Modified kinetic rate equation model for cooling crystallization

Modeling of Nucleation and Growth in the Synthesis of PbS Colloidal Quantum Dots Under Variable Temperatures

“…Currently, there exists three temperature-changing strategies for CQDs synthesis: the traditional linear or exponential cooling crystallization, the constant temperature approach, and the temperature plateau manipulation . The general constant temperature synthesis was more thoroughly studied through experimental and theoretical investigation. , In the temperature plateau approach for PbS CQD synthesis via the hot-injection method, the reaction mixture is maintained at the nucleation temperature for a sufficiently long time; this intermediate temperature is determined by using the saturated temperature of solubility data, , and there are twice or more cooling or heating processes.…”

“…The original KRE model proposed by Mazzotti et al assumed that the reaction system was under isothermal condition, and its solution supersaturation was set as an initial condition . Compared with this model, a size- and temperature-dependent cluster solubility to describe the post-cooling crystallization process was introduced into the original KRE model by Jin et al The growth and dissociation frequencies, respectively, are given by Here, c = 1/3, α is the capillary length of the crystal, and k a is the attachment rate constant. Though the modified KRE model was validated by comparison with the experimental data of metal nanoparticle crystallization, it should be further developed to describe the PbS CQD formation via hot-injection synthesis and to understand the knowledge of kinetic parameters consisting of different variables.…”

“…Based on deMello’s improvement, the detachment rate of the monomer on cluster or another monomer surface in this paper is assumed to be proportional to the number of sites on the surface; thus, this value is proportional to the surface area of a CQD Here, the empirical value α is estimated from the properties of the solute dissolved into the solvent, and it was set between 1/2 and 1/3 . There is a main assumption that the monomer detachment that occurs rapidly becomes available to other CQDs without having to diffuse out of the surfactant layer …”

“…Rempel et al employed the discrete KREs to describe the model at the smaller nanoparticle sizes while utilized the continuous Fokker–Planck equations to approximate the kinetic model at the larger nanoparticle sizes. ,, This method covers the early growth stage and the ripening stage, which can be solved by the numerical technique with reasonable computational effort and great accuracy. The ordinary differential formulations in the discrete part of the model can be calculated by standard integration solvers in MATLAB ode15s that are effective in solving a system of stiff differential equations; , the variable-coefficient Fokker–Planck equations in the continuous part of this model can be solved explicitly through a discretization method first presented by Chang and Cooper, which has already been applied to simulate the atmospheric cluster dynamics and the sol cluster dynamics; the discretized KREs are again solved with the MATLAB ode15s. Our computer simulation is according to the method proposed by deMello et al In these cases, the method of characteristics can be used to calculate the analytically solution.…”

“…Because of characteristic differences between reacting particles and a temperature-changed reaction system, the original KRE model should be developed to match more comprehensive parameters to describe the whole CQD formation under different temperature strategies. It was inspired by the use of size-dependent growth rate incorporating the Gibbs–Thomson expression into the PBE model; the KRE model was modified to delineate post-cooling crystallization . For using the hot-injection method to obtain high quality and large-quantities CQDs, especially under variable temperatures, it is indispensable that burst nucleation, size-focusing growth, and delayed OR growth must be meet.…”