Displacement disorder in a liquid crystalline phase with cubic symmetry

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Displacement disorder in a liquid crystalline phase with cubic symmetry

Y. Rançon, J. Charvolin

To cite this version:

Y. Rançon, J. Charvolin. Displacement disorder in a liquid crystalline phase with cubic symmetry.

Journal de Physique, 1987, 48 (6), pp.1067-1073. �10.1051/jphys:019870048060106700�. �jpa-00210515�

Displacement ^{disorder in a} liquid crystalline phase ^{with cubic} symmetry

Y. Rançon and J. Charvolin

Laboratoire de Physique ^des solides, ^{associé au CNRS} (LA 2), bâtiment 510, Université Paris-Sud, 91405 Orsay, LURE, ^bâtiment 209, Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France

(Requ ^{le 16 décembre} 1986, accepté ^{le 17} février 1987)

Résumé.

On étudie la phase cubique ^du système lyotrope ^mono n-dodécyl ^éther hexa-éthylèneglycol (C12EO6)/eau par diffusion des rayons X ^aux petits angles (rayonnement synchrotron ^de LURE). ^L’indexa-

tion de 15 réflexions de Bragg ^nous conduit à proposer ^{une structure} cubique ^centrée dont le groupe d’espace

est Ia3d. La présence de diffusions diffuses dans l’espace réciproque démontre l’existence d’importantes

fluctuations de position dans le cristal. L’organisation ^{de ces} diffusions diffuses nous montre que ^ces fluctuations correspondent à ^un désordre linéaire dû à des déplacements ^le long des directions 111>. ^Ces déplacements ^sont discutés dans le cadre du modèle structural proposé habituellement pour les cristaux

liquides ^{de structure} cubique Ia3d, ^{et en relation avec les} propriétés mécaniques classiques des cristaux

liquides.

Abstract.

The cubic phase ^{of the} lyotropic system hexa-ethyleneglycol ^mono n-dodecyl ^ether (C12EO6 )/water is studied by ^small angle X-ray scattering, ^{at the} synchrotron ^radiation facility of LURE. The indexation of 15 Bragg reflections is strongly in favor of a body-centred ^{cubic structure} with space group Ia3d. The presence of diffuse scatterings ⁱⁿ reciprocal space demonstrates the existence of large fluctuations of

position ^{in the} crystal. ^The organization of these diffuse scatterings indicates that these fluctuations correspond

to a linear disorder caused by displacements along 111 > directions. These displacements ^are discussed within the framework of the structural model usually proposed for Ia3d cubic phases. Their existence can be understood considering the usual mechanical properties ^of liquid crystals.

1. Introduction.

Recent structural studies of monocrystalline ^lamellar phases ^of amphiphile/water systems have shown that the classical description ^{of these} phases ^{has to be} completed, by the introduction of various types ^of structural fluctuations [1]. ^{An obvious} question ^is

that of whether this is limited to lamellar phases ^or applies ^{also to} phases ^{with different} symmetries.

This is the reason why ^{we have} investigated ^the

structure of the viscous isotropic phase, presumably

of cubic symmetry, ^{of the} lyotropic system

hexa-ethyleneglycol ^mono n-dodecyl ^ether (Cl2EO6)/water. We chose this system, although

the structure of the phase ^{is not} precisely known,

because we noticed its ability ^to grow quite large monocrystals, ^a very favourable situation for a

complete structural study including the search for lattice fluctuations through ^the examination of dif- fuse scatterings. ^{We show here that this} phase ^has

indeed a body-centred cubic structure, with space group Ia3d, ^{and we} demonstrate the existence of an

original type of fluctuations in this system, ^corres- ponding ^to displacements parallel ^{to one} crystallo- graphic ^{axis. The static or} dynamic ^{nature of these}

fluctuations cannot be specified, owing ^{to the} very nature of scattering ^of X-rays by ^matter.

The phase diagram ^{of the} system ^is given ⁱⁿ [2]

and represented ⁱⁿ figure 1. The viscous isotropic phase lies in between the lamellar and hexagonal phases, ^at intermediate temperatures ^{and concen-} trations. It can be identified easily by observation under a polarizing microscope. ^The stage ^{of the}

microscope ^is black because of the isotropy ^{of the} phase which does not allow the observation of a

texture. Its crystalline ^{nature, as well as} the fact that it spontaneously ^forms quite large monocrystals, ^is strongly suggested by the observation presented ⁱⁿ figure ^{2. This} phase, ^{which is} optically isotropic ^and crystalline, ^{has most} certainly ^{a cubic} symmetry.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048060106700

1068

Fig. ^1.

^Phase diagram ^of C12E06/H20, from reference

[2] ; ^{the arrow} represents ^the sample ^under study.

We studied its structure by ^small angle X-ray scattering ^{at the} synchrotron ^radiation facility ^of

LURE (Laboratoire pour l’Utilisation du Rayonne-

ment Electromagndtique, Orsay). ^Our study ^was

made in two steps. ^{We first} determined the _space group from ^the analysis ^{of the} powder diagrams

obtained with samples prepared ^to present ^{a texture}

as polycrystalline ^as possible. ^{Then we} identified and characterized the diffuse scatterings by analysing ^the diagrams obtained with monocrystalline samples.

2. Experimental.

The non-ionic surfactant, hexa-ethyleneglycol ^mono n-dodecyl ^ether CH3- (CH2)11- (0-CH2-CH2)6-OH ^or C12E06, ^was obtained from NIKKO Chemicals Co., Ltd. A sample containing ^{38 % of water in} weight

Fig. ^2.

Photograph of lamellar monocrystals (illumi-

nated area) growing in the viscous isotropic phase (black area) under slow heating. ^{The common} orientations of all the edges ^{of these} monocrystals suggests that their growth

takes place ^{in a} large monocrystal ^{of cubic} phase.

was prepared by weighing ^the required ^{amounts of}

surfactant and water in a glass ^{vial which, after} sealing, ^{was left in an oven at 40 °C} during ^several

weeks for homogeneization. The concordance of our

phase sequence with that given ^{in the} original paper

[2], hexagonal -+ ^viscous isotropic -+ lamellar fluid

isotropic, ^{as the} temperature increases, ^{was checked} by observation of the textures under a polarizing microscope equipped ^{with a} heating stage, ^and by following ^{in a NMR} experiment ^{the deuteron} quad- rupolar splitting ^{of the doublet of} D20, ^which

Fig. ^3.

^{On the} left, ^« powder ^» diagram of the viscous isotropic phase. The first 8 circles are the only ^ones clearly

visible on this photograph. ^The scattering around the beam stop, ^{at the centre of the} photograph, ^{is due to} imperfect

collimation. On the right, indexation of the 15 circles ; ^{each one} is identified by h2 + k2 + ^{l 2.}

sequence ^was once more controlled by observing

them under the polarizing microscope.

The X-ray diagrams ^{were obtained on the} high

flux source of the synchrotron radiation of LURE with the small angle scattering spectrometer D16.

The equipment is that of a classical « monochromatic Laue camera » [3], ^with point beam collimation

(0.5 ^mm diameter). ^A helium caisson is placed

between sample and detector to eliminate parasitic scattering ^{from the air. A} photographic ^{film is used}

as detector to record bi-dimensional sections of the

reciprocal space. The parameters ^{of the} experiments

are : A ⁼ 1.564 A, _AA/A ^{= 2 x} 10-3, ^the sample ^to

film distance D ⁼ 250 _mm, and the range of scatter-

ing ^vectors (s

^{2 sin} 8 / À) ^{covered :} 7 x 10-3 A-’ --- s 80 x 10- 3 A-’

with As ⁼ 3 x 10-4 A-’. The temperature ^{of the} sample, 28 °C, ^is regulated by ^{a water} circulation in the sample ^holder, and measured with a ther-

mocouple ^{close to the} capillary.

The powder diagram ^{needed for} the determination of the space group ^are recorded with samples ^as polycrystalline ^as possible, grown ^{in their} capillaries by rapid heating ^{from the low} temperature phase,

and maintained by ^{a motor in constant slow rotation} (3 r.p.m.) around the long ^{axis of their} capillaries.

The crystal diagram ^{needed for} the search and study

of the diffuse scatterings ^are recorded with mono-

crystalline samples, grown by ^slow cooling ^{from the}

lamellar phase, ^{and oriented in the beam at defined} angles (read ^{on a} goniometer) by rotation of the

capillaries around their long ^axis.

3. Bragg reflections : structure [4].

Figure ^{3 shows the} powder diagram ^{of a} sample

heated rapidly from the low temperature phase ^into

the viscous isotropic phase ^{and contained in a} capillary rotating slowly ^{around its} axis normal to the beam. It exhibits a large number of spots distributed on 15 concentric circles. This is obviously

the diagram ^{of a} polycrystal ^{made of several} crystals

of rather large size, ⁱⁿ spite ^{of our efforts to decrease}

this size. The radii of these 15 circles give ^{access to}

the moduli (s ⁱⁿ Å - 1) ^{of 15 vectors of the} reciprocal

lattice of the crystal ^{which are} classified in table I with the estimations of the scattered intensities

(lobs) and their indexations (h, k, l ) ^{in a cubic}

lattice. This indexation has also been checked by measuring ^the angles between the vectors of the

reciprocal ^{lattice on} diagrams ^of monocrystals.

The direct lattice has a parameter a ^{= 118} A

reflections forbidden by symmetry ^are indicated by ^abs. lobs ^are

the observed intensities which were visually ^estimated

(vvs : extremely strong, ^{vs :} very strong, ^{s :} strong,

m : medium, ^{w :} weak, ^{vw :} very weak, ^{vvw :} extremely weak ; ^see Fig. 3).

It can not be face-centred F, because the first

observed spacing ^ratios (1, B/4/3, /7/3, B/8/3, J 10/31- are ^different from those of a lattice F

(1, J 4/3, B/8/3, /11/3). ^{It is most likely not a simple}

lattice P, ^{because no} reflection with h + k + 1 ^{= .} 2 n + 1 is observed _among the 15 reflections (in ^all

known P lattices, ^the reflections with h + k + 1

2 n + 1 appear for s values smaller than our maximal

one [5]). ^{It is therefore} body-centred I, ^{and as no} reflection violates the conditions Okl : k, ¹

² n;

hhl : 2 h + 1

4 n ; ^{h00 : h}

⁴ n, the space group ^is

Ia3d (see [6], space group ^No. 230, p. 707). ^Indeed

1070

for every other space group of the body-centred

cubic system, ^{there are} violations of reflection conditions.

This is the space group ^most commonly ^recorded

for cubic liquid crystalline phases. Anhydrous [7] ^or hydrated [8] phases ^of amphiphilic molecules have been already ^{studied and a model of structure was} proposed by ^{V. Luzzati. It} consists of two inter- woven, but not connected, networks of rods con-

nected three by ^three. These rods are the axes of water channels for low degrees ^of hydration, ^{and the}

two networks are separated by ^{a film of} amphiphilic

molecules. For higher degrees ^of hydration, ^{the rods}

are the axes of cylinders ^of amphiphilic ^molecules

and the two networks are separated by ^{a film of}

water. The model of structure in the latter case is

represented ⁱⁿ figure ^{4. We} adopt it here because of the location of the cubic phase ^{on the} hydrated ^side

of the lamellar phase ^{in the} phase diagram ^and

because of the qualitative agreement existing ^be-

tween the intensities of our reflections (Table I) ^and

those reported for the cubic phases ^of KC12/H20

and C16 T AB /H20 [8].

4. Diffuse scatterings : fluctuations [9].

Figure ^{5 shows two} diffraction patterns ^{of a mono-}

crystal obtained with a sample ^cooled slowly ^from

the lamellar phase into the cubic phase. ^The patterns

are recorded for two different orientations of the

capillary around its long axis normal to the beam. In both cases we observe a limited number of Bragg spots organized ^{in a} very symmetric ^manner. They

can be indexed considering their distances from the centres of the diagrams ^{and the relative} angles ^of

their polar vectors, which ^are the s vectors of the

reciprocal lattice. The two patterns ^{are indeed two} sections of the reciprocal space of the monocrystal

with a common axis close to axis [152], ^and making ^a

dihedral angle ^{of 42°.}

As seen in figure 5, ^{and more} generally ^whatever

the section of the reciprocal space considered, elongated ^diffuse scatterings, ^or straight ^diffuse streaks, parallel ^{to one same} direction, ^are apparent in the vicinity ^{of three sets of} Bragg spots :

(21 1 ) , {332} ^and (431 ) . They reveal the existence of lattice imperfections ^{in the} structure. As no

diffuse scattering ^goes through ^{the centre of the}

patterns, ^{we can} say that these imperfections ^corre- spond ^to fluctuations of position ^{of the reticular} planes ^{of the} crystal, ^{not to} fluctuations of density (see [9], ^p. 573).

A qualitative study of these diffuse scatterings ⁱⁿ reciprocal space ^can be undertaken. If we move the

sample ^{out of a} Bragg reflection position, ^for

instance by rotating ^the capillary by ^{a few} degrees,

some Bragg spots disappear ^{from the} pattern, ^{while ’} their neighbouring diffuse streak is still apparent.

This indicates that the diffuse scatterings ^{do not have} only ^{a linear} extension, ^{but also a} planar ^{one. In}

other words, ^the parallel ^{diffuse streaks, observed}

on each pattern, ^are indeed sections of parallel

diffuse planes ⁱⁿ reciprocal space. The orientation ^of these diffuse reciprocal planes ^can be evaluated by geometrical construction from the orientations of the diffuse streaks in each ,pattern, i.e. in each section of the reciprocal space. Such ^an analysis

shows that the diffuse scatterings ^{are indeed} parallel

diffuse planes ^{normal to} (111 ) directions in recipro-

cal _space.

Quantitative informations about these diffuse

planes ⁱⁿ reciprocal space ^can be obtained from the

Fig. ^4.

Perspective view of the model of structure proposed ^{for a} body-centred ^cubic phase with space group

Ia3d, ^{from reference} [7]. Two cubic cells lie along ^{the vertical} axis. The square in full line represents the limits of the horizontal face of the cell (parameter a ^{= 118} A). The black and white rods are the axes of amphiphilic cylinders. ^Dotted

lines are projections of the rods on the horizontal plane. ^{On the} right, ^a knot built with 3 amphiphilic ^rods.

Fig. ^5.

^{On the} left, ^two patterns ^{of a} monocrystalline ^{domain of cubic} phase ^which correspond ^{to two} sections of the

reciprocal space. On the right, the indexation of the patterns ; ^the dotted lines simulate the diffuse streaks and the white spots correspond ^to Bragg’s ^{spots in the} vicinity of the Ewald’s sphere.

patterns : the transversal extension /- 1 of the streaks, which is the thickness of the diffuse planes,

is not larger than the resolution and therefore _very small compared ^to a *, parameter ^{of the} reciprocal

cubic lattice (a * ⁼ a -1 ) ; ^the longitudinal ^extension L -1 of the streaks, which is the extension of the diffuse planes, ^{is not} infinite but is about a * ; finally

these {111} ^diffuse planes ^{are stacked} along ^their

normal with a periodicity, equal ^to 2 a */ B/3 ⁱⁿ

reciprocal space.

Such an arrangement ^of parallel ^diffuse planes ⁱⁿ reciprocal space, without such ^a plane ^{around the}

centre of the pattern, reveals the existence of a

linear disorder in real space (see [9], p. 508), ^caused by displacements along parallel periodic (111) ^rows

of the crystal.

Quantitative informations about this disorder in real space ^can be deduced from above : the displace-

ments along ^a (111) ^{row are} correlated over a

length I > a (1 ^{$: 400} A ), ^{and from one} (111) ^row

to another parallel ^one, they ^are quasi-uncorrelated

since L - a ; finally ^the periodicity along ^{these rows}

is equal ^to (2 a * / J3 )-1 = a 3/2. ^{At this} stage ^of the study, ^{we do not} estimate the amplitude ^{of the}

fluctuations. Indeed, ^{this would} require ^measure-

ments of scattered intensities which can not be accurate here because we were forced to overexpose

some Bragg spots ^{in order to} observe the diffuse

scatterings.

We now discuss the possible ^{nature of these}

fluctuations within the framework of the structural

model proposed ⁱⁿ figure ^4.

1072

5. Discussion.

We have analysed the observation of diffuse planes periodically ^stacked along (111) directions in re-

ciprocal space ^as showing ^{the existence in real} space of parallel displacements ^of periodic (111) ^rows

with periodicity a B/3/2. The static or dynamic

nature of these fluctuations cannot be specified, owing ^{to the} very ^nature of scattering ^of X-rays by

matter. ,

An inspection of the model of structure presented

in figure ⁴ shows indeed that ( 111 ) ^{axes of the} crystal play a particular ^{role in the structure and that}

the stacking ^of amphiphilic ^rods along them consti- tute the rows experiencing ^the displacements. ^As

shown in figure 6, ^{these axes} go through ^{the knots}

where the amphiphilic ^{rods of each network are}

connected three by three, ^and they ^are perpendicular

to the planes containing ^{these knots ;} along ^{each of}

these _axes, the knots belong alternatively ^{to one}

network or the other and the periodicity ^{of this} stacking, ^{half a} diagonal of the cubic cell, ^is exactly

the one we found for the periodic ( 111 ) ^rows.

This leads us to see these rows as the periodical stackings ^{of knots of three} amphiphilic ^rods.

Figure ^{7 shows the} organization ^{of the rows of knots}

in the cubic cell. It is quite visible that one particular

row, for instance that along ^the diagonal ^{of the cell,}

can slip within the entanglements ^formed by ^the

others. However, ^{as the rows are connected} together by ^the amphiphilic ^{rods, there must be a} restoring

force for such a displacement, associated with the deformations of the amphiphilic ^{rods under the}

shear caused by the relative displacements ^{of two} neighbouring ^rows. Estimates of relative strengths ^of

the interaction within one row and between parallel

rows can be obtained from the above correlation

lengths 1 and L. The correlation length ^{of the} displacements along ^{one row} is rather large, ¹ > a,

suggesting ^a rather low compressibility along ( 111 )

directions, ^which implies ^a rigid stacking ^{of the}

knots. The correlations between displacements ^of parallel ^{rows are} quasi-inexistent, ^{L = a,} suggesting

a rather low resistance ^to shear in planes parallel ^to ( 111 ) directions, ^which implies ^some flexibility ^of

the amphiphilic ^rods.

The first point, ^low compressibility along (111) directions, ^{can be} compared ^to the solid-like behavi-

our of lamellar or hexagonal phases ^{submitted to a}

stress normal to the amphiphile/water interfaces _as, here, in the cubic phase, ^the ( 111 ) directions are

normal to the interfaces at the knots. The second

point, ^{low shear modulus in} planes parallel ^to ( 111 ) directions, ^{can be} similarly compared ^{to the} liquid-like behaviour of bent lamellae or cylinders ^of

the above phases as, here, the shear of the rows

involves a bending of the rods in first ^{order. We}

therefore find a local mechanical behaviour quite

Fig. ^6.

Perspective view of the structure shown in

figure ⁴ along ^{an axis} [111]. ^{The knots are located by their}

coordinates in the unit cell.

Fig. ^7.

Perspective view of the cubic cell with the

( 111 ) ^{rows of knots} represented ^as heavy ^{lines. « a » and}

« b ^» are 2 knots of coordinates (3/8, 3/8, 3/8) ^and (5/8, 5/8, 5/8). The dotted lines, ^{which connect them to knots of} neighbouring ^rows, simulate the amphiphilic ^rods.

compatible ^{with that found in} phases ^with simpler

structures.

Acknowledgments.

We are particularly ^{indebted to A.} M. Levelut

(Orsay) for her critical guidance ^{in the} analysis ^of

the diffuse scatterings. ^We thank A. Yeliaskhova

(Sofia) ^{and Y. Hendrikx} (Orsay) ^{for their} help ^{in the} early stages ^{of the} study. ^{Thanks are also due to S.}

Megtert (Orsay) for advice and experimental ^assist-

ance on spectrometer D16 ^at LURE. This work was

partially supported by ^{PIRSEM of} CNRS,

« GRECO microemulsions ».

KEKICHEFF, P., CABANE, ^{B. and} RAWISO, M., ^J.

Physique ^{Lett. 45} (1984) ^L-813.

HENDRIKX, Y., CHARVOLIN, J., KEKICHEFF, ^{P. and} ROTH, M., ⁱⁿ preparation.

[2] ^{MITCHELL, D.} J., TIDDY, G., WARING, L., BOSTOCK,

T. and McDONALD, ^M. P., ^J. Chem. Soc.

Faraday Trans. 1 79 (1983) ^975.

[3] COMÈS, R., LAMBERT, M., LAUNOIS, ^{H. and} ZELLER,

H. R., Phys. ^{Rev. B 8} (1973) ^571.

[4] ^{For this} paragraph ⁱⁿ general, ^see LUZZATI, V., ⁱⁿ Biological Membranes, ^edited by ^D. Chapman (Academic Press, ^New York) ^1968.

(1983) ^612.

[6] International Tables for Crystallography, ^Vol. A, edited by ^{T. Hahn} (D. ^Reidel Publishing ^Com- pany) ^1983.

[7] LUZZATI, V. and SPEGT, ^P. A., Nature 215 (1967) ^701.

[8] LUZZATI, V., TARDIEU, A., GULIK-KRZYWICKI, T., RIVAS, ^{E. and} REISS-HUSSON, F., Nature 220

(1968) ^485.

[9] ^{For this} paragraph ⁱⁿ general, ^see GUINIER, A.,

Théorie et technique ^{de la} Radiocristallographie,

Chap. ^XIII (Dunod, ^2nd ed.) ^1956.