Order Parameters for Two-Dimensional Networks

ORAL

Abstract

We derive methods that explain how to quantify the amount of order in ``ordered'' and ``highly ordered'' porous arrays. Ordered arrays from bee honeycomb and several from the general field of nanoscience are compared. Accurate measures of the order in porous arrays are made using the discrete pair distribution function (PDF) and the Debye-Waller Factor (DWF) from 2-D discrete Fourier transforms calculated from the real-space data using MATLAB routines. An order parameter, OP$_{3}$, is defined from the PDF to evaluate the total order in a given array such that an ideal network has the value of 1. When we compare PDFs of man-made arrays with that of our honeycomb we find OP$_{3}$=0.399 for the honeycomb and OP$_{3}$=0.572 for man's best hexagonal array. The DWF also scales with this order parameter with the least disorder from a computer-generated hexagonal array and the most disorder from a random array. An ideal hexagonal array normalizes a two-dimensional Fourier transform from which a Debye-Waller parameter is derived which describes the disorder in the arrays. An order parameter \textbf{S,} defined by the DWF, takes values from [0, 1] and for the analyzed man-made array is 0.90, while for the honeycomb it is 0.65. This presentation describes methods to quantify the order found in these arrays.

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Authors

  • Forrest Kaatz

    Owens Community College, Toledo, OH

  • William H. Rauckhorst

    Gothenburg University, Denison University, Bowling Green State University, NASA Glenn Research Center, Ohio Wesleyan University, Nitronex Corporation, 2305 Presidential Drive, Durham, NC, AFRL/MLPS, Wright-Patterson AFB, OH, Miami University, Ursinus College, Purdue University Calumet, GSI, Florida State University, Carnegie Mellon University, Dept. of Health Sciences, Cleveland State University, Physics Dept., Cleveland State University, University of Rochester, OMI College of Applied Science, University of Cincinnati, Davis and Elkins College, WV, Schlumberger-Doll, Korea Institute of Science and Technology, University of Cincinnati, Lock Haven University of Pennsylvania, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6372, Albion College, Albion, MI 49224, University of Toledo, Toledo, OH 43606, Youngstown State University, Joint Institute for Nuclear Research (Dubna), Naval Research Laboratory, ManTech SRS Technologies, Inc., Ecopulse, Inc., Army Research Laboratory, University of Surrey, The Ohio State University, Univ. Akron, Owens Community College, Toledo, OH

  • William H. Rauckhorst

    Gothenburg University, Denison University, Bowling Green State University, NASA Glenn Research Center, Ohio Wesleyan University, Nitronex Corporation, 2305 Presidential Drive, Durham, NC, AFRL/MLPS, Wright-Patterson AFB, OH, Miami University, Ursinus College, Purdue University Calumet, GSI, Florida State University, Carnegie Mellon University, Dept. of Health Sciences, Cleveland State University, Physics Dept., Cleveland State University, University of Rochester, OMI College of Applied Science, University of Cincinnati, Davis and Elkins College, WV, Schlumberger-Doll, Korea Institute of Science and Technology, University of Cincinnati, Lock Haven University of Pennsylvania, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6372, Albion College, Albion, MI 49224, University of Toledo, Toledo, OH 43606, Youngstown State University, Joint Institute for Nuclear Research (Dubna), Naval Research Laboratory, ManTech SRS Technologies, Inc., Ecopulse, Inc., Army Research Laboratory, University of Surrey, The Ohio State University, Univ. Akron, Owens Community College, Toledo, OH