Analytic description of competitive grain growth

We describe an analytic model for polycrystalline thin film growth based on competition between grains with conic geometries. This model is valid for all film thicknesses and is verified using level set simulations in $2+1$ dimensions. We study the effects of nonuniform initial grain distributions on growth statistics. These results provide a possible explanation for discrepancies between experimentally measured and theoretical scaling laws.

Figure 1

(Color online) Initial grain orientations distributions from left to right: biased toward fastest growth direction, uniform, and biased away from fastest growth direction. Fastest growth direction and rings showing the direction of integration are plotted on each spherical pole figure.

Figure 2

(Color online) A deletion event between two conic grains at time $t$ in the time interval $\Delta t$. Isocontours are shown at different heights. Note that the primary axis of both cones are the same length.

Figure 3

(Color online) Geometry of a cross-section of the cone with tilt angle ${\theta }^{\prime }$ in the deletion plane shown in Fig. 2.

Figure 4

(Color online) Examples of competitive conic grain growth from level set simulations at various thicknesses. Surfaces are shaded by $x$-direction slope and cross-sectional slices also shown. Potential deletion zones marked in red. Animated version available online in supplementary materials (Ref. [20]).

Figure 5

(Color online) Comparison of the grain distribution at four different film thicknesses between analytic growth model and level set simulation for the cases where the initial distribution is (a) biased toward the fastest growth direction, (b) uniform, and (c) biased away from the fastest growth direction.

Figure 6

(Color online) (a) rms surface roughness and (b) probability of grain survival calculated from the analytic growth model for the three cases described in Fig. 1. Power laws are shown in both graphs as dotted lines.