lecture molecular physics/solid state physics winterterm ... · lecture "molecular...
TRANSCRIPT
Lecture "Molecular Physics/Solid State physics"Winterterm 2013/2014Prof. Dr. F. Kremer
Outline of the lecture on 7.1.2014
• 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?
Dcurl H jt
∂= +
∂
0D Eε ε∗= j Eσ ∗=
The linear interaction of electromagnetic fields with matter is described by one of Maxwell‘s equations
(Current-density and the time derivative of D are equivalent)
(Ohm‘s law)
0iσ ωε ε∗ ∗=
( ) iε ω ε ε∗ ′ ′′= − ( ) iσ ω σ σ∗ ′ ′′= +
Basic relations between the complex dielectric function ε* and the complex conductivity σ*
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 FieldElectric Field
Effects of an electric field on an electric dipole μ :
An electric field tries to orient a dipole μ; but the thermal fluct-uations 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 varybetween 10-12s – 1000s and longer.
-e
+e
Electric field
Effects of an electric field on (ionic) charges:
Charges (electronic and ionic) are Charges (electronic and ionic) are displaced in the direction of theapplied field. The latter gives rise to a resultant polarisation of the sample as a whole.
-
-
-
-
--
-
-
+
+
++
+
+
+ +
Electric fieldElectric 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 . In many real systems a Kohlrausch law with and a “streched exponential” β is observed.exp( / )t βτ−
exp( / )t τ−
What is the principle of Broadband Dielectric Spectroscopy?
( )2( )1
sε εε ω ω τω τ
∞−′′ =
+
( )2( )1
sε εε ω ε
ω τ∞
∞
−′ = +
+
*
(1 )s
iε εε ε
ωτ∞
∞−
= ++
2,0
2,4
2,8
3,2
3,6
100 101 102 103 104 105 106
0,0
0,2
0,4
0,6
0,8
ε'=εs
ε'=ε?
Δε=εs-ε?
ε'
ε''max
ωmax
ε''
ω [rad s-1]
0 20 40 60
0
1
2
orientational polarization
induced polarization
Δε = εS- ε00
εS
ε00
ε(t)=
(P(t)
- P
∞) /
ΔE
Time
ΔE
E (t
)Capacitor with N permanent
dipoles, dipole Moment μ Debye relaxation
complex dielectric function
( )
A closer look at orientational polarization:
2( )1
sε εε ω ω τω τ
∞−′′ =
+
*( , )Tε ω
( ) ( *( , ) 1)oP T Eω ε ε ω= −
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 systemin response to the external electric field is neglected.
The counterbalance between thermal and electric ennergy
Capacitor with N permanent Dipoles, Dipole Moment μ
Polarization :∞∞ +=+= ∑ Pμ
VNPμ
V1P i
Mean Dipole Moment
0 20 40 60
0
1
2
orientational polarization
induced polarization
Δε = εS- ε00
εS
ε00
ε(t)=
(P(t)
- P
∞) /
ΔE
Time
ΔE
E (t
)
CH
CH
CH
3
2
3
CO
O
C CH2
Dipole moment
Mean Dipole Moment: Counterbalance
EWel ⋅μ−=kTWth =
Thermal Energy Electrical Energy
∫
∫
π
π
Ω⋅μ
Ω⋅μ
μ
=μ
4
4
d)kT
Eexp(
d)kT
E(expBoltzmann Statistics:
The factor exp(μE/kT) dΩ gives the probability that the dipole moment vector has an orientationbetween Ω and Ω + dΩ.
Spherical Coordinates:
Only the dipole moment component whichis parallel to the direction of the electric field contributes to the polarization θθ
θμ
θθθμ
θμ=μ
∫
∫π
π
dsin21)
kTcosE(exp
dsin21)
kTcosE(expcos
0
0
x = (μ E cos θ) / (kT)a = (μ E) / (kT)
)a(a1
)aexp()aexp()aexp()aexp(
dx)xexp(
dx)xexp(x
a1cos a
a
a
a Λ=−−−−+
==θ
∫
∫
−
−
Langevin function
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0Λ(a)=a/3
Langevin function
a
Λ(a
)
Λ(a)≈a/3 ETk3
2μ=μ
Debye-FormulaVN
Tk31 2
0S
με
=ε−ε ∞
1.0kT
Ea <μ
=
μ<
Tk1.0E
ε0 - dielectric permittivity of vacuum = 8.854 10-12 As V-1 m-1
The Langevin-function
10-2 10-1 100 101 102 103 104 105 106
100
101
102
103
104
105
235 K220 K
205 K
190 K
propylene glycol
ε´´
frequency [Hz]
10-2 10-1 100 101 102 103 104 105 106
100
101
102
103
104
ε´´
frequency [Hz]
Analysis of the dielectric data
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 otherspectroscopies (NMR, PCS, dynamic mechanic spectroscopy).
1. Dielectric relaxation
(rotational diffusion of bound chargecarriers (dipoles) as determined from
orientational polarisation )
10-2 10-1 100 101 102 103 104 105 106
100
101
102
103
104
105
235 K220 K
205 K
190 K
propylene glycol
ε´´
frequency [Hz]
10-2 10-1 100 101 102 103 104 105 106
100
101
102
103
104
ε´´
frequency [Hz]
Analysis of the dielectric data
Relaxation time distribution functions accordingto Havriliak-Negami
3.0 3.5 4.0 4.5 5.0 5.5 6.0-2
0
2
4
6
8
10
experimental data: propylene glycol bulk VFT-fit: 1/τ=A exp(DT0/(T-T0))
log 1
0(1/ τ m
ax [H
z])
1000/T [K-1]
Analysis of the dielectric data
Summary concerning Broadband Dielectric Spectroscopy (BDS) as applied to dielectric relaxations
1.: BDS covers a huge spectral range of about 15 decadesfrom 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 distributionfunction 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-Fulcher-Tammann (VFT)) can be deduced.
2. (ionic) charge transport
(translational diffusion of chargecarriers (ions))
Dcurl H jt
∂= +
∂
0D Eε ε∗= j Eσ ∗=
The linear interaction of electromagnetic fields with matter is described by Maxwell‘s equations
(Current-density and the time derivative of D are equivalent)
(Ohm‘s law)
0iσ ωε ε∗ ∗=
( ) iε ω ε ε∗ ′ ′′= − ( ) iσ ω σ σ∗ ′ ′′= +
Basic relations between the complex dielectric function ε* and the complex conductivity σ*
Dielectric spectra of MMIM Me2PO4 ionic liquid
2
4
6
-20246
0 2 4 6 8-12-10-8-6-4
0 2 4 6 8-12-10-8-6-4
268 K 258 K 248 K 238 K 228 K 218 K 208 Klo
g ε'
log ε''
log
σ' [S
/cm
]
log σ'' [S/cm]
log f [Hz]
Strong temperature dependence of the charge transport processes and electrode polarisation
for the ionic liquid (OMIM NTf2)
2 4 6 8-12
-10
-8
-6
-4
300 K 260 K 240 K 220 K 210 K 200 K 190 K
log
(σ' /
S c
m-1)
log (ω /s-1)
σ0
OMIM NTf2
ωc
-4
0
4-4 0 4
log (ω/ωc)
log
(σ'/σ
0)
( , )Tσ ω′
There are two characteristic quantities σ0 and ωc which enable one to scale all spectra!
Basic relations between rotational and translational diffusion
Stokes-Einstein relation:
Einstein-Smoluchowski relation:
η =τη G∞
Maxwell‘s relation:
D kT ζ= D: diffusion coefficient, ζ (=6π η a) :frictional coefficient,T: temperature, k: Boltzmann constant, a: radius of moleculeG∞: instantaneous shear modulus (~ 108 -1010 Pa)η: viscosity,τη: structural relaxation time
2
2cD λ ω
=λ : characteristic (diffusion) length ωc: characteristic (diffusion) rate
Basic electrodynamics and Einstein relation:2
0 0 cq Dq n nkT
σ μ σ= = ⇒ ∝ωσ0: : dc conductivity, μ: mobility ; q: elementary charge, n: effective number density of charge carriers
Predictions to be checked experimentally:
2 2s
kTP2 G aλπ ∞
=
1.:
2.:
3.:
2
2cD λ ω
=
ωc: characteristic (diffusion) rateωη: structural relaxation rateG∞: instantaneous shear modulus (~ 108 Pa for ILs), as : Stokes‘ hydrodynamic radius ~λ,k: Boltzmann constant, λ: characteristic diffusion length ~ .2 nmD : molecular diffusion coefficient
σ0 ~ ωc
ωc = P ωη
(Barton-Nakajima-Namikawa (BNN) relation)
Measurement techniques required: Broadband Dielectric Spectroscopy (BDS);Pulsed Field Gradient (PFG)-NMR; viscosity measurements;
1. Prediction: with P ~ 1ωc = P ωη
ωc: characteristic (diffusion) rateωη: structural relaxation rateG∞: instantaneous shear modulus (~ 108 Pa for ILs),
Broadband Dielectric Spectroscopy
(BDS)
Mechanical Spectroscopy
Random Barrier Model (Jeppe Dyre et al.)
• Hopping conduction in a spatially randomly varying energy landscape
• Analytic solution obtained within Continuous-Time-Random Walk (CTRW) approximation
• The largest energy barrier determines Dc conduction• The complex conductivity is described by:
( ) ( )0 ln 1e
e
ii
ωτσ ω σωτ
⎛ ⎞= ⎜ ⎟⎜ ⎟+⎝ ⎠
is the characteristic time related to the attempt frequency to overcome the largest barrier determining the Dc conductivity. „Hopping time“.
eτ
Random Barrier Model (RBM) used to fit the conductivity spectra of ionic liquid (MMIM Me2PO4)
1/c eω τ=The RBM fits quantitatively the data;
2 4 6 8-12
-10
-8
-6
-4
300 K 260 K 240 K 220 K 210 K 200 K 190 K
log
(σ' /
S c
m-1)
log (ω /s-1)
σ0
OMIM NTf2
ωc
Scaling of invers viscosity 1/η and conductivity s0 with temperature
The invers viscosity 1/η has an identical temperature dependence as σ0 and scales with Tg.
3,2 3,6 4,0 4,4 4,8 5,210-13
10-11
10-9
10-7
10-5
10-3
10-1
HMIM Cl HMIM Br HMIM I HMIM PF6
1/η
(Pa-1
s-1)
1000/T (K-1)
Calorimetric Tg
0,8 1,0-12
-8
-4
0
log
[1/η
] (P
a-1 s
-1)
Tg/T0.6 0.8 1.0
10-13
10-9
10-5
10-1
10-14
10-10
10-6
10-2
HMIM BF4
HMIM PF6
HMIM I HMIM Br HMIM Cl
(1
/η) (
Pa-1
s-1)
Tg/T
Full symbols: 1/η
σ 0 ( S/
cm)
Correlation between translational and rotational diffusion
Stokes-Einstein, Einstein-Smoluchowski and Maxwell relations:
ηc Pω ω=
Typically: G∞ ≅ 0.1 GPa; λ ≅ .2 nm; as ≅ .1 nm;
3,6 4,0 4,4 4,8 5,2 5,610-2
100
102
104
106
108
J. R. Sangoro et al.(2009) Phys. Chem. Chem. Phys.
ωc ωη
BMIM BF4
HMIM PF6
HMIM BF4
ωc (
s-1),
ωη (s
-1)
1000/T (K-1)
Measured Ginf
BMIM BF4 (0.07 GPa)HMIM PF6 (0.07 GPa)HMIM BF4 (0.04 GPa)
2 2s
k TP2 G G
1aπ λ∞ ∞
=
Correlation between translational and rotational diffusion
0
2
4
6
0 2 4 6
log ωη (s-1)
log
ωc (
s-1)
Stokes-Einstein, Einstein-Smoluchowski and Maxwell relations:
2 2s
k TP2 G G
1aπ λ∞ ∞
=
ηc Pω ω
Typically: G∞ ≅ 0.1 GPa; λ ≅ .2 nm; as ≅ .1 nm;
J. R. Sangoro et al.Phys. Chem. Chem. Phys, DOI: 0.1039/b816106b,(2009) .
=
Prediction checked experimentally:
2 2s
kTP2G G
1aπ λ∞ ∞
=
1.: ωc: characteristic (diffusion) rateωη: structural relaxation rateG∞: instantaneous shear modulus
(~ 108 Pa for ILs), as : Stokes‘ hydrodynamic radius ~λ,k: Boltzmann constant, λ: characteristic diffusion
length ~ .2 nm
ωc = P ωη
2. Prediction: 2
( ) ~2 ( ) c
e
D TT
λτ
ω=
Pulsed-Field-Gradient NMRBroadband Dielectric Spectroscopy
(BDS)
Comparison with Pulsed-Field-Gradient NMR and deter-mination of diffusion coefficients from dielectric spectra
3,5 4,0 4,5 5,0-20
-18-16
-14
-12
-10
DNMR
DE
log
D [m
2 s-1]
1000/T (K-1)
2
( ) ~2 ( ) c
e
D TT
λτ
ω=
Based on Einstein-Smoluchowski relation and using PFG NMR measurements of the diffusion coefficients enables one to determine the diffusion length λ :
Quantitative agreement betweenPFG-NMR measurements and the dielectric determination of diffusion coefficients. Hence mass diffusion (PFG-NMR) equalscharge transport (BDS)..
3,5 4,0 4,5 5,0-20
-18-16
-14
-12
-10
DNMR
DE
log
D [m
2 s-1]
1000/T (K-1)
2
0
2( )( ) ~2
( )( ) ( ) ( )( ) c
e
D Tq n T qq n T qkT k
T TT
n TT
σ μ ωτλ
= = =
From Einstein-Smoluchowski and basic electrodynamic definitions it follows:
(n(T):number-density of charge carriers; µ(T):mobility of charge carriers)
Comparison with Pulsed-Field-Gradient NMR and deter-mination of diffusion coefficients from dielectric spectra
1. The separation of n(T) and µ(T) from σ0 (T) is readily possible.
2. The empirical BNN-relation is an immediate consequence.
Separation of n(T) and µ(T) from σ0 (T)
3,5 4,0 4,5 5,0-20
-18
-16
-14
-12
-103,5 4,0 4,5 5,0
-20-18-16-14-12-10 DNMR
DE
log
D [m
2 s-1]
1000/T (K-1)
4.0 4.4 4.8
25
26
27
log N [1/m
3]
1000/T [1/K]
μ
log
µ [m
2 V-1s-1
]
MMIM Me2PO4
1.: µ(T) shows a VFT temperature dependence
2.. n(T) shows Arrhenius-type temperature dependence
3.: the σ0 (T) derives its dependence from µ(T)
J. Sangoro, et al., Phys. Rev. E 77, 051202 (2008)
Predictions checked experimentally:
2.:
3.:
2
2cD λ ω
=ωc: characteristic (diffusion) rateλ: characteristic diffusion length ~ .2 nmD : molecular diffusion coefficientσ0 : Dc conductivity
σ0 ~ ωc (Barton-Nishijima-Namikawa (BNN) relation)
Applying the Einstein-Smoluchowski relation enables one to deduce the root mean square diffusion distance λ from the comparison between PFG-NMR and BDS measurements. A value of λ ~ .2 nm is obtained. Assuming λ to be temperature independent delivers from BDS measurements the molecular diffusion coefficient D.Furthermore the numberdensity n(T) and the mobility μ(T) can be separated and the BNN-relation is obtained.
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 ion-conducting systems.
2.: Based on dielectric measurements the self-diffusion coefficientof the ionic charge carriers and their temperature dependencecan be deduced.
3.: It is possible to separate the mobility μ(T) and the effective numberdensity n(T). The former has a VFT temperature-dependence, 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 ersich 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.