Continuum Model Applied to Granular Analogs of Droplets and Puddles

We investigate the growth of aggregates made of adhesive frictionless oil droplets, piling up against a solid interface. Monodisperse droplets are produced one by one in an aqueous solution and float upward to the top of a liquid cell where they accumulate and form an aggregate at a flat horizontal interface. Initially, the aggregate grows in 3D until its height reaches a critical value. Beyond a critical height, adding more droplets results in the aggregate spreading in 2D along the interface with a constant height. We find that the shape of such aggregates, despite being granular in nature, is well described by a continuum model. The geometry of the aggregates is determined by a balance between droplet buoyancy and adhesion as given by a single parameter, a “granular” capillary length, analogous to the capillary length of a liquid.

Figure 1

(a) Schematic side view of the experimental chamber. A glass pipette produces monodisperse oil droplets with a radius of $\sim 10\mu \mathrm{m}$ in an aqueous solution. Droplets float to the top and form an aggregate that spreads on the top glass slide over time. (b)–(e) Optical microscopy images of the aggregate at different stage of the growth (scale bars are $100\mu \mathrm{m}$). (f) Images are binarized to measure the area covered by the droplet aggregate $A$. (g) Evolution of the aggregate area as a function of the number of droplets $N$ (here $R=17.8\mu \mathrm{m}$, $C=0.10\mathrm{mol}/\mathrm{l}$). The black dashed line is a linear fit of the data for $N>30$. Left inset: early growth of the aggregate with best fit of the model presented in the main text for $N_{\text{droplets}}\le 30$. Right inset: the stepwise increase of the area corresponds to collapsing events.

Figure 2

(a) Normalized aggregate area as a function of the number of droplets for different droplet radii, while the strength of adhesion is kept constant ($C_m=0.10\mathrm{mol}/\mathrm{l}$). (b) Normalized aggregate area as a function of the number of droplets for different adhesion strengths, while the droplet radius ($R=15.3\pm 0.4\mu \mathrm{m}$) is kept constant. The strength of adhesion increases with SDS concentration. The purple curves correspond to the same data shown in (a).

Figure 3

(a) Adhesion force measurement: two droplets are pushed into contact by the left pipette, held constant, and then pulled apart. The dashed line marks the equilibrium position of the right droplet. The deflection of the sensing pipette (right) is converted to a force. The scale bar is $100\mu \mathrm{m}$. (b) Force measurement between two droplets of oil ($R=39.8\mu \mathrm{m}$, $C_m=0.13\mathrm{mol}/\mathrm{l}$). The force rises as the droplets are pushed together, and then decreases as the droplets are pulled apart (negative slope). A negative force results from adhesion between the droplets. The contact between droplets is broken when the force suddenly jumps to 0. (c) Unbinding force as a function of the droplet radius for various concentrations in SDS: red circles, $C_m=0.06\mathrm{mol}/\mathrm{l}$; blue squares, $C_m=0.13\mathrm{mol}/\mathrm{l}$; green diamonds, $C_m=0.20\mathrm{mol}/\mathrm{l}$. (d) Collapse of the unbinding force by normalizing by the concentration in micelles. The black dashed line corresponds to the linear fit including all three datasets.

Figure 4

Double-logarithmic plot of the renormalized aggregate area versus the renormalized number of droplets for various droplet sizes $R$ and adhesion strength $\mathcal{A}$. The aggregate growth transitions from ${(N/N^{*})}^{2/3}$ to $N/N^{*}$ at $(A^{*},N^{*})$ (nonrenormalized data shown in inset). The data correspond to those presented in Fig. 2 using the same color code, and three further datasets are included: black ($R=24.7\mu \mathrm{m}$, $C_m=0.20\mathrm{mol}/\mathrm{l}$), gray ($R=24.7\mu \mathrm{m}$, $C_m=0.20\mathrm{mol}/\mathrm{l}$), and yellow ($R=18.7\mu \mathrm{m}$, $C_m=0.20\mathrm{mol}/\mathrm{l}$).