Elastic Energies and Director Fields in Liquid Crystal Droplets, I. Cylindrical Symmetry

— The nature of point defects in nematic liquid crystals has been studied by placing droplets on a water substrate and viewing these between crossed polarizers. Three configurations are observed. These configurations are accounted for by minimizing the Frank elastic energies with suitable boundary conditions. The second order nonlinear simultaneous partial differential equations that describe systems with cylindrical symmetry are solved by relaxation methods using computer facilities. The importance of the cancellation of several contributions to the splay (divergence) is emphasized. 1. Droplets. — The purpose of the present research droplets between crossed polarizers is that three is to obtain descriptions of the minimum energy confidifferent optical patterns occur as shown in figure 1. gurations of the molecular alignment in liquid crystal The configurations are named the right-handed, the droplets. The energy of a drop depends upon surface left-handed and the normal solutions. In what follows tension, the angle the molecular alignment makes with it is shown that these configurations can be accounted the surfaces, and the Frank elastic energy in the for by considering the energy in a cylinder about the volume [1]. The molecular alignment is described in axis of the droplets. The problem of minimizing.the terms of a director field n. A problem of some interest energy of the entire droplet is not carried to compleis to find «(r) in a free falling drop when the director at tion. The energetically equivalent right-handed and the surface is prescribed. One can show from dimensional arguments that the drop will remain very close to B W ^ B P ^ ^ W i -*£• main purpose the treatment of a droplet which is l l fehSNVi experimentally somewhat more accessible than the free jj falling drop. Nematic liquid crystal droplets suspended fij from the surface of water take the shape of thin lenses. This geometry provides a preferred axis about which ^ ^ • • • ^ ^ ^ ^ • ^ r ^ ^m m^^^^^^^^^f the droplets are cylindrically symmetric. The somewhat FlG. j _ _ T h r e e d o p l e t s o f M B B A o n a w a t e r substrate viewed surprising result of microscopic observations of these between crossed polarizers. The diameters are ~ 100 nm. JOURNAL DE PHYSIQUE Colloque Cl , supplément au n° 3, Tome 36, Mars 1975, page Cl-177 Résumé. — La nature des singularités ponctuelles dans des cristaux liquides nématiques a été étudiée en plaçant des gouttelettes sur une surface d'eau et en les regardant entre niçois croisés. On observe trois configurations. Ces configurations sont explicables par minimisation des énergies élastiques de Frank, avec des conditions de surface appropriées. Les équations aux dérivées partielles du second ordre, non linéaires, simultanées, qui représentent des systèmes de symétrie cylindrique sont résolues par des méthodes de relaxation, sur calculateur. On discute l'importance de l'annulation de diverses contributions au splay (divergence). Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975134 Cl-178 M. J. PRESS AND A. S. ARROTT left-handed solutions have slightly lower energy than the normal solution for suitable choices of the elastic constants. This difference is quite small compared to the saving of energy which occurs due to a phenomena which is here termed splay cancelling. If the boundary conditions force a variation of the director in one direction then a variation of the director in another direction can lead to a cancellation of contributions to a divergence or splay. This principle is clearly illustrated in the example treated below. It is likely that this is a general principle in the elasticity theory of liquid crystals. In Section 2 expressions are given for the Frank elastic energies in cylindrical coordinates for the case of complete rotational symmetry about the axis. From these one obtains two torque equations to be satisfied in the volume. In Section 3 the boundary conditions are discussed and the particular problem to be solved is described. In Section 4 solutions as determined by relaxation methods using a computer are given. The paper concludes with a discussion of the results and a view of work in progress. 2. Energies and torque equations. For cylindrical symmetry the director field is G(p, z) = sin O(p, z) cos p(p, z) p ̂ + + sin O(p, z) sin p(p, z) + cos O(p, z) ' 2 (2) A A where p, @ and ẑ are unit vectors in this cylindrical coordinate system and 8 and p are the usual polar angles with cp referred to the axis. The Frank elastic energy is given by E = n: dp dzp { S(V.;)' + T(n.V X n)' + 5 The notation S, T, and B refers to splay, twist, and bend for which the notation K,,, KZ, and K,, is usual. The strength of these constants will be of the order of 10-6 erg. For the director as defined in (2), the integrand of (3) becomes + 2 sin 0 cos QO,[S cos2 p + T sin2 p] + 2 sin2 0 sin p cos pp,[S + cos^ 0 + B sin2 O] 2 sin2 0 cos 406, S + 2 sin3 0 cos 0 sin pcp,[B T] + 2 p sin 0 cos O sin q cos p$, p,[T S] + 2 p sin 0 cos 0 cos cp6, @,[B S] + 2 p sin2 0 sin pQ, p, S 2 p sin2 8 sin pep I$, T + 2 p sin" cos Q cos pp, p,[B T] The subscripts denote partial derivatives. Also included is a term for the effect of a magnetic field H in the vertical direction. AX is the diamagnetic anisotropy of the molecules. The variation of the energy with respect to 9 and p produces two torque equations 1 O = 3 sin O cos OIS cos2 p + cos^ 0 sin2 0) sin2 p i2 B sin2 0 sin2 p] + D + sin 8 cos 0 cos2 pe:[s B] + 2 sin cp cos 430, q,[S cos2 0 T + B sin2 01 ELASTIC ENERGIES AND DIRECTOR FIELDS Cl-179 + sin 8 cos 8 c p i [ ~ cosZ cp + c sin' cp 2 sinZ 8 cos2 c p ) + 2 B sin2 8 cosZ p] + sin 8 cos 88:[8 S ] 8,,[S sinZ 8 + B cosZ B] -t sin 8 cos 8cp2[2 sin' 8 + cos' 8 sin2 8)] 1 + sin 8 cos 8 sin cp cos cpcp,[S + T(4 cos2 8 5) -k 4 B sin2 81 P 1 + -sin2 8 sin cpcp,[S + T(2 4 cos2 8) + B(3 cos2 8 sin2 O ) ] P + sin 8 cos 0 sin cp cos cpcp,,[S T ] 1 + sin 8 cos 8 cos cpO,[S B] P + sin 8 cos 8 sin cp8, q,[B S ] + 2 sin 8 cos 8 cos cpO,,[S B] f [cos2 8 sin2 81 cos ye, 8,[S B] sin 0 cos 0 sin 98, y,[S B] + sin2 8 cos cpcp, v,[S + 2 sin' 8 cos2 8) + B(3 cos2 19 sin2 8)] + sin2 8 sin cpcp,,[T S ] + sin ti cos 8 A ~ H ' , (5) and 1 0 = -I sin cp COS cp[I-S + 7'cos2 8 + B sin2 81 + P + sin cp cos cpei[S + 2 7' B] -t sin cp cos c p c p % [ S + T C O S ~ 8 + B sin2 01 2 col 88, cp,[S sin2 cp c sin^ 0 cos2 8) cos2 cp + 2 B sin2 8 cos2 cp] cot 88,cpz[4 sin' 8 + 2 cos' 8 sin2 @>l y,,['Tsin2 8 + B cos2 01 1 + cot 8 sin cp cos cp8,[S + T(3 sin2 8 cos2 8) 4 B sin2 01 P 1 + sin cp8,[sin2 0 3 cosZ 81 [ B T ] P f cot 8 sin cp cos rpB,,[S T ] + cot 8 sin cp8, O,[S + 2 T B1 + sin cp8,,iT S ] + sin 8 cos 8 sin qcp, q,[B T ] + [3 cos2 8 sin2 81 cos q[8, cp, + 8, cp,] [T B ] + 2 sin 8 cos 9 cos cpcp,,[T B ] 1 + sin 8 cos 8 cos cpcp,[T B ] . P Cl-180 M. J. PRESS AND A. S. ARROTT These two simultaneous second order partial differenmation will not account for the right and left-handed tial equations as well as the expression for the integrand solutions, it will give the normal solution and illustrate of eq. (3) simplify considerably if S = B = T = K. the principle of splay cancelling. In the one constant This is called the one constant approximation with approximation elastic constant K. Though the one constant approxi+ + 2 COS q0, + 2 p sin p(0, cp, 0, 9,) + p S + ptl; + p8: + 2 sin B cos 98, (7) and the torque equations become 1 Q, + 0, + Q,, = sin 0 cos 0 P S