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Transcript of Dielectric spectroscopy of glass-forming liquids under high pressure Marian Paluch Institute of...
Dielectric spectroscopy of glass-forming liquids under high pressure
Marian PaluchInstitute of PhysicsSilesian UniversityKatowice, POLAND
Marian Paluch Silesian University
Crystallization and vitrification
glass
crystal
liquid-glasstransition
Tg Tm
Temperature
liqui
d
Vol
um
e
: 1010 s 102 s 10-12 s
crystallization
Tga Tgb
Temperature
TemperatureV
olu
me
Liquid
glass
p=
(ln
V /
T) p
En
thal
py
(H)
Hea
t ca
pac
ity
(Cp)
Marian Paluch Silesian University
Crystallization and vitrification
Tg Tm
10-4
1
10-8
c
Substance dT / dt (K sec-1)
SiO2 7 104
GeO2 7 102
Salol 50
Water 107
Ag 1010
cn
nm
c
TT
dtdT
Temperature
; c
Pg
Pressure
Vol
um
e
glass
supercooled liquidMarian Paluch Silesian University
Pressure
T=
(ln
V /
p) T
Liquid – glass transiton induced by pressure
P3
P2
Temperature
P1 < P2 < P3 < P4
Tg
P1
P4
Vol
um
e
F[Hz], C[pF], R[]
Impedance Analyzer
Thermal bath
T[°C]
P[bar]
Pressure meter Hydraulic press
High pressurechamber
Tensometric sensor
Valve
10-2Hz – 107 Hz
fiff '''*
0
'C
C
02
1''
fRC
Schematic illustration of the high pressure dielectric set-up
Pressure range: up to 1 GPa
Marian Paluch Silesian University
Force
Pressure range: up to 2GPa
force
force
Pressure range: up to 10GPa
Bakelite block
Bakelite block
Steel block
Steel block
tungsten
carbide
anvil
The gaskets were:
Epoxy-fiber laminates (<5GPA)
Sheets of polystyrene (>5GPA)
The sample is confined between
the carbidge anvils by gasket
made of plastic
G. P. Johari and E. Whalley
Faraday Symp. Chem. Soc.1971
Marian Paluch Silesian University
Relaxation dynamics of supercooled liquids
O OO
O CH3
CH3
di-ethyl phthalate
100 102 104 106 10810-2
10-1
100
''
frequency [Hz]
-process
-process
100 102 104 106 10810-2
10-1
100
Temperature; Pressure
''
frequency [Hz]
-process
-process
max2
1
fa
Marian Paluch Silesian University
Marian Paluch Silesian University
2.8 3.2 3.6 4.0 4.4
-9
-6
-3
0
32.8 3.2 3.6 4.0
-9
-6
-3
0 3.0 3.6 4.2 4.8 5.4
-9
-6
-3
0
log
10[<
>/ (
s)]
1000/T [K-1]
log
10[
/(s)]
1000/T [K-1]
log
10[/
(s)]
1000/T [K-1]
Van der Waals liuidDIBP
H-bonded liquidXylitol
PolymerPMPS, Mw=10k
0 20 40 60 80 100 120 140
-6
-4
-2
00 200 400 600 800 1000
-6
-4
-2
00 50 100 150 200 250
-6
-4
-2
0
log
10[
/(s)]
P [MPa]lo
g10
[ /(s
)]
P [MPa]
log
10[
/(s)]
P [MPa]
0 20 40 60 80 100 120 140
180
240
300
3600 200 400 600 800 1000
18
24
30
36
420 50 100 150 200 250
90
120
150
V [c
m3 /m
ol]
P [MPa]
V [c
m3 /m
ol]
P [MPa]
V [c
m3 /m
ol]
P [MPa]
Temperature VFT law:
0
0loglogTT
TDP
Pressure VFT law:
PP
PDT
00loglog
Activation volume:
dPd
RTVlog
303.2
PT 1
RT
VP 0loglog
Marian Paluch Silesian University
Temperature dependence of activation volume
1.00 1.04 1.08 1.12 1.16 1.20 1.24 1.28
320
340
360
380
400
420
440
460
480
500
520
540
PMPS
PTMPS
V [
cm3 /m
ol]
Tg(P)/T
g(1bar)
1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18200
210
220
230
240
250
260
270BMMPC
BMPC
V [
ml/m
ol]
T / Tg
Si
CH3
O n
CH3
Si
CH3
O n
Tg=246 K
PTMPSPMPS
Tg=261 K
OCH3
CH3
CH3
OCH3
BMMPC
Tg=261 K
OCH3
H3CO
BMPC
Tg=240 K
Marian Paluch Silesian University
80 100 120 140 160 18030
40
50
60
70
80
90
sorbitol
xylitol
threitol
glycerol
V [c
m3 /m
ol]
Mw [g/mol]
CH2OH
OHH
HHO
OHH
HHO
CH2OH
CH2OH
H OH
HO H
H OH
CH2OH
CH2OH
OHH
HHO
CH2OH
CH2OH
H OH
CH2OH
Sorbitol Xylitol
Threitol Glycerol
dP
dTRmV g
p303.2
dTg/dP mp Tg (at 100s)
glycerol 35±3 57 188.4
threitol 33±5 79 224
xylitol 34±2 94 247
sorbitol 40±5 128 267
Marian Paluch Silesian University
Isobaric fragility
0.0 0.2 0.4 0.6 0.8 1.0-14
-12
-10
-8
-6
-4
-2
0
2
4
glycerol threitol xylitol sorbitol
=100 s
l1/2
=10-6s
log 1
0[ /
(s)]
Tg/T
Definitions of fragility:
gT
gp
TT
d
dm
log
lTTF g 25.02 2/12/1
g
gap RT
TEm
What is the effect of pressure on mp?
Marian Paluch Silesian University
Effect of pressure on fragilityIt is usually observed that fragility decreases with increasing pressurein the case of Van der Waals liquids.
Van der Waals liquids
-200 0 200 400 600 800 1000 1200 1400 1600 180050
55
60
65
70
75
80
85
90
95
100
279 K
198 K
Effect of pressure on fragilty is often much more complex for H-bonded than for Van der Waals liquids.
0 500 1000 1500 2000 2500 300060
70
80
90
DS
DLS
PDE
kruc
hość
m
P [bar]
0 40 80 120 160 200 24052
56
60
64
68
72
BMPC BMMPC
mT
P [MPa]
Marian Paluch Silesian University
Effect of pressure on glass transition temperature
0 500 1000 1500 2000 2500 3000290
300
310
320
330
340
350
360
370
T=Tg(=100s)
T=Tg(=1s)
PDE
Tg
[K]
P [bar]
Material dTg/dP (K/GPa)
polystyrene 303 [DTA]
polymethylphenylsiloxane 290 [Dielectric]
Polyvinylchloride 189 [PVT]
Polyvinylacetate 210 [DTA]
1,2-polybutadiene 240 [Dielectric]
BMPC 240 [Dielectric]
o-terphenyl (OTP) 260 [DTA]
Salol 204 [Dielectric]
PDE 280 [Dielectric]
glycerol 35 [Dielctric]
cyclohexanol 40 [DTA]
m-fluoroanilina 81 [Dielectric]
21.0 aPaPMPaTPT gg
21
2
110k
gg Pk
kTPT
Andersson-Andersson relation:
Marian Paluch Silesian University
270
280
290
300
3100
50
100150
200250
300
-5
0
5
10
15
The -relaxation time in P-T plane
fv
Bexp0Doolitle equation:
2/1200 CTTTTTv f
2/1200
0loglogCTTTTT
B
kevB m /log2 0
kvC a /4 0
A A’
B
Cohen-Grest model
free volume:
where:
Marian Paluch Silesian University
0 500 1000 1500 2000 2500-8
-6
-4
-2
0
2
4
log
[ /(s
)]
P [bar]
PDE
P 00 aavkTkT 10
P
k
vTPT a
00
2/1
0
2
00
00
00
144
1
loglog
PCTP
CTTP
CTT
PB
Marian Paluch Silesian University
300 320 340 360 380 400 420-10
-8
-6
-4
-2
0
2
PDE
log
[ /(
s)]
T [K]
280 300 320 340 360 380 400 4201
2
3
4
5
6
7
8
9
T0
T
T [K] O
O
OCH3
OCH3
Tg=294 K
0
,K
T PP
c fusPT
C T VS T P S dT dP
T T
melt crystal
P PP
V VV V
T T T
TKCP
T
KS
T
K
T
KdT
T
KTS
K
T
T
c
K
''2
TTS
A
cAG exp
0
expTT
AAG
TB
PCTVPTV 1ln10,,
Adam-Gibbs model
at P =0.1MPa
VFT law:Tait equation:
Marian Paluch Silesian University
0 50 100 150 200 250
-8
-6
-4
-2
0
2
-6 -4 -2 0 2
-1.0
-0.5
0.0
2.4 2.6 2.8 3.0 3.2 3.4
-8
-6
-4
-2
0
2
B
a
PDE
log(
)
P [MPa]
log(
)
1000/T[K]
B
b
B
0.1MPa 363.1K 349.5K 337.7K 327.8K 317.5K
log()
Marian Paluch Silesian University
TBPPTBPTBPSTTT
APT
1ln1ln11exp,
*0
0
0 200 400 600 800 1000
210
240
270
300
T0
*(P)
T0[
K]
P0[MPa]
0 50 100 150 200 250
-6
-5
-4
-3
-2
-1
0
1
3.0 3.3 3.6 3.9
-7
-6
-5
-4
-3
-2
-1
0
1
2
a) 300200
1000.1
log
([s]
)
log
([s]
)
1000/T[K]
b) 298.1277.3287.6
P[MPa]
TBPPTBPTBPS
PTPT
1ln1ln111
000
Marian Paluch Silesian University
);exp(0 kTE
;expexp1
mm EEE
EW mEkT
mE
exp0
mE
dEEWE0
constdEEWEWRZ
SmE
ln2 0
00 ln
2 RZ
SS
ZR
SS 00
2exp30exp
P
P T
T
T
P dPP
SdT
T
CSPTS
00
0,V
T
V
P
S
PT
PT
V
V P
0
1
0
00
0 lnln,P
PV
T
TCSPTS P
Avramov modelAssumption:
Marian Paluch Silesian University
Marian Paluch Silesian University
Equation of state: model of Avramov
PTTr 130exp
p
mm
CV
ZRV 002
Where:
0 is volume expansion coefficient at ambient pressure,Cp is specific heat capacity, Vm is the molar volume and is a constant parameter
Predictions of the Avramov model
Non-linear increase of Tg with pressurePressure independence of fragilty
ZRCP2with
0log2 m
Pe
TT rg 1log2
log301
0
Marian Paluch Silesian University
0 50 100 150 200 250 300
-7
-6
-5
-4
-3
-2
-1
0
1
303 K 294 K 284 K 274 K
log
[ (s
)]
P [MPa]
0 50 100 150 200 250 300
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
303 K 294 K 284 K 274 K
P [MPa]
P1log
Master curve
Cooperative rearrangement canbe visualized as a collective displacement, involving more than two molecules, along the trajectory to form a closed loop
Consequently, the sum of the displacements of all molecules involved in the process is zero
T. Pakula, J. Mol. Liq. 86, 109 (2000)
Unsuccessful attemt whenneighboring elements tryto move in opposite dire-ction
Unsuccessful attempt because the element in the center will not be replaced by any of the neighbors
The Dynamic Liquid Lattice model of Pakula model
Marian Paluch Silesian University
Marian Paluch Silesian University
The probability that a given molecule participates in the collective displacement determines
1
30
1 )(
n
ns
h pnB
In order to obtain an explicit temperature dependence of the relaxation times, Pakula assumed:
A local volume v is assigned to each molecule.This volume can fluctuate, assuming values not smaller than a minimum volume v0. The excess
volume has an exponential distribution:
0
0
0 vv
vvexp
vv
1v
Molecular transport is driven by a thermallyactivated process with potential energy barriers E(v) dependent on the local density of the system. The probability for a molecule to take part in a local rearrangement is given by the Boltzman factor:
kT
vEexpT,vp
1
v
1
0
dvT,vpvT,vpv
Marian Paluch Silesian University
Herein we consider a linear decrease of the activationenergy from Ea1 to Ea2in the range between v0 and vc,
as depicted schematically in Figure Ea1
Ea2
VV0 VC
1
210 expexp
1
1 221
wewee
kTw
EEkT
E
kT
E
kT
E
)/()( 00 vvvvw c
Marian Paluch Silesian University
0.83 0.84 0.85 0.86 0.87 0.88-7
-6
-5
-4
-3
-2
-1
0
1
2
Polymer PMPS
313 K293 K
273 K
263 K
lo
g10
[s
]
V [cm3/g]
Isobar at 0.1MPa
252.5 K
0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.9442000
42250
42500
42750
43000
43250
43500
43750
44000
44250
44500
44750
Isobar: P = 0.1MPa
Isotherms: T = 252.5 K T = 263 K T = 273 K T = 293 K T = 313 K
range of measured V
Polymer PMPS
E/k
[K
]
V [cm 3/g]
Marian Paluch Silesian University
The secondary relaxation process
The molecular mechanism underlying the secondary relaxation in various glass formers can be very different.
Intermolecular origin(motion of the entire molecule
as a whole)
Trivial intramolecular origin(rotational motion of a smallisolated group of the entire
molecule)
KWWKWW
cJG t
1The Johari-Goldstein process
A prediction concerningthe JG relaxation time JG
comes from the coupling model of Ngai
10-210-1100 101 102 103 104 105 106
100
101
secondary process
die
lect
ric
loss
frequency [Hz]
Marian Paluch Silesian University
10-2 10-1 100 101 102 103 104 105 106 10710-3
10-2
10-1
100
101
''~f -
''~f -
T=-112oC
''
frequency [Hz]O O
CH3
OPC
Tg=159 K
”Excess wing”
Excess wing
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106
10-2
10-1
100
101
PC
''frequency [Hz]
AgingLunkenheimer, et. al., PRL
Type A – „excess wing”
•KDE•PDE•BMMPC•Salol•Glycerol•Propylen carbonate, PC
•Sorbitol•Xylitol•Di-butyl phthlate•Di-ethyl phthalate•BMPC
Type B – well resolved peak
Two types of glasses:
Marian Paluch Silesian University
-2 -1 0 1 2 3 4 5 6-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5p=1 bar
340 K331 K325 K318 K
die
lectr
ic lo
ss
log10
[f/Hz]
2.7 2.8 2.9 3.0 3.1 3.2
-6
-5
-4
-3
-2
-1
0
1
2
log 10
[ /(
s)]
1000/T [K-1]
O
O
OCH3
CH3
OCH3
H3C
KDE
Tg=311 K
' ' ' ' ' '
i1
Im''
ttdtd
dt K
K sinexp''
0
”Excess wing”
Log fJG
Marian Paluch Silesian University
Effect of pressure on „excess wing”
PTtPT KWWKWW
cJG ,, 1
10-2 100 102 104 106
10-2
10-1
100
101
10-2
10-1
100
101
"excess wing"
1.8 GPa, 254K0.1 MPa, 158K
propylene
f [Hz]
carbonate
1.78 GPa, 274K0.1 MPa, 162K
''
''
10-2 100 102 104 106
10-3
10-2
10-1
100
101
''
T = 363 K, P = 1371 bar T = 325 K, P = 1bar
f [Hz]
Marian Paluch Silesian University
10-3 10-2 10-1 100 101 102 103 104 105 106 107 10810-2
10-1
100
101
102
103
frequency [Hz]
"
T=223K T=217K T=211K T=205K
10-1
100
101
" T=253K T=247K T=241K T=235K
CH
OCH
OH
3
CHCH3
Iso-eugenol
10-3 10-2 10-1 100 101 102 103 104 105 106 107 108
10-2
10-1
100
f
KWW
= 0.62
"
frequency [Hz]
10-3 10-1 101 103 105 107
10-2
10-1
100
10-3 10-2 10-1 100 101 102 103 104 105 106 107 108 109
1E-3
0.01
0.1
1
f0
KWW
= O.68
"
frequency [Hz]
T=197 K T=195 K T=193 K T=191 K
10-3 10-1 101 103 105 107
10-2
10-1
100
Two secondary relaxation processes
CH
OCH
OH
3
2CH CH
2
eugenol
Marian Paluch Silesian University
10-3 10-2 10-1 100 101 102 103 104 105 106 10710-3
10-2
10-1
frequency [Hz]
below Tg
''
T=187K T=183K T=179K T=175K
10-2
below Tg
''
T=217K T=213K T=209K T=205K
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5
-8
-6
-4
-2
0
2
4
relaxation
-relaxation (the JG process)
-relaxations
log
1000/T [K-1]
CH
OCH
OH
3
CHCH3
Iso-eugenol
Behavior of excess wing below Tg
Relaxation map
Marian Paluch Silesian University
Behavior of the JG process during physical aging
10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107
10-2
10-1 the JG process
aging at T=209 K
"
frequency [Hz]
1 s 5.5 h 16.6 h 27.7 h 50 h
0 10 20 30 40 50
0.004
0.006
0.008
0.010
0.012
0.014
0.016
[s]
time [hour]
(t) dependence
Marian Paluch Silesian University
10-3 10-2 10-1 100 101 102 103 104 105 106 107 10810-2
10-1
100
diel
ectr
ic lo
ss
frequency [Hz]
-84oC (DBP)
-90oC (DBP)
-74oC (DOP)
-80oC (DOP)
Primary and secondary relaxation in DBP and DOP
O OO
O CH3
CH3
DBP O
O
O
O
DOP
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10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107
0.01
0.1
1
10-3 10-2 10-1 100 101 102 103 104 105 106 107
0.01
0.1
1
- process
"excess wing"
''
f [Hz]
- process
"
f [Hz]
Aging at T=-96 oC
10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107
0.01
0.1
"excess wing"
die
lect
ric
loss
f [Hz]
2.7 hour 8.3 hour 13.8 hour 42.4 hour
The excess wing in DOP is the JG process
Two secondary relaxation processes in DBP and DOP
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0.002 0.003 0.004 0.005 0.006 0.007 0.008
-10
-8
-6
-4
-2
0
2
- process
- process
"excess wing"(the JG process)
lo
g[
/ (s
)]
1/T [K-1]
The relaxation map in di-octyl phthalate
Marian Paluch Silesian University
10-2 10-1 100 101 102 103 104 105 106 107
0.01
0.1
1
process
''/'' m
ax
frequency [Hz]
T=190.2 K; P= 0.1 MPa T=293.6 K; P=1.05 GPa
1.2 decade
process
10-2 10-1 100 101 102 103 104 105 106
0.1
1
process
process
''/'' m
axfrequency [Hz]
T=305K; p=1.61 GPa T=180K; p=0.1 MPa
O OO
O CH3
CH3
di-ethyl phthalateO O
O
O CH3
CH3
di-butyl phthalate
Effect of pressure on secondary relaxation processes
Marian Paluch Silesian University
10-2 10-1 100 101 102 103 104 105 106 107
0.0
0.1
0.2
0.3
0.4
0.5
decreasing temperature
P=1 bar
diel
ectr
ic lo
ss "
10-2 10-1 100 101 102 103 104 105 106 107
0.00
0.04
0.08
0.12
0.16
0.20
decreasing temperature
P=0.5 GPa
frequency [Hz]
diel
ectr
ic lo
ss '
'
10-1 101 103 105 107
1
frequency [Hz]
T=183Kp=1bar
T=232Kp=0.5GPa
KWW
=0.4
''
-process
10-1 101 103 105 107
0.1
1
T=232 KP=0.5GPa
T=294 KP=0.6GPasorbitol
DHIQ
''/'' m
ax
frequency [Hz]
NH
DHIQ
Relaxation dynamics in DHIQ at ambient and elevated pressure
R. Richers, et. al. J. Chem. Phys. 2004.
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10-1 100 101 102 103 104 105 106
10-2
diel
ectr
ic lo
ss "
freqency [Hz]
3 4 5 6 7 8 9 10
-10
-8
-6
-4
-2
0
2
4
EA/k=1799.6K
EA/k=4774.3K
Tg=180.4K
m=168
at p=0.5 GPa
JG
log
/s
1000/T [K-1]
Tg=230.3K
m=177.6
101 102 103 104 105 106
0.00
0.05
0.10
0.00
0.05
0.10
0.15
-process
increasing pressure
T=293K
''frequency [Hz]
increasing pressure
-process
T=252K
''
Two secondary modes in DHIQ: which one is the JG process
10-1 100 101 102 103 104 105 106
10-2
the JG relaxation
diel
ectr
ic lo
ss "
freqency [Hz]
DHIQ
NH
E=~40 KJ/mol
NHN
H
2
8
8a4a .....
8a 4a
8
2
trans-DHIQ cis-DHIQ
N
H
H H
H
H
N
H
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