Curvature-Controlled Topological Defects

Abstract

Effectively, two-dimensional (2D) closed films exhibiting in-plane orientational ordering (ordered shells) might be instrumental for the realization of scaled crystals. In them, ordered shells are expected to play the role of atoms. Furthermore, topological defects (TDs) within them would determine their valence. Namely, bonding among shells within an isotropic liquid matrix could be established via appropriate nano-binders (i.e., linkers) which tend to be attached to the cores of TDs exploiting the defect core replacement mechanism. Consequently, by varying configurations of TDs one could nucleate growth of scaled crystals displaying different symmetries. For this purpose, it is of interest to develop a simple and robust mechanism via which one could control the position and number of TDs in such atoms. In this paper, we use a minimal mesoscopic model, where variational parameters are the 2D curvature tensor and the 2D orientational tensor order parameter. We demonstrate numerically the efficiency of the effective topological defect cancellation mechanism to predict positional assembling of TDs in ordered films characterized by spatially nonhomogeneous Gaussian curvature. Furthermore, we show how one could efficiently switch among qualitatively different structures by using a relative volume v of ordered shells, which represents a relatively simple naturally accessible control parameter.

Topological Sound and Flocking on Curved Surfaces

“…The frustration associated with the interplay of curvature and order [30,68,69] has many consequences for crystals [70][71][72][73][74], tethered membranes [75,76], liquid crystalline membranes [77][78][79], and jammed and glassy systems [80,81]. The presence of activity adds an entirely new nonequilibrium dimension to the whole story, allowing for completely new physics arising from competing order, curvature, and the active drive.…”

Budding and Fission of Membrane Vesicles: A Mini Review

“…Here, ⊗ represents a tensor product and {n, n ⊥ } are the eigenvectors of Q corresponding to the eigenvalues of {λ, −λ} [94,99]. In Equation 1, n represents the nematic director field, i.e., the direction of molecules, which exhibits head-to-tail invariance [92].…”

The role of membrane vesiculation and encapsulation in cancer diagnosis and therapy

“…We model a membrane as a homogeneous two-dimensional film (fluid) exhibiting in-plane nematic ordering. To mathematically describe the local shape of the membrane surface, we introduce a curvature tensor [69,75,88,89]:…”

“…To generate such shapes, we define the shell profile curve, which is then rotated about the z-axis by an angle of 4 ¼ 2p. In the Cartesian coordinates (e x ; e y ; e z ), the position vector of a generic point on an axisymmetric surface is given as [88,90]:…”

“…4 represent the nematic ordering amplitude. At the core of topological defects, the nematic order is lost [75,88,90]. Therefore, topological defects are located at the points on the surface exhibiting l ¼ 0, which are marked with red color in Fig.…”