### Abstract

We present a theoretical and computational framework to compute the symmetry number of a flexible sphere cluster in R
^{3}
, 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.

Original language | English (US) |
---|---|

Journal | Journal of Nonlinear Science |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Flexible sphere clusters
- Self-assembly
- Symmetry

### ASJC Scopus subject areas

- Modeling and Simulation
- Engineering(all)
- Applied Mathematics

### Cite this

**Calculating the Symmetry Number of Flexible Sphere Clusters.** / Zappa, Emilio; Holmes-Cerfon, Miranda.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Calculating the Symmetry Number of Flexible Sphere Clusters

AU - Zappa, Emilio

AU - Holmes-Cerfon, Miranda

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We present a theoretical and computational framework to compute the symmetry number of a flexible sphere cluster in R 3 , 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 R 3 , 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

UR - http://www.scopus.com/inward/record.url?scp=85061985959&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061985959&partnerID=8YFLogxK

U2 - 10.1007/s00332-019-09537-4

DO - 10.1007/s00332-019-09537-4

M3 - Article

AN - SCOPUS:85061985959

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

ER -