Transcript Document

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:
a) the nature of the orientation order in polar fluids,
b) the molecular basis for a sample-independent dielectric
constant.
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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).
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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 r in 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.
W
V
Fig.5.1 The dielectric sample of volume V is embedded in a dielectric
sample of volume W with dielectric constant o
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The case o = 1. Polarization occurs in W and the macroscopic filed is
E ( r )  E 0 ( r )   T ( r , r ' )  P ( r ' ) dr '
(5.1)
V
where T(r,r') is the dipole-dipole tensor
 
3( r  r' )( r  r' )
1
1
T ( r , r' )  
r  r' 

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3
r r'
r  r'
r  r'
The term T(r,r')·P(r') gives the contribution to the macroscopic field
at r from the polarization density at r'.
The case o  1 (5.1) must be replaced by
E ( r )  E 0 ( r )   D( r , r' )  P ( r' )dr'
(5.2)
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where
D( r ,r' )  T( r ,r' )  RW ( r ,r' ; 0 )
(5.3)
The reaction-field tensor RW ( r ,r' ; 0 )
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 at r'.
For o = 1 ,
RW ( r ,r' ; 0 ) vanishes and D(r,r')=T(r,r').
Because of the long range of D(r,r') and the dependence of RW ( r , r' ; 0 )
on surroundings, the eq. 5.3 shows that E(r) depends on both sample
shape and surroundings. Also, the dielectric polarization at point r
should result from the actual macroscopic field E(r) rather than from
Eo(r). Thus, we expect that P(r) also depends on sample shape and
surroundings.
Consequently, the relationship between P(r) and Eo(r) is sampledependent and for arbitrary sample geometries can be very
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complicated.
One assumes
P( r ) 
 1
E( r )
4
(5.4)
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).

 1 
P( r ) 
 E0   D( r , r' )  P( r' )dr' 
4 
V

(5.5)
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
N
(5.6)
P(r)   μ( i )( r  ri )
i 1
E0
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In this equation (ri,i) describes the position ri and orientation i of
molecule i, while μ( i )( r  ri ) 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).
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 g( r11 ;r22 )
One finds




P(r) 
d

d

'
dr
'
μ
(

)

(
r

r'
)

(



'
)

g
(
r

;
r
'

'
)
μ( ' )  E0 (r' )



kT V



(5.7)
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 82 for nonlinear molecules, because for orientation of
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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 <cosij> which is the average cos in
of the angle between two representative molecular dipoles. The
explicit relation is
 M2 
1
P( r ) 
E0 ( r ) 
 2 ( 1    cos ij  )E0 ( r )
3VkT
3 kT
(5.8)
In the preceding analysis no 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.
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For example, for spherical sample in vacuum, one finds from
electrostatics:
P(r) =
3  1
E0 (r)
4   2
(5.9)
Comparison (5.8) and (5.9) gives the following expression for <M2>:
 M2 
1
P( r ) 
E0 ( r ) 
2 ( 1    cosij  )E0 ( r )
3VkT
3kT
(5.8)
In this case we can make the conclusion that <M2>, <cosij> 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
 M 2  ( 2  1 )(   2 )

2
 M 0
9
which is about 20 for water
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Main relationships in static dielectric theory
Non-polar systems
  1 4

N Clausius-Mossotti equation
2
3
Polar diluted systems

2
  1 = 4N  
3kT

Polar systems
z

9kT (    )(2    )
 
4N
 (   2) 2
2

a


 Debye equation

Onsager Equation
Polar systems, short range interactions




9 kT    2   
g 
2
4
   2
2
Kirkwood-Fröhlich equation
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