Editing Emulsion polymerization (section)

==More detailed treatment of Smith-Ewart theory==

=== Interval 1 ===
When radicals generated in the aqueous phase encounter the monomer within the micelle, they initiate polymerization. The conversion of monomer to polymer within the micelle lowers the monomer concentration and generates a monomer concentration gradient. Consequently, the monomer from monomer droplets and uninitiated micelles begin to diffuse to the growing, polymer-containing, particles. Those micelles that did not encounter a radical during the earlier stage of conversion begin to disappear, losing their monomer and surfactant to the growing particles. The theory predicts that after the end of this interval, the number of growing polymer particles remains constant.

=== Interval 2 ===
This interval is also known as steady state reaction stage. Throughout this stage, monomer droplets act as reservoirs supplying monomer to the growing polymer particles by diffusion through the water. While at steady state, the ratio of free radicals per particle can be divided into three cases. When the number of free radicals per particle is less than {{frac|1|2}}, this is called Case 1. When the number of free radicals per particle equals {{frac|1|2}}, this is called Case 2. And when there is greater than {{frac|1|2}} radical per particle, this is called Case 3. Smith-Ewart theory predicts that Case 2 is the predominant scenario for the following reasons. A monomer-swollen particle that has been struck by a radical contains one growing chain. Because only one radical (at the end of the growing polymer chain) is present, the chain cannot terminate, and it will continue to grow until a second initiator radical enters the particle. As the rate of termination is much greater than the rate of propagation, and because the polymer particles are extremely small, chain growth is terminated immediately after the entrance of the second initiator radical. The particle then remains dormant until a third initiator radical enters, initiating the growth of a second chain. Consequently, the polymer particles in this case either have zero radicals (dormant state), or 1 radical (polymer growing state) and a very short period of 2 radicals (terminating state) which can be ignored for the free radicals per particle calculation. At any given time, a micelle contains either one growing chain or no growing chains (assumed to be equally probable). Thus, on average, there is around 1/2 radical per particle, leading to the Case 2 scenario. The polymerization rate in this stage can be expressed by

<math display="block">R_p = k_p[\mathrm{M}][\mathrm{P}^\bullet]</math>where <math display="inline">k_p</math> is the homogeneous propagation rate constant for polymerization within the particles and <math>[\mathrm{M}]</math> is the equilibrium monomer concentration within a particle. <math display="inline">[\mathrm{P}^\bullet]</math> represents the overall concentration of polymerizing radicals in the reaction. For Case 2, where the average number of free radicals per micelle are <math>1/2</math>, <math display="inline">[\mathrm{P}^\bullet]</math> can be calculated in following expression:

<math display="block">[\mathrm{P}^\bullet] = \frac{N_\mathrm{micelles}}{2N_\mathrm{A}}</math>where <math>N_\mathrm{micelles}</math> is number concentration of micelles (number of micelles per unit volume), and <math>N_\mathrm{A}</math> is the [[Avogadro constant]] ({{val|6.02|e=23|u=mol-1}}). Consequently, the rate of polymerization is then

<math display="block">R_p = k_p[\mathrm{M}]\frac{N_\mathrm{micelles}}{2N_\mathrm{A}}.</math>

=== Interval 3 ===

Separate monomer droplets disappear as the reaction continues. Polymer particles in this stage may be sufficiently large enough that they contain more than 1 radical per particle.