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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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PhysicalandEngineeringPropertiesofPolymers
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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PhysicalandEngineeringPropertiesofPolymers
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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PhysicalandEngineeringPropertiesofPolymers
, , ,
, , ,
, , ,
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)