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Ultrahardness: Measurement and Enhancement Bo Xu, and Yongjun Tian J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b00017 • Publication Date (Web): 19 Feb 2015 Downloaded from http://pubs.acs.org on February 24, 2015

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The Journal of Physical Chemistry

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: One is how the hardness can be reliably measured; the other is 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 the 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 smaller than the compressive strength of the diamond indenter. Also, we suggest nanostructuring (nanocrystalline and nanotwinning) as an effect way for hardness enhancement.

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INTRODUCTION Superhard materials, defined as materials with Vickers hardness higher than 40 GPa, are of vital importance for both industrial and scientific applications, such as cutting tools for 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, such as the poor thermal stability of diamond, the relatively low hardness and fracture toughness of cBN. Therefore, numerous efforts have been made in the past half century to search novel superhard materials for improved comprehensive performance, i.e. the simultaneously improved hardness, toughness, and thermal stability.1-4 Meanwhile, identifying synthetic materials harder than nature 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 methods (e.g. scratch, indentation, and rebound). For superhard materials, Vickers and Knoop scales (differing from each other in the indenter shape) 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 by fitting the experimental Vickers/Knoop hardness data of polar covalent single crystals.8-10 These models provide an atomic-level understanding of materials hardness and enable the possibility of hardness prediction. Also, several key factors to


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achieve superhardness are revealed from these microscopic models, namely the threedimensional network structure, short chemical bond, high bond and valence electron densities, and low bond 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 to form a permanent indentation through plastic deformation during an indentation hardness measurement? Previously, it was proposed that “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 which 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 the traditional HC criterion is incorrect, and provide an updated strength-comparison (SC) criterion that the shear strength of the tested sample is smaller than the compressive strength of the diamond indenter. In addition, we emphasize the effectivity of nanostructuring for hardness enhancement based on our semi-empirical hardness model of polycrystalline covalent materials. MATERIALS AND METHODS The Vickers hardness, 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 by H V (GPa) = 1854.4 L d 2 , where L (in Newton) is the applied load and d (in µm) is the average diagonal length of indentation determined by an optical microscope equipped on the microhardness tester. Hardness of each sample was appointed as the value from the asymptotic-hardness (load-invariant) region. An indentation formed on a nanotwinned


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diamond sample under a load of 9.8 N was measured with an atomic force microscopy (AFM, Solver P47-PRO). The diagonal length of the indentation determined by AFM provided a calibration to the optical-determined one. The compressive strengths of Cco-C812 and M-carbon13 phases were determined with the same method used previously for diamond.14 The first-principles calculations were performed with CASTEP based on the density functional theory.15 The norm-conserving pseudopotential with a cutoff energy of 770 eV was used. The exchange-correlation functional was treated by local density approximation (LDA-CAPZ),16-17 which is a better functional than GGA-PBE for group IVA elements and IIIA-VA compounds as suggested by a recent publication.18 A k-point spacing (2π × 0.04 Å−1) was used to generate the Monkhorst-Pack k-point grids for Brillouin zone sampling.19 The 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.

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 nature diamond. However, these materials are exactly what we are looking for in superhard materials research. If the HC criterion were correct, it would be impossible to characterize the hardness for these materials. We do notice many experimental counterexamples to the HC criterion. For example, Vickers hardness of annealed CVD diamond as high as 170 GPa was reported previously,20 significantly higher than


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the suggested upper limit of 120 GPa. A scratch made by ReB2 on the surface of diamond was verified by AFM even through ReB2 is much softer than diamond.21 Another example contradict to the HC criterion is the soft impressor method developed by Brookes et al.,22 where 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 is that the resolved shear stress exceeds those require for dislocation initiation and multiplication.24 To clarify the controversy, we have to 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 well-defined “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 about 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 sample during an indentation hardness measurement are analyzed below. Figure 1 schematically shows the indentation process with an exemplary Vickers indenter. When the symmetric indenter is exactly perpendicularly pushed into the tested sample, the horizontal force components from opposite facets (F2 and F2’) of the indenter are offset, leaving only the compressive stress (σc) to apply on the indenter due to the perpendicular force components (F1 and F1’). The tip of the diamond indenter is thus subjected to a compressive


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stress field. In this case, the force components of F2 and F2’ cannot result in any deformation of diamond indenter no matter how high they might be. Note the difference (F2 - F2’) between two horizontal force components may become large enough to break the tip of the indenter if the indenter is obliquely pushed into the sample surface. The stress state of the deformation zone on the sample surface is different from that of the indenter though. 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 smaller than the compressive strength of the indenter diamond.26 In other words, the SC criterion for a reliable indentation hardness measurement is

σ cindenter > τ sample ,

(1)

where σ cindenter is the compressive strength of diamond indenter, and τ sample is the shear strength of tested material. The compressive strengths of diamond are 223 GPa in the weakest direction, and about 470 GPa along and .14 Table 1 lists the shear strength of typical hard and superhard materials.27-32 It is well known that diamond has highest shear strength among the known materials. Because the lowest compressive strength of 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 indentation hardness measurement. This criterion indicates that the formation of a permanent indentation on sample does not depend on the relative hardness of the indenter and sample as claimed previously.11


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Our SC criterion can explain some previous experimental results, such as the observed indentation marks on diamond anvils by the cold-compressed carbon nanotubes33 and graphite34. Theoretical simulations suggested that the cold-compressed carbon nanotubes transforms into Cco-C8 phase,12 and the cold-compressed graphite into M-carbon phase13 under high pressure. During the compression process with diamond anvil cell at high pressure,33-34 these new phases (Cco-C8 and M-carbon) can be looked as the indenter and the flat facet of diamond anvil as the tested sample surface of a hardness measurement. The calculated theoretical hardness of both Cco-C8 and M-carbon is slightly lower than that of diamond.12 In order to understand the origin of indentation marks, we calculate the compressive strengths of Cco-C8 and M-carbon along selected crystal directions, which are demonstrated in Figure 2. The compressive strength in the weakest direction is about 420 GPa for Cco-C8 and 260 GPa for M-carbon, much larger than the shear strength of anvil diamond. The formation of the observed indentation marks on diamond anvils can thus be well accounted for by our SC criterion.

Error of hardness measurement 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, how to assess the measurement error becomes an important and generally concerned issue. In fact, the measurement error mainly comes from the determination of two diagonal lengths of an indentation. The diagonal lengths are typically measured with an optical microscope installed on a standard hardness tester. For the tough superhard materials such as nanotwinned diamond, pipe-up ridges higher than the basal plane of sample surface (see the dash line in Figure 1) are usually formed during indentation process. In


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this case, the deformation area now including the pile-up zone is significantly larger than the ideal indentation. In an optical-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 since they are easy to be distinguished under an optical microscope. The ideal diagonal length, however, should be determined from two opposite corners of the indentation in the basal plane (the black triangles in Figure 1). The ideal diagonal length can be accurately calibrated by AFM.35 Such kind 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 H V = 234.6 GPa ) and 9.75 µm ( H V = 191.1 GPa ), respectively. Obviously, the optical-based measurement overestimates the diagonal length, thus provides a conservative measure of Vickers hardness (underestimated about 18.5 %). In either case, the measured hardness is significantly higher than that of natural diamond indenter. It should be noted that, to perform a reliable hardness measurement, the indenter size effect should be avoid, which gives higher hardness values for smaller indentations due to greater strain gradient.36 Hardness should be determined from the asymptotic-hardness region of a well-controlled indentation process (without formation of cracks).

Nanostructuring for hardness enhancement The above analyses clearly indicate 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 higher than that of diamond. This is truly a challenge to materials science, and researchers are pessimistic to fulfill this goal.37 Generally, there are two pathways for this ambition: One is to design novel superhard single crystals; the other is to enhance the hardness of known materials through forming ultrafine microstructures. Although a recent


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research indicates that 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

where hardness are greatly enhanced compared with corresponding single crystals due to the

microstructures of nanotwins and nanocrystals. Figure 4 compares the measured hardness of nanotwinned diamond,2 nanotwinned cBN,1 and diamond indenter. The superior hardness of nanotwinned diamond and nanotwinned cBN over that of diamond indenter is obvious. Our previous theoretical consideration suggested 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 = H 0 + K HP

D + K qc D ,

(2)

where H 0 is the hardness of the bulk single crystal, K HP is the Hall−Petch hardening coefficient,41 K qc = 211N e1/3e −1.191 fi is the quantum confinement hardening coefficient (Ne and fi are the valence electron density of crystal and 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 nanocrystalline cBN bulks as a function of average grain size (or twin thickness) from several experiments.1,

3-4

The calculated hardness from Eq. 2 (the black curve) is also shown for

comparison. For cBN, H 0 and K HP are 39 GPa and 126 GPa·nm1/2 respectively;3 K qc 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 nanometre scale for nanocrystalline cBN bulks, in contrast to what happens in metals where the Hall–Petch


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hardening mechanism becomes invalid and hardness decreases significantly at the scale of ca. 10−15 nm.43-44 Similar softening (an inverse Hall−Petch effect) was not observed in nanotwinned cBN. Such a difference indicates that polar covalent materials are different from metals in hardening behavior at nanoscale. We do notice a hardness deviation for nt-cBN with an average twin thickness of 3.8 nm ( H V = 108 GPa ) from the calculated one by using Eq. 2. In the range smaller than 14.5 nm (the inset to Figure 5), hardness is also estimated without considering the Hall−Petch effect (the blue curve). Obviously, the Hall−Petch effect is still functional at this length scale, at least partially as suggested by the red dash line. The deviation may come 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 nt-cBN as the model system,1 the minimal twin thickness is λmin = 3 × d111 = 0.626 nm . At such a small length scale, the Hall−Petch effect may not be applicable,46-47 which needs to be further confirmed by experimental and theoretical studies. An extraordinary HUA of 256 GPa is predicted for nanotwinned cBN from Eq. 2 (without contribution of the Hall−Petch effect). It is a great challenge to synthesize nanotwinned microstructures with required twin thickness to achieve such an exceptional hardness. By refining the precursor (such as onion-like boron nitride and carbon) with smaller nanoparticle size, further decrease of twin thickness and consequential increase of hardness are possible for the synthetic nanotwinned bulks. In addition, it is also possible to tune a variety of hard materials ( H V > 15 GPa , for example) into superhard materials through formation of ultrafine nanostructures. We expect hardness enhancement as well as broadening of superhard materials family through nanostructuring: The smaller the grain size (or twin thickness), the higher the hardness.


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CONCLUSIONS In summary, we clarify the criterion for the formation of an indentation in hardness measurement through 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. With the nanostructuring pathway, harder than diamond is not a dream anymore. A rapid growth of superhard materials family is forthcoming.

AUTHOR INFORMATION Corresponding Author *Email: [email protected], Tel: 86-139-3356-2858 Notes The authors declare no competing financial interests. ACKNOWLEDGMENT 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).


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Table 1. Ideal shear strength (τ, in GPa) of typical hard and superhard crystals. Compound

diamond

cBN wBN AlN TiN

SiC ReB2 FeB4 B6O

τ

93

58

62

20

29

28

34

24

38

Reference

27

28

28

29

29

30

31

32

31


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Figure 1. An indentation process with a Vickers indenter. Red lines schematically represent the slip systems in the polycrystals with dislocations signed by ⊥. d1 and d2 shows the diagonals of the formed indentation. The black and red triangles mark the positions to determine the ideal diagonal and optical measured one, respectively. The dash line emphasizes the basal plane of sample surface. See the main text for detail.


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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 Cco-C8 and M-carbon, respectively.


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


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Figure 4. The 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 are determined as 200 GPa, 108 GPa, and 92 GPa, respectively, from the asymptotichardness region.


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Figure 5. The 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 empty circles show hardness estimated from Eq. 2. The blue curve and empty circles show hardness estimated from H = H 0 + K qc D . The red dash line is a guide for the eyes. See the main text for detail.


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TOC GRAPHICS


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Ultrahardness: Measurement and Enhancement - ACS Publications

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