# Measurement of Sub-Surface Microstructures Based on a Developed Ultrasonic Atomic Force Microscopy
## 2 Materials and Methods
### Theoretical Model  
In this study, variations in subsurface morphology would directly change the contact stiffness at each scan point and lead to changes in the probe-sample contact resonance frequency. As described above, the change of contact resonance frequency of the probe-sample system is the focus of our research. The relevant equations are organized and derived in this part to demonstrate the effect of the change in contact stiffness on the resonant frequency of the probe-sample contact. The main physical model used in this study is shown in Figure 2.  
A simplified model diagram of the contact between the tip of the probe and the sample surface is shown in Figure 2a. Note that the wavelength of ultrasonic excitation signal \(\lambda\) is much larger than the contact radius \(a\) at the ultrasonic frequency, the contact process between the probe tip and the sample surface can be studied as a quasi-static physical process. Thus, the contact between the spherical tip and the sample surface can be explained by [PERSON] contact theory [24; 25; 26] as:  
\[a=\sqrt[3]{\frac{3 FR}{4E^{*}}} \tag{1}\]  
\[d=\sqrt[3]{\frac{9F^{2}}{16 RE^{*^{2}}}} \tag{2}\]  
\[k^{*}=2 aE^{*}=\sqrt[3]{6E^{*^{2}}FR} \tag{3}\]  
\[E^{*}=\frac{\left(1-\gamma_{\rm s}^{2}\right)}{E_{\rm s}}+\frac{\left(1-\gamma _{\rm f}^{2}\right)}{E_{\rm f}} \tag{4}\]  
where \(F\) is the contact force between tip and sample, \(R\) is the tip radius, \(k^{*}\) is the equivalent contact stiffness between probe tip and sample, \(E^{*}\) is the equivalent elastic modulus between probe tip and sample, \(d\) is the pressing depth of probe, which is equal to the sum of tip and sample surface deformation, \(\gamma_{s}\) and \(\gamma_{t}\) represent the Poisson's ratio of sample and tip, respectively, and \(E_{s}\) and \(E_{t}\) represent the elastic modulus of sample and tip, respectively. Figure 2: (**a**) The simplified model of contact between the probe tip and the sample. (**b**) The simplified model for cantilever probe in contact with sample surface. Equation (4) reveals the influencing factors of the equivalent contact stiffness between the probe and the sample. When there are microstructures (such as pores, heterogeneous particles, etc.) exit on the subsurface of the sample, the elastic modulus and Poisson's ratio of the sample will change, resulting in a change in the equivalent contact stiffness between the probe and the sample. Since AFM cannot directly measure the elastic modulus of the material, but can extract the probe's oscillation signal information during the scanning process, the change of the equivalent elastic modulus can be converted into a detectable probe vibration frequency change signal. The subsurface topography is characterized by the change of the contact resonance frequency of the probe. The simplified probe vibration model is shown in Figure 2b. When the equivalent contact stiffness \(k^{*}\) of the probe-sample system is equal to zero, the model is reduced to a simple free vibration model with a fixed end of a cantilever beam, and the partial differential Equation of its free vibration is written as:  
\[EI\frac{\delta^{4}y}{\delta x^{4}}+\rho A\frac{\delta^{4}y}{\delta t^{4}}=0 \tag{5}\]  
where \(E\) is the elastic modulus of cantilever, \(I\) presents the area moment of inertia of cantilever beam, which is equal to \(\mathrm{tb}^{3}/12\) when we suppose that the cantilever beam has length \(L\), density \(\rho\), cross-sectional area \(A\), cross-sectional area width \(t\), and thickness \(b\). The boundary conditions of Equation (5) are:  
\[x=0:\;y(x)=0,\;\frac{\delta y(x)}{\delta x}=0 \tag{6}\]  
\[x=L:\;\frac{\delta^{2}y(x)}{\delta x^{2}}=0,\;\frac{\delta^{3}y(x)}{\delta x^ {3}}=0 \tag{7}\]  
Thus, the characteristic equation of Equation (5) can be calculated as:  
\[cosk_{n}Lcosh\;k_{n}L+1=0 \tag{8}\]  
where \(n\) represents modal order of the vibration, \(k\) stands for wave number of cantilever oscillation, and \(k_{n}L\) is the normalized modulus of infinite bending vibration modes of the cantilever.