Figure 1

Cross section of a spherical droplet with radius $R$ lying on a flat, homogeneous substrate. The angle $\vartheta$ denotes the colatitude on the droplet’s surface, and ${\vartheta }_c$ is the contact angle—that is, the angle between the droplet’s and substrate’s conormals.Reuse & Permissions

Figure 2

Graphical solution of system (8), for a given value of ${\vartheta }_c^0$. The admissible region $\mathcal{A}$ (in gray) has been plotted together with the curve ${\epsilon }_1({\vartheta }_c,{\vartheta }_c^0)$ and several curves in the family ${\epsilon }_2({\vartheta }_c,{\tau }_{*})$. When ${\tau }_{*}∊[0,{\tau }_{*c}\left({\vartheta }_c^0\right)\mathbf{)}$ (e.g., the curve ${\tau }_{*}={\tau }_{*1}$), two acceptable equilibria exist, while no equilibrium exists for ${\tau }_{*}>{\tau }_{*c}\left({\vartheta }_c^0\right)$ (e.g., the curve ${\tau }_{*}={\tau }_{*2}$). At ${\tau }_{*}={\tau }_{*c}\left({\vartheta }_c^0\right)$ the two equilibria coalesce into a single equilibrium solution. When ${\tau }_{*}$ is negative (e.g., the curve ${\tau }_{*}={\tau }_{*0}$), only one equilibrium exists. In $\mathcal{A}$, $\epsilon$ ranges within the interval $[-1.3,1.3]$.Reuse & Permissions

Figure 3

Complex values of $\nu$ relevant to our analysis. The real and imaginary axes of the complex $\nu$ plane are dashed lines; the set $\mathcal{I}$ is the solid line, whose thicker part represents the subset $\mathcal{U}$ corresponding to negative values of the multiplier $\mu$ and, hence, to unstable equilibria.Reuse & Permissions

Figure 4

Stability analysis for negative line tension. The dotted curve delimits the admissible region $\mathcal{A}$ in Fig. 2. The solid line is the marginal curve ${\epsilon }_1^2\left({\vartheta }_c\right)$, along which the minimum value of $\epsilon$ is $\epsilon \approx -0.52$. In the region $\mathsf{U}\mathsf{s}$ below ${\epsilon }_1^2\left({\vartheta }_c\right)$, equilibria are unstable regardless of the magnitude of the line tension. In the region $\mathsf{R}\mathsf{s}$ above ${\epsilon }_1^2\left({\vartheta }_c\right)$, equilibria are residually stable. The dashed curves in $\mathsf{R}\mathsf{s}$ are marginal curves corresponding to $m=3,4,5,6$, ordered upwards. For every point $\mathsf{p}$ in $\mathsf{R}\mathsf{s}$, there exists a value $m_{\mathrm{rs}}$ of $m$ such that all marginal curves with $m⩽m_{\mathrm{rs}}$ lie below $\mathsf{p}$, while those with $m>m_{\mathrm{rs}}$ lie above it. Consequently, $\mathsf{p}$ represents an equilibrium stable against modes with $m⩽m_{\mathrm{rs}}$ and unstable against modes with $m>m_{\mathrm{rs}}$. In the limit where $m\to \infty$, the marginal curves tend towards the ${\vartheta }_c$ axis.Reuse & Permissions

Figure 5

The index $m_{\mathrm{rs}}$ of residual stability against the dimensionless line tension ${\tau }_{*}$, when the bare contact angle ${\vartheta }_c^0$ is $\pi ∕4$ (dashed line), $\pi ∕2$, or $3\pi ∕4$ (thick line). Two features are worth noting. First, when the magnitude of the (negative) line tension is large enough, $m_{\mathrm{rs}}=2$, meaning that all modes are unstable. Second, on approaching ${\tau }_{*}=0$, $m_{\mathrm{rs}}$ diverges, implying that the range of residual stability increases.Reuse & Permissions

Figure 6

Curves corresponding to different values of the index $m_{\mathrm{rs}}$ of residual stability. From the lower to the upper curve, $m_{\mathrm{rs}}=2,5,10,20,30,40,50,70,100$. The vertical axis reports $-{\log }_{10}\mid {\tau }_{*}\mid$, so that going upwards the line tension passes from higher, negative values to smaller, negative values. The curve $m=2$ marks the onset of residual stability: below it, no stable mode exists at all.Reuse & Permissions

Figure 7

The experimental data obtained in [14] are plotted against the curves with constant index of residual stability illustrated in Fig. 6. The experimental points are represented by crosses and $m_{\mathrm{rs}}=2,5,10,20,30,40,50,70,100$, from the lower to the upper curve. Most experimental points fall across the curve with $m_{\mathrm{rs}}=50$.Reuse & Permissions

Figure 8

The marginal curve ${\epsilon }_1^0\left({\vartheta }_c\right)$ divides the admissible set $\mathcal{A}$ into two parts. Only the part below the graph of ${\epsilon }_1^0\left({\vartheta }_c\right)$ is stable. The thin lines are graphs of the functions ${\epsilon }_1({\vartheta }_c,{\vartheta }_c^0)$ and ${\epsilon }_2\mathbf{(}{\vartheta }_c,{\tau }_{*c}\left({\vartheta }_c^0\right)\mathbf{)}$ defined in Eqs. (8), for several values of ${\vartheta }_c^0$. When ${\tau }_{*}={\tau }_{*c}\left({\vartheta }_c^0\right)$, the two equilibria coalesce into one another and so ${\epsilon }_1({\vartheta }_c,{\vartheta }_c^0)$ and ${\epsilon }_2\mathbf{(}{\vartheta }_c,{\tau }_{*c}\left({\vartheta }_c^0\right)\mathbf{)}$ have a common tangent. Here we see that the marginal curve ${\epsilon }_1^0\left({\vartheta }_c\right)$ passes through all these tangent points.Reuse & Permissions

Figure 9

Bifurcation diagram showing the stability of equilibria in terms of the dimensionless line tension ${\tau }_{*}$, when the bare contact angle ${\vartheta }_c^0$ is $\pi ∕3$. Solutions are parametrized by their contact angle ${\vartheta }_c$. Dashed lines correspond to unstable equilibria. Solid lines correspond to stable equilibria: both locally stable (thin line) and globally stable (thick line). The dotted line marks the first-order drying transition.Reuse & Permissions

Figure 10

(Color online) A spherical cap perturbed by a marginal mode with $m_{\mathrm{rs}}=10$. The perturbed contact line exhibits a periodic structure with ten fingers.Reuse & Permissions