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.