_{1}

^{*}

Spherical indentations that rely on original date are analyzed with the physically correct mathematical formula and its integration that take into account the radius over depth changes upon penetration. Linear plots, phase-transition onsets, energies, and pressures are algebraically obtained for germanium, zinc-oxide and gallium-nitride. There are low pressure phase-transitions that correspond to, or are not resolved by hydrostatic anvil onset pressures. This enables the attribution of polymorph structures, by comparing with known structures from pulsed laser deposition or molecular beam epitaxy and twinning. The spherical indentation is the easiest way for the synthesis and further characterization of polymorphs, now available in pure form under diamond calotte and in contact with their corresponding less dense polymorph. The unprecedented results and new possibilities require loading curves from experimental data. These are now easily distinguished from data that are “fitted” to make them concur with widely used unphysical Johnson’s formula for spheres (“
*P* = (4/3)*h*^{3/2}*R*^{1/2}*E*^{∗}”) not taking care of the
*R/h* variation. Its challenge is indispensable, because its use involves “fitting equations” for making the data concur. These faked reports (no “experimental” data) provide dangerous false moduli and theories. The fitted spherical indentation reports with radii ranging from 4 to 250 μm are identified for PDMS, GaAs, Al, Si, SiC, MgO, and Steel. The detailed analysis reveals characteristic features.

The spherical indentations were not described by H. Hertz, who only deduced that the pressure of a contacting sphere is related to “impact” area^{3/2}, but without any depth (h) upon indentation [_{max} = 20 nm) or from >480 to 50 (for h_{max} = 3.6 µm). Every researcher should have immediately seen that any R/h term is missing in Johnson’s formula and that a cone or a pyramid behaves different from a sphere. The second obvious error is claiming “Young’s modulus” that is a unidirectional property, totally different from an indentation modulus. A very complex “equation for fitting” of the depth values is published as equation (9) in (6):

δ − δ contact = a 0 2 / R { [ 1 + ( 1 − P / P adh ) 1 / 2 / 2 ] } 4 / 3 − 2 a 0 2 / 3 R { [ 1 + ( 1 − P / P adh ) 1 / 2 / 2 ] } 1 / 3

The δ in [_{0} and P_{adh}. This data-falsification is published in [

The loading data of the materials are taken from the published curves that were enlarged to A4 size. The cone depths were checked with h_{cone} = R(1 − sinβ) where β is the half angle of the cone and R the sphere radius. When pop-ins was present these were repaired [_{indent} results from the integrated Formula (2) [

Material | F_{Nkink} (mN) | h_{kink}_{ } (µm) | W_{indent}_{ } (mNµm)_{ } | W_{applied} (mNµm) | full W_{appl} | W_{transition} (mNµm) | Areas: flat/cap πr^{2}/2πRh | p_{transition}_{-onset}_{ }mN/µm^{2} (GPa) |
---|---|---|---|---|---|---|---|---|

Ge^{a}^{)} | 10.7029 | 0.1243 | 0.4756 | 6.80929 | 10.9251 | 4.1158^{b)} | 3.226/3.272 | 2.331/2.298 |

ZnO | 22.3800 | 0.22059 | 2.16661 | 7.54011 | 14.6431 | 7.1030^{c)} | 5.6684/5.8212 | 3.7325/3.6344 |

ZnO | 56.6290 | 0.51716 | 11,0579 | 15.8916 | 26.4440 | 10.552^{d)} | 12,807/13.648 | 4.0438/3.7948 |

GaN | 38.8406 | 0.1455 | 2.12467 | 12.3876 | 22.8151 | 10.428^{e)} | 3.7731/3.8397 | 10.294/10.116 |

GaN | 118.397 | 0.3854 | 24.7506 | 33.8652 | 59.7898 | 25.925^{f)} | 9.7038/10.171 | 12.201/11.641 |

^{a)}Data taken from [^{b)}up to 50 mN; ^{c)}up to 56.6 mN; ^{d)}up to 100 mN; ^{e)}up to 117.5 mN; ^{f)}up to 250 mN.

F Ns = k s π h s 3 / 2 ( R / h s − 1 / 3 ) + F as (1)

W indent = 2 / 3 ⋅ k s π R h s 3 / 2 − 2 / 15 ⋅ k s π R h s 5 / 2 + Δ F as h s (2)

F Npy = k py h py 3 / 2 + F apy (3)

The indices in the Equations (1) (2) (3) are N for normal, s for spherical, a for axis cut when not zero, and py for pyramidal or conical.

The area of the immersed calotte for the onset of the phase-transition pressure calculations is given by its flat surface (πr^{2}) or by its cap surface (2πRh). The radius r is easily obtained by the combination of sinα = (R − h)/R and cosα = R/r when looking at the geometric situation for the penetration of the calotte from the sphere with radius R [

The spherical indentation analysis of germanium [_{N} vs πh^{3/2}(R/h − 1/3) plot and by using Equations (1) and (2). It shows the penetration resistances k_{1} and k_{2} as the slopes for the two phases up to 50 mN load.

The phase-transition onset of _{indent} = 0.4756 mNµm and the transition energy W_{transition} = 4.1157 mNµm, as calculated up to 50 mN load. The transformation pressure is also calculated to give the good correspondence of 2.3 GPa with the anvil pressurizing phase-transition at 2.5 GPa. A nevertheless published trial plot in [_{N} vs h^{3/2} (Equation (3)) for excluding h^{3/2} as prescribed by ISO standards and false Johnston’s formula for spherical indentations gave a convex plot

instead of linearity. A further endothermic phase-transition is already indicated at the end of the second straight line.

The spherical indentation onto ZnO with wurtzite structure follows Equation (1) and it reveals two phase-transitions. The analysis had to be performed after repair [

The spherical indentation onto ZnO exhibits two phase-transitions, distinguishing 3 polymorphs in the force range up to 100 mN load, as revealed with the plot by application of Equation (1) to the published original load-depth data. The inserted linear regression equations are the basis for the calculation of the energetic terms and the pressure data in

The spherical indentation onto a GaN epilayer (R = 4.2 µm) was reported in 2002 [

The more recent spherical indentation onto a single crystal of GaN [

The results with Ge, ZnO, and GaN are compared in

These materials cover maximal loads that are 50 mN for Ge, 100 mN for ZnO, and 250 mN for GaN. More phase-transition onsets are to be expected at higher loads. The force for the first phase-transition onset describes the sensitivity of the materials for their stability with respect to mechanical interactions. It is equally reflected by the sequence of the penetration resistance values k_{1} (physical hardness) in Figures 1-3. The penetration depth values are not in the same sequence and neither so the first indentation work that are required for reaching the transition onset. But the transition work values are for the first and second transition of ZnO and GaN in the same sequence as the F_{Nkink} values. The data reflect the situation of spherical indentations with the same radius covering one or two phase-transitions per sample. The depths are always very low. The results are calculated from 0 to kink, from kink to kink, and from kink to the maximal force. The transition-energy and the onset pressure values can also be calculated for every force of interest, but they cannot be normalized per force as in the pyramidal case. The energy law requires multiplication of the phase-transition F_{Nkink} values with W_{indent}/W_{applied} = F_{Nindent}/F_{Napplied} [

The pressure data of germanium has already been compared with anvil pressure as combined with X-ray diffraction date. The low anvil pressure of 2.5 GPa required enormous effort for being detected and was long disregarded, as outlined and discussed in [_{2}, “using electron cyclotron resonance plasma source to excite high density oxygen plasma with low-ion energy of 10 - 20 eV”. The lattice constant of 4.463 ± 0.015 Å was obtained from the RHEED (high-energy electron diffraction) pattern [

The GaN B4 phase (wurtzite) transforms upon hydrostatic pressurizing at 47 GPa into the GaN B1 phase (rock-salt) [

It appears that after publication of the false Johnson’s formula requiring an F N = 4 / 3 ⋅ h 3 / 2 R 1 / 2 E ∗ relation, the ISO standard 14,577, several textbooks, and publications believed in it (Section 1). A first glance on that Formula (from the beginning in 1985) should have evidenced that it does not take into account the self-evident change of the R/h ratio during penetration. As the experimental data did not concur with the assumed F_{N} − h^{3/2} relation for spheres, Authors did not hesitate to simulate spherical loading curves by using Young’s modulus and Poisson’s ratio of the material with e.g. the JKR procedures [

They so avoided the inefficient formula for spheres from [

The “fitting” of depth data is the creation of fake data for making them obey the falsely prescribed F_{N} vs h^{3/2} relation. That relation is however only valid for cones and pyramids [^{2} = 0.9999 (no phase-transition) [_{N} µ h^{3/2} relation for spherical indentations in [_{N} vs πh^{3/2}(R/h − 1/3) plot according to Equation (1) is applied to fitted spherical indentations onto materials, one obtains concave curves. That is imaged below for GaAs, Al, and Si. It is also typical for all the further analyzed materials in this Section 4 (including the fitted curve for GaN from [

It appears that most published spherical indentations were “fitted” to obey the incorrect Johnson’s formula with its false promise to obtain “Young’s moduli” values that are however incorrect fake values, not to speak of the fact that Young’s moduli are unidirectional moduli. Importantly, we can easily distinguish valid from fitted invalid spherical indentation reports by simply checking their loading curves with plots according to the Equations (1) and (3).

We do not further deal here with the details of the JKR simulations [

The spherical indentation (R = 10 µm) of GaAs in [

Conversely, the application of Equation (3) for cones and pyramids as falsely claimed by Johnson to the data of [

exponent 3/2 on h for fitted spherical indentations. However, the putative exothermic event does not represent the endothermic phase-transitions of GaAs, and also the slopes are simulation and iteration artefacts without any value.

This data check proved the already complained data fitting. It appeared therefore necessary to compare the values in the caption of

As we present strong arguments, we must check whether further fitted spherical indentation exhibit the corresponding behavior.

The analysis of the spherical indentation onto aluminum [_{cone} = 2.93 µm. This is considerably larger than the maximal depth of <1 µm) for the indentation data of pure Al. Again, the analysis with Equation (1) for spheres does not give a straight line but the concave curve of

The trial plot with Equation (3) to the spherical indentation in

Again, the calculations of elastic moduli according to false Johnson’s formula and its discussion are useless and misleading. The data fitting is again safely confirmed with ^{3/2} loading parabola retain only the information that there must be a phase-transition at a similar force, as characterized with a Berkovich indentation.

Data fitting destroys the value of spherical indentation and excludes any use of them. “Fitted” data points must not be told or suggested as being “experimental” ones.

Spherical indentations (R = 8.5 µm) of silicon were published in [_{N} vs h^{3/2} relation for spherical indentations. This must again be severely challenged. Our analysis of the published “experimental” data pair crosses from their

The data-fitting is again additionally secured with the trial Kaupp-plot (F_{N} vs h^{3/2}) in

As shown with GaAs and aluminium, the residual information of exothermic unsteadiness tells only that phase-transitions will be found by Berkovich indentation. The steepness of the preceding lines in

that of the following ones. This seems to be indeed typical for the undue fitting procedure. The genuine silicon phase-transitions are all endothermic: the Berkovich indentation onsets of Si (100) at 4, 15 and 25 (data taken from [

Unfortunately, the more recent spherical indentation with R = 5 µm of [

Datye et al. in [^{3/2} or 3742 (R = 25 µm) mN/µm^{3/2} are obtained with correlation coefficients of 0.9997 or 0.9999, respectively. These do however not describe any materials’ property, but only reflect the fitting efficiency. We nevertheless determined these slopes despite the data-fitting, for provisionally checking the influence of the tip radius influence. Interestingly, despite the fitting treatment of the depths, the ratio of these slopes (3.7) is similar to the ratio of the radii (3.3). It should be further studied whether such a relation holds also for spherical indentations with untreated experimental depth data. The mayor errors of the simulating procedure appear to be the modulus E* as calculated with the false Johnson formula.

The phase-transition pressures of SiC have been calculated to 102 and105 GPa (hexagonal 6H to 1B) or (cubic 3C to 1B = sodium chloride) phase, respectively, and the experimental shock data have them at about 100 GPa [

There seems to be the same data fitting techniques in all of the here analyzed cases, but we still need further analyses with a crystalline oxide.

The authors of [

For rounding up our knowledge of the falsifying effects of data-fitting spherical indentations we also need the analysis of a technical multi-component material. A spherical indentation onto a standard microhardness steel block (500 HV30; H/E = 0.04), using a sphero-conical tip with radius of 7.2 µm [_{cone} = 2.11 µm) appeared appropriate. Actually, the nominal radius of 5 µm ± 6.6 nm (we calculate h_{cone} = 1.46 µm) was increased by a “well-fitting simulation” to 7.2 µm. And the “nominal values of 210 GPa and 0.3 were assumed for Young’s modulus and Poisson’s ratio in all simulations”. The load-depth data were taken from _{cone} value is not surpassed. But the depth values for a certain force are strongly dependent on the tip radius (cf. Section 4.4.). Again, concave curves (not shown here) but not straight lines are obtained both for nominal 5 µm or iterated 7.2 µm radius. These analyses tell that our already multiply complained data-fitting was again performed in [

The unsteadiness onsets in

Very strange is the publication of

absurd. The Authors of [

Furthermore, both simulated curves for the spherical indentation “of a steel standard hardness block (900 HV30 nominal” with H/E = 0.4 in

worthless) with a pyramidal or conical indentation that would give a totally different slope (yet unknown for steel 900 H/E 0.4). It appears rather strange that the paper [

The plot in

A very recent report deals with the spherical indentation of several steels with a ball of radius 250 µm, in e.g. _{N} vs h^{3/2} plot in

A prerequisite for the analysis of spherical indentations is the use of the correctly deduced force-depth relation (Equation (1) that takes into account that the R/h ratio changes strongly during the penetration. Equation (1) describes experimental (not fitted) spherical indentation loading curves. Unfortunately, data-treatment with simulations and fittings are still (2020) used by ISO 14577 prescriptions with the false Johnson Formula (here as an inequation F N ≠ ( 4 / 3 ) h 3 / 2 R 1 / 2 E ∗ ) that does not care for the R/h changes. Typical loading curves from spherical indentations with (untreated) experimental data for Ge, ZnO, and GaN are successfully analyzed. The unprecedented results demonstrate the unexpected wealth of spherical indentations. The plot of the experimental data according to Equation (1) is linear with kinks at the phase-transition onset points (one, or two within the loading ranges). In addition to the onset force and onset pressure one obtains the phase-transition energy. These values are of great value for the rating of the materials’ compliances and for avoiding phase-transitions with their dangerous polymorph interfaces by overloading. These are mayor advances of experimental spherical indentations. The transition onset pressures can be compared with available anvil pressure onsets, because we are close to hydrostatic conditions. In the case of germanium, our calculated onset pressure favorably supports the results of the anvil experiment that had formerly been questioned. It turns out that low-pressure phase-transitions under anvil pressurizing are either not resolved, or too rapidly overrun, or simply overlooked. Our detected polymorphs under the sphere calotte are also reasonably attributed. The most favorable uses of experimental spherical indentations are the expansion of the mechanical characterization of materials and the controlled synthesis of the various polymorphs that is much easier than by any other technique. The polymorphs are located at a most favorable site under the sphere calotte cap, clean and next to their preceding less dense polymorph. That opens new horizons for their structure elucidation by X-ray diffraction and spectroscopy. This should become the method of choice for the characterization of other solid materials with their polymorphs.

Any trust in the historical concepts and formulas is unsuitable and dangerous. Despite their apparently general use, one must strongly reject the false Johnson formula and all connected false theories that neglect the R/h dependency. It should have been seen before by Authors, Reviewers and Editors when looking at the abounding printed circles in most of the relevant papers. There is no excuse when black-box routines in their instruments might have automatically simulated, iterated, and fitted. Furthermore, the technical users who apply the JKR technique for the evaluation of adhesion properties (cf [

We finally state that valid reported spherical indentations are very useful for complementing the highly demanding and less sensitive hydrostatic pressurizing experiments. They reveal also the lower-force phase-transition pressures that might have been hydrostatically overlooked under the anvil. It will be possible now to recognize and stop the widespread data falsifying techniques not only for regaining the scientific reputation in the field of indentations. Peer Reviewers must no longer support data falsifying fake papers. It is not enough when historical authors are cited with their paper titles, but without referring to their antiquated content, or when black-box manipulations produce exact coherency with erroneous equations. The risk of false technical materials’ properties will be removed by the sorting out of falsified data and by urgent repetition of the corresponding indentations, if the original experimental data are no longer available for revised publication. The various new possibilities with experimental spherical indentations provide all of the further important characteristics of phase-transitions. They open new horizons for creation and structural characterization of yet unknown polymorphs of materials. For technically used materials they tell how to avoid dangerous cracking, originating from polymorph interfaces, which often continue to disastrous crashes [

The author declares no conflicts of interest regarding the publication of this paper.

Kaupp, G. (2020) Real and Fitted Spherical Indentations. Advances in Materials Physics and Chemistry, 10, 207-229. https://doi.org/10.4236/ampc.2020.1010016