Bds Tut 10 Schoenhals

Broadband Dielectric Spectroscopy and its Relationship to Other Spectroscopic Techniques

A. Schönhals

BAM Federal Institute for Materials Research and Testing, Unter den Eichen 87, 12205 Berlin

6th International Conference on Broadband Dielectric Spectroscopy and its Applications

Satellite Courses, 6.09.2010 Madrid, Spain

Content

Introduction, Dielectric Spectroscopy

Basics of Relaxation Spectroscopy

Examples of Relaxation Spectroscopy

Thermal spectroscopyNeutron spectroscopy (Scattering)

Application of Relaxation Spectroscopy

Dielectric Relaxation Spectroscopy

Maxwell´s Equations:

Bt

Erotrr

∂∂

−=

Dt

jHrotrrr

∂∂

+= ρ=Ddivr

0Bdiv =r

Electric field Magnetic induction

Magnetic field Dielectric displacement

Current density

Density of charges

Material Equation:

E*D 0rr

εε=

Complex Dielectric Function

Small electrical field strengths:

ε0 – permittivity of vacuum

ε* is time dependent if time dependent processes taking place within the sample

E)1*(DDP 00rrrr

ε−ε=−= Polarization

Dielectric Relaxation Spectroscopy

System under investigation

Input Output

E(t) P(t)ε(t)

Time

ΔE

E(t)

; Inp

ut

Time

PS

P∞

P(t);

Out

put

Relaxational Contribution

Instantaneous response

Simplest Case : Step-like Input The polarization measured as response to a step-like disturbance

A Special Case of Linear Response Theory

EP)t(P)t(

Δ−

=ε ∞

Dielectric Relaxation SpectroscopyA Special Case of Linear Response Theory


Input Output

E(t) P(t)ε(t)

Arbitrary input Approximation of the Input by step-like changes

E(t)

; Inp

ut

Time

t*tE

ΔΔΔ

∑ ΔΔΔ

+≈ t*tE)0(E)t(E

dt*dtdx

'dt'dt

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

0 ∫∞−

∞ −εε+=

Basics of Linear Response Theory

Linearity

Causality


Special input Periodical input ( ) )tsin(EtiexpE)t(E 00 ω=ω−=

)tsin(P)t(P 0 δ+ω=

δPhase shift

Complex Numbers

In-Phase with E(t) 90°-shifted to E(t)

Complex Dielectric Function

ω - angular frequency

E(t)

; P

(t)

Time

)(E]1´´)i´[()(P 0 ω−ε−εε=ω

)(''i)(')(* ωε−ωε=ωε

Complex Compliance


Complex Compliance

Real Part Imaginary or Loss Part

Related to the energy stored reversibly during one cycle

δ

J´

J´´

J*

Complex Plane '''tan

εε

=δ

Phase shift

)(''i)(')(* ωε−ωε=ωε

Related to the energy dissipated during one cycle

ε´´

ε´

ε*

Real PartRelated 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

ξω−ξξε

π=ε−ωε ∫∞ d)(''1)('

ξω−ξε−ξε

π=ωε ∫ ∞ d)('1)(''



'dt'dt

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

0 ∫∞−

∞ −εε+= Linear Equation Can be inverted

Relationship between ε(t) and M(t)


InputOutput

E(t) P(t)M(t)

Electrical Modulus

ε(t)

)t('dt)'t(M)'tt( δ=−ε∫−∞

∞−δ(t) – Dirac function

1)ω(*)(*M =εω

Fourier Transformation


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

>μ<

>τμμ<+>τμμ<=τΦ

∑∑∑∑

<2

ii

i jiji

iii

)(

)()0(2)()0()(>Δ<

>ΔτΔ<=τΦ 2P

)0(P)(P)(

∑= iμV1P rr

Fluctuation Dissipation Theorem

τωττΦπ

>Δ<=Δ ∫

∞

∞−ω d)i(exp)(

2P)P(

22

Spectral DensityMeasure for the frequency distribution of the fluctuations

ε1)τ(ε

Tk1)τ(

Δ−

−=Φ

πω−ωε

=Δ ω1)(''

kT1)P( 2

Links microscopic fluctuations and macroscopic reactions

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

-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

ε''

log ( f [Hz])

Dielectric Spectroscopy…..…PPG, Mw= 4000 g/mol

Time Domain Spectrometer

Frequency Response Analysis

Coaxial Reflectometer

Normal Mode Relaxation

Dynamic Glass Transition, α-relaxation

Dielectric Spectroscopy …..

-4 -2 0 2 4 6 8 10-3.0

-2.5

-2.0

-1.5

-1.0

log

ε''

log (f [Hz])

Alpha-AnalyzerHP4191A

γβ∞ ωτ+εΔ

+ε=ωε))i(1(

)(HN

*HN

Mean Relaxation Rate fp

Dielectric Strength Δε

…PMPS, Mw= 1000 g/mol

3.0 3.5 4.0 4.5 5.0-4

-2

0

2

4

6

8

lo

g(f p [H

z])

1000 / T [K-1]

0pp TT

Aflogflog−

−= ∞

VFT- equation:

?

Linear Response Theory

System under Investigation

Input Output

Disturbancex(t)

Reactiony(t)Material function

Molecular Motions

Electric field E Polarization P

Shear tension σ Shear angle γ

Magnetic field H Magnetization M

Temperature T Entropy S

Basics of Relaxation Spectroscopy


Input Output

Disturbancex(t)

Reactiony(t)

Molecular Motions

Material functionDisturbance 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 Strain Elastic Compliance

Relaxation

Retardation


Input Output

x(t) y(t)

Type ofExperiment

Disturbancex(t)

Responsey(t)

ComplianceJ(t) or J*(ω)

ModulusG(t) or G*(ω)

Dielectric Electric field E

PolarizationP

DielectricSusceptibilityχ*(ω)=(ε*(ω)-1)

DielectricModulus

MechanicalShear

Shear tension σ

Shear angel γ

Shear complianceJ(t)

Shear ModulusG(t)

IsotropicCompression

Pressurep

VolumeV

Volume complianceB(t) V

Compression ModulusK(t)

Magnetic Magnetic fieldH

Magnetization M

Magnetic susceptibilityα*(ω)=(μ*(ω)-1)

-

Heat Temperature T

EntropyS

Heat capacitycp(t)

Temperature ModulusGT(t)


Input Output

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

Specific Heat SpectroscopyA Special Case of Linear Response Theory

FurnaceThermocouples

Tem

pera

ture

Time

Temperature Modulated DSC

)t*sin(At*T)t(T ω+= &

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(ω)= C´p(ω) – i C´´p(ω)

Specific Heat SpectroscopyA Special Case of Linear Response Theory

Example Temperature Modulated DSC

340 360 380 400-0.05

0.00

0.05

0.10

1.50

1.75

2.00

Gaussfit

Tg(q)

Tα(ω)

cp''

cp'total cp

Spe

cific

Hea

t Cap

acit

y in

J g

-1 K

-1

Temperature in K

Polystyrene

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

Due to heating rate

Due to modulation

Glass Transition: Tg = 211 K

Specific Heat Spectroscopy

Example Temperature Modulated DSC Glass forming liquid crystal E7

A Special Case of Linear Response Theory

Mixture of different LC´s

Phase Transition: TN/I = 333 K

-100 -50 0 50 1001

2

3

4

Hea

t Flo

w [m

W]

T [°C]

Phase Transition

Glass Transition

200 205 210 215 2201.0

1.2

1.4

1.6

1.66 mHz 3.33 mHz 6.66 mHz 13.3 mHz 26.6 mHz 53.2 mHz

c p´ [J

g-1K

-1]

T [K]

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

200 205 210 215 220

1.66 mHz 3.33 mHz 6.66 mHz 13.3 mHz 26.6 mHz 53.2 mHz

corr

ecte

d ph

ase

angl

e / r

ad

T [K]

Real Part Loss factorFrequency Frequency

Frequency range : 10-3 Hz …. 5 *10-2 HzHeating and cooling rates must be adjusted to meet stationary conditions !

Specific Heat Spectroscopy A Special Case of Linear Response Theory


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 coolingDifferential setup Increase of sensitivity

Specific Heat Spectroscopy A Special Case of Linear Response Theory

Example AC calorimetry

PMPS, Mw= 1000 g/mol

180 200 220 240 260 280 30070

75

80

85

90

95

T [K]

C' [

au]

0.0

0.5

1.0

1.5

δC

orr [deg]f=240 Hz

Frequency range : 10 Hz …. 5000 Hz

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

PMPS, Mw= 1000 g/mol

Combination ofDielectric Spectroscopy …..

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 [Hz]

)

1000 / T [K-1]

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]

and Specific Heat Spectroscopy

Both methods measure the same process….… Glassy dynamics of PMPS

Dielectric

Specific Heat

AC Chip Calorimetry

TMDSC

-2 0 2 4 6 8 10

-2

-1

0

1

log

ε''

log (f [Hz])

δ-relaxation

Tumbling-mode

Nematic Isotropic

A.R. Bras et al. Phys Rev. E 75 (2007) 61708

Questions:

Temperature dependence of both modes?

Which mode corresponds to glassy dynamics?


Glass forming liquid crystal E7


2.5 3.0 3.5 4.0 4.5 5.0-2

0

2

4

6

8

10

log

(f p [Hz]

)

1000 / T [K-1]

TI/N

2.5 3.0 3.5 4.0 4.5 5.0-2

0

2

4

6

8

10

log

(f p [Hz]

)

1000 / T [K-1]

δ-relaxation

2.5 3.0 3.5 4.0 4.5 5.0-2

0

2

4

6

8

10

log

(f p [Hz]

)

1000 / T [K-1]

Tumbling-mode

Tg

The temperature dependence of the relaxation rates of both the δ- and the tumbling mode is curved vs. 1/T.

Close to Tg the temperature dependencies of both mode seems to collapse.

Derivative analysis



VFT- equation:

0loglog

TTAff pp −

−= ∞

Derivative

( )02/1

2/11log

TTATd

fd p −=⎥⎦

⎤⎢⎣

⎡−

Straight Line 160 180 200 220 240 260 280 300 3200

2

4

6

8

10

{d lo

g (f p [H

z])

/ dT

[K] }

-1/2

T [K]160 180 200 220 240 260 280 300 320

0

2

4

6

8

10

{d lo

g (f p [H

z])

/ dT

[K] }

-1/2

T [K]

δ-relaxation

160 180 200 220 240 260 280 300 3200

2

4

6

8

10

{d lo

g (f p [H

z])

/ dT

[K] }

-1/2

T [K]160 180 200 220 240 260 280 300 320

0

2

4

6

8

10

{d lo

g (f p [H

z])

/ dT

[K] }

-1/2

T [K]

Tumbling-mode

The temperature dependence of the relaxation rates of both the δ- and the tumbling mode follow the VFT-equation.

T0,δ T0,Tumb

Difference in ΔT0 = 30 K !!!!

Similar to the decoupling of rotational and translation diffusion in glass-forming liquids

Similar to the decoupling of segmental and chain dynamics in polymers

Can be explained by different averaging of modes with

different length scales

Which mode is responsible for glassy dynamics?




2.5 3.0 3.5 4.0 4.5 5.0-4

-2

0

2

4

6

8

10

log

(f p [Hz]

)

1000 / T [K-1]

Relaxation Map

TI/N

Tg

δ-relaxation

Tumbling-mode

2.5 3.0 3.5 4.0 4.5 5.0-4

-2

0

2

4

6

8

10

log

(f p [Hz]

)

1000 / T [K-1]

Stars – Thermal data

Derivative Plot

160 180 200 220 240 260 280 300 3200

2

4

6

8

10

{d lo

g (f p [H

z])

/ dT

[K] }

-1/2

T [K]

δ-relaxationTumbling-mode

160 180 200 220 240 260 280 300 3200

2

4

6

8

10

{d lo

g (f p [H

z])

/ dT

[K] }

-1/2

T [K]

Thermal data

The thermal data agree perfectly with that of the tumbling with regard to the absolute values and its temperature dependence Tumbling mode is responsible

for glassy dynamics!A.R. Bras et al. Phys Rev. E 75 (2007) 61708


and Specific Heat Spectroscopy


Neutrons are parts of the atomic nuclei

Isotopes: Elements with the same number of protone but with a different number of neutrons.

Examples: Hydrogen, Deuterium, Tritium

Neutrons

Neutrons can be obtained by nuclear fission processes

Atomic reactor, Neutron spallation source

Have the same mass than protons

Electrically neutral

Have a spin

Neutron Scattering

Theoretical Consideration – Neutron Cross Sections, Correlation Functions

Nuclei

kf

ki ki – wave vector of the incoming neutron

kf – wave vector of the scattered neutron

Scattering vector: q = kf - ki

Momentum transfer: ħq = ħ( kf – ki)

Energy transfer: ħω = ħ2 ( kf2 – ki

2)2θ - scattering angle

Elastic scattering: | kf | ≈ | ki | θλπ

=||= sin4qq0

r

Inelastic scattering: | kf | ≠ | ki | Time dependence

q

Neutron Scattering

Time and spatial information

10 -12

10 -10

10 -8

10 -6

10 -4

10 -2

10 0

10 2

10 4

10 -3 10 -2 10 -1 10 0 10 1 10 2

10 -110 010 110 210 3

10 -3

10 -1

10 1

10 3

10 5

10 7

10 9

10 11

E /m

eV

Q / Å-1

r / Åt / psNMR

inelastic

X-R

Laser PCS X-R

PCS

Raman

Brillouin scatt.

Comparison of Scattering Vector and Energy Ranges of

Different Inelastic Scattering Methods

The intensity of the scattered neutrons is described by a double differential cross section:

[ ]{ }∫ ∑∞

∞−−ω−

π=

ω∂Ω∂σ∂ N

k,jkjkj

i

f2

)0(r)t(rqiexpbb)tiexp(dtkk

21 rrr

hh

Solid angle Position of atom j

at time t

Energy change Position of atom k

at time t =0Scattering lengths

[ ]{ }),q(Sb),q(SbbNk

k 2inc

22

i

f2

ω+ω−=ω∂Ω∂

σ∂ rr

hh

∫ ∑∞

∞−−ω

π=ω

N

jjjinc )]t(rqiexp[)]0(rqiexp[)tiexp(dt

N21),q(S rrr

Self Correlation

∫ ∑∞

∞−−ω

π=ω

N

k,jkj )]0(rqiexp[)]t(rqiexp[)tiexp(dt

N21),q(S rrr

Pair Correlation

[ ]∑⎭⎬⎫

⎩⎨⎧ −−=

N

j

2jjinc )0(r)t(r

6qexp

N1)t,q(S

rr

Time Domain, Incoherent scattering

Scattering lenght:Hydrogen bH = -0.374 * 10-12 cm

Deuterium bD = 0.667 * 10-12 cm

Neutron ScatteringTheoretical Consideration – Neutron Cross Sections, Correlation Functions

Neutron Scattering A Special Case of Linear Response Theory ?


Input Output

x(t)=? y(t)=?J(t)=?

[ ]∑⎭⎬⎫

⎩⎨⎧ −−=

N

j

2jjinc )0(r)t(r

6qexp

N1)t,q(S

rr

van Hove Correlation function Density correlation function

Thermodynamics (t=0):

>ρρ=< )q()t,q()t,q(Sincrrr

W. Götze in Hansen et al. "Liquids, Freezing and the Glass Transition", North Holland 1990

Correlation function Fluctuation Dissipation Theorem Modulus, Compliance ?

NTk)q(S)q( =κ Isothermal compressibility

S(q,t) is related to a q-dependent Compression Modulus

IN 16

IN 6

Time of Flight - IN 6 / ILL

Energy scale: meV Time scale: 0.2 .. 20 ps

Backscattering - IN 16 / ILL and BSS Jülich

Energy scale: μeV Time scale: 0.1 .. 4 ns

Neutron Scattering

Neutron Scattering…..… Examples

IN6

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

S Inc(ω

, 54°

) [a.

u.]

Energy Transfer [meV]

Resolution of the spectrometer

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

S Inc(ω

, 54°

) [a.

u.]


PMPS 212 KQuasielastic Scattering

-4 -2 0 2 4

0.00

0.02

0.04

0.06

0.08

0.10

S Inc(ω

, 54°

) [a.

u.]


PMPS 212 K

Spectral Shape of IN16 data similar but energy resolution μeV

Fourier Transformation of each data set

Comparison in time domain

0 1 2 3 40.5

0.6

0.7

0.8

0.9

1.0

PMPS - Bulk

1.81 Å-1

1.64 Å-1

1.42 Å-1

1.16 Å-1

0.87 Å-1

0.54 Å-1

Sin

c(Q,t)

log(t [ps])

T=120 K

IN 6 IN 16

incoherent intermediate scattering function

Sinc(Q,t)

Broadband Neutron Spectroscopy

Overview of the molecular Dynamics of PMPS…… by elastics Scans

Elastic scans on a backscattering instrument give an efficient overview over the temperature dependent dynamics.

Mean square displacement ueff:

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ><−=

3uQexpI/I eff

220el

Iel – elastically scattered intensity

I0 – totally scattered intensity

0 50 100 150 200

0.0

0.1

0.2

0.3

0.4

0.5

<u2 > ef

f [Å2 ]

T [K]

Harmonic Vibrations

Increase of Mobility

Methyl Groups

0 100 200 300

0

1

2

3

<u2 > ef

f [Å2 ]

T [K]

Methyl Groups

Glass Transition

Analysis Methyl group rotation

Analysis Glass transition

3 protons in the CH3 -group

5 protons in the phenyl ring

8 protons3/8 of the scattering is due to localized rotation of the

CH3 -group

Analysis of the CH3 – group rotation !

0 1 2 3 40.5

0.6

0.7

0.8

0.9

1.0

PMPS - Bulk

1.81 Å-1

1.64 Å-1

1.42 Å-1

1.16 Å-1

0.87 Å-1

0.54 Å-1

Sin

c(Q,t)

log(t [ps])

T=120 K

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

τ−−=

β

Atexp)A1(*DWF)t,Q(S3CH

3CHInc

Debye-Waller-factor

Elastic incoherent structure factor (EISF)

KWW-function

β=0.7 !

Data Analysis Methyl Groups PMPS

IN 6 IN 16

Above Tg the scattering results from the

Fast Processes (DWF)

CH3 –group rotation

Segmental relaxation

-1 0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

S

inc (Q

,t)

log (t [ps])

PMPS Bulk

T=250 KQ=1.16 Å-1

Segmental

CH3

DWF

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛τ

−+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

τ−−=

ββ seg3CH

3 segCHInc

texpAtexp)A1(*DWF)t,Q(S

From CH3-Analysis FromDielectric

-1 0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Contribution segmental

S

inc (Q

,t)

log (t [ps])

PMPS Bulk

T=250 KQ=1.16 Å-1

Segmental

CH3

DWF

Contribution CH3 -group

⎟⎠

⎞⎜⎝

⎛β

Γβ

τ=>τ<

1KWW

Comparison with other experiments

Data Analysis Segmental Relaxation PMPS

Segmental

Thermal

Neutrons

Dielectric

Overall Molecular Dynamics PMPS

2 3 4 5 6 7 8 9-4

-2

0

2

4

6

8

10

12

log(

f p [Hz]

)

1000 / T [K-1]

2 3 4 5 6 7 8 9-4

-2

0

2

4

6

8

10

12

log(

f p [Hz]

)

1000 / T [K-1]

2 3 4 5 6 7 8 9-4

-2

0

2

4

6

8

10

12

log(

f p [Hz]

)

1000 / T [K-1]

Q

2 3 4 5 6 7 8 9-4

-2

0

2

4

6

8

10

12

log(

f p [Hz]

)

1000 / T [K-1]

Q

CH3-Rotation

Precise determination of glassy dynamics in a wide temperature range using different probes

Analysis of the Temperature dependence…

3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8-4

-2

0

2

4

6

8

10

lo

g (f p [

Hz]

)

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

lo

g (f p [H

z])

1000 / T [K-1]

Dielectric data…VFT- equation:

0loglog

TTAff pp −

−= ∞

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

Derivative

( )02/1

2/11log

TTATd

fd p −=⎥⎦

⎤⎢⎣

⎡−

Straight Line, T0>0

Analysis of the Temperature dependence…

160 180 200 220 240 260 280 3000

2

4

6

8

{ log

(fp [H

z])

/ dT

[K] }

-1/2

T [K]160 180 200 220 240 260 280 300

0

2

4

6

8

{ log

(fp [

Hz]

) / d

T [K

] }-1

/2

T [K]

Derivative-Plot

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 3000

2

4

6

8

{ log

(fp [

Hz]

) / d

T [K

] }-1

/2

T [K]

Thermal and dielectric data have the same temperature dependence

red : dielectric data

blue: thermal data

Temperature dependence……of the dielectric relaxation strength

VN

Tkg

31

B

2

0

με

=εΔ

Dipole moment

Debye Theory

Interaction parameter

3.2 3.6 4.0 4.4 4.8

20

40

60

80

T*Δε

1000 / T [K-1]

3.2 3.6 4.0 4.4 4.8

20

40

60

80

T*Δε

1000 / T [K-1]

TB

Temperature dependence of εis much stronger than predicted

Cooperative effects

Change of the temperature dependence of ε at TB

Temperature dependence… …of the dielectric relaxation strength

The change in the temperature dependence of ε is not much pronounced

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

Plot of ε vs. log fp

-2 0 2 4 6 8 10 12

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Δε

log (fp [Hz])

TBSharp transition fro a low temperature to a high temperature branch

The crossover frequency coresponds to TB

Segmental

Thermal

Neutrons

Dielectric


2 3 4 5 6 7 8 9-4

-2

0

2

4

6

8

10

12

log(

f p [Hz]

)

1000 / T [K-1]

Q

CH3-Rotation

Dynamics PMPS: Q-Dependence Segmental Relaxation

-0.4 -0.2 0.0 0.2 0.40

1

2

3

4

5

T=250 K T=270 K T=300 K T=330 K

log

(τKW

W [p

s])

log (Q [Å-1])

Bulk

240 260 280 300 320 340 3601

2

3

4

5

- Slo

pe

T [K]

2 / βKWW

2

Theory :

⎥⎦⎤

⎢⎣⎡

⎟⎠⎞

⎜⎝⎛

τ−−=−=

texp)DtQexp()t,Q(S 2inc 2Q~ −τ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

τ−−=−=

ββ texp)AtQexp()t,Q(S 2

inc β−τ /2Q~

heterogeneous

homogeneous

Segmental

Thermal

Neutrons

Dielectric


2 3 4 5 6 7 8 9-4

-2

0

2

4

6

8

10

12

log(

f p [Hz]

)

1000 / T [K-1]

Q

CH3-Rotation

Temperature dependence

Arrhenius-like

EA=8.3 kJ/mol

In agreement with values found for other polymers

2 3 4 5 6 7 8 9-4

-2

0

2

4

6

8

10

12

log(

f p [Hz]

)

1000 / T [K-1]

Methyl Group Rotation in PMPS

Threefold Potential

3/8 of the scattering is due to localized rotation of the CH3 -group

fixfixApp c)Q(A)c1()Q(A +−=

Structure: cfix=5/8=0.624

0.0 0.5 1.0 1.5 2.00.4

0.6

0.8

1.0

EIS

F Met

hyl g

roup

s

Q [Å-1]0.0 0.5 1.0 1.5 2.0

0.4

0.6

0.8

1.0

EIS

F Met

hyl g

roup

s

Q [Å-1]0.0 0.5 1.0 1.5 2.0

0.4

0.6

0.8

1.0

EIS

F Met

hyl g

roup

s

Q [Å-1]

Experiment: cfix=0.55

⎟⎟⎠

⎞⎜⎜⎝

⎛+=− rQ3

)rQ3sin(2131)Q(A jump3

0.0 0.5 1.0 1.5 2.00.4

0.6

0.8

1.0

EIS

F Met

hyl g

roup

s

Q [Å-1]

Methyl Group Rotation in PMPS …

…Q - dependence of the Relaxation Time

-0.4 -0.2 0.0 0.2 0.4

1.2

1.6

2.0

2.4

log

(τC

H3 [

ps])

log(Q [Å-1])

Below Tg the relaxation times are independent of Q

T=120 K

-0.4 -0.2 0.0 0.2 0.4

1.2

1.6

2.0

2.4

log

(τC

H3 [

ps])

log(Q [Å-1])

T=212 K = Tg

Slope: - 1.5

At Tg the relaxation times become Q dependent

… Onset of segmental dynamics

Methyl Group Rotation in PMPS …

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.5

0.6

0.7

0.8

0.9

1.0

EIS

F

Q [Å-1]

…Temperature dependence of the EISF

120 K170 K

< Tg

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.5

0.6

0.7

0.8

0.9

1.0

EIS

F

Q [Å-1]

212 K = Tg

Decrease of the EISF above Tg

100 150 200 250 300 3500.3

0.4

0.5

0.6

0.7

0.8

TB

EIS

F Met

hyl g

roup

s

T [K]

Tg

Additional mobility above Tg

Dynamical Heterogeneities

Islands of Mobility

Higher quasielastic scatteringQ=1.81 Å-1

Phenylring flip ???

Conclusion….

Thanks to the ILL and Research Center Jülich for enabling the neutron experiments.

Precise Determination of the molecular dynamics of PMPS over 17 orders of magnitude in time

α-Relaxation (glassy dynamics)The temperature dependence of the relaxation times shows a crossover from a low to a high temperature range.

Crossover temperature TB= 250K.

In both ranges the temperature dependence of the relaxation rates is VFT-like with different Vogel temperatures

Also the dielectric strength shows a crossover behavior at the same TB

The dependence of the relaxation times on the momentum transfer indicates that the molecular dynamics is heterogeneous even above TB

Methyl group Rotation

The methyl group rotation can be described by jumps in a threefold potential considering the elastic scattering of the frozen chain.

Above Tg the EISF becomes temperature dependent and the relaxation times depends on momentum transfer .

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