The equation of state for dense fluids has been derived within the framework
of the Sutherland and Katz potential models. The equation quantitatively agrees
with experimental data on the isothermal compression of water under
extrapolation into the high pressure region. It establishes an explicit
relationship between the thermodynamic experimental data and the effective
parameters of the molecular potential

Bardik, V. Yu.

Nerukh, D.

Pavlov, E. V.

Zhyganiuk, I. V.

English

arXiv

The equation of state for dense fluids has been derived within the framework of the Sutherland and Katz potential models. The equation quantitatively agrees with experimental data on the isothermal compression of water under extrapolation into the high pressure region. It establishes an explicit relationship between the thermodynamic experimental data and the effective parameters of the molecular potential

Bardik, V.Yu.

Pavlov, E.

Zhyganiuk, I.

Aston Publications Explorer

Equation of state for water in the small compressibility region  V.Yu. Bardik1, E.V. Pavlov1, I.V. Zhyganiuk2, D. Nerukh3  1Kiev National Taras Shevchenko University, Physics Faculty  (Prospect Acad. Glushkova , 4, Kiev 03127, Ukraine ) 2Odessa Mechnikov National University (Dvoryanskaya str., 2, Odessa 65026, Ukraine) 3Aston University Non-linearity and Complexity Research Group (Birmingham, B4 7ET, UK)   We derive the equation of state for supercritical fluids in the framework of the Sutherland and Katz potential models using the free energy perturbation expansion. The derived equation of state quantitatively agrees with experimental data on isothermal compression of water in the high pressure region.  It establishes an explicit relationship between the thermodynamic experimental data and the parameters of the molecular potential.  This can be used in calibration protocols for simulation forcefields for the high pressure regions.   1. Introduction   Deriving the equation of state (EoS) for water in a wide range of pressure and temperature remains a challenging open problem, especially in the high pressure region. In the recent papers [1-4] devoted to the water EoS this problem was solved by fitting multiparameter formulae with such a large number of adjustable parameters that it approaches the number of experimental points. These methods are not based on reliable statistical mechanic foundations and the applicability of these EoS is restricted. If the functional form of the EoS and their parameters are applicable to other substance or solutions is an open question.   In comparison with the majority of one-component liquids, water reveals many unusual properties in its normal and supercooled states. The analysis of the diffusion peak of quasi-elastic incoherent neutron scattering and kinematic shear viscosity of water has shown that the global H-bond network disintegrates into an ensemble of weakly interacting clusters: dimers, trimers, tetramers etc [9,10,11,12,13]. It was also shown [14] that in the supercritical region the properties of water are determined by the averaged spherically symmetrical potential. Therefore, it is reasonable to use for water in the high pressure region such well-known models as Lennard-Jones, Buckingham, Sutherland or Katz potentials.   In this paper we derive the EoS for a supercritical fluid within the framework of the Sutherland or Katz potentials. In our previous work []we used a new version of the thermodynamic perturbation theory (TPT) originally proposed by Sysoev []. The main feature of the proposed TPT is in the assumption that the functional form of the perturbed potential is identical to the potential of the reference system and, therefore, the deviation of the potential of the more compressed system from the potential of the less compressed system is considered as the perturbation. On this basis the concept of a reference thermodynamic state has been developed. A functional expansion of the free energy gave the possibility to derive, at a certain choice of the parameter expansion, two EoS modifications within the framework of a realistic and a “soft” sphere potential models. As it was shown in [] these EoS correctly described the isothermal compression for supercritical fluids of inert gases.   Following [], we here use the free energy perturbation expansion  ( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( )lmijlmijmljmejiiijijjiiV rereVFrererdrdrdrdreVFrerdrdVFVFδδδδδ 020000 !21⋅∆⋅∆⋅+⋅∆⋅+= ∑∫∑∫<<<, (1) where ( ) ( ) ( )( ) ( )kTrkTqrijijijijijeererereϕϕ−−−=−=∆ 0 , qrijφ    is the potential of the perturbed system and q is a scale factor, 30, Vr qr qV→ = .   Equation (1) can be transformed into the expression for pressure (the details are given in []): ( ) ( ) ( )( )[ ] ( ) ( ) ( ) rrddrqrdrgrrddrqrdrqrrgVVVVNkTPPVVΦ+Φ⋅Φ+Φ−−∆=− ∫∫ 22020200 6exp6ρβρ . (2)  This expression was obtained within the framework of a realistic potential model that can be presented in the general form ( ) ( ) ( )rrr ψϕ +Φ= ,           (3) where ( )rΦ  is the repulsive part of the potential and ( )rψ  is its attractive part.  2. The equation of state within the framework of Sutherland and Katz potential models  In the case of short-range potentials the expression for pressure can be rewritten in the form ( ) ( ) ( ) ( ) ( )2 20 0 0 02 2 06 6V VNkT V VP g r f r r dr g r r drV V Vρ ρυ υ= − −∫ ∫ ,     (4) where ( ) ( ) ( )( )[ ] 1−= −− rqrerf ϕϕβ , ( ) ( )r qr rυ ϕ= −∇  is the virial of intermolecular forces and  ( ) ( ) ( )0 1qr qr r r rυ ϕ ϕ== −∇ = −∇  is the virial of intermolecular forces in the reference state. Taking into account the expression for pressure in the reference state  ( ) ( ) rdrrgVNkTPV022000 6υρ∫−= .         (5), We rewrite (4) as ( ) ( ) ( ) ( ) ( )2 20 0 0 0 0 02 26 6V VV P V VP g r f r r dr g r r drV V Vρ ρυ υ= − − ∆∫ ∫ .     (6) Eq.(6) can be expressed in the terms of 00PPP −=Π  or 00VVV −=∆ , using of the approximate equality ∆+≈ 10VV  ( )( )KL +∆++∆=Π 1 ,           (7) where ( ) ( ) ( ) rdrrfrgPLVυρ∫−= 20206,          (8) ( ) ( ) rdrrgPKVυρ∆−= ∫ 20206.          (9) Now the problem of deriving EoS reduces to the evaluation of the integrals (8) and (9). We first consider the Sutherland potential  ( )>−<∞= −00,,drcrdrr mϕ ,         (10) d0 is the molecular diameter and the potential well depth ε is defined by mdc0=ε . Then ( )qrϕ  is written as ( )>−<∞= −0300,,drrVVcdrqr mmϕ ,        (11) where 03 0 dVVd = . We calculate the singular force ( )drrdϕ−  following [18] and obtain ( ) ( ) ( )drrVVcmrdrkTr mm−Θ+−−= −30δυ ,      (12) ( ) ( ) ( )[ ] ( ) ( )−Θ−−Θ+−−−−=∆ − 0300 drdrVVcmrrdrdrkTrmmδδυ  ,  (13) ( )dr −δ  is the Dirac delta function and ( )dr −Θ  is the Heaviside step function. The expression for L takes the form ( ) ( ) ( ) ( )+↓−−= ∫∞+− drrfrrgVVcmddrfdkTgPLdmm22303202032πρ ,   (14) in which ( ) ( )rfdrfdr 0lim+→=↓  [18]. The integral in (14) can be evaluated if we assume ( ) ( )rerg βϕ−≈2 , then in view of ∆+≈ 10VV  we have  ( ) ( ) ( )( ) ( )−−= −∞ +−∫3030320332230 ,,mmmmmmdmmVVyxVVyxVVcdrfrrgVVcm γγβ .  (15) Here ( )yx,γ  is the upper incomplete gamma function with β3−=mx  , mdcy0β−= . Using the same assumptions for the integral (9) we obtain ( ) ( )[ ] ( )( ) ( )−−−↓−=−yxVVyxVVcddrgddgkTPKmmmmm,,32 303203330232020 γγβπρ ,   (16) where ( ) ( )rgdrgdr 2002lim+→=↓ . Taking into account the smallness of the parameter ∆ , we can expand ( )dg 2  at a point d0 using the Taylor series ( ) ( ) ( )2 2 0 1g d g r d C T≈ ↓ + ∆   ,        (17) where 00 0)(ln3)( +→∂∂= drrrgdTC is the function of the reference state. Since in the 1<<∆  approximation ( ) 13exp0− ∆=↓ mdcmdrf β  and 1<<Π∆ . In the case of small compressibility Eq.(7) takes the form ( ) ( ) ( )( ) ( )( ) ( )( ) 20 0 1 1 2 1 2 1A T TP P B T P e C T C T C T∆⋅   = + + ⋅ − + ∆ + − + ∆ + + ∆ + Γ           ,(18)  Γ is the term which comprises the incomplete gamma function  ( )( ) ( )−−=Γ−yxVVcPmmmm,132 32033020 γβπρ .       (19)  For the Sutherland model it is expressed through the second virial coefficient B2(T):  ( ) +−=Γ32130232000 dTBVVPkTmπρ .       (20) The experimental data for some dense fluids (water, argon, neon, krypton) show that this term can be neglected at the condition 310≈Π  with the accuracy of 1%. There is a wide range of values of thermodynamic variables where the isothermal compressibility is low (∆<< 1) corresponding to the pressure interval 100 – 2200 MPa. The term ( )( ) 212 ∆+ TC  can also be ignored with the same accuracy. Finally, we arrive at the equation of state in the framework of the Sutherland model: ( ) ( ) ( ) ( )( )0 0 1 1 (2 1A T TP P B T P e C T C T∆⋅  = + + ⋅ − + ∆ + − ∆ +            .   (21)  The expression (21) is a three-constant equation of state with the adjustable parameters B(T), C(T) and A(T). The parameter B(T) depends on temperature. It is related to the pressure caused by attractive forces ( ) ( )00000300202032VNkTPPPPddrkTgTB ar −=−=−↓=πρ ,     (22) Where rP0  is the pressure caused by the repulsive forces in the reference state and aP0  is the part of pressure due to the attractive forces. The parameter A(T), within the framework of the Sutherland model, is expressed by the formula 03( )md kA Tmc⋅ ⋅= .          (23)  If we use the Katz potential model  ( )>−<∞=Φ −0300,4,0 dreaRdrr rRπ,        (24) where R0 is the interaction radius of the attractive forces, π430aR  is the interaction constant, Eq.(21) retains its functional form, but the parameter A(T) is defined by the formula  ( )0 0 040 012 exp( )md R d kA Td R aπ⋅ ⋅= .        (25) The parameters B(T) and C(T) are defined in a similar way.  3. Experimental PVT data analysis  We used the technique from the previous paper [] to process the PVT data and evaluate the EoS parameters. It turns out that Eq.(21) yields good agreement with the experimental PVT data under the extrapolation to the high pressure region. The results of the comparison are presented in Fig. 1. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30-10001002003004005006007008009001000T=500 T=550T=450T=400T=350Pressure,MPaVolume, (cm3/mol) experiment equationT=300Fig.1. The experimental and theoretical PVT data for supercriticlal water.  Table 1.  The values of the EoS parameters. T,K 300 350 400 450 500 550 B(T), MPa       C(T)       A(T), K-1 0,0042 0,0037 0,0032 0,0027 0,0024 0,002   Of special interest is the values of the parameter A(T) because it is related to the parameters of the potential, formulae (23), (25) for the Sutherland and Katz models. On the basis of (23) we evaluated the values of the potential well depth ε at a fixed value m =12 common-used in the Lennard-Jones model (Tab.2).   Table 2.  The values of ε (Sutherland potential) at m=12 in the  temperature interval 300-550K. T,K 300 350 400 450 500 550 ε, J·103/mole 0,493 0,562 0,650 0,769 0,866 1,038  If we fix the value of ε = 0,650 kJ/mole, which corresponds to the SPC/E model [], we obtain the variation of the softness parameter m with temperature. In general, fixating  ε at the values for well-known potential models, such as  ,[] , [] , [], leads to the variation of the softness parameter m with temperature within the framework of the Sutherland potential.   4. Conclusion  The approach developed on the basis of the free energy perturbation expansion and a new version of TPT resulted in some universality (generalisation) for the EoS statistical foundation of low weight molecular supercritical fluids.  The concept of the thermodynamic reference state implies that an initial state (P0 , V0) on the isotherm corresponds to the reference system with unperturbed potential and every subsequent point on the isotherm (P1,V2),… (Pn,Vn ) corresponds to the system with the perturbed potential at isothermal compression of the system. This modification of TPT allowed obtaining the EoS which exhibits good results under the extrapolation to the high pressure region. 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Equation of state for water in the small compressibility region

Equation of State for Water in the Small Compressibility Region

Equation of State for Water in the Small Compressibility Region

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