THEORETICAL STUDY OF ANTIPHASE BOUNDARIES IN FCC ALLOYS

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THEORETICAL STUDY OF ANTIPHASE BOUNDARIES IN FCC ALLOYS

A. Finel, V. Mazauric, F. Ducastelle

To cite this version:

A. Finel, V. Mazauric, F. Ducastelle. THEORETICAL STUDY OF ANTIPHASE BOUND- ARIES IN FCC ALLOYS. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-139-C1-143.

�10.1051/jphyscol:1990120�. �jpa-00230278�

THEORETICAL STUDY OF ANTIPHASE BOUNDARIES IN FCC ALLOYS

A. FINEL, V. MAZAURIC and F. DUCASTELLE

Office National d l E t u d e s et d e Recherches Adrospatiales (ONERA), BP.

7 2 , F-92322 Chdtillon Cedex, France

RCsum6 - Nous avons CtudiC le comportement des antiphases non conservatives (100) dans les alliages ordonnks L12 au voisinage de la transition du premier ordre L1?-dksordre. Des calculs CVM tr6s prkcis dans l'approximation du t6trakdre montrent l'existence d'un phenomkne de mouillage avec divergences logarithrniques, ceci par l'intermuiaire d'une s6rie infinie de transitions du premier ordre.

Abstract

-

We have investigated the behaviour of non-conservative (100) antiphase boundaries in L12- ordered alloys, close to the congruent fust order order-disorder transition. Very accurate inhomogeneous CVM calculations in the tetrahedron approximation show that wetting does occur with logarithmic divergences through an infinite series of layering transitions.

1- INTRODUCTION

From a very general point of view, an order- disorder transition is associated with a symmetry breaking: the ordered phase has less symmetry than the disordered one. Hence, when the ordering pro- cess starts in different locations of the disordered lattice, different variants (or domains) of the same ordered phase can appear and coexist. An antiphase boundary (APB) forms wherever two such do- mains contact.

Whether the ordering transition is first or second order is a very important factor for the quali- tative behaviour of the APB. At the critical tempera- ture of a second order transition, the ordered phase become identical to the disordered one and, conse- quently, the APB disappears.

On the other hand, for a first order transition, the ordered domains retain always a certain degree of long-range order and the APB's exist up to the tran- sition temperature Tc. In that case, the free energy of the APB ( i.e. the excess free energy with respect to a situation with no APB's) does not vanish at Tc

.

However, it may happen that, precisely at Tc and in order to minimize the excess free energy, the APB takes a particular configuration: a large layer of dis- ordered phase develops in-between the two ordered domains. In other words, the APB splits into two order-disorder interfaces : this is the complete wet- ting of the APB by the disordered phase.

Many experimental and theoretical studies have been devoted to the wetting phenomena [l]. We will consider here the case of a binary alloy, on the FCC lattice, which orders in the L12 structure. Wet- ting in that case has recently been observed in the system C O ~ $ ~ ~ ~ [2] , and had in fact been predicted before by Kikuchi and Cahn using inhomogeneous cluster variation method (CVM) calculations [3]. The

accuracy of their calculation however was not suffi- cient for studying in detail what happens close to T,.

We present here a detailed and precise study of a non-conservative APB in L12 as a function of temperature. The numerical method used is the CVM in the tetrahedron approximation. But, first, we pre- sent briefly a simple argument which gives the gen- eral behaviour of an APB in case of wetting.

2. OUALITATIVE DISCUSSION

When the ordered phase can be discribed by a one-dimensional order parameter, there is a simple qualitative argument which suggests that, as the temperature increases up to T,, the width R of the APB diverges as -Log(%- T).

Consider the simple case where the ordered phase has only two variants. Moreover, suppose that these variants are characterized by two opposite val-, ues of a scalar order parameter (7) , ^-q.) In that case and for a first order transition, the free energy, as a function of 7) and for T just below T,, has the quali- tative behaviour presented in Fig.1. To first order, the difference 6F of free energy between the disor- dered phase (q=O) and the ordered one (1120) (see Fig.1) is proportional to Tc- T

.

Fig. l Free energy F(q) for afirst order transition

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990120

COLLOQUE DE PHYSIQUE

Consider now an inhomogeneous situation with domain 1 (q < 0 ) at z = - 00 and domain 2 (11

> 0 ) at z =

+

(Fig. 2a). Obviously, the order parameter q , as a function of z, has to go through zero. Suppose (this is a qualitative discussion !) that the APB is sufficiently large and, consequently, that q stays in the neighborhood of zero within a finite region. Then the order parameter profile should look as in Fig.2a and the excess free energy as in Fig.2b

where AS =

-

aAF/aT is the APB entropy.

Therefore, this crude approach predicts that, with a scalar order parameter, complete wetting does occur, and the width R and the APB entropy diverge logarithmically as the temperature T increases up to

l-

l c -

We consider in the following a situation where the ordered phase is discribed by a three di- mensional order parameter. We would like to point out that, in such a case, wetting as described above may not occur or may be modified because of the three dimensional character of the order parameteror because of lattice effects neglected in the above con- tinuous approximation. In particular layering transi- tions corresponding to discontinuous variations of the concentration Gofiles as a function of tempera- ture may well occur. This is precisely what has been found in this study, as we show below.

3. STUDY OF THE APB IN L13 : CVM RESULTS We now consider the case of APB's in the FCC lattice. To describe ordering effects, we use the Ising model with positive first neighbor interactions :

- W

Fig.2 Exemple of inhomogeneous system

a

: ( a ) order pa- rameter and (b) excessfree energy profiles.

The APB free energy

AF,

given by

+m

AF

= F(z) dz ,

-00

can be approximately analysed as the sum of three .terms :

a) a constant term, equal to twice the energy of an order-disorder interface,

b) an attractive term, due to the excess free en- ergy in the central region, and proportional to R (Tc- T), where R is the size of disordered region (see Fig. 2b),

c) and finally a repulsive term due to the overlap of the two interface tails, proportional to exp(-a 2)'.

In short, the APB free energy, AF(2) writes : AF(R)

-

^c

⁺

bR (Tc- T)

+

^exp(-a R )

.

The competition between the last two terms leads to an equilibrium width R given by

~ A F ( R ) I ~ R =

o

Hence, the equilibrium is characterized by : 2

- ^-

Log&-T)

AF

-

^- (T,- T) Log CT,- T)

+

Ct AS

-

-Log (Tc- T)

'1t is easy to show, in a continuous Landau theory, that the tails of the excess energy profile of a single order-disorder interface is exponentially decaying.

where the prime means that the sum is over first neighbor pairs. The spin-like variables on take values +l depending on the type A or B of the atom at site n

.

The corresponding phase diagram presents three ordered phases at stoichiomemes AB, A3B and AB3

.

Its precise shape close to the AB- AB3 (or A3B) boundary is still controversial [4], but this is not the case near the congruent point of the A3B structure.

This is precisely the region we will consider in the following. More precisely, we fix the field to h = 7.5 J

.

The corresponding critical temperature is Tc/J = 1.9426375061, which is very close to the congruent point. At zero temperature, the A3B equi- librium state is infinitely degenerate but, at finite temperature, this degeneracy is lifted and the stablest structure is the simplest A3B one, namely the L12 phase.

In L12, one simple cubic sublattice of the FCC structure is predominantly occupied by B atoms, and the three others by A atoms. As there is a choice for the sublattice occupied by B atoms, the L12 ordering leads to four different variants or domains. The L12 phase can also be analysed as an infinite sequence of (001) planes of stoichiometries alternatively pure A and AB (Fig.3a). Hence, the contact of two different ordered domains leads either to a conservative APB (Fig.3b) or to a non-conservative APB (Fig.3~).

With first neighbor interactions only, the energy of a

n = - 2 -1

Fig.3 Full (empty) circles represent B ( A ) atoms. Large (small) circles are the sites on [n 0 O],planes with n integer (halfinteger). The index i=0,1,2 and 3 labels the four simple cubic sublattices, and n the tetrahedra along [l001 (one of them is represented by dotted lines).

3.1 REMARKS ON THE CVM ALGORITHM

Our aim now is to calculate the equilibrium state of an inhomogeneous system with boundary condi- tions. We will use the CVM method, which is known to be a very precise method and is in principle fairly easy to implement, but accuracy problems rapidly arise when we try to display a logarithmic (i.e. a rather weak) singularity.The free energy minimization is performed on a finite system con- taining typically one or two hundred tetrahedra along [OOl], and therefore involve a few thousand varia- tional parameters. The APB free energy per site is of the order of 0.1 J close to Tc , to be compared to a total energy of about a few hundred J. On the other hand, the splitting of the APB's involves energies less than 10 - J .Different numerical minimiza- tion procedures can be used. Due to the linear geom- etry of our system, the most natural method seems to be a local relaxation algorithm, which consists in visiting succe.ssively each tetrahedron: at step t we only minimize with respect to the correlation func- tions of tetrahedron number t and then we proceed on tetrahedron number t + l

.

In other words, we re- place the global minimization by local minimizations involving the correlation functions of a single tetra- hedron, coupled with iterations through the lattice.

We have found that in fact a much more efficient method i s the brute force global NewtonTRaphson procedure provided advantage is taken of the nearly block-diagonal structure of the hessian matrix. The latter method has proved to be hundred times faster

TO THE TRANSITION

The index n will now label the nth tetrahe- dron along direction [001] and the index i=0,1,2 or 3 the four simple cubic sublattices (see Fig.3~). Each tetrahedron shares one site with each sublattice. Let c,!, be the concentration in minority atoms of site i in tetrahedron number n (see Fig.3~). In a perfectly or- dered L12 phase, these concentrations do not depend on the index n and one of them is e ual to one, the three others to zero (say, $=l, cl= c$= c3 = 0).

We consider here the case of a non-conserva- tive APB along direction [OOl], i.e. we assume that minority atoms principally occupy sublattice 0 (or 3) at z =

-

and 1 (or 2) at z = -t- (see Fig.3~).

As this i s an inhomogeneous situation, the concentrations c,!, now depend explicitely on indices n and i. More precisely? for a given tetrahedron n, the four concentrations c:, , i=0,1,2 and 3, can be all different. Therefore, besides the mean concentration (c!+cf, C: +C: )/4, a three dimensional order parame- ter is needed to describe completely the degree of order. We then introduce q, = (q: ^T$, q3) with ql = c O + c l - c 2 - ~ 3 , , q; = c ~ , - c f , + c : - c f a n d q! = c! - c: - c!

+

^c;

.

These quantities measure respec- tively the local amplitudes of the waves of q-vectors (loo), (010) and (001).

We now discuss the equilibrium state of the APB at a temperature very close to the transition temperarure: T=Tc - 0.778 I O - ~ J

.

In Fig.4a are shown the concentration profiles c6 , i = 0,1,2 or 3 which clearly display a large wet region (the profiles at lower temperature are similar to those presented by Kikuchi and Cahn [3]). A remarkable feature of these profiles is that the (001) planes in the central layer are strictly disordered when they are consid- ered as two-dimensional square lattices. This is still more striking when we draw the order parameter profiles q;, a = 1,2,3

.

The order parameters then strictly vanish in the wetting layers (Fig.4b).

We also show in F i g . 4 ~ the free energy profile ob- tained by decomposing the total free energy into a sum of tetrahedron free energies.

COLLOQUE DE PHYSIQUE

Tetrahedron number

Fig.4 Calculated profiles for the APB at T =Tc- 0.778 ~ o - ~ J (h=7.5J):

a) sublattice concentrations cO, cl, c2, c3

b) order parameters q 1 , q2,,,q3 (note the region where qlond q2slrictly vanish)

c) excess localfree energy

3.3 WElTING AND LAYERING MECHANISMS We have then investigated in more detail the wetting process by computing the variation with tem- perature of the APB free energy and entropy (we have checked that the latter can be obtained either from the temperature derivative of the free energy or directly from the CVM formula). Typical results concerning the APB entropy are presented in Fig.5. The remark- able point is that we have found an infinite series of first order transitions associated with entropy jumps.

We have verified that each jump corresponds to the disordering of a mixed plane in the central region . More precisely, as the temperature increases up to T,, the width of the wetting layer increases discontinu- ously by jumps of two atomic planes (see Fig.6 for an illustration of this wetting process; completely similar results have been obtained using the Bragg- Williams approximation). Hence, the width of the APB, defined as the number of consecutive disor- dered atomic planes in the central layer, is always even and can be noted 21. Let then TR be the transi- tion temperature between phases 2(R-1) and 21. We have computed the first twenty temperatures TJ (see Table 1); Fig.7 clearly shows that R behaves asymp- totically as -In(Tc-TR). As a technical remark, it may also be noted that, due to the first order character of the transitions, many metastable states can exist close to Tc : as example, all the phases 2 1 with R > 6 are metastable at T=Tc (this is depicted in Table 1, where we report the free energies of these phases at T=Tc) . Hence, the true equilibrium state cannot be obtained through a single minimization process: for a given temperature, one has to compare the free energies of all phases 2.2.

To summarize, complete wetting of non-con- servative (100) APB's has been shown to occur close to coexistence with the disordered phase, through an infinite series of layering transitions. This layering mechanism can be explained from a discussion of the mean field (Bragg-Williams) equations [S]. The anal- ysis also shows that layering is not expected for other orientations of the APB plane. Calculations are currently under progress to check this point and to include further interactions, which is necessary to ac- count for experimental situations.

Fig.5 Excess entropy S as a function of temperature T. The dotted lines mark the first order transitions.

I I

O ~ O ~ ~ . o . @ e @ ; ~ O ~ O ~ . O e O * ; @ o O Q @ ~ I . O . " .

1 2 * 3 1

l l ) T = T 4

0 ~ 0 ~ 0 ' ~ @ ~ 0 ~ @ ~ U ~ 0 ~

- o a o * j e ~ @ ~ s O @ ~ ; a ~ a

2 * 4

T=T5

REFERENCES

: ^>

O ~ ~ l ~ @ ~ @ ~ @ ~ [l] @ ~Diemch S, in Phase Transitions and Critical @ ~ o ~

* O * ~ @ O @ * @ O @ ~ @ O ~ ~ O O Phenomena, vol. 12, edited by C. Domb and J.

I 2 ' 5 I Lebowitz (Academic Press, London, 1988), p. l

Fig.6 I11usfration of the wetting process: as the temperature increases up to Tc, the width of the wetting layer increases dis- continuously (by jumps of W O atomic planes) through an inf- nite series offirst order transitions (see Fig3 for the site rep- resentations).

'%

-' m

Fig.7 2 , half-width of the wetting layer, against LoglO (T,-Tl) : note that the logarithmic behaviour i s ob- served on more than four decades.

[2] Ch. Leroux , A.Loiseau , M.C.Cadeville and F.Ducastelle, submitted to Euro Phys. Lett.

Ch. Leroux and A. Loiseau, In siru ordering and disordering processes in CoPt3 , film ONERA no 1202 (1988)

Ch. Leroux, A. Loiseau and M.C.Cadeville, EUREM 88, Proceedings of the 9th European Congress on Electron Microscopy, Inst. Phys. Conf.

Ser. no 93, 2, (Institute of Physics, Bristol and Philadelphia 1988), p.485

A. Loiseau et al., this conference.

[3] R. Kikuchi and J.W. Cahn, Acta Metall.

27,1337 (1979); see also J.M. Sanchez, S. Eng,

Y.P.

Wu and J.K Tien, Mat. Res. Soc. Symp. Proc.

81,57 (1987)

[4] A.Fine1 and F.Ducastelle, Europhys.Letters 1, 135 (1986); erratum 1, 543 (1986)

kKikuchi, Progr.Theor.Phys.Supp. 87, 69 (1986)

[S] A.Fine1, V. Mazauric and F. DucastelIe, to be published.