Phase field crystal modeling as a unified atomistic approach to defect dynamics

Material properties controlled by evolving defect structures, such as mechanical response, often involve processes spanning many length and time scales which can not be modeled using a single approach. We present a variety of results that demonstrate the ability of phase field crystal (PFC) models to describe complex defect evolution phenomena on atomistic length scales and over long, diffusive time scales. Primary emphasis is given to the unification of conservative and nonconservative dislocation creation mechanisms in three-dimensional fcc and bcc materials. These include Frank-Read–type glide mechanisms involving closed dislocation loops or grain boundaries as well as Bardeen-Herring–type climb mechanisms involving precipitates, inclusions, and/or voids. Both source classes are naturally and simultaneously captured at the atomistic level by PFC descriptions, with arbitrarily complex defect configurations, types, and environments. An unexpected dipole-to-quadrupole source transformation is identified, as well as various complex geometrical features of loop nucleation via climb from spherical particles. Results for the strain required to nucleate a dislocation loop from such a particle are in agreement with analytic continuum theories. Other basic features of fcc and bcc dislocation structure and dynamics are also outlined, and initial results for dislocation-stacking fault tetrahedron interactions are presented. These findings together highlight various capabilities of the PFC approach as a coarse-grained atomistic tool for the study of three-dimensional crystal plasticity.

Figure 1

The bcc $a/2\langle 111\rangle$ screw dislocation from a single-peaked XPFC model. A cross section of $n\left(\vec{r}\right)$ is shown on the left, and the differential displacement map [41] around the core is shown on the right. Results were generated using Eqs. (1), (3), and (4) with model parameters $w=1.4$, $u=1$, $n_0=0.05$, ${\alpha }_1=0.25$, $\sigma =0.12$, ${\rho }_1=1$, and ${\beta }_1=8$. Nearly identical results are obtained at ${\alpha }_1=1$ and 2.

Figure 2

Schematic of the simulation setup used for Frank-Read source operation.

Figure 3

Operation of a Frank-Read dipole source under shear strain ${\epsilon }_{\mathrm{zy}}$. (a) A dislocation loop composed of four $a/2\langle 110\rangle$ edge dislocations. The two mobile horizontal segments are dissociated in the $\left(1\overline{1}1\right)$ plane, the two vertical segments are relatively immobile. (b) 3D view of the source near pinch off at $560t$. (c), (d), (e), and (f) show $xy$-plane views at $t=375$, 500, 595, and 625, respectively. For analysis and visualization purposes, local peaks in $n\left(\vec{r}\right)$, which represent the most probable atomic positions, are taken to correspond to atomic sites. Density peaks with hcp coordination (stacking faults) are shown in orange (light) or red (dark) depending on position along the out-of-plane $z$ axis. Those with irregular coordination (dislocation core) are shown in gray. See Supplemental Material [42] for the associated animation.

Figure 4

Operation of a Frank-Read dipole-to-quadrupole source. The initial loop is shorter in the $x$ direction and longer in the $z$ direction than that of Fig. 3. (a), (b), (c), (d), (e), (f), and (g) show $xy$-plane views at $t=450$, 520, 580, 680, 750, 840, and 1030, respectively. A 3D view at $t=750$ is also shown in (e). The source emits two initial loops near $520t$, then reorients and converts into a quadrupole source before being restored to a closed-loop form near $1030t$. Density peaks with hcp coordination (stacking faults) are shown in white, yellow (light gray), orange (gray), or red-orange (dark gray), depending on position along the out-of-plane $z$ axis. Those with irregular coordination (dislocation cores) are shown in gray. (h) An unzipped stacking fault tetrahedron acting as a Frank-Read–type source. Density peaks with hcp coordination (stacking faults) are shown in red (dark gray), those with irregular coordination (dislocation cores) are shown in gray. See Supplemental Material [42] for the associated animations.

Figure 5

Emission of dissociated $a/2\langle 110\rangle$ dislocations and $a/6\langle 121\rangle$ partials from grain boundaries. (a)–(d) Nanopolycrystalline fcc sample with average grain size $\overline{d}\simeq 46a$ under tensile strain ${\epsilon }_{\mathrm{zz}}=0.082$, $0.086$, $0.092$, and $0.1$, respectively. Both full dislocations [leading and trailing partials with stacking faults, two of which are tracked by green (light gray) arrows] and leading partials with stacking faults are emitted from or absorbed into grain boundaries, beginning near the yield point. Density peaks with hcp (stacking faults), irregular (dislocation cores), and fcc coordination are shown in red (dark gray), gray, and green (light gray), respectively. (e) Stress-strain curves for fcc polycrystals with various average grain sizes $\overline{d}$ and applied strain rates (constant strain rate tensile deformation). Model parameters are the same as those of Figs. 3 and 4. (f) As (e), but for bcc polycrystals with model parameters identical to those reported in Sec. 5, except ${\alpha }_1=2$, and Eq. (5) was employed with $\beta =0.01$, $\alpha =1$. See Supplemental Material [42] for the associated animations.

Figure 6

Critical strain ${\epsilon }_{ii}^{*}$ for Bardeen-Herring climb source activation vs sphere radius $R_0$. The points represent simulation data at different effective strain rates, and the lines are predictions of the theory of Brown et al. [58], without and with a finite minimum critical strain. The fits employ fixed parameters $b=1/\sqrt{2}$ and $\nu =\frac{1}{3}$, and adjustable parameters ${\epsilon }_{ii}^{\min }$ and $B_S/B_M$, which are in all cases close to $0.016$ and 4, respectively.

Figure 7

Operation of a Bardeen-Herring–type spherical climb source with $R_0=11.3a$ in a bcc crystal under uniaxial compression ${\epsilon }_{\mathrm{xx}}$. Time evolution shown at (a) $t=110$, (b) $t=126$, (c) $t=135$, (d) $t=145$, (e) $t=160$, and (f) $t=180$. Only density peaks with irregular coordination (interface and dislocation core sites) are displayed. Those at the sphere-matrix interface are shown in gray (dark), those inside the dislocation core are shown in blue-green (light) depending on position along the $y$ axis. See Supplemental Material [42] for the associated animation.

Figure 8

Operation of a Bardeen-Herring–type spherical climb source with $R_0=28.3a$ in a bcc crystal under uniaxial compression ${\epsilon }_{\mathrm{xx}}$. Time evolution shown at (a) $t=110$, (b) $t=138$, (c) $t=150$, and (d) $t=162$. Only density peaks with irregular coordination (interface and dislocation core sites) are displayed. Those at the sphere-matrix interface are shown in gray (dark), those inside the dislocation core are shown in blue-green (light) depending on position along the $y$ axis. See Supplemental Material [42] for the associated animation.

Figure 9

Examples of various other Bardeen-Herring–type climb sources in bcc crystals. (a) Linked, concentric, biplanar loop pairs under ${\epsilon }_{\mathrm{zz}}$ with $R_0=11.3a$. (b) Offset nonconcentric loop pairs under ${\epsilon }_{\mathrm{yy}}$ with $R_0=28.3a$. (c) Offset concentric loop pairs under ${\epsilon }_{\mathrm{xx}}$ with $R_0=9a$ and $\alpha =2$. (d) Dislocation network/tangle formation from an array of 10 sources with $R_0=9a$ in a bicrystal under constant volume tension ${\epsilon }_{\mathrm{zz}}$. Only density peaks with irregular coordination (interface and dislocation core sites) are shown. Those at the sphere-matrix interface are shown in gray (dark), those inside the dislocation core are shown in blue-green (light) depending on position along the out-of-plane axis. See Supplemental Material [42] for the associated animations.

Figure 10

Interaction between a SFT and a gliding dissociated edge dislocation in a fcc crystal, ($\text{ED/Down},4/13,0.0001/t$). Perspective views of the SFT and dislocation are shown in (a). The upper image shows both at $t=10100$, while the lower left and right images show the SFT at $t=15800$ and $20100$, respectively. Only the leading partial has passed the SFT at $t=15800$, both partials have passed by $t=20100$. $xy$-plane views are shown in (b), (c), (d), (e), and (f) at $t=12100$, $15100$, $16600$, $18600$, and $19600$, respectively. (g) The damaged SFT following the ($\text{SD/Edge},4/13,0.0001/t$) interaction. Density peaks with hcp coordination (stacking faults) are shown in blue (dark gray), those with irregular coordination (dislocation cores) are shown in gray (light gray). See Supplemental Material [42] for the associated animations.