Debye equation, rotational motion

Existence of a high degree of orientational freedom is the most characteristic feature of the plastic crystalline state. We can visualize three types of rotational motions in crystals free rotation, rotational diffusion and jump reorientation. Free rotation is possible when interactions are weak, and this situation would not be applicable to plastic crystals. In classical rotational diffusion (proposed by Debye to explain dielectric relaxation in liquids), orientational motion of molecules is expected to follow a diffusion equation described by an Einstein-type relation. This type of diffusion is not known to be applicable to plastic crystals. What would be more appropriate to consider in the case of plastic crystals is collision-interrupted molecular rotation. 

The time dependence of the anisotropy r(t) depends on the underlying dynamics of reorientational motion. For rotational diffusion (tumbling) of a spherical object, the expected anisotropy decay is exponential with a rotational diffusion time given in the hydrodynamic limit by the Stokes-Einstein-Debye equation. For nonspherical molecules, more complex time dependence may be detected. (For more on these topics, see the book by Cantor and Schimmel in Further Reading.)... 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)—(11) may be obtained from our fractional generalizations of the Fokker-Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate 4> (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored. 

The translational and rotational motion of a Brownian particle immersed in a fluid continuum is well described by the Stokes-Einstein and Debye equations, respectively. 

Classic Brownian motion has been widely applied in the past to the interpretation of experiments sensitive to rotational dynamics. ESR and NMR measurements of T and Tj for small paramagnetic probes have been interpreted on the basis of a simple Debye model, in which the rotating solute is considered a rigid Brownian rotator, sueh that the time scale of the rotational motion is much slower than that of the angular momentum relaxation and of any other degree of freedom in the liquid system. It is usually accepted that a fairly accurate description of the molecular dynamics is given by a Smoluchowski equation (or the equivalent Langevin equation), that can be solved analytically in the absence of external mean potentials. 

Equations (7.3.19) and (7.3.20) are applicable to scattering from very dilute solutions of cylindrically symmetric macromolecules. Such systems usually satisfy the assumptions in the derivation of these equations (a) dilute solutions, (b) independence of molecular rotation and translation, (c) translational motions described by the translational diffusion equation, and (d) rotational motions described by the rotational diffusion (Debye) equation. 

The basic derivations of the van der Waals forces is based on isolated atoms and molecules. However, in many particle calculations or in the condensed state major difficulties arise in calculating the net potential over all possible interactions. The Debye interaction, for example is non additive so that a simple integration of Equation (4.27) over all units will not provide the total dipole-induced dipole interaction. A similar problem is encountered with the dipole-dipole interactions which depend not only on the simple electrostatic interaction analysis, but must include the relative spatial orientation of each interacting pair of dipoles. Additionally, in the condensed state, the calculation must include an average of all rotational motion. In simple electrolyte solutions, the (approximately) symmetric point charge ionic interactions can be handled in terms of a dielectric. The problem of van der Waals forces can, in principle, be approached similarly, however, the mathematical complexity of a complete analysis makes the Keesom force, like the Debye interaction, effectively nonadditive. 

Commonly, the aggregates are not uniformly aligned. Moreover, their orientation is steadily changing due to migration as well as shear and thermal motion (Brownian rotation). For these reasons, usually the orientation average of Eq. (4.38) is considered, which can be derived by means of the Debye equation (Debye 1915) ... 

The measured frequency dependences of e"(m) and e (cu) in real fluids do not always fit the Debye-Pellat equations, and many methods [1] have been proposed to analyse skewed or displaced Cole-Cole plots. Debye s theory of dipole relaxation assumes that rotational motion can be described in terms of a single relaxation time. In a real system, fluctuations in the local structure of a molecule or its environment may result in a distribution of relaxation times about the Debye value, and such a situation can be described by a modification to Eq. (10)... 

Equation (31) assumes an existence of two kinds of molecules showing quite different reorientational dynamics in the considered time interval. The water molecules with strong retardation of rotational motion and with broad distribution of relaxation times (low value of P) should be considered strongly bound. Such kind of water molecules with similar low values of the stretching exponent were observed in neutron-scattering experiment for combined rotational-translational motion of water in hydrated myoglobin (P 0.3) [643] and in simulations of water near mica surface (P k 0.25) [652]. Those water molecules, which show simple one-term exponential relaxation, like in the bulk, should be considered as weakly bound. Fractions of these two kinds of molecules ((1 - a) and a, respectively) should depend on the hydration level. Note, that weakly bound water molecules with Debye rotational relaxation were not distinguished in other simulation studies [610, 644]. 

It is clear from a variety of spectroscopic techniques that biomembranes are dynamic not static structures. It is also known that certain membrane functions depend critically on the fluidity of the membrane lipids. Spin-labelled and fluorescent-labelled lipid probes are found to perform rotational motions in the nanosecond timescale in fluid lipid bilayers and membranes. For diffusive rotation the characteristic correlation times are given by the Debye equation (t = n V/kT) and correspond to effective viscosities in the range q 0.1-1 poise. A spin-labelled steroid analogue of cholesterol for instance rotates rapidly about its long... 

Intcrmolecular dipole-dipole relaxation depends on the correlation time for translational motion rather than rotational motion. Intermolecular dipole-dipole interactions arise from the fluctuations which are caused by the random translational motions of neighboring nuclei. The equations describing the relaxation processes are similar to those used to describe the intramolecular motions, except is replaced by t, the translation correlation time. The correlation times are expressed in terms of diffusional coefficients (D), and t, the rotational correlation time and the translational correlation time for Brownian motion, are given by the Debye-Stokes-Einstein theory ... 

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... 

Debye extended the foregoing arguments in order to establish the Smoluchowski equation [Eq. (5)] for the rotational Brownian motion of a dipolar particle about a... 

CONCEPTS More about the effect of collisions on distribution functions microscopic theory of dielectric loss The Debye theory can define a distribution function which obeys a rotational diffusion equation. Debye [22, 23] has based his theory of dispersion on Einstein s theory of Brownian motion. He supposed that rotation of a molecule because of an applied field is constantly interrupted by collisions with neighbors, and the effect of these collisions can be described by a resistive couple proportional to the angular velocity of the molecule. This description is well adapted to liquids, but not to gases. 

It can be concluded that remanent polarization and hence the piezoelectric response of a material are determined by Ae this makes it a practical criterion to use when designing piezoelectric amorphous polymers. The Dielectric relaxation strength Ae may be the result of either free or cooperative dipole motion. Dielectric theory yields a mathematical approach for examining the dielectric relaxation Ae due to free rotation of the dipoles. The equation incorporates Debye s work based on statistical mechanics, the Clausius-Mossotti equation, and the Onsager local field and neglects short-range interactions (43) ... 

---