Modified phase-field-crystal model for solid-liquid phase transitions

A modified phase-field-crystal (PFC) model is proposed to describe solid-liquid phase transitions by reconstructing the correlation function. The effects of fitting parameters of our modified PFC model on the bcc-liquid phase diagram, numerical stability, and solid-liquid interface properties during planar interface growth are examined carefully. The results indicate that the increase of the correlation function peak width at $k=k_m$ will enhance the stability of the ordered phase, while the increase of peak height at $k=0$ will narrow the two-phase coexistence region. The third-order term in the free-energy function and the short wave-length of the correlation function have significant influences on the numerical stability of the PFC model. During planar interface growth, the increase of peak width at $k=k_m$ will decrease the interface width and the velocity coefficient $C$, but increase the anisotropy of $C$ and the interface free energy. Finally, the feasibility of the modified phase-field-crystal model is demonstrated with a numerical example of three-dimensional dendritic growth of a body-centered-cubic structure.

Figure 1

Calculated phase diagrams for liquid-bcc system in one-mode approximation with two different $\varepsilon :\varepsilon =0$ and 0.3.

Figure 2

One-mode correlation functions ${\widehat{C}}_2\left(k\right)$ for bcc lattices with different peak widths: ${\alpha }_1=0.5$ (solid), ${\alpha }_1=1.0$ (dashed), ${\alpha }_1=1.5$ (dotted), ${\alpha }_1=2.0$ (dash-dotted), and ${\alpha }_1=2.5$ (dash-dot-dotted).

Figure 3

Calculated phase diagrams for liquid-bcc system in one-mode approximation with different correlation peak widths $\alpha :\alpha =0.5,1.0,1.5,2.0,2.5,\mathrm{and}3.0$.

Figure 4

Effect of the $k=0$ mode illustrated using the bcc ${\widehat{C}}_2\left(k\right)$. (a) Correlation functions with different amplitudes in the $k=0$ mode; (b) effect of the $k=0$ mode on the free-energy density curves for the liquid and bcc phases at $\sigma =0$ for different amplitudes: $A_0=0,1,2,3.$

Figure 5

Effect of the $k=0$ mode on the phase diagram with different amplitudes: $A_0=0,1,2,3.$

Figure 6

Effect of the $k=0$ mode on the coexistence gap with different amplitudes: $A_0=0,1,2,3.$

Figure 7

Schematic illustration of the correlation functions truncated at different positions.

Figure 8

bcc (100) plane with different initial conditions: (a1) $(a,dt,\widehat{x})=(0,1.0,1.414)$, (a2) $(a,dt,\widehat{x})=(0,1.0,1.56)$, (a3) $(a,dt,\widehat{x})=(0,1.0,1.8)$, (a4) $(a,dt,\widehat{x})=(0,1.0,\infty )$, (b1) $(a,dt,\widehat{x})=(1,1.0,1.414)$, (b2) $(a,dt,\widehat{x})=(1,1.0,1.56)$, (b3) $(a,dt,\widehat{x})=(1,1.0,1.8)$, (b4) $(a,dt,\widehat{x})=(1,1.0,\infty )$, (c1) $(a,dt,\widehat{x})=(1,0.1,1.414)$, (c2) $(a,dt,\widehat{x})=(1,0.1,1.56)$, (c3) $(a,dt,\widehat{x})=(1,0.1,1.8)$, (c4) $(a,dt,\widehat{x})=(1,0.1,\infty )$.

Figure 9

The bcc density profiles along the direction normal to the (100) plane. Correlation functions with different high-frequency approximations: ${\widehat{C}}_2\left(\infty \right)=-\infty$ (solid), ${\widehat{C}}_2\left(\infty \right)=-4$ (dashed), ${\widehat{C}}_2\left(\infty \right)=-1$ (dotted). The density profiles simulated by different free-energy equations, $a=1$ (the third-order term in the ideal free energy was retained) and $a=0$ (the third-order term was not retained).

Figure 10

Schematic illustration of the correlation functions truncated at different positions.

Figure 11

The bcc density profiles along the direction normal to the (100) plane. Correlation functions with different high-frequency approximations: ${\widehat{C}}_2\left(1.8\right)=0$ (solid), ${\widehat{C}}_2\left(\infty \right)=-1.0$ (dashed), and ${\widehat{C}}_2\left(\infty \right)=-5.0$ (dotted). The density profiles simulated by different free-energy equations, $a=1$ (the third-order term in the ideal free energy was retained) and $a=0$ (the third-order term was not retained).

Figure 12

(a) Schematic illustration of the two-mode direct correlation function; (b) the bcc density profile along the direction normal to the (100) plane.

Figure 13

Density profile across the $\left\{100\right\}$ and $\left\{110\right\}$ interfaces with different peak widths: for (a),(b) $\alpha =1.0$; for (c),(d) $\alpha =1.5$; for (e),(f) $\alpha =2.0$. The length of the arrow shows the value of the interface width.

Figure 14

Atom number density of (100), (110), and (111) planes with different peak widths: for (a1),(a2),(a3) $\alpha =1.0$; for (b1),(b2),(b3) $\alpha =1.5$; for (c1),(c2),(c3) $\alpha =2.0$.

Figure 15

(a) Interface position vs $\tau$ for the three low-index orientations with different ${\alpha }_1:{\alpha }_1=1.0,1.5,\mathrm{and}2.0$. (b) The smoothed bcc density profiles along the direction normal to the (100) plane. Correlation functions with different peak widths: ${\alpha }_1=1.0$ (solid), ${\alpha }_1=1.5$ (dashed), and ${\alpha }_1=2.0$ (dotted).

Figure 16

The ratio of velocity coefficients vs ${\alpha }_1$.

Figure 17

The anisotropy parameter ${\epsilon }_4({\epsilon }_4=({\gamma }_{100}-{\gamma }_{110})/({\gamma }_{100}+{\gamma }_{110}))$ vs $\varepsilon$ calculated with a constant peak width ${\alpha }_1:{\alpha }_1=2$.

Figure 18

The crystal morphology and the smoothed density in the $\left\{110\right\}$ plane with different initial conditions: for (a1),(a2) $(\sigma ,\psi )=(0.15,-0.34)$; for (b1),(b2) $(\sigma ,\psi )=(0.15,-0.33)$; for (c1),(c2) $(\sigma ,\psi )=(0.15,-0.3278)$.

Figure 19

The profile of the smoothed density at solid-liquid fronts with different initial conditions: (a) $(\sigma ,\psi )=(0.15,-0.34)$, (b) $(\sigma ,\psi )=(0.15,-0.33)$, and (c) $(\sigma ,\psi )=(0.15,-0.3278)$.