HAL Id: jpa-00226294
https://hal.archives-ouvertes.fr/jpa-00226294
Submitted on 1 Jan 1987
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
GROWTH OF ICE CRYSTALS FROM THE VAPOUR PHASE. INTERACTION OF BASAL AND PRISM FACES THROUGH THE DIFFUSION PROCESS AND
THE SURFACE KINETIC PROCESS
E. Yokoyama, T. Kuroda
To cite this version:
E. Yokoyama, T. Kuroda. GROWTH OF ICE CRYSTALS FROM THE VAPOUR PHASE. INTER- ACTION OF BASAL AND PRISM FACES THROUGH THE DIFFUSION PROCESS AND THE SURFACE KINETIC PROCESS. Journal de Physique Colloques, 1987, 48 (C1), pp.C1-347-C1-353.
�10.1051/jphyscol:1987148�. �jpa-00226294�
Colloque supplbment au n o 3, Tome 48, mars 1987
GROWTH OF ICE CRYSTALS FROM THE VAPOUR PHASE .
INTERACTION OF BASAL AND PRISM FACES THROUGH THE DIFFUSION PROCESS AND THE SURFACE KINETIC PROCESS
E. YOKOYAMA and T. KURODA
The Institute of Low Temperature Sciences, Hokkaido University, Sapporo 060, Japan
& s d - Deux dcanismes se rapportent & la croissance des cristaux de glace 5 partir de la phase vapeur : i) processus de diffusion des mo~kcules dleau, ii) processus cin6tique de surface pour incorporer les mol6cules dans le rbseau cristallin. Le but de liCtude pr6sent6e est d16tudier l'interaction des faces basales et prismatiques en consid6rant divers aspects prenant en compte de manisre coherente les deux processus. Les r6sultats obtenus sont les suivants : 1) la vitesse de croissance des faces prismatiques &cro?t avec llaccroissement du rapport axial c/b, c'est-A-dire l'accroissement de la surface de la face prismatique, le
&rim&tre du cristal restant constant. Par ailleurs, la vitesse de croissance des faces basales croit car les d e w faces sont en ccxqStition vis-2-vis des mo16des de vapeur d'eau. 2) L'interaction des deux faces d6croit avec l'augmentation de la constante de diffusion. 3) La vitesse de croissance en masse est quasi i&pendante du rapport axial pour un @rimstre du cristal constant et pour diffgrentes constantes de diffusion.
A b s t r a c t - T h e f o l l o w i n g t w o p r o c e s s e s a r e r e l e v a n t t o t h e g r o w t h o f i c e c r y s t a l s f r o m t h e v a p o u r p h a s e : i) d i f f u s i o n p r o c e s s o f w a t e r m o l e c u l e s , ii) s u r f a c e k i n e t i c p r o c e s s f o r i n c o r p o r a t i n g t h e m o l e c u l e s i n t o c r y s t a l l a t t i c e . T h e p u r p o s e o f t h e p r e s e n t s t u d y i s t o i n v e s t i g a t e t h e i n t e r a c t i o n o f b a s a l a n d p r i s m f a c e s f r o m v a r i o u s p o i n t s o f v i e w by t a k i n g i n t o a c c o u n t b o t h p r o c e s s e s s e l f - c o n s i s t e n t l y . T h e o b t a i n e d r e s u l t s a r e as f o l l o w s : ( 1 ) The g r o w t h r a t e o f p r i s m f a c e d e c r e a s e s w i t h i n c r e a s i n g a x i a l r a t i o c / b , i.e. i n c r e a s i n g a r e a o f p r i s m f a c e , u n d e r t h e c o n d i t i o n o f c o n s t a n t p e r i p h e r y l e n g t h o f a c r y s t a l . On t h e . o t h e r h a n d , t h e g r o w t h r a t e o f b a s a l f a c e i n c r e a s e s , s i n c e b o t h f a c e s s c r a m b l e f o r w a t e r m o l e c u l e s i n a i r . (2) The i n t e r a c t i o n o f b o t h f a c e s d e c r e a s e s w i t h i n c r e a s i n g d i f f u s i o n c o n s t a n t . ( 3 ) The m a s s g r o w t h r a t e i s a l m o s t i n d e p e n d e n t o f t h e a x i a l r a t i o f o r a c o n s t a n t p e r i p h e r y l e n g t h o f a c r y s t a l a n d f o r v a r i o u s d i f f u s i o n c o n s t a n t .
1 . I n t r o d u c t i o n
The f o l l o w i n g t w o p r o c e s s e s are r e l e v a n t t o t h e g r o w t h o f i c e c r y s t a l s f r o m t h e v a p o u r p h a s e ; i ) D i f f u s i o n p r o c e s s o f w a t e r m o l e c u l e s t o w a r d s c r y s t a l s u r f a c e , i i ) S u r f a c e k i n e t i c p r o c e s s f o r i n c o r p o r a t i n g t h e m o l e c u l e s i n t o c r y s t a l l a t t i c e . T h e l a t t e r p r o c e s s i n c l u d i n g g e n e r a t i o n o f s t e p s a n d t h e i r l a t e r a l m o t i o n c o n t r i b u t e s t o r e t a i n i n g f l a t n e s s o f t h e b a s a l a n d p r i s m f a c e s w h i c h a r e c o n s t i t u e n t s u r f a c e s of a g r o w i n g h e x a g o n a l p r i s m - s h a p e d i c e . T h e b a s a l a n d p r i s m f a c e s o f a g r o w i n g i c e c r y s t a l a r e e x p e c t e d t o i n t e r a c t t o e a c h o t h e r b e c a u s e o f c o u p l i n g o f b o t h p r o c e s s e s . I n t h e m o s t o f t h e p r e v i o u s t h e o r e t i c a l s t u d i e s on v a p o u r g r o w t h o f i c e [ l ] , h o w e v e r , t h e s u r f a c e k i n e t i c p r o c e s s was n e g l e c t e d . The p u r p o s e s o f t h e p r e s e n t s t u d y a r e as f o l l o w s : ( 1 ) t o c l a r i f y how t h e b a s a l a n d p r i s m f a c e s , i n t e r a c t t o e a c h o t h e r by t a k i n g i n t o a c c o u n t b o t h t h e d i f f u s i o n p r o c e s s a n d t h e s u r f a c e k i n e t i c p r o c e s s s e l f - c o n s i s t e n t l y , ( 2 ) t o i n v e s t i g a t e t h e d e p e n d e n c e o f g r o w t h r a t e s of b o t h f a c e s o n
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987148
Cl-348 JOURNAL DE PHYSIQUE
a x i a l r a t i o c / b a n d o n d i f f u s i o n c o n s t a n t D, ( 3 ) t o i n v e s t i g a t e t h e d e p e n d e n c e o f m a s s g r o w t h r a t e o n t h e a x i a l r a t i o c / b a n d (4) t o p u r s u e t h e c h a n g e o f a x i a l r a t i o w i t h t i m e . To s i m p l i f y t h e problem, we t r e a t e d two-dimensional r e c t a n g u l a r c r y s t a l w h i c h c o n s i s t s o f a ( 1 2 1 0 ) f a c e l i m i t e d a t i t s i n t e r s e c t i o n s w i t h b a s a l a n d p r i s m f a c e s (Fig.1). P r e c i s e l y s p e a k i n g , t r a n s p o r t p r o c e s s of s u b l i m a t i o n h e a t may a l s o c o n c e r n s t h e i n t e r a c t i o n of b o t h f a c e s . T h i s p r o c e s s was assumed t o b e n e g l i g i b l e , h o w e v e r , u n l e s s t h e s u b l i m a t i o n h e a t g e n e r a t e d a t g r o w i n g i n t e r f a c e i s t o o l a r g e , i.e. g r o w t h r a t e i s t o o l a r g e .
2. D i f f u s i o n f i e l d
L e t u s d e f i n e t h e s u p e r s a t u r a t i o n 0 a s
a = ( p - p e ) / p e (1)
w h e r e p i s t h e p r e s s u r e o f w a t e r v a p o u r and pe t h e e q u i l i b r i u m v a p o u r p r e s s u r e o f i c e . When a p o l y h e d r a l c r y s t a l g r o w s r e t a i n i n g i t s m a c r o s c o p i c a l l y f l a t s u r f a c e , t h e s u p e r s a t u r a t i o n d e t e r m i n e d by L a p l a c e 1 s e q u a t i o n is n o t u n i f o r m o v e r t h e c r y s t a l s u r f a c e [ 2 , 3 ] . Using a c o n f o r m a l mapping and F o u r i e r t r a n s f o r m [ 2 ] , i t i s shown t h a t s u p e r s a t u r a t i o n usb a t t h e c e n t e r of b a s a l f a c e and asp a t t h e c e n t e r of prism f a c e a r e g i v e n a s
usb=u,-(b+c){ B ~ ~ P + B ~ ~ ~ I asP=u,-(b+c){ P ~ ~ P + P ~ ~ ~ I (2) where 0, i s t h e s u p e r s a t u r a t i o n a t some d i s t a n c e L a p a r t from t h e c e n t e r of c r y s t a l , and qb=(3a/an)basal and a r e n o r m a l g r a d i e n t s of s u p e r s a t u r a t i o n a t b a s a l and p r i s m f a c e s which a r e unknown p a r a m e t e r s g i v e n a s boundary c o n d i t i o n s f o r L a p l a c e ' s e q u a t i o n a n d d e t e r m i n e d l a t e r [ 3 ] . P1, P 2 , B1 a n d B2 a r e c o n s t a n t s w h i c h d e p e n d on a x i a l r a t i o c / b a n d o n r a t i o L / ( b + c ) , a n d a r e o b t a i n e d by n u m e r i c a l i n t e g r a l s . F i g . 2 s h o w s t h e d e p e n d e n c e o n t h e a x i a l r a t i o o f P I , P 2 , B1 a n d B2 f o r L/(b+c)=50. It should be noted t h a t t h e v a l u e s of q b and qP c o r r e s p o n d t o t h e g r o w t h r a t e Rdb of b a s a l f a c e and t h e g r o w t h r a t e RdP of prism f a c e determined by d i f f u s i o n f l u x t o w a r d s each s u r f a c e , i.e.
R ~ ~ = ( P , / ~ T ) v , D ~ ~ R ~ P = ( P ~ / ~ T ) v , D ~ P (3)
H e r e D i s t h e d i f f u s i o n c o n s t a n t o f w a t e r m o l e c u l e , vc t h e v o l u m e o f a m o l e c u l e i n c r y s t a l , k t h e B o l t z m a n n c o n s t a n t a n d T t h e a b s o l u t e t e m p e r a t u r e . It s h o u l d b e n o t i c e d t h a t a s p a t t h e p r i s m f a c e d e p e n d s on qP r e p r e s e n t i n g t h e s i n k a c t i o n f o r w a t e r m o l e c u l e s a t t h e p r i s m s u r f a c e and a l s o on qb a t b a s a l f a c e and v i c e versa.
C 1 [x:
0 0.2 a 5 a x i a l r a t i o F i g . 1. A h e x a g o n a l prism-shaped
i c e and a r e c t a n g u l a r c r y s t a l . c / b
Fig.2. PI, P 2 , B and B2 v e r s u s 1
c / b f o r ~ / ( b + c ) = 5 0 .
I f s p i r a l g r o w t h o c c u r s w i t h t h e a i d o f a s c r e w d i s l o c a t i o n e m e r g i n g a t c e n t e r o f t h e s u r f a c e , t h e g r o w t h r a t e Rk(os) d e t e r m i n e d by t h e s u r f a c e g r o w t h k i n e t i c s i s a f u n c t i o n o f s u p e r s a t u r a t i o n as a t c e n t e r o f t h e r e f e r r e d s u r f a c e . The g r o w t h r a t e Rk i s e x p r e s s e d by Hertz-Knudsen e q u a t i o n
Rk=a(aS)vcPeas/J2'S;;mi;T ( 4 ) w i t h a s o - c a l l e d s u b l i m a t i o n c o e f f i c i e n t a ( a s ) ( o r c o n d e n s a t i o n c o e f f i c i e n t )
a ( a s ) = a i ( o s / ~ i ) t a n h ( ~ i / ~ s ) ( 5 ) a i a s / o i ( o s < < a i )
={ a1 ( o s > > a i ) ( 5 ) '
w h e r e a1=9.5£ o ~ / k T x ~ ( 6 )
I n e q . ( 4 ) m i s t h e m a s s o f a w a t e r m o l e c u l e . I n e q s . ( 5 ) a n d (5)', a1 i s t h e s t i c k i n g c o e f f i c i e n t , i.e. t h e r a t i o o f m o l e c u l e s w h i c h s t i c k o n t h e s u r f a c e a t t h e i n s t a n c e w h e n t h e y h a v e i m p i n g e d o n i t , f o t h e s u r f a c e a r e a o c c u p i e d by a w a t e r m o l e c u l e , K t h e e d g e f r e e e n e r g y p e r u n i t l e n g t h i n a s t e p and xs t h e mean d i f f u s i o n d i s t a n c e o f a n a d m o l e c u l e o n a s u r f a c e . We m u s t d i s c r i m i n a t e b e t w e e n v a l u e s o f 01
a t b a s a l a n d p r i s m f a c e s b e c a u s e o f t h e d i f f e r e n c e o f K a t b o t h f a c e s . o l b=9.5f O K ~ / ~ T X ~ olP=9.5f O K P / ~ T X ~ ( 7 )
I t i s t o b e n o t e d t h a t a ( o s ) w h i c h r e p r e s e n t s t h e a c t i v i t y o f t h e s u r f a c e f o r i n c o r p o r a t i n g w a t e r m o l e c u l e s i n t o i c e c r y s t a l d e p e n d s on as, s i n c e t h e d e n s i t y o f s t e p s b e i n g s i n k o f m o l e c u l e s i s c o n t r o l l e d by o s [ 4 ] .
From eqs.(2) and ( 4 ) , it i s c l e a r t h a t t h e g r o w t h r a t e Rkb o f b a s a l f a c e depends n o t o n l y on q b b u t a l s o qP t h r o u g h t h e s u p e r s a t u r a t i o n osb and RkP a l s o d e p e n d s b o t h o n qP a n d q b . N a m e l y , b o t h f a c e s i n t e r a c t t o e a c h o t h e r t h r o u g h t h e d i f f u s i o n p r o c e s s a n d t h e s u r f a c e k i n e t i c pr.ocess.
4 . D e t e r m i n a t i o n o f Unknown p a r a m e t e r s q b and q P
Unknown p a r a m e t e r s qb a n d qP, and c o n s e q u e n t l y t h e g r o w t h r a t e s R~ a n d RP, a r e d e t e r m i n e d by mass b a l a n c e e q u a t i o n s a t b o t h s u r f a c e s ; t h e g r o w t h rate Rdb(eq.3) o f b a s a l f a c e d e t e r m i n e d by v o l u m e d i f f u s i o n i s e q u a l t o t h e g r o w t h r a t e Rkb(eq.4) d e t e r m i n e d by t h e s u r f a c e k i n e t i c s , i.e.
R ~ ~ ( ~ ~ ) = R ~ ~ ( ~ ~ ~ ( ~ P , ~ ~ ) ) = R ~ ( 9 ) T h e s a m e e q u a t i o n h o l d s a l s o f o r t h e p r i s m f a c e , 1.e.
R ~ P ( ~ P ) = R ~ P ( ~ ~ P ( ~ P , ~ ~ ) ) = R ~ (10) Eqs.(9) a n d (10) a r e s i m u l t a n e o u s e q u a t i o n s f o r q b a n d qP.
5 . R e s u l t s
I n t h i s s e c t i o n we i n v e s t i g a t e t h e i n t e r a c t i o n o f b a s a l a n d p r i s m f a c e s f r o m v a r i o u s p o i n t s o f v i e w . I n o r d e r t o s h o w i t c l e a r l y , we c o m p a r e t h e t w o c a s e s ; ( I ) T h e s c r e w d i s l o c a t i o n s e m e r g e b o t h a t c e n t e r s o f b a s a l a n d p r i s m f a c e s . ( 1 1 ) T h e s c r e w d i s l o c a t i o n e m e r g e s o n l y a t c e n t e r o f p r i s m f a c e , w h i l e b a s a l f a c e d o e s n o t g r o w . T h e v a l u e s o f t h e f o l l o w i n g q u a n t i t i e s a r e k e p t c o n s t a n t i n t h i s s t u d y :
C1-350 JOURNAL DE PHYSIQUE
a l = 0 . 1 [ 5 , 6 ] , ci,=15%, a l b = 2 . 9 % , a l p = 2 . 5 % , T=253.15K, ~ , = 0 . 7 7 6 T o r r , ~ ~ = 3 . 2 5 ~ 1 0 - ~ ~ c m ~ , m = 3 . 0 ~ 1 0 - ~ ~ ~ , 2 ( b + c ) = O . l c m ( c o n s t a n t ) a n d L=2.5cm. T h e v a l u e s o f a x i a l r a t i o c / b
and d i f f u s i o n c o n s t a n t D a r e changed.
5.1 An example f o r i n t e r a c t i o n o f b o t h f a c e s
T a b l e 1 shows a r e s u l t o b t a i n e d f o r ~ = 0 . 2 2 c m ~ / s e c ( c o r r e s p o n d i n g t o 1 a t m o f a i r p r e s s u r e ), a n d c/b=0.2. T h e g r o w t h r a t e R ~ P o f p r i s m f a c e i n t h e c a s e ( 1 1 ) i s n e a r l y t w i c e t h e g r o w t h r a t e R ~ P i n t h e c a s e o ( I ) , s i n c e OsP i n t h e c a s e ( 1 1 ) i s l a r g e r t h a n t h a t i n t h e c a s e ( I ) b e c a u s e of no d i f f u s i o n f l o w t o w a r d s t h e b a s a l f a c e i n t h e c a s e ( I 1 ) . I t c l e a r l y s h o w s t h a t t h e g r o w t h r a t e o f a s u r f a c e i s s t r o n g l y i n f l u e n c e d by t h a t o f t h e o t h e r s u r f a c e t h r o u g h t h e d i f f u s i o n p r o c e s s a n d t h e s u r f a c e k i n e t i c p r o c e s s .
T a b l e 1 2 (b+c)=O. lcm
a x i a l r a t i o c/b=O. 2 , D = 0.22cm / s e c 2
5.2 Dependence on a x i a l r a t i o
F i g . 3 s h o w s t h e d e p e n d e n c e o n a x i a l r a t i o c / b o f t h e s u p e r s a t u r a t i o n a t t h e c e n t e r s o f t h e p r i s m a n d b a s a l f a c e s i n t h e a b o v e t w o c a s e s . Fig.4 r e p r e s e n t s t h e d e p e n d e n c e o n c / b o f t h e g r o w t h r a t e s o f b o t h f a c e s c o r r e s p o n d i n g t o F i g . 3 . T h e s u p e r s a t u r a t i o n asp on p r i s m f a c e d e c r e a s e s w i t h i n c r e a s i n g a x i a l r a t i o c / b i n t h e c a s e s ( I ) a n d ( I I ) , s i n c e t h e s u p p l y o f w a t e r m o l e c u l e s t o t h e p r i s m f a c e b e c o m e s i n s u f f i c i e n t b e c a u s e o f i n c r e a s e i n t h e a r e a o f p r i s m f a c e a c t i n g a s s i n k o f m o l e c u l e s . T h e r e f o r e , t h e g r o w t h r a t e RP of p r i s m f a c e d e c r e a s e s w i t h i n c r e a s i n g
0.2 0.5 1 2 5
axial ratio c/b
F i g . 3 . Us o f b a s a l a n d p r i s m f a c e s v e r s u s c / b .
F i g . 4 . R o f b a s a l a n d p r i s m f a c e s c o r r e s p o n d i n g t o F i g . 3
i n t h e c a s e ( I ) n e c e s s a r i l y i n c r e a s e s w i t h i n c r e a s i n g c / b s i n c e b o t h f a c e s s c r a m b l e f o r w a t e r m o l e c u l e s i n t h e a i r .
I t is a l s o shown i n Fig.4 t h a t t h e d i f f e r e n c e b e t w e e n R2P and R ~ P d e c r e a s e s w i t h i n c r e a s i n g a x i a l r a t i o c / b . The r e s u l t i s i n t e r p r e t e d a s f o l l o w s . S i n c e t h e a r e a o f b a s a l f a c e d e c r e a s e s w i t h i n c r e a s i n g c / b , t h e f l u x o f w a t e r m o l e c u l e s i n c o r p o r a t e d i n t o c r y s t a l a t t h e b a s a l f a c e i n t h e c a s e ( I ) d e c r e a s e s w i t h i n c r e a s i n g c / b . T h e r e f o r e , a s c / b i n c r e a s e s , t h e d i f f e r e n c e b e t w e e n t h e c a s e ( I ) and t h e c a s e (11) i n w h i c h no w a t e r m o l e c u l e s a r e i n c o r p o r a t e d a t t h e b a s a l f a c e d e c r e a s e s .
5.3 Dependence on d i f f u s i o n c o n s t a n t
F i g . 5 s h o w s t h e s u p e r s a t u r a t i o n o n b o t h f a c e s v e r s u s a x i a l r a t i o f o r h i g h e r d i f f u s i o n c o n s t a n t ~ = 4 4 7 c m ~ / s e c ( c o r r e s p o n d i n g t o 0 . 3 T o r r o f a i r p r e s s u r e ) . T h e s u p e r s a t u r a t i o n on b o t h f a c e s d o e s n o t d r o p l a r g e l y f r o m t h e s u p e r s a t u r a t i o n Om a n d h a r d l y c h a n g e f o r v a r i o u s a x i a l r a t i o c / b i n t h e c a s e s ( I ) a n d (11). T h i s i s d u e t o e a s i e r s u p p l y o f w a t e r m o l e c u l e s by d i f f u s i o n t o t h e s u r f a c e s u n d e r t h e c o n d i t i o n s of h i g h e r d i f f u s i o n c o n s t a n t D. T h e r e f o r e , t h e g r o w t h r a t e o f b o t h f a c e s i s a l m o s t i n d e p e n d e n t o f c / b ( F i g . 6 ) . N a m e l y , t h e i n t e r a c t i o n o f b o t h f a c e s d e c r e a s e s w i t h i n c r e a s i n g D.
5.4 Mass g r o w t h r a t e
I t i s o f i n t e r e s t t o i n v e s t i g a t e t h e d e p e n d e n c e o n a x i a l r a t i o c / b o f t h e mass g r o w t h r a t e d M / d t w h i c h i s d e t e r m i n e d b y t h e t o t a l f l u x o f w a t e r m o l e c u l e s i n c o r p o r a t e d i n t o a n i c e c r y s t a l a t b a s a l a n d p r i s m f a c e s . A s s h o w n i n F i g . 7 w i t h t h i c k l i n e s , d M / d t i n t h e c a s e ( I ) i s a l m o s t i n d e p e n d e n t o f c / b f o r a c o n s t a n t p e r i p h e r y l e n g t h o f c r y s t a l a n d f o r v a r i o u s D. S i n c e t h e g r o w i n g b a s a l a n d p r i s m f a c e s s c r a m b l e f o r t h e w a t e r m o l e c u l e s w h i c h a r e s u p p l i e d by d i f f u s i o n w i t h f i n i t e r a t e , t h e g r o w t h r a t e o f e a c h s u r f a c e s t r o n g l y d e p e n d s o n t h e a x i a l r a t i o , i . e . t h e r a t i o o f a r e a s o f b o t h f a c e s , w h i l e t h e t o t a l m a s s f l u x i n c o r p o r a t e d i n t o a c r y s t a l t h r o u g h b o t h f a c e s d o s e n o t d e p e n d o n t h e a x i a l r a t i o . On t h e o t h e r h a n d , d M / d t i n t h e c a s e ( 1 1 ) s l i g h t l y d e c r e a s e s w i t h d e c r e a s i n g c / b ( s e e d a s h e d l i n e s i n F i g . 7 ) , s i n c e t h e a r e a o f b a s a l f a c e w h i c h d o e s n o t g r o w a t a l l i n c r e a s e s w i t h d e c r e a s i n g c / b .
5.5 Change o f a x i a l r a t i o w i t h t i m e
The g r o w t h r a t e s o f b a s a l a n d p r i s m f a c e s a r e s e l f - c o n s i s t e n t l y d e t e r m i n e d f o r a g i v e n a x i a l r a t i o c / b a c c o r d i n g t o t h e method d e s c r i b e d i n s e c t i o n 2,3 and 4. Then a n i n c r e a s e o f t h e l e n g t h s o f i n d i v i d u a l a x e s c a n d b f o r a t i m e i n t e r v a l A t i s e v a l u a t e d by t h e g r o w t h r a t e s o f b a s a l a n d p r i s m f a c e s . T h e r e f o r e , we c a n p u r s u e t h e c h a n g e o f a x i a l r a t i 0 , i . e . t h e c h a n g e o f c r y s t a l s h a p e , w i t h t i m e a s shown i n Fig.8.
The d e t a i l s w i l l b e d i s c u s s e d e l s e w h e r e s h o r t l y .
J O U R N A L DE PHYSIQUE
2 (b+c) =O. lcrn 2
V) D = 447cm /sec
- 0.2 0.5 1 2
Fig .5.
as of both faces versus c h .
0.2 0.5 1 2 5
a x i a l r a t i o c/b
F i g . 7 . dM/dt versus c/b.
Fig. 6. c / b
R o f basal and prism faces corresponding to Fig .5 .
Fig.8. Change of crystal shape with time for a time interval At=665sec. initial size 2(c,+
b,)=O.Olcm, initial axial ratio c./b.=0.3 and ~=0.22cm~/sec.
References
[ l ] B.J. Mason, Quart. J. Roy. Meteor.Soc. 79 (1953) 104-111
[ 2 ] A. Seeger, Phil. Mag. 44 (1953) 1-13
[ 3 ] T.Kuroda, T.Irisawa and A. Ookawa, J. Crystal Growth 42 (1977) 41-46
[ 4 ] W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. Roy. Soc. London A 2 4 3 (1951)
299-358 . . .-.
[ 5 ] W. Beckmann and R. Lacmann, J. Crystal Growth 58 (1982) 433-442 [ 6 ] T. Kuroda and T. Gonda, J. Meteor. Soc. Japan, 62 (1984) 563-572
N. FUKUTA
Your t r e a t m e n t is based on screw d i s l o c a t i o n mechanism. Screw d i s l o c a t i o n is known t o be d i f f i c u l t t o n u c l e a t e . H a l l e t t and h i s c o l l e a g u e s f a i l e d t o d e t e c t screw d i s l o c a t i o n on n a t u r a l snow c r y s t a l s by X-ray a n a l y s i s . Your t r e a t m e n t , t h e r e f o r e , may b e based o n n o n - e x i s t i n g screw d i s l o c a t i o n . What is y o u r r e s p o n s e t o t h i s p o i n t ?
Answer :
Yes, t h e y r e p o r t e d t h a t snow c r y s t a l s were d i s l o c a t i o n f r e e i n most c a s e s . I n my o p i n i o n , i t i s n o t always t h e c a s e , o f c o u r s e , we c a n e a s i l y a p p l y t h i s t r e a t m e n t t o two dimensional n u c l e a t i o n growth o f p e r f e c t snow c r y s t a l s and s i m i l a r l y i n v e s t i g a t e t h e i n t e r a c t i o n between b a s a l and prism f a c e s .
K. ROESSLER
Does t h e kind end q u a l i t y o f t h e s u r f a c e p l a y a r o l e i n your e q u a t i o n s , o r , where would you i n t r o d u c e i t .
Answer :
A s t o basal and prism f a c e s , i n f l u e n c e o f roughening 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 can be t a k e n i n t o account through t h e d e c r e a s e i n s t e p f r e e e n e r g y which c o n t r o l s t h e s t e p d e n s i t y . On t h e o t h e r hand, a d h e s i v e growth o c c u r s on t h e o t h e r s u r f a c e s c o r r e s p o n d i n g t o h i g h v a l u e o f M i l l e r - i n d i c e s . However, s u c h s u r f a c e s q u i c k l y d i s a p p e a r because of t h e i r v e r y f a s t growth i n comparison w i t h l a t e r a l growth.
