Classical scattering theory

Motivations

Matter is often studied by scattering with beams of particles such as photons, electrons, neutrons, or others. The quantities of interest are the cross-sections which give the effective surface offered by the target for the realization of a certain scattering event. Scattering processes are usually conceived in a statistical approach. For instance, a cross-section cannot be determined by a single collision but by a statistical ensemble of collisions with a uniform distribution of the incoming impact parameters. In this regard, a natural relation appears between scattering theory and the Liouvillian dynamics.

Many different processes may be considered in scattering theory, for instance elastic or inelastic collisions (Joachain 1975). Among the latter, the reaction processes between molecules or nuclei are of particular importance because they play a crucial role in the transformation of matter. Beside the cross-sections, other important quantities are the reaction rates which characterize the time evolution of statistical ensembles during reactions. The rates have the inverse of a time as unit. We may thus expect that reaction rates belong to the same class of properties as the relaxation rates of Liouvillian dynamics. This is the case, in particular, for unimolecular reactions which are dissociation processes (Gaspard and Rice 1989a, 1989b). The reaction rates can here be assimilated with the lifetimes of the metastable states of the transition complex, i.e., of the transient states formed when the fragments of the reactions are still in interaction. Here also, these lifetimes are essentially statistical properties of the time evolution instead of properties of individual trajectories of the system.

Nonequilibrium systems in Liouvillian dynamics

Most systems in nature are maintained out of equilibrium either by incident fluxes of particles or by external fields. The earth bathed by sunlight1 is an illustration of such out-of-equilibrium systems. From this viewpoint, the systems may be considered as subjected to some scattering processes, which leads us to the scattering theory of transport of Chapter 6. In this context, the fact that most classical scattering processes are chaotic has important consequences in our understanding of nonequilibrium states and the methods of the previous chapters are thus required for the investigation of out-of-equilibrium systems.

Works on out-of-equilibrium systems have revealed that such systems remain in a thermodynamic state which is the continuation of the equilibrium state under weak nonequilibrium constraints. Beyond a certain threshold, the thermodynamic branch becomes unstable and new states emerge by bifurcation with spatial or temporal inhomogeneities, called dissipative structures (Prigogine 1961; Glansdorff and Prigogine 1971; Nicolis and Prigogine 1977, 1989). Turing structures in reaction–diffusion systems and convection rolls in fluids are examples of such nonequilibrium structures (DeWit et al. 1992, 1993, 1996; Cross and Hohenberg 1993). The transitions to dissipative structures appear sharp from a macroscopic viewpoint which ignores the thermodynamic fluctuations due to the atomic structure of matter. These fluctuations can be modelled by stochastic dynamical systems like Langevin processes, birth-and-death processes, or lattice-gas automata, which show that transitions may be rounded in systems with finitely many particles (Nicolis and Malek Mansour 1978, Malek Mansour et al. 1981, Dab et al. 1991, Lawniczak et al. 1991, Kapral et al. 1992, Baras and Malek Mansour 1997).

Ideas from the theory of critical phenomena have been of great importance in the modelling of polymers ever since the Nobel prize winner P. G. de Gennes showed (in 1972) how the two subjects can be connected. In the 25 years that have passed since then, almost every major development in the understanding of criticality has led to parallel progress in the study of polymer models. We can think of the renormalisation group, the introduction of ideas from fractal geometry, conformal invariance …. As a result of all this work, the equilibrium behaviour of a polymer in a diluted regime is by now very well understood. That's why I considered the time ripe to write an overview of this field of research.

There already exist excellent books on the statistical mechanics of polymers and it may therefore be important to say a few words about the ‘niche’ in which this book should be placed. I have put the emphasis on models on a lattice and have therefore said very little about models, and methods to treat them, which work in the continuum. For completeness, important results obtained in the continuum are mentioned, but it would take much more space (and expertise on the part of the author) to treat them in all detail. Moreover, they have been very well described, for example in. This book also deals almost exclusively with the very dilute regime in which we can study one isolated polymer and can neglect the influence of any other polymers which may be present.

In the previous three chapters we considered the behaviour of polymers in bulk. In reality systems are never infinite and one always has to consider the presence of surfaces. When the polymer is close to or even attached to a surface its critical properties may change. When there is an interaction between the monomers and the surface interesting adsorption effects can occur. We now turn to a discussion of these phenomena.

Surface magnetism

Consider a (d > 1)-dimensional lattice, in which a polymer is restricted to be in a semi-infinite region, e.g. the region with x ≥ 0. We imagine a wall at x = 0 which is impenetrable. When the polymer is very far from this surface, i.e. in the bulk, its properties are hardly changed by the presence of the wall. As the polymer is placed closer and closer to the surface its properties may be modified. These effects can be expected to have a scaling behaviour depending on the ratio of the distance xcm of the centre of mass of the polymer from the surface and its radius RN. As we are interested in properties of the polymer on a coarse grained scale, let us immediately attach one of the monomers to the surface (figure 5.1). There are two ways to discuss the properties of such a polymer. The first one works directly with SAWs, while the other one uses, through the O(n)-connection, known properties of surface magnetism. Since we don't expect the general reader to know about surface critical behaviour, we will give a brief overview of the main ideas from this field (for a general review, see).

So far we have studied SAWs in the low fugacity regime where z ≤ μ-1. When z > μ-1, the grand partition function diverges. We can still give a meaning to it by making an analytic continuation. On the other hand, we know that SAWs appear in the high temperature expansion of the O(n)-model and that μ-l corresponds to the critical temperature of that model. The regime where z > μ-1 therefore corresponds to the low temperature phase of the O(n)-model. A spin model has of course a well defined low temperature regime and one might ask what this phase means for polymers. It is these questions which we study in the present chapter. The polymers in this phase are usually referred to as dense polymers. Our discussion will mostly be limited to the two-dimensional case.

The low temperature region of the O(n)-model

The study of polymers in the high fugacity regime can be performed in essentially two ways. The first way is to study the low temperature properties of the O(n)-model. This will be done using the techniques known from previous chapters; the Coulomb gas, exact solutions using the Bethe Ansatz, and so on. On the other hand we can study immediately the properties of walks themselves, in the regime where z > μ-1. Sure enough, in that region the grand partition function diverges, but the trick is to study the properties of walks in finite systems, e.g. in a finite box of volume Λ. A typical size for such a volume is Λ1/d. The finite volume leads to a cutoff for the grand partition sum.

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.