Transcript Document
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.
1
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:
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)
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?
2
Let us consider the derivation of Kubo result from classical statistical
mechanics.
In this theory one considers the dipole moment j of the j-th molecule
and looks for the response μ j ( t ) 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.
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.
In classical statistical mechanics the behavior of a system is described
with the help of generalized coordinates qi and conjugated moment pi,
where the index i ranges over all degrees of freedom of all particles.
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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.
The Hamiltonian for N interacting molecules, where N is a very large
number, placed in a uniform electric field E0(t) is
1
H H0 dRm( R) E0 (t) dRE0 (t) E0 (t) a ( R) E0 (t)
2
(8.1)
where H0 is the Hamiltonian in the absence of the electric field, E0 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)
m( R) m ( R)
j
j
a( R ) a j ( R)
(8.3)
j
4
m j ( R) j ( R q j ) k Tkj j ( R q j )
k
k Tkl l Tlj j ( R q j ) .......,
(8.4)
k ,l
1
a j ( R) j ( R q j ) k Tkj j ( R q j )
2 k
1
k Tkl l Tlj j ( R q j ) .......,
6 k ,l
(8.5)
where qj is the position of the j-th molecule, Tjk 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
E0; the dipolar interaction terms dependent have been explicitly
represented, other intermolecular interactions are incorporated
implicitly in j, j and H0. The polarization per unit volume P(R,) at
point R is
5
P(R , ) dt exp(it )[m(R , t ) a(R , t )] E0 (t )]
(8.6)
0
where the bar over a quantity indicates an instantaneous ensemble
average at time t and position R in the presence of electric field E0(t).
By means of linear response theory, we can rewrite equation (8.6) as:
P(R , ) ( kT ) 1 dR' m( R,0) m( R' ,0)
i dt dR' m( R,0) m( R' ,0) exp(it ) a ( R, t ) E0 ( )
0
(8.7)
where k is the Boltzmann constant, temperature , <> indicates an
equilibrium ensemble average in the absence of the field E0. 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 (qj)=V-1. The correlation function dR' m( 0, t)m( R',0 ) > is
then proportional to (N/V)(t), where (t) is the normalized
macroscopic dipole correlation function:
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N
M(0)M(t )
(t )
M(0)M(0)
N
(0) (t )
i
i
j
N
N
i
j
j
(8.8)
(0) (0)
i
j
where M(t) is a vector sum of N polar molecules with dipole moment
I (t) in unit volume at time t. The terms i (0) j (0) in (8.8) express the
equilibrium orientation correlation between dipoles i and j in the
sample. It means that
N
N
is equal to .
(0) (0)
i
i
j
j
M 2 (0) N 2 [1 ( N 1) cos ] N 2 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:
N 1 i 1
N
(t )
(0) (t ) 2 (0) (t )
i 1
i
i
N
i 1
2
i
i 2 j 1
N 1 i 1
i
2 i (0) j (0)
i 2 j 1
j
(8.9)
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Let us consider the autocorrelation function
i (0)i (t ) i2 ui (0)ui (t )
where ui 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:
i (0) i (t )
N
2
(
0
)
(
t
)
N
(
0
)
(
t
)
N
i i
i
i
i ij (t )
i 1
where
ij (t ) ui (0)ui (t )
(8.10)
(8.11)
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 ui(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:
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ui (0)ui (t ) cos (t )
(8.12)
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.
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Complex dielectric permittivity
and Dipole correlation function
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:
* ( ) 1
(8.13)
P( )
E( )
4
The relationship between the polarization and external electric field
Eo(), in terms of linear response theory leads to:
P( ) 0* ( )E0 ( )
(8.14)
where *o() is the quasi susceptibility and connected with
macroscopic DCF (t) by the following relation:
2
2
M
d
(
t
)
M
d (t )
*
0 ( )
exp( it )dt
L[
]
kTV 0 dt
kTV
dt
(8.15)
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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 Eo() 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 Eo() is:
E( )
* ( ) 2
3
E 0 ( )
(8.16)
so that
* ( ) 1
3
*
*
0 ( )
4
( ) 2
(8.17)
The value <M2(0)>=<M(0)M(0)> can be related to the static dielectric
permittivity according to the Clausius-Massoti relation:
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3VkT s 1
M (0) 0
4 s 2
2
(8.18)
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:
* ( ) 1 s 2
L[ (t )]
*
( ) 2 s 1
(8.19)
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:
3s
* ( ) 1
L[ (t )]
s 1 2s * ( )
where
m(0) m(t )
(t )
m(0) m(0)
(8.20)
(8.21)
is a normalized autocorrelation DCF of the small spherical sample
embedded in a infinite dielectric continuum.
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The macroscopic DCF
deconvolution equation:
(t)
connected
with
(t)
by
integral
t
(t ) (t ) A (t ') (t t ')dt '
2( 1)
2
(8.22)
0
s
where A
is a time independent constant. Bob Cole used
(2 s 1)( s 2)
the same approach to obtain the similar equation for the case of the
polarizable molecules:
* ( )
3s
L[ (t )]
*
s 2s ( )
(8.23)
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:
* ( ) 1 [2 * ( ) 1]s
L[ (t )]
s 1 [2s 1] * ( )
(8.24)
13
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:
[ * ( ) ][2 * ( ) ]s
L[ (t )]
*
[s ][2s ] ( )
(8.25)
where
(t )
μ( 0 ) μ( t )
μ( 0 ) μ( 0 )
(8.26)
is a molecular DCF.
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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.
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.
15
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:
[ * ( ) 1][2 * ( ) 1] 4 2
L g (t )
* ( )
kT
(8.27)
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 (t0 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.
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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
3[ * ( ) 1]
L[a (t )]
L[ g (t )]
*
2 ( ) 1
(8.28)
where a(t) is the long range correlation function.
The generalization of these results for a particular situation with
polarizable molecules gave the following expression
[ * ( ) 1][2 * ( ) 1] 4 2
L[ g(t )]
*
kT
( )
(8.29)
where the connection between the short and long correlation functions
can be found from:
3[ * ( ) ]
L[a(t )]
L[ g(t )]
*
2 ( )
(8.30)
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As it was mentioned above, the equation (8.29) represents the
dynamical extension of the Kirkwood-Frohlich formula in the classical
limit (t0 and ). Making combination with Kirkwood-Frohlih
expression, we can get the following:
[ * ( ) ][2 * ( ) ] 4 2
g
L[ g(t )]
*
kT
[s ][2s ] ( )
(8.31)
If g=1 the equation (8.31) is equal to (8.25).
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