Critical nucleus size for crystallization of supercooled liquids in two dimensions

Using molecular dynamics simulations, we study the crystallization of supercooled liquids in two dimensions in which particles interact with other particles via the Lennard-Jones-Gauss potential. We first prepare supercooled liquids at various temperatures by rapid quenching from the melt. The simulations are performed with a crystalline seed inserted at the center of the initial system. We investigate the time evolution of the inserted nucleus and its surroundings and determine the critical nucleus size $n_c$ defined as the smallest nucleus which survives. The results show that $n_c$ scales as $\sim {(T_m-T)}^{-2}$ with the melting temperature $T_m$, as expected in the classical nucleation theory. We also obtain the crystallization time at various temperatures as a function of nucleus size and show that the presence of a crystalline seed significantly affects the crystallization time when the temperature is higher than the characteristic temperature $T^{*}$ at which the crystallization time becomes the shortest. This indicates that the crystallization is controlled by thermodynamics in this temperature range. When the temperature is lower than $T^{*}$, the effect of the inserted nucleus on crystallization is less significant, which indicates that crystallization is controlled by emergence and merging of small crystalline nuclei.

Figure 1

The LJG potential for the parameters $r_G=1.47r_0, \epsilon =1.5{\epsilon }_0, {\sigma }^2=0.02r_0^2$.

Figure 2

The initial configuration, before quenching, of a liquid sample with a crystal nucleus at the center. The local structure can be classified by a few basic tiles which are drawn by lines between the first neighbor particles. Pentagon, triangle, and square tiles are shown in blue, orange, and purple, respectively.

Figure 3

(a) Three characteristic tiling structures: PPPT, PPP, and PPTS, adapted from Ref. [15]. (b) The time evolution of the number of characteristic tiling structures at $T=0.34$. The curves (PPPT), (PPP), and (PPTS) correspond to the structures shown in (a). The curve (all) represents the sum of the numbers of those three tiling structures.

Figure 4

The time evolution of the number of crystal tilings $n_x$ for $T=0.34$, 0.30, and 0.26 in three regions: (i) $r\le R$ (purple solid line), (ii) $R<r<R+5$ (green dashed line), and (iii) $R+5\le r\le R+10$ (blue dotted line), where $R$ is the radius of the initially inserted nucleus. The abscissa represents time in reduced units. The sizes of the inserted nuclei are (a) $R=6$ and (b) 12. $n_x$ is normalized by the initial density of crystal tilings in the seed ${\rho }_0$ by multiplying the area of the region of interest $A_j(j=1,2,3)$.

Figure 5

The average fraction $f\left(R\right)$ of crystal tilings in region (i) $r\le R$, where $R$ is the radius of the initially inserted nucleus. The calculations were performed over the period from $t=9000$ to 10 000. Solid lines are fitting functions of $f\left(x\right)=a{(x-b)}^c$, where $a, b$, and $c$ are fitting parameters.

Figure 6

Temperature dependence of the critical nucleus size. The solid line is a fitting function of Eq. (4).

Figure 7

The time-temperature-transformation diagram for various sizes of crystal nuclei. The crystallization time $t^{\prime }$ is defined by Eq. (5). $T^{*}$ is the temperature at which the crystallization time becomes the shortest without an inserted nucleus. The melting temperature of bulk system is 0.44.

Figure 8

Crystallization process at $T=0.26$. The local structures are classified by a few basic tiles: blue is pentagon, orange is triangle, and purple is square. The circles are drawn for eye guide.