NPL Report MATC(A)149
A Comparison of the
Linear Intercept and
Equivalent Circle
Methods for Grain Size
Measurement in WC/Co
Hardmetals
B Roebuck, C Phatak
and I Birks-Agnew
March 2004
NPL Report MATC(A)149
April 2004
A Comparison of the Linear Intercept and
Equivalent Circle Methods for Grain Size
Measurement in WC/Co Hardmetals
B Roebuck, C Phatak and I Birks-Agnew
Engineering and Process Control Division
National Physical Laboratory
Teddington, Middlesex TW11 0LW, UK
ABSTRACT
A computer model (POLYCHOP; developed at NPL) was used to calculate the equivalent
circle diameter (ECD) and linear intercept (LI) grain size for a range of different crystal
shapes, including a cube, a tetrakaidekahedron and a truncated trigonal prism. The effects of
differences in size distribution and shape factors were studied and the results were compared
with experimental measurements of the ECD and LI on sintered WC/Co hardmetals. Both the
model and experimental results showed a linear correlation between ECD and LI for materials
with grain sizes ranging from 0.5 to 5 mm. However, the model calculations, although
showing the same trend, did not agree with the measurements. Possible reasons for this
discrepancy, such as the difficulty of measurement of small grains, are discussed.
[ECDvLIreport/BM]
NPL Report MATC(A)149
© Crown copyright 2004
Reproduced by permission of the Controller of HMSO
ISSN 1473 2734
National Physical Laboratory
Teddington, Middlesex, UK, TW11 0LW
Extracts from this report may be reproduced provided
that the source is acknowledged
Approved on behalf of Managing Director,
NPL, by Dr C Lea, Head, NPL Materials Centre
[ECDvLIreport/BM]
NPL Report MATC(A)149
CONTENTS
1
INTRODUCTION............................................................................................................ 1
2
MATERIALS AND EXPERIMENTS ........................................................................... 4
3
RESULTS AND DISCUSSION ...................................................................................... 5
3.1
MEASUREMENTS................................................................................................... 5
3.2
MODELLING ............................................................................................................ 6
3.2.1 Effects of Size, Distribution and Sample Number................................................. 6
3.2.2 Effect of TTP Truncation and Shape ................................................................... 26
3.2.3 Effect of Measurement Resolution ...................................................................... 29
4
SUMMARY .................................................................................................................... 33
5
ACKNOWLEDGEMENTS .......................................................................................... 33
6
REFERENCES............................................................................................................... 34
[ECDvLIreport/BM]
NPL Report MATC(A)149
1
INTRODUCTION
Crystal grain size is one of the most important factors in controlling the properties of
materials. For example, the strength, toughness and hardness are all important engineering
properties that are strongly influenced by this parameter. For this reason it is important to
have standard methods for its measurement with commonly used and agreed terminology.
WC/Co hardmetals are manufactured from powders with the objective of producing hard and
strong tool materials by controlling the grain (crystal) size of the WC phase. A typical
micrograph of WC/Co is shown in Fig 1. The grey crystals are the WC phase. The black
contrast regions between the WC grains is the cobalt-tungsten-carbon alloy binder phase.
There are currently two methods in use to estimate the grain size of WC/Co hardmetals [1-6]
– one is based on the measurement of an equivalent circle diameter (ECD) and another on the
use of linear intercepts (LI). Variability between the different methods impedes the
comparability of data, and inhibits trade and new product development through the lack of a
common set of results. This report investigates the relation between the two methods, both by
modelling and experiment.
The LI method is based on the number average lengths of intercepts through each crystal/
grain along a line drawn across the material surface. The ECD method is based on the number
average area of the grain/crystals. The average area is converted to an equivalent circle
diameter (ECD) as a measure of size. Both methods are justified, but find different values.
Fig 1 WC/Co structure.
1
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NPL Report MATC(A)149
The factors which may affect estimates of the grain size include the size and shape of the
grains, their height to width ratios and the distribution of the size of the grains. In particular,
determining the effect of the distribution of grain size in the material may be important. These
factors can affect the two measurement methods in different ways, thus perhaps changing the
relationship between the values found.
The grain size distribution data can be plotted as cumulative probability plots with the
abscissa (x) on a linear or logarithmic scale. Historically, this has been the preferred method
for plotting WC size distribution data, where a number probability is used for the ordinate and
the size is plotted as log size on the abscissa. It has been found that linear intercept
measurements plotted in this way have a lognormal distribution, giving a straight line when
lognormal probability paper is used. The equation for lognormal probability is:
(
é - ln(x ) - ln (m g )
f (x ) =
exp ê
2s 2
xs 2p
êë
1
) 2 ùú
úû
where s is a measure of the distribution width and mg is the geometric mean. This does not
plot conveniently mathematically, but many software packages have numerical
approximations to the function, permitting s to be determined. The use of probability paper
generally gives a straight line for data for current commercial hardmetals, indicating that the
parameter, s, is the width of the lognormal distribution. Linear intercept measurements of
WC grains have lognormal number distributions.
Modelling
The computer programme POLYCHOP was used to model the crystals and to calculate the
average area and intercept from a sample containing a fixed number of crystals. POLYCHOP
simulates the crystals in a material by taking intercept and cross-sectional area measurements
of the crystal in different random orientations [7].
Three shapes were investigated; cubic, tetrakaidecahedral (TKC) and truncated triangular
prism (TTP) crystals, Figs 2 and 3. Different size distributions of the crystals were used to
simulate the variation of crystal size within a material. A lognormal distribution was chosen
with s values of 0.0, 0.1, 0.3 and 0.5, where s is a measure of distribution width. The sizes of
the crystal were also varied with scaling factors of 1, 2, 4 and 8, doubling the size of crystal
each time. The effect of varying the number of grains counted was examined between about
100 and 100,000.
When comparing the modelled results with measured results the TTP shaped model was used
as it is assumed to approximate most accurately to the shape of crystals in WC/Co materials.
For the modelling exercise the dimensions of the TTP were edge, height and truncation 0.5.
(Fig 3).
[ECDvLIreport/BM]
2
NPL Report MATC(A)149
TTP
Cube
TKC
Fig 2
POLYCHOP shapes with typical random cross sections and intercept lengths.
3
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NPL Report MATC(A)149
Fig 3
2
Diagram showing TTP measurements
MATERIALS AND EXPERIMENTS
Linear intercepts and equivalent circle diameter measurements were obtained on a range of
WC/Co hardmetals (Table 1). All the materials investigated were manufactured in the twophase, WC/Co region, and magnetic data are also given in Table 1. The coercivity data were
supplied by Marshalls Hard Metals Ltd. The magnetic moments were measured at NPL using
LDJÒ equipment to support these observations.
Table 1
Hardmetal Properties
Grade
wt%
Co
Volume
Fraction
WC
Coercivity
kA m-1
T25
D10
T
UC9
MA11
K3520
24.2
10.0
9.1
9.8
11.7
20.3
0.55
0.82
0.83
0.82
0.78
0.62
6.8
9.8
8.4
4.0
3.8
5.1
[ECDvLIreport/BM]
Magnetic
moment
mT m3
kg-1
2.82
1.35
1.10
1.35
1.64
2.88
4
Grade
Wt%
Co
Volume
Fraction
WC
G10
CW25C
CW20C
TC222
K3560
M15
9.90
24.44
19.68
6.09
9.69
15.2
0.82
0.54
0.62
0.88
0.82
0.7
Coercivity
kA m-1
6.5
4.4
4.1
2.5
4.6
5.1
Magnetic
moment
mT m3
kg-1
1.48
3.52
2.87
0.98
1.37
2.11
NPL Report MATC(A)149
The samples were polished and etched in Murakami’s reagent and images for subsequent
measurement of grain size were obtained either using an optical microscope at ´1600 or in a
scanning electron microscope at different magnifications.
One image of each sample was analysed to calculate the average ECD. The weight % Co was
converted to a volume % Co(F) and then the area of the image was multiplied by a factor
(1-F) to get the total area occupied by WC. The number of grains was counted. Grains that
were on the edge of the image were counted as half grains. The average area of each grain
was calculated and then the ECD calculated from this value.
3
RESULTS AND DISCUSSION
3.1
MEASUREMENTS
Samples of 11 hardmetals with a range of grain size were polished and imaged using a
Scanning Electron Microscope (SEM). These images were then analysed by 2 different
operators, (A) and (B) to calculate ECD values for each sample. The ECD values were then
compared to the LI data taken from recent NPL reports [8,9]. These results show a clear linear
relationship (Fig 4) between the two parameters. The individual results are given in Table 2.
Table 2
ECD/LI Measurements
Grade
T25
D10
T
UC9
MA11
K3520
G10
CW25C
CW20C
TC222
K3560
M15
Operator A
LI
0.77
1.05
1.68
4.97
5.16
2.30
1.76
2.14
2.17
3.59
2.75
2.04
ECD
1.10
1.27
2.00
5.75
6.38
2.62
2.66
2.73
2.50
4.71
3.30
2.60
Grade
T25
D10
T
UC9
MA11
K3520
G10
CW25C
CW20C
TC222
K3560
M15
Operator B
LI
0.77
1.05
1.68
4.97
5.16
2.30
1.76
2.14
2.17
3.59
2.75
2.04
ECD
1.06
1.16
1.80
5.51
5.98
2.23
2.29
2.30
2.25
4.13
3.26
2.30
The results show good agreement between the relation between ECD and LI, Fig 4 illustrating
this through the similarity of the slopes of the two lines. There is a clear linear relationship,
but there is a systematic difference between the results of operators (A) and (B) where the
ECD calculated by (A) is always bigger than that of (B). The mean value of the slope is 1.15.
This correlation is in good agreement with previous work cited by De Hoff and Rhines in the
book “Quantitative Microscopy” [10] where it is stated than an empirical relation had been
found in measurements in aluminium and ferrous alloys where
5
[ECDvLIreport/BM]
NPL Report MATC(A)149
LI =
A
where Ā is the average grain area (Jefferies method). Thus since A =
ECD =
4
p
p
(ECD) 2
4
A = 1.13 A = 1.13 LI
Clearly the measurements on WC/Co alloys on the comparison between ECD and LI agree
well with previously published work on other engineering alloys.
Equivalent circle diameter (ECD)
mm
7
6
ECD =1.19 LI
5
ECD =1.13 LI
4
3
2
1
Measured data - operator A
Measured data - operator B
0
0
1
2
3
Intercept (LI)
4
5
6
mm
Fig 4 Measured ECD against LI data
3.2
MODELLING
3.2.1 Effects of Size, Distribution and Sample Number
Cumulative distribution graphs for computed linear intercept and area plotted against size are
shown in Figs 5-8 and 9-12 respectively for each crystal shape, size and size distribution. The
data are plotted on a linear scale for the y ordinate and a log scale for the x ordinate to enable
uniform comparisons to be made. A set of parallel frequency distribution plots are shown in
Figs 13-20. The sample size used for generating these graphs was N=1000.
The cumulative plots show similar trends for all the shapes. There was no effect of scaling
factor, confirming internal consistency of the program and generating information for
comparing ECD and LI averages. The frequency distributions show similar trends with the
expected peaks corresponding to face edge lengths in the distributions for s=0 (i.e. all crystals
the same size). For higher values of s these peaks systematically reduce, as expected.
[ECDvLIreport/BM]
6
NPL Report MATC(A)149
Intercept distribution sigma=0 scaling=1
% Cumulative Frequency
80
Intercept distribution sigma=0.1 scaling=1
100
Cube
TKC
TTP
80
% Cumulative Frequency
100
60
40
20
Cube
TKC
TTP
60
40
20
0
0.01
0.1
0
1
0.1
Intercept
110
100
Cube
TKC
TTP
90
80
% Cumulative Frequency
% Cumulative Frequency
80
10
Intercept
Intercept distribution sigma=0.3 scaling=1
100
1
60
40
Intercept distribution sigma=0.5 scaling=1
Cube
TKC
TTP
70
60
50
40
30
20
20
10
0
0
0.1
1
0.1
10
Intercept
Intercept
Fig 5
1
Computed cumulative number distribution plots for intercepts through each crystal
shape showing variation with distribution width at a scaling factor of 1.
7
[ECDvLIreport/BM]
10
NPL Report MATC(A)149
Intercept distribution sigma=0 scaling=2
100
Intercept distribution sigma=0.1 scaling=2
100
Cube
TKC
TTP
80
% Cumulative Frequency
% Cumulative Frequency
80
Cube
TKC
TTP
60
40
20
60
40
20
0
0
0.01
0.1
1
0.1
Intercept
1
10
Intercept
110
Intercept distribution sigma=0.3 scaling=2
100
Intercept distribution sigma=0.5 scaling=2
100
Cube
TKC
TTP
80
80
% Cumulative Frequency
% Cumulative Frequency
Cube
TKC
TTP
90
60
40
20
70
60
50
40
30
20
10
0.1
1
0
10
0.1
10
Intercept
Intercept
Fig 6
1
Computed cumulative number distribution plots for intercepts through each crystal
shape showing variation with distribution width at a scaling factor of 2.
[ECDvLIreport/BM]
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NPL Report MATC(A)149
Intercept distribution sigma=0 scaling=4
100
Cube
TKC
TTP
Cube
TKC
TTP
80
% Cumulative Frequency
80
% Cumulative Frequency
Intercept distribution sigma=0.1 scaling=4
100
60
40
20
60
40
20
0
0.1
1
0
10
0.1
1
Intercept
10
Intercept
110
Intercept distribution sigma=0.3 scaling=4
Intercept distribution sigma=0.5 scaling=4
100
Cube
TKC
TTP
Cube
TKC
TTP
90
80
% Cumulative Frequency
80
% Cumulative Frequency
100
60
40
70
60
50
40
30
20
20
10
0
0
0.1
1
0.1
10
10
Intercept
Intercept
Fig 7
1
Computed cumulative number distribution plots for intercepts through each crystal
shape showing variation with distribution width at a scaling factor of 4.
9
[ECDvLIreport/BM]
100
NPL Report MATC(A)149
Intercept distribution sigma=0 scaling=8
100
Intercept distribution sigma=0.1 scaling=8
100
Cube
TKC
TTP
80
% Cumulative Frequency
% Cumulative Frequency
80
Cube
TKC
TTP
60
40
20
60
40
20
0
0.1
1
0
10
0.1
1
Intercept
10
Intercept
110
Intercept distribution sigma=0.5 scaling=8
Intercept distribution sigma=0.3 scaling=8
100
100
Cube
TKC
TTP
90
80
% Cumulative Frequency
% Cumulative Frequency
80
Cube
TKC
TTP
60
40
20
70
60
50
40
30
20
10
0
1
10
0
100
1
Intercept
Fig 8
10
Intercept
Computed cumulative number distribution plots for intercepts through each crystal
shape showing variation with distribution width at a scaling factor of 8.
[ECDvLIreport/BM]
10
100
NPL Report MATC(A)149
Area distribution sigma=0 scaling=1
100
Area distribution sigma=0.1 scaling=1
100
Cube
TKC
TTP
80
% Cumulative Frequency
% Cumulative Frequency
80
Cube
TKC
TTP
60
40
60
40
20
20
0
0
0.01
0.1
1
0.1
1
10
Area
Area
110
Area distribution sigma=0.3 scaling=1
Area distribution sigma=0.5 scaling=1
100
100
90
80
% Cumulative Frequency
% Cumulative Frequency
80
60
Cube
TKC
TTP
40
70
Cube
TKC
TTP
60
50
40
30
20
20
10
0
0
1
10
1
100
100
Area
Area
Fig 9
10
Computed cumulative distribution for areas from each crystal shape showing
variation with distribution width for a scaling factor of 1.
11
[ECDvLIreport/BM]
NPL Report MATC(A)149
Area distribution sigma=0 scaling=2
100
Area Distribution sigma=0.1 scaling=2
100
Cube
TKC
TTP
80
% Cumulative Frequency
80
% Cumulative Frequency
Cube
TKC
TTP
60
40
60
40
20
20
0
0
0.1
1
1
10
10
100
Area
Area
110
Area distribution sigma=0.5 scaling=2
Area distribution sigma=0.3 scaling=2
100
100
90
80
% Cumulative Frequency
% Cumulative Frequency
80
Cube
TKC
TTP
60
40
70
Cube
TKC
TTP
60
50
40
30
20
20
10
0
0
10
100
10
1000
100
1000
Area
Area
Fig 10 Computed cumulative distribution for areas from each crystal shape showing
variation with distribution width for a scaling factor of 2.
[ECDvLIreport/BM]
12
NPL Report MATC(A)149
Area distribution sigma=0 scaling=4
100
Cube
TKC
TTP
80
Cube
TKC
TTP
80
% Cumulative Frequency
% Cumulative Frequency
Area distribution sigma=0.1 scaling=4
100
60
40
20
60
40
20
0
0
1
10
100
10
Area
100
1000
Area
110
Area distribution sigma=0.5 scaling=4
Area distribution sigma=0.3 scaling=4
100
100
90
Cube
TKC
TTP
80
% Cumulative Frequency
% Cumulative Frequency
80
60
40
70
Cube
TKC
TTP
60
50
40
30
20
20
10
0
10
0
10
100
1000
100
1000
Area
Area
Fig 11 Computed cumulative distribution for areas from each crystal shape showing
variation with distribution width for a scaling factor of 4.
13
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NPL Report MATC(A)149
Area distribution sigma=0 scaling=8
100
Area distribution sigma=0.1 scaling=8
100
Cube
TKC
TTP
80
% Cumulative Frequency
% Cumulative Frequency
80
Cube
TKC
TTP
60
40
60
40
20
20
0
0
1
10
100
10
100
Area
100
Cube
TKC
TTP
Area
110
Area distribution sigma=0.3 scaling=8
100
90
80
Cube
TKC
TTP
Area distribution sigma=0.5 scaling=8
80
% Cumulative Frequency
% Cumulative Frequency
1000
60
40
20
70
60
50
40
30
20
10
0
0
100
1000
10000
100
Area
1000
10000
Area
Fig 12 Computed cumulative distribution for areas from each crystal shape showing
variation with distribution width for a scaling factor of 8.
[ECDvLIreport/BM]
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NPL Report MATC(A)149
180
180
Intercept distribution sigma=0 scaling=1
140
160
Cube
TKC
TTP
140
120
120
100
100
Frequency
Frequency
Intercept distribution sigma=0.1 scaling=1
Cube
TKC
TTP
160
80
80
60
60
40
40
20
20
0
0
0.01
0.1
0.1
1
1
10
Intercept
Intercept
300
Intercept distributin sigma=0.3 scaling=1
350
Cube
TKC
TTP
250
Cube
TKC
TTP
300
250
Frequency
200
Frequency
Intercept distribution sigma=0.5 scaling=1
150
200
150
100
100
50
50
0
0
0.1
1
0.1
10
1
Intercept
Intercept
Fig 13 Computed frequency distribution plots of intercepts through each crystal shape
showing variation in distribution width at a scaling factor of 1.
15
[ECDvLIreport/BM]
10
NPL Report MATC(A)149
160
160
Intercept distribution sigma=0 scaling=2
Cube
TKC
TTP
140
120
120
100
100
80
60
Cube
TKC
TTP
140
Frequency
Frequency
Intercept Distribution sigma=0.1 scaling=2
80
60
40
40
20
20
0
0
0.01
0.1
0.1
1
1
10
Intercept
Intercept
250
350
Intercept distribution sigma=0.3 scaling=2
Intercept distribution sigma=0.5 scaling=2
Cube
TKC
TTP
200
Cube
TKC
TTP
300
250
Frequency
Frequency
150
100
200
150
100
50
50
0
0.1
1
0
10
0.1
Intercept
Fig 14
1
10
Intercept
Computed frequency distribution plots of intercepts through each crystal shape
showing variation in distribution width at a scaling factor of 2.
[ECDvLIreport/BM]
16
NPL Report MATC(A)149
200
Intercept distribution sigma=0 scaling=4
180
Intercept distribution sigma=0.1 scaling=4
180
Cube
TKC
TTP
160
Cube
TKC
TTP
160
140
140
120
Frequency
Frequency
120
100
80
60
100
80
60
40
40
20
20
0
0.1
1
0
10
0.1
Intercept
1
10
Intercept
250
350
Intercept distribution sigma=0.3 scaling=4
Intercept distribution sigma=0.5 scaling=4
Cube
TKC
TTP
200
Cube
TKC
TTP
300
250
Frequency
Frequency
150
100
200
150
100
50
50
0
0.1
1
0
10
0.1
Intercept
Fig 15
1
10
Intercept
Computed frequency distribution plots of intercepts through each crystal shape
showing variation in distribution width at a scaling factor of 4.
17
[ECDvLIreport/BM]
100
NPL Report MATC(A)149
140
Intercept distribution sigma=0 scaling=8
Intercept distribution sigma=0.1 scaling=8
180
Cube
TKC
TTP
120
Cube
TKC
TTP
160
140
100
120
Frequency
Frequency
80
60
100
80
60
40
40
20
20
0
0.1
1
0
10
0.1
1
Intercept
Intercept
350
Intercept distribution sigma=0.3 scaling=8
250
Intercept distribution sigma=0.5 scaling=8
Cube
TKC
TTP
Cube
TKC
TTP
300
200
250
Frequency
Frequency
10
150
100
200
150
100
50
50
0
0
1
10
100
10
Intercept
Intercept
Fig 16
1
Computed frequency distribution plots of intercepts through each crystal shape
showing variation in distribution width at a scaling factor of 8.
[ECDvLIreport/BM]
18
100
NPL Report MATC(A)149
260
180
Area distribution sigma=0 scaling=1
240
Cube
TKC
TTP
220
200
Area Distribution sigma=0.1 scaling=1
Cube
TKC
TTP
160
140
180
120
Frequency
Frequency
160
140
120
100
100
80
60
80
60
40
40
20
20
0
0
0.01
0.1
1
0.1
1
Area
Area
Area distribution sigma=0.3 scaling=1
180
Area distribution sigma=0.5 scaling=1
250
Cube
TKC
TTP
160
10
140
Cube
TKC
TTP
200
Frequency
Frequency
120
100
80
150
100
60
40
50
20
0
0
1
10
100
1
Area
10
100
Area
Fig 17 Computed frequency distribution plots of areas through each crystal shape showing
variation in distribution width at a scaling factor of 1.
19
[ECDvLIreport/BM]
NPL Report MATC(A)149
200
200
Area distribution sigma=0 scaling=2
Area distribution sigma=0.1 scaling=2
180
180
Cube
TKC
TTP
160
140
140
120
120
100
100
Frequency
Frequency
160
Cube
TKC
TTP
80
80
60
60
40
40
20
20
0
0
0.1
1
10
-20
Area
1
10
100
Area
200
Area distribution sigma=0.3 scaling=2
180
160
Area distribution sigma=0.5 scaling=2
Cube
TKC
TTP
250
Cube
TKC
TTP
200
140
Frequency
Frequency
120
100
80
150
100
60
40
50
20
0
0
10
100
10
1000
1000
Area
Area
Fig 18
100
Computed frequency distribution plots of areas through each crystal shape showing
variation in distribution width at a scaling factor of 2.
[ECDvLIreport/BM]
20
NPL Report MATC(A)149
Area distribution sigma=0 scaling=4
200
Area distribution sigma=0.1 scaling=4
Cube
TKC
TTP
180
160
200
150
120
Frequency
Frequency
140
Cube
TKC
TTP
100
80
100
60
50
40
20
0
0
1
10
100
10
100
Area
1000
Area
250
Area distribution sigma=0.5 scaling=4
Area distribution sigma=0.3 scaling=4
160
Cube
TKC
TTP
140
Cube
TKC
TTP
200
120
150
Frequency
Frequency
100
80
100
60
40
50
20
0
0
10
100
10
1000
1000
Area
Area
Fig 19
100
Computed frequency distribution plots of areas through each crystal shape showing
variation in distribution width at a scaling factor of 4.
21
[ECDvLIreport/BM]
NPL Report MATC(A)149
180
Area distribution sigma=0 scaling=8
160
180
Cube
TKC
TTP
140
Area distribution sigma=0.1 scaling=8
Cube
TKC
TTP
160
140
120
100
Frequency
Frequency
120
80
60
100
80
60
40
40
20
20
0
0
1
10
100
10
100
Area
1000
Area
180
Area distribution sigma=0.3 scaling=8
Area distribution sigma=0.5 scaling=8
250
Cube
TKC
TTP
160
140
Cube
TKC
TTP
200
100
Frequency
Frequency
120
80
60
150
100
40
50
20
0
0
100
1000
10000
100
Area
Fig 20
1000
10000
Area
Computed frequency distribution plots of areas through each crystal shape showing
variation with distribution width at a scaling factor 8.
[ECDvLIreport/BM]
22
NPL Report MATC(A)149
Arithmetic mean values for LI and ECD were obtained from the number distribution plots of
intercepts and areas (Figs 21-22) and the computed relationship between intercept and ECD
appears to be more or less independent of the spread of crystal size (sigma). The pattern that
seems to be emerging is that the ECD is about 1.7 times the intercept value. This holds for all
values of sigma. Fig 22 also includes a graph showing the affect of computed sample size
(between N=100 to 100,000) on the relation between ECD and LI for the TTP shape. There is
some evidence that the measured slope is different, slightly higher, for the smallest sample
size (N=100), but all the results are quite close.
20
65
Sigma=0
60
Cube
TKC
TTP
best-fit line cube
best-fit line TKC
best-fit line TTP
18
16
55
50
14
45
12
40
Cube
TKC
TTP
best-fit line cube
best-fit line TKC
best-fit line TTP
Sigma=0.1
ECD
ECD
35
10
8
Y =0.0292+1.63162 X
25
Y =0.34412+1.42067 X
6
30
Y =-0.0193+1.58614 X
4
2
20
Y =0.07362+1.64748 X
15
Y =-0.29992+1.49519 X
10
Y =-0.08169+1.65692 X
5
0
0
0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
Intercept
Intercept
65
60
55
Sigma=0.3
60
55
45
40
40
35
35
30
15
10
30
25
Y =0.06709+1.83087 X
20
Y =-0.74834+1.70614 X
Y =-0.15527+1.54971 X
15
Y =0.05298+1.64981 X
10
Y =0.10387+1.74674 X
5
5
0
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
Intercept
Intercept
Fig 21
Sigma=0.5
Y =0.23107+1.69522 X
25
20
Cube
TKC
TTP
best-fit line cube
best-fit line TKC
best-fit line TTP
50
45
ECD
ECD
50
65
Cube
TKC
TTP
best-fit line cube
best-fit line TKC
best-fit line TTP
Effect of crystal shape and distribution width on the relation between ECD and LI.
23
[ECDvLIreport/BM]
NPL Report MATC(A)149
In addition to the relationship between LI and ECD being almost independent of the size
distribution of crystals, it appears that it is also almost independent of the shape of the crystal.
This is shown by the similarity of the slopes of the plots of ECD against LI for the different
shapes with different distributions.
26
14
N=100
N=1000
N=10000
N=100000
best-fit line N=100
best-fit line N=1000
best-fit line N=10000
best-fit line N=100000
12
Cube
24
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit line sigma=0
best-fit line sigma=0.1
best-fit line sigma=0.3
best-fit line sigma=0.5
22
20
18
10
16
ECD
ECD
8
6
Y =-0.11841+1.701 X
Y =0.05298+1.64981 X
4
12
10
Y =0.0292+1.63162 X
8
Y =0.07362+1.64748 X
Y =0.23107+1.69522 X
6
Y =-0.00842+1.6713 X
Y =0.06709+1.83087 X
4
Y =0.01602+1.66111 X
2
14
2
0
0
0
1
2
3
4
5
6
7
8
0
2
4
6
Intercept
8
10
12
14
Intercept
70
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit line sigma=0
best-fit line sigma=0.1
best-fit line sigma=0.3
best-fit line sigma=0.5
60
50
TKC
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit line sigma=0
best-fit line sigma=0.1
best-fit line sigma=0.3
best-fit line sigma=0.5
14
12
TTP
10
8
ECD
ECD
40
30
6
20
Y =-0.0193+1.58614 X
Y =-0.08169+1.65692 X
Y =0.34412+1.42067 X
4
Y =0.05298+1.64981 X
Y =-0.29992+1.49519 X
10
Y =0.10387+1.74674 X
Y =-0.15527+1.54971 X
2
Y =-0.74834+1.70614 X
0
0
0
4
8
12
16
20
24
28
32
36
0
Intercept
Fig 22
1
2
3
4
5
6
7
8
Intercept
Effects of sample size on shape TTP and distribution width on the relation between
ECD and LI for all crystal shapes.
[ECDvLIreport/BM]
24
NPL Report MATC(A)149
The plot of the slopes of the ECD/LI graphs against sigma for each of the shapes illustrates
the similarity between the values. It also shows a trend that as sigma increases the slope of the
plot increases (Fig 23). This is true for all the shapes. The anomalous point in the TTP plot is
most probably due to statistical variation. Although the plot shows that as sigma increases the
slope of the ECD against LI graph increases for all three shapes, it seems to indicate that the
relationship between sigma and the slope is slightly different for each shape. The TKC plot
could be approximated by a straight line, whereas the TTP and cube plots, ignoring
anomalous points, might be better approximated by a curve, possibly exponential in shape.
1.85
Cube
TKC
TTP
1.80
1.75
1.70
Slope
1.65
1.60
1.55
1.50
1.45
1.40
0.0
0.1
0.2
0.3
0.4
0.5
Sigma
Fig 23
Dependence of shape of ECD/LI plot on distribution width
25
[ECDvLIreport/BM]
NPL Report MATC(A)149
3.2.2 Effect of TTP Truncation and Shape
A further set of results were obtained by changing the height/length of the TTP crystal for
different amounts of truncation (Fig 24). The results are shown in Figs 25-27, including linear
best fits.
Fig 24
Some examples of different shaped TTPs
[ECDvLIreport/BM]
26
NPL Report MATC(A)149
TTP height=0.2 truncation=0
TTP height=0.2 truncation=0.5
10
10
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit line sigma=0
best-fit line sigma=0.1
best-fit line sigma=0.3
best-fit line sigma=0.5
8
8
6
ECD
ECD
6
4
Y =-0.00435+1.82667 X
Y =-0.04391+1.85057 X
Y =-0.04074+1.9199 X
Y =0.09003+2.00642 X
4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0
2
Y =0.01838+1.71304 X
Y =-0.03558+1.79844 X
0
Intercept
Fig 25
1
2
3
4
5
Intercept
Effect of TTP truncation for a height of 0.2
16
16
TTP height=0.5 truncation=0.5
TTP height=0.5 truncation=0.
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit line sigma=0
best-fit line sigma=0.1
best-fit line sigma=0.3
best-fit line sigma=0.5
14
12
14
12
10
8
8
ECD
10
6
6
Y =-0.00166+1.77019 X
Y =0.06921+1.70426 X
Y =-0.03964+1.832 X
Y =0.16455+1.90293 X
4
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit line sigma=0
best-fit line sigma=0.1
best-fit line sigma=0.3
best-fit line sigma=0.5
Y =0.00757+1.59489 X
Y =-0.04405+1.62398 X
4
2
Y =-0.02767+1.6988 X
Y =-0.16247+1.87581 X
2
0
0
1
2
3
4
5
6
7
0
8
0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.0
Intercept
Fig 26
Y =0.06108+1.72914 X
Y =-0.00818+1.98478 X
2
0
ECD
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit line sigma=0
best-fit line sigma=0.1
best-fit line sigma=0.3
best-fit line sigma=0.5
Intercept
Effect of TTP truncation for a height of 0.5
27
[ECDvLIreport/BM]
NPL Report MATC(A)149
TTP height=1.4 truncation=0
20
18
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit sigma=0
best-fit sigma=0.1
best-fit sigma=0.3
best-fit sigma=0.5
18
16
14
14
12
12
ECD
ECD
16
TTP height=1.4 truncation=0.5
20
sigma=0
sigma=0.1
sigma=0.3
sigma=0.5
best-fit line sigma=0
best-fit line sigma=0.1
best-fit line sigma=0.3
best-fit line sigma=0.5
10
10
8
8
Y =-0.09645+1.95035 X
Y =-0.07806+1.86915 X
Y =0.29418+1.80086 X
Y =0.52791+1.90565 X
6
4
Y =-0.02301+1.59681 X
Y =-0.00751+1.62243 X
Y =-0.12477+1.71847 X
Y =-0.00724+1.77636 X
6
4
2
2
0
0
0
0
2
4
6
8
10
4
6
8
10
Intercept
Intercept
Fig 27
2
Effect of TTP truncation for a height of 1.4
These simulation studies showed that the shape of the crystal had only a small effect on the
relationship between LI and ECD, and also showed that the amount of truncation and the
relative dimensions of the TTP shape crystal had little effect. The range of values for the
slopes were similar to the range for the different shapes, indicating the similarity between
changing the shape and the relative dimensions of the TTP. They both seem to amount to the
same change in LI/ECD relationship. This is not unsurprising since changing the height and
truncation of the TTP does effectively change its shape, thus you would expect a similar
pattern to be found.
The graph of slope of ECD/LI plot against sigma (Fig 28) doesn’t show quite the same pattern
as that found between the 3 initial shapes, that the slope increases as sigma increases, but it
does indicate that as the truncation decreases the slope increases.
[ECDvLIreport/BM]
28
12
NPL Report MATC(A)149
2.0
Slope
1.9
1.8
1.7
height=0.2 truncation=0
height=0.2 truncation=0.5
height=0.5 truncation=0
height=0.5 truncation=0.5
height=1.4 truncation=0
height=1.4 truncation=0.5
1.6
0.0
0.1
0.2
0.3
0.4
0.5
Sigma
Fig 28
Effect of crystal shape on relation between ECD and LI and distribution width
3.2.3 Effect of Measurement Resolution
When measurements of crystal area and linear intercept are taken from images of sectioned
and polished samples they are generally found to follow a lognormal distribution [2,3,11].
However the model produced by POLYCHOP didn’t generally follow this trend (Fig 29),
where on the left the cumulative frequency is plotted on a probability scale showing deviation
from linearity for computed data (taken from Fig 5). On the right hand side experimental data
taken from NPL data [8,9] is shown with a good linear fit and no curvature at small values.
This difference could be attributed to the accuracy with which practical measurements can be
taken compared to that with which the simulation can model. When measurements are taken
practically the smallest crystal visible is related to the resolution of the image. This means
that when the number of crystals in a given area are counted, the smaller crystals may be
under-represented in the distribution as they often cannot be seen. This leads to the average
area or intercept being greater than expected, as fewer crystals are counted.
In order to verify the accuracy of the model, the effect of the resolution problem was added to
the model, and the resulting distributions plotted. In order to simulate the effects of the
resolution the area and intercept data for a TTP were taken from POLYCHOP and grouped
into number frequency sets. The smallest bins were then removed, starting with 0, then
2,4,6,8 and 10 bins being removed from the data. -The cumulative frequency plots of the
remaining data was then plotted (Fig 30). The pattern exhibited by these plots is that the more
bins that are ignored, so simulating a lower resolution of image, the more linear the lognormal
29
[ECDvLIreport/BM]
NPL Report MATC(A)149
plot becomes. This means that although the data produced by POLYCHOP appears not to
give the same pattern as practical data, it is really quite similar. However it exhibits a slightly
different distribution due to the lack of practical limitations in measuring small grains or
intercept lengths.
99.999
Intercept distribution sigma=0.3 scaling=2
Cube
TKC
TTP
99.5
Probability % Cumulative Frequency
Probability % Cumulative Frequency
99.999
95
70
40
10
1
99.5
95
70
40
10
1
0.01
0.01
1E-4
0.1
1
1E-4
10
0.1
Intercept
Fig 29
Polychop Model
Experimental Data
1
Intercept
Cumulative frequency plotted on a probability scale.
The knowledge gained from the comparison between the distribution of practical values, and
the distribution of modelled values, where no practical limitations apply, can be used as a
measure of the accuracy of the practical method. In order for the POLYCHOP data to form a
linear pattern 10 bins had to be removed, representing the smallest 10 bin sizes out of 100
bins (10%). This would indicate that the smallest 10% of grain sizes are not measured using
the practical method. This value seems higher than expected, and also seems unrealistic in
terms of the resolution being probably dependent on the imaging equipment, not on a property
of the material being imaged.
The plot of ECD against intercept in Fig 30 for different numbers of bins removed shows that
although removing bins affects the distribution of the data it has only a small effect on the
relationship between ECD and intercept, which remains linear, and of almost constant
gradient.
[ECDvLIreport/BM]
30
10
NPL Report MATC(A)149
99.999
99.5
95
Cum Freq %
TTP sigma=0.3 scaling=2
0 bins removed
2 bins removed
4 bins removed
6 bins removed
8 bins removed
10 bins removed
70
40
10
1
0.01
1E-4
0.01
99.999
99.5
Cum. Freq. %
95
0.1
Intercept
1
TTP sigma=0.3 scaling=2
0 bins removed
2 bins removed
4 bins removed
6 bins removed
8 bins removed
10 bins removed
70
40
10
1
0.01
1E-4
0.1
Fig 30
1
Area
10
Effect of systematically removing smallest bins from computed distributions of
intercept and area
31
[ECDvLIreport/BM]
NPL Report MATC(A)149
0 bins removed
2 bins removed
4 bins removed
6 bins removed
8 bins removed
10 bins removed
best-fit line 0 bins removed
best-fit line 2 bins removed
best-fit line 4 bins removed
best-fit line 6 bins removed
best-fit line 8 bins removed
best-fit line 10 bins removed
12
10
ECD
8
6
4
Y =-0.29988+1.7423 X
Y =-0.29551+1.74075 X
Y =-0.28506+1.74059 X
Y =-0.26549+1.74025 X
Y =-0.25179+1.74302 X
Y =-0.224+1.74041 X
2
0
0
1
2
3
4
5
6
7
Intercept
Fig 31
Effect of very small intercepts on the relation between ECD and LI
[ECDvLIreport/BM]
32
8
NPL Report MATC(A)149
4
SUMMARY
A study of the relation between the equivalent circle diameter (ECD) and linear intercept (LI)
methods for measuring the grain size of WC/Co materials has been conducted through an
extensive modelling exercise supported by a limited number of experimental data.
The experimental data indicated that
ECD » 1.15 LI
(1)
This is in good agreement with the empirical reaction ECD = 4 / p LI found in
measurements on aluminium and ferrous alloys [10]. It would be reasonable therefore to use
this expression for converting measurements of linear intercept grain size to equivalent circle
diameter grain size and vice versa.
However, the computed data indicated that
ECD » 1.7 LI
(2)
At present no satisfactory reason for this difference is known. Further, more detailed
experiments will be conducted to check expression (1) above. The modelling studies showed
that the relationship between ECD and LI was only slightly sensitive to:
·
Crystal shape, for the three shapes studied (cube, truncated triagonal prism and
tetrakaidecahedron).
·
Amount of truncation and aspect ratio of truncated triagonal prisms.
Modelling studies indicated that a lack of measurement resolution may affect the detailed
characterisation of distribution plots of grain size at smaller values.
5
ACKNOWLEDGEMENTS
The work reported in this document was performed as part of the DTI research programme on
the characterisation of powder route materials (MPP6100 – Durability of Hard Materials) in
support of the ISO standardisation process for WC/Co hardmetals. Members of the British
Hardmetal Research Association are thanked for the supply of materials and discussion of
preliminary results. Former NPL colleague Austin Day is thanked for developing the
POLYCHOP programme.
33
[ECDvLIreport/BM]
NPL Report MATC(A)149
6
REFERENCES
1
B. Roebuck, E.G. Bennett and M.G. Gee. Grain size measurement methods for WC/Co
Hardmetals, 13 International Plansee Seminar, May 24-28 1993, Reutte, Austria, V2,
273-292.
2
B. Roebuck. Measurement of Grain Size and Size Distribution in Engineering Alloys.
RMS/IoM Conference on Quantitative Microscopy of High Temperature Materials,
November 1999, Sheffield Hallam University and Mater. Sci. and Techn., Oct 2000, 16,
1167-1174.
3
E.G. Bennett and B. Roebuck. The Metallographic Measurement of WC Grain Size.
NPL Good Practice Guide 22, August 1999.
4
T. Werlefors and C. Eskilsson. On-line Computer Analysis of WC-Co Structures
Imaged in an SEM Metallog., 12, 1979, 153-173.
5
K. M. Friederich and H. E. Exner. Metallographic Investigations on WC Powders. Prakt.
Met., 21, 1984, 334-341.
6
A. Nordgren. Microstructural Characterisation of Cemented Carbides using SEM based
Automatic Image Analysis. Int. J. Refract. and Hard Materials, 10, 1991, 61-81.
7
B. Roebuck. Measuring WC Grain Size Distribution, Metal Powder Report, April 1999,
20-24.
8
B. Roebuck and E. G. Bennett. NPL MATC(A)101, August 2002. Microstructure and
Toughness of Various Baseline and Wide Grained Hardmetals.
9
B. Roebuck, E.G. Bennett, W.P. Byrne and M.G. Gee. NPL CMMT(A)172, April 1999.
Characterisation of Baseline Hardmetals using Property Maps.
10
R. T. de Hoff and F. N. Rhines. Quantitative Microscopy, McGraw-Hill, USA, 1968,
239-241.
11
H.E. Exner, Powder Metall., 13 (26), 1970, 429-448.
[ECDvLIreport/BM]
34