Hydrodynamic theory ofp-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 nematics (p = 2) and hexatics (p = 6) are the two best known examples. Previous hydrodynamic theories of p-atics are characterized 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 nonuniversal power of the applied shear rate at small shear rates.

Long-Ranged Order and Flow Alignment in Shearedp‐atic Liquid Crystals

“…However, for all p > 2, except p ¼ 4, as shown in Ref. [20], η ðaÞ vanishes. Hence, kinetic energy is isotropically dissipated throughout the flow.…”

“…As demonstrated in Ref. [20], in the Stokesian limit (i.e., where inertial effects are negligible) and when ρK p =η 2 ≪ 1, a limit that applies extremely well to all known liquid crystals, Eqs. (3) suitably supplemented by noise terms required by the fluctuation-dissipation theorem at finite temperature can be cast in a single linear equation for the microscopic orientation ϑ, i.e.,…”

“…is the elastic stress arising in response to a static deformation of a fluid patch, and σ ðrÞ and σ ðvÞ are, respectively, the reactive (i.e., energy-preserving) and viscous (i.e., energydissipating) stresses originating from the reversible and irreversible couplings between p-atic order and flow. Calculating the entropy production rate [20] allows one to express the former as…”

“…Hence, kinetic energy is isotropically dissipated throughout the flow. For p ¼ 4, on the other hand, ðaÞ ¼ ϱ 4 ⟦n ⊗4 ⟧, where −2η ≤ ϱ 4 ≤ 2η is an anisotropic viscosity, and the inequalities follow from the second law of thermodynamics [20]. Lastly, in the absence of topological defects or other singular features, the phase θ of the complex order parameter is the only hydrodynamic variable arising from the broken rotational symmetry, since jΨ p j relaxes to its equilibrium value in a finite time.…”

“…From the exact solution of Eq. ( 11) [20], one can then compute the correlation function of the complex p-atic order parameter in the form hψ Ã p ðrÞψ p ð0Þi ¼ exp½−p 2 gðrÞ, with…”