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Publication Detail
Mathematical modelling of the formation of gold nanoparticles via the citrate synthesis method
This work presents a new model for predicting the evolution of the size of gold nanoparticles (GNPs) in the citrate synthesis method. In this method, the precursor is an acid solution of tetrachloroauric acid, while the reducing agent is a base solution of sodium citrate. The acid-base properties of the solutions influence how the size of the particles evolves during the synthesis. In the literature, various mechanistic theories have been proposed to explain this evolution. Turkevich et al. (1951), who pioneered this synthesis method, suggested the “organizer theory” also known as “nucleation-growth” mechanism. Recently, however, Wuithschick et al. (2015) proposed a “seed-mediated” mechanism, a nucleation-aggregation-growth mechanism. In investigating the synthesis, while Turkevich et al. (1951) used the conventional techniques such as the transmission electron microscopy (TEM), the scanning electron microscopy (SEM) and the UV-vis spectroscopy, Wuithschick et al. (2010) used a combination of X-ray absorption near edge spectroscopy (XANES) and small angle X-ray scattering (SAXS) along with the conventional techniques. This setup provides time-resolved in situ information on the formation of GNPs, thereby yielding reliable accounts of the synthesis mechanism. Nevertheless, only one mathematical model has been developed, that advanced by Kumar et al. (2007), which is based on the nucleation-growth theory proposed by Turkevich et al. (1951). This model had not been thoroughly tested. In a part of this work, we investigate the model of Kumar et al. (2007) for different conditions of pH, temperature and initial reactant concentrations. To solve the model, we use the numerical code Parsival, which is developed for solving population balance equations. We test the model for different synthesis conditions studied experimentally by various researchers, for which experimental data are available in the literature. The model poorly predicts these data, because the Turkevich organizer theory does not account for the acid-base properties of chloroauric acid and sodium citrate. Thereafter, we present a novel kinetic model based on the synthesis seed-mediated mechanistic description proposed by Wuithschick et al. (2015). In this description, the precursor concurrently reduces into gold atoms and hydroxylates into a passive form. The gold atoms then aggregate into seed particles, which finally react with the passive form of the precursor in a growth step. We validate the model using experimental data from the literature obtained for conditions in which the seed-mediated mechanism is valid. The predicted GNP final sizes closely agree with those obtained experimentally. Finally, we present a modelling approach for the aggregation process in metal nanoparticles syntheses based on the theory proposed by Polte (2015). In this theory, metal atoms formed by reducing the precursor solution aggregate to larger sizes due to the Van der Waals’ forces of attraction. Then, due to the electrostatic forces of repulsion induced by the “potential determining” ions, the nanoparticles eventually stop aggregating and become stabilized. Based on this theory, we develop a model for the aggregation process resulting from the interplay of the attractive and repulsive forces in the evolution of nanoparticles. Using this model, we describe how gold atoms aggregate into seed particle in the citrate synthesis method. Then, we couple this aggregation model with the model developed for the seed-mediated mechanism. To validate the model predictions, we employ the experimental data used to validate the seed-mediated mechanism. In addition to the GNP final sizes, this integrated model correctly predicts the size polydispersity and completely describes the final GNP particle size distribution.
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