Dynamic frequency

The Imass Dynastat (283) is a mechanical spectrometer noted for its rapid response, stable electronics, and exact control over long periods of time. It is capable of making both transient experiments (creep and stress relaxation) and dynamic frequency sweeps with specimen geometries that include tension-compression, three-point flexure, and sandwich shear. The frequency range is 0.01—100 H2 (0.1—200 H2 optional), the temperature range is —150 to 250°C (extendable to 380°C), and the modulus range is 10" —10 Pa. 

Arai et al. (1997) used EQCM to study iodide absorption on polycrystalline gold in IM NaC104 containing different concentrations of Nal. Eigure 27.21 shows dynamic frequency change-potential A/-i curves for 0.1M NaC104 containing different concentrations of Nal. The frequency at potentials more positive than -0.8 V was less than that found without Nal. The frequency increased with... 

Specifically, following the rate expression of QTST in Eq. (4-1) and assuming the quantum transmission coefficients the dynamic frequency factors are the same, the kinetic isotope effect between two isopotic reactions L and H is rewritten in terms of the ratio of the partial partition functions at the centroid reactant and transition state... 

Vibrational spectroscopy can help us escape from this predicament due to the exquisite sensitivity of vibrational frequencies, particularly of the OH stretch, to local molecular environments. Thus, very roughly, one can think of the infrared or Raman spectrum of liquid water as reflecting the distribution of vibrational frequencies sampled by the ensemble of molecules, which reflects the distribution of local molecular environments. This picture is oversimplified, in part as a result of the phenomenon of motional narrowing The vibrational frequencies fluctuate in time (as local molecular environments rearrange), which causes the line shape to be narrower than the distribution of frequencies [3]. Thus in principle, in addition to information about liquid structure, one can obtain information about molecular dynamics from vibrational line shapes. In practice, however, it is often hard to extract this information. Recent and important advances in ultrafast vibrational spectroscopy provide much more useful methods for probing dynamic frequency fluctuations, a process often referred to as spectral diffusion. Ultrafast vibrational spectroscopy of water has also been used to probe molecular rotation and vibrational energy relaxation. The latter process, while fundamental and important, will not be discussed in this chapter, but instead will be covered in a separate review [4],... 

Rheological Properties Measurements. The viscoelastic behavior of the UHMWPE gel-like systems was studied using the Rheometric Mechanical Spectrometer (RMS 705). A cone and plate fixture (radius 1.25 cm cone angle 9.85 x 10" radian) was used for the dynamic frequency sweep, and the steady state shear rate sweep measurements. In order to minimize the error caused by gap thickness change during the temperature sweep, the parallel plates fixture (radius 1.25 cm gap 1.5 mm) was used for the dynamic temperature sweep measurements. 

A schematic of the system is illustrated in Figure 1. For dynamic frequency sweeps (refer to Figure 2), the polymer is strained sinusoidally and the stress is measured as a function of the frequency. The strain amplitude is kept small enough to evoke only a linear response. The advantage of this test is that it separates the moduli into an elastic one, the dynamic storage modulus (G ) and into a viscous one, the dynamic loss modulus (G"). From these measurements one can determine fundamental properties such as ... 

Consider now the influence of the high-frequency fluctuations in the environment only (is 3> B). Since the frequencies of the fluctuations are much higher than the typical spin-dynamics frequencies, one may eliminate these high-frequency fluctuations using the adiabatic (Born-Oppenheimer) approximation, as described, e.g., by Leggett et al. [8]. 

Between the static (time-dependent) and the dynamic (frequency-dependent) behaviour the following correlation exists ... 

Wu et al. (73) studied the viscoelastic properties, viz. storage modulus (GO and complex viscosity (r 0 with respect to frequency (co) of PLA-carboxylic-acid-functionalized MWCNTs nanocomposites using a rheometer (HAAKE RS600, Thermo Electron Co., USA). The dynamic frequency sweep measurements were carried out at the pre-strain level of 1%. They observed that the addition of carboxylic-acid-functionalized MWCNTs weakened the dependence of G on go, especially at higher loading levels (Figure 9.12). This indicates... 

Figure 9.12. (a) Dynamic storage modulus (GO and (b) complex viscosity (r) ) for pure PLA and PLA- carboxylic-acid-functionalized MWCNTs nanocomposites obtained in dynamic frequency sweep. Reprinted with permission from D. Wu et al., Polymer Degradation and Stability, Vol. 93, p. 1577,2008, 2008, Elsevier Science Ltd. 

The equation offers a rough prediction of peak loss factor temperature at dynamic frequency, thus enabling the material scientist to select candidate viscoelastic damping materials from DSC T data. 

The reference frequency for DSC was assigned a low value (fDSC = 0.0001 Hz). The value does not have any physical significance and was chosen for best data fit. This equation requires the value of the activation energy for the polymers which generally lies between 200-900 kJ/mole. An intermediate value of 400 kJ/mole is suggested as an approximation, if the activation energy of the polymer is unknown. The T at specific dynamic frequencies can now be calculated if the DSe T at the same heating rate is known. ... 

Double logarithmic plots of rj versus dynamic frequency (a>, rad s ) and % versus shear rate (y) of a 4% tapioca starch dispersion heated at 70°C for 30 min, were described by Equations 4.52 and 4.53, respectively. 

Figure 5-15 Magnitudes of C and G" of Three Tomato Coneentrates from Juice Using a 0.84 mm Sereen as a Function of Dynamic Frequencies.

Integral of time-temperature history, t -First normal stress coefficient. Pa s Second normal stress coefficient. Pa s Dynamic frequency, rad s ... 

The core element of LR-TDDFT is the relation between the dynamic (frequency dependent) polarizability of the system under investigation and the quantities derivable from Kohn-Sham equations. LR-TDDFT has been subject to many reviews (for fundamental aspects see the recent review by Gross and Marquard112, for the original description of the adaptation of LR-TDDFT to molecular systems see the review by Casida111). Before discussing the response of an embedded electron density, the key elements of LR-TDDFT will be provided here1. 

Interestingly, it has been shown experimentally for many systems (Cox, 1958) that the profile of the complex viscosity ( / ) as a function of the dynamic frequency (o) is equivalent to the profile of the steady-shear viscosity tj) with respect to the shear rate (y ) for the same system. That is. 

Dynamic-shear measurements are of the complex viscosity rj ) as a function of the dynamic oscillation rate (o), at constant temperature. These tests are defined as isothermal dynamic frequency sweeps. Since the dynamic frequency sweeps are conducted at a given amplitude of motion, or strain, it is necessary to ensure that the sweeps are conducted in the region where the response is strain-independent, which is defined as the linear viscoelastic region. This region of strain independence is determined by an isothermal strain sweep, which measures the complex viscosity as a function of applied strain at a given frequency. This ensures that a strain at which the dynamic frequency sweep may be conducted in the linear viscoelastic region is selected. 

The complex viscosity as a function of frequency, maximum strain and temperature is generally determined with one rheometer. Standard ASTM 4440-84/90 defines the measurement of rheological parameters of polymer samples using dynamic oscillation. This standard reiterates the importance of determining the linear viscoelastic region prior to performing dynamic frequency sweeps. 

Figure 60. Influence of space velocity on the conversion of CO, HC and NO.v, reached over a three-way catalyst in the fresh state and after engine aging, at fixed exhaust gas temperature and exhaust gas composition (monolith catalyst with 62 cells cm , three-way formulation with Pt 1.42gl->, Rh 0.28gr engine bench test at 723 K exhaust gas temperature exhaust gas composition lambda 0.995 dynamic frequency 1 Hz amplitude 1 A/F engine bench aging during 200 h). Reprinted with permission from ref. [76], ...

Figure 66. Influence of the washcoat loading of a ceramic monolith on the conversion of NO t (monolith catalyst with 62cells cm" three-way formulation with Reprinted with permission from ref. [34], 1991 Society of Automotive Engineers, Inc. Pt 1.42gl" , Rh 0.28gl" after aging on an engine bench 20 h engine bench test space velocity 60000N1E h exhaust gas temperature 723 K exhaust gas composition lambda 0.999 dynamic frequency 1 Hz amplitude 1 A/F). Reprinted with permission from ref [34], 1991 Society of Automotive Engineers, Inc.

Figure 67. Influence of the washcoat formulation on the conversion of CO, HC and NO , reached over various engine aged three-way catalysts as a function of the exhaust gas lambda value (monolith catalyst with 62cells cm three-way formulation with Pt 1.16gC, Rh 0.23gl" engine bench test A/F scan at a space velocity of 60000 Nl 1 h exhaust gas temperature 723 K dynamic frequency 1 Hz amplitude 1 A/F engine bench high temperature aging cycle for lOOh). Sample A baseline formulation, samples B-D increasing amounts of stabilizers.

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