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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
System under investigation
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
Dielectric Relaxation SpectroscopyA Special Case of Linear Response Theory
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
Dielectric Relaxation SpectroscopyA Special Case of Linear Response Theory
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)(''
Dielectric Relaxation SpectroscopyA Special Case of Linear Response Theory
Dielectric Relaxation SpectroscopyA Special Case of Linear Response Theory
'dt'dt
)'t(Ed)'tt(P)t(Pt
0 ∫∞−
∞ −εε+= Linear Equation Can be inverted
Relationship between ε(t) and M(t)
System under investigation
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
Dielectric Relaxation SpectroscopyA Special Case of Linear Response Theory
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
System under Investigation
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
System under Investigation
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)
System under investigation
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
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 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?
Combination ofDielectric Spectroscopy …..
Glass forming liquid crystal E7
A.R. Bras et al. Phys Rev. E 75 (2007) 61708
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
Combination ofDielectric Spectroscopy …..
Glass forming liquid crystal E7
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?
A.R. Bras et al. Phys Rev. E 75 (2007) 61708
Combination ofDielectric Spectroscopy …..
Glass forming liquid crystal E7
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
Combination ofDielectric Spectroscopy …..
and Specific Heat Spectroscopy
Glass forming liquid crystal E7
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 ?
System under Investigation
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.]
Energy Transfer [meV]
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.]
Energy Transfer [meV]
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
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]
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
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]
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 .