In this chapter we make a side step to a subject which is not directly concerned with polymers but which is closely connected to it. It is the study of percolation. Together with problems such as SAWs and lattice animals percolation forms a subject which is sometimes called geometrical critical phenomena. As we will see in chapter 8, percolation (in d = 2) is closely related to the behaviour of polymers at the ‘θ-point’ Moreover, percolation plays a role in the collapse of branched polymers. We will also discuss how percolation is related to a spin model (the Potts model) just as polymers are related to the O(n)-model. This Potts model turns out to be of importance in the description of dense polymer systems (chapter 7). The Potts model can also be used to describe the so called spanning trees. These are in turn interesting in the study of branched polymers (chapter 9).

In this book we have to limit ourselves to a discussion of those properties of percolation and Potts models which are necessary for the sequel of this book. There exists excellent reviews and books about percolation and for more information we refer to these.

Percolation as a critical phenomenon

To introduce percolation, think of a regular lattice, e.g. the hypercubic lattice in d dimensions. Take a real number 0 ≤ p ≤ 1 which we call the occupation probability. We occupy either the vertices (sites) or the edges (bonds) of this lattice with probability p.

Self avoiding walks

In the discussion in the previous chapter we neglected the fact that polymers are almost always immersed in a solvent. A good solvent is defined as a solvent in which it is energetically more favourable for the monomers of the polymer to be surrounded by molecules of the solvent than by other monomers. As a consequence, one can imagine that there exists round each monomer a region (the excluded volume) in which the chance of finding another monomer is very small. This will lead to a more open, more expanded structure for the polymer than if the excluded volume effects were absent.

The most popular model to describe this effect is the self avoiding walk. Here one considers only the subset of random walks which never visit the same site again. An example is given in figure 2.1. When one compares this figure with that of a random walk, the excluded volume effect is obvious.

Thus, the equilibrium properties of a polymer with excluded volume effects are studied by making averages over the set of all N-step self avoiding walks (SAW) (we will encounter a ‘continuum version’ of the SAW model in chapter 4). All energy effects are taken into account by limiting the set of allowed configurations to the self avoiding ones. For the moment all SAWs therefore have the same energy and thus when we calculate averages, we weight all configurations equally. Note that the self avoidance constraint doesn't come from the fact that no two monomers can be in the same place, as is often stated.

In the previous chapter we learned the importance of the O(n)-model for the study of polymers. In this chapter we will see how in two dimensions the critical behaviour of the O(n)-model has been determined exactly. The critical exponents of this model were first conjectured from renormalisation group arguments by Cardy and Hamber. These conjectures were then confirmed by Nienhuis using an approximate mapping onto the ‘Coulomb gas’. Since the Coulomb gas can be renormalised exactly, this lent support to the belief that the exponents of Cardy and Hamber were indeed exact. In 1986, Baxter succeeded in solving the O(n)-model exactly on the hexagonal lattice. His results, which were obtained in the thermodynamic limit, were later extended to finite systems by Batchelor and Blöte. In more recent years an O(n)-model on the square lattice has received a lot of attention since it has a very rich critical behaviour.

We conclude this chapter with a discussion of the SAW on fractal lattices.

The Coulomb gas approach to the SAW in d = 2

In this section we discuss how the O(n)-model (2.17) can be related to the Coulomb gas. This relation holds for the O(n)-model on the hexagonal lattice. It will give exact results for exponents ν and γ which because of universality should also hold on other lattices. Furthermore an exact value for μ follows from this mapping. When we talk about an exact solution we must note that the result is derived by nonrigorous means but that nevertheless it is generally believed that the result is the exact one. All numerical calculations furthermore support these conjectured values.

In this chapter we study self avoiding surfaces on a lattice. These surfaces are not immediately relevant for the study of polymers, although they could be of interest in the study of β-sheet polymers, which are important building blocks in proteins. Surfaces are used as models in the study of membranes or interfaces. Moreover, as we will see below, they allow the generalisation of vesicles to d = 3. The reason why we discuss surfaces in this book is mainly to show how the methods introduced in the statistical mechanics of polymers can be used in the study of other, but related, problems.

Several kinds of surfaces have been introduced in the literature. A distinction has to be made between surfaces which are models of polymerised membranes and those which describe liquid membranes. In the first case, the number of nearest neighbours of a given monomer is fixed. An interesting model is that of so called tethered surfaces introduced by Kantor, Kardar and Nelson. For liquid surfaces, on the other hand, the number of neighbours is not fixed. In this chapter, we will limit ourselves to a study of a lattice model of liquid surfaces, the ‘plaquette’ surfaces.

The critical behaviour of these surfaces is closely related to that of branched polymers. This is one of the many relations between surfaces and polymers.

Let us begin by defining the objects which we will study in this chapter and which will be referred to as plaquette surfaces or as self avoiding surfaces (SAS). An example of such a surface is shown in figure 11.1. The surface is built out of plaquettes of the cubic lattice.

The world of polymers

Polymers are long chain molecules consisting of a large number of units (the monomers), which are held together by chemical bonds. These units may all be the same (in which case we speak of homopolymers) or may be different (heteropolymers).

Chemists spend most of their time developing polymers with specific chemical or physical properties. Such properties are often determined by the characteristics of the monomers and their mutual binding. In other words, they are determined on a local scale. In contrast, physicists work in the spirit of Richard Feynman and “have a habit of taking the simplest example of any phenomenon and calling it ‘physics’, leaving the more complicated examples to become the concern of other fields.” This attitude is taken to the extreme in the statistical mechanics of polymers, where one is interested mainly in universal properties, i.e. those properties that depend only on the fact that the polymer is a long linear molecule, and are determined by ‘large scale quantities’ such as the quality of the solvent in which the polymer is immersed, the temperature, the presence of surfaces (on which a polymer can adsorb) and so on.

Having this in mind, we can introduce a description of polymers in terms of random and self avoiding walks. When we look at the polymer on a microscopic scale we remember from our chemistry courses that one of the binding angles between successive monomers is essentially fixed (like the well known 105° angle between the two H–O bonds in a molecule of water), leaving one rotational degree of freedom (figure 1.1) for the chemical bond.

Our discussion of lattice-gas models now takes a qualitative turn. We continue to study fluid mixtures as in the previous chapter, but now they will exhibit some surprising behavior—they won't like to mix!

This change in direction also steers us towards the heart of this book: models for complex hydrodynamics. The particular kind of complexity we introduce in this chapter relates to interfaces in immiscible fluids such as one might find in a mixture of oil and water. We are all familiar with the kind of bubbly complexity that that can entail. So it seems all the more remarkable that only a revised set of collision rules are needed to simulate it with lattice gases. Indeed, the models of immiscible fluids that we shall introduce are so close to the models of the previous chapters that we call them immiscible lattice gases.

This chapter, an introduction to immiscible lattice-gas mixtures, is limited to a discussion of two-dimensional models. In the next chapter, we introduce a lattice-Boltzmann method that is the “Boltzmann equivalent” of the immiscible lattice gas. That then sets the stage for our discussion of three-dimensional immiscible lattice gases in Chapter 11.

Color-dependent collisions

In the miscible lattice gases of the previous chapter, the collision rules were independent of color. The diffusive behavior derived instead from the redistribution of color after generic colorblind collisions were performed. Aside from some diffusion, the color simply went with the flow.

In this chapter we give a full derivation of the Navier-Stokes equation for the lattice gas. The first step, the Boltzmann approximation, is an approximation of the exact Liouville dynamics. The Boltzmann approximation should not be confused with the lattice-Boltzmann method of Chapter 6 and Chapter 7, but results that we obtain here for the Boltzmann approximation and the Navier-Stokes equations are also useful for the Boltzmann method. One of these results is the H-theorem for lattice gases. In this chapter we also reopen the tricky issue of the spurious invariants of lattice-gas or Boltzmann dynamics. We specifically discuss non-uniform global linear invariants for which, unlike the general nonlinear invariants, some theoretical results are known. Among them we find the staggered momentum invariants. We discuss their effect on hydrodynamics, which leads us to corrections to the Euler equation of Chapter 2.

General Boolean dynamics

It is useful to express the Boolean dynamics as a sequence of Boolean calculations, as we did in Section 2.6. In this section we shall denote by s or n local configurations. We further define a field of “rate bits”, which are random, Boolean variables, defined independently on each site and denoted by ass′(x, t). They are equal to one with probability 〈ass′〉 = A(s, s′). Thus if the pre-collision configuration is n the post-collision configuration is that single s′ for which ans′ = 1.

Our objectives for this chapter are twofold. First, we review some elementary aspects of fluid mechanics. We include in that discussion a classical derivation of the Navier-Stokes equations from the conservation of mass and momentum in a continuum fluid. We then discuss the analogous conservation relations in a lattice gas. Finally, we briefly describe the derivation of hydrodynamic equations for the lattice gas, but defer our first detailed discussion of this subject to the following chapter.

Molecular dynamics versus continuum mechanics

The study of fluids typically proceeds in either of two ways. Either one begins at the microscopic scale of molecular interactions, or one assumes that at a particular macroscopic scale a fluid may be described as a smoothly varying continuum. The latter approach allows us to write conservation equations in the form of partial-differential equations. Before we do so, however, it is worthwhile to recall the basis of such a point of view.

The macroscopic description of fluids corresponds to our everyday experience of flows. Figure 2.1 shows that a flow may have several characteristic length scales li. These lengths scales may be related either to geometric properties of the flow such as channel width or the diameter of obstacles or to intrinsic properties such as the size of vortical structures. The smallest of these length scales will be called Lhydro.