Analysis of the Indentation Size Effect in the Microhardness Measurements in B₆O

Abstract

The Vickers microhardness measurements of boron suboxide (B₆O) ceramics prepared by uniaxial hot-pressing was investigated at indentation test loads in the range from 0.10 to 2.0 kgf. Results from the investigation indicate that the measured microhardness exhibits an indentation load dependence. Based on the results, we present a comprehensive model intercomparison study of indentation size effects (ISEs) in the microhardness measurements of hot-pressed B₆O discussed using existing models, that is, the classical Meyer's law, Li and Bradt's proportional specimen resistance model (PSR), the modified proportional specimen resistance model (MPSR), and Carpinteri's multifractal scaling law (MFSL). The best correlation between literature-cited load-independent Vickers microhardness values, the measured values, and applied models was achieved in the case of the MPSR and the MFSL models.

1. Introduction

With hardness values reported to range between 24 GPa and 45 GPa [1, 2], boron suboxide (B₆O) is sometimes considered to be the third hardest material after only diamond (from ~70 to ~100 GPa) and cubic boron nitride (~60 GPa) [2]. The boron-rich icosahedral ultrahard material belongs to the 𝛼-rhombohedral boron type structure (R3m) [2] which is similar to those of other ultrahard boron-rich materials namely boron carbide B₄C [3], aluminium magnesium boride AlMgB₁₄ [4], and the newly synthesized boron subnitride B₁₃N₂ [5].

B₆O exhibits a rather unusual and wide range of superior properties; among these are high hardness with low density, high mechanical strength, and high chemical inertness [2, 6]. Along with the other boron-rich ultrahard materials, the potential applications of B₆O have been an object of intense interest in recent years [2]. However, despite the intensive research efforts, the commercial applications are yet to be realized partly because of (a) the low fracture toughness of polycrystalline B₆O [7] and considerable practical challenges in the densification of that material [2, 6, 7] by hot-pressing, (b) stoichiometric B₆O samples that are not easy to synthesize [1, 2], (c) poor crystallinity [1, 7], and hence (d) numerous mechanical properties of the material that are still poorly understood [8]—indentation size effect (ISE) in microhardness measurements is one such property.

Until now, the investigation of the mechanical properties of B₆O by indentation has been neglected in favour of the improvement of densification and fracture toughness of the composites [2, 6, 7]. The hardness reported for the material by different workers often remains a topic of debate, with literature-cited values varying from 24 GPa to 45 GPa. To the best of our knowledge, the indentation load-size dependence in the microhardness of hot-pressed B₆O is being comprehensively examined for the first time in this paper. The aim of this paper is twofold: (1) to report experimental data on the microhardness of hot-pressed B₆O determined by Vickers pyramidal indentation and (2) to critically examine the ISE phenomenon by means of a comprehensive intermodel comparison of various approaches and models proposed in the literature. The nature and the origins of ISE are investigated and discussed.

2. Materials and Experimental Method

B₆O powder was prepared by the reaction of B and B₂O₃ as detailed by Andrews et al. in [9] thatwere uniaxiallyhot-pressed in hBN pots under argon environment at 1800°C and 50 MPa for 20 minutes. The hot-pressed compacts were then metallographically prepared using a method prescribed by Machaka et al. in [10]. The density of the hot-pressed compacts measured 2.44  g/cm³. The polished samples were analyzed for phase composition using X-ray diffraction, with Cu K_α radiation. The powder diffraction patterns were collected using the Bragg-Brentano backscattering geometry over a 10°–90° 2𝜃 range, with a 0.02° step size.

The measurements were performed at room temperature by a Future-Tech microhardness tester at different applied test load between 0.1 and 2.0 kgf. At least ten hardness measurements were carefully taken for each load at different locations on the middle of each sample, and the average values were taken as the hardness of the sample at that load. The Vickers indentation impressions were observed using optical microscopy.

3. Results

3.1. XRD Measurements

Figure 1 shows the full profile X-ray powder diffraction pattern measured on the starting hot-pressed B₆O samples. The diffraction pattern is characteristic of a nominally pure polycrystalline B₆O. All the dominant diffraction peaks can be perfectly indexed to the XRD pattern of B₆O (JCPDS File Card no. 31-210) reported elsewhere [9]. The lattice parameters, 𝑎=5.39 Å and 𝑐=12.34 Å, obtained from XRD analysis are in good agreement with the single crystal data [2].

Figure 1 

The powder diffraction pattern of a sample of hot-pressed B6O showing the predominance of the B6O crystalline phase.

3.2. Microindentation Measurements

To investigate ISE, microindentation measurements were conducted at test load between 0.1 and 2.0 kgf using a Vickers diamond indenter. Apparent Vickers microhardness values 𝐻_𝑣 were calculated at each load using the conventional approach shown in:𝐻_𝑣=1.8544⋅𝑃𝑑²,(1) where 𝑑 is the average diagonal length of the Vickers indentation impressions and 𝑃 is the indentation test load in kgf. Figure 2 shows the dependenceof the measured Vickers microhardness of hot-pressed B₆O on the applied indentation load. Each of the plotted data point is an average of at least ten measurements at each applied test load. The insert in Figure 2 depicts an optical micrograph of a typical Vickers indentation impression.

Figure 2 

The dependence of apparent Vickers microhardness of hot-pressed B6O with applied load. The figure insert shows the optical micrograph of a typical Vickers indentation impression.

Evidently, the Vickers hardness is a function of the applied load at low applied indentation test load. The existence of ISE makes it unsatisfactory here to quote a single hardness value when Vickers hardness is used for material characterization. By observation, at applied indentation test loads ≳1 kgf, a single load-independent Vickers hardness value of 30.0 GPa seems to exists. The load-independent hardness is also sometimes deemed the intrinsic or “true” hardness. The observed value is consistent with literature reported values [1, 2, 11].

3.2.1. Meyer's Law

The classical power law shown in (2a), also known as the Meyer's law, is frequently used to indicate the existence of ISE for ceramic and other materials [12]. 𝐵 is a power-fit constant, and the exponent 𝑛 is the Meyer's index, also known as the size-effect index. The Meyer's index is generally used as a measure of ISE: log𝑃=𝑃lo=g𝐵𝐵⋅+𝑑𝑛^𝑛,⋅log𝑑.((22ba))The regression analysis of the experimental data using (2b) is shown in Figure 3. The slope of the linear fit is the Meyer's index, and observed to be 𝑛=1.75. When 𝑛=2, the microhardness is expected to be independent of the applied load. However, 𝑛≤2 indicates the existence of an ISE trend, the measured microhardness decreasing with applied load [13, 14].

Figure 3 

Linear regression analysis of log𝑃 against log of the indentation diagonal length, 𝑑 according to the Meyer's law (2b).

3.2.2. PSR Model

Several studies [14, 15] agree that the Meyer law is probably insufficient to describe the origin of the ISE, although it is a good indicator of its existence. According to Li and Bradt [14], the applied indentation load is related to the average indent diagonal length through a polynomial relationship (3a): 𝑃==𝑎𝑎₁₁⋅𝑑+𝑎₂⋅𝑑²(3a)⋅𝑑+𝐻^psr1.8544⋅𝑑².(3b)The parameters 𝑎₁ and 𝑎₂ are constants for a given material. The parameters are related to the elastic and the plastic deformation properties of the test material, respectively. The parameter 𝑎₁ is a measure of the surface effects during microhardness indentation which is directly related to the ISE contribution, whilst 𝑎₂ is directly related to the load-independent microhardness value 𝐻_psr as demonstrated in (3b).

From (3a), a plot of 𝑃/𝑑 against 𝑑 can be constructed to yield a straight line, where best-fit parameters 𝑎₁ and 𝑎₂ can easily be obtained by regression analysis. As shown in Figure 4, a plot of 𝑃/𝑑 against 𝑑 is significantly linear, implying that (3a) gives an indication of the existence of ISE in hot-pressed B₆O.

Figure 4 

A plot of 𝑃/𝑑 against 𝑑 constructed from the experimentally measured data and a best linear fit of the plot.

As noted previously, the PSR analysis can be extended to determine 𝐻_psr directly from 𝑎₂ using (3b). In this study 𝐻_psr was found to be 26.8 ± 0.3 GPa.

3.2.3. Modified PSR Model

Gong et al. in [16] modified the PSR model to investigate the ISE behaviour in various materials. The resulting MPSR model is defined by (4a):𝑃==𝑎𝑎₀₀+𝑎₁⋅𝑑+𝑎₂⋅𝑑²(4a)+𝑎₁⋅𝑑+𝐻^psr1.8544⋅𝑑²,(4b)where 𝑎₀ is a constant related to the residual surface stresses associated with the surface grinding and polishing processes during sample preparation. The parameters 𝑎₁ and 𝑎₂ are constants as defined in the PSR model (refer to Section 3.2.2). The values of 𝑎₀, 𝑎₁, and 𝑎₂ can be evaluated by plotting the 𝑃 data against 𝑑. The plot of 𝑃 versus 𝑑 (and polynomial curve fitting thereof according to the MPSR model) is shown in Figure 5.

Figure 5 

A plot of 𝑃 versus 𝑑, and curve fitting thereof, according to the MPSR model.

As is the case for the PSR model, a load-independent hardness value, 𝐻_mpsr, can be determined directly from best-fit value of 𝑎₂ from Figure 5 using (4b). In this study, 𝐻_mpsr was found to be 28.2 ± 0.9 GPa.

The estimated best-fit values of the 𝑎₀, 𝑎₁, and 𝑎₂ parameters corresponding 𝐻_mpsr values are listed in Table 1. The residual surface stresses contribution to the observed ISE is negligible compared to the contribution of bulk material's plastic deformation characteristics.

Table 1 

A summary of the best-fit parameters calculated according to the PSR and MPSR models.

The PSR and MPRS parameters estimated by regression analysis are summarized in Table 1. Based on the values and their physical significance according to the energy-balance relation [17], we suggest that the residual surface stresses' contribution (𝑎₀) to the observed ISE is negligible compared to the contribution of bulk material's indentation elastic recovery (𝑎₁) and the indenter/specimen interface friction resistance coupled with elastic resistance of the specimen and the mixed elastic/plastic deformation response of material (𝑎₂).

3.2.4. MFSL Approach

As illustrated in Figure 2, the variation of microhardness with indentation load is a two-parameter problem exhibiting at least two totally different size-dependent regimes: a load-dependent regime and a load-independent regime. The variation of microhardness with indentation load in other words exhibits self-similarity; that is, by observing the property in different scales, a similar, but not exactly the same structure, is obtained—a property of multifractals.

According to Carpinteri and Puzzi in [18], ISE in materials follows the multifractal scaling law shown in (5). However, the indentation size effect property is observed to vanish in the limit of the applied load tending to infinity. From the later, an asymptotic value of hardness which resembles a load-independent or intrinsic value can be determined as illustrated in Figure 6. In this study, we apply the multifractal scaling law in the analysis of observed indentation size effect in hot-pressed B₆O. Further in-depth discussions on the fractal approach to indentation size effect are refereed to Carpinteri's comprehensive review on the subject [18].

Figure 6 

A bilogarithmic plot of the hardness calculated according to (5) plotted against the measured indentation diagonal length. The slope of the plot equals 𝐷𝐻, a fractal dimension for hardness with respect to 𝑑. In the insert, the fractal dimensional increment 𝐷𝐻 is plotted as a function of log𝑑. Nonlinearity of the scaling fractal dimensional suggests multifractal behaviour.

On the basis of this approach, the multifractal scaling law (MFSL) for ISE can be written in the following analytical form:𝐻_𝑣=𝐻_mfsl⋅1+𝑑^∗𝑑^0.5,(5) where 𝐻_mfsl is the intrinsic hardness in the limit of infinite (a huge) applied load and 𝑑^∗ is the critical material characteristic length which individuates the transition between the two regimes, that is, the fractal one and the nonfractal (Euclidean) one.

In order to study the fractal characteristic of the size effect dependence of microhardness depicted in Figure 2, a log-log plot of the Vickers microhardness and the indentation diagonal length was constructed based on the multifractal scaling law shown in (5) (and presented as Figure 6). The result shows that log𝐻 as a function of the log𝑑 is approximately linear in two parts of the scale dependence.

In this study, both 𝑑^∗ and 𝐻_mfsl were determined simultaneously by means of nonlinear least squares analysis of the experimental data using Mathematica. 𝐻_mfsl and 𝑑^∗ were determined to be 28.9 GPa and 1.41 μm, respectively. The fractal dimension in the first region can be depicted as the following equation.

The slope of the bilogarithmic plot shown in Figure 6 represents the fractal dimensional increment 𝐷_𝐻, which is a measure of the variable influence of ISE on the measured Vickers hardness. The variation of fractal dimension of multifractality 𝐷_𝐻 with log𝑑, as determined from log𝐻 versus log𝑑, is presented as an insert in Figure 6. Two limiting conditions have to be satisfied: 𝐷_𝐻=0 for large indents (Euclidean response) and 𝐷_𝐻=−0.5 for small indents (multifractal behaviour). The last situation corresponds to the highest possible indentation size effect in hardness, which is a theoretical upper bound.

According to the insert in Figure 6, 𝐷_𝐻 tends to zero and coincides with the occurrence in a crossover between fractal and Euclidean behaviour implying the progressive vanishing of fractality in the measured Vickers microhardness with increasing indentation load/size.

4. Summary and Conclusions

Figure 2 shows that the variation of 𝐻_𝑣 with 𝑃 indicates that the hot-pressed B₆O ceramic material exhibits ISE. Based on the observation, a comprehensive study of ISE in microhardness measurements of B₆O was conducted using a number of existing models, that is, the classical Meyer's law, Li and Bradt's proportional specimen resistance model (PSR), the modified proportional specimen resistance model (MPSR), and Carpinteri's multifractal scaling law (MFSL). A summary of the hardness values calculated is shown in Table 2.

Table 2 

A summary of the intrinsic microhardness values calculated according to Li and Bradt's PSR model, the Gong's MPSR model, and Carpinteri's MFSL model.

The average load-independent hardness of hot-pressed B₆O from this study has been observed to be 28.0±0.3 GPa.

Based on the values of the PSR and MPSR parameters estimated by regression analysis, the origins of ISE in hot-pressed B₆O can be attributed to either the bulk material's indentation elastic recovery, the indenter/specimen friction resistance coupled with elastic resistance of the specimen, or/and the mixed elastic/plastic deformation response of material. The residual surface stresses' contribution to the observed ISE is observed to be negligible compared to the aforementioned.

The linear relationship of the double logarithm between measure (the microhardness) and scale (the applied load) shows indentation self-similarity, both in indentation shape and in the results of hardness. We can conclude this to be fractal behaviour. The load dependence of hardness shows dual fractal behaviour given the two different regions of hardness.

We can conclude and suggest that the variation of fractal dimension of multifractality 𝐷_𝐻 with log𝑑 is a more resourceful indicator of the existence of ISE than the Meyer's index, for example.

Acknowledgments

The authors gratefully acknowledge the useful contributions of O. T. Johnson and the financial support from the DST/NRF Centre of Excellence in Strong Materials and the University of the Witwatersrand Mellon Award.