Tutorial 1 – Static DipolesTutorial 1 – Static Dipoles

1.1 - Dipoles1.1 - Dipoles1.2 - Electric potentials arising from 1.2 - Electric potentials arising from an isolated dipolean isolated dipole1.3 - Point dipole1.3 - Point dipole

1.4 - Field of an isolated point dipole1.4 - Field of an isolated point dipole1.5 - Force exerted on a dipole by an 1.5 - Force exerted on a dipole by an external electric fieldexternal electric field1.6 - Dipole-dipole interaction1.6 - Dipole-dipole interaction1.7 - Torques on dipoles1.7 - Torques on dipoles1.8 - Dipole moment and dielectric 1.8 - Dipole moment and dielectric permittivity Molecular versus permittivity Molecular versus macroscopic picturemacroscopic picture1.9 - Local field. The Debye static 1.9 - Local field. The Debye static theory of dielectric permittivitytheory of dielectric permittivity

1.10 - Drawback of the Lorentz local 1.10 - Drawback of the Lorentz local fieldfield1.11 - Dipole moment of a dielectric 1.11 - Dipole moment of a dielectric sphere in a dielectric mediumsphere in a dielectric medium1.12 - Actual dipole moments, 1.12 - Actual dipole moments, definition and statusdefinition and status

1.13 - Directing field and the Onsager 1.13 - Directing field and the Onsager

equationequation1.14 - Statistical theories for static 1.14 - Statistical theories for static dielectric permittivity. Kirkwood dielectric permittivity. Kirkwood theories theories 1.15 - Fröhlich's statistical theory1.15 - Fröhlich's statistical theory1.16 - Distortion polarization in the 1.16 - Distortion polarization in the Kirkwood and Fröhlich theoriesKirkwood and Fröhlich theories

Evaristo Riande, Ricardo Diaz-Calleja. Evaristo Riande, Ricardo Diaz-Calleja. Electrical Properties Electrical Properties of Polymersof Polymers. Marcel Dekker, NY, 2004. Marcel Dekker, NY, 2004

These tutorials are based on the following book:These tutorials are based on the following book:

Due to the electrical neutrality of the matter, dipoles, or multipoles are basic Due to the electrical neutrality of the matter, dipoles, or multipoles are basic

elements of the electric structure of the many material. elements of the electric structure of the many material.

A Dipole can be defined as an entity made up by a positive charge A Dipole can be defined as an entity made up by a positive charge qq separated a separated a

relatively short distance relatively short distance ll an equal negative charge. an equal negative charge.

The dipole moment is a vectorial quantity defined as The dipole moment is a vectorial quantity defined as

By convention, its positive direction points towards the positively charged end.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 For more complicated systems, it is necessary to take into account the geometry of

the molecules and interaction with other surrounding molecules. the molecules and interaction with other surrounding molecules.

q l +q

-q

l

1.1 – DIPOLES.1.1 – DIPOLES.

1.2. ELECTRIC POTENTIALS ARISING FROM 1.2. ELECTRIC POTENTIALS ARISING FROM AN ISOLATED DIPOLEAN ISOLATED DIPOLE

zz

-q-q

+q+q

d/2d/2

d/2d/2

( , ) (1.2.1)q q

rr r

Electrostatic potential at the Electrostatic potential at the generic point generic point PP is is

According with the cosine According with the cosine theorem,theorem,

By using series expansionBy using series expansion

12 21 1

1 cos (1.2.2)2

d d

r r r r

2( 1)1 1 ..... (1.2.3)

2!n n n

x nx x

P

r+

r-

r

Where Where PP, is the Legendre polynomials, is the Legendre polynomials

2

1 32

1( , ) (cos ) (cos ) .... (1.2.4)

4

qd dr P P

r r

20 1 2

33

11, cos , = (3cos 1),

21

(5cos 3cos ),... (1.2.5)2

P P P

P

( , ) (1.2.1)q q

rr r

12 21 1

1 cos (1.2.2)2

d d

r r r r

2( 1)1 1 ..... (1.2.3)

2!n n n

x nx x

Note that in equation 1.2.4, Note that in equation 1.2.4, q·dq·d cos cos θθ = = q·dq·d · r/ · r/rr (where (where r/r/rr = u= u), ), and the quantity (and the quantity (μμ = = qq·d·d is called dipolar moment. is called dipolar moment.

The S.I units for dipoles is The S.I units for dipoles is C·mC·m, although the Debye, , although the Debye, DD, , which is the dipole moment corresponding to two which is the dipole moment corresponding to two electronic charges separated by 0.1 nm, is a commonly electronic charges separated by 0.1 nm, is a commonly used unit.used unit.

23

2

1 1( , ) cos (5cos 3cos ) .... (1.2.4)

4 2

qd dr

r r

Dipolar moment Cuadripolar moment

1.3 - POINT DIPOLE1.3 - POINT DIPOLE

Some times it is convenient to refers at dipoles as point dipole, that is: a point Some times it is convenient to refers at dipoles as point dipole, that is: a point in the space, with moment and direction in the space, with moment and direction μμ

1.4 - FIELD OF AN ISOLATED POINT 1.4 - FIELD OF AN ISOLATED POINT DIPOLEDIPOLE

The dipole potential can be written as

The electric field is

After some calculations, one obtains

The field, expressed in matrix is

2 2 3

·cos · · 1· (1.4.1)grad

r r r

u r

r

3

· (1.4.2)

rE

r

5 3

3( ) (1.4.3)E

r r

μ·r r μ

2 2

5 5 5

2 2

5 5 5

2 2

5 5 5

3 3 3

3 3 3

(1.4.4)

3 3 3

x x

y y

z z

x r xy xz

r r rE yx y r yzE r r r

zx zy z rEr r r

1.5 - FORCE EXERTED ON A DIPOLE BY AN1.5 - FORCE EXERTED ON A DIPOLE BY ANEXTERNAL ELECTRIC FIELDEXTERNAL ELECTRIC FIELD

Consider a dipole located in an external electric field Consider a dipole located in an external electric field EEoo. The total force acting on the dipole is . The total force acting on the dipole is

the sum of the forces acting on each charge, that isthe sum of the forces acting on each charge, that is

(1.5.1)(1.5.1)

If If dd is very small in comparison with is very small in comparison with rr, the first term on the right-hand side of , the first term on the right-hand side of EqEq..((1.5.11.5.1) can ) can

be expanded into a Taylor series around be expanded into a Taylor series around rr, giving, giving

(1.5.2)(1.5.2)

substituting Eq ( substituting Eq ( 1.5.21.5.2) into Eq ( ) into Eq ( 1.1.55.1.1) and neglecting terms of order ) and neglecting terms of order dd22 and higher, we obtain and higher, we obtain

(1.5.3)(1.5.3)

The electrical force can also be derived from the The electrical force can also be derived from the

potential electrical energy potential electrical energy UU, , noting thatnoting that

(1.5.5)(1.5.5)

For a single charge For a single charge qq, the ratio of Eq (1, the ratio of Eq (1..55..5) to 5) to qq gives gives

EoEo = - = - ΦΦ. . By comparing Eqs (1By comparing Eqs (1..55..3) and (13) and (1..55..5), the 5), the

potential for a single dipole is given bypotential for a single dipole is given by

(1.5.6)(1.5.6)

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

0 0 00

0 0 00

0 0 00

(1.5.7)

x x xx x y z

y y yy x y z

z z zz x y z

E E EF

x y z

E E EF

x y z

E E EF

x y z

1.6. DIPOLE - DIPOLE INTERACTION1.6. DIPOLE - DIPOLE INTERACTION

The energy of a system formed by two The energy of a system formed by two

dipoles dipoles µµ11 and and µµ22 is the work done on is the work done on

one of these dipoles placed in the field one of these dipoles placed in the field

of the other dipole. According to Eqof the other dipole. According to Eq..

(1(1..44..3), the energy is:3), the energy is:

the force exerted on one of these the force exerted on one of these

dipoles owing to the field produced by dipoles owing to the field produced by

the other isthe other is

2 1 1 2

2 1 1 23 2

· ( ) · ( )

1 3( · ) ( · )( · ) (1.6.1)

iU E E

r r

r r

1 2 1 23 5

11 2 23 5

21 2 15 5

1 2 1 2 2 1 1 25 2

· ( · )( · )3

( · )1· 3 ( · )

( · ) 13 ( · ) 3( · )( · )

3 5· · ( · ) · ( · )

i iF Ur r

r r

r r

r r

r r

rr

rr r r

r r r r r r

(1.6.3)

SSpecial cases pecial cases

If If µµ11==µµ22 = = µµ, , the mutual force will bethe mutual force will be

For parallel dipoles (For parallel dipoles (µµ11==µµ22) perpendicular to ) perpendicular to rr

For antiparallel dipoles (For antiparallel dipoles (µµ11=-=-µµ22) perpendicular to ) perpendicular to rr

Finally, for perpendicular dipoles that are both also perpendicular to Finally, for perpendicular dipoles that are both also perpendicular to

rr

1.7. TORQUES ON DIPOLES1.7. TORQUES ON DIPOLES

The total torque isThe total torque is

Neglecting the term of high order and Neglecting the term of high order and

considering the dipole definition,considering the dipole definition,

( ) ( ) (1.7.1)2 2

q q

0 0

d dE d r E r

(1.7.2)q 0 0d E μ E

+q

-q Eo

Fo

Fo

d/2

d/2

The torque is null when The torque is null when µµ and and EEoo are parallel. are parallel.

As a consequence of the torque, the dipole tends to reorient As a consequence of the torque, the dipole tends to reorient

along the field, even in the presence of a uniform field. This is along the field, even in the presence of a uniform field. This is

a consequence of the not cancellation of the torque of two a consequence of the not cancellation of the torque of two

opposite non collinear forces. opposite non collinear forces.

The work of reorientation from a position perpendicular to the The work of reorientation from a position perpendicular to the

field to another forming an angle field to another forming an angle θθ is given byis given by

2 2

' sin ' ' cos (1.7.3)W d d

0 0Γ μ E μ E

1.8. DIPOLE MOMENT AND DIELECTRIC1.8. DIPOLE MOMENT AND DIELECTRICPERMITTIVITY. MOLECULAR VERSUSPERMITTIVITY. MOLECULAR VERSUS

MACROSCOPIC PICTUREMACROSCOPIC PICTURE

IIn real situationsn real situations continuous medium continuous medium

enormous number of elementary dipolesenormous number of elementary dipoles..

Dipolar substances contain permanent dipoles arising from the asymmetrical Dipolar substances contain permanent dipoles arising from the asymmetrical

locating of electric charges in the matter. locating of electric charges in the matter.

Under an electric field, these dipoles tend to orient as a result of the action of Under an electric field, these dipoles tend to orient as a result of the action of

torques, torques,

The macroscopic result is the orientation polarization of the material The macroscopic result is the orientation polarization of the material

TheThe effect of the applied electric field is also to induce a new type effect of the applied electric field is also to induce a new typess of polarization of polarization

by distortion of the electronic clouds (electronic polarization)by distortion of the electronic clouds (electronic polarization)

and the nucleus (atomic polarization) of the atomic structureand the nucleus (atomic polarization) of the atomic structure

TTiime scale me scale to reach equilibrium 10 to reach equilibrium 10--1010~10~10--1212 s s..

Under an applied field of frequency 10Under an applied field of frequency 101010~10~101212 Hz, the orientation polarization Hz, the orientation polarization

scarcely contributes to the total polarizationscarcely contributes to the total polarization. . HHowever, this contribution increases owever, this contribution increases

wwhen the frequency diminishes (time increases), and this is the essential feature of hen the frequency diminishes (time increases), and this is the essential feature of

the dielectric dispersionthe dielectric dispersion..

For a static case, the fieldFor a static case, the field EE acting on a dipole is, in general, different from the acting on a dipole is, in general, different from the

applied field applied field EEoo. . The average moment for an isotropic material can be written asThe average moment for an isotropic material can be written as

where where αα isis total polarizability total polarizability ((includes the electronic, includes the electronic, ααee, , the atomic, the atomic, ααaa, , and the and the

orientational, orientational, ααoo, , contributionscontributions))

· (1.8.1)m E

1 (1.8.2)N

P NV

m m

Equation (1Equation (1..88..1)1) is strictly true only if is strictly true only if EE is small is small..

This means This means :: no saturation effectsno saturation effects relationship between the induced moment and therelationship between the induced moment and the electric field is electric field is

of the linear typeof the linear type..

The total polarization ofThe total polarization of NN11 dipolar molecules per unit volume, each of dipolar molecules per unit volume, each of

them having an average moment them having an average moment mm, is, is

where where NN is the total number of molecules is the total number of molecules..

The average field caused by the induced dipoles of the dielectric placed The average field caused by the induced dipoles of the dielectric placed

between two parallel plates is given bybetween two parallel plates is given by

The mean electric field (The mean electric field (EEzz) must be added to the field arising from the charge ) must be added to the field arising from the charge

densities in the platesdensities in the plates..

31 4( ) 4 (1.8.3)z z

NmE E d P

V V

r r

· (1.8.1)m E

This field is given byThis field is given by

((DD electric electric

displacementdisplacement))

The dielectric susceptibility is defined asThe dielectric susceptibility is defined as

εεrr is the relative permittivity and is the relative permittivity and εεoo (= 8,854x 10 (= 8,854x 10-l2-l2 CC22kgkg-1-1mm-3-3ss22) is) is

the permittivity ofthe permittivity of the free spacethe free space.. In c In c..gg..ss.. units, units, εεoo = 1= 1 and and εεrr= = εε

4 (1.8.4) 0E P D

1 (1.8.5)

4

0

P

E1 4 (1.8.6)

0

D

E

0

r0

(1.8.7a)

(1.8.7b)

r

0 0D E E

1.9. LOCAL FIELD. THE DEBYE STATIC 1.9. LOCAL FIELD. THE DEBYE STATIC THEORY OF DIELECTRIC PERMITTIVITYTHEORY OF DIELECTRIC PERMITTIVITY

Until now the dipolar structure of the dielectric was considered, but the Until now the dipolar structure of the dielectric was considered, but the

electric field acting on the dipoles was not determined. electric field acting on the dipoles was not determined.

For the evaluation of the static dielectric permittivity in terms of the dipole For the evaluation of the static dielectric permittivity in terms of the dipole

moment moment μμ of the molecules of the dielectric require the determination of the of the molecules of the dielectric require the determination of the LOCAL LOCAL

FIELDFIELD acting upon the molecules and the ratio of the dipole moment of the acting upon the molecules and the ratio of the dipole moment of the

molecule and either the polarizability molecule and either the polarizability ααoo or the average moment or the average moment mm. .

The concept of The concept of INTERNAL FIELDINTERNAL FIELD has received considerable attention has received considerable attention

because it relates because it relates macroscopic propertiesmacroscopic properties to to molecular onesmolecular ones. The problem involved . The problem involved

in the determination of the field acting on a single dipole in the dielectric is its in the determination of the field acting on a single dipole in the dielectric is its

dependence on the polarization of the neighboring molecules. dependence on the polarization of the neighboring molecules.

The first approach to the analysis in this problem is due to Lorentz. 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, The basic idea is to consider a a spherical zone containing the dipole under study,

immersed in the dielectric. immersed in the dielectric.

The sphere is small in comparison with the dimension of the condenser, but large The sphere is small in comparison with the dimension of the condenser, but large

compared with the molecular dimensions. compared with the molecular dimensions.

We treat the properties of the sphere at the microscopic level as containing many 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. 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 The field acting at the center of the sphere where the dipole is placed arises from the

field due to field due to

(1) the charges on the condenser plates (1) the charges on the condenser plates

(2) the polarization charges on the spherical surface, and (2) the polarization charges on the spherical surface, and

(3) the molecular dipoles in the spherical region. (3) the molecular dipoles in the spherical region.

The field due to the polarization charges on the The field due to the polarization charges on the

spherical surface, spherical surface, EEspsp, can be calculated by , can be calculated by

considering an element of the spherical surface considering an element of the spherical surface

defined by the angles defined by the angles θθ and and θθ+d+dθθ. .

The area of this elementary surface is: The area of this elementary surface is:

22ππrr22sin sin θθddθθ. .

The density of charge on this element is given The density of charge on this element is given

by by P·cosP·cosθθ, and the angles between this , and the angles between this

polarization and the elementary surface is polarization and the elementary surface is θθ. .

Integrating over all values of angle formed by Integrating over all values of angle formed by

the direction of the field with the normal the direction of the field with the normal

vector to spherical surface at each point and vector to spherical surface at each point and

dividing by the surface of the sphere we obtaindividing by the surface of the sphere we obtain

E

+ +

+

++

+

- -- -

- -

d

The polarization was previously defined(1.8.6) as:The polarization was previously defined(1.8.6) as:

Therefore, the total field will beTherefore, the total field will be

This is the This is the Lorentz fieldLorentz field..

By combining Eqs (1.9.2),(1.9.3), and (1.8.1), the By combining Eqs (1.9.2),(1.9.3), and (1.8.1), the

following equation relating the permittivity and the following equation relating the permittivity and the

total polarizability is obtained:total polarizability is obtained:

Here, the assumption that Here, the assumption that m=m=μμ (dilute systems) is (dilute systems) is

made. If the material has molecular weight made. If the material has molecular weight MM, and , and

density density ρρ, then , then NN11==ρρNNAAMM, where , where NNAA is the Avogadro’s is the Avogadro’s

number, in this case the equation 1.9.4 becomes: number, in this case the equation 1.9.4 becomes:

This is the This is the Claussius – MossottiClaussius – Mossotti equation valid for equation valid for

nonpolar gases at low pressure. nonpolar gases at low pressure.

This expression is also valid for high frequency limit.This expression is also valid for high frequency limit.

The remaining problem to be solved is the calculation The remaining problem to be solved is the calculation

of the dipolar contribution to the polarizability. of the dipolar contribution to the polarizability.

The first solution to this problem for the static case was proposed by Debye. 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 In absence of structure, the potential energy of a dipole of μμ is given by: is given by:

In this case, In this case, EEii is the internal field, instead of the external applied field is the internal field, instead of the external applied field EEoo, because , because

EEii is the field acting upon the dipole. is the field acting upon the dipole.

If a Boltzmann distribution for the orientation of the dipoles axis is assumed, the If a Boltzmann distribution for the orientation of the dipoles axis is assumed, the probability to find a dipole within an elementary solid angle probability to find a dipole within an elementary solid angle dΩdΩ is given by is given by

The average value of the dipole moment in the direction of the field is The average value of the dipole moment in the direction of the field is

After integration by partsAfter integration by parts

Where Where L(y)L(y) =coth y – y=coth y – y-1-1 denotes the Langevin function given bydenotes the Langevin function given by

For y<<1, the Langevin function is approximately For y<<1, the Langevin function is approximately L(y)y/3, and the orientational , and the orientational polarization is given bypolarization is given by

If the distortional polarizability If the distortional polarizability d is added, the total polarization isis 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 At very high frequencies, dipoles do not have time for reorientation and the dipolar contribution to permittivity is negligible. Then,contribution to permittivity is negligible. Then,

Where Where dd==∞∞.. Since the infinite frequency permittivity is equal to the square of the Since the infinite frequency permittivity is equal to the square of the

refraction index, refraction index, nn, Eq (1.9.15) becomes, Eq (1.9.15) becomes

Lorentz-Lorenz equationLorentz-Lorenz equation

Note that equation 1.9.16, can be rewritten asNote that equation 1.9.16, can be rewritten as

where where =1/=1/, is the specific volume, and , is the specific volume, and C=4C=4NNAA/3M/3M. By plotting the . By plotting the

C(nC(n22+2)/(n+2)/(n22+1) vs T+1) vs T-1-1, the isobaric dilatation coefficient can be obtained, and, , the isobaric dilatation coefficient can be obtained, and, eventually, the glass transition temperature could be estimated. eventually, the glass transition temperature could be estimated.

Note that from the temperature dependence of the index of refraction of glassy systems, Note that from the temperature dependence of the index of refraction of glassy systems, physical aging and related phenomena could be analyzed.physical aging and related phenomena could be analyzed.

1.10. DRA1.10. DRAWWBACK OF THE LORENTZ LOCAL BACK OF THE LORENTZ LOCAL FIELDFIELD

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) Ferro-electricity is uncommon in nature, and predictions made by Eq (1.10.1)

are not experimentally supported. are not experimentally supported.

The failure of this theory arises from considering null the contribution to the The failure of this theory arises from considering null the contribution to the

local field of the dipoles in the cavity. local field of the dipoles in the cavity.

This fact emphasizes the inadequacy of the Lorentz field in a dipolar dielectric 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. and leaves open the question of the internal field in the cavity.

1.11. DIPOLE MOMENT OF A DIELECTRIC 1.11. DIPOLE MOMENT OF A DIELECTRIC SPHERE IN A DIELECTRIC MEDIUMSPHERE IN A DIELECTRIC MEDIUM

Claussius-Mossotti equation represent the first attempt to relate a macroscopic Claussius-Mossotti equation represent the first attempt to relate a macroscopic

quantity, the dielectric permittivity, to the microscopic polarizability of the substances. quantity, the dielectric permittivity, to the microscopic polarizability of the substances.

The equation only is valid for the displacement polarization. The equation only is valid for the displacement polarization.

To solve the same problem for the orientational polarizability is much more To solve the same problem for the orientational polarizability is much more

complicated. complicated.

The starting point of the solution to the problem of the local field in the context of 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 the theory of the orientational polarization is to consider a spherical cavity of radius aa

and relative permittivity and relative permittivity 11, that contains at the center a rigid dipolar molecule with , that contains at the center a rigid dipolar molecule with

permanent dipolepermanent dipole . .

The radius of the cavity is obtained The radius of the cavity is obtained

form the relationform the relation

Where Where NN is the number of molecules in is the number of molecules in

a volume a volume VV. In many cases . In many cases aa can be can be

considered as the “molecular radius”. considered as the “molecular radius”.

This cavity is assumed to be surrounded This cavity is assumed to be surrounded

by a macroscopic spherical shell of by a macroscopic spherical shell of

radius radius b>>ab>>a, and the relative , and the relative

permittivity permittivity 22. .

The ensemble is embedded in a The ensemble is embedded in a

continuum medium of relative continuum medium of relative

permittivity permittivity 33. .

According to Onsager, the internal field in the cavity has two According to Onsager, the internal field in the cavity has two

components:components:

1 –1 – The cavity field, The cavity field, GG, (the field produced in the empty cavity by , (the field produced in the empty cavity by

the external field.)the external field.)

2 -2 - The reaction field, The reaction field, RR (the field produced in the cavity by the (the field produced in the cavity by the

polarization induced by the surrounding dipoles).polarization induced by the surrounding dipoles).

Dielectric or conducting shells are technologically important, and Dielectric or conducting shells are technologically important, and

biological cells are also examples of layered spherical structures.biological cells are also examples of layered spherical structures.

The general solution of the Laplace equation (The general solution of the Laplace equation (22фф=0=0), in spherical coordinates and for ), in spherical coordinates and for

the case of axial symmetry, isthe case of axial symmetry, is

Where the Where the i i subscript refers to each one of the three zones under consideration. For our subscript refers to each one of the three zones under consideration. For our

particular geometry, the following boundary condition holdparticular geometry, the following boundary condition hold

These conditions arise from the fact that the only charges existing in the cavity These conditions arise from the fact that the only charges existing in the cavity

contributing to contributing to фф11 belong to the dipole. The contribution of the external field to belong to the dipole. The contribution of the external field to фф33, for , for

r>>>br>>>b is is EEoor·cosr·cosθθ..

Owing the boundary conditions, only the first term of the polynomial (Owing the boundary conditions, only the first term of the polynomial (cos cos θθ) appears in ) appears in

the Eq. 1.11.2, so that series of spherical harmonics is greatly simplified. Consequently the Eq. 1.11.2, so that series of spherical harmonics is greatly simplified. Consequently

the potentials arethe potentials are

Solving the system of equation 1.11.3 and 1.11.4, the resulting potentials are:Solving the system of equation 1.11.3 and 1.11.4, the resulting potentials are:

(1.11.5)

Special Cases:Special Cases:

1 – Empty cavity, in this case 1 – Empty cavity, in this case 11=1=1, and all the terms in Eq 1.11.7 containing , and all the terms in Eq 1.11.7 containing as a as a

factor vanish. The resulting potential are given byfactor 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 2 – Dipole point in the center of the empty cavity. Absence of external field. In this case, again 11=1=1, and the terms containig , and the terms containig EEoo vanish. The potentials are given by vanish. The potentials are given by

It could be also useful to consider the inner cavity in a continuum, that is, without spherical shell. It could be also useful to consider the inner cavity in a continuum, that is, without spherical shell. In this case, In this case, 22= = 33, and consequently , and consequently GG3232=1=1, and , and RR3232=0=0. Thus the Equations 1.11.8 and 1.11.9 . Thus the Equations 1.11.8 and 1.11.9

become respectivelybecome respectively

These results indicate that the field in an empty cavity embedded in a dielectric medium with These results indicate that the field in an empty cavity embedded in a dielectric medium with permittivity permittivity 22 is is

On the other hand, dipoles induce on the surface of the spherical cavity (even in absence of 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 external field) an electric field opposing that of the dipole himself, called the reaction fieldreaction field, ,

3 – Finally, for 3 – Finally, for 11= = 33=1=1, Eq (1.11.7) leads to, Eq (1.11.7) leads to

1.12. ACTUAL DIPOLE MOMENTS, DEFINITION 1.12. ACTUAL DIPOLE MOMENTS, DEFINITION AND STATUSAND STATUS

The potential of a rigid non-polarizable dipole in a medium of relative permittivity The potential of a rigid non-polarizable dipole in a medium of relative permittivity 11 is is

given bygiven by

((ee subscript means external) subscript means external)

Usually, dipoles are associated to molecules having an electronic cloud that shields them. Usually, dipoles are associated to molecules having an electronic cloud that shields them. Thus let us consider a dipole Thus let us consider a dipole in the center of a sphere representing the molecule. in the center of a sphere representing the molecule.

The electronic polarizability of the sphere gives rise at a macroscopic level, to an The electronic polarizability of the sphere gives rise at a macroscopic level, to an instantaneous permittivity instantaneous permittivity ..

In this case, the potential is given byIn this case, the potential is given by

By identifying the potentials in Eqs. (1.12.1) and (1.12.2), one findBy identifying the potentials in Eqs. (1.12.1) and (1.12.2), one find

Where Where eeis called the external moment of a molecule in a medium with relative permittivity is called the external moment of a molecule in a medium with relative permittivity 11. .

The dipole molecule in vacuum will beThe dipole molecule in vacuum will be

However, the total moment of the molecule, also called the internal moment, is the vector sum of 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 isits vacuum value and the value induced in it by the reaction field, that is

Which can be written asWhich can be written as

If the polarizability If the polarizability is known, is known,

Then, one obtainThen, one obtain

Internal dipole momentInternal dipole moment

Note that it expression is independent of the size of the spherical specimen. So, if a very large 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 sphere containing a dipole with internal moment ii, in a infinite dielectric medium of relative , in a infinite dielectric medium of relative

permittivity permittivity 11 is considered, the dipole moment of the large sphere is also, is considered, the dipole moment of the large sphere is also, ii. For . For = = 1 1 , the , the

following expression holdsfollowing expression holds

Note that the factor appearing in Eq (1.12.10), also appear in Eq. (1.11.12c)Note that the factor appearing in Eq (1.12.10), also appear in Eq. (1.11.12c)

The internal moment can be also written asThe internal moment can be also written as

The internal moment can also be calculated as the geometric sum of the external moment of the 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 and the moment of a sphere with the same permittivity as the medium surrounding the dipole. In this conditionsdipole. In this conditions

Although this integral can be obtained by a simple electrostatic calculation, it can also be 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 isdetermined by comparing Eqs (1.12.11) and (1.12.12), that is

1.13 1.13 DIRECTING FIELD AND THE ONSAGER DIRECTING FIELD AND THE ONSAGER EQUATIONEQUATION

previously, the internal dipole moment has been calculated in absence of an external field. When 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 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 the cavity is now the superposition of the cavity field cavity field GG with the with the field due to the dipolefield due to the dipole

From whichFrom which

Where Where is given by Eq. (1.12.8) is given by Eq. (1.12.8)

The mean value of the dipole moment parallel to the external field is calculated form the 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). 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 However, the energy of the dipole is calculated from the torque acting on the molecule, which

according to Eq (1.13.1) is given byaccording to Eq (1.13.1) is given by

From Eq. (1.13.2) into Eq. (1.13.3)From Eq. (1.13.2) into Eq. (1.13.3)

From Eq. (1.7.3) it follows that (From Eq. (1.7.3) it follows that (U=U=∫∫dd))

Equations (1.9.8) and (1.13.5) lead toEquations (1.9.8) and (1.13.5) lead to

By substituting this equation into Eq. (1.13.2) By substituting this equation into Eq. (1.13.2)

On the other hand, the polarization per unit of volume isOn the other hand, the polarization per unit of volume is

Substitution of the Eq. (1.13.7) into the Eq. (1.13.8) givesSubstitution of the Eq. (1.13.7) into the Eq. (1.13.8) gives

Where the quantity Where the quantity

Is called Is called DIRECTING FIELDDIRECTING FIELD, , EEdd..

The directing field is calculated as the sum of The directing field is calculated as the sum of cavity fieldcavity field and the and the reaction fieldreaction field caused by a caused by a fictive dipole fictive dipole EEdd..

Note: Directing field must be not confused with the internal fieldNote: Directing field must be not confused with the internal field E Eii. .

EEii= E= Edd+R+R

According to these results, Eq. (1.13.9) can be written as 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 Rearrangement of Eq (1.13.9), taking into account Eq. (1.12.8), gives the Onsager Onsager

expressionexpression

It is interesting to compare the Debye results (Eq 1.9.11) and the Onsager theories. The 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 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 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 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 theories in which Lorentz field is used. Note that, after some rearrangements, the

Onsager equation can be written asOnsager equation can be written as

It is clear that Onsager equation does not predict ferro-electricity as the Debye It is clear that Onsager equation does not predict ferro-electricity as the Debye

equation does. equation does.

In fact, the cavity field tends to In fact, the cavity field tends to 3E3Eoo/2/2, when , when εε11 tends to infinite. tends to infinite.

Onsager takes into account the field inside the spherical cavity that caused by Onsager takes into account the field inside the spherical cavity that caused by

molecular dipoles, which is neglected in the Debye theory. molecular dipoles, which is neglected in the Debye theory.

However, following the Boltzmann-Langevin methodology, the Onsager theory However, following the Boltzmann-Langevin methodology, the Onsager theory

neglects dipole – dipole interactions, or equivalently, local directional forces neglects dipole – dipole interactions, or equivalently, local directional forces

between molecular dipoles are ignored. between molecular dipoles are ignored.

The range of applicability of the Onsager equation is wider than that of the Debye The range of applicability of the Onsager equation is wider than that of the Debye

equation. equation.

For example, it is useful to describe the dielectric behavior on non-interacting 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.dipolar fluids, but in general this is not valid for condensed matter.

1.14. STATISTICAL THEORIES FOR STATIC 1.14. STATISTICAL THEORIES FOR STATIC DIELECTRIC PERMITTIVITY.DIELECTRIC PERMITTIVITY.

KIRKWOOD’SKIRKWOOD’S THEORY THEORY

Onsager treatment of the cavity differs from Lorentz’s because the cavity is Onsager treatment of the cavity differs from Lorentz’s because the cavity is

assumed to be filled with a dielectric material having a macroscopic assumed to be filled with a dielectric material having a macroscopic

dielectric permittivity. dielectric permittivity.

Also Onsager studies the dipolar reorientation polarizability on statistical Also Onsager studies the dipolar reorientation polarizability on statistical

grounds as Debye does. grounds as Debye does.

However, the use of macroscopic argument to analyze the dielectric However, the use of macroscopic argument to analyze the dielectric

problem in the cavity prevents the consideration of local effects which are problem in the cavity prevents the consideration of local effects which are

important in condensed matter. important in condensed matter.

This situation led Kirkwood first, and Fröhlich later on the develop a fully This situation led Kirkwood first, and Fröhlich later on the develop a fully

statistical argument to determine the short – range dipole – dipole statistical argument to determine the short – range dipole – dipole

interaction. interaction.

Making use of statistical mechanics, Kirkwood obtained the expression for the average Making use of statistical mechanics, Kirkwood obtained the expression for the average

dipolar moment asdipolar moment as

ThenThen

Taking Taking mm en Eq (1.8.2) as en Eq (1.8.2) as <µe><µe>, and the cavity field as given Eq (1.B.14), , and the cavity field as given Eq (1.B.14),

Kirkwood Equation for non-polarizable dipolesKirkwood Equation for non-polarizable dipoles , , gg is the correlation parameter, which is the correlation parameter, which is a measure of the local order in the specimen. is a measure of the local order in the specimen.

g g will be different fromwill be different from 11 when there is correlation between the orientations of when there is correlation between the orientations of neighboring molecules.neighboring molecules.

When the molecules tend to direct themselves withWhen the molecules tend to direct themselves with parallel dipole momentsparallel dipole moments, , will be positive and g>1will be positive and g>1..

When the molecules prefer an ordering withWhen the molecules prefer an ordering with anti-parallel dipoles, g <1anti-parallel dipoles, g <1.

g =1g =1 in the case of no dipolar correlation between neighboring molecules, or in the case of no dipolar correlation between neighboring molecules, or equivalently a dipole does not influence the position and orientations of the equivalently a dipole does not influence the position and orientations of the neighboring ones. neighboring ones.

g g depends on the structure of the material, and for this reason it is a parameter depends on the structure of the material, and for this reason it is a parameter

that fives information about the forces of local type.that fives information about the forces of local type.

1.15. FR1.15. FRÖÖLHICH’S STATISTICAL THEORYLHICH’S STATISTICAL THEORY

Like Lorentz, Fröhlich consider a macroscopic spherical region within an infinite Like Lorentz, Fröhlich consider a macroscopic spherical region within an infinite

continuum material. continuum material.

For the representation of a dielectric with dielectric permittivity For the representation of a dielectric with dielectric permittivity , consisting , consisting

of polarizable molecules with a permanent dipole moment, of polarizable molecules with a permanent dipole moment, FröhlichFröhlich

introduced a introduced a continuum with dielectric constant continuum with dielectric constant in which point dipoles with in which point dipoles with

a moment a moment dd are embedded. are embedded.

In this model each molecule is replaced by a point dipole In this model each molecule is replaced by a point dipole dd having the same having the same

non-electrostatic interactions with the other point dipoles as the molecules non-electrostatic interactions with the other point dipoles as the molecules

had, while the had, while the polarizability polarizability of the molecules can be imagined to be smeared of the molecules can be imagined to be smeared

out to form a continuum with dielectric constant out to form a continuum with dielectric constant ..

Fröhlich analysis leads toFröhlich analysis leads to

<M<M22>>oo. .

1.16. DISTORTSION POLARIZATION ON THE 1.16. DISTORTSION POLARIZATION ON THE KIRKWOOD AND FRKIRKWOOD AND FRÖÖHLICH THEORIESHLICH THEORIES

Kirkwood deals with the distortional polarization by postulating that the polarizability in Kirkwood deals with the distortional polarization by postulating that the polarizability in Eq. (1.9.12) is also affected by the local filed given byEq. (1.9.12) is also affected by the local filed given by

Onsager cavity fieldOnsager cavity field Local Field for a Local Field for a vacuum spherevacuum sphere

Then , the Kirkwood equation that includes the distortional polarizability is given byThen , 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 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)).the moment in vacuum (Eq(1.12.10)).

Fröhlich takes into account the distortion polarization by assuming the dipoles Fröhlich takes into account the distortion polarization by assuming the dipoles

embedded in a polarizable continuum of permittivity embedded in a polarizable continuum of permittivity ..

Cavity FieldCavity Field

Increment of the permittivity that is due to reorientation is given byIncrement 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 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 In Fröhlich theory, the dipoles are not themselves polarizable and the distortional

polarizability corresponds to that of the continuum medium surrounding the dipoles. polarizability corresponds to that of the continuum medium surrounding the dipoles.

For this reason, when cells in the Fröhlich theory are single molecules, For this reason, when cells in the Fröhlich theory are single molecules, mm, is related , is related

to the vacuum moment byto the vacuum moment by

Generalization of the Onsager equation.Generalization of the Onsager equation.

Claussius – Mossotti: Only valid for non polar Claussius – Mossotti: Only valid for non polar gases, at low pressuregases, at low pressure

Debye: Include the distortional polarization.Debye: Include the distortional polarization.

Onsager: Include the orientational polarization, Onsager: Include the orientational polarization, but neglected the interaction between dipoles. but neglected the interaction between dipoles. describe the dielectric behavior on non-describe the dielectric behavior on non-interacting dipolar fluidsinteracting dipolar fluids

Kirkwood: include correlation factor (interaction Kirkwood: include correlation factor (interaction dipole-dipole) dipole-dipole)

Fröhlich – Kirkwood – OnsagerFröhlich – Kirkwood – Onsager