- Email: [email protected]
Theoretical study of vibrational contribution on cluster formation in a binary alloy system
Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171 www.elsevier.com/locate/calphad
Theoretical study of vibrational contribution on cluster formation in a binary alloy system Koretaka Yugea,∗, Shigeto R. Nishitanib, Isao Tanakaa a Department of Materials Science and Engineering, Kyoto University, Japan b Department of Informatics, Kwansei Gakuin University, Japan
Received 11 June 2004; received in revised form 16 August 2004; accepted 23 August 2004 Available online 15 September 2004
Abstract Lattice vibrational effects in the process of precipitation in a binary alloy system has been studied theoretically. We derived a cancellation condition of vibrational free energy change due to cluster formation within the first order approximation, in terms of the nearest bond pair interaction. Vibrational contribution to the free energy is calculated for a specific example of cluster formation from isolated atoms, and it is almost perfectly vanished when the unlike atom potential is assumed to be the geometric mean of their constituent potentials. © 2004 Elsevier Ltd. All rights reserved. PACS: 61.46.+w; 63.20.-e; 63.22.+m; 64.60.Qb; 65.40.Gr; 65.80.+n; 81.30.Mh Keywords: Nucleation; Precipitation; Vibration; Cluster formation; Cancellation; Phonon; Vibrational entropy; Binary alloy
1. Introduction In the first half of the 1990’s, phase diagram calculations based on the first principles theory typically take account of the configurational entropy, while the contribution of lattice vibration to the free energy had been neglected though this justification of assumption had no foundation. The most popular approach to calculating the phase diagram, cluster expansion method, using the ab initio calculation to obtain the sets of effective pair or cluster interactions has been introduced by Connolly and Williams [1] and later in 1993, Sanchez and Becker [2] suggested that the vibrational contribution to the free energy would be included through the Debye–Grüneisen analysis, which gives a reasonable thermal property comparing to the experimental data [3,4]. Thus, the method of treating the vibrational effects from ab initio calculation has been established. Recent experiments indicate that the vibrational effects in fact play important roles as well as configurational entropy ∗ Corresponding author. Tel.: +81 757 535 435; fax: +81 757 535 447.
E-mail address: [email protected] (K. Yuge). 0364-5916/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2004.08.006
in dealing with the phase stability in many systems [5–8]. Following these stimulating experimental results, in 1996 Garbulsky and Ceder [9] suggested the relation for substitutional alloy between its transition temperature and vibrational interaction in terms of the cluster expansion method: the importance of vibrational effects on transition temperature has been pointed out. Very recently the study of vibrational effects on phase stability based on the first principles calculation has been actively carried out for real binary systems: van de Walle and Ceder [10] found the vibrational entropy difference to be zero for Ni3 Al system with their calculation, contrary to the previous study’s expectation, and Ozolinš et al. [11] found that the vibrational entropy lowers the transition temperatures by 15% for Cu–Al alloy. Then the systematic study of the vibrational effect on substitutional alloy dynamics had been performed by van de Walle and Ceder [12] in 2002: the ‘bond proportion model’, which gives a simple interpretation of the vibrational entropy differences between order and disorder states in terms of the nearest-neighbor bond proportion effect, has been introduced. This model provides a derivation of expressing the nearest-neighbor
168
K. Yuge et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171
The phonon LDOS in the α (α = x, y, z) direction at the j -th atom, n j α (ω) is expressed as follows: 2ω G j αj α (ω2 + i 0) π 1 2ω |α, j , lim α, j | 2 =− π →0 (ω + i )I − D
n j α (ω) = −
Fig. 1. Schematic illustration of precipitation. Empty circle denotes the solvent atom ‘A’, and filled circle the solute ‘B’.
where G j αj α (ω2 ), the Green’s function, denotes the diagonal matrix element of the resolvent operator [ω2 I − D]−1 , I is the unit matrix, and |α, j is the state vector corresponding to the vibration at site j in the α direction. D in Eq. (3) is the dynamical matrix described as 1
effective cluster interactions, which indicates the intuitive interpretation of the vibrational effect on an alloy’s stability. In the present study, we will focus on the investigation of ‘configurational-dependent’ vibrational free energy change. The derived expression is applied on newly proposed treatment of nucleation free energy change in the process of precipitation, where the configurational contribution to the free energy has been included but the vibrational contribution has not [13]. We will derive a universally approved relation between vibrational free energy change and spring constants between like and unlike atom pairs. 2. Present treatment of vibrational dynamics and its application 2.1. Approximation of evaluating vibrational free energy with interatomic force constants We first consider the given binary alloy system at constant temperature T , with initial and final atomic configurations illustrated schematically in Fig. 1. The total free energy change of isolated atoms for cluster formation, Ftot , is defined as Ftot ≡ Ffin (T ) − Fini (T ),
(1)
where Ffin (T ) and Fini (T ) are the total free energy for final and initial states respectively. Since Ftot can be divided into three terms of static E stat, configurational Fconfig and vibrational Fvib contributions, Eq. (1) is transformed into the following: Ftot = E stat(T ) + Fconfig (T ) + Fvib (T ).
(2)
E stat(T ) is directly obtained through ab initio calculation for a given atomic configuration, though in fact the static contribution for equilibrium state is difficult to be precisely estimated due to correlation between Ftot and the stable atomic configuration. Fconfig (T ) is typically treated under the ideal or regular solution approximation. Here in after we will focus on the term Fvib (T ). In order to discuss vibrational effects for such atomic configurations where perfect periodicity is lacking, the phonon local density of states (LDOS) will be introduced.
(3)
1
D = M− 2 Φ M− 2 ,
(4)
where M is the atomic mass tensor and Φ is the interatomic force constant matrix. The Green’s function G j αj α (ω2 ) can be expressed in a continued fraction expansion form as 1
G j αj α (ω2 ) = ω2
b12
− a0 −
,
(5)
b22 ω 2 − a2 − · · · where coefficients {an } and {bn } are evaluated from the orthonormality property of the new basis, with starting vector |α, j . This method, called a ‘recursion method’ [14, 15], has been applied to the calculations of the electronic and vibrational DOS of surface and amorphous materials [16]. Expanding the Green’s function G(ω2 ) for a higher order n of coefficients an and bn corresponds to including a higher moment of lattice vibrational mode. Computationally, this recursion method consists of calculating all coefficients up to the maximum order rmax and approximate higher order coefficients {an } and {bn } for n ≥ rmax + 1 in a proper way to terminate a continued fraction. In the present study, we will expand Green’s function G(ω2 ) up to the first order rmax = 1, and terminate G(ω2 ); ω 2 − a1 −
G j αj α (ω2 ) =
1 . ω2 − a − b 2 G j αj α (ω2 )
(6)
Applying this operation of Eq. (6), which is the effective medium approximation of the 2nd order moment expansion of Eq. (5), to Eq. (3) naturally leads to the semi-elliptic form of expressing the local density of states g j α (ω2 ) whose center of gravity and bandwidth are specified by the coefficients a and b respectively. The derived semi-elliptic DOS becomes identical to that of Sutton’s [17]. When we determine the coefficient b so that the minimum vibrational frequency of LDOS corresponds to zero, g j α (ω2 ) = −1/πG j αj α (ω2 + i 0) becomes 1/2 2 (µ j α )2 − (ω2 − µ j α )2 g j α (ω2 ) = , (7) πµ j α where µ j α equals to D j αj α . Note that Eq. (7) can only hold for ω2 ∈ [0, 2µ j α ]. The above expression satisfies that the DOS n(ω) is proportional to the square of frequency ω2 at the lower band edge, which is the desirable behavior
K. Yuge et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171
169
of the real DOS. One interesting point of Eq. (7) is that one can construct the semi-elliptic density of states using only the diagonal elements of the dynamical matrix. This improvement affects so much on the lower or intermediate temperature beyond the simple high temperature limit models [9,18]. Using the expression of Eq. (7), the local vibrational contribution in the α direction to the free energy from site j , F j α is expressed as ∞ ω 2 dω 2ωg j α (ω ) ln 2 sinh F j α = kB T 2kB T 0 c y 16kB T 1 2 jα = y (1 − y 2 )1/2 ln 2 sinh dy, (8) π 2 0 where c jα =
(2µ j α )1/2 , kB T
y=
ω . (2µ j α )1/2
(9)
In the above derivation of Eq. (8), it is clear that the local vibrational free energy F j α can be uniquely determined by one parameter µ j α . When we assume that the interatomic potentials are described by the usual pairwise interaction, then the parameter µ j α is determined by the sum of the surrounding bonds’ contribution:
k j l Cαα M −1 (10) µ jα = j , l
where k j l denotes the scalar force constant between the j -th and l-th atom, C is the dimensionless matrix depending only on a lattice symmetry, and M j is the atomic mass of the j -th atom. In terms of the numerical calculation, Eqs. (8)–(10) can be combined with the interatomic force matrix obtained through first principles calculation, and lead to the vibrational free energy within the first-order approximation of the phonon DOS. Combining Eqs. (8)–(10) with Seko’s [13] calculational method can be easily achieved through ab initio calculation directly, which enables us to estimate the nucleation free energy including the vibrational effects precisely. 2.2. Expression of vibrational free energy change due to cluster formation Following the above discussion, vibrational free energy on atom j can be estimated in a given atomic configuration when scalar force constant between a like- and unlike-atom pair is determined. Here going back to the vibrational free energy change with cluster formation in an A–B binary alloy system shown in Fig. 1. In order to discuss the configurational dependent vibrational free energy, we first assume that the lattice symmetry of the bulk retains in the process of precipitation. Consider the spring constant at atom j in the α direction, k j α = l k j l Cαα , changes into k j α (1 + δk j α ) due to the surrounding atomic configuration changing. Then the vibrational free energy change at atom j
Fig. 2. Taylor coefficient f (µ j α , T ) as a function of µ j α at specific temperatures.
in the α direction is expressed as
F j α = f (µ j α 1 + δk j α , T ) − f (µ j α , T ) f (µ j α , T )δk j α + O[(δk 2j α )],
(11)
where f corresponds to the expression in Eq. (8), and f (µ j α , T ) is the first coefficient of the Taylor series for f (µ j α (1 + δk j α ), T ) respectively. Therefore, the total vibrational free energy change is described as
F j α Fvib = j,α
f (µ j α , T )δk j α .
(12)
j,α
Fig. 2 shows the Taylor coefficient f (µ j α , T ) as a function of µ j α at specific temperatures. It is clear from Fig. 2 that at the higher temperature than T 1000, the first Taylor coefficient becomes fairly constant. Analytically, this Taylor coefficient becomes (1/2)kB T in the high temperature limit, in terms of the Taylor expansion of F j α in Eq. (8) with c j α . Therefore, the vibrational free energy change due to cluster formation becomes Fvib f (µ j α , T )
δk j α =
j,α
1 kB T δk j α . 2
(13)
j,α
When one considers the cancellation of Fvib , the summation of the proportion for k j α , j,α δk j α = 0 is required. This requirement is equivalent to the condition where the summation of the difference for k j α , j,α k j α = 0 is satisfied within the first approximation of δk j α , which is already satisfied in Eq. (12). Here, the relation between δk j α and k j α is as follows: k j α (1 + δk j α ) = k j α + k j α .
(14)
170
K. Yuge et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171
Then the cancellation of Fvib can be rewritten as
k j = 0,
(15)
j
where k j = α k j α . Under the symmetry where the nearestneighbor bond pair is equivalent for all lattice sites, the sets of diagonal elements of dimensionless matrix C become equivalent and we now consider such a case. Therefore, the following cancellation condition of Fvib is immediately derived from Eq. (15): kAA + kBB → Fvib 0, (16) 2 where ki j denotes the spring constant between i and j atoms, respectively. Here it should be noted that in terms of the cancellation condition, the arithmetic mean of the spring constant of like-atom pair, namely (kAA + kBB )/2,√cannot be distinguished from the geometric mean, namely kAA kBB , because Eq. (16) derived from Eq. (15) is based on the firstorder Taylor expansion for the change of spring constants in Eq. (12). For the purpose of understanding the above cancellation condition in Eq. (16) more intuitively, we will show an example of vibrational free energy change Fvib due to cluster formation. We consider an initial and a final state of cluster formation process in a square lattice illustrated in Fig. 1. Fig. 3 shows the vibrational free energy change Fvib at T = 1000 K of these two states calculated from Eqs. (8)–(10), as a function of ln(kAA /kBB ) at specific r , where r = kAB /((kAA + kBB )/2), the ratio of spring constants between the unlike pair kAB and the geometrical average of their constituent pairs (kAA + kBB )/2. Though Fvib can be roughly estimated by using Eq. (13), we use Eqs. (8)–(10) alternatively because of more quantitative estimation. It is clear from Fig. 3 that when the spring constant of an unlike-atom pair is equal to the arithmetic mean of its constituent spring constants, namely r = 1, the cancellation of Fvib is achieved for the wide range of the ratio (kAA/kBB ). Additionally, Fvib becomes a negative value when the spring constant of an unlike-atom pair, kAB excesses the arithmetic mean of the spring constants of a like-atom pair, and vice versa. The same tendency for other cluster structures of different size and shape and for various temperatures is found through our calculation, and the validity of the derived cancellation condition has been seen. We will point out one more important derivation from Eq. (16), for the determination of the empirical interatomic potentials. The widely used empirical interatomic pair potential, Morse potential, is written as kAB =
ψ(Ri j ) = D(exp(−2 p(Ri j − R0 )) −2 exp(− p(Ri j − R0 ))),
(17)
where Ri j denotes the distance between the i -th and j -th atom, R0 the equilibrium atomic distance, and parameter D and p specifies the depth and curvature of potential,
Fig. 3. Vibrational free energy change due to cluster formation at T = 1000 K illustrated in Fig. 1, as a function of ln(kAA /kBB ) at specific values of r, where r = kAB /((kAA + kBB )/2), the ratio of spring constants between the unlike pair kAB and the arithmetic average of their constituent pairs (kAA + kBB )/2.
respectively. These potential parameters for a like-atom pair are typically determined by fitting the calculated physical property to the experimental data, while the potential for an unlike-atom pair is usually determined by the geometric mean, namely (18) ψAB = ψAA ψBB . For such a case, because the interatomic force constant is obtained by the second derivative of the potential, kAB is written by the arithmetic mean, namely kAA + kBB . (19) 2 In order to obtain Eq. (19), we have to assume that parameters D and R0 are identical for all the atomic pairs. For more general potentials, this assumption is not valid, but at least it is true for the normalized hardness [19]. On the contrary, when one employs the assumption of Eq. (18), the vibrational contribution to the free energy hardly appears due to Eq. (16). kAB =
3. Conclusions We have investigated the vibrational contribution on the free energy change. We have shown the expression of the vibrational free energy derived from the effective medium approximation of the 2nd order moment expansion of Green’s function of a phonon, whose semi-elliptic DOS is determined only by the diagonal element of the dynamical matrix. Then we derived a cancellation condition of vibrational free energy change due to the bond pair recombination at fixed temperature. At the first order approximation, the vibrational free energy change should be
K. Yuge et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171
vanished if taking the unlike-atom pair spring constant to be the arithmetic mean of those of their constituent pairs. We have also seen the validity of the cancellation condition from the calculation for a given square lattice model of cluster formation in the process of precipitation. Another important notification has been pointed out that if the unlike atom pair potential is assumed to be the geometric mean of their constituent potentials, the vibrational free energy change vanishes almost perfectly. References [1] J.W.D. Connolly, A.R. Williams, Phys. Rev. B 27 (1983) 5169. [2] J.M. Sanchez, J.D. Becker, Mater. Res. Soc. Sympos. Proc. 291 (1993) 115. [3] V.L. Moruzzi, J.F. Janak, K. Schwarz, Phys. Rev. B 37 (1988) 790. [4] J.M. Sanchez, J.P. Stark, V.L. Moruzzi, Phys. Rev. B 44 (1991) 5411.
171
[5] L. Anthony, J.K. Okamoto, B. Fultz, Phys. Rev. Lett. 70 (1993) 1128. [6] L. Anthony, L.J. Nagel, J.K. Okamoto, B. Fultz, Phys. Rev. Lett. 73 (1994) 3034. [7] B. Fultz, L. Anthony, L.J. Nagel, R.M. Nicklow, S. Spooner, Phys. Rev. B 52 (1995) 3315. [8] B. Fultz, L. Anthony, J.L. Robertson, R.M. Nicklow, S. Spooner, M. Mostoller, Phys. Rev. B 52 (1995) 3280. [9] G.D. Garbulsky, G. Ceder, Phys. Rev. B 53 (1996) 8993. [10] A. van de Walle, G. Ceder, Phys. Rev. Lett. 80 (1998) 4911. [11] V. Ozolinš, C. Wolverton, A. Zunger, Phys. Rev. B 58 (1998) 5897. [12] A. van de Walle, G. Ceder, Rev. Mod. Phys. 74 (2002) 11. [13] A. Seko, N. Odagaki, S.R. Nishitani, I. Tanaka, H. Adachi, Mater. Trans. 45 (2004) 1978. [14] R. Haydock, V. Heine, M. Kelly, J. Phys. C 5 (1972) 2845. [15] R. Haydock, V. Heine, M. Kelly, J. Phys. C 8 (1975) 2591. [16] P.E. Meek, Phil. Mag. 33 (1976) 897. [17] A.P. Sutton, Phil. Mag. A 60 (1989) 147. [18] G.D. Garbulsky, G. Ceder, Phys. Rev. B 49 (1994) 6327. [19] D.G. Pettifor, Bonding and Structure of Molecules and Solids, Clarendon Press, Oxford, 1995, p. 85.
Theoretical study of vibrational contribution on cluster formation in a binary alloy system Koretaka Yugea,∗, Shigeto R. Nishitanib, Isao Tanakaa a Department of Materials Science and Engineering, Kyoto University, Japan b Department of Informatics, Kwansei Gakuin University, Japan
Received 11 June 2004; received in revised form 16 August 2004; accepted 23 August 2004 Available online 15 September 2004
Abstract Lattice vibrational effects in the process of precipitation in a binary alloy system has been studied theoretically. We derived a cancellation condition of vibrational free energy change due to cluster formation within the first order approximation, in terms of the nearest bond pair interaction. Vibrational contribution to the free energy is calculated for a specific example of cluster formation from isolated atoms, and it is almost perfectly vanished when the unlike atom potential is assumed to be the geometric mean of their constituent potentials. © 2004 Elsevier Ltd. All rights reserved. PACS: 61.46.+w; 63.20.-e; 63.22.+m; 64.60.Qb; 65.40.Gr; 65.80.+n; 81.30.Mh Keywords: Nucleation; Precipitation; Vibration; Cluster formation; Cancellation; Phonon; Vibrational entropy; Binary alloy
1. Introduction In the first half of the 1990’s, phase diagram calculations based on the first principles theory typically take account of the configurational entropy, while the contribution of lattice vibration to the free energy had been neglected though this justification of assumption had no foundation. The most popular approach to calculating the phase diagram, cluster expansion method, using the ab initio calculation to obtain the sets of effective pair or cluster interactions has been introduced by Connolly and Williams [1] and later in 1993, Sanchez and Becker [2] suggested that the vibrational contribution to the free energy would be included through the Debye–Grüneisen analysis, which gives a reasonable thermal property comparing to the experimental data [3,4]. Thus, the method of treating the vibrational effects from ab initio calculation has been established. Recent experiments indicate that the vibrational effects in fact play important roles as well as configurational entropy ∗ Corresponding author. Tel.: +81 757 535 435; fax: +81 757 535 447.
E-mail address: [email protected] (K. Yuge). 0364-5916/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2004.08.006
in dealing with the phase stability in many systems [5–8]. Following these stimulating experimental results, in 1996 Garbulsky and Ceder [9] suggested the relation for substitutional alloy between its transition temperature and vibrational interaction in terms of the cluster expansion method: the importance of vibrational effects on transition temperature has been pointed out. Very recently the study of vibrational effects on phase stability based on the first principles calculation has been actively carried out for real binary systems: van de Walle and Ceder [10] found the vibrational entropy difference to be zero for Ni3 Al system with their calculation, contrary to the previous study’s expectation, and Ozolinš et al. [11] found that the vibrational entropy lowers the transition temperatures by 15% for Cu–Al alloy. Then the systematic study of the vibrational effect on substitutional alloy dynamics had been performed by van de Walle and Ceder [12] in 2002: the ‘bond proportion model’, which gives a simple interpretation of the vibrational entropy differences between order and disorder states in terms of the nearest-neighbor bond proportion effect, has been introduced. This model provides a derivation of expressing the nearest-neighbor
168
K. Yuge et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171
The phonon LDOS in the α (α = x, y, z) direction at the j -th atom, n j α (ω) is expressed as follows: 2ω G j αj α (ω2 + i 0) π 1 2ω |α, j , lim α, j | 2 =− π →0 (ω + i )I − D
n j α (ω) = −
Fig. 1. Schematic illustration of precipitation. Empty circle denotes the solvent atom ‘A’, and filled circle the solute ‘B’.
where G j αj α (ω2 ), the Green’s function, denotes the diagonal matrix element of the resolvent operator [ω2 I − D]−1 , I is the unit matrix, and |α, j is the state vector corresponding to the vibration at site j in the α direction. D in Eq. (3) is the dynamical matrix described as 1
effective cluster interactions, which indicates the intuitive interpretation of the vibrational effect on an alloy’s stability. In the present study, we will focus on the investigation of ‘configurational-dependent’ vibrational free energy change. The derived expression is applied on newly proposed treatment of nucleation free energy change in the process of precipitation, where the configurational contribution to the free energy has been included but the vibrational contribution has not [13]. We will derive a universally approved relation between vibrational free energy change and spring constants between like and unlike atom pairs. 2. Present treatment of vibrational dynamics and its application 2.1. Approximation of evaluating vibrational free energy with interatomic force constants We first consider the given binary alloy system at constant temperature T , with initial and final atomic configurations illustrated schematically in Fig. 1. The total free energy change of isolated atoms for cluster formation, Ftot , is defined as Ftot ≡ Ffin (T ) − Fini (T ),
(1)
where Ffin (T ) and Fini (T ) are the total free energy for final and initial states respectively. Since Ftot can be divided into three terms of static E stat, configurational Fconfig and vibrational Fvib contributions, Eq. (1) is transformed into the following: Ftot = E stat(T ) + Fconfig (T ) + Fvib (T ).
(2)
E stat(T ) is directly obtained through ab initio calculation for a given atomic configuration, though in fact the static contribution for equilibrium state is difficult to be precisely estimated due to correlation between Ftot and the stable atomic configuration. Fconfig (T ) is typically treated under the ideal or regular solution approximation. Here in after we will focus on the term Fvib (T ). In order to discuss vibrational effects for such atomic configurations where perfect periodicity is lacking, the phonon local density of states (LDOS) will be introduced.
(3)
1
D = M− 2 Φ M− 2 ,
(4)
where M is the atomic mass tensor and Φ is the interatomic force constant matrix. The Green’s function G j αj α (ω2 ) can be expressed in a continued fraction expansion form as 1
G j αj α (ω2 ) = ω2
b12
− a0 −
,
(5)
b22 ω 2 − a2 − · · · where coefficients {an } and {bn } are evaluated from the orthonormality property of the new basis, with starting vector |α, j . This method, called a ‘recursion method’ [14, 15], has been applied to the calculations of the electronic and vibrational DOS of surface and amorphous materials [16]. Expanding the Green’s function G(ω2 ) for a higher order n of coefficients an and bn corresponds to including a higher moment of lattice vibrational mode. Computationally, this recursion method consists of calculating all coefficients up to the maximum order rmax and approximate higher order coefficients {an } and {bn } for n ≥ rmax + 1 in a proper way to terminate a continued fraction. In the present study, we will expand Green’s function G(ω2 ) up to the first order rmax = 1, and terminate G(ω2 ); ω 2 − a1 −
G j αj α (ω2 ) =
1 . ω2 − a − b 2 G j αj α (ω2 )
(6)
Applying this operation of Eq. (6), which is the effective medium approximation of the 2nd order moment expansion of Eq. (5), to Eq. (3) naturally leads to the semi-elliptic form of expressing the local density of states g j α (ω2 ) whose center of gravity and bandwidth are specified by the coefficients a and b respectively. The derived semi-elliptic DOS becomes identical to that of Sutton’s [17]. When we determine the coefficient b so that the minimum vibrational frequency of LDOS corresponds to zero, g j α (ω2 ) = −1/πG j αj α (ω2 + i 0) becomes 1/2 2 (µ j α )2 − (ω2 − µ j α )2 g j α (ω2 ) = , (7) πµ j α where µ j α equals to D j αj α . Note that Eq. (7) can only hold for ω2 ∈ [0, 2µ j α ]. The above expression satisfies that the DOS n(ω) is proportional to the square of frequency ω2 at the lower band edge, which is the desirable behavior
K. Yuge et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171
169
of the real DOS. One interesting point of Eq. (7) is that one can construct the semi-elliptic density of states using only the diagonal elements of the dynamical matrix. This improvement affects so much on the lower or intermediate temperature beyond the simple high temperature limit models [9,18]. Using the expression of Eq. (7), the local vibrational contribution in the α direction to the free energy from site j , F j α is expressed as ∞ ω 2 dω 2ωg j α (ω ) ln 2 sinh F j α = kB T 2kB T 0 c y 16kB T 1 2 jα = y (1 − y 2 )1/2 ln 2 sinh dy, (8) π 2 0 where c jα =
(2µ j α )1/2 , kB T
y=
ω . (2µ j α )1/2
(9)
In the above derivation of Eq. (8), it is clear that the local vibrational free energy F j α can be uniquely determined by one parameter µ j α . When we assume that the interatomic potentials are described by the usual pairwise interaction, then the parameter µ j α is determined by the sum of the surrounding bonds’ contribution:
k j l Cαα M −1 (10) µ jα = j , l
where k j l denotes the scalar force constant between the j -th and l-th atom, C is the dimensionless matrix depending only on a lattice symmetry, and M j is the atomic mass of the j -th atom. In terms of the numerical calculation, Eqs. (8)–(10) can be combined with the interatomic force matrix obtained through first principles calculation, and lead to the vibrational free energy within the first-order approximation of the phonon DOS. Combining Eqs. (8)–(10) with Seko’s [13] calculational method can be easily achieved through ab initio calculation directly, which enables us to estimate the nucleation free energy including the vibrational effects precisely. 2.2. Expression of vibrational free energy change due to cluster formation Following the above discussion, vibrational free energy on atom j can be estimated in a given atomic configuration when scalar force constant between a like- and unlike-atom pair is determined. Here going back to the vibrational free energy change with cluster formation in an A–B binary alloy system shown in Fig. 1. In order to discuss the configurational dependent vibrational free energy, we first assume that the lattice symmetry of the bulk retains in the process of precipitation. Consider the spring constant at atom j in the α direction, k j α = l k j l Cαα , changes into k j α (1 + δk j α ) due to the surrounding atomic configuration changing. Then the vibrational free energy change at atom j
Fig. 2. Taylor coefficient f (µ j α , T ) as a function of µ j α at specific temperatures.
in the α direction is expressed as
F j α = f (µ j α 1 + δk j α , T ) − f (µ j α , T ) f (µ j α , T )δk j α + O[(δk 2j α )],
(11)
where f corresponds to the expression in Eq. (8), and f (µ j α , T ) is the first coefficient of the Taylor series for f (µ j α (1 + δk j α ), T ) respectively. Therefore, the total vibrational free energy change is described as
F j α Fvib = j,α
f (µ j α , T )δk j α .
(12)
j,α
Fig. 2 shows the Taylor coefficient f (µ j α , T ) as a function of µ j α at specific temperatures. It is clear from Fig. 2 that at the higher temperature than T 1000, the first Taylor coefficient becomes fairly constant. Analytically, this Taylor coefficient becomes (1/2)kB T in the high temperature limit, in terms of the Taylor expansion of F j α in Eq. (8) with c j α . Therefore, the vibrational free energy change due to cluster formation becomes Fvib f (µ j α , T )
δk j α =
j,α
1 kB T δk j α . 2
(13)
j,α
When one considers the cancellation of Fvib , the summation of the proportion for k j α , j,α δk j α = 0 is required. This requirement is equivalent to the condition where the summation of the difference for k j α , j,α k j α = 0 is satisfied within the first approximation of δk j α , which is already satisfied in Eq. (12). Here, the relation between δk j α and k j α is as follows: k j α (1 + δk j α ) = k j α + k j α .
(14)
170
K. Yuge et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171
Then the cancellation of Fvib can be rewritten as
k j = 0,
(15)
j
where k j = α k j α . Under the symmetry where the nearestneighbor bond pair is equivalent for all lattice sites, the sets of diagonal elements of dimensionless matrix C become equivalent and we now consider such a case. Therefore, the following cancellation condition of Fvib is immediately derived from Eq. (15): kAA + kBB → Fvib 0, (16) 2 where ki j denotes the spring constant between i and j atoms, respectively. Here it should be noted that in terms of the cancellation condition, the arithmetic mean of the spring constant of like-atom pair, namely (kAA + kBB )/2,√cannot be distinguished from the geometric mean, namely kAA kBB , because Eq. (16) derived from Eq. (15) is based on the firstorder Taylor expansion for the change of spring constants in Eq. (12). For the purpose of understanding the above cancellation condition in Eq. (16) more intuitively, we will show an example of vibrational free energy change Fvib due to cluster formation. We consider an initial and a final state of cluster formation process in a square lattice illustrated in Fig. 1. Fig. 3 shows the vibrational free energy change Fvib at T = 1000 K of these two states calculated from Eqs. (8)–(10), as a function of ln(kAA /kBB ) at specific r , where r = kAB /((kAA + kBB )/2), the ratio of spring constants between the unlike pair kAB and the geometrical average of their constituent pairs (kAA + kBB )/2. Though Fvib can be roughly estimated by using Eq. (13), we use Eqs. (8)–(10) alternatively because of more quantitative estimation. It is clear from Fig. 3 that when the spring constant of an unlike-atom pair is equal to the arithmetic mean of its constituent spring constants, namely r = 1, the cancellation of Fvib is achieved for the wide range of the ratio (kAA/kBB ). Additionally, Fvib becomes a negative value when the spring constant of an unlike-atom pair, kAB excesses the arithmetic mean of the spring constants of a like-atom pair, and vice versa. The same tendency for other cluster structures of different size and shape and for various temperatures is found through our calculation, and the validity of the derived cancellation condition has been seen. We will point out one more important derivation from Eq. (16), for the determination of the empirical interatomic potentials. The widely used empirical interatomic pair potential, Morse potential, is written as kAB =
ψ(Ri j ) = D(exp(−2 p(Ri j − R0 )) −2 exp(− p(Ri j − R0 ))),
(17)
where Ri j denotes the distance between the i -th and j -th atom, R0 the equilibrium atomic distance, and parameter D and p specifies the depth and curvature of potential,
Fig. 3. Vibrational free energy change due to cluster formation at T = 1000 K illustrated in Fig. 1, as a function of ln(kAA /kBB ) at specific values of r, where r = kAB /((kAA + kBB )/2), the ratio of spring constants between the unlike pair kAB and the arithmetic average of their constituent pairs (kAA + kBB )/2.
respectively. These potential parameters for a like-atom pair are typically determined by fitting the calculated physical property to the experimental data, while the potential for an unlike-atom pair is usually determined by the geometric mean, namely (18) ψAB = ψAA ψBB . For such a case, because the interatomic force constant is obtained by the second derivative of the potential, kAB is written by the arithmetic mean, namely kAA + kBB . (19) 2 In order to obtain Eq. (19), we have to assume that parameters D and R0 are identical for all the atomic pairs. For more general potentials, this assumption is not valid, but at least it is true for the normalized hardness [19]. On the contrary, when one employs the assumption of Eq. (18), the vibrational contribution to the free energy hardly appears due to Eq. (16). kAB =
3. Conclusions We have investigated the vibrational contribution on the free energy change. We have shown the expression of the vibrational free energy derived from the effective medium approximation of the 2nd order moment expansion of Green’s function of a phonon, whose semi-elliptic DOS is determined only by the diagonal element of the dynamical matrix. Then we derived a cancellation condition of vibrational free energy change due to the bond pair recombination at fixed temperature. At the first order approximation, the vibrational free energy change should be
K. Yuge et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 167–171
vanished if taking the unlike-atom pair spring constant to be the arithmetic mean of those of their constituent pairs. We have also seen the validity of the cancellation condition from the calculation for a given square lattice model of cluster formation in the process of precipitation. Another important notification has been pointed out that if the unlike atom pair potential is assumed to be the geometric mean of their constituent potentials, the vibrational free energy change vanishes almost perfectly. References [1] J.W.D. Connolly, A.R. Williams, Phys. Rev. B 27 (1983) 5169. [2] J.M. Sanchez, J.D. Becker, Mater. Res. Soc. Sympos. Proc. 291 (1993) 115. [3] V.L. Moruzzi, J.F. Janak, K. Schwarz, Phys. Rev. B 37 (1988) 790. [4] J.M. Sanchez, J.P. Stark, V.L. Moruzzi, Phys. Rev. B 44 (1991) 5411.
171
[5] L. Anthony, J.K. Okamoto, B. Fultz, Phys. Rev. Lett. 70 (1993) 1128. [6] L. Anthony, L.J. Nagel, J.K. Okamoto, B. Fultz, Phys. Rev. Lett. 73 (1994) 3034. [7] B. Fultz, L. Anthony, L.J. Nagel, R.M. Nicklow, S. Spooner, Phys. Rev. B 52 (1995) 3315. [8] B. Fultz, L. Anthony, J.L. Robertson, R.M. Nicklow, S. Spooner, M. Mostoller, Phys. Rev. B 52 (1995) 3280. [9] G.D. Garbulsky, G. Ceder, Phys. Rev. B 53 (1996) 8993. [10] A. van de Walle, G. Ceder, Phys. Rev. Lett. 80 (1998) 4911. [11] V. Ozolinš, C. Wolverton, A. Zunger, Phys. Rev. B 58 (1998) 5897. [12] A. van de Walle, G. Ceder, Rev. Mod. Phys. 74 (2002) 11. [13] A. Seko, N. Odagaki, S.R. Nishitani, I. Tanaka, H. Adachi, Mater. Trans. 45 (2004) 1978. [14] R. Haydock, V. Heine, M. Kelly, J. Phys. C 5 (1972) 2845. [15] R. Haydock, V. Heine, M. Kelly, J. Phys. C 8 (1975) 2591. [16] P.E. Meek, Phil. Mag. 33 (1976) 897. [17] A.P. Sutton, Phil. Mag. A 60 (1989) 147. [18] G.D. Garbulsky, G. Ceder, Phys. Rev. B 49 (1994) 6327. [19] D.G. Pettifor, Bonding and Structure of Molecules and Solids, Clarendon Press, Oxford, 1995, p. 85.