Linear viscoelasticity measurement

The spectrum method. Since we are dealing with linear viscoelastic measurements, the following equation might provide a very general approach for an estimate of G (o) ... 

Whilst obtaining this is the ultimate goal for many rheologists, in practice it is not possible to develop such an expression. However, our mechanical analogues do allow us to develop linear constitutive equations which allow us to relate the phenomena of linear viscoelastic measurements. For a spring the relationship is straightforward. When any form of shear strain is placed on the sample the shear stress responds instantly and is proportional to the strain. The constant of proportionality is the shear modulus... 

These two mathematical Equations (4.59) and (4.60) illustrate an important feature about linear viscoelastic measurements, i.e. the central role played by the relaxation function and the compliance. These terms can be used to describe the response of a material to any deformation history. If these can be modelled in terms of the chemistry of the system the complete linear rheological response of our material can be obtained. 

The most surprising result is that such simple non-linear relaxation behaviour can give rise to such complex behaviour of the stress with time. In Figure 6.3(b) there is a peak termed a stress overshoot . This illustrates that materials following very simple rules can show very complex behaviour. The sample modelled here, it could be argued, can show both thixotropic and anti-thixotropic behaviour. One of the most frequently made non-linear viscoelastic measurements is the thixotropic loop. This involves increasing the shear rate linearly with time to a given... 

Linear viscoelastic measurements using infrared dichroism on the compatible blend polyethylene oxide) and poly(methyl methacrylate) were reported by Zawada et al. [139]. Unlike Monnerie and coworkers [127], who reported seeing only orientation in the PMMA component, and none in the PEO, Zawada et al. observed alignment in the PEO. However, since the PEO was of lower molecular weight (as was the case for Monnerie and coworkers), its relaxation timescales were substantially faster than the PMMA. This may explain the lack of any measurable orientation by Monnerie and coworkers, who studied quenched samples, since their preparation may have allowed the PEO to relax prior to testing. 

Static and dynamic linear viscoelastic measurements are used to gain insights into the relationships between cheese structure and rheological behavior. Non-linear viscoelastic measurements have been used to a relatively small degree to measure the response of cheese to large deformations. 

The linear viscoelastic measurements of the end-tethered nanocomposites reveal several features unique to these materials. Time-temperature superposi-... 

Linear viscoelastic measurements detect a low frequency plateau in the shear modulus whenever the yield stress appears. The magnitudes of both correlate... 

We will discuss in this section the variations of the viscoelastic parameters derived from linear viscoelastic measurements all these parameters may be derived from any t3rpe of measurement (relaxation or creep experiment, mechanical spectroscopy) performed in the relevant time or frequency domain. The discussion will be focused however on the complex shear modulus which is the basic function derived from isothermal frequency sweep measurements performed with modem rotary rheometers. 

A simple linear viscoelastic measurement that has become very easy to implement with the advent of modern electronics is oscillatory shear. A sinusoidal strain with angular frequency oj is applied to a sample in simple... 

Xing et al. (2000) compared five different techniques for the measurement of interfacial tension in a model polystyrene (PS)/polyamide-6 (PA-6) system at a constant temperature. The techniques include three dynamic methods (the breaking thread, the imbedded fiber retraction, and the retraction of deformed drop), one equilibrium method (the pendant drop), and a rheological method based on linear viscoelastic measurements. The advantages, the limitations, and the difficulties of each technique were discussed and compared. 

Linear viscoelastic measurements can provide surprising insight into the steady shear behavior of many polymer melts through an empiricism known as the Cox-Merz rule. According to this relation, a plot of t] = [( ) + versus fre-... 

A final consideration is that the Argon theory essentially regards yield as nucleation controlled, analogous to the stress-activated movement of dislocations in a crystal produced by the applied stress, aided by thermal fluctuations. The application of the Eyring theory, on the other hand, implies that yield is not concerned with the initiation of the deformation process, but only that the application of stress changes the rate of deformation until it equals imposed rate of change of strain. The Eyring approach is consistent with view that the deformation mechanisms are essentially present at zero stress, and are identical to those observed in linear viscoelastic measurements (site model analyses in Section 7.3.1). Here, a very low stress is applied merely to enable detection of the thermally activated process, without modification of the polymer structure. 

For the generalized linear Maxwell model (GLM) and the upper convected model (UCM), the only material parameters needed are contained in the relaxation time spectrum of the material which can be obtained from simple linear viscoelastic measurements. For the Giesekus model, one needs in addition the mobility factors which Christensen and McKinley obtained by fitting the stress-strain curves of the adhesive. The advantage of the Giesekus model was that it provided them with a better description of the stress-strain curves. This, of course, is to be expected since those curves were used to deduce the parameters of the model. 

Therefore, with the exception of the Giesekus model, the parameters for all of these constitutive equations can be deduced from the relaxation time spectrum of the material which can be obtained from the small strain linear viscoelasticity measurements alone. There are various numerical methods in the literature which allow the determination of this spectrum from measured viscoelastic master curves, such as dynamic modulus, relaxation modulus, and creep compliance. 

A few of these interrelationships can be derived from postulates that are general enough to be valid for most polymeric fluids. For example, the steady state shear viscosity and first normal stress coefficient at low shear rate can be derived from low frequency linear viscoelastic measurements ... 

The fixtures in contact with the sample, which constitute the rheometer geometry , may be either a cone and a plate or two parallel plates. For linear viscoelasticity measurements, parallel disks are nearly always used, as the loading of samples and setting of the gap are much simplified. Factors that must be taken into consideration in order to obtain rehable data are ... 

Figure 11.10 Predictions of shear stress and first normal stress growth coefficients by the CCR model of Likhtman etal. modified to include chain stretch by Graham eta/. [37] in start-up of steady shearing compared to data [53] for a 7% solution of nearly monodisperse polybutadiene (M = 350,000) at the shear rates shown, where the parameters = 4.156 -10" s,M = 51779, and Gg = (5/4) G 5 = 51770 Pa are obtained from linear viscoelastic measurements, and = 0.1 sets the rate of constraint release.The longest Rouse orientational relaxation time should be given by the theoretical relationship to =Z however, the retraction rate is artificially doubled (equivalent to taking xj, =0.5 Z T )tooffsetthe error introduced by a closure approximation and to give a better fit to the data. From Graham et al. [37].

Figure 11.12 Comparison of predictions of shear and extensional stress growth coefficients rf (f, y) and (f, e) according to the CCR model of Likhtman etal. [28] modified to include chain stretch by Graham et al. [37] with experimental data for 7% solutions in tricresylphosphate at 40 °C of nearly monodisperse polystyrenes of molecular weights (a) 2.89 million and (b) 8.42 million at the rates shown.These samples are the same as described in Figs. 11.8 and 11.9 the shear data are from Pattamaprom and Larson [36] and the extensional data from Ye et al. [52] shifted to 40 °C. The parameters Tg = 1.911 -10 s, = 270,000, and Gg = (5/4) G 5 = 8075 Pa are obtained from linear viscoelastic measurements and c = 0.1 sets the rate of constraint release. The Rouse relaxation time is set to be tj, = 0.5 Tg, as...

The distribution of relaxation time and that of retardation time are quantitatively related. Also, the result from any linear viscoelastic experiment is quantitatively related to distribution of relaxation or retardation time. Therefore, from the result of one type of linear viscoelastic measurement, the data of any other type of measurement can be calculated, provided the data are available for a wide range of time scales. For example, dynamic shear storage modulus, G (a>) and dynamic shear loss modulus, G"(co), may be calculated from relaxation modulus, G(t). The method of calculation is described in textbooks [6] and outside of scope of this book. The importance here is to recognise that the distribution of the relaxation time is related to every data of linear viscoelasticity. 

The formation of a continuous network involving carbon nanotubes was recognized by Potschke et al. using linear viscoelastic measurements made in shear, and has been explored by this author and others. The... 

Thus, the aim of linear viscoelastic measurements (for incompressible materials) is to experimentally determine the relaxation modulus G(t) or quantities equivalent to G(t). In most cases of actual linear viscoelastic measurements, a sinusoidal shear strain y(t) =yosin(Bt (co is the angular frequency and is equal to 2rr/ with / being the frequency in the unit of Hertz) with the amplitude yo 1 is applied to a material. From eqn [22], the resulting shear stress is expressed as... 

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