Defect dynamics in crystalline buckled membranes

Abstract

We study the dynamics of defect annihilation in flexible crystalline membranes suffering a symmetry-breaking phase transition. The kinetic process leading the system toward equilibrium is described through a Brazovskii-Helfrich-Canham Hamiltonian. In membranes, a negative disclination has a larger energy than a positive disclination. Here we show that this energetic asymmetry does not only affect equilibrium properties, like the Kosterlitz-Thouless transition temperature, but also plays a fundamental role in the dynamic of defects. Both unbinding of dislocations and Carraro-Nelson "antiferromagnetic" interactions between disclinations slow down the dynamics below the Lifshitz-Safran regime observed in flat hexagonal systems.

Long-scale dynamics of crystalline membranes

“…Then, for ω → 0 and k → 0, the integral (26) is reduced to a residue in the pole ω = 0, and the real part of the self-energy functionˆ reads as αβμν = − T 384π 2 κ 2 (δ αβ δ μν + δ αμ δ βν + δ αν δ βμ )ξ. (27) As it should be, the expression (27) reproduces the static renormalization of the Lame coefficients λ and μ [Eq. (9)].…”

“…This and other nonelectronic graphene properties make it a promising candidate for various nanoelectromechanical device applications, e.g., vibrational mass detectors. 13 Theoretical studies of thermodynamic (static) properties of crystalline membranes started actually long before the "graphene era" [14][15][16][17][18] and are still an active area of research nowadays [19][20][21][22][23][24][25][26][27] (these papers will be partially commented in what follows, and much more publications could be added to this list). Surprisingly, to our knowledge, no work was devoted to dynamics of crystalline membranes.…”

Wrinkles and splay conspire to give positive disclinations negative curvature

“…Likewise, if a 2D crystal is allowed to buckle out of the plane, the elastic energy associated with isolated disclinations can be strongly reduced by screening their strain fields through curvature, trading off stretching for bending energy (9,10). Depending on the sign of the topological charge, isolated defects can deform the membrane into cone-or saddle-shape configurations, acting as sources of Gaussian curvature (9,11). Here we demonstrate experimentally, theoretically, and through simulations that the molecular splay distortions associated with disclinations in a free-standing smectic membrane act as sources of Gaussian curvature, resulting in a pattern of wrinkles in the membrane that form perpendicular to the underlying smectic layers.…”

“…Combining the Helfrich-Canham model of a fluid membrane and the classical Brazovskii model describing striped phases accounts for both smectic order and membrane elasticity (11). The equilibrium configuration of unstructured fluid membranes has been extensively studied since the pioneering papers by Helfrich (17) and Canham (18).…”

“…where η ψ ðr, tÞ and η h ðr, tÞ are random Gaussian noise fields satisfying the fluctuation dissipation theorem (11). As in the experiment, ψðr, tÞ is first evaluated in a planar system (h = 0) to obtain the desired defect density.…”