Hedgehog structure in nematic and magnetic systems N. Schopohl, T.J. Sluckin To cite this version: N. Schopohl, T.J. Sluckin. Hedgehog structure in nematic and magnetic systems. Journal de Physique, 1988, 49 (7), pp.1097-1101. <10.1051/jphys:019880049070109700>. <jpa-00210792> HAL Id: jpa-00210792 https://hal.archives-ouvertes.fr/jpa-00210792 Submitted on 1 Jan 1988 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
J. Phys. France 49 (1988) 1097-1101 JUILLET 1988, 1097 Classification Physics Abstracts 61.30-61.70G Short communication Hedgehog structure in nematic and magnetic systems N. Schopohl and T.J. Sluckin(*) Institut Laue-Langevin, 156X, 38042 Grenoble Cedex, France (Re u Ie 8 mars 1988, accepti le 6 mai 1988) Résumé.2014 On étudie la structure des points singuliers (hérissons) dans les systèmes nématiques et magnétiques, dans le cadre d une théorie de type Landau. Nous avons trouvé des solutions exactes avec la symétrie sphérique et les coeurs isotropes. Pour le hérisson nématique la solution est partout uniaxiale. Les hérissons nématiques sont beaucoup plus grands que les hérissons magnétiques, les longueurs étant mesurées par rapport à la longueur de relaxation du paramètre d ordre. Abstract.- The core structure of hedgehog point disclinations in magnetic and nematic liquid crystal systems is investigated within Landau theory. Exact spherically symmetric solutions are found, with isotropic cores. In the nematic hedgehog the solution is uniaxial everywhere. Nematic hedgehogs are very much larger than magnetic hedgehogs when measured with respect to the order parameter healing length. In many ordered statistical mechanical systems the equilibrium order parameter space is a continuous manifold. In such systems there are often defects; lines or points in real space at which the order parameter direction cannot consistently be defined. The large scale of such defects, it is now realized, can be understood using homotopic group classification [1, 2]. Close to a defect the order parameter escapes from the equilibrium order parameter space. To understand the details of such behaviour it is necessary to go beyond the usual hydrodynamical theories [3] to (for instance) a Landau theory of an inhomogeneous ordered system. defects. In particular we make some rather elementary observations on the internal structure of the so-called hedgehog defect in magnetic and nematic systems [1, 2]. In a (Heisenberg magnetic system the order parameter is a vector m=mn, where n is a unit vector. A hedgehog defect at r = 0 gives rise to a far field n = ± r ; it is thus twofold degenerate. Now in the Landau theory the appropriate free energy density is In a recent paper we have made such a study of a defect line in a nematic liquid crystal [3]. In this note we turn our attention to point (*) On leave from Department of Mathematics, University of Southampton, Southampton S09 5NH, G.B. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070109700
1098 t is a scaled temperature and ferromagnetism occurs for t 0. This free energy density can be divided in a natural way into bulk and gradient contributions. The appropriate Landau-de Gennes free energy is now [4] The resulting Euler-Lagrange equations are In what follows we scale m with respect to the bulk magnetization, mb == (Itl /b)1/2, and measure lengths in terms of the correlation A solution to length e = [(Ki + K2) / ltl]1/2. (2) of the form m(r)r exists, providing that m(0) = 0, m( 00) = 1 and and the resulting Euler-Lagrange equations [3] are where primes denote differentiation with respect to r. The consequence of scaling is that all solutions to (2) with spherical symmetry are selfsimilar ; all magnetic hedgehogs look the same. The free energy of the defected system can now be expressed as where Ebulk is the (extensive) contribution of the uniform asymptotic system and Elastic (- system dimension) is the contribution of the hydrodynamic elastic terms [1]. The next largest contribution is independent of system size; we define this as E core. In scaled units this is with L23 = (L2 + L3. As in our previous work we adopt [4] scaling in which b = -1, c 1/9; = in these units the first order isotropic-nematic transition takes place at t 1/3 = to a nematic state in which state becomes Q(t 1/3) = = 1. The isotropic mechanically unstable at t = 0. At this stage it is convenient for us to measure lengths in terms of a (temperature independent) correlation length A hedgehog solution of the form Q(r) = Q (r) [3r @rt -!oj now exists, by analogy to the magnetic case above, providing that Q(O) = 0 and a By way of comparison the bulk (i.e. asymptotic) free energy density in these units is - 1/4. In the nematic liquid crystal the order parameter, by contrast, is a traceless symmetric tensor Q in uniform systems Q = Q (3fi (g ft _ 1) - The nematic hedgehog at r = 0 also yields a far field n = ± r, but is unique because of the equivalence of ± n configurations. Unlike in the magnetic case the solutions to this equation are not self-similar and depend qualitatively on temperature; this follows because the bulk phase transition is first order and the bulk free energy density consequently contains an (unscaleable) term in Q3. A separation into bulk, elastic and core energies can be carried out in the same way as for the magnetic case, we find (again in scaled units)
1099 where Qb is the bulk value of the order parame- is the non-elastic contribution to ter, and f ( Q) the free energy density. We now discuss the results for the disclination structure. In order to make precise comparisons we now measure lengths with respect to the healing length a for perturbations of the order parameter in the bulk phase. For magnetic hedgehogs a = 1 /J2, whereas for the nematic systems a = (t - 2Q + 2Q2) -1/2 - a diverges at t = 3/8, the limit of stability of the nematic phase. It is also convenient the normalised or- In this way we eliminate effects purely attributable to the bulk behaviour. to discuss Q (r) Q(r)/Qb, = der parameter(mb = 1 by construction). We first discuss the asymptotic behaviour of 8m - m - 1 and 8 Q Q - = 1. From equations (3) and (8) we find From this point of view (but only from this point of view) all nematic hedgehogs are identical. However, the amplitude of 8Q is three times larger than that of b m : nematic hedgehogs decay more slowly, or equivalently, are larger. This is emphasised by examining the low r behaviour, for which analysis of equations (3) and (8) again yields Results (10) and (11) both depend crucially on the behaviour of the 1/r term in equations (3) and (8). In general we expect the coefficient of this term to be governed by the intrinsic angular momentum I and to be of the form - 1 (I + 1) /r. For magnets 1 = 1 and for nematic liquid crystals 1 = 2. The intrinsic angular momentum provides the analogue of a centrifugal energy term; high angular momentum thus increases effective disclination size. In figure 1 we plot Q(r) and m(r), demonstrating more precisely these trends. The nematic hedgehogs are indeed larger than the magnetic hedgehogs, particularly so close to the nematicisotropic phase transition. A measure of the size of a defect is given by with an analogous formula for magnetic systems. Fig. 1.- Order parameter (normalized) as a function of distance from core centre, in units of nematic healing length; (-) magnet, (... ) t = tni = 1/3. Below t = 0 w is more or less constant and w = 4.7 : disclination size more or less scales with effective correlation length a. This statement is rigourously true, of course, for the magnetic systems. Close to the transition temperature w increases rapidly, reaching a value of N 8.4 at TNI. This effect is even more dramatic experimentally because a also is large in this region. It is interesting also to compare the magnetic energy density and the nematic energy density We show the magnetic energy density in figure 2, and the nematic energy density at tni in figure 3. The most striking features are the maximum at r = 0 in the magnetic case, contrasting with the maximum which develops at finite r for the nematic case. This latter feature corresponds to the development of a structure resembling an interface between the isotropic phase in the core of the hedgehog and the nematic phase on the exterior. As the temperature is lowered this feature becomes less significant as the bulk isotropic phase loses its stability, although it never entirely disappears.
1100 Fig. 2.- Magnetic energy density (natural units) as a function of distance from core centre. core consists of normal fluids [6]. The analogy fails, however, essentially because the number of spin components n = 3 is not equal to the number of space components d 2, whereas = in the quantum vortex (or X Y model) case n = d. As we have shown elsewhere [3] the consequence is a biaxial, rather than isotropic disclination line core. A better analogy to the quantum vortex line is the case we consider here, the hedgehog disclination point. Here n = d and an isotropic core once again becomes plausible. Fig. 3.- Nematic energy density (natural units) as a function of distance from core centre at transition temperature tni 1/3. At this temperature en (r),..., r-4 for large r; otherwise = en ( r) - en ( 00),..., r - 2. The picture we develop in this paper can be, of course, understood in a simple phenomenological way by writing down the following free energy for a nematic hedgehog where r is hedgehog size. The first term is the elastic free energy lost by imposing an isotropic core; in our units A = 144rQ. The second term is a surface tension term from the free energy cost of surrounding the core by a nematic-isotropic interface; in our units the surface tension 7 = 1.15 [4]. the cost of having a bulk isotropic The last term counts core with free energy density Ae greater than that of the surrounding nematic medium. This phenomenological free energy gives rather good results for hedgehog size, and gives results for Ecore within 25 % of those obtained from equation (9). Analogous calculations for the magnetic hedgehog (in which the surface tension term is now missing) slightly overestimate core size. Fan [5] developed a model for disclination line cores in nematic systems involving a core of isotropic fluid, based on earlier models of quantum vortices in superfluid He 4in which the The profiles we find in this paper are exact solutions of the relevant Euler-Lagrange equations. They maintain everywhere the asymptotic spherical symmetry of the system. We do not however show that they are the minimum energy solutions of these equations, although we have not in practice been able to find others. Nevertheless there are a number of reasons for believing that such solutions may, at least under some circumstances, exist. First, the symmetric solutions involve the existence of a shell of nematic-isotropic interface, at which the director at that interface lies perpendicular to it. However it is known that if L23 > 0(which is indeed the experimental regime) the director at a nematic-isotropic interface lies in the plane of the interface [4] (and experimentally, often at angle of N 60 to it). Second, we have found that the core of a disclination line is biaxial and this has a local energy density lower than that of an isotropic region. Furthermore a disclination line of index 1/2 arranged in a ring can have an asymptotic director configuration identical to that of a hedgehog point core. Such an arrangement would save energy density at the cost of greater extent. Third, Landau-de Gennes theory predicts that the Franck splay and bend constants will have the same value; a hedgehog disclination involves only splay. In a more sophisticated theory with a lower splay than bend constant deformation of the central core regime might be favoured. Finally there is one observation [7] that a hedgehog core does not consist of single defect point but a small twisted region, although more detailed measurements are not available. Thus a search for a locally nonspherically symmetric hedgehog does not seem an entirely unfruitful exercise. In conclusion we have made a study of the internal structure of the hedgehog point discli-
1101 nation in magnetic and nematic systems, using Landau theory. Exact spherically symmetric solutions can be found in both cases. Nematic hedgehogs are much larger than magnetic hedgehogs, particularly close to the nematic-isotropic phase transition. The core in both cases consists of isotropic material, in contrast to the core of disclination lines in nematic liquid crystals. A simple phenomenological picture accounts for the results semi-quantitatively. Acknowledgment. We thank P. Nozieres for providing the opportunity of collaborating Laue-Langevin us with at the Institut References [1] CHANDRASEKHAR, S. and RANGANATH, G.S., Adv. Phys. 35 (1986) 507. [2] See e.g. MICHEL, L., The Physics of Defects, Eds. R. Balian, M. Kléman and J.P. Poirier, Proc. of Les Houches, session XXXV (North-Holland Amsterdam) 1981, pp. 361-383. [3] See e.g. HORNREICH, R.M., KUGLER, M. and SHTRIKMAN, S., Phys. Rev. Lett. 48 (1982) 1404; SCHOPOHL, N. and SLUCKIN, T.J., Phys. Rev. Lett. 59 (1987) 2582. [4] DE GENNES, P.G., Mol. Cryst. Liq. Cryst. 12 (1971) 193. [5] FAN, C.P., Phys. Lett. A 34 (1971) 335. [6] See e.g. FETTER, A.L., The Physics of Liquid and Solid Helium, Eds. K.H. Bennemann and J.B. Ketterson (John Wiley, New York) 1976, pp. 207-307. [7] CANDAU, S., LE ROY, P. and DEBEAUVAIS, F., Mol. Cryst. Liq. Cryst. 23 (1973) 283.