nonlinear dielectric effects in simple...
TRANSCRIPT
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
R. Richert, Rev. Sci. Instrum. 78 (2007) 053901
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
R. Richert, Rev. Sci. Instrum. 78 (2007) 053901
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
K. Duvvuri, R. Richert, J. Chem. Phys. 118 (2003) 1356
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
K. Duvvuri, R. Richert, J. Chem. Phys. 118 (2003) 1356
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
K. Duvvuri, R. Richert, J. Chem. Phys. 118 (2003) 1356
-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
K. Duvvuri, R. Richert, J. Chem. Phys. 118 (2003) 1356
-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
K.R. Jeffrey, R. Richert, K. Duvvuri, J. Chem. Phys. 119 (2003) 6150
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
K.R. Jeffrey, R. Richert, K. Duvvuri, J. Chem. Phys. 119 (2003) 6150
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
R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (2006) 095703
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)
R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (2006) 095703
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
*
R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (2006) 095703
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]
R. Richert, S. Weinstein, Phys. Rev. Lett. 97 (2006) 095703
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)
L.-M. Wang, R. Richert, Phys. Rev. Lett. (submitted)
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
L.-M. Wang, R. Richert, Phys. Rev. Lett. (submitted)
( ) ( )
( ) ( )[ ]
( )
( ) ( )[ ]
( )
⎟⎠⎞
⎜⎝⎛ 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
L.-M. Wang, R. Richert, Phys. Rev. Lett. (submitted)
-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)