Abstract. In this paper we use a new method based on Einstein’s equation to calculate the diffusion coefficient of impurity atoms in the disorder
systems.Simulation results show that the diffusion coefficient D obeys the
Arrhenius law. The dependence of the diffusion coefficient D on temperature for regular disordered lattice is also examined and discussed. This
study also investigated the influence of density and size effect to the diffusion process. Calculated results are the basis helping us to understand the
diffusion mechanism in amorphous materials and predict their structure.

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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2010, Vol. 55, No. 6, pp. 37-43
SIMULATION THE DIFFUSION PROCESS
IN DISORDERED SYSTEM
Trinh Van Mung(∗), Pham Khac Hung and Pham Ngoc Nguyen
Hanoi University of Science and Technology
(∗)E-mail: mungtv76@gmail.com
Abstract. In this paper we use a new method based on Einstein’s equa-
tion to calculate the diffusion coefficient of impurity atoms in the disorder
systems.Simulation results show that the diffusion coefficient D obeys the
Arrhenius law. The dependence of the diffusion coefficient D on temper-
ature for regular disordered lattice is also examined and discussed. This
study also investigated the influence of density and size effect to the diffu-
sion process. Calculated results are the basis helping us to understand the
diffusion mechanism in amorphous materials and predict their structure.
Keywords: diffusion, disorder system, Intertitial diffusion, amorphous sys-
tem
1. Introduction
When studying diffusion mechanisms in disordered system such as amorphous
alloys (AMA) and metallic glasses (MG) [1–4], these researchers found the linear
behavior of Arrhenius plots of deviation from it depending on the concrete model of
site and transition energy. However, many problems with the interstitial diffusion
in disordered media need to be clarified such as density and size effect, activation
energy.... To give an insight into these problems, it is convenient to use a regular
disordered lattice where the ordered arrangement of sites is retained, but the site
and transition energies are different at different sites. Using this approach we show
in earlier works [5, 6] that the presence of broad distribution of site and transition
energies leads to two specific effects. The first consists in fact that the real path
of diffusing particles is enriched with lower barriers compared to general sets of
barriers in the entire system. This effect reduces the time of site occupation by a
particle and enhances the diffusion coefficient. The second effect concerns the fact
that instead of ordinary expression for mean square displacement of a particle after
n hops x2n = na
2 we obtain:
x2n = Fna
2, (1.1)
where a is a length of a single hope; F is the correlation factor.
37
Trinh Van Mung, Pham Khac Hung and Pham Ngoc Nguyen
The presence of these two effects is also noted in [7]. Due to the poor accuracy,
the method employing in [5, 6] allows to calculate the diffusion constant only for
a small range of temperature and cannot be used for more realistic models. In
this present article we show a new calculation method for amorphous atomistic
model when the impurity–matrix atom potential (IMAP) is known. In terms of
illustrating the applicability of that new method, it is necessary to present the
interstitial diffusions in LJ models for different impurities (different IMAP).
2. Calculation Method
For example demonstrating the new calculation technique, consider the diffu-
sion of a single particle in a linear finite chain of sites with given site and transition
energy εi, εi+1. The probability of finding particles at ith site after n hops ci(n)
relates to one at previous hop by:
ci(n) = ci+1(n− 1)pi+1 + ci−1(n− 1)qi−1 (2.1)
where pi and qi are the probability of particles moving from ith site to left and right
neighbour site; pi+qi = 1. In the case of constant probability pi = 0.5, if the particle
locates initially at sth site; then cs(0) = 1 and ci(0) = 0 for j#s. Next hop (n = 1)
leads to cs(1) = 0, cs−1(1) = cs+1(1) = 0.5, and cj(0) = 0 for j#s − 1 and s + 1.
Obviously, after n/2 hops the particle moves out of the chain; hence, the number
n must be less than N/2 for the chain without any special boundary condition.
Here N is number of sites in the chain. Equation (2.1) allows to calculate ci(n) at
each step and thus study the diffusion process. The mean square displacement of
particles after n hops is given by the expression
〈
x2n
〉
=
N∑
i
ci(n)(xi − xs)2 (2.2)
The sum with respect to i is over all sites in the chain; xi is the coordinate
of ith site. Since the time period between hop of particle from site i to site (i + 1)
is τi,i+1 = τ0 exp
(εi,i+1−εi)
kT
(τ0the period of atomic vibration at a lattice site; k –
Boltzmann constant, T – temperature; εi,i+1 and εi – transition and site energy),
the average time that particle stays at ith site can be given as:
τi = piτi,i−1 + qiτi,i+1 =
2τ0 exp
(− εi
kT
)
exp
(−εi,i+1
kT
)
+ exp
(−εi,i−1
kT
) (2.3)
The average time period for realizing n hops is:
tn =
n∑
k
N∑
i
ci (k) τi (2.4)
38
Simulation the diffusion process in disordered system
The diffusion constant D can be determined by the slope of the 〈x2n〉 versus tn
curve. The method described above is very simple, but the number of sites linearly
increases with number of hop n (for one–dimension system) and it works badly for
three-dimension systems where the number of sites must be bigger than n3. Now
we introduce a new calculation method (external force method) based on Einsteins
equation
ν =
Dg
kT
(2.5)
where v is the drift velocity and g characterizes the external force. Applying the
external force g leads to change in site and transition energies by
εfi = εi + gxi; ε
f
i,i+1 = εi,i+1 + gxi,i+1 (2.6)
Here the index f indicates the site and transition energy upon applying ex-
ternal force g; xi,i+1 is coordination of saddle-point for adjacent ith and (i + 1)
th
sites.
Initially, the site energies εi are set to every ith site and the transition energies
εi,i+1 to two adjacent ith and (i+1)
th sites. The transition energies εij can have one
of two values: ε1 and ε2 (two–level distribution; ε2 > ε1).
Figure 1. The radial distribution function
To examine the interstitial diffusion in the system with available IMAP, we
construct six models with the same atomic density, each model containing atomic
numbers respectively from 1024 to 5000 atoms and three models with the different
atomic density. All models have a form of cubic box with boundary periodic condi-
tions. Figure 1 depicts the radial distribution function of LJ model and we can see
the characteristic splitting of second peaks which is observed for most AMA. The
potential between diffusing particle and matrix atoms is adopted by:
ϕ(r) = ε0
(r0
r
)m
(2.7)
To calculate the set of sites and set of energies εi and εij, we insert into sim-
ulation box a simple cubic lattice with sides 151× 151× 151 and 3375000 diffusing
39
Trinh Van Mung, Pham Khac Hung and Pham Ngoc Nguyen
particles at lattice nodes (model containing 5000 atoms). Then all diffusing particles
move step by step at the distance of 0.01 A˚ towards the direction of force acting on
the impurity from all matrix atoms. This movement is repeated many times until
all diffusing particles reach the equilibrium positions, which correspond to the local
minimum of energy. These finding positions create a set of interstitial sites and the
energy of diffusing particles at these sites are the site energy εi. To determine the
transition energy between ith and jth sites εij we calculate the energetic profile by
moving diffusing particles step by step on the line connecting two neighbour sites i
and j.
3. Results and discussion
A well straight line is observed with slope determining the diffusion coefficient
and drift velocity.
For computing convenience we use τs =
(εtrans−εsite)
kT
, τ ∗ = τ0 exp
(
εtrans−εsite
kT
)
,
D∗ = γa
2
τ∗
– the diffusion coefficient for regular ordered lattice with constant site and
transition energies; where εtrans, εsite are the transition and site energy (εsite = ε1
in the case of two–level distribution of transition energy); γ –geometrical factor
(γ = 1/6 for simple cubic lattice). From Figure 2 we can see that the diffusion
constant D is well described by Arrhenius Law with the same atomic density LJ
model containing 1024 atoms to 5000 atoms. It means that the diffusion coefficient
D independence on the number of impurity atoms in the system.
Figure 3 shows that three lines are nearly parallel to teach with other. It
means that the increasing concentration w of low energy ε in considered range,
increases the diffusion constant D, but it only slightly affects the activation energy
Q. Therefore, in this case, the pre-exponential factor D0 also increases with w. The
diffusion behavior for disordered cubic lattice is very complicated, in this case, to
diffuse from site i to site j, the particle has a number of diffusion paths, whereas
conversely there is only one diffusion path in the linear chain. Therefore, if w is large
enough then particles always find a path where it must overcome only low transition
energy. It can be seen from Figure 3 that the Arrhenius dependence occurs when
the concentration w is less than 0.250 or bigger than 0.236.
The corresponding activation energy Q changes from ε2−εsite to ε1−εsite with
increasing the concentration w. The variation of activation energy can be explained
by the fact that whereas the impurity must overcome high barriers ε2 − εsite in the
system with small w, there is a diffusion path with only low barriers ε1 − εsite in
system with large enough w. The particle prefers to diffuse along these paths and
it leads to decrease the activation energy Q. Thus, the diffusion in disordered cubic
lattice with small w is similar to one for linear chain due to absence of diffusion path
with only low transition energy (percolation problem).
40
Simulation the diffusion process in disordered system
Figure 2. The dependence of − ln( D
D∗
) as a function of Ts
with the same atomic density.
Figure 3. The dependence of − ln( D
D∗
) as a function of Ts
with the different atomic density.
Table 1. The number of sites per matrix atom in LJ model
Number of matrix atoms 1024 2000 2500 3000 4000 5000
m = 6 Number of sites 2.252 2.244 2.196 2.204 2.224 2.184
m = 9 Number of sites 2.714 2.679 2.664 2.667 2.673 2.637
Table 1 presents the number of sites per matrix atom in LJ model, we see
that the number of sites as well as sites positions changes with parameter m. With
stronger repulsive potential (m = 9) the number of sites increases.
Figure 4 and Figure 5 present the distribution of site and transition energy
εi, εi,ij for LJ model. Two pronounced peaks appear for site energy distribution,
indicating the existence of two regular configurations in LJ structure. Probably,
they are distorted octahedral and tetrahedral interstice. Therefore, our simulation
supports the assumption in [8] that impurity in amorphous structure might occupy
41
Trinh Van Mung, Pham Khac Hung and Pham Ngoc Nguyen
the interstices, which are similar to octahedral and tetrahedral interstitial sites in
crystalline alloys. Regarding the influence of IMAP, the diffusion for impurity with
potential m = 6 is significantly faster than one for impurity with potential m = 9
(Figure 5). However, the activation energy of different diffusing impurity (different
IMAP) is very close to each other. It means that transport processes of different im-
purity with potential type is characterized by significantly different pre-exponential
factor D0 and very close activation energy Q.
Figure 4. The distribution of site energy for LJ.
Figure 5. The dependence of − ln( D
D∗
) as a function of Ts
with the different atomic density.
42
Simulation the diffusion process in disordered system
4. Conclusion
The diffusion constant D is well described by Arrhenius Law with the same
atomic density LJ model containing 1024 atoms to 5000 atoms. It means that the
diffusion coefficient D independence is on the number of impurity atoms in the
system.
The increasing concentration w of low energy ε1 in considered range increases
the diffusion constant D, but it only slightly affects the activation energy Q. There-
fore, in this case, the pre-exponential factor D0 also increases with w.
The calculation results for the LJ model, we obtain a broad continuous dis-
tribution of site and transition energies. The distribution of energy εi has two pro-
nounced peaks which correspond to distorted octahedral and tetrahedral interstices
in LJ structure.
The diffusion for impurity with potential m = 6 is significantly faster than
one for impurity with potential m = 9. However, the activation energy of different
diffusing impurity (different IMAP) is very close to each other.
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