Studying diffusion mechanism and dynamics slowdown in iron liquid

Nguyễn Thị Thanh Hà và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 135(05): 167 - 172 167 STUDYING DIFFUSION MECHANISM AND DYNAMICS SLOWDOWN IN IRON LIQUID Nguyen Thi Thanh Ha * , Le Van Vinh, Pham Khac Hung Hanoi University of Science and Technology SUMMARY The dynamic properties of iron liquid (Fe) are studied by molecular dynamics (MD) simulation. We trace the evolution of local density fluctuations (LDFs) in Fe liquid over the simulation time and in the 300-2300 K temperature range. The result simulation reveals that atomic diffusion is realized through the LDFs and the high localization LDFs at low temperature in the iron liquid is the cause of the anomalous dynamics slowdown. We find that the diffusion depends on both rate of LDFs and the averaged square displacement of particles Fe as one LDF occurs. As the temperature decreases, both quantities reduce. Keywords: Molecular dynamics simulation, iron liquid, dynamics slowdown, diffusion, local density fluctuations. INTRODUCTION * This transition to a disordered solid known as the glass transition is accompanied with the drastic increase in the viscosity and a subtle change in the structure. Understanding the microscopic mechanism governing glass transitions is one of the most important problems in statistical physics [1-3]. To tackle this problem, several working hypotheses have been proposed. The studies from refs.[4- 8] focus on the dynamics heterogeneity, the percolation in real space and properties of energy landscapes. They found the existence of mobile and immobile regions which migrate in the space over time. Authors in [9- 10] put forward the mechanism by which the small modification of statistic density correlations can produce an extremely large dynamical change. The essential result in this direction is the mode coupling theory [9] that predicts a freezing of dynamics from the non- linear feedback effect. The theoretical and experimental investigations on universal mechanisms controlling slow dynamics have been done for long time, however as mention in [11] many open questions are still remained. Iron is an important element and has many industrial applications. Therefore, knowledge * Tel: 0983 012387, Email: ha.nguyenthithanh1@hust.edu.vn about their microstructure and dynamical properties would be essential to understand this material [12-14]. In this paper, MD simulation is conducted to examine the dynamics in iron liquid. Our purpose is to clarify the diffusion mechanism and the cause of slowdown in the iron liquid near glass temperature. CALCULATION PROCEDURE MD simulation is conducted for 10 4 atom models with periodic boundary conditions using Pak–Doyama potential [15]. To integrate the equation of motion Verlet algorithm is used with MD step of 0.67 fs. Initial configuration is obtained by randomly placing all atoms in a simulation box. Then this sample is equilibrated at temperature of 6000 K and cooled down to desired temperature. Next, a long relaxation has been done in ensemble NPT (constant temperature and pressure) by 10 5 MD steps to obtain the equilibrium sample. We prepare six models (M1, M2... M6) have been constructed at ambient pressure and at temperature of 300 K, 800 K, 1200 K, 1500 K, 1800K and 2300 K. To study dynamical properties the obtained samples are relaxed in ensemble NVE (constant volume and energy) over 5x10 6 steps. Nitro PDF Software 100 Portable Document Lane Wonderland Nguyễn Thị Thanh Hà và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 135(05): 167 - 172 168 Obviously, the diffusivity in system is impossible if no exchanging the coordinated Fe occurs. Therefore, we trace the evolution of local density fluctuations (LDF) in Fe liquid over the simulation time. To calculate the coordination number we use the cutoff distance RO=3.35 Å chosen as a minimum after first peak of PRDF. The local density around i th particle can be quantified as: Oi i O n V   (1) where VO= 4RO 3 /3; nOi is the number of particles in a coordination sphere of i th particle; RO is the radius of the coordination sphere. If the number nOi changes, then the local density around i th particle varies. It means that the change of nOi at some moments represents the local density fluctuation (LDF) act. The existence of non-mobile and mobile regions is originated from the density fluctuation in the liquid. RESULTS AND DISCUSSION To test the validity of MD model one usually determines the pair radial distribution functions (PRDF). They are very close to simulation result reported in ref. [14, 16] and in good agreement with experimental data. 0 2 4 6 8 10 12 0 1 2 3 4 Simulation Experiment [14] B g (r ) r, Å A 0 1 2 3 Simulation Experiment [16] Fig 1. The pair radial distribution functions for amorphous solid iron at 300K (A) and liquid iron at 1500K (B) Fig.2 The schematic illustration of local density fluctuations for selected particle Nitro PDF Software 100 Portable Document Lane Wonderland Nguyễn Thị Thanh Hà và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 135(05): 167 - 172 169 The schematic illustration of LDF for selected particle is presented in Fig.2. One can see that LDFs happen four times for a selected particle. In MD simulation the diffusion coefficient is usually determined via Einstein equation: 2 2( ) ( ) lim lim 6 6 .t n MD R t R t D t n t        (2) Where <R(t) 2 > is mean square displacement (MDS) over time t, n is step, tMD =0.67fs. If we define: MLDF is a number of LDFs happening with i th particle during n steps,  is a rate of LDF and  is the averaged square displacement of particles Fe as one LDF occurs. lim LDF n M n    (3) 2( ) lim LDFM LDF R t M      (4) The equation (3) can be reduced to 2( ) 1 lim . . . . 6 . 6.n MD MD R t D A n t t           (5) The dependence of MLDF vs. n and <R(t) 2 > vs. MLDF is shown in Fig.3 and 4, respectively. Well straight lines are seen and the quantities determined from these lines are presented in Table 1. We see that both  and  monotonously increase in the temperature interval of 300-2300 K. Table 1. Dynamical characteristics of simulated liquids: here D, D* is the diffusion coefficient callculated by (5) and Einstein equation, respectively Model M1 M2 M3 M4 M5 M6 Temprature 300 800 1200 1500 1800 2300 υ 0.0002 0.0006 0.0012 0.0018 0.0022 0.0027 δ ( Å2/ one LDF) 0.0001 0.0004 0.0149 0.0538 0.0934 0.1820 D×10 5 (cm 2 /s) 0 0.0038 0.2597 1.4054 3.0082 7.2125 D * ×10 5 (cm 2 /s) 0 0.0058 0.2608 1.4224 3.0103 7.2025 Fig 3. The dependence of MLDF as a function of MD steps n 0 50000 100000 150000 200000 0 50 100 150 200 300K 600K 1200K 0 100 200 300 400 500 < M L D F > MD steps, n 1500K 1800K 2300K Nitro PDF Software 100 Portable Document Lane Wonderland Nguyễn Thị Thanh Hà và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 135(05): 167 - 172 170 Fig.5 shows the temperature dependence of dynamical quantities for simulated liquids. As the temperature decreases from 2300 to 1200 K, δ decreases by 12.2 times that significantly larger than the change in υ equal to 2.2. It means that the major contribution to diffusion belongs to the averaged square displacement of particles Fe as one LDF occurs (δ). Fig.5. The temperature dependence of the quantities υ and  500 1000 1500 2000 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 Temperature, K 0.00 0.05 0.10 0.15 0.20 ra te o f L D F T h e a v e ra g e d M S D /o n e L D F 0 30 60 90 120 0.00 0.05 0.10 0.15 0.20 0.25 0.30 M LDF T h e m e a n s q u a re d is p la c e m e n t o f p a rt ic le s , Å 2 M LDF 300K 800K 0 100 200 300 400 500 0 20 40 60 80 100 1200K 1500K 1800K 2300K Fig.4. The dependence of <R(t) 2 > as a function of Nitro PDF Software 100 Portable Document Lane Wonderland Nguyễn Thị Thanh Hà và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 135(05): 167 - 172 171 LDFs happen rarely in the immobile regions and occur frequently in the mobile ones. Hence, the examining of the spatial distribution of LDFs happened in the liquid should give new insight into the mechanism governing slow dynamics. We now measure the distribution of MLDF through particles for samples at temperature of 1200K and 2300 K in order to identify the cause of slowdown in the iron liquid near glass temperature. For each run the number of steps n is adopted so that the total number of LDFs, Fig.6 shows the distribution of MLDF through particles for considered samples. The curves have a Gauss form but distribution of MLDF for low- temperature sample is spread in much wider range than for high-temperature sample. There is a pronounced peak which location is almost unchanged with temperature. Its height for low-temperature sample is lower than for high-temperature one. In our simulation the non-mobile regions are the places where LDFs happen rarely or not occur. Further, as the temperature approached to the glass transition point, the density reduces and the non-mobile regions expand. As a result, they percolated over whole system. Therefore, the anomalous dynamics slowdown near the glass transition temperature can be explained by the high localization LDFs in the iron liquid. CONCLUSION The diffusion mechanism in iron liquids is studied by mean of molecular dynamic simulation and the activated LDFs. We establish an expression for diffusion coefficient via the rate LDFs. We find that  - the averaged square displacement of particles Fe as one LDF occurs and  - rate of LDF monotonously decreases with temperature. But  rapidly decreases to zero and mainly contributes to the slow dynamics. The result shows that the localization LDFs near the glass transition point is the reason of the anomalously slow dynamics in iron liquid. REFERENCES 1. A. Heuer (2008), J. Phys.: Condens. Matter 20, 373101. 2. H. Tanaka, T. Kawasaki, H. Shintani, K. Watanabe (2010), Nat.Mater. 9, 324. 3. L. Berthier, G. Biroli (2011), Rev. Mod. Phys. 83, 587. 4. J. S. Langer and S. Mukhopadhyay (2008), Phys. Rev. E 77, 061505. 5. G. Lois, J. Blawzdziewicz, and C. S. O'Hern (2009), Phys. Rev. Lett. 102, 015702. 6. F. Sausset, G. Tarjus (2010), Phys. Rev. Lett. 104, 065701. 7. A. Cavagna, T.S. Grigera, P. Verrocchio (2007), Phys. Rev. Lett. 98, 187801. 8. D. Rodney and T. Schrøder (2011), Eur. Phys. J. E 34: 100. 90 120 180 240 300 360 0.0 0.1 0.2 0.3 0.4 F ra c ti o n o f ir o n p a rt ic le s The number of LDFs 2300 K 1200 K Fig.6. The distribution of LDF in iron liquid Nitro PDF Software 100 Portable Document Lane Wonderland Nguyễn Thị Thanh Hà và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 135(05): 167 - 172 172 9. W. G¨otze, Complex Dynamics of Glass- Forming Liquids: A Mode-Coupling Theory (Oxford University Press, Oxford, 2008). 10. L Berthier (2007), Phys. Rev. E 76, 011507. 11. G. Tarjus, in Dynamical Heterogeneities in Glasses, Colloids, and Granular Media, edited by L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, W. van Saarloos (Oxford University Press, Oxford, 2011). 12. Y. Limoge (1997), Materials Science and Engineering A226-228, 228. 13. A.V. Evteev et. al. (2006), Phys. Solid State 48, 815. 14. Vo Van Hoang (2009), Nguyen Hung Cuong, Physica B 404,340. 15. H.M. Pak, M. Doyama (1969), J. Fac. Eng. Univ. Tokyo B 30, 111. 16. X.Y. Fu, M.L. Falk, D.A. Rigney (2001), Wear 250, 420. TÓM TẮT NGHIÊN CỨU CƠ CHẾ KHUẾCH TÁN VÀ ĐỘNG HỌC CHẬM TRONG SẮT LỎNG Nguyễn Thị Thanh Hà*, Lê Văn Vinh, Phạm Khắc Hùng Đại học Bách khoa Hà Nội Các tính chất động học của sắt lỏng (Fe) được nghiên cứu bằng phương pháp mô phỏng động lực học phân tử. Chúng tôi theo dõi sự tiến hóa của sự thăng giáng mật độ địa phương trong Fe lỏng theo thời gian mô phỏng và trong khoảng nhiệt độ 300-2300 K. Kết quả mô phỏng chỉ ra rằng sự khuếch tán nguyên tử được thực hiện thông qua sự thăng giáng mật độ và sự thăng giáng mật độ địa phương định xứ cao ở nhiệt độ thấp trong Fe lỏng là nguyên nhân của động học chậm dị thường. Chúng tôi phát hiện ra rằng sự khuếch tán phụ thuộc vào cả tốc độ thăng giáng mật độ địa phương và khoảng cách dịch chuyển bình phương trung bình của các hạt Fe khi một thăng giáng mật độ địa phương xảy ra. Khi nhiệt độ giảm, cả hai đại lượng này đều giảm. Từ khóa: Mô phỏng động lực học phân tử, sắt lỏng, động học chậm, sự khuếch tán, thăng giáng mật độ địa phương Ngày nhận bài:25/7/2014; Ngày phản biện:30/8/2014; Ngày duyệt đăng: 31/5/2015 Phản biện khoa học: TS. Phạm Hữu Kiên – Trường Đại học Sư phạm - ĐHTN * Tel: 0983 012387, Email: ha.nguyenthithanh1@hust.edu.vn Nitro PDF Software 100 Portable Document Lane Wonderland