Effective intermolecular potential energy function. Hydrogen-bonded structure of water

1 August 1994 PHYSICS LETTERS A

Physics Letters A 190 (1994) 480-482

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Effective intermolecular potential energy function. Hydrogen-bonded structure of water A.K. Karmakar,

R.N. Joarder

Department of Phystcs, Jadavpur Umversuy, Calcutta 700 032, Indta Received 16 May 1994, accepted for pubhcatxon 26 May 1994 Cornmumcated by J Flouquet

Abstract

The average effective pmr potential energy function between two ne~ghbourmg molecules has been computed from the centre structure factor data of water and this shows an unusual feature at short range due to strong hydrogen bonding The effective potentml is highly temperature dependent and at elevated temperature ~t turns out to be approximately hke that for simple hqmds The effecUve potentml can be represented satisfactorily by the square well model

It is well known that liquid water is c o m p o s e d o f molecules that are strongly influenced by hydrogen b o n d i n g Though a tetrahedral o r d e n n g o f neighbourlng molecules is generally accepted, the detailed features o f the local structure have not been decisively established as yet from experimental studies [ 1 ] C o m p u t e r simulations [2 ] based on a n u m b e r o f recently p r o p o s e d site-site potential models give general agreement with both X-ray and neutron measurements but reveal systematic differences that are d e p e n d e n t on the type o f the interaction potential used Further, the b e h a v l o u r at high t e m p e r a t u r e and pressure where mostly all the H - b o n d s are broken, though conceptually simple, the detailed form o f the two-body potential is not fully defined [ 3 ] It is also recognised for long that the double m a x i m a in the Xray structure function o f water [4] is a unique characteristic o f low t e m p e r a t u r e water not normally found for any insulating liquid The indication is clear that the effective pair potential energy function for low t e m p e r a t u r e water molecules is probably very different [4 ] and to this p r o b l e m we have addressed our attention

We have used the c e n t r e - c e n t r e structure factors c o m p u t e d earher [ 5 ] from the c o m b i n e d analysis o f N a r t e n ' s X-ray data [4] to determine the effective intermolecular potential using the accurate m o d i f i e d hypernetted chain ( M H N C ) theory [6] The ass u m p t i o n o f the existence o f an effective pair potential, ~eff(r) allows one to write the pair distribution function, g(r) as

g(r)=exp[-fl~¢ff(r)+h(r)-c(r)+B(r)]

(1)

where h ( r ) = g ( t ) - 1, c(r) and B(r) are the direct correlation function and "bridge function". We have g ( r ) and c(t ) obtainable from centre structure factor data. B(r) is not known It is however known that B(r)~-Bhs(r) for small r where Bhs(r, r/) iS the "bridge" function o f the hard sphere system F o r large r, since c(r) ~ -flO¢ff(r) it is easy to see that

B(r) ~--Bhs(r, q) + ½[ h 2 ( r ) -h~s(r) ]

(2)

In the M H N C for all r one assumes B(r) to be equal to Bhs ( r, r/) with a packing density r/given by the Lado criterion [ 7 ],

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A K Karmakar, R N Joarder /Phystcs Letters A 190 (1994) 480-482

[ g ( r ) --ghs(r, 17) ]

8Bhs(r, q) 517

dr=O

(3)

Thus, In the M H N C Ceff(r) IS given by cell(r) = --kB T[ln g(r) +c(r) -h(r) --Bh~(r, rl) ] (4) The computed O~ff(r) is shown in Fig. 1 For comparison we have given results through other approximate theories e.g., the Percus-Yevlck (PY), the hypernetted chain ( H N C ) , the mean spherical approximation (MSA) [6] and the Rao-Joarder (RJ) method based on the consistency of pressure between the pressure and compressibility equations [8 ] The basic feature of the potential is the same in all the methods having a positive m l m m u m at r ~ 2.8 A. Beyond r ~ 4 A all the methods yield almost similar results It is however very different from the effective pair potential in simple liquids. The double minima in the potential are possibly due to a short-range hydrogen bonding interaction between nelghbouring molecules The X-ray data on deuterated water [ 9 ] also yields a more or less similar result. To see the effect of H-bond breaking at high temperature we have obtained ¢~ff(r) at two higher temperatures and these are shown in the same figure. The potential en0 003

0 002

0001

"~ 0 000

/ /

-0

001

- 0 002 2

4

6 r(/~)

8

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Fig 1 Calculated ¢~r(r) by different approximate theory Solid line MSA, crosses HNC, plusses PY, open dots MHNC, solid dots R-J (25"C), long-dashed hne MSA (150"C), short-dashed line MSA (200"C)

ergy curve is highly temperature dependent and this is primarily because the H-bonded interaction is temperature dependent. The effective core diameter gradually decreases with temperature. At higher temperature the potential energy function gradually turns to be more like a normal insulating liquid though there a substantial difference exists. In order to test the general nature of the effective pair potential we approximate the H-bonding feature of the effective pair interaction of room temperature water by a model square-well (SW) function (repulsive) The representation of the H-bonding effect by the SW model (attractive) is however not new [ 10] In the simplest sense we can write the O - O direct correlation between two nelghbounng molecules (since oxygen is at the centre of the molecule this is also the centre-centre c(r) ) as the sum of a reference hard core direct correlation and a SW term due to Hbonding. Thus in Q-space we write

Coo(a)=chs(Q)+csw(Q) exp( -

!.42 2 ~ s w ~ /")2~,

(5)

Chs(Q) is taken as one given by the PY solution Csw(Q) represents the strength of H-bonding given by -flEsw(Q) where E~w(Q) is for the repulsive SW potential operative in the region tr
482

A K Karmakar, R N Joarder / Physws Letters A 190 (1994) 480-482

ture water [ 12 ] It is therefore possible that for associated hqulds, where H-bonding plays a major role, the structure and thermodynamics should be described well by the appropriate SW model potential.

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References I

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q(~l-') Fig 2 (a) Oxygen-oxygen structure factor S(Q) at room temperature Sohd hne calculated, dotted hne Narten's X-ray data (b) Neutron molecular structure functxon HN(Q) Sohd hne calculated, dotted hne Ref [ 11 ]

the effect of the SW part improves the hard-sphere values m the right dlrecUon. It Is also interesting to note that for hquld methanol at room temperature the centre structure shows double peak maxima qmte similar to room tempera-

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