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Kinetic equations for a modeling system with type-I radical-mediated and type-II oxygen-mediated pathways are derived and numerically solved for the photopolymerization efficacy and curing depth, under the quasi-steady state assumption, and bimolecular termination. We show that photopolymerization efficacy is an increasing function of photosensitizer (PS) concentration ( C  0 ) and the light dose at transient state, but it is a decreasing function of the light intensity, scaled by [ C  0 / I  0 ] 0.5  at steady state. The curing (or cross-link) depth is an increasing function of  C  0  and light dose (time × intensity), but it is a decreasing function of the oxygen concentration, viscosity effect, and oxygen external supply rate. Higher intensity results in a faster depletion of PS and oxygen. For optically thick polymers (>100 um), light intensity is an increasing function of time due to PS depletion, which cannot be neglected. With oxygen inhibition effect, the efficacy temporal profile has an induction time defined by the oxygen depletion rate. Efficacy is also an increasing function of the effective rate constant,  K  =  k ′/k    T    0  .  5, defined by the radical producing rate ( k ′) and the bimolecular termination rate ( k  T ). In conclusion, the curing depth has a non-linear dependence on the PS concentration, light intensity, and dose and a decreasing function of the oxygen inhibition effect. Efficacy is scaled by [ C  0 / I  0 ] 0.5  at steady state. Analytic formulas for the efficacy and curing depth are derived, for the first time, and utilized to analyze the measured pillar height in microfabrication. Finally, various strategies for improved efficacy and curing depth are discussed.

frontiers in chemistry

Modeling the Kinetics, Curing Depth, and Efficacy of Radical-Mediated Photopolymerization: The Role of Oxygen Inhibition, Viscosity, and Dynamic Light Intensity