Rate of radical entry

Smith and Ewart [4] first proposed that the transfer of free radical activity into the interior of a polymer particle takes place by the direct entry of a free radical into a polymer particle. They pointed out that the rate of radical entry into a polymer particle is given by the rate of diffusion of free radicals from an infinite medium of concentration into a particle of diameter dp with zero radical concentration. 

On the other hand, Nomura and Harada [14] proposed a kinetic model for the emulsion polymerization of styrene (St), where they used Eq. 7 to predict the rate of radical entry into both polymer particles and monomer-swollen micelles. In their kinetic model, the ratio of the mass-transfer coefficient for radical entry into a polymer particle kep to that into a micelle kem> K lk,... 

On the other hand, several reports have been published that point out that when a polymeric surfactant acting as an electrosteric stabilizer is used, the rate of radical entry into a polymer particle should decrease due to a diffusion barrier of the hairy layer built up by the polymeric surfactant adsorbed on the surface of the polymer particles [34-36]. Coen et al. [34] found that in the seeded emulsion polymerization of St using a PSt seed latex stabilized elec-trosterically by a copolymer of acrylic acid (AA) and St, the electrosteric stabilizer greatly reduced the radical entry rate p compared to the same seed latex... 

Case A The rate of radical entry into micelles that results in the formation of new particles is approximately equal to the rate of radical generation in the water phase (p ), as long as emulsifier micelles are present in other words. 

It is important to note that, even in this present limiting case of a transfer-dominated system, the chain-stoppage mechanism can be changed by compartmentalization. Thus, the MWD formed in the polymerization of styrene appears to be transfer-dominated in some emulsion systems (Piirma et al., 1975) but to be combination dominated in bulk or solution (George, 1967). This difference occurs because, in serene emulsion systems, the rate of radical entry into a particle is slow, and most particles usually contain either zero or one free radical. In the state one particles (Section I,B), the growing free radical has time to undergo several transfer reactions before a further entry causes radical annihilation. 

The parameter c is a measure of the rate of radical entry relative to the rate of bimolecular termination within loci. Similarly, the parameter m is a measure of the rate of radical exit relative to the rate of bimolecular teimiDatioD within loci. The ratio tjm = ojk is a measure of the rate of radical entry relative to radical exit. The so-called Case I, Case 2, and Case 3 of Smith and Ewart (1948) correspond to the following circumstances Case I m large relative to a ... 

Let us consider a seeded emulsion polymerization where no particle formation occurs and hence the number of polymer particles is constant. At a steady state, the overall rate of radical entry into the particles is expressed by... 

Smith and Ewart proposed the steadynstate equation for in terms of the rate of radical entry into the particles, the rate of radical desorption... 

However, when radical desorption from particles takes place m the interval of particle formation, p in Eqs. (108) and (111) should be changed to Pa, the overall rate of radical entry into micelles and particles, and the volumetric growth rate of a particle can no longer be a constant. 

The rational for the use of finishing initiators that have appreciable solubility in the polymer particle follows. At high conversions, the concentration of monomer in the aqueous phase is very low and water-phase termination of hydophilic radicals becomes excessive. The rate of radical entry into polymer particles is thus greatly reduced and the polymerization rate falls to a very low level prematurely. 

The early kinetic model by Smith and Ewart was based on Harkin s mechanistic understanding of the batch process. The particle population balances were written for a stationary state assuming that the rate of formation of particles with n radicals equals the rate of their disappearance (see equation at the bottom of this page). Where / , is the rate of radical entry into a particle (m /sec) is the rate constant for radical exit (m/sec) S is the particle surface area (m ) ktp is the rate constant for bimolecular termination in the particles (m /sec) and o is the particle volume. According to Smith and Ewart three limiting cases can be identified ... 

Case / Where the rate of radical exit from the particle is greater than the rate of radical entry... 

Rate of polymerization. Under general conditions employed in emulsion polymerization, a typical value of the rate of generation of free radicals (Rr) in the aqueous phase is 10 per second per milliliter and a typical value of the number of polymer particles is 10 per milliliter. If all the radicals generated eventually enter M/P particles, since the micelles will have already disappeared, the rate of radical entry in a particle will average out to about one every 10 seconds, which means that the free radicals will generally enter the particles singly. 

When a soaplike free radical enters a M/P particle, polymerization takes place. However, when another free radical enters the same particle, it terminates the growing chain radical by combining with it. (Calculation using known kt values predicts that two radicals cannot coexist in the same polymer particle and they would terminate mutually within a few thousands of a second.) So the particle remains inactive till another free radical enters and initiates the polymerization. Thus, if a radical enters a polymer particle every 10 seconds as calculated above, the particle will grow in alternating periods of activity and inactivity, each of 10 seconds duration. In other words, each particle will remain active for half of the total time (and inactive for the other half). This situation will be unchanged even if the rate of radical entry into the particle is decreased or increased. This can... 

Figure 6.19 Effect of rate of radical entry into particle (i ) on rate of polymerization per particle (Epp).

Figure 6.16 A ba r ffia-gram showing the effect of rate of radical entry into particle on rate of polymerization per particle iRpp). The total period of activity (sum of shaded regions) is not changed even if the rate of radical entry into particle is increased from Ryp to 2Rrp and further to 3Rrp. (After Williams, 1971.)...

The above controversy regarding the physical mechanism of radical entry is reflected in the theoretical expressions which were developed for describing the rate of radical entry by various workers. Hansen and Ugelstad [27,41] summarized the various dependencies of the rate of radical capture on the dimensions of the particles or micelles as follows. Depending on what is the rate-determining step in the absorption process, radical capture rate may be proportional to either the radius of the micelles or particles (for diffusion control in the water phase, when the water solubility of the radicals is low and/or the particles are large), their surface area (for diffusion control in the monomer/polymer phase where the diffusion constant in this phase is low), or their volume (when the particles... 

Fitch and Shih [8S] elaborated Garden s calculation of the rate at which radicals generated in the aqueous phase would collide with latex particles. They found that the electrostatic barrier between a latex particle stabilized only by ionic end-groups and a similarly charged radical was negligible. But experimental results indicated that the rate of radical entry was not proportional to the cross-section of the particles as required by the collision theory instead it was proportional to particle radius as predicted for diffusional entry. 

If X is less than three, the smallest particles grow faster than the laiger ones. In a bimodal system, the dimensionless parameters o and m for each of the two particle sizes are not independent variables. The authors, therefore, made the reasonable assumption that the rate of radical entry (pAa) into particles with diameter da was proportional to the product Neglecting taminaticHi in the aqueous phase, the following equation was obtained [65] ... 

The qualitative approach of Harkins was put on a quantitative basis bv Smith and Ewart. Because 10 radicals are produced per second and can enter between 10 " and 10 particles. Smith felt that a free radical can enter a particle once every 10 to 100 seconds. It can cause the polymerization to occur for 10 to 100 seconds before another free radical would enter and terminate chain growth. A period of inactivity would follow that would last 10 to 100 seconds and then the process would repeat itself. Such a stop and go mechanism implies that a particle contains a free radical approximately half of the time. It can also be said that the average number of radicals per particle is 0.5. This is predicted on condition that (a) the rate of chain transfer out of the particle is negligible, and (b) the rate of termination is very rapid compared with the rate of radical entry into the particle. 

For Case 1, n 0.5, and it corresponds to a system in which the radical desorption rate is much faster than the rate of radical entry. In Case 2, n = 0.5 corresponding to a system in which the radical desorption rate is zero, and instantaneous termination occurs when a radical enters a polymer particle already containing one radical. In Case 3, the concentration of radicals in the polymer particle approaches that of bulk polymerization (n 0.5). For Case 2, the polymerization rate is proportional to the number of particles and the molecular weight also increases with Np. For Cases 1 and 3 the polymerization rate is independent of the number of polymer particles if radical termination in the aqueous phase is negligible, and increases with Np when it is significant. In Case 1, the molecular weights are determined by chain transfer, and in Case 3, the molecular weights are similar to those in bulk. 

Case (ii) is of interest in that it allows account to be taken of, inter alia, the possibilities that the rate of radical entry may vary because (a) the initiator becomes depleted and (6) radicals which have exited from loci into the external phase are available for re-initiation. Case (iii) applies to reaction systems in which the rate of generation of new radicals within the external phase of the reaction system is suddenly reduced to zero it predicts the manner in which the distribution of locus populations will decay from that which pertained at the instant when the rate of radical generation was reduced to zero. 

The rate of radical entry into particles is given bys... 

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