Probing the composition dependence of residual stress distribution in tungsten-titanium nanocrystalline thin films

Nanocrystalline alloy thin films offer a variety of attractive properties, such as high hardness, strength and wear resistance. A disadvantage is the large residual stresses that result from their fabrication by deposition, and subsequent susceptibility to defects. Here, we use experimental and modelling methods to understand the impact of minority element concentration on residual stresses that emerge after deposition in a tungsten-titanium film with different titanium concentrations. We perform local residual stress measurements using micro-cantilever samples and employ machine learning for data extraction and stress prediction. The results are correlated with accompanying microstructure and elemental analysis as well as atomistic modelling. We discuss how titanium enrichment significantly affects the stress stored in the nanocrystalline thin film. These findings may be useful for designing stable nanocrystalline thin films.

parameters for further analysis and the method can't be automatized.Gaussian blur is implemented to remove noise from the images which helps in better edge and line detection.
As shown in Fig. 2 (main text), the deflection marker is at the tip of the cantilever, which is used to calculate deflection from image analysis.This reference marker is detected by line detection.To perform efficient line detection, Gaussian blur is implemented.Once blur is applied, canny edge detection is applied to detect edges around the deflection marker.Then detected edges are used in Hough transformation to detect lines around the deflection marker.Lines are detected around the deflection marker.The number of pixels in the give image is calculated as a result of the difference in the detected lines of the beam and the reference marker on the fixed side of the beam.Using the image pixel size and the number of pixels, deflection for the single sublayer step is calculated.This process is automatized for all the images for a sample.Therefore, to measure the layer thickness same procedure is carried out with the difference of line detection position.In the end of image analysis, we obtain the deflection profile as a function of layer thickness.This deflection profile is then used to calculate the stresses analytically as well as finite element simulations.

Supplementary Note 3: DFT
For DFT calculations we used the Vienna ab initio simulation package with periodic boundary conditions and the projector augmented wave method (PAW) [1][2][3][4] .The detailed methodology used for the DFT calculations of Bulk WTi is given in Ref 5 .For the grain boundary calculations we employed the same input parameters as for the bulk calculations with the cells shown in main text Figure 5a.
We calculated segregation energies in the dilute limit according to: With GB and B standing for the total energy from a grain boundary and bulk cell, respectively, and the super script denotes the number of Ti atoms in the cell.For higher Ti bulk concentrations, the segregation energy was calculated from the formation energies as The formation energy is usually defined as where   is the total energy of an atomic compound (bulk, GB, or other structure),    is the bulk reference energy of element  in its ground state, and N is the number of atoms in the compound.So the formation energy is the energy it takes to assemble a specific atomic compound from its elemental ingredients.The GB is a somewhat special case as it is an extended defect where periodic boundary conditions enforced in the DFT calculations have to be considered.For the chosen GB setup with a GB slab that is separated on the opposite ends (vacuum on top setup), the formation energy for the GB reads Here,   ,  is the total energy of a cell containing a slab with a GB and two surfaces with   Ti atoms at the GB, whereas   , is the total energy of a similar slab but without a GB and containing only W atoms.For   = 0, the first two terms already constitute the formation energy of the GB.The last term gives the correction in case the GB contains Ti atoms by the reference energies for Ti and W, respectively.The formation energy is then normalized by the number of atoms in the GB, which in our case are all atoms in the central GB layer and the two layers next to it on both sides of the GB.The segregation energy obtained from the formation energies is an averaged quantity as the formation energies are already an averaged quantity.
Based on the segregation energies and the Ti concentration in the bulk, the concentration at the GB can be computed as a function of the temperature using the McLean isotherm 6 : The GB enrichment or interfacial enrichment (IFE) is then computed as With  the area of the GB and   the number of atoms in the GB.This quantity denotes how much more Ti atoms are present at the GB in comparison to the bulk.If the segregation energy depends on the GB concentration, i.e.   (   ), this equation needs to be solved iteratively.
Note that for reasons of simplicity, this aspect has been usually neglected in most DFT investigations of GB segregation so far.Another simplification often encountered is the assumption that there is only one average segregation energy for the whole GB.In reality the GB structure is made up of multiple distinctive sites with different segregation energies (see Fig. 5a) and for an accurate treatment, the spectral nature of the segregation energy needs to be considered 7,8 .This can be achieved by adding an index i for the GB sites so that we have a specific segregation energy and GB concentration at each site,    and   , .The overall concentration at the GB is computed by averaging the site concentrations at the GB.A comparison of the approach with spectral segregation energies and averaged segregation energy is given in Supp.
Fig. 8, which shows that in this specific case, the effect is negligible, as long as the dependency of the bulk concentration is taken into account.
STEM diffraction results a Orientation map for 30 at% Ti b Orientation map for 20 at% Ti.Ti Enrichment of the GB when using Averaged Eseg, Spectral Eseg and a segregation energy of -0.5eV at 700 K temperature.Supplementary Figure 9| Meshing FEM model using Contact Elements: (a) A 2d mesh is shown at the interface from which a 3d volumes will be extracted.ILR region is meshed finely and then gradually mesh becomes coarser towards the end support.(b) A finished 3D model is shown with the zoomed mesh difference at the interface.(c) Interface meshing is shown using constraint equations, coupled degree of freedom and contact elements method.The best result is obtained using contact elements (area to area) as the volumes are not overlapped or underlapped.As shown in Supp.Fig 5a, histogram equalization is performed as the first step to make all the images on the same gray scale.If not performed, different images might need different