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. The Kosterlitz-Thouless superfluid onset temperature in the film is found to occur just as the Casimir dip in the film thickness from critical fluctuations becomes evident. Additionally, a new film-thickening effect is observed precisely at $T_\lambda$ when the temperature is swept extremely slowly. We propose that this is a non-universal Casimir effect arising from the viscous suppression of second sound modes in the film.