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.

Curvature-Controlled Topological Defects

“…Such structures are among others observed in biological membranes [14,[22][23][24], where v could be relatively easily varied by changing the osmotic pressure of a system. Furthermore, in biological membranes there are several different origins of in-plane ordering [12]. In our simulations, stability ranges of different structures are as follows: oblates are stable within the window v ∈ [ v 1 ∼ 0.59, v 2 ∼ 0.65], prolates for v > v 2 , and stomatocytes for v < v 1 .…”

“…For this purpose, the effective topological charge cancellation [12] mechanism can be used. Its predicting power works well in the cases where the so-called intrinsic geometry contributions [1][2][3]29] are dominant in the free energy terms coupling geometry and orientational ordering.…”

“…Kd 2 r = m tot (1) Recently, it has been shown [12] that one can assign the effective topological charge ∆m eff to a surface patch ∆ζ of an ordered film characterized by its spatially averaged Gaussian curvature K. In the absence of "impurities" it can be expressed as ∆m eff = ∆m + ∆m K [12]. Here, ∆m stands for the total charge of "real" TDs within ∆ζ and the hypothetical smeared Gaussian curvature charge [12,17] is defined as…”

“…Here, ∆m stands for the total charge of "real" TDs within ∆ζ and the hypothetical smeared Gaussian curvature charge [12,17] is defined as…”

“…In this sense, one could say that any closed system is strictly neutral. The neutralization tendency ∆m eff (∆ζ) → 0 [4,17] is presented also in each finite (unclosed) patch of a surface exhibiting spatial non-homogeneities in K, which is referred to as the effective topological charge cancellation (ETCC) mechanism [12]. However, if a neutralization within a patch cannot be achieved via redistribution of existing TDs, it is necessary that a large enough local curvature exists that has the potential to trigger formation of pairs {defect, antidefect} of TDs.…”