Fluctuations in membranes with reduced symmetry

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Fluctuations in membranes with reduced symmetry

L. Peliti, Jacques Prost

To cite this version:

L. Peliti, Jacques Prost. Fluctuations in membranes with reduced symmetry. Journal de Physique,

1989, 50 (12), pp.1557-1571. �10.1051/jphys:0198900500120155700�. �jpa-00211014�

Fluctuations in membranes with reduced symmetry

L. Peliti (1) and J. Prost (2)

(1) Dipartimento di Scienze Fisiche and Unità GNSM-CISM, ^Associato INFN, Sezione di

Napoli, Università di Napoli, ^Mostra d’Oltremare, ^Pad. 19, I-80125 Napoli, Italy

(2) Groupe ^de Physicochimie Théorique, ^Ecole Supérieure ^de Physique ^{et Chimie} Industrielles, 10 rue Vauquelin, F-75231 Paris Cedex 05, ^France

(Reçu le 5 décembre 1988, accepté ^{le 8 mars} 1989)

Résumé.

Nous discutons les effets de l’ordre local des membranes d’amphiphiles ^{sur la} statistique de leurs fluctuations de forme. Nous considérons en particulier les effets de la flexo- et ferro-électricité et de différentes espèces d’ordre orientationnel. Nous discutons enfin les ondulations d’une membrane immergée ^{dans un fluide} anisotropie.

Abstract.

We discuss the effects of in-plane ordering ^of amphiphilic ^{membranes on the}

statistics of their shape fluctuations. Effects of flexo- and ferroelectricity ^{as well as} of several kinds of orientational order are considered. We close by ^a discussion of the fluctuations of a membrane immersed in an anisotropic ^fluid.

Classification

Physics ^Abstracts

68.10Cr

68.15+e

82.65Dp

1. Introduction.

Thermal fluctuations have an essential influence on the behavior of fluid amphiphilic membranes, since their curvature moduli are comparable ^{to kT} [1-3]. ^One specific

consequence is that isotropic fluid membranes are crumpled ^{at all nonzero} temperatures [4, 5]. ^{On the other} hand, hexatic membranes should exhibit a low temperature phase ^{in which}

correlations in the normals decay slowly [6, 7]. Polymerized membranes should even exhibit a

low temperature ^flat phase [6, 8-11]. ^Real amphiphilic ^{membranes cannot} be described as

simple geometrical surfaces : they possess in general ^electric properties, ^or special kinds of in-

plane order, which have not been considered in the investigations ^{we have} mentioned. It is natural to wonder whether these properties modify ^the fluctuating behavior of the membrane.

This is the question ^we address in the present paper.

Amphiphilic ^{membranes are in} general flexoelectric : ^when they ^are warped, they possess ^a local electric dipole proportional ^{to the mean curvature and directed} along the normal [12].

Moreover, ^several amphiphiles possess ^a nonvanishing ^electric dipole, ^and exhibit therefore

ferroelectric properties [13]. It is therefore interesting ^to investigate the consequences of the

couplings arising ^between undulations and induced or permanent ^electric dipoles.

We have found that flexoelectricity ^{does not} qualitatively modify ^{the behavior of fluid} membranes, although ^it renormalizes the bending rigidity. ^Since flexoelectricity ^{is a} general property ^{of real} amphiphilic membranes, ^a different result would have cast doubts over the effective interest of the current theory of undulations.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500120155700

The effects of ferroelectricity ^{are more} complicated. We have found that if the electric

polarization ^is tangent ^{to the} membrane, undulations decouple ^from polarization modes in the

long wavelength ^limit. On the other hand, ^{if the} dipoles ^{are normal to the} membrane, ^the

undulation spectrum is modified in a nontrivial way and ^one may observe ^a novel behavior.

In realistic situations, ^the embedding fluid will often be an electrolyte : therefore, ^electric

interactions will be screened beyond ^the Debye length. This introduces a new length scale,

and our results, derived in the purely dielectric _case, will hold for distances shorter than it.

We have also considered different kinds of orientational order in the membrane. The

coupling ^between undulations and orientational modes in nematic or smectic-C membranes is screened at finite lengths. ^These systems behave therefore like isotropic fluid membranes.

One may also consider the hypothetical ^{case of «} tetratic » membranes, corresponding ^to

hexatics for the case of square symmetry. ^{We show that} they behave like hexatic membranes.

One may also consider the case in which the solvent surrounding the membrane is

anisotropic. ^{For the case of a} nematic solvent we find an effective interaction which stabilizes the flat phase : ^{we thus} expect ^the crumpling transition of the membrane to be driven by ^the nematic-isotropic transition of the solvent.

The plan of the paper is ^as follows : section 2 contains a reminder of the theory ^of shape

fluctuations in fluid membranes. The effects of flexoelectricity ^are discussed in section 3.

Ferroelectricity is considered in sections 4 and 5. Namely, section 4 contains the discussion of tangent polarization, and section 5 that of normal polarization. ^Section 6 contains the discussion of the effects of in-plane orientational order, and in section 7 the effects of nematic order in the solvent are investigated. ^A brief discussion is contained in section 8.

2. Fluid membranes.

We recall in this section the basic tools used in the description of undulations in fluid membranes. We represent the membrane as a two-dimensional surface embedded in ambient,

three-dimensional space. If ^s

(sl, s2) ^{is a} system of internal coordinates on the surface, ^the embedding ^{is defined} by

The local geometric properties ^{of the membrane are defined} by two tensors [14]. ^{The metric}

tensor gij, ^which defines the arc element dl2 - _gij dSi dsj, ^is given by

It also defines the surface element dS

-,fg-d2s, where g

^Det (gij ), and the covariant derivative Di. ^The curvature tensor Kij is defined by

where rb ^{is a} Christoffel symbol. ^Each component Kij ^{of this tensor is a vector in} R3 normal to the surface. Therefore, ⁱⁿ particular, ^{its trace} may be expressed by

where k is the normal unit vector to the surface and the scalar H is the mean curvature. The

Gaussian curvature K is defined by

The elastic energy Fe ^{of an} undulating ^{membrane is derived} [15,16] ^{if one assumes that it} depends only ^{on local} geometric properties ^{of the surface, which are invariant under}

Euclidean transformations in ambient space, and under coordinate transformations in the internal space of the membrane. If ^one further assumes that only derivatives _{of X up} to the second _{order may} appear ⁱⁿ it, ^{one obtains the} following expression ^for Fe :

The quantities ^{r, rc, K} appearing ^{in this} expression ^{are the bare} surface tension, rigidity, ^and

Gaussian rigidity respectively. ^For membranes, ^it is known that the effective value of ^r is

small, ^{so that} fluctuations are controlled by ^{the curvature terms}

^{the two last ones in the} equation above. The spontaneous ^curvature Ho ^{vanishes for} symmetric ^{surfaces. The last term} plays ^{no role for} fluctuations which do not change ^the topoligical genus of the surface. The elastic energy ^one effectively considers reduces therefore to the simple ^{form :}

The behavior of membranes having this form of the elastic energy is well known.

Undulations are so intense that the effective area of a given ^membrane region ^is substantially

reduced with respect ^{to its true area} [17]. ^Moreover, the effective rigidity K ^{becomes smaller}

and smaller as the size of the membrane increases [4, 5]. Therefore, ^the unit normals k to the membrane are correlated only up ^to the de Gennes-Taupin [18] persistence length §, given, ^for large ^{values of K,} by

where a is a microscopic length, of the order of the membrane thickness (a =10- 6 cm). ^The

existence of large undulations leads to a long-range ^effective repulsion among membranes

[19].

The key point ^{we want to address} in this paper is whether it is warranted to take equation (2.7) ^{as a} starting point ^{for the} description ^{of membranes with reduced} symmetry. ^{The main} doubts arise because of the nonlocality of electric interactions. We shall dwell elsewhere on

the effects of a net charge distribution on the membrane. But dipolar interactions are already sufficiently long-range ^{to make the} arguments leading ^to equations ((2.6), (2.7)) worthy ^{of a}

closer look. Other sources of effective long-range interactions are different kinds of local order

analogous ^to the hexatic order considered in references [6, 7].

We shall only consider small deviations from a planar shape. ^{In this} case, the shape ^{of the}

membrane can be represented ^{in the} Monge ^{form :}

Equation (2.7) ^{assumes, to} lowest order in _u, the form

where V2 is the ordinary Laplacian ^with respect ^{to the two} coordinates s

(x, y ). Equation

(2.10) implies a q4 dispersion ^{law for} undulations, with which we shall compare other ^terms

appearing in different cases. In the Monge ^form, the unit normal k to the membrane has the

expression

where Vu

(oxu, Oyu).

3. Flexoelectric membranes.

We discuss in this section the effects of curvature-induced polarization (flexoelectricity) [12].

As we have seen in equation (2.4), ^the symmetry ^{of the mean curvature} is that of a vector.

Therefore a curved membrane bears in general ^{an electric} polarization proportional (at ^the

lowest order) ^to Kij. ^{Such a} phenomenon has been first recognized in nematic liquid crystals [12], ^{where its} microscopic ^{roots can be} traced back to either dipolar ^or quadrupolar ^effects [20]. ^{In the first case, a} macroscopic polarization ^{arises from the} packing ^of asymmetric

molecules in a splayed (or bent) configuration. ^{The second case} stêms from simple

electrostatics : the divergence ^{of a} quadrupole density ^{is a} dipole, just ^{like the} divergence ^{of a} dipole ^{is a} charge. ^A third contribution to flexoelectricity arises in the case of membranes

[21]. Space charges often accumulate in proximity ^of highly polar, ^or zwitterionic interfaces.

The charge unbalance due to the membrane curvature results in a net polarization. ^{When this}

mechanism is present, ^{it is the most} important. ^The flexoelectric coefficient e is defined by

It has the dimensions of a charge and may be ^as large ^{as several} elementary charges.

In nematics, flexoelectricity renormalizes the director propagator by introducing ^a

nontrivial anisotropy, but without affecting ^the global q 2 dependence of the elastic term [22].

Since two-dimensional systems ^are notoriously ^{more sensitive to} perturbations, ^{it is} legitimate

to investigate the membrane case. In presence of ^an electric field E, the Helmholtz free energy F ^can be written

where Fe ^is given by equation (2.7), ^{and e} is the dielectric permittivity ^{of the} embedding ^fluid

- which we assume to be isotropic. Equation (3.2) yields ^the following expression ^{for the}

dielectric displacement ^{D :}

The delta function expresses the fact that the polarization ^{field is} concentrated on the membrane.

For an almost planar membrane, lying ^{in the z}

⁰ plane ^and represented ^{in the} Monge form, ^{we have}

where ez is the unit ^vector in the ^z direction. The condition V D

0 of charge neutrality

yields ^a relation between the displacement ^u (s) ^{and the electric} potential V (X) :

where A is the three dimensional Laplacian. ^{We now} take the Fourier transform of this

equation and obtain the following relation between the Fourier components uq1 ^{of the} displacement ^and those, Vq, of the electric displacement :

We have set q

(q1, qZ ) and q = Iq.L 1. Obviously, Mq ^{does not} depend ^on q,,. Letting ^this equation ^into equation (3.2) yields

If we now integrate ^over the qz component up ^{to a} cutoff wavevector A (where ll-1 ^1B is of the order of the membrane thickness a), ^{we obtain a} renormalization of the curvature term Fe :

By considering ^the expression (2.10) ^of Fe ^we may write the above expression ^{as follows :}

where we have defined ^an effective rigidity :

The contribution to K eff arising ^from flexoelectricity ^{can be as} large ^{as several times}

kT [23]. This contributes to a considerable stiffening ^{of the} membrane, ^{but does not} change ^its qualitative ^behavior.

The negative q5 term, arising ^from polarization charge-polarization charge interactions, ^is

in principle irrelevant in the long wevelength limit. One should however be careful not to dismiss it without further consideration : since ^K gets renormalized to smaller and smaller values when the length scale is increased [4, ^{5, 24,} 25], it may well lead ^{to a} finite wavelength instability of the kind of the one discussed in reference [25], but of different origin. ^{Even at}

the purely ^{harmonic level, one} may witness ^an undulation instability : although ^{the second}

term in equation (3.7) ^is always strictly positive, ^K may well be negative (because ^of impurity

effects [26] ^{or molecular} packing ^{reasons, for} example) and therefore equation (3.8) might

exhibit an instability ^{at finite} wavelengths.

When the membrane is embedded in ^an electrolyte, ^{the same} calculation can be made, ^with

the only modification that the 1/ (q2 ⁺ q2z ) propagator ⁱⁿ equation (3.6) ^{should be} replaced by

1/ (q2 + qz + À D- 2), ^where À D ^{is the} Debye screening length. ^{As a} result, ^one obtains, ^instead

of equation (3.9) :

In most cases the Debye screening length ^is large compared ^to the inverse cutoff, ^{and the}

correction is correspondingly ^small. Moreover, ^{one sees from the} above that the leading

behavior of the inverse propagator ^{is still} proportional ^to q4.

These results may be easily generalized ^{to the case of a} D-dimensional membrane embedded in a d-dimensional space. The steps leading ^from equation (3.2) ^to equation (3.8)

may be exactly followed, ^{with the} only difference that the integration leading ^from equation (3.7) ^to equation (3.8) ^{must be} performed ^over (d - D ) components. ^One would then find :

(i) ^{a constant} contribution _{to K eff} proportional ^to e2 Ad - D / e; (ii) analytical corrections, proportional ^to q 2 kas long ^{as 2 k d -} D, ^{in case with a} nonanalytic ^term proportional ^to

q4+d-D.

4. Ferroelectric membranes : tangent polarization.

Chiral smectic-C* (Sc*) and related phases ^{are in} general (improper) ferroelectrics [13].

Indeed, ^one may consider ^a smectic-C membrane as a fluid film, in which the molecules are

tilted on average with respect ^{to the} plane. ^{Such a structure is} characterized by ^a point group symmetry C2Z h. ^{However, il the} building ^{blocks are chiral} molecules, ^the only symmetry element left is a C2 ^{on an axis} parallel ^to the film and normal to the tilt direction (Fig. 1).

Hence, ^no symmetry argument ^{can rule out an electric} polarization parallel ^{to the} C2 axis. This very low symmetry has the consequence that three linear terms can be retained in the expression of the elastic energy of Sc. ^films (Ref. [22], pp. 311-312). ^{The first one is} proportional ^{to the} divergence ^of polarization : ^{since it is a scalar} quantity, ^{it can enter} linearly in the free energy. However, ^{it is} just ^a boundary ^{term :} hence, ^{it either} plays ^{no role}

in bulk thermodynamics, ^{or leads to defect} structures [27]. ^{We shall} only consider here the first possibility. The second linear term can be written _p Va, where p is the unit ^vector

parallel ^{to the} polarization, and a is the membrane thickness. It drops ^{out for an} incompressible membrane : in any ^case, since the fluctuations in the layer thickness do not

Fig. ^1.

^{In an} Sc* ^membrane, the molecules are tilted with respect ^{to the} (x, y ) plane, in which their

centers of mass assume a liquid-like ^{order. Because of} chirality ^the only remaining point symmetry ^{is a}

C2 ^axis along x, if the tilt is directed along the y ^axis.

have long-range correlations, ^{it will not} especially affect the fluctuation spectrum ^{of the}

polarization. ^{The third term describes a} spontaneous tendency of the membrane to twist, ^and

may be written [(c.V)k].p, ^{where c} is the tilt direction and k is the normal to the membrane. It forces membranes with Sc* symmetry ^{to assume the} shape of helical ribbons

[28]. This introduces a characteristic length f, typically of the order of microns. The considerations we develop ^{in the} following ^are meaningful ^{to the extent that this} length ^is larger ^{than the de} Gennes-Taupin [18] persistence length e ^mentioned in section 2. The

opposite ^case ffirelevant for the morphology discussed in reference [28].

Taking ^{into account the} observations just made, ^we may write down the free energy of ^an almost planar, fluctuating Sc* membrane, represented ^{in the} Monge ^{form :}

Here, Fe ^{is the elastic curvature} energy of the membrane, which, because of the

C,2 symmetry, ^{assumes the} anisotropic ^{form :}

We take the x axis along ^the unperturbed polarization p, and the y axis in the direction of the tilt (Fig. 1). Since p2 =1, only the py ^{component needs to be} considered to lowest order. We omit the subscript y ^to lighten ^the following ^formulae.

The other three terms involve the polarization ^field [29]. ^In particular, Fep ^{describes the}

Frank-Oseen elastic energy of the polarization ^{field :}

The elastic constants K 1, K 3 correspond ^to splay and bend deformations respectively. ^The

term Fc represents the Coulomb interaction of the polarization charges :

Note that, contrary ^{to the} flexoelectric case, we do not retain the full polarization energy, but

only ^{the extra} energy due ^to fluctuations. The background energy is assumed ^to be contained in an independent part, which determines the polarization modulus po. The last term,

Fcn? describes the coupling ^{between curvature and} polarization :

Its existence stems from the fact that u and y behave identically ^{under the} C2 operation.

After a straightforward ^Fourier transformation, ^{we obtain :}

where

We obtain therefore the following behavior of the correlation functions of p ^{and u :}

where

In these relations one can identify ^a length ^scale

beyond which electric interactions dominate over the elastic ones [30]. When qd .-, 1, ^the C 2 ^term in à (q, ), ^{which is} proportional ^to q6 ^is negligible ^with respect ^{to the AB} term, ^which is proportional ^to q 5. ^{Then the} p and u fluctuations decouple :

Thus, aside from an important anisotropy, ^{we recover the} q 4 undulations spectrum of the fluid membrane case. Hence, ferroelectric Sc* ^{membranes are} crumpled.

Iri the presence of ^an electrolyte, ^the polarization ^{term is modified as follows :}

One now has three lengths ^{in the} problem. ^If 5 - ç - k Dl the membrane is crumpled. ^If 5, À D - ç, ^{the situation is the same as in a} non-chiral Sc. These conclusions hold as long ^as d - ç. If § 8 , ^the situation is analogous ^{to that of a} nonchiral smectic-C (Sc ), ^{which is} investigated in section 6 [31].

5. Ferroelectric membranes : normal polarization.

If the fluid membrane lacks mirror symmetry (i.e. ^{its two sides are} nonequivalent), symmetry does not rule out the appearance of any ^vector quantity parallel ^to its normal. Two instances

are relevant : the spontaneous ^curvature Ho, which appears when the ^states with lowest energy ^{are not} planar [16], ^{and an electric} polarization, arising because such membranes are

often made of polar molecules. such asymmetric ^{membranes are} often films of amphiphilic

molecules separating ^{two different} media, ^{such as water and air} (Langmuir films), ^{or water}

and oil (microemulsions). Langmuir ^{films are} usually flat, because of the density ^difference

between water and air, ^{and the} spontaneous ^curvature appears ^{as a} boundary ^{term which does}

not affect the polarization spectrum. In the microemulsion case, although ^the spontaneous

curvature does play ^a role, it may be made ^to vanish by ^a suitable choice of the surfactant and cosurfactant concentrations. We shall consider in the following the effects of the electric interaction alone.

Similarly ^{ot the} previous case, ^we do not take into account that part of the electric energy which arises from the uniform background polarization. ^We again ^assume that it is taken into account in the energy balance which determines the modulus po ^{of the} polarization. ^{Hence the}

contribution due to the fluctuations is given by Fc, equation (4.3) :

where _p is the unit vector in the direction of the polarization. ^In equation (5.1) ^{we have set :}

where _El and E2 ^are the dielectric constants of the two different fluids which are separated by

the membrane. In practice, ^{one of the} fluids is often an electrolyte, and in the long wavelength ^{limit one} may simply ignore ^the corresponding contribution to the electrostatic energy. The only difference with the previous ^{case lies} in the relation between p and the layer configuration : ^{in the} present ^{case, since the} polarization ^{is normal to the} membrane, ^we may choose the normal unit vector k to coincide with p. We obtain therefore for small

undulations :

where V 2 is the two-dimensional Laplacian. We obtain therefore

Introducing ^the integration in ambient space in ^a way analogous ^to equation (3.3), ^{it is} possible ^to express equation (5.3) ⁱⁿ three-dimensional Fourier space :

After integrating ^out the qz component, ^{we obtain}

Therefore the undulation spectrum ^{of the membrane} is modified in a nontrivial _{way. The}

physical consequences depend ^{on the nature of the} system.

For Langmuir films, ^we may write the full undulation free energy in Fourier _space as

follows :

Here, ^{K is the} rigidity, ^r is the interfacial tension of the water-air interface in presence of the

amphiphile, ^and p mis the density difference between water and air. One may identify ^several important length scales in equation (5.6) :

is the capillary length, usually of the order of millimeters ;

separates ^the regime controlled by interface tension from that dominated by ^electric interactions ;

is the length below which the usual undulation behavior (described ^{in Sect.} 2) ^{takes over. In}

most cases, SI ^is significantly larger ^than S2 ^and S3 [32]. ^If S2 > S3 ^{one observes a} regime ⁱⁿ

which the electric interactions dominate. On the other hand, ^if S2 d3, ^one directly ^switches

over from the undulation regime described in section 2 to one dominated by the interfacial tension at longer lengths.

For the case of the free floating ^{membrane or of the} microemulsion, the relevant length

scales to be compared ^{are the de} Gennes-Taupin persistence length e ^and d3. ^If

e > S3, the film is crumpled ^for all length scales shorter than d3, ^{which is} implicitly ^defined by

This length is smaller than d 3, because of the thermal softening ^of undulations, which makesd

K eff(£) become smaller when the length ^{scale f} increases. Beyond 6’, ^the q3 behavior takes over, and the film is flat. Whether or not electric interactions play ^a role in microemulsions

near the inversion point depends precisely ^on the relative values of 53 ^and e. ^{We remark}

moreover that in such floating ^{membranes we} expect ^{a finite} temperature crumpling

transition to take place.

6. Effects of smectic-C or « tetratic » order.

Membranes with in-plane ^hexatic order have a peculiar ^low temperature behavior, ^{as it was}

shown in references [6, 7]. ^Other two-dimensional systems belong, ^when they ^are planar, ^to

the same universality ^{class as hexatics. The} question ^{arises whether these} systems also exhibit the peculiar ^behavior of hexatics if their shape is allowed to fluctuate.

Hexatic order is manifested in the long-range correlations of the directions of the bonds

linking neighboring molecules. Near any given molecule in a closely packed planar fluid,

there are six special directions, forming ^{with one another an} angle multiple ^of 7r /3 . ^{The local}

order may be defined by ^a two-dimensional unit vector t, defined up ^to rotations by multiples

of ± ’TT /3. ^The system possesses therefore an unbroken C6 ^symmetry, corresponding ^{to these}

rotations. This symmetry entails in fact the invariance of the effective Hamiltonian under rotations of t by ^an arbitrary angle. ^{The reason is that all} couplings which reduce the full rotation symmetry ^{to its} C6 subgroup ^{involve at least six factors of t and are therefore}

irrelevant by power counting. This invariance still holds if the membrane shape is allowed to

fluctuate.

We now recall the origin ^{of the} coupling between undulations and orientational modes. In

planar ^hexatics, orientational rigidity ^is expressed by ^the following ^additive contribution to the free energy :

where KA is the hexatic stiffness constant. It is natural to generalize ^{this term to} undulating

membranes by replacing ^d2s by ^{the area element} dS, the hexatic director t by ^{the vector field} t such that t = t aiX, and the two-dimensional gradient ^V by the covariant derivative

Di :

In fact Q

Di ti Di tj ^{is, up to total} derivatives, ^the only nontrivial scalar which may be built with the unit vector field t ⁱ [7].

Equation (6.2) ^{encodes a} long-range interaction among undulations, as one can see from the following argument. Whenever the Gaussian curvature K of the membrane does not

vanish, ^{it is not} possible ^{to define a vector} field t such that Q ^vanishes everywhere. This is due to the fact that a vector, ^{which is} parallel transported along ^{a closed} path encircling ^a region

with nonvanishing ^Gaussian curvature, undergoes ^{a rotation} proportional ^{to K} [6, 7]. ^This

effect may be represented by introducing ^{a vector} potential fi whose ^{rotor is} proportional ^to

K.

This interaction is long-range. Imagine ^{in fact to} modify slightly ^the shape of the membrane in a localized area. The readjustement ^{of the field ti} will be felt far away, because the field is massless : and this is a direct consequence of the rotational symmetry ^we have discussed above.

We can now turn to consider other kinds of in-plane rotational order.

(i) In nonchiral smectic-C membranes the amphiphilic ^{molecules are tilted on} average with respect ^{to the} surface normal. The direction of the tilt is identified by ^{the unit} tangent ^{vector t.}

Rotations of this vector by any angle around the normal produce ^{a different} physical

situation : therefore there is no residual symmetry.

(ii) Nematic membranes ^are probably ^{hard to come} by. ^{In this case,} the nematic director t is defined only up ^{to a} sign : hence there is a residual symmetry corresponding ^{to rotations} by

± 03C0

(iii) ^{One can} envisage ^a situation in which there are locally four special directions, along mutually orthogonal ^{axes. These} systems may be called tetratics in analogy with hexatics. The residual symmetry ^{is the} C4 subgroup of rotations by multiples ^of 7T /2.

The elastic free energy of planar systems of this kind can be written in general [22] :

Introducing ^the angle 0 by ^means of the substitution

equation (6.4) may be rewritten in the form [33] :

We see that the second terms breaks invariance upon rotations by 7T /2. ^{For tetratics,}

therefore, K 1 = K 3 . On the other hand, ^{it was shown in reference} [33] ^that (K3 - K 1 ) ^{is an}

irrelevant variable in the long-wavelength ^limit. Therefore, ^the effective Hamiltonian for all these systems ^is given, ^when they ^are planar, by ^the expression

If the membrane is warped, ^one may define other scalars involving ^{the curvature matrix}

Kij. One may have in fact [7] :

If the ambient dimension is equal ^to three, Kij ^is actually proportional ^to the membrane normal k. We may write therefore

and by suitably choosing the coordinates we may locally diagonalize the matrix Kij :

where _{rl, r2} are the principal ^curvature radii. Then equations ((6.7)-(6.9)) ^{reduce to the form} [7] :

We see that all these terms break the C4 invariance characteristic of tetratics. Therefore none

of them may appear in the effective Hamiltonian for tetratics. As a consequence, their effective Hamiltonian, and thus their behavior, ^{is the same as} for hexatics. On the other hand, the reduced symmetry ^{of smectics and} nematics is insufficient to prevent ^{them from} appearing.

They play ^{the role} of curvature-induced anisotropy ^terms for orientational fluctuations, tending ^to align ^t along ^the principal ^curvature directions. This induces a strong coupling

between undulations and orientational modes. Nevertheless, ^this coupling ^is short-range, ^and

its only ^{effect is a} renormalization of the elastic constants. That this is the case may be ^{seen as} follows.

If a warped membrane is deformed in a small region, the directions of the principal

curvature axes are not modified outside this region : hence the configuration ^{of the}

orientational order parameter, just because of the anisotropy, ^is only locally ^{affected. Indeed,}

the couplings will make the orientational modes massive, ^{with masses} proportional ^to

(H 2> . ^A similar mechanism produces ^the screening ^{of the} coupling among undulations due ^to the incompressibility constraint [34]. ^In this way, the ^mass introduced by undulations

effectively ^screens orientational correlations, making ^{nematic and smectic-C membranes}

behave like isotropic ^ones.

7. Membrane immersed in an anisotropic ^fluid.

The case of an undulating ^{membrane immersed in an} anisotropic solvent is not as academic as

it might appear ^{at first} sight : lyotropic nematics, ^{in which} floating sheets may be included, ^do exist. We shall consider in the following only ^{uniaxial nematics.}

Short-range interactions will in general align the nematic axis at a definite angle ^with respect ^{to the membrane} plane : ^the simplest ^{case is that in which} the nematic director

n is aligned along the normal k to the surface. Since we shall show that the membrane is rigid

in this _case, we can perform ^the analysis just ^{to second} order in the déformation field u.

The condition that the nematic director n is aligned along the normal k to the surface, slightly crumpled ^{near the z}

⁰ plane, may be written in the form

where we have set n = (nl ,1 ), ^{and V is the two} dimensional gradient. ^This equation

expresses the coupling between undulations and nematic fluctuations, ^{due to the} fluctuating boundary conditions. The corresponding energy may be evaluated in the following way.

To the distortion 8 nl (x, ^{y, z}

0 ) corresponds the director field nl (x, y, z ) satisfying ^the

nematic Euler-Lagrange équations :

where K 1, K 3 ^are the splay ^and bend elastic constants respectively. By ^{means of a two-}

dimensional Fourier transformation, ^and solving ^{for the} z-dependence, ^{we obtain :}

The associated nematic _{energy is}

By performing ^a Fourier transformation in the parallel direction, ^and integrating ^{out the z} component, ^we finally ^{obtain :}

We obtain therefore a q3 dispersion law, ^as in the normal ferroelectric case. Therefore, ^the

membrane is not crumpled. This holds as long ^as the solvent keeps ^its nematic order. We expect ^{therefore the} crumpling transition of the membrane to be triggered by the nematic-

isotropic transition of the solvent.

8. Discussion.

We found several regimes worthy ^{of a} closer examination : in particular ^the cross-over to a

fluid membrane behavior for the flexoelectric _case, and the different possible regimes ^{for the}

normal ferroelectric order. The anisotropy ^induced by ^the tangent polarization ^{can also} produce interesting physical effects which one might ^{like to} discuss in greater detail. But we

have found in most cases that the current theory of membrane undulations has found its

validity vindicated also in these potentially ^« dangerous » situations.

Acknowledgments.

This work was performed while L.P. was visiting ^the Groupe ^de Physicochimie Théorique,

Ecole Supérieure ^de Physique ^et Chimie Industrielle and the Laboratoire de Physique Théorique, Ecole Normale Supérieure de Paris. He is grateful ^to both Institutions for their

hospitality. He also thanks D.R. Nelson for useful observations and V. Marigliano ^{for his}

kind help.

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p0 ~ 10-4 esu

which is the case of many phospholipids

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