The Plateau-Rayleigh instability in solids is a simple phase separation

Abstract

A long elastic cylinder, radius $a$ and shear-modulus $\mu$, becomes unstable given sufficient surface tension $\gamma$. We show this instability can be simply understood by considering the energy, $E(\lambda)$, of such a cylinder subject to a homogenous longitudinal stretch $\lambda$. Although $E(\lambda)$ has a unique minimum, if surface tension is sufficient ($\Gamma\equiv\gamma/(a\mu)>\sqrt{32}$) it looses convexity in a finite region. We use a Maxwell construction to show that, if stretched into this region, the cylinder will phase separate into two segments with different stretches $\lambda_1$ and $\lambda_2$. Our model thus explains why the instability has infinite wavelength, and allows us to calculate the instability's sub-critical hysteresis loop (as a function of imposed stretch), showing that instability proceeds with constant amplitude and at constant (positive) tension as the cylinder is stretched between $\lambda_1$ and $\lambda_2$. We use full nonlinear finite-element calculations to verify these predictions, and to characterize the interface between the two phases. Near $\Gamma=\sqrt{32}$ the length of such an interface diverges introducing a new length-scale and allowing us to construct a 1-D effective theory. This treatment yields an analytic expression for the interface itself, revealing its characteristic length grows as $l_{wall}\sim a/\sqrt{\Gamma-\sqrt{32}}$.