Transcript Document

Lecture 4
1.
Statistical-mechanical approach to dielectric theory.
2. Kirkwood-Fröhlich's equation.
3. The Kirkwood correlation factor.
4. Applications: pure dipole liquids, mixtures, dipolar solids.
1
The methods of statistical mechanics provide a way of obtaining
macroscopic quantities when the properties of the molecules and the
molecular interactions are known.
All statistical-mechanical theories of the dielectric constant start from
the consideration that the polarization P, given by:
4P  D  Ε 
is equal to the dipole density P, when the influence of higher multipole
densities may be neglected. By definition, we may write the dipole
density P of a homogeneous system:
PV  M 
(4.1)
where V is the volume of the dielectric under consideration an <M> is
its average total (dipole) moment (the brackets < > denote a
statistical mechanical average). If we assume the system to be
isotropic, we also have
D  E
2
Thus we find:
  1E = 4
V
(4.2)
M
Since the dielectric is isotropic, <M> will have the same direction as E
and it will be sufficient to calculate the average component of M in the
direction of E. Using e to denote a unit vector in the direction of the
field, we may therefore rewrite Eqn (4.2) in scalar form:
  1E = 4
V
M e 
4
M e
V
(4.3)
Another way of writing Eqn.(4.3) results form the fact that P and <M>
contain in general also terms in higher powers of E than the first (see
eqn.(1.19)). Thus, (-1)E/4 is the first term in a series development of
P in powers of E, and must be set equal to the term linear in E of the
series development of <Me>/V in powers of E. Since variations in V
due to electrostriction do not appear in the leaner term we may develop
<Me> in a Taylor series, finding:
  1 = 4  

M e 
V  E
 E 0
3
Rewriting with the external field Eo instead of the Maxwell field E as the
independent variable we obtain:

4  E0   
   1 = 
M e 
 
(4.4)
V  E  E 0  E0

E0  0
For the average of a quantity like Me, which is a function of the
positions and orientations of all molecules, one can write:
M e
dXM  e exp( U / kT )


 dX exp(U / kT )
(4.5)
where X denotes the set of position and orientation variables of all
molecules. This expression for the average can be obtained from the
general ensemble average by integrating over the momenta.
The integration over the moment results in a weight factor, which is
contained in our notation dX. For example, if we take spherical
coordinates r,, for the position of a molecule and integrate over the
conjugated momenta, we obtain a weight factor r2sin, in this
integration over the coordinates of that molecule. Thus, in this case
dX=r2sindrdd, which is the expression for a volume element in
spherical coordinates.
4
Before the expression for the dielectric constant in terms of molecular
quantities can be obtained two problems have to be solved.
1. E, the Maxwell field, has to be expressed as a function of the
external field Eo. In some special cases electrostatic theory leads
to a simple relation between Eo and E, which can be used
immediately. In general case, however, E is given by the sum of
the external field and the average field due to all molecules of the
dielectric. It must be remarked that even for homogeneous Eo the
Maxwell field E will not be homogeneous.
2.<Me>, expressed as a function of Eo, has to be calculated as the
average of the sum of the dipole moments of all molecules for a
given value for the external field. When the total number of
particles in the dielectric is N, and the instantaneous dipole
moment of the i-th particle is mi, we may write for the
instantaneous total moment M:
N
M =  mi
i=1
(4.6)
5
Let us take a region with N molecules which are treated explicitly;
the remaining N-N molecules are considered to form a continuum and
are treated as such. The approximations in this method can be made
as small as necessary by taking N sufficiently large. If this value of N
is still manageable in the calculations, the method can be used to
introduce the molecular interactions into the calculation of the dielectric
constant of polar liquids.
Non-polarizable molecules (rigid dipoles)
Let us consider the idealized case that the polarizability of the
molecules can be neglected, so that only the permanent dipole
moments have to be taken into account. We are taking a sphere of
volume V, containing N molecules.
For convenience in the calculations we suppose that it is embedded
in its own material, which extends to infinity. The material outside
the sphere can be treated as a continuum with dielectric constant .
In this case the external field working in the sphere is the cavity field
(eqn.2.14):
6
3
E0 = EC 
E
2  1
(4.7)
where E is the Maxwell field in the material outside the sphere. We can
substitute (4.6) in the general expression for the dielectric constant of
homogeneous, isotropic dielectric:
E0   1
1

2

e

A

e



M


0
0
 E  E 0  N
3kTN

  1 = 4N 
(4.8)
where N=N /V is the number density, and the tensor A plays the role
N
2
of a polarizability;
 M  0     i  j  0 Because we defined that the
i , j 1
molecules are non polarizable, <eAe>o=0. In this case after
substitution of (4.7) into (4.8), we obtain:
4 3  M 2  0
   1 =
V 2  1 3kT
(4.9)
The average of the square of the total moment can be calculated as
follows in the case of non-polarizable molecules:
7
N
M =  i
(4.10)
i=1
so that we may write:
N
M2
0

i 1
N
dX
 i  M exp( U / kT )
 dX
N
exp( U / kT )
(4.11)
In this equation the superscript N to dX to emphasize that the
integration is performed over the positions and orientations of N
molecules. The integration in the numerator of eqn.(4.11) can be
carried out in two steps.
Since i is a function of the orientation of the i-th molecule only, the
integration over the positions and orientations of all other molecules,
denoted as N -i, can be carried out first. In this way we obtain (apart
from a normalizing factor) the average moment of the sphere in the
field of the i-th dipole with fixed orientation. The averaged moment,
denoted by Mi*, can be written as:
8
M i* 
N 1
dX
M exp( U / kT )

 dX
N 1
exp( U / kT )
(4.12)
The average moment Mi* is a function of the position and orientation
of the i-th molecule only. Expression (4.12) for Mi* can be substituted
into eqn. (4.11). Denoting the position and orientation coordinates of
the i-th molecule by Xi and using a weight factor p(Xi)
p(X i ) 
N 1
dX
exp( U / kT )

N
dX
 exp( U / kT )
(4.13)
We obtain:
M
N
2
0
   p( X i  i  M i* dX i N  p( X i  i  M i* dX i
(4.14)
i 1
since after integration over the positions and orientations of molecule i,
the resulting expression will not depend on the value of i.
9
Before we substitute the expression for <M2>o into eqn.(4.9) , we note
that it is possible to rewrite expression (4.14) in suggestive form.
According to eqn.(4.12) and (4.13) Mi* can be written as the sum of
moments j, averaged with the orientation of the i-th dipole held fixed.
denoting the angle between the orientation of the i-th and the j-th
dipole by ij, this leads to:
N
i  M i*    
j=1
N 1
dX
cosij exp( U / kT )

 dX
and thus to:
M2
N
0
 N 2   p( X i )
N 1
exp( U / kT )
N -i
dX
cos ij exp( U / kT )

j 1
 dX
(4.15)
N -i
exp( U / kT )
(4.16)
dX i
This expression for <M2>o can be abbreviated by introducing the
average of cosij, defined as:
 cos ij   p( X i )
N -i
dX
cos ij exp( U / kT )

 dX
N -i
exp( U / kT )
dX i
(4.17)
10
We then write:
 M 0  N
2
2
N
  cos
j 1
ij

(4.18)
If we now substitute Eqn. (4.14) or its another form (4.18) into (4.9)
we find after some rearrangements, and using N=N/V for the number
density:
( - 1)(2 + 1)
N

12
3kT

N
N
p( X i  i  Mi* dX i 
 2   cosij 
3kT
j 1
(4.19)
Let us compare this expression with (3.34) from the previous lecture,
for a special case of a pure dipole liquid with non-polarizable molecules
(=0):
( -1)(2 +1)
N 2
(4.20)
12

3kT

This expression was derived in previous lecture with the help of the
continuum approach, in which a sphere containing only one
molecule is used. Therefore the (4.20) is a special case of (4.19)
when the sphere containing N molecules is restricted to one, Mi* =i
and cosij is equal to 1.
11
When the sphere contains more than one molecule, the value of Mi*
can be different from i . When the number of molecules included in
the sphere increases, Mi* reaches a limiting value, so that Mi* will be
independent of N as long as N exceeds a certain minimum value.
The dipole moment M of a sphere in the field of an arbitrary charge
distribution within it, is given by:
 1
M
m
 2
(4.21)
where m is the dipole moment of the charge distribution. Since this
expression for M does not depend on the radius of the sphere, the
dipole moment of a spherical shell in the field of a point dipole within
the inner sphere must be zero.
This conclusion will also hold if the sphere is not in vacuum, but
embedded in a dielectric, even with same dielectric constant as the
sphere itself.
12
Thus the addition of a number of molecules contained in a spherical
shell to original number N will not change the moment of the sphere
as long as the spherical shell can be treated macroscopically.
From this argument we conclude that the deviations of Mi* from the
value i are the result of molecular interactions between the i-th
molecule and its neighbors.
It is well known that liquids are characterized by short-range order and
long-range disorder. The correlations between the orientations (and
also between positions) due to the short-range ordering will lead to
values of Mi* differing from i. This is the reason that Kirkwood
introduced a correlation factor g which accounted for the deviations of
N
i
*
i
2
p
(
X



M
dX


  cosij 
i
i

j 1
from the value
2:
13
g
1
2
 p( X
N
i
 i  M dX    cosij 
*
i
i
j 1
(4.22)
With the help of this definition, eqn. (4.19) may be written as:
( - 1)(2 + 1)
N

g 2
12
3kT
(4.23)
When there is no more correlation between the molecular orientations
than can be accounted for with the help of the continuum method, one
has g=1 we are going to Onsager relation for the non-polarizable
case, for rigid dipoles with =1.
An approximate expression for the Kirkwood correlation factor can be
derived by taking only nearest-neighbors interactions into account. In
that case the sphere is shrunk to contain only the i-th molecule and its
z nearest neighbors. We then have:
14
Mi* 
 dX
N 1
z


 i    j  exp( U / kT )


j=1
N 1
dX
exp( U / kT )

(4.24)
with N=z+1. Substitution this into (4.22) and using the fact that the
material is isotropic, we obtain:
z
g  1 +   p( X i )dX i
j 1
N -i
dX
cosij exp( U / kT )

 dX
N -i
exp( U / kT )
(4.25)
Since after averaging the result of the integration will be not depend on
the value of j, all terms in the summation are equal and we may write:
g  1  z  cosij 
(4.26)
Since cosij depends only on the orientation of the two molecules, all
other coordinates can be integrated out and we may write:
15
 cos ij
d d cos  exp(   U
 / kT )


 d d exp(   U  / kT )
i
j
i
i  j
ij
j
(4.27)
i  j
where i and j denote the orientation coordinates of the i-th and the
j-th molecules, and U   is a rotational intermolecular interaction
energy, averaged over all positions and the orientations of all other
molecules.
i
j
It is clear from eqn. (4.27) that
g will be different from 1 when <cosij>0, i.e. when there is
correlation between the orientations of neighboring molecules.
When the molecules tend to direct themselves with parallel dipole
moments, <cosij> will be positive and g>1.
When the molecules prefer an ordering with anti-parallel dipoles, g <1.
16
Polarizable Molecules
Let as consider the system of N identical molecules with permanent
dipole strength  and scalar polarizabilities . The i-th molecule is
located at a point with radius vector ri and has an instantaneous dipole
moment mi. This dipole moment will be given by:
mi  μi  ( E1 )i
(4.28)
where (E1)i, the local field at the position of the i-th molecule for a
specified configuration of the other molecules, is given by:
N
E1 i  E0  Tij  m j
(4.29)
j i
In this equation Eo is the external field and -Tij · mj is the field at ri
due to a dipole moment mj at ri. We can also define the 33dimensional dipole-dipole interaction tensor Tij, connected with the
17
molecules i and j.
In the case of polarizable molecules the total moment of the sphere in
Kirkwood approximation will be given by:
N
M =   i  pi 
(4.30)
i=1
where pi is the induced moment of the i-th molecule. The induced
moment pi is a function of the positions and orientations of all other
molecules. Taking into account (4.28) we can write:
pi  mi - μi  ( E1 )i
(4.31)
The local field depends on the positions and orientations of all other
molecules. Therefore it is not possible to perform the integrations in
<M2>o in two steps , as we did in the non-polarizable case.
18
Approximation of Fröhlich
For the representation of a dielectric with dielectric permittivity ,
consisting of polarizable molecules with a permanent dipole moment,
Fröhlich introduced a continuum with dielectric constant  in which
point dipoles with a moment d are embedded. In this model each
molecule is replaced by a point dipole d having the same nonelectrostatic interactions with the other point dipoles as the molecules
had, while the polarizability of the molecules can be imagined to be
smeared out to form a continuum with dielectric constant .
Let us also split polarization P in two parts:
the induced polarization Pin and the orientation polarization Por
The induced polarization is equal to the polarization of the continuum
with the , so that we can write
  1
Pin 
E
4
(4.32)
19
The orientation polarization is given by the dipole density due to the
dipoles d. If we consider a sphere with volume V containing N
dipoles (as we did in non-polarizable case), we can write:
where:
1
Por   M d  e 
V
(4.33)
N
M d =   d  i
(4.34)
i=1
<Md·e>, the average component in the direction of the field, of the
moment due to the dipoles in the sphere, is given by an expression :
< M d  e 
N 1
dX
M d  e exp( U/kT )

 dX
N 1
exp( U/kT )
(4.35)
20
Here U is the energy of the dipoles in the sphere. This energy consists
of three parts:
1. the energy of the dipoles in the external field
2. the electrostatic interaction energy of the dipoles
3. the non-electrostatic interaction energy (London-Van der Waals
energy) between the molecules which is responsible for the shortrange correlation between orientations and positions of the
molecule.



The external field in this model is
equal to the field within a spherical
cavity filled with a continuum with
dielectric constant , while the cavity
is situated in a dielectric with dielectric
constant  (Fröhlich field EF ).
The field of this cavity field with
dielectric will be:
3
EF 
E
2  
(4.36)
21
Thus the energy of the dipoles in the external field can be written
as -Md·EF. Taking in consideration that the first derivative of P with
respect to E was identified with (-1)/4 we can write:
  1 = 
4

Pin  Por 

 E
 E 0
after substituting (4.32) and (4.33) and rearrangement we can obtain:
 1
P 
E
V 4
41 




e 
Pinor  V EMdM ed  
 E 0
(4.32)
(4.33)
We now rewrite with EF instead of E as the independent variable:

4  E F   
   
 M d  e  

 
V  E  E 0  E F
 EF 0
(4.37)
This equation has the same form as eqn (4.4). The differentiation with
respect to EF in expression (4.37) gives:
22
4  EF   M d2 0
   


V  E  E 0 3kT
With the help of (4.36) for EF we can write :
4 3  M d2 0
   
V 2    3kT
(4.38)
3
EF 
E
2  
(4.39)
or, after rearrangement:
kTV (     )( 2     )
 M 0 
4

2
d
(4.40)
The average in (4.39) and (4.40) can be evaluated in the same way as
has been done for <M2 >o in (4.9). Instead of eqn. (4.29) we now
obtain:
( -   )(2 +   )
N

g d2
12
3kT
(4.41)
The moment d can be connected with the moment  of the molecule
in the gas phase in the following way.
23
In terms of the simplified model, evaporation consists in the
disengagement of small spheres with dielectric constant  and a
permanent dipole moment d in the center. The moment m of such a
sphere in vacuum consists of the permanent moment d and the
moment induced by d in the surrounding dielectric:
  1
3
m  μd 
μd 
μd
  2
  2
(4.42)
Obviously, m must be set equal to the moment  of the molecule in
the gas phase. In this case we find:
μd 
  2
μ
3
(4.43)
Substituting eqn.(4.43) into eqn.(4.41), we obtain after simple
rearrangement:
9 kT    2   
g 
2
4
   2
2
Equation (4.44) is called the Kirkwood-Fröhlich equation.
(4.44)
24
This equation gives the relation between , dielectric permittivity, ,
the dielectric permittivity of induced polarization, the temperature, the
density, and the permanent dipole moment, for those cases where the
intermolecular interactions are sufficiently well known to calculate
Kirkwood correlation factor g.
If there is no specific correlations one has g=1.
If the correlations are not negligible, detailed information
about the molecular interactions is required for the
calculations of g.
For associating compounds, where the occurrence of hydrogen bonds
makes relevant the assumption that only certain specific angles
between the dipoles neighboring molecules are possible the molecular
interactions may be represented by simplified models.
Let us derived this extended equation for those compounds where the
molecules or the polar segments form clusters of limited size.
Consider each kind of polymers or multimers as a separate compound
and neglecting any specific correlation between the total dipole
moments of the polymers or multimers we can try the general
equation for the polar fluids:
25




N 0 0
Nn
( - 1)(2 + 1)
1
n
 n 

=

12
1  f 0 0 n1 1  f n  n 
3kT 1  f n  n 



(4.45)
where index o refers to the non-polar solvent, and index n refers to
2
the polymers or multimers containing n polar units. The average  n is
taken over all conformations of the polymer or multimer. The upper
limit of the summation can be extended to infinity since Nn, the number
of n-mers per cm3, becomes zero when n becomes large. The
polarizability o can be calculated from the Clausius-Mossotti equation
for the pure solvent.
(4.46)
3 0 1 M 0
0 
4  0  2 d0 N A
Combining the Clausius-Mossotti equation and the Onsager
approximation for the radius of the cavity, we have
1
2  1 0  2

1  f 0 0 2    0 3
(4.47)
26
We assume that the polarizabilities and the molecular volumes of the
n-mers are proportional to n, so that:
(4.48)
n  n1
where
3    1 M1
1 
4     2 d1 N A
(4.49)
Here, d is the density and  is the dielectric constant of induced
polarization of the polar compound in the pure state. In the same way
we find, using Onsager's approximation for the radius of the cavity:
1
2  1    2
(4.50)

1  f n n 2    3
We now substitute equations (4.46) - (4.50) into (4.45) and divide both
members by (2+1), obtaining:

N 0 (  0  1 )M 0
 1
(    1 )M 1


nN n 

12 4( 2   0 )d 0 N A 4( 2    )d1 N A n 1
( 2  1 )(    2 )2

27kT( 2    )2

N
n 1
n

2
n
(4.51)
27
The average 
2
n

can be calculated as
2
n

 n
  n
   μn ,i     μn , j 
 i 1
  j 1

(4.52)
where n,i is the moment of the i-th dipolar unit of the n-mer. By
taking the i-th dipolar unit as the representative unit and averaging
over all possible positions of the i-th unit in the chain, we may write
eq.(4.52) as follows:

2
n
 n

 n  n ,i     n , j   ng n 2
 j 1

(4.53)
where  has been used to denote the dipole strength of a single unit.
The factor gn represents the average value of the ratio between the
component of the moment of the whole n-mer in the direction of the
permanent moment of an arbitrary segment and the dipole strength of
the segment. Since we assumed that there is no correlation between
different chains, gn represents the total correlation between the
permanent moment of a segment in a n-mer and its surroundings.
28
To obtain the Kikwood correlation factor g, we must average gn over all
values of n, with a weight factor equal to the chance that a segment

forms part of a n-mer. This chance is given by nN /
nN
n

n 1
n
if Nn is the number of n-mers per cm3. Thus, we find with the help of
eqn. (4.53):

g
 nN
n 1


n
gn
 nN
n 1
n

N
n 1

2
n
 2n

 nN
n 1
(4.54)
n
We now use molar fractions xo and xp for the nono-polar component
respectively, regarding each segment as a separate molecule:
xo N A
N0 

xp N A
nN n 

n 1

(4.55)

(4.56)
29
In these equations =(xoMo+xpM1)/d denotes the molar volume of the
mixture. Substituting eqns. (4.54)-(4.56) into eqn.(4.51) we find:
x p (    1 ) M1
 1
x0 (  0  1 ) M 0



12  4 ( 2    0 )d0 4 ( 2     )d1
( 2   1 )(    2 )2 x p N A 2

g
2
27 kT ( 2     ) 
(4.57)
From this it follows:
2
9
kT
(
2



)

g 2 

4N A x p (2  1)(  2) 2
  (  1) 3x0 M 0 (0  1) 3x p M1 (  1) 





(
2



)
d
(
2



)
d
0
0

1 

(4.58)
This equation makes possible the calculation of the Kirkwood
correlation factor g from experimental data for solutions of associating
or polymeric compounds in non-polar solvents.
30