Simulation study of tracer self - Diffusion mechanism in cobalt amorphous solids

JOURNAL OF SCIENCE OF HNUE Natural Sci., 2010, Vol. 55, No. 6, pp. 28-36 SIMULATION STUDY OF TRACER SELF-DIFFUSION MECHANISM IN COBALT AMORPHOUS SOLIDS Pham Huu Kien(∗) and Pham Khac Hung Hanoi University of Science and Technology Vu Van Hung Hanoi National University of Education (∗)E-mail: huukienpham@yahoo.com Abstract.The statistical relaxation (SR) simulation has been conducted to study the behavior of simplexes in cobalt amorphous solid containing 2×105 atoms. The simulation reveals that the fraction of 4-simplex increases and of N-simplex with N > 4 decreases upon relaxation degree. The simulation finds a large number of vacancy-simplexes, which plays a role of diffusion vehicle for cobalt atom and varies with relaxation degree. A new diffusion mechanism for tracer diffusivity in cobalt amorphous solid is supposed as follows: The elemental atomic movement includes a jump of neighbouring atoms into the vacancy-simplexes and then collective displacement of a large number of atoms. Keywords: Amorphous solid, Microscopic bubble, Vacancy-simplex, Diffu- sion mechanism, Statistical relaxation, Molecular dynamic. 1. Introduction Metal has been known as an indispensable material in the lives of the whole of mankind. So, it has attracted the attention of many researchers in all fields: ex- periments, theory and computer simulation [1-10]. In all of the nature of the metal, atom diffusion mechanism in amorphous metals (AM) is the topical area of high nature and should be particularly interesting to research. Many works have studied the atomic diffusion mechanisms in amorphous solids [1, 8, 9]. For example, the ex- perimental researches [4-7], said that in amorphous solids there exists quasi-vacancy and diffusion mechanism of the atom is indirect through quasi-vacancy. However, the definition of quasi-vacancy is still not well understood and is not a question to be answered. On the other hand, computer simulation has proved the existence of the vacancy in amorphous solids [1, 2, 8], the number of vacancies level changes upon relaxation degree. More comprehensive research of computer simulation is to detect the existence of voids and simulation results have found a continuous spectrum of these voids in amorphous solid. Size of the voids found smaller radius of the atom, 28 Simulation study of tracer self-diffusion mechanism in cobalt amorphous solids but the radius of the atoms is determined also containing a uniformity. Recently, Vo Van Hoang and colleagues have discovered in models of liquid and amorphous Fe, the temperature dependence of the diffusion coefficient showing two properties: from not obeying the Arrhenius law at high temperature regions to obeying the Arrhenius law in a low temperature region. However, they have not yet determined exactly the diffusion coefficient of Fe atoms in these temperature regions [2]. Therefore, atomic diffusion mechanisms in amorphous solids are still a question to be addressed. From the evidence of existence of the vacancy-simplexes in Co amorphous solids, we have discovered the role of the vacancy-simplexes for atomic diffusion in amorphous met- als. Therefore, in this paper, three models of amorphous Co have been built to study the existence of the vacancy-simplexes and their role for the atomic diffusion mechanism. The dependence on the relaxation degree of the vacancy-simplexes and diffusion coefficient of Co atoms are also addressed and discussed in this article. 2. Computation procedure The simulation has been conducted for the model consisting of 2× 105 cobalt atoms in a cube box (size 132.7 × 132.7 × 132.7 A˚) with periodic boundary condi- tions. We use the Pak - Doyama potential, it is given as follows: U(r) = −0.12812(r − 1.82709)4 + 1.15421(r− 2.50849)2 − 0.13448, (2.1) for 0 ≤ r ≤ 3.44 A˚, which has a cut-off between the second and third nearest- neighbour distance of cobalt. From this pair interatomic potential was initialty obtained from the data in [1]; here r is the interatomic distance in A˚ and U(r) is in eV. The density is set to be the value of real amorphous cobalt (8.666g/cm3). Initial configuration is generated by randomly placing all atoms in the simulation box. Then the sample is relaxed over several thousand steps until the system reaches the equilibrium, using the statistic relaxation (SR) technique. The SR movement length is equal to 0.002 A˚. The equilibrium model obtained corresponds only to one among the possible metastable states and a more stable state can be prepared by the shaking procedure. This procedure includes relaxing the system over 50-200 SR steps with movement length of 0.4 A˚ and further treatment of the system towards equilibrium using movement length of 0.002 A˚. To obtain a good relaxed model, the shaking procedure must be performed many times. We consider four nearest neighbour atoms forming a tetrahedron. If we draw a circum-sphere (CSP) of this tetrahedron, the CSP could contain some atoms inside. The simplex denotes a tetrahedron of four atoms of which the CSP has no atoms inside. There are also one or more atoms nearby the surface of CSP. Let RCSP , Nsa be the radius of CSP and the number of atoms which locate from CSP centre at a distance of RCSP ± 0.1 A˚. In convenience, hereafter, we call the simplex having Nsa atoms nearby the CSP surface, the Nsa-simplex. Figure 1 illustrates 4- , 5- and 6-simplex. As shown from 29 Pham Huu Kien, Pham Khac Hung and Vu Van Hung Figure 1. The schematic illustration of simplexes: a) four-simplex; b) six-simplex; c) five-simplex and neighbouring atoms; d) the new five-simplex and neighbouring atoms after RDA. The dash circle presents the CSP of four atoms and the arrow is a path along which the MA jumps into CSP Figure 1, the additional atoms located nearby the CSP surface prevent the efficient parking of four atoms forming the tetrahedron. Therefore, the more the number Nsa is, the bigger the CSP becomes. Importantly, the big simplex can accumulate an atom inside its CSP. If one among Nsa atoms jumps towards the CSP centre, then as a result, the simplex disappears and it involves the collective movement of many atoms. Such atomic motion is unlike the movement of vacancy and atom on opposite directions in crystal lattice, but like the collapse of the ”microscopic bubble” in amorphous matrix (see Figure 1c and d). 3. Results and discussions Figure 2. The radial distribution function of amorphous models Co Figure 2 shows that the radial distribution function (RDF) g(r) of amorphous 30 Simulation study of tracer self-diffusion mechanism in cobalt amorphous solids Co models. As shown in Figure 2, although the energy per atom of considered models changes from -0.9336 to -0.9534 eV, the RDFs g(r) for all considered models is identical indicating the non-sensitivity of the function g(r) to the change in local microstructure of amorphous metal. To test the validity of the constructed models, we have compared our obtained RDF with experimental data. As shown from Figure 3, the structural characteristics of our models agree well with experiment data [5, 6]. In addition, the function g(r) has a splitting second peak which is often thought to be related to the icosahedrons in system [2-7]. Figure 3. The radial distribution function of amorphous Co model obtained compared with the experimental in [5] Table 1. The fraction of simplexes in the models of amorphous cobalt Model The mean potential Fraction of simplexes energy per atom (eV) Total Nsa 4 5 6 ≥ 7 1 -0.9336 1298224 0.9608 0.0374 0.0017 0.0001 2 -0.9462 1288298 0.9626 0.0358 0.0017 0.0000 3 -0.9534 1281292 0.9645 0.0339 0.0016 0.0000 Table 1 presents the numbers of simplexes and fractions of simplexes found in the obtained models. One can see that the number of total Nsa decreases from 1298224 to 1281292 upon decreasing of the potential energy. In addition, the fraction of 4-simplexes increases as the models potential energy decreases. i.e. the more stable state is, the bigger the number of 4-simplex is. For other kinds of simplexes we observe the opposite trend. The formation of AM generally follows the ”deficient parking rule”. It means that due to fast quenching from liquid the parking of atoms 31 Pham Huu Kien, Pham Khac Hung and Vu Van Hung in AM is not efficient and there is always an amount of structural defects like large void (free volume unit). The relaxation is accompanied by annihilation of those defects. As mentioned above, the bigger number Nsa is, the bigger the size of simplex and the void inside is. Therefore, the monotony decrease in the number of N -simplex with N > 4 from less relaxed (model 1) to well relaxed state (model 3) shown in Table 2 evidences the annihilation of structural defects in amorphous matrix and the ”deficient parking rule”. This result also can be seen in the radius distribution of simplexes in Table 3, here radius of simplexes is equal to RCSP . From Table 3 one can see clearly the significant decrease in the number of large simplexes (RCSP > A˚) in the model 1 as compared to the model 3. In summary we can make the following conclusion: The relaxation of AM is accompanied by partial annihilation of N -simplex with N > 4 and increases the number of 4-simplex. We Figure 4. The typical PEP of neighbour atom moving into CSP of the simplexes now turn to discuss the role of large simplex for diffusivity in AM. For each simplex we examine the potential energy profile (PEP) for neighbor atom moving into CSP of the simplex. The PEP is determined as follows: Each among Nsa atoms of the simplex is taken and then we move this atom step by step towards the center of CSP. The step length is set to be 0.05 A˚. At each step the potential energy of the moving atom (MA) is recorded. Finally, we obtain the PEP for MA. Figure 4 displays the 32 Simulation study of tracer self-diffusion mechanism in cobalt amorphous solids typical PEPs for several simplexes detected in our models. The curve g shows the monotonous increase in the MA energy indicating that the MA could not jump into the CSP due to very high potential barrier. Other PEPs in Figure 4 (a, b, c, d, e and f), in converse, have a pronounced maximum. Obviously, the barrier height is determined by the maximum and initial points in PEP. They like the PEP for tracer atom jumping in vacancy in crystalline lattice. Therefore, the MA can jump into CSP and the simplex play a role of the diffusion vehicle assisting tracer atom to move in amorphous matrix. The simplex having PEP like curves a, b ,c, d, e and f, and the corresponding MA are called the vacancy-simplex and the diffusing atom (DA), respectively. The number of vacancy-simplexes is found in Table 4. As show in Table 4, the number of vacancy-simplexes decreases upon relaxation degree. The dependence of numbers of DA versus their corresponding barrier height is shown in Figure 5. Most frequent barrier height lies in the interval of 0.6-1.8 eV that is observed in practice for cobalt diffusivity in certain AM. The diffusion coefficient in general is proportional to the concentration of vacancy-simplexes, Nvasimp, it is given as follows: D = 1 6 fν exp( sm k )Nvasimpexp(−Em kT ), (3.1) where d is the jump length; Sm, Em is the effective migration entropy and energy for diffusion in disordered media, respectively; k is the Boltzmann’s constant; ν is the attempt frequency; f is the correlation factor. Note that the parameter represents the mean square displacement of different jump lengths. Because the SR model is in fact the MD model in the limit of zero temperature, hence the number of simplexes found in the models is independent of temperature and it varies only with the relaxation degree. As a result, the activation energy is the migration energy, Em and the pre-exponential factor D0 is given as: D0 = 1 6 fν exp( sm k )Nvasimp. (3.2) Furthermore, the decrease in diffusion coefficient under relaxation is ascribed by the annihilation of the vacancy-simplexes. Therefore, the decrease in iron diffu- sivity is found to be 1679/62 ≈ 27 times for model 1 and 3. Several experimental observations are interpreted as a result of direct diffusion mechanism occurred in relaxed amorphous sample, because for indirect diffusion mechanism the change DR should be retarded by the times requited for establishing a new equilibrium concentration of diffusion vehicles. Our simulation shows possible diffusivity via vacancy-simplexes and it is consistent with experiments because the concentration of vacancy-simplexes is independent of temperature and it varies only with relax- ation degree. Moreover, the activation energy for diffusion via vacancy-simplex is equal to the migration energy, Em. 33 Pham Huu Kien, Pham Khac Hung and Vu Van Hung Table 2. The radii distribution of simplexes R 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 ≥ 2.3 1 2525 454295 489092 180162 142758 18606 7742 2210 528 306 2 1817 458658 501510 178301 133753 10686 2885 550 73 65 3 1536 463522 506531 174185 127165 6741 1465 125 9 13 First row presents the radii RCSP in A˚, next rows indicate the numbers of simplexes with radius RCSP in the interval RCSP ± 0.5 A˚ Figure 5. The distribution of barrier height of diffusing atom Figure 6. The distribution of averaged square displacement 34 Simulation study of tracer self-diffusion mechanism in cobalt amorphous solids To estimate the parameter in (3.1) we replace DA into CSP and then relax the system until a new equilibrium is attained. This procedure for convenience is called the RDA. The distribution of mean square displacements for 100 RDAs is displayed in Figure 6. For the model 3 the parameter of most RDAs is less than 10 A˚2 (92 atoms). Because the jump length of DA lies in the interval of 2.0-2.5 A˚, whereby the contribution of DA to will be essential and the RDA locates only in the small region nearby the vacancy-simplex. In the case of model 1 we observe a very large value (> 10 A˚2) which represents a collective movement of large numbers of atoms. Obviously, the atomic movement is spread over a large volume inside the amorphous matrix as shown in Figure 1d. Therefore, one may see two distinct diffusion mechanisms occurred in the as quenched (model 1) and well relaxed metastable state (model 3). The first one likes the hoping mechanism via vacancy-simplexes. The second one relates to the collective mechanism involved a large number of atoms in each elemental diffusion movement. Assuming ν = 1012s−1; f = exp(sm/k) ≈ 1; d2 ≈ 10 → 100 A˚2; and Nvasimp ≈ 62/(2× 105)→ 1679/(2× 105) = 3.1× 10−4 → 8.3× 10−2, according to the equation (3), the pre-exponential factor will be 3.1× 10−8 → 8.3× 10−7m2s−1. This calculation result is consistent with experimental data for amorphous models Co89Zr11(D0 ≈ 8 × 10−7m2s−1) [9], the 58Co self-diffusion in relaxed amorphous Co79Nb14B7(D0 ≈ 3.3× 10−6m2s−1) [10]. Table 3. The number of Nvasimp vacancy-simplex Models Model 1 Model 2 Model 3 Nvasimp vacancy-simplex 1679 268 62 4. Conclusion The microstructure for the cobalt model containing 2× 105 atoms is in good agreement with experimental data. The simulation result reveals that a large num- ber of vacancy-simplexes which varies with relaxation degree in amorphous cobalt models. The relaxation has been accompanied with increased fraction of 4-simplex and decreased fraction of N -simplex concentration with N > 4. A new diffusion mechanism is proposed for the tracer diffusivity in amorphous solids of which the elemental atomic movement includes a jump of neighboring atom into the vacancy- simplex and then collective displacement of a number of atoms. The activation energy and pre-exponential factor are found in reasonable agreement with experi- ments. It may be seen two distinct diffusion mechanisms occurred in the as-quenched and well relaxed samples of AM. With this model the decrease in self-diffusion coef- ficient upon relaxation is ascribed to the partial annihilation of vacancy-simplexes. The pre-exponential factor D0 of Co atoms is determined via the vacancy-simplexes. The calculation result shows that the pre-exponential factor D0 decreases on the re- 35 Pham Huu Kien, Pham Khac Hung and Vu Van Hung laxation degree, this result agrees well with the experimental data. Acknowledgment The authors are grateful for support by the NAFOSTED of Ministry of Science and Technology Vietnam (Grant No 103.01.32.09). REFERENCES [1] P.K. Hung, H.V. Hue, L.T. Vinh, J. Non-Cryst. Solids, 352 (2006) 3332. [2] Vo Van Hoang, Nguyen Hung Cuong, Physica B, 404 (2009) 340. [3] T. Ichikawa et al., Phys. Status Solidi, A 19 (1973) 707. [4] Y.Waseda, H.S.Chen, Solid Stat.Comm., 27 (1978) 809. [5] S. Dalgyc et al., Journal of Optoelectronics And Advanced Materials, 2 (2001) 537. [6] Y. Waseda, S. Tamaki, Philos. Mag., 32 (1975) 273. [7] P.K. Leung, J.G. Wright, Philos. Mag., B 30 (1974) 185. [8] P.K. Hung, P.H. Kien and L.T. Vinh, J. phys.: Condens. Matter, 22 (2010) 035401. [9] V. Naundorf et al., Journal of Non-Crystalline Solids, 224 (1998) 122. [10] J. Pavlovsky and J. Cermak, Scripta Metallurgica et Materialia, 10 (1994) 1317. 36