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Submitted on 1 Jan 1962

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On the nearest neighbour interaction model with a concentration dependent interaction energy

Mats Hillert

To cite this version:

Mats Hillert. On the nearest neighbour interaction model with a concentration dependent interaction

energy. J. Phys. Radium, 1962, 23 (10), pp.835-840. �10.1051/jphysrad:019620023010083500�. �jpa-

00236691�

(2)

ON THE NEAREST NEIGHBOUR INTERACTION MODEL WITH A CONCENTRATION DEPENDENT INTERACTION ENERGY

By MATS HILLERT,

Royal Institute of Technology, Stockholm, Sweden.

Résumé.

2014

Pour pouvoir être appliqué

aux

systèmes alliages possédant des propriétés thermo- dynamiques assymétriques, le modèle d’interaction des plus proches voisins doit avoir

une

énergie

d’interaction dépendant de la concentration. On introduit alors

au

moins

un nouveau

paramètre qui peut être utilise pour réaliser l’accord entre l’expérience et la théorie. Les trois systèmes Au-Ni, Al-Zn, et Al-Ag sont examinés

sur

cette base.

Dans le système Al-Ag, l’influence de la concentration est très forte. Une conséquence est

que le diagramme de phase contient

une

région de miscibilité métastable et

une

région d’ordre

métastable qui s’interpénètrent. Ceci peut expliquer comment deux types de

zones

peuvent

se

former pendant le durcissement par vieillissement dans

ce

système.

Abstract.

2014

In order to apply to alloy systems with asymmetric thermodynamic properties,

the nearest neighbor interaction model must have

a

concentration dependent interaction energy.

At least

one new

parameter is then introduced which

can

be used to make experiment agree with

theory. The three systems Au-Ni, Al-Zn and Al-Ag

are

examined

on

this basis.

In the Al-Ag system the concentration dependence is very strong. As

a

consequence the phase diagram contains

a

metastable miscibility gap and

a

metastable ordering region which overlap

each other. This may explain how two types of

zones can

form during age-hardening in this system.

PHYSIQUE 23, 1962,

1. Introduction.

-

Already in the twenties,

when the X-ray method was first applied to the study of atomic arrangements in solid metallic solutions and made possible the study of ord er- disorder transformations, one discussed the possi- bility of explaining the observed phenomena by regarding all nearest neighbor pairs as molecules, independent of each other. An exchange of posi-

Lion between two différent atoms in a binary alloy

was regarded as a chemical reaction where some

bonds were formed at the expense of others [1] :

The heat of reaction v, which is now called the

"

interaction energy ", depends on the three dif- ferent bond energies, v g

=

EAB - 1 2 EAA EBB . ( --I- )

For a completely disordered solution this model

predicts a molar heat of mixing Hm

=

ZN v XA XB

where N is the Avogadro number and. Z is the

number of nearest neighbour to each atom.

When discussing this approach in 1928, Borelius

and al. [1] concluded that it would be too crude an

approximation to neglect the effect of the other

neighbouring atoms and in a later paper [2]. Bore-

lius extended the treatment to take into account

larger groups of atoms than two. He thus arrived at an expression for the heat of mixing

which he has subsequently used in a large number

of papers, for the purpose of calculating the ther- modynamic properties of supersaturated solid solu- tions from phase diagrams, for instance. In this connection it is quite obvious that the simple expression HM

=

NZ v XA’XB can account for a miscibility gap or an ordering region only when it

is symmetric in the phase diagram, which is very seldom the case even approximptely.

Despite this, the simpler nearest neighbor inter-

action model was developed in détail by Gorsky [3]

in 1928 and by a great number of authors in later years [4]. Recently the validity of this model has been tested extensively by comparison of different

thermodynamic measurements and X-ray deter-

minations of the local atomic arrangement in solid solutions. The latter can be related to the ther-

modynamics, for instance by the first approxi-

mation [4] of the nearest neighbour interaction model (the so-called quasi-chemical theory). In a

few cases the agreement is rather satisfactory but

in most cases large discrepancies have been found.

As a consequence, most people now might tend to

agree with Borelius that the nearest neighbour inter-

action model is too crude. The present paper will deal with the possibility of improving the model by allowing to vary with alloy composition. The possibility that v may also vary with the degree of

order in the alloy will not be considered.

2. Concentration dépendent interaction energy.

-

In the f ormal theory one usually treats the

interaction energy v as a constant. However, it

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010083500

(3)

836

was pointed out by Guggenheim [4] that v is actually a free energy and may vary with tempe-

rature. As a consequence, the local atomic , arrangement in a solid solution should be related to the excess free enèrgy (or more exactly

rather than to the heat of mixing Hm, a ,fact that

has been overlooked many times. Except for this,

the temperature dependence of

v

does not cause

any difficulties in the formal theory or in the com- parison with experiments.

Far worse is the fact that a concentration depen-

dent v must be used in order to account for the behavior of actual alloy systems. The assumption

of a concentration dependence is itself a violation

of the nearest neighbour interaction model because it implies that the bond energies do not only de- pend on the identity of the two atoms under consi- deration but also on the average composition of thé surroundings. However, it has been shown [5]

that the quasi-chemical theory can be derived on a purely formaI basis without the usual assumption concerning the constancy of the bond energies.

Instead, the bond energies are considered as partial

free energies related to the integral free energy of the solution Gm by the following relation

in equivalence to the usual partial free energies of

the chemical components :

The so-called first approximation according to Guggenheim’s terminology then yields

and

where :

and

It is demonstrated by eq. (4) that the free energy of mixing GM is not only a function of v but also

of GEAA and G B, a fact that was pointed out by

Rudman [6] and has been discussed on several occasions [7, 5, 8]. As a consequence, as soon as one accépts a concentration dependent v, one has thide parameters which can be used in order to

give àgreemeht between thëôèy and experiment.

The chances to account for the behavior of an

actual alloy system in a formai way is thus much

eater than is usually realized, but, on the other

and, the chances to prove or disprove the nearest neighbor interaction model in this form is greatly

reduced. The purpose of this paper will thus be to demonstràte how the experimental results can be

accounted for in a few cases j not to prove that the

interaétion model itsélf is valid. The local atomic

arrangement Will be accountéd for in terms of

eq. (3) but otherwise the zeroth approximation will

be used, which implies that XAB = 2XA XB and

Sconf = - R(XA ln Xl + XB ln XB) fôr a disor- dered solution. The excess free energy of the

AA-bonds is zero for pure A and as a first approxi-

mation it may thus be convenient write

and, by the same reason, GB = b . Xg . N. The free eriergy for a disordered solution can then be

represented by the following expression :

The appearance of a miscibility gap in a phase diagram, thus depends on the quantity (v + c).

Let us now consider an ordered system where

every other atomic plane has the composition XB + y and the rest of the planes XB - y.

According to the zeroth approximation we can apply eq. (8) to each atômic plane but in an expres- sion for the whole system we must also take into account the " interfacial energy " between each

pair of planes [5]. We thus arrive at an expression

where z is the number of nearest neighbours in one plane to a given atom in the next plane. Thé degree of order can be calculated from dGM fdy

=

0

and thus depends on the quentity v in the second

term, whereas (v + c) in the first term is of no

importance in this connection. As a consequence, the appearence of miscibility gaps and ordering regions are controlled by différent factors and the

existence of one doës not necessarily exclude the

existence of the other, as is usually believed.

(4)

837 There are two kinds of local atom arrangement,

called " clustering" and " short range order ". In the first case there are less AB bonds than expected

in a random solution, XAB 2XA XB. In the

other case there are more, i.e. XAB > 2XA XB.

According to eq. (3) the local atomic arrangement

is controlled by v and this may be true in the case

of short range order and for small clusters. For

large clusters, on the other hand, our model predicts

that the free energy of the bonds will change with

concentration.

In the following seétions a few actual systems will be discussed. We shall take into account the concentration and température dependence of the

interaction energy by using an expression with a

minimum number of parameters, i.e.

An equivalent expression was used by the présent author [9] in an attempt to evaluate thermo-

dynamic quantities from phase diagrams and was found to yield surprisingly reliable information

particularly regarding the excess entropy. For

that purpose it seems to be particularly important

to usé as few parameters as possible in order not to

rely too much upon the accuracy of experimental

data concerning phase diagrams.

3. The Au-Ni system.

---

In accordance with the

preceding discussion we shall represent the free

energy of homogeneous Au-Ni alloys by an expres-

sion (see eq. (9))’:

using x = XB = 1

2013

XA. In the previous

work [9] the following data were selected from the

miscibility gap in the phase diagram : Xl = 0.238

and x. = 0.962 at 873 DK and the peak tempe-

rature Tpeak

=

1 103 PK. The resulting parameter

values were ZN, (c + d)

=

3 300 R, ZNe = 1 660 R and ZNf

=

2.27 R. The last parameter value

indicates a considerable oxcess free energy, in

agreement with the emf measurements’ by Seigle [10].

X-ray measurements [11] at x

=

0.5 and

T = 1 173 oK, i.e. above the miscibility gap, has revealed a short range order to a degree that corres- ponds to v jkT = - 0.06 according to eq. (3). If the calculation of v jkT from the thermodynamics of the solid solution is based upon HM, as is usually

done (i.e. on the terms ,c + d + ex in the above expression) one obtains vjkT == + 0.3. From

this result one concluded [11] that the nearest neighbor inter.action model did not at all apply t9 the Au-Ni system. However, if the calculation

is based on GE (i.e. on c + d + ex + f T ) the result is l’ == -)- 0.1. More than half of ’the discre- pancy previously reported thus vanishes. The remoiping discrepancy pan now be taken into

account by the parameter c and one obtains

ZNc

=

2 320 R and ZNd

=

980 R. It is interesting

to note that the two parameters c and e are of about the same order of magnitude which is very reasonable because they both depend on tbe con-

centration dependence of the free energy of the

bonds.

Figure 1 shows the phase diagram for solid Au-Ni alloys according to the parameter values just calcu-

FIG. 1.

-

Au-Ni phase diagram.

lated. The dashed curve is the so-called spinodal

where homogeneous alloys become thermodyna- mically unstable. Its position depends upon the value of c -E- d + ex

-

fT. The dotted curve is

a similar curve calculated from v = d -E- ex

-

f T.

One might say that it defines the region of insta- bility for clusters so small that v is-unaff ected. It is interesting to note that it lies far below the ther- modynamic spinodal. It may be tempting to re-

late this fact to the experimental observation that Au-Ni alloys do not easily decompose by the for-

mation of modulated structures or zones or by any

general précipitation mechanism. Instead, there

is a grain boundary nucleated discontinuous precii pitation process. However, a more straight for-

ward explanation of this observation has recently

been suggested by Cahn [12], base on direct straind

energy considerations.

(5)

838

4. The Al-Zn system.

-

In the previous work [9 ]

the following data were selected from the misci-

bility gap in the phase diagram : xl

=

0. I6 and

X2

=

0.59 at 548 OK and the peak temperature Tpeak

=

626 OK. The resulting parameter values

were ZN(c + d)

=

1050 R, ZNe = -- 570 R

and ZN f

= -

0. 57 R. It should be observed that these parameter values hold for the solid solutions of Zn in A.1, refering to standard states of pure Al and Zn of the same crystal structure, i.e. FCC.

If v jkT is again erroneously calculated from Hm,

i.e. from c + d

-

ex, one obtains v jkT = + 0.1 at x

=

0.5 and T

=

673 op. X-ray measu-

rements [13] of the local atomic arrangement carried out for this composition and temperature

indicate clustering to a degree that corresponds

to v jkT = + 0.3. Although these two values

have the same sign, their différence is just as appreciable as in the Au-Ni system. If we again account for the discrepancy in terms of the para-

meter c we obtain ZNc

= -

1 800 Rand ZNd

=

2 850 R. In this case c is appreciably larger than e, a fact that casts some doubt on the validity of this analysis.

Figure 2 shows the’phase diagram for solid Al-Zn

alloys of FCC structure according to the parameter

FiG. 2.

-

Al-Zn phase diagram.

values just calculated. Again the dashed curve

is the thermodynamic spinodal.

Hilliard and al. [14] carried out emf measu-

rements on solid Al-Zn alloys and evaluated the

thermodynamic properties for solid solutions of Zn in Al. However, these data refer to a standard state of HCP Zn and FCC Al, these being the stable

structures of the pure elements. In order to allow

a calculation of the interaction energy and thus a

comparison with the X-ray measurements of the local atomic arrangement, one must correct for the free energy of the transformation

This was not donc originally and the agreement

between the X-ray and emf results, which was then found [13], was purly fortuitous, After esti-

mating [9, 7] the free energy of transformation of pure Zn to 500 R - 0.5 RT the present author

could show that the emf measurements are in good agreement with the thermodynamic data eva-

luated from the phase diagram and they are thus

in disagreement with the X-ray data unless the c

parameter is introduced.

5.. The AI-Ag system. - There is no stable misci-

bility gap in the AI-Ag system. However, the shape of the solubility for the intermetallic com-

pound y in AI has been used by the present

author [9] and by Borelius and al. [15] and in both

cases a metastable miscibility gap was found for the FCC structure. Thermodynamic properties

evaluated from emf measurements gave the same result [9, 7]. It was suggested that this miscibility

gap might play a decisive role in the so-called pre-

precipitation stage of the age-hardening in this sys-

tem.

,

A characteristic feature of FCC alloys in the AI- Ag system is the pronounced difference between Al-rich and Ag-rich alloys. Al-rich alloys show clustering and a positive heat of mixing, Ag-rich alloys show short range order and a negative heat

of mixing. The corresponding values of v /kT are +0.6 and + 0.1 on the Al side and

-

0.4 and

-

0.1 on the Ag side [13, 7]. The agreement is

not very impressive although it is satisfying that

the two sets of data agree with respect to the sign.

If one instead uses the excess free energy for the

comparison, as one should, one obtains + 0.6

and 0 on the Al side and

-

0.4 and

-

0.4 on the

Ag side. The agreement is thus improved on the Ag side but slightly worse on the AI side. It is

apparent that a rather high value of the parameter

c must be used in order to remove the discrepancy.

A high value of c also seems plausible in view of the very strong variation of v with composition in this system.

The phase diagram as well as the emf measu-

rements indicate that the excess free energy in the Al-rich side is close to 0 and negative in the

rest of the system. This may be of great impor-

tance for the pre-precipitation process in the Al- rich alloys, as suggested by the present author [7]

and by Borelius and al. [15]. In order to examine

this effect further we shall now neglect the effect of all parameters except e in the expression for the

free energy of FCC alloys. In order to clearly

indicate that e is negative we shall write e = - K and thus v

= -

.Kx and

The negative value of the excess free energy indi-

(6)

cates the existence of an ordering region. In order

to simplify the calculation of the phase diagram we

shall thus choose a crystal structure where the ordering region can be easily calculated. A conve-

nient choice is the BCC structure of Cu Zn and the number z in eq. (10) is then equal to (1 /2) Z. The

first two terms in eq. (10) are thus reduced to

The phase diagram resulting from this calcu- lation is presented in figure 3. A remarkable fea- ture of this diagram is the shape of the «i + x2

FIG. 3.

-

Phase diagram for

a

binary BCC system with

an

interraction energy

v = -

Kx.

miscibility gap. In the upper part the two

oc

phases are disordered but in a quite narrow tempe-

rature region oc2 becomes ordered to a considerable

degree and at the same time the width of the misci- bility gap increases markedly.

It is intei esting to compare this result with the

experimental results on the pre-precipitation in Al-Ag alloys. Guinier has reported the existence of two kinds of zones in this system, the second one

possibly having an ordered structure [16].

Hardy [17] has suggested the notation G.-P. 1 for

the disordered zones which form first and G.-P. I I for the ordered zones. Borelius and al. [15, 18]

have been able to distinguish betwéen these two stages by resistometric and calorimetric measu- rements of the ageing process. They call the first kind e and the second kind -1 (or y" in ref. [18]).

The -1 zones only form below a fairly well defined temperature which seems to be rather independent

of the alloy composition. According to a recent study by Baur and Gerold [19] the Ag content of

the zones increases markedly at about the same temperature. Their results are shown in figure 4.

FIG. 4.

-

The miscibility gap in Al Ag according to Baur

and Gerold [19].

0, our measures,

of measures of

+, after G, Borelius [2], extrapolation of measures of resistance,

The similarity with the bypothetical diagram pre- sented in figure 3 is striking. It is thus tempting

to suggest that the equilibrium diagram for FCC alloys in the Al-Ag system in fact resembles figure 3

and also that the zones form as the first stages of decomposition inside such a miscibility gap. This

would be in agreement with a hypothesis suggested indepentenly by the present author and Borelius

and al. Without actually calculating the phase diagram, they concluded from the shape of the

free energy curye for FCC alloys that a cluster (or zone), which attains a sufficient size and silver content should start to order and thus transform from G.P. 1 to G.P. II.

One may now wonder how common is such a de-

composition sequence, i.e. :

supersaturated solution- disordered zones -+ordered zones.

According to the above hypothesis it should only

occur when the phase diagram resembles figure 3

and this in turn may only occur under quite special

conditions. These conditions are not present in

the Al-Zn system according to our calculations in section 4 and, in agreement with this, there does

not seem to be any experimental evidence of

two different kinds of zones in the Al-Zn system.

The thermodynamics of the Al-Cu system, on the

other hand, resembles that of the Al-Ag system [20]

(7)

840

Here the conditions may thus be right and it is interesting to note that in this system there are experimental indications of the two kinds of zones [16, 18].

It should be emphasized that, according to the

above hypothesis, the zones form inside a misci- bility gap (which may be either stable or metas-

tablê), According to a thepry by Dehlinger and

Knapp [21], on the other hand, zones pr clusters may form above p. miscibility gap. Howpver, in the opinion of the présent author [5] their respit de-

pended on a comparison of the frpe energy of the

clustered system with that of a completely random solution and is only an indication of thé fact that there will be less A-B pairs than in the rapdom

case, i.e. XAB 2XA XB. That phenomenon is

much better described by the quési-chemical

theory, eq. (3), and should not be confused with the

formation of zones. However, we should now

consider another possibility pf forming zones above

a miscibility gap :

As shown by eq. (9), the position of a miscibility

gap depends on the quantity v + c, whereas the local atomic arrangement and the formation of

very,small clusters dépend only on v. In the case

v

» v + c, small clusters may thus form and grow in size even above the miscibility gap, Remem- bering that the free energies of the bonds are assumed to vary with the average composition

evaluated for some region around the twp atoms under consideration, one should conclude that the size of such clusters is limited by the size qf that region. There should thus be an equilibrium size

for such clusters, the characteristic value probably yarying with temperature. Whether such clusters

occur in any actual alloy system, is an open ques- tion but should be examined experimentally.

It may also be interesting to discuss the opposite

case, i.e.

v

« v + c, which Was found in the Au-Ni system. As mentioned in section 3, hère one might expect that the formation of small clusters should

be prevented although large clusters should be

stable and able to grow indefinitely, if they could only nucleate. At very low temperatures, on the

other hand, v has .higher values (compare fig. 1)

and there should be no difficulties forming zones.

Here zones have indeed been reported [22].

6. Inhomogenpous systeips.

:

In a recent work [5, 231 the présent author applied the nearest neighbour interaction model to inhomogeneous sys- tems and demonstrated how the structure of anti-

phase domain boundaries, cohérent phase boun-

daries and critical nuclei could be caleulated, as

well as the décomposition process for homogeneous Systems inside ordering regions or miscibility gaps.

In that work qnly symmetric ordering regions and miscibility gaps were considered and v was thus

treated as a const#nt. When similar calculations

were âttempted for the present case where v is

allowe4 to vary with composition, considerable difficulties were encountered, mainly because the

free energy. of the bonds dépend on an civerage

composition evaluated for some region aroqnd the

two atoms under consideration. When calculating

the equilibrium diagram this fact can be formally neglected as we have seen in the preceding sections,

the refison being that the equilibrium phases are

assumpd to have homogeneous compositions.

Even thpugh the interaction energy model ceases

to be q. nearest neighbour model as soon as v varies with composition, this fact does pot seem to have

any fonmal conséquences for homogeneous sys- tems. As a conséquence, in this paper we have not been able to test ouf. model but only to show how some experimental data can be accounted for

in terms of the model. For inhomogeneous sys- tems, on the other hand, the true nature of the

model should become apparent. It is therefore probable that the most important informations in

the future will corne from the study of inhomo- geneous systems rather than homogeneous.

REFERENCES [1] BORELIUS (G.), JOHANSSON (C. H.) and LINDE (J. O.),

Ann. Physik, 1928, 86, 291.

[2] BORELIUS (G.), Ann. Physik, 1934, 20, 54.

[3] GORSKY ,Z. Physik, 1928, 50, 64.

[4] GUGGENHEIM (E. A.), Mixtures, Oxford, 1952.

[5] HILLERT (M.), Sc. D. Thesis, M. I. T., Cambridge, Mass., 1956.

[6] RUDMAN (P. S.), Sc. D. Thesis, M. I. T., Cambridge, Mass., 1955.

[7] HILLERT (M.), AVERBACH (B. L.) and COHEN (M.), Acta Metall., 1956, 4, 31.

[8] ORIANI (R. A.) and MURPHY (W. K.), N. P. L. Sym- posium No 9, June 1958.

[9] HILLERT (M.), S. M. Thesis, M. I. T., Cambridge,

Mass., 1954.

[10] SEIGLE (L. L.), COHEN (M.) and AVERBACH (B. L.),

Trans. A. I. M. E., 1952, 192, 1320.

[11] AVERBACH (B. L.), FLINN (P. A.) and COHEN (M.),

Acta Metall., 1954, 2, 92.

[12] CAHN (J. W.), Acta Metall., 1962, 7.

[13] RUDMAN (P. S.) and AVERBACH (B. L.), Acta Metall., 1954, 2, 576.

(14] HILLIARD (J. E.), AVERBACH (B. L.) and COHEN (M.),

Acta Metall., 1954, 2, 621.

[15] BORELIUS (G.) and LARSSON (L. E.), Arkiv Fysik, 1956, 11, 137.

[16] GUINIER (A.), C. R. Acad. Sc., 1950, 231, 655.

[17] HARDY (H. K.) and HEAL (T. J.), Progress in Metal Physics, 1954, 5, 143.

[18] BORELIUS (G.) and LARSSON (L. E.), Arkiv Fysik, 1962, 21, 213.

[19] BAUR (R.) and GEROLD (V.), Z. Metall., 1961, 52, 671.

[20] MEIJERING (J. L.), Rev. de Métallurgie, 1952, 49, 906.

[21] DEHLINGER (U.) and KNAPP (H.), Z. Metall., 1952, 43, 223.

[22] TUKANO (Y.), J. Phys. Soc., Japan, 1961, 16, 1195.

[23] HILLERT (M.), Acta Metall., 1961, 6, 525.

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