As long as possible we have postponed a general study of elasticity and its partial differential equations. Here they are!

The elastic equilibrium of a solid is generally treated in the continuum approximation. The strain satisfies certain equations in the bulk, and other equations at the surface. This set of equations has an infinite number of solutions, and the correct one is that which minimizes a given thermodynamic potential or free energy. This minimization is not needed for a semi-infinite solid because the good solution in this case is the one which vanishes at infinity.

The power of continuous elasticity theory is limited. In particular it is not appropriate to investigate the surface relaxation, i.e. the change in the atomic distance near the surface. Nevertheless, the continuum approximation allows for spectacular predictions, for instance the Asaro-Tiller-Grinfeld instability, which is one of the major obstacles to layer-by-layer heteroepitaxial growth.

Memento of elasticity in a bulk solid

In this section, the theory of linear elasticity in a homogeneous solid away from the surface will be recalled.

In order to write the condition for mechanical equilibrium, one has to consider the forces acting on a volume δV of the solid (Fig. 16.1). There may be an external force δfext, and there is a force produced by the part of the solid outside δV.

Why cellular automata are useful in physics

The purpose of this section is to show different reasons why cellular automata may be useful in physics. In a first paragraph, we shall consider cellular automata as simple dynamical systems. We shall see that although defined by very simple rules, cellular automata can exhibit, at a larger scale, complex dynamical behaviors. This will lead us to consider different levels of reality to describe the properties of physical systems. Cellular automata provide a fictitious microscopic world reproducing the correct physics at a coarse-grained scale. Finally, in a third section, a sampler of rules modeling simple physical systems is given.

Cellular automata as simple dynamical systems

In physics, the time evolution of physical quantities is often governed by nonlinear partial differential equations. Due to the nonlinearities, solution of these dynamical systems can be very complex. In particular, the solution of these equation can be strongly sensitive to the initial conditions, leading to what is called a chaotic behavior. Similar complications can occur in discrete dynamical systems. Models based on cellular automata provide an alternative approach to study the behavior of dynamical systems. By virtue of their simplicity, they are potentially amenable to easier analysis than continuous dynamical systems. The numerical studies are free of rounding approximations and thus lead to exact results.

Crudely speaking, two classes of problem can be posed. First, given a cellular automaton rule, predicts its properties.

While the first chapter of this book dwelt on the confirmation of the fundamental ideas of the philosophers of the fifth century BC, this last chapter will be devoted to the twentieth century and to the transformation of our materialistic everyday life brought about by electronics.

The early developments employed electronic tubes which owed nothing to surface physics and crystal growth. Then came the transistor and the realm of silicon, of which huge, dislocation-free crystals can be grown from the melt. In this end of the twentieth century, electromagnetic waves play an increasingly important part. Their production and detection require more and more complex semiconducting materials, often grown by molecular beam epitaxy.

Introduction

The modernity of surface science lies for a large part in the fact that new experimental techniques now make surfaces accessible to investigation. But the present interest in surface physics is also much motivated by the technological importance of crystals grown with a well-controlled composition and a high degree of perfection, and this depends very much on the state of the crystal surface during growth.

The purpose of this chapter is to outline the technological importance of materials with a controlled purity and morphology.

We shall mainly discuss semiconductors. It is not very easy to find simple articles or books explaining semiconductor electronics to the ignorant, but the book by Parker (1994) accomplishes this function.

Introduction

The concept of phase transition plays a very important role in many domains of physics, chemistry, living matter and biology. The simplest case is the one in which a phase transition arises between two equilibrium states when varying a control parameter. Examples of control parameters are temperature, pressure or and concentration. The most familiar phase transitions are the paramagnetic—ferromagnetic transition, the liquid—gas transition and the supraconducting—normal phase transition in a metal. Equilibrium statistical mechanics gives a coherent description of the physics of these states. The determination of the presence of a phase transition and the study of its properties is not generally an easy problem, however there exists a well-defined framework to approach it. When, at the transition some quantities have a discontinuity, one speaks of a first-order phase transition. If the physical quantities vary continuously one speaks of second-order phase transitions. This latter case is particularly tricky because one has to face a multiscale problem. Fluctuations of all wavelengths (from a few angström to the size of the system) play an equivalent important role at the transition. However, even in these complicated situations, such tools as the renormalization group method have been developed to solve these problems. Although several analytical methods exist in equilibrium statistical mechanics, one often has to have recourse to numerical simulations. In particular, the Monte-Carlo method is widely used. As we have seen in chapter 2, Monte-Carlo algorithms can be viewed as probabilistic cellular automata providing that some care is taken with the evolution rules (checkerboard invariants).

Introduction

Systems in which different chemical species diffuse (for instance in a solvent or a gel) and then react together are quite common in science. Lattice gases provide a very natural framework to model such phenomena. In the previous chapter, we discussed how diffusion can be implemented as a synchronous random walk of many particles, using a velocity shuffling technique.

The model can be extended so that several different “chemical” species coexist simultaneously on the same lattice. It just requires more bits of information to store the extra automaton state. Then, it is easy to supplement the diffusion rule with the annihilation or creation of particles of a different kind, depending on the species present at each lattice site and a given reaction rule.

In agreement with the cellular automata approach, chemical reactions are treated in an abstract way, as a particle transformation phenomena rather than a real chemical interaction. Only the processes relevant at the macroscopic level are taken into account.

Systems in which reactive particles are brought into contact by a diffusion process and transform, often give rise to very complex behaviors. Pattern formation, is a typical example of such a behavior in reaction-diffusion processes.

In addition to a clear academic interest, reaction-diffusion phenomena are also quite important in technical sciences and still constitute numerical challenges. As an example, we may mention the famous problem of carbonation in concrete.

Introduction

Diffusive phenomena play an important role in many areas of physics, chemistry and biology and still constitute an active field of research. There are many applications involving diffusion for which a particle based model, such as a lattice gas dynamics, could provide a useful approach and efficient numerical simulations.

For instance, processes such as aggregation, formation of a diffusion front, trapping of particles performing a random walk in some specific region of space, or the adsorption of diffusing particles on a substrate are important problems that are difficult to solve with the standard diffusion equation. A microscopic model, based on a cellular automata dynamics, is therefore of clear interest.

Diffusion is also a fundamental ingredient in reaction-diffusion phenomena that will be discussed in detail in the next chapter. Reaction processes such as A + B → C, as well as growth mechanisms are naturally implemented in the framework of a point particle description, often with the help of threshold rules. Consequently, microscopic fluctuations are often relevant at a macroscopic level of observation because they make symmetry breaking possible and are responsible for triggering complex patterns.

Cellular automata particles can be equipped with diffusive and reactive properties, in order to mimic real experiments and model several complex reaction-diffusion-growth processes in the same spirit as a cellular automata fluid simulates fluid flow: these systems are expected to retain the relevant aspects of the microscopic world they are modeling.