Viscoelastic loss functions

A summary of analytic expressions obtained in this manner for all the viscoelastic functions is presented in Table 4 and 5 for the linear and cubic arrays. The well-known phenomenological analogy (8) between dynamic compliance and dielectric permittivity allows the formal use of Eqs. (T 5), (T 6), and (T 11), (T 12) for the dielectric constant, e (co), and loss, e"(co), of the linear and cubic arrays, respectively (see Table 6). The derivations of these equations are elaborated in the next section and certain molecular weight trends are discussed. 

For a specific food, magnitudes of G and G are influenced by frequency, temperature, and strain. For strain values within the linear range of deformation, G and G" are independent of strain. The loss tangent, is the ratio of the energy dissipated to that stored per cycle of deformation. These viscoelastic functions have been found to play important roles in the rheology of structured polysaccharides. One can also employ notation using complex variables and define a complex modulus G (o) ... 

The appropriate viscoelastic functions are the dynamic rheological properties (storage modulus G and the loss modulus G", and the dynamic viscosities f and T ") extrapolated to infinite dilution and are called the intrinsic dynamic rheological properties ... 

When CO 0, the loss compliance function for viscoelastic solids is given by... 

Accordingly, the loss compliance function presents a maximum in the frequency domain at lower frequency than the loss relaxation modulus. This behavior is illustrated in Figure 8.18, where the complex relaxation modulus, the complex creep compliance function, and the loss tan 8 for a viscoelastic system with a single relaxation time are plotted. Similar arguments applied to a minimum in tan 8 lead to the inequalities... 

The curves showing the frequency dependence of loss functions [tan 5, G"(g)), or / (to)] permit the detection in the frequency domain, at temperatures just slightly above the glass transition temperature, of a prominent absorption or a process. The unavailability of experimental devices to measure mechanical viscoelastic functions at high frequencies impedes the detection of a fast process or P relaxation in the high frequency region. This latter process is usually detected in the glassy state at low frequencies. 

We note that when the losses are too large, free oscillations cannot be excited. For this reason it is compulsory to use the elastic auxiliary element in order to get information on the viscoelastic functions. A stiff elastic element, with constant k, can be added to reduce the loss. When the loss of the system is sufficiently small, the discrepancies between the results obtained from the former theory and the solution based on the classical second-order differential equation [see Eq. (7.49), for example]... 

It has been remarked that time (frequency) - temperature reduced data on carbon black filled rubbers exhibit increased scatter compared to similar data on unfilled polymers. Payne (102) ascribes this to the effects of secondary aggregation. Possibly related to this are the recent observations of Adicoff and Lepie (174) who show that the WLF shift factors of filled rubbers giving the best fit are slightly different for the storage and loss moduli and that they are dependent on strain. Use of different shift factors for the various viscoelastic functions is not justified theoretically and choice of a single, mean ar-funetion is preferred as an approximation. The result, of course, is increased scatter of the experimental points of the master curve. This effect is small for carbon black... 

Thus, once the four parameters of Eq 7.42 are known, the relaxation spectrum, and then any linear viscoelastic function can be calculated. For example, the experimental data of the dynamic storage and loss shear moduli, respectively G and G , or the linear viscoelastic stress growth function in shear or uniaxial elongation can be computed from the dependencies [Utracki and Schlund, 1987] ... 

The dynamic tests at small amplitude in parallel plates or cone-and-plate geometry are simple and reproducible. From the experimental values of storage and loss shear moduli, G and G", respectively, first the yield stress ought to be extracted and then the characteristic four material parameters in Eq. (2.13), rjo, r, mi, and m2, might be calculated. Next, knowing these parameters one may calculate the Gross frequency relaxation spectrum (see Eqs. (2.31) and (2.32)) and then other linear viscoelastic functions. 

A fundamental quantity relating the basic viscoelastic functions (i.e., storage, loss modulus and compliance, shear viscosity) is the monomeric friction coefficient, which is a measure of the frictional resistence per monomer unit encountered by a moving chain segment. This co-... 

The loss tangent determines such macroscopic physical properties as the damping of free vibrations, the attenuation of propagated waves, and the frequency width of a resonance response. It can often be more conveniently measured than any other viscoelastic function, by observations of these phenomena, and is of considerable practical interest. It is less susceptible of direct theoretical interpretation than the other functions, however. 

The appropriate viscoelastic functions are ordinarily the complex modulus or the complex viscosity, and the corresponding quantities extrapolated to infinite dilution are the intrinsic storage and loss moduli... 

Integration over equations 41 and 42 with appropriate limits by equations 19, 23, and 24 of Chapter 3 and addition of vRT/2 in the first two cases provides the viscoelastic functions G(t), G, and G" for the Chompff-Duiser theory. The corresponding curve for the loss compliance J" is included in Fig. 10-7. It extends farther to the low-frequency side than the others, as would be expected from the additional slow relaxation mechanisms. 

The differences in spectral shape are of course reflected in the other viscoelastic functions, especially the loss tangent, which is always the most sensitive. The loss tangents for poly(/t-butyl methacrylate) and three of its solutions are plotted in Fig. 17-11 with the frequency scale reduced in the same way as the abscissa in Fig. 17-10. They show some degree of broadening with increasing dilution, but their maxima remain considerably above the theoretical value of 1.0 which would be predicted throughout the rq ition zone by the flexible chain theory in its simple form. Similar behavior has bwn reported for concentrated solutions of polystyrene in tricresyl phosphate, by Wasser and Kurath. ... 

We finally observe that delayed response phenomena akin to creep and relaxation occur in other areas of Mechanics and Physics, and are attributable to the same fundamental cause, namely (usually internal) frictional losses. The mathematical techniques used for analyzing such phenomena are similar to those used in analyzing the properties of the viscoelastic functions. Such a close analogy exists between certain phenomena in the theory of Dielectrics and Linear Viscoelasticity, as emphasized by Gross (1953). 

Rheometric Scientific markets several devices designed for characterizing viscoelastic fluids. These instmments measure the response of a Hquid to sinusoidal oscillatory motion to determine dynamic viscosity as well as storage and loss moduH. The Rheometric Scientific line includes a fluids spectrometer (RFS-II), a dynamic spectrometer (RDS-7700 series II), and a mechanical spectrometer (RMS-800). The fluids spectrometer is designed for fairly low viscosity materials. The dynamic spectrometer can be used to test soHds, melts, and Hquids at frequencies from 10 to 500 rad/s and as a function of strain ampHtude and temperature. It is a stripped down version of the extremely versatile mechanical spectrometer, which is both a dynamic viscometer and a dynamic mechanical testing device. The RMS-800 can carry out measurements under rotational shear, oscillatory shear, torsional motion, and tension compression, as well as normal stress measurements. Step strain, creep, and creep recovery modes are also available. It is used on a wide range of materials, including adhesives, pastes, mbber, and plastics. 

CR 3nd tp are the contributions from chain recoiling and interfacial dynamics (i.e. drag forces and disentanglement), respectively, and / ve is the viscoelastic loss function which has interfacial and bulk parts. / is a characteristic length of the viscoelastic medium, t is the contact time and n is the chain architecture factor. Fig. 21 illustrates the proposed rate dependency of adhesion energy. 

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