Nucleation. Thermodynamical and Stochastical Descriptions

Abstract

Nucleation is a non-equilibrium pro cess through which a system evolves from an old thermo dynamic phase to a new (energetically stable) one in what we call a first-order phase transition. This process is widely spread in natural phenomena like condensation, sublimation, evaporation or crystal growth, and it is also of technological relevance [1]. In this work, the main theories behind nucleation developed throughout the last century by researchers like Gibbs, Zeldovich and Kashchiev will be reviewed. We will start with the study of the Van der Waals equation (VDWE), as it provides a general insight of the main conditions required for a phase transition to occur. Next, the Classical Nucleation Theory (CNT), the core of this work, with its central paradigm of well-defined clusters is introduced. With the aid of the capillary model and the assumption of large spherical clusters, a first approach to estimate the minimum work required for a cluster to form (also known as the potential or nucleation barrier) is obtained, with a general description of how the cluster grows depending on its initial size. Moreover, the CNT also provides a dynamical theory of cluster formation based on the attachment and detachment of n-mers (clusters formed by ’n’ molecules), which leads to Fokker-Planck, Langevin, Kramers-Moyal and Master equations, due to the stochastic nature of the time-dependent concentration of n-mers in the system. With the help of the BDT (Becker-D¨oring-Tunitskii) model and the Zeldovich’s ideas, an analytical expression for the nucleation rate is derived, which constitutes the main quantity that allows to corroborate the theoretical predictions experimentally. With this framework, we will focus on the study of homogeneous nucleation of water droplets in condensate vapor phases, with the goal of obtaining the cluster size equilibrium distribution functions and the performance of a comparison between the theoretical and experimental nucleation rates measured in diffusion cloud chambers. This procedure will allow us to study the limitations and validity of the classical theory, and its study will conclude with a section where an oversight of its main controversies and deficiencies is provided. Finally, following this, we will make a brief introduction presenting the framework and some fundamental approximations of the modern paradigm of the Density Functional Theory, which is based on variational calculus. This theory provides a completely new framework that improves the classical theory and opens the door to new models that, in concordance with the experiments, allows us to deepen in this field of study, helping us to uncover the mysteries of phase transitions and matter itself.