Elasto-capillary adhesion: Effect of deformability on adhesion strength and detachment

We study the interaction between capillary forces and deformation in the context of a deformable capillary adhesive: a clamped, tense membrane is adhered to a rigid substrate by the surface tension of a liquid droplet. We find that the equilibrium adhesive force for this elasto-capillary adhesive is significantly enhanced in comparison to the capillary adhesion between rigid plates. In particular, the equilibrium adhesion force is orders of magnitude greater when the membrane is sufficiently deformed to contact the substrate. From a dynamic perspective, however, the formation of a fluid-filled dimple slows this approach to contact and means that stable attachment is only achieved if adhesion is maintained for a minimum time. The inclusion of a variable membrane tension (as a means of modifying the deformability) gives additional control over the system, allowing new detachment strategies to be explored.

Figure 1

A liquid droplet (blue) of volume $V$ bridges the gap between a rigid plate (gray) and a deformable membrane (red). The membrane (of thickness $\tau$) is clamped around a circle of radius $L$ with an applied tension $T$ and at a height $h_{\infty }$ above the plate. The free surface of the droplet has a surface tension $\gamma$ and makes a contact angle $\theta$ with each surface.

Figure 2

The dimensionless adhesive force $F(\mathrm{\Gamma },H_{\infty })$. Results are shown for (a) fixed deformability $\mathrm{\Gamma }=1,10,{10}^2,{10}^3,{10}^4$ and varying edge height $H_{\infty }$ and (b) fixed edge height $H_{\infty }=2,5,10,20$ and varying membrane deformability $\mathrm{\Gamma }$. The computation stops when the system is flooded, i.e., $R_M=1$ (solid circles). In panel (a) the force is compared to Eq. (7) for a perfectly rigid membrane $\mathrm{\Gamma }=0$ (black dotted line) and in both panels different thickness curves are used to distinguish different states, as described in the legend of panel (a). We note that the adhesion force is significantly larger when the membrane is in contact with the base (thicker solid curves) but even out of contact (thinner solid curves) the adhesive force remains larger than in the rigid case.

Figure 3

(a) The equilibrium meniscus radius $R_M$ (blue) and contact point $C$ (red) vary with gap width $H_{\infty }$, when $\mathrm{\Gamma }$ is fixed (here $\mathrm{\Gamma }=100$). There is a noncontacting stable solution (thin solid curve), an unstable solution (dashed curve), and a contacting stable solution (thick solid curve). Each of these three states only exists over a specific range of values of $H_{\infty }$. Inset: Membrane profiles of the contacting and noncontacting stable solutions when $H_{\infty }=8$ (blue circles). (b) The variation of the meniscus height $H_M$ with $H_{\infty }$ (again shown for $\mathrm{\Gamma }=100$). The dotted line denotes $H_M=H_{\infty }$ for comparison.

Figure 4

The number and type of equilibria varies with the two parameters $\mathrm{\Gamma }$ and $H_{\infty }$. The hatching in each region denotes the number of contacting or noncontacting solutions, as described in the legend.

Figure 5

Schematic of the experimental setup. A dyed droplet of mineral oil (blue) was confined between a PVS-coated glass plate and a clamped PVS sheet (red). This system rested on a mass balance that recorded the adhesive force. The height of the clamp (gray) was varied using a computer-controlled linear actuator, and the tension in the sheet could be increased by withdrawing air from an annular chamber (light blue) in the clamp, which sucked the outer edge of the sheet into the chamber, pulling the whole membrane taut. A camera imaged the experiment from above.

Figure 6

The measured dimensionless (a) force and (b) radius (symbols) compared with the theoretical predictions without any fitting parameter (curves) for three different experiments. The arrows illustrate the progression of the experiment: an initial lowering of the membrane, followed by a period at fixed height and then retraction at constant speed. Dark blue squares: $T=4.9\mathrm{N}{\mathrm{m}}^{-1}$, $V=7.8\mu \mathrm{L}$, $\mathrm{\Gamma }=1100$. Blue-green crosses: $T=5.1\mathrm{N}{\mathrm{m}}^{-1}$, $V=8.2\mu \mathrm{L}$, $\mathrm{\Gamma }=970$. Yellow triangles: $T=5.2\mathrm{N}{\mathrm{m}}^{-1}$, $V=9.9\mu \mathrm{L}$, $\mathrm{\Gamma }=650$.

Figure 7

Increasing the tension, while maintaining a fixed separation, results in the switching off of strong adhesion. The tension is increased from $T=2.5\mathrm{N}{\mathrm{m}}^{-1}$ to $T=7.1\mathrm{N}{\mathrm{m}}^{-1}$ over the duration of the highlighted region and is constant otherwise. Examples of the droplet spread before and after the tension change are shown, with the meniscus position denoted by a red dashed circle. Here the clamp was fixed at a height $h_{\infty }=1.1\mathrm{mm}$, with a droplet volume $V=7.8\mu \mathrm{L}$.

Figure 8

Schematic of the dynamic adhesive test. (a) The clamp is lowered to a set distance from the glass plate. (b) After a time $t_{\text{wait}}$, the clamp is raised at a constant speed, lifting the glass plate and load. (c) After some time $t_{\text{hold}}$, detachment may occur and the load falls back. (d) Experimental results showing the relationship between the time the system is left to equilibrate, $t_{\text{wait}}$, and the adhesion time, $t_{\text{hold}}$, for $h_{\infty }=0.3\mathrm{mm}$, $T=2\mathrm{N}{\mathrm{m}}^{-1}$, $V=10\mu \mathrm{L}$ and lifting a mass of $2.7\mathrm{g}$. Experiments were performed with $t_{\text{wait}}\le 4\mathrm{s}$ but no noticeable lift-off was observed. When $t_{\text{wait}}\gtrsim 10$ s, the load would remain attached on the timescale of hours, making it difficult to obtain an accurate value for $t_{\text{hold}}$, and for $t_{\text{wait}}=15\mathrm{s}$, the load remained attached overnight (over 12 h).

Figure 9

The dynamic approach of the dimensionless adhesive force, $F\left(t\right)$, to its equilibrium value $F_{\infty }$ in both (a) noncontacting and (b) contacting cases. (a) In the noncontacting case, the force decays exponentially to its equilibrium value, consistent with a linear stability analysis of the equilibrium (dashed lines show the expected decay rate, i.e., slope, only). (b) In the contacting case, the force instead appears to decay according to a power law. In each case, the equilibrium force, $F_{\infty }$, is calculated from the static theory, and $H_{\infty }=5$.

Figure 10

Snapshots of the cross-sectional profile of the membrane and droplet during the approach to contact of a highly deformable membrane (here $\mathrm{\Gamma }=100,H_{\infty }=5$). Note that some of the fluid is trapped in a dimple in the membrane and drains very slowly. The initial condition is shown in the top left panel with the behavior at five subsequent times also shown. Note that the vertical scale is exaggerated here (by the different scales used to nondimensionalize horizontal and vertical lengths)—in reality the droplet is thin and wide.

Figure 11

When approaching contact, a dimple forms beneath the membrane. To understand the evolution of the dimple, we split the fluid into 3 distinct regions: region I is the dimple at uniform positive pressure with dimple height $H_0\left(t\right)$, region III is the meniscus region at uniform negative pressure, and region II is a small annular region surrounding the narrow gap, $H_{\text{min}}$, that controls the fluid flux, and is at a radial position close to the edge of the equilibrium contact region, $C_{\infty }$.

Figure 12

The adhesion force decaying to its equilibrium value, $F_{\infty }$ (calculated from the analysis of Sec. 3), in the contacting regime. The numerics (solid curves) approach the asymptotic dimple theory (dashed lines) at late times. Inset: the absolute error (solid curve) between the numerically calculated force $F\left(t\right)$ and the first two terms of the asymptotic expansion is approximately linear (black line for comparison) in $t^{-1/2}$ at late times (here $\mathrm{\Gamma }=10,H_{\infty }=2.5$).

Figure 13

Work of separation from different methods of detachment. (a) Pull directly away from the surface (yanking), $H_{\infty }\to \infty$ with $\mathrm{\Gamma }$ fixed. The solution follows the equilibrium value (blue solid curves) and jumps from contact to noncontact at the fold (ii). The work of separation is the integral $\int Fd{H}_{\infty }$ and is illustrated by the shaded dark gray area. (b) If we instead first increase the tension (decrease the deformability $\mathrm{\Gamma }$) at fixed $H_{\infty }$ and then yank, we expect that the work of separation may be significantly reduced. The different stages are (i) initially decrease $\mathrm{\Gamma }$ (to green solution) before (ii) increasing $H_{\infty }$ to pull away through the snap-off transition. The total work is the sum of the dark gray area and the work done to increase the tension in stage (i) and this method of detachment will be preferable to yanking provided that the energy needed to stretch the membrane in panel (b) stage (i) is less than the yanking energy saving compared to panel (a), illustrated by the light gray region.

Figure 14

Starting from an initial contacting state $({\mathrm{\Gamma }}_0,H_0)$, there are many possible paths in parameter space that result in detachment. The simplest of these is a direct pull off at constant tension (yanking, dashed path). We investigate various quasistatic paths that involve decreasing $\mathrm{\Gamma }$ by $\mathrm{\Delta }\mathrm{\Gamma }$ before taking $H_{\infty }\to \infty$ (solid path); we find that the optimal (smallest work of separation) is to decrease $\mathrm{\Gamma }$ until contact is lost before pulling away (dotted path). Note that here the area labeled contact possible contains the three-solution region seen in Fig. 4, and we have omitted the region with no solutions for simplicity.

Figure 15

(a) Pulling the membrane off at a faster rate results in a larger adhesion force over a wider range of gap widths, $H_{\infty }$. (b) The energy required to detach ($H_{\infty }\to \infty$) when retracting at constant speed increases with the retraction speed, ${\dot{H}}_{\infty }$. Dynamic simulations (blue “x”) are performed at different constant retraction rates and compared to the quasistatic change in energy at fixed $\mathrm{\Gamma }$ (black dashed) and the optimal quasistatic strategy (red dash-dotted) in which $\mathrm{\Gamma }$ is decreased until contact is lost before pulling away at fixed $\mathrm{\Gamma }$. In both figures, the initial condition for dynamic simulations is close to the contacting equilibrium with parameters $\mathrm{\Gamma }=100,H_{\infty }=5$.