### Abstract

Particles interacting with short-ranged potentials have attracted increasing interest, partly for their ability to model mesoscale systems such as colloids interacting via DNA or depletion. We consider the free-energy landscape of such systems as the range of the potential goes to zero. In this limit, the landscape is entirely defined by geometrical manifolds, plus a single control parameter. These manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system - equilibrium or nonequilibrium - can be computed for arbitrary potentials. To consider dynamical quantities we compute the asymptotic limit of the Fokker-Planck equation and show that it becomes restricted to the low-dimensional manifolds connected by "sticky" boundary conditions. To illustrate our theory, we compute the low-dimensional manifolds for n ≤ 8 identical particles, providing a complete description of the lowest-energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom. The results can be directly tested on colloidal clusters. This limit is a unique approach for understanding energy landscapes, and our hope is that it can also provide insight into finite-range potentials.

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

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 110 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2 2013 |

### Fingerprint

### Keywords

- Self-assembly
- Sticky Brownian motion
- Sticky spheres
- Transition rates

### ASJC Scopus subject areas

- General

### Cite this

**A geometrical approach to computing free-energy landscapes from short-ranged potentials.** / Holmes-Cerfon, Miranda; Gortler, Steven J.; Brenner, Michael P.

Research output: Contribution to journal › Article

*Proceedings of the National Academy of Sciences of the United States of America*, vol. 110, no. 1. https://doi.org/10.1073/pnas.1211720110

}

TY - JOUR

T1 - A geometrical approach to computing free-energy landscapes from short-ranged potentials

AU - Holmes-Cerfon, Miranda

AU - Gortler, Steven J.

AU - Brenner, Michael P.

PY - 2013/1/2

Y1 - 2013/1/2

N2 - Particles interacting with short-ranged potentials have attracted increasing interest, partly for their ability to model mesoscale systems such as colloids interacting via DNA or depletion. We consider the free-energy landscape of such systems as the range of the potential goes to zero. In this limit, the landscape is entirely defined by geometrical manifolds, plus a single control parameter. These manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system - equilibrium or nonequilibrium - can be computed for arbitrary potentials. To consider dynamical quantities we compute the asymptotic limit of the Fokker-Planck equation and show that it becomes restricted to the low-dimensional manifolds connected by "sticky" boundary conditions. To illustrate our theory, we compute the low-dimensional manifolds for n ≤ 8 identical particles, providing a complete description of the lowest-energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom. The results can be directly tested on colloidal clusters. This limit is a unique approach for understanding energy landscapes, and our hope is that it can also provide insight into finite-range potentials.

AB - Particles interacting with short-ranged potentials have attracted increasing interest, partly for their ability to model mesoscale systems such as colloids interacting via DNA or depletion. We consider the free-energy landscape of such systems as the range of the potential goes to zero. In this limit, the landscape is entirely defined by geometrical manifolds, plus a single control parameter. These manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system - equilibrium or nonequilibrium - can be computed for arbitrary potentials. To consider dynamical quantities we compute the asymptotic limit of the Fokker-Planck equation and show that it becomes restricted to the low-dimensional manifolds connected by "sticky" boundary conditions. To illustrate our theory, we compute the low-dimensional manifolds for n ≤ 8 identical particles, providing a complete description of the lowest-energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom. The results can be directly tested on colloidal clusters. This limit is a unique approach for understanding energy landscapes, and our hope is that it can also provide insight into finite-range potentials.

KW - Self-assembly

KW - Sticky Brownian motion

KW - Sticky spheres

KW - Transition rates

UR - http://www.scopus.com/inward/record.url?scp=84871943759&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871943759&partnerID=8YFLogxK

U2 - 10.1073/pnas.1211720110

DO - 10.1073/pnas.1211720110

M3 - Article

C2 - 23248296

AN - SCOPUS:84871943759

VL - 110

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 1

ER -