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PAIRED DISCLINATIONS OF OPPOSITESTRENGTHS AND DISCLINATIONS OF THE

STRENGTH O IN PLANAR CHOLESTERIC WEDGELAYERS

A. Stieb

To cite this version:A. Stieb. PAIRED DISCLINATIONS OF OPPOSITE STRENGTHS AND DISCLINATIONS OFTHE STRENGTH O IN PLANAR CHOLESTERIC WEDGE LAYERS. Journal de Physique Collo-ques, 1979, 40 (C3), pp.C3-94-C3-97. <10.1051/jphyscol:1979321>. <jpa-00218717>

PAIRED DISCLINATIONS OF OPPOSITE STRENGTHS AND DISCLINATIONS OF THE STRENGTH O

IN PLANAR CHOLESTERIC WEDGE LAYERS (*)

A. E. STIEB

Institut f. Ang. Festkorperphysik der FhG., 7800 Freiburg, W. Germany

Abstract. — In wedge shaped cells cholesteric liquid crystals of planar alignment often show disclination lines whose direction is predominantly perpendicular to the usual Grandjean-Cano lines. In one case, neighboured Grandjean-Cano lines turn together to a parallel direction, form a pair, and change sides before separating again. The missing annihilation of the pair in spite of the opposite strengths of the two lines is explained by a simple model.

In the other case, a new type of defect is stretched between two neighboured Grandjean-Cano lines. The disclination strength of this defect is found to be zero. Directorfield models for different possible cases are given.

1. Introduction. — In unidirectionally rubbed wedge cells, cholesteric liquid crystals tend to align with their helical axis perpendicular to the glass plates of the cell [1, 2]. Due to the uniform boundary condi­tions, the helical turns in the layer are quantized and the overall turn angle can be only a multiple of n [3]. In the wedge the free helix is compressed, respecti­vely stretched, until the number r of half director turns in changed by | Ar | = 1 at a discontinuity line [4, 5]. The strength s = Ar/2 of these Grandjean-Cano disclinations is one half, but also nonsingular integer strength disclinations are observed [6]. The disclinations appear as regularly spaced straight lines with a direction perpendicular to the wedge gradient.

Additional defects which have a pairlike character and which are stretched between the Grandjean-Cano lines are found in cholesteric wedge layers [7]. The common property of these defects is that their total strength is zero. This means that along a Burger's circuit around the defect an overall director turn angle of zero is found, respectively that the defor-

(*) This work was supported by the Bundesministerium fur Forschung und Technologie under contract No. NT 694.

mation states in the two areas adjoining to the defect are identical [8]. The latter property was proved by microscopically observing the colours of the different areas after inserting a A-phase plate for the first red between the crossed polarizers. The colour, resulting from a deformation dependent elliptization of the polarized light, is the same on both sides of the defects in question, while it is changing abruptly at a Grandjean-Cano disclination.

According to more detailed observations, there are to be distinguished two different groups of discli­nation pairs of the total strength zero. The first type, shown in figure 1, consists of two individual and neighboured Grandjean-Cano lines which turn away from their usual direction to become parallel to the wedge gradient, to form a pair, and to change sides. The defects of the second type, as shown in figure 3, are not only a simple addition of two Grandjean-Cano lines, they have a new structure of their own.

2. Experimental. — The wedge cells, used for the observations, consisted of two gfass plates which had been cleaned and rubbed in one direction before mounting. The wedge gradient, obtained by using

JOURNAL DE PHYSIQUE Colloque C3 , supplément au n° 4, Tome 40, Avril 1979, page C3-94

Résumé. — Il est fréquent d'observer dans des coins d'échantillons cholestériques, en alignement planaire, des lignes de disclination essentiellement perpendiculaires aux lignes de Grandjean-Cano. Dans un cas, des lignes de Cano voisines se rejoignent en une paire, et interchangent leur position pour se recouper à la fin. Nous expliquons par un modèle simple la non-annihilation des deux lignes de disclination dont les rangs sont de signes opposés.

Dans un autre cas, un nouveau type de défaut s'étend entre deux lignes de Cano voisines. On trouve que le rang de disclination est zéro en ce cas. Des champs directeurs sont proposés pour différents types de défaut.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979321

PAIRED DISCLINATIONS OF OPPOSITE STRENGTHS AND DISCLINATIONS OF STRENGTH 0 C3-95

their relative positions with respect to the wedge edge. These effects have been observed previously by Bouligand for integer strength lines [7]. By microscopic examination it is found that the two lines of the pair are situated in different planes of the layer as to be seen clearly, for example, in figure lb. The tendency to form a pair against the line tension of the discli- nations decreases with increasing vertical distance of the lines (compare Figs. l a and b).

The missing annihilation of the attracting Grand- jean-Cano lines in the pair in spite of their opposite strengths can be explained by a simple model. In figure 2a, a schematic cross section of a pair consisting of two half strength disclinations is sketched. The

a 1 dots, nails, and bars indicate the direction of the

FIG. 1. - Microscopic pictures of overcrossing Grandjean-Cano disclinations in wedge layers with planar boundary alignment (rubbing direction and wedge gradient horizontally, ordinary ray of light) ; a) disclinations of half integer strength with small vertical distance; b) disclinations of integer strength with large vertical distance.

spacers of different width, was parallel to the rubbing direction in most of the experiments. The cholesteric liquid crystal was a mixture of cholesteryl chloride and nematic ester compounds (ZLI 518, Merck). The pitch p of the mixture was about

where the concentration c of cholesteryl chloride was about 1 % of weight. Because of the large pitch no reflection colours were visible. A Leitz Orthoplan polarizing microscope was used for the optical observations.

3. Overcrossing disclimation pairs. - As shown in the figures la and b, two neighboured Grandjean- Cano disclinations of equal strength can turn away from their direction parallel to the wedge edge with opposite curvatures to form a pair. After turning to an antiparallel direction, the strengths of the two lines, seen in the same direction, are of different signs [9]. The lines, forming the pair, interchange

optical axis as perpendicular, oblique, and parallel to the figure plane. The small circles represent the singular cores of the half strength lines, each of which is forming the edge of a layer across which the director turns through an angle of n. From left to right, the number of half turn layers decreases by one at the first disclination (s = - 1/2), and increases again at the second line (s = + 112). The two lines are separated by an intermediate half-turn layer which interchanges its position with respect to the boundary plates. Behind the overcrossing point of the disclinations, the two outer half-turn layers begin to overlap and to form the area with a higher number of half turns, while in front of the figure the lower ranked area is formed.

In figure 2a, a planar structure of the disclinations according to the de Gennes model [lo, 111 was sup- posed. In this case, the free elastic energy per unit length is not depending on the direction of the discli- nation with respect to the rubbing direction, and therefore no cause for changing the level in the curved and straight parts of the line is given. Additionally, the line tension tends to keep the lines in their planes.

........................... i-l i-I+1;*;::** 44-l+4 I b) ------ ----- I-,-+k t - ~ . \ ~ ~ > * . ........ 9. + + + :.::;Q :;:-: ----- ---- k t . r c c c c

FIG. 2. - Directofield structures of paired disclinations with opposite strengths which are separated by a layer of a half director turn, seen in a direction parallel to the wedge.gradient ; a) disclina- tions of half integer strength with singular cores (0) ; b) disclina-

tions of integer strength with continuous cores (encircled).

C3-96 A. E. STIEB

The apparent lateral attraction is not a property of the lines, but due to the excess elastic torque in the area with a wrong number of half director turns, left between the two lines of the pair. The vertical distance of the two lines can be the n-fold of p/2, but then the excess torque in the enclosed area is dissipated over a larger width and becomes less dominant in comparison to the line tension.

The disclinations forming the pair can also be of integer strength, then being the nonsingular boun- daries of layers with an integer turn of the director (Fig. lb). A schematic cross section of the defect pair is given in figure 2b. At the kinks the disclinations change their position vertically by the amount OF one pitch [7].

4. Zero strength disclinations. - Besides the over- crossing Grandjean-Gano line pairs, another type of defect, as first described by Bouligand [7], is found (see Fig. 3). The interaction of the two lines, composing this type of defect, is much stronger than in the cases described above. By focusing to a thin layer it can be detected that the two lines of the pair are situated in the same plane. The character of the composant lines is very similar to the integer

FIG. 3. - Microscopic pictures of a zero strength disclination, stretched between Grandjean-Cano lines of half and integer strength in a wedge layer with planar boundary alignment (rubbing direction and wedge gradient horizontally, pitch 20 pm) ; a) ordinary ray of

light ; b) extraordinary ray of light.

strength disclinations in nematics. They have a constant distance with no respect to the width of the wedge. Strong lense effects and focal lines are to be seen in the extraordinary ray of light. The diameter of the whole structure is about one pitch. The refractive index shows two minima across the pair, indicating vertical director alignments.

If the Grandjean-Cano lines are both of integer strength, the composant lines of the pair overcross themselves at one end and turn away, from each other at the other end. At the open end of the pair the bulk twist deformation of the adjoining area seems to be extended into a narrow stripe between the two lines of the pair. This residual twist in the center of the defect is responsible for a colour shift

1 against the surrounding areas, when a phase plate is used for the microscopic observation. If both Grand- jean-Cano lines are singular, the two composant lines merge at one end of the pair to a singular point on the Grandjean-Cano disclination. At the other end, the two lines of the pair split up into singular lines to join directly the Grandjean-Cano line. Sometimes the end of such a pair is found floating in the cho- lesteric layer, then bearing a singular point at the tip.

Never singular core lines have been found in these defect pairs whatever their direction with respect to the rubbing direction has been. Therefore disclination cores of the z-type [12] are not the appropriate model to describe these defects. The model of Meyer [I31 and Nehring [9] for the structures of nonsingular integer strength disclinations in nematics seems to be well suited to describe also these cholesteric structures. The smooth cores of this model can have two equivalent structures which are changing dis- continuously at singular points [9, 141. Besides this singular change, the core structure is also inverted by a smooth U-turn of the disclination to an anti- parallel direction [8]. Both effects together lead to parallel core polarities in the parallel branches of the U while the strengths of the two branches are opposite [9]. This is shown in figure 4 where the core

FIG. 4. - Two possible configurations of zero strength disclina- tions (horizontal) stretched between neighboured Grandjean-Cano lines (vertical) of the strength one half; the parallel core structures of the composant integer strength disclinations are indicated by arrows; a) singular connection at the higher ranked Grandjean- Cano line and lowered residual twist in the center; b) reversed

connections and residual twist increased by 2 n.

PAIRED DISCLINATIONS OF OPPOSITE STRENGTHS AND DISCLINATIONS OF STRENGTH 0 C3-97

states of the integer strength lines are marked by small arrows. Between the two lines of the pair a residual twist, differing from its surroundings by a full helical turn, is left. Because of its excess elastic twist energy it is responsible for the apparent attrac- tion of the two integer strength lines.

In spite of their opposite strengths, the two lines of the pair cannot annihilate with each other because the director turns in their cores are of the same sense, so that they are piling up instead of disap- pearing in the case of a further approach. This can be seen clearly in figure 5 where schematical cross

FIG. 5. - Directorfield cross sections of zero strength disclinations as observed in cholesteric wedge layers, seen in a direction parallel to the wedge gradient ; a) boundary alignment perpendicular, and

b) parallel to the wedge gradient.

sections of our model of the new defect structure are sketched for defect line directions perpendicular and parallel to the rubbing direction. The nonsingular cores of the composant lines which are the places of vertical director alignment are merged to a new structure. This new defect structure can be called a zero strength disclination since the deformation states on both sides of it are identical and because a Burger's circuit around the defect yields a value of zero. The structure of this defect type can b e calculated by minimizing the free elastic energy. For the sake of simplicity a nematic liquid crystal with a uniform alignment in infinity is supposed in the following paragraph.

[I] CHATELAIN, P., C. R. Hebd. Sian. Acad. Sci. 66 (1943) 105. [2] GRANDJEAN, F., C. R. Hebd. Sian. Acad. Sci 172 (1921) 71. [3] CANO, R., Bull. Soc. Fr. Min. Cristallogr. 91 (1968) 20. [4] FRIEDEL, G., Ann. Phys. 18 (1922) 273. [5] FRANK, F. C., Disc. F'araday Soc. 25 (1958) 19. [6] GROUPE D'ORSAY, J. Physique Colloq. 30 (1969) C4-38. [7] BOULIGAND, Y., J. Physique 35 (1974) 959. [8] STIEB, A. E., LABES, M. M., Mol. Cryst. Liq. Cryst. 45 (1978)

21.

5. Directorfield of nematic zero strength disclinations. Let us suppose that the elastic constants K for splay, twist, and bend are equal and that the defect structure is independent of the Z coordinate. At large distances from the Z axis the director may be aligned uni- formly parallel to it. The director N can be expressed by

sin 8. cos (p

N = sine-sinq [ O e ) where 8 is the angle between the director and the Z axis and (p the angle between the X axis and the projection of the director parallel to the Z axis. Using the Frank formula [5], the density of the free elastic energy is then

where the subscripts indicate the respective deriva- tives. The condition for the free elastic energy to be a minimum is given by the Euler-Lagrange equations :

((p,, + (p,,).sin 8 + 2 cos 8-(8, (p, + 8, (p,) = 0 .

The equations are solved [13] by :

8 = 2 arc tan ( R I J X ~ + y2)

(p = arc tan (YIX) + x/2

where R is a constant. This solution is similar to the directorfield pattern in the center of figure 5b. Equi- valent patterns can be obtained by a constant local rotation of the director N through an angle of n/2. For example, the rotation about the Y axis gives a structure similar to the center of figure 5a :

8* = arc cos 2 RY

R2 + X 2 + Y 2

- 2 RX (p* = arc tan

~2 - 1 2 - y2'

The common property of these directorfields is that, in contrast to the Frank disclinations [5], the free elastic energy per unit length is finite, in this case 4 nK. If a spontaneous twist and boundary plates of a cell are taken into consideration the solutions will change quantitatively but the topological properties of the directorfield will be preserved in a cholesteric layer.

[9] NEHRING, J., Phys. Rev. A 7 (1972) 1737. [lo] DE GENNES, P: G., Mol. Cryst. Liq. Cryst. 7 (1969) 325. [11] SCHEFFER, T. J., Phys. Rev. A 5 (1972) 1327. [I21 KLEMAN, M., FRIEDEL, J., J. Physique Colloq. 30 (1969) C443. [13] MEYER, R., Phil. Mag. 27 (1972) 405. [I41 NABARRO, F., J. Physique 33 (1972) 1089.