Scripta METALLURGICA Vol. 27, pp. 1283-1288, 1992 Pergamon Press Ltd. et MATERIALIA Printed in the U.S.A. All rights reserved

A M E A N - F I E L D T H E O R Y A P P R O A C H T O S O L U T E A T O M S E G R E G A T I O N A T

T H E D I S L O C A T I O N C O R E IN A D I L U T E B I N A R Y A L L O Y

Aki ra Sek i A d v a n c e d T e c h n o l o g y R e s e a r c h Labo ra to r i e s ,

S u m i t o m o Meta l Indus t r ies , L i m i t e d , 1-8 F u s o c h o , A m a g a s k a i 660,

J A P A N

(Received July 24, 1992) (Revised September 8, 1992)

Introduction

The Monte Carlo simulations I1,2, 3] utilizing embedded atom method (EAM) potentials [4, 51 have shown that the

solute atoms in alloys can segregate to the cores of dislocations. The core is the region where the treatment based on

linear elasticity theory, such as the concept of a Cottrell atmosphere, cannot be applied. The simulations using EAM

potentials include all factors that may induce the segregations, such as the chemical effects and the atomic relaxation due

to the size-mismatch between the solute and the solvent atoms. In the simulation, however, it is not easy to extract one

factor that causes the segregation fi'om the calculated results.

A number of phenomenological models to describe the surface and grain boundary segregation have been presented

by many authors [6, 7, 8]. They have discussed the influences of the chemical effects and size-mismatch between the

solute and solvent atoms on the segregation. The aim of this study is to propose an atomic level model of the core

segregation to clarify the contribution of the chemical effects, i. e., the tendency of ordering of alloys and the surface

tensions. The approach that Treglia and Legrand [9] used to construct the surface segregation model on atomic plane-by-

plane basis is employed. This is the bond-breaking model based on a mean-field theory considering the number of bonds

associated with each atom in the surface region. In the present model, a detailed structure of the dislocation core after

relaxation is taken into account.

F o r m a l i s m

A dis locat ion core model

Let us consider an infinite screw dislocation with Burgers vector b of (a /2) [110] in an FCC structure. Fig. 1 is a

schematic representation of the dislocation after the structural relaxation occurs (see Fig. 4 in Reference [ 1 ]), The atomic

relaxation due to the segregation can take place but is ignored here. The big s~uctural change after segregation may not

occur in dilute alloys. The figure shows two (002) planes, which are viewed from [0011 direction. These planes contain

a dislocation core. The set of four rows of atoms enclosed by rectangles is defined as a core. In the present model, it is assumed that the atoms in the core have different coordination numbers from that of bulk atoms. As shown in the figure,

the atoms on the row, a01, have four bonds with the atoms on the same (002) plane and three bonds with the atoms on the

other (002) plane. They also have four bonds with the atoms on the adjacent (002) plane at the bulk side, which is not

shown in the figure. Thus the coordination number is 11 in total, which is smaller than 12 of bulk atoms.

In the bond-breaking models for the surface segregation, the absence of some bonds in the surface is the origin of the

1283 0956-716X/92 $ 5 , 0 0 + . 00

Copyright (c) 1992 Pergamon Press Ltd.

1284 SOLUTE SEGREGATION V o l . 27 , No. 10

inhomogeneous profiles of the solute. This implies that the difference in the coordination number between atoms at the

core and in the bulk in the present model results in the core segregation of solute atoms.

Calculation of the Ising Hamiltonian and the free energy

The lsing Hamiltonian of the alloy AI_cB c (A: solvent, B: solute) is given by

1 ~ i j r t i n j (]) H = E F n v m , n,m,i,j

where e q is the interaction energy between an atom of type i and an atom of type j (i, j = A, B). Here only the nearest

neighbor interaction is considered and no modification near the core is assumed, p i n is the occupation number which is

equal to 1 (0) if the site n is (not) occupied by an atom of type i . p An+ p Bn=l for a binary alloy, so that Eq.(1) can be

written in terms of only one variable, pn=p Bn:

H = H o + ( ' r - V ) ~ p n Z n + V ~ p . p r a , n n, m

(n * m)

with

(2)

n, ra

(n ~ m)

where Z n is the coordination number of site n. The quantity r is called surface tension term and can be related to the

difference in surface energy between A and B metals. V is the pair interaction which characterizes the tendency of the

system to order (segregate) in the bulk. Note that with the present sign convention, V >0 (V < 0) indicates a tendency of

the alloy to order (segregate).

The internal energy is obtained by averaging the Hamiltonian (2) over all configurations. The average of the two site

correlation functions <Pn Pm> is assumed to be factorized into the product of one site correlation functions ,<pn > <pro >.

This assumption is justified when an alloy is disordered. The concentrations, col and c02,are defined at the core, aol and

a02. In addition, the concentrations, c l , are defined on the I-th shell (1=1,2,3,"-), which encloses the core (Fig. l(b)). It

is assumed that the concentrations are the same at the same shell but can be different for the different shells. The internal

energy is expressed in terms of Col, c02, Cl, c 2, .." considering the number of interacting atoms at the cores and on the

shells, as follows:

IH)=Ho + N ( r - V)(22Col + 24c02 + 144 cl) + 2NV{co l (3Co l + 3c02 +5Cl ) +c02(3c01 +2c02 +7c l )

+Cl (5c01 +7 (702 + 34cl +26c2 )}+ 4 N Z !12{r- V )(21+l)Cl + V c l { ( 7 1 - l)Cl-i +(101+ 7)cl + (71+ 6) C l + 1 ~ ,

I->2

(3)

where N is the number of atoms on one row of atoms.

Equation (2) does not include a size effect, i.e., size-mismatch between solute and solvent atoms. This effect is

introduced by adding a term, AH~ to the Harniltonian:

AHse = 247r K G ro ri (ro - rt )z - 3 K r l + 4 G r o ' (4)

where K is the bulk modulus of the solute, G is the shear modulus of the solvent and r 0 and r 1 are appropriate radii tbr

Vol. 27 , No. i0 SOLUTE SEGREGATION 1285

solvent and solute, respectively. Size-mismatch causes relaxation of the structure after the solute atoms segregate. But in the present model, the effects of atomic relaxation is not be incorporated. Thus, in this paper, we do not consider the

size-mismatch and we put AHse=0.

The free energy is obtained by adding the entropy term to the Hamiltonian.

F=',H',+ N /kBT Z lct lnct+(1-ct) ln(1-ct) i ' -N! Z clp , I l

(5)

where N 1 is the number of atoms on the I -th shell (or at the core) and p is the chemical potential. The minimization of

the free energy leads to the coupled equations whose solutions give the solute atom concentrations. It is enough to let the

concentrations, col, c02, c I and c 2, vary assuming c 3 = c 4 = c 5 . . . . . c B, where c B is the bulk concentration. This assumption will be justified by the calculation results. The chemical potential ~ can be determined by the condition

c~=c13. Finally we get the following coupled equations which must be solved self-consistently.

c01 = c a exp~_{_(.r_V)+V(6col+6tb2+lOcl_24cM} 1 -c01 1 - c B

c02 = cB expi - (6c01+4c02 + 1 4 c l - 2 4 ~ ) i I - C 0 2 I - C B

Cl = c a exp ~_ V i - c l 1 - c B 3 ( 5 C 0 1 + 7 t b 2 + 3 4 c l +26Ca-72CB)!~

C2 = c a exp i -~ / (26c l+54c2 -80cMi 1 - c 2 1 - c B ~

(6}

Here, the parameters, r and V are normalized by kBT.

R e s u l t s

Effect o f "r

The effect of the r (V ---0, AHse=0) on the solute segregation is exhibited in Fig. 2. In this figure, the solute atom

concentrations at the core and on the shells enclosing it (Fig. 1 (b)) are plotted for various values of the r. c 8 is set to be 1

at.%. For x < 0, the depletion of the solute atoms takes place whereas for r > 0, the segregation of the solute is observed.

In both cases, segregation or desegregation occurs only at aol (core) and the concentrations at a02, s 1, s2 are kept at the

bulk concentration, c B. The degree of the segregation and the desegregation becomes higher as the absolute value of r

increases.

E f f e c t o f V

The dependence of the solute atom distribution profile at and near the core for CB=l.O at. % on the values of the

ordering parameter, V (z ---0, AHse--O) is shown in Fig. 3. For V < 0, the solute atoms segregate at a01 and the solute

concentration monotonically decreases to the bulk value for outer shells. For V = -1.0, the c0! and Co2 are 2.98 and 1.14

at. %, respectively. On the other hand, when V > 0, the desegregation is observed. For V =1.0, the col, c02, c 1 and c 2

are 0 .39,1.03, 1.01 and 1.0 at.% respectively. In this case, the depletion of the solute occurs at aol, but at a02 the solute

atoms are enriched slightly. This seems to be similar to the oscillatory profiles of the solute atoms observed for the

surface and grain boundary segregation [ 10].

D i s c u s s i o n

In this model the effect of atomic relaxation after the segregation occurs are included only in ~/4se- This relaxation

may occur due to the size-mismatch between the solute and solvent atoms. In this model, if the size-mismatch effects is

taken into consideration (AHse~ 0), the solute atom segregation is promoted no matter what the values of ' r and V are. It has been known that in the particular case, such as Fe-Zr |9] and Pt-Ni [10], the size effect prevails unambiguously. The

1286 SOLUTE SEGREGATION Vol. 27, No. I0

present results show that the parameters, r and V, can also induce inhomogeneous distribution of the solute atoms around the dislocation.

The effect of the difference in surface tension between the components (r) has been given much attention by many authors in the field of surface segregation [7]. In this case, the segregation behavior can be easily understood. That is, the atoms with the lower surface tension in a pure metal segregate to the surface. In the present sign convention, r > 0 means that the solute atom has a surface energy lower than the other component. That is, for r > 0, the segregation occurs whereas for r < 0, the desegregation occurs. The results from the present model show that the atoms with lower surface tension are enriched at the core like the surface segregation. This can be explained in a similar way to the surface

segregation, although we should be careful since the structure is more complicated than the surface.

The system with the negative V has a large miscibility gap in the equilibrium phase diagram. In such a system, clustering takes place easily. In the present case, this means that the segregation occurs at the core. For V> 0, oscillatory profiles of the solute atoms have been reported in the surface and grain boundary segregation. In the present results, desegregation at a 0 and enrichment of the solute at a I are observed. This seems to manifest that the oscillatory distribution of the solute can be obtained in the core segregation.

Acknowledgment: I am very indebted to Dr. Y. Maehara for many enlightening discussions.

References

1. A. Seki, D. N. Seidman, Y. Oh and S. M. Foiles, Acta Met. Mater., 39, 3167, (1991). 2. A. Seki, D. N. Seidman, Y. Oh and S. M. Foiles, Acta Met. Mater., 39, 3179, (1991).

3. S. M. Foiles, Mater. Res. Soc. Symp. Proc. 63, 61 (1985). 4. M. S. Daw and M. I. Baskes, Phys. Rev. Lett. 50, 1285, (1983). 5. S. M. Foiles, M. I. Baskes and M. S. Daw, Phys. Rev. B33, 1285, (1983). 6. Interfacial Se~re~,ation, edited by W. C. Johnson and J. M. Blakely (Am. Soc. Metals, Metals Park, Ohio, 1977). 7. Surface Segregation Phenomena, edited by P. A. Dowben and A. Miller (CRC Press, Boca Raton, Florida, 1990).

8. D. McLean, Grain Boundaries in Metals, Oxford University Press, (1957). 9. G. Treglia and B. Legrand, Phys. Rev., B35, 4338, (1987).

10. S. M. Kuo, A.Seki, Y. Oh and D. N. Seidman, Phys. Rev. Lett. 65, 199, (1990).

V o l . 27 , No. 10 SOLUTE SEGREGATION 1287

02

%%%%'-, % % % % i ! ~ ,

• o'o °o°~ I • • o'o'o%,, , %'o%%*', - , • oOo'o'o~ , OoOoOoOo,,:, • oOoOo,oWw~.

a01

*0%%% %%%% %%%% %%%% %%%% %%%% %%%% %%%%

i i

a°l l (a)

S

s; sl l #S3 [0011

(002) O a n e " ~ : ~ - ~ , i ~ . . O . . , 'O * ,

a02// / ~..e..e..~..o..@..o..~..~..~

[110] (b)

FIG. I. Schematic representation of the dislocation core used to construct the Ising Hamiltonian. (a) A view from [001] direction. Filled and open circles denote atoms on different (002) planes. The atoms enclosed by the rectangular are defined as core atoms. (b) A view from I 1101 direction. Circles are the rows of atoms arranged in [ 110] direction. Sl is for the/-th shell.

o~. 1-1

v

.=_o 0.9

C

o~0.7 C O O

o E 0.5

-8 0.3 I I

a01 a02

R o w of a t o m s

+ T= -I.00 • "--'-O-- T = -0.75

T = -0.50 T = -0.25

I I I

s l s2 a01

v

C ._o

¢-

0 0 C 0 eO

E 0 t~ 0

-6 0 if)

v

, = 1.00 -----O-- r = 0,75 -----o---- T = 0.50

~" T = 0.25

i I

a02 sl

R o w of a t o m s

i

s2

(a) (b)

FIG. 2. Solute atom concentrations at and near the dislocation core for different values of T. (a) T < 0,

(b) T > O.

1288 SOLUTE SEGREGATION Vol. 27, No. i0

3

v t-"

._o 2

tO

0

1 E o

o 0

- - . -

V = -1.00 V = -0.75 V = -0.50 V = -0.25

o~ 4 "

¢--

0

"5

1.1

Q v

0.9

0 ,7

0.5

0.3 , J ,

a01 a02 s l

R o w of a t o m s

(b)

v = 1.00 - ~ o - - v= 0.75 -.-o.--.- v= 0.50 .---a---- v = 0.25

I I I I I , O r )

a01 a02 s l s2 s2

R o w of a t o m s

(a)

FIG. 3. Solute atom concentrations at and near the dislocation core for different values of V. (a) V < 0, (b) V > 0.