From Close-Packed to Topologically Close-Packed: Formation of Laves Phases in Moderately Polydisperse Hard-Sphere Mixtures

Appendix

All simulations employed a finite $N$ approximation to the various sphere diameter ($d$) distributions, where $N$ is the number of particles in the simulation. Practically, each probability distribution $P(d)$ was divided into $N$ contiguous, equi-probability chunks with the midpoints taken as the particle diameters.

With the above finite population approximation scheme, we performed standard NVT Monte Carlo simulations with the addition of particle swap moves. Translation and swap moves were selected at random with an $80:20$ weighting. Translation involved randomly selecting a particle and attempting a random translation with a maximum displacement chosen to achieve an average acceptance rate between $17-25\%$. In the case of a swap move, two particles were chosen at random, and their diameters were exchanged if no overlaps were generated by the swap. For the power law and flat top distributions, $N=1000$ was employed, while $5000$ particles were used for the Gaussian distribution to provide adequate sampling of the tails of the distribution. A simulation with $N=100$ was used for the pressure calculation in Fig. 1a of the main text–useful primarily to identify the approximate density at which freezing takes place ($\eta=0.58-0.6$).

Initial configurations were generated via a random sequential addition protocol at $\eta=0.25$ and quickly compressed to $\eta=0.595$. Simulations were run until the sample crystallized, after which swap moves were disabled and the system was quickly compressed to $\eta=0.66$ for the flat top and power law distributions and $\eta=0.65$ for the Gaussian, at which point the latter mixture jammed and could not be compressed further. The final compression was utilized to help lock the particles into their idealized lattice positions for identification purposes. Only very small scale particle movements occurred during this step.

.0.2 Description of Template-Matching Procedure

For the template-matching procedure referenced in the main text, we first visually analyze a configuration to find an orientation that displays the 10-fold coordination characteristic of Laves phases such as those shown in Fig. 1b,c of the main text. We rotate the configuration so that a large ordered domain is oriented properly with respect to the templates shown in Fig. 1d,e. For each particle in the simulation box, we assign it to an ideal lattice position in the template and then search for neighboring particles in the simulation that are closest to the other ideal lattice positions prescribed by the template, all while using periodic boundary conditions consistent with the original simulation box. The C14 template has 41 particles; the C14 unit cell only has 12 particles, but we have orthogonalized the unit cell (which doubles the number of particles) and replicated particles that fall on the edges and corners as dictated by the periodicity of the lattice. The C15 template has 34 particles after replicating the particles on the corners and edges of the 24 particle unit cell. The lattice constants for the template are determined by averaging over several approximate realizations of the template in the simulation box.

We calculate the root-mean-squared deviation (RMSD) of the local environment of the selected particle in the simulation box with respect to the template for every lattice position in the template, using the minimum image convention to place the selected particle as close to the center of the template as possible. The reported RMSD value for a selected particle is given by the lowest value of the RMSD over all of the template sites, and its lattice site assignment corresponds to the template site that has the lowest RMSD as well. Therefore, the assignments based on the template-matching scheme are only interdependent insofar as they are all calculated from the same structure; there is no check for consistency of the lattice type assignment among groups of particles. The largely correct arrangements of particle types observed in a large domain of Fig. 3a with respect to an ideal C14 lattice is a consequence of the reasonable accuracy of the template-matching scheme, and, as described in the main text, deviations from the ideal arrangement can indicate mixing of other Laves phases (C15 in the case of Fig. 3a). They could also indicate a local defect in the crystalline region.

Because the template-matching is sensitive to the orientation of the simulation box, it is not guaranteed to be an exhaustive characterization of all of the particles in the system. If the large majority of the particles form a single ordered domain (as seems to be the case in Fig. 3a), then the scheme should provide a reasonable description of all of the particles. However, if multiple domains exist that are not oriented consistently with one another, it is possible that the template-matching scheme will only accurately describe some of the domains. Therefore, we intend that the template-matching scheme is a tool to characterize the composition of large ordered domains that can be visualized within a simulation box, and not as a fully automated and exhaustive means to search for Laves phases in any arbitrary system.

.0.3 Analysis of Template-Matching for Additional Structure

Here, we examine the structure shown in Fig. 1c of the main text. Visually, it seems that the ordered domain in Fig. 1c is largely composed of C15 motifs. Therefore, we performed the analysis below with the C15 template instead of the C14 template. In Fig. A1a,b, see the configuration color-coded such that darker regions indicate C15 crystalline regions (low RMSD values) and lighter regions appear disordered (high RMSD values). Fig. A1c gives the distributions of RMSD values for the particles, and, as in the main text, a cut-off of 0.25 was used as the upper bound for the Laves crystalline region.

The configuration, color-coded by particle type, is shown in Fig. A1d,e. Visually it is possible to discern that the red (C site) particles tend to be larger than the blue (A site) particles. This trend is born out in the particle diameter distributions shown in Fig. A1f for the A, C, and total particle size distributions. Identically with the results in the main text, $d_{\text{L}}/d_{\text{S}}=1.24$.

.0.4 Visualizations of Structures Corresponding to Other Size Distributions

In Fig. 4b-d of the main text, we reported the particle size distributions for the A (or A+B) and C sites for a sixth-order power law, a uniform, and a Gaussian distribution, respectively. In Figs. A2- A4, we show the corresponding structures, color-coded by both RMSD and particle type, as well as the probability distribution of the former. For the sixth-order power law, the C14 template was used and for the uniform and Gaussian distributions, the C15 template was used. The respective values for $d_{\text{L}}/d_{\text{S}}$ were: 1.23, 1.22, and 1.23.

As mentioned above, we found that we needed to use a larger simulation box ($N=5000$) for the Gaussian distribution. Since the particles that comprise the Laves phases are generally selected from the ends of the distributions, the Gaussian distribution furnished fewer particles of the appropriate size ratio to form Laves phase. For simulations using $N=1000$, some degree of ordering was apparent, but larger Laves phase domain were not visible. In the $N=5000$ simulations, some Laves phase domains self-assembled as can be seen in Fig. A4, possibly in co-existence with other crystalline motifs–an interesting avenue for future work.

.0.5 Evaluation of Finite-Size Effects

To test for the importance of finite-size effects, we performed an identical analysis as that shown in Figs. 2–3 of the main text after increasing $N$ from 1000 to 10000. Laves phase formation occurred in both simulations, and the size distributions were highly similar. We observed quantitative agreement of $d_{\text{L}}/d_{\text{S}}\approx 1.24$ in both cases.

References