broadband dielectric spectroscopy - basics and...

BROADBAND DIELECTRIC SPECTROSCOPY -

BASICS AND SELECTED APPLICATIONS

Andreas Schönhals 9th International Conference on Broadband Dielectric Spectroscopy and its Applications September 11-16, 2016 in Pisa (ITALY)

11.09.2016

Table of Content

Basics of Dielectric Spectroscopy

Electrostatics

Dynamics

Analysis

Scaling of the dynamic glass transition (-relaxation)

Chain dynamics – Normal mode

Conclusion

11.09.2016 Broadband Dielectric Spectroscopy – Basics and Applications 2

Dielectric Spectroscopy


Maxwell´s Equations:

Bt

Erot

Electric field Magnetic induction

0Bdiv

Dt

jHrot

Magnetic field

Current density

Dielectric displacement

Ddiv

Density of charges

Material Equation:

Small electrical field strengths:

E*D 0

E)1*(DDP 00

Polarization

Complex Dielectric Function * is time dependent if time dependent processes taking place within the sample

0 – permittivity of vacuum



System under investigation

Input Output

E(t) P(t) (t)

Time

E

E(t

); I

nput

Time

PS

P

P(t

); O

utp

ut Relaxational Contribution

Instantaneous Response

Simplest Case : Step-like Input E

P)t(P)t(




Input Output

E(t) P(t) (t)

Arbitrary input Approximation of the Input by step-like changes

E(t

); I

np

ut

Time

t*t

E

t*

t

E)0(E)t(E

dt*dt

dx

'dt'dt

)'t(Ed)'tt(P)t(P

t

0

Basics of Linear Response Theory

Linearity

Causality




Input Output

E(t) P(t) (t)

E(t

) ;

P(t

)

Time

Phase shift

)tsin(EtiexpE)t(E 00

)tsin(P)t(P 0

- Angular Frequency

Complex Numbers

)(''i)(')(* Complex Dielectric Function


Phase angle

14.10.2015 Molecular Dynamics of plasma polymerized layers by relaxation spectroscopy 7


Input Output

E(t) P(t)

)(''i)(')(* Complex Dielectric Function

Real Part (in Phase) Imaginary or Loss Part

(90 ° Phase shifted)

´

´´

*

Complex Plane '

''tan


Kramers-Kronig relationship


Real Part Related to the energy stored reversibly during one cycle

Loss Part Related to the energy dissipated during one cycle

Law of energy conservation

Relationship between both quantities

Kramers – Kronig Relationships

d

)(''1)('

d)('1

)(''


Modulus


'dt'dt

)'t(Ed)'tt(P)t(P

t

0

Linear Equation

Relationship between (t) and M(t)


Input Output

E(t) P(t)

M(t)

Electrical Modulus

(t)

)t('dt)'t(M)'tt(

(t) – Dirac function

1)ω(*)(*M

Fourier Transformation

Can be inverted


Molecular quantities


Thermodynamic quantities are average values. Because of the thermal movement of the molecules these quantities fluctuate around their mean values.

Correlation function of Polarization Fluctuations

2

i

i

i ji

ji

i

ii

)(

)()0(2)()0(

)(

2P

)0(P)(P)(

iμV

1P

Fluctuation Dissipation Theorem

d)i(exp)(2

P)P(

22

Spectral Density

Measure for the frequency distribution of the fluctuations

ε

1)τ(ε

Tk

1)τ(

1)(''

kT

1)P( 2

Links microscopic fluctuations and macroscopic reactions

For a small (linear) disturbance a system reacts only in the way as it fluctuates


Propylene glycol


-4 0 4 8

0

20

40

60

80

-4 -2 0 2 4 6 8 10

0

10

20

30

40

''

'

313 K258.5 K215 K185.4 K168.5 K

log ( f [Hz])

Time Domain

Fourier Correlation Analysis

Bridges

Coaxial Reflectometer

Network Analysis

A. Schönhals, F. Kremer, E. Schlosser, Phys. Rev. Lett. 67, 999 (1991)

Linear Response Theory


System under Investigation

Input Output

Disturbance x(t)

Reaction y(t)

Material function

Molecular Motions

Electric field E Polarization P

Shear tension Shear angle

Magnetic field H Magnetization M

Temperature T Entropy S



System under Investigation

Input Output

Disturbance x(t)

Reaction y(t)

Molecular Motions

Material function Disturbance Reaction

Intensive - independent of V Extensive ~ V Generalized Modulus G(t)

Tensile Strain Strain Elastic Modulus

Extensive ~ V Intensive - independent of V Generalized Compliance J(t)

Strain Tensile Stress Elastic Compliance

Relaxation

Retardation



Type of Experiment

Disturbance x(t)

Response y(t)

Compliance J(t) or J*()

Modulus G(t) or G*()

Dielectric Electric field E

Polarization P

Dielectric Susceptibility *()=(*()-1)

Dielectric Modulus

Mechanical Shear

Shear tension

Shear angel

Shear compliance J(t)

Shear Modulus G(t)

Isotropic Compression

Pressure p

Volume V

Volume compliance B(t) V

Compression Modulus K(t)

Magnetic Magnetic field H

Magnetization M

Magnetic susceptibility *()=(*()-1)

-

Heat Temperature T

Entropy S

Heat capacity cp(t)

Temperature Modulus GT(t)


Input Output

x(t) y(t)


Information of a measurement


log ( /p)

log

''

p = 2 f

p=1/

S

= S -

'

Three essential measurement information

Frequency position fp

Dielectric relaxation strength

Shape of the relaxation time spectra




Time

S

(t

)=P

/E

Induced dipoles

Orientational polarization

Capacitor with N permanent Dipoles, Dipole Moment

Polarization :

PμV

NPμ

V

1P

i

Mean Dipole Moment

Mean Dipole Moment: Counterbalance

EWel

kTWth

Thermal Energy Electrical Energy

4

4

d)kT

Eexp(

d)kT

E(exp

The factor gives the probability that the dipole moment vector has an orientation between and + d.




Spherical Coordinates:

Only the dipole moment component which is parallel to the direction of the electric field contributes to the polarization

dsin2

1)

kT

cosE(exp

dsin2

1)

kT

cosE(expcos

0

0

x = ( E cos ) / (kT) a = ( E) / (kT)

)a(a

1

)aexp()aexp(

)aexp()aexp(

dx)xexp(

dx)xexp(x

a

1cos

a

a

a

a

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0(a)=a/3

Langevin function

a

(a

) 1.0kT

Ea

Tk1.0E

Langevin function

(a)a/3 ETk3

2

V

N

Tk3

1 2

0S

Debye-Formula




Shielding effects:

E

eff

= + Eloc

Polar Molecules; Onsager – Theory, Reactions Field

V

N

TkF

3

1 2

0S

)2(3

)2(F

S

2S

Onsager Factor F= 2 … 3

eff

Interaction effects:

inter

2

2

.Interact

2

jiji

i

N1g

V

N

TkgF

3

1 2

0

S

z nearest neighbors: cosz1g

Kirkwood / Fröhlich Correlation Factor g= 0.5 … 5


Dynamics - Debye model


2

ii

i jiji

iii

)(

)t()0(2)t()0(

)t(Dynamics is due to the dipole – dipole correlation function

Debye model: Change of the polarization is proportional to the actual value )t(P1

td

)t(Pd

D

D

texp)t(

Time domain Frequency domain

Di1)(*

2

D )(1'

2

D

D

)(1''

Rotational Diffusion Model Double Potential Model

)t(P2nn2dt

dn

dt

dn

dt

Pd21

21

A rigid isolated fluctuating dipole is regarded in viscous media under the influence of a stochastic force

)]t,r(fkT

)t(E)t,r(f[D

t

)t,r(fRot

)tD)1m(m(exp))t((cosP))0((cosP Rotmm


Model functions


In most cases the half width of measured loss peaks is much broader than predicted by Debye equation and in addition their shapes are asymmetric.

Di1)(*

Debye function:

Cole/Cole function:

)i(1)(

CC

*

CC 0<1

symmetric

symmetric broadening

Cole/Davidson function:

)i1()(

CD

*

CD 0<1 asymmetric broadening

Havriliak/Negami function:

))i(1()(

HN

*

HN 0<; 1 asymmetric and symmetric broadening


Data Analysis


-2 0 2 4 6-2

-1

0

'' = 0 / (2 f

0)

-1

log

''

log (f [Hz])

min)f(**w2

iModelii

i

PVAC

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1819

2021

22

23

24

25

2627

28

2930

31

3233

34

35

C o

l e

- D

a v

i d

s o

n

K W W

C o l e - C o l e

D e b y e

Most general function is the formula of Havriliak / Negmai

Overview about the dynamics of

polymers


Polymers are complex systems For an isolated macromolecule a large number of atoms (100 to 1000 000) are covalently bonded.

Almost unlimited number of conformations in space and time.

Most properties depend on this large conformation.

Chain flexibility

Mean End-to-End vector (shape)

Mean Square dipole moment

… and many more

Behavior is complex ….

Lo

g (

Sh

ea

r M

od

ulu

s G

)

Temperature

glass-like behavior

viscoelastic

behavior liquid-like

behavior

glass transition G ~109

Pa

G ~106

Pa flow transition

Tg

Shear modulus vs. Temperature

… and related to the molecular mobility

Molecular dynamic in polymers


CH

CH

CH

3

2

3

C

OO

Local

CCH

2

Seg Chain

r

Local Motion

Localized bond rotations

Fluctuations in a side group

LOCAL < 1 nm

Segmental Motion

SEG ~ 1 .. 2 nm

Chain Motion

Chain < = r

“Glassy State” Glass Transition, -Relaxation

Flow Transition


Scaling of segmental dynamics

Dynamic glass transition

Dynamics in glass-forming materials


Thermal Response

Dynamic Response

-relaxation, dynamic glass transition

Slow-relaxation

Fast-relaxation

Boson-peak

Activation Diagram

Aexp

ET

kTSlow-relaxation

0

0

1( ) exp

2 ( )

DTT v

T T T

-relaxation

15 orders of magnitude

The temperature dependence of the dynamical relaxation times correlates with the glass transition temperature

at ca. 102 s.

Dynamics in glass-forming materials


Crossover Phenomena form a high to a low temperature regime at TB ~ 1.2 .. 1.3 Tg

Non-Debye behavior of the Relaxation function

Formation of dynamic heterogeneities below TB

t

exp~ KWW-function

Size : few nanometers


Poly(methyl phenyl siloxane) (PMPS) ; Mw=1000 g/mol


-4 -2 0 2 4 6 8 10

-3.0

-2.5

-2.0

-1.5

-1.0

log

''

log (f [Hz])

Alpha-Analyzer

HP4191A

?


Poly(methyl phenyl siloxane) (PMPS) ; Mw=1000 g/mol


Mean Relaxation Rate fp

Dielectric Strength 3.0 3.5 4.0 4.5 5.0

-4

-2

0

2

4

6

8

log(f

p [

Hz])

1000 / T [K-1]

-2 0 2 4 6-2.5

-2.0

-1.5

-1.0

-0.5 161.7 K155.7 K

log

''

log (f [Hz])

152. 7 K

))i(1()(

HN

*HN

VFT- equation:

0pp

TT

Aflogflog

?

Specific Heat Spectroscopy



Input Output

T(t) S(t) cp(t)

Furnace

Thermocouples

Te

mp

era

ture

Time

Temperature Modulated DSC

)t*sin(At*T)t(T

Ch. Schick, Temperature Modulated Differential Scanning Calorimetry - Basics and Applications to Polymers in Handbook of Thermal Analysis and Calorimetry edited by S. Cheng (Elsevier, p 713. Vol. 3, 2002).

The modulated temperature results in a modulated heat flow

The heat flow is phase shifted if time dependent processes take place in the sample

c*p()= c´p() – i c´´p()


Example - Polystyrene


340 360 380 400-0.05

0.00

0.05

0.10

1.50

1.75

2.00

Gaussfit

Tg(q)

T( )

cp''

cp'

total cp

Sp

ec

ific

He

at

Ca

pa

cit

y i

n J

g -1

K -1

Temperature in K

Hensel, A., and Schick, C.: J. Non-Cryst. Solids, 235-237 (1998) 510-516.

Due to heating rate

Due to modulation



System under

investigation

Input Output

T(t) S(t) cp(t)

AC Chip calorimetry Pressure gauge, Xsensor Integration

10 mm 2 mm 0.1 mm

H. Huth, A. Minakov, Ch. Schick, Polym. Sci. B Polym. Phys. 44, 2996 (2006).

Fast heating and cooling




Input Output

T(t) S(t) cp(t)

H. Huth, A. Minakov, Ch. Schick, Polym. Sci. B Polym. Phys. 44, 2996 (2006).

Differential setup

U*

Increase of sensitivity

Complex differential Voltage


Example - PMPS


180 200 220 240 260 280 300

70

75

80

85

90

95

T [K]

UR [V

]

0.0

0.5

1.0

1.5

C

orr [d

eg

]

f=240 Hz

Frequency range : 10 Hz …. 5000 Hz

A. Schönhals et al. J. Non-Cryst. Solids 353 (2007) 3853

3.0 3.5 4.0 4.5 5.0

-4

-2

0

2

4

6

8

log

(f p

[H

z])

1000 / T [K-1]

Dielectric

3.0 3.5 4.0 4.5 5.0

-4

-2

0

2

4

6

8

log

(f p

[H

z])

1000 / T [K-1]

Specific Heat

AC Chip Calorimetry

TMDSC

Both methods measure the same process….

… Glassy dynamics of PMPS

Analysis of the temperature

dependence of the relaxation rate


3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8

-4

-2

0

2

4

6

8

10

log

(f p

[H

z])

1000 / T [K-1]

3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8

-4

-2

0

2

4

6

8

10

log

(f p

[H

z])

1000 / T [K-1]

VFT- equation:

… cannot be described by the VFT-equation in the whole temperature range

0

loglogTT

Aff pp

Derivative: 02/1

2/11log

TTATd

fd p

Straight Line, T0>0


dependence of the relaxation rate


160 180 200 220 240 260 280 300

0

2

4

6

8

{ lo

g (

f p [

Hz])

/

dT

[K

] }-1

/2

T [K]

160 180 200 220 240 260 280 300

0

2

4

6

8

{ lo

g (

f p [

Hz])

/

dT

[K

] }-1

/2

T [K]

low and high temperature regime

two VFT - dependencies

T0,Low

T0,high Crossover temperature

TB = 250 K

TB

160 180 200 220 240 260 280 300

0

2

4

6

8

{ lo

g (

f p [

Hz])

/

dT

[K

] }-1

/2

T [K]

Thermal and dielectric data have the same temperature dependence

red : dielectric data

blue: thermal data


dependence of the dielectric strength


V

N

Tkg

3

1

B

2

0

Dipole moment

Debye Theory:

Interaction parameter

3.2 3.6 4.0 4.4 4.8

20

40

60

80

T*

1000 / T [K-1]

3.2 3.6 4.0 4.4 4.8

20

40

60

80

T*

1000 / T [K-1]

TB

Temperature dependence of is much stronger than predicted

Cooperative effects

Change of the temperature dependence of at TB


dependence of the dielectric strength


The change in the temperature dependence of is not much pronounced

A. Schönhals Euro. Phys. Lett. 56 (2001) 815

Plot of vs. log fp

-2 0 2 4 6 8 10 12

0.1

0.2

0.3

0.4

log (fp [Hz])

TB Sharp transition from a low temperature to a high temperature branch

The crossover frequency corresponds to TB


Chain dynamics

Normal mode

Dipole moments of polymers


rrrrr

Type A Type B Type C

Dipole Moment “Parallel” to the Chain

Dipole Moment “Perpendicular” to the Chain

Dipole Moment in a Side Group

Poly(cis-1,4-isoprene)

Poly(propylene glycol)

Polystyrene

Poly(vinyl chloride)

Poly(methyl phenyl siloxane)

Poly(methyl methacarylate)

Segmental Dynamics Chain Dynamics


Polypropylene glycol; Mw=4000 g/mol


-4 -2 0 2 4 6 8 10

-1.5

-1.0

-0.5

0.0

0.5

303.3 K

273.2 K

253.3 K233.7 K332.8 K204.5 K

log

''

log ( f [Hz])Time Domain Spectrometer Frequency Response

Analysis

Coaxial Reflectometer

Dynamic Glass Transition, - relaxation

Normal Mode Relaxation

Data analysis


-4 -2 0 2 4 6

-2.0

-1.5

-1.0

-0.5

0.0

0.5233.7 K204.5 K

log

''

log ( f [Hz] )

The two model function of Havriliak and Negami are fitted to the data.

Relaxation rate fp

Dielectric strength

fp,

fp,n

… for each process

Theory of chain dynamics

Rouse motion, Reptation


r

A

A

A

A

A

A

N-1

N-2

0

1

2

3

l

l

l

lN1

2

3

lN-1

Fig. 7.2

Beads

Springs

Monomeric friction

xi

dxi

dt+ 3 kT 2xi − xi−1 − xi+1 = 0

Rouse Model:

Friction Chain connectivity (Entropy springs)

Solution:

) t / ( exp p

1

8 =

> r <

> ) t ( r ) 0 (r <)t( p2

N

1p22

Short chains:

Long chains (Entanglements):

3... 1, = p , M~p Tk 3

b N = 2

22

22

p

3

222

43

p M~p a kT

b N =

Molecular weight dependence

Temperature dependence of the relaxation rates of segmental and chain dynamics should be similar

Predictions:


Poly(cis-1,4 isoprene)


200 250 300 3500.00

0.01

0.02

0.03

0.04

200 250 300 3500.00

0.01

0.02

0.03

0.04

200 250 300 3500.00

0.01

0.02

0.03

0.04

H

T [K]

H

´´

f=1000 Hz

MW 1400 3830 8400 g/mol

Schönhals, A. Macromolecules 26 (1993) 1309


Poly(cis-1,4 isoprene)


3.0 3.5 4.0 4.5 5.0

-8

-6

-4

-2

0

Anstieg = 2 ( Rouse )

Anstieg = 3.7

log

( n

[s])

log (MW

[g/mol])

Reptation

T=320 K

M~p Tk 3

b N = 2

22

22

p

3

222

43

p M~p a kT

b N =

Critical molecular weight for the formation of entanglements


Self diffusion


p kT 3

b N =

22

22

p

Relaxation times:

Diffusion coefficient: N

kTDSelf

p

22

2

Selfp3

rD

2.5 3.0 3.5 4.0-15

-14

-13

-12

-11

log

(D

Se

lf [

m2/s

])

1000 K / T

Blue - dielectric

Red - NMR

Poly(propylene glycol) MW=4000 g/mol DSelf can be measured by

Pulsed Field Gradient NMR

Both quantities agree in both there absolute values and in there temperature dependence

Normal mode corresponds to translatorial diffusion

Appel, M.; Fleischer, J. Kärger, G.; Chang, I.; Fujara, F.; Schönhals, A.; Colloid and Polymer Sci. 275 (1997) 187

Chain dynamics

Temperature dependence relaxation rates


2.0 2.5 3.0 3.5 4.0 4.5 5.0

-4

-2

0

2

4

6

8

10

2 3 4 5

-4

0

4

8

Blue: PPG2000

Green: PPG3000

Red: PPG4000

Black: PPG5300

log (

f p [

Hz])

1000 K / T

log (

f p [H

z])

1000 K / T

Segmental dynamics

Chain dynamics

The relaxation rates of the segmental dynamics are independent of the molecular weight.

The relaxation rates of the chain dynamics depends of the molecular weight.

The temperature dependence of both relaxation rates seems to follow the Vogel/Fulcher/Tammann formula.

0

loglogTT

Aff pp

Both processes seems to merge close to Tg.

Chain dynamics

Temperature dependence relaxation rates


200 225 250 275 300 3251.0

1.5

2.0

2.5

3.0

3.5

PPG2000

PPG3000

PPG4000

log

(f p

/

fp

n)

T [K]

Detailed investigation of the temperature dependence of the relaxation rates!

Ratio of both relaxation rates Plateau

Decrease


Crossover of segmental dynamics


-2 0 2 4 6 8 10 12 140

1

2

3

4

log (f [Hz])

Polypropylene glycol; Mw=4000 g/mol

A low temperature branch

A high temperature branch

TB =240 K

=0

150 200 250 300 350 4000

2

4

6

8

244 K

(d lo

g f

p / d

T)-1

/2

T [K]



150 200 250 300 350 400

0

2

4

6

8

10

(lo

g f

pN

/ d

T)

-1/2

T [K]

02/1

2/11log

TTATd

fd p

Derivative:

The temperature dependence of the -relaxation rate shows a change at T=244 K as it was observed also for the dielectric strengths

No change of the temperature dependence of the relaxation rate of the normal mode at TB.

The reason for the different temperature dependencies of - and normal mode relaxation is the crossover in the dynamic glass transition.

Decoupling of rotational and

translatorial diffusion


Segmental dynamics – rotational diffusion

Chain dynamics – translatorial diffusion

3.0 3.5 4.0 4.5 5.0 5.5

1.0

1.5

2.0

2.5

3.0

3.5 PPG2000

PPG3000

PPG4000

log (

f p /

fp

n)

~ l

og

(DR

ot/D

Tra

ns)

T [K]

Different temperature dependencies of - and normal mode relaxation is due to the decoupling of rotational

and translatorial diffusion at TB.

Decoupling of Rotation and translation in low molecular weight glass-forming liquids

Rössler, E.; Phys. Rev. Lett. 69 (1992) 1620, 65 (1990) 1595

Can be explained by different averaging of modes with different length scales

Conclusion


Dielectric spectroscopy is a powerful tool to study the structure and dynamics of matter.

Extremely broad dynamical range.

Extremely high sensitivity.

Careful data analysis is needed.

This can be done by model functions.

Employing theoretical approaches.

Combination with other spectroscopic methods.

Mechanical spectroscopy

Thermal spectroscopy

Neutron spectroscopy

NMR spectroscopy

broadband dielectric spectroscopy - basics and...

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