Free energy and specific heat of critical films and surfaces

Based on the field-theoretical renormalization-group theory for O(N) symmetric systems in 4-ɛ dimensions, the free energy of critical films confined between two parallel plates at distance L is analyzed. For five different boundary conditions and for temperatures above and at the bulk critical temperature ${\mathit{T}}_{\mathit{c},}$b, the analytic expressions of the universal scaling functions are derived and discussed. At bulk criticality, the scaling functions reduce to universal amplitudes commonly known as Casimir amplitudes. In addition it is shown that the scaling functions can be expanded in a series of powers of L/${\xi }_{\mathrm{bulk}}$, in which all expansion coefficients are universal numbers with the Casimir amplitudes characterizing the leading terms. Special attention is paid to the specific heat of the films at bulk criticality as a function of L. One of those universal expansion coefficients of the scaling functions enters the experimentally accessible amplitude of the specific heat. In order to identify the finite-size contributions to the free energy of the films, the surface free energies are discussed too and expressed in terms of universal surface amplitudes. In view of experimental applications such as, e.g., for ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He mixtures, tricritical films are also analyzed. The implications of these results for surface forces and wetting films are pointed out.