Excitations and thermodynamics for liquid-helium films

We derive a hydrodynamic theory of the long-wavelength excitations of helium films adsorbed in cylindrical pores and on planar substrates. In the case of cylindrical geometry, the spectrum exhibits an instability for a particular value of the ratio of the film surface radius to the pore radius. This instability condition is shown to coincide with a thermodynamic instability condition determined by the relative magnitude of the surface energy and the energy of interaction with the substrate. Calculated adsorption isotherms show a corresponding hysteresis in good agreement with measured values. We suggest the possibility that the soft modes present for films thinner than the unstable limit mediate an attractive interaction between quasiparticles of a ${\mathrm{He}}^3$ film adsorbed on the ${\mathrm{He}}^4$, leading to a new kind of ${\mathrm{He}}^3$ superfluid. Similar calculations are presented for the spectrum of ${\mathrm{He}}^4$ on a planar substrate. In this case, as well as in the cylindrical case, surface excitations (ripplons) dominate the thermodynamic properties at low temperature ($T\lesssim 0.1$ K). In this regime the properties are a simple function of the dimensionality of the surface. At higher $T$, short-wavelength modes contribute, and the results depend in a complicated fashion on the finite dimensions of the system. We discuss the available data, where possible, and suggest further experiments.