Transport properties and entropy-scaling laws for diffusion coefficients in liquid Fe_0.9Ni_0.1 up to 350 GPa

Abstract

Transport properties and entropy-scaling laws for diffusion coefficients in liquid Fe_0.9Ni_0.1 alloy under high pressure conditions have been studied by molecular dynamics simulations based upon the Quantum Sutton and Chen potential. We find that the entropy-scaling laws proposed independently by Rosenfeld and Dzugutov for diffusion coefficients under ambient pressure, approximating the excess entropy by the pair correlation entropy, can be fruitfully extended to liquid Fe–Ni alloy under high pressure conditions to understand and predict the transport properties of the Earth's core. In addition, our results suggest that the temperature dependence of the self-diffusion coefficient and viscosity follow the Arrhenius-type relation, and the activity energies for diffusion and viscosity increase with increasing pressure. The viscosity of liquid Fe_0.9Ni_0.1 alloy is slightly greater than that of pure liquid Fe, and lower than that of pure liquid Ni, at a given temperature and pressure. This result indicates that Ni has a slightly positive influence on the viscosity of liquid Fe–Ni alloy.

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1. Introduction

Transport properties of liquids are of immense importance in various disciplines and have been investigated from time to time using the latest probes or tools available at each stage.¹ However, the experimental data for transport under high temperature and high pressure conditions reported in the literature are quite scarce, since experimental determination currently seems to be extremely expensive and difficult. It is well known that transport properties are inextricably bound up with structural properties and the structural quantity can be easier to predict or measure than transport quantity.^2,3 Therefore, finding the formulation for bridging relations between the transport properties of a system and its structural properties would be of great importance and practical value. Rosenfeld was among the first to propose a scaling law to relate transport coefficients to excess entropy, which can be computed from diffraction measurements.^4,5 Apart from the Rosenfeld approach, another scaling law was proposed by Dzugutov to connect structural properties of a system to its dynamics.⁶ Such scaling laws are of the form D* = A exp(BS_ex), where D* are dimensionless diffusion coefficients with either macroscopic (Rosenfeld) or microscopic (Dzugutov) reduction parameters, A and B are scaling parameters. According to the scaling laws, the diffusion coefficients of simple liquid can be expressed as single-valued functions of the excess entropy (S_ex), we call those laws the entropy-scaling laws. While the entropy-scaling laws were initially formulated for simple liquids, recent simulations show that the scaling laws are applicable to a much wider variety of liquids than originally assumed, including water,⁷ core-softened fluids,^8–10 model polymeric melts,¹¹ molecular fluids,^12,13 and multi-component ionic melts.¹⁴ More recently, the entropy-scaling laws have been examined extensively and found to be valid for monatomic liquid metals over a wide range of pressure.^15–17 However, to our best knowledge, the transport properties in the context of the entropy-scaling law for liquid metallic alloy under high pressure and high temperature conditions have not been reported previously. That is to say, the validation of the entropy-scaling law for liquid metallic alloys remains an open question under high pressure conditions.

We know that the most abundant component of the Earth's core is iron (Fe), and the core must also contain some nickel (Ni) and light elements. According to the geochemical models, the amount of Ni in the core is about 5 to 15 at%,¹⁸ and the outer core may contain slightly more Ni than the inner core, since the partition coefficient of Ni is slightly less than unity.¹⁹ The transport properties of liquid Fe and Fe-based alloy determine the generation of the earth's magnetic field, natural oscillations and rotation of the earth. During the last decade, the scientific investigations for the transport properties of liquid Fe and Fe–S alloy have been made extensively.^15,20–26 However, it has not been cleared about the transport properties of liquid outer core, and the transport and structural properties of liquid Fe–Ni alloy under high pressure conditions reported in the literature are quite scarce, except the molecular dynamics (MD) calculation of the viscosity of liquid Fe–Ni alloy at the temperatures and pressures of the Earth's outer core.²⁷ It is important to point out that the potential parameters used in the MD simulation by Zhang et al. were obtained only by fitting the potential with experimental data in solid states.²⁷ Meanwhile, another group of potential parameters for iron, obtained by fitting the FPLMTO energy-volume data for both hcp and liquid iron, was proposed by Belonoshko et al.,²⁸ and it was suggested to be a reliable potential for simulating melting curve, transport and structural properties of iron under high pressure conditions.^15,28,29 So, further accurate simulations about the transport properties of liquid Fe–Ni alloy under high pressure conditions are required.

In this work, MD simulations were applied to calculate the transport and structural properties including the diffusion coefficient, viscosity, the pair correlation function g(r), and the pair correlation entropy S₂ of liquid Fe_0.9Ni_0.1 alloy under high pressure conditions. Moreover, the entropy-scaling laws for diffusion coefficients are investigated. The rest of the paper is organized as follows: in Section 2, the potential and computational details are described; in Section 3, the results are presented and discussed; finally, the conclusions are given in Section 4.

2. Potential and computational details

It is very important to choose a reliable interatomic potential for the MD simulation. In this work, the Quantum Sutton and Chen (Q-SC) potential was used to describe the atomic interaction in the Fe–Ni alloy

The first term on the right hand side describes the pairwise interactions and the second term represents the many-body cohesive term, r_ij represents the distance between atom i and j, and the density term ρ_i is given by

For the interaction between iron and nickel, the following mixing rules were employed:³⁰

Potential parameters used in our simulations are listed in Table 1. The parameters for nickel were suggested to be a reliable potential for simulating the melting curve of nickel and the transport properties of liquid nickel under high pressure condition.^16,31 As motioned above, the parameters for iron are fitted well to reproduce both the experimental data of hcp Fe as well as the structural and transport properties of liquid Fe under high pressure conditions.^28,29

Table 1 Q-SC potential parameters used in this work

Interaction	ε (eV)	a (Å)	C	m	n
a Take from ref. 27.b Take from ref. 30.
Fe–Fe^a	0.017306	3.47139	24.93900	4.78770	8.13738
Ni–Ni^b	0.031774	3.13230	33.57410	3.63100	8.97500
Fe–Ni^a	0.023450	3.30185		4.20935	8.55619

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Using the large-scale atomic/molecular massively parallel simulator (LAMMPS code), all MD simulation were performed for systems of 13 500 atoms with periodic boundary conditions in three directions. As experimental and theoretical results suggested that Fe_0.9Ni_0.1 alloy remain in the hexagonal close-packed (hcp) structure throughout the reported pressure range at 300 K. The MD simulation was performed in the isothermal and isobaric (NPT) ensemble, and the Nosé–Hoover thermostat and barostat were applied to control the temperature and pressure. The simulation time step was set as 1 fs. The cutoff distance for calculating atomic interaction forces was 12 Å, the cutoff distance for calculating the pair correlation entropy was 15 Å. At each temperature step, the samples are aged for 1 ns.

3. Results and discussion

3.1 Equations of state and pair distribution function

One key to accurate predictions of the structural and transport properties of material is the quality of the description of atomic interactions. To assess the quality of the potentials used in this work, the equations of state (EOS) of Fe–Ni alloy and the pair distribution functions g(r) of liquid nickel under high pressure high temperature conditions are calculated using MD simulation with Q-SC potential, and the comparisons were made between our calculated data and experiments or other calculations. EOS is fundamentally important to understand the Earth's core, and the EOS of Fe_0.9Ni_0.1 alloy up to 272 GPa was reported by the laser annealing technique.³² Our calculated compression curve of Fe_0.9Ni_0.1 alloy at 300 K is plotted in Fig. 1, in which we also compare the results with the experiments. We can find that our data is in agreement with experiments in the whole pressure range. The comparisons of our calculated pair distribution functions of liquid nickel at two pressure–temperature points (68 GPa, 4000 K and 47 GPa, 4000 K) with first-principles MD results were also made to check the validity of the potential with the parameters as listed in Table 1 (see Fig. 2). As shown in the figure, our calculated pair distribution functions are in agreements with the first-principles MD results.³³ As mentioned in above section, the Q-SC potential with the parameters as list in Table 1 can accurately reflect the structural and transport properties of liquid Fe under high pressure condition. The good matching further confirms the reliability of the Q-SC potential using in this work.

Fig. 1 Comparison of compression curve at 300 K with that measured by laser-heated diamond anvil cell experiments.

Fig. 2 Comparison of the pair distribution functions at two pressure–temperature points with those calculated using first-principles MD simulations.

3.2 Melting curve and transport coefficients

In order to get liquid Fe_0.9Ni_0.1 alloy in equilibrium state under high pressure conditions, the melting temperatures of hcp Fe_0.9Ni_0.1 alloy in the pressure range from 50 to 350 GPa have been calculated using the hysteresis approach. The resulting melting curve for Fe_0.9Ni_0.1 is shown in Fig. 3. The lack of experimental or simulation melting point for the Fe–Ni alloy at high pressure does not allow us to make any comparison. The melting curve of pure iron calculated with the Q-SC potential at high pressure conditions is also plotted in Fig. 3.¹⁵ We can find that, the melting curve of Fe_0.9Ni_0.1 and that of pure Fe are close to each other. This is expected since Fe and Ni have a quite similar electronic structure. The comparison of melting curve of pure iron under high pressure with experiment was described in detail in ref. 15.

Fig. 3 Comparison of melting curve of Fe_0.9Ni_0.1 with that of pure Fe up to 350 GPa.

According to the Einstein formula, for sufficiently long time interval, diffusion coefficient D can be extracted from the equation

where r(t) denotes an atom position at time t. The angular brackets denote an average over all the particles as well as all time origins. Besides, the viscosity η can be obtained from the Green–Kubo equationwhere the σ_αβ is the component of the stress in the αβ direction, V is the volume of the system, k_B is the Boltzmann's constant, and T is the temperature of the system.

The temperature dependence of self-diffusion coefficients of Fe (D_Fe, filled symbols) and Ni (D_Ni, open symbols) in liquid Fe_0.9Ni_0.1 are reported in Fig. 4(a). We can find that, at given temperature and pressure, the value of D_Fe is always greater than that of D_Ni. Over the investigated temperature range, D_Fe and D_Ni follow an Arrhenius-type law for all the investigated pressure. The activation energies for self-diffusion of Fe and Ni in liquid Fe_0.9Ni_0.1 alloy increase monotonously as increases pressure, and Fe and Ni atoms in liquid Fe_0.9Ni_0.1 alloy display similar activation energies as shown in Table 2. Fig. 4(b) shows the viscosity of liquid Fe_0.9Ni_0.1 (gray symbols) as a function of temperature, and the temperature dependence of the viscosity of Fe_0.9Ni_0.1 also shows an Arrhenius-type relation at given pressure. Activation energies for viscosity of liquid Fe_0.9Ni_0.1 alloy (^AE_η) are also presented in Table 2. In order to address the influence of Ni on transport properties of liquid Fe_0.9Ni_0.1, the viscosity of pure liquid Fe (solid line) and pure liquid Ni (dash line) at 50 GPa are also plotted in Fig. 4(b), the activation energies for viscosity of liquid pure Fe (^PE^Fe_η) and Ni (^PE^Ni_η) under high pressure conditions are also presented in Table 2. We note that the viscosity of Fe_0.9Ni_0.1 is always slightly greater than that of liquid pure Fe and smaller than that of pure Ni, at given temperature. At given pressure, ^AE_η is always greater than that of liquid pure Fe and smaller than that of liquid pure Ni. That is to say, the Ni has slightly positive effect on the viscosity of liquid Fe_0.9Ni_0.1.

Fig. 4 Temperature dependence of diffusion coefficients (a) of Fe and Ni in liquid Fe_0.9Ni_0.1, and viscosity (b) of liquid Fe_0.9Ni_0.1, pure liquid Fe as well as pure liquid Ni under high pressures (50 GPa, square; 100 GPa, circle; 150 GPa, up triangle; 200 GPa, down triangle; 250 GPa, diamond; 300 GPa, left triangle; 350 GPa, right triangle). The filled symbols refer to the D_Fe, the open symbols denote D_Ni, the gray symbols are viscosity of Fe_0.9Ni_0.1, and the solid line and dash line correspond to the viscosity of pure liquid Fe and pure liquid Ni at 50 GPa, respectively.

Table 2 Comparison of the self-diffusion activation energies of Fe and Ni in liquid Fe_0.9Ni_0.1 alloy (^AE^Fe_D and ^AE^Ni_D), the viscosity activation energies of liquid Fe_0.9Ni_0.1 alloy (^AE_η) with the self-diffusion activation energies of pure liquid Fe (^PE^Fe_D) and Ni (^PE^Ni_D) and the viscosity energies of pure liquid Fe (^PE^Fe_η) under high pressure conditions.^15,16 Pressure and activation energy in units of GPa and eV, respectively

Pressure	^AE^FeD	^AE^NiD	^AEη	^PE^FeD	^PE^Feη	^PE^Ni
50	0.533	0.546	0.370	0.516	0.280
100	0.789	0.782	0.537	0.789	0.453	1.01
150	1.039	1.051	0.767	1.013	0.578
200	1.228	1.283	0.880	1.194	0.667
250	1.402	1.460	0.990	1.364	0.774
300	1.551	1.683	1.099	1.530	0.916
350	1.721	1.747	1.272	1.682	1.018

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3.3 Entropy and entropy-scaling law

We now consider the scaling of diffusion coefficients with the excess entropy S_ex in liquid Fe–Ni alloy under high pressure conditions. In the Rosenfeld approach,^4,5 the reduced particle diffusion coefficient D^R_α of α components in liquid alloy is expressed in the formwhere D_α and m_α are the particle diffusion coefficient and particle mass of α components, respectively. The reduced diffusion coefficients were shown to be related to the excess entropy in quasi-universal behaviour

D^R_α ≈ 0.6e^0.8S_ex, (8)

In the above equation, the excess entropy can be approximately by the pair correlation entropy S₂. For binary alloys, S₂ is given by

where g_αβ(r) is the pair correlation function between α and β components, χ_α is the mole fraction of α components, and ρ is the number density. Fig. 5 shows the pair correlation entropy S₂ of liquid Fe_0.9Ni_0.1 as a function of the temperature up to 350 GPa. As expected, S₂ increases with increasing temperature at given pressure, which indicates that the order of the liquid Fe_0.9Ni_0.1 alloy decreases with increasing temperature at given pressure.

Fig. 5 Temperature dependence of pair correlation entropy S₂ of liquid Fe_0.9Ni_0.1 under high pressure conditions.

Fig. 6 shows the Rosenfeld reduced diffusion coefficients, scaled as eqn (7), of Fe and Ni in liquid Fe_0.9Ni_0.1 as a function of the pair correlation entropy S₂ up to 350 GPa. We find that a fitting of simulated data yields to slope of −0.691 and −0.708 for D^R_Fe and D^R_Ni, respectively. And, the values are close to the ideal value −0.8 in Rosenfeld scaling equation eqn (8).

Fig. 6 The Rosenfeld reduced diffusion coefficients of Fe (a) and Ni (b) in liquid Fe_0.9Ni_0.1 as a function of the pair correlation entropy under high pressure condition. The solid lines, being the best fit to the data up to 350 GPa, represent the present scaling law of equations given in the figure.

In Dzugutov approach,^6,34 the diffusion coefficient of the liquid alloy is expressed in the from

Where D_α and D_β represent the self-diffusion coefficients of the respective components with mole fractions χ_α and χ_β, and the scaling factor Γ_α is defined ashere σ_αβ (α, β = Fe or Ni) is the first peak position of the corresponding partial pair correlation function g_αβ(r), and m_α is the atomic mass of α component. Then the reduced diffusion coefficients can be expressed by

D^D ≈ 0.049e^S_ex. (12)

As mentioned above, the excess entropy S_ex can be approximately by the pair correlation entropy S₂. Fig. 7 shows the scaled diffusion coefficient D^D [eqn (10)] as a function of S₂ as computed from eqn (9) up to 350 GPa. The solid line in the figure which represents the best fit to the data, within the numerical uncertainty, is consistent with the scaling prediction. In other words, the Dzugutov scaling law is valid for Fe–Ni alloy under high pressure condition.

Fig. 7 The Dzugutov reduced diffusion coefficients of liquid Fe_0.9Ni_0.1 as a function of the pair correlation entropy under high pressure condition. The solid lines, being the best fit to the data up to 350 GPa, represent the present scaling law of equations given in the figure.

4. Conclusions

In summary, the molecular dynamic simulations were used to study transport properties and the entropy-scaling laws for diffusion coefficients of liquid Fe_0.9Ni_0.1 alloy up to 350 GPa. Our calculated equations of state of Fe_0.9Ni_0.1 alloy at 300 K up to 350 GPa are found to agree very well with experimental data, and the pair distribution functions of liquid nickel at two pressure–temperature points are in agreements with the first-principle MD simulation results. Melting curve of Fe_0.9Ni_0.1 alloy and that of pure Fe are close to each other. Diffusion coefficients and viscosity are calculated with Einstein formula and Green–Kubo equation, respectively. We show that the temperature dependencies of the self-diffusion coefficients of Fe and Ni in liquid Fe_0.9Ni_0.1 alloy, and the viscosity of liquid Fe_0.9Ni_0.1 alloy follow the Arrhenius-type relation, at given pressure. The activation energy for both diffusion and viscosity increases monotonously as increases pressure. Through comparisons between viscosity/activation energy of liquid Fe_0.9Ni_0.1 alloy and that of pure liquid Fe as well as the pure liquid Ni, we find that Ni in the liquid Fe_0.9Ni_0.1 alloy has slightly positive influence on the viscosity. Moreover, the results presented here suggested that the entropy-scaling laws, proposed by Rosenfeld and Dzugutov for diffusion coefficients, still hold well for liquid Fe_0.9Ni_0.1 alloy under high pressure and high temperature conditions.

Acknowledgements

Supported by the Science and Research Foundation of Sichuan Educational Committee of China (Grant no. 13ZB0211), the Open Research Fund of Computational Physics Key Laboratory of Sichuan Province, Yibin University (No. JSWL2014KF06), and Scientific Research Key Project of Yibin University (No. 2011B08).

References

1. A. Samanta, S. M. Ali and S. K. Ghosh, Phys. Rev. Lett., 2001, 87, 245901.
2. N. Jakse and A. Pasturel, Sci. Rep., 2016, 6, 20689.
3. A. Pasturel and N. Jakse, J. Phys.: Condens. Matter, 2015, 27, 325104.
4. Y. Rosenfeld, Phys. Rev. A, 1977, 15, 2545.
5. Y. Rosenfeld, J. Phys.: Condens. Matter, 1999, 11, 5415.
6. M. Dzugutov, Nature, 1996, 381, 137–139.
7. M. Agarwal, M. Singh, R. Sharma, M. Parvez Alam and C. Chakravarty, J. Phys. Chem. B, 2010, 114, 6995–7001.
8. W. P. Krekelberg, T. Kumar, J. Mittal, J. R. Errington and T. M. Truskett, Phys. Rev. E, 2009, 79, 031203.
9. J. R. Errington, T. M. Truskett and J. Mittal, J. Chem. Phys., 2006, 125, 244502.
10. J. Mittal, J. R. Errington and T. M. Truskett, J. Chem. Phys., 2006, 125, 076102.
11. T. Goel, C. N. Patra, T. Mukherjee and C. Chakravarty, J. Chem. Phys., 2008, 129, 164904.
12. E. H. Abramson and H. West-Foyle, Phys. Rev. E, 2008, 77, 041202.
13. E. H. Abramson, Phys. Rev. E, 2009, 80, 021201.
14. S. Jabes and C. Chakravarty, J. Chem. Phys., 2012, 136, 144507.
15. Q.-L. Cao, P.-P. Wang, D.-H. Huang, J.-S. Yang, M.-J. Wan and F.-H. Wang, J. Chem. Phys., 2014, 140, 114505.
16. C. Qi-Long, W. Pan-Pan, H. Duo-Hui, Y. Jun-Sheng, W. Ming-Jie and W. Fan-Hou, Chin. Phys. Lett., 2015, 32, 086201.
17. Q.-L. Cao, J.-X. Shao, F.-H. Wang and P.-P. Wang, et al., J. Appl. Phys., 2015, 117, 135903.
18. L. Dubrovinsky, N. Dubrovinskaia, O. Narygina, I. Kantor, A. Kuznetzov, V. Prakapenka, L. Vitos, B. Johansson, A. Mikhaylushkin and S. Simak, et al., Science, 2007, 316, 1880–1883.
19. J. H. Jones and M. J. Drake, Geochim. Cosmochim. Acta, 1983, 47, 1199–1209.
20. Y. D. Fomin, V. Ryzhov and V. Brazhkin, J. Phys.: Condens. Matter, 2013, 25, 285104.
21. L. Koči, A. B. Belonoshko and R. Ahuja, Geophys. J. Int., 2007, 168, 890–894.
22. C. Desgranges and J. Delhommelle, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 172102.
23. K.-I. Funakoshi, High Pressure Res., 2010, 30, 60–64.
24. J.-P. Perrillat, M. Mezouar, G. Garbarino and S. Bauchau, High Pressure Res., 2010, 30, 415–423.
25. D. Alfè and M. J. Gillan, Phys. Rev. B: Condens. Matter Mater. Phys., 1998, 58, 8248.
26. M. D. Rutter, R. A. Secco, T. Uchida, H. Liu, Y. Wang, M. L. Rivers and S. R. Sutton, Geophys. Res. Lett., 2002, 29, 38.
27. Y.-G. Zhang and G.-J. Guo, Phys. Earth Planet. Inter., 2000, 122, 289–298.
28. A. B. Belonoshko, R. Ahuja and B. Johansson, Phys. Rev. Lett., 2000, 84, 3638.
29. L. Koči, A. B. Belonoshko and R. Ahuja, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 224113.
30. Y. Qi, T. Ģağın, Y. Kimura and W. A. Goddard III, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 3527.
31. M. Pozzo and D. Alfè, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 024111.
32. T. Sakai, S. Takahashi, N. Nishitani, I. Mashino, E. Ohtani and N. Hirao, Phys. Earth Planet. Inter., 2014, 228, 114–126.
33. L. Y. Yin and M. H. Sun, Chin. Sci. Bull., 2013, 58, 426–431.
34. J. Hoyt, M. Asta and B. Sadigh, Phys. Rev. Lett., 2000, 85, 594.

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