Transcript PowerPoint-Präsentation

Lecture "Molecular Physics/Solid State physics"
Winterterm 2013/2014
Prof. Dr. F. Kremer
Outline of the lecture on 7.1.2014
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The spectral range of Broadband Dielectric Spectroscopy
(BDS) (THz - <=mHz).
What is a relaxation process?
Debye –relaxation
What is the information content of dielectric spectra?
What states the Langevin equation
Examples of dielectric loss processes
The spectrum of electro-magnetic waves
UV/VIS IR
Broadband Dielectric
Spectroscopy (BDS)
What molecular processes take place in the spectral
range from THz to mHz and below?
Basic relations between the complex dielectric function *
and the complex conductivity *
The linear interaction of electromagnetic fields with matter is
described by one of Maxwell‘s equations
curl
D
H  j
t
j  E
D   0 E


(Ohm‘s law)
(Current-density and the time derivative of D are equivalent)


      i 
        i 
  i 0


Effect of an electric field on a unpolar atom or molecule:
In an atom or molecule the electron cloud is deformed with respect
to the nucleus, which causes an induced polarisation; this response is
fast (psec), because the electrons are light-weight
+
-
-
+
-
Electric Field
Effects of an electric field on an electric dipole m :
An electric field tries to orient a dipole m; but the thermal fluctuations of the surrounding heat bath counteract this effect; as
result orientational polarisation takes place, its time constant is
characteristic for the molecular moiety under study and may vary
between 10-12s – 1000s and longer.
Electric field
+e
-e
Effects of an electric field on (ionic) charges:
Charges (electronic and ionic) are displaced in the direction of the
applied field. The latter gives rise to a resultant polarisation of the
sample as a whole.
-
- + +
+
+ +
+
- + +
-
-
+
+
+
+
+ +
+ +
Electric field
What molecular processes take place in the spectral
range from THz to mHz and below?
1. Induced polarisation
2. Orientational polarisation
3. Charge tansport
4. Polarisation at interfaces
What is a relaxation process?
Relaxation is the return of a perturbed system into equilibrium. Each
relaxation process can be characterized by a relaxation time τ. The
simplest theoretical description of relaxation as function of time t is
an exponential law exp(t /  ) . In many real systems a Kohlrausch
b
law with exp(t /  ) and a “streched exponential” b is observed.
What is the principle of
Broadband Dielectric Spectroscopy?
A closer look at orientational polarization:
*
Debye relaxation     
Capacitor with N permanent
dipoles, dipole Moment m
s  
(1  i )
2
3,6
'=s
3,2
E (t)
1
E
'
2,8
 ( ) 
60
40
20
0,8
s  

2
1    
S
''max
0,6
''
0
=s-?
'=?
2,0
0
(t)=(P(t) - P) / E
2,4
0,4
0,2
 = S- 00
orientational polarization
induced polarization
0
10
00
1
10
2
10
3
10
4
10
5
10
6
10
-1
 [rad s ]
Time
complex dielectric function
max
0,0
 *( , T )
P( )   o ( *(, T )  1) E
 ( )    
 s  
1    
s  
 ( ) 

2
1    
2
P. Debye, Director (1927-1935) of the Physical Institute
at the university of Leipzig (Nobelprize in Chemistry 1936)
What are the assumptions of a Debye relaxation
process?
1. The (static and dynamic) interaction of the
„test-dipole“ with the neighbouring dipoles is
neglected.
2. The moment of inertia of the molecular system
in response to the external electric field is
neglected.
The counterbalance between thermal and electric ennergy
x
Local
CH3
Capacitor with N permanent Dipoles, Dipole Moment m
Polarization :
2
 1   N  
P   μ i  P  μ  P
V
V
C
CH2
C
O
O
CH2
CH3
Mean Dipole Moment
E (t)
1
E
Mean Dipole Moment:
Counterbalance
Wth  kT
Dipole moment
Local motion
localized bond rotations
fluctuations in a side group
x local < 1nm  
Wel  m  E
0
Thermal Energy
(t)=(P(t) - P) / E
0
20
40
60
S
 = S- 00
orientational polarization
induced polarization
Time
00
Boltzmann Statistics:
Electrical Energy
 

m E
m exp (
) d

kT

m  4
 
m E
 exp( kT ) d
4
The factor exp(mE/kT) d gives the probability that
the dipole moment vector has an orientation
between  and  + d.
The Langevin-function
Spherical Coordinates:

m E cos  1
) sin  d
kT
2
m E cos  1
exp (
) sin  d
kT
2
 m cos  exp (
Only the dipole moment component which
is parallel to the direction of the
electric field contributes to the polarization
m 0


0
x = (m E cos ) / (kT)
a = (m E) / (kT)
a
 x exp(x ) dx exp(a )  exp(a ) 1
1 a
cos  

   (a )
a a
exp(a )  exp(a ) a
 exp(x ) dx
1.0
(a)=a/3
0.8
a
(a)
0.6
a
0.4
Langevin function
mE
 0.1
kT
Langevin function
0.1k T
E
m
0.2
(a)a/3
0.0
0
2
6
4
a
8
10
Debye-Formula
0 - dielectric permittivity of vacuum = 8.854 10-12 As V-1 m-1
m2
m 
E
3k T
1 m2 N
S   
3 0 k T V
Analysis of the dielectric data
propylene glycol
5
10
235 K
220 K
10
4
10
3
10
2
10
1
10
0
4
´´
10
3
´´
10
205 K
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
frequency [Hz]
2
10
190 K
1
10
0
10
-2
10
-1
10
0
10
1
10
frequency [Hz]
2
10
3
10
4
10
5
10
6
10
6
Brief summary concerning the principle of Broadband
Dielectric Spectroscopy (BDS):
1. BDS covers a huge spectral range from THz to mHz and
below.
2. The dielectric funcion and the conductivity are comlex because
the exitation due to the external field and the response of the
system under study are not in phase with each other.
3. The real part of the complex dielectric function has the
character of a memory function because different dielectric
relaxation proccesses add up with decreasing frequency
4. The sample amount required for a measurement can be
reduced to that of isolated molecules.
(With these features BDS has unique advantages compared to other
spectroscopies (NMR, PCS, dynamic mechanic spectroscopy).
1. Dielectric relaxation
(rotational diffusion of bound charge
carriers (dipoles) as determined from
orientational polarisation )
Analysis of the dielectric data
propylene glycol
5
10
235 K
220 K
10
4
10
3
10
2
10
1
10
0
4
´´
10
3
´´
10
205 K
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
frequency [Hz]
2
10
190 K
1
10
0
10
-2
10
-1
10
0
10
1
10
frequency [Hz]
2
10
3
10
4
10
5
10
6
10
6
Relaxation time distribution functions according
to Havriliak-Negami
Analysis of the dielectric data
10
8
log10(1/max [Hz])
6
4
2
experimental data: propylene glycol bulk
VFT-fit: 1/=Aexp(DT0/(T-T0))
0
-2
3.0
3.5
4.0
4.5
1000/T [K-1]
5.0
5.5
6.0
Summary concerning Broadband Dielectric Spectroscopy
(BDS) as applied to dielectric relaxations
1.: BDS covers a huge spectral range of about 15 decades
from THz to below mHz in a wide range of temperatures.
2.: The sample amount required for a measurement can be reduced
to that of isolated molecules.
3.: From dielectric spectra the relaxation rate of fluctuations of a
permanent molecular dipole and it´s relaxation time distribution
function can be deduced. The dielectric strength allows to
determine the effective number-density of dipoles.
4.: From the temperature dependence of the relaxation rate
the type of thermal activation (Arrhenius or Vogel-FulcherTammann (VFT)) can be deduced.
2. (ionic) charge transport
(translational diffusion of charge
carriers (ions))
Basic relations between the complex dielectric function *
and the complex conductivity *
The linear interaction of electromagnetic fields with matter is
described by Maxwell‘s equations
curl
D
H  j
t
j  E

D   0 E

(Ohm‘s law)
(Current-density and the time derivative of D are equivalent)


      i 
        i 
  i 0


Dielectric spectra of MMIM Me2PO4 ionic liquid
log 
2
-4
-6
-8
-10
-12
-4
-6
-8
-10
-12
log '' [S/cm]
log ' [S/cm]
4
6
4
2
0
-2
log ''
'
6
268 K
258 K
248 K
238 K
228 K
218 K
208 K
0 2 4 6 8
0 2 4 6 8
log f [Hz]
Strong temperature dependence of the charge transport processes
and electrode polarisation
 (T ,  ) for the ionic liquid (OMIM NTf2)
OMIM NTf2
-6
log (/c)
-4 0 4
-8
-10
log ('/0)
-1
log (' /S cm )
-4
0
300 K
260 K
240 K
220 K
210 K
200 K
190 K
4
c
0
-12
2
4
-1
log ( /s )
-4
6
8
There are two characteristic quantities 0 and c which enable
one to scale all spectra!
Basic relations between rotational and
translational diffusion
Stokes-Einstein relation:
D  kT 
Maxwell‘s relation:
  G
D: diffusion coefficient,  (=6  a) :
frictional coefficient,T: temperature, k:
Boltzmann constant, a: radius of molecule
G: instantaneous shear modulus (~ 108 1010 Pa)
: viscosity,: structural relaxation time
Einstein-Smoluchowski relation:
 2c
D
2
 : characteristic (diffusion) length
c: characteristic (diffusion) rate
Basic electrodynamics and Einstein relation:
q2 D
 0  qm n  n
  0  c
kT
0: : dc conductivity, m: mobility ; q:
elementary charge, n: effective number
density of charge carriers
Predictions to be checked experimentally:
1.:   P 
c
P
kT
2
2
G
2as

 c
2.: D 
2
2
3.: 
0
~

c
c: characteristic (diffusion) rate
: structural relaxation rate
G: instantaneous shear modulus (~ 108 Pa
for ILs),
as : Stokes‘ hydrodynamic radius ~,
k: Boltzmann constant,
: characteristic diffusion length ~ .2 nm
D : molecular diffusion coefficient
(Barton-Nakajima-Namikawa (BNN) relation)
Measurement techniques required: Broadband Dielectric Spectroscopy (BDS);
Pulsed Field Gradient (PFG)-NMR; viscosity measurements;
1. Prediction: c  P 
Broadband Dielectric
Spectroscopy
(BDS)
with P ~ 1
Mechanical Spectroscopy
c: characteristic (diffusion) rate
: structural relaxation rate
G: instantaneous shear modulus (~ 108 Pa for ILs),
Random Barrier Model (Jeppe Dyre et al.)
• Hopping conduction in a spatially randomly varying
energy landscape
• Analytic solution obtained within Continuous-TimeRandom Walk (CTRW) approximation
• The largest energy barrier determines Dc conduction
• The complex conductivity is described by:
 i e

     0 

 ln 1  i e  
 e is the characteristic time related to the
attempt frequency to overcome the
largest barrier determining the Dc
conductivity. „Hopping time“.
Random Barrier Model (RBM) used to fit the
conductivity spectra of ionic liquid (MMIM Me2PO4)
OMIM NTf2
-1
log (' /S cm )
-4
0
300 K
260 K
240 K
220 K
210 K
200 K
190 K
-6
-8
-10
c
-12
2
6
4
-1
log ( /s )
8
The RBM fits quantitatively the data; c  1/  e
Scaling of invers viscosity 1/ and conductivity s0 with
temperature
s )
-1
-3
-1
10
-1
10
-5
10
-9
10
log [1/  (Pa
-1
-5
-1
10
1/ (Pa s )
-1 -1
(1/) (Pa s )
10
-7
10
-9
10
-11
10
10
-13
-13
0
Full symbols: 1/
-4
HMIM BF4
HMIM Cl
HMIM
PF6Br
HMIM
-8
HMIM
HMIM
I I
HMIM
HMIM
Br PF6
10
-2
10
-6
10
-10
10
-14
0 ( S/cm)
-1
HMIM Cl
-12
10 0.6
3,2
Calorimetric Tg
0.8 4,4
1.0 5,2
4,0
4,8
0,8
1,0
Tg/T T /T
-1
g
K )identical temperature
The invers viscosity 1/1000/T
has (an
3,6
dependence as 0 and scales with Tg.
Correlation between translational and rotational
diffusion
Measured Ginf
8
10
BMIM BF4 (0.07 GPa)
HMIM PF6 (0.07 GPa)
HMIM BF4 (0.04 GPa)
6
-1
-1
 c (s ),   (s )
10
4
ωc  Pω η
10
2
10
Stokes-Einstein, EinsteinSmoluchowski and Maxwell
relations:

c


BMIM BF4
0
10
P
HMIM PF6
HMIM BF4
-2
kT
2G  2 G   2 a s
10
3,6
4,0
4,4
4,8
5,2
5,6
-1
1000/T (K )
Typically: G  0.1 GPa;   .2 nm; as  .1 nm;
J. R. Sangoro et al.(2009) Phys. Chem. Chem. Phys.
1
Correlation between translational and rotational diffusion
-1
log  (s )
0
-1
log c (s )
6
4
2
2
4
6
Stokes-Einstein, EinsteinSmoluchowski and Maxwell
relations:
ωc  Pω η
P
kT
2G  2 G   2 a s
1
0
Typically: G  0.1 GPa;   .2 nm; as  .1 nm;
J. R. Sangoro et al.Phys. Chem. Chem. Phys, DOI: 0.1039/b816106b,(2009) .
Prediction checked experimentally:
1.: c  P 
P
kT
2G  2 G   2 a s
1
c: characteristic (diffusion) rate
: structural relaxation rate
G: instantaneous shear modulus
(~ 108 Pa
for ILs),
as : Stokes‘ hydrodynamic radius ~,
k: Boltzmann constant,
: characteristic diffusion
length ~ .2 nm
2

2. Prediction: D(T ) 
~ c
2 e (T )
Pulsed-Field-Gradient NMR
Broadband Dielectric Spectroscopy
(BDS)
Comparison with Pulsed-Field-Gradient NMR and determination of diffusion coefficients from dielectric spectra
Based on Einstein-Smoluchowski relation and using PFG NMR
measurements of the diffusion coefficients enables one to determine the
diffusion length  :
2
D(T ) 
-10
Quantitative agreement between
PFG-NMR measurements and
the dielectric determination of
diffusion coefficients. Hence mass
diffusion (PFG-NMR) equals
charge transport (BDS)..
2 -1
log D [m s ]
-12
-14
-16
DNMR
-18
DE
-20
3,5
4,0
2 e (T )
~ c
4,5
-1
1000/T (K )
5,0
Comparison with Pulsed-Field-Gradient NMR and determination of diffusion coefficients from dielectric spectra
From Einstein-Smoluchowski and basic electrodynamic definitions it
follows:
2
2
qD(T ) n(T )q 
 0 (T )  n(T )qm (T )  n(T )q

~ c
kT
kT 2 e (T )
(n(T):number-density of charge carriers; µ(T):mobility of charge carriers)
-10
2 -1
log D [m s ]
-12
1. The separation of n(T) and µ(T)
from 0 (T) is readily possible.
-14
-16
DNMR
-18
DE
2. The empirical BNN-relation is an
immediate consequence.
-20
3,5
4,0
4,5
-1
1000/T (K )
5,0
Separation of n(T) and µ(T) from 0 (T)
MMIM Me2PO4
4,0
4,5
DNMR
4.0
-14
27
-18
26
-20
3,5
4,0
-18
3
25
-16
log N [1/m ]
-16
-20
4,5
5,0
1.: µ(T) shows a VFT
temperature
dependence
-1
1000/T [1/K]
4.4
4.8
-12
-1
m
2
DE
2
-1
log D [m s ]
-12
-14
5,0
-10
log µ [m V s ]
-10
3,5
2.. n(T) shows Arrheniustype temperature
dependence
3.: the 0 (T) derives its
dependence from µ(T)
-1
1000/T (K )
J. Sangoro, et al., Phys. Rev. E 77, 051202 (2008)
Predictions checked experimentally:
 2c
2.: D 
2
c: characteristic (diffusion) rate
: characteristic diffusion length ~ .2 nm
D : molecular diffusion coefficient
0 : Dc conductivity
Applying the Einstein-Smoluchowski relation enables one to deduce the root mean
square diffusion distance  from the comparison between PFG-NMR and BDS
measurements. A value of  ~ .2 nm is obtained. Assuming  to be temperature
independent delivers from BDS measurements the molecular diffusion coefficient D.
Furthermore the numberdensity n(T) and the mobility m(T) can be separated and
the BNN-relation is obtained.
3.: 
0
~

c
(Barton-Nishijima-Namikawa (BNN) relation)
Summary concerning Broadband Dielectric Spectroscopy
(BDS) as applied to ionic charge transport
1.: The predictions based on the equations of Stokes-Einstein and
Einstein-Smoluchowski are well fullfilled in the examined ionconducting systems.
2.: Based on dielectric measurements the self-diffusion coefficient
of the ionic charge carriers and their temperature dependence
can be deduced.
3.: It is possible to separate the mobility m(T) and the effective
numberdensity n(T). The former has a VFT temperaturedependence, while the latter obeys an Arrehnius law.
4.: The Barton-Nishijima-Namikawa (BNN) relation turns out to
be a trivial consequence of this approach.
Kontrollfragen 7.1.2014
103. Was ist ein Relaxationsprozess? Wie unterscheidet er
sich von einer Schwingung?
103. Welche Annahmen liegen der Debye Formel zugrunde?
104. Was besagt die Langevin Funktion?
105. Was ist der Informationsgehalt dielektrischer Spektren?
106. Nennen Sie Beispiele für dielektrisch aktive Verlustprozesse.