Dynamic Measurements of Thermal Transport Coefficients and Boundary Resistance. II.Model for ³He-Superfluid ⁴He Mixtures

Abstract

This is the second of a three-part study of the ac response of liquid helium. We derive the temperature response function, ΔT(ω), of a ³ He-superfluid ⁴ He mixture from the equations of superfluid hydrodynamics in the presence of two interfacial boundary resistances,R_b.Specifically, we consider the response ΔT(ω), across a fluid layer of thickness,d, to an ac heat flux,Q(t) = Q_o exp(iωt).ΔT(ω) depends on the effective thermal conductivity, κ _eff , Griffin's diffusion coefficient, Γ _o (i.e. the thermal diffusivity of ³ He impurities, D_iso in the low ³ He concentration limit) and the thermal boundary resistance, 2R_b. This analysis provides the basis for experiments to determine these parameters. Although past experiments to measure these properties have been carried out using dc and transient techniques, an ac technique offers significant noise reduction over these techniques. By sweeping the frequency, it is possible for an experimenter to clearly identify different components of the system response to the heat flux. For instance, if τt is the slowest fluid thermal response time, conventional Kapitza boundary effects dominate at frequencies, ωτ≫1. These calculations reveal an interesting analogy to the “Piston Effect” for near-critical classical fluids. In Part I of this work, we used normal liquid ⁴ He as a testing ground for developing models of ac heat transport. In Part III of this work, we will present results in which we apply this technique to measurements on dilute mixtures of ³ He in superfluid ⁴ He.