1. Tutorial 1 – Static Dipoles
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
Drawback of the Lorentz local field
Dipole moment of a dielectric sphere in a dielectric medium
Actual dipole moments, definition and status
Directing field and the Onsager equation
Statistical theories for static dielectric permittivity. Kirkwood theories
Fröhlich's statistical theory
Distortion polarization in the Kirkwood and Fröhlich theories

2. These tutorials are based on the following book:
Evaristo Riande, Ricardo Diaz-Calleja. Electrical Properties of Polymers. Marcel Dekker, NY, 2004

3. 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.
The dipole moment is a vectorial quantity defined as
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. +q -q l

4. 1.2. ELECTRIC POTENTIALS ARISING FROM AN ISOLATED DIPOLE
Electrostatic potential at the generic point P is
According with the cosine theorem,
By using series expansion z P r+ r +q
d/2  r-
d/2 -q

5. Where P, is the Legendre polynomials

6. 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.
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.
Dipolar moment
Cuadripolar moment

7. 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

8. 1.4 - FIELD OF AN ISOLATED POINT DIPOLE
The dipole potential can be written as
The electric field is
After some calculations, one obtains
The field, expressed in matrix is

9. 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)

10. 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)

11. For an arbitrary external electric field, Eq (1 5
For an arbitrary external electric field, Eq (1 5.3) can be written in terms of the components as

12. 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 of the other dipole. According to Eq.(1.4.3), the energy is:
the force exerted on one of these dipoles owing to the field produced by the other is

13. 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

14. 1.7. TORQUES ON DIPOLES Eo The total torque is Fo d/2
Neglecting the term of high order and considering the dipole definition, Fo +q -q Eo
d/2
d/2 Fo

15. 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

16. 1. 8. DIPOLE MOMENT AND DIELECTRIC PERMITTIVITY
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

17. 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
where α is total polarizability (includes the electronic, αe, the atomic, αa, and the orientational, αo, contributions)

18. Equation (1.8.1) is strictly true only if E is small.
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
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
The mean electric field (Ez) must be added to the field arising from the charge densities in the plates.

19. This field is given by
(D electric displacement)
The dielectric susceptibility is defined as
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

20. 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.

21. 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.

22. E + - d
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 

23. The polarization was previously defined(1.8.6) as:
Therefore, the total field will be
This is the Lorentz field.

24. By combining Eqs (1. 9. 2),(1. 9. 3), and (1. 8
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 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.

25. 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

26. 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

27. Debye equation for the static permittivity
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

28. 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 , 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.

29. 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.

30. Ferro-electricity is uncommon in nature, and predictions made by Eq (1
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.

31. 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 .

32. 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.

33. 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.

34. 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.

35. Owing the boundary conditions, only the first term of the polynomial (cos θ) appears in the Eq , so that series of spherical harmonics is greatly simplified. Consequently the potentials are
Solving the system of equation and , the resulting potentials are:
(1.11.5)

37. Special Cases:
1 – Empty cavity, in this case 1=1, and all the terms in Eq 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

38. 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 and 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,

39. 3 – Finally, for 1= 3=1, Eq (1.11.7) leads to

40. 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

41. 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,

42. 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 ( ), also appear in Eq. ( c)
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

43. Although this integral can be obtained by a simple electrostatic calculation, it can also be determined by comparing Eqs ( ) and ( ), that is

45. 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

46. 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

47. 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

48. Rearrangement of Eq (1. 13. 9), taking into account Eq. (1. 12
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 ) and the Onsager theories. The Lorentz local field (Eq ) change Eq.(1.9.11) to
Comparison of Eqs. ( ) and ( ), 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

49. 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.

50. 1. 14. STATISTICAL THEORIES FOR STATIC DIELECTRIC PERMITTIVITY
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.

51. 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.

52. 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.

53. 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 .

54. Fröhlich analysis leads to
<M2>o.

55. 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( )).

56. 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

57. 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.

59. 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