Optimizing Hyperuniformity in Self-Assembled Bidisperse Emulsions

We study long range density fluctuations (hyperuniformity) in two-dimensional jammed packings of bidisperse droplets. Taking advantage of microfluidics, we systematically span a large range of size and concentration ratios of the two droplet populations. We identify various defects increasing long range density fluctuations mainly due to organization of local particle environment. By choosing an appropriate bidispersity, we fabricate materials with a high level of hyperuniformity. Interesting transparency properties of these optimized materials are established based on numerical simulations.

Figure 1

Top: Scheme of the microfluidic chip. Bottom: Experimental images of droplet assemblies. Gray scale represents the orientational order parameter ${\psi }_6$. White particles are crystallized in a hexagonal lattice. $\mathrm{SR}\approx 0.8$ and $\mathrm{NF}\approx 0.88$ (left, polycrystal) or $\mathrm{NF}\approx 0.38$ (right, disordered).

Figure 2

(a) Fraction of crystalline particles and rattlers for simulated and experimental binary mixtures of solid particles for different fraction of big ones NF. (b) Spectral density of a binary disk mixture for different fractions of big particles NF. Linear dependence $\chi \sim q$ characteristic of MRJ systems is shown as a guide for eyes. Inset: linear fit of experimental data for $\mathrm{NF}\approx 0.38$, $y$ intercept $(2\pm 3)\times {10}^{-4}$. (c) Spectral density at minimum experimentally available wave vector ($q{D}_m/2\pi \approx 0.1$) and optical thickness of the simulated 2D jammed assemblies $L/\ell$ as a function of NF. The refractive index is 1 for the host medium and 1.3 for the droplets. A strong correlation between the fraction of crystal particles, the minimum spectral density ${\chi }_{\min }$ and optical thickness $L/\ell$ is clearly observed. All figures correspond to $\mathrm{SR}\approx 0.80$.

Figure 3

(a) Spectral density ${\chi }_{\min }$ at minimum experimentally available wave vector ($q{D}_m/2\pi \approx 0.1$). Peaks and valleys are signatures of complex internal organization of different defects. Two examples of overpacked defects: a bidisperse crystal domain with fourfold symmetry $\mathrm{SR}\approx 0.40$ (left), a cluster of a big particle surrounded by seven small particles $\mathrm{SR}\approx 0.70$ (right). (b) Fraction of small particles having a given coordination number as a function of SR. (c) Fraction of big particle having a given coordination number as a function of SR. (d) Variance for cluster local fraction. All figures correspond to $\mathrm{NF}\approx 0.50$.

Figure 4

Geometrical resonance for a cluster of one small and five big particles. Three size ratios are shown: slightly before, close, and slightly after the resonance size ratio $\mathrm{SR}\approx 0.7$ corresponding to a compact cluster. The compact cluster has a density ${\phi }_{cl}$ well above the loosely packed ones.