Determination Of The Mean First Passage Time Of A Vacancy In Nial Binary Alloy By Analytical And Adibs Simulation Algorithm And Finding Its Diffusion Coefficient

The mean first passage time (MFPT) and effective diffusion coefficients of a vacancyrndiffusing in NiAl compound via three main vacancy diffusion mechanisms are studiedrnboth numerically and analytically. These mechanisms are: the next-nearest-neighborrn(NNN), the six-jump, and the triple defect mechanisms. The vacancy diffusion in thernthree dimensional crystal structure of NiAl is mapped onto a one-dimensional latticernsites, allowing the vacancy to hope to its nearest-neighbor with local transitions rates,rnk(xi ± | xi) = ±D0rn 2 e Urn0rn(xi) rn2kBT , that is calculated from its potential profile U(x). Therntime evolution of the vacancy in this one dimensional lattice site is governed by arnmaster equation in which it is evaluated numerically using Algorithms for Brownianrnfirst passage time estimation of Artur B. Adib where the lifetime of the vacancy inrnany state xi is drawn from an exponentially distributed random number i with meanrnequal to the reciprocal of the sum of the outgoing rates, < i >−1=rnPrnx0=n.n k(x0 | xi)rnand the nearest neighboring site where the vacancy is going next is chosen with arnprobability w(x0) = k(xrn0rnP |x)rnxrn0rn=n.nrnk(x0 |xi)rn. The sum of the times < i > until the vacancyrntouches the absorbing sites is the first passage time of that mechanism. For eachrnmechanism, we calculate the mean first passage time of the vacancy and our resultrnpredict that the triple defect mechanism takes less time compared with the othersrnand it is the major contributor to vacancy diffusion in NiAl binary alloy.rnirnIn the above one dimensional lattice sites, instead of the local transition rates, wernallowed the vacancy to hope from any site to its nearest neighbor with local jumprnprobability, pi or qi. The mean first passage times (MFPTs) for a vacancy thatrndiffuses via the three mechanisms are evaluated using the properties of random walksrnon networks technique. These analytical result show that the MFPT of the vacancyrnin those mechanisms can be expressed in terms of the local jump probabilities (pirnand qi) which in turn they are given by local MFPTs. Finally from these localrnMFPTs, we came up the vacancy’s MFPTs to complete its mechanisms and theserntimes are a functions of vacancy’s local (Ei) as well global barrier heights (Eg), thernbackground thermal energy (T), and the lattice space ( ) of these mechanisms, i 'rn4rnD0rn rn rn (E2+E3)rn 2rnexp[ Eg]. Fixing the background temperature at T = 1200K andrnusing computed local and global energies and other related parameters, we evaluaternanalytically MFPT of these mechanisms. This result also favored the triple defectrnmechanism as the main diffusion path of a vacancy in NiAl compounds, moreover,rnthe analytical values of six-jump and triple defect mechanisms are nearly identicalrnwith the one computed by Adib’s method.rnThe three local energy barrier heights of the six-jump and triple defect mechanismsrnwhere the vacancy crosses along its diffusion paths are summed which we callrnthem global energy barrier heights (U(x) = E1 − E2 + E3). From this new potentialrnstructure which is globally varying periodic potential U(x) = U(x + g) with periodrn g = 3 i, we postulated a one dimensional random walk of the vacancy with latticernstep size g which is the potential period and the lattice sites are centered atrnthe global potential minima. Assuming this diffusion of the vacancy in these globalrniirnenergy barrier heights as over-damped Brownian particle in a symmetric global periodicrnpotential, we calculated the effective diffusion coefficient(D) of the vacancy inrnthese mechanisms. Our result indicates that the effective diffusion coefficient of thesernmechanisms are reduced to the mean first passage time as well as the global latticernspace, D = grn2 . Since this, , is described by the local as well as global energy barrier,rnlattice space and temperature, therefore, taking the background temperature atrnT=1200K and values of computed local and global energy barrier heights and experimentalrnvalues of the free diffusion coefficients (D0) of those mechanisms, we evaluaternthe effective diffusion coefficient of each mechanism and we have found that the valuernof triple defect is 107 and 103 greater than the NNN and the six-jump mechanismsrnrespectively. Moreover, the temperature dependence of the effective diffusions coefficientsrn(D) of a vacancy in NiAl by these mechanisms was found to obey the Arrheniusrnlaw in the temperature interval from 1200 to 1500K

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