-rich) ^can coexist with the interface between the phases having ^a macroscopic

surface tension. Of _course, the interface and its tension are only ^{well defined on this manifold. As}

soon as one leaves this manifold, say by varying

Fig. ^1.

Proposed ^schematic phase diagram ^{of our}

random-film model (3) ^for Ko

^{0. All} phase ^boundaries

converge ^at the multicritical point ^{W at} ro

1/ KO

^{0 as} 1 / K 0 - 1 log r 0 1- 1, thus their high degree ^of tangency ^to

one another. The random isotropic phase ^is fully ^dis-

ordered ^« high-temperature ^» phase, while the ordering

increases as one moves either towards the smectic lamellar

phase ^or the dilute droplet phase. The various phases ^are

described in some detail in the text. The tense bicontinuous to dilute droplet phase transition, ^shown dashed, ^{is a} percolation transition with no thermodynamic consequ-

ences. The random isotropic ^{to tense} bicontinuous tran-

sition, ^shown dotted, is the breaking ^{of the « oil »-« wa-}

ter » symmetry. ^{This is not a} symmetry of real oil-water- surfactant systems and thus this transition will not be present generally ^{unless one} adjusts the relative chemical

potential of the bulk components, go in (1), ^{to a} special

critical value.

go ^or Ho, ^{there is} only ^one thermodynamically ^stable phase.

As ro decreases the droplets ^{of the} minority ^bulk component ^{in the} droplet phase grow in size in order to increase the total film entropy, ^{and start} forming larger ^connected domains. The domains of minority

component percolate ^before the volume fraction

occupied by it reaches 1/2. This leads to a tense, bicontinuous phase ^for intermediate positive ^values

of ro. In this phase ^{the « oil-water »} symmetry ^{is still} spontaneously broken and the macroscopic ^surface

tension is still non-zero, although ^infinite percolating

domains of both bulk components ^are present.

Again ^{this « tense »} phase of the film corresponds ^to

a manifold of two-phase coexistence in the larger

space of ILo =F ^{0 and} H0 =F ^0.

For large positive Ko and sufficiently negative

ro the system ^{wants to} put ^{in as} much film as

possible, ^{but not to} bend the film. A possible way ^to do this is to stack films parallel ^{to one} another and very close together. ^This produces the smectic lamel- lar phase, ^{which has} long-range order in the orien- tation of the films, ^and quasi long-range order in the

positions of the film planes along ^{the direction}

normal to the films, ^as in normal smectic liquid crystals [39]. ^This quasi long range positional ^order gives ^{rise to a} power law divergence (« quasi-Bragg peak ») ^{of the} scattering intensity S (q ) ^at the lowest- order reciprocal ^lattice point. There is also the

possibility ^{that at} intermediate negative ^{values of}

ro ^a lamellar nematic phase appears, with long-range

order in the orientation of the films, ^but only ^short-

range positional order. Note that the full rotational symmetry ^{of our} Hamiltonian is spontaneously

broken in the lamellar phases. ^{We are} using ^term

« lamellar ^» to refer to phases in which there is long-

range order in the orientations of the films, ^so they

are on average parallel ^{to one} another. Thus

although the lamellar nematic phase has many defects in the stacking ^{of the} parallel ^{films and thus}

no quasi-Bragg peak ^{in its} scattering, ^{the basic} anisotropy, ^{as seen for} example ⁱⁿ birefringence

measurements [40], is established in this phase.

Finally, ^at values of ro ^{near zero,} between the lamellar and tense phases, ^there should be a dis- ordered, ^random isotropic phase, ^{where no} sym- metries of our Hamiltonian (3) ^are spontaneously

broken. The random isotropic phase ^{is a} high-tem- perature disordered phase, while the lamellar phases

are low-temperature ^ordered phases. ^The phase

transition between the random isotropic ^{and tense}

bicontinuous phase is shown dotted in figure ¹

because the « oil ^{»-« water »} symmetry ^{that is broken} there is not a true symmetry of real oil-water- surfactant systems. ^{In the} bigger space of ILo =F ^{0 and} Ho =F ^{0 this} transition is likely ^{to be a} line of critical

points terminating ^a manifold of two-phase ^coexist-

ence. These critical points ^occur only ^{at a} special

value of u0. ^For Ho = 0, the critical point ^must, by symmetry, ^be at tk 0 = 0, ^so is in the special sym- metric space ^{we are} focussing ^on. However, ^{in a real} system ^with Ho =F 0, the critical point ^{occurs at a}

non-zero value of go that is ^a function of Ho,

ro, K0, ^etc. For ro > 0 the system ^is quite analogous

to a ferromagnetic Ising ^model, with ro the ^nearest

neighbour coupling ^and K0 ^a higher-order (4-spin ^or more) interaction that does not break the global Ising symmetry. ^Then the random isotropic phase ^is

the high-temperature paramagnetic phase, ^{while the}

tense bicontinuous and dilute droplet phase ^{are the} ferromagnetic phase. ^The symmetry-breaking ^field

go is then analogous ^{to a uniform} magnetic field,

while Ho ^{is a} higher-order (3-spin ^or more) ^interac-

tion that also breaks the Ising symmetry.

For large K0 and thus small curvatures a pertur- bative renormalization group calculation ^can be

performed [41]. ^At length ^scale L, ^to leading ^order

in kB T / K o, the renormalized parameters ^{are :}

where a is a microscopic ^cutoff parallel ^{to the film}

and both values ^a = 1 and ^a

3 have been propos- ed [2, 41, 42]. ^{Thus we have a} multicritical fixed

point (W) ^{at the} origin ⁱⁿ figure 1. This is effectively

a zero-temperature ^fixed point, ^{since it occurs for} kB T / K o

^{0. In} this paper ^we mostly ^{focus on the}

behaviour near this fixed point, ^where K0 is large

and the perturbation theory leading ^to (4) ^{can serve}

as a reliable guide. One should stress however that the results (4) ^are perturbative : ⁱⁿ fact, they ^{are in} principle ^valid only ^{for small} curvatures, and all the effects of self-interactions are completely neglected.

,

Let us also mention that the same perturbation

calculation leads, ^for higher dimensional films, ^to

the prediction ^{of a} crumpling transition [41], ^which

separates ^{two distinct} regimes of the behaviour of the film, namely ^a crumpled ^{film and a} macroscopi- cally flat film. In fact, ^a crumpling transition _may also take place ^for two-dimensional crystalline ^or polymerized ^films [43, 44]. Here, ^{as mentioned be-} fore, ^{we are} interested in the phase behaviour of bidimensional fluid films only, ^{and thus} crumpling

transitions do not occur in our model. Note that we

will use the results (4) ^{only on} length scales for which

K (L ) > kB T, ^{in which case} the renormalization of r is small, [r(L) - ro] : roo ^{Thus, for} simplicity, ^we

will henceforth ignore ^the renormalization of _r,

assuming r (L )

ro. This can readily be corrected in what follows, leading ^{to no} substantive changes.

An important length ^{in this} system ^{is the} persist-

ence length ^{of an isolated film,} 03BEK [5, 41]. This is the

length ^{scale at} which the renormalized elastic con-

stant for the mean curvature, K (L), ^{becomes of}

order kB ^{T. From} equation (4a) ^{we see that for} Ko> kB T

For an isolated film gK is the correlation length ^of

the local film orientation. For the ensemble of self-

avoiding ^{film we are} considering here, 6K is ^{one of}

the important length scales, ^{as is} emphasized ^below.

For example, ^the typical local radii of curvature and

spacings ^{between films} in the random isotropic phase ^{are of order} 6K-

In the film Hamiltonian (3) ^{we have} explicitly

included the Gaussian rigidity K0. ^{Even when the} bare elastic constant for the Gaussian curvature

vanishes, K0

0, the renormalized _one, on longer scales, may ^not vanish, being given by :

where the value ii = - 10/3 and «

0 have been

recently proposed [45, 46, 47]. ^{Note that the} physical

case of three bulk and two film dimensions is the

marginal ^{dimension where the « reduced » curvature}

elastic constants {3 ^{K and} {3 ^{K are} dimensionless ; ^this

results in the logarithmic renormalizations given

above. Since, ^{in our} opinion, the value of « has not

yet ^{been well} established, ^we shall consider (while discussing ^the droplet phase) ^{various cases corre-} sponding ^to different values of « and study ^some

consequences. After having first studied the case

where K can be neglected (e.g. ^{for «}

K0

0) ^we

will consider more general situations of non-vanish-

ing ^{Gaussian term.} However, ^for simplicity ^and

definiteness we assume &

0 in many of the dis- cussions that follow.

If we allow the bare elastic constant for the Gaussian curvature, K0, ^{to be} non-zero, then other

phases ^{not shown in} figure ^{1 can} be stabilized. By

the Gauss-Bonnet theorem [18] ^the integral ^{of the}

Gaussian curvature is dS K

4 ^7T (nc - nh ), ^where

nc and nh ^are the number of disconnected _compo-

nents of the film and the number of handles, respectively. ^Thus K0 only couples ^{to the} topology ^of

the film : Ko > 0 favours more handles and fewer components, ^while Ko ^{0 favors more} components and fewer handles.

Figure ^{2 shows a} possible phase diagram ^{for our}

model with K 0 =F 0 ^{as a} function of the area coef- ficient ro and the rigidity K o ^at K o > kB T ^{fixed. It we}

first move on the ro axis of this diagram, putting K o

0, ^{we obtain} again ^{the same sequence of} phases

as in figure 1, namely : ^{lamellar smectic}

^lamellar

nematic

random isotropic (relaxed)

^tense

bicontinuous

droplets. However, ^as K o ^{is in-}

Fig. ^2.

Proposed ^schematic phase diagram ^{of our model} (3) ^{as a} function of ro and Ko, ^for K0 > k, ^T. The abbreviation denote : NL, nematic lamellar ; RI, random isotropic ; ^{and TB, tense} bicontinuous. The « elastic constant »

K0 couples ^{to the} topology ^{of the film,} favouring many handles and thus the plumber’s nightmare phase ^for Ko sufficiently positive ^and favouring ^many droplets ^which crystallize ^for Ko sufficiently negative.

creased from zero we expect that each of these

phases will transform itself into a new ordered

phase, ^{which we} call here by ^the generic ^name

« plumber’s nightmare » [48, 49]. The ideal plum-

ber’s nightmare phases ^are fully connected, periodic

and topologically nontrivial surfaces that have zero mean curvature H everywhere. For this last reason

they ^are often called minimal surfaces [49], ^{but this}

name is (in ^our opinion) misleading since the total

area of such surfaces is not in general ^minimal (as opposed ^{to the case of} finite ^surfaces spanning ^a

closed loop, which also have zero mean curvature

everywhere [50]). Four unit cells of a simple ^cubic plumber’s nightmare phase ^are depicted ⁱⁿ figure ^3.

Such structures have many handles (of ^{order one} per unit cell) and, ideally, only ^one component, ^{so for} K o > 0 the Gaussian energy ^term favours them. We

Fig. ^3.

Four unit cells of an ideal simple ^cubic plumber’s nightmare phase. The film is shown, the interior of the

shape ^{shown is} occupied by ^{one bulk} component, say

« oil », while the exterior, which has the same shape ^{as the}

interior but displaced by ^{the vector} 1 11 1 1 2 ^{2 2}

^is

occupied by the other bulk component, « water ». See reference [49].

do not address here the question which of the many

possible [49, 51] plumber’s nightmares ^{are chosen} by

the system, although ^this question ^{will be} quite interesting ^to study ^both analytically and numeri-

cally. ^In fact, ^{even our} simple model may have ^some structural phase transitions (inside ^the region ^marked

Plumber’s Nightmare ⁱⁿ Fig. 2) ^between periodic

structures of different symmetries. ^{Such different} phases ^{and the} transitions between them have indeed been observed in some amphiphilic systems [52].

Note, however, ^{that in some real} systems ^the plumber’s nightmare phases appear ^to arise as the result of competition ( frustration) ^{between the curva-}

ture energy ^term and the energy of stretching [51, 53]

of the hydrocarbon chains, the latter being ^an

« internal » degree of freedom not explicitly ^treated

here.

When K o ^is sufficiently negative ^{the surface does}

not want to have many « handles », ^{as in the} plumber’s nightmare phase, but rather prefers ^to

form many disconnected components. ^{This can be} done in presumably ^{the most} efficient way by forming many roughly spherical droplets. ^When

these droplets ^are very closed-packed they ^should

freeze into a droplet crystal. ^We expect ^{that this}

crystalline droplet phase is stable at sufficiently negative Ro. Since it arises from the system’s ^desire

to have as many droplets (components) ^as possible

in a given volume, ^{it will} presumably ^{be one of the} close-packed crystals, ^{either f.c.c. or} h.c.p.

We now proceed ^{to discuss in more detail the}

various phases ^and phase transitions shown in fig-

ures 1 and 2. Note that in drawing figures ^{1 and 2 we}

have assumed what we feel is the simplest scenario,

with most of the phase transitions being ^either

continuous ^or weakly ^first order. Some phases, particularly the nematic lamellar and tense bicon-

tinuous, could be absent over some or all of the

phase diagrams ^if they ^are preempted by strongly

first-order transitions. Of _course, strongly first-order transitions occur in many real experimental systems either due to effects not included in our model (3) ^or

due to effects within our model that we have overlooked.

3. Phase behaviour of the model.

3.1 THE DROPLET PHASE.

Let us first consider the dilute limit of the droplet phase. ^{This occurs for} sufficiently large ro, where, ^{in order to avoid} having large ^{amounts of film} present, ^the system spon-

taneously breaks the « oil »-« water » symmetry, forming ^a phase with dilute droplets ^{of the} minority

component and the remainder of space occupied by

the majority component. ^{We will} begin by discussing

the statistics of dilute near-spherical droplets.

We consider droplets of radius R such that the

mean curvature elastic constant renormalized to

length ^scale R, equation (4a), satisfies K (R) > kB ^T.

In terms of the persistence length (5) ^{this means} R , 6 K [54]. ^{We also assume} that ro R 2-c K (R) ^so

that the bare surface tension, ro, ^can be neglected ⁱⁿ estimating the fluctuations of the droplet ^{about a} spherical shape by ^{amounts of order}

as discussed in some detail by ^Helfrich [47].

AR is the measure of the « roughness » ^{of the} fluctuating film. The radius and the position ^{of the} droplet ^are naturally ^{viewed as uncertain to this}

extent. Therefore a sensible partition function for this droplet is obtained by integrating ^{over all} configurations with the average radius within AR of R and the centre-of-mass position within AR of a

given point. ^The resulting ^Boltzmann weight ^{for the} droplet should be :

where C is a constant, ^and Ko

^{0 is assumed} [55].

Assuming ^the droplets ^{are dilute so we can} neglect

excluded volume effects, ^the density ^of droplets ^with

average radius within AR of R is then

where C is another numerical constant. Let us now

introduce a new lengthscale

It is a characteristic length associated with ro > 0 ; droplets ^{of size} g ^{have area} energy

The total volume fraction occupied by ^the droplets

is then

Now, ^{if 2 « + a were} negative, ^{which has not been} proposed ^{in the} literature, ^this integral ^{would con-}

verge for small ro (large g r), yielding

cP ~ (Ko/kB T)2 exp (- ⁸ -iT /3 K 0) ^{and thus} only ^a

small volume fraction occupied by spherical topology droplets ^for K o > kB ^T. However, ^{as we} shall discuss later droplets ^{of other, more} complicated topologies

with handles would proliferate ^{if a} 0. There is every indication that 2 a + a > 0, ^{in which case we}

have :

as the volume fraction occupied by ^near spherical droplets. ^This expression ^{is valid} only ^for K (03BEr) > kB T, or 03BEr 03BEK , otherwise AR becomes of order R or greater for the the droplets ^{of size}

R -- ç ^and they ^{are far from} spherical. ^{In terms of} 6K, ^the total volume fraction occupied by ^near spherical droplets for 6K >> 6, ^{is :}

Let us now examine this result for various values of a and see what would be the consequences for the behaviour of the droplet phase.

i) ^a > 0. (This ^{case has not been} proposed.) ^The

volume fraction 0 becomes of order unity ^for ç r ç K. ^{This means that the} droplets ^{become dense}

and thus interacting when their radii ^are still well below the persistence length ^and they ^{are therefore}

all near spherical. ^{Thus the} system ^{retains a} very strong preference, ^{due to K} (6,) ^{0, for} spherical droplets, ^{even when the} droplets ^are dense. This scenario sounds somewhat implausible ^{because one}

would think that the film could gain ^much entropy by reconnecting ^{into more} complicated, topologically

nontrivial shapes, rather than just simple spheres.

(Remember ^{that we} neglect here the molecular interaction term Jeinter ^{which in real} systems ^could stabilize dense droplet phases [56].)

ii) ^{« 0} (as proposed, ^for instance, ^{in Ref.} [45]).

The system ^at long length ^scales prefers ^{in this case} topologically nontrivial droplets ^with handles, e.g.

tori and more complicated objects. The minimal curvature energy ^{of a} droplet ^{with N handles and}

linear size R is 41T[e(N) ^{K +} (1- N ) K], ^where e (0) = 2 ^{and we} expect e(N) - ^{N2/3 for} large

X. The minimal curvature energy droplets ^with large ^{JV are} presumably pieces ^{of a} plumber’s nightmare structure, ^closed (capped) ^on the surface.

Only the surface ^« caps ^» contribute to the mean curvature energy, the interior having ^{zero mean}

curvature, this results in the proposed 2/3-power

law. The coefficient of K is precisely (1 - X) by ^the

Gauss-Bonnet theorem [18].

Let us define s (X) such that the total surface area

of the minimum curvature energy shape ^with

X handles and average ^local radius of curvature R is 4 1TR2S2(X), ^where s (0) ^{= 1 and} s (N) - ^{Xl/2 for} large ^{N. Then such a} droplet with total area energy

kB ^{T has R} = gr/s(X). ^{If we then} apply ^{the same} analysis ^as given ^{above for} near-spherical droplets,

we find that the total volume fraction occupied by

N -handled droplets ^is (ignoring prefactors ^of (K (gr)/kB T)) : ^·

Therefore, ^if Ko and Çr ^are large enough, ^{the total}

volume fraction occupied by droplets ^{with one or}

more handles could greatly exceed that occupied by near-spherical droplets, ^even when that total is much less than unity ^and 03BEr 6K ^{so that the individual} droplets ^{are not} very rough. ^{This is a rather} surpris- ing situation and to our knowledge ^{has not been} suggested.

iii) ^a

0, ^{in this} case, proposed by ^Helfrich [47],

the volume fraction occupied by ^each type ^of droplet

becomes of order unity when ç becomes of order of

ÇK. ^{Thus when} K0

^{0 the} system ^{does not} develop ^a preference ^{for a} given topology ^{and all} types ^of droplets proliferate as K (ç r) decreases. This scenario strikes us as the most plausible ^{and we will}

assume a

0 in most of the remainder of this paper.

An actual calculation of the free energy ^{as a} function of size for droplets ^{of various} topologies (e.g. spheres, cylinders, multi-handled torii, etc.) ^is clearly ^{needed to} resolve the issue of « and

« and to check the above expectations ^{for the}

statistics of dilute droplets.

It is important ^{to stress that in the above discussion}

we have neglected the role of the spontaneous

curvature Ho. A ^nonzero spontaneous ^{curvature will} clearly ^favor spherical droplets ^{of that curvature.}

There have been a variety ^of treatments of the

droplet phase ^{within «} mean-field ^» models of the

sort originated by ^{Talmon and} Prager [12]. ^{A recent}

version that includes the renormalization of K has been proposed by Safran et al. [15]. ^{Their model} has, within the droplet phase, monodisperse droplets (modeled ^as cubes) ^{of linear} size g ^and density

n = 0/03BE3 . The model free energy per unit volume of such phase is, ^{for W « I and K} =&=ro=O:

where the first term is a phenamenological entropy of mixing. ^For the choice a = 1, ^taken by ^the

authors [15], ^the minimum free energy is attained by making g ^{as small as} possible, g

^{a, and thus} only having microscopic droplets ^with density na3 = _exp (- ^{8 ’7TB K} o). ^{Thus, for a : 3/2 and}

K o > kB T, ^the droplet phase remains dilute even for

ro - 0 in this model (16). This results in a strongly

first-order transition from the dilute droplet phase ^to

the random isotropic phase. ^Such strongly first-order transitions do occur in some microemulsion systems.

This is very similar ^to what would occur in our model if the coefficient ^a were negative. ^{If «}

3, ^{on the}

other hand, the model free _energy (16) ^{does not}

have a dilute droplet phase for ro -> 0. Moreover, ^if

ro =1= 0 is considered, ^one finds that the droplet ^{size is}

of order 03BEr, and the volume fraction occupied by droplets ^is 4> = (g r/ g K)2 a. This differs from our

result (14) only by ^the factor of (R/ åR)4 =

[K (6,)IkB T]2 originating ^{from the} roughness ^{of the} droplets. Thus the behaviour of the droplet phase

for ^a = 1 is rather different from what one obtains if

«

3 is instead used in their model (16). ^The

difference with our results for « 3/2 arises because the monodispersity assumption requires approximat- ing ^the droplet ^size distribution (9) N(R) ^{with a}

delta function at R = g. ^{Now we have}

n (R) -- R2 a - 3, ^{so for a 3/2 the most} probable droplet size, which dominates the total free _energy

(16), ^is microscopic, 6

^R

^{a. The} monodisperse

model naturally ^chooses this size. However, ^{if one} calculates, ^{within our} approach, the total volume fraction 0 occupied by droplets (12), this is domi- nated by ^the large droplets ^{for all a} > 0. Thus the

assumption ^of monodispersity ^{leads to} misleading

results for 0 « 3/2.

3.2 BICONTINUOUS TENSE PHASE.

As 6r ^increases

and becomes of order 6K, the ^{volume fraction} occupied by ^the minority component approaches

1/2. Assuming 0 ^increases continuously with 6r ^r (arguments supporting ^{this are} given below), ^{there is}

a percolation transition in the minority ^{domains at}

some volume fraction Op ^{less than 1/2. This is not a} thermodynamic phase transition only ^a change ^{in the}

statistics of instantaneous connectivity of the min-

ority domains. This transition is presumably ^{in the} universality ^{class of} three-dimensional percolation,

with §K serving ^{as the} microscopic length ^{at the}

transition. For large 6K, the transition should occur

at a value of ø = CP p ^{which is} independent ^of 6 K. ^{For 1/2} :> ø:> øp ^{there is an infinite connected}

domain of minority (as ^{well as} majority) component present ^{so the} system ^{is now} bicontinuous. The

« oil »-« water » symmetry ^{is still} broken, ^since 0 =A 1/2, ^so the surface tension a is still non-zero.

To differentiate this phase from bicontinuous phase

with 45 = 1/2 and unbroken « oil »-« water » sym- metry, ^this phase ^{is denoted as tense. As mentioned} above, ^{this tense} phase represents ^{a manifold in the} larger parameter space including ILo -:1= ^{0 and} H0 =F 0

on which the bulk phases ^coexist.

3.3 THE TENSE-RELAXED TRANSITION.

As gr ^is

increased still further (or equivalently, as ro is

decreased) the « oil ^»-« water » symmetry ^{must be} restored at some point. ^At this transition the macro-

scopic surface tension [38] ^{vanishes. We assume that}

in absence of attractive interactions this transition is continuous and, ^like percolation, ^{occurs at} g r ^of

order §K, though ^the percolation transition clearly

occurs first as 6, is increased. These assumptions ^are supported by ^the following suggestive arguments :

Let us consider putting ^our model for small K0 ^{on a} simple ^cubic lattice, ^so that the film must lie

on the plaquettes of the lattice. If _K0

0 the model is then equivalent ^{to a} ferromagnetic Ising ^model

with nearest neighbour couplings ^J

ro a2/2, ^where

a is now the lattice spacing. ^{The tense} phase ^{is the} ferromagnetically ^ordered low-temperature phase,

where there is a surface tension, ^{and the relaxed} phase ^{is the} paramagnetic high-temperature phase,

with no surface tension. This system ^{then has a} continuous transition at i3J =-= 0.22, ^or gr ==

(4 Tr P ro)- 1/2 -= ^{0.4 a. The} curvature, ^of course, is not well defined on a lattice, ^{but we can} argue that the typical ^{local curvature} of the interface confined to a lattice like this is of order 1/a. Furthermore, ^the persistence length of the interface normal is of order of a. Thus if we consider K0>0, ^{the curvature}

energy per unit volume ^near critical point is of order K o, and should not radically change ^{the nature and} position of the transition for K0 kB T. ^{If we} keep

K o of order kB T, ^{and remove the} system ^{from the} lattice, short-wavelength fluctuations that were pre- vented before by the lattice constraint are still

suppressed ^{but now} by ^{the curvature} energy term, Ko. Thus we argue that for K o of order kB T ÇK ^{of order} a), ^the system ^{has a} continuous transition in the Ising universality ^{class at} g r of order a. In similar way, ^for larger K0 the ^curvature energy furnishes an effective cutoff at CK ^{and the transition}

should occur at g ^{of order} g K and should still remain in the Ising universality ^class.

This transition separates ^the bicontinuous tense

phase, where the « oil »-« water » symmetry ^is broken, from the random isotropic phase, ^{where no} symmetries ^{of the} system ^are broken. The Ising

order parameter ^is (1/2 - 0 ). ^For large )K, ^the

volume fraction occupied by ^the minority phase

should scale as

where the scaling function, P (x), ^{behaves as}

for x - xc . ^The transition occurs at 03BEr/ 03BEK

^{xc, and}

(3 ^=--x 0.31 is the usual Ising ^order parameter ^ex- ponent. For x > xc, P (x ) = 1/2. For x -+ 0, by (4), (5) ^and (14) ^{we have} (assuming ^{here a}

0)

Similarly, the surface tension, a, should scale as

with the scaling ^function behaving ^as

for x --> xc , with tL 1.26, ^and S(x) -> 1 for x -> 0.

The scaling of the correlation functions is determined

by noting ^that the energy is ^« stored ^» in the films so

that the surfactant concentration is an energy-like variable, ^{while the « oil » and « water »} play ^{the role}

of « up ^» and « down » spins. ^Thus surfactant-surfac- tant correlation functions scale as energy-energy correlation functions, while « oil ^»-« oil _», « oil ^»-

« water » or « water »-« water » correlations scale as

spin-spin correlations in usual Ising model. In these

scaling functions ÇK ^{serves as the} microscopic length.

The correlation length ÇI ^which diverges ^{at the}

transition scales as

where

for x - xc, v ^{= /-t} /2, ^and X(x ) -- ^x, for x -+ 0. For

example, the truncated ^« water »-« water » corre-

lation function should, ^for R > ç K, ^{scale as}

where G(y)-yle-Y ^{for y --+ oo,} G (1 )

0(l) ^for

y - 0, and q = ^0.03.

We have argued ^{that this} tense-relaxed transition

is continuous in our model (3). ^However, many

microemulsion systems ^{are known to have a} strongly