Pure Rotational Diffusion

1 Pure Rotational Diffusion Here we consider the rotational motion of the rodhke molecule in details. We do not pay attention to the center-of-mass position. Let us define by u(t) the unit vector along the rod axis at time t and place the rodlike molecule in the spherical polar coordinate system (Fig. 3.59). The orientation vector u(0 is represented by the polar angle d and the azimuthal angle p. We define the probabihty density iK0, P, 0 for the distribution of u(t) in the same way as the concentration represents the population of solute molecules per volume. The probabihty to find u(0 between 6 and 6 + dO and between p and + d is Osin0d0d.  

The rotational part of the motion is described by the rotational diffusion equation for p(0, p, t)  

In some systems, ip(0,(p t) does not depend on p. For instance, when we consider how the probability density i/ (u t) evolves for a rod with u(0) parallel to the polar axis, the distribution is a function of 0 and t only. Another example is a rodlike molecule that has a permanent dipole moment along the axis in an electric field. The natural choice of the polar axis is the direction of the electric field. When ip depends on 6 and t only, the rotational diffusion equation is simplified to 

We apply Eq. 3.260 to consider how fast the correlation of the rod orientation is lost. We place the polar axis in the direction of u(0) and consider how (u(t) u(0)) = (cosff) = (Piicosd)) changes with time. The statistical average is calcnlated with a weight of sin0. From Eq. 3.260, 

Rotational motion of the rodUke molecule can be viewed as the motion of its end point on the surface of a sphere with the rod as its diameter (Fig. 3.60). Over a short period of time (Dj 1), 6 1 and therefore (P/(cos0)) = (cos0) = (1 - (P/2) = 1 (dV2). The right-hand side of Eq. 3.264 is = 1 t/ri = 1 - 2D. Thus, ( ) = 4Djt. The end point makes a two-dimensional diffusion. Over a longer period of 

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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. 

Theoretically, similar dimer structures appear near all rotational transition frequencies of nearly any molecule, for example if N2 is substituted for H2 in such measurements. However, the narrowly spaced N2 So(J) lines are much more numerous and the dimer structures are thus more difficult to resolve. Just a few measurements exist in the purely rotational induced bands of molecules other than H2. The phenomena described here are, however, nearly universal and should in general be more complex if other molecules than H2 are involved than the examples shown. Note that in Fig. 3.26 the diffuse induced H2 So(0) fine was suppressed. Similar structures have been seen in the other H2 So(J) fines [268]. 

In Figure 2.10 we show a selection of results, in which experimental and calculated spectra are compared at 292 and 155K. The results are quite satisfactory, especially when considering that no fitted parameters, but only calculated quantities (via QM and hydrodynamic models) have been employed. The overall satisfactory agreement of the spectral line shapes, particularly at low temperatures, is a convincing proof that the simplified dynamic modelling implemented in the SLE through the purely rotational stochastic diffusive operator f, and the hydrodynamic calculation of the rotational diffusion tensor, is sufficient to describe the main slow relaxation processes. 

Four different models for the molecular dynamics have been tested to simulate the experimental spectra. Brownian rotational diffusion and jump type diffusion [134, 135] have been used for this analysis, both in their pure forms and in two mixed models. Brownian rotational diffusion is characterized by the rotational diffusion constant D and jump type motion by a residence time t. The motions have been assumed to be isotropic. In the moderate jump model [135], both Brownian and jump type contributions to the motion are eou-pled via the condition Dx=. ... 

The time-dependent expression of photo-orientation is derived by considering the elementary contribution per unit time to the orientation by the fraction of the molecules dC (Ll), whose representative moment of transition is present in the elementary solid angle dQ near the direction Q(0, ) relative to the fixed laboratory axes (see Figure 3.4). This elementary contribution results from orientational hole burning, orientational redistribution, and rotational diffusion. The transitions are assumed to be purely polarized, and the irradiation light polarization is along the Z axis. The elementary contribution to photo-orientation is given by ... 

Photoisomerization was studied from a purely photochemical point of view in which photo-orientation effects can be disregarded. While this feature can be true in low viscosity solutions where photo-induced molecular orientation can be overcome by molecular rotational diffusion, in polymeric environments, especially in thin solid film configurations, spontaneous molecular mobility can be strongly hindered and photo-orientation effects arc appreciable. The theory that coupled photoisomerization and photo-orientation processes was also recently developed, based on the formalism of Legendre Polynomials, and more recent further theoretical developments have helped quantify coupled photoisomerization and photo-orientation processes in films of polymer. 

Relative integrated intensities of light scattered from optically isotropic rigid rods. S is the total relative integrated intensity, So the intensity of the pure translational part, Si the first non-zero term whose spectral width contains the rotational diffusion coefficient, and Si, the sum of intensities of all other terms. 

By way of introduction let us note that the depolarized spectrum Ivh(co) calculated in Section 7.5 for independent rotors consists of a superposition of Lorentzian bands all centered at zero frequency. In the simplest case of symmetric top rotors the spectrum consists of a single band with a width [q2D + 6<9] which depends only on the translational self-diffusion coefficient D and on the rotational diffusion coefficient 0. This should be compared and contrasted with the depolarized spectrum Ivh(co) of certain pure liquids (e.g., aniline, nitrobenzene, quinoline, hexafluorobenzene) shown schematically in Fig. 12.1.1. The spectrum appears to be split. This entirely novel fea-... 

Another contribution to the correlation function, which cannot be neglected for large particles, is their rotational motion. This problem has not been treated theoretically so far but intuitively one can expect that when a large particle rotates, the mean-square displacement of embedded flurophores would be larger than that without rotation. Hence, a higher apparent diffusion coefficient than that for a pure translational diffusion should be observed. Recently, we also addressed this problem in our computer simulations. Because the use of PCS in polymer science is very limited, we will describe its advantageous features in Sect. 3.5. 

Fig. 21.5. Characteristic values related to proton conduction (a) , self-diffusion (b) and reorientation (c) in sulphuric acid aqueous solutions at 290 K as a function of H2SQ, concentration expressed in weight % (top scale) or molarity (bottom scale). For pure HjO, the QNS study yields the same D value as measured by NMR spin-echo or tracer methods . In the same graph (b) is reported the residence time xp measured by QNS and defined in Eqn (21.5). The rotational diffusion constant >, defined in Eqn (21.8) is reported in graph (c) together with the characteristic rotational time x, = fi/6 >,.

Fig. 6.13. The theoretical width of the broad line (in units of 2a) as a function of k x y. The solid curved line corresponds to pure bound translational diffusion. The dashed line corresponds to a combination of bound translational and free rotational diffusion. The straight line corresponds to a linewidth of 2k D, obtained for free diffusion. The insert shows the theoretical width of the narrow line in units of F (the natural linewidth) as a function of 2k D/r, for spheres of radius of 20 nm participating in bound translational and free rotational diffusive motions. (Ofer et ai, 1984.)...

Fig. 5. Dynamic electrostatic attachment of Fremy s salt dianion to PDADMAC. (a) The CW ESR spectrum (=9.6 GHz) of a 0.5-mA/ solution of in pure water, (jb) Spectrum after addition of PDADMAC with a concentration of 10-mM repeat units, (c) Model of the site-bound state derived from the rotational diffusion tensor and from N ESEEM measurements...

That a large increase in the flow viscosity may make little difference in the microscopic viscosity is exemplified by the fact that the translational and rotational diffusion of small molecules in gels is much the same as in the pure solvent. 

Fig. 3.12 Ratio of the hydrodynamic radius Rh to (in our notation), plotted versus Here the hydrodynamic radius is not defined as a purely geometric quantity, but rather as the Stokes radius of a sphere which would have the same sedimentation velocity as the polymer. The latter is obtained via static dynamics , also taking rotational diffusion into account (Zimm s approach. ). Data are shown for star polymers on the cubic lattice and/= 1,3,4,6 (from top to bottom) for different effective monomeric Stokes radii a = 1/4 (full symbols) and 0=1/2 (open symbols) (from Ref. 106).

Overall, this treatment predicts that the scalar term for rotational diffusion is independent of double quantum relaxation (W2) and depends only on zero quantum relaxation. The coupling factor can assume values between 0.5 and —1.0 for pure dipolar and pure scalar relaxation, respectively. Moreover, the curves in Fig. 4 clearly show the expected field dependence with low values for the coupling factor, and therefore low enhancements for high magnetic fields. 

These experimental findings on the rotational dynamics of anthraquinone dyes in liquid hosts are also interesting as they show deviations from hydro-dynamic theories usually used for describing the rotational diffusion of dye molecules in liquid solvents [6-12]. These theories are commonly gathered in the Uterature under the name of Stokes-Einstein-Debye theory (SED). SED models treat the solvent as a macroscopic continuum, in which the diffusional rotation of the solute is only affected by the viscosity and temperature of the hosting solvent. However, as expected, the validity of this continuum description breaks down when the size of the solute molecules approaches that of the solvent molecules or becomes smaller. In this regime that includes the important case of pure materials, the specific intermolecular interactions between the solute and the solvent molecules start to play a fundamental role in their rotational dynamics [9,11-16]. 

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