Chemistry for Sustainable Development Calculation of BCC Phase Diagram by mainskweeze


									Chemistry for Sustainable Development# %!!`!&                                                   133

Calculation of BCC Phase Diagram Using the Cluster-Site
Approximation and First Principle Calculations


Theoretical Physics Laboratory, Department of Physics,
Faculty of Science University of Batna, Batna 05000 (Algeria)

    A combin ation of First Princi ple Calculations (FPC) and statistical thermodyn amics, i.e., the Cluster-
Site Approximation (CSA), is applied to describe the bcc-based Fe–Al phase diagram. The formation energies
of ordered compounds are calculated using Full-Potential Linearised Augmented Plane Wave (FP-LAPW)
results, and the entropy term is evaluated using the so-called modified Cluster Variation Method (CVM).
The CSA model has been used to model the bcc bases in the Fe–Al system. The results obtained from this
combin ation are compared with those obtained from the irregular tetrahedron approximation of the CVM,
with the same FP-LAPW total energies.

INTRODUCTION                                                tion, i.e., the computation simplicity. And it
                                                            can take into account both long and short
   The Cluster Site Approximation (CSA) model               range order, thus retaining the ability of
was used to model fcc phases in several sys-                CVM method [8]. This makes the CSA ideal
tems [1–3], this model is based on two approxi-             for multicomponent phase diagram calcula-
mations: Bragg–Williams [4, 5], and physically              tions [9].
sounder Cluster Variation Method (CVM) [6, 7].                  The configurational entropy in the cluster-
Since the clusters in the CSA are non-interfer-             site approximation was derived for the fcc phases
ing (see Fig. 1, b), the independent variables              by Yang and Li [10, 11]; but the application of
are the site probabilities, thus retaining the              this method to the calculation of the bcc phase
advantage of the Bragg–Williams approxima-                  equilibria has never appeared in the literature.
                                                            Then in the present study, formulations of the
                                                            CSA for bcc structure were performed, based
                                                            on the non-interfering irregular tetrahedron clus-
                                                            ter in the CVM method.
                                                                Fe–Al is the most important bin ary system
                                                            for both Al and Fe alloys, because it shows ex-
                                                            cellent corrosion and sulphidation resistance
                                                            even at high temperature, reduced density com-
                                                            pared to other ferrous alloys. Addition al engi-
                                                            neering advantages are low raw material and
                                                            processing cost, all these advantages make the
                                                            alloys of this system interesting engineering
                                                            materials [14]. The set of phases used in the
                                                            present work is formed by the A2–Al, A2–Fe,
Fig. 1. Interfering (a) and non-interfering (b) irregular   DO3–Fe3Al, DO3–FeAl3, B2–FeAl and B32–FeAl
tetrahedrons in the bcc structure.                          compounds.
134                                      S. BOURKI a nd M. ZEREG

    The basis set is kept limited here on pur-           The derivation of the CVM entropy formula
pose since only a simplified thermodyn amic de-       was thoroughly outlined in several reviews [20]
scri ption of the bcc system based on the irreg-      and shall not be discussed here. In the bcc lattice,
ular tetrahedron approximation of the cluster         the configurational entropy is written as:
site method is targeted.
    The bin ary phase diagram Fe–Al was pre-
viously investigated experimentally [15], and                                             
theoretically using the combin ation between
                                                                       ∏ (Nz ) ! ∏ (Nx ) !                                                
                                                                                                                                              
                                                                                                    ijk                   i

                                                                                           
                                                                                              ijk                     i

First Princi ple Calculations (FPC) and statisti-      /K
                                                      S B = log             6           4                                                   3 
                                                                                                                                       
                                                                 ∏ (Nρ ) ! ∏ (Ny ) ! ∏ (Ny
cal model CVM [16]. With these new data of                                             αβγδ
                                                                                                                                         ) ! 
the FPC, the present work aims to investigate                               ijkl                ik                      ij             
the ability of the CSA to describe the order-
                                                                       2               αβγδ
disorder transition in the bcc phases and show        where xi , y1 , yij , zijk and ρijkl are the cluster
how the cluster-site approximation can be com-        probabilities of finding the atomic configurations
bined successfully with the First Princi ple Cal-     specified by the subscri pt at a point, nearest-
culations.                                            neighbor pair, second nearest-neighbor pair, at
                                                      a triangle and at a tetrahedron cluster,
                                                      respectively. N presents the number of lattice
                                                         To obtain the phase equilibrium conditions in
    The cluster variation method is based on the
                                                      this method, the grand potential function is used:
concept of a basic cluster defined as a set of
lattice points, chosen in such a way that it                                                         2

contains the maximum correlation length to be          Ω(T, µ1 , µ2 ) = E – TSconf – ∑ µ* xi
                                                             *    *
                                                                                                    i =1
considered [17, 18]. In the present instance the                                                                                         (4)
                                                              + λ(1– ∑ ρijkl )
irregular tetrahedron (IT) is considered to                                  ijkl
describe the superstructure of the cubic-
centered structures [19]. It is the simplest tree     where xi is the mole fraction of component i, λ
dimension al cluster to take into account the         is the Lagrange multi plier to the constraint
first αγ, αδ, βγ and βδ and the second αβ, γβ          ∑ρ     ijkl   = 1, and µ* called the effective chemical
nearest pair interactions in the bcc lattice (see      ijkl

Fig. 1, a).                                           potential is defined as µ* = (µ A – µB )/ 2 where
    Defining the basic cluster we may write the
                                                      µi is the absolute chemical potential of element i.
corresponding thermodyn amic functions. Firstly
                                                          The equilibrium values of the grand potential
the intern al energy for any bcc phase can be
                                                      Ω and corresponding configurations of clusters
described as:
                                                      are obtained by minimizing this function with
E = 6N ∑ ωαβγδρijkl
                                               (1)    respect to ρijkl :

In this expression  ρijkl represents the proba-         ∂Ω
                                                                             =0                                                          (5)
bility of finding {αβγδ} IT cluster with               ∂ρijkl        T ,µ*

configuration {ijkl}, and  ωαβγδ are the eigenen-

ergy associated with this configuration               MODIFIED CVM
 αβγδ  1             1
 ijkl = (ω2 + ωkl ) + (ω1 + ω1 + ω1 + ω1 ) (2)
                        ik   il   jk   jl
       4             6                                    In spite of its successes, a major disadvan-
where ω(1) and ω(2) presents respectively the         tage of the CVM is the large number of inde-
nearest and next nearest-neighbour pair               pendent variables in the free energy function al
interactions.                                         when it is applied to multicomponent solutions
                                                      [8]; i.e., if an alloy contains N components, the

number of independent variables is Nn, where                CLUSTER-SITE APPROXIMATION
n is the number of atoms in the cluster chosen.
But in the CSA method it is in the order of                     The cluster-site approximation is an adap-
N × n, because the independent variables are                tation of the generalized quasi-chemical meth-
                                                            od, introduced many years ago by Fowler for
the site probabilities xiq , q = α, β, γ, δ , instead of
                                                            treating atom-molecule equilibria in gases, and
the cluster probabilities ρijkl in the CVM ap-              used for clusters in solid solutions by Yang and
proximation.                                                Li [10, 11].
    The basic idea of the CSA method is to de-                   The free energy in the CSA approximation
fine the CVM free energy with non-interfering               takes the same form as that in the modified
clusters: they are permitted only to share cor-             CVM in the above section, intern al energy (E)
ners; therefore sub-cluster energies do not en-             is given by Eq. (6) and Eq. (7) and the configu-
ter into the energy equation, i.e. as a correction          ration al entropy (Sconf) by Eq. (8). Therefore we
term. This method was used to calculate differ-             can rewrite the molar free energy as:
ent fcc-based phase diagrams with great suc-                                4
cess [1–3, 9], but not for the bcc-based alloys.             Fm = γRT( ∑ PA xA – ln ϕ)
                                                                          i i

                                                                           i =1
    In this paper we focus our attention on the
bcc alloys, and since the CSA is based on the                                                                    (9)
                                                                   – (4 γ – 1)RT∑          ∑ fi xiq ln xiq )
cluster variation method, the main objective is                                        q   i

to define the free energy of the system with
                                                            here ϕ is the cluster partition function related
non-interfering clusters, i.e., the so-called mod-
                                                            to the cluster energies ωijkl mentioned above by:
ified CVM (see Fig. 1, b). Using the generaliza-
                                                                    2n            4
tion proposed by C. Colinet [21], the intern al
                                                             ϕ = ∑ exp[( ∑ Pni )ijkl – ωijkl                    (10)
energy for a system of N site is:                                   ijkl        i =1

E = γN ∑ ωαβγδ ρijkl
                                                            In Eqs. (9) and (10) the xiq values are the sub-
                                                            lattice species concentrations, and the PA val-
where γ is the number of non-interfering
                                                            ues, are new parameters related to the species
clusters per site, and the tetrahedron energies
                                                            chemical potentials. Since they are related to
will be rewriten as:
                                                            the x iq values, only one of them is required as
ωαβγδ = ω2 + ω2 +ω1 + ω1 + ω1 + ω1
 ijkl    ij   kl  ik   il   jk   jl                 (7)
                                                            independent variables.
non-interfering clusters always result in two
                                                                The first step in the search for equilibrium
term for the entropy, as:
                                                            between two phases is to minimize the grand
                        ( n γ – 1)                        potential (Ω), with respect to the site probabil-
              (Nxi )!
              ∏                                          ities under the constraints of constant temper-
             i                   
S / KB = ln                    γ 
                                                            ature, and effective chemical potential (µ*). The
                  αβγδ 
                                                  (8)     minimization process was performed by the
             ∏ (Nρijkl )! 
              ijkl
                             
                                                          Natural Iterative (NI) algorithm, developed by
                                                            Kikuchi [22], and the equilibrium parameters
   Of course, this method decreases the com-
                                                             Piq values in each sub-lattice q = α,β,γ,δ can be
putation al time, because we can see in Eq. (8)
                                                            done as:
that the number of dependent variables de-
                                                                           γ – 0.25
creases, i.e. the pair and triangle probabilities              q xiq        γ
                                                                                           µ* – µ*    
                                                             P = q                   exp i     B
                                                                                                               (11)
                                                              i                            4γRT       
                                                                 xB 
 (1)   (2)
y ,y
 ij    ij    and tijk do not appear in the entropy                                                    
term, but the number of independent param-                  if the site is occupied by the species B (i = B),
eters is still Nn. However, it gives a good start-          these parameters are equal to unity, i.e.
ing point for generalized cluster site approxi-               q
                                                             PB = 1, q = α, β, γ, δ                             (12)
mation to all alloy structures.
136                                                  S. BOURKI a nd M. ZEREG

and the cluster probabilities can be calculated                    TABLE 2
explicitly as follows:                                             The formation energies and equilibrium lattice
                                                                   parameters of the various phases in the Fe–Al system
          Piα Pjβ Pkγ Plδ
Pijkl =                     exp(–ωijkl / RT)             (13)      Alloy              Ef, J/mol          a, Å
                                                                   Fe                       0            2.801
    The number of independent variables de-
                                                                   Fe3Al (DO3)        –21273.23          2.914
creases to four instead of 16 in the tetrahe-
dron-CVM approximation, that makes the CSA                         FeAl (B32)         –24911.12          2.933

very promising for the multicomponent phase                        FeAl (B2)          –36282.55          2.910
diagram calculations.                                              FeAl3 (DO3)        –9275.42           3.042
    Among various techniques of the first prin-                    Al                      0             3.257
ciple investigation, it is recognized that the com-
bin ation of the electronic structure total ener-
                                                                   to the bcc structure, only the bcc phases were
gy calculation with statistical mechanics calcu-
                                                                   modeled. Thus the phases that will be studied
lations by the CVM [23–26] provides a reliable
                                                                   in this paper are two disordered phases (A2),
                                                                   and four ordered phases with DO3, B32 and B2
    In the present study, we adopted FP-LAPW
electronic structure calculations to obtain the
                                                                       Figure 2 shows the formation energies as a
total energies of a set of selected ordered com-
                                                                   function of composition for all the compounds
pound. These data are plugged into the CSA in
                                                                   of this bin ary system, calculated from the FP-
order to obtain phase boundaries in a bcc alloys
                                                                   LAPW results [16]. From this Figure we can
                                                                   expect by plotting the ground state line the
                                                                   appearance of the B2–FeAl and the DO3–Fe3Al
RESULTS FOR THE Fe–Al SYSTEM                                       superstructure in the phase diagram at low tem-
                                                                   perature, and the B32–FeAl end DO3–FeAl3 will
    Table 1 shows the total energies of the six                    always be metastable and will not appear in
compounds forming the basis for the cluster-                       the equilibrium phase diagram.
site approximation applied to the Fe–Al sys-                           The Fe–Al phase diagram shown in Fig. 3 is
tem, and Table 2 presents the formation ener-                      calculated using the cluster site approximation,
gies of the bcc compounds, as well as the cor-                     and it illustrates the ability of the present mod-
responding equilibrium lattice constant. The                       eling to describe the bcc phases.
an alysis of the formation energies shows that                         The CSA in conjugation with First princi ple
the most stable bcc-based compound in this basis                   calculations is capable of fitting the order-dis-
set is B2.                                                         order equilibrium, the phase diagram calculat-
    In the Fe–Al system there are different
structural phases, since the objective of this
study was to explore the application of the CSA

Fe–Al cohesive energies of the bcc phases
obtained by FP-LAPW [16]

Alloy                Space group        Structure   E, eV/at.
Fe                   Im3m               A2          –6.696
Fe3Al                Fm3m               DO3         –6.127
FeAl                 Fd3m               B32         –5.353

FeAl                 Pm3m               B2          –5.474

FeAl3                Fm 3m              DO3         –4.378
                                                                   Fig. 2. Formation energies as a function of composition
Al                   Im3m               A2          –3.477         for all the compounds studied.

                                                             Kikuchi and Jindo [27] demonstrated that the
                                                             temperature scale of prototype phase diagrams
                                                             can be reduced to about 40 % of the rigid lat-
                                                             tice model value by explicitly taking into account
                                                             the positional degree of freedom of the atoms
                                                             with the continuous displacement cluster varia-
                                                             tion method.
                                                                  It is worth pointing out that Ormeno el al.
                                                             [14] calculated the phase diagram of the Fe–Al
                                                             system using the first princi ple method with-
                                                             out spin polarization (Fig. 4, a) and with spin
                                                             polarization (see Fig. 4, b). The calculated
                                                             phase diagrams are topologically similar to
                                                             that shown in Fig. 3. The transition tempera-
Fig. 3. Fe–Al phase diagram calculated using the CSA model   ture between B2 and A2 states is considerably
for γ = 1.5.                                                 higher in this calculation than that calculat-
                                                             ed in the present work.
ed indicates that the B2–FeAl phase is stable
                                                                 Figure 5 shows the order-disorder transition
over an extended composition range up to the
                                                             (B2→A2) calculated using γ = 1 and γ = 1.5, re-
order-disorder transition temperature of 2650 K
                                                             spectively. An important characteristic of these
and the DO3–Fe3Al compound, by contrast is
                                                             calculations is the relationship between the adjust-
found to be stable up to 950 K, where it un-
                                                             able parameter γ and the critical temperature Tc
dergoes a peritectoid reaction to B2 + A2; these
results indicated that the previous predictions
of the phases that will be appearing in the
phase diagram from the formation energies are
    The transition temperature obtained exper-
imentally is 1583 K [15]. Hence, the present re-
sults of 2650 K are overestimated by 1067 K.
The overestimation origin ates from the reason
that our calculations are based on a rigid lattice
model, which neglects important contributions
to the alloy free energy, like the vibrational term.

                                                             Fig. 5. Phase diagram of bcc Fe–Al alloys as obtained from
Fig. 4. Calculated order-disorder transition A 2 →B 2        ab initio calculation [14]: a – without spin polarization,
for γ = 1.5.                                                 b – with spin polarization.
138                                            S. BOURKI a nd M. ZEREG

of the order-disorder transition, which makes the             3 W. Cao, Y. A. Chang, J. Zhu et al., Ibid., 53 (2005) 331.
determination of γ value more accurate.                       4 W. L. Bragg, E. J. Williams, Porc. Roy. Soc, A145 (1934) 699.
                                                              5 W. L. Bragg, E. J. Williams, Ibid., A151 (1935) 540.
                                                              6 R. Kikuchi, Phys. Rev., 81 (1951) 988.
                                                              7 R. Kikuchi, Acta Metallurgica, V 25 (1977) 195.
                                                              8 Y. A. Chang, S. Chen et al., Progr. Mat. Sci., 49 (2004) 313.
                                                              9 W. Cao, J. Zhu,Y. Yang et al., Acta. Mater., 53 (2005) 4189.
    In the present study, a general formula of               10 C. N. Yang, J. Chem, J. Phys., 13 (1945) 66.
the CSA for the bcc structure was derived,                   11 C. N. Yang, Y. Li, Ibid., 7 (1947) 59.
based on the entropy expressions reported in                 12 L. G. Ferreira, A. A. Mbaye, A. Zunger, Phys. Rev.,
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                                                             13 F. Zhang , W. A. Oates, S. L . Chen, Y. A. Chang,
The model was applied to the bcc phases in the
                                                                Intermetallics, 9 (2001) 5.
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determined using the FPC results.                            15 M. Hansen, Constitution of Bin ary Alloys, McGraw-
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encouraging because all the predicted phases                 16 P. G. G. Ormeno, Determin ation of the Fe–Al Phase
                                                                Diagram Using the First Princi ple Calculation, PhD
appear in the calculated phase diagram. In ad-
                                                                thesis, Sao Paulo University, 2002, Brazil.
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ab initio method.                                            18 J. M. Sanchez and D. de Fontaine, Phys. Rev. B,
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                                                             19 R. Kikuchi and G. M. Van Baal, Scri pta Metallurgica,
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                                                             22 R. Kikuchi, Prog. Theor. Phys. Suppl., 115(1994).
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