STRENGTH , DEFORMATION AND COMPRESSION BEHAVIOR OF TUNGSTEN CARBIDE, KRYPTON, AND XENON UNDER QUASI - STATIC LOADING By Benjamin Lee Brugman A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for t he degree of Geological Sciences Doctor of Philosophy 202 1 ABSTRACT STRENGTH , DEFORMATION AND COMPRESSION BEHAVIOR OF TUNGSTEN CARBIDE, KRYPTON, AND XENON UNDER QUASI - STATIC LOADING By Benjamin Lee Brugman Strength and deformation are fund amental material responses to stress that are importa nt to multiple scientific disciplines . The response of materials to thermodynamic conditions affects geologic and planetary processes and the structure, chemical behavior, and rheology of planetary inte riors , as well as the outcome of accretionary impacts . These same properties are applied in defense science and are important industrially for the development of novel materials and the manufacture of strong too ls and parts . In t his dissertation , I deter mine the high - pressure strength, compressio n and deformation behavior of tungsten carbide, and solid i f i e d krypton and xenon simple materials that exhibit very different strength and compression propertie s . Material properties are examined in this work wi th synchrotron radiation in the diamond anvil cell , which allows a variety of X - ray and complementary optical techniques to be used on materials under quasi - static stress load s . Synchr otron radiation is brilliant, tunable, and hig hly focused, allowing pre cision measurements of small samples . P ressure media may be used to minimize the effects of non - hydrostatic stress , or if non - hydrostaticity is requi red , the sample may be compressed without a medium . By using both the axial and radial diffraction geomet ries, materials may be probe d under a range of stress conditions . Tu ngsten carbide is a hard, ultra - incompre ssible ceramic used widely in industry . The reported bulk modulus of WC was discrepant by more than 125 GPa . T his is attributed to grain size dep endence , though measurement techniques may also be impo rtant . Q uasi - static compressive strength and deformation behavior of WC at high pressure have not been previously studied . I compressed b ulk and nano - crystalline hexagonal WC to 66 GPa in the diamond anvil cell . N ano - WC is softer than bulk WC . Yielding at ~30 GPa is accommodated by prismatic slip up to 40 - 50 GPa, at which pressure pyramidal slip along a different direction also becomes activate d . WC supports ~ 12 - 16 GPa differential stress , the diff erence between stress ali gned with the load axis and stress in the direction of the gasket, at yielding but its strength is anisotropic and the ( 001 ) plane supports a ~ 6 8 - 70 % higher differential stress at yielding . Solidified noble gases are prototypical solids that crystallize with simple structures and low strength . Heavy rare gas solids Ar, Kr , and Xe undergo a martensitic fcc - hcp phase transition as pressure is in creased . At 300 K the metastable fcc phase persists over a wide pressure range . I comp ressed Kr and Xe in the diamond anvil cell to 115 GPa to determi ne p hase equilibria , strength, and deformation . The phase transition progresses more quickly in Xe than in Kr . Bo th Kr and Xe crystallize as large fcc crystals and develop preferred crystall ite orientation (texture) in the fcc and hcp phases . Xe peaks are highly textured and broad to at least 101 GPa . Non - hydrostaticity is observed at 15 - 20 GPa in both Kr and Xe and increases with pressure , with both materials support ing at least 5 - 7 GPa di fferential stress above 40 GPa . This work examines the strength, compression and deformation behavior of WC, Kr, and Xe , extremes in the range of possible mechanical responses to stress . S trength in WC is overestim ated when determined by lattice strain d ue to plastic deformation and is anisot ropic , possibly due to t he position of C atoms blocking s l i p s y s t e m s dislocation motion . Grain size may affect incompressibility . The phase transition progresses more slowly in both K r and Xe than previous ly report ed . R adial diffraction can reveal more about material properties than axial diffraction and may be especially useful when multiple phases are present because texture and s tress orientation can bias axial patterns . iv v ACKNOWLEDGEMENTS Completing the PhD has been quite a journey. I have had an inestimable amount of support from many wond erful peo ple during my tenure at Michigan State University ; and I have met and collaborated with amazing researchers , from faculty , staff, and students at MSU and other universities, to scientists at national lab oratorie s and government organizations . I w ould not have arrived at this milestone without so much support. First and most of all , I want to offer my si ncere thanks and gratitude t o my advisor Dr. Su ki Dorfman . Suki involved in me experiments immediately after taking me on as a student , encourage d me to p resent my work at every possible opportunity , and sent me to conferences right from the start . I would not have been able to produce the quality of work that I have without her investment of time , advice, moral and financial support . Suki your guidance and support were critical to my growth as a researcher . I especially want to thank you for giv ing me the space and time I needed, when I needed it, to persist in this work. It means a lot. T o my current dissertation committee members: Dr. Micha el A. Vel bel, Dr. Martin A. Crimp, and Dr. Allen K. McNamara, thank you for reading my proposals and dra fts and offering guidance and support throughout my tenure at MSU (and to Dr. Velbel and Dr. Crimp thanks for sticking around!). To Dr. Velbel, thank you for giving me the space to figure out what I really wanted to do, and supporting me in that endeavor f or all these years. To Allen thank you for your help over the last 10 years (can you believe it?) , thanks for undergraduate mentoring, advice when applying To my former and informal committee members Dr. David T. Long , Dr. Masako Tominaga , and Dr. Damian Swift thank you for your help, support, encouragement, and guidance throu ghout my tenure at MSU. vi Mingda Lv, Garrett Die drich, Allison Peas, Alison Farmer, Dr. Jiachao Liu, and Dr . Shah Najiba. T hese experiments would not have come together the way they did without your help and support. Thanks also go to Dr. Lowell Miyagi and Dr. Feng Lin at Universit y of Utah. Your collabora tion and EVPSC si mulations on the WC project made the paper much better . I want to recognize the support I received from CDA C in its various incarnations (i.e. the Carnegie/Capital/Chicago DOE Alliance Center). I would especially like to thank Dr. Steve G ramsch for his su pport, multiple beamtime allocations, funding to visit LLNL, and o verall guidance . I would also like to than k Dr. Rus s Hemley for his insight and guidance . This work would not have been possible without the beamline scientists at AP S like to thank th e scientists at Sector 16 (HPCAT): Dr. Dmitry Popov, Dr. Changyong Park, and Dr. Curtis Kenney - Benson; and a t Sector 13 (GSECARS): Dr. Vitali Prakapenka and Dr. Eran Greenberg (now at Soreq Nuclear Research Center, Israel). also like t o thank the staff and scientists at the JLF at LLNL and Trident when it was at Los Alamos , esp. Dr. Bob Cauble at JLF and Dr. Randy Jo hnson at LANL for support, training, and access to the various facilities and equipment needed for the seamless onboarding experience I had with dynamic compression experiments. I would also like to thank my colleagues in those experiments Dr. Damian Sw ift (again) and Dr. Laura Chen. I received a great deal of further training and assistance from Dr. Marius Millot and Dr. Federica Coppari at LLNL. Thank you for teaching me how to analyze V ISAR data , for helping me get (and keep) a handle on my data and their limitations, and for your support throughout my collaboration with LLNL. vii Thank s to my graduate colleagues and t h e faculty and staff over the years in the EES . To Pamela Robinson, it would not have been the same without all the wo nderful con versation s in the department office , t hank yo u . T o Dr. Tyrone Rooney , I am very grateful for your w illingness to take time and help me figure out what I needed to do to persevere , thank you so much . To Dr. Danita Brandt, t hank you for always listening an d always being willing to help me find a solution . I espec ially want to thank Erin Haacker, Trish Sm recak, Susan Kran s, Kaitlyn Trestrail, Brandon C hiasera, Chris Svoboda, Andrew Lavigne, Alex Kuhl, Au tumn Parish, Kayla Cotterman, Lin Liu, Dan Burk, Jeremy Rapp, T yler Ham pton, and Joe Lee - Cullen . All of you have provided so muc h support , input , friendship , and ins piration over the years I am very grateful to each of you . Thank you! Finally, thanks to my friends and family to my parents and brother, the Brunne meyer family, and especially to Dave for supporting me , organizing papers, proofre ading , and putting up with me traveling on anniversaries , birth days, and holiday s - y ou re the very best . viii TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ........................... x L IST OF FIGURE S ................................ ................................ ................................ ....................... xi KEY TO ABBREVIATIONS ................................ ................................ ................................ ..... xvii C HAPTER 1: I ntroduction ................................ ................................ ................................ .............. 1 1.1 Significance of elasticity, str ength and plasticity deformation in Earth ........................ 1 1.2 Significance of the materials in this work ................................ ................................ ...... 3 1.2.1 Tungsten carbide ................................ ................................ ................................ ... 4 1.2.2 Heavy rare g ases krypton and xenon ................................ ................................ .... 6 1.3 Dissertation purpose and structure ................................ ................................ ................. 8 C HAPT ER 2: Metho ds for m easur ing strength and deformation at high pressure ....................... 11 2 .1 Introduction ................................ ................................ ................................ ........................ 11 2.2 Quasi - static compression in t he d iamond a nvil c ell ................................ .......................... 12 2.3 The st ress state in the DA C ................................ ................................ ................................ 15 2.4 Pressure c alibration, stress markers, a n d ruby fluorescence ................................ .............. 16 2.5 Synchrotron ra diation ................................ ................................ ................................ ......... 19 2.6 X - ray di ffraction, d iffraction geometry, and texture eva luation ................................ ........ 20 2.6.1 Axia l g eom etry ................................ ................................ ................................ .......... 21 2.6.2 Radial g eometry ................................ ................................ ................................ ........ 22 2.7 Conclusions ................................ ................................ ................................ ........................ 24 C HAPTER 3: Strength, defo r mation, and equation of state of tungsten c arbide to 66 GPa ......... 25 A b stract ................................ ................................ ................................ ................................ .... 25 3.1 Introduction ................................ ................................ ................................ ........................ 25 3. 2 Experimental d etails ................................ ................................ ................................ ........... 29 3.2.1 Sample pre paration and lo ading ................................ ................................ ................ 29 3 .2.2 X - ray diffra ction in the DAC ................................ ................................ .................... 30 3.3 Results and i nterpretation ................................ ................................ ................................ .. 32 3.3.1 Equation of s tate and linear incompressibility ................................ .......................... 33 3.3.2 Differential stress and el astic stiffness co efficients ................................ .................. 39 3.3.3 P lastic deformation ................................ ................................ ................................ ... 47 3.3. 3.1 Texture an alysis ................................ ................................ ............................... 48 3.3. 3. 2 Plasticity simulation ................................ ................................ ......................... 49 3.3. 3.3 Crystallite size and microstrain ................................ ................................ ........ 51 3.4 Discussion ................................ ................................ ................................ .......................... 53 3.4.1 Equation of s tate ................................ ................................ ................................ ....... 53 3.4.2 Strength, e lasti city, and d eformation ................................ ................................ ........ 55 3.5 Conclusions ................................ ................................ ................................ ........................ 56 Acknowledgements ................................ ................................ ................................ .................. 56 ix CHAPTER 4: Phase e quilibria, strength, and deformation behavi or of Kr and Xe to Mbar pressures ................................ ................................ ................................ ................................ ......... 58 Abstra ct ................................ ................................ ................................ ................................ .... 58 4.1 Introduct ion ................................ ................................ ................................ ........................ 59 4.2 Methods ................................ ................................ ................................ .............................. 62 4.2.1 X - ray diffraction ................................ ................................ ................................ ....... 63 4.2.2 Ruby fluoresce nce ................................ ................................ ................................ ..... 65 4.3 Results and disc ussion ................................ ................................ ................................ ....... 66 4.3.1 The fcc - hcp phase transit ion in Kr and Xe ................................ ............................... 66 4.3.2 Texture in Kr and Xe due to phase transition an d deformation ................................ 75 4.3.3 Str ength of Kr and Xe across the phase transition ................................ .................... 78 4.3.3.1 Differential stress measured by ruby fluorescence ................................ .......... 78 4.3.3.2 Differential stress measured by lattice strain in Pt and Ag .............................. 80 4.3.3.3 Str ain i n Kr and Xe ................................ ................................ .......................... 84 4.4 Conclusion s ................................ ................................ ................................ ........................ 89 Acknowledgements ................................ ................................ ................................ .................. 90 CHAPTER 5: Conclusions and implications of this work ................................ ............................. 91 5.1 Some reflections and recommended best practices for experimental measurements of physical properties at high pressure ................................ ................................ ......................... 91 5.2 Grain size effects on elastic deformation behavior ................................ ............................ 93 5.3 Effects of anisotropy and pl asticity on material strength ................................ ................... 94 5.4 Impact of sluggish pha se transitions on physical properties ................................ ...... ........ 96 REFERENCE S ................................ ................................ ................................ .............................. 98 x LIST OF TABLES Table 1 : Experimental and theoretical values for the equation of state parameters for hexagonal tungsten monocarbide . US = ultrasonic, SW = shock wave, DAC = diamond anvil cell, MAP = multi - anvil press, XRD = X - ray diffraction, LD A = local - density approximation, GGA = generalized gradient approximation, PWP = plane wave potential, PBE = Perdew, Burke, and Ernzerhof, LMTO = linear muffin - tin orbital. ................................ ................................ ............... 34 Table 2 : CRSS and pressure dependence values for active slip systems in WC under non - hydrostatic compressive stress. ................................ ................................ ................................ ...... 35 Table 3 : Summary of experimenta l runs for Kr and Xe . ................................ .............................. 63 xi LIS T OF FIGURES Figure 1 : Strength of geologic and selected materials with pressur e. Black squares, diamond, (Wang, He and Duffy, 2010) , Gray circles, heterodiamond (Dong et al., 2009) , blue squares, ringwoodite (Kavner and Duffy, 2001) , green diamonds, olivine (Uchida et al., 1996) , open gold triangles, MgO (Uchida et al., 1996) , closed gold triangles, MgO (Merkel et al., 2002) , red hourglasses NaCl (open with annealing, closed without annealing, (Funamori, Yagi and Uchida, 1994) , orange triangles SiO 2 (Shieh, Duffy and Li, 2002) , pink diamonds, n eon (Dorfman et al. , 2012) . ................................ ................................ ................................ ..................... 5 Figure 2 : Schematic of t he fcc - hcp phase transition . The two structures can be represented as displaced layers of atoms, where A, B, and C are n ot equivalent closest - packed layers. The fcc structure is ordered ABCABC and the hcp structure is ordered ABABAB. Closest - packed plan es (represented here by the orange diamond) shared between the two structures, result in the continuation of some diffrac tion lines throughout the transition . ................................ .................... 8 Figure 3 : Standard symmetric diamond anvil cell . Bellville washers provide cushioning as hex - screws are tightened . X - rays or optics are admitted through an aperture, are scattered by the sample, and exit an aperture on the opposite end. ................................ ................................ ......... 13 Figure 4 : Diamond anvil cell schematic . Two opposed anvils press on a cell chamber defined by the anvil culets and a gasket. The sample is loaded inside the chamber along wi th a pressure - transmitting medium or other internal markers (if desired). ................................ .......................... 14 Figure 5 : View of the DAC chamber through the anvils before loading compressed gas . The cell is pre - loaded with an Ag pressure standard and a ruby ball f or gas - loading of Kr or Xe gas . The anvils are 300 um culets appropriate for pressures up to ~70 GPa . The Re gasket has a 150 ................................ ........................ 17 Figure 6 : Diffraction geometries using the diamond anvil cell. In the axial geometry (top) the X - ray beam is parallel to the loading axis the angle between the plane normal and the load axis is a function of the diffraction a ngle 2 . Due to the structure of the DAC, most di ffracted planes which reach the detector have small 2 and is close to 90°, the minimum strain condition. In the radial geometry (bottom) the beam is normal to the loading axis, and varies from 0° to 9 0°, so the full range of strain is sampled. ................................ ................................ ............................ 23 Figure 7 : Representative synchrotron X - ray diffraction patterns of bulk and nano - crysta lline WC compressed in Ne pressure medium with Au pressure standard and Re gasket in the axial diffraction geometry ................................ ................................ ................................ ....................... 32 xii Figure 8 : Compression of bulk (blue circles, with 3 rd order Birch - Murnaghan equation of state fit in blue line) and nanocrystalline (yellow circles, with EoS fit in yellow line) WC in Ne compared with other studies. Pressure was determined from the EOS of Au, using the (111), (200), and (220) Au peaks and t he pressure scale of Dewaele (Dewaele, Loubeyre and M ezouar, 20 04) . Data from previous studies were obtained in the multi - anvil press ( Litasov et al., 2010) (black open circles) and in the DAC for bulk (red open squares ( Amulele et al., 2008) ) and nano - crystalline WC (green open triangles ( Lin et al., 2009) ). Ultrasonic measurements ( Amulele et al., 2008) are displayed in red solid line and extrapolated with red dashed line ........................... 33 Figure 9 : Ellipses representing 95.3% confidence in K 0 and K 0 ' obtained from Birch - Murnaghan equation fit to pressure - volume data for bulk (blue, this study and black, room temperature data from multi - anvil (Litasov et a l., 2010) ), and nano - grained WC (yellow). Dashed ellipses with open circles are w ith V 0 fixed to ambient XRD measurements, solid ellipses with solid circles are with V 0 fit. The open gray circle indicates reported values from multi - anvil, high - temperatu re/pressure EOS (Litasov et al., 2010) . Solid gray squares are from other DAC stu dies on nano - crystalline (Lin et a l., 2009) and bulk (Amulele et al., 2008) . Red triangles indicate values obtained from th eoretical calculations (Lin et al., 2009; Li et al., 2010, 2014; Cheng et al., 2012; Liu et al., 2013) . The solid red diamond is the adiabatic bulk modulus from ultrasonic experiments (Amulele et al., 2008) . Dashed lines are from shock wave (black line (Dandekar, 2002) ) and ultrasonic interferometry experiments (gray line (Zhukov and Gubanov, 1985) ), which only constrain K 0 but not K 0 ' ................................ ................................ 37 Figure 10 The ratio c/a vs P for bulk - and nano - grain WC from experimental measurements . Values obtained using nonhydrost atic media (open circles) are systematically higher than with hydrostatic media (filled circles, this study). ................................ ................................ ................. 38 Figure 11 : X - ray diffraction pattern data (lower half of each image) and full - profile refinements (upper half of each image) for selected pressures a) 16 GPa, b) 34 GPa, c) 48 GPa and d) 66 GPa . The Debye - Scherrer rings are transformed to azimuth vs 2 and Miller indices for WC are labeled in each pattern . WC peaks exhibit increasing sinusoidal curvature with pressure due to non - hydrostatic strain . Systematic variation in intensity in individual diffraction lines is indicative of plastic deformation and also increases with pressure in WC . Diffracting planes from gasket materials Be and BeO at ambient conditions are also obse rved and exhibit no strain. ................................ ................................ ................................ ................................ ........................ 40 Figure 12 : Strain obtained for WC lattice planes Q ( hkl ) at selected pressures plotted vs . the orientation function B ( hkl ) (eqn 8 ) relative to the stress axis (a - d) and for all planes as pressure increases (e) . In (a - d), red curves are quadratic fits to strain Q ( hkl ) vs . B ( hkl ) . Strain anisotropy increases with pressure (scaling is constant for Figures a - d) . Error in Q(hkl) at individual pressures repre sents the error of the refinement to the experimentally observed curvature . In e), the mean strain (red circles and dashed line between points for emphasis) increases monnotonically , and the shape of the Q vs P curve is similar for all Q(hkl) , however values diverge in Q as pressure increases, indicating an increase in anisotropy as pressure incre ases. ... 42 xiii Figure 13 : Differential stress t in WC ob tained from lattice strain analysis and EVPSC model of experimental measurements . Figure 12 a (left): Average Voigt (red triangles) and Reuss (teal circles) values for differential stress computed from Q(hkl) and elastic constants obtained from theoretical calculations (Liu et al., 2013) , with values obtained using aggregate shear modulus (gold squares (Amulele et al., 200 8) ). Also shown is differential stress obtained from EVPSC simulation incorporating texture and plasticity (solid black line) . Stress accounting only for elast ic strain diverges from stress accounting for both elastic and plastic strain at the yield p oint near 30 GPa . Figure 12 b (right): Reuss stresses computed for individual lattice planes hkl using purely elastic strain . (001) supports the largest differenti al stress ~28 GPa, ~68 - 70% larger than differential stress values of ~16 - 20 GPa at yielding, which use the aggregate and Reuss - limit shear modulus, respectively, and 65 - 89% larger than the stress from the EVPSC model ............ 43 Figure 14 : Differential stress observed in WC and other hard ceramics compressed uniaxially with no pressure - transmitting medium . Stress in WC is computed with both lattice strain theory (solid circles) and EVPSC simulation (dashed line) . Dynamic yield strength from shock wave data are open gold circle, calculated after (Feng, Chang and Lu, 2016) . Previous studies on all other hard ceramics (open symbols) use lattice strain theory . B lue squares are WB (Dong et al., 2012) , green triangles are - Si 3 N 4 (Kiefer et al., 2005) , black diamon ds are TiB 2 (Amulele, Manghnani and Somayazulu, 2006) , teal hexagons are B 6 O (He, Shieh a nd Duffy, 2004) , and magenta triangles are BC 2 N heterodiamond (Dong et al., 2009) . Uncertainty is calculated as ± the standard deviation in mean Q at each pressure, propagated through equation 10. Differential stress increase s with uniaxial load until yielding, where t he change in slope of the t(P) indicates strain is accommodated by both elastic and plastic deformation . In WC, yielding at 30 GPa is supported by the development of texture at the same pressure ( Figure 16 ) . The f low stress of WC above yielding is higher than flow stresses observed in WB and Si 3 N 4 . Flow stress obtained from EVPSC simulation is systematically lower than that derived from lattice strain analysis only, thou gh the results of the two methods remain wi t hin uncertainty of each other. ... 45 Figure 15 : Ela stic stiffness coef ficients for experimental data as calculated from e q uation s 5 - 10 , and 12 (closed symbols: this study) . Open symbols are experimental zero - pressure values (Gerlich and Kennedy, 1979) . Solid and dashed lines are from theoretical calculations (Liu et al ., 2013; Li et al., 2014) . Our values for c 11 and c 13 agree well with theoretical predictions . Other c ij s deviate from predicted values rapidly as plastic deformation increases . Above yielding at 30 GPa, values for c 44 is subst antially higher than pred icted, and c 33 and c 12 diverge from predicted values . Values for c 33 appear to converge with c 11 , while c 12 varies only slightly from a constant value of 200 GPa throughout the experimental pressure interval. .......... 47 Figure 16 : Inverse pole figures reconstructed from experimental ODF data fit with fiber texture at selected pressures . A texture ma ximum is observed at ~30 GPa near the 2 - 1 - 10 planes (symmetrically equivalent to 100 planes in hkl notation), indicating plastic deformation . This value increases to ~1.9 multiples of a random distribution at the maximum pressure of 66 GPa. .... ................................ ................................ ................................ ................................ ........................ 48 xiv Figure 17 : Strain and texture in WC . Left : Experimental and modeled strain (Q - factors) for selected planes of WC vs . pressure . Right : Experimental (top) and modeled (bottom) inverse pole figures illustrating non - random texture at 66 GPa ................................ ................................ . 50 Figure 18 : Modeled slip system activities as determined from the EVPSC simulation . Based on experimental texture, below 8 GPa, defo rmation is entirely elastic, and no plastic deformation occurs . Between 8 - 30 GPa, a small amount of plastic strain is accommodated by prismatic slip activation on At 30 GPa, bulk plastic yielding occurs, accommodated by continued prismati c slip on and activation of prismatic slip on A third (pyramidal) slip system is activated between 40 - 50 GPa on ....................... 52 Figure 19 : Crystallite size and microstrain vs pressure from ful l - profile refinement in WC under non - hydrostatic compression . Crystallite size decreases and microstrain increases up to ~30 GPa, the pressure at which lattice strain suggests yielding and texture indicates activation of prismatic slip on and Reduction in crystallite size below plasticity onset is attributed to lattice - bending, which reduces the size of the coherently diffracting regions contributing to crystallite size in MAUD software . Above 30 GPa, microstrain drops and th en resumes increasing, while grain size remains ~constant at ~80 - 90 nm . A second dip in microstrain at ~50 GPa follows the activation of pyramidal slip on ................................ ................................ ................................ ............................... 52 Figure 20 : Photomicrogra phs of diamond anvil cell sample chambers inside Re gaskets loaded with ruby grids prior to gas - loading . The cell in figure A (left) was loaded with Xe and the cell in figure B (right) was loaded with Kr . Fluorescence measurements for each cell are plott ed in Figure 26 and compared with similar measurements for other RGS ................................ ............. 65 Figure 21 : Axial s ynchrotron X - ray diffraction patterns for a) Xe (left) and b) Kr (right) with Ag and Pt internal pressure sta ndards, respectively . Intensities of peaks shared between the fcc and hcp phase : fcc (111)/hcp (002), fcc (220)/hcp (110), and fcc (311)/h cp (112) , continue to change to 115 GPa, suggesting conversion to hcp is not complete at this pressure . In Kr, stack ing faults (SF) remain the only indicator of the hcp phase to 94 GPa. ................................ .......................... 68 Figure 22 : Unrolled radial 2D diffraction pattern s for Xe at 2.3 GPa (a) and 101 GPa (b) illustrating the appearance of stacking faults associated with hcp ph ase and the subsequent hcp peak development with pressure . The first appearance of diagonal features indica ting stacking faults occurs at 2.2 GPa . By 101 GPa, stacking faults have transformed into hcp Xe (100) and hcp Xe (101) diffraction lines. ................................ ................................ ................................ ....... 69 Figure 23 : Radial diffraction pattern s for Kr at 10 GPa . Be, Be2, Be3, and BeO peaks are gasket materi al. Kr hcp peaks appear as stacking faults and increase in occurrence and intensity with pressure . Texture in fcc and hcp Kr peaks is interpreted primarily as crystalliza tion texture . No e v idence of significant non - hydrostatic strain in fcc or hcp Kr at 10 GPa. ................................ ... 70 Figure 24 : Radial diffraction patterns for Kr at 94 GPa . Be , Be2, Be3, and BeO peaks are gasket materi al. Kr hcp peaks appear as stacking faults and incr e ase in occurrence and intensity with xv pressure . Deformation texture is prevalent in both fcc and hcp Kr peaks and strain is evident as curvature in both fcc and hcp peaks. ................................ ................................ .............................. 71 Figure 25 : Radial diffraction patterns for Xe at 12 GPa . Be , Be2, and BeO peaks are gasket materi al. Xe h cp peaks initially appear as stacking faults similar to onset of hcp Kr ( Figure 23 and Figure 24 ) and increase in intensity with pressure . Texture at 12 GPa is interpreted as resulting from crystallization of fcc and hcp Xe. There is slight curvature in some diffraction lines, corresponding to qualitatively low non - hydrostatic strain in fcc and hcp Xe at 12 GPa . .... 72 Figure 26 : Radial diffraction patterns for Xe at 101 GPa . Be and BeO peaks are gasket materi al. S tacking faults have completely transformed into hcp peaks, and significant strain is evident in hcp and shared fcc/hcp diffraction lines. Th e fcc 200 diffraction line is broad and highly textured, and strain cannot be reliably assessed for this plane. ................................ ................................ ................................ ................................ ........................ 73 Figure 27 : Volume % hcp in Kr and Xe with pressure . The transition proceeds more quickly in Xe than i n Kr b ut is not complete in either phase at Mbar conditions. Rosa et al. (2018) report 20% conversion to hcp Kr at 20 GPa, which is not reached until 94 GPa in our radial analysis. Xe is not fully transformed to hcp at 101 GPa, 30 GPa greater than the estimate of full conversion to hcp at ~70 GPa (Cynn et al., 2001) . Peaks from the low pressure fcc phase in Xe are seen at 101 GPa in Figure 26 b. ................................ ................................ ................................ 74 Figure 28 : Texture evolution in Kr (left) and Xe (right) . At low pressure, texture in fcc is probably due to crystallization . There is more initial texture in fcc Kr th a n in fcc Xe . Deformation texture becomes evident with increasing p ressure as a texture maximum develops near in both fcc Kr and fcc Xe . Initial texture in hcp Xe differs from hcp Kr, with initial Xe texture near in Xe and near in K r . Deformation texture on increases with pressure in hcp Kr, but texture also develops near By the maximum pressure in both phases, texture in both hcp Kr and hcp Xe is concentrated around ......................... 76 Figure 29 : Crystallite size in Kr (squares) and Xe (circles) for fcc (main plot) and hcp (inset) . Refined grain size decreases with pressure and is ~3 orders of magnitude larger in the fcc phase than in hcp phases . The crystall ite size for fcc Xe is ~2x the size of fcc Kr, but hcp Xe crystallites size parameters refine to smaller values than hcp Kr . Crystallites in the hcp phase for both materials exhibit a slight increase before decreasing again, suggesting grain growth follow ed by compression . ................................ ................................ ................................ ................................ ........................ 77 xvi Figure 30 : Difference in ruby R1 - R2 peak positions vs pressure for rub ies surrounded by RGS media, a measure of hydrostaticity of stress supported by the medium . Ruby peak splitting measured in He medium is inferred to represent ~hydrostatic stress applied to the ruby, and peak splitting values higher than this trend may be observed due to nonhy drostatic stress and anisotropy of the fluorescence behavior for these two fluorescence lines . Ruby peak splitting averaged from 8 or 9 rubies in Ar , Kr, and Xe exhibits an increase in slope indicative of stiffening above 15 - 20 GPa . The difference R1 - R2 for both Kr and Xe is generally larger than for Ar, Ne, and He, indicating that Kr may become less hydrostatic than Ar at ~35 GPa, but Xe is less hydrostatic than all other RGS at all pressures studied . ................................ .......................... 80 Figure 31 : Plots of a(hkl) from axial diffraction lines vs 3×(1 - 3cos 2 r Pt compressed in Kr (top) and Xe (bottom) . The slope and intercept of the gamma plots are used with elastic anisotropy to calculate differential stress . A slope of zero corresponds to hydrostatic conditions, and for materials with positive elastic anis otropy such as Pt, the slope becomes increasingly negative as differential stress on the sample increases. ................................ ................................ . 83 Figure 32 : Differential stress t of metals within RGS media compared to strength of met als, i.e. maximum t observed with no medium. a) Pt in Xe (green circles, this study) and Kr (blue and gray squares, this study) media, and no media: red, gold , pink, and black tr iangles (Dorfman, Shieh and Duffy, 2015) , and b) Ag in Xe (green and gray circles, this study), and no media: red and gold triangles (Liermann et al., 2010) . Under hydrostatic conditions, t is 0. The differential stress supported by Pt in both media remains lower than the strength of Pt below 20 GPa, but then increases rapidly above 20 GPa to match the strength of Pt by ~40 G Pa. Ag has been observed to be weaker, and s tress supported by Ag in Xe is consistent with the flow stress in Ag throughout the studied pressure range. A change in the EOS of Kr at 20 GPa in previous work (Rosa et al., 2018) was attributed to stiffening induced by the fcc - hcp phase transition. ............. 85 Figure 33 : Refined strain Q for hkl from fcc (top , A and B ) and hcp (bottom , C and D ) Kr (left , A and C ) and Xe (right , B and D ) . Strain in peaks from planes shar ed between the fcc and hcp structure fcc (111)/hcp (002), fcc (220)/hcp (110), fcc (311)/hcp (112) generally increases with pressure, with some non - monotonic behavior. For hcp 100 in both Kr and Xe , strain is refin ed successfully, but does not increase sy stematically with P . ................................ ............................ 87 Figure 34 : Compression curve (a) for Kr (blue circles and yellow circles, this study) with EOS (red line, (Rosa et al. , 2018) ), and F - f plot (b) with Kr data from this study and red triangles and gray triangles from (Rosa et al. , 2018) . Red tr iangles represent F - f values below observed stiffening at f E = 0.4 (corresponding to ~20 GPa pressure), and gray triangles above stiffening . Their assessment of deviation from normal behavior at 20 GPa is supported by our data, though the change in slope at f E = 0.4 is less pronounced in our data . The change at 4 0 GPa may be attributed to plastic flow in the Pt pressure standard. ................................ ................................ .... 88 xvii KEY TO ABBREVIATIONS ANL Argonne National Laboratory APS Advanced Photon Source bcc body - centered cubic bct body - centered tetragonal CDAC Chicago - DOE Alliance Ce nter COMPRES Consortium for M aterials P roperties R esearch in Earth S cience s CRSS Critical resolved shear stress DAC Diamond anvil cell DFT Density functional theory EOS Equation of state EVPSC Ela sto - viscoplastic self - consistent fcc face - centered cubi c fc o face - c entered orthorhombic GSECARS GeoSoilEnviro Center for Advanced Radiation Sources hcp hexagonal closest - packed HPCAT High Pressure Collaborative Access Team LLNL Lawrence Livermore N atio nal Laboratory LPO Lattice preferred orientation m.r.d. Mu ltiples of a random distribution pPv post - Perovskite RGS Rare gas sol id XRD X - Ray Diffraction Z Atomic number 1 CHAPTER 1: I ntroduction S trength and deformation behavior are important prope rtie s for understanding the behavior of naturally occurring mate rials under the extreme pressure conditions of planetary interiors, the high stresses involved in accretionary impacts, detonations and defense applica tions, and in industry for the improvemen t of tools and parts . Strength is a fundamental material pr oper ty describing resistance of a material to deformation under applied str ess . It is a result of chemical bonding and the internal atomic structure of th e materi al. Deformation may be elastic an d/ or plastic, and results from the orientation of the applie d st resses and the physica l properties of the materi al. Elastic d eformatio n is reversible, and includes hydrostatic compres sion, which is described by the pressure - volume Equation of State (EOS) of t he material, and elastic strain in response to loading f orce s. The new experimental results presented in this dissertation characte rize elasticity, strength, and plasticity of soli ds . Accurate measurements o f m aterial strength and deformation behavior at high pressure are important for a thorough and fundament al u nderstanding of the mechanics of solids under varying thermodynamic con ditions. 1.1 Significance of elasticity, strength , and plastic deformation in Earth Elastic properties of Earth materials gov ern th e de nsity structure of the planet . Inferences about structures within the Earth, including the layers of the Earth and regional structures in the mantle, subducting tectonic plates (slabs) and th ermo chemical heterogeneities associated with isto ry come predominantly from seismic waves . These properties vary with t hermodynamic conditions , in particular pressure and temperature which reach up t inte rior and range up to hundreds of GPa and 2000 - 10000 K (Melosh, 1989; Kraus et al. , 2012) during the accretion and evolution of planets and other rocky bodies . In Earth, the mechanical properties of pro minent 2 mineral phases at these extreme pressure - temperat ure conditions affect mantle viscosity (Meade and Jeanloz, 1990) . This has important implications f or the evolution of the Earth as it affects mantle convection, but also acts on pla te tectonics by modifying slab subduct ion. Pressure in planetary interiors increases with depth an d ma terials inside planets are subjected to compressive deformation, result ing in changes in the density or in some cases the atomic structure of the mater ial ( e.g . phase transitions) . Elastic v olum e tric compression is described by the EOS . Phase transi tion s may or may not induce a change in the density of a material but may a lter the elastic properties substantially . Density changes, whether from volume tric compression or phase equilibria , aff ect t ransport of materials within planetary bodies and are a maj or component of planetary differentiation and accretion. Yield s trength , the maximum elastic stress supported by a material before it deforms plastical ly, controls the occurrence of earthquake s, t h e boundary between the lithosphere and asthenosphere, a nd t he deformation of planetoids during accretionary impacts . Accretion is the result of impact processes . Strength is thus important to the dynamics and evolution of the interior of the Earth a nd o t her planets . Strength and EOS measurements may be inc orpo rated into hydrocode models for planetary impacts along with other type s of collisions and detonations, an area of overlap between planetary and defens e science . The mechanical response of ma teri a ls is relevant to the development of energetic material s (e xplosives) and damage - resistant materials such as armor and building ma terials . Material strength may also play a role in hazard mitigation . The use of the yield strength and the elastic pro pert i es of metals and silicate minerals in asteroid impact h azar d mitigation models is an ongoing project at US national labs and NASA (Bruck Syal, Michael Owen and Miller, 2016; Cotto - Figueroa et al. , 2016) . 3 The plastic flow of solid minerals controls the dynamics and and possibly the s olid inner core (Wenk et al. , 2011; e.g. L incot et al. , 2016) . Changes in the d eformation mechanisms ( i.e . changes in dominant slip systems during plastic deformation ) in forsterite dri ve thermal convection and development of lattice preferred orientation (LPO) in the upper mantle (Raterron et al. , 2011) , associated with seismic anisotropy . In the lower mantle, sli p systems in Mg O affect viscosity and can res ul t in stagnation of subduc ting slabs (Marquardt and Miyagi, 2015; Mao a nd Zhong, 2018) , and the relative strengths of dominant mineral phases in the lower mantle may prom ote the perseverance of geochemical reser voir s by localizing deformation and mixing to boundary layers (Girard et al. , 2016) . The rheology of hcp iron has also b een used to infer viscosit y and rotation of the inner core (Ritterbex and Tsuchiya, 2020) . Plasticity also occurs under short timescales of accretionary i mpacts on Earth and other rocky bodies . T he yi eld strength of impactors is important for understanding asteroid collisions and crater development on Earth and other rocky bodies (Nolan et al. , 1996; e.g. Kimberley and Ramesh, 2011; Timms et al. , 2012; Rae et al. , 2019) . The rheology of planetary interiors and the mechanics of accretion and impacts are controlled by the strength of a material and the mechanics of its deformation. 1.2 Significance of materials in this work Mechanical properties of materials at high pressure are poorly understood . The strength and de formation properties of even high - symmetry, simple materials such as metals and sal ts are difficult to determi ne strain rates that cannot be replicated in the laboratory . Equation o f state me asurements at Mbar ore suffer from uncertainty in the stress state and calibration at ~few percent level (Dorfman et al. 2012) . Onset of plasticity in materials 4 at high pressure confounds mea surements of elasticity (Weidner et al. , 2004) . Many materials exhibit heterogeneity and co mplex crystal structures, making experimental determination of their properties challenging . WC as well as Xe and Kr cover a wide range of materials properties, from hard t o soft . A ll are simple materials ; the rare gas solids ( RGS ) are considered the pro totypical van der Waals solids and WC is a hard ceramic commonly used in industry . Observing strength and deformation of simple materials at high pressure provides a founda tion for w ork on more complex materials, whether geologic or industrial ( Figure 1 ) . 1.2.1 Tungsten carbide Tungsten carbide is a transition metal carbide and a hard, ultra - incompressible ceramic . Hard ceramics are relevant to the manufacture of strong and durable parts in indust ry . Transition metal ca the deep Earth . In particular, carbides such as Fe 3 C and Fe 7 C 3 are found in superdeep diamond s (Kaminsky, 2012) and u nderstanding their elastic properties may help constrain the density and seismic velocity observed in Ea (Nakajima et al. , 2011; Chen et al. , 2018) . These carbides, along with rare r naturally occurring carbide phases like qusongite (WC) have been found in exhumed ophiolites (e.g. Q. Fang et al. , 2009; Shi et al. , 2009; Kamins ky, 2012) . Transition metal carbides are important catalysts in redox reactions (Levy and Boudart, 1973; e.g. Abdelkareem et al. , 2020) , and WC is used as a redox buffer in high - pressure experiments (Taylor and Foley, 1989) . High hardness and incompressibility help resist surface damage (Yeung, Mohammadi and Kaner, 2016) and WC has widespread use in in dustry for this purpose it is used in the manufacture of protective coatings for tools and chemical containers, as well as in the manufacture of durable cutting tools, and st rong components . Because of these same useful physical properties, WC is also u sed in high - pressure experiments for support and as an 5 anvil material, and as such is fundamental to scientific state - of - the - art in understanding physical properties of mantle minerals . New constraints on the high - pressure elasticity, strength and deforma tion mechanisms of WC will help to develop techniques for measuring these physical properties, and to improve design of anvils for experimental mineral physics . Figure 1 : Strength of geologic and selected materials with pressur e. Black squares, diamond, (Wang, He and Duffy, 2010) , Gray circles, heterodiamond (Dong et al., 2009) , blue squares, ringwoodite (Kavner and Duffy, 2001) , green diamonds, olivine (Uchida et al., 1996) , open gold triangles, MgO (Uchida et al., 1996) , closed gold triangles, MgO (Merkel et al., 2002) , red hourglasses NaCl (open with annealing, closed without annealing, (Funamori, Yagi and Uchida, 1994) , orange triangles SiO 2 (Shieh, Duffy and Li, 2002) , pink diamonds, n eon (Dorfman et al. , 2012) . 6 WC exhibits impressive physical properties that make it a popular choice in industrial applications and important for research technology . It is very strong, ultra - incompressible, and has a high hardness . Nevertheless the incompres sibility ( 1 - bar bulk modulus) reported in the literature is discrepant and may be affected by grain size, with nano - crystalline WC reported to have incompressibility on par with diamond (Lin et al. , 2009) . Despite widespread use of WC , its compressive yield stress and deformation mechanisms at high pressur e have not been studied . Accurate constraints on the equa tion of state and knowledge of the strength and deformation mechanisms of WC under high stress conditions will lead to a better understanding of its response to extreme pressure and stre ss condition s , as well as support improvements in durable parts made f rom WC . 1.2.2 Heavy rare gases krypton and xenon Like carbides, the inert gases (a . k . a . rare gases and noble gases) are important to a range of disciplines, including physics, materials sci ence, and Ea rth and planetary science . Rare gases are extremely impor tant geochemical and cosmochemical tracers (e.g. Jephcoat, 1998; Mandt et al. , 2015) , offerin g insight into the evolution of Earth and the solar system . They are present in Earth and other bodies such as asteroids and comets . The gas planets (e.g . Jupiter and Saturn) are comp o sed of hydr ogen and helium, and understanding the strength and deform ation of gas solids at high pressure will be necessary for understanding the interiors of these planets as well . On Earth, RGS exist both dissolved in magma, where their solubility is affected by magma chemistry, temperature, and pressure (Chamorro - Pérez et al. , 1998; Shibata, Takahashi and Matsuda, 1998) as trapped gases and possibl y as solid phases or bonded with iron and other elements within the Earth (e.g. Ono, 2020; Peng et al. , 2020) , or may be stored withing the lattice structure of other materials (Ono, 2020; Rosa et al. , 2020) . The role of noble gases in the 7 is an active area of research . Xe has been shown both computationally and experimentally to form stable alloy stru ctures with Fe and Ni at conditio n (Zhu et al. , 2014; Dewaele et al. , 2017; Stavrou et al. , 2018) , which has been proposed as a potential reservoir for Xe in the Earth, approximately 90% of which is missing relative to A r and Kr (Anders and Owen, 1977; Jephcoat, 1998) , based on solar and chondritic measurements (e.g. Caldwell, 1997; Lee and Steinle - Neumann, 2006) . More recently, th e core has been proposed as a reservoir for other noble gases, including He and Ne (Bouh if d et al. , 2020) . Rare gas solids are prototypical simple solids and determining their physical properties may improv e our understanding of the compression of solids and the effects of pressure on interatomic potentials in van der Waals materials . Lig hter rare gas solids He, Ne, and Ar are often used as pressure - transmitting media in high - pressure experiments (e.g. Angel et al. , 2007; Klotz et al . , 2009) . Ar, Kr, and Xe are known to undergo a kinetically inhibited martensitic fcc - hcp phase transition ( Figure 2 ) , in which the Gibbs free energy of the two phases are very similar, but the activation energy required to drive the transition is relatively high, resulting in a wide pressure interval in whi ch both the fcc and hcp phases co - exist (Caldwell, 1997; Boehler et al. , 2001; C ynn et al. , 2001; Errandonea et al. , 2002, 2006; Rosa et al. , 2018) . The quality of Kr or Xe as a pressure medi um will depend on their respective strengths an d on the strength of the two - phase fcc - hcp mixture during the ir respective phase transition s . S tudying the strength, deformation, and phase equilibria of Kr and Xe may provide insight into thei r behavior in the deep Earth, while improving our understanding of solid s and discovering novel alloys. 8 Figure 2 : Schematic of t he fcc - hcp phase transition . The two structures can be represented as displaced layers of atoms, where A, B, and C are n ot equivalent closest - packed layers. The fcc structure is ordered ABCABC and the hcp structure is ordered ABABAB. Closest - packed plan es (represented here by the orange diamond) shared between the two structures, result in the continuation of some diffrac tion lines throughout the transition . 1.3 Dissertation purpose and structure In this dissertation, I consider a range of strength and def ormation responses in simple materials by examining a hard, ultra - incompressible ceramic, WC, and two soft solidified ine rt gases, Kr and Xe. WC is a strong ceramic which is widely used in indus try and research technology, yet its quasistatic yield stren gth and plastic deformation mechanisms are not known. The reported EOS in the literature varies substantially. The pres sure - dependence of the single - 9 crystal elastic constants c ij for WC has neve r been measured experimentally. Experiments are needed to r esolve differences in the EOS and to determine strength, deformation behavior and to test theoretical elasticity. Rare g as solids are simple, inert van der Waals materials which are important tra cers of geological processes (e.g. Burnard, Zimmermann and S ano, 2013). They are prototypical solids, but are known to form compounds, become metallic, and undergo sluggish phase t ransitions at high pressure (Caldwell, 1997), behavior which is important t o determining the role of RGS in planetary interiors and for understanding bonding in solids (Jephcoat, 1998; Sanloup, 2020). RGS are frequently used in high - pressure experiments a s pressure - transmitting media. Despite their importance across a range of disciplines, the strength of Kr and Xe as a function of pres sure has not been studied, and the effect of the martensitic fcc - hcp phase transition on strength is not known. Understa nding simple, high - symmetry materials lays the groundwork for measuring str ength and deformation properties of lower - symmetry mineral p hases prominent in the Earth. Precision in measurements of physical properties for materials with simple structures will make determination of these properties in more complex phases, such as ort horhombic bridgmanite and post - perovskite (pPv) in the lower mantle less challenging . The results presented in this dissertation will improve the results of high - pressure experimen ts. Improved loading methods described are important for more successful e xperiments at higher pressures. Oriented WC anvils may exte nd the capabilities of modern high - pressure apparatuses. The effects of phase equilibria on RGS improve our understandin g of quasi - hydrostatic pressure media. In Chapter 2, I review methods used in quasi - static high - pressure experiments, highlighting the axial and radial diffraction geometries, which provide different means of assessing stress and strain at high pressure . In Chapter 3, I determine the compressive yield 10 strength, deformation properties, and Equation of State of tungsten carbide, and assess the e ffects of crystallite size on its EOS . Dr. Lowell Miyagi and Dr. Feng Lin at the University of Utah provide an Ela sto - viscoplastic self - consistent (EVPSC) model to determine plastic deformation mechanisms, which provides insight into the effects of plasticity on strength . Chapter 4 explores of the strength and deformation of solid noble gases Kr and Xe determined fro m multiple complementary methods and examines the effect of the martensitic fcc - hcp phase transition on their respective strengths. In Chapter 5, I discuss the implications of the f indings in Chapter 3 and Chapter 4 . I consider the importance of grain si ze effects, elastic ity , strength, and deformation behavior for future work in Earth science and materials research. 11 CHAPTE R 2: M e thods for m easuring strength and deformation at high pressure 2.1 Introduction The mechanical response of materials to th ermodynamic conditions such as pressure and stress can be either studied experimentally or inferred through theoretical calculations fr om first principles . Experimental measurements may be conducted under quasi - static conditions, where the thermodynamic c onditions remain stable throughout the experiment, or they may be dynamic (time - resolved) . Quasi - static experiments such as the use of X - ray diffraction or optical spectroscopy in the diamond anvil cell provide information on the material u nder approximat ely consistent stress conditions, whereas dynamic experiments include strain - rate effects . The diamond anvil cell is the only method b y which conditions at Mbar pressure s, relevant to the lower mantle and core of Earth , may be obtaine d . Ga skets are used to stabilize the anvils and provide confining stress . Pressure - transmitting media are needed to produce quasi - hydrostatic conditions i n the diamond anvil cell . Pressure within the cell chamber is calibrated by a material within the chamber or from the an vil culets . Synchrotron radiat ion is ideal for X - ray diffraction measurements in the diamond anvil cell because it is bright, focused and tunable . The a xial diffraction geometry may allow for a higher maximum pressure, but strain and texture evolution ar e more clearly observed in the radial geometry . Measurements in the radial geometry may be more sensitive to phase transitions in plas tically deforming materials because a range of stress orientations and texture intensities are observed, some of which ma y be invisible in the axial geometry . A smaller gasket hole helps stabilize the cell chamber for radial experiments to Mbar conditions . In this work, quasi - static compression methods are used to probe the strength in a range of materials. 12 2.2 Quasi - stat ic compression in the diamond anvil cell Diamond anvil cells (DACs) of various configurations have been used to generate pressures up t o ~10 Mbar (1000 GPa), approximately the stability of the diamond phase of carbon (Ruoff, Xia and Xia, 1992; Dubrovi nsky et al. , 2015; Dubrovinskaia et al. , 2016; Dias and Silvera, 2017; Dewaele et al. , 2018) relevant to gas giants and larger rocky planets s uch as super - Earths . The DAC is one of a family of opposed, rigid anvil devices, some of which employ other materials for anvils, such as sapphire or WC (Furuno, Onodera and Kume, 1986; e.g. Besson et al. , 1992; Rüetschi and Jaccard, 2 007) , and which date back to the Bridgman cell (Bridgma n, 1952) . The DAC is also currently the only method by which quasi - static stress at Mbar pressures may be achieved . T he underlying principl e of the DAC includes a cell defined by a chamber constrained by the anvil culets and a gasket, and contains the sample, any internal pressure standards, and the medium (if any) used to transmit pressure . Pressure in the DAC is gener ated by applying a for ce to a sample or pressure - transmitting medium with opposed diamond anvils . Diamond anvils are optimal for high - pressure experiments because of their high strength and optical transparency, which allows a range of techniques to be u sed on materials insid e the cell, including diffraction, spectroscopic analyses, and laser heating . The surface of the anvil which app lies force to the cell chamber is called the culet . The maximum attainable pressure for a given DAC experiment is limite d primarily by the sur face area of the diamond culets et al. , 2018) and the strength of the anv ils . Single - th a beveled edge have been used in experiments approaching 500 GPa (Dias and Silvera, 2017) , where as ppropriate for experimental pressures below 15 GPa . The a nvils are glued to tungsten carbide or X - ray transparent cubic boron nitride (cBN) seats with 13 a strong epoxy and placed in an opposed piston - cylinder arrangemen t, which maintains alignment while all owing the application of force by tightening the cell with screws . Force in the DAC is moderated by spring - cushioning provided by Bellville washers, and balanced by opposed threading (right - and left - handed) in the sc re ws ( Figure 3 ) . Figure 3 : Standard symmetric diamond anvil cell . Bellville washers provide cushioning as hex - screws are tightened . X - rays or optics are admitted through an aperture, are scattered by the sample, and exit an aperture on the opposite end. A gasket stabilizes the anvil alignment further , prevents anvils from coming into contact directly . D iamonds are brittle , superhard , and highly incompressible. Stress from contact is is olated to a very small surface area and would cause them to fracture and fail. The gasket distributes stress across a broader area and provides confining pressure for materials inside the cell . Gasket s materials are typically made from metals that exhibi t a high - strength such as rhenium or tungsten (Duffy, 2007) , or from X - ray transparent m aterials such as beryllium (Hemley, 1997) , 14 cubic boron - nitride (cBN), boron - mica, and/or with Kapton epoxy for radial experiments (Merkel and Yagi, 2005; Miyagi et al. , 2013) . Materials such as Be and cBN may be hazardous, more difficult to use, or add additional signal to collected data . However, they are the only means by which radial experiments can be conducted in the DAC . Stronger materials provide better stability and allow access to higher pressures but are not X - ray transparent and can only be employed i n axial experiments where access through the gasket is not required . Regardless of experimental geometry, gaskets provide support for anvil alignment at high pressure and exert confining pr essure for materials inside the DAC as the chamber is compressed ( Figure 4 ) . Figure 4 : Diamond anvil cell schematic . Two opposed anvils press on a cell chamber defined by the anvil culets and a gasket. The sample is loaded inside the chamber along wi th a pressure - transmitting medium or other internal markers (if desired). 15 2.3 The stress state in the DAC Uniaxial forces generated by the opposed anvils combined with radial confining stress from the gasket induces differential and shear stresses withi n the cell chamber which increase with pressure. The stress state in the DAC can be described in tensor notation (Ruoff, 1975; and e.g. Singh and Takemura, 2001) : ij P + D ij , (1) where 11 is the radial stress component, 33 is the axial stress component , P is the hydrostatic component of stress defin ed by: (2) where mean normal stress , and the stress deviator D ij is given by : , (3) with differential stress t = ( 33 11 ), such tha t: . ( 4 ) This results in both a stress gradient throughout the chamber, and a non - hydrostatic stress condition in all material s loaded in the DAC . Depending on the goal of the experiment, this may or may not be desired . For experiments in which o nly volumetric compression is desired, hydrostatic stress conditions are preferable . Samples compressed in the DAC in a liquid medium will experience h ydrostatic stress . Common media for this purpose include methanol - ethanol mixtures, silicone oil, or He g as . No media are liquid above the room - temperature freezing pressure of helium (~12 GPa) however, so 16 for pressures exceeding this range, pressure - tra nsmitting medium are selected for their ability to maintain quasi - hydrostatic, or low non - hydrostatic stre ss . Since the non - hydrostaticity of pressure - media like methanol - ethanol and silicone oil rises rapidly above 12 - 15 GPa (Angel et al. , 2007; Klotz et al. , 2009) , nobl e ga ses such as He , Ne , and Ar are used for experiments approaching or extending into the Mbar regime . He and Ne are known to diffuse into anvils and sam ple materials and can cause anvil failure and measurement errors (Dewaele et al. , 2006; Sato, Funamori and Yagi, 2011; Shen et al. , 2011) , and the strength of Ar increases rapidly above 15 - 20 GPa (Mao et al. , 2006) . All solid pressure media are non - hyd rostatic and s ubject the sample to non - uniform stresses, which introduces uncertainty into DAC measurements, so care ful decisions about appropriate pressure - transmitting media must be made depending on the needs of the experiment . To observe the hydrostat ic compression of a sample material, pressure transmitting media are required . If deviatoric stresses are desired, as is the case in experiments probing the yield strength and plastic deformation behavior of materials, then no pressure medium needs to be employed, and non - hydrostatic stress will be maximized . 2.4 Pr e ssure calibration, stress markers, and ruby fluores cence Because the stress - state inside the DAC is non - uniform , t he pressure at a given point within the DAC results from a combination of the loading insid e the cell, the stress gradient, mechanical properties of the gasket material, and the distribution of the force over screws and washers that drive uniaxial load . The non - uniform stress conditions in the DAC are a source of systematic experi mental uncerta inty in pressure calibration (e.g. Meng, Weidner and Fei, 1993; Wu a nd Bassett, 1993) , so obtaining an accurate measurement of pressure inside the cell chamber requires an internal barometer . Such a pressure - standard could be sample itself, if the equation of state (EOS) is known, or an internal pressure standard may be loaded with the 17 sample (e.g . A g and ruby in Figure 5 ) . L ine shift analy sis of Raman scattering from the diamon d culet may also be used to determine pressure within the cell , as can the EOS of the pressure - transmitting medium . Internal measurements of pressure may be accomplished in several ways . Figure 5 : View of the DAC chamber through the anvils before loading compressed gas . The cell is pre - loaded with an Ag pressure standard and a ruby ball f or gas - loading of Kr or Xe gas . The anvils are 300 um culets appropriate for pressures up to ~70 GPa . The Re gasket has a 150 Each method of pressure - determination has benefits and limitations , depending on the individual experiment . Direct determination from the EOS of the sample or pressure transmit ting medium obviates the need for loading other materials and can reduce signal not associated with the sample . In many cases, the P - V relations hip of the sample is not known however, or the sample 18 may undergo a structural transition to a differe nt phase, making pressure determination in this way unreliable . Standards are therefore usually a well - characterized material loaded with the sample . P ressure scales from internal standards are based on the equation of state (EOS) of a material loaded w ith the sample . This EOS is determined from independent measurements of volume or density and pressure, elasticity, or compressibility . These measuremen ts can come from temperature - reduced shock wave (e.g. Nellis, 2007) measurem ents , or the combination of measurements of elasticity with ultrasonic interferometry or Brillouin measurements and compressibility measurements from neut rons or X - ray diffraction (Poirier, 2000) . The ruby pressure - scale is particularly well - known and frequently reviewed (Forman et al. , 1972; Mao, Xu and Bell, 1986; Dorogokupets and Oganov, 2006; Syassen, 2008a; Shen et a l. , 2020) . The ruby p ressure scale has been benchmarked against reduced shock isotherms and ultrason ic measurements of several metals (Dewaele, Loubeyre and Me zouar, 2004; e.g. Dewaele et al. , 2008) , and a metal with a well - characterized EOS may be loa ded in s mall quantities with the sample and its EOS used to calculate pressure . Foils are often employed for this purpose as their thickness can be constrained a nd the chances of bridging the anvils is minimized . Internal barometers provide a reliable me ans of c alculating pressure as it occurs inside the DAC. Pressure can be considered the hydrostatic component of stress . To constrain the non - hydrostatic stresse s inside the DAC chamber, internal stress markers are required as well . If the elastic deform ation of a material inside the cell can be constrained, it may be used to determine the stress state of the surrounding medium . Internal barometers such as metal foils loaded with the sample may double as an internal stress marker in this way, provided th ey do no t deform plastically . An effective stress marker must therefore be stronger than the pressure - transmitting medium and bridging of the marker must be avoi ded . If the internal standard generates an X - ray 19 diffraction pattern, a stress analysis may b e perfor med on the pattern of the stress marker similarly to the sample and this provides a constraint on the stresses exerted on the marker . Because diamonds a re optically transparent, spectroscopic methods such as ruby fluorescence (Mao et al., 1978, 1986) and Raman (Dubrovinskaia et al., 2010) can be used in DAC experiments as a complementary analysis to X - Ray diffraction or stand - alone experiments to characte rize the stresses affecting the sample (e.g . Piermarini, 1973; Klotz et al., 2009 ) . T his can be acco mplished with fluorescence line shift analysis, or pressure gradients analysis . 2.5 Synchrotron r adiation Synchrotron X - rays are ideal for diamond anvi l cell experiments because the y provide a high density of coherent photons focused to a small beam siz e , and desired energy can be selected . A small beam size is critical as the surface area of the anvils is small so that pressure is maximized and samples must not only be smaller than the anvil culets but must also be small enough to fit in side the gasket chamber, which must 1/3 1/2 the diameter of the culet to avoid closure of the gasket hole while stabilizing anvil alignment . Synchrotron radiation pro vides sufficient brilliance such that well - resolved patterns may be collected from very small samples, using short collection times (as low as ~30s) . In a synchrotron, c harged particles (electrons or positrons) are accelerated around the accelerator ring by bending magnets and insertion devices, resulting in emission of photons that are col lected at stati ons tangent to the ring . Each station has instrumentation suitable to the preferred or specialized techniques of the work conducted at that beamline , suc h as powder or single - crystal diffraction, time - resolved experiments, or low angle/surf ace diffraction . Synchrotron radiation provides X - rays with a beam diameter small enough to probe the inside of the chamber, and brilliant enough to produce a signal fr om small sample volumes. 20 2.6 X - Ray diffraction, diffraction geometry, and texture evalu ation X - rays tr ansmitted through the diamond anvils and the sample chamber are diffracted by the sample . P hotons are scattered according to the Bragg law: , where n is a positive integer, is the X - ray wavelength, d is the lattice - spacing , a function of hkl , and is the scattering angle . The high photon intensity at characteristic angles may be analyzed as thermodynamic conditions are vari ed to observe changes in material properties such as unit cell volume or structure . Because the dia mond anvil cell restricts optical access to the sample and limits sample size , a ll of the diffraction experiments in this work utilize high - energy, high - bri lliance synchrotron angle - dispersive diffraction at a fixed energy ~ 25 - 40 keV , resulting in typical Bragg angles <~ 15° . If the sample is a powder with random crystallite orientation, d iffracted X - rays produce Debye - Scherrer rings when collected on a 2D detector such as an image plate . Each ring is representative of an individual lattice plane hkl , whi ch can be rel ated to the lattice parameter of the sample through the diffraction angle and X - ray wavelength . Changes in d iffraction rings may be analyzed to determine physical properties, for example the unit cell volume of a material under a specific the rmodynamic co ndition, such as temperature or pressure , may be determined from lattice parameter (a function of the diffraction ring position) at that condition . The analysis used will vary depending on the diffraction geometry employed. X - ray diffraction is used in DA C studies to characterize structure and determine changes in the lattice spacing of a sample as thermodynamic conditions such as temperature or pressure are varied . radial - geometry . In axial diffraction, the X - ray probe is transmitted through the diamon d anvils ( Figure 6 ) . There is no need for an X - ray transparent 21 gasket, and maximum pressure s may be achieved . In this configuration, the beam is approximately parallel to the loading axis . The angle between the diffracting planes and the loading axis ( Figure 6 ) is maximized in this case and the direct ion of minimum strain is sampled. Diffraction pat terns collected in the axial geometry are integrated to 1D line patterns and fit with Voigt or Lorentzian line profiles . Peak positions from the line fits are used to determine lattice parameters . Volumes for EOS calculations may be determined from latti ce parameters of diffraction patterns collected under hydrostatic conditions . In the absence of a pressure medium, non - hydrostatic stress is maximized the material being studied is effectively its own pre ssure - transmitting medium . Variability in the la ttice parameter for different hkl arises from non - hydrostatic compression and can be used to derive strength from lattice strain theory if the elastic anisotropy (a function of the single - crystal elastic con stants) of the material is known (Singh, 1993) . This is theoretically possible for any crystal system, however in practice it is difficult to apply to many materials in axial patterns because knowledge of the lattice parameter under hydrostatic stress conditions is required . This can be determined from non - hydrostatic compression data for cubic materials because of the linear relationship between orientation and lattice parameter for these materials (Singh, 1993) , but a priori knowledge of this parameter is required to apply the lattice strain theory to lower - symmetry materials using the axial diffraction geometry, and this may not be readily available for the material in question . 2.6.1 Axial g eometry In the axial diffra ction geometry, the X - ray probe is transmitted through the length of the DAC and the angle of conical openings in the seats to which the diamonds are mounted limits k - space availability . Typical angular ranges in th . This can be alleviated somewhat by small rotations of the DAC relative to the beam . This limitation 22 also ensures that most of the diffracted planes are ~parallel to the beam and load axis, which is the direction of minimum strain . Sampling only planes in the direction of minimum strain may cause structural changes to be detected at incorrect pressures . Development of preferred orientation (texture) during plastic deformation of the sample may induce minimize dif fracted intensity in any direction relat ive to the stress load . If these minima occur in the direction of minimum strain (i.e . parallel to the stress load) , it is possible peaks may be absent from the collected pattern. 2.6.2 Radial g eometry In the radia l geometry ( Figure 6 ), the beam is transmitted through an X - ray transparent gasket, and is norm al to the loading axis . Radial diffraction generates diffraction patterns that sample st r ain in all orientations relati ve to the stress axis . This allows characterization of the deviatoric strain and observation of texture development ( i.e . development of variation in diffracti on line intensity) as pressure is increased . Nevertheless, t he maximum pressure accessible wit h the diamonds may be further limited in the radial configuration by the need for expensive and sometimes toxic X - ray transparent gaskets, which are weaker and m ore brittle than gaskets used in the axial geometry . P anoramic DAC s are optimized for radial d iffraction experiments . The piston and cylinder are separated by long post s, so the X - ray transparent gasket is the only detectable signal between the cell cham ber and the detector . This provides wide access to k - space, but panoramic DACs can be very dif ficult to align . Radial experiments to high - pressure can therefore be especially challenging . For this work, use of a smaller gasket he culet diameter, rather than ½) helped stabilize the panoramic cells to reach Mbar pressure s. 23 Figure 6 : Diffraction geometries using the diamond anvil cell. In the axial geometry (top) the X - ray beam is parallel to the loading axis the angle between the plane normal and the load axis is a function of the diffraction a ngle 2 . Due to the structure of the DAC, most di ffracted planes which reach the detector have small 2 and is close to 90°, the minimum strain condition. In the radial geometry (bottom) the beam is normal to the loading axis, and varies from 0° to 9 0°, so the full range of strain is sampled. In this work, radial diffraction is analyzed with full - profile Rietveld refinement of 2D diffraction patterns . Each pattern is divided into 5° wedges using Fit2D software (Hammersley, 1997, p. 2 ) . The wedges are aggregated in Materials Analysis Using Diffraction (MAUD) 24 software (Lutterotti, Matthies and Wenk, 1999) . Instrument parameters such as detector distance and beam position are calibrated from CeO 2 or LaB 6 standards . In addition to refinement of structural parameters, crystallite size, and microstrain, MAUD allows for refinement of deviatoric strain by fitting eccentricity in the diffraction rings with Q - values . Lattice strain theory can thus be applied more readily to r adial patterns as diffraction ring Q - values can be fit for individual hkl . For radial lattice strain analysis, the shear modulus of the material must be known, either from previous experiments or from theoretical computations. Upon compression, plast icall y deforming materials may exhibit a preferred crystallite orientation or texture . Texture is a function of orientation and may cause diffraction line intensity to be biased in the axial geometry . In the radial geometry, texture produces intensity va riati on around the diffraction rings . Various models for fitting and analyzing texture in MAUD are available . Texture in this work is fit using the Entropy - modified Williams - Imhof - Matthies - Vinel (E - WIMV) texture model (Lutterotti et al. , 2004; Chateigner, Lutterotti and Morales, no date) . 2.7 Conclusion Experimental high - pressure techniques provide a means to constrain the physical response of materials to varyi ng thermodynamics, which is important for advancing scientific research and finding novel materials and applications for industry . The diamond anvil cell enables these properties to be examined under quasi - static conditions to Mbar pressures, using a vari ety of X - ray and optical techniques . In this dissertation, I use these techniques to study mechanical properties in a range of materials . 25 CHAPTER 3: S trength, deformation , and equation of state of tungsten carbide to 66 GPa This chapter has been su bmitted and is under minor revision s B enjamin Lee Brugman, F . Lin, M . Lv, C . Kenney - Benson, D . Popov, L . Miyagi, and S.M . Dorfman Abstract Strength, texture, and equatio n of state of hexagonal tungsten monocarbide (WC) have been determined under quasi - hydrostatic and non - hydrostatic compression to 66 GPa using angle - dispersive X - ray diffraction in the diamond anvil cell . Quasi - hydrostatic compression in a Ne pressure med ium demonstrates that nanocrystalline WC is slightly less incompressible than bulk - scale WC, with respective bulk moduli of K 0 = 397 ± 7 and 377 ± 7 GPa and pressure derivatives K 0 3.7 ± 0.3 and 3.8 ± 0. 3 . This decrease in incompressibility with grain size is similar to behavior observed in other ceramics . Under nonhydrostatic compression, WC supports a mean differential stress of ~ 12 - 16 GPa at plastic yielding, which occurs at ~30 GPa . Strength in WC is anisotropic, with the (001) plane supporting 6 8 - 70 % higher stress than stresses calculated from mean strain . Simulations using an Elasto - ViscoPlastic Self - Consistent model (EVPSC) indicate that strength inferred from lattice strain theory may be an ove restimate due to effects of plastic deformation . Plastic deformation generates a texture maximum near in the compression orientation, initially through prismatic slip on the and slip systems, followed by activation of p yramidal slip on at ~40 - 50 GPa . 3.1 Introduction Tungsten monocarbide is a transition metal carbide used extensively in industrial and research technology because of an abundance of useful physical propert ies, including high strength and hardness, ultra - incompressibility, wear resist ance, and high melting temperature 26 (Amulele et al. , 2008; Lin et al. , 2009; Gol ovchan, 2010; Roebuck, Klose and Mingard, 2012; Yeung, Mohammadi and Kaner, 2 016) . Because of its high strength, synthetic WC is used as a sturdy backing for abrasives on modern industrial cutting tools, in wear - resistant coatings (Cook and Bossom, 2000; Lee and Gilmore, 1982; Li et al. , 2010; Benea and Ben ea, 2015) , and in the manufacture of anvils and support structures in high - p ressure apparatuses 1,10 and e.g. 11 . Improvin g the strength of parts made from WC and extending the pressure range accessible by high - pressure devices are active areas of research (Yamazaki et al. , 2019; e.g. Silvestroni et al. , 2020) . WC is also one of the least compressible materials known, wi th a bul k modulus comparable to other incompressible materials such as Os - borides, cBN, and cRuO 2 (Lee and Gilmore, 1982; Haines, Léger and Bocquillon, 2001; Gilman, Cumberland and Kaner, 2006; Gu, Kra u ss and Steurer, 2008) . WC has also been useful to high - pressure/temperature redox chemistry: it was originally discovered by reduction of tungsten oxide (Z. Z. Fang et al. , 2009; Borovinskaya, Vershinnikov and Ignatieva, 2017) , a reac tion that defines the WC - WO redox buffer used in geochemistry (Taylor and Foley, 1989) the rare mine ral qusongite (Q. Fang et al. , 2009) . Despite these remarkable properties and widespread applications, the strength and deformation mechanisms of WC under extreme quasi - static stress have not been studied. Constraints on the equation of state (EOS) of WC are important for understanding its response to extr eme conditions and chemistry (Cheng et al. , 2012) . Experimental wor k on WC has reported values of the ambient bulk modulus, K 0 , ranging from 329 - 452 GPa (Brown, Armstrong and Kempter, 1966; Lin et al. , 2009) depending on method and grain size of WC . EOS measurements for WC based on X - ray diffraction of samples compressed in a multianvil device under hydrostatic condition s with high - temperature annealing have been reported to 30 27 GPa (Litasov et al. , 2 010) . Relative to these measurements, experiments conducted in the diamond anvil cell (DAC) have yielded systematically higher volumes and incompressibility under pre ssure, possibly due to nonhydrostatic stress . Nano - grain - size WC was also suggested t o be much more incompressible than bulk - grain - size WC, with K 0 ~452 GPa, similar to diamond (Lin et al. , 2009) . In general, effects of nanoscale grain sizes on bulk incompressibility are not clearly systematic : a few - 10s - nm grain size cubic BN (Le Godec et al. , 2012) , Al 2 O 3 (Chen et al. , 2002) and TiO 2 (Al - Khatatbeh, Lee and Kiefer, 2012) have been observed to be less incompressible than bulk samples, while nano - grain - sized noble metals Au, Ag, and Pt appear more incompressible than micron - scale grain sizes (Gu, Krauss and Steurer, 2008; Mikheykin et al. , 2012; Hong et al. , 2015) . For other materials such as Fe, TiC, and TiN, observations suggest that grain sizes have either no effect or nonmonotonic effects on bulk modulus (Chen et al. , 2001; Q F Gu et al. , 2008; Wang et al. , 2014) . Recent first - principles studies pr ovide values for K 0 for WC that mostly cluster in the center of the experimental range for bulk WC ~380 - 390 GPa . Additional experiments are needed to reconcile these differences in observed and predicted bulk compression behavior. Anisotropic elasticity o f WC has also been studied by both experiments and theory, but limited high - pressure constraints are available . The elastic stiffness coefficients c ij for WC have been studied at ambient conditions experimentally (Gerlich and Kennedy, 1979) and computationally [2,8,32 - 36] . Theoretical predictions for c ij s as a function of pressure have been computed to 100 GPa (Liu et al. , 2013; Li et al. , 2014) , and agree well with previous experimental values at ambient conditions, with the exception of c 13 , which is consistently predicted to be ~100 GPa lower than the experimental value (Gerlich an d Kennedy, 1979) . The 28 pressure - dependence of the c ij s of WC has not been measured experimentally, and experimental tests of theory are required. The high strength, i.e . maximum stress before transition from elastic to plastic deformation of WC and other strong metal - light element compounds is linked to covalent bonding which impedes deformation mechanisms common in metals . In hexagonal WC, carbon atoms are positioned as interstitial layers in what would be an otherwise softer (though among the st rongest of all metals) hexagonally closest - packed sub - lattice of W atoms (Goldschmidt, 1967; Liu, Wentzcovitch and Cohen, 1988) . This interstitial positioning combined with the density of valence electrons promotes strong covalent W - C bonding (Liu, Wentzcovitch and Cohen, 1988; Yeung, Mohammadi and Kaner, 2016) . In addition, the incomplete 5 d band in W atoms promotes replacement of the softer metallic W - W bonds by W - C covalent bonds, increasing the hardness and inc omp ressibility of WC relative to WN, which has similar structure but different valence states . The interstitial C atoms also impede the movement of dislocations within the lattice during strain and act to prevent basal slip, which is common ly observed in hexagonal materials (Bolton and Redington, 1980) . Slip at ambient conditions activates in the closest - packed directions and is prismatic on and , and Burgers vector has been noted as dislocation decomposition of (Takahashi and Freise, 1965; Bolton and Redington, 1980) . T his blocking of common slip systems and of dislocation motion in general increase hardness and strength by resisting plasticity (Yeung, Mohammadi and Kaner, 2016) . Ultim ately, there is still sufficient metallic character such that WC only reaches a Vickers hardness of ~30 GPa (Teter, 1998; Haines, Léger and Bocquillon, 2001) , making WC harder than many industrial ceramics, but substantially soft er than superhard (Vickers hardness > 40 GPa) material s like 29 diamond (e.g. Irifune et al. , 2003) and cubic boron nitride (Sumiya, Uesaka and Satoh, 2000; Dubrovinskaia et al. , 2007) . However, at high pressures, the strength, hardness, and slip mechanisms of WC have not been studied . Because the high - pressure compressi ve yield strength is related to both hardness and bulk modulus, WC is expected to yield at lower stress relative to superhard materials, but comparable or higher stress than yielding in other ultra - incompressible ceramics . Elastic and plastic anisotropy i nduced by interstitial carbon layers may translate into slip strength anisotropy in the WC lattice. To characterize the strength, deformation, and the equation of state of WC with pressure, we compressed hexagonal WC powder of bulk (microcrystalline) and n anocrystalline grain size to 66 GPa with X - ray diffraction in the diamond anvil cell (DAC) . Complementary Elasto - ViscoPlastic Self Consistent (EVPSC) simulations on textures and lattice strains were carried out to infer the plastic deformation mechanisms and strength at high pressures . Our results extend the pressure range of the quasi - hydrostatic EOS of WC to 59 and 64 GPa for bulk and nanocrystalline WC, respectively, and offer new constraints on strength and plastic deformation mechanisms of WC . 3.2 Experimental d etails 3.2.1 Sample p reparation and l oading Microcrystalline (Alfa Aesar) and nanocrystalline (Inframat) hexagonal WC powders were used as sample materials . Initial grain sizes of these materials were determined to be 1.2 d 54 nm using hkl - de pendence of size and strain broadening (Popa, 1998) based on peak widths for ambient X - ray diffraction (XRD) patterns collected using a Bruker DaVinci D8 Characterization. 30 Volumetric com pression under hydro static c onditions and strain and texture development under non - hydrostatic conditions were investigated in WC in diamond anvil cells . For hydrostatic experiments, WC powder was loaded with internal standards Au (Alfa Aesar) and ruby wi thin a Ne medium using the C OMPRES/GSECARS gas - loading apparatus (Rivers et al. , 2008) . Each sample was enclosed by a rhenium gasket pre - indented to ~40 - ickness with ~150 - - . For nonhydrostatic experiments, WC powder was packed without a medium and an Au foil s tandard was placed on top . An X - ray transparent beryllium gasket pre - 100 - - diameter sample chamber hole was used with a 2 - pin panoramic DAC with 300 - anvil culets . Gaskets were machined using the HPCAT laser cutting facility (Hrub iak et al. , 2015) . Samples were compressed in 2 - 10 GPa steps up to m aximum pressure of 66 GPa, with pressure at each step calculated using the equation of state for Au (Dewaele, Loubeyre and Mezouar, 2004) . 3.2.2 X - ray diffraction in the DAC Upon compression, synchrotron X - ray diffraction was obtained using b oth axial diffraction geometry in a symmetric DAC in which the X - ray p robe was parallel to the loading axis (both grain sizes), and the radial diffraction geometry in a panoramic DAC in which the incident X - rays were perpendicular to the loading axis (bulk WC only) . Angle - dispersive X - ray diffraction (ADXD) was conducted at the High - Pressure Collaborative Access Team (HPCAT) beamline at Argonne National Lab, Sector 16 - BM - D . X - rays monochromatized to 40 keV (axial experiments) or 37 keV (radial experiments ) were focused to 4 - 6 - - Baez focusing mir rors and collimated using a 90 - . Diffraction patterns were collected 31 for 60 - 80s on a MAR2300 image plate detector and detector geometry was calibrated using a CeO 2 standard . Diffraction patterns were masked to eliminate saturated intensity and integrated to 1 - D profiles using Fit2D (Hammersley et al. , 1996) or Dioptas software (Prescher and Prakapenka, 2015) . For data collected in the axial geometry, diffraction peaks were fit to Voigt lineshapes using the IgorPro MultipeakFit mod ul e . For analysis of data collected in the radial geometry, each pattern was divided into 5° azimuthal wedges over the full 360° azimuthal range for full - profile Rietveld refinement with Materials Analysis Using Diffraction (MAUD) software (Lutterotti et al. , 1997; Lutterotti, Matthies and Wenk, 1999) . The synchrotron instrument parameters in MAUD were refined using the CeO 2 standard . Sample parameters, including polynomial backgrounds, lattice co nstants, grain size, and microstructure were refined at each pressure step . . To include the maximum number of diffraction lines from WC in our calculations and to m inimize the effects of peak overlap, Q - values for higher - order parallel planes were fixed equal to the lowest order plane to which they were parallel . Be and BeO phases (at 1 bar) were included in the refinement to model diffraction from the gasket peak s . Texture in all phases was fit using the Entropy - modified Williams - Imhof - Matthies - Vinel (E - WIMV) texture model (Lutterotti et al. , 2004; Chateigner, Luttero tti and Morales, no date) with an imposed fiber symmetry . The orientation distribution function (ODF) was exported from MAUD and inverse pole figures were plotted using the BEARTEX software (Wenk et al. , 1998) . Pressure was calculated from unit cell volumes of Au determined by fitting the (111) diffraction peak in the 5° azimut h . 32 3.3 Results and i nterpretation Representative diffraction patterns for bulk and nano - crystalline WC compressed in Ne are presented in Figure 7 . All observed d iffraction peaks correspond to the WC sample, Ne medium, Au pressure standard, and Re gasket . Ne peaks (highly textured spots) and diamond Figure 7 : Representative synchrotron X - ray diffraction patterns of bulk and nano - crysta lline WC compressed in Ne pressure medium with Au pressure standard and Re gasket in the axial diffraction geometry . spots were masked to remove overlap with WC sample . Only non - overlapped WC and Au diffraction lines were used to determine unit cell pa rameters . Lattice spacings for WC ( 001 ) , ( 100 ) , ( 101 ) , ( 110 ) , and ( 111 ) and Au (111 ) , ( 200 ) , and ( 220) were fit by least squares with 33 UnitCe ll Software (Holland and Redfern, 1997) . The resulting unit cell volumes for both nano - crystalline WC and bulk WC are pr esented in Figure 8 . Figure 8 : Compression of bulk (blue circles, with 3 rd order Birch - Murnaghan equation of state fit in blue line) and nanocrystalline (yellow circles, with EoS fit in yellow line) WC in Ne compared with other studies. Pressure was determined from the EOS of Au, using the (111), (200), and (220) Au peaks and t he pressure scale of Dewaele (Dewaele, Loubeyre and M ezouar, 20 04) . Data from previous studies were obtained in the multi - anvil press (Litasov et al., 2010) (black open circles) and in the DAC for bulk (red open squares (Amulele et al., 2008) ) and nano - crystalline WC (green open triangles (Lin et al., 2009) ). Ultrasonic measurements (Amulele et al., 2008) are displayed in red solid line and extrapolated with red dashed line. 3.3.1 Equation o f s tate and linear compressibility Volume - pressure data collected in the axial geometry for WC compressed in Ne medium ( Figure 8 ) were fit to a 3 rd order Birch - Murnaghan equation of state (BME), yielding EOS paramet ers of V 0 3 , K 0 = 397 ± 7 GPa, and K 0 V 0 = 3 , K 0 = 377 ± 7 GPa, and K 0 - crystalline WC . With V 0 fixed 3 3 for bulk and nano - WC respectiv ely, the fit 34 results in higher K 0 and lower K 0 K 0 = 412 ± 4 GPa, and K 0 K 0 = 388 ± 5 GPa, and K 0 . These values are tabulated with previous work in Table 1 . Previous studies in the DAC report h igher K 0 but lower K 0 our work or the work by (Litasov et al. , 2010) . In addition to non - hydrostatic stress, the trade - off between K 0 and K 0 the values for the EOS parameters ( Figure 9 ) . Comb ined with independent measurements of elasticity from ultrasonic (Lee and Gilmore, 1982; Amulele et al. , 2008) and shock wave (Dandekar, 2002) studies, the consensus value for K 0 is ~380 - 400 GPa, which is consistent with our data particularly with the bulk fit where V 0 3 . The range of K 0 our data and the consensus K 0 is ~3.6 - 4.3 . In comparison, density functional theory (DFT) predictions using both the local density approximation (LDA) and generalized gradient approximation (GGA) all obtain a higher V 0 than that measured in our samples at 1 bar, and accordingly suggest K 0 sis tent within uncertainty with our results ( Figure 9 ) . Our experiments indicate that the bulk modulus of nano - crystalline WC is lower than that of the bulk material, and consistent with the consensus of ultrasonic, s hock wave, and DFT EOS. The ratio of the hexagonal lattice parameters c / a can indicate a convolution of anisotropic elasticity and anisotropic stress . Our experimental values for c/a in bulk WC compressed hydrostatically in Ne medium indicate a systemati cally lower ratio than oth er DAC XRD studies ( Figure 10 ) . Again , note that the axial XRD in the DAC samples crystallites oriented near the direction of minimum stress . Anisotropic stress combined with anisotropic elasticity will result in systematic differences in lattice parameters c and a calculated from diffraction lines at the minimum stress orientation . Systematically higher c/a ratio from studies of WC under non - hydrostatic compression in the axial geometry indicates anisotropy in linear compressibility . 35 Table 1 : Experimental and theoretical values for the equation of state parameters for hexagonal tungsten monocarbide . US = ultrasonic, SW = shock wave, DAC = diamond anvil cell, MAP = multi - anvil press, XRD = X - ray diffraction, LD A = local - density approximation, GGA = generalized gradient approximation, PWP = plane wave potential, PBE = Perdew, Burke, and Ernzerhof, LMTO = linear muffin - tin orbital. V 0 (Å 3 ) K 0 (GPa) K 0 ' Grain Size Method Reference 20.4667 329 - not specified US (Brown, Armstrong and Kempter, 1966) 20.707 - 20.747 383 - not specified SW (Dandekar, 2002) - 390.3 - not specified US (Lee and Gilmore, 1982) 20.806 ± 0.020 383.8 ± 0.8 2.61 ± 0. 07 Bulk US (Amulele et al. , 2008) 20.806 ± 0.020 411.8 ± 12.1 5.45 ± 0.73 Bulk DAC XRD, NaCl, silicone oil, and 4:1 methanol - ethanol solution (Amulele et al. , 2008) 20.749 452.2 ± 7.8 1.25 ± 0.53 Nano DAC XRD, silicone oil (Lin et al. , 2009) 20.750 ± 0.002 384 ± 4 4.65 ± 0.32 Bulk MAP XRD, MgO (Litasov et al. , 2010) 20.75± 0.00 387 ± 5 4.38 ± 0.40 Bulk MAP XRD, MgO BM - EOS fit to (Litasov et al. , 2010) 20.77 ± 0.01 397 ± 7 3.7± 0.3 Bulk DAC XRD, Ne This study 20.74 (fixed) 412 ± 4 3.3± 0.2 Bulk DAC XRD, Ne This study 20.75 ± 0.01 377 ± 7 3.8 ± 0.3 Nano DAC XRD, Ne This study 20.72 (fixed) 388 ± 5 3.5± 0.2 Nano DA C XRD, Ne This study 36 Table 1 (continued) V 0 (Å 3 ) K 0 (GPa) K 0 ' Exchange - corre lation functional Reference - 655 - not specified (Zhukov and G ubanov, 1985) 404 - GGA (Christensen, Dudiy and Wahnström, 2002) 20.5267 382.4 - GGA (Suetin, Shein and Ivanovskii, 2008) - 382.4 - GGA (Shein, Suetin and Ivanov 2008) - 392.5 - LDA (Su et al. , 2009) 20.749 390.2 ± 0.5 4.19 ± 0.04 LDA (Lin et al. , 2009) 20.6558 393 - GGA (Li et al. , 2010) 20.6558 400.9 4.06 GGA (Li et al. , 2010) 21.240 356 - GGA (Kong et al. , 2010) 21.33 373 4.40 GGA (Cheng et al. , 2012) - 389.4 4.16 GGA (Liu et al. , 2013) 20.99 389.6 4.27 GGA (Li et al. , 2014) 37 Figure 9 : Ellipses representing 95.3% confidence in K 0 and K 0 ' obtained from Birch - Murnaghan equation fit to pressure - volume data for bulk (blue, this study and black, room temperature data from multi - anvil (Litasov et a l., 2010) ), and nano - grained WC (yellow). Dashed ellipses with open circles are w ith V 0 fixed to ambient XRD measurements, solid ellipses with solid circles are with V 0 fit. The open gray circle indicates reported values from multi - anvil, high - temperatu re/pressure EOS (Litasov et al., 2010) . Solid gray squares are from other DAC stu dies on nano - crystalline (Lin et a l., 2009) and bulk (Amulele et al., 2008) . Red triangles indicate values obtained from th eoretical calculations (Lin et al., 2009; Li et al., 2010, 2014; Cheng et al., 2012; Liu et al., 2013) . The solid red diamond is the adiabatic bulk modulus from ultrasonic experiments (Amulele et al., 2008) . Dashed lines are from shock wave (black line (Dandekar, 2002) ) and ultrasonic interferometry experiments (gray line (Zhukov and Gubanov, 1985) ), which only constrain K 0 but not K 0 ' . 38 Figure 10 The ratio c/a vs P for bulk - and nano - grain WC from experimental measurements . Values obtained using nonhydrost atic media (open circles) are systematically higher than with hydrostatic media (filled circles, this study). The linear compressibilities a and c may be determined from their relations to the bulk modulus and the pressure dependence of the c/a ratio in a hexagonal material (Singh, Mao, et al. , 1998; Duffy et al. , 1999) : ( 5 ) ( 6 ) Under hydrostatic conditions, the c/a ratio of WC increases non - linearly with pressure, so the slope of its pressure dependence cannot be accurately represented wit h a constant value . To determine the pressure dependence of c/a , we determined lattice para meters a and c and K(P) from the quasi - hydrostatic diffraction data and fit a least - squares 3 rd order BME to parameters a 39 and c to obtain parameters a(P) and c(P) a nd computed numerical derivatives of the ratio c/a at each pressure step. Experiments and th eoretical computations agree that the a direction of WC is more compressible than c . Our BME fit of lattice parameters for WC yields linear ambient bulk moduli of K a = 366 GPa and K c = 456 GPa for bulk WC and K a = 359 GPa and K c = 407 GPa for nanocrystall ine WC . The value of K c for nanocrystalline WC compressed in Ne medium is lower than for bulk WC, and lower than the value reported in previous experiments on nano crystalline WC (Lin et al. , 2009) . 3.3.2 Dif ferential s tres s and e lastic s tiffness c oefficients Without a hydrostatic medium, a sample in an opposed anvil device such as the DAC 3 parallel to the direction of the compression by the diamonds 1 (Ruoff, 1975) . T he difference between these stresses is termed the differential stress . In order to characterize the effects of non - hydrostatic stress on deformation of anisotropic materials, the radial diffraction geometry allows obs ervation of strains at a wide range o f orientations relative to the orientation of maximum stress . upon compression of WC are presented in Figure 11 . Diffraction lines of WC under anisotro pic strain exhibit varying d - spacing along the azimuthal angle . The measured d - spacing d m deviates from the hydrostatic d - spacing d p as a function of t 40 Figure 11 : X - ray diffraction pattern data (lower half of each image) and full - profile refinements (upper half of each image) for selected pressures a) 16 GPa, b) 34 GPa, c) 48 GPa and d) 66 GPa . The Debye - Scherrer rings are transformed to azimuth vs 2 and Miller indices for WC are labeled in each pattern . WC peaks exhibit increasing sinusoidal curvature with pressure due to non - hydrostatic strain . Systematic variation in intensity in individual diffraction lines is indicative of plastic deformation and also increases with pressure in WC . Diffracting planes from gasket materials Be and BeO at ambient conditions are also obse rved and exhibit no strain. 41 by the non - hydrostatic lattice strain Q ( hkl ) for individual lattice planes hkl (Singh and Balasingh, 1994; Singh, Balasi ngh, et al. , 1998) : . ( 7 ) Our detection limit for Q ( hkl ) is ~8 - 9×10 - 4 , with typical uncertainty up to 6 - 7×10 - 4 , exemplified by the strain observed in the (201) plane at 16 GPa . For materials in the hexagonal crys tal system such as WC, Q ( hkl ) is a quadratic function (Singh and Balasingh , 1994; Singh, Balasingh, et al. , 1998; Singh, Mao, et al. , 1998) of lattice plane orientation B ( hkl ), relative to the loading axis: , ( 8 ) where , in which a and c are the me asured lattice parameters at pressure, and the m i are the coefficients of the quadratic relationship between Q and B . In the elastic regime, the strain Q ( hkl ) is a function of the differential stress, t , the elastic shear moduli G R and G V under isostress (Reuss bound) and isostrain (Voigt bound) conditions, and Reuss co nditions, i.e . stress vs . strain continuity at grain boundaries (Singh, 1993; Singh and Balasingh, 1994; Singh, Balasingh, et al. , 1998) . , ( 9 ) The mean strain and range of Q(hkl) for different diffraction lines indicate lattice strain due to increasing anisotropic stress, change in anisotropic elasticity, or both . Above the yield stress, in the vis coelastic regime, Q ( hkl ) will be modified by plasticity as well . We used full - pro file refinement in MAUD ( Figure 11 ) with Q - factors for each hkl ( Figure 12 ) . With increasing pressure 42 Figure 12 : Strain obtained for WC lattice planes Q ( hkl ) at selected pressures plotted vs . the orientation function B ( hkl ) (eqn 8 ) relative to the stress axis (a - d) and for all planes as pressure increases (e) . In (a - d), red curves are quadratic fits to strain Q ( hkl ) vs . B ( hkl ) . Strain anisotropy increases with pressure (scaling is constant for Figures a - d) . Error in Q(hkl) at individual pressures repre sents the error of the refinement to the experimentally observed curvature . In e), the mean strain (red circles and dashed line between points for emphasis) increases monnotonically , and the shape of the Q vs P curve is similar for all Q(hkl) , however values diverge in Q as pressure increases, indicating an increase in anisotropy as pressure incre ases. 43 (and differential stress), Q(hkl) increases for all diffraction lines, and the range of Q(hkl) observed increases, with maximum lattice strain in WC at (001) and (100) directions, and minimum lattice strain near (101) and (112) . Up to ~30 GPa, strai n is increasingly anisotropic for WC ( Figure 12 e) . At ~30 GPa, the effect of pressure on Q tapers off, and anisotropy in Q values is due to both elastic and plastic deformation . Figure 13 a illustrates the range of differential stress values obtained for analysis assuming Reuss and Voigt bounds . In the Reuss limit ( = 1, implying stress continuity across Figure 13 : Differential stress t in WC ob tained from lattice strain analysis and EVPSC model of experimental measurements . Figure 12 a (left): Average Voigt (red triangles) and Reuss (teal circles) values for differential stress computed from Q(hkl) and elastic constants obtained from theoretical calculations (Liu et al., 2013) , with values obtained using aggregate shear modulus (gold squares (Amulele et al., 200 8) ). Also shown is differential stress obtained from EVPSC simulation incorporating texture and plasticity (solid black line) . Stress accounting only for elast ic strain diverges from stress accounting for both elastic and plastic strain at the yield p oint near 30 GPa . Figure 12 b (right): Reuss stresses computed for individual lattice planes hkl using purely elastic strain . (001) supports the largest differenti al stress ~28 GPa, ~68 - 70% larger than differential stress values of ~16 - 20 GPa at yielding, which use the aggregate and Reuss - limit shear modulus, respectively, and 65 - 89% larger than the stress from the EVPSC model . crystallite boundaries), mean strai n and prior constraints on the shear modulus G (Singh, 1993) may be used to determine t : 44 ( 10 ) where f(x) . Across a range o f materials and crystal systems f has been shown to have a value close to 1 (e.g. Singh, 2009) , so we adopt f =1 in analysis of WC . Aggregate shear modulus G(P) was constrained by extrapolation of a linear fit of ultrasonic data obtained up to 14 GPa (in Amulele et al. , 2008) . Based on these assumptions, elastic differential stress sustained by WC is reported in Figure 14 , with error bars computed based on the standard deviation of Q ( hkl ) at each pressure . Average values of differential stress obtained from lattice strain increase with pressure GPa at the maximum pressure measured, 66 GPa . The slope of t ( P ) decreases at ~30 GPa, at which pressure the observed differential stress is ~12 GPa . A decrease in slope of t ( P ) is consistent with expected behavior at initiation of plastic flow . Figure 13 b illu strates Reuss stresses for individual lattice planes in which t(hkl) is calculated using equation 10 with Q(hkl) for (001), (100), (110), (101), and (111) and the X - ray shear modulus given by (Singh, Balasingh, et al. , 199 8) : , ( 11 ) where the Sij are the elastic compliances . Differential stress t (001) is substantial ly higher than t(hkl) for other planes, supporting 2 6 GPa of differential stress at the yield stress, 40 - 43 % higher than the Reuss bound differential stress calculated from with theoretical G R , 6 8 - 70 % higher than the differential stress determi ned from and the aggregate shear modulus . Below the yield pressure of ~30 GPa, in the elastic regime, the strain anisotropy from Q ( hkl ) can also be used to compute elastic compliances S ij . S ij at a giv en pressure may be 45 Figure 14 : Differential stress observed in WC and other hard ceramics compressed uniaxially with no pressure - transmitting medium . Stress in WC is computed with both lattice strain theory (solid circles) and EVPSC simulation (dashed line) . Dynamic yield strength from shock wave data are open gold circle, calculated after (Feng, Chang and Lu, 2016) . Previous studies on all other hard ceramics (open symbols) use lattice strain theory . B lue squares are WB (Dong et al., 2012) , green triangles are - Si 3 N 4 (Kiefer et al., 2005) , black diamon ds are TiB 2 (Amulele, Manghnani and Somayazulu, 2006) , teal hexagons are B 6 O (He, Shieh a nd Duffy, 2004) , and magenta triangles are BC 2 N heterodiamond (Dong et al., 2009) . Uncertainty is calculated as ± the standard deviation in mean Q at each pressure, propagated through equation 10. Differential stress increase s with uniaxial load until yielding, where t he change in slope of the t(P) indicates strain is accommodated by both elastic and plastic deformation . In WC, yielding at 30 GPa is supported by the development of texture at the same pressure ( Figure 16 ) . The f low stress of WC above yielding is higher than flow stresses observed in WB and Si 3 N 4 . Flow stress obtained from EVPSC simulation is systematically lower than that derived from lattice strain analysis only, thou gh the results of the two methods remain wi t hin uncertainty of each other. 46 determined by the vector product of the inverted coefficient matrix of the lattice strain equations (Singh, Balasingh, et al. , 1998) with their solution matrix for the hexagonal system (Singh, B alasingh, et al. , 1998; Singh, Mao, et al. , 1998) : = ( 12 ) Equations ( 5 - 10 ) and ( 12 ) may thus be used to combine Q ( hkl ) wi th independent cons traints on the linear inc ompressibilities in a and c a c derived from hydrostatic data, and an average G ( P ) or t ( P ), to determine S ij s . The elastic stiffness coefficients ( c ij s) are obtained from equivalence relations b etween c ij s and S ij s (Nye, 1985) . Calculated c ij values are s hown in Figure 15 . Values for c 11 and c 13 increase monotonically throughout the pressure regime . The values for c 33 and c 12 decrease with pressure until the yield stress . Values for c 44 increase with pressure to the yield strength . Above 3 0 GPa, apparent c i j are modified by convolv ed effects of plasticity with elasticity on strain behavior . Our calculated values for c 11 and c 13 agree well with theoretical predictions from 18 to 66 GPa , but c 12 , c 33 , and c 44 diverge from theory rapidly as plasticity progress es ( Figure 15 ) . Specifically , c 12 decreases to ~30 GPa, and then remains approximately constant with pressure, c 33 decreases until it becomes similar to c 11 at ~30 GPa, and c 44 increases rapidly and remains ~200 GP a higher than predicted . For similar behavior observed in rhenium, Duffy et al. (1999) note the c ij s in agreement correspond to basal planes, but this is not the case in WC. Our experimental c ij s are in best agreement with theory below the plastic y ield pressure at ~30 GPa, but significant discrepancies remain between experimental and theoretical c 33 and c 44 even in the elastic regime , and more work is needed to resolve these discrepancies . 47 Figure 15 : Ela stic stiffness coef ficients for experimental data as calculated from e q uation s 5 - 10 , and 12 (closed symbols: this study) . Open symbols are experimental zero - pressure values (Gerlich and Kennedy, 1979) . Solid and dashed lines are from theoretical calculations (Liu et al ., 2013; Li et al., 2014) . Our values for c 11 and c 13 agree well with theoretical predictions . Other c ij s deviate from predicted values rapidly as plastic deformation increases . Above yielding at 30 GPa, values for c 44 is subst antially higher than pred icted, and c 33 and c 12 diverge from predicted values . Values for c 33 appear to converge with c 11 , while c 12 varies only slightly from a constant value of 200 GPa throughout the experimental pressure interval. 3.3.3 Plastic Deformation Plasticity may be evaluat ed based on the texture (non - random orientation distribution of crystallites) of the sample and lattice strains of a series planes as observed as systematic azimuthal variations in diffraction intensity and d - spacing variation with azimuth ( Figure 11 ) . The E - WIMV model implemented in MAUD software fits intensity variation (texture) in the Debye - Scherrer rings by generating an orientation distribution function that describes the frequency of crystallite orientations within the sample coordinate system (Wenk, Lutterotti and Vogel, 2010) . in the MAUD software fits the 48 d - spacing variation with a zimuth to obtain lattice strains . Deformation mechanisms can be investigated using EVPSC simulations, which model lattice strains and texture as a function of slip system activities and strength. 3.3.3.1 Te xture a nalysis To determine crys tallite orientation in WC , the E - WIMV texture model was applied to each phase at each pressure step . Upon compression of WC up to 16 GPa, texture remained random . At 16 GPa, weak texture develops ( Figure 11 and Figure 14 ) . Texture strength scaled in multiples of random distribution (m.r.d.) is observed to increase with pressure, particularly above 30 GPa, the pressure at which yielding was inferred from latti ce strain . The development of texture s upports the onset of plasticity at ~30 GPa ( Figure 16 ). Figure 16 : Inverse pole figures reconstructed from experimental ODF data fit with fiber texture at selected pressures . A texture ma ximum is observed at ~30 GPa near the 2 - 1 - 10 planes (symmetrically equivalent to 100 planes in hkl notation), indicating plastic deformation . This value increases to ~1.9 multiples of a random distribution at the maximum pressure of 66 GPa. 49 At the maximu m pressure examined in this study, 66 GPa, the texture maximum in the inverse pole figure of the c ompression direction is near the pole, which is the pole to the (100) in 3 - coordinate hkl notation ( Figure 16 ) . In the case of WC, (001) is the lattice plane supporting the highest strain and exhibiting the highest strength . Note that WC is a layered structure, with layers of C - atoms (graphene) orth ogonal to 001, between hexagonal W la yers (Goldschmidt, 1967) . The covalent C - C bonds within the layer are very strong, making deformation in the direction extremely difficult . To determine wh ich deformation mechanism(s) is con sistent with generating t his preferred orientation in WC, modeling elasto - viscoplastic response of a polycrystalline WC aggregate is necessary. 3.3.3.2 Plasticity s imulation Plasticity was simulated with an elasto - viscoplastic self - consistent (EVSPC) (Wang et al. , 2010) model, mod ified for application to high - pressure experiments (Lin et al. , 20 17) . The model simultan eously reproduces refined Q values (lattice strain) and texture development at each pressure step and accounts for both elastic and viscoplas tic deformation ( Figure 17 ) . For our models we u sed theoretical elastic p roperties for WC (Liu et al. , 2013) . The EVPSC model treats individual grains in a polycrystalline materi al as inclusions in an anisotropic homogeneous effective medium (HEM) . The average of contributions from all grain inclusions determines the properties of the HEM matrix . Plasticity of a grain in the HEM matrix is then described by rate - sensitive constit utive equation for multiple slip systems: ( 13 ) where is the plastic strain rate, is the re ference shear strain rate and is the critical resolved shear stress (CRSS) of the slip system s at the reference strain rate under conditions in 50 Figure 17 : Strain and texture in WC . Left : Experimental and modeled strain (Q - factors) for selected planes of WC vs . pressure . Right : Experimental (top) and modeled (bottom) inverse pole figures illustrating non - random texture at 66 GPa . the HEM . The grains are subject to local stress tensor , the symmetric Schmid factor descr ibes the straining direction of slip system s . When the stress resolved onto a given slip system is close to or above the threshold value, plastic deformation will occur on that slip system . T he empirical stress exponent n descri bes strain rate sensitivi ty to applied stress, where infinite n implies rate - insensitivity . Deformation of WC appears to be rate insensitive (Mandel, Ra dajewski and Krüger, 2014) and consequently we assume a high stress exponent of n = 30 (Li n et al. , 2017) . The parameter represents the effective polycrystal CRSS and includes both strain hardening and pressure hardening . Pressure hardening and strain hardening effects on CRSS cannot be separated because both pressure and strain incre ase simultaneously in DAC experiments . Both are included in the pressure dependence of CRSS calculated by: (1 4 ) 51 where is the initial CRSS value, and is its pressure - dependence . Values of CRSS and its pressure dependence for WC are presented in Table 2 . The CRSS effectively controls slip Table 2 : CRSS and pressure dependence values for active slip systems in WC under non - hydrostatic compressive stress. system activity and different active slip systems (Lin et al. , 2019) result in different lattice strains and texture and must be matched to experimental observations. Lattice strain and texture evolution in WC are modeled simultaneously to determine deformation mechanisms such as slip system activity and slip system streng th and are used to calculate yield stress from reproduced Q - values an d texture ( Figure 13 and Figure 16 ) . Slip is activated at ~30 GPa on the prismatic slip system . From 30 - 40 GPa, this system converges towards ~50% of the slip system activity with the other 50% supported by prismatic slip ( Figure 18 ) . Above 50 GPa, these systems each account for ~45% of the slip system activity, with the remaining 10% contributed fro m pyramidal system ( Figure 18 ), which activates at ~40 GPa, and increases to 10% activity by 50 GPa . This slip system is needed to induce yielding on Q(001) and occurs in rather than in as described in previ ous work (Takahashi and Freise, 1965; Bolton and Redington, 1980) . 3.3.3.3 Crystallite s ize and m icrostrain Refined values of grain size and microstrain in radial XRD patterns of bulk WC support the observed texture and modeled defo rmation mechanisms ( Figure 19 ) . Mean anisotropic grain Slip System Slip Mechanism CRSS (GPa) d(CRSS)/dP Prismatic 4.0 0.065 Prismatic 2.6 0.065 Pyramidal 14.0 0.08 52 Figure 18 : Modeled slip system activities as determined from the EVPSC simulation . Based on experimental texture, below 8 GPa, defo rmation is entirely elastic, and no plastic deformation occurs . Between 8 - 30 GPa, a small amount of plastic strain is accommodated by prismatic slip activation on . At 30 GPa, bulk plastic yielding occurs, accommodated by continued prismati c slip on and activation of prismatic slip on A third (pyramidal) slip system is activated between 40 - 50 GPa on . Figure 19 : Crystallite size and microstrain vs pressure from ful l - profile refinement in WC under non - hydrostatic compression . Crystallite size decreases and microstrain increases up to ~30 GPa, the pressure at which lattice strain suggests yielding and texture indicates activation of prismatic slip on and . Reduction in crystallite size below plasticity onset is attributed to lattice - bending, which reduces the size of the coherently diffracting regions contributing to crystallite size in MAUD software . Above 30 GPa, microstrain drops and th en resumes increasing, while grain size remains ~constant at ~80 - 90 nm . A second dip in microstrain at ~50 GPa follows the activation of pyramidal slip on . 53 size decreases rapidly until plastic yielding, after which the grain size decreases sl owly . Anisotropic crystallite size represents the size of coherently diffracting regions within the sample . Lattice bending can reduce the refined grain size by reducing the size of these regions, which can explain grain size reduction below plastic yiel ding . Microstrain increases with pressure until yielding, where it drops sharply and then begins to increase again . A second drop in microstrain may follow activation of slip on . Both microstrain and elastic macrostrain beha vior as a function of pressure support elastic stress release in WC through plastic slip. 3.4 Discussion 3.4.1 Equation of s tate Observed volumes for bulk WC obtained in this study under quasi - hydrostatic conditions are similar to data obtained in multi - anvil experiments on annealed WC, but systematically lower than volumes observed in previous DAC studies (Amulele et al. , 2008; Lin et al. , 2009) ( Figure 8 ) . Prev ious DAC studies employed less hydrostatic pressure - transmitting media: NaCl, methanol - ethanol solutions, and/or silicone oil are known to sustain significantly non - hydrostatic stress p articularly at pressures above ~10 GPa (Funamori, Yagi and Uchida, 1994; Angel et al. , 2007; Klotz et al. , 2009) . Under nonhydrostatic axial compression, diffraction in axial geometry samples the crystallites near the orientation of minimum compression, and so obtains systematically larger calculated volumes and a correspondingly higher apparent incompressibility . The neon medium used in this study supports < ~1 GPa diff erential stress through the 64 GPa maximum pressure investigated here (e.g. Dorfman et al. , 2012) , resulting in reliable quasi - hydrostatic volumes for constraining the EOS of WC . 54 Although previous work had suggested nano - WC is highly incompressible (Lin et al. , 2009) , data obtained under quasi - hydrostatic compression in this study demon str ate that nano - WC is not more incompressible than bulk WC . Observed volumes for bulk and nano - crystalline samples are indistinguishabl e at ambient conditions and remain similar upon compression . With increasing pressure, volumes obtained for nano - WC di ver ge to slightly smaller volumes relative to those for bulk WC . Previous work on nano - WC used silicone oil pressure medium (Lin et al. , 2009) , and as for bulk WC, may have ove restimated incompressibility due to effects of non - hydrostatic stress . Based on our results for both bulk - and nano - WC co mpressed in Ne medium, we conclude there is no significant stiffening due to grain size; if anything, nano - WC is slightly less incompr essible than bulk WC . This decrease in incompressibility with decreasing grain size in the nano - regime is consistent w ith observations for other ceramics cBN, TiO 2 and Al 2 O 3 (Chen et al. , 2002; Al - Khatatbeh, Lee and Kiefer, 2012; Le Godec et al. , 2012) . Understanding the effects of grain size on incompressibility is important for assessing overall elasto - viscoplastic responses of polycrystalli n e materials . Our bulk modulus value of 397 ± 7 GPa is in agreement with both theory and other hydrostatic experimental studies on WC . For nanocrystalline WC, our bulk modulus value of 377 ± 7 is lower than the bulk value, and substantially lower than pre vious values reported for nano WC . A growing body of evidence indicates that while nano - scale grain size increases strength (e.g . the Hall - Petch effect), it decreases incompressibility for multiple incompressible materials, now including tungsten carbide . WC is among the least compressible materials, with incompressibility on par with cBN and cRuO 2 (cf. Haines, Léger and Bocquillon, 2001) , but neither the bulk nor the nano - crystalline phase is as incompressible as diamond or higher - K 0 osmium borides (Haines, Léger and Bocquillon, 2001; Yeung, Mohammadi and Kaner, 2016) . 55 3.4.2 Strength, e lasticity, and d eformation The strength of bulk WC determined from lattice strain is comparable to other hard ceramics bel ow 3 0 GPa pressure and ~ 12 - 16 GPa differential stress is supported at the yield point of 30 GPa . The strength of WC determined by lattice strain is similar to that of TiB 2 and B 6 O (He, Shieh and Duffy, 2004; Amulele, Manghnani and Somayazulu, 2006 , p. 2) ( Figure 14 ) . It supports less differential stress than doped diamond (Dong et al. , 2009) , but is stronger than tungsten boride (Dong et al. , 2012) . Reuss stresses provide information on strength anisotropy in WC, with (001) supporting the highest strength . Oriented WC crystals may provide a means of producing stronger p arts without the need for binders . Lattice strain assumes purely elastic deformation however, and the de termination of strength based on EVPSC modeling suggests a lower overall yield strength and flow stress when plasticity is considered . Plasticity affe cts the experimental results, and as noted by previous studies on other materials (Weidner et al. , 2004; Raterron and Me rkel, 2009) strength from inferred elasticity may be overestimated in previous studies when not accoun ting for plasticity. Deformation of WC above the yield stress includes both plastic and elastic components . The elastic stiffness coefficients calcula ted from our results only agree in part with theoretical calculations . This is consistent with observati ons of other materials in which plasticity is expected to occur . Previous experimental studies of elasticity based on radial diffraction have similarl y observed that only some elastic constants agree with density functional theory predictions, while other s diverge (e.g. Duffy et al. , 1999; Merkel and Yagi, 2006) . This is attributed to the effects of plasticity (Weidner et al. , 2004; Merkel, Tomé and Wenk, 2009) . In the case of rhenium (Duffy et al. , 1999) , c 11 and c 12 , which describe stress and strain in the basal plan es of the hexagonal system, agree well with computations . In hcp cobalt (Merkel and Yagi, 56 2006; Merkel et al. , 2006) , only c 12 and c 13 are in modest agreement with theory . In WC, c 11 and c 13 , representing stress an d strain in both the basal and meridional planes, agree with theory but c 12 , c 33 , and c 44 do not . Q - values are a function of both plasticity and elasticity and more work is needed to successfully solve for elastic stiffnesses in X - ray diffraction experime nts on materials undergoing plastic deformation . Experimental values for the bulk and shear moduli at pressures > 15 GPa are needed to minimize error in calculations of c ij s measured in X - ray diffraction experiments and provide additional constraints for theoretical predictions of these parameters . New theoretical computations accounting for experimental measurements of c/a with pressure are necessary to better constrain the pressure - dependence of the c ij s, and to assess the effect of non - hydrostatic stress on hexagonal materials like Re and WC. 3.5 Conclusion Our results demonstrate the mechanical response of WC under quasi - hydrostatic and non - hydrostatic compressive loads up to 66 GPa . As determined by our data and modeling, the strength of the (001) plane in WC is ~ 6 8 - 70 % larger than the mean strength of WC . Plastic deformation in WC above yielding at 30 GPa is accommodated by prismatic slip on and , a nd pyramidal slip on . WC anvils should be oriented to the strongest direction to maximize strength performance under pressure . The new constraints provided by this study on the strength, deformation, and EOS of WC can help inform product ion of WC parts, and potentially applications of polycrystal line materials more broadly, for research and industry. Acknowledgements and support The authors would like to acknowledge Garrett Diedrich, Wanyue Peng, and Gill Levental for their support and as sistance during data collection . This work was supported by the 57 U.S . Department of Energy National Nuclear Security Administration (Capital - DOE Alliance Center DE - NA0003858) and National Science Foundation (EAR 1663442 and EAR 1654687) . This work was performed at HPCAT (Sector 16), Advanced Photon Source (APS) , Argonne National Laboratory . HPCAT operations are supported by DOE - Experimental Sciences . The Advanced Photon Source is a U.S . Department of Energy (DOE) Office of Science Us er Facility operated for the DOE Office of Science by Argonn e National Laboratory under Contract No . DE - AC02 - 06CH11357 . Use of the Advanced Photon Source was supported by the U . S . Department of Energy, Office of Science, Office of Basic Energy Science s, under Contract No . DE - AC02 - 06CH11357. 58 CHAPTER 4: P hase equilibria, strength, and deformation of K r and X e to M bar pressures B enjamin L ee Br ugman, M . Lv, J . Liu, A. Farmer, E . Greenberg, V.B . Prakapenka, D.Y . Popov, C . Park, and S.M . Dorfman Abstract Phase transformations, strength, and plastic deformation of heavy rare gas solids krypton and xenon have been investigated using compressi on in the diamond - anvil cell . Strength determined from stress measurements in Pt and ruby standards at pressures up to 115 GPa is complemented by observations of strain and texture measurements obtained by synchrotron angle dispersive x - ray diffraction in the radial geometry to 100 GPa . Stacking faults indica tive of the martensitic fcc - hcp phase transition are observed at pressures at and above 2.3 and 2.2 GPa in Kr and Xe, respectively . The strength and pressure dependence of Kr and Xe are compared with other rare gas solids He, Ne, and Ar . Non - hydrostatici ty determined by calibrant materials within Kr and Xe media indicates stiffening at 15 - 20 GPa . Above 15 - 20 GPa, deviation from ideal fcc d - spacing ratios is observed in both Kr and Xe, indicating str ess computed for these materials with the cubic lattice strain theory is not reliable . Lattice strain computed for Pt compressed in Kr and Xe indicates a differential stress of at least 5 - 7 GPa above 40 GPa, in agreement with non - hydrostaticity observed i n ruby R1 - R2 separation . Texture obtained from radial diffraction data indicate s the persistence of broad highly - textured fcc diffraction lines to 101 GPa in Xe, suggesting axial measurements may underestimate the metastable persistence of the fcc phase d ue to preferred crystallite orientation. 59 4.1 Introdu ction Rare gas solids (RGS) are prototypical van der Waals solids, composed of atoms with filled electron shells which impart an approximately spherical atomic geometry and make them chemically inert at amb ient conditions . Another consequence of this elec tronic configuration is the relatively high freezing pressures of RGS, which crystallize into simple, high - symmetry structures with low strength and high compressibility . The simple structures and systemat ic behavior of RGS with atomic number make them pa rticularly useful for understanding the role of interatomic potentials and many - body forces in van der Waals bonding under increasing pressure (Hama and Suito, 1989; Polian, Itie, et al. , 1989; Boehler et al. , 2001; Errandonea et al. , 2006) . He is hcp at 300 K above its freezing pressure (Franck and Daniels, 1980; Mao et al. , 1988) , while Ne cr ystallizes in an fcc structure and is expected to remain fcc to hundreds of terapascals (He, Tang and Pu, 2010) , though partial fcc - hcp structure has been proposed based on Ram an experiments (Schuberth, Creuzburg and Müller - Lierheim, 1976) . Ar, Kr, and Xe undergo a kinetically inhibited, martensitic fcc - hcp phase transition, which may include intermediate st ructures, as pressure increases (Asaumi, 1984; Jephcoat et al. , 1987; Caldwell, 1997; Kim et al. , 2006) . B oth He and Ne retain low strength to multi - Mbar pressures (Takemura, 2001; Takemura et al. , 2010; Dorfman et al. , 2012; Singh, 2012a) , but the progression and mechanism of the fcc - hcp phase transformation of heavy RGS affects physical properties at high - pressure (Rosa et al. , 2018; Stavrou et al. , 2018; Sanloup, 2020) . Effects of pressure on bonding and reactivity in rare gas solids are also important to constraining the roles of these elements in planetary interiors (Jephcoat, 1998; Sanloup, 2020) . Phys ical properties such as low strength, high freezing pressures, and inert chemistry (Angel et al. , 2007) make RGS attractive for experimental purposes, such as pressure - 60 transmi tting media for DAC experiments . The utility of He, Ne, and Ar as pressure media is well - established (Bell and Mao, 1981; Meng, Weidner and Fei, 1993; Klotz et al. , 2009; Dorfman et al. , 2012; Singh, 2012b and references therein) , and Xe has been proposed as a pressure medium because of apparent hydrostaticity observed in ruby fluorescence R1 - R2 peak splitting (Asaumi and Ruoff, 1986) . Relatively little is known about the str ength of either Kr or Xe, or the effect of the fcc - hcp transition on the physical properties of bul k RGS, though stiffening in Kr at ~20 GPa has been observed (Rosa et al. , 2018) . The strength and deformation mechanisms of rare gas solids (RGS) with pressure can provide important tests of theory for c omp ression of simple solids and yield insight into the nature of bonding and phase equilibria in conde nsed matter (Kittel, 1996; Errandonea et al. , 2006) . The fcc - hcp tr ansition in heavy RGS is anticipated at lower pressures with higher Z (Caldwell, 1997) , but the ene rgy difference between the two structures is small (Kwon et al. , 1995) , and the onset pressures fo r the transition in the available literature are not in good agreement due to hysteresis along a de compression pathway or heating, which accelerates the phase transition (Eremets et al. , 2000; Cynn et al. , 2001; Errandonea et al. , 2002) . At room temperature, stacking faults associated with onset of the fc c - hcp phase transition have been observed at 2.7 GPa in Kr (Rosa et al. , 2018) and at 3.7 GPa in Xe (Cynn et al. , 2001) . These stacking faults have also been observed in Ar after high - temperature annealing at 49.6 GPa (Errandonea et al. , 2006) . The transition may be further complicated by multiple phase transformation pathways (e.g. Jephcoat et al. , 1987; Kim et al. , 2006) , though transition mechanisms such as fcc - fco - hcp have not been obse rve d in all models (Li et al. , 2017) , and the reported XeII phase w as not matched to any structures (Jephcoat et al. , 1987) . The pressure ranges associated with onset of transition have bee n r eported from 5 to 29 GPa for Xe (Cynn et 61 al. , 2001; Dewhurst et al. , 2002) and 0.8 to 3.2 GPa for Kr (Cynn et al. , 2001; Chen et al. , 20 14) , with theoretical prediction of the transition occurring in Kr at 130 GPa (Kwon et al. , 1995) . The transition in Xe was observed in experiments to be complete at ~70 GPa at 300 K on the basis of di sappearance of fcc diffraction lines (Cynn et al. , 2001) , but is not complet e in Kr at pressures up to 140 GPa at 300 K (Rosa et al. , 2018) . The initiation of the phase transition in Ar is observed with annealing at 49.6 GPa (Errandonea et al. , 2006) , but experiments have bee n conducted to at least 80 GPa without observation of hcp Ar (e .g. Mao, Xu and Bell, 1986; Ross et al. , 1986) and theory places the upper bound on initiation of the transition at 230 GPa (McMahan, 1986) , while completion is expected at 310 GPa (Errandonea et al. , 2006) . The low - pressure onset of hcp Kr and Xe means that the utility of Kr or Xe as a pressure - transmitting medium in DAC experiments will need to include careful consideration of the stren gth of both fcc and hcp phases, and the effect of the incomplete phase transition on net physical properties. Modeling deformation and phase equilibria of the heavy RGS is difficult relative to their lower - Z counterparts because they exhibit increasingly c omp lex electronic interactions as pressure increases (Tsuchiya and Kawamura, 2002; Troitskaya, Val. V. Chabanenko, et al. , 2012; Tian et al. , 2012) . Strength of RGS is generally expected to increase with Z because more electrons are involved in the dipole moment of the respective atoms, which would act to strengthen van der Waals forces within the materi al. Kr and Xe ar e therefore expected to be stronger than Ar . The slow kinetics of the fcc - hcp phase transition allow for the presence of both fcc and hcp phases, as well as intermediate structures, across a wide pressure regime in the heavier RGS, and the strength of a m ixture of these phases may differ from that of single phases . Deformation and phase equilibria of solid noble gases can be studied with X - ray diffraction and 62 spectroscopic methods in the dia mond anvil cell (DAC) . Experimental results may be used to te st theoretical models and constrain systematics of the mechanical properties of RGS. To date, no experimental studies have addressed the effect of the fcc - hcp phase transition on the strength of a multi - phase aggregate of RGS . In this study, we examine th e s trength of fcc Kr and Xe, and the effect of the fcc - hcp phase transition using analysis of X - ray diffraction and ruby fluorescence in the DAC, supported by theoretical elasticity from density functional theory (DFT) predictions (e.g. Cohen, Stixrude and Wasserman, 1997; Tsuchiya and Kawamura, 2002) for behavior of RGS under high pressure. 4.2 Methods Compressi on of sample materials Kr (Praxair or Airgas) or Xe gas (Ingas) was performed in diamond anvil cells . Gases were loaded using the Consortium for Materials Properties Research in Earth Sciences ( COMPRES ) / GeoSoilEnviro Center for Advanced Radiation Sources ( GS ECARS ) gas loading device at the Advanced Photon Source (APS) at Argonne National Lab (Rivers et al., 2008) . Rhenium gaskets pre - indented to thicknesses from ~30 - for diffraction and ruby fluorescence experiments conducted in the axial g eometry a nd X - ray transparent beryllium gaskets pre - experimen ts . Gaskets were drilled onsite using either the GSECARS or the High - Pressure Collaborative Access Team ( HPCAT ) laser drill facilities (Hrubiak e t al. , 2015 ) , the GSECARS laser drill, or at MSU using a Hylozoic electronic discharge machine . Pt (Goodfellow) and Ag (Fisher Scientific), and ruby were loaded as internal pressure standards, and Pt and Ag were also used as stress markers . Table 3 prov ides a su mmary of experimental runs . 63 4.2.1 X - ray diffraction Phase transitions, strain and texture development wer e measured using angle - dispersive X - ray diffraction obtained in both the axial diffraction geometry in symmetric DACs with X - rays Run Sample G eometry Pressure Range Step size (GPa) Standard Kr A1 Kr Axial 1 b ar - 94 GPa 0.5 - 1 Pt 0.4246 Kr A3 Kr A xial 1 b ar 47 GPa 3 - 5 Pt 0.31 Kr P1 Kr Axial 1 b ar - 52 GPa 1 - 3 Ruby 450 Kr R2 Kr Radial 1 b ar - 100 GPa 1 - 3 Ruby 0.31 Xe A1 Xe Axial 1 b a r - 83 GPa 3 - 5 Pt 0.2952 Xe A2a Xe Axial 1 b ar - 48 GPa 3 - 5 Ag 0.3344 Xe A2b Xe Axial 55 - 115 GPa 3 - 5 Ag 0.31 Xe A3 Xe Axial 1 Bar - 17 GPa 0.5 - 1 Ag 0.31 Xe P1 Xe Axial 1 Bar - 33 GPa 1 - 3 Ruby 532 Xe R3 Xe Radial 1 Bar - 100 GPa 1 - 3 Ruby 0.31 parallel to the load axis, and in the radi al geometry in a 2 - post panoramic DAC with the X - ray beam normal to the loading axis . Axial and radial experiments were conducted at the HPCAT beamline at Argonne National Lab, Sector 16 - BM - D . Axial experiments were also conducted at GeoSoilEnviro Center for Advanced Radiation Sources (GSECARS) at Argonne National Lab, Sector 13 - ID - D . The X - ray beam at HPCAT was monochromatized to 40 keV (axia l experiments) or 37 keV (radial experiments) . X - rays were focused to 4 - 6 - Kirkpatrick - Baez fo cusing mirrors and collimated using a 90 - 64 diameter and minimize diffraction from gasket materials . Diffraction patterns were collected for 60 - 300s on a MAR23 00 image plate detector, which was calibrated using a CeO 2 standard . At GSECARS, a monochromatized 37 keV X - ray beam was focused to 2.5 - 3 - ted on either the MARCCD 165 or Pilatus detector, calibrated with an LaB 6 standard. Saturated intensity in each diffraction pattern was masked and diffraction patterns were integrated using Dioptas software (Prescher and Prakapenka, 2015) for axial experiments and Fit2D (Hammersley et al. , 1996) for radial data . Axial diffraction peaks were fit with Lorentzian lineshapes using the IgorPro MultipeakFit module (Wave Metrics Inc., Lake Oswego, OR, USA) . Radial patterns were divided into 5° azimuthal wedges across a 360° azimuthal range with Fit2D and Fit2D2MAUD for Rietveld refinement . Full - profile refinement of radial patterns was conducted with Materials Analysis U sing Diffraction (MAUD) software (Lutterotti, Matthies and Wenk, 1999) . Be and BeO phases from the gasket were included in the refinement in addition to sample and internal standards . Global background intensities, as well as phase fr action, unit cell parameters, microstructure and texture for all phases were fit with MAUD . Strain was fit for Xe, . Texture in all phases was modeled with the a refinable orientation dist ribution function (ODF) using the extended WIMV (E - WIMV) texture model (Lutterotti e t al. , 2004) . Pressure in axial diffraction experiments was calculated from the equation of state of Ag (Dewaele et al. , 2008) or Pt (Dewaele, Loubeyre and Mezouar, 2004) and unit cell volumes for th ese metals derived from the (111), (200), (311), and (222) planes for Pt, and the (111) and (200) planes for Ag . Pressure for radial diffraction ex periments was determined from fcc Kr and Xe directly by fitting the volumes determined from the refined latt ice parameters of the fcc phases to EOS of fcc Kr (Polian, Besson, et al. , 1989) or fcc Xe (Cynn et al. , 2001) . 65 4.2.2 Ruby f luorescence . Ruby is a well - charact erized stress sensor (Piermarini, 1973; Mao, Xu and Bell, 1986; Syasse n, 2008b) . - 9 rubies ( Figure 20 ) and the cell chamber was filled with Kr or Xe gas at APS . Xe was compressed in Figure 20 : Photomicrogra phs of diamond anvil cell sample chambers inside Re gaskets loaded with ruby grids prior to gas - loading . The cell in figure A (left) was loaded with Xe and the cell in figure B (right) was loaded with Kr . Fluorescence measurements for each cell are plott ed in Figure 26 and compared with similar measurements for other RGS . 1 - 3 GPa pressure increments to 33 GPa . Kr was compressed in 1 - 3 GPa steps to 52 GPa . Fluorescence spectra for each ruby at each press ure ste p were collected at either the offline ruby system at 16 - BM - D (HPCAT) or with an Optiprexx ruby fluorescence system at MSU . The R1 and R2 ruby fluorescence peaks were fit using Lorentzian lineshapes and the local pressure for each ruby was calculat ed from the R1 and R2 peak positions at ambient temperatures, using the ruby pressure scale of Dewaele et al. (2004) . 66 4.3 Results and d iscussion 4.3.1 The fcc - hcp phase transition in Kr and Xe Based on our X - ray diffraction data, we identified the fcc - hcp transition proceeding over a wide pressure range as observed in previous studies of Kr and Xe compressed at room temperature (Jephcoat et al. , 1987; e.g. Cynn et al. , 2001; Errandonea et al. , 2 0 02; Rosa et al. , 2018) and peaks from both phases are present through the majority of our diffraction data . All observed diffra ction peaks in axial diffraction patterns correspond to either sample (fcc and hcp Kr or Xe) or internal pressure standards (P t or Ag) . In radial diffraction data, additional peaks were observed that correspond to either gasket materials (Be and BeO) or t o a previously reported Xe peak related to an as yet undetermined high - pressure phase (Jephcoat et al. , 1987) . Selected axial 1D diffraction patterns and lattice spacings for Kr and Xe are presented in Figure 21 a and Figure 21 b and 2D radial diffraction patterns are presented in Figure 22 - Figure 26 . In our radial data, the onset of the tr a nsition to hcp occurs as diffusely s cattered intensity which may extend over a range of diffraction angles, up to ~1° - peaks as pressure increases . This signal appears as weak and broad hcp peaks in 1D patterns, and are interpreted as stacking faults and d islocations as in previous reports (e.g. Rosa et al. , 2018) . The appearance of these stacking faults on compression is observed at 1.9 and 2.3 GPa in Kr and Xe respectively in axial patterns and at 2.2 GPa in both Xe and in Kr in the radial diffraction patterns . Stacking fault s have transformed comple tely to hcp peaks by 101 GPa ( Figure 22 a - b). No stacking faults were observed in fcc Xe at 0.7 GPa in the axial geometry or 1.5 GPa in the radial geome try, indicating onset of hcp occurs be tween 1.5 and 2.2 GPa in Xe, with pressure calibrated on the Xe EOS (Cynn et al. , 2001) . In Kr, 1.9 GPa was the lowest pressure at which 67 fcc Kr peaks could be identified in the axial geometry, and 2.2 GPa was the lowest pressure pattern collected for Kr in th e radial geometry, with pressure calibrated on the Kr EOS (Polian, Besson, et al. , 1989) . Our observations are consistent with the onset of stacking fau lts associated with the martensitic fcc - hcp phase transition previous ly observed at 2.1 - 2.7 GPa in Kr (Cynn et al. , 2001; Rosa et al. , 2018) and ~ 1 GPa lower than observations at 3.7 GPa in Xe (Cynn et al. , 2001) . Raman shifts associated with the hcp phase in Xe have been observed at 5 GPa (Sasaki et al. , 2009) . On decompression, coexistence of the hcp phase ha s been observed to persist down to 3.2 GPa in Kr and down to 1.5 GP a in Xe (Errandonea et al. , 2 002) . A weak extra peak that matches neither fcc nor hcp structures appears in Xe at ~57 GPa and increases in intensity to 101 GPa ( Figure 26 ) . This peak appears to be consistent with previous identification of a Xe - II phase, though no structure was matched to this peak (Jephcoat et al. , 1987) . The peak does not match dhcp or bct structures, nor does it match the bcc st ructure indicated by (Krainyukova, 2011) or the fco structure indicated by (Kim et al. , 2006) to be possible intermediate metastable structures formed during the Xe fcc - hcp transition . D - spacing ratios for twinning in the fcc structure (cf. Pashley and Stowell, 1963) also do not explain this peak . It has been proposed that this peak would not occur under quasi - hydrostatic compression (Belonoshko, Ahuja and Johansson, 2001) . No extra pe ak s are present in Kr. The phase transition from fcc to hcp proceeds more completely at lower pressures in Xe than in Kr ( Figure 21 , Figure 23 - Figure 26 ) . The progress of the transition from fcc to hcp is quantified based on increasing int ensity of hcp peaks with decreasing intensity of fcc peaks, with completion of the transition indicated b y the disappearance of the fcc 200 diffract ion peak . 1D patterns emphasizing the appearance of 100 and 101 hcp peaks and their development with 68 Figure 21 : Axial s ynchrotron X - ray diffraction patterns for a) Xe (left) and b) Kr (right) with Ag and Pt internal pressure sta ndards, respectively . Intensities of peaks shared between the fcc and hcp phase : fcc (111)/hcp (002), fcc (220)/hcp (110), and fcc (311)/h cp (112) , continue to change to 115 GPa, suggesting conversion to hcp is not complete at this pressure . In Kr, stack ing faults (SF) remain the only indicator of the hcp phase to 94 GPa. 69 Figure 22 : Unrolled radial 2D diffraction pattern s for Xe at 2.3 GPa (a) and 101 GPa (b) illustrating the appearance of stacking faults associated with hcp ph ase and the subsequent hcp peak development with pressure . The first appearance of diagonal features indica ting stacking faults occurs at 2.2 GPa . By 101 GPa, stacking faults have transformed into hcp Xe (100) and hcp Xe (101) diffraction lines. pressu r e ( Figure 21 a - b) illustrate more growth of the hcp phase at lower pressures in Xe relative to Kr . The hcp stacking fault peaks grow dramatically in Xe beginning at pressure ~30 GPa . Previous work on Kr (Rosa et al. , 2018) indicated a 20% volume fraction of hcp a t 25 - 29 GPa, and complete transformation to hcp in Xe at ~75 GPa (Cynn et al. , 2001) . Refined volume fractions from radial patte r ns indicate only ~4.8% hcp conversion at 22 GPa in Kr, due to the 70 Figure 23 : Radial diffraction pattern s for Kr at 10 GPa . Be, Be2, Be3, and BeO peaks are gasket materi al. Kr hcp peaks appear as stacking faults and increase in occurrence and intensity with pressure . Texture in fcc and hcp Kr peaks is interpreted primarily as crystalliza tion texture . No e v idence of significant non - hydrostatic strain in fcc or hcp Kr at 10 GPa. 71 Figure 24 : Radial diffraction patterns for Kr at 94 GPa . Be , Be2, Be3, and BeO peaks are gasket materi al. Kr hcp peaks appear as stacking faults and incr e ase in occurrence and intensity with pressure . Deformation texture is prevalent in both fcc and hcp Kr peaks and strain is evident as curvature in both fcc and hcp peaks. 72 Figure 25 : Radial diffraction patterns for Xe at 12 GPa . Be , Be2, and BeO peaks are gasket materi al. Xe h cp peaks initially appear as stacking faults similar to onset of hcp Kr ( Figure 23 and Figure 24 ) and increase in intensity with pressure . Texture at 12 GPa is interpreted as resulting from crystallization of fcc and hcp Xe. There is slight curvature in some diffraction lines, corresponding to qualitatively low non - hydrostatic strain in fcc and hcp Xe at 12 GPa . 73 Figure 26 : Radial diffraction patterns for Xe at 101 GPa . Be and BeO peaks are gasket materi al. S tacking faults have completely transformed into hcp peaks, and significant strain is evident in hcp and shared fcc/hcp diffraction lines. Th e fcc 200 diffraction line is broad and highly textured, and strain cannot be reliably assessed for this plane. 74 lack of sharp hcp peaks and presence of diffuse stacking faults ( Figure 26 ) . We observe only ~77% con v ersion in Xe at 101 GPa and at 94 GPa, only about 18% of Kr is hcp ( Figure 27 ) . The Figure 27 : Volume % hcp in Kr and Xe with pressure . The transition proceeds more quickly in Xe than i n Kr b ut is not complete in either phase at Mbar conditions. Rosa et al. (2018) report 20% conversion to hcp Kr at 20 GPa, which is not reached until 94 GPa in our radial analysis. Xe is not fully transformed to hcp at 101 GPa, 30 GPa greater than the estimate of full conversion to hcp at ~70 GPa (Cynn et al., 2001) . Peaks from the low pressure fcc phase in Xe are seen at 101 GPa in Figure 26 b. continued presence of intensity from the fcc 200 peak indicates the transition remains inco mplete in b oth phases at the maximum pressures (95 - 100 GPa) . The fcc Xe 200 peak disappears in the axial orientation at ~70 GPa ( Figure 21 a), however radial data show persistence of fcc 200 peaks to at least 101 GPa . The tr a nsition is more sluggish in Kr, and weak peaks that develop at lower pressures remain the only evidence of hcp Kr above 90 GPa ( Figure 22 b) in axial patterns . 75 Recent work has shown that the transition in Kr remain s incomplete, with fcc peaks present up to 140 GPa ( Rosa et al. , 2018) . Our radial data indicate the pressure range observed for the fcc - hcp transition may be underestimated in previous studies due to preferred orientation and sampling bias of axial d iffraction geometry . Axial geometry samples diffracti o n near the minimum stress direction (equivalent to the azimuthal wedge from radial diffraction containing the 90° angle) . At this minimum stress orientation, our radial diffraction data for Kr and Xe indicate that intensity of hcp (100) is near a texture maximum, while fcc (111) is near a texture minimum ( Figure 23 - Figure 26 ) . As a result, previous studies which applied diffraction in the axial geometry to Kr and Xe may be e xpected to overestimate the volume fract ion of hcp and underestimate the pressures required to initiate and complete the phase transition . 4.3.1.1 Texture in Kr and Xe due to phase transition and deformation The radial diffraction geometry enables observations of preferred orientation developed at all orientations relative to the load axis and analysis of textures generated by phase transitions and plasticity . Representative radial diffraction patterns are presented in Figure 23 - Figure 26 . Azimuthal variation in intensity (texture) in Kr and Xe are determined with full - profile Rietveld refinement . Texture orientations are determined by analysis of inverse pole figures for Kr and Xe ( Figure 28 ). 76 Figure 28 : Texture evolution in Kr (left) and Xe (right) . At low pressure, texture in fcc is probably due to crystallization . There is more initial texture in fcc Kr th a n in fcc Xe . Deformation texture becomes evident with increasing p ressure as a texture maximum develops near in both fcc Kr and fcc Xe . Initial texture in hcp Xe differs from hcp Kr, with initial Xe texture near in Xe and near in K r . Deformation texture on increases with pressure in hcp Kr, but texture also develops near . By the maximum pressure in both phases, texture in both hcp Kr and hcp Xe is concentrated around . At the lowest pressures, Kr and Xe crys t allize as a few large fcc grains, obse rved as several diffraction spots . The initial grain size of fcc Xe is larger than the initial Kr grains ( Figure 29 ), and compression rapidly decreases the fcc grain size for b oth as the weak RGS deform plastically . Significant deformation texture develops by 10 GPa in fcc Kr ( Figure 28 ) . Texture in fcc Xe at 12 GPa is weak . At the highest pressures in Kr and Xe, the texture in 77 Figure 29 : Crystallite size in Kr (squares) and Xe (circles) for fcc (main plot) and hcp (inset) . Refined grain size decreases with pressure and is ~3 orders of magnitude larger in the fcc phase than in hcp phases . The crystall ite size for fcc Xe is ~2x the size of fcc Kr, but hcp Xe crystallites size parameters refine to smaller values than hcp Kr . Crystallites in the hcp phase for both materials exhibit a slight increase before decreasing again, suggesting grain growth follow ed by compression. the last remaining metastable fcc crys tallites is , indicating this orientation is the most resistant to transformation. The hcp phase has also begun to crystallize by 10 - 12 GPa in both Kr and Xe . At 10 GPa in hcp Kr, a tex ture is evident, which increases with pressure, but also devel ops near as pressure increases and is then converted back to . In hcp Xe, initial texture is near , but is also converted to at higher pressures . We infer that te xture evolution occurs at lower pressures in hcp Xe, following the more co mplete transformation of fcc to hcp in Xe . 78 4.3.2 Strength of Kr and Xe across the phase transition Effects of the gradual transformation of fcc to hcp structure on the strength and stiff ness of Kr and Xe can be evaluated indirectly through the stress transmitt ed by the Kr and Xe media to enclosed stress sensors and directly through strain in the Kr and Xe structures . Stress applied by the diamond anvil cell is nonhydrostatic, as opposed rigid anvils exert a ~uniaxial stress modified by resistance to flow from the gasket cylinder (Ruoff, 1975) . The difference between stress parallel to the load axis 3 and radial stress in the gasket plane 1 is the differential stress, given by The differential stress sustained by a sample within the DAC is b ounded from above by the yield strength or flow stress of the sample or surrounding pressure - transmitting medium (Ruoff, 1975; Singh, 1993) . A purely hydrostatic medium supports no differential stress (t = 0) while a quasi - hydrostatic medium supports a lo w differential stress . A sample that is stronger than the surrounding weaker medium will exhibit elastic strain reflecting the differential stress transmitted by the medium; if the sample is weaker than the surrounding medium, the sample will fail and its strain will be a combination of elastic and plastic strains . In this study, differential stresses supported by Kr and Xe media through the transitions to hcp are assessed in ruby, Pt, and Ag stress sensors . Strain in Kr and Xe media are also examine d th rough analysis of deviation of compression behavior from hydrostatic ideal, observed through diffraction. 4.3.2.1 Non - hydrostatic stress measured by ruby fluorescence. While the ruby R1 and R2 fluorescence peaks both shift to higher wavelengths with pressure, anis otropy of the ruby crystal results in different positions of the R1 and R2 peaks as non - hydrostatic stress increases . The R2 line is sensitive only to pressure, while the R1 line is 79 sensitive to both pressure and non - hydrostaticity of the applied stress (Chai and Brown, 1996; Syassen, 2008b) . The difference between wavelengths of these peaks thus varie s systematically with the stress conditions applied by the DAC and medium and orientation of the ruby crystal relative to the load axis . Non - hydrostatic stress in the DAC may then be determined in three ways ; pressure gradients using standard deviation of the pressure of multiple ruby sensors, peak - R2 (Forman et al. , 1972; Piermarini, 1973; Adams, Appleby and Sharma, 1976) . To account for unknown crystallographic orientation of spherical ruby sensors, run - to - run differences in ruby position, spectral and spatial resolution of measurement instrum entation, and variation of stress state across a sample chamber under nonhydrostatic stress, we used the average wavelength difference R1 - R2 for a population of rubies positioned acro ss the cell chamber . Ruby fluorescence was obtained from 8 rubies arran ged across the sample chamber within Kr and Xe to 52 and 33 GPa respectively . Similar data have been reported for rubies surrounded by He, Ne and Ar media (Klotz et al. , 2009) . Rubies compressed in He, Ne, and Ar media report approximately hydrostatic conditions below 15 - 20 GPa . The increase in the average of the difference R1 - R2 for all rubies deviates from the ~hydrostatic tre nd followed by rubies in Ne and He media above this pressure ( Figure 30 ), supporting stiffening of the media . Rubies compressed in Kr and Xe exhibit more non - hydrostatic stress than those in He, Ne, or Ar below 15 - 20 GPa, as well, suggesting they are stiffer in this pressure range as well . The R1 - R2 values observed in Kr to 52 GPa appear to converge with the R1 - R2 value reported in Ar (Klotz et al. , 2009) , indicating differential stress supported by Kr may be comparable to that supported by Ar. 80 Figure 30 : Difference in ruby R1 - R2 peak positions vs pressure for rub ies surrounded by RGS media, a measure of hydrostaticity of stress supported by the medium . Ruby peak splitting measured in He medium is inferred to represent ~hydrostatic stress applied to the ruby, and peak splitting values higher than this trend may be observed due to nonhy drostatic stress and anisotropy of the fluorescence behavior for these two fluorescence lines . Ruby peak splitting averaged from 8 or 9 rubies in Ar , Kr, and Xe exhibits an increase in slope indicative of stiffening above 15 - 20 GPa . The difference R1 - R2 for both Kr and Xe is generally larger than for Ar, Ne, and He, indicating that Kr may become less hydrostatic than Ar at ~35 GPa, but Xe is less hydrostatic than all other RGS at all pressures studied . 4.3.2.2 Differential stress measure d by lattice strain in Pt and Ag Lattice strain theory describes the correspondence between crystal lattice spacings observed by way of diffraction and applied nonhydrostatic stress . L attice spacings vary ng le between the diffracting plane normal and the loading direction: 81 In this equation d m is the measured spacing between lattice planes with Miller indices hkl at d p is the la ttice - spacing under hydrostati c stress (S in gh, Balasingh, et al. , 1998) . Strain is given by the strain Q(hkl) . For purely elastic strain, Q is related to the differential stress t , the aggregate Voigt and Reuss shear moduli G V and G R a constant that describes the weight b et ween isostress and is ostrain conditions at grain boundaries (Singh, Balasingh, et al. , 1998; Singh and Takemura, 2001) : These equations can be simplified for the determination of stress and strain in cubic standards Pt and Ag to constrain the strength of the Kr and Xe media . For a cubic crystal structure, Q can be wr it ten (Singh, Balasingh, et al. , 1 998) : where S is the elastic anisotropy of a cubic material, given by S = ( S 11 S 12 S 44 )/2, where the S ij are elastic compliances (Singh, 1993) , and crystal plane orientation is determined by . In the axial diffraction geometry, equation ( 18 ) is a linear function, expressed in terms of the lattice paramete r (Singh, 1993; Singh and Takemura, 2001) : where, M 0 = Q(hkl)a P , and M 1 = . Observed values of a(hkl) vs the orientation function in equation ( 19 ) are plotted at selected pressures for Pt compressed in Kr and Xe ( Figure 31 ) . 82 If the elastic anisotropy S is known as a function of pressure, differential stress t can be determined based on observed cubic lattice spacings . Elasticity of Pt obtained as a function of P has previou sly been calculated by first principles (Menéndez - Proup in and Singh, 2007) . Elasticity of Ag at high pressure was extrapolated accord ing to finite strain (Birch, 1978) incorporating ultrasonic EOS data (Holzapfel, Hartwig and Sievers, 2001) , ultrasonic measurements of elastic constants c ij (0) (Neighbours and Alers, 1958) , and ultrasonic measurements of their de rivatives c ij (Daniels and Smith, 1958) . Differential stress observed in Pt compressed in Kr and Xe increases steadily up to 40 GPa, consistent with significant differential stress sus tained in the media, but less differenti al stress than would be observed with no medium for grain sizes ranging from 70 - 300 nm ( Figure 32 a) . While Pt with similar grain size would exhibit ~4 GPa differential stress at 10 GPa (Dorfman, Shieh and Duffy, 2015) , ~1 GPa differential stress is observed at 10 GPa in Pt within Kr and Xe . At ~40 GPa, the trend of differential stress with pressure in Pt changes slope, consistent with yielding and the onset of plastic deformation in Pt . Above this pressure, the measured t in Pt is 5 - 7 GP a ( Figure 32 a), similar to flow stress in Pt in no m edium (Kavner a nd Duffy, 2003; Dorfman, Shieh and Duffy, 2015) . These conditions correspond to rapid growth of hcp Kr and Xe peaks . Ag is a softer metal than Pt (particularly the ~nano - grained Pt used in these experiments), and thus yields at lower differential stres ses . For Ag compressed in Xe, differential stress remains below 2 GPa to 115 GPa pressure . The strength of Ag determined from lattice strain in 83 Figure 31 : Plots of a(hkl) from axial diffraction lines vs 3×(1 - 3cos 2 r Pt compressed in Kr (top) and Xe (bottom) . The slope and intercept of the gamma plots are used with elastic anisotropy to calculate differential stress . A slope of zero corresponds to hydrostatic conditions, and for materials with positive elastic anis otropy such as Pt, the slope becomes increasingly negative as differential stress on the sample increases. 84 this work is in good agreement with previous work on the strength of silver (Liermann et al. , 2010) with grain size ~300 nm up to 40 GPa ( Figure 32 b) . This indicates that the yield strength of Ag is exceeded throughout the experimental pressure range, and thus Ag and cannot be use d to determine the strength of Xe . The formation of a connected network of hcp within the fcc matrix may result in stiffening of the heavy RGS, which become stronger than Pt and Ag and induce plastic deformation in the metal s . 4.3.2.3 Strain in Kr and Xe Strai n and differential stress can also theoretically be assessed directly in Kr and Xe using axial and radial diffraction data through lattice strain theory, but the broad phase transition and overlapping diffraction peaks between these related structures ( Figure 21 ) and background from the gasket impede separation of the strain of the fcc and hcp structures in the analysis . The lattice strain analysis used above to constrain stress and strain in Pt and Ag based on axia l diffraction data depends on relative differences in lattice parameter s determined by multiple different diffraction lines e.g . ( 111 ) and ( 200 ) . However, once hcp Kr and Xe begins to form, the fcc ( 111 ) line overlaps with the analogous c lose - packed plan e in hcp, 002, and similarly all fcc diffraction peaks except fcc 200 are overlapped with corresponding hcp peaks . With radial diffraction data, observation of lattice spacings at a range of orientations relative to the load axis allows me asurement of str ain and potentially stress based on a single line, if it does not overlap with diffraction from the gasket. Deviatoric strain Q ( hkl ) is observed as curvature in diffraction lines in radial 2D patterns . Q ( hkl ) was refined or fixed manually for ( 111 ) , ( 200 ) , ( 220 ) , and ( 311 ) in fcc and ( 100 _ for hcp Kr and Xe . Due to overlapping diffraction lines between multiple related structures and the Be 85 Figure 32 : Differential stress t of metals within RGS media compared to strength of met als, i.e. maximum t observed with no medium. a) Pt in Xe (green circles, this study) and Kr (blue and gray squares, this study) media, and no media: red, gold , pink, and black tr iangles (Dorfman, Shieh and Duffy, 2015) , and b) Ag in Xe (green and gray circles, this study), and no media: red and gold triangles (Liermann et al., 2010) . Under hydrostatic conditions, t is 0. The differential stress supported by Pt in both media remains lower than the strength of Pt below 20 GPa, but then increases rapidly above 20 GPa to match the strength of Pt by ~40 G Pa. Ag has been observed to be weaker, and s tress supported by Ag in Xe is consistent with the flow stress in Ag throughout the studied pressure range. A change in the EOS of Kr at 20 GPa in previous work (Rosa et al., 2018) was attributed to stiffening induced by the fcc - hcp phase transition. 86 gasket, some constraints were needed to successfully refine strain for additional hkl . For both Kr and Xe, strain in the shared fcc/hcp planes was fixed equal, allowing refinement of hcp ( 002 ) with fcc ( 111 ) , hcp ( 110 ) with fc c ( 220 ) , and hcp ( 112 ) with fcc ( 311 ) . Higher order planes parallel to lower order planes were fixed equal to the corresponding lower order plane, allowing refinement of fcc ( 222 ) with fcc ( 111 ) , hcp ( 200 ) with hcp ( 100 ) , and hcp ( 004 ) with hcp ( 002 ) . Only the hkl values for which strain w as fit at all pressures were used in the calculation of flow stress . Because of overlap in peaks from shared planes, as well as stacking faults, broadening, and high texture in non - overlapped diffraction lines, we conclude that calculation of flow stress from radial patterns is not quantitatively reliable . Qualitative trends in Q(hkl) which may prove meaningful for comparison to observations of stress markers are discussed below . Strain in fcc and hcp Kr and Xe i s plotted in Figure 33 a - d . In both Kr and Xe strain in fcc ( 111 ) /hcp ( 002 ) and fcc ( 311 ) /hcp ( 112 ) increases and then levels off or decreases . In fcc ( 220 ) /hcp ( 110 ) strain increases monotonically above ~20 GPa . Strain in hcp 100 in both Kr and Xe rises at low pressure and then decreases at ~20 GPa to values similar to other hkl . Strain in hcp ( 101 ) is poorly fit in both Kr and Xe due to intensity and peak width variation arising from stacking faults . The fcc ( 200 ) line in Kr and Xe is highly textured, and in Xe it broadens substantially with pressure . It is affected by deviation from the ideal d - spacing ratio in both Kr and Xe, and trades intensity with the adjacent hcp ( 101 ) peak . As a result, fitting Q for this line is not reliable. The strength of fcc Kr and Xe generally increases with pressure . Stiffening in the EOS of Kr is observed at 20 GPa (Rosa et al. , 2018) and predicted in Xe at 20 GPa, though the effect in Xe is attributed to a transition to a face - centered orthorhombic (fco) structure (Kim et al. , 2006) . 87 Figure 33 : Refined strain Q for hkl from fcc (top , A and B ) and hcp (bottom , C and D ) Kr (left , A and C ) and Xe (right , B and D ) . Strain in peaks from planes shar ed between the fcc and hcp structure fcc (111)/hcp (002), fcc (220)/hcp (110), fcc (311)/hcp (112) generally increases with pressure, with some non - monotonic behavior. For hcp 100 in both Kr and Xe , strain is refin ed successfully, but does not increase sy stematically with P . 88 We compared the compression curves and F - f plots ( Figure 34 ) from our axial Kr data to those of Rosa et al. (2018) , and observe a similar deviation from the normal compression behavior at ~20 GPa . Our radia l data suggest only about 7% hcp conversion at this pressure however, and we infer the stiffening is probably not caused by a substantial volume of hcp phase . While strain generally appears to increase in fcc Kr and Xe in our radial data, there is no nota ble qualitative change in slope at 20 GPa . Non - hydr ostaticity at 20 GPa is observed in Kr and Xe by an Figure 34 : Compression curve (a) for Kr (blue circles and yellow circles, this study) with EOS (red line, (Rosa et al. , 2018) ), and F - f plot (b) with Kr data from this study and red triangles and gray triangles from (Rosa et al. , 2018) . Red tr iangles represent F - f values below observed stiffening at f E = 0.4 (corresponding to ~20 GPa pressure), and gray triangles above stiffening . Their assessment of deviation from normal behavior at 20 GPa is supported by our data, though the change in slope at f E = 0.4 is less pronounced in our data . The change at 4 0 GPa may be attributed to plastic flow in the Pt pressure standard. increase in ruby R1 - R2 separation however ( Figure 30 ) . A deviati on in ideal ratios of fcc ( 111 ) - ( 200 ) peak positions in Xe, which could indicate lattice strain due to strengthening of the Xe, which becomes prominent above 20 GPa is also observed in (Jephcoat et al. , 1987) and 89 attributed to texture, however our radial data demonstrate that while preferred orientation is present in both fcc and hcp Xe, it does not explain the shift in peak p osition . 4.4 Conclusions Phase equilibria, strength, and texture of heavy RGS Kr and Xe have been studied in the diamond anvil cell and have been evaluated in the context of the systematics of the RGS . The onset of the fcc - hcp phase transition is o bserved a t 2. 2 and 2. 3 GPa in Kr and Xe respectively . Above 15 - 20 GPa, the d - spacing ratio ( 111 ) / ( 200 ) deviates from the ideal fcc structure . Overall, the transition proceeds to a larger phase fraction of hcp at lower pressures in Xe than in Kr, but is not comple te in Xe to 101 GPa, and progresses over a wider pressure range in Kr than previously reported . Flow stress determined from lattice strain was not accurately determined in the axial geometry, however it must exceed 5 GPa in order to explain defo rmation ob served in Pt compressed in Kr and Xe . This suggests Kr and Xe are similar in strength below 40 GPa and that both are stronger than Ar within this pressure range . This suggests that Kr and Xe are not ideal pressure - transmitting media, as they ar e remarkab ly stronger than the lower - Z RGS He and Ne . Strain in fcc diffraction lines remains lower than for fcc Ar, however . The strength of the hcp phase in Xe appears low indicating the strength calculated from these lines is not capturing the overal l strength of the two - phase mixture . This is possibly due to difficulty fitting the fcc ( 200 ) line to the non - ideal d - spacing conditions and/or stacking faults confounding accurate fitting of the hcp ( 100 ) / ( 101 ) lines . There is no significant difference in the ove rall non - hydrostatic stress supported by Kr or Xe, based on Pt and ruby standards within the medium . Differences in stress reported by R1 - R2 ruby fluorescence peaks indicate that non - hydrostaticity in Kr and Xe is higher than other RGS below 52 GPa, with Xe exerting more non - hydrostatic stress than Kr, and both exhibiting stiffening above 15 - 20 GPa. 90 Acknowledgements 91 CHAPTER 5: Conclusions and implications of this work WC, Kr, and Xe exhibit simple crystal structures a nd cover a range of important p hysical behavior , including elastic compression , failure , and plastic deformation. In t his dissertation , I have used multiple complementary experimental methods in the diamond anvil cell to determine the ir strength and deformation behavior . In this final chapter, I discuss the conclusions and implications of chapters 2 - 4 for Earth and materials science . I begin with best practices in the se tup of high - pressure experiments . I then discuss the implications of grain size, strength anisotropy, plasticity - mo dified strength calculations , and the impact of deformation mechanisms and phase transitions on material properties . U nderstanding the properties of simple materials at high - pressure provides a foundation for work on more complex materials and can guide d iscovery of similar properties in materials found in the interiors Earth and other planets . 5.1 Some reflections and recommended best practices for experimental measurements of physical properties at high pressure Technique is important for obtaining the bes t results from experiments at high pressure. This includes appropriate experimental setup, DAC pre paration, and loading conditions. Careful consideration of diffraction geometry, gasket dimensions, and choice of pressure medium can improve results dramat ically. Experiments on Kr and Xe in the radial diffraction geometry clearly show the fcc - hcp phase transition occurs over a much wider pressure range than was detected in the axial geometry. Because experiments in the axial geometry sample minimum strain , calculated unit cell volume may be too high in highly strained materials. Texture minima oriente d in the direction of minimum strain could make diffraction peaks invisible in the axial geometry and will affect peak intensity ratios used to calculate pha se fraction. 92 The effects of non - hydrostatic stress on EOS measurements discussed in Chapter 3 is important for measurements of EOS in the DAC. For pressures below ~12 GPa, liquid media may be used, and this will generate truly hydrostatic stresses. For pressures above 12 GPa, Ne remains the ideal candidate for quasi - hydrostatic pressure generation. Ne retains its low strength to upwards of 2.5 Mbar (Dorfman et al. , 2012) . Ne atoms are larger than He and are less likely to diffuse into samples and affect measurements (e.g. Sato, Funamori and Yagi, 2011; Shen et al. , 2011) , or diffuse into the anvils and weaken the DAC (Dewaele et al. , 2006) . Kr and Xe have not been compressed hydrostatically in any DAC experime n ts . Compression under quasi - hydrostatic conditions might improve the EOS of both materials. It has also been suggested that hydrostatic compression of Xe could eliminate the extra peak (Belonoshko, Ahuja and Johansson, 2001) . Better EOS measurements would improve our unders t an ding of the behavior of heavy RGS in planetary interiors and might help reconcile important differences between experiments and theoretical predcitions (e.g. Sasaki et al. , 2008; Troitskaya, V. V. Chabanenko, et al. , 2012) . Kr and Xe become sufficiently non - hydrostatic to induce yielding in the Pt marker at ~40 GPa, above which pressure differential stress supported by Pt in Kr and Xe is at le ast 5 - 7 GPa. The high strength and additional diffraction intensity from stacking faults and peaks from the onset of the fcc - hcp phase transition occurring below 3 GPa make Kr and Xe poor candidates for pressure transmitting media in DAC experiments un les s non - hydrostatic stress is desirable, and the presence of diffraction peaks is not a concern (for example in experiments using Raman or ruby to characterize the sample). 93 5.2 Grain size effects on elastic deformation behavior The mineral physics commu ni ty is actively addressing effects of grain size on plasticity of the crust and mantle (e.g. Karato, 1984; Dannberg et al. , 2017 ; Maierová et al. , 2017; Mohiuddin, Karato and Girard, 2020) , but the possibility of grain size effects on elasticity is not commonly addressed and thu s not well understood. The effects of grain size in Earth science deal with size differences on the or d er of - mm in the mantle (So l omato v, El - Khozondar and Tikare, 2002; Dannberg et al. , 2017) m - k m in the inner core (Bergman, 1998) . The behavior of nano - grain materials may be very different than that of microcrystalline or larger materials , as has bee n obse rved in the Hall - Petch effect (Hall, 1951; Petch, 1953; and e.g. Schiøtz, 2001) , yet inferences about p lan et - sca le dynamics are routinely made from experiments on materials with ~100 nm grain sizes . We observe a difference of ~20 GPa in the bulk modulus between nano - crystalline (~ 50 nm particles) and bulk (micron - scale particles) WC determined under quasi - hydr ostati c loading conditions. Similar differences in elasticity have been observed in other materials, but effects of grain size are not systematic. Nano - crystalline WC is less incompressible than its bulk counterpart, similar to many other ceramics (Chen et al. , 2002; Al - Khatatbeh, Lee and Kiefer, 2012; Le Godec et al. , 2012) , but differing from many other materials, including ceramics and metals (Chen et al. , 2001; Q F Gu et al. , 2008; Q. F. Gu et al. , 2008; Mikheykin et al. , 2012; Wang et al. , 2014; Hong et al. , 2015) . Changes in the EOS resulting from nano - crystalline grain size may be important to industry for tailored composites (Lin et al. , 2011) an d the manufacture of hard components. More work is needed to understand these effects in ceramics and other materials. Grain size is als o known to affect material hardness (e.g. Hall, 1954; Rice, Wu and 94 Boichelt , 1994 ) . The e ffects of size on the bulk modulus (as a parameter of hardness) may be important for better understanding both the incompressibility and the hardness of materials. Grain size effects include the Hall - Petch relationship for strength and chang es in deformation mechanisms (e.g. Yamakov et al. , 2003; Cheng et al. , 2013) . F uture work analyzing the effects of nano - crystalline grain size on strength and slip system activ ities in WC might yield a big ger pi cture of the interplay between grain size, elasticity, yield strength , and rheolog ical properties in WC , which can guide similar lines of inquiry on other materials. A thorough understanding of the relationships between these properties would allow for op timization of materials in industrial applications, and perhaps help constrain the dynamics of planetary interiors. 5.3 Effects of anisotropy and pl asticity on material strength Very few strength measurements obtained in the DAC provide information on i ndivid ual lattice planes. In WC the (0001) plane support s remarkably higher differential stress than other hkil , consistent with the absence of basal slip predicted by the EVPSC simulation . This could be quite important for understanding mantle dynamics in the Earth and for the production of oriented components which take advantage of this property . Oriented anvils which take advantage of strength anisotropy may be useful in extending the pressure range of large - volume presses. Multi - anvil devices allow compre ssion of substantially larger volumes of material and generate homogeneous P - T distributions in the sample material (Ishii et al. , 2017) . This is useful for the produc tion of materials. The use of sintered diamond anvils has greatly improved the pressure reg ime attainable, but these anvils are much more expensive than WC anvils (Ishii, Liu and Katsura, 2019) . Oriented (0001) cubes should be used to maximize pressure in opposed anvil devices using WC anvils. 95 Aniso tropic strength has also been observed in a few bori de materials, WB (Lei et al. , 2018) , ZrB 12 and YB 12 (Lei et al. , 2019) . Differences in the strength of various hkil in WB are attributed to differences in the W - B bond structure. Likewise, our assessment for WC is that covalent W - C bonds and positioning of C - atoms block basal slip and dislocation propagation (Bolton and Redington, 1980) . Anisotropic strength is related to elastic anisotropy, which is common in Earth ma terials. Carbides and borides are probably most similar to alloys or intermetallic of Fe and FeNi alloys, as well as compounds such as Fe 3 C and Fe 7 C 3 would provide informat ion on important materials and could help illuminate seismic anisotropy in the core. In WC, p yramidal slip activation on at 40 - 50 GPa was needed to induce any yielding of (0001) in the EVPSC simulation . Development of secondary slip syste ms with pressure have been observed in oliv i ne (Raterron et al. , 2011) , and are important for accurately describing rheology and viscosity in the mantle . Further work using radial diffraction to isolate strength values for indi vidual hk i l could provide similar information o n other materials , including prominent mineral phases in the deep Earth , and industrial materials used in the manufacture of strong components. In Chapter 3, EVPSC simulations are used to obtain a more compl ete assessment the strength of WC calculated from the mean of all hkil by including the effects of plasticity. Strength determined from elasticity theory from all planes is the standard reported by most studies on strength in the DAC. It is known that p l astic deformation affects strength calculations obtained in this way (Weidner et al. , 2004; Raterron and Merkel, 2009) . For WC , strength as determined from lattice strain is an overestimate when compared with strength derived from EVPSC simulation . More studies using EVPSC simulations are needed to better constrain the 96 strength of other materials in which lattice strai n is used to determine strength. This may lead to an improved understanding of slip and strength in industrial materials but is eq ually relevant to materials in the interiors of planets. Strength determined from EVPSC simulations for MgO at ~5 GPa is hig her than all previous experimental measurements (Lin et al. , 2017) , though still in good a greement with work by Merkel et al. ( 2002) , so assumptions that strength is systematically overestimated may not be accurate for all materials. However, these comparis ons support the importance of modeling plasticity to support accurate measurement of the yield stress. 5.4 Impact of sluggish pha se transitions on physical properties Gradual phase transitions from element partitioning or metastability are important to plasticity and convection of the mantle over a range of P - T conditions . Partitioning of Al from garnet to bridgmanite (We idner and Wang, 1998) and of Fe between bridgmanite and pPv (Hernlund, Thomas and Tackley, 2005) near th transition gradients composed of two - phase regimes with different bulk properties than either single phase. Metastable, lower - density minerals which persist in the transition zone may increase mantle viscosity and contribute to stagnation of sinking slabs (Faccenda and Dal Zilio, 20 1 7) , while the kinetics of the forsterite - wadsleyite - ringwoodite phase transition , which are notably also associated with grain size, may weaken the slabs and alter convection properties (Mohiuddin, Karato and Girard, 2020) . Plasticity is associat e d with the sluggish fcc - hcp transition in Kr and Xe as well . Texture orientations for Kr and Xe vary with pressure in both the fcc and hcp structures. In the fcc - phase, low - pressure textures are inferred to be associated with crystallization of the fcc phase and high - pressure textures are associated with deformation. For hcp, texture development is more complicated and the LPO changes from the initial c rystallization texture on 97 , proceeds through texture near and converts back to . This change in the LPO may be important for understanding the development of LPOs in planetary interiors , and whether they are associated with nucleation or defo rmation of the minerals involved . It may also be significant for deformation in the interiors of gas planets , where large regions of the planetary interior could exist in a two - phase mixture, potentially weaker than either pure phase because the ongoing t ransition could accommodate strain . M ore work is needed to determine strain in the individual phases during the transition , and to separate strength in individual phases and relate them to strength in the two - phase bulk . De termination of the thermodynamic conditions of phase transformations is important for understanding the effects of pressure on bonding within the material . Changes in bonding can lead to changes in elasticity and conductivity (e.g. metallization). Elastic ity affects the transport and storage (Rosa et al. , 2020) of Kr and other volatiles in the deep Earth. Conductivity and metallization are important for reactions leading to intermetallic Fe - Xe and Ni - compounds (Dewaele et al. , 2017) which are controversial proposed mech anisms for Xe storage in the deep Earth (Caldwell, 1997; Lee and S teinle - Neumann, 2006; Zhu et al. , 2014; Dewaele et al. , 2017; Stavrou et al. , 2018) . Future work on the phase diagram of Xe should also include temperature effects. Theoretical P - T behavior of Xe indicate s a triple - point with a bcc phase phase at ~2700 - 4000 K and 25 - 50 GPa ( Belonoshko, Ahuja and Johansson, 2001; Belonoshko et al. , 2002; Lukinov et al. , 2015) . Understanding the phase equilibria of Kr and Xe is important for determining t heir physical properties at high pressure and their reactions with other materials in th e deep Earth. 98 R EFERENCES 99 REF ERNCES Abdelkareem, M. A., Wilberforce, T., Elsaid, K., Sayed, E. T., Abdelghani, E. A. M. and Olabi, International Journal of Hydrogen Energy , p. S036 0319920333061. doi: 10.1016/j.ijhydene.2020.08.250. Spectroscopy at very high pressures. X. Use of ruby R - Journal of Physics E: Sci entific Instruments , 9(12), pp. 1140 1144. doi: 10.1088/0022 - 3735/9/12/034. Al - Khatatbeh, Y., Lee, K. K. 2 The Journal of Physical Chemistry C , 116(40), pp. 21635 21639. doi: 10.102 1/jp3075699. Amulele, G. M., Manghnani, M. H., Marriappan, S., Hong, X., Li, F., Qin, X. and Liermann, H . - 6%Co up to 50 GPa determined by synchrotron x - Journal of Appl ied Physics , 103(11), p. 113522. doi: 10.1063/1.2938024. Amulele, G. M., Manghnani, M. H. and Somayazulu - ray Journal of Applie d Physics , 99(2), p. 023522. doi: 10.1063/1.2164533. Science , 198(4316), pp. 453 465. doi: 10.1126/science.198.4316.453. Angel, R. J., Bujak, M., Zhao, J., Gatta, G. D. and limits of pressure media for high - pressure crystallograph Journal of Applied Crystallography , 40(1), pp. 26 32. doi: 10.1107/S0021889806045523. - pressure x - ray diffraction st udy of solid xenon and its equation of state Physical Review B , 29(12), p. 7026. alkali iodides when used as pressure med Physical Review B , 33(8), p. 5633. Bell, P. M. and Mao, H. - He, Ne and Ar pressure - Carnegie Institution of Washington Year Book , 80, pp. 404 -- 406. Belonoshko, A. B., Ahuja, R. and Johansson, and fcc - Physical Review Letters , 87(16). doi: 10.1103/PhysRevLett.87.165505. 100 of phase transit The Journal of Chemical Physics , 117(15), p. 7233. doi: 10.1063/1.1507775. Benea, M. L. and IOP Conference Series: Materials Science and Engineering , 85, p . 012004. doi: 10.1088/1757 - 899X/85/1/012004. Geophysical Research Letters , 25(10), pp. 1593 1596. doi: https://doi.org/10.1029/98GL01239. Besson, J. M., Nelmes, R. J., Hamel, G., Love Physica B: Condense d Matter , 180 181, pp. 907 910. doi: 10.1016/0921 - 4526(92)90505 - M. ystalline NaCl Journal of Geophysical Research: Solid Earth , 83(B3), pp. 1 257 1268. doi: 10.1029/JB083iB03p01257. - Pressure Melting Curves of Argon, Krypton, Physical Review Letters , 86(25), pp. 5731 5734. doi: 10.1103/PhysRevLett.86.5731. Journal of Materials Sci ence , 15(12), pp. 3150 3156. doi: 10.1007/BF00550388. Borovinskaya, I. P., Vershinnikov, V. I. and Ignat Concise Encyclopedia of Self - Propagating High - Temperature Synthesis . Elsevier, pp. 406 407. doi: 10.1016/B978 - 0 - 12 - 804173 - 4.00162 - 9. Bouhifd, A. M., Jephcoat, A. P., Porcelli, D., Kelley, S. P. and Marty, B. (2020) Geochemical Perspectives Letters , pp. 15 18. doi: 10.7185/geochem let.2028. Kg/Cm 2 , Proceedings of the American Academy of Arts and Sciences . American Academy of Arts & Sciences, 81(4), pp. 165 251. doi: 10.2307/20023677. Brown, H. L. The Journal of Chemical Physics , 45(2), pp. 547 549. doi: 10.1063/1.1727602. Bruck Syal, M., Michael Owen, J. and Miller, P. L. (2016) Icarus , 269, pp. 50 61. doi: 10.1016 /j.icarus.2016.01.010. Science , 277(5328), pp. 930 933. doi: 1 0.1126/science.277.5328.930. 101 - hydrostatic stress on the R lines of ruby Geophysical Research Letters , 23(24), pp. 3539 3542. doi: https://doi.org/10.1029/96GL03372. Chamorro - Pérez, E. solubility in silicate melts at h Nature , 393(6683), pp. 352 355. doi: 10.1038/30706. is and International Tables for Crystallography , p. 560. Available at: http://onlinelibrary.wiley.com/iucr/itc/Ha/ch5o3v0001/sec5o3o2o3o6/? (Accessed: 11 August 2020). Chen, B., Lai, X., Li, J., Liu, J., Zhao, J., Bi, W., Ercan Alp, E., Hu, M. Y. and Xiao, Y. (2018) Earth and Planetary Science Letters , 494, pp. 164 171. doi: 10.1016/j.epsl.2018.05.002. Chen, B., Pen - size effect on the compressib Physical Review B , 66(14), p. 144101. doi: 10.1103/PhysRevB.66.144101. Chen, B., Penwell, D., Kruger, M. B., Yue, A. Journal of Applied Physics , 89(9), pp. 4794 4796. doi: 10.1063/1.1357780. Chen, J. - Y., Yoo, C. - S., Evans, W. J., Liermann, H. - P., Cynn, H., Kim, M. and Jenei, Z. (2014) c to metastable hcp phase transition in krypton under variable co mpression Physical Review B , 90(14). doi: 10.1103/PhysRevB.90.144104. Effect on Deform Materials Resear ch Letters . Taylor & Francis, 1(1), pp. 26 31. doi: 10.1080/21663831.2012.739580. - principles thermal equation of state Computational Materials Science , 59, pp. 4 1 47. doi: 10.1016/j.commatsci.2012.02.028. - principles simulations of metal - Physic al Review B , 65(4), p. 045408. doi: 10.1103/PhysRevB.65.045408. C - binding computations of elastic Physical Review B . American Physical Society, 56(14) , pp. 8575 8589. doi: 10.1103/PhysRevB.56.8575. Cook, M. W. and B manufacture and cutting tool application of polycrystalline diamond and polycrystalline cubic 102 International Journal of Refractory Metals and Hard Materials , 18(2 3), pp. 14 7 152. doi: 10.1016/S0263 - 4368(00)00015 - 9. Cotto - Figueroa, D., Asphaug, E., Garvie, L. A. J., Rai, A., Johnston, J., Borkowski, L., Datta, S., e - dependent measurements of meteorite strength: Implications for Icarus , 277, pp. 73 77. doi: 10.1016/j.icarus.2016.05.003. Cynn, H., Yoo, C. S., Baer, B., Iota - Herbei, V., McMahan, A. K., Nicol, M. and Carlson, S. tic fcc - to - , Physical Review Letters , 86(20), pp. 4552 4555. doi: 10.1103/PhysRevLett.86.4552. AIP Conferenc e Proceedings . Shock Compression of Condensed Matter - 2001: 12th APS Topical Conference , Atlanta, Georgia (USA): AIP, pp. 783 786. doi: 10.1063/1.1483654. Sil ver, and Gold t Physical Review , 111(3), p. 713. Geochemistry, Geophysics, Geosystems , 18( 8), pp. 3034 3061. doi: https://doi.org/10.1002/2017GC006944. - ray topographic study of Journal of Applied Ph ysics , 99(10), p. 104906. doi: 10.1063/1.2197265. Physical Review B , 70(9). doi: 10.1103/PhysRevB.70.094112. Dewaele, A., Loubeyre, P., Occelli, F., Marie, O. and Mezouar, M Nature Communications . Nature Publishing Group, 9(1), p. 2913. doi: 10.1038/s41467 - 018 - 05294 - 2. Dewaele, A., Pépin, C. M., Geneste, G. and Garba rino, G. (2017) High Pressure Research , 37(2), pp. 137 146. doi: 10.1080/08957959.2016.1267165. transition metals in the Mbar range: Experiments and projector augmented - Physical Review B , 78(10). doi: 10.1103/PhysRevB.78.104102. Phy sical Review Le tters , 88(7). doi: 10.1103/PhysRevLett.88.075504. - Huntington transition to metallic Science , 355(6326), pp. 715 718. doi: 10.1126/science.aal1579. 103 Dong, H., Dorfman, S. M., Chen, Y ., Wang, H., Wang, J., Qin, J., He, D. and Duffy, T. S. (2012) Journal of Applied Physics , 111(12), p. 123514. doi: 10.1063/1.4728208. Dong, H., H e, D., Duffy, T nanocrystalline cubic BC 2 N from x - Physical Review B , 79(1), p. 014105. doi: 10.1103/PhysRevB.79.014105. Dorfman, S. M., Prakapenka, V. B., Meng, Y Journal of Geophysical Research , 117(B8). doi: 10.1029/2012JB009292. and texture of Pt compressed to Journal of Applied Physics , 117(6), p. 065901. doi: 10.1063/1.4907866. Doklady Earth S ciences , 410(7) , pp. 1091 1095. doi: 10.1134/S1028334X06070208. Dubrovinskaia, N., Dubrovinsky, L., Solopova, N. A., Abakumov, A., Turner, S., Hanfland, M., Bykova, E., Bykov, M., Prescher, C., Prakapenka, V. B., Petitgirard, S., Chuvashova, I., Gasharova, B., Mathis, Y. - Science Advances , 2(7), p. e1600341. doi: 10.1126/sciadv.1600341. Dubrovinskaia, N., Solozhenko, V. L., Miyajima, N., Dmitriev, V., Kurakevych, O. O. and Applied Physics Letters , 90(10), p. 101912. doi: 10.1063/1.2711277. D ubrovinsky, L., Dubrovinskaia, N., Bykova, E., Bykov, M., Prakapenka, V., Prescher, C., Glazyrin, K., Liermann, H. - P., Hanfland, M., Ekholm, M., Feng, Q., Pourovskii, L. V., le metal osmium Nature . Nature Publishing Group, 525(7568), pp. 226 229. doi: 10.1038/nature14681. Shock Compression of Condensed Ma tter - 2007: Proceedings of the Conference of the American Physical Society Topical Group on Shock Compression of Condensed Matter (2007 APS SCCM) Part One (AIP Conference Proceedings Volume 955) , pp. 639 644. doi: 10.1063/1.2833175. Duffy, T . S., Shen, G., Heinz, D. L., Shu, J., Ma, Y., Mao, H. - K., Hemley, R. J. and Singh, A. K. Physical Review B , 60(22), p. 15063. Eremets, M. I., Gregoryanz, E. A ., Struz hkin, V. V., Mao, H., Hemley, R. J., Mulders, N. and Physical review letters , 85(13), p. 2797. 104 Errandonea, D., Boehler, R., Japel, S., Mezouar, M. and Benedetti, L. R. (2006) transformation of compressed solid Ar: An x - Physical Review B , 73(9). doi: 10.1103/PhysRevB.73.092106. xenon to 50 G Physical Review B , 65(21). doi: 10.1103/PhysRevB.65.214110. solid phase transitions in mantle Lithos , 268 271, pp. 198 224. doi: 10.1016/j.lithos.2016.11.007. Fang, Q., Bai, W., Yan American Mineralogist , 94(2 3), pp. 387 390. doi: 10.2138/am.2009.3015. intering , and mechanical properties of nanocrystalline cemented tungsten carbide International Journal of Refractory Metals and Hard Materials , 27(2), pp. 288 299. doi: 10.1016/j.ijrmhm.2008.07.011. imental research on HEL and failure properties of Defence Technology , 12(3), pp. 272 276. doi: 10.1016/j.dt.2016.01.007. Made by the Utilizat ion of Ruby Sharp - Science , 176(4032), pp. 284 285. doi: 10.1126/science.176.4032.284. - fcc He 4 Phase Diagram to about 9 Physical Review Letters , 44(4), pp. 259 262. do i: 10.1103/PhysRevLett.44.259. Journal of Applied Physics , 75(9), pp. 4327 4331. doi: 10.1063/1.355975. Fu runo, K., Onodera, A. and Kume, S. (198 - Japanese Journal of Applied Physics . IOP Publishing, 25(8A), p. L646. doi: 10.1143/JJAP.25.L646. pressure derivatives for tungsten carb Journal of Applied Physics , 50(5), pp. 3331 3333. doi: 10.1063/1.326273. International Journal of Refractory Metals and Hard M aterials , 24(1 2), pp. 1 5. doi: 10.1016/j.ijrmhm.2005.05.015. 105 bridgmanite and magnesiowüstite aggregates at lower mantle c Science , 351(6269), pp. 144 147. doi: 10.1126/science.aad3113. Goldschmidt, H. J. (1967) Interstitial Alloys . 1st edn. London, UK: Butterworth & Co., Lt.d. Available at: DOI 10.1007/978 - 1 - 4899 - 5880 - . International Journal of Refractory Metals and Hard Materials , 28(2), pp. 250 253. doi: 10.1016/j.ijrmhm.2009.10.006. microcrysta Journa l of Physics: Condensed Matter , 20(44), p. 445226. doi: 10.1088/0953 - 8984/20/44/445226. P hysical Review Letters , 100(4), p. 0455 02. doi: 10.1103/PhysRevLett.100.045502. - Advanced Materials , 20(19), pp. 3620 3626. doi: 10.1002/adma.200703 025. Haines, J., Léger, J. and Bocquill Annual Review of Materials Research , 31(1), pp. 1 23. doi: 10.1146/annurev.matsci.31.1.1. haracteristics of the lüders deformatio Proceedings of the Physical Society. Section B , 64(9), p. 742. Nature . Nature Publishing Group, 173(4411), pp. 948 949. doi: 10.1038/173948b0. Physics Letters A , 140(3), pp. 117 121. doi: 10.1016/0375 - 9601(89)90503 - 3. Hammersley, A. P. (1997) Fit2D: An Introduction and Overview . ESRF Internal Report. ESRF Internal Report. Available at: htt p://www.esrf.eu/computing/scientific/FIT2D/FIT2D_INTRO/fit2d.html (Accessed: 19 November 2019). Hammersley, A. P., Svensson, S. O., Hanfland, M., Fitch, A. N. and Hausermann, D. (1996) - dimensional detector softwa re: From real detector to idealised ima ge or two - High Pressure Research , 14(4 6), pp. 235 248. doi: 10.1080/08957959608201408. from radial x - ray dif fraction in a diamond cell under nonhyd Physical Review B , 70(18), p. 184121. doi: 10.1103/PhysRevB.70.184121. 106 First - Physica B: Condensed Matter , 405(20), pp. 4335 4338. doi: 10.1016 /j.physb.2010.07.037. X - ray Imaging of Stress and Strain of Diamond, Iron, and Tungsten at Science , 276(5316), pp. 1242 1245. doi: 10.1126/science.276.5316.1242. Hernlund, J. W., Thomas, C. and Tackley, P. J. (2005 - perovskite phase b Nature . Nature Publishing Group, 434(7035), pp. 882 886. doi: 10.1038/nature03472. powder diffraction data: the use of regress Mineralogical Magazine , 61(1), pp. 65 77. for Wide Ranges in Temperature and Pressure up to 500 GPa an Journal of Physical and Chemical R eference Data , 30(2), pp. 515 529. doi: 10.1063/1.1370170. - induced stiffness of Au nanoparticles to 71 GPa under quasi - Jou rnal of Physics: Condensed Matter , 27(48), p . 485303. doi: 10.1088/0953 - 8984/27/48/485303. - machining system for diamond anvil cell experiments and general precision machining applica tions at the High Pressure Collaborative Acc Review of Scientific Instruments , 86(7), p. 072202. doi: 10.1063/1.4926889. Natu re , 421(6923), p. 599. doi: 10.1038/421599b. - Type Multi - Engineering , 5(3), pp. 434 440. doi: 10.1016/j.eng.2019.01.01 3. Ishii, T., Yamazaki, D., Tsujino, N., Xu, F., Liu, Z., Kawazoe, T., Yamamoto, T., Druzhbin, D., GPa in a Kawai - type multi - anvil apparatus with tungsten carbide High Pressure Research , 37(4), pp. 507 515. doi: 10.1080/08957959.2017.1375491. - Nature , 393(6683), pp. 355 358. Jephcoat, A. P., Mao, H. - k, Finger, L. W., Cox, D. E., Hemley, R. J. and Zha, C. - s (1987) - induce Physical review letters , 59(23), p. 2670. - Earth - Scienc e Reviews , 110(1 4), pp. 127 147. doi: 10.10 16/j.earscirev.2011.10.005. 107 Karato, S. - - Tectonophysics , 104(1 2), pp. 155 176. doi: 10.1016/0040 - 1951(84)90108 - 2. Kavner, A. and Duffy, T. S. ( e at upper mantle Geophysical Research Letters , 28(14), pp. 2691 2694. doi: https://doi.org/10.1029/2000GL012671. ssure and Physical R eview B , 68(14), p. 144101. doi: 10.1103/PhysRevB.68.144101. state of the nanocrystalline cubic silicon nitride Physical Review B , 72(1), p. 014102. doi: 10.1103/PhysRevB.72.014102. Kim, E., Nicol, M., Cynn, H. and Yoo, C. - - to - hcp Transformations in Solid Xenon under Pressure: A First - Physical Rev iew Letters , 96(3). doi: 10.1103/PhysRevLett .96.035504. Meteoritics & Planetary Science , 46(11), pp. 1653 1669. doi: 10.1 111/j.1945 - 5100.2011.01254.x. Kittel, C. (19 96) Introduction to Solid State Physics . 7th edn. New York: Wiley. Available at: http://www.wiley.com/WileyCDA/WileyTitle/productCd - EHEP000803.html (Accessed: 10 October 2017). Klotz, S., Chervin, J. - C., Munsch, imits of 11 Journal of Physics D: Applied Physics , 42(7), p. 075413. doi: 10.1088/0022 - 3727/42/7/075413. Kong, X. - S., You, Y. - W., Xia, J. H., Liu, C. S., Fang, Q. F., Luo, G. - N. and Huang, Q. - Y. (2010) Journal of Nuclear Materials , 406(3), pp. 323 329. doi: 10.1016/j.jnucmat.2010.09.002. ion in rare - Low Temperature Phy sics , 37(5), pp. 435 438. doi: 10.1063/1.3606459. Kraus, R. G., Stewart, S. T., Swift, D. C., Bolme, C. A., Smith, R. F., Hamel, S., Hammel, B. D., Spaulding, D. K., Hicks, D. G., Eggert, J. H. and Collins, G. W. silica and th Journal of Geophysical Research: Planets , 117(E9). doi: https://doi.org/10.1029/2012JE004082. - p rinciples study of solid Ar and Kr under hig Physical Review B , 52(21), p. 15165. 108 Le Godec, Y., Kurakevych, O. O., Munsch, P., Garbarino, G., Mezouar, M. and Solozhenko, V. Journal of Superhard Materials , 34(5), p p. 336 338. doi: 10.3103/S1063457612050085. Lee, K. K. M. and Steinle - - pressure alloying of iron and xenon: Journal of Geophysical Research: Solid Earth , 111(B2), p. B02202. doi: 10.1029/2005JB003 781. Journal of Materials Science , 17(9), pp. 2657 2660. doi: 10.1007/BF00543901. Lei, J., Akopov, G., Yeung, M. T., Y an, J., Kaner, R. B. and Tolbert, S. H. (201 - Ray Advanced Functional Materials , 29(22), p. 1900293. doi: https://doi.org/10.1002/adfm.201900293. Lei, J., Yeung, M. T., Robinson, P. J., Mohammadi, R., Turner, C. L., Yan, J., Kavner, A., Controls Strength Anisotropy in Hard Materials by Comparing the High - Pressure B ehavior o f The Journal of Physical Chemistry C . American Chemical Society, 122(10), pp. 5647 5656. doi: 10.1021/acs.jpcc.7b11478. - Like Behavior of Tungsten Carbi de in Sur face Science , 181(4099), pp. 547 549. doi: 10.1126/science.181.4099.547. - to - The Journal of Chemical P hysics , 1 46(21), p. 214502. doi: 10.1063/1.4983167. - principles calculations of structural stability and mechanical properties of tungsten carbide under high pressur Journ al of Physics and Chemistry of Solids , 75(11), pp. 1234 1239. doi: 10.1016/j.jpcs.2014.06.011. stability, elasticity, hardness and electronic structure s of W C Journal of Alloys and Compounds , 502(1), pp. 28 37. doi: 10.1016/j.jallcom.2010.04.184. Liermann, H. - diamond anvil cell: Pressure dependences of str ength and grain size from X - Journal of Physics and Chemistry of Solids . (SMEC 2009), 71(8), pp. 1088 1093. doi: 10.1016/j.jpcs.2010.03.012. Lin, F., Hilairet, N., Raterron, P., Addad, A., Immoor, J., Marquardt, H., Tomé, C. N., Miyag i, L. and - viscoplastic self consistent modeling of the ambient temperature Journal of Applied Physics , 122(20), p. 205902. doi: 10.1063/1.4999951. 109 Lin, Giannetta, Jugle, Coupe r, Dunlea Minerals , 9(11), p. 679. doi: 10.3390/min9110679. Lin, Z. J., Zhang, J. Z., Li, B. S., Wang, L. P., Mao, H. - K., Hemley, R. J. and Zhao, Y. (2011) Applied Physics Letters . American Institute of Physics, 98(12), p. 121914. doi: 10.1063/1.3570645. incompres Applied Physics Letters , 95(21), p. 211906. doi: 10.1063/1.3268457. - core Geophysical Researc h Letters , 43(3), pp. 1084 1091. doi: https://doi.org/10.1002/2015GL067019. - volume - Jou rnal of A pplied Physics , 108(5), p. 053513. doi: 10.1063/1.3481667. Physical Review B , 38(14), pp. 9483 9489. doi: 10.1103/PhysRevB.38.9483. Liu, L., Bi, Y., Xu, J Physica B: Condensed Matter , 413, pp. 109 115. doi: 10.1016/j.physb.2012.11.026. Lukino v, T., Ro of the fcc Computational Materials Science , 107, pp. 66 71. doi: 10.1016/j.commatsci.2015.04.055. Luttero tti, L., structural analysis of thin films using a combined X - Thin Solid Films , 450(1), pp. 34 41. doi: 10.1016/j.tsf.2003.10.150. Lutterotti, L., Matthies, S. a nd Wenk, H. - CPD News! , pp. 14 15. Lutterotti, L., Matthies, S., Wenk, H. - texture and structure analysis of deformed limeston e from time - of - flight neutron diffraction Journal of Applied Physics , 81(2), pp. 594 600. doi: 10 .1063/1.364220. of deformation mechanisms and grain si ze evolution in granulites Implications for the rheology Earth and Planetary Science Letters , 466, pp. 91 102. doi: 10.1016/j.epsl.2017.03.010. 110 - rate dependence of the compressi ve strength of WC Materials Science and Engineering: A , 612, pp. 115 122. doi: 10.1016/j.ms ea.2014.06.020. Mandt, K. E., Mousis, O., Bockelée - Space Science Reviews , 197(1), pp. 5 7. doi: 10.1007/s11214 - 015 - 0215 - 2. Mao, H., Badro, J., Shu, J., Hemley Journal of Physics: Condensed Ma tter , 18(25), pp. S963 S968. doi: 10.1088/0953 - 8984/18/25/S04. Mao, H. K., Hemley, R. J., Wu, Y., Jephcoat, A. P., Finger, L. W., Zha, C. S. and Bassett, W. A. - pressure phase diagram and equation of state of solid helium from single - crystal X - Physical review letters , 60(25), p. 2649. Mao, H. - k., Xu, J. and Bell, P. M. Under Quasi - Journal of Geophysical Research , 91(B5), pp. 4673 46 76. doi: 10.1029/JB091iB05p04673. ayer beneath the Nature Geoscience . Nature Publishing Group, 11(11), pp. 876 881. doi: 10.1038/s41561 - 018 - 0225 - 2. Marquardt increase in mantle visc Nature Geoscience , 8(4), pp. 311 314. doi: 10.1038/ngeo2393. Physical Review B , 33(8), p. 5344. icates at high pressures and room Nature , 348(6301), pp. 533 535. doi: 10.1038/348533a0. Melosh, H. J. (1989) Impact cratering: A geologic process . Available at: http://adsabs.harvard.edu/abs/1989icgp.book.....M. Menéndez - Ab initio calculations of elastic properties of Physical Review B , 76 (5). doi: 10.1103/PhysRevB.76.054117. Meng, Y., Weidner, D - hydrostatic diamond anvil cell: Effect on the volume - Geophysical Research Letters , 20(12), pp. 1147 1150. doi: 10.1 029/93GL01400. Merkel, S., Miyajima, N., Antonangeli, D., orientation and stress in polycrystalline hcp - Journal of Applied Physics , 100(2), p. 023510. doi: 10.1 063/1.2214224. 111 Merkel, S., Tomé, C. and Wenk, H. - R. (2009) on high pressure deformation of hcp - Physical Review B , 79(6), p. 064110. doi: 10.1103/PhysRevB.79.064110. Merkel, S., Wenk, H. R., Shu, J., Sh en, G., Gillet, P., Mao, H. and Hemley, R. J. (2002) Journal of Geophysical Research: Solid Earth , 107(B11), p. 2271. doi: 10.1029/2001JB000920. - ray tr ansparent gasket for diamond anvil cell high pressure expe Review of Scientific Instruments . American Institute of Physics, 76(4), p. 046109. doi: 10.1063/1.1884195. e strains in polycrystalline materials deformed under high pressure: Application to hcp - Journal of Physics and Chemistry of Solids , 67(9 10), pp. 2119 2131. doi: 10.1016/j.jpcs.2006.05.025. Mikheykin, A. S., Dmitriev, V. P., Chagovets, S. V., Kurigano va, A. B., Smirnova, N. V. and Applied Physics Letters , 101(17), p. 173111. doi: 10.1063/1.4758000. Miyagi, L., Kanitpanyacharoen, W., Raju, S. V., Kaercher, P., Knight, J., MacDowell, A., Wenk, H. - ined resistive and laser heating technique for in situ radial X - Review of Scientific Instruments . American Institute of P hysics, 84(2), p. 025118. doi: 10.1063/1.4793398. Mohiuddi Nature Geoscience . Nature Publishing Group, 13(2), pp. 170 174. doi: 10.1038/s415 61 - 019 - 0523 - 3. Nakajima, Y., Takahashi, E., Sata, N., Nish ihara, Y., Hirose, K., Funakoshi, K. and Ohishi, Y. - pressure stability of Fe7C3: Implication for iron - American Mineralogist . De Gruyter, 96(7), pp. 1158 1165. doi: 10.2138/am.2011.3703. Physical Review , 111(3), p. 707. - High Pressur e Research , 27(4), pp. 393 407. doi: 10.1080/0895795070165 9734. Icarus , 124(2), pp. 359 371. doi: 10.1006/icar.19 96.0214. Nye, J. F. (1985) Physical Properties of Crystals : Their Representation by Tensors and Matrices . 2nd edn. Clarendon Press. 112 Culet diameter and the ach ievable pressure of a diamond anvil cell: Implications for the upper Review of Scientific Instruments , 89(11), p. 111501. doi: 10.1063/1.5049720. Scientif ic Reports . Nature Publishing Group, 10(1), p. 1393. doi: 10.1038/s41598 - 020 - 58252 - 8. Philosophical Magazine , 8(94), pp. 1605 1632. doi: 10.1080/14786436308207327. Peng, F., Song, X., Nature Communications . Nature Publishing Group, 11(1), p. 5227. doi : 10.1038/s41467 - 020 - 19107 - y. Journal of the Iron and Steel Institute , 174, pp. 25 28. Journal of Applied Physics , 44(12), p. 5377. doi: 10.1063/1.1662159. Poirier, J. - P. ( 2000) . Cambridge University Press. of state and ela Physical Review B , 39(2), p. 1332. Polia - ray absorption Physical Review B , 39(5), p. 3369. Dependence of Diffraction - Line Broadening Caused by Strai n and Journal of Applied Crystallography , 31(2), pp. 176 180. doi: https://doi.org/10.1107/S0021889897009795. Prescher, C. and Prakapenka, V. B. (20 DIOPTAS : a program for reduction of two - dimensional X - High Pressure Research , 35(3), pp. 223 230. doi: 10.1080/08957959.2015.1059835. Rae, A. S. P., Collins, G. S., Poelchau, M., Riller, U., Davison, T. M., Grieve, R. A. F., Osinski, G. R., Morgan, J. V. and Scien tists, I. - - Strain Evolution During Peak - Journal of Geophysical Research: Planets , 124(2), pp. 396 417. doi: h ttps://doi.o rg/10.1029/2018JE005821. slip systems: Implications for upper - Physics of the Earth and Planetary Interiors , 188(1 2), pp . 26 36. doi: 10.1016/j.p epi.2011.06.009. 113 In situ rheological measurements at extreme pressure and temperature using synchrotron X - Journal of Synchrotron Radiation , 16(6), pp . 748 756. d oi: 10.1107/S090904950903 4426. Grain - Journal of the American Ceramic Society , 77(10), pp. 2539 2553. doi: https://doi.org/10.1111/j.1151 - 2916.1994.tb04641.x. Ritterbex, S Scientific Reports . Nature Publishing Group, 10(1), p. 6311. doi: 10.1038/s41598 - 020 - 63166 - 6. Rivers, M., Prakapenka, V., Kubo, A ., Pullins, COMPRES/GSECARS gas - loading system for diamond anvil cells at the Advanced Photon High Pressure Research , 28(3), pp. 273 292. doi: 10.1080/08957950802333593. Roebuck, B., Klose, P. and Mingard, K. P. (2012 Acta Materialia , 60(17), pp. 6131 6143. doi: 10.1016/j.actamat.2012.07.056. Rosa, A. D., Bouhifd, M . A., Morard, G., Briggs, R., Garbarino, G., Irifune, T., Matho n, O. and Pa Earth and Planetary Science Letters , 532, p. 116032. doi: 10.1016/j.epsl.2019.116032. Rosa, A. D., Garbar ino, G., Briggs, R., Svitlyk, V., Morard, G., Bouhifd, M. A., J acobs, J., I - hcp martensitic transition on Physical Review B , 97(9). do i: 10.1103/PhysRevB.97.094115. Ross, M., Mao, H. K., Bell, P. M . and Xu, J. Rüetschi, A. - to liquid pressure Review of Scientific Instruments . American Ins titute of Physics, 78(12), p. 123901. doi: 10.1063/1.2818788. Journal of Applied Physics , 46(3), p. 138 9. doi: 10.1063/1.321737. Ruoff, A. L., Xia, H. and Xia, Q. (19 from a sample with a pressure gradient: Studies on three samples with a maximum pressure of Review of Scientific Instruments , 63(10), pp. 4342 4348. doi: 10.1063/1.1143734. Sanloup, C. (2 Frontiers in Physics , 8, p. 157. doi: 10.3389/fphy.2020.00157. 114 - pressure brillouin study on solid Journal of Physics: Conference Series , 121 (4), p. 0420 13. doi: 10.1088/1742 - 6596/121/4/042013. - pressure Brillouin study of the elastic properties of rare - gas solid xenon at pr Journal of Raman Spectroscopy , 40(2), pp . 121 127. d oi: 10.1002/jrs.2087. Nature Communications , 2, p. 345. doi: 10.103 8/ncomms1343. e metals at the atomic scale. What can we do? arXiv:cond - mat/0109320 . Available at: http://arxiv.org/abs/cond - mat/0109320 (Accessed: 23 February 2021). Schuberth, E., Creuz burg, M. and Müller - FCC HCP Pha physica status solidi (b) , 76(1), pp. 301 306. rbide, nitride, and boride ceramics with WC - T echnical Phy sics Letters , 34(10), pp. 841 844. doi: 10.1134/S106378500810009X. Shen, G., Mei, Q., Prakapenka, V. B., Lazor, P., Sinogeikin, S., Meng, Y. and Park, C. (2011) Proceedings of the Nati onal Academy of Sciences , 108(15), pp. 6004 6007. Shen, G., Wang, Y., Dewaele, A., Wu, C., Fratanduono, D. E., Eggert, J., Klotz, S., Dziubek, K. Asimow, P. D., Mashimo, T., Wentzcovitch, R. M. M., and other members of t A proposal for an IPPS ruby gauge (IPPS - High Pressure Research , 40(3), pp. 299 314. doi: 10. 1080/08957959.2020.1791107. Shi, N., Bai, W., Li, G., Xiong, M. , Fang, Q., Acta Geologica Sinica - English Edition , 83(1), pp. 52 56. doi: https://doi.org/10.1111/j.1755 - 672 4.2009.00007.x. Shibata, T., Takahashi, E. and Matsuda, J. (199 ty of neon, argon, krypton, and xenon Geochimica et Cosmochimica Acta , 62(7), pp. 1241 1253. Shieh, S. R i O 2 acros s the Stishovite C a C l 2 - Physical review letters , 89(25), p. 255507. Silvestroni, L., Gilli, N., Migliori, A., Sciti, D., Watts, J., Hilmas, G. E. and Fahrenholtz, W. G. h and toughn Journal of the European Ceramic Society , 40(6), pp. 2287 2294. doi: 10.1016/j.jeurceramsoc.2020.01.055. 115 ubic system) compressed nonhydrostatically in an opposed anvil Jou rnal of Applied Physics , 73(9), p. 4278. doi: 10.1063/1.352809. - pressure diffraction data (cubic system): Assessment of various Journal of Applied Physics , 106(4), p. 043514. doi: 10.1063/1.3197213. Journal of Physics: Conference Series , 377, p. 012007. doi: 10.1088/1742 - 6596/377/1/0120 07. Jo urnal of Physics: Conference Series , 377, p. 012007. doi: 10.1088/1742 - 6596/377/1/012007. The lattice strains in a specimen (hexagonal system) compressed nonhydrostatically in an opposed anvil high pressur Jo urnal of Applied Physics , 75(10), pp. 4956 4962. doi: 10.1063/1.355786. Singh, A. K., Balasingh, C., Mao, H., Hemley, R. J. and S Journal of Applied Physics , 83( 12), pp. 756 7 7575. doi: 10.1063/1.367872. - crystal elastic modu li from polycrystalline X - Physical Review Letters , 80(10), p. 2 157. strain component in niobium to 145 GPa under various fluid pressure - Journal of Applied Physics , 90(7), pp. 3269 3275. doi: 10.106 3/1.1397283. Solomatov, V. S., El - constraints from numerical modeling of grain growth in two - Physics of the Earth and Planetary Interiors , 129(3), pp. 265 282. doi: 10.101 6/S0031 - 9201 (01)00295 - 3. Stavrou, E., Yao, Y., Goncharov, A. F., Lobanov, S. S., Zaug, J. M., Liu, H., Greenberg, E. and Prakapenka, V. B. (2 - Nickel Intermetallic Compounds at Physica l Review Let ters , 120(9), p. 096001. doi: 10.1103/PhysRevLett.120.096001. latively low temperature synthesis of hexagonal tungsten carbide films by N doping and its effect on the preferred orientation, Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films , 27(2), pp. 167 173. doi: 10.1116/1.3058721. roperties of hexagonal and cubic polymorphs of tungsten monocarbide WC and mononitride WN from first - physica status solidi (b) , 245(8), pp. 1590 1597. doi: 10.1002/pssb.200844077. 116 nical proper ties of high purity Journal of Materials Science , 35, pp. 1181 1186. High Pressure Research , 28(2), pp. 75 126. doi: 10.1080/0895 795080223564 0. High Pressure Research , 28(2), pp. 75 126. doi: 10.1080/08957950802235640. Takahash Phil osophical Ma gazine , 12(115), pp. 1 8. doi: 10.1080/14786436508224941. - press ure medium with powder x - Journal of Applied Physics , 89(1), pp. 662 668. doi: 10.1063/ 1.1328410. T akemura, K., Watanuki, T., Ohwada, K., Machida, A., Ohmura, A. and Aoki, K. (2010) - ray diffraction study of Ne up to 24 Journal of Physics: Conference Series , 215, p. 012017. doi: 10.1088/1742 - 6596/215/1/012017. Taylor, W. R. a nd Foley, S. - buffering techniques for C - O - H fluid - Journal of Geophysical Research: Solid Earth , 94(B4), pp. 4146 4158. doi: 10.1029/JB094iB04p04146. emy: The Sea MRS Bulletin , 23(1), pp. 22 27. doi: 10.1557/S0883769400031420. Tian, C., Wu, N., Liu, F., Saxe - body interaction energy for The Journal of Chemical Physics , 137(4), p. 044108. doi: 10.1063/1.4737183. Timms, N. E., Reddy, S. M., Healy, D., Nemchin, A. A., Grange, M. L. , Pidgeon, R. T. and Hart, - related microstructures in lunar zircon: A shock - deforma tion mechani Meteoritics & Planetary Science , 47(1), pp. 120 141. doi: https://doi.org/10.1111/j.1945 - 5100.2011.01316.x. Troitskaya, E. P., Chabanenko, V. V., Zhikharev, I. V., Gorbenko, I. I. and Pilipenko, E. A. - gas Physics of the Solid State , 54(6), pp. 1254 1262. doi: 10.1134/S106 3783412060340. Troitskaya, E. P., Chabanenko, Val. V., Zhikharev, I. V., Gorbenko, Ie. Ie. and Pilipenko, E. A. - gas Physics of the Solid State , 54(6), p p. 1254 1262. doi: 10.1134/S1063783412060340. 117 - principles study of systematics of high - pressure The Journal of Chemical Physics , 117(12), p. 5859. doi: 10.1063/1 .1502241. in High Pressure Science and Technology: Proceedings of th e XV AIRAPT and XXXIII EHPRG International Conference , pp. 183 185. viscoplastic self - Journal of the Mechanics and Physics of So lids , 58(4), pp. 594 612. doi: 10.1016/j.jmps.2010.01.004. Journal of Applied Physics . American Insti tute of Physics, 108(6), p. 063521 . doi: 10.1063/1.3485828. Wang, Q., He, D., Peng, F., Xiong, L., Wang, J., Wang, P., Xu, C. and Liu, J. (2014) Solid State Communications , 182, pp. 26 29. doi: 10.1016/j.ssc.2 013.12.015. Weidner, D. J., Li, L. Geophysical Research Letters , 31(6), p. n/a - n/a. doi: 10.1029/2003GL019090. - and Clapeyro n - induced buoyancy at the 660 km d Journal of Geophysical Research: Solid Earth , 103(B4), pp. 7431 7441. doi: https://doi.org/10.1029/97JB03511. Wenk, H. - R., Cottaar, S., Tomé , C. N., McNamara, A. and Romanowicz, B. (2011) Earth and Planetary Science Letters , 306(1 2), pp. 33 45. doi: 10.1016/j.epsl.2011.03.021. Wenk, H. - R., Lutterotti, L. Powder Diffra ction , 25(3), pp. 283 296. doi: 10.1154/1.3479004. Wenk, H. - - based program syste Journal of Applied Crystallography , 31(2), pp. 262 269. doi: 10 .1107/S002188989700811X. Wu, T. - radiation with two diffraction ge Experimental Techniques in Mineral and Rock Physics . Springer, pp. 509 519. Available a t: http://link.springer.com/chapter/10.1007/978 - 3 - 0348 - 5108 - 4_16 (Accessed: 9 October 2016). Yamakov, V., Wolf, D., Phillpot, S. R., Mukherjee, A. K. and G mechanism crossover and mechanical behaviour in nanocrystalline materi Philosophical Magazine Letters , 83(6), pp. 385 393. doi: 10.1080/09500830031000120891. 118 Yamazaki, D., Ito, E., Yoshino, T., Tsujino, N., Yoneda, A., G omi, H., Vazhakuttiyakam, J., - pressure ge neration in the Kawai - type multianvil apparatus equipped with tungsten - carbide anvils and sintered - diamond anvils, and X - ray observation on CaSnO3 and (Mg, Comptes Rendus Geoscience , 351(2 3), pp. 253 259. doi: 10.1016/j.crte.2018.07.004. Yeung, M Annual Review of Materials Research , 46(1), pp. 465 485. doi: 10.11 46/annurev - matsci - 070115 - 032148. ns of xenon with iron and Nature Chemistry . doi: 10.1038/nchem.1925. Zhukov, V. P. and Gubanov, V. A. (198 - mechanical properties of tungsten and tungsten carbides as studi ed by the LMTO - Solid State Communications , 56(1), pp. 51 55. doi: 10.1016/0038 - 1098(85)90532 - 0.