Controlling Droplets at the Tips of Fibers

TL;DR

This work demonstrates that a liquid bridge between two tilted fibers exhibits a sharp, angle-driven transition between post-rupture states, governed by a bifurcation structure tied to a projected-area criterion $K= (R_2/R_1)^2\cos\phi$. By combining precise experiments, perturbative theory (Lyapunov–Schmidt) and mesoscale simulations, the authors show how tilt, volume, and geometry control liquid partition and enable near-complete droplet transfer via contact-line depinning. The findings offer a geometric route to manipulate fluids on deformable, architected surfaces and hint at practical implementations, such as ruck-guided droplet transport across fiber trains in soft metamaterials.

Abstract

Many complex wetting behaviors of fibrous materials are rooted in the behaviors of individual droplets attached to pairs of fibers. Here, we study the splitting of a droplet held between the tips of two cylindrical fibers. We discover a sharp transition between two post-rupture states, navigated by changing the angle between the rods, in agreement with our bifurcation analysis. Depinning of the bridge contact line can lead to a much larger asymmetry between the volume of liquid left on each rod. This second scenario enables the near-complete transfer of an aqueous glycerol droplet between two identical vinylpolysiloxane fibers. We leverage this response in a device that uses a ruck to pass a droplet along a train of fibers, a proof-of-concept for the geometric control of droplets on deformable, architected surfaces.

Figures (9)

• Figure 1: Symmetry breaking in splitting a liquid bridge between two identical rods. (a) Snapshots of a liquid bridge pinching off from two separating rods into droplets of volumes $V_1$ and $V_2$, for a range of i) separation velocities $U$ and ii) total volumes $V$. Symmetry is gradually broken when $U$ decreases or $V$ increases. (b) Fractional volumes of the separate droplets in the post-rupture state, versus $U$. (c) Fractional volumes of the separate droplets in the post-rupture state, versus $U$. Inset: corresponding last stable states.
• Figure 2: Splitting a droplet between two angled rods. (a) Snapshots of a liquid bridge pinching off for a range of inclination angles $\phi$. A sharp transition occurs near $\phi \approx 20^{\circ}$. (b) Fractional volumes of the separate droplets in the post-rupture state, versus $\phi$. (c) Fractional volumes versus the projected area ratio $K$ for the same data shown in (b) (transparent), as well as for a pair of rods at $\phi=0$ with varying $R_2$ (dots). Open markers: last stable states for both $\phi=0$ (circles) and $\phi>0$ (squares/triangles).
• Figure 3: Threshold angles for different droplet volumes and rod sizes. (a) i: Two possible outcomes near the threshold angle. ii: Superimposed outlines of the bifurcated states at threshold angles, as the total volume $V$ is gradually reduced. (b) Measured $\phi$ near the bifurcations, for three sets of rods size ratios. Green: the bottom rod receives a larger droplet. Pink: the bottom rod receives a smaller droplet. Horizontal dashed lines: perturbation theory Eq. \ref{['eq:firstorder']}. Arrows: corresponding Plateau limit $V=V_0$ where the perturbation theory is asymptotically accurate. (c) Data collapse in $\delta \phi$-$v$ space. Solid orange line: higher order perturbation theory.
• Figure 4: Bifurcation analysis of tilted bridges. (a) Photograph of a liquid bridge spanning two rod faces with radii $R_{1}$ and $R_{2}$, separated by $L$ and tilted by $\phi$. Red arrow: position of the bridge neck. (b) The amplitude, $a$, of the antisymmetric deformation as a function of the length parameter, $\lambda$, for $\phi=0$ (blue) and $\phi>0$ (red). The tilt angle $\phi$ controls the two unfoldings (i and ii, red) of a perfect subcritical pitchfork bifurcation (i, blue) with the change occurring across $\phi_{\text{thresh}}$. The location of the neck changes drastically for the two unfoldings (inset schematic). Stable (unstable) branches are depicted as solid (dashed) curves, with a switch occurring at turning points $(\lambda_{\text{max}},a^{*})$ (black circles). The angled bridge bifurcations are plotted with $h=0.053$ and $v=-0.1$.
• Figure 5: Depinning the contact line allows a larger range of splitting ratios. (a) Image overlays of the profiles of a liquid bridge between the tips of two VPS polymer fibers at an inclination $\phi = 49.6^{\circ}$. (b) The receding contact angle $\theta\textsubscript{r}$ as the contact line slides across fiber face, for two rod angles. (c) Post-rupture states between two VPS fibers with a range of $\phi$. (d) Post-rupture volumes versus $\phi$. The smaller droplet reduces its size by 4 orders of magnitude as $\phi$ increases, facilitating a near-complete transfer. Solid markers: pinned contact line at low $\phi$. Open markers: depinned contact line at large $\phi$. Solid and dashed lines: MDPD simulations for two effective contact angles $\theta$. (e) Snapshots of the MDPD simulation with $\theta=80^{\circ}$, exhibiting contact-line depinning.