Exploring the hardness and high-pressure behavior of osmium and ruthenium-doped rhenium diboride solid solutions

Materials with superior hardness are desirable in the machining and manufacturing industries, where higher hardness affords higher wear resistance and longer lifetimes of cutting tools and abrasives.^1–3 Traditional superhard materials (Vickers hardness, H_v ≥ 40 GPa), such as diamond and cubic boron nitride (c-BN), are both difficult and expensive to synthesize due to the required high pressure and high temperature conditions.^4–6 These shortcomings have motivated the search for alternative superhard materials that are readily synthesized and capable of cutting materials at lower costs.^7–10 

Transition metal borides are a class of materials that have attracted significant interest among materials researchers due to their remarkable magnetic, thermal, and mechanical properties.^11–13 More specifically, several transition metal boride systems exhibit exceptionally high hardness, making them an attractive alternative to traditional hard materials for industrial applications.¹⁴ Rhenium diboride (ReB₂) is an example of an exceptionally hard transition metal boride system, with Vickers hardness reaching as high as 40.5 GPa under a low applied load (0.49N).¹⁵ Since the investigation of ReB₂, the scope of hard metal boride research has widened toward higher borides, such as tungsten tetraboride (WB₄)^16,17 and metal dodecaborides (MB₁₂),¹⁸ along with their complementary alloys and solid solutions.^19,20 Studies based on the metal boride systems demonstrate that both atomistic or intrinsic routes to increasing hardness (e.g., solid solution formation) and composite or extrinsic routes to hardness enhancement (e.g., grain boundary strengthening and precipitation hardening) can lead to improved hard materials.^21–24 

The ReB₂ structure is one of three structurally related yet distinct diboride structure types: AlB₂, ReB₂, and RuB₂ (Fig. 1).^25,26 A vast majority of diborides (e.g., TiB₂ and ZrB₂) adopt the AlB₂-type structure (P6/mmm) of planar boron sheets that alternate with 12-coordinate metal layers. In the case of the ReB₂ and RuB₂ systems, the boron sheets corrugate in “chair” and “boat” conformations for the ReB₂ (P6₃/mmc) and RuB₂ (Pmmn) structures, respectively. Among these three diboride structure types, only ReB₂ has been experimentally measured as superhard at low load. A theoretical investigation of the diboride structures shows that only ReB₂ has the right electron count to show only bonding metal–boron (M–B) interactions with no antibonding M–B bonds.²⁷ Moreover, the same study indicates that the hypothetical “chair” OsB₂ configuration should be harder than its existing “boat” OsB₂ counterpart, emphasizing the significance of the boron configuration in determining hardness. These results accurately describe the hardness of the known phases and identify specific chemical bonding environments that facilitate high hardness in metal borides.

Previous studies have examined the ReB₂ structure type in other diboride solid solutions (e.g., W_0.5Ru_0.5B₂ and W_0.5Os_0.5B₂).^28,29 With the exception of solid solutions of tungsten in ReB₂ (Re_1−xW_xB₂)³⁰ and a limited number of Re–M–B phase diagrams, no other ReB₂-based solid solutions with the ReB₂ structure have been extensively studied or reported for their mechanical properties. Given the high incompressibility of both osmium (Os) and ruthenium (Ru) and their close proximity to rhenium (Re) on the Periodic Table, the Re–Os–B and Re–Ru–B diboride systems merit further examination of their solubility and hardness.

In this study, we report the Vickers hardness of two ReB₂ solid solutions (Re_0.98Os_0.02B₂ and Re_0.98Ru_0.02B₂) and probe changes in bond length and strength within the solid solutions using high pressure diffraction. Although both Os and Ru exhibit limited solubility in the ReB₂ structure (<5 at. % for both metals), our results indicate that substituting low concentrations of these metals for Re in the ReB₂ lattice significantly increases both the hardness and resistance to deformation upon pressure.

Samples were prepared using elemental powders of rhenium (99.99%), ruthenium (99.9%), osmium (99.8%), and amorphous boron (99+%). All powders were purchased from Strem Chemicals, U.S.A. Solid solutions of rhenium diboride were weighed and mixed according to the nominal composition: Re_1−xM_xB_2.3, where x = 0, 0.02, and 0.05 and M = Ru or Os with the total M:B ratio maintained at 1:2.3. The extra boron was added to compensate for its loss during synthesis and to ensure the complete formation of the compounds. The mixed powders were pressed into pellets under a 10 ton load using a hydraulic jack press (Carver). Each pellet (∼1 g) was arc-melted in an argon atmosphere (I ≥ 70 A; t = 1–2 min) until molten, three times, flipping in between each melt.

The cooled ingots were then bisected, where one-half was crushed into powder (−325 mesh) for powder X-ray diffraction (PXRD) analysis. The other half was mounted in epoxy using an epoxy/hardener set (Buehler, U.S.A.) before being polished on a semi-automated polishing station (Buehler, U.S.A.). Each sample was polished using the following abrasives: silicon carbide disks (120–600 grit size, Buehler, U.S.A.) and polishing cloths coated in polycrystalline diamond suspensions (15–0.25 µm, Buehler, U.S.A.). Once an optically flat surface was achieved, the polished samples were analyzed through scanning electron microscopy (SEM), energy-dispersive x-ray spectroscopy (EDS), and Vickers hardness testing.

PXRD patterns of the powder samples were collected on an X’Pert Pro powder x-ray diffraction system (PANalytical, Netherlands) using a Cu_Kα x-ray beam (λ = 1.5418 Å) in the 5–100° 2θ range. The collected data were compared to reference patterns in the Joint Committee on Powder Diffraction Standards (JCPDS) database, now known as the International Center for Diffraction Data (ICDD), for phase identification in each sample. Rietveld refinement was performed on GSAS-II in order to determine unit cell parameters.³¹ The surface morphology and elemental composition of the polished samples were determined on an FEI Nova 230 high resolution scanning electron microscope (FEI Company, U.S.A.) with a backscattered electron detector (BSED) and an UltraDry EDS Detector (Thermo Scientific, U.S.A.).

The Vickers hardness of each sample was measured using a multi-Vickers hardness tester (Leco, U.S.A.) with a pyramid diamond tip. Ten indentations per load were made on polished samples under applied force loads ranging from low to high: 0.49, 0.98, 1.96, 2.94, and 4.9N. The indentation diagonal lengths were measured with a high-resolution Axiotech 100 HD optical microscope (Carl Zeiss Vision GmbH, Germany) under 500× magnification. Vickers hardness (H_v in GPa) was calculated and averaged at each load using Eq. (1),

where F is the applied force in Newtons (N) and d is the average diagonal length for each indent in micrometers (μm).

Nonhydrostatic in situ high pressure radial x-ray diffraction was performed in a diamond anvil cell at synchrotron beamline 12.2.2 of the Advanced Light Source (ALS, Lawrence Berkeley National Laboratory). The crushed powder of the samples (Re_0.98Os_0.02B₂ and Re_0.98Ru_0.02B₂) was loaded into a laser-drilled hole (∼60 µm in diameter, ∼130 µm in depth) in a ∼400 µm diameter boron gasket made of amorphous boron and epoxy. A small piece of Pt foil (∼25 µm diameter) was placed on top of the sample to serve as an internal pressure standard. A monochromatic x-ray beam (λ = 0.4959 Å, spot size = 20 × 20 µm²) was passed through the sample, which was compressed between two diamond tips up to 60 GPa of pressure, and 2-dimensional (2-D) diffraction data were collected using an MAR-345 image plate and FIT2D software. A cerium dioxide (CeO₂) standard was used to calibrate the detector distance and orientation.

The angle-dispersive diffraction patterns were converted from elliptical to rectangular coordinates using FIT2D. The integrated “cake” patterns, azimuthal angle (η) vs diffraction angles (2θ), were then analyzed using Igor Pro (WaveMetrics, Inc.). Peak positions were manually picked for four easily resolvable diffraction peaks (002, 100, 101, and 110). All peaks of the 1-dimensional (1-D) diffraction patterns of x-ray intensity as a function of 2θ obtained at the magic angle (φ = 54.7°, effectively hydrostatic condition) were indexed to hexagonal phases with no indication of first-order phase transition throughout the measured pressure range. The pressure for each compression step was determined from the equation-of-state of the Pt standard using its d-spacing at φ = 54.7°.

The stress in the sample under uniaxial compression is described by Eq. (2),

where Σ₁ is the minimum stress along the radial direction, σ₃ is the maximum stress in the axial direction, σ_P is the hydrostatic stress component, and t is the differential stress, which gives a lower-bound estimate of yield strength.³² The d-spacing is calculated by

where d_m is the measured d-spacing, d_p is the d-spacing under the hydrostatic component of the stress, φ is the angle between the diffraction normal and axial directions, and Q(hkl) is the lattice strain under the uniaxial stress condition.³³ The differential stress, t, is directly related to the differential strain, t/G(hkl), by

where G(hkl) is the shear modulus of the specific lattice plane. Incompressibility was determined using the generalized Birch–Murnaghan equation-of-state (EOS), which can be written as

where P is the pressure, K_0T is the bulk modulus at ambient pressure, V is the volume, V₀ is the undeformed unit cell volume, and K_0t′ is the derivative of K_0T with respect to P.³⁴ 

Here, we investigate the effects of osmium and ruthenium substitutions in ReB₂ (Re_1−xOs_xB₂ and Re_1−xRu_xB₂) on Vickers hardness and deformation of the ReB₂ structure upon pressure. According to the Hume-Rothery rules, a thermodynamically favorable substitutional solid solution forms when the host and dopant atoms are: (1) < 15% different in atomic radius; (2) similar in the crystal structure; (3) similar in oxidation state (valency); and (4) similar in electronegativity.^35–37 Although both Os (1.35 Å) and Ru (1.34 Å) are similar to Re (1.37 Å) in atomic radius, valence electron count, and electronegativity, they form solid solutions with ReB₂ only at low percentages, with solubility limits of less than 5 atomic percentage (at. %) for both metals.³⁸ The low solubility of Os and Ru in the hexagonal ReB₂ (P6₃/mmc) structure can be largely explained by the fact that both of these metals form metal diborides of the orthorhombic RuB₂-structure type (Pmmn).^26,39 When ReB₂ is substituted with >2 at. % Os or Ru, OsB₂ (Pmmn) and RuB₂ (Pmmn) appear as secondary phases, respectively (Fig. S1). In order to isolate intrinsic solid solution effects from the presence of any potential extrinsic effects (e.g., secondary phase precipitation hardening), we discuss the properties of only the two low concentration solid solutions, Re_0.98Os_0.02B₂ and Re_0.9Ru_0.02B₂, in further detail below.

Rietveld refinement of Re_0.98Os_0.02B₂ and Re_0.9Ru_0.02B₂ (hereon noted as “ReOsB₂” and “ReRuB₂,” respectively) was performed through comparison with the ReB₂ structure using GSAS-II software. Table I compares the refined unit cell parameters of ReOsB₂ and ReRuB₂ with those of pure ReB₂ synthesized under the same conditions. While Os substitution shrinks the unit cell along the c-axis, it expands ReOsB₂ along the a-axis, resulting in the lowest c/a axial ratio between the three compositions and less volume compression than that observed with Ru substitution in the ReB₂ unit cell. In the ReRuB₂ system, both the a and c lattice parameters decrease in comparison to unsubstituted ReB₂, indicating an overall unit cell compression and decrease in volume as a result of Ru substitution. These materials are known to slip along the layering direction,⁴⁰ and so the decreased c-axis spacing in the ReOsB₂ is noteworthy.

Subsequent Vickers micro-indentation hardness measurements were performed on both ReOsB₂ and ReRuB₂ to study the effects of Os and Ru on ReB₂ hardness. Hardness measurements of unsubstituted ReB₂, also synthesized in this study, are in agreement with literature values for ReB₂ and serve as a basis for comparison between the two solid solutions.^15,30 Measured values from low to high load (0.49–4.9N) for the three compositions are given in tabular form and compared to literature values for OsB₂ and RuB₂ in Table S1 and are plotted in Fig. 2. While both Os and Ru adopt the same orthorhombic diboride structure, studies have measured higher hardness for OsB₂ than RuB₂, suggesting that the increase in hardness for OsB₂ is due to the increased bond strength for Os–B bonds compared to Ru–B bonds.^41–43 In the current study, a similar trend of higher hardness in the Os-based system is observed. The hardness increases from 40.3 ± 1.6 GPa for unsubstituted ReB₂ to 47.4 ± 1.5 GPa with 2 at. % Os addition under an applied load of 0.49N, while the same amount of Ru addition (2 at. % Ru) results in a hardness of 43.0 ± 2.8 GPa (at 0.49N).

SEM images and elemental maps of ReOsB₂ and ReRuB₂ indicate homogeneous elemental distributions across the samples with no secondary phase formation (Fig. S2). It should be noted that the two compositions exhibit no significant difference from one another in surface grain morphology. Therefore, any difference in hardness between the two compositions likely arises solely due to solid solution effects, with no extrinsic contributions from precipitation hardening or grain size reduction. Although Os and Ru are similar to one another in their atomic properties and their effects on ReB₂ morphology, our results suggest that the two metals differ in how they bond to other atoms in the ReB₂ lattice and enhance Vickers hardness.

In general, superhard materials show high resistance to volume change.^44,45 The bulk modulus (incompressibility) of a material reflects its resistance to volume change upon compression and correlates with valence electron count and structure. Radial x-ray diffraction was conducted under non-hydrostatic compression up to 56 GPa to study the deformation mechanisms of ReOsB₂ and ReRuB₂ upon pressure. Figure 3(a) shows representative “cake” patterns and the integrated 1-D diffraction patterns of ReRuB₂ recorded at the lowest and highest pressures (∼2 and 56 GPa, respectively). The nearly straight diffraction lines of the “cake” patterns at low pressure (∼2 GPa) are due to the hydrostatic stress state. However, at high pressure, sinusoidal variations of the diffraction lines occur due to nonhydrostatic stress, so that the diffraction lines deviate to a higher angle (2θ) in the high stress direction (φ = 0°) and to a lower angle (2θ) in the low stress direction (φ = 90°). The waviness of the diffraction lines indicates the lattice-supported strains, which will be further discussed in the next paragraph. In the 1-D diffraction patterns, a clear shift of the peaks to higher angles at higher pressure indicates a decrease in the lattice spacing with greater compression, and the peak broadening implies strain inhomogeneity. Similar representative diffraction patterns for ReOsB₂ can be found in Fig. S3.

The d-spacing data as a function of φ-angle from Figs. 3(a) and S3.3 can then be plotted as (1 − 3 cos² φ) to generate a linear trend [Eq. (3) and Figs. 3(b) and 3(c)], with the minimum d-spacing in the high stress direction (φ = 0° and 180°), and the maximum d-spacing in the low stress direction (φ = 90° and 270°). The zero-intercept of the linear regression at φ = 54.7° (the magic angle) gives the d-spacing under hydrostatic condition, which we use to determine the hydrostatic lattice parameters and volume, as summarized in Table S2 and Figs. S4(a) and 4(a). For all lattice planes, a steady decrease in d-spacings as a function of increasing pressure is observed [Fig. 3(d)]. Moreover, we observe a continuous increase in the c/a ratio upon compression, as shown in Fig. S4(b), indicating that no phase transitions occur in either solid solution.⁴⁶ The general trend of increasing the c/a ratio with increasing pressure has been observed previously across many members of this family of compounds.⁴⁷ 

We then analyzed volume change upon pressure to determine the bulk modulus using the Birch–Murnaghan EOS [Eq. (5)]. Interestingly, the best fit to the overall data shows very high bulk moduli for both solid solutions, with K_0T = 413.2 ± 22.8 (K_0T′ = 1 ± 0.7) for ReOsB₂ and K₀ = 447.5 ± 14.8 (K_0T′ = 1.5 ± 0.5) for ReRuB₂ [Fig. 4(a)]. We note that, based on the Birch–Murnaghan EOS, the bulk modulus (K_0T) is significantly affected by the pressure derivative of the bulk modulus (K_0T′), with a lower K_0T resulting as K_0T′ increases. This interplay has been mapped out in Fig. S5 to emphasize the potential for error in reported K_0T values. In comparison, the bulk modulus of ReB₂, obtained from various experiments and calculations, falls between 317 and 383 GPa.⁴⁷ Both solid solutions present similarly high bulk moduli, indicating great incompressibility attributed to higher valence electron concentration with Ru or Os substitution.

The differential strain (t/G) is obtained by taking the ratio of the slope of the linear fit to the zero-intercept in Figs. 3(b) and 3(c), according to Eq. (4). The measured differential strain for each lattice plane increases linearly with pressure and then appears to level off, as shown in Figs. 4(b) and 4(c). This plateau implies the onset of plastic deformation where the lattice plane can no longer sustain additional differential strain and the deformation is no longer reversible. The plane with the lowest plateau value, or lowest differential strain (t/G), supports the strain before the onset of plastic deformation, while planes with higher differential strain plateau values support more strain before plastic deformation ensues. For all materials studied here, the (002) plane, which is parallel to the boron layers, is prone to slip and, therefore, supports the least lattice deformation, while the (100), (101), and (110) planes, which cut through the metal–boron layers, exhibit higher differential strain [Figs. 4(b) and 4(c)]. Due to the puckered boron sheets in the ReB₂ structure, ReB₂ exhibits greater resistance to dislocation slip than other MB₂ materials, but the (002) plain remains the primary slip system.

In the current study, we investigate the effects of metal substitution on enhancing resistance to dislocation by doping 2 at. % transition metals (Os and Ru) in ReB₂ solid solutions. Both ReOsB₂ and ReRuB₂ in the ReB₂ structure show higher differential strain values than that of pure ReB₂. For example, the plateaued t/G values of all lattice planes in the ReOsB₂ system are higher than those of the strongest plane in ReB₂, the (110) plane, which is found to be ∼0.03, or ∼3%, at 50 GPa. The t(hkl)/G(hkl) plateau values for ReOsB₂ are in the range of 3.4% to 4.3%, while ReB₂ shows a much lower range from 1.5% to 2.8%. Unlike ReB₂, where the plateaued t(002)/G(002) value is lower than that of other ReB₂ lattice planes by almost 50%, the plateaued t(002)/G(002) value of ReOsB₂ (∼0.034) is almost as high as t(110)/G(110), indicating that the (002) plane is significantly strengthened in ReOsB₂. In addition, the differential strain in ReOsB₂ does not plateau until higher pressures than those observed in ReB₂. The (100), (101), and (110) planes in ReOsB₂ do not plateau until 30 GPa, while the (002) plane does not reach its plateau value of t/G until 50 GPa [Figs. 4(b) and 4(c)]. Overall, ReOsB₂ shows both higher plateau values and higher plateau pressures for all lattice planes, indicating significantly improved mechanical properties from doping a small percentage of Os in ReB₂, in good agreement with the measured enhancement in Vicker’s hardness. These effects likely arise in part from the compressed c-axis distance in the material, which should facilitate metal–boron bonding between the layers.

In ReRuB₂, only the (002) plane is significantly enhanced. All other lattice planes exhibit similar trends in t/G to ReB₂. In the elastic region, the t(hkl)/G(hkl) values for all lattice planes in ReRuB₂ increase similarly to those for ReB₂ planes. Starting from the onset of plastic deformation for ReB₂ at ∼20 GPa, t(002)/G(002) of ReRuB₂ begins to deviate from the trend of the differential strain of ReB₂, while t(101)/G(101) and t(110)/G(110) remain similar to those of ReB₂. Above 30 GPa, the differential strain of the (002) plane in ReRuB₂ yields a plateau value ∼48% higher than that of ReB₂. Therefore, while the differential strain data in ReRuB₂ is not as dramatically different from ReB₂ as it is in ReOsB₂, the strength of the primary slip system is enhanced, likely resulting in the observed increase in Vicker’s hardness.

It is noteworthy that solid solution effects are significant at even 2 at. % dopant and can change strain anisotropy and enhance differential strain in the ReB₂ system. The addition of metal dopant atoms that differ from rhenium in atomic size and valence electron count leads to stronger dislocation barriers that enhance the solid solution’s resistance to slip. Because the effect is observed at such low doping levels, however, it is likely to be an electronic structure effect rather than a size effect. As suggested by a recent computational study on bonding in ReB₂ and OsB₂, both ReB₂ and OsB₂ show strong metal–boron bonds through a d → σ_2px state.²⁷ Because of the higher electron count in Os (and presumably Ru) compared to Re, however, these materials also have an occupied d → π* state that is not occupied in ReB₂.²⁷ In other words, Os has enough electrons to give the B–B bonds some π* character and to form additional Os–B interactions; this makes the B–B bonds weaker and the Os–B bonds stronger. The corollary to this is that because the π* back bond is not present in the ReB₂, B–B bonds are much stronger, and in comparing pure materials, this leaves ReB₂ as much harder than OsB₂. By introducing only a small amount of Os into the ReB₂ system, however, the majority of the B–B bonds remain strongly covalent, but a small percentage of charge density flows from B–B to Os–B bonds through the d → π* donation. This net strengthens the bonding in this solid solution system relative to pure ReB₂. This finding further confirms that doping enhances the bond strength between the boron and metal interlayers of ReB₂ and consequently improves the Vickers hardness. Interestingly, doping Os into ReB₂ enhances resistance to dislocation slip in all lattice planes, whereas doping Ru enhances only the weakest plane in the ReB₂ system. These results offer insight into the lattice specific strengthening mechanisms that lead to higher hardness with solid solution formation.

ReB₂-structured solid solutions (ReOsB₂ and ReRuB₂) were synthesized via arc melting and characterized for their mechanical properties. Vickers micro-indentation measurements demonstrate a greater increase (>17%) in hardness for ReOsB₂ than for ReRuB₂. To understand the lattice specific changes induced by metal doping in ReB₂, the high-pressure behavior of ReOsB₂ and ReRuB₂ were examined and compared to that of ReB₂ using synchrotron based x-ray diffraction under non-hydrostatic compression. The equations of states for these two solid solutions were determined from the hydrostatic volume data measured at the magic angle (φ = 54.7°). Both ReOsB₂ (K_0T = 413.2 ± 22.8, K_0T′ = 1 ± 0.7) and ReRuB₂ (K₀ = 447.5 ± 14.8, K_0T′ = 1.5 ± 0.5) exhibit a higher bulk modulus than pure ReB₂, indicating enhanced incompressibility upon doping. Finally, lattice-dependent strength anisotropy suggests that Os doping enhances resistance to slip in all lattice planes, while Ru doping only strengthens the slip plane (002) in the ReB₂ structure.

See the supplementary material for load dependent hardness data, crystallographic parameters as a function of pressure, powder diffraction as a function of composition, elemental mapping, examples of raw and integrated high pressure data, pressure dependent c/a data, and an exploration of the relationship between the bulk modulus and its pressure derivative.

The authors thank M. Kunz and A. MacDowell for technical support at the Lawrence Berkeley National Laboratory (LBNL) Beamline 12.2.2. We also thank Professor H.-R. Wenk for his equipment support. This work was financially supported by the National Science Foundation Division of Materials Research under Grant No. DMR-2004616 (R.B.K. and S.H.T.). Additional support was provided by the Dr. Myung Ki Hong Endowed Chair in Materials Innovation (R.B.K.), SuperMetalix (R.B.K.), and a UCLA Graduate Division Dissertation Year Fellowship (L.E.P.). Radial x-ray diffraction experiments were performed at the Advanced Light Source at Lawrence Berkeley National Laboratory at Beamline 12.2.2. Beamline 12.2.2 at the Advanced Light Source is a DOE Office of Science User Facility supported under Contract No. DE-AC02-05CH11231. This research was partially supported by COMPRES, the Consortium for Materials Properties Research in Earth Sciences, under NSF Cooperative Agreement No. EAR 1606856.

C.L.T. and R.B.K. have an equity interest in SuperMetalix, Inc.

S.H. and L.E.P. contributed equally to this work.

Shanlin Hu: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (lead). Lisa E. Pangilinan: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Methodology (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Christopher L. Turner: Conceptualization (supporting); Methodology (supporting); Writing – review & editing (supporting). Reza Mohammadi: Writing – review & editing (supporting). Abby Kavner: Methodology (supporting); Project administration (supporting); Supervision (supporting); Writing – review & editing (supporting). Richard B. Kaner: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Sarah H. Tolbert: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.