super-arrhenius behavior of the structural relaxation times viscosity measurements under high...

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 ?


“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


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


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


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



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


metal sphere0.05 mm diameter

0.25 mm

0.5 mm

diamond

stainless steel gasket

ruby chip

Frequencycounter

Laser

Mirror

F=2R


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:


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]


Volume dependence of the -relaxation times


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


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



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:


53.0P

V

EE




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)



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



gPP

V

TE

E

1

1

For polymer Boyer-Spencer rule:

PTg=0.2


P

V

P

V

m

m

E

E


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

super-arrhenius behavior of the structural relaxation times viscosity measurements under high...

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marian paluchtemperature

volume dependence

thermal energy

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light scattering

thermal effects

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