text: The LaMer model for the kinetics of the formation of hydrosols is widely applicable for production of monodisperse systems, and it was originally hypothesized that the Stöber process followed this monomer addition model. This model includes a rapid burst of nucleation forming all of the particle growth sites, then proceeds with hydrolysis as the rate-limiting step for condensation of triethylsilanol monomers to the nucleation sites. The production of monodisperse particle sizes is attributed to monomer addition happening at a slower rate on larger particles as a consequence of diffusion-limited mass transfer of TEOS. However, experimental evidence demonstrates that the concentration of hydrolyzed TEOS stays above that required for nucleation until late into the reaction, and the introduction of seeded growth nuclei does not match the kinetics of a monomer addition process. Consequently, the LaMer model has been rejected in favour of a kinetic model based around growth via particle aggregation.
  Under an aggregation-based model, nucleation sites are continually being generated and absorbed where the merging leads to particle growth. The generation of the nucleation sites and the interaction energy between merging particles dictates the overall kinetics of the reaction. The generation of the nucleation sites follows the equation below:
  Where J is the nucleation rate, k and k are rate constants based on the concentrations of HO and NH and g is the normalization factor based on the amount of silica precursor. Adjusting the concentration ratios of these compounds directly influences the rate at which nucleation sites are produced.
  Merging of nucleation sites between particles is influenced by their interaction energies. The total interaction energy is dependent on three forces: electrostatic repulsion of like charges, vanderWaals attraction between particles, and the effects of solvation. These interaction energies (equations below) describe the particle aggregation process and demonstrate why the Stöber process produces particles that are uniform in size.
  The van der Waals attraction forces are governed by the following equation:
  Where A is the Hamaker constant, R is the distance between the centers of the two particles and a, a are the radii of the two particles. For electrostatic repulsion force the equation is as follows:
  :       where       
  Where &epsilon; is the dielectric constant of the medium, k is Boltzmanns constant, e is the elementary charge, T is the absolute temperature, &kappa; is the inverse Debye length for a 1:1 electrolyte, x is the (variable) distance between the particles, and &phi;' is the surface potential. The final component of the total interaction energy is the solvation repulsion which is as follows:
  Where A is the pre-exponential factor (1.5 &times; 10 J m) and L is the decay length (1 &times; 10 m).
  This model for controlled growth aggregation fits with experimental observations from small-angle X-ray scattering techniques and accurately predicts particle sizing based on initial conditions. In addition, experimental data from techniques including microgravity analysis and variable pH analysis agree with predictions from the aggregate growth model.