BROADBAND DIELECTRIC SPECTROSCOPY - BASICS AND SELECTED APPLICATIONS

Transcription

1 BROADBAND DIELECTRIC SPECTROSCOPY - BASICS AND SELECTED APPLICATIONS Andreas Schönhals 9 th International Conference on Broadband Dielectric Spectroscopy and its Applications September 11-16, 16 in Pisa (ITALY)

2 Table of Content Basics of Dielectric Spectroscopy Electrostatics Dynamics Analysis Scaling of the dynamic glass transition (-relaxation) Chain dynamics Normal mode Conclusion Broadband Dielectric Spectroscopy Basics and Applications

3 Dielectric Spectroscopy Maxwell s Equations: rot E t B divb Electric field rot H j D t Magnetic induction divd Density of charges Magnetic field Current density Dielectric displacement Material Equation: Small electrical field strengths: D* E P DD ( * 1) E Polarization Complex Dielectric Function * is time dependent if time dependent processes taking place within the sample permittivity of vacuum Broadband Dielectric Spectroscopy Basics and Applications 3

4 E(t); Input P(t); Output Dielectric Spectroscopy Input E(t) Simplest Case : Step-like Input System under investigation (t) (t ) Output P(t) P(t ) P E P S Relaxational Contribution E Instantaneous Response P Time Time Broadband Dielectric Spectroscopy Basics and Applications 4

5 E(t); Input Dielectric Spectroscopy Input System under investigation Output Arbitrary input E(t) (t) Approximation of the Input by step-like changes P(t) E * t t dx dt E(t) *dt E() E * t t P(t) P t (t d E(t') t') dt' dt' Time Basics of Linear Response Theory Linearity Causality Broadband Dielectric Spectroscopy Basics and Applications 5

6 E(t) ; P(t) Dielectric Spectroscopy Input System under investigation Output Phase shift E(t) (t) E(t ) P(t) it E sin( t ) E exp - Angular Frequency P(t ) P sin( t ) Complex Numbers Time Complex Dielectric Function * ( ) ' ( ) i ' ' ( ) Broadband Dielectric Spectroscopy Basics and Applications 6

7 Dielectric Spectroscopy Phase angle Input E(t) System under investigation Output P(t) Complex Dielectric Function * ( ) ' ( ) i '' ( ) Real Part (in Phase) Imaginary or Loss Part (9 Phase shifted) Complex Plane * ' ' tan ' Molecular Dynamics of plasma polymerized layers by relaxation spectroscopy 7

8 Dielectric Spectroscopy Kramers-Kronig relationship Real Part Loss Part Related to the energy stored reversibly during one cycle Related to the energy dissipated during one cycle Law of energy conservation Kramers Kronig Relationships Relationship between both quantities '( ) 1 ''( ) d ''( ) 1 '( ) d Broadband Dielectric Spectroscopy Basics and Applications 8

9 Dielectric Spectroscopy Modulus P(t) P t (t d E(t') t') dt' Output E(t) dt' Linear Equation (t) System under investigation M(t) Electrical Modulus Input P(t) Can be inverted Relationship between (t) and M(t) (t) Dirac function ( t t')m(t')dt' (t) M*( ) *(ω) 1 Fourier Transformation Broadband Dielectric Spectroscopy Basics and Applications 9

10 Dielectric Spectroscopy 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 P( ) P() ( ) P Spectral Density ( P ) P P ( )exp(i) d 1 V μ i ( ) i () ( ) i i ( i i i ) ij () ( ) Measure for the frequency distribution of the fluctuations i j Fluctuation Dissipation Theorem (τ ) ( P ) 1 ε k T 1 kt (τ ) 1 ε ''( ) 1 Links microscopic fluctuations and macroscopic reactions For a small (linear) disturbance a system reacts only in the way as it fluctuates Broadband Dielectric Spectroscopy Basics and Applications 1

11 Dielectric Spectroscopy Propylene glycol K K 15 K 58.5 K 313 K 3 6 '' 4 ' log ( f [Hz]) Bridges Network Analysis Coaxial Reflectometer Fourier Correlation Analysis Time Domain A. Schönhals, F. Kremer, E. Schlosser, Phys. Rev. Lett. 67, 999 (1991) Broadband Dielectric Spectroscopy Basics and Applications 11

12 Linear Response Theory Molecular Motions Input Disturbance x(t) System under Investigation Material function Output Reaction y(t) Electric field E Polarization P Shear tension Shear angle Magnetic field H Magnetization M Temperature T Entropy S Broadband Dielectric Spectroscopy Basics and Applications 1

13 Linear Response Theory Molecular Motions Input Disturbance x(t) System under Investigation Output Reaction y(t) Disturbance Reaction Material function Intensive - independent of V Extensive ~ V Generalized Modulus G(t) Tensile Strain Strain Elastic Modulus Relaxation Extensive ~ V Intensive - independent of V Generalized Compliance J(t) Strain Tensile Stress Elastic Compliance Retardation Broadband Dielectric Spectroscopy Basics and Applications 13

14 Linear Response Theory Input x(t) System under investigation Output y(t) 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 c p (t) Temperature Modulus G T (t) Broadband Dielectric Spectroscopy Basics and Applications 14

15 log '' ' Dielectric Spectroscopy Information of a measurement p = f p =1/ Three essential measurement information S Frequency position f p = S - Dielectric relaxation strength Shape of the relaxation time spectra log ( / p ) Broadband Dielectric Spectroscopy Basics and Applications 15

16 (t)=p/e Dielectric Spectroscopy Dielectric relaxation strength Capacitor with N permanent Dipoles, Dipole Moment Polarization : 1 N P μ i P μ P V V Mean Dipole Moment Mean Dipole Moment: S W th kt Counterbalance W el E Orientational polarization Induced dipoles Thermal Energy Electrical Energy 4 E exp( )d kt E exp( )d kt 4 Time The factor gives the probability that the dipole moment vector has an orientation between and + d Broadband Dielectric Spectroscopy Basics and Applications 16

17 (a) Dielectric Spectroscopy Dielectric relaxation strength Spherical Coordinates: Only the dipole moment component which is parallel to the direction of the electric field contributes to the polarization Ecos cos exp( ) kt Ecos 1 exp( ) kt sin 1 sin d d x = ( E cos ) / (kt) a = ( E) / (kt) (a)=a/3 Langevin function E a.1 kt.1kt E cos 1 a (a)a/3 a a a x exp(x)dx exp(a) exp( a) 1 (a) exp(a) exp( a) a exp(x) dx a Langevin function E 3kT a Debye-Formula Broadband Dielectric Spectroscopy Basics and Applications 17 S 1 3 kt N V

18 Dielectric Spectroscopy Dielectric relaxation strength Shielding effects: Polar Molecules; Onsager Theory, Reactions Field E S eff = + E loc 1 3 F kt N V Onsager Factor F= 3 S( F 3( S ) ) eff Interaction effects: S 1 3 gf k T N V g 1 i i j N i j Interact. Kirkwood / Fröhlich Correlation Factor g=.5 5 inter z nearest neighbors: g1 z cos Broadband Dielectric Spectroscopy Basics and Applications 18

19 Dielectric Spectroscopy Dynamics - Debye model Dynamics is due to the dipole dipole correlation function i (t) i () (t) i ( Debye model: Change of the polarization is proportional to the actual value P(t) Time domain t (t) exp D Frequency domain i d P(t) d t i i j i) 1 D i () (t) D *( ) ' 1 i 1 ( D D) 1 ( D) '' j Double Potential Model d P dn1 dn dt dt dt n n P(t) 1 Rotational Diffusion Model A rigid isolated fluctuating dipole is regarded in viscous media under the influence of a stochastic force f (r, t) D t Rot E(t) [f (r, t) f (r, t)] kt P m (cos())p m (cos(t)) exp( m(m 1)D Rot t) Broadband Dielectric Spectroscopy Basics and Applications 19

20 Dielectric Spectroscopy 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. Debye function: *( ) 1 i D symmetric Cole/Cole function: * CC ( ) 1 (i CC ) <1 symmetric broadening Cole/Davidson function: ( ) * CD <1 asymmetric broadening (1 i CD) Havriliak/Negami function: ( ) * HN <; 1 asymmetric and (1 (i HN ) ) symmetric broadening Broadband Dielectric Spectroscopy Basics and Applications

21 log '' C o l e - D a v i d s o n Dielectric Spectroscopy Data Analysis w i i * * (f ) min i Model i Most general function is the formula of Havriliak / Negmai '' = / ( f ) PVAC 1. D e b y e.8 C o l e - C o l e log (f [Hz]) K W W Broadband Dielectric Spectroscopy Basics and Applications 1

22 Log (Shear Modulus G) Overview about the dynamics of polymers Polymers are complex systems For an isolated macromolecule a large number of atoms (1 to 1 ) are covalently bonded. Shear modulus vs. Temperature Almost unlimited number of conformations in space and time. glass-like behavior Most properties depend on this large conformation. G ~1 9 Pa glass transition Chain flexibility T g viscoelastic behavior liquid-like behavior Mean End-to-End vector (shape) Mean Square dipole moment and many more G ~1 6 Pa Temperature flow transition Behavior is complex. and related to the molecular mobility Broadband Dielectric Spectroscopy Basics and Applications

23 Molecular dynamic in polymers Glassy State Glass Transition, -Relaxation Flow Transition Local CH 3 Seg Chain C CH O C O CH CH 3 Local Motion Localized bond rotations Fluctuations in a side group LOCAL < 1 nm Segmental Motion SEG ~ 1.. nm r Chain Motion Chain < = r Broadband Dielectric Spectroscopy Basics and Applications 3

24 Scaling of segmental dynamics Dynamic glass transition Broadband Dielectric Spectroscopy Basics and Applications 4

25 Dynamics in glass-forming materials Dynamic Response -relaxation, dynamic glass transition Slow -relaxation Fast -relaxation Boson-peak 15 orders of magnitude Slow -relaxation -relaxation Activation Diagram E T kt A exp 1 ( T) exp DT v ( T) T T Thermal Response The temperature dependence of the dynamical relaxation times correlates with the glass transition temperature at ca. 1 s Broadband Dielectric Spectroscopy Basics and Applications 5

26 Dynamics in glass-forming materials Non-Debye behavior of the Relaxation function t exp ~ KWW-function Crossover Phenomena form a high to a low temperature regime at T B ~ T g Formation of dynamic heterogeneities below T B Size : few nanometers Broadband Dielectric Spectroscopy Basics and Applications 6

27 log '' Dielectric Spectroscopy Poly(methyl phenyl siloxane) (PMPS) ; M w =1 g/mol ? log (f [Hz]) Alpha-Analyzer HP4191A Broadband Dielectric Spectroscopy Basics and Applications 7

28 log(f p [Hz]) log '' Dielectric Spectroscopy Poly(methyl phenyl siloxane) (PMPS) ; M w =1 g/mol K K K -1. * HN ( ) (1 (i HN ) ) log (f [Hz]) 4 VFT- equation: logf A p logfp? T T - Mean Relaxation Rate f p -4 Dielectric Strength / T [K -1 ] Broadband Dielectric Spectroscopy Basics and Applications 8

29 Temperature Specific Heat Spectroscopy Input T(t) Temperature Modulated DSC System under investigation c p (t) Output S(t) The modulated temperature results in a modulated heat flow Thermocouples Furnace The heat flow is phase shifted if time dependent processes take place in the sample T(t) T*t Asin( *t) c* p ()= c p() i c p() Time 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, ) Broadband Dielectric Spectroscopy Basics and Applications 9

30 Specific Heat Spectroscopy Example - Polystyrene Due to heating rate Specific Heat Capacity in J g -1 K T g (q) 1.5 total c p c p ' Due to modulation.1.5 c p '' T ( ). -.5 Gaussfit Temperature in K Hensel, A., and Schick, C.: J. Non-Cryst. Solids, (1998) Broadband Dielectric Spectroscopy Basics and Applications 3

31 Specific Heat Spectroscopy AC Chip calorimetry Input T(t) System under investigation c p (t) Output S(t) Pressure gauge, Xsensor Integration 1 mm mm.1 mm Fast heating and cooling H. Huth, A. Minakov, Ch. Schick, Polym. Sci. B Polym. Phys. 44, 996 (6) Broadband Dielectric Spectroscopy Basics and Applications 31

32 Specific Heat Spectroscopy Input T(t) System under investigation c p (t) Output S(t) U* Complex differential Voltage Differential setup Increase of sensitivity H. Huth, A. Minakov, Ch. Schick, Polym. Sci. B Polym. Phys. 44, 996 (6) Broadband Dielectric Spectroscopy Basics and Applications 3

33 log (f (f p p [Hz]) U R [V] Specific Heat Spectroscopy Example - PMPS Both methods measure the same process Glassy dynamics of PMPS 8 75 f=4 Hz.5 Corr [deg] 8 6 Specific Heat AC Chip Calorimetry T [K] 4 Dielectric Frequency range : 1 Hz. 5 Hz - TMDSC A. Schönhals et al. J. Non-Cryst. Solids 353 (7) / T [K -1 ] Broadband Dielectric Spectroscopy Basics and Applications 33

34 log (f p [Hz]) Analysis of the temperature dependence of the relaxation rate VFT- equation: log f p A log f p T T 4 d log f d T 1/ p Derivative: T T A 1 1/ / T [K -1 ] Straight Line, T > cannot be described by the VFTequation in the whole temperature range Broadband Dielectric Spectroscopy Basics and Applications 34

35 Analysis of the temperature dependence of the relaxation rate 8 low and high temperature regime T B { { log log (f (f p / dt [K] } -1/ p [Hz]) / dt [K] } -1/ 6 two VFT - dependencies 4 T,Low T,high Crossover temperature T B = 5 K Thermal and dielectric data have the same temperature dependence T [K] red : dielectric data blue: thermal data Broadband Dielectric Spectroscopy Basics and Applications 35

36 T* Analysis of the temperature dependence of the dielectric strength Debye Theory: Dipole moment 1 3 g k B T N V Interaction parameter 8 T B Temperature dependence of is much stronger than predicted Cooperative effects Change of the temperature dependence of at T B / T [K -1 ] Broadband Dielectric Spectroscopy Basics and Applications 36

37 Analysis of the temperature dependence of the dielectric strength The change in the temperature dependence of is not much pronounced Plot of vs. log f p.4 Sharp transition from a low temperature to a high temperature branch.3 T B The crossover frequency corresponds to T B..1 A. Schönhals Euro. Phys. Lett. 56 (1) log (f p [Hz]) Broadband Dielectric Spectroscopy Basics and Applications 37

38 Chain dynamics Normal mode Broadband Dielectric Spectroscopy Basics and Applications 38

39 Dipole moments of polymers Type A Type B Type C Chain Dynamics Segmental Dynamics rr r rr Dipole Moment Parallel to the Chain Poly(cis-1,4-isoprene) Poly(propylene glycol) Dipole Moment Perpendicular to the Chain Polystyrene Poly(vinyl chloride) Poly(methyl phenyl siloxane) Dipole Moment in a Side Group Poly(methyl methacarylate) Broadband Dielectric Spectroscopy Basics and Applications 39

40 log '' Dielectric Spectroscopy Polypropylene glycol; M w =4 g/mol K 33.8 K Dynamic Glass Transition, - relaxation 33.7 K 53.3 K 73. K 33.3 K Normal Mode Relaxation Time Domain Spectrometer log ( f [Hz]) Frequency Response Analysis Coaxial Reflectometer Broadband Dielectric Spectroscopy Basics and Applications 4

41 log '' Data analysis K 33.7 K The two model function of Havriliak and Negami are fitted to the data. f p,n -.5 f p, -1. Relaxation rate f p Dielectric strength -1.5 for each process log ( f [Hz] ) Broadband Dielectric Spectroscopy Basics and Applications 41

42 Theory of chain dynamics Rouse motion, Reptation Rouse Model: Springs Beads x i dx i dt + 3 kt x i x i 1 x i+1 = A 3 l 3 A A l 1 Predictions: l A 1 Monomeric friction r A N-1 l N l N-1 A N- Friction Chain connectivity (Entropy springs) Solution: < r( ) r ( t ) > 8 N 1 ( t) = exp ( t / p ) < r > p1 p Short chains: p = 3 N b kt p ~M, p =1,3... Molecular weight dependence Temperature dependence of the relaxation rates of segmental and chain dynamics should be similar Long chains (Entanglements): p 3 4 = N b kt a p ~ M 3 Fig Broadband Dielectric Spectroscopy Basics and Applications 4

43 Chain dynamics Normal mode Poly(cis-1,4 isoprene).4 M W g/mol f=1 Hz T [K] Schönhals, A. Macromolecules 6 (1993) Broadband Dielectric Spectroscopy Basics and Applications 43

44 log ( n [s]) Chain dynamics Normal mode Poly(cis-1,4 isoprene) p = 3 N b k T p ~ M p 3 4 = N b kt a p ~ M 3 - Critical molecular weight for the formation of entanglements Anstieg = Reptation -6 Anstieg = ( Rouse ) T=3 K log (M W [g/mol]) Broadband Dielectric Spectroscopy Basics and Applications 44

45 log (D Self [m /s]) Chain dynamics Normal mode Self diffusion Relaxation times: Diffusion coefficient: D Self can be measured by Pulsed Field Gradient NMR p = D Self N b 3 kt p kt N D Self r 3 p p Poly(propylene glycol) M W =4 g/mol Both quantities agree in both there absolute values and in there temperature dependence -13 Normal mode corresponds to translatorial diffusion -14 Blue - dielectric Red - NMR -15 Appel, M.; Fleischer, J. Kärger, G.; Chang, I.; Fujara, F.; Schönhals, A.; Colloid and Polymer Sci. 75 (1997) K / T Broadband Dielectric Spectroscopy Basics and Applications 45

46 log (f p [Hz]) log (f p [Hz]) Chain dynamics Temperature dependence relaxation rates Segmental dynamics Blue: Green: Red: Black: PPG PPG3 PPG4 PPG53 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 K / T K / T Chain dynamics The temperature dependence of both relaxation rates seems to follow the Vogel/Fulcher/Tammann formula. log f p A log f p T T Both processes seems to merge close to T g Broadband Dielectric Spectroscopy Basics and Applications 46

47 log (f p / f pn ) Chain dynamics Temperature dependence relaxation rates Ratio of both relaxation rates Plateau Decrease PPG PPG3 PPG T [K] Detailed investigation of the temperature dependence of the relaxation rates! Broadband Dielectric Spectroscopy Basics and Applications 47

48 Chain dynamics Normal mode Crossover of segmental dynamics Polypropylene glycol; M w =4 g/mol 4 3 T B =4 K A low temperature branch 1 A high temperature branch log (f [Hz]) = Broadband Dielectric Spectroscopy Basics and Applications 48

49 Chain dynamics Normal mode (d log f p / dt) -1/ 8 Derivative: 6 4 d log f d T p 1/ A 1 1/ T T 44 K The temperature dependence of the -relaxation rate shows a change at T=44 K as it was observed also for the dielectric strengths (log f pn / d T) -1/ T [K] 8 No change of the temperature dependence of the relaxation rate of the normal mode at T B. 6 4 The reason for the different temperature dependencies of - and normal mode relaxation is the crossover in the dynamic glass transition. T [K] Broadband Dielectric Spectroscopy Basics and Applications 49

50 log (f p / f pn ) ~ log(d Rot /D Trans ) Decoupling of rotational and translatorial diffusion Segmental dynamics rotational diffusion Chain dynamics translatorial diffusion PPG PPG3 PPG4 Decoupling of Rotation and translation in low molecular weight glass-forming liquids Rössler, E.; Phys. Rev. Lett. 69 (199) 16, 65 (199) T [K] Different temperature dependencies of - and normal mode relaxation is due to the decoupling of rotational and translatorial diffusion at T B. Can be explained by different averaging of modes with different length scales Broadband Dielectric Spectroscopy Basics and Applications 5

51 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 Applications 51