Hydrodynamic theory of $p-$atic liquid crystals

Abstract

We formulate a comprehensive hydrodynamic theory of two-dimensional liquid crystals with generic p−fold rotational symmetry, also known as p−atics, of which mematics (p = 2) and hexatics (p = 6) are the two best known examples. Previous hydrodynamic theories of p−atics are characrerized by continuous O(2) rotational symmetry, which is higher than the discrete rotational symmetry of p−atic phases. By contrast, here we demonstrate that the discrete rotational symmetry allows the inclusion of additional terms in the hydrodynamic equations, which, in turn, lead to novel phenomena, such as the possibility of flow alignment at high shear rates, even for p > 2. Furthermore, we show that any finite imposed shear will induce long-ranged orientational order in any p−atic liquid crystal, in contrast to the quasi-long-ranged order that occurs in the absence of shear. The induced order parameter scales like a non-universal power of the applied shear rate at small shear rates.

Fractional defect charges in $p$-atic liquid crystals on cones

“…Finally, it will be interesting to examine the dynamics of these defects both near and well away from equilibrium. The tensor hydrodynamics of general p-atics in flat space has recently been investigated theoretically [66,67], while p = 2 nematic defects in the presence of activity has been thoroughly studied both in flat space and spherical surfaces [11][12][13]. It would be intriguing to generalize these studies to study the non-equilibrium active dynamics of p-atics on conic surfaces, realizable in epithelial monolayers [16][17][18] and living tissue [19] embedded in conic geometries.…”

Epithelia are multiscale active liquid crystals

“…(1) gives, up to a normalization constant, the traceless part of the standard rank−2 shape tensor [8, 9]. Regardless of its rank, the tensor G p has only two linearly independent components in two dimensions [13, 14], from which one can extract information about the cells’ orientation and anisotropy. In particular, using a generalization of the spectral theorem to tensors with arbitrary rank [15, 16], one can show that all elements of G p are proportional to either the real or the imaginary part of the complex order parameter where ϕ i = arctan( y i /x i ) the angular coordinate of the i− th vertex of a given cell (Fig.…”

“…Top (bottom) panel shows a pair of nematic (hexatic) defects of winding number s = ±1 / 2 ( s = ±1 / 6) f Magnitude of Ψ 2 and Ψ 6 versus the coarse graining radius R measured from experimental and numerical mpf data. Both data sets fit the power law , with η p a non-universal exponent [13, 14], with the following fitting parameters: (experiments) η 2 = 0.41 ± 0.01, η 6 = 0.49 ± 0.01; (mpf) η 2 = 0.43 ± 0.02, η 6 = 0.48 ± 0.01. In both experiments and multiphase field simulations, the |Ψ 2 | and |Ψ 6 | order parameters crossover at the length scale R × , with: (experiment) R × /R cell = 4.6 ± 1.0; (mpf) R × /R cell = 5.0 ± 1.2.…”

Hydrodynamics and multiscale order in confluent epithelia

“…Motivated by these recent discoveries, in this Letter we propose a continuum theory of confluent epithelia rooted in the hydrodynamics of liquid crystals with generic p−atic rotational symmetry (hereafter p−atic liquid crystals). To this end, we exploit preliminary work toward extending the classic hydrodynamic theory of hexatic liquid crystals [32,33] to account for arbitrary discrete rotational symmetry [34,35]. Our analysis suggests that collectively migrating epithelial layers, such as that theoretically depicted by the SPV model in Eqs.…”

“…The order parameter tensor Q p , the mass density ρ and the momentum density ρv, with v the local velocity field, comprise the set of hydrodynamic variables describing the dynamics of a generic p−atic fluid, which in turn is governed by the following set of partial differential equations [34,35]:…”