Frequency dependence elastic moduli

Techniques for measuring the complex sound speeds and moduli of polymers are described in the section on test methods. The data shows that the real and imaginary components of the elastic moduli are frequency dependent. The frequency dependence is strongest for materials with high values of the loss factor r. Materials with frequency-dependent elastic moduli are called dispersive, and measurements and theory show that sound absorption mechanisms lead to dispersion. The real and imaginary part of an elastic modulus are related by the Kramers-Kronig relations, which are presented in the next section. 

Kramers-Kronig Relations. The Kramers-Kronig (KK) relations are derived from the basic causality condition that the output strain cannot precede the input stress in any physical material (13-15). These relations apply to the complex, frequency-dependent elastic moduli of any material, and relate the real and imaginary components of the modulus. For example, for the complex shear modulus, G (co) = G (co) + iG" co), the Kramers-Kronig relations are... 

Fig. 6 Frequency-dependent elastic modulus G and viscous modulus G" of solution of glycerol 85% (A), Sterocoll FD 1% (B), Sterocoll D 1% (C). Open symbols were obtained with MPT, solid symbols with mechanical rheology...

In a frequency sweep, measurements are made over a range of oscillation frequencies at a constant oscillation amplitude and temperature. Below the critical strain, the elastic modulus G is often nearly independent of frequency, which is a characteristic of a structured or solid-like material. On the other hand, frequency-dependent elastic modulus is a characteristic of a more fluid-like material. 

The peculiarities of dynamic properties of filled polymers were described above in connection with the discussion of the method of determining a yield stress according to frequency dependence of elastic modulus (Fig. 5). Measurements of dynamic properties of highly filled polymer melts hardly have a great independent importance at present, first of all due to a strong amplitude dependence of the modulus, which was observed by everybody who carried out such measurements [3, 5]. 

Though due to the fact that it is difficult to interprete amplitude dependence of the elastic modulus and to unreliable extrapolation to zero amplitude, the treatment of the data of dynamic measurements requires a special caution, nevertheless simplicity of dynamic measurements calls attention. Therefore it is important to find an adequate interpretation of the obtained results. Even if we think that we have managed to measure correctly the dependences G ( ) and G"( ), as we have spoken above, the treatment of a peculiar behavior of the G (to) dependence in the region of low frequencies (Fig. 5) as a yield stress is debatable. But since such an unusual behavior of dynamic functions is observed, a molecular mechanism corresponding to it must be established. 

Fig. 14. Amplitude dependences (y0 is the amplitude of cyclic deformations) of the elastic modulus for frequency a) = 63 s 1 13% dispersion of acetylene carbon black in low- (/) and high-molecular (2) poly(isobutylene)s...

Chain stretching is governed by the covalent bonds in the chain and is therefore considered a purely elastic deformation, whereas the intermolecular secondary bonds govern the shear deformation. Hence, the time or frequency dependency of the tensile properties of a polymer fibre can be represented by introducing the time- or frequency-dependent internal shear modulus g(t) or g(v). According to the continuous chain model the fibre modulus is given by the formula... 

Dynamic rheological measurements have recently been used to accurately determine the gel point (79). Winter and Chambon (20) have determined that at the gel point, where a macromolecule spans the entire sample size, the elastic modulus (G ) and the viscous modulus (G") both exhibit the same power law dependence with respect to the frequency of oscillation. These expressions for the dynamic moduli at the gel point are as follows ... 

We can consider the friction coefficient to be independent of the molecular weight. At times less than this or at a frequency greater than its reciprocal we expect the elasticity to have a frequency dependence similar to that of a Rouse chain in the high frequency limit. So for example for the storage modulus we get... 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

The attenuation may be expressed by making the wavenumber complex (this would be k — ia in eqn (6.12)), and the velocity (= w/k) may also be written as a complex quantity. This in turn corresponds to a complex modulus, so that the relationship v - /(B/p) is preserved indeed the acoustic wave equation may be written as a complex-valued equation, without the need for the extra term in (6.11). Complex-valued elastic moduli are frequency-dependent, and the frequency-dependent attenuation and the velocity dispersion are linked by a causal Kramers-Kronig relationship (Lee et al. 1990). 

The thus obtained high-density Mn-Zn ferrite was investigated in detail from the view of physical and mechanical properties, that is, the relationships between the composition of metals (a,) ) and <5 the magnetic properties such as temperature and frequency dependence of initial permeability, magnetic hysteresis loss and disaccommodation and the mechanical properties such as modulus of elasticity, hardness, strength, and workability. Figures 3.13(a) and (b) show the optical micrographs of the samples prepared by the processes depicted in Fig. 3.12(a) and (b), respectively. The density of the sample shown in Fig. 3.13(a) reached up to 99.8 per cent of the theoretical value, whereas the sample shown in Fig. 3.13(b) which was prepared without a densification process, has many voids. 

Here pg and p f are the mass densities of the gel and the solvent, respectively, K is a bulk modulus, c0 is the speed of sound, and i s is the solvent shear viscosity. The solvent bulk viscosity has been neglected. The terms proportional to / arise from an elastic coupling in the free energy between the density deviation of gel and that of solvent The p in Eq. (6.1) coincides with the shear modulus of gels treated so far. We neglect the frequency-dependence of the elastic moduli. It can be important in dynamic light scattering, however, as will be discussed in the next section. 

Thus a measurement of the ultrasonic properties can provide valuable information about the bulk physical properties of a material. The elastic modulus and density of a material measured in an ultrasonic experiment are generally complex and frequency dependent and may have values which are significantly different from the same quantities measured in a static experiment. For materials where the attenuation is not large (i.e., a ca/c) the difference is negligible and can usually be ignored. This is true for most homogeneous materials encountered in the food industry, e.g., water, oils, solutions. 

The relaxation spectrum H(0) completely characterizes the viscoelastic properties of a material. H(0) can be found from the measured frequency dependence of the dynamic modulus of elasticity G (co) by means of the following integral equation ... 

The evolution of the dynamic viscosity rp (co, x) or of the dynamic shear complex modulus G (co.x) as a function of conversion, x, can be followed by dynamic mechanical measurements using oscillatory shear deformation between two parallel plates at constant angular frequency, co = 2irf (f = frequency in Hz). In addition, the frequency sweep at certain time intervals during a slow reaction (x constant) allows determination of the frequency dependence of elastic quantities at the particular conversion. During such experiments, storage G (co), and loss G"(co) shear moduli and their ratio, the loss factor tan8(co), are obtained ... 

Another indirect method for the estimation of Gibbs elasticity modulus is based on the determination of the surface dilatation modulus E in experiments in which the surfaces of the surfactant solutions undergo small amplitude deformations of oscillatory nature [100-102], It is shown [100, see also Chapter 7] that the concentration dependence of a Gibbs elasticity modulus at constant film thickness should be nearly the same as the concentration dependence of (twice) the surface elastic modulus E when film thickness and frequency are related by... 

More detailed calculations of the elastic properties of model networks have confirmed Phillips model. The coordination dependence of the elastic modulus is shown in Fig. 2.12 (He and Thorpe 1985). Both the modulus Cn and the number of zero frequency vibrational modes, /, drop to zero at the critical coordination of 2.4, as predicted by Eq. (2.17). The properties are explained in terms of percolation of rigidity. The coordination of 2.4 represents the lowest network coordination for which locally rigid structmes are fully connected, so that the entire network is rigid, but only just so. The elastic modulus is therefore non-zero and continues to increase as the network becomes more connected. The four-fold amorphous silicon network is far from the critical coordination and is very rigid. 

Fig. 2.12. Calculated dependence of the elastic modulus, C , of a random network on coordination number Z . The inset shows the average number of zero frequency modes per atom (He and Thorpe 1985).

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