Shifted shear frequency

Figure 14.11 Elastic (<30 and loss (G 0 components of the dynamic modulus of the first seven generations of bulk PAMAM dendrimers as a function of shifted shear frequency [aT co] at 40°C. Solid lines indicate a fit for each curve to the generalized Maxwell model...

The increased shift in frequency by a thin viscoelastic film may increase the frequency response to added mass as compared to eqn [1], thus acting as an acoustic amplifier. The change in motional resistance upon addition of a film with a thickness h, a density Pfj and a complex shear modulus G = G + iG" can be estimated as... 

Acoustic shear waves and shear-wave resonators, in particular, have a long tradition in interfacial sensing. Typically one infers the thickness and softness of an adsorbate layer from the shifts of resonant frequency and bandwidth. Laterally heterogeneous samples (vesicles, cells, adsorbed particles) can be modeled numerically. The shifts of frequency and bandwidth are proportional to the stress-velocity ratio (the "load impedance") at the interface, and this load impedance can be calculated by (for instance) the finite element method (FEM). 

Figure 8.2 Tangential displacement close to the resonator surface (u(z), thick blue line). Any object inside the tail of the shear wave will deviate the flow In one way or another. For a given geometry, the flow field can be calculated numerically. The shifts of frequency and bandwidth are proportional to the (area-averaged) ratio of tangential stress and tangential velocity at the resonator surface, where the latter quantities are to be understood as complex amplitudes of oscillatory variables...

When oscillatory shear data are being shifted, the frequency shift factor is simply a, and the reduced frequency is ... 

The Doppler shift of frequency due to motion of the wave source relative to the point of observation can be an explanation of the phenomenon observed. A theoretical prediction of wave emission from the tip of a crack penetrating through ice floe was made by Slepyan (1993). The asymptotic solution presumes the emission of shear bending waves with frequency of about 0.58 Hz, wave length 38.5 m and phase velocity equal to 22.4 m/s for the crack moving faster than Kelvin s phase velocity. Smirnov and Shushlebin (1988) observed variations in frequency of seismic waves related, they assume, to movement of the crack tip. 

The procedure described above is an application of the time-temperature correspondence principle. By shifting a set of plots of modulus (or compliance) versus time (or frequency) at any temperature (subscript 1) along the log t axis, we obtain the value of that mechanical property at another time and temperature (subscript 2). Using the shear modulus as an example, the time-temperature correspondence principle states... 

The shear-mode acoustic wave sensor, when operated in liquids, measures mass accumulation in the form of a resonant frequency shift, and it measures viscous perturbations as shifts in both frequency and dissipation. The limits of device operation are purely rigid (elastic) or purely viscous interfaces. The addition of a purely rigid layer at the solid-liquid interface will result a frequency shift with no dissipation. The addition of a purely viscous layer will result in frequency and dissipation shifts, in opposite directions, where both of these shifts will be proportional to the square root of the liquid density-viscosity product v Pifti-... 

Here t is the resulting shear stress, 6 is the phase shift often represented as tan(d), and (O is the frequency. The term 6 is often referred to as the loss angle. The in-phase elastic portion of the stress is To(cosd)sin(wt), and the out-of-phase viscous portion of the stress is To(sind)cos(complex modulus and viscosity, which can be used to extend the range of the data using the cone and plate rheometer [6] ... 

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). 

If one side of the quartz is coated with material, the spectrum of the resonances is shifted to lower frequencies. It has been observed that the three above mentioned modes have a somewhat differing mass sensitivity and thus experience somewhat different frequency shifts. This difference is utilized to determine the Z value of the material. By using the equations for the individual modes and observing the frequencies for the (100) and the (102) mode, one can calculate the ratio of the two elastic constants Cgg and C55. These two elastic constants are based on the shear motion. The key element in Wajid s theory is the following equation ... 

Fig. 2.5. Steady-state and dynamic oscillatory flow measurements on a 2 wt. per cent solution of polystyrene S 111 in Aroclor 1248 according to Philippoff (57). ( ) steady shear viscosity (a) dynamic viscosity tj, ( ) cot 1% from flow birefringence, (A) cot <5 from dynamic measurements, all at 25° C. (o) cot 8 from dynamic measurements at 5° C. Steady-state flow properties as functions of shear rate q, dynamic properties as functions of angular frequency m. Shift factor aT which is equal to unity for 25° C, is explained in the text, cot 2 % and cot 8 are expressed in terms of shear (see eqs. 2.11 and 2.22)...

Bueche-Ferry theory describes a very special second order fluid, the above statement means that a validity of this theory can only be expected at shear rates much lower than those, at which the measurements shown in Fig. 4.6 were possible. In fact, the course of the given experimental curves at low shear rates and frequencies is not known precisely enough. It is imaginable that the initial slope of these curves is, at extremely low shear rates or frequencies, still a factor two higher than the one estimated from the present measurements. This would be sufficient to explain the shift factor of Fig. 4.5, where has been calculated with the aid of the measured non-Newtonian viscosity of the melt. A similar argumentation may perhaps be valid with respect to the "too low /efi-values of the high molecular weight polystyrenes (Fig. 4.4). 

In addition to knowing the temperature shift factors, it is also necessary to know the actual value of ( t ) at some temperature. Dielectric relaxation studies often have the advantage that a frequency of maximum loss can be determined for both the primary and secondary process at the same temperature because e" can be measured over at least 10 decades. For PEMA there is not enough dielectric relaxation strength associated with the a process and the fi process has a maximum too near in frequency to accurately resolve both processes. Only a very broad peak is observed near Tg. Studies of the frequency dependence of the shear modulus in the rubbery state could be carried out, but there... 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

A comparative study of the readout options for the SAW sensor with additional film has shown that for a single SAW sensor the highest signal-to-noise ratio is obtained from the amplitude measurement (Wohltjen and Dessy, 1979). Voltage output related to the phase-shift as discussed above works well for dual delay lines. There are also inherent advantages in measurement of the change of the resonant frequency. The frequency shift due to deposited film of low elastic shear modulus p is... 

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