1. Introduction This paper has three parts. The first part is a simplified presentation of the basic ideas of the renormalization group and the ε expansion applied to critical phenomena , following roughly a summary exposition given in 1972 1. The second part is an account of the history (as I remember it) of work leading up to the papers in I971-1972 on the renormalization group. Finally, some of the developments since 197 1 will be summarized, and an assessment for the future given.

The renormalization group and critical phenomena

New force fields for carbon dioxide and nitrogen are introduced that quantitatively reproduce the vapor–liquid equilibria (VLE) of the neat systems and their mixtures with alkanes. In addition to the usual VLE calculations for pure CO2 and N2, calculations of the binary mixtures with propane were used in the force-field development to achieve a good balance between dispersive and electrostatic (quadrupole–quadrupole) interactions. The transferability of the force fields was then assessed from calculations of the VLE for the binary mixtures with n-hexane, the binary mixture of CO2/N2, and the ternary mixture of CO2 /N2/propane. The VLE calculations were carried out using configurational-bias Monte Carlo simulations in either the grand canonical ensemble with histogram–reweighting or in the Gibbs ensemble.

Vapor–liquid equilibria of mixtures containing alkanes, carbon dioxide, and nitrogen

We review the current state of knowledge of phase separation and phase equilibria in porous materials. Our emphasis is on fundamental studies of simple fluids (composed of small, neutral molecules) and well-characterized materials. While theoretical and molecular simulation studies are stressed, we also survey experimental investigations that are fundamental in nature. Following a brief survey of the most useful theoretical and simulation methods, we describe the nature of gas‐liquid (capillary condensation), layering, liquid‐liquid and freezing/melting transitions. In each case studies for simple pore geometries, and also more complex ones where available, are discussed. While a reasonably good understanding is available for phase equilibria of pure adsorbates in simple pore geometries, there is a need to extend the models to more complex pore geometries that include effects of chemical and geometrical heterogeneity and connectivity. In addition, with the exception of liquid‐liquid equilibria, little work has been done so far on phase separation for mixtures in porous media.

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Phase separation in confined systems

Critical phenomena and renormalization-group theory

Colloids display intriguing transitions between gas, liquid, solid and liquid crystalline phases. Such phase transitions are ubiquitous in nature and have been studied for decades. However, the predictions of phase diagrams are not always realized; systems often become undercooled, supersaturated, or trapped in gel-like states. In many cases the end products strongly depend on the starting position in the phase diagram and discrepancies between predictions and actual observations are due to the intricacies of the dynamics of phase transitions. Colloid science aims to understand the underlying mechanisms of these transitions. Important advances have been made, for example, with new imaging techniques that allow direct observation of individual colloidal particles undergoing phase transitions, revealing some of the secrets of the complex pathways involved.

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Insights into phase transition kinetics from colloid science

We report an extensive Monte Carlo study of the density and energy fluctuations in a 2D Lennard-Jones fluid, within the grand-canonical ensemble. The results are analyzed within a finite-size-scaling theory incorporating the mixed character of the scaling fields that manifests particle-hole asymmetry. The limiting critical distribution of the density matches that of the magnetization in the 2D Ising model, illuminating the configurational basis of universality; the corrections to the limiting behavior contain an antisymmetric component which reflects the critical Ising energy operator, corroborating the anticipated structure of the scaling fields.

Scaling fields and universality of the liquid-gas critical point.

The authors develop a finite-size-scaling theory describing the joint density and energy fluctuations in a near-critical fluid. As a result of the mixing of the temperature and chemical potential in the two relevant scaling fields, the energy operator features in the critical density distribution as an antisymmetric correction to the limiting scale-invariant form. Both the limiting form and the correction are predicted to be functions that are characteristic of the Ising universality class and are independently known. The theory is tested with extensive Monte Carlo studies of the two-dimensional Lennard-Jones fluid, within the grand canonical ensemble. The simulations and scaling framework together are shown to provide a powerful way of identifying the location of the liquid-gas critical point, while confirming and clarifying its essentially Ising character. The simulations also show a clearly identifiable signature of the field-mixing responsible for the failure of the law of rectilinear diameter.

https://iopscience.iop.org/article/10.1088/0953-8984/4/12/008/pdf

Density fluctuations and field mixing in the critical fluid

The fundamental inconsistencies inherent in classical theories of structural phase transitions are discussed as a prelude to a review of the contemporary theory, emphasizing the concepts and techniques that constitute the renormalization group theory of static critical phenomena. The theory's predictions for critical behaviour at structural phase transitions are surveyed and compared with experiment.

Structural phase transitions. II. Static critical behaviour

We present a Monte Carlo method for evaluating the difference between the free energies of two crystal structures. The method uses a biased sampling of atomic displacements to favor configurations of one structure that can be replaced by corresponding configurations of the other through a Monte Carlo switch of the lattice. The configurations of both structures can be sampled in a single process, and the difference between their free energies evaluated from their measured probabilities. The method is applied to the free energies of the fcc and hcp phases of hard spheres.

Alastair Bruce

Papers

Scaling fields and universality of the liquid-gas critical point.

Density fluctuations and field mixing in the critical fluid

Structural phase transitions. II. Static critical behaviour

Free energy of crystalline solids: A lattice-switch Monte Carlo method

Coupled order parameters, symmetry-breaking irrelevant scaling fields, and tetracritical points