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Tutorial 1 – Static Dipoles
1.1 - Dipoles
1.2 - Electric potentials arising from an
isolated dipole
1.3 - Point dipole
1.4 - Field of an isolated point dipole
1.5 - Force exerted on a dipole by an
external electric field
1.6 - Dipole-dipole interaction
1.7 - Torques on dipoles
1.8 - Dipole moment and dielectric
permittivity Molecular versus
macroscopic picture
1.9 - Local field. The Debye static
theory of dielectric permittivity
1.10 - Drawback of the Lorentz local
field
1.11 - Dipole moment of a dielectric
sphere in a dielectric medium
1.12 - Actual dipole moments,
definition and status
1.13 - Directing field and the Onsager
equation
1.14 - Statistical theories for static
dielectric permittivity. Kirkwood
theories
1.15 - Fröhlich's statistical theory
1.16 - Distortion polarization in the
Kirkwood and Fröhlich theories
These tutorials are based on the following book:
Evaristo Riande, Ricardo Diaz-Calleja. Electrical Properties
of Polymers. Marcel Dekker, NY, 2004
1.1 – DIPOLES.
Due to the electrical neutrality of the matter, dipoles, or multipoles are basic
elements of the electric structure of the many material.
A Dipole can be defined as an entity made up by a positive charge q separated a
relatively short distance l an equal negative charge.
-q
The dipole moment is a vectorial quantity defined as
  q l
l
+q
By convention, its positive direction points towards the positively charged end.
For more complicated systems, it is necessary to take into account the geometry of
the molecules and interaction with other surrounding molecules.
1.2. ELECTRIC POTENTIALS ARISING FROM
AN ISOLATED DIPOLE
z
P
Electrostatic potential at the
generic point P is
q q
(r ,  )  
(1.2.1)
r r
r+
+q
d/2

d/2
According with the cosine
theorem,
r
r-
1 1 d
d 
 1
cos    
r r  r
 2r 
2




1
2
(1.2.2)
By using series expansion
-q
1  x   1  nx 
n
n(n  1) 2
x  ..... (1.2.3)
2!
(r ,  ) 
q q

(1.2.1)
r r
1 1 d
d 
 1
cos    
r r  r
 2r 
1  x   1  nx 
n
qd
(r , )  2
r
2




1
2
(1.2.2)
n(n  1) 2
x  ..... (1.2.3)
2!
2


1d 
 P1 (cos  )    P3 (cos  )  .... (1.2.4)
4 r 


Where P, is the Legendre polynomials
P0  1, P1  cos  , P2 =
P3 
1
(3cos 2   1),
2
1
(5cos3   3cos  ),...
2
(1.2.5)
Note that in equation 1.2.4, q·d cos θ = q·d · r/r (where r/r
= u), and the quantity (μ = q·d is called dipolar moment.
2


qd
1d  1
3
 (r ,  )  2 cos     (5cos   3cos  )  .... (1.2.4)
r 
4 r  2

Dipolar moment
Cuadripolar moment
The S.I units for dipoles is C·m, although the Debye, D,
which is the dipole moment corresponding to two
electronic charges separated by 0.1 nm, is a commonly
used unit.
1.3 - POINT DIPOLE
Some times it is convenient to refers at dipoles as point dipole, that is: a point
in the space, with moment and direction μ
1.4 - FIELD OF AN ISOLATED POINT DIPOLE
The dipole potential can be written as
The electric field is
 ·cos 
r2

E    
After some calculations, one obtains
The field, expressed in matrix is

 ·u
r2
 ·r
r
3

 ·r
r3
   ·grad
1
(1.4.1)
r
(1.4.2)
 3(μ·r )r μ 
E 
 3  (1.4.3)
5
r 
 r
3x 2  r 2
3 xy
3 xz
r5
r5
r5
Ex
3 yx
3y2  r 2
3 yz
Ey  r 5
r5
r5
3 zx
3 zy
3z 2  r 2
Ez
r5
r5
r5
x
y
z
(1.4.4)
1.5 - FORCE EXERTED ON A DIPOLE BY AN
EXTERNAL ELECTRIC FIELD
Consider a dipole located in an external electric field Eo. The total force acting on the dipole
is the sum of the forces acting on each charge, that is
(1.5.1)
If d is very small in comparison with r, the first term on the right-hand side of Eq.(1.5.1) can
be expanded into a Taylor series around r, giving
(1.5.2)
substituting Eq ( 1.5.2) into Eq ( 1.5.1) and neglecting terms of order d2 and higher, we obtain
(1.5.3)
The electrical force can also be derived from the
potential electrical energy U, noting that
(1.5.5)
For a single charge q, the ratio of Eq (1.5.5) to q gives
Eo = -  Φ. By comparing Eqs (1.5.3) and (1.5.5), the
potential for a single dipole is given by
(1.5.6)
For an arbitrary external electric field, Eq (1 5.3) can be written in terms of
the components as
E0 x
E0 x
E0 x
F0 x   x
 y
 z
x
y
z
E0 y
E0 y
E0 y
F0 y   x
 y
 z
x
y
z
E0 z
E0 z
E0 z
F0 z   x
 y
 z
x
y
z
(1.5.7)
1.6. DIPOLE - DIPOLE INTERACTION
The energy of a system formed by two
dipoles µ1 and µ2 is the work done on
one of these dipoles placed in the field U    ·E (  )    ·E (  )
i
2
1
1
2
of the other dipole. According to
1
3
Eq.(1.4.3), the energy is:
the force exerted on one of these
dipoles owing to the field produced by
the other is



(

·

)

(

·
r
)(

·
r
)
1
2
 2 1

r3 
r2

(1.6.1)
(  ·r )(  ·r ) 
  ·
Fi  U i     1 3 2  3 1 5 2 
r
r


( 1 ·r )


1

·



3

(

·
r
)


2
 1 2  r3 

r5



 
 ( 2 ·r )
 1 

3

(

·
r
)

3(

·
r
)(

·
r
)


1
2
1
 5 
r5
 r 


3
5


·

r


·
r


(

·
r
)


 ·r ( 2 ·r )r 




1
2
1
2
2
1
5 
2  1 
r 
r

(1.6.3)
Special cases
If µ1=µ2 = µ, the mutual force will be
For parallel dipoles (µ1=µ2) perpendicular to r
For antiparallel dipoles (µ1=-µ2) perpendicular to r

Finally, for perpendicular dipoles that are both also perpendicular to r
1.7. TORQUES ON DIPOLES
Fo
The total torque is
-q
d/2
d
 d
   qE0 (d  r )        q  E0 (r ) (1.7.1)
2
 2
d/2
Fo
Neglecting the term of high order and
considering the dipole definition,
  d  qE0  μ  E0 (1.7.2)
+q
Eo
The torque is null when µ and Eo are parallel.
As a consequence of the torque, the dipole tends to reorient
along the field, even in the presence of a uniform field. This is
a consequence of the not cancellation of the torque of two
opposite non collinear forces.
The work of reorientation from a position perpendicular to the
field to another forming an angle θ is given by


W   Γ d '   μ E0 sin  ' d '  μ E0 cos 


2
2
(1.7.3)
1.8. DIPOLE MOMENT AND DIELECTRIC
PERMITTIVITY. MOLECULAR VERSUS
MACROSCOPIC PICTURE
In real situations
continuous medium
enormous number of elementary dipoles.
 Dipolar substances contain permanent dipoles arising from the asymmetrical
locating of electric charges in the matter.
 Under an electric field, these dipoles tend to orient as a result of the action of
torques,
 The macroscopic result is the orientation polarization of the material
The effect of the applied electric field is also to induce a new types of polarization
 by distortion of the electronic clouds (electronic polarization)
 and the nucleus (atomic polarization) of the atomic structure
Time scale to reach equilibrium 10-10~10-12 s.
Under an applied field of frequency 1010~1012 Hz, the orientation polarization
scarcely contributes to the total polarization. However, this contribution increases
when the frequency diminishes (time increases), and this is the essential feature of
the dielectric dispersion.
For a static case, the field E acting on a dipole is, in general, different from the
applied field Eo. The average moment for an isotropic material can be written as
m   ·E
(1.8.1)
where α is total polarizability (includes the electronic, αe, the atomic, αa, and the
orientational, αo, contributions)
Equation (1.8.1) is strictly true only if E is small.
m   ·E
(1.8.1)
This means :
 no saturation effects
 relationship between the induced moment and the electric field is
of the linear type.
The total polarization of N1 dipolar molecules per unit volume, each of
them having an average moment m, is
N
P  N1m  m
V
(1.8.2)
where N is the total number of molecules.
The average field caused by the induced dipoles of the dielectric placed
between two parallel plates is given by
1
4 Nm
3
Ez   Ez (r )d r  
 4 P
V
V
(1.8.3)
The mean electric field (Ez) must be added to the field arising from the charge
densities in the plates.
This field is given by
E0  4 P  D
(1.8.4)
(D electric displacement)

The dielectric susceptibility is defined as
 1 P


4
E0
(1.8.5)
D   E 0   0 r E 0
(1.8.7a)

r 
0
(1.8.7b)
  1  4 
D
E0
(1.8.6)
εr is the relative permittivity and εo (= 8,854x 10-l2 C2kg-1m-3s2) is
the permittivity of the free space. In c.g.s. units, εo = 1 and εr= ε
1.9. LOCAL FIELD. THE DEBYE STATIC
THEORY OF DIELECTRIC PERMITTIVITY

Until now the dipolar structure of the dielectric was considered, but the
electric field acting on the dipoles was not determined.

For the evaluation of the static dielectric permittivity in terms of the dipole
moment μ of the molecules of the dielectric require the determination of the LOCAL
FIELD acting upon the molecules and the ratio of the dipole moment of the molecule
and either the polarizability αo or the average moment m.

The concept of INTERNAL FIELD has received considerable attention
because it relates macroscopic properties to molecular ones. The problem involved in
the determination of the field acting on a single dipole in the dielectric is its
dependence on the polarization of the neighboring molecules.
The first approach to the analysis in this problem is due to Lorentz.
The basic idea is to consider a a spherical zone containing the dipole under study,
immersed in the dielectric.
The sphere is small in comparison with the dimension of the condenser, but large
compared with the molecular dimensions.
We treat the properties of the sphere at the microscopic level as containing many
molecules, but the material outside of the sphere is considered a continuum.
The field acting at the center of the sphere where the dipole is placed arises from the
field due to
(1) the charges on the condenser plates
(2) the polarization charges on the spherical surface, and
(3) the molecular dipoles in the spherical region.
The field due to the polarization charges on the
-
spherical surface, Esp, can be calculated by
considering an element of the spherical surface
defined by the angles θ and θ+dθ.
The area of this elementary surface is:
2πr2sin θdθ.
The density of charge on this element is given
by P·cosθ, and the angles between this
polarization and the elementary surface is θ.
Integrating over all values of angle formed by
the direction of the field with the normal vector
to spherical surface at each point and dividing
by the surface of the sphere we obtain
-
E
d
-

-
+
+
+
+
+ +
The polarization was previously defined(1.8.6) as:
Therefore, the total field will be
This is the Lorentz field.
By combining Eqs (1.9.2),(1.9.3), and (1.8.1), the
following equation relating the permittivity and the
total polarizability is obtained:
Here, the assumption that m=μ (dilute systems) is
made. If the material has molecular weight M, and
density ρ, then N1=ρNAM, where NA is the Avogadro’s
number, in this case the equation 1.9.4 becomes:
This is the Claussius – Mossotti equation valid for
nonpolar gases at low pressure.
This expression is also valid for high frequency limit.
The remaining problem to be solved is the calculation
of the dipolar contribution to the polarizability.
The first solution to this problem for the static case was proposed by Debye.
In absence of structure, the potential energy of a dipole of μ is given by:
In this case, Ei is the internal field, instead of the external applied field Eo, because
Ei is the field acting upon the dipole.
If a Boltzmann distribution for the orientation of the dipoles axis is assumed, the
probability to find a dipole within an elementary solid angle dΩ is given by
The average value of the dipole moment in the direction of the field is
After integration by parts
Where L(y) =coth y – y-1 denotes the Langevin function given by
For y<<1, the Langevin function is approximately L(y)y/3, and the orientational
polarization is given by
If the distortional polarizability d is added, the total polarization is
By substituting Eq (1.9.3) into Eq (1.9.12) and taking into account Eq(1.9.2), one obtain
Debye equation for the static permittivity
At very high frequencies, dipoles do not have time for reorientation and the dipolar
contribution to permittivity is negligible. Then,
Where d=∞. Since the infinite frequency permittivity is equal to the square of the
refraction index, n, Eq (1.9.15) becomes
Lorentz-Lorenz equation
Note that equation 1.9.16, can be rewritten as
where =1/, is the specific volume, and C=4NA/3M. By plotting the
C(n2+2)/(n2+1) vs T-1, the isobaric dilatation coefficient can be obtained, and,
eventually, the glass transition temperature could be estimated.
Note that from the temperature dependence of the index of refraction of glassy systems,
physical aging and related phenomena could be analyzed.
1.10. DRAWBACK OF THE LORENTZ LOCAL
FIELD
For highly polar substances, the dipolar contribution to the polarizability is clearly
dominant over the distorortional one.
From previous equations and substituting o=2/3kT, the susceptibility becomes infinite
at a temperature given by
This temperature , is called Curie temperature.
This equation suggest that at temperature , spontaneous polarization should occur and
the material should become ferroelectric, even in absence of an electric field.
Ferro-electricity is uncommon in nature, and predictions made by Eq (1.10.1)
are not experimentally supported.
The failure of this theory arises from considering null the contribution to the
local field of the dipoles in the cavity.
This fact emphasizes the inadequacy of the Lorentz field in a dipolar dielectric
and leaves open the question of the internal field in the cavity.
1.11. DIPOLE MOMENT OF A DIELECTRIC
SPHERE IN A DIELECTRIC MEDIUM
Claussius-Mossotti equation represent the first attempt to relate a macroscopic
quantity, the dielectric permittivity, to the microscopic polarizability of the substances.
The equation only is valid for the displacement polarization.
To solve the same problem for the orientational polarizability is much more
complicated.
The starting point of the solution to the problem of the local field in the context of the
theory of the orientational polarization is to consider a spherical cavity of radius a and
relative permittivity 1, that contains at the center a rigid dipolar molecule with
permanent dipole .
The radius of the cavity is obtained
form the relation
Where N is the number of molecules in
a volume V. In many cases a can be
considered as the “molecular radius”.
This cavity is assumed to be surrounded
by a macroscopic spherical shell of
radius b>>a, and the relative
permittivity 2.
The ensemble is embedded in a
continuum medium of relative
permittivity 3.
According to Onsager, the internal field in the cavity has two
components:
1 – The cavity field, G, (the field produced in the empty cavity by
the external field.)
2 - The reaction field, R (the field produced in the cavity by the
polarization induced by the surrounding dipoles).
Dielectric or conducting shells are technologically important, and
biological cells are also examples of layered spherical structures.
The general solution of the Laplace equation (2ф=0), in spherical coordinates and for
the case of axial symmetry, is
Where the i subscript refers to each one of the three zones under consideration. For our
particular geometry, the following boundary condition hold
These conditions arise from the fact that the only charges existing in the cavity
contributing to ф1 belong to the dipole. The contribution of the external field to ф3, for
r>>>b is Eor·cosθ.
Owing the boundary conditions, only the first term of the polynomial (cos θ) appears in
the Eq. 1.11.2, so that series of spherical harmonics is greatly simplified. Consequently
the potentials are
Solving the system of equation 1.11.3 and 1.11.4, the resulting potentials are:
(1.11.5)
Special Cases:
1 – Empty cavity, in this case 1=1, and all the terms in Eq 1.11.7 containing  as a factor
vanish. The resulting potential are given by
2 – Dipole point in the center of the empty cavity. Absence of external field. In this case, again
1=1, and the terms containig Eo vanish. The potentials are given by
It could be also useful to consider the inner cavity in a continuum, that is, without spherical shell.
In this case, 2= 3, and consequently G32=1, and R32=0. Thus the Equations 1.11.8 and 1.11.9
become respectively
These results indicate that the field in an empty cavity embedded in a dielectric medium with
permittivity 2 is
On the other hand, dipoles induce on the surface of the spherical cavity (even in absence of
external field) an electric field opposing that of the dipole himself, called the reaction field,
3 – Finally, for 1= 3=1, Eq (1.11.7) leads to
1.12. ACTUAL DIPOLE MOMENTS, DEFINITION
AND STATUS
The potential of a rigid non-polarizable dipole in a medium of relative permittivity 1 is
given by
(e subscript means external)
Usually, dipoles are associated to molecules having an electronic cloud that shields them.
Thus let us consider a dipole  in the center of a sphere representing the molecule.
The electronic polarizability of the sphere gives rise at a macroscopic level, to an
instantaneous permittivity .
In this case, the potential is given by
By identifying the potentials in Eqs. (1.12.1) and (1.12.2), one find
Where eis called the external moment of a molecule in a medium with relative permittivity 1.
The dipole molecule in vacuum will be
However, the total moment of the molecule, also called the internal moment, is the vector sum of
its vacuum value and the value induced in it by the reaction field, that is
Which can be written as
If the polarizability  is known,
Then, one obtain
Internal dipole moment
Note that it expression is independent of the size of the spherical specimen. So, if a very large
sphere containing a dipole with internal moment i, in a infinite dielectric medium of relative
permittivity 1 is considered, the dipole moment of the large sphere is also, i. For = 1 , the
following expression holds
Note that the factor appearing in Eq (1.12.10), also appear in Eq. (1.11.12c)
The internal moment can be also written as
The internal moment can also be calculated as the geometric sum of the external moment of the
dipole and the moment of a sphere with the same permittivity as the medium surrounding the
dipole. In this conditions
Although this integral can be obtained by a simple electrostatic calculation, it can also be
determined by comparing Eqs (1.12.11) and (1.12.12), that is
1.13 DIRECTING FIELD AND THE ONSAGER
EQUATION
previously, the internal dipole moment has been calculated in absence of an external field. When
this force field is applied, it is necessary to take into account this effect. In fact the total field in
the cavity is now the superposition of the cavity field G with the field due to the dipole
From which
Where  is given by Eq. (1.12.8)
The mean value of the dipole moment parallel to the external field is calculated form the
Boltzmann distribution for the orientational polarization given by Eq. (1.9.6) and (1.9.7).
However, the energy of the dipole is calculated from the torque acting on the molecule, which
according to Eq (1.13.1) is given by
From Eq. (1.13.2) into Eq. (1.13.3)
From Eq. (1.7.3) it follows that (U=∫d)
Equations (1.9.8) and (1.13.5) lead to
By substituting this equation into Eq. (1.13.2)
On the other hand, the polarization per unit of volume is
Substitution of the Eq. (1.13.7) into the Eq. (1.13.8) gives
Where the quantity
Is called DIRECTING FIELD, Ed.
The directing field is calculated as the sum of cavity field and the reaction field caused by a
fictive dipole Ed.
Note: Directing field must be not confused with the internal field Ei.
Ei= Ed+R
According to these results, Eq. (1.13.9) can be written as
Rearrangement of Eq (1.13.9), taking into account Eq. (1.12.8), gives the Onsager
expression
It is interesting to compare the Debye results (Eq 1.9.11) and the Onsager theories. The
Lorentz local field (Eq. 1.9.3) change Eq.(1.9.11) to
Comparison of Eqs. (1.13.13) and (1.13.11), leads to the conclusion that the Onsager
theory takes for the internal and directing fields more accurate values that older theories
in which Lorentz field is used. Note that, after some rearrangements, the Onsager
equation can be written as
It is clear that Onsager equation does not predict ferro-electricity as the Debye
equation does.
In fact, the cavity field tends to 3Eo/2, when ε1 tends to infinite.
Onsager takes into account the field inside the spherical cavity that caused by
molecular dipoles, which is neglected in the Debye theory.
However, following the Boltzmann-Langevin methodology, the Onsager theory
neglects dipole – dipole interactions, or equivalently, local directional forces
between molecular dipoles are ignored.
The range of applicability of the Onsager equation is wider than that of the Debye
equation.
For example, it is useful to describe the dielectric behavior on non-interacting
dipolar fluids, but in general this is not valid for condensed matter.
1.14. STATISTICAL THEORIES FOR STATIC
DIELECTRIC PERMITTIVITY.
KIRKWOOD’S THEORY
Onsager treatment of the cavity differs from Lorentz’s because the cavity is
assumed to be filled with a dielectric material having a macroscopic
dielectric permittivity.
Also Onsager studies the dipolar reorientation polarizability on statistical
grounds as Debye does.
However, the use of macroscopic argument to analyze the dielectric
problem in the cavity prevents the consideration of local effects which are
important in condensed matter.
This situation led Kirkwood first, and Fröhlich later on the develop a fully
statistical argument to determine the short – range dipole – dipole
interaction.
Making use of statistical mechanics, Kirkwood obtained the expression for the average
dipolar moment as
Then
Taking m en Eq (1.8.2) as <µe>, and the cavity field as given Eq (1.B.14),
Kirkwood Equation for non-polarizable dipoles , g is the correlation parameter, which is
a measure of the local order in the specimen.
g will be different from 1 when there is correlation between the orientations of
neighboring molecules.
When the molecules tend to direct themselves with parallel dipole moments,
will be positive and g>1.
When the molecules prefer an ordering with anti-parallel dipoles, g <1.
g =1 in the case of no dipolar correlation between neighboring molecules, or
equivalently a dipole does not influence the position and orientations of the
neighboring ones.
g depends on the structure of the material, and for this reason it is a parameter
that fives information about the forces of local type.
1.15. FRÖLHICH’S STATISTICAL THEORY
Like Lorentz, Fröhlich consider a macroscopic spherical region within an infinite
continuum material.
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
non-electrostatic 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 .
Fröhlich analysis leads to
<M2>o.
1.16. DISTORTSION POLARIZATION ON THE
KIRKWOOD AND FRÖHLICH THEORIES
Kirkwood deals with the distortional polarization by postulating that the
polarizability in Eq. (1.9.12) is also affected by the local filed given by
Onsager cavity field Local Field for a
vacuum sphere
Then , the Kirkwood equation that includes the distortional polarizability is given by
Note that the dipolar moment appearing in the Eq (1.16.2) is the internal moment
related to the moment in vacuum (Eq(1.12.10)).
Fröhlich takes into account the distortion polarization by assuming the dipoles
embedded in a polarizable continuum of permittivity .
Cavity Field
Increment of the permittivity that is due to reorientation is given by
The mean square dipole moment of the spherical region in the absence of the field is
In Fröhlich theory, the dipoles are not themselves polarizable and the distortional
polarizability corresponds to that of the continuum medium surrounding the dipoles.
For this reason, when cells in the Fröhlich theory are single molecules, m, is related
to the vacuum moment by
Generalization of the Onsager equation.
Claussius – Mossotti: Only valid for non polar
gases, at low pressure
Debye: Include the distortional polarization.
Onsager: Include the orientational polarization,
but neglected the interaction between dipoles.
describe the dielectric behavior on non-interacting
dipolar fluids
Kirkwood: include correlation factor (interaction
dipole-dipole)
Fröhlich – Kirkwood – Onsager