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\author{H. Tun\c{c} \c{C}ift\c{c}i}
\affiliation{
  Advanced Research Center for Nanolithography (ARCNL),
  Science Park 106, 1098~XG Amsterdam, The Netherlands
}
\author{Ozan \c{S}ahin}
\affiliation{
  Advanced Research Center for Nanolithography (ARCNL),
  Science Park 106, 1098~XG Amsterdam, The Netherlands
}
\author{Thibault Roch}
\affiliation{
  Van der Waals-Zeeman Institute, Institute of Physics,
  University of Amsterdam, Science Park 904, 1098~XH Amsterdam, The Netherlands
}
\author{Bart Weber}
\email{b.weber@arcnl.nl}
\affiliation{
  Advanced Research Center for Nanolithography (ARCNL),
  Science Park 106, 1098~XG Amsterdam, The Netherlands
}
\alsoaffiliation{
  Van der Waals-Zeeman Institute, Institute of Physics,
  University of Amsterdam, Science Park 904, 1098~XH Amsterdam, The Netherlands
}



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\title[Visualizing Sub-10 nm Transfer Films in Metal–Ceramic Fretting Using Atomic Force Microscopy]{Supplementary Information: Formation and Compaction of Sub-10 nm Transfer Films in Metal–Ceramic Fretting}

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% \section{Introduction}

% In this document, we provide some additional experimental methods and results related to the preprint "Amorphous \ce{CaSO4} nanocrystal deposits for friction and wear reduction on silicon wafers". We especially provide images which are interesting but not important enough to be included in the main paper.


\newpage



\section{Hertzian Contact Calculations}

To quantify the mechanical response at the sapphire--chromium interface under normal loading, we implemented a classical Hertzian contact model for a sphere-on-flat configuration. The goal was to compute the contact radius, peak contact pressure, and dimensionless ratios relevant for interpreting elastoplastic behavior. The model is based on the following assumptions:
\begin{itemize}
    \item Frictionless, elastic, isotropic contact between a spherical sapphire lens and a flat chromium-coated wafer.
    \item Normal load \( F = 0.15~\mathrm{N} \), consistent with the experimental conditions.
    \item Sphere radius \( R = 1.6~\mathrm{mm} \), matching the sapphire lens curvature.
\end{itemize}

\subsection*{Material Properties}
The elastic properties were set as:
\begin{itemize}
    \item Sapphire: Young’s modulus \( E_1 = 335~\mathrm{GPa} \), Poisson’s ratio \( \nu_1 = 0.25 \)~\cite{shinkosha}
    \item Chromium: Young’s modulus \( E_2 = 279~\mathrm{GPa} \), Poisson’s ratio \( \nu_2 = 0.21 \)~\cite{ALAKIOZIDIS2025, WebElementsCr}
\end{itemize}

The effective modulus \( E^* \) is computed using\cite{CHAUDHRY2013}:
\[
\frac{1}{E^*} = \frac{1 - \nu_1^2}{E_1} + \frac{1 - \nu_2^2}{E_2} \quad \Rightarrow \quad E^* \approx 1.61 \times 10^{11}~\mathrm{Pa} = 161~\mathrm{GPa}.
\]

\subsection*{Hertzian Quantities}
The following Hertzian parameters are computed\cite{Johnson1985}:
\begin{align*}
a &= \left( \frac{3FR}{4E^*} \right)^{1/3} \approx 10.2~\mu\mathrm{m} \quad \text{(contact radius)} \\
A &= \pi a^2 \approx 3.3 \times 10^{-10}~\mathrm{m}^2 \quad \text{(contact area)} \\
p_{\text{max}} &= \frac{3F}{2\pi a^2} \approx 693~\mathrm{MPa} \quad \text{(maximum contact pressure)} \\
p_{\text{avg}} &= \frac{F}{\pi a^2}  \approx 462~\mathrm{MPa} \quad \text{(average contact pressure)} \\
%\delta &= \frac{a^2}{R} \approx 69.6~\mathrm{nm} \quad \text{(elastic indentation depth)}
\end{align*}

\subsection*{Dimensionless Ratios and Plasticity Threshold}
Here, a representative hardness of \( H = 1~\mathrm{GPa} \) was assumed for the chromium film. To assess the tendency for plastic deformation, we calculated:
\begin{align*}
\frac{p_{\text{max}}}{E^*} &\approx 0.0043 \quad \text{(normalized maximum contact pressure)} \\
\frac{p_{\text{avg}}}{E^*} &\approx 0.003 \quad \text{(normalized average contact pressure)} \\
\frac{H}{E^*} &\approx 0.0062 \quad \text{(hardness-to-modulus ratio)} \\
\frac{p}{H} &\approx 0.5-0.7 \quad \text{(pressure-to-hardness ratio)}
\end{align*}



\subsection*{Interpretation}
The calculated contact pressure approaches the predicted threshold for plastic deformation based on hardness comparisons, implying that yielding occurs within the contact zone. The values of \( p_0/E^* \) and \( p_0/H \) support a transition from elastic to elastoplastic deformation under the applied load, aligning with the simulation results discussed in the main text (Figure~2).

\section{Elasto-plastic contact computation}

\begin{figure}[!h]
\begin{center}
\includegraphics[scale=1]{FigSI1.png}
\caption{Map of the sub-surface plastic zones obtained from an elasto-plastic numerical simulation of the sapphire-chromium contact. The areas with plastic strains is spreading over the entire Hertzian contact radius (inner black circle).} 
\label{fig:SIplasticstrains}
\end{center}
\end{figure}

We use an open-source implementation of a volume integral formulation of equilibrium for the
solution of elastic-plastic rough contact \cite{Frerot2020,Frerot2019} for solving the sapphire-chromium contact problem. We solve a coupled problem between (i) the elastic contact problem and (ii) the bulk elastic-plastic problem.
First, (i) the elastic contact between the sapphire sphere and the chromium coated wafer is modeled as a single rough surface in contact with a rigid flat surface, with the roughness from both surfaces mapped onto the single rough surface. The contact problem is solved with a compound roughness $h = h_1 - h_2$, with $h_1$ the roughness of the sapphire sphere and $h_2$ the one of the chromium coated wafer measured with AFM. The surface topography size used for the computation is 34 × 34 $\mu\text{m}^2$ and is discretized into a 512 × 512 grid, with a pixel resolution of 66.4 × 66.4 nm. We use the effective mechanical properties of the sapphire-chromium contact, as given previously: Effective Young's modulus: \(E^* = 161 \pm 3 \, \text{GPa}\). Solving this problem for a given contact load $F_N$ provides the local pressures and displacements at the surface. The resulting pressures are used as the boundary conditions of (ii) the bulk elastic-plastic problem. The volume of the deformable material is discretized over a depth of 9.9 $\mu\text{m}$ with 26 points. The bulk behavior is modeled with Von Mises plasticity and isotropic hardening, with a yield stress $\sigma_y = 0.333 GPa$ and a hardening modulus $E_h = E_2/100$. The plastic strains obtained with the bulk elastic-plastic problem are then used to modify the surface of the rough contact problem. The coupled problem is solved until the residual surface displacements has converged \cite{Frerot2019}. In Fig.\ref{fig:SIplasticstrains}, we show the extent of the plastic strains just below the surface for the sapphire-chromium contact problem under a normal load $F_\mathrm{N}=150$~mN, emphasing the choice of an elasto-plastic computation.



\section{Peripheral Ring Calculations}

To escape from the contact, a debris particle must reach a region where the vertical gap between the sapphire lens and the wafer equals its height, as illustrated in Figure~7. Assuming a debris particle height of \(z = 25~\mathrm{nm}\) and a sapphire lens radius of curvature \(R = 1.59~\mathrm{mm}\), we estimate the radial distance \(r_{\text{ring}}\) from the apex at which the gap equals \(z\), accounting for the Hertzian indentation depth \(\delta\).

For a spherical surface of radius \(R\), the geometric separation at a radial distance \(r\) is given by
\[
d(r) = \frac{r^2}{2R}.
\]
Solving for \(r\) yields
\[
r_\text{ring} = \sqrt{2Rd}.
\]

The Hertzian indentation depth can be approximated as
\[
\delta = \frac{a^2}{2R},
\]
where \(a\) is the Hertzian contact radius. Using \(a = 10.2~\mu\mathrm{m}\) and \(R = 1.59~\mathrm{mm}\), we obtain
\[
\delta
= \frac{(10.2\times10^{-6})^2}{2(1.59\times10^{-3})}
= \frac{104.04\times10^{-12}}{3.018\times10^{-3}}
\approx 34.5\times10^{-9}~\mathrm{m}
= 34.5~\mathrm{nm}.
\]

The total characteristic gap required for a particle of height \(z\) to escape is therefore
\[
d = \delta + z
= 34.5~\mathrm{nm} + 25~\mathrm{nm}
= 59.5~\mathrm{nm}.
\]

The corresponding radial distance from the apex is then
\[
r_{\text{ring}}
= \sqrt{2R(\delta+z)}
= \sqrt{2(1.59\times10^{-3})(59.5\times10^{-9})}
= \sqrt{1.796\times10^{-10}}
\approx 1.34\times10^{-5}~\mathrm{m}
= 13.4~\mu\mathrm{m}.
\]

This lies outside the Hertzian contact radius (\(a=10.2~\mu\mathrm{m}\)) by
\[
r_{\text{ring}}-a \approx 13.4-10.2 \approx 3.2~\mu\mathrm{m}.
\]

As the sapphire topography includes roughness peaks in reality, the chromium debris particles may be pushed outward by a larger distance than $r_{\text{ring}}$.




\section{Results from Ball-2 from the same three-ball-puck}




\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.5]{FigSI2.png}
\caption{
AFM height maps of the sapphire apex of Ball-2 (a) before and (b) after fretting (512 x 512 pixel). 
The yellow dashed circle indicates the Hertzian contact area, the dark blue arrow indicates the fretting direction. 
A thinner transfer film forms within the contact region in (b), consistent with the predicted Hertzian contact size, while the formed peripheral ring reveals a thicker profile as indicated by $\textit{r}_{\text{L}}$ and $\textit{r}_{\text{R}}$. 
Trenches visible in both images are filled by transferred material.
(c--e) Height profiles represent the black and red arrows from raw topographies before and after fretting (a an b). (f--h) Height difference plots taken from the gray colored areas between the black and the red profiles (c--e).}
\label{fig:comparisonSI1}
\end{center}
\end{figure}

In the main Results section, we presented experimental outcomes from Ball~1 of the three-ball puck configuration (Figure~1c). Here, we present complementary results obtained from Ball~2, which was brought into contact with a different location on the same chromium-coated wafer. To further quantify the topographic changes and assess the spatial extent of material transfer, we employed a topographical difference method based on cross-sectional line profiles extracted from identical regions of AFM scans acquired before and after fretting. Figure~\ref{fig:comparisonSI1}a and b show the AFM height maps of the sapphire apex before and after fretting, respectively.



\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.45]{FigSI3.png}
\caption{(a) AFM topograph of the chromium-coated wafer after fretting against Ball-2 (1024 $\times$ 1024 pixels). The post-fretting image shows a well-defined wear scar, which highlights local fretting-induced features. (b) Height profiles and the corresponding height-difference profile revealing the surface modulation associated with fretting.
}
\label{fig:comparisonSI2}
\end{center}
\end{figure}


Height profiles extracted along the arrows in Figure~\ref{fig:comparisonSI1}(c--h) quantitatively compare the topographic evolution. Raw and difference profiles from profile~1 and profile~2 (Figure~\ref{fig:comparisonSI1}a and b) reveal a median height increase of approximately 5~nm within the Hertzian contact area (Figure~\ref{fig:comparisonSI1}c--g). Both profile~1 and profile~2 exhibit two peripheral ``horns'' in the difference profiles, denoted as $r_\mathrm{L}$ and $r_\mathrm{R}$, corresponding to debris accumulation beyond the Hertzian contact area, with peak heights of 15--20~nm at the outer ring, located 7--9~$\mu$m outside the Hertzian zone. Furthermore, both profiles show that the broader grooves on the sapphire surface become progressively filled by transferred debris, as indicated by local height increases in the difference maps. profile~3, taken outside the contact region, shows a nearly flat difference ($\sim$0.2~nm), confirming the stability and repeatability of the topographic measurement.

We also assessed complementary changes on the chromium-coated wafer, providing further evidence of interfacial material exchange and surface restructuring. Figure~\ref{fig:comparisonSI2}(a and b) show corresponding AFM height maps of the chromium-coated wafer. After fretting, a deformed area within and around the Hertzian contact is clearly visible. The scar diameter and topography are consistent with the material transfer area observed on the sapphire (Figure~\ref{fig:comparisonSI2}b). Notably, the extended features within the wear scar suggest the possibility of a pre-slide phase before stable fretting motion was established, potentially due to slight nanometric tilt or bending of the wafer or stage system at the point of contact initiation.




\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.45]{FigSI4.png}
\caption{Adhesion maps of (a) the sapphire apex of Ball-2 and (b) the wear scar on chromium surface after fretting measured in PeakForce QNM mode using the same AFM tip (Adama Innovations, AD-2.8-AS). (c) Adhesion profiles extracted along the corresponding marked positions (in a and b). The dark red profile along the transferred material at the ball apex exhibits lower adhesion compared to the dark gray profile from surrounding pristine sapphire regions. The red profile along the wear scar on the chromium-coated wafer shows higher adhesion compared to the black profile from surrounding flat chromium film. The contrast between reduced adhesion on the sapphire and enhanced adhesion on the chromium confirms asymmetric interfacial evolution during material transfer, correlating with the topographic profiles shown in Figure~\ref{fig:comparisonSI1} and  Figure~\ref{fig:comparisonSI2}.
}
\label{fig:comparisonSI3}
\end{center}
\end{figure}


Lastly, the adhesion mapping results shown in Figure~\ref{fig:comparisonSI3} complement the height data and reveal spatially resolved contrast. On the sapphire apex (Figure~\ref{fig:comparisonSI3}a), the transferred chromium region exhibits significantly lower adhesion compared to the pristine sapphire areas outside the wear scar, indicating a distinct surface composition within the transfer film. On the chromium-coated wafer (Figure~\ref{fig:comparisonSI3}b), the wear scar displays a higher adhesion signal relative to the surrounding smooth chromium surface. Importantly, the same AFM tip was used for all adhesion measurements, both before and after fretting. The observed adhesion contrast between the sapphire and chromium surfaces confirms complementary interfacial evolution at both sides of the contact.

\section{Additional results}

Figures~\ref{fig:comparisonSI4} and \ref{fig:comparisonSI5} show the results of an independent repeat of the fretting experiment reported in the main text. Overall, the results are similar. Variation in transfer film and peripheral ring thickness is likely sensitive to the age and cleaning history of the chromium sample. This will be investigated in more detail in future work. 

\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.45]{FigSI5.png}
\caption{
AFM height maps of the sapphire apex of Ball-1 (from the repeat experiment) (a) before and (b) after fretting (512 x 512 pixel). A thinner transfer film forms within the contact region in (b), consistent with the predicted Hertzian contact size, while the formed peripheral ring reveals a thicker profile as indicated by $\textit{r}_{\text{L}}$ and $\textit{r}_{\text{R}}$. 
Trenches visible in both images are filled by transferred material.
(c--e) Height profiles represent the black and red arrows from raw topographies before and after fretting (a an b). Height difference plots (right) taken from the gray colored areas between the black and the red profiles (c--e).}
\label{fig:comparisonSI4}
\end{center}
\end{figure}

\begin{figure}[!t]
\begin{center}
\includegraphics[scale=0.45]{FigSI6.png}
\caption{Adhesion maps of (a) the sapphire apex of Ball-1 after fretting measured in PeakForce QNM mode using the same AFM tip (Adama Innovations, AD-2.8-AS). (b) Adhesion profiles extracted along the corresponding marked positions (in a). The red profile along the transferred material at the ball apex exhibits lower adhesion compared to the black profile from surrounding pristine sapphire regions. The contrast in adhesion correlates with the topographic profiles shown in Figure~\ref{fig:comparisonSI4}.}

\label{fig:comparisonSI5}
\end{center}
\end{figure}
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\end{document}
