Distribution of Topological Types in Grain-Growth Microstructures

An open question in studying normal grain growth concerns the asymptotic state to which microstructures converge. In particular, the distribution of grain topologies is unknown. We introduce a thermodynamiclike theory to explain these distributions in two- and three-dimensional systems. In particular, a bendinglike energy $E_i$ is associated to each grain topology $t_i$, and the probability of observing that particular topology is proportional to $[1/s(t_i)]e^{-\beta E_i}$, where $s(t_i)$ is the order of an associated symmetry group and $\beta$ is a thermodynamiclike constant. We explain the physical origins of this approach and provide numerical evidence in support.

Figure 1

Two grains with eight faces but different topologies.

Figure 2

Distribution of $n$-sided grains and faces in two- and three-dimensional steady-state grain growth [22], respectively, compared with Eqs. (3) and (5), with $\beta =1.62$ and $\beta =1.29$, for the two systems. Although these equations are defined only for integer values, we illustrate them as continuous functions to aid visualization. Error bars showing standard errors of the mean are smaller than the data points.

Figure 3

The product of the observed probability $p(t_i)$ and the symmetry group order $s(t_i)$ as a function of energies (a) $E_f$ and (b) $E_v$ for each observed topology $t_i$, as suggested by Eq. (7). Data are taken from three-dimensional front-tracking simulations of steady-state grain growth [22, 23].

Figure 4

The product of the observed probability $p(t_i)$ and the symmetry group order $s(t_i)$ as a function of (a)–(c) $E_f$ and of (d)–(f) $E_v$ for each observed topology $t_i$ with fixed numbers of faces. Probabilities $p(t_i)$ are normalized so that they sum to 1 for each number of faces. Dashed curves show 5 standard deviations of the sample mean for the relevant sample size. Data are taken from simulations of three-dimensional steady-state grain growth [22, 23].

Figure 5

Three examples for which a fixed set of faces provides a distribution of topological types. Probabilities are relative to other samples in the limited dataset. Error bars show standard errors of the mean.