Modelling nanocrystal growth via the precipitation method

https://doi.org/10.1016/j.ijheatmasstransfer.2020.120643Get rights and content

Highlights

  • Provides a mathematical model for nanocrystal growth from a colloidal solution.

  • With just one

    tting parameter shows excellent agreement with the growth of Cadmium Selenide nanocrystals.

  • Proves that an analytical solution for the growth of a single crystal is a good approximation for the average radius in a large system.

  • Provides Provides a simple model for Ostwald ripening.

  • Provides a method for optimising nanocrystal growth in a multi-step in-jection process.

Abstract

A mathematical model for the growth of a single nanocrystal is generalised to deal with an arbitrarily large number of crystals. The basic model is a form of Stefan problem, describing diffusion of monomer over a moving domain. Various levels of approximation (an analytical solution, an ordinary differential equation model and an N particle model) are compared and shown to agree well. The N particle model and analytical solution are then shown to have excellent agreement with experimental data for the growth of CdSe nanocrystals. The theoretical solution clearly shows the effect of problem parameters on the growth process and, significantly, that there is a single controlling group. By increasing the value of N it is shown that in the absence of Ostwald ripening the single particle model may be considered as representing the average radius of a system with a large number of particles. Consequently a system with N=2 may represent either a two particle system or a bimodel initial distribution. The solution of the N=2 model provides an understanding of Ostwald ripening. In general if Ostwald ripening is expected some form of the N particle model should be employed. Finally it is shown how the analytical solution may be employed to represent a multi-stage growth process which can then guide and optimise crystal growth.

Introduction

Nanoparticles (NPs) are small units of matter with dimensions in the range 1-100nm. They exhibit many advantageous, size-dependent properties such as magnetic, electrical, chemical and optical, which are not observed at the microscale or larger [4], [11], [18], [30]. Consequently the ability to produce monodisperse particles that lie within a controlled size distribution is critical.

There exist a number of NP synthesis methods, including gas phase and solution based synthesis techniques. Although the first method can produce large quantities of nanoparticles, it produces undesired agglomeration and nonuniformity in particle size and shape. Precipitation of NPs from solution avoids these problems and is one of the most widely used synthesis methods [17]. The typical strategy is to cause a short nucleation burst in order to create a large number of nuclei in a short space of time, and the seeds generated are used for the latter particle growth stage. The resulting system consists of varying sized particles. Small NPs are more unstable than larger ones and tend to grow or dissolve faster. Thus at relatively high concentrations size focussing occurs (leading to monodispersity). When the concentration is depleted by the growth some smaller NPs shrink and eventually disappear while larger particles continue to grow, thus leading to a broadening of the size distribution (which involves the process of Ostwald ripening). Ostwald ripening is the process whereby smaller crystals dissolve and the material from these crystals is redeposited onto the larger ones. Hence, below a certain size the crystals start to decrease in size until they disappear, while larger crystals increase in size. Perhaps the most well-known example of Ostwald ripening is the coarsening of crystals in ice cream, giving a different texture to old ice cream. Depending on the system Ostwald ripening can be rapid or very slow, as in the famous experiment of Faraday in the 1850’s using colloidal gold which is still optically active.

The particle size distribution (PSD) can be refocused by changing the reaction kinetics. For example, Peng et al. [24] observed size focusing during Cadmium Selenide growth following the injection of additional solute. Bastús et al. [2], [3] were also able to induce size focusing of gold and silver nanoparticles by the addition of extra solute and adjusting the temperature and pH. This type of technique for size focussing is still rather ad hoc in that the precise relationships between particle growth, system conditions and the final PSD are not fully understood [26]. Hence, in practice, the optimal reaction conditions are usually ascertained empirically or intuitively.

In the 1960’s Lifshitz and Slyozov [14] and, independently, Wagner [35] were amongst the first to provide theoretical descriptions of Ostwald ripening. Their classical theory, hereafter referred to as LSW theory, consisted of a system of three coupled equations: a growth equation for a single particle, a continuity equation for the PSD and a mass conservation expression for the concentration. They solved the model to obtain pseudo-steady-state asymptotic solutions for the average particle radius and PSD. Lifshitz and Slyozov [14] focused on diffusion-limited growth, where growth is limited by the diffusion of reactants to the particle surface, while Wagner [35] considered growth limited by the reactions at the particle surface. In fact recent work described by Myers and Fanelli [20] has shown that, within the restrictions of the steady-state assumption, the models cannot distinguish between diffusion or reaction driven growth, so both approaches are equally valid. For this reasons authors using either mechanism, or both, have been equally successful in approximating experimental data.

Experimental studies on NP growth [6], [22], [27] show that LSW theory may provide good predictions for the particle size but the observed PSDs are typically broader and more symmetric. Possible explanations for this disparity is that LSW theory does not account for the finite volume of the coarsening phase ϕ, and that it assumes a particle’s growth rate is independent of its surroundings. In addition, LSW theory does not indicate how long it takes to reach the final state. A further issue is that it purports to describe the dynamics in the initial stages of the growth process. In [20] it is proven that the pseudo-steady solution does not hold for small times.

Many studies have modified and built on the pioneering analysis of LSW theory. Ardell [1] and Sarian and Weart [25] extended LSW theory to systems where the mean distance between particles is finite. Several authors [5], [33], [34] have addressed the shortcomings of LSW theory by statistically averaging the diffusional interaction of a particle of a given size with its surroundings to demonstrate that the resulting PSD becomes broader and more symmetric with increasing ϕ. The inclusion of stochastic effects, due to temperature and changes in concentration, in the modified population balance model of Ludwig et al. [16] led to broader PSDs in line with experimental data. The population balance approach of Iggland and Mazzotti [12] was used to examine the evolution of non-spherical particles at the beginning of growth.

Most of the above studies were in relation to micron or larger-sized particles. As measurement techniques have advanced many researchers have applied LSW theory and the related modifications to the study of nanoparticle growth. Talapin et al. [29] used a Monte Carlo approach to simulate the evolution of a nanoparticle PSD subject to diffusion-limited growth, reaction-limited growth and mixed diffusion-reaction growth. In contrast to other treatments, their simulations gave PSDs narrower than those predicted by LSW theory. This was explained by the fact that they considered much smaller particles. Their main conclusion was that Ostwald ripening occurs much more rapidly for nanoparticles while PSDs are narrower than in their microscale counterparts. Similarly, Mantzaris [17] used a population balance formulation and a moving boundary algorithm to study the diffusion and reaction-limited growth regimes.

Another issue which is particularly relevant in the context of nanoparticles is the applicability of the Ostwald-Freundlich condition which relates the radius of the particle, rp*, to its solubility, s*. This condition can be written ass*=s*exp(2σVMrp*RGT)s*exp(αrp*),where s* is the solubility of the bulk material, σ the interfacial energy, RG the universal gas constant, T the absolute temperature. The capillary length α=2σVM/(RGT) defines the length scale below which curvature-induced solubility is significant [29]. This equation shows that the particle solubility increases as the size decreases (which promotes Ostwald ripening). One approximation to the Ostwald-Freundlich condition is to assume that the exponential term in (1) can be linearised to give the two term expression s*s*(1+α/rp*) [14], [15], [28], [35]. Obviously this expansion, which is based on α/rp*, is invalid for nanoparticles where the capillary length is of the same order of magnitude as the particle radius [20]. Mantzaris [17] used an expansion for the exponential term in the Ostwald-Freundlich condition with n terms and showed that increasing n led to higher average growth rates and a narrowing of the PSD. However, when comparing his simulation to experimental data for CdSe nanoparticles from [23], he applied a linear version for the solubility. Talapin et al. [29], noting that for nanoparticles of the order 1-5nm the linearised Ostwald-Freundlich condition may be incorrect, applied the full condition.

In the following we begin by analysing the growth of a single particle. This is the basic building block for more complex models. The treatment leads to equations similar to those of standard LSW theory, however we arrive at them following a non-dimensionalisation which highlights dominant terms and those which may be formally neglected. In this way we can ascertain which standard assumptions are appropriate and, more importantly, which are not. Under conditions which appear easily satisfied for nanocrystal growth the governing ordinary differential equation has an explicit solution, in the form rp=rp(t) and also shows that the growth is controlled by a single parameter which may be calculated by comparison with experiment. This section closely follows the work described in [20]. The single particle model is obviously incapable of reproducing Ostwald ripening, where larger particles grow at the expense of smaller ones. Consequently we then generalise the model to deal with a large number of particles. In the results section we compare the analytical solution with that of a full numerical solution and experimental data for the growth of a single particle and show excellent agreement between all three. By setting the number of particles to two in the general model we are able to clearly demonstrate Ostwald ripening. Simulations with N= 10 and N= 1000 particles demonstrate that increasing N leads to increasingly good agreement between the average radius and that predicted by the single particle model. The single particle model may thus be considered as a viable method for predicting the evolution of the average radius of a group of particles.

Section snippets

Growth of a single particle

As shown in Fig. 1, we initially focus on a single, spherical nanoparticle, with radius rp* in a system of particles. The * notation represents dimensional quantities. The assumption is that particles are separated at large but finite distances compared to their radius. Their morphologies remain nearly spherical and particle aggregation is neglected. Thus, the mass flow from each particle can be represented as a monopole source located at the center of the particle [32] and the problem becomes

Evolution of a system of N particles

We now extend the above single particle model to a system of N particles where N is arbitrarily large and may decrease with time due to Ostwald ripening. The particle radii, initial radii and solubilities are denoted ri*, ri,0* and si*, respectively, where i=1,,N represents the ith particle. We nondimensionalise via (11) with the only difference being that the length scale rp,0* is replaced by the mean value r¯i,0*=i=1N(ri,0*/N). It has to be noted that this affects the concentration scale

Comparison of model with experiment

Using the full numerical solution, defined by (12)–(14), on the N particles model would be prohibitively expensive. For this reason the first goal of this section is to demonstrate that the pseudo-steady state model of Eq. (20) is a good approximation to (12)–(14), so justifying its use in our N particles model. We validate the models using experimental data on CdSe nanocrystals synthesis taken from [23]. Certain parameter values concerning the experiment and CdSe are provided in that paper,

Conclusions

We have developed a model for the growth of a system of N particles, where N may be arbitrarily large. The model involves a system of first order nonlinear ordinary differential equations, which are easily solved using standard methods. The basis of the N particle model is the pseudo-steady approximation presented in [20] which was shown to be an excellent approximation to the full numerical solution. This model incorporates the particle solubility variation which then permits the model to

Author Statement

The work has not been published previously and it is not under consideration for publication elsewhere. Its publication is approved by all authors. If accepted, it will not be published elsewhere in the same form, in English or in any other language, including electronically without the written consent of the copyright-holder.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Claudia Fanelli acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Maria de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445). Tim G. Myers acknowledges the support of Ministerio de Ciencia e Innovacion Grant No. MTM2017-82317-P. Francesc Font and Vincent Cregan acknowledge financial support from the Obra Social ”la Caixa” through the programme Recerca en Matemática Col·laborativa. The authors have been partially funded by the

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