MODEL OF OSCILLATING INTERFACE AT FIRST ORDER PHASE TRANSITION AND AT GLASS TRANSITION - NONVANISHING AMPLITUDE APPROXIMATION

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MODEL OF OSCILLATING INTERFACE AT FIRST ORDER PHASE TRANSITION AND AT GLASS TRANSITION - NONVANISHING AMPLITUDE

APPROXIMATION

P. Korpiun, R. Tilgner, D. Schmidt

To cite this version:

P. Korpiun, R. Tilgner, D. Schmidt. MODEL OF OSCILLATING INTERFACE AT FIRST ORDER PHASE TRANSITION AND AT GLASS TRANSITION - NONVANISHING AMPLI- TUDE APPROXIMATION. Journal de Physique Colloques, 1983, 44 (C6), pp.C6-43-C6-53.

�10.1051/jphyscol:1983607�. �jpa-00223166�

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Colloque

supplement au nO1O, Tome 44, octobre 1983 page

43

MODEL OF OSCILLATING INTERFACE AT F I R S T ORDER PHASE TRANSITION AND AT

GLASS TRANSITION - NONVANISHING AMPLITUDE APPROXIMATION

P . Korpiun, R. ~ i l ~ n e r * and D . Schmidt

Technische Universitat Miinchen, Physik-Department, 0-8046 Garching,

F.R.G.

*Siemens AG,

ZFA,

F.R.

Resume - Un modsle t h e o r i q u e pour l a v a r i a t i o n de l ' a m p l i t u d e e t de l a phase du s i g n a l photoacoustique au cours d'une t r a n s i t i o n de phase du l e r ordre, e s t gen6ralisS pour t o u t e v a l e u r des 6paisseurs thermiques de 1 1 6 c h a n t i l l o n . L ' a p p l i c a t i o n aux t r a n s i t i o n s du second o r d r e e s t possible. Des r e s u l t a t s experimentaux au p o i n t de f u s i o n de 1 'indium e t pour l a t r a n s i t i o n v i t r e u s e du polystyrene confirment l e modele.

A b s t r a c t - A t h e o r e t i c a l model f o r PA amplitude and phase angle a t f i r s t order phase t r a n s i t i o n i s generalized t o a r b i t r a r y thermal thickness o f the sample. I t can be a p p l i e d t o second o r d e r phase t r a n s i t i o n s , too. Results o f measurements a t t h e m e l t i n g p o i n t o f indium and i n the glass t r a n s i t i o n r e g i o n o f p o l y s t y r e n e c o n f i r m t h e model.

I - INTRODUCTION

Amplitude o f pressure 6p and phase a n g l e c p measured i n a gas-microphone c e l l de- pend on

1. The o p t i c a l absorption c o e f f i c i e n t o f t h e sample and t h e conversion process f o r heat;

2. The thermal p r o p e r t i e s o f t h e sample determining t h e t r a n s p o r t o f the heat t o t h e surrounding gas: d e n s i t y

thermal c o n d u c t i v i t y A, s p e c i f i c heat c a p a c i t y c and l a t e n t heat L.

~ h e r l f o r e , th e photoacoustic e f f e c t (PAE) should be s e n s i t i v e t o phase t r a n s i t i o n s . F i r s t measurements were performed by P e l z l and coworkers / I / , o t h e r f o l l o w e d 12-81.

I 1 - THE PAE AT FIRST ORDER PHASE TRANSITIONS - MODEL OF OSCILLATING INTERFACE The model 1 7 1 i s sketched i n f i g . 1 f o r a s o l i d l i q u i d t r a n s i t i o n . The sample i s i l l u m i n a t e d by modulated l i g h t . The power p e r u n i t surface area converted t o heat i n an opaque absorber i s assumed t o be

I t causes a steady s t a t e l i n e a r temperature d i s t r i b u t i o n i n t h e c e l l expressed by a temperature g r a d i e n t e2k ( f i g . I ). I t i s superposed by a temperature o s c i l l a - t i o n . I n t h a t r e g i o n o f t h e sample where t h e t r a n s i t i o n temperature Tm i s reached t h e boundary between s o l i d and l i q u i d phase o s c i l l a t e s w i t h amplitude ~ ( x , t ) around a mean value no 171.

The p o s i t i o n o f t h e i n t e r f a c e between two phases i s rl(To,t)

no(To)

S(To,t)

c(To,t)

&(TO) e jut .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983607

JOURNAL DE PHYSIQUE

oscillating interface

l t l = y b t t t l +Cea''

-- - -- 1 =light

--

&oPPed

light freq.

Fig. 1 - Scheme of t h e sample c e l l assembly and temperature d i t r i b u t i o n ; model of o s c i l l a t i n g i n t e r f a c e .

Increasing i n an experiment t h e ambient temperature To, t h e o s c i l l a t i n g i n t e r f a c e and thus

i s s h i f t e d more t o t h e i n s i d e of t h e sample, and v i c e versa. Vanishing ampl i tude approximation means t h a t t h e ampl i t u d e

of t h e i n t e r f a c e o s c i l l a t i o n i s small compared t o t h e length of t h e sample o r

nonvanishi ng ampl i tude approximation means

see a l s o s e c t . 11.2

Experiments t o i n v e s t i g a t e phase t r a n s i t i o n s a r e performed i n t h a t way, t h a t am- p l i t u d e and phase angle of t h e PA s i g n a l a r e measured a s f u n c t i o n of ambient tem- p e r a t u r e using a microphone. This PA signal i s proportional t o t h e p r e s s u r e v a r i - a t i o n 6 P of t h e gas surrounding t h e sample. Using t h e modeTof Rosencwaig and Gersho 6P again is proportional t o t h e average temperature v a r i a t i o n <AT> i n t h e gas /9,10/. <6T> i s t h e s p a t i a l average over t h e temperature d i s t r i b u t i o n i n t h e gas, t h a t is obtained a s a s o l u t i o n of the equation of h e a t conduction. The po- s i t i o n

o f t h e phase boundary and t h e l a t e n t heat L e n t e r some boundary condition f o r temperature and heat flux.

11.1 - Temperature d i s t r i b u t i o n

The problem i s described by one dimensional d i f f e r e n t i a l equation of conduction

of h e a t f o r region k

where

'$

k ( x 5 t ) = T k ( x 3 t ) - To

w i t h theambient temperature To and k = 0: backing, k

1: sample ( s o l i d ) , k = 2:

sample ( l i q u i d ) , k

3: gas.

The most general form f o r t h e temperature d i s t r i b u t i o n according t o Rosencwaig and Gersho /9/ i s

w i t h

o k

( l + j ) a k , - 1

at

(2Ak/cpk pk w ) ' / ~ : thermal d i f f u s i o n l e n g t h o f medium k.

The r e a l constants e i k and complex constants Uk, Vk a r e determined from t h e boun- dary c o n d i t i o n s : c o n t i n u i t y o f temperature and h e a t f l u x . The l a t e n t heat L comes i n by the boundary c o n d i t i o n f o r t h e heat f l u x a t the o s c i l l a t i n g boundary between t h e two thermodynamic phases /7/:

d

5 / d t = v e l o c i t y o f t h e o s c i l l a t i n g i n t e r f a c e . We f i n d s o l u t i o n s o f eq.

16) f o r the temperature d i s t r i b u t i o n w i t h boundary c o n d i t i o n (10) f o r samples w i t h a r b i t r a r y thermal l e n g t h a l l 1 i f t h e amplitude 15) o f t h e i n t e r f a c e i s small com- pared t o t h e thermal d i f f u s i o n length, ] s l a l

1 . (11)

11.2 - Pressure V a r i a t i o n , Complex Temperature and C h a r a c t e r i s t i c Q u a n t i t i e s As mentioned above, t h e pressure v a r i a t i o n i n t h e gas /9,10/ i s connected by

t o t h e complex temperature amplitude i n t h e p e r i o d i c p a r t o f t h e s o l u t i o n (8)

V3 = I ^{V 3 /} exp[ j(p3-n/4)1 (13)

a t t h e sample t o gas boundary.

The r e a l valued q u a n t i t i e s amplitude [ V 3 [ and t h e phase angle ~3 a r e f u n c t i o n s o f a l l thermal p r o p e r t i e s

, Xk, c t h e l a t e n t h e a t o f t h e sample L, the geometry o f t h e sample c e l l assembly, e.g!$engths l k , t h e heat Qo, t h e ambient temperature To, and t h e d i f f e r e n c e 6 between t h e temperature i n t h e sample and t h e m e l t i n g temperature

The e x p l i c i t expression f o r jV3) i s given elsewhere /7,8/ and i n a paper t h a t w i l l be published.

The amplitude of t h e i n t e r f a c e o s c i l l a t i o n i n t h e v i c i n i t y o f x = 0 i s

E =

-

1 V 3 (qO=O)/

2 e 21 (15)

C6-46

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w i t h t h e temperature g r a d i e n t

i n t h e low temperature s t a t e (k

1) o f t h e sample. The c o n d i t i o n ( 4 ) f o r vanishing i n t e r f a c e approximation becomes now

This c o n d i t i o n can be f u l l f i l l e d e x p e r i m e n t a l l y f o r t h e r m a l l y t h i c k samples a l l 1 ?, 1. Nonvanishing amplitude approximation ( 5 ) means now

2 1 s / / 1 1 1 -. Qo/alll 2L(A2a2+A3a3)/rcp- ( l + s ) (18)

w i t h

r = a /a _{2 1'} s

( I - A a 0 0 1 1

a )/(l+Aoao/hlal) .

Care should be taken t o t h i s i n e q u a l i t i y w i t h respect t o the l i g h t i n t e n s i t y per- forming measurements on t h e r m a l l y very t h i n samples, a l l l << 1.

Some q u a n t i t i e s should be i n t r o d u c e d t h a t a r e used i n t h e f u r t h e r formal des- c r i p t i o n o f amplitude and phase angle o f t h e temperature o s c i l l a t i o n .

The d i f f e r e n c e between ambient temperature To and temperature a t t h e sample surface x = O i s

i f t h e average p o s i t i o n o f t h e boundary qo = 0. Here

Rk

(20)

i s t h e thermal r e s i s t a n c e o f medium k. The r e s u l t s w i l l be expressed as f u n c t i o n s o f a reduced temperature d e f i n e d as

6 = 0 means no

0 .

11.3 - V a r i a t i o n o f Ambient Temperature

Experiments on samples t h a t undergo a phase t r a n s i t i o n a r e performed by v a r y i n g t h e ambient temperature To. The expression (13) f o r V3 o n l y describes the tempe- r a t u r e amplitude i n t h a t temperature r e g i o n where an ~ n t e r f a c e and t h e r e f o r e two d i f f e r e n t phases o f t h e sample e x i s t . On t h e scale o f t h e reduced temperature (21) i t i s t h e range 62

where a t t h e t o p s i d e o f t h e sample, x = 0, the tem- p e r a t u r e never i s lower,or a t t h e bottom o f t h e sample, x

-11, never i s h i g h e r than t h e t r a n s i t i o n temperature Tm, r e s p e c t i v e l y . For temperatures s u f f i c i e n t l y f a r below and above t h e t r a n s i t i o n temperature T, t h e r e e x i s t s no i n t e r f a c e . The whole volume o f t h e sample i s i n s t a t e k = 1 o r 2, r e s p e c t i v e l y . The temperature d i s t r i b u t i o n i s a s o l u t i o n of eq. ( 6 ) w i t h o u t t h e presence of a phase t r a n s i t i o n :

w i t h k

1,2 r e s p e c t i v e l y . vB1) describesamplitude and phase angle i n t h e

t e m p e r a t u r e range 8 5 81 where t h e temper t r e a t x

0 n e v e r i s h i g h e r t h a n . T, The h i g h t e m p e r a t u r e phase d e s c r i b e d by V12Y e x i s t s f o r 8 2

i f t h e t e m p e r a t u r e a t x = - I 1 i s n e v e r l o w e r t h a n T,. One o b t a i n s

There a r e two more c h a r a c t e r i s t i c temperatures e s s e n t i a l f o r t h e m o d e l . A t 8 = 0 t h e average p o s i t i o n o f t h e i n t e r f a c e i s a t t h e t o p s i d e o f t h e sample, no

0. The i n t e r f a c e i s a t t h e bottom o f t h e sample,

= -11, a t t h e reduced temperature

V a r y i n g c o n t i n u o u s l y t h e ambient t e m p e r a t u r e t h e r e a r e t h e r e g i o n s 8 2 8 2 82 and 83 5 ₈ 5 85 where e x i s t s an i n t e r f a c e o n l y f o r a f r a c t i o n a l p a r t wtm/n of a h e a t i n g c y c l e . F o r a f i r s t approach we suppose t h e e p e r a t u r e v a r i a t i o n i n t i m e t o be composed o f t h e s o l u t i o n s v3, eq. (13), and vJ~T, eq. ( 2 2 ) as sketched i n f i g . 2 . bleasurements a r e performed u s i n g t h e l o c k - i n technique. A l o c k - i n d e t e c t o r measures t h e f i r s t harmonic of a f u n c t i o n . The complex F o u r i e r

F i g . 2 - Temperature o s c i l l a t i o n a t t o p s i d e o f t h e sample i n i n t e r - m e d i a t e r e g i o n

el 5 8 5 0.

I

-wt, 0 wt,

w t -

c o e f f i c i e n t o f t h e f i r s t harmonic o f t h e f u n c t i o n shown i n f i g u r e 2 has t h e f o r m

t , t

Both, amp1 i t u d e I V 3 1 and phase a n g l e y 3 a be e v a l u a t e d w i t h t h e e x p l i c i t ex- p r e s s i o n s / 7 / f o r V3, eq. (13) and V 3 f k y , eq. (22) i n t h e t e m p e r a t u r e range 81 5 8 5 82 f o r k = 1 and w i t h i n 83 5 8 5 85 f o r k

2. The c h a r a c t e r i s t i c tempe- r a t u r e s 82 and 83 a r e

O 2 =

V3(vo=O) (Ro+R1+R3X2/A1 )/QoR3(Ro+R1 and

83

e 4 - e 2 exp(-a211) .

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The e x p r e s s i o n (26) f o r V$ i s a f i r s t approach t o d e s c r i b e t h e a m p l i t u d e and phase a n g l e o f t h e p e r i o d i c t e m p e r a t u r e and p r e s s u r e v a r i a t i o n i n t h e gas o f a PA c e l l f o r samples o f a r b i t r a r y t h e r m a l t h i c k n e s s a l l 1 as l o n g as t h e o s c i l l a t i o n o f t h e i n t e r f a c e does n o t exceed t h e sample t h i c k n e w , eq. ( 5 ) .

The r e s u l t s o b t a i n e d agree w i t h e x p e r i m e n t a l data. T h i s i s shown i n s e c . I I 1 f o r t h e m e l t i n g o f i n d i u m and i n sec.IV f o r a g l a s s t r a n s i t i o n o f a t a c t i c p o l y s t y r e n e . I 1 1 - MELTING OF INDIUM

The p h o t o a c o u s t i c s i g n a l has been measured on i n d i u m samples i n t h e v i c i n i t y o f t h e m e l t i n g t e m p e r a t u r e T,

156.6 OC f o r samples w i t h t h e r m a l l e n g t h s a l l 1

1.7, 0.3 and 0.02 on a b a c k i n g o f copper. The measurements were performed w i t h a gas microphone c e l l - T h e samples were i l l u m i n a t e d by w h i t e l i g h t o f a 250 Watt halogen lamp. I t s i n t e n s i t y was modulated by a mechanical chopper (PAR 192). A condensor microphone ( B r u e l & K j a e r , Type 4166) was used. A m p l i t u d e and phase were d e t e c t e d u s i n g a l o c k - i n ampl i f i e r ( I thaco, Dynatrac 393). The e x p e r i m e n t a l r e s u l t s a r e shown i n t h e f i g s . 3, 5 and 7. A m ~ l i t u d e 3 3 ~ ~ and phase a n g l e y3ex a r e p l o t t e d v e r - sus temperature. The c o r r e s p o n d i n g t h e o r e t i c a l quan

i e s a r e shown i n t h e f i g s . 1, 6 and 8. The normal i r e d ampl i t u d e d3

~ 3 ~ 1 , l V 3 I i j / and phase a n g l e y 3 t a r e p l o t t e d versus t h e reduced t e m p e r a t u r e

eq. (21). The t h e r m a l d a t a o f i n d i u m used f o r t h e c a l c u l a t i o n s a r e l i s t e d i n t a b l e 1.

From t h e f i g s . 3 t o 8 i t becomes o b v i o u s t h a t t h e c h a r a c t e r i s t i c b e h a v i o u r o f am- p l i t ~ d e and phase a n g l e w i t h t e m p e r a t u r e i s d e s c r i b e d s a t i s f a c t i v e l y by t h e model o f o s c i l l a t i n g i n t e r f a c e . Comparing t h e e x p e r i m e n t a l r e s u l t s w i t h c a l c u l a t e d d a t a one has t o t a k e i n t o account a t e m p e r a t u r e g r a d i e n t i n t h e sample p e r p e n d i c u l a r t o t h e x - d i r e c t i o n . It i s v e r y d i f f i c u l t t o r e a l i z e i n a p h o t o a c o u s t i c c e l l i s o - thermes p a r a l l e l t o t h e y-z-plane. The t e m p e r a t u r e g r a d i e n t w i l l smear o u t t h e s i g n a l on t h e t e m p e r a t u r e s c a l e . I t i s o b v i o u s t h a t w i t h d e c r e a s i n g thermal l e n g t h

of t h e sample t h e t e m p e r a t u r e o s c i l l a t i o n

~3~ s h o u l d become more s e n s i t i v e t o t h e l i g h t i n t e n s i t y expressed by

o r 6

T a b l e 1 - Thermal P r o p e r t i e s o f I n eq. (19), r e s p e c t i v e l y . A t a t h e r m a l l y v e r y t h i n s a m ~ l e a v a r i a t i o n o f 8, even

429.76 K

156.6

may- change es;entially t h e c h a r a d t g r i s t i c

b e h a v i o u r o f t h e phase a n g l e w i t h tem- L

28.62 J g - I p e r a t u r e , P i g . 8b.

hl

0.729 Watt cm-lK-I h2

0.360 Watt c m - l ~ - I C p l

0.2657 J g - l ~ - l c = 0.2669 J g-'K-'

~2 -3

p1

7.31 g cm pZ

7.13 g cm -3

F i g s . 3,5,7 - a ) PA a m p l i t u d e 8 3 e X and b ) phase a n g l e v 3 e X measured on i n d i u m samples versus dmbient t e m p e r a t u r e T. 11: sample t h i c k n e s s ,

mo- d u l a t i o n frequency, a ? 1 1 : thermal t h i c k n e s s . Backing: copper.

F i g s . 4,6,8 - a ) Normalized a m p l i t u d e 8 3 and b ) phase a n g l e

c a l c u l a t e d f o r

i n d i u m samples w i t h thermal t h i c k n e s s a l l 1 on copper v e r s u s reduced

t e m p e r a t u r e

6 ,

5K.

a F i g . 3

1

F i g . 4

F i g . 5

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F i g . 6

2

-

alll ~ 0 . 0 2

-

.

a

-

alll

= 0 02

0

-

I l l

150 155

T I ° C

-

TI0C-

a F i g . 7 b )

a _{Fig. 8 b}

I V - PAE AT THE GLASS TRANSITION OF ATACTIC POLYSTYRENE

The PAE has been measured on an amorphous polymer i n t h e g l a s s t r a n s i t i o n r e g i o n . The sample was a t a c t i c p o l y s t y r e n e (168 N, BASF). The sample was t h e r m a l l y w e l l annealed. I t s g l a s s t r a n s i t i o n r e g i o n i s between 100 and 110 OC. The sample geo- m e t r y was 12 x 5 x 3 mm3, i t s t h e r m a l t h i c k n e s s all1

10 > l , ~

29 Hz. The PA- c e l 1 , i l l u m i n a t i o n and d e t e c t i o n w e r e t h e same as f o r t h e measurements on indium.

Ambient t e m p e r a t u r e i n c r e a s e d a t a r a t e o f 5 K m i n - I . PA-amplitude and phase a n g l e measured as a f u n c t i o n o f t h e mean t e m p e r a t u r e T o f t h e sample a r e p l o t t e d i n f i g . 9.

F i g . 9 - a) PA a m p l i t u d e and b) phase a n g l e measured on p o l y s t y r e n e i n g l a s s t r a n s i t i o n r e g i o n 100 - 110

S o l i d l i n e : f i r s t measurement;

dashed l i n e : second measurement. T

ambient temperature.

A r r i v i n g a t t h e t e m p e r a t u r e o f

120

t h e sample was c o o l e d a t a r a t e o f 8 K min-I down t o 60 OC. Measuring a g a i n t h e PA-signal a t i n c r e a s i n g t e m p e r a t u r e a l o n g t h e g l a s s t r a n s i t i o n r e g i o n no e f f e c t c o u l d be observed.

I V . l - I n t e r p r e t a t i o n o f Experimental R e s u l t s

Experiments show t h a t a m p l i t u d e and phase s h i f t A y o f t h e PA-signal have a minimum i n t h e g l a s s t r a n s i t i o n r e g i o n . The e x p e r i m e n t a l r e s u l t s can be i n t e r p r e t e d w i t h t h e model o f o s c i l l a t i n g i n t e r f a c e , f i g . 10, i f

I. i n eq. (101 pL d g / d t

0,

C o n d i t i o n 2 h o l d s g e n e r a l l y f o r amorphous polymers /I I/.

L d g / d t = 0 means 1.1 L

0, d < / d t # 0: Second o r d e r phase t r a n s i t i o n o r any k i n e t i c t r a n s i t i o n :

a =

A/cp p changes.

1.2 L # 0, d c / d t

0: Phase boundary does n o t o s c i l l a t e ; phase t r a n s i t i o n i s i r r e - v e r s i b l e w i t h i n t h e t i m e 27r/w.

1.3 L

0, d < / d t

0: I r r e v e r s i b l e second o r d e r o r k i n e t i c t r a n s i t i o n .

The e x p e r i m e n t a l r e s u l t t h a t no e f f e c t i s observed i f t h e measurement i s r e p e a t e d

(sample n o t annealed) i s c o n s i s t e n t w i t h measurements o f t h e s p e c i f i c h e a t c P'

C6-52

JOURNAL DE

01 0.1 I 0.2 I

-a 1 ^b

a 1

^-

-

Fig. 1.0 - a ) Normalized amplitude and b) phase angle versus reduced temperature f o r various values of

( X C ~ P ) ~ i q u i d / ( X c p ~ ) g l a s s .

IV.2 -Temperature Range of Kinetic Transition

The t r a n s i t i o n can be observed with PAE only i f i t occures in the sample within a region <<al-1. Since the temperature gradient in t h e sample can be determined,

%

5 - 1 0 - ~ K y m - I , the temperature ( t r a n s i t i o n ) region AT where

o r

c p and X respectively vary e s s e n t i a l l y can be estimated t o be AT < 10-2 K.

V - CONCLUSION

The PAE i s s e n s i t i v e not only t o f i r s t order phase t r a n s i t i o n s b u t a l s o t o higher order and kinetic t r a n s i t i o n s . Those t r a n s i t i o n s can be observed,if the thermal properties vary considerably in the temperature gradient along a distance t h a t i s small compared t o the wavelength of the temperature wave.

From the q u a n t i t i e s t h a t a r e measured i t i s especially the behaviour of the phase angle with temperature t h a t depends in a c h a r a c t e r i s t i c manner on the thermal thickness of the sample, the s p a t i a l variation of t h e thermal pro- perties and a moving o r not moving interface. Though the description of the temperature signal in the intermediate temperature region i s r e l a t i v e l y rough,i t describes f a i r 1 y we1 1 the mean features of amp1 itude and phase angle along the t r a n s i t i o n region.

References

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