Faraday Discuss. Chem. SOC., 1985 79, 215-224 Defects and their Relationship to Molecular Configurations in Nematic Polymers BY MAURICE KLEMAN Laboratoire de Physique des Solides (associ6 au C.N.R.S.), Universitk de Paris-Sud, 91405 Orsay Cedex, France Received 15th January, 1985 Some recent observations of defects in main-chain nematic and chiral polymers are reviewed in order to obtain a better understanding of the nature of molecular configurations. Important parameters are (a) the chain length, the density of free ends and the length of flexible spacers and (b) the orientational correlations between chains. The first topic, (a), is discussed with respect to observations of defects in uniaxial nematics; the second, (b), is discussed in the light of observations of biaxial nematics and of cholesteric textures of rigid polymers of biological interest. The elements of a geometrical (structural) model which includes local competitions between coiling and orientational correlations and which uses the methods developed for the representation of frustration in curved spaces are given.Defects in small-molecule liquid crystals (SMLC; typical molecular length 30 A) have been studied actively for more than ten years, and had already attracted the attention of physicists by the beginning of this century.' It was the observation of defects with the help of the polarizing microscope'.* which started a long era of fruitful research that has since benefited the whole field of condensed-matter physics. Liquid-crystal polymers (LCP) are vastly different from SMLC and also very different from one another.If they do present any analogies with SMLC and among them- selves, it is at the structural level (the same symmetries and the same type of order parameters). This implies the same topological properties for the defects but not the same energetical properties. Defects are indeed characteristic breaks in local symmetry3 of the order parameter and can be distinguished one from another by (a) their dimensionality (point defects, line defects or surface defects) and (6) the particular type of symmetry they break. The same structure in nematic SMLC and nematic LCP therefore imply equal topological stabilities for disclination lines of the same strength, S, or point defects of the same topological charge.Similarly, dislocation lines of the same Burgers vector are equally topologically stable in SMLC smectics and LCP smectics etc. However, things may be very different with regard to energetical stability; i.e. (a) the relative occurrence of defects of a given type (each defect, which is a metastable configuration of the order parameter, carries a positive energy which depends on the stiffness constants of the material); (6) the nature of the core of the defect, i.e. of this region, whose size is generally a coherence length 5, and where the order of the liquid crystal is replaced by the (dis)order of the higher-temperature phase, but which can suffer various types of arrangements; (c) the mobility of the defect, which always involves molecular processes etc. Also, mutual organization of defects (textures) depends on energetical considerations and physicochemical conditions (boundaries).The same ingredients enter the observa- tional study of instabilities, flow properties etc4 While the observations of defects in SMLC has quickly proved rewarding' and although many observations have already been made on LCP, very few conclusions 215216 DEFECTS AND MOLECULAR PROPERTIES CONFIGURATIONS of a general nature have been reached, except the discovery that the observed defects pertain to the expected topological classes. The experiments are difficult because of the high temperature range in which many of the compounds exist and because of their high viscosity: equilibrium textures are obtained long after the sample has been brought to the temperature of observation.Also, most of the compounds are polydisperse in a way which is not usually known, so that the transition temperatures between the various phases are badly defined and two-phase regions often occur. Hence we are often led to make investigations on ill-defined products. This situation makes all the more interesting future studies on defects in mixtures of mesomorphic polymers in their parent monomer which are dilute enough for us to consider that the polymers behave as independent macromolecules. This paper will mainly discuss observations which have been made on defects and textures in main-chain polymers (either thermotropic or lyotropic) and will leave aside important questions relating to side-chain polymers and block copoly- mers, where the problems of molecular structure are of a different nature.In block copolymers the preferred affinity of each segment along a chain for a segment of the same nature, or for some specific solvent, makes this class of chemicals more akin to lyotropic In side-chain polymers the mesomorphic properties are carried by the small side-chains: the question of the nature of the order parameter is then similar to that in SMLC (observations’ show that, at first sight, most of the properties of defects met in SMLC are found in side-chain LCP), while the viscoelas- tic properties’ are ruled by the conformational properties of the polymeric backbone, which are still quite obscure, as well as the nature of the coupling between the backbone and the side-chain.This paper is divided as follows: in the first section we review recent observations onbdefects in nematics and interpret them, as far as possible, in terms of classical geometries for the director (splay, bend and twist). Frank’ has recently insisted on the fact that the experimentally defined optical axis n ( r ) is not necessarily the best physical molecular axis; this is true in SMLC, and even truer in LCP, where the essential new question is the nature of the orientational coupling between semi- flexible chains. The recent discovery of a biaxial LCP nematic illustrates Frank’s remark, and we stress the interest of a detailed study of defects in these new media. In the second section we discuss observations in cholesterics made of rigid chains in solutions and interpret them in terms of a competition between two incompatible tendencies: local two-dimensional ordering and cholesteric order.This brings us in the third section to a more general discussion of the concept of ‘frustration’ in systems of flexible chains; in particular we propose a geometrical representation in a curved space of the competition between local coiling and two-dimensional (or nematic) ordering, which might be of interest as a basic tool in studying molecular correlations not only in nematic LCP but also in polymer melts. DISCLINATIONS AND WALLS SUMMARY OF THE THEORY OF DISCLINATIONS AND WALLS IN NEMATICS The free energy of a disclination line of strength S, S being the number of times the director rotates by an angle 2.n about the line, is typically of the form R rc W = vKiS2 In -+ W,M. KLEMAN 217 S = * ' / 2 S=+l S=+l s=-1 s= +3/2 Fig.1. Wedge disclination lines: two-dimensional representations in planes perpendicular to the lines. c * t - - _= 1 .... t ......... --I 4 4 4 4 Fig. 2. Twist disclination lines (use is made of the nail convention to represent directors at an angle to the plane of the cut3). Fig. 3. Cut in a plane containing the line S = + 1 without a singularity. (a) Singularity present; (b) no singularity after escape in the third dimension. where Ki is the Frank constant involved in the deformation due to the line ( i = 1,2,3 or Ki is some function of K , , K2 and K 3 ) , R is a typical distance between defects, r, is the core radius and W , is the core energy per unit length of line defect.Fig. 1 shows, after Frank," typical arrangements of wedge disclinations (here wedge qualifies the fact that the axis of rotation of the molecules is about the line itself) and fig. 2 gives a typical arrangement of a twist disclination (twist for a director rotating about an axis perpendicular to the line). The free energy varies as the square of the strength, which favours lines of half-integral strength S = *$. In fact in SMLC the most frequent disclinations have S = *l. This happens because of a possible 'escape in the third dimension' (R. B. Meyer) of the director, in such a way that the singularity of the order parameter in the core vanishes. This218 DEFECTS AND MOLECULAR PROPERTIES CONFIGURATIONS phenomenon is topologically possible for S integral, but forbidden for S half- integral.3 Fig.3 represents part of a S=+l wedge line where the deformation involves splay ( Kl), bend ( K 3 ) but no twist ( K 2 ) . Other geometries exist.3 The free energy per unit length of an integral line reads W=27i-KlSl (2) and can be definitely smaller than eqn (1) if K1, K2 and K3 are of the same order of magnitude ( K is some function of K1, K2 and K3). This is what happens in SMLC, but lines of half-integral strength can be stable against three-dimensional perturbations if some inequalities are reached. More precisely," if K2 > ;( K1 + K3), wedge lines S = *; (as in fig. 1) are stable; if K,<$(K,+ K 3 ) , twist lines IS(=; (as in fig. 2) are stable. The core of a half-integral line is a region where the order parameter is broken, and is described as a true (isotropic) liquid in SMLC.Its radius is a few molecular lengths far from the clearing point T, but increases without limit at T'. When K1 and/or K3 are much larger than KZ, a singular core can be favoured in an integral line. r,, by definition, is always of the order of the coherence length 6, which is ultimately the only characteristic length for an isolated disclination line with a singular core. For a non-singular disclination the energy per unit length does not depend on R [eqn(2)]. Application of the theory to the special case of LCP has been carried out by Meyer;'* in particular, he discusses how the chain length, the density of free. ends and the flexibility affect the magnitude of the coefficients K I , K2 and K3.Walls are not topologically stable defects. They generally occur when surface anchoring is in competition with the bulk tendency to homogeneity. A typical wall width is therefore a penetration length 6 = K / W s (3) where W, is a surface anchoring energy. When the sample thickness is larger than 6 the wall generally splits into surface disclinations. 0 BS E RVATIO NS The first systematic observations were made in sulphuric acid solutions of various aromatic polyamids by Skoulios and They observed a large number of disclinations in the form of very mobile thin threads, displaying frequent sharp- cornered points. This is certainly related to the large anisotropy of the elastic constants. The authors interpret their observations as demonstrating a large value of K3 (as expected for rigid chains with a large persistence length), a reasonably small value of K1 (splay deformations being easily achieved by the diffusion of chain ends) and claim most of the defects to be of integral strength with a radial singular core structure (fig.1). KlCman et dL5 have investigated thermotropic nematic polymers belonging to a series of polyesters differing in the length of the flexible alkyl group -[CH21n- inserted in the monomer. For large n ( n = 14 say) the viscosity is extremely high and the sample does not anneal at all during the observations; the local orientation does not vary even after long reheating in the isotropic phase. However, X-ray diffraction gives evidence of the existence of the nematic phase; the molecules are probably tightly correlated by coiling around one another (hence there is no anneal- ing), and these coilings subsist in the isotropic phase while the extent of the correlations decreases.For n = 5 the texture varies with the degree of polymerization.M. KLEMAN 219 For shorter chains ( M == 1000) one observes a typical SLMC thread texture, with thin lines of strength IS1 = $ and thick lines of strength IS( = 1. In the course of time the threads have a tendency to disappear and give way to well resolved Friedel's nuclei (pluge 6 noyuux, with only integral lines). However, for longer chains ( M = 10000 and more) there are very few integral lines or nuclei, and most of the defects are half-integral lines, either loops floating in the bulk and tending to collapse in a few minutes after their appearance (in which case they have mostly a twist character, as in fig.2) or lines attached to either the glass covers or (in free droplets) the free boundary. In this last case one can observe directly that the size of the cores of S = +$ is much larger than the size of the cores of S = -$ lines.7 The interpretation is qualitatively as follows: Kl is large in this product, and the core of S = +$ lines are clusters of chain ends (all the necessary splay deformation tends to be concentrated here); chain ends accumulate differently near a S = -$ line, where the chains align parallel to the core on a three-branched star (plate 1). Electron-microscopy observations would be useful in understanding this question better. Surface lines are also observed, and in their vicinity are seen clear phenomena of crystallization induced by the boundary conditions.Fayolle et all6 have studied a slightly different polyester and corroborate some of these observations. The same polyester^'^ also display 90" Bloch walls (to use terminology borrowed from the study of ferromagnetic walls) in which the molecules rotate about an axis perpendicular to the wall (plate 2). In these walls the gradient of the molecular directions is practically pure twist, which indicates that K2 is small (relative to K 1 and K3). Finally, walls can be observed in the geometries of the Freederickx tran~iti0n.I~ They have been studied in the C5 polyester recently;'* in the K , geometry they bind domains of elliptic shape; the ratio of the major to the minor axes of the ellipse gives K , / K2 = 10.This ratio is confirmed by direct Freederickx measurements ( K1 == 3.10-6 dyn; K2 == 3 x lov7 dyn). K1 is an order of magnitude larger than K2 in this compound. K3 is more difficult to measure (because of chemical degradation) but is larger than K2 and very probably smaller than K1 [see ref. (18)]. BIAXIAL NEMATICS Investigations of a random copolyester (B-ET) have seemingly shown the existence of two nematic above T,, = 340 "C and below Tc2 =r 350 "C it is an uniaxial nematic, with numerous IS1 = --; and less numerous IS] = 1, as in the above polyesters; below 340 "C only IS1 = 1 defects and walls are present. Much evidence indicates that this second phase is biaxial, i. e. that orientational correlations between molecules exist not only between their long axes but also between their short axes; hence benzene rings in adjacent molecules tend to stack parallel to each other.It is reasonable to assume that, above some temperature Tcl, freedom of rotation along the major axis is recovered, and that the existence of longer flexible spacers along the chain lowers Tc1. Note that B-ET has very small spacers (C, compared with the C5 polyester mentioned above). The question of defects is very different in uniaxial and biaxial nematics;2' each of the three axes of the molecule plays the role of a director and has associated defects, but only defects of even integral strength can 'escape in the third dimension'; defects of odd integral strength are all topologically equivalent, which means that any defect of this type associated with a given 'axis' can turn continuously towards the more favourable 'axis' configuration.This is important essentially for the core region. Such a possibility does not exist for the half-integral disclinations, which220 DEFECTS AND MOLECULAR PROPERTIES CONFIGURATIONS are of three different types. This might be related to the fact that half-integral disclinations have not yet been observed. (Conversely, the absence of integral defects in the C5 polyester^'^ points towards the existence of a true uniaxial nematic.) Also, defects in a biaxial nematic are topologically isomorphic to the elements of the quaternion group ; without entering into details, this property implies that two defects of half-integral strength belonging to two different classes cannot cross without the appearance of a third defect joining them.This obstruction to crossing should play an important role in the rheology of these phases, which must appear effectively more viscous when half-integral defects are present. DEFECTS AND TEXTURES IN CHOLESTERIC RIGID POLYMERS In solution DNA, PBLG, xanthan, collagen and other polymers of biological interest display characteristic cholesteric phases whose defects have been extensively studied recently. Most of these defects are similar to those observed in usual thermotropic SMLC cholesterics. However, there are a number of situations where particular defects or textures are observed; this is the case for Dinoflagellate chromosomes, decondensed chromatin in water, precholesteric phases of sonicated DNA and some large-scale arrangements (self-assemblies) of these molec~les.~~-*~ This point is worth considering in more detail, since it relates directly to how the local correlation of molecular conformations extends at large distances ; in particular, all these molecules, which can be very long (indeed ‘infinite’ for DNA in chromatin), display local two-dimensional order (seen by X-ray diffraction), which is crystal- lographically incompatible on a large scale with cholesteric order.This ‘frustration’ is relieved for distances of the order of the pitch of the defects and textures described below. COLLAGENz4 AND DECONDENSED DNA2372S The long molecules are arranged in bundles of lines twisted along their length (fig.4). The central molecule is straight, while the others rotate helically about it with a constant pitch equal to the pitch of the cholesteric phase. There is also helical rotation of the molecules along any radius of the bundle. Therefore the configuration is doubly twisted and, as can easily be shown, splayless. It is therefore favoured where K , is very large, as one might expect for molecules which are extremely long (having no free ends available in large numbers) and rigid (hairpins have a very high energy); it is also favoured (because of double twist) when KZ4, the saddle-splay constant, is large and positive. (It can be shown that a large K24 favours the nucleation of double-twist configurations.) The analogy with the blue phases (local configurations of SMLC) is striking:26 blue phases of SMLC are stable when the ratio t / p is large, 5 being some correlation length which scales necessarily with the length of the molecules (for SMLC) or the persistence length Zp (for LCP). The question of the presence of two-dimensional ordering is more novel and would require a larger development than allowed here.Let us just indicate that it is possible to show that the molecules are at the intersection of two sets of orthogonal surfaces, so that there is locally two-dimensional ordering; these surfaces are not equidistant (except in the vicinity of the central molecule) and the two-dimensional ordering is strained. It can be any two-dimensional local order, on short distances (see previous section), or nematic order (which is a particular case of two- dimensional ordering).In this respect the blue phases of SMLC appear as resulting from competition between nematic and cholesteric ordering.M. KLEMAN 22 1 f ' 1 IZ Fig. 4. The cylindrical geometry in decondensed DNA, collagen, or in a blue phase. 'i I I Fig. 5. Geometrical model for I I I I Z I the Dinoflagellate chromosome. Molecular configuration drawn in a straight section. THE DINOFLAGELLATE23*27 CHROMOSOME This is also a twisted cylindrical configuration, but orthogonal to the former one (fig. 5). On the axis the molecular directions are horizontal (perpendicular to the axis); they rotate about the axis with a pitch p equal to the cholesteric pitch and generate a helicoid; the other directions of helicity are along the normal to this helicoid.The total configuration is therefore doubly twisted, as in the previous example, but presents two helical defects on the periphery of the cylinder which are two S = ++ disclinations (along C, and C,, fig. 5 ) , corresponding to cuspidal lines on the focal surface of the helicoid. The two sets of orthogonal surfaces along which the molecular directions lie are (a) the helicoid and the surfaces parallel to it and (6) a set of hyperbolic paraboloids. The geometry is limited to the cylinder by the addition of other peripheral disclinations (fig. 6). Although experimentally the chromosome of Dinoflagellate (Prorocentrum micans) has an aspect ratio (pitch over radius) which differs from the mathematical model described here, this model probably offers a good basis for understanding the chromosome configuration (fig 6), which has also been proposed independently by F~-iedel*~ and which fits with the crude model first proposed by Bo~ligand.~~.~' In particular, the mathematical model contains splay, in the form of hairpins, for an infinitely long molecule and implies that K3 and K1 are large compared with222 DEFECTS AND MOLECULAR PROPERTIES CONFIGURATIONS L- J Fig.6. Two-dimensional vertical cut. The disclinations D, and D2 have been added in order to obtain a bounded geometry. KZ. Also it can be shown that the nucleation of the configuration is favoured if there is a strong local tendency to true two-dimensional ordering, with an elastic shear modulus p large compared with K2/ bp, where b is the mean distance between molecules.DNA in chromatin is a complex chemical species whose interactions with the proteins of the matrix can satisfy the above requirements. Moreover, local helix-coil transitions can explain the ease of formation of hairpins. Finally, a description of the chromosome in terms of defects might be relevant to the still unknown processes which, on a semi-macroscopic level, occur during cell division. SELF-ASSEMBLIES The structural elements described above can pack together and form ordered or disordered arrangements on higher scales.22 For example, the cylinders of the blue phase assemble into either cubic crystals or amorphous systems, the regions between the cylinders being filled with disclinations. There are many reasons to believe that similar processes exist for polymeric structural elements.What has been described in the previous sections is a cholesteric self-assembly of local hexagonal packings. Self-assemblies with very large pitch of large hexagonal packings of PBLG have also been observed. One can expect, when the self-assembly is disordered, that the process is hierarchical ; at each scale a geometrically well characterized assembly of elements of the lower scale is built. Such concepts of hierarchical ‘frustration’ appear naturally in a theory which introduces a curved-space des~ription:~~ the local competitions (two-dimensional versus cholesteric) which are incompatible with three-dimensional homogeneous euclidean space-filling are reconciled in a three-dimensional curved space, where they build a ‘crystalline’ arrangement. The local arrangement is a projection in flat space of a small piece of the crystal in curved space ; the disclinations which separate the structural elements assembling at a higher scale are the projections of the disclinations in the curved-space crystal.Such a theory is well established for amorphous metals;30 we will now summarize some recent attempts in the same direction made for long flexible molecules. FRUSTRATION IN ASSEMBLIES OF LONG FLEXIBLE MOLECULES3’ In the frame of the theory, the blue phase is an euclidean projection of a sphere S3 (the three-dimensional sphere in four-dimensional euclidean space, i. e. a space of positive curvature R ) endowed with a regular (crystalline) arrangement of directors possessing intrinsically the property of double twist.This same arrange- ment can be given, with a trivial addition, the property of global or local two- dimensional ordering, in many different ways, because the lines of force of theM. KLEMAN 223 directors in S3 are equidistant lines. Now, place long flexible molecules along these lines of force; in S3 they form a set of equidistant and mutually twisted great circles of pitch p = * 2 r R (right or left great circles, according to choice; in spherical geometry these equidistant great circles are called Clifford parallels or paratactic lines). Locally the arrangement is similar to the local arrangement in decondensed DNA or in a Dinoflagellate chromosome. However, a detailed study of all possible local arrangements in S3 enables us to enlarge our point of view. We give only the results.S3 can be described as a fibrous bundle of equidistant right or left great circles, each attached at a different point of an ordinary sphere S2. This is the celebrated Hopf theorem. The structural arrangement of the flexible molecules can therefore be depicted by the mutual arrangement of a set of points on S2. The mapping is such that all the molecules located at a distance d = R8 (0 8 d r / 2 ) of a given molecule map onto a small circle at a distance 28 from the pole of S2, this pole representing the central molecule. Therefore (a) the densest regular packings of molecules in S3 are represented by regular deltahedra on S2, Le. the vertices of an equilateral triangle on a great circle of S2 of a tetrahedron, an octahedron or an icosahedron, (6) other regular packings, represented by the vertices of a cube or a dodecahedron, are not dense packings and ( c ) all other packings of equidistant molecules have no long-range order, since they are not represented by the vertices of platonician solids.Note that the hexagonal packing appears as a special case of regular deltahedra, for six neighbours, i.e. R infinite. The number of neighbours is always (6 for all the cases (a) and (6). It can be any number in case ( c ) , which also includes anisotropic packings with orientational correlations on any given length. Note that our description of frustration in assemblies of long flexible molecules does not have to be specialized to chiral molecules.In fact it also applies to molecules whose flexion has an entropic origin and is locally either left- or right- handed. In particular it leads to a new way of thinking of local arrangements in molten polymers, at a scale 5 larger than the persistence length I,, but smaller than the radius of gyration. Assume for example that each flexible molecule has n = 5 neighbouring molecules and that d is their diameter. In S3 the condition of dense packing with n = 5 leads to p = 11.234. . . d (these figures would be slightly different when projecting in this ‘spherical’ set of strands in flat space); p is therefore this scale 6, which defines the length along which the molecules assemble in a string whose lateral size is of the same order of magnitude. This is not in contradiction to Flory’s conception of molten polymers, but rather superimposed on its model at a scale 6, by achieving local density requirements and using entropic effects which appear at all scales.The same concepts may apply to the isotropic phase of semi-flexible mesogenic polymers as long as the persistence length lp is smaller than the length of the stretched molecules. An isotropic phase would then be described by a local order of one of the types discussed, and locally ordered domains separated by disclinations or walls (for domains of opposite chirality) ; the mobility of the liquid phase would therefore involve the mobility of these defects. Note that in all cases considered (except the true uniaxial nematic case with no positional correlations) there is always obstruction to the crossing of line defects.Finally the biaxial nematic might also be locally described in the framework of our case (c), since a finite coherence length of biaxial correlations has been found. I thank Dr A. M. Donald, Prof. J. Friedel, Dr M. R. Mackley, Dr A. Skoulios, Prof. M. Veyssik and Dr A. H. Windle for discussions.224 DEFECTS AND MOLECULAR PROPERTIES CONFIGURATIONS ' G. Friedel, Ann. Phys. (Paris), 1922, 18, 273. 0. Lehmann, Hussige Kristalle (W. Engelmann, Leipzig, 1904). M. KlCman, Points, Lines and Walls (Wiley, Chichester, 1983). M. 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Cryst., 1984, 111, 321. G. Toulouse, J. Phys. (Paris) Lett., 1977, 38, L-67. 21 31 M. KlCman, work in preparation. 32 J. P. Sethna, D. C. Wright and N. D. Mermin, Phys. Rev. Lett., 1983, 51, 467.Plate 1. Half-integral lines in a free droplet of a C5 polyester (courtesy of G. Mazelet). The two terminating configurations of each line on the free surface are, respectively, +; and -;. There is a clear three-fold starred contrast at each --+ (circularly polarized light). Plate 2. A 90" wall in the C5 polyester separating homeotropic region from a planar one; ( b ) schematic drawing of the configuration in the wall. [facing page 224