TY - JOUR

T1 - Calculating the Symmetry Number of Flexible Sphere Clusters

AU - Zappa, Emilio

AU - Holmes-Cerfon, Miranda

N1 - Funding Information:
We thank Louis Theran for useful discussions. This material is based upon work supported by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under award DE-SC0012296. M.H.-C. acknowledges support from the Alfred P. Sloan Foundation.
Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - We present a theoretical and computational framework to compute the symmetry number of a flexible sphere cluster in R3, using a definition of symmetry that arises naturally when calculating the equilibrium probability of a cluster of spheres in the sticky-sphere limit. We define the sticky symmetry group of the cluster as the set of permutations and inversions of the spheres which preserve adjacency and can be realized by continuous deformations of the cluster that do not change the set of contacts or cause particles to overlap. The symmetry number is the size of the sticky symmetry group. We introduce a numerical algorithm to compute the sticky symmetry group and symmetry number, and show it works well on several test cases. Furthermore, we show that once the sticky symmetry group has been calculated for indistinguishable spheres, the symmetry group for partially distinguishable spheres (those with nonidentical interactions) can be efficiently obtained without repeating the laborious parts of the computations. We use our algorithm to calculate the partition functions of every possible connected cluster of six identical sticky spheres, generating data that may be used to design interactions between spheres so they self-assemble into a desired structure.

AB - We present a theoretical and computational framework to compute the symmetry number of a flexible sphere cluster in R3, using a definition of symmetry that arises naturally when calculating the equilibrium probability of a cluster of spheres in the sticky-sphere limit. We define the sticky symmetry group of the cluster as the set of permutations and inversions of the spheres which preserve adjacency and can be realized by continuous deformations of the cluster that do not change the set of contacts or cause particles to overlap. The symmetry number is the size of the sticky symmetry group. We introduce a numerical algorithm to compute the sticky symmetry group and symmetry number, and show it works well on several test cases. Furthermore, we show that once the sticky symmetry group has been calculated for indistinguishable spheres, the symmetry group for partially distinguishable spheres (those with nonidentical interactions) can be efficiently obtained without repeating the laborious parts of the computations. We use our algorithm to calculate the partition functions of every possible connected cluster of six identical sticky spheres, generating data that may be used to design interactions between spheres so they self-assemble into a desired structure.

KW - Flexible sphere clusters

KW - Self-assembly

KW - Symmetry

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U2 - 10.1007/s00332-019-09537-4

DO - 10.1007/s00332-019-09537-4

M3 - Article

AN - SCOPUS:85061985959

SN - 0938-8974

VL - 29

SP - 2021

EP - 2053

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 5

ER -