**5. Results and Discussion**

#### *5.1. Gaussian Process Regression and Visualization*

For the above-described initial dataset created from a design of experiments approach, different GPR models were trained. Before training the different models, the dataset was scaled to only contain values between 0 and 1. This was especially useful for GPR, to reduce training effort and stabilize the optimization of the model parameters. The main difference between the different GPR models was the used kernel function for the gaussian processes. The used GPR supports a variety of different kernel functions which were optimized during the training of the GPR model. It was found that with a dot product kernel with some additional white noise the prediction capabilities of the model were enhanced to reach a mean absolute error of around 440 MPa. Moreover, the root mean squared error was around 387 MPa. This results in an CoP of around 90%, which means that the prediction quality and quantity is acceptable to classify this model for a prediction model. For model training, a train-test-split of 80–20% was used and the training data was shuffled before training. The overall prediction quality is a notable finding since the dataset used for training is relatively small. Here also GPR with little white noise show their strengths on sparse datasets. However, model performance can further benefit from more data. This prediction model is also capable of visualizing the prediction space, see Figure 3.

**Figure 3.** Predicted space in a 20-color colormap for better differentiation between the different areas of resulting hardness for minimum combined gas flow.

The striped pattern emerges from the usage of a 20-color-based colormap for drawing. This is done to further show the different sections of the predicted data. The whole plot can be viewed as a process map. In order to find the ideal coating properties, the tribology experts need to look for their color in indentation hardness and then easily see the bias voltage and sputtering power needed. For tuning purposes, the gas flow can be changed via the slider at the bottom. The plot for the maximum combined gas flow is depicted in Figure 4.

**Figure 4.** Predicted space for maximum combined gas flow.

The space for lower indentation hardness is getting bigger and the highest indentation hardness of around 4.2 GPa vanished. This correlates with the experience made from initial experimental studies. It was expected that the gas flow—especially the C2H2 gas flow [15]—influenced the hydrogen content and thus the mechanical properties and further affected the tribologically effective behavior. Based on these visualizations, it can be easily seen which parameters lead to the desired indentation hardness. This visualization technique benefits the process of where to look for promising parameter sets for ideal indentation hardness.

For validation of our model, we performed another experimental design study based on a Box–Behnken design with 3 factors and two stages (see Table 4). Initially, the indentation hardness was predicted using our GPR model. Subsequently, the GPR model was evaluated—after coating the specimens—by determining the indentation hardness experimentally. For illustrative purposes, the prediction of the central point, which was deposited at a sputtering power of 3 kW, a bias voltage of 200 V, and a combined gas flow of 108 sccm, is shown in Figure 5. In this context, it should be noted that the prediction space included a significant extension of the training space and thus could be influenced by many factors.

**Figure 5.** Predicted extended space for probe points.


**Table 4.** Summary of main deposition process parameters and predictions for a-C:H on UHMWPE, prediction of *H*IT by the GPR model as well as experimental determination of *H*IT based on the average values and standard deviations of the different a-C:H coatings (*n* = 10).

As shown in Figure 5 and Table 4, the *H*IT values of the previously performed prediction of the GPR model largely coincided with the experimentally determined *H*IT values. Especially with regard to the standard deviation of the experimentally determined *H*IT values, all values were in a well-usable range for further usage and processing of the data. Despite a similar training space, the prediction for the coating variations P1–P4 showed a slightly lower accuracy than for the coating variations beyond the training space, but this could be attributed to the difficulty of determining the substrate-corrected coating hardness. Thus, during the indentation tests, the distinct influence of the softer UHMWPE substrate [54,55] was more pronounced for the softer coatings (P1–P4), which were coated with lower target power than for the harder coatings (P5–P13). However, the standard deviation of the hardness values increased with hardness, which could be attributed to increasing coating defects locations and roughness. In brief, the predictions match with the implicit knowledge of the coating experts. This is the only physical conceivable conceptual model that can be considered when looking at the results presented, as the coating deposition is a complex and multi-scale process.

Though the visualization of the prediction space in Figure 5 differed slightly from the prediction spaces in Figure 3 and in Figure 4 due to steeper dividing lines, the prediction space in Figure 5 spanned larger coating process parameter dimensions.

Generally, the prediction quality and especially the quantity of the model was very good, so the model can be used for further coating development processes and adjustments of the corresponding coating process parameters. An extension of the GPR model to other coating types, such as ceramic coatings, e.g., CrN, or solid lubricants, e.g., MoS2, or different coating systems on various substrates is conceivable.
