The Orientation Factor

In most reactions, molecules must be oriented in a certain way during collision for a reaction to occur. The relative orientations of the molecules during collision determine whether the atoms are suitably positioned to form new bonds. For example, consider the reaction 

FIGURE 14.16 Energy is needed to overcome a barrier between initiai and fmal states. 

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The K factors in (C3.4.1) represent another very important facet of tire energy transfer [4, H]. These factors depend on tire orientations of tire donor and acceptor. For certain orientations tliey can reduce tire rate of energy transfer to zero—for otliers tliey effect an enhancement of tire energy transfer to its maximum possible rate. Figure C3.4.1 exhibits tire angles which define tire mutual orientation of a donor and acceptor pair in tenns of Arose angles the orientation factors and are given by [6, 7]... 

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

Dale, R., Eisinger, J. and Blumberg, W. (1979). The orientational freedom of molecular probes. The orientation factor in intramolecular energy transfer. Biophys. J. 26, 161-94. 

Importantly, in most applications the measured (change in) FRET efficiency cannot be translated directly into an average distance between donor and acceptor fluorophore because the fraction of donor molecules involved in FRET is unknown (i.e., all molecules display 25% FRET or 50% of the molecules display 50% FRET), and the orientation factor (k2) is unknown (see also Chapter 7). 

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. 

Fig. 4.17. Angles involved in the definition of the orientation factor k2 (left) and examples of values of tc2 (right).

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) ... 

Dale R. E., Eisinger J. and Blumberg W. E. (1975) The Orientational Freedom of Molecular Probes. The Orientation Factor in Intramolecular Energy Transfer, Biophys. J. 26, 161-194. 

Figure 1.20. (a) Angles 0 0y, and y, describing the relative orientation of the electronic transition dipole moments s between two dye molecules, (b) Relative orientations of the electronic transition dipole moments between two equal dye molecules in the channels of zeolite L. (c) Angular dependence of the orientation factor k2 under the anisotropic conditions (b) and averaged over y. 

Figure 13.3 also shows the orientation factors of the crystalline and amorphous regions as a function of take-up speed, which is pronounced in the case of a branched PET polymer. The shift towards increased freezing temperatures in branched polymer samples seems to be an indicator of higher elasticity (Figure 13.4). 

With the exception of the orientation factor, all the parameters in this equation may be obtained within reasonable error by direct experimental measurement or by estimation. The problem of setting reasonable values for k2, which may vary from 0 to 4 for orientations in which the dipole moments are orthogonal or parallel, respectively, is nontrivial. A value of , which is an unweighted average over all orientations, is often used. Dale et al.(53) have examined this problem in great detail and have shown that a k2 value of is never justified for energy transfer in macromolecules because it is impossible for the donors and acceptors to achieve a truly isotropic distribution. They do provide an experimental approach, using polarized emission spectroscopy, to estimate the relative freedom of motion for the donor and acceptor that allows reasonable limits to be set for k2. 

An important parameter required for the calculation of R0 is the orientation factor k2 which takes into account the angular dependence of dipole-dipole energy transfer, as described by eq 22... 

Since B-C has equal probability of all orientations over a hemisphere, we simply integrate cos2 0, weighted with the area of the hemisphere, to derive the orientation factor... 

N, is the number of surface atoms, A, is the amplitude of vibration of the sth atom of mass m,. The expression inside the brackets is called an amplitude factor, written by Stretton33 as (A2). In general the orientation factor may be taken as N,/6, since for each atom we need to consider orientation over a complete sphere. 

The rate of EET between a pair of weakly coupled donor (D) and acceptor (A) molecules, according to Forster theory, [76] depends on the interchromophoric distance R, expressed in units of cm, their relative orientation (through the orientation factor k), and the spectral overlap I between donor emission and acceptor absorption spectra. The rate expression is ... 

The orientation factor provides the projection of the molecular P tensor onto the crystal or laboratory axis (note that the crystal will be oriented intentionally with respect to the applied electric field by the act of depositing electrodes on the crystal). In Eq. (8), the indices I, J, K and i, j, k refer, respectively, to the principal directions of the crystal and molecular coordinate systems. Nc is the number of equivalent positions in the unit cell. 

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