Stretching helical nano-springs at finite temperature

Using dynamic simulations and analytic methods, we study the elastic response of a helical filament subject to uniaxial tension over a wide range of bend and twist persistence length. A low-pitch helix at low temperatures exhibits a stretching instability and the force-extension curve consists of a sequence of spikes. At elevated temperature (i.e. small persistence lengths) the helix melts and a pronounced force plateau is obtained in the fixed-extension ensemble. The torque boundary condition significantly affects the resulting elastic properties. The elasticity of flexible filaments has been the subject of intense research efforts, responding to the growing need to understand mechanical and thermodynamic properties of biopolymers such as DNA or filamentous proteins [1]. Various theoretical approaches, ranging from linear elasticity theory [2] to quantum-chemical modelling [3], have been successfully used to describe experimental forceversus-extension curves of synthetic and biological polymers. Yet, less is known about mechanical properties of filaments with more complicated ground-state molecular architectures. Helices are ubiquitous motifs in nature [4, 5] and provide potential applications in a wide spectrum of engineering and scientific fields [6]. Anorganic nanosprings, like SiC nanowires or single-crystal ZnO nanobelts, are promising key components in nanotechnology [7]. Organic self-assembled helical ribbons are potentially useful for drug delivery system or as biological probes [8]. From the theoretical point of view, the mechanics of elastic helices is intriguing due to the coupling of elasticity and geometry. An analysis at zero-temperature (i.e., in the absence of shape fluctuations) revealed a discontinuous multi-step transition of a helical spring with increasing stretching force [9]. Such tension-induced instabilities have been experimentally observed for organic self-assembled helical ribbons using a micromanipulator [8] and for the helical polysaccharide xanthan with the atomic force microscope [10], exhibiting a pronounced force plateau. For experiments on nanoscopic helices at room temperature, shape-fluctuations are expected to modify the resulting elastic response in a crucial way. However, only few theoretical works investigated the interplay of thermal fluctuations and helix elasticity in the presence of external forces [11, 12]. In this paper we first present a simple analysis of the force-stretching relation for a helix at zero temperatures, based upon previous theoretical approaches [9, 11]. Next, employing dynamic simulations, we systematically study thermal effects on the helix elasticity. For elevated temperatures (low bending persistence length) a force plateau is obtained in the fixed extension ensemble; the characteristic plateau force obeys a simple scaling relation with a numerical prefactor that is determined by simulations. For very high temperatures the helical structure melts and simple worm-like-chain elasticity is recovered. The helix becomes stiffer when terminal rotation is prohibited via an externally applied torque. To proceed, consider an inextensible filament (or ribbon) with contour length L, parameterized by the arclength s. A generalized Frenet orthonormal basis {ê1, ê2, ê3} is defined along the filament centerline r(s), where ê3 points along the tangent and ê1, ê2 correspond to the principle axes of the cross section. The strain rate vector Ω(s) = (Ω1,Ω2,Ω3) characterizes the shape of the filament through the kinematic relation ∂sêj = Ω × êj , where κ = (Ω21 + Ω 2 2) 1/2 is the curvature, Ω3 the twist density, and ∂s denotes the partial derivative with respect to s. According to linear elasticity theory, the bending and twisting energy of an inextensible filament reads