High-Throughput First-Principles Prediction of Interfacial Adhesion Energies in Metal-on-Metal Contacts

Adhesion energy, a measure of the strength by which two surfaces bind together, ultimately dictates the mechanical behavior and failure of interfaces. As natural and artificial solid interfaces are ubiquitous, adhesion energy represents a key quantity in a variety of fields ranging from geology to nanotechnology. Because of intrinsic difficulties in the simulation of systems where two different lattices are matched, and despite their importance, no systematic, accurate first-principles determination of heterostructure adhesion energy is available. We have developed robust, automatic high-throughput workflow able to fill this gap by systematically searching for the optimal interface geometry and accurately determining adhesion energies. We apply it here for the first time to perform the screening of around a hundred metallic heterostructures relevant for technological applications. This allows us to populate a database of accurate values, which can be used as input parameters for macroscopic models. Moreover, it allows us to benchmark commonly used, empirical relations that link adhesion energies to the surface energies of its constituent and to improve their predictivity employing only quantities that are easily measurable or computable.

Cohesive Energy (eV) Cohesive Energy  Figure S1: Comparison with the reference values for the lattice parameter, bulk modulus, cohesive energy, and surface energy. The reference values for the lattice parameter, the cohesive energy and bulk modulus are from Ref., S1 whereas the surface energies are taken from the Materials Project database. S2 S-2 Figure S2: Raw data for charge redistribution calculated for the heterogeneous interfaces. The data are ordered following the value of E adh of the correspondent homogeneous interfaces. Moving along the columns, the colour scale helps understanding how the charge transfer varies when the particular surface is matched to all the other ones. S-3

Coefficients from the SISSO algorithm
We report here the correlation coefficients, the root mean square errors and the best fit coefficients for the formulas Φ 0 , Φ 1 , Φ 2 found by the SISSO alghoritm in order to express E adh as algebraic functions of single components properties. Model Training score Cross-validation score Figure S4: Learning curves for the three relationships identified with the SISSO algorithm, Φ 0 (left panel), Φ 1 (center panel), Φ 2 (right panel). The red (green) curves represent the training (validation) score for each model.
We can test the effectiveness of the SISSO models generated in Eqs. 2-4 of the main manuscript by computing the learning curves, as shown in Fig. S4. These plots show the normalised training and validation scores when we vary the size of the training dataset.
Moreover, this approach helps to identify whether our dataset is large enough to obtain reliable results. We generated the learning curves by employing the Scikit-learn library available in Python. S4 The graphs in Fig. S4 exhibit a similar behaviour for all the proposed model, with a small gap between the training and validation score, which confirms the accuracy of our models. Furthermore, both scores remain unchanged when a training set half the size of the original dataset is considered, meaning that there is no need to increase our dataset to improve the training.

Testing SISSO algorithm without specific outliers
A further analysis in creating predicting models for the adhesion of heterogeneous interfaces is to groups the different materials in more homogeneous sets. For example, we can consider only the transition metals and excluding Al, Mg and Ti. We made this choice because Al and Mg are the only simple metals present in our set, whereas Ti requires an exceedingly large number of atomic layers to get a converged surface energy influencing the calculation S-5 of adhesion performed on thinner slabs. When we apply the SISSO algorithm in this new subset, we obtained the following relationships: Model These prediction, compared to those present in the main manuscript, are significantly better: the correlation coefficient is above 0.9 for all the feature spaces and the RMSE is reduced between 20% and 24%.
7 Considering additional descriptors for SISSO predictions As suggested in the main manuscript, we can increase the number of descriptors to further reduce the RMSE. In particular, we identified additional descriptors about the surface com-S-6 mensurability by using the geometric average of the ratios between the slab and supercell areas defined as follows: We also identified the geometric average of the fractional part in the ratio between the supercell and slab areas as a term of commensurability between the elements present in an interface. We defined this quantity as: By using these new descriptors, we obtained the following relationships: R 2 Φ 1 = 0.92 RMSE Φ 1 = 0.37 J/m 2 R 2 Φ 2 = 0.93 RMSE Φ 2 = 0.35 J/m 2 Table S4: Coefficients with corresponding units for Eq. 6, 7 and 8. Model