Supporting information for: Heterogeneous Crystallization on Pairs of. Pre-Structured Seeds

Transcription

1 Supporting information for: Heterogeneous Crystallization on Pairs of Pre-Structured Seeds Swetlana Jungblut and Christoph Dellago Faculty of Physics, University of Vienna, Boltzmanngasse, 9 Wien, Austria swetlana.jungblut@univie.ac.at Phone: +43 () Fax: +43 () To whom correspondence should be addressed S

2 Initial State From the straightforward MD simulations in the initial state, used to compute the flux out of the initial state, we extracted the distribution of the distances between the seeds and the corresponding average distances for a giveize of the crystalline and for the initial state as a whole. The results are presented in Figure S. Evidently, the distribution of the distances is relatively broad but the average at each size does not change as long as the systems stays in the initial state. As the crystalline s grow and the system leaves the initial state, the average inter-seed separatiohifts to smaller values for cubic seeds and to larger values for ahedral seeds. The analysis of the transition paths also confirms these shifts. The density of the reactive flux out of the initial state in the presence of one and two seeds are shown in Table S. As referred to in the main text, the initial flux densities are determined by the interactions between the seeds in the smaller system in the presence of two seeds and by the competition of homogeneous and heterogeneous crystal nucleation in larger systems in the presence of single seeds. Essentially, the presence of ahedral seeds does not modify the flux density, which is similar to the homogeneous flux density and independent on the simulation box volume. Transition Paths Cluster-to-Seed Distance In Figures S2, S3, and S4, we collected the data for the distances between the seeds and the crystalline. Figures S2 and S3 present, for a giveize of the crystalline, the fraction of configurations with a certain distance between the particles of the crystalline and of the seed closest to this. Figure S4 includes data on the distribution of distances to the second mobile seed, which is located farther away from the. S2

3 Table S: Average Flux Density f first /V in the Presence of One and Two Freely Moving Seeds with Different Structures in Systems of Different Sizes. The Last Column Represents the Ratio of these Flux Densities. f one seed two seeds first /V ffirst /V #2/#. L26 (4.86 ±.7) 6 (9.37 ± 9) 6.93 L22 (8.2 ±.34) 6 (.7 ±.7) 2.9 L8 (.44 ±.6) (4.9 ±.3) L26 (9.29 ±.34) 6 (.8 ±.6).94 L22 (.44 ±.) (3.4 ±.8) 2. L8 (2.7 ±.) (6.67 ±.32) L26 (.64 ±.) 6 (2.83 ±.4) 6.73 L22 (2.2 ±.6) 6 (.6 ±.3) L8 (4.7 ±.3) 6 (9. ± ) L26 (6.29 ± 6) 6 (.7 ±.4).86 L22 (9.43 ±.43) 6 (.99 ±.6) 2. L8 (.82 ±.9) (4. ±.3) L26 (6.9 ±.6) 6 (6.72 ±.3) 6.97 L22 (6.4 ±.6) 6 (7. ± 2) 6. L8 (6.68 ± 4) 6 (6.69 ± 4) 6..9 L26 (6.93 ±.3) 6 (6.7 ±.4) 6.97 L22 (6.79 ±.7) 6 (6.6 ±.8) 6.97 L8 (6.29 ± 3) 6 (6.87 ± 8) 6.9 no seed L26 (6.7 ±.3) L22 (7.2 ± 2) L8 (6.62 ± 6) As indicated in the main text, crystalline s nucleate on one of the seeds with and structure while the second seed is included into the only in the smaller system. The average distance between the crystalline and ahedral seeds is non-zero for all sizes considered, although becomes smaller as the crystals grow. The second ahedral seed may be included into the larger crystalline s but the probability to find such configurations in the course of the transition is very low. In contrast, the inclusion of the second seed with cubic structure into the crystalline is more likely, particularly S3

4 in the smaller systems and at late stages of the transformation. Cluster Structure In Figures S, S6, S7, and S8, we present the structural composition of the crystallizing s. For every particle of the crystal, we identify the type of the crystalline environment around this particle by considering a combination of the w 4 and w 6 invariants of the Steinhardt bond order parameters as described in Ref. S To optimize the recognition of the crystalline environments, we include the second shell of neighbors into the analysis. S2 We differentiate between particles in ahedral, face-centered cubic (), hexagonally close-packed (), body-centered cubic (), and x- environments. The latter (x-) corresponds to a distorted phase, which was recently discovered in a LJ system. S3 Furthermore, we separately look at the particles in the of the, which are identified as those without fluid neighbors. The structures of the crystals formed in the course of the transition are similar to those obtained by the crystallization in the presence of single seeds. S4 Due to the optimized set of thresholds S used in the ten Wolde, Ruiz-Montero and Frenkel S6 scheme to recognize crystalline s, the fraction of particles in the of the crystalline is slightly reduced. Still, we draw the same conclusions as for the crystallization transition in the presence of single structured seeds: The presence of and seeds initially increases the corresponding fraction of these particles in the crystallizing and the effect is more pronounced if the seeds have regular structures. The evolution of structural composition with size for regular ahedral seeds is indistinguishable from the homogeneous system. For squeezed ahedral seeds, one recognizes the peaks in the and fractions appearing at small sizes, which we previously identified as an indication of the S4

5 ahedrally structured crystals formed on these seeds. S7 References (S) Jungblut, S.; Dellago, C. Crystallization of a binary Lennard-Jones mixture. J. Chem. Phys. 2, 34, 4. (S2) Lechner, W.; Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters. J. Chem. Phys. 28, 29, 477. (S3) Eshet, H.; Bruneval, F.; Parrinello, M. New Lennard-Jones metastable phase. J. Chem. Phys. 28, 29, 26. (S4) Jungblut, S.; Dellago, C. Crystallization on prestructured s. Phys. Rev. E 23, 87, 23. (S) Jungblut, S.; Singraber, A.; Dellago, C. Optimising reaction coordinates for crystallisation by tuning the crystallinity definition. Mol. Phys. 23,, (S6) ten Wolde, P. R.; Ruiz-Montero, M. J.; Frenkel, D. Numerical calculation of the rate of crystal nucleation in a Lennard-Jones system at moderate undercooling. J. Chem. Phys. 996, 4, (S7) Jungblut, S.; Dellago, C. Heterogeneous crystallization on tiny s. EPL 2, 96, 66. S

6 probability distribution D s-s2 D s-s2 D s-s2 D s-s2 D s-s2 D s-s2 L L22 L Figure S: Distribution of the distances between the central particles of the seeds (in the initial state) color-coded according to their relative occurrence probability for a given size. Red solid lines indicate the average values in the respective system for a given size, while the black dotted lines denote the average distance between the seeds in the initial state. Vertical dotted lines in the background indicate the positions of the borders of the initial states (n = n = 2 and n no seed = n = ). Here, the sizes were sampled iteps of and the distances iteps of.. S6

7 probability distribution D cl-s() D cl-s() D cl-s() D cl-s() D cl-s() D cl-s() L26.4 L22 L Figure S2: Distribution of the distances between the particles of the seed closest to the crystalline and of this color-coded according to their relative occurrence probability for a given size. If at least one of the seed particles is part of the crystalline this distance is set to zero. In order to improve the statistics, the sizes were sampled iteps of 2 and distances iteps of.. S7

8 probability distribution.4 D3 D D9 D cl-s().9 D cl-s().9 D cl-s().9 Figure S3: Same as Figure S2 but for seeds with fixed relative distances. S8

9 probability distribution.4 L26 L22 L Figure S4: Same as Figure S2 but for the second seed. S9

10 ..9.4 x-.4 x- Figure S: Fractions of crystalline structures in the largest and its in the presence of squeezed. (left) and regular.9 (right) seeds in L26 (dark red and faded blue), L22 (red and blue), and L8 (orange and dark blue) systems. In order to improve the statistics, the sizes were sampled iteps of x-.4 x- Figure S6: Same as Figure S but in the presence of squeezed. (left) and regular.9 (right) seeds x-.4 x- Figure S7: Same as Figure S but in the presence of squeezed. (left) and regular.9 (right) seeds. S

11 x-.4 x-.9 no seed.4 x-.4 x- Figure S8: Same as Figure S but for homogeneous systems (bottom right) in L26 (dark red and faded blue), L22 (red and blue), and L8 (orange and dark blue) simulation boxes and in the presence of regular.9 (top left),.9 (top right) and.9 (bottom left) seeds with relative distances fixed at D3 (dark red and faded blue), D (red and blue), and D9 (orange and dark blue). S