Spinodal decomposition in the context of Nucleation

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics, probability theory, information theory, neural networks, finance, and marketing.

The concept of diffusion is widely used in many fields, including physics (particle diffusion), chemistry, biology, sociology, economics, statistics, data science, and finance (diffusion of people, ideas, data and price values). The central idea of diffusion, however, is common to all of these: a substance or collection undergoing diffusion spreads out from a point or location at which there is a higher concentration of that substance or collection.

Phase separation is the creation of two distinct phases from a single homogeneous mixture. The most common type of phase separation occurs between two immiscible liquids, such as oil and water. This type of phase separation is known as liquid-liquid equilibrium. Colloids are formed by phase separation, though not all phase separations form colloids - for example, oil and water can form separated layers under gravity rather than remaining as microscopic droplets in suspension.

A common form of spontaneous phase separation is termed spinodal decomposition; Cahn–Hilliard equation describes it. Regions of a phase diagram in which phase separation occurs are called miscibility gaps. There are two boundary curves of note: the binodal coexistence curve and the spinodal curve. On one side of the binodal, mixtures are absolutely stable. In between the binodal and the spinodal, mixtures may be metastable: staying mixed (or unmixed) in the absence of some large disturbance. The region beyond the spinodal curve is absolutely unstable, and (if starting from a mixed state) will spontaneously phase-separate.

The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, spinodal decomposition, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. If ${\displaystyle c}$ is the concentration of the fluid, with ${\displaystyle c=\pm 1}$ indicating domains, then the equation is written as

where ${\displaystyle D}$ is a diffusion coefficient with units of ${\displaystyle {\text{Length}}^{2}/{\text{Time}}}$ and ${\displaystyle {\sqrt {\gamma }}}$ gives the length of the transition regions between the domains. Here ${\displaystyle \partial /{\partial t}}$ is the partial time derivative and ${\displaystyle \nabla ^{2}}$ is the Laplacian in ${\displaystyle n}$ dimensions. Additionally, the quantity ${\displaystyle \mu =c^{3}-c-\gamma \nabla ^{2}c}$ is identified as a chemical potential.

In thermodynamics, the limit of local stability against phase separation with respect to small fluctuations is clearly defined by the condition that the second derivative of Gibbs free energy is zero.

The locus of these points (the inflection point within a G-x or G-c curve, Gibbs free energy as a function of composition) is known as the spinodal curve. For compositions within this curve, infinitesimally small fluctuations in composition and density will lead to phase separation via spinodal decomposition. Outside of the curve, the solution will be at least metastable with respect to fluctuations. In other words, outside the spinodal curve some careful process may obtain a single phase system. Inside it, only processes far from thermodynamic equilibrium, such as physical vapor deposition, will enable one to prepare single phase compositions. The local points of coexisting compositions, defined by the common tangent construction, are known as a binodal coexistence curve, which denotes the minimum-energy equilibrium state of the system. Increasing temperature results in a decreasing difference between mixing entropy and mixing enthalpy, and thus the coexisting compositions come closer. The binodal curve forms the basis for the miscibility gap in a phase diagram. The free energy of mixing changes with temperature and concentration, and the binodal and spinodal meet at the critical or consolute temperature and composition.