Photochemical characterization of rate laws, rate constants and photonicities in optically–dense, multiphoton–reactive systems

• Averaged intensities account for Beer’s Law–induced decreases in intensity. • Averaged intensities facilitate analysis of optically–dense, multiphoton reactive systems. • Incident intensities yield unreliable rate constants. • Averaged intensities yield reliable rate constants and photonicities. • Averaged intensities apply well to multiphoton synthesis and photodynamic therapy. A novel method for characterizing the rate constants k and photonicities n of multiphoton–induced reactions in optically dense media is presented. Conventionally, k and n are calculated by irradiating samples of identical absorbance with different incident initial intensities I 0,0 and characterizing the rates of change in absorbance, with incident rate constants k 0,0 being proportional to I 0 , 0 n for n– photon processes. In our method, I 0 , 0 n is replaced by a longitudinally–and–temporally–averaged n th –order intensity I x , t n ¯ = 1 - 10 - n A ¯ 2.303 n A ¯ I 0 , 0 n which accounts for the exponential decrease in intensity across the optical pathlength due to the average absorbance A ¯ over an irradiation interval of duration t . We demonstrate the utility of I x , t n ¯ by contrasting longitudinally–and–temporally–averaged rate constants k x,t and photonicities n x,t with incident rate constants k 0,0 and photonicities n 0,0 for the 532 nm, two–photon induced photodegradation of β–carotene in CCl 4 solvent. Because the actinic intensity declines more rapidly with x in high ( A ∼ 1.0) than low ( A ∼ 0.2) absorbance samples, the incident rates R 0 , 0 = k 0 , 0 I 0 , 0 n are faster in the low absorbance samples: i.e., k 0,0 ( A 532 = 0.2) > k 0,0 ( A 532 = 1.0). In significant contrast, we find that k x,t ( A 532 = 1.0) ∼ k x,t ( A 532 = 0.2). indicating that k x,t provides better estimates of the rate constant than k 0,0 . As with k x , t , we find that the photonicities are minimally affected by A 532 ; i.e ., n 0,0 ∼ n x , t for both low and high absorbance samples. Hence, the dependence of the rate on the actinic absorbance is incorporated principally into the incident rate constants k 0,0 ; k x , t , n 0,0 and n x , t are largely unaffected by A 532 . We detail potential applications of our method to both multiphoton–induced chemical synthesis and multiphoton–induced photodynamic therapy scenarios, as high absorbance samples are advantageous for both of these applications.