2.1.3 Liquid–Liquid Phase Transition

Binary liquid mixtures with a miscibility gap

Having discussed the classification of phase transitions, we now consider the liquid–liquid phase transition in binary liquid mixtures in greater detail.

Binary liquid mixtures exhibit diverse behavior and are strongly affected by external influences and changes in thermodynamic parameters. As the term “binary” indicates, such a system consists of two components, A and B. Its composition is described by the particle-number fraction x = n(A)/(n(A) + n(B)). Examining the system as a function of x and temperature T or pressure P reveals regions in which a single homogeneous phase exists and others in which two distinct phases are present. The latter range is termed the miscibility gap.

The boundary between homogeneous and heterogeneous states is described by the phase-equilibrium curve, also known as the coexistence curve or binodal. Its shape is system-specific and depends on the chemical and physical properties of components A and B. Figure 2.2 illustrates familiar possible forms of the binodal (solid line) in phase diagrams. Another feature of binary systems is the stability-limit curve, or spinodal (dashed line). Together, the binodal and spinodal define a metastable region in which two phases may coexist, but need not necessarily do so.

Above the binodal, the system is homogeneous and thermodynamically stable; below the spinodal, it is heterogeneous and thermodynamically unstable. As shown in Fig. 2.2(a–f), various types of mixing behavior are possible. The system studied in this thesis corresponds to case (b), the most common form. Other systems, such as cyclohexane–carbon tetrachloride, are miscible over the entire concentration and temperature range. Case (c) depicts a mixture with a lower critical point, such as triethylamine–water. Case (d) has two miscibility gaps and is fully miscible in the temperature interval between them. Overlapping cloud-point curves yield the hourglass-shaped case (e), whereas case (f) displays a closed miscibility gap.

When phases I and II are in thermodynamic equilibrium under isobaric conditions, the following condition for the chemical potentials μ_A and μ_B must hold:

Equation (2.10) expresses an equilibrium condition that is linked to the stability criteria discussed on the following page.