Shear modulus storage

Dynamical Shear Modulus (Storage and Loss Moduli) and Viscosity. 177... 

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 case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

Filler-filler interaction (Payne effect) - The introduction of reinforcing fillers into rubbery matrices strongly modifies the viscoelastic behavior of the materials. In dynamic mechanical measurements, with increasing strain amplitude, reinforced samples display a decrease of the storage shear modulus G. This phenomenon is commonly known as the Payne effect and is due to progressive destruction of the filler-filler interaction [46, 47]. The AG values calculated from the difference in the G values measured at 0.56% strain and at 100% strain in the unvulcanized state are used to quantify the Payne effect. 

The dynamic mechanical measurements were performed with a Rheometrics IV apparatus in a geometrical arrangement of parallel plates. The complex shear modulus G (= G + fG", where G and G", respectively, are the storage and loss moduli) at a constant frequency of 1 Hz was determined [30]. 

Isothermal measurements of the dynamic mechanical behavior as a function of frequency were carried out on the five materials listed in Table I. Numerous isotherms were obtained in order to describe the behavior in the rubbery plateau and in the terminal zone of the viscoelastic response curves. An example of such data is shown in Figure 6 where the storage shear modulus for copolymer 2148 (1/2) is plotted against frequency at 10 different temperatures. 

FIG. 13.5 Storage shear modulus for polyethylene terephthalate) with different degrees of crystallinity temperature. From McCrum, Buckley and Bucknall, 1988. Courtesy Oxford University Press. 

This section examines the dynamic behavior and the electrical response of a TSM resonator coated with a viscoelastic film. The elastic properties of viscoelastic materials must be described by a complex modulus. For example, the shear modulus is represented by G = G + yG", where G is the storage modulus and G" the loss modulus. Polymers are viscoelastic materials that are important for sensor applications. As described in Chapter S, polymer films are commmily aj lied as sorbent layers in gas- and liquid-sensing applications. Thus, it is important to understand how polymer-coated TSM resonators respond. 

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49] K = K jA and G = G -I- jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K and G are referred to as storage moduli) the imaginary parts represent the component of stress 90° out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). 

Figure 13.18 Storage shear modulus G for lamellar samples of PEP-PEE 50,000) at 40°C. The...

Estimate the frequency dependence of the storage and loss modulus of semiflexible chains at intermediate frequencies tg a To from Eq. (8.117), keeping in mind that Young s modulus is three times the shear modulus (E(t) — 3G(t)). What is the value of the loss tangent tan 6 in... 

There are four popular measures of liquid elasticity the first normal stress difference, the extrudate swell, B, the Bagley entrance-exit pressure drop correction, P, and the storage shear modulus, G. For multiphase systems there Is no simple correlation between and G, although the Sprigg s theoretical relation (51) ... 

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