Indian Journal of Pure & Applied Physics Vol. 46, November 2008, pp. 771-775
Study of thermal properties of nickel using embedded-atom-method S S Hayat*, M A Choudhry, S A Ahmad, J I Akhter+ & Altaf Hussain Departmenrt of Physics, The Islamia University of Bahawalpur, Bahawalpur 63120, Pakistan Physics Division, Pakistan Institute of Nuclear Science & Technology, PO Nilore, Islamabad, Pakistan
+
*E-mail:
[email protected] Received 12 May 2008; accepted 28 August 2008 Thermal properties of metals and alloys have been studied for a long due to their importance in the materials design. Molecular dynamics simulation technique is applied to investigate thermal properties of Ni. Semi-empirical potentials based on the Embedded Atom Method (EAM) have been employed to calculate lattice parameter, energy per atom, mean square displacements and radial distribution function. Thermal properties like specific heat, thermal coefficient of linear expansion and melting temperature are deduced from the calculated parameters. The results are found to compare well with the experimental results. Keywords: Molecular dynamics simulation; Thermal properties; Nickel metal
1 Introduction There has been considerable interest in the estimation of thermal properties of metals owing to their importance in the design of various components and production of new materials1. Molecular Dynamics (MD) and the related method of Monte Carlo2 have been successfully used to calculate phase diagrams of model systems3 as well as to study melting and freezing phenomena4-7. While these are powerful techniques for dealing with the statistical mechanics of a many-body system, their applications to real materials have limitations imposed by the interatomic potential used and the finite system size and time duration of simulation. MD computer simulation method, based on many-body interatomic potentials, has become an established tool in materials science to evaluate many properties8-10. This is because one can investigate and understand structures of complex system at atomic level and can follow the evolution of the system at each step. Structure and surface energies have been evaluated by tight binding total energy method for transition and noble metals8 and results were found reasonably well for some metals and agreement was poor for others. In another study9 using tight binding interatomic potential, results for thermal properties of Ag were comparable but the agreement was not good for Au. Thermal and mechanical properties of some fcc transition metals were studied10 using Sutton-Chan potential and increase in deviation from experimental values of linear expansion was found at higher temperatures11.
Application of tight-binding potentials for transition metals and alloys has been explored by extending the interactions to fifth nearest neighbour distance12. EAM potentials13,14 have been developed for fcc metals by fitting experimental parameters like lattice constants, cohesive energy, vacancy formation energy and have been produced very useful and reliable results15,16. Akhter et al.17 employed EAM potential to study the three low index surfaces of Pd, namely (100), (110) and (111) by the MD technique and reported considerable pre-melting by measuring the mean-square vibrational amplitudes and the structure factors. The potential with cut-off distance between the third and fourth nearest neighbour produced results on melting temperature, specific heat, linear thermal expansion coefficient and diffusion coefficient of Pd comparable to the experimental values18. Ahmad et al.19 and Akhter et al.20 have reported thermal properties of noble metals Ag and Au by MD simulations using EAM potential which reasonably agreed with the experimental results. In case of Ag, melting temperature was found to be 12% higher than the experimental value. Noorullah et al21. have studied the structures and vacancies of closed packed vacancy clusters in Cu at 0 K using these potentials. Recently, Park et al.22 have studied the deformation of fcc nanowires by twinning and slip using EAM potential. In this paper, the thermal properties of Ni using the embedded atom potential with interactions extending to third nearest neighbour distance have been studied.
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INDIAN J PURE & APPL PHYS, VOL 46, NOVEMBER 2008
We have calculated the lattice parameter and energy per atom at various temperatures and deduced the melting temperature, specific heat and thermal coefficient of linear expansion. The results are compared with experimental and simulated data available in the literature. 2 Computational Techniques MD techniques in detail can be found elsewhere17-20. However, the salient features of the technique as applied in our case are given here. Nordsieck’s algorithm23 with a time step of 10-15 s was used in order to solve classical equations of motion for atoms interacting by EAM interatomic potential13,14. The computational cell used was consisted of 256 atoms arranged on fcc lattice. All simulations to be described in this case were performed using periodic boundary conditions in the x, y and z directions of the cubic simulation cell and the cut-off distance was kept between the third and fourth nearest. Simulation was carried out in different statistical ensembles. We used the condition of constant number of atoms; pressure and temperature (NPT) ensemble to calculate the lattice parameters at various temperatures. The computational cell was generated at a specific temperature while keeping the pressure constant. The system was allowed to evolve till the cell edges and volume becomes constant. The lattice constants, thus, obtained are used to generate the bulk fcc crystals at various temperatures. In all the other calculations, the system was equilibrated for sufficient number of time steps under the condition of constant volume and temperature (NVT). Once the required average temperature was attained (in about 10 ps) then, simulation was switched over to constant volume and energy run (NVE). The velocities and positions of particles were recorded after every 0.02 ps for about 60 ps. In order to guarantee that the system was equilibrated properly, the initial few configurations were ignored. All the relevant information about the structure and dynamics of the bulk was obtained from this data. 3 Results and Discussion The internal energy per atom was calculated as a function of temperature from 300 to 2500 K in steps of 100 K using NPT simulation. The results are plotted in Fig. 1 which shows a sudden jump between 2080 and 2095 K. The data for the energy per atom is fitted to a third order polynomial in the temperature range below melting temperature (400-2000 K) as given in the following equation:
T E (T ) = E 0 + A1 Tm
T + A2 Tm
2
T + A3 Tm
3
where E(T) is energy per atom at temperature T and Tm = 2080 K is the melting temperature obtained in the present study. The values of constants E0, A1, A2 and A3 are presented in Table 1. The specific heat at constant pressure is calculated by taking the temperature derivative of E(T) i.e, Cp =
dE (T ) dT
The calculated and experimental24 values of Cp at different temperatures are plotted in Fig. 2. The difference between the present computed and experimental values is very small. Deviation is 4.7% at room temperature, 4% at 1500 K and 5% at 2000 K. However, the calculated values of Cp largely deviate from experimental values in the temperature range 400-700 K, because of the quantum effects which have not been taken into account by the classical MD simulation as pointed out by Mei et al.25 for Cu and Ahmed et al.19 for Au. This difference at
Fig. 1 — Lattice parameter α(T) and Energy per atom E(T) as a function of temperature Table 1 — The constants of expressions for energy per atom and lattice parameters Energy (eV) E0 A1 A2 A3
−4.44123 0.48921 0.08078 0.07444
Lattice parameter (Å) a0 B1 B2 B3
3.5183 0.12238 0.02939 0.06006
HAYAT et al.: THERMAL PROPERTIES OF NICKEL USING EMBEDDED-ATOM-METHOD
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Fig. 2 — Specific heat at constant pressure (Cp) as a function of temperature
Fig. 3 — Comparison of α (T) at various temperatures
higher temperature is attributed to the presence of anharmonic effects in the potential at higher temperature. However, experimental data plotted in Fig. 2 from the source25 demonstrate scatter in its values as well. The large deviations in the experimental data may be due to impurities in the samples or other experimental limitations. The NPT simulation was also used to calculate the lattice parameter at various temperatures. The computational cell is generated at a particular temperature and then keeping the pressure and temperature constant, the system is allowed to evolve till the cell edges and the volume becomes constant. The lattice parameter is calculated from a = (4Ω)1/3, where Ω is the calculated average atomic volume at each temperature. A plot of calculated lattice parameter as a function of temperature is also shown in Fig. 1. A transition can again be noted at the same temperature as it was observed for energy per atom curve. The data for lattice parameter is also fitted to a third order polynomial in the temperature range below melting temperature (400-2000 K) as given by the following equation:
thermal expansion is calculated by using the expression : 1 da (T ) α(T ) = a (T ) dT
T a (T ) = a0 + B1 Tm
T − B2 Tm
2
T + B3 Tm
3
The values of constants a0, B1, B2, and B3 obtained are also given in Table 1. No experimental data for lattice parameter is available above the room temperature to compare with the computed values up to my observation. Therefore, the coefficient of linear
The calculated values of coefficient of thermal expansion at various temperatures along with the experimental results26,27 are plotted in Fig. 3. There is a good agreement between the calculated and the experimental values of α(T) from 600 to 1500 K. The maximum deviation between the calculated and experimental values of thermal coefficient in the present calculations is only about 2.34% at 1273 K. This deviation may be due to many reasons, such as, limitation in the semi-empirical description of the material. Melting temperature is an important parameter of the material as it indicates its stability against heating. The fundamental concept of melting is based on the co-existence of the solid with the liquid phase when the free energies of the two phases are equal. The energy per atom and the lattice parameter, plotted in the Fig. 1 as a function of temperature clearly reveal a sudden jump between 2080 and 2095 K. Therefore, it is concluded that the melting temperature of Ni lies between 2080 and 2095 K. When the system was cooled from higher temperature, the transition from liquid to solid state was observed in the temperature range 2000-1950K. This indicates the presence of super-cooling for Ni. In order to investigate the melting transition, radial distribution function g(r) is also calculated using the following expression.
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g (r
INDIAN J PURE & APPL PHYS, VOL 46, NOVEMBER 2008
)=
1 N
∑
n i (r
)
4 π r 2 ρ (∆ r
)
where ni(r) are the number of particles situated at a distance r from a given particle and the angular brackets denote a time average over all N recorded configurations, which in our case are 3000. The results of radial distribution at 300, 1200 and 2200 K are shown in Fig. 4. The peaks in Fig. 4 show 1st, 2nd, 3rd, etc. nearest neighbours. The graph also shows that higher order peaks are disappearing as long range order in the crystal vanishes at higher temperatures. Structural transformation from solid phase to liquid phase can be predicted from the disappearance of higher order peaks in the radial distribution function. The peaks in the radial distribution function are broadening out when we increase the temperature of system. This plot also confirms that Ni has liquid phase at 2200 K. Another thermal parameter which can indicate melting transition is the mean square displacements (MSD) of bulk atoms. At melting, these displacements increase enormously and the magnitudes become a few tenths of the interatomic distances28. The MSD values at different temperatures are calculated using the following equation. U ix2
=
1 Nj
N
j
∑
[ rix ( t ) − 〈 rix ( t − τ ) 〉 τ ]
i =1
where rix is the instantaneous position of atom i in the direction x and τ is the time interval after which
Fig. 4 — Radial distribution at various temperatures
the configurations were recorded. The angular brackets 〈 〉 denote the time average taken over all Nj recorded configurations. The average values of MSD at selected temperatures are given in Table 2. There is an abrupt jump in MSD at temperature 2080 K, which is the melting temperature of the Ni obtained from the energy per atom, lattice parameter and radial distribution function. As the mean square displacements increase by order of magnitude after melting so a log scale plot is given in Fig. 5. It is clear that mean square displacements increase linearly with temperature up to the point of transition and then, there is a sharp increase in values. The value of melting point obtained for Ni in the present calculation is 20% higher as compared to experimental value29 of 1726 K. The present value of melting temperature is higher compared to that calculated by tight-binding potential12 where cut-off distance was extended up to fifth nearest neighbour distance. In the present calculations, the potential includes interactions up to third nearest distance. Apart from the already discussed limitations in the Table 2 — Mean square displacement values at various temperatures Temperature (K) 300 600 900 1200 1500 1800 2080 2095
〈U2〉 (0.1Å)2 0.3865067 0.842437 1.387999 2.0963390 2.8863690 4.1482128 7.3819756 1207.2396700
Fig. 5 — Mean square displacement as a function of temperature
HAYAT et al.: THERMAL PROPERTIES OF NICKEL USING EMBEDDED-ATOM-METHOD
semi-empirical calculation of the atomic interaction, deviations from the experimental value of melting temperature of Ni can be due to the hysteresis effects associated with the necessity to wait for a seed of liquid phase to appear by a spontaneous fluctuation. The liquid phase can start to grow at the expanse of solid one only when the liquid seed containing few atoms is formed. That is why on typical MD scale, one can generally, expect melting at a temperature above the experimentally determined values. However, it is an excellent way for the judgement of the exact melting point of a material by observing the variation in lattice parameter, energy per atom and mean square displacements as a function of temperature. The present results of thermal properties are satisfactory, in the sense that the atomic relaxations which occur are all consistent with what might be anticipated using hard sphere model. The results presented in this paper can be relied upon as they have been obtained using a well established suit of computer programs, which have already produced plausible results for other fcc metals17-21. 4 Conclusions Embedded atom method many-body potential was employed to determine the thermal properties, including coefficient of thermal expansion, specific heat, mean square displacements and melting temperature of Ni. The results for specific heat at constant pressure reasonably agreed with the experimental values except in the temperature range 400-700 K. The maximum deviation between the calculated and experimental values of thermal coefficient in present calculations is about 2.34%. The melting temperature was computed in three different ways, i.e., by looking at variation in lattice parameters, energy per atom and mean square displacements as a function of temperature. The actual formation of the liquid like phase was checked by computing the radial distribution function. Our calculation estimates Tm ≅ 2080 K. This value lies
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