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Cholesteric Liquid Crystals: Defects
Cholesteric Liquid Crystals: Defects$ M Kleman, Institut de Physique du Globe de Paris, Paris, France r 2016 Elsevier Inc. All rights reserved.
1 2 3 4 4.1 4.2 4.3 5 6 7 8 References
The Classification of Defects Based on Symmetries Disclinations k, χ, s Flexibility of Quantized Line Disclinations: Continuous Defect Densities Effects of the Layer Structure Dislocations Focal Conic Domains, Polygonal Textures, Oily Streaks Robinson Spherulites Double Twist and Frustration The Classification of Defects Based on Topology Local Chiral Axis Concluding Remarks
1 1 3 5 5 5 5 6 7 7 8 8
For a long time the only defects to be seriously considered in condensed matter physics were dislocations in solids, because they control the non-linear (plastic) deformation properties of metals. But this limited point of view has been superseded by more general and profound considerations, which tie the nature of the defects in any material endowed with a non-trivial order parameter, (a solid, a liquid crystal, even an isotropic phase like superfluid He), to its various symmetries and/or to the topological properties of its order parameter (Michel, 1980). One speaks of stable topological defects (Toulouse and Kleman, 1976). These extensions have particularly benefited from the researches into cholesterics, which are rich in defects of different types. Although discovered as early as the beginning of the last century (Friedel, 1922), long before dislocations in solids, it is only since the early 1970s that defects in liquid crystals have been vigorously investigated.
1
The Classification of Defects Based on Symmetries
Dislocations are topological line defects: the symmetries of translation are broken along a linelike region. Nematics display two types of topological defects, disclinations (line defects) and hedgehogs (point defects), both breaking the symmetries of rotation of the phase. The cholesteric phase, as to it, possesses line defects of two types: disclinations, as in nematics, and dislocations, as in solids. This classification, which rests on the cholesteric symmetries (Friedel and Kleman, 1969), is a direct extension of the Volterra process for dislocations in solids (Friedel, 1960). Topological considerations (Volovik and Mineev, 1977) show further that there are no point defects, contrarily to nematics. A cholesteric phase is characterized, at the length scale which is of interest here, by its pitch p, i.e. the distance over which the direction of the director n rotates by an angle of 2p about the helicity axis, noted χ, Figure 1(a). A nematic phase is a cholesteric whose pitch is infinitely large. An easy way to obtain a pitch p large compared to the wavelength of light is to mix a nematic phase with a small proportion of Canada balsam, a chiral molecule. In these conditions the cholesteric structure is visible with the simple use of a polarizing microscope, Figure 1(b). The direction of the molecule, or more precisely the local optical axis, will be noted n; χ is the helicity axis; the third direction is noted s ¼ n χ. These 3 directions are directors, since changing any of them to its opposite does not change the cholesteric orientation. It follows that the repeat distance measured along χ is p/2, the half pitch. The variation of n, χ, s can be described by vector fields, which are called l, χ, t fields, respectively. Defects in cholesterics are relating to the symmetries of the structure, as follows: (1) due to the existence of three directors n, χ, s, there are three types of disclinations, showing great similarities with disclinations in nematics, and (2) the layered structure (of periodicity p/2) entails properties analogous to those of smectics: one expects to find dislocations, which break this periodicity. This is indeed what is observed but the actual situation is somewhat more subtle, and shows interplays between the different types of defects. These interplays are better understood when described with the two languages, Volterra and topological. Besides, cholesterics defects, due to the 1D periodicity of the phase, have some close relation with focal conic domains, which are typical defects of lamellar phases. ☆ Change History: March 2015. M. Kleman added Sections 3 and 7 and slightly modified the other sections. Figures 4 and 5 are newly added and Figure 1 has been slightly modified.
Reference Module in Materials Science and Materials Engineering
doi:10.1016/B978-0-12-803581-8.03128-3
1
2
Cholesteric Liquid Crystals: Defects
Figure 1 (a) Frank representation of the cholesteric orientation. The director n is represented as a nail whose head is supposed to be on the side of the reader and points to the back. The length of the nail is the projection on the plane of the sketch. (b) Cholesteric sample observed between crossed polarizers (8CB plus Canada balsam). Through each stripe the director n rotates by an angle of p about the helicity axis χ, which is constant and perpendicular to the stripe. Width of the stripe p/2E2300 nm. Photograph: courtesy Dr. C. Blanc.
2
Disclinations k, χ, s
According to the orientation of the line defect with respect to the direction of the broken rotation, one distinguishes wedge disclinations, which are parallel to the broken rotation, twist disclinations, which are Perpendicular to it, and mixed disclinations. A defect which makes singular the two sets of directors, t and χ, but leaves continuous the l field, is called a ‘l disclination’. Such a situation is pictured Figure 2(a) for a wedge disclination of strength k¼ 1/2, featuring a breaking of symmetry of angle p for the χ and the t vector fields: both rotate by an angle of p when traversing a loop surrounding the disclination. Consequently there is some place inside the loop where the χ and the s directors are not defined and where the defect line is located. This disclination is noted l. The l þ line, of opposite strength k¼ þ 1/2, is pictured Figure 2(b). Figure 2(c) pictures a t, i.e. a wedge line of strength k¼ 1/2, singular for the χ and l fields and continuous for the t field. One can also feature χ lines, i.e. disclination lines which preserve the continuity of the χ field; in fact, χ lines are directly related to dislocations and will be discussed apart. The resulting configurations are reminiscent of lines in nematics; do they also escape in the third dimension, as observed in nematics (R. B. Meyer)? The answer is negative for a k ¼ 1, also noted l2 þ , Figure 2(d), which cannot be made continuous for the two fields t and χ altogether. The same result holds for a t2 þ or a χ2 þ . On the other hand a l4 þ , a t4 þ and a χ4 þ can be made entirely continuous. More generally, any line l, t and χ with strength k¼ 2m, mAZ, can be made non-singular by escape in the third dimension. This result can be fully justified in the frame of the topological theory of defects, see below. Figure 3(a) pictures a χ þ wedge disclination (the χ field is continuous, and the l and t fields rotate by an angle of p about the line defect). The same object can as well be considered as a screw dislocation, since a p-rotation along the χ axis and a p/2 translation along the same axis add up to the same operation of symmetry. In the final configuration each cholesteric plane yields a 2D k¼ 1/2, Figure 3(a), whose configuration rotates helically along the χ line with a pitch p, the resulting Burgers vector is b¼ p/2. Figure 3(b) extends to edge dislocations vs twist disclinations the equivalence just demonstrated between screw dislocations and wedge χ disclinations. Dislocations and χ disclinations are fully equivalent, whatever the shape of the line. In the general case, one gets: b ¼ kp
½1
Cholesteric Liquid Crystals: Defects
3
Figure 2 Wedge disclinations in a cholesteric: (a) l, (b) l þ , (c) t , and (d) l2 þ (after Kleman and Lavrentovich, 2003).
Figure 3 Equivalences: (a) χ wedge disclinationscrew dislocation, (b, c) χ twist disclinationedge dislocation; in (c), the core of the χ is split into a l and a t þ (after Kleman and Lavrentovich, 2003).
3
Flexibility of Quantized Line Disclinations: Continuous Defect Densities
A wedge disclination L of rotation axis t along L, of rotation angle O is, formally, equivalent to a set of infinitesimal dislocations db, see Figure 4(a), spread over the cut surface S. And indeed the Volterra process for the construction of the line defect consists in
4
Cholesteric Liquid Crystals: Defects
Figure 4 (a) Wedge disclination L, of rotation vector X ¼Ot and its (formal) edge dislocation content. OM¼OM þ ¼OM; M þ M ¼2(OM)sinO2 , (b) displacement of the wedge disclination segment AB; appearance of twist segments AA0 , BB0 and of dislocation segments attached between AA0 and BB0 .
Figure 5 Curved disclination line L with X constant in magnitude and direction. Attached infinitesimal dislocations that relax the curvature, adding to a Burgers vector dbP. M is the running point on the cut surface of the disclination. The total Burgers vector of the dislocations attached between P and Q is 2sinO2 t MQ 2sinO2 t MP ¼ 2sinO2 t dP, according to the Volterra process (Friedel, 1960; Kleman and Friedel, 2008).
opening a dihedral void and filling it with matter (for simplicity we consider a disclination of negative strength), or equivalently in introducing in the present case a density of dislocations of Burgers vector (Kleman and Friedel, 2008) dbM ¼ 2sin
O t dM 2
½2
The energy of this wedge disclination would become prohibitive if the rotation vector X ¼ Ot is not along L; thus the displacement of the disclination from a position L to a position L0 is necessarily attended by the emission or absorption of a certain amount of dislocation densities. If L is displaced to L0 along a finite segment AB, two kinks AA0 , BB0 appear that join L to L0 and are twist disclination segments: they are perpendicular to W, which has been transported with the displacement of AB. There are now (real) dislocations attached to the twist parts, Figure 4(b). These considerations can be extended to a curved disclination line L (Kleman and Friedel); Figure 5 indicates the principle of the calculation of the infinitesimal Burgers vector segment dbP attached between P and Q on L, by application of the Volterra process: dbP ¼ 2sin
O t dP 2
½3
Thus the curvature of a disclination line L of any shape, of rotation vector X ¼ Ot, can be attributed to a density of continuous dislocations attached to the line. The flexibility of L is related to a variable such density. Whereas we have presented these relations between a curved disclination and its attached continuous dislocations as a formal process, it may have a physical reality. Consider a medium whose symmetry group contains a subset of continuous translations, as it is the case with liquid crystals. If the infinitesimal dislocation Burgers vectors attached to L belong to the group of symmetry of the medium, then the Volterra process that employs such Burgers vectors has a physical reality. We have restricted our discussion to the case of infinitesimal translational symmetries; the reasoning extends to infinitesimal rotational symmetries. If such symmetries exist, then infinitesimal disclinations are possible; they would relax a variation of X along the line.
Cholesteric Liquid Crystals: Defects
5
The continuous symmetries belonging to the symmetry group of a cholesteric are (1) any translation in a plane perpendicular to the helicity axis χ, (2) continuous helical rotations δX; pδX 2p , with δX ¼ δO χ an infinitesimal rotation along the helicity axis, pδX 2p an infinitesimal translation along the same axis. The corresponding infinitesimal lines are called dispiration densities. (1) Consider then a curved disclination line L with X constant. The associated attached Burgers vectors are along the direction X dP. Therefore there is no topological obstruction to the flexibility of a line L in a plane perpendicular to s if L is a l disclination, or in a plane perpendicular to k if L is a t disclination, but no other types of flexibility are allowed for these lines. On the other hand, a χ line could curve in any plane, which is a property one expects from a dislocation of Burgers vector b ¼ kpχ, which a χ disclination line is also. (2) The disclination lines L to which dispiration densities are attached are very remarkable: they are either l lines, with rotation p , are such that the rotation vector is along the vector along s, (or t lines, with rotation vector along k), have a constant torsion 2p binormal and that the tangent to the line is a s direction (or a k direction). These disclinations are of pure twist character. Among the solutions, the simplest ones are circular helices. For the demonstration of these results, see Kleman and Friedel (2008).
4 4.1
Effects of the Layer Structure Dislocations
An important property of dislocations in cholesterics is their possible splitting into l or t disclinations of opposite signs, at a distance multiple of p/4. Figure 3(c) pictures the same dislocation as in Figure 3(b), whose core is now split into a l and a t þ at a distance p/4 one from the other. The Burgers vector is b ¼ p/2. The splitting of a dislocation (equivalently: of a χ disclination) into two disclinations l or t of strength |k| ¼ 1/2, i.e., of rotation 7p, relates to the fact that the product of two opposite p rotations along two parallel axes is a translation (Friedel and Kleman). Split edge dislocations have been observed and studied in Grandjean–Cano wedge samples. The singularity (observed end-on) of Figure 1(b) is an edge dislocation.
4.2
Focal Conic Domains, Polygonal Textures, Oily Streaks
Most frequently, cholesterics present domains analog to the focal domains of smectics, as one might expect from the existence of a 1D periodicity. However the layers can suffer large thickness distortions (contrarily to smectics), since it is possible to let the pitch vary in a large range at the moderate expense of some twist energy K2. As a rule, the cholesteric layers are saddle-shaped in focal conic domains. Polygonal textures are sets of domains where the layers, observed on the side, are practically parallel, separated by boundaries marked by line defects (Figure 6). Oily streaks are complex aggregates of edge dislocations of large, opposite, Burgers vectors, present in samples whose boundaries are parallel to the cholesteric layers (Figure 7). They partition the sample in ideal domains of flat layers.
4.3
Robinson Spherulites
Semiflexible cholesteric biopolymers (but also some thermotropic cholesterics, Figure 8), have other types of layered textures, the so-called Robinson spherulites: (Pryce and Frank, 1958). The layers are approximatively along concentric spheres. The molecular orientation is singular, since it is impossible to outline a continuous field of directors on a sphere. The total strength of these
Figure 6 Polygonal texture in 8CB plus Canada balsam, crossed polarizers, p/2E2000 nm. (Courtesy Dr. C. Blanc.)
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Cholesteric Liquid Crystals: Defects
Figure 7 Oily streaks in a small pitch cholesteric, crossed polarizers, p/2E3000 nm. (Courtesy Pr. O.D. Lavrentovich.)
Figure 8 Robinson spherolites of 8CB plus Canada balsam, polarizers slightly uncrossed. Disclination line k ¼2 (a) in the plane of observation and (b) perpendicular to the plane of observation. (Courtesy Dr. C. Blanc.)
singularities is k¼ 2. And indeed observations tell us that either a k¼ 1 singularity goes through the whole spherulite along a diameter, or a k¼ 2 line defect extends from the surface to some point inside (generally the center of the spherulite). The k ¼ 2 line is believed to relax to a non-singular configuration, by escape (Bouligand and Livolant, 1994).
5
Double Twist and Frustration
The helicity of a cholesteric liquid crystal is due to the presence of a chirality in the individual molecules: the helical stacking is present in the direction of the χ axis, but in the direction perpendicular to this axis the molecules are constrained to parallel stacking. One expects that if the helical stacking could be achieved in both directions, the energy would be reduced. And indeed, namely when the sign of the elastic modulus K24 is positive, and K1cK3, the nucleation of this helical packing is favored. However this double twist cannot extend to a radius large compared to the natural pitch: it gets frustrated. In the thermodynamically stable blue phases, such finite double-twisted cylinders arrange in a 3D structure, separated by k¼ 12 line defects (Meiboom et al., 1981). One speaks of a frustrated phase. Notice that the pure cholesteric phase is also frustrated, as there is only one direction of twist. There is a geometry that relieves frustration totally, in which the molecules are aligned with the great circles of a 3D sphere of p radius q1 ¼ 2p in four dimensions; these great circles are in fact double twisted with a pitch p, and therefore represent an ideal unstrained, unfrustrated, cholesteric structure (Sethna, 1985). Another double twisted geometry is met in the DNA arrangement of the chromosome of a microscopic alga, Prorocentrum Micans, whose cholesteric structure is well documented (Livolant and Bouligand, 1978). The double-twisted region is bound by
Cholesteric Liquid Crystals: Defects
7
two k¼ 12 disclination lines which rotate helically about the chromosome axis (Kleman, 1987), and so limit the size of the chromosome.
6
The Classification of Defects Based on Topology
The topological classification of defects in cholesterics rests on the topology of the order parameter, whose geometry is that of the orthogonal set {n,χ,s} The rotations of this trihedron generate the group SO(3), which is also the 3D sphere S3 with antipodal points identified. The translations, which are equivalent to χ rotations, can be discarded. A loop surrounding a 2p rotation defect (k¼ 71) maps on SO(3) along a path which joins two antipodal points. Such a path – a loop in SO(3) – being non-trivial in the sense that it cannot be shrunk smoothly on SO(3) to a point, a k¼ 71 defect necessarily owns a singular core. On the other hand, a loop in SO(3) representing a 4p rotation defect (k¼ 72) can be shrunk to a point on SO(3), from which we infer that a 4p disclination requires no singular core. This important result is also true for 3He and biaxial nematics (Anderson and Toulouse, 1977) whose order parameter has the same geometry. The whole group of defects is isomorphic to the first homotopy group of the order parameter space R, which is SO(3) factorized by the four-element point group D2 of p-rotations about the directions {n,χ,s}: ½4
R ¼ SOð3Þ=D2
The fundamental group p1(R)¼ Q is the eight-element quaternion group, whose elements fall into five conjugacy classes C0 ¼ {I}, C 0 ¼ fJg, Cx ¼ {r, r}, Cy ¼ {s, s} and Cz ¼ {t, t}, each class corresponding to an irreducible Volterra defect, see Table 1. For example if the x direction is along the molecular director n, the elements {r} and { r} represent a l þ and l, or any l with an odd number of 7p rotations. According to the properties of the topological classification, it is possible to go smoothly from a {r} to a { r}, by the rule { r}¼ {t}{r}{ t}, which employs the non-commutativity properties of the elements of the group of quaternions, namely {r}2 ¼ { 1}, {r}{s}¼ { 1}{s}{r}¼ {t}, etc. The non-existence of point defects follows from the fact that the second homotopy group is trivial: p2(R)¼ 0. The non-commutativity of Q yields interesting entanglements effects between defects (Poénaru and Toulouse, 1977). They can also be discussed, although less systematically, via the Volterra process (Kleman and Lavrentovich, 2003). Since the n director field alone has a material reality, it is tempting to consider its topology independently of the two other directors. This approach has led to the notion of distortions with double topological character and to a classification of topologically stable configurations (without singularities) based on the Hopf mapping (Bouligand et al., 1978).
7
Local Chiral Axis
The local director n (the local optical axis) is well defined in a distorted cholesteric, but not the two other basis vectors χ, s. Similarly the cholesteric χ planes are well defined in the ground state of the cholesteric phase; do they transform continuously into well-defined surfaces in a distorted cholesteric, do they lose their individuality or, more simply, can one define χ surfaces in the distorted state, not necessarily correlated with the χ planes of the ground state? This last statement, as we show, makes sense. The analysis of the χ surfaces in distorted states starts from the consideration of the congruence of the lines of force of the director n, (a congruence is a set of lines depending on two parameters a and b). According to well-known results of differential geometry, there are partitions Pi of this set into subsets depending on one parameter, such that in each subset the lines of force have an envelope Eı,r; to this requirement corresponds a particular relation a¼ a(b), each subset being a surface Sı,r. If the envelopes are P pointlike, the full 3D envelope is a line of singularity Lı ¼ ⋃ rEı,r that crosses all the Sı,rs of the partition Pı; if they are linelike the singularity is a surface. Let us consider the first case. Since in general there are several possible partitions Pı, there are several Lı. One can show that a line Lı is contained in the subsets of the other partitions ja ı, in such a way that all the subsets of the same partition j merge along Lı, cf. Kleman (1973). The concept of congruence has been employed with the same purpose of analyzing the geometry of a cholesteric phase by Beller et al. (2014). Table 1 Correspondence between the Volterra (l, t, and χ) classification and the topological classification of line defects in cholesteric liquid crystals (m is an integer). The χ lines are equivalent to dislocations of Burgers vectors respectively equal to b¼ (2m þ 1)p for the C 0 class, b¼ (m þ 12)p for the Cχ class C0
C0
Cl(¼ Cx)
l(2m) t(2m) χ(2m)
l(2m þ 1) t(2m þ 1) χ(2m þ 1)
l(m þ 12)
Ct( ¼Cy) t(m þ 12)
Cχ( ¼Cz)
χ(m þ 12)
8
Cholesteric Liquid Crystals: Defects
These properties provide us with local frames of reference; at each point of a surface S (we drop the index ı that labels the partition) there is a triple n, N, N n, where N is the normal to S at the point. Two close triples are related by an infinitesimal rotation dop ¼ Kqp dxq
½5
dpr ¼ erst Kus pt dxu
½6
and one has
for a vector p that has constant components in the moving triple. Clearly, N plays the role of an helicity axis χ and S of a χ plane. According to eqn [5], Or ¼ KrsNs is the local rotation, the local pitch P obeys the relation 2p ¼ POrNr. Thus q¼
2p ¼ Krs Nr Ns ¼ Krr P
½7
the second equality stemming from the relation N ∇ N ¼ 0, which expresses the fact that the Ns are orthogonal to a set of surfaces χ. There are as many χ planes of this type as there are line singularities Lı, i.e., as there are partitions ı. Clearly, Lı is a wedge line for Pı, a twist line for Pj , ja ι.
8
Concluding Remarks
The concept of chirality has taken a great extension in soft matter physics, due to the manifold presence of chiral molecules. Defects and textures in cholesterics, apart their fundamental physical interest, are also an active field of research in cholesteric phases of biopolymers, like DNA, xanthan, etc. (Bouligand et al., 1998). Chirality appears under different ordered realizations (e.g., in the many twist grain boundaries phases) in which the notion of defect plays a crucial role. One can therefore expect that investigations on chirality cannot but increase in the future. And since cholesterics are in many respects the simplest chiral ordered systems, there is all the more reason to pursue the effort to fully understand them.
References Anderson, P.W., Toulouse, G., 1977. Phase slippage without vortex cores: vortex textures in superfluid 3He. Phys. Rev. Lett.. 508–511. Bouligand, Y., Livolant, F., 1994. The organization of cholesteric spherulites. J. Phys. 45, 1899–1923. Bouligand, Y., Derrida, B., Poénaru, V., Pomeau, Y., Toulouse, G., 1978. Distortions with double topological character: the case of cholesterics. J. Phys. 39, 863–867. Beller, D.A., Machon, T., Čopar, S., et al., 2014. Geometry of the cholesteric phase. Phys. Rev. X 4, 031050. Bouligand, Y., 1998. Defects and textures. In: Demus, D., et al. (Eds.), Handbook of Liquid Crystals. Fundamentals, vol. 1. Weinheim, Germany: Wiley-VCH. Friedel, G., 1922. Les états mésomorphes. Ann. Phys. (Paris) 18, 273–374. Friedel, J., Kleman, M., 1969. Lignes de dislocation dans les cholestériques. J. Phys. 30 (C4), 43–53. Friedel, J., 1960. Dislocations. Oxford: Pergamon. Kleman, M., 1973. Local chiral axis in a directional medium and wedge coponent of a disclination. J. Phys. 34, 931–935. Kleman, M., 1987. Effects of frustration in liquid crystals and polymers. Phys. Scr. T19, 565–572. Kleman, M., Friedel, J., 2008. Disclinations, dislocations and continuous defects: a reappraisal. Rev. Mod. Phys. 80, 61. Kleman, M., Lavrentovich, O.D., 2003. Soft Matter Physics, an Introduction. New York: Springer. Livolant, F., Bouligand, Y., 1978. New observations on the twisted arrangement of dinoflagellate chromosomes. Chromosoma 68, 21–44. Meiboom, S., Sethna, J.P., Anderson, P.W., Brinkman, W.F., 1981. Theory of the blue phase of cholesteric liquid crystals. Phys. Rev. Lett. 46, 1216–1219. Michel, L., 1980. Symmetry defects and broken symmetry. Configurations. Hidden symmetry. Rev. Mod. Phys. 52, 617–651. Poénaru, V., Toulouse, G., 1977. The crossing of defects in ordered media and the topology of 3-manifolds. J. Phys. 38, 887–895. Pryce, M.H.L., Frank, F.C., 1958. Liquid crystalline structures in solutions of a polypeptide, Part 2. Discuss. Faraday Soc. 25, 29–42. Sethna, J.P., 1985. Frustration, curvature, and defect lines in metallic glasses and the cholesteric blue phase. Phys. Rev. B 31, 6278. Toulouse, G., Kleman, M., 1976. Principles of a classification of defects in ordered media. J. Phys. Lett. (Paris) 37, L149–L151. Volovik, G.E., Mineev, V.P., 1977. Investigation of singularities in superfluid 3He and liquid crystals by the homotopy topology method. Sov. Phys. JETP Lett. 45, 1186–1196.
1 2 3 4 4.1 4.2 4.3 5 6 7 8 References
The Classification of Defects Based on Symmetries Disclinations k, χ, s Flexibility of Quantized Line Disclinations: Continuous Defect Densities Effects of the Layer Structure Dislocations Focal Conic Domains, Polygonal Textures, Oily Streaks Robinson Spherulites Double Twist and Frustration The Classification of Defects Based on Topology Local Chiral Axis Concluding Remarks
1 1 3 5 5 5 5 6 7 7 8 8
For a long time the only defects to be seriously considered in condensed matter physics were dislocations in solids, because they control the non-linear (plastic) deformation properties of metals. But this limited point of view has been superseded by more general and profound considerations, which tie the nature of the defects in any material endowed with a non-trivial order parameter, (a solid, a liquid crystal, even an isotropic phase like superfluid He), to its various symmetries and/or to the topological properties of its order parameter (Michel, 1980). One speaks of stable topological defects (Toulouse and Kleman, 1976). These extensions have particularly benefited from the researches into cholesterics, which are rich in defects of different types. Although discovered as early as the beginning of the last century (Friedel, 1922), long before dislocations in solids, it is only since the early 1970s that defects in liquid crystals have been vigorously investigated.
1
The Classification of Defects Based on Symmetries
Dislocations are topological line defects: the symmetries of translation are broken along a linelike region. Nematics display two types of topological defects, disclinations (line defects) and hedgehogs (point defects), both breaking the symmetries of rotation of the phase. The cholesteric phase, as to it, possesses line defects of two types: disclinations, as in nematics, and dislocations, as in solids. This classification, which rests on the cholesteric symmetries (Friedel and Kleman, 1969), is a direct extension of the Volterra process for dislocations in solids (Friedel, 1960). Topological considerations (Volovik and Mineev, 1977) show further that there are no point defects, contrarily to nematics. A cholesteric phase is characterized, at the length scale which is of interest here, by its pitch p, i.e. the distance over which the direction of the director n rotates by an angle of 2p about the helicity axis, noted χ, Figure 1(a). A nematic phase is a cholesteric whose pitch is infinitely large. An easy way to obtain a pitch p large compared to the wavelength of light is to mix a nematic phase with a small proportion of Canada balsam, a chiral molecule. In these conditions the cholesteric structure is visible with the simple use of a polarizing microscope, Figure 1(b). The direction of the molecule, or more precisely the local optical axis, will be noted n; χ is the helicity axis; the third direction is noted s ¼ n χ. These 3 directions are directors, since changing any of them to its opposite does not change the cholesteric orientation. It follows that the repeat distance measured along χ is p/2, the half pitch. The variation of n, χ, s can be described by vector fields, which are called l, χ, t fields, respectively. Defects in cholesterics are relating to the symmetries of the structure, as follows: (1) due to the existence of three directors n, χ, s, there are three types of disclinations, showing great similarities with disclinations in nematics, and (2) the layered structure (of periodicity p/2) entails properties analogous to those of smectics: one expects to find dislocations, which break this periodicity. This is indeed what is observed but the actual situation is somewhat more subtle, and shows interplays between the different types of defects. These interplays are better understood when described with the two languages, Volterra and topological. Besides, cholesterics defects, due to the 1D periodicity of the phase, have some close relation with focal conic domains, which are typical defects of lamellar phases. ☆ Change History: March 2015. M. Kleman added Sections 3 and 7 and slightly modified the other sections. Figures 4 and 5 are newly added and Figure 1 has been slightly modified.
Reference Module in Materials Science and Materials Engineering
doi:10.1016/B978-0-12-803581-8.03128-3
1
2
Cholesteric Liquid Crystals: Defects
Figure 1 (a) Frank representation of the cholesteric orientation. The director n is represented as a nail whose head is supposed to be on the side of the reader and points to the back. The length of the nail is the projection on the plane of the sketch. (b) Cholesteric sample observed between crossed polarizers (8CB plus Canada balsam). Through each stripe the director n rotates by an angle of p about the helicity axis χ, which is constant and perpendicular to the stripe. Width of the stripe p/2E2300 nm. Photograph: courtesy Dr. C. Blanc.
2
Disclinations k, χ, s
According to the orientation of the line defect with respect to the direction of the broken rotation, one distinguishes wedge disclinations, which are parallel to the broken rotation, twist disclinations, which are Perpendicular to it, and mixed disclinations. A defect which makes singular the two sets of directors, t and χ, but leaves continuous the l field, is called a ‘l disclination’. Such a situation is pictured Figure 2(a) for a wedge disclination of strength k¼ 1/2, featuring a breaking of symmetry of angle p for the χ and the t vector fields: both rotate by an angle of p when traversing a loop surrounding the disclination. Consequently there is some place inside the loop where the χ and the s directors are not defined and where the defect line is located. This disclination is noted l. The l þ line, of opposite strength k¼ þ 1/2, is pictured Figure 2(b). Figure 2(c) pictures a t, i.e. a wedge line of strength k¼ 1/2, singular for the χ and l fields and continuous for the t field. One can also feature χ lines, i.e. disclination lines which preserve the continuity of the χ field; in fact, χ lines are directly related to dislocations and will be discussed apart. The resulting configurations are reminiscent of lines in nematics; do they also escape in the third dimension, as observed in nematics (R. B. Meyer)? The answer is negative for a k ¼ 1, also noted l2 þ , Figure 2(d), which cannot be made continuous for the two fields t and χ altogether. The same result holds for a t2 þ or a χ2 þ . On the other hand a l4 þ , a t4 þ and a χ4 þ can be made entirely continuous. More generally, any line l, t and χ with strength k¼ 2m, mAZ, can be made non-singular by escape in the third dimension. This result can be fully justified in the frame of the topological theory of defects, see below. Figure 3(a) pictures a χ þ wedge disclination (the χ field is continuous, and the l and t fields rotate by an angle of p about the line defect). The same object can as well be considered as a screw dislocation, since a p-rotation along the χ axis and a p/2 translation along the same axis add up to the same operation of symmetry. In the final configuration each cholesteric plane yields a 2D k¼ 1/2, Figure 3(a), whose configuration rotates helically along the χ line with a pitch p, the resulting Burgers vector is b¼ p/2. Figure 3(b) extends to edge dislocations vs twist disclinations the equivalence just demonstrated between screw dislocations and wedge χ disclinations. Dislocations and χ disclinations are fully equivalent, whatever the shape of the line. In the general case, one gets: b ¼ kp
½1
Cholesteric Liquid Crystals: Defects
3
Figure 2 Wedge disclinations in a cholesteric: (a) l, (b) l þ , (c) t , and (d) l2 þ (after Kleman and Lavrentovich, 2003).
Figure 3 Equivalences: (a) χ wedge disclinationscrew dislocation, (b, c) χ twist disclinationedge dislocation; in (c), the core of the χ is split into a l and a t þ (after Kleman and Lavrentovich, 2003).
3
Flexibility of Quantized Line Disclinations: Continuous Defect Densities
A wedge disclination L of rotation axis t along L, of rotation angle O is, formally, equivalent to a set of infinitesimal dislocations db, see Figure 4(a), spread over the cut surface S. And indeed the Volterra process for the construction of the line defect consists in
4
Cholesteric Liquid Crystals: Defects
Figure 4 (a) Wedge disclination L, of rotation vector X ¼Ot and its (formal) edge dislocation content. OM¼OM þ ¼OM; M þ M ¼2(OM)sinO2 , (b) displacement of the wedge disclination segment AB; appearance of twist segments AA0 , BB0 and of dislocation segments attached between AA0 and BB0 .
Figure 5 Curved disclination line L with X constant in magnitude and direction. Attached infinitesimal dislocations that relax the curvature, adding to a Burgers vector dbP. M is the running point on the cut surface of the disclination. The total Burgers vector of the dislocations attached between P and Q is 2sinO2 t MQ 2sinO2 t MP ¼ 2sinO2 t dP, according to the Volterra process (Friedel, 1960; Kleman and Friedel, 2008).
opening a dihedral void and filling it with matter (for simplicity we consider a disclination of negative strength), or equivalently in introducing in the present case a density of dislocations of Burgers vector (Kleman and Friedel, 2008) dbM ¼ 2sin
O t dM 2
½2
The energy of this wedge disclination would become prohibitive if the rotation vector X ¼ Ot is not along L; thus the displacement of the disclination from a position L to a position L0 is necessarily attended by the emission or absorption of a certain amount of dislocation densities. If L is displaced to L0 along a finite segment AB, two kinks AA0 , BB0 appear that join L to L0 and are twist disclination segments: they are perpendicular to W, which has been transported with the displacement of AB. There are now (real) dislocations attached to the twist parts, Figure 4(b). These considerations can be extended to a curved disclination line L (Kleman and Friedel); Figure 5 indicates the principle of the calculation of the infinitesimal Burgers vector segment dbP attached between P and Q on L, by application of the Volterra process: dbP ¼ 2sin
O t dP 2
½3
Thus the curvature of a disclination line L of any shape, of rotation vector X ¼ Ot, can be attributed to a density of continuous dislocations attached to the line. The flexibility of L is related to a variable such density. Whereas we have presented these relations between a curved disclination and its attached continuous dislocations as a formal process, it may have a physical reality. Consider a medium whose symmetry group contains a subset of continuous translations, as it is the case with liquid crystals. If the infinitesimal dislocation Burgers vectors attached to L belong to the group of symmetry of the medium, then the Volterra process that employs such Burgers vectors has a physical reality. We have restricted our discussion to the case of infinitesimal translational symmetries; the reasoning extends to infinitesimal rotational symmetries. If such symmetries exist, then infinitesimal disclinations are possible; they would relax a variation of X along the line.
Cholesteric Liquid Crystals: Defects
5
The continuous symmetries belonging to the symmetry group of a cholesteric are (1) any translation in a plane perpendicular to the helicity axis χ, (2) continuous helical rotations δX; pδX 2p , with δX ¼ δO χ an infinitesimal rotation along the helicity axis, pδX 2p an infinitesimal translation along the same axis. The corresponding infinitesimal lines are called dispiration densities. (1) Consider then a curved disclination line L with X constant. The associated attached Burgers vectors are along the direction X dP. Therefore there is no topological obstruction to the flexibility of a line L in a plane perpendicular to s if L is a l disclination, or in a plane perpendicular to k if L is a t disclination, but no other types of flexibility are allowed for these lines. On the other hand, a χ line could curve in any plane, which is a property one expects from a dislocation of Burgers vector b ¼ kpχ, which a χ disclination line is also. (2) The disclination lines L to which dispiration densities are attached are very remarkable: they are either l lines, with rotation p , are such that the rotation vector is along the vector along s, (or t lines, with rotation vector along k), have a constant torsion 2p binormal and that the tangent to the line is a s direction (or a k direction). These disclinations are of pure twist character. Among the solutions, the simplest ones are circular helices. For the demonstration of these results, see Kleman and Friedel (2008).
4 4.1
Effects of the Layer Structure Dislocations
An important property of dislocations in cholesterics is their possible splitting into l or t disclinations of opposite signs, at a distance multiple of p/4. Figure 3(c) pictures the same dislocation as in Figure 3(b), whose core is now split into a l and a t þ at a distance p/4 one from the other. The Burgers vector is b ¼ p/2. The splitting of a dislocation (equivalently: of a χ disclination) into two disclinations l or t of strength |k| ¼ 1/2, i.e., of rotation 7p, relates to the fact that the product of two opposite p rotations along two parallel axes is a translation (Friedel and Kleman). Split edge dislocations have been observed and studied in Grandjean–Cano wedge samples. The singularity (observed end-on) of Figure 1(b) is an edge dislocation.
4.2
Focal Conic Domains, Polygonal Textures, Oily Streaks
Most frequently, cholesterics present domains analog to the focal domains of smectics, as one might expect from the existence of a 1D periodicity. However the layers can suffer large thickness distortions (contrarily to smectics), since it is possible to let the pitch vary in a large range at the moderate expense of some twist energy K2. As a rule, the cholesteric layers are saddle-shaped in focal conic domains. Polygonal textures are sets of domains where the layers, observed on the side, are practically parallel, separated by boundaries marked by line defects (Figure 6). Oily streaks are complex aggregates of edge dislocations of large, opposite, Burgers vectors, present in samples whose boundaries are parallel to the cholesteric layers (Figure 7). They partition the sample in ideal domains of flat layers.
4.3
Robinson Spherulites
Semiflexible cholesteric biopolymers (but also some thermotropic cholesterics, Figure 8), have other types of layered textures, the so-called Robinson spherulites: (Pryce and Frank, 1958). The layers are approximatively along concentric spheres. The molecular orientation is singular, since it is impossible to outline a continuous field of directors on a sphere. The total strength of these
Figure 6 Polygonal texture in 8CB plus Canada balsam, crossed polarizers, p/2E2000 nm. (Courtesy Dr. C. Blanc.)
6
Cholesteric Liquid Crystals: Defects
Figure 7 Oily streaks in a small pitch cholesteric, crossed polarizers, p/2E3000 nm. (Courtesy Pr. O.D. Lavrentovich.)
Figure 8 Robinson spherolites of 8CB plus Canada balsam, polarizers slightly uncrossed. Disclination line k ¼2 (a) in the plane of observation and (b) perpendicular to the plane of observation. (Courtesy Dr. C. Blanc.)
singularities is k¼ 2. And indeed observations tell us that either a k¼ 1 singularity goes through the whole spherulite along a diameter, or a k¼ 2 line defect extends from the surface to some point inside (generally the center of the spherulite). The k ¼ 2 line is believed to relax to a non-singular configuration, by escape (Bouligand and Livolant, 1994).
5
Double Twist and Frustration
The helicity of a cholesteric liquid crystal is due to the presence of a chirality in the individual molecules: the helical stacking is present in the direction of the χ axis, but in the direction perpendicular to this axis the molecules are constrained to parallel stacking. One expects that if the helical stacking could be achieved in both directions, the energy would be reduced. And indeed, namely when the sign of the elastic modulus K24 is positive, and K1cK3, the nucleation of this helical packing is favored. However this double twist cannot extend to a radius large compared to the natural pitch: it gets frustrated. In the thermodynamically stable blue phases, such finite double-twisted cylinders arrange in a 3D structure, separated by k¼ 12 line defects (Meiboom et al., 1981). One speaks of a frustrated phase. Notice that the pure cholesteric phase is also frustrated, as there is only one direction of twist. There is a geometry that relieves frustration totally, in which the molecules are aligned with the great circles of a 3D sphere of p radius q1 ¼ 2p in four dimensions; these great circles are in fact double twisted with a pitch p, and therefore represent an ideal unstrained, unfrustrated, cholesteric structure (Sethna, 1985). Another double twisted geometry is met in the DNA arrangement of the chromosome of a microscopic alga, Prorocentrum Micans, whose cholesteric structure is well documented (Livolant and Bouligand, 1978). The double-twisted region is bound by
Cholesteric Liquid Crystals: Defects
7
two k¼ 12 disclination lines which rotate helically about the chromosome axis (Kleman, 1987), and so limit the size of the chromosome.
6
The Classification of Defects Based on Topology
The topological classification of defects in cholesterics rests on the topology of the order parameter, whose geometry is that of the orthogonal set {n,χ,s} The rotations of this trihedron generate the group SO(3), which is also the 3D sphere S3 with antipodal points identified. The translations, which are equivalent to χ rotations, can be discarded. A loop surrounding a 2p rotation defect (k¼ 71) maps on SO(3) along a path which joins two antipodal points. Such a path – a loop in SO(3) – being non-trivial in the sense that it cannot be shrunk smoothly on SO(3) to a point, a k¼ 71 defect necessarily owns a singular core. On the other hand, a loop in SO(3) representing a 4p rotation defect (k¼ 72) can be shrunk to a point on SO(3), from which we infer that a 4p disclination requires no singular core. This important result is also true for 3He and biaxial nematics (Anderson and Toulouse, 1977) whose order parameter has the same geometry. The whole group of defects is isomorphic to the first homotopy group of the order parameter space R, which is SO(3) factorized by the four-element point group D2 of p-rotations about the directions {n,χ,s}: ½4
R ¼ SOð3Þ=D2
The fundamental group p1(R)¼ Q is the eight-element quaternion group, whose elements fall into five conjugacy classes C0 ¼ {I}, C 0 ¼ fJg, Cx ¼ {r, r}, Cy ¼ {s, s} and Cz ¼ {t, t}, each class corresponding to an irreducible Volterra defect, see Table 1. For example if the x direction is along the molecular director n, the elements {r} and { r} represent a l þ and l, or any l with an odd number of 7p rotations. According to the properties of the topological classification, it is possible to go smoothly from a {r} to a { r}, by the rule { r}¼ {t}{r}{ t}, which employs the non-commutativity properties of the elements of the group of quaternions, namely {r}2 ¼ { 1}, {r}{s}¼ { 1}{s}{r}¼ {t}, etc. The non-existence of point defects follows from the fact that the second homotopy group is trivial: p2(R)¼ 0. The non-commutativity of Q yields interesting entanglements effects between defects (Poénaru and Toulouse, 1977). They can also be discussed, although less systematically, via the Volterra process (Kleman and Lavrentovich, 2003). Since the n director field alone has a material reality, it is tempting to consider its topology independently of the two other directors. This approach has led to the notion of distortions with double topological character and to a classification of topologically stable configurations (without singularities) based on the Hopf mapping (Bouligand et al., 1978).
7
Local Chiral Axis
The local director n (the local optical axis) is well defined in a distorted cholesteric, but not the two other basis vectors χ, s. Similarly the cholesteric χ planes are well defined in the ground state of the cholesteric phase; do they transform continuously into well-defined surfaces in a distorted cholesteric, do they lose their individuality or, more simply, can one define χ surfaces in the distorted state, not necessarily correlated with the χ planes of the ground state? This last statement, as we show, makes sense. The analysis of the χ surfaces in distorted states starts from the consideration of the congruence of the lines of force of the director n, (a congruence is a set of lines depending on two parameters a and b). According to well-known results of differential geometry, there are partitions Pi of this set into subsets depending on one parameter, such that in each subset the lines of force have an envelope Eı,r; to this requirement corresponds a particular relation a¼ a(b), each subset being a surface Sı,r. If the envelopes are P pointlike, the full 3D envelope is a line of singularity Lı ¼ ⋃ rEı,r that crosses all the Sı,rs of the partition Pı; if they are linelike the singularity is a surface. Let us consider the first case. Since in general there are several possible partitions Pı, there are several Lı. One can show that a line Lı is contained in the subsets of the other partitions ja ı, in such a way that all the subsets of the same partition j merge along Lı, cf. Kleman (1973). The concept of congruence has been employed with the same purpose of analyzing the geometry of a cholesteric phase by Beller et al. (2014). Table 1 Correspondence between the Volterra (l, t, and χ) classification and the topological classification of line defects in cholesteric liquid crystals (m is an integer). The χ lines are equivalent to dislocations of Burgers vectors respectively equal to b¼ (2m þ 1)p for the C 0 class, b¼ (m þ 12)p for the Cχ class C0
C0
Cl(¼ Cx)
l(2m) t(2m) χ(2m)
l(2m þ 1) t(2m þ 1) χ(2m þ 1)
l(m þ 12)
Ct( ¼Cy) t(m þ 12)
Cχ( ¼Cz)
χ(m þ 12)
8
Cholesteric Liquid Crystals: Defects
These properties provide us with local frames of reference; at each point of a surface S (we drop the index ı that labels the partition) there is a triple n, N, N n, where N is the normal to S at the point. Two close triples are related by an infinitesimal rotation dop ¼ Kqp dxq
½5
dpr ¼ erst Kus pt dxu
½6
and one has
for a vector p that has constant components in the moving triple. Clearly, N plays the role of an helicity axis χ and S of a χ plane. According to eqn [5], Or ¼ KrsNs is the local rotation, the local pitch P obeys the relation 2p ¼ POrNr. Thus q¼
2p ¼ Krs Nr Ns ¼ Krr P
½7
the second equality stemming from the relation N ∇ N ¼ 0, which expresses the fact that the Ns are orthogonal to a set of surfaces χ. There are as many χ planes of this type as there are line singularities Lı, i.e., as there are partitions ı. Clearly, Lı is a wedge line for Pı, a twist line for Pj , ja ι.
8
Concluding Remarks
The concept of chirality has taken a great extension in soft matter physics, due to the manifold presence of chiral molecules. Defects and textures in cholesterics, apart their fundamental physical interest, are also an active field of research in cholesteric phases of biopolymers, like DNA, xanthan, etc. (Bouligand et al., 1998). Chirality appears under different ordered realizations (e.g., in the many twist grain boundaries phases) in which the notion of defect plays a crucial role. One can therefore expect that investigations on chirality cannot but increase in the future. And since cholesterics are in many respects the simplest chiral ordered systems, there is all the more reason to pursue the effort to fully understand them.
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