The state of different condensed matter systems is characrerized by different quantities: density, symmetry of a crystal, magnetization, electric polarization, etc. Many such states can have a certain ordering. Different types of ordering can be characterized by order parameters.

Examples of order parameters are, for instance: for ferromagnets – the magnetization M; for ferroelectrics – the polarization P; for structural phase transitions – the distortion uαβ, etc. Typically the system is disordered at high temperatures, and certain types of ordering may appear with decreasing temperature. This is clear already from the general expressions for thermodynamic functions, see Chapter 1: at finite temperatures the state of the system is chosen by the condition of the minimum of the corresponding thermodynamic potential, the Helmholtz free energy (1.8) or the Gibbs free energy (1.10), and from those expressions it is clear that with increasing temperature it is favourable to have the highest entropy possible, i.e. a disordered state. But some types of ordering are usually established at lower temperatures, where the entropy does not play such an important role, and the minimum of the energy is reached by establishing that ordering.

The general order parameter η depends on temperature, and in principle also on other external parameters – pressure, magnetic field, etc. Typical cases of the dependence of the order parameter on temperature are shown in Fig. 2.1.

Introduction

The modeling and simulation of granular materials is important to our understanding of their behavior and the wealth of phenomena they exhibit [1–3]. Many phenomena and practical applications, such as the design of industrial processes, remain out of reach of traditional simulation methods due to the large numbers of grains involved. A liter of fine sand, mgrain ≈ 0.1 mg, may contain 107 grains. An industrial process such as mixing or hopper flow can easily involve 101 to 103 liters. Geophysical processes, such as sand dunes and earthquake faults, involve even larger numbers of grains. Traditional simulation techniques are currently unable to deal with such large numbers of grains; however, cellular automata models are able to simulate larger numbers of grains for longer times and show promise in the simulation of large, real-world granular flows.

Molecular dynamics, where Newton's laws are applied to individual grains and the resulting motion is determined from the forces, has made dramatic progress in the past two decades through improved techniques and more powerful computers. Molecular dynamics is particularly effective for gases of simple grains interacting through hard-sphere collisions. In principle, even the physics of complex grain interactions can be included. However, as the complexity of the interactions increases or the number of grains increases, the computational demands increase as well. As a result, molecular dynamics is primarily used for relatively small numbers of grains (102–105), whereas real granular flows often contain many more.

As we have seen in the previous chapters, interactions, if they are not too strong, preserve many features of the electronic system which are present in the noninteracting case (Fermi gas, Chapter 7). In general, however, the interactions are not at all small: for instance in typical metals, rs ∼ 2–3, and not rs « 1 as was implicitly assumed in Chapter 9 and which was actually the condition for the applicability of perturbation theory used there. Nevertheless we know that the description of normal metals using the concepts developed for the free Fermi gas or Fermi systems with weak interactions (such asDrude theory, for example) is very successful.

An explanation of the success of the conventional theory of metals, and the generalization of the corresponding description to a more general situation, was given by Landau in his theory of Fermi liquids. This theory is very important conceptually, although in the usual metals there are only few special effects which indeed require this treatment. However, there exist also systems (3He, or rare earth systems with mixed valence and heavy fermions) for which this approach is really vital. Also the emerging new field of non-Fermi-liquid metallic systems requires first an understanding of what is the normal Fermi liquid.

The foundations of the Fermi-liquid theory

Themain assumptions of the Landau Fermi liquid theory are completely in linewith our general approach.

Introduction

Jammed materials are ubiquitous in nature and share several defining characteristics. They are disordered, yet solid-like with a nonzero static shear modulus. Jammed systems typically exist in metastable states with structural and mechanical properties that depend on the procedure used to create them. There are a number of different routes to the jammed state, including compressing systems to densities near random close packing [1], lowering the applied shear stress below the yield stress [2], and quenching temperature below the glass transition for the material [3]. Examples of jammed and glassy particulate systems include dense colloidal suspensions [4], attractive glasses and gels [5], static packings of granular materials [6], and quiescent foams [7] and emulsions [8]. Due to space constraints, we will limit our discussion to athermal jammed systems in which thermal energy at room temperature is unable to induce local rearrangements of particles. We note though that there are deep connections [9] between athermal jammed systems and thermal, glassy systems [10]. An important open problem in the field of jammed materials is identifying universal features that are not sensitive to the particular path in parameter space taken to create them.

In this contribution, we will review the computational techniques used to generate athermal jammed systems and characterize their structural and mechanical properties. We will focus on frictionless model systems that interact via soft, pairwise, and purely repulsive potentials. (Computational studies of frictional granular materials will be the focus of Chapter 5.) The methods for generating jammed particle packings discussed here are quite general and can be employed to study both two- and three-dimensional systems; both monodisperse and polydisperse systems; a spectrum of particle shapes, including spheres, ellipsoids, and rods; and a variety of boundary conditions and applied stress.

Introduction

Optical measurement techniques are ubiquitous across many disciplines of science. Since so much of our everyday interaction with the world around us is based on vision, optical investigations feel very natural and, when possible, are often the preferred interrogation technique.

While the sophistication of optical measurement has certainly increased with time, the basic idea remains the same: the physical system of interest is illuminated with visible light (which is assumed to interact with the system only passively) and is imaged by a photosensitive detector. In the early days of optical measurement, film was the preferred medium, and optical studies were by and large static. The few dynamic measurements that were made involved long or multiple exposures and tedious reconstructions done by hand [1]. With the advent of digital imaging and large-scale digital storage, however, optical measurements can now easily address dynamic questions, with enough temporal and spatial resolution to measure a wide range of quantities and sufficient storage to gather enough samples for well-converged statistics. In physics, we are often interested in velocities and accelerations, which give us access to momentum, energy, and force; such dynamic quantities are now accessible with imaging techniques.

Introduction

Fluids deform irreversibly under shear; in other words, they flow. In contrast, solids deform elastically when subjected to a small shearing force and recover their original shape when the force is removed. The behavior of what is termed soft matter is somewhere in between. Soft matter systems are typically viscoelastic, that is they display a combination of viscous (fluid-like) and elastic (solid-like) behavior. Measuring the flow behavior and the mechanical response to deformation of viscoelastic materials provides us with information that can be interpreted in terms of their small-scale structure and dynamics.

The mechanical properties of soft materials depend on the length scale probed by the measurements due to the fact that the materials are structured on length scales intermediate between the atomic and bulk scales [1]. For example, a colloidal suspension has structure on the scale of the spacing between the colloidal particles; a concentrated polymer system, on the scale of the entanglements between large molecules. As a result, their bulk properties can be quite different from properties on length scales smaller than or comparable to the structural scale. Making measurements on both macroscopic and microscopic length scales can help us to develop a better understanding of the relationship between microstructure and bulk properties in soft materials.

Following a brief introduction to viscoelasticity, this chapter will focus on two methods of measuring the viscoelastic properties of soft matter. On the macroscopic scale, rotational shear rheometry provides a well-established set of techniques for determining the mechanical properties of complex fluids.

Introduction

“Soft materials” is a loose term that applies to a wide variety of systems we encounter in our everyday experience, including:

1. • Colloids, which are microscopic solid particles in a liquid. Examples include toothpaste, paint, and ink.

2. • Emulsions, which are liquid droplets in another immiscible liquid, for example milk and mayonnaise. Typically a surfactant (soap) molecule or protein is added to prevent the droplets from coalescing.

3. • Foams, which are air bubbles in a liquid. Shaving cream is a common example.

4. • Sand, composed of large solid particles in vacuum, air, or a liquid; examples of the latter include quicksand and saturated wet sand at the beach.

5. • Gels are cross-linked polymers such as gelatin, or sticky colloidal particles. Usually the components of a gel (the polymers or particles) are at low concentration, but the gel still is elastic-like due to strong attractive forces between the gel components.

One common feature to all of these materials is that they are all comprised of objects of size 10 nm–1 mm; that is, objects much larger than atoms. In fact, it is these length scales that gives them their softness, as a typical elastic modulus characterizing these sorts of materials is kBT/a3, where kB is Boltzmann's constant, T is the absolute temperature, and a is the size of the objects the material is made from [1].

Introduction

The behavior of dense granular materials, which consist of large collections of individual grains, is an example of a complex system. Despite the relative simplicity of the constituents, the large number of frictional contacts leads to indeterminacy, history dependance, and jamming. We still lack a general set of macroscopic equations to describe their flow. A continuum description of the relevant state variables is desirable, and early studies in soil mechanics focused on characterizing bulk stress/strain relationships and failure. However, it was determined through experiments using photoelastic materials [1–3] that forces transmitted through granular assemblies are carried through an inhomogeneous network of stress chains in which the majority of force is carried through chains of particles comprising a minority of grains (e.g. Figure 9.1(b)). The creation and failure of these chains are central to the fluctuations that can dominate in measurements of dense, granular systems [4].

To visualize internal stresses, these experiments used grains composed of photoelastic materials, which exhibit stress-induced birefringence. When placed between crossed polarizers, in a polariscope, the intensity of transmitted light varies with the local principal stress difference, allowing visualization of the internal stresses in the system. Regions of differential stress appear as a series of bright and dark fringes. The resulting pattern offers both an immediate insight into the spatial stress distribution and the opportunity to measure quantitative local force data in the sample.

Using the techniques described in the previous chapter, we can in a unified way discuss properties of the electron gas with Coulomb interaction and consider such effects as optical response, screening, plasmons, etc. In many textbooks these properties are obtained using a variety of methods. The virtue of the Green function method is its universality and, I would say, not much simpler, but standardized form. This method permits one to obtain all the properties mentioned above in the form of one general expression, and it also gives the possibility to generalize the results quite easily to the cases of low-dimensional (1d, 2d) systems, or to take into account the details of the band structure of the material, etc. But more important is the fact that it leads naturally to a number of special interesting consequences which would be rather difficult to obtain with the usual classical methods. In this and in the next two chapters I will demonstrate how to reproduce, using this method, the familiar results such as Debye screening or the plasmon energy, but I will mostly concentrate on less frequently discussed effects which are quite naturally obtained using this technique.

Dielectric function, screening: random phase approximation

We start by studying the form of the effective electron–electron interaction in metals. The ordinary Coulomb interaction V(q) = 4πe2/q2 is modified by the reaction of the electronic system. The first, well-known effect is just the screening of the electric charge. But there are other interesting effects as well.

Introduction

Fast and affordable computing power, especially in the form of personal computers and workstations, has enabled the expansion of the study of soft condensed matter physics over the last two decades. The use of computing power to not only analyze image data but acquire it from high-resolution and high-speed digital sources has also made many significant investigations in soft condensed matter experiments accessible to students while still pursuing their undergraduate degrees. Unlike students just a generation before, contemporary undergraduates are well versed in the use of computing power, even operating systems such as Linux, when they first arrive on campus. Because of outreach initiatives such as REU programs, intentions to recruit future graduate students, and an increasingly competitive trend in the careers of undergraduate majors, the opportunities to engage undergraduates in research has flourished over nearly the same period of time.

Granular systems, with simple hard sphere interactions and inter-particle friction, tend to be investigated in experiments that are tabletop in scale. As a subset of soft condensed matter systems, the macroscopic nature of granular physics makes the systems conceptually accessible to students as early as the sophomore or junior year of their baccalaureate careers. This no way trivializes the investigations or minimizes the advances in knowledge that a properly trained undergraduate can contribute to the larger scientific community when mentored well.

In the previous section we have already seen that in the case of strong electron–electron interactions, when the average interaction energy becomes larger than the corresponding kinetic energy, one can expect drastic changes of the properties of the system. Notably, the electrons will have a tendency to localize, so as to minimize their repulsion at the expense of a certain increase in kinetic energy. Materials and phenomena for which this factor plays an important role are now at the centre of activity of both experimentalists and theoreticians; this interest was especially stimulated by the discovery of high-Tc superconductivity in which electron correlations play a very important role. But even irrespective of the high-Tc problem, there are a lot of other interesting phenomena which are connected with strong electron–electron interactions. These phenomena include electron localization, orbital ordering and certain structural phase transitions, insulator–metal transitions, mixed valence and heavy fermion behaviour. The very existence of localized magnetic moments in solids, both in insulators and in metals, is actually determined by these correlations. That is why this is one of the most actively studied classes of phenomena at present.

Real materials to which one applies the models and the treatment presented in this chapter are mostly transition metal and rare earth compounds, although general ideas developed in this context are now applied to many other systems, including organic materials, nanoparticles or supercooled atoms. The typical situation in transition metal compounds is the one with partially filled d-shells.

Introduction

Though the movement of a single, isolated bacterium is reasonably well understood, when a large number of interacting bacteria are put together they produce beautiful and often complex phenomena. “Large” here typically means from 106 to 1012 individual cells: a small population by thermodynamic standards, but certainly unwieldy for any except statistical descriptions. Even restricting ourselves to the simple system of bacteria moving on or in solidified agar plates, the colony structures produced are surprisingly rich. In dilute solutions, swimming cells move independently, interacting with each other through their common consumption of a reservoir of nutrient. A point source of swimming cells expands in concentric rings as successive waves of bacteria chase gradients of nutrients, sometimes condensing into regular geometric patterns by chasing self-generated gradients. More concentrated solutions of swimming cells interact hydrodynamically through the fluid, producing large-scale swirls reminiscent of turbulence. This swirling occurs in two-dimensional surface motility as well, where uncorrelated motion turns into large-scale swirling as surface density increases. At extremely high density, bacteria jam and stop moving, as occurs in colloids. These high densities occur naturally on hard surfaces, where colony expansion is driven by cell growth rather than by motion; as the surface property becomes softer and wetter and bacteria begin to move, the resulting colonies change from fractal-like to radially symmetric and finally to a form dominated by a fast-growing, single-cell-thick outer layer.