Lecture 5. The modern theories of the static dielectric permittivity (Böttcher, Nienhuis and Deutch, Ramshaw, Omini, Wertheim etc). The microscopic theory of dielectric constant There are two closely dependent issues: the nature of the orientation order in polar fluids,
The microscopic theory dealing with these issues have been developed in three stages.
1. the theory was developed for the rigid polar fluid model (Nienhuis and Deutch);(fluids composed of hypothetical polar molecules with zero polarizability).
2. The theory was extended to the case of real nonpolar fluids (Wertheim)(fluids composed of polarizable but nonpolar molecules).
3. A general theory for polarizable and polar fluids was presented (Wertheim, Ramshaw, Høe and Stell, Omini).
Let us consider an arbitrarily shaped molecular dielectric sample with volume V embedded in surroundings of volume W composed of a continuum dielectric with dielectric constant o. If we apply now the electric field Eo(r), a polarization P(r) will be induced in the sample.
The applied field Eo(r) is defined as the electric filed in point rin the absence of the sample. The actual macroscopic field E(r) which the sample experiences is a superposition of the applied field, the fields produced by the polarization in W, which is induced by the polarization in the sample.
Fig.5.1 The dielectric sample of volume V is embedded in a dielectric sample of volume W with dielectric constant o
The case o = 1. Polarization occurs in W and the macroscopic filed is
where T(r,r')is the dipole-dipole tensor
The term T(r,r')·P(r')gives the contribution to the macroscopic field at rfrom the polarization density at r'.
The case o 1 (5.1) must be replaced by
The reaction-field tensor depends on the shape of region W, as well as its dielectric constant o. It gives the contribution to the macroscopic field at r caused by the part of the polarization in W induced by the polarization atr'.
For o = 1 , vanishes andD(r,r')=T(r,r').
Because of the long range of D(r,r') and the dependence of
on surroundings, the eq. 5.3 shows that E(r)depends on both sample shape and surroundings. Also, the dielectric polarization at point rshould result from the actual macroscopic field E(r) rather than from Eo(r). Thus, we expect thatP(r) also depends on sample shape and surroundings.
Consequently, the relationship between P(r) and Eo(r) is sample-dependent and for arbitrary sample geometries can be very complicated.
where is a sample-independent dielectric constant. Combining equations (5.1)-(5.4) gives an integral equation for P(r), which can be explicitly solved for certain sample geometry and surroundings. This gives a sample-dependent relationship between P(r)and Eo(r).
A relationship between P(r) and Eo(r) can also be found from a microscopic calculation.
To carry out the microscopic calculations let us define polarization P(r).This is
To order Eo(r) the polarization can readily be computed in terms of pair distribution function of the fluid in the absence of the field
In this equation (ri,i) describes the position ri and orientation i of moleculei, while
is the contribution of molecule i to the
dipole moment density at point r. The sum runs over all N molecules in the field, and the average is taken over the canonical distribution function in the presence of the field Eo(r).
where =N/V is the uniform particle density in the system, and =d - is an angular-phase space volume equal to 4 for linear molecules and 82 for nonlinear molecules, because for orientation of molecular one needs three Eulerian angles.
The polarization P(r) can be also described in terms of <M2> - the mean-square total dipole moment of the fluid in the absence of the field. This is in turn is related to <cosij>which is the average cos in of the angle between two representative molecular dipoles. The explicit relation is
In the preceding analysisno mention is made of sample surroundings or geometry, and a general microscopic relation between P(r) and Eo(r) is found.
The macroscopic relationship between P(r) and Eo(r), however, is sample-dependent.
For example, for spherical sample in vacuum, one finds from electrostatics:
Comparison (5.8) and (5.9) gives the following expression for <M2>:
In this case we can make the conclusion that <M2>, <cosij>and therefore the pair distribution function must depend on sample shape and surroundings. That this shape dependence in no negligible can be seen by considering the ratio
which is about 20 for water
Polar diluted systems
Polar systems, short range interactions
Main relationships in static dielectric theory