Unexpected Twinning and Phase-Transition of the Indentation Standards, Their Transition Energies, and Scientific Dichotomy ()
1. Introduction
The still enforced ISO-ASTM 14577 standards for hardness and elastic indentation modulus (falsely called “Young’s modulus”) have not been changed after overwhelming empirical proof [1] and after the mathematical proof in [2] 2010 and [3]. Also, the huge number of FN vs h3/2 plots for all very different inorganic and organic crystals, including anisotropies [4] did not help. All of these were (without any iteration) mathematically calculated for own and primarily from published loading curves of others. All of their experimental aluminium indentation loading curves follow the undeniable FN h3/2 law for one-point indentations Equation (1) [2]-[9] that does not violate the energy-law (it takes care of the 20% (for correct h3/2) of the force and thus energy that is not used for the penetration depth formation) [8]. It is the present author who deduced the physical law Equation (1) for all conical, pyramidal (one-point) and wedged indentations into all types of solid materials Equation (1), where FN is normal force, h penetration depth and Fa the axis cut force upon plotting of this function, because it corrects for all initial surface errors including tip rounding, difficulties for finding the zero point upon the tip approach to the surface that might not be perfectly flat or free from adhesion or repulsion effects surface layers, etc. Also, the off zero starting linear branches of the transformed polymorphs are so corrected with the axis-cut.
(1)
And we can even determine the energy of activation for the phase-transitions for indentations at different temperatures (to be distinguished from the pressure induced phase transitions and different phase-transitions at different temperatures). The certification of industries that involve indentations for being permitted (e.g. Hysitron Inc. and its companies in Europe must accept the worldwide false standards of ISO-ASTM 14577). They had thus no chance for different views, and when selling instruments with their Handbooks to academic researchers they have no reason to check the physical reality or correct the formulas and software necessary for the running of their instruments. Thus, academic researchers also accepted these and helped themselves on that basis with more and more complicated unnecessary additional sub-theories for actually easily understandable results with misleading vocabulary or disallowed data-fittings, or faked iterations and simulations.
For believing in calibration curves that must be used (after violating the energy law) with two iterations the first one with three free parameters and the second one with 8 free parameters with ether positive or also negative sign for a “perfect fit” to the standard. That still produces only relative values but not absolute ones and the indentation field suffers from being no physical science. Unfortunately, it is still the world-wide way to obtain ISO-ASTM-hardness and -modulus values coming out of the instrumental computer. Fortunately, all academic Teachers in Germany have the legally guaranteed freedom for research and teaching, but why don’t they use it? They must not get stuck with obviously false standards. Thus, the present Author as natural scientist professor was able to check, whether the black-box output of ISO-ASTM-H and ISO-ASTM-Er to be transformed into ISO-ASTM-E values and then falsely called “Young’s moduli” are faked. He can even tell it when he proved that these are faked. Nobody can forbid him to tell the mathematical truth. He is even obliged to do so for avoiding daily risks, even if that is often not making friends. I thus worried about the observance of the correct exponent of the normal force FN vs penetration depth h loading curves and checked the exponent of such parabolas and did not find the prescribed exponent 2 of the depth h but found that it is 3/2 instead. I deduced undeniably that it must be 3/2 [Equation (1)] for conical, pyramidal, and wedged indenters [1] [2] [3] [4] [5] instead of prescribed 2. And why should the non-directed indentation modulus ISO-E be a Hook’s law Young’s modulus? It is not! And what about the standard curves of fused quartz and aluminium for the iterations with 3 + 8 free parameters. It was not easy to stand alone with such undeniable objections and all my early publications were repeatedly blocked and rejected by anonymous Reviewers. So, I had to rebuttal and resubmit elsewhere always after long blocking times. These took much too long with very late rejections of anonymous Reviewers (in one case a Reviewer used in his own publication important new content from my rejected publication and I could therefore convince the Editor to immediately publish my rejected paper). As my strict mathematics cannot be disproved, other colleagues tried with inventing quacking “counter-proof” of the exponent 2 on h by starting with the energy-law violation (33.33% of the work should be created from nothing). The physical proof in [2] [3] challenges the common indentation hardness (ISO-ASTM-H = “Fmax/projected hc2”). But the Quackers tried to reintroduce that exponent 2 of “hc” into their question for after a mathematical reformulation series for reinventing “h2” now as answer for the loading curve. Clearly, h2 was put in the question for receiving h2 as the answer. This quacking trick very badly prevented the timely publication of my particularly important very clear publication [10]. It is formulated in the rejection letter and my detailed explaining of the quacking trick [11] from the Reviewer to the Editor of the Swiss Journal Crystals did not help (see more below in this Section). The triple disaster in aviation could thus not be prevented by FAA (Federal Aviation Administration), due to this quacking publication rejection of [10]. So, [10] could only appear in an accepting Journal, appearing unfortunately after the fatal crashing of 3 airliners. This badly needed publication [10] was also before blocked, for more than 2 years by the American Journal Scanning, as the biased Reviewers and Editors waited excessively long with their rejections. And the undoubtedly Reviewer citing [11] tried to disqualify my analytical FN vs h3/2 plot again as “Kaupp-fitting” instead of correctly “Kaupp-plot analysis”, which is much more revealing for the exponent detection than the logarithmic technique. But I have been always convinced that it would help to prevent all types of crashing events and kept on with submitting. It is now also very clear who is jointly responsible for the three fatal airliner crashes that could and would have been avoided by the timely publication of [10]. The FN vs h3/2 plot must since the appearance of [11] correctly be called “Kaupp-plot with linear regression”. It appears that I am still the only natural scientist worldwide, who dares and openly defends the physical FN h3/2 law for cones and pyramids in publications and lectures. And it is the indentation Dichotomists, who maintain and spread inexcusable dangerous to fatal risks worldwide. Conversely, my mathematically correct FN vs h3/2 analyses have additionally the practical advantage of allowing for regressions, and providing the physical hardness (indentation resistance) of all detected polymorphs from the loading curve. That is a world-wide breakthrough (very important in the safety of airliners, see more below, or for the definition of the physical hardness). For example, the authors of [11] would obtain the twinning and structural kinks as I already checked with the plot of their Sapphire and NCAl indentations. And they are invited to check all of that themselves. But they must check and correct their axes descriptions, because these vary enormously in the literature, when compared with dozens of published Sapphire loading curves of others. As I already proved that their figure 2 in [11] is an obviously correct untreated experimental curve, they will find a low force twinning and at higher force a structural kink unsteadiness, each in the regression lines (e.g. by using Excel) and for the calculation of the important normalized transition energies they should use the algebraic formulas that I repeatedly published (at first in [5]) and cited in my more recent publications [12]. Actually, I also need their help for convincing ISO-ASTM to thoroughly correct ISO-ASTM 14577, because they are University Teachers and thus free to do so, unlike the industries that must still agree with ISO-ASTM, due to their certification procedure, for being admitted.
I challenge the continuous violation of the energy law, due to my responsibility towards everyone (daily life and particularly important now due to emerging artificial intelligence AI). And I use the correct analyses for the development of various unprecedented applications of (nano) indentations that trust in pure mathematic rather than believing in more than questionable 3 + 8 free-parameter-iterations. Most important are the now possible rapid detection and localization of phase-transitions with calculation of the transition energy. And it helps to immediately sort out not-experimental loading curves when following h2, or linear plots without anisotropies when these do in fact occur at related work of others. Dangerous phase-transitions create polymorph interfaces with micro-cracks (only seen at 5000-fold enlarged images) at the generated polymorph interfaces. And these are also the proved (and imaged in [10]) nucleation sites (due to polymorph interfaces) for catastrophic crashes upon further mechanical loads at much lower force than nucleation at local defect sites. Catastrophic crashes from these stable micro cracks started only at higher force, but cracks from a point defect required much higher force. Importantly, half a year after the final appearance of my paper [10], when the FAA (Federal Aviation Administration) had a chance to see the images in my more than 2 or 3 years blocked paper, by having a chance to read it. It’s to despair; my publication could have been appeared in USA (Scanning) or Switzerland (Crystals), well before the three catastrophic crashes with all passengers and crew dead! I could only lecture on it and I did so before, but it is forbidden to submit a manuscript to several Journals at the same time. And one cannot judge which Journal chooses biased reviewers who block a publication as long as possible. When FAA had a chance to read [10] (and the following 6 months for the regular check of all airlines), it all at once grounded 250 of the huge airliners for 18 months; because all of these had such micro cracks at their pickle forks (these connect the wings with the fuselage). These till then had passed the unavoidable half yearly safety checks of all airliners, or they remained undetected. The manufacturer and seller of the grounded airliners admitted his overall damaging costs to 100 billion Dollar. But that are well spent 1011 Dollar for the now worldwide safer flying, due to the publication [10], which was unfortunately blocked for more than 2 or 3 years by anonymous reviewers and editors. But these editors ask me now for further manuscripts. It’s their fault and also their responsibility. What a pity that [10] could not appear two years earlier! The micro cracks on the pickle forks were either not searched for, or not judged risky at all, or not seen on the metal surface at the half-year safety checks of all airliners before. I detected them on a transparent surface where it is easier seen and imaged. And my helpful publication could have been appeared well before the three airliner crashes happened with the deplorable life’s losses of all passengers and crew in China, Indian Sea, and Ethiopia (cf the Sections 3.1 and 4).
It is absolutely impossible that one could have imagined such a disaster with the thousands and thousands of published ISO-ASTM hardness and ISO-ASTM modulus data with their violating of the energy-law and impossibility to detect phase-transitions due to the mechanical load upon instrumental indentations! The rather frequent phase-transitions must be detected and energetically characterized by the easiest means of (nano) indentations without iterations. It is impossible to simulate phase-transition onsets and their transition energies even if these would now use h3/2 alone. The detailed understanding is not possible without the chemical and crystallographic aspects, in strict combination with the basic energy law arithmetic.
The risk of phase-transitions for the light-weight airplane materials aluminium or AlTi alloy in comparison to super alloys had already been stressed and warned of in [4]. Their comparably low onset depth and particularly very low endothermic phase-transition energies is frightening. And the variations of the calculated data for aluminium require more precision with more reliable more recent reports. This in connection with our warning from comparably low-force phase-transition and strongly variable reports is responsible for unclear onset and transition energy values of ill characterized aluminium alloys for the airliner light metals [3] [4]. It requires the reanalyzing of the indentations to aluminium alloys and improving these. Also, the report, that pure fcc-aluminium does not form deformation twins at room temperature by mechanical impact but only so interpreted 77K results [13] require scrutiny. It must be rechecked with in-situ techniques and we take it over in this publication.
It is not honest to think that ISO-ASTM hardness H and ISO-ASTM modulus E from indentation would “at least be better than nothing”, despite of their energy-violation. But the finding and taking care for phase transitions with their onsets and transition energies are indispensable. The faulty situation will now become more dangerous with the evolving AI (artificial intelligence) that must not be trained with false data. I therefore must dare to tell that the iterative fittings to both ISO-ASTM standards are unsuitable. That appears important enough for being included here. We consider for example the calibration plot of “aluminium” from the Triboscope Manual p. 60 version of Hysitron Inc. from 1999. We find as their figure 3 an experimental FN vs h indentation curve, termed as indentation onto “aluminium”, that analyzes as FN vs h3/2 parabola (that is with exponent 3/2 even though the whole Manual with all text and all formulas used FN vs h2). Clearly, only h3/2 gives here one linear plot from zero to 1 µm impression depth up to its end at 10 mN load. It is therefore not h2 as claimed exclusively all over the whole Handbook. And now comes the present Author’s experimental proof. Such linearity with h3/2 instead of h2 would be all right for the imaged FN vs h curve. However, that linearity cannot be correct for an indentation onto aluminium, because aluminium experiences at least one kink anisotropy (2 crossing straight lines forming a kink where they connect) within such load region. That is well-known (see also Table 1 and Table 4 below in Section 3.1 and Section 3.3), due to a twinning-transition. Thus, this FN vs h calibration image with iterated ISO-ASTM-H and -Er values is a falsification. It is a loading curve from a different material (but not from aluminium) with its first phase transition kink far below 10 mN loads (Table 1 and Table 4). Apparently, these Authors knew that Berkovich indentations must lead to FN vs h3/2 parabolas from my lectures at their meetings and cooperation, while on the other hand still disputing phase-transitions. So, that is not only dichotomy but also falsification. A corresponding data-plot with h2 provided a continuous curve, not a straight line. So, this is the first example with a standard material of ISO-ASTM enforcement that cannot be reproduced by experiment. And iterations of other material’s “contact height hc and Ahc”, using 3 + 8 free parameters as prescribed by ISO-ASTM-H and -Er for fitting with this standard are completely obsolete. But can aluminium be nevertheless a suitable calibration standard? The answer is NO! Even when the twinning of aluminium occurs at low force and structural transitions remain unnoticed at very high indentation force scales, it does influence the results. Importantly, one must never integrate or extrapolate over unsteadiness’ within a curve. But most indentations are far beyond 10 mN load. So, we had and have to analyze them by stepwise integration. This contradiction with the exponent question is a clear scientific dichotomy (in combination with a falsification). But we must not follow such a deception and look at the Table 1 below. We can only suppose that all involved companies had to present a good looking “standardization curve” for their certification procedure to agree with the ISO-ASTM standards. But still fighting against phase-transitions and twinning is connected to it.
Instrumental indentations may reach forces up to the Newton ranges, some up to 100 N (e.g. the curves in [10] are up to 50 N). The tip-quality is microscopically checked, and we do never iterate, and never integrate over unsteadiness in composed parabolas. All integrations are from kink to kink (the last one is kink to the end) for the different polymorphs in the intersecting branches. One detected such important features only with the linear FN vs h3/2 plot. We could so identify six phase-transitions up to 50 N loads with NaCl, following each other. Clearly, phase-transitions cannot be seen without such “Kaupp-plot” from the experimental FN vs h curves that always look quite continuous. One must analyze FN vs h curves with h3/2 but not with quackingly deduced “h2” [11], and the FN vs h3/2 plot is the in [11] so disdainfully called “Kaupp-plot”. There were numerous internal discussions after my Nanoscope purchase in 1995 and at world-wide lectures, also those that were organized by Hysitron Inc., including the ones at the yearly Hysitron Nanoscope conferences (number1of these was 2010 at the city of Saarbrücken), and including the cooperation’s culmination in a joint publication with one of their Coworkers (U.D. Hangen) [14]. Unimaginably, the black-box indenter computers determine the iterated values (3 + 8 free parameters!) as if these would follow h2, but the analyses follow h3/2, So what? My loading curve analyses from the most cited [15] were reported in [1] [4]. But this standard of ISO-14577-ASTM must no longer be used, because it relies on iterating rather wavy indentation loading curves (either missing force linearity or instability and not considering phase-transition onsets). But, industrial users, as certified by ISO-ASTM, must remain inconvincible believers and do not openly trust in our calculation rules. They cannot change their views and must continue to use the energy law violation with h2 instead of h3/2 and 100% instead of 80% of the applied work for the indentation work (factor 0.8 is true). Thus, ISO-ASTM violate the energy-law and lose the physical indentation hardness as the regression slope k (mN/µm3/2) for pyramidal, conical, and wedged indentations. We see now, University Teachers should have known [or they know] it, but they teach and calculate against (better knowledge). They must have known it after the mathematical deduction of the present Author’s strict FN h3/2 deduction [2] [3]. But they are refusing to use it for new developments (e.g. regression with Excel, detection of phase-transition onsets and energies under load, etc.).
We had been using cube corner and Berkovich indentations onto numerous different materials all from loading curves of believers in and fighters for false h2. But exclusively found h3/2 for experimental loading curves but not for simulated FN-h curves. Nobody admitted the use of the FN vs h3/2-plot for the evaluation of phase-transitions, not to speak of phase-transition energies. But nobody could falsify the physical and mathematical deduction in [2] [3]. Also, the phase-transition analyses from the older loading curves onto aluminium all provide linear FN vs h3/2 plots (all were from believers using h2 and not trusting in calculation rules requiring h3/2). These undoubtedly believed in the false “FN h2” relations with variable reports. According to our precise regression analyses we now prefer the experimental indentation onto aluminium of [16] (their figure 7a) up to 100 mN load, where an additional third high-force phase transition can be safely detected and analyzed Table 1). But the phase-transitions remained undetected by these authors on the basis of the still world-wide believed false “FN h2 relation”. We therefore continue to detect and publish them also now, as the experimental curves always confirm (and must confirm) the linear FN vs h3/2 physical law. All deviations from exponent 3/2 would be experimental error. For example, poorly aligned instrument with not perfectly linear load increase would deviate at the same load for all different materials. And the present calibration standards have either twinning and or structural phase-transitions, so that our correct standard-free analysis has multiple advantages over the ISO-ASTM enforced 14577 techniques. The force linearity of the indentation standards from [15] is not given above 80 or 90 mN loading force. The obvious high quality of the diamond Berkovich and cube-corner indenter side-face flatness is remarkable. But cracks of materials can disturb. Tip rounding is part of the initial effects and corrected by axis-cut Fa together with all surface effects. All deviation from not linear regression without sharp phase-transition kink unsteadiness at sufficient force are faked and we must exclude such “loading curve” as not experimental (as AI will probably not be able to do it by itself, as long as it is world-wide appreciated, or will it by itself sort-out and prefer physical laws?). Nevertheless, the physically false standards ISO-ASTM 14577 were not corrected, even though several ISO representatives were repeatedly informed directly and with my lectures and publications of these physical and mathematical facts, and after my petition for correction of their standard 14577. Only the correct physical data provide important yet unexpected new qualifications of materials, all from the loading curves of indentations. These are most easily available and free from iterations, simulations, or further guesses. And the qualities of aluminium for the standardization of hardness, modulus and from those depending numerous further pseudo “properties” must urgently be replaced by physical ones such as ZerodurR. For example, the physical indentation hardness is not ISO-ASTM 14577-H but the linear regression penetration resistance (slope k of force/depth3/2). One should therefore also stop to use the unphysical exponent 2 on h for incorrect simulations. ISO-ASTM require so, but even the instrument’s Handbook’s standardization curves also confirm now that it must be h3/2 instead of h2 for conical and pyramidal indenters [2].
It was requested that deformation twinning upon indentations of the standard aluminium was never occurring except at very low temperatures. Such statement reads: “Even under extreme conditions, deformation twins have never been observed in coarse-grained aluminium” [13]. We have twinning for fcc-gold (initial flat start) [17] but no initial twinning for fcc-copper for low-load indentation (see [12] and Section 3.3). And the exothermic phase transitions upon indentation onto (100) of copper at 150 K with transition energy of −4.928 and −26.243 mNµm/Δµm must be a very low temperature specialty and not twinning (Table 2 of [12]), which must be studied by in-situ-on-site spectroscopy and diffraction.
We will discuss the 0.486 mNµm/Δµm phase transition of the Berkovich indentation onto (100) of aluminium as twinning in Section 3.2. And initial twinning upon mechanical load disqualifies aluminium further as an iteration standard, it appears now important to study the Berkovich indentations of aluminium in more detail with the physical and algebraically correct analyses for avoiding huge mechanical errors for aluminium as the prevailing component in light-metal alloys for airplanes. We recall the above discussed three recent airliner crashes and 250 airliners that were grounded for 18 months about half a year after [10] had appeared. That concern holds also for all other metallic and non-metallic materials, the mechanical qualities of which were “qualified” by using the unsuitable aluminium standard (when still believing in ISO-ASTM H instead of physical k also called penetration resistance).
2. Experimental and Methods
Our aluminium and fused quartz test samples were from Hysitron Inc., and the polycrystalline gold sample from a commercial goldsmith. We deal here only with pyramidal indentations but the results are also valid for conical indentations. All of the recent strictly mathematically deduced arithmetic formulas are repeatedly printed in [12], so that their repetition appears not necessary here again. All calculations were with a Rebell SC2030 scientific pocket calculator always using all 10 decimals with reasonable shortenings only at the final results. The cited loading curves were plotted as FN vs h3/2 lines by Excel with steepness regression (k-value = penetration resistance = physical indentation hardness [mN/µm3/2]). Also, the axis cut Fa of the linear branches is so obtained. The calculated properties are collected in Table 1 through Table 4. Simulated falsified curves are easily recognized and disregarded if not challenged here. The transition energies are reported per Δµm and mostly not with weighed rule of three corrections for the preceding contribution, for simplifying. But that can be easily done by everybody, if requested. Our analyses confirm that the loading curve published as fig. 1(b) in [18] is in fact a valid experimental (FN = kh3/2 + Fa) Equation (1) curve, which proves that it is purely experimental. However, the authors of [18] wrote on page 3542 of their paper that their indentations would follow their (false) equation “P = Ch2” for their Berkovich indentation. Their fig. 1(b) could thus be used for physical analyses in Table 1 (entry 3) and we included its analysis despite some problems at the lower force (probably surface treatment effects). It shows by similarity reasons that the later phase transition at the 30 mN range of [16] (who was not clear whether a Berkovich or Vickers indenter was used) must have been a Berkovich, due to different side areas and angles that are not equal between these. In the cases of large numbers from µN and nm plots we better calculate with mN and µm units. And as the energetic values are free from exponents after their calculation that required exponent and an integration one can easily change the units when multiplying both the force and the depth units by the same decimal. Thus, [mNµm/Δµm] and [µNnm/Δnm] expressions give equal numbers for the normalized conversion energy values. And different values occur when e.g. mN and nm3/2 are plotted giving mN nm/Δnm that are only more complicated transformed into different units. We choose therefore for the calculations the easier suited unit pairs for force and depth. The regression calculations are absolutely necessary for obtaining precise slopes (these k-values are the penetration resistance = physical indentation hardness), phase-transition onsets, and the axis cut FNa or briefly Fa. Any deviation of it from zero before a phase transition are (and must be) corrected in all of the arithmetic calculations that contain it. But these cannot be published, because they are not materials’ constants but depend on the particular instances of the particular measurement (e.g. material surface preparations and roughness, indenter rounding, zero-finding errors, etc.). The equation for the phase-transition conversion energy is Wtransition = fullWapplied − (Wapplied) and all the equations for the calculation of these values are published in several preceding publications (first in [5] [10], and most recently in [12]. The dimensions of the k-values [Nm−3/2] have to be carefully chosen, as the published depth values might read in nm, or µm units and can due to the exponent not be linearly transformed from each other, which is however possible for the normalized phase-transition conversion energy [mNµm/Δμm]. In Table 2, we stay with our original choice of 2019 with µN and nm units and normalize per ΔµN, whereas we use for Table 1, Table 3, and Table 4 mN and µm units. The conversion energies can be linearly converted if required, but only the force unit can be linearly changed after fixing the depth unit before the exponentiation with 3/2 for k and Wconv. Unfortunately, we still have to live with relative calibration standards.
3. Results and Discussion
3.1. Twinning of Indentation Standard of fcc Metals
A recent paper claims the first twinning of (fcc)-aluminium at 77 K by “high strain rate impact” [13] as a new technique and calls it twinning as deformation-activated recrystallization. But pure fcc metals have been claimed to have difficulties with twinning. But how pure must it be? Conversely, twinning in alloys appears quite common (e.g. [19]).
The (fcc)-copper does not twin upon nanoindentation up to 1 mN, but our search for copper twinning [12] by indentation at room temperature was perhaps incomplete, as the published loading curve ended already at 1 mN load. That should be urgently tested up to 2 mN loading force, but we expect no twinning. Furthermore, the first conversion energy values of 6.5 and 10.2 mNµm/Δµm (entries 2, 4 in Table 2 of [12]) are at least questionable. They could already be structural transitions of copper, and safely structural conversions of copper follow. All of that requires in-situ analyses. It remains the unsolved questions of slipping transformation twinning of copper at room temperature.
A metallic calibration standard must use cheap and very stable crystals in flat polished sheets such as cubic face centered fcc ones with (100) surface. These were known to be stable towards twinning when pure. They must be stable in moist ambient atmosphere. Thus, the choice should be between aluminium or copper that both can be used in ambient atmosphere, because they are protected with oxide layer (Al) or patina (Cu). The oxide layer of aluminium is corundum Al2O3 that forms on a freshly scratched surface within seconds in air and grows within 1 month up to 5 - 10 nm thickness and stays constant thereafter, colorless transparent under ambient conditions. These are only a few highly protecting molecular layers, which is an extremely efficient protection with close to nothing. Conversely, the patina protection of copper consists of copper salts (basic carbonates, sulfates, chlorides) that form very slowly or must be produced by patination. Such patination protection is of course so much inferior that the choice had to be aluminium for a worldwide uniform standard material. The question of twinning and structural phase transitions was not evident for ISO-ASTM, because the normal force versus depth parabolas looks always with continuing shape. Any composition of these parabolas became only evident with the deduction of the physical law for conical and pyramidal indentations Equation (1) [2] and its “Kaupp-plot”, that reveals the phase-transition under load with sharp unsteadiness kinks. All the other mechanical properties of aluminium did not exclude aluminium so that a calibration plate of it goes with every commercial indentation instrument. But we must inform now that there is twinning under load depending on ppm quantities of impurities that cannot be controlled and leads to variable twinning onsets that is different results in time and location.
Deformation activated twinning of not alloyed fcc aluminium was not yet known at room temperature. The only “new twinning path” appeared to be cryogenic “rapid compression application at very low temperature” [13]. Our results with copper nourished the hope that the standard fcc-aluminium might nevertheless twin upon indentation. And we could not understand that the FN vs h3/2 plot for figure 3 in the Hysitron Inc. Handbook of 1999 was a straight line from zero loads up to 10 mN load, which seemed to be no twinning as the only reasonable possibility. But that is a fake. We have a twinning-kink of aluminium at 2 mN load and compared therefore with the measurements of others with 8 and 12 mN kink loads for Al, all with only small phase-transition energies in Table 1. The twinning unsteadiness energies are much smaller than the also found structural phase-transition in Section 3.4. All of that is only visible after the regression analysis. These twinning onsets are far from being reproducible (as these are twinnings) and that disqualifies aluminium as indentation standard, because such twinning standard depends on facilitating impurities at the ppm level content that is not controllable. Weren’t there any calibration specialists? Clearly any twinning disqualifies a material for being a calibration standard, because the onset, and now also the detectable transformation energies, depend on inevitable local lattice faults due to impurities in the ppm ranges. Hysitron Inc. did however not also replace this and the other experimental actually h3/2 providing images in their 1999 Handbook by the also available simulated h2 images and all formulas use “h2” instead of h3/2. As I repeatedly complained that my in 1995 purchased TriboscopeR instrument did presumably not work well, because it provided only FN h3/2 data but not the prescribed “FN h2” ones (a chemist must always check all aspects of the results that he obtains, and so here also the exponent). I thus went with my coworker to the German dealer for an instruction of how to run my purchased instrument, but we had done everything correctly and at home we still obtained h3/2 and I complained again. Furthermore, I lectured at the same place at their workshops, where all other physicist lecturers pretended “h2” (always Berkovich) and fought against the FN h3/2 plot for the exponent detection (with “this happens only with the chemists’ site”). They disdainful called my FN vs h3/2 “Kaupp-plot”, and did not see that it is a much better determination of an exponent than their preferred rather poor logarithmic check. Only the former is connected to a regression analysis and as a surplus it also reveals unsteadiness values for twinning onsets with their (here endothermic) twinning energy. And I lectured at the yearly German indenter-instrument dealer conferences. In the open discussion of my talk at the one in 2000 the leading US Hysitron manager thanked me in the open discussion “for my talk” and donated me a diamond indenter tip in front of all attendees. But he could not say something like you are right, it is h3/2 for cone, pyramid, and wedge. Also in USA, his Company has to be certified by ASTM for admittance with respect to the ISO-ASTM standards. On the other hand, he was in a dilemma, as he cannot deny the infallible calculation rules that I correctly use. But it is a shame that the academia people with its physicists, who are not in a dilemma, believe in history but do not trust in the infallible sacrosanct calculation rules. I must tell this story and I cannot avoid to use these hard words, because it is a severe exaction to a Chemist for using formulas that violate the energy law as has been physically proved in [2] and long before quoted with numerous shown empirically, also with the experimental loading curves despite the vehement claims of physical Colleagues and Reviewers that their indentations “would follow h2”. But only their simulated or manipulated curves were faking with “h2” ! It proves that ISO-ASTM and Hysitron Inc. with the authors of their revised Handbook of 1999 had been known the truth and also knew and know the truth from my yearly lectures at the latter’s working groups, not to speak of my submitted urgent proposals for a revision of ISO 14577, of my important successful regressions. Not to speak of the fakes with the quacking catastrophically claim that the exponent of area a2 for the basic surface of a Vickers indenter (start of the question) would prove after various reformulation equations that the same exponent “h2” must be valid for pointed impression (clearly, the answer already is put into the question).
We must repeat again: This quacking of the reviewers in US (Scanning) and Swiss (Crystals) Journals prevented the early enough publication for years, so that it could not appear before the fatal crashing of 3 huge airliners with all passengers and crew dead, so that these catastrophes could not be avoided. It is for howling with rage: The FAA would certainly more than one year before the first of these crashes in China have grounded the 250 airlines with such pre-cracks at their pickle forks for18 months, as it did half a year after appearance of [10], which was after these crashes. But due to the culpable delay of the acceptance for publication of [10], FAA could only read it with the convincing microscopic images of stable only 1 - 1.5 micrometer long micro-cracks at polymorph interfaces as produced by phase-transition and the crashing from these at higher force. That was after these accidents when [10] had appeared. [10] describes and images with microscopic photos how the “new mechanism of crashing” works. The grounding by FAA occurred suddenly, half a year after the appearance of [10], which is an admirable rapidity, as all airliners are checked within six months. All of the 250 right away grounded airliners exhibited stable microscopic pre-cracks at their pickle forks. We should be worried about biased or faking reviewers and whether AI (alternative intelligence) will be able to sort out all the multi-thousands of energy law violations of the industrial and alas also academia-indentation test claims. Teachers are free in using their hidden (not analyzed) own information that all published experimental conical, pyramidal, wedged (e.g. sensing Vickers) follow FN h3/2 but not “h2”. And despite the printed but not analyzed h3/2 curves one very often reads in the explaining texts always next to them that these curves would confirm “h2” (but not the actual h3/2). We use such analysis with the still scolded “Kaupp-plot” to distinguish experimental from simulated or manipulated curves. The experimental not manipulated ones contain all of the unavoidable sharply occurring phase-transitions and reveal their onsets with calculation of their transition energy. To say it again: only the physically correct FN vs h3/2 relation can detect and calculate them. In the present case the detectable twinning events must not be wiped out by manipulation of the experimental loading curve. Please note the hidden acceptance of my FN vs h3/2 plot by the scientific Dichotomists with its undeniable deduction against also believing in and defending and using of the ISO-ASTM standard procedures.
One had to conclude from the earlier claims with faked curves, that the room-temperature indentations of aluminium would not be complicated by twinning. In the case of fcc copper (equally directed slip-planes and channels) the indentation was probably not deep enough for detecting a probably corresponding twinning upon indentation at room temperature and more than 1 mN load (e.g. up to 10 or 20 mN) upon Berkovich indentation. The two phase-transitions of copper at 150 K and exothermic Wconv = −4.9276 and −26.2426 mNµm/Δµm (Table 2 in [12]) are probably not twinning and must be checked by in-situ spectroscopy or diffraction.
We check now the room temperature indentation onto aluminium in Table 1 and find for the first time the twinning of (100) aluminium at room temperature upon indentation (probably by the “slipping transformation”) in Table 1. The channel structure of the fcc (face centered cubic) metal structures presents the well-arranged channels. Also, the soft gold has that same fcc crystal structure and is compared in Table 1 [20]. But the twin structures require still scrutiny with local in-situ techniques. And that is of systematic importance, because this calibration standard is not aluminium itself but after the twinning onset the twinned aluminum. That is evidenced by two intersecting straight regression lines at low force in the FN vs h3/2 plot. These lines intersect as kink-unsteadiness at the onset-depth and -force, and the endothermic phase-transition energy requirement can be calculated from the regression data (Table 1). This again underlines the fertility of the Kaupp-plot analyses of indentations, leading to otherwise not attainable unexpected applications (notwithstanding the well-shaped indentation instrument calibrations for linear force supply, but please not with 3 + 8 free iteration parameters). The calculated aluminium (100) values from the supplied calibration sheet at low load from Table 1 have their kink load, kink depth, and kink conversion values at quite different values. All are small, as is typical for twinning events (we have here mNµm/Δµm units). Pile-up and sink-in effects as claimed in [21] are not part of the twinning. They belong to the not-penetrating use from the indentation force. The strong variations at entries 1 and 2 are also typical for the twinning of metals: they depend on the always very small amounts of impurities that facilitate the twinning due to isolated crystal lattice distortions with foreign atoms. The sample of entry 1 was the calibration foil as bought from Hysitron Inc. in 1995 with apparently less purity so that the phase-transition occurred at lower force and depth with smaller k1-value and less conversion energy than in entry 2. However, the k2 value of the second branch remains very close to the same as in entry 2, where more energy is required at deeper penetration for the conversion.
Aluminium and copper [12] have the same fcc crystal structure at slightly different a-values (crystal constants). It is nevertheless important to analyze them separately, because next to the geometric factors there are also chemical ones. The impurity level may be comparable at the ppm level. The chemical differences between the different atoms will play their roles at that level and the types of impurities will probably be different. And they do so for the differences in the values of the corresponding mechanical data, cf. [12].
The data of aluminium in Table 1 show, that the kink positions are at rather low load and depth and their poor reproducibility is seen by comparison of entries 1 to 3. It is well understood for twinning events, which were actually not expected for fcc-crystals [13] [19]. But as even ppm contents of impurities within these indentation standards enables the twinning under load they do exist and have their influence also on the strong load structural phase-transition onsets and energies in Section 3.3 below. After the twinning we no longer indent aluminium but the aluminium twin. As the very low impurity level cannot be controlled, there is the spread of the k-values understandable. And we learn from the endothermic transition-energy Wconv that entry 1 had more impurity content than entry 2, and that the most resilient borderline entry 3 had the least concentration or efficiency requiring more force energy for the twinning. That is a very unsuitable situation and one should no longer use such unsuitable calibration standard because we should be interested into the neat material but not into varying results due to twinning of the calibration standard. All materials’
Table 1. Low force twinning of aluminium and gold with berkovich.
Entry |
Metal |
kink load (mN) |
kink depth (µm) |
k values (mN/µm3/2) |
Wconv (mN µm/Δµm) |
References for the original loadings |
1 up to 13 mN |
Al (100) |
1.92598 |
0.38851 |
k1 7.8255 k2 9.6392 |
0.59779 |
[8]
|
2 up to 120 mN |
Al (100) |
8.33375 |
1.10115 |
k1 8.2765 k2 9.6431 |
2.10457 |
[16]
|
3 |
Al (100) |
12.092 32.48 |
1.210.8 |
k1 9.7181 k2 11.619 k3 14.333a |
5.27933a about 13a,b |
[18]
|
4 |
Au (100) AFMc |
0.78922 µN |
0.86017 nm |
k1 0.8702 k2 1.3281 (µN/nm3/2) |
0.18161
(µN nm/Δnm) |
[20] c |
5 |
Au (111) AFMc |
0.58825 µN |
0.61685 nm |
k1 1.2877 k2 1.7355 (µN/nm3/2) |
0.82942
(µN nm/Δnm) |
[20] c |
aThese are from structural phase-transitions, cf Section 3.3; bWe do not publish a more precise value for the conversion energy here, because the FNmax is too close to FNmax; cµN nm units; atomic force microscopic indentation, AFM tip as conical etched tungsten indenter R = 70 nm, see explaining text; it is useful for the twinning detection. The non-equivalence of cones and pyramids [22] is here not relevant.
indentations are matched with 3 + 8 free iteration parameters by ISO-ASTM 14577 to this totally unsuitable standard, which leads to unreliable measurements. Clearly, such unbelievable black-box iteration must be urgently terminated in favor of the correct algebraic analysis with the physical law of Equation (1). We must get rid of the widespread use of the aluminium standard and similarly also of fused quartz, below in Section 3.2. I already complained it, and suggested to choose ZerodurR (a ceramic glass for e.g. cooking plates) as a viable calibration standard. This is the heavy-duty material for nano-positioning, measuring instruments, lenses, telescopes, precision optics space-flight-instruments, high-standard gauges efficiency lasers, flat layers, nanomachines all from nano to macro, cooking plates etc. All very stable and free from temperature extensions. The phase transitions are at 1.5 µm and 250 mN (Berkovich) or 3.3 µm and 250 mN cube corner and the respective transition energies are 87 and 96.5 mNµm/µm. The indentation volumes for Berkovich and Cubecorner are closely identical from 100 to 600 mN load so that we can also check the good quality of these indenters. It was exhaustively checked in [23]. This standard tells the actual force and its linearity so that force-depth will become worldwide uniform when officially standardized and we do not need unimaginable iterations, but will obtain absolute values directly. Unfortunately, the present values are only relatively standardized towards aluminium or fused quartz, not yet with absolute standardization of the applied force and depth. That absolute standardization should be executed by a calibration agency and then transformed to ISO-ASTM for general use.
For the low-force indentations in the 1 mN range, the copper (100) indentations without twinning [12] should be measured and certified against the ZerodurR standard for obtaining absolute values also for these low-force ranges that are required for soft materials.
Structural phase-transitions of aluminium require loading forces above 30 mN loads in Section 3.3. Entry 3 in Table 1 gives a first hint for it.
For revealing the twinning anisotropy of fcc crystals we choose gold (100) and gold (111). The publication [20] provided AFM (atomic force microscopic) indentations, the respective FN vs h loading curves are again analyzed according to the physical law Equation (1). The (100) and (111) (also not distinguishable (110) faces reveal marked anisotropies (entries 4 and 5) in Table 1. The twinning onsets are at smaller forces, penetration depths, k-values, and conversion energies than those of aluminium. Gold is just the softer material (lower physical indentation hardness k, here AFM tip and probably also low for most other indenter tip geometries. The lattice constant of fcc-gold is a = 4.0782 Å. Entry 4 onto (100) has a more facilitated conversion by its vertical a/2 channels under (100). That is a first advantage for the penetration when compared with the (111) face of gold (entry 5) of Table 1, where the channels are much smaller. And the kink depth is indeed 1.4 times deeper, the kink load 1.34 times higher under (100). Thus, the k1 value is 1.5 times smaller and k2 1.3 times smaller under (100). It follows that the resulting endothermic conversion energy is 4.57 times lower under (100) with respect to the (111) face for the twinning. As to the energetic part we have to consider that the displaced and changed materials can be better accommodated under (100). The indenter loses under (100) less of its total kinetic energy that is delivered from the indenter drive. This necessary part of its energy for the phase-transition work is lost at deeper penetration under (100) than under (111). The remaining Wconv value under (100) is thus considerably smaller (4.57-fold!) than under (111). We can thus reliably interpret and explain the twinning anisotropy with the crystal structure and the energetic contributions, when applying the consequences of the physical law Equation (1).
Table 1 shows again that the FN vs h3/2 plots locate any twinning phase-changes, at rather low transformation energies. This is reported by [20] for the (001), (101), and (111) oriented faces. However, the (101) face loading curve was within the scatter range of (111) and is thus not calculated here. The (101) channels of fcc crystals are different but more comparably small as with (111), so that a similar value would have to be expected.
The different gold surfaces were passivated before the indentations with a self-assembled monolayer of hexadecylthiol to avoid strong adhesive interaction with the penetrating electrochemically etched tungsten AFM cone (radius 70 nm) in an IFM (Indentation Force Microscope) [20]. Penetrations were up to 4.5 µN loads and 3 nm depths. Under these conditions there must have been either very sharp conical spikes, or wedges at the indenter tip, because the loading curves provide the FN h3/2 relation as mathematically proved in [2] with excellent regression from zero to 3 nm depth with sharp twinning onsets. That is only possible with a very sharp conical indenter but not with a cone of radius 70 nm. Under these length conditions that cannot be a spherical indentation with the exponent 3/2. And instead of the paraboloid equation of Johnson that has been challenged as faulty in [24] [25], as it lacks the R/h ratio, one would have to consider the paraboloid up to 3 nm depths as a sphere-cap The correct physical equation of which is FN = h3/2π (R/h − 1/3). This is also not a possibility for a linear FN = h3/2 result with almost maximal confidence level. The only possibility to understand these experimental results that remains is that the etched tungsten wire that served as AFM indentation tip, and which had been characterized “by field emission scanning electron microscopy”, had unresolved spikes or a sharp wedge on its surface that could not have been resolved at these few nm ranges. We are thus confident and include these important data of [20] into Table 1. The onset forces are less sensitive to the indenter volume than the other mechanical properties [23].
This invention might be helpful for the further applications of the AFM uses, when nano-cones and or nano-wedges must be preferred. It is a further wake-up of biased believers to stop the false Johnson equation for being ever used again. My infallible mathematically deduced formula Equation (1) that cannot be disproved has long been published and used [2] [3]. It presents numerous very revealing applications of the correct mathematical formula. Fortunately, we prove without knowing the precise nano-tip data the twinning of gold upon conical or wedged indentation and can so far use these published loading curves. A real spherical or parabolic indentation test would require a normal-sized spherical tip [24] [25].
Importantly, [20] could only report rather uncertain “elastic modulus” and “shear-stress” calculations; with enormous error ranges (±10% ranges). But our physical analysis provides important new results on the basis of excellent regressions per loading curve (here R2 = 0.9998):
a) The proof of fcc-gold twinning upon indentation.
b) The physical hardness (penetration resistance k1 and k2 (µN/nm3/2) that are 1.5 times and 1.3times larger at (111) with respect to (100).
c) The phase-transition kinks are located and the endothermic phase transition energies (µNnm/Δnm) for the twinning are 4.57 times larger under the denser packed (111) surface in Table 1.
After tip contact a fully elastic FN-h3/2 relation is obtained. Anisotropy will be also here shown, but the “(101)” indentation trace is within the scatter of the measurements next to (111) so that we cannot reasonably analyze the slightly flatter loading curve, despite the shifting of it by 3 nm. Only the impression image differences are evident by the permanent impression images with pile-up after “permanent” indentations with “repulsive forces”. It used indentions down to 2 or 3 nm depths, onto each of the three surfaces. It produced square (100) holes, triangular (111) holes and roughly hexagonal (with beginning octagon appearance) (110) holes, all with pileup on the indentation periphery. The involved planes are the most prominent slip planes and there is pile-up. These can however also be judged from crystal model programs with tip-rotation for the case of pyramidal indenters [4]. This shows the clear anisotropy between the (111) and (101) surfaces due to the different angles at almost identical loading curves with the still very close to conical AFM indenter tip. The very complicate iterative calculations missed the twinning and they suffer from high error ranges. Our purely closed mathematic analyses with regressions of loading curves, be they fully elastic or plastic, is much easier and leads to more direct material’s qualities due to their high precision with the never before reaching or even thinkable phase-transition energies. And that requires only the not falsified (without flattening!) experimental loading curves. Structural phase-transitions could not be found at these low-force indentations onto gold. But it appears not impossible to find such phase-transitions of gold at sufficiently high indentation loads. For example, the fcc to hpc phase-transition of gold is predicted to require a hydrostatic stress of 100 GPa and the transformed material is more compliant according to the theoretical analysis but without providing a loading curve [26]. Experimentally seen was bcc-gold formation starting from 162 GPa and complete between 377 and 690 GPa [27]. That is hydrostatic verification of this polymorph without an indentation loading curve and would require verification by such experiments, where we have much shear-force.
We published in 2004 [17] a steep short linear start of a sphero-conical indention onto gold from a goldsmith, ending at about 2 mN force and about 100 nm depth. This might indicate that this started with twinned gold that exothermic produced plain gold that proceeded linear up to 10 mN load. It is however not precise enough for a mathematical analysis. We rather needed now the 5 µN scans with etched 100 µm tungsten wire cones (radius 70 nm) and AFM (atomic force microscopy) indentation onto crystalline fcc gold [20] for the search and characterization of gold twinning with the Kaupp-plot. A thorough description of AFM tips is described in [23]. And as expected, we find anisotropy also for the endothermic twinning on (100) at very low load with endothermic transition energy.
We could even use it for the analysis of indentations onto (111) and (110) of (fcc) gold from [20]. The different crystal packing under these lattices is both dense. But the angles of the channels with respect to the indenter angle are different [12]. The transformation energies depend on it. That was and is, of course, totally unthinkable by the enforced 11 free-parameters (!) iterations to match with unsuitable standards and with the violation of the energy-law. My absolute algebraic technique is free from an indentation standard. To say it again: the production of work can never be achieved with zero energy supply [3]. But such silly zero energy claims for work are still multi-thousand-fold claimed and used. Sorry that I must say it again and again.
3.2. The Twinning of Fused Quartz and α-Quartz upon Indentation, New Insights
The most suggested and used nanoindentation standard is fused quartz. Their flat plates consist of amorphous SiO2, which yields endothermic phase-transition unsteadiness under indentation load, which is however a disadvantage for being a standard. But ISO-ASTM 14577 did not know Equation (1) and do still refuse to use it.Table 2 contains the phase-transition information together with crystalline quartz for additionally sorting out the false claims of [15] that are part of the ISO-ASTM 14577-standard, both with the introduction of inconceivable energy-law violation. That is only possible based on Equation (1) because both forms of quartz behave totally different with their twinning behaviors and with quite different onset forces and onset depths. As poor reproducibility is typical for twinning, and fused quartz twins, it can also not be a reliable standard. So, we still have to live with indentation standards, which will become a severe problem in the wake of AI (artificial intelligence) uses. It is well-known that twinning is facilitated by hardly controllable impurities at the ppm concentration range. But conversely, we compare with the strict physical law of Equation (1) algebra for conical, pyramidal and wedged force vs depth plots linearly regressed indentation data from indentations loading curves, as analyzed with Equation (1). Again, the axis cut Fa corrects for any initial effects and it is part of the different slope after phase-transition onsets. Such phase-transition of the standard is clearly part of the standardization but it is as yet variable and not known. But nevertheless, the iteration procedure uses 3 + 8 free parameters for
Table 2. Physical parameters from the cube corner indentations onto four different surfaces of α-quartz (rock crystal) up to 5000 µN load [5].
Entry and (hkl) |
k1 µN/nm3/2 |
k2 µN/nm3/2 |
FNkink (µN) |
hkink (nm) |
Wapplied/ 5000 (µNnm/µN) |
Wconv (µN nm/ΔµN) |
1 (011) |
2.5443 |
1.861 |
2097.594 |
85.7560 |
97.6134 |
−15.744 |
2 (010) |
2.1574 |
1.717 |
2237.789 |
105.8103 |
101.3803 |
−11.048 |
3 (1 − 10) |
2.2037 |
1.648 |
2264.184 |
101.5669 |
104.5936 |
−14.663 |
4 (101) |
2.2147 |
1.677 |
2241.625 |
100.3592 |
103.4883 |
−14.032 |
adjusting to the loading curve of fused quartz, whatever its twinning transition onsets are. One is not taking into account of it and lives with very broad worldwide error limits, with overwhelming systematic ones. But it is unscientific to argue that poor standards are tolerable if we also accept energy law violation and unimaginable double iterations with 3 and 8 free parameters, as adjusted by the instrument computer. That is as yet done by iterating and storing H and Er to give E values and ISO-ASTM even call the latter falsely “Young’s moduli”. That is far from being scientific.
We must again deal with the twining situation, the instruments’ calibration and the detection of faked reports, as in Section 3.1. We can do it correctly, that is exclusively by regression and arithmetic calculation of physical quantities including the phase-transition energies. What we need and do not have after the proof of quartz’ twinning is only an improved standard for checking on-site every instrument’s linear force generation. For that purpose, we again recommend a ceramic such as ZerodurR, which had already been checked in great detail [23]. Clearly, that standard must still be independently fixed for that purpose. Fortunately, Equation (1) is an undeniable physical law [2] [3], even though these publications were for two years blocked before they could appear in press. These were finally only published, because I had complained at the editor that his anonymous reviewer had used some details in his publication that he could only know from my rejected paper. I have and had been using the correct exponent 3/2 on h and profited from our also experimentally found FN vs h3/2 relation for the one-point and wedged indentations upon all the different types of material. That was not everywhere liked, but I developed automatically unprecedented as bargain the detection of phase-transitions. These produce repeatable sharp unsteadiness kinks (see Equation (1)). One can precisely obtain their onsets (force and depth3/2) by linear regression and detect anisotropies with highest precision. The decision to use fused quartz with an early and variable phase-transition as an indentation standard has been a bad mistake of ISO-ASTM 14577. It still further falsifies thousands of ISO-ASTM indentation calibrations. And the appearance of [2] [3] with the deduction of the physical law (therefore 3/2 and not “1.5”) in the form of Equation (1) had been blocked for years by anonymous reviewers, who tried with variable exponents. But all deviations from the exponent 3/2 on h are experimental error. Unavoidable tip rounding is corrected out together with all other surface effects by the axis cut Fa in the formulas that have been multiply published (most recently in [12]). These must therefore not be repeated here.
For the better understanding of the dilemma with quartz we remind that all experts in crystallography know that α-quartz is almost inevitably covered by twins, and that fused quartz is amorphous. Thus, we must distinguish any twins that are already present on the material surface, or that are formed upon the indentation. Certainly, twinning is a process at lower energy than a structural change and that is in accord with the data. Any such iteration to an uncontrolled changing onset, changes the property values from the unbearable exaction of natural scientists. But most people did not believe in what we told in [2] [3] [7] [8] and numerous of them are still Dichotomists. They still tried and try to stay with the false ISO-ASTM formulas.
Rock crystals α-quartz (P3121) has a well-known twin surface all around. Indentation upon 4 different crystal surfaces (Table 2) leads to strongly exothermic initial transformation from the twinning regions to neat α-quartz [5]. We normalized the conversion energies in 2019 per µN load. They vary between −11.048 (010) and −15.744 (011) [µNnm/µN] at 2237.8 and 2027.6 µN loads. The other two faces are in between. It is surprising that so high-energy twins survive so obstinate. One indents onto twinned α-quartz up to the kink at 100 nm depth to see pure α-quartz at first after the de-twinning and new endothermic twinning occurs at higher force. We used 4 different dry polished crystal surfaces but that did apparently not remove the twinned surface layer on them. The exothermic transition energies suggest their presence and we see that the silicon-twins on the different faces lose their different energies relative to neat silicon faces. That is a clear twinning anisotropy that should also be spectroscopically clarified. Crystallographic, the packing differences of neat α-quartz are well known. The visual analyses of the imaged crystal packing projections in [5] show: less energy is obtained at smaller channels of the neat crystal, upon pressure application, as expected already at this restricted level. And more transition energy is obtained at the faces over the largest channels. And intermediate channels give the intermediate transition energy. This has been judged when the crystal presents larger channels in more directions upon cube corner indentation. A complete computer analysis would compare it with more detail by calculating all directions and also with respect to the cube corner angle directions that penetrates vertically. But we can use the preliminary content with this short crystal structure analysis. But we learn, there are also exothermic conversions of twin structures with marked anisotropies.
Fused quartz behaves totally different from α-quartz with respect to twinning. It is free from twin layers, which would appear favorable for a standard, but it also undergoes twinning upon indentation (Table 3) and that at much lower loading force (2 mN region) and with much lower phase-transition energy (0.2 to 0.6 mN µm/Δµm) range, which suggest that these might have a different structure than the ones of α-quartz. Clearly, a viable indentation standard must be free from twinning. Needed is everywhere an official onsite standard for controlling the applied force values and their linearity up to high loads. But that is neither fused quartz nor aluminium. I repeat my suggestion from Section 3.1 to use ZerodurR (for hot-plates and heavy-duty applications) as such standard that had been tested for Berkovich and cube corner indentations with very high phase-transition onset forces and depths [23]. It compares Berkovich and cube corner up to very high loads with a single late phase-transition in all respects. And we also revealed in this paper the exothermic start of the indentation upon α-quartz with twinned surface layers at rather low forces in [7]. But all our then (2005) in [7] published H and Er values are now again retracted. This paper found but limited interest, because it still used in 2005 the ISO-ASTM formalism and its parameters and looked for improved ones. It started with linear FN vs h3/2, found phase-transition kinks, and challenged fused quartz as standard, and excluded h2. But we did not have the regression analyses at that time. The fused quartz indentation of the most cited reference paper [15] is actually a repetition of a soda lime glass indentation. And the extreme errors when a chosen standard for instrument calibration with two 3 + 8 free parameters iterations (these are rapidly performed by the software of the commercial instruments and thus not seen) used to an always strongly variably twinning material that must be indented for being matched. So, these unknown changing properties of the unstable standard are transformed into the material’s properties for all so indented materials. Clearly, everything can be matched by such a technique. This black-box behavior also helps for the longevity of the rapidly aging fused quartz standard, because the 3 + 8 free parameter iterations keep on matching, irrespective of its own changing properties. There is the often-occurring encouragement for performing hundreds of indentations per sample for statistics purposes, but no offer for a frequent replacing of the standard fused quartz plates. We thus used the mathematic technique with the Kaupp plot that does not require such type of standardization, but only a well-aligned indentation instrument. And I just told above, how that can be easily achieved with a ZerodurR indentation. We found force jumps (non-linearity) in [15] of quartz with unproved (001) surface (that must be from another material), “fused quartz”, sapphire, aluminium, and tungsten; all with wavy plots providing unclear kinks in the “Kaupp-plot”. Such additional pseudo-kinks were not found from the indentations of such materials by others. The reason for these wavy plots from the curves in [15] could have been sudden segmental force jumps, or alternatively a Berkovich indenter with defective side protrusions. These errors are for the five “reference materials”. Only their figure 6 for sodium lime glass in [15] does not provide additional kinks at the same loading range. Clearly, this indentation must have profited from a new alignment of the force linearity or by changing to a reliable Berkovich indenter. The final nominal loading force was 120 mN.
The poor reproducibility of the endothermic twinning is now confirmed in Table 3. The rapid aging of the poor “standard” fused quartz is thus very clear. The twinning of fused quartz is revealed starting from the very short initial effect to a pronounced transition onset kink. But the varying of such unsteadiness onset of the indentation parabola clearly confirms the twinning and totally disqualifies fused quartz as a fitting standard. Only the unprecedented physical and mathematical correct analyses open the possibility for using the energetics of the indentation for the phase-transition detection and the calculations of polymorph qualities, including the conversion energies. Table 3 shows: the kink load of the entries 1 - 3 for Berkovich is virtually unchanged. But entries 4 and 5 with data from the most cited publication [15] are most questionably: The conversion energy shows that entry 4 cannot be fused or crystalline quartz and entry 5 cannot be fused quartz with such large conversion energy. Only entry 6 of Table 3 is reliably the result from plain crystalline quartz. Any twin layer was obviously removed, and it provides two structural phase transitions. We might hope that one of these new polymorphs turns out for being coesite(?) These results prove the unprecedented importance and power of the conversion energy calculations. The structures of these new polymorphs should be urgently studied with the now available in-situ techniques. And the requirement of the energy conservation law regard is stressed again [23]. The Reference [15] is still the most cited one in the nanoindentation field, because ISO-ASTM-14577 and thus also their standards took them over and the involved industries are certified and enforced upon it. Their instruments are thus equipped with computer software that automatically performs the 3 + 8 free iterations mostly to fused quartz and aluminium and the corresponding advice. It appears that one was therefore eager to include loading curves for fused quartz (and aluminium) in their Handbooks (e.g. Hysitron and CSIRO), without their checking these in comparison with those of [15] in Table 1 and Table 3. But it was not easy on the basis of the erroneous views of 1992 to see the mistakes that we now discovered. For example, the authors of [15] published in their figure 5 a crystalline material; the conversion energy result is far away from the correct value (entry 6 in Table 3). And our strange conversion energy for the “fused quartz curve in its figure 8 is also far away from both sorts of quartz as that is not a mix-up situation all in a poorly controlled load homogeneity. They wrote “(001) quartz single crystal)” to the caption of their figure 5 without telling who determined, produced and aligned such face after removing of the twin layer and used something else (but not fused quartz). As we did so for Table 2, they could have asked their crystallographers. We must come to that important conclusion with these hard words, because the often-cited authors of [15] point out in their text, that their Figures 5 and 8 would prove “the differences” in hardness and modulus of quartz and fused quarts, as seen by different penetration depth at the same loading force upon Berkovich indentation. Such fake must be urgently retracted, as people were and still are severely misled. And fused quartz has been taken as iteration standard for the multi-thousands of published and tabulated ISO-ASTM hardness and modulus values with increased uncertainty due to the violation of the energy law. I do also not calculate the conversion energy of entry 5, as nobody
Table 3. Twinning of quartz upon Berkovich indentation.
Entry |
Quartz |
Kink load (mN) |
Kink depth (µm) |
k values (mN/µm3/2) |
Wconv (mN µm/Δµm) |
References |
1 |
SiO2 Fused |
1.9796 |
0.1214 |
k1 50.186
k2 61.955 |
0.5648 |
Hysitron, p.59 Manual 1999 |
2 |
SiO2 Fused |
1.69 |
0.113 |
k1 45.734 k2 62.129 |
0.5085 |
Hysitron, p.57 Manual 1999 |
3 |
SiO2 Fused |
2.1014 |
0.989 |
k1 63.864 k2 71.875 |
0.1743 |
CSIRO-UMIS Manual (2003) |
4 |
“SiO2 Fused” [5] shows it is Soda lime glass |
21.8586 46.3582 |
0.41522 0.63834 |
k1 88.225 k2 108.12 k3 125.11? |
19.6a Soda lime glass |
[15]-figure 8, not from fused quartza |
5 |
“SiO2(001)”b is excluded by [5] but guess is impossible |
25.3693 64.2641 |
0.23025 0.59650 |
k1 124.72 k2 168.78 k3 195.04 |
not calculatedb |
[15]-figure 5, unclear material |
6 |
SiO2 (dry) up to 400 mN |
45.7538 200.000 |
0.52833 1.14854 |
k1 122.01 k2 171.63 k3 222.93 |
66.7389c 41.5567c |
[29] without further kinks up to 500 mN loadd |
aThis value is from [5] and the so declared fused quartz is identical with the curve for soda lime glass; b”SiO2 (001)” was mixed up with something else (cf [5]), but we must not guess what it was; cStructural phase-transition values; dFrom crystalline quartz without a twin-layer.
knows, which of the entries 4 or 5 in Table 3 are for poor not reasonably aligned instrument. Furthermore, every scientist who does not read my correct publications will continue being misled and believe that there are only minor differences between amorphous and crystalline quartz. Also, the present Author was in 2006 for a short time misguided by such fake in the most cited publication [15].
Unfortunately, I must discuss here how that still works until recently (2023) culminating in an inexcusable way of scientific fraud and extremely strange behavior. I was at first extremely happy when I found the open access paper [28] with indentation onto crystalline quartz up to 400 mN load using a Berkovich indenter. I immediately copied their loading curves and chose for analysis the last curve each that could be sufficiently separated from 2 (up to 100 mN) and 6 (up to 400 mN load) close together presented loading curves. The text was as usual extensively outlined with almost all formulas of [15] and thus ISO-ASTM14577. They call it “Oliver-Pharr model” [15] and list 6 of the necessary parameters there from as the “indentation footprint” with the iterations. These are used for all calculations depending on the energy-law violation of normal force versus depth2. Their aim is the search and characterization of polymorphs with AFM and confocal Raman microscopy that should be correlated with structural properties. I thus expected experimental relation to the loading curves, but only the unloading curves and pop-outs are discussed. But how should it work without the FN vs h3/2 plots? In fact, despite enormous discussions of Raman shifts and their splitting, the results remain very uncertain. I hurried and hoped to perform my plot for telling these authors all of them to be expected onset forces that would be necessary for the detection of the polymorph’s Raman spectra. After all, these authors concluded only that there were no coesite peaks, but probably the formation of a “ferroelastic material”, and they discarded amorphization.
Now, what does my FN vs h3/2 analysis from the curves of [28] tell? These printed loading parabolas are not experimental but manipulated curves! Without manipulation these straight lines would exhibit several phase transition kinks, the onset of which cannot be simulated. Their single straight lines with h3/2 are from constructed h3/2 parabolas but not from a FN-h2curves (according to their iterations with the ISO-ASTM 14577 technique). Clearly, these authors do no longer accept the violation of the energy-law (unfortunately without citing [2] and [3]) and know that all experimental loading curves also those of prominent fighters for h2 analyze and must analyze with h3/2. Or they must have heard from our Equation (1) that cannot be negotiated as a physical law. But on the other hand, they clearly violate with their iterations to support [15] and cite a long list of the believers in h2 and do not check the exponent of their experimental curves with what is disdained as “Kaupp-plot”. So, they decided for scientific dichotomy. They drew several h3/2 parabolas as loading curves and wrote as “indentation footprint” a list of 6 iteration parameters with error limits “according to the Oliver-Pharr model” (that is [15], which is taken so by ISO-ASTM). They published hf, hc, Ac, hs, B, and m values all with mean errors for 5 different loads as obtained from the iteration with 3 + 8 free parameters. And they restricted their discussion to 6 parallel unloading drawings all with equal “pop-out” at 10 mN residual load, and all with 239 nm length. These are all on one line with all entries to and exits from it. However, it is well-known that pop-outs (also pop-ins) are instrumental or environmental disturbances. And both are rare events to be avoided. And they never indicate phase-transitions. These drawings are freely constructed and can never so occur experimentally. It does not help that these authors need support for a “sharply appearing new denser phase” from their Raman spectra. Again: Pop-outs do not indicate phase-transitions. These authors might have successfully pleased certain reviewers and editors. But they abused the physical law of Equation (1) with faked h3/2 parabolas, which is severe fraud. One may really ask, why could such a faking paper be published? Able reviewers and editors should at least have required cutting out the faked indentation part of the paper, if they were convinced of the confocal Raman part of the paper.
The easiest way for obtaining such close to h3/2 curves despite actually present phase-transitions that analyze with kink unsteadiness (that is their removal from the experimental load-depth curves) would be to run an everywhere available flattening program over it until almost an uncomposed FN vs h3/2 curve is obtained. But such action is a strongly forbidden falsification. But why are faking people interested in wiping out the so important phase-transition information? They would obviously try to stay on both sides of their dichotomy: At a) they want to gain positive reviews from leading reviewers, and at b) they will also be able to state that they already knew of the physical law that is Equation (1) (only not it’s various consequences and applications). But in [28] we have pure fraud that made their publication for all time totally worthless. Even worse, they became convicted of their data manipulation. Clearly, these authors were totally misled by their still believing in [15], which does violate the energy law [2] [3] [8]. However, they obviously thought that they should better also try to not violate the energy law by using a devious or sneaky back stage. But such self-made dilemma must never be solved with severe fraud. Physical laws and calculation rules cannot be negotiated. The strictness of my deduced physical law Equation (1) is evident, so that the frauds of [28] could be easily detected. Good and reliable results on dry and wet quartz provided [29]. Dichotomists loose the straight possibilities of the correct analyses (e.g. finding phase transition onsets with calculation of the conversion energies, or indenter geometry influences, etc.) But it is not easy to recognize the faking from the very complicated text of [28]. It’s a pity that we do still not have more viable experimental loading curves up to 100 mN and 400 mN load of α-quartz upon different faces with or without their twin-covers. But Table 2 above shows the effect of the twin cover.
The paper [5] is certainly less frequently read and cited, because my mathematically founded physical law Equation (1) [2] was widely available by my worldwide lectures and after the papers [2] and [3] could finally appear. Due to its importance, we repeated Table 2 again with the calculated transition energies. It shows the enormous algebraic superiority of the correct physical-mathematical analysis based on the physical law of Equation (1) over the widespread iterative and energy-law violating procedure. Why is it still only the present author who dares to speak that out? At least all German university professors are free in their research and teaching of science, according to the German Constitutional Law from May 23, 1949, § 5(3). He or she must thus not agree to any dangerous and insecure ISO-ASTM standards. And the eye-catching energy-law violation is extremely evident: even any wood cleaver does not only penetrate his axe down, but the axe also separates the wood sidewise into parts and the angle of the axe counts for its efficiency. I therefore urge to use the truth for easiest detection of phase-transitions with their onsets and conversion energies. The risk of polymorph interfaces for crashing of light-weight airliner alloys has been discussed in Chapter 3.1.
Table 3 reveals several interesting unexpected results. Their twinning of fused quartz is endothermic and the necessary energy is provided by the applied pressure. The three leading loading curves (entry1 through 3) of Table 3 for fused quartz’ physical indentation hardness k1 and k2 and for the resulting conversion-energies give too different values for a standard of the physical hardness and the conversion energy, even though one suspects that the impurity contents were the same but not the age and the number of pre-indents of the test sample. The considerably larger differences between the Handbook curves of a different company (entries 1 through 3 of Table 3) are at first glance the result of different miniscule impurities or different concentrations of them. Again, such twinning makes the prescribed standard fused quartz totally unsuitable. The corresponding entries 4 and 5 require highest alertness with the basic publication for the construction of the ISO-ASTM 14577 standards. Unfortunately, industries must obey their certification procedures that are against the physical law of Equation (1). It is described in the text and captions of Table 3 why these are invalid and impossible. Entry 6 in Table 3 refers to reliable phase-transition energies of neat α-quartz without twin layer. Twin-layers on 4 different surface indentations upon rock crystal are listed above in Table 2. In these cases, we obtain from the beginning at first exothermic transitions. It is long known that twins are covering such surfaces. And for fused quartz we expect endothermic structural phase transition only at much higher indentation force, in analogy to soda lime glass. For α-quartz indentation (entry 6 in Table 3, obviously after removal of the twin layer) we have only high-load structural phase-transitions. Nevertheless, fused quartz remained the most used indentation standard, and it is thus used for the iterations of loading curves with 3 + 8 free parameters for the so-called contact depth of the ISO-ASTM 14577 procedure. The twinning onsets of fused quartz (entries 1 - 3 in Table 3) are around 2 mN load and 1 µm depth (or alternately of aluminium entries 1 - 3 in Table 1 of 2-12 mN and 0.4 - 2 µm) are present in all such indentations. That clearly characterizes them as twinning transitions. So, these materials are by no means valid calibration standards. Getting along with that by using 3 + 8 free iteration parameter iterations should never be enforced to the certification of instrument builders by ISO-ASTM. The structural phase-transitions of α-quartz are very much larger with 67 and 42 mNµm/Δµm when compared to the twinnings. The application of the brand-new in-situ local spectroscopy with the “PI cryoR equipment” of Hysitron (cf. [12]), which can also be used at room temperature, should urgently check, whether coesite formation is one of these transitions.
We have in Table 3 the calculations from the two different Handbooks that give, as experimental curves, linear FN vs h3/2 plots. That supports the author’s still often insulted plot according to Equation (1). But the present author is a scientist who stays highly responsible to correct science, even though that still requires highest fortitude. But his bargain is completely unprecedented new scientific knowledge and development that helps to bear public harm.
A comparison of fused quartz with aluminum is now required. A positive result from the publications [21] and [30] is again the presentation of really experimental (untreated) force-depth loading curves, because they follow the physically correct FN = kh3/2 law Equation (1) [2]. That gives straight lines with a kink for the calculation of a phase-transition for the calculation of the phase-transition work. While only [21] prints and discusses initial treatments but without spectroscopic surface analyses there might be chemical surface changes. And this might also make the aluminium surface hydrophilic. It is well known (cf. quartz twinning), that the twinning is very sensitive to facilitating effects of the surface qualities that might be changing between different phases, and impurities (e.g. oxides for aluminium and also hydroxides and hydrates of SiO2). In particular, the initial “pop-ins” of [21] might derive from its electro-polishing in HClO4 solution and washings with water and ethanol. Conversely, [30] tells only “electro polishing”. But an axis cut correction is always part of all types of initial effects. We thus don’t see in Table 1 huge variations in the k1 and k2 values. But these authors did not realize it and published unphysical ISO-hardness and indentation modulus. Only the present author analyzed their loading curves physically and he found for the first time the new detection of an aluminium twinning by the low force (10 mN) indentations that got eventually lost in the previous 100 mN indentations within the initial effects. Such twinning at low force and phase-transition energy and its detection for aluminium is a real breakthrough, as it could yet only be detected at 77 K with application of strong force [13]. Special effects at 77 K have been discussed in [31]
When the neat α-quartz indentation was loaded with rather high loads, two consecutive structural phase-transitions occurred at 46 and 200 mN endothermic with 66 and 41.56 mNµm/Δµm with entry 6 of Table 3. That indentation urgently requires the use of the in-situ PI cryoR equipment for structure elucidations. This will reveal whether the as yet never found coesite (with still four-fold Si-O coordination) can be generated by indentation upon α-quartz.
A much higher energetic quartz modification is the tetragonal (P42/mm) SiO2 polymorph stishovite (extraterrestrial from a Martian meteorite or from Moon or from nuclear explosion remains, or even in some diamonds and now also synthetically produced) with six-fold Si-O coordination. The Kaupp-plot of the Berkovich indentation onto (110) of stishovite (after pop-in repairs) provided two further six-fold coordinated polymorphs, the orthorhombic seifertite (Pbcn) and the monoclinic post-seifertite (P21/c). A still further polymorph (beyond the maximal force of 23 mN) is only calculated. These high energy modifications do not give up their six-fold coordination of stishovite (density 4.287 g/cm3) for transforming down to four-coordinated coesite (X-ray density 2.92 g/cm3) that would be exothermic. But stishovite retains its very high density and transforms at higher forces endotherm to polymorphs with practically unchanged further density changes [32]. Its first endothermic transformation is into orthorhombic seifertite (Pbcn) and from there it proceeds to post-seifertite (Pb2n) again by keeping with the six-fold Si-O coordination. That is well studied and analyzed with phase-transition energies of 1.77 and 6.14 mNnm/Δnm in [33]. Also, SiO2 (P21/c) was identified in a meteorite, and might be found by indentation at higher temperature. The highly energetic six-coordinated polymorphs of SiO2 do on compression not exothermally form the various more stable polymorphs coesite and none of the further lower-energetic polymorphs all with four-coordination and thus much lower density.
3.3. The Structural Phase-Transitions of Aluminium and Copper
We recall our indentation curve analysis of the heavy-duty Aluminium7075 alloy with 89.66% Al, 5.6 % Zn and 7 further minor components that starts at room temperature with endothermic phase-transition energy of 3.4566 mNµm/Δµm. And such twinning energy does not significantly change at temperatures up to 170˚C [31]. But at 77K and 22.3N load it switched from +0.7123 to enormous exothermic −14.672 Nµm/Δµm, which gives with the rule of three for only counting the negative part −54.363 Nµm/Δµm exothermic phase-transition energy, and the production of so much super stable polymorphs far beyond thermodynamic rules has been amply discussed in [31]. This epochal news deserves much further investigations, as do the further low temperature indentations of pure aluminium at 77K that require also in-situ analysis rather than heating to room temperature followed by Raman spectroscopy showing decomposition to polymers.
The metals aluminium and copper crystallize in space group (Fm-3m) with the face-centered cubic (fcc) structure. Their respective crystal constants (a-values) are aAl = 4.0495 Å and aCu = 3.6149 Å, which is not excessively different. We thus expect similar anisotropies upon Berkovich indentations at different crystal faces. And the reason for these anisotropies have been convincingly explained for copper on the basis of the crystal packing effects with respect to the vertical penetration angle of the three indenter side faces (in the Chapter 3.2.2 of [12]). It includes less penetration onto (100) and deeper penetration onto (111), together with the there from following energetic terms and considerations. The indentations up to 10 mN load of [21] and [30] are at much lower maximal force, as compared to the copper indentions up to 90 mN in [12]. We obtain for (100) with Berkovich only one phase transition at a kink force of rather low 2.66 mN at the k1-value (physical hardness) of 5.945 mN/µm3/2. That compares with 29.791 mN and for k1 18.6 mN/µm3/2 for the first phase-transition of copper in [12]. We furthermore obtain at the Vickers indentation for (100) [23] only one phase-transition with an almost equal kink force of 2.69 mN. But we have at first sight of Table 4 within experimental error no anisotropy, neither at Berkovich, nor at Vickers indentation in the first six entries. That is absolutely impossible for experimental loading curves. These are faked curves from both research groups [21] and [30]. That is totally different from the copper results in [12] with the same fcc crystal structure and further proved by the comparison with the further data in Table 4. Such strongly required correspondence (at a different number scale) must be independent from the type of phase-transition, and such faking cannot be tolerated. One must conclude that the authors of both publications always indented upon the most easily available (100) = (001) (for fcc cubes), but never on (101), and (111). It appears that the indenting personnel was either not sufficiently trained to distinguish the claimed (001), and the skew (101), and (111) faces, or alternatively, that there must have been a strong desire to “support” these “non-anisotropy results”, so that all of these authors missed to cooperate with experts in crystallography for the choosing of the correct surfaces for their indentations in Munich [21] and also in Australia [30]. We must complain here an obvious ill-trust in FEM simulations that can, however not take into account the anisotropic crystal packing influences of crystals (for example h3/2 but not h2, directions and widths of channels, cleavage planes, and indenter angles, etc.). These in combination with the absolutely valid general first energy law dominates the mechanics’ field. They are applied for indentations in [2] [3] [12] [25] [33], but not for the present ISO-ASTM standards or FEM simulations. Importantly, there are no information in both papers [21] and [30] concerning who cut the aluminum crystal with what tool, at what angle with what precision, and who controlled it. That is particularly strange in view of numerous excessively detailed descriptions. The so created inherent claim, that crystal packing would not be responsible for anisotropy of indentations should at least have worried the chief authors. Why did these not protest to the data of their coworkers? They might perhaps feel safe, as anisotropy is also largely not appreciated by others in the field. For example, the authors of [15] write in their 1992-paper on page 1582 that aluminium “is nearly elastically isotropic”, not considering that all faces around the cube are identical at fcc crystals, but not the skew ones. Sorry that we have to remind it here, those cubic crystals are very anisotropic and their packing properties such as (101) and (111) must be considered. Furthermore, ISO-ASTM 14577 calls indentation elastic moduli unpardonably “Young’s moduli”. Correct Young’s moduli are from a 6 × 6 matrix with symmetry reductions, for cubic crystals 3 independent ones. These must be unidirectional measured by using Hook’s law ΔL/L or with ultrasound. Sorry for being enforced to write this down. But that had been already challenged by the present author at least in 2017 with [8]. We must conclude that the authors of [21] and [30] were not aware of having indented three times the same most easily obtainable (100) surface. Also, our own entry 1 in Table 1 and entry 8 in Table 4 support this view. As the published loading curves labeled (110 and (111) were experimental in [21] and [30] it follows, that all of these curves were from indentations onto the (100) face of an aluminium cube. It is impossible to claim a chemical interaction between the diamond indenter and the aluminium or copper, that would wipe out the anisotropy. And the 11.4% smaller crystal constant of aluminium (aAl = 4.05Å; aCu = 3.59Å) can also not establish such a wipe-out. But we must inform of the dangerously faked claims of missing anisotropy for helping the AI to sort out such fakes from consideration.
As the experimental anisotropy upon indentation onto copper could be convincingly interpreted with only packing and energy law considerations [12] the same must be applied to aluminium. It is thus clear that the experimental anisotropies must be at a similar ratio in both cases primarily due to the different crystal constants for the Berkovich indenter. We therefore continue with calculation of only the (100) labeled curves and cancel the not existing values for (110) and (111) that are noted with question marks in Table 4. The comparison with copper in Table 4 would expect changes from copper to aluminium from (100) to (111) (smaller hardness, deeper penetration, and smaller endothermic transition energy) by about 36% and 28% as a first judgment instead of no significant increase. We therefore repeat the copper values from [12] in Table 4. Similar anisotropies are expected between the (100) and (111) faces for the Vickers indentations.
Table 4 challenges the faked false claims of anisotropy absence that must not be used in the just now awaking AI applications. It reveals several interesting topics. The second entry nicely confirms entry 1. The indentation measurements are thus correct. Of higher interest is the comparison of the Berkovich with Vickers indentation (the wedge risk decreases the reproducibility between single items) onto Al (100). For this Vickers indentation onto Al (100) are the k1 value 4.1 % larger, k2 31.2% smaller, the twinning onset hkink 1.7% smaller, the FNkink 2.8% larger and the endothermic Wconv 36.6% larger than for the Berkovich indenter. One remarks from these values that the physical hardness k1 is most related with the phase-transition. One has to consider that, against diverting claims of renowned scientists [15], the penetrated volume of whatever pyramidal, wedged or conical indenter must be identical at the same loading force, again just because of the basic energy law and the volumes with respect to the depth are described in and used in [22] and [23] for further use. Despite the almost identical depths and k1 values (within experimental error) between Berkovich (entry 2) and Vickers (entry 5), the k2 values for twinning are slightly larger for Berkovich and the conversion energy is smaller. That is confirmed by entry 8 in Table 4. There is both the influence of geometry and phase-transition energy Wconv after the twinning. The larger k2 gives a smaller conversion energy Wconv.
Table 4. Berkovich and Vickers indentations onto aluminium and Berkovich indentations onto copper from literature loading curves.
Entry Reported Face |
Kink Depth
(µm) |
Kink Force
(mN) |
k1 (mN/µm3/2) |
k2
(mN/µm3/2) |
Wconv
(mNµm/Δµm |
Reference for experiment curve |
1 AlBerkovich (100) up to 13 mN |
0.3885 |
1.926 |
7.8255 |
9.6392 |
0.5978a |
[8] |
2 AlBerkovich (100) up to 10 mN |
0.5955 |
2.660 |
5.9447 |
11.854 |
0.486 |
[30] |
3 AlBerkovich (101)?b |
---- |
---- |
5.1833?b |
---- |
---- |
[30] |
4 AlBerkovich(111)?b |
---- |
---- |
6.1556?b |
---- |
---- |
[30] |
5 AlVickers (100) |
0.5864 |
2.6896 |
6.2004 |
8.1544c) |
0.767a |
[21] |
6 AlVickers (110)?b |
---- |
---- |
6.0209?b |
---- |
---- |
[21] |
7 AlVickers (111)?b |
---- |
---- |
6.2696?b |
---- |
---- |
[21] |
8 AlBerkovich (100) up to 13 mN |
0.3885 |
1.9260 |
7.8255 |
9.6392 |
0.5978a |
[8] |
9 AlBerkovich (100) up to 100 mN |
1.1011 2.4549 3.9795 |
8.3375 34.387 79.678 |
8.2765 9.6431 11.092 |
9.6431 11.092 11.896 |
2.1046a 6.3973 11.0708 |
[16] |
10 AlBerkovich(100) up to 40 mN |
1.2108 |
12.092 |
9.7181 |
11.619 |
about13 |
[18] |
11 Copper (100) |
1.2325 |
36.9647 |
28.980 |
33.748 |
10.244 |
[12] |
12 Copper (111) |
1.3648 |
29.7906 |
18.641 |
24.149 |
6.502 |
[12] |
aThese repeat the twin values from Table 1 for easier comparison; bWe cannot trust these values, because the actual anisotropy is missing and these are practically equal (experimental error) with the (100) value, which is clearly shown by comparison with the experimental examples and the therefore repeated copper values.
For penetrations of different indenter geometries (always same force = same volume!) it is well known that the deeper indenter indicates a smaller physical indenter hardness k and smaller phase-transition energy Wconv as it occurs within the same applied force.
We have geometrically for the ideal Vickers volume (without wedge)
, and for ideal Berkovich (without apex rounding)
and
[22] [23]. Clearly, equal volume penetration is not equal depth difference of penetration (but with that the Vickers depth calculates to be only 0.0264% deeper). That is too small for safe predictions in the case of aluminium, because the indenters are not ideal. The onset depths are so close that we can confirm that Vickers and Berkovich provide closely the same results (But for the comparison with cubecorner it is the explanation of the huge distinctions, when applicable). However, the indenter must always be mentioned in publications, because of differences between Berkovich and Vickers that arise from differences of the side-face areas and indenter angles. We also see that correctly calculated loading curves of averaged results reproduce these precisely. And please keep in mind that systematic errors must be avoided. They do not at all fall within the rounding of the experimental data values. The conceptual errors with energy-law violation, false exponent and data manipulations introduce huge falsifications, and can lead to catastrophes and mistrust of the iterated or simulated indentation field. So please calculate on a physical basis but do not produce or use iterated indentation data according to the ISO-ASTM 15477 standards.
Differences are seen in the physical hardness values k1 and k2 and thus also in the conversion energy Wconv. When comparing the twin formations of the entries 1, 2 and 5 of Table 4 one must not forget that the calibration standards aluminium and fused quartz are very poor, unsuitable, and unreliable.
Furthermore, the different size and areas of three sided Berkovich and four-sided Vickers and their different slop angles (65.3˚ and 68˚) are responsible for the different values under any crystallographic face differences between their indentations [22] [23]. These interact with the slopes of channels and cleavage planes within the crystal. For example, calibrated ZerodurR should be made the official indentation standard as suggested and cited from [23].
Clearly, the entries 2 and 8 in Table 4 deviates because of the different twinning onsets. So, the entry 8 standard was cleaner at the ppm level or with less efficient impurities than the entry 2 standard, because these impurities enhance the twinning onset. To say it again: Aluminium is an unreliable standard.
The entries 3 through 7 in Table 4 do not show the required anisotropy outside the experimental error for twinning onsets and in comparison, with the entries in case of copper with the same space group (corresponding packing). Anisotropy is totally missing in the questioned not calculated entries. In fact, only an (100) face was indented in all of these published cases.
The lattice constants of Cu and Al are 3.524 Å and 4.0479 Å, respectively, and their difference is 13%, whereas the tabulated or Web atomic radii have the aluminium atoms’ radius 10% larger than the one of copper atoms. Thus, the widths of the slip-planes and channels are not very close. And their directions are identical. The reason for the anisotropy of copper has been convincingly explained on the crystal structure and energy-law bases in [12] and that need thus not require repetition here for aluminium. (100) in fcc crystals would be advantaged over (111) in both cases. The conversion energies in Table 4 are therefore not calculated for an average with the not printed values of Table 4. Such averaging has been done onsite for the over there existing calibration conditions. Calibrated ZerodurR as standard would improve that situation.
Unfortunately, there exist now these two publications [21] and [30] that deny anisotropy for the indentations onto aluminium, which must be excluded for being used in AI uses. And only the detailed independent reading of [21] and [30] provided the important hints. None of them tells who performed and controlled the sawing of the crystal for obtaining the skew (110) and (111) surfaces into parallel plates and their final precision X-ray checks. The molecular characterization of the electro polished surfaces (but only on (100) certainly removed all oxides onto aluminium. But the washings with water and alcohol rapidly reformed a final layer for an initial effect that we had to correct out with our axis-cut correction of the experimental loading curves. We could thus still detect the twinning transition, thanks to our correction capabilities. But their floppiness ([21] and [30]) with the sample preparation contrasts with the overloading of these papers with very detailed descriptions of photos, theories, simulations, pop-ins, pile-up, slip system directions for (111) Al. But (111) Al was at best available to them by simulation. Very strangely, their table 5 in [30] compares “experimental” versus simulated “Young’s moduli” with impossibly exact matching for (001): 63.18/63.14, for (101): 1.79/1.56, for (111): 75.10/74.98. These cannot be obtained by the poorly matching “tested” and simulated curves in their figure 4. And their so called “experimental” curves do falsely not show the required indentation anisotropy (but the authors could not know how easy it is to blame that). Importantly, [30] tells in its Experimental Setup Section: “The materials used in the nanoindentation tests was single-crystal Al disks with a purity of 99.9999 wt.%, provided by MaTecK. The detailed information of the raw materials is shown in their Table 4.” But their Table 4 shows only (100), (101), (111) under “Orientation”, <0.1˚ under “Orientation Accuracy”, and <10 nm under “Roughness of Surface”. And all samples were “electro-polished”. There is no further information of who sawed the crystal at what angles into parallel plates, or who controlled it, and what were the details of this electro-polishing? Similarly, [21] tells in its Materials and Methods section; “Subsequently, three sections along (111), (110), and (100) lattice planes were cut by a gentle lapping saw.” Who did it at what sawing angle, and who checked the precision of that angle and the parallelism of the plates? And the electro-polishing of [20] was in an aqueous HClO4 bath followed by washings “with distilled water and ethanol”. No further hints whatsoever are provided. Such behaviors are suspect, but only to a well-informed reader, not eventually to AI.
Why did the described faults not provoke protest by the research leaders? The meaning of Miller Indices might be hard to understand by the indentation personal and the chiefs are also in need to rapidly publish as many papers as possible for confirming their simulation efforts, not to speak of the judgments of flattered referees. But the simulation techniques of [30] {CPFEM (crystal plasticity finite element method)} predict the false “non-anisotropy” of aluminium fcc with “extremely high precision”. We prove them here as being wrong and useless efforts that must be totally disregarded. They failed visibly by “predicting” the obviously faked experimental assertions. Also, that should become known as a failure risk to the AI, if not being reported with the full details. We conclude that the project leaders believed in proper experiments of their indentation personal with rather poor experience with Miller indices from both sides. They could therefore not judge that the absolutely necessary anisotropy of their indentations should have been much larger than the experimental error. All the experimental claims, except with (100) Al, in both publications must be withdrawn by these Authors/Editors and the indentations onto (110) and (111) properly repeated for the protection of the AI from false data and for preventing potentially disastrous situations. AI can probably not by itself exclude faked data from its judgments and advices. And we must therefore use these strong words here, as our suggestions will otherwise most likely not be widely acknowledged.
But there are also positive aspects in these papers [21] and [30].
While the imaging and azimuthally rotation statistics should no longer be used for unphysical ISO-hardness and ISO-modulus with angle matrices by [21] [30], the azimuthally fit function of [21] seems to be well-fitting the automatically obtained rotations that provide 1.8% and 2.6% variation (part of the experimental error when not investigated) of indentations, requiring the average of several indentation parameters. But here we had only one, almost certainly the outer (100) cube face, and such effects are small and they do not image the slit planes and channels of the crystal packing that are responsible for the effect. Such azimuthally rotation had been profitably developed in 2019 for the indentation and crystallography of α-iron, though not automatically [4]. Rather the changes of the channel directions at azimuthal angles with respect to the penetrated tip sides’ angle were judged at different azimuthal angles with the help of projected representations of the crystal packing. These more time-consuming studies can be facilitated by the reported automatic fittings. They will be helpful to find the most interesting rotation angles for the visual technique of [4], which provides the direct crystallographic understanding of the effect (why is a certain angle crystallographic favorable and another angle inactive?). The now automatically obtained flat band views are not only valid for error ranges of single measurements, but it is more helpful for explaining, using, and understanding the indentations onto (100) and (110) of bcc α-iron and Fd-3m silicon onto (100) = (10-10) and (010) = (01 - 10) [4], as well as indentations onto (P3121) α-quartz onto (011) = (01-11) and (010) = (01-10) in [5]. That has been started by [4] and [5] with crystal packing imaging of slip planes’ and of channels’ angles with respect to indenter angles for various crystal structures.
All of the published force-depth curves in [21] [30] follow the physically correct FN = kh3/2 law Equation (1) as deduced by [2] (but not cited by [21] [30]. But their ISO-ASTM 14577 hardness and moduli iterations are requiring the disproved “h2”, which is dichotomy behavior, far from serious or decent behavior.
3.4. Some Remarks for Artificial Intelligence and Data Bases
Scientific dichotomy puts an extreme risk to AI. One will have to check AI-advices with huge data-bases that will retain their importance. Data bases must be interpreted by AI and how will artificial intelligence come along with dichotomy? We must therefore use hard and very direct words in this publication to help AI for finding and removing the world-wide scientific dichotomy within the (nano) indentation community. At least it must always ask for the detection of phase transitions under load when it gives advices. No doubt, huge data bases and Wikipedia are the first to be used when asking for the mechanical properties of materials. But these do not present the force vs depth loading curves, or any images in Wikipedia might be not precise enough for the determination of phase-transition onsets and the conversion energies. One must thus also check the original publication. What can be found are physically not correct “H”, “Er”, and on these depending “factors”, but often without FN vs h curves. AI systems will right now, depending on the developers, start in various directions within the (nano) indentation field, because the world-wide dichotomy prevails strongly. However, they will hopefully be constructed in a way that their development of self-supporting advices will be possible to them. AI routines must be endowed to recognize and refuse the obvious errors. Some of these basic ones are told here and in our cited publications. AI must not blindly accept data bases’ entries and learn by efficient training how databases’ faults and fakes can be determined/eliminated. Thus, AI must be trained to perform the well-founded calculations with mathematic solutions. AI’s performance will stay poor, when it is trained on the basis of iterative that is unscientific results. It is therefore important for Scientists to spread the here found and clear-worded errors to the AI developers (e.g. strictly forbid any violation of the energy conservation law). There are numerous hints in the present and our cited previous publications. If so, AI can warn of all fighting against face-transition onset detections under load and the already disclosed further errors and fraud. AI can so ask future anonymous reviewers for not rejecting publications that found and describe these. Such AI advice would have prevented the blocking by Reviewers and Editors of the publication [10] that describes and images how a new found and described crashing mechanism leads to increased crashing probabilities via microcracks that are formed because of phase-transition. That would have prevented three airliner crashes with all inmates deplorably deceased, and 1011 Dollar overall costs of the airplane’s provider. Such dangerous microcracks were available in 250 of such airliners. And so were the warnings in [5] and [10], also providing the arithmetic background. But why not train AI with algebra (calculation rules, including illicit integration over unsteadiness of a loading curve), so that it will be able to use and calculate with such formulas itself? The present difficulties will be that all the ISO-ASTM hardness and modulus (falsely called Young’s modulus) values H and E cannot be transformed into the values of physical hardness k, kink onset-force and -depth, and Wconv. AI must be trained to analyze loading curves (exponent3/2 on h, linear Kaupp-plot) with the mathematical formulas as cited in [12] and several of our earlier publications. At first AI must detect and sort out faked claims, warn from falsified claims and suggest re-indentations onto important materials with the necessary instructions. And it must ask the Dichotomists to correctly analyze their own published experimental loading curves for the sake of daily safety. But AI must never advice on the basis of faulty and faked claims. If AI is asked for H, E and from those derived “data” it must also respond: did you detect or exclude phase-transitions within your loading area? Alternatively, it should ask its clients: did you check for loading curves in the original publication and analyze these for phase-transition onsets and energies with the algebraic formulas? Hopefully, AI will become able soon to undertake such calculations by itself in the future.
4. Conclusions
The unprecedented correct physical and mathematical analyses open the possibility for using the energetics of the indentation for the phase-transition detection and the very precise calculations of phase-transition conversion energies also for distinction of polymorphism at different indented crystal-faces. ISO-ASTM must instantly replace fused quartz and aluminium by an everywhere available ceramic stable and reliable ZerodurR standard [23]. The Zerodur standard’s precise composition and preparation must be worldwide standardized for that purpose. And it must be fully standardized for providing absolute values of force and depth to the very instrument. This paper sorts out faked indentation results that rely upon not experimental and manipulated FN vs h loading curves. It reveals how these can be recognized and that is important, because their use by AI applications and recommendations is dangerous for daily life. The twinning of the most used indentation standards was denied or disregarded, but we proofed it now for the first time for the basic ISO-ASTM 14577 standard aluminium and for fused quartz. The twinning onsets at force and depth depend on impurities that facilitate twinnings at concentrations of the not controllable ppm level. Therefore, present standardization varies locally by different impurities and timely by the dependence on the standard plates’ usage history. As these twinning onsets influence all of the indentation results, these prescribed calibration and indentation results vary from where and when these had been obtained. That must be urgently avoided. Therefore, present structural phase-transition-onsets and thus energies of materials are burdened with an undetected variable systematic error easily surmounting the experimental ones. We thus tested and supposed official certification of a ceramic (unrivalled will be standardized ZerodurR) as indentation standard. Reliable indentation results with sufficiently high precision for the study of anisotropies and their understanding with respect to crystal-structure packing and energy law, rely on the physical law (FN = kh3/2 + Fa) (1). That implies the correct exponent 3/2 on h (but not 2) and exclusively arithmetic calculations with consideration of axis cut. That is amply and multiply published with simple closed arithmetic formulas. It allows unprecedented very precise calculation of twinning energy and structural phase-transition energy with multiple applications. It so avoids all iterations with respect to a varying standard. The calculated phase transition energies are reproducible material’s properties, but in fact only by calibration with twinning standards and thus not providing absolute values of force depth and energy, but only relative to the used standard ones. Both transition type onsets are sharp kink unsteadiness with the possibility for linear regression of the intersecting branches in the plot of Equation (1). Everything is much easier if we do not forget and apply the energy conservation law also for the penetration of the indenter tip with its geometric details, the crystal packing, and also here the basic energy conservation law. We can thus explain and foresee the anisotropies in various directions and between polymorphs. Phase-transitions are never realized by iterations and they had even been denied in favor of “work hardening” approaches, introducing further unbelievable exponential approaches without a practical understanding basis. The next highly rewarding step must be the in-situ spectroscopy or diffraction. That is close below the onset forces, where the maximal concentration of the transformed material is accumulated, and still under pressure. Such instrumentation is now available (e.g. the “in-situ PI cryoR equipment” of Bruker Co.) and its use is emerging and opening important new field of very important research.
False unphysical theories for indentation results must be halted for protecting of the AI from false recommendations. Numerous typical examples are therefore challenged and clearly named. That involves well known laws of physics and opens unprecedented new possibilities. All of that is inherent in experimental indentation loading curves, also in hardly recognizable double mix of materials that belong to the incorrect foundation of instrumental indentations. It is part of and the reason for world-wide scientific dichotomy within the indentation crowd. Authors who still strongly fight for the incredible energy-law violation of false ISO-ASTM standards, handbooks/textbooks, and almost all of perhaps innocent colleagues who are still proclaiming the false “FN vs h2 relation” are Dichotomists. Some of them even tried with quacking “deduction” of h2 for point-indentation [11] by putting h2 in the question for after that reinventing energy-law violating “FN h2”, while their own published loading curves continuously prove the physically correct FN h3/2 as in Equation (1). Unfortunately, even anonymous Reviewers blocked the publication [10] that uncovers the new crash mechanism involving micro-cracks with such very quacking. Such scientific DICHOTOMY is not only horrid but also extremely dangerous. The for years rejection of [10] was well-timed before the 3 in short sequence crashes of huge airliners, so that FAA (Federal Aviation Administration) could not punctually ground the then 253 airliners for 18 months exhibiting micro cracks at the pickle-forks (wing to trunk connections). The sharply intersecting linear branches according to Equation (1) were not acknowledged by the quackery. But it proves that every phase-transition onset creates an interface between two polymorphs interphase, which is a favorable site for the nucleation of micro cracks (1 to 2 µm long) [10] and from these upon further force impact catastrophic crashes. One must thus choose technical materials such, that the maximal expectable forces to the component part must never reach the phase-transition onset force and keep the endothermic transition energy as low as possible. All of that can be measured and calculated. It must be improved by chemical change of the material (the old topic of “strain hardening” denies phase-transitions, leads in the false direction, and is meaningless empty). Fortunately, half a year after [10] had finally appeared in another Journal, the FAA very rapidly (it takes routinely 6 months to check all existing airliners) and right away grounded 250 airliners of that type for 18 months, because these had micro-cracks at their wing to trunk connections, which before did not attract attention. What a despair? It could have been avoided!
We sincerely hope that natural scientists will finally stop “producing” 33.33% not penetrating indentation work with zero energy. And that they stop defining the indentation hardness as “H = normal force/iterated contact depth
”, instead of determining the regressed penetration resistance k = normal force/h3/2 as the plot of the physical law Equation (1). They can find it with their own published experimental loading curves for conical, pyramidal, or wedged indentations, and the indent work/applied work ratio is 80/100 due to h3/2 [3].
5. Outlook
For making an exact physical Indentation Science, it appears urgently important that indentation instrument Builders/Providers and Dichotomists will arrange with ISO and ASTM to install the absolute standardization of ZerodurR by a proper calibration Institute. That is particularly important because of AI. The present publication with its clear claim for absolute (not relative) indentation results paves the way for such arrangements with the call for: “We need now absolute indentation values!” that can and should unite these parties. The aim must be the termination of the continuing disgraceful situation of prescribing and obeying the ISO-ASTM 14577 standards. These are opposing correct physics and are thus (of course) worldwide not “obeyed” by the experimental indentation experiments. Instrument providers have and must use their possibilities to create the algorithms for the automatic calculation of the mathematically deduced formulas’ sequence, as published in [12] and several times before for conical, pyramidal and wedged indentations. That is also necessary for spherical indentations, as deduced and described in [24] and [25]. And all of the unprecedented applications of the previously not foreseen applications and developments with these values will so become unshakable physical facts, but no longer falsely obtained relative guesses. The three described airliner crashes and the FAA-groundings of 250 airliners with microcracks at the pickle forks only after appearance of the blocked [10] (that could have appeared more than one year before these crashes) are horrible results of the existing disregarding of physical truth. The violation of the energy law denied phase-transitions and required the unimaginable iterations instead of detecting the dangerous phase-transition onsets and energies by plotting with the physical exponent 3/2 on h and arithmetic calculations. ISO-ASTM thus missed numerous further unprecedented insights and applications, subject to further developments that are to be foreseen and that could be viewed for more than 20 years by the open-access publications of the present author. The required calculation algorithms of the undeniable published algebraic formulas must replace the present iterating routines. Only these arithmetic calculations will provide the unprecedented absolute values, provided the absolute calibration for new indentations had been performed and that must also be the case for a twinning-less very stable and reliable standard that can presently only be ZerodurR. These new algorithms must become open available to cure the still suffering field of nanoindentation. The Dichotomists can give up their unfavourable characteristic by easy and rapid calculation of their already published loading curves from their archives and tell the absolute values to the corresponding data bases that are asked by AI. All must rely on the absolute standard. As long as the absolute calibration of ZerodurR is not yet available, they can use its present relative value and correct the so obtained results subsequently.