Basic features of the kinetics of diffusion-controlled two-species annihilation, A + B → 0, as well as that of single-species annihilation, A + A→ 0, and coalescence, A + A → A, under diffusion-controlled and ballistically controlled conditions, are reviewed in this chapter. For two-species annihilation, the basic mechanism that leads to the formation of a coarsening mosaic of A- and B-domains is described. Implications for the distribution of reactants are also discussed. For single-species annihilation, intriguing phenomena arise for ‘heterogeneous’ systems, where the mobilities (in the diffusion-controlled case) or the velocities (in the ballistically controlled case) of each ‘species’ are drawn from a distribution. For such systems, the concentrations of the different ‘species’ decay with time at different power-law rates. Scaling approaches account for many aspects of the kinetics. New phenomena associated with discrete initial velocity distributions and with mixed ballistic and diffusive reactant motion are discussed. A scaling approach is outlined to describe the kinetics of a ballistic coalescence process which models traffic on a single-lane road with no passing allowed.

Introduction

There are a number of interesting kinetic and geometric features associated with diffusion-controlled two-species annihilation, A + B → 0, and with single-species reactions, A + A → 0 and A + A → A, under diffusion-controlled and ballistically controlled conditions.

In two-species annihilation, there is a spontaneous symmetry breaking in which large-scale single-species heterogeneities form when the initial concentrations of the two species are equal and spatially uniform.

Editor's note

The first three chapters of the book cover topics in reactions and catalysis. Chemical reactions comprise a vast field of study. The recent interest in models in low dimension has been due to the importance of two-dimensional surface geometry, appropriate, for instance, in heterogeneous catalysis. In addition, several experimental systems realize 1D reactions (Part VII).

The classical theory of chemical reactions, based on rate equations and, for nonuniform densities, diffusion-like differential equations, frequently breaks down in low dimension. Recent advances have included the elucidation of this effect in terms of fluctuation-dominated dynamics. Numerous models have been developed and modern methods in the theory of critical phenomena applied. The techniques employed range from exact solutions to renormalization-group, numerical, and scaling methods.

Models of reactions in 1D are also interrelated with many other 1D systems ranging from kinetic Ising models (Part II) and deposition (Part IV) to nucleation (Part III). Chapter 1 reviews the scaling theory of basic reactions and summarizes numerous results. One of the methods of obtaining exact solutions in 1D, the interparticle-distribution approach, is reviewed in Ch. 2. Other methods for deriving exact results in 1D are not considered in this Part. Instead, closely related systems and solution techniques based on kinetic Ising models and cellular automata are presented in Chs. 4, 6, 8. Coagulation models in Ch. 9 employ methods that have also been applied to reactions.

More complicated models of catalysis, directed percolation, and kinetic phase transitions, are treated in Ch. 3.

Exact solutions for the phase-ordering dynamics of three one-dimensional models are reviewed in this chapter. These are the lattice Ising model with Glauber dynamics, a nonconserved scalar field governed by time-dependent Ginzburg-Landau (TDGL) dynamics, and a nonconserved 0(2) model (or XY model) with TDGL dynamics. The first two models satisfy conventional dynamic scaling. The scaling functions are derived, together with the (in general nontrivial) exponent describing the decay of autocorrelations. The 0(2) model has an unconventional scaling behavior associated with the existence of two characteristic length scales—the ‘phase coherence length’ and the ‘phase winding length’.

Introduction

The theory of phase-ordering dynamics, or ‘domain coarsening’, following a temperature quench from a homogeneous phase to a two-phase region has a history going back more than three decades to the pioneering work of Lifshitz, Lifshitz and Slyozov, and Wagner. The current status of the field has been recently reviewed.

The simplest scenario can be illustrated using the ferromagnetic Ising model. Consider a temperature quench, at time t = 0, from an initial temperature TI, which is above the critical temperature TC to a final temperature TF, which is below TC-At TF there are two equilibrium phases, with magnetization ±M0. Immediately after the quench, however, the system is in an unstable disordered state corresponding to equilibrium at temperature TI. The theory of phase-ordering kinetics is concerned with the dynamical evolution of the system from the initial disordered state to the final equilibrium state.

Editor's note

Nucleation, phase separation, cluster growth and coarsening, ordering, and spinodal decomposition are interrelated topics of great practical importance. While most experimental realizations of these phenomena are in three (bulk) and two (surface) dimensions, there has been much interest in lattice and continuum (off-lattice) 1D stochastic dynamical systems modeling these irreversible processes.

The main applications of 1D models have been in testing various scaling theories such as cluster-size-distribution scaling and scaling forms of orderparameter correlation functions. Exact solutions are particularly useful in this regard, and the focus of all three chapters in this Part is on exactly solvable models. Additional literary sources are cited in the chapters, including general review- articles as well as other studies in 1D.

Chapter 7 reviews exact solutions of three different models of phaseordering dynamics, including results based on the Glauber-Ising model introduced in Part II. Chapter 8 review's a model with synchronous (cellularautomaton) dynamics and relations to reactions (Part I). In both chapters exact results for scaling of the two-point correlation function are obtained. Finally, Ch. 9 describes models of coagulating particles and associated results for cluster-size-distribution scaling.

The aim of this chapter is to summarize briefly recent results on directed walks and provide a guide to the literature. We shall restrict consideration to the equilibrium properties of directed interfaces and polymers, focusing particularly on their collapse and binding transitions. The walks will lie in a nonrandom environment.

Directed walks and polymers

A clear introduction to the physics of directed walks is given by Privman and Švrakić in a book published in 1989. This summarizes the work up to that time and therefore here we shall aim to describe more recent progress after a brief description of the relevant models.

Many of the interesting results for nonrandom systems have been obtained for walks that should strictly be labeled partially directed. In these movement is allowed along either the positive or negative x-direction but only along the positive t-direction, as shown in Fig. 16.1. Hence the position ratof the walk in column t= i is unique.

Also shown in Fig. 16.1 for comparison is a fully directed walk, each step of which must have a nonzero component in the positive t-direction. This is a simpler model, which has been very useful in studying the behavior of interfaces in a random environment (not reviewed here; see). The partially directed walk reduces to the fully directed one if the constraint is imposed.

An exact solution of a lattice spin model of ordering in one dimension is reviewed in this chapter. The model dynamics is synchronous, cellularautomaton- like, and involves interface diffusion and pairwise annihilation as well as spin flips due to an external field that favors one of the phases. At phase coexistence, structure-factor scaling applies, and the scaling function is obtained exactly. For field-driven, off-coexistence ordering, the scaling description breaks down for large enough times. The order parameter and the spin-spin correlation function are derived analytically, and several temporal and spatial scales associated with them analyzed.

Introduction

Phase separation, nucleation, ordering, and cluster growth are interrelated topics of great practical importance. One-dimensional (1D) phase separation, for which exact results can be derived, is discussed in this chapter. The emphasis will be on dynamical rules that involve simultaneous updating of the 1D-lattice ‘spin’ variables. Such models allow a particularly transparent formulation in terms of equations of motion the linearity of which yields exact solvability.

The results are also related to certain reaction-diffusion models of annihilating particles (see Part I of this book), and to deposition-with-relaxation processes (Part IV). Some of these connections will be reviewed here as well. While certain reaction and deposition processes have experimental realizations in 1D (see Part VII), 1D models of nucleation and cluster growth have been explored mainly as test cases for modern scaling theories of, for instance, structure-factor scaling, which will be reviewed in detail.

The dynamics of the deposition and evaporation of k adjacent particles at a time on a linear chain is studied. For the case k = 2 (reconstituting dimers), a mapping to the spin-½ Heisenberg model leads to an exact evaluation of the autocorrelation function C(t). For k ≥ 3, the dynamics is more complex. The phase space decomposes into many dynamically disconnected sectors, the number of sectors growing exponentially with size. Each sector is labeled by an irreducible string (IS), which is obtained from a configuration by a nonlocal deletion algorithm. The IS is shown to be a shorthand way of encoding an infinite number of conserved quantities. The large-t behavior of C(t) is very different from one sector to another. The asymptotic behavior in most sectors can be understood in terms of the diffusive, noncrossing movement of individual elements of the IS. Finally, a number of related models, including several that are many-sector decomposable, are discussed.

Introduction

Problems related to random sequential adsorption (RSA), initially studied several decades ago, have aroused renewed interest over the past few years. The reason for this is the growing realization that the basic process of deposition of extended objects, which is modeled by RSA, has diverse physical applications. In turn, this has led to the examination of a number of extensions, including the effect of interactions between atoms on adjacent sites, and the diffusion and desorption of single atoms.

In this chapter we give a brief review of one-dimensional (1D) kinetic Ising models that display nonequilibrium steady states. We describe how to construct such models, how to map them onto models of particle and surface dynamics, and how to derive and solve (in some cases) the equations of motion for the correlation functions. In the discussion of particular models, we focus on various problems characteristically occurring in studies of nonequilibrium systems such as the existence of phase transitions in 1D, the presence or absence of the fluctuation-dissipation theorem, and the derivation of the Langevin equations for mesoscopic degrees of freedom.

Introduction

The Ising model is a static, equilibrium, model. Its dynamical generalization was first considered by Glauber who introduced the single-spin-flip kinetic Ising model for describing relaxation towards equilibrium. Kawasaki then constructed a spin-exchange version of spin dynamics with the aim of studying such relaxational processes in the presence of conservation of magnetization. Other conservation laws were introduced soon afterwards by Kadanoff and Swift and thus the industry of kinetic Ising models wTas born.

The value of these models became apparent towards the end of the 1960s and the beginning of the 1970s when ideas of universality in static and dynamic critical phenomena emerged. Kinetic Ising models were simple enough to allow extensive analytical (series-expansion) and numerical (Monte Carlo) work, which was instrumental in determining critical exponents and checking universality.

Random sequential adsorption (RSA) and cooperative sequential adsorption (CSA) on 1D lattices provide a remarkably broad class of far-fromequilibrium processes that are amenable to exact analysis. We examine some basic models, discussing both kinetics and spatial correlations. We also examine certain continuum limits obtained by increasing the characteristic size in the model (e.g., the size of the adsorbing species in RSA, or the mean island size in CSA models having a propensity for clustering). We indicate that the analogous 2D processes display similar behavior, although no exact treatment is possible here.

Introduction

In the most general scenario for chemisorption or epitaxial growth at single crystal surfaces, species adsorb at a periodic array of adsorption sites, hop between adjacent sites, and possibly desorb from the surface. Such processes can be naturally described within a lattice-gas formalism. The microscopic rates for different processes in general depend on the local environment and satisfy detailed-balance constraints. The net adsorption rate is determined by the difference in chemical potential between the gas phase and the adsorbed phase. In many cases, thermal desorption can be ignored for a broad range of typical surface temperatures, T. Furthermore, for sufficiently low T, thermally activated surface diffusion is also inoperative, so then species are irreversibly (i.e., permanently) bound at their adsorption sites. Henceforth, we consider the latter regime exclusively. Clearly the resultant adlayer is in a far-from-equilibrium state determined by the kinetics of the adsorption process.

Continuous phase transitions from an absorbing to an active state arise in diverse areas of physics, chemistry and biology. This chapter reviews the current understanding of phase diagrams and scaling behavior at such transitions, and recent developments bearing on universality.

Introduction

Stochastic processes often possess one or more absorbing states—configurations with arrested dynamics, admitting no escape. Phase transitions between an absorbing state and an active regime have been of interest in physics since the late 1950s, when Broadbent and Hammersley introduced directed percolation (DP). Subsequent incarnations include Reggeon field theory, a high-energy model of peripheral interest to most condensed matter physicists, and a host of more familiar problems such as autocatalytic chemical reactions, epidemics, and transport in disordered media. For the simpler examples—Schlögl's models, the contact process, and directed percolation itself—many aspects of critical behavior are well in hand. In the mid-1980s absorbing-state transitions found renewed interest due to the catalysis models devised by Ziff and others, and to a proposed connection with the transition to turbulence. A further impetus has been the ongoing quest to characterize universality classes for these transitions. Parallel to these developments, probabilists studying interacting particle systems have established a number of fundamental theorems for models with absorbing states.

Interest in the influence of kinetic rules on the phase diagram has spawned many models over the last decade; the majority must go unmentioned here.

Recent results for the Glauber-type kinetic Ising models are reviewed in this chapter. Exact solutions for chains and simulational results for the dynamical exponents for square and cubic lattices are given.

Introduction

A study on the dynamical behavior of the Ising model must begin with the introduction of a temporal evolution rule, because the Ising model itself does not have any a priori dynamics naturally introduced from the kinetic theory. Various kinds of dynamics are possible and some are useful to describe and predict physical phenomena or to make simulation studies of the equilibrium state. The Ising model with an appropriately defined temporal evolution rule is called the kinetic Ising model.

The statistical mechanical studies of the dynamical behavior in and around the equilibrium state started in the 1950s. During that decade, theoretical and computational developments provided a breakthrough and advanced such studies. The Kubo theory and its successful application established the linearly perturbed regime around the equilibrium state generally treated by methods of statistical mechanics. It gave a means of investigating the dynamic behavior of macroscopic systems. Another great advance in that decade was the application of computing machines to statistical physics. Dynamical Monte Carlo (MC) simulation on computers gave rise to the problem of computational efficiency, which is related to the dynamical behavior of the system, although this aspect became clear rather recently, in the 1980s.

It is often thought that superplasticity is only found at relatively low strain rates, typically about 10–4 to 10–3 s–1. Several recent studies have indicated, however, that superplasticity can exist at strain rates considerably higher than 10–2 s–1. This high-strain-rate superplasticity (HSRS) phenomenon has now been observed in metal-matrix composites, mechanically alloyed materials, and even the more conventionally produced metallic alloys. We will discuss the phenomenon in detail in the following.

Experimental observations

Metal-matrix composites

The phenomenon of HSRS was initially observed in Al-based metal-matrix composites and has continued to be studied mainly in Al-based alloys. Composite reinforcements include SiC and Si3N4 whiskers and SiC particles; matrix alloys include 2000, 6000, and 7000 series Al. A list of published HSRS results is presented in Table 9.1. Despite the differences in the type of reinforcement and matrix composition, all of these composites are noted to exhibit approximately similar deformation and microstructural characteristics. In the following, we use a powder-metallurgy 20%SiC whisker-reinforced 2124Al composite (SiCw/2124Al) as an example to reveal the key experimental observations of HSRS. This composite was the first material observed to exhibit HSRS.

To the present time, reports on HSRS are found in aluminum composites mainly produced by powder-metallurgy methods. High-temperature deformation investigations of the SiCw/2124Al indicated that the material was not superplastic in as-extruded conditions; over the conventional strain-rate range of 1.7×10–3 to 3.3×10–1 s–1, elongation-to-failure values of 30 to 40% were recorded.

Despite extensive studies of superplastic behavior in metallic systems since the 1960s, work on superplasticity in ceramics and ceramic composites is of very recent origin. This is primarily because ceramics normally fracture intergranularly at low strain values, as a result of a weak grain-boundary cohesive strength. The low grain-boundary cohesive strength is a result of inherent high grain-boundary energy. Research in superplastic ceramics began actively only in the late 1980s but has expanded very rapidly since then.

The ceramics and ceramic composites made superplastic to date are essentially based on the principles developed for metallic alloys. Existing data indicate that for polycrystalline ceramics, however, a grain size of less than 1 μm is necessary for superplastic behavior. This is in contrast to superplastic metals, in which grain sizes are typically only required to be less than 10 μm. To highlight the dominant effect of grain size on the deformation behavior of ceramics, Figure 6.1 shows the modulus-compensated flow stresses measured from a number of studies on tetragonal zirconia as a function of diffusivity-compensated strain rate. It is evident that for a given stress, the strain rate increases dramatically as grain size decreases. (Or, conversely, that for a given imposed strain rate the stress required decreases dramatically as grain size decreases.) Figure 6.1 illustrates the importance of grain-boundary-sliding (GBS) mechanisms in the deformation of fine-grained ceramics.

Interest in superplasticity is extremely high. The major areas include superplasticity in metals, ceramics, intermetallics, and composites. Superplasticity at very high strain rates (i.e., approximately 0.1–1 s−1) is an area of strong emphasis that is expected to lead to increased applications of superplastic-forming technology.

Historically, there has been no universally accepted definition for superplasticity. After some debate, the following version was proposed and accepted at the 1991 International Conference on Superplasticity in Advanced Materials (ICSAM-91) held in Osaka, Japan:

It is anticipated that there will continue to be some modifications to this definition, but it should serve as a working definition for a phenomenon that was scientifically reported in 1912 and, indeed, may have a far longer history, as described in the following chapter.

During the course of the ICSAM-91 Conference, many different superplastic materials were described. A list of those mentioned is presented in Table 1.1. It is reasonable to infer from the broad range of superplastic materials listed that there is now a good basic understanding of the requirements for developing superplastic structures.