Shear modulus complex

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). 

FIGURE 1.8 In-phase (jx ) and out-of-phase (pf ) components of the complex shear modulus of uncross-linked namral ruhher versus log (frequency) at various temperatures. (From Payne, A.R. and Scott, J.R., Engineering Design with Rubber, Interscience Pubhshers, New York, 1960.)... 

FIGURE 29.5 Effects contributing to the complex shear modulus. 

Like any dynamic strain instrument, the RPA readily measures a complex torque, S (see Figure 30.1) that gives the complex (shear) modulus G when multiplied by a shape factor B = iTrR / ia, where R is the radius of the cavity and a the angle between the two conical dies. The error imparted by the closure of the test cavity (i.e., the sample s periphery is neither free nor spherical) is negligible for Newtonian fluids and of the order of maximum 10% in the case of viscoelastic systems, as demonstrated through numerical simulation of the actual test cavity." ... 

Contrary to the phase separation curve, the sol/gel transition is very sensitive to the temperature more cations are required to get a gel phase when the temperature increases and thus the extension of the gel phase decreases [8]. The sol/gel transition as determined above is well reproducible but overestimates the real amount of cation at the transition. Gelation is a transition from liquid to solid during which the polymeric systems suffers dramatic modifications on their macroscopic viscoelastic behavior. The whole phenomenon can be thus followed by the evolution of the mechanical properties through dynamic experiments. The behaviour of the complex shear modulus G (o)) reflects the distribution of the relaxation time of the growing clusters. At the gel point the broad distribution of... 

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] ... 

Figure 3.5 Demonstration of correlation between the stickiness of protein-coated droplet pair encounters in shear flow (left ordinate axis) and viscoelasticity of concentrated emulsions (right ordinate axis) with the strength of protein-protein attraction as indicated by the second virial coefficient A2 determined from static light scattering , percentage capture efficiency (0%) A, complex shear modulus (G ) for emulsions stabilized by asl-casein or (3-casein (pH = 5.5, ionic strength in the range 0.01-0.2 M).

Figure 7.10 Effect of the thermodynamic incompatibility of otsi/p-casein + high-methoxy pectin (pH = 7.0, / = 0.01 M) on phase diagram of the mixed solutions and elastic modulus of corresponding casein-stabilized emulsions (40 vol% oil, 2 wt% protein), (a) (O) Binodal line for p-casein + pectin solution with critical point ( ) ( ) binodal line for asi-casein + pectin solution with critical point ( ). (b) Complex shear modulus G (1 Hz) is plotted against the pectin concentration (O) p-casein ( ) o i -casein. Dotted lines indicate the range of pectin concentration for phase separation in the mixed solutions. The pectin was added to the protein solution before emulsion preparation. Data are taken front Semenova et al. (1999a).

Figure 7.19 Influence of pH on the complex shear modulus G (at 1 Hz) of emulsions (20 vol% soybean oil, 0.5 wt% p-lactoglobulin) prepared with untreated (open symbols) and high-pressure-treated (800 MPa for 30 min filled symbols) protein in the absence (O, ) and presence (A, ) of 0.5 wt% pectin. Reproduced from Dickinson and James (2000) with permission.

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]. 

Sensitivity to shear would also be a function of the extent of molecular entanglements in the swollen granules. Svegmark and Hermansson150,437 have found that warm cereal starch pastes are less sensitive to shearing than similarly treated potato starch pastes (Figure 8.14) the complex shear modulus of the sheared (HS) 10%... 

Figure 1. Phase angle for complex shear modulus.

Figure 15. Complex shear modulus in the modified Cole-Cole model.

The complex shear modulus of the viscoelastic damping polymer is obtained directly from measurements on the sample using the relationship... 

Thus, FTMA determines complex modulus as the transfer function between input strain and output stress. A prerequisite is that the Fourier transform of y(t) must exist. White no se should suffice since it contains all frequencies. Note that G (jw) in Equation 10 will be the complex Young s modulus if a(t) and 7(t) are the normal stress and normal strain, respectively and the complex shear modulus if they are the shear stress and shear strain. 

A particularly useful expression for the complex dynamic moduli was introduced using operators and fractional derivatives for the viscoelastic constitutive equations ( ). For a rheologically simple system the complex shear modulus is... 

The complex shear modulus, which is the complex stress to complex strain ratio ... 

Although the (Simplex shear modulus is not the most appropriate function to use in all c ses, we wUl describe the linear viscoelastic behaviour in terms of this last function, which is tiie most referred to experimentally furthermore, molecular models are mostly linked to the relaxation modulus, which is the inverse Fourier transform of the complex shear modulus. 

We will discuss in this section the variations of the viscoelastic parameters derived from linear viscoelastic measurements all these parameters may be derived from any t3rpe of measurement (relaxation or creep experiment, mechanical spectroscopy) performed in the relevant time or frequency domain. The discussion will be focused however on the complex shear modulus which is the basic function derived from isothermal frequency sweep measurements performed with modem rotary rheometers. 

Figure 2 Schematic of the variations of the complex shear modulus of linear polymers (dotted lines molecular weight M2>Mi).

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