A mathematical model to describe the growth of an arbitrarily large number of nanocrystals from solution is presented. First, the model for a single particle is developed. By non-dimensionalising the system we are able to determine the dominant terms and reduce it to the standard pseudo-steady approximation. The range of applicability and further reductions are discussed. An approximate analytical solution is also presented. The one particle model is then generalised to $N$ well dispersed particles. By setting $N=2$ we are able to investigate in detail the process of Ostwald ripening. The various models, the $N$ particle, single particle and the analytical solution are compared against experimental data, all showing excellent agreement. By allowing $N$ to increase we show that the single particle model may be considered as representing the average radius of a system with a large number of particles. Following a similar argument the $N=2$ model could describe an initially bimodal distribution. The mathematical solution clearly shows the effect of problem parameters on the growth process and, significantly, that there is a single controlling group. The model provides a simple way to understand nanocrystal growth and hence to guide and optimise the process.

中文翻译：

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通过沉淀法模拟纳米晶体生长

提出了描述从溶液中任意大量纳米晶体生长的数学模型。首先，开发单个粒子的模型。通过对系统进行无量纲化，我们能够确定主要项并将其减少到标准的伪稳态近似值。讨论了适用范围和进一步的减少。还提供了近似解析解。然后将单粒子模型推广到 $N$ 良好分散的粒子。通过设置 $N=2$，我们能够详细研究 Ostwald 成熟的过程。将各种模型、$N$ 粒子、单粒子和解析解与实验数据进行比较，均显示出极好的一致性。通过允许 $N$ 增加，我们表明单粒子模型可以被视为代表具有大量粒子的系统的平均半径。遵循类似的论点，$N=2$ 模型可以描述最初的双峰分布。数学解决方案清楚地显示了问题参数对生长过程的影响，而且很明显，存在一个单一的控制组。该模型提供了一种理解纳米晶体生长的简单方法，从而指导和优化过程。