# Electrical resistivity discontinuity of iron along the melting curve

Electrical resistivity discontinuity of iron along the melting curve Summary Discontinuous changes of electrical resistivity ρel (increase), density ϱ and isothermal compressibility βT (decrease) occur across the melting temperature of metals and can be directly related by Ziman's theory in the long-wavelength approximation. By evaluating experimental data at ambient pressure, we show that Ziman's approximation holds for iron and other simple and transition metals. Using a thermodynamic model to determine βT for γ-, ε- and liquid Fe and a previously published model for ρel of liquid Fe, we apply Ziman's approximation to calculate ρel of solid Fe along the melting curve. For pure Fe, we find the discontinuity in ρel to decrease with pressure and to be negligibly small at inner core boundary conditions. However, if we account for light element enrichment in the liquid outer core, the electrical resistivity decrease across the inner core boundary is predicted to be as large as 36 per cent. Electrical properties, High pressure behaviour, Core 1 INTRODUCTION Electrical resistivity ρel of liquid metals and alloys under extreme conditions is a critical parameter for the stability and evolution of planetary dynamos. In particular, the magnetic fields of the Earth, Mercury and Ganymede are thought to be generated by a self-sustained dynamo in the convecting liquid portions of their cores, which are composed of mainly iron and a variety of lighter elements (Merrill et al.1996; Sarson et al.1997; Anderson et al.2011). In that context, the role of the inner core (IC) in magnetic field generation remains a subject of controversy (Olson & Amit 2014). Initial geodynamic studies (Hollerbach & Jones 1993; Glatzmaier & Roberts 1995) reported evidence that finite electrical resistivity in the Earth's solid IC has a stabilizing effect on the magnetic field and therefore leads to a decrease in the frequency of global field reversals. Subsequently, however, there have been conflicting results regarding this argument, ranging from support of the initial inference (Dharmaraj & Stanley 2012), no influence (Wicht 2002), to an increased number of excursion in the presence of a conducting IC when compared to results with an insulator at the centre of the planet (Lhuillier et al.2013). For conducting ICs, these studies have relied on conductivity values of the IC being the same as for the outer core (OC), while one would expect that resistivity changes across the IC boundary (ICB) for two reasons: (i) The structure of solid and liquid are quite different and resistivity changes across the melting temperature (Tm) and (ii) light elements will not equally partition between the OC and the IC. In the current manuscript we look at these two effects with the goal to provide values of ρel for use in geodynamo simulations that may help to better assess the effect of the IC on magnetic field generation. In general, metals exhibit a distinct increase in electrical resistivity upon melting. For iron, it increases by ∼8 per cent at ambient pressure (P; van Zytveld 1980; Table 1), a value that remains approximately constant up to 6 GPa as determined in large volume press experiments (Secco & Schlössin 1989; see Supporting Information Fig. S1). In similar experiments, Ezenwa & Secco (2017b) recently determined a jump of similar magnitude for another transition metal, Co, while for the closed d-shell metals, Zn and Cu, the discontinuity is significantly larger (Ezenwa & Secco 2017a; Ezenwa et al.2017). These data are of very high quality and determine the jump across Tm with great precision. At higher P, static experiments using the diamond anvil cell (DAC) are more scarce and the small sample size and potential temperature (T) gradients inside a DAC make high P–T experiments challenging, especially close to Tm (Dobson 2016), resulting in large uncertainties of the data and consequently the determination of a potential resistivity discontinuity. Ohta et al. (2016) recently reported T-dependent resistivities for the solid and the liquid phase of iron up to 51 GPa measured in the DAC, and they describe an increase by as much as ∼20 per cent across melting (see Supporting Information Fig. S2). At higher P, Ohta et al. (2016) relied on resistivity measurements for the solid hcp (ε) phase of Fe only, and in order to estimate ρel for the liquid, they used a Bloch–Grüneisen fit—taking resistivity saturation into account—to their data to extrapolate to Tm and added 20 per cent to account for the increase of ρel upon melting in an ad-hoc fashion. Gomi et al. (2013) performed measurements of ρel for hcp Fe up to 100 GPa at room T only, and argue along the theory of Mott (1972) that iron at core conditions is close to saturation resistivity and therefore the increase of ρel on melting should be negligible. In the development of their model resistivity and its application to the Earth's core, Gomi et al. (2013) do not distinguish between the solid and liquid phase. Table 1. Thermophysical data for pure iron at ambient pressure at the melting point (Tm = 1808 ± 5 K)a. Resistivities have been measured by Cezairliyan & McClure (1974)a and van Zytveld (1980)b. If not reported directly, values measured by Dever (1972)c, Basinski et al. (1955)d, Tsu et al. (1985)e, Drotning (1981)f and compiled by Desai (1986)g have been extrapolated to Tm while using the misfit as uncertainty. For the liquid phase, a scenario including values for ϱ and α from Assael et al. (2006)h has been tabulated for comparison. A column of thermodynamic properties for liquid Fe from the model of Komabayashi (2014)i has been included. Compressibilities βS and βT have been calculated using eqs (6) and (7).   δ-Fe  Liquid Fe      Drotning  Assael  Thermodynamic modeli  ρel [μΩ cm]  127.0 ± 0.6a  137.6 ± 1.0b  vp [km s−1]  4.73 ± 0.07c  3.98 ± 0.03e  3.98  vs [km s−1]  2.49 ± 0.06c        ϱ [g cm−3]  7.29 ± 0.02d  7.017 ± 0.002f  7.04 ± 0.06h  7.09  α [10−5 K−1]  6.5 ± 0.1d  8.8 ± 0.1f  13.2 ± 0.1h  9.0  cp [J g−1K−1]  0.80 ± 0.06a  0.84 ± 0.05g  0.82  βS [GPa−1]  1/(103 ± 6)  1/(111 ± 2)  1/(112 ± 2)  1/112  βT [GPa−1]  1/(91 ± 5)  1/(88 ± 2)  1/(70 ± 2)  1/87  $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}}\left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^2\left(\frac{\beta _T^{{\rm liquid}}}{\beta _T^{{\rm solid}}}\right)^{-1}$$    0.97 ± 0.05  0.78 ± 0.05  0.98 ± 0.05    δ-Fe  Liquid Fe      Drotning  Assael  Thermodynamic modeli  ρel [μΩ cm]  127.0 ± 0.6a  137.6 ± 1.0b  vp [km s−1]  4.73 ± 0.07c  3.98 ± 0.03e  3.98  vs [km s−1]  2.49 ± 0.06c        ϱ [g cm−3]  7.29 ± 0.02d  7.017 ± 0.002f  7.04 ± 0.06h  7.09  α [10−5 K−1]  6.5 ± 0.1d  8.8 ± 0.1f  13.2 ± 0.1h  9.0  cp [J g−1K−1]  0.80 ± 0.06a  0.84 ± 0.05g  0.82  βS [GPa−1]  1/(103 ± 6)  1/(111 ± 2)  1/(112 ± 2)  1/112  βT [GPa−1]  1/(91 ± 5)  1/(88 ± 2)  1/(70 ± 2)  1/87  $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}}\left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^2\left(\frac{\beta _T^{{\rm liquid}}}{\beta _T^{{\rm solid}}}\right)^{-1}$$    0.97 ± 0.05  0.78 ± 0.05  0.98 ± 0.05  View Large Complementary to experiments, electronic transport properties at high P have been calculated by evaluating the Kubo–Greenwood (KG) formula for the Onsager kinetic coefficients on results of density functional theory (DFT)-based molecular dynamics (MD) simulations (Vlček et al.2012; de Koker et al.2012; Pozzo et al.2012, 2013, 2014), a computationally expensive approach. For pure liquid iron at conditions of the Earth's ICB, these studies agree on values of resistivity slightly above 60 μΩ cm (de Koker et al.2012; Pozzo et al.2012), while for ε-Fe at similar densities, T-dependent values of $$\rho _{{\rm el}}^{{\rm solid}}=53$$–57 μΩ cm have been reported (Pozzo et al.2014). These results suggest that a change in ρel on melting—albeit small—persists to core conditions. The determination of electronic transport properties of metals under extreme P and T is challenging both experimentally and computationally, and remains a topic of a controversial discussion. In this study, we address the question of electrical resistivity of solid Fe close to Tm along the melting curve by a different approach: We use Ziman's theory to relate electronic and elastic properties upon melting, which we test on experimental data of Fe at ambient pressure. We then apply this method along the melting curve and infer resistivities of the solid phase at high P based on previously published values for the liquid and discuss the influence of light element alloying and partitioning at crystallization of the IC, as it is relevant for cores of terrestrial planetary bodies. 2 METHODS The Ziman formula for electrical resistivity (Ziman 1961)   $$\rho _{{\rm el}}=\frac{a_0\hbar }{e^2}\frac{4\pi ^3Z}{a_0k_F}\frac{1}{(2k_F)^4}\int \limits _{0}^{2k_F}S(q)|U(q)|^2q^3\mathrm{d}q$$ (1)has been widely applied to determine ρel for metallic liquids up to the warm dense matter regime (e.g. Burrill et al.2016). In this model, quasi-free conduction electrons of momentum ℏk scatter off screened ionic potentials U(q), where q = |k − k΄| is the scattering wavenumber. The spatial arrangement of scattering centres is described by the static ion–ion structure factor S(q) in reciprocal space and incorporates an implicit dependence on density ϱ and T. The pre-factor $$\frac{a_0\hbar }{e^2}\approx 21.74$$ μΩ cm is the atomic unit of resistivity (with a0 being the Bohr radius, ℏ the reduced Planck constant and e the elementary charge), Z the number of valence electrons and kF the Fermi wavenumber. Due to conservation of momentum, scattering takes place from and into states at the Fermi level. Therefore, the largest possible change of momentum upon an elastic collision is 2kF for a backscattered electron (k → k΄ = −k) which determines the upper integration boundary. In the derivation of eq. (1), several approximations have been made. (i) The Fermi surface is assumed to be isotropic. For liquids and amorphous solids, this is always the case. For many crystalline solids it is a good approximation as has been shown for bcc and fcc metals (Papaconstantopoulos 1986). In addition, if one is interested in resistivity of a polycrystal, the Hashin–Shtrikman (HS) bounds (Hashin & Shtrikman 1963) for ρel are very narrow for the group 8 elements Ru and Os, that crystallize in the hcp phase at ambient P (Volkenshteyn et al.1978; Schriempf 1968) (see Supporting Information Table S2). (ii) Higher frequency (ω) moments of S(q, ω) have been omitted, since they are dominated by interionic contributions which are generally very small (Cheung & Ashcroft 1978). Furthermore, for T above the Debye temperature (θD), the ω-dependence due to the distribution function $$\frac{\hbar \omega }{k_BT}/(e^{\frac{\hbar \omega }{k_BT}}-1)=\frac{\hbar \omega }{k_BT}/[(1+\frac{\hbar \omega }{k_BT}+\ldots )-1]\approx 1$$ (with the Boltzmann constant kB) can be neglected. (iii) The spread of the Fermi–Dirac distribution (kBT) is larger than the maximum energy transfer by phonons (kBθD) at these conditions. Therefore, it is reasonable to treat electron scattering in solids at high T quasi-elastically. By means of eq. (1), Ashcroft & Lekner (1966) calculated resistivities for a number of liquid metals by using different model potentials and an analytical expression for the hard sphere structure factor Shs(q) (Wertheim 1963). Their results agree reasonably well with experimental values at ambient P. Taking ϱ-dependence of Shs(q) into account, the Ziman formula has been applied to P of the Earth's OC by Jain & Evans (1971), who constrained the resistivity of liquid iron at the core–mantle boundary to be 104 ± 6 μΩ cm at 3473 K. The KG approach yields values of about two thirds of this number at comparable P and T, which may indicate a breakdown of the hard sphere approximation at high densities. As the ionic potential U(q) does not change across Tm, the increase of ρel upon melting occurs due to the transition from a crystalline into a disordered structure, which is reflected by changes in S(q) and ϱ. As a first order approximation, Ziman (1961) proposed to replace the integral in eq. (1) by the integrand's value at q = 0. While S(q) contains implicit information on density, other factors in eq. (1) depend on ϱ explicitly: U by its normalization to −2/3· EF at q = 0 (EF being the Fermi energy), and the Fermi radius kF. For the free electron gas, they scale as EF ∝ ϱ2/3 and kF ∝ ϱ1/3, respectively. While Fe is not a free electron metal, its Fermi level shows the same dependence on volume (Supporting Information Fig. S3). Combining $$k_F^{-5}$$ in the pre-factor with $$E_F^{-2}$$ from the integrand, resistivity decreases ∝ϱ−3. If one considers the ratio of resistivities in the liquid and the solid phase at Tm, in the Ziman approximation the following relation holds:   $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}}\simeq \frac{S(0)^{{\rm liquid}}}{S(0)^{{\rm solid}}}\left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^{-3}.$$ (2) On the other hand, the distribution of atoms in condensed matter determines bulk elastic parameters. Since the thermodynamic limit   $$\lim _{q\rightarrow 0}S(q)=\varrho k_BT\beta _T$$ (3)is known from fluctuation theory (March 1990), the structure factor ratio in eq. (2) can be reformulated in terms of isothermal compressibilities, thermodynamically defined as βT = −ϱ∂(1/ϱ)/∂P, and Ziman's approximation becomes   $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}}\simeq \frac{\beta _T^{{\rm liquid}}}{\beta _T^{{\rm solid}}}\left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^{-2}$$ (4)or   $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}} \left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^2 \left(\frac{\beta _T^{{\rm liquid}}}{\beta _T^{{\rm solid}}}\right)^{-1}\simeq 1.$$ (5) Jain & Evans (1971) inserted a model for ϱ, T and βT of the Earth into eq. (3), which fully defines the variation of Shs(q) with P and T. While we do not calculate absolute resistivity values, we use the long-wavelength approximation to eq. (1) as a method to estimate the relative increase of ρel from $$\rho _{{\rm el}}^{{\rm liquid}}$$ to $$\rho _{{\rm el}}^{{\rm solid}}$$ across the melting curve based on thermodynamic parameters. The application of eq. (4) is not limited to this direction. High P experiments such as those performed by Gomi et al. (2013) or Ohta et al. (2016) for the solid could make use of it to convert $$\rho _{{\rm el}}^{{\rm solid}}$$ to $$\rho _{{\rm el}}^{{\rm liquid}}$$ or cross-check their data. 3 RESULTS AND DISCUSSION 3.1 Ambient pressure We assess the validity of relation (5) for iron at ambient P by compiling experimental data for ρel, ϱ and βT right above and below Tm. Electrical resistivity of iron in the solid state has been measured up to 1800 K (Cezairliyan & McClure 1974) and for the liquid starting at 1808 K (van Zytveld 1980), with a ratio of $$\rho _{{\rm el}}^{{\rm liquid}}/\rho _{{\rm el}}^{{\rm solid}}=1.08\pm 0.01$$ (Table 1). To obtain compressibilities, we rely on ultrasonic measurements of the longitudinal and transverse acoustic velocities (vp and vs, respectively), which are related to isentropic compressibility βS via   $$v_p^2-\frac{4}{3}v_s^2=\frac{1}{\varrho \beta _S},$$ (6)with vs = 0 for the liquid. To convert βS to βT, one has to apply the thermodynamic relation   $$\beta _T=\beta _S+\frac{\alpha ^2T}{\varrho c_p},$$ (7)with cp being the heat capacity at constant P and α the coefficient of thermal expansion. If not directly reported, we extrapolated the thermophysical quantities in eq. (7) to Tm (see Table 1). To the best of our knowledge, no data exist on acoustic velocities for the δ-phase of iron (in the bcc structure) which is in equilibrium with the liquid at ambient P. Instead, we use single crystal elastic constants of the α-phase (also in the bcc structure) from Dever (1972) above 1050 K, calculate vp and vs from a Voigt–Reuss–Hill polycrystalline average and linearly extrapolate to Tm (Table 1). This approach is well justified as any magnetic contribution to βT vanishes above the Curie temperature (1043 K), and the α- and δ-phase stability region can be viewed as belonging to the same stability field that is connected at negative pressure (Komabayashi & Fei 2010). In order to apply eq. (7), we use values for ϱ and α from the X-ray diffraction study of Basinski et al. (1955) and cP from Cezairliyan & McClure (1974). Thermodynamic parameters for liquid iron are more controversial. Measured sound velocities range from 3820 to 4052 m s−1 (Nasch et al.1994; Casas et al.1984) and ϱ from 6937 to 7120 kg m−3 (Blumm & Henderson 2000; Hixson et al.1990). For vP we choose the measurements by Tsu et al. (1985) (Table 1) that are most consistent with values computed with the thermodynamic model of Komabayashi (2014). Density values are closely associated with those of thermal expansivity α which has the strongest influence on the determination of βT from βS, as it enters in quadratic form in eq. (7). As for the solid phase, values of ϱ and α should be chosen consistently, i.e. come from the same underlying data of ϱ(T). Thermal expansivity of liquid iron has been the topic of a long-standing controversy (Williams 2009) with either values below 0.9 × 10−4 K−1 (e.g. Drotning 1981; Nasch & Steinemann 1995; Blumm & Henderson 2000) or larger than 1.1 × 10−4 K−1 (e.g. Kirshenbaum & Cahill 1962; Saito et al.1969; Hixson et al.1990). With the exception of the dilatometer work of Blumm & Henderson (2000), the former data stem from γ-ray attenuation studies and the latter from constant mass setups (e.g. Archimedean, maximum bubble pressure, levitation techniques). Following Assael et al. (2006), Williams (2009) argued for the larger α on the basis of potential systematic errors due to the sample geometry in γ-ray attenuation. However, constant mass methods may underestimate effects of wetting, surface tension and viscosity (Drotning 1981; Nasch et al.1994), which could also account for the discrepancy. Using the values recommended by Assael et al. (2006), βT = 1/(70 ± 2) GPa−1, with ϱ and α from Drotning (1981), βT = 1/(88 ± 2) GPa−1. Only for the βT value based on ϱ and α of Drotning (1981), the Ziman approximation (eq. 5) holds (Table 1). The thermodynamic limit of S(q) (eq. 3) provides an alternative route to the determination of βT independent of α. We have fitted measured S(q) of liquid iron (Waseda & Ohtani 1974) by means of a Percus–Yevick hard-sphere expression in the long wavelength limit (see Supporting Information Figs S4 and S5) and find, despite considerable uncertainty, a compressibility value that is consistent with the acoustic-velocity-based value using ϱ and α of Drotning (1981), supporting the validity of Ziman's approximation for iron at ambient P. Model parameters for liquid iron from Komabayashi (2014) also agree well with the S(q)-based βT and satisfy eq. (5) in combination with the experimental data of δ-Fe (Table 1). A similar evaluation of eq. (5) for other metallic elements can be found in the Supporting Information (Tables S3– S5). We find that the resistivity ratio is well represented by the right hand side of eq. (4) for the simple metals Na and Al, and the 3d transition metals Co and Ni. For Zn and the noble metals Cu, Ag and Au—all metals with closed d-shell—it is systematically underestimated by a factor of ∼1.5, that is, the increase in ρel across melting (left-hand side of eq. 4) is significantly larger than the ratio of thermodynamic properties (right-hand side of eq. 4). For Co, Cu and Zn, the resistivity jump at ambient P is consistent with the experiments up to 5 GPa (Ezenwa & Secco 2017a,b; Ezenwa et al.2017). 3.2 High pressure In order to obtain the right hand side of eq. (4) at high P for Fe, we compute ϱ and βT on both the liquid and the solid side of the melting curve with the model of Komabayashi (2014) (Fig. 1a). For the liquid, the model yields a value for ϱliquid which is consistent with experimental work by Tateyama et al. (2011) at 4.3 GPa and shock experiments up to ∼440 GPa (Brown & McQueen 1986; Brown et al.2000) (see Supporting Information Fig. S6). The thermodynamic model by Komabayashi (2014) has been designed to reproduce results from DAC and multi-anvil experiments for the solid phases. The applicability of this model in the geophysical context is further supported when comparing its ratio of adiabatic compressibilities at the ICB ($$\beta _S^{{\rm liquid}}/\beta _S^{{\rm solid}}=1.02$$) with that of PREM ($$\beta _S^{{\rm liquid}}/\beta _S^{{\rm solid}}=1.03$$) (Dziewonski & Anderson 1981). Figure 1. View largeDownload slide (a) Liquid to solid compressibility/density ratio for Fe along the melting curve from the thermodynamic model of Komabayashi (2014) used in Ziman's approximation (right-hand side of eq. 4). The lower x-axis label and ticks show pressure, the upper ones the corresponding melting temperature. Pressure intervals indicated correspond to core P in the terrestrial bodies of our solar system (Ganymede, Mercury, Mars, Venus and Earth). The vertical line represents the γ-ε-liquid triple point in the model of Komabayashi (2014; 96 GPa and 3300 K). (b) Electrical resistivities in the liquid (red curve) and the solid (black curve) phases of Fe along the melting curve. For the liquid phase, the modified Bloch–Grüneisen fit from de Koker et al. (2012) has been evaluated along the melting curve, while the resistivity in the solid has been calculated using the ratios shown in panel (a) and applying eq. (4). The band widths take fitting uncertainties of the original Kubo-Greenwood results by de Koker et al. (2012) into account. Laboratory data (filled symbols) are by Secco & Schlössin (1989) (S89) and Ohta et al. (2016) (O16) from static experiments, and by Bi et al. (2002) (B02), Keeler (1971) (K71) and Matassov (1977) (M77) from shock wave experiments. For $$\rho _{{\rm el}}^{{\rm liquid}}$$ and $$\rho _{{\rm el}}^{{\rm solid}}$$, data by Secco & Schlössin (1989) and Ohta et al. (2016) up to 51 GPa have been fitted linearly in the liquid and solid regions, respectively, and extrapolated towards Tm from both sides (see Supporting Information Figs S1 and S2). For the shock wave experiments, the lowest P point by Bi et al. (2002) and the highest P point each by Keeler (1971) and Matassov (1977) have been used, all for the solid phase. Temperatures along the Hugoniot at these pressures (Brown & McQueen 1986) are significantly below the melting point. Open circles in the P-range between 100 and 160 GPa show values calculated from combined Bloch–Grüneisen/resistivity-saturation fit parameters given in Ohta et al. (2016) and evaluated at Tm of Komabayashi (2014). G13 (open diamond) represents the high T extrapolation of a room temperature DAC experiment reported in Gomi et al. (2013), also taking resistivity saturation into account. At inner core boundary P, the Kubo–Greenwood results by Pozzo et al. (2012, 2014) (P12 and P14) are included for liquid and solid Fe, respectively. (c) Negative logarithmic derivative −∂(ln ρel)/∂P for liquid and solid iron along its melting curve. While the P-gradient is significant in the liquid, it is negligible in the solid, particularly for ε-Fe. Figure 1. View largeDownload slide (a) Liquid to solid compressibility/density ratio for Fe along the melting curve from the thermodynamic model of Komabayashi (2014) used in Ziman's approximation (right-hand side of eq. 4). The lower x-axis label and ticks show pressure, the upper ones the corresponding melting temperature. Pressure intervals indicated correspond to core P in the terrestrial bodies of our solar system (Ganymede, Mercury, Mars, Venus and Earth). The vertical line represents the γ-ε-liquid triple point in the model of Komabayashi (2014; 96 GPa and 3300 K). (b) Electrical resistivities in the liquid (red curve) and the solid (black curve) phases of Fe along the melting curve. For the liquid phase, the modified Bloch–Grüneisen fit from de Koker et al. (2012) has been evaluated along the melting curve, while the resistivity in the solid has been calculated using the ratios shown in panel (a) and applying eq. (4). The band widths take fitting uncertainties of the original Kubo-Greenwood results by de Koker et al. (2012) into account. Laboratory data (filled symbols) are by Secco & Schlössin (1989) (S89) and Ohta et al. (2016) (O16) from static experiments, and by Bi et al. (2002) (B02), Keeler (1971) (K71) and Matassov (1977) (M77) from shock wave experiments. For $$\rho _{{\rm el}}^{{\rm liquid}}$$ and $$\rho _{{\rm el}}^{{\rm solid}}$$, data by Secco & Schlössin (1989) and Ohta et al. (2016) up to 51 GPa have been fitted linearly in the liquid and solid regions, respectively, and extrapolated towards Tm from both sides (see Supporting Information Figs S1 and S2). For the shock wave experiments, the lowest P point by Bi et al. (2002) and the highest P point each by Keeler (1971) and Matassov (1977) have been used, all for the solid phase. Temperatures along the Hugoniot at these pressures (Brown & McQueen 1986) are significantly below the melting point. Open circles in the P-range between 100 and 160 GPa show values calculated from combined Bloch–Grüneisen/resistivity-saturation fit parameters given in Ohta et al. (2016) and evaluated at Tm of Komabayashi (2014). G13 (open diamond) represents the high T extrapolation of a room temperature DAC experiment reported in Gomi et al. (2013), also taking resistivity saturation into account. At inner core boundary P, the Kubo–Greenwood results by Pozzo et al. (2012, 2014) (P12 and P14) are included for liquid and solid Fe, respectively. (c) Negative logarithmic derivative −∂(ln ρel)/∂P for liquid and solid iron along its melting curve. While the P-gradient is significant in the liquid, it is negligible in the solid, particularly for ε-Fe. Fig. 1(b) shows both $$\rho _{{\rm el}}^{{\rm liquid}}$$ and $$\rho _{{\rm el}}^{{\rm solid}}$$ for pure Fe as a function of P, covering conditions up to 360 GPa, the pressure in the Earth's centre. Resistivities in the liquid phase have been calculated from the modified Bloch–Grüneisen model by de Koker et al. (2012) and $$\rho _{{\rm el}}^{{\rm solid}}$$ has been computed by means of eq. (5). We predict $$\rho _{{\rm el}}^{{\rm solid}}$$ for iron in the range of 69–71 μΩ cm at conditions of the cores of Mars, Mercury and Ganymede, while $$\rho _{{\rm el}}^{{\rm liquid}}\approx 78-88$$ μΩ cm. The KG results by de Koker et al. (2012) underestimate $$\rho _{{\rm el}}^{{\rm liquid}}$$ at ambient conditions and low P compared to experiments, and this mismatch suggests that $$\rho _{{\rm el}}^{{\rm solid}}$$ should also be considered with caution there. Two effects contribute to this discrepancy. (i) The underlying equation of state overestimates density at ambient P by ∼20 per cent, similar to the small volume at zero pressure reported for liquid iron from DFT–MD simulations by Ichikawa et al. (2014). Smaller volumes lead to reduced resistivity values. (ii) Recent results of Drchal et al. (2017) indicate that there is a possible contribution of spin disorder to ρel at high T, which has not been taken into account by previous computational studies. At higher P, however, resistivity values of de Koker et al. (2012) are in good agreement with shock wave data (Keeler 1971; Matassov 1977; Bi et al.2002). Stacey & Anderson (2001) have argued that ρel remains constant along the melting curve. In their derivation of ∂(ln ρel)/∂P = 0, they express the Grüneisen parameter in terms of an average lattice frequency, and they do not distinguish between the liquid and the solid phase. Since the melting point defines an extreme case of anharmonicity, it is not clear to what degree their conclusion is applicable to the liquid phase. Indeed, in contrast to a relatively strong change of −∂(ln ρel)/∂P along the melting curve for liquid Fe (Fig. 1c), the corresponding slopes for both the fcc (<10−3 GPa−1) and hcp phases (<4 × 10−4 GPa−1) are small, supporting the hypothesis of Stacey & Anderson (2001) to first order for the solid phases. An interesting feature of the model is the predicted decrease of $$\rho _{{\rm el}}^{{\rm solid}}$$ at the γ-ε-liquid triple point. Resistivity generally decreases with increasing charge carrier density: ρel scales with ϱ between ∝ ϱ−3 (eq. 2) and ∝ ϱ−2 (eq. 4), depending on the implicit ϱ-dependence of S(q). In Ziman's long wavelength approximation (eq. 2), this behaviour is captured both directly (increase in ϱ) and indirectly, by βT decreasing from the γ to the ε phase. Although measurements of Ohta et al. (2016) confirm an increase of ρel upon melting up to 51 GPa within large uncertainties, their absolute values suggest ρel to increase as a function of P for the γ-phase along Tm. This is neither consistent with an expected decrease of ρel ∝ ϱ−2… − 3, nor in quantitative agreement with KG results. Resistivities computed with the Bloch-Grüneisen/resistivity saturation model of Ohta et al. (2016) for ε-Fe at P > 100 GPa, where ε-Fe coexists with the liquid along the melting curve, show the expected decrease of $$\rho _{{\rm el}}^{{\rm solid}}$$ with P within the uncertainties. Absolute values, however, are significantly smaller than KG results, estimates from Ziman's approximation and shock wave experiments (Bi et al.2002; Keeler 1971; Matassov 1977; Fig. 1). Gomi et al. (2013) reported ρel of ε-Fe based on DAC experiments at 300 K. Although their model also takes resistivity saturation into account, their predicted value at core mantle boundary pressure plots significantly higher than those of Ohta et al. (2016) and is consistent with our model. The resistivity contrast across the ε-liquid phase boundary decreases gradually towards higher P. For ICB pressure and Tm = 6382 K (Komabayashi 2014), Ziman's approximation yields a value of 58 ± 2 μΩ cm for solid ε-iron, which is only marginally different from the 62 ± 2 μΩ cm in the liquid phase (increase on melting by 7 per cent). At comparable T (6350 K), Pozzo et al. (2014) computed solid resistivity values slightly lower than our result for pure iron (57 μΩ cm), which—in combination with their value for liquid Fe of 64 μΩ cm (Pozzo et al.2012)—yields a discontinuity of 12 per cent. As no reliable thermodynamic model for Co is currently available, we are not able to test whether the recently reported resistivity data by Ezenwa & Secco (2017b) follow the relation of eq. (5) at high P as well as it does at ambient pressure (Supporting Information Table S4). 3.3 Influence of light element partitioning With the addition of light element impurities, such as Si, O, S or C, resistivity will increase with impurity concentration. Although Matthiessen's rule will be violated close to saturation resistivity (Gomi et al.2016), this general behaviour continues to hold for compositions in the Fe–O–Si system for both the liquid (de Koker et al.2012; Pozzo et al.2013) and the solid (Pozzo et al.2014). Since light elements can dissolve in higher concentration in the liquid than in the solid and lead to a depression of the liquidus T in the binary system (Anderson & Ahrens 1994; Alfè et al.2002), our method cannot be easily applied to the Earth's core. We can, however, compare the estimated $$\rho _{{\rm el}}^{{\rm solid}}$$ of pure Fe and $$\rho _{{\rm el}}^{{\rm liquid}}$$ of selected alloys. As a result of light element segregation into the liquid OC, the resistivity contrast is significantly enhanced compared to pure iron. When we combine our estimate for ρel of solid Fe at the ICB pressure of ∼58 μΩ cm with that of liquid Fe0.82Si0.10O0.08 of 79 μΩ cm (Pozzo et al.2014) or 75 μΩ cm for liquid Fe7Si (de Koker et al.2012), resistivity in the OC would be larger than in the IC by 29-36 per cent. Although the influence of IC resistivity on the frequency of global field reversals is controversial as we discuss in the Introduction, a change of ρel at the ICB provides an important constraint on the boundary conditions of dynamo simulations. By using the discontinuity of ρel, its possible effect on the magnetic field could be explored for terrestrial planets up to the pressure of the Earth's IC. It is worth noting that the applicability of the model is neither limited by the size of the IC nor the P at the ICB, which makes it viable for models that include a growing IC. 4 CONCLUSIONS Having analysed data for isothermal compressibility and electrical resistivity at ambient P, we find that the increase of electrical resistivity of iron upon melting can be represented by a change of ϱ and S(q) in the long-wavelength limit to first order. High P experiments (Secco & Schlössin 1989; Deng et al.2013; Ohta et al.2016) and the computational work of Pozzo et al. (2012, 2014) indicate that a change of electrical resistivity persists along and across the melting curve. Knowing five out of six quantities on both sides of the melting curve in Ziman's approximation (eq. 5), this observation allows for a first order estimate of the remaining quantity. We combine DFT-MD results for $$\rho _{{\rm el}}^{{\rm liquid}}$$ (de Koker et al.2012) and the compressibility/density ratio in eq. (4) from a thermodynamic model of Fe (Komabayashi 2014) to compute $$\rho _{{\rm el}}^{{\rm solid}}$$ along the melting curve of iron up to 360 GPa. For planetary cores, the difference in electrical resistivity is likely to be amplified by differences in chemical composition across an ICB, since light elements prefer to remain in solution in the OC. This difference might be as large as 36 per cent for the Earth's core. As long as the pressure of a growing IC does not cross a solid-solid phase boundary, $$\rho _{{\rm el}}^{{\rm solid}}$$ at the ICB remains approximately constant, as suggested by Stacey & Anderson (2001). Acknowledgements Data to produce the results of this manuscript are contained in this manuscript, the Supporting Information or in the corresponding references. This work was supported by the German Science Foundation (DFG) in the Focus Program Planetary Magnetism (SPP 1488, STE1105/10-1). We greatly appreciate helpful discussion on Ziman's theory with Vojtěch Vlček and Philipp Sperling and on implications for planetary dynamos with Johannes Wicht. Comments of anonymous reviewers helped to significantly improve the manuscript. REFERENCES Alfè D., Gillan M.J., Price G.D., 2002. Composition and temperature of the Earth's core constrained by combining ab initio calculations and seismic data, Earth planet Sci. Lett.  195 91– 98. https://doi.org/10.1016/S0012-821X(01)00568-4 Google Scholar CrossRef Search ADS   Anderson B.J. et al.  , 2011. The global magnetic field of Mercury from MESSENGER orbital observations, Science  333 1859– 1862. https://doi.org/10.1126/science.1211001 Google Scholar CrossRef Search ADS PubMed  Anderson W., Ahrens T., 1994. An equation of state for liquid iron and implications for the Earth's core, J. geophys. Res.  99 4273– 4284. https://doi.org/10.1029/93JB03158 Google Scholar CrossRef Search ADS   Ashcroft N.W., Lekner J., 1966. 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Electrical resistivity as a function of T for pure iron at high P as reported by Ohta et al. (2016). Thick vertical lines indicate Tm from the model of Komabayashi (2014). Dashed lines represent the linear fit in both liquid and solid, which has been used to extrapolate towards Tm and determine the discontinuity there. Figure S3. Fermi level from DFT-MD simulations of liquid Fe as a function of (V/V0)−2/3. The offset at zero density is due to the ambiguity of the energy zero in DFT-MD. Figure S4.S(q) of liquid Fe in the long wavelength region at ambient P and three different T, measured by Waseda & Ohtani (1974). The solid lines represent a fit to the data up to 1.5 Å−1 with a Percus-Yevick expression of the structure factor of a hard sphere liquid. Figure S5. Isothermal bulk modulus KT as a function of T. Red circles have been obtained by fitting a Percus-Yevick expression of the hard-sphere structure factor up to 1.5 Å−1 to data by Waseda & Ohtani (1974), and extrapolating to the thermodynamic limit limq → 0S(q) = ϱkBT/KT (Figure 4). They agree well with the model of Komabayashi (2014) (red line) within the error. Considerable uncertainties due to the extrapolation towards q → 0 propagate to a large uncertainty for KT at Tm. However, the S(q)-based data provide strong support for a thermal expansivity α of ∼0.9 · 10−4 K−1, as long as two physical constraints are met: (a) ∂KT/∂T < 0 in the liquid and (b) $$K_T^{\rm liquid}<K_T^{\rm solid}$$ (black square) at Tm. The other red symbols refer to values of $$\beta _T^{\rm liquid}$$ (Table 1 in the paper), obtained by different datasets for density ϱ and thermal expansivity α of liquid Fe at Tm and ambient P. Figure S6. Relative deviations [%] of densities from shock wave experiments (Brown & McQueen 1986; Brown et al. 2000) from the liquid Fe model of Komabayashi (2014). For P larger than 260 GPa, the Hugoniot lies in the liquid stability field of iron. Table S1. Electrical resistivities of Fe along the melting curve based on extrapolation of high P data, as shown in Figures 1 and 2. Lower P values correspond to data of Secco & Schlössin (1989), higher P values to data of Ohta et al. (2016). Uncertainties have been determined from errors of the linear fit towards Tm. Table S2. Single crystal electrical resistivity of the group 8 elements Ru (Volkenshteyn et al. 1978) and Os (Schriempf 1968) at room temperature, Hashin-Shtrikman bounds (Hashin & Shtrikman 1963) for their polycrystal resistivity and maximum relative error due to anisotropy. Table S3. Thermophysical properties of simple metals at Tm. The resistivity ratio is in good agreement with the long-wavelength approximation to Ziman's formula (equation 5). References are labeled as follows: Sobolev (2011)a, Faber (1972)b, Fritsch et al. (1973)c, Blairs (2006)d, Peng et al. (2015)e, Mills (2002)f, Tallon & Wolfenden (1979)g. Table S4. Thermophysical properties of transition metals with partially occupied d −bands at Tm. The resistivity ratio is in good agreement with the long-wavelength approximation to Ziman's formula (equation 5). References are labeled as follows: Faber (1972)b, Hess et al. (1994)h, Tsu et al. (1985)i, Schramm (1962)j, Brillo & Egry (2003)k, Owen & Yates (1936)l, Desai (1987)m, Alers et al. (1960)n. Table S5. Thermophysical properties of metals with fully occupied d-bands at Tm. The resistivity ratio is systematically underestimated by Ziman's long-wavelength approximation (equation 5). References are labeled as follows: Faber (1972)b, Blairs (2006)d, Owen & Yates (1934)l, Chang & Himmel (1966)o, Kanai & Tsuchiya (1993)p, Grønvold & Stølen (2003)q, Ledbetter (1977)r. Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the paper. © The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

# Electrical resistivity discontinuity of iron along the melting curve

, Volume 213 (1) – Apr 1, 2018
7 pages

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The Royal Astronomical Society
ISSN
0956-540X
eISSN
1365-246X
D.O.I.
10.1093/gji/ggx526
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### Abstract

Summary Discontinuous changes of electrical resistivity ρel (increase), density ϱ and isothermal compressibility βT (decrease) occur across the melting temperature of metals and can be directly related by Ziman's theory in the long-wavelength approximation. By evaluating experimental data at ambient pressure, we show that Ziman's approximation holds for iron and other simple and transition metals. Using a thermodynamic model to determine βT for γ-, ε- and liquid Fe and a previously published model for ρel of liquid Fe, we apply Ziman's approximation to calculate ρel of solid Fe along the melting curve. For pure Fe, we find the discontinuity in ρel to decrease with pressure and to be negligibly small at inner core boundary conditions. However, if we account for light element enrichment in the liquid outer core, the electrical resistivity decrease across the inner core boundary is predicted to be as large as 36 per cent. Electrical properties, High pressure behaviour, Core 1 INTRODUCTION Electrical resistivity ρel of liquid metals and alloys under extreme conditions is a critical parameter for the stability and evolution of planetary dynamos. In particular, the magnetic fields of the Earth, Mercury and Ganymede are thought to be generated by a self-sustained dynamo in the convecting liquid portions of their cores, which are composed of mainly iron and a variety of lighter elements (Merrill et al.1996; Sarson et al.1997; Anderson et al.2011). In that context, the role of the inner core (IC) in magnetic field generation remains a subject of controversy (Olson & Amit 2014). Initial geodynamic studies (Hollerbach & Jones 1993; Glatzmaier & Roberts 1995) reported evidence that finite electrical resistivity in the Earth's solid IC has a stabilizing effect on the magnetic field and therefore leads to a decrease in the frequency of global field reversals. Subsequently, however, there have been conflicting results regarding this argument, ranging from support of the initial inference (Dharmaraj & Stanley 2012), no influence (Wicht 2002), to an increased number of excursion in the presence of a conducting IC when compared to results with an insulator at the centre of the planet (Lhuillier et al.2013). For conducting ICs, these studies have relied on conductivity values of the IC being the same as for the outer core (OC), while one would expect that resistivity changes across the IC boundary (ICB) for two reasons: (i) The structure of solid and liquid are quite different and resistivity changes across the melting temperature (Tm) and (ii) light elements will not equally partition between the OC and the IC. In the current manuscript we look at these two effects with the goal to provide values of ρel for use in geodynamo simulations that may help to better assess the effect of the IC on magnetic field generation. In general, metals exhibit a distinct increase in electrical resistivity upon melting. For iron, it increases by ∼8 per cent at ambient pressure (P; van Zytveld 1980; Table 1), a value that remains approximately constant up to 6 GPa as determined in large volume press experiments (Secco & Schlössin 1989; see Supporting Information Fig. S1). In similar experiments, Ezenwa & Secco (2017b) recently determined a jump of similar magnitude for another transition metal, Co, while for the closed d-shell metals, Zn and Cu, the discontinuity is significantly larger (Ezenwa & Secco 2017a; Ezenwa et al.2017). These data are of very high quality and determine the jump across Tm with great precision. At higher P, static experiments using the diamond anvil cell (DAC) are more scarce and the small sample size and potential temperature (T) gradients inside a DAC make high P–T experiments challenging, especially close to Tm (Dobson 2016), resulting in large uncertainties of the data and consequently the determination of a potential resistivity discontinuity. Ohta et al. (2016) recently reported T-dependent resistivities for the solid and the liquid phase of iron up to 51 GPa measured in the DAC, and they describe an increase by as much as ∼20 per cent across melting (see Supporting Information Fig. S2). At higher P, Ohta et al. (2016) relied on resistivity measurements for the solid hcp (ε) phase of Fe only, and in order to estimate ρel for the liquid, they used a Bloch–Grüneisen fit—taking resistivity saturation into account—to their data to extrapolate to Tm and added 20 per cent to account for the increase of ρel upon melting in an ad-hoc fashion. Gomi et al. (2013) performed measurements of ρel for hcp Fe up to 100 GPa at room T only, and argue along the theory of Mott (1972) that iron at core conditions is close to saturation resistivity and therefore the increase of ρel on melting should be negligible. In the development of their model resistivity and its application to the Earth's core, Gomi et al. (2013) do not distinguish between the solid and liquid phase. Table 1. Thermophysical data for pure iron at ambient pressure at the melting point (Tm = 1808 ± 5 K)a. Resistivities have been measured by Cezairliyan & McClure (1974)a and van Zytveld (1980)b. If not reported directly, values measured by Dever (1972)c, Basinski et al. (1955)d, Tsu et al. (1985)e, Drotning (1981)f and compiled by Desai (1986)g have been extrapolated to Tm while using the misfit as uncertainty. For the liquid phase, a scenario including values for ϱ and α from Assael et al. (2006)h has been tabulated for comparison. A column of thermodynamic properties for liquid Fe from the model of Komabayashi (2014)i has been included. Compressibilities βS and βT have been calculated using eqs (6) and (7).   δ-Fe  Liquid Fe      Drotning  Assael  Thermodynamic modeli  ρel [μΩ cm]  127.0 ± 0.6a  137.6 ± 1.0b  vp [km s−1]  4.73 ± 0.07c  3.98 ± 0.03e  3.98  vs [km s−1]  2.49 ± 0.06c        ϱ [g cm−3]  7.29 ± 0.02d  7.017 ± 0.002f  7.04 ± 0.06h  7.09  α [10−5 K−1]  6.5 ± 0.1d  8.8 ± 0.1f  13.2 ± 0.1h  9.0  cp [J g−1K−1]  0.80 ± 0.06a  0.84 ± 0.05g  0.82  βS [GPa−1]  1/(103 ± 6)  1/(111 ± 2)  1/(112 ± 2)  1/112  βT [GPa−1]  1/(91 ± 5)  1/(88 ± 2)  1/(70 ± 2)  1/87  $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}}\left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^2\left(\frac{\beta _T^{{\rm liquid}}}{\beta _T^{{\rm solid}}}\right)^{-1}$$    0.97 ± 0.05  0.78 ± 0.05  0.98 ± 0.05    δ-Fe  Liquid Fe      Drotning  Assael  Thermodynamic modeli  ρel [μΩ cm]  127.0 ± 0.6a  137.6 ± 1.0b  vp [km s−1]  4.73 ± 0.07c  3.98 ± 0.03e  3.98  vs [km s−1]  2.49 ± 0.06c        ϱ [g cm−3]  7.29 ± 0.02d  7.017 ± 0.002f  7.04 ± 0.06h  7.09  α [10−5 K−1]  6.5 ± 0.1d  8.8 ± 0.1f  13.2 ± 0.1h  9.0  cp [J g−1K−1]  0.80 ± 0.06a  0.84 ± 0.05g  0.82  βS [GPa−1]  1/(103 ± 6)  1/(111 ± 2)  1/(112 ± 2)  1/112  βT [GPa−1]  1/(91 ± 5)  1/(88 ± 2)  1/(70 ± 2)  1/87  $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}}\left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^2\left(\frac{\beta _T^{{\rm liquid}}}{\beta _T^{{\rm solid}}}\right)^{-1}$$    0.97 ± 0.05  0.78 ± 0.05  0.98 ± 0.05  View Large Complementary to experiments, electronic transport properties at high P have been calculated by evaluating the Kubo–Greenwood (KG) formula for the Onsager kinetic coefficients on results of density functional theory (DFT)-based molecular dynamics (MD) simulations (Vlček et al.2012; de Koker et al.2012; Pozzo et al.2012, 2013, 2014), a computationally expensive approach. For pure liquid iron at conditions of the Earth's ICB, these studies agree on values of resistivity slightly above 60 μΩ cm (de Koker et al.2012; Pozzo et al.2012), while for ε-Fe at similar densities, T-dependent values of $$\rho _{{\rm el}}^{{\rm solid}}=53$$–57 μΩ cm have been reported (Pozzo et al.2014). These results suggest that a change in ρel on melting—albeit small—persists to core conditions. The determination of electronic transport properties of metals under extreme P and T is challenging both experimentally and computationally, and remains a topic of a controversial discussion. In this study, we address the question of electrical resistivity of solid Fe close to Tm along the melting curve by a different approach: We use Ziman's theory to relate electronic and elastic properties upon melting, which we test on experimental data of Fe at ambient pressure. We then apply this method along the melting curve and infer resistivities of the solid phase at high P based on previously published values for the liquid and discuss the influence of light element alloying and partitioning at crystallization of the IC, as it is relevant for cores of terrestrial planetary bodies. 2 METHODS The Ziman formula for electrical resistivity (Ziman 1961)   $$\rho _{{\rm el}}=\frac{a_0\hbar }{e^2}\frac{4\pi ^3Z}{a_0k_F}\frac{1}{(2k_F)^4}\int \limits _{0}^{2k_F}S(q)|U(q)|^2q^3\mathrm{d}q$$ (1)has been widely applied to determine ρel for metallic liquids up to the warm dense matter regime (e.g. Burrill et al.2016). In this model, quasi-free conduction electrons of momentum ℏk scatter off screened ionic potentials U(q), where q = |k − k΄| is the scattering wavenumber. The spatial arrangement of scattering centres is described by the static ion–ion structure factor S(q) in reciprocal space and incorporates an implicit dependence on density ϱ and T. The pre-factor $$\frac{a_0\hbar }{e^2}\approx 21.74$$ μΩ cm is the atomic unit of resistivity (with a0 being the Bohr radius, ℏ the reduced Planck constant and e the elementary charge), Z the number of valence electrons and kF the Fermi wavenumber. Due to conservation of momentum, scattering takes place from and into states at the Fermi level. Therefore, the largest possible change of momentum upon an elastic collision is 2kF for a backscattered electron (k → k΄ = −k) which determines the upper integration boundary. In the derivation of eq. (1), several approximations have been made. (i) The Fermi surface is assumed to be isotropic. For liquids and amorphous solids, this is always the case. For many crystalline solids it is a good approximation as has been shown for bcc and fcc metals (Papaconstantopoulos 1986). In addition, if one is interested in resistivity of a polycrystal, the Hashin–Shtrikman (HS) bounds (Hashin & Shtrikman 1963) for ρel are very narrow for the group 8 elements Ru and Os, that crystallize in the hcp phase at ambient P (Volkenshteyn et al.1978; Schriempf 1968) (see Supporting Information Table S2). (ii) Higher frequency (ω) moments of S(q, ω) have been omitted, since they are dominated by interionic contributions which are generally very small (Cheung & Ashcroft 1978). Furthermore, for T above the Debye temperature (θD), the ω-dependence due to the distribution function $$\frac{\hbar \omega }{k_BT}/(e^{\frac{\hbar \omega }{k_BT}}-1)=\frac{\hbar \omega }{k_BT}/[(1+\frac{\hbar \omega }{k_BT}+\ldots )-1]\approx 1$$ (with the Boltzmann constant kB) can be neglected. (iii) The spread of the Fermi–Dirac distribution (kBT) is larger than the maximum energy transfer by phonons (kBθD) at these conditions. Therefore, it is reasonable to treat electron scattering in solids at high T quasi-elastically. By means of eq. (1), Ashcroft & Lekner (1966) calculated resistivities for a number of liquid metals by using different model potentials and an analytical expression for the hard sphere structure factor Shs(q) (Wertheim 1963). Their results agree reasonably well with experimental values at ambient P. Taking ϱ-dependence of Shs(q) into account, the Ziman formula has been applied to P of the Earth's OC by Jain & Evans (1971), who constrained the resistivity of liquid iron at the core–mantle boundary to be 104 ± 6 μΩ cm at 3473 K. The KG approach yields values of about two thirds of this number at comparable P and T, which may indicate a breakdown of the hard sphere approximation at high densities. As the ionic potential U(q) does not change across Tm, the increase of ρel upon melting occurs due to the transition from a crystalline into a disordered structure, which is reflected by changes in S(q) and ϱ. As a first order approximation, Ziman (1961) proposed to replace the integral in eq. (1) by the integrand's value at q = 0. While S(q) contains implicit information on density, other factors in eq. (1) depend on ϱ explicitly: U by its normalization to −2/3· EF at q = 0 (EF being the Fermi energy), and the Fermi radius kF. For the free electron gas, they scale as EF ∝ ϱ2/3 and kF ∝ ϱ1/3, respectively. While Fe is not a free electron metal, its Fermi level shows the same dependence on volume (Supporting Information Fig. S3). Combining $$k_F^{-5}$$ in the pre-factor with $$E_F^{-2}$$ from the integrand, resistivity decreases ∝ϱ−3. If one considers the ratio of resistivities in the liquid and the solid phase at Tm, in the Ziman approximation the following relation holds:   $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}}\simeq \frac{S(0)^{{\rm liquid}}}{S(0)^{{\rm solid}}}\left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^{-3}.$$ (2) On the other hand, the distribution of atoms in condensed matter determines bulk elastic parameters. Since the thermodynamic limit   $$\lim _{q\rightarrow 0}S(q)=\varrho k_BT\beta _T$$ (3)is known from fluctuation theory (March 1990), the structure factor ratio in eq. (2) can be reformulated in terms of isothermal compressibilities, thermodynamically defined as βT = −ϱ∂(1/ϱ)/∂P, and Ziman's approximation becomes   $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}}\simeq \frac{\beta _T^{{\rm liquid}}}{\beta _T^{{\rm solid}}}\left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^{-2}$$ (4)or   $$\frac{\rho _{{\rm el}}^{{\rm liquid}}}{\rho _{{\rm el}}^{{\rm solid}}} \left(\frac{\varrho ^{{\rm liquid}}}{\varrho ^{{\rm solid}}}\right)^2 \left(\frac{\beta _T^{{\rm liquid}}}{\beta _T^{{\rm solid}}}\right)^{-1}\simeq 1.$$ (5) Jain & Evans (1971) inserted a model for ϱ, T and βT of the Earth into eq. (3), which fully defines the variation of Shs(q) with P and T. While we do not calculate absolute resistivity values, we use the long-wavelength approximation to eq. (1) as a method to estimate the relative increase of ρel from $$\rho _{{\rm el}}^{{\rm liquid}}$$ to $$\rho _{{\rm el}}^{{\rm solid}}$$ across the melting curve based on thermodynamic parameters. The application of eq. (4) is not limited to this direction. High P experiments such as those performed by Gomi et al. (2013) or Ohta et al. (2016) for the solid could make use of it to convert $$\rho _{{\rm el}}^{{\rm solid}}$$ to $$\rho _{{\rm el}}^{{\rm liquid}}$$ or cross-check their data. 3 RESULTS AND DISCUSSION 3.1 Ambient pressure We assess the validity of relation (5) for iron at ambient P by compiling experimental data for ρel, ϱ and βT right above and below Tm. Electrical resistivity of iron in the solid state has been measured up to 1800 K (Cezairliyan & McClure 1974) and for the liquid starting at 1808 K (van Zytveld 1980), with a ratio of $$\rho _{{\rm el}}^{{\rm liquid}}/\rho _{{\rm el}}^{{\rm solid}}=1.08\pm 0.01$$ (Table 1). To obtain compressibilities, we rely on ultrasonic measurements of the longitudinal and transverse acoustic velocities (vp and vs, respectively), which are related to isentropic compressibility βS via   $$v_p^2-\frac{4}{3}v_s^2=\frac{1}{\varrho \beta _S},$$ (6)with vs = 0 for the liquid. To convert βS to βT, one has to apply the thermodynamic relation   $$\beta _T=\beta _S+\frac{\alpha ^2T}{\varrho c_p},$$ (7)with cp being the heat capacity at constant P and α the coefficient of thermal expansion. If not directly reported, we extrapolated the thermophysical quantities in eq. (7) to Tm (see Table 1). To the best of our knowledge, no data exist on acoustic velocities for the δ-phase of iron (in the bcc structure) which is in equilibrium with the liquid at ambient P. Instead, we use single crystal elastic constants of the α-phase (also in the bcc structure) from Dever (1972) above 1050 K, calculate vp and vs from a Voigt–Reuss–Hill polycrystalline average and linearly extrapolate to Tm (Table 1). This approach is well justified as any magnetic contribution to βT vanishes above the Curie temperature (1043 K), and the α- and δ-phase stability region can be viewed as belonging to the same stability field that is connected at negative pressure (Komabayashi & Fei 2010). In order to apply eq. (7), we use values for ϱ and α from the X-ray diffraction study of Basinski et al. (1955) and cP from Cezairliyan & McClure (1974). Thermodynamic parameters for liquid iron are more controversial. Measured sound velocities range from 3820 to 4052 m s−1 (Nasch et al.1994; Casas et al.1984) and ϱ from 6937 to 7120 kg m−3 (Blumm & Henderson 2000; Hixson et al.1990). For vP we choose the measurements by Tsu et al. (1985) (Table 1) that are most consistent with values computed with the thermodynamic model of Komabayashi (2014). Density values are closely associated with those of thermal expansivity α which has the strongest influence on the determination of βT from βS, as it enters in quadratic form in eq. (7). As for the solid phase, values of ϱ and α should be chosen consistently, i.e. come from the same underlying data of ϱ(T). Thermal expansivity of liquid iron has been the topic of a long-standing controversy (Williams 2009) with either values below 0.9 × 10−4 K−1 (e.g. Drotning 1981; Nasch & Steinemann 1995; Blumm & Henderson 2000) or larger than 1.1 × 10−4 K−1 (e.g. Kirshenbaum & Cahill 1962; Saito et al.1969; Hixson et al.1990). With the exception of the dilatometer work of Blumm & Henderson (2000), the former data stem from γ-ray attenuation studies and the latter from constant mass setups (e.g. Archimedean, maximum bubble pressure, levitation techniques). Following Assael et al. (2006), Williams (2009) argued for the larger α on the basis of potential systematic errors due to the sample geometry in γ-ray attenuation. However, constant mass methods may underestimate effects of wetting, surface tension and viscosity (Drotning 1981; Nasch et al.1994), which could also account for the discrepancy. Using the values recommended by Assael et al. (2006), βT = 1/(70 ± 2) GPa−1, with ϱ and α from Drotning (1981), βT = 1/(88 ± 2) GPa−1. Only for the βT value based on ϱ and α of Drotning (1981), the Ziman approximation (eq. 5) holds (Table 1). The thermodynamic limit of S(q) (eq. 3) provides an alternative route to the determination of βT independent of α. We have fitted measured S(q) of liquid iron (Waseda & Ohtani 1974) by means of a Percus–Yevick hard-sphere expression in the long wavelength limit (see Supporting Information Figs S4 and S5) and find, despite considerable uncertainty, a compressibility value that is consistent with the acoustic-velocity-based value using ϱ and α of Drotning (1981), supporting the validity of Ziman's approximation for iron at ambient P. Model parameters for liquid iron from Komabayashi (2014) also agree well with the S(q)-based βT and satisfy eq. (5) in combination with the experimental data of δ-Fe (Table 1). A similar evaluation of eq. (5) for other metallic elements can be found in the Supporting Information (Tables S3– S5). We find that the resistivity ratio is well represented by the right hand side of eq. (4) for the simple metals Na and Al, and the 3d transition metals Co and Ni. For Zn and the noble metals Cu, Ag and Au—all metals with closed d-shell—it is systematically underestimated by a factor of ∼1.5, that is, the increase in ρel across melting (left-hand side of eq. 4) is significantly larger than the ratio of thermodynamic properties (right-hand side of eq. 4). For Co, Cu and Zn, the resistivity jump at ambient P is consistent with the experiments up to 5 GPa (Ezenwa & Secco 2017a,b; Ezenwa et al.2017). 3.2 High pressure In order to obtain the right hand side of eq. (4) at high P for Fe, we compute ϱ and βT on both the liquid and the solid side of the melting curve with the model of Komabayashi (2014) (Fig. 1a). For the liquid, the model yields a value for ϱliquid which is consistent with experimental work by Tateyama et al. (2011) at 4.3 GPa and shock experiments up to ∼440 GPa (Brown & McQueen 1986; Brown et al.2000) (see Supporting Information Fig. S6). The thermodynamic model by Komabayashi (2014) has been designed to reproduce results from DAC and multi-anvil experiments for the solid phases. The applicability of this model in the geophysical context is further supported when comparing its ratio of adiabatic compressibilities at the ICB ($$\beta _S^{{\rm liquid}}/\beta _S^{{\rm solid}}=1.02$$) with that of PREM ($$\beta _S^{{\rm liquid}}/\beta _S^{{\rm solid}}=1.03$$) (Dziewonski & Anderson 1981). Figure 1. View largeDownload slide (a) Liquid to solid compressibility/density ratio for Fe along the melting curve from the thermodynamic model of Komabayashi (2014) used in Ziman's approximation (right-hand side of eq. 4). The lower x-axis label and ticks show pressure, the upper ones the corresponding melting temperature. Pressure intervals indicated correspond to core P in the terrestrial bodies of our solar system (Ganymede, Mercury, Mars, Venus and Earth). The vertical line represents the γ-ε-liquid triple point in the model of Komabayashi (2014; 96 GPa and 3300 K). (b) Electrical resistivities in the liquid (red curve) and the solid (black curve) phases of Fe along the melting curve. For the liquid phase, the modified Bloch–Grüneisen fit from de Koker et al. (2012) has been evaluated along the melting curve, while the resistivity in the solid has been calculated using the ratios shown in panel (a) and applying eq. (4). The band widths take fitting uncertainties of the original Kubo-Greenwood results by de Koker et al. (2012) into account. Laboratory data (filled symbols) are by Secco & Schlössin (1989) (S89) and Ohta et al. (2016) (O16) from static experiments, and by Bi et al. (2002) (B02), Keeler (1971) (K71) and Matassov (1977) (M77) from shock wave experiments. For $$\rho _{{\rm el}}^{{\rm liquid}}$$ and $$\rho _{{\rm el}}^{{\rm solid}}$$, data by Secco & Schlössin (1989) and Ohta et al. (2016) up to 51 GPa have been fitted linearly in the liquid and solid regions, respectively, and extrapolated towards Tm from both sides (see Supporting Information Figs S1 and S2). For the shock wave experiments, the lowest P point by Bi et al. (2002) and the highest P point each by Keeler (1971) and Matassov (1977) have been used, all for the solid phase. Temperatures along the Hugoniot at these pressures (Brown & McQueen 1986) are significantly below the melting point. Open circles in the P-range between 100 and 160 GPa show values calculated from combined Bloch–Grüneisen/resistivity-saturation fit parameters given in Ohta et al. (2016) and evaluated at Tm of Komabayashi (2014). G13 (open diamond) represents the high T extrapolation of a room temperature DAC experiment reported in Gomi et al. (2013), also taking resistivity saturation into account. At inner core boundary P, the Kubo–Greenwood results by Pozzo et al. (2012, 2014) (P12 and P14) are included for liquid and solid Fe, respectively. (c) Negative logarithmic derivative −∂(ln ρel)/∂P for liquid and solid iron along its melting curve. While the P-gradient is significant in the liquid, it is negligible in the solid, particularly for ε-Fe. Figure 1. View largeDownload slide (a) Liquid to solid compressibility/density ratio for Fe along the melting curve from the thermodynamic model of Komabayashi (2014) used in Ziman's approximation (right-hand side of eq. 4). The lower x-axis label and ticks show pressure, the upper ones the corresponding melting temperature. Pressure intervals indicated correspond to core P in the terrestrial bodies of our solar system (Ganymede, Mercury, Mars, Venus and Earth). The vertical line represents the γ-ε-liquid triple point in the model of Komabayashi (2014; 96 GPa and 3300 K). (b) Electrical resistivities in the liquid (red curve) and the solid (black curve) phases of Fe along the melting curve. For the liquid phase, the modified Bloch–Grüneisen fit from de Koker et al. (2012) has been evaluated along the melting curve, while the resistivity in the solid has been calculated using the ratios shown in panel (a) and applying eq. (4). The band widths take fitting uncertainties of the original Kubo-Greenwood results by de Koker et al. (2012) into account. Laboratory data (filled symbols) are by Secco & Schlössin (1989) (S89) and Ohta et al. (2016) (O16) from static experiments, and by Bi et al. (2002) (B02), Keeler (1971) (K71) and Matassov (1977) (M77) from shock wave experiments. For $$\rho _{{\rm el}}^{{\rm liquid}}$$ and $$\rho _{{\rm el}}^{{\rm solid}}$$, data by Secco & Schlössin (1989) and Ohta et al. (2016) up to 51 GPa have been fitted linearly in the liquid and solid regions, respectively, and extrapolated towards Tm from both sides (see Supporting Information Figs S1 and S2). For the shock wave experiments, the lowest P point by Bi et al. (2002) and the highest P point each by Keeler (1971) and Matassov (1977) have been used, all for the solid phase. Temperatures along the Hugoniot at these pressures (Brown & McQueen 1986) are significantly below the melting point. Open circles in the P-range between 100 and 160 GPa show values calculated from combined Bloch–Grüneisen/resistivity-saturation fit parameters given in Ohta et al. (2016) and evaluated at Tm of Komabayashi (2014). G13 (open diamond) represents the high T extrapolation of a room temperature DAC experiment reported in Gomi et al. (2013), also taking resistivity saturation into account. At inner core boundary P, the Kubo–Greenwood results by Pozzo et al. (2012, 2014) (P12 and P14) are included for liquid and solid Fe, respectively. (c) Negative logarithmic derivative −∂(ln ρel)/∂P for liquid and solid iron along its melting curve. While the P-gradient is significant in the liquid, it is negligible in the solid, particularly for ε-Fe. Fig. 1(b) shows both $$\rho _{{\rm el}}^{{\rm liquid}}$$ and $$\rho _{{\rm el}}^{{\rm solid}}$$ for pure Fe as a function of P, covering conditions up to 360 GPa, the pressure in the Earth's centre. Resistivities in the liquid phase have been calculated from the modified Bloch–Grüneisen model by de Koker et al. (2012) and $$\rho _{{\rm el}}^{{\rm solid}}$$ has been computed by means of eq. (5). We predict $$\rho _{{\rm el}}^{{\rm solid}}$$ for iron in the range of 69–71 μΩ cm at conditions of the cores of Mars, Mercury and Ganymede, while $$\rho _{{\rm el}}^{{\rm liquid}}\approx 78-88$$ μΩ cm. The KG results by de Koker et al. (2012) underestimate $$\rho _{{\rm el}}^{{\rm liquid}}$$ at ambient conditions and low P compared to experiments, and this mismatch suggests that $$\rho _{{\rm el}}^{{\rm solid}}$$ should also be considered with caution there. Two effects contribute to this discrepancy. (i) The underlying equation of state overestimates density at ambient P by ∼20 per cent, similar to the small volume at zero pressure reported for liquid iron from DFT–MD simulations by Ichikawa et al. (2014). Smaller volumes lead to reduced resistivity values. (ii) Recent results of Drchal et al. (2017) indicate that there is a possible contribution of spin disorder to ρel at high T, which has not been taken into account by previous computational studies. At higher P, however, resistivity values of de Koker et al. (2012) are in good agreement with shock wave data (Keeler 1971; Matassov 1977; Bi et al.2002). Stacey & Anderson (2001) have argued that ρel remains constant along the melting curve. In their derivation of ∂(ln ρel)/∂P = 0, they express the Grüneisen parameter in terms of an average lattice frequency, and they do not distinguish between the liquid and the solid phase. Since the melting point defines an extreme case of anharmonicity, it is not clear to what degree their conclusion is applicable to the liquid phase. Indeed, in contrast to a relatively strong change of −∂(ln ρel)/∂P along the melting curve for liquid Fe (Fig. 1c), the corresponding slopes for both the fcc (<10−3 GPa−1) and hcp phases (<4 × 10−4 GPa−1) are small, supporting the hypothesis of Stacey & Anderson (2001) to first order for the solid phases. An interesting feature of the model is the predicted decrease of $$\rho _{{\rm el}}^{{\rm solid}}$$ at the γ-ε-liquid triple point. Resistivity generally decreases with increasing charge carrier density: ρel scales with ϱ between ∝ ϱ−3 (eq. 2) and ∝ ϱ−2 (eq. 4), depending on the implicit ϱ-dependence of S(q). In Ziman's long wavelength approximation (eq. 2), this behaviour is captured both directly (increase in ϱ) and indirectly, by βT decreasing from the γ to the ε phase. Although measurements of Ohta et al. (2016) confirm an increase of ρel upon melting up to 51 GPa within large uncertainties, their absolute values suggest ρel to increase as a function of P for the γ-phase along Tm. This is neither consistent with an expected decrease of ρel ∝ ϱ−2… − 3, nor in quantitative agreement with KG results. Resistivities computed with the Bloch-Grüneisen/resistivity saturation model of Ohta et al. (2016) for ε-Fe at P > 100 GPa, where ε-Fe coexists with the liquid along the melting curve, show the expected decrease of $$\rho _{{\rm el}}^{{\rm solid}}$$ with P within the uncertainties. Absolute values, however, are significantly smaller than KG results, estimates from Ziman's approximation and shock wave experiments (Bi et al.2002; Keeler 1971; Matassov 1977; Fig. 1). Gomi et al. (2013) reported ρel of ε-Fe based on DAC experiments at 300 K. Although their model also takes resistivity saturation into account, their predicted value at core mantle boundary pressure plots significantly higher than those of Ohta et al. (2016) and is consistent with our model. The resistivity contrast across the ε-liquid phase boundary decreases gradually towards higher P. For ICB pressure and Tm = 6382 K (Komabayashi 2014), Ziman's approximation yields a value of 58 ± 2 μΩ cm for solid ε-iron, which is only marginally different from the 62 ± 2 μΩ cm in the liquid phase (increase on melting by 7 per cent). At comparable T (6350 K), Pozzo et al. (2014) computed solid resistivity values slightly lower than our result for pure iron (57 μΩ cm), which—in combination with their value for liquid Fe of 64 μΩ cm (Pozzo et al.2012)—yields a discontinuity of 12 per cent. As no reliable thermodynamic model for Co is currently available, we are not able to test whether the recently reported resistivity data by Ezenwa & Secco (2017b) follow the relation of eq. (5) at high P as well as it does at ambient pressure (Supporting Information Table S4). 3.3 Influence of light element partitioning With the addition of light element impurities, such as Si, O, S or C, resistivity will increase with impurity concentration. Although Matthiessen's rule will be violated close to saturation resistivity (Gomi et al.2016), this general behaviour continues to hold for compositions in the Fe–O–Si system for both the liquid (de Koker et al.2012; Pozzo et al.2013) and the solid (Pozzo et al.2014). Since light elements can dissolve in higher concentration in the liquid than in the solid and lead to a depression of the liquidus T in the binary system (Anderson & Ahrens 1994; Alfè et al.2002), our method cannot be easily applied to the Earth's core. We can, however, compare the estimated $$\rho _{{\rm el}}^{{\rm solid}}$$ of pure Fe and $$\rho _{{\rm el}}^{{\rm liquid}}$$ of selected alloys. As a result of light element segregation into the liquid OC, the resistivity contrast is significantly enhanced compared to pure iron. When we combine our estimate for ρel of solid Fe at the ICB pressure of ∼58 μΩ cm with that of liquid Fe0.82Si0.10O0.08 of 79 μΩ cm (Pozzo et al.2014) or 75 μΩ cm for liquid Fe7Si (de Koker et al.2012), resistivity in the OC would be larger than in the IC by 29-36 per cent. Although the influence of IC resistivity on the frequency of global field reversals is controversial as we discuss in the Introduction, a change of ρel at the ICB provides an important constraint on the boundary conditions of dynamo simulations. By using the discontinuity of ρel, its possible effect on the magnetic field could be explored for terrestrial planets up to the pressure of the Earth's IC. It is worth noting that the applicability of the model is neither limited by the size of the IC nor the P at the ICB, which makes it viable for models that include a growing IC. 4 CONCLUSIONS Having analysed data for isothermal compressibility and electrical resistivity at ambient P, we find that the increase of electrical resistivity of iron upon melting can be represented by a change of ϱ and S(q) in the long-wavelength limit to first order. High P experiments (Secco & Schlössin 1989; Deng et al.2013; Ohta et al.2016) and the computational work of Pozzo et al. (2012, 2014) indicate that a change of electrical resistivity persists along and across the melting curve. Knowing five out of six quantities on both sides of the melting curve in Ziman's approximation (eq. 5), this observation allows for a first order estimate of the remaining quantity. We combine DFT-MD results for $$\rho _{{\rm el}}^{{\rm liquid}}$$ (de Koker et al.2012) and the compressibility/density ratio in eq. (4) from a thermodynamic model of Fe (Komabayashi 2014) to compute $$\rho _{{\rm el}}^{{\rm solid}}$$ along the melting curve of iron up to 360 GPa. For planetary cores, the difference in electrical resistivity is likely to be amplified by differences in chemical composition across an ICB, since light elements prefer to remain in solution in the OC. This difference might be as large as 36 per cent for the Earth's core. As long as the pressure of a growing IC does not cross a solid-solid phase boundary, $$\rho _{{\rm el}}^{{\rm solid}}$$ at the ICB remains approximately constant, as suggested by Stacey & Anderson (2001). Acknowledgements Data to produce the results of this manuscript are contained in this manuscript, the Supporting Information or in the corresponding references. This work was supported by the German Science Foundation (DFG) in the Focus Program Planetary Magnetism (SPP 1488, STE1105/10-1). 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Electrical resistivity as a function of T for pure iron at high P as reported by Ohta et al. (2016). Thick vertical lines indicate Tm from the model of Komabayashi (2014). Dashed lines represent the linear fit in both liquid and solid, which has been used to extrapolate towards Tm and determine the discontinuity there. Figure S3. Fermi level from DFT-MD simulations of liquid Fe as a function of (V/V0)−2/3. The offset at zero density is due to the ambiguity of the energy zero in DFT-MD. Figure S4.S(q) of liquid Fe in the long wavelength region at ambient P and three different T, measured by Waseda & Ohtani (1974). The solid lines represent a fit to the data up to 1.5 Å−1 with a Percus-Yevick expression of the structure factor of a hard sphere liquid. Figure S5. Isothermal bulk modulus KT as a function of T. Red circles have been obtained by fitting a Percus-Yevick expression of the hard-sphere structure factor up to 1.5 Å−1 to data by Waseda & Ohtani (1974), and extrapolating to the thermodynamic limit limq → 0S(q) = ϱkBT/KT (Figure 4). They agree well with the model of Komabayashi (2014) (red line) within the error. Considerable uncertainties due to the extrapolation towards q → 0 propagate to a large uncertainty for KT at Tm. However, the S(q)-based data provide strong support for a thermal expansivity α of ∼0.9 · 10−4 K−1, as long as two physical constraints are met: (a) ∂KT/∂T < 0 in the liquid and (b) $$K_T^{\rm liquid}<K_T^{\rm solid}$$ (black square) at Tm. The other red symbols refer to values of $$\beta _T^{\rm liquid}$$ (Table 1 in the paper), obtained by different datasets for density ϱ and thermal expansivity α of liquid Fe at Tm and ambient P. Figure S6. Relative deviations [%] of densities from shock wave experiments (Brown & McQueen 1986; Brown et al. 2000) from the liquid Fe model of Komabayashi (2014). For P larger than 260 GPa, the Hugoniot lies in the liquid stability field of iron. Table S1. Electrical resistivities of Fe along the melting curve based on extrapolation of high P data, as shown in Figures 1 and 2. Lower P values correspond to data of Secco & Schlössin (1989), higher P values to data of Ohta et al. (2016). Uncertainties have been determined from errors of the linear fit towards Tm. Table S2. Single crystal electrical resistivity of the group 8 elements Ru (Volkenshteyn et al. 1978) and Os (Schriempf 1968) at room temperature, Hashin-Shtrikman bounds (Hashin & Shtrikman 1963) for their polycrystal resistivity and maximum relative error due to anisotropy. Table S3. Thermophysical properties of simple metals at Tm. The resistivity ratio is in good agreement with the long-wavelength approximation to Ziman's formula (equation 5). References are labeled as follows: Sobolev (2011)a, Faber (1972)b, Fritsch et al. (1973)c, Blairs (2006)d, Peng et al. (2015)e, Mills (2002)f, Tallon & Wolfenden (1979)g. Table S4. Thermophysical properties of transition metals with partially occupied d −bands at Tm. The resistivity ratio is in good agreement with the long-wavelength approximation to Ziman's formula (equation 5). References are labeled as follows: Faber (1972)b, Hess et al. (1994)h, Tsu et al. (1985)i, Schramm (1962)j, Brillo & Egry (2003)k, Owen & Yates (1936)l, Desai (1987)m, Alers et al. (1960)n. Table S5. Thermophysical properties of metals with fully occupied d-bands at Tm. The resistivity ratio is systematically underestimated by Ziman's long-wavelength approximation (equation 5). References are labeled as follows: Faber (1972)b, Blairs (2006)d, Owen & Yates (1934)l, Chang & Himmel (1966)o, Kanai & Tsuchiya (1993)p, Grønvold & Stølen (2003)q, Ledbetter (1977)r. Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the paper. © The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.

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Published: Apr 1, 2018

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