Plateau-Rayleigh instability in solids is a simple phase separation

A long elastic cylinder, with 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 gamma/(a mu) > root 32] it loses 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 subcritical 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 = root 32 the length of such an interface diverges, introducing a new length scale and allowing us to construct a one-dimensional effective theory. This treatment yields an analytic expression for the interface itself, revealing that its characteristic length grows as l(wall) similar to a/root Gamma -root 32.