Effective modes

Economic and process considerations usually dictate that agitated thin-film evaporators be operated in single-effect mode. Veiy high temperature differences can then be used many are heated with Dowtherm or other high-temperature media. This permits achieving reasonable capacities in spite of the relatively low heat-transfer coefficients and the small surface that can be provided in a single tube [to about 20 m" (200 ft")]. The structural need for wall thicknesses of 6 to 13 mm (V4 to V2. in) is a major reason for the relatively low heat-transfer coefficients when evaporating water-like materials. 

The efficiency of an evaporator can be increased by operating the equipment in single or multiple-effect modes. 

In typical cause-and-effect mode, where chlorides penetrate the deposit or where a localized overconcentration of hydroxyl ions occurs, the magnetite film is disrupted and particular forms of very damaging corrosion occurs. In addition, where localized heat flux exceeds design limits within a boiler and may be accompanied by departure from nucleate boiling (DNB) conditions, overheating and metal failure may also occur. 

One of Balsamo s most effective modes of warfare was to paint Masonic and biblical emblems and prophecies on the walls of his cell—the Sicilian had, it seems, returned to one of his earliest talents. Most of these murals were smeared on the plaster with a paint he created by blending his urine with rust from the bars of the cell. Sometimes he also used excrement. He made a brush by ripping away pieces of wood from his bed with his teeth. To this handle he somehow sewed a straw and cotton tip. He could use his improvised brush with uncanny speed on one occasion Marinis spyhole was unattended for only an hour, during which time Balsamo managed to fill an entire wall with loathsome drawings. ... 

The waveguides were optically characterized at X = 632.8 nm. The effective mode indices were determined by the m-line technique, based on a standard two prism coupling set-up. The refractive index depth profiles of the fabricated waveguides were reconstructed by the means of the inverse WKB procedure . 

The propagation losses and their dependence on the effective mode index were estimated with the scattering detection method at X = 632.8 nm. 

Hernandez-Romero Y Rojas J-I, Castillo R, Rojas A, MataR, Spasmolytic effects, mode of action, and structure-activity relationships of stilbenoids from Nidema boothii, J Nat Prod 67 160—167, 2004. 

Small radicals such as tert-butylperoxy and ethylperoxy can, however, react via 1,4 H-transfer only the strain energy involved in O-heterocycle formation is 28 kcal. per mole. In this case, k.4(x — 106 sec."1 whereas krta = 10r> 4 sec. 1 and when [02] = 200 mm. of Hg, ko[02] = 105,3 sec. 1, so that k.4ct < < (tkr,a + k [02]). The result is that in the oxidation of small alkyl radicals, the route via alkylperoxy radicals will be blocked because reverse Reaction —4 competes successfully with Reaction 5. Reaction 2 will thus be a more effective mode of reaction of alkyl radicals with oxygen and the conjugate alkene will be a major product. 

Third, ultrashort laser pulses can provide for selective electronic or vibrational excitation of molecules, which will probably allow one to effect a high selective excitation of certain vibrational degrees of freedom, that is, to effect mode-selective photochemical reactions. 

Vibronic coupling in many dimensions conical intersections and effective modes... 

The LVC model further allows one to introduce coordinate transformations by which a set of relevant effective, or collective modes are extracted that act as generalized reaction coordinates for the dynamics. As shown in Refs. [54, 55,72], neg = nei(nei + l)/2 such coordinates can be defined for an electronic nei-state system, in such a way that the short time dynamics is completely described in terms of these effective coordinates. Thus, three effective modes are introduced for an electronic two-level system, six effective modes for a three-level system etc., for an arbitrary number of phonon modes that couple to the electronic subsystem according to the LVC Hamiltonian Eq. (7). In order to capture the dynamics on longer time scales, chains of such effective modes can be introduced [50,51,73]. These transformations, which are briefly summarized below, will be shown to yield a unique perspective on the excited-state dynamics of the extended systems under study. 

The residual Hamiltonian Hles contains the remaining (X —3) modes, and their bilinear coupling to the effective modes,... 

Using this transformation, it has been shown in Refs. [54,72] that the effective-mode Hamiltonian Heg by itself reproduces the short-time dynamics of the overall system exactly. This is reflected by an expansion of the propagator, for which it can be shown that the first few terms of the expansion - relating to the first three moments of the overall Hamiltonian - are exactly reproduced by the reduced-dimensional Hamiltonian Heg. 

The effective-mode transformation described here is closely related to earlier works which led to the construction of so-called interaction modes [75, 76] or cluster modes [77, 78] in Jahn-Teller systems. The approach of Refs. [54,55,72] generalizes these earlier analyses to the generic form - independent of particular symmetries - of the linear vibronic coupling Hamiltonian Eq. (8). 

In Refs. [55, 79], the truncation at the level of Heg has been tested for several molecular systems exhibiting an ultrafast dynamics at Coin s, and it was found that this approximation can give remarkably good results in reproducing the short-time dynamics. This is especially the case if a system-bath perspective is appropriate, and the effective-mode transformation is only applied to a set of weakly coupled bath modes [55,72]. In that case, the system Hamiltonian can take a more complicated form than given by the LVC model. 

For the polymer systems studied here, the approximation defined by Heg is not necessarily sufficient, as demonstrated below (Sec. 5). We therefore resort to a strategy which generalizes the effective-mode approach in such a way that a chain of effective modes is generated, which successively unravel the dynamics as a function of time. 

To this end, an additional orthogonal coordinate transformation is introduced, by which the bilinear couplings occurring in Eq. (13) are transformed to a band-diagonal form that only allows a coupling to the (three) nearest neighbors. By concatenating the effective-mode construction described in the previous section with this additional transformation in the residual-mode subspace,... 

Fig. 5 Schematic illustration of the HEP construction. In addition to the transformation which identifies the three effective modes that couple directly to the electronic subsystem, further transformations are introduced for the residual bath in such a way that the chain-like representation of Eqs. (14)-(15) is obtained.

If the phonon distribution of the model Eq. (8) spans a dense spectrum - as is generally the case for the extended systems under consideration, which are effectively infinite-dimensional - the dynamics induced by the Hamiltonian will eventually exhibit a dissipative character. However, the effective-mode construction demonstrates that the shortest time scales are fully determined by few effective modes, and by the coherent dynamics induced by these modes. The overall picture thus corresponds to a Brownian oscillator type dynamics, and is markedly non-Markovian [81,82],... 

A generalization of the effective-mode construction to three or more electronic states is straightforward, using a set of neff = nei(nei + l)/2 effective modes as mentioned above. In Refs. [52,53], we have thus employed a three-state representation with six effective modes, based on the following form of the Hamiltonian which generalizes Eq. (14),... 

We now briefly summarize the key results of the analysis of Refs. [50,51] for a reduced XT-CT model of the TFB F8BT heterojunction, using explicit quantum dynamical (MCTDH) calculations for a two-state model parametrized for 20-30 phonon modes. At this level of analysis, an ultrafast ( 200 fs) XT state decay is predicted, followed by coherent oscillations, see Fig. 8 (trace exact in panel (a)). Further analysis in terms of an effective-mode model and the associated HEP decomposition (see Sec. 4.2) highlights several aspects ... 

Fig. 7 For the two-state XT-CT model, the projection of the three effective modes (Xi, X2, X3) (shown in red, blue, and green, respectively) onto the primitive phonon modes xi is illustrated, for a model comprising 14 primitive high-frequency modes and 14 primitive low-frequency modes (i.e., 28 modes overall). Even though the projection involves contributions from both phonon bands, the low-frequency contributions are small, and all three effective modes are of high-frequency type. (Note the change in scale between the l.h.s. and the r.h.s. of the figure.) Furthermore, since the primitive modes are localized on the individual molecular units, the effective modes can be shown to be partially localized as well. Thus, two of the effective modes (shown in blue and green) are dominated by local contributions coming from either the F8BT chain or the TFB chain. The third mode (shown in red) exhibits contributions from both chains. (Reproduced from Ref. [93].)...

Figs. 7 and 8 illustrate the HEP analysis for the XT-CT model in more detail. Fig. 7 shows the decomposition of the first three modes of the hierarchy (Xi, X2, X3) into the primitive phonon modes [93], As mentioned above, these modes constituting Heg, are all of high-frequency type. Two of these effective modes can be seen to be localized on the TFB vs. F8BT moieties, whereas the third mode is delocalized over both chains. 

Fig. 8 shows time-dependent state populations as obtained from quantum dynamical (MCTDH) calculations. While the full (here, 24 dimensional) model exhibits an ultrafast XT decay, no net decay is observed for the reduced 3-mode model truncated at the lowest level of the effective mode hierarchy. The dynamics is strongly diabatic if confined to the high-frequency subspace (Heff ) and involves repeated coherent crossings [51]. The dynamical interplay between the high-frequency and low-frequency modes is apparently a central feature of the process. To account for these effects, a treatment at the level of is necessary, i.e., a six-mode model including the low-frequency modes. At the level of the dynamics is found to be essentially exact. 

A qualitatively correct picture of the dynamics can indeed be obtained from a two effective-mode model - one high-frequency mode plus one low-... 

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