Diffusion of interstitial atoms in interstitial alloys FeSi and FeH with BCC structure under pressure

48 TRNG I HC TH  H NI 
DIFFUSION OF INTERSTITIAL ATOMS IN INTERSTITIAL ALLOYS 
FeSi AND FeH WITH BCC STRUCTURE UNDER PRESSURE 
Nguyen Quang Học1(1), 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 [1-10]. 
In previous paper [10], by the statistical moment method (SMM) [5-7, 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) Nhận bài ngày 19.8.2016; gửi phản biện và duyệt đăng ngày 15.9.2016 
 Liên hệ tác giả: Bùi Đức Tĩnh; Email: bdtinh@hnue.edu.vn 
TP CH KHOA HC − S
 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 n-m interaction potemtial [7] 
0 0( ) ,
n m
r rd
r m n
n m r r
ϕ
    = −    −       
(1) 
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. 
The parameters 0 , ,r d m and n of the n-m potental (1) for the interaction potetials Fe-Fe and 
Si-Si are given in Table 1. 
Table 1. The parameters 0 , ,r d m and n of the interaction potentialsFe-Fe and Si-Si 
 m n d (10−16 erg) r0(10−10m) 
Fe 7 11.5 6416.448 2.4775 
Si 6 12 45128.34 2.295 
We use the following approximation 
( )Fe-Si Fe-Fe Si-Si
1
.
2
ϕ ϕ ϕ≈ +
(2) 
 For the interstitial alloy FeH, we use the Morse potential [10] 
( ) ( )0 02( ) 2 ,r r r rr D e eα αϕ − − − − = −  (3) 
where 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 
potential for the alloy FeH are given in Table 2. 
Table 2. The parameters of the Morse for the interstitial alloy FeH 
r0 (Ǻ) D (eV) α (Ǻ) 
1.73 0.32 1.34 
50 TRNG I HC TH  H NI 
For the interstitial alloy FeSi, we use the potential (1) for the interaction potentials Fe-
Fe and Si-Si with the potential parameters in Table 1and use the appoximation (2) for the 
interaction potential Fe-Si. Using the formulae in the previous paper, we find the 
expressions of the cohesive energy 0BU and the alloy parameters ,
B
Bk γ of the atom Si in 
the position 1 in the interstitial alloy FeSi as follows 
8 10 8 10
01 11,5 7 12 6
1.755118523.10 5.586962213.10 4.969164799.10 8.453440955.10
,BU
r r r r
− − − −
= − + −
(4) 
6 7 6 8
1
13,5 9 14 8
2.451815336.10 2.693657436.10 7.543922121.10 2.808431783.10
,Bk
r r r r
− − − −
= − + −
(5) 
5 7 4 7
1 15,5 11 16 10
5.17258208.10 2.117826188.10 1.735930771.10 1.671565741.10
.B
r r r r
γ
− − − −
= − + −
(6) 
Analogously, we can obtain the expressions of the cohesive energy 0BU and the alloy 
parameters ,B Bk γ 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 
Fe-Fe, Fe-H, H-Fe,H-H with the potential parameters 0 ,r D and α in Table 2. The 
expressions of the cohesive energy 0BU and the alloy parameters ,
B
Bk γ of the atom Si in 
the position 1 in the interstitial alloy FeH have the form: 
11 2.68 11 1.34 10 3.790092346
01 5.289022639.10 1.0414153.10 1.057804528.10
r r r
BU e e e
− − − − − −= − + − 
11 1.8950461732.0828306.10 ,re− −− (7) 
1 10 2.68 11 1.34 10 5.9926621783.79878762.10 1.869965313.10 3.039030097.10B r r rk e e e− − − − − −= − + −
10 3.790092346 11 1.895046173
11 2.996331089 2.004588422.10 1.973530079.101.49597225.10 ,
r r
r e ee
r r
− − − −
− −− − − (8)
9 2.68
10 2.68 12 1.34
1
11 1.34 10 3.790092346
11 1.895046173 1
1.018075082.10
4.547402034.10 5.59618286.10
2.505753519.10 7.198877938.10
1.771835304.10 7.59757524.10
r
r r
B
r r
r
e
e e
r
e e
r r
e
r
γ
− −
− − − −
− − − −
− − −
= − − +
+ − +
+ +
0 2.68 11 1.34
2 2
3.739930626.10r re e
r r
− − −
+ +
10 2.68 11 1.34
3 3
2.834916134.10 2.790993004.10
,
r re e
r r
− − − −
+ +
(9) 
Analogously, we can obtain the expressions of the cohesive energy 0BU and the alloy 
parameters ,B Bk γ of the atom Si in the positions 2 and 3 in the interstitial alloy FeH. 
TP CH KHOA HC − S
 8/2016 51 
 In the case of applying the n- m potential (1), the cohesive energy between atoms in 
the clean metal A has the form [6] 
0 0
0
1 1
,
n m
A n m
A A
r rd
U mA nA
n m r r
    
 = −   −      
 (10) 
where 1Ar is the neraest neighbour between atoms A at temperature T, 01Ar is the neraest 
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: 
01 0 .
n
n mA
m
A
r r
A
−=
(11) 
Then, the metal parameters 1 2, ,
A
A Ak γ γ and Aγ 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 Edecreases, the coefficient D0 changes 
negligibly and the diffusion coefficient D increases. In the same pressure, in D is a 
monotonously decreasing function of 1/T. In the same temperature, when the pressure 
increases, the activated energy E increases, the coefficient D0 increases, the diffusion 
coefficient D and ln D decreases. 
The dependences of the diffusion coefficient 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. 
100 200 300 400 500 600 700 800 900 1000
29
30
31
32
33
34
35
36
37
38
39
40
 P = 0 (GPa)
 P = 30(GPa)
 P = 70(GPa)
E
(k
ca
l/m
ol
)
T(K)
 Fe-Si
Fig 1. E(T) of FeSi at P = 0, 30 and 70 GPa 
100 200 300 400 500 600 700 800 900 1000
17
18
19
20
21
22
23
24
25
26
27
28
 P= 0 (GPa)
 P= 30 (GPa)
 P= 70 (GPa)
D
0
(1
0
-2
 c
m
2
/s
)
T(K)
 Fe-Si
Fig 2. D0(T) of FeSi at P = 0, 30 and 70 GPa 
52 TRNG I HC TH  H NI 
0.002 0.004 0.006 0.008 0.010
-200
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
 P = 0 (GPa)
 P = 30 (GPa)
P = 70 (GPa)
ln
D
1/T
 Fe-Si
Fig 3. lnD (1/T) at P = 0, 30 and 70 GPa 
for FeSi 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
6
7
8
9
10
11
12
13
14
15
16
 P= 0 (GPa)
 P= 30 (GPa)
 P= 70 (GPa)
D
0
( 
1
0-
2 c
m
2
/s
)
Csi (%)
 Fe-Si
 T=300K 
Fig 4. D0(cSi) of FeSi at P = 0, 30, 70 GPa 
and T = 300K 
 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 P = 0, T = 1150oC and cSi = 4.9%, the alloy FeSi has D 
= 1.4.10-6 cm2/s according to the experimental data [8]. Accordinng to our calculated 
result, at P = 0, T = 1000K, cSi = 5%, the alloy FeSi hasD = 0.08. 10-6 cm2/s. 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
6
7
8
9
10
11
12
13
14
15
16
 P= 0 (GPa)
 P= 30 (GPa)
 P= 70 (GPa)
D
0
( 
1
0
-2
cm
2
/s
)
Csi (%)
 Fe-Si
 T=900K 
Fig 5. D0(cSi) of FeSi at P = 0, 30, 70 GPa 
and T = 900K 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
5.0
5.2
5.4
5.6
5.8
6.0
6.2
6.4 P= 0,
 T=300K 
D
 (
1
0-
26
 c
m
2 /
s)
CSi(%)
 Fe-Si
Fig 6. D(T) of FeSi at P = 0 and T = 300K 
TP CH KHOA HC − S
 8/2016 53 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
7.6
7.8
8.0
8.2
8.4
 P= 0,
 T=900K 
D
 (
10
-9
 c
m
2
/s
)
CSi(%)
 Fe-Si
Fig 7. D(T) of FeSi at P = 0 and T = 900K 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.70
1.75
1.80
1.85
1.90
1.95
2.00
2.05
2.10
 P= 30,
 T=300K 
D
 (
10
-2
5 
cm
2
/s
)
CSi(%)
 Fe-Si
Fig 8. D(T) of Fe-Si at P = 30 GPa 
and T = 300K 
At P = 0 and from 200 to 780oC, the alloy FeH has D0 = 1.4.10-3 cm2/s, E = 0.139eV 
and at T = 700oC, the alloy FeH có D = 2.45.10-4 cm2/s according to the experimental data 
[9]. Accordinng to our calculated result, at P = 0, T = 1000K, the alloy FeH has 
D0 = 0.9.10-3 cm2/s, E = 0.19479 kcal/mol. Figure 3 shows the dependence of ln D on 1/T 
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. 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.70
1.75
1.80
1.85
1.90
1.95
2.00
2.05
2.10
 P= 30,
 T=900K 
D
 (
1
0
-7
 c
m
2
/s
)
CSi(%)
 Fe-S i
Fig 9. D(T) ofFeSiat P = 30 GPaand T = 900K 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
 P= 70,
 T=300K 
D
 (
1
0
-2
4
 c
m
2 /
s)
CSi(%)
 Fe-S i
Fig 10. D(T) ofFeSiat P = 70 GPa 
and T = 300K 
54 TRNG I HC TH  H NI 
1.0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0
4 .3
4 .4
4 .5
4 .6
4 .7
4 .8
4 .9
5 .0
5 .1
 P= 70 ,
 T =900K 
D
 (
1
0
-5
 c
m
2
/s
)
C S i(% )
 Fe-S i
Fig 11. D(T) of FeSi at P = 70 GPa 
and T = 900K 
100 200 300 400 500 600 700 800 900 1000
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
 P= 0 (G Pa)
 P= 30 (G Pa)
 P= 70 (G Pa)
E
 (
kc
a
l/m
o
l)
T (K)
 Fe-H
Fig 12. E(T) of FeH at P = 0, 30 
and 70 GPa 
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5 P= 0 (GPa)
 P = 30(Gpa)
 P= 70(GPa)
ln
D
1/T
 Fe-H
Fig 13. lnD (1/T) at P = 0, 30 and 70 GPa 
for FeH 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
P = 0 (GPa)
P =30 (GPa)
 P =70 (GPa)
D
0(
10
-2
cm
2
/s
)
CH(%)
 Fe-H
T=300K
Fig 14. D0(cH) of FeH at P = 0, 30, 70 GPa 
and T = 300K 
1 .0 1 .5 2.0 2.5 3 .0 3 .5 4 .0 4 .5 5 .0
6 .0
6 .5
7 .0
7 .5
8 .0
8 .5
9 .0
9 .5
1 0 .0
1 0 .5
1 1 .0
1 1 .5
1 2 .0
1 2 .5
1 3 .0
P = 0 (G P a )
P = 3 0 (G P a )
 P = 7 0 (G P a )
D
0(
10
-2
cm
2 /
s)
C H (% )
 F e -H
T = 90 0K
Fig 15. D0(cH) of Fe-H at P = 0, 30, 70 GPa 
and T = 900K 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.0422
1.0423
1.0424
1.0425
1.0426
1.0427
1.0428
1.0429 P= 0 GPa
D
(1
0
-7
cm
2
/s
)
CH(%)
 Fe-H
T=300K
Fig 16. D(cH) of FeH at P = 0 
and T = 300K 
TP CH KHOA HC − S
 8/2016 55 
1 .0 1 . 5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0
6 .5 5
6 .6 0
6 .6 5
6 .7 0
6 .7 5
6 .8 0
 P = 3 0 G P a
D
(1
0
-6
cm
2
/s
)
C H (% )
 F e - H
T = 3 0 0 K
Fig 17. D(cH) of FeH at P = 30 GPa 
and T = 300K 
1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0
5 .2 0
5 .2 2
5 .2 4
5 .2 6
5 .2 8
5 .3 0
 P = 7 0 G P a
D
(1
0-
2 c
m
2 /
s)
C H (% )
 F e - H
T = 3 0 0 K
Fig 18. D(cH) of FeH at P = 70 GPa 
and T = 300K 
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.600
1.605
1.610
1.615
1.620
1.625
1.630
 P= 70 GPa
D
(1
0
-1
cm
2
/s
)
CH(% )
 Fe-H
T=900K
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 n-m 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 TRNG I HC TH  H NI 
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NGHIÊN CỨU SỰ KHUẾCH TÁN CỦA NGUYÊN TỬ XEN KẼ 
TRONG CÁC HỢP KIM XEN KẼ Fe-Si VÀ Fe-H VỚI CẤU TRÚC 
LẬP PHƯƠNG TÂM KHỐI DƯỚI TÁC DỤNG CỦA ÁP SUẤT 
Tóm tắt: Trong bài báo trước [10], chúng tôi rút ra biểu thức giải tích đối với năng 
lượng tự do của nguyên tử xen kẽ, khoảng cách lân cận gần nhất giữa hai nguyên tử xen 
kẽ, các thông số hợp kim đối với nguyên tử xen kẽ, các đại lượng khuếch tán như tần số 
bước nhảy của nguyên tử xen kẽ, độ dài bước nhảy hiệu dụng, thừa số tương quan, hệ số 
khuếch tán và năng lượng kích hoạt cùng với phương trình trạng thái của hợp kim kim 
xen kẽ AB với cấu trúc lập phương tâm khối dưới tác dụng của áp suất bằng phương 
pháp mômen thống kê. Trong bài báo này, chúng tôi áp dụng các kết quả lí thuyết này 
cho các hợp kim xen kẽ Fe-Si và Fe-H trong vùng nồng độ nguyên tử xen kẽ từ 0 đến 5%, 
vùng nhiệt độ từ 100 đến 1000K và vùng áp suất từ 0 đến 70GPa. Kết quả tính toán phù 
hợp khá tốt với số liệu thực nghiệm hoặc dự báo thực nghiệm 
Từ khoá: Hợp kim xen kẽ, tần số bước nhảy, độ dài bước nhảy hiệu dụng, nhân tố tương 
quan, hệ số khuyếch tán, năng lượng kích hoạt