Ultrahardness: Measurement and Enhancement - ACS Publications

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Ultrahardness: Measurement and Enhancement Bo Xu and Yongjun Tian* State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China ABSTRACT: In the quest for novel superhard materials with hardness comparable to or even higher than that of natural diamond, two issues are of central concern: how hardness can be reliably measured and how hardness can be enhanced. Here, we analyze the specific stress states of the indenter and the tested sample during an indentation hardness measurement and reveal that the traditional prerequisite that the indenter should be harder than the measured sample is not necessary to ensure a reliable hardness measurement. In fact, the indentation hardness can be reliably measured as long as the shear strength of the sample is less than the compressive strength of the diamond indenter. Also, we suggest nanostructuring (nanocrystallinity and nanotwinning) as an effect means of hardness enhancement.

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INTRODUCTION Superhard materials, defined as materials with Vickers hardness values greater than 40 GPa, are of vital importance for both industrial and scientific applications, such as cutting tools for the machinery industry, drilling bits for oil production, and diamond anvil cells for high-pressure science. This class of materials is represented by diamond and cubic boron nitride (cBN), each of which shows some inherent drawbacks, including the poor thermal stability of diamond and the relatively low hardness and fracture toughness of cBN. Therefore, numerous efforts have been made in the past half century to find novel superhard materials for improved comprehensive performance, that is, simultaneously improved hardness, toughness, and thermal stability.1−4 Meanwhile, identifying synthetic materials harder than natural diamond has always been a pursued goal of superhard materials research.5 For both purposes, an in-depth understanding of hardness is necessary. Unlike other mechanical properties (such as bulk and shear moduli), hardness has to be classified as an engineering quantity,6 and distinct hardness scales have been developed experimentally depending on the specific measurement method (e.g., scratch, indentation, and rebound). For superhard materials, the Vickers and Knoop scales (differing from each other in the shape of the indenter) of indentation hardness are most widely used. Indentation hardness measures a material’s resistance to permanent plastic deformation due to a compressive load from a sharp indenter, or microscopically speaking, it can be defined as the combined resistance of chemical bonds in a material to indentation.7 Based on this microscopic consideration, several hardness models have been established recently through the fitting of experimental Vickers/Knoop hardness data for polar covalent single crystals.8−10 These models provide an atomic-level understanding of material hardness and enable the possibility of hardness prediction. Also, several key factors for achieving superhardness are revealed by these microscopic models, namely, a three-dimensional network structure, short chemical bonds, high bond and valence electron densities, and low bond © 2015 American Chemical Society

ionicity, which greatly promote the theoretical investigations of novel superhard crystals.7 Along with the progress in understanding hardness, there exists a practical issue in hardness measurement: What is the criterion for forming a permanent indentation through plastic deformation during an indentation hardness measurement? Previously, Brazhkin et al. stated: “It is not yet possible to express the hardness of a material ‘harder than diamond’ by a single number. We recommend that values higher than 120 GPa should not be called ‘hardness’ to avoid confusion.”11 This statement is hereinafter referred to as the hardness-comparison (HC) criterion. By reviewing the definition of indentation hardness and analyzing the specific stress states of the indenter and tested sample during an indentation hardness measurement, we illustrate here that the traditional HC criterion is incorrect and provide an updated strength-comparison (SC) criterion that the shear strength of the tested sample must be less than the compressive strength of the diamond indenter. In addition, we emphasize the effectivity of nanostructuring for hardness enhancement based on our semiempirical hardness model of polycrystalline covalent materials.

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MATERIALS AND METHODS The Vickers hardness values, HV, of a nanotwinned cBN bulk sample,1 a nanotwinned diamond bulk sample,2 and a diamond Vickers indenter were measured with a microhardness tester (KB 5 BVZ). HV was calculated as HV(GPa) = 1854.4L/d̅2, where L (in newtons) is the applied load and d̅ (in micrometers) is the average diagonal length of indentation determined by an optical microscope equipped on the microhardness tester. The hardness of each sample was determined as the value from the asymptotic-hardness (loadinvariant) region. An indentation formed on a nanotwinned diamond sample under a load of 9.8 N was measured by atomic force microscopy (AFM, Solver P47-PRO). The diagonal Received: January 2, 2015 Revised: February 12, 2015 Published: February 19, 2015 5633

DOI: 10.1021/acs.jpcc.5b00017 J. Phys. Chem. C 2015, 119, 5633−5638

Article

The Journal of Physical Chemistry C

sample during an indentation hardness measurement are analyzed below. Figure 1 schematically shows the indentation process with an example Vickers indenter. When the symmetric indenter is

length of the indentation determined by AFM provided a calibration for the optically determined value. The compressive strengths of Cco-C812 and M-carbon13 phases were determined by the same method as used previously for diamond.14 First-principles calculations were performed with CASTEP software, which is based on density functional theory.15 A norm-conserving pseudopotential with a cutoff energy of 770 eV was used. The exchange-correlation functional was treated by the local density approximation as parameterized by Perdew and Zunger (LDA-CAPZ),16,17 which is a better functional than the generalized gradient approximation with the Perdew−Burke−Ernzerhof exchange-correlation functional (GGA-PBE) for group IVA elements and IIIA− VA compounds, as suggested by a recent publication.18 A kpoint spacing (2π × 0.04 Å−1) was used to generate the Monkhorst−Pack k-point grids for Brillouin zone sampling.19 A 1 × 1 × 1 unit cell was used for compressive deformation calculations. During the calculations, increasing compressive strains were applied in the selected direction. For each compressive strain, the crystal structure was relaxed until the stress orthogonal to the applied strain was less than 0.02 GPa. A compressive strain−stress relationship was thus determined.

Figure 1. Indentation process with a Vickers indenter. Red lines schematically represent the slip systems in the polycrystals with dislocations indicated by ⊥. d1 and d2 show the diagonals of the formed indentation. The black and red triangles mark the positions for determining the ideal diagonal and the optical measured one, respectively. The dashed line emphasizes the basal plane of the sample surface. See the main text for details.

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RESULTS AND DISCUSSION Criterion to Form a Permanent Indentation. The HC criterion implies that the test sample must be softer than the natural diamond indenter (with maximum hardness of 120 GPa) to ensure a hardness measurement.11 In other words, hardness cannot be measured for materials harder than natural diamond. However, such materials are exactly what is being sought in superhard materials research. If the HC criterion were correct, it would be impossible to characterize the hardness of these materials. However, many experimental counterexamples to the HC criterion are available. For example, a Vickers hardness value of annealed chemical vapor deposition (CVD) diamond as high as 170 GPa was reported previously,20 which is significantly higher than the suggested upper limit of 120 GPa. A scratch made by ReB2 on the surface of diamond was verified by AFM even though ReB2 is much softer than diamond.21 Another example that contradicts the HC criterion is the soft impressor method developed by Brookes and Green,22 whereby a plastic deformation can be formed even though the hardness of the impressor is much lower than that of the tested sample.23,24 The successful application of the soft impressor method requires that the resolved shear stress exceed those necessary for dislocation initiation and multiplication.24 To clarify the controversy, one must recall the definition of indentation hardness and examine the formation mechanism of a permanent indentation on the tested sample during hardness measurement. The indentation hardness of a material is determined by the indenter load divided by the contact (or projected) area of the permanent indentation formed on the sample surface.25 According to this definition, indentation hardness is a welldefined “engineering” quantity and can be reliably measured as long as a permanent indentation through plastic deformation can be left on the surface of the tested sample with no visible plastic deformation of diamond indenter. Obviously, the question of the criterion for a reliable indentation hardness measurement comes down to the condition of forming a permanent indentation on the surface. To identify this condition, the specific stress states of the indenter and tested

pushed into the tested sample exactly perpendicularly to its surface, the horizontal force components from opposite facets (F2 and F2′) of the indenter are offset, leaving only the compressive stress (σc) applied to the indenter as a result of the perpendicular force components (F1 and F1′). The tip of the diamond indenter is thus subjected to a compressive stress field. In this case, force components F2 and F2′ cannot result in any deformation of the diamond indenter no matter how high they might be. Note that the difference between the two horizontal force components (F2 − F2′) might become large enough to break the tip of the indenter if the indenter is pushed into the sample surface obliquely. The stress state of the deformation zone on the sample surface is different from that of the indenter, however. In the sample tested zone surrounding the indenter, dislocation initiation and multiplication cause slips and plastic strain when the applied stress exceeds the shear strength of the sample, leading to a plastic deformation and the formation of a permanent indentation on the sample surface. Therefore, indentation hardness can be measured reliably as long as the shear strength of the sample is lower than the compressive strength of the indenter diamond.26 In other words, the SC criterion for a reliable indentation hardness measurement is σcindenter > τ sample

(1)

σindenter c

where is the compressive strength of the diamond indenter and τsample is the shear strength of the tested material. The compressive strengths of diamond are 223 GPa in the weakest ⟨100⟩ direction and about 470 GPa along the ⟨110⟩ and ⟨111⟩ directions.14 Table 1 lists the shear strengths of typical hard and superhard materials.27−32 It is well-known that diamond has the highest shear strength among the known materials. Because the lowest compressive strength of the indenter diamond is significantly higher than the highest shear strength (diamond, 93 GPa), the SC criterion of eq 1 is naturally satisfied when a diamond crystal is used as an indenter in an indentation hardness measurement. This criterion 5634


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determination of the two diagonal lengths of an indentation. The diagonal lengths are typically measured with an optical microscope installed on a standard hardness tester. For tough superhard materials such as nanotwinned diamond, pile-up ridges higher than the basal plane of the sample surface (see the dashed line in Figure 1) are usually formed during the indentation process. In this case, the deformation area now including the pile-up zone is significantly larger than the ideal indentation. In an optically based measurement, the diagonal length of an indentation is determined as the distance between two pile-up positions (indicated by the red triangles in Figure 1) of opposite corners because they are easy to distinguish under an optical microscope. The ideal diagonal length, however, should be determined from two opposite corners of the indentation in the basal plane (black triangles in Figure 1). The ideal diagonal length can be accurately calibrated by AFM.35 This type of calibration is presented in Figure 3 for our ultrahard nanotwinned diamond sample.2 The mean diagonal lengths determined from AFM and optical measurements are 8.80 μm (corresponding to HV = 234.6 GPa) and 9.75 μm (HV = 191.1 GPa), respectively. Obviously, the optically based measurement overestimates the diagonal length, thus providing a conservative measure of the Vickers hardness (underestimated by about 18.5%). In either case, the measured hardness is significantly higher than that of the natural diamond indenter. It should be noted that, to perform a reliable hardness measurement, the indenter size effect should be avoided, which gives higher hardness values for smaller indentations because of the greater strain gradient.36 Hardness should be determined from the asymptotic-hardness region of a well-controlled indentation process (without the formation of cracks). Nanostructuring for Hardness Enhancement. The above analyses clearly indicate that hardness exceeding that of natural diamond indenter does have physical meaning and can be reliably measured. The next question is how to achieve materials with hardness greater than that of diamond. This is truly a challenge to materials science, and researchers are pessimistic about achieving this objective.37 Generally, there are two pathways toward this goal: One is to design novel superhard single crystals, and the other is to enhance the hardness of known materials by forming ultrafine microstructures. Although recent research indicates that the design of novel single crystals harder than natural diamond seems unrealistic,38 the feasibility of the second pathway is evidenced by solid experimental results,1−4,39,40 in which hardness has

Table 1. Ideal Shear Strengths (τ) of Typical Hard and Superhard Crystals compound

τ (GPa)

ref

diamond cBN wBN AlN TiN SiC ReB2 FeB4 B6O

93 58 62 20 29 28 34 24 38

27 28 28 29 29 30 31 32 31

indicates that the formation of a permanent indentation on a sample does not depend on the relative hardness of the indenter and sample, as claimed previously.11 Our SC criterion can explain some previous experimental results, such as the observed indentation marks on diamond anvils by cold-compressed carbon nanotubes33 and graphite.34 Theoretical simulations suggested that cold-compressed carbon nanotubes transform into the Cco-C8 phase12 and that coldcompressed graphite transforms into the M-carbon phase13 under high pressure. During the compression process with a diamond anvil cell at high pressure,33,34 these new phases (CcoC8 and M-carbon) can be considered as the indenter and the flat facet of the diamond anvil as the tested sample surface of a hardness measurement. The calculated theoretical hardness values of both Cco-C8 and M-carbon are slightly lower than that of diamond.12 To understand the origin of the indentation marks, we calculated the compressive strengths of Cco-C8 and M-carbon along selected crystal directions, which are presented in Figure 2. The compressive strength in the weakest direction is about 420 GPa for Cco-C8 and 260 GPa for M-carbon, much higher than the shear strength of anvil diamond. The formation of the observed indentation marks on diamond anvils can thus be well explained by our SC criterion. Error of Hardness Measurements and Indentation Calibration. Recently, we successfully synthesized nanotwinned polycrystalline diamond at high pressure and high temperature.2 The measured Vickers hardness of the nanotwinned diamond can reach 200 GPa, about twice that of natural diamond crystals.2 Although such a high hardness value can be reliably and repeatedly measured, assessment of the measurement error becomes an important issue of general concern. In fact, the measurement error mainly comes from the

Figure 2. Calculated compressive strain−stress curves of (a) Cco-C8 and (b) M-carbon along selected directions. The insets show the crystal structures of the corresponding materials. 5635


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Figure 3. AFM calibration of an indentation formed on a nanotwinned diamond sample at a load of 9.8 N for hardness and fracture toughness tests. The line profiles shown at the right are colored corresponding to the lines in the AFM image. An optical microscope photograph is shown as an inset of the AFM image. The mean diagonal is 8.80 μm from our AFM calibration and 9.75 μm from our optical measurements (cyan box).

the quantum confinement hardening coefficient (Ne and f i are the valence electron density of the crystal and the Phillips ionicity of the chemical bond, respectively),7,42 and D is the average grain size (d) or twin thickness (λ). Next, we discuss the hardness enhancement of cBN. Figure 5 shows the Vickers hardness of bulk nanocrystalline cBN as a

been greatly enhanced compared with that of the corresponding single crystals through the formation of microstructures such as nanotwins and nanocrystals. Figure 4 compares the

Figure 4. Vickers hardness as a function of applied load for selected samples. The hardness values of a nanotwinned diamond bulk sample, a nanotwinned cBN bulk sample, and a diamond indenter were determined to be 200, 108, and 92 GPa, respectively, from the asymptotic-hardness region.

Figure 5. Vickers hardness as a function of average grain size (d) or twin thickness (λ) for nanocrystalline cBN bulks. The inset shows the range of 0−14.5 nm. The solid black, blue, and red circles are data points taken from refs 1, 3, and 4, respectively. The black curve and open circles show hardness estimated from eq 2. The blue curve and open circles show hardness estimated from H = H0 + Kqc/D. The red dashed line is a guide for the eyes. See the main text for details.

measured hardness values of nanotwinned diamond,2 nanotwinned cBN,1 and a diamond indenter. The superior hardness of nanotwinned diamond and nanotwinned cBN over that of the diamond indenter is obvious. Our previous theoretical consideration suggested that this hardness enhancement should originate from the joint contributions of the quantum confinement effect (valid only for covalent materials) and the Hall−Petch effect.7 For well-sintered nanocrystalline covalent bulk samples, hardness can be estimated as H = H0 + KHP/ D + Kqc/D

function of average grain size (or twin thickness) from several experiments.1,3,4 The calculated hardness from eq 2 (black curve) is also included for comparison. For cBN, H0 and KHP are 39 and 126 GPa·nm1/2 , respectively,3 and Kqc is calculated as 136 GPa·nm. The consistency between the experimental and calculated data is satisfactory. A continuous hardening with decreasing D is revealed at deep nanometer scale for bulk nanocrystalline cBN, in contrast to what happens in metals, where the Hall−Petch hardening mechanism becomes invalid

(2)

where H0 is the hardness of the bulk single crystal, KHP is the Hall−Petch hardening coefficient,41 Kqc = 211Ne1/3e−1.191f I is 5636



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and hardness decreases significantly at a scale of ca. 10−15 nm.43,44 Similar softening (an inverse Hall−Petch effect) was not observed in nanotwinned cBN. This difference indicates that polar covalent materials are different from metals in terms of their hardening behavior at the nanoscale. We do notice a hardness deviation for nanotwinned cBN with an average twin thickness of 3.8 nm (HV = 108 GPa) from the value calculated using eq 2. In the range smaller than 14.5 nm (inset to Figure 5), hardness was also estimated without considering the Hall− Petch effect (blue curve). Obviously, the Hall−Petch effect is still functional at this length scale, at least partially as suggested by the red dashed line. The deviation might arise from the residual boron oxides in the starting onion BN nanoparticles,45 which can degrade hardness to some extent. It is instructive to estimate the ultimately achievable hardness (HUA) of cBN. Taking {111} twins in nanotwinned cBN as the model system,1 the minimal twin thickness is λmin = 3d111 = 0.626 nm. At such a small length scale, the Hall−Petch effect might not be applicable,46,47 although this issue needs to be further confirmed by experimental and theoretical studies. An extraordinary HUA of 256 GPa is predicted for nanotwinned cBN by eq 2 (without the contribution of the Hall−Petch effect). It is a great challenge to synthesize nanotwinned microstructures with the required twin thickness to achieve such an exceptional hardness. By refining the precursor (such as onionlike boron nitride and carbon) with a smaller nanoparticle size, a further decrease of the twin thickness and consequential increase of the hardness are possible for synthetic nanotwinned bulks. In addition, it is also possible to tune a variety of hard materials (HV > 15 GPa, for example) into superhard materials through the formation of ultrafine nanostructures. We expect the enhancement of hardness and the broadening of the family of superhard materials to be achievable through nanostructuring: The smaller the grain size (or twin thickness), the higher the hardness.

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CONCLUSIONS In summary, we have clarified the criterion for the formation of an indentation in hardness measurements by analyzing the stress states of the indenter and the tested sample during an indentation process: The shear strength of the sample has to be smaller than the compressive strength of the diamond indenter. Once this criterion is satisfied, a permanent indentation can be formed on the sample surface, and hardness can be reliably measured even if the sample is harder than diamond. Through nanostructuring, materials that are harder than diamond are no longer just a dream. Indeed, rapid growth of the family of superhard materials should be forthcoming.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 86-139-3356-2858. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation of China (51421091 and 51332005) and the Natural Science Foundation for Distinguished Young Scholars of Hebei Province of China (E2014203150). 5637


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