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Surface segregation and bond strength-atomic size representation
Scripta M E T A L L U R G I C A
Vol. 13, pp. 989-991, 1979 Printed in the U.S.A.
SURFACE SEGREGATION AND BOND STRENGTH-ATOMIC
Pergamon Press Ltd. All rights reserved
SIZE R E P R E S E N T A T I O N
Jong K. Lee D e p a r t m e n t of M e t a l l u r g i c a l Engineering Michigan Technological University Houghton, MI 49931 (Received July 9, 1979)
Recently, Abraham, Tsai and Pound (I) derived a relationship b e t w e e n bond strength and atomic size differences, w h i c h predicts the surface segregation of solute in very dilute b i n a r y solid alloys. One of their surprising results is that atomic r e l a x a t i o n causes little change in the r e l a t i o n s h i p between bond strength ratio (£*) and atomic size ratio (o*). The amounts of elastic strain energy change due to atomic r e l a x a t i o n for the solute in the bulk and at the surface are b e l i e v e d to be equal such that the combined relaxation c o n t r i b u t i o n to the segregation driving force is small. In view of this interesting result, the e*-a* r e p r e s e n t a t i o n w i t h o u t atomic r e l a x a t i o n should be useful in predicting surface segregation of solutes. A l t h o u g h their expression for the e*-~* r e p r e s e n t a t i o n is complete, it is not a convenient form to be used because it involves lattice summation. In the Letter, we present their e*-a* representation in closed forms. The c r i t e r i o n
for surface segregation was defined as
(I)
Z f(o/ri) An(r i) i=ii e* < [ f{(l+o,)o/2ri}dn(ri)] 2, i=l
[i]
where ~ and e are parameters in the form of an interatomic potential ~(r) = ef(~/r), o* and e* are the ratios of the solute's to the solvent's a and e, respectively, and An(ri) is the d i f f e r e n c e between the number of the ith n e a r e s t - n e i g h b o r s for a bulk atom n_(r.), and that for a surface atom ns(r i) A s s u m l n g L e n n a r d - J o n e s 12-6 potenti~l and no atomlc relaxation as in A b r a h a m et al (i), we have for an fcc crystal (2,3),
F f(o/r) An(ri) i=l
s
= (pb 2 - P12) (
)
_ (p
_ P6 ) (
o
)6,
[2]
o
and b
i=l
s
f { ( l + o * ) ~ / 2 r i } A n ( r i) = (P12-P12) b s (P6-P6)
-o
__ ,
o
989 0036/9748/79/110989-03502.00/0 Copyright
(c) 1979 Pergamon
Press Ltd.
[3]
990
SURFACE SEGREGATION
w h e r e r O is the first n e a r e s t - n e i g h b o r distance,
PI2 =
Z n ( / ~ ro )i-6' i=l
P6 =
Vol.
13, No.
r i = /f ro,
~ n ( / ~ ro)i -3 i=l
[4]
and s u p e r s c r i p t s "b" and "s" refers to bulk and surface atom, respectively. The lattice s u m m a t i o n constants, P12 and P6' are o b t a i n e d through a c o m p u t e r c a l c u l a t i o n and listed in Table I for bulk, (i00), (iii), and (ii0) atoms. For (ii0) surface, there are two kinds of surface atoms: a surface atom w i t h 5 b r o k e n n e a r e s t - n e i g h b o r bonds is i n d i c a t e d w i t h a s u b s c r i p t "e" while a surface atom w i t h one b r o k e n n e a r e s t - n e i g h b o r bond is m a r k e d with a s u b s c r i p t
by
Since the e q u i l i b r i u m i n t e r a t o m i c d i s t a n c e ro, at a b s o l u t e zero is given (2)
(a/ro) 6
b b = P6/2P12,
[5]
the e x p r e s s i o n for a critical value,
E*=[ c
e~, becomes 2
I-i 6
12 ] ,
[6]
where
1
=
b b s 2P12(P6 - P6 )
b(pb 2
P6
[7]
s
- PI2 )
The values of I are listed in T a b l e I. We note that except for (ii0)~, the other three surfaces (100), (iii) and (ii0), yield I a p p r o x i m a t e l y equal to 2. The higher value of I for (iI0)~ is caused 5y the fact that a surface a t o m on (ll0) e is m o r e like a "bulk" at6m b e c a u s e it has only one b r o k e n nearestneigbJSor bond. A typical surface a t o m may be c o n s i d e r e d to retain about o n e - h a l f of the b _b 2 n e i g h b o r i n g atoms of a bulk atom. Therefore, p~s = P6/2 and p s _ 12 = p 12 / . S u b s t i t u t i n g these r e l a t i o n s into Eq. [7], I yields 2 for a typical surface. When ~ = 2, Eq. [6] becomes ~,= c
[
1
6
2 12 ] .
[8]
e* is p l o t t e d against ~* in F i g u r e 1 w h e r e the solid curve r e p r e s e n t s the case o9 i = 2, the b r o k e n curve stands for a (100) surface, w h i c h was p r e v i o u s l y studied by A b r a h a m et al (i) and the b r o k e n - d o t curve e x p r e s s e s the e*-~* map for a (iii) surface. Since the I v a l u e for (i10)~ is n e a r l y two, t h e C s o l i d
Ii
Vol.
13, No.
ii
SURFACE S E G R E G A T I O N
991
curve r e p r e s e n t s e s s e n t i a l l y a (110) surface. We note that as the number of b r o k e n bonds (nearest-neighbor) for a surface atom d e c r e a s e s and thus I increases, the e * - ~ * c u r v e s h i f t s t o w a r d the lower r i g h t - h a n d - s i d e corner direction, C i n d i c a t i n g a d e c r e a s i n g s e g r e g a t i o n tendency. In a p r a c t i c a l use of the £*-~* map, A b r a h a m et al i n d e n t i f i e d e* as the ratio of the heats of s u b l i m a t i o n and o* as the ratio of atomic radii. Therefore, for a d i l u t e b i n a r y alloy system, if the v a l u e of e* o b t a i n e d from Eq. [8] is larger than e* of the b i n a r y system, the solut~ atom will segregate to a surface of the alloy. As p o i n t e d out by A b r a h a m et al, however, there are several b i n a r y systems of early t r a n s i t i o n m e t a l s for w h i c h the c r i t i e r i o n does not work, and the c o n s i d e r a t i o n of atomic r e l a x a t i o n increases the a s y m m e t r y of the e*-o* curve e s p e c i a l l y in the lower o* side. Nevertheless, Eq. [8] should be ~seful, to a first approximation, in p r e d i c t i n g the surface s e g r e g a t i o n of solute in a d i l u t e b i n a r y alloy. Acknowled@ements The author w i s h e s to thank P r o f e s s o r T. H. C o u r t n e y for his helpful c o m m e n t s on the m a n u s c r i p t and the NSF for the financial support under Grant DMR78-05741. References i. 2. 3.
F. F. Abraham, N-H Tsai and G. M. Pound, Surface Science, 83, 406 (1979). T. K i h a r a and S. Koba, J. Phys. Soc. Japan, [, 348 (1952). C. Kittel, I n t r o d u c t i o n to Solid State Physics, 3rd ed., p. 87, Wiley, N e w York (1967). TABLE I:
FCC L A T T I C E S U M M A T I O N C O N S T A N T S
Position bulk
PI2
P6
12.1319
14.4539
(100)
8.0980
9.5564
-
(iii)
9.0708
10.4148
2.215
(Ii0)~
7.0846
8.4623
1.993
(110) 8
11.0904
12.6775
2.863
2.038
,l - tA
,3
i H i
j
FIG. 1 - The e*-o* representation; the • C solld curve (--) for a case of I = 2, the b r o k e n (---) for a (I00) surface and the b r o k e n - d o t (-.-) for a (iii) surface. S u r f a c e s e g r e g a t i o n occurs if, at a given o*, e* is less than e*.
I/ - 1.2
C
1!2 - ag
Vol. 13, pp. 989-991, 1979 Printed in the U.S.A.
SURFACE SEGREGATION AND BOND STRENGTH-ATOMIC
Pergamon Press Ltd. All rights reserved
SIZE R E P R E S E N T A T I O N
Jong K. Lee D e p a r t m e n t of M e t a l l u r g i c a l Engineering Michigan Technological University Houghton, MI 49931 (Received July 9, 1979)
Recently, Abraham, Tsai and Pound (I) derived a relationship b e t w e e n bond strength and atomic size differences, w h i c h predicts the surface segregation of solute in very dilute b i n a r y solid alloys. One of their surprising results is that atomic r e l a x a t i o n causes little change in the r e l a t i o n s h i p between bond strength ratio (£*) and atomic size ratio (o*). The amounts of elastic strain energy change due to atomic r e l a x a t i o n for the solute in the bulk and at the surface are b e l i e v e d to be equal such that the combined relaxation c o n t r i b u t i o n to the segregation driving force is small. In view of this interesting result, the e*-a* r e p r e s e n t a t i o n w i t h o u t atomic r e l a x a t i o n should be useful in predicting surface segregation of solutes. A l t h o u g h their expression for the e*-~* r e p r e s e n t a t i o n is complete, it is not a convenient form to be used because it involves lattice summation. In the Letter, we present their e*-a* representation in closed forms. The c r i t e r i o n
for surface segregation was defined as
(I)
Z f(o/ri) An(r i) i=ii e* < [ f{(l+o,)o/2ri}dn(ri)] 2, i=l
[i]
where ~ and e are parameters in the form of an interatomic potential ~(r) = ef(~/r), o* and e* are the ratios of the solute's to the solvent's a and e, respectively, and An(ri) is the d i f f e r e n c e between the number of the ith n e a r e s t - n e i g h b o r s for a bulk atom n_(r.), and that for a surface atom ns(r i) A s s u m l n g L e n n a r d - J o n e s 12-6 potenti~l and no atomlc relaxation as in A b r a h a m et al (i), we have for an fcc crystal (2,3),
F f(o/r) An(ri) i=l
s
= (pb 2 - P12) (
)
_ (p
_ P6 ) (
o
)6,
[2]
o
and b
i=l
s
f { ( l + o * ) ~ / 2 r i } A n ( r i) = (P12-P12) b s (P6-P6)
-o
__ ,
o
989 0036/9748/79/110989-03502.00/0 Copyright
(c) 1979 Pergamon
Press Ltd.
[3]
990
SURFACE SEGREGATION
w h e r e r O is the first n e a r e s t - n e i g h b o r distance,
PI2 =
Z n ( / ~ ro )i-6' i=l
P6 =
Vol.
13, No.
r i = /f ro,
~ n ( / ~ ro)i -3 i=l
[4]
and s u p e r s c r i p t s "b" and "s" refers to bulk and surface atom, respectively. The lattice s u m m a t i o n constants, P12 and P6' are o b t a i n e d through a c o m p u t e r c a l c u l a t i o n and listed in Table I for bulk, (i00), (iii), and (ii0) atoms. For (ii0) surface, there are two kinds of surface atoms: a surface atom w i t h 5 b r o k e n n e a r e s t - n e i g h b o r bonds is i n d i c a t e d w i t h a s u b s c r i p t "e" while a surface atom w i t h one b r o k e n n e a r e s t - n e i g h b o r bond is m a r k e d with a s u b s c r i p t
by
Since the e q u i l i b r i u m i n t e r a t o m i c d i s t a n c e ro, at a b s o l u t e zero is given (2)
(a/ro) 6
b b = P6/2P12,
[5]
the e x p r e s s i o n for a critical value,
E*=[ c
e~, becomes 2
I-i 6
12 ] ,
[6]
where
1
=
b b s 2P12(P6 - P6 )
b(pb 2
P6
[7]
s
- PI2 )
The values of I are listed in T a b l e I. We note that except for (ii0)~, the other three surfaces (100), (iii) and (ii0), yield I a p p r o x i m a t e l y equal to 2. The higher value of I for (iI0)~ is caused 5y the fact that a surface a t o m on (ll0) e is m o r e like a "bulk" at6m b e c a u s e it has only one b r o k e n nearestneigbJSor bond. A typical surface a t o m may be c o n s i d e r e d to retain about o n e - h a l f of the b _b 2 n e i g h b o r i n g atoms of a bulk atom. Therefore, p~s = P6/2 and p s _ 12 = p 12 / . S u b s t i t u t i n g these r e l a t i o n s into Eq. [7], I yields 2 for a typical surface. When ~ = 2, Eq. [6] becomes ~,= c
[
1
6
2 12 ] .
[8]
e* is p l o t t e d against ~* in F i g u r e 1 w h e r e the solid curve r e p r e s e n t s the case o9 i = 2, the b r o k e n curve stands for a (100) surface, w h i c h was p r e v i o u s l y studied by A b r a h a m et al (i) and the b r o k e n - d o t curve e x p r e s s e s the e*-~* map for a (iii) surface. Since the I v a l u e for (i10)~ is n e a r l y two, t h e C s o l i d
Ii
Vol.
13, No.
ii
SURFACE S E G R E G A T I O N
991
curve r e p r e s e n t s e s s e n t i a l l y a (110) surface. We note that as the number of b r o k e n bonds (nearest-neighbor) for a surface atom d e c r e a s e s and thus I increases, the e * - ~ * c u r v e s h i f t s t o w a r d the lower r i g h t - h a n d - s i d e corner direction, C i n d i c a t i n g a d e c r e a s i n g s e g r e g a t i o n tendency. In a p r a c t i c a l use of the £*-~* map, A b r a h a m et al i n d e n t i f i e d e* as the ratio of the heats of s u b l i m a t i o n and o* as the ratio of atomic radii. Therefore, for a d i l u t e b i n a r y alloy system, if the v a l u e of e* o b t a i n e d from Eq. [8] is larger than e* of the b i n a r y system, the solut~ atom will segregate to a surface of the alloy. As p o i n t e d out by A b r a h a m et al, however, there are several b i n a r y systems of early t r a n s i t i o n m e t a l s for w h i c h the c r i t i e r i o n does not work, and the c o n s i d e r a t i o n of atomic r e l a x a t i o n increases the a s y m m e t r y of the e*-o* curve e s p e c i a l l y in the lower o* side. Nevertheless, Eq. [8] should be ~seful, to a first approximation, in p r e d i c t i n g the surface s e g r e g a t i o n of solute in a d i l u t e b i n a r y alloy. Acknowled@ements The author w i s h e s to thank P r o f e s s o r T. H. C o u r t n e y for his helpful c o m m e n t s on the m a n u s c r i p t and the NSF for the financial support under Grant DMR78-05741. References i. 2. 3.
F. F. Abraham, N-H Tsai and G. M. Pound, Surface Science, 83, 406 (1979). T. K i h a r a and S. Koba, J. Phys. Soc. Japan, [, 348 (1952). C. Kittel, I n t r o d u c t i o n to Solid State Physics, 3rd ed., p. 87, Wiley, N e w York (1967). TABLE I:
FCC L A T T I C E S U M M A T I O N C O N S T A N T S
Position bulk
PI2
P6
12.1319
14.4539
(100)
8.0980
9.5564
-
(iii)
9.0708
10.4148
2.215
(Ii0)~
7.0846
8.4623
1.993
(110) 8
11.0904
12.6775
2.863
2.038
,l - tA
,3
i H i
j
FIG. 1 - The e*-o* representation; the • C solld curve (--) for a case of I = 2, the b r o k e n (---) for a (I00) surface and the b r o k e n - d o t (-.-) for a (iii) surface. S u r f a c e s e g r e g a t i o n occurs if, at a given o*, e* is less than e*.
I/ - 1.2
C
1!2 - ag