Introduction

The study of rheology is the study of the deformation of matter resulting from the application of a force. The type of deformation depends on the state of matter. For example, gases and liquids will flow when a force is applied, whilst solids will deform by a fixed amount and we expect them to regain their shape when the force is removed. In other words we are studying the 'handling properties of materials'. This immediately reminds us that we must consider solutions and dispersions and not simply pure materials. In fact, the utility of many of the materials we make use of every day is due to their rheological behaviour and many chemists are formulating materials to have a particular range of textures, flow properties, etc. or are endeavouring to control transport properties in a manufacturing plant. Interest in the textures of materials such as a chocolate mousse or a shower gel may be of professional interest to the chemist in addition to natural curiosity. How do we describe their textures quantitatively? What measurements should we make? What is the chemistry underlying the texture so that we may control it? All these questions make us focus on rheology.

The aim of this text is to enable the reader to gain an understanding of the physical origins of viscosity, elasticity and viscoelasticity. The route that we shall follow is to introduce the key concepts through physical ideas and analogues that are familiar to chemists and biologists. Ideas from chemical kinetics, and infrared and microwave spectroscopy are invariably covered in some depth in many science courses and so should aid the understanding of rheological processes. The mathematical content is kept to the minimum necessary to give us a quantitative description of a process, and we have taken care to make any manipulations as transparent as possible.

There are two important underlying ideas that we shall return to throughout this work. Firstly, we should be aware that intermolecular forces control the way in which materials behave. This is where the chemical nature is controlling the physical response. The second is the importance of the timescale of our observations, and here we may become aware of different physical responses if our experiments are carried out at different times. The link between the two arises through the structure that is the consequence of the forces and the timescale for changes by microstructural motion resulting from thermal or mechanical energy. What is so exciting about rheology is the insights that we can gain into the origins of the behaviour of a wide variety of systems in our everyday mechanical world.

1.1 DEFINITIONS

1.1.1 Stress and Strain

The stress is simply defined as the force divided by the area over which it is applied. Pressure is a compressive bulk stress. When we hang a weight on a wire, we are applying an extensional stress and when we slide a piece of paper over a gummed surface to reach the correct position, we are applying a shear stress. We will focus more strongly on this latter stress because most of our instruments are designed around this format. The units of stress are Pascals.

When a stress is applied to a material, a deformation will occur. In order to make calculations tractable, we define the strain as the relative deformation, i.e. the deformation per unit length. The length that we use is the one over which the deformation occurs. This is illustrated m Figures 1.1 and 1.2.

There are several features of note in Figures 1.1 and 1.2:

1. The elastic modulus is constant at small stresses and strains. This linearity gives us Hooke's Law, which states that the stress is directly proportional to the strain.

2. At high stresses and strains, non-linearity is observed. Strain hardening (an increasing modulus with increasing strain up to fracture) is normally observed with polymeric networks. Strain softening is observed with some metals and colloids until yield is observed.

3. We should recognise that stress and strain are tensor quantities and not scalars. This will not present any difficulties in this text but we should bear it in mind because the consequences can be both dramatic and useful. To illustrate the mathematical problem, we can think about what happens when we apply a strain to an element of our material. The strain is made up of three orthogonal components which can be further subdivided into three elements, each of which is lined up with one of our axes. This is shown in Figure 1.3.

Figures 1.2 and 1.3 show how, if we apply a simple shear strain, γ, in our rheometer this is formally made up of two equal components, γxy and γyx. By restricting ourselves to simple and well-defined deformations and flows, i.e. simple viscometric flows, most algebraic difficulties will be avoided but the exciting consequences will still be seen.

1.1.2 Rate of Strain and Flow

When a fluid system is studied by the application of a stress, motion is produced until the stress is removed. Consider two surfaces separated by a small gap containing a liquid, as illustrated in Figure 1.4. A constant shear stress must be maintained on the upper surface for it to move at a constant velocity, u. If we can assume that there is no slip between the surface and the liquid, there is a continuous change in velocity across the small gap to zero at the lower surface. Now in each second the displacement produced is x and the strain is

γ = x/z (1.1)

and us u - dx/dt, we can write the rate of strain as

dγ/dt = u/z (1.2)

The terms rate of strain, velocity gradient and shear rate are all used synonymously and Newton's dot is normally used to indicate the differential operator with respect to time. For large gaps the rate of strain will vary across the gap and so we should write

γ = du/dz (1.3)

When the plot of shear stress versus shear rate is linear, the liquid behaviour is simple and the liquid is Newtonian with the coefficient of viscosity, η, being the proportionality constant.

When a flow is used which causes an extension of a liquid, the resistance to this motion arises from the extensional viscosity, ηε, and the extension rate is ε. Extensional flows require an acceleration of the fluid as it thins and so steady flows are never achieved. This means that microstructural timescale is particularly important. Many practical applications involve extensional flows, frequently with a shear component. For example spraying, spreading and roller coating are common ways of applying products from the food, pharmaceutical, paint and printing industries. Although the analysis may be carried out as though the materials are continua with uniform properties, the control comes from an understanding of the role of molecular architecture and...