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

E

+ +

+

++

+

- -- -

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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