Modulus Wagner

Hauri EH, Wagner TP, Grove TL (1994) Experimental and natural partitioning of Th U Pb and other trace elements between garnet clinopyroxene and basaltic melts. Chem Geol 117 149-166 Hazen RM, Finger LW (1979) Bulk Modulus-volume relationship for cation-anion polyhedra. J Geophys Res 84 6723-6728... 

Finding pore resistance effects from experiment. Here we have a simple trick to help us. Define another modulus which only includes observable and measurable quantities. This is known as the Wagner-Weisz-Wheeler modulus My (lucky for us that the three researchers who first dealt with this problem had last names all starting with the same letter). 

The diffusion modulus diffusion effect are entirely determined by the magnitude of the modulus v>, involving size (72), diffusivity (7)en), and intrinsic activity (fc,) of the catalyst. In many practical cases of experimentation the intrinsic activity constant fc, will not be directly known, but instead it will be desirable to estimate tp from the diffusivity and size of the solid, and the actvxilly observed reaction rate dnjdt. Use is made in such cases (Wagner, 16) of the definition of the modulus p and the basic activity equation (first-order reaction being used here) ... 

Maxwell model A mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus E) in series with a dashpot of coefficient of viscosity (ri). It is an isostress model (with stress 8), the strain (e) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as stress relaxation and creep with Newtonian flow analysis. Also called Maxwell fluid model. See stress relaxation viscoelasticity. Maxwell-Wagner efifect See dielectric, Maxwell-Wagner effect. 

Assuming (as it is reasonable) that for conditions in which the approximation ko 5> 1 is valid, the dynamic mobility also contains the (1 — Cq) dependence displayed by the static mobility (Equation (3.37)), one can expect a qualitative dependence of the dynamic mobility on the frequency of the field as shown in Figure 3.14. The first relaxation (the one at lowest frequency) in the modulus of u can be expected at the a-relaxation frequency (Equation (3.55)) as the dipole coefficient increases at such frequency, the mobility should decrease. If the frequency is increased, one finds the Maxwell-Wagner relaxation (Equation (3.54)), where the situation is reversed Re(Cg) decreases and the mobility increases. In addition, it can be shown [19,82] that at frequencies of the order of (rj/o Pp) the inertia of the particle hinders its motion, and the mobility decreases in a monotonic fashion. Depending on the particle size and the conductivity of the medium, the two latter relaxations might superimpose on each other and be impossible to distinguish. 

FIGURE 6.6 (see facing page) Values (x) for results (left) and relative residuals (right) from calculations of the reduced bulk modulus, pkgTKj-, for the pure LJ fluid at reduced temperatures T = 0.85, T = 1.0, r = 1.5, and T = 2.5. Lines derived from EOS of Mecke et al. are also shown. (From M. Mecke, A. Muller, J. Winkelmann, J. Vrabec, J. Fischer, R. Span, and W. Wagner, 1996, An Accurate Van der Waals-Type Equation of State for the Lennard-Jones Fluid, International Journal of Thermophysics, 17, 391, with permission from Springer.)... 

Evaluation of this expression generates a spectrum rather similar to that for the Maxwell-Wagner model, but with a different weighting of the two phases. Figure 4.1.12 shows a modulus spectrum for the same input parameters as those that were used to produce the spectrum in Figure 4.1.9fc. 

Impedance spectra of such crystals, reported by Bonanos and Lilley [1981], displayed only bulk and electrode arcs. The same data plotted in the modulus plane (Figure 4.1.45) revealed two overlapping arcs a low frequency arc ascribed to the matrix and a high frequency one ascribed to the dispersed phase. Using the Maxwell-Wagner effective medium relation (Eq. 20), the modulus spectra were modeled and the microscopic conductivities of the two phases were evaluated for... 

A correlation between complex viscosity t and the storage modulus G was derived [61,64] by analogously following the mettiod of Wagner [87,88] for relating the elastic response to the viscous response of the material (equation (2.61) in section 2.3.4). The expression obtained [61,64] was as follows ... 

H. D. Wagner, 0. Lourie (1998) Evaluation of Young s modulus of carbon nanotubes by micro-Raman spectroscopy, J. Mater. Res. 13,2418. 

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