Effective Topological Charge Cancelation Mechanism

Abstract

Topological defects (TDs) appear almost unavoidably in continuous symmetry breaking phase transitions. The topological origin makes their key features independent of systems' microscopic details; therefore TDs display many universalities. Because of their strong impact on numerous material properties and their significant role in several technological applications it is of strong interest to find simple and robust mechanisms controlling the positioning and local number of TDs. We present a numerical study of TDs within effectively two dimensional closed soft films exhibiting in-plane orientational ordering. Popular examples of such class of systems are liquid crystalline shells and various biological membranes. We introduce the Effective Topological Charge Cancellation mechanism controlling localised positional assembling tendency of TDs and the formation of pairs {defect, antidefect} on curved surfaces and/or presence of relevant "impurities" (e.g. nanoparticles). For this purpose, we define an effective topological charge Δmeff consisting of real, virtual and smeared curvature topological charges within a surface patch Δς identified by the typical spatially averaged local Gaussian curvature K. We demonstrate a strong tendency enforcing Δmeff → 0 on surfaces composed of Δς exhibiting significantly different values of spatially averaged K. For Δmeff ≠ 0 we estimate a critical depinning threshold to form pairs {defect, antidefect} using the electrostatic analogy.

Figure 1. Schematic representation of closed shells and their electrostatic analogue.

(a) prolate ellipsoid, (b) oblate ellipsoid, (c) dumb-bell shape. Typical spatial variation of the Gaussian curvature is depicted at the right side of these shapes. (d) The electrostatic analogue of structures. The capacitor plates located at ρ₁ and ρ₂ host effective topological charges Δm_eff (Δζ₊) > 0 and Δm_eff (Δζ₋) < 0, respectively, where |Δm_eff (Δζ₋)| = Δm_eff (Δζ₊).

Figure 2. Schematic representation of ETCC limit structures on ellipsoidal shells.

Approximate positions of TDs are presented on (a) spherical, (b) oblate, and (c) prolate ellipsoids. In cases (I), (II) and (III) ellipsoids host one, two, and three NPs bearing m = 1, respectively.

Figure 3. Calculated order parameter profiles in the (u, v) plane of the ellipsoidal shells.

Column (a): spherical shapes, a/b = 1.0; Column (b): oblate shapes, a = R and (I) b/a = 1.5, (II) b/a = 1.5, (III) b/a = 2.0, (IV) b/a = 2.0; Column (c): prolate shells, b = R and (I) a/b = 7.0, (II) a/b = 5.0, (III) a/b = 4.5, (IV) a/b = 3.0. Black circles and ellipses indicate shell shapes. Nanoparticles are labelled NP. Nematic ordering was calculated for: R/ξ = 3.5, k_e = 0, R = min{a, b}.

Figure 4. Trajectories of topological defects as the functions of η = a/b.

By v_d we denote the v coordinate of the defect origin and by v_1/4 the point to which the surface integral of the Gaussian curvature from the equator divided by 2π equals 1/4 m_tot. Cases with (a) one, (b) two and (c) three nanoparticles are presented. R/ξ = 3.5, k_e = 0, R = min{a, b}.

Figure 5. Depinning threshold in dumb-bell configurations.

Panels (a,b) show nematic ordering in the (u, s) plane just below and above the threshold calculated for ρ₂/ξ = 7, k_e = 0. Here, s is the arc length of the profile curve (see Eq. (14)). The shape, calculated at the threshold, is presented in panel (c), where ρ₀ determines the value of ρ(s) where K = 0. In panel (d) we plot the penalty Δf^(p) (left-hand side of Eq. (13), dashed red line (ρ₂/ξ = 3.5), dotted dashed blue line (ρ₂/ξ = 7)) and gain Δf ^(g) (right-hand side of Eq. (13), full line) contributions as the functions of ρ₂/ρ₁. In the inset we plot as the function of ρ₂/ρ₁. Squares reveal points for which was calculated and the full line is obtained as the corresponding best fit.