thermal gradient

8/8/2019 Thermal Gradient

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Journal of Thermal Analysis, Vol. 6 (1974) 345--354

T H E R M A L G R A D I E N T S I N D I F F E R E N T I A L S C A N N I N G

C A L O R I M E T R Y

M . J . R I C H A R D S O N a n d P . BURRINGTON

Division of Materials Applications, National Physical Laboratory,Teddinyton, Middlesex, U.K.

(Received O ctober 26 , 19 72; in revised fo rm D ecem ber 2 , 1972)

A combined s ta t ic and dynamic temperature ca l ibra t ion is descr ibed. The s ta t ic

calibration co rrects the instrumental dial temp eratu re reading. The dyn am ic calibrationhas ins trumenta l and mater ia l components and therefore var ies f rom specimen tospecimen. It is obtained from individual DSC curves and so removes uncertaintiesin sample temperature due to varying mass , geometry , and heat ing ra te . The instru-menta l performance is improved and specif ic heats may be obta ined to an accuracyo f ~ 1 % .

T~,Tf

A T = r ~ - r ~ ;TDT

Tm

6r~

6 T ( : 6 T 0 - [- -6Treat)

6To, 6Tn~.~

6 T d ( = 6 T ~ + 6 T )

R

H ( T 3 , H ( T O

6p

C p , T f

L i s t o f s y m b o l s u s e d

s t e a d y i n i t i a l a n d f i n a l t e m p e r a t u r e s , r e s p e c t i v e l y , i n a

c a l o r i m e t r i c e x p e r i m e n t ;

d i a l t e m p e r a t u r e r e a d i n g ;

t r u e t e m p e r a t u r e c o r r e s p o n d i n g t o T o a f t e r a ll c o r r e c t i o n s

h a v e b e e n m a d e ;

m e l t i n g p o i n t ;

c o r r e c t i o n t o T D t o g i v e T u n d e r i s o t h e r m a l , ' s t a t i c ' c o n d it i o n s;

a d d i t i o n a l c o r r e c t i o n t o T D , d u e t o t h e r m a l l a g, to g i v e T

u n d e r p r o g r a m m e d t e m p e r a t u r e c o n d i t i o n s ;

c o n t r i b u t i o n s t o 6 T o r i g i n a t in g i n t h e i n s t r u m e n t a n d s a m p l e

m a t e r i a l r e s p e c t i v e l y ;

t o t a l c o r r e ct i o n t o T D t o g i ve T u n d e r p r o g r a m m e d t e m -

p e r a t u r e c o n d i t i o n s ;

c a l o r i m e t e r h e a t i n g r a t e ;

e n t h a l p y a t T~ a n d T f r e s p e c t i v e l y ;

m e a n h e a t c a p a c i t y o v e r t h e t e m p e r a t u r e r a n g e T i t o T ~;

h e a t c a p a c i t y a t T f .

D i f f e r e n t i a l t h e r m a l a n a l y s i s h a s l o n g b e e n a u s e f u l t o o l i n t h e f i el d s o f g e o l o g y

a n d m e t a l l u r g y [ 1] . T h e r e c e n t i n t r o d u c t i o n o f c o m m e r c i a l a p p a r a t u s , b a s e d

o n e i t h e r d i f f e r e n t i a l t h e r m a l o r d i f f e r e n t i a l e n t h a l p i c m e t h o d s , h a s l e d t o m a n y

a p p l i c a t i o n s f a r r e m o v e d f r o m t h e s e a re a s . I n p a r t i c u l a r t h e t e c h n i q u e h a s p r o v e d

J . T h e r m a l A n a l . 6 , 1 9 7 4



346 R I C H A R D S O N , B U R R I N G T O N : T H E R M A L G R A D I E N T S IN D S C

o f i m m e n s e v a lu e f o r d e t e r m i n in g t h e p u r i ty o f o r g a ni c c o m p o u n d s a n d f o r

c h a r a c t e r i s i n g p o l y m e r s [ 2]. B o t h t h e s e a p p l i c a t i o n s i n v o l v e m a t e r i a l s o f g e n e r a l l y

p o o r t h e r m a l c o n d u c t i v i ty . T h i s p a p e r w i ll e x a m i n e t h e e f f ec ts o f th e r m a l l a g

w h i c h h a v e b e e n e n c o u n t e r e d i n th e d e v e l o p m e n t o f a d a t a t r e a t m e n t s y s t e m

a n d s h o w h o w t h e y m a y b e e l i m i n a t e d b y s u i ta b l e u se o f t h e e x p e r i m e n t a l d a t a .

Experimental

A l l e x p e r i m e n t s w e r e c a r r i e d o u t u s i n g a P e r k i n E l m e r M o d e l 1 B d i f f e r e n t i a l

s c a n n i n g c a l o r i m e t e r ( D S C ) . T h e o u t p u t s i g n a l f r o m t h e D S C w a s m o n i t o r e d

o n a d i g i t a l v o l t m e t e r a n d r e c o r d e d o n p a p e r t a p e .

C a l i b r a t i o n o f th e D S C h a s b e e n d e s c r i b e d i n d e t a il e ls e w h e r e [3 ]. H e r e i t is

e m p h a s i s e d t h a t t h e t e m p e r a t u r e c a l i b r a t i o n w a s c a r r i e d o u t a t a z e r o r a t e o fh e a t i n g b y t h e s l o w s t e p w i s e m e l t i n g ( u s in g t e m p e r a t u r e i n c r e m e n t s o f -< _0.1 K )

o f p u r e m a t e ri a ls . T h e c o r r e c t i o n 6T~ w h i c h g iv e s t h e t r u e t e m p e r a t u r e T c o r r e -

s p o n d i n g t o a n y d i a l r e a d i n g T D is t h e r e f o r e f r e e o f th e r m a l l a g ( ' s t a t ic ' , s u b -

sc r ip t ' s ' ) and T = TD + f i ts; 6T~ m a y b e o f e it h e r s ig n .

E m p Y ~ * ~ i

I _ I b ; ii ~ . . -1 T t !: L I T m eL ~ t e a d y j ~ T e m p e r a , . . . i s t a g ~ , ~ . s~eadg ~+ + t " + I

b )

I x+ ~ I1 4 s t e a d ~ m . ~ T e r n p e r a I . . . . i s i n g m . . T f ~ e l a d y . .~ -

Fig. I. Treatment of calorimetric data; (a) original curves (displaced for clarity), (b) materialcurves normalised to unit mass

T h e c a l o r i m e t e r w a s a l w a y s u s e d in t h e " s p ec i fi c h e a t m o d e " [4 ]. T h e t e m -

p e r a t u r e w a s r a i se d f r o m o n e s t e a d y v a l u e Ti t o a n o t h e r T f i n t h e s e q u e n c e :

( i) e m p t y p a n , ( ii) r ef e r e n c e m a t e r i a l , (i ii ) s a m p l e ( F i g . l a ) . A c o m p u t e r p r o g r a m m e

s u b t r a c t e d r e s ul ts f o r t h e e m p t y , p a n f r o m t h o s e f o r t h e r e f er e n c e m a t e r i a l a n d

s a m p l e t o g iv e c o m p o s i t e c u r v e s , n o r m a l i s e d t o u n i t m a s s ( F i g . l b ) , t h e a r e a s

o f w h i c h w e r e p r o p o r t i o n a l t o t h e s p ec if ic e n t h a l p y c h a n g e s H ( T f ) - H ( T i )

f r o m T i t o T ~, T h e m e a n h e a t c a p a c i t y e p i s H ( T f ) - H ( T i ) / 8 9 + Tf). A fea ture

J . T h e r m a l A n a l . 6 , 1 9 74



R I C H A R D S O N , B U R R I N G T O N : T H E R M A L G R A D I E N TS t N D S C 347

o f th e p r o g r a m m e w a s t h e a b il it y t o c o m p e n s a t e f o r e v e n g ro s s m i s m a t c h e s i n

t h e i s o t h e r m a l p o r t i o n s o f t h e c u r v e ( i. e. P i # q i # 0 , w h e r e i = I o r 2 ) . A d e q u a t e

b a l a n c i n g o f t h e i s o t h e r m a l b a s e li n e s is n o r m a l l y a d i ff ic u l t a n d t i m e c o n s u m i n g

o p e r a t i o n .

Thermal lag

T h e r e s p o n s e o f a n i d e a l c a l o r i m e t e r , w i t h n o t h e r m a l l ag , t o a l in e a r i n c re a s e

i n t e m p e r a t u r e f r o m T t o T r is t h e c o m p o s i t e c u r v e A B C D o f F i g. l b . W h e n

t h e r e a r e n o m a j o r t h e r m a l e v e n t s, s u c h a s m e l t i n g o r p h a s e c h a n g e s , t h e o r d i n a t e

i s p r o p o r t i o n a l t o t h e sp e c if ic h e a t c p. I n p r a c t i c e t h e o n s e t o f s t e a d y s t a t e

c o n d i t i o n s ( E F ) is d e l a y e d a n d a c u r v e A E F G i s o b t a i n e d . T h e t o t a l a r e a , X ,

~ 5 ~M /' " "'"'"

[ i 1 I335 340 345 350 355 350

' , \ ,"'...

~ -..

Temperature ~K

Fig. 2. Norm alised curves for synthetic sapphire d iscs at several heating rates

o f b o t h c u r v e s i s e q u a l a n d i s p r o p o r t i o n a l t o ? p . A T , w h e r e A T = T f - T i.

D i s t h e p o i n t w h e r e t h e f i n a l s t e a d y t e m p e r a t u r e , T f , i s a t t a i n e d a t t h e s e n s o r a n d

t h e s h a d e d " t a i l " a r e a , Y , r e p r e se n t s t h e a d d i t i o n a l e n e r g y n e e d e d t o b r i n g t h e

t e m p e r a t u r e o f th e s p e c im e n a n d c a l o r i m e t e r t o t h is v a l ue . Y is p r o p o r t i o n a l

to cp, T~ 6 T , w h e r e c p, T f i s t h e s p e c i f i c h e a t a t Zf o r , s t r i c t l y , a m e a n v a l u e o v e r

t h e r a n g e 6 T , w h e r e 6 T is t h e t e m p e r a t u r e l a g i n t h e sy s t e m . I t i s g i v e n b yY A T d p / X c p , T r T h e r a t i o ? p/C p, x~ i s n o r m a l l y o n l y a f e w p e r c e n t l es s t h a n u n i t y

a n d m a y g e n e r a ll y b e n e g le c t ed . H o w e v e r i t is im p o r t a n t w h e n a p o l y m e r p a s se s

f r o m a g la s s t o a r u b b e r a n d t h e r e i s a d i s c o n t i n u i t y i n Cp. A g o o d a p p r o x i m a t i o n

is g iv e n b y t h e r a t i o o f t h e a v e r a g e v a l u e o f th e o r d i n a t e t o t h e v a l u e w h e n t h e

p r o g r a m m e t e m p e r a t u r e f i r s t r e a c h e s T f .

T h e D S C o u t p u t is b a si c al ly a c o m b i n a t i o n o f ti m e ( a b sc is sa ) a n d v o l t a g e

( o r d in a t e ). T h e f o r m e r is e a si ly t r a n s f o r m e d t o t h e t e m p e r a t u r e s ca le s h o w n

i n F i g . 2 ( t h e h e a t i n g r a t e , R , i s a l i n e a r f u n c t i o n o f TD ). W h e n t h i s is d o n e t h e

o r d i n a t e m u s t b e c o r r e s p o n d i n g l y n o r m a l i se d . W i t h o u t t h is , a d o u b l i n g o f R

f o r a f i x e d t e m p e r a t u r e i n c r e m e n t , A T , w o u l d l ea v e th e t e m p e r a t u r e a x is u n c h a n g e d

a . T h e r m a l A n a l , 6 , 1 9 7 4



3 4 8 R I C H A R D S O N ,U R R I N G T O N :H E R M A LRADIENTSN D SC

( a l t h o u g h h a l v i n g t h e o r i g i n a l time s c a l e ) b u t d o u b l e t h e o r d i n a t e v a l u e s . D i s p l a y

o f t h e e x p e r i m e n t a l d a t a i n t h is w a y a l l o w s a d i r e c t c o m p a r i s o n o f th e e f f ec t s

o f h e a t in g r a t e a n d o f s a m p l e s iz e a n d g e o m e t r y .

R e s u l t s f o r a n a l u m i n a d i s c ( s y n t h e t i c s a p p h i r e 6 m m d ia n a x 5 m m t h i c k )

a r e s h o w n i n F i g . 2. T h e a l u m i n a w a s p l a c e d d i r e c t l y o n t h e D S C s p e c i m e n

h o l d e r . A l l a r e a s a g r e e t o w i t h i n + 1 ~ . T h e i n c r e a s e d d e l a y i n r e a c h i n g t h e

s t e a d y s t a t e r e g i o n (EF, Fig . l b ) , a s the he a t in g ra te i s ra i sed , i s c lea r ly v i s ib le 9

C a l c u l a t e d v a l u e s o f t h e t h e r m a l I a g a r e s h o w n i n F i g . 3 w h i c h a l s o g i v es re s u l ts

f o r a d i s c o f t h e s a m e d i a m e t e r b u t a p p r o x i m a t e l y t w i ce th e t h i c k n e ss ,

Z

E

i0 ~ f / / J / / I / C ' P L [ I , / ~ " I0 10 20 3(1

HeQtincJ ra te ~K rain 4

Fig. 3. Thermal lag as a func tion of sample ma ss and heating ra te

T h e b r o k e n l i n e o f F ig . 3 (6To) w a s o b t a i n e d b y p l o t t in g 6 T , a t a c o n s t a n t

h e a t i n g r a t e , a g a i n s t s a m p l e m a s s a n d e x t r a p o l a t i n g t o z e r o m a s s ( i d e n t i c a l ,

r e p r o d u c i b l e t h e r m a l c o n t a c t w i t h t h e D S C i s e s se n ti a l a n d a g i v e n e x t r a p o l a t i o n

a l w a y s i n v o l v e d d is c s o f c o n s t a n t d i a m e t e r ) . T h e b r o k e n l i n e is a n i n s t r u m e n t a l

c o n s t a n t , th e th e r m a l l ag be t w e e n t h e b o t t o m p l a n e o f t h e s a m p l e a n d t h e

t e m p e r a t u r e s e n s o r . I t i n c l u d e s t h e t h e r m a l r e s i s t a n c e b e t w e e n t h e s a m p l e a n dt h e D S C p a n h o l d e r ( al l re s u l ts s o f a r r e f e r t o d i sc s p l a c e d d i r e c t l y i n t h e i n s t r u -

m e n t ) . I t i s d i ff i cu l t t o v a r y t h e s a m p l e m a s s ( t h ic k n e s s ) b y m o r e t h a n a f a c t o r

o f a b o u t t w o a n d a g i v en e x t r a p o l a t io n t o z e r o m a s s is b a se d o n o n l y t w o o r

t h r e e p o i n t s 9 T h e v a l i d i t y m u s t t h e r e f o r e b e j u d g e d b y t h e v a l u e s o f 6To f o u n d

u s i n g m a t e r i a l s o f v e r y d i f f e re n t t h e r m a l c o n d u c t i v it i e s . T h u s t h e f o l l o w i n g r e s u l ts

w e r e o b t a i n e d u s i n g t h e s u b s t a n c e s s h o w n i n b r a c k e t s : 6To = .133 R (s i lver) ,

91 3 0 R ( l e a d ) , . 1 2 6 R ( s a p p h i r e ) , . 1 2 7 R ( b e n z o i c a c i d ) . T h e e x t r a p o l a t i o n i s c l e a r l y

v a l i d a n d g i v e s a t r u e i n s t r u m e n t a l c o n s t a n t .

U s i n g a n a l u m i n u m p a n a s a sa m p l e h o l d e r i n cr e as e s 6T 0 b y a p p r o x i m a t e l y

5 0 ~ . T h e l a r g e r f ig u r e i n c l u d es t r a n s f e r o f h e a t a c r o s s t w o i n t e r f a c e s , p a n h o l d e r

.1". The r~za l An al . 6 , 1 97 4



R I C H A R D S O N , B U R R I N G T O N : T H E R M A L G R A D I E N T S IN D S C 349

t o p a n a n d p a n t o s a m p l e , a n d t h e s e a r e t h e c o n d i t i o n s e n c o u n t e r e d i n n o r m a l

o p e r a t i o n o f th e i n s t r u m e n t . T h e s i m p l i f ie d c o n d i t i o n s n o t e d e a r l ie r w e r e d e si r a b l e

t o r e d u c e u n c e r t a i n t i e s d u e t o d if f e re n c e s i n s a m p l e p a n s . S a m p l e s a r e u s u a l l y

f a r r e m o v e d i n f o r m f r o m t h a t o f i d e al d is cs a n d 5 T m a y b e c o n s i d e r a b l y m o r e

t h a n i n d i c a t e d a b o v e .

A l l r u n s w i t h t h e D S C u s e s y n t h e t i c s a p p h i r e d i sc s a s r e f e r e n c e m a t e r i a l s

( t h e y w e r e c h o s e n b e c a u s e h i g h p u r i t y m a t e r ia l i s r e a d i l y o b t a i n a b l e a n d t h e

t h e r m o d y n a m i c p r o p e r t i es a r e w e l l d e f i n e d [ 5 ]) . M a n y r e s u lt s a r e t h e r e f o r e

a v a il a b le f o r 6 T a s a f u n c t i o n o f h e a ti n g r a t e , m a s s, a n d w h e t h e r a l u m i n i u m

Z 3.6H

; /o6 r n g s a p p h i r e d l s c

3 4 1 6 K m in 1 h e a t ~

3 2

3.0

s

2 . 8 ~

2 6 i [ I _ _i ~

300 400 5 00 600 70 0 8 0 0

Tempercaure) K

Fig. 4. Thermal lag as a function of temperature

s a m p l e p a n s a r e p r e s e n t o r n o t . F o r a g i v e n s e t o f c o n d i t i o n s 6T i s r e p r o d u c i b l e

t o + 0 . l K f o r R < 1 6 K m i n - 1 a n d _ +0 .2 K f o r R = 3 2 K m i n - 1 . T h e p o i n t s

o f F i g. 4 g i v e a g o o d i d e a o f t h e e x p e r i m e n t a l s c a t te r . L a r g e r v a r i a t i o n s m a y b e

f o u n d w h e n p a n s a r e u s e d b u t t h e s e i n v a r i a b l y i n v o l v e i n c r e a s e d v a l u e s o f 6T

a n d a r e d u e t o t h e u s e o f p a n s w h i c h a r e n o t p e r f e c t l y f la t. S i m i l a r r e m a r k s

a p p l y t o o t h e r m a t e r i a l s .B e c a u s e 5 T v a r ie s f r o m o n e s p e c i m e n t o a n o t h e r , t e m p e r a t u r e s d e t e r m i n e d

f r o m a c a l i b r a t i o n c u r v e m u s t o n l y b e f o r t u i t o u s l y c o r r e c t . T h e c o n s e q u e n c e s

o f e r r o r s i n t h e s p e c i m e n t e m p e r a t u r e w i l l b e c o n s i d e r e d i n t h e f o ll o w i n g s e c t io n .

F i r s t i t w i ll b e s h o w n h o w t h e y m a y b e r e m o v e d . B e f o r e th i s is d o n e t h e c h a n g e

o f 6T w i t h t e m p e r a t u r e m u s t b e c o n s i d e r e d . S o m e r e s u lt s f o r s a p p h ir e a r e s h o w n

i n F i g . 4 . T h e r e i s a s l o w i n c r e a s e w i t h t e m p e r a t u r e b u t e v e n a t 7 0 0 K 6 T o i s

o n l y a b o u t 3 0 % g r e a t e r t h a n t h e v a l u e a t r o o m t e m p e r a t u r e . T h e o v e r a l l v a r i a t io n

o f 6T i s a c o m p l e x f u n c t i o n o f t h e s e v e ra l m a t e r i a ls a n d i n t e r f a c e s e n c o u n t e r e d

e n r o u t e f r o m t h e h e a t e r s t o t h e s p e c i m e n a n d i t i s n o t p o s s i b l e t o d e r i v e a n y

i n f o r m a t i o n a b o u t t h e t h e r m a l c o n d u c t i v i t y o f t h e s p e c im e n .

3 ". T h e r m a l A n a l . 6 , 1 9 7 4



350 RICHARDSON, BURRINGTON: TH ERMA L GRADIENTS IN DSC

T h e t r u e t e m p e r a t u r e , T , o f a s a m p l e i n a D S C is o b t a i n e d f r o m T = T D +

+ f i T s + f i T . 6 T ~ i s a n i n s t r u m e n t a l c o n s t a n t ; f i T v a r i e s f r o m s a m p l e t o s a m p l e

a n d , v e r y s t r o n g l y , w i t h h e a t i n g r a t e . T h e c a l c u l a t i o n o f f i T w a s d e s c r i b e d e a r l i e r

a n d , a s f iT s w a s d e t e r m i n e d i n t h e " s t a t i c " c a l i b r a t i o n i t i s m a t h e m a t i c a l l y a

t r iv i a l o p e r a t i o n t o c o r r e c t a l l d ia l t e m p e r a t u r e s , T D . W h e n a r e f e r e n c e m a t e r i a l

a n d s a m p l e a r e r u n i n s u c c e s s i v e e x p e r i m e n t s f i T w i l l i n g e n e r a l b e d i f f e r e n t f o r

t h e t w o m a t e r i a l s . T h e c o m p u t e r p r o g r a m m e m u s t i n t e r p o l a t e o r d i n a t e v a l u e s

f o r b o t h a t c o m m o n , c o r r e c t e d te m p e r a t u r e s , T .

T h e t r e a t m e n t a b o v e r e f e r s t o c u r v e s ( F i g . 2 ) w h i c h a r e u n p e r t u r b e d b y p h a s e

c h a n g es . W h e n t h e se o c c u r t h e c a l c u l a t io n f o r 6 T is in v a li d . F o r t u n a t e l y p h a s e

c h a n g e s n o r m a l l y i n v o l v e l a r g e q u a n t i t i e s o f h e a t ( r e l a ti v e to s p e ci fi c h e a t ) a n d

o n t y s m a l l s a m p l e s a r e r e q u i r e d , i n t h i s c a s e 6 T m a y b e r e p l a c e d b y 6 T o .

Appl icat ions

(a ) T e m p e r a t u r e c a l i b r a t i o n

T h i s s e c t i o n w i ll c o n s i d e r s o m e o f t h e c o n s e q u e n c e s o f t h e t h e r m a l g r a d i e n t s

d i s cu s s ed a b o v e . A p r o b l e m o f b as i c i m p o r t a n c e c o n c e r n s t h e t e m p e r a t u r e s c a le

o f th e i n s t r u m e n t . T h i s is n o r m a l l y [ 6] c a l i b r a t e d d y n a m i c a l l y ( s u b s c ri p t ' d ' )

b y o b s e r v i n g t h e m e l t i n g p o i n t s , T ~ , o f p u r e m e t a l s a t t h e r e q u i r e d h e a t r a t e

a n d u s i n g t h e r e l a t i o n T m = Z D - -}-f iT where 6 T d i s t h e d y n a m i c c o r r e c t i o n

t o t h e d i a l r e a d i n g T D . T h e m e l t i n g p o i n t i s t a k e n a s t h e i n t e r s e c t i o n o f t h e l e a d i n g

s t r a i g h t e d g e ( e x t r a p o l a t e d ) o f t h e m e l t in g c u r v e w i t h t h e i n s t r u m e n t a l b a s e li n e .T h e p r o c e d u r e d e s c r i b e d i n t h e p r e s e n t p a p e r g i ve s T ( = T ~ a t a m e l t in g

p o i n t ) = T D + 6 T s + 6 T o a n d t h e tw o m e t h o d s o f c a l i b r a ti o n s h o u l d t h e r e f o r e

b e e q u i v a l e n t i f 6 T , 1 = f i T s + 6 T o . T h i s r e l a t i o n s h i p w a s f o u n d t o h o l d t o b e t t e r

t h a n 0 . 3 K f o r th e m e l t i n g p o i n t s o f i n d i u m , t in , b i s m u t h a n d l e a d f o r h e a t i n g

r a t e s f r o m 0 .5 t o 3 2 K r a i n - 1 . T w o a s p e c t s o f th e d y n a m i c c a l i b r a t i o n s h o u l d

b e e m p h a s i s e d : i t i s e s s e n t i a l t o u s e s m a l l s a m p l e s a n d t o u s e t h e m e l t i n g p o i n t

d e s c r i b e d a b o v e . W i t h l a r g e s a m p l e s 6 T o m u s t b e re p l a c e d b y 6 T a n d t h e c a li -

b r a t i o n is a f u n c t i o n o f s a m p le m a s s ; p e a k t e m p e r a t u r e s f r o m a D S C c u r v e a r e

m u c h m o r e s e n s i t i v e t o h e a t i n g r a t e a n d m a s s t h a n i s c a l l e d f o r b y t h e r e l a t i o n

6T H = f i t s + ~ ST o .A l t h o u g h " s t a t i c " a n d " d y n a m i c " m e t h o d s a re e q u iv a le n t t he f o r m e r m u s t

b e u s e d i f a n a c c u r a t e c a l i b ra t i o n , c o v e r i n g t h e w h o l e i n s t r u m e n t a l t e m p e r a t u r e

r a n g e , i s r e q u i r e d . T h e r e a r e n o t e n o u g h m e t a l s w i t h s u i t a b l y s p a c e d m e l t i n g

p o i n t s . O t h e r m a t e r i a l s h a v e t o b e u s e d ; t h e i r p u r i t y is g e n e r a l l y o r d e r s o f

m a g n i t u d e l e ss t h a n t h a t o f m e ta l s. T h e e x t r a c t i o n o f T m f r o m a " d y n a m i c "

D S C c u r v e t h e n b e c o m e s a r a t h e r s u b j e c t i v e o p e r a t i o n . A t r u e m e l t i n g p o i n t

e x is t s o f c o u r s e , a l b e i t d e p r e s s e d , a n d t h i s c a n b e d e t e r m i n e d b y c o n v e n t i o n a l

m e t h o d s . T h e f ig u r e c a n t h e n b e u n a m b i g u o u s l y r e l a t e d to a " s t a t i c " v a l u e f o u n d

i n t h e D S C - - t h e r e a r e n o u n c e r t a i n t i e s d u e t o t h e r m a l l a g a t t h i s s t a g e . F i n a l l y ,

t h e i n s t r u m e n t a l c o r r e c t i o n 6 T o i s a p p l i e d . A s s e e n a b o v e t h is is n o r m a l l y

. L T h e r m a l A n a L 6 , 1 9 7 4



RICHARDSON, BURRINGTON: THERMAL GRADIENTS IN DSC 351

sufficient for the small (milligram) quantities of material required in fusion

measurements.

Larger samples are needed for accurate determination of the heat capacity

and thermal lag within the sample itself becomes important.

(b) H e a t c a p a c i t y

The ordinate displacement in a DSC is proportional to the heat capacity and

heating rate. The highest accuracy will therefore be obtained from a large mass

and/or heating rate. Increasing either of these increases the thermal lag in the

system. Neglect of this affects % by amounts which depend on the relative

magnitude of 6 % / 5 T for the sample and reference material. Calculated values

of cp are in error because they refer to an apparent temperature, Ta, which isgreater than the true sample temperature, T. The error is cp(Ta) - cp(T), assuming

that cp is a constant for the reference material. This is normally a good approx-

imation since the latter is deliberately chosen to be thermally inert - numerical

values are given below for synthetic sapphire. The error will be negligible if

6 c p / f T is small but errors of several percent may occur when there is appreciable

curvature as, for example, in the approach to a phase change. Here "lattice

loosening" or "premelt ing" may lead to considerable increases in cp over a

temperature range which, for a polymer, may cover several decades prior to

the actual melting point. Thus the percentage change in %, per K, for poly-

ethylene crystals (Tm = 401 K) is [71 .47(240), .48(360), ..93(390), .14(410),.14(500 K). Corresponding figures for synthetic sapphire, the most usual reference

material, are .47, . 18, .14, . 12, and .07 ~.

Conventional operation of the instrument assumes a sample temperature

T = TD + 6Td . It was shown above that 5Td = fiTs + 5To. (Thermal lag is

always a negative quantity, 6T ~ may be positive or negative.) For large samples

6 T o should be replaced by 6 T = 6 T o + 6Treat. The additional term is defined

in Fig. 3 for sapphire discs. Organic compounds with this favourable shape

give comparable values for 6 T ma say 1- 2 K at a 16 K min -~ heating rate.

This is because the generally higher heat capacities permit the use of thinner

discs and this compensates for the poorer thermal conductivity. Calculatedspecific heats of the polymer crystals described above will be low by 0.5 to 1

50 K below Tm. All are systematic errors and, although 1 ~o is within the antic-

ipated performance of the instrument, it was the low temperature data which

first drew our attention to the problems of thermal lag. Polymer crystals were

usd as control substances in a quite separate series of experiments in the range

330-360 K. The mean ep from 10 runs was only trivially different from the

value determined by adiabatic calorimetry [7] whereas the "dynamic" figure

at 89 T + T0 was 1.1 ~ lower (curvature in the %-Tcurve is small and would

not account for the difference). The internal inconsistency was completely removed

when effects of thermal lag were considered.

6 J . T h e r m a l A n a l . 6 , 1 9 7 4



352 RICHARDSON, BURR1NGTON: THERMAL GRADIENTS IN DSC

The examples discussed above indicate that thermal lag generally has a fairly

small effect on apparent values of cp when samples of suitable geometry are

used and the instrument has been calibrated at the relevant heating rate. The

qualification with regard to geometry refers to idealised conditions. It is rareto meet with samples having regular shape, let alone thin discs. Indeed, a major

advantage of the DSC is the ability to monitor the enthalpy change H(Tf )- H(Ti)

irrespective of sample size and shape. It is often possible to form polymers into

discs by the application of pressure. Unfortunately, the morphology may be

completely changed thereby and subsequent measurements will refer to an

atypical sample. The values given above as possible errors in ep for polymer

crystals must therefore be regarded as minimum values, they may as much as

double for unfavourable geometries. It is evident from Fig. 3 that doubling

the sample mass gives a smaller increase in thermal lag than doubling the

heating rate. As a general rule, therefore, specific heat determinations shouldbe carried out with large samples at low heating rates.

(c) T r a n s i t i o n t e m p e r a t u r e s

Reasons have been given above for preferring the "sta tic" method of temperature

calibration. When several heating rates are used the technique also involves

the least effort in calibration. The only requirements in addition to the static

correction 6T~ are values of 6T o at a few temperatures and heating rates. Few

are required since 6To is effectively a linear function of both T and R of theform 67"o = (a + b T )R . The true sample temperature is obtained from T = T• +

+ ~ T s + ( a + b T ) R and this is generally sufficient for small samples. However,

even here a note of caution should be sounded. The melting temperatures of

metals, although independent o f sample mass in the range up to a few milligrams

are not independent of geometry. For example, a 2.5 mg pellet of indium melted

at TD = 437.8 (leading edge) or 441.4 K (peak) for R = 16 K min -~. These

were reproducible values (the indium showed no tendency to wet the aluminium

sample pan) which dropped to 436.4 and 438.0 K respectively when the pellet

was flattened by an aluminium disc. Only the latter pair of temperatures were

independent of sample mass.The above effects can be recognised and treated in metals but the problem

is more intractable when considering the melting point of a polymer. These

must oftenbe scannedthrough the melting region at ahigh rate (to avoid irreversible

annealing effects). Since there is no well-defined "leading edge" (polymers melt

over a wide range of temperature) it is normal to take the "peak" temperature

as the melting point. This is not valid even for pure metals - especially when

in an unfavourable geometry. Polymeric samples are usually of irregular shape

and the small differences in melting point which are frequently described in the

literature and attributed to subtle structural differences should be treated with

some reserve.

J . T h e r m a l A n a l . 6 , 1 9 7 4



RICHARDSON, BURRINGTON: THERMAL GRADIENTS IN DSC 353

When larger samples are used 8To is replaced by 5T which is computed for

each run and thus automatically compensates for any specimen peculiarities.

It was found, for example, that the uncorrected glass temperature, Tg, of a

polystyrene disc dropped from 380 to 379 K (R = 8 K rain -1) after leaving

overnight at 450 K. The polymer had the same thermal treatment prior to each

measurement and no change in molecular weight was detectable by gel permeation

chromatography after holding at 450 K. The original disc was cut from a moulded

sheet to fill the aluminium specimen pan as well as possible but there was still

the 1~ change in Tg as the sample effectively flowed and made better contact

with the pan. (The weight-average molecular weight was 310,000 and the long

treatment time at 450 K was needed because of the high viscosity.) This

"instrumental" effect was entirely removed when the correction fiT was applied

and the two curves could be superimposed with no temperature shift. This

example illustrates the care which must be taken in analytical treatment of thermallag using the flow of heat through thin slabs [8]. No allowance can be made

for differences in thermal contact between specimen and calorimeter.

CGItclusf~l lS

An experimental analysis has been made of the factors which affect the sample

temperature in a differential scanning calorimeter. The importance of both

instrumental and material parameters has been shown. The former include the

non-linear response of the platinum resistance thermometer (which accountsfor the parabolic shape of the normal dial temperature correction curve) and

thermal lag within the instrument. The lat ter encompass sample size, geometry

and packing in addition to the physical constants characteristic of the material

itself; all combine to give a response for which an a priori calculation is impos-

sible.

A procedure has been described which considers all the above variables and

gives accurate sample temperatures. Initially a calibration gives the true tem-

perature corresponding to any static dial temperature. A second correction is

applied which gives thermal lag within the instrument as a function of heating

rate and temperature. The final correction is obtained from individual thermo-grains and so automatically compensates for the several material variables.

Examples have been given which show how the instrumental accuracy is

improved by the above treatment.

Re f e r e n c e s

1 . W . J . SMOTHERS an d Y. CHIANG,H a n d b o o k o f D i ff e re n ti a l T h e r m a l A n a l y si s , C h e m i c a l

P u b l i s h i n g C o ., N e w Y o r k , 1 9 66 .

2 . R . S . P OR TE R a n d J . F . J o h N s o n , E d s , A n a l y t i c a l C a l o r i m e t r y , P l e n u m P r e ss , N e w Y o r k ,

1968 .

6* J. Thermal Anal. 6, 1974



3 5 4 R I C H A R D S O N , B U R R I N G T O N : T H E R M A L G R A D I E N T S I N D S C

3 . M . J . R I C H A R D S O N , J . P o l y m e r S c i. , C 3 8 ( 1 9 7 2 ) 2 5 1 .

4. ]~. WUNDERLICH,J . P h y s . C h e m . , 6 9 ( 1 9 6 5 ) 2 0 7 8 .

5 . G . T . F U R U K A W A , T . ] 3. D O U G LA S , R . E . M c K O S K EY a n d D . C . G I N r ~I N G S, J . R e s . N a t l .

B u r . S t d . , 5 7 ( 1 9 6 5 ) 6 7 .

6 . E . P E L LA a n d M . N E B U L O N I , J . T h e r m a l A n a l . , 3 ( 19 7 1 ) 2 2 9 (d i s cu s s m a n y o f t h e p r o b l e m s

o f t h e d y n a m i c t e m p e r a t u r e c a l ib r a t io n ) .

7 . C . M . L . A T K IN S O N a n d M . J . R I C H A R D S O N , T r a n s . F a r a d a y S o c . , 6 5 ( 1 9 6 9 ) 17 7 4 .

8 . S . STRELLA an d P . F . E R H A R D T , J . A p p l . P o l y m e r S c i . , 1 3 ( 1 9 6 9 ) 1 3 7 3 .

R ~SU Mr~ - - O n d 6 c r i t u n m o d e d ' 6 t a l o n n a g e d e l a t e m p 6 r a t u r e p o u r l e s e s s a is s t a t i q u e s e t

d y n a m i q u e s . L ' d t a l o n n a g e s t a t i q u e p e r m e t d e c o r r i g e r l a v a l e u r d e l a t e m p e r a t u r e l u e s u r

l e c a d r a n d e l ' i n s t r u m e n t . L ' 6 t a l o n n a g e d y n a m i q u e v a r i e a v e c l es 6 c h a n t i l lo n s 6 tu d i d s ca r i l

u t il is e d e s a c c e s s o i r es i n s t r u m e n t a u x e t d e s m a t d r i a u x a d a p t 6 a u x 6 t u d e s p a r ti e u li g r es . I 1 e s t

f o u r n i p a r d e s e n r e g i s t r e m e n t s i n d i v i d u e l s q u i 6 1 im i n e n t l es i n c e r t it u d e s d u e s a u x v a r i a t i o n s

d e m a s s e , d e g 6 o m ~ t r i e e t d e v i t e ss e d e c h a u f f a g e . L e s c a r a c t 6 r i s t i q u e s i n s t r u m e n t a l e s s o n t

a i n s i a m 6 1 i o r d e s e t l e s c h a l e u r s s p 6 c i f i q u e s p e u v e n t ~ t r e o b t e n u e s a v e c u n e e x a c t i t u d e d e+ 1 p . c .

ZU SA MM EN FA SS UN G - - L i n e k o m b i n i e r t e s t a t i s c h e u n d d y n a m i s c h e T e m p e r a t u r e i c h u n g w i r d

b e s c h r i e b e n . D i e s t a t is c h e E i c h u n g k o r r ig i e r t d i e T e r n p e r a t u r a b l e s t m g a n d e r l n s t r u m e n t e n -

s c h ei b e. D i e d y n a m i s c h e E i c h u n g b e s t e h t a u s i n s t ru m e n t e l l en u n d S t o f f - K o m p o n e n t e n u n d

i s t d e s h a l b i n v e r s c h i e d e n e n F ~ il le n u n t e r s c h i e d l i ch . S i e w i r d a u s i n d i v i d u e l l e n D S C K u r v e n

e r h a l t e n t r o d b e s e i ti g t s o m i t U n s i c h e r h e i t e n d e r T e m p e r a t u r d e s M u s t e r s , w e lc h e d u r c h

, ~ n d e r u n g e n d e r M a s s e, d e r G e o m e t r i e u n d d e r A u f h e i z u n g s g e s c h w i n d ig k e i t b e d in g t s in d .

D i e L e i s t u n g d e s I n s t r u m e n t e s k a n n e r h S h t w e r d e n u n d d ie B e s t i m m u n g s p e z if i sc h e r W ~ i r m e n

i s t m i t e i n e r G e n a u i g k e i t y o n - { - 1 % m S g l i c h .

P e 3 t o M e - - O n a c a H a K O M 6 1 4 n H p O B a n H a f l C T a T n ~ e cI ~ a ~ H ~ rt 14 a M 1 4q e cK a n K a ~ n 6 p O B K a T e M n e p a -

TypI, I. CT aT~qec ra r [ Kan146poBKa BHOCHT r ronpaB ~y B ~Tea ~e TeM nepaTypbi no m ra n e ~p n6 op a .

~H H a~ ra ~e cr as KaYin6pOBKa BK~IoqaeT 14p146opHbI~ H M aTepHaflbnLI~ KOMIIOHeHTt,I !,I HO3TOMyI , 3 M e H ~ Ie T C ~I B 3 aB H e kI M O C T I/ I O T o ~ p a 3 I / a . O H a H p O B e ~ e H a H a O C H O B e I' IH ~ I, B I, , ~ y a ~ l b H b I X T e p M 1 4 -

q e c K H x K p 14 BIa IX I4 , c s Ie ,/ IO B a T eY l bI tO , y c T p a 1 4 g e T r o n e 6 a n r ~ f l B T e M n e p a T y p e 0 6 p a 3 t [ a , o6ycnoBztei~-

H~ie 14BMeI-IeHI ,~ IM I I M a C C Ia I, F e O M e T p H H Y I C K O p O C T I'I H a r p e B a . l ( I l i c T p y M e I - I T a r l b H O e B I a lI IO Y I H eH I ,e

y ~ y q m e n o n y , ~ e Y l b H y l o T e Y I~ IO e M K O C TI ~ M O T K H O o r t p e ~ e y l ~ T t , c T 0 q H 0 C T I a I O ~ O ~___ 1 ~ oo ,

J . T h e r m a l A n a l . 6 , 1 9 7 4

thermal gradient

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