Dielectric spectroscopy of glass-forming liquids under high pressure

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

liquid

Vol

ume

: 1010 s 102 s 10-12 s

crystallization

Tga Tgb

Temperature

TemperatureV

olum

e

Liquid

glass

p=

(ln

V /

T) p

Ent

halp

y (H

)H

eat c

apac

ity (C

p)


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

TTdtdT

Temperature

; c

Pg

Pressure

Vol

ume

glass

supercooled liquidMarian Paluch Silesian University

Pressure

T=(

lnV

/ p

) T

Liquid – glass transiton induced by pressure

P3

P2

Temperature

P1 < P2 < P3 < P4

Tg

P1

P4

Vol

ume

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

'CC

02

1''fRC

Schematic illustration of the high pressure dielectric set-upPressure range: up to 1 GPa


Force

Pressure range: up to 2GPa

force

force

Pressure range: up to 10GPa

Bakelite block

Bakelite block

Steel block

Steel block

tungstencarbide anvil

The gaskets were:Epoxy-fiber laminates (<5GPA)

Sheets of polystyrene (>5GPA)

The sample is confined betweenthe carbidge anvils by gasket

made of plastic

G. P. Johari and E. WhalleyFaraday Symp. Chem. Soc.1971


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

max21fa



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

0loglogTTTDP

Pressure VFT law:

PPPDT

00loglog

Activation volume:

dPdRTV log303.2

PT 1

RTVP

0loglog


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 [c

m3 /m

ol]

Tg(P)/Tg(1bar)

1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18200

210

220

230

240

250

260

270 BMMPC

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


80 100 120 140 160 18030

40

50

60

70

80

90

sorbitol

xylitol

threitol

glycerol

V [c

m3 /m

ol]

Mw [g/mol]

CH2OHOHHHHOOHHHHO

CH2OH

CH2OH

H OH

HO H

H OH

CH2OH

CH2OHOHHHHO

CH2OH

CH2OHH OH

CH2OH

Sorbitol Xylitol

Threitol Glycerol

dPdT

RmV gp303.2

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


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 10

[ /(s

)]

Tg/T

Definitions of fragility:

gT

gp

TTd

dm log

lTTF g 25.02 2/12/1

g

gap RT

TEm

What is the effect of pressure on mp?


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 K198 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]


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

T g [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 PkkTPT

Andersson-Andersson relation:


270

280

290

300

3100

50100

150200

250300

-5

0

5

10

15

The -relaxation time in P-T plane

fvBexp0Doolitle 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:


0 500 1000 1500 2000 2500-8

-6

-4

-2

0

2

4

log

[ /(s

)]

P [bar]

PDE

P 00 aavkTkT 10

P

kvTPT a

00

2/1

0

2

00

00

00

144

1loglog

PCTPCTTPCTT

PB


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 dPT T

melt crystal

P PP

V VV VT T T

TKCP

TKS

TK

TKdT

TKTS

K

T

Tc

K

''2

TTSA

cAG exp

0

expTT

AAG

TBPCTVPTV 1ln10,,

Adam-Gibbs model

at P =0.1MPa

VFT law:Tait equation:


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()


TBPPTBPTBPSTTTAPT

1ln1ln11exp, *

00

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

PTPT1ln1ln111

000


);exp(0 kTE ;expexp

1

mm EEEEW mEkT

mEexp0

mE

dEEWE0

constdEEWEWRZSmE

ln2 0

00 ln

2 RZSS

ZRSS 0

02exp30exp

P

P T

T

T

P dPPSdT

TCSPTS

00

0, VTV

PS

PT

PT

VV P

01

00

00 lnln,

PPV

TTCSPTS P

Avramov modelAssumption:



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

PeTT rg 1log2log30

1

0


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



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 excessvolume has an exponential distribution:

0

0

0 vvvvexp

vv1v

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


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

kTwEE

kTE

kTE

kTE

)/()( 00 vvvvw c


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 K263 K

lo

g 10[

s]

V [cm3/g]

Isobar at 0.1MPa252.5 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]


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)

KWWKWWcJG 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

diel

ectri

c lo

ss

frequency [Hz]


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:


-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

diel

ectr

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''

ttdtddt K

K sinexp''

0

”Excess wing”

Log fJG


Effect of pressure on „excess wing”

PTtPT KWWKWWcJG ,, 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]


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

2 CH CH2

eugenol


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


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


10-3 10-2 10-1 100 101 102 103 104 105 106 107 10810-2

10-1

100

diel

ectri

c 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


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"

diel

ectri

c lo

ssf [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


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


10-2 10-1 100 101 102 103 104 105 106 1070.01

0.1

1

process

''/

'' max

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


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

ectri

c 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

ectri

c 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.


10-1 100 101 102 103 104 105 106

10-2

diel

ectri

c 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

ectri

c lo

ss "

freqency [Hz]

DHIQ

NH

E=~40 KJ/mol

NHN

H2

8

8a4a .....8a 4a

8

2

trans-DHIQ cis-DHIQ

N

H

H H

H

H

NH


Documents

Dielectric spectroscopy of glass-forming liquids under high pressure