G* Complex shear modulus

G Complex shear modulus, with real (elastic) and imaginary (viscous) components. 

Ef Evaporation at film surface Eq. 8.162 G Complex shear modulus Eq. 3.24... 

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). 

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]. 

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. 

Experimental evidence of this hypothesis is given in Fig. 18. Two monodisperse polystyrene samples (N = 50 and P = 5) are blended with different concentrations and the complex shear modulus is measured in a wide range of fi quendes allowing us to scan the relaxation of the two components. The reduced imaginaiy part of the complex viscosity ( n" = G /o)) shows, at intermediate concentrations, two maxima connected to the relaxation time of each component (x=l/tOmax)- The most striking point is the large decrease of the relaxation time of the longest N-chains (increase of (Omax ) ... 

The relaxation modulus is the core of most of the viscoelastic descriptions and the above expression can be checked from experimental viscoelastic functions such as the complex shear modulus G (co) for instance. In addition to the molecTilar weight distribution function P(M), one has to know a few additional parameters related to the chemical species the monomeric relaxation time x,... 

Figure 27 Experimental complex shear modulus of unentangled polystyrene (M 8 500 g.mol-i) compared to the Rouse model [37].

Abstract Grease lubrication is a complex mixture of science and engineering, requires an interdisciplinary approach, and is applied to the majority of bearings worldwide. Grease can be more than a lubricant it is often expected to perform as a seal, corrosion inhibitor, shock absorber and a noise suppressant. It is a viscoelastic plastic solid, therefore, a liquid or solid, dependent upon the applied physical conditions of stress and/or temperature, with a yield value, ao- It has a coarse structure of filaments within a matrix. The suitability of flow properties of a grease for an application is best determined using a controlled stress rheometer for the frequency response of parameters such as yield, a, complex shear modulus, G phase angle, 5, and the complex viscosity, rj. ... 

Fig. 14.2 Frequency dependence of the complex shear modulus, G (x), phase angle, 5 (+), and complex viscosity, rj (v), for a lithium grease at 25°C...

Equations 11.8 and 11.9 are isomorphous to equations 9.9 and 9.10 which define the storage and loss components of the complex dielectric constant . Similar equations are also used to define the complex bulk modulus B, the complex shear modulus G, and the complex Poisson s ratio v, in terms of their elastic and viscous components. The physical mechanism giving rise to the viscous portion of the mechanical properties is often called "damping" or "internal friction". It has important implications for the performance of materials [8-15],... 

For viscoelastic materials, the response is as in Figure 5.9d. Here crve = crei + complex shear modulus G can be derived, and the following relations hold ... 

The nomenclature of complex moduli and compliances is also often used. Here the out-of-phase component is made the imaginary part of a complex parameter thus the complex shear modulus G and the complex shear compliance J are defined as... 

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