Singular values, nematic disclinations, and emergent biaxiality

Abstract

Both uniaxial and biaxial nematic liquid crystals are defined by orientational ordering of their building blocks. While uniaxial nematics only orient the long molecular axis, biaxial order implies local order along three axes. As the natural degree of biaxiality and the associated frame, that can be extracted from the tensorial description of the nematic order, vanishes in the uniaxial phase, we extend the nematic director to a full biaxial frame by making use of a singular value decomposition of the gradient of the director field instead. New defects and degrees of freedom are unveiled and the similarities and differences between the uniaxial and biaxial phase are analyzed by applying the algebraic rules of the quaternion group to the uniaxial phase.1 arXiv:1302.3159v2 [cond-mat.soft]

Umbilic Lines in Orientational Order

“…We demonstrate a general connection with the shape operator for a surface or vector field, from which their chirality pseudotensor can be derived, and show that natural benefits are gained by considering both operators. Analogous geometrical structures to the umbilics we describe have been discussed by Thorndike et al [68], Čopar et al [69], and Beller et al [70], although not with the topological aspect that we give here.…”

“…6 for some standard blue phase textures. The umbilics coincide with structural motifs that have been identified previously, either through symmetry considerations [106] or by taking a singular value decomposition of the Q-tensor [69]. These motifs are the axes of double twist cylinders [6,7].…”

“…1). However, in a Q-tensor description of BPII these are not degeneracies where the local structure is axisymmetric, as in the standard idealised picture of a double twist cylinder; rather they are 2-fold lines where the Qtensor is maximally biaxial [69,107,108]. Second, it is perhaps worth emphasising that the umbilics identified in all other blue phase textures also differ from the standard idealised picture of a double twist cylinder.…”

Fast algorithms for liquid crystal modelling

“…Dopants may posses permanent electric 13,14 or magnetic dipoles [15][16][17][18][19] leading to ferronematic systems, or their influence may be due to mechanical anchoring on the dopant surface that can lead to remarkable defect structures. 20,21 Here we focus on metallic, e.g. gold, nanoparticles.…”

Defects in Planar Cell Polarity of Epithelium

“…Consequently, their essential properties are independent of system's microscopic details, and they exhibit several universalities. Major advances in physics of TDs have been done in the field of liquid crystals (LCs) [24]. Namely, rich variety of LC phases and structures reached via symmetry-breaking phase transitions enables realization of almost all qualitatively different TDs in nature.…”