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PHYSICAL REVIEW E 92, 012124 (2015)
Thermophysical properties of supercritical water and bond flexibility
I. Shvab1 and Richard J. Sadus2,*
1School of Civil Engineering and Geosciences, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom2Centre for Molecular Simulation, Swinburne University of Technology, PO Box 218, Hawthorn, Victoria 3122, Australia
(Received 15 April 2015; published 17 July 2015)
Molecular dynamics results are reported for the thermodynamic properties of supercritical water using examplesof both rigid (TIP4P/2005) and flexible (TIP4P/2005f) transferable interaction potentials. Data are reported forpressure, isochoric and isobaric heat capacities, the thermal expansion coefficient, isothermal and adiabaticcompressibilities, Joule-Thomson coefficient, speed of sound, self-diffusion coefficient, viscosities, and thermalconductivity. Many of these properties have unusual behavior in the supercritical phase such as maximum andminimum values. The effectiveness of bond flexibility on predicting these properties is determined by comparingthe results to experimental data. The influence of the intermolecular potential on these properties is both variableand state point dependent. In the vicinity of the critical density, the rigid and flexible potentials yield very differentvalues for the compressibilities, heat capacities, and thermal expansion coefficient, whereas the self-diffusioncoefficient, viscosities, and thermal conductivities are much less potential dependent. Although the introductionof bond flexibility is a computationally expedient way to improve the accuracy of an intermolecular potential,it can be counterproductive in some cases and it is not an adequate replacement for incorporating the effects ofpolarization.
DOI: 10.1103/PhysRevE.92.012124 PACS number(s): 65.20.Jk, 31.15.xv, 61.25.Em, 05.20.Jj
I. INTRODUCTION
The central role of water in almost every aspect of lifeon Earth has been the motivation for many experimentaland theoretical studies [1]. Early theoretical work focused ondeveloping equations of state [2] and an accurate referenceequation is available [3] that reproduces the thermodynamicproperties of water to a high degree of accuracy. More recently,molecular simulation [4] has become the preferred method forstudying the properties of both fluids in general and water inparticular. The major advantage of molecular simulation is theclose nexus between the underlying theoretical assumptionsand the results observed. Changing an aspect of the model,and observing the change in the molecular simulation resultsrelative to experimental data, provides direct and unambiguousinformation regarding its influence on the system beinginvestigated. The most important assumption is the nature ofthe intermolecular potential [5], which characterizes both thestrength and nature of interactions between particles.
In principle, it would be highly desirable to use anab initio potential because this would correspond to themost accurate theoretical understanding of the propertiesof the molecule. Although some promising results for anab initio water potential have been reported [6], the highcomputational cost, theoretical complexity, and limited trans-ferability of such potentials severely restricts their wider use. Incontrast, there has been considerable success [7] in developingsemiempirical models, typically involving a three-site simplepoint charge (SPC [8], SPC/E [9]) or transferable interactionpotential with n points (TIPnP [10]) as the underlying theoret-ical framework. Most of the intermolecular potentials use rigidbonds [11–15] but increasingly models [16,17] with flexiblebonds have been proposed. The advantage of introducingbond flexibility is that it allows us to mimic the effect of
*Corresponding author: [email protected]
polarization without incurring the increased computationalcost. This is important because polarization has a profoundimpact [18–22] on the properties of water. Raabe and Sadus[23,24] showed that accounting for intramolecular degrees offreedom improves the prediction of the dielectric constant [23],diffusion coefficient [24], and viscosity [24]. Lopez-Lemuset al. [25] demonstrated that the surface tensions and coexistingdensities of water predicted by a flexible model are closer tothe experimental data than those of a rigid model. Mizan et al.[26] investigated the structural properties, self-diffusion, anddielectric constant of supercritical water using flexible andrigid SPC models and obtained superior performance for theflexible model, particularly at higher densities.
Except for studies of vapor-liquid equilibrium [4], somehigh-temperature thermodynamic properties [22], and super-cooled water and ice [7], the application of intermolecularpotentials for water has been confined largely to ambient con-ditions, i.e., temperatures (T ), pressures (p), and densities (ρ)close to 298 K, 0.1 MPa, and 1000 kg/m3. In contrast, molec-ular simulation studies of water intermolecular potentialsat supercritical conditions are much less common. Previousinvestigations of supercritical water have been largely confinedto the hydrogen bond network and structural properties[27], and orientational and dielectric properties [28,29]. Dataobtained using polarizable potentials for supercritical water arevery scarce [30,31]. Ideally, the systematic analysis of waterpotentials should include a comprehensive set of properties(structural, thermophysical, and transport) over a much widertemperature and density range. For example, Shvab and Sadus[21] recently investigated thermodynamic response functionsof liquid water in the temperature range of 298–650 K.
A pure fluid is supercritical at temperatures and pressuresabove its critical point, which for water [3] is Tc = 647.096 K,ρc = 322 kg/m3, and pc = 22.064 MPa. The properties ofsupercritical water are very different from those at ambientconditions. At conditions in the vicinity of the critical point,water is highly compressible and expandable, having a very
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I. SHVAB AND RICHARD J. SADUS PHYSICAL REVIEW E 92, 012124 (2015)
high self-diffusion coefficient and low viscosity, thermalconductivity, and dielectric constant [32]. Near-critical waterloses its ability to dissolve salt and prefers to mix with organicmatter and nonpolar gases [32]. The properties of supercriticalwater have been exploited as a medium for chemical reactionssuch as pyrolysis, hydrolysis, oxidation, material synthesis,etc. [26]. Experimental studies have provided insights for thestructure of near-critical water [33,34]; however, there has beenrelatively little theoretical work on thermophysical propertiesof bulk water near the critical point.
In this work we report molecular dynamics (MD) sim-ulations for the rigid [35] TIP4P/2005 and flexible [17]TIP4P/2005f water models for state points covering the densityrange of 100–1000 kg/m3 at a temperature of 670 K whichis above the critical temperature of 640 K predicted by theTIP4P/2005 potential [36]. Our choice of water models is moti-vated by the fact that TIP4P/2005 is one of the most accurate [7]and computationally efficient rigid four-site models available.The flexible TIP4P/2005f potential is a recent modificationof the TIP4P/2005 potential, which preserves many goodqualities of its predecessor and adds intramolecular degrees offreedom, which can potentially mimic the effect of polarizationwithout great additional computational cost. The aim of thiswork is twofold. First, to investigate the thermodynamic andtransport properties of water as a function of density at asupercritical temperature. Second, to study the effect of bondflexibility on the thermophysical properties. In so doing, thiswork provides insights into the strengths and limitations of theTIP4P/2005 and TIP4P/2005f intermolecular potentials andguidance for the future development of water models.
II. THEORY
A. Intermolecular potentials for water
This work is focused on two widely used water models,namely the TIP4P/2005 [35] and TIP4P/2005f [17] potentials.These potentials are computationally efficient and provide avery good representation of many properties of liquid waterat ambient conditions [7]. As such, they are an appropriatestarting point for investigating water over a wider range oftemperature and pressure. In both cases the intermolecular partof the potentials combines contributions from both Lennard-Jones interactions and an electrostatic term:
uinter(�r) = 4ε
N∑i
N∑j �=i
{(σ
rooij
)12
−(
σ
rooij
)6}+ 1
4πε0
qiqj
rij
,
(1)
where rij and rooij are the distances between charged sites and
oxygen atoms, respectively, and ε0 is the permittivity of thevacuum. The TIP4P/2005 potential uses a rigid four-site watermodel with an O-H distance of 0.9572 A and the H-O-H angleof 104.52◦. Values of the parameters are σ = 3.1589 A andε = 0.7749 kJ/mol with a charge of +0.5564 on both H atoms.A characteristic feature of the TIP4P/2005 water model is thatthe negative charge is not located on the O atom but is insteaddisplaced by dOM = 0.1546 A on a bisector between the Hatoms.
The TIP4P/2005f water model of Gonzalez and Abascal[17] differs from TIP4P/2005 by having additional potentialterms, which account for molecular flexibility and intramolec-ular interactions:
uintra(�r) =2∑
i=1
Dr
{1 − exp
[−β(rOHi
− req
)]2}
+ 1
2Kθ (θ − θeq)2. (2)
In Eq. (2) the values of the parameters are Dr =432.581 kJ/mol, Kθ = 367.81 kJ/(mol rad2), the equilibriumO-H distance req = 0.9419 A, and the equilibrium H-O-Hangle θeq = 107.4◦. Since the TIP4P/2005f molecule is notrigid, the location of the displaced oxygen charge is defined as
dOM = d relOM
(zOH1 + zOH2
), (3)
where d relOM = 0.13194 A and zOHi
= rOHicos(θ/2).
B. Molecular simulation details
All MD simulations were performed in NpT and NV T
ensembles using LAMMPS [36]. The number of water moleculeswas 1728 at a temperature of 670 K and a density range of100–1000 kg/m3. The Nose-Hoover temperature thermostatand pressure barostat [4] were used to keep temperature andpressure stable during the simulation. Isometric coupling anddamping parameters at intervals of 50 and 100 time stepswere used for temperature and pressure, respectively. Thesimulations were commenced from an initial face centeredcubic lattice in the NV T ensemble. The total simulation timein the NV T ensemble was 1.5 ns, which includes 0.5 ns forequilibration and 1 ns for calculating the pressure of the systemat a given density. Thereafter, simulations were commenced inthe NpT ensemble from prestored coordinates. The total NpT
simulation time was not less than 4 ns. All properties reportedhere were obtained in the NpT ensemble.
The standard particle-particle particle-mesh (PPPM) sum-mation method and Verlet integrator were used to evaluatethe long-range part of the Coulomb potential and integrateequations of motion [4]. The Lennard-Jones forces weretruncated at 13 A. For the calculations of all of the properties,we used time steps of 1 fs and 0.2 fs for the rigid TIP4P/2005and flexible TIP4P/2005f potentials, respectively.
C. Calculation of thermodynamic properties
The isothermal compressibility (βT ), thermal expansion co-efficient (αp), and isobaric heat capacity (Cp) were calculatedfrom the fluctuation of the properties of the NpT ensemblevia the following relationships [4]:
βT = 〈V 2〉 − 〈V 〉2
kT 〈V 〉 ,
Cp = 〈H 2〉 − 〈H 〉2
kT 2, (4)
αp = 〈V H 〉 − 〈V 〉〈H 〉kT 2〈V 〉 ,
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where k is the Boltzmann constant, H is enthalpy, and theangled brackets denote ensemble averages. The adiabatic com-pressibility (βS), isochoric heat capacity (CV ), Joule-Thomsoncoefficient (μJT ), and speed of sound (w0) were obtained usingthe following standard thermodynamic relationships [37]:
βS = βT − V T α2p
Cp
,
CV = Cp
βS
βT
,
μJT = V (αpT − 1)
Cp
,
wo =√
1
ρβS
. (5)
The self-diffusion coefficient was calculated using theEinstein equation [4] for the center of mass,
D = limt→∞
〈|r(t) − r(0)|2〉6t
, (6)
where r(t) is the position of ith molecule at time t .The dynamic (shear) viscosity (η) and thermal conductivity
(λ) were calculated using the corresponding Green-Kuborelations [4]:
η = V
3kT
∫ ∞
0[Pxy(0)Pxy(t) +Pxz(0)Pxz(t) +Pyz(0)Pyz(t)]dt,
(7)
λ = 1
3kV T 2
∫ ∞
0〈J(t)J(0)〉dt, (8)
where Pαβ are the off-diagonal components of the pressuretensor and J(t) is the heat current at time t .
III. RESULTS AND DISCUSSION
Ideally, results from molecular simulation should be com-pared directly to experimental data. However, in practice thisplaces a limitation on the extent of the comparison becauseexperimental data are unlikely to be available at all of thestate points of interest. There is also the issue of selectingamong conflicting data sets. However, in the present casethe International Association for the Properties of Waterand Steam (IAPWS-95) software [38] provides an accuratealternative. IAPWS-95 is based on a highly accurate empiricalequation of state [3] for water, which was formulated followinga very comprehensive and critical evaluation of the availableexperimental data. Except for the self-diffusion coefficient,all simulation data reported in the figures are compared withIAPWS-95 reference data for pure water [38]. Experimentalvalues of the self-diffusion coefficient were taken from Lambet al. [39]
The standard errors were obtained by dividing the simula-tion time into ten blocks and taking averages from each block.The standard errors are shown in every figure unless they arecomparable to the symbol size used in the figures.
FIG. 1. (Color online) Pressure as a function of density predictedby the TIP4P/2005 (blue ) and TIP4P/2005f (red ) intermolecularpotentials and compared to IAPWS-95 reference data [38] (black �).Lines through the data points are given only for guidance.
A. Pressure
The pV T properties are arguably the most basic test ofthe accuracy and reliability of an intermolecular potential.The pressures predicted by the TIP4P/2005 and TIP4P/2005fpotentials are compared with IAPWS-95 reference data in Fig. 1.Values of ρ from 100 to 1000 kg/m3 correspond to referencep values of approximately 25–750 MPa. For ρ < 500 kg/m3,p obtained from either intermolecular potential are in goodagreement with the reference data. However, a noticeabledifference in the quality of agreement is observed for ρ >
500 kg/m3. The TIP4P/2005f results deviate significantly fromthe reference data whereas good agreement is maintained forthe TIP4P/2005 potential. These observations are in contrast tothe near-perfect agreement [22] with reference data reported atsubcritical temperatures involving a polarizable intermolecularpotential for water.
B. Isothermal and adiabatic compressibilities
βT and βS (Fig. 2) are properties that measure changesin volume under pressure. Values obtained from both theTIP4P/2005 and TIP4P/2005f water models are compared withIAPWS-95 reference data. It is apparent that these two propertiesbehave very differently in the ρ range up to 500 kg/m3. Thevalues of βT [Fig. 2(a)] exhibit a prominent peak in thevicinity of ρc whereas the values of βS decrease progressivelywith increasing ρ without passing through a maximum. Thevalues of βT for both water models attain a maximum valueat ρ = 250 kg/m3 that compares with a ρ of approximately280 kg/m3 for the IAPWS-95 data. The existence of such aprominent peak is undoubtedly influenced by the proximity ofthe critical point [32].
The rigid TIP4P/2005 water model typically overestimatesthe βT [Fig. 2(a)] reference data by approximately 20% atρ < ρc, whereas TIP4P/2005f considerably underestimatesthe reference data for ρ � 500 kg/m3. At other ρ values,the two water potentials are in close agreement with both
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I. SHVAB AND RICHARD J. SADUS PHYSICAL REVIEW E 92, 012124 (2015)
FIG. 2. (Color online) Isothermal (a) and adiabatic (b) compress-ibilities as functions of density predicted by the TIP4P/2005 (blue )and TIP4P/2005f (red ) intermolecular potentials and compared toIAPWS-95 reference data [38] (black �). Lines through the data pointsare given only for guidance.
each other and the reference data, almost merging as densityincreases [see inset in Fig. 2(b)]. Similarly, there is verygood agreement between the potentials and the referencedata for βS when ρ < 500kg/m3 [Fig. 2(b)]. However, bothpotentials substantially overestimate βS at lower values of ρ.In contrast to the βT case [Fig. 2(a)], the values obtained forthe TIP4P/2005f potential are in better agreement with thereference data than the rigid TIP4P/2005 calculations.
The inadequacy of the calculated βS and βT in the low-density region can be at least partly attributed to the inabilityof the intermolecular potentials to describe the structure ofnear-critical water. It is apparent from Eqs. (4) and (5) that βS
and βT are directly related to volume fluctuations. It has beenreported previously [7,21] that nonpolarizable water models,whether rigid or flexible, do not capture significant temperatureand density induced changes in the structure of water.
The radial distribution functions (RDFs) obtained fromexperiment [Fig. 3(a)] and the TIP4P/2005 [Fig. 3(b)] andTIP4P/2005f [Fig. 3(c)] potentials at different densities areillustrated in Fig. 3. It is apparent from Fig. 3 that the calculatedRDFs are much less sensitive to density than is observedexperimentally. Experimental [32,40] RDFs [Fig. 3(a)] shownoticeable decreases in the first oxygen-oxygen (O-O) peakwith decreasing density, indicating smaller O-O coordination
FIG. 3. (Color online) Comparison of the oxygen-oxygen radialdistribution functions at 673 K obtained from (a) experiment [40], andcalculations with the (b) TIP4P/2005 and (c) TIP4P/2005f potentialsat densities of 870 kg/m3 (blue), 730 kg/m3 (green), 660 kg/m3 (red),and 580 kg/m3 (black).
number and reduced attraction. In contrast, RDFs of nonpolar-izable rigid models [Figs. 3(b) and 3(c)] preserve too highfirst and second O-O peaks even at reduced densities andhigh temperatures [41]. The TIP4P potentials predict a morestructured first solvation shell than the experimental data, butunderestimate the long-range structure as seen on experimentalgOO(r) after 5.5 A. However, the position of the second peaksuggests that the long-range structure does not correspondto a tetrahedral arrangement because it occurs at separationsgreater than 4.5 A [42]. This discrepancy can be attributed toa too high electrostatic interaction due to the nonpolarizablenature of examined water models. In effect, the TIP4P dipolemoments correspond to that of water at ambient conditions,which is higher than that of less dense phases.
The large values of TIP4P/2005 βT in Fig. 1(a) for ρ �350 kg/m3 indicate high compressibility of the rigid watermodel in this region. In contrast, the flexible TIP4P/2005fmodel shows much lower values of βT . In view of the fact thatthe two potentials have the same parametrization, a possiblereason for such different behavior can be attributed to thedensity dependence of the internal degrees of freedom ofthe TIP4P/2005f model [17]. The TIP4P/2005f O-H bondlength and H-O-H angle distributions are illustrated Fig. 4.It is apparent from the data in Fig. 4 that both the O-Hbond length and H-O-H widen as density decreases. Thisindicates the presence of an increasing number of molecules
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THERMOPHYSICAL PROPERTIES OF SUPERCRITICAL . . . PHYSICAL REVIEW E 92, 012124 (2015)
FIG. 4. (Color online) Distributions of TIP4P/2005f H-O-H angles and O-H bonds at densities of (a) 1000 kg/m3, (b) 600 kg/m3,(c) 400 kg/m3, and (d) 200 kg/m3.
in a slightly deformed state, which attempt to preserve theH-bond network due to strong O-O attraction. Therefore wecan expect that, in the vicinity of the critical density, theflexible TIP4P/2005f potential will yield considerably smallervalues of thermodynamic response functions than its rigidTIP4P/2005 counterpart.
C. Isobaric and isochoric heat capacities
Results obtained for Cp and CV are illustrated in Fig. 5.Similar to βT [Fig. 2(a)], values of Cp [Fig. 4(a)] exhibit a highpeak close to ρc. This behavior [43,44] is common in fluids atsupercritical temperatures. A maximum in CV is also observed[43,45] for some fluids whereas more commonly the maximumis only observed in Cp. A line of Cp maxima often extendsfrom above the critical point to supercritical temperatures, withthe maxima becoming progressively smaller with increasingtemperature. It has been suggested [43] that the line ofmaxima meets a line of minima ending at a common transitiontemperature.
Both intermolecular potentials significantly underestimatethe reference data for ρ � ρc. IAPWS-95 yields a maxi-mum value of Cp = 615.6 J/mol K (ρ = 311 kg/m3) whereasthe Cp maximum for the TIP4P/2005 (ρ = 250 kg/m3) and
TIP4P/2005f (ρ = 300 kg/m3) potentials are smaller by 29%and 63%, respectively. The density of the Cp maximum is T
dependent and shifts to lower ρ with increasing temperatures[32]. After the peak, Cp values decline rapidly and thevalues predicted by the two potentials become progressivelycloser. At ρ = 1000 kg/m3, Cp = 52.75 and 52.70 J/mol Kfor TIP4P/2005 and TIP4P/2005f, respectively, compared with65.53 J/mol K for IAPWS-95.
Mairhofer and Sadus [43] determined the criteria forobserving a Cp maximum in the supercritical phase; i.e.,
(∂2p
∂T 2
)VM
+ 2αpV
(∂2p
∂V ∂T
)+ α2
pV 2
(∂2p
∂V 2
)TM
= 0,
(∂3p
∂T 2∂V
)+ 2βT V
(∂2p
∂V ∂T
)2
+ 2αpV
[(∂3p
∂V 2∂T
)+ 2V βT
(∂2p
∂V ∂T
)(∂2p
∂V 2
)TM
]
+α2pV 2
[2V βT
(∂2p
∂V 2
)2
TM
+(
∂3p
∂V 3
)TM
]< 0. (9)
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I. SHVAB AND RICHARD J. SADUS PHYSICAL REVIEW E 92, 012124 (2015)
FIG. 5. (Color online) Isobaric (a) and isochoric (b) heat capac-ities as functions of density predicted by the TIP4P/2005 (blue )and TIP4P/2005f (red ) intermolecular potentials and compared toIAPWS-95 reference data [38] (black �). Lines through the data pointsare given only for guidance.
An analysis [43] of the terms in Eq. (9) indicates that αp
made the most important contribution in the vicinity of the Cp
maximum. Therefore, we would expect the inaccuracies of theintermolecular potentials in the vicinity of the Cp maximumto be at least partly linked to this factor (see Fig. 5).
The calculated CV [Fig. 5(b)] exhibits a much smaller rangeof values and is in overall better agreement with the referencedata. All three CV curves show only a moderate peak for ρ =250–300 kg/m3 after which a gradual decrease is observeduntil ρ = 700 kg/m3. For ρ = 700–1000 kg/m3, IAPWS-95 CV
data exhibit a shallow minima at approximately 750 kg/m3.The flexible TIP4P/2005f potential does not accurately predictthe reference data but it at least qualitatively reproducesboth the maximum and minimum in CV . In contrast, theTIP4P/2005 potential slightly overestimates reference data atρ < ρc, closely following the reference data at higher valuesof ρ, but it fails to predict the CV minimum.
The large disparity in the magnitudes of Cp and CV nearρc can be explained by the thermodynamic relationship forCV in Eq. (5). It is apparent from Eq. (5) that the differencebetween Cp and CV is directly proportional to βT , which hasa prominent peak near ρc [Fig. 2(a)].
FIG. 6. (Color online) Thermal expansion coefficient as a func-tion of density predicted by the TIP4P/2005 (blue ) and TIP4P/2005f (red ) intermolecular potentials and compared to IAPWS-95reference data [38] (black �). Lines through the data points are givenonly for guidance.
Results presented on Fig. 5(a) clearly indicate the failure ofboth rigid and flexible TIP4P water potentials to predict super-critical Cp. This supports the conclusion that improvements inheat capacity prediction should be focused on including twomajor missing factors, namely polarization [7,21] and nuclearquantum effects [46].
D. Thermal expansion coefficient
αp is a property of the fluid that measures changes in volumewith varying temperature. In common with data for βT , CV ,and Cp, it is apparent from Fig. 6 that reference values of αp
have a pronounced maximum in the vicinity of ρc. Both theintermolecular potentials for water significantly underestimateIAPWS-95 reference data in this ρ region. The referencedata have a maximum value αp = 0.0524 K–1 at 300 kg/m3
whereas values for the TIP4P/2005 and TIP4P/2005f aresmaller by 20.7% and 64.1%, respectively. The diminishedαp values directly impacts on the Cp maximum [Fig. 5(a)]via its important role in Eq. (9). The deficiencies in theintermolecular potentials can be again attributed (see Fig. 3)to overly structured first and weak second hydration shells ofTIP4P molecules, especially at low ρ. In particular the toolow values of αp for the TIP4P/2005f potential in the criticalregion may be caused by the high flexibility of this potential.It appears that the TIP4P/2005f molecules adjust the H-bondnetwork for the high thermal and density fluctuations in anattempt to preserve original the bulk volume (see Fig. 4).
E. Joule-Thomson coefficient
μJT is the rate of change of temperature with respectto pressure measured at constant enthalpy. The sign of theJoule-Thomson coefficient μJT at any given state determineswhether the fluid is cooled (μJT > 0) or heated (μJT < 0) for asmall change in pressure at constant enthalpy. Joule-Thomsonheating of water is of particular interest in industry becauseit has a significant influence on temperature in and aroundinjection wells.
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FIG. 7. (Color online) Joule-Thomson coefficient as a function ofdensity predicted by the TIP4P/2005 (blue ) and TIP4P/2005f (red
) intermolecular potentials and compared to IAPWS-95 referencedata [38] (black �). Lines through the data points are given only forguidance.
Calculations (Fig. 7) using both the TIP4P/2005 andTIP4P/2005f potentials are in semiquantitative agreementwith reference data for the whole ρ range. Values of μJT
decrease almost linearly until ρ = 600 kg/m3, after whichall curves almost flatten out and cross over to negativevalues at ρ ≈ 750 kg/m3. The sign of μJT at any given statedetermines whether the fluid is cooled (μJT > 0) or heated(μJT < 0) for a small change in p at constant enthalpyH . It is apparent from Fig. 7 that, in common with allother properties calculated from Eqs. (5)–(9), the standarderrors are much smaller than those associated with propertiescalculated directly from either volume or enthalpy fluctuations[Eq. (4)].
F. Speed of sound
The reference values for wo exhibit a shallow minimum(398.24 m/s) at ρ ≈ 300 kg/m3. MD simulations for wo as afunction of ρ are compared to reference data in Fig. 8. Boththe rigid TIP4P/2005 and flexible TIP4P/2005f potentials arein qualitative agreement with IAPWS-95 data. By definition wo
is proportional to the square root of βS [Eq. (6)], which meansdiscrepancies in βS will be at least partly reflected in wo. There-fore, it is not surprising that the largest discrepancies are at lowρ. The wo minima calculated for both potentials are less pro-nounced and shifted to lower ρ than the reference data, havingvalues of 361 m/s (TIP4P/2005f), and 322 m/s (TIP4P/2005).For ρ > ρc, wo increases steadily reaching values of2481.6 m/s (TIP4P/2005f), 2364.4 m/s (TIP4P/2005), and2285.6 m/s (IAPWS-95) at ρ = 1000 kg/m3. It is of interestto note that the TIP4P/2005f potential yields values thatare consistently 5%–10% higher than the rigid TIP4P/2005potential for the whole ρ range. Taking account of the identicalparametrization for the two potentials, this trend can beattributed to the effect of the internal degrees of freedom onthe TIP4P/2005f potential.
FIG. 8. (Color online) Speed of sound as a function of densitypredicted by the TIP4P/2005 (blue ) and TIP4P/2005f (red )intermolecular potentials and compared to IAPWS-95 reference data[38] (black �). Lines through the data points are given only forguidance.
G. Self-diffusion coefficient
Values of D for the TIP4P/2005 and TIP4P/2005fpotentials are compared with the experimental datareported by Lamb et al. [39] in Fig. 9. The flexiblepotential is in good overall agreement with experimentaldata for the whole ρ range while the rigid water modelconsistently underestimates IAPWS-95 reference data byapproximately 2–10% in the first half of the density range.Starting from values of 289.27×10–9 (TIP4P/2005f) and273.91×10–9 m2/s (TIP4P/2005) at ρ = 100 kg/m3, D
decreases progressively attaining values of 23.03×10–9 and23.17×10–9 m2/s, respectively, at ρ = 1000 kg/m3. Thisrepresents a greater than tenfold difference in D over thisρ range. The higher self-diffusion coefficient of TIP4P/2005f
FIG. 9. (Color online) Self-diffusion coefficient as a function ofdensity predicted by the TIP4P/2005 (blue ) and TIP4P/2005f (red
) intermolecular potentials and compared to experimental data [39](black �). Lines through the data points are given only for guidance.
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I. SHVAB AND RICHARD J. SADUS PHYSICAL REVIEW E 92, 012124 (2015)
compared to TIP4P/2005 can be attributed to the elongatedO-H bond, which in turn also results in the increased dipolemoment μTIP4P/2005f = 2.319 D. This is consistent with workat ambient conditions [23,24,26], which typically reporta 2%–6% difference in dipole moment between rigid andflexible water potentials.
The good agreement along the 670 K isotherm comes asa surprise when compared to the significant underestimationof D at high temperatures along the 998 kg/m3 isochore[42,47]. In the latter case, a major discrepancy betweencalculated D values occurs when temperature passes through400 K. This corresponds with the onset of changes in thestructure of the first solvation shell, i.e., from tetrahedralto dodecahedral arrangements. It has been argued [42] thatalong the 998 kg/m3 isochore diffusion behavior changesfrom being cage dominated because of the hydrogen bondnetwork (T < 400 K), to collision dominated (T > 450 K)[42]. Comparing this temperature dependence of D to ourcase, we can infer a single mechanism (cage effects) governingsupercritical diffusion at ρ = 100−700 kg/m3. However, ifthe self-diffusion data of Krynicki et al. [48] are extrapolatedto 670 K (998 kg/m3), it could be argued that the simulatedD values again significantly underestimate the experimentalone at normal density. Arriving at a definite conclusion ishindered by the scarcity of experimental data away fromambient conditions. To the best our knowledge supercriticalself-diffusion data provided by Lamb et al. [39] are the onlyone available in the literature.
H. Viscosity
The behavior of the dynamic (η) and kinematic (ν = η/ρ)viscosities is illustrated in Figs. 10(a) and 10(b), respectively.The IAPWS-95 data indicate that values of η gradually increasewith increasing ρ. The MD values for both TIP4P/2005 andTIP4P/2005f show a similar increasing trend but η values aresubstantially underestimated at ρ = 750 kg/m3. For example,at ρ = 100 kg/m3 the η predicted by the TIP4P/2005 andTIP4P/2005f potentials are approximately 63% smaller thanthe IAPWS-95 value of 2.585 × 10–5 Pa s. The underestimationof viscosity in the low-density region by both water modelscan be attributed to the same causes as in the βT and αp
discussion, namely too strong O-O attraction, higher densityin the first hydration shell, and weak long-range structure. Incontrast, at ρ = 750 kg/m3 the TIP4P/2005 and TIP4P/2005fcalculations cross the IAPWS-95 data and η is overestimated athigher ρ. Recent work [12,23,24,49] has also reported that η isoverestimated from many three- and four-site nonpolarizablewater models. The close agreement between TIP4P/2005 andTIP4P/2005f in Fig. 10 indicates that the internal degrees offreedom have relatively little influence on viscosity.
The IAPWS-95 curve for ν exhibits [Fig. 10(b)] a steep initialdescent at low ρ, followed by a shallow but nonetheless well-defined minimum at ρ ≈ 520 kg/m3. Thereafter, ν slowlyincreases with increasing ρ. The TIP4P/2005 and TIP4P/2005fwater potentials fail to accurately predict both the elevatedvalues of ν in the low-ρ region and the minimum in ν. We havealso observed this failure for the SPC/E potential [not shownin Fig. 10(b)]. Ryltsev and Chtchelkatchev [50] interpret thepresence of the ν minima as indicating a division between
FIG. 10. (Color online) Dynamic (a) and kinematic (b) viscosityas functions of density predicted by the TIP4P/2005 (blue ) andTIP4P/2005f (red ) intermolecular potentials and compared toIAPWS-95 reference data [38] (black �). Lines through the data pointsare given only for guidance.
“gaslike” and “liquidlike” behavior in the supercritical fluid.This interpretation is based on different regimes of long-rangeoscillations of the velocity autocorrelation function. The p-Tsequence of such division points is known as a Frenkel line[51]. It has been suggested [50,51] that the Frenkel line is amore appropriate measure of different regimes of supercriticalfluid behavior than the Widom line, i.e., the p-T line of themaxima of certain thermodynamic response functions such asCp [43,52,53].
The TIP4P/2005 and TIP4P/2005f potentials exhibit dis-tinct maxima for βT , Cp, and αp [see Figs. 2(a), 5(a), and 6] andthus have a well-defined Widom line in the supercritical region[54]. However, Fig. 10 illustrates that both the TIP4P potentialsfail to reproduce the dynamic behavior of supercritical water.In view of the fact that ν minima are observed for pureLennard-Jones fluids [50,51,52], we can attribute this failure toinadequate charge interactions at low densities [see Eq. (1)],which again leads us to the question of potential functionchoice.
I. Thermal conductivity
The IAPWS-95 data for λ are compared with MD cal-culations for the TIP4P/2005 and TIP4P/2005f potentials
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FIG. 11. (Color online) Thermal conductivity as a function ofdensity predicted by the TIP4P/2005 (blue ) and TIP4P/2005f (red
) intermolecular potentials and compared to IAPWS-95 referencedata [38] (black �). Lines through the data points are given only forguidance.
in Fig. 11. The TIP4P/2005 and TIP4P/2005f data aregenerally in close agreement, although the values obtainedfor TIP4P/2005 are slightly higher at some values of ρ. Itis apparent from Fig. 10 that good agreement between MDand the IAPWS-95 is confined to a narrow range of ρ below300 kg/m3. In this range of ρ, λ is slightly underestimated. Incontrast, the quality of agreement between MD calculationsand IAPWS-95 data deteriorates rapidly for ρ > 300 kg/m3.In this region, the IAPWS-95 data begin to taper off whereasthe MD data continue to increase steadily, resulting inprogressively greater overpredictions. Mao and Zhang [49]reported that many rigid nonpolarizable water potentials alsooverestimate λ at ambient conditions.
The limited (D,η,λ) and in some cases counterproductive(p,βS,T ,αp,CV,p,μJT ,w0) influence of bond flexibility onthermodynamic properties in the low-density region is unex-pected in view of previous work that indicated improved agree-ment for the dielectric constant [23] and vapor-liquid equilibria[55]. At the same time, our MD data at ρ > 800 kg/m3 arein agreement with earlier findings [24] which show that theslowing down of the dynamics via bond flexibility meanshigher η and lower D values and improved agreement withexperiment. In contrast, calculations [15–22] that explicitlyinclude polarization consistently report improved agreementfor all thermodynamic properties investigated.
IV. CONCLUSIONS
The thermophysical properties of supercritical water atT = 670 K and ρ = 100–1000 kg/m3 (p ≈ 25–750 MPa)have been determined using MD with both the rigidTIP4P/2005 and flexible TIP4P/2005f intermolecular poten-tials. In general, there is good agreement between MD data andIAPWS-95 reference data for βS,T , CV,p, and αp for high valuesof ρ (500–1000 kg/m3). However, there are considerablediscrepancies at low values of ρ (100–500 kg/m3) and clear
differences are apparent between the intermolecular potentials.The TPI4P/2005 overpredicts the βT maximum whereas itis substantially underpredicted by the TIP4P/2005f potential.Both potentials underpredict the Cp maximum, whereas theTIP4P/2005 potential both overpredicts the CV maximumand fails to detect the CV minimum. The maximum in αp isunderpredicted by both potentials with the TIP4P/2005 beingin closer agreement with the reference data. Good results areobtained for μJT , w0, and D, which are much less sensitiveto the nature of the intermolecular potential. η and λ arenot greatly affected by the choice of intermolecular potentialand the agreement with reference data for λ is worse athigh ρ.
The variation in the quality of predictions using the rigidTIP4P/2005 and flexible TIP4P/2005f potentials at differentdensities can be attributed to inadequacies in reproducing thestructure of water. In the vicinity of the critical temperature,water undergoes remarkable structural and thermodynamicchanges while moving from low to high densities. At near-critical densities (250–315 kg/m3) water is both almost 100%more compressible and expandable, and has an isobaric heatcapacity that is 89.5% higher than at a density of 1000 kg/m3.In addition, near-critical water has approximately 4.9 timeshigher self-diffusivity and approximately 3.3 times smallerviscosity than water at 1000 and 850 kg/m3, respectively.Experimental RDFs exhibit much smaller first and distinctsecond oxygen-oxygen peaks even at reduced densities, whilethe TIP4P potentials give too high first and very shallow secondpeaks with almost no sensitivity to change in density. Thenonpolarizable TIP4P/2005 and TIP4P/2005f water potentialsdo not reproduce the full spectrum of the structural andphysical changes in water in the critical and near-criticalregions. Comparing the performance of the rigid TIP4P/2005and the flexible TIP4P/2005f potentials we can concludethat the inclusion of harmonic H-O-H angle bending andMorse-type O-H bond vibrations in the TIP4P/2005f does notimprove the overall agreement with IAPWS-95 reference datafor a wide density range. In some cases, such as the detectionof the Cp and αp maximum, it is actually counterproductive.
It is well known that nonpolarizable models largely failto adequately predict changes in water structure (hydrogenbond network) for state points far from conditions at whichthe given water model was parametrized [7,21,42]. Recentsimulation work shows that the use of polarizable potentialsor any other way of accounting for dynamic changes inwater interaction (continuous parametrization, Gaussian, ordisplaceable charges) results in improvement in predictingsome water properties for a wider range of state points[20,22,30]. Although our analysis strictly only applies to theTIP4P/2005 and TIP4P/2005f potentials, it is reasonable toinfer that similar conclusions would be reached for alternativeflexible intermolecular potentials. Taking account of previouswork, the use of bond flexibility appears to mostly benefit theprediction of properties such as vapor-liquid equilibria anddielectric constants [24,55]. The improvement for transportproperties is relatively small and it is counterproductive forsome aspects of thermodynamic behavior. It would appearthat focusing on polarization rather than bond flexibility isa more promising route to improving the predictions of thethermodynamic properties of water.
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