Determination of volume distribution function by lineal analysis

Metallography

161

Determination

of Volume

Distribution

Function by Lineal Analysis

K. G. STJERNBERG Department

of Physics, Chalmers

University

of Technology,

Gothenburg,

Sweden

A linear method for calculating the volume frequency distribution function is described. The factors affecting lineal analysis are defined, and a diagrammatic method of obtaining the size frequency distribution function is presented.

Introduction Information about grain size and grain shape in metals is often obtained by studying a polished and etched section in a metallographic microscope. It is usually sufficient to determine a mean grain size by counting either the number of grains per unit area (Jeffries’ method’) or the number of grains per unit length (Heyn’s method2). The latter method has the advantage that the mean grain size is directly related to the specific grain surface through a simple relation.3 To obtain more information

about the structure,

the sectioned areas of the

individual grains are measured (area1 analysis), or a straight line is drawn intersecting many grains and the distances between the pairs of points (intercepts) in which this line cuts each grain are measured (lineal analysis). In the former case the area1 distribution distribution function. Using the assumption mathematical

function

is determined,

and in the latter the lineal

of spherical grains, Bockstiege14 has derived a simple

relation between the spatial grain size distribution function and the

lineal distribution function. In this paper a method is described to determine by lineal analysis the volume frequency function, g(Z), defined by g(Z) dZ equals the fraction of volume occupied by grains with diameters between 1 and 1 + dl. It is also shown that corrections General

must be made when it is impossible to measure along one single line. Theoretical

Considerations

In the following presentation it is assumed that thematerial grains. Starting from Bockstiegel’s

consists of spherical

equation, some quantities will be calculated. Metallography,

Copyright

0

1969 by American

Elsevier Publishing

2 (1969)

161-170

Company,

Inc.

K. G. Stjernberg

162 The following

notation

d(h) dA = number

is used:

of intercepts

per unit

length

with

lengths

between

h and

X + dh. N’

zzz

total number

of intercepts

silnV) ~

= probability

F(Z) =

Nl distribution

n3(h) dA = number

per unit length. that an intercept

is less than

1 (the lineal

function). of grains

per unit

volume

with

diameters

between

h and

X + dh. jjl3

=

total number s:, n3(h) dh

F3(1) =

N3 (grain

of grains

= probability

size distribution

G(1) = fraction

of volume

(volume

distribution

occupied

of a grain is less than 1

by grains

with

diameters

less than

I

function).

l/N1

= mean

TX

l/N3

= mean volume

intercept

length. of the grains.

we introduce

f”(Z) = $F3(1);

f’(Z) = &F’(Z);

According

that the diameter

function).

I=

In addition,

per unit volume.

g(l) = $

G(j)

to Bockstiegel,4 = ; I jW n”(D) dD 2

p(1)

Derivation

and rearrangement

Integration

of equation

(1)

gives

2 gives $‘3(4

=

1 _

2 ~_N1

nN3

f’(l)

1

(3)

Lineal Analysis Multiplication

Determination by 1 and derivation F(Z)

Putting

163 gives

+ 1 &F(Z)

=

1 -

J&

-F(Z) ;a

1 = 0 gives (4)

Thus

7ri

v=

(5)

2(d2/dZ2) F(0)

By definition,

g(Z) = Iv327+f which

“(I)

(6)

Z

(7)

can be written g(z)

-Adfl(l)

=

31 Definition Suppose

dZ

of x that a point

is randomly

chosen

p(Z) dl, that the point

probability,

on a straight

is on an intercept

line in a specimen.

with length

between

The 1 and

Z + dZ is p(Z) dZ = +(Z) We now chosen

define point.

x as the mean

length

dZ

of intercept

(8) drawn

through

a randomly

Thus

j-aZ f

ii =

f l(Z) dZ = f [” Z2f1(Z) dZ

0

YO

The definition of x is valid for any grain shape, and so is equation 9. The value of x can always be calculated from lineal analysis data. Thus x is a general structure parameter. The mean value,

& , of Z with respect

to g(Z), for spherical

grains

is

i, = _f* t-g(Z) dZ = $ j-m Z”f’(Z) dZ = ; ii 0

0

(10)

Lineal Analysis Some general function

relations

between

have been derived.

Before

different

quantities

we use these

and the lineal distribution

relations

for numerical

calcula-

K. G. Stjernberg

164 tions, a method of determining

the function F’(Z) will be described.

As some

corrections have to be made, it will be shown in detail which quantities can be determined by lineal analysis and how they must be corrected. In the following discussion we consider a material consisting of two phases, a and j3. The a-phase consists of separate grains which are embedded in continuous B-phase. The grain shape is such that only a slight error is introduced by regarding it as spherical. Furthermore,

we are interested only in the lineal distribution

of the a-phase. As to the p-phase, intercept length.

we are content with determining

We also assume the lineal analysis to be done on photographs. on photographs,

the method

function the mean

(By measuring

can also be used with an electron

microscope.)

All quantities connected with the quantity I (for example, L and A) are lengths in the specimen. The corresponding lengths on the photograph are denoted by analogous notation connected with the letter x (for example, L and 1 correspond to X and 5, respectively).

Thus if the magnification

is G,

x = Gl On the photographs length X.

Total

the intercepts

accumulated

are measured

lengths

along N lines of average

in the 01- and B-phases

are measured

separately. The total lengths also include the lengths of those fractions of intercepts that are at the beginning and the end of every measuring line. By lineal analysis it is possible to determine the following quantities: Xtot = total measured length on the photographs. X,

= total measured length in a-phase on the photographs.

Xs = total measured length in B-phase on the photographs. Ni”

= number

of measured

a-intercepts

along Xt,t

which are less than xi ,

where xi; i = 0, I, 2 ,..., p, is the grain class boundaries. N,”

= number of measured a-intercepts

along Xt,,t .

IV,” = number of measured B-intercepts

along Xt,,t .

(The upper indices, m, in the last three quantities mean that the quantities are those initially determined. It will be shown below that they need to be corrected. The corresponding corrected quantities.)

quantities

without

upper

indices

With this notation it is possible to calculate some important

represent

the

quantities:

c, = volume fraction of a-phase5 = X,/Xtot .

(114

cs = volume fraction of B-phase = 1 - c, .

Ulb)

j

a

_

1

x, _ mean __

G N,

intercept length in a-phase.

Lineal Analysis Determination 1

4

=LxB=

G NB

&NE--___

N,

165

mean intercept length in p-phase.

= continuity.6

To get a correct value of the total number of intercepts

found on the length

Xtot , it is necessary to add half the number of end points of the measuring lines which are in the or-phase. The probability that an end point is in the a-phase is c,.~ Thus the following equations are valid: N, = N,” + 42Nc, NB = N,”

= N,” + NC,

(124

+ NcB

(12b)

Determination of Fl(x) In practice, Fl(x) is determined

by measuring the different intercepts

along

lines on photographs. This is most easily done by using a device which has certain reference lengths xi; i = 0, 1, 2 ,..., p, built in. The number of intercepts less than xi is determined

for each value xi,& The

probability,

F”(xJ,

that a

measured intercept is less than x is given by

It is now convenient to discuss the way of choosing the xI’s. In the literature the relation xi+i = kxi is often seen. This choice is suitable when n3(Z) is to be determined because n”(Z) often seems in reality to be logarithmiconormaldistributed.ssg

Experience

shows that it is better to use a lineal dividing when

determining g(Z), because g(Z) is almost

symmetric

even if n3(Z) seems to be

logarithmiconormal (see Fig. 1). Thus xi is chosen here as xi

=

ih;

i = 0, 1) 2 )..., p

where h is the difference between consecutive values of xi . Suppose that the total measuring line, Xtot , is divided into N parts (they need not be of the same length). Those intercepts that have one end point outside the

& The author has used an instrument where the reference lengths are contact bars. After measuring on the required number of photographs, N,“, No”‘, and N,“; i = 1, 2, 3,..., p, are read on electrical counters, and the lengths Xtot and X,g on mechanical counters.

K. G. Stjernberg

166 photograph

cannot be measured.

The probability

C, ;&) An expression between

that an end point is on an

of length between x and x + dx is

a-intercept

can be written

x and x + dx which

(compare with equation 8)

dx

for the number

of intercepts

would have been

counted

with lengths

on a continuous

measuring line of length Xtot :

N&f’(x) dx = Nmmfm(x) dx + 2Nc, ; ;jl(z)

dx

This can be written as

f’(x) =

-%I&)fm(x)

N,

If X = Xt,,t/N and N,/Nem = K, the equation becomes

Multiplying

by K(X

-

X

~-

f’cx) =

K(X

_

x)

f

“(‘)

(13)

x) and integrating gives

KX ,; f ‘(x) dx -

K s; xf’(x)

dx = X c

f “(x) dx

Thus (14) The value of K can be calculated from equation 13:

sco

K=

Multiplying

o

equation 13 by xK(X

-

-+&f

m(4dx

(15)

x) and integrating gives

f_X-& where l = GA and xm = s,” xf”(x) dx. Equations numerical calculations of K and A, respectively.

(16) 15 and

16 are used

for

Numerical Calculation of g(Z) We now assume that we have an array of experimentally the function F”(xi)

determined values of

= Nim/Nam, where xi = ih; i = 0, 1, 2,..., p. The values of

Lineal Analysis P(x,)

Determination

167

are considered to be so exactly determined

derivation

and the dividing so finethat

can be done as follows:

;Fqxi+J= fyXi+$)

A new array of values offm( x ) is obtained. Equation

f’@i+J = jr(x)

is approximated

fqx

i’

1

13 gives

f”(Xi+J

xi+;)

by a polynomialfl(x)

0, 1,2 )..., p -

i =

= ; [F”(x,+,) - FV’(Xi)l

= ax2 +

bc +

c,

where a, 6, and c

are determined in such a way that fr(x,+J when j = i -

= ax:++ + bxj+* + c

1, i, and i + 1. Using equation 7 gives

_--

By solving the system of equations above, (ax2 the quantitiesfr(xj+;);

g(xi+J

=

j = i -

1 x. (axf+r 3% z+t 1

c)

c) can be expressed in terms of

1, i, and i + 1, and it is possible to write

& I”i+ifl(x$++) ++

[fl(Xi-J

-fY%+*lli

The following notation is now introduced:

X

Yi+t =

X

-

xi+'

f

*(xi+*)

I

G and

Q =

6X(K

-

1)

It is easily shown that g(Z,++) is given by

g(Z,++) = Q

Estimation

[2xi++yi++ + kXf++(J’i-+- yi+s)]

(17)

of the Errors

The number of intercepts with length between two given values is a stochastic variable that according to statistical laws is binomically the probability 3

distributed.

Let SF, be

that the intercept length is between the two given values. The

168

K. G. Stjernberg

standard deviation, ogF, of SF, when one attempts to determine SF by measuring N intercepts, is 06F

=

I

6F,(l - SF,)

(18)

2-N

If only the error due to the statistical uncertainty is considered (this is supposed to be the largest part of the total error), yi+* can be written:

u

the standard deviation of the value

X

y*+:=

_y - xi+’ I

Differentiation of equation 15 gives

Assuming that the numerical values of all dy’s are equal and that the combination of signs is unfavorable, the standard deviation of g(1) can be roughly estimated to be

ug - TX

(1+ 5)-&

J

f”

(:,-f-) (19)

N

Diagram for g(1) The function g(Z) is represented

diagrammatically

in the usual manner (see

Fig. 1). In the diagram is also included the function g = cl3 for different values of c. According to equation 6, c = (n/6) n”(l). Thus along a curve g = c13, n”(l) = constant = 6c/~r. This fact can be used when plotting the function n”(1). The values of 1 are read for the intersections between g(1) and the curves g = c13, and the corresponding 1 and n3(l) values are plotted in a diagram. There is also another case where the curves g = cl3 are of great importance. This is in studying grain growth when it is expected that all grains of the same size grow at the same rate. Then all grains of a certain size will grow or shrink by equal amounts. This means that n3 is a constant. In the diagram this means that the points on g(l) move along curves g = ~13. As an example, the functions g(l) and n3(1) for a specimen tungsten-titanium-carbide

consisting

of

sintered with cobalt are shown in Fig. 1. Some 1000

grains are measured. The value of g(l) is calculated by means of equation 17. The errors in g(l) are estimated by means of equation 19. The errors in n”(l) are easily obtained because the relative errors of g(1) and n3(l) are equal.

Lineal Analysis

Determination

1

2

3

169

L

5

Diameter

6

7-

i-9

10

[micron]

FIG. 1. The volume frequency distribution function and the spatial grain size frequency distribution function for the same material. (The curve parameters are the values of the spatial grain size frequency distribution function.)

Discussion From equations 4 and 5 it is clear that it is not possible to determine accurately the total number of grains per unit volume or the mean volume of the grains, because the determination of these quantities involves the determination of the second derivative at the origin of the lineal distribution function. Smith and Guttman3 showed that it is possible to obtain a value for the specific grain surface, s, from lineal analysis data from the formula s = 4/i independent of structure.

Thus the mean diameter according to Heyn (that is, I) is in reality

not a length but a ratio between a volume and an area. The mean diameter as defined by Jeffries has even less meaning. In this paper another structureindependent parameter, A, is introduced which in fact is a certain length in the structure.

Thus for any structure it is possible by lineal analysis to calculate two

parameters of importance, namely i and A. Not much will be said about the corrections.

Experimentally

it is found that

the correction in i is about 10 o/owhen NIN, is between 0.1 and 0.3. As a consequence the corrections cannot be neglected even though there are ten intercepts on the average per line. In Fig. 1 it is easy to see the difference between g(Z) and n”(l). The part of n3(Z) to the right which is usually called a tail occupies in fact more than half the volume. Furthermore

in this case half the number of the smallest grains occupies

only a few percent of the volume.

170 The

K. G. Stjernberg position

of the maximum

in g(l) is of importance

when

studying

grain

growth. At least in this case it is found that grains with this diameter neither grow nor shrink. Another important fact is that the mean value of I with respect to the functiong(l)

is related

in a simple

manner

to x (equation

10).

Summary Some

structure

parameters

and

distribution

functions

have

been

defined.

Assuming spherical grains, some valuable relations have been derived. Lineal analysis has been treated in detail. Specifically it has been shown corrections

have to be made when

A method described.

for calculating A type of diagram

one is not measuring

the volume

frequency

has been introduced

along a continuous

distribution which

help when studying grain growth. It has also been shown distribution function can be obtained from the diagram.

function

is found

which line.

has been

to be of great

how the size frequency

References 1. 2. 3. 4. 5. 6. 7. 8.

Z. Jeffries, A. H. Kline, and E. B. Zimmer, Trans. &ME, 54 (1917) 594. E. Heyn, Metallogruphist, 6 (1903) 37. C. S. Smith and L. Guttman, Trans. AIME, 197 (1953) 81. G. Bockstiegel, Z. MetaUk., 57 (1966) 647. A. Rosiwal, Bull. GeoZ. Sot., 14 (1903) 466. J. Gurland, Trans. Met. Sot. AZME, 212 (1958) 452. A. A. Glagolev, Eng. Mineral. J., 135 (1934) 399. E. J. Meyers, First International Conference on Stereology, Congressprint, Austria, 1963. 9. H. E. Exner and H. Fischmeister, Arch. Eisenhiittenw., 37 (1966) 418.

Accepted May 6, 1969

Vienna,