Clustering and percolation transitions in helium and other thin films

A theory of the growth of thin films and the evolution of bulk characteristics is developed from a thermodynamic basis. The work has been prompted by recent experiments on ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, and ${\mathrm{N}}_2$ films adsorbed on uniform basal-plane graphite surfaces, which indicate that as coverage increases the characteristics of the films may progress from a monolayer regime to a two-layer regime, and then at coverages which are specific to each system and temperature, to the relatively abrupt appearance of nearly bulklike properties. It is shown that the nature of film growth with increasing adsorption is controlled by three interfacial coefficients and the spreading pressure, which permit three general classes of behavior: (I) uniform deposits at all thicknesses; (II) bulk nucleation after a thin uniform deposit; (III) bulk nucleation with no preadsorption. Metal deposits on inert insulator and semiconductor surfaces typically belong to classes II and III. Van der Waals films are conventionally considered class I, although several class II examples are known. The recent experiments strongly suggest that He and ${\mathrm{N}}_2$ on graphite are members of class II, and quantitative estimates of the heliumgraphite parameters are consistent with this possibility. Further aspects of class-II growth are also considered. On ideal surfaces the clustering transition is guarded by a high nucleation barrier but in real systems the barrier is reduced or entirely removed by substrate heterogeneities. This usually leads to the formation of many isolated clusters which grow into "thick" islands coexisting with ultrathin film. The islands are shown to be markedly flattened by the substrate field, and as more material is deposited they grow principally in the lateral direction. When the coverage is sufficiently high the islands establish connectivity in a percolation transition. Percolation transitions appear to take place in metal films; we propose that they occur also in helium films. In ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ films below $T_{\lambda }$ the percolation transition can take place via Josephson tunneling between droplets, at coverages too low for "geometric" connectivity: this causes the superfluid onset temperature to depend on coverage. A number of persistent puzzles associated with onset phenomena are understandable in terms of the theory. It seems likely that models of superfluid onset in uniform slab geometry have little to do with real films.