Treatment of Gel Effect

According to the free volume theory, all species in the reaction mixture occupy a certain amount of free volume. A small monomer molecule provides more free volume than a long-chain polymer molecule. The total amount of free volume of the system Vf is the summation of the free volumes provided by all species in the reaction mixture, and can calculated by the following equation  

Based on the free volume theory, Vfj° and txj (thermal expansion coefficient) are assumed to be constant for different molecules. Vfj° has a value of 0.025 and clj has a value of 0.001 for small molecules such as solvent or monomer and 0.00048 for polymer long chains. The subscript j stands for species in the reaction mixture such as monomer, solvent, polymer, etc.. 

After polymerization has started, the free volume of the reaction system will continuously decrease. The smaller the amount of free volume, the lower is the mobility of the long-chain molecules. At a certain point, the rate constant of termination becomes diffusion-controlled and autoacceleration takes place. The diffusion-controlled rate constant kt will become smaller than the original chemically controlled rate constant kt. To keep track of the diffusion-controlled kt, one can use the following equation  

As the reaction continues, the system becomes more viscous and hence the free volume of the system decreases further. Depending on the glass transition temperature of the reaction mixture, when the free volume is less than a critical value the rate constant of propagation kp will also become diffusion controlled and it could be calculated by the following equation  

By using the free-volume theory, one can successfully monitor the whole course of free-radical polymerization. It is commonly accepted that the presence of autoacceleration and limiting conversion in the polymerization reaction are due to the diffusion-controlled kt and kp, respectively. The initiation efficiency, f, will behave in a way similar to kp. 

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