Modulus, high frequency

It is appropriate to focus on some general parameters that could characterize a relaxation and to see how these are reflected in the experimental data. These parameters would include an unrelaxed, low temperature, high frequency modulus, Ejj, and a relaxed, high temperature,... 

Interestingly, the same power law governs both the relaxation and the dynamic moduli. Further use of this expression is given in Chapter 5 with a microstructural description. It is unrealistic to expect the gel equation to apply over the whole of the frequency and time ranges. For example it predicts an infinite high frequency modulus. However, it can apply over a wide range of frequencies and provide a useful description of the system. 

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.)...

In the ordered state, the suspension can have a substantial modulus (see Fig. 6-31). Buscall et al. (1982b, 1991) have derived an expression for the high-frequency modulus from the interparticle potential by calculating the work required for particles to move affinely with the flow. The result of Buscall et al. follows from the Zwanzig-Mountain expression, Eq. (6-20), where g r ) can be replaced by a delta function since the suspension is crystalline. Neglecting the osmotic term, the result is (Evans and Lips 1990 Buscall 1991)... 

Figure 6.31 High-frequency modulus Goo versus particle volume fraction (p for charged polystyrene latices of radius a = 26.3 nm (O). 34.3 nm ( ), 39,2 nm (A), and 98.3 nm (A) in aqueous solutions with 5 X lO" M NaCl. The lines are the predictions of the theory of Buscall et al. (1982b) for an FCC lattice with = 12 and (pm = 0.74, and a best-fit value of increasing from 50 to 89 mV as the particle radius increases from 26.3 to 98.3 nm. These predictions differ from Eq. (6-70) only in that a prefactor (3/32) in Buscall et al. (1982b) was corrected to 1 /(5n) in Buscall (1991), (From Buscall et al. 1982b, reproduced with permission of the Royal Society of Chemistry.)...

Problem 6.4(a) Estimate the high-frequency modulus of a suspension of hard spheres of radius 1 /xm at a concentration 0 of 0.4, at room temperature in a solvent with viscosity 1 P. 

Geo(e). Here tire storage modulus dominates over the loss one, which drops like G"(< 0) (0. The high frequency modulus GL = 0°) is characteristic... 

Measurement of the high frequency modulus, c0, as a function of the equilibrium surface pressure, tt, should provide a sensitive criterion for interaction for monolayers that are quite soluble by normal standards, which involve much longer time spans than the inverse frequency of the compression/expansion experiment. A numerical example of the greater sensitivity of an e0 vs. tt plot, compared with that of the ir vs. log c relationship is shown in Figure 1 for a hypothetical case. The specific defini-nition of surface interactions used here to arrive at numerical values includes all mechanisms that produce deviations from Szyszkowski-Langmuir adsorption behavior. Ideal behavior, with zero surface interactions, then is represented by zero values of In fis in the equation of state ... 

Table 2. Experimental limiting current, high frequency modulus, and ratio values for... 

Figure 5 Prediction of high frequency modulus vs. (he relative magnitude of the electrostatic to thermal contributions Xn, Af)- 7il2)eQS 2af/kgT at the particle volume fraction 0.35 and 0-40. p=0.75. Reproduced with permission from P.M. Adriani, and A.P. Gast, Phys. Fluids, 31(1988)2757.

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