The polymerization rate in this stage can be expressed by where is the homogeneous propagation rate constant for polymerization within the particles and is the equilibrium monomer concentration within a particle. represents the overall concentration of polymerizing radicals in the reaction. For Case 2, where the average number of free radicals per micelle are , can be calculated in following expression: where is number concentration of micelles (number of micelles per unit volume), and is the Avogadro constant (). Consequently, the rate of polymerization is then 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. Process considerations Emulsion polymerizations have been used in batch, semi-batch, and continuous processes. The choice depends on the properties desired in the final polymer or dispersion and on the economics of the product. Modern process control schemes have enabled the development of complex reaction processes, with ingredients such as initiator, monomer, and surfactant added at the beginning, during, or at the end of the reaction. Early styrene-butadiene rubber (SBR) recipes are examples of true batch processes: all ingredients added at the same time to the reactor. Semi-batch recipes usually include a programmed feed of monomer to the reactor. This enables a starve-fed reaction to ensure a good distribution of monomers into the polymer backbone chain. Continuous processes have been used to manufacture various grades of synthetic rubber. Some polymerizations are stopped before all the monomer has reacted. This minimizes chain transfer to polymer. In such cases the monomer must be removed or stripped from the dispersion. Colloidal stability is a factor in design of an emulsion polymerization process. For dry or isolated products, the polymer dispersion must be isolated, or converted into solid form. This can be accomplished by simple heating of the dispersion until all water evaporates. More commonly, the dispersion is destabilized (sometimes called "broken") by addition of a multivalent cation. Alternatively, acidification will destabilize a dispersion with a carboxylic acid surfactant. These techniques may be employed in combination with application of shear to increase the rate of destabilization. After isolation of the polymer, it is usually washed, dried, and packaged. By contrast, products sold as a dispersion are designed with a high degree of colloidal stability. Colloidal properties such as particle size, particle size distribution, and viscosity are of critical importance to