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Lecture 9. Models of dielectric relaxation i. Rotational diffusion; Dielectric friction. ii. Forced diffusion of molecules with internal rotation iii. Reorientation by discrete jumps iv. Memory-Function Formalism v. The fractal nature of dielectric behavior. (9.1).

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Lecture 9

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lecture 9
Lecture 9

Models of dielectric relaxation

i. Rotational diffusion; Dielectric friction.

ii. Forced diffusion of molecules with internal rotation

iii. Reorientation by discrete jumps

iv. Memory-Function Formalism

v. The fractal nature of dielectric behavior.



According to Frenkel the molecular rotational motion is usually only the rotational rocking near one of the equilibrium orientation. They are depending on the interactions with neighbors and by jumping in time they are changing there orientation.

In this case the life time  of one equilibrium orientation have to be much more then the period of oscillation 0=1/( >>0). And the relationship between them can be written in the following way:

where H is the energy of activation that is required for changing the angle of orientation. The small molecules can be rotated on comparatively big angles. The real Brownian rotational motion can be valid only for comparatively big molecules with the slow changing of orientation angles. In this case the differential character of rotational motion is valid and the rotational diffusion equation can be written.



Debye was the first who applied the Einstein theory of rotational Brownian motion to the polarization of dipole liquids in time dependent fields.

According to Debye the interaction of molecules between each other can be considered as the friction foresees with the moment proportional to the angle velocity=P/, where is the rotational coefficient of friction that can be connected with Einstein rotational diffusion coefficient(DR = kT /) andPis the moment of molecule rotation. In the case of small macroscopic sphere with radius a, the coefficient of rotational motion according to Stokes equation can be defined as:

where  is the coefficient of viscosity.



Here ri(t) and ui(t) are, respectively, the position and orientation of moleculeiat time t and the sum goes over all the molecules. The average value of C is (1/4)0, where 0is the number density of the fluid. In this equation the operator is related to


where DTand DR are, respectively, the transnational and rotational diffusion coefficients, is the gradient operator on the space (x,y,z) and is the rotation operator . In this equation C(r,u,t)d2ud3r is the number of molecules with orientation u in the spheroid angle d2uand center of mass in the neighborhood d3rof the point r at time t. The microscopic definition of C is

Let us start with the diffusion equation:



the dimensional angular momentum operator of

quantum mechanics; that is

It should be recalled

that the spherical harmonics Ylm(u) are eigenfunctions of

corresponding to eigenvalue ofl(l+1).

The solution of the equation (9.3) can be done by expanding of C(r,u,t) in the spherical harmonics {Ylm(u)}. In the case of dipole moment rank lis equal to one. In the case of magnetic moment l=2. For the spherical dipole moment in viscous media the result of equation (9.3) can be obtained in the following way:



This is Debye’s expression for the molecular dielectric relaxation time. According to Debye, this formula valid if:

(a) There is an absence of interaction between dipoles.

(b) Only one process leading to equilibrium(e.g. either transition over a potential barrier, or frictional rotation).

(c) All dipole can be considered as in equivalent positions, i.e. on an average they all behave in a similar way.

The molecular dipole correlation function in this case will be the simplest exponent:

This result was generalized to the case of prolate and oblate ellipsoids by Perrin and Koenig:




a) Prolate ellipsoid: =b/a <1



b) Oblate ellipsoid: >1




In the case of ellipsoid of revolution the dipole correlation function can be written in the following way:


Let us now consider the influence of long-range forces such as Coilomb, or dipolar forces on the results of the Debye theory. In this case each molecule not only experiences the usual frictional forces which give rise to a diffusion equation, but also must respond to the local electric field which arises from the permanent multiple moments on the neighboring molecules.


One of the ways to include these interactions into Debye theory is to add forces and torque’s in a generalized diffusion equation and to solve this equation self-consistently with the Poisson equation. In this case the generalized diffusion equation can be written as a following:


where F(r,t) and N(r.t) are the force and torque respectively that acting on a molecule at (r,t). They are arise from the Coulomb interactions between molecules and can be expressed as:



Here linear molecule centered at r with orientation u is considered. (r+su) is the position of a distance s from the molecular center along the molecular axis. Then E(r+su) is the electric field at the point due to all charges in the system. Z(s) is the linear charge density and dsZ(s)E(r+su) is the electric force exerted on this charge by the surrounding fluid. Likewise sudsZ(s)E(r+su) is the corresponding torque.


To make the equations (9.12-9.14) self-consistent the Poisson equation has to be used:


where (r,t) is the charge density and (r,t) is the electrostatic potential at r,t. In the case of polarizable molecules 4 in Poisson equation have replace by 4/, where is dielectric constant due to the polarizability [(-1)/( +2)=o]. Also the dipole moment of the linear molecules might be taken as an effective dipole moment.

In the absence of net molecular charges, the only multipole moment that contributes to the orientation relaxation is the dipole moment.

The solution of diffusion equation taking into account dipolar forces gives the correlation function (t) that decays on two different time scales specified by the relaxation times:




where DR is the rotational diffusion coefficient, and


Correlation function can be written in the following way:


Two relaxation times for a single component polar fluid was found also by Titulaer and Deuthch, Bordewijk and Nee- Zwanzig. If Berne discussed the two correlation times asdecay of transverse and longitudinal fluctuations, Nee and Zwanzig considering dielectric friction in diffusion equation. Considering the diffusion equation they made the assumption that by some reasons the frictional forces on the particle is not developed instaneously, but lags its velocity. Considering the correlation function of angular velocities they came to the frequency dependent friction coefficient in diffusion equation:



In this case in the theory of rotational Brownian motion, the position of the particle is replaced by its orientation, specified by the unit vector u(t). The translational velocity is replaced by an angular velocity (t) and the force is replaced by a torque N(t). The frictional torque is proportional to the angular velocity:


or in Fourier components,


The total friction coefficient () consists of two parts. The first is due to ordinary friction, e.g. Stokes’ law friction 0 independent on frequency. The other part is due to dielectric friction and is denoted by D(). The sum is


Using the Onsager reactive field and calculating the transverse angular velocity and torque in terms of time dependent permanent dipole moment, they obtained an explicit expression for the dielectric friction coefficient:



This expression is valid for spherical isotropic Brownian motion of a dipole in an Onsager cavity. To obtain the molecular DCF it is necessary to average over distribution of orientations at time t, for a given initial orientation and then to average over an equilibrium distribution of initial orientations.

The average of (t) can be found from knowledge of the distribution function C(u,t)of orientations as a function of time. This distribution function obeys the diffusion equation for spherically isotropic Brownian motion. The solution of this equation leads to a very simple relation between dielectric friction and DCF:


It is convenient to introduce in this case the frequency dependent relaxation time () defined by



One can now write for molecular DCF the following relation:


From comparison of (9.27) with the Debye behavior we are coming to the simple relationship between macroscopic and molecular correlation times:


which is different from the relationship obtained by Bordewijk for the same molecular DCF


where k=s/


Non-exponential relaxation


Cole-Cole law

1941 year

is the relaxation time

Character of





(1-)  / 2

is a phenomenological parameter



The Memory function for Cole-Cole law

the memory function

 = df

a fractional derivation

Fractal set

L. Nivanen, R. Nigmatullin, A. LeMehaute,Le Temps Irrevesibible a Geometry Fractale, (Hermez, Paris, 1998)

R.R. Nigmatullin, Ya.E. Ryabov,Physics of the Solid State, 39 (1997)


Scaling relations

N,  are scaling parameters

dGis a geometrical fractal dimension

is the limiting time of the system self-similarity in the time domain

is the self-diffusion coefficient

is the constant depends on relaxation units transport properties


Polymer water mixtures



are electrolyte polymers



Is a nonelectrolyte with strong interaction

between hydroxyl groups and water




are nonelectrolyte polymers

N. Shinyashiki, S. Yagihara, I. Arita, S. Mashimo, Journal of  Physical Chemistry, B 102 (1998) p. 3249


Composite polymer structure

T is not Constant

The samples with Kevlar fibers

have the longer relaxation time

H. Nuriel, N. Kozlovich, Y. Feldman, G. MaromComposites: Part A 31 (2000) p. 69


Water absorbed in the porous glass

T is not Constant

Samples are separated in two groups according to the humidity value h.

A. Gutina, E. Axelrod, A. Puzenko, E. Rysiakiewicz-Pasek, N. Kozlovich, Yu. Feldman, J. Non-Cryst. Solids,235-237 (1998) p. 302



IThe Cole-Cole scaling parameter depends on the features of interaction between the system and the thermostat.

IIThe Cole-Cole scaling parameter and the relaxation time are directly connected to each other.

IIIFrom the dependence of the parameter on the relaxation time, the structural parameters can be defined.