Simulation the diffusion process in disordered system

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. REFERENCES [1] N. Eliaz, D. Fuks, D. Eliezer, Acta Mater. 47 (10) (1999) 2981. [2] J.S. Langer, S. Mukhopadhyay, Phys. Rev. E 77 (2008) 061505. [3] C. Mller, E. Zienicke, S. Adams, J. Habasaki, P. Maass, Phys. Rev. B 75 (2007) 014203. [4] V.K. de Souza, D.J. Wales, Phys. Rev. B 74 (2006) 134202. [5] D.K. Belashchenco, P.K. Hung, Izv. Vysh. Chernaya Metallurgiya 11 (1986) 89. [6] D.K. Belashchenco, P.K. Hung, Izv. Akad. Nauk. SSSR Metally 2 (1986) 57. [7] Y. Limoge, J.L. Bocquet, Phys. Rev. Lett. 65 (1990) 60. [8] R. Kirchheim, F. Sommer, G. Schluckebier, Acta Metall. 30 (1982) 1059. 43