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, $E\left(\lambda \right)$, of such a cylinder subject to a homogenous longitudinal stretch $\lambda$. Although $E\left(\lambda \right)$ has a unique minimum, if surface tension is sufficient $[\mathrm{\Gamma }\equiv \gamma /\left(a\mu \right)>\sqrt{32}]$ 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 $\mathrm{\Gamma }=\sqrt{32}$ 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 $l_{\text{wall}}\sim a/\sqrt{\mathrm{\Gamma }-\sqrt{32}}$.

Figure 1

Equilibrium shapes of agar-gel [11] and finite-element cylinders (radius $a$, shear modulus $\mu$) ordered by their elastocapillary number $\mathrm{\Gamma }=\gamma /\left(\mu a\right)$ and showing Plateau-Rayleigh instability for $\mathrm{\Gamma }\gtrsim 6$. The finite-element instability adopts the longest possible wavelength and is colored by pressure.

Figure 2

(a) Energy and (b) tension (its derivative) of an elastic cylinder stretched by $\lambda$ at various elastocapillary numbers $\mathrm{\Gamma }$. As $\mathrm{\Gamma }$ increases, the minimum-energy and zero-tension stretch moves below 1 and, for $\mathrm{\Gamma }>\sqrt{32}=5.67...$, a region of negative curvature appears in the energy between ${\lambda }_{-}$ and ${\lambda }_{+}$, corresponding to an unstable region where tension falls with extension. In (a) the dashed line indicates the energy's common-tangent chord, which spans the concave region meeting the energy at ${\lambda }_1<{\lambda }_{-}$ and ${\lambda }_2>{\lambda }_{+}$. The cylinder can pay the lower common-tangent energy by phase-separating into ${\lambda }_1$ and ${\lambda }_2$ regions and [dashed line in (b)] thus stretches from ${\lambda }_1$ to ${\lambda }_2$ at constant tension. (c) Stretch ratios ${\lambda }_1, {\lambda }_2$, and ${\lambda }_{\pm }$ plotted as a function of $\mathrm{\Gamma }$.

Figure 3

(a) Tension and (b) amplitude of surface-tension-driven phase separation in an elastic cylinder as a function of elastocapillary number $\mathrm{\Gamma }$. In (b) we compare theory (solid line), finite elements (red diamonds), and experimental data (blue circles) from [11]. (c) Comparison of theoretical (solid lines) and finite-element (points) hysteresis loops showing amplitude as a function of imposed stretch at $\mathrm{\Gamma }=10$; the constant amplitude is the hallmark of phase separation.

Figure 4

Domain walls between separated phases. (a) Finite-element calculation of stretch ratio $\lambda \left(z\right)$ along a $\mathrm{\Gamma }=10$ cylinder during phase separation, showing ${\lambda }_1$ and ${\lambda }_2$ regions separated by a domain wall that moves along the cylinder as the average stretch, $〈\lambda 〉$, is increased. (b) Finite-element domain walls (red dots) in cylinders at various elastocapillary numbers $\mathrm{\Gamma }$. The domain wall spreads out and becomes more symmetric as $\mathrm{\Gamma }\to \sqrt{32}$, agreeing with the black theoretical curve [Eq. (14)] by $\mathrm{\Gamma }=5.7$. (c) Comparison of theory (curve) and finite elements (points) for the domain wall length, $l_{\text{wall}}$, as a function of $\mathrm{\Gamma }$.

Figure 5

Top: The actual simulation mesh is a triangulated rectangular mesh between $0<r<a$ and $0<z<L/2$, which covers only one-fourth of the total cylinder. For visualization purposes, the mesh shown here has only $15\times 4$ nodes, and hence 84 triangles, which is many fewer than those used in the paper. Bottom: The same mesh at equilibrium after a surface tension $\mathrm{\Gamma }=9$ has been applied. Color indicates pressure.