# Simple Deconvolution Models for Evaluating the True Microhardness of Thin Nanostructured Coatings Deposited via an Advanced Physical Vapor Deposition Technique

^{1}

^{2}

^{*}

## Abstract

**:**

_{1}or Me

_{2}= Cr, Hf, Nb, W, and Zr. The resulting NTC coatings were deposited onto 100Cr6 steel substrates using an advanced physical vapor deposition (PVD) technique, referred to here as high-power ion-plasma magnetron sputtering (HiPIPMS). The comprising crystalline nanometer-scale TiAlSiMe

_{1}-N/TiMe

_{2}-CN nanoparticles strengthened by Me additives significantly increased the NTC microhardness to over 3200 HV. The primary focus of this research was to determine the true microhardness of the NTC film samples. The apparent microhardness (H

_{a}) of the film/substrate system for various NTC samples was measured during microindentation testing using the Vickers method. Nine NTC samples were tested, each generating a corresponding microindentation dataset containing between 430 and 640 imprints, depending on the specific NTC sample. These datasets were analyzed using three distinct empirical approaches: (i) the inverse power-law model (IPL-Model), (ii) the sigmoid-like decay model (SLD-Model), and (iii) the error function model (ERF-Model). The observed solid correlation between the proposed models and experiments suggests that the true microhardness estimates (H

_{f}) obtained through the empirical mathematical modeling approach are reliable.

## 1. Introduction

_{2}/Ar gas mixtures, have enjoyed sustained interest. This approach enables the creation of coatings with heightened tribological performance and micromechanical properties, surpassing those inherent to equivalent monolayered coatings [11,13,14,15].

## 2. Materials and Methods

#### 2.1. Preparation of the NTC Samples

^{2}within the erosion zone of the magnetron sputtering target (MST), which is a crucial factor for the equal sputtering rate of different materials like transition metals, which have different sputtering yields during PVD processes [10].

^{2}threshold for the discharge power of magnetron sputtering devices (MSDs). In contrast to conventional PVD methods, HiPIPMS deposition systems feature magnetically intense MSDs (with a magnetic B-field strength exceeding 800 mT) equipped with MSTs devoid of backing plates. This process was complemented by highly efficient bubble-free water cooling with a high flow rate directly beneath the MST copper membrane. The water pressure for direct cooling of the MSTs was maintained during the PVD process at approximately 4–5 bars at the inlet of the MSDs.

_{b}= 0.131 mPa; (ii) Total operating gas pressure of the sputter gas mixture (Ar/N

_{2}), p

_{sg}= 0.32 Pa; and (iii) Partial gas flow rate, q

_{gN2}= 11.9 L/h, (iv) before film deposition, substrate heating, and in situ sputter cleaning were performed using a collimated linear ion beam etching at 1750 V × 135 mA; (v) The distance between the MSD and substrate L = 78 mm; (vi) The substrate temperature during the PVD deposition process was kept at 350 °C; and (vii) The electrical discharge power regime of the MSDs was varied within (640–680) V × (7.5–10.5) A depending on the sputter gas mixture ratio, the design and configuration of the MSTs, and their total working surface. The DC discharge power density of the operating MSDs exceeded the 60 W/cm

^{2}threshold value needed for the equal-rate sputtering conditions of the mosaic-type sputtering targets; (viii) The NTC film sample deposition rate at 2D rotation was kept at 180 nm/min, and (ix) the negative bias voltage during the deposition process was maintained at 90 V, causing a bias current of about 320–480 mA depending on the sputter discharge power. Collectively, these parameters were carefully controlled to enable the precise execution of the HiPIPMS-PVD process, facilitating the sputtering of MSTs and the growth of NTCs with the targeted properties.

#### 2.2. Microindentation Hardness Testing Experiment

^{2}units but in GPa units. These GPa units, although accurate, may not be intuitively interpretable for most engineers and technicians. As ASTM guidelines recommended, a “soft” metric approach in alignment with the Vickers hardness testing experiment was adopted. This approach facilitates an ease of comprehension and enhances the practical utility of the hardness measurements within the engineering and technical community.

#### 2.3. Nanoindentation Hardness Testing Experiment

^{1}. Additionally, a superimposed, harmonically oscillating force frequency of 45 Hz was applied. Before conducting the nanoindentation experiments, the G200 tester was calibrated using a reference sample of fused silica. Subsequently, the load–displacement measurements, elastic modulus, hardness, and standard deviations were calculated based on experimentally obtained loading–unloading datasets. These calculations were performed using the well-established Oliver–Pharr method [29] and facilitated by MTS TestWorks V4.1 software.

#### 2.4. Empirical Mathematical Models Applied for The Calculation of the Apparent Microhardness, Ha(h)

_{a}(h), of the NTC samples using three distinct empirical approaches: (i) the inverse power-law model (IPL-Model), (ii) the sigmoid-like decay model (SLD-Model), and (iii) the error function model (ERF-Model). Each model exhibits a characteristic sigmoidal behavior characterized by two distinct asymptotic domains bridged by an intermediary transitional regime. This pattern effectively encapsulates the micromechanical properties of the hard-film/soft-substrate system or vice versa. During the indentation process, underneath the loaded indenter, a transition occurs from the film surface layer (the upper asymptotic value or “true” hardness of the film) to the underlying substrate bulk volume (the lower asymptotic value or “true” hardness of the substrate). These models are relevant in empirically approximating the indentation measurement data relating to the film-dominated region at shallow indentation depths, the film/substrate-governed transitional region at modest indentation depths, and the substrate-dominated region at large indentation depths. These models were optimized with best-fit parameters, which were subsequently leveraged to extrapolate and estimate the true hardness, H

_{f}= H

_{a}(h => 0), of the NTC at its surface when h => 0. The utilization of multiple models facilitated an assessment of the robustness of the true hardness estimates. A strong concurrence among the different models would indicate a heightened level of reliability in the estimated values. Thus, the IPL-Model was defined as follows:

_{a}(h) is the apparent hardness of the film/substrate system as a 4-parameter (4P) function of the indentation depth, h. Parameters H

_{f}and H

_{s}control the upper (i.e., the film surface) and lower (i.e., the bulk substrate) asymptotic hardness levels of the empirical film/substrate model. Furthermore, the parameter “c” controls the inflection point of the model curve regarding the upper and lower asymptotic domains with respect to the transition region of the film/substrate system, while the parameter “k” is related to the steepness of the model curve, “k” value is the slope at the point “h = c”.

_{ai}; h

_{i}} of the apparent microhardness, H

_{ai}, at the indentation depth, h

_{i}, was calculated as an averaged value of 20–40 imprints, depending on the test load and sample surface roughness, RMS-value, to obtain safe and reliable experimental data. The smaller the test load, P, the more indentations needed to be made. The total number of indentations exceeded 600 Vickers pyramidal imprints, d = (d

_{1}+ d

_{2})/2, per sample. Special attention was paid to outliers, which were discarded so as not to affect the apparent hardness’s mean values. The fitting was conducted using the non-linear least squares method implemented by the Python scientific computing library SciPy [30]. The non-linear least squares method minimizes the squared residuals to obtain optimal parameters for each model. To ensure convergence during the optimization, reasonable initial parameter guesses were provided for each model.

#### 2.5. Determination of the Robustness of the Predicted True Hardness, H_{f}, of the Coated Film

_{f}, values at its surface H

_{a}(h = 0) [31]. This is a well-established approach for determining confidence intervals of modeling estimates, whereby the measurement data are repeatedly sampled with replacement, and the resampled datasets are fitted via the empirical models discussed above. Given each empirical mathematical model described in Section 2.4 n = 1000 resampled datasets for each measured and averaged dataset obtained during the microindentation testing experiment were created.

_{f}/H

_{s}-values based on our previous knowledge of the NTC samples. The same goes for the two other parameters, the “c” and “k” values—our prior knowledge about the behavior of the materials under indentation was used and some preliminary analysis was performed. That way, the best-fit parameters were found for each resampled dataset. The best-fit parameters were then used to estimate the true hardness at the film surface H

_{f}= H

_{a}(h = 0). The predicted true hardness of each empirical model, H

_{f}, was then determined using the median (50th percentile) estimate of H

_{a}(h = 0) across all n resampled datasets of a given hardness measurement; the 95% confidence interval was estimated using the 2.5th (H

_{f}

^{2.5}) and 97.5th (H

_{f}

^{97.5}) percentiles.

## 3. Results and Discussion

#### 3.1. Preparation and Characterization of the NTC Film Samples

_{1}N/TiMe

_{2}-CN} through the incorporation of refractory metals Me

_{1}, Me

**= Cr, Hf, Nb, W, and Zr as alloying agents. This augmentation aimed to strengthen the NTC’s core nitride/carbonitride bilayer architecture. Bespoke mosaic-type magnetron sputtering targets (MSTs) were crafted to facilitate the implementation of these bilayer structures. These MSTs featured integrated Me**

_{2}_{1}, Me

_{2}= Cr, Hf, Nb, W, and Zr inserts positioned within the annular erosion zone of the circular planar MSTs.

_{f}, of the NTCs, thus the information about other coatings’ characteristics can be found in Leitans et al.’s work [16]. Microindentation testing experiments using the Vickers method to measure various NTC samples’ apparent hardness, H

_{a}, were conducted. These measurements were a basis for acquiring the experimental datasets for calculating the true hardness, H

_{f}, for nine NTC samples (Table 1). Three distinct four-parameter sigmoidal empirical mathematical models were used to fit the apparent hardness curve related to the film/substrate system’s transition region. These models were instrumental in extracting the true hardness of the film from experimentally measured datasets {H

_{ai}; h

_{i}}. Through these methodologies, we aimed to gain a comprehensive understanding of the true hardness of the NTCs based on a simple mechanical model of the elastic–plastic deformation processes grounded in Hertzian elastic contact theory [32].

#### 3.2. Micromechanical Properties of the NTC Film Samples

#### 3.2.1. Microindentation Response Analysis Using the Load–Displacement Curves

^{n}(Figure 4 and Figure 5). Here, n represents a constant exponent referred to as Meyer’s index, and C is a constant amplitude dependent on the elastic–plastic properties of the coating and, to some extent, on the indenter geometry. The constant “C” represents the proportionality factor between the applied load and the indentation depth, raised to the power of “n”. It can be thought of as the “hardness coefficient” in the power-law relationship. This coefficient indicates the inherent hardness behavior of the material being tested. Each NTC sample’s load–displacement, P–h, curve represents averaged datasets from nine microhardness testing experiment series containing 430–640 imprints, depending on the specific NTC sample. While point markers and error bars have been omitted for clarity, it is essential to note that the relative standard deviation for the resulting curves typically fell within the range of 6–29%. This deviation tended to increase at lower loads and shallower penetration depths.

_{max}= 1.3115) nor the hard substrate sample set (n

_{max}= 1.6851) is lower than that of the 100Cr6 steel substrate without any coating, which displays a steeper load–displacement (P-h) curve with an n value of 1.7617. For bulk materials, such as steel, silicon, sapphire, and glass substrates, Meyer’s index is a valuable indicator of the indentation size effect (ISE). When the exponent n is less than 2, it signifies normal ISE behavior. Conversely, when n exceeds 2, it indicates the presence of reverse ISE behavior. At n = 2, ISE is absent for the given material and the applied indentation depths. The range of “n” values observed between 0.9228 and 1.6851 indicates a significant influence of these size-dependent effects. The fact that “n” is less than 2 is in contrast to what is typically observed in bulk materials, where “n” is close to 2 due to the expected quadratic relationship between the projected contact area, A(h), and the indentation depth h [34,35]. This suggests that the film/substrate system’s mechanical behavior is influenced by nanoscale or microscale effects that deviate from classical bulk material behavior.

#### 3.2.2. Microindentation Response Analysis Using the Hardness–Displacement Curves

_{0}), where h

_{0}signifies some initial penetration depth when searching the film surface, as opposed to the Cauchy strain, defined as ε = h/t, where t is representing the film thickness. Consequently, Figure 6, Figure 7 and Figure 8 illustrate the hardness–displacement, H-h, curves presented in a semi-logarithmic scale. This semi-log scale visualization of the hardness–displacement, H-h, curves facilitate a more meticulous examination of the undulating patterns, peak values, and potential plateaus associated with the intrinsic material properties, especially within shallow indentation ranges.

#### 3.3. Mathematical Modelling of the Apparent Microhardness, H_{a}(h)

_{Ti}= 120 GPa) and after that by the stiffer bulk steel substrate (E

_{steel}= 190–210 GPa, Figure 3). Accordingly, the substrate’s elastic modulus first significantly impacted the apparent elastic modulus of the coated film and then intermittently impacted its apparent hardness.

_{c}, and encompasses the interface between the core (the intensely deformed homogeneous region beneath the indenter) and the attenuating stress–strain field tail beyond the core [40]. For the present investigation involving the utilization of a sharp, three-faceted Berkovich indenter for the bulk substrate, the heavily deformed core radius beneath the indenter is approximated as a

_{c}≈ 2.8 h. For instance, in the case of the NTC-1 sample subjected to a test load of 10 gf (100 mN), the indenter’s penetration depth reached about 500–700 nm, corresponding to a contact radius, a

_{c}, of approximately 2000 nm. Consequently, it is generally advisable to employ loads smaller than 100 mN to investigate the positive skin effect, also known as the reverse ISE. This reasoning only heightens the concern that the measured apparent microhardness of the NTC samples might be influenced not only by the strain-hardening process occurring within the core region itself but also by the interplay of the underlying adhesion layer and the elastic–plastic reverberations from the bulk substrate.

_{f}= H

_{a}(h = 0). Each model exhibits sigmoidal behavior approximating the indentation measurement data related to the film-dominated region at shallow indentation depths, the film/substrate governed transitional region at intermediate indentation depths, and the substrate-dominated region at large indentation depths. We show the resulting model fits of the apparent microhardness, H

_{a}(h), of the representative samples NTC-1, NTC-5, and NTC-8, in Figure 9, Figure 10 and Figure 11, respectively, whereas the fits for the remaining samples are shown in Appendix A (Figure A1, Figure A2, Figure A3, Figure A4, Figure A5 and Figure A6). The NTC sample set of nine items can be split into three subsets depending on the alloying refractory metals, Me = Cr, Hf, Nb, W, and Zr: (i) NTC-1, NTC-2, and NTC-3 belong to those doubly alloyed with Cr, Nb, or W; (ii) NTC-4, NTC-5, and NTC-6 belong to those alloyed with Nb only; (iii) NTC-7, NTC-8, and NTC-9 belong to those alloyed with Hf only.

_{f}, of the coated film using the IPL-Model for each of our samples, as well as the 2.5th and 97.5th percentile estimates obtained via the bootstrapping resampling technique that constitute the 95% confidence interval. Similarly, Table 4 and Table 5 characterize the H

_{f}values estimated using the SLD and ERF models, respectively.

_{f}, of the coated film of three empirical mathematical models. It was found that the three models are in good agreement. The difference between the maximum and minimum predicted true hardness H

_{f}between the three models is around 6.9% across all nine samples. The best agreement between the models was shown for the NTC-5 sample, with only a 1.2% difference, where the SLD-Model estimated the maximum predicted true hardness H

_{f}= 2106.5 HVN and the IPL-Model estimated the lowest predicted true hardness H

_{f}= 2081.4 HVN. Meanwhile, the largest variation between the models was found for the NTC-4 sample, with a difference of 14.1%, where the ERF-Model estimated the maximum predicted true hardness H

_{f}= 2558.1 HVN and the IPL-Model estimated the lowest predicted true hardness H

_{f}= 2242.6 HVN. Overall, the agreement among the different models indicates a heightened level of reliability in the estimated values of predicted true hardness, H

_{f}, of the coated film.

_{f}

_{,}of the coated film samples obtained via the proposed empirical mathematical modeling approach can be deemed reliable.

## 4. Conclusions

_{f}, values for the examined hard nitride/carbonitride NTC samples between the proposed models averages around 6.9% across all nine samples.

_{f}, as 2106.5 HV, while the IPL-Model estimates the lowest predicted true hardness, H

_{f}, as 2081.4 HV, for this sample. In contrast, the most significant variation between the models was observed for the NTC-4 sample, with a difference of 14.1%. Here, the ERF-Model predicts the highest, H

_{f}, as 2558.1 HV, while the IPL-Model predicts the lowest, H

_{f}, as 2242.6 HV.

_{f}, of the considered NTC film samples. Additionally, the 95% confidence intervals for the predicted true hardness of the proposed models largely overlap for each sample. It is worth noting that the IPL-Model tends to have much larger confidence intervals than the SLD-Model and ERF-Model, suggesting a high sensitivity of the IPL-Model to the bootstrapping resampling technique. This sensitivity can be attributed to the fact that the IPL-Model also exhibited the best model fit for most of the samples, as assessed via the mean-squared error. The mathematical modeling of the apparent microhardness, H

_{a}(h), with three distinct empirical models, yielded concurrent estimates of the true microhardness, H

_{f}, of the nine examined NTC film samples representing hard nitride/carbonitride structures, with the differences of the H

_{f}between the models being approximately 6.9%, on average, across the nine samples.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Microindentation response as an apparent microhardness–displacement curve of the NTC-2 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

**Figure A2.**Microindentation response as an apparent microhardness–displacement curve of the NTC-3 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

**Figure A3.**Microindentation response as an apparent microhardness–displacement curve of the NTC-4 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

**Figure A4.**Microindentation response as an apparent microhardness–displacement curve of the NTC-6 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

**Figure A5.**Microindentation response as an apparent microhardness–displacement curve of the NTC-7 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

**Figure A6.**Microindentation response as an apparent microhardness–displacement curve of the NTC-9 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

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**Figure 1.**Microindentation hardness testing matrix of 10 measurements at each applied load, P (gf), within the load interval [10, 600] in gram-force units, gf [16]. The applied test load increment was 10 gf within the load interval [10, 100], and 50 gf within the load interval [100, 600]. Imprints’ spacing within the testing matrix in the XY plane was 300 µm to avoid any mutual impact of neighboring imprints. For each test load value, 10 imprints were performed.

**Figure 2.**Nanoindentation response as a nanohardness–displacement, H-h, curve and an elastic modulus–displacement, E-h, curve of the 100Cr6 steel substrate of the NTC-4 sample in semi-log scale plot, measured after the PVD process and calculated via the Oliver–Pharr method [29].

**Figure 3.**Nanoindentation response as a nanohardness–displacement, H-h, curve and an elastic modulus–displacement, E-h, curve of the 100Cr6 steel substrate of the NTC-4 sample in semi-log scale plot, measured before the PVD process and calculated via the Oliver–Pharr method [29].

**Figure 4.**Microindentation responses as load–displacement, P-h, curves of the NTC-1, NTC-2, NTC-3, NTC-5, NTC-6, and NTC-7 samples on the hard-annealed-bearing 100Cr6 steel substrates in a log–log scale plot to demonstrate Meyer’s well-known power-law [33].

**Figure 5.**Microindentation responses as load–displacement, P-h, curves of the NTC-4, NTC-8, and NTC-9 samples on the soft-annealed-bearing100Cr6 steel substrate in a log – log scale plot to demonstrate Meyer’s well-known power-law [33].

**Figure 6.**Microindentation response as apparent hardness–displacement, H-h, curves of the NTC-1, NTC-2, NTC-3, NTC-5, NTC-6, and NTC-7 samples on the hard-annealed-bearing 100Cr6 steel substrate in a semi-log scale plot.

**Figure 7.**Microindentation response as apparent hardness–displacement, H-h, curves of the NTC-4, NTC-8, and NTC-9 samples on the soft-annealed-bearing 100Cr6 steel substrate in a semi-log scale plot.

**Figure 8.**Microindentation response as apparent hardness–displacement, H-h, curves of the NTC-7 sample deposited on the hard-annealed-bearing 100Cr6 steel substrate, and NTC-8 and NTC-9 samples deposited on the soft-annealed-bearing steel 100Cr6 substrate in a semi-log scale plot.

**Figure 9.**Microindentation response as an apparent microhardness–displacement curve of the NTC-1 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

**Figure 10.**Microindentation response as an apparent microhardness–displacement curve of the NTC-5 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

**Figure 11.**Microindentation response as an apparent microhardness–displacement curve of the NTC-8 sample calculated using two measured diagonals of the Vickers imprints at each specified test load within the 10–700 gf range (0.098–6.86 N) in a semi-log scale. The fitted microhardness–displacement curves relevant to the proposed three sigmoidal decay models are shown as solid lines.

**Figure 12.**Determination of the robustness of the predicted true hardness, H

_{f}, of the coated film. The markers denote the predicted true hardness, H

_{f}, of the coated film using three different empirical mathematical models. The bars denote the range between the 2.5th and 97.5th percentile estimates obtained via the bootstrapping resampling technique, constituting the 95% confidence interval of the predicted true hardness.

Sample Label | Coating’s Bilayered Substructure ^{1} | Coating Thickness, t (nm) ^{2} | Indentation Depth, h (nm) at 100 gf ^{3} | Microhardness, HVN at 100 gf ^{4} |
---|---|---|---|---|

NTC-1 | {TiAlSiZr-N/W-CN} | 6200 ± 315 | 1720 ± 137 | 1244 ± 103 |

NTC-2 | {TiAlSiNb-N/Ti-CN} | 5400 ± 270 | 1652 ± 132 | 1355 ± 109 |

NTC-3 | {TiAlSiCr-N/Si-CN} | 5060 ± 250 | 1731 ± 138 | 1228 ± 98 |

NTC-4 | {TiAlSi-N/TiNb-CN} | 4100 ± 190 | 2509 ±201 | 586 ± 47 |

NTC-5 | {TiAlSi-N/TiNb-CN} | 3300 ± 160 | 1880 ± 150 | 1131 ± 91 |

NTC-6 | {TiAlSi-N/TiNb-CN} | 2200 ± 110 | 1930 ± 154 | 1777 ± 142 |

NTC-7 | {TiAlSi-N/TiHf-CN} | 4700 ± 130 | 1540 ± 123 | 1861 ± 149 |

NTC-8 | {TiAlSi-N/TiHf-CN} | 6200 ± 315 | 1783 ± 142 | 1371 ± 110 |

NTC-9 | {TiAlSi-N/TiHf-CN} | 6100 ± 300 | 1937 ± 154 | 1294 ± 104 |

^{1}Chemical composition of the bilayered structure of the NTC superlattice was stated according to the MSD-MST configuration used in the relevant PVD process.

^{2}A Calo tester (CSM Instruments, Peseux, Switzerland) and digital microscope KH-7700 (Hirox, Tokyo, Japan) were used to measure the thickness of the deposited coatings.

^{3}Indentation depth, h, was calculated using two measured diagonals of the Vickers imprints at the 100 gf (0.98 N) test load.

^{4}Vickers hardness number (HVN in kgf/mm

^{2}units) was calculated using two measured diagonals of the Vickers imprints at the 100 gf (0.98 N) test load.

Sample | IPL-Model | SLD-Model | ERF-Model |
---|---|---|---|

NTC-1 | 107.9 | 248.4 | 284.2 |

NTC-2 | 393.5 | 398.3 | 425.9 |

NTC-3 | 130.9 | 453.8 | 531.3 |

NTC-4 | 589 | 1753.6 | 2250.6 |

NTC-5 | 85.6 | 177.5 | 201.4 |

NTC-6 | 135.2 | 68.6 | 64.8 |

NTC-7 | 1106.3 | 1373.3 | 1392.8 |

NTC-8 | 4529.3 | 1991.5 | 1810.9 |

NTC-9 | 3886.2 | 1780.5 | 1619.1 |

^{1}The model with the smallest mean-squared error is highlighted in bold for each NTC sample.

**Table 3.**Predicted true hardness, H

_{f}, of the coated film using the IPL-Model, and the 2.5th and 97.5th percentile estimates obtained via the bootstrapping resampling technique.

Sample | H_{f} (HVN) | H_{f}^{2.5} (HVN) | H_{f}^{97.5} (HVN) |
---|---|---|---|

NTC-1 | 2397.4 | 2129 | 4848.3 |

NTC-2 | 2462.7 | 2192.9 | 2898.9 |

NTC-3 | 2738.8 | 2262.3 | 4650 |

NTC-4 | 2242.6 | 2198.8 | 3773.1 |

NTC-5 | 2081.4 | 1925.6 | 3612 |

NTC-6 | 2029.7 | 1716.4 | 2143 |

NTC-7 | 3137.2 | 2997.7 | 4635.3 |

NTC-8 | 3297 | 2821.1 | 3490.9 |

NTC-9 | 3162.4 | 2523.9 | 3403.5 |

**Table 4.**Predicted true hardness, H

_{f}, of the coated film using the SLD-Model, and the 2.5th and 97.5th percentile estimates obtained via the bootstrapping resampling technique.

Sample | H_{f} (HVN) | H_{f}^{2.5} (HVN) | H_{f}^{97.5} (HVN) |
---|---|---|---|

NTC-1 | 2378.9 | 2235.1 | 2431.7 |

NTC-2 | 2617.5 | 2501.1 | 2683.2 |

NTC-3 | 2587 | 2334.8 | 2669.8 |

NTC-4 | 2479.2 | 2337.6 | 4322.7 |

NTC-5 | 2106.5 | 1990.6 | 2147.8 |

NTC-6 | 2303.3 | 2050.6 | 2363.2 |

NTC-7 | 3408.1 | 3230.4 | 3650.5 |

NTC-8 | 3418.1 | 3075.1 | 3553 |

NTC-9 | 3324 | 2924.5 | 3458.4 |

**Table 5.**Predicted true hardness, H

_{f}, of the coated film using the ERF-Model, and the 2.5th and 97.5th percentile estimates obtained via the bootstrapping resampling technique.

Sample | H_{f} (HVN) | H_{f}^{2.5} (HVN) | H_{f}^{97.5} (HVN) |
---|---|---|---|

NTC-1 | 2357.5 | 2213.8 | 2418.3 |

NTC-2 | 2601.2 | 2478.6 | 2664.6 |

NTC-3 | 2558.1 | 2303.4 | 2652.4 |

NTC-4 | 2559 | 2343.2 | 4146.2 |

NTC-5 | 2092.3 | 1956.5 | 2131.4 |

NTC-6 | 2314.1 | 2120.6 | 2341.8 |

NTC-7 | 3421.7 | 3254.7 | 3603.2 |

NTC-8 | 3413.2 | 3144.3 | 3545.7 |

NTC-9 | 3313 | 3039.6 | 3450.3 |

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**MDPI and ACS Style**

Kanders, U.; Kanders, K.; Jansons, E.; Lungevics, J.; Sirants, R.; Leitans, A.; Boiko, I.
Simple Deconvolution Models for Evaluating the True Microhardness of Thin Nanostructured Coatings Deposited via an Advanced Physical Vapor Deposition Technique. *Lubricants* **2023**, *11*, 501.
https://doi.org/10.3390/lubricants11120501

**AMA Style**

Kanders U, Kanders K, Jansons E, Lungevics J, Sirants R, Leitans A, Boiko I.
Simple Deconvolution Models for Evaluating the True Microhardness of Thin Nanostructured Coatings Deposited via an Advanced Physical Vapor Deposition Technique. *Lubricants*. 2023; 11(12):501.
https://doi.org/10.3390/lubricants11120501

**Chicago/Turabian Style**

Kanders, Uldis, Karlis Kanders, Ernests Jansons, Janis Lungevics, Raimonds Sirants, Armands Leitans, and Irina Boiko.
2023. "Simple Deconvolution Models for Evaluating the True Microhardness of Thin Nanostructured Coatings Deposited via an Advanced Physical Vapor Deposition Technique" *Lubricants* 11, no. 12: 501.
https://doi.org/10.3390/lubricants11120501