## 2 Materials and Methods
### Computational Model for Photopolymerization Process  
#### 2.2.1 Photopolymerization Model  
Photopolymerization plays a central role in stereolithography AM processes including PuSL. Photopolymerization occurs in three steps; initiation, propagation, and termination [20]. When UV light is projected on a photocurable resin, free radicals are generated from photoinitiator (initiation). Free radicals readily react with monomer molecules, connecting monomers to form long polymer chains (propagation). Propagation continues until two large chains of polymer cross-link with each other (termination). The liquid resin is converted into a solid when cross-linked polymer network is formed. Environmental conditions are also an important factor that influences the reaction kinetics. Oxygen acts as an inhibitive agent because free radicals react not only with monomer molecules, but also with oxygen molecules when present, forming peroxides. These peroxides do not take part in the polymerization process, thereby inhibiting the overall conversion of liquid resin to solid. This photopolymerization process can be modeled as follows. Light irradiation intensity (\(I\)) which decays as light travels through the resin can be modeled as  
\[\frac{dI}{dz}=\ -(\alpha[\mathrm{PI}]+\alpha_{\mathrm{a}}[\mathrm{PA}])I, \tag{1}\]  
where [PI] and [PA] represent photoinitiator and photo absorber concentrations, \(\alpha\) is molar absorptivity of photoinitiator, and \(\alpha_{\mathrm{a}}\) is the molar absorptivity of photo-absorber [21]. The term \((\alpha[\mathrm{PI}]+\alpha_{\mathrm{a}}[\mathrm{PA}])\) represents overall absorption coefficient of the resin which follows the Beer-Lambert law [22]. The light intensity of the projected beam is modeled as a convolution of unit light intensity profile which is modeled as a Gaussian function [17]. In Equation (2), \(w\) is the beam width of the Gaussian function, \(I_{0}\) is the peak light intensity, and r is the radial position. For \(n\) number of activated pixels on DMD, the light intensity distribution on the surface of resin is therefore given by Equation (3). \[I\ =\ I_{0}\times\mathrm{e}^{\frac{-2(r)^{2}}{w^{2}}}\, \tag{2}\]  
Figure 2: PuSL experimental setup. \[I\ =\ I_{0}\times\sum_{0}\mathrm{e}^{-\frac{2(r-\mathrm{i}\pi\times n)^{2}}{\omega^ {2}}}, \tag{3}\]  
As descripted in the photopolymerization principle, initiation process consumes photoinitiator molecules which split into free radicals upon irradiation. Accounting for diffusion flux, photoinitiator concentration can be written as [21]  
\[\frac{\partial[\mathrm{PI}]}{\partial\mathrm{t}}=\
abla(D_{\mathrm{PI}}\
abla[ \mathrm{PI}])\ -\ \frac{1}{2}\ \varphi\alpha\beta[\mathrm{PI}]\,I, \tag{4}\]  
where \(D_{\mathrm{PI}}\) is diffusivity of photoinitiator, \(\varphi\) is quantum yield of free radicals, and \(\beta=1/3.27\times 10^{5}\) mol/J is the amount of energy contained in one photon [23]. Upon UV exposure, photoinitiator molecules split to generate free radicals which react with monomers and activate their functional groups. These active monomers react with other monomers and begin a propagation reaction [24], given by  
\[\frac{\partial[\mathrm{M}]}{\partial\mathrm{t}}\ =\ -\ k_{p}[\mathrm{M}][ \mathrm{R}], \tag{5}\]  
where [M] is monomer concentration, \(k_{p}\) is propagation rate constant and [R] is radical concentration. A negative sign indicates that monomer concentration decreases as they are converted into polymer. The radical concentration is given by [25]  
\[\frac{\partial R}{\partial\mathrm{t}}=\
abla(D_{r}\
abla[\mathrm{R}])\ +\ R_{g}\ -\ R_{c}, \tag{6}\]  
where the first term represents radical diffusion with diffusion coefficient \(D_{r}\), and \(R_{g}\) and \(R_{c}\) account for generation and consumption of free radicals, respectively. \[R_{g}=\ \varphi\alpha\beta[\mathrm{PI}]\,l_{0}\exp(-\alpha[\mathrm{PI}]z), \tag{7}\]  
\[R_{c}=\ k_{t}[\mathrm{R}]^{2}+\ k_{o}[\mathrm{O}][\mathrm{R}], \tag{8}\]  
[O] is oxygen concentration, \(k_{t}\) is termination rate constant, and \(k_{o}\) is oxygen rate constant.