The hyperuniformity concept provides a unified means to classify all perfect crystals, perfect quasicrystals, and exotic amorphous states of matter according to their capacity to suppress large-scale density fluctuations. While the classification of hyperuniform point configurations has received considerable attention, much less is known about the classification of hyperuniform two-phase heterogeneous media, which include composites, porous media, foams, cellular solids, colloidal suspensions, and polymer blends. The purpose of this article is to begin such a program for certain two-dimensional models of hyperuniform two-phase media by ascertaining their local volume-fraction variances ${\sigma }_{{}_V}^2\left(R\right)$ and the associated hyperuniformity order metrics ${\overline{B}}_V$. This is a highly challenging task because the geometries and topologies of the phases are generally much richer and more complex than point-configuration arrangements, and one must ascertain a broadly applicable length scale to make key quantities dimensionless. Therefore, we purposely restrict ourselves to a certain class of two-dimensional periodic cellular networks as well as periodic and disordered or irregular packings of circular disks, some of which maximize their effective transport and elastic properties. Among the cellular networks considered, the honeycomb networks have minimal values of the hyperuniformity order metrics ${\overline{B}}_V$ across all volume fractions. On the other hand, for all packings of circular disks examined, the triangular-lattice packings have the smallest values of ${\overline{B}}_V$ for the possible range of volume fractions. Among all structures studied here, the triangular-lattice packing of circular disks have the minimal values of the order metric for almost all volume fractions. Our study provides a theoretical foundation for the establishment of hyperuniformity order metrics for general two-phase media and a basis to discover new hyperuniform two-phase systems with desirable bulk physical properties by inverse design procedures.

中文翻译：

---

表征有序和无序两相介质的超均匀性

超均匀性概念提供了一种统一的方法，可以根据其抑制大规模密度波动的能力，对所有完美晶体，完美准晶体和奇特的非晶态进行分类。尽管超均匀点构型的分类已引起广泛关注，但对超均匀两相异相介质的分类知之甚少，其中包括复合材料，多孔介质，泡沫，多孔固体，胶体悬浮液和聚合物共混物。本文的目的是通过确定局部体积分数方差来为某些超均匀两相介质的二维模型启动这样的程序${\sigma }_{{}_V}^2（\mathrm{[R}）$ 以及相关的超均匀性订单指标 ${\overline{乙}}_V$。这是一项极富挑战性的任务，因为相的几何形状和拓扑通常比点配置的布局更丰富，更复杂，并且必须确定广泛适用的长度范围以使关键量无量纲。因此，我们有目的地将自己限制在特定类别的二维周期性蜂窝网络以及圆盘的周期性和无序或不规则堆积中，其中一些最大化了其有效传输和弹性特性。在所考虑的蜂窝网络中，蜂窝网络具有超均匀度阶度度量的最小值${\overline{乙}}_V$所有体积分数。另一方面，对于所检查的所有圆盘填料，三角晶格填料的最小值为${\overline{乙}}_V$体积分数的可能范围。在这里研究的所有结构中，圆盘的三角形晶格堆积对于几乎所有体积分数都具有最小的阶次度量值。我们的研究为建立一般两相介质的超均匀性阶度量提供了理论基础，并为通过逆设计程序发现具有理想整体物理性质的新的超均匀两相系统奠定了基础。