A Method of Measuring Stored Energy Macroscopically
Using Statistically Stored Dislocations in Commercial Purity
Aluminum
MITRA TAHERI, HASSO WEILAND, and ANTHONY ROLLETT
Stored energy from plastic deformation in rolled aluminum has been quantified with both macroscopic and microscopic methods. Differential scanning calorimetry (DSC) and Microhardness tests
were used to determine a value for stored energy based on energy released during recrystallization
and resistance to plastic flow from the accumulated dislocation content, respectively. For a value of
stored energy based only on geometrically necessary dislocations, orientation imaging microscopy
(OIM) within a scanning electron microscope (SEM) was used and supported by transmission electron microscopy (TEM) observation of subgrain cell structure. A value for the average misorientation angle that could be associated with the TEM was obtained from the OIM data. The values of
stored energy derived from the various analyses were found to be similar with slight overestimation
from the OIM technique. Thus, the difference between the macroscopic and microscopic methods
represented the statistically stored dislocations.
I. INTRODUCTION
IN order to measure the mobility of grain boundaries during recrystallization, it is imperative to quantify accurately the
driving force for migration in the functional relationship V M P. It has long since been understood that the driving force
for migration of boundaries during the process of recrystallization is the stored energy of cold work. In recent publications, stored energy has been measured as a driving force for
recrystallization (in units of stress) in two primary ways: either
directly via differential scanning calorimetry (DSC)[1,2,3] or
indirectly with orientation imaging microscopy (OIM)[4–9] where
stored energy is derived from some measure of orientation gradients. Though both methods have proved useful, measurements of the geometrically necessary dislocation content do
not take into account all dislocations[9,10] that might contribute
to the overall driving pressure, and, specifically, cannot measure the statistically stored dislocation density because of limitations in accuracy of orientation measurement and spatial
resolution inherently not possible by OIM.
Calorimetry[11,12] detects an energy change upon reaction,
enthalpy, and so includes all the free, or statistically stored,
and geometrically necessary dislocations that are eliminated
during annealing. Orientation imaging microscopy, however,
does have the capability of determining subgrain size and
orientation within the limitations of the technique. Typical
spatial and angular resolutions in automated electron backscattered diffraction (EBSD) systems are 0.1 m and 0.5 deg,
MITRA TAHERI, formerly with the Department of Materials Science and
Engineering, Carnegie Mellon University, is Postdoctoral Fellow, United States
Naval Research Laboratory. HASSO WEILAND, Scientist, is with the Alcoa
Technical Center, Alcoa Center, PA 15069. ANTHONY ROLLETT, Professor, is with the Department of Materials Science and Engineering, Carnegie
Mellon University, Pittsburgh, PA 15213. Contact e-mail: mitra.taheri@
gmail.com
This article is based on a presentation made in the symposium entitled
“Processing and Properties of Structural Materials,” which occurred during
the Fall TMS meeting in Chicago, Illinois, November 9–12, 2003, under
the auspices of the Structural Materials Committee.
METALLURGICAL AND MATERIALS TRANSACTIONS A
respectively.[4,5] This article attempts to quantify the stored
energy values not only with these two methods, but also with
transmission electron microscopy (TEM) for a more precise
characterization of subgrain size. Vicker’s microhardness was
used to check the accuracy of the calorimetry results.
Alloy 1050 was chosen for this study specifically for its low
solute content. This, coupled with the high stacking fault energy
of aluminum, and a moderately high homologous temperature
for cold working, promotes the formation of subgrains in a
single-phase microstructure. These characteristics are nearly
optimum for the comparison of methods undertaken here.
II. EXPERIMENTAL METHODS
A. Differential Scanning Calorimetry
Measurements of the total stored energy (i.e., stored energy
arising from all dislocations) were performed using a PerkinElmer DSC-7 calorimeter (Perkin-Elmer, Wellesley, MA).
This particular model is a power-compensated differential
scanning calorimeter, with a feedback loop to maintain equal
sample and reference temperatures during a heating experiment. The difference between the powers supplied to the two
independently heated samples is recorded as a function of
the reference temperature.[11]
In these particular experiments, uniform samples of 0.2
mm3 were punched out of 52-mm (0.2-in.) thick cold-rolled
slabs of A11050. These samples were placed directly in aluminum sample holders and heated from 50 °C to 500 °C at
rates of 40 °C/min, 50 °C/min, and 60 °C/min.
B. Microhardness (Hv) Measurements
Hardness measurements were performed using a ZWICK*
*ZWICK is a trademark of Zwick/Roell, Zwick USA, Kennesaw, GA.
Vicker’s microhardness indenter; a total of six hardness indents
were performed at different locations along the rolled sheet
VOLUME 37A, JANUARY 2006—19
Table I. Vicker’s Microhardness Parameters
Calculation/Measurement
Table II. TSL Software Diameter and Misorientation
Input/Output
Value
TSL Parameter
Average of diamond indent
diameters
Alpha (constant)
Taylor factor, M
Angle
Load
Resulting hardness
Estimated flow stress
Shear modulus at 300 K
Calculated stored energy
0.5(d1 d2) d 30 m
0.5
3.0
130 deg
200.0 g
395.0 MPa
131.7 MPa
26 GPa
0.296 MPa
sample for the average value of hardness, where F is the applied
load and d is the average diameter of the diamond indent:
2F sin
HV 130 deg
2
d2
[1]
An approximate flow stress, which is taken as Hv /3, was
then used in the following relationship to yield an overall
value stored energy, ED, where G is the shear modulus of
the aluminum, M is the Taylor factor, and the constant is
approximated as 0.5.[9,10]
ED (Hv /3 M a)2
G
[2]
Table I shows the parameters that were used in Eqs. [1]
and [2].
C. Orientation Imaging Microscopy (SEM)
In order to characterize the driving pressure microscopically, the average subgrain diameter was calculated
using EBSD within a PHILIPS* XL-40 field emission gun–
*PHILIPS is a trademark of Philips Electronic Instruments, Mahwah, NJ.
scanning electron microscope (FEG–SEM); TSL** OIM soft
**TSL is a trademark of EDAX/TSL, Draper, UT.
ED u
u
b a1 ln a bbd
um
um
D
0.150 m
4X4
15 deg
0.723 m
0.824 deg
*A detailed view of subgrain delineation within the TSL software is
shown in Fig. 7.
Within these areas of similar orientation are orientation
gradients, equivalent to low-angle boundaries. To calculate
an average misorientation along these low-angle boundaries
(subgrain boundaries), the TSL software was adapted to calculate the average misorientation for only those grains
entirely bounded by misorientations greater than 15 deg; the
misorientation determination simply refers to the minimum
crystallographic misorientation associated with a line segment separating two measurement points in a scan; however, since the orientations at both points A and B separated
by the line segment are known, the misorientation, g, associated with a line segment can be calculated using Eq. [4],
where application of appropriate symmetry to find the minimum physical rotation is implied:
g gAgBT
[4]
The TSL software was used to calculate the circle equivalent diameter from the area of each grain. Table II provides a summary of the input and output data using the TSL
software.
D. Transmission Electron Microscopy
Because of the (known) limits on resolving subgrains
using OIM,[4–6,17] TEM was used to supplement the SEM
results. Though orientations were not measured with TEM,
a more accurate subgrain size was obtained. Bright-field
images taken with a JEOL* transmission electron micro-
scope showed both geometrically necessary and statistically stored dislocations; the GNDs formed subgrains that
were readily identifiable, facilitating the analysis of subgrain
sizes. The boundaries in the images of the subgrain structures were hand traced and imported into SCION IMAGE**
**SCION IMAGE is a trademark of Scion Corporation, Frederick, MD.
[3]
where m and m are the values of boundary energy and misorientation characteristic of high-angle boundaries. Previous
calculations following the work of Murr[13] yielded a value
of 0.324 J/m2, which was used in the preceding relationship. Specifically, the EBSD scan reveals orientation gradients. Areas located within a single “grain” or macroscopic
orientation also contain results in a number of discontinuous
boundary segments, and are discussed at length in this analysis with respect to their contribution to stored energy.
20—VOLUME 37A, JANUARY 2006
Scan step size
Binned pattern size
Angle cutoff value
Calculated subgrain diameter*
Calculated average misorientation angle
*JEOL is a trademark of Japan Electron Optics Ltd., Tokyo.
ware was used to derive grain sizes. The method is based
on that of Humphreys and co-workers,[4–7] where an average
misorientation, u, and an average subgrain diameter, D, are
used in the Read–Shockley relationship for the grain boundary energy, ED:
cgm a
Value
(http://www.scioncorp.com), where they were skeletonized
and then analyzed to extract the average grain diameter
(Figure 1).
The Scion image subgrain determination is not fully automated. The user may calculate area under the analyze particles option by number of pixels within a hand-traced grain;
this area is in units calibrated according to the scale bar set
by the actual micrograph (e.g., 67 pixels per 500 nm). In
this experiment, the average grain area was obtained from
three different micrographs (192 grains total), and an average
circle-equivalent diameter of 0.685 m was obtained.
METALLURGICAL AND MATERIALS TRANSACTIONS A
(a)
(b)
(c)
Fig. 1—(a) TEM image (b) traced by hand and then (c) skeletonized using SCION IMAGE.
III. RESULTS
Nonisothermal heating of samples in the DSC yielded
exothermic peaks upon recrystallization from which the
stored energy was calculated from the peak area.[1–3,11,12]
Peak areas were calculated by finding the area under the
curve, where the cutoff temperatures were located at the
most nearly linear areas before and after the peaks along
the “baseline.” Note that the enthalpies associated with
recrystallization are small by comparison to typical phase
transformations. All the peaks associated with recrystallization were close to 350 °C, as shown in the example in
Figure 2, and their areas in MilliJoules were calculated using
the baseline correction described in Figure 3; their respective standard deviations (difference from mean) are given
in Table III.
To check that the peak was indeed associated with recrystallization, DSC disk samples were analyzed using EBSD
at temperatures of 25 °C, 275 °C, and 350 °C, respectively, to
mimic the heating evolution of as-rolled AA1050, recovery,
and recrystallization. The results supported the DSC curve
well as shown subsequently by confirming a recrystallization temperature of 350 °C. Figure 4.
The values in Table III were then averaged and divided
by 1.11 mm3, the volume of each DSC sample, to obtain a
stored energy of 0.293 MPa, which agrees well with the
value of 0.296 MPa obtained from the microhardness tests.
With respect to microscopy, Figure 5 shows that the subgrains are not easily outlined in an EBSD image using the
naked eye, nor can they be traced because of the frequent
occurrence of incomplete boundaries around some subgrains.
The TEM image (Figure 6), however, shows that each subgrain is easily identified and its border traced because of the
different contrast mechanism(s).
The values obtained using OIM yielded larger subgrain
values than those observed in the TEM, as might be expected.
Using the algorithm provided in the TSL software, the subgrain diameter was calculated as 0.723 m; this contrasts
with the value of 0.685 m obtained with the Scion Image
software analysis of the TEM images. The small difference is
most likely a consequence of the limited spatial resolution of
the EBSD technique, as discussed previously. Though the
TSL software subgrain determination is fully automated
whereas the Scion Image subgrain determination requires
operator judgment in the preliminary identification and
METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 2—Calorimetry scan at 60 °C/min. Exothermic recrystallization peak
(denoted by the arrow) is seen at 358 °C.
Fig. 3—Peak area calculation (magnified view of peak identified with arrow
in Fig. 2). The shadowed area under the peak is enclosed by the three thin
straight lines and flanked by the upper and lower temperature bounds of
the exothermic reaction.
Table III. Peak Area Values
Heating rate
40 °C/min
50 °C/min
60 °C/min
Peak area (mJ) 3.37 (0.12) 3.03 (0.22) 3.35 (0.10)
tracing of subgrains, both methods use grain area (calculated
using the number of data points within a grain) to calculate
circle-equivalent diameters. Note that the average misorientation angle was also determined by EBSD. The value of
0.824 deg reported previously excludes high-angle
VOLUME 37A, JANUARY 2006—21
Fig. 4—Inverse pole figure maps of AA1050 DSC samples (from left to right in the upper row: as-rolled, 275 °C, 350 °C) showing partial recrystallization
only at the highest temperature; magnified portions of two images in the lower row show the reduction of speckle in the images with increased temperature, denoting recrystallized grains interspersed in the deformed matrix for the specimen annealed to 350 °C.
Fig. 5—OIM boundary rotation angle map, showing only 15 to 180 deg
misorientations (black lines) surrounding networks of low angle grain boundaries (LAGBs) (gray lines). The LAGBs tend not to form complete subgrain shapes and the subgrain structure is diffuse.
boundaries and corresponds to the average subgrain diameter
value of 0.723 m reported in Table II. The difference in
subgrain size between the two methods is simply a consequence of the fact that all subgrain boundaries are distin22—VOLUME 37A, JANUARY 2006
Fig. 6—TEM bright-field micrograph of cold-rolled AA1050. Subgrains
vary in size from 500 m to 2 m.
METALLURGICAL AND MATERIALS TRANSACTIONS A
Table IV. Comparison of Stored Energy Values (MPa)
Microhardness
DSC
OIM
TEM
0.296
0.293
0.096
0.101
guishable in the TEM, whereas EBSD can only resolve
boundaries with misorientations greater than about 1 deg.
The overall stored energy values, in MPa, using the methods described previously, are compared in Table IV.
A. Sources and Calculation of Error
1. Microhardness
An overall stored energy average value of 0.296 MPa was
calculated using microhardness, and the variability was
assessed from the measured indent lengths, d1 and d2. The
value of 30 m that was used as d was the average of six
indents with individual average d values of 29.75, 30.00,
31.50, 30.30, 30.00, and 29.50 m.
2. Differential scanning calorimetry
Errors in the calorimetry experiments were dominated by
the calculations of peak area and were easily estimated (Table
III). The choice of the initial and final temperatures, which
equal the cutoff points for the automated integral calculation,
requires non-utomatic operator intervention within the Pyris
software (as shown in Figures 3 and 4), leading to variability
in the peak area standard deviation. The operator must choose
the temperatures that lie at the most nearly linear areas preceding and following the recrystallization peak, as stated in
Section II. The values of the peak area did not fluctuate
greatly, however; therefore, the three peak area values extrapolated were averaged to obtain one final peak area, which in
turn was divided by the constant sample volume to obtain a
stored energy. Because the sample volume remained constant
throughout the experiments, the stored energy is 0.2927 0.0002 MPa, where the variability was calculated from the
upper and lower bounds of the peak area values.
3. Electron backscattered diffraction
A distribution of misorentation angles and grain diameters was calculated from EBSD measurements using the TSL
software; the plots of the number fraction (for a 70 70 m
EBSD scan) of misorentation angles and grains of a given
size are shown in Figure 7. Rotation angles 15 deg and higher
were left out of the calculation of the distribution for the
number fraction of boundaries with a given misorentation
angle.
The distributions reveal a large fraction of small grains
with extremely low angle boundaries. The spread in misorientation angle is much smaller than that of the grain size,
with practically no angles greater than 2.0 deg, and an average misorientation angle of 0.824 deg. The misorientation
distribution is skewed to small sizes with 65 pct of boundaries having a misorientation angle of 0.375 deg or less,
27 pct between 0.375 deg and 1.13 deg, but a maximum
measured misorientation angle of 14.625 deg. By contrast,
the spread in diameter was substantial, even though the
majority of grains had a diameter between 0.27 and 5.53 m;
the average grain diameter measured was 0.723 m, and
50 pct were between 0.275 and 0.359 m.
METALLURGICAL AND MATERIALS TRANSACTIONS A
(a)
(b)
Fig. 7—Number fraction of (a) misorientation angles and (b) grain diameters for an EBSD scan of an as-rolled AA1050 sheet.
4. Transmission electron microscopy
The variability of subgrain size in the TEM measurements is
similar to that of a standard lognormal distribution. Figure 8
shows the distribution of grain Areas for the 192 grains
that contributed to an average subgrain area of 1.472 m2
(0.685-m diameter). The plot shows large variations, with
a maximum area of 5.334 m2 (1.30-m diameter) and a minimum of 0.167 m2 (0.231-m diameter). The average value
corresponds with the mode of the distribution, at approximately 1.50 m2.
IV. DISCUSSION
The results for the stored energy determined microscopically are consistent in magnitude with those of Huang and
Humphreys,[7] who used a similar approach for an AlSi alloy
with a higher, 70 pct reduction and determined a stored
energy value of 0.13 MPa based on the average subgrain
diameter. The energy values derived from DSC and hardness
are significantly higher, however, than those based on subgrain measurements using EBSD and TEM. If the density
VOLUME 37A, JANUARY 2006—23
Fig. 8—Distribution of grain areas for 192 skeletonized grains from TEM
(top) and normalized distribution (bottom).
of statistically stored dislocations were significant, one might
expect the scanning electron microscopy to yield a lower
stored energy than the microhardness or DSC methods
because scanning electron microscopy/TEM can only measure the geometrically necessary dislocations (GNDs). When
one considers the absence of alloying in AA1050 that promotes the formation of subgrains, however, the statistically
stored (“free”) dislocation density should be minimal. Values
of stored energy calculated from microscopy that were more
than 60 pct lower than those calculated from DSC or hardness,
however, suggest that a non-negligible density of statistical
dislocations was indeed present, prompting further analysis.
The validity of the various methods for calculating diameter
coupled with the Read–Shockley equation is discussed thoroughly in the literature[4–7] and is a well-accepted method of
estimating (sub-) grain size and therefore stored energy. Concerning EBSD data, there are many factors that can affect the
subgrain size, including angular resolution limits, grain tolerance angle within the TSL software, and imaging capabilities specific to each microscope. As discussed by Humphreys,
et al.,[4,5,6] the occurrence of a nonindexed pattern at a grain
boundary can skew the data significantly due to these points
being lost and therefore an entire grain being lost, especially
if the grain size is small. In this study, the average confidence
index for the scan used was approximately 0.8, which is well
above the widely accepted minimum of 0.1. The grain size
was obtained by averaging all grain diameters; with a maximum grain size well above the calculated average in the TEM,
24—VOLUME 37A, JANUARY 2006
the data were considered to be sufficiently accurate. The grain
size distribution shows few grains above the calculated average of 0.723 m. This suggests that a reasonable estimate of
the grain size and average misorientation can be obtained with
EBSD, provided that the spatial resolution and (average) confidence index are high enough. Recent research[4] suggests
that in an FEG–SEM/EBSD system, spatial resolution is
roughly 50 to 150 nm. In this study, spatial resolution limits
appeared to be 100 nm.
As previously mentioned, an important restriction of the
EBSD measurement is the misorientation limit. The sample used in this study was cold rolled, therefore containing
a high density of subgrain-forming dislocations, surrounded
by high angle grain boundaries (HAGBs). The lower limit
of misorientation measurement using OIM is widely stated
as 1 deg, which means that the subgrain size will be overestimated (and the stored energy underestimated) if the actual
misorientations are smaller. The TEM measurements yield
a 5 pct smaller grain diameter. Though this could be attributed to the small area that is examined in a TEM foil as well
as the processing that a sample must undergo in order to be
investigated via TEM, resolution is much higher in such an
instrument. In this particular study, three TEM images, with
a total of only 192 grains, were taken from different sections
of the rolled AA1050 sample and then analyzed using Scion
Image to calculate an average subgrain diameter. It was
assumed that TEM is a much more accurate way to measure
subgrain size simply because spatial resolution is much greater
than that using an SEM. However, the average misorientation
used in the context of EBSD maps may be a slight overestimation for the boundaries in the TEM images since it is
reasonable to suppose that one can resolve boundaries at
smaller misorientations then is practicable in EBSD/SEM.
As shown in Figure 5, however, the 0–15 deg boundaries were visible in an EBSD scan, and comprised more
than 80 pct of the total boundaries in the map. The algorithm provided in the TSL software is intended for analysis of annealed grain structures, which means that each grain
must be completely enclosed by boundaries of the specified minimum misorientation. Boundaries contained within
a grain that terminate within the grain, as commonly occur
for low-angle boundaries, are therefore not included in the
grain size estimate (as they would be for a line-intercept
method). This effect is seen in Figure 9, where higher angle
boundaries, heavy lines, surround areas containing irregular
networks of low-angle boundaries.
V. CONCLUSIONS
1. The methods laid out in this article are all reasonable methods of calculating stored energies in AA1050, as the relationship between the resulting values of stored energy and
the amount of dislocations present were as expected: measurements of stored energy arising from both statistically
stored dislocations and geometrically necessary dislocations (macroscopic) yielded a value approximately 67 pct
larger than those of stored energy comprising solely geometrically necessary dislocations (microscopic).
2. The main limitation of EBSD appears to be the 1 deg misorientation lower limit on misorientation detection, which
in turn affects the measurement of an accurate subgrain
METALLURGICAL AND MATERIALS TRANSACTIONS A
size; data cannot be collected for subgrains with a misorientation smaller than this limit. This limitation accounts for
the discrepancy in the stored energy measured by both the
TEM and SEM methods, and could account for the 5 pct
larger value found in the TEM measurements. Direct measurement of an average misorientation angle is needed for
TEM experiments for better comparison with diameter
measured via Scion Image.
ACKNOWLEDGMENTS
This work was supported by a grant from the Alcoa Technical Center and by the MRSEC program of the National
Science Foundation under Award No. DMR-0079996. The
authors thank Paula Kolek, Alcoa, for many useful discussions. The transmission electron microscopy images were
acquired with the generous help of Paul Bagethun.
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