3.1. The Still Believed H/E (“Elastic Index”) ClaimsThe methods of [1] are for various reasons questionable and useless. The authors do not explain why they claim proportionality between ISO-H and ISO-Er (here for Berkovich). The according to [16] defined ISO-H is FN per contact area Ahc that is geometrically 27.15hc2. The dimension is (force/depth2) usually reported as GPa. The elastic property is experimentally measured as stiffness S = ΔFNmax/Δh with the different dimension (force/depth). To obtain an ISO-Er with the same dimension as the ISO-H, one multiplies the stiffness S with 0.5 π1/2 Ahc−1/2. This provides the further hc−1 for the dimension of ISO-Er as force per area (GPa). But the Ahc value requires one iteration with 3 and another iteration with up to 8 free parameters (also + or − sign selection) according to [16]. The plots of log H vs log Er numbers, suggesting a “≈0.05 ratio of H/Er” for uncountable published Berkovich indentations would at best indicate very poor worldwide measurements of indentations over the years if that would be reality. It cannot be used for the calculation of Er from H numbers with a “statistical confidence of 95%” and R2 = 0.96. For example a hardness number H of 0.6 GPa in figure 1 of [1] contains a spread from 6-30 GPa for the moduli Er of the densely overlapping entries. Or an entry at H = 7 × 10−5 GPa has a data triangle value of about 1.05 × 10−2 GPa for Er, while the corresponding H value on the H/Erline for that Er is at 4.8 × 10−4 GPa, which is an about 6.9-fold higher hardness number. These examples show drastically that the claimed linear relation between the ISO hardness H and ISO modulus Er numbers is not correct. And it will be shown in Section 3.2 why it cannot be correct. The claimed statistic confidence of 95% for the log-log plot is useless and dangerous. In the figures 1 and 4a, 4b of [1] there must be selective choices of data from old papers and tables (often not the more recent ones) with questionable reliability. And even in the more recent papers it was never considered or known which polymorph of the sample had been probed under load, because their onsets could never been seen or excluded. We also must complain that the used entries in figure 1 of [1] are not cited and the materials not named. It appears that numerous of these are unpublished own values. This must also be concluded from the caption of figure 4 in [1], where, unlike the H/Er plot, straight linear plots are imaged. These are for spherical and for Berkovich indentations without any visible deviations. Beware from the risk of H vs Er plots in view of figure 1 of [1] and beware from predictive uses from there!
Particularly risky and dangerous are the use of H/Er plots or values for the evaluation of brittleness characterizations, critical load ratios, strengthening, toughness, and “true hardness”. For example figure 4a in [1] describes linear correlations of the brittleness parameter against the load ratio, which rests on (H/Er)2 numbers, and the normalized characteristic indentation dimension vs critical normalized cracking load ratios are plotted in figure 4b of [1]. Only some wet and dried materials are named here and the authors of [1] cite hardly checkable data collections and unpublished own data (which ones?). For example they did not cite the soda lime glass values from [16], which are still a present ISO-standard for HI and EI, even though they repeatedly invoked the “Oliver-Pharr analysis” or “-model”. And only a few entries are directly cited, but almost none of these disclosed published original loading curves that could be checked and used for the calculation of real properties like physical hardness, iteration-free elastic moduli and phase-transitions under load.
Furthermore, the authors of [1] try to define and calculate a so-called “true hardness” Htrue = H/{1 − (H/E)1/2 (2/tan ß)1/2}2, where ß is the cone angle of the indenter. This shall be the “resistance to plastic deformation” or “resistance to irreversible deformation”, which “depends on the ratio between indentation hardness and indentation modulus”. It is strangely claimed that “a large indentation hardness does not imply a large resistance to irreversible deformation per se”. This hard to understand basis by using the H/Er fraction is exemplified in [1] as follows: H = 1.07 GPa and Er = 10.5 GPa shall imply “true hardness” of 17.8 GPa”; or H = 3.12 GPa and Er = 87.02 GPa shall imply “true hardness” of 10.8 GPa”. The H/Er fraction is contained in such calculations. And the other formulas are in the appendix of [1]. These need not be depicted here, due to the incorrect physical basis from the beginning. Such “true hardness” with exorbitantly misleading high values adds further to misleading confusion without any physical merit.
3.2. The Physical Errors of the H/E Ratio Claims with Their UsesThe proportionality claims of ISO-H with ISO-Er in [1] and the therein appropriate citations are physically wrong. They cannot be valid for basic physical reasons! The only correct physical hardness of conical or pyramidal indentations is the penetration resistance k [force/depth3/2] from the slope of the so named Kaupp-plot FN vs h3/2. None of the cited and used HI and EI values tells which polymorph of the material was probed, because their onset forces cannot be found with the wrong exponent 2 on h. One needs such linear plots for the detection of polymorph formation onsets [4] [10] - [15]. And every polymorph has its own mechanical properties. With other words:
All of these H and Er values are unphysical and so are their ratios, because they rely on the disproved exponent 2 on h (instead of the FN vs h3/2 relation) and require data-fitting iterations [16]. For correct analysis of loading curves for pyramidal and conical indentations see [3] and [6]. Spherical indentations are more complicated [6], but some of the examples in figure 1 of [1] are from unphysical interpreted spherical indentations or Vickers hardness as in e.g. [17] with similar H/E ratios containing log/log plots. Unfortunately, Er numbers are not well defined and depend strongly on the details of their detection, as outlined in [9], so that one has to choose from sometimes extremely different values for Er when comparing different methods.
The Oliver-Pharr technique and the still present ISO 14577 standard assume the physically disproved exponent 2 on h for the loading curves instead of undeniably physically deduced h3/2 [3], and they violate the energy law [6] [8]. Therefore the authors of [1] cited H and Er values that are entirely unphysical parameters. If indentation hardness have to be compared with indentation modulus one should only take physically sound values from the so named Kaupp plots (FN vs h3/2) that most easily provide penetration resistance onsets and differentiate the properties of every polymorph under load. And it provides directly measured indentation moduli (Ephys) without any iteration. All of the trouble in Section 3.2 originates from the widespread refusal to check the exponent of their loading curves. These FN vs h loading curves follow always the physically correct relation FN = kh3/2. All of the respective authors stay with the physically disproved h2, apparently until ISO and the authors of [16] correct their basic errors with public announcements.
3.3. The Wealth of Penetration Resistance for the Physical Analysis as Exemplified for the Marine Mollusks CaseThe claims of linear relations between ISO-hardness H and ISO-modulus Er include nacre, eggshell, aragonite, calcite, hydroxyapatite, enamel, dentine, bone, etc in the unphysical and incorrect log/log plots in [1]. But their data are selective and none of them reveals which polymorph was probed under what applied load. Furthermore, it is not told which of the triangle data points around the regression line in figure 1 of [1] belong to which materials. These data are without any value.
We show now that hitherto unthinkable materials’ properties are straightforwardly obtained on the physical analysis of indentations from correctly cited publications. This will be exemplified for the case of two mollusk varieties with their aragonite shells, including the distribution of the organic materials. The physical analysis of the indentation loading curve onto the polar pteropod Limacina Helicina Antarctica shell, as recently published with figure 4 in [2], yields the FN vs h3/2 diagram of Figure 1. One recognizes an initial surface effect up to about 300 µN load (probably due to the water content) and three linear branches that are connected by smooth transition zones between them. This was not seen in their FN vs h curve and it does never show up in the unphysical FN vs h2 relation with its false exponent 2 on h. The authors of [2] did thus not see that their calculation of ISO-H and ISO-Er values [16] do not at all relate to properties of their sample but to the third polymorph of it that is present at their maximal force. Our three linear regression lines with (R2 = 0.9994, 0.9993, and 0.9993, respectively) have the equations that are inserted in Figure 1. The kink positions, as obtained by equation of two adjacent regression equations, at 1182.08 and 2958.895 µN loads are phase-transition onsets. With these experimental values we calculate the phase transition energies with the simple well-known arithmetic equations that are most recently comprehensively published in [6] and [14], and earlier publications of the present author. The obtained normalized per µN values are 0.01496 µNµm/µN and 0.14879 µNµm/µN. Clearly, there
are three different polymorphs up to a loading range of 5 mN load with a Berkovich. Interestingly these values are similar to the ones of calcite in table 1 of [14] that are at 0.01599 and 0.10692 µNµm/µN, respectively, although at their mN ranges. It is clear that we have the phase transitions with the Limacina Helicina Antarctica shell.
The smooth transition zones rather than sharp kinks that are here only seen by the intersecting regression lines reveals a gradual change of the strengthening organic material between the different polymorphs. This is certainly a bionics model for avoiding the crack increasing risk when unavoidable polymorph interfaces contact smoothly [11]. This appears to be further studied and used for modifying the negative effect of phase-transitions also with technical materials.
The precise distribution of the about 5 wt% of organic material is certainly worth further studies. Conversely, the unphysical H and Er measurements led to the claim of “essentially homogenous distribution throughout the shell for “strengthening the cell” [2]. Furthermore, the rather strong variation of the averaged ISO-H and ISO-Er values in figure 5 of [2], or the report that some indentations were going down to 700 nm depth and others down to only 200 nm depth, or the observation of rough and smooth areas would also suggest a thorough new and physically analyzed indention. All of these strongly deviating results must be reproduced and separately analyzed rather than averaged as in [2]. Comparison of the so available penetration resistance k-values (mN/µm3/2) of the different polymorphs up to the same load maximum would tell, whether there are zones with more or less organic material also laterally distributed. In the case of micro-caverns empty or filled with water, these would show-up as spurious pop-ins [15]. The physical analysis technique is the only mans for the distinction of soft and hard regions that cannot be visibly traced.
It is particularly unsuitable that the authors of [2] did not compare their findings with the physical analysis of the Berkovich indentation onto the lamellar pteropod structure of the red abalone Haliotis rufescens in [5] . This pteropod shell exhibits distinct well defined stepwise organic layers between the aragonite lamellas and the indentation curves of them are physically correct analyzed in [5]. The non-appreciation of the well documented different strengthening bionics model is misleading, even though the introduction of paper [2] cites numerous old and very old papers on different pteropod varieties with prismatic helical and lamellar aragonite structures, but unduly questioned the paper of [18]. We must therefore resume our analyses here with the calculations on the correct physical basis. These add to a much better understanding of the pteropods stability with reliable parameters instead of iterated H and Er that are against physics.
The averaged Berkovich indentation curve of the red abalone Haliotis rufescens shell from Baja, California [18] was physically analyzed in [14]. The exterior nacre shell of 250 - 300 nm thickness with kfirst-aragonite-shell = 0.9058 µN/nm3/2 is sharply distinguished by the organic layer with korganic = 0.274 µN/nm3/2 and the following inner apatite layer with k inner-aroganite-shell = 1.1495 µN/nm3/2. The stepwise behavior is also shown in the original FN vs h plot from figure 7 in [18]. Clearly both aragonite layers are sharply separated by the soft organic layer. The FN vs h3/2 plot in Figure 2 with the inserted regression equations, as calculated up to 400 nm depth (8000 nm3/2) with the inserted regression lines is totally different from Figure 1. After a minor initial surface effect that is not part of the regression (5 points) two aragonite layer lines are separated by the organics line. The larger k-value of the first inner aragonite layer starts at a 26.9 per cent higher load and at higher h3/2. Therefore at least some of this k-value increase represents the shift relative to the end of the first aragonite layer. The thinner strengthening organic layer is considerably softer. The 16.8% difference between the k-values of the two aragonite layers cannot solely be responsible for their displacement. The second aragonite layer should also be phase-transformed at the increased force with an onset right above the organic layer at 4312 µN. The regression line values of the hard branches correlate both with R2 = 0.9997.
However, there are difficulties for the calculation of the phase-transition
energy, because we do not have a kink-point between the displaced aragonite layers. We must try to secure that the inner aragonite layer is a polymorph by a phase-transition. Figure 2 indicates that the first branch ends very close to the lower end of the organic layer at 4000 µN load and 287 nm depth (4853 nm3/2). The second aragonite branch starts directly at the upper end of the organic layer at 4312 µN. This data point is already part of the regression line for the steeper (harder) branch. The 4000 µN load would thus be the phase transition onset point if the organic layer was not there. Unlike the repair of pop-ins [15], our phase-transition energy calculation technique is not applicable for such separation by a different material. A shift of the upper layer line for joining with the lower line at 4000 µN by formally removing the organic layer is not allowed. And it would not solve the problem: the steepness of the second hard layer is influenced both by higher FN and by higher h3/2 values at its start. This influence on the steepness cannot be undoubtedly judged and also minor corrections in that sense would strongly influence our precise and highly sensitive calculations. A phase transition part from about 4500 µN load of the hard nacre shell is however most likely. That judgment is in view of the k1 and k2 values of the softer Limacina Helicina Antarctica shell that experiences the phase transition and the k1 and k2 values of the Haliotis rufescens shell that are in the same order of magnitude even though the shells of Limacina Helicina Antarctica are softened by the embedded organic layers. Final proof would require comparison with an indentation of pure aragonite at forces up to about 7000 µN load. Unfortunately we did not find accessible reliable Berkovich indentation curves of pure aragonite at such a loading range with smooth loading curves that are not interrupted by continuously repeated unloads. There is however a phase-transition within a 500 µN loading range of pelletized aragonite from [19] that occurs with an endothermic phase-transition at 348 µN load We interpret it as an endothermic twinning of aragonite and calculate the normalized conversion energy to 0.02922 µNµm/µN. Clearly this interesting twinning transition cannot be remarked at an indentation range of 7000 µN of the mollusks indentations.
We must report here how one can identify and exclude experimentally false reported data by the calculation of transition energies. The reported Berkovich indentation onto (001) of aragonite up to 1000 µN load from the figure 1 of [20] revealed a minor exothermic transition at FNkink = 408.6 µN when physically analyzed. Thus, figure 15b in [5] would exhibit exothermic transition energy of −0.00276 µNµm/µN [14]. This putative twinning of aragonite appeared to be a specialty of the (001) face of aragonite. It was hardly resolved in the indentations of nacre up to 7000 µN load and must be cancelled now in Table 1 of [14]. This failure is an important application of the calculation of phase-transition energies, because it helps to eliminate undue measurements. The work of [20] failed, because the loading curves were averaged and pop-ins was also reported with three imaged individual curves on the probed surface. Their inclusion in the curves averaging constructed the exothermic event. It is still not widely recognized that pop-ins are instrumental errors due to distortions that must be repaired or eliminated from further use [15]. The pop-in generation in the present case was probably by touching of tip sides with terrace steps on the probed surface (further reasons of pop-in distortions are listed in [15] ). Any averaging of experimental curves must be strictly avoided. Only the results from all individual undistorted curves should be averaged. The curve for (001) in figure 1 of [20] and its analysis in table 1 of [14] must be disregarded.
The behavior of aragonite must now be compared with the other ambient modifications of CaCO3. Hexagonal calcite, orthorhombic aragonite, and hexagonal vaterite crystallize in the respective space groups R3−c, Pbm6n, and P63/mmc. Their X-ray densities are 2.71, 2.93, and 2.93 g/cm3, respectively. The most frequent twins of calcite occur along (10 - 11) by mechanical stress on (01 - 12) and those of aragonite on (110) by mechanical stress on (010) [21]. So there is the possibility of mechanical twinning by pressure. Such twins of aragonite are orthorhombic Pmcn. Vaterite twins have been found in pearls [22]. All three modifications occur as minerals. Aragonite and vaterite (with some organic material) are primarily of biological origin as in pearls or mollusk shells, and vaterite in gallstones and nephritic stones and plants (e.g. [23] ). An important point is the repair of deformed Gastropodes’ aragonite shells with vaterite attachments [23]. The loading curves of calcite up to 10 mN [24] and up to 40 mN [25] have been analyzed in [14] to give conversion energies of 0.01599 mNµm/mN and 0.1069 µNµm/µN for the second phase transition.