Mean-Field Theories

The most widely used analytic models for representing thermodynamic behavior in supercritical ßuids are of the mean-Þeld variety. In addition to the practical interest in studying this topic, this class of models is also the conceptual starting point for any microscopic discourse on critical phenomena. In this chapter we take up the basic ideas behind this approach, studying different physical models, showing how their mean-Þeld approximations can be constructed as well as investigating their critical behavior. A useful conceptual model for understanding mean-Þeld ideas is the Ising model whose properties we consider in some detail, especially its mean-Þeld approximation. The Ising model has the advantage of belonging to the same critical universality class as so-called simple fluids, deÞned as ßuids with short-range intermolecular potentials. Most supercritical ßuid solvent systems of practical interest fall within this class; hence results developed using the Ising model have important implications for understanding the critical behavior of this entire universality class. While we discuss universality and related ideas in more detail in subsequent chapters, sufÞce it to say here that the Ising system belongs to arguably the most important critical universality class from a process engineering standpoint. In its simplest form, the Ising model considers N spins arranged on a lattice structure (of 1, 2, or 3 dimensions) with each spin able to adopt one of two (up or down) orientations in its lattice position. A speciÞc state of the system is determined by a given conÞguration of all the spins. The model can be made more complex by considering additional degrees of freedom to the spin orientations. For example, the Heisenberg model considers a 3-dimensional lattice with the spin orientation at each lattice site described by a 3-dimensional vector quantity. All that is required to facilitate the use of statistical mechanics with this model is the deÞnition of the Hamiltonian (the systemÕs energy function) associated with a particular lattice state υ. This Hamiltonian usually consists of spinÐspin interaction terms, as well as a term representing the presence of a magnetic Þeld, which serves to orient the spins in its direction.