Figure 1

(Color online) Schematic view of the investigated system. The substrate is topographically flat, invariant in the $\mathit{y}$ direction, chemically heterogeneous, and $2\mathit{L}$-periodic in the $\mathit{x}$ direction. A liquid ridge forms above the lyophilic stripe (gray) of width ${2\mathit{x}}_0$ centered on the $\mathit{y}$ axis and a wetting film covers the surrounding lyophobic regions (black). A body force aligned with the $\mathit{y}$ axis (caused, e.g., by a substrate tilt) generates flow along the stripe. The local film thickness at time $\mathit{t}$ is given by $\tilde{H}\left(x,y,t\right)$. Depending on the characteristics of the chemical stripe, the ridge may be subject to pearling instabilities (exaggerated).Reuse & Permissions

Figure 2

Present model of the chemical step of width $w=1$ with $A_{\text{out}}=4$ and $A_{\text{in}}=1$ as used in the numerical analysis. In the limiting case of infinite lateral extent $2\mathit{L}$, Eq. (6) reduces to $f\left(x\right)=2\frac{x-x_0}{w}$. (a) Effective interface potential $\Phi$ and disjoining pressure $\Pi$ at the channel center $x=0$ [see Eqs. (6, 8, 10)]. (b) Laterally varying amplitude $A\left(x\right)$ as defined in Eq. (5). (c) Contour plot of the effective interface potential $\Phi$ near the channel edge. Note that the position $\tilde{H}=1$ of the minimum of $\Phi$ as a function of $\tilde{H}$ is independent of $\mathit{x}$.Reuse & Permissions

Figure 3

(Color online) (a) Stationary cross-section profiles of the liquid ridge on a chemical channel of half-width $x_0=50$ and with edges of effective width $w=10$ for excess cross sections $V_{\text{ex}}=0$, 500, 1000, 2000, 3000, and 4000 [see Eq. (1)]. Also shown is the lateral dependence of the Hamaker constant $\mathit{A}$ in units of $A_{\text{in}}$ and its main features as given by Eq. (5). $A/A_{\text{in}}=1$ and $A/A_{\text{in}}=A_{\text{out}}/A_{\text{in}}=4$ correspond to ${\theta }_{\text{in}}=\frac{\sqrt{3}}{2}\delta$ and ${\theta }_{\text{out}}=\sqrt{3}\delta$, respectively ($\delta =a/\lambda$ is the scale factor for the slopes, with the lateral scale $\lambda =a^2/\sqrt{\sigma /A_{\text{in}}}$). (b) $\left(P,V_{\text{ex}}\right)$ for the same values of $A_{\text{in}}$ and $A_{\text{out}}$ as in (a) for $w=10$ [dashed line, with dots indicating those systems that the profiles in (a) correspond to], 20 (dashed-dotted line), and 30 (solid line). The thinner lines correspond to the power laws $P\propto V_{\text{ex}}^{-1/2}$ given by Eqs. (5, 15) for ridges resting on homogeneous substrates corresponding to the outside of the channel (dashed-double-dotted) and to the inside of the channel (dotted). For $w\gtrsim 30$, $P\left(V_{\text{ex}}\right)$ is monotonic while for $w\lesssim 30$ there is a range of cross sections for which $\frac{\text{d}P}{\text{d}{V}_{\text{ex}}}>0$.Reuse & Permissions

Figure 4

Part (a) shows the $\left(P,V_{\text{ex}}\right)$ diagram of the stationary solutions for a given substrate heterogeneity ($A_{\text{in}}=1$, $A_{\text{out}}=4$, $w=10$, $x_0=50$; compare Fig. 3). In this case, the chemical steps are sharp enough for a change of sign of $\frac{\text{d}P}{\text{d}{V}_{\text{ex}}}$ in a finite interval of excess cross sections $V_{\text{ex}}$. As a function of $V_{\text{ex}}$, (b) shows the growth rate $\text{Re}\omega \left(k;V_{\text{ex}}\right)$ [divided by ${\mathit{k}}^2$ and multiplied by $P\left(V_{\text{ex}}\right)$] of long-wavelength varicose modes for $g=0$ and for several small values of $\mathit{k}$. The stability window of the varicose mode corresponds to the range where $\frac{\text{d}P}{\text{d}{V}_{\text{ex}}}>0$. Note that for small wave numbers $\mathit{k}$ and large $V_{\text{ex}}$, $P\omega \left(V_{\text{ex}},k\right)/k^2$ approaches a plateau [with a value predicted by Eq. (35) to be $\frac{9}{35}=0.26$; for large $V_{\text{ex}}$ one has ${\overline{\theta }}_{\text{eq}}^2=3A_{\text{out}}/4=3$, which is consistent with the observed value]. On the present scales, the curves for $k\lesssim {10}^{-3}$ would be barely distinguishable from the one for $k={10}^{-3}$. For $k=0.01$, a typical stabilization can be seen for $V_{\text{ex}}\gtrsim 17000$, as the wavelength corresponding to $\mathit{k}$ becomes short with respect to the ridge width. For $k=0.1$, a finite-size effect is observed for $V_{\text{ex}}\gtrsim 2500$: in this case, the mode no longer corresponds to a bulging of the ridge but to system-wide perturbations of the film, which are stable.Reuse & Permissions

Figure 5

For reference purposes, part (a) corresponds to the rotated Fig. 4b, obtained for the case without driven flow, with the curve for $k=0.1$ removed. Part (b) traces the two zeros of $\text{Re}\omega \left(k\right)$ as a function of the driving force $\mathit{g}$ by showing the isolines of $\text{Re}\omega \left(k;V_{\text{ex}},g\right)=0$ for selected small values of $\mathit{k}$. Ridges with cross sections between the horizontal lines are stable. Other ridges are unstable with respect to pearling, with the marginally stable wavelength depending on $\mathit{g}$. The channel parameters are the same as in Fig. 4.Reuse & Permissions

Figure 6

(a) Curve of marginal stability, $\text{Re}\omega \left(k;g,V_{\text{ex}}\right)=0$, in the $g\text{−}k$ plane (full line) for $V_{\text{ex}}={10}^4$ (i.e., outside the stability region for $g=0$, see Fig. 5) and the same channel parameters as in Figs. 4, 5. For large $\mathit{g}$, the body force required to stabilize modes with large wavelengths is proportional to $1/k$. The straight dashed line is a fit $k\propto 1/g$ to the curve for $g>{10}^3$. (b) The growth rate $\text{Re}\omega \left(k;g,V_{\text{ex}}\right)$ as a function of $\mathit{gk}$ for the same ridge and several values of $\mathit{g}$. For large $\mathit{g}$, the curves approach a limiting curve ${\omega }_{\infty }\left(gk;V_{\text{ex}}\right)$, which depends on $\mathit{g}$ and $\mathit{k}$ only via the product $\mathit{gk}$.Reuse & Permissions