Spatial correlations in metal structures and their analysis

Spatial Correlations in Metal Structures and Their Analysis D. Stoyan* a n d K. W i e n c e k t * Fachbereich Mathematik, Bergakademie Freiberg, Bernhard-v.-Cotta-Str. 2, D-0-9200 Freiberg, Germany; and "k Institute of Metallurgy, Academy ~?(Mining and Metallury, y, al. Mickiewicza 30 paw. A2, 30-059 Krakow, Poland From the stochastic geometry point of view, the dispersed phases of materials can often be regarded as a homogeneous marked point process. The second-order characteristics of that process, the pair (PCF) and mark (MCF) correlation functions, are measures of the spatial distribution of particles. The correlation functions were estimated by using of a kernel estimator for two examples of two-dimensional dispersions, i.e., the Ag particles on an iron substrate and the FegC particle sections of a tempered steel. The empirical correlation functions are then discussed and interpreted. A stereological analysis based on the PCF of Fe3C particle sections indicates that the particle follows the Stienen model, i.e., the FeBC particle centers form a Poisson process while the particle sizes are dependent and determined by the nearest neighbor distances. The coarsening process of the Fe3C particles is discussed on basis of the correlation analysis. Vom stochastisch-geometrischen Standpunkt k6nnen disperse Phasen von Werkstoffen dutch homogene markierte Punktprozesse beschrieben werden. Die Eigenschaften 2. Ordnung solcher Prozesse, wie die Paarkorrelationsfunktion (PCF) und die Markenkorretationsfunktion (MCF), dienen zur Charakterisierung der r~umlichen Teilchenanordhung. Diese Korrelationsfunktionen werden ffir zwei zwei-dimensionale Dispersionen, namlich Ag-Teilchen auf einer Fe-Unterlage sowie f~r Fe3C-Teilchenschnitte in Stahl, mit Hilfe eines Kern-SchMzers ermittelt. Diese empirischen Korrelationsfunktionen werden dann diskutiert und interpretiert. Eine stereologische Analyse der PCF f6r Fe3C-Teilchenschnitte ergibt, dal3 die Fe3C-Teilchenanordnung im Raum dem Stienen-Modell (ein Modell f~r nichtfiberlappende Kugeln) entspricht. In diesem Modell bilden die Teilchenmittelpunkte einen Poisson-Prozel3, wahrend die TeilchengrSI3en voneinander abhfingig sind und von den Abstanden zum nfichsten Nachbarn bestimmt werden. Der VergrOberungsprozef3 (Ostwald Reifung) der FeBC-Teilchen wird diskutiert. Du point de vue de la g6ometrie stochastique les phases dispers6es des materiaux peuvent 6tre trait6es comme des processus ponctuels marqu6s homog6nes. Les characteristiques de second ordre de ces processus, comme la fonction de corr61ation des paires (PCF) et la fonction de corr61ation de marque (MCF), sont mesur6s de l'arrangement des particules dans l'espace. Une 6stimation quantitative de la fonction de corr61ation a 6t6 faite a l'aide d'un estimateur special (nomme kernel estimator) pour deux dispersions bidimensionnelles, c'est a dire pour des particules Ag form6es sur une surface de Fe et pour des sections planes de particules Fe3C dans un acier r6venu h temperature 61ev6e. Les fonctions de corr61ation empiriques 6taient compar6es aux fonctions de correlation de modbles connus. Pour des particules dispers6es de Ag le mod61e est le processus softcore. L'analyse st6r6ologique de la fonction PCF pour sections planes des particules Fe3C 167 :
MATERIALS CHARACTERIZATION 26:167 176 (199l) 1044 5803,'91:$3 5(I

168

D. Stoyan and K. Wiencek

a demontr6e que l'ensemble de particules Fe3C est conforme avec le module de Stienen (modUle pour boules qui ne sont pas superpos6es). Dans ce mod61e les centres de particules forment le processus de Poisson et leur tailles (diam~tres) d6pendant les unes des autres et sont determin6es par les distances entre les particules les plus proches. Le processus de coagulation des particules Fe3C est analys6.

INTRODUCTION

In many metal alloy structures one of the crystalline phases appears in the form of dispersed particles. Such systems are relatively well described by parameters such as particle density Nv, volume fraction Vv, specific surface area Sv etc. Therefore, these parameters have been widely applied in metallography. However, they describe average properties of the structure only. But in more detailed metallographic studies, it is necessary to take into account also the problem of spatial distribution or arrangement of particles as well as correlations of various parameters corresponding to particles, because one has to expect that these properties are also in some relation to materials properties (see, e.g., Stoyan and Schnabel [1]). There are two fundamental ways of describing the spatial distribution of particles: an integral one and a differential one. The integral description is based on the distribution of particle numbers in test areas of given shape and dimension. Its results depend on the properties (size and shape) of the test areas. This counting method was applied in some metallographic studies on microinhomogenities of carbide particles distribution in steel [2,3]. The differential description is based on measurements of some parameters (distance, diameter, etc.) of pairs of particles and determination of two correlation functions: pair correlation function (PCF) and mark correlation function (MCF). These functions give an average characterization of interaction among particles. They are not dependent on the observation and measurement conditions as in the case of

the integral description. They can be applied in studies of interactions between particles in many physical-metallurgical processes, e.g., particle coarsening, precipitation, etc. The PCF was introduced into stereological studies by Hanisch and Stoyan [4]. They gave formulae that enable the determination of the PCF of sphere centers in space using information available from planar sections. Although they assumed independence of sphere diameters, their method is complicated; even in the particular case of spheres of constant diameter, it is necessary to solve an integral equation with similar (bad) properties as those in Wicksell's corpuscle problem. As an application, they gave a stereological analysis of the distribution of graphite particles in spheroidal cast iron using the PCF. Stoyan et al. [5] studied the general case of dependent sphere diameters and planar sections. Their formulae are rather complicated; even in the particular case of a discrete diameter distribution, they lead to a system of coupled integral equations. Nevertheless, this theory is not worthless, because it is possible to calculate the PCF of the point process of section circle centers for given stochastic models of sphere systerns. The theory of MCF is presented in Stoyan [6] and Penttinen and Stoyan [7]. The aim of this article is giving some metallographical applications of this idea.

PAIR CORRELATION AND MARK CORRELATION FUNCTIONS

The theory of correlation is presented here for a set of spheres in space. The problem

Spatial Correlations in Metal Structures

169

of statistical estimation of correlation functions is discussed for the two-dimensional case, e.g., for planar sections. The set of spherical particles u n d e r s t u d y is described by a h o m o g e n e o u s and isotropic m a r k e d point process (see Stoyan et al. [8]):

~3 = {[Xi, D(xi)]}. Here xi is the center of the ith particle, and the mark D(x,) d e n o t e s the diameter of this particle. The xi form a simple point process in space R 3 with the point density Nv. The particle diameters are r a n d o m variables with m e a n value E(D). The values Nv, and E(D) belong to the so called first-order characteristics of xP3. In general the relative positions of the points as well as the values of marks are not i n d e p e n d e n t , which means that between the elements of.W3 there exist some correlations. These correlations are characterized by the so-called second-order characteristics of the process. To them belong the PCF gv, and the MCF kl~r~. A heuristic interpretation of these characteristics is as follows. Consider in space R 3 a test sphere T of infinitesimal v o l u m e dV. The probability that T contains a point of ~ is NvdV. The mean value of the c o r r e s p o n d i n g mark is E(D). Next in the space two nonoverlapping test spheres T~ and T2 with infinitesimal volumes dV and distance r of centers are considered. The probability p(r) that in both spheres there is each a point of the set xP~ is connected with the PCF ~gv(r) as follows: ,~¢v(r) = p(r)(NvdV)

2

(1)

Consider n o w the marks of the two points in T1 and T2. Let K[)L)0") be the m e a n of the product of these marks, u n d e r the condition that T~ and T2 i n d e e d contain each a point. A normalized version of K>>(r) is the mark correlation function kDD(r):

kt,lffr) = K>D(r)(E(D))

2

(2)

While the PCF describes the relative positions of the points (as it is well k n o w n in

physics), the mark correlation function describes relations b e t w e e n the marks. For very great values of r, the function kD~,(r) tends to 1 as in the case of i n d e p e n d e n c e . Values of k>>(r) smaller than 1 indicate inhibition (small values of D for m e m b e r s of point pairs of a distance of r) and great ones excitement (great values of D for m e m b e r s of point pairs of a distance ot r). See the examples later for interpretation of empirical PCFs and MCFs. In the two-dimensional case all definitions are analogous. A set of circular particles distributed on a plane is described as a h o m o g e n e o u s and isotropic marked point process xl'2 {[xi, d(xi)]}. N.~ denotes the density of centers, while E(d) is the m e a n diameter. The c o r r e s p o n d i n g seco n d - o r d e r characteristics are the functions gA(r) and k,,,dr) defined analogously to the three-dimensional case.

ESTIMATION OF THE

CORRELATION FUNCTIONS A direct estimation can be p e r f o r m e d in the two-dimensional case, analyzing images resulting, e.g., from planar sections. The functions gA(r) and k,~,¢(r) can be estimated by using edge-corrected kernel estimators of the functions p(r) and p,l,dr) defined bv p(r) = N~2S,,~(r)

(3)

and p,~,dr) - k,~d(r)gA(r)(E(d)) 2

(4)

An edge-corrected estimator of p,,dr) has the form

f~(r,,

1

p,~,dr) - 2Tr Ix:d(x)ll~t:d(tDl ~ ~

r)d(x)d(y)

r,,,A(W, A W,,)

~, ~ ' (5) The s u m m a t i o n goes over all marked points [x; d(x)] and [y; d(y)] of the pattern in the w i n d o w of observation W; rx,, is the distance b e t w e e n points x,y; W, is the set of all points of the plain having the form z = w + x, w in W (Fig. 1); A (.) is the area;

170

D. Stoyan and K. Wiencek

yd

FIG. 1. The sampling-window W and definition of WxOWv for estimation of f~,i~(r)in Eq. (5).

is a naive estimator for NA [KA (r + ~) - KA (r -- ~)]/2~. If W is not very large it has not a good quality because of edge effects: For a point x near the boundary of W, a considerable part of points y of ~I/2 with r -< rxy <-r + ~ may be outside of the window. This edge effect can be compensated by replacing 1 / A ( W ) by the greater factor 1/A (WK A Wy); see Stoyan et al. [8] for theoretical justification. Further improvement of the estimator p*(r) is possible by replacing ka by a smoother function f~ with similar properties as k~: fa(x)>0, f~( - x)

f~ is a kernel function. A possible kernel

= fa(x) and J ~

0

O

k_~ .'.x,xx~ff~' ) ~ O (9° ,.\\~/',~ O ~ . Wxfl Wy O O C~ ( o ( O O k, X 0 .--. ~"~ o o o o O Y W 0~

~

function is I 3 fa(r) = ~

(

r2 ) 1 - ~ , ]rI < 8~/5 _

0,

otherwise,

(6) where 8 is a band-width parameter. The function p(r) is estimated analogously to Eq. (5); one has only to put d(x) = d(y) = 1. The estimator for p(r) can be explained as follows. Consider the integrated version of p(r), ~ NA2KA(r) = 2~r

xp(x)dx,

(7)

where KA(r) is the so-called Ripley's K function. The quantity NA2KA(r) can be interpreted as the mean of the number of ordered pairs of distinct points less than distance r apart with the first point in a given set of unit area [6]. By setting {~,x,<~, ka(r) =

(8) ,

otherwise,

the double-sum p*(r) = ~ ~ ka(r - rxy) xy ~ x A(W)

(9)

f~(x)dx

1.

Usually (but this is not necessary) fa(x) = 0 outside some domain. A good choice is the fa above, the so-called Epanechnikov kernel. For the parameter 8, we suggest a value near to O.05/NA 1/3. When applying the estimation method described above, one has to determine all planar particle centers and diameters in W. This time-consuming work can be avoided by using the Image C-MATAN software of ROBOTRON. It determines automatically first the particle centers and their diameters (as marks) and then estimates the PCF and MCF. Of course, if the spatial (x,y,z)coordinates of spheres and their diameters are available, then estimators analogous to the above yield g v ( r ) e t c .

THE MODELS

This section briefly describes simple models for systems of spheres in space. The Boolean and the Stienen models are closelyrelatedtoaPoissonpointprocess of intensity (density) N v , which yields the sphere centers. In the Boolean model the sphere diameters are mutually independent; it is kDD(r) ~ 1. There is also no correlation in the pattern of centers, gv(r) =1. Consequently, there are overlappings of spheres, and the model is not a realistic

Spatial Correlations in Metal Structures

171

model for systems of n o n o v e r l a p p i n g spheres, with the exception of very sparse systems. It was successfully applied to more irregular structures that are formed bv very complicated figures that can be approximated by u n i o n s of spheres. T h e o r y and formulae for the Boolean model are p r e s e n t e d in M a t h e r o n [9], Serra [10], Stoyan et al. [8], and Hall [111. For stereology it is i m p o r t a n t to k n o w that the planar section of a Boolean model is again a (planar) Boolean model. The point process of the centers of the c o r r e s p o n d i n g section circles is again a Poisson process, Stienen [12, 13] has p r o p o s e d some models for n o n o v e r l a p p i n g spheres for describing dispersed carbide phase in steels. The Stienen model described here is his first model [12, 13], which is easy to treat mathematically. In this model the sphere diameters are d e p e n d e n t and d e t e r m i n e d as follows, Let x be an arbitrary point of the Poisson point process P of centers. Denote by n(x) the distance from x to its nearest neighbor in P. Then the sphere centered at x has the diameter D(x) = ¢xn(x). The p a r a m e t e r a, 0 < c¢ < 1, is related to the v o l u m e fraction Vv by = 2V~ 1'3

(10)

In the original version [13, 14], only the case with maximal v o l u m e fraction Vv 0.125 or c¢ = 1 was considered. The sphere diameters' distribution F(D) coincides with the nearest n e i g h b o r distance distribution of the Poisson point process given by

F(D) = 1 - exp (

~6 NVv v D3 )

(11)

A stereological formula for the particle density Nv is: Nv = 0.857 NA 32 Vv 1..2

(12)

For this model, the MCF has a nontrivial form (see Stoyan [14]). In contrast the PCF has the simple form of a Poisson prosess,

g~(r) ~ 1. Planar sections of the Stienen model produce patterns of n o n o v e r l a p p i n g circles. Their centers form a stationary and isotropic point process. This process is not a

! 10~ [

f ~ ' ~ -c 08 ~ //// -go6 i /J~ ! ii~ [ ~i ~- 0~ //// ~ 2 ~...... ~/J, , .'i,,;r, ~ I ...... i ........ o2 '..~ 0 o 02 ~ 06 08 1I0 W2 1~~ DISTA,CE.r F~c. 2. Volume fraction dependence of the pair--correlation functions gA(r) associated with the Stienen model. Poisson point process! By simulation Wiencek and H o u g a r d y (unpublished results) have studied the distribution of planar sections for the Stienen model using the counting m e t h o d and f o u n d that the n u m b e r of circles in test quadrats can be a p p r o x i m a t e d by a binomial distribution. Using the general theory in Stoyan et al. [5], the planar pair correlation function gA(r) can be calculated (see Stoyan 114]). Figure 2 shows gA(r) for some values of v o l u m e fractions, Vv, and Table 1 gives numerical values for N v = 1 and Vv = 0125 obtained by numerical integration. The form of gA(r) is similar to that of point processes with a very weak repulsion for small distances r (see Stoyan [61). The small values of gA (r) for small r result from the rareTable 1 Values of g,~(r) for the Stienen model (Nv - 1, ~ - 1) r

g.4(r)

0.025 0.05 0.10 0.20 0.30 0.40 0.50 0.60

0.008

0.70 0.80

0.966 0.993

0.90 1.00

0.999 1.000

0.027 0.086 0.255 0.452 0.642 0.797 0.905

172

D. Stoyan and K. Wiencek

ness of events such as intersecting two very small spheres close together or of two spheres nearly touching the section plane and with centers in different half spaces determined by the plane. We point out that there are no a priori reasons for using the Stienen model. Perhaps a more natural model would be a system of spheres with volumina proportional to the corresponding Voronoi cells with coalescence of overlapping spheres. But such a model would be, at least at the moment, mathematically not tractable because of very complicated spatial correlations. Even the distribution of the volume of Voronoi cells is not known theoretically. The Stienen model may be interpreted as an approximation of such a model, since there are correlations between the volumina of the Stienen spheres and the Voronoi cells,

TWO EXAMPLES

We studied two qualitatively different metallic structures with the aim to get information on their spatial distribution by means of the pair correlation function and the mark correlation function.

THE DISPERSION OF SILVER PARTICLES ON IRON SUBSTRATE The dispersion consists of silver particles distributed on an iron substrate. It was produced by evaporation of silver on a flat, well-polished iron specimen (silver layer thickness about 0.5 ~m). Next, the sample was heated in vacuum up to 970°C, held for 5 min and cooled slowly. After melting of silver-layer, a dispersed phase is formed (spheroidization). Figure 3 shows the microstructure taken by scanning electron microscopy. The projections of silver particles on the substrate plane are nearly circular and uniformly distributed. Visual inspection already shows three aspects of the arrangement of the particles: (1) they are not positioned completely at random; (2) there is inhibition at small distances, and

Fro. 3. SEM microstructure of the Ag-dispersion. particles do not overlap; and (3) there is no long-range order. A quantitative analysis of the microstructure was performed in 120 x 120-mm test quadrats W at x 2,280 magnification. For each particle in W the diameter was measured and the particle-center was determined by Opton's semiautomatic particle size analyzer TGZ-3. The system of particles is modeled as a marked point process ~2, where the points are the centers and the marks the diameters. Fundamental first-order parameters of the set of particles are as follows: the mean diameter ~ = 3.85 × 10 - 3 mm, the den-

NA = 18.39 x 103 mm -2, and the area fraction AA = 0.25. The diameters and coordinates of the particle-centers were measured for 531 particles in 10 quadrats, and for these particles the functions gA (r) and k,~,dr) were determined using Eq. (5). Figure 4 gives information on the pair correlation functions. It shows the mean value of the 10 statistically determined PCFs and the upper and lower standard deviation regions. The form of the curves is different from that of Poisson process and is a typical one, which was often observed in the analysis of planar point patterns (see, e.g., the figures in Stoyan [6] and Stoyan and Schnabel [1]). The curves suggest the following statements can be made about the microstructure. There is a minimalinterpoint distance of ca. 2 x 10 3 sity

Spatial Correlations in Metal Structures

12i

TI

[ 10~. E 08-

/

c2~ /~

S

THE DISPERSION OF Fe~C PARTICLES IN CARBON STEEL

,

. / V ~ I ~I T T ,I ~/- ~ ~ t~ ' 1~

e_ 06 0~

0

_

173

,

i ~

Ag: gA(r)

Y . 10

.

. 20 DISTANCE

r×

. 30 p .ram

40

FIG. 4. Experimental pair correlation f u n c t i o n ~ a ( r ) of the Ag-particles.

m m (4.6 m m for the m a g n i f i e d d i m e n s i o n of r in Fig. 4). The p r e f e r r e d interpoint distance is ca. 6 x 10 -3 m m (13.7 m m in Fig. 4). (The last s t a t e m e n t results from Stoyan [6].) Figure 5 gives i n f o r m a t i o n on the m a r k correlation functions. It s h o w s the m e a n of the 10 statistically d e t e r m i n e d MCFs and the c o r r e s p o n d i n g u p p e r a n d lower standard deviation regions. As the figure shows, for distances greater t h a n 9 x 10 3 m m (20.5 m m in Fig. 5), it is kd,l = 1, which s u p p o r t s the a s s u m p t i o n of noninteraction for distances greater t h a n 9 × 10 3 m m . For shorter distances, k,~a(r) is smaller t h a n one. This c o r r e s p o n d s to the obvious fact that silver particles v e r y close together can only coexist if their d i a m e t e r s are small.

,

~ 08-

~

m x 06~

i

04 -

A h a r d e n e d s p e c i m e n of carbon steel of 0.6% C w a s a n n e a l e d in v a c u u m at 700°C for 600 h. Figure 6 p r e s e n t s the microstructure of a planar section of the specim e n . Visual inspection is possible onh, for these planar images. The t w o - d i m e n s i o n a l Fe3C profiles are a p p r o x i m a t e l y convex a n d r a n d o m l y distributed in the ferritic matrix. H o w e v e r , m o s t of the particles are on the grain b o u n d a r i e s of the matrix and s o m e of the particles are attached to each other [20]. The quantitative estimation of the microstructure was p e r f o r m e d in 80 x 80-mm q u a d r a t s W at ×2,000 magnification. The m e a s u r e m e n t of the particle-section diameters a n d the m a r k i n g of their centers were m a d e a n a l o g o u s l y for the silver particles u s i n g the analyzer TGZ-3. The dia m e t e r s as well as the coordinates of the circle centers w e r e m e a s u r e d for 727 particles a n d 20 test quadrats. The m e a n value p a r a m e t e r s of the t w o - d i m e n s i o n a l dispersion are as f o l l o w s :

AA = 0.095, NA = 2.3

x 104 m m 2, d = 2.1 × 10 ~ m m

The functions gA(r) and kJ,dr ) w e r e determ i n e d b y Eq. (5). Figure 7 gives s o m e in.2

~ ,~O x j

'..

)

d

"

~Z i

0

C

0

~0 Ag: kdd(r)

"..,,f

o

02-

@

o

¢' m

o DISTANCE

rxp,

mm

Fro. 5. Experimental m a r k (diameter) function k~(r) of t h e Ag-particles.

~

correlation

&0

I

FIG. 6. Microstructure of the Fe3C-particle sections (in q u e n c h e d a n d t e m p e r e d for 600 h at 700°C steel).

D. Stoyan and K. Wiencek

174 II

ments. Particles close together tend to be

~.2-,

T ~T~TTITTIITr2TI ! T

lO. T [ Z 08 ~ l i

l

T

~ J ' ~ I
~_ 06 o~

l]

F% C : gA(r)

02- / 0 "

0

-

1~0

-

i,.

2'0 3'0 dO DISTANCE r* n .m~ Fic. 7. Experimental pair correlation function gA(r)of the FeBC-particle sections,

formation on the pair correlation functions. It s h o w s the mean of the 20 statistically determined PCFs and the corresponding u p p e r and lower standard deviation regions. The qualitative form of the curves is b e t w e e n that of PCFs in Poisson process and a process with w e a k shortrange order. Obviously, there is very little order in the pattern; if there is interaction, then it is only for distances less than 5 x 10 3 mm, ( - 2 . 5 d) (10 m m for the magnified dimension of r in Fig. 7). For very short distances there is inhibition that simply results from the fact that the section profiles do not overlap. The curves for k~(r), s h o w n in Fig. 8, support these state-

I 12q

T

T

i

Ti

T

T

:T

'

relatively small. All these statements are valid only for planar sections. We also tried some stereological analysis. If one uses the value of Nv, it is possible to compare the empirical PCF gA(r) of particle profile centers with that of the Stienen model. (Remember that this model has only two parameters, the intensity Nv of the generating Poisson process and the coefficient c~.) The Nv-value calculated by Eq. (12) for Vv = 0.095 (~ = 0.9) is Nv = 0.95 X 107 m m -3 while estimated by Saltykov's formula (Eq. (11.4.1) in Stoyan et al. [8]) Nv = 1.10 x 107 m m - 3. Figure 9 also shows the empirical PCF and the region of gA (r) for the centers of section circles for the Stienen model for volume fractions Vv > 0.05. The approximation is satisfactory. This statement is also supported by some results obtained via the integral approach in Wiencek and H o u g a r d y [2]. They found that the n u m b e r of FesC-particles in test quadrats corresponds to the binomial distribution for the Stienen model. In order to strengthen the belief that the FegC-particles structure follows the model, a more precise analysis of the empirical size analysis was performed. The empirical frequency distribution function of section diameters was determined from 1,520 FeBC-particles, and is s h o w n in Fig. 10. In the two-dimensional representation it is possible to compare the empirical distribution with the theoretical distribution of I

~

o.B

u

THEORETICAL oc =1 m-THEORETICAL ¢¢.:0'75 ~ - EXPERIMENTAL o{- : 09

k~

~F- D~ i O/+-

i l

•

~J 13. 0.~.

I

•

Fe 3c: kdd r)

02 0

00

±

0

~

1~O

2'0

DISTANCE

30 r * p ,

-

,.o-

/li~

-

2

4~

/

4'0

mR

Flo. 8. Experimental mark (diameter) correlation function kdd(r) of the Fe3C-particle sections,

6

8

10

12

l',t+

DISTANCE , rxnlmm) FIG.

9.

Comparison

of

the

experimental

pair

corre-

lation functiongA(r) of Fe3C-particle sections with the theoretical functions of the Stienen model.

Spatial Correlations in Metal Structures 1 i 20 °~-

b z

W

i

I/ °,"

i 10t

~,,

°'" ,i

9

Oz'~'~-\ . . . .

~, ',_ o', \

Fe3(

175

EMPIRICAL

S/PENES MODEL

,{,, x,,~,. ~ . 00 , "'.-2, "25 ~ ~ . .3.0 05 1'0 115 2'0 RELATIVE SECTIONS O I A M E T E R , ( X } d Fro. 10. Comparison of the experimental Fe3C-particle section diameters distribution with the model distribution, circle section diameters of the Stienen model. The frequency distribution of section diameters for the model was determined on basis of the data given in Table 2 of Stoyan's article [14], for AA = 0.095 (c~ - 0.9) and Nv calculated by Eq. (12). Figure 10 shows that there is a systematic difference between the two distributions, However, these differences are not very great. In general the distribution function for the model is inside the error intervals for the empirical values, Thus, we have considered two quite different geometrical characteristics of the FeBC structure and both behave as expected for the Stienen model. This we take as justification of the assumption that the particle configurations follow in principle the Stienen model. Consequently, one can assume that the particle centers form a Poisson process (are "completely rand o m l y " arranged in space), while the particle sizes are d e p e n d e n t and determined by the nearest neighbor distances. Of course, a direct proof of this statement is possible only by means of serial sections, which yield the (x, y, z) coordinate of partide centers. The dispersion of Fe3C-particle was formed by a particle coarsening process (Ostwald ripening), by growing of bigger particles at the expense of smaller ones, at constant volume fraction, Vv. This process (for dilute systems, Vv < < 1) is described in the so-called LSW theory established by L i f s h i t z a n d Slyozov [151 a n d Wagner [16[.

The LSW theory predicts a stationary asymptotic particle size distributions. For example, Fig. 11 shows the asymptotic distribution (A) for the diffusion-controlled kinetics of coarsening. For the Fe~C-dispersion u n d e r this study and also in the studies of carbide coarsening in steels [17], it was found that the empirical size distributions are substantially different from the distributions predicted by" the LSW theorv; " they are broader and of an opposed (positive) skewness, in the LSW theory the particle correlations are not taken into account. In the last few years, the effect of volume fraction, Vv, on the coarsening kinetics have been intensively investigated, both theoretically and by simulation. A review of some results is given bv Enomoto et al. [181]. By increasing Vv, the probability of a direct diffusional interaction between the neighboring particles increases and the behavior of these particles is correlated. (For the size correlations by this diffusional constraint of particles, Marder [19] introduces in his theory the MCF kL)D(r).) The asymptotic size distributions in these models are more symmetric and broader than that of the LSW theory. Figure 11 shows a typical size distribution (B) for Vv - 0.1, given by Enomoto et al. [18]. The observed FeBC-particle size distribution is closer to that of the Stienen model (distribution (C) in Fig. 11) than to that from the LSW theory and the models of

oTs t = I /~ /A; I~'i ~ '-,~ ENOMCTO g~-- 20! ~ ~=~et ,at'.',,,.'~....rJl U C~; S" ENEN MIC~L El& :11 ~ ,' I ii g // I,~ ~1o / //Y-"x / ~'~',', ~.% ~ I ~ 0. 05 10 15 20 RELATIVEPARTICLEDIAMETER(D/D) Fro. 11. Dependence of the asymptotic size distribution functions on the particle volume fraction and the distribution function for the Stienen model.

D. Stoyan and K. Wiencek

176

finite v o l u m e fractions. It is w o r t h o b s e r v i n g t h a t t h e size d i s t r i b u t i o n f u n c t i o n for the Stienen model results only from some g e o m e t r i c a l c o n s t r a i n t s i m p o s e d o n the particles. T h e p r e s e n t a n a l y s i s s u g g e s t s that, i n t h e case w h e n t h e c o a r s e n i n g p r o c e s s of Fe3C particles p r o c e e d s a c c o r d i n g to the g e n e r a l a s s u m p t i o n of t h e t h e o r y of O s t w a l d r i p e n i n g , s o m e a d d i t i o n a l geom e t r i c a l c o n s t r a i n t s h a v e to b e t a k e n i n t o consideration. O n e m a y suppose that t h e s e c o n s t r a i n t s m a y b e a n a l o g o u s to t h e

7.

8.

9. 10.

11.

space-filling condition i m p o s e d on grain growth in polycrystalline metals,

12.

The authors wish to thank the a n o n y m o u s referees for their m a n y valuable suggestions,

13.

14.

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Received June 1990; accepted January 1991.