Diffusion‐controlled reaction. V. Effect of concentration‐dependent diffusion coefficient on reaction rate in graft polymerization

The effect of diffusion on radiation‐initiated graft polymerization has been studied with emphasis on the single‐ and two‐penetrant cases. When the physical properties of the penetrants are similar, the two‐penetrant problem can be reduced to the single‐penetrant problem by redefining the characteristic parameters of the system. The diffusion‐free graft polymerization rate is assumed to be proportional to the v power of the monomer concentration C, in which the proportionality constant a = kpR /k , where kp and kt are the propagation and termination rate constants, respectively, and Ri is the initiation rate. The values of v, w, and z depend on the particular reaction system. The results of our earlier work were generalized by allowing a non‐Fickian diffusion rate, obtained from an extension of the Fujita free‐volume theory, which predicts an essentially exponential dependence on the monomer concentration of the diffusion coefficient, D = D0 [exp(δC/M)], where M is the saturation concentration. It was shown that a reaction system is characterized by the three dimensionless parameters v, δ, and A = (L/2)[aM(v−1)/D0]1/2, where L is the polymer film thickness. Graft polymerization tends to become diffusion controlled as A increases. Larger values of δ and v cause a reaction system to behave closer to the diffusion‐free regime. The transition from diffusion‐free to diffusion‐controlled reaction involves changes in the dependence of the reaction rate on film thickness, initiation rate, and monomer concentration. Although the diffusion‐free rate is w order in initiation rate, v order in monomer, and independent of film thickness, the diffusion‐controlled rate is w/2 order in initiator rate and inverse first‐order in film thickness. The dependence of the diffusion‐controlled rate on monomer is dependent in a complex manner on the diffusional characteristics of the reaction system.