Polymer Characterization - ACS Publications - American Chemical

12 Thermogravimetry Applied to Polymer Degradation Kinetics BRIAN D I C K E N S and J O S E P H H . F L Y N N 1

Downloaded by NORTH CAROLINA STATE UNIV on October 13, 2012 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0203.ch012

National Bureau of Standards, Polymer Science Division, Washington, DC 20234

The kinetics of polymer degradations (and oxidations) may be represented in the simple general form dα/dt = f(α)Ae-E/RT, where α is the extent of reaction, and A and Ε are Arrhenius parameters. The various attempts to rep resent f(α) in a simple way are discussed, with the conclusion that none is satisfactory for polymer degrada tion studies. Therefore, four methods of thermogra vimetry have been devised and implemented to avoid any need to model f(α). Three of these methods give val ues for the activation energy, E, and through it shed some light on the dominant contributors to the kinetic form. The fourth method can be used in favorable cases to examine the importance of competing or successive reactions in the degradation mechanism. The methods are (1) factor-jump thermogravimetry, a series of isothermals requiring only a single sample; (2) isoconversional diagnostic plots, a variable heating rate method applied to a series of samples; (3) analysis of the initial stage of reaction, a variable heating rate method re quiring only one sample; and (4) variable heating rate analysis, applied to several samples to examine any change in component reactions in f(α).

HERMAL

JL

METHODS

OF ANALYSIS

find

wide

use

for

three

reasons:

phase changes c a n b e s t u d i e d v i a the heats o f transition; t e m p e r a t u r e affects t h e rate o f r e a c t i o n o f m o s t c h e m i c a l p r o c e s s e s ; a n d c o m p l e x molecules can be studied from fragments p r o d u c e d b y pyrolysis. T h i s chapter is c o n c e r n e d w i t h the use of t h e r m o g r a v i m e t r y , w h e r e the s a m p l e is h e a t e d a n d w e i g h e d c o n t i n u o u s l y , t o s t u d y p o l y m e r d e g r a dations. T h e r m o g r a v i m e t r y is, i n p r i n c i p l e , a s i m p l e t e c h n i q u e that 1

T o whom correspondence should be sent.

T h i s chapter not subject to U . S . copyright P u b l i s h e d 1983, A m e r i c a n C h e m i c a l Society

In Polymer Characterization; Craver, C.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.

210

POLYMER CHARACTERIZATION

provides k i n e t i c information o n the degradation, oxidation, evaporat i o n , or s u b l i m a t i o n of samples i n any c o n d e n s e d form. T h e m a t e r i a l is p r e s e n t e d here i n the f o r m of a n o v e r v i e w . F o r m o r e c o m p l e t e treatm e n t s , see R e f e r e n c e s 1 a n d 2 a n d o t h e r r e f e r e n c e s c i t e d i n t h i s c h a p t e r .

Kinetic Analysis

of Thermogravimetric

Data

T h e m a t h e m a t i c a l m o d e l t h a t is e v o k e d to d e s c r i b e t h e k i n e t i c s o f a system u n d e r g o i n g c h e m i c a l change is u s u a l l y e x p r e s s e d i n the f o r m


daldt

=f(a)k(T)

(1)

w h e r e t h e rate o f c h a n g e o f t h e c o n v e r s i o n o r f r a c t i o n r e a c t e d , a , w i t h r e s p e c t t o t i m e , t, i s e q u a t e d t o s e p a r a b l e f u n c t i o n s o f a a n d t h e a b s o lute temperature, T. E q u a t i o n 1 represents a d e q u a t e l y the k i n e t i c s of m a n y r e a c t i n g s y s t e m s i n t h e g a s e o u s p h a s e a n d a l s o a p p l i e s to s o m e reactions o c c u r r i n g i n h o m o g e n e o u s s o l u t i o n s . H o w e v e r , e v e n for homogeneous systems, i f the overall k i n e t i c process involves several e l e m e n t a r y s t e p s s u c h as o p p o s i n g , c o n s e c u t i v e , p a r a l l e l , o r c h a i n reactions, t h e n E q u a t i o n 1 is p r o b a b l y not s u f f i c i e n t to d e s c r i b e t h e rate o f r e a c t i o n . T h e s e c o m p l e x i t i e s c a n b e t a k e n i n t o a c c o u n t f o r m a l l y b y t h e a d d i t i o n o f a t e r m , g ( a , T ) (3), t o E q u a t i o n 1 t o o b t a i n : daldt

=

f(a)k(T)g(a,T)

(2)

w h e r e g(a,T) i n c l u d e s a l l c o n v e r s i o n - t e m p e r a t u r e c r o s s - t e r m s . A l t h o u g h this case m a y a p p e a r to b e tractable, a s i n g l e r e a c t i o n c o o r d i n a t e , a, i s i n s u f f i c i e n t i n m a n y c a s e s s u c h as w h e n t h e a m o u n t o f residue or char d e p e n d s o n p r e v i o u s t h e r m a l treatment. T h e u s u a l a p p r o a c h to f i t t i n g E q u a t i o n 2 i s to e m p l o y s p e c i f i c a n a l y t i c a l e x p r e s sions that u s u a l l y w e r e o b t a i n e d from m o d e l s b a s e d o n p h y s i c a l a n d chemical evidence, intuition, and/or experience. Heterogeneous kinetics pose additional problems and complicat i o n s o v e r h o m o g e n e o u s k i n e t i c s b e c a u s e p r o c e s s e s s u c h as d i f f u s i o n , sorption, e v a p o r a t i o n , a n d c h e m i c a l reactions a m o n g separate phases a r e i n v o l v e d . I f d u e c a r e is n o t t a k e n , r a t e s m a y d e p e n d o n p h y s i c a l f a c t o r s s u c h as c o n t a c t a r e a s a n d t h e m o v e m e n t o f s p e c i e s t h r o u g h f i x e d m a t r i c e s o r h i g h l y v i s c o u s fluids. T h e r e f o r e , a n o t h e r t e r m m u s t b e a d d e d to E q u a t i o n 2 t o o b t a i n : daldt

=f(a)

k(T) g(a,T)

h(X Y . u

h

. .)

(3)

w h e r e t h e t e r m , h(X Y . . .) i s a n a n a l y t i c a l e x p r e s s i o n o f t h e f u n c t i o n a l r e l a t i o n s h i p s b e t w e e n t h e c h e m i c a l rate a n d a l l o t h e r r a t e a f f e c t i n g factors a n d t h e c r o s s - t e r m s a m o n g t h e f a c t o r s t h e m s e l v e s as i9

jy


12.

DICKENS A N D F L Y N N

211

Thermogravimetry


w e l l as w i t h t e m p e r a t u r e a n d c o n v e r s i o n . T h e f a c t o r s , X,Y,. . ., r e p r e s e n t t h e e f f e c t s o f s u c h v a r i a b l e s as p r e s s u r e , gas flow r a t e , gas c o m p o sition, physical a n d geometrical properties o f the sample, catalytic impurities, labile groups introduced d u r i n g preparation, a n d m e c h a n i c a l stresses. M a n y factors t h a t affect t h e rate o f w e i g h t l o s s m a y c h a n g e i n a noncontrollable manner d u r i n g the course of a n experiment. F o r e x a m p l e , t h e t e m p e r a t u r e i n t h e i n t e r i o r o f a large s p e c i m e n i s affected significantly b y endothermic or exothermic reactions; the pressure o f a gaseous species c a n b e different i n t e r n a l l y a n d externally d e p e n d i n g o n its r a t e o f p r o d u c t i o n o r c o n s u m p t i o n a n d r e m o v a l . S u c h effects set u p p o t e n t i a l gradients i n t h e systems, a n d t h e r e s u l t a n t fluxes affect t h e o v e r a l l r e a c t i o n r a t e . W h e n t h e rate i s a f f e c t e d o r l i m i t e d b y s u c h d i f f u s i o n a l p r o c e s s e s , i t is e x t r e m e l y d i f f i c u l t to m o d e l the kinetics adequately. I n practice, these complications c a n b e ameliorated c o n s i d e r a b l y b y decreasing the d r i v i n g potential, i.e., b y m a i n t a i n i n g c o n s t a n t i n t e n s i t i e s o f e x t e r n a l factors s u c h as p r e s s u r e a n d a t m o s p h e r i c c o m p o s i t i o n , o r b y s l o w i n g d o w n t h e r e a c t i o n rate so that i n t e r n a l gradients b e c o m e less significant. S o m e r a t e - a f f e c t i n g f a c t o r s are n o t c o n t r o l l e d e a s i l y . S t a n d a r d i z a t i o n o f t h e p h y s i c a l a n d g e o m e t r i c a l p r o p e r t i e s o f t h e s p e c i m e n s so as to r e n d e r h(X Y . . .) i n E q u a t i o n 3 a c o n s t a n t t e r m t h a t m a y b e i g n o r e d i n the k i n e t i c treatment often is extremely difficult. C o n t r o l o f c h e m i c a l factors i s e q u a l l y i m p o r t a n t . T r a c e a m o u n t s o f a d d i t i v e s a n d r e s i d u a l catalysts, solvents, a n d m o n o m e r s m u s t either b e r e m o v e d or t h e i r effects o n t h e k i n e t i c s m u s t b e t a k e n i n t o a c c o u n t . i9

i9

T h e p r o c e d u r e m o s t o f t e n f o l l o w e d i s t o k e e p t h e factors j u s t m e n t i o n e d at c o n t r o l l e d l e v e l s t o i n c o r p o r a t e t h e i r effects i n t o t h e e x p r e s s i o n s f o r f(a) a n d k(T). I n t h a t c a s e , g(ct,T) h(X Y . . .) = 1, a n d E q u a t i o n 3 r e d u c e s t o E q u a t i o n 1, b e c a u s e o n l y m a t h e m a t i c a l m o d e l s for t e m p e r a t u r e a n d c o n v e r s i o n w i l l b e n e c e s s a r y . u

The Conversion Function,

h

f(a)

T h e c o n v e r s i o n f u n c t i o n , f(a), i n g e n e r a l i s e x t r e m e l y c o m p l i cated. I n attempting to characterize it, isothermal experiments must b e u s e d t o s e p a r a t e o u t t h e effects o f t e m p e r a t u r e c h a n g e . A s u r v e y o f the literature o n the isothermal weight-loss kinetics o f polymers e m phasizes t h e complexity o f t h e kinetics o f these systems. Plots o f w e i g h t - l o s s rate v s . c o n v e r s i o n f r a c t i o n d o n o t f o l l o w s i m p l e r e a c t i o n orders; also, their overall shape u s u a l l y changes w i t h temperature. T h u s , a p a r t i c u l a r c o n v e r s i o n f u n c t i o n i s v a l i d o n l y for a l i m i t e d r a n g e of experimental conditions. Because p o l y m e r degradations are often c h a i n reactions,/(a) represents the net result o f a series o f e l e m e n t a r y


212


steps. E a c h e l e m e n t a r y step has its o w n a c t i v a t i o n e n e r g y ,

which

m a k e s e a c h s u c h step r e s p o n d d i f f e r e n t l y to t e m p e r a t u r e c h a n g e . A g o o d e x a m p l e is the e x i s t e n c e of a c e i l i n g t e m p e r a t u r e , a b o v e w h i c h d e g r a d a t i o n takes p l a c e a n d b e l o w w h i c h the m a t e r i a l p o l y m e r i z e s . T h e t r a d i t i o n a l r a d i c a l c h a i n m o d e l for the m e c h a n i s m o f v i n y l p o l y m e r degradation illustrates m a n y of these

complexities.

This

m o d e l i n v o l v e s t h r e e t y p e s o f p r o c e s s e s : (1) i n i t i a t i o n r e a c t i o n s

by

w h i c h r a d i c a l s a r e f o r m e d , (2) c h a i n p r o p a g a t i o n r e a c t i o n s t h a t p r o d u c e s c i s s i o n o f t h e p o l y m e r b a c k b o n e , a n d (3) t e r m i n a t i o n r e a c t i o n s


that r e m o v e c h a i n - p r o p a g a t i n g radicals from the system. Initiation may result from labile linkages already i n the polymer. I f s u c h i n i t i a t i o n is f o l l o w e d b y a d e p r o p a g a t i o n ( u n z i p p i n g ) to l o w m o l e c u l a r w e i g h t f r a g m e n t s , t h e n the rate o f v o l a t i l i z a t i o n w i l l

de

c r e a s e c o n t i n u o u s l y as t h e r e a c t i o n p r o c e e d s . O n t h e o t h e r h a n d , i f t h e c h a i n b r e a k s c o m e a b o u t m o r e or less r a n d o m l y , e i t h e r b e c a u s e

of

r a n d o m initiation or more u s u a l l y because of /3-scission of radicals f o r m e d o n the p o l y m e r b a c k b o n e b y h y d r o g e n abstraction (transfer), m o s t e a r l y fragments w i l l b e too large to v o l a t i l i z e , b u t t h e rate w i l l i n c r e a s e w i t h c o n v e r s i o n as g r e a t e r n u m b e r s o f s m a l l e r f r a g m e n t s are formed.

T h i s process produces

a n a p p a r e n t a u t o c a t a l y t i c rate

of

w e i g h t l o s s . I f n e w l a b i l e l i n k a g e s s u c h as u n s a t u r a t e d e n d g r o u p s a r e f o r m e d f r o m transfer a n d d i s p r o p o r t i o n a t i o n r e a c t i o n s , t h e rate o f i n itiation w i l l increase. T h u s , i n o n l y a f e w cases of p o l y m e r d e g r a d a tion,

s u c h as t h e

thermal degradation

of

polytetrafluoroethylene

( w h i c h o f t e n is u s e d as a n i l l u s t r a t i v e e x a m p l e ) , d o t h e w e i g h t - l o s s k i n e t i c s c o n f o r m r e a s o n a b l y c l o s e l y to s i m p l e n

t h

Temperature Dependence: The Arrhenius

order kinetic models.

Equation

T h e A r r h e n i u s e q u a t i o n is the m o d e l u s e d a l m o s t u n i v e r s a l l y to e x p r e s s t h e t e m p e r a t u r e d e p e n d e n c e o f t h e rate o f r e a c t i o n , v i z . , k(T)

=A

exp [ - E ( R T ) " ] 1

(4)

I n E q u a t i o n 4, R is t h e gas c o n s t a n t a n d Τ i s t h e a b s o l u t e t e m p e r a t u r e . T h e e n e r g y o f a c t i v a t i o n ( £ ) a n d t h e p r e - e x p o n e n t i a l f a c t o r (A) are p a r a m e t e r s d e t e r m i n e d b y f i t t i n g e x p e r i m e n t a l rate d a t a . T h e a p p l i c a t i o n o f the A r r h e n i u s e q u a t i o n to c o n d e n s e d

phase

kinetics received considerable recent adverse criticism because often v a l u e s for Ε c h a n g e w i t h r e s p e c t to t e m p e r a t u r e a n d c o n v e r s i o n a n d show poor agreement w h e n compared with values obtained using different t e c h n i q u e s or u n d e r different c o n d i t i o n s . T h e r e f o r e , it is p e r t i n e n t to g i v e s o m e j u s t i f i c a t i o n f o r its u s e . M u c h o f t h i s c r i t i c i s m results f r o m m i s c o n c e p t i o n s of the t h e o r e t i c a l basis for the e q u a t i o n a n d its r o l e as a m a t h e m a t i c a l f u n c t i o n for f i t t i n g a n d m o d e l i n g d a t a .


12.


Thermogravimetry

213

M o s t m o d e l s for a k i n e t i c step postulate a n e n e r g y b a r r i e r b e t w e e n t h e i n i t i a l state (reactants) a n d t h e f i n a l state ( p r o d u c t s ) . T h e y assume that o n l y a s m a l l fraction o f the species has sufficient e n e r g y to c r o s s t h i s b a r r i e r . T h i s a s s u m p t i o n i s t h e c a s e w h e t h e r t h e m o d e l i s based o n statistical or m o l e c u l a r concepts. T h e relationship b e t w e e n the energy d i s t r i b u t i o n o f the reacting species a n d temperature is represented b y a Boltzmann exponential function, A exp (—BT ). ( M a n y m o d e l s p r e d i c t s o m e sort o f m i l d p o l y n o m i a l f u n c t i o n a l i t y w i t h temperature for A , but, i n practice, e v e n slight uncertainties i n t h e e x p o n e n t i a l p a r a m e t e r , E , o v e r w h e l m a n d r e n d e r f u t i l e efforts t o m e a sure a t e m p e r a t u r e d e p e n d e n c e for A . ) T h u s , one finds A r r h e n i u s - t y p e temperature d e p e n d e n c e w h e n c o l l i s i o n , transition-state, a n d other m o d e l s are a p p l i e d not o n l y to h o m o g e n e o u s c h e m i c a l p r o c e s s e s b u t a l s o t o m a n y p h y s i c a l p r o c e s s e s s u c h as t h e v i s c o u s f l o w o f l i q u i d s a n d diffusion i n solids.


- 1

H o w e v e r , the basic p r o b l e m i n the application o f the A r r h e n i u s e q u a t i o n to c o n d e n s e d phase p o l y m e r systems is the c o m p l e x i t y of the k i n e t i c s . A h e t e r o g e n e o u s s y s t e m o f t e n i s c o m p o s e d o f at l e a s t s e v e r a l e l e m e n t a r y p r o c e s s e s , e a c h w i t h its o w n set o f A r r h e n i u s p a r a m e t e r s . Whereas i n homogeneous systems, these processes can b e c o n d e n s e d into a small n u m b e r o f discrete prototype elementary processes, i n m a n y h e t e r o g e n e o u s s y s t e m s , s u c h as p o l y m e r d e g r a d a t i o n r e a c t i o n s , the combination of a l l these processes m a y approach a c o n t i n u u m . B e c a u s e o f t h i s fact, t h e m o r e s u c c e s s f u l a p p r o a c h e s t o m o d e l i n g t h e k i netics o f p o l y m e r degradation reactions have b e e n statistical i n na ture, a s s u m i n g average values for the values o f the parameters o f the rate c o n s t a n t s f o r a n a r r a y o f s i m i l a r p r o c e s s e s . T h e r a d i c a l c h a i n d e p o l y m e r i z a t i o n m o d e l s for v i n y l p o l y m e r d e g r a d a t i o n fit the d e s c r i p t i o n just g i v e n . T h e r e , w h o l e series o f reactions i n v o l v i n g h o m o l o g o u s s p e c i e s a r e g i v e n a n i d e n t i c a l rate c o n s t a n t i r r e s p e c t i v e o f m o l e c u l a r size. A s a result, those m o d e l s c a n u s u a l l y predict only t h e general c h a r a c t e r i s t i c s o f i s o t h e r m a l rate c u r v e s . T h e p a r a m e t e r s o f t h e t e m p e r a t u r e f u n c t i o n , k(T), i n E q u a t i o n 1 must, o f course, b e d e t e r m i n e d from experiments i n v o l v i n g different temperatures. T h e parameter most amenable to precise determination is E, t h e a c t i v a t i o n e n e r g y . M e t h o d s e x i s t (see l a t e r d i s c u s s i o n ) t o d e t e r m i n e Ε w i t h o u t k n o w l e d g e off(a). T h e p a r a m e t e r Ε i s u s e f u l i n that i t gives some i n d i c a t i o n o f change i n t h e r a t e - d e t e r m i n i n g p r o cesses d u r i n g the degradation a n d , b e i n g a n energy, can b e r e l a t e d to p r o c e s s e s , s u c h as d i f f u s i o n o r b r e a k i n g o f a p a r t i c u l a r t y p e o f b o n d , found i n m o d e l systems. A l t h o u g h / ( a ) n e e d n o t b e specified to d e t e r m i n e E, a m o d e l forf(a) m u s t b e u s e d t o d e c o m p o s e t h e a c t i v a t i o n e n e r g y o f the o v e r a l l process into the values o f the c o m p o n e n t steps. T h e o v e r a l l v a l u e of Ε is a p p l i c a b l e o n l y to the same range o f e x p e r i -


214



m e n t a l c o n d i t i o n s t h a t f(a) a p p l i e s t o , b u t t h e i n d i v i d u a l Ε v a l u e s f o r the component reactions have more universal application. T h e o t h e r , l e s s t e m p e r a t u r e - d e p e n d e n t f a c t o r i n E q u a t i o n 4, t h e p r e e x p o n e n t i a l f a c t o r , A, a l s o m a y b e d e t e r m i n e d a l o n g w i t h t h e a c t i v a t i o n energy. F o r c o n d e n s e d phase k i n e t i c s , h o w e v e r , it is i m p o s s i b l e to u n c o u p l e A a n d f(a) e x p e r i m e n t a l l y , a n d b e c a u s e / ( a ) m a y c o n t a i n p h y s i c a l a n d g e o m e t r i c a l p r o p e r t i e s o f t h e r e a c t i n g s p e c i e s as w e l l as o t h e r f a c t o r s , i t i s i m p o s s i b l e t o m o d e l i t s a t i s f a c t o r i l y so as t o separate it a n a l y t i c a l l y f r o m t h e p r e e x p o n e n t i a l factor.

Kinetic

Forms Used in Polymer

Thermogravimetry

T h e simplest a n d most w i d e l y u s e d (and usually incorrect) m o d e l f o r f(a) i s t a k e n f r o m h o m o g e n e o u s c h e m i c a l k i n e t i c s , v i z . , f(a)

= (1 - « ) »

(5)

w h e r e η i s t h e a p p a r e n t o r d e r o f r e a c t i o n . S u b s t i t u t i n g f o r / ( a ) a n d k(T) i n E q u a t i o n 1 gives daldt

= (1 - a) Ae~ n

(6)

EIRT

I n c a s e s w h e r e t h e s a m p l e i s h e a t e d c o n t i n u o u s l y so t h a t t h e v a r i a t i o n , β, o f t e m p e r a t u r e w i t h t i m e i s β = dT/dt, daldt

= β daldT

w e have

= (1 - a) Ae~ n

EIRT

(7)

or, after t a k i n g l o g a r i t h m s , I n (daldt)

= η I n (1 - a) + I n A - E/RT

(8)

D i f f e r e n t i a l M e t h o d s . T h e s i m p l e s t m e t h o d o f a n a l y s i s is to ex t e n d t h e m e t h o d o f v a n ' t H o f f (4) t o t h e n o n i s o t h e r m a l c a s e , i . e . , s o l v e E q u a t i o n 8 for I n A, n, a n d E/R f r o m t h r e e sets o f d a t a p o i n t s f o r t h e r a t e , f r a c t i o n r e a c t e d , a n d t e m p e r a t u r e . L e t o r t (5) e x t e n d e d v a n ' t H o f f ' s m e t h o d for t h e i s o t h e r m a l c a s e to a n y n u m b e r o f d a t a p o i n t s b y p l o t t i n g I n (daldt) v s . I n (1 — a) t o o b t a i n a s l o p e o f η a n d a n i n t e r c e p t o f I n k. F r e e m a n a n d C a r r o l l (6) m a d e a s i m i l a r e x t e n s i o n f o r t h e r m o g r a v i m e t r i c d a t a at c o n s t a n t r a t e o f h e a t i n g b y e x p r e s s i n g E q u a t i o n 8 as a d i f f e r e n c e e q u a t i o n to r e m o v e t h e q u a n t i t y I n A , v i z . , Δ Ι η (daldt) a n d d i v i d i n g b y A(l/T)

= η Δ1η (I - a) - (E/R) Δ ( 1 / Γ )

(9)

t o o b t a i n v a l u e s f o r η a n d E/R f r o m t h e s l o p e

and intercept of a plot of


12.


Δ Ι η (daldt)


215

Thermogravimetry

v s . Δ1η (1 - a) ^ ( l / T ) ] "

[A(IIT)]-

1

1

T h i s p r o c e d u r e has b e c o m e t h e m o s t p o p u l a r m e t h o d for t h e a n a l y s i s of n o n i s o t h e r m a l data. T h e d e t e r m i n a t i o n o f the order, n , is formally e x p l i c i t , a n d gross changes i n η a n d £ w i t h c h a n g i n g a m a y b e s e e n . U n f o r t u n a t e l y , E q u a t i o n 4, o n w h i c h t h i s m e t h o d i s b a s e d , does n o t a p p l y to the c o m p l e t e range o f a i n p o l y m e r degradations. F o r example, activation energies determined from F r e e m a n — C a r r o l l plots for the degradation o f p o l y ( m e t h y l a-phenylacrylate) c o m p a r e d p o o r l y w i t h t h o s e d e t e r m i n e d f r o m i s o t h e r m a l e x p e r i m e n t s (7). T h e i s o t h e r m a l k i n e t i c s s u g g e s t (7) t h a t a r a n d o m s c i s s i o n - t y p e m o d e l i n w h i c h t h e rate g o e s t h r o u g h a m a x i m u m i s m o r e a p p r o p r i a t e t h a n E q u a t i o n 5 forthis system. M a n y differential methods make use o f the point o f inflection o f the thermogravimetric curve. I f the derivative o f E q u a t i o n 6 w i t h re spect to t i m e i s set e q u a l to zero, w e c a n factor o u t t h e q u o t i e n t E/(nR), i . e . , E/(nR)

= T (daldt) ^ (l 2

m

m

-

m

(10)

a)

m

w h e r e t h e s u b s c r i p t , m , r e f e r s t o t h e v a l u e at t h e m a x i m u m . T o o b t a i n £ , w e m u s t d e t e r m i n e o r e s t i m a t e n . W e stress t h a t t h e v a l u e s o f η a n d £ are d e p e n d e n t o n t h e m o d e l chosen for t h e kinetics. M e t h o d s i n w h i c h Ε a n d η are d e t e r m i n e d i n this w a y h a v e l i m i t e d a p p l i c a b i l i t y to p o l y m e r d e g r a d a t i o n k i n e t i c s . N o test c a n b e m a d e o f the v a l i d i t y o f the kinetic parameters obtained. I n i t i a l k i n e t i c parameters are s i g n i f i c a n t b o t h i n e s t a b l i s h i n g t h e m e c h a n i s m a n d i n investigating the slow aging o f p o l y m e r i c materials. M e t h o d s at c o n t i n u o u s rates o f t e m p e r a t u r e c h a n g e h a v e a n a d v a n t a g e o v e r i s o t h e r m a l m e t h o d s i n t h e d e t e r m i n a t i o n o f i n i t i a l rates as t h e y m i n i m i z e t h e p r o b l e m s o f e s t a b l i s h i n g t h e t r u e t i m e o r i g i n , t = 0, d u r i n g the w a r m u p time. S u c h methods o f analysis, based o n a single experi ment, c a n b e adapted u s e f u l l y to t h e d e t e r m i n a t i o n o f initial A r rhenius parameters. E q u a t i o n 7 may b e differentiated w i t h respect to a ( r e m e m b e r i n g t h a t T da/dT = -da/d(l/T) a n d that Γ is a f u n c t i o n o f a b e c a u s e t h i s m e t h o d i s n o n i s o t h e r m a l ) t o g i v e (8) 2

d_

j>2

da

da dT

Ε = — +

Τ

df(a)

da

f(a)

da

dT

2

9 T +

R

—

w h i c h simplifies to d_ da

J»2

da ~dT

£

= — 4- 2 Γ (a « R

1, β = c o n s t a n t )


(11)

216


S i m i l a r a p p r o x i m a t i o n s o f t h e i n t e g r a t e d rate e q u a t i o n g i v e , at l o w conversion (9—II), ln(l

-a)/T

2

o r (1 - a)IT


2

= -E/RT

+ c o n s t a n t (a «

1, β = c o n s t a n t )

(12)

F i g u r e 1 s h o w s t h e r m o g r a v i m e t r i c c u r v e s for —dald(HT) vs. a for l i n e a r p o l y e t h y l e n e (8). T h e i n i t i a l a c t i v a t i o n e n e r g y is c a l c u l a t e d f r o m t h e s l o p e , —da/d(l/T), t o b e —60 k c a l / m o l f o r a s a m p l e t h a t w a s p r e h e a t e d i n a v a c u u m at 2 0 0 °C for 1 h . A s a m p l e w i t h n o p r e t r e a t m e n t y i e l d e d Ε = 30 k c a l / m o l o v e r the first 2 % c o n v e r s i o n range. T h i s v a l u e c o r r e s p o n d s to t h e l a t e n t heat o f v a p o r i z a t i o n o f h y d r o c a r b o n s b u t is also n e a r the a c t i v a t i o n e n e r g y for o x i d a t i o n of h y d r o c a r b o n s . I n t e g r a l M e t h o d s . W h e n / ( a ) a n d k(T) are c o n s i d e r e d t o b e s e p a r a b l e a n d k(T) i s r e p r e s e n t e d b y t h e A r r h e n i u s e q u a t i o n , E q u a t i o n 4, t h e n E q u a t i o n 1 m a y b e i n t e g r a t e d to o b t a i n

1

[

1

1

1

•

Δ Ο

Figure 1. An example of initial activation energy determination: —da/ άΥΙ/Τ,) vs. a (10 mg of polyethylene at 1.8°/min in vacuum). Key: Δ , untreated sample; preheated in vacuum at 200 °C for 1 h; and O, a theoretical case.


12.


F (α) = f

dalf(a)

= (Α'β')

0

T h e parameter Τ temperature

α

217

Thermogravimetry

f 0

T e x p (-EIRT)dT a

(13)

is i n c l u d e d i n this equation to represent both t h e

dependence

of the preexponential

factor, A , ( i f t h e

p r e e x p o n e n t i a l factor is c o n s i d e r e d to b e t e m p e r a t u r e d e p e n d e n t ) a n d a n y t e m p e r a t u r e d e p e n d e n c e o f t h e h e a t i n g r a t e , β. If a is zero, t h e i n t e g r a t i o n o f t h e r i g h t side o f E q u a t i o n 13 i n


volves the exponential integral j

(e /x)dx, x

w h e r e χ = -E/RT

—x

T h i s integral has b e e n e v a l u a t e d for various values o f χ a n d a p p r o x i m a t e d i n m a n y w a y s (12-18). I f a i n E q u a t i o n 13 e q u a l s t w o , t h e n t h e temperature integral i s integrated easily i n a c l o s e d form. If A is sus pected o f b e i n g temperature dependent, o r o n e w i s h e s to a v o i d a p proximating the integral, then a temperature-dependent heating rate m a y b e u s e d to p r o v i d e a total e x p o n e n t o f 2 for T. I n practice, h o w e v e r , this step is n o t necessary b e c a u s e t h e t e r m e x p (-E/RT) dominates the results, a n d , because of experimental uncer t a i n t i e s i n rate a n d t e m p e r a t u r e m e a s u r e m e n t s , a n y o f t h e m a n y a p proximations O f the temperature integral w i l l u s u a l l y suffice. A large n u m b e r o f methods o f k i n e t i c analysis w e r e d e v e l o p e d to use t h e i n t e g r a t e d f o r m o f E q u a t i o n 4, i . e . , " F («) = f i v

;

da (1 - a)

= l n ( l - a )

(1 - a) - 1 ( n ^ l o r 0 1 - η 1

=-

Γ

n

71

1

v

(n = 1)

(14)

b u t , as w i t h d i f f e r e n t i a l m e t h o d s , t h e m e t h o d s b a s e d o n t h i s e q u a t i o n are g e n e r a l l y i n a p p r o p r i a t e f o r t h e c o m p l e x k i n e t i c s y s t e m s f o u n d i n polymer degradation. T h e inability of both differential methods based o n E q u a t i o n 6 a n d i n t e g r a l m e t h o d s b a s e d o n E q u a t i o n 14 t o d e t e r m i n e r e l i a b l e kinetic parameters has b e e n demonstrated b y m a n y investigators (12-14, 19, 20). S i m p l e o r d e r o f r e a c t i o n t y p e k i n e t i c s c a n n o t r e p r e s e n t a m a x i m u m i n r e a c t i o n rate at o t h e r t h a n 0 % r e a c t i o n , a n d f i t t i n g such a process w i t h a combination o f order o f reaction kinetics a n d A r r h e n i u s - t y p e temperature d e p e n d e n c e r e q u i r e s a large effect f r o m the A r r h e n i u s e q u a t i o n to force t h e order o f reaction k i n e t i c b e h a v i o r into the right pattern. T h e essential p r o b l e m is illustrated i n F i g u r e 2 w h e r e r e s i d u a l f r a c t i o n , 1 — a, i s p l o t t e d a g a i n s t t e m p e r a t u r e f o r t h r e e calculated curves: C u r v e 1 for η = 1 i n E q u a t i o n 7 a n d Ε = 80,000 c a l / m o l , a n d C u r v e s 2 a n d 3 for t y p i c a l p o l y m e r degradation reactions



218


i n w h i c h t h e rate g o e s t h r o u g h a m a x i m u m at 5 0 % c o n v e r s i o n , a n d Ε = 40,000 a n d 35,000 c a l / m o l , r e s p e c t i v e l y . T h e closeness of fit o f these three curves shows w h y investigators w h o use methods b a s e d o n the a s s u m p t i o n of a n order, n , for a n a l y z i n g p o l y m e r i c systems m a y e n d u p w i t h absurdly large calculated activation energies. E f f o r t s h a v e b e e n m a d e to fit m o r e r e a l i s t i c m o d e l s f o r / ( a ) to t h e r m o g r a v i m e t r i c d a t a f o r d e g r a d i n g p o l y m e r i c s y s t e m s . O z a w a (21 ) t o o k t h e c l a s s i c a l r a n d o m d e p o l y m e r i z a t i o n e q u a t i o n o f S i m h a a n d W a l l (22) a n d c o n s t r u c t e d f a m i l i e s o f m a s t e r c u r v e s for v a r i o u s v a l u e s o f L , t h e l e n g t h of the smallest c h a i n of c a r b o n atoms that does not evaporate. A w e i g h t - l o s s c u r v e m a y b e c o m p a r e d w i t h s u c h a f a m i l y o f c u r v e s to d e t e r m i n e a b e s t v a l u e for L . H o w e v e r , f e w s y s t e m s , i f a n y , f o l l o w s i m p l e r a n d o m d e p o l y m e r i z a t i o n k i n e t i c s . A l s o , it is d i f f i c u l t to d e c i d e o n t h e a p p r o p r i a t e f o r m off (a).


12.


Thermogravimetric Activation

219

Thermogravimetry

Methods

for

Determining

Energies


M e t h o d s w e r e d e v e l o p e d at t h e N a t i o n a l B u r e a u o f S t a n d a r d s f o r h a n d l i n g t w o possible ways o f taking temperature-dependent data: e x p e r i m e n t s that consist o f a series o f isothermals, a n d other e x p e r i ments i n w h i c h the temperature is i n c r e a s e d c o n t i n u o u s l y . T h e g e n eral a i m w a s to determine activation energies w i t h o u t the n e e d to define the f u n c t i o n / ( a ) . A n o t h e r m e t h o d o f processing the data gives s o m e i n s i g h t i n t o t h e p r o c e s s e s t h a t f(ct) i s s u p p o s e d t o r e p r e s e n t . These methods w i l l now be described. Factor-Jump

Thermogravimetry,

a Series

of

Isothermals

I n t h e f a c t o r - j u m p m e t h o d , a p o l y m e r s p e c i m e n i s s u b j e c t e d t.o a series o f temperature isothermals w h i l e t h e temperature a n d its w e i g h t a r e r e c o r d e d c o n t i n u o u s l y . T h e rates o f w e i g h t l o s s a n d t h e temperatures for adjacent isothermals are extrapolated to halfway b e t w e e n the isothermals i n terms o f t i m e or i n terms o f the associated parameter, extent o f reaction. T h e activation energy t h e n is estimated f r o m t h e A r r h e n i u s e q u a t i o n , E q u a t i o n 4 , as RTJ r Ε = ^ Γ ΐ η ~ 2

2

(15)

w h e r e t h e r a n d Τ v a r i a b l e s a r e e x t r a p o l a t e d rates a n d t e m p e r a t u r e s f r o m t w o a d j a c e n t p l a t e a u s a n d Δ Τ =T - T . B e c a u s e b o t h rates a n d b o t h t e m p e r a t u r e s a r e e s t i m a t e d at t h e s a m e e x t e n t o f r e a c t i o n , t h e term containing the extent o f reaction a n d other temperature-inde p e n d e n t factors c a n c e l o u t (see R e f e r e n c e 2 3 ) . 2

t

T h e strong points o f the m e t h o d are that a c t i v a t i o n e n e r g i e s are determined u s i n g only one specimen (whereas i nmultisample tech n i q u e s o n e m u s t a s s u m e that t h e r m a l h i s t o r i e s are u n i m p o r t a n t ) , that an activation energy is p r o v i d e d for roughly every 5 % o f conversion, that t h e e x p e r i m e n t i s c o n d u c t e d o v e r a n a r r o w range o f rates o f w e i g h t l o s s so t h e c o n c e n t r a t i o n s o f r e a c t a n t s w i t h i n t h e s p e c i m e n a n d the products above t h e s p e c i m e n are r o u g h l y constant, a n d that t h e q u a n t i t i e s u s e d to c a l c u l a t e the a c t i v a t i o n e n e r g y are o b t a i n e d f r o m a s m a l l ( 6 - 1 0 ° C ) t e m p e r a t u r e r a n g e so t h a t t h e A r r h e n i u s e q u a t i o n i s p r o b a b l y v a l i d a n d the p r e e x p o n e n t i a l factor is u n d o u b t e d l y i n d e p e n dent o f temperature over this small interval. O n the other h a n d , the i n i t i a l activation e n e r g y cannot b e d e t e r m i n e d b e c a u s e t h e first d e t e r m i n a t i o n is m a d e b e t w e e n the first a n d s e c o n d i s o t h e r m a l s , u s u a l l y at — 5 % w e i g h t l o s s . A l s o , t h e m e t h o d i s n o t c o m p u t a t i o n a l l y r o b u s t . I t



220


r e q u i r e s that t h e w e i g h t - t i m e t r e n d b e f i t t e d to a p o l y n o m i a l that t h e n is d i f f e r e n t i a t e d a n d e x t r a p o l a t e d — 1 5 % b e y o n d the range o f data. T h e loss of volatiles d u r i n g the d e g r a d a t i o n of m a n y p o l y m e r s results i n the bursting of b u b b l e s i n the sample. I n a d e r i v a t i v e c a l c u l a t i n g m e t h o d s u c h as t h i s o n e , o n l y s l i g h t p e r t u r b a t i o n s i n t h e s a m p l e w e i g h t are n e e d e d t o m i t i g a t e t h e s u c c e s s f u l c a l c u l a t i o n o f t h e d e r i v a t i v e . T h i s p o i n t is m a d e b e c a u s e w i l d v a l u e s c a n h a v e e n o r m o u s e f f e c t s ( p r o p o r t i o n a l to t h e s q u a r e o f t h e i r w i l d n e s s ) o n t h e l e a s t s q u a r e s c u r v e f i t t i n g , w h i c h t r i e s to m i n i m i z e t h e s u m o f s q u a r e d deviations. T h e f a c t o r - j u m p m e t h o d i s a u t o m a t e d (24—28) a n d c o m p u t e r c o n t r o l l e d i n o u r i m p l e m e n t a t i o n at N B S (29). S u c h a u t o m a t i o n i s h i g h l y d e s i r a b l e i n t h a t c o n d i t i o n s m u s t b e c h a n g e d o f t e n ( e v e r y 10 m i n ) a n d e q u i l i b r a t i o n t i m e s a l l o w e d (—3 m i n ) b e f o r e t h e m e a s u r e m e n t s a r e continued. T h e computer program determines the activation energy d u r i n g t h e e x p e r i m e n t so t h a t v i s u a l m o n i t o r i n g o f t h e p r o c e s s as w e l l as c o m p u t e r i z e d f e e d b a c k i s p o s s i b l e . T h e a u t o m a t i o n p r o v i d e s a m e t h o d of m e a s u r i n g the sample w e i g h t a n d also p r o v i d e s control of s a m p l e t e m p e r a t u r e , flow r a t e o f N a n d 0 o v e r t h e s a m p l e , a n d t h e pressure i n the sample chamber. 2

2

T h e g e n e r a l s c h e m e i s g i v e n i n F i g u r e s 3 a n d 4. A l l m o d u l e s except the furnace are c o m m e r c i a l l y a v a i l a b l e . T h e c o m p u t e r sends

CLOCK

DIGITAL VOLTMETER Τ Π Γ Τ

CRT TERMINAL

INTERFACE

EXPERIMENT

DIGITAL-TO-ANALOG CONVERTERS

COMPUTER ¥

FLEXIBLE DISCS Figure

LINE PRINTER 3. A generalized, block-diagram thermogravimetry

outline of the apparatus.

factor-jump



12.


Thermogravimetry

221

COMPUTER

Figure 4. The linkage between the controller and the experiment. Abbreviations used: AS, analog scanner, BCD I/O, digital input/output; DAC, digital to analog converter; DT, drying tubes; DVM, digital voltmeter; EB, electrobalance; F, furnace; FC, furnace controller; FPS, furnace power supply; IPRC, ice point reference cell; MFC, mass flow controller; PC, pressure controller; PS, pressure sensor; SC, stopcock; SSS, saturated salt solution; SV, servo-driven valve; and TC, thermocouple. c o m m a n d s to the interface that generates voltages u s e d to specify the t e m p e r a t u r e , p r e s s u r e , a n d flow rates a r o u n d t h e s a m p l e as w e l l as r e a d i n g voltages generated b y the apparatus. T h e programs a n d data a r e s t o r e d o n a d i s k a n d t h e p r o g r e s s o f t h e e x p e r i m e n t is d i s p l a y e d o n the cathode ray tube c o m p u t e r t e r m i n a l . Furnace T h e furnace s u p p l i e d b y the manufacturer o f the thermobalance w a s r e p l a c e d b y a r a p i d - r e s p o n s e f u r n a c e ( F i g u r e 5) t h a t u s e s m a n y strands o f bare n i c h r o m e w i r e strung b e t w e e n c i r c u l a r forms to heat the environment a r o u n d the specimen. F o r applications w h e r e a f l o w i n g gas s t r e a m is u s e d , a n a d d i t i o n a l h e a t e r heats t h e gas s t r e a m t h r o u g h o u t its entire cross-section, thus a v o i d i n g the c o m m o n practice o f heating t h e s t r e a m o n l y at its e d g e . T o i n c l u d e t h e gas s t r e a m heater, w e h a d to r e v e r s e



Figure 5. A rapid response furnace featuring in-stream heaters for the gas stream (flow is left to right). Abbreviations used: 28 NW, 28 gauge nichrome wire; EB, electrobalance; ISH, in-stream heater; PRM, Pyrex reaction manifold; SR, support rod; SSS, saturated salt solution.


12.


223

Thermogravimetry


t h e d i r e c t i o n o f gas flow, w h i c h n o w e n t e r s o n t h e s i d e o f t h e s a m p l e remote from the balance h o u s i n g a n d exits before the balance h o u s i n g through a tube i n the borosilicate reaction manifold. T h e balance is a horizontal beam type. T h e t h e r m o c o u p l e i n the apparatus is type-Ε, w h i c h gives a large ( — 8 0 / A V / ° C ) change o f electromotive force (emf) w i t h temperature i n t h e r a n g e o f i n t e r e s t a n d has a r e a s o n a b l y l i n e a r r e s p o n s e . T y p e - K i s not particularly suitable for o u r a p p l i c a t i o n ; i t has r o u g h l y h a l f t h e t e m p e r a t u r e c o e f f i c i e n t o f t y p e - Ε a n d suffers f r o m n o n l i n e a r i t y b e c a u s e o f a n o r d e r — d i s o r d e r t r a n s i t i o n (see R e f e r e n c e 3 0 a n d r e f e r ences therein). It c a n b e s h o w n (26) t h a t t h e f a c t o r - j u m p m e t h o d a l l o w s a n a p p r e c i a b l e offset b e t w e e n t h e r e a l t e m p e r a t u r e o f t h e s a m p l e a n d t h e value measured b y the thermocouple. T h e fractional error i n the acti v a t i o n e n e r g y d u e t o t h i s t h e r m a l offset, T , i s g i v e n (26) a p p r o x i m a t e l y b y 2To/T, w h i c h f o r Τ — 5 0 0 Κ a n d a n e r r o r o f 1 % i n Ε a l l o w s a t e m p e r a t u r e offset o f 2 . 5 ° C . F o r a m o r e g e n e r o u s e r r o r o f 1 k c a l i n a n a c t i v a t i o n e n e r g y o f 3 0 k c a l / m o l , t h e t o l e r a b l e t e m p e r a t u r e offset at 5 0 0 Κ is 8 . 3 ° C . 0

T h i s s u r p r i s i n g l y l a r g e offset m a y b e v i e w e d i n t w o w a y s : ( 1 ) t h e t h e r m o c o u p l e n e e d n o t b e i n c o n v e n i e n t l y c l o s e t o t h e s a m p l e , a n d (2) appreciable furnace inhomogeneity c a n b e tolerated. W e p l a c e d the t h e r m o c o u p l e as n e a r t o t h e s a m p l e c u p as i s c o n v e n i e n t ( < 2 m m a w a y ) , a n d , at o u r m a x i m u m i n h o m o g e n e i t y o f 3 ° c m , e x p e c t a m a x i m u m t e m p e r a t u r e offset o f < 1 . 5 ° b e t w e e n t h e t h e r m o c o u p l e a n d the m i d d l e of the sample c u p , w h i c h w o u l d i n t r o d u c e m a x i m u m errors o f — 0 . 6 % into the calculated values o f E . - 1

Sample

Considerations

T h e sample is c o n t a i n e d i n a quartz spoon o n the e n d o f a quartz rod, w i t h t h e t h e r m o c o u p l e centered u n d e r t h e spoon. T h e spoon b o w l is made i n the form o f a right c y l i n d e r , about 4 m m i . d . T y p i c a l l y , 1 5 — 3 0 m g o f s a m p l e a r e s p r e a d e v e n l y o v e r t h e floor o f t h e s p o o n . S a m p l e s u s u a l l y are p r e c o n d i t i o n e d u n d e r p r o g r a m c o n t r o l for 1 0 m i n at t h e t e m p e r a t u r e o f t h e f i r s t i s o t h e r m a l so t h a t t h e t e m p e r a t u r e s t a b i l i z e s a n d t h e s a m p l e a t t a i n s a m o d e s t rate o f w e i g h t l o s s v i a t h e degradative process o f interest.

Computer

Program

T h e p h i l o s o p h y u s e d i n d e s i g n i n g the c o m p u t e r program to oper ate t h e a p p a r a t u s w a s t o u s e t h e c o m p u t e r as a s o u r c e o f a c t i v e c o n t r o l , so t h a t i t c a n assess t h e c o u r s e o f t h e e x p e r i m e n t d u r i n g t h e e x p e r i m e n t a n d take appropriate action ( F i g u r e 6 ) . T h e c o m p u t e r program is written almost entirely i n F O R T R A N , is modular, a n d can b e adapted


224


INITIALIZE PROGRAM VARIABLES

CALCULATE Ε STATISTICS FOR 10 LATEST Ε VALUES


OPTIONAL CHECK OF FACTOR RESPONSE TIMES

OPTIONAL M E A S U R E M E N T OF BIAS VOLTAGES

RELATIVE PRECISION A D E Q U A T E OR DATA COLLECTION TIME RAN OUT OR WORSENING PRECISION

INITIALIZE A P P A R A T U S

READ INITIAL A N D FINAL S A M P L E WEIGHTS OPTIONAL S A M P L E BAKEOUT

C A L C U L A T E RELATIVE PRECISION Ej/olEi)

C O M P U T E FACTOR LEVELS

INADEQUATE PRECISION A N D IMPROVING PRECISION A N D TIME LEFT

RELATIVE PRECISION A D E Q U A T E OR D A T A COLLECTION TIME RAN OUT OR WORSENING PRECISION

COLLECT Ν POINTS e.g., w, σ (w) FROM M

READINGS

w

Τ, σ (Τ) FROM M

T

READINGS

Ρ, a (P) FROM M

p

READINGS

F . o ( F ) FROM M 0

0

F

F , σ (Fjyj) FROM Mp N

q 2

READINGS

FIT FUNCTION. EXTRAPOLATE. C A L C U L A T E RELATIVE PRECISION,F/o(F), IN FACTORS, AND flATES

READINGS

IWEIG WEIGHT TOO llMPFRECISE RELATIVE PRECISION INADEQUATE I STOP )

Figure

6. Flow diagram of program for automated jump thermogravimetry.

control

of

factor-


12.


Thermogravimetry

225

readily to other configurations o f apparatus a n d to other computers. It c o n t a i n s r e g r e s s i o n a n d e x t r a p o l a t i o n r o u t i n e s so t h a t i t c a n p r o c e s s r a w data to o b t a i n final d e r i v e d quantities a n d c a n calculate standard deviations i n the d e r i v e d quantities to estimate the attained precision. T h e c o m p u t e r p r o g r a m a l s o sets u p t h e i n i t i a l c o n d i t i o n s , g i v e s t h e o p e r a t o r t h e o p p o r t u n i t y t o m a k e f u r t h e r c h a n g e s after d i s p l a y i n g a l l his currently chosen values, a n d later changes some user chosen o p tions i f necessary.


Weight-Loss

Polynomial

E x t e n d e d tests (27) s h o w e d that t h e s m a l l p o r t i o n s o f t h e w e i g h t l o s s — t i m e c u r v e u s e d h e r e are b e s t f i t t e d b y a s e c o n d - d e g r e e p o l y n o m i a l , a n d that s u c h a p o l y n o m i a l c a n b e extrapolated successf u l l y b e y o n d t h e r a n g e o f its d e t e r m i n a t i o n t o g i v e s u f f i c i e n t l y v i a b l e e s t i m a t e s o f i t s d e r i v a t i v e , t h e rate o f w e i g h t l o s s w i t h r e s p e c t t o time.

Temperature

Polynomial

Although the temperature—time behavior is ideally lineara n d i n d e p e n d e n t of t i m e , i n practice there is a complex d a m p e d oscillation a p p r o a c h t o e q u i l i b r i u m . T h e r e f o r e , w e f i t its t r e n d w i t h t i m e w i t h a first-degree p o l y n o m i a l . G e n e r a l l y , h i g h e r coefficients t h a n first are not statistically significant a n d confer instability rather t h a n i m p r o v e the precision.

Polynomials for Other Factors T h e r e m a i n i n g factors, t h e p r e s s u r e a n d f l o w rates, u s u a l l y are h e l d at c o n s t a n t l e v e l s d u r i n g t h e e x p e r i m e n t . T h u s , i t i s a p p r o p r i a t e to f i t t h e m w i t h a z e r o - o r d e r p o l y n o m i a l , t h a t i s , t o a c o n s t a n t v a l u e .

Data

Collection

T h e r e q u i r e m e n t s o f data c o l l e c t i o n are that isothermals s h o u l d b e f a i r l y s h o r t so t h a t t h e w e i g h t — t i m e c u r v e c a n b e f i t t e d w e l l w i t h a second-degree p o l y n o m i a l , the behavior of the sample s h o u l d b e c o m e s t e a d y u n d e r a n e w set o f c o n d i t i o n s b e f o r e d a t a c o l l e c t i o n i s s t a r t e d , a n d that the p r o g r a m s h o u l d fit p o l y n o m i a l s to the w e i g h t a n d factor ( t e m p e r a t u r e , p r e s s u r e , gas f l o w s ) t r e n d s w i t h t i m e d u r i n g t h e s a m p l e equilibration time. A compromise must b e struck b e t w e e n m a n y closely spaced data a n d l o n g computational time-outs and, o n t h e other hand, f e w w i d e l y spaced data w i t h imprecise but r a p i d p o l y n o m i a l fits.

Applications T h e m e t h o d provides estimates, E o f the activation energy for w h a t e v e r c o m b i n a t i o n o f p r o c e s s e s i s rate l i m i t i n g . T h e d i s t r i b u t i o n o f u


226


Ει w i t h t i m e a n d e x t e n t o f r e a c t i o n i s e x a m i n e d for r e g i o n s i n w h i c h the same process is d o m i n a n t . F r o m a p p l y i n g the p r o p a g a t i o n o f error p r o c e d u r e w i t h no correlations b e t w e e n the errors i n Τ a n d r, the p r o g r a m a l s o p r o v i d e s e s t i m a t e s o f of., t h e v a r i a n c e o f E * . T h e s e q u a n t i t i e s , E a n d σ ., a r e u s e d i n t h e s t a t i s t i c a l d a t a h a n d l i n g p r o g r a m s t o c o m p a r e t h e d i s t r i b u t i o n o f (Ε - Ει)/σ . w i t h t h e n o r m a l d i s t r i b u t i o n to s e e k o u t a b e r r a n t v a l u e s o f E ; h e r e Ε i s t h e a v e r a g e v a l u e o f E * . Satisfactory v a l u e s o f E a n d σ . are u s e d to p r o v i d e r e l i a b l e estimates o f t h e a v e r a g e a c t i v a t i o n e n e r g y a n d its s t a n d a r d d e v i a t i o n f o r r e g i o n s of conversion, a , w h e r e the same rate-limiting process applies. T h e t e c h n i q u e a n d the m e t h o d s o f statistical e v a l u a t i o n w e r e u s e d to s t u d y t h e t h e r m a l d e g r a d a t i o n a n d o x i d a t i o n o f p o l y s t y r e n e (30) a n d i n o t h e r i n v e s t i g a t i o n s n o w i n p r e p a r a t i o n for p u b l i c a t i o n . T h e s t a t i s t i c a l t e c h n i q u e s i n p a r t i c u l a r are d e s c r i b e d i n d e t a i l w i t h e x a m p l e s i n R e f e r ence 29. t

Ε

Ε

f


t

Dynamic Heating

Ε

Rate Methods

T h e s e methods fall into two groups, one w h e r e the activation e n e r g y of the i n i t i a l d e g r a d a t i o n process is e s t i m a t e d f r o m one ex p e r i m e n t , a n d the other w h e r e c o m p a r i s o n of a series of experiments is n e c e s s a r y b e f o r e a n a c t i v a t i o n e n e r g y c a n b e e s t i m a t e d . T h e c o n d i tions o f a p a r t i c u l a r e x p e r i m e n t are k e p t constant, e x c e p t for t h e t e m p e r a t u r e , w h i c h is i n c r e a s e d c o n t i n u o u s l y , a n d t h e p r o c e s s i n g o f t h e d a t a is s t r a i g h t f o r w a r d . T h e r e f o r e , i n c o n t r a s t to o u r a p p l i c a t i o n o f t h e factor-jump m e t h o d , t h e r e is n o n e e d of a u t o m a t i o n to c o n s t a n t l y c h a n g e e x p e r i m e n t a l c o n d i t i o n s a n d little n e e d o f f e e d b a c k to c h a n g e the course of the experiment.

A Thermogravimetric Instrument for Isothermal and Heating Rate Experiments

Constant

A t h e r m o g r a v i m e t r i c i n s t r u m e n t for d e t e r m i n i n g w e i g h t c h a n g e o f p o l y m e r s h e a t e d at p r o g r a m m e d rates w a s d e s i g n e d a n d b u i l t at t h e N a t i o n a l B u r e a u of Standards a n d utilizes over 30 years of experiences i n t h i s a r e a . I t m a y b e o p e r a t e d u n d e r v a c u u m o r m i l d f l o w o f gas a n d is e q u i p p e d w i t h a c c e s s o r i e s f o r m e a s u r e m e n t a n d c o n t r o l o f f l o w r a t e , p r e s s u r e , a n d v a c u u m . C o l d traps a n d outlets for the c o l l e c t i o n o f v o l a t i l e s are a v a i l a b l e . W e i g h t i s d e t e r m i n e d b y a g l a s s - e n c l o s e d e l e c t r o b a l a n c e o f 1 0 ~ g s e n s i t i v i t y so t h a t w e i g h t c h a n g e i n a 1 0 - m g polymer specimen may be determined w i t h good precision. E x p e r i m e n t s i n f l o w i n g gas are c a r r i e d o u t at s l o w rates o f v o l a t i l i z a t i o n , so t h e r a t e o f gas f l o w s u f f i c i e n t t o r e m o v e v o l a t i l e p r o d u c t s c a n b e k e p t s l o w e n o u g h so as n o t t o d i s t u r b t h e w e i g h i n g p r o c e s s . L o n g - t e r m b a s e - l i n e s t a b i l i t y o f t h e e l e c t r o b a l a n c e p e r m i t s e x p e r i m e n t s at v e r y 6


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s l o w h e a t i n g rates (97day) so that t h e range o f t e m p e r a t u r e f o r t h e i n v e s t i g a t i o n o f t h e d e g r a d a t i o n o f a t y p i c a l p o l y m e r i s o v e r 200°. T h e k e y t o the p y r o l y s i s p o t e n t i a l o f t h e apparatus i s its f u r n a c e . I t consists of t w o concentric 1.6-mm t h i c k steel tubes, the outer one, 38.0 m m i n diameter, a n d t h e i n n e r , 31.5 m m . T h e tubes are p o s i t i o n e d v e r t i c a l l y b y t w o 1 2 . 7 - m m t h i c k p y r o p h y l l i t e d i s k s at t h e t o p a n d b o t t o m . N a r r o w v e r t i c a l slits i n t h e stainless steel tubes are staggered b e t w e e n t h e i n n e r a n d outer t u b e s so that t h e h e a t i n g w i r e s d o n o t transmit direct radiation to the pyrex envelope, w h i c h may b e c o o l e d w i t h l i q u i d nitrogen. T h e assembly is d e s i g n e d to p e r m i t volatiles to diffuse r a p i d l y f r o m the hot z o n e ; most are e n t r a p p e d o n the sides o f the envelope. T h e N o . 2 0 nichrome heating wires are w o u n d tightly between the p y r o p h y l l i t e d i s k s , w h i c h are m o u n t e d o n the s t e e l t u b e s w i t h six 1.6-mm I n c o n e l springs to m a i n t a i n the t e n s i o n o n the h e a t i n g w i r e s as t h e y e x p a n d o r c o n t r a c t . T h e f u r n a c e a t t a i n s 3 0 0 ° C i n < 2 m i n , 4 0 0 °C i n 3 m i n , 5 0 0 °C i n 5 m i n , a n d 6 0 0 °C i n 12 m i n , starting at a m b i e n t t e m p e r a t u r e s . T h u s , t h e s y s t e m c o m b i n e s t h e fast r e s p o n s e necessary for quick establishment of isothermal temperatures w i t h sufficient t h e r m a l i n e r t i a for m a i n t e n a n c e o f i s o t h e r m a l t e m p e r a t u r e s w i t h i n 1 °C. T h e temperature of the type Ε thermocouple seated 1 - 2 m m b e l o w the sample bucket was calibrated b y comparison w i t h ther mocouples embedded i n specimens of polyethylene a n d polytetraf l u o r o e t h y l e n e o v e r t h e t e m p e r a t u r e r a n g e 2 0 0 - 6 0 0 ° C at a h e a t i n g rate o f 2 7 m i n . A t e m p e r a t u r e c o r r e c t i o n o f a b o u t 3° w a s e s t a b l i s h e d to b e necessary for the f i x e d t h e r m o c o u p l e . Isoconversional D i a g n o s t i c Plots. I n this m e t h o d , w e assume that the A r r h e n i u s e q u a t i o n is v a l i d a n d that the same f o r m , / ( a ) , of the d e p e n d e n c e o f rate o n e x t e n t o f r e a c t i o n , a , i s m a i n t a i n e d t h r o u g h o u t . E x p e r i m e n t s are c o n d u c t e d at a series o f h e a t i n g rates, β . G i v e n t h e c o n d i t i o n s j u s t m e n t i o n e d , w e c a n formulate (21,31 ) for a g i v e n extent of reaction, a , the equation ΔΙη β = 1.05(E/R) Δ(1/Τ)

(16)

T h e parameter Ε t h e n can b e estimated from the slope o f a plot o f In β v s . 1/T a t a g i v e n e x t e n t o f r e a c t i o n f r o m r u n s at s e v e r a l d i f f e r e n t h e a t i n g rates. T h i s m e t h o d c a n b e e x t e n d e d t o g i v e Ε f o r s e v e r a l extents o f reaction, t y p i c a l l y every 10%. A s does t h e factor-jump m e t h o d , t h i s m e t h o d o b v i a t e s t h e n e e d t o k n o w f(a). A l s o , i t c o v e r s a w i d e temperature range a n d is computationally robust because w e i g h t - l o s s c u r v e s are u t i l i z e d d i r e c t l y . T h e c o n s i s t e n c y o f the a c t i v a tion energy throughout the range o f reaction a n d throughout a w i d e


228



t e m p e r a t u r e r a n g e is s h o w n v i s u a l l y b y p a r a l l e l l i n e s o n t h e p l o t o f I n β v s . 1/T. T h e m e t h o d has s o m e d i s a d v a n t a g e s i n t h a t t h e effects o f e r r o r s are c u m u l a t i v e , that i s , e a r l y errors are p a s s e d o n to later r e s u l t s . A s e r i o u s r e s t r i c t i o n is that o n e m u s t u s e m o r e t h a n o n e s p e c i m e n a n d m u s t a s s u m e that the w e i g h t - l o s s k i n e t i c s are i n d e p e n d e n t o f the d i f f e r i n g thermal histories a n d of any difference i n the p h y s i c a l character of the s p e c i m e n s . T h i s r e s t r i c t i o n m e a n s that, r e g a r d l e s s o f t h e t h e r m a l h i s t o r y o f t h e s p e c i m e n , t h e t e m p e r a t u r e , a n d t h e effects o f d e g r a d a t i o n , t h e d e p e n d e n c e o f rate o f w e i g h t loss o n the extent o f r e a c t i o n is a s s u m e d to b e t h e s a m e i n a l l s p e c i m e n s for a g i v e n e x t e n t o f r e a c t i o n . A l s o , as is the case w i t h a l l d y n a m i c h e a t i n g experiments, the p o s s i b i l i t y exists t h a t t h e t e m p e r a t u r e w i l l n o t b e a b l e t o e q u i l i b r a t e at t h e f a s t e r h e a t i n g rates. I f t h e s l o p e s o f t h e i s o c o n v e r s i o n a l l i n e s o f a p l o t o f I n β v s . 1/T a r e e q u a l , t h e n a s i n g l e g l o b a l a c t i v a t i o n e n e r g y exists for the ranges o f a a n d T c o v e r e d b y t h e e x p e r i m e n t . H o w e v e r , for p o l y m e r w e i g h t - l o s s k i n e t i c s s u c h c a s e s are t h e e x c e p t i o n . A m o r e t y p i c a l e x a m p l e i s i l l u s t r a t e d i n F i g u r e 7 i n w h i c h l o g β i s p l o t t e d a g a i n s t 1/T f o r t h e d é g r a d a -


12.


229

Thermogravimetry

-

251243

320 -

-

-

237 230

233

230

233

228

300 -

211

226


T(°C) 280

234

-

196

200

205

189 176

184 182

179

260 -171 -

172 172

-

175

170

-

171

240 -

-

162

-

"152 .1

.2

.3

.4

.5

.6

.7

.8

.9

FRACTIONAL WT.-LOSS (5) Figure 8. Grid of activation energy kilojoules per mole (kjimol) polymethyl methacrylate in nitrogen as a function of temperature and conversion (a.) from data in Figure 7.

for (Ύ)

t i o n o f p o l y m e t h y l m e t h a c r y l a t e i n n i t r o g e n . T h e r a n g e o f h e a t i n g rates from 1 0 " to 1 0 K/s permits a n observation of the kinetics over t h e t e m p e r a t u r e r a n g e f r o m 2 1 0 t o 3 4 0 ° C (32). T h e c u r v a t u r e o f t h e i s o c o n v e r s i o n a l s i n d i c a t e s that the a c t i v a t i o n e n e r g y is c h a n g i n g c o n t i n u ously. H o w e v e r , t h e w i d e range o f temperatures that these reactions cover gives o n e t h e p r e c i s i o n necessary to d e t e r m i n e t h e activation energies from the slope b e t w e e n each successive pair o f points o n e a c h i s o c o n v e r s i o n a l . F r o m t h e s e v a l u e s a g r i d is set u p as i n F i g u r e 8, w h e r e t h e a c t i v a t i o n e n e r g i e s ( i n k j / m o l ) a r e g i v e n as a f u n c t i o n o f a a n d a n a v e r a g e t e m p e r a t u r e , T. C o m p a r i s o n o f a c t i v a t i o n e n e r g i e s p a r a l l e l to t h e ordinate shows that Ε increases w i t h i n c r e a s i n g t e m p e r a t u r e at c o n s t a n t c o n v e r s i o n , b u t p a r a l l e l t o t h e a b s c i s s a , t h e a c t i v a t i o n e n e r g y a p p e a r s t o b e n e a r l y i n d e p e n d e n t o f c o n v e r s i o n at c o n stant t e m p e r a t u r e . 4

_ 1

T h i s m e t h o d o f t r e a t m e n t o f w e i g h t - l o s s d a t a at c o n s t a n t h e a t i n g rate i s n o t o n l y a g o o d d i a g n o s t i c t o s e e i f t h e a c t i v a t i o n e n e r g y i s


230


c o n s t a n t for a c e r t a i n r e g i o n o f c o n v e r s i o n - t e m p e r a t u r e s p a c e b u t , i n some cases, c a n also give some i n s i g h t into the k i n e t i c s from the w a y t h a t Ε c h a n g e s w i t h a a n d T. A c t i v a t i o n E n e r g y of I n i t i a l Stage of D e g r a d a t i o n . T h e i n i t i a l a c t i v a t i o n e n e r g y i s o b t a i n e d , as i n d i c a t e d i n E q u a t i o n 1 1 , f r o m a p l o t o f T da/dT a g a i n s t a , w h i c h g i v e s a s l o p e o f E/R + 2 Γ at l o w e x t e n t o f r e a c t i o n (a ^ 0.05). T h i s c a l c u l a t i o n c a n b e d o n e s i m p l y f r o m a s i n g l e t h e r m o g r a v i m e t r i c t r a c e o f w e i g h t a g a i n s t t i m e o r t e m p e r a t u r e (8, 3 3 ) . T h i s m e t h o d is i n d e p e n d e n t o f t h e f o r m o f f(a) b e c a u s e t h e e x t e n t o f r e a c t i o n is a l w a y s s m a l l . H e n c e , t h e f o r m o f t h e d e p e n d e n c e o f t h e rate o f r e a c t i o n o n t h e e x t e n t o f r e a c t i o n m a y b e i g n o r e d s a f e l y . A n o t h e r a d v a n t a g e i s t h a t o n l y o n e s p e c i m e n i s r e q u i r e d , so t h a t p r o b l e m s a r i s i n g f r o m d i f f e r i n g s a m p l e h i s t o r i e s are a v o i d e d . W e c a n e s t i m a t e o t h e r a s p e c t s o f t h e k i n e t i c s f r o m t h e c h a n g e i n s l o p e as a i n c r e a s e s . I f t h e s l o p e i n c r e a s e s w i t h i n c r e a s i n g a, t h e n t h e d e p e n d e n c e o n w e i g h t l o s s h a s a n a u t o c a t a l y t i c c h a r a c t e r as i n r a n d o m l y i n i t i a t e d degradations of p o l y m e r s . W h e n the slope decreases w i t h i n c r e a s i n g a , t h e k i n e t i c s b e h a v e as a p o s i t i v e o r d e r r e a c t i o n s u c h as i n p o l y m e r degradations that occur m a i n l y b y u n z i p p i n g . A slope i n d e p e n d e n t of a i m p l i e s zero-order k i n e t i c s , one p o s s i b i l i t y for w h i c h is e v a p o r a t i o n of preformed molecules.


2

O n e p r o b l e m is that the extent of c o n v e r s i o n enters i n t o b o t h q u a n t i t i e s p l o t t e d , so o n e m u s t b e m o r e t h a n u s u a l l y c o n s i s t e n t i n p i c k i n g the b e g i n n i n g of the p o l y m e r d e g r a d a t i o n . T h e i n i t i a l rates of w e i g h t loss are e s p e c i a l l y s e n s i t i v e to v o l a t i l e c o n t a m i n a n t s i n c l u d i n g m o n o m e r , solvent, plasticizer, a n d other small m o l e c u l e s that c a n be lost w i t h o u t d e g r a d i n g the p o l y m e r . S o m e pretreatment trials w i l l be n e c e s s a r y i f t h e e f f e c t o f t h e s e m a t e r i a l s i s to b e m i n i m i z e d . H o w e v e r , i f o n e w i s h e s t o s t u d y t h e k i n e t i c s o f t h e i r l o s s ( u s u a l l y at s o m e e l e vated temperature) t h e n this technique may be appropriate. T h e same r e a s o n i n g a p p l i e s to c h e m i c a l effects f r o m r e s i d u a l catalysts, a n t i o x i dants, stabilizers, a n d l a b i l e linkages i n t r o d u c e d d u r i n g synthesis a n d s t o r a g e , a l l o f w h i c h affect t h e i n i t i a l k i n e t i c s . W h e n t h e k i n e t i c s o f t h e m a i n - c h a i n s c i s s i o n p r o c e s s are to b e i n v e s t i g a t e d , the effects j u s t n o t e d m a y m a s k the e a r l y phases o f t h e r e a c t i o n b y t h e e r r a t i c e v o l u t i o n o f l o w m o l e c u l a r w e i g h t s p e c i e s as i s d e m o n s t r a t e d b y t h e c u r v e f o r t h e u n t r e a t e d s a m p l e i n F i g u r e 1. S u c h c o m p l i c a t i o n s m a y b e a v o i d e d i n t w o w a y s : (1) t h e e x p e r i m e n t s m a y b e c o n d u c t e d at v e r y s l o w h e a t i n g r a t e s , w h e n s u c h l o w Ε p r o c e s s e s w i l l o f t e n u n c o u p l e as s e p a r a t e l o w t e m p e r a t u r e e v e n t s , a n d (2) a n i s o t h e r m a l p r e t r e a t m e n t i n v a c u u m at a t e m p e r a t u r e — 1 0 0 ° C b e l o w t h a t at w h i c h t h e r e a c t i o n o f i n t e r e s t b e g i n s t o t a k e p l a c e at a m e a s u r a b l e rate w i l l often r e m o v e these c o m p l i c a t i o n s . T h e r e m o v a l o f part of t h e s p e c i m e n i n the faster h e a t i n g rate e x p e r i m e n t s i n t h i s case r e -


12.


231

Thermogravimetry

quires the calculation of a n e w fraction reacted, α', based o n a n initial w e i g h t f r o m w h i c h the w e i g h t lost d u r i n g t h e pretreatment has b e e n subtracted. A n error i n t h e estimate o f this latter quantity results i n a curve whose slope w i l l nearly parallel the correct slope b u t w i l l not extrapolate through the origin. Varied H e a t i n g Rate Analysis.

T h i s m e t h o d is d e s i g n e d to s h e d

light o n some of the elementary reactions m a k i n g u p the composite f(a). I t c o n s i s t s o f e x a m i n i n g t h e s h i f t o f p e a k s i n p l o t s o f daldT v s . Τ as t h e h e a t i n g r a t e i s v a r i e d . T h e o r e t i c a l c o n s i d e r a t i o n s (34) s h o w t h a t


peaks c o r r e s p o n d i n g to i n d e p e n d e n t reactions w i t h w i d e l y d i f f e r i n g a c t i v a t i o n e n e r g i e s c a n b e r e s o l v e d at s o m e a t t a i n a b l e h e a t i n g r a t e . F o r c o m p e t i t i v e r e a c t i o n s , o n e p e a k o r t h e o t h e r w i l l d o m i n a t e as t h e h e a t i n g rate is c h a n g e d . T h e weight-loss process can sometimes b e resolved into simple or c o m p l e x cases, a n d c o m p l i c a t e d cases c a n b e r e s o l v e d i n t o c o m p e t i n g and i n d e p e n d e n t reactions. T h e necessary conditions i n c l u d e a w i d e r a n g e o f t e m p e r a t u r e s a n d h e a t i n g r a t e s , so t h e c o m p l e t e r a n g e o f t h e d e g r a d a t i o n r e a c t i o n is e x a m i n e d . H o w e v e r , t h e d i f f e r e n c e i n a c t i v a t i o n e n e r g i e s m u s t b e r a t h e r l a r g e (—20 k c a l / m o l ) t o o b t a i n g o o d r e s o l u t i o n b e c a u s e t h e largest r a n g e o b t a i n a b l e i n h e a t i n g rates is — 1 0 . 4

R e a c t i o n s o c c u r r i n g at r o u g h l y c o m p a r a b l e r a t e s i n t h e s a m e t e m p e r ature range, a n d w h i c h are realistic alternatives to o n e another, are o f t e n n o t as d i f f e r e n t as 2 0 k c a l / m o l i n a c t i v a t i o n e n e r g y . E x p e r i m e n t s at v e r y s l o w h e a t i n g r a t e s m a y p r o v i d e s o m e i n d i c a t i o n o f w h i c h r e a c t i o n w i l l d o m i n a t e at s e r v i c e c o n d i t i o n s , b u t t h e d u r a t i o n o f t h e i n v e s t i g a t i o n b e c o m e s v e r y l o n g . N o n e t h e l e s s , t h e v a r i e d h e a t i n g rate m e t h o d w a s a p p l i e d o v e r a r a n g e o f h e a t i n g r a t e s f r o m 1 0 " K / s (—97 4

day) to 1 0 In

- 1

K / s ( 6 7 m i n ) t o s e v e r a l p o l y m e r d e g r a d a t i o n r e a c t i o n s (32).

some cases, d e r i v a t i v e peaks w e r e

uncoupled b y reduction of

h e a t i n g rate. I n others, d i s p e r s i o n effects i n p e a k a m p l i t u d e s o f rate curves w e r e u s e d to diagnose changes i n m e c h a n i s m , a n d t h e e x p e r i m e n t a l temperature range w a s e x t e n d e d s i g n i f i c a n t l y to l o w e r t e m peratures (J, 2, 3 2 , 35, 36).

Conclusions T h r e e o f the techniques described i n this chapter have a c o m m o n e l e m e n t — t h e d e t e r m i n a t i o n o f activation energy for p o l y m e r degrada tion reactions w h i l e avoiding the pitfall of a p p l y i n g the more c o m m o n t e c h n i q u e s o f k i n e t i c a n a l y s i s to t h e r m o g r a v i m e t r y — t h a t o f h a v i n g to i n v o k e m a t h e m a t i c a l m o d e l s for t h e e l u s i v e / ( a ) , t h e func t i o n a l i t y o f t h e rate w i t h r e s p e c t to f r a c t i o n a l c o m p l e t i o n . T h e f o u r t h t e c h n i q u e sheds some l i g h t o n the c o m p o n e n t reactions that enter i n t o f(ct). T h e t e c h n i q u e s w e r e u s e d s u c c e s s f u l l y i n s e v e r a l i n v e s t i g a t i o n s , most o f w h i c h are c i t e d here.



232


Literature Cited 1. Flynn, J. H. In "Aspects of Degradation and Stabilization of Polymers"; Jellinek, H. H. G., Ed.; Elsevier: Amsterdam, 1978; Chap. 12, pp. 573-603. 2. Flynn, J. H. In "Thermal Analysis in Polymer Characterization"; Turi, E., Ed.; Heyden: London, 1981; pp. 43-59. 3. Flynn, J. H. In "Thermal Analysis"; Schwenker, R. F., Jr.; Garn, P. D., Eds.; Academic: New York, 1969; Vol. 2, p. 1111. 4. van't Hoff, J. H. In "Studies in Chemical Dynamics" (revised by Cohen, E.; translated by Ewan, T.); Chem. Pub. Co.: Easton, 1896. 5. Letort, M. J. Chim. Phys. Phys.-Chim. Biol. 1937, 34, 206. 6. Freeman, E. S.; Carroll, B. J. Phys. Chem. 1958, 62, 394. 7. Cameron, G. G.; Kerr, G. P. Vac. Microbalance Tech. 1970, 7, 27. 8. Flynn, J. H.; Wall, L. A. Polym. Lett. 1967, 5, 191. 9. Reich, L. J. Polym. Sci. 1965, B3, 231. 10. Coats, A. N.; Redfern, J. P. J. Polym. Sci. 1965, B3, 917. 11. Piloyan, G. Ο.; Ryabchikov, I. D.; Novikova, O. S. Nature (London) 1966, 212, 1229. 12. Flynn, J. Η.; Wall, L. A. J. Res. Natl. Bur. Stand. (U.S.) 1966, 70A, 487. 13. Cameron, G. G.; Fortune, J. D. Eur. Polym. J. 1968, 4, 333. 14. MacCallum, J. R.; Tanner, J. Eur. Polym. J. 1970, 6, 1033. 15. Šesták, J. Thermochim. Acta 1971, 3, 150. 16. Beigen, J. R.; Czanderna, A. W. J. Therm. Anal. 1972, 4, 39. 17. Simon, J. J. Therm. Anal. 1973, 5, 271. 18. Škvará, F.; Šesták, J. Chem. Listy 1974, 68, 225. 19. Audebert, R.; Aubineau, C. Eur. Polym. J. 1970, 6, 965. 20. Ozawa, T. J. Therm. Anal. 1975, 7, 601. 21. Ozawa, T. Bull. Chem. Soc. Jpn 1965, 38, 1881. 22. Simha, R.; Wall, L. A. J. Phys. Chem. 1952, 56, 707. 23. Flynn, J. H.; Dickens, B. Thermochim. Acta 1976, 15, 1. 24. Dickens, B.; Pummer, W. J.; Flynn, J. H. In "Analytical Pyrolysis"; Jones, C. E. R.; Cramers, C. Α., Eds.; Elsevier: Amsterdam, 1977; pp. 383-91. 25. Dickens, B.; Flynn, J. H. In "Proceedings of the First European Sympos ium on Thermal Analysis"; Dollimore, D., Ed.; Heyden: London, 1976; pp. 15-18. 26. Dickens, B. Thermochim. Acta 1979, 29, 41. 27. Dickens, B. Thermochim. Acta 1979, 29, 87. 28. Dickens, B. Thermochim. Acta 1979, 29, 57. 29. Dickens, B. "User's Manual for Factor-Jump Thermogravimetry Appara tus and Associated Computer Programs, Including a General Plotting Program," National Bureau of Standards Internal Report NBSIR 80— 2102, 1981. 30. Dickens, B. Polym. Degrad. Stabil. 1980, 2, 249. 31. Flynn, J. H.; Wall, L. A. Polym. Lett. 1966, 4, 323. 32. Flynn, J. H. In "Thermal Methods in Polymer Analysis"; Shalaby, S. W., Ed.; Franklin Inst.: Philadelphia, 1978; pp. 163-86. 33. Flynn, J. H.; Pummer, W. J.; Smith, L. E. Polym. Prepr., Am. Chem. Soc., Div. Polym. Chem. 1977, 18 (1), 757. 34. Flynn, J. H. Thermochim. Acta 1980, 37, 225. 35. Flynn, J. H.; Dickens, B. In "Durability of Macromolecular Materials"; Eby, R. K., Ed.; ACS SYMPOSIUM SERIES No. 95, ACS: Washing ton, D.C., 1979; pp. 97-115. 36. Flynn, J. H. Polym. Eng. Sci. 1980, 20, 675. RECEIVED for review October 14, 1981. ACCEPTED December 28, 1981.


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