Non-universal Casimir Effect in Saturated Superfluid $^4$He Films at T$_\lambda$

Measurements of Casimir effects in $^4$He films in the vicinity of the bulk superfluid transition temperature $T_\lambda$ have been carried out, where changes in the film thickness and the superfluid density are both monitored as a function of temperature. A new Casimir film-thickening effect is observed precisely at $T_\lambda$ when the temperature is swept extremely slowly. We believe this arises from the viscous suppression of any second sound modes in the superfluid film, while thermally excited second sound still propagates in the bulk superfluid to within microkelvins of $T_\lambda$, giving rise to a free energy difference between the bulk and film. At $T_\lambda$ this difference drops abruptly to zero, leading to a step increase in the film thickness that we have observed. The magnitude of the step increases rapidly with the equilibrium film thickness, in agreement with a calculation of the Casimir energy balance. From the amplitude of the increase we can extract the first measurement of the second-sound free energy at $T_\lambda$, found to be about 2.6 ergs/cc. This is at least roughly consistent with a Debye-type calculation of the free energy.