Modeling of rate-dependent phase transition in bacterial flagellar filament

Recent experiment of Darnton and Berg [34] showed that phase transition of bacterial flagellar filament is loading rate-dependent. The object of this paper is to describe the observed loading rate-dependent phase transition responses of the filament by using time dependent Ginzberg–Landau continuum model. We developed a finite element method (FEM) code to simulate the phase transition under a displacement-controlled loading condition (controlled helix-twist) by using viscosity-type kinetics. Our FEM simulation captures the main features of the rate-dependence: under slow loading (i.e., loading time ≫ the relaxation time) the filament phase transition is an equilibrium process and each phase grows via interface propagation on the Maxwell line; under rapid loading (i.e., loading time ≪ the relaxation time), the phase transition does not occur and the response is elastic. Our FEM model provides a tool to study the effects of loading-rate dependent phase transition for bio-filament with viscous kinetics.