Nanoparticle Superlattices as Quasi-Frank-Kasper Phases

I show that all phases reported experimentally in binary nanoparticle superlattices can be described as networks of disclinations in an ideal lattice of regular tetrahedra. A set of simple rules is provided to identify the different disclination types from the Voronoi construction, and it is shown that those disclinations completely screen the positive curvature of the ideal tetrahedral lattice. In this way, this study provides a generalization of the well-known Frank-Kasper phases to binary systems consisting of two types of particles, and with a more general type of disclinations, i.e., quasi-Frank-Kasper phases. The study comprises all strategies in nanoparticle self-assembly, whether driven by DNA or hydrocarbon ligands, and establishes the universal tendency of superlattices to develop icosahedral order, which is facilitated by the asymmetry of the particles. Besides its interest in predicting nanoparticle self-assembly, I discuss the implications for models of the glass transition, micelles of diblock polymers, and dendritic molecules, among many others.

Figure 1

(a1) Example of a $(2\pi )/3$ disclination transforming a tetrahedra into an octahedra. (a2) The fcc lattice consists of edges (in green) sharing two tetrahedra (blue) and octahedra (red). (b) Some constitutive elements (surface around a given NC) of quasi-FK phases (full list provided in Supplemental Material [36]). (c) Disclinations considered in this study, see also Supplemental Material [36].

Figure 2

Twelve lattices and their respective disclination lines, drawn with the convention of Fig. 1. The radical Voronoi tesselation as implemented in voro++ [39] is used.

Figure 3

Plots of $L(\gamma )$ and $q_l(\gamma )$, see Eq. (7) (normalized to $L({\gamma }_c)$ and $q_l({\gamma }_c)$, where ${\gamma }_c$ is the maximum of the packing fraction, see Ref. [16]). $L(\gamma )$ is strongly dependent on $\gamma$, but $q_l$ is basically an invariant. The value of $\psi$ (${\mathrm{NaZn}}_{13}$) is provided in [18].

Figure 4

Reported experimental phases over the range of $\gamma$ for hydrocarbon [18] and DNA [10] NCs. The continuous curve is a guide to the eye illustrating the general trend. SC stands for single component, see discussion in text.