Introduction - Rheology for Chemists: An Introduction - page 6

Figure 1.5(a) shows a steady shear-thinning response and the experimental
points can be fitted to a simple equation:
s ¼ A c g n

ð 1 : 6 Þ

where the two fitting parameters are A c , the ''consistency'', and n, the ''power-
law index''. This equation is often presented in its viscosity form:
Z ¼ A c g n 1 ð 1 : 7 Þ

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

s ¼ Z p g þ s B ð 1 : 8 Þ

Here, the fitting parameters are the slope of the line, (the plastic viscosity, Z p )
and the Bingham or dynamic yield stress, (the intercept, s B ). 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 be able to compare the values from batch to
batch of a product or reformulation. This ability to collapse more or less
complicated curves down to a few numbers is of great value whether we are
engaged in 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 wide 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 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:

k r ¼ A exp ð E a = RT Þ ð 1 : 9 Þ

Here k r is the rate of a reaction measured at temperature T. E a is the
activation energy and R is the gas constant. Now RT is the value of thermal