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THE POSSIBLE ORDERED STRUCTURES OF ICE Ih

R. Howe

To cite this version:

R. Howe. THE POSSIBLE ORDERED STRUCTURES OF ICE Ih. Journal de Physique Colloques, 1987, 48 (C1), pp.C1-599-C1-604. �10.1051/jphyscol:1987182�. �jpa-00226447�

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JOURNAL D E PHYSIQUE

C o l l o q u e C1, s u p p l 6 m e n t au n o 3, Tome 48, m a r s 1987

THE POSSIBLE ORDERED STRUCTURES OF I C E Ih

R . HOWE

D e p a r t m e n t of P h y s i c s , U n i v e r s i t y of B i r m i n g h a m , G B - B i r m i n g h a m B15 ZTT, re at-~ritain

R6sum6 - En considerant l a r e c e n t e decouverte de l a g l a c e X I , l a forme ordonnee de l a g l a c e I h , il e s t i n t e r e s s a n t d S 6 t u d i e r l e s p o s s i b i l i t e s dlarrangement protoniques dans c e t t e phase Ih. Ce t r a v a i l p r e s e n t e une analyse syst6matique des 17 s t r u c t u r e s ordonn6es q u l i l e s t p o s s i b l e de former & p a r t i r de l a g l a c e Ih en gardant l a c e l l u l e de base & 8 mol6cules. Les p o s s i b i i t 6 s avec une c e l l u l e de 12 mol6cules ef quelques c a s p a r t i c u l i e r s d l o r d r e p a r t i e l s o n t a u s s i envisag6s.

Abstract - I n view of t h e r e c e n t discovery of i c e X I , t h e ordered form of i c e I h , it i s of i n t e r e s t t o consider t h e possible arrangements of protons i n t h e phase. This paper p r e s e n t s a systematic a n a l y s i s of t h e 17 ordered s t r u c t u r e s which it i s p o s - s i b l e t o form from i c e I h assuming a u n i t c e l l of 8 molecules. The p o s s i b i l i t i e s w i t h a c e l l of 12 molecules and s p e c i a l cases of p a r t i a l ordering a r e a l s o considered.

INTRODUCTION

The arrangement of t h e protons i n i c e I h has been of i n t e r e s t f o r many y e a r s , and s e v e r a l attempts have been made t o propose ordered s t r u c t u r e s , e i t h e r a s an a l t e r n a t i v e t o t h e s t a t i s t i c a l model a t high temperatures, o r a s a d i s t i n c t phase which should be a t t a i n e d a t low temperatures. , t h i s i n t e r e s t has been s t i m u l a t e d by t h e observation by c a l o r i m e t r i c me%zt%] of a f i r s t - o r d e r t r a n s i t i o n a t 72 K, c a t a l y s e d by doping with KOH. The b e l i e f t h a t t h i s i s t h e long-sought t r a n s i t i o n t o t h e ordered phase, i c e X I 121, i s supported by d i e l e c t r i c measurements (31 and a l s o by a neutron d i f f r a c t i o n study 141 of p a r t i a l l y

transformed D20 i c e t h e r e s u l t s o f which a r e c o n s i s t e n t with a p o l a r s t r u c t u r e of space group Cmcll a s proposed by Kamb [5] f o r t h e ordered phase.

The aim o f t h i s paper i s t o p r e s e n t a systematic a n a l y s i s of t h e p o s s i b l e ordered s t r u c t u r e s with u n i t c e l l s o f f o u r , e i g h t , and ( i n summary) twelve molecules. The Bernal-Fowler r u l e s [6] - two protons p e r oxygen and one p e r bond

- a r e assumed t o be s t r i c t l y obeyed, b u t no c o n s i d e r a t i o n s of energy o r s t a b i l i t y have been taken i n t o account.

THE FOUR-MOLECULE CELL

I f t h e protons a r e ignored, t h e s t r u c t u r e of i c e I h has t h e hexagonal symmetry o f t h e space group ~ 6 ~ / m m c , with f o u r oxygen atoms i n t h e u n i t c e l l . I n t h e s t a t i s t i c a l model, t h i s symmetry i s n o t broken by t h e a d d i t i o n o f t h e protons, a s t h e two proton s i t e s on each bond a r e c r y s t a l l o g r a p h i c a l l y e q u i v a l e n t and each is considered t o be half-occu&ed. Any attempt t o o r d e r t h e protons i n t h e four- molecule c e l l w i l l r e s u l t i n t h e l o s s both o f t h e c e n t r e of symmetry and, u n l e s s o r d e r i n g occurs i n t h e c-axis d i r e c t i o n only, of t h e hexagonal symmetry of t h e d i s o r d e r e d s t r u c t u r e .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987182

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C1-600 JOURNAL DE PHYSIQUE

It i s p o s s i b l e t o f i n d arrangements of protons which r e t a i n t h e t r a n s l a t i o n a l symmetry of t h e four-molecule c e l l while s a t i s f y i n g t h e Bernal-Fowler r u l e s . There i s only one arrangement of the protons i n one puckered (0001) l a y e r of t h e c e l l which maintains t h e t r a n s l a t i o n a l symmetry. I n such an ordered l a y e r , t h e u n i t c e l l can be chosen i n any of three o r i e n t a t i o n s , a s shown i n Figure 1. '?'he a l t e r n a t e l a y e r can a l s o have any of t h r e e o r i e n t a t i o n s . There a r e t h e r e f o r e two d i s t i n c t s t r u c t u r e s possible with the four-molecule c e l l , of which one, with t h e mirror-axes of t h e l a y e r s a n t i p a r a l l e l , has t h e orthorhombic space group CmcZ1. The o t h e r , where t h e mirror axes of the l a y e r s make an angle of 60°, i s monoclinic, with space group Cc. Both these s t r u c t u r e s a r e conventionally represented on an eight-molecule centred c e l l (Figure 2 ) .

It i s i n t e r e s t i n g t o note t h a t t h e four-molecule c e l l permits only p o l a r ordered s t r u c t u r e s . I t i s t h e r e f o r e misleading t o attempt, a s has been done f o r example by Minagawa [7], t o p r e d i c t on the b a s i s of c a l c u l a t i o n s using t h i s c e l l the conditions f o r s t a b i l i t y of an a n t i f e r r o e l e c t r i c s t r u c t u r e .

Figure 1 Ordered l a y e r of four-molecule s t r u c t u r e with t h r e e choices of u n i t c e l l .

Figure 2 Oxygens and bonds involved i n t h e e i g h t - molecule c e l l .

THE EIGHT-MOLECULE CELL

I f t h e orthorhombic t r a n s l a t i o n a l symmetry of t h e eight-molecule c e l l i s preserved, o t h e r p o s s i b l e ordered s t r u c t u r e s can be found, a s shown i n Figure 3 , i n which each rectangle represents t h e upper (001) l a y e r . For t h e purposes of t h i s a n a l y s i s , it i s convenient t o consider t h e c-axis a s v e r t i c a l and t h e puckered s h e e t s of t h e i c e s t r u c t u r e a s horizontal l a y e r s , of which t h e r e a r e two i n t h e u n i t c e l l . Here, and subsequently, t h e oxygen'site a t t h e l e f t of t h e 'upper' l a y e r i s considered t o be a t t h e f o o t of a c-axis bond connected t o t h e l a y e r above, while t h e oxygen a t t h e r i g h t i s a t t h e upper end of a c-axis bond connected t o t h e l a y e r below. I n t h e 'lower' l a y e r t h i s i s reversed and t h e s l a n t of each oblique bond i s opposite t o t h a t of t h e bond immediately above it. Those protons which a r e

independently f i x e d a r e shown i n Figure 3 by small f i l l e d c i r c l e s : t h e small open c i r c l e s represent protons whose p o s i t i o n s a r e f i x e d a s t h e r e s u l t of t h e choice of t h e o t h e r s . The f i r s t column shows t h e choice of a proton on the c-axis bond s i t e a t l e f t : t h e second column shows t h e t h r e e possible choices f o r t h e p o s i t i o n of t h e second occupied c-axis s i t e i n t h e l a y e r . In t h e t h i r d column t h e p o s s i b l e ways of completing t h e molecule on t h e l e f t a r e shown f o r each configuration of t h e c-axis bonds. I n t h r e e cases, l a b e l l e d 3 , 6 and 9, t h e Bernal-Fowler r u l e s and t h e boundary conditions of t h e c e l l f i x t h e p o s i t i o n s of a l l t h e remaining protons i n t h e l a y e r . I n t h e o t h e r cases, two o r i e n t a t i o n s of t h e molecule a t r i g h t a r e possible: once one of these has been chosen a l l o t h e r protons a r e fixed. The configurations r e s u l t i n g from these choices a r e shown i n t h e fourth column. The diagram is s o arranged t h a t each of t h e t h r e e s e c t i o n s defined by t h e choice of c-axis bonds has complete mirror symmetry.

There a r e thus 13 f i n a l arrangements of a s i n g l e layer: t h e r e a r e of course a f u r t h e r 13 p o s s i b l e , each corresponding t o one of those shown, with a l l occupied proton s i t e s exchanged with empty s i t e s . These 'converse' arrangements do not however increase t h e number of unique arrangements of t h e extended l a y e r , of which

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there a r e five: 2 , 3 and 4 a r e a l l equivalent by r o t a t i o n through 120°; 1 and 5, 10 and 1 3 , and 11 and 1 2 a r e mirror-image p a i r s ; 6 , 7 , 8 and 9 a r e a l l r e l a t e d by r o t a t i o n and r e f l e c t i o n .

10

11

12

13

Figure 3 Generation of 1 3 upper layers of t h e eight-molecule c e l l .

For each upper layer shown, a lower layer may be constructed i n projection by the same process as the converse layers, remembering however t h a t i n three dimensions the bonds s l a n t i n the opposite sense. Each lower layer so constructed w i l l be designated by t h e same number a s i t s corresponding upper layer. Provided t h a t t h e v e r t i c a l bonds a r e compatible, any upper layer may be combined with any lower layer.

There a r e thus 5 x 5 + 4 x 4 + 4 x 4 = 57 possible combinations of t h e upper layers shown with the corresponding lower layers. However, r o t a t i o n s and

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Cl-602 JOURNAL DE PHYSIQUE

r e f l e c t i o n s s e v e r e l y r e s t r i c t t h e number o f unique s t r u c t u r e s : f o r example, t h e combination o f upper l a y e r s 2, 3 and 4 w i t h t h e i r a p p r o p r i a t e lower l a y e r s a l l g i v e t h e same s t r u c t u r e , denoted by F i n Table 1. S i m i l a r arguments apply t o o t h e r groups o f combinations: t h e 1 7 unique s t r u c t u r e s and t h e ways i n which t h e y may b e formed a r e shown i n Table 1, where t h e upper l a y e r s a r e shown a l o n g t h e t o p l i n e and t h e lower l a y e r s down t h e l e f t - h a n d column. The s t r u c t u r e s a r e denoted by l e t t e r s A-S. No combinations o f l a y e r s shown i n F i g u r e 3 w i t h lower l a y e r s from t h e

'converse' s e t a r e p e r m i t t e d , a s by t h e n a t u r e o f t h e s e l a y e r s t h e c - a x i s bonds w i l l n o t be compatible.

Table 1 Combinations o f t h e upper and lower l a y e r s 1-13 o f t h e eight-molecule c e l l t o q i v e t h e 1 7 s t r u c t u r e s A-S.

1 A B C D E 6 H J K L l O M N P Q

2 B F G G D 7 J H L K 1 1 N R S P

3 C G F G C 8 K L H J 1 2 P S R N

4 D G G F B 9 L K J H 1 3 Q P N M

5 E D C B A

The 1 7 s t r u c t u r e s , w i t h t h e i r space groups and p o l a r o r a n t i f e r r o e l e c t r i c n a t u r e , a r e summarized i n Table 2 , where t h e l e t t e r s A-S r e f e r t o Table 1.

F, t h e s i m p l e s t and most symmetrical o f t h e s e s t r u c t u r e s , i s t h e C m ~ 2 ~ s t r u c t u r e d i s c u s s e d above. G i s t h e o n l y o t h e r p o s s i b l e c e n t r e d s t r u c t u r e .

M i s t h e s t r u c t u r e proposed by Owston [8] on t h e grounds t h a t t h e p r o t o n s o f each molecule a r e a s f a r a s p o s s i b l e from t h o s e o f t h e neighbouring molecules. I n t h a t p a p e r , t h e s t r u c t u r e was i n c o r r e c t l y a s s i g n e d t o t h e monoclinic space group PC.

The same s t r u c t u r e , w i t h i t s c o r r e c t space roup Pnaal, h a s a l s o been proposed on s i m i l a r grounds by Davidson and Morokuma [9(1.

Q and R a r e i n t e r e s t i n g i n t h a t , a l t h o u g h n o t centrosymmetric, t h e y a r e g e o m e t r i c a l l y non-polar : a l l t h e o t h e r s t r u c t u r e s belong t o p o l a r space groups.

THE TWELVE-MOLECULE CELL

The o r d e r e d s t r u c t u r e proposed by Bernal and Fowler [63 h a s a c e l l o f twelve molecules, w i t h t h e hexagonal space group PE3cm. This i s t h e s m a l l e s t c e l l on whichan o r d e r e d s t r u c t u r e w i t h hexagonal symmetry can be found. I t i s p o s s i b l e t o c a r r y o u t a n a n a l y s i s o f o t h e r s t r u c t u r e s , w i t h t h e t r a n s l a t i o n a l b u t n o t t h e r o t a t i o n a l symmetryof t h e twelve-molecule c e l l , i n a s i m i l a r manner t o t h a t g i v e n above f o r t h e e i g h t - molecule c e l l . This y i e l d s 5 t y p e s o f l a y e r , o f which one i s i d e n t i c a l t o 2/3/4 i n Figure 3 and one i s t h a t used by Bernal and Fowler. The remaining t h r e e l a y e r s havetwo c-axis bonds i n one o r i e n t a t i o n f o r every one i n t h e o p p o s i t e o r i e n t a t i o n : two a r e l a y e r s with no symmetry while t h e t h i r d h a s a m i r r o r a x i s . The p o s s i b l e combinationsof t h e s e l a y e r s g i v e a t o t a l o f 1 5 d i s t i n c t s t r u c t u r e s i n c l u d i n g F, G and t h e Bernal- Fowler s t r u c t u r e : most o f t h e r e s t a r e o f low symmetry.

PARTIAL ORDERING

T h e p o s s i b i l i t y o f p a r t i a l o r d e r i n g a l s o h a s t o b e t a k e n i n t o a c c o u n t . T h i s can t a k e a v a r i e t y o f forms. Most simply, and probably most r e a l i s t i c a l l y , t h e r e i s t h e p o s s i b i l i t y t h a t , w h i l e some s i t e s which would b e occupied i n a f u l l y o r d e r e d s t a t e a r e v a c a n t , and v i c e v e r s a , t h e r e i s a b i a s towards t h e ' c o r r e c t ' s i t e s .

C r y s t a l l o g r a p h i c a l l y , t h i s would r e s u l t i n any s u p e r l a t t i c e r e f l e c t i o n s a p p r o p r i a t e t o t h e f u l l y o r d e r e d s t a t e b e i n g p r e s e n t b u t weak.

I t i s a l s o c o n c e i v a b l e t h a t some s i t e s should be completely o r d e r e d , f o r example t h e c - a x i s bonds o n l y , w h i l e t h e o t h e r s a r e d i s o r d e r e d s u b j e c t t o t h e Bernal-Fowler r u l e s . Such a c a s e was proposed by Rundle [lo], i n an a t t e m p t t o

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account f o r t h e apparently polar behaviour of some i c e specimens. In a s i m i l a r ' fashion, a d i r e c t i o n i n t h e plane may be chosen t o be completely ordered while leaving d i s o r d e r i n t h e c-axis bonds and i n t h e o t h e r d i r e c t i o n s i n t h e plane. Such configurations would have much lower r e s i d u a l entropy than t h e f u l l y disordered s t a t e .

Table 2 Symmetryandpolar p r o p e r t i e s of t h e 17 eight-molecule s t r u c t u r e s .

Dipole Dipole moment moment p a r a l l e l perpendicular t o c t o c

A Pcazl Yes No

P1 Yes Yes

C PC Yes yes

Pl Yes Yes

E PnaZl Yes Yes

F CmcPl Yes No

G Cc yes yes

H P21 No No

J P21 No Yes

K PC NO yes

L PC No Yes

N P21 No Yes

P P21 No Yes

S PnaZl No Yes

F i n a l l y , i t i s p o s s i b l e t o imagine a s t r u c t u r e i n which t h e bonds i n two d i r e c t i o n s a r e f u l l y ordered, dividing t h e l a t t i c e i n t o an a r r a y of l i n e a r chains each of which can be independently o r i e n t e d i n e i t h e r of two senses. Such a s i t u a t i o n would leave only a very small r e s i d u a l entropy. Some, though n o t a l l , such s t r u c t u r e s have a higher symmetry than t h e corresponding f u l l y ordered s t a t e . CONCLUSION

The possible arrangements of protons i n i c e X I $ssuming c e r t a i n u n i t c e l l s of four, e i g h t o r twelve molecules have been i n v e s t i g a t e d and c l a s s i f i e d . The simplest f u l l y ordered s t r u c t u r e , denoted by 'F' i n t h i s paper, i s t h a t proposed by Kamb [6].

However, t h e p o s s i b i l i t y t h a t some of t h e o t h e r s t r u c t u r e s suggested might have lower energy cannot y e t be ruled out.

ACKNOWLEDGEMENTS

This work was c a r r i e d out during t h e tenure of a research studentship from t h e Science and Engineering Research Council. The author wi,shes t o acknowledge many h e l p f u l discussions with D r R. W. Whitworth, D r A. I . M. Rae, D r R. C. Ward and Professor A. J. Leadbetter.

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JOURNAL DE PHYSIQUE

REFERENCES

[l] TAJIMA, Y . , MATSUO, T., and SUGA, H . , Nature, Lond. 299 (1982) 811-812.

TAJIMA, Y . , MATSUO, T., and SUGA, H., J. Phys. Chem. S o l i d s 5 (1984) 11351144.

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MATSUO, T., TAJIMA, Y., and SUGA, H., J. Phys. Chem. S o l i d s 5 (1986) 165-173- KAWADA, S., and DCHATA, H . , J. Phys. Soc. Japan 54 (1985) 477-479

LEADBETTER, A. J . , WARD, R. C., CLARK, J. W., TUCKER, P. A., MATSUO, T., and SUGA, H., J. Chem. Phys. 82 (1985) 424-428.

[s] KAMB, B., ' P h y s i c s and Chemistry o f I c e ' , Eds E. Whalley, S. J. Jones and L. W. Gold, Royal S o c i e t y o f Canada, Ottawa f1973) 28-41.

BERNAL, J. D., and FOWLER, R. H., J. Chem. Phys. 1 (1933) 515-548.

MINAGAWA, I . , J. Phys. Soc. Japan 50 (1981) 3669-3676 OWSTON, P. G., J. Chim. Phys. 50 (1953) C13-~18.

DAVIDSON, E. R., and MOROKUMA, K . , J. Chem. Phys. 81 (1984) 3741-3742.

RUMILE, R. E., J. ~ h y s . Chem. 59 (1955) 680-682.

COMMENTS

Q u e s t i o n of T. MATSUO

It seems t h a t an i n f i n i t e l y l o n g c h a i n of d i s p l a c e d hydrogen atoms is r e q u i r e d i f one wants t o i n t r o d u c e p a r t i a l d i s o r d e r i n t h e ordered s t r u c t u r e s . Is it p o s s i b l e t o do t h a t without changing t h e i r s p a c e groups ?

Answer :

I n t h e case of some of t h e s t r u c t u r e s d e s c r i b e d , it would be p o s s i b l e t o i n c r e a s e t h e symmetry of t h e u n i t c e l l by having series of much c h a i r s a l l s t r a i g h t and p a r a l l e l t o one s i d e of t h e c e l l , w i t h a random series of o r i e n t a t i o n s . I n t h e c a s e o f structure "FW, however, t h e space group would n o t be changed as its symmetry is a l r e a d y as high as p o s s i b l e f o r t h i s u n i t cell.

A c h a i n o r c h a i n s of r e v e r s e d bonds which w a s n o t s t r a i g h t would have no e f f e c t on t h e o v e r a l l s y m m e t r y o f t h e s t r u c t u r e e x c e p t s l i g h t l y t o w h e a k e n a n y c r y s t a l l o g r a p h i c r e f l e c t i o n s due t o t h e lowering o f symmetry caused by t h e o r d e r i n g .

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