2 Theoretical Foundations of Critical Phenomena in Liquids

The purpose of this chapter is to examine the theoretical description of the critical behavior of binary liquids exhibiting a miscibility gap. The most important properties of critical systems and the physical quantities required for the present investigation are introduced. Particular attention is paid to the key predictions of the dynamic scaling theory of Bhattacharjee and Ferrell, which plays a central role in this thesis.

The development of scaling theories and of the associated characteristic quantities—the critical exponents—begins with a discussion of phase transitions in thermodynamic systems.

2.1 Phase Transitions in Thermodynamic Systems

2.1.1 Phase Transitions in General

Consider a many-particle system S described thermodynamically by a potential G(A₁, A₂, …, Aₙ). The partial derivatives αᵢ := ∂G/∂Aᵢ correspond to phenomenological thermodynamic observables, such as temperature T, pressure p and volume V. The total differential of the general potential is

For a many-particle system S, or for an individual phase, Eq. (2.1) is a fundamental equation: knowing the Gibbs free energy G makes it possible to derive all thermodynamic properties of the system. If the system parameters change, a phase transformation or phase transition may occur under conditions that depend on the particular system.

Original footnote 1: ∂_AᵢG = ∂G/∂Aᵢ.