GRAIN BOUNDARY DIFFUSION AND FRACTAL INTERFACE CONCEPT

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GRAIN BOUNDARY DIFFUSION AND FRACTAL INTERFACE CONCEPT

P. Guiraldenq

To cite this version:

P. Guiraldenq. GRAIN BOUNDARY DIFFUSION AND FRACTAL INTERFACE CONCEPT. Jour-

nal de Physique Colloques, 1990, 51 (C1), pp.C1-495-C1-499. �10.1051/jphyscol:1990177�. �jpa-

00230345�

GRAIN BOUNDARY DIFFUSION AND FRACTAL INTERFACE CONCEPT

P. GUIRALDENQ

Ecole Centrale de Lyon, DBpartement ~atdriaux-Mdcanique physique (URA-CNRS 447) BP 163, F-69131 ~ c u l l y Cedex, France

!egg@

-

Cette Btude porte sur une nouvelle approche de la description de la diffusion aux joints des grains s'appuyant sur le concept de surfaces fractales, a travers les resultats quantitatifs obtenus par radiotraceurs.

Le concept de surface fractale applique aux joints tient compte ici de l'influence des impuretCs et des Blements d'alliage B haute temperature, qui participent directement aux modifications des sauts 616mentaires associBs sux d6fauts intergranulaires.

Ab2tgac; - This study concerns a new way for a generalization concerning the movements of atoms along grain boundaries taking into account quantitative values of intergra- nular diffusion and bulk diffusion obtained by radiotracer techniques.

The concept of fractal interface applied to grain boundaries, particularly under the influence of impurity contents or alloying additions, gives a new description of these phenomena by the determination of fractal dimension of grain boundaries (GBs) at high temperatures

In previous papers, the author has suggested the possibility to introduce the new fractal sur- face concept in the description of the intergranular diffusion mechanism ( 1 , 2 )

However, no direct determination of fractal dimension of the grain boundaries was reported.

This paper aims to apply the fractal concept to grain b o u n d a r i e s q u a n t i t a t i v e r e s u l t s from the literature.

Some years ago, Mandelbrot et a1 ( 3 ) have given interesting interpretation of the Mandelbrot model (4) to fracture surfaces of metals (maraging steel) under the influence of heat treatment at different temperatures.

Here, we give a transposition of the surface roughness analyzed by the fractal concept, using the mobility of atoms in grain boundaries during high temperature treatments, taking into ac- count the diffusion parameters measured by classical radiotracer methods.

2

-

BRIEF RECALL OF GRAIN BOUNDARY DIFFUSION

...

: DEFINITION OF THE DYNAMIC FRACTAL GB DIMENSION In pratice, the intergranular diffusion parameters studied by radiotracer techniques are defi- ned by the magnitude of the parameter :

PGB = DGB

. b . ^a

where

5

is the GB width a n d a is the segregation term of the moving species.

In spite of an important work described in the literature ( 5 ) it is often difficult to analyze clearly the different parameters to know :

a) The mechanism of the mobilityoftheckfects in the boundary which concerns the interface zone dimension.

b) The mechanism of segregation in the boundaries

( a )

which concerns reorganization on short distances in and near the G B.

We can consider and 0( as consequences of the fractal properties of the GB during the mi- gration and segregation of atoms at the same temperature, through the picture presented on Fig 1. But, it is necessary to consider at the same time the bulk diffusion coefficient (DB) which is an unavoidable term in the mathematical solutions of the modeis describing the GB diffusion. Recently SAPOVAL ( 6 ) have shown that the front of the bulk diffusion (by computing simulation) has fractal properties and have found a value of 1.75 for this dimension. This pro- perty is to be introduced in the grain boundary diffusion because the contribution of bulk dif- fusion from the GB is always present.

So, the fractal characteristic of a grain boundary may be used to analyze the change of the parame te r P GB/DB versus concentrations concerning GB zone and bulk medium (CGB and CB).

To correlate these different factors, we use the relationship proposed by KIRKPATRICK ( 7 ) which defines the percolation threshold, necessary to obtain the diffusion in an alloy versus con- centration.

l f a G B is the GB dimension corresponding to DGB, the penetration parameter PGB can be des-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990177

Cl-496 COLLOQUE DE PHYSIQUE

c r i b e d a s :

ozf

P - = ( C G B ) GB (bC= p r o p o r t i o n n a l ) GB

I f

3

B i s t h e b u l k d i m e n s i o n c o r r e s p o n d i n g t o D B . we can w r i t e , i n t h e same way : D,-(C B )

oa

B

S o , t h e p a r a m e t e r

-

^'GB t a k e s t h e form : D-

The r e l a t i v e c o n c e n t r a t i o n

-

c o n c e r n s t h e moving e l e m e n t s t u d i e d by r a d i o t r a c e r s , The r e l a t i o n ( 1 ) i s e q u i v a l e n t t o t h e d e f i n i t i o n o f a measured l e n g t h o f a p a t h N measured w i t h a u n i t s c a l e k c h a r a c t e r i z e d by a f r a c t a l d i m e n s i o n 2

l o g N (

*

^{) =}

s

^{l o g}

,

1

The u n i t l e n g t h 4 h a s t h e same r o l e a s t h e e f f e c t o f c o n c e n t r a t i o n a c t i n g on jumps o f atoms t h r o u g h t h e l a t t i c e s i t e s and t h e boundary d e f e c t s .

The d i m e n s i o n o f t h e d i f f u s i o n i n t h e GB c a n be c o n s i d e r e d w i t h two a p p r o c h e s : a ) T a k i n g i n t o a c c o u n t t h e s e g r e g a t i o n

factors

, we o b t a i n d i r e c t l y t h e v a l u e b ) To c h a r a c t e r i z e t h e d i f f u s i o n o n l y t h r o u g h t h e d e f e c t s o f G B

w i t h o u t t h e f a c t o r d , we t a k e :

which g i v e s :

NEAR THE MELTING POINT 3 -

4P_P_LIC_IXIO_N_-TO_-EE-D_I_FEU_S_LO_N_-I!-E!EE-MET_4&

- - -

We c o n s i d e r t h e s i n g u l a r c a s e o f p u r e m e t a l s n e a r t h e m e l t i n g p o i n t where DB and PGB a r e w e l l d e f i n e d by t h e l i t e r a t u r e ( 8 , 9 , 1 0 ) , p a r t i c u l a r l y f o r c u b i c m e t a l s .

I n t h i s c a s e , we s u b s t i t u t e t h e c h e m i c a l t e r m s by t h e vacancy c o n c e n t r a t i o n s , ( ) , a t t h e m e l t i n g p o i n t :

(

D

S. 102 ( l i q u i d s t r u c t u r e w i t h c l u s t e r s )

-

^PCB S 5x10 -18 m3 S-' with

2

⁼ 5x10-l0 m.

<

^{= 1 and DGB = 1 0 - ~ m2 s-l}

- D, 10- 12 m2 s-l

Taking for

dB

the value of 1.75 given by SAPOVAL et a1 ( 6 ) . we note thatoaCB is 2.92 at the melting point. This means that the movements of a oms along grain boundaries, at the mel- ting point, concern widely a tridimensionnal space

&=

³⁾

Now, it is interesting to see what happens at lower temperatures with impurities or chemical effects at grain boundaries.

We present some cases concerning polycrystals and high angle grain boundaries, limited to rare experimental cases where the terms

have been correctly measured by different workers. (or the present author) First example : G diffusion of iron in Fe-S alloys (11 12)

- - - B - - - L - - -

The GB segregation of sulfur has been determined at 1123 K for different bulk concentrations and at the same time, D and P have been measured with iron tracer (Fe53). Table I gives, for the self-diffusion gf ironFIBthe dimensionalityaGB

- -4

For low sulfur content,

dGB

is similar to a pure metal (1 + 1.75 = 2.75)+ For greater sulfur concentrations, we note a hlgh value fo?

2

^GB, concerning iron moveme-nts along grain boun- daries and near the GB (

dGB=

3.55, 3.35)

.

These high lralues f o r d G B mean that the iron diffusion concerns a wsder G B , without any reorganization of iron and sulfur atoms.

Second example : GB diffusion of sulfur in Cu alloys (13) ---

AUFRAY has studied two different systems : Cu (Fe,S) and Cu (Ni,S), Diffusion coefficients and segregation of sulfur have been measured at 873 K .

m "

r ~ 3 L ~ 3

The representation of

-

^versus

^-

silows an important difference of the two systems (Fig. 2 ) :

D~

For Cu (Fe ,S) alloys

DGB - ^dB

approximately 0.3, which gives for

aGB

^{= 2.05}

"

For Cu (Ni,S) alloys,

dCB- dB

is unrealistic because O( = G B has a quasi-constant value, and cannot be taken as a correct parameter. CB

It may be concluded that the self-diffusion of S in (Cu-Fe-S) system concerns all the sites in the space of the grain boundaries, because

a

is 2. In other words, moving atoms have used paths without no directive ways in the plane GBand with an important interaction between sulfur and iron particles during diffusion and cosegregation.

5 - CONCLUDING REMARKS

---

The description of the grain boundary diffusion by the fractal concept is a new approch in the study of elementary mechanism. It gives a characterization of the space in which moving atoms are present. Particularly, segregation effeccs which _{- -} lay - - an important role in diffusion kine- tics can be used to describe a dynamic fractal dimension of grain boundaries at high tempera- tures. In futnre,it remains to analyse by this way the GB diffusion in bicrystals with selected cristallographic orientations under the influence of typical impurities interactions.

"""""E"""-

( 1 ) P. GUIRALDENQ, " Physical chemistry of the solid state : application to metals and their compounds", Elsevier (1984) 215-224.

Cl-498 COLLOQUE DE PHYSIQUE

( 2 ) P. GUIRALDENQ, T r a n s J a p I n s t M e t a l s s u p p l . 27 ( 1 9 8 6 ) 509-516.

( 3 ) B. MANDELBROT, D PASSOJA, A. PAULLAY, N a t u r e , 308 n o 5961 ( 1 9 8 4 ) 721-722

(4) B . MANDELBROT, "The F r a c t a l Geometry of N a t u r e " , Freeman, New York, N Y ( 1 9 8 2 ) ( 5 ) P. GUIRALDENQ, J o u r n a l d e P h y s i q u e . C o l l o q u e n " 6 , Tome 4 3 , d 6 c . 1982 137-146 ( 6 ) B. SAPOVAL, M. ROSSO, J F . GOUYET, J Phys L e t t e r s 46 ( 1 9 8 5 ) L 1 4 9 .

( 7 ) S . KIRKPATRICK, Rev Modern Phys 45 ( 1 9 7 3 ) 574-588

( 8 ) A . HEURTEL, C. COTTE, P . GUIRALDENQ, Met. C o r r 572 ( 1 9 7 3 ) a p r i l , 3-11.

( 9 ) N . A GIOSTEIN, i n " D i f f u s i o n " Am S o c . f o r M e t a l s , M e t a l P a r k , Ohio ( 1 9 7 3 ) 241-274

( 1 0 ) J . PHILIBERT " D i f f u s i o n e t T r a n s p o r t de M a t i e r e s d a n s l e s s o l i d e s " , Ed d e P h y s i q u e , P a r i s ( 1 9 8 5 ) 246.

( 1 1 ) C . PICHARD, These d ' E t a t , P a r i s V 1 - Ao 1 2 6 0 ( 1 9 7 6 ) ( 1 2 ) D TREHEUX, Acta M e t a l l . 30, ( 1 9 8 2 ) 563-570

( 1 3 ) B AUFRAY, These d l E t a t , M a r s e i l l e ( 1 9 8 2 ) TABLE I - -

-

- -

-

GRAIN BOUNDARY DIFFUSION OF I r o n i n ( F e - S ) a l l o y s ( 1 1 ) ( 1 2 )

F i g . 1 : P i c

0 f

t u r e o f t h e g r a m boundary d l f f u s x o n f r o n t o f r a d l o t r a c e r s w l t h d l f atoms i n t h e G B p l a n e a n d I n t h e b u l k .

C w t . pprn s u l f u r ) B

(

10

3 5 51

' f e r e n t C w t ppm s u l f u r )

GB 1 3 8

m o b i l i t y CGB ( % i r o n

99 97 92

p G ~ ( m 3 / ~ - 1 ) 2 . 1 0 - l 9 5 , I O - ~ O

1.5

~ O - ~ O

DB (m's-') 1 . 5 ~ 1 0 - ~ 1 ~ 1 0 - l ~ - 1 ~ 1 0 - l 5

A

03GB

-4

1 1.8 1 . 6

( 1 3 ) D~ < R