Our experience with the EW and the KPZ equations has already taught us an important lesson: nonlinear terms may be present for certain growth models and, if present, may control the scaling behavior. For the problem of MBE dominated by surface diffusion, there is no reason to ignore the possibility of nonlinear terms, so we must carefully examine when nonlinear terms are relevant and explore the resulting scaling behavior.

In order to address the relevance of certain nonlinear terms that can affect the growth equation (13.9), it will be necessary to use the dynamic RG method, as we did with the KPZ equation. We shall see that the dynamic RG method leads to exact exponents in any dimension, in contrast to the case of the KPZ equation where results are found only for d = 1.

Surface diffusion: Nonlinear effects

If the relevant relaxation process is surface diffusion, then the growth equation must satisfy the continuity equation (13.2). If the surface current is driven by the gradient of the local chemical potential (13.4), we argued that the diffusive growth process in the linear approximation is described by the growth equation (13.9).

For d = 1, 2, Eq. (13.9) predicts α = 3/2, 1 respectively. However, continuum growth equations are valid in the ‘small slope’ approximation, |∇h| « 1, which means that the local slopes are small at every stage of the growth process.

The formation of interfaces and surfaces is influenced by a large number of factors, and it is almost impossible to distinguish all of them. Nevertheless, a scientist always hopes that there is a small number of basic ‘laws’ determining the morphology and the dynamics of growth. The action of these basic laws can be described in microscopic detail through discrete growth models – models that mimic the essential physics but bypass some of the less essential details.

To this end, we introduce a simple model, ballistic deposition (BD), which generates a nonequilibrium interface that exemplifies many of the essential properties of a growth process. We shall use the BD model to introduce scaling concepts, a central theme of this book.

Ballistic deposition

Ballistic deposition was introduced as a model of colloidal aggregates, and early studies concentrated on the properties of the porous aggregate produced by the model. The nontrivial surface properties became a subject of scientific inquiry after the introduction of vapor deposition techniques.

It is simpler to define and study the BD model on a lattice, as in Fig. 2.1, but off-lattice versions have been investigated as well. A particle is released from a randomly chosen position above the surface, located at a distance larger than the maximum height of the interface. The particle follows a straight vertical trajectory until it reaches the surface, whereupon it sticks.

The elementary processes taking place at crystal surfaces suggest that numerical simulations of discrete models might be helpful in understanding the collective behavior of the atoms during growth. In recent years, a number of models have been proposed to describe the evolving morphology of crystal surfaces.

There are two main sources of interest in motivating the study of simple discrete models. First, there is the hope that the studies on simple models might serve as a guide in understanding the experimentally-observable morphologies and scaling behavior. In this respect, discrete models might serve as an intermediate step between experiments and the continuum theories discussed in previous chapters. The evident advantage of the models is the separability of different secondary effects always present in experiments, thereby offering the possibility of studying the influence of selected mechanisms, such as surface diffusion or desorption, on the growth process.

Second, apart from the connection with MBE, models with surface diffusion represent potentially new universality classes in the family of kinetic growth models, distinct from both the KPZ and EW universality classes. Thus their study poses exciting theoretical questions for statistical mechanics and condensed matter physics.

The models that we discuss in this chapter can be classified into three main categories.

This book is intended to serve as an introduction to the multidisciplinary field of disorderly surface growth. It is a reasonably short book, and is not designed to review all of the recent work in this rapidly developing area. Rather, we have attempted to provide an introduction that is sufficiently thorough that much of the current literature can be profitably read. This literature spans many disciplines, ranging from applied mathematics, physics, chemistry and biology on the one hand to materials science and petroleum engineering on the other.

It is envisaged that this book may be of use in courses in many different departments, so no specific background on the part of the reader is assumed. Part 1 of the book is an introduction that should bring readers from a variety of disciplines to a common place of discourse. Thus the first chapter illustrates the range of natural examples of disorderly surface growth, and mentions without any use of mathematics a few of the key new ideas that serve to provide some insights. The second chapter introduces the scaling approach to describing surface growth, by focusing on a single tractable model system – ballistic deposition. No prior exposure to scaling concepts is assumed. The third and last chapter of Part 1 introduces the key fractal concepts of self-similarity and self-affinity.

Part 2 comprises five chapters devoted to the general topic of nonequilibrium roughening.

One of the main methods used to study the roughening of nonequilibrium interfaces is to construct discrete models and study them using computer simulations. To simulate the models on a computer and to study their scaling properties, we must use a number of numerical methods. If the model is simple, the corresponding simulation is also simple, e.g., a program to simulate RD or BD requires only a few lines. However, the analysis of the simulation results is far from being simple, requiring special care due to finite size effects, slow crossover behavior, and other complications. In this chapter we collect some additional methods that are useful in obtaining a thorough analysis of the scaling properties.

Measuring exponents for self-affine interfaces

As noted, α and β are universal exponents, which do not depend on the particular details of the model. Thus obtaining these scaling exponents helps to identify the universality class of the growth process. The method used for the determination of α and β for a particular interface depends on the information available. If the interface is obtained from numerical simulation, every detail of the interface and its dynamics can be extracted from the computer. However, experiments may not be able to follow every detail of the time evolution of the growth, and one can analyze only the final interface.

We briefly present five methods for the numerical analysis of a rough interface.

Thus far, we have seen that many surfaces are self-affine, and their scaling can be characterized using a single number – the roughness exponent α. This is because the statistical properties characterizing the surface are invariant under an anisotropic scale transformation (3.4). For characterizing the scaling properties of some surfaces, however, the roughness exponent by itself is not sufficient.

In this chapter, we discuss a class of surfaces whose scaling properties are describable only in terms of an infinite set of exponents. Since this class is an extension of self-affine surfaces, we will call them multiaffine surfaces. We shall see that the multi-affine behavior is reflected in the existence of an entire hierarchy of ‘local’ roughness exponents, i.e., the roughness exponent for multi-affine interfaces changes from site to site.

A number of models studied recently in the literature lead to multi-affine surfaces. For example, if the noise in the system has a power-law distribution, the interface is multi-affine for length scales shorter than a characteristic length scale ℓx, set by the noise (see §23.3). Similarly, the temporal fluctuations of the interface width in the saturated regime leads to a multi-affine function. Quenched noise, present in many experimental systems, leads naturally to power-law distributed noise, so one expects that multi-affine behavior might be present in such systems as well. Indeed, Sneppen and Jensen studied the temporal properties of the SOD model discussed in §10.2, and found this type of multi-affine behavior.

In this chapter, we describe a few models that have had a key impact on our knowledge about specific aspects of interface roughening. Due to intractable mathematical difficulties, numerical methods are commonly used to determine the scaling exponents for systems with d > 1. Most growth models originate from specific physical or biological problems, and only recently have been investigated using the methods described in this book.

Ballistic deposition

The ballistic deposition model introduced in Chapter 2 is the simplest version – termed the nearest-neighbor (NN) model because falling particles stick to the first nearest neighbor on the aggregate. If we allow particles to stick to a diagonal neighbor as well, we have the next-nearest neighbor (NNN) model (Fig. 8.1). Since the nonlinear term is present for both models (λ ≠ 0), the scaling properties for both models are described by the nonlinear theory. These two models therefore belong to the same universality class, since they share the same set of scaling exponents, α, β, and z. Their non-universal parameters, however, are different. For example, for the velocity V0 (see (A.13)), we find v0 = 2.14, 4.26 for the NN and NNN models, respectively. The coefficient λ of the nonlinear term differs as well, with λ = 1.30, 1.36, respectively.

The origin of the nonlinear term in the model is the lateral sticking rule, leading to the presence of voids.

The exact results presented in the previous chapter allow us to obtain the scaling exponents for d = 1, and reduce the number of independent scaling exponents to one. The same results can be obtained using the dynamic renormalization group method, which we now develop and use to study the scaling properties of the KPZ equation. In particular, we analyze the ‘flow equations’ and extract the exponents describing the KPZ universality class for d = 1. We also discuss numerical results leading to the values of the scaling exponents for higher dimensions.

Basic concepts

So far, we have argued that we can distinguish between various growth models based on the values of the scaling exponents α, β and z. The existence of universal scaling exponents and their calculation for various systems is a central problem of statistical mechanics. A main goal for many years has been to calculate the exponents for the Ising model, a simple spin model that captures the essential features of many magnetic systems. A major breakthrough occurred in 1971, when Wilson introduced the renormalization group (RG) method to permit a systematic calculation of the scaling exponents. Since then the RG has been applied successfully to a large number of interacting systems, by now becoming one of the standard tools of statistical mechanics and condensed matter physics. Depending on the mathematical technique used to obtain the scaling exponents, the RG methods can be partitioned into two main classes: real space and k-space (Fourier space) RG.

It is a truism to remark that no one – not even a theoretical physicist – can predict the future. Nonetheless, after asking the beleaguered reader to indulge in the rather extensive ‘banquet’ of the preceding 27 chapters, it seems only fair to offer a light ‘dessert’ that affords some outlook and perspective on this rapidly-evolving field.

What concepts loom above the details is a question worth addressing at the end of any large meal. Charles Kittel wrote his first edition of Introduction to Solid State Physics almost 50 years ago. He surely realized that solid state physics was a rapidly-evolving field, so his book ran the risk of becoming dated in short order. Therefore the first chapter systematically discusses the various crystal symmetries – and the group theory mathematics that describes these symmetries. The topics comprising solid state physics have changed rather dramatically, and most chapters of Kittel's 7th edition hardly resemble the chapters of the first edition. Nevertheless, the opening chapter of the first edition could serve as well today as an introduction to the essential underpinnings of the subject.

Inspired by Kittel's example, we have attempted in this short book to highlight where possible what seems to us to be the analog for disorderly surface growth of the various symmetries obeyed by crystalline materials. These newer ‘symmetries’, described using terms that may frighten the neophyte – such as scale invariance and self-affinity – are as straightforward to describe as translation, rotation, and inversion.

The increasing interest of researchers in the basic properties of growth processes has provided the initiative for a number of experimental studies designed to check the applicability of various theoretical ideas to experimental systems. While many experimental studies have been inspired by the KPZ theory, most have failed to provide support for the KPZ prediction that α = ½. Instead, most data suggest that α > ½. These experimental results initiated a closer look at the theory, and led to the discovery that quenched noise affects the scaling exponents in unexpected ways. In this chapter, we discuss some of these key experiments, including fluid-flow experiments, paper wetting, propagation of burning fronts, growth of bacterial colonies and paper tearing. Atom deposition in molecular beam epitaxy, which is one important class of experiments on kinetic roughening of interfaces, will be discussed in Chapter 12. The new theoretical ideas needed to understand the effect of atomic diffusion on the roughening process will be developed at that time.

Fluid flow in a porous medium

Two-phase fluid flow experiments have long been used to study various growth phenomena. The Hele–Shaw cell, well-known from studies on growth instabilities in viscous fingering, has proved to be a useful experimental setup for the study of the growth of selfaffine interfaces. A trypical setup used in these experiments (Fig. 11.1) is a thin horizontal Hele–Shaw cell made from two transparent plates.

Most of this book deals with local growth processes, for which the growth rate depends on the local properties of the interface. For example, the interface velocity in the BD model depends only on the height of the interface and its nearest neighbors. However, there are a number of systems for which nonlocal effects contribute to the interface morphology and growth velocity. Such growth processes cannot be described using local growth equations, such as the KPZ equation; if we attempt to do so, we must include nonlocal effects. In this chapter we discuss phenomena that lead to nonlocal effects, and we also discuss models describing nonlocal growth processes.

Diffusion-limited aggregation

Probably the most famous cluster growth model is diffusion-limited aggregation (DLA). The model is illustrated in Fig. 19.1. We fix a seed particle on a central lattice site and release another particle from a random position far from the seed. The released particle moves following a Brownian trajectory, until it reaches one of the four nearest neighbors of the seed, whereupon it sticks, forming a two-particle cluster. Next we release a new particle which can stick to any of the six perimeter sites of this two-particle cluster. This process is then iterated repeatedly. In Fig. 19.2, we show clusters resulting from the deposition of 5 × 105, 5 × 106, and 5 × 107 particles.