Deformation and organization of droplet-encapsulated soft beads

TL;DR

An experimental approach to encapsulate a controlled number of soft beads within aqueous droplets in oil by manipulating droplet-encapsulated gels in a deformable microfluidic device.

Abstract

Many biological, culinary, and engineering processes lead to the co-encapsulation of several soft particles within a liquid interface. In these situations the particles are bound together by the capillary forces that deform them and influence their biological or rheological properties. Here we introduce an experimental approach to encapsulate a controlled number of soft beads within aqueous droplets in oil. These droplet-encapsulated gels are manipulated in a deformable microfluidic device to merge them and modify the liquid fraction. In the dry limit the contact surface between the hydrogels is found to be determined by the elastocapillary number $E_c$, with the contact radius scaling as $E_c^{1/3}$, indicating that the deformation increases for soft or small particles. When multiple beads are co-encapsulated within a single droplet they can be arranged into linear or three-dimensional aggregates that remain at a local energy minimum.

Figures (5)

• Figure 1: (a) Droplets are produced at flow-focusing junctions (i), trapped in capillary anchors within the observation chamber (ii), and the chamber ceiling is lowered by evacuating air from side cavities (iii). (b) Ultraviolet exposure polymerizes the droplets into gel beads; subsequent ceiling actuation compresses the bead pairs and the expelled liquid coalesces into an outer droplet. Initial gel diameter $2R_0=179~\mu\mathrm{m}$. (c) Additional compression decreases the encapsulating liquid volume. All scale bars, $100~\mu\mathrm{m}$.
• Figure 2: (a) Effective elasticity $E^{*}$ of five PEGDA formulations, measured by micropipette indentation. The table presents the bead stiffness color-coding, the PEGDA molecular weight and the PEGDA:PEG volume ratio. (b) Image-based measurement of the liquid volume fraction $\phi$ in Fiji schindelin_fiji_2012. Cross-sectional areas ($A_{\mathrm{liquid}}$, $A_{\mathrm{total}}$) and centroidal distances to the symmetry axis ($L_{\mathrm{liquid}}$, $L_{\mathrm{total}}$) are obtained from traced polygons under an axial-symmetry assumption, and volumes are computed via Pappus’ theorem. Scale bar $100~\mu\mathrm{m}$.
• Figure 3: DEGs morphologies shown for different values of $\phi$ and $\mathrm{E_c}$. (a) Soft gel beads ($E^{*} = 1.3 \pm 0.5 \times 10^{3}$); (b) Intermediate gel beads ($E^{*} = 1.0 \pm 0.3 \times 10^{4}$); (c) Stiff gel beads ($E^{*} = 3.5 \pm 1.1 \times 10^{4}$). In each row, increasing $\phi$ from left to right reduces the contact length. Scale bar: $100~\mu\mathrm{m}$. Scale bar, $100~\mu\mathrm{m}$. Cases for the blue and white gels are shown in SI Fig. S2. (d) Normalized contact radius $a/R_0$ vs. liquid volume fraction $\phi$ for five values of $E^{*}$. The analysis was performed on 8 DEGs. Lines are least-squares linear fits for each gel type within the explored range.
• Figure 4: (a) Definition of the geometric parameters $R$, $\delta$ and $a$ and their calculation from experimental images of the DEGs in the dry limit. Scale bar, $100~\mu\mathrm{m}$. (b) Comparison of measured $a/R$ versus $\delta/R$ with the geometric relation $a^2 = 2R\delta - \delta^2$. (c) Dimensionless energy landscape $\hat{U}(\hat{\delta})$ with $\hat{\delta}=\delta/R$ exhibiting a single minimum for each value of $E_c$. (d) Dimensionless contact length $a/R$ versus elastocapillary number $E_c$ in the dry limit. Data points correspond to experiments with two bead diameters ($n=8$ for large beads, $n=5$ for small beads). The green solid line represents the model of Eq. \ref{['eq:uhat']} ($c=0.49$). The black dashed line shows the scaling law $a/R=3.00\mathrm{E_c}^{1/3}$. The data points for the yellow gels were excluded while fitting the $E_c^{1/3}$ law.
• Figure 5: (a) Very soft beads ($E_c > 0.1$) showing buckling in the dry limit ($\phi = 0$). The right image is the same as the left image but with the interface between the two beads highlighted with the colors to help visualization. (b) Pair with unequal stiffnesses: Yellow bead (i) is soft, and black bead (ii) is stiff, shown before merging (left) and in the dry limit (right). Only the soft bead buckles. (c) Pair with unequal sizes shown before merging (left) and in the dry limit (right). (d) $N=3$–6: DEGs in compact arrangements at high $\phi$. (e) Air injection into the cavities raises the chamber ceiling, enabling reconfiguration. (f) $N=3$–6: DEGs at low $\phi$. All scale bars, $100~\mu\mathrm{m}$.