Heat Treatment-Driven Structural and Morphological Transformation Under Non-Parametric Tests on Metal–Ceramic-Sputtered Coatings

Abstract

The present study analyzes experimental data using qualitative and quantitative methods to identify significant statistical changes. These methods were employed to evaluate the results from the structural characterization of annealed TiWN and TiWC coatings elaborated by magnetron sputtering. The as-grown coatings were thermally treated at 500 °C in a furnace under an Ar atmosphere. Structural characterization was performed by X-ray diffraction and optical and electronic microscopy. The chemical composition was determined by energy dispersive X-ray spectroscopy. The data were analyzed using the Kruskal–Wallis (K-W) and Spearman correlation tests as non-parametric methods, employing free statistical software. The response variable—the crystallite size calculated through the Scherrer formula—is statistically tested. The data of the crystallite size of each sample were forecasted using the simple moving average (SMA) method to increase the number of data points of each sample to 12. The crystallite size of each sample remained unchanged before and after thermal treatment. However, microscopy analyses revealed strong surface cracking. The average crystallite size before and after the thermal treatment was analyzed by the K-W correlation, revealing significant changes considering a reliability level of 95% and a significance error of 5%. The analysis revealed a strong correlation between experimental data and statistical treatment results.

1. Introduction

1.1. TiWN and TiWC Transition Metal–Ceramic Coatings

Titanium (Ti) is a light metal with a high melting point and excellent corrosion resistance; however, the compromise between flexibility and malleability needs to be considered upon improvement of structural properties by alloying with other elements [1,2]. Tungsten (W) is considered a high-melting temperature transition metal with solid atomic bonds in crystalline lattices, providing high strength and heat resistance [3]. Furthermore, tungsten nitride (WN) has strong mechanical properties, as high hardness with thermal stability is of great technological interest for cutting tools and other industrial applications [4]. Finally, tungsten carbides (WCs) have many advantageous characteristics, such as low thermal expansion, high thermal conductivity, and dimensional stability, that have a widespread use as a hard-ceramic reinforcement in modern functional applications [5]. In most cases, post-growth thermal treatments (TTs) can improve the coating properties by the reduction of cracks, pores, and residual stress [6], i.e., improving the surface microstructure by increasing the crystallite size. Rong et al. [7] reported that carbide coatings annealed at 500 °C exhibited an increase in compact and ordered grains in comparison with their microstructure before thermal treatment; the carbides undergo a solid phase transformation through atomic diffusion at high temperatures. These results demonstrate that the coating is in a non-equilibrium state before thermal treatment, which promotes recrystallization processes within grain refinement [8]. However, despite the TT promoting positive effects, it can also promote the formation of cracks attributed to the exerted strain on the layer [9]. It is well known that carbides with high W concentration are subjected to stress at the interface, promoting crack initiation and reducing the mechanical properties on materials [10]. However, this stress concentration can be reduced by enrichment with Ti atoms incorporated in the WC lattice [11]. For example, coatings exposed to high-temperature TT can promote strong atomic diffusion by increasing the crystallite size [12,13]. Consequently, the peaks of the X-ray diffraction (XRD) patterns became more intense [14].

1.2. Relevance of Statistical Treatment on Materials Science

A thorough review of the literature revealed the use of statistical analysis in various studies on materials science. For example, Taghipour et al. [15] investigated the effect of different TTs on aluminum coatings deposited by gas phase aluminizing process. The study used experimental methods to evaluate mechanical and chemical properties and statistical analyses to determine the significance of the results. In another study [16], it was reported the effect of TT on Ti coatings grown by D.C. sputtering. They used analysis of variance (ANOVA) to statistically compare properties such as hardness and corrosion resistance. Martínez et al. [17] analyzed the influence of different TTs on copper coatings elaborated by D.C. sputtering. The authors conducted a comparative analysis using both experimental and statistical approaches, highlighting the importance of the temperature of the TT and its effect on the modification of the microstructure and mechanical properties of the coating. In [18], they conducted a comparative study between experimental methods and statistical analyses on niobium (Nb) coatings, also grown by D.C. sputtering. The study highlighted the effectiveness of both approaches in evaluating wear resistance and adhesion of the coatings. A recent study [19] reported the impact of heat treatment on the adhesion of ceramic coatings. An experimental approach was employed to measure adhesion of the coating and statistical analysis to assess the significance of the results. It was also found that the friction coefficient in each film was reduced by TT, being the reason why wear resistance was improved. It is important to note that before conducting experiments, the experimental design will reduce interpretation errors. Randomized controlled trials (RCTs) are the gold standard in this context due to their ability to reduce bias and provide robust evidence of causality. Bickman and Rog [20] determined that statistical significance is crucial in quantitative research to ensure that the results are not due to chance. According to conventional criteria, a result is significant if the p-value is less than 0.05, which indicates that there is less than 5% probability that the observed differences are due to random variation. In materials science, discerning the significant differences between the results becomes a complex task, which is why this work suggests the use of statistical methods to facilitate research. Understanding the differences between experimental and statistical approaches, researchers can make better decisions about the design and analysis of future studies. Integrating both methodologies, one can enhance the robustness and reliability of the results [21].

1.3. Statistical Techniques

Statistical tools like ANOVA, linear regression, and t-tests are commonly used for evaluating significant differences of experimental results [22]. This type of analytical tool is widely used in natural and exact sciences for parametric statistics in addition to the realization of mean comparisons between groups of interest [23]. This evaluation is crucial to validate the effectiveness of the TT realized in this work [24]. Knowing the parameters of hypothesis tests, one can determine if a difference is statistically significant or not, depending on the parameters of the experiment. Previous studies have shown that a combination of experimental and statistical approaches provides a more comprehensive understanding of the phenomena being investigated. A simple moving average (SMA) statistical forecasting method finds practical applications in various disciplines, including medicine, psychology, engineering, and materials science. This versatility underscores its relevance and applicability in research. The SMA model is beneficial for calculating forecasts for the upcoming period, whether based on previous quarterly data or a specific period [25]. An SMA of order k, SMA (k), is the value of k consecutive observations [26]. Normality is crucial in many parametric and nonparametric statistical methods. The nonparametric tests are attractive because they do not require an assumption of the normal distribution; even though the data come from normal distributions, these nonparametric tests do not sacrifice reliability in comparison to tests based on the normality assessment [27]. As the statistician Kolmogorov [28] mentioned, there are four statistical tests that are commonly used for evaluating normality from the sampled data: the Kolmogorov–Smirnov [29,30], Anderson–Darling [31], Cramer-von Mises [32], and Shapiro–Wilk tests [33]. It is well established that the null hypothesis of the normality test state appears when the data are sampled from a normal distribution. Therefore, if the p-value is greater than the predetermined critical value (α = 0.05), then the null hypothesis is accepted, and thus, it concludes that the data are normally distributed. The important factor to consider is the sample size because it influences the outcome of the statistical tests. The Shapiro–Wilk (S-W) test requires the sample size to be between 3 and 50 [33,34]. The Kruskal–Wallis (K-W) test, proposed in 1952, is a rank-based non-parametric test that can be used to determine if there are statistically significant differences between two or more groups of an independent variable on a continuous or ordinal dependent variable. The K-W test is a more generalized form of the Mann–Whitney U-test and the non-parametric of the one-way ANOVA test [33]. As described by Nwobi and Akanno [35], the K-W test is an alternative to the ANOVA F-Student test, as it has been demonstrated to have a greater potential than the F-Student test when the population distribution is not normal [36,37]. According to [38], the K-W test with ranked data can be employed particularly when the following are true:

a.

The data are ordinal and do not meet the precision of interval data;

b.

There are serious concerns about extreme deviation from normal distribution;

c.

There is a considerable difference in the number of subjects for each comparative group.

The K-W test’s hypotheses are as follows: The null hypothesis (H₀): the population median is equal between groups. The alternative hypothesis (H₁): the population median is not equal [39]. To calculate the K-W value test, the procedure follows the next steps [40].

1.

Determine the ranks (use average rank in the case of ties).

2.

Determine the overall average rank.

3.

Determine the number of cases in each category, determine the average rank per category, and square the difference with the overall average rank, then multiply by the number of cases in the category.

4.

Sum up the last column.

5.

For each rank, subtract the overall rank average and square the result.

6.

Sum up the results of step 5.

7.

Determine the H-value statistics. The exact distribution of H is complicated. It depends on the sample size, n₁, n₂, …, n_k, and so it is not practical to tabulate its values beyond a small number of samples. When k or n is large, the exact distribution of H under H₀ can be approximated by the ${\chi }^2$ distribution with (k − 1) degrees of freedom. For this purpose, we state the K-W theorem without proof.

8.

Determine the degrees of freedom.

9.

Use a ${\chi }^2$ bilateral distribution to determine the critical value.

To determine if H₀ is accepted or rejected, one evaluates the critical value if it is greater than the calculated H; thus, H₀ is kept, and there is no difference; in other words, reject H₀ if the H value > critical value.

Now, the Spearman rank correlation coefficient (also called Spearman’s ρ) is a non-parametric measure. It is used when the data are generally not distributed between two variables. The Spearman correlation (rs) is used for the correlation of ranks; it assumes that the variable A has rank (RA) and that the variable B has the rank (RB), and assuming that (d) represents the difference between the two ranks (d = RA − RB), from the smallest to the largest ones [41]. The differences between the positive and negative ranks RA, RB, and rs are calculated from the Pearson correlation coefficient (R). rs takes a value from +1 to −1. If rs = +1, then it indicates a perfect positive correlation. If rs = −1, it indicates a perfect negative correlation [42,43].

When the study produces little data (<30 elements), the non-parametric method is used. However, when the sample is extremely small, <12 elements, the statistical treatment can produce misleading bias [44]. In materials science research, statistical methods are employed for mathematical modeling data projecting for the generation of simulated data forecasts. The statistical method SMA for simulating data forecasts is very common and is one of the widely known technical indicators used to predict future data in a time series analysis. During its development, researchers have made many variations and implementations [45].

Thus, the present work has the aim of studying if the TT induces phase changes in the TiWN and TiWC sputtered layers considering structural, elemental composition, and morphological properties. The coating properties and experimental growth methods were previously published in [46].

2. Materials and Methods

2.1. Coating Growth

The base pressure of the sputtering system was 4 × 10⁻⁴ Pa; the target–substrate distance was 100 mm and the substrate temperature was 300 °C. For improvement of the mechanical adhesion, a pure Ti layer was deposited with a power of 450 W, applying a voltage bias of −10 V also at 300 °C. The Ar plasma was created with a mass flow of 50 sccm, and for the nitride and carbide compounds, the mass flow for N₂ and CH₄ was 12 and 16 sccm, respectively. The rf power of 420 and 350 W was applied on Ti and W targets, respectively. The thickness measurement was realized by profilometry; the thicknesses were 3.54 ± 0.04 μm and 2.95 ± 0.08 μm for the TiWN and TiWC coatings, respectively.

2.2. Preparation of Sample and Thermal Treatment

The coatings on AISI 1065 substrates (~2.5 × 2.5 cm²) commercial steel carbon (Alambiques 2981, Alamo Industrial, Tlaquepaque, Jalisco, México), were prepared as follows. The samples were cut using a precision diamond saw (VC 50, LECO, St. Joseph, MI, USA), using a cutting speed of 8 m/min in dimensions to obtain sections of 1.25 × 1.25 cm². Later, the samples were cleaned with isopropanol prior the TT. The samples were set into a quartz tube furnace (70 mm in diameter and 1000 mm in length) with a 450 mm heating zone (OTF-1200X, MTI Corporation, Richmond, CA, USA). The samples were subjected to TT under an Ar flow of 3 sccm and at a temperature of 500 °C under a base pressure of 6.7 Pa for 120 min and left cooling freely. The coatings will be labeled hereinafter as TiWN-AS, TiWN-TT, TiWC-AS, and TiWC-TT, where AS and TT represent the as-sputtered and thermally treated samples, respectively.

2.3. Characterization

The micrographs were acquired from portable, Dino-Lite Digital Microscope, JNYZ59419 model (Chongxin Road, Sanchong Dist., New Taipei City 241513, Taiwan, China); focus ranges from 15 to 40 mm using the minimum magnification of 50×. The images of the surfaces after TT were acquired with a field emission-scanning electron microscope (FE-SEM) (JSM-7401F, JEOL, Akishima, Tokyo, Japan) incorporating a cold cathode field emission gun, using a 30 kV acceleration voltage. Secondary and retractable backscatter electron detectors (Akishima, Tokio 196-8558, Japan) were also used. Energy dispersive X-ray spectroscopy (EDS), (EDAX, a semiquantitative technique, was employed to evaluate the evolution of the elemental composition (weight percent, wt.%) before and after TT. The X-ray diffraction (XRD) patterns (PANalytical, X´Pert PRO, Eindhoven, The Netherlands) were recorded in Bragg-Brentano geometry using a diffractometer Philips Panalytical Xpert equipment (PANalytical, Eindhoven, The Netherlands) with Cu-kα (λ = 1.5405 Å) at room temperature (RT) collecting patterns in the 2θ range (20° < 2θ < 80°) with a scanning step size of 0.05° and a time count of 1.5 s for each point.

2.4. Statistical Analysis

To determine the significance level of the TT on the structural properties, the entering variable of crystallite size (response variable) in TiWN-AS and TiWC-AS coatings was determined by the Scherrer formula; the statistical treatment was carried out using the t-test and Spearman correlation coefficient method. The statistical analysis was carried out using © 2019 Minitab-19 (Chicago, IL, USA) software free version. Statistical analysis selected the sample-dependent T-Student mode and Mann–Whitney tests with a small sample size considering the experimental data by XRD high deflection (preferential orientation plane). The performance of treatment statistics was carried out using a reliability of 95% and a significance of 5%. For a crystallite grain size study, we used the SMA forecast method.

3. Results and Discussion

3.1. Data Experimental Analyses

3.1.1. XRD Patterns

Figure 1 shows the XRD pattern of the sputtered coatings realized at room temperature and annealed at 500 °C. In both coatings, there is a slight increase in the peak intensity after TT. The sharp peaks in the XRD pattern indicate the excellent crystallinity of the annealed coating. This behavior can be attributed to the process of grain refinement promoted by TT. According to previous research work published by [46], the XRD pattern of the TiWN-AS layer before deposition shows two secondary phases corresponding to T₂N and W₂N identified as tetragonal cell (PDF 01-080-3438), and simple cubic (ICDD PDF 00-025-1257), respectively. After TT, the positions of the diffraction peaks are similar. However, all the peaks of the XRD pattern show a right shift. This is attributed to residual stress and strain [47]. Considering the intensity of peaks in each TT, it is possible to infer that a not-significant difference exists using statistical methods, such as the non-parametric one. One can observe that the peaks of the substrate are located at 2θ = ~45° (110) and 2θ = 65° (200) indexed within the ICDD Data Card 04-014-0360 [48].

3.1.2. Determination of Crystallite Size by Scherrer Formula

The crystallite size is estimated using the full width at half maximum (FWHM) of the diffraction peaks. The crystallite sizes of the AS and TT samples are calculated using the Scherrer formula [49], written as follows:

$$D={\displaystyle \frac{0.9\lambda }{{\beta }_{hkl}cos{\theta }_B}}$$

(1)

where ${\beta }_{hkl}$ is the FWHM of the peak in radians, ${\theta }_B$ is the Bragg angle, and λ is the wavelength of incident X-ray (λ = 1.54 Å).

The micro-strain is the root mean square of variants in the lattice parameters across the sample, expressed by the following Equation (2):

$$\epsilon ={\displaystyle \frac{\beta }{4 tan \theta }}$$

(2)

where $\epsilon$ is the micro-strain in radians, β is the line broadening at FWHM in radians, and θ is the Bragg angle in degrees. The dislocation density ($\delta$) is a measure of the number of dislocations in a unit volume of a crystalline material, i.e., the length of dislocation lines per unit volume (m/m³), written as follows in Equation (3):

$$\delta ={\displaystyle \frac{1}{D^2}}$$

(3)

where D is the average crystallite size in nm.

In

Table 1. Crystallite size, micro-strain, and dislocation density.

Material	Peak Intensity 2θ (°)	FMHW β (°)	Crystallite Size (nm)	Microstrain (ε)	Dislocation Density (δ)
TiWN-AS	42.56	0.40092	22.20 ± 0.63	4.49 × 10−3	2.03 × 10−3
TiWN-TT	42.54	0.38287	23.25 ± 0.72	4.29 × 10−3	1.85 × 10−3
TiWC-AS	36.88	1.15049	7.60 ± 0.45	1.51 × 10−2	1.73 × 10−3
TiWC-TT	37.22	1.10597	7.91 ± 0.26	1.43 × 10−2	1.60 × 10−3

, one can observe the crystallite size obtained by the Scherrer formula. The crystallite size increases slightly on sample TiWN from 22.2 to 23.2 nm upon annealing at 500 °C. Also, similar behavior was shown in the sample TiWC, where a slight increase was found from 7.6 to 7.9 nm. Also, a shift to the right of the peaks of both materials that were TT is observed. These results show that differences in crystallite size are like those found by [50]. They described that the grain sizes of Ni-W alloys increased to 6.3 to 9.9 nm after annealing at 500 °C.

This phenomenon is due to crystallinity improvement [51], i.e., the grains were refined through atomic diffusion due to an increase in temperature [52]. The increase in crystallite size is due to the kinetic energy and reconfiguration of the atoms caused by the grain boundary migration and sub-grain growth during annealing [53]. The micro-strain ε also has evolved with a reduction from 4.49 × 10⁻³ to 4.29 × 10⁻³ in sample TiWN after TT.

The TiWC-AS sample showed similar behavior upon annealing at 500 °C. This trend is also found in [54], where ε depends on the crystallite size indicated by the stoichiometry of the coating, which in turn causes volumetric expansion. At low δ, the grain boundary blocks the dislocation displacement. TiWN-AS δ exceeds a critical value; thus, dislocations are absorbed at grain boundaries, causing an increase in the surface dislocation density and the misorientation between adjacent grains [55]. As described by Lakhotkin and Kukushkin [56], δ shows the relationship between the increase in grain size and the reduction of dislocation density.

Crystalline Size Forescated Data by SMA

An SMA of order k, MA (k), is the value of k consecutive observations. According to [29], the calculation of forecasting is expressed as follows (see Equation (4)):

$${\mathrm{F}}_{\mathrm{t}+1}={\displaystyle \frac{{\mathrm{y}}_{\mathrm{t}}+{\mathrm{y}}_{\mathrm{t}-1}+{\mathrm{y}}_{\mathrm{t}-2}+\dots +{\mathrm{y}}_{\mathrm{t}-\mathrm{k}+1}}{\mathrm{k}}}$$

(4)

where k is the number of terms in the moving average. In an SMA the value of F_t+1 depends on k. An SMA is a particular case of a weighted moving average in which the exact weight is assigned to all the average data.

The crystallite size available in Table 1 can forecast an increase in sample size of n = 12 using the SMA; the results are shown in

Table 2. Crystallite size forecasted by SMA.

Sample/Data	Experimental Data			Forecasting Data									Statistic Parameter
	1	2	3	4	5	6	7	8	9	10	11	12	Av.	S.D.	S.E.
TiWN-AS	21.69	22	22.91	22.2	22.37	22.49	22.36	22.41	22.42	22.39	22.41	22.41	22.34	0.29	0.08
TiWN-TT	23.72	22.41	23.61	23.25	23.09	23.32	23.22	23.21	23.25	23.23	23.23	23.24	23.23	0.31	0.09
TiWC-AS	7.63	8.02	7.13	7.59	7.59	7.44	7.54	7.52	7.5	7.52	7.52	7.51	7.54	0.20	0.06
TiWC-TT	7.62	8.03	8.1	7.91	8.01	8.01	7.98	8	8	7.99	8	7.99	7.97	0.12	0.03

. It can be observed that the average crystallite size forecast decreases from 0.14 to 0.07 nm in the TiWN-AS and TiWC-TT samples. In contrast, the TiWC-AS sample has a smaller decrease of 0.02 and 0.06 nm, respectively. This phenomenon can be attributed to the increase in sample size, where the variance was reduced.

Figure 2 shows a column/bar + error plot from experimental and forecasting data using the SMA method. An analysis of experimental data in sample TiWN shows that when the coating is thermally treated, it promotes an increase in the average crystallite size, and their variance is reduced. Sample TiWC shows a similar trend. Now, when the experimental data are forecasted at n = 2, the meaning of each one is similar, exhibiting a closer variance, sample TiWN AS vs. TT data. Sample TiWC also shows similar performance when considering the average means, but it observes that the variance is kept between response variables. The size of the sample has influenced the standard deviation. This characteristic is because the population means of the distribution of sample means are the same as the population means of the distribution being sampled. Therefore, the means of the distribution of the means never change. The standard deviation of the sample means, however, is the population standard deviation from the original distribution divided by the square root of the sample size. Thus, as the sample size increases, the standard deviation of the means decreases, and as the sample size decreases, the standard deviation of the sample means increases. Consequently, larger sample sizes increase power and decrease estimation error [57].

3.1.3. Top View by Optical Microscopy

Figure 3 shows the top view of TiWN and TiWC surfaces observed by optical microscopy before and after TT. The surface color of the TiWN-AS sample is light gray. However, when the sample is thermally treated at 500 °C, the surface color turns goldish. The change in the surface color is due to the enrichment of Ti atoms, indicating the formation of TiN [57]. The goldish color is validated with the EDS analysis corresponding to 4.81 wt.% of Ti before annealing at 500 °C and 10.59 wt.% after. Sample TiWC exhibits a color change from gray to black, indicating an increase in carbon concentration in the regions corresponding to the W enrichment from 61 to 74 wt.% after thermal annealing [58] (see

Table 3. Elemental composition (wt.% by EDS).

Figure/ Sample	Point Selected	C	N	O	W	Ti
(a) TiWN-AS	1	3.75	10.82	9.20	75.54	0.69
	2	3.54	11.95	11.09	70.01	3.41
	3	-	-	18.35	67.75	13.9
(b) TiWN-TT	1	33.0	-	29.35	31.39	6.26
	2	-	-	54.91	29.15	15.94
	3	-	-	9.35	74.41	16.24
(c) TiWC-AS	1	-	15.3	12.74	63.53	8.43
	2	19.06	-	20.15	58.72	2.07
(d) TiWC-TT	1	4.07	-	12.45	72.6	10.88
	2	-	6.15	12.40	78.46	2.99

).

3.1.4. Scanning Electron Microscopy

Figure 4 is the top view of the surface of as-deposited coating (TiWN-AS, TiWC-AS, and those of coatings annealed at 500 °C (TiWN-TT, TiWC-TT samples). Figure 4a shows the TiWN-AS sample as a dense coating with a smooth texture with few surface defects (nodules). However, the thermal treatment in the TiWN-TT sample promotes an irregular surface (cracks and flower-like nodules). The TiWC-AS coating shows defects such as cracks and small nodules (Figure 4c). However, when the coating is subjected to thermal treatment (Figure 4d), it reveals a higher crack density and crack width. It has been reported that cracking along grain boundaries can be attributed to improper heat treatment and material contamination, i.e., carbide precipitation along grain boundaries [59]. The increased crack growth can be associated with a higher release rate of tensile stress with increasing annealing time [60].

The elemental composition revealed 18.9 wt.% of C and 15.3 wt.% of W from 58.56 to 63.53 wt.% on the surface. However, sample TiWC-TT exhibits larger cracks and the apparition of circular nodules while the elemental composition corresponds to a mixture of 4.07 wt.% C, 12.45 wt.% O, 72.60 wt.% W, and 10.88 wt.% Ti. According to [61], smoothness is probably due to an increase in the tungsten concentration. The creation of nodules is generally associated with phase transformations due to TT, grain growth, and grain coarsening [61]. The cracks on samples of TiWN induced by the TT are a consequence of high stress localized at the crack edges. Higher temperatures are still necessary to generate a thermal gradient and stress to reduce the material ductility [62]. In general, considering the optical and electronic microscopy of each sample, one can infer that there exists a significant difference before and after TT.

3.1.5. Chemical Composition by EDS

Table 3 shows the elemental composition expressed as wt.%. In TiWN there is a significant evolution of N, O, and Ti between the AS and the TT samples. Its elemental composition is 3.75 wt.% of C, 10.82 wt.% N, 9.2 wt.% of O, and 75.54 wt.% of W, corresponding to point 1 in Figure 4a and Table 3. Also, it is observed that small nodules–top content wt.% similar (point 2). After thermal treatment, the N content disappears due to diffusion into the substrate [63], and the O and Ti concentrations were drastically modified as well. The selected points show an increase of O from 9.35 to 54.91 wt.%.

Conversely, the TiWC-AS coating has an average composition consisting of 19.06 wt% of C, 15.3 wt.% of N, and from 58.72 to 74.41 wt.% of W. However, in the TiWC-TT sample, the Ti concentration is increased, probably due to the diffusion process after annealing. It is important to mention that the compositions of Ti, W, and N (TiWN-AS) and Ti, W, and C (TiWC-AS) are not stoichiometric; the mismatch can be attributed to atomic diffusion towards the interface with the substrate or towards the coating surface. In general, the surface has an elemental composition corresponding to the mixture composition compound from N (from 6.15 to 15.30 wt.%), O (from 9.20 to 54.91 wt.%), C (from 3.54 to 33.0 wt.%), W (from 29.15 to 78.46 wt.%), and Ti (from 2.07 to 16.24 wt.%) according to elemental composition by EDS (Table 3). Therefore, there exists a significant composition difference between AS and TT samples.

3.2. Statistical Results

3.2.1. Normality Test

Considering the crystallite sizes in Table 2, the data are probably normally distributed because of our meticulous testing process. The data are rigorously tested using the Shapiro–Wilk method, ensuring the accuracy of our findings. The proposed hypotheses (H_i) are as follows:

H₀. 

TiWN-AS = TiWN-TT (the crystallite sizes have a normal distribution).

H₁. 

TiWN-TT ≠ TiWN-AS; TiWN-AS ≠ TiWN-TT (the crystallite sizes do not have a normal distribution).

3.2.2. Dependent Relationship Paired Sample Test

As shown in Table 2, the crystallite size expressed in nm is executed by a hypothesis test to elucidate if the results have a significant influence using statistical methods. According to Uchiyama et al. [13], the crystallite size has a direct relationship with the TT temperature; the crystallite size increases as temperature increases. For this reason, the hypotheses are expressed as follows (where µ indicates the average crystallite size):

H₀. 

µTiWN-TT > µTiWN-AS (The crystallite size is larger).

H₁. 

µTiWN-TT ≤ µTiWN-AS (The crystallite size is smaller).

H₀. 

µTiWC-TT > µTiWC-AS (The crystallite size is larger).

H₁. 

µTiWC-TT ≤ µTiWC-AS (The crystallite size is smaller).

3.2.3. K-W Hypothesis

The theory indicates that thermal treatments induce crystal grain growth. In contrast, it can be inferred the opposite behavior (no changes in shift and intensity of the peaks of the XRD patterns). Thus, the hypotheses should be described as follows:

H₀. 

µTiWN-TT = µTiWN-AS (The crystallite size is equal).

H₁. 

µTiWN-TT ≠ µTiWN-AS (The crystallite size is different).

H₀. 

µTiWC-TT = µTiWC-AS (The crystallite size is equal).

H₁. 

µTiWC-TT ≠ µTiWC-AS (The crystallite size is different).

3.2.4. K-W Procedure Step-by-Step

The procedure of the calculation of H and C is shown from Tables S1–S4. The experimental data of the crystallite size are reorganized using new nomenclature: A = TiWN-AS; B = TiWN-TT; C = TiWC-AS, and D = TiWC-TT samples (see Table S1). Later, the statistical treatment was carried out considering the groups A and B (Table S3). Groups C and D follow on Table S4. In all three cases, the critical value (C-value) and H value were calculated; meanwhile, the K-W test parameters are summarized in

Table 4. Statistical parameters of normality distribution, relationship dependence, and K-W tests.

Sample	Normality Test	Dependent (rs)	K-W
	Shapiro–Wilk (p-Value)	Spearman Rank (p-Value)	H-Value (Bilateral)		C-Value
			N = 48	N = 12	N = 48	N = 12
TiWN-AS	0.034	0.252	42.30	15.19	7.81	3.84
TiWN-TT	0.010
TiWC-AS	0.010	−0.343		12.40
TiWC-TT	0.010

Significance, α = 5%.

.

Table 4 shows a normal distribution relationship with K-W tests, considering the crystal size from all samples. In reference to the normality test, all series have an abnormal distribution (0.034, <0.010, <0.010, and <0.010, for the groups A, B, C, and D, respectively); we suggest that the data are not normally distributed [64]. Now, the relationship between TiWN-AS and TiWN-TT samples exhibits a weak Spearman correlation coefficient of 0.252. However, when the TiWC-AS and TiWC-TT samples are related, the rs has a weak negative coefficient (−0.343), indicating that there is a poor dependence between the variables from negative to positive. This behavior describes the inverse relationship between both responses [65]. Furthermore, the K-W test gives the criteria to determine if H₀ is accepted; the condition H < C must be satisfied. Here, H = 42.30 (when N = 48) vs. C = 7.81. Thus, H₀ is rejected. A similar behavior exhibited statistical treatment considering two groups, i.e., groups A and B show H = 15.19 and C = 3.84 when N = 12, and groups C and D show H = 12.40 vs. C = 3.84. Then, with a confidence of 95% and a significant error of 5%, it is statistically inferred that all samples contrasted with the K-W test; the crystallite sizes are different between them in the same group as well as individual samples.

4. Conclusions

Thermal treatments (TTs) on TiWN and TiWC coatings were performed at 500 °C under an Ar atmosphere. The phases identified by XRD analysis do not evidence any significant change upon TT. The metal–ceramic systems are quite stable, even secondary phases: W₂N, T₂N, Ti₃C₂, and W₂C remain stable. It is verified that an increase in temperature promotes the increase in crystallite size. Optical microscopy revealed color change after TT in both coatings. The surface observed by scanning electron microscopy evidenced experimentally the significant change in the crack width after TT. However, the statistical analysis using non-parametric methods on the series data, specifically the crystalline grain size, showed a normal distribution. The Spearman analysis indicated a weak positive rs = 0.252 and negative correlation rs = −0.343, in the relationship analysis between TiWN-AS vs. TiWN-TT and TiWC-AS vs. TiWC-TT, respectively. The T-Student test further confirmed a proportional relationship, suggesting that if the TT temperature increases, so does the crystallite size on the TiWC-TT sample. The K-W test showed that with a confidence level of 95% and a significant error of 5%, it is statistically inferred that all samples contrasted with the K-W test; the values of crystallite sizes are different between them despite the group or individual sample. The statistical methodology proposed in this work is a reference to determine whether in a set of quantitative data, from different groups (treatments), they are equal or different on the response variable when the sample size is small (n < 30).