Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals

A continuum field theory approach is presented for modeling elastic and plastic deformation, free surfaces, and multiple crystal orientations in nonequilibrium processing phenomena. Many basic properties of the model are calculated analytically, and numerical simulations are presented for a number of important applications including, epitaxial growth, material hardness, grain growth, reconstructive phase transitions, and crack propagation.

Figure 1

The points correspond to an experimental liquid structure factor for ${}^{36}\mathrm{Ar}$ at 85 K taken from 24. The line corresponds to a best fit to Eq. (22).

Figure 2

One-dimensional phase diagram in the one-mode approximation. The solid line is the boundary separating constant (i.e., liquid) and periodic (i.e., crystal) phases. The hatched section of the plot corresponds to regions of liquid-crystal coexistence.

Figure 3

In (a) the minimum of the free energy is plotted as a function of $r$ for $\overline{\psi }=\sqrt{-r}∕2$. The solid line is Eq. (38) and the points are from numerical simulations. In (b) the bulk modulus is plotted as a function of $r$ for $\overline{\psi }=\sqrt{-r}∕2$. The solid line is an analytic calculation $[{\left(q_t{A}_t\right)}^2∕6]$ and the points are from numerical simulations.

Figure 4

Two-dimensional phase diagram as calculated in a one-mode approximation. Hatched areas in the figure correspond to coexistence regions. The small region enclosed by a dashed box is superimposed on the argon phase diagram in Fig. 5. In this manner the parameter of the free-energy functional can be chosen to reproduce certain aspects of a liquid-crystal phase transition.

Figure 5

The phase diagram of argon. The hatched regions correspond to the coexistence regions. The points are from the PFC model.

Figure 6

Heterogeneous nucleation of three crystallites in a supercooled liquid. The grey scale in (a), (b), and (c) corresponds to the density field $\psi$ and in (d), (e), and (f) to the smoothed local free energy. The configurations are taken at times $t=300$, 525, and 3975 for $\left(\mathrm{a}\right)+\left(\mathrm{d}\right),\left(\mathrm{b}\right)+\left(\mathrm{e}\right)$, and $\left(\mathrm{c}\right)+\left(\mathrm{f}\right)$, respectively. (Note that only a portion of the simulation is shown here.)

Figure 7

Vacancy diffusion times. In this figure the grey scale is proportional to the $\psi (\vec{r},t)-{\psi }_{\mathit{eq}}$. The times shown are (a) $t=0$, (b) $t=50$, (c) $t=100$, and (d) $t=150$.

Figure 8

Vacancy diffusion. In this figure the average of the standard deviation in the $x$ and $y$ directions is plotted as a function of time.

Figure 9

Schematic of a grain boundary.

Figure 10

The grey scale corresponds to the magnitude of the field $\psi$ for a grain boundary mismatch of $\theta =5.8°$ and $\theta =34.2°$ in (a) and (b), respectively. In (a) squares have been placed at defect sites.

Figure 11

The grain boundary energy is plotted as a function of mismatch orientation. The points correspond to numerical simulations of the PFC model and the solid line corresponds to Eq. (50).

Figure 12

The grain boundary energy of the PFC model is compared with experiments on tin [30], lead [30], and copper [31].

Figure 13

Epitaxial growth. (a), (b), (c), and (d) correspond to times $t=150$, 300, 450, and 600, respectively. The grey scale is proportional to the local density (i.e., $\psi$) in the film and liquid. The substrate is highlighted by a darker grey background. To highlight nucleated dislocations, small white dots were placed on atoms near the two dislocation cores that appear in this configuration.

Figure 14

Epitaxial growth. (a), (b), (c), (d), and (e) correspond to times $t=150$, 300, 450, 600, and 750, respectively. The grey scale is proportional to the free-energy density. To highlight the excess strain energy in the film the grey scale near the defect was saturated. The region enclosed by dashed lines corresponds to the configuration shown in Fig. 13.

Figure 15

Epitaxial growth. In (a) and (b) the average film-liquid interface height and width is shown as a function of time. Both the width and height have been scaled by $a*$, which is the one-mode approximation for the distance between layers in the appropriate direction.

Figure 16

Epitaxial growth. In this figure a histogram of the number of defects is shown as a function of height above the substrate. (a), (b), (c), and (d) correspond to $t=300$, 450, 600, and 1000.

Figure 17

Epitaxial growth. In this figure $H_c$ is the critical height as defined in the text and $ϵ$ is the mismatch strain between the film and substrate.

Figure 18

In this figure the grey scale corresponds to the local energy density before a strain is applied. The dark black regions on the left and right of the figure are the regions that are coupled to the external field.

Figure 19

The same as Fig. 18, except at a strain of 7.8%.

Figure 20

In (a) the stress is plotted as a function of strain for a system with an average grain radius of 35 particles. The solid lines from top to bottom in (a) correspond to strain rates of $24\times {10}^{-6},12\times {10}^{-6}$, and $6\times {10}^{-6}$, respectively. In (b) the solid lines from top to bottom correspond to systems with average grain sizes of 70, 50, 35, and 18 particles, respectively. In both (a) and (b) the dashed line corresponds to a unit slope.

Figure 21

The yield stress is plotted as a function of average grain radius. The solid, open, and starred points correspond to strain rates of $24\times {10}^{-6},12\times {10}^{-6}$, and $6\times {10}^{-6}$, respectively. The dashed lines are guides to the eye.

Figure 22

The elastic moduli $Y_0$ (see text) are plotted as a function grain radius. The solid, open, and starred points correspond to strain rates of $24\times {10}^{-6},12\times {10}^{-6}$, and $6\times {10}^{-6}$, respectively.

Figure 23

Heterogeneous nucleation and grain growth. In this figure the grey scale corresponds to the smoothed local free energy. (a), (b), (c), (d), (e), and (f) correspond to times 50, 200, 1000, 3000, 15 000, and 50 000, respectively.

Figure 24

A portion of a simulation is shown where the grey scale corresponds to the local energy density. The size of both figures is $2048\Delta x\times 1024\Delta x$, where $\Delta x=\pi ∕3$. (a) and (b) are at times $t=25000$ and 65 000 after the rip was initiated, respectively.

Figure 25

The grey scale corresponds the field $\psi$. (a), (b), (c), and (d) correspond to times $t=2$, 20, 40, and 180, respectively. In (d) the solid lines are guides to the eye.