Minimal Energy Clusters of Hard Spheres with Short Range Attractions
Fall 2010 Anna Wang
Arkus, N., Manoharan, V. N., & Brenner, M. P. (2009). Minimal Energy Clusters of Hard Spheres with Short Range Attractions. Physical Review Letters, 103(11), 118303-4
The thermodynamics of a cluster of small particles have been studied for a range of potentials using global optimisation methods. These methods, however, are poorly suited for strictly non-overlapping or ‘hard’ particles. Arkus et al are able to analytically solve for the ground states of ‘hard sphere clusters’ which satisfy minimal rigidity constraints by using graph theory and geometry.
The system consists of identical spheres, with repulsive, non-overlapping cores. The range of the attractive potential is much smaller than the particle diameter, so that the total potential energy of a cluster is linearly proportional to the number of contacts.
There are two types of packings:
- Iterative packings: these are packings such that all the m-particle subsets with <math>\ge 3m-6</math> contacts are also minimally rigid.
- Non-iterative packings: packings such as octahedra and tetrahedral fall into this category. They are termed ‘seeds’.
An <math>n \times n</math> adjacency matrix <math>A</math> is used to represent all possible configurations of <math>n</math> particles - <math>A_ij</math> <math>= 1</math> if particles <math>i</math> and <math>j</math> touch, <math>A_ij</math> <math>= 0</math> if they do not. The degeneracy of the particle labelling is then taken into account to obtain a set of non-redundant <math>A</math>’s.
Minimal rigidity constraints are then imposed:
- each particle has at least 3 contacts
- there is at least one contact for each internal degree of freedom (ie total contacts is <math>3n-6</math>)
Distance matrices <math>D</math> are then found for each <math>A</math>. For each <math>A_ij</math> <math>= 1</math>, <math>D_ij</math> <math>= 2r</math> ie the distance between the centres of two touching spheres. If <math>A_ij</math> <math>= 0</math>, <math>D_ij</math> <math>\ge 2r</math>. If any <math>D_ij</math> <math>< 2r</math>, the solution is unphysical and discarded. If a continuous set of <math>D</math> is found for a given <math>A</math>, the structure is not rigid. If a packing is chiral, one <math>D</math> will correspond to two enantiomers.
Subgraphs of <math>A</math> are examined to determine whether they correspond to lower &n-seeds. If so, <math>A</math> is considered an iterative packing. Since the <math>D</math> matrices corresponding to these lower &n-seeds are known, simple geometrical analysis is used to find the remaining distances and the triangle inequality is used to check for valid packings.
If an <math>A</math> can not be solved in this fashion, it is potentially a new seed. As very few packings are non-iterative for <math>n < 10</math>, the distances can be geometrically solved.
For <math>n</math> up to 9, all packings have <math>3n-6</math> contacts ie they have the same potential energy. For <math>n=10</math>, three packings with <math>3n-5</math> contacts were found. These special cases were all subunits of the hexagonally close-packed lattice. By building on these, a <math>3n-4</math> structure and <math>n = 11</math> and <math>3n-3</math> structure at <math>n = 12</math> were found.
It was also found that the icosahedron is not the ground state at <math>n = 12</math>. This is in contrast to the van der Waal cluster situation where icosahedra are the energy minima. If the hard sphere cluster behaviour is indeed qualitatively different from many other potentials, previous experiments and simulations may benefit from being re-examined.
The first example of a non-rigid packing with <math>3n-6</math> contacts was found. The <math>n=9</math> packing was still minimally rigid but had an internal rotational mode. This is expected to have a higher vibrational entropy than the more rigid structures; further richness in thermodynamic properties are expected for clusters with <math>n > 9</math>.