Nonuniformity of chain-length distributions in photopolymerized layers

For free-radical photopolymerization with a photobleaching initiator, constant chain propagation and termination rate constants k_p and k_t, and termination occurring only by recombination, we account for Beer-Lambert attenuation and initiator consumption to predict how spatial variation of the final chain length distribution (CLD) depends on k_p, k_t, incident light intensity I_o, layer thickness L, photoinitiator absorption coefficient α_A and initial concentration C_A,0, and quantum yield of photoinitiator consumption φ, for a typical value (10^-4) of the initial ratio of initiator and monomer concentrations C_A,0/C_M,0. We show how spatial variation of the final CLD depends on initial absorbance γ = α_AC_A,0L and a parameter β = k_p[fC_A,0/(φα_A I_ok_t)]^1/2, where f primary radicals are produced per photoinitiator molecule consumed. For small γ, the number-averaged mean chain length increases with depth at each β and with β at each depth. The chain length at which the CLD achieves its maximum value, along with a measure of polydispersity (half the CLD width at half-maximum, divided by the number-averaged mean), increase with depth at small β and decrease with depth at large β, with the CLD having its minimum nonuniformity at intermediate β. Front-to-rear CLD variation increases as γ increases. At small β, nonuniformity is confined to a progressively smaller portion of the front of the layer as γ increases, while for large β, spatial variation is more evenly distributed. The results are discussed in terms of spatiotemporal variation of initiation and monomer conversion. Examples from the literature are used to illustrate the degree of CLD nonuniformity that can be expected in experiments.