Ballooning, bulging and necking: an exact solution for longitudinal phase separation in elastic systems near a critical point

Abstract

Prominent examples of longitudinal phase separation in elastic systems include elastic necking, the propagation of a bulge in a cylindrical party balloon and the beading of a gel fiber subject to surface tension. Here we demonstrate that, if the parameters of such a system are tuned near a critical point (where the difference between the two phases vanishes) then the behaviour of all systems is given by the minimization of a simple and universal elastic energy familiar from Ginzburg-Landau theory in an external field. We minimize this energy analytically, which yields not only the well known interfacial tanh solution, but also the complete set of stable and unstable solutions in both finite and infinite length systems, unveiling the elastic system's full shape evolution and hysteresis. Correspondingly, we also find analytic results for the the delay of onset, changes in criticality and ultimate suppression of instability with diminishing system length, demonstrating that our simple near-critical theory captures much of the complexity and choreography of far-from-critical systems. Finally, we find critical points for the three prominent examples of phase separation given above, and demonstrate how each system then follows the universal set of solutions.