Report

X-ray Imaging of Stress and Strain of Diamond, Iron, and Tungsten at Megabar Pressures

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Science  23 May 1997:
Vol. 276, Issue 5316, pp. 1242-1245
DOI: 10.1126/science.276.5316.1242

Abstract

Synchrotron x-ray imaging and stress measurements of diamond-anvil cell gaskets revealed large elastic strains at the diamond tip at a pressure of 300 gigapascals. The diamond, generally considered a rigid body, bent 16 degrees over a distance of 300 micrometers without failure. To complement these measurements, a technique was developed that permits x-ray diffraction to be measured through a beryllium gasket. Measurements on tungsten and iron revealed the strain anisotropy, deviatoric stress, and texture and showed that the yield strengths of these materials increase by up to two orders of magnitude at confining pressures of 200 to 300 gigapascals. The results allow identification of the maximum amount of strain accommodated by the anvil tips before failure. Further development of ultrahigh pressure techniques requires relieving stress concentrations associated with this large elastic deformation.

The ability to confine and study materials at pressures in the multimegabar range (>200 GPa) is a result of continued refinements in the diamond-anvil cell (1), including the development of the long piston-cylinder megabar cell design (2), the introduction of beveled anvils (3), the extension of the range of optical (4-6) and x-ray (5-8) pressure calibration, the advent of new classes of megabar devices (9, 10), and the development of increasingly sensitive and accurate microanalytical techniques for probing materials under high-pressure conditions (11). Despite these advances, however, our knowledge of material behavior under extreme pressures has been limited by a lack of information on the three-dimensional distribution of stress and strain and the ultimate strength of materials under these conditions. Here we report the development and application of techniques that permit imaging and measurement of stress-strain distributions and deformation of materials at multimegabar pressures.

The behavior of materials in highly stressed states can differ considerably from that near ambient conditions. For example, otherwise strong and hard materials can develop texture and plastically deform at megabar pressures (>100 GPa) (12-15). To date, however, estimates of the stress-strain states and yield strength of materials under these conditions have been inferred from indirect measurements and extrapolation from lower pressure data (16-23), or from higher pressure experiments using limited stress-strain geometries (8, 14, 24). In addition to the fundamental interest in these properties (25), such information on high-strength materials is crucial for extending the range of static compression techniques to still higher pressures. This extension requires direct determination of the stress-strain distributions of all load-bearing components of the high-pressure device, including elastic deformation of the anvils and elastic and plastic deformation of the gaskets. Elastic deformation of diamond anvils has been inferred from optical observations (15) and modeled theoretically (26), but it has not been measured directly above 200 GPa (27).

Two experiments were carried out with diamond-anvil cells modified for improved stability and alignment in the multimegabar pressure range (10). First, the strain distribution in gaskets and the elastic deformation of the diamonds were determined by direct imaging of the topography of the diamond-anvil surface in situ to pressureP ∼ 300 GPa. These observations showed that diamond can accommodate remarkably large strains localized over small areas in the anvil tip. Second, using various diffraction geometries (Fig.1) together with high-strength but x-ray–transparent gaskets, we directly measured the deviatoric stress of the sample and gasket materials associated with the diamond’s large deformations. We used intense and highly collimated synchrotron x-ray beams that permit diffraction and transmission measurements with micrometer-scale spatial resolution in three dimensions within the samples (28).

Figure 1

Three geometries for the diamond-cell x-ray experiments reported here. (A) Conventional (axial) geometry for x-ray diffraction and transmission. (B) Radial diffraction geometry with ψ = 0° and (C) radial diffraction geometry, ψ = 90°. A, axial direction; R, radial direction; ψ, angle between the diffraction vector and the load axis; σ1 and σ3 are the radial and normal stresses, respectively. In studies of the behavior of gasket material itself at ultrahigh pressures, the gasket is continuous across the tip [that is, the sample chamber is not drilled out (3, 5,12)].

The first set of experiments was carried out to determine the pressure distribution and macroscopic strain of the anvil at ultrahigh pressure using the conventional geometry shown in Fig. 1A. Single-beveled diamond anvils with small central flats of 10 μm were used together with rhenium (Re) gaskets. A small grain of tantalum (Ta) was placed on the central flat, and x-ray diffraction of the Ta and Re were used for pressure determination, on the basis of their shock-wave equations of state (29). Initially—that is, at zero pressure—the combined bevel angle was 17°. The central sample thickness was only 3 μm after gasket indentation, whereas the edges of the culets of the two diamonds were separated by 45 μm of Re (cc′ in Fig. 2C). Because the Re gasket highly absorbs x-rays, the magnitude of the transmission at a point measured with the 5 μm by 5 μm x-ray beam gives the thickness of the gasket as determined by plastic flow of the metal and the concomitant elastic deformation of the anvil (Fig. 2B). With increasing load, the 17° combined angle at the anvil tip begins to decrease; that is, the anvil tip begins to flatten. At the highest loads, however, the originally straight slope of the bevel transforms to a cup, with the bevel angle reversed at the edge (c in Fig. 2C). The corresponding pressure distribution obtained from diffraction of the gasket with the same micro x-ray beam initially shows the sharp peaked structure associated with the concentration of pressure at the anvil tip (Fig. 2A). With increasing load, the bevel angle decreases, and the peak broadens as the pressure on the culet edge increases, indicating the gradual loss of the central pressure concentration initially created by the bevel.

Figure 2

(A) The pressure distribution and (B) x-ray transmission profiles as a function of radial distance across (C) the diamond culet for the Re-Ta experiment. Each curve gives the distribution at selected loading increments (numbered 1 to 5), where the pressure was determined from x-ray diffraction of Re. (C) shows the original shape of the anvil tips (blue). The dimensions of the central flat, culet, and bevel angles of the two diamonds are 10 μm, 300 μm, and 9°, and 10 μm, 300 μm, and 8°, respectively, giving a total bevel angle of 17°. The yellow region shows the gasket shape at the highest load and the extent of diamond deformation such that the average bevel angle is close to zero. A small grain of Ta (∼5 μm) was placed on top of the Re gasket at the center before loading; its diffraction was consistent with that of Re, and it did not contribute measurably to the transmission profile.

Rastering the sample position in two dimensions perpendicular to the x-ray beam provides more detailed information (30). With the spatial resolution provided by the small x-ray beam, detailed features of the anvil surface, including the 16 facets of the brilliant cut of the diamond, are apparent (Fig. 3). The image of the diamond tip shows that a large amount of macroscopic strain can be accommodated by the cupped anvil tip. The pressure distribution determined from gasket diffraction (Fig. 3) shows that the pressure falls off rapidly from the central region to the edge of the culet (the rim of the cup) and then decreases smoothly with radial distance from the culet edge. When the bevel collapses [giving a shape essentially like that of the flat culet anvils used in the early lower pressure experiments (1, 2)], the pressure advantage provided by the bevel is consumed, and the capability for further pressure increase at the center is limited. Numerous experiments show that diamond failure results when the cupping reaches a critical value such that the transmission at the rim is about one half of that within the cup.

Figure 3

(A) Three-dimensional view of the diamond deformation for the Re-Ta experiment at the highest load (load 5 in Fig. 2A). The Re gasket casts a “mold” to give the deformation at the tips of the two opposed anvils. The horizontal scales show the radial distance from the center of the diamond tips (as in Fig. 2), and the vertical scale is the distance from the tip of an anvil under load for a single diamond. The values correspond to one-half the thickness of the gasket determined from the x-ray transmission and therefore represent the average deformation of the two (symmetrical) opposed anvils. The superimposed colors give the pressure distribution in the gasket measured with the micro x-ray beam at the interface with the diamond. (B) Magnified view of the central depression in (A) (the “blue lagoon”).

Complementary information is obtained from x-ray diffraction experiments carried out with radial diffraction geometries (Fig. 1, B and C). With the axially symmetric geometry of the diamond cell, the stress-strain field is defined by the radial and normal stresses σ1 and σ3 and their associated strains Δɛ1 and Δɛ3. The hydrostatic stress component is σP = (2σ1 + σ3)/3 = σ1 + t/3, where t is the differential stress (or uniaxial component) t = σ3 − σ1 (31). With a conventional geometry, the x-ray beam is nearly coaxial with the load axis, and the diffracted beam is collected at either fixed or variable 2θ angle. This approach measures diffraction planes that do not correspond to the maximum strains in the sample (32). In fact, the maximum stress-strain states reached on materials at multimegabar confining pressures are unknown. Although such effects were measured at much lower pressures, they have only been estimated theoretically at multimegabar pressures (8). There are no direct measurements because of the limited access to the sample allowed by the conventional geometry. The ideal geometry requires passing the x-rays through the gasket to access the sample. However, typical high-strength gaskets such as Re strongly absorb x-rays and preclude such measurements. To solve this problem, we used x-ray–transparent beryllium (Be) gaskets, which permit direct measurements of deviatoric strains at multimegabar pressures.

Polycrystalline W and Fe were loaded in a specially prepared Be gasket (33) mounted at the 10-μm tips of single-beveled diamonds. Tungsten remains cubic [body-centered-cubic (bcc) structure] at megabar pressure (8), whereas Fe is bcc at low pressure but transforms to the hexagonal closed-packed (hcp) structure at 13 GPa (13). The lattice strains were measured in two directions by x-ray diffraction with all three geometries. Diffraction data obtained through the Be gasket (Fig. 4A) show that the relative intensities of diffraction peaks differ drastically between the two directions, indicating a strong preferred orientation under uniaxial deformation. As in previous measurements on ɛ-Fe with the conventional geometry (13), the characteristic 100 and 101 hexagonal reflections were observed, with the 002 reflection nearly absent. The diffraction pattern from the radial x-ray measurement with ψ = 90° is therefore identical to that obtained by axial diffraction (ψ is the angle between the diffraction vector and the load axis). In the radial measurement with ψ = 0°, however, the 002 reflection is the strong peak, and the 100 reflection is not observed, indicating alignment of crystallites with the c axis, parallel to the load direction (13).

Figure 4

(A) X-ray diffraction patterns measured in the three geometries shown in Fig. 1 at the highest load [σP(Fe) = 290 GPa; σP(W) = 265 GPa]. (B) Differential stress of W (circles) and Fe (squares) as a function of pressure. The solid symbols are the values obtained for t; the error bars reflect the experimental error and the uncertainties in shear modulus G at high pressure (40, 41), and the hatched region reflects the range in measured t. The open symbols are the stress differences (Δσ) determined by assuming that the strains measured at ψ = 0° and ψ = 90° correspond to volumetric strains with the stress given by the P-V equation of state (13, 29). The values of σPfor the two samples are not the same for the same load because we determine the stress conditions (σ1, σ2, and t) for each component separately. Bridgman’s (39) data for the yield strengths σy of both materials are shown by the single bold line at low pressures, and the dashed line shows the extrapolation of his data for W. The dashed line for Fe shows an estimate of the theoretical strength for the hcp phase calculated from the scaling σy = 2τ ≈ 0.2G(25), where G was determined from recent calculations (41).

The difference in d-spacings obtained from the ψ = 0° and ψ = 90° patterns gives the deviatoric strains F= (d d 90°)/3d P, whered P = (d + 2d 90°)/3 is the d-spacing under hydrostatic pressure (34). The pressure was determined from the average d P using the equations of state of W and Fe (13, 29). These data can be used to determine the deviatoric stress component t, which can be writtent = σ1 − σ3 = 6GF(hkl)〉 (21), whereG is the aggregate shear modulus and 〈F(hkl)〉 is the average of F over the measured diffraction peaks with indices hkl. We find that t increases with increasing P, reaching values of ∼20 GPa for W and Fe at 200 to 300 GPa (Fig. 4B). In Fig.4B, we also show the stress difference between orientations assuming that the stress is equivalent to the pressure obtained from the equations of state of each material, an approach that significantly overestimates the deviatoric stress (35).

The maximum uniaxial stress t supported by a material is determined by its strength; that is, t ≤ σy = 2τ, where σy and τ are the yield and shear strengths of the material, respectively. It is often assumed in high-pressure experiments that t = σy; however, in general t varies with sample environment, and the equality (von Mises condition) holds only if the sample deforms plastically under pressure. Our results indicate that the yield strengths of W and Fe exceed 20 GPa at P > 200 to 300 GPa (36), a significant increase relative to ambient conditions [where σy(W) = 0.6 GPa (37) and σy(α-Fe) = 0.03 GPa (38)]. Moreover, above ∼50 GPa, t is considerably less than σybecause diamond cupping arrests the flow of the sample. This difference is confirmed by spatially resolved x-ray diffraction and transmission measurements. Our finding that the yield strengths of W and Fe are higher than the uniaxial stress component is also supported by the theoretical estimates based on scaling to the shear modulus (25) and by the extrapolation of Bridgman’s data (39). Together with our observations for diamond, these results demonstrate that the strength of these materials is dramatically enhanced at ultrahigh pressures.

  • * To whom correspondence should be addressed. E-mail: hemley{at}gl.ciw.edu

  • Present address: Consortium for Advanced Radiation Sources, University of Chicago, Chicago, IL 60637, USA.

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