Orientational phase transition in cubic liquid crystals with positional order

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Orientational phase transition in cubic liquid crystals with positional order

V.L. Pokrovsky, P.A. Saidachmetov

To cite this version:

V.L. Pokrovsky, P.A. Saidachmetov. Orientational phase transition in cubic liquid crystals with positional order. Journal de Physique, 1988, 49 (5), pp.857-860. �10.1051/jphys:01988004905085700�.

�jpa-00210762�

Orientational phase transition in cubic liquid crystals ^with positional

order

V. L. Pokrovsky (*) ^{and P. A.} Saidachmetov

L.D. Landau Institute for Theoretical Physics, ^{Moscow V-334,} Kosygina 2, ^U.S.S.R.

(Reçu ^{le 27 mai 1987,} accepté ^sous forme définitive ^{le 29} janvier 1988)

Résumé.

Un champ électrique peut produire ^une déformation de cisaillement dans des cristaux liquides cubiques possédant ^{un ordre à} longue distance des positions qui ^{est ancré sur deux} plaques ^{aux limites de}

l’échantillon. La valeur critique ^du champ ^ne dépend pas de la taille de l’échantillon, ^mais dépend ^de façon

cruciale de son orientation.

Abstract.

An electric field can give ^{rise to a} shear deformation of a cubic liquid crystal ^with long-range positional order fixed by ^two plates. The critical value of the field does ^not depend ^on the size of the system and depends crucially ^on the orientation.

Classification

Physics ^Abstracts

61.30 - 61.50

1. Introduction.

The orientation of cubic liquid crystals (CLC) by ^an applied electric field has been suggested by ^{one of}

the authors [1] and has been experimentally ^ob-

served by Pieranski et al. [2] ^{in blue} phases ^{of some}

cholesteric liquid crystals. ^CLC considered in [1]

have been assumed to have no positional long-range

order (PO). ^{However, all the} experimentally ^investi- gated CLC possess PO, including ^D phases [4], ^blue phases (see, ^{for instance, reviews} [3]) ^{and cubic} phases of micellar solutions [5]. ^{The PO} changes drastically the character of orientational phase ^tran-

sitions driven by ^electric field. In this work we

present ^a theory ^{of such} phase transitions.

2. Elastic and orientational forces.

We consider CLC placed ^{in a} space between ^two

parallel plates. ^These plates play ^an important ^role

for our problem. Firstly, they ^{fix a CLC} crystal ^{in a}

way ^to align ^{one of the} symmetry ^axes normally ^to

the surface. In the experiment [2] ^these preferential

directions were 2, 3 and 4-fold axes as well as the direction [211]. For the sake of simplicity ^{we assume}

(*) ^{Visitor at the} Physics Department ^{of the} University

of Toronto.

the 4-fold axis to be normal to the surface. If PO

exists, ^the limiting plates ^also play another role :

they pin the cubic lattice of the crystal by ^{their relief.}

We neglect ^the possibility ^{of a} sliding ^{of CLC} along

the surface of the plates. ^We accept ^zero displace-

ments at the surface as proper boundary conditions.

This type ^of boundary condition is not commonly

considered by ^the theory ^of elasticity ^{where a stress}

tensor at the boundary ^is regularly ^fixed.

A source of deformation in CLC is an external electric field E directed arbitrarily ^with respect ^to the surface. Since CLC is rigidly pinned by ^the plates, ^{both the} polar angle 0 and azimuthal angle ^cp defining the orientation of the field with respect ^to normal to the surface n are relevant. Usually ^the

electric field causes a striction effect, proportional ^to

the square of the field. However, ^{for the} special boundary conditions of our problem the electrostric- tion is absent. The orientational effect of the electric field proportional ^to the fourth power of the field E is described by ^a Hamiltonian [1] :

where ni is the triade of unit ^vectors directed along

the 4-fold axis, and g ^{is a} coupling ^{constant. For a}

mixture of 50 and 42.5 CB 15 in E 9 Pieranski et al.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004905085700

858

have shown g ^{to be} negative [2]. ^{As a} consequence, the orientation of one of the 4-fold ^axes along ^the

electric field is energetically favourable. Principally

the positive ^value of g ^is possible. Then the field tends to align ^a 3-fold axis parallel ^to its direction.

The competition ^{of the} orientational effect of a

field and the rigid pinning ^{of CLC} by ^the plates

results in the appearance of shear stresses starting

from a critical value of the electric field. To calculate this effect the elastic energy of CLC has to be taken into account. It has the usual form (see ^{Landau and}

Lifshitz [6]) :

1/8M aup B

1 ( aUa aup j

where uaJ3 = _2 ^{( - +} - is the deformation

2 axp axa

tensor, and Ua is the displacement ^vector. The elastic energy is written for ^a special frame of reference with the axis coinciding with the 4-fold axis of CLC.

For the blue phases, ^{elastic constants were calculated} by Dmitrienko [7]. ^Further only small deformations and rotations will be considered. Hence local values of the vector ni ^can be expressed ^{in terms of local}

values of the rotation vector (a

The local rotation vector W is defined as

We neglect the energy of orientation distortions which is quadratic ^{over the} derivative of ni, ^as being

small by ^{the ratio} (a/d )2 compared with the elastic energy (2) (a ^{is a lattice} constant, d ^{is the} interplate distance). ^{In CLC without} positional ^order, displace-

ments and deformations are not well defined, ^hence

the orientational distortions play the crucial role for them.

3. Orientational phase transition.

Here an analogue of the Fredericks phase ^transition

will be considered. This is also the boundary prob-

lem. However, the result occurs to be independent

of the thickness of a liquid crystal.

The problem ^{is to} minimize the total energy

consisting of the interaction energy (1) and elastic energy (2) ^{with a} boundary ^{condition u}

^{0 at the}

surface of a crystal. ^{Let z be} perpendicular ^{to the} plate while x and y are directed along ^{the other two}

4-fold axes of CLC. Put the coordinates z of plates ^to

be equal ^{to ±} d/2. ^{We assume the} displacement ^to depend ^{on z} only. Then the elastic energy takes the form :

The energy of interaction with the electric field with the same assumption ^{and the} preciseness ^to quadratic

terms of ro is

Hint = Cst. + aúJ 1 + búJ2 + AúJl +

+2BúJlúJ2+CúJi; (6)

where

The equations ^of equilibrium ^{are :}

The coefficients a and b do not enter equations (8)

and further will be omitted. A nonzero solution of

equations (8) satisfying the necessary boundary ^con-

ditions exists only ^{when the} following equality ^is

satisfied :

Equation (9) ^{defines a} critical value of the field

Ec (0, cp ) for a fixed direction. This equation ^for

g 0 is convenient to rewrite in the following ^{form :}

where E is measured in the units (JL / I 9 I ) 1/4 , ^and Ã, B, ^{C can} be obtained from A, ^{B and C} respectively by omitting ^{the factor} gE4. So, Ã, B, ^4f depend ^{on the} angles 0 ^{and cp} only :

Ã

2(6 ^{sin 2 0} COS2 0 sin 2 _cp

-

sin4 0 sin 4 cp

COS4 0 F3

sin20 sin cp ^cos (p (sin 2 0-6 ^COS2 0

4f

2 (6 ^{sin2 (J COS2 0 COS2 (p}

- sin 4 (J COS4 cp ^{- cos4} 0 (11)

Displacements ^are defined with the exactness of

constant factors :

and

If E > Ec (0, ^{cP ),} the elastic anharmonicity ^{as well as}

the cubic and fourth-order terms over W in the electric coupling energy have ^to be taken into account. After substitution of the solution (12) ^into

anharmonic terms in the elastic energy and minimiz-

ing the energy ^over the amplitude,we ^{find as usual a}

to be proportional ^to (E - Ee)1/2. We would like to remark that cubic anharmonicity ^which principally

could lead to the first-order phase transition turns out to be zero due to the symmetry ^{of the} arising instability (12).

When the field E overcomes the critical value

Ec (0, cp ), deformations _appear to be maximal in the centre of a layer ^{and are} equal ^to ( (E - Ee)/ Ee)1/2

by ^{the order of} magnitude. ^{When E} is increased by ^a

value of the order Ec ^{above the} threshold, ^{the linear} elasticity theory becomes invalid. This fact interferes the further analysis ^of bifurcations.

4. Anisotropy of the critical field.

The critical field as a function of the orientation can

be explicitly found from equation (10)

The sign ^{of the} quadratic ^{root has to be chosen in a}

way ^to obtain a letter value E, ^at fixed 0 and cp.

An investigation ^of equation (14) shows that a real value of Ec ^{is not} possible ^for any 0 ^{and cp. The}

shadowed regions ⁱⁿ figure ¹ correspond ^to complex

values of Ec (g 0 ). Everywhere along ^the boundary

of this region Ec ^becomes infinitely large. ^{It means}

that no transition proceeds in the shadowed region.

The values 0

0 and 0

7T /2, ^cp

⁰ correspond

to the orientation of the field along ^one of the 4-fold

axes. It is obvious that ^a field oriented in this way

Fig. ^1.

^The plot of forbidden (shadowed) ^and permitted (light) directions of the field for the orientational phase

transitions.

does not cause any rotation ^nor any deformation associated with it. For the appearance of ^a shear

deformation, ^not only ^a sufficiently strong field, ^but

also a sufficiently large deviation from symmetric

orientation are necessary. The crystal is also stable when the field is oriented along ^a diagonal ^{of the}

face 0

7T /2, cp

^7T /4.

Graphs ^of Ec ^{vs. 0 for two} fixed values of _cp are

depicted ⁱⁿ figure ^{2. The minimal value} E, = (u/g)1/4 correspond ^{to 0}

7T /4, cp

^{0. The}

cusps ^on the curves 1, ² correspond ^to transitions from one sign ^{of the} equation (14) ^to another. The dashed line in figure ¹ is the line of the cusps e

arc sin /6/7

^67.8°.

Fig. ^2.

Graphs of the critical field Ec ^{vs. 0 for two values}

of cp. (1) cp

7T/8 ; (2) ^cp

^{3 7T/16.}

The angular dependence of the critical field is

universal ; ⁱⁿ particular ^{it does not} depend ^{on the}

elastic coefficients. This fact can be used for exper- imental observation of the effect.

5. Discussion.

We have neglected ^the possibility ^of sliding ^or removing ^the crystal ^{from the} plates. This assump- tion can be invalid when approaching the lines of the infinite critical field. Obviously, ^the theory may be restricted by ^a trivial electric breakdown if it pro- ceeds before the crystal will be removed from the

plates. Principally ^{we cannot} fully exclude the pos-

sibility ^of sliding ^{at a} very low value of the field.

However, ^{we do not see} any ^reason for this

phenomenon.

Below the transition point only ^the birefringence

associated with a quadratic Kerr effect arises. The direction of the main axis of the dielectric tensor

depends ^on the direction of the electric field only

and does not depend ^on the value of the field.

Above the critical field the direction of this axis

changes. ^{This fact can} be checked by ^the ellipsometry

methods.

860

The estimation of the critical electric field shows that Ec ~ ³ 000 V/cm for Blue phases. This value is close to the fields applied ^{in the} experiments by

Pieranski et al. [2]. Generally, the estimate of the critical field is Ec ~ (Te/a3)1/2, where Tc ^{is the} temperature necessary for the existence of CLC. For smectics D and lyotropic CLC the value of a is

approximately ^ten times less than the same value for blue phases. Therefore, the critical field would be 30 times more in these crystals.

The theory ^{of the} phase transition induced by ^a

weak electric field in bulk Blue Phases has been considered by Hornreich et al. [8]. They ^concluded

that the electric field can stabilize a hexagonal ^{or a} tetragonal phase. However, ^the phase diagram, predicted by ^these authors, contains sections T

Cst. _{of pure} cubic phases. Our calculations differ from [8] by ^the assumption ^{of a restricted} geometry.

The experimental situation is controversial.

Porsch and Stegemeyer [9] ^reported the observation of a new phase in the electric field 30 kV/cm which

they ^call E-phase. ^{The structure of the} suggested

phase ^{is not known.} They ^{also do not} observe any orientational effect in contrast with Pieranski et al.

[2].

Heppke et ^al. [10] ^{have observed a} shift of the maximum of reflection to the shorter wavelength ^for

BP2 which can be treated as a dilatation. For BPI the two maxima of reflectivity ^shifted towards each other. This motion has been treated as a defor- mation. Authors claimed the deformations to

change gradually ^{with the} voltage. However, ^the region of the fields E 5 10 kV/cm has not been studied.

Acknowledgements.

One of us (V. P.) thanks Allan Jacobs, ^Allan Griffin, Michael Walker and Erik Fawcett for the kind hospitality during ^{his visit to the} University ^of Toronto, and Michele Morrison for the preparation

of the manuscript. ^{We are} grateful ^{to 0.} Martynova

for a help.

We are grateful ^{to Prof. H.} Stegemeyer ^for sending ^{us the} reprints of relevant works.

References

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[2] PIERANSKI, P., CLADIS, ^P. E., GAREL, T., ^BARBET- MASSIN, R., J. Phys. ^{France 47} (1986) ^139.

[3] BELYAKOV, ^V. A., DMITRIENKO, ^V. E., UFN, ^Sov.

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STEGEMEYER, H., BLUMEL, ^{Th. et} al., Liquid Cryst.

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