Crystallization induced by multiple seeds: Dynamical density functional approach

Using microscopic dynamical density functional theory, we calculate the dynamical formation of polycrystals by following the crystal growth around multiple crystalline seeds imposed to an undercooled fluid. Depending on the undercooling and the size ratio as well as the relative crystal orientation of two neighboring seeds, three possibilities of the final state emerge, namely no crystallization at all, formation of a monocrystal, or two crystallites separated by a curved grain boundary. Our results, which are obtained for two-dimensional hard disk systems using a fundamental-measure density functional, shed new light on the particle-resolved structure and growth of polycrystalline material in general.

Figure 1

(a) Schematic view of the initial nuclei of hard spheres with diameter $\sigma$ separated by a distance $d$ and surrounded by a homogeneous fluid, at area fraction $\eta$. Here the left nucleus consists of $N_1=61$ particles, whereas the right nucleus is built of $N_2=37$ particles and additionally rotated by an angle $\varphi$. (b)–(f) The Voronoi construction of an exemplary growth at times $t=0,0.24,1.35,1.9,2.5{\tau }_{\mathrm{B}}$, where blue (medium gray) cells have six neighboring particles and defects with more or less neighboring particles are coded in different colors (light or dark gray) all surrounded by a fluid. A grain boundary grows, first on a straight and finally on a curved line between the two crystalline regions. For a movie of the growth see Supplemental Material [41].

Figure 2

Long-time situations of the interaction of two nuclei at distance $d=10\sigma$ rotated with respect to each other by an angle $\varphi$ at area fraction $\eta$. The resulting cases are (a) a fluid, (b) a monocrystal, and (c) a crystal with a grain boundary where only cutouts of the full quadratic simulation box are shown. (d)–(f) The phase diagrams for different nuclei sizes where blue triangles indicate a fluid, red (gray) diamonds denote a monocrystal, and dark green (dark gray) diamonds label the crystalline region with a stable grain boundary. The nuclei consist of (d) $N_1=N_2=19$, (e) $N_1=61$, $N_2=37$, and (f) $N_1=37$, $N_2=19$ particles. For movies of the situations (a)–(c) as well as a movie for $N_1=N_2=19$, $\varphi =5^{\circ }$, and $\eta =0.754$, see Supplemental Material [41].

Figure 3

Analysis of the grain boundaries between two grown crystalline regions after a time $t=4{\tau }_{\mathrm{B}}$, where the initial right nucleus is rotated by $\varphi ={20}^{\circ }$ for different nuclei sizes: (a) $N_1=61$, $N_2=37$, (b) $N_1=37$, $N_2=19$, and (c) $N_1=61$, $N_2=19$. The area fractions are chosen such that grain boundaries occur. The shown Voronoi cells are colored in dark blue (medium gray) for six neighbors in contrast to defects depicted with other colors (light or dark gray). Black circles indicate the spherically modeled initial nuclei and the green (light gray) line displays the analytically calculated interface. (d) The extracted radii of curvature for cases (a)–(c) obtained at different distances $d$ between the initial nuclei. For movies of the growth, see Supplemental Material [41].

Figure 4

Time series of the crystal growth dynamics originated from three nuclei of 19 particles. Shown is a Voronoi constructon of (a) the initial, (b) an intermediate ($t=0.3{\tau }_{\mathrm{B}}$), and (c) the configuration at $t=4{\tau }_{\mathrm{B}}$. The outer nuclei are positioned symmetrically around the center nucleus at distances $d=10\sigma$, while the center nucleus is displaced vertically by $1\sigma$. In addition, the latter is rotated by $\varphi =5^{\circ }$.

Figure 5

Similar to Fig. 4 but for a rotation angle $\varphi ={20}^{\circ }$.

Figure 6

Similar to Fig. 4 but with distance $d=15\sigma$ between the center and the outer nuclei.