Xem mẫu
- JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 112-118 This paper is available online at http://stdb.hnue.edu.vn LOCAL DENSITY FLUCTUATIONS IN SIMULATED LIQUID IRON Nguyen Thi Thao1 , Pham Khac Hung2 and Le Van Vinh2 1 Faculty of Physics, Hanoi National University of Education 2 Faculty of Computational Physics, Hanoi University of Science & Technology Abstract. We present a result of a molecular dynamics (MD) simulation of liquid iron. A series of MD models containing 104 particles under periodic boundary conditions at temperatures from 290 to 2300 K have been constructed. We focus on local density fluctuations (LDFs) which enable the diffusion of iron particles. We found that LDFs operate as a diffusion vehicle for Fe particles. The diffusion coefficient is found to be a product of the rate of LDF and mean square displacement of Fe particles per LDF. The diffusion is realized by two types of activated LDFs. First type is accompanied by frequent back-and-forth movements of Fe particles. The second type causes the random movement of Fe particles. Keywords: Density fluctuations, simulated liquid, iron particles. 1. Introduction It is commonly accepted that density fluctuation enables the collective movement of particles in the liquid. The local density around the ith particle can be quantified as nOi ρi = (1.1) VO 3 where VO = 4πRO /3; nOi is the number of particles in a coordination sphere of the ith particle; RO is the radius of the coordination sphere. If the number nOi changes, then the local density around the ith particle varies. This means that the change of nOi at some moments represents the local density fluctuation (LDF). The schematic illustration of LDF for selected particle is presented in Figure 1. It can be seen that LDFs happen four times for a selected particle. Therefore, in the present study we investigate LDFs happening in Fe MD models. The structure of obtained liquids and amorphous solids has been also analyzed through the pair radial distribution function (PRDF). Received March 19, 2014. Accepted September 29, 2014. Contact Nguyen Thi Thao, e-mail address: ntthao.hnue@gmail.com 112
- Local density fluctuations in simulated liquid iron Figure 1. The schematic illustration of local density fluctuations for a selected particle The dash and solid circles represent the coordination sphere of selected particle and particles, respectively Iron, an element of great interest for industrial application has been under intensive investigation using both computer simulation and experiments [3, 4]. In particular, much attention has been paid to the structure and diffusion of liquid Fe. To obtain proper results from MD simulation, various inter-atomic potentials have been proposed [2, 4]. The pair potential is employed to describe interatomic interactions between particles because this simulation consumes much less time while reproducing well the experimental results. For example, the MD simulation based on the Pak-Doyama potential has been successfully used to explore the structure and dynamics of iron within a wide temperature range [5]. 2. Content 2.1. The simulation method The Pak-Doyama potential is given as follows [1]: U (r) = −0.188917(r−1.82709)4 +1.70192(r−2.50849)2 −0.198294; r < 3.44A ˚ (2.1) here r is the inter-atomic distance in A˚ and U (r) in eV . We performed the simulation in a cube containing 104 particles under periodic boundary conditions. The equations of motion were solved numerically using the Verlet algorithm. The initial random configuration was equilibrated at a constant density of 7.0 g/cm3 by relaxation for 106 MD steps at 5000 K (i.e. NVT ensemble). From this melt we prepare eight other samples at temperatures from 2300 to 290 K by cooling down to the desired temperature and density. Then each obtained sample was relaxed for 1.2 - 2.5 × 107 MD steps until reaching equilibrium. The MD step is equal to 0.4 fs. To collect the dynamical and structural data we also performed a run for each equilibrated sample within 5 × 106 MD steps in the ensemble NVE. To calculate the coordination number we use the cutoff distance RO = 3.35 A ˚ chosen as a minimum after the first peak of PRDF. 2.2. Results and discussion Figure 2 shows the PRDFs and the comparison with experimental data for liquid and amorphous samples. There is good agreement with experiment data indicating that 113
- Nguyen Thi Thao, Pham Khac Hung and Le Van Vinh the Pak-Doyama potential reproduces well the structure for both liquid and amorphous iron. As shown in Figure 2, the height of the first peak of PRDF increases with a decrease in temperature. It follows that a nearest neighbor coordination is raised during the solidification of materials. Further, the positions of the first peak and the PRDF minimum are almost unchanged with temperature. They are located at 2.55 and 3.35 A, ˚ respectively. It should be noted that at a temperature below 1000 K, the second PRDF peak is clearly split into two sub-peaks which is a signature of a transition from a liquid to amorphous state. The splitting of the second PRDF peak is also observed by other researchers investigating different materials. This may be related to the existence of a local icosahedral order in materials [5] and could be considered a common feature of amorphous matter [5, 6]. Figure 2. Pair radial distribution functions for liquid and amorphous iron During simulation runs the diffusion coefficient is determined via the Einstein relation < r(t)2 > D = lim (2.2) t→∞ 6t here < r(t)2 > is the mean square displacement of particles over time t. The curves < r(t)2 > vs. MD steps n shown in Figure 4 represent well-straight lines for high temperatures. Their slopes are used to calculate diffusion coefficient D. One can see that when the temperature drops below 830 K, the graph becomes a horizontal line. This means that the diffusion coefficient drops to zero, because the material transforms to the amorphous state. During simulation runs we determine for each particle a number of local density fluctuation acts. Let mi (n) be the number of LDFs happening with the ith particle during n steps. The quantity of interest is the mean number of LDF which is given as ∑ N mi (n) i < m(n) >= (2.3) N 114
- Local density fluctuations in simulated liquid iron here N is the number of particles in the sample. The data of < m(n) > presented in Figure 3 shows well-straight lines with a slope equal to the rate of LDF, ξ. Hence the number < m(n) > is given as < m(n) >= ξn + A (2.4) here A is independent of n. The diffusion mechanism is realized as follows: LDFs happen at different places in the system and at different moments over time. As a LDF happens, the particles perform a collective movement causing the diffusivity. We denote the mean square displacement of particles per one LDF to δ. It follows that the mean square displacement of particles over time t is equal to < r(t)2 >=< m(t) > δ =< m(n) > δ (2.5) Now Eq. (2.2) can be rewritten as 1 < r(t)2 > < m(t) > δ (ξn + A)δ D= lim = B lim = B lim = Bδξ 6t0 n→∞ n n→∞ n n→∞ n (2.6) where t = nt0 ; B is a constant equal to 1/6t0 ; t0 is the time consumed for one MD step. We have calculated ξ and δ from Figures 3 and 4. Their values plotted vs. temperature are presented in Figure 5. The rate of LDF monotonously reduces when the temperature decreases. Moreover, the value of ξ is large enough even at 293.7 K. Compared to ξ at 826.22 K, it is only two times smaller. Meanwhile, δ rapidly reduces to zero with a decrease in temperature (see Figure 5). This result clearly indicates that the main contribution of δ is to the slow dynamics near the glass transition point. Figure 3. The dependence Figure 4. The mean square displacement of versus steps n of all particles < r(n)2 > 115
- Nguyen Thi Thao, Pham Khac Hung and Le Van Vinh Figure 5. The temperature dependence Figure 6. The distribution of LDF of the quantities ξ and δ and number of visiting particles The significant variation of δ indicates that LDFs cause the different collective movement of particles at different temperatures. To identify this property we perform two runs for samples, one at temperature 1200.38 K and one at 2225.53 ∑ K. For each run the number of steps n is adopted ∑ so that the total number of LDFs, mi (n) is close to a given value. The chosen mi (n) is equal to 233174 and 2331573 for samples at 1200.38 and 2225.53 K, respectively. Fig. 6 shows the distribution of mi (n) through particles for considered samples. There is a pronounced peak, the location being almost unchanged with temperature. The curves have a Gauss form. The peak height for the low-temperature sample is lower than that for the high-temperature sample. Moreover, the graph for the low-temperature sample is much wider. This result can be explained if one assumes that (i) LDFs are realized by surmounting different activation energy barriers and (ii) the energetic barrier set for both samples is little changed. Hence, at low temperature, the LDF happens frequently for particles where the energy barrier is small. For particles where the energy barrier is large, the LDF in converse happens rarely. As a result, the distribution of mi (n) spreads wider as the temperature is lowered. The significant change of δ may be explained on basis of the percolation of non-mobile regions. In our simulation the non-mobile regions are the places where LDFs happen rarely or not at all. Furthermore, as the temperature approached the glass transition point, the density decreases and the non-mobile regions expand. As a result, they percolated over the whole system. However, we observe a homogeneous spatial distribution of LDF for both considered samples. This means that the reason for the observed change of δ has not yet been mentioned. This will be discussed below. 116
- Local density fluctuations in simulated liquid iron Figure 7. A schematic illustration of two types Figure 8. The dependence of number of LDFs: type I (A) and type II (B) of visiting particles vs. on steps n Figure 7 presents a schematic illustration of LDF occurring in the constructed samples. It can be seen that there are two types of LDF. For the first type (type I), after a LDF happens, particle 6 leaves the coordination sphere and then it come back as the next LDF occurs. As a consequence, the list of particles visiting the coordination sphere is unchanged after two LDFs (it includes particles 1, 2,... 6). In the case of the second type (type II), particle 7 replaces particle 6 and the list of visiting particles also includes particle 7, i.e. the number of visiting particles increases by one. Because with the type I LDF particle 6 moves back and forth in the system, the type II LDF causes a larger square displacement of particles compared to the type I LDF. It follows that, given the same number of LDFs, if the fraction of type I LDFs increases, then the mean square displacement of particles decreases. In order to estimate this effect, we determine a list of visiting particles for each ith particle during simulation runs. Let the number of visiting particles during n steps be si (n). The quantity of interest is the mean number of visiting particles given as ∑ N si (n) i < s(n) >= (2.7) N The distribution of si (n) is shown in Figure 8. Unlike the distribution of mi (n), the curve for a high-temperature sample is spread wider and the location of the main peak shifts significantly to the left. In particular, the peak location is 26 and 54 for low and high-temperature samples, respectively. This result clearly indicates that the fraction of type I LDF increases significantly with decreasing temperature. To this end we determine the dependence of < s(n) > vs. on n for samples at different temperatures. The obtained result is presented in Figure 8. Here it can be seen that the graph is well-straight lines, 117
- Nguyen Thi Thao, Pham Khac Hung and Le Van Vinh and at low temperature the curve becomes horizontal indicating all of the LDFs are type I, i.e. most of the particle movement at LDF is a back and forth motion. The slope of lines shown in Figure 8 is used to define quantity ζ which characterizes the fraction of type I LDF. Table 1 presents value ζ and it can be seen that it monotonously increases with temperature. Table 1. The parameters of ξ Sample 1 2 3 4 5 6 7 8 Temperature 293.72 655.80 826.22 1200.38 1564.36 1823.03 2049.49 2225.53 ξ 0 0 0 0.00006 0.00016 0.00025 0.00035 0.00043 The significant change in δ is related to the change in the fraction of type I LDF with the lowering of the temperature. Density fluctuations happen in the liquid so that there are the places where the density is high and LDF happens rarely. In other places LDFs happen frequently. 3. Conclusion We have constructed a series of MD models of liquid and amorphous Fe at temperatures ranging from 290 to 2300 K. A distinctive result of this research is that the diffusivity in liquid iron is realized by two types of activated LDFs. The fraction of type I LDFs increases significantly as the temperature decreases. The diffusion coefficient is found to be a product of the rate of LDF ξ and mean square displacement per LDF δ. As the temperature decreases, both quantities ξ and δ become smaller, but the decrease in δ occurs much more rapidly. REFERENCES [1] H. M. Pak, M. Doyama, 1969. The polyhedron and cavity analyses of a structural model of amorphous iron. J. Fac. Eng. Univ. Tokyo B 30, 111. [2] L. Koci, A. B. Belonoshko, R. Ahuja. 2006. Molecular dynamics study of liquid iron under high pressure and high temperature. Phys. Rev. B 73, 224113. [3] M. Aykol, A. O. Mekhrabov, M.V. Akdeniz, 2009. Nano-scale phase separation in amorphous Fe- B alloys: Atomic and cluster ordering. Acta Materialia 57, 171. [4] M.W. Finnis, J. E. Sinclair. 1984. A simple empirical N- body potential for transition metals. Philos. Mag., A 50 (1) 45. [5] Vo Van Hoang, Nguyen Hung Cuong, 2009. Local icosahedral order and thermodynamics of simulated amorphous Fe. Physica B 404,340. [6] X. Y. Fu, M. L. Falk, D. A. Rigney. 2001. Sliding behavior of metallic glass: Part II. Computer simulations. Wear 250, 420. 118
Chính trị học
Báo chí - Truyền thông
Xã hội học
Giáo dục học
Tâm lý học
Lịch sử - Văn hoá
Triết học
Ngôn ngữ học
Thư viện thông tin
Văn học nước ngoài
Ngư nghiệp
Hành chính - Pháp luật
Địa lý - Địa danh
Văn học Việt nam
Lịch sử Đảng
CNXH - KH
Tư Tưởng HCM
Ngụ ngôn - Cổ tích
Ca dao - Tục ngữ
Hoá học
Sinh học
Y khoa - Dược
Kinh tế học