Predicting the crystal structure of a self-assembling system of particles is an outstanding challenge in physics, chemistry, and material science. Because the microscopic structure of a system is often inextricably linked to its resulting macroscopic properties, engineering material properties amounts to controlling the system's propensity to self-assemble into a desired target structure. As a first approach, one can imagine trying to enumerate all possible arrangements of atoms or particles on a regular grid. Simply evaluating the energy of each arrangement and searching for the lowest candidate is one way to make such a prediction. However, the number of possibilities quickly undergoes what is known as a “combinatorial explosion” surpassing Avogadro's number for even relatively coarse grids. In fact, searching one million structures per second would still require more than the age of the universe to complete the search. As a result, stochastic or approximate methods are generally used instead, or searches may be are performed over databases of millions of known structures. Unfortunately, with new systems, novel crystals may emerge that do not belong to such databases, or cannot be found by chance. This is problem is further compounded when one wants to understand how systems with many different components crystallize.

Figure 1: All symmetry groups in two dimensions with their fundamental domains outlined in black.

In this work, we devised a new method to find thermodynamically stable structures that assemble from multicomponent mixtures of colloids, or microscopic particles. Colloidal science has provided many routes to tuning the interactions between such colloids over the past several decades, including surface functionalization, shape, and charge. This leads to the question: “what will an arbitrary mixture of such colloids form?” Here, we considered two dimensional films of colloidal particles. In both two and three dimensions there are a finite number of different symmetries a crystal can have. Symmetry groups contain all the information describing the mathematical operations that have to be applied to the smallest piece of the puzzle, or fundamental domain, to create a crystal of that symmetry. Depending on the lattice, this imposes certain constraints on what can be placed at the edges of the fundamental domain. By creating a lattice on the fundamental domain such that its points intersect the edges, these constraints manifest in a mathematical way that allows the crystal structure prediction to be recast as a constraint satisfaction problem. This is the general type of problem that crossword or sudoku puzzles fall into. By finding all solutions to the “colloidal sudoku puzzle” we find all possible crystal structures.

Our approach greatly reduces the number of candidate structures by many orders of magnitude, making it possible to enumerate them in a matter of minutes or hours instead of ages-of-the-universe. These can then be feasibly screened to construct phase diagrams which predict how an arbitrary mixture of many components will combine to form different structures as a function of solution composition.