Diffusion-induced nonuniformity of photoinitiation in a photobleaching medium

Absorption of incident light in photopolymerizations gives rise to instantaneous distributions of initiator concentration and initiation rate that are nonuniform along the beam path. Absent diffusion, however, the time-integrated production of primary radicals is uniform if the initial initiator concentration is uniform and all initiator is consumed, since each initiator molecule is photolyzed in place. Here, we consider the effects of diffusion of a photobleaching initiator for finite values of the ratio of the diffusive time scale l(2)/D to the reaction time scale 1/(phiI(o)alpha(A)), where l and I-o are the layer thickness and incident light intensity at the optical entrance, and D and alpha(A) are the diffusion coefficient and molar absorption coefficient of photoinitiator, respectively, whose quantum yield of consumption is phi. Compared to the limiting case in which diffusion is negligible, diffusion has the effect of shifting the instantaneous initiation rate profiles forward in the layer, where initiator is relatively depleted. On the other hand, for any nonzero initial absorbance, the overall (i.e., time-integrated) consumption of initiator becomes more nonuniform as the ratio of the rates of diffusion and reaction, expressed in the dimensionless ratio delta = D/(l(2)phiI(o)alpha(A)), increases. When diffusion is fast (large delta), the front-to-back difference in the time-integrated primary radical production varies quadratically with the initial concentration of initiator. Implications of the results for conversion of monomer and for chain-length distributions are discussed.