1 Lecture 8 The relationship between macroscopic and molecular dielectric relaxation behavior I. The dipole correlation function. ii. The relationship between response function and the dipole correlation function. Complex dielectric permittivity and dipole correlation function. iii. The relationship between the macroscopic and the microscopic correlation function. iv. Short-range and long range correlation functions. Fulton's Theory.

2 The main problem of the dielectric relaxation theory is establishing a relationship between macroscopic, phenomenological characteristics and molecular parameters and kinetical properties of the system being tested. To resolve this problem a statistical mechanics of time dependent processes should be used. On the base of leaner response theory of Kubo the dynamical properties of a substance can be expressed in quantities pertaining to the dielectric in the absence of the field. Two steps are required in this case: macroscopic response functionscorrelation functions for fluctuating quantities in the equilibrium system (the electric dipole moment) I. The derivation of the relation between the macroscopic response functions introduced in lecture 6 and correlation functions for fluctuating quantities in the equilibrium system (the electric dipole moment) correlation functions for macroscopic systems to correlation functions on a molecular scale. II. Reduction of the correlation functions for macroscopic systems to correlation functions on a molecular scale. III. What functional forms of molecular dipole correlation function (DCF) and consequently macroscopic (DCF) are most appropriate for the study of dipolar systems? By another words : What kind of models of molecular motions can be presented in terms of molecular correlation DCF?

3 Let us consider the derivation of Kubo result from classical statistical mechanics.  j j-th field must be smallterms linear in the field need be retained  In this theory one considers the dipole moment  j of the j-th molecule and looks for the response of this dipole, averaged over the equilibrium ensemble of the entire sample in the presence of a small perturbing electric field, a field which is unaltered by the presence of the sample. The field must be small so that only terms linear in the field need be retained; equivalently,  and  are field-independent. It was shown by Nienhuis and Deutch and Titulaer and Deutch that for a macroscopic spherical sample embedded in a infinite continuous medium with the same dielectric permittivity as the sample, the effect of the external matter can be neglected. h  /kT The resulting expressions are sufficient for the interpretation of dielectric relaxation in view of the rather low values of h  /kT involved. If higher frequencies are at issue (in the infrared and optical regions), a quantum mechanical correction should be applied. q i p i i In classical statistical mechanics the behavior of a system is described with the help of generalized coordinates q i and conjugated moment p i, where the index i ranges over all degrees of freedom of all particles.

4 Hamiltonian function H(p,q) For such a system the Hamiltonian function H(p,q) describes its evaluation in time. At each moment, the Hamiltonian is equal to the total energy (kinetic and potential) of the system. N interacting moleculesN E 0 (t) The Hamiltonian for N interacting molecules, where N is a very large number, placed in a uniform electric field E 0 (t) is (8.1) H 0 E 0 R dRR Vm(R) Ra(R) ma where H 0 is the Hamiltonian in the absence of the electric field, E 0 is the electric field in the absence of the particles, R is a space-point and dR is the three-dimensional volume element corresponding to R, the integrals are taken over the volume V of the sample, m(R) is the dipole density at point R, and a(R) is the corresponding polarizability density tensor. m and a are defined by : (8.2) (8.3)

5 (8.4) (8.5) q j j-thT jk  j molecular dipole momentpolarizability tensor E 0  j,  j H 0 P(R,  ) R where q j is the position of the j-th molecule, T jk is the dipolar interaction tensor between molecules j and k.  j and  j are the molecular dipole moment and polarizability tensor, respectively. The Hamiltonian given by (8.1)-(8.5) is complete through second order in E 0 ; the dipolar interaction terms dependent have been explicitly represented, other intermolecular interactions are incorporated implicitly in  j,  j and H 0. The polarization per unit volume P(R,  ) at point R is

6 (8.6) t RE 0 (t). where the bar over a quantity indicates an instantaneous ensemble average at time t and position R in the presence of electric field E 0 (t). By means of linear response theory, we can rewrite equation (8.6) as: (8.7) k<> E 0 {R,R’}.P(R,  ) R R=0(q j )=V -1 (N/V)  (t),  (t) where k is the Boltzmann constant, temperature, <> indicates an equilibrium ensemble average in the absence of the field E 0. The average is taken over phase points, i.e. coordinates of the molecules, and not over space points {R,R’}. In uniform fields P(R,  ) is independent of R. Therefore, in this case equation (8.7) can be written with R=0 and  (q j )=V -1. The correlation function is then proportional to (N/V)  (t), where  (t) is the normalized macroscopic dipole correlation function:

7 M(t)N  I (t)t ij where M(t) is a vector sum of N polar molecules with dipole moment  I (t) in unit volume at time t. The terms in (8.8) express the equilibrium orientation correlation between dipoles i and j in the sample. It means that is equal to. g(0) Where g(0) have a meaning of the Kirkwood correlation factor. For the more than one type of the dipoles the relation (8.8) can be presented as follows: (8.8) (8.9)

8 Let us consider the autocorrelation function u i  where u i is a unit vector, associated with dipole moment  I. For a pure polar system in which all the dipoles are equivalent, all are equal and we can write: (8.10) where (8.11)  ij (t)  ij (t)=1t=0  0 i t=0u i (0) t  (t). The dipole auto-correlation function  ij (t) has the limiting values  ij (t)=1 at t=0 and it  0 when t . It means, that if we consider the dipole i at t=0 whose orientation may be represented as u i (0) and as time develops, because of the molecular motion the dipole reorients in space, so that at a later time t it will have some average direction with angle  (t). In this case, we can write:

9 (8.12) average projection of the vector on the original direction decreases  ij (t) rate and the shape of the decay of the function  ij (t) As time develops so the average projection of the vector on the original direction decreases, so  ij (t) decreases and eventually reaches zero. The rate and the shape of the decay of the function  ij (t) are connected with the structure and kinetic properties of the system being tested. It seems that in the terms of the molecular DCF it is possible to get an excellent tool for the molecular rotational motion investigation by dielectric spectroscopy method. However, there are several difficulties in this direction that have to be taken in to account. They are: 1) Local field effects 2) Orientation correlation’s between dipoles.

10 Complex dielectric permittivity and Dipole correlation function  (t)  * (  )  (t). P(  )E(  ), The molecular dipole correlation function  (t) can be associated with experimental parameters  * (  ) only through the macroscopic correlation function  (t). Let us consider the main relationship between the polarization P(  ) and macroscopic electric field E(  ), in the sample: (8.13) E o (  ), The relationship between the polarization and external electric field E o (  ), in terms of linear response theory leads to: (8.14)  * o (  )  (t) where  * o (  ) is the quasi susceptibility and connected with macroscopic DCF  (t) by the following relation: (8.15)

11  * o (  )  (t) E(  )E o (  )  * o (  )  * (  ) E(  ) Comparison of (8.13) and (8.15) immediately permits one to conclude that  * o (  ), hence the autocorrelation function  (t) must depend on sample shape and surroundings. The reasoning is as follows. Since the relation between E(  ) and E o (  ) depends on the sample shape, it is necessary for  * o (  ) to contain geometry-dependent factors to compensate for this dependence if the computed  * (  ) is to be a single function independent of the sample configuration. For example, in the case of a spherical sample consisting from rigid dipoles in vacuum, the relation between E(  ) and E o (  ) is: so that (8.16) (8.17) = The value = can be related to the static dielectric permittivity according to the Clausius-Massoti relation:

12 (8.18)  (t)  * (  ) Substituting (8.18) in to (8.15) and taking into account (8.17) we can obtain the equation for  (t) associated with  * (  ) for the simplest case of spherical sample in vacuum: (8.19) spherical region embedded in an infinite dielectric medium: Glarum was the first who obtained this equation using the linear response theory. Using the approximations of Kirkwood theory, he also obtained the result for a spherical region embedded in an infinite dielectric medium: (8.20) where (8.21) DCF is a normalized autocorrelation DCF of the small spherical sample embedded in a infinite dielectric continuum.

13 macroscopicDCF  (t)  (t) The macroscopic DCF  (t) connected with  (t) by integral deconvolution equation: (8.22) where is a time independent constant. Bob Cole used the same approach to obtain the similar equation for the case of the polarizable molecules: (8.23) Cole and Glarum Fatuzzo and Mason Obtained by Cole and Glarum relations (8.20) and ( 8.23) were criticized by Fatuzzo and Mason. For the same spherical geometry of the sample embedded into the infinite continuum, they obtained another relationship: (8.24)

14 spherical sample embedded into the continuum with the same complex dielectric permittivity  * (  ).  * (  )  (t) m(t), embedded in spherical sample embedded into dielectric continuum: This equation was obtained by direct applying of the linear response theory to the spherical sample embedded into the continuum with the same complex dielectric permittivity  * (  ). For the case of the induced as well as permanent dipole moments, this relation was generalized by several authors (Klug et.al, Nee and Zwanzig, Rivail) independently. Using different approaches all of them derived the same equation for  * (  ) associated with correlation function of dipole  (t) of single polar molecule embedded in small sphere with dipole moment m(t), which by itself embedded in spherical sample embedded into dielectric continuum: (8.25) (8.26) where is a molecular DCF.

15 Titulaer and Deutch that the Fatuzzo-Mason result is correct for the case of a spherical sample embedded in its own medium case of a spherical sample embedded in a medium with a frequency-independent dielectric constant equal to the static dielectric content of the sample. The result of the conflict between these two approaches was the paper of Titulaer and Deutch where they presented an analysis of these conflicting theories of dielectric relaxation. Their principal result was to find that the Fatuzzo-Mason result is correct for the case of a spherical sample embedded in its own medium, and that the Glarum expression is incorrect for this case. The Glarum equation refers to the case of a spherical sample embedded in a medium with a frequency-independent dielectric constant equal to the static dielectric content of the sample. molecular DCF concentric spheres model  (t) It means that the equation (8.25) is the most accurate one for pure polar system to associate molecular DCF with complex dielectric permittivity in the framework of concentric spheres model. However, the model that was used for the derivation of this expression and the reactive field approximation is not realistic and  (t) can be considered as a single molecular DCF only in the first approximation. To overcome all the difficulties in this direction one have to develop the molecular theory of dielectric relaxation that can distinguish the short and long rang corrections and to associate them with response function or complex dielectric permittivity.

16 Robert Fulton from macroscopic electrodynamics not involve consideration of spherical specimens non-polarizable molecules Robert Fulton proposed one of such theories. Using very general arguments, that follows from macroscopic electrodynamics Fulton build a dielectric relaxation theory by means which does not involve consideration of spherical specimens, nor connection between the correlation functions of two distinct spatial regions. The result for an isotropic dielectric composed of non-polarizable molecules with permanent dipole moment was written as following: (8.27)   g (t) is the short rang orientation correlation function.  r  -3 (t  0 and  ) where  is the number of molecules per unit volume,  is the dipole moment of one molecule, and g (t) is the short rang orientation correlation function. The short-range correlation’s fall faster than  r  -3 and the equation (8.27) represents the dynamical extension of the Kirkwood-Frohlich formula in the classical limit (t  0 and  ) for the non-polarizable molecules. This expression agrees substantially with the results of Fatuzzo-Mason and others but disagrees with the results obtained by Glarum-Cole.

17 short ranged orientation correlation’s long ranged correlation’s In addition to (8.27) Fulton showed that the connection between the short ranged orientation correlation’s (those which fall off faster than the inverse cube of the separation) and the long ranged correlation’s (which fall off as the inverse cube of the separation) is found from (8.28) a(t) long range correlation function. where a(t) is the long range correlation function. polarizable molecules The generalization of these results for a particular situation with polarizable molecules gave the following expression (8.29) where the connection between the short and long correlation functions can be found from: (8.30)

18 g =1 If g =1 the equation (8.31) is equal to (8.25). dynamical extension of the Kirkwood-Frohlich formula in the classical limit (t  0 and  ). As it was mentioned above, the equation (8.29) represents the dynamical extension of the Kirkwood-Frohlich formula in the classical limit (t  0 and  ). Making combination with Kirkwood-Frohlih expression, we can get the following: (8.31)