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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Addis Ababa University


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



The Three Dimensional Crystal Structure