Advanced Functional Properties 6PC20

1
Advanced Functional Properties 6PC20

• Lecture 3 Dielectric response

2
Dielectric response for water
3
Can we use linear response theory to account for
features in the dielectric spectra ?
Fourier transformation
G(t)
c(w)c1(w)ic2(w)
Causality implies that
G(t) 0 for tlt0.
4
Polarization as a response to an applied field
Materials respond to an applied electric field by
redistributing bound charge in such a way that
the applied electric field is opposed.
(compare principle of Le Chatellier and Van t
Hoff) Redistributed, bound charge is referred to
as polarization P How can this be accomplished
at the molecular level ?

• If a molecule/ion is charged
• 0. Translational motion (salt solutions)
• If molecule has a dipole moment
• Rotational motion (polar solvents H2O, MeOH)
• If a molecular vibration changes the dipole
  moment
• of the molecule
• 2. Vibrational motion (often asymmetric
  vibrations)
• If the electron motion has dipolar character
• 3. Electronic motion (electron moving to another
  orbital)

5
Rotational motion of polar molecules in an
oscillating field
Rotational motion of molecules with a permanent
dipole moment can contribute to the polarization
provided that the frequency of the oscillating E
field is low enough for the molecules to follow
he changes in the oscillating field. In a solid
molecules can not rotate infinitely rapidly
because they experience friction from
interactions with their neighbors
This friction can be expressed by the rate with
which Polarization P(t) decays back to zero after
the applied filed has been switched off.
For instance
Field pulse
P(t)
Linear response P0 ? E t independent of E
P0
P0/e
0
t
6
Time domain response function G(t)
?
7
Fourier transform of the Polarization decay
Is this consistent with Kramers-Kronig? See
exercise
8
Fourier transform of a derivative
Integration by parts

zero


Required for Fourier transform to exist
9
Magnitude of the polarization in a static
electric field E
Polarization P ? Average component of the
dipole moment m of the molecule in the direction
of E
Average over all orientations
E
Probability of orientation with angle ?
m
?
10
Average of cos2? over all orientations
z
Pythgoras
r1
f
?
y
Average over all orientations
?
x
Thus
11
Magnitude of the static polarization Pstat
Where average component of
the dipole moment m of the molecule n the
direction of E
N number of polar molecules per unit volume
In order to be compatible with the static limit
must have
Debye model
12
Dielectric function in the Debye approach
13
Debye dielectric function a.k.a Maxwell-Wagner
dispersion
t 1 ?(0) 1
e1 (w)
e2 (w)
Full width half maximum (FWHM) 1 decade
w
dispersion the way it is dispersed over the
frequency range
14
What is the interpretation of e2(w) ? (1)
Classical RC circuit
R
C
?
I
e2(w) corresponds to conductivity ( reciprocal
resistance)
15
What is the interpretation of e2(w) ? (2)
Complex plane t 0
Im
In phase with V(t)
I2
I1
V
Re
90o out of phase with V(t)
Electrical power dissipated in a circuit
16
What is the interpretation of e2(w) ? (3)
e10 e20.9
e10.9 e20
Power goes in and out of the system storage
modulus elastic
Power is absorbed or dissipated in the system
loss modulus inelastic viscous
17
What is the interpretation of e2(w) ? (4)
e2(w) describes the absorption and dissipation
of energy in the material dielectric loss
This turns out to be a general result. Also for
optical and mechanical frequency domain response
functions it holds that
The imaginary part ic2 of the response function
describes dissipation of power
Why is this ? Friction in the system (described
by t) results in a response that lags behind the
stimulus. This results in part of the response
being out of phase with the stimulus This is
described by the imaginary part of the response
function ic2(w) e.g. note that e2(w) ?0 if t ?0
18
What is the interpretation of t ?

• - t is independent of the applied electric field.
  
• 1/t w0 maximum frequency that the
  molecules are able to follow with their
  rotational motion due to the friction. t is a
  rotational correlation time that describes how
  quickly molecule can reorient.
• Due to thermodynamical fluctuations, a (small)
  polarization P(t) could arise spontaneously
  without applied field. It would decay with the
  same time constant t as the field induced
  polarization. t describes spontaneous
  fluctuations.
• t describes friction and its value determines
  the rate at which power is dissipated in the
  material

Fluctuation - dissipation theorem Dissipation
of energy in a systems is described by a friction
coefficient that also describes the rate with
which spontaneous fluctuations rise and decay in
the material.
19
How large is the dielectric correlation time t ?
Experimental data on ethanol
e2
e1
exp
20
How large is the dielectric correlation time t ?
Experimental data on water
21
How large is the dielectric correlation time t ?
For strongly hydrogen bonded solvents (water ,
ethanol) t is in the range 10-9 10-11 sec.
22
What is the importance of t ?
t describes the shortest time scale at which
polar solvent molecules can solvate an
electrically charged reaction product
HBr ? H Br-
Without solvation the free energy for this
reaction is very unfavourable The rate of
solvation can therefore in some cases control the
speed of a reaction This lower limit for the time
required for the reaction is set by t
23
Is the Debye model always accurate ? Experiments
of cyclooctanol (1)
Dielectric response fitted with Fourier
transform of
Kohlrausch Williams Watts decay or stretched
exponential
If b1 then
Exponential decay re-obtained Debye response !
Birge et al. Phys. Rev. B 50, 13250 (1995)
24
Is the Debye model always accurate ? Experiments
of cyclooctanol (2)
Kohlrausch Williams Watts Distribution of
relaxation times. Not all molecules in the same
environment. Rapidly frozen alcohol disordered
solid glass
208 K
b
167 K
glass
Debye liquid
Loid et al. Phys Rev B 56, R 5713 (1997)
25
The glass transition
26
A glass shows ageing very slow crystallization
dynamics
Time evolution of e2 after rapid cooling from
room temperature to T 213 K
e2
Loid et al. Phys Rev B 56, R 5713 (1997)
27
Propylene carbonate
28
Mechanical and dielectric response co-polymer of
isoprene and butadiene
Mechanical response function G(w) G1(w)iG2(w)

Shear Force
?
Motions of the polymer segments and chains
contribute both to the mechanical and the
dielectric response
Pakula et al. Rheol. Acta 35, 631 (1996)
29
Dielectric function for a layer of C60
Dresselhaus Annu. Rev. Mater. Sci. 1995.25487-523
30
Polarization due to vibrational and electronic
motionLorentz oscillator
Charge (atom, electron ) bound harmonically to
the molecule Natural frequency w0
Electric field oscillating at w drives the charge
Fdrive q E0 eiwt
Fspring -kx kmw02
-q, M (Mgtgtm)
q, m
Molecule
Ffriction - (m/t) (dx/dt)
Friction damps the motion of the charge
Causal relation Newton F m a
31
Lorentz oscillator
Damped harmonic oscillator
Ffriction
ma
Fdrive
Fspring
with
gives
32
Lorentz oscillator
Oscillating dipole moment per molecule
Polarization Nq number of oscillators per unit
volume
Assuming one oscillator per molecule
NA Avogadro P density Mw molecualr weight
33
Lorentz oscillator
34
Polarization due to vibrations and electronic
motion
w0 1 t0.5 a1
w0 1 t0.25 a1
e1(w)
e1(w)
e2(w)
e2(w)
w
w
35
Lorentz compared to Debye
w0 1 t5 a1
The Lorentz harmonic oscillator model starts to
resemble the Debye model for t/w gtgt 1
e1(w)
The Debye model represents an overdamped harmonic
oscillator
e2(w)
w
36
Experimental data on water
37
Features explained
In experimental e(w) spectra we observed the
following features
Gradual Transition possible
Debye response Overdamped motion Rotation of
polar molecules
Lorentz response damped Harmonic
oscillator Molecular vibrations Electronic
excitations