Fang Zhong

Thermal transport properties in helium near the superfluid transition. I.4He in the normal phase

The thermal conductivity κ and the associated relaxation time τ to reach steady-state conditions are reported for the normal phase of several very dilute mixtures of3He in4He (X<4 × 10−6) at saturated vapor pressure near Tλ. The measurements were made over the reduced temperature range 2.5 × 10−6<ɛ<2×10−1, where ɛ ≡ (T−Tλ)/Tλ, and are representative for pure4He. The spacing between the cell plates was 0.147 cm. The systematic uncertainty in the conductivity data is estimated to increase from ∼2% for ɛ=0.2 to ∼4% for ɛ=3 × 10−6. The random scatter due to finite temperature resolution increases to ∼7% at the smallest ɛ. The data are in agreement within the combined uncertainty with recent ones by Tam and Ahlers (cell F, spacing 0.20 cm) and with previous ones in this laboratory taken with a different plate spacing. The thermal diffusivity coefficientDT = κ / ϱCp obtained from τ is found to agree within better than 15% with the calculated one using data for κ, the density ϱ, and the specific heatCp. Measurements of the effective boundary resistivityRb in the superfluid phase are described.Rb is found to depend on the thermal history of the cell when cycled up to 77 K and above. Also,Rb shows the beginning of an anomalous increase for ¦ɛ¦≲10−4. The possible reasons for this anomaly are discussed, and their impact on the analysis of conductivity data in the normal phase is appraised.

Transport properties in the superfluid phase of dilute3He-4He mixtures nearT ?

A cell for the simultaneous measurements of temperature and3He concentration gradients, induced by a heat currentQ across a fluid helium layer is described. This cell is operated over the temperature range 1.7≤T<2.5 K. Measurement of the anomalous boundary resistance ΔRb in superfluid4He(X(3He)∼2 ppb) near Tλ for heat currents between 8 and 47 µW/cm2 are described. The results for both the weakly divergent and the heat-dependent, more strongly divergent contributions to ΔRb are presented. The observed amplitude of the linear part is found to be larger than predictions by Onuki, by Ginzburg and Sobaynin and by Frank and Dohm, and also larger than recent data by Duncan and Ahlers (DA). These discrepancies are discussed in the light of the effective boundary area in the cell. The nonlinear part of ΔRb is consistent with the data by DA. It is found that the maximum observed ΔRb at the superfluid transition is independent ofQ. Analysis of the conductivity data of mixtures leads to the conjecture that ΔRb will decrease will the addition of3He. Our measurement of both the effective thermal diffusion ratiokT* and of the thermal conductivity κeff over the range 10−6<X<5×10−2 show departures from the predictions by Khalatnikov. In particular forX<10−3,kT* is found to be a function of X, and κeff deviates from the predicted κeff∝X−1. This last result confirms measurements with a previous cell. In the appendices, determinations and tabulations of auxiliary thermodynamic derivatives, such as (∂V/∂X)T,P and (∂S/∂X)T,P are presented.

Relaxation times and mass diffusion in superfluid dilute3He-4He mixtures

Relaxation times τ are reported from the transients observed during thermal conductivity κeff and thermal diffusionkT* measurements in superfluid mixtures of3He in4He with a layer thickness of 1.81 mm. The experiments extend from 1.7 K toTλ and over a3He concentration range 10−6≤X<5×10−2. The agreement between the measured and the predicted τ from the two-fluid thermohydrodynamic equations is satisfactory forX>10−3 but deteriorates for smaller3He concentrations. This situation is similar to that for κeff andkT* results and indicates that the transport properties in very dilute mixtures with layers of finite thickness are not well understood. ForX>10−3, the mass diffusion coefficientDiso for isolated3He in4He has been determined from τ and from κeff measurements. There is an inconsistency by a constant numerical factor between these determinations. This problem might be related to the observations that in the superfluid phase, the relaxation times for different cell heightsh do not scale withh2. FromDiso derived via the κeff data, the3He impurity-roton scattering cross section is determined. Comparisons with previous work are made.

Thermal response of a fluid near its critical point: {sup 3}He at T > T{sub c}

The local density response is studied in a simple fluid near the liquid-vapor critical point, subjected to temperature oscillations of its container. This investigation provides a new approach in the study of the adiabatic energy transfer (“piston effect”) in the fluid. The density response functionZF(ω, ε,z) is calculated for3He in the absence of stratification, where ω is the angular frequency, ε=(T−Tc)/Tc the reduced temperature,Tc=3.316 K the critical temperature, andz the vertical position in the container. Experiments are described where the density is measured by two superposed capacitive sensors in a cell of 3.5 mm height, and where the temperature oscillation frequencyf=ω/2π is varied between 10−4 and 2 Hz. Over the experimental range 5×10−4<ε<5×10−2 there is in general reasonable agreement between predictions and experiments. The systematic departures might be accounted for by deviations from 1D geometry, which were not included in the calculations. Over the frequency and reduced temperature ranges, the damping effect from the critical bulk viscosity is predicted to be too small to be detectable. The observed effect of the stratification and its frequency dependence inZF are briefly discussed. In the appendix, the predicted critical acoustic attenuation from the bulk viscosity is compared with published data, the effect from finite thermal conductivity of the fluid container plates and also the corrections toZF for the effects of the cell sidewalls are calculated.

Thermal transport properties in the normal phase of dilute3He-4He mixtures

We present steady-state measurements of the thermal diffusion ratiokT and of the heat conductivity κ for three dilute mixtures of3He in4He with concentrations 9×10−3≤X(3He)≤5×10−2 at saturated vapor pressure in the normal phase close to the superfluid transition. The data are compared with predictions by Dohm and Folk from the renormalization group (RG) theory. From auxiliary determinations of thermodynamic derivatives for these mixtures, we obtain the separation factor Ψ=−(kT/T)×(∂ρ/∂X)T,P/(∂ρ/∂T)X,P above Tλ over the range wherekT is positive. Here ρ is the mass density. From the transients of δX(t) as a function of time, we obtain an estimation for the mass diffusion coefficientD and compare the results with predictions by Dohm and Folk and with results from other experiments.

Thermal transport properties of the helium near the superfluids transition II. Dilute /sup 3/He-He mixtures in the superfluid phase

Measurements of the average thermal conductivity κexp ≡hQ/ΔT and of the thermal relaxation time τ to reach steady-state equilibrium conditions are reported in the superfluid phase for dilute mixtures of3He in4He. Hereh is the cell height,Q is the heat flux, andΔT is the temperature difference across the fluid layer. The measurements were made over the impurity range 2×10−9<X(3He)<3×10−2 and with heat fluxes 0.3<Q<160 µW/cm2. Assuming the boundary resistanceRb, measured forX<10−5, to be independent ofX over the whole range ofX, a calculation is given for κexp. ForQ smaller than a well-defined critical heat fluxQc (X) ∝X0.9, κexp is independent of Q and can be identified with the local conductivity κeff, which is found to be independent of the reduced temperature ɛ = (T−Tλ)/Tλ for −ɛ≤10−2. Its extrapolated value at Tλ is found to depart forX≲10−3 from the prediction κλ ∝X−1, tending instead to a weaker divergence κλ ∝X−a witha≈0.08. A finite conductivity asX tends to zero is not excluded by the data, however. ForQ >Qc(X), a nonlinear regime is entered. ForX≲10−6, the measurements with the available temperature resolution are limited to the nonlinear conditions, but can be extrapolated into the linear regime forX≳2×10−7. The results for κexp(Q),Qc(X), and κeff(XX) are found to be internally consistent, as shown by comparison with a theory by Behringer based on Khalatnikovs transport equations. Furthermore, the observed relaxation times τ(X) in the linear regime are found to be consistent forX>10−5 with the hydrodynamic calculations using the measured κeff(X). ForX<10−5, a faster relaxation mechanism than predicted seems to dominate. The transport properties in the nonlinear regimes are presented and unexplained observations are discussed.

Transport Properties of Very Dilute Superfluid Mixtures of 3He in 4He

The theory of transport properties of superfluid helium mixtures, based on the two fluid model, has been developed more than thirty years ago 1–3). One of the tests of this theory was by heat conductivity experiments 4), performed with dilute mixtures of 3He in the 4He with 1.4×10−4 ≤ X ≤ 1.3×10−2, where X is the 3He mole fraction. For the range X ≤ 10−2 the predictions become particularly simple and were confirmed by Ptukha’s experiments4). However more recent experiments with more dilute mixtures 5–7) showed the heat conductivity keff to depart from predictions for X ≤ 5×10−4. Furthermore the thermal relaxation times τ also showed an anomalous behavior 6,7). This stimulated further measurements of keff, τ and also of the thermal diffusion ratio kT* which we report in this paper.

The singular boundary resistance of superfluid 4He

Experimental investigations on the singular boundary resistance of Helium 4 are described.(AIP)

Dynamics of density equilibration near the liquid-vapor critical point of3He

The density change σρ(t) was measured as a function of time at two locations in the upper and the lower half of a flat cell, after a temperature step ΔT≪(T−Tc), whereTc is the critical temperature. The series of measurements were made along the critical isochore. Direct observation of the sharp response from adiabatic energy transfer into the bulk fluid (“piston effect”), and the effect of slow diffusive relaxation to equilibrium are reported. The relaxation time is found to diverge asTc is approached. The onset of convection for a step ΔT beyond a critical value could be observed, with a dramatic change of the temporal profile of σρ(t). Computer simulations of this profile in the absence of convection have been carried out and are compared with the experiment.

Density equilibration near the liquid-vapor critical point in helium-3

Measurements of the density equilibration in3He after a step change in temperature are reported along the critical isochore as a function of reduced temperature e above and below the critical point Tc, with 10-6<z.sfnc;ez.sfnc;<10-1. The equilibrium process was followed separately in the liquid and the vapor phases below Tc where we observe different equilibration profiles. The relaxation time above Tc is discussed in terms of the thermal diffusivity.