The effect of inter-cluster interactions on the structure of colloidal clusters

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Abstract

Colloidal systems present exciting opportunities to study clusters. Unlike atomic clusters, which are frequently produced at extremely low density, colloidal clusters may interact with one another. Here we consider the effect of such interactions on the intra-cluster structure in simulations of colloidal cluster fluids. A sufficient increase in density leads to a higher population of clusters in the ground state. In other words, inter-cluster interactions perturb the intra-cluster behaviour, such that each cluster may no longer be considered as an isolated system. Conversely, for dilute, weakly interacting cluster fluids little dependence on colloid concentration is observed, and we thus argue that it is reasonable to treat each cluster as an isolated system.

Introduction

Clusters are a distinct state of matter which exhibit different structural ordering and phase behaviour, relative to bulk materials [1]. Of particular relevance to, for example, many biological systems such as viruses, is their tendency to exhibit five-fold symmetry such as icosahedra and decahedra [2]. Recently there has been a surge of interest in clusters formed in colloidal systems [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], leading to the development of ‘colloidal molecules’ [4], [9], [12], [14], [15]. These may in turn provide novel functionalised materials [4], [9], [10], [12], [14], [15]. In addition to this technological relevance, the visualisation of colloidal clusters allows direct access to their free energy landscape [16].

Part of the attraction of studying colloidal dispersions is that, although in principle they are rather complex multicomponent systems, the spatial and dynamic asymmetry between the colloidal particles (10 nm–1 μm) and smaller molecular and ionic species has led to schemes where the smaller components are formally integrated out [17]. This leads to a one-component picture, where only the effective colloid–colloid interactions need be considered.

Given that the structure of ground state clusters of simple liquid models is known, along with local energy minima [2], it seems natural to investigate the prevalence of such structures in colloidal systems, in particular those with depletion attractions such as colloid–polymer mixtures where, at fixed real temperature, an effective temperature may be interpreted as the inverse of the attraction strength between the colloids. These colloidal systems exhibit similar [18], [19], [20] though not identical [19], [21] structures to clusters of simple liquid models. In general, as the (effective) temperature is reduced, we expect more clusters in the ground state, unless kinetic frustration comes into play.

Unlike atomic clusters, which are often considered in isolation, colloidal clusters may themselves form fluids [5], [6]. Colloidal cluster fluids are found in systems with competing interactions with short-ranged attractions and long-ranged repulsions [5], [6]. The attractions drive clustering while the repulsions prevent aggregation and phase separation, leading to a characteristic cluster size [22]. At sufficient concentration, these cluster fluids form gels [6], [23], [24], [25], while for sufficiently strong repulsions, the cluster fluid can undergo dynamical arrest [26], [27] or crystallisation [28]. Furthermore, the separation in length- and time-scales allows us to treat the system in a hierarchical manner. For example, dynamical arrest within clusters [29] and of a fluid comprised of clusters [26], [27] can in general be decoupled. This hierarchy means that one may consider each cluster as an isolated system [30]. Alternatively, one may operate at the cluster–cluster level [31]. Here instead we consider the influence of the inter-cluster interactions on the intra-cluster behaviour. In other words, how is the free energy landscape of a given cluster perturbed by its neighbours?

In a recent experimental study on a cluster fluid [30], we observed that a rather small number of clusters were found in the expected ground state, around 20%. However, after a careful mapping of experimental parameters to conventional spherically symmetric interactions, Brownian dynamics simulations showed that isolated clusters reached their ground state, unless geometric frustration led to ergodicity breaking for deep quenches [29].

Here we consider an experimentally relevant set of conditions [30] and use Brownian dynamics simulations to investigate the effects of inter-cluster interactions on the intra-cluster behaviour. In particular we consider the population of clusters in the ground state. Our system is characterised by the colloid packing fraction ϕ, the strength of the attractive interaction βεM and the strength of the repulsion βεY, where β=1kBT, kB is the Boltzmann constant and T is the temperature. We first determine the range of parameters that correspond to a cluster fluid, before investigating the effect that varying these parameters has on the intra-cluster structure.

Section snippets

Simulation details and model

Although a great many improvements have since been made [32], [33], the theory of Asakura and Oosawa (AO) [34] is generally accepted to capture the essential behaviour of polymer-induced depletion attractions between colloids. This AO model ascribes an effective pair interaction between two colloidal hard spheres in a solution of ideal polymers. However, the hard core of the AO potential leads to difficulties with Brownian dynamics simulations. The Morse potential is a variable range

Phase diagram

Here we are interested in cluster fluids, however at low density and/or low attraction, the system is dominated by unbound particles which we term a monomer fluid (Fig. 1(a)) while at high density we find aggregation into much larger, often elongated [18], [22] clusters and ultimately gelation (Fig. 1(c)) [6], [23]. Between the aggregation/gel regime and the monomer fluid lies the cluster fluid seen in Fig. 1(b).

We begin our presentation of the results by determining those state points we

Discussion

We have considered the effect of cluster–cluster interactions on the intra-cluster structure in the Brownian dynamics simulations of a cluster fluid. The overall behaviour is broadly similar to isolated clusters. That is to say, upon increasing the strength of the attraction clusters are able to reach their ground states, unless geometrically frustrated from doing so [29]. For weak cluster–cluster interactions around 0.15 kBT between individual clusters, the intra-cluster behaviour depends

Conclusions

The effect of cluster–cluster interactions on the intra-cluster structure of a model colloidal system has been studied. In the case of weak cluster–cluster interactions at low colloid concentration, the yield of structures is similar to if the clusters were isolated. This is the ‘cluster gas’ limit and interactions between clusters may be neglected. By increasing colloid density and/or increasing the strength of the Yukawa repulsion between colloids, cluster–cluster interactions become stronger

Acknowledgements

CPR thanks the Royal Society for funding, AM acknowledges the support of EPSRC grant EP/5011214. This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol — http://www.bris.ac.uk/acrc/.

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