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A Classical Density-Functional Theory for Describing Water Interfaces

Jessica Hughes, Eric Krebs, David Roundy

arXiv:1209.0035v1cond-mat.soft

TL;DR

Calculating solvent effects accurately is challenging because liquid structure and the cost of ab initio molecular dynamics limit accessible system sizes. This paper develops a classical water density functional combining SAFT with hard-sphere FMT. It models water at small and large length scales and is applied to hydrophobic rods and spheres, with sphere results agreeing well with SPC/E simulations.

  • Problem

    Predicting free energies of molecules in solvent remains challenging because liquid structure and ab initio molecular-dynamics costs restrict calculations to a few hundred water molecules per unit cell.

    Source: §I · p. 1

  • Method

    The functional combines SAFT with hard-sphere FMT, adds one empirical parameter to fit experimental surface tension, and expresses its terms as density convolutions.

    Source: §IV · p. 7

  • Results

    The functional models water at small and large length scales; hard-sphere density profiles agree reasonably with SPC/E simulations, while rod systems show transitions between vapor-filled and liquid-filled states.

    Source: §IV · p. 7, D. Hydration energy of hard-sphere… · p. 7

  • Takeaways & Limitations

    The functional is computationally efficient, with computational cost comparable to FMT itself, and its hard-sphere solute predictions agree with SPC/E simulations.

    Source: §IV · pp. 7–8

    The largest disagreement with SPC/E density profiles is at contact, where the contact-value theorem precludes agreement because the free energies differ.

    Source: D. Hydration energy of hard-sphere… · p. 7

Abstract

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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 on the Statistical Associating Fluid Theory (SAFT-VR). This functional reproduces the properties of water at both long and short length scales over a wide range of temperatures, and is computationally efficient, comparable to the cost of FMT itself. We demonstrate our functional by applying it to systems composed of two hard rods, four hard rods arranged in a square and hard spheres in water.

I. INTRODUCTION

Solvent effects are difficult to calculate accurately and efficiently, motivating a more efficient approach for nanoscale and larger solutes. Classical DFT offers a flexible framework for water, while existing water functionals capture different properties and have important limitations.

I. INTRODUCTION

  • Ab initio molecular dynamics is limited to a few hundred water molecules per unit cell, while standard classical molecular dynamics without van der Waals corrections over-structures water.These computational costs and modeling issues motivate a more efficient approach for nanoscale and larger solutes.Source: §I · p. 1

A. Classical density-functional theory

  • Classical DFT evaluates fluid free energy and thermally averaged density in arbitrary external potentials, offering a flexible framework for water–solute interactions.In this framework, cavity size reflects a balance among cavity-creation free energy, water–solute attraction, and steric repulsion.Source: A. Classical density-functional theory · p. 1
  • White Bear FMT uses convolutions of density to describe hard-sphere fluids and reproduces the Carnahan–Starling equation of state in the homogeneous limit.It also reproduces exact free energy in the strongly confined limit of a small cavity.Source: A. Classical density-functional theory · p. 1
  • Existing classical water functionals capture some qualitative behavior, but reported limitations include room-temperature-only applicability and inaccurate pressure or missing temperature dependence.SAFT-based functionals reproduce temperature dependence, while many use a local density approximation that limits molecular-scale density variations.Source: A. Classical density-functional theory · pp. 1–2

B. Statistical associating fluid theory

SAFT models complex fluids with significant hydrogen bonding and decomposes their free energy into molecular and associative contributions. The cited SAFT-based functionals capture temperature trends but can overestimate surface tension and be unsuitable for molecular-scale density variations.

B. Statistical associating fluid theory

  • SAFT describes complex fluids with significant hydrogen bonding and models pure-fluid and mixture equations of state across wide temperature and pressure ranges.It is based on Wertheim’s first-order thermodynamic perturbation theory (TPT1), which accounts for strong associative interactions.Source: B. Statistical associating fluid theory · p. 2
  • SAFT free energy comprises ideal-gas, hard-sphere, dispersion, association, and chain terms, with the first three representing monomer contributions.The association term describes hydrogen bonds, and the chain term describes chain formation in polymer fluids.Source: B. Statistical associating fluid theory · p. 2
  • SAFT-based functionals overestimate surface tension by about 50%, and most are unsuited to molecular-scale density variations because they use a local density approximation.The cited SAFT model predicts an enthalpy of vaporization of 39.41 kJ/mol at 100°C versus the experimental 40.65 kJ/mol; its critical point is poorly described.Source: B. Statistical associating fluid theory · p. 2
  • Figure 1 compares pressure–density curves across temperatures with NIST data and shows theoretical and experimental coexistence curves.The caption identifies calculated pressures as solid colored lines and NIST data as dotted colored lines.Source: Figure 1 · p. 2
  • Local-density-approximation functionals fail the contact-value theorem, which links hard-wall pressure to contact density and constrains predictions for small hard solutes.The theorem’s small-solute limit makes excess chemical potential proportional to solvent-excluded volume.Source: B. Statistical associating fluid theory · p. 2

II. THEORY AND METHODS

The functional begins with a purely local ideal-gas free-energy term. Because its remaining terms are nonlocal, the total functional retains the contact-value theorem and its small-solute limit.

II. THEORY AND METHODS

  • The water functional reduces in the homogeneous limit to Clark et al.’s optimal SAFT model and uses its ideal-gas, hard-sphere, dispersion, and association terms.The paper introduces these contributions as the functional’s first four terms.Source: §II · p. 2

A. Ideal gas functional

  • The ideal-gas free-energy functional is purely local and forms the first term of the water functional.The density is the density of water molecules, and nQ denotes the quantum concentration.Source: A. Ideal gas functional · p. 3
  • The total functional retains the contact-value theorem and its small-solute limit because all terms beyond the ideal-gas contribution are purely nonlocal.The ideal-gas functional itself satisfies both properties.Source: A. Ideal gas functional · p. 3

B. Hard-sphere repulsion

Hard-sphere repulsion is modeled with the White Bear version of FMT, whose fundamental measures represent local geometric and density properties. The functional uses a hard-sphere radius of 3.03420 Å.

B. Hard-sphere repulsion

  • The hard-sphere excess free energy uses the White Bear FMT functional, expressed through integral contributions from fundamental measures.These measures provide local quantities such as filling fraction, density of spheres touching a point, and mean curvature.Source: B. Hard-sphere repulsion · p. 3
  • FMT represents the hard-sphere fluid using fundamental-measure densities.These densities are defined in the following equations.Source: B. Hard-sphere repulsion · p. 3
  • The water model uses a hard-sphere radius of 3.03420 Å, identified as optimal by Clark et al.The FMT measures include n3 for filling fraction and n2 for the number of spheres touching a point.Source: B. Hard-sphere repulsion · p. 3

C. Dispersion free energy

The dispersion free energy models van der Waals and other orientation-independent attractions using a SAFT-VR term. For inhomogeneous densities, a Gaussian-averaged effective filling fraction supplies the density input.

C. Dispersion free energy

  • The SAFT-VR dispersion term includes van der Waals attraction and other orientation-independent interactions.It uses interaction energy ϵd and length scale λdR.Source: C. Dispersion free energy · p. 3
  • The dispersion free energy combines a mean-field interaction term with a fluctuation correction tied to compression of the fluid.The correction uses the local compressibility approximation, relating energy fluctuations to the compressibility of a hard-sphere reference fluid.Source: C. Dispersion free energy · p. 3
  • For inhomogeneous densities, a Gaussian convolution constructs an effective local filling fraction ηd for evaluating the dispersion free energy.ηd represents filling fraction averaged over the effective range of the dispersive interaction; sd adjusts the correlation length scale.Source: C. Dispersion free energy · p. 3

D. Association free energy

The association free energy represents hydrogen bonding with four attractive sites per hard sphere. Its modified Yu–Wu functional incorporates density inhomogeneity into the contact correlation function and uses SAFT-VR association terms.

D. Association free energy

  • Hydrogen bonding is modeled with four attractive patches representing two protons and two electron lone pairs on each hard sphere.An interaction occurs when one molecule’s proton overlaps another’s lone pair; ϵa and κa set the interaction energy and volume.Source: D. Association free energy · p. 3
  • The association term’s empirical parameters are fit to water’s experimental equation of state.These two parameters are taken from Clark et al. 34.Source: D. Association free energy · p. 4
  • The association functional modifies Yu and Wu’s formulation to include density inhomogeneities in the hard-sphere contact correlation function.It adapts the formulation from SAFT-HS to SAFT-VR by adding a correction term to the correlation function.Source: D. Association free energy · p. 4
  • The four-site association energy uses n0(r), the density of hard spheres touching a given point, in the standard SAFT-VR association expression.The fraction X denotes unbonded association sites, and ζ(r) measures density inhomogeneity.Source: D. Association free energy · p. 4

E. Determining the empirical parameters

The functional adopts most empirical parameters from Clark et al.’s SAFT model for water and adds a fitted density-averaging length parameter. The fit targets experimental surface tension near room temperature because the model overestimates the critical temperature.

E. Determining the empirical parameters

  • Most empirical parameters are taken from Clark et al.’s optimal SAFT model for water. 34That model has five parameters: a hard-sphere radius and energy and length scales for dispersion and association.Source: E. Determining the empirical parameters · p. 4
  • The added dimensionless parameter sd is fitted to 0.353 to set the density-averaging length scale for the dispersion free energy and its derivative.It is fitted to match experimental surface tension near room temperature.Source: E. Determining the empirical parameters · p. 4, Figure 2 · p. 4
  • Because the SAFT model overestimates the critical temperature, the surface-tension fit is restricted to temperatures at and around room temperature.The theoretical and experimental surface tensions are compared across temperature in Figure 2; the experimental data are from NIST.49Source: E. Determining the empirical parameters · p. 4, Figure 2 · p. 4

III. RESULTS AND DISCUSSION

For a single hydrophobic rod, the excess chemical potential per area approaches the bulk surface-tension value at large radius and is proportional to volume for very small rods. Density profiles and contact-value predictions agree, with contact density tending from bulk density to vapor density as rod radius grows.

III. RESULTS AND DISCUSSION

  • 75 mN/m is the maximum excess chemical potential per area for rods larger than about 0.5 nm, slightly above the bulk surface tension.At very large radii it decreases toward the bulk surface value; Figure 3 plots the quantity against rod radius and shows the small-radius linear relationship.Source: A. One hydrophobic rod · p. 4, Figure 3 · p. 5
  • For rods smaller than about 0.5 Å, excess chemical potential is proportional to volume, as required by the contact-value theorem.Figure 3’s inset shows the small-radius linear relationship.Source: A. One hydrophobic rod · p. 5, Figure 3 · p. 5
  • Agreement between density profiles and contact-value predictions confirms that the functional satisfies the contact-value theorem and that minimization is well converged.Figure 4 shows density profiles for different radii alongside contact-value predictions derived from the free energies.Source: A. One hydrophobic rod · p. 5, Figure 4 · p. 5
  • As rod radius grows, contact density tends from bulk density toward vapor density.Figure 4 marks saturated liquid density with a dotted line and shows expected contact density as points.Source: A. One hydrophobic rod · p. 5, Figure 4 · p. 5

B. Hydrophobic interaction of two rods

Two hard hydrophobic rods exhibit a vapor-to-liquid transition in the water between them as separation increases. The density-functional interaction energies agree well with a macroscopic model at close separations, with reasonable agreement on the critical separation for rods wider than 0.5 nm.

B. Hydrophobic interaction of two rods

  • As separation increases, the vapor region between two rods expands until liquid water fills it; for 0.6 nm-radius rods, the critical separation is about 0.65 nm.The critical separation depends on rod radius, and attraction between rods and water would shift it.Source: B. Hydrophobic interaction of two rods · p. 5
  • At small separations, water around the rods forms a stadium-shaped boundary, and the macroscopic model neglects pressure-volume work when the length scale is small relative to γ/p, about 20 µm.In the micron-scale limit, the water boundary bows inward and density falls toward vapor near the closest point.Source: B. Hydrophobic interaction of two rods · pp. 5–6
  • The density-functional and macroscopic-model interaction energies agree well on force at close separations and reasonably on critical separation for rods larger than 0.5 nm.Figure 6 plots free energy per unit length against separation for rod radii from 0.2 to 1.2 nm.Source: B. Hydrophobic interaction of two rods · p. 6, Figure 6 · p. 6
  • Walther et al. observed drying at a smaller nanotube radius, but disabling nanotube-water attraction moved drying to longer range, in agreement with these results.Their SPC water model underestimates water surface tension by about 24%, limiting quantitative comparison with real water.Source: B. Hydrophobic interaction of two rods · p. 6

C. Hydrophobic interactions of four rods

Four hydrophobic rods show a drying transition, with free energies agreeing with a macroscopic approximation for radii above 0.5 nm. For hard-sphere solutes, the functional gives reasonable agreement with SPC/E simulation density profiles, though contact densities differ when free energies do.

C. Hydrophobic interactions of four rods

  • For four rods, the free energy agrees well with the macroscopic approximation above 0.5 nm radius, and the association profile has no significant barrier.The system exhibits the drying transition also predicted by Lum, Chandler and Weeks [10], while its free-energy curve is qualitatively similar to LCW theory [10].Source: C. Hydrophobic interactions of four rods · p. 6

D. Hydration energy of hard-sphere solutes

  • For hard-sphere solutes, the effective surface tension exceeds the bulk value near a 1 nm radius, approaches it at large radii, and has the analytically correct small-solute behavior.The free energy agrees quite well with SPC/E water simulation, although that model significantly underestimates room-temperature bulk surface tension.Source: D. Hydration energy of hard-sphere… · pp. 6–7
  • Hard-sphere density profiles agree reasonably with SPC/E simulation, but contact densities differ because the free energies do not agree.The contact value theorem prevents agreement at contact when the free energies differ.Source: D. Hydration energy of hard-sphere… · p. 7

IV. CONCLUSION

The authors present an efficient water density functional combining SAFT and FMT, designed to describe both small and large length scales. Applications show vapor–liquid transitions around hydrophobic rods and good agreement with SPC/E simulations for spherical solutes.

IV. CONCLUSION

  • The functional combines SAFT with hard-sphere FMT, uses one additional empirical parameter to match experimental surface tension, and is efficient to compute and minimize.It describes water at both small and large length scales without a local density approximation.Source: §IV · p. 7
  • For a single hydrophobic sphere, excess chemical potential per area is compared across radii with SPC/E results and experimental and simulated bulk surface tensions.Source: Figure 9 · p. 7
  • For spherical solutes, the functional shows good agreement with SPC/E simulations of density profiles around hard spheres of different radii.Source: Figure 10 · p. 7, §IV · p. 8
  • Two- and four-rod systems show a transition between vapor-filled and liquid-filled states; the critical-separation model works well for rods larger than 1 nm.Source: §IV · p. 7
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