Low-mode-number limit

At longer times, f 0.05(6 z )V Z s> the influence of normal modes with numbers p N 6k becomes essential. This limit is called the low-mode-number limit. In this case both contributions in the denominator of the integrand are important and the following result for the normal mode relaxation time can be obtained... 

The difference between the RRM and the TRRM becomes more pronounced in the low-mode-number limit, pnormal-mode relaxation time of order p reads... 

In the low-mode-number limit, the TRRM predicts for the mean squared segment displacement... 

Below it will be shown that field-cycling NMR relaxometry studies unambiguously reveal a crossover between high-frequency and low-frequency dispersion regimes that can be identified with the high-mode-number and low-mode-number limits of the renormalized Rouse models. Moreover, the variation of the power law exponents closely corresponds to that predicted by the renormalized Rouse models. These dynamic regimes cannot be ex-... 

Fig. 28. Proton spin-lattice relaxation times in the laboratory system (Tj) and in the rotating frame (Tip) of polyisobutylene (PIB) melts as a function of the frequency (v, Lar-mor frequency in the laboratory frame Vi=yBi/(27i), rotating-frame nutation frequency) [125]. The data refer to the molecular weight independent chain-mode regimes I (high-mode-number limit) and II (low-mode-number limit) [49], The arrow indicates the crossover frequency between regions I and II...

High- and Low-Mode-Number Limits (Dispersion Regions I and II)... 

Based on the generalized Langevin equation, the renormalized Rouse models suggest dynamic high- and low-mode-number limits as an implicit structural feature of this equation of motion. This is a stand-alone prediction of paramount importance independent of any absolute values of power law exponents that arise and are measured in the formalism and in experiment, respectively. The two limits manifesting themselves as power law spin-lattice relaxation dispersions were clearly identified in bulk melts of entangled polymers of diverse chemical species. 

The appearance of dispersion regions I and II in experiments is an exultant confirmation of the high- and low-mode-number, short-time limits predicted by the twice renormalized Rouse models (Tables 2 and 3). The exponents of the power laws suggested by the experimental data even match the theoretical predictions almost perfectly [47, 49]. Nevertheless, the good coincidence of the numerical values of these exponents is not considered to be the decisive finding backing up the renormalized Rouse models. The problem is that the theoretical exponents are slightly affected by the renormalization... 

The time dependence of the mean squared segment displacement in the laboratory frame was derived on the relevant time scale as low-mode-number, short-time limits of the renormalized and twice renormalized Rouse models as and (R ,(f)) oc (limit (II)trr)> respectively... 

Due to the different working principles of WDXRF and EDXRF, the applications differ strongly (Table 8.43). Simultaneous WDXRF with ten channels (elements) and increased sensitivity for the low atomic number elements (e.g. a few ppm of phosphorous in a low atomic number matrix) has been used for QC of polymer granules [252], To detect elements at trace levels (ppm-ppt), generally the special XRF modes, mainly EDXRF techniques, are applied like TXRF, SR-XRF or pXRF. Detection limits with SR-XRF are now at the attogram level. 

An alternative carrier-gas system uses a condensable gas, such as steam, as the carrier sweep fluid. One variant of this system is illustrated in Figure 9.7(d). Low-grade steam is often available at low cost, and, if the permeate is immiscible with the condensed carrier, water, it can be recovered by decantation. The condensed water will contain some dissolved organic and can be recycled to the evaporator and then to the permeate side of the module. This operating mode is limited to water-immiscible permeates and to feed streams for which contamination of the feed liquid by water vapor permeating from the sweep gas is not a problem. This idea has been discovered, rediscovered, and patented a number of times, but never used commercially [37,38], If the permeate is soluble in the condensable... 

To improve sensitivity, selected ion monitoring (SIM) mode may be used for the detection of routine YOCs and SVOCs and in low resolution dioxin/furan analysis (EPA Method 8280). In the SIM mode, only specific ions from the analyte s spectrum are scanned for the detector s dwelling time on each ion is increased resulting in higher sensitivity. A mass spectrometer operated in the SIM mode is approximately ten times as sensitive as one in the full scan mode. The SIM mode has limitations, such as the capacity to monitor only a limited number of ions and the need to monitor multiple ions for each compound to improve the degree of confidence in compound identification. That is why typically no more than 20 compounds can be analyzed simultaneously in the SIM mode. 

The main limitation of the TAB model is that it can only keep track of one oscillation mode, while in reality more than one mode exists. The model keeps track only of the fundamental mode, corresponding to the lowest order harmonic whose axis is aligned with the relative velocity vector between droplet and gas. This is the most important oscillation mode, but for large Weber numbers other modes are also contributing significantly to drop breakup. Despite this limitation, a rather good agreement is achieved between the numerical and experimental results for low Weber numbers. 

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