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The Elastic Properties of Iron Alloys and their Measurement
The measurement of sound wave propagation through elastic media has always been foundational to the understanding of the elastic and mechanical properties of materials, and the materials present in the Earth’s interior remain as no exception. The velocity proles of P and S waves which propagate through the Earth’s core are strongly constrained by seismic observations, and are instrumental to the determination of the structure and properties of the Earth’s deep interior.
On the basis of VP and density measurements of a variety of rocks up to 1 GPa, [Birch, 1961] suggested an approximately linear evolution of VP with density, an approximation still widely used in the study of the Earth’s deep interior. Initial studies of sound wave propagation in minerals and metals at high static pressures relied on the use of ultrasonics. While ultrasonics remains the gold standard for the measurement of elastic properties of materials, it is a technique which is very limited in P-T space with respect to the temperatures and pressures expected in the Earth’s core. Furthermore, most materials relevant to the Earth’s mantle and core undergo phase transitions which can signicantly alter their properties at relevant conditions of pressure and temperature ( [Takahashi and Bassett, 1964], [Saxena et al., 1996]). As a consequence, with the advent of improved synchrotron radiation sources, new techniques were developed to extend VP (and VS) measurement to everhigher conditions of P and T in order to place stronger constraints on the properties of geomaterials at conditions more comparable to that of the Earth’s deep interior.
Ambient Conditions X-ray Diraction
Most alloys measured in this thesis were measured at ambient temperature and pressure by grazing-incidence XRD (GI-XRD) or transmission XRD. The instrument used for GI-XRD exploited the Co K line = 1:7888A.
The samples were either loose ribbons or lms deposited on a glass substrate. The samples were placed on Si (100) oriented pastelles for measurement. The diraction patterns were measured with a line-detector, with the angle of the incoming Xray beam and detector changing to scan in 2. Diraction patterns were collected both on stationary and rotating samples in order to detect potential textural eects and phase purity. There was little change in diraction peak linewidth or intensity between diraction patterns of stationary or rotated samples. Due to the absence of a well-dened signal from the glass substrate, the focussing of the X-ray beam on the surface of the sample was a challenge, resulting in systematic calibration errors. All alloys where an EoS were measured were characterized further in transmission geometry either at IMPMC or at a synchrotron source in an empty DAC.
Synchrotron X-ray Diraction
Ambient temperature, high pressure XRD experiments were performed on PSICHE at Synchrotron-SOLEIL and ID27 at ESRF, and simultaneous high-pressure hightemperature experiments on ID27 at ESRF. At both sources, a monochromatic X-ray beam of wavelength = 0:3738A was used. For all Synchrotron XRD, the sample-detector distance was calibrated with a CeO2 standard (NIST SRM 674b), and diraction images were processed into radially averaged diraction patterns using the software package DIOPTAS [Prescher and Prakapenka, 2015].
Physical Vapor Deposition
Physical vapor deposition (PVD) is a technique by which atoms are deposited directly on a substrate, allowing for the synthesis of alloys in which the components are immiscible in the liquid phase (e.g. Fe-Si-O alloys, [Hirose et al., 2017]). Additionally, due to the availability of low synthesis temperatures (500-550 K for this Thesis), this technique can also be used to suppress precipitation of ordered phases, and in some cases induce amorphization of the alloy (however this has only been observed for Fe27Si in the alloys studied in this Thesis). The alloys of this Thesis were synthesized by Dephis Company. In the PVD synthesis of Fe-Si alloys, atoms of Fe and Si are sputtered onto a glass base from chemically uniform Fe and Si targets. The extraction of Fe and Si from the targets is enhanced by a magnetic eld generated in a chamber of ionized argon plasma. This process results in a chemically homogeneous sample, with a nearly uniform thickness [Miozzi et al., 2018].
Crystal structure and unit cell volume of bcc and ‘bcc- like’ Fe-Si alloys at ambient pressure
Due to the signicant technological and commercial value of Fe-Si alloys, they have been the subject of scientic inquiry for more than a century [Phragmen, 1926]. In spite of this, the role of order, disorder, and the crystal structure of these alloys has been a major scientic challenge. In the words of Bridgman: « [from 0.39 to 5.75 at% Si.] The metallurgical phase diagram shows only a solid solution in this range, although at higher concentrations the [phase] diagram becomes exceedingly complicated » [Bridgman, 1957]. At present, the low temperature (< 1000 K) ambient pressure phase diagram between Fe and stoichiometric FeSi is considered to be composed of 4 primary phases: bcc Fe-xSi (Space group: Im-3m), B2 Fe-xSi (Space group: Pm-3m), DO3 Fe-xSi (Space group: Fm-3m), and stoichiometric B20 FeSi (Space group: P213). While end-member B20 FeSi is approximately stoichiometric, for a wide range of more Fe-rich compositions, a given sample may be a mix of domains of B2, bcc and DO3 Fe-xSi, depending on both Si content and the thermal history of the material [Shin et al., 2005]. At very low Si contents (approx. less than 3 wt% [Jayaraman et al., 2018]) the Fe-xSi alloys are primarily composed of the bcc phase, a simple solid solution with Si atoms randomly replacing Fe in the bcc structure. Figure 3.1 shows a schematic of the disordered bcc structure.
Elastic properties of Fe-Si alloys at ambient pressure
In the 1970s and 1980s ( [Alberts and Wedepohl, 1971], [Routbort et al., 1971], [Rausch, 1976], [Machova and Kadeckova, 1977], [Buchner and Kemnitz, 1981], [Kotter et al., 1989]) much work was performed studying the link between the elastic properties of Fe-Si alloys and their composition. Scatter in the reported Vp above 7 wt% Si is a direct consequence of the scatter in measured single crystal elastic moduli (Cij) (Vp shown in Figs 3.7a and 3.7b). This may be due to porosity in Si-rich samples, as the two main systematic studies [Buchner and Kemnitz, 1981] and [Machova and Kadeckova, 1977] both use similar methods for synthesis and measurement of the single-crystals, but the former uses the theoretical density derived from SEM and XRD, while the latter measured density directly (which was observed to deviate by up to 1.1% from theoretical densities where both were measured). In these two studies, while the error bars on elastic constants are about 1-1.5%, the derived bulk moduli (Figure 3.11 in next section) vary by up to 15% for Fe15Si. By a combination of picosecond acoustics and prolometry on PVD Fe-Si samples, it is shown in Figures 3.7a (Vp vs. Density) and 3.7b (Vp vs. Si Content) that Vp of our samples are generally close to the Reuss (isostress) bound derived from single crystal ultrasonics. In the samples measured for this Thesis, we could not detect void space within the samples by SEM, indicating that the eects of porosity in the present work is small. It is notable however that both Fe9Si and Fe10Si measured by PA are below Reuss bound. Due to the large variation of unit cell volume with Si content for Fe8Si, Fe9Si and Fe10Si, the densities of these alloys are nearly identical.
Within the ultrasonics literature, it is unambiguous that there is a change in the alloying eect of Si at the onset of the bcc/DO3 transition region ( [Machova and Kadeckova, 1977], [Buchner and Kemnitz, 1981], indicating that both Si ordering and Si composition change the elasticity of these alloys. While Fe-Si alloys of 8, 12, 17 and 26 wt % Si measured by PA are in generally good agreement with trends observed in ultrasonics literature, there is signicant discrepancy for alloys of Fe9Si and Fe10Si. At low Si contents, Si addition does not aect Vp signicantly up to the disorder-order transition, and so this discrepancy between 8 and 12 wt%Si may be related to the suppression of Si ordering, as Vp of Fe-8,9,10Si synthesized by PVD have the same or lower Vp than that of pure iron [Guinan and Beshers, 1968]. The signicant jump in Vp observed between Fe10Si and Fe12Si is likely related to an abrupt change in elastic moduli. Fe26Si was observed to be a mix of Fe-rich B2 FeSi (Volume = 21.42 A3, for stoichiometric B2 FeSi Volume 21.30 A3 [Ono, 2013] at ambient conditions) and an amorphous phase. As the density of the material was derived from the lattice parameter of the B2 unit cell volume, its Vp is generally consistent with that of an interpolation between Fe17Si and B20 FeSi [Petrova et al., 2010] when plotted as Vp vs. Density, while disagreement in Vp vs. Si content may be attributed to a possible enrichment of the amorphous phase with Si.
Table of contents :
1 Introduction
1.1 High Pressure Physics and the Diamond Anvil Cell
1.2 The Evolution of the Early Earth and the Composition of the Earth’s Core
1.3 Mineral Physics Constraints from X-Ray Diraction in a Diamond Anvil Cell
1.4 The Elastic Properties of Iron Alloys and their Measurement
2 Methods
2.1 High Pressure Generation
2.1.1 Diamonds
2.1.2 Sample Preparation and Loading
2.1.3 Pressure Measurement
2.2 X-ray Diraction
2.2.1 Ambient Conditions X-ray Diraction
2.2.2 Synchrotron X-ray Diraction
2.2.3 Equations of State
2.3 Picosecond Acoustics
2.3.1 Compressional Sound Velocity and the Thermodynamics of Fe-alloys
2.3.2 Instrument
2.4 Prolometry
2.5 Scanning Electron Microscopy
2.6 Synthesis of Fe Alloys
2.6.1 Rapid Melt-Spinning
2.6.2 Physical Vapor Deposition
3 Velocity-density systematics of bcc and ‘bcc-like’ Fe-alloys 51
3.1 Ambient Pressure Behaviour of bcc and ‘bcc-like’ Fe-Si alloys
3.1.1 Crystal structure and unit cell volume of bcc and ‘bcc-like’
Fe-Si alloys at ambient pressure
3.1.2 Elastic properties of Fe-Si alloys at ambient pressure
3.2 Velocity-density systematics of bcc and ‘bcc-like’ Fe-alloys: Properties of Fe-Si alloys at High Pressures
3.2.1 EoS of bcc and ‘bcc-like’ Fe-Si alloys at high pressures .
3.2.2 Compressional velocity-density relations at high pressures .
3.2.3 Derivation of shear properties at high pressures for bcc-Fe-Si alloys
3.3 On the eects of Si ordering and Si content in bcc Fe-Si alloys
3.3.1 Shear and Bulk Moduli of Fe-Si alloys: Si ordering revisited .
3.3.2 Evolution of compressional and shear velocities with Si content at high pressures
4 On the hcp phase of Fe-Si alloys: Constraints on Earth’s core com- position and anisotropy
4.1 On the bcc-hcp transition in Fe-Si alloys at high pressures
4.1.1 The bcc-hcp transition by X-ray Diraction
4.1.2 Elasticity in the vicinity of the bcc-hcp transition of Fe-Si alloys
4.1.3 XRD of Si-rich hcp alloys under quasihydrostatic conditions .
4.2 Velocity-Density Systematics of Fe-5wt.% Si at Extreme Conditions: Constraints on Si content in the Earth’s Inner Core
4.2.1 Velocity-Density Systematics of Fe-5wt%Si at Extreme Conditions
4.2.2 Shear velocities and derived quantities
4.2.3 The Inuence of Thermoelastic Parameters on Theory and Experiment
4.3 High pressure behaviour of PVD hcp Fe-Si alloys
4.4 Eect of Ni alloying in Fe-Ni-Si alloys and the c/a axial ratios of hcp
Fe-Si and Fe-Ni-Si alloys at high P-T conditions.
4.4.1 The eect of Ni on dilute Fe-Si alloys at high P-T conditions .
4.4.2 c/a axial ratios of Fe-Si and Fe-Ni-Si alloys at high pressures and high temperatures
5 Conclusions
6 Bibliography
A Other technical aspects of Picosecond Acoustic measurements
A.1 Further Experimental Details of Picosecond Acoustics
A.2 Sample Preparation and Technical Observations
A.2.1 Sample Preparation: Sample Loading and General Observations
A.2.2 Instrument Setting Tests
A.2.3 Deformation Tests
A.3 Error Analysis
B Benchmarking Velocity Measurements at High Pressures via Pi- cosecond Acoustics
B.1 Eects of non-hydrostatic stress on PA measurements
B.1.1 Eect of PTM on PA measurements
B.1.2 Non-hydrostatic eects due to sample-gasket contact
B.2 On IXS and NRIXS as high pressure sound velocity measurement techniques
C Tabulated Datasets
C.1 Foreword
C.2 Fe5Si
C.3 Fe5Ni5Si
C.4 Fe8Si
C.5 Fe10Si
C.6 Fe12Si
C.7 Fe17Si
