**3. Results**

Figure 3a shows a typical BSE micrograph taken from a location approximately 14 mm away from the pure titanium end of the sample strip aged at 500 °C. In this micrograph, the lamellated hcp (hexagonal closest packing) α-Ti and bcc (body-centered cubic) β-Ti are visible as the darker and brighter regions, respectively. The EDS measurements also show less than 1 wt.% of manganese for the darker phase and about 15 wt.% of manganese for the brighter phase, thereby identifying these regions as α and β titanium, respectively. A map sum spectrum was also taken, measuring the average manganese content at 5.8 wt.% for this scan. Similar measurements were carried out at each location identified in Figure 1b for each sample strip (i.e., each composition–post-heat treatment combination). The results are presented in Figure 3b. As expected, it is seen that the variation of the Mn content along the strip is highly consistent between the different strips. The manganese composition rises from the pure titanium end but peaks at about 20 mm and stays at about 12 wt.%. Note that this 12 wt.% Mn is a little lower than the target composition of 15 wt.%. The maximum manganese composition is present over a few millimeters at the end of the build, as programmed into the original motion control source code. The deviations between the obtained local composition and the targeted composition are attributed to variations in the local elemental powder being fed or, as is the case here, due to an intentional extension of a region with the maximum Mn concentration. Since our primary interest in the present study is the development of a framework for establishing the correlations between the processing parameters, microstructure statistics and the properties, we have not iterated with different starting powder mixtures to attain specific compositions in the produced samples. Instead, our focus will be on the protocols needed to acquire, efficiently, the material data needed for the targeted correlations.

℃ **Figure 3.** (**a**) Back-scattered electron (BSE)-SEM image for the sample strip aged at 500 °C for four hours and at the location where the Mn content was 5.8 wt.%. It depicts the dual-phase microstructure of the sample, where the darker phase is α-Ti and the brighter phase is β-Ti. (**b**) Means and standard deviations from the energy dispersive spectroscopy (EDS) measurements of the Mn content at the five locations for all three high-throughput (HT) sample strips produced for this study. For clarity, all 500 ◦C and 700 ◦C values are intentionally shifted slightly in the negative and positive x directions, respectively. All points in each group correspond to the same nominal distance indicated by the axis ticks. 

 – Multiple BSE micrographs were obtained corresponding to each combination of manganese composition and post-heat treatment temperature. The volume fractions estimated from the segmented images are shown in Figure 4. It is seen that the β volume fraction increased with Mn content and with the temperature of the post-build aging treatments. This is because post-build aging at a higher temperature pushes the microstructure to be close to its equilibrium state. Note that the high manganese locations subjected to the low 500 ◦C treatment (see the bottom left micrograph in Figure 5) produced a small-scale (10–100 nm) secondary α phase [33] in addition to the bigger (~2 µm) primary α laths. Such secondary α is expected, especially at these lower temperatures, and results when new nucleation events become more favorable and accelerate the rate of transformations. – 

deviations of the percentage volume fractions of the β phase obtained **Figure 4.** Means and standard deviations of the percentage volume fractions of the β phase obtained for the different Mn contents and post-build aging heat treatments.

deviations of the percentage volume fractions of the β phase obtained

 **Figure 5.** Segmented SEM-BSE images for the sample library produced and studied in this work. The left, middle and right columns correspond to aging heat treatments of 500, 600 and 700 ◦C, respectively. The rows correspond to different locations exhibiting different manganese compositions (see Figures 1b and 3b). The black phase in these micrographs represents α-Ti, while the white phase represents β-Ti. 

 For the computation of the averaged CLs, all the chords in the micrograph were collected at intervals of 2.5 degrees to avoid imaging orientation bias. The averaged value of all the collected chord lengths for each phase at each of the five sample locations identified is reported in Figure 6. The averaged CL of the dominant β phase decreased consistently with an increase in the Mn content. By contrast, the averaged chord length of the β phase increased with a higher manganese content, with the higher aging temperature promoting a more drastic change. 

 **Figure 6.** Averaged chord lengths (CLs) of (**a**) α phase and (**b**) β phase at the selected five locations for all three high-throughput (HT) sample strips studied in this work.

Spherical indentation tests were performed on a grid of twenty-five sites for each of the five sample locations (see Figure 1b,d) for all three sample strips. Figure 7 summarizes the measured values of elastic moduli, indentation yield strengths and indentation hardening rates at each of the five locations on all three strips studied in this work. It is observed that the measured indentation moduli did not show significant variations between different locations and between different sample strips. On the other hand, a strong positive correlation was observed between the Mn content (which also correlated well with the beta volume fraction (see Figure 4)) and the indentation yield strength as well as the indentation hardening rate.

– Young's modulus, ( **Figure 7.** Mechanical properties estimated from the spherical indentation stress–strain protocols: (**a**) Young's modulus, (**b**) indentation yield strength and (**c**) indentation initial hardening rate. The blue, green and red boxes correspond to the 500, 600 and 700 ◦C aged strips, respectively.

 It is clearly seen that the various microstructural features (β volume fraction and the averaged CLs of the α and β regions) and the resulting mechanical properties are highly correlated with each other. In order to analyze the effects of the process conditions (i.e., the aging temperature and Mn content) on the microstructural features and the resulting mechanical properties, it is necessary to conduct a statistical analysis. Gaussian process regression (GPR) was employed in this work for this purpose. As mentioned before, the hyperparameters of the kernel function provide reliable insights into the sensitivities of the different inputs to the outputs of interest.

A separate GP was built for each of the six outputs listed in Table 1, while using the post-build aging temperature and Mn content as features (i.e., independent variables). Traditionally, GP models are built to provide predictions for new inputs. However, in the present application, the size of the dataset is too small to formally establish a reliable predictive model with rigorous cross-validation. Therefore, it was decided to use the GP models to provide reliable insights into the sensitivities between

the various measured quantities in this study. The interpolation length scale parameters established by these GP models and summarized in Table 1 are ideally suited for extracting such insights. As a specific example, it is seen that the interpolation length scale hyperparameter for the aging temperature in the GP model for the averaged CL for the α phase is very large, especially compared to the corresponding values obtained for the GP models for the other five outputs. This indicates a much lower sensitivity of the averaged CL of the α phase to the aging temperature. In fact, the averaged β-CL and the indentation hardening rate are found to exhibit the highest levels of sensitivity to the aging temperature. The table also indicates that all the microstructural parameters exhibited strong sensitivity to the Mn content, with the β volume fraction showing the highest sensitivity.

**Table 1.** Gaussian process regression (GPR) interpolation length hyperparameters and the mean absolute percentage error (MAPE) for each of the six outputs selected for these models. CL denotes the averaged chord length, VF is the volume fraction, Y is the indentation yield strength, and E is the Young's modulus.


A comparison of the output scaling factor σ*<sup>f</sup>* , which controls the overall spread of the output values in the entire dataset with the output noise parameter σ*n*, provides insight into the combined overall predictive capability of the GPR model. The ratio σ*<sup>f</sup>* /σ*<sup>n</sup>* is referred to as the output-to-noise ratio and reflects the capability of the selected inputs in influencing the predicted output. For example, a very high value of σ*<sup>f</sup>* /σ*<sup>n</sup>* obtained in a specific GPR model indicates that the selected inputs (i.e., the Mn content and the aging temperature) are able to reliably account for most of the observed variations in the selected output in the collected dataset. In other words, the GP models with high values of σ*<sup>f</sup>* /σ*<sup>n</sup>* are indeed more mature and can be used reliably in making predictions for new inputs. In Table 1, it is seen that the GPR models for the elastic modulus and the β volume fraction show very high values of σ*<sup>f</sup>* /σ*n*, indicating that these models are able to account for almost all of the measured variations in these quantities in the data aggregated in this work. Similarly, a low value of σ*<sup>f</sup>* /σ*<sup>n</sup>* might suggest a lack of adequate correlations between the selected inputs and the output. This could suggest that there is inherently more noise in the measured values of the selected output, the possible existence of as-yet-unidentified inputs influencing the output variable, or both. In Table 1, the lowest value of σ*<sup>f</sup>* /σ*<sup>n</sup>* was obtained for the averaged CL for the β regions. In this study, we believe this is because of the inherent noise resulting from the protocols used to estimate this attribute from the micrographs (i.e., the segmentation and CL protocols). In other words, if one intends to establish more accurate correlations for the averaged CL for the β regions, it would be prudent to improve the protocols used to extract this value.

Table 1 also summarizes the mean absolute percentage error (MAPE) using a leave-one-out cross-validation strategy. This entails obtaining a model by setting aside one data point at a time in establishing the GPR model and subsequently testing the obtained model on the excluded point. The process is then systematically repeated for all available data points, and the MAPE is computed based on the obtained errors. It is seen from Table 1 that the GPR model for the elastic modulus exhibits the highest accuracy, while the GPR models for the averaged CL for the β regions exhibited the lowest accuracy. It is also seen that this is consistent with the σ*<sup>f</sup>* /σ*<sup>n</sup>* values.
