Capillary Slinky: Equilibrium and Dynamics of Droplets in a Soft Spring

TL;DR

This work investigates capillarity-driven deformation and motion of droplets in a soft helical spring (a capillary slinky) under a small elastocapillary number $N_{\mathrm{EC}}=k/\gamma$. Through experiments with a soft polyester spring and water/glycerol droplets, it maps static equilibrium shapes (annulus, Eruciform, spherical) and dynamic flow regimes, revealing that the spring geometry controls both interfacial shape and internal flow, and providing a predictive scaling $\frac{\Delta z}{V^{1/3}}\sim\frac{\gamma V^{2/3}}{k \lambda^2}\sim\left( N_{\mathrm{EC}}\frac{V^{1/3}}{\lambda} \right)^2$ that links capillary forces to elastic compression. The study further demonstrates actuation by converting capillary energy into spring work and enables active flow control by tuning the spring pitch, showing reversible switching between Eruciform and spherical regimes to adjust droplet speed. These findings have potential implications for softRobotics and microfluidics, where capillary actuation and controlled droplet transport can enable new functionalities with flexible, deformable structures.

Abstract

Springs can be found in many applications and biological systems, and when these are soft, they easily deform. At small scales, capillarity can induce a force leading to spring deformations when the elastocapillary number is small. We demonstrate through experiments the non-trivial equilibrium shape liquid droplets adopt in these soft springs, which form an annulus, Eruciform, and spherical shapes. When these droplets are set in motion, they display different flow regimes with significant dissipation generated by the internal rotational flow. The static and dynamics of droplets in such a capillary slinky is also used to demonstrate how surface tension can actuate springs by stretching/compression, while providing a way for active flow control in soft springs.

Figures (17)

• Figure 1: Soft helical spring: (a) schematic of the experimental system (top and side view), with a droplet of volume $V$ and soft helical spring of pitch $\lambda$ and radius $R$, and axis of the helical spring is aligned in z-direction with $g$ gravity and $r$ the fiber radius; (b) from left to right represents the static configurations; Eruciform (helicosymmetric) to spherical droplet in the on helical spring as increase with $\uplambda/2R$.
• Figure 2: Equilibrium shapes: (a)-(c) optical images at different times from regime I ($N_{\mathrm{EC}}$=0.008) to regime III ($N_{\mathrm{EC}}$=0.06); labeled as annulus, catenoid/cylinder, and rotund, based on their curvature in lateral direction; (d) total compression of the spring in the z-direction ($\Delta z$) with the $\uplambda$, scaled plot of the compression shows dependency on the $\lambda$, $V$, $k$, $\gamma$ (inset) using Equation 1. The scale bar is the same for all the optical images.
• Figure 3: Droplet on a fiber; (a) Eruciform-to-spherical droplet transition with $\uplambda/2R$ for $V=$10 $\upmu$l, where $\lambda/2R=0$ is for capillary tube. The first row corresponds to the droplet equilibrium shapes on different $\lambda/2R$ and curvature (red circle) of the droplet front while moving downward, and the second row shows the droplet front ahead of the contact line with $\lambda/2R$. The green dashed line indicates that the droplet front and contact line positions are the same, while the blue dashed line represents the droplet front and the red dashed line represents the contact line position, if they are different. The scale bar is the same for all the optical images. (b) Downward velocity $v_z$ for an Eruciform and spherical droplet with $\uplambda$, (c) azimuthal velocity $v_{\mathrm{\phi}}$ for an Eruciform and spherical shape droplet. The yellow shaded region is the bistable region.
• Figure 4: Flow inside helical spring; (a) x,z-position of the advancing front of an Eruciform and spherical droplet for $\uplambda/2R=0.65$ and $V$=10 $\upmu$l, (b) PIV images for internal velocity $\overline{v}=(v_z^2+v_x^2)^{1/2}$ for an Eruciform and spherical shape droplets for $V$=10 $\upmu$l, and $\uplambda/2R=0.65$. A helical spring schematic is added over the PIV images to guide the droplet shape.
• Figure 5: Droplet flow dynamics; (a) $v_z$ of the Eruciform and spherical droplet for $\upmu\in[0.001-1.412]$ Pa$\cdot$s, (b) scaled $v_z$ of the Eruciform and spherical shaped droplets with $\upmu\in[0.001-1.412]$ Pa$\cdot$s, $\lambda/2R\in[0.15-0.9]$ and $V\in[10-25]\upmu l$. The black dashed line is the scaling prediction in Equation 2 (slope 1), and the vertical red dashed line indicates the transition where $Re$=1, where to the right of the line $Re>1$.