At fixed frequency

Another resonant frequency instmment is the TA Instmments dynamic mechanical analy2er (DMA). A bar-like specimen is clamped between two pivoted arms and sinusoidally oscillated at its resonant frequency with an ampHtude selected by the operator. An amount of energy equal to that dissipated by the specimen is added on each cycle to maintain a constant ampHtude. The flexural modulus, E is calculated from the resonant frequency, and the makeup energy represents a damping function, which can be related to the loss modulus, E". A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

It is clear that this data treatment is strictly valid providing the tested material exhibits linear viscoelastic behavior, i.e., that the measured torque remains always proportional to the applied strain. In other words, when the applied strain is sinusoidal, so must remain the measured torque. The RPA built-in data treatment does not check this y(o )/S (o)) proportionality but a strain sweep test is the usual manner to verify the strain amplitude range for constant complex torque reading at fixed frequency (and constant temperature). 

Thirdly, we need to appreciate how the current term in the Levich equation represents a faradaic current, and hence the stipulation that we remove all dissolved oxygen from the solution before our analyses commence. Furthermore, the current is a limiting one, so we will commonly perform a few sample experiments before the analysis (usually at fixed frequency) by slowly increasing the potential until a limiting current is reached. 

The impedances of a cell s leads, contacts and electrolyte solution can be approximated to a situation of impedances in series. Typical values will be Zieads — 1, Zcontacts — 6 and ZQi t).Qiyt0 solution 40 (with all Z values beinq cited at fixed frequency). Calculate the total impedance of these three components when combined together. 

Worked Example 8.6. The three impedances, Z = 12.5, Z2 = 100 and Z3 = 0.5 (at fixed frequency), are placed in parallel. What is their combined impedance, i.e. Ztotai ... 

An electrode bears a layer of indium-tin oxide (ITO) having an impedance of 25 Q, on which is a layer of adsorbed chromophore having an impedance of 1.0 (all values of Z being cited at fixed frequency). In addition, between the chromophore layer and the bulk electrolyte is the Helmholtz double-layer (see Section 5.1.2), which has an impedance of 120 By assuming that these three layers act as impedances in parallel, calculate the total impedance, Ziotai-... 

Fig. 2.6 Temperature dependence of ultrasonic attenuation at fixed frequency

As the screening effect increases, the nuclei are said to be shielded on a continuous wave instrument operating at fixed frequency, the intensity of the field B0 has to be increased in order to obtain resonance. Signals to the right of the spectrum are said to be resonant at high field. Signals observed to the left of the spectrum correspond to deshielded nuclei and are said to be resonant at low field (Fig. 9.11). 

The important point to note here is that the capacitive terms determine the overall shape of the transfer function, whereas the frequency f and the sheet resistance Rg occur only within the parameter a, and only as a product. Therefore, an increase in frequency at fixed sheet resistance can cause the same change in transfer function as an increase in sheet resistance at fixed frequency. This property can be exploited in visualizing the transfer function by plotting the amplitude of Equation 1 against its phase shift, using... 

Fig. 5. Selected experimental (full line) and nearly identical underlying theoretical spectra (dotted line) for [Gd(H20)8]3+ at fixed frequency and temperature values versus variable external magnetic field. The difference between experimental and theoretical line shape is shown on the bottom of each figure...

Figures 2.37 and 2.38, show the isochronal curves of the permittivity and loss factor for P2NBM and P3M2NBM as a function of temperature at fixed frequencies. A prominent relaxation associated with the dynamic glass transition is observed in both polymers. Clearly the effect of the methyl substitution in position 3 of the norbornyl group is to decrease the temperature of this relaxational process.

The DMTA operates at fixed frequencies over a broad temperature range. Sixteen discrete frequencies from 0.01 Hz to 200 Hz are available. The very low frequencies, below about 0.1 Hz, require a long time to complete, while frequencies above 30 Hz are often near or above the system resonance and require special consideration. Though the system is capable of a... 

Thus, dynamic mechanical viscoelastic properties may be measured in tests with sinusoidal strain input at fixed frequency. Such tests have to be repeated at different frequencies over the range of interest to completely characterize the material. 

A sample of (Poly methyl methacrylate)(PMMA), known as Acrylite GP (a product of Cyro Ind.), was characterized using the DMA 983. The data were collected at fixed frequencies of 0.01,... 

To acquire an SFG vibrational spectrum of adsorbate molecules on a metal catalyst, two (picosecond) laser pulses are spatially and temporally overlapped on the sample (Fig. 5). One input beam is in the visible range at fixed frequency (covis), and the second one is tunable in the mid-IR region (giir) to probe the vibrational... 

The initial pulse of polymer solution which was injected into the column entry becomes diluted and attenuated as the different species are separated on the gel packing. The column effluent is monitored by detectors which respond to the weight concentration of polymer in the flowing eluant. The most common detector is a differential refractometer. Spectrophotometers, which operate at fixed frequencies, are also used as alternative or auxiliary detectors. Some special detectors which are needed particularly for branched polymers or copolymers are mentioned in Section 3.4.4. 

Fig. 3. The derivative of the Pt NMR absorption signal in a slightly alloyed platinum sample. (In this experiment the field is swept slowly while the sample is continuously irradiated at fixed frequency. Lock-in detection records the derivative.) Most of the Pt nuclei resonate at a field/frequency ratio of approximately 1.14 G/kUz, oft scale to the right (dashed curve). Those Pt atoms that are first neighbors to an alloying impurity resonate at 1.127 G/kUz, and second and third neighbors give separate resonances as well. The results show that the susceptibility is site dependent. [Reproduced with permission from Weisman and Knight (77). Copyright 1968 American Physical Society.]...

Above relation (1) between cr and y is exact in linear response, where nonlinear contributions in 7 are neglected in the stress. The linear response modulus (to be denoted as g (f)) itself is defined in the quiescent system and describes the small shear-stress fluctuations always present in thermal equilibrium [1, 3]. Often, oscillatory deformations at fixed frequency co are applied and the frequency dependent storage- (G (m)) and loss- (G"((u)) shear moduli are measured in or out of phase, respectively. The former captures elastic while the latter captures dissipative contributions. Both moduli result from Fourier-transformations of the linear response shear modulus g (f), and are thus connected via Kramers-Kronig relations. 

The elasticities and viscosities calculated from the oscillations as a function of the concentration are given in Figures 9 and 10. The two proteins show a completely different behaviour. The concentration dependencies for j -CS are fully in line with what we would expect for a slightly soluble surfactant. At fixed frequency, the elasticity increases while the viscosity decreases. This behaviour is observed for systems with a characteristic relaxation frequency larger than the applied frequency. Note, there is a peculiarity at the concentration where the isotherm shows a kink in the slope. 

These books remain the reference works in measuring dielectric and magnetic constants of homogeneous materials. Methods used at that time were limited in frequency band measurements, permitting only the determination of the complex permittivity and permeability at fixed frequencies and sometimes versus temperature variations. 

Fig. 3. Diagram of continuous wave (cw) laser sources suitable for metalloprotein resonance Raman spectroscopy. The best quality spectra are provided by Ar, Kr, He-Ne, and He-Cd lasers operating at fixed frequencies (the lengths of the lines indicate the relative output for a given laser) throughout the visible and near-UV region. An intracavity frequency-doubled (ICFD) Ar laser has been developed with five useful cw excitation wavelengths in the far-UV region (257, 248, 244, 238, and 228.9 nm). The high-powered Ar and Kr lasers can also be used to pump dye lasers which are tunable between the near-UV and near-IR region. The cw Nd YAG laser with a fundamental at 1064 nm is the primary excitation source in FT Raman spectrometers.

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