Topological defects in two-dimensional orientation-field models for grain growth

Standard two-dimensional orientation-field-based phase-field models rely on a continuous scalar field to represent crystallographic orientation. The corresponding order parameter space is the unit circle, which is not simply connected. This topological property has important consequences for the resulting multigrain structures: (i) trijunctions may be singular; (ii) for each pair of grains there exist two different grain boundary solutions that cannot continuously transform to one another; (iii) if both solutions appear along a grain boundary, a topologically stable, singular point defect must exist between them. While (i) can be interpreted in the classical picture of grain boundaries, (ii) and therefore (iii) cannot. In addition, singularities cause difficulties, such as lattice pinning in numerical simulations. To overcome these problems, we propose two formulations of the model. The first is based on a three-component unit vector field, while in the second we utilize a two-component vector field with an additional potential. In both cases, the additional degree of freedom introduced makes the order parameter space simply connected, which removes the topological stability of these defects.

Figure 1

(a) Due to the $2\pi$ periodicity of the orientation field $\theta$, there exist two continuous paths connecting the orientations ${\theta }_A=0.9\pi$ and ${\theta }_B=1.6\pi$. The small-turn connection is shown by the thick blue line, while the large-turn connection is shown by the thin red line. (b) These paths correspond to two different grain boundary solutions. Due to topological reasons, there is no continuous transformation between these two paths or solutions. Please note that as 0 and $2\pi$ are equivalent orientations, the red profile on the right is also continuous.

Figure 2

Schematic view of trijunctions where the orientation field is (a) singular and (b) nonsingular. The solid lines may equally correspond to abrupt changes in the orientation (such as in sharp interface models) or may indicate thin regions where the change in orientation is continuous (such as in orientation-field-based phase-field models). The sums of orientational increments around the trijunctions are $(0.7+0.8+0.5)\pi =2\pi$ (winding number 1) and $(0.4-0.9+0.5)\pi =0$ (winding number 0), respectively. As the winding number is an integer quantity, it cannot be changed by continuous transformations in the orientation field.

Figure 3

Schematic view of a grain boundary containing two defects where the small-turn and large-turn solutions meet. The two different grain boundary profiles are represented by the thick blue and thin red isolines that correspond to steps of $0.2\pi$ in $\theta$. In the magenta region the orientation field is significantly distorted as the different grain boundary profiles try to match each other, but it is still assumed to be continuous. The integral of $\mathbf{\nabla }\theta$ along the black paths are $+2\pi$ ($-2\pi$) for the left (right) circle and 0 for the ellipse. This means that the magenta regions must contain topologically stable defects with winding numbers $+1$ and $-1$, but the larger area that includes both defects can be made defect free.

Figure 4

The two different grain boundary solutions of the HMP model (solid lines) and the respective two solutions of the US model (symbols). For the latter, ${\theta }_3$ remained zero and the scalar $\theta$ plotted is the polar angle of the vector $({\theta }_1,{\theta }_2)$. The results produced by the two models are visually indistinguishable. The triangles (circles) and the solid lines underneath them correspond to the lower- (higher-) energy solutions. The open black symbols show the phase field and the closed red and blue symbols show the orientation field. (a) Profiles in the real space and (b) profiles in the order parameter space. Neighboring symbols correspond to neighboring cells in the simulation domain.

Figure 5

Transition from the unstable higher-energy grain boundary solution to the stable lower-energy grain boundary solution in the unit sphere model. The nine lines shown correspond to grain boundary profiles at different simulation times, mapped to the order parameter space. Time is increasing from left to right. To trigger the transition, a small value was added to ${\theta }_3$ in the beginning of the simulation.

Figure 6

Grain boundary solutions obtained by the LDG model for $\nu =50$. With this value of $\nu$ the barrier is high enough for two separate solutions to exist: (a) the $\varphi$ and $\theta$ profiles in real space and (b) the $\mathbf{\theta }$ profiles in the order parameter space. The background shading is according to $f_s\left(\mathbf{\theta }\right)$: White corresponds to low values of the potential and black to high values. The black solid line is the unit circle.

Figure 7

Grain boundary solution obtained by the LDG model for $\nu =10$. With this value of $\nu$ the barrier is low and only one solution, the one corresponding to the lower-energy solution in Fig. 6, exists. For further description of the figure elements, see the caption of Fig. 6.

Figure 8

Transition of the $\mathbf{\theta }$ profile from the profile corresponding to the higher-energy solution of the HMP model to the solution of the two-component vector field model with $\nu =10$, shown in the order parameter space. As in Fig. 4, time increases from left to right.

Figure 9

Structure of the orientation field around the equilibrium defect in the HMP model with (a) $\mathrm{\Delta }x=0.05$, (b) $\mathrm{\Delta }x=0.025$, and (c) $\mathrm{\Delta }x=0.0125$. The top row shows the color map of $\theta$ in a small region around the defect, strongly enlarged for better visibility. We used a circular color map that assigns the colors red, blue, yellow, and again red to the orientations $\theta =0,2\pi /3,4\pi /3,2\pi$ and interpolates the RGB values of these colors for orientations in between. The bottom row shows the respective 1D profiles. The four lines shown correspond to the leftmost (black triangles), rightmost (green circles), and the two middle (blue squares and red pentagons) columns of the simulation domain.

Figure 10

Structure of the orientation field around the equilibrium defect in the unit sphere model. (a) Color map of the scalar orientation field $\theta$. For further explanation, see the caption of Fig. 9. (b) Order parameter space map of the vector orientation field $\mathbf{\theta }$. Points corresponding to neighboring pixels are shown connected.

Figure 11

Structure of the orientation field around the equilibrium defect in the LDG model with $\nu =10$, when two stable grain boundary solutions and therefore a stable defect exist. (a) Color map of the scalar orientation field $\theta$. (b) Order parameter space map of the vector orientation field $\mathbf{\theta }$. For further explanation, see the captions of Figs. 9 and 10.

Figure 12

Defect position vs time in an asymmetric setup in the (a) HMP and (b) LDG models. Time and space coordinates are in arbitrary units. The reference grid spacing is $\mathrm{\Delta }{x}_0=0.05$. In the case of the HMP model we see the pinning of the defect as the spatial resolution increased, while in the case of the LDG model a convergence to a constant drift velocity is observed.

Figure 13

Snapshots of the orientation field corresponding to 0, 5, 10, 20, 30, and 40 equal units of time of a grain coarsening simulation with the HMP model. The multigrain configuration the simulation was started with was obtained by simulating solidification with the same HMP model. All snapshots show the same $100\times 400$ pixel region of the full $4096\times 4096$ pixel simulation domain. The color coding is the same as in Fig. 9. The defects, which are the points where the purple and yellow segments of the grain boundary (the two different solutions) meet, are mobile on the smaller-angle grain boundary on the lower half of the snapshots. However, the defects on the larger-angle grain boundary on the upper half are pinned and do not allow the grain boundary to move, resulting in an unphysical kink (shown by the white arrowhead) on the red-blue grain boundary.

Figure 14

Snapshots of the orientation field corresponding to 0, 1, 2, 4, 6, and 8 equal units of time of a grain coarsening simulation with the HMP model. To start from a defect-free configuration, the initial multigrain structure was obtained by using the LDG model when simulating solidification. All snapshots show the same $120\times 480$ pixel region of the full $4096\times 4096$ pixel simulation domain. The color coding is the same as in Fig. 9. The signed misorientations between the grains marked by the white circles A, B, and C are 0.37 (B-A), 0.13 (C-B), and slightly below 0.5 (A-C). Thus, the winding number corresponding to the enclosed trijunction on the leftmost snapshot is 1. The second snapshot shows that as the red grain shrinks, the trijunction point (defined as the meeting point of the orange, red, and gray grains) is decoupled from the defect (defined as the point with winding number 1) and the defect stays visible long after the original red grain disappeared. The white arrowhead on the fourth snapshot shows a defect pinned by the lattice, being responsible for the kink in the grain boundary.

Figure 15

Orientation-field maps from simulations by the (a) US and (b) LDG models. The snapshots show randomly chosen $1200\times 600$ pixel parts of the total $4096\times 4096$ pixel simulation domain from a later stage of the simulation, when grains are relatively large. The color coding is the same as in Fig. 9. No defects can be seen on the grain boundaries.

Figure 16

Grain boundary character distributions obtained by the different models. The length of grain boundaries with large misorientation is significantly larger in the HMP model than in the US and LDG models. We attribute this to the pinning of the defects that is most effective on grain boundaries with misorientation $\mathrm{\Delta }\theta =\pi$.

Figure 17

Limiting grain size distributions obtained by the different models. The line labeled as HMP-LDG represent a variant of the HMP simulation, where the initial configuration of the HMP model was taken from the LDG simulation.