# Dispersion of OH Radicals in Applications Related to Fear-Free Dentistry Using Cold Plasma
## 2 Materials and Methods  
The human mandibular jaw (i.e., the lower jaw) was modeled. Average dimensions of the main parts of the mandibular teeth were taken from previous measurements [48] and are shown in Table 1.  
The modeled mandibular jaw with teeth is shown in Figure 1, in which CAP discharge from a nozzle is also schematically shown. For modeling purposes, the mandibular jaw was placed in a rectangular chamber with the dimensions \(90\times 90\times 45\) mm\({}^{3}\). The chamber was filled with air and all surfaces were set to be outlets (i.e., exposed to open air). The nozzle was assumed to be filled with helium (He) gas (i.e., the CAP carrier gas). The air in the chamber was assumed to be initially at rest and the plasma carrier gas was injected into the air domain at a specific flow rate \(Q\). The plasma carrier gas (He) was discharged out of the nozzle with a radius of 1 mm and with a flowrate of 1 L/min (\(Q=1\) L/min) [47]. Following our previous work [47], the propagation of the carrier gas (He) in the air chamber was described using the level set method, since the flow resembled a two-phase flow system. The level set function was  
\[\frac{\partial\phi}{\partial t}+\mathbf{u}\cdot\
abla\phi=\gamma\
abla\left(\varepsilon \
abla\phi-\phi(1-\phi)\frac{\
abla\phi}{\left|\
abla\phi\right|}\right) \tag{1}\]  
where \(\phi\) was the level set variable, \(\mathbf{u}\) was the velocity field, \(t\) was time and \(\gamma\) was introduced for an enhanced numerical stability [49]. The level set function (i.e., Equation (1)) was fully coupled with Equation (2) which described the diffusion and convection of OH radicals:  
\[\frac{\partial c_{OH}}{\partial t}+\
abla\cdot(-D_{OH}\
abla c_{OH})+\mathbf{u} \cdot\
abla c_{OH}=S_{OH}+R_{OH} \tag{2}\]  
where \(c_{OH}\) was the concentration of OH radicals, \(D\) was the diffusion coefficient, \(\mathbf{u}\) was the velocity field, \(S_{OH}\) and \(R_{OH}\) were the source and reaction terms of the OH radicals [47]. The fluxes of OH radicals were defined in the nozzle to ensure the flow of these species from the nozzle along the axis perpendicular to the target (tooth under CAP treatment). The largest inflow concentration of OH radicals (i.e., at the OH source) was set to be 1 mol/m\({}^{3}\), which simplified the task in obtaining the OH concentrations in a normalized fashion over the target. In addition, variations in concentrations of OH radicals as a result of their reactions were modeled through a decay rate formulation. Interested readers are referred to our previous work [47] for more modeling details and for benchmarking with previous CAP discharge experiments. While OH radicals contributed to the bleaching effect, they might not be the main effectors. Short-lived reactive species in plasma-air-water system were analyzed [50] and different reactive species formed in CAPs were found to be responsible for tooth bleaching. In conventional dental bleaching, H\({}_{2}\)O\({}_{2}\) was used (and still is used) as a bleaching agent. It is formed as a result of the recombination of OH radicals and via other reactions. Therefore, the transportation and dispersion of H\({}_{2}\)O\({}_{2}\) species were also considered in our computational model in addition to the OH radicals. The diffusion coefficient data for H\({}_{2}\)O\({}_{2}\) was taken from the previous work of [PERSON] et al. [51]. \begin{table}
\begin{tabular}{c c c} \hline
**Mandibular Teeth** & **Crown Length (mm)** & **Root Length (mm)** \\ \hline Central incisor & 8.80 & 12.6 \\ \hline Lateral incisor & 9.40 & 13.5 \\ \hline Canine & 11.0 & 15.9 \\ \hline First premalor & 8.80 & 14.4 \\ \hline Second premalor & 8.20 & 14.7 \\ \hline First molar & 7.70 & 14.0 \\ \hline Second molar & 7.70 & 13.9 \\ \hline Third molar & 7.50 & 11.8 \\ \hline \end{tabular}
\end{table} 
Table 1: Average dimensions of crown and root length of mandibular teeth. The CAP discharge tube was set to be perpendicular to the tooth under treatment, and the distance between the tube and the tooth was set to be 2 or 4 mm.