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12

Some Aspects of Polymer Dispersed and Polymer Stabilized Chiral Liquid Crystals Gregory P. Crawford, Daniel SvensÏek, and Slobodan ZÏumer

Systems consisting of a low molar mass liquid crystal and a polymer are currently of great interest with respect to applications and due to the intriguing ®nite size eëcts. This chapter describes some aspects of liquid crystals embedded in dense polymer binders and low concentration polymer networks that modify the bulk liquid crystal phase, with added emphasis on chirality. The introduction to the phenomenological description is followed by the modeling of ®eld eëcted chiral nematic droplets in polymer-dispersed liquid crystal systems. Next the orientational ordering induced by polymer networks is described, and ®nally the usefulness of these materials for directview re¯ection displays, bistable displays, and light valves is reviewed.

12.1

Introduction

The nature of con®ning surfaces on most materials (e.g., solids and liquids) has little in¯uence on the internal properties, and, for the most part, can be safely ignored even for small-scale samples. Liquid crystalline materials, on the other hand, are an exception to this rule. The con®nement of a liquid crystal material can modify ordering over macroscopic distances; in some cases, the bulk order of a liquid crystal material is determined in millimetersized samples. This unique ability to manipulate the bulk liquid crystal structure via properties of the boundary, along with the ability to modify and control that structure using applied electric and/or magnetic ®elds, is the main reason why liquid crystal materials are ideal for a host of electro-optical applications. In addition to the direct eëct of the surface in real liquid crystal samples, the eëcts of the con®ning geometry are also important. The simplest example of a particular geometry is the liquid crystalline cell with the con®nement between parallel plates, separated by a few micrometers, which is widely used in ¯at panel liquid crystal display applications. In recent years there has been a move to study more complex con®ned liquid crystal systems with a large surface-to-volume ratio [1], [2]. Fundamentally, the eëct of geometry and surface-induced ordering on liquid crystalline phases 375

376

G.P. Crawford, D. SvensÏek, and S. ZÏumer

Figure 12.1. Schematic presentation of the operation of a PDLC shutter.

is intriguing and conceptually challenging. On the other hand, it is perhaps even more important that the applicability of these complex microcon®ned liquid crystals is gaining widespread attention for scattering, re¯ective, and bistable display applications. In the mid-1980s the usefulness of con®ning liquid crystals to spherical droplets became readily apparent [3]. This came almost a century after the ®rst identi®cation of the liquid crystal phase in 1887 by Reisser and over 80 years after the ®rst liquid crystal droplets were observed by Lemman in a viscous liquid medium. Figure 12.1 presents a simple schematic of the operation of a polymer-dispersed liquid crystal (PDLC) device [2], [3], which depicts the ®rst practical demonstration of liquid crystal droplets. A rigid polymer binder permanently supports the liquid crystal droplets. In the passive state (no applied voltage), the symmetry axes of the liquid crystal con®gurations within the droplets are randomly oriented. The droplets in Figure 12.1 show the well-known bipolar con®guration, which is most common in nematic liquid crystal and polymer systems. This randomly oriented droplet system scatters light because of the mismatch between the average index of refraction of the droplet and the polymer binder. In its active state when a voltage of su½cient magnitude is applied, the droplets will reorient their symmetry axes parallel to the applied ®eld direction for materials with a positive dielectric anisotropy. If care is taken to select a liquid crystal with an ordinary index of refraction n0 , that approximately matches that of the polymer np , the material is optically homogenous and is therefore transparent. There have been a number of reviews on conventional PDLCs in the literature [2], [4]. In the spirit of this book, we will focus our attention to liquid crystal polymer systems that utilize chiral liquid crystal materials. Con®ned chiral nematic (N*) liquid crystals have also been the subject of numerous investigations, just as their nematic counterpart described above

12. Polymer Dispersed and Stabilized Chiral Liquid Crystals

377

[5], [6], [7]. Consequently, the development of PDLCs has been extended to chiral liquid crystals. Crooker and Yang [9] and Kitzerow and Crooker [10] recognized the usefulness of con®ned chiral materials for re¯ective display applications. These papers, in the early 1990s, rejuvenated interest in these systems beyond the scope of applications, and stimulated intense basic research in the area of diërent con®ned chiral liquid crystals [7], [8], [11], [12], [13], [14], [15], [16]. During the same time-frame of the ®rst reports of chiral PDLCs, another polymer-based liquid crystal system was reported by Hikmet [17] (1990). In contrast to PDLCs, these systems contained only a small volume fraction of a highly cross-linked mesogenic polymer network dispersed in the liquid crystal. Shortly after the report by Hikmet on lowconcentration networks in nematic liquid crystals, Yang and coworkers [18] (1992) adapted these materials to chiral nematic liquid crystal systems. These polymer-stabilized chiral liquid crystals were immediately realized to be useful for normal and reverse mode light shutters and re¯ective displays with bistable memory [19]. In this chapter we will present selected aspects important for the understanding of polymer-dispersed and polymer-stabilized chiral nematic liquid crystals. In addition to a brief review of applications of chiral nematic± polymer composites, we have in particular selected two seldom-discussed topics: chiral nematic structures in spherical cavities and polymer-induced ordering in the liquid crystal. They must be taken as rather simple but instructive examples of ordering in these systems. The chapter is organized in the following way: it begins with a brief introduction to materials and with a phenomenological description of nematic and chiral nematic ordering, it then treats in detail ®eld-induced structural transition in chiral droplets embedded in a polymer matrix. The next section is devoted to polymer-induced pretransitional orientational ordering and to the network structure determination. Finally, several optical display applications are described.

12.2

Polymer-Liquid Crystal Dispersions

Polymer liquid crystal dispersions have been extensively reviewed elsewhere [2], [4], [19], therefore we will only provide a sample of the rich variety of polymer morphologies that are possible. These complex systems are developed using a phase separation process. Separations caused by free radical or photo-induced polymerization in a mixture of monomers and liquid crystal are the most common. Thermally induced phase separation is used with a solution of thermoplastic polymers and liquid crystals. Solvent-induced phase separation based on solvent evaporation is usually used with thermoplastics which decompose before melting. Morphologies of the polymer matrix on submicrometer scale are most directly observed by scanning electron microscopy (SEM) after removal of the liquid crystal. Figure 12.2 shows three SEM photographs that are radically diërent. Figure 12.2(a) shows a con-

378

G.P. Crawford, D. SvensÏek, and S. ZÏumer (a)

Figure 12.2. Polymer morphology recorded with scanning electron microscopy: (a) classical PDLC display material and (b) and (c) polymer network morphologies formed under diërent initial conditions (courtesy of Y.K. Fung).

(b)

(c)

ventional PDLC matrix where the polymer content was highÐapproximately 50% by weight. When only low concentrations of polymer are used, as shown in Figure 12.2(b) and (c), fascinating polymer network morphologies are formed that strongly depend on conditions under which they were phase separated. The structure shown in Figure 12.2(b) was formed under homeotropic alignment conditions in zero ®eld, while that of Figure 12.2(c) was formed under homogenous surface anchoring conditions in the presence of a strong electric ®eld. It is very di½cult to classify the morphologies because of their strong dependence on initial conditions, but it can be said that conventional PDLCs have droplets that are separated from each other, while the low-concentration polymer networks are a bicontinuous medium with the interconnected cavities.


379

Conventional chiral PDLCs are formed using a liquid crystal material with a negative dielectric anisotropy, doped with a chiral component [9], [13], [19]. A typical mixture is composed of the commercially available Merck material ZLI-2806, which has a negative dielectric anisotropy, and a chiral dopant such as CE2, also available from Merck. These materials are combined with a thermosetting polymer, polyvinyl butyral (PVB) in the proportions ZLI-2806 (33 wt.%), CE2 (20 wt.%), and PVB (47 wt.%). The solution is completely mixed in chloroform and decanted onto conducting substrates at room temperature. Chloroform evaporates, the sample is heated to 140 C, compressed with another conduction substrate, and allowed to cool. Micrometer spacers are typically used to control the cell gap between the two substrates. The droplet size is determined by the cooling rate [10]. This is only one example of a cholesteric PDLC formulation that re¯ects light at approximately 520 nm. The SEM photographs of these materials reveal isolated droplets, reminiscent of the photograph in Figure 12.2(a). This matrix promotes the planar anchoring of liquid crystal molecules and thus the formation of spherical and oblate chiral structures discussed in Section 12.4. The low-concentration networks, presented in Figure 12.2(b) and (c), are fabricated by mixing small concentrations of reactive monomer (0.5±5 wt.%) into a liquid crystal [17], [18], [19]. Due to the similarity of the molecular structure of the reactive monomer, it typically aligns within the liquid crystal con®guration through steric hindrance. Reactive monomers are typically acrylate based and therefore they are polymerized by ultraviolet (UV) light. Many diërent structures have been synthesized and used for polymerstabilized liquid crystals [19]. The polymer morphologies strongly depend on initial conditions prior to the UV polymerization, monomer concentration, and UV curing conditions [20], [21]. Studies by Dierking have revealed that mean pore size in the polymer networks strongly depends on the curing temperature [20] and UV curing conditions [21], and that this network is responsible for two diërent switching regions [22] upon application of an electric ®eld. A detailed description of the investigation of the network structure via pretransitional orientational ordering, induced by a polymer in an isotropic liquid crystal, is given in Section 12.7.

12.3

Description of Nematic and Chiral Nematic Ordering

In this section we brie¯y discuss the most important aspects of the phenomenological description of nematic and chiral nematic systems, needed for the understanding of our particular examples in Sections 12.4 and 12.5. More details on this subject can be found in the general references [25], [26], [27], [28], [29], [30], [39], and references therein. Unlike ordinary liquid, a nematic liquid crystal is macroscopically aniso-

380


tropic, meaning that it exhibits a long-range order of molecular orientation. On average its molecules (essentially being rod- or disk-like) point in a certain direction, which is represented by a unit vector called the director ~ n. In an undeformed homogeneous nematic, ~ n is constant throughout the sample. In an undeformed chiral nematic (also called cholesteric) however, the director rotates about a constant perpendicular helical axis as we go along it. Locally the cholesteric appears like the ordinary nematic, only over larger scales characterized by the pitch length (the length of one revolution), spiraling becomes manifest. The pitch length in the range of wavelengths of visible light is responsible for the remarkable optical properties of cholesterics (see [25], for instance).

12.3.1

The Order Parameter

To describe the distribution of molecular orientations, a second-rank symmetric traceless tensor is introduced as an order parameter [27, p. 168]: Qij hdi dj i

1 3 dij ;

12:1

where h i stands for the statistical average over all possible molecular orientations represented by the unit vectors ~ d. A properly weighted unit tensor has been subtracted so that the order parameter is zero in the isotropic phase. The order parameter (12.1) is nothing but a traceless quadrupole tensor of the distribution of molecular orientations. It is the ®rst nontrivial nonzero moment, too, since ~ n represents the same orientation as ~ n (the director is a ``headless'' vector) and therefore the dipole moment vanishes. A symmetric traceless tensor contains ®ve independent scalar quantities, hence it follows that one needs that many scalars to describe nematic ordering in a complete manner. Most conveniently, the tensorial order parameter is viewed in its eigensystem 3 2 b s 7 6 3 7 6 b s 7 6 12:2 Q6 7; 7 6 3 5 4 2s 3 the z-axis was chosen to coincide with the director ~ n. In (12.2), two quantities have been introduced, s being the scalar order parameter s

3hcos 2 Qi 2

1

;

cos Q ~ n~ d;

12:3

and b giving the degree of biaxiality, i.e., nonuniformity of the distribution projected onto the plane perpendicular to the director, ranging from b s 1 (reached when hdy2 i 0) to b s 1 (achieved for hdx2 i 0),


381

whereas b 0 corresponds to the uniaxial distribution, of course. The scalar order parameter s can be both positive or negative, as seen from (12.3). In the latter case, molecular axes align perpendicularly to the director, so they tend to lie in a plane, as opposed to orienting along a single direction when s > 0. Now the need of ®ve scalar parameters can be understood more clearly, namely, three of them are necessary to de®ne the eigensystem, with the remaining two specifying the degree of order (s) and biaxiality b.

12.3.2

The Free Energy

When dealing with systems at constant temperature, the free energy F is the proper thermodynamic potential minimized in equilibrium, provided that there is no work done on the system (dF U 0 is valid when approaching equilibrium). The (nonequilibrium) free energy density must be expressed as a functional of the order parameter pro®le. Then its minimum will correspond to the equilibrium order parameter ®eld. According to Landau, near the phase transition the free energy density f is expressed in powers of the order parameter and its derivatives. In the presence of electric and magnetic ®elds also their contribution to the free energy density must be taken into account. In the case of the nematic liquid crystal, the order parameter (12.1) is a tensor, so the free energy density expression must be composed of scalar invariants formed by Q, its spatial derivatives, and external ®eld vectors. Including terms of the fourth order, the part of the free energy density not depending on inhomogenities (the so-called bulk contribution) and external ®elds reads (see [28, p. 156], for instance): fb f0 12 AT

T Tr Q 2

3 1 3 B Tr Q

14 CTr Q 2 2 ; 12:4

where 2 Tr Q 4 Tr Q 2 2 holds for Q's of the form (12.2), so only one of the two needs to be included in (12.4). The free energy density of the isotropic phase has been denoted f0 . The constants A and C are positive, while B may be either positive or negative. The cubic term is necessary, because Q and Q describe diërent states, which results in a discontinuous phase transition. The temperature T is the isotropic phase supercooling temperature. The part of the free energy density resulting from spatial variation of the order parameter (the so-called deformation terms) is expressed as [28, p. 156]: . . 2 2 2 fd L1 `Q..`Q L2 ` Q ` Q L3 `Q..`Qy . . 2 2 L1 ` ` Q L5 `` Q..Q L6 ` 2 Q..Q higher order terms; n Li . are

12:5

temperature-independent generalized elastic constants of the where nth order, .. denotes contraction to a scalar, and qi Qjk y qj Qik . In the ®rst row of (12.5) terms quadratic in Q containing ®rst derivatives are listed, in the second one those containing second derivatives have been collected,

382


either linear or quadratic in Q. All terms given by (12.5) are invariant against rotations as well as inversion of the coordinate system. To describe a chiral nematic phase, however, a term has to be added to (12.5) that is not invariant against inversion in order to re¯ect the symmetry of chiral structures. In the lowest order this gives [37]: 2

fc L4 eijk Qil qk Qjl :

12:6

On inversion of the coordinate system fc changes sign, hence it is a pseudoscalar. It is this term what makes the director rotate in a helical manner and causes a nematic to become a cholesteric. Such a situation occurs if a substance consisting of chiral molecules is dissolved in an ordinary nematic sample. The same structure has been found with pure cholesterol esters as well, so this is where the name ``cholesteric'' comes from. With electric or magnetic ®elds present, additional energy contributions must be taken into account. To be more precise, the free energy of dipoles in external ®elds is to be considered, e.g., the free energy density due to magnetization in a magnetic ®eld H0 ~0 m0 ~ H ~ d H; ~ M 12:7 f m H 0

~ is the magnetic ®eld strength and M ~ magnetization, depending where H ~ In the lowest order, construction of invariants from the tensor linearly on H. order parameter and the ®eld vectors, not including any spatial derivatives, yields fem

1 ~ 2 e0 Xe E

~ QE

1 ~ Q B; ~ Xm B 2m0

12:8

where Xe and Xm are electric and magnetic susceptibility anisotropies, re~ For simplicity, terms con~ m0 H. spectively, to be speci®ed below, and B taining derivatives were not included in (12.8), because of this some possible energy contributions were not taken into account, like the energy of a dipole in an inhomogeneous ®eld and, on the other hand, the so-called orderelectric [44] and ¯exoelectric [25, p. 135] eëcts, i.e., polarization as a result of inhomogenities of the order parameter, thus resulting in an energy contribution when in external ®eld. The total free energy density is the sum of individual contributions f fb fd fc fem ;

12:9

thus being a functional of the order parameter and its spatial derivatives ! qQ qQ 2 ; ;... : 12:10 f f Q; q~ r qr 2 As discussed above the free energy is minimal when the system reaches


383

equilibrium, or in other words, when the order parameter takes up its equilibrium pro®le. This requires that the variation of the free energy be zero dF d f dV 0; 12:11 from which a set of Euler±Lagrange equations is obtained. In general, this set is di½cult to solve, therefore usually the equilibrium con®guration is obtained from a simpli®ed free energy. A homogeneous undeformed nematic exhibits a uniaxial structure. In this case, the order parameter (12.1) can be written in a simpler form as Qij sni nj

1 3 dij ;

12:12

where s is the scalar order parameter, de®ned in (12.3), measuring the degree of molecular alignment with the director ~ n, or, when expressed in the eigensystem 3 2 1 6 Q s4

3

1 3

7 5;

12:13

2 3

as before (12.2) ~ n points along the z-axis. Now the temperature-dependence of the scalar order parameter s can be studied, rewriting (12.4): f f0 12 aT

T s 2

1 3 3 bs

14 cs 4 ;

12:14

where a new set of temperature-independent constants has been introduced. For a typical liquid crystal like the 5CB it has been found that experimental results are best described with a 0:13 10 6 J/m 3 K, b 3:89 10 6 J/m 3 , c 3:92 10 6 J/m 3 , and T 307 K. As soon as a deformation is present, however, s is no more constant in space, and, in general, the biaxiality b is no longer zero. This can be easily understood from a symmetry point of view; as soon as a deformation occurs a new direction in space is de®ned beside the director and, hence, there is no particular reason for the uniaxial state to have the lowest energy. To get an insight into the case when the degree of order s is inhomogeneous, it is instructive to consider a situation where, in the tensor (12.12), only left s is spatially dependent. Taking into account only the ®rst term in (12.5) the 2 deformation free energy density reduces to fd L`s 2 =2, with L1 written simply as L. After combining fd with the bulk free energy (keeping only the quadratic term in (12.14)) one ®nds that a variation of the order parameter s decays with a characteristic length s L : 12:15 x aT T This quantity is referred to as the nematic correlation length. For 5CB at the phase transition temperature it measures approximately 10 nm.

384


12.3.3

The Frank Elastic Theory

In the case of weak deformation, it is possible to consider b to be zero together with s constant, while keeping only spatial variation of the director. Then the deformation free energy (12.5) can be rewritten as the so-called Frank elastic energy [25]: n 2 12 K22 ~ n ` ~ n q0 2 12 K33 ~ n ` ~ n 2 fd 12 K11 ` ~ 1 2 K24 `

~ n` ~ n ~ n ` ~ n K13 ` ~ n` ~ n:

12:16

Frank elastic constants were introduced giving the energy cost of individual deformation modes: K11 for splayed, K22 for twisted, and K33 for bent director ®elds. The K24 and K13 terms can be converted to a surface free energy density contribution on the sample boundary, as indicated by writing them as a divergence. Only in the case of ®xed boundary conditions can they be n dropped. The elastic constants Kij can be expressed in terms of Li and s, thus they are functions of temperature [37]. It should be mentioned that cubic terms are required in (12.5) in order to have K11 0 K33 , a situation that is usually observed experimentally. In the twist term a part of fc (12.6) has 2 been included, with q0 K22 s 2 L4 . Apparently, the twist term is minimized when ~ n ` ~ n q0 , resulting in a spontaneously twisted director ®eld characteristic for cholesterics. The director rotates by p on the pitch length p ; 12:17 lc jq0 j typically ranging from 0.1 mm to 100 mm. The parameter q0 is referred to as the chirality. Finally, the electric and magnetic free energies (12.8) are expressed in terms of s and ~ n: fem

1 ~ ~2 2 e0 Xe sE n

1 2 3E

1 ~~ Xm sB n 2 2m0

1 2 3 B ;

12:18

where Xe and Xm are microscopic susceptibility anisotropies, i.e., diërences between susceptibilities along the molecular axis and susceptibilities in the 0 0 0 0 we? , Xm wmk wm? ). Multiplied by s perpendicular direction (Xe wek they represent diërences in susceptibility along the director and in perpendicular direction as observed macroscopically (for a cholesteric this is true only on small scales compared to the length lc (12.17)): wea wek

we? Xe s;

wma wmk

wm? Xm s:

12:19

It is worth mentioning that in (12.18) only those contributions have been retained that depend on the order parameter, whereas the order parameter 0 0 0 0 2we? E 2 and 1=2m0 13 wmk 2wm? B 2 , independent terms e0 =2 13 wek respectively, have not appeared. This is due to the fact that invariants of the form E 2 and B 2 had not been included in (12.8).


385

Frequently we want to keep the calculations as simple as possible. In this case the so-called one-constant approximation is introduced, setting all elastic constants Kii in (12.16) equal and dropping the surface terms. In this way a simple expression for the elastic distortion energy is obtained fd 12 Kqi nj qi nj :

12.3.4

12:20

Anchoring and Characteristic Lengths

Until now we have been discussing solely the bulk properties of the nematic liquid crystal, paying no attention to what is going on at its boundaries, e.g., container walls, contact with air, etc. Generally, the value of the order parameter at the boundary is in¯uenced by properties of the bounding surface. It can favor diërent director arrangements, e.g., perpendicular (homeotropic), parallel orientation, etc., what is more, the degree of ordering can be suggested as well. These eëcts are known as the surface anchoring. Usually anchoring is modeled by a short-range surface±nematic interaction, expressed simply in terms of a delta function [35], [36]: r 12 W0 1 fs ~

~ n ~ k 2 d~ r

~ R;

12:21

k the preferred direction. with W0 being the zenithal anchoring strength and ~ If ~ k is normal to the surface then (12.21) describes homeotropic anchoring, whereas degenerate planar anchoring is described by r 12 W0 ~ n ~ k 2 d~ r fs ~

~ R:

12:22

In general the azimuthal (in plane) anchoring must also be taken into account (see, for instance, [31]). Close to the con®ning surfaces on distances comparable to the nematic correlation length (12.15) the degree of order s and biaxiality b are also aëcted by the presence of the interface [32]. To describe such eëcts the anchoring free energy usually is expressed in terms of scalar invariants formed from the tensor order parameter Q and parameters characterizing the anchoring. To visualize this possibility a simple model [33] should be mentioned, favoring a (uniaxial) degree of order s0 as well as a surface director ~ k: fs 12 w0 TrQ

Q0 2 ;

12:23

with w0 as a coupling constant Q0ij s0 ki kj 13 dij . Often we call for simplicity and try to neglect the free energy contribution of the anchoring. In this case two possibilities emerge; either we neglect the anchoring free energy, which results in no anchoring (or better, the weak anchoring limit), or, we consider it having an in®nite value as soon as the director deviates from the favored arrangement, implying strong anchoring with ®xed con®guration at the surface. The relevance of the anchoring is often estimated in terms of the so-called extrapolation length, de®ned as [25, p. 113]:

386


l

K ; W0

12:24

where K is one of the elastic constants or their combination. Evidently, l, ranging from 100 nm to 100 mm, is a measure of the anchoring strength compared to the energy of elastic distortion. Eëctively, strong anchoring corresponds to l f R, where R is a typical dimension over which elastic deformation takes place and, conversely, l g R means that the weak anchoring limit is a reasonable approximation. In the chiral nematic liquid crystal, however, the extrapolation length has to be compared to the pitch length lc (12.17) as well, in order to estimate the importance of anchoring. Similarly, the strength of external ®elds is usually characterized by corresponding lengths, where ®eld eëcts prevail over the elastic resistance of the nematic against distortion. In this way, electric (xe ) and magnetic (xm ) coherence lengths can be de®ned [25, p. 123], saying that they are typical lengths, over which electrically or magnetically induced order is restored, if the director is in some place brought out of alignment with the ®eld s s K m0 K ; xm ; 12:25 xe e0 jwea jE 2 jwma jB 2 where electric and magnetic susceptibility anisotropies have been de®ned in (12.19). Again, a small coherence length means that the interaction with the ®eld is strong, and vice versa. For a typical liquid crystal these lengths are about 1 mm in ®elds B 4 T or E 1 V/mm.

12.3.5

Defects in Nematics and Cholesterics

Let us ®rst try to give a useful de®nition of defects. Naively we can say that a defect is an irregularity in the order parameter ®eld, i.e., a discontinuity. This can happen in a single point, a line, or a plane, resulting in zero-, one-, and two-dimensional defects. Their fundamental properties depend on the order parameter, or more precisely, on its symmetry. We can say that defects are a ``registration mark'' of systems with broken symmetry. Of course, physically it is hardly possible to speak about any discontinuities, so there may be a problem with our initial de®nition of the defect. In a nematic, a discontinuity is present only as long as the director description is considered, with s kept ®xed; if this restriction is abandoned and changes of degree of order are allowed, a continuous solution is obtained [38]. Therefore a more general de®nition of a defect must be searched for. Figure 12.3 shows a point defect in two dimensions or a cross-section of a line defect in three dimensions present in the center, arising as a result of a homeotropic con®nement. If a loop is imagined around the defect and then traversed counterclockwise so as to return to the starting point, a so-called winding number n can be de®ned as a measure of the total angle the director is rotated by on this trip, n f=2p. Since the loop passes over a defectless structure only, the con-


387

Figure 12.3. A point defect as a result of a homeotropic boundary condition. The ®gure can be interpreted as a cross-section of a line defect, as well. A loop is placed around it to de®ne its winding number n. For the defect shown, the winding number is n 1.

tinuity of the director ®eld imposes that the angle of rotation must be an integer multiple of p, so n 0; G12; G1; G32; . . . :

12:26

The winding number does not depend on the size or actual shape of the loop but solely on the type of the defect encircled. Therefore the winding number n identi®es the defect completely and is sometimes referred to as the strength of the defect. Even if the singularity in the center (called the core of the defect) is somehow smeared (e.g., by melting, to be the case below) the winding number does not change. In fact, to determine the strength of the defect n we do not need any information whatsoever on what the central con®guration is. Therefore the core region is not of importance for the macroscopic description of the so-called topological defects, i.e., defects that cannot be converted to a defectless structure by means of any continuous transformation of the director ®eld [39]. Topologically speaking, all defects transformable into each other by continuous transformations are identical. This means that those which can be so transformed to a defectless structure, are not defects in the topological sense. From now on only topological defects will be considered. Usually topological defects are energetically stable, although they do not correspond to states of the lowest free energy [27]. Namely, when trying to transform them to a defectless structure a high energy barrier occurs due to discontinuities, which inevitably take place at such a transformation. 12.3.5.1

Point Defects in a Two-Dimensional Nematic

We begin with a not particularly realistic example, a two-dimensional nematic. Here the classi®cation of defects is quite illustrative, and in the oneconstant approximation (12.20) a calculation of structures with point defects is simple. The equilibrium condition for a point defect, located at the center of the coordinate system, reads

388


` 2 y 0;

12:27

where y is the polar angle of the director ~ n cos y; sin y:

12:28

The solution to (12.27), depending only on the polar angle f, satisfying the continuity condition for the director, follows immediately as y n 0; G12; G1; G32; . . . : 12:29 y nf n arctg ; x The half-integral number n is the strength of the defect exactly as de®ned above. Note that in a medium with an ordinary vector order parameter ~ v with ~ v0 ~ v, e.g., a ferromagnet, the half-valued winding numbers are not allowed. The elastic energy of such a con®guration is obtained by integration of (12.20) for the solution (12.29): 2 2 ! 2p K y qy qy R r dr df 12:30 pKn 2 ln ; Fd 2 a qx qy r0 0 R is a typical size of the nematic con®guration, whereas r0 is a microscopic cut-o¨ needed to avoid nonphysical divergence. This means that at distances near r0 the director con®guration (12.29) cannot possibly be correct. In this region the deformation becomes large, so that the Frank elastic theory ceases to be valid, and changes in the scalar order parameter s or even the biaxiality b must be taken into account. In ®rst approximation a core of radius r0 is invented with the nematic in the isotropic (melted) state, having then 0; r < r0 , 12:31 s s0 const; r > r0 . Of course, this is not the con®guration with minimal free energy. The core energy Fc due to melting is Fc pr02 D f ;

12:32

where D f is the diërence in free energy densities of isotropic and ordered phase. By minimizing the total energy Fd Fc the radius of the core is set to s Kn 2 ; 12:33 r0 2D f so it increases linearly with the strength of the defect n. The size of the core is comparable to the nematic correlation length x (12.15). It is very small if compared to the wavelength of light, so one can conclude that optics cannot be used for the investigation of defect cores. However, a better description of the defect core is gained by a precise numerical calculation based on the tensor order parameter [38]. In the case of


389

an n 1 defect, it demonstrates a uniaxial center with s < 0 and ~ n pointing out of the plane, followed by a biaxial ring around it, where the so-called eigenvalue exchange takes place, i.e., the central con®guration is transformed into a uniaxial one, but this time with s > 0 and ~ n lying in the plane, which at great enough distance from the core can be described by a radial director ®eld and s constant. In the case where multiple defects with strengths ni are present, on account of (12.27) being linear, the equilibrium con®guration is obtained simply by summing the solutions (12.29) for a single defect X X y yi ni fi ni arctg : 12:34 y x xi i i The elastic distortion energy of two defects is then [27, p. 529]: Fd Fd1 Fd2 2pKn1 n2 ln

R ; r

12:35

where r is the distance between the centers of the defects. The ®rst two terms stand for the energies (12.30) of single defects, whereas the third term represents the interaction energy. Evidently, defects with equally signed strengths repel each other, while those with opposite strengths are attracted. In this sense, the strength of the defect resembles the electric charge, indeed, in the one-constant approximation the analogy is almost perfect. Neglecting the dependence of the core radius r0 on the winding number n it is possible to write (12.35) in a slightly diërent way [27, p. 529]: Fd pKn1 n2 2 ln

R r0

2pKn1 n2 ln

r : r0

12:36

Let us now discuss the basic properties of point defects in the twodimensional nematic. Our main interest will be in how topologically diërent defects are characterized and then how they combine. Point defects (or line defects in a three-dimensional nematic) have already been characterized by their winding number or strength n. For the two-dimensional nematic it turns out that defects of diërent strengths are topologically diërent [39], i.e., they cannot be continuously transformed into each other. What is more, two defects with strengths n1 and n2 can combine to form a defect with strength n1 n2 , i.e., in combining, the winding numbers are simply summed. Particularly it follows that two defects with opposite winding numbers can combine to form a defectless structure with zero winding number n 0. Even if the defects remain unannihilated, (12.36) shows that the logarithmic divergence is eliminated if n1 n2 0, so the distortion energy of two opposite defects is small, provided, of course, that they are not very far apart. This is not di½cult to understand, because using a loop that encircles both defects a winding number n n1 n2 0 is determined, which re¯ects a defectless structure with a low distortion energy outside the loop. Generally, defects with oppositely signed (not necessarily equal in magni-

390


tude) strengths will combine in order to reduce the distortion energy. On the other hand, it is energetically favorable for a defect with large strength to decay into defects with lower strengths, which then can move apart reducing the distortion energy. One has to be careful when applying (12.36) to this case, because it does not include the core energies and, particularly, disregards the fact that the size of the core varies with the winding number. Nevertheless, if the newly created defects are able to move apart su½ciently the total energy is decreased. A question of defect characterization arises if it is located at the bounding surface of the nematic sample (Figure 12.4(c)), because now the loop cannot surround the defect. In this case a convention is necessary to determine the winding number: the director ®eld is to be arti®cially continued beyond the surface (i.e., as a mirror image) and then the loop is continued as if there were no wall. The corresponding surface defect strength n 0 adequately describes the density of the director ®eld deformation, which is like that shown in Figure 12.4(a). Nevertheless, n 0 is twice as large as the bulk defect strength n corresponding to the situation where the defect is displaced in®nitesimally from the wall (Figure 12.4(b)) so that the encircling loop lies in the medium completely, n 0 2n. 12.3.5.2

Line Defects in a Three-Dimensional Nematic

As mentioned, the solution and distortion energies obtained in two dimensions can also be applied to straight-line defects of in®nite length in the three-dimensional nematic, provided that the director ®eld is constrained to a planar con®guration (e.g., by an external ®eld in the case of negative susceptibility anisotropy (12.19)). In this case, the distortion energies ((12.30), (12.32), (12.35) and (12.36)) must be interpreted as the energies per unit length of the system. Regarding topological stability, however, linear defects in three dimensions behave quite diërently compared to the point defects in two dimensions, because continuous transformations changing the winding number by an integer are now possible [39]. This implies that topologically all defects with integer strengths are no defects at all, since they can be continuously transformed to a defectless structure. For the defect on Figure 12.3 such a transformation (a so-called escape in third dimension) is achieved by a p=2 out-of-plane rotation of directors when going from the boundary toward the center. Similarly, defects with half-integer strengths are continuously transformable into each other, thus being identical. Ultimately this means that in the three-dimensional nematic there exists only one topological line defect, we choose it to be the defect with winding number n 12 (Figure 12.4(b)) [39]. Again the combination law is simply the addition of winding numbers. With only one topological defect, though, there is little possibility left: two defects with strengths n 12 can combine to form a defectless structure with n 0.


(a)

(b)

391

(c)

Figure 12.4. Defects with (a) n 1 and (b) n 12 in the bulk. The director ®eld of the defect located at (c) the surface is the same as in (a) and is therefore assigned a surface strength n 0 1. Topologically, however, it is equivalent to the defect in (b) with n 12 (courtesy of J. Bajc).

12.3.5.3

Line Defects in a Cholesteric

In cholesterics, defects are much more complicated due to the broken translational symmetry of the system. An undeformed helical structure in the cholesteric can be described by ~ n e^x cosqz e^y sinqz;

12:37

with the z-axis as the axis of rotation. In a deformed structure, (12.37) is valid only locally with the z-axis directed along the local twist axis ~ q. Thus, the local coordinate system is de®ned by three mutually perpendicular vectors ~ n, ~ q, and ~ q ~ n. Now three types of line defects are possible [30]. The so-called w defects are the same as in the two-dimensional nematic, only the directors are rotating as we move in the third dimension (along ~ q). With the w lines the ~ q ®eld is regular. With l and t defects (Figure 12.5), however, the ~ q ®eld is singular along the line of defect, while ~ n and ~ q ~ n are regular along the l and t lines, respectively. Because ~ n is regular along the t-line, its energy is lower than that of the other two lines. Topologically the situation is much more diverse now than it was in the nematic case. In cholesterics, four topological defects exist: defects of all three types w; l; t with odd strengths n 2k 1 (designated by C 0 ), plus defects with half-integer strengths n k 12 of types w, l, and t, respectively. Con®gurations of all three types with even strengths (labeled C0 ) are not topological defects. Combination rules for all these defects are collected in Table 12.1 [41]. For the ®rst time, nonuniqueness upon combination is encountered. According to Table 12.1, two l, t, or w defects of half-integer strengths can combine to form either a defectless structure or a defect with odd strength. The ®rst process can be visualized as a mergence of defects

392


(a)

(b)

(c)

(d)

Figure 12.5. Cross-sections of director ®elds corresponding to the l and t lines: l with (a) n 12 and (c) n 12; t with (b) n 12 and (d) n 12. Points indicate the director is perpendicular to the page, whereas marked ends represent tilted directors. The line defect is indicated by a heavy dot (courtesy of J. Bajc). Table 12.1. Results of combining two topological defects. Note that in some cases they are not unique, but depend on the path along which the defects are brought together. The l, t, and w defects carry half-integer strengths, while the C 0 defects are characterised by old strengths and can be of either kind l; t; or w. Topologically, the con®gurations l, t, and w with even strengths are defectless structures C0

C0 C0 l t w

C0

C0

l

t

w

C0 C0 l t w

C0 C0 l t w

l l C0 or C 0 w t

t t w C0 or C 0 l

w w t l C0 or C 0

with n1 12 and n2 12 (topologically identical defects) to an n 0 structure. On the other hand, the second outcome can be seen as combining defects with n1 n2 12 to a defect with n 1. In truth, in both cases just mentioned, either outcome is possible; starting with two n 12 defects, one of them can be easily transformed when approaching the other, and then anni-


393

hilated to n 0 (and vice versa with n 12 and n 12 defects). What will actually happen depends on the path over which the two defects are brought together.

12.4

Chiral Nematic Liquid Crystal Droplets

The ordering of chiral nematics in general con®ning geometries is a complex problem. Therefore we decide to choose the spherical droplet as an instructive but nontrivial example. To a great extent, this section is a summary of research covered by [34], [13], [14]. First, we give some general features of chiral structures con®ned in spherical cavities, enforcing a planar anchoring with no preferred direction (easy axis). As a result of such anchoring, a spherical structure is obtained with the helical axes (~ q's) pointing radially. It can be described by spherical chiral surfaces, de®ned as surfaces perpendicular to ~ q, thus joining points in the director ®eld with the same phase of rotation. Every chiral surface contains a two-dimensional director ®eld to be rotated due to chirality as the surfaces succeed. It is known that the director ®eld on a sphere cannot be defectless but inevitably features defects of total strength n 2 [40]. As a result of this a line defect of type w emerges, running from the center to the surface of the droplet. Figure 12.6 shows the two most known structures drawn on sequential spheres. The structure with a radial defect line of

Figure 12.6. Director ®eld in the radial (above) and the diametrical (below) structures, shown for three successive chiral surfacesÐspheres. The ®rst structure contains one n 2 w-line along the radius, whereas the second structure shows a vertical n 1 w-line along the diameter (courtesy of J. Bajc).

394


(a)

(b)

Figure 12.7. Two con®gurations of lowest energy on a nematic disk with strong tangential anchoring at the boundary: (a) bipolar with two n 12 defects and (b) monopolar with an n 1 defect. Because the defects appear at the surface they could have also been given double strengths n 0 , as discussed above (from [14]).

strength n 2, known also as the Frank±Pryce structure, is observed most often, while structures with a diametrical defect line of strength n 1 are observed less frequently (see [30], [13], [8], and references therein). The eëct of the applied electric ®eld on the droplet structure will be discussed; here only substances with negative dielectric anisotropies wea (12.19) will be of interest. In this case perpendicular alignment is favored (12.18), so that the helical axes tend to align with the ®eld (applying an electric ®eld to a cholesteric with positive wea would result in destabilizing the chiral order). In a strong ®eld limit helical axes become completely aligned with the ®eld, or, in other words, chiral surfaces are planes perpendicular to the ®eld. Now strong anchoring implies a defective structure with a total strength of n 1 on each chiral plane. Con®gurations of lowest energy are those with two n 12 defects (bipolar planar structure) or a single n 1 defect (monopolar planar structure), all of them pushed to the surface (Figure 12.7). According to our previous discussion of surface defects they could also have been given surface strengths n 0 1 and n 0 2, respectively. Again the w defect lines appear as Figure 12.8 suggests, this time spiraling on the surface, however. Consequently, their length is greater than that of the inner w lines in spherical structures, which results in the stability of the latter if there is no electric ®eld applied. For electric ®elds which are not extreme, however, intermediate structures must exist between the spherical and planar solutions. The transition induced by the ®eld has been observed to be continuous. In ®rst approximation the helical structure of the cholesteric is considered to be unaltered, so new chiral surfaces have to be found that would then give the intermediate structure in a weak electric ®eld. Flattened ellipsoids qualify as a natural generalization of the spheres (Figure 12.9(a)), but unfortunately such chiral surfaces are not equidistant, resulting in pitch variations. In ®rst approxi-


(a)

395

(b)

Figure 12.8. Strong electric ®eld limit: director ®eld on a sequence of chiral surfaces in the case of strong anchoring for (a) planar bipolar and (b) planar monopolar structures (courtesy of J. Bajc).

(a)

(b)

(c)

Figure 12.9. Modeling of intermediate chiral surfaces: (a) ¯attened rotational ellipsoids are not equidistant, so they are substituted by (b) equidistant disk-like surfaces, referred to as the oblate chiral surfaces. The length d is a measure of how much they are ¯attened, i.e., how much they deviate from spherical shape. In (c) their threedimensional layout is presented (from [14]).

mation, however, as mentioned above, this must not be allowed on account of the high energy costs. A disk-like surface with a rounded side (corresponding to the outer part of a toroid) is chosen as the proper chiral surface instead (Figure 12.9(b), (c)), yielding a structure with constant pitch. It shall be referred to as the oblate chiral surface. The length d is a measure of the extent to which the oblate surfaces diër from the spherical ones. The latter are obtained from the former by putting d 0, while the planar surfaces result when d > R. The transition from the structures in zero ®eld to structures in nonzero ®eld can thus be described by the parameter dE. Comparing the new intermediate chiral surfaces to the spherical ones, there are two important distinctions. First, spheres become degenerate to a point, whereas the oblate surfaces become ¯attened to a circle carrying a

396


planar director structure. Second, from a topological point of view, two different types of oblate chiral surfaces exist; for d h < R they are closed, thus topologically identical to spheres, whereas for d h > R they are cut by the bounding surface to what are topological circles. From the way the chiral surfaces have been set up it is clear that the rim of the central circular chiral surface corresponds to a l-line defect of strength 12 (discontinuous ®eld of helical axes), which eëctively results in strong tangential anchoring on the rim. Hence, the bipolar and the monopolar structures (Figure 12.7) are stable on the degenerate central chiral surface, just as in the case of planar chiral surfaces (strong electric ®eld) with in®nite anchoring (Figure 12.8), only that now the eëctive `ìn®nite anchoring'' is a topological consequence of how the oblate chiral surfaces have been set up. Recently, two models have been constructed [13], [14] trying to describe the intermediate con®guration in a weak electric ®eld by means of oblate chiral surfaces, each in its own approximation. An intuitive topological model that was ®rst developed does not go into determining the director con®guration on each of the chiral surfaces [13]. Instead it assumes in®nite anchoring on the droplet boundary and predicts the existence of inner and surface w-line defects using purely topological arguments. Within an improved model, the director ®eld is constructed directly from the known con®guration on the central chiral surface, while for simplicity only weak anchoring is considered [14].

12.4.1

Topological ModelÐStrong Anchoring Limit

The director ®eld on closed oblate surfaces (obtained for d h < R) must still contain defects with total strength n 2, just like the director ®eld on a sphere (Figure 12.6). Again this results in inner w lines with a total strength of n 2, starting from defects on the central chiral surface. From an n 0 2n planar defect, w lines of total strength 2n must necessarily emerge. Sooner or later these lines arrive at chiral surfaces that are not closed, being topologically equivalent to disks (Figure 12.9). On account of the in®nitely strong anchoring assumed in this model, the director ®eld on these surfaces must contain defects of total strength n 1. To reduce the free energy they are pushed to the droplet surface, which results in w defect lines, spiraling on the surface. In this way, w lines are obtained both in the droplet and on its surface, as shown in Figures 12.10 and 12.11, thus both the features of spherical structures in the zero ®eld as well as those of planar structures in the strong ®eld are encountered in a single intermediate oblate structure. If the dielectric energy contribution is calculated for both spherical structures in a weak electric ®eld it is found that it is minimal when: in the diametrical spherical structure the diametrical w-line of strength n 1 is oriented along the ®eld; and in the radial spherical structure the radial w-line of strength n 2 is perpendicular to the ®eld.


(a)

397

(b)

Figure 12.10. Line defects in the (a) NDO and (b) RO structures. A low chirality qR 4p has been chosen in order to preserve clarity, d R=2. The l-line defect is shown dotted, whereas the inner and the surface w-line defects are represented by dashed and solid lines, respectively (courtesy of J. Bajc).

(a)

(b)

Figure 12.11. Line defects in the (a) PDO II and (b) PDO I structures. Again qR 4p, d 0:6R. The meaning of line textures is the same as in Figure 12.10 (courtesy of J. Bajc).

Two possible w-line orientations in the spherical structures (normal and parallel to the external electric ®eld) and two possible director con®gurations on the degenerate oblate chiral surface suggest four possible oblate structures: (1) The central circle carries the monopolar con®guration with an n 0 2 defect, continued by a w-line with n 2 perpendicular to the ®eld as shown in Figure 12.10(b). This structure is known as the RO structure; clearly it originates from the radial spherical structure. (2) Two n 1 defect lines emerge from the n 0 2 defect of the monopolar con®guration, both in the `ùpward'' and ``downward'' direction (paral-

398


lel to the ®eld) as shown in Figure 12.11(b). This structure is formed from the diametrical spherical structure, its diametrical n 1 line moves to the side when the size of the central circle increases. It is known as the PDO I structure. (3) Two n 0 1 defects of the bipolar planar structure are continued by two diametrical n 1 lines perpendicular to the ®eld (Figure 12.10(a)). This structure is referred to as the NDO structure, being a result of deforming the diametrical spherical structure, with the w-line perpendicular to the ®eld (a metastable orientation). (4) Two parallel n 12 lines emerge out of each n 0 1 defect of the bipolar structure, oriented along the ®eld (Figure 12.11(a)). The so-called PDO II structure is evolved from the diametrical spherical structure by splitting its n 1 line into two n 12 lines. It is beyond our scope to pursue the total energy estimation of the oblate structures in a precise manner. Nevertheless, it has to be noted that only two energy contributions are taken into account in this model. These are the dielectric free energy (obtained by integration of (12.18)) and the elastic distortion energy or, to be more precise, merely an estimate of it, based on the distortion energy of straight defect lines in an in®nite medium (12.30), since the exact director ®eld con®guration is unknown. For a surface line defect the right strength entering (12.30) is that of n 0 , but approximately one-half of the director ®eld is cut away by the surface, yielding only one-half of the distortion energy. By means of total energy minimization the dependence dE is calculated, i.e., the measure of the extent to which the oblate structures are ¯attened on applying the electric ®eld (Figure 12.12). The only oblate structure that has been observed experimentally is the RO structure [43]. Comparison of the experimental values for d=R with predictions of the topological model shows a qualitative agreement, whereas quantitatively the correspondence is poor (Figure 12.13). The main disadvantage of the topological model is the assumption of in®nitely strong anchoring. Namely, on account of the strong anchoring, the surface w-line defects necessarily exist regardless of large energy contributions due to their considerable length. Indeed, with decreasing anchoring strength the surface w lines gradually disappear because a deviation from the preferred surface alignment costs less energy than the lines themselves. In a chiral system, ordering induced by the surface is particularly costly if it tends to disrupt the spontaneous twist deformation. For planar anchoring this happens as soon as the chiral axis ceases to be normal to the surface. Evidently this is the case in the oblate structures. The strong anchoring is a valid approximation only when l f lc is satis®ed, i.e., when the extrapolation length l (12.24) is small compared to the pitch length lc of the chiral helix (12.17), of course, l f R must still hold (see Figure 12.14). If l A 50 nm, which corresponds to a very strong anchoring, the approximation with in®nite anchoring is questionable for pitch lengths less than 0.5 mm.


399

Figure 12.12. Dimensionless measure of ¯atness of oblate structures (d=R) as a function of dimensionless electric ®eld strength E=E0 , E0 is chosen to be the ®eld at which the electric coherence length xe equals R. The dependencies for the four oblate structures are shown: RO (heavy solid line), NDO (solid), PDO I (dashed), and PDO II (dotted). Two sets of curves are shown, the upper one corresponds to qR 10p, the lower one to qr 40p (courtesy of J. Bajc).

Figure 12.13. Comparison of theoretically predicted transition from spherical to planar structure via the intermediate RO structure with experimental observations [43]. Dimensionless quantities Q qR are attached to the calculated curves, whereas the experimental data corresponds to Q A 20p. Two sets of experimental values are shown, obtained by observations of droplet textures parallel (squares) and perpendicular (circles) to the ®eld (courtesy of J. Bajc).

400


Figure 12.14. Director ®eld on successive planar chiral surfaces in the case of in®nite (left) and very weak planar anchoring. With the left con®guration the condition R A lc g le is ful®lled, whereas R g lc A le is valid for the right case. As the chirality is increased, the con®guration on a planar chiral surface approaches the undistorted one (right), and the surface w lines disappear (from [14]).

12.4.2

Director ModelÐWeak Anchoring Limit

As promised above in this model the director ®eld con®guration on oblate chiral surfaces will be constructed, whereas the anchoring on the droplet surface is considered to be very weak. According to the estimate made at the end of Section 12.4.1 the weak anchoring may give better results than the in®nite one assumed in the topological model (Figure 12.14). The director con®guration on the central circle of radius d (Figure 12.9) is the same as before, with the l n 12 line defect of strength on the edge eëctively implying in®nitely strong anchoring. In one-constant approximation (12.20) the two most probable planar con®gurationsÐthe bipolar and the monopolar structure (Figure 12.7)Ðcan be easily obtained in an analytical form. The bipolar director ®eld expressed in a cylindrical coordinate system reads ~ nbi

d 2 r 2 sin f d 2 r 2 cos f ^ f; r^ 4 2 d r 4 2d r 2 cos2f d r 4 2d 2 r 2 cos2f 4

12:38

with the defects lying in the y-axis at f p=2 and f 3p=2, respectively (see Figure 12.15(a)). The monopolar solution is obtained simply by considering the director ®eld solution around a defect of strength n 2 in an in®nite medium (12.29). In the one-constant approximation it consists of circles of diërent radii having one point in common. If this point is put on the edge of the central circle and the structure is properly oriented so that the circle of radius d matches the rim of the central circle, both the distortion energy is minimized and the boundary condition is obeyed (Figure 12.7(b)), hence this must be the solution for the monopolar structure r 2 d 2 cosf r 2 d 2 sin f 2dr ^ ~ nmo q r^ q f: d 2 r 2 2d sin f d 2 r 2 2dr sin f The defect of strength n 0 2n 2 lies on the y-axis at f p=2.

12:39


(a)

401

(b)

Figure 12.15. (a) A cylindrical coordinate system with coordinates r; f; z is used to describe the director ®eld on the central part of the chiral surfaces, while for the outer part a toroidal coordinate system with coordinates r; y; f is particularly suitable (from [14]).

Now the director ®eld on all chiral surfaces can be constructed from the solutions (12.38) and (12.39) on the central circle. The chirality is to be kept ®xed, so the surfaces in Figure 12.9(b) and (c) serve as suitable chiral surfaces. As mentioned, the anchoring on the droplet boundary is said to be weak, so that in the ®rst approximation the director ®eld, anywhere in the droplet, is determined solely by the central circle con®guration. Most conveniently, the director ®eld on the central (planar) part of the chiral surfaces is expressed in a cylindrical coordinate system (Figure 12.15(a)), whereas for the outer (bent) part a toroidal coordinate system is used (Figure 12.15(b)). By de®nition, chiral axes are perpendicular to chiral surfaces, so their con®guration is like the one presented in Figure 12.16(a). Now only the constant chirality requirement has to be satis®ed. Directors in points B and B 0 (Figure 12.16(b)) are uniquely determined by the directors in A and A 0 , respectively. The director in B, for example, is rotated with respect to the director in A by an angle j q AB, depending on the chirality q and the distance between the points; and analogously for points B 0 and A 0 . In the central part the director ®eld is expressed as ~ n

sinf qz %p2 sinf qz cosf qz %p2 cosf qz ^ 12:40 q r^ q f; 1 %p4 2%p2 cos2f 1 %p4 2%p2 cos2f

using the cylindrical system with %p r=d being a dimensionless radial coordinate. For the set of points fr d; z; f Gp=2g the director ®eld given by (12.40) is singular, resulting in two inner w lines of strength n 12 parallel to the z-axis. Evidently, the con®guration so obtained appears like the PDO II structure. In the outer part, the director ®eld expressed in the toroidal system depends only on the distance from the edge of the central circle ~ n

^ sinqry^ cosqrf:

12:41


402

(a)

(b)

Figure 12.16. (a) Layout of chiral axes. The ®eld of chiral axes is homogeneous in the central part, whereas in the outer part it is splayed. (b) The directors in points B and B 0 are simply rotated directors of A and A 0 ; chirality is kept constant (courtesy of J. Bajc).

The PDO I structure is constructed starting with the monopolar planar con®guration by the same procedure as the PDO II structure. In the central part the solution is ~ n

%p2 cosf qz 1

cosf %p2

qz 2%p sinqz

2%p sin f

r^

%p2 sinf qz sinf qz 2%p cosqz ^ f; 1 %p2 2%p sin f

12:42

whereas in the outer part the solution is ~ n sinqry^

^ cosqrf:

12:43

Now a singularity is obtained for fr d; z; f p=2g, representing the inner w-line with n 1. In what follows solutions for the other two structures (the RO and NDO) with inner w lines perpendicular to the ®eld must be sought. The NDO structure can be constructed from the PDO II structure by a continuous transformation, because both have the same structure on the central circle. All that has to be done is to combine both pairs of w lines of strength n 12 oriented parallel to the ®eld to two single w lines with n 1 lying in the central plane, as shown in (Figure 12.17). Topologically such transformation is allowed (Table 12.1, page 392). Every chiral plane can be regarded as an elastic membrane, since the director ®eld on it is governed by elastic forces. Now the desired transformation can be viewed as stretching of the membrane, as Figure 12.17 suggests, so that the director ®eld of the central part only is drawn over the whole chiral plane of the NDO structure so obtained.


(a)

403

(b)

Figure 12.17. Continuous transformation of (a) the PDO II structure to (b) the NDO structure. Above a three-dimensional director ®eld sketch is shown for a speci®c chiral surface, whereas below transformation of defect lines is presented schematically; their strength is represented by line thickness (from [14]).

For details see [14]. The director ®eld on the central part of the NDO structure is ~ n

sinf qz %n2 sinf qz cosf qz %n2 cosf qz ^ f; 12:44 p r^ q 1 %n4 2%n2 cos2f 1 % 4 2% 2 cos2f 2

n

where %n hzr=d hz%p and hz d=d pjzj=2 is a shrinking factor. The director ®eld on the outer part of chiral surfaces is expressed as sinf qr h 2 r; y sinf qr ~ n p y^ 1 h 4 r; y 2h 2 r; y cos2f cosf qr h 2 r; y cosf qr ^ p f; 1 h 4 r; y 2h 2 r; y cos2f

12:45

with the same shrinking factor hr; y d ry=d rp=2, written in toroidal coordinates this time. The director ®eld is singular in fr > d, y p=2; f Gp=2g, giving the w-lines of strength n 1 that lie in the equatorial plane. Analogously the RO structure is created, with the solution for the central part ~ n

%n2 cosf qz cosf qz 2%n sinqz r^ p 1 %n2 2%n sin f

%n2 sinf qz sinf qz 2%n cosqz ^ p f; 1 %n2 2%n sin f

12:46

404


Figure 12.18. Director ®eld on three successive chiral planes of the RO structure, d R=2, qR 2p, whereas values of h are h 0, h d=4, and h d=2, passing from left to right (from from [14]).

and the one for the outer part ~ n

h 2 r; y cosf qz cosf qz 2hr; y sinqz ^ y p 1 h 2 r; y 2hr; y sin f

h 2 r; y sinf qz sinf qz 2hr; y cosqz ^ f: p 1 h 2 r; y 2hr; y sin f

12:47

The director ®eld on three successive chiral planes of the RO structure is shown in Figure 12.18. With the help of Figure 12.19 we summarize schematically how continuous transformations between the four oblate structures can be performed. As in the case of the topological model the stability of the four calculated con®gurations is assessed by their total energies, only that with the director ®elds known the energies can be calculated in a much more precise way. The energy contributions to include are the elastic distortion energy, the core energy of defect lines, the dielectric energy, and the surface energy due to deviations from the favored anchoring direction. The minimum of the total energy with respect to d=R for diërent values of the electric ®eld E yields a function d=RE, shown in Figure 12.20 for all four structures in a high chirality limit. All curves are quite alike, the diërences appear to be relevant only at low electric ®elds, i.e., at early stages of the transition. As observed, the transition via the PDO I structure starts at a remarkably higher ®eld strength than the others. The PDO I structure originates from the diametrical spherical structure (like the PDO II), then being transformed to what resembles the RO structure. The latter, however, starts directly from the radial spherical structure. For this reason the PDO I structure undergoes stronger elastic deformation at the beginning of the transition, so higher ®elds are needed in order to induce the transformation. In the low chirality cases, a step-like behavior is observed (Figure 12.21), although globally it has little signi®cance.


405

Figure 12.19. Schematic presentation of continuous transformation between the oblate structures. The strength of line defects is shown by line thickness (courtesy of J. Bajc).

On the other hand, the strength of surface anchoring is more important. As expected, with stronger anchoring, larger ®elds are needed to induce the transition (Figure 12.22). The free energy estimate [14] gives the lowest value for the PDO II structure and the highest one for the RO structure. This is consistent with earlier estimates [8] yielding lower free energy for the spherical chiral structure with diametral defect line in comparison to the one with radial n 2 defect line. It also agrees with a simple topological consideration, according to which an n 2 line has higher energy than four n 12 lines. However, the fact that solely the RO intermediate structure has been observed experimentally comes as a surprise. A partial source of error is the simple modeling of line defects, i.e., by a core of isotropic phase (12.31); their free energy is too high, and, in particular, the interaction between the w lines is not properly described for small separations. Further, possible reduction of the free energy not taken into account is the fact that for integral strengths the core can disappear to form an escaped structure [7], [8].

406


Figure 12.20. Stability diagram, showing the electric ®eld-induced transition from spherical to planar structures via the four intermediate oblate structures in the case of a high chirality qR 60p. For all structures the behavior is similar, except for the low ®eld region at the beginning of the transition (from [14]).

Figure 12.21. In¯uence of chirality on the form of d=RE for the RO (on the left) and the PDO II structure. The chiralities are: qR 10p (heavy), qR 20p (thin), and qR 60p (dashed) (courtesy of J. Bajc).

Although the model reported cannot provide relevant free energy values it can well explain the observed transformation of the spherical chiral structure to a nearly planar one. Experimental data for d=RE can be best ®tted with an anchoring strength W0 of 0.2±0.4 mJ/m 2 (Figure 12.22). An even more direct assessment of theoretical results has been established by simulating the polarizing microscope textures for the RO structure and comparing them to


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Figure 12.22. In¯uence of surface anchoring strength on the transition via the RO structure and comparison with experimental data [43], observed parallel (squares) and perpendicular (circles) to the ®eld. The theoretical curves have been calculated taking Kii K 5 10 12 N, wea 5, qR 20p, and R 10 mm. The anchoring strength W0 (measured in mJ=m 2 ) is given with the curves. If compared to the experiment the right value of W0 should be between 0.2 and 0.4 mJ/m 2 (from [14]).

(a)

(b)

Figure 12.23. Comparison of simulated (on the left) and observed polarizing microscope patterns for the RO structure, viewed from the direction (a) normal to both the electric ®eld and the w-line and (b) along the ®eld. On the photograph in (a) one of the chiral planes is emphasized in order to perceive the others. The radial w-line is visible in (b), whereas in (a) it causes a left±right asymmetry (from [14]).

textures observed experimentally [43]. The textures have been calculated considering only the rotation of polarization due to the anisotropy wea , whereas di¨raction as well as refraction eëcts have been neglected. Both calculated and observed microscope images are shown in Figure 12.23, demonstrating a nearly perfect agreement. Of course, one cannot expect an exact matching of bright and dark patterns, because they depend on the composition of light. In the simulated images only three diërent wavelengths have been used to simulate ordinary white light worked with in experiments. For more details, see [14]. In Section 12.6, dedicated to applications, there can be found more information on the electrically controlled re¯ectivity from the PDLC made of chiral droplets.

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12.5

Polymer Networks Dispersed in Liquid Crystals

Polymer networks which can memorize the orientational order of the nematic liquid crystal environment where they are assembled [71], [72], [73], [74] are particularly attractive because of their potential for a variety of electrooptic technologies. We postpone this subject to the last section and here concentrate our attention on the ordering and structures of these composite materials. These systems have many physical properties analogous to liquid crystals con®ned to diërent submicrometer-sized cavities [75], [76] and random porous matrices [77], [78]. Large surface-to-volume ratios enable a strong in¯uence of the polymer network on nematic ordering in the liquid crystalline solvent and thus govern optical properties of the composites. The information about these systems obtained from diamagnetic and viscosity measurements [79], birefringence [72], [74], small angle neutron scattering measurements [73], nuclear magnetic resonance and relaxation [80], and scanning electron microscopy [81], patched together, led to a consistent picture about the polymer structure on all scales. In the following we focus our attention to the most complete study of birefringence so far [74] instead of completely reviewing all research on polymer dispersions in liquid crystals. We show how one can, with a rather simple experimental method combined with the phenomenological description of liquid crystalline ordering, point out important details about the polymer network structure on both the micro and macro levels.

12.5.1

Measurements of Optical Anisotropy

The monomer BMBB-6 (4 0 4-bis-f4-[6-(methacryloyloxy)-hexyloxy] benzoateg-1,1 0 biphenylene) and very small amounts of photoinitiator BME were mixed with 5CB to perform a study of the polymer network-induced birefringence in the isotropic phase of a nematic liquid crystal. Mixtures of 5CB in the nematic phase with 1, 2, 2.5, 3, and 4 wt.% of monomer were ®lled into planar cells and irradiated by UV light (4 W/cm 2 ) at constant ambient temperature for 1 hour. In some cells liquid crystal was substituted by hexane which, after the cells were opened, completely evaporated so that only the polymer network was left on the glass substrate. The examination by SEM (Figure 12.24) shows a ®ber-like polymer network perpendicular to the substrate. The thickness of ®bers and aggregates of ®bers ranges from 0.1 mm to 1 mm. Although the substitution of the liquid crystal with the solvent hexane [72] does not aëct the network this is certainly not true for the evaporation of the solvent. The rather dense SEM-detected structures are the result of shrinking and partial collapse of the network after the solvent was removed. The structure close to the surface which was less eëcted by the drying process indicates that ®bers of thickness @ 0:1 mm after evaporation combine in thicker tree-trunk-like structures.


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Figure 12.24. Polymer network originally formed in a homeotropically oriented liquid crystal and examined with an electron microscope after the removal of the liquid crystal (from [74]).

The cells of all ®ve polymer concentrations were used for the birefringence study with the light of an He±Ne laser. The polarizer and analyzer were crossed, and the cell was aligned so that the rubbing direction (director ®eld ~ n) was tilted by 45 with respect to the polarizer and analyzer. The intensity of light with wavelength l transmitted through a cell of thickness d is described by I I0 sin 2 pDnd=l:

12:48

This expression allowed us to determine the temperature-dependence of the eëctive birefringence Dn within a precision of 10 mK. Experimental results can be summarized as follows [74]. Isotropic Network in a Liquid Crystal Phase. The photopolymerization in the isotropic phase yields in a polymer network exhibiting high light scattering when cooled to the nematic phase but it was completely transparent and did not exhibit birefringence above the NI transition temperature. This experiment reveals that the polymer network assembled in the isotropic phase does not possess any long-range order. If there is a local order, its range is small compared to the wavelength of light. Therefore its eëct is completely averaged out by a light beam sampling randomly oriented areas of local order.

410

G.P. Crawford, D. SvensÏek, and S. ZÏumer Figure 12.25. Experimentally determined temperature-dependence of birefringence of (a) a polymer network in an isotropic liquid crystal and (b) a polymer network in an isotropic solvent for several polymer concentrations (from [74]).

Ordered Network in an Isotropic Liquid Crystal. The strong pretransitional increase of the eëctive birefringence (Figure 12.25(a)) for all examined concentrations of the polymer suggests that in addition to the direct contribution of the polymer network, there is a temperature-dependent contribution from the paranematic order induced in the isotropic liquid crystal phase by internal surfaces of the network. Ordered Network in an Isotropic Solvent. Substituting the liquid crystal which surrounds the network with the isotropic ¯uid chlorobenzene and assuming that the birefringence DnPIL can solely be attributed to the polymer network, it was estimated to be between 5 10 4 to 3 10 3 depending on the network concentration h (See Figure 12.25(b)). The weak temperaturedependence indicates that networks are practically rigid and stable up to


411

Figure 12.26. Schematic presentation of the polymer network in a two-scale (®bril-bundle) model showing both the local and macrodirector. The local distribution of ®brils is represented by a square array of polymer ®brils. The relevant distances are also illustrated (from [74]).

100 C. Not detecting any appreciable change in the volume of the dispersion when the liquid crystal was replaced with the chlorobenzene, one can assume that the eëctive order parameter of the network does not change as well.

12.5.2

Model Structure of the Network

The polymer network dispersed in a liquid crystal is modeled by an array of thin polymer ®brils formed along the local director ®eld. The ®brils are described locally as parallel cylindrical rods characterized by radius R and packed into a two-dimensional square array with inter-®brile distance d (Figure 12.26(a)). The local order parameter sp of the polymer network is also assumed to be equal to the order parameter sn of the bulk nematic liquid crystal where it was formed. Further, assumed to be the polymer-induced paranematic order is uniaxial with the director ~ nloc parallel to the polymer ®brils. This allows us to describe the paranematic ordering in the inter®brile space by a simple scalar order parameter ®eld s (see on (12.3)). To take into account the observations [74], [73], [79], [81], indicating the existence of ®berlike objects with diameters around 0.1 mm, the thin ®brils with typical diameter D are assumed to form bundles of polymer-rich material where the concentration h 0 pR 2 =d 2 is larger than the average polymer concentration h. Further, a bundle of parallel ®brils forming a large (micron) scale network is simply represented by an average interbundle distance B (Figure 12.26(b)). In the space between bundles the polymer concentration is low and thus does not contribute to the paranematic ordering. It should be stressed that in the nematic phase the behavior of these parts of liquid crystal is characterized by

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constraints on the scale of the interbundle distance B while the behavior of the liquid crystal in the bundles is characterized by the much smaller inter®brile distance d. Therefore only liquid crystal in the polymer-poor regions can be easily aëcted by an applied electric ®eld and is thus useful for electro-optic applications. This ®bril-bundle picture is denoted as a two-scale model of the polymer network. To reduce the freedom (four parameters) the bundle thickness D will be taken as 60 nm using the results of Jakli et al. [73], [79] which is also in agreement with the SEM study [81]. Further, will be used the ratio h 0 =h of the polymer concentration in the bundle (h 0 ) to the average concentration (h) as a parameter instead of the interbundle distance B. Thus the two-scale model is controlled by only two independent structural parameters R and h 0 =h.

12.5.3

Optical Anisotropy

When a light beam passes through a sample with nonuniform direction and degree of ordering it is refracted and scattered. The birefringence of a uniaxial medium is proportional to the orientational order parameter Dn s Dn0 if n g Dn0 (see, for instance, [25]). The refraction which is linear in the order parameter is expected to be crucial in the isotropic phase, while scattering eëcts which are quadratic in order parameter variations should be more carefully examined. To estimate the relevance of scattering on small domains with size D f l one can use the Rayleigh or Rayleigh±Gans description [83] for the scattering cross section of such a domain yielding s @ Dn0 s 2 D 6 =l 4 . Taking D 3 for the density of randomly oriented domains and following ZÏumer et al. [82] one can, for small domains D U 0:1l, ®nd that the extinction coe½cient is negligible even in a nematic liquid crystal phase where Dn0 s @ 0:2 yields s=D 3 @