Effective Heat Conductivity and Relaxation Processes in Superfluid Mixtures of 3He-4He

The effective thermal conductivity coefficient'ceff in superfluid ³He-⁴He mixtures with concentration of 9.8% ³He has been studied experimentally between 100 and 500 mK, where the main contribution to the kinetic processes is made only by phonons and ³He impurity excitations. In this case the effective thermal conductivity is a combination of diffusivity, thermal conductivity and thermal diffusion. The κ _eff value was found from stationary measurements of the temperature gradients caused by the thermal flow and from the temperature relaxation kinetics. Both the methods provide consistent resugts which also agree with those on effective thermal conductivity calculated in terms of the kinetic theory of phonon-impuriton system.     

Transport properties in3He-4He mixtures near the superfluid transition

We report measurements of the temperature, density, and concentration gradients in³He-⁴He mixtures, induced by a vertical heat flux. The flat horizontal cell included two superposed capacitors and the density was determined by means of the dielectric constant method. The experiments were carried out on mixtures with mole fractionsX ₃=0.37, 0.15, and 0.05 at saturated vapor pressure, with special emphasis on the region near the superfluid transition. Our measurements under steady-state conditions give the conductivity , the thermal diffusion ratiok _T , and the coefficient of thermal expansion. We describe the singular behavior of these quantities in the neighborhood ofT (X). In the superfluid phase, we test with fair success a relation by Khalatnikov between gradX/ gradT and several static properties. From the relaxation times needed to attain steady-state conditions, and in combination with measured static and transport properties, we obtain in the normal phase the mass diffusionD, which diverges strongly asT is approached. In the superfluid phase, we test successfully a scaled relation that results from the solution of Khalatnikov's hydrodynamic equations. From our data the dispersion relations for scattered light are calculated: _o/q ² in both the normal and the superfluid phases and ₂/q ² in the normal phase.Research supported by NSF grant DMR 8024056.     

Critical and tricritical dynamics in3He-4He mixtures

A new formalism is presented to study the critical and tricritical dynamics of³He-⁴He mixtures near the superfluid transition for arbitrary concentration. In this fluid two conserved variables, the molar concentrationX and the entropys, are twofold coupled to a complex order parameter ψ first in the dynamic equations reversibly and second in the free energy dissipatively. However, at an intermediate concentrationX=X _D (which is 0.37 at the saturated vapor pressure) a linear combination ofX ands is found to be asymptotically decoupled from ψ both reversibly and dissipatively. There, dynamic renormalization group equations reduce to those of pure⁴He (or those of the F model) and some dynamic properties are common to those of pure⁴He. For example, atX≈X _D , the gradient ?(T?T _λ ) under heat flow goes to zero without³He mass flow, whereT _λ is the critical temperature, dependent onX, and the thermodiffusion ratio loses the singularity with the critical exponent α. Our dynamic renormalization group equations take into account the above two nonlinear couplings and can be used for any concentrations. Furthermore, using a linear response scheme, general relations are obtained among the kinetic coefficients. As a result the thermal conductivity on the λ line is found to be exactly proportional toX ^?1 at smallX. The coefficient in front ofX ^?1 can be expressed in terms of the diffusion constant of an isolated³He molecule in⁴He.     

The Damping of a Quartz Tuning Fork in Superfluid 3He-B at Low Temperatures

We have measured the damping on a quartz tuning fork in the B-phase of superfluid ³He at low temperatures, below 0.3T _c. We present extensive measurements of the velocity dependence and temperature dependence of the damping force. At the lowest temperatures the damping is dominated by intrinsic dissipation at low velocities. Above some critical velocity an extra temperature independent damping mechanism quickly dominates. At higher temperatures there is additional damping from thermal quasiparticle excitations. The thermal damping mechanism is found to be the same as that for a vibrating wire resonator; Andreev scattering of thermal quasiparticles from the superfluid back-flow leads to a very large damping force. At low velocities the thermal damping force varies linearly with velocity, but tends towards a constant at higher velocities. The thermal damping fits very well to a simple model developed for vibrating wire resonators. This is somewhat surprising, since the quasiparticle trajectories through the superfluid flow around the fork prongs are more complicated due to the relatively high frequency of motion. We also discuss the damping mechanism above the critical velocity and compare the behaviour with other vibrating structures in superfluid ³He-B and in superfluid ⁴He at low temperatures. In superfluid ⁴He the high velocity response is usually dominated by vortex production (quantum turbulence), however in superfluid ³He the response may either be dominated by pair-breaking or by vortex production. In both cases the critical velocity in superfluid ³He-B is much smaller and the high velocity drag coefficient is much larger, compared to equivalent measurements in superfluid ⁴He.     

The Tricritical Region for Helium Mixtures in 0.87 Porosity Aerogel

We have used ultrasonic velocity measurements to study ³ He- ⁴ He mixtures in aerogel with a porosity 0.87. The phase diagram resembles that of bulk mixtures, with a single transition for ³ He-rich mixtures, in contrast to the detached phase separation curve seen in 0.98 porosity aerogel. A kink in the lambda line at a 3He concentration of X _C =0.51 suggests that the phase separation line meets it at a tricritical point. We have measured the amount of superfluid which decouples both at low temperature and close to the superfluid transition, as functions of ³ He concentration. Each showed a sudden change at the concentration where the kink appeared in the lambda line, suggesting an abrupt change in the morphology of the superfluid phase in the mixtures. Similar measurements were made for pure 4He films on the same aerogel. We discuss the nature of ³ He-rich mixtures in aerogels based on these experiments.     

Second-sound velocity and superfluid density in3He-4He mixtures near superfluid transition points—Applicability of universality and scaling laws

From second-sound velocity measurements, superfluid densities near superfluid transition points were determined in ³ He- ⁴ He mixtures under saturated vapor pressure. The critical exponent of the superfluid density thus obtained increases about 15% with increasing ³ He concentration up to 40 mole percent, which contradicts the universality concept. Furthermore, the critical exponents obtained here and the exponents of the specific heat do not satisfy the scaling laws. A new concept of universality is introduced and discussed.Submitted in partial fulfillment of the requirements for the Ph.D. degree at The University of Tokyo.     

Depression of the superfluid transition in4He

We study superfluid⁴He near T in a homogeneous metastable state where a finite superfluid velocity v_s is present. Neglecting vortices we perform a renormalization-group calculation of the critical velocity v_sc(T) at which the superfluid state becomes unstable. We apply this result to the situation where the superfluid velocity is induced by a finite heat current Q. A critical heat current Q_c(T) corresponding to v_sc(T) is found which implies a transition temperature T (Q)=T[1–A_oQ^x]. We determine the exact exponent x=[(d–1)v] ^–1 0.744 in d=3 dimensions and calculate A_o in one-loop order. Our results for A_o and x are compared with recent experimental data.     

Mass and spin diffusion near the λ line in dilute3He-4He mixtures

We examine the possibility of using NMR and other measurements on very weak solutions of ³ He in liquid ⁴ He to investigate the superfluid phase transition. It is found that even for these very weak solutions the mass (D _m) and spin (D _s) diffusion coefficients associated with the ³ He behave in radically different ways:D _m is predicted to diverge asT approaches T from above as (T–T) ^–1/3 , while the behavior ofD _s depends on the type of experiment, and for the ordinary spin-echo type has no particular singularity.Based on a D.Phil. thesis submitted to the University of Sussex by M. A. Eggington.     

Transport properties of helium near the liquid-vapor critical point. II. Thermal conductivity of a3He-4He mixture

Measurements of the thermal conductivity are reported for an 80%³He-20%⁴He mixture above the critical point along several isotherms and near-critical isochores, using the same techniques and apparatus described for a study of³He. Using again the assumption that the observed conductivity can be decomposed into a sum of a regular and a singular contribution _reg and _sing, it is shown that along two near-critical isochores, _sing diverges. In particular, along the isochore showing the largest at the phase transition, the divergence is nearly the same as for³He and can be roughly characterized by a simple power law (T-T_c)^– with 0.58. This observation is contrary to predictions that foresee _sing0 asT _c is approached. The relaxation times characterizing the attainment of steady state conditions after switching the heat flux on and off show a similar behaviour as a function of reduced temperature as do those for pure³He. This result might indicate a substantial coupling between concentration and entropy diffusion. In the Appendix, the correlation length for³He nearT _c is calculated from heat conductivity, viscosity, and specific heat data.Work supported by Grant DMR 8024056 of the National Science Foundation.     

Thermal convection and thermal conductivity of nondilute3He-4He mixtures

We present measurements of the critical temperature difference for the onset of thermal convection and the effective thermal conductivity in two³He-superfluid-⁴He mixtures. The mixtures were 6.8% and 9.8% by molar volume of³He in⁴He and the measurements were made from 0.65 K to just above the superfluid transition temperature for each mixture. The measurements were made as part of an effort to visualize convective flow patterns in helium mixtures using optical shadowgraph techniques. We discuss the implications of our results for this effort.     

Theory of helium under heat flow near the λ point. II. Dynamics of phase change

The He I-He II interface is a crucial aspect in the transformation processes between the superfluid and normal fluid phases. Its motion is investigated when temperatures and heat flows at boundaries deviate from those of a stationary coexistence state. As a unique feature, the heat flow to the interface from the He I side can be mostly transmitted to the He II side by thermal counterflow, and the latent heat generation (or absorption) at the interface becomes negligibly small. In any case the interfacial motion is so slow that the temperature on the He II sideT is still given by the stationary relationT –T Q ^3/4, whereT is the critical temperature and Q is the heat flow. The temperature profile and the interfacial position are calculated in some nonstationary cases. To this end a simple approximation scheme is developed. First, the interface can propagate with a constant velocity and the superfluid region can expand as a shock wave. Second, if the heat flow at the warmer boundaryQ _w and that in the He II regionQ _– are fixed at different values, the length of the He I region y_i changes in time as (d/dt)y _i ^1+p = const Q _w – Q_–, wherep=1/(1–x), and x is the critical exponent of the thermal conductivity. In particular, ify _i=0 att=0 andQ _w>Q_–, the normal fluid region emerges asy _it ^1/(1+p) at the warmer boundary. Third, ifQ _– and the temperature at the warmer boundary are fixed, the interfacial position approaches an equilibrium position exponentially in time. The uniqueness of the problem arises from the superfluidity on the He II side and the strong critical singularity of the thermal conductivity on the He I side.     

The Thermal Conductivity of Superfluid 3He in Aerogel: A Measurement of the Energy Gap

Transport properties of superfluid ³ He in aerogel are governed by a fixed mean free path set by the typical dimensions of the strand separation. We describe preliminary measurements of the thermal conductivity of superfluid ³ He confined in 98% silica aerogel. The majority of the measurements are made in relatively high magnetic fields where the superfluid within the aerogel is in the A-phase like state. Since the quasiparticle mean free path is fixed, the temperature dependence of the thermal conductivity depends only on the superfluid energy gap and on the texture in the case of the A-phase. The results so far obtained are broadly consistent with the aerogel-confined superfluid having a temperature dependent energy gap close to the BCS prediction although there are significant deviations at the lowest temperatures.     

Thermal conductivity study of thin He films adsorbed in porous glasses with well controlled pore sizes

Thermal conductivity study of thin He films adsorbed in porous glasses with well controlled pore sizes^[2] is reported. The measurements are performed in a cell where a torsional oscillator monitors the superfluid density change, for the same films for comparison. Since the heat flux through the superfluid is proportional to the superfluid velocity, we discuss the possible observation of the intrinsic critical velocity v_sl = h/ml, inherent in superfluid He films in such porous systems with unit pore length l, as discussed by Minoguchi and Nagaoka^[3], which marks the velocity at which phase slippages start to occur over macroscopic scales.on leave from: Inst. Physics near Prahaon leave from: Lukin's Research Inst. Phys. Problems, Zelnograd, Moscow.     

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

Relaxation times are reported from the transients observed during thermal conductivity _eff and thermal diffusionk _T ^* measurements in superfluid mixtures of³He in⁴He with a layer thickness of 1.81 mm. The experiments extend from 1.7 K toT and over a³He concentration range 10^–6X<5×10^–2. The agreement between the measured and the predicted from the two-fluid thermohydrodynamic equations is satisfactory forX>10^–3 but deteriorates for smaller³He concentrations. This situation is similar to that for _eff andk _T ^* 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 coefficientD _iso for isolated³He in⁴He has been determined from and from _eff 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 withh ². FromD _iso derived via the _eff data, the³He impurity-roton scattering cross section is determined. Comparisons with previous work are made.     

Onset of Convection in Superfluid 3He-4He Mixture Heated from Below

The thermal instability of ³He-⁴He mixtures caused by heating the liquid from below has been studied experimentally. The temperature gradients were measured which appeared in the mixture with initial concentration 9.8% of ³He below 0.5 K in the presence of different heat flows from the heater at the cell bottom. At a certain critical heat flow the effective thermal conductivity of the liquid was observed to increase sharply which was naturally attributed to the convective heat transfer. It is shown that the thermal convection develops at high temperature gradients. In this case the Rayleigh numbers exceed many orders of magnitude those for heating from above. Thus the convective instability develops in a system in which the light liquid is at the top and where no prerequisite for instability is seemingly available. The resugts obtained are analyzed in terms of the theory of convective instability in binary mixtures. It is suggested that the phase separation, of superfluid mixtures caused by a heat flow could be a destabilizing factor initiating convection. The vortex formation in superfluid helium and the related turbulent flows appearing at high temperature gradients can be another factor favourable for instability of the liquid.     

Temperature coefficient of optical path of tellurite glasses doped with gold nanoparticles

This work reports on the thermo optical parameters of TeO₂–PbO–GeO₂ glasses as a function of the concentration of gold nanoparticles. Techniques such as thermal lens and heat capacity measurements allowed the determination of the thermal diffusivity D, thermal conductivity K and the temperature coefficient of optical path (ds/dT). It is shown an expressive decrease in ds/dT as a function of gold nanoparticles concentration, while the opposite effect was observed for the thermal diffusivity D.     

Quantitative Ratio of Heat Flows Due to Second Sound to Dissipation in Superfluid 3He-4He Mixtures

In this paper we compare two hydrodynamic modes in superfluid ³He-⁴He mixtures, focusing on which mode gives greater contribution to propagation of heat in superfluid liquids. Starting from the two-fluid hydrodynamic model, the quantitative contributions of second sound mode and dissipative thermal conductivity mode to the total heat flow are studied. The problem is considered in a cell with heat source of a form Q ₀cos²(ωt) at one side. Analytical expressions for space-time temperature and heat flow dependence are obtained and discussed.     

Wetting of3He-4He mixtures on alkali metal substrates

From available information concerning the wetting behavior of pure³He and pure⁴He on alkali metal substrates, as well as the known properties of bulk³He-⁴He mixtures, the complete wetting phase diagrams for such mixtures, showing prewetting, isotopic separation and lambda transitions for the film phases have been derived. We predict new phenomena such as a triple-point induced dewetting transition, and the absence of a superfluid film wetting Cs, Rb and K walls under concentrated³He solutions.     

Transport properties of helium near the liquid-vapor critical point. IV. The shear viscosity of3He and4He

Shear viscosity measurements with a precision of 0.05% are reported for³He and⁴He along near-critical isochores 0.85 c <1.12, where _c is the critical density. The temperature range was –10^–4<<1, where =(T – T _c)/T _c is the reduced temperature. The experiments were carried out with a torsional oscillator operating at 158 Hz, driven at resonance in a phase-locked loop. The absolute value of the viscosity was obtained by calibration at the superfluid transition of⁴He, based on published values and from direct calculations using the free decay time constant of the oscillations. The data are analyzed in terms of a model using the recent mode-coupling (MC) expressions by Olchowy and Sengers, and where account is taken of the earth's gravity effects. The theory could be fitted very well to the experiment with a single free parameter, the cutoff wave numberq _D, which was found to be 3.0×10⁶ and 7.0×10⁶ cm^–1 for³He and⁴He, respectively. We have used for the critical exponent the MC predicted value of z=0.054, which permits a fit superior to that using z=0.064 predicted by dynamic renormalization group (DRG) theories. Detailed comparisons are made between the model calculations and data for various isochores and isotherms and good agreement is obtained. The effects of gravity are described in some detail. The predicted frequency effect in viscosity measurements is calculated for³He and is shown to be obscured by gravity effects. Using the Olchowy-Sengers formulas, we have also fitted the MC theory to the critical thermal conductivity data of³He, again withq _D as the only free parameter. This fit gaveq _D=6 × 10⁷ cm^–1, which in the ideal situation should have been the same asq _D from viscosity. We also discuss a representation of the³He viscosity data along the critical isochore by a power law and first correction-to-scaling erm. Using the viscosity and the critical conductivity data for³He, we have calculated the dynamic amplitude ratio and obtained =1.05±0.10, in agreement with predictions from MC and DRG theories. Also, agrees with data of classical fluids. Finally, a comparison is made of recent shear viscosity data for CO₂ by Bruschi and Torzo with those on He. The CO₂ data are also analyzed in terms of the MC theory, and the discrepancies are discussed. In the Appendices, we present the results of new compressibility measurements on³He along the critical isochore, as used in the MC analysis. We also present a brief analysis of the fluid hydrodynamics in the torsional oscillator leading to relations for the viscosity as a function of the measured quantities. Finally, we give a short outline of the vertical density profile calculations from the earth's gravity field for the calculations of the viscosity nearT _c.