Broadband Dielectric Spectroscopy (BDS) in soft matter research

F.Kremer

The spectrum of electro-magnetic waves

UV/VIS IR Broadband Dielectric Spectroscopy (BDS)

Questions to be addressed:

1. What molecular processes take place in the spectral range from THz to mHz and below?

2. What is the principle of Broadband Dielectric Spectroscopy?

3. Excursion: What is the dynamics in a glass?

4. Example: Glassy dynamics in low molecular weight and polymeric amorphous materials.

5. Example: Glassy dynamics in nanoporous confinement, or „how many molecules form a liquid?“

6. The counterbalance between surface and confinement effects

What molecular processes take place in the spectral range from THz to mHz and below?

Dcurl H j

t

0D E j E (Ohm‘s law)

The linear interaction of electromagnetic fields with matter is described by one of Maxwell‘s equations

(Current-density and the time derivative of D are equivalent)

0i

i i

Basic relations between the complex dielectric function * and the complex conductivity *

Effect of an electric field on a unpolar atom or molecule:

In an atom or molecule the electron cloud is deformed with respect to the nucleus, which causes an induced polarisation; this response is fast (psec), because the electrons are light-weight

+

-

+

-

-

Electric FieldElectric Field

Effects of an electric field on an electric dipole :

An electric field tries to orient a dipole but the thermal fluct-uationsof the surrounding heat bath counteract this effect; as result orientational polarisation takes place, its time constant is characteristic for the molecular moiety under study and may vary between 10-12s – 1000s and longer.

-e

+e

Electric field

-e

+e

Electric field

Effects of an electric field on (ionic) charges:

Charges (electronic and ionic) are Charges (electronic and ionic) are displaced in the direction of theapplied field. The latter gives rise to a resultant polarisation of the sample as a whole.

-

-

-

-

--

-

-

+

+

++

+

+

+ +

Electric fieldElectric field

+

+

+

+

+

+

+

+-

-

-

-

-

-

-

-

What molecular processes take place in the spectral range from THz to mHz and below?

1. Induced polarisation

2. Orientational polarisation

3. Charge tansport

4. Polarisation at interfaces

What is the principle of Broadband Dielectric Spectroscopy?

2( )1

s

2( )1

s

*

(1 )s

i

2,0

2,4

2,8

3,2

3,6

100 101 102 103 104 105 106

0,0

0,2

0,4

0,6

0,8

'=s

'=?

=s-

?

'

''max

max

''

[rad s-1]

0 20 40 60

0

1

2

orientational polarization

induced polarization

= S-

00

S

00

(t)=

(P(t

) -

P)

/

E

Time

E

E (

t)Capacitor with N permanent

dipoles, dipole Moment Debye relaxation

complex dielectric function

A closer look at orientational polarization:

P. Debye, Director (1927-1935) of the Physical Institute at the university of Leipzig (Nobelprize in Chemistry 1936)

The counterbalance between thermal and electric ennergy

Capacitor with N permanent Dipoles, Dipole Moment

Polarization : PμV

NPμ

V

1P i

Mean Dipole Moment

0 20 40 60

0

1

2

orientational polarization

induced polarization

= S-

00

S

00

(t)=

(P(t

) -

P)

/

E

Time

E

E (

t)

Dipole moment

Mean Dipole Moment: Counterbalance

EWel

kTWth

Thermal Energy Electrical Energy

4

4

d)kT

Eexp(

d)kT

E(exp

Boltzmann Statistics:

The factor exp(E/kT) d gives the probability that the dipole moment vector has an orientationbetween and + d.

Spherical Coordinates:

Only the dipole moment component whichis parallel to the direction of the electric field contributes to the polarization

dsin2

1)

kT

cosE(exp

dsin2

1)

kT

cosE(expcos

0

0

x = ( E cos ) / (kT)a = ( E) / (kT)

)a(a

1

)aexp()aexp(

)aexp()aexp(

dx)xexp(

dx)xexp(x

a

1cos

a

a

a

a

Langevin function

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0(a)=a/3

Langevin function

a

(a)

(a)a/3 ETk3

2

Debye-FormulaV

N

Tk3

1 2

0S

1.0kT

Ea

Tk1.0E

0 - dielectric permittivity of vacuum = 8.854 10-12 As V-1 m-1

The Langevin-function

10-2

10-1

100

101

102

103

104

105

106

100

101

102

103

104

105

235 K220 K

205 K

190 K

propylene glycol

´

frequency [Hz]

10-2

10-1

100

101

102

103

104

105

106

100

101

102

103

104

´

frequency [Hz]

Analysis of the dielectric data

Brief summary concerning the principle of Broadband Dielectric Spectroscopy (BDS):

1. BDS covers a huge spectral range from THz to mHz and below.

2. The dielectric funcion and the conductivity are comlex because the exitation due to the external field and the response of the system under study are not in phase with each other.

3. The real part of the complex dielectric function has the character of a memory function because different dielectric relaxation proccesses add up with decreasing frequency

4. The sample amount required for a measurement can be reduced to that of isolated molecules.

(With these features BDS has unique advantages compared to other spectroscopies (NMR, PCS, dynamic mechanic spectroscopy).

Excursion: What is the dynamics in a glass?

Single molecule relaxation Liquid-like relaxation (dynamic glass transition)

The temperature dependence of the relaxation rate enables one to distinguish between a single molecule and a liquid-like relaxation.

Dynamic glass transition: a gradual decrease over many orders of magnitude of the structural relaxation time upon cooling.

The (dynamic) glass transition

1000/T [1/K]2.0 3.0 4.0 5.0

-2

0

2

4

6

8

-4

-6

10

Calorimetric

Tg

log/

s]

180 200 220 240 280

heat

ca

paci

ty C

Temperature /K

Differential scanning calorimetry

Tg

Heat capacity

340 360 380 400 42098

100

102

104

106

108

110

340 360 380 400 420

2.0

2.5

3.0

3.5

C''(

) [a.u.]

T [K]

C'(

) [a

.u.]

Frequency-dependent calorimetry (20 Hz)

Heat capacity320 340 360 380 400

2

2

C

T

C

T

C',

C"

[a.u

.]

Temperature /K

0.96

0.97

0.98

0.99

1.00

320 340 360 380 400

Tg

C (

norm

aliz

ed)

(capacitive scanning) dilatometry

Thermal expansion

320 340 360 380 400

1.586

1.578

2

2

( )n

T

n', n

'' [a

.u.]

Temperature /K

320 340 360 380 400

( )n

T

Tg

n (5

40nm

)

1.570

Ellipsometry

Optical constants n and d

10-1 100 101 102 103 104 105

0.01

240 K235 K230 K225 K220 K215 K

''

f [Hz]

Broadband Dielectric Spectroscopy

Molecular motion

Summary concerning glassy dynamics

1. Single-molecule relaxations are Arrhenius-like. Glassy dynamics is characterized by the empirical Vogel-Fulcher-Tammann (VFT) dependence.

2. The dynamic glass transition is assigned to relaxations between structural substates. It corresponds to a continuous slowing down of the molecular dynamics upon cooling which is usually described by the empirical Vogel-Fulcher-Tammann (VFT) dependence.

3. The dynamic glass transition scales with the calorimetric glass transition. At Tg typically a relaxation rate of .01 Hz is reached.

4. There are manifold ways to measure the dynamic glass transition (Ac- and Dc-calorimetry, dilatometry, viscosimetry, scattering techniques, ellipsometry, Broadband Dielectric Spectroscopy etc). The latter has the advantage that it measures also the relaxation-time distribution.

Glassy dynamics in low molecular weight and polymeric amorphous materials

Vogel-Fulcher-Tammann (VFT)-equation:

0

0

1( ) exp

2 ( )

DTT v

T T T

Arrhenius-type temperature dependence:

AexpB

ET

k T

Types of thermal activation:

2

00

log1 log

1/

TdDT e

d T T

Aloglog

(1/ ) B

d Ee

d T k

(EA is the activation energy, kB the Boltzmann constant and the relaxation rate in the high temperature limit)

(D is a constant and T0 denotes the Vogel temperature)

Activation- and derivative-plot for relaxation-processes having different types of thermal activation

Activation- and derivative-plot for glycerol

The dynamics can not be described by a single VFT-function

Activation- and derivative-plot for propylene-glycol and poly(propylene glycol)

The dynamics can not be described by a single VFT-function

Activation- and derivative-plot for salol

The dynamics can not be described by a single VFT-function

Does time-temperature superposition holds in general?

No!

Does time-temperature superposition holds in general?

No!

Is there a characteristic change in glassy dynamics?

Yes! Between about .1 GHz - 1 GHz the temperature dependence of glassy dynamics changes

Summary concerning glassy dynamics in low molecular weight and polymeric amorphous materials

1. The dynamic glass transition is not at all a „transition“.

2. The dynamic glass transition scales usually with the calorimetric glass transion Tg.

3. At about 1 GHz a change in the temperature dependence of glassy dynamics takes place.

4. Time-temperature superposition does not hold in general.5. The scaling of the dynamic glass transition shows in derivative-

plots a signature which is characteristic for the material under study.

Example: Glassy dynamics in nanoporous confinement, or

„how many molecules form a liquid?“

5. Glassy dynamics of low molecular-weight systems in the geometrical confinement of zeolitic hosts systems - or how many molecules form a liquid?

Quantitative analysis of the molecular arrangement of the guest molecules in host systems, as deduced by computer simulations (Cerius2). bulk liquid zeolite beta silicalite distance between molecules [nm]

0.420.01 0.410.01 0.420.01

average length of H-bonds [nm]

0.230.02 0.250.02 0.240.02

density [g/cm3] 1.113 1.00.1 1.00.1 Neither for the distance between molecules nor the length of hydrogen-bonds or the density a significant change is found.

EG in AlPO4-5 shows a VFT-dependence

as in the bulk and for EG in zeolite beta.The coordination number at r = 0.7 nm is 5.

Ethylene glycol in ALPO4-5

0.3 0.4 0.5 0.6 0.70

2

4

6

8

10

12

EG bulk EG in zeolite beta EG in silicalite EG in AlPO

4-5

coor

dina

tion

num

ber

radius r [nm]

Simulation

3 4 5 6 7 8

0

2

4

6

8

10

EG bulk liquid EG in zeolite beta EG in ALPO

4-5

log

10(1

/ max

[H

z])

1000/T [K-1]

Experiment

Summary for ethylenglycol being in the confinement of zeolitic host systems

1. A single molecule dynamics is characterized by an Arrhenius temperature dependence - a liquid-like dynamics by a Vogel – Fulcher –Tammann (VFT)-like temperature dependence

2. For ethylenglycol in zeolites a sharp transition from a single molecule to a liquide-like dynamics is observed

3. An ensemble as small as 6 ethylenglycol molecules is sufficient to perform a liquid-like dynamics

R.Stannarius et al. PRL 75,4698,(1995); M.Arndt et al. PRL 79,2077,(1997); A.Huwe et al. PRL 82,2338,(1999); F.Kremer et al. J.Phys.Cond.Matter 11:A175 (1999); F.Kremer et al.; Chap.6 in:

„Broadband Dielectric Spectroscopy“ (Eds.:F.Kremer and A.Schönhals),Springer (2002)

The counterbalance between surface and confinement effects

Salol in coated and non-coated Sol-Gel glasses

In uncoated nanopores two relaxation processes are observed due to the adsorbed and “free” salol molecules. In silanized nanopores the suface effect is fully

removed.

Salol in silanized nanopores

Confinement effect:The molecular dynamics becomes faster with increasing con-finement.An inherent lengthscale exists for the dynamic glass transtion. M.Arndt et al.PRL 79,2077(1997), F.Kremer et al. J.Phys.Cond.Matter 11:A175 (1999)

Attractive vs.repulsive interactions between guest and host (e.g. Propylenglycol in uncoated and silanised SiO2 surfaces of

Sol-gel glasses)

The attractive interaction between guest and host causes a slowing down ofthe molecular dynamics (Surface effect).Due to the silanization this effect is fully

removed and a dynamics becomes comparable to that of a bulk liquid .

2,5 3,0 3,5 4,0 4,5 5,0-4

-2

0

2

4

6

8

10

d1 < d

2

confinement effect(decrease of T

g)

bulk

1000/TG

bulk

(T=Tg)

=100 s

surface effect(increase of T

g)

log

(1/)

1000/T [1/K]

Molecular dynamics in confinement – a result of the counterbalance between surface and confinement effects.

2,5 3,0 3,5 4,0 4,5 5,0-15

-10

-5

0

confinement effect(decrease of T

g)

surface effect(increase of T

g)

bulkd

1 < d

2

d(l

og

(1/

)) /

d(1

000/

T)

1000/T [1/K]

surface effect (Increase of Tg)

confinement effect(decrease of Tg)

surface effect (Increase of Tg)

confinement effect(decrease of Tg)

Surface effect: Due to the attractive interaction between host and guestsystem the molecular dynamics is slowed down Confinement effect: The growth of the inherent lengthscale of the dynamic glass transition is limited by the external confinement. This causes a change from a VFT- to an Arrehnius temperature dependence

Summary for low molecular weight systems

The dynamic glass transition has an inherent lengthscale which increases with decreasing temperature. At the calorimetric glass transition temperature it has a value in the nanometer range which varies strongly from system to system.The growth of this inherent lengthscale is the physical origin for the Vogel-Fulcher-Tammann (VFT)-dependence.

If due to an external confinement the inherent lengthscale canno longer grow the Vogel-Fulcher-Tammann-dependence turnsinto an Arrhenius-dependence.

R.Stannarius et al. PRL 75,4698(1995); M.Arndt et al.PRL 79,2077(1997); A.Huwe et al. PRL 82,2338 (1999); F.Kremer et al. J.Phys.Cond.Matter 11:A175 (1999);F.Kremer et al. Chap.6 in: „Broadband Dielectric Spectroscopy“(Eds.:F.Kremer and A.Schönhals),Springer (2002)

The molecular dynamics in confinement is determined by thecounterbalance between surface- and confinement effects

Thanks to

...DFG for financial support and you for attention