^{1}, Alex Malins

^{2}, Stephen R. Williams

^{3}and C. Patrick Royall

^{4,a)}

### Abstract

We revisit the role of attractions in liquids and apply these concepts to colloidal suspensions. Two means are used to investigate the structure; the pair correlation function and a recently developed topological method. The latter identifies structures topologically equivalent to ground state clusters formed by isolated groups of 5 ⩽ *m* ⩽ 13 particles, which are specific to the system under consideration. Our topological methodology shows that, in the case of Lennard-Jones, the addition of attractions increases the system's ability to form larger (*m* ⩾ 8) clusters, although pair-correlation functions are almost identical. Conversely, in the case of short-ranged attractions, pair correlation functions show a significant response to adding attraction, while the liquid structure exhibits a strong decrease in clustering upon adding attractions. Finally, a compressed, weakly interacting system shows a similar pair structure and topology.

J.T. and C.P.R thank the Royal Society for funding, A.M. is supported under EPSRC Grant No. EP/E501214/1. The authors are grateful to Jens Eggers, Bob Evans, Rob Jack and an anonymous reviewer for helpful discussions and suggestions.

I. INTRODUCTION

II. SIMULATIONS AND INTERACTION POTENTIALS

A. Interaction potentials

B. Comparing different systems

C. The topological cluster classification

D. Systems studied

III. RESULTS AND DISCUSSION

A. The Lennard-Jones triple point: Long-ranged interactions

B. High-temperature systems: Short-ranged interactions

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Ground states
- 16.0
- Colloidal systems
- 10.0
- Cluster analysis
- 7.0
- Crystallization
- 7.0
- Topology
- 7.0

## Figures

Clusters found in bulk systems using the topological cluster classification. For *m* ⩽ 7, where *m* is the number of particles in a cluster, all studied ranges of the Morse potential Eq. (3) form clusters of identical topology. In the case of larger *m* the cluster topology depends on the interaction range. Here we follow the nomenclature of Doye *et al.* (Ref. 14) where *A* corresponds to long-ranged potentials and *B*… to minimum energy clusters of shorter-ranged potentials.

Clusters found in bulk systems using the topological cluster classification. For *m* ⩽ 7, where *m* is the number of particles in a cluster, all studied ranges of the Morse potential Eq. (3) form clusters of identical topology. In the case of larger *m* the cluster topology depends on the interaction range. Here we follow the nomenclature of Doye *et al.* (Ref. 14) where *A* corresponds to long-ranged potentials and *B*… to minimum energy clusters of shorter-ranged potentials.

Interaction potentials used. (a) Long-ranged potentials: Morse (dark green) and truncated Morse (bright green) with range parameter . (b) Lennard-Jones (red) and WCA (pink). (c) Short-ranged potentials: Morse (blue) and truncated Morse (turquoise) with range parameter . Dashed cyan line in (c) denotes the hard sphere interaction. denotes the effective hard sphere diameter as defined in Eq. (7) and listed in Table I.

Interaction potentials used. (a) Long-ranged potentials: Morse (dark green) and truncated Morse (bright green) with range parameter . (b) Lennard-Jones (red) and WCA (pink). (c) Short-ranged potentials: Morse (blue) and truncated Morse (turquoise) with range parameter . Dashed cyan line in (c) denotes the hard sphere interaction. denotes the effective hard sphere diameter as defined in Eq. (7) and listed in Table I.

Pair-correlation functions. (a) Long-ranged potentials: Morse with (dark green, dashed) and without (bright green) attractions. Here . (b) Lennard-Jones (red, dashed) and WCA (pink) for a well depth of (the triple point).

Pair-correlation functions. (a) Long-ranged potentials: Morse with (dark green, dashed) and without (bright green) attractions. Here . (b) Lennard-Jones (red, dashed) and WCA (pink) for a well depth of (the triple point).

Population of particles in a given cluster. is the number of particles in a given cluster, *N* the total number of particles sampled. Here we consider only ground state clusters for each system. (a) Morse () (dark green) and truncated Morse (bright green). (b) Lennard-Jones at the triple point (red) and corresponding WCA (pink). Note the semilog scale.

Population of particles in a given cluster. is the number of particles in a given cluster, *N* the total number of particles sampled. Here we consider only ground state clusters for each system. (a) Morse () (dark green) and truncated Morse (bright green). (b) Lennard-Jones at the triple point (red) and corresponding WCA (pink). Note the semilog scale.

Ratio of cluster populations in systems mapped to the Lennard-Jones triple point. (a) Morse and truncated Morse (). (b) Lennard-Jones and WCA. These plot the same data as Fig. 4 expressed to emphasize the difference between the systems.

Ratio of cluster populations in systems mapped to the Lennard-Jones triple point. (a) Morse and truncated Morse (). (b) Lennard-Jones and WCA. These plot the same data as Fig. 4 expressed to emphasize the difference between the systems.

Population of particles in a given cluster at parameters mapped to the Lennard-Jones triple point. is the number of particles in a given cluster, *N* the total number of particles sampled. Here we consider ground state clusters for all ranges of the Morse potential (Ref. 14). Colors are Lennard-Jones (red), corresponding WCA (pink), Morse () (bright green) and truncated Morse (dark green). Those clusters which are ground states are labeled as ‘both’ when both potentials share the same ground state, and ‘LJ’ and ‘M’ corresponding to the Lennard-Jones and Morse cases accordingly. Note the semilog scale.

Population of particles in a given cluster at parameters mapped to the Lennard-Jones triple point. is the number of particles in a given cluster, *N* the total number of particles sampled. Here we consider ground state clusters for all ranges of the Morse potential (Ref. 14). Colors are Lennard-Jones (red), corresponding WCA (pink), Morse () (bright green) and truncated Morse (dark green). Those clusters which are ground states are labeled as ‘both’ when both potentials share the same ground state, and ‘LJ’ and ‘M’ corresponding to the Lennard-Jones and Morse cases accordingly. Note the semilog scale.

Pair-correlation functions. (a) Long-ranged potentials: Lennard-Jones (red) and WCA (pink) for a well depth of . (b) Short-ranged potentials: Morse (blue) and repulsive Morse (turquoise) according to Eq. (4). Here the well depth . Cyan denotes the Hard Sphere interaction.

Pair-correlation functions. (a) Long-ranged potentials: Lennard-Jones (red) and WCA (pink) for a well depth of . (b) Short-ranged potentials: Morse (blue) and repulsive Morse (turquoise) according to Eq. (4). Here the well depth . Cyan denotes the Hard Sphere interaction.

Population of particles in a given ground state cluster. is the number of particles in a given cluster, *N* the total number of particles sampled. (a) Lennard-Jones with (red) and corresponding WCA (pink). (b) Morse () (turquoise) truncated Morse (light blue) and hard sphere (dark blue). Note the semilog scale.

Population of particles in a given ground state cluster. is the number of particles in a given cluster, *N* the total number of particles sampled. (a) Lennard-Jones with (red) and corresponding WCA (pink). (b) Morse () (turquoise) truncated Morse (light blue) and hard sphere (dark blue). Note the semilog scale.

Ratio of cluster populations in high temperature systems. (a) Lennard-Jones and WCA. (b) Truncated Morse and Morse. This plot has the same data as in Fig. 8 expressed to emphasize the difference between the two systems. Note that in (b) we invert the ratio to consider the truncated Morse divided by the attractive Morse potential and plot on a different scale.

Ratio of cluster populations in high temperature systems. (a) Lennard-Jones and WCA. (b) Truncated Morse and Morse. This plot has the same data as in Fig. 8 expressed to emphasize the difference between the two systems. Note that in (b) we invert the ratio to consider the truncated Morse divided by the attractive Morse potential and plot on a different scale.

Population of particles in a given cluster, at parameters mapped to the Morse potential (, ). is the number of particles in a given cluster, *N* the total number of particles sampled. Here we consider ground state clusters for all ranges of the Morse potential (Ref. 14). Colors are Lennard-Jones (red), corresponding WCA (pink), Morse () (turquoise) truncated Morse (light blue) and hard sphere (dark blue). Those clusters which are ground states are labeled as ‘both’ when both potentials share the same ground state, and ‘LJ’ and ‘M’ corresponding to the Lennard-Jones and Morse cases respectively. Note the semilog scale.

Population of particles in a given cluster, at parameters mapped to the Morse potential (, ). is the number of particles in a given cluster, *N* the total number of particles sampled. Here we consider ground state clusters for all ranges of the Morse potential (Ref. 14). Colors are Lennard-Jones (red), corresponding WCA (pink), Morse () (turquoise) truncated Morse (light blue) and hard sphere (dark blue). Those clusters which are ground states are labeled as ‘both’ when both potentials share the same ground state, and ‘LJ’ and ‘M’ corresponding to the Lennard-Jones and Morse cases respectively. Note the semilog scale.

## Tables

State points studied. LJ high temp. and triple correspond to the two temperatures at which Lennard-Jones and WCA simulations were carried out. Trunc. Morse denotes the truncated Morse interaction [Eq. (4)].

State points studied. LJ high temp. and triple correspond to the two temperatures at which Lennard-Jones and WCA simulations were carried out. Trunc. Morse denotes the truncated Morse interaction [Eq. (4)].

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