On the spectrum of fluctuations of a liquid surface: From the molecular scale to the macroscopic scale

We show that to account for the full spectrum of surface fluctuations from low scattering vector qd << 1 (classical capillary wave theory) to high qd > 1 (bulk-like fluctuations), one must take account of the interface's bending rigidity at intermediate scattering vector qd = 1, where d 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.Comment: 18 pages, 11 figures, accepted for publication in The Journal of Chemical Physic

The existence of a bending rigidity for a hard sphere liquid near a curved hard wall: Helfrich or Hadwiger?

In the context of Rosenfeld's Fundamental Measure Theory, we show that the bending rigidity is not equal to zero for a hard-sphere fluid in contact with a curved hard wall. The implication is that the Hadwiger Theorem does not hold in this case and the surface free energy is given by the Helfrich expansion instead. The value obtained for the bending rigidity is (1) an order of magnitude smaller than the bending constant associated with Gaussian curvature, (2) changes sign as a function of the fluid volume fraction, (3) is independent of the choice for the location of the hard wall.Comment: 19 pages, 5 figures, to appear in Physical Review

On the spectrum of fluctuations of a liquid surface: From the molecular scale to the macroscopic scale

We show that to account for the full spectrum of surface fluctuations from low scattering vector qd 1 (bulk-like fluctuations), one must take account of the interface's bending rigidity at intermediate scattering vector qd = 1, where d 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.Comment: 18 pages, 11 figures, accepted for publication in The Journal of Chemical Physic

Density Functional Theory of a Curved Liquid-Vapour Interface: Evaluation of the rigidity constants

It is argued that to arrive at a quantitative description of the surface tension of a liquid drop as a function of its inverse radius, it is necessary to include the bending rigidity k and Gaussian rigidity k_bar in its description. New formulas for k and k_bar in the context of density functional theory with a non-local, integral expression for the interaction between molecules are presented. These expressions are used to investigate the influence of the choice of Gibbs dividing surface and it is shown that for a one-component system, the equimolar surface has a special status in the sense that both k and k_bar are then the least sensitive to a change in the location of the dividing surface. Furthermore, the equimolar value for k corresponds to its maximum value and the equimolar value for k_bar corresponds to its minimum value. An explicit evaluation using a short-ranged interaction potential between molecules, shows that k is negative with a value around minus 0.5-1.0 kT and that k_bar is positive with a value which is a bit more than half the magnitude of k. Finally, for dispersion forces between molecules, we show that a term proportional to log(R)/R^2 replaces the rigidity constants and we determine the (universal) proportionality constants.Comment: 28 pages; 5 figures; accepted for publication in J. Phys.: Condens. Matter (2013

Tolman lengths and rigidity constants of multicomponent fluids: Fundamental theory and numerical examples

acceptedVersio

Description of the fluctuating colloid-polymer interface

To describe the full spectrum of surface fluctuations of the interface between phase-separated colloid-polymer mixtures from low scattering vector q (classical capillary wave theory) to high q (bulk-like fluctuations), one must take account of the interface's bending rigidity. We find that the bending rigidity is negative and that on approach to the critical point it vanishes proportionally to the interfacial tension. Both features are in agreement with Monte Carlo simulations.Comment: 5 pages, 3 figures, 1 table. Accepted for publication in Phys. Rev. Let

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Tolman lengths and rigidity constants of multicomponent fluids: Fundamental theory and numerical examples

The curvature dependence of the surface tension can be described by the Tolman length (first-order correction) and the rigidity constants (second-order corrections) through the Helfrich expansion. We present and explain the general theory for this dependence for multicomponent fluids and calculate the Tolman length and rigidity constants for a hexane-heptane mixture by use of square gradient theory. We show that the Tolman length of multicomponent fluids is independent of the choice of dividing surface and present simple formulae that capture the change in the rigidity constants for different choices of dividing surface. For multicomponent fluids, the Tolman length, the rigidity constants, and the accuracy of the Helfrich expansion depend on the choice of path in composition and pressure space along which droplets and bubbles are considered. For the hexane-heptane mixture, we find that the most accurate choice of path is the direction of constant liquid-phase composition. For this path, the Tolman length and rigidity constants are nearly linear in the mole fraction of the liquid phase, and the Helfrich expansion represents the surface tension of hexane-heptane droplets and bubbles within 0.1% down to radii of 3 nm. The presented framework is applicable to a wide range of fluid mixtures and can be used to accurately represent the surface tension of nanoscopic bubbles and droplets

Correspondence between the pressure expressions and van der Waals theory for a curved surface

We investigate the apparent contradiction between the pressure expressions, or ‘‘mechanical expressions,’’ and the van der Waals squared-gradient expressions for the curvature coefficients k/R0 , k, and k¯. We show that, in the context of the mean-field theory discussed, both types of expression are indeed equivalent, with the differences only being caused by the thermodynamic conditions used to vary the curvature