Controlling disorder in two-dimensional networks
Journal of Physics: Condensed Matter
Continuous random networks are commonplace in nature and found in many chemical, biological and physical systems, ranging from grains in crystallites     to cellular structures      to foams         and even the Giant's Causeway  . There has been renewed interest in materials of this type due to the synthesis and visualisation of near-two-dimensional (2D) network-forming materials such as silica and aluminosilicate bilayers [18, 19]
... These advances potentially facilitate the controlled growth of a range of technologically-useful materials. (e.g. catalysis, gas separation etc-see  and references therein.) When classifying 2D networks a primary descriptor is the distribution of ring sizes, p n . The size of a ring, n, is defined by the number of constituent edges, which may represent bonds, crystal edges or the interfaces between soap bubbles, for example. The heart of the synthetic problem is the ability to exert true atomistic control over the material growth, which equates to controlling p n . It is clear upon inspecting a sample however, that the ring statistics alone are not sufficient to describe the full system topology and that information may be available on a given network at a number of 'levels'. Euler's law defines the mean ring size as n = 6 and we may know the second moment, µ = n 2 − n 2 , or even the full ring distribution, p n . However, even a full knowledge of p n is insufficient. For example, figure 1 shows three networks which have the same ring distributions but fundamentally different Abstract Two-dimensional networks are constructed by reference to a distribution of ring sizes and a parameter (α) which controls the preferred nearest-neighbour spatial correlations, and allows network topologies to be varied in a systematic manner. Our method efficiently utilizes the dual lattice and allows the range of physically-realisable configurations to be established and compared to networks observed for a wide range of real and model systems. Three different ring distributions are considered; a system containing five-, six-and seven-membered rings only (a proxy for amorphous graphene), the configuration proposed by Zachariasen in 1932, and a configuration observed experimentally for thin (near-2D) films of SiO 2 . The system energies are investigated as a function of the network topologies and the range of physicallyrealisable structures established and compared to known experimental results. The limits on the parameter α are discussed and compared to previous results. The evolution of the network structure as a function of topology is discussed in terms of the ring-ring pair distribution functions.