The kinetics of short-range order in some noble metal alloys

^ THE KINETICS OF SHORT-RANGE ORDER

fc IN SOME NOBLE METAL ALLOYS

f

\

F

f

THE KINETICS OF SHORT-RANGE ORDER

IN SOME NOBLE METAL ALLOYS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. CJ.D.M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 27 SEPTEMBER 1967

TE 14.00 UUR DOOR

SI]BRAND RADELAAR

NATUURKUNDIG INGENIEUR GEBOREN TE 'S-GRAVENHAGE 1967 "BRONDER-OFFSET" ROTTERDAM

Dit proefschrift is goedgekeurd door de promotor P R O F . D R . M.J.DRUYVESTEYN

This work is p a r t of the r e s e a r c h p r o g r a m m e of the R e s e a r c h group "Metals F.O.M. - T . N . O . " of the "Stichting voor Fundamenteel Onderzoek d e r M a t e r i e " (Foundation for Fundamental R e s e a r c h of Matter - F . O . M . ) and was also made possible by financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (Netherlands Organization for pure R e s e a r c h

C O N T E N T S

1. INTRODUCTION AND SUMMARY 9

2. SHORT-RANGE ORDER AND ITS INFLUENCE ON PHYSICAL

PROPERTIES 13 2 . 1 . Introduction 13 2 . 2 . Short-range o r d e r p a r a m e t e r s 13 2 . 2 . 1 . Definitions 14 2 . 2 . 2 . Bethe's s . r . o . p a r a m e t e r 16 2 . 2 . 3 . W a r r e n ' s s . r . o . p a r a m e t e r 17 2 . 3 . The nature of the ordering forces 18 2 . 4 . Statistical t r e a t m e n t s of s . r . o . 18

2 . 4 . 1 . Quasi-chemical approach 18 2 . 4 . 2 . Other statistical t r e a t m e n t s of s . r . o . 24

2 . 5 . Influence of s . r . o . on physical p r o p e r t i e s 26

2 . 5 . 1 . Introduction 26 2 . 5 . 2 . The influence of s . r . o . on e l e c t r i c a l resistivity 26

2 . 5 . 3 . The Zener-effect 31

3 . VACANCIES IN ALLOYS WITH SHORT-RANGE ORDER 35

3 . 1 . Introduction 35 3 . 2 . Formation energy of vacancies in binary alloys 35

3 . 2 . 1 . Random distribution of atoms 36 3 . 2 . 2 . Vacancy concentration in alloys with s . r . o . 37

3 . 3 . Migration of vacancies in binary alloys 40 3 . 3 . 1 . Probability distribution of the migration energy 40

3 . 3 . 2 . Average mobility of atoms in b . c . c . s t r u c t u r e s 43 3 . 3 . 3 . Average mobility of atoms in f . c . c . s t r u c t u r e s 45

3 . 4 . Theory of the kinetics of s h o r t - r a n g e o r d e r in binary alloys 49

3 . 4 . 1 . Fundamental approximations 49 3 . 4 . 2 . The kinetics of s . r . o . in b . c . c . s t r u c t u r e s 50

3 . 4 . 3 . The kinetics of s . r . o . i n f . c . c . s t r u c t u r e s 56 3 . 4 . 4 . The relation between ordering kinetics and selfdiffusion 57

3 . 4 . 5 . Discussion 60

4 . EXPERIMENTAL TECHNIQUE 62

4 . 1 . Introduction 62 4 . 2 . Measurements below 250 C (thermostat) 63

4 . 3 . Measurements above 250 C 65 4 . 3 . 1 . Standard-method 65 4 . 3 . 2 . Dummy-method 66 4 . 3 . 3 . Description of the specimen holder 68

4 . 3 . 4 . Description of the furnaces 69 4 . 3 . 5 . Description of the helium container 70

4 . 4 . Materials 70

5. RESULTS AND COMPARISON WITH OTHER DATA 72

5 . 1 . CuAl (85,15) 72 5 . 2 . AuCu (86,14) 76 5 . 3 . AuCu(75,25) 78 5 . 4 . AuAg-alloys 80 5 . 5 . Summary of the r e s u l t s 86 6. DISCUSSION 87 6 . 1 . Introduction 87 6 . 2 . Vacancy equilibrium during the m e a s u r e m e n t s 87

6 . 3 . Discussion of quenching and annealing experiments 88 6 . 4 . The relation between the kinetics of s . r . o . and Zener-eff ect 90

6 . 5 . The relation between the kinetics of s . r . o . and selfdiffusion 91 6 . 6 . The kinetics of s . r . o . n e a r the critical t e m p e r a t u r e 94

CONCLUSION 97 SAMENVATTING 98 LIST OF SYMBOLS 101 REFERENCES 104

C H A P T E R I

INTRODUCTION AND SUMMARY

Numerous studies have been devoted to the a r r a n g e m e n t s of atoms in h o -mogeneous (single phase) metallic solid solutions. It a p p e a r s , that the atoms a r e in general not randomly distributed among the various lattice sites.If t h e r e is a tendency of the atoms to surround themselves with like n e a r e s t neighbours clustering is said to occur, if t h e r e exists a preference for unlike neighbours the alloy is said to be ordered. If the l a t t e r correlation is confined to the i m -mediate environment of a p a r t i c u l a r atom the alloy exhibits short-range order ( s . r . o . ) . When a correlation exists between the occupation of lattice sites which a r e far apart long-range order ( l . r . o . ) o c c u r s .

Much work has been devoted to the investigation of the kinetics of the establishment of long-range o r d e r . Ordering proceeds by a redistribution of the constituent a t o m s , which occurs by means of vacancies. The kinetics of l . r . o . have been investigated both with equUibrivim and non-equilibriimi concen-trations of v a c a n c i e s . With a single exception [ l ] the kinetics of s . r . o . have been studied with non-equilibrium concentrations only. In this thesis we p r e s e n t resistivity m e a s u r e m e n t s of the kinetics of s . r . o . at elevated t e m p e r a t u r e s , where the number and mobility of the vacancies is sufficiently large to establish

4 S . r . o . - e q u i l i b r i u m in reasonably m e a s u r a b l e times (1-10 s e c ) .

In chapter 2 we s t a r t with a discussion of the various p a r a m e t e r s used in the l i t e r a t u r e to describe the state of s . r . o . After a brief outline of the t h e o -r i e s , which explain the existence of o-rde-ring f o -r c e s , we discuss some of the statistical t r e a t m e n t s of s . r . o . Special emphasis is laid on the quasi-chemical theory, which approximation will be used throughout this t h e s i s . A survey is given of existing t h e o r i e s , which deal with the influence of s . r . o . on e l e c t r i c a l 9

r e s i s t i v i t y . In the l a s t section we give a brief introduction to the theory of the Zener relaxation-effect.

Chapter 3 is entirely devoted to the kinetics of vacancies in alloys which exhibit s . r . o . In the f i r s t p a r t we d i s c u s s the formation energy of vacancies in binary alloys. In pure metals the neighbours of a vacancy a r e all of the same type. In AB-alloys z-i-1 different surroundings a r e possible (z= coordination number of the lattice) v i z . 0 , 1 . . . z neighbouring A - a t o m s . It i s c l e a r that the energy of the vacancy w ü l depend on its p a r t i c u l a r environment. Schaplnk [ 2 ] , using the quasi-chemical approximation, derived an expression for the vacancy concentration in t e r m s of the binding energies V . . , V . „ and V _ _ .

The second p a r t of this chapter is devoted to the mobility of vacancies in binary alloys. Again the probability that a given atom jumps into a vacancy will depend on the occupation of the neighbouring lattice s i t e s . We derive e x -p r e s s i o n s for the average mobilities of A - and B - a t o m s . Combination of these equations with the formula for the vacancy concentration r e s u l t s in an e x p r e s -sion for the selfdiffu-sion constants of the components of the alloy.

In the r e s t of this chapter the kinetics of s. r . o . is studied. The t r e a t m e n t is based on the p a p e r of Kidin and Shtremel' [ 3 ] , a s i m i l a r t r e a t m e n t for the kinetics of l . r . o . was already given by Vineyard [ 4 ] . Instead of computing the t i m e a v e r a g e of the contribution to the s . r . o , of successive jumps of a p a r t i c -ular vacancy V, these authors replace this t i m e - a v e r a g e by an average over all the contributions of the rate of interchanges of VA- and VB-pairs.Consequently the correlation between successive jumps is not taken into account. If two A -atoms jump consecutively into the vacancy, the contribution to the s . r . o . of the f i r s t jump is largely destroyed. Two equations r e s u l t from t h i s approximate t r e a t m e n t for the time r a t e of change of A B - and VA-pairs r e s p . In the equi-librium state these r a t e s must vanish, from which condition the equüibriimi values of c . „ (the concentration of AB-pairs) and p „ . (the probability to find an A-atom next to a vacancy) r e s u l t . It appears that the value of c . „ is equal to the value obtained from the quasi-chemical theory discussed in chapter 2. Moreover we show the value of p „ . is equal to the value implicitly contained in Schapink's t h e o r y . In o r d e r to simplify the solution Kidin and Shtremel' assume that the equilibrium environment of the vacancy is established much faster than s . r . o . equilibrium, thus p ^ . is equal to its equilibrium value P-y^«. hi the theoretical context this assumption implies that t h e r e exists a relation between the selfdiffusion coefficients D . and D_ of the components. Kidin and Shtremel' used this relation to compute the ordering r a t e ( c ) , t h e i r expression contains

accordingly only one of the diffusion coefficients. Although the assumption that p „ . = p ^ . may be a fair approximation if used to calculate the ordering r a t e , we believe that the resulting relation between D and D is unvalid. We derive therefore a relation for the ordering r a t e , which contains both diffusion coeffi-c i e n t s . Although the basicoeffi-c assumptions a r e r a t h e r d r a s t i coeffi-c this relation p r e d i coeffi-c t s that the activation energy for s . r . o . can be lower than the activation energy for selfdiffusion of the fastest component.

The experimental technique used to determine the relaxation time of s . r . o . is presented in chapter 4 . The method is essentially a s t e p - r e s p o n s e method. The t e m p e r a t u r e of the specimen is suddenly changed (within 1 s e c . ) and the time dependence of the r e s i s t a n c e of the specimen after this t e m p e r a -t u r e jump is o b s e r v e d . Vacancy equilibrium is apparen-tly es-tablished r a -t h e r fast, since the r e s u l t s of u p - and downquenches a r e indisttt^uishable.

The following alloys w e r e studied: CuAl (85,15), t h r e e AuAg-alloys (with 45,50 and 56 at% Au) and t h r e e AuCu alloys (with 14,25 and 75 at% Cu). The l a t t e r (Au„Cu and Cu-Au) a r e known to exhibit l . r . o .

The r e s u l t s of the m e a s u r e m e n t s a r e presented in chapter 5 and compared with other data. It a p p e a r s that for the alloys which do not exhibit l . r . o . : 1) the activation energy for s . r . o . is approximately equal to the simi of the

formation and migration energy of v a c a n c i e s .

2) the relaxation time for s . r . o . is only slightly l a r g e r than the corresponding relaxation time for the Z e n e r relaxation-effect. The activation e n e r g i e s a r e the same within experimental e r r o r .

Moreover:

3) The relaxation time for s . r . o . shows an anomalous r i s e in the vicinity of the critical t e m p e r a t u r e for l . r . o .

Especially the second experimental fact is of i n t e r e s t . The problem of the exact relationship between the relaxation time of the Zener-eff ect and the selfdiffusion constants of the alloy h a s Intrigued many investigators, but no s o l u -tion has hitherto been p r e s e n t e d . This is partly due to the fact, that no p r e c i s e knowledge about the origin of the Zener effect e x i s t s , hence the atomic m o v e -ments during the relaxation p r o c e s s a r e more or l e s s unknown. The study of the kinetics of s . r . o . is much m o r e useful since s . r . o . p a r a m e t e r s can be deduced from diffuse X - r a y scattering and detailed t h e o r i e s on the influence of s . r . o . on e l e c t r i c a l resistivity exist. The close relationship between the kinet-ics of s . r . o . and of the Zener-eff ect makes it possible to explain the kinetkinet-ics of the l a t t e r effect a l s o .

hl chapter 6 we wül discuss this matter more fully and compare the

ex-perimental data on AgAu with the theoretical values. The agreement between

the computed relaxation times and the experimental ones is not quite

satisfac-tory. The computed values ly about a factor 5 below the experimental results.

This is due to the fact, that the equation for the ordering rate overestimates the

efficiency of the ordering process, since the influence of a successive jump is

not taken into accoimt. The computed activation energy is larger than the

acti-vation energy for selfdiffusion of süver, contrary to experiment. As wül be

shown in chapter 6 the süver-gold system is not well suited for a test of the

theory. Alloy systems like AgZn or CuZn are much better suited and in this

case our expression correctly predicts a lower activation energy for s . r . o .

than for diffusion of zinc (for low concentrations of zinc) in agreement with

ex-periment.

C H A P T E R II

SHORT-RANGE ORDER AND ITS INFLUENCE ON PHYSICAL PROPERTIES

2 . 1 INTRODUCTION

In this chapter we w ü l give a survey of the existing l i t e r a t u r e on those aspects of s h o r t - r a i ^ e o r d e r ( s . r . o . ) , which a r e relevant to the discussions in following c h a p t e r s . In s e c t . 2 . 2 we p r e s e n t the various p a r a m e t e r s used in the l i t e r a t u r e to d e s c r i b e the state of s . r . o . in binary alloys. After a brief outline of the theories which explain the existence of ordering forces in s e c t . 2.3^ we d i s c u s s some of the statistical t r e a t m e n t s of s . r . o . ( s e c t . 2 . 4 ) . These theories enable us to make quantitative predictions of the o r d e r p a r a m e t e r s discussed in s e c t . 2 . 2 and of the values of the thermodynamica! variables associated with a certain state of s . r . o . The influence of s . r . o . on some p h y s -ical p r o p e r t i e s is t r e a t e d in s e c t . 2 . 5 .

2.2 SHORT-RANGE ORDER PARAMETERS

Let u s consider a homogeneous binary alloy AB with concentration c . and Cp. of A - and B - a t o m s r e s p . ÏÏ these atoms a r e randomly distributed among the various lattice s i t e s , the probabüity to find an A- r e s p . B-atom at a given lattice site is c . r e s p . c_ ( c . -H c_ = 1). X - r a y experiments [ 5 - 1 0 ] , however,

A xi A J3

show that the distribution is in general not random. If we consider a B-atom at a certain lattice site (all lattice sites a r e assumed to be equivalent), the con-ditional probabüity p _ . to find an A-atom at a neighbouring lattice site w ü l in general deviate from the random value c . .

2 . 2 . 1 . D e f i n i t i o n s

S h o r t r a i ^ e o r d e r occurs if P j , . > c . , clustering if p < c . . In the f o r -m e r case t h e r e i s a tendency of the ato-ms to surround the-mselves with unlike neighbours, whüe in the l a t t e r c a s e like atoms a r e p r e f e r r e d . In either case t h e r e exists a correlation between the occupational probabüity of neighbouring lattice points. This correlation is not confined to the occupation of neighbouring lattice s i t e s , but w ü l in general also exist for atoms which a r e farther a p a r t .

Let u s define, following Klein [ 1 1 ] .

)[ B/A(i)]

_PBA<i)

which is the conditional probabüity that if a given site is occupied by a B - a t o m , then a site in the i-th shell around the B-atom is occupied by an A - a t o m . In the s a m e way the conditional probabUities p B/B(i) , p A/B(i) J and pj A/A(i) a r e defined. These probabUities a r e related to p . (i) by:

p [ B / B ( i ) ] = l - p g ^ ( i )

' ^ B / ' ^ A P B A ^ ^ ) (2.1)

^ R / ' ^ A P B A ^ * ) A / A ( i ) ] =

3 [ A / B ( i ) ]

In o r d e r to simplify the foUowing discussion we w ü l consider only those alloys ( c . s c_) which, in the fully o r d e r e d s t a t e , satisfy the following r e q u i r e m e n t s : 1) All n e a r e s t neighbours of B - a t o m s a r e A - a t o m s .

2) A fraction c „ / c . of the n e a r e s t neighbours of A-atoms a r e B - a t o m s , the remaining a r e A - a t o m s .

3) The shells aroimd a B-atom a r e altematingly occupied by A - o r B - a t o m s . We WÜ1 discuss now the behaviour of the probabüity Pr)A(i) as a fimction of t e m p e r a t u r e and i.At infinite t e m p e r a t u r e all PT>A{i) w ü l be equal to the random value c . . If the t e m p e r a t u r e is lowered we w ü l find a tendency towards local o r d e r , either s . r . o . o r clustering. We wül discuss only the f o r m e r c a s e .

If s . r . o . occurs the odd shells i = odd around the B-atom tend to be occu-pied by A-atoms (p_ . (o) > c . ) , whUe even shells a r e preferentially occuoccu-pied by B-atoms (p„ . (e) < c . ) . These correlations a r e of finite range, the range becomes l a r g e r the lower the t e m p e r a t u r e . Beyond this range the occupational

probabüity is s t ü l essentially random, p .(i) = c . (Fig. 2 . 1 ) .

Let us now assume that the alloy has a transition t e m p e r a t u r e T . Below this t e m p e r a t u r e , the c r i t i c a l t e m p e r a t u r e for o r d e r i n g , the pattern described

PBA'° C A f-BA'^' 0 X T > Tc Ü T < T c ^ ^ ^ - 1 3 • ^ ^ " * ~ X °"

\

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Fig. 2 . 1 The probability Pgy\(i) for two temperatures, above and below the critical temperature T resp.

above is essentially changed. Even if i goes to infinity Pp.4(1) does not approach its random value. Instead p _ . (i = even) and p _ . (i = odd) tend to different limiting v a l u e s , PTJACB) and P Ü A ( ' ' ) r e s p . (Fig. Ï . I . ) . In this case we say that l o i ^ - r a n g e o r d e r e x i s t s . As the t e m p e r a t u r e is further decreased the düference between the limiting values p . (e) and p _ . (o) i n c r e a s e s u n t ü , for perfect o r -d e r ,

PBA^'' " ^^'> " PfiA^"^ = 1 and

P g ^ ( i = e v e n ) = p g ^ ( e ) = 0.

F o r a discussion of m o r e complicated c a s e s , like AB(50,50) alloys with f . c . c . s t r u c t u r e , we r e f e r the r e a d e r to the l i t e r a t u r e [12-15].

We w ü l now discuss the various s . r . o . p a r a m e t e r s , which occur in the l i t e r a t u r e .

2 . 2 . 2 . B e t h e ' s s . r . o . p a r a m e t e r

Let us introduce the quantity N . _ , w h i c h is equal to the number of n e a r e s t neighbour A B - p a i r s in the l a t t i c e . The Bethe p a r a m e t e r is defined by [ l 6 ] :

^ N A B - N A B ^ ^ ^ ^ " ' " )

N ^ g ( m a s ) - N^g(random) • ^^-^l

where N.—(random) a n d N . _(max) a r e the values of N . _ for complete d i s o r d e r and perfect o r d e r r e s p . The B e t h e - p a r a m e t e r has the convenient property, that CT = 1 for perfect o r d e r and CT = 0 for complete d i s o r d e r . It is c l e a r that a can only depend on the probabüity p _ . (i).

The total number of p a i r s in the l a t t i c e , n e g l e c t ü ^ surface effects, is equal to 5 Nz,where N is the total number of a t o m s . Let us define the fractions of U - p a i r s by:

N 2 N

Cjj = j ; ^ (I 5^ J) and c ^ = ^ ; ^ (I = A, J = B or J = A, I = B) (2.3)

This somewhat a r b i t r a r y distinction between AB and BA-pairs simplifies the notation since the fractions c „ a r e now related to the probabUities P j j by:

Cjj = C j P j j (2.4)

The c--.'s a r e related by:

'^ij + ' ' n = ^i ''^

^AA ^ ' ' A B = ' ' A ^-^ (2.5)

"BA

"^

''BB '^B

c

CT =

AB - ^AB^^^^'''") PBA - «A

c^gCmax) - c^g(random) Pg^(max) - c^

Bethe's definition can be easüy generalized for other than neighbouring shells:

CT(i)

c^g(i) - c^g(i) (random)

c^g{i) (max) - c^g(i) (random)

where c.-,(i) is the fraction of i-th neighbour pairs of type AB. Beal [17-19]

uses a simUar definition in her study of resistivity effects due to s . r . o . viz.:

PAA<^) - P A A ^ ^ ^ ' ^ O " ^ )

'^•r.(i)'=-^ r^^ ; 3 r (2.6)

2 . 2 . 3 . W a r r e n s . r . o . p a r a m e t e r

In the study of X-ray scattering the alloys exhibiting s. r . o . or clustering

the following parameters, introduced by Warren, are very convenient [20,21]:

PAB(^)

a^l--f— (2.7)

'^B

One can show, that these a. are the Fourier coefficients in the expansion of the

diffuse intensity in terms of the reciprocal lattice coordinates. The pair

proba-bUities are easüy expressed as a function of a.:

PAA<^) = ''A ^ ^B°'i

PAB(^> = '^B - '^B'^i

(2.8)

PBA(^) = ' ' A " ^A^'i

PBB^^) = '^B •" '=A°'i

2 . 3 . THE NATURE OF THE ORDERING FORCES

Several t h e o r i e s have been proposed, which t r y to explain the tendency towards o r d e r o r clustering on a quantum-mechanical b a s i s . F r i e d e l [ 2 2 , 2 3 ] predicted that s . r . o . wUl occur if a polyvalent metal is solved in a metal with lower valency, whüe clustering is favoured when the solvent has a higher valency than the solute. Flinn [24] deduced from his calculations that ordering o c -c u r s if the ele-ctron/atom ratio is s m a l l e r than 1.5, whüe otherwise -clustering WÜ1 r e s u l t . T h e r e a r e a few c a s e s where both theories predict opposite r e s u l t s e . g . Mg-rich MgIn alloys, F r i e d e l ' s theory predicts s . r . o . , F l i n n ' s clustering. X - r a y experiments show, that s . r . o . o c c u r s in this alloy, in agreement with F r i e d e l ' s theory.

H a r r i s o n and Paskin [ 2 6 , 2 7 ] using the "polar" model of Mott [28] have calculated the ordering energy of B-CuZn and obtained good agreement with specific heat m e a s u r e m e n t s . They showed that the ordering energy is longrange (the eighthneighbour interaction energy is s t ü l 5% of the n e a r e s t n e i g h -bour interaction energy).

Apart from the valency effects mentioned above another cause of s . r . o . may exist, which is due to a difference in size of the two components of the alloy. It is c l e a r that the elastic s t r a i n s due to size differences can be m o r e o r l e s s relieved by surrounding l a r g e atoms with s m a l l e r ones. Rudman [29] has calculated that this tendency might give a substantial contribution to the o r -dering energy in the CuAu-system.

Summarizing it can be said, that the nature of the ordering forces is now m o r e o r l e s s understood, but the theoretical r e s u l t s obtained a r e mostly only qualitative.

2.4 STATISTICAL TREATMENTS OF S . R . O .

2 . 4 . 1 . Q u a s i - c h e m i c a l a p p r o a c h

Let u s assume that the crystal with coordination number z is held together by forces between the neighbouring atoms only. Each of these bonds U has a definite energy V.,. associated with it, the p a i r energy V... being independent of the surroundings of the p a i r . If a mole of AB contains N . . , N andN . _ p a i r s

A A D O A B of type AA, BB and AB respectively, the configurational energy is given by:

E f = N. . V. . + N . ^ V . „ + N „ „ V „ „ (2.9) conf AA AA AB AB BB BB ^ '

The numbers N. . etc. are not independent but are related by:

2 N A A ^ N ^ B = N A ^

2 N B B " N ^ B = V

(2.10)

With the aid of these equations the configurational energy can be written as:

^ A A ^ ^ ^ B B ^ N^B [ ^ B - i ( ^ A A ' ^ B B ) ] <2-") N , z N„z

E = - ^ conf 2

The first two terms on the right-hand side represent resp. the energy of N. A-atoms and N„ B-atoms in the pure state, so the energy of mixing is given by

M = ^AB [ ^AB - i^^AA ^ ^ B B ) ] = ^ A B ^ (2.12)

For reasons which wUl become clear below the term

^ = ^AB

4 ( V A A

^

^ B B )

(2.13)

is usually called the "ordering energy".

From an inspection of e.g. (2.11) it is clear that the energy of the crystal is lowered by an increase of N . „ ü W < 0 and by a decrease of N . _ if W > 0. In the first case there is a tendency for ordering whüe in the latter case clus-tering is favoured. An increase in the number of AB-pairs above the random value wUl necessarily result in a decrease of the configurational entropy. The difficulty is to calculate this contribution to the free energy, the expression for the energy is exact within the limits of the model. We wül now follow the treat-ment of Fowler and Guggenheim [30-34] to obtain an expression of the free energy for the nonrandom case.

The configurational partition function can be written as

^conf = N _ g ( N A ' ^ B ' N A B ) ^ - p [ - i f ^ ] (2.14) AB

The summation extends over all possible configurations of nearest-neighbour p a i r s , subject to the restriction that both N . and N a r e constant.

Let u s a s s u m e that the number of possible configurations g(N. , N ,N ) is given by

[

Nz/2

1'.

g ( N ^ , N 3 . N ^ 3 ) = n(N^,N3) N , '. N ^ ^ ' . N , , ' . N , , ' . (2.15) AA BB AB BA

The second t e r m on the right-hand side of this equation gives the number of distinguishable ways to a r r a n g e the numbers N . . , e t c . of the entities

AA

AA, e t c . on ( N . + N _ ) z / 2 p l a c e s , so we assume in fact that the p a i r s of atoms a r e randomly distributed. In this way we neglect of course the correlation which e x i s t s between the occupation of neighbouring p a i r s of s i t e s . If a given p a i r is occupied e . g . by an AA-pair, the number of possible a r r a n g e m e n t s of those p a i r s w h i c h s h a r e a site with this AA-pair d e c r e a s e s . T h e factor n(N. ,N_) is a normalization factor, which is determined by the condition that the sum of all S(NAfN.D>N ) over all possible configurations with different N . „ must be equal to the number of ways of arranging N . A-atoms and N_ B-atoms on N A Jo lattice s i t e s :

^ ^ g g ( N A ' ^ B ' N A B ) = N ^ T l V (2-16)

As usual in statistical mechanics this sum can be very closely approximated by the maximum value of g, which we will call g (N. , N _ , N ) . N . „ is the

A a A B A B

value of N . - which maximizes g. The maximum value of g occurs when the distribution of A - and B-atoms is random:

* [ N Z / 2 ]

g = n ( N , N ) ^ f J ,^ j

-N . . • -N „ „ ' -N . „ ' -N„ '

N! .. N ' N ' AA- BB- AB- BA- ^ A ' B" Solving (2.17) for n and substituting this value in (2.15) gives:

g ( N , , N ^ , N , ^ N'. N ^ - ^ B B - ^ A B - ^ B I ' - (2.18) A' B ' AB' N . ' . N „ ' . N . .'. N „ „ ' . N . „ ' . N „ . ' .

A B AA BB AB BA

The numbers N . . , N _ e t c . a r e e a s ü y found to be

N * = c2 N z / 2 AA A

N 3 * = C | N Z / 2 (2.19)

N B I = N^B = '=A^BN^/2

Since we have now an explicit expression for g we can evaluate the partition function. Once again we replace the summation by its maximum t e r m , which is attained of c o u r s e for the equüibrium value of N . _ , N . _

S In Z . J , ,., . , o 5-T=— = 0 for N . _ = N . _ d N , „ AB AB

AB

Substituting (2.18) and (2.19) in (2.14) and differentiating the logarithm of (2.14) with r e s p e c t to N . _ we find, with the aid of Stirling's approximation:

1 XT o , 1 X T O 1 XT O 1 XT O 2 W . In N ^ ^ + m N g g - In N ^ ^ - In N ^ ^ - j ^ = 0 o r : ^ ^ ' ^ - - P ( - ^ ) (2.20) N^A • ^ B B W

Let us put w = exp (j-=;) and use the pair concentrations c . - (eqs. 2.4 and 2.5):

(^A-'^AB) C ' B - ^ B )

This equation is r e a d ü y solved for c . _ : ^o2

^ ^ - w ' 2 (2.21)

o - 1 + V 1 + 4 C c (w2-l)

' A B = , , 2 ,^ (2-22) 2(w - 1)

Only the positive sign is retained before the root, since c . _ must obey the condition:

O ^ ' ^ I B ^ I

It is e a s ü y shown that if W = 0, c reduces to c c the random value. K W < 0, t h e r e a r e m o r e unlike p a i r s ( c . „ > c . c ) than in the random c a s e , short-range order occurs.If W > 0 like atoms a r e favoured and clustering occurs

F o r high temj be approximated by

2

F o r high t e m p e r a t u r e s the t e r m (w -1) is s m a l l , for this case (2.22) can

o ,, 2W. ,„ „ „

^ A B = " V B ( I - V B W ^

(2-23)

We see that at high t e m p e r a t u r e s the deviation of c from the random value is proportional to T . The same r e m a r k holds for the Bethe s . r . o . p a r a m e t e r

' ' A B • °A'=B ( ' ' A ' ^ B ) ^ 2W/kT

^ ^ ^ A B A B ^ A B ^ c (max)-c c c (max)-c c

We WÜ1 now derive the values of the activity coefficients in the quasi-chemical approach. As is well known the free energy is given by:

F = - k T l n Z (2.25)

a r e defined by /5F , , ,aF , ^^A = ( Ö N T ) N ^ , T ^ " ^ ^B - ( 5 Ï C ) N , , T A ID r> A F = N(c^u,^ + CgM,g), with

^ A 4 [ F ^ ' = B ( % ) T ]

('•'^>

^^B = N [ ^ - « A ( | f : ) T

_A

The activity coefficients y» and YT. of A and B in the solution are defined by:

^x^ = k T l n Y ^ c ^

Ug = kT In Yj^Cj^

(2.27)

Using formulae (2.25), (2.26) and (2.27) we fmd: r,„„ , Z , _ ^ A - " A B I

^.^ = k T L l n c ^ + 2l'^ ^ J

^A - ' ' A B

F r o m this equation the activity coefficient in the Q. C.-approximation follows v i z . :

= | ^ A - ^ I B | "/2 ^ I ^ A - ^ I B J ^/2 ^ ^PAAJ ^/2

'^A-'^AB ""1 ""^

In the same way

<é-)

/ 2

T h i s equation was f i r s t derived by Guttman [ 3 5 ] , who applied this r e s u l t to the AuCu-system where detaüed m e a s u r e m e n t s of the activity coefficients have been made by Oriani [ 3 6 ] . Guttman [14] applies his theory to these m e a s u r e -m e n t s and shows that although the total free energy and enthalpy of for-mation a r e fairly consistent with the observed s . r . o . [ 5 , 3 7 ] , the individual activity coefficients deviate r a t h e r much. This comparison is made in F i g . 2 . 2 , taken from Guttman [ 1 4 ] . The curves a r e calculated for W =-0.015 and - 0 . 0 2 5 e V / a t o m .

o - 0 . 4 -o.e

n,

o o -" -1.2 -1.6 - 2 . 0 O 0.2 0 . 4 076 O.B I.O ATOM FRACTION, Au OR Cu O

Fig. 2 . 2 Activity coefficients of copper and gold in their disordered solid solution at 427 C. Points from observations of Oriani [ 3 6 ] . Curves are calculated from eq. 2. 28 with W = - 0 . 015 eV/atom (upper curve) and - 0 . 025 eV/atom (lower curve). After [ l 4 ] .

This is one of the many instances, where the quasichemical theory p r o -vides a qualitative picture that is in agreement with experiments, but where quantitative r e s u l t s have not much value. F o r a detaüed discussion of the m e r i t s and shortcomings of the quasichemical theory we r e f e r the r e a d e r to the l i t e r -ature [ 3 8 - 4 0 ] .

2 . 4 . 2 O t h e r s t a t i s t i c a l t r e a t m e n t s of s . r . o .

The quasichemical theory discussed above is based on the following a p -proximations :

1) Only interactions between nearest-neighbours a r e considered.

-roundings.

3) The partition function is calculated assuming non-interference of p a i r s i . e . the probabUities for a p a i r of lattice sites to be occupied in any one of several possible ways is independent of the manner of occupation of all other lattice s i t e s .

The approximative c h a r a c t e r of the first assumption is clear from the d i s -cussion i n s e c t . 2. 3 where the long-range c h a r a c t e r of the quantum-mechanical-ly calculated interactions was mentioned. The second approximation implies that the relevant energies a r e independent of concentration. One can often o b -tain b e t t e r agreement with experiment by dropping assumption 2, working formally with concentration dependent p a i r energies [ 4 l ] . It is a p r i o r i not easy to see how bad the third approximation of non-interference of p a i r s i s .

Apart from the above approximations the quasi-chemical theory has the drawback that it makes no predictions about other than nearest-neighbour c o r r e l a t i o n s . X r a y e x p e r i m e n t s , however, have shown that in general the o c -cupational probabUities of even r a t h e r distant s i t e s a r e c o r r e l a t e d . T h e r e a r e two causes for this correlation over l a r g e r d i s t a n c e s . F i r s t l y the occupation of the nearest-neighbour s i t e s of a given atom influences the occupation of its next nearest-neighbour s i t e s e t c . , the o r d e r is "propagated" [ 4 2 ] . The other cause is the p r e s e n c e of interactions between atoms which a r e farther a p a r t . An exact solution of the statistical problem for a three-dimensional lattice, even with nearest-neighbour interactions only, does not yet e x i s t .

Approximate calculations of m o r e distant correlations have been made by ZemIke [ 4 l ] , who considered only first neighbour interactions and by Cowley [ 4 3 4 5 ] , who took also m o r e distant interactions into account. Cowley's d e r i -vations have been criticized by s e v e r a l authors [ 4 6 - 4 8 ] , but the fact r e m a i n s that his equations d e s c r i b e the physical situation remarkably well.

Hall and coworkers [ 4 6 , 4 7 ] claimed to have derived m o r e exact expressions but Clapp [49] has shown that this is not the c a s e . The problem, which a p -proximations a r e basic to Cowley's equations seem to have been solved by the recent work of Clapp and Moss [ 5 0 ] . T h e s e authors derived an exact expression for the correlation functions of an alloy of a r b i t r a r y composition and range of in-teraction. They showed that the Cowley and Z e m i k e theories a r e approximate solutions of their expression. Moreover they presented a third approximation which was already used independenüy by other authors [ 1 7 , 1 9 , 2 7 , 5 1 ] .

2.5 INFLUENCE OF S . R . O . ON PHYSICAL PROPERTIES

2 . 5 . 1 I n t r o d u c t i o n

Changes in s . r . o . effect in principle all physical p r o p e r t i e s , but in con-t r a s con-t con-to l . r . o . con-the effeccon-ts a r e generally nocon-t very speccon-tacular.

Among the most useful techniques for the investigation of s . r . o . a r e m e a s u r e m e n t s of the diffuse scattering of X r a y s , electrons or neutrons, s p e -cific heat and electrical resistivity. The information that can be obtained from studies of the diffuse scattering effects is most fundamental, since they allow a d i r e c t determination of the p a i r probabUities (or W a r r e n s . r . o . p a r a m e t e r s ) [52]. On the other hand the high sensitivity of resistivity m e a s u r e m e n t s is a g r e a t advantage, since the effect of s . r . o . is usually r a t h e r s m a l l . Mechanical p r o p e r t i e s a r e also influenced by s . r . o . F o r a review of these effects we r e f e r the r e a d e r to the article by Cohen and Fine [ 5 3 ] . Since we used resistivity m e a s u r e m e n t s for the study of the kinetics of s . r . o . , we wUl give a brief d i s -cussion of the influence of s . r . o . on e l e c t r i c a l r e s i s t i v i t y . The last section is devoted to the Zener relaxation-effect, which s e e m s to be closely related to s . r . o .

2 . 5 . 2 T h e i n f l u e n c e of s . r . o . o n e l e c t r i c a l r e s i s t i v i t y

A number of papers [ 1 8 , 5 4 - 5 8 ] have been published, dealing with the in-fluence of s . r . o . on electrical r e s i s t i v i t y . The basic approach of these p a p e r s is the s a m e , but various approximations a r e used for the wave functions of the e l e c t r o n s , scattering potential, e t c . The basic calculations a r e very s i m U a r to the theory of the diffuse s c a t t e r i n g of X - r a y s [ 5 2 ] .

Let V , (r) denote the potential in a cell occupied by an A-atom and V_(r) the potential in a B - c e l l . We now construct the average potential:

V(r) = c^V^(r) + CgVg(r)

w h e r e V(r) denotes the average potential in each cell of the l a t t i c e . The dif-ference between the actual potential in an A-cell and the average potential i s :

UA(^

= V ^ - ^(i^ = *=B

[^A(^

-

^B(^

]

and simüarly

Ugd) = Vg(r) - V(r) = - c^ [ V^(r) - Vg(r)]

Let us define the potential

U(r) = ^ U (r - r ),

_{^-' n TT— - n "}

where r denotes the position of a lattice point n and U (r ) is equal to U.

resp.U— if an A-atom resp. B-atom is at r . Let us assume that the Schrödinger

a —n

equation has already been solved for the aversie potential V(r) in the form of

Bloch functions:

' ' ' k " V W \ ( ^ exp(ik.r),

where u, (r) is a periodic function of r. We consider now the potential U(r) as a

perturbation. This is obviously a gross oversimplüation of the problem since

U(r) is not small with respect to V(r). However a correct approach would reqiüre

the solution of the Schrödinger equation for concentrated solid solutions, a

hitherto unsolved problem.

The matrix element of U(r) with respect to the states k and k' is given by:

U(k',k) =< k'l U(r)| k > = liji* U(r) if^dr

Thus the absolute square of the element U(k',k) is given by:

|u(k',k)|2 = < k ' l E U ( r - r )| k x k ' l s , U , (r-r ,) I k > (2.29)

I ^ ' 'I In n n'l | n' n' ^ n' |

I i I [ ^

= E 2 < k ' U ( r - r ) k >< k' U ,(r-r ,) k >

n n'

It is easüy shown that

TV n ' n'^— n '

< k ' |Uj^(r-r^)| k > = exp [ i ( k ' - k ) . r ^ ] < k' 1 U^(r)| k > = exp (JK.r^) U„(k',k)

w h e r e we have posed K = k' - k . Equation (2.29) becomes now:

| u ( k ' , k ) I 2 = 2 2^ exp [ i K . ( r ^ - r ^ , ) ] Uj^(k',k)U*,(k',k) n n

We w ü l now c h a i s e the o r d e r of the summation by taking together in the double sum all t e r m s belonging to a common vector r = r - r , (m = n' - n):

"^ ^ - m -n - n ' ^ ' | u ( k ' , k ) l 2 = E 2 U (k',k) U* (k',k) exp ( i K . r )

I ^ • ' I m n n^ ' n+va. ^ ' ' '^ ^ ^m'

At this stage we introduce the assumption that the crystal is homogeneous, that is to say we assume that the sum over n may be replaced by N t i m e s its a v e r -age value:

| u ( k ' , k ) | ^ = NE^ exp (iK.r^^) V^{k\k) U*^j^(k',k) (2.30)

In the following table we give the value of the fimction U U for dif-ferent types of p a i r s : c..(m) 1] U n ( ' ^ ' ' ' ^ ) \ . m ( ' ^ ' ' ' ^ ) AA AB BA BB ^ A P A A ( ™ ) ' ' A P A B ( ™ ) (m) (m) ' = B P B A ( " ^ ) ' ^ B P B B c | [ V ^ ( k ' , k ) - V g ( k ' . k ) ] 2 - c ^ C g [ V ^ ( k ' , k ) - V g ( k ' , k ) ] 2 -VA'^'^A('''''')"^B('''''')^^ c2 [ V ^ ( k ' , k ) - V g ( k ' , k ) ] 2

Expressing the p a i r probabUities p . _ ( m ) in o/ ( e q s . 2.8) we can e a s ü y

4( A J J I I I

compute the average of U U :

Substitution of this result in eq. (2.30) gives:

I U(k',k) I 2 = Nc^Cg [v^(k',k) - Vg(k',k) ]2 ^ a^ exp (iK. r^^^)

In the fully disordered state all a vanish except o- , which is unity. In

this case the extra resistivity p is proportional to:

Po "

V B [ V ' ' * " )

- VB(k'.k)]^ (2.31)

This result was already obtained by Nordheim [59]. Equation (2.31) shows that

the resistivity depends parabolically on the concentration. This is indeed

ob-served for alloys of the normal metals (Fig. 2.3). For alloys of the transition

elements large deviations occur [62].

10-

8-

te-P ((incm) 4 2 -Ag ^ = A u ^ ^ Fig. 2.3 Resistivity at 0°C as a function of concentration for AgAu-alloys [62].

Hi the alloy exhibits s. r . o . at least some of the a (m 7^ 0) do not vanish.

This results in the following difficulty: Even if we assume that the

matrix-I matrix-I 2

elements depend only on the magnitude of K = k - k ' , U(k',k) wUl depend

on the direction of K with respect to the crystal-axes as a consequence of the

factor exp (iK.r ). A simüar problem is encountered in the theory of Umklapp

processes [60] .Umklapp-processes give rise to complicated geometrical

lems in the calculation of electrical resistivity.We wUlnotdiscuss these

prob-lems here but refer the reader to the papers by Gibson [56] and Asch and Hall

[57].

F o r numerical values of the resistivity changes a knowledge of the Warren s . r . o . p a r a m e t e r s is r e q u i r e d . Two possibUities a r i s e , either one u s e s e x perimental values of a , which a r e known only for a very limited number of a l -l o y s , o r one u s e s a theoretica-l approximation for the s . r . o . p a r a m e t e r s . The f i r s t approach was chosen by Asch and Hall [57] and Gibson [ 5 6 ] , the l a t t e r by Krivoglaz and Matysina [58] and Beal [ 5 5 ] .

Asch and Hall [ 5 6 ] , using r a t h e r sophisticated models, obtained an e x -p r e s s i o n of the form:

P = P f l + 5n » z H 1 (2.32)

^s ^o L m?^0 m m J ^ '

w h e r e p is proportional to c . c .

H is a complicated function of the s e v e r a l p a r a m e t e r s used to describe the s c a t t e r i n g potential, wave functions, e t c . and z is the number of atoms in the m - t h shell.

Krivoglaz and Matysina [58] considering only nearestneighbour i n t e r -actions found:

PS = 4 . ' = A ' = B - ^ ( ' = A B - V B ) ]

(2-33)

F o r high t e m p e r a t u r e s eq. (2.23) is valid:

/i 2W,

^ A B " V B ( l - V B k T )

Substituting this expression in formula (2.33) we find:

r zwi

p = p 1 + 2c . c_, i-=- , (p = K c , c_) '^s "^o L A B kT J ' ^*^o A B'

Beal [ 5 5 ] , using the free electron approximation and the B o m approxima-tion, finds:

where A is a complicated function of the wave vector at the F e r m i s u r f a c e kp, and the interatomic distance a. It is easy to show that, a p a r t from the different functions A and H, formula (2.34) is a special case of (2.32) for n e a r e s t n e i g h -bour correlations only and high t e m p e r a t u r e s . At high t e m p e r a t u r e s :

- 1 = 1 BA 2W

'^A'^B kT (2.35)

The numerical r e s u l t s of the various t r e a t m e n t s depend largely on the kind of scattering potential and wave functions used. Most theories predict, in ^ r e e -ment with e x p e r i m e n t s , an increase in resistivity with increasing s . r . o . for alloys like CuAu, where the valency of the constituant atoms is the s a m e . If the valency of the components is different one observes in general a d e c r e a s e in resistivity with increasing s . r . o . ( e . g . CuZn, CuAl).

Beal has compared h e r r e s u l t s with the m e a s u r e m e n t s of Korevaar [ 6 l ] , who determined the quenchedin r e s i s t i v i t y as a function of quenching t e m p e r a -t u r e for a number of gold-rich AuCu-alloys (Fig. 2 . 4 ) . Béal's -theory c o r r e c -t l y predicts both the t e m p e r a t u r e dependence (as T ) and the concentration d e -pendence (as c . c_) of the o r d e r dependent r e s i s t i v i t y .

A p .

100 2CX) 3 0 0 4 0 0 5 0 0 °C

Fig. 2 . 4 Resistivity increase due to s . r . o . as a function of annealing temperature and copper content (at.'Sl) after quenching from 450°C [ 6 l ] .

2 . 5 . 3 T h e Z e n e r - e f f e c t

In a following chapter we wUl compare our m e a s u r e m e n t s of the kinetics of s . r . o . with m e a s u r e m e n t s of the Zener-effect. F o r this r e a s o n we discuss briefly this effect and its relation with s . r . o .

In a perfectly elastic m a t e r i a l the relation between the s t r e s s S and s t r a i n e is independent of t i m e . S and e a r e related by:

S = Me

where M is the 4th rank t e n s o r of the elastic constants. We wUl limit the d i s -cussion to the case of an unaxial s t r e s s in an isotropic solid, in which case only one constant M r e m a i n s .

• t i m e

Fig. 2.5 Strain vs time diagram of an elastic after-effect experiment.

A m o r e general relationship between s t r e s s and s t r a i n is given by: S + T ^ S = M ^ ( e +T^e),

where T., and T_ a r e the relaxation t i m e s of s t r e s s and s t r a i n respectively. If the above equation is obeyed we deal with a "standard linear solid". If we apply at time t., (Fig. 2.5) a constant s t r e s s S , the solution of equation (2.36) is given by:

- o , o , 2 r r

(2.36)

The s t r a i n r i s e s instantaneously to the value e , thereafter a gradual i n c r e a s e in s t r a i n is observed untü e r e a c h e s its final value (Fig. 2 . 5 ) . The s a m e b e -haviour is observed ü we unload the specimen at t = t - . The variation of the s t r e s s observed after applying a constant s t r a i n shows a s i m ü a r p i c t u r e . The proportionality between s t r e s s and s t r a i n after complete relaxation has taken place (t » T 2 o r T £/ is described by the modulus M , which is therefore called the relaxed modiüus. On the other hand after an instantaneous variation of the s t r e s s the s t r a i n is inversely proportional to: M = (T A )M , M is called the unrelaxed modulus. The relaxation strength 6x/r is defined by:

M

M - M u r

It is easy to show that for a periodically varying s t r e s s the relaxation causes a phase lag cp of the s t r a i n with respect to the s t r e s s . The phase shifts depends of course on T and T .

J- di

l-HU T T - l + U ) T

The maximum value of cp is obtained for (irr = 1 and is given by:

. „, , AT * M

tg9(max) = ^ = g

-It is easy to show [63] that Q , the inverse quality factor, is equal to sin 6. Since usually 6 x^ is s m a l l :

Q - ' . ' #

At very low frequencies (lu « — ) the relaxation is established sufficiently fast,

'^ 1 so one wUl m e a s u r e the relaxed modulus. At very high frequencies (u) » —) the

relaxation cannot follow the rapid variations in s t r e s s and a m e a s u r e m e n t of the modulus will yield the value M . So there a r e in principle t h r e e ways to o b -s e r v e a relaxation phenomenon viz:

1) Observation of the elastic c r e e p curve (elastic after-effect) ( F i g . 2.5). 2) Measurements of the damping a s a function of frequency around u) x = 1. 3) Measurements of the modulus as a function of frequency. In p r a c t i c e how-ever one makes use of the fact that the relaxation time T is strongly dependent on t e m p e r a t u r e . In the case of the Zener-effect this dependence is given by:

^ = ^ o ^ ^ P ( k | ) (2-38)

-14

T is a constant of the o r d e r of 10 sec and Q is an activation energy, a p -proximately equal to the activation energy for self dUf us ion. So instead of varying the frequency one m e a s u r e s the damping or modulus at constant frequency as a fimction of t e m p e r a t u r e [ 6 4 ] .

There a r e quite a number of relaxation phenomena. Many of them a r e

related to imperfections in the crystal s t r u c t u r e . It has been shown however that the Zener-effect, which occurs only in substitutional solid solutions,is in-dependent of these imperfections and is related to the r e a r r a n g e m e n t of atoms under the influence of an applied s t r e s s . It is s t ü l uncertain which kind of r e -distribution o c c u r s . F r o m a comparison with diffusion data it can be shown that the number of jumps p e r atom is very smaU, of the o r d e r of one.

Zener [65] originaUy proposed the following theory (pair reorientation model). He supposed that (isolated) p a i r s of solute atoms reorientate t h e m -selves under the influence of an applied s t r e s s . Several objections have been raised against this theory [ 6 6 - 6 9 ] , the most important being, that the isolated pair concept has no meaning for concentrated solid solutions. The theory of LeClaire and Lomer [66] r e l a t e s the Zener-effect to a change in s . r . o . due to an applied s t r e s s (hence the other name of the Zener-effect: s t r e s s - induced o r d e r i i ^ .

Both theories make incorrect predictions about the orientation dependence of the magnitude of the Zener-effect for b . c . c . alloys [ 6 9 ] . Moreover for f . c . c . alloys the predicted anisotropy is only qualitatively c o r r e c t . These e r r o r s a r e probably due to the fact that both theories consider only n e a r e s t neighbour interactions. Possibly the discrepancy between theory and e x p e r i -ment wUl disappear if second nearest-neighbour interactions a r e also taken into account. F o r both b . c . c . and f . c . c . s t r u c t u r e s the n e x t - n e a r e s t neighbours ly in the (100) d i r e c t i o n s , in which directions 6„ is maximum [ 6 9 ] .

C H A P T E R I I I

VACANCIES IN ALLOYS WITH SHORT-RANGE ORDER

3 . 1 INTRODUCTION

In this chapter we wUl discuss the formation and migration of vacancies in alloys which exhibit s h o r t - r a n g e o r d e r . We w ü l f i r s t show how the vacancy concentration in a binary alloy c a n b e calculated by m e a n s of the q u a s i c h e m i -cal theory (sect. 3.2).

After discussing the migration of vacancies in binary alloys (sect. 3.3), we derive a theory of the kinetics of s h o r t - r a n g e o r d e r (sect. 3.4). A s i m ü a r t r e a t m e n t has already been given by Kidin and Shtremel' [ 3 ] , but by dropping some of t h e i r hypotheses we derive an expression for the relaxation time of s . r . o . in t e r m s of the t r a c e r diffusion coefficients of the components of the alloy.

3.2 FORMATION ENERGY OF VACANCIES IN BINARY ALLOYS

In a pure element the energy of a vacancy does not depend on its position in the lattice (except for, of c o u r s e , the sites at or near the surface of the c r y s t a l ) . In a binary alloy, however, this is no longer t r u e , the energy of a vacancy at a certain lattice site wUl depend on the environment of that s i t e . In general there a r e (z+1) different configurations. We will follow now the t r e a t -ment of Schaplnk [ 2] who calculated the vacancy concentration in binary alloys using the quasi-chemical approximation.

It is well known that absolute values of the formation energy of vacancies obtained from the quasichemical theory deviate r a t h e r much from the e x

perimental r e s u l t s . However, the r e s u l t s obtained a r e usually qualitatively c o r r e c t , so the theory may be useful to explain e . g . the concentration depen-dence .

3 . 2 . 1 R a n d o m d i s t r i b u t i o n of a t o m s

Let us consider an alloy consisting of N . A-atoms and N B - a t o m s , A B which a r e randomly distributed among the avaUable lattice sites (N'). When we introduce N-. vacancies the number of lattice sites i n c r e a s e s from N = N , + N-,

V A B to N' = N , + N_ + N , , . We introduce the m a s s - f r a c t i o n s _{A B V}

Nl

C j = ^ , (i = A , B , V ) .

which r e m a i n constant for the A - and B - a t o m s . The average energy of an atom p a i r in the vacancy-free alloy i s :

^ = ' = A V A A ^ 2 c ^ C 3 V ^ 3 . c | v 3 g (3.1)

If vacancies a r e introduced the probabüity to find an A-atom next to a given atom is no l o i t e r equal to

Stirling's approximation:

atom is no l o i t e r equal to c but c / ( I + c..). The free energy b e c o m e s , using A A V

^ = N T Ï T I ^ V - kT [ ( N + N^) ln(N + N^) - N ^ In N ^ + N ^ In N ^ - N ^ l n N ^

^ (3.2)

The equilibrium concentration of vacancies is obtained by minimizing the free energy with r e s p e c t to N „ . This procedure yields:

"'' = in ( ^ ^ ) 2kT(l+c.^)2 1 + Cy

Neglecting c.^ with r e s p e c t to 1 gives:

The formation energy of vacancies is in this approximation equal to:

Qf = - | ( ' = > A A ^ 2 C A C B V A B ^ <=|Vg3) (3.4)

3 . 2 . 2 V a c a n c y c o n c e n t r a t i o n i n a l l o y s w i t h s . r . o .

In the t r e a t m e n t given above we assumed that the atoms w e r e randomly distributed. However, this in general not the c a s e , but the r e s u l t s obtained in s e c t . 3 . 2 . 1 can be considered as the zeroth approximation.

Using the procedure of Fowler and Guggenheim [ 3 2 ] , Schaplnk [ 2 ] d i s -cussed the problem of the vacancy concentration in a binary alloy. He extended t h e i r t r e a t m e n t to a t e r n a r y alloy with the vacancies as the third component. He derived the following equations:

c2 V I n ^ ^l = - j ^ (3.5) AA" W "^^ , '^BV _ ^ B B ,„ „, ^'^c c kT~ (3-^) c c V , AV BV AB ,„ „, AB VV ^^

^

''W

=

^^"^ ^Y

"

V^"

(1 "" V *

h~^^

''V (3-

8)

where the c..,, have been defined in the following way (2.3):

^TT 2N

'=ij = N ? ( i ^ J ) ^ ^ ° n = ^ (3-9)

N . - i s the number of p a i r s of type U . The c^^'s satisfy the consistency relations:

J

The vacancy concentration is small so we may use, as a first approximation,

the solution for the vacancy-free case (sect. 2.4). Note that eq.2.21 stUl holds

if vacancies are present, as can be easüy shown from eqs. (3.5)-(3.7):

^ c - % - = - W (3-10)

''AA-'^BB ^^

The assumption that we may use the solution of eq. (3.10), even If vacancies

are present, corresponds to the assumption, that the fractions of AA, BB and

AB-pairs are not altered, when vacancies are introduced. We can now determine

c „ from (3.5) and (3.6), making use of the consistency relation c . ^ -i- c_„ = c „

(neglecting c.^^).

V. . , V

ƒ è AA A B B \ z ,„ , , ,

y = [o^l exp 2 i ^ - C g | exp ^ ^ j (3.11)

The average formation energy is derived from the equation:

din c,,

Q f = - d ( I 7 k ¥ ) (3-12)

which yields for the equiatomic composition:

c A f - ^ . VAA^^P(^AA/2kT) - V33exp(V33/2kT)

^f 2Lw + l exp(V^^/2kT) + exp(Vgg/2kT) J ^'^'^'^'

Since W is smaller than either V . or V„_ the main contribution

comes from the second term of eq.(3.13). Eq.(3.13) shows that the formation

energy is temperature dependent. This temperature dependence is larger the

larger the düference between V land V „I .The concentration dependence

of Qf is shown in Fig. 3.1 for three arbitrary values of V. .fY .-r, and V„^.If

I AA A B B B

V. . = V__, Q, decreases with increasing temperature irrespective of the sign

of W. This is evident since at low temperatures due to the increase in s . r . o .

or clustering the atoms are on the average at a lower energy level and

there-fore more energy is required to form a vacancy.

In connection with results to be derived in sect. (3.4) we calculate here

e A Y

A-1.5 Qf CeV) 1.0 0.8

^N

\

\

V ^ ^ = - 0 . 2 2 eV V ^ B = - 0 . 3 2 e V V g B = - 0 . 3 8 «V ^ V ^ x 1 5 0 0 'K 5 0 0 ' K ^ 1 O B 20 40 60 80 100 a f / . A A

Fig. 3 . 1 Average formation energy Q , for a b . c . c . alloy as a function of concentration for two temperatures.

atom next to a vacancy. Thus p „ . is a m e a s u r e of the clustering of A-type atoms aroimd a vacancy. The expression for p „ . i s obtained by solving (3.5) and (3.6) for c . „ and c.^.^. respectively and eliminating the l a t t e r with the aid of the consistency relation:

'^V '^AV "^ '^BV PvA = • "AV 1/2 ^AA ^ ^ P ( V A A / 2 ' ^ ' ^ )

VA oy c^/^e.v(y^/2^)-c^^^^eKpiV^^/2kT)

(3.14) 2 2 F o r the case of a random distribution of A - and B - a t o m s e . . = c . andc_,_ = c_.,

so _{" A e x p ( V ^ / 2 k T )} PvA = c ^ e x p ( V ^ / 2 k T ) + c ^ exp(Vgg/2kT) °''' ^VA V - V ^ / B B '^AA, ''A

"^

''B ^ ^ P ( ^

)

If V

AAl V__, the vacancies a r e preferentially surrounded by A-atoms BBl ( p ^ > c . ) . This result holds equally weU for the case of a nonrandom d i s -tribution of A - and B - a t o m s . The concentration dependence of p.^.. is shown in

F i g . 3.2 for two s e t s of energies V . . ,V and V . 1 . O B V A B -0.32 -0.32 V A A -0.30 - 0 2 2 VBBI -0.3O - a 3 8 2 0 4 0 6 0 8 0 10O a t ' / o A A

Fig. 3.2 The probability p y ^ as a function of concentration for two sets of energy values (W = constant) and two temperatures.

3.3 MIGRATION OF VACANCIES IN ALLOYS

3 . 3 . 1 P r o b a b i l i t y d i s t r i b u t i o n of t h e m i g r a t i o n e n e r g y

The migration energy of a p a r t i c u l a r atom in an alloy depends on its e n v i ronment. We wUl assume in the following calculations that the migration e n e r -gy of a p a r t i c u l a r atom depends only on its own environment (not of the environ-ment of the vacancy). The migration energy is given by the energy difference V„-V, (Fig. 3.3), where V., is the energy of an atom next to a vacancy a n d V - i s the energy of the atom in the saddle point position. We assume for the moment that V„ is constant for a given type of atom, this condition wUl be relaxed in a fol-lowing section (sect. 3 . 3 . 3 . ) . V.. is in general dUferent for A - and B - a t o m s . If the A-atom has n neighbouring B - a t o m s

V ^ = ( z - l - n ) V ^ ^ + nV

AB (3.15)

1 2 3

Fig. 3 . 3 Potential barrier for motion.

Qm^^) = V l - V^ = V ^ [(z-1) V^^ . n(V^3 - V^^) ]

Since we have assumed that V„ is independent of concentration we may write:

Q > ) = [K - (^-1) ^AA] - "(^AB - ^AA) = Qm - -""A (3-1^)

The equivalent expression for B-atoms is:

Qm(^) = [ v 2 - (^-1> V B B ] - "(^AB - ^ B B ) = ^ m " ' ' ^ B (3-17)

We have defined:

UA = V A B - V A A ^ ^

U B = V A B - V B B

(3.18)

A

B

whUe Q and Q are the migration energies in pure A and B resp.

Clearly:

W = V A B 4 ( V A A ^ V g B ) = i ( " A ^ U B )

-(W = ordering energy)

The probabüity that a given Aatom, next to a vacancy, is surrounded by n B -atoms i s

«ÏAC») = ( ' n ' ) P A B " (1 - P A B ) " ' • " (^-i^)

F r o m e q s . (3.16, 3.17 and 3.19) we see that there exist in an alloy two (bino-mial) distributions of activation e n e r g i e s , one for A - and one for B - a t o m s . An example of such a distribution of migration energies is given in F i g . 3 . 4 . for one kind of atoms in a b . c . c . s t r u c t u r e (z = 8).

This spectrum of activation energies may be responsable for the peak broadening of the Zenerdamping. Nowick and B e r r y [ 7 0 , 7 1 ] assumed that the logarithm of the relaxation time is normally distributed with halfwidth g. They show that this distribution corresponds to a normal distribution of activation energies Q with halfwidth P„ which is related to P by:

» = ^

Nowick and B e r r y explain the existence of a distribution of Qvalues by a s -suming that local variations in composition occur. The value of Q is concentra-tion dependent and from this dependence the number of atoms in a group with different concentration can be estimated. It appears that this number is as 350 a t o m s .

In our opinion the explanation of the existence of the energy spectrum given above i s m o r e straightforward, and it is well known that the computed peak broadening is not very sensitive to the kind of probabüity distribution assumed.

V"'

05-1 0 4 (n) 0 3 0.2 0.1 N ' A B = " I \ y ^ A ^ ~ N i

/ A \ \

S . _ 0 * - ^ 0 1 2 3 4 5 6 7 n

Fig. 3.4 Probability distribution of the migration energy of an A-atom for different values of p

3.2.2. A v e r a g e m o b i l i t y of a t o m s in b . c . c . s t r u c t u r e s

Thus far we have discussed the migration energy of an atom in a particular

configuration. We wül now calculate the mobUity of a particular type of atom,

from which an average migration energy can be calculated. The mobUity of

A-atoms in the crystal is determined by the following factors:

(i) The probabüity p..y. that the atom has a vacancy as one of its nearest

neighbours:

PvA

PAV = V

^ ^^-^'^

(ii) The probabüity qA^) that the atom is surroimded by n B-atoms:

^AC») = (n) ( P A B ) " ( 1 - P A B ) ' "

(iii) The jump rate of A-atoms with given environment n:

Q'^(n) Q^

'^AC') " V ^ ^ P ( - T ^ ) = V ^ P ( - l ^ ) ( V ' (3-21)

where we have defined: u . = exp(U./kT)

Ug = exp(Ug/kT)

and V is a frequency factor, approximately equal to the Debye frequency.

A

The average number of jumps of A-atoms next to a vacancy per unit of time

is thus given by:

Q^d»)

^A = g " A V " ) ^ ^ P ( - T Ö ^ ) =

Q 7

= v ^ e x p ( - ^ ^2^ (I) (PAfiV'^d-PAB)^""' °^

Q ^

^ A v ^ e x p ( - ^ ) { l . p ^ g ( u ^ - l ) } ^ (3.22)

F o r B-atoms one derives e a s ü y : .B Q

^ B = v ^ e x p ( - ^ ) { l + P g ^ ( U 3 - l ) } 7 (3.23)

We w ü l now consider two limiting c a s e s for the average mobUity M.

(1) High t e m p e r a t u r e s , U . « kT — I A I

In this case we may replace the l a s t factor of M . by: f 17 ' ^ P A B ^ A

{I^PAB("A-I)}

- - P ( - ^ - ^ )

u. u.

A A U

since u . = exp -r^ sj 1 + ^j-=- and (1 + x) ss exp nx (x « 1).

So M . becomes: A

Q i - 7P,^U

^ A f ^m. '^AB"A>, ,„ „ . , ^A ^^P ^ - kT J (^• 24)

This equation shows that at high t e m p e r a t u r e s the average migration e n e r -gy v a r i e s linearly with p . _

< Q m > = < - ^ P A B U A

(2) Low t e m p e r a t u r e s , | u . I » kT

In this case we must distinguish between U > 0 and U . < 0. (i) U > 0

I I 7 Since U . » kT we may neglect the other t e r m s in the factor ( ) , thus

I A '

^ A = ^ A P A B ^ ' ^

C Qm -

^^A^

V" kT J (3.25)

(ii) U ^ < O

7 7 The factor ( ) may now be written as (1 - PAT,) > since u . can be

neg-lected.

Q

^ A = " A P A A « ^ ( - T S - ) (3.26)

Both r e s u l t s can be s u m m a r i z e d by stating, that for U. « kT the lowest

I A I

value of the distribution of activation energies determines the mobUity M . , A B an obvious r e s u l t . The concentration dependence of (Q ) and <Q ) is

'^ m ^ m

shown in F i g . 3 . 5 . for two s e t s of p a i r energies V , V . ^ and V .

0 20 40 60 8 0 100

A a t % B B Fig. 3 . 5 Average migration energies of A - and B-atoms as a function of concentration.

3 . 3 . 3 A v e r a g e m o b i l i t y of a t o m s i n f . c . c . s t r u c t u r e s

The formulae (3.22) and (3.23) derived in s e c t . 3 . 3 . 2 for b . c . c . s t r u c -t u r e s can be derived in -the s a m e way for f . c . c . s -t r u c -t u r e s :

It is possible to

<

r

^ A = ^ - P ( - k T ) L l ^ P A B ( ' ^ A

^ = ^ B - P ( - k T ) [ ^ - ^ P B A ( ^

take account of the fact, that in f.c. and the atom under consideration have four common probabüity that

given by:

The probabüity,

an AV-pair has m common nearest

*1AVB("') = (m> ( P A B P V B > " ' ( 1 " P A B P

that n of the other neighbours of the A

W " ) = (m)PAB''(l-PAB>''

-1)P

-Dp

(3.27)

c.structures the vacancy nearest ne neighbours 4-m V B ' -atom are -n ighbours. The , of tyi)e B is (3.28) B-atoms, is: (3.29)

Thus the average number of jumps per sec. of A-atoms next to a vacancy with given environment (m,n) is given by:

Q ^ 4 7 (m+n)U ^ A = " A ^ ^ P ( - k T ) m = 0 n=0 ^AVB(°^>'1AB("> ^"^ HO^

° ' A

^A ^A ^'^P ( - k T ) ( l ^ PABPVB(^A - 1 ) } H 1 -^ PABC^A - 1)} ' (3-30)

Formula (3.30) shows that the occurence of common nearest-neighbours in-troduces the extra factor:

^ " P A B P V B ( " A - 1 ) } '

Q ^

M^ = v^ exp ( ^ ) {l .

PBAPVA("B

" ^^l^i^ "

PBA("B

" '''Y (^'^D

It is possible to generalize the above equations by introducing a concentration dependence of V ^ and V , which were considered thus far as constants. Let us assume that the energy V„ can be written a s a sum of p a i r interaction energies of the jumping atom in the saddle point position. Let us further assume that only interactions with the common n e a r e s t neighbours a r e signüicant. We write therefore:

V^(m) = m S ^ + (4 - m ) S ^ (3.32)

and s i m ü a r l y :

V^(m) = m S g ^ + (4 - m ) S g 3 (3.33)

It is not n e c e s s a r y to assume that the e n e r g i e s S . T, andS_ . a r e the s a m e . U s i n g the expressions (3.32) and (3.33) we can calculate the migration energy for given m and n: Q ^ ( m , n) = V^(m) - V^(m, n) = m S ^ ^ + ( 4 - m ) S ^ - (m+n) V ^ g - { l l - (m+n)} V AA Q ^ ( m , n ) = ( 4 S ^ - I I V ^ ^ ) . MS^^ - S ^ ) - (m+n) ( V ^ ^ - V^^> = = Q ^ + n i S ^ - ( m + n ) U ^ , (3.34) w h e r e we have written: ^A ' ^ A B " ^AA Simüarly: Q^ (m,n) = Q ^ + mS_ - (m+n) U_ (3.35) 47

^i<^ % = ^ B A - ^BB

The r a t e of exchange of AV-pairs with environment (m,n) is .A Q •.r , \ I m . / , m m+n , _ _ _ . M^(m, n) = v ^ exp (- ^ ^ ) (s^) u ^ ( 3 . 36) ^A with s ^ = e x p ( j ^ ) .

A calculation of the average jump r a t e w ü l r e s u l t in: .A

P^A Q

=A

^A = ^A V ^ ^^P(-kr) {l "

PAB(^A

- l ) } n i ^

PABPVB("A«A

" ^

B (3.37)

^B =

"B

^ v ^ ^'^P(- W-){i ^

P B A ( ^

- i)}Hi ^

PBAPVA(^^B

- 1)^ *

B

F o r m u l a e (3.37) show that the introduction of a concentration dependence in this special form affects only the factor | | .

It is well known [72] that the selfdiffusion constants D . and D of A - and B - a t o m s r e s p . a r e related to F . and T r e s p ;

I 3 A = - r r A > ° B = # r 3 (3.38)

3 . 3 . 4 A v e r a g e m o b i l i t y of v a c a n c i e s

B e r r y [73] determined the annealing time of very small supersaturations of vacancies by means of the "delayed c r e e p " method. In alloys the movement of a vacancy is in general not random. The fastest diffusing component wUl jump m o s t frequently into the vacancy. So the vacancy moves (in nondUute alloys) m o r e o r l e s s through a network of the fastest diffusing component.

We can define the average mobility of the vacancies by: