Organisers

• Andrew Masters (University of Manchester)
• Michael Allen (University of Warwick)

Supports

 CECAM

Description

This workshop is to be organised jointly by Dr A J Masters (Manchester, UK) and Professor M. P. Allen (Warwick, UK).

The main techniques traditionally used to calculate the equilibrium properties of homogeneous, isotropic fluids are: (a) a virial expansion; (b) integral equation theory; (c) perturbation methods around a known reference system.

By the end of the 1970s, these methods had been developed to a very sophisticated level, so that it was possible, for the most part, to calculate very accurate thermodynamic and structural properties for fluids of spherical particles.

Since that time, many further significant advances have been made. While space does not permit anything like a comprehensive list, we give just a few developments. One is Wertheim’s associating fluid theory, which, for example, provides an accurate and computationally straightforward way to treat chain molecules. A second is the development of efficient numerical algorithms to solve integral equation theories for the isotropic phase of axially symmetric molecules. A third is the advent of reference interaction site theories (RISM, PRISM, etc). A fourth is the growth of density functional theories which aim to describe inhomogeneous fluids. Fundamental Measure Theory has had a major impact in this field in recent times. Finally we note that methodologies have recently been developed to calculate the 8th to 10th order virial coefficients, re-invigorating this particular theoretical approach.

The recent explosion of interest in colloidal systems has given liquid state theory a fresh impetus. Compared to simple liquids the potentials of interaction between the colloidal particles are relatively straightforward, so all the liquid state methods can be employed without the need to agonize over the accuracy of molecular pair potentials of interaction. These particles also form ordered phases, such as liquid crystals and various forms of crystal, and again there is a need to describe these phases theoretically. Another related development is in the study of systems whose particles interact via soft, integrable pair potentials. Such potentials appear in dissipative particle dynamics simulations and in configurationally averaged potentials that describe the interactions of two polymers. Theories such as the high temperature approximation and random phase approximation, developed in the 1970s, work very well for these systems and this has again engendered much theoretical work.

Scientific Objectives

In the light of the recent developments and the upsurge of liquid state research brought on by colloidal studies, deficiencies in the theories come rapidly to light. Here are just a few outstanding problems in this general area:

• To go systematically beyond the hnc closure, one needs a reliable and soundly-based approximation for the bridge function. How do we achieve this, especially for aspherical particles?
  
• The hypernetted chain(hnc) equations and the equations resulting from related theories have an aggravating tendency to have regions in which there is no real solution. Often this lack of solution sets in quite far away from any instability. How can this be cured?
  
• What are the prospects for extending integral equation methods into ordered phases?
  
• What are the prospects for using integral equations (such as hnc) to study inhomogeneous fluids?
  
• What are the pros and cons of using interaction-site approaches compared to “full molecule” methods?
  
• How far may one push the virial expansion to predict the properties of both isotropic and anisotropic phases?
  
• What are the possibilities for constructing soundly based density functionals for a wide variety of interaction potentials (e.g. to deal with attractive potentials)?
  
• How does one deal with many body potentials (e.g. polarizable particles or the many body potentials currently used in DPD simulations)?
  
• How does this theoretical work impact on the search for reliable equations for state for real fluids and the possibly anisotropic phases they may form?
  
• How might computer simulation best be used to advance this field?
  

The computational challenges lie mainly in devising efficient algorithms. For example one needs robust methods to solve integral equations and density functional equations in complex geometries and in anisotropic phases. Similarly efficient Monte Carlo schemes are needed to calculate high order virial coefficients and bridge diagrams. On the computer simulation side, there are computational issues involving the best ways to calculate quantities such as the direct correlation function and bridge function

The aims of this activity are

• to review recent advances in the area and discuss openly the drawbacks and limitations of the various techniques
  
• to seek a cross-fertilisation of ideas from people with expertise in different strands of theory
  
• to propose new methods and computational algorithms to get past current bottle-necks
  
• to foster new collaborations between theoreticians and simulators
  

References

1. An excellent review of the fundamental theory is given in J. P. Hansen and I. R. MacDonald, Theory of Simple Liquids (3rd edition), Academic Press, 2006.

2. P. Paricaud, A. Galindo and G. Jackson, “Recent Advances in the use of the SAFT approach in describing electrolytes, interfaces, liquid crystals and polymers”, Fluid Phase Equilibria, 194 – 197, 87 (2002).

3. P. H. Fries and G. N. Patey, “The solution of the hypernetted-chain approximation for fluids of non-spherical particles. A general method with application to dipolar hard spheres.” J Chem. Phys. 82, 429 (1985)

4. M. S. Wertheim, “Fluids with highly directional attractive forces”, J. Stat. Phys. 35, 19 (1984).

5. R. Kjellander and S. Marčelja, “Inhomogeneous coulombic fluids with image interactions between planar surfaces. I.”, J. Chem. Phys. 82, 2122 (1985)

6. N. Clisby and B. M. McCoy, “Ninth and tenth order virial coefficients for hard spheres in D dimensions”, J. Stat. Phys. 122, 15 (2006)

7. Y. Rosenfeld, “Free-energy model for the inhomogeneous hard sphere fluid mixture and density functional theory of freezing”, Phys. Rev. Lett. 63, 980 (1989)

8. A. A. Louis, P. G. Bolhuis and J. P. Hansen, “Mean-field behaviour of the Gaussian core model”, Phys. Rev. E. 62, 7961 (2000)

9. L. Belloni, “Inability of the hypernetted chain integral equation to exhibit a spinodal line”, J. Chem. Phys. 98, 8080 (1993)