## 3 Results
### Correction Factor for Euler Beam Equation for Plan-View Trapezoidal Cantilevers
Firstly, we require \(k_{z}\) for a triangular cantilever. From Equation A2a and A2b from [PERSON] and [PERSON] [1], we can express \(k_{z}\) for a triangular cantilever as:  
\[K=(E_{c}t^{3}w_{0}/(4L^{3}))\cdot\{(w/w_{0}-1)^{3}/(3w/w_{0})\;((1-w/w_{0})-(1- w/w_{0})^{2}/(2w/w_{0})+(w/w_{0})\;ln(w/w_{0}))^{-1}\} \tag{6}\]3  
The term \(w_{0}\) is now the width of the base of the triangular cantilever. Alternatively, we could use Equation (6) or simply refer to Figure 2 since when \(w/w_{0}\) approaches zero, we see that the \(K\) correction factor will approximate to 1.5. If we substitute 1.5 into Equation (7), we obtain the expression shown in Equation (11). After extensive searching, the authors could not find a simple expression for \(k_{z\theta}\) for a triangular cantilever in the literature. However, in the article by [PERSON] [40], they derive an equation for \(k_{z\theta}\) for a V-shaped cantilever. This Equation is 3b in the article [40] and is as follows:  
\[ \begin{array}{c}k_{z\theta}=E_{c}t^{3}w_{0}/(12L^{2})\left[2w_{a}/w_{0}- \Delta L/L(1+log((2w_{a}/w_{0})\left(\Delta L/L\right))\right)\\ \left.+\Pi\left(w_{0}/2w_{a}-1\right)(1/2+w_{0}/2w_{a}-\Delta L/L)\right]\end{array}  \tag{12}\]  
where \(w_{a}\) is now the width of one of the arms of the V-shaped cantilever, and \(w_{0}\) is the overall base width of the cantilever. In Sader [40], the term \(\Pi\) is expressed as:  
\[\Pi=(1+\
u)/((\Gamma^{2}(2-[1-\
u]\Gamma^{2})) \tag{13}\]  
where \(\
u\) is the Poisson's ratio for the cantilever material and \(\Gamma\) is given by  
\[\Gamma=(1+w_{a}{}^{2}/(4L^{2}))^{-0.5} \tag{14}\]  
This can be simplified further if we assume the length, \(L\), of the cantilever is from the base to the tip apex and we ignore the additional length of the cantilever from the tip apex to the very end of the beam. As mentioned, this is reasonable as the extra length has no impact on the torque since the AFM tip apex is where the force or torque is being applied. This means we can effectively assume \(\Delta L=0\) and \(L=L_{tip}\), and therefore, Equation (12) can be simplified to:  
\[k_{z\theta}=E_{c}t^{3}w_{0}/12L_{tip}{}^{2}\left[2w_{a}/w_{0}+\Pi\left(w_{0}/ 2w_{a}-1\right)(1/2+w_{a}/w_{0})\right] \tag{15}\]  
As the width of the arms of the cantilever, \(w_{a}\), approaches \(w_{0}/2\), the V-shaped cantilever approaches a closed triangular shape as shown in Figure 5.  
Therefore, if we assume that \(w_{a}=w_{0}/2\), and substitute this into Equation (14), it will simplify to:  
\[k_{z\theta}=E_{c}t^{3}w_{0}/(12L_{tip}{}^{2}) \tag{16}\]  
We now have equations for \(k_{z}\) and \(k_{z\theta}\) for a triangular cantilever. We can now determine \(\rho_{z}\) for a triangular cantilever using Equation (9), and then using that value determine \(T_{tri. }\) for the triangular cantilever using Equation (8). Doing this results in the expression:  
\[T_{tri.}=1-(tan\alpha)(2(D/L_{tip})) \tag{17}\]  
We can assume that the torque for a trapezoidal cantilever will be between \(T_{rect. }\) and \(T_{tri. }\), or to simplify this, we shall simply average Equations (3) and (16). We, therefore, approximate the torque produced by the AFM tip for a trapezoidal cantilever as:  
\[T_{trap.}\approx 1-(tan\alpha)(7/4)(D/L_{tip}) \tag{18}\]