Curvature controls beading in soft coated elastic cylinders: Finite wavemode instability and localized modulations

Axisymmetric beading instabilities in soft, elongated cylinders have been observed in a plethora of scenarios, ranging from cellular nanotunnels and nerves in biology to swollen cylinders and electrospun fibers in polymer physics. One of the common geometrical features that can be seen in these systems is the finite wavelength of the emerging pattern. However, modelling studies often predict that the instability has an infinite wavelength, which can be associated with localized necking or bulging. In this paper, we consider a soft elastic cylinder with a thin coating that resists bending, as described by the Helfrich free energy functional. The bending stiffness and natural mean curvature of the coating are two novel features whose competition against bulk elasticity and capillarity is investigated. For intermediate values of the bending stiffness, a linear stability analysis reveals that the mismatch between the current and natural mean curvature of the coating can lead to patterns emerging with a finite wavelength. This analysis creates a continuous bridge between the classical solutions of the shape equation obtained from the Helfrich functional and a curvature-controlled zero-wavemode instability, similar to the one induced by the competition between bulk elasticity and capillarity. The weakly non-linear analysis predicts that the criticality of the bifurcation depends on the controlling parameter, with both supercritical and subcritical bifurcations possible. When capillarity is introduced, the criticality of the bifurcation changes in a non-trivial way.