# Recombination of Poly(Acrylic Acid) Radicals in Acidic Aqueous Solutions: A Pulse Radiolysis Study
## 3 Results and Discussion
### Radical Decay Kinetics at pH 2  
#### 3.4.1 Kinetic Traces  
In our further study we concentrate on decay of radicals on fully protonated PAA chains at pH 2. These kinetics have some interesting features. We shall consider that while at moderate pulses of ionizing radiation, such as used in this study, it is improbable to find more than one radical at the same OAA or PA molecule, it is quite probable to find several radicals generated by the same pulse on a PAA chain. As a consequence, we may have two different modes of recombination: intermolecular (between two radicals on two different macromolecules) and intramolecular (between two radicals located on the same chain). For flexible polymer chains, intramolecular recombination is expected to be fast, facilitated by the initial proximity of radicals located at the same chain.  
Recombination of two radicals is, in general, expected to follow second-order kinetics. However, it has been demonstrated in a number of experimental [55; 88; 89; 90; 91], Monte-Carlo [92; 93], and deterministic [94; 95] simulation studies that recombination of polymer-bound radicals in solution may deviate from classical second-order equations, in contrast to chemically identical radicals of their low-molecular-model compounds. The apparent momentary rate constant is often observed to decrease in the course of the reaction. This behavior seems to be typical for intramolecular recombination, when initially many radicals are located on each chain. A model that has been found to describe such cases well is dispersive kinetics (Equation (6), which for reactions of the A+A type takes the form of Equation (7)), where \(k(t)\) is the momentary rate constant, \(B\) and \(\alpha\) are parameters, \(c(t)\) is the substrate concentration at time \(t\), and \(c_{0}\) is the initial substrate concentration [96; 97]. For \(\alpha=1\) this model is reduced to the classical homogeneous kinetics with \(k\) independent of time. Deviation of \(\alpha\) from 1 is a measure of reactivity dispersion in the system. It has been demonstrated that this dispersion stems from the initial distribution of interradical distances along the chain (this distance being the main factor governing the reaction rate) and subsequent change in this distribution during reaction [92].  
\[k(t)=Bt^{\alpha-1} \tag{6}\]  
\[\frac{1}{c(t)}-\frac{1}{c_{0}}=\frac{Bt^{\alpha}}{\alpha} \tag{7}\]  
While systematic analysis of PAA radical reactivity in terms of dispersive kinetics theory is beyond the scope of our study, we have attempted to test if Equation (7) describes the observed decay of PAA radicals more adequately than the regular second-order kinetics,  
Figure 4: Pulse radiolysis of aqueous solution containing (**a**) 25 mmol dm\({}^{-3}\) OAA, (**b**) 150 mmol dm\({}^{-3}\) PA at various pH, dose per pulse around 60 Gy. Normalized absorbance at \(\lambda=280\) nm as a function of time after pulse. Solutions at pH 2 saturated with Ar, at pH 5 and 9 saturated with N\({}_{2}\)O.  
in line with earlier observations for doses per pulse much higher than in this work [15; 55]. An exemplary test is shown in Figure 5. A typical kinetic trace of PAA radical decay is fit with first-order and second-order kinetic functions, as well as with Equation (7). While it is clear that experimental data fit much better to the second-order than to the first-order kinetic model, under our experimental conditions (molecular weight, pH, concentration range, moderate doses per pulse), fitting with Equation (7) was only slightly better than with the classical second-order kinetics, the \(\alpha\) parameter being not far from 1.