THE "LOCALLY-STRUCTURED TRANSIENT GEL" MODEL OF WATER STRUCTURE

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MODEL OF WATER STRUCTURE

H. Stanley, R. Blumberg, A. Geiger, P. Mausbach, J. Teixeira

To cite this version:

H. Stanley, R. Blumberg, A. Geiger, P. Mausbach, J. Teixeira. THE ”LOCALLY-STRUCTURED TRANSIENT GEL” MODEL OF WATER STRUCTURE. Journal de Physique Colloques, 1984, 45 (C7), pp.C7-3-C7-12. �10.1051/jphyscol:1984701�. �jpa-00224262�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n09, Tome 45, septembre 1984 page C7-3

THE "LOCALLY-STRUCTURED TRANSIENT G E L O M O D E L O F WATER STRUCTURE

H . E . S t a n l e y , R.L. Blurnberg, A . ~ e i ~ e r * , P . Mausbach* and J . ~ e i x e i r a * * Center for PoZymer S t u d i e s and Department o f Physics, Boston U n i v e r s i t y , L$ston,MA 02215, U.S.A.

I n s t i t u t für PhysikaZische Chemie, Rheinisch-WestfiiZische Technische HochschuZe, 0-51 00 Aachen, F.R. G.

** Laboratoire Léon BriZZouin, CEN SacZay, 91 191 @if-sur-Yvette Cedex, France Résumé

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c a r a c t é r i s é p a r l a présence de r é g i o n s localement s t r u c t u r é e s de dimension c a r a c t é r i s t i q u e l i n é a i r e de l ' o r d r e de 8 A .

A b s t r a c t - This t a l k r e v i e w s b r i e f l y s o m e o f t h e i n f o r m a t i o n t h a t c o m e s t o l i g h t c o n c e r n i n g t h e s t r u c t u r e of l i q u i d water when one pays a t t e n t i o n t o t h e con- n e c t i v i t y p r o p e r t i e s of t h e hydrogen bonds. The evidence g e n e r a l l y s u p p o r t s t h e p i c t u r e t h a t l i q u i d water i s a " t r a n s i e n t g e l " c h a r a c t e r i z e d by t h e presence of l o c a l l y - s t r u c t u r e d r e g i o n s of a c h a r a c t e r i s t i c l i n e a r dimension of about 8 A.

It i s n o t easy t o g i v e an elementary opening t a l k t o a meeting populated by experts! Hence I've chosen t o g i v e a r a t h e r b r i e f and p a r o c h i a l overview of t h e p i c t u r e of water s t r u c t u r e t h a t has been evolving--1argely i n c o l l a b o r a t i o n with those who k i n d l y consented t o j o i n me a s CO-authors. The d e t a i l s t h a t 1 w i l l omit can be found i n t h e o r i g i n a l papers.l-14 It i s a p l e a s u r e t o acknowledge a t t h e o u t s e t i n t e r a c t i o n s w i t h L. Bosio, P. G. de Gennes, M. Mezei, P. Papon, F. H.

S t i l l i n g e r , and e s p e c i a l l y C. A. Angell.

1 s h a l l organize t h e t a l k a s follows. F i r s t 1 w i l l p r e s e n t , very b r i e f l y , a review of those puzzling f e a t u r e s of l i q u i d water t h a t most a t t r a c t t h e f a s c i n a t i o n of a p h y s i c i s t . Then 1'11 p r e s e n t some c l u e s concerning t h o s e f e a t u r e s . T h i r d l y 1'11 d e s c r i b e t h e " t r a n s i e n t g e l " model, and l a s t l y 1'11 g i v e some t e s t s of t h e t r a n s i e n t g e l nodel on computer water and on r e a l water.

The o v e r a l l p i c t u r e t h a t w i l l emerge i s t h a t water may be viewed a s a g e l on time s c a l e s s m a l l e r than t h e bond l i f e ; i . e , , i f we took a photograph w i t h a very s h o r t s h u t t e r speed, water and j e l l o would have something i n common. This g e l i s c h a r a c t e r i z e d by many s m a l l , l o c a l l y - s t r u c t u r e d r e g i o n s , whose c h a r a c t e r i s t i c l i n e a r dimension i s roughly 8 A.

1. PUZZLE

We s h a l l s t a r t a t t h e beginning w i t h t h e puzzle of l i q u i d water. Three s t a t i c p r o p e r t i e s of water a r e p a r t i c u l a r l y puzzling t o a p h y s i c i s t . The f i r s t of t h e s e concerns t h e f l u c t u a t i o n s i n s p e c i f i c volume ( " d e n s i t y " ) , which a r e p r o p o r t i o n a l t o the i s o t h e r m a l c o m p r e s s i b i l i t y . For most l i q u i d s t h i s f u n c t i o n d e c r e a s e s a s t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984701

temperature is lowered. This makes intuitive sense: the temperature "drives"

density fluctuations. For water, the reverse is true: as T decreases, KT increases! This is not a 1% effect: KT increases by almost a factor of two from its minimum at about 50C to its value at the lowest attainable temperatures, about -30C. Thus the explanation of this effect requires some mechanism that operates from +50 to -30!

Analogous behavior occurs for the entropy fluctuations, which are proportional to the constant-pressure specific heat Cp. These are anomalous in the same two respects: the specific heat is larger in magnitude than for most liquids (giving rise to its utility as a coolant), and when one lowers the temperature, the entropy fluctuations increase. This is also counter-intuitive, since we always imagine that the temperature "drives" entropy fluctuations. Again, this is not a small effect:

the specific heat at the lowest attainable temperatures is almost a factor of two larger than its value at high temperatures.

One might expect that the thermal expansivity dp = (dV/dT)p could not be negative, since it is proportional to the cross fluctuations of entropy and specific -- volume. When one considers a local region in which the specific volume is larger, then one expects that there are more "arrangements" of the molecules and hence a larger local entropy. Thus the cross fluctuations should be positive. For water the thermal expansivity is anomalous in two respects: firstly, at high temperatures it is only a fraction of its value for typical liquids. Secondly, below about 4C, it is negative!

The dynamic properties are also anomalous. For example, if we examine the dependence upon 1/T of the logarithms of characteristic times such as the

dielectric relaxation time, inverse diffusion coefficient, or shear viscosity, then we find that these are roughly linear at high T but increase much faster than linearly as T is decreased. Pressure increases the viscosity at high temperatures as for most liquids, but has the opposite effect for T below about 20C.

II. CLUE?

The list of strange properties could occupy the entire talk. Perhaps even more striking than the anomalies themselves is the fact that they seem to be "amplified"

as one reduces T below the melting temperature. What could be the physical mechanism underlying this amplification?

At first sight, one might seek something special about the supercooled state of matter. This approach has been taken by some workers in an effort to explain the

"mysterious behavior of supercooled water." However it is somewhat misleading, since the acceleration in the anomalous behavior in the supercooled regime is already manifest in the normal regime: if one, e.g., fits the behavior of a function in the normal regime to a polynomial (as Kell has done) and then substitutes temperature values corresponding to the supercooled regime then one finds values of KT remarkably close to the observed behavior (C. A. Angell, private discussions). Thus a major constraint on any possible explanation of the

supercooled behavior i s t h a t g n u s t be based on phenomena t h a t a r e o p e r a t i v e i n t h e normal regime. The same physics i s c o n t r o l l i n g t h e 50C minimum i n KT a s t h e

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f a c t o r of two i n c r e a s e i n KT i n t h e supercooled regime.

What could t h i s mechanism be? S e v e r a l c l u e s p r e s e n t themselves. Perhaps t h e foremost c l u e i s t h a t a d i s t i n g u i s h i n g f e a t u r e of water i s t h e presence of a remarkably high degree of hydrogen bonding when compared t o o t h e r l i q u i d s . I t has been known s i n c e t h e c l a s s i c o b s e r v a t i o n s of Pauling t h a t when i c e m e l t s a t OC, only a s m a l l f r a c t i o n (perhaps 20%) of t h e hydrogen bonds break. If one c o n t i n u e s t o i n c r e a s e t h e temperature, t h e f r a c t i o n of i n t a c t bonds, pg, d e c r e a s e s very slowly--a handy mnemonic formula s u g g e s t s t h a t f o r every 2.5C of h e a t i n g , an a d d i t i o n a l 1% of t h e bonds break. S i m i l a r l y , i f one s u p e r c o o l s water, pB

i n c r e a s e s very slowly--the same 1% f o r every 2.5C of supercooling. The problem, t h e n , is t o go from a v e r y smoothly and slowly varying microscopic parameter--PB--to the d r a m a t i c a l l y varying macroscopic parameters mentioned above. We Say t h a t t h e hydrogen bonds l i n k t h e water molecules i n t o a network o r "gel" because t h e g e l a t i o n threshold f o r a 3-dimensional system i s w e l l below 80%. How can we be c e r t a i n t h a t the hydrogen bond network i s r e s p o n s i b l e f o r t h e anomalous p r o p e r t i e s of water? We can do t h i n g s t o water t h a t a r e known t o weaken t h e network, and s e e t h e anomalies go away. For example, we can apply h y d r o s t a t i c p r e s s u r e , o r i n t r o d u c e

hydrogen-bonding i m p u r i t i e s such a s H202. A i t e r n a t i v e l y , we may r e p l a c e p a r t o r a l 1 of t h e Hz0 by D20 and f i n d t h a t t h e anomalies become s t r o n g e r .

III. LOCALLY-STRUCTUKED TRANSIENT GEL MODEL

The q u a l i t a t i v e p i c t u r e of water s t r u c t u r e f i r s t proposed i n Refs. 1-3 i s receiving increased q u a n t i t a t i v e s u p p o r t . In i t s s i m p l e s t v e r s i o n , one imagines t h a t water molecules a r e l i n k e d t o g e t h e r by hydrogen bonds. T y p i c a l l y t h e r e a r e a maximum of f o u r bonds per molecule. The mean l i f e t i m e of a bond i s about 1 p s , but a t any g i v e n i n s t a n t t h e r e w i l l be about 80% of t h e bonds i n t a c t .

Ail e s t i m a t e s of t h e g e l a t i o n t h r e s h o l d f o r 3-dimensional systems a r e w e l l below 80% ( f o r example, i n Flory t h e o r y t h e g e l a t i o n t h r e s h o l d occurs a t 33% f o r f o u r - f u n c t i o n a l monomers). Hence water i s c e r t a i n l y above i t s g e l a t i o n t h r e s h o l d ! This f a c t has been noted over t h e y e a r s , but by i t s e l f does n o t e x p l a i n any

anomalies s i n c e thermodynamic f u n c t i o n s a r e not s i n g u l a r a t t h e g e l a t i o n t h r e s h o l d . In f a c t , t h e anomalies i n water occur w e l l above t h e g e l a t i o n t h r e s h o l d . C l e a r l y we need a d d i t i o n a l i n s i g h t . This a d d i t i o n a l i n s i g h t a r i s e s from t h e o b s e r v a t i o n t h a t the l o c a l environment of a water molecule i s c o r r e l a t e d w i t h i t s degree of

bondedness. For example, a n o l e c u l e s i t u a t e d i n a r e g i o n of t h e g e l where a l 1 t h e molecules have f o u r i n t a c t bonds s e e s a d i f f e r e n t l o c a l environment than a molecule i n a r e g i o n of t h e network where most of the molecules have broken bonds, Indeed, the statement t h a t a molecule has f o u r i n t a c t bonds means t h a t t h e r e a r e f o u r o t h e r water molecules l o c a t e d a t t h e c o r r e c t d i s t a n c e and a n g l e r e q u i r e d f o r hydrogen bonding. The l o c a l s p e c i f i c volume would t h u s be expected t o be l a r g e r i n t h e v i c i n i t y of t h i s molecule, compared t o t h e g l o b a l s p e c i f i c volume. S i m i l a r l y , t h e

l o c a l e n t r o p y s h o u l d b e s m a l l e r t h a n t h e g l o b a l e n t r o p y . Hence we h a v e a mechanism t h a t makes a p o s i t i v e c o n t r i b u t i o n t o t h e f l u c t u a t i o n s i n s p e c i f i c volume and e n t r o p y and a n e g a t i v e c o n t r i b u t i o n t o t h e c r o s s f l u c t u a t i o n s . M o r e o v e r , t h i s e f f e c t becomes more pronounced a s t h e t e m p e r a t u r e i s l o w e r e d . The

l o c a l l y - s t r u c t u r e d t r a n s i e n t g e l model would t h e r e f o r e seem t o e x p l a i n t h e p u z z l i n g f e a t u r e s of w a t e r we m e n t i o n e d a b o v e . It a l s o e x p l a i n s t h e e f f e c t o f h y d r o s t a t i c p r e s s u r e o r h y d r o g e n b o n d i n g i m p u r i t i e s : t h e f o r m e r s e r v e t o r e d u c e p g a n d h e n c e t o r e d u c e t h e p r o b a b i l i t y of f i n d i n g l o c a l l y - s t r u c t u r e d r e g i o n s w h i l e t h e l a t t e r s e r v e t o r e d u c e t h e " s t r u c t u r a t i o n " o f t h e l o c a l r e g i o n s s i n c e t h e y l a c k t h e s t r o n g d i r e c t i o n a l i t y o f t h e t e t r a h e d r a l bonds i n w a t e r ( T a b l e 1 ) . One m o l e c u l e t h a t d o e s have t h i s s t r o n g d i r e c t i o n a l i t y i s D20, a n d f o r heavy w a t e r o n e f i n d s t h a t t h e a n o m a l i e s a r e more pronounced.

TABLE 1 [ f r o m Ref. 31

I V . TESTS

I n t h e r e m a i n d e r o f t h i s t a l k , 1 w i l l m e n t i o n some o f t h e r e c e n t work d e v o t e d t o o t h e r

i m p u r i t i e s

I C 1

t e s t i n g t h e o v e r a l l p i c t u r e o u t l i n e d a b o v e . T h i s work n a t u r a l l y f a l l s i n t o two i m p u r i t y D20

t

t f

K r

c, - ^%P

c l a s s e s , computer s i m u l a t i o n s c a r r i e d o u t f o r r e a l i s t i c m o d e l s of w a t e r and P r e s s u r e

1

+

1

e x p e r i m e n t s c a r r i e d o u t o n r e a l w a t e r .

TESTS ON "COMPUTER WATER"

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A l t h o u g h t h e r e h a v e b e e n many s t u d i e s o f t h e thermodynamic p r o p e r t i e s o f w a t e r , t h e r e h a s b e e n c o n s i d e r a b l y l e s s work o n t h e c o n n e c t i v i t y p e r o p e r t i e s . The most s t r a i g h t f o r w a r d p r o p e r t y t o m e a s u r e i s f j , t h e f r a c t i o n o f m o l e c u l e s w i t h j i n t a c t bonds. T h i s f u n c t i o n d o e s n o t p r o v i d e a c o m p l e t e d e s c r i p t i o n of t h e

c o n n e c t i v i t y , b u t i s n o n e t h e l e s s u s e f u l . We h a v e found t h a t f j i s w e l l d e s c r i b e d by a s i m p l e b i n o m i a l f o r m u l a , i n d i c a t i n g t h e r e f o r e t h a t t h e h y d r o g e n bonds a r e t o a good a p p r o x i m a t i o n randomly i n t a c t o r b r o k e n . T h i s f i n d i n g seems t o b e i n d e p e n d e n t of what model i s u s e d . F o r e x a m p l e , Our work h a s used t h e famous R a h m a n - S t i l l i n g e r t a p e s o n ST2 w a t e r a s i n p u t , w h i l e Mezei and B e v e r i d g e h a s c a r r i e d o u t a n a l o g o u s c a l c u l a t i o n s of t h e f j u s i n g o t h e r models--and e v e n o t h e r d e f i n i t i o n s o f a hydrogen bond (we u s e a n e n e r g e t i c d e f i n i t i o n , w h i l e t h e y u s e b o t h e n e r g e t i c a n d g e o n e t r i c d e f i n i t i o n s . I n d e e d , i n t h e l i t e r a t u r e o f MD ( a n d MC) c a l c u l a t i o n s o n w a t e r , one f i n d s many e x a m p l e s o f h i s t o g r a m s o f f j p l o t t e d a g a i n s t j . We f i n d t h a t t h e s e d a t a a r e w e l l f i t by s i m p l e b i n o m i a l e x p r e s s i o n s ( c f . F i g . 1).

--

"HE

FIG. 1 [Ref. 141

Suppose we a c c e p t t h e p o s s i b i l i t y t h a t t o a z e r o t h o r d e r approximation, we can d e s c r i b e t h e hydrogen bonding a s random--cooperativity e f f e c t s c e r t a i n l y a r e p r e s e n t , but a r e perhaps not so s t r i k i n g l y important a s t o r e q u i r e c o n s i d e r a t i o n a t the p r e s e n t time. Then we can a c t u a l l y o b t a i n a completely q u a n t i t a t i v e "network a n a l y s i s " of t h i s t r a n s i e n t g e l , by simply c a r r y i n g over t o water t h e i d e a s developed by Flory and Stockmayer f o r p o l y f u n c t i o n a l condensation. Here t h e water molecules play t h e r o l e of f o u r - f u n c t i o n a l monomers i n Flory t h o e r y , and t h e

s h o r t - l i v e d hydrogen bonds a r e r e s p o n s i b l e f o r t h e network p r o p e r t i e s . Flory worked out simple a n a l y t i c formulae f o r t h e complete s t a t i s t i c a l d i s t r i b u t i o n - - t h e weight f r a c t i o n W(M,p) of molecules belonging t o an 1I-molecule "network" when a f r a c t i o n pg of t h e hydrogen bonds i s i n t a c t ,

W$~[Flory th e o r y ] = M A ( M ) ~ ~ - ~ ( ~

-

where

i s a combinational f a c t o r . Flory's c a l c u l a t i o n has no a d j u s t a b l e parameters, s o i t s p r e d i c t i o n s can be d i r e c t l y compared w i t h MD c a l c u l a t i o n s on ST2 water. One f i n d s e x c e l l e n t agreement f o r s m a l l networks. For l a r g e r networks, d i s c r e p a n c i e s a r i s e , presumably due t o t h e f a c t t l i a t Flory t h e o r y n e g l e c t s " c y c l e s , " t h e p o s s i b i l i t y t h a t one can form a l o o p o r c y c l e of hydrogen-bonded water molecules.

'Ihese c y c l e s a r e r a r e f o r s m a l l networks but they become i n c r e a s i n g l y important f o r l a r g e r networks--indeed, i t i s d i f f i c u l t t o imagine a network o f , Say, 100

molecules without t h e presence of a t l e a s t a few c y c l e s . In s h o r t , Flory t h e o r y assumes a t r e e - l i k e c o n n e c t i v i t y and t h e r e f o r e should f a i l f o r l a r g e networks. TO o b t a i n b e t t e r agreement f o r l a r g e r networks, we have c a r r i e d o u t e x t e n s i v e

c a l c u l a t i o n s f o r t h e network p r o p e r t i e s f o r g e l a t i o n on an i c e Ih l a t t i c e . We f i n d much b e t t e r agreement w i t h t h e MD s i m u l a t i o n s , but of c o u r s e t h i s does not mean t h a t t h e water molecules a c t u a l l y r e s i d e on a l a t t i c e ! Rather, i t only

s u g g e s t s t h a t t h e c o n n e c t i v i t y p r o p e r t i e s a r e s i m i l a r t o t h o s e of a l a t t i c e . T h i s i s a t o p o l o g i c a l m a t t e r , n o t a g o e m e t r i c a l m a t t e r . There h a s been some c o n f u s i o n on t h i s p o i n t i n p r e v i o u s d e s c r i p t i o n s o f Our work, s o one cannot make t h i s p o i n t t o o e m p h a t i c a l l y : we a n a l y z e t h e t o p o l o g i c a l p r o p e r t i e s o f w a t e r by making r e f e r e n c e t o t h e i c e l a t t i c e , which s a y s n o t h i n g a t a l 1 a b o u t t h e g e o m e t r i c a l p r o p e r t i e s of water. We found e x c e l l e n t agreement between F l o r y t h e o r y and t h e MD s i m u l a t i o n s provided M i s n o t t o o l a r g e ; f o r l a r g e r M t h e s i m u l a t i o n s a g r e e d b e t t e r w i t h c a l c u l a t i o n s o n a l a t t i c e w i t h t h e t o p o l o g i c a l c o n n e c t i v i t y of i c e Ih. Aithough we a r e l i m i t e d by t h e s m a l l s i z e of t h e MD w a t e r s y s t e m , we may p r e d i c t t h a t f o r even l a r g e r M t h e agreement w i t h t h e i c e l a t t i c e c a l c u l a t i o n s w i l l c e r t a i n l y b r e a k d o m . The main p o i n t i s t h a t on a s c a l e of a few A t h e hydrogen bond network o f water i s t o p o l o g i c a l l y s i m i l a r t o t h e I h l a t t i c e .

LOCALLY-STRUCTLRED REGIONS

We mentioned above t h a t t h e t r a n s i e n t g e l of connected w a t e r m o l e c u l e s i s n o t s u f f i c i e n t t o e x p l a i n t h e u n u s u a l b e h a v i o r of l i q u i d w a t e r below about 50C. We a l s o s u g g e s t e d t h a t t h e a n o m a l i e s c o u l d be understood i n terms of l o c a l l y - s t r u c t u r e d r e g i o n s c o n s i s t i n g of four-bonded m o l e c u l e s . For t h i s r e a s o n , we have c a r r i e d o u t t h e a n a l o g o u s " q u a n t i t a t i v e network a n a l y s i s " on t h e four-bonded m o l e c u l e s .

AIthough t y p i c a l l y t h e hydrogen bonding i s s o e x t e n s i v e t h a t w a t e r i s w e l l above t h e g e l a t i o n t h r e s h o l d , t h e f r a c t i o n of four-bonded m o l e c u l e s i s much s m a l l e r - - i f

P = 0.8, t h e n f 4 = 0.4096. Thus t h e four-bonded m o l e c u l e s c a n be n e a r o r even B

below t h e i r p e r c o l a t i o n t h r e s h o l d . We have s t u d i e d i n d e t a i l t h e a n a l o g o u s d i s t r i b u t i o n f u n c t i o n s W: g i v i n g t h e weight f r a c t i o n of four-bonded m o l e c u l e s belonging t o a "patch" o f t h e g e l , a l 1 of whose members a r e connected t o one a n o t h e r and a l 1 of whose members a r e four-bonded. We g e n e r a l i z e d t h e F l o r y t h e o r y t o t h i s problem, w i t h t h e r e s u l t

W: [ F l o r y t h e o r y ] = s ~ ( s ) ~ ~ ~ + l ( l - ^{p3)2s+2. ( 2 )}

In analogy w i t h Our s t u d y o f t h e network f u n c t i o n s Wll, we found e x c e l l e n t agreement between F l o r y t h e o r y and t h e MD s i m u l a t i o n s provided s i s n o t t o o l a r g e ; f o r l a r g e r s t h e s i m u l a t i o n s agreed b e t t e r w i t h c a l c u l a t i o n s on a l a t t i c e w i t h t h e t o p o l o g i c a l c o n n e c t i v i t z of i c e Ih. Thus t h e MD s i m u l a t i o n s s u p p o r t t h e p i c t u r e t h a t w a t e r i s a l o c a l l y - s t r u c t u r e d hydrogen-bonded network w e l l above i t s g e l a t i o n t h r e s h o l d .

What i s needed n e x t i s e v i d e n c e t h a t t h e l o c a l r e g i o n s o f four-bonded m o l e c u l e s have a d i f f e r e n t s p e c i f i c volume. T h i s q u e s t i o n was f i r s t a d d r e s s e d i n t h e

f o l l o w i n g f a s h i o n : a n imaginary b a l l o o n was i n f l a t e d around each w a t e r m o l e c u l e , and t h e number of o t h e r m o l e c u l e s r e s i d i n g i n s i d e t h e b a l l o o n was c a l c u l a t e d . I f t h e l o c a l s p e c i f i c volume around four-bonded m o l e c u l e s were i n d e e d l a r g e r ( a s c o n j e c t u r e d a b o v e ) , t h e n one would e x p e c t t o f i n d fewer m o l e c u l e s i n s i d e t h e

b a l l o o n s c e n t e r e d on four-bonded m o l e c u l e s . T h i s was indeed found t o be t h e c a s e (Fig. 2 ) , w i t h some c a v e a t s f o r t h o s e c a s e s i n which t h e b a l l o o n r a d i u s corresponded t o t h e s e p a r a t i o n d i s t a n c e between n e a r e s t n e i g h b o r s i n an i c e l a t t i c e ( t h e s e d e t a i l s a r e d e s c r i b e d i n Ref. 10).

FIG. 2 [ f rom Ref. IO]

It would be n i c e t o make t h e above c o n c l u s i o n s independent of any d e f i n i t i o n of hydrogen bond, s i n c e t h i s i s somewhat a r b i t r a r y i n I D c a l c u l a t i o n s . Hence we have s t u d i e d t h e r e l a t i o n s h i p between t h e number n of n e i g h b o r s j found w i t h i n a s p h e r e of r a d i u s r c around some r e f e r e n c e molecule i , and t h e sum u i of t h e c o r r e s p o n d i n g p a i r i n t e r a c t i o n e n e r g i e s v i j ,

Here u i i s t h e b i n d i n g e n e r g y of t h e r e f e r e n c e molecule i w i t h r e s p e c t t o i t s n neighbors i n t h e s p h e r e of r a d i u s r c ; i t i s a measure of t h e l o c a l c o n n e c t i v i t y , which i s more g e n e r a l t h a n t h e hydrogen bond p i c t u r e and which a v o i d s t h e a r b i t r a r y d e f i n i t i o n s of a hydrogen bond.

Due t o t h e f l u c t u a t i o n of t h e l o c a l d e n s i t y , we o b s e r v e a r a n g e of numbers n , t h e n e i g h b o r s i n t h e s p h e r e . We c a l c u l a t e d t h e a v e r a g e b i n d i n g e n e r g y a s a f u n c t i o n of t h e number of n e i g h b o r s n

R e s e v a l u e s a r e shown i n Fig. 3 f o r f o u r d i f f e r e n t c h o i c e s r c ; t h e v e r t i c a l b a r s i n d i c a t e t h e mean s q u a r e d e v i a t i o n s from t h e a v e r a g e s u.

These d e v i a t i o n s a r e s m a l l e s t i n t h e c e n t r a l p a r t of t h e g r a p h s , because we f i n d t h o s e numbers of n e i g h b o r s n most f r e q u e n t l y and t h e r e f o r e we have many c o n t r i b u t i o n s t o t h e c o r r e s p o n d i n g a v e r a g e s . Through t h e s e more r e l i a b l e p o i n t s a dashed g u i d a n c e l i n e h a s been drawn. Averages from l e s s t h a n 100 c o n t r i b u t i o n s , which o c c u r a t t h e outermost wings of t h e d i s t r i b u t i o n s ( v e r y h i g h and v e r y low n ) a r e n o t c o n s i d e r e d i n t h e s e g r a p h s .

We s e e t h a t f o r r c = 5.5A t h e a v e r a g e b i n d i n g e n e r g y u d e c r e a s e s w i t h a n i n c r e a s i n g number of n e i g h b o r s w i t h i n t h e s p h e r e of r e g a r d e d i n t e r a c t i o n s . T h i s i s what we would e x p e c t from a "normal" l i q u i d l i k e a Lennard-Jones l i q u i d a t n o t t o o high packing d e n s i t i e s , b e c a u s e t h e a d d i t i o n of a n o t h e r i n t e r a c t i o n p a r t n e r w i l l add a n e g a t i v e ( a t t r a c t i v e ) c o n t r i b u t i o n v i j . However, i n t h e c a s e s rc = 3.5A, 4.5A

and 6 . 5 A , we observe e x a c t l y t h e o p p o s i t e behavior: u i n c r e a s e s w i t h i n c r e a s i n g l o c a l d e n s i t y . This means t h a t a l e s s dense l o c a l arrangement of t h e water

molecules i s e n e r g e t i c a l l y f a v o r a b l e over more dense s t r u c t u r e s ; a behavior t h a t we regard a s t y p i c a l l y " w a t e r l i k e " i s r e l a t e d t o t h e occurrence of t h e anomalies.

FIG. 3 [from Ref. 141

The o b s e r v a t i o n t h a t f o r some c h o i c e s of r c we g e t t h e p i c t u r e of a normal l i q u i d can be explained by t h e o s c i l l a t o r y n a t u r e of t h e p a i r c o r r e l a t i o n f u n c t i o n s , wilich d e s c r i b e t h e l o c a l s t r u c t u r e of water. Figure 3 i n d i c a t e s a decreased l o c a l d e n s i t y around four-bonded water molecules when u s i n g rc = 3.5, 4 . 5 o r 6 . 5 A , wilereas f o r t h e c h o i c e r c = 5 . 5 A no such d i f f e r e n c e c a n be observed.

Thus t h e p r e s e n t r e s u l t s shown i n F i g . 3 confirm and a l s o g e n e r a l i z e our

previous f i n d i n g of 5 c o r r e l a t i o s between i n c r e a s e d c o n n e c t i v i t y and d e c r e a s e d local

d e n s i t y .

--

Furthermore, c o n c e n t r a t i n g on t h e graph f o r rc = 3 . 5 A (a v a l u e which had been used b e f o r e a s t h e l i m i t i n g d i s t a n c e f o r hydrogen bonds), Fig. 3 i n d i c a t e s a marked minimum of u a t n=4. This i n d i c a t e s a g a i n a s t r o n g e n e r g e t i c p e r f e r e n c e f o q four-coordinated l o c a l s t r u c t u r e s .

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TESTS ON REAL WATER

By now you may be convinced t h a t "computer water" conforms q u i t e n i c e l y t o t h e p i c t u r e presented above. What about r e a l water? This i s t h e r e a l c h a l l e n g e !

It i s not easy t o d e s i g n an experiment t h a t i s s e n s i t i v e t o t h e d e t a i l e d hydrogen bond c o n n e c t i v i t y of water. However one can t e s t t h e consequences of t h e e x i s t e n c e of a l o c a l l y - s t r u c t u r e d hydrogen-bonded r e g i o n s of t h e t r a n s i e n t g e l : t h e

c h a r a c t e r i s t i c s i z e of s u c h r e g i o n s i s p r e d i c t e d t o be a b o u t 8A by m o l e c u l a r dynamics c a l c u l a t i o n s , and t h e l o c a l d e n s i t y of w a t e r i s s m a l l e r i n t h e v i c i n i t y of such r e g i o n s . A c c o r d i n g l y , we might a n t i c i p a t e t h a t t h e l e n g t h s c a l e of d e n s i t y f l u c t u a t i o n s would a l s o be a b o u t 8A. T h i s q u a n t i t y can be measured by s m a l l - a n g l e x-ray s c a t t e r i n g . The s t r u c t u r e f a c t o r S ( q ) i s known t o g i v e an a c c u r a t e e s t i m a t e of t h e l e n g t h %(T) f o r f l u i d s n e a r t h e i r c r i t i c a l p o i n t , where 3 (T) becomes q u i t e l a r g e . Indeed, d a t a a r e o f t e n f i t by a s i m p l e L o r e n t z i a n f u n c t i o n

Thus t h e w i d t h o f t h e L o r e n t z i a n i s a d i r e c t measure of t h e i n v e r s e c o r r e l a t i o n l e n g t h , and can be r e a d i l y o b t a i n e d from a n "Ornstein-Zernike-Debye p l o t " i n which 1/S(q) i s p l o t t e d a g a i n s t q2.

E a r l y e x p e r i m e n t s o f S ( q ) i n w a t e r d i d n o t a t t a i n e x t r e m e l y low q o r e x t r e m e l y low T, and no i n c r e a s e of S ( q ) n e a r q=O was observed. Bosio e t a l succeeded i n o b t a i n i n g a c c u r a t e measurements of S ( q ) f o r low q and low T and t h e d a t a i n d i c a t e a c l e a r i n c r e a s e , a s i n d i c a t e d i n F i g . 4. The observed b e h a v i o r can be f i t t o a L o r e n t z i a n , and t h e c h a r a c t e r i s t i c v a l u e of ? from t h e d a t a i s 8A, i n agreement w i t h t h e MD s i m u l a t i o n s t u d i e s .

FIG. 4 [from Ref. 61

In summary, a v a i l a b l e MD s i m u l a t i o n s and d i r e c t e x p e r i m e n t s s u p p o r t t h e p o s s i b i l i t y t h a t l i q u i d w a t e r i s a hydrogen bonded network, c h a r a c t e r i z e d by many t i n y l o c a l l y - s t r u c t u r e d r e g i o n s whose d e n s i t y i s l e s s t h a n t h e g l o b a l d e n s i t y and whose c h a r a c t e r i s t i c l i n e a r s i z e i s r o u ~ h i y 8A a t low t e m p e r a t u r e s .

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