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Deconstructing classical water models at interfaces and in bulk

Richard C. Remsing,1 Jocelyn M. Rodgers,2 and John D. Weeks3, ∗

1Chemical Physics Program and Institute for Physical Science and Technology,

University of Maryland, College Park, MD 207422Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720

3Institute for Physical Science and Technology and Department of Chemistry and Biochemistry,

University of Maryland, College Park, MD 20742

Using concepts from perturbation and local molecular field theories of liquids we divide the po-tential of the SPC/E water model into short and long ranged parts. The short ranged parts definea minimal reference network model that captures very well the structure of the local hydrogenbond network in bulk water while ignoring effects of the remaining long ranged interactions. Thisdeconstruction can provide insight into the different roles that the local hydrogen bond network,dispersion forces, and long ranged dipolar interactions play in determining a variety of propertiesof SPC/E and related classical models of water. Here we focus on the anomalous behavior of theinternal pressure and the temperature dependence of the density of bulk water. We further utilizethese short ranged models along with local molecular field theory to quantify the influence of theseinteractions on the structure of hydrophobic interfaces and the crossover from small to large scalehydration behavior. The implications of our findings for theories of hydrophobicity and possiblerefinements of classical water models are also discussed.

I. INTRODUCTION

Classical empirical water potentials involving fixedpoint charges and Lennard-Jones (LJ) interactions wereintroduced in the first computer simulations of waterforty years ago and modern versions are widely used eventoday in many biomolecular and materials-based simula-tions. Two recent reviews [14, 37] have focused on thiswide class of model potentials and assessed their per-formance for a broad range of different structural andthermodynamic properties, some of which were used astargets in the initial parameterization of the models. De-spite known limitations associated with the lack of molec-ular flexibility and polarizability, they qualitatively andoften quantitatively capture a large number of propertiesof water and often represent a useful compromise betweenphysical realism and computational tractability.Given the simple functional forms of the intermolecular

potentials it may seem surprising that such good agree-ment is possible. But recent work has shown that evensimpler models where particles interact via isotropic re-pulsive potentials with two distinct length scales are ableto qualitatively reproduce certain characteristic dynamicand thermodynamic anomalies of bulk water [8, 42, 43].Similarly in dense uniform simple liquids a hard-sphere-like repulsive force reference system can give a good de-scription of the liquid structure, and this in turn permitsthermodynamic properties to be determined by a simpleperturbation theory [15, 40].This suggests it should be useful to analyze the con-

struction and predictions of empirical water potentialsfrom the perspective of perturbation theory of uniformfluids and the related Local Molecular Field (LMF) the-ory [10, 11, 29, 30, 39, 40]. LMF theory provides a more

∗ email: jdw@umd.edu

general approach applicable to both uniform and nonuni-form fluids and gives strong support to the basic idea ofperturbation theory that in a uniform fluid slowly vary-ing long ranged parts of the intermolecular interactionshave little effect on the local liquid structure.

To apply these ideas to water we divide the intermolec-ular interactions in a given water model into appropri-ately chosen short and long ranged parts. In this con-text, it is conceptually useful to consider separately theslowly varying long ranged parts of both the LJ inter-actions and the Coulomb interactions. This deconstruc-tion of the water potential via LMF theory provides ahierarchical framework for assessing separately the con-tributions of (i) strong short ranged interactions leadingto the local hydrogen bonding network, (ii) dispersive at-tractions between water molecules, and (iii) long rangeddipolar interactions between molecules. Disentanglingthese contributions without the insight of LMF theoryis very difficult due to the multiple contributions of thepoint charges and the LJ interactions in standard molec-ular water models

In uniform systems, the long ranged forces on a givenwater molecule from more distant neighbors tend to can-cel [40, 41]. The remaining strong short ranged forces be-tween nearest neighbors arise from the interplay betweenthe repulsive LJ core forces and the short ranged attrac-tive Coulomb forces between donor and acceptor charges.These forces determine a minimal reference model thatcan accurately describe the local liquid structure – thehydrogen-bond network for bulk water. The slowly vary-ing parts of the intermolecular interactions are not im-portant for this local structure and could be varied es-sentially independently to help in the determination ofother properties as is implicitly done in the full model.Based on previous work with LMF theory [28, 29, 31],we examine two basic areas where we expect the differ-ent contributions to play varying but important roles –

2

FIG. 1. (Color online) Schematic diagram of the SPC/E watermodel listing its various geometric parameters and interactionparameters. The O-H bond length and H-O-H angle are fixed,such that the molecule is rigid. The LJ well depth is ǫLJ =0.65 kJ/mol. The oxygen site is depicted as a large red circle,while the hydrogen atoms are shown as smaller, gray circles.

bulk thermodynamics and nonuniform structure. Theshort ranged interactions responsible for the hydrogen-bonding network are clearly necessary in all cases. LMFtheory allows us to determine the relative importanceof dispersive attractions and long-ranged dipolar attrac-tions in these applications using simple analytical correc-tions for thermodynamics and an effective external fieldfor nonuniform structure.

In the next section we discuss the separation of the wa-ter potential into short and long ranged parts and showthat a minimal short ranged reference model can veryaccurately describe atom-atom correlation functions andother properties of the hydrogen bond network in bulkwater. In Sec. 3 we examine the thermodynamic impli-cations of this picture, focusing on two anomalous prop-erties of bulk water qualitatively well described by thefull water model, the density maximum at one atmo-sphere pressure and the behavior of the “internal pres-sure” (∂E/∂V )T , which has a temperature and densitydependence opposite to that of most organic solvents.Then in Sec. 4 we look at the effects of the unbalancedlong-ranged electrostatic and dispersion forces at aque-ous interfaces. LMF theory reveals strong similaritiesbetween the behavior of water at the liquid-vapor in-terface and a planar hydrophobic wall, consistent withprevious work, and provides new insight into the relativeimportance of electrostatic and dispersion forces and thetransition from small to large scale hydrophobicity as theradius of a nonpolar solute is increased.

FIG. 2. (Color online) Optimal hydrogen bonding config-uration of water taken from two molecules in ice Ih. LJcores are depicted as gray transparent spheres with a diam-eter σLJ = 3.16 A, while the hydrogen bond between waterswith oxygens separated by 2.75 A is illustrated by a dashed,blue cylinder. Oxygen and hydrogen atoms are colored redand white, respectively.

II. LOCAL HYDROGEN BONDS IN FULL AND

TRUNCATED WATER POTENTIALS

In this paper we consider one of the simplest and mostwidely used water models, the extended simple pointcharge (SPC/E) model [5], but similar ideas and conclu-sions apply immediately to most other members of thisclass. As shown in Fig. 1, SPC/E water consists of a LJpotential as well as a negative point charge centered atthe oxygen site. Positive point charges are fixed at hy-drogen sites displaced from the center at a distance of 1 Awith a tetrahedral HOH bond angle. It is a remarkablefact that this simple model can reproduce many struc-tural, thermodynamic, and dielectric properties of bulkwater as well as those of water in nonuniform environ-ments around a variety of solutes and at the liquid-vaporinterface.

In the following we use the perspective of perturbationand LMF theory to help us see how this comes about. Weuse these ideas here not to suggest more efficient simu-lations using short ranged model potentials but ratheras a method of analysis that provides physical insightinto features of the full model as well. Since a detaileddescription and justification of LMF theory is given else-where [30], we will focus on qualitative arguments andjust quote specific results when needed.

Fig. 2 gives some insight into why a perturbation pic-ture based on the dominance of strong short ranged forcesin uniform environments could be especially accurate forbulk SPC/E and related water models. This shows twoadjacent water molecules with a separation of 2.75 A thatform an optimal hydrogen bond as seen in the structureof ice Ih. Hydrogen bonding in this model is driven bythe very strong attractive force between opposite chargeson the hydrogen and oxygen sites of adjacent properlyoriented molecules. Proper orientation permits similar

3

strong bonds to form with other molecules, leading toa tetrahedral network in bulk water. The gray circlesdrawn to scale depict the repulsive LJ core size as de-scribed by the usual parameter σLJ = 3.16 A. The sub-stantial overlap indicates a large repulsive core force op-posing the strong electrostatic attraction, finally result-ing in a nearest neighbor maximum in the the equilibriumoxygen-oxygen correlation function of 2.75 A.It is interesting to note that the first BNS water

model introduced in 1970 used a smaller core size σLJ =2.82 A [4]. However a much larger LJ core with strongcore overlap at typical hydrogen-bond distances is acommon property of almost every water model intro-duced since then and seems to be a key feature neededto get generally accurate results from simple classicalpoint charge models. Evidentially the highly fluctuat-ing local hydrogen-bond network in these models arisesfrom geometrically-frustrated “charge pairing”, wherethe strong LJ core repulsions and the presence of othercharges on the acceptor water molecule oppose the closeapproach of the strongly-coupled donor and acceptorcharges.We can test the accuracy of this picture by considering

various truncated or “short” water models where slowlyvarying long ranged parts of the Coulomb and LJ inter-actions in SPC/E water are completely neglected. Wefirst consider a Gaussian-truncated (GT) water model,already studied by LMF theory [28, 29, 31]. Herethe Coulomb potential is separated into short and longranged parts as

v(r) =1

r=

erfc(r/σ)

r+

erf(r/σ)

r= v0(r) + v1(r), (1)

where erf and erfc are the error function and comple-mentary error function, respectively. The short-rangedv0(r) is the screened electrostatic potential resulting froma point charge surrounded by a neutralizing Gaussiancharge distribution of width σ. Hence v0(r) vanishesrapidly at distances r much greater than σ while at dis-tances less than σ the force from v0(r) approaches thatof the full 1/r potential.In GT water, depicted in Fig. 3a, the Coulomb poten-

tial associated with each charged site in SPC/E water isreplaced by the short-ranged v0 with no change in the LJinteraction. As suggested by Fig. 2, important featuresof the local hydrogen-bond network should be well cap-tured by such a truncated model if the cutoff distancecontrolled by the length parameter σ in Eq. (1) is cho-sen larger than the hydrogen bond distance. FollowingRefs. [28] and [31], here we make a relatively conservativechoice of σ = 4.5 A, but values as small as 3 A give es-sentially the same results. The circles are drawn to scalewith diameters σ and σLJ .The basic competition between very strong short

ranged repulsive and attractive forces in the hydrogenbond depicted in Fig. 2 should be captured nearly aswell by an even simpler reference model where the LJpotential is truncated as well, and replaced by the re-

FIG. 3. (Color online) Schematic diagrams of (a) GT and (b)GTRC water models. Truncated interactions are indicatedby dashed lines, while full interaction potentials are indicatedby solid lines. LJ interactions are represented by black lines,while oxygen and hydrogen electrostatic interaction potentialsare shown as red and gray lines, respectively.

pulsive force reference potential u0(r) used in the WCAperturbation theory for the LJ fluid [40]. The result-ing Gaussian truncated repulsive core (GTRC) model isschematically shown in Fig. 3b.

As discussed in perturbation theories of simple liq-uids [15, 40], a well-chosen reference system should accu-rately reproduce bulk structure present in the full systemat the same fixed density and temperature. As illustratedby the pair distribution functions in Fig. 4, bulk GT andGTRC water models have a liquid state structure virtu-ally identical to that in the full SPC/E model. This verygood agreement is also reflected in other properties ofthe hydrogen-bond network. We directly examined thehydrogen bonding capabilities of GT and GTRC watermodels through the calculation of the average number ofhydrogen bonds per water molecules, 〈nHB〉, as well asthe probability distribution of a water molecule takingpart in nHB hydrogen bonds, P (nHB), using a standarddistance criterion of hydrogen bonds, ROO < 3.5 A andθHOO′ < 30◦, where ROO is the oxygen-oxygen distanceand θHOO′ is the angle formed by the H-O bond vectoron the hydrogen bond donating water molecule and theO-O′ vector between the oxygen on the donor water (O)and the acceptor oxygen (O′) [25]. For both GT andGTRC water models, 〈nHB〉 and P (nHB) were calcu-lated at temperatures ranging from 220-300 K, and werefound to be nearly identical to the analogous quantitiesin the full SPC/E model (data not shown). These find-ings give credence to the idea that these two truncatedmodels reproduce the hydrogen-bond network of the fullmodel to a high degree of accuracy.

These truncated models offer a minimal structural rep-resentation of bulk water as a fluctuating network ofshort ranged bonds determined mainly by the balancebetween the very strong electrostatic attraction betweendonor and acceptor charges and the very strong repul-sion of the overlapping LJ cores. We can view them asprimitive water models in their own right, analogous toother simplified models recently proposed, which capturevery well arguably the most important structural fea-

4

0

0.5

1

1.5

2

2.5

3

3.5

-0.03

-0.01

0

0.01

0.03

1 2 3 4 5 6 71 2 3 4 5 6 7

FIG. 4. (Color online) (a) Oxygen-oxygen, oxygen-hydrogen, and hydrogen-hydrogen site-site pair distribution functions,gOO(r), gOH(r), and gHH(r), respectively, for the three water models under study at T = 300 K and v = 30.148 A3. gHH andgOH have been shifted by 0.5 and 1 units, respectively, for clarity. (b) Differences between gOO(r) of the full model and thatof the designated reference systems, ∆gOO(r).

ture of bulk water, the hydrogen bond network, and itis instructive to see what other properties of water suchminimal network models can describe. But correctionsfrom neglected parts of the intermolecular interactionsare certainly needed for bulk thermodynamic and dielec-tric properties and for both structure and thermodynam-ics of water in nonuniform environments. LMF theoryprovides a more general framework where the truncatedmodels are viewed as useful reference systems that canbe systematically corrected to achieve good agreementwith full water models. We will use both viewpoints inthe next section.

III. THERMODYNAMIC ANOMALIES OF

BULK WATER USING FULL AND TRUNCATED

WATER POTENTIALS

A. Density maximum

Now we turn our attention to the thermodynamics ofbulk water. For a fixed volume V , temperature T , andnumber of molecules N , the pressure and other thermo-dynamic properties of the GT and GTRC systems willnot generally equal those of the full system. However,because of the accurate reference structure, we can cor-rect the thermodynamics using simple mean-field (MF)arguments. Thus we can define the pressure in the fullsystem to be the sum of the short-ranged reference pres-sure and a long-ranged correction, P = P0 + P1.Simple corrections to the energy and pressure of

the GT model from this perspective were recently de-rived [31]. With σ = 4.5 A, these corrections are rela-tively small and were ignored in most earlier work usingtruncated water models but they are conceptually impor-

tant in revealing the connections between truncated mod-els and perturbation theory and are required for quanti-tative agreement. The pressure correction P1 = P q

1 forthe GT model arises only from long-ranged Coulomb in-teractions and is given as

P q1 = −

kBT

2π3/2σ3

ǫ− 1

ǫ, (2)

where ǫ is the dielectric constant.In the case of the GTRC model, the need for a thermo-

dynamic correction is much more obvious since we haveto correct for the absence of LJ attractions as well. Herewe adopt the simple analytic correction used in the vander Waals (vdW) equation derived from WCA theory forthe LJ fluid, as discussed in Ref. [39]. Thus P1 = P q

1 −aρ2

for the GTRC potential, where

a ≡ −1

2

∫

dr2 u1(r12) (3)

and u1 is attractive part of the LJ potential. This sim-ple approximation does not give quantitative results butdoes capture the main qualitative features and we use ithere to emphasize the point that both the long rangedCoulomb and dispersion force corrections to bulk GTRCwater can be treated by simple perturbation methods.We can test the accuracy of these corrections by using

them to help determine the temperature TMD at whichthe density maximum of the full SPC/E water model ata constant pressure of 1 atm should occur. This canalternatively be defined as the temperature at which thethermal expansion coefficient, αP , is zero. Accordingly,we seek to evaluate αP using the relation

αP ≡1

v

(

∂v

∂T

)

P

= −1

v

(

∂P

∂T

)

v

(

∂v

∂P

)

T

, (4)

5

0.985

0.99

0.995

1

1.005

1.01

1.015

-0.6

-0.4

-0.2

0

0.2

0.4

220 230 240 250 260 270 280 220 230 240 250 260 270 280

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

FIG. 5. (Color online) The dependence of (a) density and (b)pressure for the corrected reference models as a function oftemperature. The analogous quantities for the models with-out MF corrections are shown in (c) and (d), respectively.ρ(T ) is calculated at a constant pressure of 1.0 atm and P (T )is calculated at a fixed volume of v = 30.148 A3. Full SPC/Edata for ρ(T ) at constant P was taken from the work of Ash-baugh et al. [3].

where v = V/N is the volume per particle. Usingthe last expression we can determine where the quan-tity (∂P/∂T )v = 0 by monitoring the corrected pres-sure of the reference systems while changing the tem-perature at a fixed density. This can be done by simula-tion in the canonical ensemble. The fixed density ensuresthat the structure of the reference and full systems arevery similar, as assumed in the derivation of the correc-tions in Eqs. (2) and (3). We can also determine TMD

through the first equality in Eq. (4) by finding where(∂v/∂T )P = 0. Thus we simulate GT and GTRC waterat constant pressures of P0 = P −P1, where P = 1.0 atmis the pressure in the full system. Note that the correc-tion P q

1 ≡ P q1 (T ; ǫ(T )) is temperature-dependent, as is

the dielectric constant ǫ, so that we are not moving alongan isobar in P0, but an isobar in P . The temperature-dependent values of ǫ were taken to be the experimentalvalues [16].Figs. 5a and 5b give the density and pressure of full

SPC/E water and the corrected reference models as afunction of temperature. As expected, the inclusion of P q

1

in the pressure of GT water quantitatively corrects thedensity and pressure of this system. However, the MFcorrection applied to GTRC water, P0 = P − P q

1 + aρ2,is not as accurate, although the dependence of ρ on T isqualitatively well captured. These remaining errors arisefrom our use of the simple van der Waals aρ2 correctionfor the long ranged part of the LJ potential. This level ofagreement is typical when this correction is used in pureLJ fluids [39] and a full WCA perturbative treatment ofthe attractive portion of the LJ potential in GTRC waterwould likely lead to quantitatively accurate results [40].We now turn to the alternate and less accurate inter-

pretation of the GT and GTRCmodels as primitive watermodels in their own right. Do these models at an uncor-rected pressure of 1 atm have a density maximum andhow well does it compare to that of the full model? Tothat end, we find where (∂v/∂T )P = 0 in each model byvarying the temperature along an isobar using MD simu-lations in the isothermal-isobaric ensemble at a constantpressure of 1 atm. By requiring the same pressure in thefull and reference models, we probe structurally differ-ent state points in general and there is no guarantee thatthe density and temperature of the reference systems ata density maxima (if present) will be similar to that inthe full system. Nevertheless Fig. 5c shows that the GTmodel does have a density maximum very similar to thatof the full model. This is because the pressure correctionto the density from the long-ranged Coulomb interactionsin Eq. (2) is very small on the scale of the graph. In con-trast, the uncorrected GTRC model does not exhibit adensity maximum at P = 1.0 atm, even upon cooling to50 K (data not shown).These results should be compared to earlier work where

the TIP4P water potential was approximated by a sim-pler “primitive model” [20]. In that work, the repul-sive LJ core was mapped onto a hard-sphere potential,hydrogen bonding was captured by a square-well poten-tial, and long-ranged dipole-dipole interactions were rep-resented with a dipolar potential. The equation of statewas found using a perturbative approach, and thermody-namic quantities were analyzed. The authors of Ref. [20]found that the inclusion of dispersion forces does notlead to a density maximum, and only when both dis-persive interactions and long-ranged dipole-dipole inter-actions were taken into account did a density maximumappear.To provide some understanding of these differing re-

sults, we analyze the structure of the uncorrected GTand GTRC reference models in comparison to the fullmodel at the common pressure of one atmosphere. Theoxygen-oxygen radial distribution functions, gOO(r), foreach of the three water models at T = 300 K are de-picted in Fig. 6a. The GT model is in good agreementwith the full model, consistent with its accurate descrip-tion of the bulk water density and the density maximum.In contrast, as shown later in Fig. 9, the coexisting liquiddensity of GTRC water is about about 15% lower thanthat of the full water model. Nevertheless the first peakof gOO(r) in GTRC water is higher than that of the fullwater model due to better formation of local hydrogenbonds. As shown in the inset, a molecule of GTRC wa-ter has slightly fewer hydrogen bonds on average thanfull and GT water models for temperatures higher than240 K. However the hydrogen bond efficiency shown inFig. 6b,

ηHB =〈nHB〉

〈nNN〉, (5)

where 〈nNN〉 is the average number of nearest-neighborssatisfying ROO < 3.5 A, indicates that GTRC water is

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0.5

1

1.5

2

2.5

3

3.5

2 3 4 5 6 7

3.4

3.5

3.6

3.7

3.8

3.9

4

220 240 260 280 300

0.65

0.7

0.75

0.8

0.85

0.9

0.95

220 240 260 280 300

FIG. 6. (Color online) (a) The oxygen-oxygen pair distribution function, gOO(r), for the three water models at T = 300K. Inset: The average number of hydrogen bonds per water molecule as a function of temperature, 〈nHB(T)〉, for full andtruncated water models. (b) Hydrogen bonding efficiency ηHB as a function of temperature. All results were obtained at aconstant uncorrected pressure of 1 atm.

about 10 percent more efficiently hydrogen bonded to itsavailable neighbors at all temperatures. In this sense thelow density GTRC water at P = 1.0 atm is structurallymore ice-like than the full water model.

These results provide some insight into earlier firstprinciples simulations of liquid water using density func-tional theory [21, 32, 38]. The standard exchange-correlation functionals used there can give a good de-scription of local hydrogen bonding, but do not includeeffects of van der Waals interactions. These simulationsproduced a decrease in the bulk density of water accom-panied by increased local structural order very similarto that seen here for GTRC water. Moreover, whendispersive interactions were crudely accounted for, theyobserved much better agreement with experiment, incomplete agreement with our findings for perturbation-corrected GTRC water.

Our results indicate that van der Waals attractionsplay the role of a cohesive energy needed to achieve thehigh density present in SPC/E water at low pressure, asdemonstrated by the qualitative accuracy of Eq. (3) andthe good agreement of the GT model. Evidentially a den-sity maximum can arise only when additional somewhatless favorably bonded molecules are incorporated into theGTRC network to produce the full water density. If thelocal hydrogen bond network of water at the correct bulkdensity is properly described, long ranged dipolar forcesare not needed to obtain the correct behavior of ρ(T ).Indeed, LJ attractions are not needed either providedthat the proper bulk density is prescribed by some othermeans. Thus we found that if GTRC water is kept at ahigh constant pressure of 3 katm, where its bulk densityis close to that of the full water model at ambient con-ditions, a density maximum is also observed (data notshown).

B. Internal pressure

We further employ the reference water models to ex-plain the anomalous “internal pressure” of water [33].For a typical van der Waals liquid, the internal pressureis given by Pi = (∂ε/∂v)T ≈ aρ2 for low to moderatedensities, where ε = E/N is the energy per molecule. Infact, it was recently shown by computer simulation thatthe portion of the internal pressure due to the attractionsin a LJ fluid displays this aρ2 dependence even at highdensities [13]. Water, on the other hand, displays a neg-ative dependence of Pi on density. It is this anomalousbehavior that we seek to explain.We begin by partitioning the internal energy of the

system as

ε = εLJ + εq, (6)

where εLJ is the Lennard-Jones contribution to the en-ergy and εq is the energy due to charge-charge interac-tions (note that the change in kinetic energy when per-turbing the volume at constant T is zero, so we onlyconsider the potential energy). We can then write theinternal pressure as

Pi =

(

∂ε

∂v

)

T

= PLJi + P q

i . (7)

This decomposition of Pi will allow us to determine whichmolecular interactions are responsible for the strange de-pendence of this quantity on ρ.Fig. 2 suggests the following qualitative picture. At

a given temperature and density the dominant hydrogenbond contribution to the energy ε is determined fromthe balance between strong repulsive forces from the LJcores and strong attractions from the more slowly vary-ing Coulomb interactions between donor and acceptor

7

0

20

40

60

80

100

-300

-250

-200

-150

-100

-50

0.8 1 1.2 1.4 1.6 1.8 2

-200

-160

-120

-80

-40

0

40

0.8 1 1.2 1.4 1.6 1.8 2

FIG. 7. (Color online) (a) The electrostatic contribution tothe internal pressure, P q

i , and (b) the analogous contributionfrom LJ interactions, PLJ

i . The total internal pressure as afunction of density is shown in the inset. Lines are guides tothe eye.

charges. The Coulomb contribution P qi to the internal

pressure Pi(T, ρ) is positive since a small positive changein volume reduces the negative Coulomb energy and sim-ilarly the LJ core contribution to PLJ

i is negative. If thedensity is now varied at constant temperature we wouldexpect the changes in Pi(T, ρ) to be dominated by therapidly varying LJ core forces.Conversely, to the extent that the repulsive LJ cores

are like hard spheres, they would contribute no tempera-ture dependence to the internal pressure at fixed density.Thus we expect the more slowly varying Coulomb forcesto largely determine how the internal pressure varies withtemperature at fixed density. The results given below arein complete agreement with these expectations.We evaluated Eq. (7) by performing MD simulations

of water in the canonical ensemble for various volumes atT = 300 K. The dependence of the internal pressure ondensity at T = 300 K is shown in Fig. 7. Note that thetotal internal pressure, Pi, becomes increasingly negativeas ρ is increased, in direct opposition to the aρ2 depen-dence given by the vdW equation of state. However, itis known that as the density of a LJ fluid is increased tohigh values so that neighboring repulsive cores begin to

-1

0

1

2

3

4

5

6

7

8

-7

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-5

-4

-3

-2

-1

240 260 280 300 320 340

-5

-4

-3

-2

-1

0

1

2

3

4

5

240 260 280 300 320 340

FIG. 8. (Color online) (a) The electrostatic contribution tothe internal pressure, P q

i , and (b) the analogous contributionfrom LJ interactions, PLJ

i , both as a function of temperatureat a fixed volume of v = 29.9 A3. The total internal pressureas a function of temperature is shown in the inset. Lines areguides to the eye.

overlap, the total Pi exhibits a maximum, after which theinternal pressure becomes increasingly negative from thedominant contribution of the repulsive interactions [13].

As shown in Fig. 2 there is substantial overlap of therepulsive LJ cores between nearest neighbors in SPC/Ewater. The repulsive interactions from these LJ coresdominate the density dependence of both ε and Pi forSPC/E and related water models, as evidenced by thesimilarity of the internal pressures of both the full andGTRC water models in Fig. 7. Although εq > εLJ forall density, εq does not exhibit very large changes uponincreasing density, a direct consequence of the ability ofwater to maintain its hydrogen bond network under theconditions studied. Thus the density dependence of theinternal pressure of SPC/E water is actually similar tothat of a LJ fluid but one at a very high effective densitywith substantial overlap of neighboring cores.

In addition to the anomalous density dependence ofPi, the temperature dependence of the internal pressureof water has also been called an anomaly [33]. For mostorganic liquids (and vdW fluids), the internal pressuredecreases with increasing temperature, but that of waterincreases when the temperature is increased, as shown in

8

Fig. 8. Using the concepts presented above, we can ra-tionalize this behavior in terms of molecular interactions.By decomposing Pi into its electrostatic and LJ compo-nents, we find that P q

i dominates the temperature depen-dence of the internal pressure, increasing with increasingtemperature, while PLJ

i is dominated by repulsive in-teractions at all temperatures studied, as evidenced byits negative value for all T . As the temperature of thesystem is increased, the number of ideally tetrahedrallycoordinated water molecules decreases, and the hydrogenbond network becomes increasingly “flexible”. Therefore,if one increases the volume of the system at high T , wa-ter will more readily expand to fill that volume. But anincrease in the electrostatic energy will also occur due toa slight decrease in the number of (favorable) hydrogenbonding interactions. This will happen to a lesser extentat low temperatures, when the hydrogen bond network ismore rigid and the thermal expansivity of water is lower.

IV. UNBALANCED FORCES IN

NONUNIFORM AQUEOUS MEDIA FROM THE

VIEWPOINT OF LMF THEORY

In contrast to uniform systems, a net cancellation oflong ranged forces does not occur in nonuniform envi-ronments, and these unbalanced forces can cause signif-icant changes in the structure and thermodynamics ofthe system [29, 39]. As shown above, the bulk struc-ture of both the GT and GTRC models are very similarto that of the full water model at a given temperatureand density. But interfacial structure and coexistencethermodynamic properties of the uncorrected referencemodels can be very different. For example, GTRC wa-ter still has a self-maintained liquid-vapor (LV) interfaceat T = 300 K as illustrated in Fig. 9, even though theLJ attractions are ignored, because of the strong chargepairing leading to hydrogen bond formation. Howeverits 90-10% interfacial width increases to w ≈ 4.9 A fromthe w ≈ 3.5 A seen in both GT water and the full watermodel, and the coexisting liquid density of GTRC wateris about about 15% lower. In contrast, the density pro-file of the GT model with LJ interactions fully accountedfor is in very good qualitative agreement with that of thefull model. This strongly suggests that if local hydrogenbonding is properly taken into account, the equilibriumstructure of the LV interface of water is governed mainlyby long ranged LJ attractions, with long ranged dipole-dipole interactions playing a much smaller role. It is theexact balance of these long ranged interactions we seekto examine in this section.LMF theory provides a framework in which the aver-

aged effects of long ranged forces are accounted for byan effective external field [30]. It has previously beenused mainly as a computational tool to permit very ac-curate determination of properties of the full nonuniformsystem while using a numerical simulation of the shortranged reference system in the presence of the effective

0

0.01

0.02

0.03

0.04

-15 -10 -5 0 5 10

FIG. 9. (Color online) Density profiles of oxygen sites at theliquid-vapor interface of Full, GT, and GTRC water models.The Gibbs dividing surface of each interface is located at z =0.

field [10, 12, 29]. But the effective or renormalized fieldalso gives a convenient and natural measure of the impor-tance of long ranged forces in different environments. Inthis section we use the renormalized external fields deter-mined directly from simulations of interfaces in the fullSPC/E water model along with simulations of truncatedwater models to quantitatively examine the relative influ-ence of the local hydrogen bond network and unbalancedlong-ranged Coulomb and van der Waals forces.

A. Water-vapor and water-solid interfaces

We first consider the LV interfaces of the SPC/E, GT,and GTRC water models shown in Fig. 9. The removalof long-ranged electrostatics in the GT model leavesthe density distribution virtually unchanged, whereas re-moval of the LJ attractions in GTRC water has a sub-stantial impact on ρ(z). To understand this behavior,we focus our attention on the impact of the averaged un-balanced forces from the long-ranged electrostatic and LJinteractions, as determined in LMF theory from the effec-tive external fields VR1 and φLJ

R1, respectively and definedbelow. The unbalanced force F acting on an oxygen sitefrom the LMF potentials is given by

FO(r) = −∇rφLJR1(r)− qO∇rVR1(r). (8)

Here qO is the partial charge on the oxygen site andVR1(r) is the slowly-varying part of the effective elec-trostatic field, given by

VR1(r) =1

ǫ

∫

dr′ρq(r′)v1 (|r− r′|) , (9)

where ρq(r) is the total charge density of the system.The other contribution φLJ

R1(r) is the field arising from

9

the unbalanced LJ attractions on the oxygen site (wherethe LJ core is centered), given by

φLJR1(r) =

∫

dr′[

ρ(r′)− ρB]

u1 (|r− r′|) , (10)

with ρ(r) indicating the nonuniform singlet density distri-bution of oxygen sites and ρB defined as the bulk densityof oxygen sites at the state point of interest [30, 39]. Sincethe hydrogen sites lack LJ interactions, the unbalancedLMF force acting on a hydrogen site is due exclusivelyto electrostatics,

FH(r) = −qH∇rVR1(r). (11)

Given its importance in the density distribution of wa-ter, it may seem natural to examine the components ofthe LMF force on the oxygen sites, FO(z), shown in theinset of Fig. 10a. Naive examination of the relative mag-nitude of these force functions would lead to the con-clusion that long-ranged electrostatics are the dominantunbalanced force at the LV interface. However, VR1 alsointeracts with hydrogen sites and one should instead con-sider the net forces from long ranged Coulomb and LJinteraction felt by an entire water molecule at these in-terfaces.This ensemble averaged net molecular force 〈F〉

(Fig. 10b) clearly indicates that the net unbalanced forceat an interface is almost entirely due to long-ranged LJattractions from the bulk, which pull water moleculesaway from the interface. The long-ranged Coulomb con-tributions to the average force on a water molecule are es-sentially negligible in comparison. This is not surprisingsince water molecules are neutral, and it has previouslybeen shown that the small net interfacial electrostaticforce simply provides a slight torque on water moleculesin this region [29]. This torque has little effect on theoxygen density distribution, as illustrated by the goodagreement of the GT model density profile with that ofthe full water model in Fig. 9. However, it plays a keyrole in determining electrostatic and dielectric properties,which are strongly affected by the behavior of the totalcharge density, and here the uncorrected GT model givesvery poor results [29, 30].It is also instructive to compare the unbalanced long

ranged forces at the LV interface to those at the liquid-solid (LS) interface between water and a model hy-drophobic 9-3 LJ wall introduced by Rossky and cowork-ers [22], as shown in Fig. 10b. Despite the large differ-ences in the density profiles shown in Fig. 10a, the netunbalanced forces on molecules at the LV and LS in-terfaces are remarkably similar for all z until moleculesencounter the harsh repulsion of the wall and an accuratesampling of 〈F(z)〉 by simulation cannot be made. Wa-ter molecules can sample all regions in the liquid-vaporinterface, leading to a smooth 〈F(z)〉 at smaller z.Indeed, the net molecular force due explicitly to a

configurational average of the attractive u1(r) acting onmolecules present at each z-position is in outstanding

0

0.01

0.02

0.03

0.04

-0.2

0

0.2

0.4

0.6

0 2 4 6 8 10 12 14 16 18

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8 10 12 14

FIG. 10. (Color online) (a) Density distributions as a functionof the z-coordinate for the hydrophobic LS interface and theLV interface of water. (b) Ensemble averaged net force on awater molecule due to VR1 (open symbols) and φLJ

R1 (closedsymbols) at the LV (circles) and LS (squares) interfaces andthe density profiles of the two systems. Solid lines indicatethe net force due to u1. The black dashed line at z = 0 Aindicates the position of the hydrophobic wall. The Gibbsdividing interface of the LV system is located at z = 2.34 A,in order to make comparison with the water-wall interface.Inset: Forces on oxygen sites only, determined by evaluatingthe gradient of the corresponding LMF potentials. Labelingfor the inset is the same as that in (b).

quantitative agreement with that arising from φLJR1(z) for

all adequately sampled regions in the liquid, as illustratedby the solid lines in Fig. 10b. This serves largely as confir-mation of the validity of the mean-field treatment inher-ent in LMF theory within the liquid slabs. Deviationsbetween the two quantities for distances less than theGibbs dividing surface are a reflection of the increasingeffect of larger force fluctuations due to long-wavelengthcapillary waves not well described by mean field theory.The relative magnitudes of the components of 〈F〉 forthe LV and LS interfaces are strikingly similar, with thenet unbalanced LJ force

⟨

F(

z;φLJR1

)⟩

reaching its max-

imum value of slightly less than kBT/A near the Gibbsdividing interface and the repulsive boundary of the wall,respectively.

The similarities of the unbalanced forces at the LV and

10

the hydrophobic LS interfaces of water and the dom-inance of the LJ attractions are completely consistentwith the analogies commonly drawn between these twosystems [7, 9, 35] and used in the LCW theory of hy-drophobicity [24, 36, 39]. A common criticism of LCWtheory is its apparent neglect of the hydrogen bond net-work of water and the use of a van der Waals like expres-sion for the unbalanced force at an interface. Althoughsome features of the network are implicitly captured byusing the experimental surface tension and radial dis-tribution function of water as input to the theory, elec-trostatic effects at the interface, including dipole-dipoleinteractions, are ignored. However, this assumption isjustified since the averaged effects of long-ranged dipole-dipole interactions, accounted for by VR1, are shown toindeed be negligible at a hydrophobic interface (Fig. 10).LCW theory correctly describes the unbalanced LJ at-tractions from the bulk, which dominates the behaviorat both the liquid-vapor and extended hydrophobic in-terfaces.

B. Crossover from small to large length scale

hydration

LCW theory combined the idea of unbalanced forceswith experimental data in water to predict a crossover inthe solvation of spherical hydrophobic solutes occurringat a radius of about 1 nm [24]. While the local hydrogenbond network can be maintained around smaller solutes,some bonds must be broken on larger length scales, lead-ing to an enthalpically dominated regime [7, 9, 35]. Herewe use LMF theory and a direct analysis of the unbal-anced long ranged forces to provide further physical in-sight into how this crossover comes about. We study thehydration of spherical hydrophobic solutes, which inter-act with the oxygen site of water via a solute-water LJpotential usw(r) with a fixed well depth of εsw = 0.19279kcal mol−1 and varying solute-water interaction length-scales, σsw, ranging from 2.0 A to 15.0 A. This size rangespans the crossover from small to large length-scale hy-drophobic hydration as determined by both simulationand theory [2, 24].As shown earlier, the unbalancing force from LJ attrac-

tions dominates that due to long ranged electrostatics atnonpolar interfaces so we focus only on the LJ forces inour analysis of the length scales of hydration. The totalrenormalized external solute-solvent field is the sum ofthe bare solute-solvent field and the slowly-varying LMFpotential φLJ

R1(r) in Eq. (10), which accounts for unbal-anced LJ forces from the nonuniform water (oxygen) den-sity distribution around the solute. It can be written as

φLJR (r) = usw(r) + φLJ

R1(r). (12)

We compare these potentials for small and large LJ so-lutes in Fig. 11. At small solute sizes, the renormalizedfield φLJ

R exhibits a repulsive core nearly identical to thatof the bare usw, and the effective attractions are hardly

-0.5

0

0.5

1

1.5

2

0.8 1 1.2 1.4 1.6 1.8

FIG. 11. (Color online) Comparison of renormalized solute-water vdW potentials for σsw = 2.0 A (squares) and σsw =15.0 A (circles). The full solute-water LJ potential usw is alsoshown for comparison (black dash-dot line). Note that thex-axis is scaled by the solute-water interaction length-scaleσsw.

altered upon renormalization of the potential, indicat-ing only a small unbalanced force around a small solute.The drive to maintain the hydrogen bond network aroundsmall solutes dominates the water structure, and solute-water and water-water LJ attractions are found to havelittle effect on the solvation structure [9].In contrast, water cannot completely preserve its hy-

drogen bond network at an interface around a large soluteand one hydrogen bond per interfacial water moleculetends to be broken on average [7, 9, 22, 35]. This cre-ates a soft fluctuating interface for which partial dryingcan be induced by unbalanced attractions from the bulk.These are re-expressed in LMF theory as an effective re-pulsion pushing water away from the solute surface, asillustrated by the renormalized potential for the large so-lute in Fig. 11. In order for water to wet such an extendedsolute surface, this large effective repulsion must be over-come. In hydrophilic surfaces this typically arises fromstrongly attractive polar groups on the surface, such ashydrogen bonding sites or charged groups, which also canstrongly perturb local hydrogen bond configurations.To illustrate the interplay between unbalanced forces

from the bulk, water-water hydrogen-bonding, and thelength-scale dependence of hydration, we study the hy-dration of a small (σsw = 2.0 A) and large (σsw = 15.0 A)LJ solute by SPC/E water and the short-ranged GTRCwater model. Since the GTRC model lacks LJ attrac-tions, the unbalancing force from the bulk is absent butthe local hydrogen bonding network of water is accuratelycaptured. The GTRC model thus properly describes thedominant crossover behavior of retaining or breaking hy-drogen bonds in the small or large length scale regimes,respectively, but the subsequent large length scale inter-face properties will be incorrect. Comparing GTRC wa-

11

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 2 4 6 8 10

0

0.01

0.02

0.03

0.04

0.05

10 12 14 16 18 20 22 24

FIG. 12. (Color online) Singlet density distributions of water oxygen sites, ρ(r), around LJ solutes with σsw = 2.0 A (a) andσsw = 15.0 A (b) for Full and GTRC water models at pressures of P = 1.0 atm and P0 = P −P1, respectively. Data for GTRCwater in the presence of the renormalized LMF solute potential is also shown for the large solute (GTRC-LMF).

ter with the full water model provides more insight intothe relative importance of the local network and the un-balancing force on the structuring of water around eachsolute.

Singlet density distributions of the water oxygen sitesaround the small and large LJ solute are shown inFig. 12a and 12b, respectively. In the small solute regime,ρ(r) for GTRC water is essentially identical to that of thefull water model. This provides dramatic confirmationof the standard physical picture that small length scalehydrophobicity is almost completely dominated by theneed to maintain local hydrogen bonding around the so-lute. Unbalanced forces from either the LJ or long-rangedCoulomb interaction play essentially no structural role inthis regime.

In contrast, although the GTRC model correctly de-scribes the necessary breaking of local hydrogen bondsaround a large solute in Fig. 12b the details of the result-ing interface profile are very different from that of the fullwater model. Removal of LJ attractions in GTRC watereliminates the large effective repulsion at the extendedsolute surface, and uncorrected GTRC water appears towet the surface of the nonpolar solute. However, the in-terface is soft and when the unbalanced force from φLJ

R1

in Fig. 10 as given by LMF theory is also taken intoaccount, depletion and the correct structuring of waterat the solute surface is very accurately described by theGTRC-LMF curve in Fig. 12b.

Although the qualitative dependence of the unbalancedforce on solvation length scale is depicted in Fig. 11, amore quantitative metric of this behavior is desirable. Tothat end, we introduce the mean solvation force acting

on water due to the renormalized LJ solute external field,

FS

[

φLJR

]

= −

∫

drρ(r)∂usw(r)

∂r−

∫

drρ(r)∂φLJ

R1(r)

∂r

= FS [usw] + FS

[

φLJR1

]

. (13)

As illustrated in Fig. 13, the unbalanced force due towater-water LJ attractions dominates in the large-scaleregime, and can only be overcome by unphysically largesolute-water LJ attractions. In the small scale regime,solute-water attractions are comparable in magnitude tothe unbalancing force, and can be larger for certain so-lute sizes (the relative magnitudes are dependent uponthe value of εsw). However, as the GTRC model shows,the important physics in this regime is simply maintain-ing the hydrogen bond network, and this imposes a nearconstant solvation structure as the water-solute attrac-tions are varied [9, 17].Upon normalizing the mean unbalanced force by the

surface area of the solute,

fS[

φLJR1

]

= FS

[

φLJR1

]

/4πσ2sw, (14)

a transition from scaling of FS with solute volume to scal-ing with surface area occurs at the small to large lengthscale solvation crossover length of roughly 10 A, as ev-idenced by the plateau in fS depicted in the inset ofFig. 13. This plateau is similar to that which occurs inthe solvation free energy [18, 24], and provides anotherindicator of the transition from small to large length scalehydrophobicity.

V. CONCLUSIONS

In this work, we have examined the different roles ofshort and long ranged forces in the determination of the

12

0

25

50

75

100

125

150

175

200

2 4 6 8 10 12 14 16

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2 4 6 8 10 12 14 16

FIG. 13. (Color online) Components of the mean solvationforce FS due to φLJ

R1 and usw as a function of the solute-waterinteraction length scale, σsw. The inset displays the solvationforce due to φLJ

R1 scaled by the surface area of the solute.

structure and thermodynamics of uniform and nonuni-form aqueous systems, using concepts inherent in classi-cal perturbation and LMF theory. In particular, we haveevaluated individually for SPC/E water the contributionsof (i) all the strong short ranged repulsive and attrac-tive interactions that lead to the local hydrogen-bondingnetwork, (ii) longer ranged dispersive LJ attractions be-tween molecules, and (iii) long ranged dipole-dipole in-teractions, and demonstrated a hierarchical ordering oftheir importance in determining several properties of wa-ter in uniform and nonuniform systems.

All of our truncated models accurately describe the lo-cal hydrogen bonding network, and as expected, this net-work alone is sufficient to match bulk structure as well assolvent structure around small hydrophobic solutes pro-vided that the bulk density and temperature are accu-rately prescribed. Furthermore, the anomalous tempera-ture and density dependence of the “internal pressure” ofwater is found to be dominated by the competing short-ranged repulsive and attractive forces determining thelocal hydrogen bonding network as well.

But local network concepts alone cannot captureall the complexities of even the simple SPC/E watermodel. While the dispersive LJ attractions between wa-ter molecules primarily provide a uniform cohesive energyin bulk systems, they strongly influence the structureand density profile of large scale hydrophobic interfaces.Their importance provides further support for analogiesbetween water at extended hydrophobic interfaces andthe liquid-vapor interface, and the unbalanced LJ forcecan be used to quantify the transition between small andlarge scale hydrophobicity for simple solutes.

Although the long-ranged dipolar interactions betweenmolecules have only small effects on most of the interfa-cial density properties considered here, we have shown

elsewhere that they are crucial in determining dielectricproperties of both bulk and nonuniform water. Indeed,as will be discussed elsewhere, we have found that elec-trostatic quantities may in fact be a sensitive structuralprobe of hydrophobicity in general environments [27].

This interaction hierarchy, wherein strong short-ranged local interactions alone determine structure inuniform environments while the longer ranged forces areneeded as well to capture other properties could provequite useful in refining simple site-site water models.Current water models incorporate a vast amount of cleverengineering and empirical fine-tuning and manage to re-produce a variety of different properties through a com-plex balance of competing interactions with simple func-tional forms. Changes in the potential that improve oneproperty generally speaking produce poorer results forseveral others.

One promising route to a more systematic proceduremay be sensitivity analysis, in which small perturbationsof potential parameters are made and the correlated re-sponse of a variety of physical observables is quantified.By perturbing the relative magnitudes of short and longranged interactions, Iordanov et al. found that thermody-namic properties of bulk water are most sensitive to smallchanges in the LJ repulsions and the short ranged electro-static interactions [19], in agreement with our findings.A new water model was then proposed by optimizingparameters to reproduce a specific bulk thermodynamicquantity (the internal energy) in an attempt to correctthe deficiencies present in a previously developed waterpotential.

However, the theoretical scheme of splitting the poten-tial described in this paper may provide a more concreteand physically suggestive path to incrementally matchvarious known physical quantities for water without ru-ining the fitting of previous quantities, and one couldcombine an approach like sensitivity analysis with theconceptual framework presented herein to systematicallyoptimize a specific water model.

In particular, it has recently been suggested that theaccuracy with which a water model can predict the exper-imental TMD correlates well with the accuracy that thesame model displays in predicting the thermodynamicsof small-scale hydrophobic hydration [3]. Arguably, theleast justified feature of current simple water models likeSPC/E is the functional form of the core LJ potentialu0(r), especially at the very short separations relevantfor describing local hydrogen bonding as illustrated inFig. 2. One could try to fine-tune a GTRC-type modelthrough alteration of the local hydrogen bond network byvarying the form of the repulsive core in order to matchthe experimental density maximum, as well as other bulkproperties like the internal pressure, in order to obtaina short-ranged system that yields accurate bulk prop-erties. Although a detailed discussion of this process isbeyond the scope of this article, one could try to use sometype of optimization procedure to determine such poten-tials [19, 26, 45]. Perhaps first principles DFT simula-

13

System N T tequil trun

Bulk Water 256 220-340 K 5 ns 5 ns

Bulk Water 1000 220-300 K 2 ns 2 ns

LV Interface 1728 298 K 1 ns 500 ps

LS Interface 2468 298 K 1 ns 500 ps

Small LJ Solutes (σsw < 10 A) 1000 298 K 2 ns 2 ns

Large LJ Solutes (σsw ≥ 10 A) 6000 298 K 1 ns 1 ns

TABLE I. Details of the MD simulations performed in thiswork. N and T refer to the number of water molecules andthe temperature of the system, respectively. The equilibrationtimes and data collection times are denoted by tequil and trun,respectively.

tions [21, 32] could be used to provide a more fundamen-tal description of the local network. Subsequently, thestructure and thermodynamics of nonuniform systems,which require dispersions and long ranged Coulomb in-teractions, could be used to parametrize the long-rangedinteractions.

ACKNOWLEDGMENTS

This work was supported by the National ScienceFoundation (grants CHE0628178 and CHE0848574). Weare grateful to Lawrence Pratt and Shule Liu for help-ful remarks. We also thank an anonymous reviewer forbringing references [21] and [32] to our attention.

Appendix: Simulation Details

All molecular dynamics simulations were performed us-ing modified versions of the DL POLY software pack-age [34] and the SPC/E water model [5] or its variantsdescribed in Section 2. The equations of motion wereintegrated using the leapfrog algorithm with a timestepof 1 fs [1] while maintaining constant temperature andpressure conditions through the use of a Berendsen ther-mostat and barostat respectively [6].

1. Bulk water simulations

The evaluation of electrostatic interactions in bulk sim-ulations of the full SPC/E water model employed thestandard Ewald summation method using a real spacecutoff of 9.5 A, unless this was larger than half of the boxlength, in which case the cutoff was set to half of the box

length [1]. Short-ranged electrostatic interactions in theGT and GTRC reference systems, as well as LJ interac-tions in all systems, were truncated at the real space cut-off length used in the analogous full system. Simulationsof bulk water were performed with N = 1000 moleculesin the isothermal-isobaric (NPT) ensemble to determinethe density maximum and with N = 256 molecules inthe canonical (NVT) ensemble to determine P (T ) andthe internal pressure. The duration of equilibration andproduction runs, as well as the temperatures sampled arelisted in Table 1. The internal pressure in Eq. (7) wascalculated by evaluating ε(v) for numerous values of vat each T . The function ε(v) was then fit to a polyno-mial, which was differentiated at the desired v to yieldthe internal pressure.

2. Simulation of nonuniform systems

In order to generate starting configurations for the LVand LS interfacial systems discussed in Section 5, wefirst equilibrated N water molecules in a cubic geome-try, where N is listed in Table 1. The z-dimension of thesystem was then elongated to more than three times thex- and y-dimensions, and in the case of the LS interface,a wall potential of the form

Uw(z) =A

|z − zw|9−

B

|z − zw|3

(A.1)

was added at zw = 0 and the parameters A and B aregiven in Ref. [23]. In order to ensure water moleculesdid not approach the wall from z < 0, a repulsive wallwas added at large z to constrain the water moleculesto the desired region of the simulation cell while still al-lowing a large vacuum region for the formation of a va-por phase. Electrostatic interactions were handled usingthe corrected Ewald summation method for slab geome-tries [44] with a real space cutoff of 11.0 A, which was alsothe cutoff distance for LJ and short-ranged electrostaticinteractions.

Molecular dynamics simulations of the hydration of LJsolutes were performed in the NPT ensemble using stan-dard Ewald summation to evaluate the electrostatic in-teractions, with a real space cutoff of 11.0 A. The shortranged v0 potential and the water-water LJ potentialwere also truncated at 11.0 A. The LJ solute was rep-resented by a fixed external potential centered at the ori-gin, and the solute-water interactions were truncated atone-half the length of the simulation cell. The number ofmolecules and simulation times are listed in Table 1.

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