INTERACTION BETWEEN SURFACE AND BULK DISCLINATIONS IN NEMATIC LIQUID CRYSTALS

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INTERACTION BETWEEN SURFACE AND BULK DISCLINATIONS IN NEMATIC LIQUID CRYSTALS

G. Ranganath

To cite this version:

G. Ranganath. INTERACTION BETWEEN SURFACE AND BULK DISCLINATIONS IN NE- MATIC LIQUID CRYSTALS. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-87-C3-89.

�10.1051/jphyscol:1979319�. �jpa-00218715�

INTERACTION BETWEEN SURFACE AND BULK DISCLINATIONS IN NEMATIC LIQUID CRYSTALS

G. S. RANGANATH

Raman Research Institute, Bangalore 560006, India

Abstract. — The problem of the interaction between a surface disclination and a bulk disclination placed symmetrically with respect to it has been considered on the basis of the continuum theory.

It is found that at large distances the bulk disclination is attracted by the surface disclination with a force that is independent of the sign of the former. The force of attraction is found to be nearly the same for bulk disclinations of strength 1/2, 1 and 2. The theory appears to qualitatively account for the experimental observations of Kleman and Williams.

1. Introduction. — Nematic liquid crystals are known to have both body and surface disclinations.

Body disclinations lie in the bulk of the material while surface disclinations reside at the boundaries. The structure and properties of body disclinations have been extensively investigated both theoretically and experimentally [1-4]. It is only recently that the inte- resting features associated with surface disclination have been elucidated [5-7]. Meyer [6] and Vitek and Kleman [7] have theoretically studied the structure of disclinations on a rubbed surface, while Kleman and Williams [5] have described some interesting experimental observations which reveal many of their properties.

It is well known that bulk disclinations of the same sign repel while those of opposite signs attract one another. The properties of a hole (air bubble) and its interaction with bulk disclinations are also fairly well understood [8, 9]. The interaction between a body disclination and rubbed surface has been worked out recently by Meyer [6]; at large distances it is repelled by the boundary and at short distances it is attracted.

However, the interaction between a surface and a body disclination does not appear to have been considered so far. The problem is important in the light of the observations of Kleman and Williams [5]. They find experimentally that the surface disclinations have a tendency to attract body disclinations. In this paper we have considered the problem of a body disclination placed symmetrically with respect to a surface dis- clination. In this geometry we do find an attractive interaction between the two types of disclinations.

2. Theory. — Let the liquid crystal occupy the half space z — 0 to oo with z = 0 plane representing the rubbed surface. The boundary condition on this surface is such that the director is along x-axis. Then everywhere else also the director will be parallel to x-axis.

If we now introduce at the origin a surface dis- clination with its singular axis along y-axis then the director pattern is given by

with

Here 2 W is the width of the surface disclination, A is a measure of the anchoring energy at the surface (energy per unit area required to introduce a tilt 6 is 1/2 ,4 sin² 9).

On the other hand if a body disclination is at x = 0 and z = d(^> W) from the surface then the director pattern is

with

Here S is the strength of the body disclination. In this particular case the body disclination will be repelled from the boundary by an image force.

JOURNAL D E PHYSIQUE Colloque C 3 , supplément au n° 4, Tome 40, Avril 1979, page C3-87

Résumé. — On considère, sur la base de la théorie continue, le problème de l'interaction entre les disclinaisons de surface et les disclinaisons en volume. On montre qu'à grande distance, les disclinai- sons en volume sont attirées par les disclinaisons en surface avec une force indépendante du signe de la première.

La force d'attraction est la même pour les disclinaisons en volume de force 1/2, 1 et 2. La théorie rend compte qualitativement des observations de Kléman et Williams.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979319

C3-88 G. S. RANGANATH

We shall now consider the situation where we have the body as well as surface disclinations. Let the surface disclination be at x = 0, z = 0 and the body disclination of strength S be at x = 0 , z = d. In this case we will have a general director pattern

n, = cos 0 cos cp

,

,

.

( 5 ) In the approximation k , , = k2, = k33 = k the elastic energy density is

+

cpl

0 . = -

ao

31 axi ' axi

We have to obtain 0 and cp by minimising F. This results in the following equations of equilibrium :

On the surface z = 0, cp = 0 and, in addition, we have to balance the torques. This yields

3. Results. - To get 0 and cp we must solve (7) subject to the boundary conditions (8). However, the problem gets considerably simplified if we invoke the superposition principle. In this limit the net distortion is nothing but the sum of the distortions due to both the singularities, i.e., 0 = 0, and cp = q0.

This approximation is valid only when d is large. Also both eq. (7) and (8) are satisfied to a good degree of approximation in this limit. Under these assumptions the elastic energy density is

If there had been no interaction whatever between the two types of disclinations we would have had only the first term. Therefore the interaction energy density is F,,, = - k [ - (02

2 ^{o , ,}

+

and the total interaction energy is

Since the second term in (10) is antisymmetric in x

It is interesting to notice that El,, does not depend on the sign of cp,. The interaction energy is independent of the sign of the body disclination. The force of inter- action is

Calculations show that f is always negative indicating an attractive force between the two types of discli- nations. Figure 1 shows the dependence of f on d for S = 112, 1 and 2. We conclude from this that the force is not too different for the three types of dis- clinations. It must be remarked that the repulsive image force between the body disclination and the surface counteracts the attractive interaction worked out above. Our calculation off is not accurate enough to give a reliable value of the equilibrium separation between the surface and body disclinations.

I I I I I I I I / I I

0 10 20

d (PI-

FIG. 1 . - Dependence of f/k on d for different values of S. f is seen to be quite insensitive to different values of S.

If the liquid crystal is confined between two parallel rubbed plates then the body disclination will prefer to be at the centre. If now a surface disclination is introduced on one of the plates then the attractive force calculated above will come into play forcing the body disclination to move towards the surface.

INTERACTION BETWEEN SURFACE AND BULK DISCLINATIONS I N NEMATIC LIQUID CRYSTALS C3-89

However, as it approaches the boundary the repulsive they attribute to the repulsive interaction between term will be more and more important. Thus the body the surface and the bulk line.

disclination will finally settle at an equilibrium point. It appears from these observations that the theory Kleman and Williams [5] in their experiments on can qualitatively account for the experimental findings.

surface disclinations confined the liquid crystal bet-

ween two parallel plates. They found the body dis- Acknowledgment. - The author is grateful to clination to be attracted by the surface disclination prof. S. Chandrasekhar for the many useful discus- lines. The two lines however did not merge. And this sions he had -with him.

References

[I] FRANK, F. C., Disc. Faraday Soc. 25 (1958) 19. [5] KLEMAN, M. and WILLIAMS, C., Phil. Mag. 28 (1973) 725.

[2] NEHRING, J. and SAUPE, A., J. Chem. Soc. Faraday Trans. 11 [6] MEYER, R. B., Solid Stare Commun. 12 (1973) 585.

68 (1972) 1. [7] VITEK, V. and KLEMAN, M., J. Physique 36 (1975) 59.

[3] SAUPE, A., MoI. Cryst. Liq. Cryst. 21 (1973) 211. [8] MEYER, R. B., Mol. Cryst. Liq. Cryst. 16 (1972) 355.

[4] KL~MAN, M., Advances in Liquid Crystals, ed. Brown, G. H. [9] RANGANATH, G. S., MoI. Cryst. Liq. Cryst. 40 (1977) 143.

(Academic Press) Vol. 1 (1975) 267.