Shear modulus, high-frequency

The contribution of this component is shown in Figures 4.16 and 4.17. One of the important features to recognise about the retardation spectrum is that it only has an indirect relationship to both the zero shear rate viscosity and the high frequency shear modulus. Both these properties are contained in the relaxation spectra. We shall see in Section 4.5.7 that, whilst a relationship exists between H and L it is somewhat complex. 

The high frequency shear modulus is proportional to the rate of change of force with distance. For colloidal systems this is dominated by the integral expression15... 

Figure 5.18 The high frequency shear modulus versus volume fraction for a polystyrene latex for three different electrolyte concentrations. The symbols are the experimental data and the solid lines are calculated fits using a cell model. The radius of the latex particles was 38 nm...

Our expression for the high frequency shear modulus is now (Section 5.5) ... 

Figure 6.27 Plot of the high frequency shear modulus G(oo) and the static modulus G(0) versus volume fraction. The points are the experimental data, the solid lines represent the models...

Here (pc is the volume fraction of the core and ac its radius. This equation has not been widely tested owing to a paucity of data. Thorough characterisation allows all the terms to be determined except (pm and/L. The packing fraction can be found by extrapolation to zero concentration of a plot of the high frequency shear modulus as a function of volume fraction since this corresponds to the volume fraction before the chains come into contact. The functionality of the link can be used as an adjustable parameter. For the system here a good fit is found with /l = 8/3, as shown in Figure 6.27. 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

Some unexpectedly complex liquid solid interactions have been detected and studied by ultrasonic impedance measurements (ultrasonic impedometry). Small amounts of water and alcohols have pronounced effects on the physical state of hydrophilic polymers specifically, the high frequency shear modulus and crystallinity index of a poly (vinyl alcohol) film increases with water content to a maximum before normal solution phenomena occur. These effects are attributed to the increased molecular order owing to water hydrogen bonded between polymer chains. The unusual effects of moisture on a novel poly(vinyl chloride)/plasticizer system and on hydrophilic polymers other than poly (vinyl alcohol) are also described. 

Fig. 9 Simulation results for monodisperse packings. The high-frequency shear modulus Goo open circles, the low-frequency shear modulus Go closed circles, and the osmotic pressure TT diamonds are all scaled with the particle contact modulus E and plotted versus the packing fraction

Both the osmotic pressure and the high-frequency shear modulus can be calculated analytically from the radial distribution function derived above. The osmotic pressure n is related to the radial distribution function and the energy potential... 

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 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 ... 

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). 

---