# Optimisation of Mechanical Properties of Gradient Zr–C Coatings

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}/TiN [5]. Moreover, in this type of coating, discontinuities of stress and deformation (step change in values of tensor components) occur at the boundaries of the layers, which significantly increases the probability of the formation of cohesive cracks and local delamination at the interfaces [5].

_{1}, low abrasive wear W

_{2}and high fracture toughness W

_{3}. Hence, one of the important problems connected with the design of the coating’s structure in the case of multilayer coatings is the selection of the chemical composition, mechanical properties and thickness of individual layers and, in the case of FGM coatings, the selection of spatial distribution of its chemical composition, in order to achieve the maximum values of W

_{1}–W

_{3}[12,13,14,15,16,17]. In this field, numerical simulations of the state of residual stresses and strains in the substrate-coating systems, arising after the deposition process, play a special role due to the significant impact of residual stresses on the mechanical and tribological properties of the coatings [18,19,20,21,22,23,24,25]. The analysis of stresses and deformations initiated as a result of external mechanical loads provides additional information about potential wear of coatings in various tribological tests, including ball-on-disc or pin-on-plate tests [26,27,28,29] and resistance to cracking in the scratch tests [30,31,32].

## 2. Materials and Methods

#### 2.1. Optimisation

#### 2.1.1. Object

_{t}), (2) exponent of the power transition function describing the continuous change in carbon concentration in the gradient layer (p). The domain of decision variables is defined as follows

_{ZrC}

_{(base)}and C

_{ZrC}

_{(top)}denote the carbon concentration at the boundaries of the adhesive layer/gradient layer and gradient layer/top layer, d

_{total}—total coating thickness, d

_{t}—gradient layer thickness. The form of the transition function defined by Equation (2) makes it possible to identify the change in the exponent p with the change in the carbon concentration in the coating. Figure 2 shows the carbon concentration profiles for the analysed set of parameters (p), for fixed thickness of gradient layer (d

_{t}).

- The steel substrate is treated as a homogeneous continuous medium;
- Substrate and layer materials are elastic-plastic bodies represented by a so-called bi-linear model with work hardening;
- Before deposition process in substrate initial stresses were 0 GPa;
- At the boundary between the substrate and the coating mesh nodes are connected—no separation allowed;
- Spatial distribution of carbon concentration in gradient layer is represented by a continuous power transition function;
- A homogeneous temperature distribution was assumed in the samples during the deposition of the coatings—the deposition temperature value was 400 °C.

#### 2.1.2. Decision Criteria

_{0}—maximum contact pressure, µ—coefficient of friction, a—radius of contact. The maximum value of the contact pressure and the contact radius determined for the elastic deformation range differ significantly when taking into account the substrate’s plastic deformation. Namely, in the case of taking into account the plastic deformations of the substrate and the coating, the contact area increases and the maximum value of the contact pressure decreases compared to those obtained for pure elastic contact [42,43,44]. Exemplary distributions of normal contact pressure ${p}_{N}\left(x,y\right)$ on the surface of the top layer, initiated by a spherical indenter with a radius of 200 µm, obtained on the basis of FEM simulation, are shown in Figure 4a elastic contact, Figure 4b elasto-plastic contact. In this paper, for the purposes of optimisation in further calculations, only the ${p}_{N}\left(x,y\right)$ distribution obtained from elasto-plastic contact will be used (Figure 4b).

_{1}criterion was the weighted sum of the fraction of positive first-principal stresses σ

_{I}and their average value in a given volume of the coating. The explicit formula of this criterion is

_{2}criterion was adopted the maximum value of I principal stress in the zone of interface of adhesive layer/gradient layer (Figure 1).

_{1}and K

_{2}criteria were already tested as potential measures of coating’s resistance to cracking (through thickness perpendicular cracks) and the mechanical integrity of the adhesive layer/gradient layer interface [35,45,46].

_{3}was the maximum value of equivalent plastic deformations in the adhesive layer

_{4}was the maximum value of the module of shear stresses σ

_{xy}in the adhesive layer/gradient layer interface (Figure 1)

_{5}criterion as the maximum value of III principal stress in the coating.

_{5}could be a significant indicator connected with the probability of coating buckling.

_{6}criterion

_{zy}stresses on the adhesive layer/gradient layer interface was adopted as the criterion K

_{7}.

_{7}criterion was introduced due to the relationship of shear stresses in the plane perpendicular to the direction of the indenter motion, and the probability of chipping and delamination due to recovery spallation after the disappearance of the external load at the indenter-coating contact boundary [32,45,46,48]. The proposed set of decision criteria is a synthesis of various relationships, known in the literature, between the stress and strain distributions in the coatings with the probability of occurrence inter alia: coating’s spallation, conformal and angular cracking, brittle tensile cracking. According to the definitions of the introduced criteria, their simultaneous minimisation is required for providing enhanced resistance for the occurrence of coating’s failure mode, thus optimal coating prototypes should be characterised by low values of the adopted criteria. The scaling of the adopted decision criteria to a dimensionless form significantly facilitates the analysis of the set of obtained solutions in the criteria space. Thus, criteria were normalised as follows

_{i}

^{min}and K

_{i}

^{max}denote, respectively, the minimal and maximal value of criteria in the defined domain (1). In the further presentation of the results, only normalized criteria will be analysed without upper index (n), i.e., K

_{i}

^{(n)}≡ K

_{i}.

#### 2.2. Experimental Details

#### 2.2.1. Coatings Deposition

_{a}= 0.023 µm, R

_{z}= 0.187 µm. (R

_{a}—Arithmetical mean deviation of the assessed profile, R

_{z}—Maximum roughness height according to ten profile points). Grinding was carried out with a semi-automatic grinder with water-cooled SiC grain sandpaper. The papers were changed gradually from granulation 100 to 2000. The surfaces were then polished with pastes with a monocrystalline diamond suspension with an average grain size of 3 µm, then 1 µm, on a polishing cloth of acetate fibers. Before the deposition process, the samples were washed with acetone at 58.08 g/mol and rinsed in an ultrasonic cleaner with EcoShine Ultrasonic K3 solution containing <5% of each individual of the three nonionic surfactants sodium 5–15 edetate (EDTA), <5% corrosion inhibitor then they were rinsed in deionized water and dried. In the working chamber, substrates were subjected to ion cleaning at a pulse voltage of a glow discharge with an amplitude of 1.5 kV at an Ar pressure of 7 Pa. Sputtering was performed from a metallic Zr target, with a diameter of 114 mm supplied by PLANSEE Composite Materials, Lechbruck, Austria. The target was made by powder metallurgy. Composition: Zr + Hf min. 99.2 wt.%, Hf max. 4.5 wt.%, Fe + Cr max. 2000 µg/g, H max. 50 µg/g, N max 250 µg/g, C max. 500 µg/g, O max. 1600 µg/g, at fixed discharge power of 500 W in a mixed Ar and C

_{2}H

_{2}atmosphere by controlling the flow rate of both gases. Different profiles of carbon concentrations in the coatings were obtained by changing the flow rate of C

_{2}H

_{2}in the range of 1.5–6.5 sccm. The magnetron source was powered with pulse voltage with a frequency of 1 kHz with a signal modulated by 100 kHz. Substrates were polarised with negative 10 V bias voltage, and their temperature during all the deposition processes was stabilised in the range of 400 ± 20 °C.

#### 2.2.2. Characterisation Methods

^{®}4.3 software. To determine the value of Young’s modulus and hardness using the Olivier–Pharr model [51], indentation tests were carried out with a given maximum force and constant penetration rate, obtaining load vs. depth curves.

_{c1}) was determined, as well as the load value at which the coating was completely removed from scratch path (L

_{c3}).

## 3. Results and Discussion

#### 3.1. Simulation Results

#### 3.1.1. Criteria Values in the Decision Variables Domain

_{0}, is necessary to calculate the values of the adopted decision criteria, using the created FEM models, for each coating prototype in the domain of decision variables. For this purpose, single-layer Zr-C coatings with a carbon concentration of 21 at.%, 51 at.%, 66 at.%, 74 at.% and 84 at.% on HS-6-5-2 steel and Si substrates were produced. Details on the methodology of sample preparation and coating deposition conditions are included in the authors’ previous works [37,38,39]. Then, experimental indentation tests were performed and the values of Young’s modulus and hardness in the function of carbon concentration were determined (Figure 5a,b) using procedures discussed in Section 2.2.

_{1}–K

_{7}on values of parameter p, of transition function, and thickness of gradient layer (d

_{t}), were determined (Figure 7a–g).

_{i}= K

_{i}(d

_{t},p), i = 1, 5, 6 indicate the presence of relatively large changes in the value of the criteria in the neighborhood of the decision variable p = 1 for each thickness d

_{t}. It follows that the prototype with p = 1 is a kind of a limit prototype in the neighborhood of which significant differences in mechanical properties of prototypes can be expected. It should be noted that in the case of the K

_{i}= K

_{i}(d

_{t},p), i = 1, 7 (criteria related to the potential resistance to tensile cracks and chipping initiation), the increment in gradient layer thicknesses (d

_{t}) results in a strong differentiation of the prototypes due to their potential anti-wear properties.

#### 3.1.2. Coating’s Prototypes

_{1}, K

_{2}, K

_{3}, K

_{4}, K

_{5}, K

_{6}, K

_{7}) in the 7-dimensional space of decision criteria, the distance d = d(d

_{t},p) was determined for each of them from the so-called utopian solution represented by the origin of the coordinate system.

_{t},p), which values are below the level of 0.3, was determined, i.e.,

_{t},p) and N = N(d

_{t},p) the quality index was defined in the form

_{t},p) was 0, the index J(d

_{t},p) was not calculated. The introduced J(d

_{t},p) quality index was the basic criterion for selecting the best solutions—coating prototypes. It contains information about the distance of the tested solution from the so-called utopian solution (the best possible theoretical acceptable solutions) in the criteria space and, at the same time, supports the selection of solutions for which individual criteria are below the value of 0.3. In further considerations, it was assumed that the level of 0.3 is a limit value below which the criterion, scaled to the fuzzy variable, is at the Low (L) level. The J(d

_{t},p) index has been defined in such a way that its minimum value corresponds to the best coating prototype. For the analysed domain of decision variables (1), the chart of J(d

_{t},p) is presented in Figure 8.

_{t}= 1 μm, the differences in the values of the criteria between the individual prototypes are low; therefore, these prototypes should have similar anti-wear properties. If the gradient layer reaches about 50% of the total thickness of the coating, an increase in the differences in the values of the criteria between individual prototypes is observed, which leads to their diversification in terms of potential anti-wear properties. Maximum differences in the values of the criteria are achieved in the case of a gradient layer with a thickness of d

_{t}= 2 μm, which is about 80% of the total thickness of the coating. Thus, in order to obtain the highest diversification of the prototypes in terms of their mechanical properties, prototypes with the highest gradient layer thickness were selected for further experimental research. In particular, prototypes were selected for which the index J(d

_{t},p) has a minimal or maximal value (Table 1). They are potentially the best and the worst solutions in the considered domain of decision variables D. The table also includes a representative of the group of solutions, with the value of curvature parameter of the power transition function equal to p = 1, characterised by a linear increase in carbon concentration in the area of the gradient layer—limit prototype (Section 3.1.1).

_{t},p) compared to prototypes with p > 1 for the same thickness of the dt gradient layer. Moreover, among the prototypes with p < 1, there is a prototype ensuring the global minimization of J(d

_{t},p), while among the prototypes with p > 1, there is a prototype with the maximum value of this quality index. Hence, in further considerations, in order to draw general conclusions about the design of optimal carbon concentration profiles, the solutions will be divided into three groups together with their representatives (Table 1), i.e., p < 1; (J(d

_{t},p)→min), p = 1 and p > 1; (J(d

_{t},p)→max). Spatial distributions of the Young’s modulus and yield strength for selected prototypes are shown in Figure 9.

#### 3.2. Experimental Results

#### 3.2.1. Scratch Test

_{c1}and L

_{c3}critical loads are presented in Figure 11a–f.

_{c1}= 13 N), the so-called Chevron (angular) cracks, belong to the group of brittle tensile cracks. These cracks arise behind the indenter as a result of tensile stresses, whose values exceed coating tensile strength [56,57,58]. With these coatings, a further indenter load increment results in the compressive spallation, initiated in front of the indenter in the high-compressive-stress zone. It should also be emphasized that in the case of this prototype, there is a zone of high internal stresses (−4 to −4.7 GPa), directly under the top layer, which add up to the compressive stresses caused by the indenter, which additionally increases the rate of energy release and the size of initiated cracks. The coating’s complete delamination occurs for the value of L

_{c3}= 37 N.

_{c1}= 11 N), with few chippings and areas of delamination along the path, are so-called conformal cracks and are typical for relatively ductile coatings. They result from the action of compressive stresses in front of the indenter, which then slides over the fracture surface and pushes them into the path. With a further increase in the indenter load, buckling spallations become larger and spread to the sides of the scratch path. Consequently, this leads to the complete coatings detachment as the formed cracks extend over the entire thickness of the coating [56,57,58].

_{c1}= 23 N) among the tested prototypes. Coating damages in the scratch path, as in the case of the prototype with p = 1, belong to the ductile failure modes group. However, in the case of the prototype with p < 1, no buckling spallations or wedging are observed, which is characteristic of highly ductile coatings possessing relatively low hardness. For this prototype, only conformal cracks are observed, with a small amount of chipping along the scratch path until the coating is fully delaminated from the substrate. The critical force L

_{c3}, for this prototype is 60 N and is about two times higher than in other prototypes.

#### 3.2.2. Wear Test

_{1}, S

_{2}, S

_{3}—cross section of wear track, mm

^{2}, r—wear track radius, mm.

^{−7}mm

^{3}/N∙m was approx. two times lower compared to the prototypes with p = 1 and p > 1, for which the values of volumetric wear indicator were, respectively, W = 2.5·10

^{−7}mm

^{3}/Nm and W = 1.9·10

^{−7}mm

^{3}/Nm. The differences in the values of wear indicators are directly correlated with the obtained graphs of the values of COF vs. sliding distance (Figure 12) for the tested prototypes. Namely, the prototype with p < 1 characterised by the lowest volumetric wear ratio, possesses the lowest value of COF and relatively small fluctuations in its value vs. sliding distance.

#### 3.3. Summary Results

_{t}), and (2) exponent of the power transition function, describing the continuous change in carbon concentration in the gradient layer. Seven decision criteria were adopted, being the functions of the stress and deformation states, which simultaneous minimisation allows for obtaining the optimal distribution of stress in terms of preventing the occurrence of cracks under a given load. In order to verify the proposed optimisation procedure, Zr-C coatings, with different spatial distributions of carbon concentration described by power functions, were produced. Table 2 presents the collective results of simulations and experimental tests for the analysed prototypes.

_{1}–K

_{7}criteria were rescaled to a fuzzy-logic variable in the form:

- ${K}_{1-7}\in \langle 0;0.1\rangle $—VL (Very Low);
- ${K}_{1-7}\in \left(0.1;\right.0.3\rangle $—L (Low);
- ${K}_{1-7}\in \left(0.3;\right.0.5\rangle $—M (Moderate);
- ${K}_{1-7}\in \left(0.5;\right.0.7\rangle $—H (High);
- ${K}_{1-7}\in \left(0.7;\right.1\rangle $—VH (Very High).

_{1}, K

_{2}, K

_{3}, K

_{4}, K

_{7,}are at the VL level, and the other two criteria, K

_{5}and K

_{6,}at the L level. At the same time, this prototype has the highest values of L

_{c1}and L

_{c3}and the lowest wear rate among the tested prototypes. For the prototype with p = 1, four of the examined criteria, K

_{1}, K

_{3}, K

_{4}, K

_{6,}are at the VH level, the K

_{7}criterion is at the H level, and K

_{5}and K

_{2}are at the L and VL levels, respectively. Along with the increase in the value of the above-mentioned criteria compared to the value of these criteria for the prototype with p < 1, a significant decrease in the value of the critical loads is observed by about 50%, as well as an almost 2.5-times increase in the wear rate.

_{3}, K

_{4}, K

_{6}are also at the VH level, and the criteria K

_{1}and K

_{7}at the H level; however, the main difference from the prototype with p = 1 is the fact that the values of the K

_{3}and K

_{4}criteria take the extreme value equal to 1 and the J(d

_{t},p) index takes the maximum value from the tested range. This means that in the case of this prototype, extreme and unfavorable effects deteriorating the anti-wear properties should be expected. Although, for the prototype with p > 1, the values of L

_{c1}and L

_{c2}are slightly higher compared to the prototype with p = 1, extensive buckling spallations are observed in the scratch test in this prototype. The obtained results clearly indicate that the prototypes with p > 1 are characterised by lower resistance to brittle cracking and delamination compared to the prototypes with p = 1 and p > 1. At the same time, prototypes with p > 1 are characterised by a much higher value of the J index than prototypes with p = 1 and p < 1, for which this index has the lowest values. It follows that the adopted index J(d

_{t},p) can be regarded as a kind of measure of the coating resistance to the occurrence of so-called catastrophic failure modes. The obtained results also prove that using only the critical load L

_{c1}as a measure of the quality of the coating adhesion to the substrate and its fracture toughness is ambiguous, because coatings with similar L

_{c1}values may have different types of cracks with the same normal force value in the scratch test.

## 4. Conclusions

_{t},p) quality index was developed, which facilitated the diversification of prototypes in terms of all criteria at the same time, creating subsets of solutions corresponding to the coatings with the best and worst possible anti-wear properties. Using the defined index, it was determined that optimal coatings should be characterised by:

- Exponent of the power transition function p < 1 (J(d
_{t}, p) → min) and the thickness of the gradient layer d_{t}at 80% of the total thickness of the coating; - The asymmetric Lorentzian distribution of Young’s modulus and hardness (Figure 9), the maxima of which occur at a distance of about 20% d
_{t}from the adhesive layer interface; - A strong decrease in hardness and Young’s modulus starting from the maximum, up to the top layer interface.

_{1}–K

_{7}criteria—the values of VL and L in the fuzzy-logic variable space.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**General structure of optimisation object (functionally graded material (FGM) coating) with mesh grid.

**Figure 4.**Distribution of normal contact pressure on the surface of the top layer initiated by a spherical indenter with a radius of 200 µm (

**a**) elastic contact, (

**b**) elasto-plastic contact.

**Figure 5.**Dependence of the hardness (

**a**) and Young’s modulus; (

**b**) of Zr–C coatings on the concentration of carbon.

**Figure 6.**Dependence of the residual stresses, after deposition process, in Zr–C coatings on the concentration of carbon.

**Figure 7.**Dependences of K

_{1}–K

_{7}(

**a**–

**g**) on values of parameter p and thickness of gradient layer (d

_{t}).

**Figure 8.**Quality criterion $J\left({d}_{t},p\right)$ as a function of parameter (p) and thickness of gradient layer (d

_{t}).

**Figure 9.**Spatial distributions of the Young’s modulus (

**a**) and yield strength (

**b**) for selected prototypes (p < 1, p = 1, p > 1).

**Figure 11.**Microscopic photos of scratches for selected prototypes: p > 1 (

**a**,

**b**), p = 1 (

**c**,

**d**), p < 1, (

**e**,

**f**).

**Figure 13.**Wear ratio with cross-section of wear tracks for selected prototypes after ball-on-disc test, sliding distance 2000 m.

p | d_{t}, µm | K_{1} | K_{2} | K_{3} | K_{4} | K_{5} | K_{6} | K_{7} | N | J |
---|---|---|---|---|---|---|---|---|---|---|

0.25 | 2 | 0.019 | 0.003 | 0.068 | 0 | 0.193 | 0.232 | 0 | 7 | 0.044 |

1 | 1.75 | 0.853 | 0.064 | 0.778 | 0.760 | 0.115 | 0.820 | 0.622 | 2 | 0.93 |

3 | 2 | 0.557 | 0.051 | 1 | 1 | 0.405 | 0.799 | 0.597 | 1 | 1.864 |

p | d_{t}, µm | K_{1} | K_{2} | K_{3} | K_{4} | K_{5} | K_{6} | K_{7} | J | L_{c1}, N | L_{c3}, N | W 10^{−7} mm^{3}/Nm | H, GPa |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.25 | 2.00 | VL | VL | VL | VL | L | L | VL | 0.05 | 23 | 60 | 1.08 | 21 |

1.00 | 1.75 | VH | VL | VH | VH | L | VH | H | 0.87 | 11 | 31 | 2.5 | 23 |

3.00 | 2.00 | H | VL | VH | VH | M | VH | H | 1.86 | 13 | 37 | 1.9 | 25 |

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**MDPI and ACS Style**

Szparaga, Ł.; Bartosik, P.; Gilewicz, A.; Mydłowska, K.; Ratajski, J.
Optimisation of Mechanical Properties of Gradient Zr–C Coatings. *Materials* **2021**, *14*, 296.
https://doi.org/10.3390/ma14020296

**AMA Style**

Szparaga Ł, Bartosik P, Gilewicz A, Mydłowska K, Ratajski J.
Optimisation of Mechanical Properties of Gradient Zr–C Coatings. *Materials*. 2021; 14(2):296.
https://doi.org/10.3390/ma14020296

**Chicago/Turabian Style**

Szparaga, Łukasz, Przemysław Bartosik, Adam Gilewicz, Katarzyna Mydłowska, and Jerzy Ratajski.
2021. "Optimisation of Mechanical Properties of Gradient Zr–C Coatings" *Materials* 14, no. 2: 296.
https://doi.org/10.3390/ma14020296