# Mathematical modelling of nanocrystal growth

Nanoparticles are units of matter with dimensions between 1 and 100 nanometers (nm) that have gained a lot of interest during the last decades, due to their wide variety of applications in biomedicine, environmental-related problems, electronics and catalysis. They have unique chemical, physical, mechanical, and optical properties that naturally occur at that scale that we cannot appreciate in any other scale. Gold nanoparticles provide an excellent example: at the nanoscale, the motion of the gold’s electrons is confined and, because of that, they react differently with light compared to larger-scale. From this comes that gold nanoparticles are not yellow as we expect, but can appear purple or red. Moreover, adjusting their size, gold nanoparticles can be tuned according to the purpose: for example, they can selectively accumulate in tumors in order to identify diseased cells and to target laser destruction of the tumor avoiding healthy cells.

Many properties of nanoparticles, such as electronic and optical properties of metals and semiconductors, luminescence, photostability, and optical radiation efficiencies among others, are size dependent. Hence, the ability to create nanoparticles of a specific size is crucial. In order to do this, we need a clear understanding for the process of growing nanoparticles.

Figure 1: Change in colour of solution containing gold nanoparticles of increasing size (Picture taken, with permission, from [1].

My project is motivated by a research area at the Institut Català de Nanociència i Nanotecnologia (ICN^2) concerning nanoparticle growth from a supersaturated solution. Typical particles are shown in Figure 2. Using the precipitation method (i.e. the creation of a solid from a solution) monodisperse spherical nanoparticles can be generated. The standard approach is to apply the classical La Mer and Dinegar synthesis strategy where nucleation and growth are separated. However, in the last phase of the growth period we can observe a phenomenon called Ostwald ripening, a process by which larger particles grow at the expense of the smaller ones which dissolve due to their much higher solubility. This process produces monomer, which is subsequently used to support growth of the larger particles. This simultaneous growth and dissolution leads to the unwanted defocusing of the particle size distribution, that can be refocused by changing the reaction kinetics.

Fig 2: Particles grown from solution, steps indicate the number of times monomer is added to the solution, see [2].

The mathematical challenge is to model the process of synthesizing nanoparticles of the required size from a liquid solution, that is analogous to one-phase Stefan problem. For this project, we focus our work on size control of nanocrystals. We look for a model that can predict the standard deviation of the radius as a function of the initial particle size distribution, which is not totally controlled, since they are largely determined by a generally unknown nucleation mechanism.
The goal in developing a theoretical model is to determine how the controlling parameters influence the process and so understand how to optimise the growth.
The model consists of a diffusion equation for the concentration of the solution and a mass balance (equivalent to a Stefan condition) for the evolving particle radius. For the far-field bulk concentration, a mass conservation expression is used. Based on a small dimensionless parameter, we propose a pseudo-steady state approximation to the model, that is consequently extended to a system of N particles. The first stage of this model building is described in [3].

By Claudia Fanelli, PhD student, Centre de Recerca Matematica

[1] File:Gold255.jpg – Wikimedia Commons, https://www.google.es/search?q=gold+nanoparticles&tbm=isch&source=lnt&tbs=sur:f&sa=X&ved=0ahUKEwiq-6qt76XcAhUDuRoKHa2XA5oQpwUIHg&biw=1920&bih=947&dpr=1#imgrc=g_xdRB5Fe6C6XM

[2] Neus G. Bastus, Joan Comenge, and Víctor Puntes, Kinetically Controlled Seeded Growth Synthesis of Citrate-Stabilized: Gold Nanoparticles of up to 200 nm: Size Focusing versus Ostwald Ripening. Langmuir 2011.

[3] Tim G. Myers and Claudia Fanelli, On the incorrect use and interpretation of the
model for colloidal, spherical crystal growth. Submitted to Nano Letters, June 2018.