Definition of order (a) microscopic level :

ℚ x , t =∫d u f x ,u , t uu ,

f x ,u , t is the molecular distribution function,

uu denotes the symmetric traceless part of the tensor u u.

ℚ =32S n n −

13

12T l l −m m,

(b) mesoscopic level :

n, l, m are three orthogonal unit vectors,

S=⟨P 2cos⟩≡12⟨ 3cos2

−1 ⟩ is the measure of uniaxial alignment,

T=⟨sin 2cos2⟩ is the measure of biaxial alignment.

Free energy and statics

The Landau-Ginzburg free energy functional is obtained from a local expansion in powers of rotationally invariant combinations of the order parameter It consists of a homogeneous contribution and an elastic contribution,

denotes the supercooling transition temperature, denote the cartesian coordinates. Higher powers of can be excluded for the description of the uniaxial phase [1].

Q x , t .

Fh ℚ =12ATrℚ 2

13BTrℚ 3

14CTrℚ 2

2E '

Trℚ 32

Fel [∂ ℚ]=12L1∂ℚ ∂ℚ

12L2∂ℚ ∂ℚ .

A= A0 1−TT ✯ ,T

✯

, ,Trℚ 3

Relaxational kinetics

In the absence of thermal fluctuations and hydrodynamic flow, the equation of motion of the order parameter is,

∂tℚ x , t=−

Fℚ

, with = [ −23 ] .

The equation of motion reads,

These equations are solved by projecting in a tensorial basis [7] The time dependent coefficients are evolved to get the symmetric traceless matrix tensor which supplies the information of the degree of uniaxiality, biaxiality and the textures.

∂tℚ x , t =−[ AC Trℚ2ℚ x , t B6 E 'Trℚ3ℚ 2 x , t −L1

2ℚ x , t − L2 ℚ x , t ] .

ℚ T ,ℚ x , t = i=15 a i x , t T

i .

Nematic droplet nucleation in an isotropic background Classical nucleation theory predicts that the nucleation rate is proportional to , where B the barrier height, has both surface and bulk contributions. Crucially, the critical nucleus need not be spherical [2]. The shape of a droplet of fixed volume cannot be obtained simply by minimising the surface area : one must minimise the surface energy including the angular dependence of the surface tension. Even at fixed volume, the volume contribution can involve non-trivial director configurations. [4-6]We have made a preliminary study of the dynamics of droplet growth in the nematic phase, allowing for anisotropic surface tension and general director configurations. Figure (a) depicts a spherical nematic nucleus with the director anchored at an angle . In the absense of elastic anisotropy, , the droplet remains circular as shown in figure (b). Figure (c) /(d) shows the orientation of the droplet along the direction of nematic order for planar anchoring, / homeotropic anchoring, . .

Textures and topological defects in nematic phasesTextures and topological defects in nematic phases Amit Kumar Bhattacharjee, Gautam I. Menon & Ronojoy Adhikari The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai – 600 113, India

Abstract

We propose an efficient numerical scheme, based on the method of lines, for solving the Landau-de Gennes coarse-grained equations describing the relaxational dynamics of nematic liquid crystals. It provides a stringent test of the de Gennes ansatz for the isotropic – nematic interface, illustrates the anisotropic character of droplets in the nucleation regime, validates dynamical scaling in the coarsening regime and accurately describes defect dynamics and core structure.

4L2=0

L20 L20

Line disclinations of uniaxial nematics in three dimensions : A system with broken continuous symmetry has topological defects, which depend on the structure of the order parameter space. These can be classified using the homotopy group of the order parameter space In the uniaxial nematic phase in 3d, there are stable linedefects of half integer charge and stable point defects of integer charge. No stable point defects exist for the biaxial phase. Rules exist for the combination of defects of different classes,determined by the homotopy groups.

Two disclinations interact, exchanging segments, to form separate rings, which shrink and finally disappear from the system. There is a reduction of uniaxial ordering in the defect core, which is surrounded by a ring of maximum biaxiality [8].

The figures shown are of 64^3 volume, volume rendered in false colour, with superimposed isosurfaces. . Figs (a) – (d) represent the degree of uniaxiality with the isosurface at a value near the minimum value, whereas (e) – (h) show the degree of biaxiality with the isosurface near the maximum value.

References :

[1] P.G. de Gennes & J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993), 2nd Ed.[2] S. Chandrasekhar, Mol. Cryst. Liq. Cryst. 2, 71 (1966)[3] A. K. Bhattacharjee, Gautam I. Menon, R. Adhikari, Phys. Rev. E 78, 026707 (2008).[4] A. Rapini and M. J. Papoular, J. Phys. (Paris) Colloq. 30, C4 (1969).[5] P. Prinsen and P. van der Schoot , Phys. Rev. E. 68, 021701 (2003).[6] A. Cuetos and M. Dijkstra, Phys. Rev. Lett. 98, 095701 (2007).[7] G. Rienacker, M. Kroger and S. Hess, Physica A, Statistical Mechanics and its Applications 315, 537 (2002).[8] N. Schopohl and T. J. Sluckin, Phys. Rev. Lett. 59, 2582 (1987).

Point defects and schlieren textures of biaxial nematics in two-dimensions

(a) t =0 (b) t = 3000

(c) t = 900 (d) t = 1500

e− B/ k BT

(i) t = 0 (j) t = 79 (k) t = 2651 (l) t = 34781

(e) t = 0

(a) t = 0 (b) t = 79

(f) t = 79

(c) t = 2651

(g) t = 2651 (h) t = 34781

(d) t = 34781

(a) t = 168 (e) t = 224

(b) t = 440 (f) t = 440

(c) t = 701 (g) t = 701

(d) t = 1155 (h) t = 1155

1S3 /D2=Q ,

1S2 /Z 2=Z2

In a uniaxial nematic in 2d, the only stable class of defects have half-integer charge, corresponding to . In a biaxial nematic, there are two stable classes of defects corresponding tothe group of quaternions, with strength 1/2 and 1.

Textures are seen between crossed polarizers and calculated as the anglebetween the optic axis and polarizer. Centers with four brushes are k=1 and two arek=½ charge. All classes of defects cannot be tracked in the texture, but areclearly seen in the uniaxial and biaxial degree of alignment profiles. Thefigures shown are taken from a small portion of a 1024^2 lattice, and plottedin false colour.