Ternary interaction parameters in Calphad solution

Ternary interaction parameters in Calphad solution models Luiz T. F. Eleno∗ Instituto de F´ısica, Universidade de S˜ ao Paulo, CP 66318, 05315-970 S˜ ao Paulo-SP, Brazil

Cl´audio G. Sch¨on† Computational Materials Science Laboratory, Department of Metallurgical and Materials Engineering, Escola Polit´ecnica da Universidade de S˜ ao Paulo, Av. Prof. Mello Moraes, 2463, CEP 05509900 S˜ ao Paulo-SP, Brazil For random, diluted, multicomponent solutions, the excess chemical potentials can be expanded in power series of the composition, as a function of pressure- and temperaturedependent parameters. For a binary system, this approach is equivalent to using polynomial truncated expansions such as the Redlich-Kister series for describing integral thermodynamic quantities. For ternary systems, an equivalent expansion of the excess chemical potentials provides a clear justification for the use of ternary interaction parameters, which arise naturally in the form of correction terms as higher-order power expansions are employed. We demonstrate this fact by truncated polynomial expansions of the excess chemical potentials up to the sixth power of the composition variables.

I.

INTRODUCTION

The Wagner formalism [1] for the thermodynamics of diluted solutions, proposed in the first half of the 20th century, is still nowadays widely used for metallurgical calculations, mainly for applications in steelmaking processes. The formalism, however, was shown by Darken [2] to be thermodynamically inconsistent and cannot be applied in the case of concentrated solutions. A correction to the Wagner formalism has been proposed [3, 4], but recently it has been shown that it is not unique [5], so that one should not be overconfident in the results of the corrected formalism. On the other hand, Calphadtype models [6], which cover the whole composition range, are always thermodynamically consistent, since these models use the integral excess properties, such as the Gibbs energy, as their starting point, and not derived quantities such as excess chemical potentials without regard to the thermodynamic consistency, as is the case of the Wagner formalism. Therefore, the reliable range of application of Calphad models is larger, and has been applied successfully in varied systems and situations [6]. Even so, the Wagner formalism still has its practical importance for its simplicity and accuracy within its ∗ †

Electronic address: [email protected] Electronic address: [email protected]

2 applicability range, even though it is inconsistent with equilibrium conditions. The aim of this work is not, however, to propose a new correction to the Wagner formalism, as is has already been shown that there is an unlimited number of ways by which it could be done [5]. Instead, in the present work we propose to derive an interpretation for ternary interaction parameters in ternary systems, using the same starting point as the one used in Wagner’s treatment of diluted solutions [1], showing that the ternary interaction parameters in Calphad models have the same origin as the binary parameters in Redlich-Kister or other polynomial expansions for binary systems. It is quite possible that Mats Hillert had the same idea in mind when he published his proposals for ternary solution models [7], and also Pelton and Bale [8], who realised that a polynomial description of integral quantities such as the excess Gibbs energy is equivalent to using a truncated polynomial expansion of partial excess quantities, such as the excess chemical potentials (or, equivalently, the activity coefficients). These authors, however, only stated this equivalence but did not express explicitly the connection between Wagner’s treatment and the Calphad solution models, except for the first-order approximation and some general guidelines for higher-order extrapolations [9]. The present authors, therefore, argue that the proof of the equivalence should be provided in all its details, at least for binary and ternary systems. The procedure adopted here to derive the required equivalent proof is not unique. For instance, postulating an integral quantity expression, as done by Pelton and Bale [8], would result in a simpler and more compact proof, though not as easily generalised to higher polynomial degrees. But Wagner theory is not written in terms of integral quantities. The whole idea of working with Wagner interaction parameters (and their extension to concentrated solutions) is about dealing with partial quantities such as the activity coefficients, which can be directly related to measurable quantities (activities, electrochemical measurements, vapour pressure measurements and so on). It is important to emphasise that we do not intend to provide here a new methodology. Our work is centred on the fundamentals of some of the existing Calphad solution models. We also limit our discussion to substitutional solution models, for simplicity’s sake. Also, we do not discuss here any of the several “geometric” extrapolation methods that exist in the literature (for a recent review, see Malakhov [10]), but focus instead solely on ternary parameters. These very convenient extrapolations are sometimes used instead of ternary interaction parameters, in order to derive parameters for a ternary system based on binary data only. The present work is organised as follows. Section II gives a general and brief overview of selected aspects of Calphad disordered solution models. We then derive series expansions for the excess chemical potentials in powers of the composition to show that they are equivalent, firstly, to Redlich-Kister series for binary systems in section IV and, secondly, to a mix of Redlich-Kister polynomials and ternary interaction parameters for ternary systems in section V. Finally, section VI brings us to our final discussion and conclusions.

3 II.

CALPHAD SOLUTION MODELS

Let us say we need to describe a solution phase φ in a multicomponent system. In one of the most widespread methodologies, one would write its molar Gibbs energy as a sum of several contributions, each of which is (or may be) a function of the temperature T , the pressure p and the molar fractions of the components (xi , xj , . . .) as [6] Gm (p, T, xi , xj , . . .) = ser Gm + id Gm + phys Gm + ex Gm The first term,

ser G

m,

(1)

is the molar Gibbs energy of a standard energy reference state, e.g., the

mechanical mixture of the pure components at the same temperature, pressure, and structure as the phase φ under consideration. The term

id G

m

is the molar Gibbs energy of an ideal solution. This

is the only term which explicitly takes into account the configurational part of the Gibbs energy and is equivalent to a Bragg-Williams-type of treatment [11]. The term

phys G

m

in Eq. (1) indicates

contributions due to physical phenomena such as ferromagnetism. The last term in Eq. (1) is called the excess molar Gibbs energy for the phase φ. It is nothing more than a measure of the inadequacy of the previous terms to describe the physical reality. Until more physically-sound methods and models are devised, one is forced to use excess quantities, which should include a parametrisation of all uncertainties inherent to the model given by Eq. (1). Even so, it is still possible to describe mathematically the excess term by, for instance, a polynomial fit, so that a complete temperature–pressure–composition dependence of the molar Gibbs energy is developed. In the present work, it will be demonstrated that the same truncated expansions, which generate Redlich-Kister [12], Legendre [8], Margules [13] or other polynomials for binary systems, also give rise to ternary interaction parameters when applied to ternary systems.

III.

EXCESS CHEMICAL POTENTIALS AS POWER SERIES

The excess term in Eq. (1) can be written in terms of the so-called excess chemical potentials µex i , so that ex

Gm =

X

xi µex i

(2)

i

The excess chemical potentials µex i are related through the Gibbs-Duhem differential equation which can be written in the form X

xi dµex i =0

(3)

i

For a completely random solution, it is possible to expand the the excess chemical potentials µex i in terms of composition, pressure, and temperature. To the best of our knowledge, Margules [14] was the first to perform such an expansion to describe the thermodynamics of a system, an idea that

4 was to be further explored by Wagner [1]. The methodology was also employed by Wohl [15], who developed expressions up to the 5th power of composition. On the other hand, Redlich and Kister [12] introduced independently their power series for integral quantities such as the excess Gibbs energy, after an idea previously used by Guggenheim [16]. Cheng and Ganguly [17] later demonstrated the equivalence between the series expansions by Wohl and by Redlich and Kister. A comparison of several models was made by Ansara [18]. However, even though Redlich and Kister cite the work of Margules, the explicit demonstration of the equivalence of the two approaches was not carried out, since it was evident to them, as it continued to be for more recent authors [7, 8, 19, 20], who preferred to work directly with integral excess quantities rather than activity coefficients, a methodology still employed today. To start our discussion, let us first consider the case of a binary disordered solution.

IV.

BINARY SOLUTION PHASES

In the case of a random, diluted solution of two species, labelled A and B, it is possible to expand the excess chemical potentials in terms of the molar fractions, so that we may write, following Wagner [1], µex i = Ωi,1 (1 − xi ) +

Ωi,2 Ωi,3 (1 − xi )2 + (1 − xi )3 + . . . 2 3

(4)

with i = A, B. In Eq. (4), the unknown parameters Ωi,η (η = 1, 2, 3,. . . ) are supposed to be functions of pressure and temperature. In this way, the proposed expression for µex A depends only upon p, T , and 1 − xA = xB . Similarly, µex B will be a function of p, T , and 1 − xB = xA . The unknown coefficients Ωi,η appear due to the fact that the relationship between the composition and the other two variables, p and T , is not known a priori. Of course, the series must be truncated at some power of (1 − xi ). Therefore, the Ωi,η coefficients should be considered as fitting parameters, in the very sense of a Calphad optimisation. It should be also clear that the truncated expansion of µex i is strictly valid for xi → 1, i.e., in the case of infinite dilution of the second component. It is not possible to go into concentrated solutions without considering interactions between groups of atoms. Replacing id Gm in Eq. (1) by a more precise entropic term (such as those given by the Cluster Variation Method [21] or the Cluster-Site approximation [22]), is an alternative that could be used instead of (or simultaneously with) increasing the order of the series expansions in Eq. (4). In the following, however, we will continue to suppose the use of the ideal term in Eq. (1). Due to the fact that the Gibbs-Duhem equation, given by Eq. (3), must be obeyed by the power expansions in Eq. (4), it follows that not all of the Ωi,η are independent. If we adopt the ΩA,η as our fitted quantities, the parameters for the second element (B), i.e., ΩB,η , should be written in terms of ΩA,η . In the appendix we prove that we must have ΩA,1 = ΩB,1 = 0 and  ηX max  λ−2 ΩB,η = (−1)η ΩA,λ (η ≥ 2) η−2 λ=η

(5)

5 in which ηmax is the maximum degree of the truncated polynomials in (1 − xi ) used in Eq. (4). It was ex assumed that the same value of ηmax was used for the expansion of µex A and of µB .

One may now rewrite the excess molar Gibbs energy, using Eq. (2) and the power expansions, so that ex

Gm = xA xB

ηX max η=2

ΩA,η xB η−1 + ΩB,η xA η−1 η

(6)

In order to arrive at Redlich-Kister polynomials, one introduces the transformations below: 1 xA − xB + , 2 2

xA ←

xB ←

1 xA − xB − 2 2

(7)

first proposed by Hillert [7] and which are valid as long as xA + xB = 1. That is, for a binary system, Eqs. (7) are identities and we can replace the arrows by equal symbols. We note that Eqs. (7) will only be used here for binary systems, since for ternary or higher-order systems they are no longer valid. With the substitutions above, we obtain ex

Gm = xA xB

ηmax X−2

(xA − xB )ν ×

ν=0

×

 max  ηX 

η=g(ν)



 



1 η−1 [ΩA,η (−1)ν + ΩB,η ]  η · 2η−1 ν

(8)

with the function g(ν) defined as g(ν) = 1 +

ν + 1 + |ν − 1| 2

(9)

The term inside curled braces in Eq. (8) is a linear combination of the coefficients ΩA,η , and is unique for each value of the running index ν. We then call it simply Lν and rewrite Eq. (8) as ex

Gm = x A x B

νX max

Lν (xA − xB )ν

(10)

ν=0

with νmax = ηmax − 2. The polynomials in (xA − xB ) in Eq. (10) are identical to the so-called Redlich-Kister polynomials [12]. Of course, if one has values for the Redlich-Kister parameters Lν , it is also possible to derive values for the Ωi,η parameters, considering them as the dependent variables, and also to other series such as Legendre or Chebyshev polynomials which, as truncated series of orthogonal polynomials, have the advantage that the addition of a new higher term has only a small effect or none at all on the values of the lower terms [7, 8]. This has a lesser importance for our interests here, but one is referred to the works of Hillert [7], Tomiska [23–25], and Pelton and Bale [8] for clues about how the conversion between different polynomial series can be done.

6 V.

TERNARY SOLUTION PHASES

A more interesting case is obtained once we move to a ternary system A–B–C. It is possible to write the excess chemical potentials of A, B and C as series expansions of the composition, in analogy with the binary case. Since the same assumptions should also apply in this case (diluted, random solution), we can write, say, µex A in terms of the concentrations of the two other components, xB and xC , as µex A =


1 A A 2 ΩBB xB 2 + 2ΩA BC xB xC + ΩCC xC 2 1 2 + ΩA xB 3 + 3ΩA BBC xB xC + 3 BBB
2 A 3 3ΩA + ... BCC xB xC + ΩCCC xC

(11)

This is just a generalisation of Eq. (4) for the case of a ternary system. The Ω coefficients have the same interpretation as for the binary case, that is, they are simply mathematical interaction parameters to be fitted to experimental data and the series should be truncated at some maximum power of the compositions. Similar equations hold for the excess chemical potentials of the other two species. In Eq. (11) we already set the linear terms Ωji xi (i, j = A, B, C) equal to zero, since they should vanish once the Gibbs-Duhem equation is employed to determine the dependent variables. With the use of Eq. (3), it is possible to derive two equations involving the coefficients in Eq. (11), namely,  ex   ex   ex  ∂µA ∂µB ∂µC xA + xB + xC =0 (12a) ∂xB xC ∂xB xC ∂xB xC

 xA

∂µex A ∂xC



 + xB

xB

∂µex B ∂xC



 + xC xB

∂µex C ∂xC

 =0

(12b)

xB

It is obvious that not all three composition variables are independent. That is the reason why, in Equations 12(a-b), we are considering xA as dependent on xB and xC through the expression xA + xB + xC = 1. This choice is not unique and is completely arbitrary, but the final result will not depend on it. It should be noted that Equations (12) could be combined with the thermodynamic identity ex ex ∂ (µex ∂ (µex B − µA ) C − µA ) = , ∂xC ∂xB

(13)

which is valid as long as xA + xB + xC = 1 and considering xB and xC as the independent variables. This would lead to Darken’s criterion for thermodynamic consistency in ternary systems [2]. From ex A B Eq. (11) and its equivalents for µex B and µC , there are three sets of parameters, consisting of Ω , Ω

and ΩC variables. But, because of Equations 12(a-b) and (13), only one of these sets are independent

7 (or a linear combination of them). Therefore, similar to the binary case, we are able to reduce the number of parameters, so that we can guarantee the thermodynamic compatibility of the proposed model.

A.

Regular solution model

Let us first consider the simpler case in which the coefficients of third or higher order are ignored, that is, let us start by assuming Ωlijk... = 0 and only the Ωkij terms are different from zero (i, j, k, l, . . . = A, B, C). In this case, Equations 12(a-b) and (13) give Ωjii = Ωijk + Ωjik (i, j, k = A, B, C)

(14)

Therefore, out of the nine Ωkij parameters, only three are independent, which we chose to be ΩA BC , C ΩB AC , and ΩAB . Using Eq. (2), one can then write the excess molar Gibbs energy for a ternary system

A–B–C as ex

 Gm =

 A  B  ΩBC + ΩC ΩA AB BC + ΩAC xA xB + xA xC + 2 2  B  ΩAC + ΩC AB + xB xC 2

(15)

which could be rewritten as ex

AC BC Gm = LAB 0 xA xB + L0 xA xC + L0 xB xC

(16)

From Eq. (15), it is evident that there is no term depending on xA xB xC . In other words, when the excess chemical potentials depend only on the second powers of composition variables, no ternary term arises, and this approximation is therefore equivalent to the Muggianu “geometrical” extrapolation [26]. This equivalence had already been proved by Saulov [27] in a multicomponent context. As stated in the introduction, the above procedure is not the only way to arrive at the equivalence proof. One could postulate Eq. (16) and arrive at Equations (14) and (15). This is a much simpler route, but not as easily extended to higher powers of the compositions in Eq. (11).

B.

Subregular solution model

We now consider terms up to third power in Eq. (11). There are now 21 Ω-parameters, but three expressions relating them are provided by the Gibbs-Duhem equation (Equations 12a-b) and Eq. (13). Therefore, out of the 21 parameters, we can pick as few as 7 as independent, including at least one of the third-order Ω’s, in order to consider ternary interactions. We chose to keep the ΩA parameters as the independent ones, and solve for the others. The result

8 is the list of equations below, A A ΩB AA = ΩBB + ΩBBB

(17a)

A A A A ΩB AC = ΩBB − ΩBC + ΩBBB − ΩBBC

(17b)

A A A A A A ΩB CC = ΩBB + ΩCC − 2ΩBC + ΩBBB − 2ΩBBC + ΩBCC

(17c)

A ΩB AAA = −ΩBBB

(17d)

A A ΩB AAC = −ΩBBB + ΩBBC

(17e)

A A A ΩB ACC = −ΩBBB + 2ΩBBC − ΩBCC

(17f)

A A A A ΩB CCC = −ΩBBB + 3ΩBBC − 3ΩBCC + ΩCCC

(17g)

A A ΩC AA = ΩCC + ΩCCC

(17h)

A A A A ΩC AB = −ΩBC + ΩCC − ΩBCC + ΩCCC

(17i)

A A A A A A ΩC BB = ΩBB + ΩCC − 2ΩBC + ΩBBC − 2ΩBCC + ΩCCC

(17j)

A ΩC AAA = −ΩCCC

(17k)

A A ΩC AAB = ΩBCC − ΩCCC

(17l)

A A A ΩC ABB = −ΩBBC + 2ΩBCC − ΩCCC

(17m)

A A A A ΩC BBB = ΩBBB − 3ΩBBC + 3ΩBCC − ΩCCC

(17n)

With the use of the reduction relations given by Equations 17(a-n), we can rewrite the excess molar Gibbs energy as given by Eq. (2), so that, after a few more algebraic steps, we have an excess molar Gibbs energy given by ex

  AB Gm = xA xB LAB 0 + L1 (xA − xB ) +   AC +xA xC LAC 0 + L1 (xA − xC ) +   +xB xC LBC + LBC 0 1 (xB − xC ) + +xA xB xC LABC

(18)

9 with the L-parameters in Eq. (18) written in terms of the ΩA coefficients as follows: A LAB = (2ΩA 0 BB + ΩBBB )/4

(19a)

LAB = −ΩA 1 BBB /12

(19b)

A LAC = (2ΩA 0 CC + ΩCCC )/4

(19c)

LAC = −ΩA 1 CCC /12

(19d)

A A LBC = (2ΩA 0 BB + 2ΩCC − 4ΩBC + A A A +ΩA BBB − ΩBBC − ΩBCC + ΩCCC )/4

(19e)

A A A LBC = (ΩA 1 BBB − 3ΩBBC + 3ΩBCC − ΩCCC )/12

(19f)

A A A LABC = (3ΩA BBC + 3ΩBCC − 2ΩBBB − 2ΩCCC )/12

(19g)

It is also possible, if required, to write the ΩA coefficients in function of the L parameters, and then proceed to write also the ΩB and ΩC coefficients in terms of the L parameters using Equations 17(a-n). Now, from Eq. (18), we conclude that, if powers up to 3 are used in the series expansions of the excess chemical potentials given by Eq. (11), all three binary systems will be described as subregular solutions. More importantly, a ternary term xA xB xC LABC arises naturally from the equations. With that in view, we can say that the ternary interaction parameter LABC is part of the subregular solution model, when extrapolated to ternary systems. In other words, if all three binaries are described as subregular solutions, the Muggianu “geometric” extrapolation [26] is not enough as a description of the ternary system. Its use would only make sense in the particular case where LABC is expected to be negligible (LABC ≈ 0), which is not the most general situation.

C.

Higher-order approximations

We can continue to add higher powers to Eq. (11) and, using the Gibbs-Duhem relations (Eqs. 12a-b), arrive at expressions for

ex G

m.

Details of the calculation, in the lines of the results of section

V B, are not presented here, due to the large amount of equations and in order to save space. The interested reader can contact the authors if he or she is interested in these details. As we follow this procedure, we arrive at an excess Gibbs energy given by ex

in which

ex G(bin) m

Gm = ex G(bin) + ex G(tern) m m

(20)

is a sum of Redlich-Kister polynomials coming from the three binaries, ex

G(bin) m

=

X X i=A,B,C j>i

xi xj

ηmax X−2

ν Lij ν (xi − xj )

(21)

ν=0

so that ηmax is the maximum power in the series given by Eq. (11). The cases ηmax = 2 and ηmax = 3 were described in sections V A andV B, respectively. As can be noticed from Eq. (21), each new power

10 ν added to Eq. (11) gives three new Lij ν Redlich-Kister coefficients, one for each binary system i–j. Of course we could write Eq. (21) differently, using, for instance, Legendre polynomials. There is also the need to add ternary terms, as indicated in Eq. (20) by the term we have seen, for ηmax = 2, there is no ternary term, that is,

ex G(tern) m (ηmax =2)

ex G(tern) . m

As

= 0. For ηmax = 3,

the ternary excess term is given by the last term in Eq. (18). Now, for a maximum power of 4 (ηmax = 4) in Eq. (11), the ternary parameter LABC is replaced by three coefficients, so that Eq. (20) is complemented by the ternary excess term ex

(tern)

Gm (ηmax =4) = xA xB xC (xA LA + xB LB + xC LC )

(22)

This is exactly the ternary term proposed by Hillert [7] and commonly used in ternary assessments, being also equivalent to the Margules-type polynomial expansion proposed by Gokcen and Baren [28]. But, as Eq. (21) makes explicit, there is also the need to introduce L2 parameters in the binary descriptions, in order to take full advantage of the series expansion. This procedure, however, is not always followed, and the ternary parameters LA , LB , and LC in Eq. (22) must (mathematically) take AC BC care of the corrections due to LAB 2 , L2 , and L2 . Whether this is a valid procedure will depend on

the particular system being modelled (the ternary and the three binaries). The next power added (ηmax = 5) has a ternary term given by ex

(tern)

(tern)

Gm (ηmax =5) = ex Gm (ηmax =4) + xA xB xC (xA xB LAB + +xA xC LAC + xB xC LBC )

(23)

plus three L3 binary interaction parameters. The authors of the present work have no information regarding the use of this degree of approximation (ηmax = 5). Probably it was never used, although it was discussed by Gokcen and Moser [29]. In any case, the importance of such high-order corrections is expected to be negligible for the vast majority of systems. The same can be said about the approximation ηmax = 6, which gives a L4 interaction parameter for each of the tree binary systems, plus a ternary correction with four additional parameters, which can be written as ex

 (tern) (tern) Gm (ηmax =6) = ex Gm (ηmax =5) + xA xB xC xA xB xC L0ABC + 2 B +x2A (xB − xC )LA ABC + xB (xA − xC )LABC +  x2C (xA − xB )LC ABC

VI.

(24)

CONCLUSION

As we ventured to show in the present work, the inclusion of ternary interaction parameters is mathematically justified on the same grounds as the binary interaction parameters. We are led to agree with the analyses of Helffrich and Wood [30] and Saulov [31], who arrived at similar conclusions as ours, but using different approaches. For the same reason, contrary to what has been said by

11 Janz and Schmid-Fetzer [32], it is not necessary to avoid the use of ternary parameters, although one should be aware of their counter-intuitive behaviour, as reported by those authors and commented upon by Saulov [31]. On the other hand, we stress the fact that the present work does not suggest that one should preclude other formulations of the excess ternary Gibbs energy, such as the use of an extrapolation scheme, as the one proposed by Chartrand and Pelton [33].

Acknowledgements

This work has been financially supported by the S˜ao Paulo State Research Funding Agency (FAPESP, S˜ao Paulo, SP, Brazil) under Procs. 2009/14532-9 and 2012/04023-2, and by the Brazilian National Research Council (CNPq) under Proc. 304445/2010-0.

Appendix A: Demonstration of Eq. (5)

Eq. (5) is simply a generalisation of the expansion proposed by Wagner [1], who gave only the first few terms. The demonstration of the general expression, using the Gibbs-Duhem equation, may be of some interest. For that, it is possible to put Eq. (3) in the form xB

dµex dµex B = xA A d xA d xB

(A1)

valid for a binary system A–B. Besides, Eq. (4) can be expressed more compactly as µex A

=

ηX max η=1

ΩA,η η xB , η

µex B

=

ηX max η=1

ΩB,η η xA η

Therefore, substituting Eq. (A2) in Eq. (A1), we are left with     ηX ηX max max xB ΩB,1 + ΩB,η xA η−1  = xA ΩA,1 + ΩA,η xB η−1  η=2

(A2)

(A3)

η=2

from which we deduce immediately that ΩA,1 = ΩB,1 = 0, as pointed out by Darken [34]. Besides, we should have the equality of the summations in Eq. (A3), so that, after some simplification by dividing both sides by xA xB , ηX max

ΩB,η xA

η−2

=

ηX max

ΩA,η xB η−2

(A4)

η=2

η=2

Using the fact that xB = 1 − xA and the binomial theorem to expand powers of xB , we arrive at   ηX ηX η−2 max max X η−2 η−2 ΩB,η xA = ΩA,η (−1)λ xA λ (A5) λ η=2

η=2 λ=0

Since the right-hand side of Eq. (A5) is a polynomial in xA , we can easily invert the order of the double sum, arriving at ηX max η=2

ΩB,η xA

η−2

=

ηmax max X−2 ηX λ=0

η=λ+2

 ΩA,η

 η−2 (−1)λ xA λ λ

(A6)

12 To have the same indexes in both sides of Eq. (A6), we could perform the formal transformations λ → η − 2 and η → λ (in the right-hand side only), which lead us to ηX max

ΩB,η xA η−2 =

ηX max max ηX

η=2

 ΩA,λ

η=2 λ=η

 λ−2 (−1)η xA η−2 η−2

(A7)

in which we used also the fact that (−1)η−2 = (−1)η . Now we are able to compare equal powers of xA in Eq. (A7), which gives us the final result, ΩB,η =

ηX max λ=η

  λ−2 ΩA,λ (−1)η η−2

(A8)

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