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Thermodynamic limit for particle monodispersity: How narrow can a particle size distribution be?

A key problem in many emerging nanotechnologies is to create a narrow distribution of particles. Here, we prove the existence of a fundamental lower bound for the width of a particle size distribution (PSD) at equilibrium. This thermodynamic limit arises from the decrease of entropy for an ideal monodisperse distribution and is determined by the effective surface energy and temperature. The existence of this limit, provided that it is energetically favourable, implies that it is not possible to generate an arbitrarily narrow PSD through any process that reaches a thermodynamic equilibrium. We show that if a narrow PSD is generated it will tend to broaden in time until it reaches an equilibrium PSD of Gaussian shape. Conversely, if the initial PSD is broad and free to evolve to its equilibrium shape, it will eventually acquire the same Gaussian shape and width. The effect of PSD broadening can be significant for functional materials built from nanoparticulate arrays. Our study provides the framework for understanding and describing this phenomenon.