Introduction and synopsis

Packaging surrounds most things we buy or do. Food is packaged, parcels through the post are packaged, and within a car or aeroplane, we ourselves are carefully packaged. It is hard to say how much is spent on it, or the worth of the goods damaged due to inadequate packaging, but the sums involved are certainly considerable, and the potential return on any improvement is large.

The essence of protective packaging is the ability to convert kinetic energy into energy of some other sort – usually, heat – via plasticity, viscosity, visco-elasticity or friction; and this must be done whilst keeping the peak force (and thus the deceleration or acceleration) on the packaged object below the threshold which will cause damage or injury. And there is more to it than that. The direction of impact may not be predictable; then the package must offer omni-directional protection, that is, it must absorb impact from any side. Since the package must be carried with the object it protects, light weight is important. And – since much packaging is discarded – it must (almost always) be cheap.

Foams are especially good at this. The energy-absorbing capacity of a foam is compared with that of the solid of which it is made in Fig. 8.1. For the same energy-absorption, the foam always generates a lower peak force. Energy is absorbed as the cell walls bend plastically, or buckle, or fracture (depending on the material of which the foam is made), but the stress is limited by the long, flat plateau of the stress-strain curve (Figs. 4.2 and 5.1).

Open problems

Long lectures like these tend to be over-optimistic, giving the impression that most physical questions are under control. The reality is different: soft interfaces are far from a happy end. Let me give a few examples:

On the dynamics of wetting: the role of molecular processes is not fully appreciated.

1. (a) Following Blake's ideas, they may sometimes be dominant (at large dynamic contact angles). The difficulty is that they are critically dependent on the atomic structure of the surface.

2. (b) When the liquid induces a real chemical reaction on the supporting solid (e.g. a silanation on the OH groups of a silica surface), the exact nature of the driving force is subtle: what fraction of the reaction enthalpy is directly transformed into heat, and what fraction pulls the contact line?

3. (c) In the case of Aztec pyramids, for instance with one molecular layer spreading out from a thicker region, the description of this layer as a two-dimensional liquid is open to some doubt: we may, in some cases, be dealing with a two-dimensional gas rather than a two-dimensional liquid.

4. (d) The role of surface rugosity is important for these thin layers. It may be that the spreading molecules follow preferentially certain channels (or steps) on the surface: the percolation properties of the channel network may be essential.

5. […]

Dirac was a man who concentrated on the difficult problems of his time. He was principally interested in the basis of quantum mechanics and the elementary particles. However, on one occasion, around 1938, he did write a paper which went to the opposite extreme and discussed the size of the cosmos and the age of the universe in terms of very simple dimensional analysis; such data is still alive and well and still food for thought.

As the centuries have gone by, physicists have of course tended always to move in those directions where great problems remain, and I think if one looks at the progress in physics until, shall we say, the 1940s, they have definitely concentrated on very small things – atoms and small molecules. In the period, in the fifties, then the sixties, it was realised that the methods of physics could be applied to other regions which lay above the atomic scale of, shall we say, everyday life – and by that I mean below the scale of what one normally thinks of as hydrodynamic phenomena. There was a mesoscopic physcs, an inter mediate scale where the ideas and the methods of physics could work and make progress. One of the leading workers in this area, one of the prophets, is Pierre Gilles de Gennes. In the last few decades he has produced an enormous number of ideas which have been directly applicable to the experimental world, and which indeed have come to dominate our study of that world. It will be one of these areas that he will talk about here.

The borders between great empires are often populated by the most interesting ethnic groups. Similarly, the interfaces between two forms of bulk matter are responsible for some of the most unexpected actions. Of course, the border is sometimes frozen (the great Chinese wall). But in many areas, the overlap region is mobile, diffuse, and active (the Middle East border of the Roman empire; disputed states between Austria and the Russians, or the Italians, …).

At a certain naive level, these distinctions can be transposed to physical interfaces between two different forms of matter.

1. (1) The hard frozen surfaces of metals, of ionic solids, or of semiconductors can be studied under conditions of high vacuum: this allows us to probe them – using electron beams, or other radiations which extract electrons from the surfaces; or even beams of neutral atoms. The net result is, in our days, a highly sophisticated knowledge of these sharp robust fortifications.

2. (2) The soft interfaces built from liquids, from polymers, from organic solids, or from detergents are much harder to probe. High vacuum is usually not acceptable. And even if it is, the probing beams can damage the interface. For many centuries, the main information on soft interfaces came from mechanical studies: adhesion, slippage, wear, … During the last fifty years, electrical properties have also been helpful – in particular for the electric ‘double layers’ at the contact region between water and a solid.

We have seen that the toughness of bulk (glassy) polymers, which craze under tension, begins to be understood through an original idea of H. Brown [18]. We shall now try to extend the Brown ideas to various systems of ‘weak junctions’. The junction may be a partly healed contact between two identical polymer blocks A/A, as in the experiments of the Lausanne group [19,20]. Alternatively, it could be a contact between two different polymers A and B.

In all our discussions, we shall assume these junctions to be perfect, with full contact between the two partners, and no gaps. Experimental arguments for the existence of these good contacts have been presented by Kausch and coworkers [19].

Our aim here is:

1. (a) to give a brief reminder of the theoretical description of the weak junctions;

2. (b) to show how some basic mechanical properties can be related to the structure.

One of the major conclusions, for the A/A case, is that chain ends play a crucial role. Thus, any attraction between a chain end and the free surface of one A block will react significantly on the A/A mechanical properties after welding.

This type of attraction was first suggested by systematic experiments on melts by D. Legrand and G. Gaines [21], showing that the surface tension γ of oligomers was often lower than the surface tension γ∞ of a high polymer, and that the correction has the form:

where N is the degree of polymerisation, and x an exponent of order 2/3. The fact that x < 1 shows that we are not dealing with a simple uniform dilution of chains ends (which would give a correction ∼N−1).

A chain of N monomers is attached to a small colloidal particle, and is pulled (at a velocity V) inside a polymer melt (chemically identical, with P monomers per chain). The main parameter for this problem is the number X(V) of P chains entangled with the N chain. Earlier estimates of X are criticised in this appendix, which is based on work by A. Ajdari, F. Brochard-Wyart, C. Gay, and J. L. Viovy (1995), and a new form is proposed: at large, we are led to a ‘Stokes’ regime, X = N½, while at smaller, we find a ‘Rouse’ regime, X = N/Ne (where Ne is the number of monomers per entanglement).

The motion of a long tethered chain (N monomers) inside a polymer melt (P) is special: the N chain cannot reptate inside the P matrix. This occurs in star polymers, and also in two recent experimental situations (figure 2):

1. (a) The N chain is grafted to a colloidal particle (of size smaller than the coil radius RN of the N chain). The particle can be driven by sedimentation or by optical tweezers.

2. (b) The N chain is grafted on a flat wall, and the P melt flows tangentially to the wall (figure 15). (In all that follows, we assume that the grafting density is very small: no coupling between different Nchains.)

Problem (b) was first considered theoretically (for the low V limit) in reference [30]. The starting point is that a certain number X(V) of P chains are entangled with the N chain.

Introduction to the density operator

Motivation and definition

It is often inconvenient, for one reason or another, to keep track of all the variables of a complex system. For example, in a many-body system it would be impracticable to consider the co-ordinates of each particle. Furthermore, such information is actually not of much interest. Thus in a spin system the behaviour of the individual magnetic moments is unimportant; the components of the total magnetisation are the variables of primary interest. Very generally, for a system with ∼ 1023 co-ordinates, one is unlikely to need more than, say, ten variables to describe its observable properties. However, it is quite clear that this reduced amount of information is no longer sufficient to write down a wavefunction and, therefore, it is no longer possible to calculate the evolution of such a system using the usual methods of quantum mechanics.

In Chapter 5 we saw that it was possible, by the introduction of probabilistic arguments, to calculate the evolution of the total magnetisation of a spin system. The calculations were performed using the machinery of quantum mechanics and meaningful and useful results were thereby obtained. In this chapter we will look at things from a rather different point of view. We will not, initially, be concerned with the expectation values of certain specified observables, rather we direct our attention to the general description of the ‘state’ of the system and the way it evolves with time – possibly towards thermal equilibrium.

This book started as a joint project between Michael Richards and myself. We discussed and planned the work in great detail but somewhere along the way Michael moved away from physics, drawing on his experience at Sussex University, to become a psychotherapist. Notwithstanding the plans we made, his withdrawal from the project has resulted in a book hardly recognisable from that originally envisaged. Since the publication of Anatole Abragam's encyclopaedic treatise on nuclear magnetism in 1961 the field has grown at such a pace that no single book could hope to cover its many aspects. However the coverage of this book is without doubt narrower and its content impoverished as a consequence of Michael's career change.

Mea culpa. I am guilty of the very sin of which I have accused others. The motivation for writing a book on NMR was that the various books available, since Abragam's, did not cover the material which I, my colleagues and research students required. It seemed that the new books comprised assorted collections of topics in NMR. Michael and I wanted to redress this, but in surveying the finished product I see no more than yet another assortment of topics. This time, however, it is my collection of topics. While the broader aim might not have been achieved my hope is that this assortment will have its appeal.

In the preface to Abragam's book in 1960 he wrote that he hoped his book would still be the book on the subject thirty years after its publication. That was prophetic; his wish has been amply fulfilled.

The CW method

Frequency and time domains

We have seen in Chapter 1 that the essential features of a resonance phenomenon may be investigated by looking at the system's response to a small sinusoidal disturbance of varying frequency. This is called the continuous wave (CW) method and is the technique traditionally used in most branches of spectroscopy. Alternatively, as we also saw in Chapter 1, we may look at the time response to a transient excitation, this being called the pulse method in NMR.

Results of the two methods have been shown to be equivalent, one response being the Fourier transform of the other. However, if we look at the history of the subject we find that studying the frequency response (CW) was the method used almost exclusively for the first ten years of NMR, i.e. 1946–56. In Andrew's book (Andrew, 1955) for instance 27 pages are devoted to CW experimental methods while pulse methods are treated in a page and a half. Erwin Hahn is usually thought of as the father of pulsed NMR due to his important paper in 1950. Other important contributors include Carr and Purcell (1954) and Torrey (1952).

Since Andrew's book was written, pulsed NMR has to a large extent eclipsed CW NMR. There are two main reasons for this. Firstly, it is much easier to measure the relaxation times T1 and T2 using pulse methods. In fact the long T2 values found in some liquids can only be measured by the pulse technique because magnet inhomogeneity masks the weaker effects of spin–spin coupling. This point will be taken up again in the following chapter.