Critical events, entropy, and the grain boundary character distribution

Mesoscale experiment and simulation permit harvesting information about both geometric features and texture in polycrystals. The grain boundary character distribution (GBCD) is an empirical distribution of the relative length [in two dimensions (2D)] or area (in 3D) of an interface with a given lattice misorientation and normal. During the growth process, an initially random distribution of boundary types reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, it is found that if the given energy density depends only on lattice misorientation, then the steady-state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest nonrandom distributions, corresponding to independent trials with respect to the energy. In this paper, we derive an entropy-based theory that suggests that the evolution of the GBCD satisfies a Fokker-Planck equation, an equation whose stationary state is a Boltzmann distribution. Cellular structures coarsen according to a local evolution law, curvature-driven growth, and are limited by space-filling constraints. The interaction between the evolution law and the constraints is governed primarily by the force balance at triple junctions, the natural boundary condition associated with curvature-driven growth, and determines a dissipation relation. A simplified coarsening model is introduced that is driven by the boundary conditions and reflects the network level dissipation relation of the grain growth system. It resembles an ensemble of inertia-free spring-mass dashpots. Application is made of the recent characterization of Fokker-Planck kinetics as a gradient flow for a free energy in deriving the theory. The theory predicts the results of large-scale two-dimensional simulations and is consistent with experiment.

Figure 1

An arc $\Gamma$ with normal $n$, tangent $b$, and lattice misorientation $\alpha ,$ illustrating lattice elements (reproduced from Ref. 16).

Figure 2

Example of an instant during the simulated evolution of a cellular network. This is part of the frame from a small simulation with constant energy density and periodic conditions at the border of the configuration.

Figure 3

The average area of six-sided cell populations during coarsening in two different cellular systems showing that the von Neumann–Mullins $n-6$ rule (12) does not hold at the scale of the network. (a) In an experiment on Al thin film, (b) a typical simulation (arbitrary units). Please refer to the first paragraph of Sec. 3 for additional explanation.

Figure 4

(a) The relative entropy ${\Phi }_{\sigma }$ of the solution $u(x,t)$ of the Fokker-Planck equation (34) for the potential $\psi (x)=1+r(x-\frac{1}{2}){}^2,r=2,$ with the choice $\lambda =\sigma =0.029\hspace{0.5em}691\hspace{0.5em}5$, computed by a routine numerical method, compared with a sequence of ${\Phi }_{\lambda }$ with the curve for $\sigma =0.029\hspace{0.5em}691\hspace{0.5em}5$ noted in red. The values of $\lambda$ correspond to ${\rho }_{\lambda }$ with $max{\rho }_{\sigma }/2⩽max{\rho }_{\lambda }⩽(3/2)max{\rho }_{\sigma }$. (b) The computed equilibrium solution, which is indistinguishable from ${\rho }_{\sigma },$ the Boltzmann distribution of (36).

Figure 5

Plots of $-\log {\Phi }_{\lambda }$ vs $t$ with $-\log {\Phi }_{\sigma }$ in red for the solution of (34), cf. Fig. 4. The plot illustrates that ${\Phi }_{\sigma }$ decreases exponentially to $0$ but that ${\Phi }_{\lambda }$ for choices of $\lambda \ne \sigma$ do not have this property.

Figure 6

Graphical results for the simplified coarsening model with potential (40). (a) Relative entropy plots for values of $\lambda$ chosen according to (41) with ${\Phi }_{\sigma }$ noted in red. The value of $\sigma =0.029\hspace{0.5em}691\hspace{0.5em}5$. (b) Empirical GBCD at simulation time $t=T_{\infty }$ in red compared with ${\rho }_{\sigma }$ in black.

Figure 7

Plots of $-\log {\Phi }_{\lambda }$ vs $t$ with $-\log {\Phi }_{\sigma },\sigma =0.029\hspace{0.5em}691\hspace{0.5em}5$, in red for the simplified coarsening model with potential (40). It shows that ${\Phi }_{\sigma }$ decays exponentially to its minimum at simulation time $t=T_{\infty }$.

Figure 8

Graphical results for the simplified coarsening model with potential (42). (a) Relative entropy plot for selected values of $\lambda$ with ${\Phi }_{\sigma }$ noted in red. The value of $\sigma =0.003\hspace{0.5em}033\hspace{0.5em}356\hspace{0.5em}683$ and is ascertained at the simulation time $t=T_{\infty }$ corresponding to $80\%$ of cells deleted. (b) Empirical GBCD at time $t=T_{\infty }$ in red compared with ${\rho }_{\sigma }$ in black.

Figure 9

The energy density $\psi (\alpha )=1+\epsilon \sin {}^{2}2\alpha ,\left|\alpha \right|<\pi /4,\epsilon =\frac{1}{2}.$

Figure 10

(a) The relative entropy of the grain growth simulation with energy density (43) for a sequence of ${\Phi }_{\lambda }$ vs $t$ with the optimal choice $\sigma \approx 0.1$ noted in red. (b) Comparison of the empirical GBCD distribution at time $t=T_{\infty }=2$, when $80\%$ of the cells have been deleted, with ${\rho }_{\sigma }$ the Boltzmann distribution of (35).

Figure 11

Plot of $-\log {\Phi }_{\sigma }$ vs $t$ with energy density (43). It is approximately linear until it becomes constant showing that ${\Phi }_{\sigma }$ decays exponentially.

Figure 12

GBCD (red) and Boltzmann distribution (black) for the potential $\psi$ of (43) with parameter $\sigma \approx 0.1$ as predicted by our theory. This GBCD is averaged over five trials.

Figure 13

(a) The relative entropy of the grain growth simulation with density (44) for a sequence of ${\Phi }_{\lambda }$ vs $t$ with the optimal choice $\sigma \approx 0.08$ noted in red. (b) Comparison of the empirical GBCD at time $t=T_{\infty }=3$, when $80\%$ of the cells have been deleted, with ${\rho }_{\sigma }$ the Boltzmann distribution of (35).