Diffusive relaxation processes in liquid³He-⁴He mixtures. I. Normal phase

Abstract

When a heat flux is switched on across a fluid binary mixture, steady state conditions for the temperature and mass concentration gradients ∇T and ∇c are reached via a diffusive transient process described by a series of terms “modes” involving characteristic times τ _n . These are determined by static and transport properties of the mixture, and by the boundary conditions. We present a complete mathematical solution for the relaxation process in a binary normal liquid layer of heightd and infinite diameter, and discuss in particular the role of the parameterA=k ² _T (∂μ/∂c) _T,P /TC _P,c coupling the mass and thermal diffusion. Herek _T is the thermal diffusion ratio, (∂μ/∂c) ⁻¹ _T,P is the concentration susceptibility, μ is the chemical potential difference between the components, andC _P,c is the specific heat. We present examples of special situations found in relaxation experiments. WhenA is small, the observable times τ(∇T) and τ(∇c) for temperature and concentration equilibration are different, but they tend to the same value asA increases. We present experimental results on four examples of liquid helium of different³He mole fractionX, and discuss these results on the basis of the preceding analysis. In the simple case for pure³He (i.e., in the absence of mass diffusion) we find the observed τ(∇T) to be in good agreement with that calculated from the thermal diffusivity. For all the investigated³He-⁴He mixtures, we observe τ(∇c) and τ(∇T) to be different whenA is small, a situation occurring at high enough temperatures. AsA increases with decreasingT, they become equal, as predicted. For the mixtures with mole fractionsX(³He)=0.510 and 0.603, we derive the mass diffusionD from the analysis of τ(∇c) and demonstrate that it diverges strongly with an exponent of about 1/3 in the critical region near the superfluid transition. As the tricritical point (T _t,X _t) is approached for the mixtureX=X _t0.675,D tends to zero with an exponent of roughly 0.4. These results are consistent with predictions and also with theD derived from sound attenuation data. We discuss the difficulties of the analysis in the regime close toT _λ andT _t, with special emphasis on the situation created by the onset of a superfluid film along the wall of the cell forX=0.603 and 0.675.