Kinetic Data for Polymers

Sergey Vyazovkin

Bulk kinetics

Polymers are materials Processes of interest:

Thermal and oxidative degradationMolding (thermoplastics)Reactive injection molding (thermosets)Fire resistance

Methods:DSC (heat release kinetics)TGA (mass loss kinetics)

Importance of TGA and DSC

ISI Web of Science ®: DSC and polymer - 3,900 TGA and polymer – 1,200 FTIR and polymer – 2,900

Application to particular processes

DSC and crystallization and kinetics – 820 Microscopy and crystallization and kinetics –

810

DSC and curing/cure and kinetics - 760 FTIR and curing/cure and kinetics - 247

TGA and polymer and degradation/decomposition – 420

FTIR and polymer and degradation/decomposition – 540

Kinetics by TGA and DSC

f

i

i

T

T

T

T

dTflowHeat

dTflowHeat

)(

)(

fi

Ti

mm

mm

T Tf

Ti

He

at

flow

mT

T

mf

mi

Ma

ss

Typical kinetic approach

Single-step treatment preferred over multiple-step

Nonisothermal conditions preferred over isothermal

Single heating rate data analysis preferred over multiple heating programs

Single step rate equation

)()( fTk

dt

d

k(T) – rate constant

f() – reaction model

Arrhenius equation (1889)

RT

EATk exp)(

Svante Arrhenius 1859-1927

Single step rate equation

)(exp f

RT

EA

dt

d

“Kinetic triplet”:

E – activation energy

A – preexponential factor

f() – reaction model, f()= (1-)n

Transition state theory

Henry Eyring 1901 - 1981

ORIGIN

AL

E activation energy

Transition state theory

Henry Eyring 1901 - 1981

1. no medium (gas phase)2. single-step reaction

Multiple reactions

)()(

)()(

)/ln(

)()(

2211

222111

1

2211

fkfk

fkEfkE

dT

dtddRE

fkfkdt

d

0.00.2

0.40.6

0.81.0150

200250

300350

100

120

140

160

180

200

T / o

C

E /

kJ m

ol-1

Diffusion

DR

DRRDefef

DRef

kk

kEkE

dT

kdRE

kkk

1

ln

111

ER

ED

ln k

T -1

Condensed phase

Reactions occur in the solid or liquid medium

Medium affects the temperature dependence of the rate

Experimental E involves physical properties of the medium

mT

T

mf

mi

Ma

ss

fi

Ti

mm

mm

Single heating rate data analysis

Coats-Redfern method (1964): ~2000 citations! TGA or DSC at 1 heating rate vs T data fit to different g() models

RT

E

E

TR

E

RA

T

g j

jj

jj

21ln

)(ln

2

Compensation effect

lnA = aE + b

E

lnA

Large uncertainty in E and lnA!

Compensation effect, lnA=aE+b

Decomposition of HMXT.B. Brill et alJ. Phys. Chem.1994, 98, 12242

50 100 150 200 250 3000

5

10

15

20

25

30solid melt gas

E / kJ mol -1

log(

A/s

-1)

N

NN

NN

O

O

NO

O

N

O

O

N O

O

Solid state reaction models

No. Reaction Model f()1 Power law 4

3/4

2 Power law 32/3

3 Power law 21/2

4 Power law (2/3)-1/2

5 One dimentional diffusion (1/2)-1

6 Mampel (first order) 1 - 7 Avrami-Erofeev 4(1 - )[-ln(1 - )]

3/4

8 Avrami-Erofeev 3(1 - )[-ln(1 - )]2/3

9 Avrami-Erofeev 2(1 - )[-ln(1 - )]1/2

10 Three dimentional diffusion 2(1 - )2/3

(1 - (1 - )1/3

)-1

11 Contracting sphere 3(1 - )2/3

12 Contracting cylinder 2(1 - )1/2

HMX: Kinetic triplets by Coats-Redfern method

No. g() E / kJ mol-1 log(A/min-1) -r 1

1/4 21.3 -0.6 0.9327

2 1/3

31.0 0.5 0.9424

3 1/2

50.4 2.7 0.9501

4 3/2

167.0 14.8 0.9582

5 2 225.3 20.7 0.9591

6 -ln(1 - ) 125.9 10.8 0.9444 7 [-ln(1 - )]

1/4 25.6 0.0 0.9205

8 [-ln(1 - )]1/3

36.7 1.3 0.9299

9 [-ln(1 - )]1/2

59.0 3.7 0.9378

10 [1 - (1 - )1/3

]2 246.1 22.2 0.9536

11 1 - (1 - )1/3

119.1 9.5 0.9508

12 1 - (1 - )1/2

116.2 9.4 0.9530

Best fits

Practical purpose: predictionsC

on

vers

ion

Temperature

A, E, g()

0

0

exp

)(

RTE

A

gt

Co

nve

rsio

n

Time

HMX: Predictions

0

0

exp

)(

RTE

A

gt

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

5

4

t / min

Experimental data

T=235oC

Thermal degradation of PMMAAtmosphere

Step 1E / kJ mol-1

Step 2 Step 3Experiment Reference

vacuum 130 – 176 Isothermal manometry 1, 2

vacuum 138 Isothermal TGA 3

vacuum 242 Isothermal TGA 3

vacuum 117 213-230 Isothermal manometry 5

N2 150 – 250 Nonisothermal TGA 8

N2 210 Nonisothermal TGA 9

N2 154 133 Nonisothermal TGA 11

N2 210 Nonisothermal TGA 12

N2 31 224 Isothermal TGA 12

N2 233 104 Isothermal heating 13

N2 113 Nonisothermal TGA 15

N2 130 – 180 Nonisothermal TGA 16

Thermal degradation of PP

Atmosphere E / kJ mol-1 Method Ref

N2 244 Nonisothermal TGA 10

N2 216 isothermal TGA 11

N2 214 Nonisothermal TGA 18

N2 160 Nonisothermal TGA 22

N2 115 – 200 Nonisothermal TGA 19

N2 130 – 200 Nonisothermal TGA 23

N2 230 Factor-jump TGA 19

Vacuum 257 Factor-jump TGA 19

Ar 98, 328 Nonisothermal TGA 25

ICTAC Kinetics Project

ICTAC Kinetics Project

Thermochim. Acta 355(2000)125 Single heating rate methods should be

avoided Use multiple heating rate methods instead Importance of detecting complex processes

“Model-free” kinetics

Rate equation

111

)(lnln)/ln(

dT

fd

dT

kd

dT

dtdd

Log derivative

)(exp f

RT

EA

dt

d

“Model-free” kinetics

R

E

dT

fd

dT

kd

dT

dtdd

111

)(lnln)/ln(

Isoconversional principle

Uses multiple heating rates

Yields a model-free estimate E

Isoconversional methodC

on

vers

ion

Temperature

Act

ivat

ion

en

erg

y

Conversion

E varies with multi-step process

Thermal degradation of PP

Atmosphere E / kJ mol-1

N2 244

N2 216

N2 214

N2 160

N2 115 – 200

N2 130 – 200

N2 230

Vacuum 257

Ar 98, 3280.0 0.2 0.4 0.6 0.8 1.0

150

180

210

240

270

E /

kJ

mol-1

Decrease in Eα suggests a shift from kinetic to diffusion control that usually associated with vitrification.

0.2 0.4 0.6 0.8 1.040

50

60

70

Diffusioncontrol

E /

kJ m

ol -1

Epoxy-amine cure: Variation of Eα with α

TTT cure diagram

In the glassy state molecular motion is largely reduced

Detecting vitrification by temperature modulated DSC

0 2 4 6 8 10

20

30

40

50

60

70

tt

AtTtT T

2sin)( 0

AT = 1oC

t = 1 min

= 5oC min-1

t / min

T /

o C

0 50 100 150

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2.5

2.6

2.7

2.8

2.9

3.0

T / oC

Hea

t flow

/ m

W

Vitrifica

tion

C* p /

J g

-1 K

-1

Decrease in Eα is actually caused by vitrification

0.2 0.4 0.6 0.8 1.040

50

60

70

2.56

2.60

2.64

2.68

E /

kJ m

ol -1

Vitrifica

tion

C* p /

J g

-1 K

-1

0.2 0.4 0.6 0.8 1.040

50

60

70

2.56

2.60

2.64

2.68

E /

kJ m

ol -1

Vitrifica

tion

C* p /

J g

-1 K

-1

Epoxy-amine cure: Variation of Eα with α

Melt crystallization kinetics Poly(ethylene terephthalate) Aldrich, MW ~18,000,

Tm=280oC

Cooled from 290 to 25C β = -3, -4, -6, -8, -12C/min

120

140

160

180

200

220

240

260

0.0 0.2 0.4 0.6 0.8 1.0-300

-250

-200

-150

-100

-50

0

50

T /

o C

E

/ kJ

mol

-1

CH2 - CH2 - O - CO - - CO - O -[-

] n

Temperature dependence of growth rate

T)T/(T fT TΔT

TfT

K

TTR

UGG

mm

g

2

exp)(

exp*

0

E > 0 E < 0

Tm

ax

Tm

Tg Temperature

Gro

wth

rat

e

Hoffman-Lauritzen theory:

Evaluating Kg and U*

420 430 440 450 460 470 480 490 500-300

-250

-200

-150

-100

-50

0

50

T / K

E

/ kJ

mol

-1

Macromol. Rapid Commun.2004, 25, 733

TTT

TTTTRK

TT

TUTE

m

mmg 2

22

2

2*

)()()(

E vs E vs T:

Bf

meg kh

TnbK

I: Kg=3.2 105 K2 , U*=4300 J/mol

II: Kg=1.9 105 K2 , U*=2300 J/mol

Model-free predictions

Assuming that kinetic triplet (E, A, reaction model) at a given does not change when changing T

0

0,

exp

)(exp

RTE

dttRT

E

t

t

t

Model-free predictionsC

on

vers

ion

Temperature

Act

ivat

ion

en

erg

y

Conversion

0

0,

exp

)(exp

RTE

dttRT

E

t

t

t

Co

nve

rsio

n

Time

Model-free predictions, HMX

0 100 200 300

0.0

0.2

0.4

0.6

0.8

1.0

Best-fit models

HMX experimental data

T=235oC

Model-free 5

4

t / min

Model-free predictions, shelf-lifeC

on

vers

ion

Temperature

Model-free predictions, shelf-life

20 25 30 35 40 45 50

0

200

400

600

800

1000

1200

1400

1600

Predicted decomposition of Aspirin

5%4%

3%

2%

1%

t / d

ay

T / oC

Model-based methods

...)()( 2211 fkfk

dt

d

Model-based methods that use multiple heating programs are being developed

By far less common than model-free methods

Co

nve

rsio

n

Temperature

Model-based Model-free

)()(

)()(

)/ln(

)()(

2211

222111

1

2211

fkfk

fkEfkE

dT

dtddRE

fkfkdt

d

Act

ivat

ion

en

erg

y

Conversion

Model-based and model-free methods are interrelated via E dependence

Conclusions

E dependence can generally be interpreted as a function of the activation energies of individual steps

E is useful in exploring reaction mechanisms

The model-free approach requires only E for kinetic predictions

E dependence provides a link to model-based methods

Model-free approach can serve as a uniform framework for creating a database of bulk polymer (and solid-state) kinetics of thermal reactions

“The 16 questions” (polymers and solid-state)

3) nonisothermal data important 4) include overall or both (overall and elementary)

reactions 7) complex reactions unavoidable 8) cannot be limited to single dif. Eq. 9) the database should include:

D) solid-state reactions H) macromolecular reactions I) polymerization reactions

“The 16 questions” (polymers and solid-state)

11) long term success via agreements w/journals 13) critical assessment is important

ThermoML

ThermoML @ J. Chem. Eng. Data

ThermoML is an XML-based format for the exchange and storage of thermophysical property data

Authors download and use the GDC software to capture the experimental property data that has been accepted for publication.

The output of the GDC Software converted into ThermoML format at TRC

Upon release of the manuscript the ThermoML files are posted on the public-domain TRC Web site

“KineticML” @ Thermochim. Acta?