Recent experiments report that the long-looked-for thermotropic biaxial nematic phase has been finally detected in some thermotropic liquid crystalline systems. Inspired by these experimental observations, we concentrate on some elementary theoretical issues concerned with the classical sixth-order Landau-de Gennes free energy expansion in terms of the symmetric and traceless tensor order parameter Q alpha beta. In particular, we fully explore the stability of the biaxial nematic phase giving analytical solutions for all distinct classes of the phase diagrams that theory allows. This includes diagrams with triple, critical, and tricritical points and with multiple (reentrant) biaxial and uniaxial phase transitions. A brief comparison with predictions of existing molecular theories is also given.
The measurement of the saddle-splay surface elastic constant K~4 in a nernatic liquid crystal is reported based on two independent deuterium nuclear-magnetic-resonance ( H-NMR) experiments. Fifty years after the pioneering work of Oseen and Zocher, these measurements were made from observations of nematic director-field configurations and a configuration transition discovered in submicrometer-sized cylindrical cavities of Nuclepore membranes under selected surface preparations and wall curvatures. The experimental difficulties in separating the effect of anchoring energy from surface elastic energy (inherent in small confining volumes) were overcome by a unique use of NMR and the ability to predict stable nematic structures with elastic theory. Direct comparison of calculated H-NMR spectral patterns to experiment is very sensitive to the details of the stable nematic director-field configuration in cylindrical cavities. Small differences in the director configuration imposed by the curvature or elastic properties of the nematic liquid crystal are strongly rejected in the shape of the spectral pattern.Different nematic structures with preferred perpendicular anchoring conditions such as the escaped radial and planar polar show markedly different patterns. Theoretical analysis reveals that a planar-polar structure is preferred in cavities with a high degree of curvature or sufficiently weak anchoring conditions at the cavity boundary. The features of the planar-polar structure are described in terms of the dimensionless parameter R 8'0/K, where R is the radius, 8'o is the molecular anchoring strength, and K is the bulk elastic constant in the one-constant approximation. At a sufficiently large radius or substantially strong anchoring conditions, the escaped-radial structure is favored and sensitive to the dimensionless surface parameter O. =R Wo/K+K24/K -1. The H-NMR technique unambiguously distinguishes between these two stable nematic director-field configurations and is sensitive to the surface parameters e and R8'0/K. Theoretical analysis also reveals that the relative magnitude of the bend (K33) and splay (K») bulk elastic constants plays a vital role in determining the molecular anchoring angle at the cavity wall of the escaped-radial configuration. As K33/K» increases, the molecular anchoring angle is shown to deviate further from its preferred perpendicular orientation in the presence of finite molecular anchoring to alleviate the expensive bend deformation. Experimentally, point defects occur in the escapedradial configuration where the direction of bend is changed, resulting is a series of alternating hyperbolic and radial defects. An analytical trial function is constructed that describes these defect structures in the region of interest. PACS number(s): 64.70.Md, 61.30. By, 61.30.Jf, 68.10.Cr I. INTRODUCTION Despite the successful application of elastic theory to uniaxial liquid crystals pioneered by Oseen [l] and Zocher [2] more than 50 years ago, the importance of the surface elastic contributions has...
Analysis of the Frank free energy for nematics confined to cylindrical regions indicates that the director pattern is dependent on the surface elastic constant KIA if there is weak normal anchoring and escape along the cylinder axis. Using deuterium nuclear-magnetic-resonance techniques on samples of submicrometer-size cavities in Nuclepore membranes, we report the first measurement of A' 24.PACS numbers: 64.70.Md, 61.30.By, 6I.30Jf, 68.10.Cr In spite of the fact that the elastic theory of uniaxial nematic liquid crystals has been studied for more than fifty years since the pioneering work of Oseen [1] and Zocher [2], there is still no generally accepted method of determining the so-called surface elastic constant Ki^. The reason for this has been that Ki^ is irrelevant unless weak anchoring occurs at a surface, and in that case it has proven to be difficult to separate the effects of the anchoring energy from the surface elastic energy. In this paper, analysis and magnetic-resonance measurements are presented which determine both the anchoring strength and K2A by studying director configurations in cylindrically shaped samples with the easy axis for the director normal to the surface. For sufficiently weak anchoring, a new planar-polar director configuration occurs in small-radius samples while at large radii a nonplanar arrangement called the escaped-radial configuration appears [3], as first discussed by Cladis and Kleman [4] and by Meyer [5].The starting point for the analysis is the elastic free energy, as described by Saupe [6], supplemented by an anchoring energy term [we have used the identity nxcurln = --(ngrad)n)], F==JyVH/^ii(divn)2 + /r22(n-curln)2 + A:33(nxcurln)^-/^24div(nxcurln4-ndivn (1) This form for the elastic free energy has been widely used, although the importance of another elastic term (with elastic constant A'13) arising due to second derivatives of n has also been studied [7,8] and Schmidt [9] has argued that the terms involving K22 and Kj^ should be regrouped. The validity of neglecting AT 13 is open to question, but is based on having weak deformation and consistently keeping terms only of first derivatives of n. If A'13 is to be included, other terms involving the square of second derivatives of n must also be kept [7,8]. The last term, proportional to the anchoring strength W^o, is of the Rapini-Papoular form [10], where (j) is the angle between n and the surface normal. In the strong anchoring limit where WQ-* 00, the surface free energy is replaced by the boundary condition 0=0; i.e., n becomes perpendicular to the surface. When a liquid crystal is confined to a cylindrical region of radius R and the surface favors ordering in the radial direction, there is competition between the bulk elastic and surface parts of the free energy. Guided by recent experiments [11] and earlier work [4,5], the four configurations shown at the top of Figs. 1 and 2 are studied here. Convenient designations for them are the planarradial (PR), planar-polar (PP), escaped-radial (ER), and escaped-radia...
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