Diffusion of interstitial atoms in interstitial alloys FeSi and FeH with BCC structure under pressure
48
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
DIFFUSION OF INTERSTITIAL ATOMS IN INTERSTITIAL ALLOYS
FeSi AND FeH WITH BCC STRUCTURE UNDER PRESSURE
1
Nguyen Quang Hꢀc1( ), Bui Duc Tinh1, Dinh QuangVinh1, Le Hong Viet2
Hanoi National University of Education
Tran Quoc Tuan University
Abstract: In our previous paper [10], the analytic expressions with free energy of
interstitial atom, the nearest neighbor distance between two interstitial atoms, the alloy
parameters for interstitial atom, the diffusion quantities such as the jumping frequency of
interstitial atom, the effective jumping length, the correlation factor, the diffusion
coefficient and the activated energy together with the equation of state for the interstitial
AB with BCC structure under pressure are derived from the statistical moment method. In
this paper, we apply these theoretical results to interstitial FeSi and FeH in the interval
of interstitial atom concentration from 0 to 5%, the interval of temperature from 100 to
1000K and the interval of pressure from 0 to 70GPa. Our calculated results are in good
agreement with experiments or predict the experimental results.
Keywords: Interstitial alloy, jumping frequency, effective jumping length, correlation
factor, diffusion coefficient, activated energy
1. INTRODUCTION
Study on the diffusion theory of metals and alloys pays attention to researchers [1ꢀ10].
In previous paper [10], by the statistical moment method (SMM) [5ꢀ7, 10]we derive the
analytic expressions of the free energy of interstitial atom, the nearest neighbor distance
between two interstitial atoms, the alloy parameters for interstitial atom, the diffusion
quantities such as the jumping frequency of interstitial atom, the effective jumping length,
the correlation factor, the diffusion coefficient and the activated energy together with the
equation of state for the interstitial AB with BCC structure under pressure. In this paper,
we apply the theoretical results in [10] to the interstitial alloys FeSi and FeHin the interval
of interstitial atom concentration from 0 to 5%, in the interval of temperature from 100 to
(1)
Nhꢁn bài ngày 19.8.2016; gꢂi phꢃn biꢄn và duyꢄt ñăng ngày 15.9.2016
Liên hꢄ tác giꢃ: Bùi Đꢅc Tĩnh; Email: bdtinh@hnue.edu.vn
TẠP CHÍ KHOA HỌC − SỐ 8/2016
49
1000K and in the interval of pressure from 0 to 70GPa. Some calculated results are
compared with experiments, where we use the Arrhenius law.
2. CONTENT
For the interstitial alloy FeSi, we use the nꢀm interaction potemtial [7]
m
d
r
r
r
n
0
0
ϕ
(r) =
m
− n
,
(1)
n − m
r
where
is the distance between two atoms corresponding to the minimum of potetial
energy, that takes the value ꢀ d, mand nare the numbers which have different values for
different atoms and are determined emperically on the basis of experimental data.
r ,d,m
The parameters
and n of the nꢀm potental (1) for the interaction potetials FeꢀFe and
0
SiꢀSi are given in Table 1.
r ,d,m
Table 1. The parameters
and n of the interaction potentialsFeꢀFe and SiꢀSi
0
m
n
d (10−16 erg)
6416.448
45128.34
r0(10−10m)
2.4775
2.295
Fe
Si
7
6
11.5
12
We use the following approximation
1
ϕFeꢀSi
≈
ϕ
(
+
ϕSiꢀSi
.
(2)
(3)
)
FeꢀFe
2
For the interstitial alloy FeH, we use the Morse potential [10]
−2
α
r−r
α r−r
0
ϕ
(
r
) = D e
) − 2e−
,
(
(
)
0
where
= ꢀ
potential for the alloy FeH are given in Table 2.
Table 2. The parameters of the Morse for the interstitial alloy FeH
αhas the dimension of distance inverse,
Dhas the dimension of energy (eV) and
D
,
the equilibrium distance of two atoms. The parameters of the Morse
r0 (Ǻ)
D (eV)
α (Ǻ)
1.73
0.32
1.34
50
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
For the interstitial alloy FeSi, we use the potential (1) for the interaction potentials Feꢀ
Fe and SiꢀSi with the potential parameters in Table 1and use the appoximation (2) for the
interaction potential FeꢀSi. Using the formulae in the previous paper, we find the
expressions of the cohesive energy 0B and the alloy parameters k B ,
γ
B of the atom Si in
U
the position 1 in the interstitial alloy FeSi as follows
1.755118523.10−8 5.586962213.10−10 4.969164799.10−8 8.453440955.10−10
U01B
=
−
+
−
,
(4)
(5)
(6)
r11,5
r7
r9
r11
r12
r6
2.451815336.10−6 2.693657436.10−7 7.543922121.10−6 2.808431783.10−8
k1B
=
−
+
−
,
r13,5
r14
r8
r10
5.17258208.10−5 2.117826188.10−7 1.735930771.10−4 1.671565741.10−7
γ1B
=
−
+
−
.
r15,5
r16
U
Analogously, we can obtain the expressions of the cohesive energy 0B and the alloy
parameters k B ,
B of the atom Si in the positions 2 and 3 in the interstitial alloy FeSi.
For the interstitial alloy FeH, we use the potential (3) for the interction potentials
γ
r , D
α
FeꢀFe, FeꢀH, HꢀFe,HꢀH with the potential parameters
and
in Table 2. The
0
and the alloy parameters k B ,
γ
B of the atom Si in
U
expressions of the cohesive energy
0B
the position 1 in the interstitial alloy FeH have the form:
U01B = 5.289022639.10−11e−2.68r −1.0414153.10−11e−1.34r +1.057804528.10−10 e−3.790092346r
−2.0828306.10−11e−1.895046173r
−
,
(7)
k1B = 3.79878762.10−10 e−2.68r −1.869965313.10−11e−1.34r +3.039030097.10−10 e−5.992662178r
2.004588422.10−10 e−3.790092346r 1.973530079.10−11 e−1.895046173r
−
−1.49597225.10−11 e−2.996331089r
−
−
,
(8)
r
r
1.018075082.10−9 e−2.68r
γ1B = 4.547402034.10−10 e−2.68r −5.59618286.10−12 e−1.34r
−
+
r
2.505753519.10−11e−1.34r 7.198877938.10−10 e−3.790092346r
+
−
+
r
r
1.771835304.10−11e−1.895046173r 7.59757524.10−10 e−2.68r 3.739930626.10−11e−1.34r
+
+
+
+
r
r2
r2
2.834916134.10−10e−2.68r 2.790993004.10−11e−1.34r
+
+
,
(9)
r3
r3
U
Analogously, we can obtain the expressions of the cohesive energy 0B and the alloy
parameters k B ,
γ
B of the atom Si in the positions 2 and 3 in the interstitial alloy FeH.
TẠP CHÍ KHOA HỌC − SỐ 8/2016
51
In the case of applying the nꢀ m potential (1), the cohesive energy between atoms in
the clean metal A has the form [6]
n
m
d
r
r
0
0
U0 A
=
mA
− nA
,
(10)
n
m
n − m
r
r
1A
1A
r
r
01A is the neraest
where 1A is the neraest neighbour between atoms A at temperature
T
,
neighbour between atoms A at temperature 0 K and is determined from the minimum
condition of the cohesive energy. Therefore, it has the following form:
An
n−m
r = r
.
(11)
01A
0
Am
Then, the metal parameters k A,γ1A,γ2A and γ A have the form as in [6]
According to figures from Figure 1to Figure 3, at the same pressure, when the
temperature increases, the activated energy decreases, the coefficient D0 changes
increases. In the same pressure, in is a
. In the same temperature, when the pressure
E
negligibly and the diffusion coefficient
monotonously decreasing function of 1/
D
D
T
increases, the activated energy
coefficient and ln decreases.
The dependences of the diffusion coefficient
E increases, the coefficient D0 increases, the diffusion
D
D
D
and the coefficient D0 on interstitial
atom concentration, temperature and pressure for the interstitial alloy FeSi are illustrated
by figures from Figure 4 to Figure 11. When the concentration of interstitial atoms Si
increases, the coefficients D0 and
D
of alloy FeSi increase. This absolutely agrees with
experiments.
28
40
FeꢀSi
P=0 (GPa)
P=30 (GPa)
P=70 (GPa)
P = 0 (GPa)
P = 30(GPa)
P = 70(GPa)
FeꢀSi
27
39
38
37
36
35
34
33
32
31
30
29
26
25
24
23
22
21
20
19
18
17
100
200
300
400
500
600
700
800
900 1000
100
200
300
400
500
600
700
800
900
1000
T(K
)
T(K)
Fig 2. D0(T) of FeSi at P = 0, 30 and 70 GPa
Fig 1. E(T) of FeSi at P = 0, 30 and 70 GPa
52
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
16
0
ꢀ20
P= 0 (GPa)
FeꢀSi
P = 0 (GPa)
P = 30 (GPa)
P = 70 (GPa)
15
FeꢀSi
P= 30 (GPa)
P= 70 (GPa)
T=300K
14
13
12
11
10
9
ꢀ40
ꢀ60
ꢀ80
ꢀ100
ꢀ120
ꢀ140
ꢀ160
ꢀ180
ꢀ200
8
7
6
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.002
0.004
0.006
0.008
0.010
C
(%)
si
1/T
Fig 4. D0(cSi) of FeSi at P = 0, 30, 70 GPa
and T = 300K
Fig 3. lnD (1/T) at P = 0, 30 and 70 GPa
for FeSi
According to our numerical results for alloy FeSi, when the interstial atom Si is in
face centres of BCC lattice of Fe at zero pressure and under pressure, this atom Si can not
diffuse through sides of lattice cells to come next cell (the first way) but only can move
from this face centre to other face centre (the second way). The interstitial atom Si changes
locally the lattice constants. In the lattice cells containing the interstitial atom Si, the lattice
constants expanse considerably. Our calculated results are in relatively good agreement
with the experimental data [8,9]. At
= 1.4.10ꢀ6 cm2/s according to the experimental data [8]. Accordinng to our calculated
P = 0, T cSi = 4.9%, the alloy FeSi has D
= 1150oC and
result, at
P
= 0,
T
= 1000K, cSi = 5%, the alloy FeSi hasD
= 0.08. 10ꢀ6 cm2/s.
16
15
14
13
12
11
10
9
FeꢀS
i
6.4
6.2
6.0
5.8
5.6
5.4
5.2
5.0
P= 0,
T=300K
P= 0 (GPa)
P= 30 (GPa)
P= 70 (GPa)
FeꢀSi
T=900K
8
7
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6
C
(%)
Si
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
C
(%)
si
Fig 6. D(T) of FeSi at P = 0 and T = 300K
Fig 5. D0(cSi) of FeSi at P = 0, 30, 70 GPa
and T = 900K
TẠP CHÍ KHOA HỌC − SỐ 8/2016
53
2.10
2.05
2.00
1.95
1.90
1.85
1.80
1.75
1.70
FeꢀS
i
FeꢀS
i
P= 30,
T=300K
P= 0,
T=900K
8.4
8.2
8.0
7.8
7.6
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
C (%)
Si
C
(%)
Si
Fig 8. D(T) of FeꢀSi at P = 30 GPa
and T = 300K
Fig 7. D(T) of FeSi at P = 0 and T = 900K
At
and at
[9]. Accordinng to our calculated result, at
D0 = 0.9.10ꢀ3 cm2/s,
= 0.19479 kcal/mol. Figure 3 shows the dependence of ln
P
= 0 and from 200 to 780oC, the alloy FeH has D0 = 1.4.10ꢀ3 cm2/s,
= 700oC, the alloy FeH có = 2.45.10ꢀ4 cm2/s according to the experimental data
= 0, = 1000K, the alloy FeH has
on 1/T
E = 0.139eV
T
D
P
T
E
D
for alloy FeSi and has a linear form. This means that in the interval of temperature from
100 to 1000K, the Arrhenius law absolutely is satisfied.
Our calculate results for alloy FeH are an analogue with ones for alloy FeSi and are
illustrated by figures from Figure 12 to Figure 19. According to our numerical results for
alloy FeH, when the interstial atom H is in face centres of BCC lattice of Fe at zero
pressure and under pressure, this atom H also can not diffuse through sides of lattice cells
to come next cell (the first way) but only can move from this face centre to other face
centre (the second way). Our calculated result can predict the experimental result.
4.9
4.8
4.7
4.6
4.5
4.4
4.3
4.2
4.1
4.0
2.10
2.05
2.00
1.95
1.90
1.85
1.80
1.75
1.70
FeꢀS
i
FeꢀS
i
P= 70,
T=300K
P= 30,
T=900K
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
C
(%)
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Si
C
(%)
Si
Fig 10. D(T) ofFeSiat P = 70 GPa
and T = 300K
Fig 9. D(T) ofFeSiat P = 30 GPaand T = 900K
54
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
5.1
5.0
4.9
4.8
4.7
4.6
4.5
4.4
4.3
0.40
FeꢀS
i
FeꢀH
P= 70,
T=900K
P= 0 (GPa)
P= 30 (GPa)
P= 70 (GPa)
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
100
200
300
400
500
600
700
800
900
1000
C
(% )
Si
T(K)
Fig 11. D(T) of FeSi at P = 70 GPa
and T = 900K
Fig 12. E(T) of FeH at P = 0, 30
and 70 GPa
13.0
5
0
FeꢀH
FeꢀH
T=300K
P= 0 (GPa)
P = 30(Gpa)
P= 70(GPa)
P = 0 (GPa)
P =30 (GPa)
P =70 (GPa)
12.5
12.0
11.5
11.0
10.5
10.0
9.5
ꢀ5
ꢀ10
ꢀ15
ꢀ20
ꢀ25
ꢀ30
ꢀ35
ꢀ40
ꢀ45
9.0
8.5
8.0
7.5
7.0
6.5
6.0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
1/T
C (%)
H
Fig 13. lnD (1/T) at P = 0, 30 and 70 GPa
for FeH
Fig 14. D0(cH) of FeH at P = 0, 30, 70 GPa
and T = 300K
13.0
FeꢀH
P
P
P
=
0
(G P a)
FeꢀH
T=300K
12.5
12.0
11.5
11.0
10.5
10.0
9.5
1.0429
P= 0 GPa
T=900K
=30 (G P a)
= 70 (G P a)
1.0428
1.0427
1.0426
1.0425
1.0424
1.0423
1.0422
9.0
8.5
8.0
7.5
7.0
6.5
6.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
C
(% )
H
C
(%)
H
Fig 15. D0(cH) of FeꢀH at P = 0, 30, 70 GPa
and T = 900K
Fig 16. D(cH) of FeH at P = 0
and T = 300K
TẠP CHÍ KHOA HỌC − SỐ 8/2016
55
6 .8 0
5 .3 0
5 .2 8
5 .2 6
5 .2 4
5 .2 2
5 .2 0
F e ꢀH
T = 3 0 0 K
F e ꢀH
T = 3 0 0 K
P = 3 0 G P a
P = 7 0 G P a
6 .7 5
6 .7 0
6 .6 5
6 .6 0
6 .5 5
1 .0
1 . 5
2 .0
2 .5
3 .0
3 .5
4 .0
4 .5
5 .0
1 .0
1 .5
2 .0
2 .5
3 .0
3 .5
4 .0
4 .5
5 .0
C
(%
)
C
(% )
H
H
Fig 17. D(cH) of FeH at P = 30 GPa
and T = 300K
Fig 18. D(cH) of FeH at P = 70 GPa
and T = 300K
1.630
1.625
1.620
1.615
1.610
1.605
1.600
FeꢀH
T=900K
P= 70 GPa
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
C
(%)
H
Fig 19. D(cH) of FeH at P = 70 GPa and T = 900K
3. CONCLUSION
Our numerial results for alloys FeX (X =Si, H) are obtained by applying the diffusion
theory builded from the SMM, using the nꢀm potential and the Morse potential and the
coordination sphere method. These results show that the diffusion mechanism of interstitial
atom in interstitial alloy depends on the size of interstitial atom and the interaction between
interstitial atom and main atom of alloy. The numerial results are in goog agreement with
experiments or can predict the experimental results because the exact determination of
diffusion quantities is a very difficult problem experimentally. Figure 13 for the
dependence of ln
D on 1/T has the linear form This mean that our obtained results are in
good agreement with the Arrhenius law in the interval of temperature below the structural
phase transition of iron.
56
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
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NGHIÊN CꢁU Sꢂ KHUꢃCH TÁN CꢄA NGUYÊN Tꢅ XEN Kꢆ
TRONG CÁC HꢇP KIM XEN Kꢆ FeꢈSi VÀ FeꢈH VꢉI CꢊU TRÚC
LꢋP PHƯƠNG TÂM KHꢌI DƯꢉI TÁC DꢍNG CꢄA ÁP SUꢊT
Tóm tꢁt: Trong bài báo trưꢂc [10], chúng tôi rút ra biꢃu thꢄc giꢅi tích ñꢆi vꢂi năng
lưꢇng tꢈ do cꢉa nguyên tꢊ xen kꢋ, khoꢅng cách lân cꢌn gꢍn nhꢎt giꢏa hai nguyên tꢊ xen
kꢋ, các thông sꢆ hꢇp kim ñꢆi vꢂi nguyên tꢊ xen kꢋ, các ñꢐi lưꢇng khuꢑch tán như tꢍn sꢆ
bưꢂc nhꢅy cꢉa nguyên tꢊ xen kꢋ, ñꢒ dài bưꢂc nhꢅy hiꢓu dꢔng, thꢕa sꢆ tương quan, hꢓ sꢆ
khuꢑch tán và năng lưꢇng kích hoꢐt cùng vꢂi phương trình trꢐng thái cꢉa hꢇp kim kim
xen kꢋ AB vꢂi cꢎu trúc lꢌp phương tâm khꢆi dưꢂi tác dꢔng cꢉa áp suꢎt bꢖng phương
pháp mômen thꢆng kê. Trong bài báo này, chúng tôi áp dꢔng các kꢑt quꢅ lí thuyꢑt này
cho các hꢇp kim xen kꢋ FeꢀSi và FeꢀH trong vùng nꢗng ñꢒ nguyên tꢊ xen kꢋ tꢕ 0 ñꢑn 5%,
vùng nhiꢓt ñꢒ tꢕ 100 ñꢑn 1000K và vùng áp suꢎt tꢕ 0 ñꢑn 70GPa. Kꢑt quꢅ tính toán phù
hꢇp khá tꢆt vꢂi sꢆ liꢓu thꢈc nghiꢓm hoꢘc dꢈ báo thꢈc nghiꢓm
Tꢕ khoá: Hꢇp kim xen kꢋ, tꢍn sꢆ bưꢂc nhꢅy, ñꢒ dài bưꢂc nhꢅy hiꢓu dꢔng, nhân tꢆ tương
quan, hꢓ sꢆ khuyꢑch tán, năng lưꢇng kích hoꢐt
9 trang
18/04/2022
1100
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