In this paper, we present a detailed deduction of the Cahn-Hilliard equation. It is a non-linear partial differential equation that continuously describes the dynamics of spinodal decomposition, i.e., spontaneous phase separation, of a binary mixture. From such an equation we can model systems that undergo phase separation toward the equilibrium, i.e., passive systems, and systems that are kept out of the equilibrium by the presence of either external forces or chemical reactions, i.e., active systems. To accomplish our main goal, we present the main concepts of non-equilibrium thermodynamics and from that, we obtain the Cahn-Hilliard equation by defining a general equation for the transport of matter and calculating the free energy for a homogeneous and non-homogeneous mixture of real fluids. By the end, we perform simulations of the Cahn-Hilliard equation and present two examples of passive and active systems that (quasi) suppress Ostwald ripening.

We present the main concepts of nonlinear dynamics and thermodynamics of irreversible processes to introduce chemistry students to the topic of self-organized phenomena. This task is performed by theoretically describing the emergence of self-sustained oscillations, waves, and stationary patterns/Turing patterns in the Belousov-Zhabotinsky (BZ) reaction, through the Oregonator model. We carefully developed such a description, which resulted in long algebraic deductions and rich supplementary material. Considering that, we encourage the use of this material in undergraduate and graduate advanced physical chemistry classes.

The Practical Stochastic Model is a simple and robust method to describe coupled chemical reactions. The connection between this stochastic method and a deterministic method was initially established to understand how the parameters and variables that describe the concentration in both methods were related. It was necessary to define two main concepts to make this connection: the filling of compartments or dilutions and the rate of reaction enhancement. The parameters, variables, and the time of the stochastic methods were scaled with the size of the compartment and were compared with a deterministic method. The deterministic approach was employed as an initial reference to achieve a consistent stochastic result. Finally, an independent robust stochastic method was obtained. This method could be compared with the Stochastic Simulation Algorithm developed by Gillespie, 1977. The Practical Stochastic Model produced absolute values that were essential to describe non-linear chemical reactions with a simple structure, and allowed for a correct description of the chemical kinetics.