& Comparison With Other Techniques” DIELECTRIC SPECTROSCOPY
• measures the dielectric and electric properties of a medium as a function of frequency (time)
• is based on the interaction of an external electric field with the electric dipole moment and charges of the medium Dielectric properties
Complex dielectric permittivity relative to vacuum
ε*(ω)= ε’(ω)-iε’’(ω)
Complex conductivity σ*(ω)= σ’(ω)+iσ’’(ω)
Complex electric modulus Μ*(ω)=Μ’(ω)+iΜ’’(ω) as a function of temperature Medium in an external electric field ¢ ¢ ¢ ¢ ¢
Electric Field ¡ ¡
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•D= ε0E+P •P= ε χ E 0 s D= ε0εsE •εs= χs+1 Medium in an external electric field
Relation between macroscopic and microscopic quantities
1 P = ∑ pi V i Dipole moments can have an induced or a permanent character
Permanent dipole moments, µ
Induced polarization, P∞
1 N P = ∑ µi + P∞ = µ + P∞ V i V
µ Is determined by: •The interaction between the dipoles •The local electric field at the location of the dipole Medium in an external electric field
Relation between macroscopic and microscopic quantities
•No interaction between dipoles and local electric field equal to the outer applied field thermal energy interaction energy between the dipole and the electric field
µ 2 1 µ 2 N µ = E ε s − ε ∞ = Langevin equation 3kT 3ε 0 kT V
•Interactions between molecules, effect of polarization of environment
2 1 µ N Onsager-Kirkwood-Froehlich equation ε s − ε ∞ = Fg 3ε 0 kT V ε (ε + 2)2 F = s ∞ “Internal field” Onsager factor 3()2ε s + ε ∞ µ 2 g = int = 1+ z cos ψ “Dipole interactions” Kirkwood-Froehlich factor µ 2 Dielectric dispersion in time domain
Perturbation Response Electric Field Electric Displacement E(t) D(t) ε(t)
E(t) D(t)
εεε εεε E 0 s 0 D(t)= ε 0ε (t)E0
ε (t)= ε ∞ + (ε s − ε ∞ )ϕ(t)
ϕ(0)= 0 ϕ(∞)= 1 E εεε εεε E 0 oo0 0
Φ(t)= 1−ϕ(t) 0 t 0 t Dielectric dispersion in frequency domain
Perturbation Response Electric Field Electric Displacement E(ω) D*(ω) ε*(ω)
E(t)=E sin(ϖϖϖt) * 0 D (ω)= D′(ω)+ iD ′′(ω) D(t)=D sin(ϖϖϖt- δδδ) 0 D′(ω)= D0 cos δ (ω)
D′′(ω)= D0 sin δ (ω)
D δδδ ϖϖϖt ε ′()ω = 0 cos δ ()ω ε 0 E0 D ε ′′()ω = 0 sin δ ()ω ε 0 E0 Time-frequency domain
Time dependent dielectric function Complex dielectric function ε(t) ε*(ω)
∞ ∞ dε t)( 1 * dε(t) = ()ε ()ω −ε exp []iωt dω ε*(ω) =ε − exp []−iωt dt ∫ ∞ ∞ ∫ dt 2π 0 0 dt
Relation between dielectric function and correlation function
ε t)( −ε∞ =(εs −ε∞)[1−Φ(t)] ∑ µi (0)µi (t) +2∑∑ µi (0)µj (t) Φ t)( = i ii< j 2 ∞ dΦ(t) ε*(ω)−ε =()ε −ε − exp []−iωt dt µ ∞ s ∞ ∫ ∑ i 0 dt i
Φ )0( =1 Φ(t →∞) =0 Dielectric dispersion in frequency domain
Debye relaxation
t 4 Φ t)( =exp − ε' τD 3
t 2 ε (t) −ε ∞ = ∆ε 1− exp − τ D 1 ε''
∆ε 0 ε (* ω) = ε ∞ + 1+ iωτ D 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 f(Hz) Dielectric dispersion in frequency domain
Kolhrauch-Williams-Watts KWW relaxation
4 t βKWW Φ t)( =exp − ε' τKWW 3 β t KWW ε (t) −ε ∞ = ∆ε 1− exp − τ 2 KWW
Havrilliak-Negami relaxation 1 ε'' 1 Φ * (ω) = HN α γ 0 []1+ ()iωτ HN 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 ∆ε f(Hz) ε (* ω) = ε + ∞ α γ []1+ ()iωτ HN
23.1 .0 387 βKWW ≈ αγ γ =1− .0 812 (1−α ) Dielectric dispersion in frequency domain Dipole moments in polymers Molecular motions accommodate dipole reorientation in polymers
Decreasing frequency of external field Dielectric dispersion in frequency domain 3x10 -1 - 60°C - 40°C
2x10 -1 10 0 '' ε ε ε ε
-1
10
-1 dielectric loss, 10 - 90°C
- 120°C
10 -1 10 0 10 1 10 2 10 3 10 4 10 5 frequency [Hz] Glassy state (localized motions) I
, relaxations
Depends on chemical structure and local molecular environment
• Motion of groups present in side chain around the main chain or bonds of the side chain
• Motion of groups in the main chain
• The relaxation of Johari-Goldstein type
Thermally activated motion between two potential wells separated by a potential barrier Glassy state (localized motions) II
Characteristics of dielectric dispersion
• Temperature dependence of relaxation rate (time) is Arrhenius
E f = f exp − act β ∞ β kT
• Dielectric strength, , increases with increasing T, more dipoles contribute µ 2 N ∆ε ∝ F g Onsager kT V
• Broad (4-6 decades) symmetric dielectric dispersion, becomes narrower with increasing T as a result of molecular environment homogenization Glassy state (localized motions) II 7 6 Glassy State 5 4 3
max 2 1 logf 0
-1 Liquid/Rubbery -2 State -3 3 4 5 6 7 8 1000/T[K-1 ] Rubbery/liquid state (segmental motions) I
relaxation (dynamic glass transition)
• relaxation time reflects the time involved in the structural rearrangement (glass transition) of the molecules
• τττ changes by many orders of magnitude from the liquid to the glass
• The ααα-relaxation in polymers involves cooperative micro-Brownian segmental motions of the chains
• Segmental motions are associated with conformational transitions taking place about the skeletal bonds Rubbery/liquid state (segmental motions) II
Characteristics of dielectric dispersion
• Temperature dependence of relaxation rate (time) described by VTF equation B fα = f∞α exp − T −T0
• Dielectric strength, , decreases with increasing T more strongly than predicted by µ 2 N ∆ε ∝ F g Onsager kT V
• Asymmetric dielectric dispersion • High f side slope, n: local chain dynamics (n increases as T increases) • Low f side slope, m: cooperative dynamics of environment (m incr eases as T increases) Rubbery/liquid state (global motions) I 7 6 Glassy State 5 4 3
max 2 1 logf 0
-1 Liquid/Rubbery -2 State -3 3 4 5 6 7 8 1000/T[K-1 ] Rubbery/liquid state (global motions) I
Normal mode relaxation (nm)
• Observed in polymers containing dipoles type A and AB
• Motions of the whole chain
• Observed at frequencies lower than the ααα relaxation
• The relaxation time of the process is strongly dependent of the molecular weight Rubbery/liquid state (global motions) II
Characteristics of dielectric dispersion 7 • T dependence of relaxation rate (time) 6 Glassy State same as this of relaxation 5
• Dielectric strength proportional to the 4 mean end-to-end vector of chain 3
max 2 1 logf 0 4πN µ 2 F A p Onsager 2 -1 ∆ε nm = r Liquid/Rubbery 3kTM -2 State -3 3 4 5 6 7 8 1000/T[K-1 ] • m=1 and n=0.7 predicted by theory, in practice n considerably smaller Rubbery/liquid state (charge motions)
Conductivity • Motion of charge carriers
* * σ (ω)= σ ′(ω)+ iσ ′′(ω)= iωε 0 (ε (ω)−1)
σ ′(ω)= ωε 0ε′′(ω)
σ ′′(ω)= ωε 0 (ε′(ω)−1)
• ’ is not affected
• ’’ increases with decreasing f (slope of log ’’vs logf equal to -1)
• ’ independent of f equal to dc conductivity, dc
• T dependence of dc Arrhenius, VTF or other gives information on the conductivity mechanism Separation of charges at interfaces
Maxwell-Wagner-Sillars relaxation (MWS) • Blocking of charges at interfaces inside inhomogeneous materials • Existence of regions with different characteristics ( , ) • Relaxation rate (time) depends on: volume fraction, shape, and values of regions
Electrode Polarization
• Blocking of charges at sample/electrode interfaces • Observed as a steep increase of ’ • Magnitude and frequency position of the process depend on the conductivity of the sample (high values of shift polarization to high f) • Time of the process depends on the sample thickness (increase of thickness results in an increase of time of process) Comparison with other techniques
• Understanding the correlation between parameters of the molecular and supermolecular structures and macroscopic properties.
• Contribution to open questions regarding the relaxation by covering the widest possible time/frequency scale range by different experimental techniques Comparison with Mechanical measurements
Material testing method & tool for studying dynamics in polymers
• Material in an external mechanical field
• insulator "elastic solid, conductor "viscous liquid
• Viscoelastic response is analogous to dielectric dispersion
• Modulus representation is used
• In order dielectric dispersion be compared to mechanical one electric modulus has to be used for data representation
1 M * ()ω = ε * ()ω Comparison with Mechanical measurements
Extension in broad f range by constructing master plots Comparison with Mechanical measurements
Comparison of relaxation time temperature dependence Comparison with Mechanical measurements
Selectivity of dielectric spectroscopy (block copolymers) Comparison with Quasielastic Neutron Scattering
• Observation of the dynamics on a microscopic length and time scale (10 -7-10 -11 s)
• S(Q, ) contains not only temporal information but also spatial information of the particle correlation function Comparison with Quasielastic Neutron Scattering
Different T Different T Comparison with Quasielastic Neutron Scattering
1 S ()Q,ω = − Im (Φ* ()ω ) ω
τ (Q,T )= α(T )τ (Q)
−1 τ (Q =1A° ,T )
Comparable to
τ diel (T ) References
A. Broadband dielectric spectroscopy , F. Kremer and A. Schoenhals eds. Chapter 1 Theory of Dielectric Relaxation A. Schoenhals and F. Kremer Chapter 7 Molecular dynamics in polymer model systems, A. Schoenhals Chapter 16 Dielectric and Mechanical Spectroscopy a Comparison, T. Pakula Chapter 18 Polymer dynamics by dielectric spectroscopy and neutron scattering-a comparison, A. Arbe, J. Colmenero and D. Richter
B. Electrical Properties of Polymers , E. Riande and R. Diaz-Calleja
C. Dielectric Spectroscopy of Polymeric materials, J. Runt and J. Fitzgerald