Phil Attard

Nanobubbles and the hydrophobic attraction.

The evidence for nanobubbles as the origin of the long-ranged attractions measured between hydrophobic surfaces immersed in water is reviewed by focusing upon several unique features of the force curves. Also covered is the morphology of nanobubbles as revealed by direct imaging with tapping mode atomic force microscopy. A discussion of the origin, thermodynamic stability and practical implications of nanobubbles is given.

Spherically inhomogeneous fluids. I. Percus–Yevick hard spheres: Osmotic coefficients and triplet correlations

For fluids in a spherically symmetric external field, it is shown that the Ornstein–Zernike convolution integral becomes a simple algebraic equation upon Legendre transformation. Applying the usual closure relations, the full inhomogeneous pair correlation functions become available. A discrete orthogonal transform pair is also derived, which, in conjunction with the Legendre factorization, makes computations feasable for these generic systems. The general method is here applied to a bulk uniform fluid of hard spheres, using the pair Percus–Yevick closure, but at the triplet level in the hierarchy of distribution functions. Consequently, the pair distribution function and the osmotic coefficient are better than the known analytic results. The method enables the accurate calculation of the triplet correlation function, and several examples are given.

Entropic Forces in Binary Hard Sphere Mixtures: Theory and Simulation

We perform extensive Monte Carlo simulations of binary hard-sphere mixtures (with diameter ratios of 5 and 10), to determine the entropic force between (1) a macrosphere and a hard wall, and (2) a pair of macrospheres. The microsphere background fluid (at volume fractions ranging from 0.1 to 0.34) induces an entropic force on the macrosphere(s); the latter component is at infinite dilution. We find good overall agreement, in both cases, with the predictions of a hypernetted chain-based theory for the entropic force. Our results also argue for the validity of the Derjaguin approximation relating the force between convex bodies to that between planar surfaces. The earlier Asakura-Oosawa theory, based on a simple geometric argument, is only accurate in the low-density limit.

Spherically inhomogeneous fluids. II. Hard‐sphere solute in a hard‐sphere solvent

The inhomogeneous Ornstein–Zernike equation and the Triezenberg–Zwanzwig expression for the density profile are solved using the Percus–Yevick closure for the inhomogeneous pair correlations of a hard‐sphere fluid in the vicinity of an isolated hard‐sphere particle. Results are presented for the solvent density profiles (solute–solvent radial distribution function) around solutes of diameter 0.01–50 times the diameter of the solvent hard spheres. At larger solute diameters, values obtained for the contact density are comparable in accuracy to those given by the scaled‐particle theory, and significantly more accurate than those given by the Percus–Yevick analytic results for a bulk asymmetric mixture. A superposition approximation is introduced and this gives the effective solvent‐mediated solute‐solute interaction. The approximation is expected to be accurate for an asymmetric mixture at low concentrations. Two lower order approximations, the Asakura–Oosawa depletion attraction, and the Derjaguin curvatur...

Hypernetted‐chain closure with bridge diagrams. Asymmetric hard sphere mixtures

Methods of calculating the first two terms in the density expansion of the bridge function are given. The closure to the Ornstein–Zernike equation is now exact to two orders in density beyond the hypernetted‐chain or Percus–Yevick approximations. The bridge function is resummed as a Pade approximant, and the results for hard spheres are relatively accurate over the whole density regime. The closure is shown to yield physically reasonable results for highly asymmetric mixtures. Infinitely dilute hard sphere solutes with diameters up to 30 times that of the hard‐sphere solvent are also considered. The Derjaguin approximation for rescaling the force between spheres to the interaction free energy between planes is examined and found to give the dominant curvature correction.

Beyond Poisson-Boltzmann: Images and correlations in the electric double layer. I. Counterions only

The general solution to the zero size mean spherical model is found for an inhomogeneous electrolyte (with specified profile) between charged planar surfaces. The analysis includes the effects of images and correlations. It provides the primary correction to the classical mean field theory of the double layer and allows the error in that theory to be easily estimated over experimental regimes. The relationship between Lifshitz theory and primitive model theories of electrolytes is made explicit. Results obtained for the one component double layer are accurate for systems with low coupling.

Nanobubbles: the big picture

Nanobubbles, whose existence on hydrophobic surfaces immersed in water has previously been inferred from measurements of long-ranged attractions between such surfaces, are directly imaged by tapping mode atomic force microscopy. It is found that the nanobubbles cover the surfaces in an irregular, interconnected or close-packed network whose morphology is dependent on pH and whose lifetimes are at least of the order of hours. Their height is of the order of 30nm and their radius of curvature is of the order of 100–300nm. It appears that the nanobubbles form from a solution supersaturated with air. A thermodynamic and statistical mechanical analysis of the homogeneous nucleation of liquid droplets from a supersaturated vapour shows that although a single droplet can be in equilibrium with a finite volume of gas, for a gas reservoir the equilibrium state is represented by a single macroscopic droplet, which grows by collisions and by Ostwald ripening. It is concluded that the electric double-layer repulsion between neighbouring nanobubbles on the hydrophobic surface plays a role in their stabilisation.

Cavitation of a Lennard-Jones fluid between hard walls, and the possible relevance to the attraction measured between hydrophobic surfaces

A Lennard‐Jones fluid confined between two planar hard walls is simulated using grand canonical Monte Carlo, and capillary evaporation is found for liquid subcritical bulk states. General methods are given for simulating a metastable fluid beyond coexistence. For the systems studied, the liquid and the gas phases coexist in equilibrium at a separation of ∼5 diam, the spinodal cavitation separation is at ∼4 diam, and the spinodal condensation separation is at ≳15 diam. The interaction pressure between the walls is found to be attractive and increases rapidly as the spinodal separation is approached. On the equilibrium liquid branch, the net pressure still appears significantly larger than the van der Waals attraction at separations of ∼10 diam. A simple analytic theory is given, which relates the force to the approach of the separation‐induced phase transition. It is suggested that this is the microscopic origin of the measured attractions between hydrophobic surfaces in water.

Beyond Poisson–Boltzmann: Images and correlations in the electric double layer. II. Symmetric electrolyte

An extension of the Poisson–Boltzmann theory of the electric double layer is applied to a symmetric electrolyte between two charged planar surfaces. This analytic treatment includes the effects of images and of ion correlations via the Debye–Huckel closure for the direct correlation function. It is demonstrated that at large separations the corrections to Poisson–Boltzmann theory appear as an effective surface charge. A means of correcting the apparent ion binding inferred from fitting experimental force data is also given. The extended theory shows good agreement with accurate numerical calculations for systems with low coupling. The generalization of the Onsager–Samaras limiting result to higher concentrations and to the case of the surface free energy of a charged dielectric interface is also presented.

Simulation of the chemical potential and the cavity free energy of dense hard‐sphere fluids

The chemical potential of dense hard‐sphere fluids, and also the work of cavity formation, are simulated directly by a force‐balance Monte Carlo technique. Here the coupling between a solute and the solvent varies in the presence of an external field. For a hard‐sphere fluid the variable is the cavity diameter, and the scaled particle theory proves sufficient for the applied field. The method is shown to be viable for densities as high as the freezing transition. A vectorizable Monte Carlo computer algorithm is also given.