Brownian motion of screw dislocations in the hexagonal Blue Phase

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Brownian motion of screw dislocations in the hexagonal Blue Phase

M. Jorand, P. Pierański

To cite this version:

M. Jorand, P. Pierański. Brownian motion of screw dislocations in the hexagonal Blue Phase. Journal

de Physique, 1987, 48 (7), pp.1197-1205. �10.1051/jphys:019870048070119700�. �jpa-00210544�

Brownian motion of screw dislocations in the hexagonal ^{Blue Phase}

M. Jorand (**) ^{et P.} Piera0144ski (*)

(*) Laboratoire de Physique ^des Solides, ^Bât. 510, Faculté des Sciences, ⁹¹⁴⁰⁵ Orsay Cedex, ^France (**) Institut des Sciences Physiques, Université de Lausanne, 1015 Lausanne-Dorigny, ^Suisse

(Requ le 13 octobre 1986, révisé le 2 f6vrier 1987, accept6 ^{le 4 mars} 1987)

Résumé.

En diminuant rapidement ^le champ électrique, la transition de la Phase Bleue hexagonale

bidimensionnelle (BPH2D) ^{à la} Phase Bleue hexagonale tridimensionnelle (BPH3D) ^fait apparaître ^des

dislocations vis dans certains cristallites. Certaines de ces dislocations s’échappent des cristallites en des temps plus ^{ou moins} longs ^{et l’on} analyse ^ce phénomène ^{en terme de mouvement brownien en} regardant l’influence du champ électrique ^{sur le} temps de diffusion.

Abstract.

By quickly decreasing the electric field, ^{we observe a} transition from the two-dimensional

hexagonal Blue Phase (BPH2D) ^to the three-dimensional hexagonal Blue Phase (BPH3D) and the appearance of screw dislocations in some crystallites. ^{After more or less} long time, ^some of these dislocations may escape from the crystallites. ^We analyse ^the phenomena through the Brownian motion and consider the influence of the electric field on the diffusion time.

Classification

Physics ^Abstracts

61.30

61.70G

61.70J

61.70L

Introduction.

In cholesteric liquid crystals ^{with a short} enough pitch, ^the transition from the isotropic liquid phase (I.L.) ^{to the usual} helicoidal phase (C) is mediated

by Blue Phases [1].

In isotropic conditions, depending upon tempera-

ture ( T) ^and chirality (K ) ^{three different Blue}

Phases can occur in the phase diagram (T, K ). ^Two

of them, BP I and BP II, ^have respectively ^cubic symmetries 08(14132) ^and 02 (P4232), ^{while the}

third _one, BP III is amorphous (isotropic). ^Hor-

nreich et al. [2-3] ^have pointed ^out theoretically ^that

in materials with a large positive dielectric anisotropy (8a

811 II

£ 1 ) ^{submitted to a} strong enough ^electric

field E, ^uniaxial anisotropic Blue Phases of, ^for example, hexagonal symmetry, ^{should be more} favorable than these cubic or isotropic Blue Phases.

This theoretical prediction ^{has been confirmed} by

studies of mixtures of a cholesterogene ^{CB15 with a}

nematic liquid crystal ^{E9, where three such uniaxial} phases have been discovered. One of them, ^called

BPX has been shown ^to have a tetragonal symmetry

Dlo(I4122) [4]. ^{The two others called BPH3D and} BPH2D are hexagonal [5].

The present paper ^{is devoted to a} study ^{of screw}

dislocations occurring ⁱⁿ monocrystals of the Blue Phase BPH3D in a mixture of 55.8 % of CB15 in E9.

We first present ^the experimental set-up ^{and the} conditions of these observations and then we propose

a qualitative analysis ^{of the motion} of these dislo- cations.

1. Experimental part.

1.1 EXPERIMENTAL SET-UP.

The sample ^was

introduced by capillarity ^{into a cell made of two} glass plates Sb ^and Su, coated with semi-transparent 1. ⁰ electrodes and separated by ¹⁰ )JLm mylar

spacers (Fig. 1). ^{The upper} glass plate ^{was 3 mm}

thick and pressed against ^{the aluminium} piece P2 by ^two metallic elastic lamellas LI ^and L2 ^which

were also used as electrodes. The lower glass plate

was a microscope ^{cover slide. The} In 0 electrode of this lower glass plate ^was connected, through ^a conducting epoxy, ^{to a narrow} strip ^of In ^{0 electrode}

in contact with L2, ^{on the} upper plate.

The temperature ^{of the} sample ^{was controlled} carefully by ^two independent regulations using ^Pel-

tier effect elements in thermal contact with the aluminium pieces PI ^and P2. Using ^this system ^a small controlled vertical temperature gradient ^was

introduced in the sample ^{in order to induce} preferen-

tial nucleation and growth of Blue Phase crystals ^on

the lower slide Su. ^{The oil immersion} objective (Mo)

was a part ^{of an inverted} metallurgical reflecting

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048070119700

1198

Fig. 1.

Experimental ^{cell. The} sample is introduced between two glass plates Sb and Su and the electric field is

produced by ^two electrodes Ll ^and L2. ^The light ^reflected by ^the crystallites is observed through ^an oil immersion

objective (Mo).

microscope. ^The monochromatic light ^{beam was}

used for illumination.

1.2 OBSERVATIONS.

1.2.1 BP II.

In the mixture of 55.8 % of CB15 in

E9, ^{BP II} crystals ^are nucleating ^{in the} isotropic liquid phase ^{when the} temperature ^{is lowered below} 25.4 °C. When the nucleation takes place ^{in the}

presence ^{of an} electric field (U « ²⁴ [V]) ^{all BP II} crystallites ^are oriented with their four fold axis, ^for example (001), parallel ^{to the field.} (As pointed ^out

in reference [5] ^this orienting ^{action of the field is}

due to an anisotropy of the nonlinear dielectric

susceptibility X afJ’Y8)’ ^{For the wave length A adjusted}

to the Bragg reflection from the (001) planes, ^the crystals ^{of BP} II appear ^as brilliant square shapes ^on

a dark background ^{of the} surrounding isotropic liquid phase (Fig. 2a). ^{The wave} length À001 ^of circulary polarized Bragg reflection was measured with the aperture diaphragm closed in order to use

only rays parallel ^{to the} optical ^{axis 0 at normal}

incidence on planes (001). ^We have observed that

A 001 ^{is field} dependent : ^{it increases} with the field

intensity.

1.2.2 BP II ^- BPH3D transition.

At the applied voltage U = U1 ^the transition from the Blue Phase II’

to the Blue Phase BPH 3D takes place. The character- istic features of this transition are the following :

1) The square shapes ^{of BP II} crystals (Fig. 2a)

are replaced by hexagonal ^ones (Fig. 2c) ^of approxi- mately ^{the same size} (about ^{20 J.Lm of} lateral dimen-

sion).

2) ^{In a narrow interval} A U -. 2 V around Ul,

coexistence of square ^and hexagonal shapes ^{is ob-}

served (Fig. 2b).

3) ^{For a} given crystal, orientations of square and

hexagonal shapes ^are related : among ^three pairs ^of opposite ^sides composing ^the hexagonal shape ^{one is} always parallel ^{to one of two} pairs composing ^the

square shape (Figs. 2a-2c).

4) ^{There is no} measurable discontinuity ⁱⁿ ÀOOI (V) at V

U, ^{for this} particular sample ^{but in}

other mixtures with higher concentration of the

cholesterogene, ^{a small} discontinuity ⁱⁿ A 001 ^occurs

at this transition.

1.2.3 BPH 3D --,. BPH2D transition.

Upon ^further

increase of the voltage, ^{the wave} length ^of (001)

reflection continues to increase but, simultaneously,

its intensity weakens. Direct observations in the

microscope reveal that this decrease of intensity corresponds ^to growing fluctuations occurring ⁱⁿ crystals. ^{At U}

U2 ^the Bragg reflection ÀOOI disap-

pears completely ^and consequently ^the crystals ^are

no more visible in reflected light. Using transmission illumination through ^a hole made in the upper aluminium plate P2, ^{we have} found that for U > U2

the crystals preserve their hexagonal shapes.

1.2.4 Creation of ^screw dislocations in BPH3D. -

When the field is lowered suddenly ^below U2, ^the Bragg reflection is immediately restored but the

original aspect ^of (perfect) ^BPH3D crystals ^{is modi-}

fied by ^{an invasion of screw} dislocations [6].

These dislocations are very mobile and interact

obviously each with the others. In crystals ^with

lateral dimensions about 20 J.Lm, adjacent ^dislo-

cations with opposite Burgers vector attract each other and form pairs ; ^{then in a} given pair ^the

distance between dislocations decreases and finally they annihilate. This first stage ^{of «} annealing ^{» is} completed ^{in about 1 min and as a result, in} crystal-

lites where the number of dislocations was odd, ^one dislocation persists (Fig. 3). ^{Such a} solitary ^dislo-

cation executes Brownian motion in the vicinity ^of

the crystal ^centre. Longer observations of its behavi-

our lead to the conclusions that the dislocation evolves effectively ^{in a} potential ^well having ^the shape ^{of a} reversed W with rounded corners (---) :

it has a minimum at the centre, ^a circular maximum for ^{/- 2013} R /2 (where ^R is the radius of the crystal) ^and

falls down for r > R/2.

The dislocation which approaches ^{the border of}

the crystal ^{seems to be} attracted toward it and,

Fig. ^2.

^These photographies ^{show some} crystallites ^of

Blue Phase (white shape ^{due to the} light reflection) ^{in an} isotropic liquid (dark ground) : a) ^square shapes ^{of the}

cubic phase ; b) coexistence of the ^two phases ; c) ^hexa- gonal shapes ^{of the BPH3D. In all cases} the lateral dimension ^{of the} crystallites is about 20 )JLm.

finally, escapes ^{from the} crystal. The escape of

solitary dislocations is the processus responsible ^for

the second stage ^of crystal annealing. ^{In contra-}

distinction with the first stage, this second stage ^is

very slow and takes typically ^{between one and few}

hours. Figures 3a, b, ^{c show the} chronological

evolution of the sample ^{with the} escape of some of the dislocations. It is clear that several solitary

dislocations succeeded ^to escape ^from their crystals

but all other are still emprisonned by ^{the central} potential ^well.

From a theoretical point ^{of view,} since other papers ^are devoted ^to the study ^{of the} symmetry properties ^{of these new} phases [3-5], ^{we restrict}

ourselves in this paper ^to the problem ^{of the} escape of the dislocations from their crystallites.

In order to get ^a qualitative explanation ^{of this} behaviour, ^we simplify ^the problem by replacing ^the

real crystal by ^{an ideal} isotropic ^{medium. Moreover,}

the frequency ^{of the} applied voltage ^was

with a distance d - 10 _J.Lm between the electrodes :

we then have lld 2 > W 2/C2 ^{where c is the velocity of}

light. The electric field wave length ^{is then much} larger ^{than the} distance d and ^we consider an

uniform electric field between the electrodes. On ^the other hand we introduce the relaxation time ^of

crystallites by ^{T = q} Ig _ ^{10- 3 s} where q and g ^are

1200 JOURNAL DE PHYSIQUE

the viscosity ^{and the elastic} modulus of the crystalline

medium (see ^Sect. 2.2.1). ^This relaxation time is

greater ^{than the} excitation time 2 7r /w ^{and we then}

consider a static electric field.

2. Theoretical part.

The observed random displacement ^{of a screw}

dislocation inside an hexagonal ^{cell is} explained by introducing ^the Brownian motion.

So we present ^{first a} general description ^{of the}

Brownian motion for a dislocation then we apply ^the general theory ^to particular ^cases by taking ^into

account the finiteness of the system. In the last section we introduce the effects of the electric field without which the hexagonal ^Blue Phase would not exist.

2.1 BROWNIAN MOTION OF A DISLOCATION.

The Brownian motion acts on a dislocation in the same

way ^{as on} any particle ^{in a} suspension. Indeed, ^as

shown for instance by ^{Hirth and Lothe} [7] ^{one can}

describe the dynamic ^{of a} dislocation by considering

it as a real object ^{with a mass m :}

where W is the dislocation _energy, Ct ⁼ (tt lp )1/2 ^is

the transverse waves velocity ^{in a medium of} density

p and shear modulus _JL.

In a infinite medium and without external forces,

the theory ^of Brownian motion (see ^{for instance Reif} [8]) ^{leads to the well known result:}

where

r : is the deviation of the dislocation from its

equilibrium position,

t : the time,

D : the diffusion coefficient :

with

K : the Boltzman constant 17 ^{: a} characteristic viscosity

h [cm] : ^{a constant} depending ^{on the} shape ^{of the} moving object.

From the experimental conditions, ^a dislocation inside a crystallite ^of BPH3D is parallel ^to the electric field E and moves perpendiculary ^to itself. We may then approximate ^it by ^a straight cylinder of height ^L

and radius a : the coefficient h in (3) ^writes then, ⁱⁿ

the limit L > a [9] :

The velocity ^{V in the} logarithm may be estimated by

the equipartition ^{theorem :}

V

(KT/m )l2 where m is given by (1) .

Relation (3) gives ^the diffusion coefficient without any external forces, ^{but a} dislocation enclosed in a

finite domain will have an interaction with the walls of this domain in order to satisfy ^the equilibrium

condition on the free boundary (the ^{total stress} equal ^to zero). So the energy of the dislocation will be modified in regard ^{to the case of an infinite}

medium.

If we replace ^the hexagonal shape by ^{a circle} of

radius R, the energy of the dislocation will depend ^of

its position inside the circle. If å W is the energy

required ^{to leave the circle,} the diffusion coefficient will write [10] :

and the diffusion time

where

2.2 DISLOCATION IN FINITE MEDIUM. - We have

some difficulties to evaluate AW because the exper- imental conditions prevent ^{us to make} simplifications (R - L). ^{Moreover, for} getting ^{an order of} mag- nitude of _T, it is necessary ^to know nearly exactly

AW. So our task is cumbersome and we simplify ^it by considering ^two limiting ^{cases :} a) RIL ..c ^{1 and} b) RIL > ^{1, from} which the intermediate case can be discussed.

2.2.1 1St case : RIL ^1.

^{We consider a} cylinder

of radius R and of height ^{L. A screw} dislocation,

whose axis is parallel ^{and at a distance A of the} cylinder axis, ^{has an} energy [11] :

where b, ^the Burgers ^{vector - lattice constant.}

As shown in figure 4, W(A ) ^{has a} maximum for A ^{max =} 0.54 R and the dislocation is in a metastable

equilibrium ^{for A = 0.} For A >. A max, ^the dislocation is attracted near the wall by ^its opposite image (see

for instance Ref. [11]) and leaves the circle. The

required energy is given by ^AW

W(A max) _ W(0).

We evaluate AW with the following values of the

parameters [12, 13]

1202

Fig. ^4.

Energy ^{of a} straight dislocation in a cylinder ^of

radius R and height ^{L as a function of} A /R, ^{where A is the}

distance from the dislocation to the cylinder ^{axis. The case} RIL -- ¹ is considered here and there is no electric field.

A is dimensionless and is proportional ^{to the} energy ^{of the} dislocation :

Since the thermal energy ^{KT - 4.3} x10-14 _erg we

have finally got the ratio: å W / kT - 230 ^{and a}

diffusion time (5) :

Under these geometrical conditions (L > R - 10- 3 cm), the model does not authorize _any dislo- cation to slip ^out.

2.2.2 2"d case : R/Z. > ^1.

^{In this limit,} Eshelby

and Stroh [15] have evaluated the energy of the dislocation when it is placed ^{at the} origin A ^{= 0.}

For 0 :0 a L and R -+ _oo, they got the result :

We first observe that this energy is independent ^{of R}

at the contrary ^{of the} preceeding ^case. This is due to the fact that the stress field around the dislocation propagates ^a distance of the order of L from the dislocation axis [15]. ^For A -- (R - L) - R, ^the

dislocation does not interact with the lateral walls, and can move randomly ^{inside the} circle in a

constant potential (Fig. 5). ^{Near these walls, the}

dislocation ^will be attracted by ^its opposite image.

The details of the potential in this domain are not known but we could imagine ^a potential ^{barrier as in}

the first case but much less high ^{since L} is smaller in that second case. Nevertheless, ^{we do not consider}

Fig. ^5.

^{As in} figure 4 but for the case RIL > ^{1. The}

dotted and continuous lines may be possible ^{near the}

walls.

this point ^{due to the} great mathematical difficulties and we evaluate a minimum diffusion time given. by (6) ^{since d WII = 0.}

With

We get ^a reasonable order of magnitude authorizing

an interpretation ^{of the} experimental observations

by the above model. By comparing ^{the cases 1 and} 2,

we see the essential role played by ^the length ^{of the} dislocation, ^or the thickness of the sample.

2.3 THE INFLUENCE OF THE ELECTRIC FIELD.

In the above discussion, ^{we do not have} introduced the electric field without which this hexagonal phase

would not exist. Since this electric field is parallel ^to

the dislocation axis, ^{in an} infinite medium the interaction energy does ^not depend ^{on the} position

of the dislocation inside this medium. On the contrary, ^{in a finite} medium, ^{the stress} field of the dislocation depends ^{on its} position in this medium and so does the electric energy of interaction, giving

then a contribution to the diffusion process.

In a electric field E, the free energy of the crystal

writes [5] :

where sq ^{is the} anisotropic dielectric tensor E ⁼ (0, ^0, E ) ^{is the constant} applied ^electric

field in Cartesian coordinates (x, y, z).

Consider a perfect crystal (without dislocation)

with the crystallographic planes oriented such that

We then introduce in this crystal ^{a screw} dislocation whose axis is parallel ^{to E.} It induces a slight ^{tilt of}

the crystallographic planes by ^an angle 0 (x, y) ^and

the electric _energy writes :

By subtracting (12) ^from (13), ^we get the modifi- cation of the electric _energy density ^{due to the}

dislocation :

with the condition _{8 -L} :: 81 in order to satisfy ^the

minimum energy condition for 0 ⁼ 0.

The problem ^{now is to} determine 0 (x, y ) ^{and we}

consider first the simple ^case (Sect. 2.2.1) ^{of a screw}

dislocation in a cylinder ^of radius Rand ^« infinite ^»

height ^{L. For a} dislocation at a distance À from the

cylinder axis, ^the displacement field writes [7, p.

69] :

with an elastic energy :

The surface of the deformed crystallographic plane

is described by

with a normal:

The tilt angle ⁹ (x, y ) ^is given by :

and

since

By introducing (20) ⁱⁿ (14) ^and (11), ^we get ^the

raising of the electric energy ^{due to} the dislocation :

which has exactly the form of (16).

The elastic modulus thus depends ^{of the field E} according ^{to :}

With the experimental ^values [5] :

The electric field changes the elastic modulus by ^an

order of magnitude ^and greatly diminishes the diffusion time since (by (16))

It is nevertheless ^more interesting ^to consider real

cases by taking ^{into account the finite} height ^{of the} sample ^{and this} will be done through ^{the two} limiting

cases of the section 2.2.

2.3.1 1st case : RIL ^1.

^{We introduce a screw}

dislocation in a cylindrical medium of radius R and

height ^{L. For a} dislocation at a distance A from the

cylinder axis, ^the displacement field writes [7, p. 61-

71]

or

We first consider the crystallographic plane ^{z = c =}

constant. After the introduction of the dislocation, its deformed surface is described by :

By following ^relations (18) ^and (19), ^we get ^{the tilt}

angle ⁸ (x, y ) :

where UZ ^is given by (23).

From (11) ^and (14), ^the variation of the electric

energy writes :

1204

The integral ^is easily performed, ^{since the}

numerator depends ^{on x and} y, with ^a result given by

Eshelby [11] and the denominator is a fonction of z

only

The self energy of the dislocation is that case is given by (7) ^{and we} get ^the variation of _energy due to the introduction of a dislocation in a electric field by summing (7) ^and (27) :

where

We still have a change of the elastic modulus given by :

On the contrary ^{of the} preceding ^{case, this} change

does not affect the energy ^{on a whole} but concerns

Fig. ^6.

^{As in} figure ^{4 for the case} RIL .c 1 but with an

electric field E. The energy of the dislocation has been

reported for various values of the parameter y (see ^section

where 81 - e, is the dielectric

anisotropy of the Blue Phase and u ^{is the} shear modulus of the crystal.

only ^{the first term of} (28). ^{This is} reasonable if we

remember that this term comes from the interaction of the dislocation with the lateral walls which

produces ^a displacement Uz parallel ^{to E and then a} non-vanishing ^{work in} the electric field. On the

contrary ^{the last term comes} from end effects which

produces ^a displacement Uo perpendicular ^{to E with}

a null electric work. This has deep consequences ^on the shape of the energy ^as shown in figure ^6.

With the numerical values given ^{in section} 2.2.1,

one finds 0 -- « ,10-1 ^or _arctg ^{a - a.} The curves

in figure ⁶ have then been drawn for different values of the parameter y

which gives the ratio between electric and elastic energy. By varying ^{y, one} gets ^two typical shapes ^for

the energy :

1) ^{For y 1, the} potential barrier found in 2.2.1

(Fig. 4), ^has completely disappeared. ^More great ^{is y}

and more quickly ^the dislocation is attracted by ^the

lateral walls.

Indeed one finds :

and the diffusion time decreases exponentially ^as y raises.

2) ^{For 0 = y 1, we} still find a potential ^barrier

but much less high than that of figure ⁴ (without E).

One gets easily ^an analytical expression ^for

Taking ^{for instance y}

0.5 and the numerical values of section 2.2.1 one finds a diffusion time (5) :

This time is still incredibly great ^{but much less} than without E (8). So in the case RIL ..r. 1, ^the

electric field changes completely ^the behaviour of

the dislocation. Our model reproduces qualitatively

the observations but the physical parameters ^{of the} sample ^{are not} sufficiently well known to authorize

more quantitative comparison.

2.3.2 2nd ^{case ;} RIL > ^1.

^{As it was} said in section

2.2, ^{the stress field around} the dislocation does not

depend ^{on its} position ^inside the medium except

near the wall. If ^we add an electric field, the electric energy will be ^{a constant} in this domain, ^{and it is}

then not active in the diffusion process (see Fig. 5).

Near the lateral walls, ^{the stress field} depends ^{on the}

dislocation position and the above conclusions are

wrong. Unfortunately, ^the mathematical difficulties

are very important ^{and we do not have} analytical

results. As a consequence ^of 3.1, ^{we can} imagine ^a sharp decrease of the potential ^near the walls and so

the electric field facilitates the disappearing ^{of the}

dislocation.

The diffusion time is given by (6) ^and section 2.2.2

but in this limit, the electric field does not appear.

2.3.3 3rd case : R - L.

The shape of the energy for this intermediate case is probably given by ^{one of}

those drawn in figure 6, ^{since the} parameter ^y may

give very different profiles.

For _y 3= 1, ^the energy is ^a monotonous decreasing

function of A and an increase of the length ^{L of the}

dislocation (or sample thickness) diminishes the diffusion time.

On the contrary, ^{for y} = 1, ^the potential ^barrier prevents ^the escape ^{of the} dislocation and the diffusion time raises when L increases.

But in all _cases, the qualitative ^{behaviour in}

function of E is the same : the diffusion time T

diminishes when the electric field E increases.

3. Conclusion.

The observation of the diffusion time of screw

dislocation in BPH3D _{may be} considered as one

among other possibilities ^to get ^the values of the elastic modulus, ^{of the} viscosity ^{or of the} anisotropic

dielectric constant. These are unfortunately ^not sufficiently well known to authorize numerical com-

parison between the experimental observations and

our crude model but the observed diffusion times of

one or few hours are in the interval given by ^the limiting ^{cases :}

So the model is not in contradiction with the observations and even without exact results, ^the phenomena ^is interesting and may be considered as an approach ^{to the more} general problems ^{of defects}

in hexagonal Blue Phase.

Acknowledgments.

We thank A.-J. Koch and F. Rothen for fruitful discussions as well as M. Babin and S. Herranz for technical assistance. We also thank the referee for his helpful ^comments.

References

[1] ^{For a review on the} subject, ^see for instance : BELYAKOV, V. A. and DMITRIENKO, ^V. E., Sov. Phys. Usp. ²⁸ (7) (1985) ^535.

H. STEGEMEYER, H., BLUMEL, Th., HILTROP, K., ONUSSEIT, ^{H. and} PORSCH, F., Liquid Cryst. ¹ (1986) ^3.

[2] HORNREICH, ^R. M., KUGLER, ^{M. and} SHTRIKMAN, S., Phys. Rev. Lett. 54 (1985) ^2099.

[3] HORNREICH, ^R. M., KUGLER, ^{M. and} SHTRIKMAN, S., « Proc. of the Winter Workshop ^{on Colloidal} Crystals ^», Les Houches 1984, ^J. Physique ^Col- loq. ⁴⁶ (1985) ^C3-47.

[4] PIERA0143SKI, ^{P. and} CLADIS, ^P. E., Phys. ^{Rev. A 35} (1987) ^355.

[5] PIERA0143SKI, P., CLADIS, P. E. and BARBET-MASSIN, R., ^J. Physique ^{Lett. 46} (1985) ^{L-973 and} Phys.

Rev. A 31 (1985) ^3912.

[6] Steps ^{are observed at} the interface of the crystallites

and the glass plates ^{and the} Burgers ^{vector b} characterizing ^these steps ^is perpendicular ^{to the} glass plates. On the other hand, ^{the dark} spot appearing ^{in the} crystallites (see Fig. 3) may be considered as the melted crystal ^{of the core of}

the defects. The defects then appear ^as 2-dim.

with an axis 03BE perpendicular ^{to the} glass plates.

We then have b~03BE which is the definition of a screw dislocation.

[7] HIRTH, J. P. and LOTHE, J., Theory of Dislocation,

2nd ed. (John Wiley ^and Sons) 1982, p. 188.

[8] REIF, F., Fundamentals of statistical and thermal

physics (McGraw-Hill) 1965, p. 560-567.

[9] See for instance : BATCHELOR, ^G. K., An Introduc- tion to Fluid Dynamics (Cambridge ^Univ. Press) 1969, p. 245-246.

[10] JOST, W., Diffusion ⁱⁿ Solids, Liquids, ^Gases (Academic Press Inc. N. Y.) 1952, p. 135.

[11] ESHELBY, ^J. D., ^J. Appl. Phys. ²⁴ (1953) ^176.

[12] ^{We have} got the value of the geometrical parameters a, b and R from direct observations while the

viscosity and the elastic modulus have been got from previous measurements [13] and from the

hydrodynamic analogy between colloidal crystals [14] and Blue Phase.

[13] CLADIS, P. E., PIERA0143SKI, ^{P. and} JOANICOT, M., Phys. Rev. Lett. 52, ^{no. 7} (1984) ^172.

[14] ^JOANICOT, M., JORAND, M., PIERA0143SKI, ^{P. and} ROTHEN, F., ^J. Physique ⁴⁵ (1984) ^1413.

[15] ESHELBY, ^{J. D. and} STROH, ^A. N., ^Philos. Mag. ⁴²

(1951) ^1401.