dielectric propertiies with barium sodium...

CHAPTER-4

DIELECTRIC PROPERTIIES WITH BARIUM SODIUM NIOBATE NANO

COMPOSITES

Abstract

Dielectric properties of BNN-PS nanocomposites were measured over a

broad frequency range (100Hz to 13 MHz) and temperature range (280C-

1300C) to explore the possibility of their use as electronic materials, and

characterize them on the basis of existing theories. The composites

revealed marked departures from the law of physical mixtures for its

dielectric properties. The dielectric constant and dielectric losses increase

with increasing BNN content. At a constant temperature, the composites

follow a linear relationship between logarithm of their dielectric constant

and volume fraction of the ferroelectric filler. The system conforms to the

Claussis Mossotti equations. Dielectric permittivity values of the

composites are intermediate between Bottcher-Bruggeman and Maxwell-

Wagner models. The presence of BNN nanofiller into polystyrene matrix

is generally responsible for an increase of glass transition temperature,

usually of about 90C, with respect to the neat polystyrene

The results of this chapter have been accepted for publication in the

International Journal of Material Science

100 Chapter IV

4.1 Introduction

The principal applications for ceramics and ceramic composites are as

capacitive elements in electronic circuits and as electrical insulation. For

these applications the properties of most concern are the dielectric constant

and dielectric loss factor.

A dielectric material has interesting electrical properties because of the

ability of an electric field to polarize the material to create electric dipoles. A

dipole is an entity in which equal positive and negative charges are separated

by a small distance, dl, the electric dipole moment is given by µ= qdl

(4.1)

The electric dipole is a vector. In its simplest model, a dipole moment

consists of two point charges of opposite sign, +q and –q separated by a

distance.

Choosing the origin of the co-ordinate system to coincide with the negative

charge, the dipole moment in this case has a magnitude equal to qdl, and is

represented by a vector pointing from the negative charge in the direction of

positive charge [1-3]

It is fundamental that the capacitance of a condenser is increased if the space

between the conductors is filled with a dielectric material. If Co is the

capacitance of the condenser with region between the conductors evacuated

and C its capacitance when the region is filled with a dielectric, then the ratio

0

rCC

εεε0

= = (4.2)

Dielectric Studies on Barium Sodium Niobate – Polystyrene Composites 101

where rε is the relative permittivity or dielectric constant of the medium and

ε and ε0 are the permittivity of the medium and free space respectively.

The dielectric constant is a measure of the extent to which the insulating

material’s surface interact with the electric field set up between the charged

plates. The constant is dependent on two molecular level properties; the

permanent ‘dipole moment’ and the ‘polarizability’ or the induced change in

dipole moment due to the presence of an electric field. The permanent dipole

moment is the average over the various dipole moments given rise to by

structural charge density differences over intramolecular distances.

Polarizability is the property which arises from changes in the molecular

electron distribution induced by the applied electric field.

Dielectrics are substances which do not possess free electric charges under

ordinary circumstances but they can modify the electric field into which they

are introduced. The most important property of dielectrics is their ability to

become polarized under the action of an external electric field. The atoms

and molecules of the dielectrics are influenced by an external field and hence

the positive particles are pushed in the direction of the field while the

negative particles in the opposite direction from their equilibrium position.

Hence dipoles are developed and they produce a field of their own. The

process of producing electric dipoles out of neutral atoms and molecules is

referred to as polarization.

Dielectrics may be broadly divided into non-polar materials and polar

materials. In non polar materials, the positive nuclei of charge q is

surrounded by a symmetrically distributed negative electron cloud of charge

–q. In the absence of an applied field the centre of gravity of the positive and

negative charge distribution coincide. When the molecule is placed in an

102 Chapter IV

external electric field the positive and negative charges experience electric

forces tending to move them apart in the direction of the external field. The

distance moved is very small (10 -10 m) since the displacement is limited by

the restoring forces which increases with increasing displacements. The

centers of positive and negative charges no longer coincide and the

molecules are said to be polarized. Dipoles so formed are known as induced

dipoles since when the field is removed the charges resume their normal

distribution and the dipoles disappear. For molecules q will be of the order

of electronic charge( 10-19C) and dl of the order of molecular dimensions(

10 -10 m) and so dipole moment will be of the order of 10 -29 Cm.

In considering a dielectric material from the macroscopic point of view, we

restrict our attention to average values over volumes which are sufficiently

small in comparison with the dimensions of the material specimen but large

enough to contain a sufficient number of molecules for the purpose of

averaging. The sum of the dipole moments in an element of volume ∆v is

_

1( )

N v

ii

N v P vµ µ∆

=

= ∆ = ∆∑ (4.3)

where _

( )µ represents the average dipole moment of each molecule and N

the number of molecules per unit volume. The vector P is the dipole moment

per unit volume and is called electric polarization.

In polar dielectrics, the molecule, which are normally composed of two or

more different atoms, have dipole moments even in the absence of an electric

field, that is, the centers of their positive and negative charges do not

coincide. Normally these molecular dipoles are randomly oriented through


out the material owing to thermal agitation, so that the average moment over

any macroscopic volume element is zero. In the presence of an externally

applied field the molecules tend to orient themselves in the direction of the

field in such a manner that the elementary volume has a net dipole moment

In some polar materials a spontaneous dielectric polarization can exist even

in the absence of an applied electric field. Such materials are known as

ferroelectrics.

The microscopic concept of polarization

Dielectric polarization is nothing but the displacement of charged particles

under the action of the electric field to which they are subjected. Devices

based on this manifestation are numerous. They range from condensers and

switch-gear equipment in power and distribution installations to rectifiers,

resonators, amplifiers and transducers- converters of electrical energy to

other forms of energy- in communication technology. They include memory

devices used for storage of information in modern computers.

The above said displacement of electric charges results in the formation of

electric dipole moment in atoms, ions or molecules of the material. The four

important types of polarization are (1) electronic polarization (2) ionic

polarization and (3) orientational polarization (4) space charge polarization.

[4-6]

4.1.1 Electronic Polarization

It is the displacement of electrons with respect to the atomic nucleus, to be

more precise the displacement of the orbits under the action of an external

electric field Electronic polarization can be observed in all dielectrics

irrespective of whether other types of polarization are displayed in the

104 Chapter IV

dielectric. When the system is subjected to an external field of intensity E,

the nucleus and the electron experience Lorentz forces of magnitude ZeE in

opposite directions. As they are pulled apart, a coulomb force develops

between them, which tends to counter the displacement and hence the actual

magnitude of displacement is very small, αe is the electronic polarizability.

The electronic dipole moment is given by

e µ eEα= (4.4)

Fig 4.1 Demonstration of electronic polarization

4.1.2 Ionic Polarization

When atoms form molecules, electronic polarization is still possible but

there may be additional polarization due to a relative displacement of the

atomic components of the molecule in the presence of an electric field.


When a field is applied to the molecule, the atoms in the molecule are

displaced in opposite directions until ionic binding force stops the

process, thus increasing the dipole moment. It is found that this induced

dipole moment is proportional to the applied field and an ionic

polarizability α i is introduced to account for the increase.

i µ i Eα= (4.5)

Of course, the individual ions experience polarization in addition. For

most materials, the ionic polarizability is less than electronic

polarizability

4.1.3 Dipole or Orientation Polarization This type of polarization only occurs in polar substances. The permanent

molecular dipoles in such materials can rotate about their axis of

symmetry to align with an applied field which exerts a torque in them.

This additional polarization effect is accounted by an orientation

polarizability term α o. With electronic and ionic polarization processes,

the force due to externally applied field is balanced by elastic binding

forces, but for orientation polarization no such forces exist. In thermal

equilibrium with no field applied the permanent dipoles contribute no net

polarization since they are randomly oriented. However, since it is

observed that the orientation polarization is of the same order as the other

forms of polarization but it is temperature dependent, since at higher the

temperature the grater is the thermal agitation and that lowers α o. The

polarizability factors, α e and α i are functions of molecular structure and

are largely independent of temperature

106 Chapter IV

4.1.4 Space charge polarization (αs) Space charges must be considered as extraneous charges which collect on

the interfaces. Space charge polarization involves a limited transport of

charge carriers until they are stopped by a potential barrier, possibly a

grain boundary or a phase boundary. This also depends on temperature

and this effect is more frequently knew as the Maxwell- Wagner effect,

arising in heterogeneous samples.

4.1.5 The Total Polaris ability is the sum of αe, αi, α0 and αs and is shown in the fig 4.2

Fig 4.2 Total contribution of polarizability


4.1.6Clausius-Mosotti Relation

Clausius- Mosotti relation relate the dielectric constant of a material to the

polarizability of atoms comprising it. The dipole moment of a single atom

is proportional to local field i.e. dipole moment = α EL where α is the

polarizability of the atom and EL is the local field.

4.1.7 Evaluation of local field

The electric field which an atom sees or the total electric field at the atom

site is called the internal field or local field. If the dielectric is placed

between two charged plates, the electric field experienced by the molecule

of the dielectric, is given by EL, and

EL = E1 + E2 + E3 (4.6)

Where E1 is the field intensity due to charge density on the plates of the

capacitor (with no dielectric), E2 is the field at the atom due to polarized

charges, and E3 is the field due to neighboring dipoles. If there are N

atoms per unit volume, the electric moment per unit volume which is

called polarization and EL given by

Now LP N Eα= (4.7)

4.1.8 Polarization in Polymers

Dry polymers are poor conductors of electricity and can be regarded as

insulators. Field-induced orientational polarization caused by permanent

or induced dipole moments is a very well known fact in dielectric theory.

Application of an electric field to a polymer can lead to polarization of the

sample, which is a surface effect, but if polymer contains groups that can

act as permanent dipoles then the applied field will cause them to align in

108 Chapter IV

the direction of the field. When the electric field is released, the dipoles

can relax back into a random orientation, but due to frictional resistance

experienced by the groups in the bulk polymer this will not be

instantaneous. The process of disordering can be characterized by a

relaxation time, but may not be easily measured. It is more convenient to

apply a sinusoid ally varying voltage to the sample and to study the dipole

polarization under steady state conditions [7].

Polystyrene, which is considered to be non polar, does in fact possess a very

very small dipole moment due to the asymmetry at the phenyl side group in

atactic polystyrene [8]. Because of their anisotropic polarizablility, phenyl

groups tend to orient with their greatest main axis of polarizabilitiy in the

direction along which the E vector of an electric field and an induced

dipole moment is produced in some phenyl groups. On the other hand, the

induced dipole moments interact with other phenyl groups present in an

ensemble of polystyrene. Both effects render it possible that the dielectric

data of Polystyrene become ac field dependent if there is an internal

degree of freedom concerning the phenyl-phenyl arrangement below the

glass transition temperature (Tg) [9]

4.2 Results and Discussions

Dielectric constants of the composites are found out and modeling is done

with different experimental predictions. The observed peak in the

dielectric loss factor is related to the glass transition temperature of PS.

The results are neatly discussed below.


4.2.1 Dielectric constant of the composites

It is perceptible that higher concentrations of ferroelectric filler lead to

higher dielectric constant composites. The dielectric constant of BNN is

greater than that of polystyrene, so the addition of BNN to the polymer

matrix will result an increase in dielectric constant. At a temperature of

about 300C, and for a frequency of 1MHz, the dielectric constant of the

composites are found out and reported in the second column of table 4.1.

Because no significant contribution for electrode polarization is observed,

the increment in dielectric constant of PS is attributed mainly to the

addition of BNN, which increases the dipoles in the system. Since the

values of ‘ε ’of the two ingredients, polystyrene and BNN are 2.55 and

430 respectively, it is also clear that this composites does not obey the law

of physical mixtures, as stated by

1 1(1 )c f pv vε ε ε= + − (4.9)

where 1v is the volume fraction of the filler, cε , pε , fε are the dielectric

constant of composites, polymer and filler respectively. The calculated

values of dielectric constants are reported in the third column of table 4.1

Table4.1 Dielectric constant of BNN-PS

Name Dielectric constant (experimental) Calculated Dielectric constant

PS 2.55 2.55

BNN10 3.75 45.29

BNN20 6.05 88.04

BNN30 10.75 130.78

BNN40 23.05 173.02

110 Chapter IV

The present system of composites is a binary phase mixture of two

dielectrically different materials where BNN is ionic and polycrystalline

and polystyrene is amorphous atactic and non-polar. A great variety of

formulae has been suggested for the calculation of permittivity of

heterogeneous mixtures. These formulae are derived on the basis of

various theoretical assumptions and experimental data [9-11].

Fig (4.3-4.7) illustrates the dielectric interaction pattern of the mixed

system with respect to composition. It is found from Fig 4.3 that cε , the

dielectric constants of the composites are non linearly dependent on

volume % of BNN. This shows that the constituent capacitors formed by

dielectrics fillers and polymer in the composites are not in parallel

combination in the composites.

0 5 10 15 20 25 30 35 400

5

10

15

20

25

Die

lect

ric c

onst

ant

Volume of BNN, %

BNN-PSLinear fit

Fig 4.3 Dielectric constant versus volume fraction of BNN


From the plot 4.4 of the inverse of the dielectric constant ( ) ( )1c cε ε− = ∫

the law of harmonic mixture), is curvilinear, which indicates that the two-

component dielectrics are not in series combination in their composites.

When two capacitors are connected in series, the resultant is 1 2

1 2

*C CC C+

and

their inverse obey harmonic series conditions

0 5 10 15 20 25 30 35 40

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

(ε∋ )−1

Volume of BNN, %

BNN-PS

Fig 4. 4 Inverse dielectric constant vs. volume fraction of filler.

Physically these composites resemble porous structures which generally

conform to (eqn 4.10), the relation assumes the form of Lichteneker and

Rother’s [12, 13] more appropriate to layered structures which are neither

parallel nor perpendicular to the electric field i.e., the valid averages are

neither arithmetic nor harmonic. Suppose we have a particulate two phase

material we could use the model of composites to predict upper and lower

112 Chapter IV

bounds of the composite capacitances. One can choose to model

composites as having capacitance in parallel (upper bound) or in series

(lower bound). In practice the answer will lie somewhere between the

two. Lichtenecker’s rule predicts that a better approximation than either of

these is given by using the logarithemic dependence of the effective

capacitance values. Fig 4.8 shows that the plots of the dielectric constant

vs. frequency and the curves are practically equidistant in logarithmic

scale for equally spaced BNN volume fractions. This indicates the

logarithm of the dielectric constant is linearly proportional to the volume

fraction of BNN at all frequencies. Hence it is believed that the

composites follow the “log law” relationship, originally proposed by

Lichtenecker in which the dielectric constant of clean two component

system can be represented by (equ 4.10) where v1, is the volume fraction

of filler. The logarithmic law of mixtures (equ 4.10) firmly confirms a

logarithmic dependence of the dielectric constant of the composite on the

volume fraction of the filler.

The log cε vs. filler volume % plot is linear for 10 to 40% of filler

volume and when extrapolating we get the log (dielectric constant) of

pure PS. This confirms the logarithmic dependence of the dielectric

values of the composites in fig 4.5. So we can apply Lichtenecker’s rule.


0 5 10 15 20 25 30 35 40

0.4

0.6

0.8

1.0

1.2

1.4

log

(ε)

Volume of BNN,%

BNN-PS Linear fit

Fig 4.5 log ε vs filler volume % of composites.

log logc pε ε= + 1v log f

p

εε

(4.10)

The plot of specific polarization 12

c

c

εε

− +

versus volume fraction as

presented in fig4.6. It is also linear with slope and intercepts of 1.4 and

0.35 respectively are as expected in accordance with Clausius-Mossotti

equation modified by Lorentz and Lorentz [14] applicable to the overall

composite dielectrics

114 Chapter IV

0 5 10 15 20 25 30 35 400.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

spec

ific

pola

risat

ion

volume of BNN, %

B Linear Fit of Data1_B

Fig 4.6 Specific polarization vs.volume fraction of the composite

It implies that not only the square of the dipole moment per unit particle

of the combination but also the polarizability of a unit particle directly

decrease proportionally to the increase of the quantity of the polystyrene.

That is why at v1 =0, the value of specific polarization term becomes the

specific polarization of polystyrene is exactly 0.34 (the diel. constant of

PS lies between 2.5 and 2.6). The Clausius-Mossotti equation itself does

not consider any interaction between filler and matrix [15]. So we used

the same equation for the calculation of dipole moment of BNN particle.

4.2.2 Calculation of Dipole Moment of BNN Particle

This discussion may be further extended to calculate dipole moments of

BNN particles. The Clausius-Mossotti equation for a single –component

system can be written as


12

c

c

εε

−+

= _

2( )3 3 B

NK Tµα

ε0

+

(4.11)

Where N is the number of molecules per unit volume, α is the

deformational polarizability (both electronic and ionic polarization.), _

2( )3 BK T

µ is the dipolar polarizability, _

( )µ is the dipole moment, BK is the

Boltzman’s constant, and T is absolute temperature.

Equation (4.11) is an appropriate one and becomes precise when polar

molecules are separated from each other, i.e. when polar molecules are

distributed in a different non polar environment. The present system more

or less conforms to this situation, where ionic BNN particles are

distributed in a polystyrene matrix. For such a system the equation

becomes

_2

0 1 1 1 2 2( ) =(1/ 3 )( )

2 3c

c B

N N NK T

ε µε α αε

−1+ +

+ (4.12)

Where ‘N1’ and ‘N2’ are the number of molecules of BNN and

polystyrene per unit volume of the composite respectively. α1 and α2 are

deformational polarizability, (electronic & ionic) and _

( )µ is the average

dipole moment of BNN particles in the polystyrene matrix.

The dipolar polarizabilitiy, _

2( )3 BK T

µ of the intensely polarized system is

usually much higher than the deformation polarizabilities α1 and α2

116 Chapter IV

.Neglecting the two terms involving subscripts 1 and 2 in (eqn 4.12) and

substituting in N1=(d1/M1) NAv1 where density of BNN d1=5950 K g cm-3,

molecular weight of BNN is M1 =1651g, and NA, Avogadro number

NA=6.06*10 23mol-1 and v1 is the volume fraction of BNN (eqn 4.12)

reduces to

_

21

11 0

1 ( ) *2 9

c A

c B

N d vK TM

ε µε ε

−=

+ (4.13)

The plot of specific polarization versus. v1 is linear and its slope is

_2

1

1 0

( )9

A

B

N dK TM

µε

ie in Cm

(_

2( )µ )1/2 =1/ 2

1

19A

B

N dK TM ε

−

0

* ( slope) ½ (4.14)

T=300K and the slope is 1.4.

The calculated dipole moment of BNN in polystyrene matrix is

17.3439x10-30 Cm. This dipole moment of BNN particles and the high

induced polarization of BNN under electric field contribute for the high

dielectric constant of the composites. This dipole moment may be

considered as the dipole moment of a unit particle of the combination.

The two phase mixtures are also represented by the Bottcher-Bruggeman

formula [16] based on the spherical particle model where the filler is

interacting with polymer. According to this formulae


( )2 1/ 21 ( 8 )4c f pH Hε ε ε= + + (4.15)

where

1 1(3 1) (2 3 )f pH v vε ε= − + −

with fε =430,the dielectric constant of the filler, and pε =2.55, the

dielectric constant of PS. The values of dielectric constant of the

composites may be calculated from this equation and plotted against 1v

.However, the Maxwell-Wagner-Sillars [17, 18] equation predicts as a

complete solution of the Wagner-Raleigh theory for a system of one

spherical particle uniformly distributed in another

1

1

2 2 ( )2 ( )

p f f pc p

p f f p

vv

ε ε ε εε ε

ε ε ε ε+ + −

=+ − − (4.16)

The behavior of the present system is in agreement with both of these

equations up to v1 =0.3 i.e. up to 30% volume fraction as revealed by the

plots of cε versus 1v as in Fig4.7 in accordance with the calculated values

from the Bottcher-Bruggeman formulae and the Maxwell-Wagner-Sillars

equation. Beyond this point, the experimentally observed values of ‘ cε ’ lie

between the two plots, Bottcher-Bruggeman and the Maxwell-Wagner-

Sillars. The experimental values are coinciding with Logarithmic laws.

This indicates that the shape, particle size, distribution and concentration

of the dispersed component do not permit a very high degree of physical

interaction as envisaged by Bottcher-Bruggeman.

118 Chapter IV

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

Die

lect

ric c

onst

ant

Volume of filler, %

Bottcher-Bruggeman Mawell-Wagner Lichteneker Experimental value

Fig 4.7 Comparison of the calculated dielectric constant by different laws governing mixing.

The composite under investigation consists of ionic BNN dispersed in

polystyrene medium. Hence it is likely that the magnitudes of both the

short range and the long range interactions possible between the ions in

the filled matrix are lessened by the imposition of a plastic environment

on the ionic BNN. However, the charge density, the dipole moment of a

unit particle of the combination, the energy of the dipole and the overall

dielectric constant are not uniquely determined by the degree of

engulfment of BNN by polystyrene only. Other balancing factors such as

particle size, degree of packing and space charge effects also gain

prominence as percentage of polystyrene in the system is varied. The best

reproduction was obtained by the use of ‘Lichteneker relations’


4.2.3 Effect of frequency on the dielectric properties of the polymer and the composites

2 3 4 5 6 70

5

10

15

20

25

Die

lect

ric c

onst

ant

Log(f) Hz

PS BNN10 BNN20 BNN30 BNN40

Fig 4. 8 Dielectric constant versus frequency of PS and the composites

When the dielectric material is subjected to an alternating field the

orientation of dipoles, and hence the polarization, will tend to reverse

every time the polarity of the field changes. At low frequencies the

polarization follows the alterations of the field without any significant lag

and the permittivity is independent of frequency and has the same value

as in static field. When the frequency is increased the dipoles will no

longer be able to rotate sufficiently rapidly so that their oscillations will

begin to lag behind those of the field. The above effect leads to a fall in

dielectric constant of the material with frequency [19, 20]

Dielectric permittivity variation can be fitted to the Maxwell- Wagner

type of interfacial polarization in many cases. In this model materials are

assumed to be composed of polarizable grains separated by poorly

120 Chapter IV

conducting grain boundaries. In nanomaterials there is an additional

chance of getting a high dielectric constant because of the large space

charge polarization owing to the large surface area of a large number of

individual grains. The interface contact area changes inversely as the

radius of the particle (1/r). In a low frequency regime electronic, ionic

dipolar and space charge polarization plays a dominant role in

determining the dielectric properties of the materials [21, 22]. In the

BNN-PS composites, there is a finite contribution from the above

mentioned polarization, which gives an initial high value for dielectric

constant and that slightly decreases with frequency due to the slight

changes in orientational polarization with frequency.

It is observed that, upto one MHz, the permanent dipoles can follow the

field quite closely and so dielectric constant is high because the dipoles

can easily align with change in polarity. This observation is evidenced

from fig4.8 that the high dielectric constant values for the prepared

samples which fall slightly with frequency for the composites [23]. This

indicates that at high frequency the mobility of polar groups in polymer

chains is unable to contribute to the dielectric constant. At low frequency

the dielectric constant of the composite strongly depends on the dielectric

properties of both polymer and ceramic contents, while at high frequency

the dielectric constant becomes dependent primarily of the ceramic filler

and its concentration. The frequency dispersion relation given by Habery

and coworkers [24], in which dielectric constant decreases with increasing

frequency and reaches a constant value for all samples.


4.2.4 Impedance-Frequency Spectra

The composites were analyzed by means of complex impedance

spectroscopy as a function of temperature and frequency. Fig 4.10 shows

the impedance spectra of each of the composites analyzed at room

temperature. A.C measurements are often made with a Wheatstone bridge

type of apparatus in which the resistance, ‘R’, and capacitance, ‘C’ of the

sample are balanced against variable resistors and capacitors as in fig 4.9.

The central problem with a.c measurements arises over the interpretation

of the data. This is because the sample and the electrode arrangement is

electrically a ‘black box’ whose equivalent circuit (i.e. its representation

by some combination of R and C elements) is often unknown.

Fig 4.9 Measurement of R and C with bridge& Vector Representation of

Capacitive and Resistive current

The impedance spectra thus obtained were then processed through

computer assisted electrochemical data analysis software that ideally fit to

the experimental data [25].

122 Chapter IV

2 2 1/ 2( )c

c

RXZR X

=+

(4.17)

where 2 2 1/ 2( )cR X+ is the vector addition of the resistance and

capacitive reactance. The impedance of a parallel RC circuit is always less

than the resistance or capacitive reactance of the individual branches. The

relative values of ‘Xc’ and ‘R’ determine how capacitive or resistive the

circuit line current is. The one that is the smallest and therefore allows

more branch current to flow is the determining factor. Thus if ‘Xc’ is

smaller than ‘R’, the current in the capacitive branch is larger than current

in the resistive branch, and the line current tends to be more capacitive.

Frequency of the applied voltage determines many of the characteristics

of a parallel RC circuit. Frequency affects the value of the capacitive

reactance and so also affects the circuit impedance, line current and phase

angle, since they are determined to some extend by the value of ‘Xc’. The

higher the frequency of a parallel RC circuit, the lower is the value of

‘Xc’. This means that for a given value of R, the impedance is also lower,

making the line current larger and more capacitive.

The impedance measurement of the composites revealed that when the

volume fraction increased from 10 to 40%, the composites remain in their

capacitive characteristics. The a.c conductivity is a measure of resistive

component and it depends on the value of ‘δ’. No significant changes are

observed for ‘δ’ value from 10 to40% volume fraction. The impedance Z-

frequency curves were parallel curved lines and the phase angle θ about

900, which are like ideal capacitors {Z depends on 1/ j (2πf C)}


3.0 3.5 4.0 4.5 5.0 5.5 6.03.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

logZ

( oh

ms)

logf ( Hz)


Fig 4.10 Impedance frequency Plot of the composites

Capacitive reactance is proportional to the frequency of the applied voltage

and so the impedance depends on the applied voltage. Furthermore,

although a voltage drop occurs when current flow through either of the

components a phase relationship between the current and the voltage drop

is established and it is a measure of the opposition to the current flow [26].

However, in the composites the impedance values were still fairly high and

showed a frequency dependence indicating that the particle-particle

contacts are very weak and there are thin dielectric layers between the

particles, which give strong capacitive effects (dominated by 1/(2πfC)).

The high normal surface resistance of the powder may also contribute to the

contact resistance between the particles, and hence to the composite

resistance. In short, it should be emphasized that temperature play an

important role in the dielectric and conducting properties

124 Chapter IV

4.2.5 Effect of Temperature on the dielectric properties of the polymer and the composites

The dielectric constant of the composites increases with increase in

temperature. It is essentially due to the different thermal expansion of the

polymer (50*10-6K-1-300*10-6K-1) on one hand and the ceramics (0.5*10-6

K-1-15*10-6K-1) on the other. The increase in dielectric constant is attributed

to the higher orientation polarization of the polymer at higher temperature

due to the greater mobility of molecules.

It is observed that the rate of variation of dielectric constant with

temperature is steeper for higher volume fraction samples [27]. This is

attributed to the internal field generated by the ceramic particles, which

favors the orientation of polymer molecules. Fig 4.11 shows the

temperature dependence of dielectric constants for polymer and the

composites. It is noteworthy that the curves for various BNN-PS

composites parallel to each other in fig 4.9 and 4.11. This indicates that the

composites follow the ‘log-law’ relationship within the entire range of

temperature studied.

The dielectric constant of BNN is slightly increased by the temperature

variations within a temperature limit of 280 C to 1200 C because of its

ferroelectric nature in the above temperature range[28] The ferroelectric

transition temperature of BNN is at 5300C (reported in chapter 9) and so

BNN is ferroelectric in the studied temperature. Electronic and ionic

polarizations are partially independent of temperature but space charge

polarization and orientation polarization depends upon temperature.


20 40 60 80 100 1200

5

10

15

20

25

30

Die

lect

ric c

onst

ant

Temperature(0C)

PolystyreneBNN10 BNN20 BNN30 BNN40

Fig 4.11 Temperature dependence of dielectric constant of PS and Composites

The number of space charge carriers governs the space charge

polarization. As the temperature increases the number of carriers

increases, resulting in an enhanced build up of space charge polarization

and hence an increase in dielectric properties. In space charge

polarization, the increases of temperature facilitate the diffusion of ions

[29]. Thermal energy may also aid in overcoming the activation barrier

for orientation of polar molecules in the direction of the field. In such

cases relative dielectric constant increases when the temperature

increases, the orientation of these dipoles is facilitated and this increases

the dielectric polarization. But at very high temperatures the chaotic

thermal oscillations of molecules are intensified and the degree of

orderliness of their orientation is diminished and thus the permittivity

passes through a maximum [30]. In orientation polarization, the

randomizing action of thermal energy decreases the tendency of the

126 Chapter IV

permanent dipoles to align themselves in the applied field at very high

temperature. These result a decrease in dielectric constant with

temperature at and above 1000C, nearly the glass transition temperature of

PS. The same effect is observed in the case of YBCO filled PS

composites and that are reported in chapter 5.

Table 4.2 Dielectric constant of the Samples at different temperatures at frequency 1 MHz

Temper(0C) PS BNN10 BNN20 BNN30 BNN40

30 2.5 3.918 6.109 9.55 23

60 3. 4.7 7.32 11.4 25.2

90 3.3 6.1 9.7356 16.25 29.3

120 3.1 5.7 9.19 15.8 27.1

The composites at lower volume fraction follow the path of polystyrene in

its dielectric properties. Fig 4.11 and table 4.2 give evidences for the

same [31-33].

4.2.6 Conductivity and Glass transition temperature

The ac conductivity (σ) of the prepared composites were calculated using

the formulae

02 tan(cfσ π ε ε δ= ) (4.18)

where ε0 is the dielectric constant of vacuum εc, the relative dielectric

constant of the composites, ‘f’ the applied frequency i.e., one MHz.


20 40 60 80 100 120 140 160 180 200 2200.000

0.005

0.010

0.015

0.020

Die

lect

ric lo

ss fa

ctor

(tan

δ)

Temperature (0C)

Polystyrene BNN10 BNN20 BNN30 BNN40

Fig4. 12 Temperature versus dielectric loss for PS and the composites

At sufficiently low temperatures all polymers are hard rigid solids. As the

temperature rises, each polymer eventually obtains sufficient thermal

energy to enable its chains to move freely enough for it to behave like a

viscous liquid (assuming no degradation has occurred). Although the

glass-rubber transition itself does not depend on polymer structure, the

temperature at which ‘Tg’ the glass transition is observed depends largely

on the chemical nature of the polymer chain i.e., chain flexibility,

molecular structure etc. The dielectric loss is responsible for conductivity

and the peak temperature is characterized as the glass transition

temperature by dielectric theory. The glass transition temperature of the

composites increases with the increment of filler content and the nature is

represented in fig 4.12. The peak values are reported in table 4.3 and are

128 Chapter IV

justified by the homogeneity of dispersion of the nanofillers into PS, as

revealed by SEM analysis, and by the enormous interfacial area of the

nanoparticles, as the strong reinforcement between the two phases reduce

the mobility of PS chains. This is schematically represented in fig 4.13.

Fig 4.13 Schematic representation of the two phase mixture

The increasing relaxation temperature of the composites with increasing

BNN concentration may be due to an interfacial or Maxwell-Wagner-

Sillars polarization [34].The relaxation arises from the fact that the free

charges, which were present at the stage of processing, are now

immobilized in the materials. At sufficiently high temperature, the

charges can migrate in the presence of an applied electrical field. These

charges are then blocked at the interface between the two media of

different conductivity and dielectric constant. In BNN-PS composites,


interfacial polarization is always present. Although this phenomenon is

clear in a conductive filler such as metal reinforced polymer composites

[35], polarizable filler-reinforced polymer composites have also been

shown to exhibit the MSW effect [36] The interpretation of the

impedance, dielectric and conductivity spectra and their electrical

characteristics support the statement that these materials, are essentially

insulating, although a slight increase in conductivity is observed with

increasing filler content. This trace of conductivity may be attributable to

an incipient tunnel effect known [37, 38] to allow the electrons to flow

from one conductor particle to the next through the polymer film

sandwiches between the particles, thus establishing an electric current.

The lesser the filler content, less likely are the electrons to leap from one

particle to the next, so for BNN10 the conductivity is minimum and the

composite exhibits purely capacitive behavior. As the filler content

increases, chances are given for electrons to flow through it and as a

consequence a drop in electrical resistivity of the polymer composite is

experienced. This fact confirms that, particle percolation is not achieved.

The filler particles cannot even be bought close enough together to give

rise to percolation condition. The particles are no longer in contact, but

surrounded by a fine polymer film and hence infinitesimal gaps among

the adjacent particles may conduct an electric current by tunneling effect.

130 Chapter IV

Table 4.3 Peak of dielectric losses at a frequency I MHz

Name of Sample Glass Transition Temp(0 C)

PS 108.7

BNN10 109.37

BNN20 112.13

BNN30 116.04

BNN40 117.12

This conductivity is dictated by nearest neighbour tunneling. The

percolation like behaviour is observed only when the radius of the particle

is superior to tunneling range.

4.2.7 Arrhenius Relation

By Arrhenius relation of activation, the conductivity of the composites is

calculated by applying the relationσ σg

B

EK Te−

0= (4.18)

where Eg is activation energy, KB is Boltzman’s constant. The results are

graphically represented in the fig 4.14.


2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.402468

101214161820222426283032

Con

duct

ivity

* 10

- 6(S

/m)

(1000/T) K

Polystyrene BNN10 BNN20 BNN30 BNN40

Fig 4.14 Arrhenius plot of PS and composites.

4.2.8Variation of conductivity with frequency at a fixed temperature

The variation of ac conductivity with frequency is shown in figure 4.15 ,

((σ)ac is a temperature and frequency dependent term, and it is attributed

to the dielectric relaxation caused by localized electric charge carriers

which obey the following power law where B and ‘n’ are composition and

temperature dependent parameters by Koop’s equation for conductivity

[39]. Also the slope of the graphs gives values of ‘n’ which determines

the conducting behaviour of the grains.

(σ)ac (f, T)= Bfn at a fixed temperature (4.19)

Log (σ)ac) = logB+ n log(f) (4.20)

132 Chapter IV

Hence the slope of log (σ)ac) versus log(f) graph provides the value of

‘n’ while the intercept gives the value of extrapolated conductivity

expected at zero applied frequency. The enhancement of conductivity

properties is explained on the basis of correlated barrier hopping model.

The zero frequency value and ‘n’ increases with increase in filler content

The conduction is due to the hopping and tunnelling of electrons. As the

frequency of the applied field increases, the polarizable grains become

more active by promoting the hopping between Nb+4 and Nb+5 ions on the

octahedral sites, thereby increasing the hopping conduction [40-43]. These

electronic exchanges determine the polarization in the composites. The

polarization decreases with increase in frequency as in fig 4.8, and then

reaches a constant value due to the fact that beyond certain frequency of the

external field, the electronic exchange between Nb+4 and Nb+5 ions cannot

follow the alternating field. Thus we observe a gradual increase in

conductivity with frequency upto a frequency of one MHz.. It is found that

the conductivity saturates at high frequencies [44-46]. This means that

electronic exchange cannot follow field variations and saturation in the

generation of charge carriers was reached at high frequency [47-50].


2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

Con

duct

ivity

(S

/m) *

1e-6

Log (f) Hz


Fig 4.15 Variation of ac conductivity with logarithmic frequency at

room temperature

4.3 Conclusion Composites do not obey the law of physical mixtures for dielectric mixing.

The logarithmic law of mixtures confirms a logarithmic dependence of the

dielectric constant of the composite on the volume fraction of the filler.

Dipole moment of BNN in polystyrene matrix is 17.3439x10-30 Cm. This

dipole moment of BNN particles and the high induced polarization of BNN

under electric field contribute for the high dielectric constant of the

composites. For 10 to 40%, volume fraction of the filler, the composites

remain in their capacitive characteristics. Dielectric permittivity variations

can be fitted to Lichteneker relations. The dielectric constant of the

composites increases with increase in temperature. The glass transition

temperature of the composites increases with the increment of filler content.

134 Chapter IV

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dielectric propertiies with barium sodium...

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