Probability distribution functions as descriptors for ... - ResearchGate

2446 J.F. Shackelford, L.P. Davila / Journal of Non-Crystalline Solids 356 (2010) 2444–2447however, such plots were distinctly non-linear. On the other hand, replottingall distributions as ring size (on a linear rather than logarithmicscale) against the cumulative percentage of the rings on a non-linearGaussian scale (Fig. 2) shows linear plots indicating the distributionsfollow a normal PDF, not lognormal.3. Discussion — a descriptor for ring statistics in vitreous silicaWhile the descriptor for interstitial site sizes in vitreous silica is thelognormal PDF analogous to the comminution of fine powders, Fig. 2indicates that the descriptor for ring statistics in this glass is thenormal or Gaussian PDF. The lognormal distribution arises when, as involume comminution, random subdivision involves a higher-poweredterm (such as volume). In that case, the logarithm of the variate (suchas powder particle size or interstitial site size in glasses) is Gaussian inits logarithm rather than the variate itself. Apparently, the randomlinkage of tetrahedral edges in defining ring sizes (n) is simplyGaussian in the variate (n) itself.It is also interesting to consider the average number of rings invitreous silica at various pressures. While the average ring size in anideal Zachariasen schematic (fully linked set of triangular buildingblocks in 2-dimensional space) is rigorously =6 by the Euler relation,Rivier has noted that the average ring size in 3-dimensional structurescan vary depending on structural details (non-planar rings, dislocations,etc.) [14]. In the two examples (soap froths and covalentglasses), Rivier found average ring sizes bnN between 5.1 and 6.28.Fig. 3 indicates that the average ring size bnN for vitreous silica overthe full pressure range considered is essentially constant at bnN=5.8.A more useful parameter from the normal PDF descriptor is thestandard deviation, σ. Fig. 4(a) shows that σ rises sharply above acompressed volume of V/V 0 =0.8 corresponding to a pressure of9 GPa. Davila et al. found that this value of V/V 0 or pressurecorresponds to the break between elastic and plastic deformation inthe associated EOS [11]. Finally, it is interesting to note that while theaverage ring size is unchanged during the elastic–plastic transition,the broadening of the distribution upon plastic deformation (withcorresponding bond breaking and topological rearrangement) indicatesa significant decrease in 5- and 6-membered rings with acorresponding increase in the number of smaller and larger rings asshown in Fig. 1.4. Summary and conclusionsWhile the lognormal PDF has been a useful descriptor for interstitialsite sizes in vitreous silica, the normal PDF appears to be the appropriatedescriptor for the ring statistics over a wide pressure range. The averagering size is slightly less than 6 (bnN=5.8) independent of pressure up to23 GPa. The standard deviation (σ) of the normal PDF is a sensitiveindicator of the elastic–plastic transition that occurs upon compressingthe glass material beyond V/V 0 =0.8 corresponding to a pressure of9GPa.AcknowledgmentsOne of us (LPD) is grateful for a SEGRF Fellowship at the LawrenceLivermore National Laboratory that supported earlier portions of thiswork and more recently for a Ford Foundation Post-doctoral Fellowshipat the University of California, Merced for funding and travel support forthis work. The other (JFS) is grateful to Professor Prabhat Gupta of OhioState University for useful discussions relative to this work.ReferencesFig. 4. The standard deviation (σ) of the normal PDF for ring statistics in vitreous silica isa sensitive indicator of the elastic–plastic transition that occurs upon compressing thematerial beyond (a) V/V 0 =0.8 corresponding to (b) a pressure of 9 GPa.[1] J.F. Shackelford, J.S. Masaryk, The Gas Atom as a Microstructural Probe forAmorphous Solids, in: R.M. Fulrath, J.A. Pask (Eds.), Ceramic Microstructures '76,Westview Press, Boulder, CO, 1977, pp. 149–159.[2] J.F. Shackelford, J.S. Masaryk, The interstitial structure of vitreous silica, Journal ofNon-Crystalline Solids 30 (1978) 127–139.[3] R.S. Wortman, J.F. Shackelford, Gas transport in vitreous silica fibers, Journal ofNon-Crystalline Solids 125 (1990) 280–286.[4] G.S. Nakayama, J.F. Shackelford, Solubility and diffusivity of argon in vitreoussilica, Journal of Non-Crystalline Solids 126 (1990) 249–254.[5] J.F. Shackelford, Gas solubility in glasses — principles and structural implications,Journal of Non-Crystalline Solids 253 (1999) 231–241.[6] F. Liebau, Structural Chemistry of Silicates, Springer, Berlin, 1985.[7] S.L. Chan, S.R. Elliott, Theoretical study of the interstice statistics of the oxygensublattice in vitreous SiO 2 , Physical Review B 43 (5) (1991) 4423–4432.[8] J.L. Finney, J. Wallace, Interstice correlation functions; a new sensitive characterizationof non-crystalline packed structures, Journal of Non-Crystalline Solids 43(1981) 165–187.[9] J.F. Shackelford, The Glassy State, in: S. Saito (Ed.), Fine Ceramics, Ohmsha, Tokyo,1987, pp. 126–146.[10] S.M. Allen, E.L. Thomas, The Structure of Materials, Wiley, New York, 1999.