Diffusion in one - Dimensional disordered lattice

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2011, Vol. 56, No. 7, pp. 78-86 DIFFUSION IN ONE-DIMENSIONAL DISORDERED LATTICE Trinh Van Mung(∗), Pham Khac Hung and Pham Ngoc Nguyen Hanoi University of Technology (∗)E-mail: mungtv76@gmail.com Abstract. The diffusion of particles in 1-d system with two types of dis- orders is studied by using both analytical method and Monte-Carlo simu- lation. The obtained result is used to construct the analytical expression for 1-d diffusion coefficient D and correlation factor F. The calculation of particular cases of two levels and uniform distributions of site and transition energies shows the applicability of the constructed expressions not only for site disorder system, but also for mixed system with both site and transition disorders. The blocking effect is studied and also discussed. Keywords: Diffusion, amorphous solid, disordered lattice, one-dimension, simulation. 1. Introduction Migration of particles (atom, molecular and ion) in disordered media has re- ceived wide attention because of a list of problems and applications that can be found [1-5]. To name, but only a few: the diffusion and conductivity of glass, poly- mer, amorphous alloy and thin film related to the subject. Special cases of interest are dynamic processes controlled by the configurational coordinate, the inter par- ticle distance, the transition (saddle) and site energies. Because the geometrical disorder has irrelevant effect of diffusion behaviour, the disordered lattice is em- ployed where the particle jumps between sites of regular lattice whose energetic properties are randomly distributed to clarify the properties of such systems. With respect to diffusion, amorphous alloys are characterized by two kinds of disorders: the site disorder corresponding a random trapping model, and transition (saddle) disorder corresponding random hopping model [3-4]. Unlike the case of electronic transport in semiconductors, where the specific cases fall into one or other model, the transport in amorphous solid involved both models giving rise to a new effect. A suitable approach to describe the particle diffusion in disordered lattice is the effective medium approach (EMA) [2, 5] which has been shown to be particularly reasonable in one-dimensional (1-d) case. Recently, this approach has been used to obtain diffusion coefficient for 1-d system in commonly encountered situation where the site energies are give by an exponential or Gauss distribution. 78 Diffusion in one-dimensional disordered lattice A useful tool to gain insight into the reliability of analytical models is the Monte-Carlo simulations (see [6]). This method is applied to simulate the motion of particle in the disordered systems and its result is used to compare with analytical expression [6, 7]. It is demonstrated that the analytical approximation based on EAM could reproduce temporal asymptotic behaviour of diffusion coefficient for some limits, especially in the case of site disorder [8]. Another approximation based on continue-time random-walk theory provides a reasonable explanation of nearly Arrhenius behaviour observed in most amorphous solids in term of the compensation between site and transition disorders [9]. Nevertheless, there are only few exact results available and the problem of determining of precise diffusion coefficient of disordered systems remained unsolved. In this study our concern is the combining of the Monte-Carlo simulation and new theoretical approaches to derive a simple analytical expression for diffusion coefficient in 1-d system. Several other important aspects of the problem, such as the temperature dependent properties of diffusion coefficient, correlation factor and averaged time between two consequent jumps, are also considered and discussed here. 2. Content 2.1. Monte-Carlo simulation The simulation is carried out for a chain consisting of 4000 sites with periodic boundary conditions. The transition or site energies for each site in the chain are calculated in accordance with the given distribution. We consider two types of energetic distributions: the uniform distribution in the range from ε1x to ε2x (ε1x < ε2x; ’x’ may be ’s’ or ’t’ corresponding site or transition disorder) and two-level distribution with the energy of ε1x and ε2x. The probability of particle jump from ith to i+1th neighbouring site and the averaged residence time of particle at ith site is given as: pi,i+1 = exp(−εi,i+1β) exp(−εi,i+1β) + exp(−εi,i−1β) , (2.1) τi = pi,i−1τ0 exp[(εi,i−1 − εi)β] + pi,i+1τ0 exp[(εi,i+1 − εi)β] = 2τ0 exp(−εiβ) exp(−εi,i+1β) + exp(−εi,i−1β) , (2.2) where εi and εi,i+1 are the site and transition energies; τ0 is frequency period; β = 1/kBT ; kB is Boltzmann constant and T is temperature. After construction of the lattice we treat the motion of particle from site to site other. The jump which carries the particle out of jth site represents a Poisson 79 Trinh Van Mung, Pham Khac Hung and Pham Ngoc Nguyen process with averaged residence time τj . Let tn be the moment that particle hops at step n and then occupies the site j. The moment of next hop is defined by tn+1 = tn − τj lnR, (2.3) here R is random number in the interval [0,1]. The term - τj lnR represents the actual time that particle spends at jth site between n and n+ 1 hops [10]. As such the motion of particle has been simulated. During each simulation the position of particle is stored and the mean square displacement is calculated. The results obtained here is obtained by averaging over 107 simulations which is found to be far more than that sufficient to obtain the result independent of number of simulations. The calculation was conducted for three types of lattices: Site disorder (SD) system where site energies are randomly chosen and transition energies are equal to ε1t; Transition disorder (TD) system with constant site energies and ran- domly distributed transition energies and Mixed system where site and transition energies are randomly chosen. 2.2. Analytic expression for the diffusion quantities At first we introduce some parameters necessary for further discussion. The time for realizing n jumps is denoted as tn and the mean time between two conse- quent jumps equals τjump = /n. Here is the averaging time tn over many simulations. The mean square displacement is equal to na 2 for SD system. Meanwhile, in the case of TD system due to forward and backward jumps the equals Fna 2. Here a is the distance between two nearest neighbour sites; F is the correlation factor which attains very small value at low temperature [1, 8]. The previous simulation demonstrates that for strongly disordered system and in short time regime the diffusion coefficient is time dependent. Nevertheless, in the long term regime it becomes independent with time. For convenience of discussion we use the quantities D∗ and τ ∗ which are diffusion coefficient and time between two subsequent jumps for ordered lattice with constant site energy of ε1s and transition energy of ε1t. The diffusion coefficient is given by D D∗ = F τ ∗ τjump . (2.4) Consider the case when particles make the random walk for a long time. In this case the time that particles spend at each ith site can be approximated by ti = tn exp(−εiβ) N∑ j exp(−εjβ) , (2.5) 80 Diffusion in one-dimensional disordered lattice here N is number of sites in the lattice. The number of particle visits for ith site is defined by: ni = ti τi = tn exp(−εi,i+1β) + exp(−εi,i−1β) 2τ0 N∑ j exp(−εjβ) (2.6) and the averaged time between two subsequent jumps in the limit of long term tn can be calculated according to: τjumpl = tn n = tn N∑ i ni = 2τ0 N∑ j exp(−εjβ) N∑ j exp(−εi,i+1β) + exp(−εi,i−1β) . (2.7) In the case of two level distributions eq. (2.7) becomes: τjumpl τ ∗ = αs + (1− αs) exp(−ξs) αt + (1− αt) exp(−ξt) , (2.8) where αs, αt is the concentration of site energy ε1s and transition energy ε1t, respec- tively; ξs = (ε2s− ε1s)β; ξt = (ε2t− ε1t)β. For uniform distribution the time τjump is given as: τjump1 τ ∗ = τ0 ε2s∫ ε1s exp(−εβ)dε ε2s−ε1s ε2t∫ ε1t exp(−εβ)dε ε2s−ε1s 1 τ ∗ = ξt(1− exp(−ξs)) ξs(1− exp(−ξt)) . (2.9) As the factor F of SD system is equal to 1. Hence the diffusion coefficient for SD system can be calculated based on eqs. (2.4) (2.8) (2.9). For two level distribution one gets: D D∗ = 1 αs + (1− αs) exp(−ξs) , (2.10) for uniform distribution: D D∗ = ξs 1− exp(−ξs) . (2.11) 81 Trinh Van Mung, Pham Khac Hung and Pham Ngoc Nguyen 2.3. Results and discussion We have calculated the temperature dependence of the diffusion quantities of interest. The values of parameters used for calculation are the same for all simula- tions: (ε2s− ε1s) = −1 (ε2t− ε1t) = 1; ξx is dimensionless and varies in the interval from 2 to 5. Figure 1. The dependence of time τjump on the concentration αx The Monte-Carlo result is presented in Figures 1, 2, 3 and 4. Here the diffu- sion coefficient and correlation factor are determined in terms of slope of the linear dependence of versus and n, respectively. It can be seen that the time τjump of SD system is significantly larger than one of TD system indicating the specific properties of trapping model in comparison with hoping model. How- ever, the diffusion coefficients for both systems are identical upon ξs = −ξt and identical form of energetic distribution, e.g. either uniform distribution or two level distribution with αs = αt (see Figure 3). In the case of TD system with two level distribution from eq. (2.8) we obtain the time τjump as: τjumpl τ ∗ = 1 αt + (1− αt) exp(−ξt) . (2.12) Because the coefficient D for both SD and TD systems are identical upon ξs = −ξt and αs = αt, one can get the diffusion coefficient for TD system from eq. (2.10) replacing ξs by −ξt and αs by αt. Subsequent substitution of the result of (2.10) (2.12) into eq. (2.4) yields: F = D D∗ τjumpl τ ∗ = 1 1 + αt(1− αt)[exp(ξt) + exp(−ξt)− 2] . (2.13) 82 Diffusion in one-dimensional disordered lattice Analysis of eq. (2.13) demonstrated that the factor correlation attains a min- imum at αt = 0.5 and it monotonously decreases with temperature. It is now easy to calculate the diffusion coefficient of mixed systems based on the eqs. (2.4) (2.8) (2.13) F = D D∗ τjumpl τ ∗ = 1 1 + αt(1− αt)[exp(ξt) + exp(−ξt)− 2] . (2.14) Figure 2. The dependence of time τjump on the temperature for site disorder (left) and transition disorder (right) Figure 3. The dependence of diffusion coefficient on the concentration αx and temperature for two level (right) and uniform (left) distribution 83 Trinh Van Mung, Pham Khac Hung and Pham Ngoc Nguyen By similar ways one can get the factor F and diffusion coefficient for uniform distribution as: F = ξ2t exp(ξt) + exp(−ξt)− 2 , (2.15) D D∗ = −ξtξs [1− exp(ξt)][1− exp(−ξs)] . (2.16) Figure 4 shows the correlation factor F as a function of ξt and αt, together with the result calculated by eqs. (2.13), (2.15). It is clear that the simulation and calculation data are almost identical. In order to check the validity of analyt- ical expression for diffusion coefficient we performe several simulations for mixed systems. The result is presented in Figure 5. Here we observe again the excellent agreement between the simulation data and result calculated by eqs. (2.14), (2.16) indicating the applicability of these expressions for diffusion coefficient of disordered 1-d system. Figure 4. The dependence of correlation factor on the concentration αt and temperature for two level (right) and uniform (left) distribution In accordance to analytical expressions (2.14), (2.16), the diffusion behavior does not follow the Arrhenius law. However, in the regime of low temperature, e.g. ξx is enough large, the diffusion coefficient is determined mainly by the exponential terms in eqs. (2.14) or (2.16), hence the diffusion almost shows the Arrhenius behaviour that can be seen in Figure 5 where the ξx is large and changes in the interval from 2.0 to 5.0. As a result the disorder gives rise to new term (ε2s− ε1s) = (ε2t − ε1t) for activation energy. 84 Diffusion in one-dimensional disordered lattice Figure 5. The dependence of diffusion coefficient for the mixed systems with two level (right) and uniform (left) distributions; the data for two level distribution presentedin left is given from system with αt = 0.3 and αs = 0.6 In the case of high concentration of diffusive particles the blocking effect be- comes essential. It gives rise to the fact that the particles prevent the movement of each other and consequently leads to a decrease in diffusion coefficient. Figure 6 shows the ratio Dθ/D1 as a function of coverage θ = Nparticle/N . Here = Nparticle is the numbers of particles; Dθ, D1 is diffusion coefficient of the systems with Nparticle and one particle, respectively. In the case of SD system the characteristic of particle motion is almost independent of temperature. Hence the ratio Dθ, D1 depends only on the coverage θ. In the case of TD system in converse the particle motion strongly depends on the temperature. It results in the dependence of Dθ/D1 on both the coverage and temperature. Note that the ratio Dθ/D1 (the strength of blocking effect) correlated with factor F (see Figure 6). Figure 6. The dependence of diffusion coefficient on the coverage 85 Trinh Van Mung, Pham Khac Hung and Pham Ngoc Nguyen 3. Conclusion In the present work we studied the diffusion in 1-d systems with two types of disorders by using the analytical method and Monte-Carlo simulation. The simu- lation is conducted for particular cases of two level and uniform distributions. The combination of the two methods enables us to propose a technique for construct- ing the expressions for diffusion coefficient and correlation factor for 1-d systems. The data obtained by simulation and calculated by the analytical expressions are identical for several example systems indicating the applicability of the analyti- cal expression not only for site disorder system, but also for mixed systems which have both site and transition disorders. Study of blocking effect shows that the ratio Dθ/D1 (strength of blocking effect) depends only on the coverage for SD sys- tem, meanwhile both temperature and coverage affect it. Furthermore, there is the monotonous increasing of the ratio Dθ/D1 with factor F . REFERENCES [1] Y. Limoge, Acta Metall. Mater., 1990. 38, 1733. [2] Y. Limoge, 1990. J. Non-Cryst. Solids 117/118, 608. [3] N.P.Lazarev and A.S.Bakai, 2002. Phys Rev Lett. 88, 4. [4] Argyrakis et al., 2008. Solid State Ionics, 179, 143. [5] G.Terranova, H.O.Mrtin and C.M.Aldao1, 2005. Phys. Rev. E 72, 061108. [6] R.Kirchheim and U.Stolz, 1987. Acra metall, 35, 281. [7] A.V.Nenashev et al., 2010. Phys. Rev. B 81, 115204. [8] J.W.Haus, 1982. Phys. Rev. B 25, 4. [9] W.D. Roos et al., 1990. Appl. Surf. Sci. 40, 303. [10] P.K. Hung, H.V. Hue, L.T. Vinh, 2006. J. Non-Cryst. Solids 352, 3332. 86