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 forces.

1.2 SIMPLE CONSTITUTIVE EQUATIONS

1.2.1 Linear and Non-linear Behaviour

It is easy to write down an algebraic relationship that describes the simpler forms of rheological behaviour. For example

for a Hookean solid σ = Gγ (1.4)

and a Newtonian liquid σ = ηγ (1.5)

These equations should fully describe the stress–strain–time relationship for the materials over the full range of response. However, the range over which such linear behaviour is observed is invariably limited. Usually large stresses and strains or short times cause deviations from Equations 1.4 or 1.5.

As the behaviour becomes more complicated, more parameters are required to fit the experimental curves. To illustrate this, consider two common equations used to describe the shear-thinning behaviour observed in viscometers. Figure 1.5 shows these two responses.

Figure 1.5a shows a steady shear-thinning response and the experimental points can be fitted to a simple equation:

σ = Acγn (1.6)

where the two fitting parameters are Ac, the 'consistency', and n, the 'power law index'. This equation is often presented in its viscosity form:

η = Acγn-1 (1.7)

Figure 1.5b shows the behaviour of a 'Bingham plastic' and the fitting equation is:

σ = η pγ + σB (1.8)

Here the fitting parameters are the slope of the line (the plastic viscosity, ηp) and the Bingham or dynamic yield stress (the intercept, σB). Other constitutive equations will be introduced later in this volume as appropriate.

1.2.2 Using Constitutive Equations

The first use that we can make of our constitutive equations is to fit and smooth our data and so enable us to discuss experimental errors. However, in doing this we have the material parameters from the model. Of course it is these that we need to record on our data sheets, as they will enable us to reproduce the experimental curves and we will then be able to compare the values from batch to batch of a product or formulation. This ability to collapse more or less complicated curves down to a few numbers is of great value whether we are engaged in the production of, the application of, or research into materials.

The corollary is that we should always keep in mind the experimental range. Extrapolation outside that range is unwise. This will become particularly clear when we discuss the yield phenomenon – an area of great interest in many practical situations. Whatever the origins claimed for these models, they all really stem from the phenomenological study of our materials and so our choice of which one to use should be based on the maximum utility and simplicity for the job in hand.

1.3 DIMENSIONLESS GROUPS

An everyday task in our laboratories is to make measurements of some property as a function of one or more parameters and to express our data graphically, or more compactly as an algebraic equation. To understand the relationships that we are exploring, it is useful to express our data as quantities that do not change when the units of measurement change. This immediately enables us to 'scale' the response. Let us take as an example the effect of temperature on reaction rate. The well-known Arrhenius equation gives us the variation

kr = A exp(-Ea/RT) (1.9)

Here kr is the rate of a reaction measured at temperature T, Ea is the activation energy and R is the gas constant. Now RT is the value of thermal energy and so the magnitude of the dimensionless group, (Ea/RT), immediately gives us a feel as to the importance of the activation process. For example, if Ea [much less than] RT, then the activation process will not slow the reaction rate significantly from the fastest possible rate A. On the other hand, if Ea [much greater than] RT, then the reaction rate will be very much slower than A. Mechanistically this reminds us of the Boltzmann energy distribution and stochastic processes. The dimensionless group, (Ea/RT), is known as the 'Arrhenius Group'.

Another example from chemical kinetics can be seen in the rate equation for first-order reactions. Here the equation relating the concentration of a species A at time t, [A](t), to the reaction time and the initial concentration, [A](00), is

[A](t) = [A](00)exp(-krt) (1.10)

The rate coefficient, kr has units of t-1 and so can equally well be thought of in terms of the characteristic time for the reaction to take place. Hence if krt [much greater than] 1, the reaction will be a long way towards completion, whereas if krt [much less than] 1, very little change will have occurred. Equation 1.10 describes the decay of radioactive elements and 1/kr could be considered as the characteristic time for the relaxation of the element from its active to its non-active state.

1.3.1 The Deborah Number

Maxwell introduced the idea of viscous flow as being the manifestation of the decay of elastically stored energy. If we follow this concept through we will see how a dimensionless group, the Deborah number, De, arises naturally. Let us consider a piece of matter in which all the molecules or particles (either small or large, it makes no difference) have had time to diffuse to some low energy state. Now if we instantaneously strain (deform) the material, we will store energy because the structure is perturbed and the molecules are in a higher energy state. As we hold the matter in this new shape, it becomes easier because the molecules diffuse until a low energy state equivalent to the initial one has been achieved, although the original shape has been lost, i.e. viscous flow has occurred. We can define the characteristic time it takes for this process to occur as the stress relaxation time, τ, of the material. Now the Deborah number is

De = τ/t (1.11)

The relaxation of the stress resulting from a step strain can be observed experimentally and we can see that it is the result of diffusive motion of the microstructural elements. Although we can have a mechanistic picture, what does this mean in terms of our measurements? We have the very striking result that our material classification must depend on the time t, i.e. the experimental or observation time. Hence, we can usefully classify material behaviour into three categories:

De [much greater than] De~O(1) De [much less than]
solid-like viscoelastic liquid-like

The most frequently quoted example to illustrate this behaviour is the children's toy 'Silly Putty', which is a poly(dimethyl siloxane) polymer. Pulled rapidly it shows brittle fracture like any solid but if pulled slowly it flows as a liquid. The relaxation time for this material is ~ 1 s. After t = 5 τ the stress will have fallen to 0.7% of its initial value so the material will have effectively 'forgotten' its original shape. That is, one could describe it as having a 'memory' of around 5 s (about that of a mackerel!). Many other materials in common use have relaxation times within an order of magnitude or so of 1 s. Examples are thickened detergents, personal care products and latex paints. This is of course no coincidence, and this timescale is frequently deliberately chosen by formulation adjustments. The reason is that it is in the middle of our, the human, timescale. Our nervous system responds over a timescale of 1 ms to 1ks and so, if a material has a relaxation time within that region, we will observe an 'interesting' or useful texture. Reiner pointed out that our observation time could be quite long with some materials, set concrete for example, and so ultimately our definition of solid-like can become one of practical rather than philosophic origin.

1.3.2 The Peclet Number

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes–Einstein equation:

D = kBT/6πηoa (1.12)

where kB is the Boltzmann constant, ηo is the viscosity of the liquid medium, and a is the radius of the diffusing moiety – molecule or particle. This has dimensions of m2s-1. We can use Equation (1.12) to estimate the time taken for the diffusing moiety to move a characteristic distance. It makes sense to choose the radius as this distance and this gives us the Einstein–Smoluchowski equation:

ta = a2/D = 6πηoa/ kBT

and so

ta = 6πηoa3/kBT (1.13)

Now the characteristic time for shear flow is the reciprocal of the shear rate. This is the time taken for a cubic element of material to be transformed to a parallelogram with angles of 45° (i.e. the time for unit strain to be applied) as shown in Figure 1.6. The Péclet number can now be written:

Pe = 6πηoa3γ/kBT (1.14)

An interesting problem arises when we consider solutions or colloidal sols where the diffusing component is much larger in size than the solute molecules. In dilute systems Equation (l.14) would give an adequate value of the Péclet number but not so when the system becomes concentrated, i.e. the system itself becomes a condensed phase. The interactions between the diffusing component slow the motion and, as we shall see in detail in Chapter 3, increase the viscosity. The appropriate dimensionless group should use the system viscosity and not that of the medium and now becomes

Pe = 6πa3σ/kBT (1.15)

where the shear stress, σ = η[??], has been used to make a clear distinction from Equation (1.14). Of course for a simple system, cyclohexane in decane for example, Equations (l.14) and (1.15) would give the same result as the intermolecular interactions between the species are similar and the viscosity of a mixture is similar to that of the two components. We shall use Equation (1.15) throughout as this indicates the importance of the interactions.

1.3.3 The Reduced Stress

The reduced stress, σr was introduced by Krieger from a dimensional analysis and has the form:

σr = a3σ/kBT (1.16)

The similarity to the Péclet number is obvious but we should also bear in mind the relationship to the Deborah number. This becomes clear when we consider the fact that the mechanism of stress relaxation is due to the diffusion of the microstructural components. For slow deformation processes – the low Deborah number, low Péclet number, or low reduced stress limit – the rate at which the structural elements can rearrange is great enough that the structure has little or no perturbation from that found in the quiescent state. Viscous deformation then occurs. Now if the straining is rapid, relaxation cannot take place and energy is stored. If the deformation is continuous, the structure must yield and breaking or 'melting' is then observed.