Modulus high frequency limit

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

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... 

Figure 6.11 The reduced high-frequency modulus G a /ksT as a function of particle concentration (p for particles STl (Q), ST2 ( ), and ST3 (0) described in the caption to Fig. 6-10. The filled symbols are estimates based on the behavior in the high-frequency limit. The solid line is a numerical prediction from the theory of Lionberger and Russel (1994), while the dashed line is their approximate result, given by Eq. (6-21). (From Shikata and Pearson, reprinted from J. Rheol. 38 601, Copyright 1994, with permission from the Journal of Rheology.)...

Another interesting result obtained for solutions of PIB in cyclohexane by Tanaka et al. (2,92) is shown in Fig. 3.4. The reduced storage modulus at infinite dilution is close to the prediction of the Rouse theory while the loss modulus is not. This result indicates the existence of an additional contribution to the intrinsic viscosity at high frequency which might be a frequency-independent high frequency limit [in contrast with the results for PMS (94,99) and for PIB in toluene (2,92)]. We will return to this subject in the following chapter. 

In this equation, Gq is the high frequency limiting vzdue of the modulus (the unrelaxed modulus), %ww is the relaxation time, and P a shape parameter. The KWW function has been found to describe various processes. Most importantly for polymers, the local segmental relaxation dynamics conform closely to form of equation 1. 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

In the limit of high frequencies the integral for the loss modulus tends to zero as the denominator in Equation 4.50 tends to infinity. The storage modulus tends to G(oo) which is just the integral under the relaxation spectrum ... 

However, in such a high concentration regime we can no longer represent the relaxation times (Equation (5.92)) in terms of the intrinsic viscosity. In the low frequency limit, because there is no permanent crosslinking present, the loss modulus divided by the frequency should equate with ... 

The contribution from the first term (reptation branch) has the same order of magnitude as the contribution from the second term at very high frequencies. However, one has to take into account that, due to distribution of relaxation times, the limit value of the first term is reached at higher frequencies than the limit value of the second term. At lower frequencies the plateau value of the dynamic modulus is determined by the second term and coincides with expression (6.52). 

The relaxation modulus G(t) is the value of the transient stress per unit strain in a step-strain experiment. This type of experiment may be achieved with modem rotary rheometers with a limited resolution in time (roughly 10 2 s). If one wishes to evaluate G(t) at shorter times, it is necessary to derive G(t) from the high frequency G (co) data by an inverse Carson-Laplace transform. 

Equations (7.26) are called dispersion relations and analogous equations can be derived for /, and J2 (see problem 7.7) and for more general models. They can also be derived for the real and imaginary parts of the dieleetrie constant (see section 9.2.4). The limiting values of G, and /, at low frequencies are called the relaxed modulus and compliance, G, and J, and the limiting values at high frequencies are called the unrelaxed modulus and compliance, G and / ... 

Figure 60.2 shows the velocity curves of butyl rubber as a function of frequency at different temperatures. The velocity increases very slowly with frequency at high temperatures where the values of the velocity are of the order of 40 m/s. The increase in velocity with frequency is much more rapid as the temperature is lowered, and at 0 °C the velocity is about 300 m/s. The corresponding modulus curves derived from the velocity curves is shown in Fig. 60.2. At all temperatures the modulus was seen to increase with frequency. From ultrasonic experiments on the same polymers, the data obtained at frequencies in the MHz range indicated a continuous rise in the modulus with frequency (measured up to 15 MHz). The dispersion over a limited frequency range can be attributed to a mechanism involving relaxation times of the order of l/w, whereas the entire dispersion range would have to be explained on the assumption of a wide distribution of relaxation times. The relaxation mechanism for these low frequency measurements...

The slope of the loss modulus (9.74) is the same integral in the limit of high frequency. We thus find a new result... 

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