tutorial n 2 – quasistatic dipoles 2.1 brownian motion2.1 brownian motion 2.2 einstein’s theory...
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TUTORIAL N 2 – TUTORIAL N 2 – QUASISTATIC DIPOLESQUASISTATIC DIPOLES
•2.1 Brownian motion2.1 Brownian motion
•2.2 Einstein’s theory of Brownian 2.2 Einstein’s theory of Brownian motionmotion
•2.3 Langevin treatment of Brownian 2.3 Langevin treatment of Brownian motionmotion
•2.4 Correlation functions2.4 Correlation functions
•2.5 Mean square displacement of a 2.5 Mean square displacement of a Brownian particleBrownian particle
•2.6 Fluctuation dissipation theorem2.6 Fluctuation dissipation theorem
•2.7 Smoluchowski equation2.7 Smoluchowski equation
•2.8 Rotational Brownian motion2.8 Rotational Brownian motion
•2.9 Debye theory of relaxation 2.9 Debye theory of relaxation processesprocesses
•2.10 Debye equations for the dielectric 2.10 Debye equations for the dielectric permittivitypermittivity
•2.11 Macroscopic theory of the 2.11 Macroscopic theory of the dielectric dispersiondielectric dispersion
•2.12 Dielectric Behavior in time 2.12 Dielectric Behavior in time dependant electric fieldsdependant electric fields
•2.13 2.13 Dissipated energy in polarization
•2.14 Dispersion relations2.14 Dispersion relations
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d
Lorentz local field
Claussius – MossottiClaussius – Mossotti equation equation
valid for nonpolar gases at low pressure. valid for 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.
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 According to Onsager, the internal field in the cavity has internal field in the cavity has two components:two components:
1 – The cavity field, G, 1 – The cavity field, G, (the field produced in the (the field produced in the empty cavity by the external empty cavity by the external field.)field.)
2 - The reaction field, 2 - The reaction field, R (the field produced in the R (the field produced in the cavity by the polarization cavity by the polarization induced by the surrounding induced by the surrounding dipoles).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 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
2.1. Brownian Motion
Many processes in the nature are stochastic. A stochastic process is a set of random time-dependent 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 BrownBritish 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 MOTIONThe 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 RELAXATIONPROCESSES
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 (condition of
equiprobability in all directions).
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 determinedOnce 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(it), 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 THEDIELECTRIC 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 THEDIELECTRIC 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)πNAi where NA is the Avogadro's number
Where is the macroscopic relaxation time, and P is the equilibrium value of the total polarization which is
related to the applied electric field E, though an similar equation to Eq.(1.8.5):
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 TIME-DEPENDENT 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(it)), 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(it), 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
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