Numerical model of solid phase transformations governed by nucleation and growth: Microstructure development during isothermal crystallization

A simple numerical model which calculates the kinetics of crystallization involving randomly distributed nucleation and isotropic growth is presented. The model can be applied to different thermal histories and no restrictions are imposed on the time and the temperature dependences of the nucleation and growth rates. We also develop an algorithm which evaluates the corresponding emerging grain-size distribution. The algorithm is easy to implement and particularly flexible, making it possible to simulate several experimental conditions. Its simplicity and minimal computer requirements allow high accuracy for two- and three-dimensional growth simulations. The algorithm is applied to explore the grain morphology development during isothermal treatments for several nucleation regimes. In particular, thermal nucleation, preexisting nuclei, and the combination of both nucleation mechanisms are analyzed. For the first two cases, the universal grain-size distribution is obtained. The high accuracy of the model is stated from its comparison to analytical predictions. Finally, the validity of the Kolmogorov-Johnson-Mehl-Avrami model [J. Chem. Phys. 7, 1103 (1939); 8, 212 (1940); 9, 177 (1941); Trans. Am. Inst. Min., Metall. Pet. Eng. 135, 416 (1939); Izv. Akad. Nauk SSSR, Ser. Fiz. 1, 355 (1937)] is verified for all the cases studied.

Figure 1

3D isothermal crystallization of amorphous silicon at $700°\mathrm{C}$ after continuous heating at $20\mathrm{K}∕\min$: our numerical method (crosses), the rough approximation of Eq. (17) (dotted line), and the quasiexact solution of Eq. (A2) (solid line). The difference between Eq. (17) and our numerical solution is shown by the dashed line (multiplied by 10). $\Delta r=5\times {10}^{-5}\mu \mathrm{m}$, $V=8\times {10}^3\mu {\mathrm{m}}^3$, and the total number of nuclei is 182.231.

Figure 2

Schematic representation of the algorithm used to calculate the microstructure.

Figure 3

Transformed fraction versus time for 3D growth of amorphous silicon at constant temperature, $T=680°\mathrm{C}$, and continuous nucleation. The solid line represents the exact solution, Eq. (14). Crosses correspond to the transformed fraction calculated from the microstructure. The time is normalized according to Eq. (20). Simulation parameters: $\Delta r=5\times {10}^{-5}\mu \mathrm{m}$, $V=12167\mu {\mathrm{m}}^3$, the number of cells is ${760}^3$, the total number of nuclei is 188 144 (89 745 phantom), and $\Delta x=0.03\mu \mathrm{m}$ $(⟨\tilde{r}⟩=0.3089\mu \mathrm{m})$.

Figure 4

Grain radius distribution for 3D growth of amorphous silicon at three different transformed fractions at constant temperature, $T=680°\mathrm{C}$, and continuous nucleation. Simulation parameters are given in the caption of Fig. 3. With the normalized radius [Eq. (20)], these distributions are universal.

Figure 5

(Color online) Simulated microstructure cross section corresponding to the crystallization of amorphous silicon at three different transformed fractions at constant temperature, $T=680°\mathrm{C}$, and continuous nucleation. Simulation parameters are given in the caption of Fig. 3. With proper time and space normalization, these grain morphologies are universal.

Figure 6

Final grain radius distribution for 2D growth of amorphous silicon at constant temperature, $T=680°\mathrm{C}$, and continuous nucleation. Simulation parameters: $\Delta r={10}^{-5}\mu \mathrm{m}$, $A=40000\mu {\mathrm{m}}^2$, the number of cells is $20{000}^2$, the total number of nuclei is 579 460 (330 840 phantom), and $\Delta x=0.010\mu \mathrm{m}$ $(⟨\tilde{r}⟩=0.2263\mu \mathrm{m})$. With the normalized radius [Eq. (20)], this distribution is universal.

Figure 7

Final grain radius distribution for 3D crystallization of amorphous silicon due to preexisting nuclei $(n_0=8.69\mu {\mathrm{m}}^{–3})$ at $T=680°\mathrm{C}$. The solid line is the corresponding Gaussian fit. With the normalized radius, the distribution is universal. Simulation parameters: $\Delta r=5\times {10}^{-5}\mu \mathrm{m}$, $V=10648\mu {\mathrm{m}}^3$, the number of cells is ${760}^3$, and $\Delta x=0.030\mu \mathrm{m}$ $(⟨\tilde{r}⟩=0.3017\mu \mathrm{m})$.

Figure 8

Final grain radius distribution for 2D crystallization of amorphous silicon due to preexisting nuclei $(n_0=64\mu {\mathrm{m}}^{–2})$ at $T=680°\mathrm{C}$. Simulation parameters: $\Delta r=5\times {10}^{-6}\mathrm{\mu }\mathrm{m}$, $A=7225\mu {\mathrm{m}}^2$, the number of cells is ${2000}^2$, and $\Delta x=0.0042\mu \mathrm{m}$ $(⟨\tilde{r}⟩=0.0705\mu \mathrm{m})$.

Figure 9

Final average grain size at constant temperature, $T=680°\mathrm{C}$, for continuous nucleation combined with preexisting nuclei. Squares are calculated from the grain-size distribution, while the solid line corresponds to the numerical solution of the theoretical prediction [Eqs. (C5, C15)]. The dashed line is the theoretical solution when crystallization is due to preexisting nuclei only [Eq. (C6)] and the dotted line is the relative standard deviation. The curves are universal.

Figure 10

Final grain radius distribution at constant temperature, $T=680°\mathrm{C}$, for continuous nucleation combined with preexisting nuclei. Gray bars, $n_0^{\prime }=0.558$; black bars, $n_0^{\prime }=3.348$.

Figure 11

Schematic representation of two scalable microstructures. New nuclei are depicted with crosses.