Micellar shape distribution in the isotropic phase near a prolate-to-oblate nematic phase transition

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Micellar shape distribution in the isotropic phase near a prolate-to-oblate nematic phase transition

C. Rosenblatt

To cite this version:

C. Rosenblatt. Micellar shape distribution in the isotropic phase near a prolate-to-oblate nematic phase transition. Journal de Physique, 1986, 47 (6), pp.1097-1102. �10.1051/jphys:019860047060109700�.

�jpa-00210285�

Micellar shape distribution in the isotropic phase ^{near a} prolate-to-oblate

nematic phase ^transition

C. Rosenblatt

Francis Bitter National Magnet Laboratory,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, ^U.S.A.

(Reçu ^le 6 janvier ^1986, accepté ^le 17 février 1986)

Résumé. 2014 Nous avons mesuré le coefficient de Cotton-Mouton en fonction de la température ^{et de la concen-}

tration dans la phase isotrope ^du système ^ternaire laurate de potassium (KL), 1-décanol, D2O. ^{Les résultats sont} analysés ^{dans le cadre d’un modèle où la solution est un} mélange ^de cylindres (tiges) ^{et de} disques, ^les interactions entre cylindres ^{étant de nature} entropique. ^On obtient ainsi les fractions relatives de cylindres ^{et de} disques ^de part ^et d’autre du point de coexistence entre la phase isotrope ^{« à} tiges » ^{et la} phase nématique ^{« à} disques ^».

Ces résultats complètent ^{les données} existantes de diffraction des rayons X obtenues dans la phase nématique.

Abstract

The Cotton-Mouton coefficient was measured as a function of temperature and concentration in the isotropic phase ^{of the} ternary system potassium ^laurate (KL), 1-decanol, ^and D2O. ^{Based on a micellar} picture

of mixed rods and disks, ^{and in the context of an} entropy ^{model for hard} cylinders, ^the data indicate the relative fraction of rods and disks ^as the KL concentration is adjusted ^{in the} vicinity ^{of the} isotropic

^{rod-like nematic}

disk-like nematic coexistence point ^{on the} concentration-temperature phase diagram. ^{This data} complements X-ray scattering results in the nematic phase.

Classification

Physics ^Abstracts

61.30G - 64.70M - 82.70K

As a function of temperature ^and amphiphile concentration, ^{Yu and} Saupe demonstrated the exis- tence of a biaxial micellar nematic (NBX) phase ^{in the} ternary system potassium ^laurate (KL), ^1-decanol

and D20 [1]. ^{At a fixed} weight fraction of decanol

they found that a biaxial phase exists between two uniaxial nematic phases, viz., ^a cylindrical prolate

nematic (Nc) phase ^at high ^KL concentrations and a

disk-like oblate nematic (ND) ^at low concentrations.

In principle the three ordered phases ^{can coexist}

with the high temperature isotropic phase (I) ^{at an}

isolated multicritical (o ^Landau ») point, ^{such that}

the ordered-to-isotropic phase transition becomes second order at this point [2-4]. ^In addition, ^other topologies ^{for the} general lyotropic phase diagram

have also been discussed For example, only ^the isotropic ^{and two uniaxial} phases ^{can coexist at a}

Landau point ^{with an} Nc - ND transition at lower temperatures [6], ^{or an} intermediate biaxial phase

can appear with ^a finite linei of first order NBx - I ^tran-

sitions along the concentration axis [6].

Subsequent ^{to the} discovery ^{of the biaxial} phase

a number of related experiments ^were performed ^to

better understand both the macroscopic properties

of this phase ^{and the structure} of the constituent

micelles. Optical birefringence measurements, ^for

example, ^{have been} interpreted ^{in terms} of the biaxial order parameter [7-9]. ^In addition, ^both light scattering [10] ^and magnetically-induced birefringence experi-

ments [11] ^{have been} performed ^{near the} ND - NBx

transition to extract the order parameter exponent

and susceptibility exponent y, with apparently ^con- flicting results : in one case classical behaviour was.

observed [10], whereas in the other (with ^much higher

temperature resolution) ^{critical behaviour was seen} [11]. ^{In the} microscopic ^{realm a number of investi-} gations of the micellar configuration ^{are also} being aggressively pursued A dozen years ago Alben and Shih [3] ^{and Alben} [2] pointed ^{out that a uniaxial-}

biaxial nematic phase transition can take place ^{in a} system composed ^of long (uniaxial) ^{rods and flat disks.}

Although this model was and remains a popular interpretation of the micellar configuration ^{in biaxial} lyotropic systems, ^recent X-ray ^{and neutron data have}

also been interpreted [12, 13] ^{in terms of} inherently

biaxial micelles preferentially fluctuating ^{about a} given axis; ^{the nature} of the fluctuations then define whether a macroscopic oblate, biaxial, ^or prolate phase ^{would obtain.} Although ^such measurements have been unable to distinguish ^{between the} models,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047060109700

1098

they have nevertheless yielded ^effective average mi- cellar dimensions in the nematic phases. ^{In an} early experiment Hendrikx et al. [14] scanned both tem-

perature ^{at a fixed} concentration and concentration at a fixed temperature, extracting average occupation numbers, diameters, ^and lengths (thicknesses) ^{for the}

rods (disks) ^{in the two uniaxial} phases. The results indicate a remarkably concentration-independent ^and temperature-independent ^effective shape within the entire _range of each uniaxial phase, although ^no report ^{was made of the} shape changes very close to the uniaxial-uniaxial transition. (The ^authors report

no intermediate NBx phase, perhaps owing ^{to the} precise region ^{of the} ternary phase diagram ^{in which}

the experiments ^were performed) Unfortunately,

within the context of the mixed rod-disk (Alben) model, ^the analysis of reference [14] precludes ^a

determination of the relative concentration of disks in the rod-like phase, and rods in the disk-like phase.

This deficiency ^{arises because} of orientational _averag-

ing in the azimuthal direction of the minority ^micellar species. Unless the X-ray experiment ^were performed

at very large q, for example, ^{one would} expect ^that

orientationally-averaged rods in the ND phase ^would

appear similar to the more ubiquitous ^disks. (This

can easily be understood by noting ^{that a disk} pos-

sesses a uniform in-plane density. ^An infinitesimally

thin rod, ^{on the other} hand, exhibits a 1 /r density

distribution when orientationally averaged ^about

one axis. This distribution is equivalent ^{to a uniform} background ^disk plus ^a cusp diverging ^{at r = 0. The} background ^term is identical to the real disk, ^whereas

the _cusp can be properly ^resolved only ^if large ^scat- tering ^{vectors are} used)

More recently, Figueiredo-Neto et ^ale reported [12]

X-ray results for the experiments performed ^{at a}

fixed concentration while scanning temperature

through ^all three nematic phases. Analysing ^their

results in terms of the biaxial micelle model (although

unable to discount the possibility ^{of the mixed rod-}

disk model), they concluded that effective local micellar order over distances ç ⁵⁰⁰ A remains similar in all three phases ^near the biaxial region, ^and

that within each of the two uniaxial phases ^the globally averaged effective micellar dimensions are nearly temperature-independent. Here ç ^{is the biaxial cor-}

relation length. Thus in both experiments relatively

constant and well-defined _average dimensions were

obtained within each uniaxial phase. ^A question ^now arises : using complementary techniques, ^{is it} possible

to investigate ^the distribution of micellar shapes ^{at a} given point ^{on the} phase diagram ? High ^field optical techniques [15, ^16] have been shown to be useful in several micellar structural studies, ^{and thus} may

provide insight ^{into the} shape distribution in the

isotropic phase. ^{To this end I} performed ^a magneti- cally-induced birefringence experiment ^{as a function}

of sample composition ^and temperature in the iso-

tropic phase ^{in the} region ^{above the} Nc phase, ^close

to the ND crossover. My central result is a rapid

evolution of the quantity 0 ^with composition, ^where

4Y * (T - T*) C, ^C is the Cotton-Mouton coefficient,

and T* is the concentration-dependent supercooling

limit of the uniform isotropic phase. These results are

analysed ^{in terms of an} entropy model ^{for hard} right-

circular cylinders [17] in the mixed rod-disk picture,

and indicate the _presence of a significant fraction of disks above the rod-like nematic phase, increasing

as the concentration W is reduced Unfortunately,

in terms of the inherently ^biaxial picture [12, 13], ^an analysis ^{was not} possible owing ^{to local biaxial}

correlations of the micelles.

Potassium laurate was obtained commercially ^from

Pfaltz and Bauer, ^{Inc. and} recrystallized ^{twice from}

absolute ethanol. Both 1-decanol (99 % purity) ^and D20 (99.8 %) ^{were obtained} from Aldrich Chemical

Company ^{and used} without further purification.

Samples ^were prepared ^{to contain a fixed} weight

percent 6.238+0-00’ of decanol, closely corresponding

to the compositions ^{used to} obtain the phase diagram

in reference [1]. ^The weight percent ^{W of KL was} varied from 25.475 % ^{to 26.895} %, ^{and the concen-}

tration of D20 ^was adjusted accordingly. Samples

were contained in ^a 0.2 cm pathlength stoppered

cuvette housed in a temperature-controlled ^oven.

The oven was in turn situated in an 11.2 T Bitter magnet possessing ^a transverse optical port ^Details of the oven and birefringence apparatus ^are given

elsewhere [118]. ^{The field H was} swept ^{from zero to} 10 T in 30 s and the induced birefringence ^{An was}

computer recorded Over the duration of the _sweep temperature ^{control was} better than 3 mK.

Well above T* the birefringence ^{was found to be}

linear in H2 and the Cotton-Mouton coefficient

C(W,1) == ð 4.n/ðH2IH=O ^was determined by ^{a linear} least-squares ^{fit to} the data. Closer to T * (T - ^T*

400 mK) ^small deviations from linearity ^were observed;

here the initial slope ð An/ðH2IH=O ^{was determined}

from a least-squares ^{fit to a} polynomial quadratic ⁱⁿ

Fig. ^1.

Extrapolated supercooling temperature ^{T* of the} isotropic phase ^{as a function of} weight percent potassium

laurate.

H2. Data were taken over a temperature range

T* T P,,: + 2 K, ^where Tg ^{is the onset of a}

small biphasic region, and in all ^cases C -1 ^was found to be linear in temperature, indicating ^{a mean field} susceptibility exponent y ⁼ 1. At all concentrations studied T,* - ^{T* was found to be a} few hundred

millikelvins ; this is unlike the case of the binary system caesium perfluorooctanoate (CsPFO) ^and H20, ^which

can exhibit values of r: - ^{T* ;g 20 mK for low} weight fractions of CsPFO [15]. ^In addition, ^the

transition remained discontinuous to fields of 10 T, contrary ^to the results of Saupe et ^al. [9, 11] ^who

observed no discontinuity ^{in An vs.} temperature ^at the Nc - I phase transition in the presence of ^a 1 T

aligning ^field.

Since the behaviour in the isotropic phase ^is apparently ^mean field, ^{for weak} ordering the Cotton- Mouton coefficient is given by [19]

where As and AX ^{are the volume} dielectric and

magnetic susceptibility anisotropies ^for fully ^saturated order, n ⁼ (8)1/2, ^and ao is the coefficient of the

quadratic ^term in the Landau free energy expansion.

The two fitted parameters T*(W) ^and O(W) ^were

obtained from the C(W, T) ^data using ^a logarithmic fitting procedure. Fitted values of T* vs. W are shown in figure ^{I and the} quantity cP[ = AB Ax/(9nao)] ^is plotted ⁱⁿ figure ^{2. As is} readily apparent 0 ^varies

Fig ^2.

0[ =- ð An/ðH2IH=O] ^vs. weight percent potassium

laurate.

smoothly ^over the entire concentration _range, with the fastest change occurring ^at lower concentrations.

To interpret ^{the data in} figure ^{2 we need a model}

for each factor in equation (1), ^{such as} ao, åe, ^etc.

We deal summarily ^{with n} by noting that it is virtually

constant over the concentration _range studied Next

we consider _ao by invoking ^a model for the entropic

part ^{of the} free energy, similar to that calcu14ted by Onsager [17] ^{but taken} only ^to é>(Tr t.J2), where Q ^is

the nematic order parameter. ^{Consider an} assembly

of rigid, right-circular cylindrical ^micelles exhibiting only repulsive excluded volume interactions and,

for the moment, monodisperse ⁱⁿ aggregation ^number

s ; ^we thus are dealing ^{with a} transition from the

isotropic phase ^to only ^{a uniform} Nc phase ^{or uniform} ND phase. ^In the context of a Landau model - Tr ao t.J2

represents ^to é>(Tr t.J2) ^an orientational entropy

change 8 per unit volume between the la ^{= 0 and} t.J =F ^{0 states. As} argued ^earlier [15] ^this entropy ^has

a direct orientational component 80 ^{and a transla-} tionally-coupled orientational component 80r. ^For

weak ordering ^{we assume} Onsager’s orientational distribution function f(0) =( rx/4n ^sinh a) cosh ( a ^cos 0) [17], ^{where rx4 oc} Tr t.J2 ^{for small a. Then to the level}

of the second virial coefficient we can obtain [15] ^the entropy 8 to é>(Tr t.J2) and thus obtain

ao[ = - 8/Tr t.J2] :

where NA ^{is the} amphiphile concentration of the

sample, kB the Boltzmann’s constant, d the micelle diameter, and I its thickness (or length). Note that the first term represents 80 ^{and the second term} ST.

Later it will be argued ^{that for an ensemble} of all disks

or all rods of similar size, ao is nearly identical in both

cases. The function 0 [ct: ^Eq. (1)] will thus depend

almost exclusively ^{on the} shape dependences ^{of the}

factor As Ax.

We consider finally ^the relationships ^between Ax

and As and the shape of the micelle. Such ^an analysis requires ^a precise knowledge ^of the molecular confi-

gurations within the micelle, still ^a subject ^{of much} investigation. ^The amphiphile conformations within

spherical ^micelles, bilayers, ^and infinitely long ^rods

have been extensively investigated theoretically [20-24], although ^the problem ^of finite, anisometric micelles has yet ^to be tackled (Recent experimental ^work by

Charvolin and Hendrikx [25] ^{on oblate} micelles will

hopefully stimulate such calculations.) ^{Thus a gross} simplification ^{will be} introduced : we consider right-

circular cylinders ^and neglect «end caps » for the rods and « rims » for the disks [26]. Moreover, ^we

assume that for disks the amphiphile chains lie

perpendicular ^to the disk face and for rods they ^lie

in a plane ^{normal to the micellar} symmetry axis;

more about this later. The discussion will now be

specialized ^{to disks. Since the} amphiphiles ^{within the}

micelle are all parallel, Ax ^{is obtained} by summing

the individual magnetic susceptibility anisotropies.

Thus the susceptibility anisotropy per micelle scales

as s åXm, ^where AXm is the molecular susceptibility

anisotropy, ^{and the} susceptibility anisotropy per

volume AX ^{scales as} NA AXm. ^{For As we can make a}

1100

similar argument, although ^an important ^difference

should be noted In this case the summation of the molecular polarizibility anisotropies Op ^{is not} straight- forward, although ^{a Lorentz term can be used to} approximate the local field corrections. It turns out that for sufficiently ^small birefringence ^{these cor-}

rections are negligible ^{and the} optical ^dielectric anisotropy per micelle scales as s dp; the volume

averaged ^{AR as seen in a} birefringence experiment,

thus scales as N A ð.p. Gathering ^{together all these} factors, ^{we find from} equation (1) that for disks above the I - ND transition,

For rods the story ^{is a} bit different. Here the molecules lie in a plane ^{normal to the} symmetry ^{axis with no}

net orientation in the plane. Along the rod axis the micellar susceptibility ^{scales as} sxl ; in the normal

plane, ^{however, the} susceptibility ^scales as -1 s(xl ⁺ X II).

It thus turns out that the micellar susceptibility ^ani- sotropy ^scales as 2 s AX. and the volume anisotropy Ax ^scales as 2 Np LBXm’ ^a value one-half that of the disks. Similar geometric arguments apply ^to A8, and we find that for rods A8 scales as 2 NA Ap, again

one-half the value of disks. Thus by substituting ^the

rod-like values for Ax and A8 into equation (1), ^we

find that the functional form for Qrod = 1/4 0 disk ^above

the I - Nc transition, such that the appropriate ^rod-like

values of 1, d, ^{and s are used in} equation (3). ^The experiment reported herein is based upon this result,

that is, ^{the measured values of} tProd ^and tPdisk ^differ (primarily) because of their respective ^{forms for}

A8 Ax. ^{If a mixture of rods and disks were} present,

an average A8 Ax would obtain. The measured value of 0 would thus correspond ^{to the number average}

of 45rod ^and 45di.,,, ^{and thus} figure ^{2 reflects} the relative concentration of the two micellar species.

Hendrikx ^et al. have used X-ray scattering [14] ^to

deduce the _average micellar dimensions and ^occu-

pation ^{number as a} function of temperature ^and concentration. They ^{found for disks I -} 23 A, d - ^64A

and s - 215; ^{for rods 1 - 63} A, ^{d -} 36 Å and s ~ 185.

Although ^these measurements were performed ^{at a} slightly ^{different set of} temperatures ^{and concentra-} tions than _my experiments, they probably represent reasonable values for use in equation (3). Upon substituting these values we find the 1- and d-dependent

factor in the denominator to be virtually ^identical

for both rods and disks and of order 1/3. ^Gelbart

et al. have also suggested [27] ^{that a} growth in rod-like micelles ^occurs in the Nc phase below the I -Nc

transition ^as the micelles attempt ^{to minimize the} loss in orientational entropy ^attendant upon align-

ment ; thus in the isotropic phase ^one might ^{find even}

slightly smaller micellar dimensions with ^{a con-} comitant reduced contribution from ST. ^The important point ^{is that} given ^the large experimental ^variation

of 4Y with W, ^to first order the actual values of tfJrod

should be approximately 4 (P disk ^{where the Ts are}

dominated primarily by ^their respective ^{forms for} AeA/, rather than the precise shape contribution

to ST.

Figure 2 shows the evolution of 4Y with W in the

region primarily above the Nc phase, ^{where the} Nc - ND transition occurs in the vicinity ^{W = 25.2} %.

This is to be compared with the data of Hendrikx

et al. [14], which indicate concentration-independent

average micellar dimensions on either side of the concentration-driven Nc - ND transition in the nematic

phases; only within the _range A W - ± ^0.1 % ^{of the}

critical concentration (at ^constant temperature) ^does

a change in dimensions occur in their data. As men-

tioned earlier, the micellar shape distribution cannot be extracted in ^a low _q X-ray scattering experiment.

In such an experiment the disks would _appear washed out due to orientational averaging about the uniaxial

direction, and thus the statistics of the rod-disk mixture

are relatively inaccessible. Only ^at higher q ^can the

X-ray ^form factors be distinguished Figure ^{2, on the}

other hand, ^{shows a slow} evolution of the quantity 0

with W. Although ^the analysis ^for ST, and therefore 4Y, is formally applicable only ^{to an} assembly ^{of either}

all rods or all disks, the virial corrections arising ^from ST ^are somewhat smaller than the experimental

variation in the quantity ^0. Thus since (AS ex)dlsk ⁼ 4(As AX)rod ^for comparably-sized micelles, ^{disks can}

be experimentally distinguished from rods by ^an

increase in 0. In the lower concentration region,

for example, ^{the data} clearly ^{indicate an} increasing

fraction of disks in the mixed rod-disk picture; ^thus

even above the Nc phase, the fraction of disks can be

quite substantial, ^and grows with decreasing ^W.

Although 0 ^is expected ^{to vary} by ^a factor of four when going from all rods to all disks, ^the experimental

variation (cf. Fig. 2) ^{is at} least this over a more limited concentration _range. Moreover, ^{the curve does not}

seem to be leveling-off ^at low W. There are numerous

possible ^{sources for this} discrepancy. ^For example, although ST ^was formally included in equation (3)

it was not deemed _very important ^{Yet this term can} ultimately ^be responsible ^{for a 20-30} % ^difference

between 0,.d ^and 45di,,,. ^{In addition, the} aggregation

number s of the micelles _may be growing ^{as W decrea-}

ses. This hypothesis, ^{however, is at} odds with the data of Hendrikx et al. [14], who found the _average dimen- sions within each nematic phase ^{to be} concentration-

independent ^{In fact, in} light of their results and those

presented ^{herein, the} shape distribution is probably bimodal, peaking ^about given positive ^and negative aspect ^{ratios with} relatively ^few spherical ^micelles.

This result is expected ^since rods and disks both

require ^{smaller surface areas per head group} [26]

than do spheres. Another ^potential difficulty ^{in 0}

arises from the neglect ^{of «} rims » and « caps ». On ^a

purely geometric basis these end corrections ^can account for nearly 50 % of the micellar volume, although ^{the volume fractions} turn out to be com-

parable for both rods and disks; it is therefore unlikely

that end corrections can account for a very large experimental variation in 4Y, although ^{these cor-}

rections can certainly ^{add to an overall} discrepancy.

A more significant ^{source of} potential problems ^is probably ^the molecular conformation within the micelles. Szleifer et ale have calculated [21] ^bond orientationaj order parameters Sk ^for spheres, ^infinite rods, ^and bilayers using representative ^bond energies Eb. They found that the Sk’s ^are largest ^for bilayers (by ^{as much as a factor of 3,} depending upon chain

length ^and Eb), indicating ^{that the} quantity Ag Ax

for disks could actually ^be considerably greater ^than four times that of rods, although ^{it is} very sensitive to the particulars of the model. Gruen, ^for example, performed ^a similar calculation [23], ^{where he} seg- mented the aggregate ^into onion-like shells and defined A., ^{as the} volume-weighted average of the

mean area of the shells. If Aav ^were comparable ^{for all}

three shapes, ^{he found} Sk relatively insensitive to

geometry. On the other hand, ^{if the core radius were}

held constant, results similar to those of reference [21]

were obtained Recent experimental ^work [25, 27-31]

on several micellar systems ^to determine bond order parameters has been helpful, although ^{more extensive}

data is necessary for ^a fully quantitative analysis ^of

the data presented herein. Charvolin and Hendrikx,

for example, have shown [25] ^{that the} shape ^{of the} Sk ^{vs. k curve} (where k is the carbon number along ^the chain) is similar in the lamellar, hexagonal, ^and

oblate nematic micellar phases ^{of the KL} system, and are all of the ^same order of magnitude, although

factors of two or three are crucial for a full analysis

of _my results. Nevertheless, I’ve been able to obtain

semiquantitative results for the shape distribution in the rod-disk model using ^the simple conformation

picture ^{outlined earlier.}

The above discussion was based _upon a model of

primarily ^rods crossing ^{over into} primarily ^disks.

As mentioned earlier, however, ^another picture ^has

been proposed ^{in which} inherently ^{biaxial micelles}

generate ^{the three} ordered phases [12, 13]. ^{This is a} particularly appealing ^{model since} experimentally

T* vs. W is relatively flat; ^{in the} mixed rod-disk

picture ^one might expect ^{a more} rapid variation in T*

as the shape distribution changes. Since local corre-

lations of the micelles play a fundamental role in such

a model, and the statistical mechanics have yet ^to be worked out, the data in figure ^{2 cannot} yet ^be discussed in terms of such a picture. ^{It is} hoped ^that

the results of references [12] ^and [13] ^{and those} pre- sented herein will provide sufficient impetus ^{for the}

necessary calculations.

To summarize, Cotton-Mouton data taken in the

isotropic phase above the rod-like nematic phase

near the Nc - ND ^{crossover indicate an} evolution in the mixture of rods and disks in the rod-disk picture, complementing average size data obtained by X-ray scattering. ^{At this} time, unfortunately, ^{the data cannot}

be analysed ^{in terms of the} inherently biaxial micelle model.

Acknowledgments.

I wish to thank Dr. P. Photinos for his critical com- ments on the manuscript. ^{This work was} supported by the National Science Foundation, ^{Division of}

Materials Research under contract number DMR- 8211416 to the Francis Bitter National Magnet Laboratory.

References

[1] Yu, ^{L. J. and} SAUPE, A., Phys. Rev. Lett. 45 (1980) ^1000.

[2] ALBEN, R., Phys. Rev. Lett. 30 (1973) ^778.

[3] SHIH, ^{C. S. and} ALBEN, R., ^{J. Chem.} Phys. ⁵⁷ (1972)

3055.

[4] PRIEST, ^{R. G. and} LUBENSKY, ^T. C., Phys. ^{Rev. B 13} (1982) 4159.

[5] GRAMSBERGEN, ^E. F., LONGA, ^{L. and DE} JEU, ^W. H., Phys. Rep. (in press).

[6] ^{ALLENDER, D. W., LEE, M. A. and} HAFIZ, N., ^Mol.

Cryst. Liq. Cryst. 124 (1985) ^45.

[7] GALERNE, ^{Y. and} MARCEROU, ^J. P., Phys. ^{Rev. Lett.}

51 (1983) ^2109.

[8] ^{GALERNE, Y. and MARCEROU, J.} P., ^J. Physique ⁴⁶ (1985) 589.

[9] SAUPE, A., BOONBRAHM, ^{P. and} Yu, ^L. J., ^{J. Chim.}

Phys. ⁸⁰ (1983) ^7.

[10] LACERDA-SANTOS, ^M. B., GALERNE, ^{Y. and} DURAND, G., Phys. Rev. Lett. 53 (1984) ^787.

[11] BOONBRAHM, ^{P. and} SAUPE, A., ^{J. Chem.} Phys. ⁸¹ (1984) ^2076.

[12] FIGUEIREDO-NETO, ^A. M., GALERNE, Y., LEVELUT,

A. M. and LIMBERT, L., ^J. Physique ^{Lett. 46} (1985) ^L-499.

[13] HENDRIKX, Y., CHARVOLIN, ^{J. and} RAWISO, M., Phys.

Rev. B (to ^be published).

[14] HENDRIKX, Y., CHARVOLIN, J., RAWISO, M., LIEBERT, ^L.

and HOLMES, ^M. C., ^J. Phys. ^{Chem. 87} (1983) ^3992.

[15] ROSENBLATT, C., Phys. ^{Rev. A 32} (1985) ^1924.

1102

[16] ROSENBLATT, C., J. Physique ^{Lett. 46} (1985) ^L-1191.

[17] ONSAGER, L., ^{Ann. NY} Acad. Sci. 51 (1984) ^627.

[18] ROSENBLATT, C., ^J. Physique ⁴⁵ (1984) ^1087.

[19] STINSON, ^T. W., LITSTER, J. D. and CLARK, ^N. A.,

J. Physique Colloq. ³³ (1972) ^C1-69.

[20] BEN-SHAUL, A., SZLEIFER, ^{I. and} GELBART, ^W. M., J. Chem. Phys. ⁸³ (1985) 3597 ; ^{Proc. Nat. Acad.}

Sci. 81 (1984) ^4601.

[21] SZLEIFER, I., BEN-SHAUL, ^{A. and} GELBART, ^W. M., J. Chem. Phys. ⁸³ (1985) ^3612.

[22] DILL, ^{K. A. and} FLORY, ^P. J., Proc. Nat. Acad. Sci.

77 (1980) 3115 ; ⁷⁸ (1981) ^676.

[23] GRUEN, ^{D. W.} R., ^J. Phys. ^{Chem. 89} (1985) 146 ;

89 (1985) ^153.

[24] OWENSON, ^{B. and} PRATT, ^L. R., ^J. Phys. ^{Chem. 88} (1984) ^2905.

[25] CHARVOLIN, ^{J. and} HENDRIKX, Y., ^Nuclear Magnetic

Resonance of Liquid Crystals, ^{J. W.} Emsley ^ed.

(P. Reidel) 1985, p. 449.

[26] MCMULLEN, ^W. E., BEN-SHAUL, ^{A. and} GELBART,

W. M., ^{J. Colloid.} Int. Sci. 98 (1984) ^523.

[27] GELBART, ^W. M., MCMULLEN, ^{W. E. and} BEN-SHAUL, A., J. Physique ⁴⁶ (1985) ^1127.

[28] MELY, M., CHARVOLIN, ^J. and KELLER, P., ^Chem. Phys.

Lipids ¹⁵ (1975) ^161.

[29] WALDERHAUG, H., SÖDERMAN, ^{O. and} STILBS, ^P. J.,

J. Phys. ^{Chem. 88} (1984) ^1655.

[30] CHARVOLIN, J., ^{J. Chim.} Phys. ⁸⁰ (1983) ^15.

[31] SEELIG, ^{J. and} SEELIG, A., ^Rev. Biophys. ¹³ (1980) ^19.

[32] SEELIG, ^{A. and} SEELIG, J., Biochem. 13 (1974) ^4839.