Nonlinear Dielectric Effects in Simple...

Nonlinear Dielectric Effectsin Simple Fluids

High-Field Dielectric Response:Coulombic stressLangevin effecthomogeneous ‘heating’heterogeneous ‘heating’

Ranko RichertLIN

EAR

RES

PON

SE -

THEO

RY

AN

D P

RA

CTI

CE

Søm

ines

tatio

nen,

Hol

bæk,

Den

mar

k1

-8 J

uly

2007

definitions of dielectric and electric quantities

‘permittivity of vacuum’

ε0 = 8.854×10-12 F m-1

= 8.854×10-12 A s V-1 m-1

= 8.854×10-14 S s cm-1

V voltage I current Q chargeD displacement E electric field P polarizationε dielectric function M electric modulus χ susceptibilityj current density σ conductivity ρ resistivity

steady state relations

( )

( ) tdtIQdVE

AQD

EEEDP

DMEED

t

′′===

−==−=−=

==

∫0

000

00

1 1

εχεχεεεε

εε

ρρρ

σσσεεε

ωερ

σρε

εωεσ

′′−′=→←′′+′=↓↓

↑↑

′′+′=→←′′−′=

=

==

=

iMiMM

ii

iM

M

i

ˆˆ

ˆˆ

0

0

/ˆˆ

ˆ/1ˆˆ/1ˆ

ˆˆ DjjE

DjjjEjAIj

&

&

==

+=+==

=

charges free

ρ

σ σεσ

dielectric relaxation techniques

0

1

0

1ε''

(ω)

ε'(ω)log(ωτ)

( )( )[ ]

Negami -

Havriliak

1 *

⎭⎬⎫

⎩⎨⎧

+

−+= ∞

∞ γαωτ

εεεωεis

( ) { } Debye1

*

ωτεεεωε

is

+−

+= ∞∞

E ε ε D 0=

fieldelectric

ntdisplacemedielectric

E dV

AQ D ≡∝≡

typical case near the glass transition

10-1 100 101 102 1030

10

20

30

40

50

60

70

ε'

ν [Hz]10-1 100 101 102 1030

10

20

30 174 K - 198 K (3K)

propylene glycol

ε''

ν [Hz]

non-linear susceptibility

( ) ( ) ( ) ( )

( )( ) ( )

( ) ( ) K

K43421321321

+++=

−−==

++++++====

4422

0

44

symmetry0

3322

symmetry0

1

causality00

:symmetry00 :causality

EE

EEEEelectret

χχχε

χχχχχχε

EP

EPEPP

EEEEEP

( ) ( ) [ ]( )

ss

s

ss

E

E

EE

λεε

ελ

λεε

−=Δ

Δ−=

−×=

2

2

22

:factorPiekara ln

10

( ) ( ) ( ) ( )∫∫ ∞−∞−−==

ttdtEdPtP θθϕθχεθ 0 :principle ionsuperposit

( ) ( ) dttJJkTVLFLJ

J

L

eee

e

FFeF

F

∫∞

===

=

==0

000

0

0

flux conjugate in the nsfluctuatio mequilibriu theoffunction n correlatio-auto

torelated are tscoefficiennsport linear tra :s)(1950' Kubo R. Green, S. M.

variety of high field techniques

0 5 10 15-60

-40

-20

0

20

40

60

volta

ge

time0 5 10 15

-60

-40

-20

0

20

40

60

volta

ge

time0 5 10 15

-60

-40

-20

0

20

40

60

volta

ge

time

high DC fieldlow AC fieldmainly 1ω

high AC fieldno bias1ω + 3ω

low AC fieldon high pulse

NDE(t)

from nonlinearity to cooperativity length scale ?

We argue that for generic systems close to a critical point, an extended fluctuation-dissipation relation connects the low frequency nonlinear (cubic) susceptibility to the four-point correlation function. In glassy systems, the latter contains interesting information on the heterogeneity and cooperativity of the dynamics. Our result suggests that if the abrupt slowing down of glassy materials is indeed accompanied by the growth of a cooperative length l, then the nonlinear, 3ωresponse to an oscillating field (at frequency ω) should substantially increase and give direct information on the temperature (or density) dependence of l. The analysis of the nonlinear compressibility or the dielectric susceptibility in supercooled liquids, or the nonlinear magnetic susceptibility in spin-glasses, should give access to a cooperative length scale, that grows as the temperature is decreased or as the age of the system increases. Our theoretical analysis holds exactly within the modecoupling theory of glasses.

Physical Review B 72, 064204 (2005)Nonlinear susceptibility in glassy systems: A probe for cooperative dynamical length scales

Jean-Philippe Bouchaud and Giulio Biroli

Coulombic stress

Coulombic stress exercise

QVEnergyd

ACCVQ

VQC

21

0

=

===εε

( )( ) ( )

[ ][ ] A

FL

Lstrainstress

E

tVtVtV

Δ=≡

=

=

0

0

modulus mechanical

GPa 1 modulus sYoung'hspacer witTeflon assume

measuredon effect calculatesin

voltagefrom resulting force calculate

εω

rFrF 3304 r

qqr

qqcgsSI

′=

′=

πε10 µm

top electrode 16 mm Øbottom electrode 30 mm Øring outer: 20 mm Øring inner: 14 mm Ø

Coulombic stress solutions

10 µm

( )

QVEdA

C

VQC

rqq

SI

21

0

30

1

4

=

==

=

′=

εε

πεrF

( )

( )

( )

( )[ ]

( ) ( )[ ]

[ ][ ] A

FL

LstrainstressE

tV

tVVVdA

tVVdA

F

VVdA

VdA

ddAVF

dAVCVVVCCVV

QVVQQVdFQVenergyFdwork

dcdc

dc

acdc

Δ==

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

−=

+==∂

∂=

∂=∂+∂=∂

=∂+∂=∂=∂

=

−

−

=

=

0

20

02

20

202

0

22

022

01

02

21

10

2212

21

0

21

21

21

0

21

21

21

modulus sYoung'

2cos12

sin22

sin2

22

ωωε

ωε

εεε

ε43421

321

( ) ( ) ( )[ ]

4

0

0

25

20

00

20

020

107.6ln

MPa 0.67N 31

(Teflon) GPa 1mm) 16-14d ring,(Teflon m 107.4

4ln

2cos142

−

−

×≤Δ

−=Δ

==

==×=

=+=Δ

−=Δ

−×==

dd

aFFYa

EaY

AaYF

dd

tEAtEAtF

s

ss

ε

εεε

ωεεεε

impedance approach to mechanical loss in a cantilever

Vac Iac

Lt

df

Vdc

dp

x

h

Vac Iac

Lt

df

Vdc

dp

x

h

Vac Iac

Lt

df

Vdc

dp

x

h

1 cm

R. Richert, Rev. Sci. Instrum. 78 (2007) 053901

ringdown of cantilever at resonance frequency

0.0 0.5 1.0 1.5

-1.0

-0.5

0.0

0.5

1.0

τ = 0.61 s

x(t)

/ xm

ax

time / s


k' and k'' across 3.5 decades in frequency

60 61 62 63 64 65 66

A(ν)

ν / Hz

10-1 100 101 102

100

101

102

A

(ν) /

A0

ν / Hz

10-1 100 101 102-270

-180

-90

0

90

ϕ (d

eg)

ν / Hz

10-1 100 101 10210-3

10-2

10-1

100

'int'

'ext'

tan

ϕ

ν / Hz

( )( )

( ) 2ext

22ext

22

0

0

1tan

1

,

μμωϕ

μμω

ρ ωω

−=

+−=

==++

D

D

AA

AexeFkxxxm titi&&&

( )( )

( ) 2

2int

22

0

0

tan

1

, ˆ

μωϕ

μω

ωω

kkk

D

AA

AexeFxkixkxmxkxm titi

−′′′

=

+−=

==′′+′+=+ &&&&

EEkkDmkD

km

′′′=′′′==

==

int

ext

0

ρ

ωωωμ


dielectric saturation

expected non-linearity: dielectric saturation, Langevin effect

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

a/3

⟨cosθ⟩ = 1

L(a) ≈ a/3

a = μE / kT

L(a)

= ⟨c

osθ⟩

( ) ( )

kTEa

akTEa

aaL

kTEa

kTEL

kTE

kTE

de

de

kTE

kTE

33cos

1or if

1cotanhcos

with

function Langevin

cos

cotanhcos

cos

cos

10

cos

0

cos

μθ

μ

θ

μ

μθ

μμθ

θ

θθθ π

θμ

πθμ

=≈

<<<<

−==

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

⇒=

−

∫

∫

saturation in water at 293 K

( ) ( )( )

( )( ) ( )222

44

30

4

2

2

2

2

222

450vanVleck

Piekara

∞∞

∞

+++

×−=−

=Δ

−=Δ

εεεεεε

εμεεε

λεε

ss

ssss

ss

kTVN

EE

E

E

H. A. Kolodziej, G. Parry Jones, M. Davies, J. Chem. Soc. Faraday Trans. 2 71 (1975) 269

J. H. van Vleck, J. Chem. Phys. 5 (1937) 556A. Piekara, Acta Phys. Polon. 18 (1959) 361

Langevin and chemical NDE's

J. Malecki, J. Mol. Struct. 436-437 (1997) 595

time resolved NDE experimens

M. Gorny, J. Ziolo, S. J. Rzoska, Rev. Sci. Instrum. 67 (1996) 4290

high field impedance

GEN

V - 1

V - 2

SI-1260 PZD-700

× 100

÷ 200

Cgeo = 136 pF

R = 500 Ω

R = 50 Ω

10 µmGEN

V - 1

V - 2

SI-1260 PZD-700

× 100

÷ 200

Cgeo = 136 pF

R = 500 Ω

R = 50 Ω

10 µm

10-1 100 101 102 1030

10

20

0

20

40

60

ε''

ν [Hz]

propylene glycol

ε'

S. Weinstein, R. Richert, Phys. Rev. B 75 (2007) 064302S. Weinstein, R. Richert, J. Phys.: Condens. Matter 19 (2007) 205128

field effect results for propylene glycol[ 71 - 282 kV/cm steady state harmonic measurement ]

S. Weinstein, R. Richert, Phys. Rev. B 75 (2007) 064302S. Weinstein, R. Richert, J. Phys.: Condens. Matter 19 (2007) 205128

( )( )

( ) ( )( )Vleckvan

222

45 222

44

30

4

2∞∞

∞

+++

×−=Δ

εεεεεε

εμε

ss

ss

kTVN

E

10-1 100 101 102 103 1040

10

20

0

20

40

60

ε''ν [Hz]

propylene glycol

ε'

-8

-6

-4

-2

0

Δ ln ε'5.6 Hz - 100 Hz

propylene glycol

1000

× Δ

ln ε

'

energy absorbtion

(less) expected non-linearity: sample heating via dielectric loss

1 0 -1 -2 -3 -40.0

0.1

0.2

0.3 βKWW = 0.5

log10(τ / τKWW)

heating

g KWW

(lnτ)( )

K 1.0

cm K J 2

cm J 22.0

MVm 2.2810

31

3

10

200

≈=Δ

≈

=

=

≈′′

′′=

−−

−

−

p

eq

p

eq

eq

cq

T

c

qE

Eq

υ

υ

ε

ωεπευ

predicted heating/temperature effects

101 102 103 1040

5

10

15

20

25

30

glycerolT = 213 K

ε''

ν [Hz]

( )

05.0ln

K 10

K 000 20

ln

exp

:dependence etemperatur

0

≈Δ

=Δ

≈Δ

≈

Δ−=Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

τττ

τ

ττ

.TkE

TkE

TT

TkET

B

A

B

A

B

A

x 1.05

features of heterogeneous dynamics

time

ξ

length scale(cooperativity)

domain life timerate exchange

R. Richert, J. Phys.: Condens. Matter 14 (2002) R703

DIELECTRIC HOLE-BURNINGIDEA AND METHOD

spectrally selective dielectric experiments

( ) ( )tEtE bb ωsin=

bωτττ 1321 ≈≈≈

( ) 20 bb EQ ωεπε ′′=Δ

1 0 -1 -2 -3 -40.0

0.1

0.2

0.3

anti-hole

hole

βKWW = 0.5

log10(τ / τKWW)

spectrallyselective'heating'

g KWW

(lnτ)

1 0 -1 -2 -3 -40.0

0.1

0.2

0.3 βKWW = 0.5

log10(τ / τKWW)

heatingg KW

W(ln

τ)

B. Schiener, R. Böhmer, A. Loidl, R. V. Chamberlin, Science 274 (1996) 752B. Schiener, R. V. Chamberlin, G. Diezemann, R. Böhmer, J. Chem. Phys. 107 (1997) 7746R. V. Chamberlin, B. Schiener, R. Böhmer, Mat. Res. Soc. Symp. Proc. 455 (1997) 117

dielectric ε(t) and electric M(t) hole-burning

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )09080302

9832

tMtMtMtMtMtMtMtM

tidle

−+−−+−

=φ

M 9

M 8

M 7

M 6

M 5

M 4

M 3

M 2

M 1

E ,

D

time -3 -2 -1 0 1 2-3

-2

-1

0

1

2

3

M1

M5

M7 M3

M2M4

M6

U(t)

/ U

(0)

log10(t/s)

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )07060504

7654

tMtMtMtMtMtMtMtM

tburn

−+−−+−

=φ

phase-cycle

-3 -2 -1 0 1 20.0

0.5

1.0original

responseP(t)

plainheating

P*(t)

selectiveheating

P*(t)

P(t)

, P*(

t)

log10(t/s)

dielectric hole-burning technique: what do we look for?

( ) ( ) ( )tPtPtPV*−=Δ

( ) ( ) ( )( ) ( )stdtdP

tPtPtPH ln

*

−−

=Δ

vertical difference

horizontal difference

DIELECTRIC HOLE-BURNINGEXPERIMENTS

dielectric relaxation and retardation in viscous glycerol

K. Duvvuri, R. Richert, J. Chem. Phys. 118 (2003) 1356

-2 -1 0 1 2 3 40.00

0.05

0.10

0.15

0.20

0.25T = 191.80 K

M''(ω)

M'(ω)

M'(ω) , M

''(ω)

log10(ω / s -1)

0

10

20

30

40

50

60

70

80

ε''(ω)

ε'(ω)

ε'(ω

) , ε

''(ω

)

0

20

40

60

80

ε'(ω

)

-4 -3 -2 -1 0 1 2 30510152025

log10(ω / s -1)

ε''(ω)

frequencydomaindata ε*(ω,T)

ε*(ω) versus M*(ω)

sample thickness: 6.4 μm

T = 183.50 KT = 187.30 KT = 191.80 KT = 195.80 KT = 200.50 K

exploiting M(t) for high frequency hole-burning


10-3 10-2 10-1 100 101 1020.0

0.2

0.4

0.6

0.8

1.0

Mn(t), Cpar = 0

Mn(t), Cpar = 5 nF

εn(t)T = 200.50 K

τb

Mn(t

) , ε

n(t)

t / s

10-3 10-2 10-1 100 101 102 103 1040.00.20.40.60.81.0

Mn(t

)

t / s

0.00.20.40.60.81.0

εn (t)

εττ ×≈801

M

variable τ: τM … τεusing parallel capacitance

1→+

+→≈ ∞∞

pars

par

s

M

εεεε

εε

ττ

ε

T = 183.50 KT = 187.30 KT = 191.80 KT = 195.80 KT = 200.50 K

DHB results for glycerol: vertical and horizontal differences


10-2 10-1 100 101 102

0

1

2

0

1

2

3 ΔM(t)

103 ×

ΔM

(t)

T = 187.30 K

t / s

ΔH(t)

102 × ΔH

(t)

vertical & horizontalsignal at:

n = 6tw = 1sfb = 0.2 HzVb = 90 VEb = 140 kV/cmT = 187.30 K

DHB results for glycerol: burn-frequency dependence


-2 -1 0 1

-2

-1

0

1

T = 183.50 K T = 187.30 K T = 191.80 K T = 195.80 K T = 200.50 K

log 10

(t max

/ s)

log10(τb / s)

-1 0 1 2 3 4 5 6 7

10-2

10-1

100

101

ΔHmax

ε''(ω)

ε''(ω

)

log10(ω τHN)

0.0

0.1

0.2

0.3 ΔHm

ax (0) Eref 2 / E

b 2

position of vertical holeversus

burn frequency

amplitude of horizontal holeversus

relative burn frequency


-2 -1 0 1 2

-1

0

1

2

τmod = 10 × τb

T = 187.30 K T = 191.80 K T = 195.80 K T = 200.50 K

log 10

(τm

od /

s)

log10(τb / s)

DHB results for glycerol: waiting-time dependence

pump - wait - probetw

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

ΔΔ

modmax

max exp0 τ

ww tM

tM

10-3 10-2 10-1 100 101 1020.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

fb = 0.02 Hz fb = 0.06 Hz fb = 0.20 Hz fb = 0.60 Hz fb = 2.00 Hz fb = 6.00 Hz

T = 187.300 K

ΔMm

ax(t w

) / Δ

Mm

ax(0

)

tw / s

DIELECTRIC HOLE-BURNINGMODEL

calculation of energy loss (heating) for RC network

R. Richert, Physica A 322 (2003) 143

C∞

C1 C2 C3 C4 CN

R1 R2 R3 R4 RN...

...

τ

g(ln

τ)

polarization

powerRC=τ

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−=

=∝

−==

TkE

TtTRtR

RCtT

ctItV

dttdT

tCVtQtP

RCtVtVtI

CdttdV

tV

B

ARB

R

p

RRR

CC

CXR

C

X

1

1

:applied voltageexternal

( ) ( )⎩⎨⎧ ≤≤

=otherwise,0

20,sin0 bbx

nttEtE

ωπω

( ) ( ) ( ) ( ) ( )[ ]( )[ ] ( )⎪⎩

⎪⎨⎧

>−−

≤≤−−+×

+′′

=bb

bbbb

b

bb

nttn

nttttEtpωπττωπ

ωπτωωτω

τωωωευε

2,2exp2exp1

20,expcossin

1 2

2

22

200

( )beq Eq ωευπε ′′= 200

bnt ωπ20 ≤≤ bnt ωπ2>

steady state case:

calculation of heating p(t) for RC network

R. Richert, Physica A 322 (2003) 143

0 2 4 6 8 10 12 14-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Vx(t)

p(t)

P(t)

ampl

itude

t / s

0 π 2π 3π 4π

bnt ωπ20 ≤≤

dielectric and thermal relaxation times in glycerol

K. Schröter and E. Donth, J. Chem. Phys. 113 (2000) 9101

N. O. Birge, S. R. Nagel, Phys. Rev. Lett. 54 (1985) 2674

C∞

C1 C2 C3 C4 CN

R1 R2 R3 R4 RN...

...

τ

g(ln

τ)

cp ∞

τ1

cp 1 ...

...τ2

cp 2

τ3

cp 3

τ4

cp 4

τN

cp N

phonon bath

τ

g(ln

τ)βKWW = 0.64 βKWW = 0.65

0

20

40

60

80

REA

L ε

-2 -1 0 1 2 30

10

20

30

LOG 10 ω

IMA

G ε

=

also assume: tau's are locally correlated

cp ∞

τ1

cp 1 ...

...τ2

cp 2

τ3

cp 3

τ4

cp 4

τN

cp N

phonon bath

C∞

C1 C2 C3 C4 CN

R1 R2 R3 R4 RN...

...

τ

g(ln

τ)

K.R. Jeffrey, R. Richert, K. Duvvuri, J. Chem. Phys. 119 (2003) 6150

Model:locally correlated structural and thermal relaxation time heterogeneity

( )

s 285 ,58.0 ,95.07.75 ,7.3

K 1021ln

cmJK 5.1

4

31

=====

×=∂∂

=

=Δ

∞

−−

HNHNHN

s

effA

p

TE

c

τγαεε

τ

DHB in glycerol: calculated vs. measured results

10-2 10-1 100 101 102

0

1

2

0

1

2

3 ΔM(t)

103 ×

ΔM

(t)

t / s

ΔH(t)

102 × ΔH

(t)

-2 -1 0 1

-1

0

1

log 10

(t max

/ s)

log10(τb / s)

vertical & horizontalsignal at n = 6

tw = 1sfb = 0.2 HzVb = 90 VEb = 140 kV/cm


10-2 10-1 100 101 1020.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

ΔMm

ax(t w

) / Δ

Mm

ax(0

)

tw / s-2 -1 0 1

-1

0

1

2

tmax vs. τb

log 10

(τm

od /

s)

log10(τb / s)

DHB in glycerol: calculated vs. measured results


FURTHER MODEL'PREDICTIONS'

DHB: insight from the model[ hole amplitude and position ]

-3 -2 -1 0 1 2 3 40

1

2

3

4

Δ V Pε, max

tw = 0

Δ V PM, max

Δ V P

max

× 1

03

log10(τb / s)-3 -2 -1 0 1 2 3 4

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3 4

-3

-2

-1

τHN

gHN(ln τ)

log 10

g(ln

(τ/s

))

log10(τ / s)

tmax = τHN

tw = 0

Δ V Pε Δ V PM Δ V PM exp.

log 10

(t max

/ s)

log10(τb / s)

S. Weinstein, R. Richert, J. Chem. Phys. 123 (2005) 224506

DHB: insight from the model[ hole recovery and accumulation ]

-3 -2 -1 0 1 2 3 40.4

0.6

0.8

1.0

-1

0

1

2

3

Δ V Pε Δ V PM

β

log10(τb / s)

τmod = τHN

Δ V Pε Δ V PM Δ V PM exp.

τmod = τb

log 10

(τm

od /

s)

1 10 100 1000

-1

0

1

2

Δ V PM

Δ V Pε

log 10

(t max

/ s)

n

0

1

2

3

4

5

tw = 0

Δ V Pε, max

Δ V PM, max

Δ V

P max

× 1

03

fb = 0.2 Hz

fb = 0.002 Hz

fb = 0.2 Hz

S. Weinstein, R. Richert, J. Chem. Phys. 123 (2005) 224506

high field impedance

model prediction for 300 kV/cm steady state harmonic field

0

5

10

15

20

25

30

-2 -1 0 1 2 3 4

0

5

10

E = 3×105 V / cm E = 0 V / cm

ε''

100

× Δ

ln ε

''log10(ωτHN)

0

20

40

60

80

-2 -1 0 1 2 3 40

5

10

15

20

25

E = 3×105 V / cm E = 0 V / cmε'

log10(ωτHN)

αHN = 0.95γHN = 0.58τHN = 285 sΔε = 72ε

∞ = 3.7

Dgeo = 20 mmsgeo = 6.4 μmΔCp = 1.5 J/K/cm3

T = 187.30 KEact = 20000 KV0 = 200 V

ε''

R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (2006) 095703

continuous harmonic (impedance) measurement(glycerol)

101 102 103 1040

5

10

15

20

25

30

glycerol

E0 = 14 kV/cm

T = 213 K

E0 = 282 kV/cm

ε''

ν [Hz]

10 µm

101 102 103 1040

5

10

15

20

25

30

glycerolT = 213 K

ε''

ν [Hz]

GEN

V - 1

V - 2

SI-1260 PZD-700× 100

÷ 200

D.U.T.

500 Ω

50 Ω


082.0 K 213

MVm 2.28D 56.2

1 =⎪⎭

⎪⎬

⎫

==

=−

kTE

TE

GLY μμ

model for high-field steady-state harmonic measurement

( )

( )

( )

( )

( ) ( ) ( )

( ) ( )( ) 22

22200

22

200

200

22

DT

200

200

0

12

12

2

!capacity heat ionalconfigurat 1

:h branch witor domain single afor

: 0 excess averaged-period22

:periodover averaged power

:periodper energy statesteady :applied sin field harmonic

τωτωεε

τωτω

τττττωεετωωεε

τττττω

τωεωε

τττ

ττ

ωωεεπω

ωεπε

ω

b

b

pb

b

p

b

p

bbR

pp

pb

bb

p

RTR

T

R

p

RR

R

bbbR

R

b

b

cE

dgdg

VcVE

CVET

dgcVC

cdg

cPTT

cP

dtdT

dtdTT

VEQP

P

VEn

QnQ

tE

+ΔΔ

=+Δ

Δ=

′′=

Δ==+

Δ=′′

==

=⇒−=

=

′′==

′′=

∞

∞

∞

-2 -1 0 1 2

0.0

0.2

0.4

0.6

0.8

1.0

m = 0

m = 1

m = 2

(ωτ)

m /

[ 1 +

(ωτ)

2 ]

log10(ωτ)


high voltage impedance results for glycerol

101 102 103 104

4

6

810

20

glycerol

E0 = 14 kV/cm

T = 213 K

Δε = 71.8τHN = 1.8 msαHN = 0.93γHN = 0.65

E0 = 282 kV/cm

ε''

ν [Hz]

( ) ( )∫∞

∞ +Δ+=

+×

ΔΔ

×−=

0

22

22200

2*

*11ˆ

12 lnln

:modelcurrent

τωτ

τεεωε

τωτωεεττ

di

g

cE

TT

p

A

( )( )[ ]γαωτ

εεεωεi

s

+

−+= ∞

∞1

:fit Negami-Havriliak

*


high voltage impedance results for glycerol[details of frequency dependent field effects ]

102 103 104

0

2

4

6 212 kV/cm 177 kV/cm 141 kV/cm

100

× Δ

ln ε

''

ν [Hz]


summary of relevant non-linear effects (~E2)

S. Weinstein, R. Richert, Phys. Rev. B 75 (2007) 064302

0 2 4 6 80

2

4

6

8Δ ln ε''

422 Hz - 42.2 kHz

100

× Δ

ln ε'

'

E0 2 [1010 V 2 cm-2 ]

-8

-6

-4

-2

0

Δ ln ε'5.6 Hz - 100 Hz

propylene glycol

1000

× Δ

ln ε'

( )( )

( ) ( )( )Vleckvan

222

45

:saturation dielectric

222

44

30

4

2∞∞

∞

+++

×−=Δ

εεεεεε

εμε

ss

ss

kTVN

E

( )

( ) ( )∫∞

∞ +Δ+=

+×=

+×

ΔΔ

=

0

220

2200

e220

220

200

*11ˆ

1

12

:absorptionenergy

τωτ

τεεωε

τωτω

τωτωεετ

di

g

Tc

ETp

e

( ) ( ) ( )tEAEAtEAtF

dACCVQQVFd

sss ωεεεεεεεε

2cos442

,,:stress Coulombic

20

0

N 31

20

020

021

−==

===

≈43421

L.-M. Wang, R. Richert, Phys. Rev. Lett. (submitted)

100 101 102 103 104

02468

E0 → 100 kV/cm

Δcp = 0.47 J K

-1 cm-3

(b)

100

× Δl

nε''

ν / Hz

106 kV/cm 141 kV/cm 177 kV/cm

100

101

102

(a) 14 kV/cm 177 kV/cm

ε''

propylene carbonateT = 166 K

situation with other liquids: propylene carbonate

loss increases

by up to 20 %

at electric field

of 177 kV/cm

ΔTcfg = 0.185 K

( )( ) ( ) ( )∫ ′′−′−Δ= fusT

T

crystp

meltpfusexc

exc

TdTCTCSTS

TS

log

:entropy excess is

On the Temperature Dependence of Cooperative Relaxation Properties in Glass-Forming LiquidsAdam G and Gibbs J H 1965 J. Chem. Phys. 43 139.

( ) ( )( ) ( )

( ) 0lim :emperatureKauzmann t is 1 :if VFT toequivalent

exp0

=−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

→

∞

TSTTTSTS

TTScT

cTTK

Kc

cfg

K

ττ

excess specific heat =configurational modes + excess vibrational modes

Adam-Gibbs: dynamics - thermodynamics link

'forward' dynamic heat capacity experiment

N. O. Birge, S. R. Nagel, Phys. Rev. Lett. 54 (1985) 2674

cp ∞

τ1

cp 1 ...

...τ2

cp 2

τ3

cp 3

τ4

cp 4

τN

cp N

phonon bath

τ

g(ln

τ)

(b)

heat capacity

configTg↓

liquid

glass

crystal

Cp(T

)

T

(a)entropy

config

↑ TK

↑Tm

Tg↓

crystal

liquid

glass

S(T

)

T

(d)

T1 < T2 < T3

Cp''(

ω)

log ω

(c)× ×ex.-vib.

crystal

liquidconfig

glass

log ωC

p'(ω)


40 60 80 100 120 1400

2

4

6

8(a)

SOR

B

PC

NM

EC

24PD

12PD

GLY

PG

E0 → 100 kV/cm

100

× Δl

nε''

m

0.0

0.5

1.0(b)

Cp,

cfg /

Cp,

exc

measuring the configurational heat capacity

cp ∞

τ1

cp1 ...

...τ2

cp2

τ3

cp3

τ4

cp4

τN

cpN

phonon bath cp ∞

τ1

cp1 ...

...τ2

cp2

τ3

cp3

τ4

cp4

τN

cpN

phonon bath

( ) ( )∫∞

∞ +Δ+=

+×

ΔΔ

×−=

0

22

22200

2*

*11ˆ

12 lnln

:modelcurrent

τωτ

τεεωε

τωτωεεττ

di

g

cE

TT

p

A


( ) ( )

( ) ( )[ ]

( )

( ) ( )[ ]

( )

⎟⎠⎞

⎜⎝⎛ Δ−=

⎟⎠⎞

⎜⎝⎛=+=

=

+=

=

+=

+=

∫

∫

ϕπδ

ϕ

ϕ

ϕωωπω

ϕ

ϕωωπω

ϕωω

ωπ

ωπ

2tantan

arctan

sin

sincos

cos

sinsin

sin

22

2

0

2

0

RIIRA

A

dttAtI

A

dttAtR

tAV

Fourier analysis of signal with field change

0.00 0.05 0.10-0.8-0.6-0.4-0.20.00.20.40.60.8

0.00 0.05 0.10-0.8-0.6-0.4-0.20.00.20.40.60.8

current

time / s

voltage

time / s


-10 0 10 20 30 40 50

0.40

0.45

0.50

νΔt

curr

ent

← 170 17 → kV/cm kV/cm

ν = 1 kHz

tan δ

time / ms

( ) ( )

( ) ( )[ ]

( )

( ) ( )[ ]

( )

⎟⎠⎞

⎜⎝⎛ Δ−=

⎟⎠⎞

⎜⎝⎛=+=

=

+=

=

+=

+=

∫

∫

ϕπδ

ϕ

ϕ

ϕωωπω

ϕ

ϕωωπω

ϕωω

ωπ

ωπ

2tantan

arctan

sin

sincos

cos

sinsin

sin

22

2

0

2

0

RIIRA

A

dttAtI

A

dttAtR

tAV

time resolved nanoscopic heat-flow

( ) ( )

( ) ( )[ ]

( )

( ) ( )[ ]

( )

⎟⎠⎞

⎜⎝⎛=+=

=

+=

=

+=

+=

∫

∫

RIIRA

A

dttAtI

A

dttAtR

tAV

arctan

sin

sin3cos

cos

sin3sin

sin

22

2

0

2

0

ϕ

ϕ

ϕωωπω

ϕ

ϕωωπω

ϕωω

ωπ

ωπ

3ω signal during field change

-20 0 20 40 60

10-4

10-3

10-2

10-1

voltage

current

A 3ω /

A ω

time / ms (periods)

Nonlinear Dielectric Effects in Simple...

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