Nguyễn Thị Thanh Hà và Đtg<br />
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Tạp chí KHOA HỌC & CÔNG NGHỆ<br />
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STUDYING DIFFUSION MECHANISM AND DYNAMICS SLOWDOWN IN<br />
IRON LIQUID<br />
Nguyen Thi Thanh Ha*, Le Van Vinh, Pham Khac Hung<br />
Hanoi University of Science and Technology<br />
<br />
SUMMARY<br />
The dynamic properties of iron liquid (Fe) are studied by molecular dynamics (MD) simulation.<br />
We trace the evolution of local density fluctuations (LDFs) in Fe liquid over the simulation time<br />
and in the 300-2300 K temperature range. The result simulation reveals that atomic diffusion is<br />
realized through the LDFs and the high localization LDFs at low temperature in the iron liquid is<br />
the cause of the anomalous dynamics slowdown. We find that the diffusion depends on both rate<br />
of LDFs and the averaged square displacement of particles Fe as one LDF occurs. As the<br />
temperature decreases, both quantities reduce.<br />
Keywords: Molecular dynamics simulation, iron liquid, dynamics slowdown, diffusion, local<br />
density fluctuations.<br />
<br />
INTRODUCTION*<br />
This transition to a disordered solid known as<br />
the glass transition is accompanied with the<br />
drastic increase in the viscosity and a subtle<br />
change in the structure. Understanding the<br />
microscopic mechanism governing glass<br />
transitions is one of the most important<br />
problems in statistical physics [1-3]. To tackle<br />
this problem, several working hypotheses<br />
have been proposed. The studies from refs.[48] focus on the dynamics heterogeneity, the<br />
percolation in real space and properties of<br />
energy landscapes. They found the existence<br />
of mobile and immobile regions which<br />
migrate in the space over time. Authors in [910] put forward the mechanism by which the<br />
small modification of statistic density<br />
correlations can produce an extremely large<br />
dynamical change. The essential result in this<br />
direction is the mode coupling theory [9] that<br />
predicts a freezing of dynamics from the nonlinear feedback effect. The theoretical and<br />
experimental investigations on universal<br />
mechanisms controlling slow dynamics have<br />
been done for long time, however as mention<br />
in [11] many open questions are still<br />
remained.<br />
Iron is an important element and has many<br />
industrial applications. Therefore, knowledge<br />
*<br />
<br />
Tel: 0983 012387, Email: ha.nguyenthithanh1@hust.edu.vn<br />
<br />
about their microstructure and dynamical<br />
properties would be essential to understand<br />
this material [12-14]. In this paper, MD<br />
simulation is conducted to examine the<br />
dynamics in iron liquid. Our purpose is to<br />
clarify the diffusion mechanism and the cause<br />
of slowdown in the iron liquid near glass<br />
temperature.<br />
CALCULATION PROCEDURE<br />
MD simulation is conducted for 104 atom<br />
models with periodic boundary conditions<br />
using Pak–Doyama potential [15]. To<br />
integrate the equation of motion Verlet<br />
algorithm is used with MD step of 0.67 fs.<br />
Initial configuration is obtained by randomly<br />
placing all atoms in a simulation box. Then<br />
this sample is equilibrated at temperature of<br />
6000 K and cooled down to desired<br />
temperature. Next, a long relaxation has been<br />
done in ensemble NPT (constant temperature<br />
and pressure) by 105 MD steps to obtain the<br />
equilibrium sample. We prepare six models<br />
(M1, M2... M6) have been constructed at<br />
ambient pressure and at temperature of 300 K,<br />
800 K, 1200 K, 1500 K, 1800K and 2300 K. To<br />
study dynamical properties the obtained<br />
samples are relaxed in ensemble NVE (constant<br />
volume and energy) over 5x106 steps.<br />
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Obviously, the diffusivity in system is<br />
impossible if no exchanging the coordinated<br />
Fe occurs. Therefore, we trace the evolution<br />
of local density fluctuations (LDF) in Fe<br />
liquid over the simulation time. To calculate<br />
the coordination number we use the cutoff<br />
distance RO=3.35 Å chosen as a minimum<br />
after first peak of PRDF. The local density<br />
around ith particle can be quantified as:<br />
<br />
n<br />
i Oi<br />
VO<br />
<br />
135(05): 167 - 172<br />
<br />
sphere. If the number nOi changes, then the<br />
local density around ith particle varies. It<br />
means that the change of nOi at some moments<br />
represents the local density fluctuation (LDF)<br />
act. The existence of non-mobile and mobile<br />
regions is originated from the density<br />
fluctuation in the liquid.<br />
RESULTS AND DISCUSSION<br />
To test the validity of MD model one usually<br />
determines the pair radial distribution<br />
functions (PRDF). They are very close to<br />
simulation result reported in ref. [14, 16] and<br />
in good agreement with experimental data.<br />
<br />
(1)<br />
<br />
where VO= 4RO3/3; nOi is the number of<br />
particles in a coordination sphere of ith<br />
particle; RO is the radius of the coordination<br />
3<br />
B<br />
<br />
Simulation<br />
Experiment [16]<br />
<br />
2<br />
<br />
g(r)<br />
<br />
1<br />
<br />
0<br />
4<br />
<br />
A<br />
<br />
Simulation<br />
Experiment [14]<br />
<br />
3<br />
2<br />
1<br />
0<br />
0<br />
<br />
2<br />
<br />
4<br />
<br />
6<br />
<br />
8<br />
<br />
10<br />
<br />
12<br />
<br />
r, Å<br />
<br />
Fig 1. The pair radial distribution functions for amorphous solid iron at<br />
300K (A) and liquid iron at 1500K (B)<br />
<br />
Fig.2 The schematic illustration of local density fluctuations for selected particle<br />
<br />
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The schematic illustration of LDF for selected<br />
particle is presented in Fig.2. One can see that<br />
LDFs happen four times for a selected<br />
particle. In MD simulation the diffusion<br />
coefficient is usually determined via Einstein<br />
equation:<br />
<br />
R(t ) <br />
R(t ) <br />
lim<br />
t <br />
n 6n. t<br />
6t<br />
MD<br />
2<br />
<br />
D lim<br />
<br />
lim<br />
n <br />
<br />
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<br />
M LDF<br />
n<br />
<br />
(3)<br />
<br />
R(t ) 2 <br />
M LDF <br />
M LDF<br />
<br />
lim<br />
<br />
(4)<br />
<br />
The equation (3) can be reduced to<br />
<br />
2<br />
<br />
R(t ) 2 <br />
1<br />
<br />
.. A..<br />
n 6n. t<br />
6. tMD<br />
MD<br />
<br />
D lim<br />
<br />
(2)<br />
<br />
Where is mean square displacement<br />
(MDS) over time t, n is step, tMD =0.67fs. If<br />
we define: MLDF is a number of LDFs<br />
happening with ith particle during n steps, is<br />
a rate of LDF and is the averaged square<br />
displacement of particles Fe as one LDF<br />
occurs.<br />
<br />
(5)<br />
<br />
The dependence of MLDF vs. n and vs.<br />
MLDF is shown in Fig.3 and 4, respectively.<br />
Well straight lines are seen and the quantities<br />
determined from these lines are presented in<br />
Table 1. We see that both and <br />
monotonously increase in the temperature<br />
interval of 300-2300 K.<br />
<br />
500<br />
1500K<br />
1800K<br />
2300K<br />
<br />
400<br />
300<br />
200<br />
<br />
<br />
<br />
100<br />
0<br />
300K<br />
600K<br />
1200K<br />
<br />
200<br />
150<br />
100<br />
50<br />
0<br />
0<br />
<br />
50000<br />
<br />
100000<br />
MD steps, n<br />
<br />
150000<br />
<br />
200000<br />
<br />
Fig 3. The dependence of MLDF as a function of MD steps n<br />
Table 1. Dynamical characteristics of simulated liquids: here D, D* is the diffusion coefficient callculated<br />
by (5) and Einstein equation, respectively<br />
Model<br />
Temprature<br />
υ<br />
δ ( Å2/ one LDF)<br />
D×105 (cm2/s)<br />
D*×105 (cm2/s)<br />
<br />
M1<br />
300<br />
0.0002<br />
0.0001<br />
0<br />
0<br />
<br />
M2<br />
800<br />
0.0006<br />
0.0004<br />
0.0038<br />
0.0058<br />
<br />
M3<br />
1200<br />
0.0012<br />
0.0149<br />
0.2597<br />
0.2608<br />
<br />
M4<br />
1500<br />
0.0018<br />
0.0538<br />
1.4054<br />
1.4224<br />
<br />
M5<br />
1800<br />
0.0022<br />
0.0934<br />
3.0082<br />
3.0103<br />
<br />
M6<br />
2300<br />
0.0027<br />
0.1820<br />
7.2125<br />
7.2025<br />
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0.30<br />
The mean square displacement of particles, Å<br />
<br />
2<br />
<br />
100<br />
<br />
1200K<br />
1500K<br />
1800K<br />
2300K<br />
<br />
0.25<br />
80<br />
<br />
300K<br />
800K<br />
<br />
0.20<br />
<br />
135(05): 167 - 172<br />
<br />
60<br />
0.15<br />
40<br />
<br />
0.10<br />
0.05<br />
<br />
20<br />
<br />
0.00<br />
<br />
0<br />
0<br />
<br />
30<br />
<br />
60<br />
<br />
90<br />
<br />
120<br />
<br />
0<br />
<br />
100<br />
<br />
MLDF<br />
<br />
200 M300<br />
LDF<br />
<br />
400<br />
<br />
500<br />
<br />
Fig.4. The dependence of as a function of <br />
<br />
The averaged MSD/one LDF<br />
<br />
0.20<br />
0.15<br />
0.10<br />
0.05<br />
0.00<br />
<br />
0.0025<br />
<br />
rate of LDF<br />
<br />
0.0020<br />
0.0015<br />
0.0010<br />
0.0005<br />
0.0000<br />
500<br />
<br />
1000<br />
<br />
1500<br />
<br />
2000<br />
<br />
Temperature, K<br />
<br />
Fig.5. The temperature dependence of the quantities υ and <br />
<br />
Fig.5 shows the temperature dependence of dynamical quantities for simulated liquids. As the<br />
temperature decreases from 2300 to 1200 K, δ decreases by 12.2 times that significantly larger<br />
than the change in υ equal to 2.2. It means that the major contribution to diffusion belongs to the<br />
averaged square displacement of particles Fe as one LDF occurs (δ).<br />
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Fraction of iron particles<br />
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135(05): 167 - 172<br />
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2300 K<br />
1200 K<br />
<br />
0.3<br />
<br />
0.2<br />
<br />
0.1<br />
<br />
0.0<br />
90<br />
<br />
120<br />
<br />
180<br />
<br />
240<br />
<br />
300<br />
<br />
360<br />
<br />
The number of LDFs<br />
<br />
Fig.6. The distribution of LDF in iron liquid<br />
<br />
LDFs happen rarely in the immobile regions<br />
and occur frequently in the mobile ones.<br />
Hence, the examining of the spatial<br />
distribution of LDFs happened in the liquid<br />
should give new insight into the mechanism<br />
governing slow dynamics. We now measure<br />
the distribution of MLDF through particles for<br />
samples at temperature of 1200K and 2300 K<br />
in order to identify the cause of slowdown in<br />
the iron liquid near glass temperature. For<br />
each run the number of steps n is adopted so<br />
that the total number of LDFs, Fig.6 shows<br />
the distribution of MLDF through particles for<br />
considered samples. The curves have a Gauss<br />
form but distribution of MLDF for lowtemperature sample is spread in much wider<br />
range than for high-temperature sample.<br />
There is a pronounced peak which location is<br />
almost unchanged with temperature. Its height<br />
for low-temperature sample is lower than for<br />
high-temperature one. In our simulation the<br />
non-mobile regions are the places where<br />
LDFs happen rarely or not occur. Further, as<br />
the temperature approached to the glass<br />
transition point, the density reduces and the<br />
non-mobile regions expand. As a result, they<br />
percolated over whole system. Therefore, the<br />
anomalous dynamics slowdown near the glass<br />
transition temperature can be explained by the<br />
high localization LDFs in the iron liquid.<br />
<br />
CONCLUSION<br />
The diffusion mechanism in iron liquids is<br />
studied by mean of molecular dynamic<br />
simulation and the activated LDFs. We<br />
establish an expression for diffusion<br />
coefficient via the rate LDFs. We find that the averaged square displacement of particles<br />
Fe as one LDF occurs and - rate of LDF<br />
monotonously decreases with temperature.<br />
But rapidly decreases to zero and mainly<br />
contributes to the slow dynamics. The result<br />
shows that the localization LDFs near the<br />
glass transition point is the reason of the<br />
anomalously slow dynamics in iron liquid.<br />
REFERENCES<br />
1. A. Heuer (2008), J. Phys.: Condens. Matter 20,<br />
373101.<br />
2. H. Tanaka, T. Kawasaki, H. Shintani, K.<br />
Watanabe (2010), Nat.Mater. 9, 324.<br />
3. L. Berthier, G. Biroli (2011), Rev. Mod. Phys.<br />
83, 587.<br />
4. J. S. Langer and S. Mukhopadhyay (2008),<br />
Phys. Rev. E 77, 061505.<br />
5. G. Lois, J. Blawzdziewicz, and C. S. O'Hern<br />
(2009), Phys. Rev. Lett. 102, 015702.<br />
6. F. Sausset, G. Tarjus (2010), Phys. Rev. Lett.<br />
104, 065701.<br />
7. A. Cavagna, T.S. Grigera, P. Verrocchio<br />
(2007), Phys. Rev. Lett. 98, 187801.<br />
8. D. Rodney and T. Schrøder (2011), Eur. Phys.<br />
J. E 34: 100.<br />
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