Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions

The creation of disordered hyperuniform materials with extraordinary optical properties (e.g., large complete photonic band gaps) requires a capacity to synthesize large samples that are effectively hyperuniform down to the nanoscale. Motivated by this challenge, we propose a feasible equilibrium fabrication protocol using binary paramagnetic colloidal particles confined in a 2D plane. The strong and long-ranged dipolar interaction induced by a tunable magnetic field is free from screening effects that attenuate long-ranged electrostatic interactions in charged colloidal systems. Specifically, we numerically find a family of optimal size ratios that makes the two-phase system effectively hyperuniform. We show that hyperuniformity is a general consequence of low isothermal compressibilities, which makes our protocol suitable to treat more general systems with other long-ranged interactions, dimensionalities, and/or polydispersity. Our methodology paves the way to synthesize large photonic hyperuniform materials that function in the visible to infrared range and hence may accelerate the discovery of novel photonic materials.

Figure 1

Structure factors for large particles $S_{11}(k)$, small particles $S_{22}(k)$, and the entire system $S_{\text{total}}(k)$ for different values of the coupling strength: (a) ${\mathrm{\Gamma }}_{22}=1$, (b) ${\mathrm{\Gamma }}_{22}=3$, (c) ${\mathrm{\Gamma }}_{22}=5$, (d) ${\mathrm{\Gamma }}_{22}=10$. Clearly, neither the point pattern associated with each component nor the entire system is hyperuniform.

Figure 2

(a) The angle $\varphi (k)$ between the complex collective density variables for small particles ${\tilde{n}}_2(\mathbf{k})$ and large particles ${\tilde{n}}_1(\mathbf{k})$ as a function of wave number $k$ under different coupling strengths. (b) A schematic plot of the complex collective density variables for small particles ${\tilde{n}}_2(\mathbf{k})$ and large particles ${\tilde{n}}_1(\mathbf{k})$ in the complex plane. Here, vector ${\tilde{n}}_1(\mathbf{k})$ can be stretched out to cancel ${\tilde{n}}_2(\mathbf{k})$ when they are added together.

Figure 3

Spectral densities ${\tilde{\chi }}_V(k)$ for binary systems with different size ratios $R_1/R_2$ for different values of the coupling strength: (a) ${\mathrm{\Gamma }}_{22}=1$, (b) ${\mathrm{\Gamma }}_{22}=3$, (c) ${\mathrm{\Gamma }}_{22}=5$, (d) ${\mathrm{\Gamma }}_{22}=10$. Insets show that the small-$k$ spectral density values for the optimal size ratio is much smaller than those for nonoptimal ratios. For ${\mathrm{\Gamma }}_{22}=10$, the hyperuniformity index $H$ is as small as 0.0004 for the optimal size ratio.

Figure 4

(a) A realization of the optimal two-phase system at ${\mathrm{\Gamma }}_{22}=5$ with volume fraction 0.15. (b) The local volume-fraction variance ${\sigma }_V^2(R)$ as a function of the spherical observation window of radius $R$ for different size ratios (optimal value 1.3). The scaling behaviors for $R^{-2}$ (nonhyperuniform) and $R^{-3}$ (hyperuniform) are included to guide the eye. (c) Optimal size ratio $Z(x_1,{\gamma }_{12})$ in terms of composition and coupling strength ratio, ${\gamma }_{12}$, as defined by Eq. (6).