# Comparison of simulated and measured grain volume changes during grain
growth

Xiaoyao Peng<EMAIL_ADDRESS>Department of Civil and Environmental
Engineering, Carnegie Mellon University  Aditi Bhattacharya  Department of
Materials Science and Engineering, Carnegie Mellon University  S. Kiana
Naghibzadeh  Department of Civil and Environmental Engineering, Carnegie
Mellon University  David Kinderlehrer  Center for Nonlinear Analysis,
Department of Mathematical Sciences, Carnegie Mellon University  Department of
Materials Science and Engineering, Carnegie Mellon University  Robert Suter
Department of Physics, Carnegie Mellon University  Department of Materials
Science and Engineering, Carnegie Mellon University  Kaushik Dayal  Department
of Civil and Environmental Engineering, Carnegie Mellon University  Department
of Mechanical Engineering, Carnegie Mellon University  Center for Nonlinear
Analysis, Department of Mathematical Sciences, Carnegie Mellon University
Gregory S. Rohrer<EMAIL_ADDRESS>Department of Materials Science and
Engineering, Carnegie Mellon University

(September 3, 2024)

###### Abstract

The three-dimensional microstructure of Ni, observed after five annealing
intervals, was compared to simulations of grain growth using the threshold
dynamics method with the assumption of capillarity as the only driving force.
A grain-by-grain comparison made it possible to identify the sources of
differences between the simulation and experiment. The most significant
difference was for grains of the smallest sizes, which the simulation
predicted would lose volume and disappear at a greater rate than observed in
the experiment. The loss of grains created errors in the numbers of neighbors
of the remaining grains, and it was found that errors in the simulated grain
volume were correlated to errors in the number of near neighbors. While
anisotropic grain boundary properties likely play a role in the differences,
the size dependence of the errors suggest that it might be necessary to
include a size dependence in the model for grain boundary migration kinetics.

††preprint: To appear in Physical Review Materials
(https://doi.org/10.1103/PhysRevMaterials.6.033402).

##  1 Introduction

The microstructures of polycrystalline metals and ceramics processed at high
temperatures are influenced by grain growth and this influences structure
sensitive materials properties. To better understand grain growth, analytic
theories [1, 2] and, more recently, Monte Carlo simulations [3, 4, 5, 6],
cellular automata models [7, 8], molecular dynamics [9, 10, 11, 12, 13, 14,
15, 16], vertex simulations [17, 18], phase field models [19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31], and other approaches [32, 33, 34, 35], have
been developed.

Recent studies of three-dimensional (3D) grain growth using simulation have
attempted to quantify and understand topological evolution [21, 36, 37, 38,
39, 40], volumetric growth rate [41], and grain boundary energy evolution in
isotropic as well as anisotropic grain growth [42, 43]. There have been
relatively few attempts in the past to compare the results of grain growth
simulations with experiments on a grain-by-grain basis. For example, in 2003
Demirel et al. [44] validated an anisotropic grain growth simulation in 2D
using GB curvature as a driving force and it explained 50% of the experimental
grain growth. They also showed that an isotropic simulation with 17%
explaining power had very poor matching with experimental evolution. McKenna
et al. [45] compared the evolution of individual grains in a 3D experiment
with grains in a 3D isotropic phase field simulation of grain growth in
polycrystalline $\displaystyle\beta$-Ti. Statistical analysis revealed good
agreement between experiment and simulation for grain growth kinetics.
However, direct comparison of individual grains revealed a poor match in grain
shapes and grain boundary widths because the simulation was unable to capture
local anisotropy in grain boundary energy and mobility. Another study by Zhang
et al. [46] compared a phase field simulation of grain growth with 3D
experimental data and assigned reduced mobilities to each grain boundary so
that the simulation reproduced the experiments. However, the assigned grain
boundary properties were independent of grain boundary crystallography in the
sense that crystallographically identical boundaries had to be assigned
reduced mobilities that varied by a factor of nine.

The prior work clearly shows that, unless inconsistent grain boundary
properties are assigned, as in [46], simulations of grain growth are poor
predictors of real grain growth. However, the mechanism of how the simulations
fail has not yet been identified and the current work is aimed at
understanding this issue. We compare the outcomes of 3D grain growth
simulations with the microstructure evolution in annealed Ni [47, 48]. For
microstructures containing 600 to 900 grains, we compare the volume changes,
changes in the numbers of near neighbors, and curvatures of grains in the
simulation and experiment. We concentrate on three places where the simulated
microstructures depart from the observations: the volume changes for the
smallest grains, grains for which the topology (number of neighbors) is
simulated incorrectly, and the grain face curvature. While the later two
observations might be connected to anisotropic boundary properties, the first
is more likely to stem from other factors.

To simulate grain growth driven by mean curvature, we have selected the
threshold dynamics (TD) method originally introduced by Merriman, Bence and
Osher in [49, 50]. The choice of using TD to tackle the 3D grain growth
problem comes naturally from its beneficial characteristics. First, we can use
experiment image/data directly without additional meshing. Second, the
representation of the interface is implicit, as in the phase field or level
set methods, so that topological changes are tracked automatically. Third, TD
has unconditional stability and high computational efficiency. Finally, the
isotropic formulation of the model can be easily extended to an anisotropic
formulation by replacing the isotropic kernel with anisotropic ones assuming
different anisotropic surface energy/mobility forms, a feature we are
leveraging in our ongoing work.

##  2 Methods

###  2.A Threshold Dynamics

The threshold dynamics model assumes that capillarity is the only driving
force for grain boundary motion. In TD, phases can be represented through the
characteristic function (order parameter)
$\displaystyle\mathbf{1}_{\Sigma_{i}}$ of each phase
$\displaystyle\Sigma_{i}$.

$\mathbf{1}_{\Sigma_{i}}=\begin{cases}1,&\text{if $\displaystyle{\mathbold
x}\in\Sigma_{i}$}.\\\ 0,&\text{otherwise}.\end{cases}$ (1)

The evolution of the interface network is reflected by the change of the phase
boundary of $\displaystyle\mathbf{1}_{\Sigma_{i}}$. The original TD scheme
from Merriman, Bence and Osher in the isotropic, two-phase setting is given as
in Algorithm 2.A from [49].

MBO

1:Initialization Given $\displaystyle\Sigma^{0}$ and time step size
$\displaystyle\delta t$

2:For iteration $\displaystyle k+1$

3: $\displaystyle\psi^{k}=K_{\delta t}*\mathbf{1}_{\Sigma^{k}}$
$\displaystyle\triangleright$ Convolution/Diffusion step

4: $\displaystyle\Sigma^{k+1}=\\{{\mathbold x}:\psi^{k}({\mathbold
x})\geq\frac{1}{2}\\}$ $\displaystyle\triangleright$ Threshold/Sharpening step

The convolution kernel $\displaystyle K:\mathbb{R}^{d}\rightarrow\mathbb{R}$,
can be any spherically symmetric kernel, and is usually chosen to be the
Gaussian:

$\displaystyle\displaystyle G_{\delta t}({\mathbold x})=\frac{1}{(4\pi(\delta
t))^{d/2}}\exp(-\frac{|{\mathbold x}|^{2}}{4(\delta t)})$ (2)

By this choice of kernel, the MBO scheme evolves the boundary
$\displaystyle\partial\Sigma$ by mean curvature motion [49] and the phase
boundary velocity is

${\mathbold v}({\mathbold x})=\mu\sigma\kappa({\mathbold x})\hat{\mathbold
n}({\mathbold x})$ (3)

where $\displaystyle\kappa$ is the mean curvature, $\displaystyle\mu$ is the
boundary mobility, $\displaystyle\hat{\mathbold n}$ is the boundary unit
normal, and $\displaystyle\sigma$ is the boundary energy.

The time step size $\displaystyle\delta t$ is a model variable that has no
effect on stability, but should be considered to reach desired accuracy. In
[51], Esedoglu and Otto introduced an extension of the two-phase MBO scheme to
a general multi-phase setting, as showing in the Algorithm 2.A.

EO

1:Initialization Given $\displaystyle\Sigma^{0}_{1},...,\Sigma^{0}_{N}$ and
time step size $\displaystyle\delta t$

2:For iteration $\displaystyle k+1$

3: $\displaystyle\psi^{k}_{i}=\sum\limits_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}K^{i,j}_{\delta t}*\mathbf{1}_{\Sigma^{k}_{j}}$
$\displaystyle\triangleright$ Convolution/Diffusion step

4: $\displaystyle\Sigma^{k+1}_{i}=\\{{\mathbold x}:\psi^{k}_{i}({\mathbold
x})\leq\min\limits_{j\neq i}\psi^{k}_{j}({\mathbold x})\\}$
$\displaystyle\triangleright$ Threshold/Sharpening step

The main difference between the MBO and the EO schemes is that in the EO
scheme the new kernel $\displaystyle K^{i,j}_{\delta t}$ is designated for the
boundary between $\displaystyle\Sigma_{i}$ and $\displaystyle\Sigma_{j}$. The
new kernel in the EO scheme is constructed to account for non-constant
$\displaystyle\sigma$ and $\displaystyle\mu$ between two phases. As mentioned
in the introduction, $\displaystyle\sigma$ and $\displaystyle\mu$ play far
more complex roles in grain growth than being constant. They are affected by
both the grain boundary normal and misorientation between neighboring grains.
Incorporating the anisotropic surface energy and mobility into $\displaystyle
K^{i,j}_{\delta t}$ yield to an anisotropic grain growth formulation.

In this work, we focus on multi-phase isotropic simulations. We use Algorithm
2.A and take the same form as in Equation (2) for the kernel $\displaystyle
K^{i,j}_{\delta t}$. In the grain growth simulations performed in this work,
Algorithm 2.A makes direct use of the observed microstructures as the input by
assigning one characteristic function $\displaystyle\mathbf{1}_{\Sigma_{i}}$
to each grain $\displaystyle\Sigma_{i}$, and evolves the grain boundary
network by first a convolution/diffusion step and then a threshold/sharpening
step. The developed code for this study is available online at
https://github.com/JadeXiaoyaoPeng/GrainGrowth_TD_iso.

###  2.B Model Validation

To validate the isotropic TD model, we compare the results of the model to the
expectations from isotropic grain growth theory. For example, for normal
isotropic grain growth, the resulting distribution of grain sizes is expected
to be self similar [52, 53]. The grain size distributions of the simulated
microstructures at different anneal states are plotted in Figure 1. The number
of grains in each simulated anneal state from anneal state 1 to anneal state 5
are 643, 666, 775, 715 and 606, respectively. The five curves from different
time steps overlap, with some small fluctuations that result from the limited
number of grains. Therefore, we conclude that the simulation produces self
similar grain size distributions once the times scale is considered.

Figure 1:  Grain size distributions of simulations at different annealing
states.

Burke and Turnbull [52] found a linear relation between time and average grain
size to the power n, where $\displaystyle 2\leq n\leq 4$, using an isotropic
curvature-driven model for both 2D and 3D systems. The most commonly reported
exponent for grain growth is n = 2. In Figure 2, the difference between the
2nd power of average grain size $\displaystyle\langle R\rangle^{2}$ and
initial average grain size $\displaystyle\langle R_{0}\rangle^{2}$ of a
simulation are plotted as a function of iteration step. The iteration steps
can be interpreted as normalized time steps. The change in the square of the
average grain size with time is well fit to a line, with a correlation
coefficient ($\displaystyle R^{2}$) equal to 0.99. Throughout this paper,
grain size is calculated as $\displaystyle R=(\frac{3}{4\pi}V)^{1/3}$ with
$\displaystyle V$ being the volume of the grain in voxels.

Figure 2:  $\displaystyle\langle R\rangle^{2}-\langle R_{0}\rangle^{2}$ as a
function of iteration step. $\displaystyle r^{2}$ is the R-squared value from
the linear fit of the data.

Taken together, Figures 1 and 2 indicate that the grain growth model produces
results consistent with the classical theory.

###  2.C Experimental data

The 3D simulations described here are instantiated with observed Ni
microstructures and compared to a second observation of the same sample at a
later time. Prior to the grain growth experiment at 800 ∘C, the high-purity Ni
was annealed at 950 ∘C for 6 [47], which led to a completely recrystallized
microstructure. The X-ray data showed that the orientation spread within the
grains was less than 0.1∘. With this spread, the geometrically necessary
dislocation density is less than 2.5$\displaystyle\times$1011/m2. The
estimated driving force for boundary migration from dislocations (the product
of the dislocation energy per length and the dislocation density) is therefore
on the order of 1 kJ/m3. The capillary driving force provided by
20$\displaystyle\times$10-6 m grains is about 100 times larger. The sample was
measured at six instances in time using near-field high energy X-ray
diffraction microscopy [47, 54] and was used to reconstruct the shapes and
orientations of the grains within six 3D volumes [55, 56]. Between each
measurement, the sample was annealed for $\displaystyle\approx 30$ min at
$\displaystyle 800^{\circ}$C. The details of the data acquisition and
interpretation have been described in previous publications [47, 55, 56]. For
this work, we segmented the grains and represented the microstructures as a
set of discrete voxels using DREAM.3D [57] as described in [48]. The Ni
microstructure has an abundance of twin boundaries that have a significantly
lower energy than other Ni grain boundaries [58, 59]. Obviously, such features
cannot be reproduced by a simulation with isotropic boundaries properties. To
create a microstructure with a narrower range of grain boundary energies that
more closely approximates the isotropic grain growth model, all neighboring
grains with the twin misorientation were merged to form a single grain using
the merge twins function in DREAM.3D [57]. After the twins were merged, the
microstructures contained 600 to 900 grains made up of voxels with dimensions
of $\displaystyle 2.3\times 2.3\times 4.0\,\mu$m. In the initial state, there
was an average of 7582 voxels per grain.

###  2.D Data processing for comparison

####  2.D.1 Establishing the common volume.

The cylindrical Ni sample in the initial state is illustrated in Figure 3(a),
where the 920 grains are colored by orientation. The experimental volume was
cylinder shaped with free surfaces around the periphery and orientations were
sampled on a fixed grid. These geometric features determined the geometry of
the simulation. The simulations volume is as rectangular parallalepiped, as
illustrated by the frame in Figure 3(a), where the transparent voxels do not
belong to any grain. The simulation directly uses the orientation map to
create characteristic functions $\displaystyle\mathbf{1}_{\Sigma_{i}}$ as in
Equation 1, and the phases $\displaystyle\Sigma_{i}^{0}$ are the input for
iterations as in Algorithm 2.A. Thus, the computational grid of the simulation
has the same size of the input microstructure, for example $\displaystyle
369\times 372\times 62$ for anneal state 0.

Because of the cylindrical shape of the sample, a special procedure must be
used to exclude the space that is inside of the computational grid but outside
of the actual sample (transparent region in Figure 3(a)). The easiest way to
accomplish this is to manipulate the characteristic function of the
transparent region $\displaystyle\Sigma_{0}$. By setting
$\displaystyle\mathbf{1}_{\Sigma_{0}}=0,\text{if $\displaystyle
x\in\Sigma_{0}$ }$ the contribution to $\displaystyle\psi_{i}$ ($\displaystyle
i=1,2,...920$) from $\displaystyle\Sigma_{0}$ is always 0 in the convolution
step in Algorithm 2.A, and the boundary between $\displaystyle\Sigma_{0}$ and
$\displaystyle\Sigma_{i}$ will not evolve during the iteration. Thus, the
volume is preserved throughout the simulation.

To compare the microstructure in two different time steps, they must be in the
same spatial reference frame and have the same volume. Because non-identical
fields of view were imaged at each time step, and the sample position was not
perfectly reproduced, it is necessary to both translate the volume and crop it
vertically to obtain a constant volume in a fixed frame of reference. We first
translated each volume until we identified the translation with the minimum
disorientation between the two time steps. The larger volume was then cropped
in the vertical direction so that both volumes had the same number of voxels
at identical locations.

####  2.D.2 Establishing the time scale.

As the simulation proceeds, the average grain size increases with the square
root of time, as illustrated in Figure 2. To compare with experiment, a
criterion is needed to decide when to stop the simulation at an equivalent
time. One was to use a voxel-by-voxel comparison of the grain identification
(ID) number, counting all voxels that matched and taking the maximum of this
as the point of best agreement. The second criterion was to determine the
average distance that the boundaries moved in the simulation and match this
with experiment. The third criterion was to stop the simulation at the time
when it had the closest average grain size as the experiments. To decide which
to use, we compared the number of grains in the simulated and experimental
microstructure. None of the methods were perfect, but the criterion based on
the nearest average grain size led to grain counts in the simulation that were
closest to the experiment.

##  3 Results

Figure 3 (a) shows the experimentally measured microstructure of the initial
anneal state of 920 grains and (b) the experimental final state microstructure
with 756 grains and (c) the simulated microstructure of the final anneal state
with 642 grains. The average grain size in (a) is $\displaystyle 26.0\,\mu$m,
in (b) is $\displaystyle 26.8\,\mu$m. and in (c) is $\displaystyle
30.2\,\mu$m. Between the grains that matched, the reconstructed initial
experimental state has an average of 11 faces per grain, and the final state
has 11 faces per grain while the simulation state has 9.6 faces per grain.
Comparing the top surface between the two, the interfacial network of the
simulated grains is smoother than the experiment. This will be addressed in
detail later.

Figure 3:  (a) Experimentally measured microstructure of the initial anneal
state of 920 grains and (b) experimental final state with 756 grains and (c)
simulated microstructure of the final anneal state with 642 grains.

Figure 4 takes a closer look at the faces of a grain for the initial and final
experimental state (a), (b) and the final simulation state (c). The number of
grain faces changes in the simulation and experiment, but not in exactly the
same way. While there are six anneal states, each state was used as an initial
state to simulate the next one. A grain that does not contact any of the
external surfaces has been visualized at all states as it evolves from Anneal
State 0 to Anneal State 5 in Figure 5. This grain shrinks during each
annealing interval in the experiment as well as simulation.

Figure 4:  A grain in experimentally measured microstructure of the (a)
initial anneal state, (b) final anneal state and (c) the corresponding grain
in the simulated microstructure of the final state. Grain faces are colored to
make them distinguishable. Figure 5:  Visualization of a bulk grain across all
Anneal States in experiment and comparison with simulation of the same grain.
The top row and bottom row shows experimental and simulation grains starting
from An0 to An5 respectively.

The experimental and simulated changes in grain radius for the grain
visualized in Figure 5 have been plotted in Figure 6 as a function of the
anneal states. Starting at a radius of $\displaystyle 42.2\,\mu$m for both,
the experimental plot shows the grain grew until An2 and then shrank to
$\displaystyle 40.6\,\mu$m. While the simulated grain also shrank in the final
state ($\displaystyle 40\,\mu$m), it did not follow the same size trajectory.
For example, while both started from the same radius, the simulated grain
shrank between An0-An1 but in the experiment it grew. Ultimately though, the
grain shrank and the simulation could captured that.

Figure 6:  The actual and simulated evolution of the grain size of the bulk
grain in Figure 5 at every anneal state.

Figure 7 depicts the grain size distribution of observed and simulated
microstructures for all the anneal states. While the simulation is mostly
coincident with the actual grain size distribution, there are significant
differences for the smaller grain sizes. The distribution shows that the
simulated microstructure has a significantly higher fraction (0.11%) of
smaller grains (<$\displaystyle 5\,\mu$m) than in experiment (0.01%). In other
words, simulation is shrinking more grains than shrink in the experiment.

Figure 7:  The grain size distribution of observed and simulated
microstructures for all the anneal states.

Figure 8 plots the histogram of fractional change in volume for observed and
simulated grains. The data includes all grains that matched from the initial
anneal state to the final anneal state as well as the grains that were smaller
than the average grain size that did not find a match and were assumed to have
disappeared (fractional volume change of -1). It is seen that approximately
the same fraction of grains have shrunk to zero in both experiment and
simulation ( 10%). In both simulation and experiment, the maximum of the
distribution is for grains that have small volume changes near zero. Also,
there are more grains in the experiment that increase in size as shown by the
right tail of the distribution.

Figure 8:  Histogram of fractional change in volume for final state
experimental and simulated grains. The markers show the bin centers and the
lines connect bin centers to guide the eye.

To understand Figure 8 in more detail we consider Figure 9, which plots the
fractional change in volume as a function of grain radius for both simulation
and experiment. The simulation tends to shrink the small grains. Note that
nearly all of the simulated grains in the size range of $\displaystyle 5$ to
$\displaystyle 25\,\mu$m have negative volume changes while grains in the same
size range in the experiment are equally probable to have positive or negative
volume changes. Counter-intuitively, most of the largest positive fractional
volume changes occur for the small grains in the experiment. When correlated
with Figure 8, it means that the lower extreme of fractional change in volume
(<-0.6) corresponds to grain sizes less than $\displaystyle 35\,\mu$m. It also
shows that in both experiment and simulation, as the grain size gets lager,
the magnitude of fractional change is smaller. This suggested that the
mismatch at the right tail of the distribution in Figure 8 is mainly caused by
incorrect volume predictions for small grains. Notice that the grains that
disappeared in the simulation are clustered in the small grain size region,
whereas grains of the same sizes exhibit significant growth in the experiment.
Based on Figure 9, we conclude that most of the differences between the volume
changes in Figure 8 arise from errors in the volume predictions for the small
grains.

Figure 9:  The fractional change in volume vs grain size in experimental and
simulated grains.

Figure 10 compares simulated and observed volume changes in the same grains.
The observed volume changes have been classified into discrete bins and the
markers in the plot are the means of the values in each bin and the bars are
the standard deviation. The negative extreme of the plot is much flatter; this
is because grains that disappeared are not included. Therefore, the sum of the
increased and decreased volumes is not zero. In other words, the volume is not
conserved because of the over prediction of grain shrinkage. The plot shows a
positive direct relationship between the experimentally observed and the
simulated change in volume. When the experimental volume change is greater
than zero, the simulated volume change also is positive, and we see a positive
monotonic trend between the two. The slope of the trend is lower than the
ideal $\displaystyle 45^{\circ}$ line, meaning simulation under-predicts the
grain growth. It was found that for all matched grains between each anneal
state pair, considered together, the simulation has correctly predicted the
sign of volume change 62% of the time.

Figure 10:  Simulated volume change as a function of observed volume change in
experiment for all matched grains.

Figure 11 shows a comparison between the unsigned triangle curvatures from the
grain boundary mesh for all matched grain boundaries of the experiment and
simulation. The y-axis displays the fraction of triangles in the simulation
and experiment. The lower curvatures have been overpredicted by the simulation
while it is underpredicting higher values of curvature. In other words, the
simulation produces smaller grain boundary curvatures. Experimental curvatures
have a higher fraction of triangles with curvatures greater than
$\displaystyle 0.008\,\mu$m-1 while simulation triangles have a higher
fraction of lower curvature values (14%) than experiment (12%). This
quantifies the visual impression one gets from comparing the experimental and
simulated microstructures, that the simulated microstructures have smoother
interfaces.

Figure 11:  Absolute triangle curvatures for all boundaries in the simulation
and final experiment state.

One source of the volume error might be errors in the predicted topology of
the grains and to test this hypothesis, we compare the volume prediction error
($\displaystyle VPE$) with the topological error ($\displaystyle TE$) for
individual grains. $\displaystyle VPE$ and $\displaystyle TE$ are defined as
follows:

$\displaystyle\displaystyle\Delta N_{s}$
$\displaystyle\displaystyle=N_{\text{sim}}-N_{\text{exp(initial)}}$ (4)
$\displaystyle\displaystyle\Delta N_{e}$
$\displaystyle\displaystyle=N_{\text{exp(final)}}-N_{\text{exp(initial)}}$ (5)
$\displaystyle\displaystyle TE$ $\displaystyle\displaystyle=\Delta
N_{s}-\Delta N_{e}$ (6) $\displaystyle\displaystyle VPE$
$\displaystyle\displaystyle=\frac{V_{s}-V_{e}}{V_{e}}$ (7)

where $\displaystyle N_{\text{exp(initial)}}$ is the number of neighbors of a
grain in the initial experiment state, $\displaystyle N_{\text{exp(final)}}$
is the number of neighbors of the same grain in the final experiment state and
$\displaystyle N_{\text{sim}}$ is the number of neighbors of the same grain in
the final simulation state. $\displaystyle VPE$ is the fractional difference
in volume predicted by simulation of final anneal state ($\displaystyle
V_{s}$) and experimental final state ($\displaystyle V_{e}$). $\displaystyle
TE$ is the difference in $\displaystyle\Delta N$ for each grain between
simulation and experiment. In other words, $\displaystyle TE$ is the error in
predicting topological evolution by the simulation. Figure 12 plots the volume
prediction error as a function of topological error. A low $\displaystyle VPE$
indicates a small difference between the final volume predicted and the actual
final volume of the grain. A high $\displaystyle TE$ value means there is a
large error in predicting the topological evolution of the grains. This plot
is approximately linear and monotonically increasing. When $\displaystyle TE$
is close to zero, $\displaystyle VPE$ is also close to zero. As $\displaystyle
TE$ increases, the error in the predicted volume also increases, in both
positive and negative directions.

Figure 12:  The volume prediction error ($\displaystyle VPE$) as a function of
topological error ($\displaystyle TE$). $\displaystyle VPE$ is the fractional
difference in predicted and observed grain volume. $\displaystyle TE$ is the
difference in grain face evolution between simulation and experiment.

##  4 Discussion

Based on Figs. 8 and 9, one of the key features of the simulation that differ
from the experiment is the behavior of the smallest grains. As an example,
during the simulation of growth during one annealing period, 101 grains
disappeared. These grains are illustrated in Fig. 13, illustrating that they
are relatively small and positioned randomly in the volume. Of these grains,
57 (47%) also disappeared in the experiment. Of the remaining grains, about
2/3 of them got smaller or did not change volume. One supposes that
eventually, these grains will also disappear. However, the remaining grains
(1/5 of the total) increased in volume, directly contrary to the simulations.
Whether a grain grows or not has more to do with its environment than its size
[48, 60] and it has been shown that small grains can form and grow, provided
they decrease the total interfacial energy [61]. However, even the grains
undergoing a negative volume change in the experiment appear to shrink more
slowly than in the simulation. This suggests the possibility of some physical
process that retards the migration of boundaries around the smallest grains.
It does not seem likely that the decreased rate of shrinkage could be a result
of the grain boundary energy anisotropy, which is not accounted for here.
While a spectrum of grain boundary energies would alter the driving force and
grain boundary migration rate, it is just as likely to increase the rate as
decrease the rate, and the grain boundary energy is not expected to change
with the grain size. One possibility is that as the grain boundary area
decreases, the local concentration of segregated impurities [62] or boundary
defects increases [63] and this influences migration kinetics. However, this
possibility cannot be tested with the available data.

Figure 13:  Visualization of the 101 grains in An4 that shrunk to zero in An5
in simulation. These grains are colored by IPF coloring and are opaque while
the rest of the microstructure is transparent.

It was noted that the grain boundaries had, on average, lower curvatures than
the experimental data set. This observation cannot explain the difference in
the behavior of the shrinking grains, as reduced curvature should reduce the
rate of grain volume change. Because the energies are anisotropic in the
experiment, the curvature is not expected to be uniform as it is in the
simulations. The variations in local curvature in the experiment may be a
result of grain boundary energy anisotropy or simply are an artifact of the
experiment’s finite spatial resolution. In any case, it is difficult to
predict how this would influence grain boundary migration; for the case of
anisotropic grain boundary energy, the migration rate depends on the grain
boundary stiffness and the anisotropy of the stiffness is greater than that of
the grain boundary energy [64]. However, grain boundary properties are not
expected to change with size, so this does not explain the difference in the
behavior of the small grains.

It was also found that errors in the volume change of each grain are tied to
errors in topological changes. In other words, if a grain’s change in its
number of neighbors is the same in the simulation and experiment, the volume
change is more accurately predicted. This is not too surprising. Once the
simulated and experimental grain have a different number of near neighbors,
the sizes of the grain faces and grain face curvatures will be different,
causing them to evolve differently. The greater the difference in the number
of neighbors, the larger the error in the volume change. The source of these
errors is difficult to identify, but one is certainly disappearance of more
grains in the simulation. Assuming the grains that disappear have a minimum of
four neighbors, each grain that disappears in the simulation but not the
experiment changes the number of neighbors of at least four grains. For the
example discussed above, where 64 more grains disappear in the simulation than
in the experiment, this changes the number of neighbors of as many as 256
grains.

There have been a number of recent grain growth simulations that assume
anisotropic grain boundary energies [65, 66, 67, 42]. It is envisioned that if
realistic models for the grain boundary energy anisotropy are used, they will
better predict the evolution of the microstructure. However, the current
results suggest that it might be necessary to include a size dependent
migration model because grains that disappear too quickly change the
environments of other grains and this leads to errors in the prediction of
volume changes. In future work, the isotropic TD scheme used here will be
extended to include anisotropic surface energy and mobility data from
experiment [68]. This platform will make it possible to explore different
forms of the anisotropic kernel that will best model the anisotropic grain
growth in Ni.

##  5 Conclusions

Experimentally observed Ni microstructures, at six time steps, were compared
in a grain-by-grain fashion to the results of isotropic grain growth
simulations. This comparison led to the following conclusions. The simulation
predicts the correct sign of the volume change for only 62% of the grains. The
errors are the most pronounced for the smaller grains, for which shrinkage is
over predicted and growth is under predicted. Improved grain growth models
might need to incorporate migration kinetics influenced by grain size. The
grain boundary curvatures in the simulation are systematically lower than in
the experimental observations. Volume prediction errors are correlated to
errors in predicting topological changes. When the simulation captures the
topological changes correctly, it can predict the grain volume change
accurately as well.

## Software Availability

A version of the code developed for this work is available at
https://github.com/JadeXiaoyaoPeng/GrainGrowth_TD_iso

###### Acknowledgements.

This work was supported by the National Science Foundation under DMREF grants
1628994 and 2118945. We acknowledge NSF for XSEDE computing resources provided
by Pittsburgh Supercomputing Center. This research used resources of the
Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science
User Facility operated for the DOE Office of Science by Argonne National
Laboratory under Contract No. DE-AC02-06CH11357.

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