William W. Anderson

An equation of state for liquid iron and implications for the Earth's core

An equation of state is presented for liquid iron based on published ultrasonic, thermal expansion, and enthalpy data at 1 bar and on pulse-heating and shock wave compression and sound speed data up to 10 Mbar. The equation of state parameters, centered at 1 bar and 1811 K (the normal melting point of iron), are density, ρ_0 = 7019 kg/m^3, isentropic bulk modulus, K_(S0) = 109.7 GPa, and the first- and second-pressure derivatives of K_S, K′_(S0) = 4.66 and K″_(S0) = −0.043 GPa^(−1). A parameterization of the Gruneisen parameter γ as a function of density ρ and specific internal energy E is γ = γ_0 + γ′(ρ/ρ_0)^n(E - E_0) where γ_0 = 1.735, γ′ = −0.130 kg/MJ, n = −1.87, and E_0 is the internal energy of the liquid at 1 bar and 1811 K. The model gives the temperature dependence of γ at constant volume as (∂γ/∂T)_(v|1bar,1811K) = −8.4 × 10^(−5) K^(−1). The constant volume specific heat of liquid Fe at core conditions is 4.0–4.5 R. The model gives excellent agreement with measured temperatures of Fe under shock compression. Comparison with a preliminary reference Earth model indicates that the light component of the core does not significantly affect the magnitude of the isentropic bulk modulus of liquid Fe but does decrease its pressure derivative by ∼10%. Pure liquid Fe is 3–6% more dense than the inner core, supporting the presence of several percent of light elements in the inner core.

Physics of interplanetary dust capture via impact into organic polymer foams

The physics of hypervelocity impacts into foams is of interest because of the possible application to interplanetary dust particle (IDP) capture by spacecraft. We present a model for the phenomena occurring in such impacts into low-density organic polymer foams. Particles smaller than foam cells behave as if the foam is a series of solid slabs and are fragmented and, at higher velocities, thermally altered. Particles much larger than the foam cells behave as if the foam were a continuum, allowing the use of a continuum mechanics model to describe the effects of drag and ablation. Fragmentation is expected to be a major process, especially for aggregates of small grains. Calculations based on these arguments accurately predict experimental data and, for hypothetical IDPs, indicate that recovery of organic materials will be low for encounter velocities greater than 5 km s^(−1). For an organic particle 100 μm in diameter, ∼35% of the original mass would be collected in an impact at 5 km s^(−1), dropping to ∼10% at 10 km s^(−1) and ∼0% at 15 km s^(−1). For the same velocities the recovery ratios for troilite (FeS) are ∼95%, 65%, and 50%, and for olivine (Mg_2SiO_4) they are ∼98%, 80%, and 65%, demonstrating that inorganic materials are much more easily collected. The density of the collector material has only a second-order effect, changing the recovered mass by <10% of the original mass.

Shock temperature and melting in iron sulfides at core pressures

The temperatures of shock-compressed FeS and FeS_2 in the pressure ranges 125–170 GPa and 100–244 GPa, respectively, are reported and used to constrain the melting curves and thermodynamic properties to core pressures. A fit of the Lindemann law parameters corresponding to the usual functional form for the lattice Gruneisen parameter gives γ_L = 1.17 ± 0.13 and n_L = 0.5 ± 0.5 for the high-pressure phase of FeS at ρ = 5340 kg/m^3 and γ_L = 2.18 ± 0.32 and n_L = 1.6 ± 0.7 for FeS_2 at ρ = 5011 kg/m^3. The entropies of fusion are ∼203 J kg−1 K−1 for FeS at 120 GPa and ∼180 J kg^−1 K^−1 for FeS_2 at 220 GPa. We find that the melting temperature of FeS is 3240±200 K, 4210 ± 700 K, and 4310 ± 750 K at 136 GPa, 330 GPa, and 360 GPa, respectively. For FeS_2, the melting temperatures are 3990 ± 300 K, 5310 ± 700 K, and 5440 ± 750 K, respectively, for the same pressures. The electronic specific heat for FeS is given by C_e = β_0 (ρ_0/ρ)γe with β0 = 0.25 ± 0.10 J kg^−1 K^−2 and γ_e = 1.34 for ρ0 = 5340 kg/m3 for the high-pressure solid phase and β_0 ≈ 0.05 J kg−1 K^−2 and γe = 1.34 for ρ_0 = 5150 kg/m^3 for the liquid phase. For FeS_2, there is no detectable electronic contribution, and the lattice specific heat is only 67% of the Dulong-Petit limit, possibly implying tight S-S binding in S_2 units. A reexamination of all shock wave melting data for Fe indicates these approximately agree, but they do not resolve the disagreement between the extrapolated static diamond anvil cell data sets. Fe should melt at ∼6600 K at 243 GPa and 6900 ± 750 K at 330 GPa (the pressure of the inner core-outer core boundary). Because the FeS melting curve falls well below that of FeS_2, FeS may eventually undergo peritectic melting at high pressures, while FeS_2 melts congruently.

Ideal FeFeS, FeFeO phase relations and Earth's core

Liquid-state and solid-state model fits to melting data for Fe, FeS and FeO provide constraints for calculating ideal phase relations in Fe-FeS and Fe-FeO systems in the pressure range corresponding to the Earths outer core. The liquid-state model fit to the Fe melting data of Williams and Jeanloz places constraints on the temperature and other properties of Fe along the liquidus beyond the range of their data. The temperature along the best-fit Fe liquidus reaches 5000 K at 136 GPa and 7250 K at 330 GPa, which is somewhat lower than that implied by the Hugoniot results (∼7800 K at 330 GPa). This discrepancy may be due to reshock in experimental targets, or some inaccuracy in the extrapolation, presuming the Hugoniot results represent the equilibrium melting behavior of Fe. Constraints on the solidi of FeS and FeO from the comparison of data and solid-state model calculations imply that FeS and FeO melt at ∼4610 and 5900 K, respectively, at 136 GPa, and ∼6150 and 8950 K, respectively, at 330 GPa. Calculations for the equilibrium thermodynamic properties of solid and liquid Fe along the coincident solidus and liquidus imply that the entropy of melting for Fe is approximately independent of pressure at a value of approximately R (where R is the gas constant), while the change in the molar heat capacity across the transition increases with pressure from ∼0.5R to 4R between standard pressure and 330 GPa. We use these constraints to construct ideal-mixing phase diagrams for Fe-FeS and Fe-FeO systems at outer core pressures, assuming that Fe and FeS, or Fe and FeO, respectively, are the solid phases in equilibrium with the liquid Fe-FeS or Fe-FeO mixtures, respectively. Calculated Fe-FeO eutectic compositions at 330 GPa (15–20 mol% O) are 25 mol% S. Combined with density considerations, these calculations imply that an O-rich outer core is more likely to lie on the FeO-rich side of the Fe-FeX eutectic, while an S-rich outer core is more likely to lie on the Fe-rich side of the Fe-FeX eutectic. In addition, eutectic temperatures in both systems are ≳5000 K at 330 GPa. Widely accepted temperature profiles for the outer core, ranging from ≲3000 K at the 136 GPa, the core-mantle boundary, to ≲4200 K at 330 GPa, the outer-inner core boundary, are ⩾800 K below this value. In the context of the outer-inner core boundary-phase boundary hypothesis, this discrepancy implies that at least one boundary layer of ⩾1000 K exists in the mantle, possibly at its base in the D″ region.

Emplacement of penetrators into planetary surfaces

We present experimental data and a model for the low-velocity (subsonic, 0–1000 m/s) penetration of brittle materials by both solid and hollow (i.e., coring) penetrators. The experiments show that penetration is proportional to momentum/frontal area of the penetrator. Because of the buildup of a cap in front of blunt penetrators, the presence or absence of a streamlined or sharp front end usually has a negligible effect for impact into targets with strength. The model accurately predicts the dependence of penetration depth on the various parameters of the target-penetrator system, as well as the qualitative condition of the target material ingested by a corer. In particular, penetration depth is approximately inversely proportional to the static bearing strength of the target. The bulk density of the target material has only a small effect on penetration, whereas friction can be significant, especially at higher impact velocities, for consolidated materials. This trend is reversed for impacts into unconsolidated materials. The present results suggest that the depth of penetration is a good measure of the strength, but not the density, of a consolidated target. Both experiments and model results show that, if passage through the mouth of a coring penetrator requires initially porous target material to be compressed to <26% porosity, the sample collected by the corer will be highly fragmented. If the final porosity remains above 26%, then most materials, except cohesionless materials, such as dry sand, will be collected as a compressed slug of material.

Shock wave equations of state of chondritic meteorites

We have obtained shock compression data for Murchison and Bruderheim chondritic meteorites. Data for Murchison suggest that the Hugoniot states are described by a smooth curve to ≥90 GPa, having ρ_0=2.656 Mg/m^3, K_(S0)=24.2±.7 GPa, K′=4.17±.10, and constant γ=1.0. The data for Bruderheim suggest more complicated behavior. A mineral mixture model consistent with the Bruderheim data suggests that the Hugoniot state is a low pressure phase below 25 GPa, with ρ_0=3.555 Mg/m^3, K_(S0)=146 GPa, K′=2.53, and constant ργ=7.11 Mg/m^3; and a high pressure phase above 65 GPa, with ρ_0=4.40 Mg/m^3, K_(S0)=225 GPa, K′=3.25, and constant ργ=7.485 Mg/m^3.

Phase relations in iron-rich systems and implications for the Earth's core

Recent experimental data concerning the properties of iron, iron sulfide, and iron oxide at high pressures are combined with theoretical arguments to constrain the probable behavior of the Fe-rich portions of the Fe-O and Fe-S phase diagrams. We infer that a solid solution exists between ϵ-Fe and S at high pressures. This is based on the similarity in the atomic radii of Fe and S (rFe/rS ≈0.97), and on the observation that S apparently does not form a solid solution with the γ-phase of Fe. We suggest that the ϵ-Fe S system may, to first order, be modeled as an ideal solid solution and, therefore, will not have a eutectic. The solid solution probably is not ideal, but there are insufficient data to constrain the non-ideal behavior. The ϵ-Fe-O system, on the other hand, probably has very little solid solution (rO/rFe ≈0.84), although there is recent evidence that a highly non-ideal solid solution does exist between γ-Fe and O. Experimental data extending to > 100 GPa suggest that solid FeO may remain a stable compound at high pressures, with a possible miscibility gap between FeO and Fe. This is based on the observation that FeO melts at higher temperatures than either Fe or O at high pressures. Theoretical extrapolation of the melting curve of FeO indicates that this behavior should continue throughout the pressure range relevant to the Earths core. Thus, the Fe-rich portion of the Fe-O phase diagram is predicted to display a eutectic between ϵ-Fe and FeO at core pressures. Comparison of the predicted composition of the Fe-FeO eutectic at inner core pressures with the amount of oxygen required to give the outer core its observed density (28 at. % O) indicates that, if the only light element in the core were oxygen, the core composition would lie significantly toward the FeO side of the eutectic and FeO would be the solid phase of the inner core. This is inconsistent with the properties of the inner core. This argument indicates that, although oxygen is probably present in non-trivial quantities, it cannot be the only light element in the core, and may not even be the most abundant light element in the core. The results of our analysis are compatible with sulfur as another major light element in the core, as the Fe-S solidus is always more Fe-rich than the coexisting liquidus.

Shock compression and isentropic release of rhyolite

A series of shock compression experiments have been conducted on rhyolite at pressure ranging from 6 to 33 GPa. A velocity interferometer (VISAR) was employed to monitor the particle velocity of an aluminum reflector with a diffused surface bonded to the rhyolite sample. In the forward ballistic experiments, a slow rise shock wave front is observed at 6 GPa. While in the forward experiments their release waves are smeared, in a reverse ballistic experiment, the particle velocity variation at the shock wave plateau and the isentropic release wave arrival have been clearly observed. Using Swegle’s mixed phase model, we simulated the experimental results with WONDY code. Like quartz and granite, the rhyolite data could be fit to a frozen release model which has some hysteric behavior. The Eulerian sound velocity at shock pressure 8.7 GPa has been determined to be 5.6 km/s.

PHYSICS OF INTACT CAPTURE OF COMETARY COMA DUST SAMPLES

The physics of hypervelocity impact into foams are of interest because of application to comet dust capture during flyby encounters. Particles much larger than the foam cells behave as if the foam were a continuum, so that standard equations of fluid mechanics describe the effects of drag and ablation. Calculations based on these arguments accurately reproduce experimental results