Kinetics of Observable with Power-law Mixing Rule in the Second-order Reaction Process

Abstract

Some biopolymer gelation process shows an inflection point (IP) in a growth curve of a rheological observable. In this note, a kinetic treatment is presented for such a second-order food reaction process dx/dt=K₂ (1-x) ² where a general observable o grows with reaction degree x through the power-law type mixing rule o^ν= (1-x) o^ν_R+xo^ν_P. For-1<ν<1, the observable-time curve could possess IP at o^*= (1-ν) ^1/ν (1+ν) -1/νo_P and K₂t^*=2^-1 (1+ν) ν^-1 (1- (o_R/o_P) ^ν) -1 with the maximum growth rate do/dt|t^*=2² (1-ν) ^{(1/ν) -1} (1+ν) ^{- (1/ν) -1}ν (1- (o_R/o_P) ν) ^-1o_PK₂. These results show that o_R dependence disappears in the product (1+K₂t^*) do/dt|_t^*. The whole o-t curve is linearized with the transformation (o^ν_P-o^ν_R) (o^ν_P-o^ν) ^-1=1+K₂t. The case of ν=0 is compensated with a logarithmic type (In o) mixing rule.