a comparison of the linear intercept and equivalent circle...

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

[ECDvLIreport/BM]

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 �m. 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]

© 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]

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


1 [ECDvLIreport/BM]

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.


[ECDvLIreport/BM] 2

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:

� �

��

�

��

�

�

�

�� 2

2g

2

)(ln)xln(exp

2x1)x(f

where � is a measure of the distribution width and �g is the geometric mean. This does not plot conveniently mathematically, but many software packages have numerical approximations to the function, permitting � to be determined. The use of probability paper generally gives a straight line for data for current commercial hardmetals, indicating that the parameter, �, 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 � values of 0.0, 0.1, 0.3 and 0.5, where � 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).


3 [ECDvLIreport/BM]

TTP

Cube

TKC

Fig 2 POLYCHOP shapes with typical random cross sections and intercept lengths.


[ECDvLIreport/BM] 4

Fig 3 Diagram showing TTP measurements

2 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 two-phase, 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 LDJ� equipment to support these observations.

Table 1 Hardmetal Properties

Grade wt% Co

Volume Fraction

WC Coercivity

kA m-1

Magnetic moment �T m3 kg-1

Grade Wt% Co

Volume Fraction

WC Coercivity

kA m-1

Magnetic moment �T m3 kg-1

T25 24.2 0.55 6.8 2.82 G10 9.90 0.82 6.5 1.48 D10 10.0 0.82 9.8 1.35 CW25C 24.44 0.54 4.4 3.52 T 9.1 0.83 8.4 1.10 CW20C 19.68 0.62 4.1 2.87 UC9 9.8 0.82 4.0 1.35 TC222 6.09 0.88 2.5 0.98 MA11 11.7 0.78 3.8 1.64 K3560 9.69 0.82 4.6 1.37 K3520 20.3 0.62 5.1 2.88 M15 15.2 0.7 5.1 2.11


5 [ECDvLIreport/BM]

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

Operator A Operator B

Grade LI ECD Grade LI ECD T25 D10

T UC9

MA11 K3520 G10

CW25C CW20C TC222 K3560 M15

0.77 1.05 1.68 4.97 5.16 2.30 1.76 2.14 2.17 3.59 2.75 2.04

1.10 1.27 2.00 5.75 6.38 2.62 2.66 2.73 2.50 4.71 3.30 2.60

T25 D10

T UC9

MA11 K3520 G10

CW25C CW20C TC222 K3560 M15

0.77 1.05 1.68 4.97 5.16 2.30 1.76 2.14 2.17 3.59 2.75 2.04

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


[ECDvLIreport/BM] 6

ALI �

where Ā is the average grain area (Jefferies method). Thus since 2)ECD(4

A �

�

LI13.1A13.1A4ECD ��

�

�

Clearly the measurements on WC/Co alloys on the comparison between ECD and LI agree well with previously published work on other engineering alloys.

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 �=0 (i.e. all crystals the same size). For higher values of � these peaks systematically reduce, as expected.

0 1 2 3 4 5 60

1

2

3

4

5

6

7

ECD =1.13 LI

ECD =1.19 LI

Equi

vale

nt c

ircle

dia

met

er (E

CD

) �

m

Intercept (LI) �m

Measured data - operator A Measured data - operator B


7 [ECDvLIreport/BM]

Fig 5 Computed cumulative number distribution plots for intercepts through each crystal

shape showing variation with distribution width at a scaling factor of 1.

0.01 0.1 10

20

40

60

80

100Intercept distribution sigma=0 scaling=1

Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept0.1 1 10

0

20

40

60

80

100Intercept distribution sigma=0.1 scaling=1

Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept

0.1 1 100

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept

0.1 1 100

10

20

30

40

50

60

70

80

90

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept


[ECDvLIreport/BM] 8



0.01 0.1 10

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept0.1 1 10

0

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept

0.1 1 10

20

40

60

80

100

Intercept distribution sigma=0.3 scaling=2

Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept

0.1 1 100

10

20

30

40

50

60

70

80

90

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept


9 [ECDvLIreport/BM]



0.1 1 100

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept0.1 1 10

0

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept

0.1 1 100

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept

0.1 1 10 1000

10

20

30

40

50

60

70

80

90

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept


[ECDvLIreport/BM] 10



0.1 1 100

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept0.1 1 10

0

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept

1 10 1000

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept1 10 100

0

10

20

30

40

50

60

70

80

90

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Intercept


11 [ECDvLIreport/BM]

Fig 9 Computed cumulative distribution for areas from each crystal shape showing

variation with distribution width for a scaling factor of 1.

0.01 0.1 10

20

40

60

80

100Area distribution sigma=0 scaling=1

Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area0.1 1 10

0

20

40

60

80

100Area distribution sigma=0.1 scaling=1

Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area

1 10 1000

20

40

60

80

100

Area distribution sigma=0.3 scaling=1

Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area

1 10 1000

10

20

30

40

50

60

70

80

90

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area





0.1 1 100

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area

1 10 1000

20

40

60

80

100Area Distribution sigma=0.1 scaling=2

Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area

10 100 10000

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area

10 100 10000

10

20

30

40

50

60

70

80

90

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area





1 10 1000

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area10 100 1000

0

20

40

60

80

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area

10 100 10000

20

40

60

80

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area

10 100 10000

10

20

30

40

50

60

70

80

90

100


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area



Fig 12 Computed cumulative distribution for areas from each crystal shape showing variation with distribution width for a scaling factor of 8.

1 10 1000

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area10 100 1000

0

20

40

60

80


Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area

100 1000 100000

20

40

60

80

100

Area distribution sigma=0.3 scaling=8 Cube TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area100 1000 10000

0

10

20

30

40

50

60

70

80

90

100

110Area distribution sigma=0.5 scaling=8 Cube

TKC TTP

% C

umul

ativ

e Fr

eque

ncy

Area



Fig 13 Computed frequency distribution plots of intercepts through each crystal shape

showing variation in distribution width at a scaling factor of 1.

0.01 0.1 10

20

40

60

80

100

120

140

160


Cube TKC TTP

Freq

uenc

y

Intercept0.1 1 10

0

20

40

60

80

100

120

140

160


Cube TKC TTP

Freq

uenc

yIntercept

0.1 1 100

50

100

150

200

250

300Intercept distributin sigma=0.3 scaling=1

Cube TKC TTP

Freq

uenc

y

Intercept

0.1 1 100

50

100

150

200

250

300


Cube TKC TTP

Freq

uenc

y

Intercept





0.01 0.1 10

20

40

60

80

100

120

140


Cube TKC TTP

Freq

uenc

y

Intercept0.1 1 10

0

20

40

60

80

100

120

140

160Intercept Distribution sigma=0.1 scaling=2

Cube TKC TTP

Freq

uenc

y

Intercept

0.1 1 100

50

100

150

200


Cube TKC TTP

Freq

uenc

y

Intercept0.1 1 10

0

50

100

150

200

250

300


Cube TKC TTP

Freq

uenc

y

Intercept





0.1 1 100

20

40

60

80

100

120

140

160

180 Intercept distribution sigma=0 scaling=4

Cube TKC TTP

Freq

uenc

y

Intercept0.1 1 10

0

20

40

60

80

100

120

140

160

180


Cube TKC TTP

Freq

uenc

y

Intercept

0.1 1 100

50

100

150

200


Cube TKC TTP

Freq

uenc

y

Intercept0.1 1 10 100

0

50

100

150

200

250

300


Cube TKC TTP

Freq

uenc

y

Intercept





0.1 1 100

20

40

60

80

100

120

140

160

180 Intercept distribution sigma=0.1 scaling=8

Cube TKC TTP

Freq

uenc

y

Intercept

1 10 1000

50

100

150

200


Cube TKC TTP

Freq

uenc

y

Intercept

0.1 1 100

20

40

60

80

100

120


Cube TKC TTP

Freq

uenc

y

Intercept

1 10 1000

50

100

150

200

250

300


Cube TKC TTP

Freq

uenc

y

Intercept



Fig 17 Computed frequency distribution plots of areas through each crystal shape showing

variation in distribution width at a scaling factor of 1.

0.01 0.1 10

20

40

60

80

100

120

140

160

180

200

220

240


Cube TKC TTP

Freq

uenc

y

Area0.1 1 10

0

20

40

60

80

100

120

140

160

180 Area Distribution sigma=0.1 scaling=1

Cube TKC TTP

Freq

uenc

y

Area

1 10 1000

20

40

60

80

100

120

140

160

180 Area distribution sigma=0.3 scaling=1

Cube TKC TTP

Freq

uenc

y

Area1 10 100

0

50

100

150

200


Cube TKC TTP

Freq

uenc

y

Area





0.1 1 100

20

40

60

80

100

120

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160

180


Cube TKC TTP

Freq

uenc

y

Area 1 10 100-20

0

20

40

60

80

100

120

140

160

180

200 Cube TKC TTP

Freq

uenc

y

Area


10 100 10000

20

40

60

80

100

120

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180


Cube TKC TTP

Freq

uenc

y

Area

10 100 10000

50

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Cube TKC TTP

Freq

uenc

y

Area





1 10 1000

20

40

60

80

100

120

140

160

180

200 Area distribution sigma=0 scaling=4

Cube TKC TTP

Freq

uenc

y

Area10 100 1000

0

50

100

150

200


Cube TKC TTP

Freq

uenc

y

Area

10 100 10000

20

40

60

80

100

120

140

160 Area distribution sigma=0.3 scaling=4

Cube TKC TTP

Freq

uenc

y

Area10 100 1000

0

50

100

150

200

250

Cube TKC TTP


Freq

uenc

y

Area




variation with distribution width at a scaling factor 8.

1 10 1000

20

40

60

80

100

120

140

160


Cube TKC TTP

Freq

uenc

y

Area10 100 1000

0

20

40

60

80

100

120

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160

180 Area distribution sigma=0.1 scaling=8 Cube TKC TTP

Freq

uenc

y

Area

100 1000 100000

20

40

60

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160


Cube TKC TTP

Freq

uenc

y

Area100 1000 10000

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Cube TKC TTP

Freq

uenc

y

Area



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.

Fig 21 Effect of crystal shape and distribution width on the relation between ECD and LI.

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

16

18

20

Y =-0.0193+1.58614 XY =0.34412+1.42067 XY =0.0292+1.63162 X

Sigma=0 Cube TKC TTP best-fit line cube best-fit line TKC best-fit line TTP

ECD

Intercept0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

0

5

10

15

20

25

30

35

40

45

50

55

60

65

Y =-0.08169+1.65692 X

Y =-0.29992+1.49519 XY =0.07362+1.64748 X

Sigma=0.1 Cube TKC TTP best-fit line cube best-fit line TKC best-fit line TTP

ECD

Intercept

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 380

5

10

15

20

25

30

35

40

45

50

55

60

65

Y =0.05298+1.64981 X

Y =-0.15527+1.54971 X

Y =0.23107+1.69522 X


ECD

Intercept

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 380

5

10

15

20

25

30

35

40

45

50

55

60

65

Y =0.10387+1.74674 X

Y =-0.74834+1.70614 X

Y =0.06709+1.83087 X


EC

D

Intercept



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.

Fig 22 Effects of sample size on shape TTP and distribution width on the relation between ECD and LI for all crystal shapes.

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

Y =0.10387+1.74674 XY =0.05298+1.64981 X

Y =-0.08169+1.65692 XY =-0.0193+1.58614 X

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

ECD

Intercept

TTP

0 4 8 12 16 20 24 28 32 360

10

20

30

40

50

60

70

Y =-0.74834+1.70614 XY =-0.15527+1.54971 X

Y =-0.29992+1.49519 XY =0.34412+1.42067 X


ECD

Intercept

TKC

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

16

18

20

22

24

26

Y =0.06709+1.83087 XY =0.23107+1.69522 XY =0.07362+1.64748 XY =0.0292+1.63162 X


ECD

Intercept

Cube

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

Y =0.01602+1.66111 X

Y =-0.00842+1.6713 XY =0.05298+1.64981 XY =-0.11841+1.701 X

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

ECD

Intercept



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.

Fig 23 Dependence of shape of ECD/LI plot on distribution width

0.0 0.1 0.2 0.3 0.4 0.5

1.40

1.45

1.50

1.55

1.60

1.65

1.70

1.75

1.80

1.85

Cube TKC TTP

Slop

e

Sigma



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



Fig 25 Effect of TTP truncation for a height of 0.2

Fig 26 Effect of TTP truncation for a height of 0.5

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.00

2

4

6

8

10

12

14

16

Y =-0.16247+1.87581 XY =-0.02767+1.6988 X

Y =-0.04405+1.62398 XY =0.00757+1.59489 X

ECD

Intercept

sigma=0sigma=0.1sigma=0.3sigma=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

TTP height=0.5 truncation=0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00

2

4

6

8

10

Y =0.09003+2.00642 XY =-0.04074+1.9199 X

Y =-0.04391+1.85057 XY =-0.00435+1.82667 X

EC

D

Intercept

sigma=0 sigma=0.1sigma=0.3sigma=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

TTP height=0.2 truncation=0

0 1 2 3 4 50

2

4

6

8

10

Y =-0.00818+1.98478 XY =0.06108+1.72914 X

Y =-0.03558+1.79844 XY =0.01838+1.71304 X

ECD

Intercept



0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

16

Y =0.16455+1.90293 XY =-0.03964+1.832 X

Y =0.06921+1.70426 XY =-0.00166+1.77019 X

ECD

Intercept


TTP height=0.5 truncation=0.



Fig 27 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.

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

Y =0.52791+1.90565 XY =0.29418+1.80086 X

Y =-0.07806+1.86915 XY =-0.09645+1.95035 X

ECD

Intercept


TTP height=1.4 truncation=0

0 2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

Y =-0.00724+1.77636 XY =-0.12477+1.71847 XY =-0.00751+1.62243 XY =-0.02301+1.59681 X

ECD

Intercept

sigma=0sigma=0.1sigma=0.3sigma=0.5 best-fit sigma=0 best-fit sigma=0.1 best-fit sigma=0.3 best-fit sigma=0.5




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

0.0 0.1 0.2 0.3 0.4 0.5

1.6

1.7

1.8

1.9

2.0

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

Slop

e

Sigma



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.

Fig 29 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.

0.1 1 101E-4

0.01

1

10

40

70

95

99.5

99.999 Polychop Model Experimental Data

Prob

abili

ty %

Cum

ulat

ive

Freq

uenc

y

Intercept0.1 1 10

1E-4

0.01

1

10

40

70

95

99.5

99.999 Intercept distribution sigma=0.3 scaling=2

Cube TKC TTP

Prob

abilit

y %

Cum

ulat

ive

Freq

uenc

y

Intercept



Fig 30 Effect of systematically removing smallest bins from computed distributions of intercept and area

0.1 1 101E-4

0.01

1

10

40

70

95

99.5

99.999

Cum

. Fre

q. %

Area

0 bins removed 2 bins removed 4 bins removed 6 bins removed 8 bins removed 10 bins removed

TTP sigma=0.3 scaling=2

0.01 0.1 11E-4

0.01

1

10

40

70

95

99.5

99.999

Cum

Fre

q %

Intercept

0 bins removed 2 bins removed 4 bins removed 6 bins removed 8 bins removed 10 bins removed

TTP sigma=0.3 scaling=2



Fig 31 Effect of very small intercepts on the relation between ECD and LI

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

Y =-0.224+1.74041 XY =-0.25179+1.74302 XY =-0.26549+1.74025 XY =-0.28506+1.74059 XY =-0.29551+1.74075 XY =-0.29988+1.7423 X

EC

D

Intercept

0 bins removed2 bins removed4 bins removed6 bins removed8 bins removed10 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



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 LI/4ECD �� 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.



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.

a comparison of the linear intercept and equivalent circle...

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