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Submitted on 1 Jan 1981
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EFFECT OF SURFACE ROUGHNESS ON THE
KAPITZA RESISTANCE
T. Shen, D. Castiel, A. Maradudin
To cite this version:
JOURNAL DE PHYSIQUE
C o l l o q u e C6, s u p p l d m e n t a u n o 12,
Tome
42, ddcembre 1981 page C6-819EFFECT OF SURFACE ROUGHNESS ON THE K A P I T Z A RESISTANCE T . J . Shen, D. C a s t i e l and A.A. Maradudin
Department o f P h y s i c s , U n i v e r s i t y o f C a l i f o r n i a , I r v i n e , C a l i f o r n i a 92717,
U.S.A.
A b s t r a c t . - The K a p i t z a r e s i s t a n c e due t o a c o u s t i c phonons c r o s s i n g a rough p l a n a r i n t e r f a c e from l i q u i d He i n t o a n i s o t r o p i c s o l i d i s i n v e s t i g a t e d , and i t s c h a r a c t e r i s t i c f u n c t i o n , v i z . t h e p e n e t r a t i o n c o e f f i c i e n t i s c a l c u l a t e d a s a f u n c t i o n o f t h e f r e q u e n c y o f t h e i n c i d e n t phonons and t h e i r a n g l e o f i n c i d e n c e .
The p r o p a g a t i o n o f a c o u s t i c phonons a c r o s s t h e i n t e r f a c e between l i q u i d He11 and a s o l i d h a s b e e n proposed a s t h e mechanism g i v i n g r i s e t o t h e t h e r m a l b o u n d a r y ( K a p i t z a ) r e s i s t a n c e a t t h i s i n t e r f a c e 1 . However, t h i s p r o c e s s p r e d i c t s a K a p i t z a r e s i s t a n c e 10-100 t i m e s h i g h e r t h a n t h e e x p e r i m e n t a l r e s u l t s . ' I h i s i n d i c a t e s t h e e x i s t e n c e o f a d d i t i o n a l h e a t c o n d u c t i o n mechanisms. Among t h e p o s s i b l e c a n d i d a t e s a r e t h e d i f f u s e l y s c a t t e r e d phonons, which a r e o b s e r v e d i n phonon r e f l e c t i o n 3 and t r a n s m i s s i o n 4 e x p e r i m e n t s . To a c c o u n t f o r t h e d i f f u s e phonons, t h e t h e o r y o f a rough i n t e r f a c e h a s been i n t r o d u c e d i n some s i m p l i f i e d
version^.^,^
Here we r e - examine t h e e f f e c t s o f r o u g h n e s s . We c o n s i d e r a system i n which t h er e g i o n x 3 > < ( x 7 ) i s f i l l e d w i t h l i q u i d H e I I , w h i l e t h e r e g i o n x3<r,(z ) i s f i l l e d by tl a n i s o t r o p i c e l a s t i c medium. We t r e a t b o t h media a s n o n - d i s s i p a t i v e . The s u r f a c e p r o f i l e f u n c t i o n <(: ) i s t a k e n t o b e a s t a t i o n a r y s t o c h a s t i c I p r o c e s s w i t h t h e f o l l o w i n g s t a t i s t i c a l p r o p e r t i e s . (1) = 0; ( 2 ) 2
<r;(z,)<(x*; )> = 6 exp(-
lZn-Z;
1 2 / a 2 ) , where 6 and a a r e t h e r o o t mean s q u a r e d e p a r t u r e o f t h e s u r f a c e from f l a t n e s s and t h e t r a n s v e r s e c o r r e l a t i o n l e n g t h , r e s p e c t i v e l y .2 The h e a t f l u x Q a c r o s s t h e i n t e r f a c e i s g i v e n by
where n i s P l a n c k ' s f u n c t i o n , co i s t h e sound v e l o c i t y i n l i q u i d He, P ( k , 8 ) i s t h e p e n e t r a t i o n c o e f f i c i e n t o f a phonon w i t h momentm%$(k= I < [ ) i n c i d e n t from t h e l i q u i d s i d e a t a n a n g l e 0 i n t o t h e s o l i d and i s g i v e n by
< S t r a m
3
>
p ( k , B ) =
sf
ncC6-820 JOURNAL DE PHYSIQUE
i nc
where S and
<sirans>
a r e t h e normal components of t h e i n c i d e n t and t h e s t a t i s - 3 t i c a l l y a v e r a g e d t r a n s m i t t e d e n e r g y f l u x e s . These e n e r g y f l u x e s depend o n t h e v e l o c i t y p o t e n t i a l $ ( z , t ) i n t h e l i q u i d and t h e d i s p l a c e m e n t f i e l d : ( z , t ) i n t h e s o l i d which a r e , f o r x3>cmax, and f o r x<
5 3 min'where we have d e n o t e d A.(;
lk+
) ( j = 1 , 2 , 3 , 4 ) by A. f o r s i m p l i c i t y . I n t h e s eJ
u
112 J 2 2
s n s = q q a 1 2 a n d a = w c q 2 f o r q < a / and
P P 4 P
irq:
-
w2/c2 ] ' l 2 f o r q >w/c,
w i t h p = o , R , t , and c c t and c a r e t h eP 1. P
l o n g i t u d i n a l , t r a n s v e r s e sound v e l o c i t i e s i n t h e s o l i d and t h e sound v e l o c i t y i n l i q u i d He. These c o e f f i c i e n t s a r e d e t e r m i n e d by t h e boundary c o n d i t i o n s , v i z . t h e c o n t i n u i t y o f t h e normal components o f t h e v e l o c i t y and t h e s t r e s s e s a c t i n g o n t h e
+
s u r f a c e x3 = ~ ( xn
), and t h e p e r t u r b a t i v e s o l u t i o n s f o r them c a n b e w r i t t e n i n t h e f o l l o w i n g forms ( 4 b ) where t h e s u p e r s c r i p t s d e n o t e t h e o r d e r o f t h e c o r r e s p o n d i n g t e r m i n c ( z1,
11 and<(ell)
i s t h e F o u r i e r t r a n s f o r m o f ~ ( c ). I n terms o f t h e s e c o e f f i c i e n t s ,w h e r e P and P a r e t h e m a s s d e n s i t i e s o f t h e s o l i d a n d l i q u i d He, a n d t h e r e s u l t 2 2
t h a t < < ( i n ) < ( $ ; ) > = 6 ( 2 n ) 6 ( d l + k * ; ) g ( k n ) h a s b e e n u s e d .
We use an adaptive
numerical integration scheme to evaluate the diffuse part of the
penetration coefficient, which is shown in
Fig. 1 for
s =
a, and will present the results
for the Kapitza resistance elsewhere. In Fig.
1, the two peaks at the angles of 6.4O and
nearly 90° come from the excitation of genera-
lized Rayleigh waves and stoneley waves,
respectively.
-.
Y lr'o 80 m,
,
m 60 YO rn ,O F i g . 1 . The d i f f u s e p a r t o f t h e p e n e t r a t i o n C (9 c o e f f i c i e n t v e r s u s t h e a n g l e o f i n c i d e n c e f o r two d i f f e r e n t f r e q u e n c i e s w(=cok). R e f e r e n c e s 1 . K a p i t z a , P. L . , zh. Eksp. T e o r . F i z .11,
1 ( 1 9 4 1 ) 2 . K h a l a t n i k o v . I . M.. T h e o r y o f S u v e r f l u i d i t y .3. Weber, J . , Sandmann, W . , D i e t s c h e , W. a n d K i n d e r H . , J. Phys. C o l l .
2,
C - 6 , 242 ( 1 9 7 8 ) .4 . T a b o r e k , P. a n d G o o d s t e i n , D. L . , P h y s . Rev.