Energy transfer rates

Here t. is the intrinsic lifetime of tire excitation residing on molecule (i.e. tire fluorescence lifetime one would observe for tire isolated molecule), is tire pairwise energy transfer rate and F. is tire rate of excitation of tire molecule by the external source (tire photon flux multiplied by tire absorjDtion cross section). The master equation system (C3.4.4) allows one to calculate tire complete dynamics of energy migration between all molecules in an ensemble, but tire computation can become quite complicated if tire number of molecules is large. Moreover, it is commonly tire case that tire ensemble contains molecules of two, tliree or more spectral types, and experimentally it is practically impossible to distinguish tire contributions of individual molecules from each spectral pool. 

Bonamy L., Thuet J. M., Bonamy J., Robert D. Local scaling analysis of state-to-state rotational energy-transfer rates in N2 from direct measurements, J. Chem. Phys. 95, 3361-70 (1991). 

When the triplet energy of the donor is 3 kcal/mole or more higher than the acceptor triplet energy, the energy transfer rate is about the diffusion-... 

Thus, E is defined as the product of the energy transfer rate constant, ku and the fluorescence lifetime, xDA, of the donor experiencing quenching by the acceptor. The other quantities in Eq. (12.1) are the DA separation, rDA the DA overlap integral, / the refractive index of the transfer medium, n the orientation factor, k2 the normalized (to unit area) donor emission spectrum, (2) the acceptor extinction coefficient, eA(k) and the unperturbed donor quantum yield, QD. 

Is net sensitized emission of the acceptor -Iad - I a)-kt energy transfer rate... 

C. Various Forms of Electron and Energy Transfer Rate Constants... 

Next we shall show that the electronic energy transfer rate can be put into the spectral overlap form. Notice that by ignoring the super-exchange term we have... 

Impact Number j qU0 2Ach TL-T ) Compare kinetic energy transfer rate to heat extraction rate Matson et al. [409]... 

In order to understand the role of spectral energy transfer in determining the turbulent energy spectrum at high Reynolds numbers, it is useful to introduce the spectral energy transfer rate Tu(jc,t) defined by... 

From this definition, it can be observed that T,(k. t) is the net rate at which turbulent kinetic energy is transferred from wavenumbers less than k to wavenumbers greater than k. In fully developed turbulent flow, the net flux of turbulent kinetic energy is from large to small scales. Thus, the stationary spectral energy transfer rate Tu(k) will be positive at spectral equilibrium. Moreover, by definition of the inertial range, the net rate of transfer through wavenumbers /cei and kdi will be identical in a fully developed turbulent flow, and thus... 

In order to understand better the physics of scalar spectral transport, it will again be useful to introduce the scalar spectral energy transfer rates T,f, and T p defined by... 

Equation (3.82) illustrates the importance of the scalar spectral energy transfer rate in determining the scalar dissipation rate in high-Reynolds-number turbulent flows. Indeed, near spectral equilibrium, 7 (/cd, 0 (like Tu(kDi, 0) will vary on time scales of the order of the eddy turnover time re, while the characteristic time scale of (3.82) is xn <[Pg.99]

When the scalar spectrum is not fully developed, the vortex-stretching term Vf will depend on the scalar spectral energy transfer rate evaluated at the scalar-dissipation wavenumber 7 (kd, O-32 Like the vortex-stretching term Vf appearing in the transport... 

The scalar spectral energy transfer rate XA m)- t) will vary on time scales proportional to the eddy turnover time re. At high Reynolds numbers, (3.131) and (3.132) quickly attain a quasi-steady state wherein... 

In homogeneous turbulence, spectral transport can be quantified by the scalar cospectral energy transfer rate Tap(ic, t). We can also define the wavenumber that separates the viscous-convective and the viscous-diffusive sub-ranges nf by introducing the arithmetic-mean molecular diffusivity Tap defined by... 

At high Reynolds number, we again find from (3.165) and (3.166) that the joint scalar dissipation rate is proportional to the cospectral energy transfer rate, i.e.,... 

Engineering models for the joint dissipation rate of inert scalars must thus provide a description of the cospectral energy transfer rate. We will look at such a model in Chapter 4. The final term in (3.151) is the joint-scalar-dissipation chemical source term S f defined... 

Despite the explicit dependence on Reynolds number, in its present form the model does not describe low-Reynolds-number effects on the steady-state mechanical-to-scalar time-scale ratio (R defined by (3.72), p. 76). In order to include such effects, they would need to be incorporated in the scalar spectral energy transfer rates. In the original model, the spectral energy transfer rates were chosen such that R(t) —> AV, = 2 for Sc = 1 and V

model parameter. DNS data for 90 < R-,. suggest that Re, is nearly constant. However, for lower... 

As described in Fox (1995), the wavenumber bands are chosen to be as large as possible, subject to die condition that the characteristic time scales decrease as the band numbers increase. This condition is needed to ensure that scalar energy does not pile up at intermediate wavenumber bands. The rate-controlling step in equilibrium spectral decay is then die scalar spectral energy transfer rate (T ) from die lowest wavenumber band. 

Note that at spectral equilibrium the integral in (A.33) will be constant and proportional to ea (i.e., the scalar spectral energy transfer rate in the inertial-convective sub-range will be constant). The forward rate constants a j will thus depend on the chosen cut-off wavenumbers through their effect on (computationally efficient spectral model possible, the total number of wavenumber bands is minimized subject to the condition that... 

The theory predicts a 1/R6 dependence of energy transfer rate on their separation distance, R, which is very steep. Deviation from this dependence is frequently observed for extended conjugated dye molecules, metal [38], and semiconductor [39] nanoparticles, the sizes of which are comparable with R, but the validity of this theory and 1/R6 dependence are confirmed in the studies of small dye molecules interacting at significant distances. 

This section deals with a single donor-acceptor distance. Let us consider first the case where the donor and acceptor can freely rotate at a rate higher than the energy transfer rate, so that the orientation factor k2 can be taken as 2/3 (isotropic dynamic average). The donor-acceptor distance can then be determined by steady-state measurements via the value of the transfer efficiency (Eq. 9.3) ... 

The energy-transfer rate constant kfT can be obtained similarly ... 

The trapping rate dPT/dt is given by the sum of energy-transfer rates from any site donor i to any trap site J. 

This equation shows that the ratio between the acceptor and donor fluorescence quantum yields is directly proportional to the energy-transfer rate constant kET. We have shown that this leads to the following linear relation between the fluorescence intensity of the acceptor 70x and that of the donor 7py, and the occupation probability pQx of the acceptor [3, 77] ... 

L channels. We assume that in all channels a situation can be prepared as illustrated in Figure 1.30, at the beginning of the experiment. Immediately after all dye molecules have entered the zeolite channels, the maximum energy transfer is observed because the donor-to-acceptor distance is short. The donor-to-acceptor distance increases, and hence the energy-transfer rate decreases, when the molecules diffuse deeper into the channels. From this, the following relation for diffusion kinetics is found under the condition that the initial distributions of the donors and the acceptors are the same, [)py+ p x+, denoted as p°. Experimental details can be found in [77],... 

---