# Determination of the mean first passage time of a vacancy in Nial Binary Alloy by Analytical and Adib’s Simulation Algorithm and Finding its Diffusion Coefficient

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## Date

2012-12

## Authors

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## Publisher

Addis Ababa University

## Abstract

The mean first passage time (MFPT) and effective diffusion coefficients of a vacancy
diffusing in NiAl compound via three main vacancy diffusion mechanisms are studied
both numerically and analytically. These mechanisms are: the next-nearest-neighbor
(NNN), the six-jump, and the triple defect mechanisms. The vacancy diffusion in the
three dimensional crystal structure of NiAl is mapped onto a one-dimensional lattice
sites, allowing the vacancy to hope to its nearest-neighbor with local transitions rates,
k(xi ± | xi) = ±D0
2 e U
0
(xi)
2kBT , that is calculated from its potential profile U(x). The
time evolution of the vacancy in this one dimensional lattice site is governed by a
master equation in which it is evaluated numerically using Algorithms for Brownian
first passage time estimation of Artur B. Adib where the lifetime of the vacancy in
any state xi is drawn from an exponentially distributed random number i with mean
equal to the reciprocal of the sum of the outgoing rates, < i >−1=
P
x0=n.n k(x0 | xi)
and the nearest neighboring site where the vacancy is going next is chosen with a
probability w(x0) = k(x
0
P |x)
x
0
=n.n
k(x0 |xi)
. The sum of the times < i > until the vacancy
touches the absorbing sites is the first passage time of that mechanism. For each
mechanism, we calculate the mean first passage time of the vacancy and our result
predict that the triple defect mechanism takes less time compared with the others
and it is the major contributor to vacancy diffusion in NiAl binary alloy.
i
In the above one dimensional lattice sites, instead of the local transition rates, we
allowed the vacancy to hope from any site to its nearest neighbor with local jump
probability, pi or qi. The mean first passage times (MFPTs) for a vacancy that
diffuses via the three mechanisms are evaluated using the properties of random walks
on networks technique. These analytical result show that the MFPT of the vacancy
in those mechanisms can be expressed in terms of the local jump probabilities (pi
and qi) which in turn they are given by local MFPTs. Finally from these local
MFPTs, we came up the vacancy’s MFPTs to complete its mechanisms and these
times are a functions of vacancy’s local (Ei) as well global barrier heights (Eg), the
background thermal energy (T), and the lattice space ( ) of these mechanisms, i '
4
D0
(E2+E3)
2
exp[ Eg]. Fixing the background temperature at T = 1200K and
using computed local and global energies and other related parameters, we evaluate
analytically MFPT of these mechanisms. This result also favored the triple defect
mechanism as the main diffusion path of a vacancy in NiAl compounds, moreover,
the analytical values of six-jump and triple defect mechanisms are nearly identical
with the one computed by Adib’s method.
The three local energy barrier heights of the six-jump and triple defect mechanisms
where the vacancy crosses along its diffusion paths are summed which we call
them global energy barrier heights (U(x) = E1 − E2 + E3). From this new potential
structure which is globally varying periodic potential U(x) = U(x + g) with period
g = 3 i, we postulated a one dimensional random walk of the vacancy with lattice
step size g which is the potential period and the lattice sites are centered at
the global potential minima. Assuming this diffusion of the vacancy in these global
ii
energy barrier heights as over-damped Brownian particle in a symmetric global periodic
potential, we calculated the effective diffusion coefficient(D) of the vacancy in
these mechanisms. Our result indicates that the effective diffusion coefficient of these
mechanisms are reduced to the mean first passage time as well as the global lattice
space, D = g
2 . Since this, , is described by the local as well as global energy barrier,
lattice space and temperature, therefore, taking the background temperature at
T=1200K and values of computed local and global energy barrier heights and experimental
values of the free diffusion coefficients (D0) of those mechanisms, we evaluate
the effective diffusion coefficient of each mechanism and we have found that the value
of triple defect is 107 and 103 greater than the NNN and the six-jump mechanisms
respectively. Moreover, the temperature dependence of the effective diffusions coefficients
(D) of a vacancy in NiAl by these mechanisms was found to obey the Arrhenius
law in the temperature interval from 1200 to 1500K

## Description

## Keywords

The Three Dimensional Crystal Structure