Rotational diffusion model, Debye

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

Now we turn to a detailed discussion of rotational motion in water. As already mentioned, the Debye rotational diffusive model was initially widely employed to describe water reorientation. As explained above, it describes the reorientation as an... 

C. Anomalous Dielectric Relaxation in the Context of the Debye Noninertial Rotational Diffusion Model... 

The correlation time, in Eq. (4) is generally used in the rotational diffusion model of a liquid, which is concerned with the reorientational motion of a molecule as being impelled by a viscosity-related frictional force (Stokes-Einstein-Debye model). Gierer and Wirtz have introduced the idea of a micro viscosity, The reorientational... 

The basic concepts of linear response theory are best illustrated by considering the rotational diffusion model of an assembly of electric dipoles constrained to rotate in two dimensions due to Debye [14] which is governed by the Smoluchowski equation... 

We have illustrated the concept of linear response using the simple rotational diffusion model of Debye. We will now illustrate how the concept applies for a rotator in an external potential. For convenience we will describe the procedure for the itinerant oscillator model given by Coffey [72],... 

The available results present something of a problem for explanation in terms of rotational diffusion models, even after recognizing that two distinct species with similar static properties and molecular sizes but different relaxation rates can relax at nearly the same rate in an environment with a common (i.e., mutual) viscosity, as Kivelson has pointed out to the writer (91). Such conditions are not met in some of the available examples. Ones from very recent work are for mixtures of methanol, ethanol, and 2-propanol with added water up to 0.5 mole fraction for the first two and at 0.15 for the last by Bertolini, Cassettori, and Salvetti (92) at frequencies from 470 MHz to 3.75 GHz. They found essentially single Debye relaxations in all cases, with indications of slight broadening at -43 0. 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

Rotational Diffusion in a Shear Field Consider for a moment rod-shaped particle of unit length. The orientation of the rod, u, can be specified by a unit vector u directed along its axis with spherical polar coordinates, Q = (0, used to describe its orientation. Because the solvent molecules are expected to frequently collide with the rod, it should exhibit a random walk on the surface of the unit sphere (i.e., r = 1.0). Debye [15] in 1929 developed a model for the reorientation process based on the assumption that collisions are so frequent that a particle can rotate through only a very small angle before having another reorienting collision (i.e., small step diffusion). 

It is evident in Fig. 25b that the ratio has a value close to 3 at high temperatures (T > 1.0) and declines steadily below T 1.0 until it reaches a value nearly equal to unity at low temperatures. While the Debye model of rotational diffusion, which invokes small steps in orientational motion, predicts the ratio ti/t2 to be equal to 3, a value for this ratio close to 1 is taken to suggest the involvement of long angular jumps [146, 147]. The ratio was observed to deviate from the Debye limit at lower temperatures in a recent molecular dynamics simulation study as well [148]. The onset temperature was thus found to mark the breakdown of the Debye model of rotational diffusion [145]. Recently, the Debye model of rotational diffusion was also demonstrated to break down for calamitic liquid crystals near the I-N phase boundary due to the growth of the orientational correlation [149]. 

The Debye model for rotational diffusion, when applied to the general case of an anisotropic diffusor, is (see Favro, 1960)... 

Another model of rotational reorientation is the jump-diffusion model first described by Ivanov (1964). In this model the molecule reorients by a series of discontinuous jumps (with an arbitrary distribution of jump angles). This should be contrasted with the Debye model, which involves infinitesimal jumps, and the Gordon model, which involves continuous free rotations between collisions. This model is probably applicable to the situation where the molecular orientation is frozen until a volume fluctuation occurs, at which time the molecular orientation jumps to a new frozen value. We present our own version of the jump model here. It is assumed that (a) the jump takes place instantaneously, (b) successive jumps are uncorrelated in time with an average time tv between jumps, and (c) the dihedral angle between the two planes defined by the orientation vector u in two successive jumps is randomized. 

There exist a number of models predicting the time evolution of r t) for small llu-orophores in the solution. They differ in accuracy and detail of physical description and in mathematical approximations. The simplest rotational model is based on the Debye hydrodynamic theory [9]. It assumes that the rotational diffusion proceeds in small steps between collisions of the fluorophore with surrounding molecules. An analytical expression for r t) as a sum of several exponentials was first derived by Favro [10] ... 

Equation 12.26 is known as Debye equation. Figure 12.3 gives the frequency dependence of the real and imaginary (loss) part of the Debye function, s shows a steplike decay with increasing frequency where s" presents a symmetric peak with a maximum cOp = 27ifp = 1/Xp and a half width of 1.14 decades. The Debye equation can be justified by different molecular models like in the framework of a simple double potential model or the rotational diffusion approach. 

Debye s original rokational diffusion model corresponds to ki = 2Dy where is the rotational diffusion coefficient which for nis model of the molecule as a sphere of volume V reorienting in a medium of shear viscosity Q ia s k T/6qV giving the Debye relaxation time... 

Stokes-Einstein-Debye hydrodynamic theory is the most commonly used model for understanding the rotational diffusion of molecular systems [76, 77]. 

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