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TUTORIAL N 2 –
QUASISTATIC DIPOLES
•2.11 Macroscopic theory of the
dielectric dispersion
•2.1 Brownian motion
•2.12 Dielectric Behavior in time
dependant electric fields
•2.2 Einstein’s theory of Brownian
motion
•2.3 Langevin treatment of Brownian
motion
•2.4 Correlation functions
•2.5 Mean square displacement of a
Brownian particle
•2.6 Fluctuation dissipation theorem
•2.7 Smoluchowski equation
•2.8 Rotational Brownian motion
•2.9 Debye theory of relaxation processes
•2.10 Debye equations for the dielectric
permittivity
•2.13 Dissipated energy in polarization
•2.14 Dispersion relations
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Lorentz local field
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.
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
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).
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.
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
2.1. Brownian Motion
Many processes in the nature are stochastic.
A stochastic process is a set of random timedependent variables.
The physical description of a stochastic problem
requires the formalization of the concept of
„probability”, and „average”
A Markov process is a stochastic process whose
future behavior is only determined by the present
state, not the earlier states. Brownian motion is the
best-know example of Markov process.
Robert Brown
British Botanist (1773 – 1858).
In 1827, while examining pollen grains and
the spores suspended in water under a
microscope, observed minute particles within
vacuoles in the pollen grains executing a
continuous jittery motion.
He then observed the same motion in
particles of dust, enabling him to rule out the
hypothesis that the motion was due to pollen
being alive.
Although he himself did not provide a theory
to explain the motion, the phenomenon is
now known as Brownian motion in his honor.
Brownian motion refers to the trajectory of a heavy particle immersed in a
fluid of light molecules that collide randomly with it.
The velocity of the particle varies by a number of uncorrelated jumps.
After a number of collisions, the velocity of the particle has a certain value v,
The probability of a certain change v in the velocity depends on the present
value of v, but not on the preceding values. An ensamble of dipoles can be
considered a Brownian system.
Einstein made conclusive mathematical perdictions of the diffusive effect
arising from the random motion of Brownian particles bombarded by other
particles of the surrounding medium.
Eintein’s idea was to combine the Maxwell-Boltzmann distribution of
velocities with the elementary Markov process known as random walk.
The random walk model is used in many branches of physics. Particularly
can be used to describe the molecular chains of amorphus polymers. The
probability distribution function for end-to-end distance R of freely jointed
chains can be obtained by solving:
(Simplest case of probability density
diffusion equation Focker – Plank
Equation)
Parabolic differential equation is the
same for other random walk phenomena
as heat conduction or diffusion
where P is the probability distribution, b is the length of each segment, and n is
the number of bonds. It is also assumed that n>>1, and R>>b. The solution
under the condition that R is at the origing when n=0,
2.2 EINTEIN’S THEORY OF BROWNIAN MOTION
If a particle in a fluid without friction collides with a molecule of the fluid, its
velocity changes.
However, if the fluid is very viscous, the change in the velocity is quickly
dissipated and the net result of the impact is a change in the position of the
particle.
Thus what is generally observed at intervals of time in Brownian motion is
the displacement of the particle after many variation in the velocity.
As a result, random jumps in the position of the particle are observed.
This is a consequence of the fact that the time interval between the
observation is larger than the time between collisions.
Accordingly, the kinetic energy of translation of a Brownian particle behaves
as a non-interactive molecule of gas, as required in the kinetic elementary
theory of gases.
Assuming very small jumps, Einstein obtained
the probability of distribution of the
displacement of particles the following partial
differential equation:
D: Diffusion coefficient, <x2> mean-square
displacement,  time interval such that the
motion of the particle at time t is independent of
its motion at time t+
Using the Maxwell distribution of velocities, the
diffusion coefficient is obtained as:
Where, T is the temperature, k, the Boltzmann
constant and  the friction coefficient
2.3 LANGEVIN TREATMENT OF THE
BROWNIAN MOTION
Langevin introduce the concept of equation of motion of a random variable and
initiated the new dynamic theory of Brownian motion in the context of stochastic
differential equations. Langevin’s approach is very useful in finding the effect of
fluctuations in macroscopic systems.
Langevin equation of motion of a Brownian particle start from the Newton’s second
law and two assumptions:
(1) the Brownian particle experiences a viscous force that represent a dynamic
friction given by
(2) a fluctuating force F(t), due to the impacts of the molecules of the surrounding
fluid on the particle in question appears. The force fluctuates rapidly and is called
white noise
The friccion force is governed by Stokes’ law. The expresion for the
friccion coefficient of a spherical particle of radius a and mass m,
moving in a medium of viscocity  is
The force F(t) is unpredictalbe, but it is clear that the F(t) may be
treated as a stochastic variable (its mean value vanishes).
2.4 CORRELATION FUNCTIONS
A correlation is an interdependence between measurable random variables. We
consider processes in which the variables evaluated at different time are such that
their stochastic properties do not change with time. This processes are said to be
stationary and the correlation function between two variables in these processes is
expressed by
An autocorrelation function is a correlation function of the same variable at different
times, that is
The normalized autocorrelation function is expressed as
Due to the fact that the random force F(t), is caused by the collisions of the
molecules of the sourrounding fluid on the Brownian particle, we can write
Where  is a contant and (t) the Dirac delta function. Eq. 2.4.4, express the idea
that the collisions are practically instantaneous and uncorrelated. It also indicates
that F(t) is a white spectrum. Eq. 2.4.4 can be written as
2.5 MEAN-SQUARE DISPLACEMENT OF A
BROWNIAN PARTICLE
By multiplying both sides of Eq. 2.3.1 by x(t), and taking into
account that
Averaging over all the particles, and assuming that the Maxwell
distribution of velocities holds, that is
Note that <Fx>=0 since the random force in uncorrelated with the
displacement.
Substituing the expression 2.5.5 in Eq 2.5.4
Whose solution is
And K is an integration constant. For long times, or friction constants
larger than the mass, the exponential term has no influence on the
motion of the particle after some time interval. This is equivalent to
excluding inertial effects, in these conditions, we have
Integrating the Eq. 2.5.8, from t=0 to t=, and assuming that x=0 for t=0,
we obtain
This is the same result obtained by Einstain.
2.6 FLUCTUATION-DISSIPATION THEOREM
The Langevin equation can also be solved for the velocity, taking into
account that x’=v.
Where v(0)=vo. By averaging this equation for an ensemble of
particles, all having the same initial velocity vo, and noting that the
noise term F(t’) is null in average and uncorrelated with velocity, we
obtain
Squaring the expr 2.6.1, and further integration on the resulting expression leads to
 is given by the expression 2.4.4 and it was taken into account that
<F(t).F(t’)>=(t-t’).
In order to identify , we note that for t →∞, eq 2.6.3 becames
The Eq. (2.6.4) relates the size of the flucuacting term , to the damping constant .
In other words, fluctuactions induce damping. This is the first version of the
fluctuation – dissipation theorem, whose main relevant aspect is that it relate the
microscopic noise to the macroscopic friction.
According to van Kampen: „The physical picture is that the random kiks tend to
spread out v, while the damping term tries to bring v back to zero. The balance
between these two opposite tendencies is the equilibrium distribution”
2.7 SMOLUCHOWSKI EQUATION
A further step in the approach to Brownian motion is to formulate a
general master equation that model more accurately the properties of
the particles in question. Eq 2.2.1 can be written as a continuity
equation for the probability density
Where j is the flux of Brownian grain or, in general, the flux of events
suffered by the random variable whose probability density has been
designed as f. Part of this flux is diffusive in origin and, according to a
first constitutive assumption, can be written as
D is the diffusion coefficient given by kT/.
Now we introduce a second constitutive assumption. Let us to suppose that
the particles are subject to an external force that derive of a potential
function, so that
Then, the current density of the particles that is due to this external force can
be expressed as
If the external foce is related to the velocity through the equation
the drift current due to this effect will be
The sum of two current fluxes (j=jdiff+jd) and the substitution into
Eq. (2.7.1) yields
Smoluchowski Eq.
diffusive
conective (transport or drift term)
2.8 ROTATIONAL BROWNIAN MOTION
The arguments used to establish the Smoluchowski equation can be applied to the
rotational motion of a dipole in a suspension.
In this case the fluctuating quantity is the angle or angles of rotation.
The Debye theory of dielectric relaxation has as its starting point a Smoluchowski
equation for the rotational Brownian motion of a collection of homogeneous sphere
each containing a rigid electric dipole , where the inertia of the spheres is neglected.
The motion is due entirely to random couples with no preferential direction.
We take at the center of the sphere a unit vector u(t) in the direction of .
The orientation of this unitary vector is described only in terms of the polar and
azimuthal angles  and . Having the system spherical symmetry, we must to take
the divergence of Eq. (2.7.1) in spherical coordinates, that is
The current density contains two terms, a diffusion term defined in Eq.
(2.7.2) given in spherical coordinates by
Where e and e, are unit vector corresponding to the  and  coordinates,
and the term f(,,t) represent the density of dipole moment orientation
on a sphere of unit radius.
On the other hand, the convective contribution to the current is due to the
electric force field acting upon the dipole. The corresponding equation
for such current is similar to Eq. (2.7.6) but now U is the potential for
the force produced by the electric field. This force can be calculated by
combining the kinematic equation for the rate of change in the unity
vector u, given by
With the noninertial Langevin equation for the rotating dipole, that is 
where F is the white noise driving torque and μ x E is the torque  due to an
external field. For this reason, between two impacts, Eq (2.8.4) can be written as
This is, the angular velocity of the dipoles under the effect of the applied field
Eq. (2.8.4) is the differential equation for the rotational Brownian motion of a
molecular sphere enclosing the dipole μ. Introduction of Eq (2.8.4) into Eq
(2.8.3) yields
Equation (2.8.6) is the Langevin equation for the dipole in the noninertial limit.
The applied electric field is the negative gradient of a scalar potential U, which,
in spherical coordinates, is given by
According to Eq (2.8.7), and neglecting the noise term, we obtain
As a consequence, the drift current density given by Eq. (2.7.6) is
where  is the drag coefficient for a sphere of radius a rotating about a fixed axis
in a viscous fluid, given by
By combining Eqs (2.8.2) and (2.8.9) and using the continuity Eq (2.7.1) the
Smoluchowski equation for Brownian rotational motion is obtained
At equilibrium, the classic Maxwell-Boltzmann distribution must hold and f is
given by
Substitution of Eq (2.8.12) into Eq (2.8.11) leads to the Einstein relationship
The constant A in Eq (2.8.12) can be determined from
the normalization condition for the distribution function
Actually
where f and U are given by Eqs (2.8.12) and (1.9.6)
respectively After integrating Eq (2.8.14) we obtain
In general, μE<< kT, so that
If we define the Debye relaxation time as
Then Eq. (2.8.11 ) becomes
2.9. DEBYE THEORY OF RELAXATION
PROCESSES
When the applied field is constant in direction but
variable in time, and the selected direction is the z axis,
we have
where use was made of Eq. (1.9.6). In this case, we
recover the equation obtained by Debye in his detailed
original derivation based on Einstein's ideas.
According to this approach
To calculate the transition probability, we assume that
U = 0. Then Eq (2.9.2) becomes
It is possible to calculate the mean-square value of
sin without solving the preceding equation First, we
note that:
Multiplying Eq (2.9.3) by 2πsin2 and further
integration over  yields:
After integrating and taking into account the
normalization condition, we obtain:
Integration of Eq (2.9.6) with the condition <sin2> =
0 for t = 0 yields:
For small values of , <sin2>  <2>. By taking the
first term in the development of the exponential in Eq
(2.9.7), the following expression is obtained:
which is equivalent to Eq (2.2.2) for translational
Brownian motion. When t   in Eq. (2.9.7), the
mean-square value for the sinus tends to 2/3
The non-negative solution of
Eq.(2.9.2) can be expressed as:
where Pn are Legendre polynomials.
However, we are only interested in a
linear approximation to the solution. In
this case, we assume that:
where  is a time-dependent function
to be determined
Once the distribution function has
been found, the mean-square dipole
moment can be calculated from:
several cases could be considered.
For a static field, E=Eo, substitution of
Eq.(2.9.10) in Eq.(2.9.2) gives:
Therefore, according to Eq.(2.9.10):
And:
For an alternating field, E=Emexp(it),
substitution of Eq.(2.9.10) into Eq. (2.9.2)
gives
Note that, () is L[-’ (t)], where L is the
Laplace transform and (t) is given by Eq
(2.9.13). The mean dipole can be written
as
Note that the difference in phase
between μ<cos> and E persists if
the real or imaginary parts of E are
taken. On the basis of Eq (2.8.10),
Debye estimate D for several polar
liquids, finding values of the order
of D = 10-1 s. Thus the maximum
absorption should occur at the
microwave region.
2.10. DEBYE EQUATIONS FOR THE
DIELECTRIC PERMITTIVITY
(2 9 11) Debye used the Lorentz field and
replaced the static permittivity with the
dynamic permittivity. From Eq (1.8.5), the
polarization in an alternating field can be
written as
From which
At low and high frequencies ( 0,),
the two following limiting equations are
obtained:
where εo and ε , are, respectively, the
unrelaxed (=) and relaxed (=0)
dielectric permittivities.
Rearrangement of Eq (2.10.2) using
Eq. (2.10.3) gives
If we define the reduced polarizability
as
we can write
Equation (2.10.5) can alternatively be
written as
where =D[(εo+2)/(ε+2)], is the Debye
macroscopic relaxation time.
Splitting the Eq.(2.10.8)
into real and imaginary
part, we obtain:
If we define the dielectric
loss angle as =tan-1("/'),
the following expression
for tan  is obtained
2.11. MACROSCOPIC THEORY OF THE
DIELECTRIC DISPERSION
Debye equation can also by obtained by considering a first order kinetics for the rate
of rise of the dipolar polarization.
When a electric field E is applied to a dielectric, the distortion polarization P, is
very quickly established (nearly instantaneously). However, the dipolar part of the
polarization, Pd, takes a time to reach its equilibrium value. Assuming that the
increase of the rate of the polarization is proportional to the departure from its
equilibrium values we have:
Pd and P are related to the polarizabilities d and  by Pi = (1/3)πNAi where NA is
the Avogadro's number
Consequently, Eq.(2.11.1) can alternatively be written as
where P= E = [(, - 1)/(4π)]E. Equation (2.11.3) can also be
written as
Integrating Eq (2.11.4) under a steady electric field, using the
initial condition P=E, for t=0, gives
The first term of Eq.(2.11.16) is the time dependent dielectric
susceptibility. The complex susceptibility is defined as:
Taking the Laplace transform, we obtain:
2.12. DIELECTRIC BEHAVIOR IN TIMEDEPENDENT ELECTRIC FIELDS
The analysis of the dielectric response to dynamic fields can be performed in terms
of the polarizability or, in terms of the permittivity.
In the first case we choose the geometry of the sample material in order to ensure a
uniform polarization.
The simplest geometry accomplishing such a condition is spherical.
The advantage of this approach is that the macroscopic polarizability is directly
related to the total dipole moment of the sphere, that is, the sum of the dipole
moments of the individual dipoles contained in the sphere.
Such dipoles are microscopic in character. Moreover, the sphere must be considered
macroscopic because it still contains thousands of polar molecules
The relationship between polarizability and permittivity in the case of spherical
geometry is given by:
where the term 3/(+2) arises from the local field factor. Accordingly, dielectric analysis
can be made in terms of the polarizability instead of the experimentally accessible
permittivity if we assume the material to be a spherical specimen of radius large
enough to contain all the dipoles under study. Additional advantages of the
polarizability representation are: (1) from a microscopic point of view, long-range
dipole-dipole coupling is reduced to a minimum in a sphere; (2) the effects of the
high permittivity values at low frequencies are minimized in polarizability plots.
Since superposition holds in linear systems, the linear relationship between electric
displacement and electric field is given by
where  is a tensor-valued function which is reduced to a scalar for isotropic materials.
Through a simple variables change t-= and, after integration by parts, one obtains
the more convenient expression
where , is  (t=0) accounts for the instantaneous response of the
electrons and nuclei to the electric field, thus corresponding to the
instantaneous or distortional polarization in the material.
For a linear relaxation system, the function  is a monotone decreasing
function of its argument, that is
For sinusoidal fields, (E=Eoexp(it)), and for times large enough to
make the displacement also sinusoidal, a decay function (t) can be
defined as
Then, Eq. (2.12.3) can be written in terms of the decay function as
If we define the complex dielectric permittivity as the ratio between the
displacement D(t) and the sinusoidal applied field Eo·exp(it), the following
expression for the permittivity is obtained:
where L is the Laplace transform. The function , accounts for the decay of the
orientational polarization after the removal of a previously applied constant field.
Williams and Watts proposed to extend the applicability of Eq.(2.10.7) by using in
Eq.(2.12.6) a stretched exponential function for , of the form
where 0 <   1.
Calculation of the dynamic permittivity after insertion of Eq.(2.12.7) into
Eq.(2.12.6) leads to an asymptotic development except for the case where =1/2.
The final result is
Note that, for some ranges of frequencies and for some values of , a bad
convergency of the series is observed. For a periodic field given by
direct insertion of the electric field into Eq. (2 12 3) leads to
Where
These expressions are closely related to the cosine and sine transforms of the
decay function. Moreover, the dielectric loss tangent is defined as
Note that ’ and ” are respectively even and odd function of the frequency, that is
2.13. DISSIPATED ENERGY IN
POLARIZATION
It is well known from thermodynamics that the power spent during a polarization
experiment is given by
By substituting Eqs.(2.12.10) and (2.12.11) into Eq (2.13.1 ), and with further
integration of the resulting expression, the following equation is obtained for the
work of polarization per cycle and volume unit
The first integral on the right-hand side is zero, because the dielectric work done on
part of the cycle is recovered during the remaining part of it.
On the other hand, the second integral is related to the dielectric
dissipation, and the total work in the complete cycle corresponds to the
dissipated energy. This value is
The rate of loss of energy will be given by
2.14. DISPERSION RELATIONS
The formal structure of Eqs (2.12.12) indicates that the real and imaginary part of
the complex permittivity are, respectively, the cosine and sine Fourier
transforms of the same function, that is, (). As a consequence, ’ and " are no
independent. The inverse Fourier transform of Eqs (2.12.12) leads to
After insertion of Eq (2.14.1b) into Eq (2.12.12a) and Eq (2.14.1a) into Eq.(2.12.12b),
the following equations are obtained
Eq. (2.14.2) becames
Kramer-Kronigs
relationships
They are a consequence of the linearity and causality of the systems
under study and make it possible to calculate the real and imaginary
parts of the dielectric permittivity one from the other.
The limit  of Eq . (2.14.4a) leads to
DIFFUSIVE THEORY OF DEBYE AND THE
ONSAGER MODEL
One of the supposition for the Onsager static theory was to assume a
Lorentz field. This fact, represent some limitation when we try to
applied an oscillating electric field.
When an alternating field is applied to a dielectric medium, the field in
an empty cavity is given by
Where /a=(-1)/(+2)
The reaction field in the cavity containing a molecule with dipole moment m can be
written as:
The total field acting on the molecule is obtained from the sum of Eqs. (2.18.2), and
(2.18.5)
which, according to Debye, determines the mean orientation of the molecules as
Then, the average dipole moment in the direction of the field is given by
The polarization by unit volume can be written as
Where
Is frequency independent
An ensemble of dipoles can be thought as a Brownian system.
A Brownian particle behaves as a Markov process (future behavior is only
determined by the present state, not the earlier states)
Einstein relates the friction with the diffusive coefficient
Smoluchowski equation
Debye Equation
KWW
Kramer – Kronigs relationship