Droplets on lubricated surfaces: The slow dynamics of skirt formation

A key question in the interaction of droplets with lubricated and liquid-infused surfaces is what determines the apparent contact angle of droplets. Previous work has determined this using measured values of the geometry of the “skirt”—the meniscus-like deformation that forms around the base of the deposited droplet. Here, we consider theoretically the equilibrium of a droplet on a smooth, impermeable lubricant-coated surface, and we argue that the small effect of gravity within the skirt and the size of the substrate are important for determining the final equilibrium. However, we also show that the evolution of the skirt toward this ultimate equilibrium is extremely slow (on the order of days for typical experimental parameter values). We therefore suggest that previous experiments on smooth lubricated surfaces may have observed only slowly-evolving transients, rather than “true” equilibria, potentially explaining why a wide range of skirt sizes have been reported.

Figure 1

Schematic illustration and notation for the analysis of the “push-and-pull” problem considered in this paper. We mimic the effect of a two-dimensional liquid droplet sitting on a thin oil film by introducing a constant pressure pushing down on the film (mimicking the droplet Laplace pressure) within a constant interval $\left|x\right|<x_c$, together with line forces pulling on the interface at $x=\pm x_c$ (mimicking the capillary force acting at the triple lines). This combination of loads leads ultimately to the formation of an equilibrium meniscus, given by $z=h\left(x\right)$. The inset figure shows a zoom of the triple line region, which, for the case of equal surface tensions considered here, gives rise to Eq. (1).

Figure 2

Equilibrium interface profiles calculated numerically from the solution of Eq. (11). (a) The effect of system size, $X_{\infty }=x_{\infty }/x_c$, with $\mathrm{Bo}={10}^{-2},{\mathcal{V}}_{\gamma }={10}^{-4}, \alpha ={10}^2$ fixed. (b) The effect of initial film thicknesses (for fixed plate size $X_{\infty }=10$ and Bond number, $\mathrm{Bo}={10}^{-2}$) is obtained by changing ${\alpha }^{-1}\in [{10}^{-3},{10}^{-1}]$ but maintaining ${\mathcal{V}}_{\gamma }={\alpha }^4/{10}^{12}$. (a) and (b) show that equilibria are significantly affected by the size of the reservoir of oil available to be sucked into the skirt. (c) The effect of van der Waals forces as encoded by the value of ${\mathcal{V}}_{\gamma }$ is much more limited, as shown by profiles calculated with fixed $\mathrm{Bo}={10}^{-2}$, initial film thicknesses (so that $\alpha ={10}^2$) and plate size $X_{\infty }=10$.

Figure 3

Dependence of (a) the equilibrium skirt height at the triple line, $H_c$, (b) the pressure in the film, $P_{\infty }$, (c) the skirt volume, defined by Eq. (18), and (d) the far-field equilibrium film thickness, $H_{\infty }$, as functions of the relative plate size, $X_{\infty }$. Asymptotic results, derived in Sec. 3, for lubricant-sated systems (dotted black lines) and lubricant-starved systems (solid black lines) are also shown, given by Eqs. (20), (22), and (25). The solid line in the inset of (b) is based on $H_c={\alpha }^2/(-8P_{\infty })$, which is given by combining Eqs. (24) and (25). All calculations use $\mathrm{Bo}={10}^{-2}$ with ${\mathcal{V}}_{\gamma }={\alpha }^4/{10}^{12}$ and $\alpha$ varying from 10 to ${10}^3$. All solid and dashed curves of the same color are calculated using the same parameters but with the dashed curves using the small slope approximation of the curvature, $|h_x|\ll 1$, and the no-rotation assumption for the pulling force, i.e., $\mathrm{\Delta }\theta =0$.

Figure 4

(a) Deviation of the meniscus angles at the triple line (as defined in Fig. 1) from $\pi /6$ as a function of the system size, $X_{\infty }$. (b) Equilibrium values of the rotation angle $\mathrm{\Delta }\theta$ defined by Eq. (16) for the scenarios in which there are only meniscus forces (“pull only”) and that in which the droplet's capillary pressure also pushes the interior meniscus downward (“push and pull”). Note that the “self-interaction” of the droplet's two menisci causes a clockwise rotation of the menisci $\mathrm{\Delta }\theta <0$, while the effect of the droplet's positive Laplace pressure works in the opposite direction, causing an anti-clockwise rotation, $\mathrm{\Delta }\theta >0$.

Figure 5

Numerical results for the early time evolution of the thin film profile. (a) Schematic of the deformation: two narrow neck regions emerge from the triple line and move away from it. The skirt volume is calculated dynamically as the volume contained in the wedge above the line $\tilde{H}=\tilde{H}(\alpha x_{\infty }/x_c)$; at early times, $\tilde{H}(\alpha x_{\infty }/x_c)\approx 1$, which is the dashed line labeled “initial.” The zoom-in view of the skirt apex illustrates the tilting of the Neumann triangle. (b) The early-time evolution of the change in skirt height, ${\tilde{H}}_c-1$. Numerical results are shown by solid curves for film aspect ratios $\alpha =100,200,400$ (curves) with different values of the system size indicated by line style: $x_{\infty }/x_c=10$ (solid gray) and $x_{\infty }/x_c=100$ (dashed orange). The dashed black line indicates the prediction, Eq. (46), that comes from the early-time similarity solution. Here $\mathrm{Bo}={10}^{-2},{\mathcal{V}}_{\gamma }={\alpha }^4/\left({10}^{12}\right)$ so that increasing $\alpha$ corresponds to increasing $h_0$.

Figure 6

Early-time evolution of (a) the minimum thickness and (b) the lateral position of the neck region that forms in the early stages of the motion. The dashed black lines are the predictions of the early time similarity solution Eq. (47), while the solid curves show numerical results with the labels “outer” and “inner” referring to the neck regions in the regions $\tilde{X}>\alpha$ and $\tilde{X}<\alpha$, respectively. Here $\alpha =100$.

Figure 7

Early-time evolution of (a) the induced rotation angle, $\mathrm{\Delta }\theta$, and (b) the volume, $V_{\mathrm{skirt}}$, of the skirt. In panel (a) the asymptotic prediction Eq. (52) is shown by the dashed line. In panel (b), $V_{\mathrm{skirt}}$ is the volume of liquid near the triple line, and is defined by $V_{\mathrm{skirt}}\left(t\right)={\int }_{x_{\min }^{-}}^{x_{\min }^{+}}[h(x,t)-h(x_{\infty },t)]dx/\left(x_c{h}_0\right)$. $V_{\mathrm{drop}}\left(t\right)$ is the rescaled volume of the liquid squeezed out beneath the drop, i.e., $V_{\mathrm{drop}}\left(t\right)={\int }_0^{x_c}[h_0-h(x,t)]dx/\left(x_c{h}_0\right)$.

Figure 8

The evolution of film thickness for an initially flat oil film subjected to the spatially nonuniform pressure profile of the droplet. Here the effect of the van der Waals parameter ${\mathcal{V}}_{\gamma }$ on the timescale of evolution may be seen as ${\mathcal{V}}_{\gamma }$ varies from ${10}^{-5}$ [in panels (a), (d)] to ${10}^{-4}$ in (b), (e) and ${10}^{-3}$ [in panels (c), (f)]. Here ${\mathrm{Bo}}^{-1}=\alpha ={10}^2$ in all calculations; in panels (a)–(c) $X_{\infty }=10$ while in panels (d)–(f) $X_{\infty }={10}^3$. Note that the film profile ultimately reaches very close to the equilibrium solution (dashed black curves), but that the time taken to do so is extraordinarily long even for $X_{\infty }=10$ ($T\gtrsim {10}^4\sim 10\mathrm{days}$ with a typical ${\mathcal{V}}_{\gamma }={10}^{-4}$). Throughout, line color is used to encode dimensionless time as in the associated color bar. Black filled and open markers are used to indicate the local minima (necks) and maxima (bumps), respectively.

Figure 9

(a) Late-time evolution of the skirt height with a typical value, ${\mathcal{V}}_{\gamma }={10}^{-4}$, and different system sizes (denoted by line color, as described in the legend). Note that dashed black curves are the results of the early-time analysis (where the true $\delta$-function forcing is implemented) while solid curves show results with the smoothed forcing (see main text). The dynamic system is not sensitive to the system size $X_{\infty }$ until $T\sim {10}^3$, at which point the film height at the outer boundary begins to change for the smallest system, $X_{\infty }=10$. (b) Late-time evolution of the skirt height with ${\mathcal{V}}_{\gamma }$ varying: ${\mathcal{V}}_{\gamma }={10}^{-3}$ (dotted), ${\mathcal{V}}_{\gamma }={10}^{-4}$ (solid), and ${\mathcal{V}}_{\gamma }={10}^{-4}$ (dashed) for various plate sizes (as indicated by line color). In all calculations $\alpha ={\mathrm{Bo}}^{-1}={10}^2$.

Figure 10

Time evolution of the properties of the local minimum (the neck region, yellow curves) and the maximum (the bump region, blue curves). Each of these regions evolves through several different behaviours. A key feature is the position of the stagnation point located between them — $X_{\mathrm{s}.\mathrm{p}.}$ is the point at which the fluid flux vanishes, $\partial P/\partial X=0$, and is shown by the cyan curve. Here, $\alpha =100$ and ${\mathcal{V}}_{\gamma }={\alpha }^4/{10}^{12}={10}^{-4}$; solid curves use $\mathrm{Bo}=0$ while dashed curves used $\mathrm{Bo}={10}^{-2}$. Different background shading is used to represent early times, early–intermediate times, late–intermediate times, and late times (from left to right). Black dashed and dotted curves are plotted using the corresponding analytical expressions, which are summarized in the first two panels of Fig. 14.

Figure 11

Evolution of the film profile at early–intermediate times in which a ‘bump’ of fixed height $H_{\mathrm{plat}}$ propagates away from the droplet. (a) The unscaled profile showing the neck at $X_{\min }$ (filled black circle) and the propagating bump (open black circle). (b) The raw data of panel (a) rescaled according to Eq. (58) agrees well with the numerical solution of the intermediate-time similarity Eq. (61) (dashed curve). (In panels (a) and (b) curves are colored according to time as in the colorbar to the right.) (c) The numerically determined film height at the center of the drop, $X=0$, as a function of $\left|\sigma \right|T$ (solid curve) compared to the the prediction of the linear stability analysis, $H(0,T)\propto exp(-\sigma T)$, with $\sigma$ given by Eq. (63) calculated with ${\mathcal{V}}_{\gamma }={10}^{-4}, \alpha ={10}^2$ (dashed line). At late times, the numerics are close to the equilibrium thickness $H_{\mathrm{eqm}}={({\mathcal{V}}_{\gamma }/\alpha )}^{1/3}$ (dotted line). (d) The time needed for the film beneath the droplet to become flat ($T_{\mathrm{flat}}$, red circles) and for the film thickness beneath the two neck regions to stop decreasing and begin growing (denoted by $T_{\mathrm{grow}}$ and shown by green diamonds for the inner neck and blue squares for the outer neck). Results are shown for ${\mathcal{V}}_{\gamma }={\alpha }^4/{10}^{12}$ (hollow symbols) and stronger van der Waals forces, ${\mathcal{V}}_{\gamma }={\alpha }^4/(2\times {10}^7)$, (filled symbols) where $\alpha$ varies from 50 to 200. The solid line shows the scaling prediction $T\propto {\sigma }^{-1}$ from the stability analysis Eq. (63) with prefactor such that $T={\left(\alpha {\mathcal{V}}_{\gamma }^2\right)}^{-1/3}$ or, in dimensional form, $t=3\mu x_c^{7/3}{\gamma }^{-1/3}{A}^{-2/3}$, independent of the initial film thickness.

Figure 12

Evolution of the film profile at late–intermediate times. (a) Raw dimensionless and (b) scaled profiles showing the location of the neck (filled black circles) and bump (open circles) in each case. The data in panel (b) are rescaled according to Eq. (65) and plotted on a semilogarithmic scale to highlight the exponential growth away from the neck region.

Figure 13

Time for the skirt height to reach 90% of its final equilibrium value, i.e., $T_{90\%}$ is defined such that $H(1,T_{90\%})=0.9H_c$. Here, numerical results are shown by points while the scaling of Eq. (68) with a prefactor 0.02 is shown by the solid line. Marker color is used to encode ${\mathcal{V}}_{\gamma }$ according to the colorbar. Open and filled markers use $\mathrm{Bo}=0$ and $\mathrm{Bo}={10}^{-2}$, respectively. Circles, squares, and diamonds denote $X_{\infty }=10,{10}^2$ and ${10}^3$, respectively. All calculations use $\alpha ={10}^2$ except pentagrams (for which $\alpha =25,X_{\infty }=100$).

Figure 14

Schematic summary of the asymptotic behavior of the thin film evolution (the yellow region) as a skirt forms in response to a droplet's capillary pressure (acting in the green region) and the triple line. (a) At early times, capillary pressure is unimportant and a symmetric wedge forms under the action of the triple line, growing according to a similarity solution. (b) At early–intermediate times, van der Waals forces stabilize thin neck regions either side of the skirt; the drainage of the dimple beneath the droplet driven by Laplace pressure is dominant here. (c) At late–intermediate times the skirt is fed mostly by liquid from the remainder of the lubricating film. The neck and bump regions are highlighted by filled and open circles, respectively; asterisks indicate the position of the stagnation point between the outer neck and bump regions. The flux of lubricating liquid into the skirt region is illustrated by blue arrows, with the lengths of these arrows representing the relative importance of flow from different regions at different instants of time.

Figure 15

Experimental data for the outer meniscus profile of a droplet on a LIS, captured from Fig. 5 of Schellenberger et al. [15]. In these experiments the value of the capillary length based on the published liquid properties, ${\ell }_c\simeq 0.97\mathrm{mm}$, is used. The position of the triple line is taken to be the highest point of the interface visible, which occurs at $r_c=0.43{\ell }_c$, with the meniscus height there, $h_c\simeq 0.23\mathrm{mm}$, used to rescale the meniscus height throughout. The prediction from the linearized Laplace–Young equation for a two-dimensional scenario is a pure exponential decay (solid line), while the more detailed, axisymmetric behavior is given by Eq. (A2) and is shown by the dashed curve.

Figure 16

Dependence of (a) the equilibrium skirt height at the triple line and the skirt volume on the strength of van der Waals forces, ${\mathcal{V}}_{\gamma }$, (with $\mathrm{Bo}={10}^{-2}$ and $\alpha ={10}^2$ fixed). Curves with the same color use the same parameters but are calculated using the fully nonlinear (solid curves) or linearized (dashed curves) curvature.

Figure 17

Evolution of the film profile at late times with (a) $\mathrm{Bo}=0$ and (b) $\mathrm{Bo}={10}^{-2}$. The colored, dashed lines in panels (a) and (b) are based on the asymptotic results Eqs. (D3) and (D5), respectively. The inset in panel (a) shows the relationship between $H_{\infty }\left(T\right)$ and $X_{\min }\left(T\right)$, with the dashed curve showing the prediction Eq. (D4). (Here $X_{\infty }={10}^3, \alpha ={10}^2$, and ${\mathcal{V}}_{\gamma }={10}^{-4}$ in all calculations.)