AC Thermal Conductivity Measurements in Dilute Mixtures of 3He in Superfluid 4He

In Part III of a three-part study, we report measurements of the thermal response, T(), of ³ He-superfluid- ⁴ He mixtures to an ac heat flux, Q(t)=Q₀e^it. These data are for dilute concentrations, X, and they show the presence of three separate thermal resistances. One of these is the bulk-fluid resistance predicted by Khalatnikov and associated with the effective conductivity, _eff . Results for this component of the resistance are in quantitative agreement with the Khalatnikov predictions. With parallel work by Murphy and Meyer, these experiments resolve a long-standing conflict between theory and experiment. One of the remaining resistances is the ordinary boundary resistance R_b. The third resistance, R ₀ , is independent of the fluid layer height, d. This resistance is presumably the same as that seen in earlier dc measurements. Both the temperature and concentration dependences of this anomalous resistance differ from that of either R _fluid or R_b. It has been ascribed recently by Murphy and Meyer to effects associated with the narrow gaps usually present in cryogenic thermal conductivity experiments. We use an ad hoc model as a convenient way to parameterize the extra thermal resistance. The present studies have been carried out with an apparatus which permits us to vary d continuously and in situ from zero to 3 mm. This feature and the ac technique are important for separating the various components of the thermal resistance. In two preceeding studies, we considered related aspects of the ac thermal response of liquid helium. Part I addresses the response of normal liquid helium. Part II, provides the theoretical backdrop for the present experimental study.     

Thermal Response of Superfluid 4He near T λ to an AC Heat Flux

We have measured the thermal response of a superfluid⁴He layer nearT _λ to a time-varying heat flux. In the low-frequency range studied, 1×10⁻³<f<0.1 Hz, we find the response to be independent of the applied frequencyf, a result consistent with the expectation that the measured response is due to the boundary resistanceR _b alone. These findings differ from those by Olafsen and Beharinger. The thermal conductivity cell used by these authors had extraneous surfaces in contact with the superfluid which were eliminated in the cell used for our experiments; we believe these surfaces were responsible for the frequency dependence observed in previous work. Furthermore, we show that AC and DC measurements ofR _b differ by no more than 1%, and that both exhibit a weak singularity nearT _λ.     

Thermal equilibration of fluids near the liquid-vapor critical point:3He and3He-4He mixtures

We study the temperature-equilibration process of fluids at constant volume in a thermal conductivity cell, where an initial temperature gradient relaxes to zero. The calculation is performed in the linear approximation for a pure fluid and a binary mixture. Near the critical point of the pure fluid, the adiabatic heating process, which takes place at constant volumeV, causes equilibration to proceed four times faster whenC _P /C _V 1 than for the process at constant pressureP. For the mixtures, the relaxation rate enhancement at constantV compared with constantP is restricted to a temperature region where the coupling between temperature and mass diffusion is small. The predictions are compared with experimental results for³He and for two³He-⁴He mixtures along their critical isochores. Finally, we discuss the thermal relaxation in the two-phase (liquid-gas) and one-phase (gas) regimes at the critical density, as measured with a conductivity and a calorimetry cell. The contrasting behavior for³He and a³He-⁴He mixture in these two regimes and under these different constraints is pointed out and discussed.     

Gravitational Effects on Nonlinear Heat Transport Near the Superfluid Transition in 4He

We have recently observed nonlinear heat transport within 30 nK of the superfluid transition temperature using heat flux, Q, in the range 0.1 < Q < 2 erg/(s cm ² ). While Haussmann and Dohm (HD) accurately predict the initial departure of the thermal conductivity, , from the linear response region, is greater than expected very close to T . We anticipate that the nature of the thermal conductivity's nonlinearity may depend upon Earth's gravity in the low heat flux limit (Q < 0.5 erg/(s cm ² )). Comparison of our data to similar data to be taken in a microgravity laboratory will provide an experimental determination of the effect of gravity on nonlinear heat transport near the superfluid transition. The microgravity measurements will also permit the first experimental test of theories that do not consider gravitational effects, such as those by HD.     

Transport properties for a dilute solution of3He in two-dimensional liquid4He

The temperature variations of the diffusion coefficientD(T), thermal diffusion ratio k _T (T) and thermal conductivity (T) in a dilute solution of³He atom in two-dimensional liquid helium are evaluated explicitly by solving the kinetic equations via phonon-phonon, phonon-roton, roton-roton, impurityelementary excitation and impurity-impurity scatterings. In the low-temperature region, the main contributions toD(T) and (T) come from the interactions between phonons and impurities, while in the high-temperature region the interactions between impurities and whole elementary excitations contribute more strongly toD(T) and (T) than those of only elementary excitations. For a dilute solution, the thermal diffusion ratio k _T (T), neglecting the internal mass counterflow, is much smaller than the effective thermal diffusion ratio k _T ^* (T), which is a function of thermostatic properties. The effective thermal conductivity _eff is much larger than the thermal conductivity and has different temperature dependence from the thermal conductivity. The behaviors of the two-dimensional diffusion coefficient and thermal conductivity are much like the bulk case, where they exhibit exponential decay with increasing temperature, although they are much smaller than those of the bulk case.     

Thermal response of a3He-superfluid-4He mixture

The response of a layer of superfluid mixture to an ac heat source,Q(t)=Q ₀ exp(it), is determined. In the low-frequency regime, the temperature response at the heated side of a superfluid layer is essentially identical to that of an ordinary fluid having a thermal conductivity _eff and a thermal diffusion coefficient ₀ /2. Here _eff is the effective conductivity of Khalatnikov, and ₀ is the diffusion coefficient of Griffin. At much higher frequencies, the results are more complicated. The low-frequency regime is defined in terms of the second sound velocityu ₂ by u ₂ ² / ₀ . The ac response function is valuable in a number of ways. It can be used to obtain the system response to more complicated time-dependent variations inQ such as step changes inQ. A knowledge of the response function in the low-frequency regime provides a mechanism for directly determining the Kapitza resistance in mixtures. Finally, a knowledge of the response function provides an additional opportunity to test two-fluid hydrodynamics. Alternative tests of superfluid hydrodynamics are of particular interest in light of recent experiments that show anomalous values for _eff in the low ³ He concentration limit     

Experiments on the Self-Organized Critical State of 4He

When a heat flux Q is applied downward through a sample of ⁴He near the lambda transition, the helium self organizes such that the gradient in temperature matches the gravity-induced gradient in T _λ . All the helium in the sample is then at the same reduced temperature and the helium is said to be in the Self-Organized Critical (SOC) state. We have made the first measurements of the ⁴He SOC state specific heat, C _{∇ T} (T(Q)). There is no measurable difference between C _{∇ T} and the static zero-gravity ⁴He specific heat for temperatures between 650 and 250 nK below T _λ . Closer to T _λ , the specific heat is depressed and reaches a maximum value at 50 nK below T _λ . This depression is similar to that predicted theoretically as reported by R. Haussmann (Phys. Rev. B 60, 12349, 1999). Contrary to the expectations of theory, however, we see another depression far below T _λ . In addition, over the heat flux range of 30 nW/cm² to 13 μW/cm², we have made improved measurements of the speed of a recently discovered propagating thermal mode, which travels only upstream against the nominal heat flux of the SOC state. We are able to accurately predict the speed of this wave by treating the helium of SOC state as a traditional fluid with a temperature dependent thermal conductivity.     

Hydrodynamic heat flow in very dilute superfluid3He-4He mixtures

The superfluid hydrodynamics of heat flow is examined for very small mass concentrationsc of³He in⁴He in an effort to better understand recent results for the effective heat conductivity _eff, which appear to be in conflict with predictions. The full hydrodynamics contains a thermal boundary layer; within this layer the temperature and concentration gradients differ from those in the bulk fluid. An examination of finite heating effects based on the ansatz _eff c ^–p for smallc shows distinctly different behavior for experimental determinations of _eff whenp<1,p=1, andp>1. Thus, finite heating can be used as a probe to evaluate the exponentp.     

The decay rate of critical fluctuations in3He-4He mixtures near the gas-liquid critical point

We have measured the critical light scattering intensity and the Rayleigh line shape for³He and for³He-⁴He mixtures with compositionX(³He)=0.95, 0.79, and 0.63 along their respective critical isochores near the plait point. The experimental linewidth of³He is compared with the calculated one from heat conductivity and equation of state measurements, and satisfactory agreement is obtained. For mixtures, gravity effects in our cell of finite height prevent us from reaching the critical point along a path at strictly constant composition and density. HenceT _c cannot be determined directly. Using the prediction that the scattered light intensity in the mixtures has the same diverging behavior as for the pure fluid, we determine the reduced temperaturet[T – T _c(X)]/T _c from the intensity. The measured Rayleigh line shape can be expressed by a single decay rate as a function oft for a given scattering angle of the light beam. Our experiments show that in the mixtures is only weakly dependent on composition. Our analysis leads to the determination of the mass diffusion coefficientD, which is found to be nearly independent of composition and nearly equal to the thermal diffusivityD _T measured for³He. The results are discussed in the light of the predictions from mode coupling theory.     

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

This is the second of a three-part study of the ac response of liquid helium. We derive the temperature response function, T(), 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 T(), across a fluid layer of thickness,d, to an ac heat flux,Q(t) = Q_o exp(it).T() depends on the effective thermal conductivity, _eff , Griffin's diffusion coefficient, _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 t is the slowest fluid thermal response time, conventional Kapitza boundary effects dominate at frequencies, 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.     

Heat capacity of3He/4He mixtures

We measured the specific heat capacity of three dilute mixtures (2.6, 1.07, and 0.44%) of³He in⁴He and of pure⁴He. The⁴He contribution to the specific heat of the mixture is subtracted, leaving only the³He part. This is fitted to a theoretical expression over the whole temperature range from 10 to 700 mK. Assuming a dispersion relation of the form =² k ²(1+k ²)/2m*, the fits yield the value of and the effective mass m* of each mixture. The average value of is –0.076±0.01 A² and the effective mass in the limit of zero concentration is (2.23±0.02)m ₃. These are compared to values deduced from other measurements.     

Heat transport in dilute mixtures of3He in superfluid4He

We report new measurements of the effective thermal conductivity K_eff and relaxation time τ in dilute mixtures of³He in superfluid⁴He, with molar concentrationsX≤10⁻³. The temperature range extended fromT≈1.4 K toT _λ. Both K_cff and τ are found to agree with theoretical predictions, in contrast to previous experiments where significant differences were observed. A new thermal conductivity cell design was used which almost completely eliminates extraneous volumes and surfaces, and the earlier results are explained in relation to these design changes.     

Time-dependent thermal convection in dilute solutions of 3He in superfluid 4He

Time-dependent thermal convection can occur in a unity-aspect-ratio Rayleigh-Beard convection cell containing a dilute solution of ³He in superfluid ⁴He when the fluid is heated from above. Results are presented primarily for a 0.24 mole % He solution at 0.925 K. Means is provided for introducing heat at the top and separately for a central plug and an outer ring such that both are at a constant temperature gDT above the bottom. A critical temperature difference T _cfor convection can be defined above which both steady and time-dependent convection occur. The time-dependent effects include a region of T. near T _cand characterized only by excessive noise, a region of somewhat higher T where there are intermittent major changes in the plug heating rate with a time distribution like that for random events, and a region at still higher T where periodic but nonsinusoidal variation of the heat flow is observed. When a long enough time, several months, has elapsed after cooling down the apparatus, time-dependent states no longer occur, and the heat flow above T _cis limited to steady convection. Briefly raising the temperature of the apparatus to 77 K is sufficient to restore the possibility of time-dependent states.     

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.     

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.     

Superfluidity of Pure 3He and Mixtures of 3He and 4He in Aerogel

We describe new torsional oscillator experiments on ³ He confined in 98.2% open aerogel. In one, we monitored the superfluid fraction of pure ³ He at T << T _c while we gradually changed the sample pressure. The resulting change in density alters ₀ of the superfluid ³ He relative to the distribution of the length scales (correlations) of silica in the aerogel. We observed a T = 0 normal-to-super fluid transition at a pressure of about 6.5 bar, in marked contrast to the bulk where liquid ³ He is superfluid at all pressures. In the second experiment, we measured the temperature dependence of the ³ He _s at a pressure of 21.6 bar with different amounts of ⁴ He present in the cell. Adding 2-3% ⁴ He slightly increases both T _c and _s . We found that for ⁴ He concentrations between 2% and 34%, the ³ He T _c increases by a very small amount. However, _s , which for pure ³ He in aerogel at 0.5T _c is no more than 11%, falls by another factor of 7. This behavior (constant T _c , reduced _s ) is similar to that observed in granular superconducting films where the long-range order is controlled by phase coherence between adjacent grains.     

Measurement of the 4He Chemical Potential in Phase-Separated 3He-4He Liquid Mixtures

As a first step towards thermodynamic measurements in highly polarised dense ³ He fluids, an accurate determination of the ⁴ He chemical potential ₄ was performed in unpolarised phase-separated ³ He- ⁴ He liquid mixtures at low pressures (0-1 bar) over a temperature range 0.12 - 0.65 K. A method introduced by H. London and relying on heat flush effects was used. Two volumes containing : a) a cold, phase-separated helium mixture and b) a warmed, pure ⁴ He liquid are connected by a narrow tube, and their temperatures are recorded under various conditions. The results agree with existing data obtained by the same technique for T >0.2K, but cannot be analysed with the simple regular solution model fitting vapour pressure data at T >0.6K. The sensitivity of the technique is shown to be sufficient to observe expected effects of nuclear polarisation on ₄.     

Localization-Fermi Liquid Transition of 3He Adsorbed on 4He-Coated Hectorite

To create a phase diagram of two-dimensional (2D) ³ He fluids, we measured heat capacity of the ³ He adatoms down to 20mK as a function of the density. The ³ He adatoms are deposited on hectorite precoated with ⁴ He in the amount of 24.70mol/m ², just above the onset density for the ⁴ He super-fluid. Isotherms of the heat capacity steeply increase with the ³ He density up to n _c =2.4mol/m ². Above n _c , the heat capacity shows properties of the 2D Fermi liquids. We pointed out that the density dependence of the heat capacity is similar to those of the insulator-metal (I-M) transitions of some electron systems, e.g. Si:P. Assuming the transition from a localized state to the Fermi liquid at the density n _c , the diameter of a localization area a _H ^* was obtained as about 3.8Å using the Mott relation that the mean distance equals 2.2 a _H ^* at the transition. Effective mass and 2D spin fluctuations of the Fermi liquids are enhanced with the density by increasing correlations in the 2D liquids.     

Sound Propagation in 3He and 4He Above the Liquid-Vapor Critical Point

A scaled plot is presented of published sound attenuation and dispersion data of ³ He and ⁴ He along the critical isochore above T_c as a function of the reduced frequency ^* . Here ^* / with the frequency and the order parameter relaxation rate, where the latter is determined from experimental transport and thermodynamic data. For a given isotope, the scaled data obtained at different frequencies lie on a single curve. These curves are different, however, for ³ He and ⁴ He. The resulting scaled plot is compared with recent predictions by Onuki, and by Folk and Moser. The crossover-term contribution in from the background thermal conductivity data is compared with the form derived by Bhattacharjee and Ferrell. In the Appendix, the calculation of for both ³ He and ⁴ He from experiments is described and discussed.     

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.