Self-assembly and entropic effects in pear-shaped colloid systems:
I. Shape sensitivity of bilayer phases in colloidal pear-shaped particle systems

Philipp W. A. Schönhöfer
College of Science, Health, Engineering and Education, Mathematics and Statistics, Murdoch University, 90 South Street, 6150 Murdoch, WA, Australia
Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
Matthieu Marechal
Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
Douglas J. Cleaver
Materials and Engineering Research Institute, Sheffield Hallam University, Sheffield S1 1WB, UK
Gerd E. Schröder-Turk
College of Science, Health, Engineering and Education, Mathematics and Statistics, Murdoch University, 90 South Street, 6150 Murdoch, WA, Australia
Department of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University, 0200 Canberra, ACT, Australia
Department of Food Science, University of Copenhagen, Rolighedsvej 26, 1958 Frederiksberg C, Denmark
Physical Chemistry, Center for Chemistry and Chemical Engineering, Lund University, Lund 22100, Sweden

The role of particle shape in self-assembly processes is a double-edged sword. On the one hand, particle shape and particle elongation are often considered the most fundamental determinants of soft matter structure formation. On the other hand, structure formation is often highly sensitive to details of shape. Here we address the question of particle shape sensitivity for the self-assembly of hard pear-shaped particles, by studying two models for this system: a) the pear hard Gaussian overlap (PHGO) and b) hard pears of revolution (HPR) model. Hard pear-shaped particles, given by the PHGO model, are known to form a bicontinuous gyroid phase spontaneously. However, this model does not replicate an additive object perfectly and, hence, varies slightly in shape from a ”true” pear-shape. Therefore, we investigate in the first part of this series the stability of the gyroid phase in pear-shaped particle systems. We show based on the HPR phase diagram that the gyroid phase does not form in pears with such ”true” hard pear-shaped potential. Moreover, we acquire first indications from the HPR and PHGO pair-correlation functions that the formation of the gyroid is probably attributed to the small non-additive properties of the PHGO potential.

In colloidal and soft matter science, the influence of particle shape on the geometry of self-assembled meso-structures and, hence, on their physical properties is well documented. To some approximation, colloids behave as hard particles that are subject to thermal Brownian motion. Similar to objects with hard-core potentials, they interact largely by volume exclusion effects, which are defined by their outline, and otherwise feel no energetic repulsion or attraction. The effect of shape is demonstrated, for instance, in dense collections of elongated nano- or microrods, which spontaneously develop a preferential particle direction and, consequently, introduce a distinguished global orientation, known as the nematic director [1, 2]. Furthermore, it has been reported that the morphology of platonic and other polyhedral colloids can be used as a tool to create complex crystalline arrangements [3, 4, 5, 6, 7, 8, 9]. Hence, the manipulation of particle shapes is an auspicious mechanism to design self-assembled materials. However, the relationship between the shape of the constituent particles and the adopted self-assembled structure is not straightforward. While particle shape is beyond doubt an important determinant of structure formation, only a handful of quantifiable shape parameters could be related to long-ranged order directly. In colloidal self-assembly it is generally accepted that nematic order only occurs in particles that are sufficiently elongated, indicated by the aspect ratio between the length and width of the particle [1, 10, 11, 6, 12]. Similarly, it has been shown that close-packed structures, like those based on the $\gamma$-brass lattice, require particles with a high isoperimetric quotient, which indicates the ratio between the particle's volume and its surface area [6].

In this article we focus on a related aspect, namely shape sensitivity upon self-assembly, which aggravates the prediction of collective behaviour in multi-particle systems by just the outline of the single constituents even further. Even if morphological parameters are identified necessary for the formation of certain mesostructures, the stability of these assemblies tend to be sensitive towards small changes in shape. The sensitivity to details of shape is presumably most clearly observed in hard-core systems. These systems are by design reduced to the shape of the inherent particles, which is defined by the hard interaction potentials. Already introducing a small degree of polydispersity into simple systems like the hard sphere fluid [The standard deviation of the diameter distribution has to be $\sigma_d\gtrapprox 0.08\bar{d}$ with the mean diameter $\bar{d}$], can destabilise the crystalline into an amorphous phase for high densities [13]. Similarly in other hard particle mixtures, where depletion attractions between hard colloidal particles are induced by a solvent of surrounding small depletants, entropic forces are highly affected by the shape of colloids [14, 15, 16, 17, 18, 19, 20] (for a more in-depth discussion about depletion see part 2 of this series [21]). The significant influence of shape becomes also apparent by comparing the phase behaviour of hard spherocylinders [1] and hard ellipsoids [2] obtained by simulations. Even though the shapes of the individual particles seem similar, the smectic phase is only assembled by spherocylinders and not by ellipsoids.

Those observations are in accordance with other hard particle systems, which have been studied by investigating the intermediate stages of interpolations between two shapes. It has been shown, for example, that in systems of hard cubes, rounded edges have a significant influence on the cubical ordering of the crystalline phase [22, 23, 4, 24]. In addition to these superballs also various families of truncated polyhedra [5, 6, 7, 8, 9], elongated and twisted triangular prisms [25], discs with adjustable thickness [26] and very recently also dimpled spheres with various dimple sizes [27] have been studied. Here, it has been indicated that especially more complex particle arrangements are stable within a narrow window of shapes which makes them even more prone to small shape changes.

Cubic structures based on triply periodic minimal surfaces are amongst the most complex representatives of such phases, which have been observed within the field of colloidal self-assembly. For instance, computational simulations of hard pear-shaped particles, reminiscent of tapered ellipsoids, indicate the spontaneous formation of highly symmetric liquid crystal phases, like the cubic and bicontinuous Ia$\bar{3}$d double gyroid [28, 29] or the Pn$\bar{3}$m double diamond phase (upon addition of a hard sphere solvent) [30]. Here, the shape of the used hard-core potential, called pear hard Gaussian overlap (PHGO) potential, is best illustrated by a pear shape, which is described by two Bézier curves [31] (see fig:ContactFuntionPears for the outline of a pear-shaped particle). However, the effective shape of the PHGO model is just a close approximation and not a perfect fit to the Bézier description. Therefore, the shape represented by the PHGO potential can be interpreted as a slight distortion of the perfect Bézier pear.

Up to this point, the influence of the distinctions between the PHGO model and the ”true” Bézier pear-shape has not been studied in detail, a fortiori, as for the ellipsoidal counterparts (the hard Gaussian overlap (HGO) ellipsoids and the hard ellipsoids of revolution (HER) ) small differences between the two models are known [32]. The phase transitions between the isotropic and orientationally ordered liquid crystal phases do not match perfectly for both ellipsoid models as the HGO interaction profile promotes the alignment of particles by a greater margin. Consequently, the phase transition of the HGO ellipsoids occurs for lower densities than for HER ellipsoids. Nevertheless, the distinct transition density does not change the characteristics of the observed phase behaviour significantly. Both models exhibit a similar nematic phase in between the isotropic and solid state without the HGO ellipsoids adding more complex phases. Thus, the two types of ellipsoids are qualitatively equivalent and their small differences in particle-shape are of only marginal consequences. However, the double gyroid phase is a much more complex structure than the “simple” nematic.

It seems plausible that higher complexity leads to an increased response and that especially the self-assembly of configurations like the double gyroid is more sensitive to the interaction of the particles. Hence, we focus in part 1 of this series on the phase behaviour of a more accurate, but computationally much more expensive Bézier pear model. In that case the hard potential is based on triangulated meshes of the pear-surface, which we address as the hard pears of revolution (HPR) model. Here, the contact is determined by testing for overlap between the triangulated surfaces and, hence, coincides with the Bézier description arbitrarily accurately.

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