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PhysicalandEngineeringPropertiesofPolymers

Lecture Notes(SS, 2012)

I Fundamental concepts1. Conductivity and permittivity

2. Parallel-plate capacitor

I I Polymer in static field

1. Molecular polarization

2. Parallel-plate capacitor with dielectrics

I I I Mechanism of polarization1. Clausius-Mossotti Equation

2. Langevin function

IV Relaxation phenomena

1. Mechanical relaxation

2. Dielectric relaxation

V Measurement and presentation of dielectric

response

1. Measurement

2. Dielectric relaxation functions

VI Thermodynamical relations

1. Fundamental thermodynamical relations

2. Piezo-, pyro- and ferroelectricity

References:

Electrical Properties of Polymer

T. Blythe and D. Bloor

Cambridge University Press 2008

Dielectric Phenomena in Solids

Kwan Chi Kao

Academic Press, Amsterdam, 2004


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I Fundamental concepts

I -1.1 Conductivity

R : resistance : resistivity

: conductivity

Good insulator: 10-16(m)-1.I -1.2 Permittivity:

The measure of the resistance that is encountered when forming an electric field in a

medium.

Determined by the ability of a material to polarize in response to the field, and thereby

reduce the total electric field inside the material.

Relates to a material's ability to transmit (or "permit") an electric field.

I -2 Parallel-plate capacitor

I -2.1 Coulombs law

Coulombs law describes the electrostatic

interaction between electrically charged

particles: The magnitude of the Electrostatics

force of interaction between two point charges

is directly proportional to the scalar

multiplication of the magnitudes of chargesand inversely proportional to the square of the

distances between them.

|| || || Or as vectors k is a constant. In vacuum, . = 8.8510

-12

Fm-1

, is permittivity of free


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PhysicalandEngineeringPropertiesofPolymersspace.

I -2.2 Electric field

An electric field surrounds electrically charged particles and time-varying magnetic

fields. It has mainly the following two properties:(a) It exerts a force on other electrically charged objects located in the field;

(b) It does work on the charged objects moving along the electric field.

Definition: : the electric force experienced by the particleq : its charge : the electric field wherein the particle is located.

Based on Coulomb's Law for interacting point charges, the contribution to the electricfield at a point in space due to a single, discrete charge located at another point in

space is given by the following

4 Superposition principle

The total electric field due to a quantity of point charges is simply the superposition of

the contribution of each individual point charge

4

I -2.3 Gauss law

Gauss law is deduced from Coulombs law in combination with superposition

principle of electric field.

Enclose a point charge +qwith a sphere

having a radius ofr. According to Coulombs

law, the electric field at any point on thesphere surface is ,

pointing out of the surface in the normal

direction. Therefore, the electric flux through

the sphere

4


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PhysicalandEngineeringPropertiesofPolymers dS 4r

According to the superposition principle ofE, above conclusion also holds when a

number of point charges (or a continuous charge distribution) are enclosed inside the

surface. (= )Gauss Law:The electric flux through any closed surface is

proportional to the enclosed electric charge.

I -2 Parallel-plate capacitor

Capacitance: Unit: 1 F=1

CV [Farad]

Homogeneous

:

Inhomogeneous : Using Gauss law

encl Qencl : Charge enclosed in the surface

right side


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PhysicalandEngineeringPropertiesofPolymersI I Polymer in static field

I I-1 Molecular polarization

I I-1.1 Molecule dipoles

Dipole moment: I I-1.2 Molecular polarization

Polarization: a vector quantity that expresses the magnitude and direction of the

density of permanent or induced electric dipole moments induced in a dielectric

material by the applied field. The SI unit is coulombs per square metre [ Cm].Macroscopic quantity:

Polarization : number density of the dipolesMicroscopically, the applied electric field induces an electric dipole on eachindividual molecule,

loc

: a constant called the polarizability of the molecule.loc : the local electric field at the molecule.(a)Electronic polarization (also called Optical polarization)

An electric field will cause a slight displacement of the electron cloud with respect

to the positive nucleus.

Vast fast process: 10-15~10-16s.


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PhysicalandEngineeringPropertiesofPolymers(b)Atomic or Ionic polarization (Vibrational polarization)

Under electric field, the arrangement of atomic nuclei in a molecule can be

distorted, while in ionic crystals the positive ions shift with respect to the negative

ones.Slower than electronic polarization: 10-12~10-13s.

(c)Orientation polarization

Molecules having a permanent dipole tend to align in the direction of the applied field,

giving a net polarization in that direction.

Much slower: 10-6~10-2s.

Molecular polarizability:

: electronic polarizability : atomic (or ionic) polarizability : orentational polarizability


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PhysicalandEngineeringPropertiesofPolymersI I-2 Parallel-plate capacitor with dielectrics

How to calculate Eint?

Dipole moment:

Total polarization charge: Number of dipoles: So that Surface charge density:

(induced surface density)

Gausss law:

encl ext int 0

int ext

Linear approx.: int : susceptibility tensor. For isotropic media, is a scalar.

int ext P

P

P=intint ext intextint 1

int

(

free )


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PhysicalandEngineeringPropertiesofPolymers freeElectric displacement: (1+) int1 freefree int (1+)=

1

1 denotes the vacuum contribution.


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PhysicalandEngineeringPropertiesofPolymersI I I Mechanism of polarization

I I I-1 Clausius-Mossotti Equation

I I I-1.1 Lorentz local electric field

loc : The field acting on an individualpolarisable entity such as an atom or molecule

Lorentz field:

loc =int M

M : the field due to the molecules inside thesphere. For cubic lattices, M = 0.loc int int int 1 int= intI I I-1.2 Clausius-Mosotti Equation

Microscopic property: polarizability

.

loc

Macroscopic property: Susceptibility . intloc int

23 int 1intClausius-Mosotti Equ.:

1 2 3Dipole density

WA

: density; W: molar mass; A: Avogadros number (6.021023).

ModelfortheLorentzlocalfield


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PhysicalandEngineeringPropertiesofPolymers 1 2W A3

(M A30 : Molar polarization)Lorenz-Lorentz equation:

If no orientation polarization is present (e.g. at very high frequency), , where is refractive index. 1 2W A3 I I I-2 Langevin function: polar molecules

Polarization: Torque: , Potential energy: Many dipoles: Average potential energy

Average cosine: According to Boltzmann statistics, probability for energy to be between Wand

W+dW:

/

1

Substitution , , ,

Orientationofamoleculardipole

Boltzmanndistribution


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1

coth Langevin function

3 45 Typically, 10Cm , even at very high 10 Vm, 10J 1meV. 25meVSo that 1 3 Orientational polarizability:

Langevinfunction


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PhysicalandEngineeringPropertiesofPolymersTotal effective polarizability: eff Substituting into Clausius-Mosotti Equation:

1 2W A3 0 23

Temperaturedependenceofmolarpolarization


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PhysicalandEngineeringPropertiesofPolymersIV Relaxation phenomena

IV-1 Mechanical relaxation

Stress: Strain: Strain tensor:

: displacement

Linear relationship: : Youngs (elastic) modulusIV-1.1 Tensile creep compliance

1IV-1.2 Stress-relaxation experiment

Time dependent tensile modulus: IV-1.3 Dynamic mechanical relaxation experiment

330

Generally Complex tensile compliance

3333 330 Or complex dynamic tensile modulus

33

33 1

33

(t)33

(t)

Time

,

Dynamicmechanicalrelaxation


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PhysicalandEngineeringPropertiesofPolymersIV-2 Dielectric relaxation

Analogy:

Mechanical stress

Electric field E

Mechanical strain

Electric displacement D(or polarization P)

Static polarization: Effect of an applied electrical field:

: relaxation timeBoltzmann statistics:

exp

: activation energyIV-2.1 Static electric field

Solution for 0 0 0 1/ ; : relaxation strength.IV-2.2 Frequency domain response

0 0 1 000 0001 01

E on

Ps

P

time

Polarization

1

2 1>

PolarizationunderstaticE


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PhysicalandEngineeringPropertiesofPolymersDebye relaxation:

or Real part: Imaginary part: and tan

, ,and tan asafunctionof ( 10, 2, 10s)


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PhysicalandEngineeringPropertiesofPolymersV Measurement and presentation of dielectric

response

V-1. MeasurementParallel-plate capacitor: Charge on capacitor: Current:

in phase advancedLoss tangent:

(a)Samplecapacitor (b)ComplexIVrelationship


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PhysicalandEngineeringPropertiesofPolymersV-2. Dielectric relaxation functions

Debye dispersion:

Real part:

Imaginary part: Eliminate from both parts, one obtains:

2 2 Real dielectric material:

Cole-cole plot:

Cole-Davison: Havriliak-Negami:


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PhysicalandEngineeringPropertiesofPolymersVI Thermodynamical relations

VI-1. Fundamental thermodynamical relations

Intensive variable: can not be added when two objects are combined, such

as temperatureT;

Extensive variable: can be added, such as mass.

, 1, , 6, 1, ,3 Intensive variable(force) Extensive variable(displacement)Thermodynamics TemperatureT Entropy S

Mechanics Stress

Strain

Electricity Electric field Displacement Magnetism Magnetic field Induction Free enthalpy (Gibbs Function) G:, , , During reversible process 0Total differential ofG for reversible processes:

Or

,, ,, ,, ,, ,,, ,,, ,,, ,,.VI-2. Piezo-, pyro- and ferroelectricity

In polymers, magnetism can often be neglected: Total differentials of

mechanical strain

, of electric displacement D and of entropy S:


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, , ,

, , ,

, , ,

Direct and inverse piezo- and pyroelectricity:

,

,

, , Piezoelectric charge coefficient for the three coordinate axes (i = 1,2, 3):

,

Inverse piezoelectric effect (strain coefficient ) ,Constant electric field :

,

Experiment: Electrode charge Q3 as a function of applied force : , Taking changes of sample dimensions into account:

, , ,


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PhysicalandEngineeringPropertiesofPolymersEmploying the elastic compliance at constant electric field E: Piezoelectrical charge or strain coefficient:

CNmVPiezoelectrical charge or stress coefficient:

Cm2 NVmPiezoelectrical field or strain coefficient:

VmN m2C Piezoelectrical field or stress coefficient:

Vm NCRelation between the four coefficients:

|

and |

(Dielectric permittivity)

| and | (Elastic modulus)Electro-mechanical coupling factor (energy ratio):

|out|in |out|in (dimensionless)

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