VISCOSITY OF ICE V

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Submitted on 1 Jan 1987

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VISCOSITY OF ICE V

C. Sotin, J. Poirier

To cite this version:

C. Sotin, J. Poirier. VISCOSITY OF ICE V. Journal de Physique Colloques, 1987, 48 (C1), pp.C1- 233-C1-238. �10.1051/jphyscol:1987132�. �jpa-00226278�

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JOURNAL DE PHYSIQUE

Colloque C1, suppl6ment au n o 3, Tome 48, mars 1987

VISCOSITY OF ICE V

C. SOTIN and J.P. POIRIER*

Laboratoire de Tectonophysique, Universitk de Nantes, Chemin de la Houssiniere, F-44072 Nantes Cedex, France

" ~ n s t i t u t de Physique du Globe de Paris, T-14, E-2, 4 Place Jussieu, F-75252 Paris Cedex 05, France

Resumb : On a d e t e r m i n e l ' b q u a t i o n r h 6 o l o q i q u e d e f l u a g e d e l a g l a c e V aux tempe-s d e -10 e t -40°c, e n s u i v a n t o p t i q u e m e n t l e d h p l a c e m e n t d e m a r q u e u r s l e l o n g du g r a d i e n t d e p r e s s i o p d a n s une c e l l u l e A e n c l u m e s d e s a p h i r . On a o b t e n u l a v i t e s s e d e f l u a g e E ( e n s- ) e n f o n c t i o n d e l a c o n t r a i n t e a, d e l a p r e s s i o n h y d r o s t a t i q u e P ( e n MPa) e t d e l a t e m p e r a t u r e T ( e n K) :

P o u r d e s c o n d i t i o n s i d e n t i q u e s d e t e m p e r t u r e e t d e c o n t r a i n t e r b d u i t e s , l a g l a c e V p o s s 6 d e une v i s c o s i t b e n v i r o n 1 0 0 f o i s p l u s B l e v b e a u e c e l l e d e l a g l a c e VI.

A b s t r a c t : The r h e o l o g i c a l e q u a t i o n f o r c r e e p o f i c e V , a t t e m p e r a t u r e s o f -10 and - m s b e e n d e t e r m i n e d by o p t i c a l l y f o l l o w i n g t h e m o t i o n o f m a r k e r s down t h e p r e s s u r e g r a d i e n t i n a s a p p h i r e a n v i l c e l l . The c r e e p r a t e t ( i n s - l ) e x p r e s s e d a s a f u n c t i o n o f s h e a r s t r e s s o, h y d r o s t a t i c p r e s s u r e P ( i n MPa) and t e m p e r a t u r e T ( i n K) is :

F o r t h e same r e d u c e d t e m p e r a t u r e and stress, i c e V h a s a v i s c o s i t y a b o u t two o r d e r s o f m a g n i t u d e h i g h e r t h a n t h a t o f i c e VI.

I n t r o d u c t i o n

I n t e r e s t i n t h e r h e o l o g i c a l p r o p e r t i e s o f h i g h p r e s s u r e p h a s e s o f i c e a r o s e a s a c o n s e q u e n c e o f t h e i n t e r e s t i n t h e t e c t o n i c s and d y n a m i c s o f t h e i c y s a t e l l i t e s o f t h e g i a n t p l a n e t s ( 1 ) . I n t h e l a s t y e a r s , s e v e r a l s t u d i e s h a v e b e e n p u b l i s h e d on t h e m e c h a n i c a l p r o p e r t i e s o f ice 11, 111 ( 2 , 3 ) and VI (4, 5 ) . The p r e s e n t a r t i c l e r e p o r t s o n t h e c r e e p p r o p e r t i e s o f ice V. A s f o r ice V I , t h e v i s c o s i t y was d e t e r m i n e d by m o n i t o r i n g t h e r a d i a l o u t w a r d f l o w o f m a r k e r s a l o n g t h e p r e s s u r e g r a d i e n t i n a s a p p h i r e a n v i l c e l l . The p a r a m e t e r s o f t h e c r e e p law were d e t e r m i n e d b y t o t a l i n v e r s i o n o f t h e d a t a (5).

E x p e r i m e n t a l p r o c e d u r e

The s a p p h i r e a n v i l c e l l wa2 d e s c r i b e d i n ( 5 ) . S u f f i c e it t o r e c a l l h e r e t h a t it c o n s i s t s o f a p a i r o f s i n q l e c r y s t a l l i n e s a p p h i r e a n v i l s ' s q u e e z i n q a n a n n e a l e d c o p p e r g a s k e t , 3 0 0 pm t h i c k , w i t h a 1.5 mm d i a m e t e r h o l e . The f o r c e on t h e a n v i l s is a p p l i e d b y an I n s t r o n t e s t i n g m a c h i n e . A d r o p o f t a p w a t e r c o n t a i n i n q s m a l l (10 pm) carborundum p a r t i c l e s is d e p o s i t e d i n t h e h o l e . The c e l l is c o o l e d t o t e m p e r a t u r e s o f -15 and -40°c i n a l i q u i d n i t r o g e n b a t h ; t h e t e m p e r a t u r e is m e a s u r e d by a c h r o m e l - a l u m e l t h e r m o c o u p l e c l o s e t o t h e g a s k e t . When t h e d e s i r e d t e m p e r a t u r e is o b t a i n e d , t h e p r e s s u r e is i n c r e a s e d u n t i l i c e V forms. The p h a s e t r a n s f o r m a t i o n s and e v e n t u a l l y t h e m o t i o n o f m a r k e r s a r e m o n i t o r e d t h o r o u g h a v i d e o c a m e r a , p h o t o q r a p h s o f t h e m a r k e r s are t a k e n a t r e g u l a r i n t e r v a l s . A t - 1 ' 3 ' ~ ~ i c e 1 m e l t s when t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987132

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C1-234 JOURNAL DE PHYSIQUE

p r e s s u r e is i n c r e a s e d and t h e water f r e e z e s i n t o i c e V ( f i g . 1 ) a t a p r e s s u r e o f 3.5'1 k b a r , e s t i m a t e d by e l a s t o p l a s t i c c a l c u l a t i o n s of t h e s t r e s s e s i n t h e g a s k e t and t h e i c e ( 5 , 6 ) . A t -&PC, i c e I f i r s t t r a n s f o r m s i n t o i c e 11, which t h e n t r a n s f o r m s i n t o i c e V , ( f i g . 2 ) .

Fig. 1. I s o t h e r m a l compression a t - ~ P c , ( u n i a x i a l f o r c e ) -.-

A : Melting o f i c e I

B : T r a n s i t i o n i c e I - ^{i q e}^V

Diameter o f t h e h o l e = 1.3 mm.

I I

For each experiment a t a given t e m p e r a t u r e , t h e v a l u e of t h e s t r a i n r a t e & ( r ) is measured f o r s e v e r a l markers a t d i s t a n c e s r from t h e c e n t e r o f t h e c e l l , t h e v a l u e s o f t h e s h e a r s t r e s s and t h e h y d r o s t a t i c p r e s s u r e o ( r ) and P ( r ) a r e c a l c u l a t e d from t h e v a l u e o f t h e a p p l i e d f o r c e and t h e g e o m e t r i c a l p a r a m e t e r s o f t h e assembly ( 5 ) . The s e t s o f d a t a E(r), o ( r ) , P ( r ) , T a r e i n v e r t e d ( 5 ) , assuming t h a t t h e c r e e p r a t e obeys t h e u s u a l law (7) :

E = ~ o " e x ~ - ^AE ^PAV*

RT ( 1

8

-.

⁶

* *

The i n v e r s i o n y i e l d s t h e p a r a m e t e r s lnA, n, AE , ^AV

Te e x p e r i m e n t a l c o n d i t i o n s were t h e f o l l o w i n g :

--

T = - 15°C

--

4.5 < ^P< ^{6 kbar}

0.5 < a <I .2 kbar

T = -15OC and -40°C 10-5 < B < ^s-l

I

5 10 15 20

TIME ( m n l

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R e s u l t s and d i s c u s s i o n

I n a d d i t i o n t o t h e numerous s t a b l e p h a s e s o f i c e , t h e r e e x i s t s e v e r a l m e t a s t a b l e p h a s e s and e v e n s t a b l e p h a s e s have b e e n known t o form o u t s i d e t h e i r s t a b i l i t y f i e l d . When p u r p o r t i n g t o d e t e r m i n e t h e v i s c o s i t y o f o n e o f t h e h i g h p r e s s u r e p h a s e s o f i c e by v i s u a l m o n i t o r i n g o f t h e m o t i o n o f m a r k e r s , it is e s s e n t i a l t o make s u r e t h a t w e a r e d e a l i n g w i t h t h e r i g h t k i n d o f i c e . AS it was n o t p o s s i b l e t o a s c e r t a i n t h e c r y s t a l s t r u c t u r e o f t h e p h a s e by X-ray d i f f r a c t i o n , we had t o r e l y on i n d i r e c t c r i t e r i a .

I n t h e e x p e r i m e n t a t - 1 5 O ~ ( f i g . I ) , i c e I was f i r s t o b s e r v e d t o m e l t and t h e w a t e r f r e e z e d a g a i n a t a c a l c u l a t e d p r e s s u r e o f a b o u t 3.5 k b a r i n agreement w i t h t h e p r e s s u r e a t which t h e m e l t i n g c u r v e o f i c e V s h o u l d b e c r o s s e d . A t -4d3C ( f i g . 2 ) , we f i r s t o b s e r v e d a t r a n s f o r m a t i o n o f i c e I i n t o a n o t h e r p h a s e , p r e s u m a b l y i c e 11, S e f o r e e n t e r i n g t h e f i e l d o f t h e p h a s e t h o u g h t t o b e i c e V , we t h e n i n c r e a s e d t h e p r e s s u r e u n t i l a n o t h e r p h a s e a p p e a r e d a t a p r e s s u r e o f a b o u t 6 k b a r , i n a g r e e m e n t w i t h t h e c r o s s i n g o f t h e p h a s e b o u n d a r y b e t w e e n ice V and VI.

The m o t i o n o f m a r k e r s i n t h e new p h a s e y i e l d a v i s c o s i t y i n good a q r e e m e n t w i t h t h e p r e v i o u s l y d e t e r m i n e d v i s c o s i t y o f i c e VI. We h a v e t h e r e f o r e good r e a s o n s t o b e l i e v e t h a t t h e p h a s e we s t u d i e d , b e i n g f o u n d i n s i d e b o u n d a r i e s c o r r e s p o n d i n g t o t h o s e o f ice V i n t h e p h a s e d i a g r a m , is i n d e e d i c e V.

Fig. 2 Isothermal compression at -40°C (uniaxial force) A : Transition in ice I - ^{ice I1}

B : Transition ice 11 - ^ice^V

C : Transition ice V

-

^{ice VI}

Diameter of he hole Y 1.3 mm.

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C 1-236 JOURNAL DE PHYSIQUE

We d i d , however, form m e t a s t a b l e i c e IV i n some c a s e s , w h i l e r e l e a s i n g t h e p r e s s u r e from t h e f i e l d o f i c e VI. I c e V f i r s t formed, b u t b e f o r e t h e p r e s s u r e d r o p p e d low e n o u g h f o r i t t o melt, i c e IV i n v a d e d t h e c e l l ( f i g . 31, s t a r t i n g from t h e e d g e o f t h e h o l e , t h e n q u i c k l y , m e l t e d , s t i l l i n s i d e t h e s t a b i l i t y f i e l d o f i c e V. A s d e p r e s s u r i z a t i o n went o n , t h e w a t e r was r a p i d l y b r o u g h t i n t o t h e f i e l d o f i c e I. The f r e e z i n g o f w a t e r a t - 1 0 ' ~ b y r a p i d d e p r e s s u r i z a t i o n o c c u r e d v e r y q u i c k l y , a r e c t i l i n e a r f r o n t p r o p a g a t i n g t h o u g h t h e c e l l .

F i g . 3 I s o t h e r m a l d e c o m p r e s s i o n o f i c e V a t -I@C A : T r a n s i t i o n i c e V - i c e I V

B : M e l t i n g o f i c e IV

C : F r e e z i n g o f w a t e r i n t o i c e I

I n v e r s i o n o f t h e d a t a o b t a i n e d from t h e m o t i o n o f m a r k e r s i n t h e f i e l d o f i c e V y i e l d e d t h e p r e e x p o n e n t i e l c o n s t a n t , t h e s t r e s s e x p o n e n t , t h e a c t i v a t i o n e n e r g y and t h e a c t i v a t i o n volume o f t h e c r e e p r a t e . These v a l u e s a r e g i v e n T a b l e 1 , t o g e t h e r w i t h p r e v i o u s l y d e t e r m i n e d v a l u e s f o r i c e VI ( 5 ) f o r c o m p a r i s o n .

T a b l e 1 : P r e e x p o n e n t i a l c o n s t a n t A , s t r e s s e x p o n e n t n ,

.__ activation e n e r g y AE* and a c t i v a t i o n volume AV* f o r i c e V a n d V I

( 0 i n MPa and & i n s i n e q u a t i o n ( I ) ) . The r e s u l t s f o r i c e VI a r e t a k e n from ( 5 ) .

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The u n c e r t a i n t y i s n o t g i v e n f o r t h e p r e e x p o n e n t i a l c o n s t a n t A , a s we c a n o n l y d e t e r m i n e b y i n v e r s i o n t h e i n c e r t a i n t y o n i t s l o g a r i t h m : 1nA = - 2.92 k 4.73.

The u n c e r t a i n t y o n n , A€* and AV* i s c o n s i d e r a b l y l a r g e r t h a n f o r i c e VI, d u e t o t h e f a c t t h a t t h e f i e l d o f i c e V is much s m a l l e r t h a n t h a t o f i c e VI and t h a t , c o n s e q u e n t l y , t h e r a n g e o f T and P e x p l o r e d was s m a l l e r . A l t h o u q h i t is n o t y e t p o s s i b l e t o g i v e a s a t i s f y i n g p h y s i c a l i n t e r p r e t a t i o n o f t h e s e p a r a m e t e r s , i n t h e a b s e n c e o f i n f o r m a t i o n on t h e l a t t i c e d e f e c t s o f i c e V , we s t i l l c a n make t h e f o l l o w i n g o b s e r v a t i o n s .

i ) Whereas i c e VI had a s t r e s s e x p o n e n t c l o s e t o 2 , i c e V h a s a s t r e s s e x p o n e n t c l o s e t o 3 , t h e " n a t u r a l " e x p o n e n t f o r c l i m b - c o n t r o l l e d c r e e p ( 7 ) .

i i ) The a c t i v a t i o n e n e r g i e s f o r i c e V and i c e VI ( a n d i c e I ) a r e q u i t e s i m i l a r , a b o u t 30 kJ/mole. T h i s m e r e f a c t i s however i n s u f f i c i e n t t o draw a n y c o n c l u s i o n a b o u t t h e p h y s i c a l mechanisms i n v o l v e d .

i i i ) The a c t i v a t i o n volume f o r i c e V i s a b o u t 1 0 cm3/mole. The volume o f a w a t e r m o l e c u l e i n i c e V a t 5 k b a r is a b o u t 1 4 cm3/mole ( u s i n g a b u l k modulus o f 1 3 9 k b a r from ( 8 ) ) . T a k i n g i n t o a c c o u n t t h e l a r g e u n c e r t a i n t y i n t h e v a l u e o f t h e a c t i v a t i o n volume, i t is f a i r t o s t a t e t h a t t h e a c t i v a t i o n volume f o r t h e c r e e p o f i c e V is r o u g h l y e q u a l t o t h e volume o f a w a t e r m o l e c u l e i n t h e s t r u c t u r e . A s i m i l a r c o n c l u s i o n had b e e n r e a c h e d f o r i c e VI ( 5 ) .

i v ) A s was t h e c a s e f o r i c e VI, we f i n d t h a t Weertman's law ( 7 , 9 ) is r a t h e r w e l l o b e y e d , i . e . t h e c r e e p r a t e c a n b e w r i t t e n :

w i t h a v a l u e o f g = 2 0 , s i m i l a r t o t h a t f o r i c e VI and f o r most m e t a l s ( 7 ) . To c a l c u l a t e g w e u s e d t h e e q u a t i o n :

w i t h dTm/dP = 6.04O/kbar ( a v e r a g e v a l u e ) from ( 1 0 ) .

Keyes' r e l a t i o n o f p r o p o r t i o n a l i t y b e t w e e n a c t i v a t i o n e n t h a l p y and a c t i v a t i o n volume ( s e e 7 ) :

" 4 "

AV = - W K

where K i s t h e b u l k modulus, i s a l s o r e a s o n a b l y w e l l obeyed : c a l c u l a t i n g AH X from A€* and AV* e x p e r i m e n t a l , a t a p r e s s u r e o f 5 k i l o b a r s , and t a k i n g K = 1 3 9 k b a r ( 8 ) ,

we f i n d Av* c a l c u l a t e d e q u a l t o 1 0 . 7 cm3/mole, q u i t e c l o s e t o t h e e x p e r i m e n t a l v a l u e : 10.1 cm3/mole (we h a v e t o u s e t h e e x p e r i m e n t a l v a l u e o f AV* t o c a l c u l a t e AH*, s i n c e o n l y A€* and AV* a r e p r o v i d e d b y t h e i n v e r s i o n , i f we assumed AH* AE*, we would f i n d AV* c a l c u l a t e d e q u a l t o 9 . 3 cm3/mole - s t i l l a good a g r e e m e n t ) .

v ) I c e V i s much more r e s i s t a n t t o d e f o r m a t i o n t h a n i c e VI : I t i s i n s t r u c t i v e t o compare t h e v i s c o s i t i e s o f i c e V and VI, c a l c u l a t e d from t h e r h e o l o g i c a l e q u a t i o n s , f o r a t e m p e r a t u r e and p r e s s u r e c o r r e s p o n d i n g t o a p o i n t o f t h e p h a s e d i a g r a m on t h e b o u n d a r y b e t w e e n V and VI.

For T = 260 K, P = 6 k b a r and a s h e a r stress o f 1 k b a r : q = 4 . 5 ~ 1 0 ~ ~ P o i s e f o r i c e V

V = 8 . 4 ~ 1 0 ~ ~ P o i s e f o r i c e VI

We c a n a l s o compare t h e v i s c o s i t y o f i c e V and VI f o r t h e same r e d u c e d tempe- r a t u r e T/Tm and r e d u c e d s t r e s s cr/p ( w h e r e Tm i s t h e m e l t i n g t e m p e r a t u r e and p t h e s h e a r m o d u l u s ) , f o r a p r e s s u r e o f 6 k b a r , c o r r e s p o n d i n g t o t h e b o u n d a r y V-VI ( p r a c t i c a l l y p a r a l l e l t o t h e t e m p e r a t u r e a x i s ) .

U s i n g t h e v a l u e s o f t h e s h e a r rn 9 d u l i o f i c e V and VI g i v e n by Gagnon e t a l . ( 8 ) we h a v e f o r T/Tm = 0 . 9 and a / p = 10- :

f o r i c e V ( p = 61 k b a r ) q = 2 . 8 x 1 0 1 3 p o i s e f o r i c e VI ( p = 7 5 k b a r ) V = 2.6 x 1 0 1 1 p o i s e

I n t h e same homologous c o n d i t i o n s o f t e m p e r a t u r e and s t r e s s , t h e c r e e p r a t e o f i c e V i s t h e r e f o r e two o r d e r s o f m a g n i t u d e s m a l l e r t h a n t h a t o f i c e VI.

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Acknowledgments

We t h a n k Dr Gagnon f o r communicating a p r e p r i n t o f h i s a r t i c l e on e l a s t i c moduli o f i c e s .

T h i s work was p a r t l y s u p p o r t e d by INSU ATP P l a n 6 t o l o g i e 1986 and by CNRS (UA 7 3 4 ) .

T h i s is IPG c o n t r i b u t i o n nOOOO.

R e f e r e n c e s

P o i r i e r , J.P., N a t u r e 299 (1982) 683.

K i r b y , S.H., Durham, W.B., Heard, H.C., I c e s i n t h e S o l a r S y s t e m , Ed. K l i n g e r , J. e t a l . ( R e i d e l , D o r d r e c h t , H o l l a n d ) 1 9 8 5 , p.89.

E c h e l m e y e r , K., Kamb, 8. Geophys. Res. L e t . 1 3 (1986) 693.

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