# Deep Learning Regressors of Surface Properties from Atomic Force Microscopy Nanoindentations
## 2 Methods  
### Hertzian and JKR Contact Models  
Foundations of contact mechanics can be traced back to the work of [PERSON] [16], where the contact between two perfectly elastic solids or surfaces was described, disregarding friction or adhesion. Considering two spherical bodies, the force obtained from such contact was derived as  
\[F=\frac{4}{3}E^{*}R^{\frac{1}{2}}\delta^{\frac{3}{2}}\, \tag{1}\]  
where \(R=(1/R_{1}+1/R_{2})^{-1}\) accounts for the radii of curvature of both solids and \(\delta\) is the contact displacement, corresponding to the indentation value in AFM studies. The effective modulus \(E^{*}\) is defined with the elastic modulus (\(E\)) and Poisson ratio (\(\
u\)) of each solid:  
\[E^{*}=\left[\frac{\left(1-\
u_{1}^{2}\right)}{E_{1}}+\frac{\left(1-\
u_{2}^{2 }\right)}{E_{2}}\right]^{-1}. \tag{2}\]  
In the case of AFM nanoindentations with a spherical tip (\(R_{1}=R_{\text{tip}}\)), the sample can be often regarded as an elastic half-space (\(R_{2}=R_{\text{sample}}\rightarrow\infty\)), so the radius of curvature is simplified to \(R=R_{\text{tip}}\). When the sample is a soft biological tissue, the tip may be treated as a rigid body, such that the effective modulus will only depend on the sample's properties (\(E_{2}=E\) and \(\
u_{2}=\
u\)): \(E^{*}=E/(1-\
u)^{2}\).  
Many works [32; 33; 17; 34; 35] have been developed to achieve an accurate relation between applied force and indentation depth, supported on Hertzian principles, for a wide range of AFM tip shapes. Despite other accurate approaches for determining this relation, a closed-form solution will be implemented, based on applying a correction factor to Expression (1):  
\[F=\frac{4}{3}\frac{E}{(1-\
u^{2})}R^{\frac{1}{2}}\delta^{\frac{3}{2}}Z\, \tag{3}\]  
with \(Z\), as suggested in [36], being approximated by  
\[Z=\left[1-\frac{1}{10}\frac{\delta}{R}-\frac{1}{840}\left(\frac{\delta}{R} \right)^{2}+\frac{11}{15120}\left(\frac{\delta}{R}\right)^{3}+\frac{1357}{6652 800}\left(\frac{\delta}{R}\right)^{4}\right]. \tag{4}\]  
These relations will allow for the generation of curves for the approach stage of AFM nanoindentations, just by defining the tip radius and material properties for each sample. Regarding the withdraw stage, JKR contact theory translates the effect of adhesion energy \(\gamma\) (in units of energy per area) upon adhesion force and contact size, reaching the following expression for the normal force of contact:\[F=\frac{R_{c}^{3}K}{R}-\sqrt{6\pi K\gamma R_{c}^{3}}\, \tag{5}\]  
in which \(K=4/3E^{*}\) and \(R_{c}\) stands for the contact radius, which is a function of indentation depth and normal force itself, so it does not provide an explicit formulation relating force and indentation. One way to tackle this problem is by approximating the contact radius as \(R_{c}=R\delta\)[37], so that a simpler closed-form solution is obtained:  
\[F=KR^{\frac{1}{2}}\delta^{\frac{3}{2}}-U_{d}K^{\frac{1}{2}}R^{\frac{3}{2}} \delta^{\frac{3}{2}}\, \tag{6}\]  
where \(U_{d}=\sqrt{6\pi\gamma}\). Accordingly, to obtain F-I withdraw curves, the tip radius must be defined, in addition to the material properties, comprising the Poisson ratio, Young's modulus and adhesion energy.