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

Abstract

A long elastic cylinder, radius $a$ and shear-modulus $μ$, becomes unstable given sufficient surface tension $γ$. We show this instability can be simply understood by considering the energy, $E(λ)$, of such a cylinder subject to a homogenous longitudinal stretch $λ$. Although $E(λ)$ has a unique minimum, if surface tension is sufficient ($Γ\equivγ/(aμ)>\sqrt{32}$) 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 $λ_1$ and $λ_2$. 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 $λ_1$ and $λ_2$. We use full nonlinear finite-element calculations to verify these predictions, and to characterize the interface between the two phases. Near $Γ=\sqrt{32}$ 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 $l_{wall}\sim a/\sqrt{Γ-\sqrt{32}}$.