Abstract. We have examined two simulation techniques for diffusion on disordered
systems: the non-force and force method. The latter enables one to study diffusion
on the system with periodic boundary conditions and size which is 20 times shorter
than the non-force method. Furthermore, the force method is an appropriate way to
study impurity diffusion in a more real model of amorphous solid constructed by
molecular dynamic or statistic relaxation techniques. Using the non-force method
we simulate diffusion on a disordered chain consisting of 2000 sites for a wide
range of temperature. Three types of disordered chains, site, saddle and site-saddle
system are considered. The influence of site and saddle disorders and Arrhenius
behavior is also investigated and discussed in this work.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 142-150
This paper is available online at
A STUDY OF DIFFUSION IN A DISORDEREDCHAIN: COMPUTER SIMULATION
Trinh Van Mung1, Pham Khac Hung2 and Nguyen Thi Thanh Ha2
1Vinh Phuc College, Vinh Phuc Province
2Department of Computational Physics, Institute of Engineering Physics,
Hanoi University of Science & Technology
Abstract.We have examined two simulation techniques for diffusion on disordered
systems: the non-force and force method. The latter enables one to study diffusion
on the system with periodic boundary conditions and size which is 20 times shorter
than the non-force method. Furthermore, the force method is an appropriate way to
study impurity diffusion in a more real model of amorphous solid constructed by
molecular dynamic or statistic relaxation techniques. Using the non-force method
we simulate diffusion on a disordered chain consisting of 2000 sites for a wide
range of temperature. Three types of disordered chains, site, saddle and site-saddle
system are considered. The influence of site and saddle disorders and Arrhenius
behavior is also investigated and discussed in this work.
Keywords: Diffusion, random walk, disordered system, simulation, non-force
method.
1. Introduction
Despite great effort over a long period of time, many aspects of diffusion in
amorphous alloys (AM) are still poorly understood [1-3]. For example, it is still not clear
why the existence of a wide, continuous spectrum of site and saddle point energy for
diffusion process could also give rise to the Arrhenius behavior experimentally observed
in certain AMs [4-6]. In comparison with crystal counterparts, the pre-exponential factor
as well as activation energy exhibits certain specific features. For instance, the factor Do
for diffusion of some impurities in AMs changes in the very wide range of 10±7cm2/s
[2-3]. Several models have been suggested to clarify those observations. One, the Fisher
random walk model, states that, for not too strong disorder, random walk in dimensions
Received October 15, 2011. Accepted September 30, 2012.
Physics Subject Classification: 62 44 01 01.
Contact Trinh Van Mung, e-mail address: mungtv76@gmail.com
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A study of diffusion in a disordered chain: computer simulation
higher than two gives rise to classical diffusion behavior at large time [5]. A simulation of
diffusion in regular disordered lattice in which the sites for target particle remain ordered
but the potential barriers vary from site to site could provide useful information [7-8].
As shown in ref. [7], the mean square displacement 〈x2n〉 of the target particle after n
jumps is proportional to fna2. Here a is the distance between the two nearest sites and
f is correlation factor which is a function of temperature and may be less than 10−3 for
strong disordered lattice. The Monte-Carlo (MC) simulation in [8] found a compensatory
effect between disorders in the site and saddle point energies which originates in the
Arrhenius behavior for diffusion of strong disordered lattice. However, it is very useful
to conduct a simulation of the diffusion process in a more real model of AMs which is
constructed by techniques like molecular dynamic (MD) or statistic relaxation method
[9-10] because these models reproduce well both the dynamic and structural properties
of AMs. The network of sites that are a link or a potential barrier between them can be
directly calculated using MD models. Nevertheless, previous MC technique reported in
refs. [8] can not be applied for that network due to the very long computation required
and the imprecise result obtained. In the case of low temperature, for example, it is very
impractical because one must run a very large number of particle hops.
The aim of this article is twofold. First, it is to find a simulation technique which
could be use to study systems with periodic boundary conditions and are of moderate
size. Two simulation techniques based on target particle probability density and the
Einstein diffusion equation have been investigated. We demonstrate the effectiveness of
suggested simulation techniques on diffusion in a definite disordered chain with and
without periodic boundary conditions, and compare this to the Monte-Carlo method.
Second, the influence of two types of energetic disorders on diffusion in disordered chains
is also considered and examined here.
2. Content
2.1. Calculation method
As mentioned above, it is useful to study the diffusion of impurities on MD model
sites. A simulation for this system can be performed by means of the MC method.
Accordingly, the target particle that is initially located somewhere in the MD model
moves from one site to another and visits a large number of sites along a random path.
The position of the target particle is recorded during every run and then an average
square displacement and diffusion time is calculated after n jumps. Such a simulation
is not accurate and also not practical at low temperatures when the particle is captured
by "deep trap" and moves many times between two neighboring sites. Now we examine
two simulation techniques which consume much less computing time. Consider a chain
of sites labeled by integer numbers 1, 2, 3, ...N , where each ith site is assigned a site
energy εi and two neighboring sites i, i + 1 the transition energy εi,i+1 (saddle point
143
Trinh Van Mung, Pham Khac Hung and Nguyen Thi Thu Ha
energy). The target particle locates initially at middle siteN/2 and then will be allowed to
randomly move to left or right sites by a thermally activated Poissonian jump. The relative
probability to jump from ith site into left and right neighboring site is given by
pi = 1− qi = exp(−εi,i−1β)
exp(−εi,i+1β) + exp(−εi,i−1β) . (2.1)
Since the time period between hops from site i to site i+1 is τi,i+1 = τ0 exp((εi,i+1−
εi)β) τi,i+1 = τ0 exp((εi,i+1 − εi)β), the average time that the particle stays at ith site can
be given as
τi = piτi,i−1 + qiτi,i+1
→ τi = 2τ0 exp(−εiβ)
exp(−εi,i+1β) + exp(−εi,i−1β) (2.2)
here τ0 is the period of atomic vibration at a lattice site.
Let us define c(i, n) as the probability of just arriving on the ith site as a result of n
jumps. The probability at the next step n + 1 is given by
c(i, n+ 1) = c(i+ 1, n)pi+1 + c(i− 1, n)qi−1. (2.3)
As a initial configuration related to step n = 0, the probability c(i, 0) is equal to
1 for site N/2 and c(i, 0) = 0 for other sites. Eq. 2.3 allows the calculation c(i, n) at
each step n and as such simulates the diffusion process. We also calculate average square
displacement 〈x2n〉 and the time period to realize n steps tn obtained as follows
= a
2
N∑
i
ci(i, n)(i−N/2)2 (2.4)
and
tn =
n∑
k
N∑
i
ci(i, k)τi (2.5)
where a is the distance between two neighboring sites.
It is now easy to calculate diffusion coefficient D that is determined as a slope
of dependence of 〈x2n〉 versus tn. The algorithm above which described and denotes the
non-force method can be used to simulate very long chain of sites, but it is not appropriate
for systems of moderate size or periodic boundary conditions. To take this into account
we employ a force method. Taking a chain withN sites and periodic boundary conditions,
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A study of diffusion in a disordered chain: computer simulation
the transition energy of site 1 and N will be equal to each other, i.e. ε0,1 = εN,N+1. Due
to periodic boundary conditions, the target particle arriving at site N can jump to site
N − 1 and 1. Analogously, after locating at the 1th site the target particle will jump into
site 2 or N at next step. For a large number of steps n, the probability c(i, n) approaches
equilibrium value, i.e. the process is stationary, given as
ceq(i, n) =
exp(−εi−1,iβ) + exp(−εi,i+1β)
2
N∑
1
exp(−εi,i+1β)
. (2.6)
If we apply external field force g, then the site and transition energies of the
considered system change according to
εfi = εi + gia ε
f
i,i+1 = εi,i+1 + g(i+ 0.5)a (2.7)
here the index f indicates the corresponding parameter of applied force g.
Obviously the probability c(i, n) varies with n in accordance to Eq. 2.1 with
parameters pfi , q
f
i determined by Eq. 2.7. Force g gives rise to drift flow in the system
for which velocity follows Einstein diffusion relation
v = Dg.β (2.8)
To calculate the drift velocity we determine the averaged mean displacement 〈xn〉
given as
〈xn〉 = a
N∑
i
c(i, n)pi − c(i, n)qi (2.9)
In the absence of force g the quantity 〈xn〉 is almost equal to zero. The drift velocity
can be obtained by determining the slope of the dependence of 〈xn〉 versus tn. As such,
the force method consists of following steps:
(i) Set up set of site and transition energies ǫi, ǫi,i+1 on the chain; i = 1, 2, ...N .
Then the initial probabilities c(i, n) are determined by Eq. 2.6.
(ii) Set up set of site and transition energies ǫfi , ǫ
f
i,i+1 using Eq. 2.7.
(iii) Calculate the probability c(i, n) at each step n according to Eq. 2.3 and using
the parameters pfi , q
f
i .
(iv) Calculate the mean displacement 〈xn〉 and compute the drift velocity, and
finally the diffusion coefficient is determined by the Einstein relation 2.8. This method can
be easily generalized for higher-dimension systems using a similar computing procedure.
Hence, for a network of sites that consists of several thousand sites and is calculated using
MDmodels, the force method is a more effective way to calculate the diffusion coefficient
of such a system than the non-force or MC method.
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Trinh Van Mung, Pham Khac Hung and Nguyen Thi Thu Ha
2.2. Results and discussion
In order to check the validity of the force method, two chains consisting of 4.105
and 2.103 sites are constructed. A simulation is carried out for the long chain using the
non-force method. The force method is applied for the short chain with periodic boundary
conditions. Both sites and saddle points of the chain will be assigned to energies chosen
at random or constant lowest value ǫ1x. We consider two types of energy distribution:
two-level distribution where site/transition energy for every site amounts to either values
ǫ1x or ǫ2x (ǫ1x < ǫ2x, x is denoted to s or t) and uniform distribution where site/transition
energies are uniformly distributed in the interval [ε1x, ε2x].
Figure 1. The mean displacement (left) and square displacement (right)
Figure 1 shows averaged square displacement 〈x2n〉 and mean displacement 〈xn〉
as a function of diffusion time tn. Here α is a fraction of low-level energies ǫ1x. Very
straight lines are observed, the slope of which determines the diffusion coefficient and
drift velocity. In what follow we shall use the quantities t1 = τ0 exp((ε1t − ε1s)β) and
D1 = a
2/2t1 = D01 exp((ε1t − ε1s)β) which is the diffusion coefficient for an ordered
chain with constant site and transition energies ǫ1s and ǫ1t, respectively. The result of the
simulation for both long and short chains is shown in Figure 2. It can be seen that while
both simulation methods provide an identical result the size of the chains employed here
is significantly different. The discrepancy obtained here is less than 0.1%. Therefore, the
non-force method allows one to attaining a precise result based on a system of relatively
small size and it can be employed for simulating the diffusion on a network of sites in
a real AM model which consists of several thousand atoms. An important point with
respect to matter diffusion is that AMs are characterized by two kinds of disorder which
have deeply different properties: on the one hand, there are site disorders corresponding to
the so-called random-trapping model, on the other hand, there are saddle point disorders,
corresponding to the random-hoping model [1].
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A study of diffusion in a disordered chain: computer simulation
Figure 2. The diffusion coefficient of a two-level distribution system at temperature
(ε2t − ε1t)β = 3 (left) in a uniform distribution system (right)
Figure 3. Temperature dependence of the diffusion coefficient
for saddle (left) and site systems (right)
In the present work these disorders are examined separately on three corresponding
chains: (1) the site system, where site energies are constant and equal to ǫ1s, but transition
energies amount to random value, (2) the saddle system, where transition energies have
constant the value of ǫ1t and site energies is chosen randomly, (3) the saddle-site system,
where both site and transition energies are chosen randomly but their type of energy
distribution is the same, i.e. it is two-level or uniform distribution with the same ε2x−ε1x.
Figure 3 presents the temperature dependence of the diffusion coefficient for site and
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Trinh Van Mung, Pham Khac Hung and Nguyen Thi Thu Ha
saddle systems. The notable feature of these plots is the linearity for a limited interval of
temperature in each case which could be fitted to following formula
ln(D/D1) = ax(ε2x − ε1x)β + bx (2.10)
We examine two temperature intervals (ε2t − ε1x)β = (1.5 - 3.5) and (7 - 9). The
fitting parameters are listed in Table 1 show that parameter at at low temperatures is higher
than at high temperatures which implies a deviation from Arrhenius behavior along the
entire interval (1.5 - 9).
Table 1. Parameters at and bt of saddle point and site systems
Two-level distribution Uniform
distributionα = 0.2 α = 0.4 α = 0.6 α = 0.8
Interval of (ε2t − ε1t)β 1.5-3.5 1.5-3.5 1.5-3.5 1.5-3.5 1.5-3.5
at -0.9773 -0.9431 -0.8798 -0.7281 -0.6867
bt 0.1395 0.2936 0.4736 0.6343 0.2044
Interval of (ε2t − ε1t)β 7.0-9.0 7.0-9.0 7.0-9.0 7.0-9.0 7.0-9.0
at -0.9995 -0.9979 -0.9953 -0.9906 -0.8743
bt 0.2184 0.4885 0.8921 1.6069 1.073
Interval of (ε2s − ε1s)β 7.0-9.0 7.0-9.0 7.0-9.0 7.0-9.0 7.0-9.0
as 0 0 0 0 0.1276
bs 1.6405 0.9243 0.5167 0.2193 1.082
Interval of (ε2s − ε1s)β 7.0-9.0 7.0-9.0 7.0-9.0 7.0-9.0 7.0-9.0
ats -0.9982 -0.9982 -0.9967 -0.9903 -0.7471
bts 1.797 1.382 1.4213 1.7954 2.1428
In the case of α = 0.8, parameter at changes from -0.7281 to -0.9906, i.e. about 1.36
times, indicating strong deviation from Arrhenius behavior. From Eq. 2.10 the diffusion
coefficient can be reduced to
D =D1 exp(ax(ε2x − ε1x)β + bx)
=D01 exp(bx) exp(ax(ε2x − ε1x)β − (ε1t − ε1s)β) (2.11)
Comparing this to an ordered chain diffusion, where site and transition energies are
equal to the ǫ1s and ǫ1t, the effective activation energy changes by −ax(ε2t − ε1x) and
the pre-exponential factor by a factor of exp(bx). Because the value of ax for the saddle
system is negative, its effective activation energy increases. Meanwhile, the parameter bx
is always positive so the pre-exponential factor increases 1.2 to 5.6 times in comparison
with the ordered chain. In the case of two-level distribution and at low temperature, the
effective activation energy approaches the value ε2t − ε1s or ε1t − ε1s for the saddle and
site system, respectively, i.e., the highest energy level of the saddle point and the lowest
energy level of the site have a dominant contribution to the activation energy in the low
148
A study of diffusion in a disordered chain: computer simulation
temperature region (see Table 1). Figure 4 presents the results for the saddle-site system.
It is noted that in the interval of (7 - 9), linearity is observed in all plots in Figure 4.
Parameters at and bt are almost equal to the sum of corresponding site and saddle system
parameters (see Table). Therefore, the pre-exponential factor D0 of a disordered chain is
bigger when both types of energetic disorders exist. The effect of compensation proposed
in ref.[1] is not observed in our simulation.
Figure 4. Temperature dependence of the diffusion coefficient in a saddle-site system
3. Conclusion
A new simulation technique is proposed which allows the study of diffusion in a
disordered system of moderate size and with periodic boundary conditions. A simulation
carried out which makes use of this method on a chain which is 20 times smaller provides
an accuracy equal to that using the usual method. The influence of both types of disorders
has been investigated in relation to disordered chain. Some features of diffusion in those
systems have been revealed as follows:
1-Despite the existence of strong energetic disorder in the system, including site and
saddle types, Arrhenius behavior is always observed within a limited temperature range,
particularly at low temperatures. Both types of disorder increases the pre-exponential
coefficient D0 by a factor of 1.2 to 9.1 depending on the actual distribution of site and
transition energy. On the other hand, along the entire temperature interval investigated we
observe strong deviation from Arrhenius law except when there was two-level distribution
with α = 0.2.
2-Concerning the effective activation energy, it approaches the value of (ε2t − ε1s)
or (ε1t − ε1s) for the saddle and site system, respectively, as temperature decreases. This
means that the highest energy level of a saddle point and the lowest energy level of a site
have a dominant contribution in determining the activation energy at low temperatures.
149
Trinh Van Mung, Pham Khac Hung and Nguyen Thi Thu Ha
3-When there exist both site and saddle disorders in the chain, parameters ats and
bts derived from Arrhenius plot are almost equal to the sum of the parameters of the site
and saddle systems. A compensation effect for pre-exponential factors was not observed
in our simulation.
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