A Classical Density-Functional Theory for Describing Water Interfaces

Jessica Hughes, Eric J. Krebs, and David RoundyDepartment of Physics, Oregon State University, Corvallis, OR 97331, USA

We develop a classical density functional for water which combines the White Bear fundamental-measure theory (FMT) functional for the hard sphere fluid with attractive interactions based onthe Statistical Associating Fluid Theory (SAFT-VR). This functional reproduces the properties ofwater at both long and short length scales over a wide range of temperatures, and is computationallyefficient, comparable to the cost of FMT itself. We demonstrate our functional by applying it tosystems composed of two hard rods, four hard rods arranged in a square and hard spheres in water.

I. INTRODUCTION

A large fraction of interesting chemistry—including allof molecular biology—takes place in aqueous solution.However, while quantum chemistry enables us to calcu-late the ground state energies of large molecules in vac-uum, prediction of the free energy of even the smallestmolecules in the presence of a solvent poses a continuingchallenge due to the complex structure of a liquid andthe computational cost of ab initio molecular dynam-ics1,2. The current state-of-the art in ab initio moleculardynamics is limited to a few hundred water molecules perunit cell3. On top of this, traditional density-functionaltheory (DFT) methods without the use of dispersioncorrections strongly over-structure water, to the pointthat ice melts at over 120C4! There has been a flurryof recent publications implicating van der Waals effects(i.e. dispersion corrections) as significant in reducing thisover-structuring5–8. However, one particular study foundthat water modeled using a hybrid functional with dis-persion corrections still has a melting point over 80C9.It has also been found that the inclusion of nuclear quan-tum effects can provide similar improvements10. Each ofthese corrections imposes an additional computationalburden on an approach that is already feasible for only avery small number of water molecules. A more efficientapproach is needed in order to study nanoscale and largersolutes.

A. Classical density-functional theory

Numerous approaches have been developed to approx-imate the effect of water as a solvent in atomistic cal-culations. Each of these approaches gives an adequatedescription of some aspect of interactions with water,but none of them is adequate for describing all theseinteractions with an accuracy close to that attained byab initio calculations. The theory of Lum, Chandler andWeeks (LCW)11, for instance, can accurately describe thefree energy cost of creating a cavity by placing a solutein water, but does not lend itself to extensions treatingthe strong interaction of water with hydrophilic solutes.Treatment of water as a continuum dielectric with a cav-ity surrounding each solute can give accurate predictionsfor the energy of solvation of ions12–17, but provides no

information about the size of this cavity. In a physicallyconsistent approach, the size of the cavity will naturallyarise from a balance between the free energy required tocreate the cavity, the attraction between the water andthe solute, and the steric repulsion which opens up thecavity in the first place.

One promising approach for an efficient continuum de-scription of water is that of classical density-functionaltheory (DFT), which is an approach for evaluating thefree energy and thermally averaged density of fluids inan arbitrary external potential18. The foundation ofclassical DFT is the Mermin theorem19, which extendsthe Hohenberg-Kohn theorem20 to non-zero temperature,stating that

Ω(T ) = minn(r)

F [n(r), T ] +

∫(Vext(r) + µ)n(r)dr

,

(1)where Ω(T ) is the grand potential of a system in the ex-ternal potential Vext at temperature T , n(r) is the den-sity of atoms or molecules, µ is the chemical potentialand F [n(r), T ] is a universal free-energy functional forthe fluid, which is independent of the external potentialVext. Classical DFT is a natural framework for creatinga more flexible theory of hydrophobicity that can read-ily describe interaction of water with arbitrary externalpotentials—such as potentials describing strong interac-tions with solutes or surfaces.

A number of exact properties are easily achieved inthe density-functional framework, such as the contact-value theorem, which ensures a correct excess chemicalpotential for small hard solutes. Much of the researchon classical density-functional theory has focused on thehard-sphere fluid21–26, which has led to a number of so-phisticated functionals, such as the fundamental-measuretheory (FMT) functionals22–28. These functionals are en-tirely expressed as an integral of local functions of a fewconvolutions of the density (fundamental measures) thatcan be efficiently computed. We will use the White Bearversion of the FMT functional27,28. This functional re-duces to the Carnahan-Starling equation of state in thehomogeneous limit, and it reproduces the exact free en-ergy in the strongly-confined limit of a small cavity.

A number of classical density functionals have beendeveloped for water29–42, each of which captures some ofthe qualitative behavior of water. However, each of thesefunctionals also fail to capture some of water’s unique

2

properties. For instance, the functional of Lischner etal38 treats the surface tension correctly, but can onlybe used at room temperature, and thus captures noneof the temperature-dependence of water. A functionalby Chuev and Skolov37 uses an ad hoc modification ofFMT that can predict hydrophobic hydration near tem-peratures of 298 K, but does not produce a correct equa-tion of state due to their method producing a high valuefor pressure. A number of classical density functionalshave recently been produced that are based on Statis-tical Associating Fluid Theory (SAFT)32–34,36,39,41,43–48.These functionals are based on a perturbative thermo-dynamic expansion, and do reproduce the temperature-dependence of water’s properties. We should give specialmention to Sundararaman et al who recently introduceda classical density functional for water using a model inwhich a water molecule is treated as a hard sphere at-tached to two tetrahedrally oriented hard spheres repre-senting voids, or orientations in which a hydrogen bondmay not be formed, with all attractive interactions be-ing lumped into a single pair potential treated in a meanfield approximation42.

B. Statistical associating fluid theory

Statistical Associating Fluid Theory (SAFT) is a the-ory describing complex fluids in which hydrogen bond-ing plays a significant role46,49. SAFT is used to ac-curately model the equations of state of both pure fluidsand mixtures over a wide range of temperatures and pres-sures. SAFT is based on Wertheim’s first-order thermo-dynamic perturbation theory (TPT1)50–53, which allowsit to account for strong associative interactions betweenmolecules.

The SAFT Helmholtz free energy is composed of fiveterms:

F = Fid + Fhs + Fdisp + Fassoc + Fchain, (2)

where the first three terms—ideal gas, hard-sphere repul-sion and dispersion—encompass the monomer contribu-tion to the free energy, the fourth is the association freeenergy, describing hydrogen bonds, and the final term isthe chain formation energy for fluids that are chain poly-mers. While a number of formulations of SAFT havebeen published, we will focus on SAFT-VR54, which wasused by Clark et al to construct an optimal SAFT modelfor water36. All but one of the six empirical parametersused in the functional introduced in this paper are takendirectly from this Clark et al paper. As an example of thepower of this model, it predicts an enthalpy of vaporiza-tion at 100C of ∆Hvap= 39.41 kJ/mol, compared withthe experimental value ∆Hvap= 40.65 kJ/mol55, with anerror of only a few percent. We show a phase diagram forthis optimal SAFT model for water in Figure 1, whichdemonstrates that its vapor pressure as a function of tem-perature is very accurate, while the liquid density shows

0.01

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

Pre

ssu

re (

atm

)

Density (g/mL)

298K

348K

398K

448K

498K

598K

FIG. 1. The pressure versus density for various temperatures,including experimental pressure data from NIST55. The solidcolored lines indicate the computationally calculated pressureand the dotted colored lines are NIST data points. The solidand dotted black lines represent the theoretical and experi-mental coexistence curves.

larger discrepancies. The critical point is very poorly de-scribed, which is a common failing of models that arebased on a mean-field expansion.

SAFT has been used to construct classical densityfunctionals, which are often used to study the surfacetension as a function of temperature31–36,39–41,47,48. Suchfunctionals have qualitatively predicted the dependenceof surface tension on temperature, but they also overesti-mate the surface tension by about 50%, and most SAFT-based functionals are unsuited for studying systems thathave density variations on a molecular length scale due tothe use of a local density approximation32–34,36,40,41,48.

Functionals constructed using a local density approx-imation fail to satisfy the contact-value theorem, andtherefore incorrectly model small hard solutes. Thecontact-value theorem relates the pressure on a hard sur-face to the contact density of the fluid at that surface:

p(rc) = n(rc)kBT, (3)

where rc is the position at which a molecule is in contactwith the hard surface, n(rc) is the density at that pointof contact, and p(rc) is the pressure that the fluid exertson the surface at the same point. This pressure is definedas a ratio of force to solvent accessible surface area. Fora solute which excludes the solvent from an arbitrarilysmall volume, the contact density will be the same as thebulk density, and therefore we can integrate the abovepressure to find that the excess chemical potential of asmall hard solute is proportional to the solvent-excludedvolume:

F = nkBTV. (4)

The contact-value theorem is violated by local classicaldensity functionals such as those using a local density

3

approximation or a square-gradient term, but is satisfiedby non-local classical density functionals, such as thoseusing a weighted-density approach.

II. THEORY AND METHODS

We construct a classical density functional for water,which reduces in the homogeneous limit to the opti-mal SAFT model for water found by Clark et al. TheHelmholtz free energy is constructed using the first fourterms from Equation 2: Fid, Fhs, Fdisp and Fassoc. Inthe following sections, we will introduce the terms of thisfunctional.

A. Ideal gas functional

The first term is the ideal gas free energy functional,which is purely local:

Fid[n] = kBT

∫n(r)

(ln(n(r)Λ3)− 1

)dr, (5)

where n(r) is the density of water molecules and Λ is the

thermal wavelength Λ =(

2π~2

mkBT

)1/2. The ideal gas free

energy functional on its own satisfies the contact valuetheorem and its limiting case of small solutes (Equations3 and 4). These properties are retained by our total func-tional, since all the remaining terms are purely nonlocal.

B. Hard-sphere repulsion

We treat the hard-sphere repulsive interactions us-ing the White Bear version of the Fundamental-MeasureTheory (FMT) functional for the hard-sphere fluid27,28.FMT functionals are expressed as the integral of the fun-damental measures of a fluid, which provide local mea-sures of quantities such as the packing fraction, densityof spheres touching a given point and mean curvature.The hard-sphere excess free energy is written as:

Fhs[n] = kBT

∫(Φ1(r) + Φ2(r) + Φ3(r))dr , (6)

with integrands

Φ1 = −n0 ln (1− n3) (7)

Φ2 =n1n2 − nV 1 · nV 2

1− n3(8)

Φ3 = (n32 − 3n2nV 2 · nV 2)n3 + (1− n3)2 ln(1− n3)

36πn23 (1− n3)2 ,

(9)

where the fundamental measure densities are given by:

n3(r) =

∫n(r′)Θ(|r− r′| −R)dr′ (10)

n2(r) =

∫n(r′)δ(|r− r′| −R)dr′ (11)

nV 2 = ∇n3 (12)

n1 =n2

4πR(13)

nV 1 =nV 2

4πR(14)

n0 =n2

4πR2. (15)

The density n3 is the packing fraction and n0 is the av-erage density at contact distance. For our functionalfor water, we use the hard-sphere diameter of 3.0342 A,which was found to be optimal by Clark et al.36

C. Dispersion free energy

The dispersion free energy includes the van der Waalsattraction and any orientation-independent interactions.We use a dispersion term based on the SAFT-VRapproach54, which has two free parameters (taken fromClark et al36): an interaction energy εd and a length scaleλdR.

The SAFT-VR dispersion free energy has the form54

Fdisp[n] =

∫(a1(r) + βa2(r))n(r)dr, (16)

where a1 and a2 are the first two terms in a high-temperature perturbation expansion and β = 1/kBT .The first term, a1, is the mean-field dispersion interac-tion. The second term, a2, describes the effect of fluctu-ations resulting from compression of the fluid due to thedispersion interaction itself, and is approximated usingthe local compressibility approximation (LCA), whichassumes the energy fluctuation is simply related to thecompressibility of a hard-sphere reference fluid56.

The form of a1 and a2 for SAFT-VR is given in ref-erence54, expressed in terms of the packing fraction. Inorder to apply this form to an inhomogeneous densitydistribution, we construct an effective local packing frac-tion for dispersion ηd, given by a Gaussian convolutionof the density:

ηd(r) =1

6√πλ3ds

3d

∫n(r′) exp

(− |r− r′|2

2(2λdsdR)2

)dr′.

(17)

This effective packing fraction is used throughout the dis-persion functional, and represents a packing fraction av-eraged over the effective range of the dispersive interac-tion. Here we have introduced an additional empiricalparameter sd which modifies the length scale over whichthe dispersion interaction is correlated.

4

D. Association free energy

The final attractive energy term is the associationterm, which accounts for hydrogen bonding. Hydrogenbonds are modeled as four attractive patches (“associa-tion sites”) on the surface of the hard sphere. These foursites represent the two hydrogen bond donor sites, andtwo hydrogen bond acceptor sites. There is an attrac-tive energy εa when two molecules are oriented such thatthe donor site of one overlaps with the acceptor site ofthe other. The volume over which this interaction oc-curs is κa, giving the association term in the free energytwo empirical parameters that are fit to the experimen-tal equation of state of water (again, taken from Clark etal36).

The association functional we use is a modified ver-sion of Yu and Wu45, which includes the effects of den-sity inhomogeneities in the contact value of the correla-tion function gHSσ , but is based on the SAFT-HS model,rather than the SAFT-VR model54, which is used in theoptimal SAFT parametrization for water of Clark et al36.Adapting Yu and Wu’s association free energy to SAFT-VR simply involves the addition of a correction term inthe correlation function (see Equation 22).

The association functional we use is constructed by us-ing the density n0(r), which is the density of hard spherestouching a given point, in the standard SAFT-VR as-sociation energy54. The association free energy for ourfour-site model has the form

Fassoc[n] = 4kBT

∫n0(r)ζ(r)

(lnX(r)− X(r)

2+

1

2

)dr,

(18)

where the factor of 4 comes from the four associationsites per molecule, the functional X is the fraction ofassociation sites not hydrogen-bonded, and ζ(r) is a di-mensionless measure of the density inhomogeneity.

ζ(r) = 1− nV 2 · nV 2

n22. (19)

The fraction X is determined by the quadratic equation

X(r) =

√1 + 8n0(r)ζ(r)∆(r)− 1

4n0(r)ζ(r)∆(r), (20)

where the functional ∆ is a measure of hydrogen-bondingprobability, given by

∆(r) = κagSWσ (r)

(eβεa − 1

)(21)

gSWσ (r) = gHSσ (r) +

1

4β

(∂a1∂ηd(r)

− λd3ηd

∂a1∂λd

), (22)

where gSWσ is the correlation function evaluated at con-tact for a hard-sphere fluid with a square-well dispersionpotential, and a1 and a2 are the two terms in the disper-sion free energy. The correlation function gSWσ is written

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300 350 400

Surf

ace

Te

nsio

n (

mN

/m)

Temperature (oC)

theoryexperiment

68

70

72

74

76

0 10 20 30 40 50

FIG. 2. Comparison of Surface tension versus temperature fortheoretical and experimental data. The experimental data istaken from NIST.55 The length-scaling parameter sd is fit sothat the theoretical surface tension will match the experimen-tal surface tension near room temperature.

as a perturbative correction to the hard-sphere fluid cor-relation function gHS

σ , for which we use the functional ofYu and Wu45:

gHSσ =1

1− n3+R

2

ζn2(1− n3)2

+R2

18

ζn22(1− n3)3

. (23)

E. Determining the empirical parameters

The majority of the empirical parameters used in ourfunctional are taken from the paper of Clark et al ondeveloping an optimal SAFT model for water36. ThisSAFT model contains five empirical parameters: thehard-sphere radius, an energy and length scale for thedispersion interaction, and an energy and length scalefor the association interaction. In addition to the fiveempirical parameters of Clark et al, we add a single ad-ditional dimensionless parameter sd—with a fitted valueof 0.353—which determines the length scale over whichthe density is averaged when computing the dispersionfree energy and its derivative. We determine this finalparameter by fitting the computed surface tension to theexperimental surface tension with the result shown inFigure 2. Because the SAFT model of Clark et al over-estimates the critical temperature—which is a commonfeature of SAFT-based functionals that do not explicitlytreat the critical point—we cannot reasonably describethe surface tension at all temperatures, and choose tofit the surface tension at and around room temperature.We note here that we could have chosen to fit the surfacetension with a square-gradient term in the free energyinstead of adjusting the length scale for the dispersive at-traction. This would result in a functional that violatesthe contact-value theorem which, among other problems,

5

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

µex/A

rea (

mN

/m)

Radius (nm)

0

2

4

6

8

0 0.025 0.05 0.075

FIG. 3. Excess chemical potential per area versus radius for asingle hydrophobic rod immersed in water. This should havean asymptote equal to the surface tension at room tempera-ture, and it agrees well with the surface tension in Figure 2.The inset for rods with a very small radius shows the linearrelationship expected based on Equation 4.

would fail to satisfy Equation 4 for the excess chemicalpotential of small solutes.

III. RESULTS AND DISCUSSION

A. One hydrophobic rod

We begin by studying a single hydrophobic rod im-mersed in water. In Figure 3 we show the excess chemi-cal potential at room temperature, scaled by the solventaccessible surface area of the hard rod, plotted as a func-tion of hard-rod radius. We define the hard-rods radiusas the radius from which water is excluded. For rods withradius larger than 0.5 nm or so, this reaches a maximumvalue of 75 mN/m, which is slightly higher than macro-scopic surface tension. In the limit of very large rods, thisvalue will decrease and approach the macroscopic value.As seen in the inset of Fig. 3, for rods with very smallradius (less than about 0.5 A) the excess chemical po-tential is proportional to volume, satisfying Equation 4,which results from the contact-value theorem.

We show in Figure 4 density profiles for different radiirods, as well as the prediction for the contact value of thedensity as a function of rod radius, as computed from theexcess chemical potentials plotted in Figure 3. The agree-ment between these curves confirms that our functionalsatisfies the contact-value theorem and that our mini-mization is well converged. As expected, as the radiusof the rods becomes zero the contact density approachesthe bulk density, and as the radius becomes large, thecontact density will approach the vapor density.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.2 0.4 0.6 0.8 1 1.2

De

nsity (

g/m

L)

Radius (nm)

r=0.05 nmr=0.15 nmr=0.30 nmr=0.50 nmr=0.80 nmr=1.00 nm

Density at contact

FIG. 4. Density profiles for single rods of different radii. Thedotted line represents the saturated liquid density and thepoints represent the expected contact density derived fromthe contact value theorem and calculated free energy data.

B. Hydrophobic interaction of two rods

We now look at the more interesting problem of twoparallel hard rods in water, separated by a distance d,as shown in Figure 5. At small separations there is onlyvapor between the rods, but as the rods are pulled apart,the vapor region expands until a critical separation isreached at which point liquid water fills the region be-tween the rods. Figure 5 shows density profiles beforeand after this transition for rods of radius 0.6 nm. Thiscritical separation for the transition to liquid depends onthe radii of the rods, and is about 0.65 nm for the rodsshown in Figure 5. The critical separation will be differ-ent for a system where there is attraction between therods and water. At small separations, the shape of thewater around the two rods makes them appear as onesolid “stadium”-shaped object (a rectangle with semi-circles on both ends).

To understand this critical separation, we consider thefree energy in the macroscopic limit, which is given by

F = γA+ pV. (24)

The first term describes the surface energy and the sec-ond term is the work needed to create a cavity of volumeV . Since the pressure term scales with volume, it can beneglected relative to the surface term provided the lengthscale is small compared with γ/p, which is around 20 µm,and is much larger than any of the systems we study. Formicron-scale rods, the water on the sides of the ‘stadium’configuration will bow inward between the rods and thedensity will reduce to vapor near the center point wherethe rods are closest to each other.

Starting from the surface energy term, we can calculatethe free energy per length, which is equal to the circum-ference multiplied by the surface tension. Working out

6

After transition

-2 -1 0 1 2

y (nm)

-1

-0.5

0

0.5

1

z (

nm

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Density (

g/m

L)

d

Before transition

-1

-0.5

0

0.5

1

z (

nm

)

dd

FIG. 5. Density profiles illustrating the transition from vaporto liquid water between the rods. The radius is 0.6 nm, the topfigure is at a separation of 0.6 nm and the bottom is 0.7 nm.Figure 6 shows the energy for these and other separations.

the circumference of the stadium-shape leads us to

F = (2πr + 4r + 2d)γ (25)

where γ is the surface tension, r is the radius of the rods,and d is the separation between rods illustrated in Fig-ure 5. The force per length is the derivative of the freeenergy with respect to the separation d, from which weconclude that the force per length is twice the surfacetension.

We plot in Figure 6 the computed free energy of in-teraction per unit length from our classical density func-tional (solid lines), as a function of the separation d, alongwith the free energy predicted by our simple macroscopicmodel (dashed lines). The models agree very well on theforce between the two rods at close separations, and havereasonable agreement as to the critical separation for rodsgreater than 0.5 nm in radius.

Walther et al57 studied the interactions between twocarbon nanotubes, which are geometrically similar to ourhydrophobic rods, using molecular dynamics with theSPC model for water. Their simulations used nanotubesof diameter 1.25 nm and separations ranging from about0.3 nm to 1.5 nm. In agreement with our findings for twopurely hydrophobic rods, Walther et al find that in theabsence of Lennard-Jones attraction between carbon andoxygen, there is a drying transition at a distance compa-rable to the diameter of the nanotube. In contrast tothis, when the attraction between nanotubes and wateris turned on, they find that the drying transition occurs

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fre

e e

ne

rgy p

er

len

gth

(kJ/m

ol n

m)

d (nm)

r=0.2 nmr=0.4 nmr=0.6 nmr=0.8 nmr=1.0 nmr=1.2 nm

FIG. 6. Free energy of interaction (also known as the poten-tial of mean force) versus separation for two hydrophobic rodsranging in radius from 0.2 nm to 1.2 nm. All were arbitrar-ily offset to zero at large separations for ease of comparison.The transition corresponds to the phase change from vaporto liquid between the rods as pictured in the density profilesin Figure 5.

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.5 1 1.5 2 2.5 3 3.5

Fre

e e

nerg

y p

er

len

gth

(kJ/m

ol n

m)

d (nm)

r=0.2 nmr=0.4 nmr=0.6 nmr=0.8 nmr=1.0 nmr=1.2 nm

FIG. 7. Free energy of interaction versus separation for fourhydrophobic rods ranging in radius from 0.2 nm to 1.2 nm.All were arbitrarily offset to zero at large separations. Thetransition corresponds to the phase change from vapor to liq-uid between the rods as pictured in the density profiles inFigure 8.

at much shorter distances, comparable to the diameterof water.

C. Hydrophobic interactions of four rods

We go on to study four parallel hard rods, as examinedby Lum, Chandler and Weeks in their classic paper on

7

After transition

-3 -2 -1 0 1 2 3

y (nm)

-3

-2

-1

0

1

2

3

z (

nm

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Density (

g/m

L)

Before transition

-3

-2

-1

0

1

2

3

z (

nm

)

FIG. 8. Density profiles illustrating the transition from vaporto liquid water between four rods. The radius is 0.6 nm,the top figure is at a separation of 1.53 nm and the bottomis 1.56 nm. Figure 7 shows the energy for these and otherseparations.

hydrophobicity at small and large length scales11. As inthe case of two rods—and as predicted by Lum et al—we observe a drying transition, as seen the density plotshown in Figure 8. In Figure 7, we plot the free energyof interaction together with the macroscopic approxima-tion, and find good agreement for rods larger than 0.5 nmin radius. This free energy plot is qualitatively similar tothat predicted by the LCW theory11, with the differencethat we find no significant barrier to the association offour rods.

D. Hydration energy of hard-sphere solutes

A common model of hydrophobic solutes is the hard-sphere solute, which is the simplest possible solute, andserves as a test case for understanding of hydrophobicsolutes in water59. As in the single rod, we begin byexamining the ratio of the excess chemical potential ofthe cavity system to the solvent accessible surface area(Figure 9). This effective surface tension surpasses the

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

µex/A

rea (

mN

/m)

Radius (nm)

CDFTSPC/E molecular dynamics

FIG. 9. Excess chemical potential per area versus radius fora single hydrophobic sphere immersed in water. This shouldhave an asymptote equal to the surface tension at room tem-perature, and it agrees well with the surface tension in Fig-ure 2. Results from a simulation of SPC/E water58 are shownas circles. The horizontal lines show the experimental andSPC/E macroscopic surface tension for water at standard at-mospheric temperature and pressure.

macroscopic surface tension at a radius of almost 1 nm,and at large radius will drop to the macroscopic value.As with the single rod, we see the analytically correct be-havior in the limit of small solutes. For comparison, weplot the free energy calculated using a molecular dynam-ics simulation of SPC/E water58. The agreement is quitegood, apart from the issue that the SPC/E model for wa-ter significantly underestimates the macroscopic surfacetension of water at room temperature60.

Figure 10 shows the density profile for several hardsphere radii, plotted together with the results of thesame SPC/E molecular dynamics simulation shown inFigure 958. The agreement with simulation is quite rea-sonable. The largest disagreement involves the densityat contact, which according to the contact value theoremcannot agree, since the free energies do not agree.

IV. CONCLUSION

We have developed a classical density functionalfor water that combines SAFT with the fundamental-measure theory for hard spheres, using one additionalempirical parameter beyond those in the SAFT equationof state, which is used to match the experimental surfacetension. This functional does not make a local densityapproximation, and therefore correctly models water atboth small and large length scales. In addition, like allFMT functionals, this functional is expressed entirely interms of convolutions of the density, which makes it effi-cient to compute and minimize.

8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r (nm)

1

1

n (

g/m

L)

1

1

FIG. 10. Density profiles around hard-sphere solutes of dif-ferent radii. Predictions from our classical density-functionaltheory are in solid red, while the dotted line shows the resultof a molecular dynamics simulation of SPC/E water58.

We apply this functional to the case of hard hydropho-bic rods and spheres in water. For systems of two orfour hydrophobic rods surrounded by water, we see atransition from a vapor-filled state a liquid-filled state.A simple model treatment for the critical separation forthis transition works well for rods with diameters largerthan 1 nm. In the case of spherical solutes, we find goodagreement with SPC/E simulations.

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