Tanner equation

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

Equation 12.2-1 has been semiquantitatively successful in predicting extrudate swelling (25). However, White and Roman (27) have shown experimentally with a number of polymers that D/Dq is not a function of Sr only. Furthermore, they demonstrated that the success of the Tanner equation depends on the method of measurement of D/Do. 

The original Phan Thien Tanner equation was written using simultaneously both modifications Gordon Schowalter derivative and segment kinetics term. The segment kinetics term (exponential form) enables a more... 

The ability of the Phan Thien Tanner equation and related models for the prediction of data in shear and elongation has been investigated. Attention has been focused on special simplified cases of the original equation which enable the imderstanding of the influence of each parameter. 

At least, using the complete Phan Thien Tanner equation, with non-affine motion and modified kinetics enables a correct description of the data in shear and in elongation. However, the parameters that can be determined for this model are bound to be some compromise. This is necessary in order to minimize the deviation to the Lodge-Meissner rule, due to the use of the Gordon-Schowalter derivative. This is also required to give adequate description of both the shear and uniaxial elongational behaviour. Additional undesirable phenomena in some flows have also been pointed out such as oscillations in transient flows. 

Though the Wagner and Phan Thien Tanner equations seem to give adequate description of the observed behaviour either in shear or in uniaxial elongation, it is worth mentioning some peculiarities and key points that should keep the attention of the user to avoid misleading conclusions. 

In its general form, the Phan Thien Tanner equation includes two different contributions of strain to the loss of network junctions, through the use of a particular convected derivative which materializes some slip of the junctions and through the use of stress-dependent rates of creation and destruction of junctions. The use of the Gordon-Schowalter derivative brings some improvement in shear and a second normal stress is predicted, whereas the... 

Table 7 gives a summary of qualitative performances and problems encountered for simple shear and uniaxial elongational flows, using the Wagner and the Phan Thien Tanner equations or more simple models as special cases of the former. Additional information may also be found in papers by Tanner [46, 64]. All equations presented hereafter can be cast in the form of a linear Maxwell model in the small strain limit and therefore are suitable for the description of results of the linear viscoelasticity in the terminal zone of polymer melts. 

Table 7 Wagner and Phein Thien Tanner equations- Problems in simple shear and uniaxial extension.

Besides measurements, there are potential benefits in the observation of other nuclides for diffusion measurements. There may be fewer problems with resonance overlap of different species or less interference from solvent or impurity resonances, for example, or the compound of interest may simply lack hydrogen as is often the case with counter-ions. In this case, one should be aware that optimum parameters for measurement may not be suitable for nuclides of lower magnetogyric ratio, 7. This is because the total effective gradient strengths employed also depend on this parameter and, as can be seen from the Stejskel-Tanner equation (Eq. (9.6)), the degree of attenuation of resonance intensity will be reduced as 7 becomes smaller. This is illustrated in Fig. 9.23 where the decay profiles for four different nuclides are shown, having been calculated with identical diffusion parameters but with the appropriate 7... 

The diffusion experiments have been performed on a Bruker Avance 500 NMR spectrometer operating at a Larmor frequency of 500 MHz for protons using a DIFF 30 probehead generating a maximum gradient strength of 12 T/m. The gradient pulse duration 8 and diffusion time A have been adjusted between 0.7 to 1A ms for 8 and 10 to 15 ms for A respectively. The narrow distribution of the molecular weight distributiOTi permits the determination of the diffusion coefficients by a linear fit to the Stejskal-Tanner equation [23]. 

Figure 2.3 Typical experimental data from a PFG-NMR diffusion experiment with the fitted Stejskal-Tanner equation fEq. 2.5 shown as the linel.

Successive experiments varying the A, or more commonly tire gradient strength, provide the data points for a non-linear least squares fit of Equation 2.5 to solve for the diffusion coefficient. Typical experimental data with the fitted Stejskal-Tanner equation (Eq. 2.5 is shown in Figure 2.3. To produce a gradient pulse of opposite sign to the first, the polarity of the gradient coil can be physically reversed. However, this does not produce the desired level of reproducibility of the pulses, and modern NMR systems use the tt-RF pulse to cause the same effect. 

The rheological responses measured at low values of strain better reflect the effects of the blend structure. For multiphase systems, there are serious disagreements between the predictions of continuum-based theories and experiments, that is, between the small and large deformation behavior. For example, the identity of zero-deformation rate dynamic and steady state viscosity is seldom found, and so is the Trouton rule. Similarly, the derived by Cogswell, relationship between the extensional viscosity and the capillary entrance pressure drop, and derived by Tanner equation for calculating the first normal stress difference from the extrudate swell, are rarely valid. 

Stejskal and Tanner devised a quantitative relationship of the echo attenuation as a function of gradient strength, the Stejskal-Tanner equation [71] ... 

For self-diffusion NMR measnrements, ME samples are filled into standard NMR tubes. Tetramethylsilane dissolved in D2O and sealed into a capillary is commonly used as reference. H-NMR spectra of the pure components and the MEs are acquired and Fourier transformed. For analysis, peaks of the ME formulation can be assigned to the individnal components by comparison to spectra obtained from the pnre ME components. To obtain the data for exponential modelling using the Stejskal-Tanner equation (Equation 9.4), one or more peaks of the component are either integrated or analysis is performed on the peak height. Selfdiffusion coefficients ranging from about 10 to 10 m s" are typically observed in solutions at ambient temperature [73]. 

Theories relating die swell and normal-stress difference are fairly successful. The phenomenon itself is a complicated one even for Newtonian liquids. The Tanner equation [34] captures the essential behavior for polymer melts ... 

The extent of extrudate or die swell of a viscoelastic liquid emerging from a long tube, see figure 3, is given by the empirical Tanner equation [20]... 

Van Lokeren and co-workers reported a study of polymer mixtures in solution using pulsed field-gradient spin eeho NMR. They proposed a quantification approach in which the fraetions of different components were obtained explicitly including relaxation effeets in the Stejskal-Tanner equation. 

In this approach the contribution to the extra stress of the matrix and the rod-matrix interaction is captured using a multi-mode viscoelastic constitutive relation. For the model predictions in the paper, we chose to use the Phan-Thien Tanner equation (PTT) [13]. The ability to indirectly capture the rod-matrix interaction is based on the presence of the fiber retarding the long relaxation... 

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