IntroToStereology-B-long.ppt

Introduction to Stereology and
Quantitative Metallography
Measurement of Grain Size
George F. Vander Voort
Consultant – Struers Inc.
Westlake, Ohio USA

Grain Requirements
 Shapes must be space filling
 Surfaces must exhibit minimum surface
area and minimum surface tension
(Plateau, 1873)

Two Common Ideal Grain Shapes
Tetrakaidecahedron (left) and Pentagonal
Dodecahedron (right)

Tetrakaidecahedron
Lord Kelvin (1887) showed that the
optimum grain shape meeting these
requirements was a polyhedron called
a tetrakaidecahedra with
14 faces
 24 Corners
 36 Edges

But, the tetrakaidecahedron does
not exhibit 120 dihedral angles
between grain boundaries where 3
adjacent grains meet at an edge –
unless the faces exhibit curvature.

Topological Relationships
Equal interfacial energies produce
face angles of 109.5 and dihedral
angles of 120
In this case:
Avg. No. Edges/Face = 5.1043
Avg. No. Faces/Grain = 13.394
Avg. No. Edges/Grain = 34.195
Avg. No. Corners/Grain = 22.78

C.H. Desch (1919)
First to study actual grain shapes by LME
of β-brass in liquid Hg. For isolated grains,
he found:
Avg. No. Faces/Grain = 14.5 (11 to 20)
Avg. No. Edges/Face = 5.14 (3 to 8)
5-sided grains were most frequent

W.M Williams and C.S. Smith
(1953)
Used stereomicroradiography to study
grains and found:
Avg. No. Edges/Face = 5.06
Avg. No. Faces/Grain = 12.48

Euler’s Law of Topology
For spacing-filling aggregates of
polyhedral grains
+ C – E + F – B = 1
C – number of point corners
E – number of lineal edges
F – number of polygonal faces
B – number of polyhedral bodies

Euler’s Law
For a single polygon, B = 1 and
C – E + F = 2
The two-dimensional form of Euler’s Law
for an array of polygons is
C – E + F = 1

5/16 Sieve 4 Sieve
8 Sieve
14 Sieve
Separation of
grains by sieving
after liquid metal
embrittlement of
 Brass in Hg
Actual Grain Shapes
3.3 mm
3.3 mm
3.3 mm
3.3 mm

Actual Grain Shapes
SEM secondary electron images of individual  Brass
grains separated by liquid metal embrittlement with Hg.
1.1 mm 1.1 mm

Grain Size Measurement
Types of Grain Sizes
• Non-twinned(ferrite, BCC metals, Al)
• Twinned FCC Metals (austenite)
• Prior-Austenite
(Parent Phase in Q&T Steels)

Grain Size Measurements
• Number of Grains/inch2 at 100X: G
• Number of Grains/mm2 at 1X: NA
• Average Grain Area, µm2 : A
• Average Grain Diameter, µm: d
• Mean Lineal Intercept Length, µm: l

Grain Size Measurement Methods
Comparison Chart Ratings
Shepherd Fracture Grain Size Ratings
Jeffries Planimetric Grain Size
Heyn/Hilliard/Abrams Intercept Grain Size
Snyder-Graff Intercept Grain Size
2D to 3D Grain Size Distribution Methods

Definition of ASTM Grain Size
n = 2 G-1
n = number of grains/in2 at 100X
G = ASTM Grain Size Number

ASTM Grain Size Equation
Introduced when E 91 – ASTM Method
for Estimating the Average Grain Size of
Non-Ferrous Metals, Other Than
Copper, and Their Alloys – was
introduced in 1951. The equation was
developed by Timken Co.

ASTM Grain Size, G
G n G n
1 1 6 32
2 2 7 64
3 4 8 128
4 8 9 256
5 16 10 512

Current ASTM Standards for Grain Size
ASTM E 112: For equiaxed, single-
phase grain structures
ASTM E 930: For grain structures
with an occasional very large grain
ASTM E 1181: For characterizing
duplex grain structures
ASTM E 1382: For image analysis
measurements of grain size, any type

Metric Equivalents
Other countries established grain size scales
using the metric system, based on the
number of grains per sq. mm at 1X, NA:
Sweden (SIS 11 11 01); Italy (UNI 3245);
Russia (GOST 5639); France (NFA04-102);
and ISO (ISO 643) according to:
M = 8 (2Gm)
where m = No. Grains/mm2 at 1X

Grain size numbers based on that
equation using the metric system
are ~4.5% higher than for the
ASTM equation, that is:
G = Gm – 0.045

German Standard SEP 1510
This standard also uses the metric system but
yields the same numbers as the ASTM
equation. The photomicrograph serial
number, K, is calculated based on the average
number of grains/cm2 at 100x, Z, by:
K = 3.7 + 3.33 log Z

Japanese Standards JIS
G0551 and G0552
These standards are also metric but
they produce numbers equal to the
ASTM numbers by:
m = 2G+3
where m is the number of grains/mm2
at 1X

ASTM Standards
Determine for any standard that generates
numerical data the precision and bias by use
of interlaboratory “round robins”.
Accuracy generally cannot be determined as
the true values being measured cannot be
determined by any referee method.

Grain Size – ASTM Standards
 E 2 was the first ASTM E-4 standard and it
described the planimetric method of Jeffries
in depth, mentioning the Heyn intercept
method briefly in an appendix
 Members claimed these were too tedious to
use and asked for a simpler method. The 1930
revision of E 2 added a comparison chart for
copper at 75X (grain contrast etch) with the
grain size expressed as d, the average
diameter in mm.

E-4 Develops More Charts!
The 1930 E 2 Cu chart was criticized as
being poorly graded and was dropped (all
grain size methods were dropped) in the
1941 revision.
A new chart with 12 micrographs was
incorporated into ASTM E 79 (Cu) when it
was introduced in 1949.
E 79 was discontinued in 1963 when all
grain size methods were incorporated in E
112 (new Plate III with 14 images)

More ASTM Grain Size Charts!
In 1933, E-4 issued E 19, ASTM
Classification of Austenite Grain Size in
Steels, based upon the McQuaid-Ehn
carburizing technique. The chart had 8
images showing both the case and core. The
grain size was listed in terms of the number
of grains/in2 at 100x, but G numbers were
not used. This chart was heavily criticized
as being poorly graded.

Still More ASTM Charts!
The 1933 E 19 chart was replaced in 1938 by
a stylized chart (artist’s rendering).
E19 was discontinued in 1961 when E 112
was developed. Plate IV of E 112 now depicts
grains outlined by cementite as by the
McQuaid-Ehn method.

And yet more ASTM Charts!
In 1950, E-4 introduced E 89, ASTM Method
for Estimating the Average Ferrite Grain
Size of Low-Carbon Steels, with grains from
ASTM 1 to 8.
This chart was also heavily criticized and
was discontinued in 1961 when E 112 was
introduced. E 112 does not have such a chart
but an idealized non-twinned grain size
chart (Plate I) – which was still being
corrected up to the end of the 1980s!

Two More Grain Size Charts!
E-4 started a study in 1947 for rating grain
size in nonferrous metals which resulted in
E 91 in 1951. E 91 had two charts, one for
twinned alloys (grain boundary etch) and
one for non-twinned alloys. Both charts
were dropped when E 112 was created.
E 112, Plate I is for non-twinned grains and
Plate II is for twinned grains – both depict
grain boundary attack.

Comparison Chart Ratings
Look at a properly etched microstructure,
using the same magnification as the chart, and
pick out the chart picture closest in size to the
test specimen. If the grain structure is very
fine, raise the magnification, pick out the
closest chart picture and correct for the
difference in magnification according to:
G = Chart G + Q
Q = 6.64Log10(M/Mb)
where M is the magnification used and
Mb is the chart magnification

German SEP 1510
SEP 1510 contains a chart depicting non-
twinned grains that are equiaxed, or
elongated (2 to 1 and 4 to 1) by cold
working – very useful for cold worked
sheet steels.

Fracture Grain Size
Comparison Method
In 1931, Ragnar Arpi of Sweden showed
that the prior-austenite grain size of high-
hardness tool steels could be rated by
comparing a fractured test specimen to a
series of 5 graded fractures.
Benjamin Franklin Shepherd in 1934
expanded the series to 10 fractures,
numbered 1 to 10, which corresponded to G
values of 1 to 10

Shepherd Fracture Grain Size
It was a remarkable coincidence that the 10
graded fractures, coded 1 to 10 by Shepherd,
correlated so well to ASTM G of 1 to 10 –
especially as Shepherd created the fracture
grain size series 17 years before G numbers
were by E-4 with E 91!

Shepherd Fracture Grain Size Specimens

The literature has claimed that all 10
fractures in the series are intergranular. This
is not the case; only specimens 1 through 6
are intergranular. For specimens of 7 to 10,
the amount of intergranular fracture drops to
zero and the fractures are cleavage with finer
and finer facets.

 Cannot rate PGS finer than 10 (eye cannot
distinguish differences in fracture appearance)
 A flat fracture face works best
 Works for martensitic structures (retained
austenite is not a problem)
 Highly tempered martensite biases results
(also presence of other constituents)
 Fractures transverse to deformation axis
should be used

First Grain Size Measurements
In 1894, Albert Sauveur published grain
sizes measured in terms of the number of
grains/mm2 but did not formally define the
measurement method.
Zay Jeffries, a graduate student of Sauveur’s
and a future E-4 member, published details
of performing the planimetric grain size
measurement method in 1916 and he
incorporated this method in E 2 when it was
developed in 1916.

Jeffries Planimetric Method
n1 = number of grains completely inside the
test circle
n2 = number of grains intercepting the
circle
NA = f[ n1 + (n2/2)]
f = Jeffries multiplier
f = magnification2/circle area

Jeffries Planimetric Method
Average Grain Area = A = ——
G = (-3.322LogA) – 2.955
1
NA

Jeffries Planimetric Method - Example
For the preceding micrograph,
n1 = 68 and n2 = 41
And
f = —— = ——— = 0.497
M2
A
1002
20106.2

NA = f[n1 + (n2/2)]
NA = (0.497)[68 + (41/2)]
NA = 44.02 mm-2

A = —— = 0.0227 mm2
1
NA
d = (A)1/2
G = 2.5

Circles Create Bias as n1 Decreases
The great Russian stereologist, Sarkis A.
Saltykov, showed that as n1 decreases, bias
results. He recommended using a square or
rectangular test figure. Intercepted grains at the
corner are not counted (assumed to be 1). n2 is
the number of grains intersecting the four sides,
but not the corners. n1 is the number of grains
inside the test figure, as before.
NA = f[n1 + (n2/2) + 1]

This is an austenitic Mn steel, solution annealed and aged to
precipitate a pearlitic phase on the grain boundaries (at 100X).
There are 43 grains within the circle (n1) and there are 25 grains
intersecting the circle (n2). The test circle’s area is 0.5 mm2 at 1X.

NA = f[n1 + (n2/2)]
NA = 2[43 + (25/2)] = 111 mm-2
f = [(1002)/5000]
G = [3.22Log10(111)] – 2.954 = 3.8
(Of course, more than one field should be measured to get
good statistical results)

Heyn/Hilliard/Abrams Intercept Method
N = number of grains intercepted
P = number of grain boundary intersections
NL = ——
N
LT
PL = ——
P
LT
where LT is the true test line length

Apply a test line over the microstructure
and count the number of grains intercepted
or the number of grain boundary
intersections (easier for a single-phase grain
structure). After you count N or P, divide
that number by the true line length to get
NL or PL.

1/2
1 1
1
1
1 1/2
The test line intercepted 5 whole grains and the line ends fell
in two grains. These are weighted as ½ an interception. So the
total is 6 intercepts (N=6).
Intercept Counts (N)

1 1 1 1 1 1
The test line has intersected 6 grain boundaries. The ends
within the grains are not important in intercept counting.
So, P=6 for the intercept count.
Intersection Counts (P)

Mean Lineal Intercept, l = — = ——
1
NL
1
PL
G = [6.644Log10(NL or PL)] – 3.288
G = [-6.644Log10(l)] – 3.288
Note: Units are in mm-1 (for NL and PL) or mm (for l)

If the grain structure is not equiaxed, but
shows some distortion of the grain shape, use
straight test lines at various angles, or simply
horizontal and vertical with respect to the
deformation axis of the specimen.
Alternatively, you can use test circles, such as
the ASTM three-circle grid (three concentric
circles with a line length of 500 mm). This test
pattern averages the anisotropy.

Example of three concentric test circles
for point counting.

To illustrate intercept counting, note that there are 41, 25 and 20 grains
intercepted (N) by the three concentric circles.

Intercept Counting Example
LT = 11.4 mm
N = 41 + 25 + 20 = 86
NL = —— = 7.54 mm-1
86
11.4
l = —— = 0.133 mm
1
7.54
G = [-6.644Log10(0.133)] – 3.288 = 2.5

Intercept Grain Size Example – Single Phase
This is a 100X micrograph of 304 stainless steel etched electrolytically with
60% HNO3 (0.6 V dc, 120 s, Pt cathode) to suppress etching of the twin
boundaries. The three circles have a total circumference of 500 mm. A count of
the grain boundary intersections yielded 75 (P=75).

Intercept Grain Size Example – Single Phase
PL = ——— = 15 mm-1
75
500/100
l = —— = 0.067 mm
1
15
G = [-6.644Log10(0.067)] – 3.288 = 4.5

Intercept Grain Size Example:
Single Phase Twinned Grain Structure
The 100X micrograph is that of a twinned FCC Ni-base superalloy, X-750, in
the solution annealed and aged condition after etching with Beraha’s reagent
which colored the grains. This is a much more difficult microstructure for
intercept counting. The three circles measure 500 mm and P is 63
(intersections with twin boundaries are ignored).

Intercept Grain Size Example:
Single Phase Twinned Grain Structure
PL = ——— = 12.6 mm-1
63
500/100
l = —— = 0.0794 mm
1
12.6
G = [-6.644Log10(0.0794)] – 3.288 = 4

Intercept Method for Two-Constituents
N = Number of  grains intercepted
LT = Test line length/Magnification
VV = Volume fraction of the  phase
l = ———
VV(LT)
N

This 500X micrograph of Ti-6242 was alpha/beta forged and alpha/beta annealed,
then etched with Kroll’s reagent. The circumference of the three circles is 500 mm.
Point counting revealed an alpha phase volume fraction of 0.485 (48.5%). 76 alpha
grains were intercepted by the three circles.

l = ———————— = 0.006382 mm
(0.485)(500/500)
76
G = [-6.644Log10(0.006382)] – 3.288 = 11.3

Because the grain size of hardened high speed tool steels is generally around
G = 9 to 12, Snyder and Graff proposed an alternate intercept method. In
this range NA changes by a factor of 10 and the mean lineal intercept length,
l, varies from 14.1 to 5 µm.
To increase the sensitivity to these small variations, they suggested doing an
intercept count at 1000X using a 5-inch (127-mm) test line. The number of
grains intercepted by the line is counted. This is repeated for 10 random
placements of the test line. The average value of the number of intercepted
grains is the S-G intercept grain size number.
ASTM G can be calculated from the NIS-G value:
G = [6.635Log10(NIS-G)] + 2.66

The 1000X micrograph above of a high speed steel in the quenched and tempered
condition has been etched with 10% nital. Two 5-inch (127-mm) lines have been
drawn and the number of intercepted grains were counted. For each line there
were two tangent hits (each weighted as (1/2). One line had 12 intercepts and the
other 13. So, N was 13 and 14, with an average of 13.5 (NIS-G = 13.5) and G=10.2.

ASTM Grain Size Round Robin
A number of ASTM E-4 members counted intercepts using the three-
circle grid and then counted the grains within a test circle, and
intersecting the test circle, on seven micrographs. Three were at
different magnifications for a ferritic stainless steel and four were at
different magnifications for another ferritic stainless steel. All images
were taken from the same region. The people did not calculate the grain
size; they only collected the raw data. Prior to that, they used a
comparison chart, plate I of E 112, to estimate the grain size of each
micrograph.
A few people digitized the images and measured the grain size with
image analysis systems.
Examples of the micrographs are shown on the next slide. For the
counting, the micrographs were enlarged to 8 x 10 inches. Random grid
placement was used for the intercept method, but for the planimetric
method, the template contained five test circles, so the placement on the
micrograph was not completely random, but forced.

Examples of the micrographs used for the round robin. There were three
magnifications for the one at left and four for the one at right. Grain
boundary delineation was excellent.

Distribution of grain size by number % and area % (preferred) for the left
image in the previous slide (image analysis results). There is a slight deviation
from a normal, Gaussian distribution in this specimen.

Distribution of grain size by number % and area % (preferred) for the right
image in the earlier slide (image analysis results). The grain size distribution is
normal, or Gaussian in this specimen unlike the other specimen.

Results for the first specimen at three magnifications.

Results for the specimen with four magnifications.

A plot of the planimetric grain size measurement vs. the intercept grain size
measurements for all specimens reveals a normal scatter around the one-to-
one trend line (except for one point) indicating no bias between the methods.
“wild” value

If the true magnification is not used, but all images are assumed to be at 100X, the
different magnifications and give a wider spread of apparent grain sizes. Note that
the comparison chart ratings are consistently lower than the measured values by
0.5 to 1 G value indicating bias in the comparison chart ratings.

Naturally, when the intercept measurements are plotted vs. the comparison chart
estimates of G (similarly to the last slide where the planimetric data was used), the
same bias in the comparison chart data is observed.

Plot of the relative accuracy for the planimetric measurements indicating that
about 1000 grains must be counted to get <10% RA.

For the intercept method, <10% RA can be obtained by counting about 400
intercepts or intersections. Counting with the planimetric method is more tedious
as the grains must be marked off to get an accurate count.

Plot of the %RA as a function of the average count per grid placement
(per field). Counting errors start to results when the count exceeds about
50-60 per field.

Influence of Etching Time
Inadequate etch times do not reveal the grain structure so bias is
created as the grain size appears to be greater than it is. The above
example was for ferrite grains in low-carbon sheet steel.

Grain Size Distributions
It is possible to make measurements of the
diameter, lineal intercept lengths, or areas of
grains and plot these data in histogram
fashion. Many procedures have been
developed to translate these measurements
on the two-dimensional sectioning plane to
develop three-dimensional grain size
information. Nearly all models utilize some
simplifying assumptions about shape, such as
spherical grain shapes.

Grain structure of 304 austenitic stainless steel etched with 60% HNO3 at 0.6
V dc, Pt cathode, 120 s (this does not bring up twin boundaries) used for the
following grain size distribution study.

A log plot of the intercept length vs. the number percent per class yields a good
representation of the distribution. Note the slight skew of the data (1) while the
kurtosis, 2, is close to the ideal value of 3 for a Gaussian distribution.

A linear plot of the data does not reveal a good distribution as it is skewed more
to the right and the kurtosis is higher.

Three specimens of an experimental 5% Cr hot-work die steel were
analyzed for their grain size distribution. This one was austenitized at 1950
°F (1066 °C). The others were austenitized at 1925 and 1975 °F (1051 and
1079 °C). The specimens were quenched to 1300 °F (704 °C), held 1 h to
precipitate a pearlitic like constituent at the grain boundaries and air
cooled. They were etched with glyceregia.

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