Inferring grain boundary structure–property relations from effective property measurements

Abstract

Grain boundaries strongly affect many materials properties in polycrystalline materials. However, very few structure–property models exist for grain boundaries, due in large part to the complicated and poorly understood way in which the properties of grain boundaries vary with their crystallographic structure. In the present work, we infer grain boundary structure–property correlations from measurements of the effective properties of a polycrystal. We refer to this approach as grain boundary properties localization. We apply this technique to a simple model system of grain boundary diffusivity in a two-dimensional microstructure, and infer the properties of low- and high-angle grain boundaries from the effective diffusivity of the grain boundary network. The generalization and use of these methods could greatly reduce the computational and experimental effort required to establish structure–property correlations for grain boundaries. More broadly, the technique of properties localization could be used to infer the properties of many microstructural constituents in complex microstructures.

Appendix: Error scaling for bicrystal experiments

In attempting to infer the parameters of the constitutive equation (Eq. 2) via bicrystal experiements, the parameter $D_1$ will be recovered exactly if at least one bicrystal that we test has a misorientation $\omega \le \omega _{\text{t}}$, assuming that there is no uncertainty in the value of $D_1$ that we measure. Likewise, $D_2$ will be recovered exactly with at least one bicrystal having $\omega > \omega _{\text{t}}$. To recover the final parameter, $\omega _{\text{t}}$, to within some specified accuracy ($\%ERR$), there must be at least one bicrystal with misorientation falling in the range $\left[ \omega _{\text{t}}-\delta ,\omega _{\text{t}}\right]$ and another bicrystal with misorientation falling in the range $\left[ \omega _{\text{t}},\omega _{\text{t}}+\delta \right]$, where $\%ERR=\delta /\omega _{\text{t}}$. For small $\delta$, the error in $\omega _{\text{t}}$ will be larger than that of both $D_1$ and $D_2$. Specifically, this will be the case for $\delta < \omega _{\text{t}}$. We wish to identify the number of bicrystal experiments, N, required to recover $\omega _{\text{t}}$ to within $\delta$ of its true value at a 95 % confidence level.

Consider a set of N bicrystals with respective misorientations, $\left\{ \omega _1,\omega _2,\dots ,\omega _N\right\}$, which are sampled from $\Omega \sim U\!\left( 0,\omega _{\text{max}}\right)$, where $\omega _{\text{max}} = {180}^{\circ}$ for the crystal system considered in this study . Let⁴ A be the event that at least one of these bicrystals falls in the interval $\left[ \omega _{\text{t}}-\delta ,\omega _{\text{t}}\right]$, and let B be the event that at least one of these bicrystals falls in the interval $\left[ \omega _{\text{t}},\omega _{\text{t}}+\delta \right]$. We are interested in finding N such that

$$\begin{aligned} \mathbb {P}\!\left( A \cap B\right) = 0.95 \end{aligned}.$$

(15)

Thus, we seek an expression for the joint probability $\mathbb {P}\!\left( A \cap B\right)$. This may be accomplished by considering the complement:

$$\begin{aligned} \begin{array}{ll} \mathbb {P}\!\left( A \cap B\right) &{}= 1-\mathbb {P}\!\left( A^C \cup B^C \right) \\ &{}= 1-\left[ \mathbb {P}\!\left( A^C\right) +\mathbb {P}\!\left( B^C\right) -\mathbb {P}\!\left( A^C \cap B^C \right) \right] \end{array} \end{aligned}.$$

(16)

The probability that none of the N samples fall within $\left[ \omega _{\text{t}}-\delta ,\omega _{\text{t}}\right]$ is given by

$$\begin{aligned} \mathbb {P}\!\left( A^C\right) = {\left( 1-\frac{\delta }{\omega _{\text{max}}}\right) }^{N} \end{aligned}.$$

(17)

Likewise, for the interval $\left[ \omega _{\text{t}},\omega _{\text{t}}+\delta \right]$ we have

$$\begin{aligned} \mathbb {P}\!\left( B^C\right) = {\left( 1-\frac{\delta }{\omega _{\text{max}}}\right) }^{N} \end{aligned}.$$

(18)

The probability that none of the samples fall in the interval $\left[ \omega _{\text{t}}-\delta ,\omega _{\text{t}}+\delta \right]$ is given by

$$\begin{aligned} \mathbb {P}\!\left( A^C \cap B^C \right) = {\left( 1-\frac{2\delta }{\omega _{\text{max}}}\right) }^{N} \end{aligned}.$$

(19)

Substituting these results into Eq. 16 we find

$$\begin{aligned} \mathbb {P}\!\left( A \cap B\right) = 1-2{\left( 1-\frac{\delta }{\omega _{\text{max}}}\right) }^{N}+{\left( 1-\frac{2\delta }{\omega _{\text{max}}}\right) }^{N} \end{aligned}.$$

(20)

Substituting $\delta = \omega _{\text{t}} \left( \%\text {ERR}\right)$, setting Eq. 20 equal to 0.95, and solving for N numerically for 100 values of $\%\text {ERR} \in \left\{ 0.001,0.002,\ldots ,0.100\right\},$ we observe the $N\!\left( \%\text {ERR}\right)$ dependence shown in Fig. 9.

Fig. 9

Solutions of Eq. 15 for various values of $\%\text {ERR} \in \left\{ 0.001,0.002,\ldots ,0.100\right\}$

These results suggest a power-law dependence, and a fit to the data results in the following:

$$\begin{aligned} N = 44.0337 {\left( \%ERR\right) }^{-1.0003} \end{aligned}$$

(21)

with a coefficient of determination equal to $R^2 = 1.0000$ and an RMS error of 1.0407. The 95 % confidence intervals for the parameters in Eq. 21 are $\left[ 44.0216,44.0459\right]$ and $\left[ -1.0002,-1.0003\right]$, respectively. For comparison with the error scaling of the localization method (See Eq. 11), we can rearrange Eq. 21 to get the error scaling law for a bicrystal approach:

$$\begin{aligned} \%ERR = 0.0227 N^{-0.9997} \end{aligned}$$

(22)