Translational diffusion process

Fig. 8.1. Stokes and free volume translational diffusion processes. Black circles solute molecules. White circles solvent molecules.

Dynamic quenching of fluorescence is described in Section 4.2.2. This translational diffusion process is viscosity-dependent and is thus expected to provide information on the fluidity of a microenvironment, but it must occur in a time-scale comparable to the excited-state lifetime of the fluorophore (experimental time window). When transient effects are negligible, the rate constant kq for quenching can be easily determined by measuring the fluorescence intensity or lifetime as a function of the quencher concentration the results can be analyzed using the Stern-Volmer relation ... 

The choice of method depends on the system to be investigated. The methods of intermolecular quenching and intermolecular excimer formation are not recommended for probing fluidity of microheterogeneous media because of possible perturbation of the translational diffusion process. The methods of intramolecular excimer formation and molecular rotors are convenient and rapid, but the time-resolved fluorescence polarization technique provides much more detailed information, including the order of an anisotropic medium. 

The rotational diffusion coefficient of the fuzzy cylinder can be formulated in a similar way. For the rotational diffusion process, it is convenient to imagine a hypothetical sphere which has the diameter equal to Lc, just encloses the test fuzzy cylinder, and moves with the translation of the fuzzy cylinder. If the test cylinder and the portions of surrounding fuzzy cylinders entering the sphere are projected onto the spherical surface as depicted in Fig. 15b (cf. [108]), the rotational diffusion process of the test cylinder can be treated as the translational diffusion process of a circle on the hypothetical spherical surface with ribbon-like obstacles. 

Diffusional motion. Many rotational and translational diffusion processes for hydrocarbons within zeolites fall within the time scale that is measurable by quasielastic neutron scattering (QENS). Measurements of methane in zeolite 5A (24) yielded a diffusion coefficient, D= 6 x lO" cm at 300K, in agreement with measurements by pulsed-field gradient nmr. Measurements of the EISF are reported to be consistent with fast reorientations about the unique axis for benzene in ZSM-5 (54) and mordenite (26). and with 180 rotations of ethylene about the normal to the molecular plane in sodium zeolite X (55). Similar measurements on methanol in ZSM-5 were interpreted as consistent with two types of methanol species (56). 

A major result of this work is that we quantitatively assessed the relevant role played by H-bond dynamics to determine the long-time diffusional properties. Indeed, both the rotational and translational diffusion processes are driven by a secondary time-dependent process the H-bond dynamics simulated by the stochastic variable %... 

Since translational diffusion process is sensitive to the microscopic structure in the solution, understanding the diffusion provides an important insight into the structure as well as the intermolecular interaction. Therefore, dynamics of molecules in solution have been one of the main topics in physical chemistry for a long time. 1 Recently we have studied the diffusion process of transient radicals in solution by the TG method aiming to understand the microscopic structure around the chemically active molecules. This kind of study will be also important in a view of chemical reaction because movement of radicals plays an essential role in the reactions. Here we present anomalous diffusion of the radicals created by the photoinduced hydrogen abstraction reaction. The origin of the anomality is discussed based on the measurments of the solvent, solute size, and temperature dependences. 

If it is further assumed that the molecule translates by a translational diffusion process, it is shown in Section (5.4 )that... 

For the DNP process to proceed from an initial polarization zone, it is essential that polarization be either transported to other nuclei via spin diffusion, as is the case in the solid state. Alternatively, in liquids unpolarized nuclei become available by translational diffusion. These two options are responsible for largely different polarization processes in (insulating) solids as compared to liquids (and metals). In liquids, this process is fundamentally different, and rotational and translational diffusion processes govern the polarization process, as originally described by Hausser et al. [37], Muller-Warmuth et al. [39] and others. Relaxation processes are driven by the time dependence of the overall Hamiltonian, arising form variations in the vector r and the hyperfine coupling constants, causing relaxation transitions between the spin states. 

Studies of the effect of permeant s size on the translational diffusion in membranes suggest that a free-volume model is appropriate for the description of diffusion processes in the bilayers [93]. The dynamic motion of the chains of the membrane lipids and proteins may result in the formation of transient pockets of free volume or cavities into which a permeant molecule can enter. Diffusion occurs when a permeant jumps from a donor to an acceptor cavity. Results from recent molecular dynamics simulations suggest that the free volume transport mechanism is more likely to be operative in the core of the bilayer [84]. In the more ordered region of the bilayer, a kink shift diffusion mechanism is more likely to occur [84,94]. Kinks may be pictured as dynamic structural defects representing small, mobile free volumes in the hydrocarbon phase of the membrane, i.e., conformational kink g tg ) isomers of the hydrocarbon chains resulting from thermal motion [52] (Fig. 8). Small molecules can enter the small free volumes of the kinks and migrate across the membrane together with the kinks. 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

In contrast to intermolecular excimer formation, this process is not translational but requires close approach of the two moieties through internal rotations during the lifetime of the excited state. Information on fluidity is thus obtained without the difficulty of possible perturbation of the diffusion process by microheterogeneity of the medium, as mentioned above for intermolecular excimer formation. 

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. 

The last point to be made is the famous Stokes-Einstein relationship that was found by Einstein by comparing the Brownian motion with common diffusion processes [66,67]. Accordingly the translational diffusion was found to depend... 

Pulsed field gradient (PFG)-NMR experiments have been employed in the groups of Zawodzinski and Kreuer to measure the self-diffusivity of water in the membrane as a function of the water content. From QENS, the typical time and length scales of the molecular motions can be evaluated. It was observed that water mobility increases with water content up to almost bulk-like values above T 10, where the water content A = nn o/ nsojH is defined as the ratio of the number of moles of water molecules per moles of acid head groups (-SO3H). In Perrin et al., QENS data for hydrated Nation were analyzed with a Gaussian model for localized translational diffusion. Typical sizes of confining domains and diffusion coefficients, as well as characteristic times for the elementary jump processes, were obtained as functions of A the results were discussed with respect to membrane structure and sorption characteristics. ... 

Diffusion is defined as the random translational motion of molecules or ions that is driven by internal thermal energy - the so-called Brownian motion. The mean movement of a water molecule due to diffusion amounts to several tenth of micrometres during 100 ms. Magnetic resonance is capable of monitoring the diffusion processes of molecules and therefore reveals information about microscopic tissue compartments and structural anisotropy. Especially in stroke patients diffusion sensitive imaging has been reported to be a powerful tool for an improved characterization of ischemic tissue. 

Classical studies of the relaxation processes, caused by translational diffusion, have been presented in the early days by Abragam (18), Torrey (136) and Pfeifer (137). Abragam (18) found, by solving the diffusion equation, the following form of the correlation function for the stochastic function Z>o under translational diffusion of two spins 1/2 ... 

Combining the above descriptions leads to a picture that describes the experimentally observed concentration dependence of the polymer diffusion coefficient. At low concentrations the decrease of the translational diffusion coefficient is due to hydrodynamic interactions that increase the friction coefficient and thereby slow down the motion of the polymer chain. At high concentrations the system becomes an entangled network. The cooperative diffusion of the chains becomes a cooperative process, and the diffusion of the chains increases with increasing polymer concentration. This description requires two different expressions in the two concentration regimes. A microscopic, hydrodynamic theory should be capable of explaining the observed behavior at all concentrations. 

As discussed in Section II, the Adam-Gibbs [48] model of relaxation in cooled liquids relates the structural relaxation times x, associated with long wavelength relaxation processes (viscosity, translational diffusion, rates of diffusion-limited... 

When the solution is dilute, the three diffusion coefficients in Eq. (40a, b) may be calculated only by taking the intramolecular hydrodynamic interaction into account. In what follows, the diffusion coefficients at infinite dilution are signified by the subscript 0 (i.e, D, 0, D10> and Dr0). As the polymer concentration increases, the intermolecular interaction starts to become important to polymer dynamics. The chain incrossability or topological interaction hinders the translational and rotational motions of chains, and slows down the three diffusion processes. These are usually called the entanglement effect on the rotational and transverse diffusions and the jamming effect on the longitudinal diffusion. In solving Eq. (39), these effects are taken into account by use of effective diffusion coefficients as will be discussed in Sect. 6.3. 

To explain the Green function method for the formulation of Dx, D and D, of the fuzzy cylinder [19], we first consider the transverse diffusion process of a test fuzzy cylinder in the solution. As in the case of rodlike polymers [107], we imagine two hypothetical planes which are perpendicular to the axis of the cylinder and touch the bases of the cylinder (see Fig. 15a). The two planes move and rotate as the cylinder moves longitudinally and rotationally. Thus, we can consider the motion of the cylinder to be restricted to transverse diffusion inside the laminar region between the two planes. When some other fuzzy cylinders enter this laminar region, they may hinder the transverse diffusion of the test cylinder. When the test fuzzy cylinder and the portions of such other cylinders are projected onto one of the hypothetical planes, the transverse diffusion process of the test cylinder appears as a two-dimensional translational diffusion of a circle (the projection of the test cylinder) hindered by ribbon-like obstacles (cf. Fig. 15a). 

When the diffusion time is short enough, the translation on the spherical surface is approximately identical with that on the tangent plane to the spherical surface. The latter is the two-dimensional diffusion process treated by the Green function method in Appendix C, and we can use Eq. (C17) again. Since the rotational diffusion coefficient Dr is related to the translational diffusion coefficient D<2) in Eq. (Cl7) by Dr = D(2)/(Le/2)2, we have... 

Translational diffusion of a particle can be described by the Green function. For simplicity, we consider here a one-dimensional diffusion process. Let x be the coordinate of the diffusing particle in its path. If no barriers are present in the path, the particle obeys the usual diffusion equation, and the unperturbed Green function associated with this diffusion equation is given by... 

The failure of the model to reproduce satisfactorily the dynamics of PeMe (Figure 3B) can be attributed to its slower dynamics. Diffusional processes then become more relevant and our rough estimation of kr, by means of the Stokes-Einstein expression, is probably not good enough. Much better agreement can be obtained when the translational diffusion coefficient is calculated with the semiempirical expression of Spemol and Wirtz [5]. 

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