Creases as elastocapillary gates for autonomous droplet control

TL;DR

Creases on soft elastomer interfaces act as long-range elastocapillary gates that autonomously gate droplets by size and surface tension. The authors combine experiments and theory to reveal a two-regime force–distance law, with a threshold radius $R_c\approx0.8$ mm that separates pass-through from stoppage, and a strain-sensitive transition at $\epsilon_c\approx0.18$. The system supports angular hysterons, pulse modulation, and basic Boolean logic (e.g., half-adder) using solely passive crease features, and can amplify signals via cascaded gates with a gain roughly proportional to $I\propto R^{6.25}$ (for $n=0.47$). This surface-native, reconfigurable mechanism enables passive, programmable interfacial information processing for microfluidics and biochemical assays, offering a low-entry, rewritable alternative to traditional DMF approaches.

Abstract

Droplets are the core functional units in microfluidic technologies that aim to integrate computation and reaction on a single platform. Achieving directed transport and control of these droplets typically demands elaborate substrate patterning, modulation of external fields, and real-time feedback. Here we reveal that an engineered pattern of creases on a soft interface autonomously gate and steer droplets through a long-range elastocapillary repulsion, allowing programmable flow of information. Acting as an energy barrier, the crease bars incoming droplets below a critical size, without making contact. We uncover the multi-scale, repulsive force-distance law describing interactions between a drop and a singular crease. Leveraging this mechanism, we demonstrate passive and active filtration based on droplet size and surface tension, and implement functionalities such as path guidance, tunable hysterons, pulse modulators, and elementary logic operations like adders. This crease-based gating approach thus demonstrates complex in-unit processing capabilities - typically accessible only through sophisticated surface and fluidic modifications - offering a multimodal, potentially rewritable strategy for droplet control in interfacial assembly and biochemical assays.

Figures (6)

• Figure 1: Size-based gating of droplets by a crease on soft elastomer surfaces.a. Schematics of crease as switchable autonomous gates for interfacial droplets. The crease gate can be toggled between on and off states by varying compressive strain $\epsilon$ above or below the instability threshold $\epsilon_c$ via the dual-layer setup shown in the middle. Symbolic representations of these two gating states are labeled on the left corners of the gel basis. b. Time-lapse images showing a small glycerin droplet, with diameter of 0.8 mm, driven by gravity and halted before reaching a crease. c. A larger glycerin droplet (diameter of 2.2 mm) overcomes the crease and continues its motion. For both these panel the soft substrate is under a compressive strain of 25$\%$. All scale bars are at 1 mm. d. Evolution of the droplet's leading edge with respect to the crease, $z$ over time for different drop radii $R$. e. (i). Schematics of three droplet responses near the the crease: stopped, pinned, pass-through. The critical radius $R_c$ marks the onset of pass-through; droplets with $R<R_c$ either stop or pin, while those with $R>R_c$ cross the barrier. (ii). Stopping distances $d_{\mathrm{stop}}$ versus droplet size, revealing a critical gating radius $R_c\simeq 0.8$ mm.
• Figure 2: Velocity profile of droplets arrested by a crease. Droplet position $z$ is rescaled by the stopping distance $d_{\textup{stop}}$, and velocity $v$ is normalized by the far-field sliding speed $v_{\infty}$. The rheological exponent of the elastomer, $n$, is measured to be at 0.47. The rate at which droplets decelerate varies with size, as shown by the two dashed lines indicating different exponents for small and large droplets. The left inset illustrates the braking strain ($\Delta=\frac{z_0-d_{\mathrm{stop}}}{z_0}$) difference between the blue (small) vs red (large) drop for the elastocapillary spring. $z_0$ is the position where the drop starts to sense the crease, an influence cutoff point.
• Figure 3: Origin of crease repulsion.a. Potential energy landscape for the drop-crease interactions. Inset schematic shows the instance of static equilibrium with the azimuthal angle definition. The green curve represents driving energy due to gravity, while the red curve represents the repulsive potential of the crease. A combination gives the potential well where the droplet stops. b. Normalized repulsive force vs. $d_{\mathrm{stop}}/R$, estimated from drop volume at $d_{\mathrm{stop}}$, for different compressive strains. Inset schematic shows the limiting cases of droplets at extreme sizes. The large drop experiences the crease deformation with significant rotation mismatch front vs. back, while the small drop feels minimal rotation. The solid line represents $m=-1/3$ and dashed line represents $m=-7/3$. Color bar denotes strain value. c. (i) Crease cross section image reconstructed from confocal microscopy. The scale bar here is 400 $\mu$m. Drop of radius $R_c$ is drawn on top for scale comparison. (ii) Normalized crease surface profiles $\bar{h}(\bar{z})$ as compressive strain $\epsilon$ varying from 0.18 to 0.29 and underlying step pattern ridge height $h_r$ varying from 0.15 to 0.82 mm.
• Figure 4: Switchable drop filtration.a. (i) Zoomed view on the drop dynamics bifurcations in the $v-z$ parameter space. (ii) Output current $I_o$ as a function of $R_i/R_c$. Inset shows current definition as $I=R^2v$. b. Upper panel shows the drop signal truth table. Lower panel shows the critical drop size $R_c$ as a function of strain. Red regime is associated with state 0 for signal; green regime is associated with state 1. c. Demonstration of the crease as an autonomous drop signal filter, with switchability. The temporal sequence is marked by ① to ⑥. Four signal readout locations are labeled from i to iv. See Supplementary Video 3. d. Signal readout via image analysis (output current $I_o$) for the four channels in c as a function of time. e. Gating based on surface tension. Glycerin drop in red is halted; FC-70 drop in grey of the same volume passes through. Upper panels show the experimental sequences (also see Supplementary Video 4), and the lower plots show the calculation of effective force per length felt from the leading edge rotation $\gamma\Delta\theta$. All scale bars here are at 1 mm.
• Figure 5: Angular hysteron. Experimental sequences and the resulting I/O map for the hysteron is shown for a 4 $\mu L$ drop. The top panel experiments demonstrate the drop residence on the right ($\mathrm{R}$) side; the bottom panels demonstrate the opposite residence on the left ($\mathrm{L}$) side. The transition occurs at the $\pm\alpha_c\approx\pm 48^{\circ}$ (shown in the experimental frames). Inner loop shows a contraction to $\pm\alpha_c\approx\pm 31^{\circ}$ via drop volume tuning. See this I/O map in Fig. S9.