Bridging Simulations and Experiments in Microstructure Evolution

Abstract

We demonstrate the importance of anisotropic interface properties in microstructure evolution by comparing computed evolved microstructures to final experimental microstructures of 5170 grains in 19 thin aluminum foil samples. This is the first time that a direct experimental validation of simulation has been performed at the level of individual grains. We observe that simulated microstructures using curvature-driven grain boundary motion and anisotropic interface properties agree well with experimentally evolved microstructures, whereas agreement is poor when isotropic properties are used.

Full text

Bridging Simulations and Experiments in Microstructure Evolution
M. C. Demirel,
1,2,3
A. P. Kuprat,
2
D. C. George,
2
and A. D. Rollett
3
1
Materials Science and Technology, MST-8, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
2
Theoretical Division, T-1, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
3
Carnegie Mellon University, Department of Materials Science & Engineering, Pittsburgh, Pennsylvania 15213
(Received 28 May 2002; published 9 January 2003)
We demonstrate the importance of anisotropic interface properties in microstructure evolution by
comparing computed evolved microstructures to final experimental microstructures of 5170 grains in
19 thin aluminum foil samples. This is the first time that a direct experimental validation of simulation
has been performed at the level of individual grains. We observe that simulated microstructures using
curvature-driven grain boundary motion and anisotropic interface properties agree well with experi-
mentally evolved microstructures, whereas agreement is poor when isotropic properties are used.
DOI: 10.1103/PhysRevLett.90.016106 PACS numbers: 68.35.–p, 68.37.Hk, 68.55.Jk, 81.40.–z
This paper seeks to extend previous statistical compari-
sons of predicted and experimentally observed grain
boundary network evolution by demonstrating agreement
at the scale of individual grains, provided that the anisot-
ropy of interfacial energy and mobility is included. With
very few exceptions [1], only statistical comparisons have
been made such as determining the exponent in the
power-law relationship between average radius and time.
In addition to this general aim, we consider coarsening in
networks of low angle grain boundaries (subgrains),
which has been the subject of some controversy. Some
authors have postulated that the interfaces are essentially
sessile because they are themselves networks of lattice
dislocations and that coarsening occurs by rotation of
individual subgrains [2,3]. Rotation is driven by minimi-
zation of interfacial energy and tends to eliminate the
misorientation between certain pairs of adjacent grains,
thus causing coalescence and coarsening to occur. Gleiter,
on the other hand, showed that low angle grain boundaries
migrate under curvature driving forces just as observed
for high angle grain boundaries [4]. Clearly, both mecha-
nisms are feasible for coarsening of subgrain networks,
and so it is of some importance to determine which one is
applicable to real materials. Since the torque that drives
grain rotation is supralinear in (inverse) grain size, it is
reasonable to expect that curvature-driven migration
dominates at the large grain sizes observed in this ex-
periment whereas rotation should dominate in nanoscale
grain sizes.
We have previously shown a strong agreement between
small-scale grain growth experiments and anisotropic
three-dimensional simulations [5] obtained from elec-
tron backscatter diffraction (EBSD) measurements [6].
Using the same technique, we obtained data for 5170
grains from 19 thin aluminum foil samples with colum-
nar grain structure (thereby avoiding serial section-
ing) and compared our computational results with
experiments.
Our simulation model uses curvature-driven motion
implemented by
GRAIN3D [7], a three-dimensional,
gradient-weighted moving finite elements code. We as-
sume that the grain boundary motion is proportional to
the local mean curvature of the interface,
v
n
 ; (1)
where v
n
is the normal velocity of the interface,  is the
grain boundary mobility, and  is the interface energy
per unit area.  is the sum of principal curvatures, i.e.,
twice the mean curvature; in these simulations, the cur-
vature is equal to the curvature observed in the plane.
The interfaces are represented as piecewise linear. For
details of the simulation method, the reader should refer
to Ref. [7].
The interfacial anisotropy is based on a previous deter-
mination of grain boundary energy, , and mobility, ,
from a statistical/multiscale analysis of triple junction
geometry and crystallography in aluminum [8–10], and
the result for energy is in very good agreement with
the Read-Shockley model [11] as expected. In the highly
textured, columnar grain structure aluminum foil we
investigated, most interfaces are low angle boundaries
(misorientation <15

). The grain boundary mobility is
low for small misorientations but undergoes a sharp
transition to high mobilities between 10

and 15

in
misorientation which is in agreement with the literature
[12–14]. For our simulations, we assume that all high
angle grain boundaries (> 15

) have the same values of
energy and mobility. The occurrence of high angle
boundaries is very low in this material, which means that
only a small error is introduced by this assumption. This
experiment permitted a verification of curvature-driven
interface motion [4], as compared with the competing
mechanism of grain rotation [3] leading to coalescence
[2]. We found a standard deviation of 0:98

in misorien-
tation angle between the initial and final experimental
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states, implying insignificant rotation of the boundary
[15] and that interface motion is indeed curvature driven.
To generate input for the simulation from EBSD ex-
perimental results, grain boundaries and triple junctions
are located from the recorded image, and the grain
boundary structure on the section plane is extruded to
create the three-dimensional mesh [16,17]. We use the
EBSD measurements to assign a single average orienta-
tion for each grain and thus a single interface misorien-
tation for each pair of grains. Using the results of Yang
et al., the interface misorientation gives a relative value
of  and  for each interface. Note that use of a constant
value of  along the boundary from one triple junction to
the other means that our model does not include boundary
inclination dependence. The total volume of the structure
is kept constant and exterior boundaries are assumed to
be quasiperiodic [5]. Figure 1(A) shows a representative
experimental input. The simulation was initialized with
this state and was run until the area match function,
explained in the next paragraph, began to decrease. The
final experimental microstructure is shown in Fig. 1(B),
and is to be compared to the computed evolved micro-
structure shown in Figs. 1(C) and 1(D). A strong cube
texture, f100gh001i, was observed for both the initial and
the final experimental microstructures. This is as ex-
pected because of the small amount of coarsening that
has taken place during the experiment.
Simulations were performed with three different as-
sumptions for the interface properties. First,  and 
were assumed constant and equal to unity (isotropic).
Second,  was allowed to depend on the misorientation
angle (anisotropic mobility). Third,  and  were al-
lowed to depend on the misorientation angle (anisotropic
mobility and energy). For quantitative comparison, we
introduce a regular grid of sampling points sx; y of
between 2500 and 10 000 points depending on the size
of the microstructure. A quantitative measure for the
comparison is then given by a normalized area match
function (NAMF). One may think of this comparison
method as overlaying the final experimental state onto
successive simulation snapshots and measuring at each
snapshot time the amount of area where the orientations
match. The higher the value, the better the match.
NAMF
i

1
Z
X
x;y
s
i
x; ys
exp
x; y; (2)
where for simulation snapshot i, s
i
is the crystal orien-
tation, and s
exp
is the crystal orientation for the final
experimental configuration,  is the delta function, and Z
is the number of sampling points. If the crystal orienta-
tion for the spatial point x; y is the same as the final
experimental crystal orientation at the same location, the
delta function is one, otherwise it is zero. NAMF
i
is then
given by summing over the set of all x; y points and
dividing by the total number of points. Perfect agreement
is denoted by NAMF  1 and perfect disagreement by
NAMF  0. Since the final experimental state evolved
from the initial experimental state, which is identical to
the initial simulation state, the comparison begins with a
high degree of match. In a representative microstructure,
we find a 62% match between the initial simulation
state and the final experimental state (NAMF  0:62).
Figure 2(A) shows the results of isotropic and anisotropic
simulations for one sample. In the isotropic case, the
maximum NAMF gives a 68% match, whereas both
anisotropic cases reach a significantly higher maximum
NAMF. Figures 1(E) and 1(F) illustrate the pixelized
comparison of experimental to simulated final states
from which NAMF is calculated. All simulations even-
tually exhibit a decreased match as they were run longer
than the equivalent final experimental time. It is clear
from these results that anisotropic interface properties
play an important part in microstructure evolution.
With regard to simulations involving anisotropic energy,
we observed violations of the wetting condition (Young’s
equation) at some triple junctions. These force imbalances
occur at triple junctions where one of the three energies is
FIG. 1 (color online). (A) Initial experimental microstructure
(top view) with dimensions 2000 m  2000 m (339 grains).
(B) Final experimental microstructure. (C) Isotropic mobility
simulation final state. (D) Anisotropic mobility simulation final
state. (E) Pixelized subtractions of (C) from (B). (F) Pixelized
subtraction of (D) from (B).
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greater than the sum of the other two and can result from
using a constant  along the entire interface. To prevent
this, we constrained the dihedral angles so that they are
always larger than 90

by adding an isotropic minimum
energy to all boundaries [15]. Because of the uncertainty
of the effects of adjusting  in this manner, we consider
only the anisotropic mobility simulations in the remain-
der of this Letter.
Scaling the 19 simulation times so that the average of
all simulations matches the experimental time of 20 min
at 550

C, we find a low standard deviation, 1.80 min,
suggesting that simulated evolution and experimental
grain growth are consistent. In Fig. 2(B), for each of
the 19 simulations using anisotropic mobility proper-
ties, we display a vector representing the amount of
experimental microstructure evolution matched by the
simulation. Each vector origin gives NAMF
0
or the
amount of match between the final experimental state
and the initial microstructure state. Each vector tip
shows the NAMF
max
or the amount of match between
the experimental and simulated final states. To com-
pare the observed experimental microstructure evolution
with the anisotropic simulated evolution, we calculate
the mean (0.5) and standard deviation (0.172) of
NAMF
max
 NAMF
0
=1  NAMF
0
, showing that our
simulation model accounts for 50% of the microstructure
evolution observed in experiment. The mean and standard
deviation for isotropic simulations are 0.17 and 0.147,
respectively.
The comparison between the experiment and compu-
tation at the individual grain level extends our knowledge
of grain evolution. First, microstructural evolution in
columnar aluminum foils can be correctly modeled using
anisotropic parameters. The results of three-dimensional
computer simulations using anisotropic mobility agree
well with experimental grain growth. Second, we have
shown that isotropic modeling has very little predictive
value. Using NAMF as a metric, we have shown that
isotropic modeling explains at most 17% of the evolution,
whereas anisotropic modeling accounts for 50% of the
evolution. The failure of our model to account for all the
evolution may be due to one or more of the following:
boundary inclination dependence, computational indeter-
minacy due to grain topology changes [7], dislocation
density, stored deformation energy, or by sample error
such as surface defects, sample preparation, local defor-
mations, and grains that are not columnar. Additionally,
in these mesoscale experiments, we observed a 0:98

standard deviation in misorientation angle between ini-
tial and final experimental states implying insignificant
grain rotation and that microstructure evolution is curva-
ture driven.
This research is support by Los Alamos National
Laboratory (DOE, W-7405-ENG-36), the MRSEC pro-
gram of NSF (DMR-0079996), and the Computational
Materials Science Network (US-DOE).
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0.6
0.7
0.8
0.9
isotropic energy(σ ) and mobility(µ)
anisotropic mobility (µ)
anisotropic mobility (µ) and energy(σ)
NAMF
Equivalent Experimental Time (minute)
0 2 4 6 8 10 12 14 16 18 20
0.5
0.6
0.7
0.8
0.9
1.0
NAMF
max
NAMF
0
NAMF
0
, NAMF
max
Microstructure number
(A) (B)
FIG. 2 (color online). (A) Results of a representative simulation with different interface properties: The isotropic case reaches a
maximum NAMF value of 0.68, accounting for 16% of the match; both anisotropic cases reach maximum NAMF values between
0.82 and 0.77, accounting for 53% and 40% of the match, respectively. (B) NAMF
max
and NAMF
0
values for 19 microstructures.
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