Analyzing microstructure relationships in porous copper using a multi-method machine learning-based approach

A need for an efficient annotation and deep learning microstructure segmentation

We conduct FIB-SEM tomography to image three different porous copper materials. For the first and second sample set, we use sinter pastes consisting of micro- and nanoparticles. Those sample sets are indicated as hybrid-paste material A (HPA) and B (HPB), respectively. The third sample set is composed of nanoparticles and is labeled as nano-paste material C (NPC), see Methods for further sample details. For the investigation of the microstructure evolution upon temperature, six sinter temperatures with 175 °C, 200 °C, 225 °C, 250 °C, 350 °C, and 400 °C are selected, see Fig.1a. Each reconstructed 3D dataset comprises about 450 images with an image size of 1120 × 640 pixels² which makes an automated analysis approach indispensable, see Methods for further details regarding the image acquisition.

Here, we aim for an accurate classification of two phases, associated with the pore and copper, respectively, using a U-Net deep learning architecture from Chollet³⁶ (see Supplementary Note 1 and 2). For an accurate prediction, the need of an efficient training for the U-Net model is apparent^25,37. Figure 1b shows a representative cross section obtained from the SEM-FIB tomography data. Here, the segmentation of the so-called shine through artifact³⁸ within the pore is challenging. Infiltration of the pore³⁸ might work but would extend the expenditure. In the following we compare various segmentation approaches. As shown in Fig. 1b, a conventional threshold algorithm (CTA)³⁹ segmentation, as shown as yellow overlay, is not able to deal with all shine through artifacts within the pore phase. The CTA assigns the pore phase correctly with darker gray values but fails for regions with similar gray values to the copper phase. Therefore, we conduct a hybrid approach to provide a more accurate training for the U-net model. For this semi-automatic hybrid approach³⁵, a two-step procedure is used. First, the selection of a threshold between the pore and copper in the second derivative histogram plot is selected, see Fig. 1c. Figure 1d illustrates with a color map the associated pore and copper phases, based on the selected threshold at the first peak of the second derivative histogram. Within the second step we conduct the simple linear iterative clustering (SLIC) algorithm for superpixel generation⁴⁰, see Fig. 1e. Each superpixel segment’s average gray value is calculated and is segmented with the threshold obtained in the first step. Figure 1f depicts the segmented pore phase utilizing the introduced hybrid method. The segmentation includes also the non-detected shine through artifacts. Figure 1g illustrates the annotation for the U-net model, see Supplementary Note 1 and 2 for further details. We are able to annotate, based on the results from the hybrid segmentation, 0 for copper, 1 for the pore detected by the conventional threshold, and 2 for the shine-through artifacts. More details with respect to the training are provided in the Methods section. Figure 1h, i shows the output for the pores in 2D and both pores as well as copper in 3D, respectively utilizing the U-Net segmentation trained with the hybrid model.

We validate the segmentation performance using known metrics⁴¹ such as Jaccard Index, precision, recall, and accuracy, see Table 1. A manually segmented dataset with the support of the Avizo software is used as the ground truth. Further details are presented in Supplementary Note 3. Overall, the U-Net using the hybrid annotation workflow provides the best performance. Our work shows that a U-Net architecture with semi-automatic annotation, is capable to provide segmentation with an accuracy of up to 94%, which is higher than previously reported with U-net architectures⁴².

Three-dimensional copper network formation upon sintering

For an improved design of the microstructure in context to the targeted property, it is essential to understand the correlation between the physical descriptors describing the microstructure and the property. Figure 2a illustrates the segmented 3D microstructure for sample HPA, HPB and NPC, utilizing the U-net model based on the semi-automatic hybrid annotation. Here, exemplary the segmented volume of interest (VOI) for 175 °C is shown. The microstructure exhibits significant differences for the three materials HPA, HPB and NPC. NPC indicates a nano-porous network with a porosity of 45.2% at 175 °C. HPA and HPB show micron-sized as well as nano-sized pores within the VOI, however, they differ in the porosity with 42.4% and 61.7% at 175 °C, respectively. Figure 2b shows the densification^43,44 of the three porous copper configurations upon sintering. Here, the relative density D is defined as the ratio of the copper volume to the total volume of the VOI. The hybrid-paste HPA and nano-paste NPC exhibits, compared to HPB, a rather similar behavior for the densification. The changes in the relative density from 175 °C to 400 °C for HPA and NPC material are about 18.5% and 20.8%, respectively. The hybrid-paste HPB shows the highest porosity and depicts a different behavior upon sintering. Here, the densification gives about 7.3%.

We inverse the specific resistivity obtained from the 4-point probe measurement to obtain the electrical conductivity σ. The superior electrical performance of the nano-paste is demonstrated in Fig. 2c. Although the relative densities D for HPA and NPC are similar, there are significant differences of 50.2 μS.cm⁻¹ and 84.3 μS.cm⁻¹ in the electrical characteristics of σ_NPC and σ_HPA at 350 °C and 400 °C, respectively.

For the evaluation of the copper strut diameter ϕ and its evolution upon sintering, we skeletonize and statistically analyze the segmented data²⁴ (Fig. 2d, e). The mean values are obtained by fitting the log-normal distribution of the strut diameter histograms (Supplementary Note 4). We observe an increase of ϕ with temperature. This behavior is linked to the continuous growth of the bonds between the sinter particles due to the increase in temperature⁴⁵. The NPC material indicates a small variation of the copper strut diameter. The 95% confidence interval lies within a range of 0.01–3.5 nm and suggests a highly homogenous distribution of the copper network as indicated for the volumetric microstructure data in Fig. 2d. For HPA and HPB, the confidence interval lies within 2.4–36.4 nm and 0.7–23.6 nm, respectively, and is significantly larger than for the NPC material.

Evolution of the copper strut-interconnectivity and investigation on the surface properties upon cycling

Another significant microstructural feature concerns the connectivity of the copper strut and its evolution upon sintering, see Methods. We use the geodesic tortuosity τ⁴⁶ as a measure for the strut interconnectivity. A high tortuosity of the copper relates to a small copper strut interconnectivity. The geodesic tortuosity is defined by the ratio of the geodesic distance to the Euclidian distance. Figure 3a illustrates the evaluated 3D tortuosity along the y-direction, which conforms to the direction from the surface to the substrate, exemplary for 175 °C, 250 °C and 400 °C. The tortuosity τ is quantified by averaging the extracted tortuosity values along the y-direction of the last 25% from the sample’s length³⁸. Further information with respect to the tortuosity is provided in Supplementary Note 5 where we calculate the overall tortuosity in five directions and average them.

As indicated by the 3D tortuosity distribution in Fig. 3a and the calculated averaged tortuosity in Fig. 3b, the HPB material indicates the highest tortuosity of the copper strut in comparison to HPA and NPC. As shown in Fig.3b, for HPA and NPC the average tortuosity decreases between 175 °C to 400 °C from 1.039 to 1.013 and from 1.030 to 1.008, respectively. The NPC material, as indicated in Fig. 3a, illustrates the lowest tortuosity distribution within the analyzed VOIs for all temperatures.

During the sintering, the surface area is reduced by the growing of bonds between the sinter particles. The driving force for the sintering decreases as the surface area is annihilated. The decline of the specific surface area SA with the sinter temperature is shown in Fig. 3c. At 175 °C, the specific surface area for HPA and NPC is about a factor of two larger than for HPB. SA_HPA decays very fast from 175 °C to 200 °C. Consequently, it starts to decay slowly after 200 °C. SA_HPB and SA_NPC decays in a more constant manner with temperature. The sample HPA and NPC reaches a similar SA at 400 °C. The value for HPB is about 30% smaller.

Further, we investigate the pore-copper interface evolution upon sintering using the Gaussian curvature G and mean curvature M⁴⁷. Both curvatures classify the local surface geometries with their joint distributions⁴⁸. Figure 3d depicts the G–M curvature joint distributions of the copper surfaces for different sinter stages. In the first (QI) and second (QII) quadrants, two tails extend to high mean curvature values. The changes of those tails with temperature indicate the change of the copper particles’ convexity. The copper particles’ local geometries at the lower sinter temperature, display mostly cup-convex surfaces resembling spheroidal structures⁴⁹, therefore, their Ms are positive. The sintering process reduces the sphericity of the particles. As a result, the emerging tails in QI are getting less pronounced by increasing the temperature for all presented materials. The observed behavior is complemented with an increased number of cup-concave geometries indicated by more pronounced tails in QII.

The intensity maximum close to the origin of the graph shows negative Gs for all samples. For HPB the magnitude of G is decreasing which suggests an increase of the neck’s radii during sintering. The sinter process also causes a reduction of the particles’ convexity leading to a decrease of the mean curvature for the higher porosity material HPB (Fig. 3d). Supplementary Note 6 provides the G and M at each sinter temperature. At 175 °C for the material HPA, G_HPA is −146 μm⁻². Its negative value indicates that necks with small radii saddle-like surfaces are prevalent. The positive M_HPA of 2.18 μm⁻¹ shows that the incidence of convex structures is quite dominant. Therefore, spherical geometries of HPA’s nanoparticles are still abundant. This type of surface illustrates that HPA is still in an early stage of sintering⁵⁰; therefore, the particles are just starting to coalesce and their necks are newly formed. As the temperature is increased, the particles are coarsened. As a result, the necks radii are flattened and its G magnitude is decreased towards zero. Subsequently, the small radii nanoparticles’ convex surfaces are diminishing and M decreases. This trend continues and G_HPA and M_HPA reaches −16 μm⁻² and −1.72 μm⁻¹ at 400 °C, respectively.

G_NPC and M_NPC at 175 °C is −102 μm⁻² and −0.93 μm⁻¹, respectively. NPC shows a more positive G than for HPA. Therefore, the necks radii are larger than for HPA. The negative value of M_NPC indicates the reduction of convex surfaces for the nanoparticles. Here, the material is in a more advanced stage than HPA at this temperature. Indeed, the electrical conductivity σ_HPA and σ_NPC at 175 °C is 2.3 μS.cm⁻¹ and 78.2 μS.cm⁻¹, respectively. This finding is in line with the tortuosity analysis and it provides further insight into the enhanced electrical property of the material NPC.

Microstructure feature importance and mathematical relationship of microstructural features and electrical behavior

The evaluation of the microstructure features, their physical analysis, as well as their correlation to the material property display crucial ingredients for accelerated material design. In the previous sections, we qualitatively tried to explain the correlation between the extracted microstructure features and the electrical behavior of the material. The understanding of such correlations, however, is challenging due the underlying complexity and multi-faceted problems. Here, we establish a mathematical relationship between the microstructure and property by applying a machine learning-based deployment in the form of a linear regression model^1,51, see Methods.

We build various multi-variable linear regression (MVLR) models that shall enable us to provide the microstructure features or independent variables as an input and the targeted electrical conductivity or dependent variable as a predicted output. As a result, the structure-property relationship can be defined arithmetically. We train the models with at least two microstructure features obtained from the segmented VOIs. The coefficient’s sign of each feature is used to indicate the dependence of the feature with respect to the electrical conductivity, see Supplementary Table 1. We use a leave-one-out cross-validation (LOOCV) to obtain reliable and unbiased results⁵² for the training, see Methods. We test different MVLR models based on different microstructural feature combinations, see Supplementary Table 2. For the prediction of the electrical conductivity, we define three different microstructure sets, as an input for the MVLR, which are not used as training data sets, see Supplementary Note 8. Subsequently, the outcomes are compared with the experimental data.

The various MVLR-based models are summarized in Table 2. Figure 4a illustrates the regression analysis for the models A, C and I. The models labeled as A, C, and I incorporate different raw features, like the relative density, specific surface area, strut diameter, average tortuosity, and Gaussian and mean curvatures. Model A, C and I provide R² values of 0.956, 0.956, and 0.960, respectively. Here, the linearity lies within the range of about 10–200 µS.cm⁻¹. Outside of this conductivity range, predictions are inaccurate. In order to enhance the prediction, we perform feature engineering to boost the model performance with mathematical transformation⁶. Therefore, we augment two additional features, labeled with α and β, taking mathematical correlations between different features into account. α incorporates the correlation between the relative density D and the tortuosity τ according to the Bruggeman equation⁵³ with τ = D^–α. The second feature is defined by β = M/(|G|. ϕ). Further details are provided in Supplementary Note 7. The corresponding models utilizing the additional features are labeled with J, N, R and Q. The results are displayed in Fig. 4b and Fig. 4c respectively.

Indeed, as illustrated in Fig. 4b and c the utilization of the additional features improve the linearity from 0 to 285 μS.cm⁻¹. In particular, the MVLR-based model Q provides the best performance, as indicated by Table 2. Further, we assess the importance of the features for the model Q utilizing a SHapley Additive exPlanations (SHAP) analysis³⁴. The global impact of the features is calculated with the mean of the absolute SHAP values. Figure 4d illustrates the impact of each feature from the highest to the lowest. The analysis indicates that the feature α, which relates to the tortuosity and relative density by the Bruggeman equation, provides the highest impact on the electrical conductivity, followed by the specific surface area SA, and β. The results provide guidelines for the microstructure design, uncovering the most critical microstructural features for the electrical conductivity.

Synthetic image reconstruction of the porous microstructures for different sinter temperatures and electrical conductivity values

A promising approach to reconstruct the microstructure for a given material parameter is by utilizing deep generative models^14,54,55. In particular, we apply a denoising diffusion probabilistic model (DDPM) architecture. A DDPM is a parameterized Markov chain and consists of forward and reverse diffusion processes¹⁰. The forward process adds different Gaussian noise levels to the images, and the reverse process denoises the images with a neural network to find the added noise distribution to each training data. Original microstructure images can be reconstructed by removing the noise⁵⁶. By applying the trained model to an image sampled from pure noise, the model can denoise it to generate images similar to the real dataset⁵⁶, see Fig. 5 and the Methods section for further details in context to the DDPM. In addition, we compare the results obtained from the DDPM with a conditional generative adversarial network (cGAN). The generator within the cGAN architecture generates synthetic images at each training cycle based on the provided input. The discriminator determines the authenticity of the reconstruction. As the training progresses, synthetic images are produced by the generator, see Fig. 5 and Methods section.

Figure 5a, d, g show the segmented real microstructure indicated by the pore and copper phases for different sinter temperatures. For the segmentation the introduced U-Net architecture, trained with the hybrid model, is used. In addition, in Fig. 5b, e, h and Fig. 5c, f, i the reconstructed synthetic microstructure images depicted from the cGAN model and DDPM, respectively, are illustrated. Clearly the change of the microstructure with temperature is represented for both models. A quantitative performance analysis is important to assess the prediction result in more detail.

Evaluation of the model performance

The next step is to assess the quality of the reconstructed synthetic images. As illustrated here¹⁶, Frechet inception distance (FID) as well as precision and recall are not suitable to measure the quality of synthetic microstructure data. Here, we assesses the quality of the synthetic images based on extracted physical descriptors of the microstructure or microstructural features⁵⁷.

In particular we validate the model performance by comparing the evaluated relative density, specific perimeter, as well as shape index for the real and synthetic microstructures in relationship to the sinter temperature, see Fig. 6. The critical descriptors are also related to the electrical conductivity. Further, we extract the R² for the three descriptors individually, as well as the average of R² for all three physical descriptors, see Table 3. The presented assessment of the synthetic microstructures in Fig. 6 and Table 3 illustrate the superiority of the DDPM over the cGAN model. The largest deviation between the two models is observed for HPA and HPB. Both exhibit a more inhomogeneous microstructure than NPC, which makes the prediction with the GAN more challenging. Nevertheless, as illustrated in Fig. 6 and Table 3 even for the NPC material, which illustrates a homogenous nano-porous structure, the DDPM predicts better than the cGAN.