# Orientational Bond order parameter

January 14, 2016

Halperin and Nelson, described a theory of melting in two dimensionsB. I. Halperin and D. R. Nelson, Theory of Two-Dimensional Melting, Phys. Rev. Lett. 41, 121 (1978) PRL 41, 121 in terms of a bond orientational order parameter, $\psi_6 = \sum_m e^{6i\theta_{m}}$. As a crystal melts, defects in the crystal lattice appear. At early stages of melting, these defects stay closely assosciated with one another, but at later times and higher temperatures they become plentiful and spread throughout the lattice. The long range orientational order disappears; the lattice dissapears as the matter becomes liquid. They predict in a liquid phase, correlations in orientational (and translational) bond order decay exponentially.

This routine begins by loading open source code into the namespace and defining a helper function to load the neighbors of a particle as determined by a Delaunay triangulation.

The above function is a simple wrapper to return the indices of neighbors in the points list originally used to make the triang object - the function’s second argument. There are many options for selecting neighboring particles to calculate $\psi_6$. W. Mickel, et al., Shortcomings of the Bond Orientational Order Parameters for the Analysis of Disordered Particulate Matter, J. Chem. Phys. 138, 044501 (2013) arXiv 1209.6180

Let’s check the results of the above function by plotting the values for each particle in a soft colloidal packing.

(3196, 2)


Below I show a typical bidisperse packing with area fraction approximately 0.84 and and a ratio of large/small particles about 1.25. There are many polycrystalline domains.

0.756

Orientational Bond order parameter - January 14, 2016 - Zoey S. Davidson