Abstract
We develop a mathematical model for the energy landscape of polyhedral supramolecular cages recently synthesized by self-assembly (Sun et al. in Science 328:1144–1147, 2010). Our model includes two essential features of the experiment: (1) geometry of the organic ligands and metallic ions; and (2) combinatorics. The molecular geometry is used to introduce an energy that favors square-planar vertices (modeling \(\mathrm {Pd}^{2+}\) ions) and bent edges with one of two preferred opening angles (modeling boomerang-shaped ligands of two types). The combinatorics of the model involve two-colorings of edges of polyhedra with four-valent vertices. The set of such two-colorings, quotiented by the octahedral symmetry group, has a natural graph structure and is called the combinatorial configuration space. The energy landscape of our model is the energy of each state in the combinatorial configuration space. The challenge in the computation of the energy landscape is a combinatorial explosion in the number of two-colorings of edges. We describe sampling methods based on the symmetries of the configurations and connectivity of the configuration graph. When the two preferred opening angles encompass the geometrically ideal angle, the energy landscape exhibits a very low-energy minimum for the most symmetric configuration at equal mixing of the two angles, even when the average opening angle does not match the ideal angle.
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Notes
The tolerance is the width of the Gaussian distribution of random test displacements applied to the vertices in each dimension; we use successively smaller tolerances, and end the annealing process when the evolution of the total energy at the lowest tolerance has a sufficiently small average slope, the threshold slope set by trial and error to balance accuracy with efficiency.
We could also use the symmetry between ‘red’ and ‘blue’ by ending the process when we have enumerated all configurations with \(m=n=12\), and finding the graph for \(m>n\) by swapping the colors of every edge; we choose instead to continue the process to \(m=2n=24\) in order to corroborate the correctness of our algorithm by comparing the results for m and \(2n-m\). The time necessary is not prohibitive; the full calculation took about 12.5 h on a 2012 MacBook Pro laptop.
In Fig. 6a, we calculate the energy for configurations with \(s=3\) only for \(m\le 8\) and \(m\ge 40\), and for a larger proportion of configurations with \(s=2\) for \(m\le 8\), \(m\ge 40\), \(m=11,13,17,31,35,37\).
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Communicated by Robert V. Kohn.
ERR acknowledges the support and hospitality of the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University, where she was a postdoctoral fellow while carrying out this work. GM acknowledges partial support from NSF grant DMS 14-11278. This research was conducted using computational resources at the Center for Computation and Visualization, Brown University.
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Russell, E.R., Menon, G. Energy Landscapes for the Self-Assembly of Supramolecular Polyhedra. J Nonlinear Sci 26, 663–681 (2016). https://doi.org/10.1007/s00332-016-9286-9
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DOI: https://doi.org/10.1007/s00332-016-9286-9