^{1,a)}

### Abstract

We show that to account for the full spectrum of surfacefluctuations from low scattering vector (classical capillary wave theory) to high (bulklike fluctuations), one must take account of the interface’s bending rigidity at intermediate scattering vector , where is the molecular diameter. A molecular model is presented to describe the bending correction to the capillary wave model for short-ranged and long-ranged interactions between molecules. We find that the bending rigidity is negative when the Gibbs equimolar surface is used to define the location of the fluctuating interface and that on approach to the critical point it vanishes proportionally to the interfacial tension. Both features are in agreement with Monte Carlo simulations of a phase-separated colloid-polymer system.

I am indebted to Dick Bedeaux for arguing with me on this intriguing topic since already 20 years. My thoughts have furthermore been shaped by discussions with giants in this field: John Weeks, Ben Widom, and Bob Evans. I would like to express my gratitude to Richard Vink for sharing unpublished simulation results and to Daniel Bonn, Didi Derks, and Joris Kuipers for discussions on the colloid-polymer system.

I. INTRODUCTION

II. THE FLUCTUATINGLIQUID SURFACE

A. Extended capillary wave model

B. Definition of the height profile

C. Bulklike fluctuations

III. COMPARISON WITH MONTE CARLO SIMULATIONS

IV. DENSITY FUNCTIONAL THEORY

A. Gradient expansion

V. DFT: SHORT-RANGED INTERACTIONS

A. Gradient expansion for short-ranged forces

VI. DFT: LONG-RANGED INTERACTIONS

A. Gradient expansion for long-ranged forces

VII. DISCUSSION

### Key Topics

- Surface tension
- 19.0
- Free energy
- 16.0
- Correlation functions
- 13.0
- Critical point phenomena
- 13.0
- Density functional theory
- 13.0

## Figures

Sketch of the fluctuating density profile as a function of ; the height is the distance over which the intrinsic density profile (dashed line) is shifted.

Sketch of the fluctuating density profile as a function of ; the height is the distance over which the intrinsic density profile (dashed line) is shifted.

MC results by Vink *et al.* (Ref. 3) for the surface structure factor (in units of ) vs (in units of ) for various values of the integration limit , 2, 3, 4. The dashed line is the CW model. In this example, , , and the colloidal particles are used to define .

MC results by Vink *et al.* (Ref. 3) for the surface structure factor (in units of ) vs (in units of ) for various values of the integration limit , 2, 3, 4. The dashed line is the CW model. In this example, , , and the colloidal particles are used to define .

MC results by Vink *et al.* (Ref. 3) for the surface structure factor (in units of ) vs (in units of ). The dotted line is the CW model, the dashed line is the combination of the CW model and the bulk correlation function, and the drawn line is the combination of the ECW model and the bulk correlation function. In this example, , , , and the colloidal particles are used to define .

MC results by Vink *et al.* (Ref. 3) for the surface structure factor (in units of ) vs (in units of ). The dotted line is the CW model, the dashed line is the combination of the CW model and the bulk correlation function, and the drawn line is the combination of the ECW model and the bulk correlation function. In this example, , , , and the colloidal particles are used to define .

MC results by Vink *et al.* (Ref. 3) for the surface structure factor (in units of ) vs (in units of ) using the colloidal particles (circles) and polymer particles (triangles) to define . The dotted line is the CW model and the drawn lines are the combination of the ECW model and the bulk correlation function. In this example, , , and .

MC results by Vink *et al.* (Ref. 3) for the surface structure factor (in units of ) vs (in units of ) using the colloidal particles (circles) and polymer particles (triangles) to define . The dotted line is the CW model and the drawn lines are the combination of the ECW model and the bulk correlation function. In this example, , , and .

Surface tension in units of vs the volume fraction difference, . In this example, ; the symbols are numerical results, the drawn line is the gradient expansion approximation, and the filled symbols are results from the MC simulations by Vink *et al.* (Ref. 3).

Surface tension in units of vs the volume fraction difference, . In this example, ; the symbols are numerical results, the drawn line is the gradient expansion approximation, and the filled symbols are results from the MC simulations by Vink *et al.* (Ref. 3).

Curvature correction to the volume fraction profile as a function of (in units of ) using the ic (circles) and the cc (squares). In this example, and ; the symbols are numerical results and the drawn lines are the analytical profiles from the gradient expansion.

Curvature correction to the volume fraction profile as a function of (in units of ) using the ic (circles) and the cc (squares). In this example, and ; the symbols are numerical results and the drawn lines are the analytical profiles from the gradient expansion.

Contributions to the bending rigidity and in units of vs the volume fraction difference, . In this example, ; the symbols are numerical results and the drawn lines are the gradient expansion approximation.

Contributions to the bending rigidity and in units of vs the volume fraction difference, . In this example, ; the symbols are numerical results and the drawn lines are the gradient expansion approximation.

Bending rigidity in units of vs the volume fraction difference, , using the ic (circles) and the cc (squares). In this example, ; the open symbols are numerical results, the drawn lines are the gradient expansion approximation, and the filled circles are the MC results by Vink *et al.* (Ref. 3); the dashed line is the fit .

Bending rigidity in units of vs the volume fraction difference, , using the ic (circles) and the cc (squares). In this example, ; the open symbols are numerical results, the drawn lines are the gradient expansion approximation, and the filled circles are the MC results by Vink *et al.* (Ref. 3); the dashed line is the fit .

Bending rigidity in units of vs the volume fraction difference, . In this example, ; the filled circles are the MC results by Vink *et al.* (Ref. 3), the drawn line is the virial expression with the sharp-profile approximation, the dashed line is the equilibrium result, and the dotted line is the Mecke and Dietrich result (Ref. 19) with .

Bending rigidity in units of vs the volume fraction difference, . In this example, ; the filled circles are the MC results by Vink *et al.* (Ref. 3), the drawn line is the virial expression with the sharp-profile approximation, the dashed line is the equilibrium result, and the dotted line is the Mecke and Dietrich result (Ref. 19) with .

Contributions to the bending rigidity and in units of vs the reduced temperature distance to the critical point, . The symbols are numerical results and the drawn lines are the gradient expansion approximation.

Contributions to the bending rigidity and in units of vs the reduced temperature distance to the critical point, . The symbols are numerical results and the drawn lines are the gradient expansion approximation.

The bending length (in units of ) vs the reduced temperature distance to the critical point, , using the ic (circles) and the cc (squares). The open symbols are numerical results and the drawn lines are the gradient expansion approximation. The dashed line is the correlation length and the dotted line is the Mecke and Dietrich result (Ref. 19) with .

The bending length (in units of ) vs the reduced temperature distance to the critical point, , using the ic (circles) and the cc (squares). The open symbols are numerical results and the drawn lines are the gradient expansion approximation. The dashed line is the correlation length and the dotted line is the Mecke and Dietrich result (Ref. 19) with .

## Tables

Listed are the simulation results (Ref. 3) for the polymer volume fraction , liquid and vapor colloidal volume fractions, and , surface tension (in units of ), bending rigidity (in units of ; in parentheses is the estimated error in the last digit), and (in units of ).

Listed are the simulation results (Ref. 3) for the polymer volume fraction , liquid and vapor colloidal volume fractions, and , surface tension (in units of ), bending rigidity (in units of ; in parentheses is the estimated error in the last digit), and (in units of ).

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