5. Electric Properties
5.1 Conductivity of polymers
Polymers are very good insulators! Most plastic materials are not conducting
and used to insulate good conductors such as metals.
1977: increase in conductivity of polyacetylene by a factor 107 due to doping
with Cl, Br or J (Nobel price in 2000 for Shirakawa, MacDiarmid and Heeger)
http://physicaplus.org.il
Application of conducting polymers in many different fields, for example:
• printed electronics
• OLEDs / displays, OFET
• photodiodes, photovoltaics (OPV)
r
v
electrical conductivity σ defined via j = σE
r
v
j
with current density and electrical field E
→ measure of ability of material to conduct electric current
Example: conductivities:
Teflon (PTFE)
polyacetylene, undoped
silver, copper
10-16 S/m
10-10 S/m
108 S/m
Conductivity requires movability of electrons (along one chain and from chain
to chain), which is different in amorphous and crystalline state of polymer.
ides.com
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Polyacetylene
Shows a small conductivity already at room temperature.
Polyacetylene is a conjugated polymer
(alternating single and double bonds).
Wikipedia.org
The trans-structure of polyacetylene has a 2-times degenerated ground state with
an A und B structure:
Single- and double bonds can exchange with no change in energy. A free radical
forms at the position at which the two ground states A and B meet. → This
defect is called a soliton.
→ Charge storage on the polymer chain leads to a structural relaxation (and vice
verse), which in turns localize the charge.
For simplicity, the chemical structure of the soliton is drawn as an abrupt change
from phase A to B; but as shown by experiments and calculations, the
structural relaxation in the vicinity of the domain boundary extends over
approx. 14 carbon atoms.
A charge can be freely moved along the polymer backbone like a soliton in a
channel. This results in a good conductivity along the polymer backbone.
Wave that propagates without dispersing are called solitary wave or soliton. In
hydrodynamics the energy dissipation process is compensated by contributions
from nonlinear processes.
∂u
∂u ∂ 3u
=0
+u
+
Basic example of soliton equation:
∂t
∂x ∂x 3
∂ 2u 1 ∂ 2u
=0
−
Differs from normal wave equation:
∂x 2 c 2 ∂t 2
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This topological defect (soliton) is mobile:
- calculated activation energy for motion ~ 2 meV
- soliton mass about 6me
Light absorption of conjugated materials:
photonicswiki.org
The π band is referred to as the valence band and the π* is called the conduction
band. This assumes a perfectly coplanar chain. The width of the valence band
and the conduction band is about 5 eV. The energy gap or band gap Eg is about
1.5 eV, this also known as the ionization potential. The bottom of the conduction
band is also known as the electron affinity.
Explain band structure with Peierls instability of a one-dimensional metal:
Assume an one-dimensional metal with one electron at each site of an atomic
chain of period a: The energy band is half-filled with the Fermi wavevector at
half of the Brillouin zone π/a. The electron density is constant in space.
Doubling the period will lead to a modulation of the electronic density, called
charge density wave (CDW), and thus to a superstructure. The doubling of the
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3
unit cell will reduce the Brillouin zone to half; which leads to a gap 2Δ right at
the Fermi energy.
→ The system undergoes a metal-to-insulator transition.
Band structure polyacetylene:
In principle quantum mechanical description necessary, with a complex
Hamiltonian function. Simplification is reached with the SSH (Su-SchriefferHeeger) Hamiltonian: theory for PA
Result: Peierls discussion of electron-lattice instability, ground state with
periodic lattice distortion due to different length of single and double bonds.
The model shows that the soliton creates a localized electronic state, which is a
non-bonding state, at a energy lying in the middle of the π-π* band gap, between
the bonding-antibonding levels of the perfect, dimerized, chain. In other words it
is located at the Fermi level.
Rev. Mod. Phys. 60, 781 (1988)
Solitons in PA in the SSH model:
Su-Schrieffer-Heeger (SSH) model describes the distribution of charge and spin
of solitons
notice the reversed
relation of solitons
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charge-spin
Q = ± |q|
Q=0
and s = 0
and s = 1/2
4
a) Neutral soliton (singly occupied state): the carbon atom at the boundary is
neutral, and there is one spin unpaired
π*
neutral, ½ spin soliton:
Q=0
s = 1/2
S0
π
b) Positive soliton (unoccupied state): carbon atom at the boundary is left with a
positive charge, and there are no unpaired spins
+
positive, spinless soliton:
Q = +q
s=0
π*
S
π
c) Negative soliton (doubly occupied state): carbon atom at the boundary is left
with a negative charge, but there are no unpaired spins
negative, spinless soliton:
Q = -q
s=0
π*
Sπ
5.2 Doping of conducting polymers
The doping of organic conductors is analogous to the doping of silicon
semiconductors, whereby a small fraction silicon atoms are replaced by
electron-rich (e.g., phosphorus) or electron-poor (e.g. boron) atoms to create ntype and p-type semiconductors, respectively.
1) Oxidation:
• analogous to the removal of an electron from the valence band (creation
of an acceptor state)
• no p-doping in is original meaning
• achieved by use of AsF5, Br2, I2, HclO4
2) Reduction:
• analogous to the addition of an electron into the valence band (creation of
an donor state)
• no n-doping in is original meaning
• achieved by electrochemical methods or alkalimetallnaphthalide
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Example:
polyacetylene, undoped 10-10 S/m
polyacetylene, doped
105 S/m
silver, copper
108 S/m
Example: Oxidation of polyacetylene with Iodine I2
Mechanism:
approaching of iodine
abstraction of an electron
I2
I2
e-
iodine (I2) abstracts an electron from the top of the valence band, becoming an
I3- ion
polaron formation - note:
superposition of charged and neutral
soliton
counter
ion
I3-
+
hole created in the polymer chain does not delocalized due to the structuralimpose localization and the coulombic binding to the iodine ion
The mobility of the polaron can be high, and then the charge is transported along
the polymer backbone. However, the mobility of the counter ion is low.
so far simple naive sketch:
Here, 2 solitons (blob with red and
blue stripes) move along a conducting
polymer chain (aqua and yellow for
hydrogen and carbon). The traditional
way to make polymer actuate is to
dope the material with an ion such as
sodium, represented by the red dot.
The soliton blob causes a localized
bend in the chain.
web.mit.edu
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6
In a crystalline inorganic material, setting a charge onto a site does not change
the surroundings, as the crystal lattice is rigid. In contrast in many disordered
organic materials charge creates a distortion of its surrounding.
Quasi-particles:
• combinations of quasi-particles forming new quasi-particles (unlike
mathematical solitons that do not interact!)
• combination of charges and lattice distortions
A polaron can be thought as a bound state of a charged soliton and a neutral
soliton whose mid-gap energy states hybridize to form bonding and anti-bonding
levels.
i) the neutral soliton contributes no charge and a single spin
ii) the charged soliton carries a charge and is spinless
Both positive and negative polarons can be viewed as quasi-particles consisting
of single electronic charge “dressed” with a local geometrical relaxation of the
bond lengths.
π*
π
P+
PQ = -|q|
s = 1/2
π
Q = +|q|
s = 1/2
π
positive polaron
negative
→ polarons have ±q charge and s=1/2
Example: polarons occur frequently, e.g. if ground state is not degenerated
PPP (polyphenylene) and PPV (poly p-phenylene vinylene)
negative polaron in poly para-phenylene (PPP)
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Poly-para-phenylene (PPP)
PPP has no degenerated ground state. The ground state is the benzenoid state A.
The first excited state is the quinoide state B with slightly higher energy.
Thus no soliton-like defects are possible in PPP, which contribute to
conductivity. The localized electron-hole pair polarize the surrounding and
relaxation into a new ground state appears by formation of polarons
(stabilization of the quinoide state).
Polarons can only move by overcoming energy barriers.
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A bipolaron is a bound state of two charged solitons of like charge (or two
polarons whose neutral solitons annihilate each other) with two corresponding
mid-gap levels.
B
-
B
Q = -2|q|
s=0
π*
+
Q = +2|q|
s=0
π*
BPπ
BP+
π
negative bipolaron
positive bipolaron
Example:
-
-
negative bipolaron in poly para-phenylene (PPP)
Example: Doping mechanism in polythiophenes
Doctoral Thesis Anna Esmeralda Javier (Carnegie Mellon University, 2010)
Heavily doped polymers:
At 6% doping with Na+ (which is considered the limit between heavy and light
doping) the neighboring intrachain soliton spacing is approx. 26 sites, a
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separation still sufficiently large that adjacent soliton overlapping is small and
conductivity is low.
At higher doping, the soliton wave functions overlap with the result that the
intergap soliton band merges with the HOMO. The soliton overlap is sufficient
for electrons (in the case of donor doping) to move freely along the chain.
→ quasi metallic transport
high
low
π
π
S
π
S
π
5.3 Photoconducting polymers
Photoconductivity is defined as an
increase of conductivity caused by
irradiation. Thus, photoconductive
polymers are insulating or poorly
conductive in the darkness and more
conductive when illuminated.
Zhang, Michigan State University
This process generates an electron
hole pair, increasing the concentration
of intrinsic carriers. The photon energy
must exceed the band gap and
consequently there is a threshold
wavelength-energy associated with the
process.
Carriers may also be generated from photon absorption in the generation of
excitons. The excitons are themselves unable to transport charge but can
produce an electron hole pair if an interface is reached or two excitons collide.
On many common polymers it is very difficult to observe photoconduction even
when the samples are irradiated by photons with energies well in excess of their
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band gaps. This is largely because the carriers have very short lifetime (<1 ns)
due to rapid recombination and deep hole trapping.
Photoconductive polymers can be p-type (hole-transporting), n-type (electrontransporting), or bipolar (capable of transporting both holes and electrons). Most
practical photoconductive charge-transporting polymers are p-type.
Example: poly(N-vinylcarbazole) (PVK)
PVK has charge in side-group.
The the polymer backbone does not contribute to the
conductivity.
PVK absorbs ultraviolet light in the 360-nm region and
forms an exciton that ionizes in an electric field. PVK takes
up a helical conformation with successive aromatic side
chains having parallel to each other in a stack along which
electron transfer is relatively easy.
Ye et al. Phys. Chem. Chem. Phys., 2010, 12, 15410-15413
Photoluminescence (PL) and electroluminescence (EL) spectra of a PVK film at
room temperature. The two possible PVK conformations, p-PVK and f-PVK
conformation, are shown in the inset.
The hole state is a radical cation and moveable with very small effective
mobility of μ = 10-6 cm²/Vs.
Example: polysilane
Polysilanes can be quasi-conjugated polymers, but their backbones are
composed of silicon atoms. The effective mobility can be high: μ = 1 cm²/Vs
Schematic drawing of p-type (hole) state
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11
Applications:
• anti-static coating for photo-films, mobile-phones
• radiation protection of electron-beam based monitor technology
• (auto)darkening windows to reduce passing light
Typical:
band gap in UV-regime
However, traps are that shallow that IR-light is sufficient to reactivate the
electrons. Color centers are not only source of free electrons but act as very
efficient traps as well.
Applications:
Photocopier working with xerography (standard for office copying)
Old: chalkogenide-glasses (Se), toxic – today: photoconducting polymers
1. Charging: cylindrical drum is electrostatically charged
by a high voltage wire called a corona wire or a charge
roller. The drum has a coating of a photoconductive
material.
2. Exposure: A bright lamp illuminates the original
document, and the white areas of the original document
reflect the light onto the surface of the photoconductive
drum. The areas of the drum that are exposed to light
become conductive and therefore discharge to ground.
The area of the drum not exposed to light (those areas
that correspond to black portions of the original
document) remain negatively charged. The result is a
latent electrical image on the surface of the drum.
3. Developing: The toner is positively charged. When it is
applied to the drum to develop the image, it is attracted
and sticks to the areas that are negatively charged (black
areas), just as paper sticks to a toy balloon with a static
charge.
Wikipedia.org
4. Transfer: The resulting toner image on the surface of
the drum is transferred from the drum onto a piece of
paper with a higher negative charge than the drum.
5. Fusing: The toner is melted and bonded to the paper by
heat and pressure rollers.
Laser printers work on the same principle. In the optics a laser is used for
sampling.
Mechanism of photoconductivity:
exciton
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12
An exciton is an excited quasi-particle in a solid, which is formed by a
Coulomb-bound electron-hole pair. Excitons may be treated in two limiting
cases, depending on the properties of the material:
• Wannier-Mott excitons: In semiconductors, the dielectric constant is
generally large. Consequently, electric field screening tends to reduce the
Coulomb interaction between electrons and holes. The result is a Wannier
exciton, which has a radius larger than the lattice spacing. As a result, the
effect of the lattice potential can be incorporated into the effective masses
of the electron and hole. Likewise, because of the lower masses and the
screened Coulomb interaction, the binding energy is usually small,
typically on the order of 0.01 eV.
Wikipedia.org
• Frenkel excitons: In materials with a small dielectric constant, the
Coulomb interaction between an electron and a hole is strong, the
screening length is large and the excitons thus tend to be small, of the
same order as the size of the unit cell. Molecular excitons may even be
entirely located on the same molecule, as in fullerenes. The typical
binding energy is on the order of 0.1 to 1 eV.
ifw-dresden.de
Spin-state of the two charges:
singlet exciton = the two spin-vectors add up to zero, singlet excitons are the
only ones which are generated upon illumination, which is due to the specific
selection rules
triplet exciton = nonzero spin vector, which is possible in three different
combinations – thus the name triplet.
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13
Singlet and triplet excitons can also be formed due to interaction following
charge injection; theoretically, this follows a one-to-three ratio, i.e., only a
quarter is of singlet type.
Köhler and Bässler, J. Mater. Chem., 2011, 21, 4003-4011
An exciplex is just an exciton which is located at the interface of its “host”
molecular material – indeed it still resides on one molecule. Due to the influence
of the surface, the exciplex experiences a different environment as compared to
a bulk exciton. This leads to photoluminescence which is slightly red shifted.
Also, the lifetime can be prolonged in comparison to the bulk exciton, as it is
stabilized by the surface states.
blog.disorderedmatter.eu
Applications:
organic photovoltaic (OPV) cell is a
photovoltaic cell that uses conductive
polymers or small organic molecules
spie.org
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The basic operational mechanism for current generation in organic
photovoltaics cells can be split into 4 steps:
www2.warwick.ac.uk
a) Absorption of a photon, hv, in the active layer leading to the formation of an
exciton which has a quantum efficiency, ηA.
b) Diffusion of the exciton to an interface characterized by a diffusion length,
LD, the quantum efficiency for exciton diffusion is ηED.
c) The exciton most probably separates at the interface between the two
materials with offset energy levels with a quantum efficiency ηCT.
d) Once separated the charges must be collected at the respective electrodes by
transporting through the organic layers with a mobility, μ, and has a quantum
efficiency ηCC.
The overall external quantum efficiency, the number of charges collected per
incident photon, is thus:
EQE = ηA ηED ηCT ηCC.
5.4 Charge-transfer complexes
A charge-transfer complex (CT complex) or electron-donor-acceptor
complex is an association of two or more molecules, or of different parts of one
very large molecule, in which a fraction of electronic charge is transferred
between the molecular entities. The resulting electrostatic attraction provides a
stabilizing force for the molecular complex. The source molecule from which
the charge is transferred is called the electron donor and the receiving species is
called the electron acceptor.
Lowering the band gap occurs due to charge-transfer interaction.
→ lowering the energy needed to produce electron-hole pair
hν charge transfer = I donor − I acceptor −
ΔΕ
{
Coulom lowering
Example: PVK (donator) and strong electron acceptor trinitrolfluorenon (TNF)
⇒ Shift of absorption from the UV-regime to the IR-regime.
⇒ doping !
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Comparison of electron-hole pair in PVK (left) and the charge-transfer state in
the donor-aceptor complex PVK-TNF (right)
5.5 Hopping transport
Localized states in the polymers which arise because of their spatial disorder
(fluctuations in inter-site distance) and energetic disorder (fluctuations in siteenergy) configurations are assumed to have a Gaussian energy distribution.
Hopping in Gaussian disorder model (GDM), pioneered by Bassler, is used to
describe transport in disordered semi-conducting polymers. In this model,
electron-phonon coupling is assumed to be sufficiently weak so that polaronic
effects are neglected. Thereby the charge carriers hop in a regular array of
hopping sites.
Wendorff, University Marburg
Drift velocity
Chapter05
v
v
v
v = μ ⋅ E with mobility μ in the electric field E
16
Further improvements in this model have been proposed where a spatially
correlated site-energy distribution was considered. This is known as the
correlated disordered model (CDM).
However, in polymers charge transport occurs via hopping of polarons instead
of the charge carriers itself. The transition rates have been modeled by Marcus,
and the field and temperature dependence of the mobility has been obtained.
In case of disordered structures such as polymers, the time-dependent current
I(t) follows a complex relaxation behavior:
Example: polysiloxane
Transient time τt determined
from bending point differs for
the two different voltages
Regime with constant slope in the log I - versus - log t presentation shows
the presence of different scaling laws.
Theoretical description:
Model of a one-dimensional dispersive transport: Transport in amorphous solid
is described by a broad distribution g(ε) of single-exponential hopping rates.
ψ(t) = probability for a charge carrier which reached a place at time t0 = 0 to
leave this place at the time t is
ψ (t ) = ∑ g (ε )⋅ W (ε )e −W (ε )t
ε
density of states of corresponding ψ(ε)↑
↑ hopping probability
important: single-exponential decaying hopping time correlation ψ(t) = r e-rt
yields a constant drift velocity vd (t) = const.
− (1+α )
but: long term behavior is ψ (t ) t →∞ ∝t
more realistic in case of polymers
→ Modeling of this long term behavior within the trap model:
• Charge carriers get trapped with a probability of 1 if they pass by a trap
• Trapped charge carriers can escape a trap with a Boltzmann-activation of the
traps
hopping probability
Chapter05
W ( ε ) = W0 e − ε / kT
with sample frequency ε
17
• For simplification assume exponential distribution of traps in the energy space
g (ε ) = e −ε / kT0
Put all together into equation for ψ(t) yields ψ (t ) ≈ ∑ e −ε / kT e −ε / kT exp− {W0 te −ε / kT }
0
ε
n n −b t
mathematically this equation can be written as ψ (t ) = ∑ a b e
type.
n
n
Such equation as a behavior ψ(t) ∝ t
→ correct long term behavior
-(1+α)
with α = T/T0 for large times.
The exponent n can be explained within a hierarchical model
Wij =e
− ΔΕij
kT
=e
− nΔ
kT
=b n with b = e
−Δ
kT
which refers to the overcoming of number of n small energy barriers Δ.
Charge transport model in polymers:
Hopping of charge carriers in the
HOMO of the charge transfer agent.
http://www.uni-muenster.de/Physik.AP/Denz/
Examples: PVK, polysilane, amorphous silicon
Experimental: Measurement of photo currents in conducting polymers with
simple set-up (device)
← semi-transparent Al-electrode
← photo-conducting polymer
← semi-transparent Al-electrode
← polyester support
Polymers differ from normal solids: (example: polysiloxane α = 0.58±0.04)
τt ∝ Ε
τt ∝ l
− 1α
1
α
≠ Ε −1
≠l
Comparison of band and hopping mechanisms of transport:
For band transport (Drude theory) electrons behave as "free particles''. They are
accelerated by the applied electric field and lose their momentum through
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scattering by impurities and phonons. As a result, an electron's motion may be
described as quantum diffusion. At low temperature, the phonon scattering
becomes weak and the conductivity increases with decreasing temperature to its
residual value. For the hopping mechanism the zero-temperature electrical
conductivity is zero because the charge carriers are localized. At finite
temperatures, electrons hop from one localized state to another by absorbing or
emitting phonons. In contrast to band transport, hopping conductivity increases
with temperature because of increased availability of phonons.
V. N. Prigodin, A. J. Epstein Europhys. Lett. 60, 750 (2002)
5.6 Electroluminescence
Since 1990 conjugated polymers are known which emit light in response to the
passage of an electric current or to a strong electric field. It is the result of
radiative recombination of electrons and holes in a material.
Polymer based LEDs are called
OLED (organic light-emitting
diode)
OLED can be flexible or big
and have only little power
consumption.
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In an OLED, an electrical current causes electrons (-) to move from the cathode
to the emissive layer (EML), creating a negative charge in the emissive layer.
The positively charged anode attracts electrons from the conductive layer
(HTL), creating a positive charge in the conductive layer, which recombine with
holes (+) in the conductive layer attract electrons from the emissive layer, which
recombine with the electron holes, lowering the energy level of the electrons and
emitting light as a by-product.
a forward bias voltage is applied to the
OLED structure
auo.com
OLED Devices:
← negative electrode
← electroluminescent film
← positive electrode
← support
negative electrode: Al, Mg, Ag-alloy, a-Si-H, C
→ low work function, electron injector
positive electrode:
Al2O3, Au, InO, InZnO (ITO), polianiline (PANI)
→ high work function, hole injector
support:
glass plate of flexible PET (transparent)
today: functional stack more sophisticated
out of ready-to-use solutions for making
hole injection layers (HIL) e.g. Plexcore®
OC
sigmaaldrich
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Examples: poly-para-phenylenevinylene (PPV)
1) IU-curves appear like Schottky-diode characteristics
Current-voltage characteristic for an electroluminescent
device having a PPV film 70 nm thick and active area
of 2 mm2, a bottom contact of indium oxide, and a top
contact of aluminum. The forward-bias regime is shown
(indium oxide positive with respect to the aluminum
electrode).
J. H. Burroughes et al. Nature 347, 539–541 (1990)
2) Electroluminescence proportional to current with yellow-green color
Integrated light output plotted against current for the
electroluminescent device giving the current-voltage
characteristic
J. H. Burroughes et al. Nature 347, 539–541 (1990)
How does an OLED work?
IU-curves can be described with the Shockley-equation of semi-conductor based
diodes.
I = I0 (eqV/nkT - 1)
However, the equivalent-circuit of a diode is still under debate.
PPP Devices:
Al
PPP
ITO
glass
• PPP has a band-gap of 2.7 eV
• Film thickness is 0.5 μm such as PPV device
• But big difference to PPV:
a) Electroluminescence (and Photoluminescence) is observed with a blockvoltage ±10V
b) operation of the device with an alternating current
⇒ description within the model of a Schottky-diode is not adequate
Special: PPP devices emit blue light → blue LED become possible!
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Further application: All-polymer laser:
Samuel, Nature 429, 709-711(2004)
A laser consists of a light-amplifying material and a resonator. A polymer
semiconductor can be used to amplify light, with a corrugated structure acting as
a resonator. The corrugation diffracts light traveling to the left back towards the
right, and vice versa. Hence, light is sent backwards and forwards through the
amplifying medium. Typically, the excitation is provided by another laser. It
would be more convenient, however, to make polymer lasers that operate using
electrical excitation (e.g. from a battery).
With a thick semiconducting polymer layer in the laser, the electric field of light
in the polymer is weak at the contacts. So absorption (hence, loss) of light at the
contacts is low and the laser works effectively (although still, for now, with laser
excitation).
Flexible conjugated polymer laser:
Using a film of a ladder-type poly(p-phenylene) (LPPP)
C. Kallinger, M. Hilmer, A. Haugeneder, M. Perner, W. Spirkl, U.
Lemmer, J. Feldmann, U. Scherf, K. Müllen, A. Gombert, and V.
Wittwer Adv. Mater. 10, 920 (1998)
→ realize optical devices with geometries which are inaccessible with
conventional materials (solid state)
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Future:
New or improved properties from the synthesis of new conjugated polymer
Conjugated polymers containing
electron-transporting, holetransporting, and blue lightemitting units
Y. Fu et al. J Polym Sci Part A: Polym Chem
46: 1349–1356, 2008
5.7 Conjugated polymers
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5.8 Dielectric behavior of polymers
Most polymers are very good insulators.
If a polymer is placed inside of a capacitor, the capacitance is changed as
compared to vacuum conditions.
Q = Cvac Uvac → Q = U C = U ε0 εr A/d
with εr = ε/ε0 relative static permittivity or dielectric constant
and the electric constant ε0 = 8.854 x 10-12 A s V-1 m-1
Example:
polyethylene
polystyrene
polyimide
silicon
water
εr = 2.25
εr = 2.4-2.7
εr = 3.4
εr = 11.68
εr = 80.1
A
dielectric
medium
showing
orientation of charged particles
creating polarization effects. Such a
medium can have a higher ratio of
electric flux to charge (permittivity)
than empty space
wikipedia.org
Not all polymers behave the same when subjected to voltage and thus polymers
can be classified as ‘polar’ or ‘non-polar’ to describe their variations in
dielectric behavior.
Examples of polar polymers are PMMA, PVC, PA (Nylon), PC
Examples of non-polar polymers are PTFE (and many other fluoropolymers),
PE, PP and PS
For polar polymers the alternating current (AC) frequency is an important factor
because of the time taken to align the polar dipoles. At very low frequencies the
dipoles have sufficient time to align with the field before it changes direction
and the dielectric constant is high. At very high frequencies the dipoles do not
have time to align before the field changes direction and the dielectric constant
is lower. At intermediate frequencies the dipoles move but have not completed
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their movement before the field changes direction and they must realign with the
changed field. Polar polymers at low frequencies (60 Hz) generally have
dielectric constants of between 3 and 9 and at high frequencies (106 Hz)
generally have dielectric constants of between 3 and 5.
For non-polar polymers the dielectric constant is independent of the alternating
current frequency because the electron polarization is effectively instantaneous.
Non-polar polymers always have dielectric constants of less than 3.
→ frequency ν dependent quantity
Similar as for relative permittivity, absolute permittivity can be decomposed into
real and imaginary parts:
ε(ν) = ε'(ν) – i ε"(ν)
with tan δ =
ε ′′(ν )
ε ′(ν )
ε'(ν) exhibits dispersion, decrease with increasing frequency - related to the
stored energy within the medium
ε"(ν) exhibits an absorption maximum at resonance frequency - related to the
dissipation (or loss) of energy within the medium
A dielectric permittivity spectrum over
a wide range of frequencies. ε′ and ε″
denote the real and the imaginary part
of the permittivity, respectively.
Various processes are labeled on the
image: ionic and dipolar relaxation,
and atomic and electronic resonances
at higher energies.
wikipedia.org
Instead of measuring the dielectric permittivity as function of frequency, it can
be probed as function of time by switching on and off an electric field.
φ (t ) =
ε (t ) − ε ∞
ε = ε at ν = ∞ or 0
ε 0 − ε ∞ ∞,0
Connecting time and frequency domain by a Fourier transformation
∞
ε (ν ) − ε ∞
= 1 − iω ∫ φ (t )e
ε0 − ε∞
0
− iωt
dt
Separation real and imaginary part
∞
ε ′(ν ) − ε ∞
= 1 − ω ∫ φ (t )sin ωtdt
ε0 − ε∞
0
∞
ε ′′(ν )
= ω ∫ φ (t )cos ωtdt
ε0 − ε∞
0
Chapter05
25
Simple assumption: function φ(t) ∝ e(-t/τ), =) one relaxation time τ
ε ′(ν ) − ε ∞
1
=
ε0 − ε∞
1 + ω 2τ 2
ε ′′(ν )
ωτ
=
(Lorentz shape)
ε 0 − ε ∞ 1 + ω 2τ 2
Single exponential relaxation in the time domain corresponds to a Lorentz shape
in the frequency or energy domain
Full-width half maximum of the loss
curve in case of a single excitation =
1.14 (log ωτ) and symmetric shape
centered at log ωτ = 0
Real: Polymers exhibit a broader loss curve
most likely with an asymmetric tail towards ωτ > 0
⇒ empirical extension in frequency domain
ε (ν ) − ε ∞
1
=
a
ε0 − ε∞
1 + (iωτ )
[
a=b=1
0 < a ≤ 1, b = 1
a = 1, 0 < b ≤ 1
0 < a ≤ 1, 0 < b ≤ 1
]
b
single excitation
Cole-Cole function
Cole-Davidson function
Havriliak-Negami function
⇒ alternative in the time domain make use of stretched exponential function
φ (t ) ∝ e(− t / τ )
β
with 0 < β ≤ 1
Kohlrausch-Williams-Watt function
All these non-exponential relaxation functions have a general application
beyond polymers !!!
Chapter05
26
Example: amorphous polymer
polyethylmethacrylate at 272 K at different pressures of p= 1(1), 340(2), 690(3)
and 1020(4) bar shows strong pressure dependence.
Very good description with KWW function ⇒ in this experiment an α-process,
meaning a long-ranged structural relaxation is observed
All amorphous polymers show several different relaxation processes
α- process: movement on large scale
β- process: localized movement (at higher frequency as α- process)
Excitation maxima at higher frequencies are labeled with γ−, δ−relaxation.
Example for connecting different relaxations with molecular movements:
Chapter05
27
α and β relaxation are very different for different polymers
a) α- process
τα the relaxation time of the α- process follows the so-called
B ⎞
⎟⎟
⎝ T − T0 ⎠
⎛
Vogel-Fulcher-Tammann (VFT)-law τ α = τ 0 exp⎜⎜
With the reference temperature T0 (typically 50°C below glass transition
tempearture). The VFT-law is equivalent to the Williams-Landel-Ferry (WLF)equation which is related to the temperature-time-superposition principle. Thus
the α- process obeys the temperature-time-superposition principle for the shiftparameter aT:
log(aT ) = log
τα
C (T − T0 )
=− 1
and C1 = B / (T − T0 );C2 = T − T0
τ0
C2 + (T − T0 )
b) β- process
τβ is single activated (Arrhenius behavior) with a weak T-dependence
At high temperatures it leads to the α- process.
Commonly the relaxation strength is Δεα > Δεβ (with some exceptions).
What is the physics behind the β-process?
• On a microscopic level not unambiguously identified.
• In polymers many different possibilities such as movement of side-groups,
local movement in main chain, partial reorientation of main chain, …
Superposition principles:
For amorphous polymers above their Tgs, there is a convenient approximation
which makes experiments easier. It is known as time-temperature
superposition, and it relates time to temperature for viscoelastic materials. If
not temperature but pressure is varied in the experiment the so called pressuretime-superposition principle is observed for polymers.
Chapter05
28
Example: normalized master-curve of ε' and ε" for polyethylacrylate •, †, „
resembles data measured at 1, 340 and 690 bar
α-process most likely fitted with KWW-law: β ~ 0.3 - 0.7
How to explain KWW-law?
• Superposition of many different relaxation times, inhomogeneous model
• Mode-coupling theory yield power-laws, which match in the α-regime well
the KWW-law
• Coupling model of Ngai
Relaxation rate
dh
= W (t ) ⋅ h
dt
with W(t) = const. would result exponential relaxation, but
with W(t) = W0(ωct)-n, meaning a time-dependent relaxation rate
results
and
Chapter05
h(t) = exp [– (t/τ)1-n]
τ = [ (1-n) ωnc τo ] (1-n)-1
29