Introduction

In the previous chapter, we saw how the long-range order of particle positions in a crystal could be described in a rather elegant manner using a space lattice that extends indefinitely. In this chapter, we examine instead disordered matter such as liquids or glasses in which a long-range repeated pattern is absent. These amorphous materials might not seem as glamorous as their crystalline counterparts, but they are increasingly prevalent in our world as they comprise the windows, computer screens and vast array of plastic components that surround us on a daily basis. In comparison with crystalline structures, these amorphous materials pose a challenge to describe, and their structure can only be defined in a statistical sense by introducing an ensemble-averaged, pair distribution function. In spite of their disordered nature, a robust pattern of particle positions emerges over short distances. This short-range order reflects the local coordination of particles and we briefly review the random close pack and the continuous random network systems as common examples of amorphous structure.

A statistical structure

Disordered or amorphous condensed matter has a clear disadvantage in that particle positions lack any long-range repeating pattern akin to that found in crystals. This is evident in Fig. 2.1, which illustrates the typical particle positions of either a glass or a liquid captured at a particular instant in time.

STRUCTURE

Picture with me an old cottage nestled in the woods. There is a small house builtof clay bricks that were thoughtfully stacked and interlaced by a masterbricklayer so as to produce a repeated interlocking pattern. The house has athatched roof consisting of bundles of straw. The straws in each bundle areoriented in a common direction to direct rainwater off the roof, and are lashedtogether with twine. Around the house is a garden enclosed by a stone wall. Likethe brick walls of the house, the stones in the wall are bonded together withmortar. But unlike the bricks, the stones lack any sense of a repeatingpattern.

In this part of the textbook, we examine the basic structures that are found incondensed matter as well as the forces (the mortar and twine) that maintainthese structures over long time periods. For our purposes, structures aredivided into two main categories: ordered (like the bricks and the straw of thehouse) and disordered (like the stones in the garden wall).

Introduction

In Chapter 5 we introduced the structure factor, S(q), as the Fourier representation of the positions of a collection of fixed, elastically scattering, particles. In reality, these particles are rarely fixed. In a solid (crystal or glass), the particles are bound together by bonds and, while unable to wander about, are able to oscillate or vibrate about a fixed center of motion. In a liquid, the particles are even less constrained and are free to wander around over considerable distances. In this chapter we develop the dynamic structure factor as a straightforward extension of the static structure factor introduced previously, and apply it to examine the dynamics of liquid-like systems. In one instance, we consider the Brownian diffusion of macromolecules in a solvent, where the motion mimics that of the random walk we discussed in the previous chapter. In another instance, we show how thermodynamically driven density fluctuations present in a simple liquid are responsible for the characteristic Rayleigh–Brillouin spectrum of light scattering. We also take this opportunity to consider the special case of slow dynamics in polymer liquids and to briefly consider the nature of the liquid-to-glass transition that separates amorphous solids from their liquid counterparts.

Dynamic structure factor

We can think of a liquid as a time-dependent amorphous structure. In many ways, the structure of a liquid resembles the structure of a glass in that, at any instant in time, a “snapshot” of its S(q) resembles that of the glass. Indeed, the only real difference between a liquid and a glass is the presence or absence, respectively, of long-range translational motion. In the liquid, the translational motion results from the incessant jostling of the particles allowing them to wander about. By virtue of this motion, particles of the liquid are able to rearrange on some characteristic time scale (related to the viscosity of the liquid) into different, but thermodynamically equivalent, amorphous configurations whose instantaneous structure resembles that of a glass. For certain glass forming liquids near their glass transition point, the characteristic time scale for these rearrangements can become exceedingly long with some unusual consequences, as we will discuss later.

Introduction

In the last two chapters we explored the behavior of phonons in a crystal. There we saw how these discrete, quantized pieces of propagating energy contributed to both the specific heat and thermal conductivity of the solid. Here we turn our attention to crystalline metals whose metallic bonding results in the formation of a sea of mobile electrons present within the crystal. Like phonons, these mobile electrons carry around energy and consequently contribute to the specific heat. But they also carry around charge and so contribute also to the electrical conductivity of the metal.

In this chapter, we begin with a simplistic model of the mobile electrons as quantum mechanical waves trapped within an infinite square well potential. This model is known as the free electron model because the interaction of the electron with the ion cores of the metal lattice is disregarded. The electron is only trapped by the confines of the crystal itself. Although this simple model is unable to capture all the experimental features of conduction in metals, it readily accounts for the smallness of the electron contribution to the specific heat and does provide a simple interpretation of such electron emission phenomena as the photoelectric effect.

Introduction

In this final chapter on the subject of structures, we turn our attention to magnetic materials. Why? Because magnetic materials are illustrative of yet another level of structure that often arises in condensed matter, beyond that of particle arrangements. As we will see, magnetic particles have a property of net spin and a magnetic moment whose orientation in space is largely unrestricted. Regardless of whether a large system of magnetic particles is positioned in an ordered or disordered manner, their spins represent an additional layer of ordering. The moments could be randomly oriented or aligned in a common direction. In ferromagnetic systems, these moments interact with one another to promote a local alignment of the moments which can eventually spread over the entire system. This is reminiscent of how pairwise bonding between particles eventually leads to crystallization of a liquid, and it is the archetype for a wide variety of phase transitions in which order appears in the form of correlated regions emerging from a disordered host.

The ordering process

So far, our discussion of structure has focused entirely on particles: their relative positions and the forces that hold them together. We have seen that arrangements of particles fall into either an ordered or disordered pattern which can be characterized by the level of symmetry present. By virtue of its disorder, the liquid has rotational invariance and an infinite symmetry (on average). The crystal, however, conforms to a space lattice and possesses only a discrete set of symmetry operations. In the process of forming a crystal from the liquid, the symmetry is often said to be “broken”.

Introduction

In this chapter, we develop some fundamental understanding of the nature ofphase transitions by examining two well-studied examples: thevapor-to-liquid transition of fluids and the paramagnetic-to-ferromagnetictransition in magnetic materials. Here, our focus is on the experimentallyobserved features of these two transitions and how to interpret and navigatethe many phase diagrams that describe them. The theoretical interpretationwill be tackled later in Chapter 17. We will find that, in general, a phasetransition is accompanied by some change in the amount of order as when, forexample, liquid water freezes into crystalline ice. Moreover, we candescribe this amount of ordering quantitatively by introducing anappropriate order parameter, whose value changessignificantly only during the transition. Based upon the manner in which theorder parameter changes, we can distinguish two different types of phasetransitions: those of first order for which the order parameter changesdiscontinuously and those of second order for which it changes continuously.Second-order transitions are possible for both the vapor-to-liquidtransition and the paramagnetic-to-ferromagnetic transition and are ofinterest due to the way in which many properties diverge near the transitionin a similar, power law manner.

DYNAMICS

A steaming cup of coffee is sitting on my desk. Aside from the steam, there islittle else that would suggest any other activity is present. The cup and itcontents appear to be “at rest”. But a little closer examinationreveals a small ripple of waves on the surface of the liquid caused by amechanical pump in the room next door. Indeed, if I rest my finger gently on thelip of the cup, I can feel the vibration. I can also feel the heat that hasdeveloped in the cup, now several minutes since I poured the coffee from the potand added some creamer.

If I could examine this evenmore closely Iwould actually see that nothing istruly “at rest”. The particles of the liquid are jostling aboutincessantly. The particles of the creamer that I added have clearly taken flightand diffused rapidly out into all regions of the coffee. The cup itself is alsoin motion. Its particles are undergoing incessant vibrations that are ultimatelyresponsible for the heat I feel when I hold it.

Perhaps most interesting is that all this microscopic motion appears to be drivenentirely by thermodynamics. The coffee and cup are sitting in aclimatecontrolled office and there is a constant flux of thermal energy (heat)entering and exiting both the coffee and the cup to keep things moving.

Introduction

What makes condensed matter “condensed”? The answer is stickiness (i.e. an attractive interaction) that exists between the particles that make up a liquid or solid. In the last two chapters we have looked at two extremes of particle arrangements in matter: ordered (crystalline) and disordered (amorphous). In this chapter, we now examine the nature of the forces that form between particles and which promote the formation of a condensed phase of matter. We begin by reviewing the five major bonds (van der Waals, covalent, ionic, metallic and hydrogen bonds) and conclude by considering the overall cohesive energy in a crystal. As thermodynamics generally favors the system with lowest energy, this cohesive energy is part of what determines why a particular crystal structure is adopted in Nature, rather than another structure.

Survey of bond types

In order for matter to condense, there must be an attractive force between the particles to promote their mutual gathering together. Of the four fundamental forces in Nature, the two nuclear forces (strong and weak) play no role in the condensation process and the gravitational force is far too weak to drive the process at ordinary terrestrial temperatures and pressures. Instead, the fundamental force that binds particles together in condensed matter arises from electrostatic interactions.