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Analysis and modelling of dielectric relaxation data using DCALC
Michael Wübbenhorst
Short course, Lodz, 8 October 2008
Part I – dielectric theory, representation of data…
Schedule
1st session (8 October 2008):1) A brief introduction into DRS including dielectric relaxation theory, presentation of dielectric
relaxation data in f- and T-domain, empirical relaxation functions and their temperature dependence (~ 45 min).( )
2) Presentation of the internal structure of DCALC, import/export of data, 2D and 3D graphical presentation capabilities and quick data analysis, example: PVAc.
2nd session (9 October 2008):2 session (9 October 2008):3) General fit-functions in the frequency domain, Fit-functions and options in DCALC, demonstration
case PVAc or P2VP 4) "Hands-on" session (Please bring your own laptop with you and have DCALC installed and tested in
advance)advance)
3rd session (date t.b.a.):5) Advanced numerical algorithm of DCALC: 1D and 2D derivatives, KK-transform, conductivity5) Advanced numerical algorithm of DCALC: 1D and 2D derivatives, KK transform, conductivity
analysis6) Fits using ε“der and fitting in temperature domain; 2D-fits and temperature depending HN-
parameters
Laboratorium voor Akoestiek en Thermische Fysica
Michael Wübbenhorst
Outline
• Electrostatic relations
• Polarization microscopic approach• Polarization – microscopic approach
• From macroscopic to macroscopic quantities
Th l itti it th f d i i t• The complex permittivity, the f-domain experiment
• Relaxation time (spectra)
• Empiric dielectric functions
• Temperature dependence of the dielectric properties
• Experimental implementation if DRS
• Representation of dielectric relaxation data
Laboratorium voor Akoestiek en Thermische Fysica
Michael Wübbenhorst
Dielectric response
Electrostatic relations on a parallel-plate capacitor:
Electrical field E between plates pin vacuum:
Charges +Q and Q on surface:
dVE =
Charges +Q and -Q on surface:
Vacuum capacitance C :
0Q EAε=ε0= 8.85×10-12 As/VmVacuum capacitance C0:
VQC =0
0
[ ] ( ) AsC F FaradV
= =
Dielectric constant of the material ε: P : polarisation
0
C Q PAC Q
ε += =
EPE
0
0
εεε +
=or
P : polarisation
October 9, 2008
Dielectric response (2)
Total electrical polarisation in a dielectric:( ) sPEP +−= 01εε
induced polarisation spontaneous polarisationall dielectric materials with ε > 1
pyroelectricity, piezoelectricityferroelectricity
passive activedielectric properties
further: χel = electrical susceptibility 1−= εχel
October 9, 2008
Polarisation – microscopic approach
molecular polarisability α
LENP α0= N0: concentration of dipolesEL: local electric field
molecular polarisability α
orientationalatomic electronic
Loae ENP )(0 ααα ++=
Electronic and atomic polarisability:
++ _ No fi ld
αaαo αeElectronic and atomic polarisability:
fast displacements of electrons+
++ _
field
-E
fast displacements of electrons and atoms at optical frequencies
(almost) T-independentE property of all nonpolar materials
⇒ refractive index n > 1
October 9, 2008
Polarisation – microscopic approach (2)
DRS – characteristic time scales
αo
αaαeαe
Relaxation phenomena IR VIS/UV
Orientational polarisability
October 9, 2008
Orientational polarisation
No field, E=0: Direction of mobile molecular dipolesarbitrary and driven by thermal energy
m=0 μ
arbitrary and driven by thermal energy kT no average orientation
E>0:
Change of angular distribution due to directing electrical field, energy is mEL E
m μ
Effective (induced) dipole moment m:
E2μkTEm L
3μ
≈ m : molecular dipole moment
October 9, 2008
Orientational polarisation
Langevin function
1( )x x
Lm e e EL x with x μ−+
1.0
( ) Lx xL x with x
e e x kTμ
μ −= = − =−
Practical (low-field) conditions:
0.6
0.8
(x)
saturationE = 1V/50μmT = 300Kμ = 2D
0.2
0.4
L(
( )3 3
Lx EL xkT
μ≈ = !102.3 5−×≈
kTEμ
μ
0 1 2 3 4 5 6 7 8x=æE/kT
0.0 linear approximation holds
October 9, 2008
From microscopic to macroscopic quantities
Loae ENP )(0 ααα ++= local field ≠ external fieldLoae )(0
local field considerations
Clausius-Mosotti relation:
1
local field considerationssee e.g. Blythe, Electrical Properties of Polymers
0
12 3
W AM Nε αε ρ ε
−=
+
MW: molecular weightρ: densityNA: Avogadro’s number
10
From microscopic to macroscopic quantities (2)
Dielectric experiment capacity C
d
dAC 0εε
= Alternating voltage:A
0CC
=ε
tieVtV ω0)( =
Current response of capacitor C= εC0ε
V
0 p p 0
)()()( 00 tVCidt
tdVCtI ωεε ==V
90° phase shift
October 9, 2008
From microscopic to macroscopic quantities (3)
Non-ideal capacitor: polarisation build-up requires finite time
"'* εεε i−=• complex dielectric constant
this leads to 2 components of the current:
imaginary (capacitive) component
VCiIC '0εω= VCIR "0εω=real (resistive) componentimaginary (capacitive) component,
I and V are out of phase real (resistive) component, energy dissipation term
• additional phase lag δ beteen V and I tanδ ε ε′′ ′=
12
Single-Time Relaxation Process
Relaxation of P0 into th l ilib i
thermal agitation
1 0
1.2
thermal equilibrium
( ) exp tP t P ⎛ ⎞⎜ ⎟
0.6
0.8
1.0
P(t)
, E(t)
E(t)
Characteristic time to attain thermal
0( ) expP t Pτ
⎛ ⎞= −⎜ ⎟⎝ ⎠
0 0
0.2
0.4P E(t)
τ=1
equilibrium:
relaxation time τ0 2 4 6 8
time
0.0
13
Single-Time Relaxation Process
Statistical nature of single-exponential relaxation
E=0 0t
m =
E>>00
tm >
time t
thermal energy: rotational diffusion of individual dipoles
time t
time τ involves many small-step rotations
October 9, 2008
Dielectric relaxations
Alternating electrical field at angular frequency ω: complex dielectric “constant”
'( ) ( ) "( )iε ω ε ω ε ω∗ = −( ) ( ) ( )iε ω ε ω ε ω=
All molecules have the same τ ⇒ Debye relaxation
10
12ε'
10 1
10 2
ωτ = 1
'2 2( )
1sε ε
ε ω εω τ
∞∞
−= +
+"
2 2( )1
sε εε ω ωτ
ω τ∞−
=+
0Cε ⋅( )0C ε ε−
RCτ =equivalent circuit:
4
6
8
10 -2
10 -1
10 0ε" (loss)
0Cε∞( )0 sC ε ε∞
R
-4 -2 0 2 4l ( )
0
2
4
10 -4
10 -3
10
15
log(ω)
Relaxation time spectra
( ) εΔ
Debye equation:deviation of peak shape from Debye type
no single relaxation time behaviour( )*
1 iεε ω εωτ∞
Δ= +
+
g
formal approach:
distribution in relaxation time L(τ)
( )L τ∫ ∫( ) ( )* ln1L
diτ
ε ω ε ε τωτ∞= + Δ
+∫ ( ) ln 1with L dτ τ =∫
October 9, 2008
Relaxation time spectra
molecular reasons:
• real distribution in molecular potentials causesdistribution in relaxation time
e g : secondary peakse.g.: secondary peaks
cooperati it man bod interactions• cooperativity – many-body interactions between relaxing units
t=t0t t Δt=t0+Δt
e.g.: α-relaxation (segmental process)
October 9, 2008
Relaxation time spectra – model functions
( )*1 i
εε ω εωτ∞
Δ= +
+Debye equation:
Cole/Davidson: ( )*CD β
εε ω ε Δ= +Cole/Davidson: ( )
( )1CD
CDi βε ω εωτ
∞ ++
asymmetric broadening
Cole-Cole: ( )( )
*
1CC
CCi β
εε ω εωτ
∞Δ
= ++ ( )1 CCiωτ+
symmetric broadening
October 9, 2008
Relaxation time spectra
Havliliak-Negami: ( )( )( )
*
1HN
HNiγβ
εε ω εωτ
∞Δ
= ++ ( )( )
2 shape parameters:β: symmetric broadeningγ: asymmetry log(ε)γ: asymmetry
• empirical function
g( )
• nevertheless, widely used to model dielectric spectra m= β n= -βγ
log(ω)
October 9, 2008
Relaxation time spectra (HN-function)
October 9, 2008
Relaxation time spectra (model-free)
distribution in relaxation time L(τ)
( ) ( )* ln1L
diτ
ε ω ε ε τωτ∞= + Δ
+∫
Computation of L(τ) byComputation of L(τ) by numerical techniques possible, e.g. Tikhonov regularizationTikhonov regularization
October 9, 2008
Temperature dependence of the relaxation time
Molecular processes usually thermally activated processes Arrhenius law
ε' αE β
αJ
β
J
T T
dielectric relaxations: mechanical relaxations:gain in orientational polarisability with temperature
gain in “softness” with temperature
22
Thermally activated processes
Temperature dependence of relaxation time: Arrheniuslawthermal activation of τ
( ) ⎟⎠
⎞⎜⎝
⎛−= ∞ kTET aexpττ6
2
4
g fα
max
(Hz)
α
β-process
2
0
log
β
T-dependence τ(T) of α and β-
β p
3 4 51000/T (1/K)
-2 T dependence τ(T) of α and βprocess in a poly(ether ester) [Mertens et al. 1999, pol. 1a]
October 9, 2008
Thermally activated processes
Physical meaning of the Arrheniuslawthermal activation of a motion over energy barrier
Energy U Barrier model: T-independent height of energy barrier Ea (activation energy)
ΔU
barrier Ea (activation energy)
rotation angle
typical Arrhenius processes in polymers:
• local (secondary) relaxations in glassy state• dynamics in liquid crystalline polymers• dynamics in liquid crystalline polymers
October 9, 2008
Dipoles in polymers
Some dipole moments of simple molecules
Compound Dipole moment × 10-30 Cm Di l t i l× 10 Cm
H2O 6.1 HF 6.4 HCl 3.6 HBr 2.6 H
H
+ -PVCC - Cl
Dipole moments in polymers
CH4 0 CCl4 0 CO2 0 NH3 4.9
CH Cl 5 3
C
H
H
C
Cl
μ
CH3Cl 5.3C2H5OH 5.7 (C2H5)2O 3.8 C6H5Cl 5.7
C6H5NO2 14 0
H
C
H
F
C F
C
F--
+ PVDF
C6H5NO2 14.0
HF
Remember: dielectric response2μ∝
October 9, 2008
Dipoles in polymers
Location of dipole moments in the polymer chain:
CH
Type A: dipole moment along chain axis
CH2 CC
CH3
CH2 CH2 H
cis-1,4-polyisopreneCl Cl cis 1,4 polyisoprene
poly(vinyl chloride)
CH2 CC CH2 H H
Type B: dipole moment perpendicular to chain axis
CH2 C
CH3
C CH2
C
CH3
OO
COO
perpendicular to chain axis
PMMA
O
CH3
O
CH3Type C: dipole moment in side group
October 9, 2008
Experimental techniques for DRS
measurement of (complex) capacitance or resistance by e.g.• bridge techniques• frequency response analyser A dielectric sample cell
phase sensitive measurement of voltage and
t
Zx V
currentA
sample configurations:
parallel plate config.:
E
comb electrode:
sample configurations:
E
Experimental techniques for DRS
RLC meter HP4284A (Agilent)
20 Hz – 1MHz
Medium resolution in lossMedium resolution in loss
Dielectric spectrometer (Novocontrol)
Alpha-Analyser (high resolution)
0 001 H 10 MH0.001 Hz – 10 MHz
Experimental techniques for DRS
Time-domain setup
Prof. Yuri Feldman, Jerusalem • Up to 10 GHz• Time domain data need
t f ti t ftransformation to frequency space
Specific application: DRS on ultrathin film "capacitors"
Top electrode
Polymer film
Lower electrodeLower electrode
Substrate Clean room for thin film preparation
1. thermal evaporation of lower metal electrode on glass substrates.
2. spin-coating of very dilute (filtred) polymer solutions
3. drying + annealing in vacuumvacuum
4. 2nd evaporationpatterned top electrode
DRS on ultra-thin films
Dielectric instrumentation:Subject of separate lecturesSubject of separate lectures
Last part of this part:Various representation of complex dielectric spectra
DRS on ultra-thin films
Representation of relaxation spectra
1) Spectra of the real (ε’) and imaginary part (ε”) of the permittivity
Generally:'( ) ( ) "( )iε ω ε ω ε ω∗ = −
2 spectra that carry (almost)2 spectra that carry (almost) equivalent information
Kramers-Kronig integral g gtransforms:
October 9, 2008
Representation of relaxation spectra
2) KK-transform – numerically computed by DCALC
ε ′′= ε= KK
Note: lacks conduction termε ′′KK
• Good reproduction of ε” in the peak region
• Additional features at lowAdditional features at low frequencies revealed
October 9, 2008
Representation of relaxation spectra
2) KK-transform – effect of frequency spacing!
less is more!less is more!
• DCALC algorithm optimized for geometric f-series (f, 2f, 4f…)
October 9, 2008
( )
Representation of relaxation spectra
3) An approximation of the KK-transform – ε”deriv
loss-free part of ε”loss free part of ε
G d d ti f ” i• Good reproduction of ε” in the peak region for broad processesF (D b )• For narrow (Debye) processes extra-sharp peak in ε”der
Additi l f t t l• Additional features at low frequencies revealed
October 9, 2008
Representation of relaxation spectra
3) ε”deriv – effect of peak shape
October 9, 2008
Representation of relaxation spectra
4) Loss tangent – a mixed quantity
εδ′′
=tan
• “normalized” dielectric loss
δε
=′
tan
o a ed d e ect c osseliminates dimensional
problems• peak in tanδ shiftedpeak in tanδ shifted
compared to ε”
October 9, 2008
Representation of relaxation spectra
5) The Cole-Cole plot
ε ε′′ ′vs
• Allows easy consistency check between e” and e’ d (KK l i f lfill d)data (KK relation fulfilled)
• Frequency not explicit displayed
• Easy graphical inspection of peak symmetry and broadening
• Commonly used in impedance spectroscopy (Nyquist-plot)
October 9, 2008
Representation of relaxation spectra
5) The Cole-Cole plot Nyquist Diagram
General Nyquist diagramGeneral Nyquist diagram
N q ist diagram for impedance
October 9, 2008
Nyquist diagram for impedance
Representation of relaxation spectra
5) Dielectric modulus
definition
Note that tanδ is the same as for ε*
M’
October 9, 2008
Representation of relaxation spectra
5) Dielectric modulus
Imaginary part shows “conduction” peak in the conduction region,
Maximum yields Ohmic relaxation time:
• Relaxation peaks appear shifted in M”-representation
• This shift is twice of the shift in tanδ beca se of M”tanδ because of
October 9, 2008
Representation of relaxation spectra
5) Dielectric modulus in CC-representation
α-peakpConduction peak
α-peak
October 9, 2008
Representation of relaxation spectra
6) Conductivity representation
σ ε ε ω′′= 0σ ε ε ω= 0
Increasing T
October 9, 2008
Representation of relaxation spectra
6) Conductivity representation – filtered data
Special function of DCALC:
Exclusive display of (nearly) frequency independent σ-values
October 9, 2008
Representation of relaxation spectra
More options, e.g.
• Admittance
• Impedance
• Capacitance
• etc.
Further reading:g• M. Wübbenhorst and J. van Turnhout, "Analysis of complex dielectric spectra. I. One-
dimensional derivative techniques and three-dimensional modelling.," J. Non-Cryst. Solids, vol. 305, pp. 40-49, 2002.
• J van Turnhout and M Wübbenhorst "Analysis of complex dielectric spectra IIJ. van Turnhout and M. Wübbenhorst, Analysis of complex dielectric spectra. II. Evaluation of the activation energy landscape by differential sampling.," J. Non-Cryst. Solids, vol. 305, pp. 50-58, 2002.
• Mario F. García-Sánchez et al., An Elementary Picture of Dielectric Spectroscopy in Solids: Physical Basis Journal of Chemical Education Vol 80 2003 pp 1062
October 9, 2008
Solids: Physical Basis, Journal of Chemical Education, Vol. 80, 2003, pp. 1062
