The local compressibility of liquids near non-adsorbing substrates: a useful measure of solvophobicity and hydrophobicity?

We investigate the suitability of the local compressibility χ(z) as a measure of the solvophobicity or hydrophobicity of a substrate. Defining the local compressibility as the derivative of the local one-body density ρ(z) w.r.t. the chemical potential μ at fixed temperature T, we use density functional theory (DFT) to calculate χ(z) for a model fluid, close to bulk liquid-gas coexistence, at various planar substrates. These range from a 'neutral' substrate with a contact angle of θ≈90°, which favours neither the liquid nor the gas phase, to a very solvophobic, purely repulsive substrate which exhibits complete drying, i.e. θ = 180°. We find that the maximum in the local compressibility χ(z), which occurs within one-two molecular diameters of the substrate, and the integrated quantity χ<inf>ex</inf> (the surface excess compressibility, defined below) both increase rapidly as θ increases and the substrate becomes more solvophobic. χ(z) provides a more pronounced indicator of solvophobicity than the density depletion in the vicinity of the surface which increases only weakly with increasing θ. For the limiting case of drying, θ = 180°, we find ln χ(l) ∼ l, where l is the thickness of the intruding film of gas which diverges in the approach to bulk coexistence μ → μ<inf>co</inf>. When the fluid is confined in a parallel slit with two identical solvophobic walls, or with competing solvophobic and solvophilic walls, χ(z) close to the solvophobic wall is altered little from that at the single substrate. We connect our results with simulation studies of water near to hydrophobic surfaces exploring the relationship between χ(z) and fluctuations in the local density and between χ<inf>ex</inf> and the mean-square fluctuation in the number of adsorbed molecules.