Translation-Rotational Diffusion

2 Translation-Rotational Diffusion We consider both the center-of-mass translation and the rotation now. The transition probability P(ro,u ro,u t) from (ro,u ) at time 0 to (ro,u) at time t follows the diffusion equation  

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Models for description of liquids should provide us with an understanding of the dynamic behavior of the molecules, and thus of the routes of chemical reactions in the liquids. While it is often relatively easy to describe the molecular structure and dynamics of the gaseous or the solid state, this is not true for the liquid state. Molecules in liquids can perform vibrations, rotations, and translations. A successful model often used for the description of molecular rotational processes in liquids is the rotational diffusion model, in which it is assumed that the molecules rotate by small angular steps about the molecular rotation axes. One quantity to describe the rotational speed of molecules is the reorientational correlation time T, which is a measure for the average time elapsed when a molecule has rotated through an angle of the order of 1 radian, or approximately 60°. It is indirectly proportional to the velocity of rotational motion. 

Chandler D. Translational and rotational diffusion in liquids. I. Translational single-particle correlation functions. J. Chem. Phys. 60, 3500-507, (1974). Translational and rotational diffusion in liquids. II. Orientational single-particle correlation functions. J. Chem. Phys. 60, 3508-12 (1974). 

We would like to point out that an order parameter indicates the static property of the lipid bilayer, whereas the rotational motion, the oxygen transport parameter (Section 4.1), and the chain bending (Section 4.4) characterize membrane dynamics (membrane fluidity) that report on rotational diffusion of alkyl chains, translational diffusion of oxygen molecules, and frequency of alkyl chain bending, respectively. The EPR spin-labeling approach also makes it possible to monitor another bulk property of lipid bilayer membranes, namely local membrane hydrophobicity. 

The time window available for FRET is dictated by the lifetime of the donor. Is there an optimal lifetime If very short , it is more difficult to measure in FLIM. If it is very long , the levels of fluorescence are low ( /-limited , [1]). In addition, the lifetime is a relevant parameter when one is interested in dynamics, either of binding, conformational change, or diffusion (translational, rotational). These processes influence FRET via the parameters K2 andrDA (Table 12.1). Long lifetimes are useful in luminescence RET (LRET) and can help to reduce background and increase signal-to-noise ratios. 

It is clearly seen that the rate constants of all the studied reactions are considerably lower in the solid phase of PE and PP than in benzene or chlorobenzene solution. At the same time, these reactions are not limited by translational or rotational diffusion. 

The above kinetic scheme of the bimolecular reaction simplifies physical processes that proceed via the elementary bimolecular act. To react, two reactants should (a) meet, (b) be oriented by the way convenient for the elementary act, and (c) be activated to form the TS and then react. Hence, not only translational but also rotational diffusion of particles in the solution and polymer are important for the reaction to be performed. So, the more detailed kinetic scheme of a bimolecular reaction includes the following stages diffusion and encounter the reactants in the cage, orientation of reactants in the cage due to rotational diffusion, and activation of reactants followed by reaction [5,13]. 

Reactions described earlier were not limited by rotational diffusion of reactants. It is evident that such bimolecular reactions can occur that are limited not by translational diffusion but by the rate of reactant orientation before forming the TS. We discussed the reactions of sterically hindered phenoxyl recombination in viscous liquids (see Chapter 15). We studied the reaction of the type radical + molecule, which are not limited by translational diffusion in a solution but are limited by the rate of reactant orientation in the polymer matrix [28]. This is the reaction of stable nitroxyl radical addition to the double bond of methylenequinone. 

The non-collective motions include the rotational and translational self-diffusion of molecules as in normal liquids. Molecular reorientations under the influence of a potential of mean torque set up by the neighbours have been described by the small step rotational diffusion model.118 124 The roto-translational diffusion of molecules in uniaxial smectic phases has also been theoretically treated.125,126 This theory has only been tested by a spin relaxation study of a solute in a smectic phase.127 Translational self-diffusion (TD)29 is an intermolecular relaxation mechanism, and is important when proton is used to probe spin relaxation in LC. TD also enters indirectly in the treatment of spin relaxation by DF. Theories for TD in isotropic liquids and cubic solids128 130 have been extended to LC in the nematic (N),131 smectic A (SmA),132 and smectic B (SmB)133 phases. In addition to the overall motion of the molecule, internal bond rotations within the flexible chain(s) of a meso-genic molecule can also cause spin relaxation. The conformational transitions in the side chain are usually much faster than the rotational diffusive motion of the molecular core. 

In fluorescence correlation spectroscopy (FCS), the temporal fluctuations of the fluorescence intensity are recorded and analyzed in order to determine physical or chemical parameters such as translational diffusion coefficients, flow rates, chemical kinetic rate constants, rotational diffusion coefficients, molecular weights and aggregation. The principles of FCS for the determination of translational and rotational diffusion and chemical reactions were first described in the early 1970s. But it is only in the early 1990s that progress in instrumentation (confocal excitation, photon detection and correlation) generated renewed interest in FCS. 

G(t) decays with correlation time because the fluctuation is more and more uncorrelated as the temporal separation increases. The rate and shape of the temporal decay of G(t) depend on the transport and/or kinetic processes that are responsible for fluctuations in fluorescence intensity. Analysis of G(z) thus yields information on translational diffusion, flow, rotational mobility and chemical kinetics. When translational diffusion is the cause of the fluctuations, the phenomenon depends on the excitation volume, which in turn depends on the objective magnification. The larger the volume, the longer the diffusion time, i.e. the residence time of the fluorophore in the excitation volume. On the contrary, the fluctuations are not volume-dependent in the case of chemical processes or rotational diffusion (Figure 11.10). Chemical reactions can be studied only when the involved fluorescent species have different fluorescence quantum yields. 

When the excitation light is polarized and/or if the emitted fluorescence is detected through a polarizer, rotational motion of a fluorophore causes fluctuations in fluorescence intensity. We will consider only the case where the fluorescence decay, the rotational motion and the translational diffusion are well separated in time. In other words, the relevant parameters are such that tc rp, where is the lifetime of the singlet excited state, zc is the rotational correlation time (defined as l/6Dr where Dr is the rotational diffusion coefficient see Chapter 5, Section 5.6.1), and td is the diffusion time defined above. Then, the normalized autocorrelation function can be written as (Rigler et al., 1993)... 

The above experimental results largely relate to spectroscopic techniques, which do not give direct information about the spatial scale of the molecular motions. The size of the spatial heterogeneities is estimated by indirect methods such as sensitivity of the dynamics to the probe size or from the differences between translational and rotational diffusion coefficients (rotation-translation paradox). It might be expected that the additional spatial information provided by neutron scattering could help to discriminate between the two scenarios proposed. 

An important theoretical development for the outer-sphere relaxation was proposed in the 1970s by Hwang and Freed (138). The authors corrected some earlier mistakes in the treatment of the boundary conditions in the diffusion equation and allowed for the role of intermolecular forces, as reflected in the IS radial distribution function, g(r). Ayant et al. (139) proposed, independently, a very similar model incorporating the effects of molecular interactions. The same group has also dealt with the effects of spin eccentricity or translation-rotation coupling (140). 

Fig. 3. Schematic representation of the topological space of hydration water in silica fine-particle cluster (45). The processes responsible for the water spin-lattice relaxation behavior are restricted rotational diffusion about an axis normal to the local surface (y process), reorientations mediated by translational displacements on the length scale of a monomer (P process), reorientations mediated by translational displacements in the length scale of the clusters (a process), and exchange with free water as a cutoff limit.

Nuclear magnetic resonance measurements of methane adsorbed to various coverages on titanium dioxide have been made by Fuschillo and Renton 16). At a coverage of 0.95 monolayer and at 20.4°K, the X-point for solid bulk methane, these authors observed an abrupt change in the proton resonance line width, presumably due to translational and rotational diffusion of methane molecules. For pure, bulk methane no change has been observed in the line width at the X-point. 

Schmitz and Lu (12) have also considered coupling of translational and rotational modes for rigid rods in congested solutions as an explanation for the extraordinary diffusion regime. They concluded that, for 350 bp dinucleosomal DNA, the coupled ion model gave better agreement with the data. Since our DNA molecules are even shorter, 150 bp, the coupled translation -rotation model can seemingly be ruled out. 

Note Rotational diffusion may be compared to translational diffusion, through which the equilibrium statistical distribution of position in space is maintained or restored. 

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