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Super-Arrhenius behavior of the structural relaxation times Viscosity measurements under high pressure
Thermodynamical scaling
Marian PaluchInstitute of PhysicsSilesian UniversityKatowice, POLAND
Marian Paluch Silesian University
1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2-5
0
5
10
15
Tg
TA
log
[P
a*s]
1000/T [K-1]
1.5 2.0 2.5 3.0
0
100
200
300
400
500E
A[kJ/mol]
Arrhenius
1000/T
1:3:5-tri--naphthylbenzene; D. J. Plazek and Magill, J. Chem. Phys.
RTEA 0loglog
The Arrhenius law:
1
log)(
T
RTEA
Activation energy:
Is the super-Arrhenius behavior near Tg at ambient pressure governed primarily by the decreasing volume, the decreasing temperature, or both ?
Marian Paluch Silesian University
“unambiguously that it is temperature, and not density, that is the overwhelmingly dominant control variable”; „This suggest that theories which focus on thermal effects and totally ignore density variations can be appropriate” (Ferrer, et al., 1999)
“relaxation processes arise from molecular motions that are driven by temporal fluctuations in thermal energy … not a time-averaged quantity such as free volume” (Williams, 1997)
Prevailing viewpoint:
Marian Paluch
Temperature dominated Volume dominated
( , ) exp
E TT V
kT
0( , ) expf
VT V C
V
T=T1
T=T2
T=T3
Log
()
T1< T2< T3
Volume
isobar
isotherm
Volume
Log
()
isotherm
isobar
Marian Paluch Silesian University
F[Hz], C[pF], R[]
Impedance Analyzer
Thermal bath
T[°C]
P[bar]
Pressure meter Hydraulic press
High pressurechamber
Tensometric sensor
Valve
fiff '''*
0
'C
C
02
1''
fRC
10-2Hz – 107 Hz
Schematic illustration of the high pressure dielectric set-up
10-3 10-2 10-1 100 101 102 103 104 105 106 107
1E-3
0.01
0.1
1
10 pressure
''
f [Hz]
10-2 10-1 100 101 102 103 104 105 106 107
1E-3
0.01
0.1
1
10
100
temperature
''
f [Hz]
O
O
OCH3
OCH3 PDE
Tg=294 K
300 320 340 360 380 400 420-10
-8
-6
-4
-2
0
2
log 1
0[ /(s
)]
T [K]
0 50 100 150 200 250-8
-6
-4
-2
0
2
4
76.1oC
63.5oC
54.8oC44.5oC35.8oC
log 1
0[ /(s
)]P [MPa]
23.6oC
max2
1
f
max2
1
f
Dielectric measurements
M. Paluch, R. Casalini, C. M. Roland, Phys. Rev. B 2002
Marian Paluch Silesian University
Schematic illustration of the light scattering techniqueused in high pressure measurements
Thermal bath
T[°C]
Membrane compressorLaser
P [bar]
Avalanch diodedetector
Valve
Correlator
Manometer
sample
High pressure chamber
Marian Paluch Silesian University
1E-4 1E-3 0.01 0.1 1 10 100 1000 10000-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
348.35 K 600 bar 700 bar 800 bar 900 bar 1000 bar 1100 bar 1200 bar 1300 bar 1400 bar 1500 bar 1600 bar 1700 bar
[g(1
) (t)]
2
t [ms]
0 200 400 600 800 1000 1200 1400 1600 1800
-5
-4
-3
-2
-1
0
1
2
3
307.35 K 317.95 K 327.75 K 338.45 K 348.35 K
log
[< K
WW
> [
s]]
P [bar]
KWW
tatg exp1
KWWKWW
KWWKWW
1
Dynamic light scattering measurements
A. Patkowski, M. Paluch, H. Kriegs, J. Chem Phys. 2002
Marian Paluch Silesian University
Marian Paluch Silesian University
0 2 4 6 8 100
2
4
6
8
10
12lo
g 10
(cP
)
Pressure (GPa)
Capillary Diamond-anvil cellCentrifuge
Diamond-anvil cellFalling ball Rolling ball
Fallingbody
Marian Paluch Silesian University
metal sphere0.05 mm diameter
0.25 mm
0.5 mm
diamond
stainless steel gasket
ruby chip
Frequencycounter
Laser
Mirror
F=2R
Marian Paluch Silesian University
The centrifugal force viscometer
TC
PTVTPV 1ln089.01,0,
2210),0( TATAATV TCCTC 10 exp
Tait equation:
M. Paluch, R. Casalini, A. Best, A. Patkowski, J. Chem Phys. 2002
PVT measurements:
Marian Paluch Silesian University
0.70 0.72 0.74 0.76 0.78-10
-8
-6
-4
-2
0
2
Glass transition
PDE
Isotherms: 296.6 K 308.8 K 317.5 K 327.8 K 337.7 K 349.5 K 363.0 K
Isobar: 0.1MPa
log
[/ (
s)]
V [cm3/g]
0.70 0.72 0.74 0.76 0.78-10
-8
-6
-4
-2
0
2
2
1
V1
isobar 0.1MPaisotherm 363K
PDE
log
[/ (
s)]
V [cm3/g]
M. Paluch, R. Casalini, C. M. Roland, Phys. Rev. B 2002
Volume dependence of the -relaxation times
Marian Paluch Silesian University
isochronal expansivity : P = V1 (V/T)P
isobaric expansivity : = V1 (V/T)
(Ferrer, Lawrence, Demirjian, Kivelson, Alba-Simonesco & Tarjus, 1998 )
First approach: Expansion coefficients
PTVP T
V
VTT
logloglog
PTP VT
loglog
TVVT TV log/log
using
>>1
~ 1 comparable
0P
T dominate
V dominate
Marian Paluch Silesian University
280 300 320 340 360 380 400 4200.70
0.72
0.74
0.76
0.78
constan pressure (P = 2 kbar)
constan pressure (P = 1 kbar)
constan pressure (P = 1 bar)
V [
cm3 /g
]
T [K]
constant (=1s)
25.1P
PDE PPGE
1bar 1.25 1.67
2 kbar 1.43 2.4
M. Paluch, R. Casalini, C. M. Roland, Phys. Rev. B 2002
Marian Paluch Silesian University
Volume Temperature (steric constraints) (thermal fluctuations)
0 EV/EP 1
1
logV
V
ET
1
logP
P
ET
M. Naoki and M. Matsushita, Chem. Soc. Jap. 56, 2396 (1983)
Second approach: Activation energies
activation energyat constant volume:
activation energyat constant pressure:
Marian Paluch Silesian University
53.0P
V
EE
M. Paluch, R. Casalini, C. M. Roland, Phys. Rev. B 2002
Marian Paluch Silesian University
Marian Paluch Silesian University
Name Tg EV/EP
BMPC 243 0.39
BMMPC 263 0.41
Salol 220 0.43
KDE 313 0.49
PDE 249 0.53
o-terphenyl 244 0.552
PMPS 246 0.46
PTMS 198 0.55
polystyrene 373 0.64
DGEBA 335 0.6
polyvinylacetate 311 0.6
PPGE 258 0.63
polyvinylmethylether 251 0.69
1,2-polybutadiene 253 0.70
polypropyleneglycol 400 276 (0.78)*
propylene glycol trimer
0.77
sorbitol 273 0.87
propylene glycol 0.89
glycerol4 189 0.94
Small molecules
Polymers
H-bonded liquidsTemperature dominantcontrol variable for H-bonded materials
density and thermalenergy have nearlythe same effect onmolecular dynamics
weaker effect of density due to intramolecularbonding
PP
V
E
E
11
R. Casalini, and C. M. Roland J. Chem. Phys. 119, 4052 (2003)
Marian Paluch Silesian University
Marian Paluch Silesian University
TV scalingParameter is material constant, and independent on P, T and V
The scaling quantity TV- can be followed from the generalized LJ potential with itsmodiefied repulsive and attractive partsproportional to r-3 and r-3/2
4log TVf - for OTP
0.0070 0.0075 0.0080 0.0085 0.0090
-7
-6
-5
-4
-3
-2
-1
0
1
PMPS
T=313K T=293K T=273K T=263K T=252.5K at 1bar
log 10
[ /
(s)]
T-1V-5.6
PMPS
4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.010-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
P = 0.1 MPa = 1.3
T = 216.8 KT = 225.6 K
T = 238.4 K
[s]
1000/(TV)
2PG
0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014
-10
-8
-6
-4
-2
0
2
4
Isotherms: 308.8 K 317.5 K 327.8 K 337.7 K 349.5 K 363.0 K
log 10
[/ (
s)]
T -1V-4.4
PDE
-0.160 -0.155 -0.150 -0.145 -0.140 -0.135
2.46
2.48
2.50
2.52
2.54
2.56
2.58
PVT data
Dielectric data
= 4.4
log 10
[Tg
/(K
)]
log10
[Vg /(cm3/g)]
gg VAT loglog 20 40 60 80 100 120 140 160
0.70
0.72
0.74
0.76
200 MPa
160 MPa
120 MPaPDE
V [c
m3 /g
]T [°C]
At T=Tg .1 constVT gg
Marian Paluch Silesian University
Marian Paluch Silesian University
0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02-8
-6
-4
-2
0
2
4
Isotherms: 308.8 K 317.5 K 327.8 K 337.7 K 349.5 K 363.0 K
log 10
[/ (
s)]
Vg/V 0.95 0.96 0.97 0.98 0.99 1.00
-7
-6
-5
-4
-3
-2
-1
0
1
lo
g10
[ /(
s)]
Vg/V
0.96 0.97 0.98 0.99 1.00 1.01 1.02-6
-4
-2
0
2
279 K 288 K 296 K 307 K
log 10
[ /(
s)]
Vg/V
BMMPC
PTMPS
PDE
0.88 0.90 0.92 0.94 0.96 0.98 1.0010-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
T = 238.4 KT = 225.6 KT = 216.8 K
[s]
Vg/V
g
g VTV
VVT 11