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An introduction to the rheology of polymers, with simple math Designed for practicing scientists and engineers interested in polymer rheology science, education.
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Rising-bubble techniques can also be included in this third category. For all the shop-floor viscometers, great care must be exercised in applying formulae designed for Newtonian liquids to the non-Newtonian case. For such instruments, the means of inducing the flow are two-fold: one can either drive one member and measure the resulting couple or else apply a couple and measure the subsequent rotation rate.

Both methods were well established before the first World War, the former being introduced by Couette in and the latter by Searle in There are two ways that the rotation can be applied and the couple measured: the first is to drive one member and measure the couple on the same member, whilst the other method is to drive one member and measure the couple on the other. For couple-driven instruments, the couple is applied to one member and its rate of rotation is measured. In Searle's original design, the couple was applied with weights and pulleys. In modem developments, such as in the Deer constant-stress instrument, an electrical drag-cup motor is used to produce the couple.

The couples that can be applied by the commercial constant stress instruments are in the range l fo f 6 to l o f 2 Nm; the shear rates that can be measured are in the range to lo3 s ' , depending of course on the physical dimensions of the instruments and the viscosity of the material. The lowest shear rates in this range are equivalent to one complete revolution every two years; nevertheless it is often possible to take steady-state measurements in less than an hour. As with all viscometers, it is important to check the calibration and zeroing from time to time using calibrated Newtonian oils, with viscosities within the range of those being measured.

Specifically, if the radii of the outer and inner cylinders are ro and r , , respectively, and the angular velocity of the inner is Q,, the other being stationary the shear rate i. For the gap to be classed as "narrow" and the above approximation to be valid to within a few percent, the ratio of r, to ro must be greater than 0.

If the couple on the cylinders is C , the shear stress in the liquid is given by. This would be the real immersed length, 1, if there were no end effects. However, end effects are likely to occur if due consideration is not given to the different shearing conditions which may exist in any liquid covering the ends of the cylinders. One way to proceed is to carry out experiments at various immersed lengths, I, keeping the rotational rate constant.

The extrapolation of a plot of C against I then gives the correction which must be added to the real immersed length to provide the value of the effective immersed length L. In practice, most commercial viscometer manufacturers arrange the dimensions of the cylinders such that the ratio of the depth of liquid to the gap between the cylinders is in excess of Under these circumstances the end correction is negligible. The interaction of one end of the cylinder with the bottom of the containing outer cylinder is often minimized by having a recess in the bottom of the inner cylinder so that air is entrapped when the viscometer is filled, prior to making measurements.

Alternatively, the shape of the end of the cylinder can be chosen as a cone. In operation, the tip of the cone just touches the bottom of the outer cylinder container. This arrangement is called the Mooney system, after its inventor. For these reasons, in many commercial viscometers the ratio of the cylinder radii is less than that stated in g2.

This is a nontrivial operation and has been studied in detail by Krieger and Maron The shear rate in the liquid at the inner cylinder is then given by. Note that the shear rate is now dependent on the properties of the test liquid, unlike the narrow-gap instrument. The shear stress in the liquid at the inner cylinder is given by. The value of n can be determined by plotting C versus 0, on a double-logarithmic basis and taking the slope at the value of Ol under consideration.

Ratio of actual eqn. The error involved in employing the narrow-gap approximation instead of the wide-gap expression, eqn. The lower limit of shear rate achievable in a rotational viscometer is obviously governed by the drive system. The upper limit, however, is usually controlled by the test liquid. One limit is the occurrence of viscous heating of such a degree that reliable correction cannot be made.

However, there are other possible limitations. Depending on which of the cylinders is rotating, at a critical speed the simple circumferential streamline flow breaks down, either with the appearance of steady Taylor vortices or turbulence.

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Since both of these flows require more energy than streamline flow, the viscosity of the liquid apparently increases. In practical terms, for most commercial viscometers, it is advisable to consider the possibility of such disturbances occurring if the viscosity to be measured is less than about 10 mPa. Rate L Couple measuring device. The cone-and-plate viscometer. Cross-sectional diagram of one possible configuration, viz. The inset shows the form of truncation used in many instruments. For a power-law liquid, the values of the shear rate and shear stress in the liquid at the rotating cylinder of radius r, are then given by.

These equations are applicable to viscometers of the Brookfield type in which a rotating bob is immersed in a beaker of liquid. The technique is restricted to moderately low shear rates: 0. The shear rate in the liquid is given by. Note that the shear rate does not depend on the properties of the liquid. The shear stress measured via the couple C on the cone is given by. How- ever, under these circumstances, 'secondary flows' may arise see, for example, Walters The secondary flow absorbs extra energy, thus increasing the couple, which the unwary may mistakenly associate with shear-thickening.

Cheng has provided an empirical formula which goes some way towards dealing with the problem. All cone-and-plate instruments allow the cone to be moved away from the plate to facilitate sample changing. It is very important that the cone and plate be reset so that the tip of the cone lies in the surface of the plate. To avoid error in contacting the cone tip which might become worn and the plate which might become indented , the cone is often truncated by a small amount. In this case, it is necessary to set the virtual tip in the surface of the plate as shown in Fig. A truncated cone also facilitates tests on suspensions.

It is this shear rate that finds its way into the interpretation of experimental data for torsional flow. It can be shown Walters , p. Cross-sectional diagram of the torsional parallel-plate viscometer. It will be noticed from eqn. In the torsional-balance rheometer, an adaptation of the parallel-plate viscometer see Chapter 4 for the details , shear rates in the l o 4 to l o 5 s-' range have been attained. If, however, the viscosity varies with shear rate the situation is more complex. Progress can be made by concentrating on flow near the wall.

Analysis shows cf. Walters , Chapter 5 that for a non-Newtonian liquid, the shear rate at the wall is modified to. The bracketed term in 2. Then finally. Since n can be as low as 0. Cutaway diagram of laminar Newtonian flow in a straight circular capillary tube. These arise from the following sources for all types of liquid:. Formulae exist which account for these effects for Newtonian liquids, i and iii being associated with the names of Hagenbach and Couette see, for example Kestin et al.

However, these effects are small if the ratio of tube length to radius is or more. The main end effects can be avoided if at various points on the tube wall, well away from the ends, the pressures are measured by holes connected to absolute or differential pressure transducers. Any error arising from the flow of the liquid past the holes in the tube wall see It is not often convenient to drill pressure-tappings, and a lengthy experimental programme may then be necessary to determine the type- i errors in terms of an equivalent pressure-drop and type- ii errors in terms of an extra length of tube.

The experiments required can be deduced from the theoretical treatment of Kestin et al. If the liquid is highly elastic, an additional entrance and exit pressure drop arises from the elasticity. The so-called Bagley correction then allows an estimate of the elastic properties to be calculated. It is also used to provide an estimate of the extensional viscosity of the liquid see From this equation we see the effect of changes in such variables as pipe radius.

For Newtonian liquids, the pressure drop for a given flow rate is proportional to the fourth power of the radius, but thls is changed if the liquid is shear-thinning. This is clearly important in any scale-up of pipe flow from pilot plant to factory operation. Viscosity [Chap. The velocity profiles for the laminar flow of power-law liquids in a straight circular pipe, calculated for the same volumetric throughput. Note the increase in the wall shear rate and the increasingly plug-like nature of the flow as n decreases. The velocity profile in pipe flow is parabolic for Newtonian liquids.

For power-law liquids this is modified to. We see the increasing plug-like nature of the flow with, effectively, only a thin layer near the wall being sheared. This has important consequences in heat-transfer applications, where heating or cooling is applied to the liquid from the outside of the pipe. The overall heat transfer is partly controlled by the shear rate in the liquid near the pipe wall. It is a two-dimensional analogue of capillary flow. The governing equations for slit flow are cf.

Walters , Chapter 5. Slit flow forms the basis of the viscometer version of the Lodge stressmeter. The stressmeter is described more fully in Chapter 4. The instrument has the advantage that shear rates in excess of lo6 s-' can be achieved with little interference from viscous heating. Of the viscometers described in this chapter, the capillary viscometer and the concentric-cylinder viscometer are those most conveniently adapted for such a purpose.

For the former, for example, the capillary can be installed directly in series with the flow: the method has attractive features, but its successful application to non-Newtonian liquids is non-trivial. Care must be taken with the on-line concentric-cylinder apparatus, since the interpretation of data from the resulting helical flow is not easy. Other on-line methods involve obstacles in the flow channel: for example, a float in a vertical tapered tube, as in the Rotameter, will arrive at an equilibrium position in the tube depending on the precise geometry, the rate of flow, the viscosity and the weight of the obstacle.

The parallel-plate viscometer has also been adapted for on-line measurement see, for example, Noltingk The word 'viscoelastic' means the simultaneous existence of viscous and elastic properties in a material cf. It is not unreasonable to assume that all real materials are viscoelastic, i. As was pointed out in the Introduction, the particular response of a sample in a given experiment depends on the time-scale of the experiment in relation to a natural time of the material. Thus, if the experiment is relatively slow, the sample will appear to be viscous rather than elastic, whereas, if the experiment is relatively fast, it will appear to be elastic rather than viscous.

At intermediate time-scales mixed, i. The concept of a natural time of a material will be referred to again later in this chapter. However, a little more needs to be said about the assumption of viscoelasticity as a universal phenomenon. It is not a generally-held assumption and would be difficult to prove unequivocally. Nevertheless, experience has shown that it is preferable to assume that all real materials are viscoelastic rather than that some are not.

Given this assumption, it is then incorrect to say that a liquid is Newtonian or that a solid is Hookean. On the other hand, it would be quite correct to say that such-and-such a material shows Newtonian, or Hookean, behaviour in a given situation. This leaves room for ascribing other types of behaviour to the material in other circumstances. However, most rheologists still refer to certain classes of liquid rather than their behaviour as being Newtonian and to certain classes of solid as being Hookean, even when they know that these materials can be made to deviate from the model behaviours.

Indeed, it is done in this book! Old habits die hard. However, it is considered more important that an introductory text should point out that such inconsistencies exist in the literature rather than try to maintain a purist approach. For many years, much labour has been expended in the determination of the linear viscoelastic response of materials. There are many reasons for this see, for example, Walters , p. First there is the possibility of elucidating the molecular structure of materials from their linear- viscoelastic response.

Secondly, the material parameters and functions measured in the relevant experiments sometimes prove to be useful in the quality-control of industrial products. Thirdly, a background in linear viscoelasticity is helpful before proceeding to the much more difficult subject of nun-linear viscoelasticity cf. Measurements of this function on low-viscosity "Newtonian" lubricants at high shear rates were made difficult by such factors as viscous heating, and this led to a search for an analogy between shear viscosity and the correspond- ing dynamic viscosity determined under linear viscoelastic conditions, the argument being that the latter viscosity was easier to measure see, for example, Dyson Many books on rheology and rheometry have sections on linear viscoelasticity.

We recommend the text by Ferry which contains a wealth of information and an extensive list of references. Mathematical aspects of the subject are also well covered by Gross and Staverman and Schwarzl The development of the mathematical theory of linear viscoelasticity is based on a "superposition principle".

This implies that the response e. So, for example, doubling the stress will double the strain. In the linear theory of viscoelas- ticity, the differential equations are linear. Also, the coefficients of the time differentials are constant. These constants are material parameters, such as viscosity coefficient and rigidity modulus, and they are not allowed to change with changes in variables such as strain or strain rate.

Further, the time derivatives are ordinary partial derivatives. This restriction has the consequence that the linear theory is applicable only to small changes in the variables. We can now write down a general differential equation for linear viscoelasticity as follows:. Note that for simplicity we have written 3. However, we emphasise that other types of deformation could be included without difficulty, with the stress and strain referring to that particular deformation process. Mathematically, this means that we could replace the scalar variables a and y by their tensor generalizations.

For example, a could be replaced by the stress tensor a i j. We now consider some important special cases of eqn. If Po is the only non-zero parameter, we have. If Pl is the only non-zero parameter, we have. This represents Newtonian viscous flow, the constant P, being the coefficient of viscosity. It is called the 'Kelvin model', although the name 'Voigt' is also used. It has the dimension of time and controls the rate of growth of strain y following the imposition of the stress a. Figure 3. The difference between the two models is that, whereas the. Hooke model reaches its final value of strain "instantaneously", in the Kelvin model the strain is retarded.

The time constant 7, is accordingly called the 'retardation time'. The word instantaneously is put in quotation marks because in practice the strain could not possibly grow in zero time even in a perfectly elastic solid, because the stress wave travels at the speed of sound, thus giving rise to a delay.

It is useful at this stage to introduce "mechanical models", which provide a popular method of describing linear viscoelastic behaviour. These one-dimensional mechanical models consist of springs and dashpots so arranged, in parallel or in series, that the overall system behaves analogously to a real material, although the elements themselves may have no direct analogues in the actual material. The correspondence between the behaviour of a model and a real material can be achieved if the differential equation relating force, extension and time for the model is the same as that relating stress, strain and time for the material, i.

In mechanical models, Hookean deformation is represented by a spring i. The analogous rheological equations for the spring and the dashpot are 3. The behaviour of more complicated materials is described by connecting the basic elements in series or in parallel. The Kelvin model results from a parallel combination of a spring and a dashpot Fig. A requirement on the interpretation of this and all similar diagrams is that the horizontal connectors remain parallel at all times.

Hence the extension strain in the spring is at all times equal to the extension strain in the dashpot. Then it is possible to set up a balance equation for the forces stresses acting on a connector. The last step is to write the resulting equation in terms of stresses and strains. Hence, for the Kelvin model the total stress a is equal to the sum of the stresses in each element. The Kelvin and Maxwell models. This is identical to eqn. The differential equation for the model is obtained by making a, and PI the only non-zero material parameters, so that.

Hence the stress relaxes exponentially from its equilibrium value to zero see Fig. The rate constant 7, is called the 'relaxation time'. Maxwell on the introduction of the concept of viscoelasticity in a fluid. Linear viscoelasticity [Chap. The pictorial Maxwell model is a spring connected in series with a dashpot see Fig. In this case, the strains, or equally strain-rates, are additive; hence the total rate of shear j.

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This equation is the same as eqn. The next level of complexity in the linear viscoelastic scheme is to make three of the material parameters of eqn. If a,, P, and P, are taken to be non-zero we have the "Jeffreys model". In the present notation, the equation is. With a suitable choice of the three model parameters it is possible to construct two alternative spring-dashpot models which correspond to the same mechanical behaviour as eqn.

One is a simple extension of the Kelvin model and the other a simple extension of the Maxwell model as shown in Fig. We note with interest that an equation of the form 3. The relaxation spectrum. The values of the constants of the elements are given in terms of the three material parameters of the model eqn. Finally, in this preliminary discussion of the successive build-up of model complexity, we draw attention to the so-called "Burgers model".

This involves four simple elements and takes the mechanically-equivalent forms shown in Fig. In terms of the parameters of the Maxwell-type representation Fig. In this equation the As are time constants, the symbol A being almost as common as T in the rheological literature. It is certainly possible to envisage more complicated models than those already introduced, but Roscoe showed that all models, irrespective of their complex- ity, can be reduced to two canonical forms. These are usually taken to be the generalized Kelvin model and the generalized Maxwell model Fig. The generalized Maxwell model may have a finite number or an enumerable infinity of Maxwell elements, each with a different relaxation time.

Linear uiscoelasticity [Chap. By a suitable choice of the model parameters, the canonical forms themselves can be shown to be mechanically equivalent and Alfrey has given methods for computing the parameters of one canonical form from those of the other. In the same paper, Alfrey also showed how a linear differential equation can be obtained for either of the canonical forms and vice versa. In other words, the three methods of representing viscoelastic behaviour the differential equation 3.

In order to generalize from an enumerable infinity to a continuous distribution of relaxation times, we choose to start with the simple Maxwell model, whose be- haviour is characterized by the differential equation 3. Considering next, a number, n, of discrete Maxwell elements connected in parallel as in Fig. The theoretical extension to a continuous distribution of relaxation times can be carried out in a number of ways. For example, we may proceed as follows.

The "distribution function of relaxation times" or "relaxation spectrum" N r may be defined such that N r d r represents the contributions to the total viscosity of all the Maxwell elements with relaxation times lying between r and r d r. We remark that we could have immediately written down an equation like 3.

It is also possible to proceed from eqn. The relationships between these functions are. In a slow steady motion which has been in existence indefinitely i. The variable 5 is the one which represents the time-scale of the rheological history. It is also easy to show from eqns. We see from eqn. Thus, the equations in 3. It is also of interest to note that q0 is equal to the area under the N r spectrum, whilst it is equal to the first moment of the H r spectrum.

It is instructive to discuss the response of viscoelastic materials to a small-ampli- tude oscillatory shear, since this is a popular deformation mode for investigating linear viscoelastic behaviour. The corresponding strain rate is given by. G' is also called the dynamic rigidity. If we now consider, for the purpose of illustration, the special case of the Maxwell model given by eqn. To some readers, the use of the complex quantity exp iwt to represent oscilla- tory motion may be unfamiliar. The alternative procedure is to use the more obvious wave-forms represented by the sine and cosine functions, and we now illustrate the procedure for the simple Maxwell model.

The part of the stress in phase with the applied strain is obtained by putting sinwt equal to zero and is written G'y. It is conventional to plot results of oscillatory tests in terms of the dynamic viscosity qf and the dynamic rigidity G'. The changes in these functions are virtually complete in two decades of frequency.

These two decades mark the viscoelastic zone. Also, in the many decades of frequency that are, in principle, accessible on the low frequency side of the relaxation region, the model displays a f - viscous response G 0. In contrast, at high frequencies, the response is elastic - q' 0. From Fig. In the literature, other methods of characterizing linear viscoelastic behaviour are to be found. By definition. The second alternative method of characterizing linear viscoelastic response is to plot G' and the 'loss angle' 6.

In t h s method, it is assumed that for an applied oscillatory strain given by eqn. At high values of the frequency, the response, as already noted, is that of an elastic solid. In these conditions the stress is in phase with the applied strain. On the other hand, at very low frequencies, where the response is that of a viscous liquid, the stress is 90 O ahead of the strain.

The change from elastic to viscous behaviour takes place over about two decades of frequency. T h s latter observation has already been noted in connection with Fig. Variation of the normalized storage modulus and phase angle with normalized frequency. Note that although the stress is 90" in advance of the shear strain for the viscous liquid, it is in phase with the rate of shear. In previous sections we have introduced a number of different functions which can all be used to characterize linear viscoelastic behaviour. These range from complex moduli to relaxation function and spectra.

They are not independent, of course, and we have already given mathematical relationships between some of the functions. For example, eqn. Equation 3. There is nothing sophisticated, therefore, in determining one viscoelastic function from another: although thls is a statement "in principle", and much work has been carried out on the non-trivial problem of inverting transforms when experimental data are available only over a limited range of the variables like frequency of oscillation.

The general problem of determining one viscoelastic function from another was discussed in detail by Gross and practical methods are dealt with by, for example, Schwarzl and Struik Two different types of method are available to determine linear viscoelastic behaviour: static and dynamic. Static tests involve the imposition of a step change in stress or strain and the observation of the subsequent development in time of the strain or stress. Dynamic tests involve the application of a harmonically varying strain. In this section we shall be concerned with the main methods in the above classification.

Attention will be focussed on the principles of the selected methods and none will be described in detail. The interested reader is referred to the detailed texts of Walters and Whorlow for further information. An important point to remember is that, whatever the method adopted, the experimenter must check that measurements are made in the linear range; otherwise the results will be dependent on experimental details and will not be unique to the material. The test for linearity is to check that the computed viscoelastic functions are independent of the magnitude of the stresses and strains applied.

In theory, the input stress or strain, whether it is an increase or a decrease, is considered to be applied instantaneously. This cannot be true in practice, because of inertia in the loading and measuring systems and the delay in transmitting the signal across the test sample, determined by the speed of sound. As a general rule, the time required for the input signal to reach its steady value must be short compared to the time over which the ultimate varying output is to be recorded.

This usually limits the methods to materials which have relaxation times of at least a few seconds. A technique for estimating whether apparatus inertia is influencing results is to deliberately change the inertia, by. The strain comprises three regions: instantaneous 0 to y, ; retardation y, to 7, ; constant rate y , to y,. In linear behaviour the instantaneous strains on loading and unloading are equal and the ratio of stress to instantaneous strain is independent of stress; the constant-rate strain y, to y, is not recovered.

In linear behaviour the instantaneous changes of stress from 0 to a, and a, to a, are equal and the ratio of instantaneous stress to strain is independent of strain. The basic apparatus for static tests is simple. Once the shape and means of holding the specimen have been decided upon, it is necessary to apply the input signal and measure, and record, the output.

It is easier to measure strain, or deformation, than stress. Hence, creep tests have been much more common than relaxation tests. The geometry chosen for static tests depends largely on the material to be tested. For solid-like materials, it is usually not difficult to fashion a long slender specimen for a tensile or torsional experiment.

Liquid-like material can be tested in simple shear with the concentric-cylinder and cone- and-plate geometries and constant-stress rheometers are commercially available for carrying out creep tests in simple shear. Plazek has carried out extensive experiments on the creep testing of polymers over wide ranges of time and temperature.

An Introduction to Rheology | Rheology | Viscoelasticity

With modern instruments it is now possible to display automatically the components of the modulus as functions of frequency. A general advantage of oscillatory tests is that a single instrument can cover a very wide frequency range. This is important if the material has a broad spectrum of relaxation times. Typically, the frequency range is from l o p 3 to lo3 s-'. Hence a time spectrum from about lo3 to l o p 3 s can be covered.

The specimen is positioned between the input motion and the output stress. The lower relaxation time limit of oscillatory methods can be extended by wave-propagation methods see 5 3. The conventional oscillatory methods involve the application of either free or forced oscillatory strains in conventional tensile and shear geometries. An advantage possessed by the free vibration technique is that an oscillator is not required and the equipment can be fairly simple. On the other hand, the frequency range available is no more than two decades.

The reason for this is that a change of frequency relies on a change in moment of inertia of the vibrating system and the scope for thls is limited. The method is readily adaptable to torsional deformation with solid-like materials. The wide frequency range quoted above is achieved with forced oscillations.

We show in Fig. The bottom member under- goes forced harmonic oscillations about its axis and this motion is transmitted through the test material to the top member, the motion of which is constrained by a torsion bar. The relevant measurements are the amplitude ratio of the motions of the two members and the associated phase lag. From this information it is relatively f simple to determine the dynamic viscosity 7' and the dynamic rigidity G , measured as functions of the imposed frequency see Walters for the details of this and related techniques.

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Kolsky has dealt with the testing of solids, Ferry has reviewed the situation as regards polymers and Harrison has covered liquids. The overall topic is usefully summarized by Whorlow Basically, the waves are generated at a surface of the specimen which is in contact with the wave generator and the evaluation of the viscoelastic functions requires the measurement of the velocity and the attenuation through the specimen. One significant advantage of wave-propagation methods is that they can be adapted to high frequency studies: they have been commonly used in the kHz region and higher, even up to a few hundred GHz.

Thls is invaluable when studying liquids which behave in a Newtonian manner in other types of rheometer. Such liquids include, as a general rule, those with a molecular weight below lo3. They include most of the non-polymeric liquids. Barlow and Lamb have made significant contri- butions in this area see, for example, Barlow et al. A relatively new group of instruments for measuring viscoelastic behaviour is based on a different principle.

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