What's the kinetics model for nanoparticle growth described by sigmoidal funcion? 

The kinetics model for nanoparticle growth described by a sigmoidal function is a complex process that has been extensively studied and modeled in various research efforts. A simple kinetic model derived to describe the formation of TiO2 particles up to a few hundred nanometers in an aqueous suspension shows that the evolution of particle size follows characteristic sigmoid-shaped kinetic curves, indicating a three-step reaction process involving rapid hydrolysis, dimerization, and further growth. This observation aligns with the general chemical mechanism for nanoparticle formation and agglomeration described by Finney and Finke, which includes steps that can be interpreted to fit "S"-shaped, sigmoidal nucleation plus growth particle-formation curves. The two-step particle synthesis mechanism, also known as the Finke-Watzky mechanism, further explains the sigmoidal decrease in precursor concentration following an induction period, which is a characteristic feature of transition metal nanoparticle synthesis. This mechanism, however, has been found to inadequately suppress continued nucleation, leading to unexpectedly large size distributions. The complexity of modeling nanoparticle formation kinetics, due to the involvement of hundreds to thousands of chemical steps, has led to the development of various models, including those based on pseudo-elementary kinetics and population balance modeling. Analytical solutions for nucleation-growth type kinetic models have shown that the final size distribution of nanoparticles is described by a monotonically decreasing function, with the average particle size primarily determined by the kernel function type and the rate constants ratio of spontaneous nucleation and particle growth. The classical model of particle coagulation and a general model combining coarsening and oriented attachment effects have also been proposed to explain growth behavior and deviations from theoretical models. Kinetic Monte Carlo approaches have been developed to study growth and evaporation of nanoparticles, providing insights into process morphology not included in phenomenological thermodynamic modeling . In summary, the kinetics model for nanoparticle growth described by a sigmoidal function encompasses a range of mechanisms, including rapid hydrolysis and dimerization steps, continuous nucleation followed by autocatalytic growth, and complex interactions modeled through various kinetic and population balance approaches. These models collectively contribute to our understanding of nanoparticle synthesis kinetics, highlighting the intricate balance between nucleation, growth, and agglomeration processes .

Full analytical solution of a nucleation-growth type kinetic model of nanoparticle formation

The kinetics model for nanoparticle growth described in the paper follows a nucleation-growth type mechanism with a monotonically decreasing function representing the final size distribution.

Kinetic Model for Hydrolytic Nucleation and Growth of TiO2 Nanoparticles

The kinetic model for nanoparticle growth described by a sigmoidal function involves three reaction steps: rapid precursor hydrolysis, primary particle formation, and subsequent growth.

Modeling Sigmoidal Transients Using Dispersive Kinetic Models to Predict Nanoparticle Size Distributions

The paper discusses using dispersive kinetic models to predict nanoparticle size distributions, including sigmoidal transients, guiding synthetic strategies for nanoparticle growth.

Fitting and Interpreting Transition-Metal Nanocluster Formation and Other Sigmoidal-Appearing Kinetic Data: A More Thorough Testing of Dispersive Kinetic vs Chemical-Mechanism-Based Equations and Treatments for 4-Step Type Kinetic Data

The kinetics model for nanoparticle growth described by a sigmoidal function is the 4-step mechanism of A → B, A + B → 2B, B + B → C, and B + C → 1.5 C.