Interface control and snow crystal growth

The growth of snow crystals is dependent on the temperature and saturation of the environment. In the case of dendrites, Reiter's local two-dimensional model provides a realistic approach to the study of dendrite growth. In this paper we obtain a new geometric rule that incorporates interface control, a basic mechanism of crystallization that is not taken into account in the original Reiter model. By defining two new variables, growth latency and growth direction, our improved model gives a realistic model not only for dendrite but also for plate forms.

Figure 1

Examples of real plate and dendrite snowflakes [2]. (a) Stellar dendrite, (b) stellar plate, (c) sectored plate.

Figure 2

Classification of cells.

Figure 3

Images generated by Reiter's model [10] for $\alpha =1$ and parameters (a) $\beta =0.3,\gamma =0.0001$; (b) $\beta =0.35,\gamma =0.001$; (c) $\beta =0.6,\gamma =0.01$; (d) $\beta =0.9,\gamma =0.05$.

Figure 4

(a) Coordinate system of hexagonal cells. (b) Definition of growth directions in the coordinate system.

Figure 5

Comparison of vapor accumulation simulations for the parameters $\alpha =1,\beta =0.4,\gamma =0.001$.

Figure 6

Comparison of growth latency simulations for the parameters $\alpha =1,\beta =0.4,\gamma =0.001$.

Figure 7

$T(0,j)-B(0,j)$ of cells $(0,j)$ along a main branch for $j=1,2,...,$ in the two-dimensional scenario. Here cell $(0,200)$ is an edge cell, and $\alpha =1, \beta =0.4, \gamma =0.001$.

Figure 8

Plots of the tips (thin curves) and envelope curves (thick curves) of the side branches from the $j$-axis main branch using the parameters of the four example images in Fig. 3. Due to symmetry, we focus on one set of side branches that grow from the right side of lattice $j$ axis. Here the black curve represents Fig. 3, blue for Fig. 3, red for Fig. 3, and magenta for Fig. 3. The axes are the horizontal and vertical axes.

Figure 9

Trace of relative orientations of source cells with respective destination cells. A destination cell becomes boundary because a source cell, which is one of the neighbors of the destination cell, becomes frozen. Legend is as follows: magenta $\blacksquare :+{30}^{\circ }$, black $☆:-{30}^{\circ }$, green + :$+{90}^{\circ }$, blue $\circ :-{90}^{\circ }$, red $·:+{150}^{\circ }$, cyan $\times :-{150}^{\circ },$ where the parameters are $\alpha =1,\beta =0.35,\gamma =0.001$. Note that not all straight paths are labeled. The axes are as in Fig. 4.

Figure 10

An arrow linking two cells indicates the source-destination relationship.

Figure 11

Two competing forces of diffusion control and interface control that determine snowflake growth, an illustration of the Mullins-Sekerka instability.

Figure 12

Surface free energy of snowflake as a function of direction and equilibrium crystal shape of snowflake derived from surface free energy plot with Wulff construction [12].

Figure 13

Equilibrium crystal shape used in the new geometric rule.

Figure 14

Snowflake images generated by the enhanced Reiter's model with the new geometric rule, where the variables are (a) $\varepsilon =0.1$; (b) $\varepsilon =0.01,\alpha =1,\beta =0.4,\gamma =0.001$.