# Engineering and Clinical Study of Surface Geometry of Clear Aligners at the Nanoscale
## 3 Results and Discussion
### Functional Parameters  
Table 3 summarizes the descriptive statistics of the functional parameters determined from the AFC. As a rule, all of the aligners under study exhibited a complex, degressive-progressive shape of the AFC, which may be considered to be an undesirable feature from the point of view of tribological properties. The reason for this is that the rapid abrasive wear of relatively few peaks sticking out of the surface during contact with other hard surfaces may change the outer geometry of the aligners at the level of hundreds of nanometers, which eventually may result in their improper corrective action. Apart from that, the results of the Shapiro-Wilk test of normality with a confidence level of \(\alpha=0.05\) are also provided in Table 3.  
Similar to the roughness parameters, the majority of the functional parameters end up with \(p\)-values less than the assumed confidence level. Hence, the null hypothesis can be rejected and the data tested do not appear to be normally distributed. The exceptions (marked with gray colors) are the thickness of the peak layer Spk in worn aligners, and both material ratios (Smrk1 and Smrk2) in the new aligners. Therefore, in order to verify the hypothesis that daily use might influence the functional parameters, a non-parametric Mann-Whitney U test was performed with a confidence level of \(\alpha=0.05\). The obtained results are shown in Table 4, and Figure 6 shows a graphical comparison of raw data and the descriptive statistics of their distributions. Figure 5: Graphical comparison of raw data and descriptive statistics of the variability of the surface shape parameters between new and worn Invisalign® clear aligners: (**A**) rms roughness Sq, (**B**) texture anisotropy ratio Str, (**C**) surface development ratio Sdr; mechanical parameters: (**D**) pseudo-Young’s modulus Y and (**E**) adhesion force Fadh. The results of the Mann-Whitney U test prove that only the parameters related to the peaks (marked with gray colors) substantially decrease between new aligners and the worn aligners. These are namely the mean peak height Spk and, correspondingly, the volume of the peak layer per unit surface Vmp, which suggests that abrasive wear of the highest parts of the surface occurs. The median of the former parameter decreased from 2.07 to 1.75 nm for the new and worn aligners, respectively. In turn, the median of the latter parameter decreased from 0.129 to 0.107 nm\({}^{3}\)/nm\({}^{2}\) for the new and worn aligners, respectively. Such a correlated trend strongly suggests the occurrence of abrasive wear within the outer part of the residual layer solely, namely the bare peaks being in contact with other surfaces--the opposing teeth and aligners, rather than chemical decomposition of the material by salivary \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multicolumn{2}{c}{**Sum of the**} & \multicolumn{2}{c}{**Sum of the**} & \multirow{2}{*}{**V**} & \multirow{2}{*}{**Z**} & \multirow{2}{*}{**\(p\) \textgreater{} 0.05**} \\  & **Ranks** & **Ranks** & **WORN** & & & \\ \hline Sk (nm) & 1000.0 & 2570.0 & 679.0 & \(-\)0.220 & 0.825 \\ Spk (nm) & 1195.5 & 2374.5 & 483.5 & \(-\)2.181 & 0.029 \\ Svk (nm) & 934.0 & 2636.0 & 658.0 & 0.431 & 0.666 \\ Smrk1 (\%) & 1133.5 & 2436.5 & 545.5 & \(-\)1.560 & 0.119 \\ Smrk2 (\%) & 1113.0 & 2457.0 & 566.0 & \(-\)1.354 & 0.176 \\ Vmp (nm\({}^{3}\)/nm\({}^{2}\)) & 1234.5 & 2335.5 & 444.5 & \(-\)2.573 & 0.010 \\ Vmc (nm\({}^{3}\)/nm\({}^{2}\)) & 1002.5 & 2567.5 & 676.5 & \(-\)0.246 & 0.806 \\ Vvv (nm\({}^{3}\)/nm\({}^{2}\)) & 871.0 & 2699.0 & 595.0 & 1.063 & 0.288 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of non-parametric Mann–Whitney U test of significant changes among the parameters explaining functional properties of the surfaces of worn and new Invisalign® clear aligners for an assumed confidence level of \(\alpha\) = 0.05.