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.

Superplasticity, the ability of certain materials to undergo very large tensile strains, was first described in 1912. It became the subject of intense research in the early 1960s following a review of Soviet work and the illustration of the potential commercial applications of superplasticity.

There have been enormous advances in the field, of superplasticity since that time. The field has clear commercial applications, but also retains fascinating scientific challenges in understanding the underpinning physical mechanisms. Recent breakthroughs include the development of superplasticity in polycrystalline ceramics, composites and intermetallics, and also the observation of superplasticity in metallic materials at high strain rates. Superplasticity at high strain rates, in particular, is expected to have a significant technological impact on promoting the commercial applications of superplastic materials.

This book emphasizes the materials aspects of superplasticity and thus was written from the materials point of view. A brief history of the development of superplasticity is first introduced. Then, the two major types of superplasticity, i.e. fine-structure and internal-stress superplasticity, and their operative mechanisms are discussed. Other possible superplastic mechanisms, such as Class I solid solutions and superplasticity at dynamic high strain rates are also described. In addition, microstructural factors controlling the ductility and fracture in superplastic materials are presented. The observations of superplasticity in metals (including Al, Mg, Fe, Ti, Ni), ceramics (including monolithics and composites), intermetallics (including Ni-, Ti-, Fe- aluminides), metal-matrix composites (including Al-, Mg- base), and laminates are thoroughly described.

We have discussed fine-structure superplasticity in various materials, including metals and ceramics, in Chapters 5–9. In these chapters, our discussions focused primarily on experimental observations, deformation mechanisms, and microstructural characteristics. Tensile ductility, which is determined by cavitation and fracture, was not emphasized. In this chapter, we will focus on the latter issue. A fracture mechanics model will be examined with available tensile elongation data for superplastic ceramics and superplastic intermetallics. The analysis permits a broad understanding of tensile elongation behavior of superplastic materials.

Tensile ductility in superplastic metals

It is well accepted that two competing processes govern the failure of superplastic materials at high temperature. One is related to macroscopic necking, and the other is related to microscopic cavitation and cracking. Macroscopic necking is governed by the strain-rate-sensitivity exponent, m, in the simplified constitutive equation σ=km, where σ is the true flow stress, is the true strain rate, and k is a material constant. A high m value usually indicates a diffuse neck development and, thus, a delay of the onset of tensile failure, which leads to high tensile elongations. The fracture profile of many superplastic metals with m≥0.4, however, reveals that there is no sharp pinpoint necking. This is because final fracture is caused by the evolution of cavities at grain boundaries, and in this sense, cavities lead to premature failure of test samples.

Creep mechanisms

Creep is a plastic deformation process that occurs in solids at high temperatures, typically, above approximately 0.5 of the homologous temperature, i.e. T/Tm, where Tm is the absolute melting point of the solid. During creep, a solid deforms permanently, under external forces, initially with negligible formation of cracks or voids. This capacity for plastic flow is associated with three discrete mechanisms that can occur at the atomic level. These mechanisms are (a) slip by dislocation movement, (b) sliding of adjacent grains along grain boundaries, and (c) directional diffusional flow. To a first approximation, the three mechanisms can be generally considered to occur independently of one another. Although, in some cases, one mechanism may be necessary to permit accommodation of another, e.g. diffusional flow or slip may be an accommodation mechanism for grain boundary sliding. For the case of large plastic strains, these mechanisms are all thermally activated and are controlled by the diffusion of atoms. They are, therefore, both temperature- and time-dependent.

Creep is commonly characterized by a strain–time curve, i.e. a creep curve. The creep rate is measured directly from the slope of such a creep curve. There are usually two basic types of creep curves: a metal type and an alloy type. As shown in Figure 4.1, for the metal type, the curve normally starts with a primary regime during which the creep rate decreases with time; this region is usually followed by a steady-state regime during which the creep rate is essentially constant; eventually, cavitation and necking begin to develop in the specimen which results in an acceleration of creep rate and leads to a tertiary region and the final failure.

Titanium alloys

Superplastic forming of titanium alloys has been widely used by the aerospace industry. A well-known example of superplastic forming carried out at Rockwell International using a Ti–6Al–4V alloy was shown in Figure 14.2. The component shown was a nacelle center-beam frame. (A number of such parts formed a proposed structure in the B–1 aircraft.) In this example, a single superplastic forming and diffusion bonding operation was designed to replace a production route which had involved the forming of eight separate pieces of the same alloy, which then had to be joined together with 96 fasteners. Estimated cost savings of 55% and weight savings of 33% were estimated using this fabrication route compared with the conventional production technique. Superplastically formed Ti–6Al–4V was also used for service doors and panels for Airbus aircraft, missile fins, turbofan blades, and turbine disks.

For non-aerospace applications, the most significant commercial product is probably golf club heads, shown in Figure 15.1. A titanium golf club head offers a light weight, large volume, and a wide sweet spot area. It is now produced in both Japan and China. In Japan, the head is made of SP–700 Ti alloy which has the nominal composition of Ti–4.5Al–3V–2Fe–2Mo. The alloy, marketed by Nippon Kokan (NKK), features a low SPF temperature (as low as 775 °C) for the maximum formability, and can be age-hardened.

A practical method for superplastic forming of bulk material is through the use of powder metallurgy methods. The approach here is to achieve net-shaped products, with high density, by compaction of both metal and ceramic powders, using fine-structure or internal-stress superplasticity (FSS and ISS) methods. Studies have been performed by Ruano et al. on the use of ISS in enhancing the densification of white cast iron powders. Caligiuri and Isonishi and Tokizane, on the other hand, used FSS to enhance the densification of ultrahigh carbon steel (UHCS) powders. In addition, Allen used FSS to consolidate Ni-based IN-100 powders (GatorizingTM* process). For ceramics, Kellett et al. used FSS to extrude fine zirconia powders and Wakai et al. to perform the bulge forming of YTZP pipes directly from powders. Panda et al. Akmoulin et al., Uchic et al., and Kwon et al. have sinter forged nanometer zirconia, titania, and alumina powders.

ISS compaction of white cast iron powders

The advent of new technologies centered on fine powders often requires development of methods of enhancing densification wherein the fine structures present in such powders are retained. Low temperatures must be used to achieve this goal, but this usually requires the application of high pressures if a high density is to be achieved. High pressures are often a limiting factor in the manufacture of powder products.

Many metal-matrix composites (MMCs) and laminates have been developed in recent years for advanced structures. Both materials are attractive for many structural applications because they exhibit unusual combinations of structural, physical, and thermal properties including high modulus and strength, good wear resistance, good dimensional stability and low thermal expansion, and low density. Many studies have shown that discontinuously reinforced MMCs can behave superplastically. These composites are mainly aluminum-based, but some magnesium-based and zinc-based composites have also been studied.

Aluminum-based metal-matrix composites

Comparative superplasticity data from some of the representative Al-based composites are summarized in Table 8.1. All of these composites are reinforced by SiC, either in whisker or particulate form. Up to the present time, superplasticity has not only been observed in MMCs produced by powder metallurgy (PM) methods but also in MMCs produced by ingot metallurgy (IM) methods. The composites listed in Table 8.1 were made by conventional PM techniques, except for the 15 vol % SiCw–7475Al, which was manufactured using SiCw layered between specially prepared foils of superplastic 7475A1 alloy. The IM composite, 10 vol % SiCp–2024Al, was produced by stir-casting. For discussion purposes, the composites listed in Table 8.1 are hereafter abbreviated as reinforcement–matrix alloy, e.g., 20 vol % SiCw–2124Al becomes SiCw/2124. As shown in Table 8.1, the reported strain rate sensitivities (m) vary significantly.

Most superplastic metal alloys exhibit large tensile elongations of about 500% to over 1000%. For the most advanced structures, however, the forming strains are typically less than 200 to 300%. Thus, these elongation values are sufficient to make extremely complex shapes using superplastic forming technology. In so doing, large cost and weight savings (through redesign) have provided the driving force for the change from conventional forming to superplastic forming technology. The principal, fine-structured alloy systems that have been commercially exploited for superplastic forming are those based on aluminum, magnesium, iron, titanium, and nickel alloys. Other alloy systems, e.g., Zn–Al, Cu–Al, and Pb–Sn, have also been widely explored. The study of these alloys, however, is usually for achieving basic understanding rather than for structural applications. Many reviews already exist to cover these alloys, so in the following sections, we will only discuss those that are important for structural applications.

Aluminum-based alloys

It is instructive to review the evolution of superplastic aluminum alloys to gain a basic understanding of how a structural alloy group is developed. For this purpose, an overview of the development of superplastic aluminum alloys from 1966 to 1984 is presented in Figure 5.1, where each box represents an individual publication. The description within each of the boxes refers to the nominal alloy composition (in wt%) or to the commercial designation, if appropriate.