Rearrangement of two dimensional aggregates of droplets under compression: Signatures of the energy landscape from crystal to glass

We study signatures of the energy landscape's evolution through the crystal-to-glass transition by compressing two dimensional (2D) finite aggregates of oil droplets. Droplets of two distinct sizes are used to compose small aggregates in an aqueous environment. Aggregates range from perfectly ordered monodisperse single crystals to disordered bidisperse glasses. The aggregates are compressed between two parallel boundaries, with one acting as a force sensor. The compression force provides a signature of the aggregate composition and gives insight into the energy landscape. In particular, crystals dissipate all the stored energy through single catastrophic fracture events whereas the glassy aggregates break step by step. Remarkably, the yielding properties of the 2D aggregates are strongly impacted by even a small amount of disorder.

Figure 1

(a) Schematic top view of the experimental chamber. The typical dimensions of the wall (dark grey) are $55\times 30\times 2.5{\mathrm{mm}}^3$. The “droplet pipette,” “pushing pipette,” and “force-sensing pipette” are labeled as (i), (ii), and (iii) respectively. (b) Schematic side view (not to scale). The buoyant droplets form a quasi-2D aggregate bounded by the top glass plate. The pushing pipette (black circle on the left) and the force-sensing pipette (red circle on the right) are placed near the average equatorial plane of the droplets so forces are applied horizontally. (c) Optical microscopy image of a typical crystal (scale bar is $50\mu \mathrm{m})$. Red dashed lines show observed fracture lines for a crystal when compressed.

Figure 2

(a) Force measurements, $F$, as a function of the distance between the pipettes, $\delta$, for seven aggregates sharing the same lattice but with different compositions of small $\left(\mathcal{R}=19.2\pm 0.3\mu \mathrm{m}\right)$ and large $\left(\mathcal{R}=25.1\pm 0.3\mu \mathrm{m}\right)$ droplets. Here $\delta$ decreases with time as the aggregate is compressed and the aggregate changes from $p=4$ to 1 as indicated at the bottom. Traces 1 to 7 correspond to $\{0;1;6;10;14;19;20\}$ large droplets with $N_{\text{tot}}=20$. The black dashed lines correspond to the positions ${\delta }_{\min }^p$ of the peak maxima for the crystal made of small droplets, while the blue dashed lines correspond to the positions ${\delta }_{\max }^p$ of the peak onsets (i.e., upon compression as $\delta$ decreases) for the crystal made of large droplets. The shaded area highlights the different transitions during the compression. (b)–(e) Optical microscopy images of the aggregates, before compression, corresponding to traces 1 to 4. Blue squares correspond to large droplets and red circles to small droplets (scale bar is $50\mu \mathrm{m})$.

Figure 3

(a) Measured total number of force peaks as a function of defect fraction, for a compression from $p_{\mathrm{ini}}$ to $p=1$. $(•)p_{\mathrm{ini}}=4$ with $q_{\mathrm{ini}}=5$ from two data sets (different colors); $\left(\blacksquare \right)p_{\mathrm{ini}}=3$ with the three rows initially made of 8, 7, and 8 droplets each, from two data sets (different colors). (b) Evolution of the normalized excess number of peaks compared to a crystal, with the black dashed line corresponding to Eq. (2). (c) Theoretical probability distribution of the dimensionless column height $h$, in an aggregate with $q\to \infty$ and $p=4$, for four defect fractions $\varphi =\{0;0.1;0.3;0.5\}$ (see Appendix pp2). Gaussian curves (blue solid lines) with same standard deviation, $\sigma$, and average, $\mu$, as the discrete distribution are overlayed as a guide to the eyes. Typical radii of large and small droplets are $\approx 22\mu \mathrm{m}$ and $\approx 18\mu \mathrm{m}$ (see Appendix pp4).

Figure 4

Normalized partial work (see definition in text) as a function of distance, for the $p=4\to p=3$ transition, for aggregates of different relative compositions (number of small droplets / number of large droplets) as indicated. Top: Experimental results corresponding to the force curves shown in Fig. 2. Bottom: Corresponding theoretical results, according to Eq. (3).

Figure 5

Schematics of the columns with $p=6$ considered in the theoretical model. The left part shows how rectangles and circles are assembled to build a column. The right part shows the different choices for circles and rectangles along with their probability of appearing.

Figure 6

(a) Probability distribution $A_n$ as a function of $\varphi$ for an aggregate made of $p=2$ rows and $q=50$ droplets per row. (b) Prediction of the average number of peaks in the force measurement as a function of $\varphi$ based on the distribution $A_n$. Note that we only plot the function for $\varphi \in [0,0.5]$ by symmetry about $\varphi =0.5$.

Figure 7

Impact of the size of the aggregate on the number of peaks in the force measurement for $p=2$ and different values of $q$.

Figure 8

Comparison between the analytical expression for the standard deviation of the height distribution and the numerically calculated values for different $\varphi$ for $p=2$.

Figure 9

Comparison between the average number of peaks predicted by the discrete calculation (for different values of $q)$ and the continuous approximation, for $p=2$. The continuous model does not take into account the number of droplets per row $q$ in the aggregate.