Behaviour of thin planar Sm C* samples in an electric field

HAL Id: jpa-00209729

https://hal.archives-ouvertes.fr/jpa-00209729

Submitted on 1 Jan 1984

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Behaviour of thin planar Sm C* samples in an electric field

J. Pavel

To cite this version:

J. Pavel. Behaviour of thin planar Sm C* samples in an electric field. Journal de Physique, 1984, 45

(1), pp.137-141. �10.1051/jphys:01984004501013700�. �jpa-00209729�

Behaviour of thin planar ^{Sm C*} samples ^{in an electric field}

J. Pavel

Institute of Physics, Czech. Acad. Sci., ^{Na Slovance} 2, ¹⁸²⁰⁰ Prague 8, Czechoslovakia

(Reçu ^le 6 juin 1983, accepté ^{le 20} septembre 1983)

Résumé.

On décrit trois types ^de structures qui peuvent exister dans les échantillons de Sm C* planaires ^en

fonction de l’épaisseur de l’échantillon. La structure caractérisée par la rotation de la molécule le long ^de l’épaisseur

de l’échantillon provoquée par l’ancrage existe pour des épaisseurs comprises ^entre deux valeurs critiques. ^Le comportement ^de cette structure

champ électrique ^{est étudié} théoriquement

supposant l’ancrage simplifié.

La valeur critique ^du champ électrique pour la transition ^entre la structure

torsion et la structure homogène

est déterminée

fonction de l’épaisseur de l’échantillon.

Abstract.

Three types ^of structures which

exist in planar ^{Sm C*} samples, depending

sample thickness,

are

described. Two critical thicknesses limiting the existence of the twisted nonhelical configuration ^{are defined.}

Behaviour of this structure in

electric field is theoretically solved for simplified anchoring conditions. The criti- cal field for the transition of the twisted structure to the homogeneous

is determined

function of sample thickness, comparing ^{the free} energies ^{of both} structures.

Classification

Physics ^Abstracts

61.30G - 77.80D

1. Introduction.

In the chiral smectic C (Sm C*) liquid crystal ^structure

the long ^{molecules are} arranged ⁱⁿ layers ^{and tilted}

with the angle ^{0 with} respect ^{to the} layer perpen- dicular. The chirality ^{of a} molecule leads to the

establishing ^{of a helical structure} with helical axis

perpendicular ^to the smectic layers. Molecules of Sm C* liquid crystals ^are characterized by ^{the mean}

component ^{of a} permanent dipole-moment [1] ^which

is perpendicular ^to the molecule and lies in the smectic layer.

The structure of a finite Sm C* sample ^differs

from the structure of an infinite one because of a

strong influence of the boundaries. Here we shall

only deal with planar samples ^{which are Sm C*}

liquid crystals ^{between two} parallel glass plates ^to

which the smectic layers ^are perpendicular. ^The glass

surfaces force the molecules to be parallel ^{to them,}

but there is no easy direction. When the smectic

layers ^are established the molecules ^can be only

in two orientations on the surface which make an

angle ² 0 and have opposite directions of dipole-

moments (Fig. lb).

Our recent experiments show that molecules of two studied Sm C* materials, namely ^DOBAMBC

and Sm C*, mixture of Sm C and cholesteric [2], ^are

oriented on the surface so that their dipole-moments point ^{from the} glass plates into the bulk of the sample (Fig. lb). This fact results in the existence of three

Fig. ^1.

^Planar anchoring of the molecules of Sm C*

the surfaces (a) ^{with the} equal orientation, (b) with diffe- rent orientations. The molecular dipole-moment ^{is denoted} by p, 0 is the tilt angle, 9 is the azimuthal angle, ^t is the pro-

jection of the molecule onto the smectic layer. ^The

dinate system ^and positive direction of electric field E

shown.

types ^of structures of planar ^{Sm C*} samples, depend- ing ^{on the} sample thickness, ^which dependence ^will

be considered in this contribution. In reference 3 the behaviour of thick planar ^{Sm C*} samples ^{in an electric} field, ^was theoretically discussed but only ^{in the case}

of equal orientation of molecules on the boundaries

(Fig. la). ^{Here we shall} discuss the influence of an

electric field on the structure of thin planar ^{Sm C*}

samples ^with different orientations of molecules anchored at the surfaces (Fig. lb).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004501013700

138

2. Planar samples of different thicknesses.

We choose the system of coordinates ^so that the x-axis is perpendicular ^{to the} layers and z-axis is

perpendicular ^{to the} glass plates. The orientation of each molecule is defined by ^two parameters 9 and 6

(see Fig. lb).

For our purposes ^we shall suppose the tilt angle

to be constant. The ideal spiral ^structure of the Sm C*

is then described by ^the equation

where p is the helical step. ^The sign is choosen accord-

ing ^{to the «} handedness » of the helix.

The structure of a planar sample ^can be obtained

by minimizing the elastic part of the free _energy.

According ^{to reference 4 we} write this _energy in the form

where Bl, B2, B3 ^{are elastic} constants, qo is related to the step of the helix by qo = 2 7r/p, V ^{is the volume}

of the sample. We suppose B1

B2 ^and cp indepen-

dent of y. ^{Then the} expression ^{2 can} be rewritten [3] ^as

The Lagrange-Euler equation whose solution mini- mizes the free energy (3) ^reads

Equation 4 has been solved numerically using ^the

net inethod for two types ^of boundary conditions :

a) ^the equal orientations of molecules on both sur-

faces, namely 9(x, ^z

dl2)

0, 9(x, ^z

dl2)

⁰ (z-coordinates ^{of the} glass ^{surfaces are} d/2, - d/2,

d is the sample thickness) (Fig. la) ; b) ^the opposite orientation, for instance 9(x, ^z

dl2)

0, 9(x, ^z

-

dl2) = ^1t (Fig. lb). Boundary conditions for cycles

in the net method were as follows. On the glasses,

i.e. for z

d/2 ^{and z}

d/2, they have been des- cribed above. In the x-direction ^we suppose ^an infinite

periodic sample ^{so that} 9(x, z) = T(x ⁺ p, z) ^then equation ⁴ is solved for one step only. ^{In order to} join

the helical structure in the middle of the sample, ^where equation ¹ holds, and the unwound structure near the

glass plates ^{it is} necessary ^to add a pair ^{of ± 2 n} disinclinations, ^so called dechiralization lines [3, 4].

For the purposes of the ^net method, ^{the 2} n-disincli- nation in the position ^x

i, ^z = j ^is given ^{in the}

these defects where curl t

0.

For every position ^{of a 2 7r} resp. - 2 7r-disinclina-

tion, the solution of equation ^{4 is} first obtained and then the free _energy (3) computed. ^{The minimum of}

this _energy determines positions ^of both defects.

Figures 2a, b, illustrate the resultant structures which

give minimum free _energy (3). ^Our computation gives ^{the same} values of elastic _energy for both cases

with different boundary conditions.

The distance h of dechiralization lines from the

nearer glass surface is approximately inversely pro-

portional ^{to the} ratio of elastic constants B3/B1. ^This

dependence is in accordance with those obtained in [3].

For two studied ferroelectric liquid crystals,

DOBAMBC and Sm C*, mixture of Sm C with a

cholesteric (see ^Ref. 2), ^{it was found} experimentally

that the case with opposite boundary conditions takes

place. ^{Molecules are attached to} the surface ^so that their dipole-moments ^are always directed from the

glass plate towards the liquid crystal [5]. This fact is not so surprising ^{when we take into account} symme-

try of the chiral molecules.

When the sample thickness d is reduced the helical part ^between dechiralization lines is also reduced.

When this part vanishes the sample becomes unwound.

We define a sample ^thickness d, ^so that for d > d1

the sample has helical structure with dechiralization lines and for d d1 ^the sample is unwound. A rough

estimation gives d1 = ^{2h = p.}

Our further considerations will be made for the unwound samples, ^{i.e. for a} sample thickness d dl.

The structure of these samples ^{does not} depend ^on

x, y, coordinates. The elastic part of the free _energy is then

Fig. ^2.

Computed structures for the ratio B31B,

^0.35 minimizing the elastic free energy in the x,

section (a)

for the equal ^surface anchoring, (b) ^for opposite ^surface

anchoring. The molecules

represented by ^{nails, the}

points ^{of which} correspond ^{to the} parts of molecules turned

toward the observer. The full circles denote

sections

of dechiralization lines. The helical step is denoted by ^p.

We _suppose anchoring energy W1 for molecules on

the surface oriented in such a way that their dipole-

moments are perpendicular ^{to the} glass plate ^and

directed towards the sample ^{bulk, and} anchoring

energy W2 > W1 for the molecules with opposite

direction of dipole-moments. ^For other orientations the anchoring energy is supposed ^{to be} infinite. When

W 1 corresponds ^to the orientation of molecules with qJ

0 on the lower glass plate (z

dl2) ^{then the}

orientation of molecules with _w = n on the _upper glass plate ^z

dl2 is described by ^{the same} energy Wl,

and with _T

0, by ^the energy W2.

For boundary conditions T(z dl2)

0, (p(z

d/2) = ⁿ the free energy is

In the free energies F 1 ^and F2 the term 4 nPs2 ^cos2 cp,

corresponding ^{to the} energy of depolarizing ^field arising from the local spontaneous polarization Ps,

is not considered. We suppose that in liquid crystals

the conductivity ^is high enough ^to compensate ^the depolarizing ^field.

The solution of the equation

minimizes the free energies F 1 ^and F2. ^{The free} energy

F 1 is minimized for

and F2 ^for

With these solutions we have

B 7T

For the thickness d - 2(W 2 1 - W 1) ^{2(W2 - Wl) the energies}

F1, F2 ^are equal. Thus the solution 8 which defines

a structure twisted along z-axis is valid ^{for d} > d2

and the homogeneous solution 9 is valid ’for d d2.

3. Influence of the electric field.

In this paragraph ^we shall restrict our considerations

to the planar ^{Sm C*} samples whose thickness satisfies

d2 ^d dl, i.e. nonhelical twisted samples ^described

by (8). ^An electric field E is applied perpendicularly

to the glass surfaces. Positive E is directed from the upper ^to the lower glass (Fig. 1).

The interaction of Sm C* crystal ^{with the external}

electric field E is described by ^{the last term in the}

free _energy expansion (cf. ^Ref. 6)

where P. is the local spontaneous polarization. ^Here

the flexo-electric term and the term quadratic ^{in E}

are neglected. ^The depolarizing energy is neglected ^as well, ^{as in the} previous paragraph. ^We put

The variation of F El ^{in equation 12 with respect}

to T gives ^the equilibrium ^condition

The first integral ^of (14) ^is

where C is the integration ^{constant. This can be}

rewritten in the form

Since the left-hand side is positive, ^{it can further be}

rewritten as

For k2 1 the solution of equation ^{14 is then}

where _zo is the integration ^{constant and w}

am(u) ^is

the inverse function to

140

The condition qJ( - d/2)

⁰ gives zo

d/2. ^The

constant C can be determined from the condition

where K (k) is the full elliptical integral of the first order. It can be easily shown that equation ^{20 has}

one solution C > r.

The structure of planar nonhelical twisted Sm C*

in the external electric field is thus described by ^the equation

(see Fig. 3).

This solution of equation ^{14 can be} approximated

for a) ^{low and} b) high intensities of external electric field.

a) ^Low intensity of the electric field is given by ^the

condition

In this case the solution can be written in the form

b) High intensity of the electric field is given by

the condition

This condition enables one to put ^C

^{r in} equation 15, whose solution is then

Fig. ^3.

^Azimuthal angle w

function of z-coordinate for different values of electric field.

Now we substitute the exact solution 21 into 12 and obtain the free _energy

where

and k is defined by equation ^18.

When we consider the homogeneous ^{structure des-}

cribed by equation ^{9, the free} energy in the electric field is

Figure ^{4 shows} dependences ^of F El ^and FE2 ^{on r,}

which is proportional ^{to E. For} the critical value rk

we have

That means that for r rk the deformed twisted structure (21) having ^the energy F El ^{is more advan-}

tageous ^{and for r} > rk the homogeneous ^structure (9) having ^the energy FE2 ^is preferred. Critical « field, » rk does not have the meaning ^{of a} coercive field as

discussed in the last paragraph.

Free energies F El ^and FE2 ^{depend on the sample}

thickness d and therefore rk ^{is also a} function of d.

This dependence ^{is shown in} figure ⁵ where rk ^has

been obtained as a solution of equation ²⁹ for different values of d. For the thicknesses d d2 the critical

value rk ^{is zero} because the structure with _energy FE

is not realized. For higher thicknesses rk ^{does not}

Fig. ^4.

^The dependence of free energy F El ^{(26) and free}

energy FE2 ⁽²⁸⁾

^E.

Fig. ^5.

^The dependence ^{of critical} «field» rk ~ Ek

on

the sample thickness d.

depend ^{on d and} approaches ^{the value}

From rk ^a value of critical electric field

can be obtained.

4. Concluding ^remarks.

The structures of planar ^{Sm C*} samples ^as computed

with equal ^surface anchoring (Fig. 2a) ^{and with}

different anchoring ^{have the same elastic} energy.

Experimental results show that the structure with different anchoring ^is realized in Sm C*, ^obtained by mixing ^Sm C and cholesteric [5], and in DOBAMBC

[5]. The alternate positions ^of upper and lower lines

as shown in figure ^{2b were} confirmed with the men-

tioned mixture. With DOBAMBC the arrangement of lower and _upper lines could not be recognized

because the lines are very dense.

In the second section two independent ^critical

thicknesses were considered : the minimum thickness

d1 for the existence of the helical structure with dechi- ralization lines and the minimum thickness d2 ^{for the} occurring ^{of a structure} with different anchoring ^at

the sample surfaces. The existence of twisted non-

helical structure with the mentioned Sm C* mixtures

[5] implies ^that di > d2 for this material. With this material the samples ^were prepared ^{down to the}

thickness d

25 _gm [5] ^{and at this thickness the}

twisted structure was realized. It ^can be expected ^that

for much thinner samples ^the homogeneous ^structure

will be observed. On the other hand the homogeneous

structure was observed with samples ^{of DOBAMBC}

1.5-2 _gm thick [7]. ^From this it follows that for

DOBAMBC d2 = ² 2 um. _gm.

The transition from nonhelical twisted structure to the homogeneous ^{structure in} applied electric field as

theoretically considered here was observed with men-

tioned Sm C* mixtures [5] ^at the coercive field of about 50 kV/cm. The coercive field found experimentally

is not exactly equal ^to the critical field Ek ^calculated

here. In the calculation a simplified model of the

anchoring ^was used, ^{which is} based on two possible

orientations of molecules on the surface which are

separated by ^an infinite energy barrier. This model does not enable us to describe the transition between these two orientations, ^the Ek is obtained from the

comparison ^{of free} energies ^{of both} structures.

Acknowledgments.

The author is indebted to Drs. M. Glogarova, ^V.

Janovec, ^J. Fousek, and V. Dvofik for valuable dis- cussions and to Dr. L. Lejcek for critical reading ^of

the manuscript.

References

[1] Meyer, ^{R. B., Mol.} Cryst. Liq. Cryst. ^{C 40} (1977) ^33.

[2] PAVEL, J., GLOGAROVÁ, M., DEMUS, D., MÄDICKE, A., PELZL, G., Cryst. Res. Techn. 18 (1983) ^915.

[3] GLOGAROVÁ, M., LEJ010DEK, L., PAVEL, J., JANOVEC, V., FOUSEK, J., ^Mol. Cryst. Liq. Cryst. ⁹¹ (1983) ^309.

[4] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Oxford Press) ^1974.

[5] GLOGAROVÁ, M., PAVEL, J., ^J. Physique (this issue).

[6] BLINC, R., 017DEK0161, B., Phys. ^{Rev. A 18} (1978) ^740.

[7] CLARK, ^N. A., LAGERWALL’, ^S. T., Appl. Phys. ^{Lett. 36}

(1980) ^899.