_{1}

Non-iterative analysis of indentation results allows for the detection of phase transitions under load and their transition energy. The closed algebraic equations have been deduced on the basis of the physically founded normal force
depth^{3/2} relation. The precise transition onset position is obtained by linear regression of the F_{N} = kh^{3/2} plot, where k is the penetration resistance, which also provides the axis cuts of both polymorphs of first order phase transitions. The phase changes can be endothermic or exothermic. They are normalized per μN or mN normal load. The analyses of indentation loading curves with self-similar diamond indenters are used as validity check of the loading curves, also from calibration standards that exhibit previously undetected phase-transitions and are thus incorrect. The phase-transition energies for fused quartz are determined from the loading curves from instrument provider handbooks. The anisotropic behavior of phase transition energies is studied for the first time. Quartz is a useful test object. The reasons for the packing-dependent differences are discussed on the basis of the local crystal structure under and around the inserting tip.

Instrumented indentations require proper calibrations and physically correct analysis. Modern instruments are highly reliable (at least since 1998) and the most used standards fused quartz, aluminum and sapphire are available in constant good quality. There are however problems with non-consideration of largely unknown phase transitions at moderately high loads of conical or pyramidal indentations [_{N})-depth square (h^{2}) proportionality [_{N}―goes with the indented volume of cones or pyramids, which is proportional to h^{3}. This deduction of the physically enforced Equation (1) can easily be repeated with simple arithmetic. Or graphically: the work that is lost for the penetration is the area between the parabola with exponent 3/2 and its secant that starts at zero. Furthermore, the applied work (W_{applied}) is the area under such secant of the parabola (down to the zero line). Furthermore, the indentation work (W_{indent}) covers the area below the parabola, and it can also be described by the area of the 0 - h_{max} - 0.8 F_{N} triangle [_{applied}/W_{indent} that was already mathematically deduced by integration of (1) in [_{N} versus h^{3/2} plots with excellent correlation. This viable analytical tool is disdainfully known as “Kaupp fitting” in the literature. We must therefore call it now “Kaupp plot (1)” to underline that it must not be degraded to a fitting technique.

F_{N} = kh^{3/2} (1)

The W_{applied}/W_{indent} = 5/4 relation means that the loss of F_{N} for the penetration depth h is for exponent 3/2 always 20% with universal mathematical precision. This is totally independent of the material. For an assumed exponent 2 it would calculate to be 33.33% [_{indent} [

For example, motorized aviation with flying machines required new physical understanding of aerodynamics and also knowledge of materials’ properties [_{ISO} and E_{r-ISO} level are unsuitable. They could therefore not prevent catastrophic failures (not only with airliners), which have been termed “failure by fatigue of materials”. The liability clearly requires that local test procedures identify phase-transitions on the physical mathematic basis. We again urge ISO-ASTM to use the undeniable strict mathematical analyses, as presented here and in our cited publications since 2005. Addition of suitable ductilizers must optimize the super-alloys, so that the first phase transition onset force will be considerably above the permitted maximal force on them (also “pop-ins” must not occur upon load). Such analytical tests are required after the common long-term stretching, bending treatments, and after the repeated thermo-mechanical stress upon application with loading curves at the prescribed service intervals. The present technique is fast and easy for obtaining the onset information.

The penetration resistance k [mN/µm^{3/2}] is literately the physical hardness with respect to the used indenter geometry. For the general applications the (effective) cone angle dependency of the self similar indenters is removed by the normalization as “penetration-resistance” hardness H_{phys} = k/π tanα^{2}, where k of Equation (1) is energy corrected with factor 0.8 when the hardness shall be related to the indentation depth [

Conversely, the still generally accepted definitions of indentation hardness as H = F_{N}/A_{projected} or H_{ISO} = F_{N}/A_{contact} use the entire maximal loading force for the depth. This seemed to verify the physically false exponent 2. But the reasoning that the area of a cone “varies as the square of the depth of contact” [^{2}tanα^{2}) variation is self evident, but the volume of the conical indenter varies with πh^{3}tanα^{2}/3. Neither is the definition of indentation hardness according to ISO and [_{Nmax} - h_{max} triangle minus the area under the loading parabola with an exponent 2 would amount to one third of the total applied work. Nevertheless, this violation of the first energy law was not allowed to be literally expressed in publications before 1997 [

A further advantage of the physical indentation resistance hardness H_{phys} = k/π tanα^{2} is its independence of the depth (self-similar indenter!). We can therefore for the first time choose from hardness with respect to the penetration act (0.8 k) or with respect to the full indentation resistance (uncorrected k). An important discussion on what should be used for what theoretical and practical use is now opened. The papers [_{Nmax} or 0.8 F_{Nmax} applies for the definition of indentation elastic moduli E_{r} from the unloading curve. But none of such indentation moduli are the still claimed “Young’s moduli”. They resemble the bulk moduli [

These important developments facilitated the easy detection of phase transitions by indentation. Previously such detection was restricted to a kink in the unloading curve. There is one in the unloading curve of silicon (though without onset information), which had been amply discussed as a particular exception [

A further application of Equation (1) with the penetration resistance k is the reliability control of published measurements on the strict physical basis [_{r} next to all other systematic errors, even with calibration standards. Some further types for disclosed errors can be found in the corresponding Section below.

Further applications use phase transition energies at different temperatures for the determination of phase transition activation energies [_{3} on (011), α-quartz on (010), InGaAs_{2} on (001) are already known [

A fully calibrated Hysitron Inc. Triboscope^{Ò} Nanomechanical Test Instrument with 2D transducer and leveling device, connected to a Nanoscope AFM was used for the own indentations. The apex radii of the cube corner (55 nm) and Berkovich (110 nm) diamond indenter were directly measured by AFM in tapping mode. The leveling to ±1˚ was in x and y direction. Loading times were 30 s up to 5000 µN final load. All our measurements were performed with the same cube corner with the effective cone angle of α = 42.28˚. The original data with about 1500 points each of our loading curves for α-quartz (rock crystal) from [

A single well developed rock crystal with smooth surfaces and excellent colorless clarity was the α-quartz sample without twins at the surface. Its indexed major faces were horizontally leveled to slopes of ±1˚ in x and y direction under AFM control at disabled plane-fit. All F_{N} and h data pairs from the loading curves were loaded to Excel^{Ò} (Microsoft; Redmond, USA, WA) for the calculation of the h^{3/2} values and the linear branches of the regression lines provided the slopes (penetration resistances) k_{1} and k_{2}, and the axis cuts F_{1-a} and F_{2-a}. They were used with all of their figures for avoiding rounding errors. The linear regression coefficients R^{2} were in all cases > 0.999 - 0.9999. The precise sharp intersection point (transition onset) was obtained by equalizing of the regression line equations and the so obtained h_{kink} and F_{Nkink} values were calculated by using Equation (2). All necessary terms are thus obtained, as h_{max} and F_{Nmax} are directly available.

F N = k h 3 / 2 + F 1-a . (2)

Table1. Physical parameters from the cube corner indentations onto four different surfaces of α-quartz (rock crystal) up to 5000 µN load.

^{(a)}For practical reasons we do not use the factor 0.8 to the µN values for the normalizations from kink to the final force.

W 1-applied = 0. 5 h kink ( F N-kink + F 1-a h kink ) . (3)

W 1-indent = 0. 8 W 1-applied . (4)

W 2-indent = 0.4 k ( h 5 / 2 − h kink 5 / 2 ) + F 2-a ( h − h kink ) . (5)

full W applied = 0. 5 F N-max h max . (6)

W transition = full W applied − Σ ( W applied ) . (7)

The used Equations (2)-(7) for the calculations contain all of the obvious corrections [_{a} corrects for axis cuts of the regression lines that are due to surface effects. Only F_{1-a is} also influenced by the apical tip rounding radius (R), giving larger penetration resistance up to h_{cone} = R (1-sinα). It varies with the surface properties (including water layers). These depend on ambient conditions, which exclude their tabulation. The W_{1-indent} is calculated from W_{1-applied} according to Equation (4). W_{2-indent} must use integration and correction with F_{2-a} as in Equation (5), and W_{2-applied} is then obtained by multiplication with 1.25 in analogy to Equation (4). The balance of Equation (6) and Equation (7) gives the transition energy W_{trans} that was reasonable rounded in

Unfortunately, all indentations of the most cited publication of Oliver-Pharr in 1992 [_{r} (from there with Poisson’s ratio elastic modulus E, which has been unduly called “Young’s modulus” [_{r} that use the maximal force are in error for all 5 examples: the faulty calibration adds to the unphysical h^{2}, the energy law violation, and the non-consideration of the phase transition onsets that occur before that load. Not the pristine material is tried to be characterized! These errors perpetuate in all further iterations that are made to converge to these values and the numerous qualities that are deduced from all of these values. They have at least influenced various reference table entries, not to speak of finite element calculations. Particularly troublesome are the errors in [_{1} = 76.044, k_{2} = 109.12, and k_{3} = 123.3 mN/µm^{3/2}, all with R^{2} > 0.999) when plotted with Equation (1). This corresponds so perfectly with the soda lime glass values from their _{1} = 77.909, k_{2} = 105.28, and k_{3} = 122.0 mN/µm^{3/2}) that the claimed “fused quartz” curve must in fact belong to the result from another soda lime glass indentation, despite the about 250 nm different maximal depths. Furthermore, their _{1} = 95.57, k_{2} = 114.50 mN/µm^{3/2}; kink ≈ 35.6 mN) and sapphire on (0001) (k_{1} = 236.58 and k_{2} = 264.68 mN/mm^{3/2}, kink ≈ 31.5 mN) (the better older curve up to 90 mN load of [

Fused quartz is the most used calibration standard for nanoindentations. We analyze therefore the corresponding Berkovich (with half angle θ of 65.3˚) loading curves of prominent instrument provider Handbooks with respect to the Equations (1)-(7) and use of the new applications without any iteration. The first point is the detection of the long known amorphous to amorphous phase transition [_{2} > k_{1}) has been denied on the basis of poor, or fitted, or too extended curves with low precision (e.g. [^{2} = 0.9999, each upon regression with the Kaupp plot (1). Their penetration resistance values are for k_{1} 1.9654 and 1.9672 and those for k_{2} are 2.4392 and 2.3936 µN/nm^{3/2} for Hysitron and CISCO, respectively. This leads after surface-cut corrections to transition onset depths of 109.6476605 and 100.682078 nm at 2.348 and 2.117 mN. These differences reflect different measurement conditions. Most likely are different force calibrations or horizontal sample leveling devices that are not specified. It is therefore not surprising that the calculated normalized per µN transition energy values also differ: we calculate 6.206 and 3.563 µNnm/µN, respectively, for the endothermic transitions. These values are remarkably large when compared with the exothermic transitions that give the negative values of crystalline quartz (

Unlike fused quartz, crystalline α-quartz in the form of rock crystal undergoes exothermic phase transition upon sufficient indentation stress. The projected images of the studied surfaces are shown in _{1} and k_{2} had already been shown to be anisotropic with different indentation works W_{indent} [_{trans} values of _{1}, h_{kink}, and W_{trans}/µN values (entries 1 and 2 are most different, 3 and 4 are in between), but not by k_{2}, F_{Nkink}, and ƩW_{applied}/µN. Clearly, there is also the force that acts normal to the surface of the cube corner with the opposite of its half angle θ = 35.26˚. It is therefore important to also consider the location of channels exiting from the side faces at the indenter surface for the transition energies. We therefore construct such surfaces at 35˚ and assume that not all of their so seen shapes will be destructed while the cube corner penetrates. They are obtained by rotation of the crystal structure around the X axis by + and − 35˚ (rX ± 35˚ ≡ 180˚ ± 35˚) and the same around an Y axis (as rY ± 35˚ ≡ 180˚ ± 35˚). The resulting images are projected (_{trans} = −15.744 µNnm/µN is produced by indentation upon the (011) surface (entry 1) of α-quartz with a cube corner indenter. The (011) packing exhibits not very favorable skew channels, as can be best seen in the center of the image where the

view goes through 5 of the interlocked pyramidal layers (_{kink} and F_{Nkink} are the least of all studied cases and the required work down to the transition onset force is lower than with the other surfaces. Furthermore,

Consistently, the least exothermic transition energy among the tested surfaces is produced under the (010) surface (entry 2) with −11.0485 µNnm/µN. It exhibits straight channels (_{Nkink} is high (_{trans} values are almost equal and between the extremes (

The universal Equations (1)-(7) are physically and mathematically deduced beyond any doubt for vertical indentations with self-similar indenters [_{ISO} and E_{r-ISO} does not consider phase transitions under load and they violate the first energy law. For example [

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

Kaupp, G. (2019) Phase-Transition Energies, New Characterization of Solid Materials and Anisotropy. Advances in Materials Physics and Chemistry, 9, 57-70. https://doi.org/10.4236/ampc.2019.94006