Defect stability in phase-field crystal models: Stacking faults and partial dislocations

The primary factors controlling defect stability in phase-field crystal (PFC) models are examined, with illustrative examples involving several existing variations of the model. Guidelines are presented for constructing models with stable defect structures that maintain high numerical efficiency. The general framework combines both long-range elastic fields and basic features of atomic-level core structures, with defect dynamics operable over diffusive time scales. Fundamental elements of the resulting defect physics are characterized for the case of fcc crystals. Stacking faults and split Shockley partial dislocations are stabilized for the first time within the PFC formalism, and various properties of associated defect structures are characterized. These include the dissociation width of perfect edge and screw dislocations, the effect of applied stresses on dissociation, Peierls strains for glide, and dynamic contraction of gliding pairs of partials. Our results in general are shown to compare favorably with continuum elastic theories and experimental findings.

Figure 1

(a) An intrinsic $N_L=35$ fcc stacking fault in a periodic simulation cell. (b) Close-up and full $xz$ views of the fault. (c) Full $xz$ view of a sheared/unfaulted crystal in the same cell. Yellow lines are guides to the eye. These images were generated using Eqs. (2) and (9) with parameter values $n_0=-0.48$, $r=-0.63$, and $B^x=1$.

Figure 2

Effect of large-$k$ modes on stacking fault stability. From top to bottom: Cross-sectional normalized $n\left(\vec{r}\right)$ maps of the XPFC fcc crystal with 2, 5, 10, and 15 fcc reflections from left to right; ${\widehat{C}}_2\left(k\right)$ of the PY hard-sphere model at ${\rho }_{\ell }=0.9445$ and of the XPFC model with 2, 5, 10, and 15 fcc peaks; log plot of the structure factor $S\left(k\right)=\langle |\widehat{n}\left(k\right){|}^2\rangle$ for faulted and sheared fcc crystals in the 15 peak XPFC model; linear $S\left(k\right)$ plot for faulted and sheared fcc crystals in the PY hard-sphere model; XPFC ${\gamma }_{\mathrm{ISF}}$ and $N_L=35$ driving force for instability, $\delta \tilde{F}={\tilde{F}}_{\mathrm{SF}}-{\tilde{F}}_{\mathrm{Shear}}$ vs $k_{\max }$. Symbols in the $S\left(k\right)$ plots are Bragg reflection maxima of perfect fcc crystals in either model. XPFC parameter values: ${\rho }_{\ell }=1.0488$, ${\alpha }_i=1$, $\sigma =0$, $H=3$, and $r=2$. $H$ and $r$ were varied from their default values, $H=1$ and $r=0$, to ensure numerical stability of the logarithmic term in Eq. (1).

Figure 3

Split fcc edge dislocation in the ${\rho }_{\ell }=0.9445$ PY hard-sphere CDFT of Eq. (1). A cutaway view of the density field $\rho \left(\vec{r}\right)$ is shown, with the color scale truncated above $\rho \left(\vec{r}\right)=1$, though $\rho \left(\vec{r}\right)\simeq 7800$ at the peak maxima. Left: Diagrammatic representation of the dissociation of a perfect dislocation $\vec{b}$ into Shockley partials ${\vec{b}}_1$ and ${\vec{b}}_2$.

Figure 4

Structure factors of fcc crystals with various types of defects. Note that the most compact structure, the planar stacking fault, produces the overall broadest line shapes. These spectra were generated using Eqs. (2) and (12) with parameter values $n_0=-0.48$, $r=-0.63$, $B^x=1$, ${\alpha }_0=1/2$, and $k_0=6.2653$.

Figure 5

Sample PFC kernels that produce stable stacking faults, with relevant fcc and hcp reflections also indicated. Inset: Close-up view of the peak maxima. Original PFC parameters: $n_0=-0.48$, $r=-0.63$, and $B^x=1$. ${\mathrm{XPFC}}^{*}$ parameters, Eq. (10): $n_0=-3/10$, $r=-9/40$, $w_2=1/50$, $w_4=49/50$, ${\alpha }_i\simeq 1$, $\sigma =0$, and $H\simeq 1.0625$. Wu et al. parameters: $n_0=-3/10$, $r=-9/40$, $R_1=1/20$, and $B^x=1$. Note that for display purposes all kernels have been shifted by $-3n_0^2$ to maintain consistency with Eq. (1), and that $k$ has been rescaled in the ${\mathrm{XPFC}}^{*}$ and Wu et al. kernels to match the equilibrium fcc reflections of the original PFC kernel.

Figure 6

Dissociated $\vec{b}=a/2\left[110\right]$ edge and screw dislocations in the (modulated) original PFC model of Eqs. (2) and (12). The lower images show cutaway views of $n\left(\vec{r}\right)$, illustrating the mixed Shockley partials connected by a single intrinsic stacking fault. The upper images show atomic representations of the peak positions in $n\left(\vec{r}\right)$: Red atoms have local hcp coordination (stacking fault), gray atoms have local noncrystalline coordination (defect cores), and atoms with fcc coordination are not shown. Parameter values $n_0=-0.48$, $r=-0.63$, $B^x=1$, ${\alpha }_0=1/2$, and $k_0=6.2653$ were used.

Figure 7

Modulated original PFC kernel of Eq. (12) where $n_0=-0.48$, $r=-0.63$, $B^x=1$, ${\alpha }_0=1/2$, and $k_0=6.2653$. $H_0=0$, $0.04$, and $0.08$ are shown. Inset: ${\gamma }_{\mathrm{ISF}}$ obtained with the given modulated kernel vs ${\tilde{F}}_{\mathrm{hcp}}-{\tilde{F}}_{\mathrm{fcc}}$ and $H_0$. ${\gamma }_1$ corresponds to $N_L=53$ and ${\gamma }_2$ to $N_L=35$. As in Fig. 5, all kernels have been shifted by $-3n_0^2$.

Figure 8

Equilibrium splitting distance between Shockley partials within dissociated perfect $\vec{b}=a/2\left[110\right]$ edge and screw dislocations, shown as a function of ${\gamma }_{\mathrm{ISF}}^{-1}$. Lines represent the predictions of isotropic and anisotropic continuum elastic theories, with no adjustable parameters. Model parameters are the same as those of Fig. 7.

Figure 9

Divergence of the equilibrium splitting distance between Shockley partials within dissociated perfect $\vec{b}=a/2\left[110\right]$ screw dislocations, shown as a function of ${\epsilon }_{zx}$, the Escaig shear strain. Lines represent the predictions of isotropic continuum elastic theory, Eq. (15), as discussed in the text. Model parameters are the same as those of Fig. 7.

Figure 10

Measured Peierls strain for glide of dissociated and undissociated edge and screw dislocations, shown as a function of applied shear strain rate. The solid and dashed lines, which are indistinguishable, are fits to the functional forms presented in the text. Model parameters are the same as those of Fig. 7.

Figure 11

Measured dissociation width of a gliding edge dislocation vs dislocation velocity for various applied shear strain rates. Steady-state widths and velocities at each shear rate are separately indicated. The solid line is a fit to the functional form of the drafting theory presented in the text, and the dashed line is an illustrative example of the behavior expected from semirelativistic effects ($d\sim \sqrt{1-v^2/{\alpha }^2}$). Model parameters are the same as those of Fig. 7.