Introduction

Accurate knowledge of the physical properties under extreme pressure–temperature conditions is of considerable interest in various fields of physics, including the Earth system1, planetary physics2, astrophysics3, and inertial confinement fusion4. The center of the Earth consists of a solid inner core, surrounded by a spherical shell of the liquid outer core. The vigorous convection in the metallic liquid outer core thus powers the dynamo that sustains the magnetic field. The geodynamo is extremely sensitive to core conditions, which is a very active research topic in Earth science. The viscosity and diffusion coefficients of the Earth’s core are the main parameters in the convection process.

In the early reported paper5, the diffusion coefficient in the diffusion equation is mostly adopted as 3×10−9 m2s−1. It is difficult to get the accurate diffusion coefficient and viscosity values under Earth’s core condition experimentally6. Now, quantum molecular dynamics (QMD) can give a direct and quantitative estimation of the transport coefficients7. The QMD results show that the self-diffusion coefficient of Fe is 5.2×10−9 m2/s, and viscosity is 8.5 mPa s at T = 4300 K and density ρ = 10,700 kg m−38. Recently, light element effects on transport properties were also considered, such as Fe–Si–O fluid9, Fe–O fluid10,11, Fe–S fluid12.

The QMD results of viscosity are far lower than the values inferred from seismic and other measurements13,14. Early in 1998, it was inferred that the viscosity of the inner core is \(1.22 \times 10^{11}\) Pa s15. Geodynamic estimation by Buffett inferred viscosity is less than \(10^{16}\) Pa s16. The theoretical value for viscosity varies largely by different methods, from ~ 1 mPa s14,17 under the outer condition to 1011 Pa s15 under the inner core condition. QMD calculation shows that the viscosity is 13 mPa s and diffusion coefficients \(5 \times 10^{ - 9}\) m2s−1 at the inner-core boundary(ICB), and 12 mPa and \(4 \times 10^{ - 9}\) m2s−1 at the core-mantle boundary (CMB)18. The viscosity is about several mPa s for MD results of FeNi fluid19. Though there were limitations in MD, the early reported MD results claim that Ni has a negligible effect on viscosity17. However, the MD results dependent on the empirical parameters of atom potential, and the self-diffusion coefficient of Fe and Ni are not fully reported. AIMD is an optimal method for the calculation of ionic transport properties without empirical parameters.

However, the self-diffusion coefficient and viscosity of Fe–Ni fluid at Earth’s core condition have never been calculated by QMD. In this paper, we calculated the diffusion coefficients and viscosity of Fe–Ni fluid under Earth's core condition by the precisely QMD methods, and give a simple analysis of the temperature effect on transport properties of Fe–Ni fluid.

Methods and calculations

Transport property

The theory equations are collected from Ref.7,20,21,22. Diffusion coefficients can be calculated by either velocity autocorrelation functions or mean-squared displacements by equilibrium molecular simulation. The self-diffusion coefficient Di of element i is calculated by the Einstein equation from velocity correlation function A(t) is

$$ \begin{gathered} A\left( t \right) = \left\langle {{\mathbf{V}}_{i} \left( t \right){\mathbf{V}}_{i} \left( 0 \right)} \right\rangle , \hfill \\ D_{i} = \frac{1}{3}\int_{0}^{\infty } {A\left( \tau \right)d\tau } , \hfill \\ \end{gathered} $$
(1)

where Vi(t) is the velocity of species i at time t. and < … > is an average of velocity correlation function over atoms. From the sight of mean square displacement (MSD) M(t), Di is the asymptotic slope of M(t) as a function of time:

$$ \begin{gathered} M\left( t \right) = \frac{1}{{N_{I} }}\sum\limits_{{N_{I} }} {\left\langle {\left| {{\mathbf{R}}_{i} \left( t \right) - {\mathbf{R}}_{i} \left( 0 \right)} \right|^{2} } \right\rangle } , \hfill \\ D_{i} = \mathop {\lim }\limits_{t \to \infty } \frac{M\left( t \right)}{{6t}}, \hfill \\ \end{gathered} $$
(2)

where Ri(t) is the trajectory of a species atom i, and NI is the number of ions.

Calculated by the Green–Kubo relation, the shear viscosity η is the integral of the autocorrelation function of the off-diagonal component of the stress tensor (ACF) S(t).

$$ \begin{gathered} C_{\alpha \beta } \left( t \right) = \left\langle {P_{\alpha \beta } \left( {t + t_{0} } \right)P_{\alpha \beta } \left( {t_{0} } \right)} \right\rangle \hfill \\ \eta = \frac{V}{{k_{B} T}}\mathop {\lim }\limits_{t \to \infty } \int_{0}^{t} {S\left( {t^{\prime}} \right)dt^{\prime}} \hfill \\ S(t) = \frac{1}{5}\left\{ {C_{xy} \left( t \right) + C_{yz} \left( t \right) + C_{zx} \left( t \right) + \frac{1}{2}\left[ {C_{xx} \left( t \right) - C_{yy} \left( t \right)} \right] + \frac{1}{2}\left[ {C_{yy} \left( t \right) - C_{zz} \left( t \right)} \right]} \right\} \hfill \\ \end{gathered} $$
(3)

where kB is the Boltzmann factor, T is temperature and V is the volume of the cell. The results are averaged from the five independent off-diagonal components of the stress tensor Pxy, Pyz, Pzx, (Pxx–Pyy)/2, and (Pyy–Pzz)/2. We adopt empirical fits to the integrals of the autocorrelation function. Both Di and η have been fit to the function in the form of \({\text{A}}\left[ {1 - exp\left( { - t/\tau } \right)} \right]\), where A and τ are free parameters. The fractional statistical error in calculating a correlation function C for molecular dynamics trajectories23 can be given by

$$ \frac{\Delta C}{C} = \sqrt {\frac{2\tau }{{T_{traj} }}} , $$
(4)

where Ttraj is the length of the trajectory and τ is the correlation time of the function. In the present paper, we generally fitted over a time interval of [0,4τ–5τ].

The pair correlation function (PCF) g(r) or radial distribution function is usually used to describe quantitatively the internal structure of fluids.

$$ g\left( r \right) = \frac{V}{{4\pi r^{2} N^{2} }}\left( {\sum\limits_{i} {\sum\limits_{j \ne i} {\delta \left( {r - r_{ij} } \right)} } } \right) $$
(5)

where r is radius distance, rij and N is atom number. The PCF quantifies how the particle of interest is surrounded by other particles.

Calculation details

Ab initio molecular dynamics were performed using the Vienna ab initio simulation package (VASP)24,25. The ion–electron interaction was represented by the projector augmented wave (PAW)26,27. The generalized gradient approximation with Perdew, Burke, and Ernzerhof corrections was employed28. The electronic states were populated following the Fermi–Dirac distribution29. For the convergence of transport property values, the size of the system is checked. Herein, 128 atoms are selected as the cell. Ni atoms were randomly distributed in the cell, with 12 Ni atoms and others are Fe atoms, and the corresponding atom ratio are 9.375 at.%. Plane wave cutoff is 400 eV enough to make sure that the pressure is converged within 1% accuracy. Time dependent mean square displacement was checked to make sure that the system is in a liquid state. The selected time step is 1 fs in all the calculations. To get the convergent transport coefficients, the total time is more than 20 ps with the beginning 2 ps discarded for equilibration. The core-mantle boundary (CMB) pressure is 136 GPa, inner-core boundary (ICB) pressure is 330 GPa, and pressures under outer core conditions were selected, such as 150 GPa, 200 GPa, 250 GPa, 300 GPa. The analytical expressions of the pressure and temperature profiles along the adiabatic curve were given by following Labrosses30.

It is assumed that the Earth’s outer core is in a well-mixed state with an adiabatic pressure–temperature profiles. The temperature and pressure profiles is the same as reported in30. The pressure P at radius (r) is

$$ P\left( r \right) = P_{c} + \frac{{4\pi G\rho_{cen}^{2} }}{3}\left[ {\left( {\frac{{3r^{2} }}{10} - \frac{{L^{2} }}{5}} \right)\exp \left( { - \frac{{r^{2} }}{{L^{2} }}} \right)} \right]_{r}^{{r_{c} }} $$
(6)

where Pc and ρcen are the pressure and density at the center of Earth, respectively; G is the Gravitational constant; L is the lengthscale; and rc is the radius at the CMB. G = 6.6873 × 10−11 m3kg−1 s−2, L = 7272 km, and rc = 3480 km. With the P(rc) = 136 GPa and P(ri) = 330 GPa, the pressures P(r) in the outer core are collected.

The adiabatic temperature Ta at radius r is

$$ T_{a} \left( r \right) = T_{cen} \exp \left( { - \frac{{r^{2} }}{{D_{L}^{2} }}} \right) $$
(7)

where Tcen is the temperature at the center of the Earth, and D is the lengthscale. Tcen = 5726 K, and DL = 6203 km.

Results and discussion

Ab initio calculated results

After getting the equilibrium information of the QMD, the pair correlation function g(t) (Fig. 1), mean-square displacement M(t) (Fig. 2a), and autocorrelation function of the off-diagonal component of the stress tensor S(t) (Fig. 2b) can be calculated. The shape of g(r) and M(t) of FeNi fluid is similar to early reported results13, and the calculated FeNi fluid is a close-packed liquid. The g(r) shapes of Fe–Ni, Fe–Fe, and Ni–Ni are similar, which implied that Fe and Ni atoms are similarly distributed. The g(r) tends towards a uniform value of 1 for a large value of r. The first peak location of g(r) at ICB is closer to the zero point than at CMB. From the QMD results, the equilibrium volume of Fe–Ni fluid is 8.274 Å3/atom at CMB and 4000 K and 6.933 Å3/atom at ICB and 5500 K, respectively. The g(r) results are consistent with the QMD calculated equilibrium volume. The peaks of g(r) decrease with temperature increase, especially at CMB conditions, which indicated that g(r) at high temperature easily converged to unity.

Figure 1
figure 1

Pair correlation functions of (a) Fe–Ni (gFe-Ni(r)), (b) Fe–Fe (gFe-Fe(r)) and (c) Ni–Ni (gNi-Ni(r)) in FeNi fluid under Earth’s core condition. The label ‘ICB’ and ‘CMB’ represent pressure at 330 GPa and 136 GPa, respectively.

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Figure 2
figure 2

(a) Mean squared displacements of Fe–Fe M(t), (b) autocorrelation function of traceless stress tensor S(t) and (c) time evoluted viscosity η(t) in FeNi fluid under Earth's core condition. The label ‘ICB’ and ‘CMB’ represent 330 GPa and 136 GPa, respectively. The S(0) is velocity-velocity autocorrelation function at t = 0.

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The mean-square displacement M(t) of Fe and Ni becomes a linear function of time after sufficiently large random steps (Fig. 2a). As the mobility of Fe and Ni are similar, only M(t) of Fe is plotted in Fig. 2a. The diffusive regime (~ t1) gives a self-diffusion coefficient which starts after a ballistic period of dozens of fs. At 136 GPa or 330 GPa, the M(t) increases nonlinearly with temperature increase. The self-diffusion coefficient is the slope of M(t), it confirmed that the self-diffusion coefficient increases with the increase of temperature. The shape of S(t), and η(t) is the same as reported31 (Fig. 2b–c). The S(t) goes to zero as time tends to infinity. The present ACF at CMB in Fig. 2 looks to decay slower than that reported data at CMB in Ref.8. Not only the temperature and pressure states are different from8, but also the thermostats in our calculation are different. The ACF convergence time and behavior may be different from previous work. The error is estimated with the analytic expression in Eq.(1). At 136 GPa or 330 GPa, the η(t → 0) decreases with the increase of temperature. Then, the viscosity of FeNi fluid decreases with the temperature.

Self-diffusion coefficient

Our QMD calculations on the self-diffusion coefficient of pure Fe liquid are lower than the early reported QMD results8,9,18 (Fig. 3a). The main difference is that the adiabatic temperatures in this calculation are lower than the reported results. From Fig. 3b, the temperature effect on the self-diffusion coefficient exhibit Arrhenius behaviors, and self-diffusion coefficients of Fe and Ni increase with the increase of temperature. Then, it is reasonable that the self-diffusion coefficient of Fe in this manuscript is a bit lower than reported in Refs.8,9,18. The self-diffusion coefficients of Fe and Ni are comparable, in the orders of 10−9 m2s−1. Our calculated self-diffusion coefficients of Fe are consistent with the ab initio calculated results of pure Fe8, Fe–Si–O fluid9, and Fe–O fluid11.

Figure 3
figure 3

The self-diffusion coefficients (DFe and DNi) of Fe and FeNi fluid under Earth's core conditions. (a) is along the core adiabatic, and (b) is temperature variation at CMB and ICB. ‘Wijs’18, ‘Alfe’8, ‘Pozzo1′9, ‘Pozzo2′9 and ‘Pozzo3′9 represent the self-diffusion coefficient of Fe in Fe, Fe0.82Si0.10O0.08 and Fe0.79Si0.08O0.13 fluids.

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Self-diffusion coefficients of Fe along the core adiabatic are in the range of [2.58,3.35] × 10−9 m2s−1. Self-diffusion coefficients of Ni along the core adiabatic are in the range of [2.47, 3.37] × 10−9 m2s−1. The self-diffusion coefficient of Fe–Ni fluid along the core adiabatic is almost constant (Fig. 3a and Table 1). The self-diffusion of Ni is a little smaller than that of Fe at the same pressure and temperature condition. From one sight, the atom mass of Ni is a little higher than that of Fe, which shows that the self-diffusion coefficient of Ni and Fe are similar, but Ni atoms move slower than Fe atoms. Considering the pressure and temperature effect (Fig. 3b), the self-diffusion coefficient is higher when at the same pressure with a higher temperature or the same temperature with lower pressure. The temperature and pressure effect on the self-diffusion coefficient in FeNi fluid can also be obtained by analyzing the of mean squared displacement (Fig. 2). Self-diffusion coefficient of Fe at CMB ranges from (2.15 ± 0.26)  × 10−9 at 3500 K to (7.47 ± 0.94) × 10−9 m2s−1 at 5000 K. The self-diffusion coefficient of Ni at CMB ranges from (1.96 ± 0.57)  × 10−9 at 3500 K to (7.14 ± 1.16)  × 10−9 at 5000 K.

Table 1 Self-diffusion coefficient and viscosity of FeNi fluid along the core adiabatic curve.
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Shear viscosity

The viscosity of FeNi fluid along the adiabatic curve is in the range of [7.78, 29.95] mPa s, and the viscosity of Fe fluid is in the range of [10.10, 22.51] mPa s (shown in Fig. 4a). The viscosity of Fe-10%Ni fluid is higher than pure Fe, under ICB, and CMB (Fig. 4b). The QMD result is different from the early reported MD results17, which point out Ni has a negligible effect on bulk viscosity of liquid iron, about ~ mPa s17. Furthermore, the QMD results of viscosity are far lower than the values inferred from seismic and other measurements13,14, and higher than those calculated by MD17. The viscosity near the CMB condition reported in this manuscript is higher than the reported results by Pozzo at the same condition (Fig. 4a). The adiabatic temperature in this calculation is lower than Ref.9. At 136 GPa or 330 GPa, the viscosity decreases as the increase of temperature (Fig. 4b). Temperature effect on shear viscosity exhibit Arrhenius behaviors. Then, a slightly low temperature also corresponds to a high viscosity at the same pressure, and the temperature effect is obvious when at a slightly low temperature.

Figure 4
figure 4

Viscosity of the pure Fe and FeNi fluid under Earth's core condition. (a) is viscosity along the adiabatic curve and (b) is viscosity under inner-core (136 GPa) and core-mantle (330 GPa) boundary pressure with different temperatures. The Fe and FeNi labels correspond to pure Fe and FeNi fluid results, respectively.

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Conclusion

Ab initio molecular dynamics estimates self-diffusion coefficient and viscosity of outer core are important for several purposes. The pair correlation function, mean square displacement, and autocorrelation function of traceless stress tensor are analyzed by the ab initio calculated equilibrium information. Self-diffusion coefficients of Ni along the core adiabatic range from 2.47 × 10−9 to 3.37 × 10−9 m2s−1. The viscosity of FeNi fluid along the core adiabatic range from 7.88 to 29.95 mPa s. At the core-mantle boundary, the diffusion coefficient increases with temperature increase, while viscosity decrease with temperature increase as expected.