# Engineering and Clinical Study of Surface Geometry of Clear Aligners at the Nanoscale
## 3 Results and Discussion
### Statistical and Mechanical Parameters
It turns out from Figure 5A that even though the rms roughness Sq does not change significantly, single results are less deviated in worn aligners to those in the new aligners, which might prove that the roughness profile at the nanoscale becomes \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Condition** & \(n\) & **Variable** & **Mean** & **Median** & **Var** & **StD** & **IQR** & **S–W Test** \\ \hline \multirow{4}{*}{REF} & \multirow{4}{*}{23} & Sq (nm) & 1.80 & 1.59 & 0.48 & 0.690 & 1.45 & 0.015 \\ \cline{3-8}  & & Str(\(\cdot\)). & 0.585 & 0.615 & 0.04 & 0.192 & 0.303 & 0.141 \\ \cline{2-8}  & & Sdr (\%) & 1.51 & 0.931 & 3.33 & 1.83 & 0.798 & \textless{}0.001 \\ \cline{2-8}  & & Y (MPa) & 1190 & 1010 & 94,900 & 308 & 528 & 0.001 \\ \cline{2-8}  & & Fadh (nN) & 4.23 & 4.42 & 0.88 & 0.936 & 0.870 & 0.014 \\ \hline \multirow{4}{*}{WORN} & \multirow{4}{*}{23} & Sq (nm) & 1.61 & 1.56 & 0.17 & 0.411 & 0.640 & 0.013 \\ \cline{2-8}  & & Str (-) & 0.513 & 0.534 & 0.04 & 0.206 & 0.381 & \textless{}0.001 \\ \cline{1-1} \cline{2-8}  & & Sdr (\%) & 0.914 & 0.867 & 0.09 & 0.297 & 0.421 & 0.221 \\ \cline{1-1} \cline{2-8}  & & Y (MPa) & 917 & 917 & 28,700 & 169 & 179 & \textless{}0.001 \\ \cline{1-1} \cline{2-8}  & & Fadh (nN) & 5.48 & 4.33 & 11.9 & 3.46 & 0.760 & \textless{}0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Surface shape parameters of worn and new Invisalign® aligners: Sq—rms roughness, Str—texture anisotropy ratio, Sdr—surface development ratio, and results of nanomechanical mapping: Y—pseudo-Young’s modulus, Fadh—adhesion force. The following statistical measures of variability are provided as well: Var—variance, StD—standard deviation, IQR—inter-quartile range, together with the results of the test of normality using the Shapiro–Wilk method, with \(\alpha\) = 0.05. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable** & \begin{tabular}{c} **Sum of the** \\ **Ranks** \\ **REF** \\ \end{tabular} & \begin{tabular}{c} **Sum of the** \\ **Ranks** \\ **WORN** \\ \end{tabular} & \
\begin{tabular}{c} **U** \\ **WORN** \\ \end{tabular} &
\begin{tabular}{c} **Z** \\ **p \textgreater{} 0.05** \\ \end{tabular} \\ \hline Sq & 1085.500 & 2830.500 & 685.5000 & \(-0.58405\) & 0.559 \\ \hline Str & 1162.000 & 2754.000 & 609.0000 & \(-1.31055\) & 0.190 \\ \hline Sdr & 1082.500 & 2833.500 & 688.5000 & \(-0.55556\) & 0.579 \\ \hline Y & 1322.000 & 2594.000 & 449.0000 & \(-2.83003\) & 0.004 \\ \hline Fadh & 977.000 & 2939.000 & 701.0000 & 0.43685 & 0.662 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of non-parametric Mann–Whitney U test of significant changes among the parameters explaining the geometric and mechanical properties of the surfaces of worn and new Invisalign® clear aligners for assumed confidence level \(\alpha\) = 0.05.  
homogeneous. Similar changes might be seen in the surface development ratio Sdr, which is directly associated with the amplitudes of surface heights. Unlike that, however, the surface texture ratio Str shown in Figure 5B remains the same, considering the median value and how the points are distributed. Because these data points span over a range from 0.2 to 0.8, such a result might reflect the random character of the manufacturing process, in which the anisotropy decay length is definitely limited by the sampling separation (no less than several micrometers), and that the aforementioned degradation does not affect the directionality of the surface patterns.