Abstract. Thermodynamic properties of some rare-earth alloys have been studied using the
statistical moment method (SMM) and analytic expressions of the Helmholtz free energy
and thermodynamic properties of some rare-earth alloys have been obtained. Present
statistical moment method results of nearest neighbor distance, linear thermal expansion
coefficient and specific heats, and pressure dependence of molecular volume for AlCe3,
Th57Ce43 and AgCe3 are compared with experimental and other calculation results. The
influence of Ce concentration on thermodynamic properties of Al1−xCex, Ag1−xCex and
Th1−xCex alloys have been studied.

11 trang |

Chia sẻ: thanhle95 | Lượt xem: 186 | Lượt tải: 0
Bạn đang xem nội dung tài liệu **Thermodynamic properties of some rare earth alloys**, để tải tài liệu về máy bạn click vào nút DOWNLOAD ở trên

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0039
Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 104-114
This paper is available online at
THERMODYNAMIC PROPERTIES OF SOME RARE EARTH ALLOYS
Dang Thanh Hai1, Vu Van Hung1 and Pham Thi Minh Hanh2
1Vietnam Education Publishing House
2Hanoi Pedagogical University No. 2
Abstract. Thermodynamic properties of some rare-earth alloys have been studied using the
statistical moment method (SMM) and analytic expressions of the Helmholtz free energy
and thermodynamic properties of some rare-earth alloys have been obtained. Present
statistical moment method results of nearest neighbor distance, linear thermal expansion
coefficient and specific heats, and pressure dependence of molecular volume for AlCe3,
Th57Ce43 and AgCe3 are compared with experimental and other calculation results. The
influence of Ce concentration on thermodynamic properties of Al1−xCex, Ag1−xCex and
Th1−xCex alloys have been studied.
Keywords: Thermodynamic properties, rare-earth alloys, statistical moment method.
1. Introduction
Study of rare earths and rare earth alloys have been attractive in recent years because of their
applications in many fields of technology. The theories about a transition state of metal such as a
theory of pseudo-potential [1-3] were used to the study rare earths of the actinide series La, Yb, Ce
and Th [4-6]. A. Rosengren, I. Ebbsjo and B. Johansson [7] studied crystalline lattice dynamics of
thori by using pseudo-potential of Krasko and Gurskii [8]. N. Singh and S. P. Singh [9] calculated
phonon scattering of La, Yb, Ce and Th. Using the calculated phonon scattering curve, phonon
density of states of thori were presented by T. C. Panelya, P. R. Vyas, C. V. Pandya and V. B.
Gohel [10]. In [11], J.K. Baria and A.R. Jani also calculated the phonon scattering curve, phonon
density of states, Debye temperature, the Gru¨neisen parameters and elastic dynamic constants of
La, Yb, Ce and Th.
Tanju Gurel and Resul Eryigit have performed an ab initio study of structural, elastic,
lattice-dynamic and thermodynamic properties of the rare-earth hexaborides, LaB6 and CeB6 [12].
The calculations have been carried out within the density functional theory and linear response
formalism using pseudopotentials and a plane-wave basis. Thermodynamic properties of LaB6
and CeB6 obtained from quasiharmonic approximation are in a good agreement with the available
experimental data. They also present complete phonon-dispersion curves, phonon density of
states and mode-Gru¨neisen parameters, and they compare well with experimental measurements.
A sizable difference between the vibrational contribution to entropy of LaB6 and CeB6 is found.
The thermal electronic contribution to entropy and specific heat is found to be important for CeB6.
Received March 10, 2015. Accepted August 17, 2015.
Contact Dang Thanh Hai, e-mail address: dthai@nxbgd.vn
104
Thermodynamic properties of some rare earth alloys
A comprehensive first principles study of structural, elastic, electronic and phonon
properties of rare-earth CeBi and LaBi is reported within the density functional theory
scheme [13]. Ground state properties such as lattice constant, elastic constants, bulk modulus and
finally the phase transition and lattice dynamic properties of rare-earths CeBi and LaBi of rock
salt and CsCl structures are determined. The electron band structures for the two phases of both
rare-earth crystals are presented.
The influence of rare earth additions on the microstructure and some properties of gold
and gold alloys have been studied [14]. Rare earth additions can refine the grain size of gold
alloys but they show a tendency to segregation, both dendritic segregation in cast alloys and grain
boundary segregation in annealed alloys. For gold alloys, rare earth additions are generally used
in trace amounts or dilute concentrations in order to avoid a large segregation of rare earth and
the potential embrittlement of gold alloys. The experimental results demonstrate that rare earth
additions can inhibit recovery softening, increase the recrystallization temperature and enhance
the strength of gold alloys. The strengthening mechanisms of rare earth additions in gold alloys are
discussed. Some gold alloys with rare earth additions have been developed and their applications
are illustrated briefly.
M.G. Shelyapina et al. [15] used the ab initiomethod combined with X-ray analysis to study
the influence of magnetic properties and V concentration in NbFV alloy on their thermodynamic
properties. The phase diagrams of Ce, Th, and Pu metals have been studied by means of the
density functional theory [16]. In addition to these metals, the phase stability of Ce-Th and Pu-Am
alloys has been also investigated from first-principles calculations. Equations of state for Ce,
Th, and the Ce-Th alloys have been calculated up to 1 Mbar pressure with good comparison to
experimental data.
This work focuses on studying the thermodynamic properties of AgCe3, AlCe3 and
Th57Ce43 alloys using the statistical moment method.
2. Content
2.1. Free energy of substitutional alloy A-B with FCC and BCC structures
Considering the free energy of the substitutional alloy A-Bmodel with fcc and bcc structure
in which there are two types of lattice point a and b corresponding to the atomic A and B types.
In the N atomics collection, NA atoms belong to the atomic A type and NB atoms belong to the
atomic B type. The concentration of A and B is determined as follows:
cA =
NA
N
; cB =
NB
N
, (2.1)
where, cA and cB are the concentration of A and B respectively. According to the definition of free
energy in statistical physics
ψAB = −kBT lnZAB , (2.2)
where ZAB is overall statistics in A-B alloy; ZAB is described in the form
ZAB =
∑
n
exp
(
− En
kBT
)
, (2.3)
105
Dang Thanh Hai, Vu Van Hung and Pham Thi Minh Hanh
where En is energy of system corresponding to the principle quantum number n. On the other
hand, En is expressed through configurative energy Et and vibration energy Em by the expression
En = Et + Em, (2.4)
here, vibration energy Em is assumed approximately, not to depend on configuration.
Substituting Eq. (2.4) to Eq. (2.3) and changing the sum depending on n to the sum
depending on i andm, we have
ZAB =
∑
i,m
exp
(
−Ei + Em
kBT
)
=
∑
m
exp
(
− Em
kBT
)∑
i
exp
(
− Ei
kBT
)
. (2.5)
In approximate calculation without correlation effect, the configuration energy Et is the
same for all the configurations and equal to E. Replace Et by E in equation (2.5) and move
exp[−E/(kBT )] to the sum of i, and the results are
ZAB =
∑
m
exp
(
−E + Em
kBT
)
W = z1W, (2.6)
z1 =
∑
m
exp
(
−E
′
m
kBT
)
, (2.7)
here E′m = E + Em is the energy of alloy which was determined by state quantum number m of
the system and the average energy configuration Eβα of the effective system (α, β) in the form
E =
∑
α, β
υαP
β
αE
β
α . (2.8)
If Em is assumed not to depend on configuration or effective systems, replacing Eq. (2.8)
with Eq. (2.7) we obtain
z1 =
∑
m
exp
−
∑
α,β
υαP
β
α (E
β
α)′m
kBT
. (2.9)
From (2.2), (2.6) and (2.9) we have
ψAB =
∑
m
υαP
β
αψ
β
α − kBT lnW. (2.10)
Notice that the second term in (2.10) is calculated by kBT lnW = TSc,where Sc is entropy
of alloy configuration
Sc = −kBN
∑
α,β
υαP
β
α lnP
β
α . (2.11)
Here, Hemlholtz free energy ψAB of an alloy is in the form
ψAB =
∑
α,β
υβP
β
αψ
β
α − TSc, (2.12)
106
Thermodynamic properties of some rare earth alloys
where P βα is the probability for atomic α (α = A,B) located on a lattice point (β = a, b) satisfying
the relations
cA = υaP
a
A + υbP
b
A; cB = υaP
a
B + υbP
b
B ;P
a
A + P
b
B = P
b
A + P
a
B = 1;N = NA +NB, (2.13)
where υβ is concentration of lattice point β. In Eq. (2.12), ψ
β
α is the free energy of the effective
system (α, β) in the form
ψβα = 3N
{
uβ0α
6
+ θ
[
xβα + ln(1− e−2x
β
α)
]}
, (2.14)
uβ0α =
∑
i
φβαi (|ai|) . (2.15)
xβα =
~ωβα
2θ
=
~
2θ
√
kβα/m∗; m∗ = cAmA + cBmB (2.16)
where uβ0α is potential interactions between atomics in the effective system (α, β);mA,mB are the
mass of atomic A or B in an A-B alloy and the second differential coefficient of (kβα) is in the form
kβα =
1
2
∑
i
(
∂2φβαi
∂u2αl
)
eq
≡ m∗
(
ωβα
)2
, (l = x, y, z). (2.17)
In case cB << cA, we can ignore the terms that are smaller than u0α and kα. This means
that uβ0A ≈ u0A, kβA ≈ kA. Therefore, the free energy of the substitutional alloys A-B which
disorders when cB << cA has distributions as a function of atomic concentration as follows:
ψAB = cAψA + cBψB − TSc. (2.18)
Expression (2.18) shows the relationship between free energy ψAB of the disordered substitutional
alloy A-B (cB << cA) and the free energy of component metals ψA and ψB as follows:
ψα = 3N
{u0α
6
+ θ
[
xα + ln(1− e−2xα)
]}
. (2.19)
2.2. Lattice parameters of alloys
To calculate the lattice parameters of alloys with fcc and bcc structures, we use
the thermodynamic equilibrium condition of system for calculating the lattice parameters of
substitutional alloy A-B. As we know, free energy ψAB Eq. (2.12) is a function of the
nearest neighbor distance in the substitutional alloy A-B so that is approximately equal to
aaA, a
b
A, a
a
B , a
b
B . Therefore, we can expand ψAB by (a − aβα) up to the second order term
approximation as follows:
ψβα (a) = ψ
β
α
(
aβα
)
+
1
2
(
∂2ψβα
∂a2
)
T
(
a− aβα
)2
. (2.20)
107
Dang Thanh Hai, Vu Van Hung and Pham Thi Minh Hanh
Using the definition of bulk modulus, we have
BABT =
1
χABT
= −V0
(
∂2ψAB
∂V 2AB
)
. (2.21)
According to the expressions (2.12), (2.20) and thermodynamic equilibrium condition((
∂ψ
∂a
)
T, P,N
= 0
)
, the nearest neighbor distance has the form
aAB ≈
(
υaP
a
Aa
a
A + υbP
b
Aa
b
A
)
BAT +
(
υaP
a
Ba
a
B + υbP
b
Ba
b
B
)
BBT
cABAT + cBB
B
T
, (2.22)
B¯T = cAB
A
T + cBB
B
T , (2.23)
where B¯T is the average value of the bulk modulus. We have
aAB ≈
(
υaP
a
Aa
a
A + υbP
b
Aa
b
A
) BT,A
B¯T
+
(
υaP
a
Ba
a
B + υbP
b
Ba
b
B
) BT,B
B¯T
. (2.24)
For the completely disordered alloy, using the approximated condition aaA ≈ abA = aA,
aaB ≈ abB = aB , we are able to easily obtain the equation to calculate a0AB and aAB as follows:
a0AB = cAa0A
B0T,A
B¯0T
+ cBa0B
B0T,B
B¯0T
at 0K (2.25)
aAB = cAaA
BT,A
B¯T
+ cBaB
BT,B
B¯T
at T (K) (2.26)
where a0α and aα(α = A,B) are the nearest neighbor distance of α metal at 0K and T (K);
B0T, α and BT, α are bulk modulus of α metal at 0K and T (K); B¯0T is the average value of the
bulk modulus of the substitutional alloys A-B at 0K which is described similarly to the Eq. (2.23).
2.3. Thermodynamic quantity of substitutional alloy A-B
According to [17], isothermal compressibility is obtained in the form
χABT =
3
(
aAB
a0AB
)3
2P +
a2
3VAB
(
∂2ψAB
∂a2AB
)
T
, (2.27)
where P is pressure, VAB is the volume of crystal including N particles: VAB = N.vAB and vAB
is the atomic volume of alloy at T temperature. Here vAB is determined by equations
vAB =
√
2
2
a3AB applied for FCC structure (2.28)
vAB =
4
3
√
3
a3AB applied for BCC structure (2.29)
108
Thermodynamic properties of some rare earth alloys
and (
∂2ψAB
∂a2AB
)
T
≈ cA∂
2ψA
∂a2A
+ cB
∂2ψB
∂a2B
. (2.30)
If we know the isothermal compressibility, we can determine the bulk moduli as follows:
BABT =
1
χABT
. (2.31)
The linear thermal expansion coefficient is defined as follows:
αABT =
kB
a0AB
· daAB
dθ
. (2.32)
Using the mathematical transformation, we determine the linear thermal expansion
coefficient by equation
αABT =
a0A
a0AB
cAα
A
T
BTA
B¯T
+
a0B
a0AB
cBα
B
T
BTB
B¯T
. (2.33)
Appling the results of thermodynamics, we can write it in the form
αABT = −
kBχ
AB
T
3
·
(
a0AB
aAB
)2
· aAB
3VAB
·
(
∂2ψAB
∂θ∂aAB
)
T
, (2.34)
where (
∂2ψAB
∂θ∂aAB
)
T
≈ cA ∂
2ψA
∂θ∂aA
+ cB
∂2ψB
∂θ∂aB
. (2.35)
This result allows us to calculate αT if we know χT . Similar to the case of metals, the energy of
an alloy is written in the form
EAB = ψAB − θ · ∂ψAB
∂θ
= cA
(
ψA − θ∂ψA
∂T
)
+ cB
(
ψB − θ∂ψB
∂T
)
. (2.36)
The specific heat at constant volume has the form
CABV =
∂EAB
∂T
= −cAT ∂
2ψA
∂T 2
− cBT ∂
2ψB
∂T 2
. (2.37)
According to the definition of specific heat at constant volume for metal, we have a simpler
equation to determine
CABV = cAC
A
V + cBC
B
V . (2.38)
The specific heat at constant pressure is determined from the well established thermodynamic
relations
CABP = C
AB
V +
9TVAB(α
AB
T )
2
χABT
. (2.39)
109
Dang Thanh Hai, Vu Van Hung and Pham Thi Minh Hanh
2.4. Results and discussions
Using the moment method in statistical dynamics, we calculated the thermodynamic
properties of rare earth alloys AlCe3, Th57Ce43 and AgCe3. For simplicity, we take the effective
pair interaction energy in rare earth metals as the power law, similar to the Lennard - Jones
ϕ(r) =
D
(n−m)
[
m
(r0
r
)n − n(r0
r
)m]
, (2.40)
where D, r0 are determined to fit the experimental data [18].
Table 1. Temperature dependence of the thermodynamic quantities
of AgCe3 alloy at P = 10GPa
T (K) 300 400 500 600 700 800 900 1000
a(A˚) 3.181 3.187 3.193 3.199 3.205 3.212 3.218 3.225
α(10−6K−1) 9.16 9.35 9.48 9.60 9.71 9.81 9.92 10.02
BT (10
10Pa) 4.422 4.314 4.208 4.104 4.000 3.897 3.795 3.693
χT (10
−12/Pa) 22.61 23.17 23.75 24.36 24.99 25.65 26.34 27.07
CV (J/mol.K) 25.30 25.85 26.28 26.68 27.06 27.45 27.84 28.24
CP (J/mol.K) 25.44 25.99 26.42 26.82 27.21 27.59 27.98 28.38
Table 2. Temperature dependence of the thermodynamic quantities
of Al2Ce3 alloy at P = 10GPa
T (K) 300 400 500 600 700 800 900 1000
a(A˚) 3.064 3.071 3.078 3.085 3.092 3.099 3.107 3.115
α(10−6K−1) 9.59 10.07 10.37 10.59 10.77 10.94 11.10 11.25
BT (10
10Pa) 4.639 4.492 4.354 4.221 4.091 3.962 3.834 3.708
χT (10
−12/Pa) 21.55 22.26 22.96 23.68 24.44 25.23 26.07 26.96
CV (J/mol.K) 24.45 25.49 26.20 26.78 27.32 27.84 28.36 28.89
CP (J/mol.K) 24.59 25.64 26.35 26.94 27.48 28.00 28.52 29.05
Table 3. Temperature dependence of the thermodynamic quantities
of Th57Ce43 alloy at P = 10GPa
T (K) 300 400 500 600 700 800 900 1000
a(A˚) 3.391 3.394 3.397 3.400 3.403 3.406 3.409 3.412
α(10−6K−1) 6.978 7.057 7.114 7.163 7.207 7.250 7.292 7.333
BT (10
10Pa) 3.970 3.918 3.866 3.814 3.763 3.712 3.661 3.611
χT (10
−12/Pa) 25.18 25.52 25.86 26.21 26.57 26.93 27.30 27.69
CV (J/mol.K) 25.15 25.50 25.77 26.01 26.24 26.47 26.69 26.92
CP (J/mol.K) 25.24 25.59 25.86 26.10 26.33 26.56 26.78 27.01
110
Thermodynamic properties of some rare earth alloys
Tables 1, 2 and 3 present the temperature dependence of the thermodynamic quantities of
AgCe3, Al2Ce3 and Th57Ce43 alloys at pressure 10GPa. We can see that the nearest neighbor
distance, linear thermal expansion coefficient, specific heat at constant volume and specific heat
at constant pressure increases when the temperature increases. The nearest neighbor distance
increased 10% and the linear thermal expansion coefficient increased 10.9% for AgCe3, 11.7% for
Th57Ce43 and 10.5% for Al2Ce3. The bulk modulusBT decreases when the temperature increases,
reduction of AgCe3 is 11.97%, Th57Ce43 is 12.5% and Al2Ce3 is 11%. The results obtained in the
above table is in accordance with the laws of experiment.
Table 4. Pressure dependence of thermodynamic quantities
of Al2Ce3 alloy at T = 300K
P (GPa) 10 20 30 40 50 60 70 80
a(A˚) 3.064 3.015 2.976 2.943 2.915 2.875 2.855 2.836
α(10−6K−1) 9.594 7.118 5.711 4.788 4.131 3.636 3.248 2.934
BT (10
10Pa) 4.639 6.430 8.176 9.895 11.59 13.27 14.95 16.61
χT (10
−12/Pa) 21.55 15.55 12.22 10.10 8.625 7.531 6.688 6.018
CV (J/mol.K) 24.45 23.86 23.38 22.97 22.61 22.27 21.96 21.67
CP (J/mol.K) 24.59 23.96 23.47 23.04 22.66 22.32 22.00 21.70
Table 5. Pressure dependence of thermodynamic quantities
of AgCe3 alloy at T = 300K
P (GPa) 10 20 30 40 50 60 70 80
a(A˚) 3.181 3.124 3.079 3.042 3.010 2.983 2.959 2.938
α(10−6K−1) 9.318 6.960 5.642 4.785 4.176 3.719 3.360 3.071
BT (10
10Pa) 4.213 5.899 7.533 9.135 10.71 12.27 13.82 15.35
χT (10
−12/Pa) 23.73 16.95 13.27 10.94 9.333 8.146 7.235 6.511
CV (J/mol.K) 25.30 24.97 24.72 24.51 24.33 24.17 24.02 23.87
CP (J/mol.K) 25.44 25.07 24.80 24.58 24.39 24.22 24.06 23.92
Table 6. Pressure dependence of thermodynamic quantities
of Th57Ce43 alloy at T = 300K
P (GPa) 10 20 30 40 50 60 70 80
a(A˚) 3.391 3.314 3.256 3.209 3.171 3.137 3.108 3.082
α(10−6K−1) 6.978 5.334 4.393 3.772 3.326 2.988 2.722 2.505
BT (10
10Pa) 3.970 5.486 6.944 8.367 9.762 11.13 12.49 13.84
χT (10
−12/Pa) 25.18 18.22 14.39 11.95 10.24 8.978 8.001 7.223
CV (J/mol.K) 25.15 24.95 24.79 24.66 24.55 24.45 24.36 24.27
CP (J/mol.K) 25.24 25.01 24.84 24.71 24.59 24.48 24.39 24.30
Tables 4, 5 and 6 present the pressure dependence of the thermodynamic quantities of
AgCe3, Al2Ce3 and Th57Ce43 alloys at T = 300K . The thermodynamic quantities such as
the linear thermal expansion coeffic ient, specific heat at constant volume and specific heat at
constant pressure decreased when the pressure increased. When pressure P increases from 10GPa
111
Dang Thanh Hai, Vu Van Hung and Pham Thi Minh Hanh
to 80GPa, the isothermal compression ratio of Al2Ce3 reduction is 72.1%, AgCe3 reduction is
72.6% and Th57Ce43 decreased 71.32%. The linear thermal expansion coefficient of Al2Ce3 is a
43% reduction, and there was a 72% reduction of AgCe3. However, the bulk modulusBT increases
when the pressure increases. When the pressure increases from 10GPa to 80GPa, the bulk modulus
BT of Al2Ce3 increases 3.5 times.
Table 7. Ce concentration dependence of thermodynamic quantities
of Ag1−xCex at P = 0 and T = 500 K
cCe 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a(A˚) 2.892 2.931 2.975 3.025 3.083 3.149 3.228 3.321
α(10−6K−1) 17.94 17.72 17.48 17.20 16.90 16.56 16.16 15.71
BT (10
10Pa) 3.618 3.401 3.186 2.973 2.762 2.554 2.348 2.147
χT (10
−12/Pa) 27.63 29.39 31.38 33.63 36.20 39.15 42.57 46.56
CV (J/mol.K) 28.05 27.90 27.76 27.61 27.47 27.32 27.17 27.03
CP (J/mol.K) 28.59 28.42 28.25 28.08 27.91 27.74 27.57 27.40
Table 8. Ce concentration dependence of thermodynamic quantities
of Al1−xCex at P = 0 and T = 500K
cCe 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a(A˚) 2.875 2.916 2.961 3.013 3.073 3.141 3.222 3.317
α(10−6K−1) 18.39 18.14 17.87 17.56 17.21 16.82 16.37 15.86
BT (10
10Pa) 3.558 3.343 3.131 2.922 2.716 2.513 2.315 2.123
χT (10
−12/Pa) 28.10 29.90 31.92 34.21 36.81 39.77 43.18 47.09
CV (J/mol.K) 27.81 27.69 27.57 27.45 27.33 27.21 27.09 26.97
CP (J/mol.K) 28.36 28.21 28.07 27.92 27.78 27.64 27.49 27.35
Table 9. Ce concentration dependence of thermodynamic quantities
of Th1−xCex at P = 0 and T = 500K
cCe 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a(A˚) 3.495 3.501 3.508 3.515 3.523 3.531 3.540 3.550
α(10−6K−1) 9.747 10.13 10.55 11.00 11.48 12.00 12.56 13.17
BT (10
10Pa) 2.532 2.450 2.366 2.282 2.197 2.111 2.025 1.938
χT (10
−12/Pa) 39.47 40.81 42.24 43.80 45.50 47.34 49.37 51.59
CV (J/mol.K) 25.97 26.05 26.14 26.22 26.31 26.40 26.48 26.57
CP (J/mol.K) 26.17 26.26 26.36 26.46 26.55 26.65 26.75 26.85
We found that the nearest neighbor distance, linear thermal expansion coefficient and
isothermal compressibility increased while bulk modulus decreased when the concentration of
Ce in the alloy increased. We also found that the concentration of Ce strongly influenced the
thermodynamic properties of the alloy when concentration of Ce in the alloy increased.
In Figure 1 we displayed the good agreement of the SMM calculations of the nearest
neighbor distance of Al2Ce3, AgCe3 and Th57Ce43 alloys. The results calculated by the statistical
moment metho