Rheology - The Fundamentals

Why Rheology?

In order to understand and control any process involving the transfer of fluids it is necessary to know how the fluid behaves under different conditions of temperature and pressure etc. An example in the plastics industry is the design of extrusion dies. Before the die can be designed, the flow properties of the polymer melt must be known, with reasonable accuracy, for different temperatures and pressures and for different rates of strain.

Rheology is "The study of the flow and deformation of matter". It crops up in all aspects of modern life. It is not by chance that the application of a gentle force to a toothpaste tube results in the delivery of just the right amount of toothpaste to the toothbrush – a great deal of analysis and experimentation has gone into making it happen that way. Similarly, when women apply a cosmetic such as lipstick, just enough remains on the lips to produce a pleasant result without smudging and smearing. Again this is no accident – the major cosmetic companies invest a great deal in rheological analysis. There are innumerable, similar examples in the oil exploration and transport industries, the plastics industry, the food processing industry – and many many more.

Everyone is familiar with Hooke’s Law and Young’s Modulus for elastic solids; i.e., stress, (s ), is proportional to strain, (e ), and the constant of proportionality is the modulus, (E).

It is obvious that the same relationship will not apply to a fluid since there is no such thing as a strain for a pure fluid – it just flows to occupy the space available to it. Some other relationship had to be found.

Sir Isaac Newton found that when a pressure was applied to a fluid it flowed at a precise rate and that the rate was proportional to the magnitude of the applied pressure. This is expressed in Newton’s Law, which states that the shear stress, ( t ), is proportional to the strain rate (the rate of change of strain with time), . In this case the constant of proportionality is called the viscosity ( m ).

                                                                                                                                                            (1)

From the last equation above, it can be seen that all that is needed is some way of obtaining the shear stress applied to the fluid and the resulting strain rate, since the ratio of these two gives the viscosity.

There are many devices available commercially which will enable this to be achieved, but, for the purposes of the current discussion , only capillary rheometry as applied to polymer melts and similar high viscosity fluids will be considered further.

The basic requirement of a capillary extrusion melt rheometer is to provide a knowledge of the rheological properties of any thermoplastic material, over as wide a range of relevant conditions as possible. This is a prerequisite towards achieving an understanding of the material’s processability.

In the capillary extrusion rheometer this is effected by heating the material under investigation in a temperature controlled chamber (the "barrel") and then forcing it through a "die" of very accurately known dimensions by means of a motor driven piston or a gear-pump. The pressure at the entry of the die is dependent on the viscosity of the melt, the geometry of the die and the piston speed. The melt viscosity is a characteristic of the material (and the effect of additives on it), the temperature and the rate of extrusion.

Repeating the process over a wide range of conditions enables a detailed "processability profile" of the material to be derived.

For capillary rheometry, the shear stress and the strain rate are given by the following equations:-

                                                                                                                                                         (2)

where D P is the pressure drop along the die, r and l are the radius and length of the die respectively.

                                                                                                                                                         (3)

where Q is the volume flow rate of the fluid, which is given by:-

where v is the piston speed and R is the radius of the piston.

Examination of these last three equations leads to the conclusion that, because r, l and R are fixed, machine parameters, all that needs to be measured is the pressure drop along the die for a series of piston speeds, i.e., :-

and,

Incidentally, combination of equations (1), (2) and (3) above leads to the well known Poiseuille’s equation, which describes the effect of the pressure drop on the volumetric flow rate for a Newtonian fluid:-

Before the advent of desk-top computers these calculations, although relatively simple to perform, were very time consuming. However, with the advantage of modern computing aids, the shear stress, strain rate and shear viscosity are calculated in fractions of a second and all the relevant flow curves can be produced with the greatest of ease.

Newtonian fluids, i.e., those that obey Newton’s Law of viscous flow, are simple to handle because the viscosity obtained is a constant at constant temperature – (if polymer melts were Newtonian then the ever popular Melt Flow Index (MFI) would be of some value). As it is, polymer melts are highly non-Newtonian, i.e., they do not obey Newton’s Law. In these materials the viscosity is not a constant, but is highly dependent on the strain rate, so the MFI is of little value because it is a one point curve measured at a value of strain rate well below that of most strain rates encountered in processing.

The only sensible thing to do is to measure the shear stress over a whole range of strain rates to obtain what is known as a "flow curve", i.e., a curve from which the viscosity (the slope of the stress/strain rate curve) can be determined for any strain rate.

Having obtained these flow curves, it is possible to suggest modifications to the simple Newtonian model so that they can be described mathematically. The simplest modification  is the "Power Law Equation"; every schoolchild knows that most curves can be straightened simply by taking logarithms. This is what the Power Law Equation crudely attempts to do. This modification is expressed mathematically as:-

Here, K is a melt consistency index, which is related to the viscosity and n is the power law index. It is obvious that for a Newtonian fluid n = 1 and K becomes m . 

Taking logarithms produces :-

which is the equation of a straight line with a slope of n and an intercept of log K. Having determined K and n it is then possible to calculate a viscosity for any strain rate for this type of non-Newtonian fluid.  An alternative model, often used to model the full flow-curve, is the Carreau Model.

Why use a Twin Bore Rheometer?

Unlike low viscosity Newtonian fluids, polymer melts and other highly viscous materials do not have a simple, linear pressure profile along the die, so it is not possible, with a single measurement, to obtain the pressure drop, DP, along the die. It is necessary to determine the die entrance and exit effects independently using two dies, one long die and a die of effectively zero length. This is where twin bore machines such as the Porpoise P7 and the new Porpoise P9 from Porpoise Viscometers come into their own - it is possible to run a test using both the long and "zero" length die at the same time and at exactly the same temperature.  The circuitry abstracts the signal from the pressure transducer at the entrance to each die and transfers it to processor of the control computer - the computer software calculates the pressure differential, i.e., the true pressure drop along the die and hence the shear stress.  Similarly, the piston speed is fed into the control computer and is used to calculate the strain rate.  The computer software analyses the data obtained for each run and calculates the values of K and n.

This use of the twin bore rheometer and the control computer has effectively changed experimental rheology from a highly specialised scientific pursuit to something that is more akin to a routine materials test.  Having said that, interpretation of the results obtained should be done with care; it is in this area that a high level of expertise is still required.

Finally, and really as an aside, there is obviously more to rheology than can be included here.  Although the shear data is of prime importance in most cases, it is possible to obtain very important data regarding the extensional properties of the melt from the shear experiment, which, in a simple case, can indicate the suitability of certain materials for different applications, e.g., melts with a high elastic component for extrusion blow moulding and melts with low elasticity for fibre spinning.  The point is that the bulk of the extensional data is obtained from the extrusion of the melt through the "zero" length die - again the computer software is included, which allows the relevant properties to be calculated.