Probability distribution functions as descriptors for long range randomness in non-crystalline solids

a r t i c l e i n f o Probability distribution functions (PDF) can be useful descriptors for the random structure of non-crystalline solids. The lognormal PDF has provided a good description of the distribution of interstitial site sizes in non-crystalline solids, both metallic and nonmetallic. The utility of this descriptor to characterize the ring statistics of vitreous silica under a wide range of pressures has been evaluated. In fact, the normal (Gaussian) PDF is a better descriptor for ring statistics for pressures up to 23 GPa. Although the average ring size is largely unchanged (slightly less than 6.0) over the entire pressure range, the standard deviation of the ring size distribution rises sharply at 9 GPa serving as a sensitive indicator of the onset of plastic deformation. Using gas transport data, Shackelford and Masaryk [1] suggested that the nature of the interstitial structure in vitreous silica could be described by a lognormal probability distribution function (PDF) [2]. Specifically, the size of individual interstitial sites (defined by inscribed spheres) varied in size from 0.1 to 0.4 nm in diameter with the size following the skewed distribution of a lognormal PDF analogous to the random subdivision of 3-dimensional space in the formation of powders by crushing and grinding. In the case of non-crystalline solids, interstitial space is randomly subdivided in the formation of the random network topology. The lognormal distribution can be defined as a normal or Gaussian distribution of the logarithm of the variate. Subsequent gas solubility measurements by Shackelford and co-workers [3,4] refined the values for the solubility site densities accessible by noble gas atoms and reinforced the lognormal model of interstitial site size distribution [5]. A distinctive structural feature of silicates is that directional, covalent bonding combined with the flexibility of the Si–O–Si bond allows the formation of various cage-like structures large enough to accommodate a solute gas atom [6]. In a crystalline silicate, the number of such cage shapes will be limited. In a silicate glass, there will tend to be a range of cage shapes both larger and smaller than those in the crystalline analog [5]. As noted earlier, the analysis of gas transport data allows one to specify the density of sites (N S) accessible to a given size solute gas atom. The one glass for which the size distribution of solubility site cages has been thoroughly determined is vitreous silica. By comparing …