Thermal transport properties of helium near the superfluid transition. II. Dilute3He-4He mixtures in the superfluid phase

Measurements of the average thermal conductivity _exp hQ/T and of the thermal relaxation time 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, andT 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 resistanceR _b , measured forX<10^–5, to be independent ofX over the whole range ofX, a calculation is given for _exp. ForQ smaller than a well-defined critical heat fluxQ _c (X) X ^0.9, _exp is independent of Q and can be identified with the local conductivity _eff, which is found to be independent of the reduced temperature = (T–T)/T for –10^–2. Its extrapolated value at T is found to depart forX10^–3 from the prediction X ^–1 , tending instead to a weaker divergence X ^–a witha0.08. A finite conductivity asX tends to zero is not excluded by the data, however. ForQ >Q _c (X), a nonlinear regime is entered. ForX10^–6, the measurements with the available temperature resolution are limited to the nonlinear conditions, but can be extrapolated into the linear regime forX2×10^–7. The results for _exp(Q),Q _c (X), and _eff(XX) are found to be internally consistent, as shown by comparison with a theory by Behringer based on Khalatnikov's transport equations. Furthermore, the observed relaxation times (X) in the linear regime are found to be consistent forX>10^–5 with the hydrodynamic calculations using the measured _eff(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.     

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

The thermal conductivity and the associated relaxation time 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 T. The measurements were made over the reduced temperature range 2.5 × 10^–6<<2×10^–1, where (T–T)/T, 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 =0.2 to 4% for =3 × 10^–6. The random scatter due to finite temperature resolution increases to 7% at the smallest . 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 coefficientD _T = / C _p obtained from is found to agree within better than 15% with the calculated one using data for , the density , and the specific heatC _p . Measurements of the effective boundary resistivityR _b in the superfluid phase are described.R _b is found to depend on the thermal history of the cell when cycled up to 77 K and above. Also,R _b shows the beginning of an anomalous increase for ¦¦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 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.     

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

We present steady-state measurements of the thermal diffusion ratiok _T and of the heat conductivity for three dilute mixtures of³He in⁴He with concentrations 9×10^–3X(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 =–(k _T /T)×(/X) _T,P /(/T) _X,P above T over the range wherek _T is positive. Here is the mass density. From the transients of X(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.     

Sound propagation in 3He-4He mixtures near the superfluid transition

Measurements of the acoustic attenuation and dispersion in liquid ³He-⁴He mixtures near the superfluid transition T (x) are reported. The frequency range is /2gp=1–45 MHz and the ³He mole fraction X of the mixtures is 0.007, 0.05, 0.15, and 0.36. Comparisons are made with the measurements of Buchal and Pobell for similar mixtures obtained in the kHz region, and on the whole, the consistency between the two experiments is very satisfactory. An analysis is then performed using both the kHz and MHz data. In the normal phase, where the energy dissipation is caused by order parameter fluctuations having a lifetime _F , the attenuation data can all be scaled according to the expression = (T )f(_F. Here (T )^1+y, with y being a function of the mole fraction X and _F(T–T )^–x, with x increasing weakly with X. In the superfluid phase, we attempt a similar scaling representation, which is found to be fairly successful, but where x(T\s-T ) is roughly 15% larger than x(T>T ). In the superfluid phase we also analyze the attenuation data, assuming the additivity of relaxation and fluctuation-dissipation mechanism, and discuss the relaxation times so derived. In contrast to the attenuation, the dispersion data cannot be brought satisfactorily into a scaling representation. However, at T , we find U()-U(0)^y as predicted by Kawasaki, where y is in good agreement with the values from attenuation experiments.Supported by a grant from the National Science Foundation.     

Concentration susceptibility of 3He-4He mixtures near the superfluid transition

Measurements of the concentration susceptibility % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabiqaaiaacaGaaeqabaWaaeaaeaaakeaacuaHhpWygaacai% abg2da9iabgkGi2kaadIfacaGGVaGaeyOaIyRaeuiLdqKaaiykamaa% BaaaleaacaWGubaabeaaaaa!3C99!\[\tilde \chi = \partial X/\partial \Delta )_T \] are reported for ³He-⁴He mixtures at saturated vapor pressure and at constant mole fraction X(³He) as a function of temperature. Here is the isotopic difference in chemical potentials. The mixtures cover the range from X = 0.60 to X = 0.677. Particular emphasis is given to the region near the lambda line for three mixtures and to the region close to the tricritical point. The method makes use of the vertical concentration gradient induced in the mixture by gravity. This difference X is measured via the dielectric constant over a height of 2 mm using a capacitance technique. The predicted peak of the susceptibility at the superfluid transition is observed and as the tricritical point is approached, this peak is progressively broadened by gravity effects. The data are compared with those from other methods and also with estimations based on calorimetric data for more dilute mixtures. The susceptibility data are transformed into results along a path at constant chemical potential . Sufficiently close to T , these can be fitted to an expression with a weakly singular term, which is consistent with the predictions on the grounds of universality. These predictions are based on the calorimetric data of more dilute mixtures. The width of this critical region is found to be consistent with theoretical estimations by Riedel, Meyer, and Behringer. The tricritical scaling scheme by these authors is tested by the new data and the resulting scaling curve is found to be in reasonable agreement with that obtained from earlier data by Goellner, Behringer, and Meyer, except in the region closest to the tricritical point. There the new data appear to be more consistent with measurements from light scattering. In addition, the concentration susceptibility for more dilute mixtures (0.05 < X < 0.4) is calculated both from calorimetric data and from saturated vapor pressure measurements and the results are found to be internally consistent.Supported by grants from the AFOSR and from the NSF.     

Velocity of second sound and superfluid density near the superfluid transition in3He-4He mixtures

Using superleak condenser transducers, the velocity of second soundU ₂ has been measured near the superfluid transition temperature T in³He-⁴He mixtures with molar concentrationsX of³He of 0.0, 0.038, 0.122, 0.297, and 0.440. We have obtained the superfluid density _s/ fromU ₂ on the basis of linearized two-fluid hydrodynamics. The results for _s/ are consistent with those obtained from the oscillating disk method, as expected from two-fluid hydrodynamics. The value of _s/ at eachX could be expressed by a single power law, _s/=k, where =1-T/R, with the experimental uncertainty. It is found that the exponent is independent of concentration forX0.44 within the experimental uncertainty. This concentration independence of is in agreement with the universality concept. From the conclusion that the values of are universal forX0.44, the concentration dependence of the superfluid component _s is expressed by an empirical equation _s(X, )=²_s(0, ). It is found that corresponds to the volume fraction of⁴He in the superfluid³He-⁴He mixture. The value of is in agreement with that obtained from the measurement of the molar volume by others.This paper is based on a thesis submitted to Tokyo University of Education in partial fulfillment of the requirements for the Ph.D. degree.     

Shear viscosity of liquid4He and3He-4He mixtures,especially near the superfluid transition

High-resolution measurements of are reported for liquid⁴He and³He-⁴He mixtures at saturated vapor pressures between 1.2 and 4.2 K with particular emphasis on the superfluid transition. Here is the mass density, the shear viscosity, and in the superfluid phase both and are the contributions from the normal component of the fluid ( _n and _n ). The experiments were performed with a torsional oscillator operating at 151 Hz. The mole fraction X of³He in the mixtures ranged from 0.03 to 0.65. New data for the total density and data for _n by various authors led to the calculation of . For⁴He, the results for are compared with published ones, both in the normal and superfluid phases, and also with predictions in the normal phase both over a broad range and close to T. The behavior of and of in mixtures if presented. The sloped/dT near T and its change at the superfluid transition are found to decrease with increasing³He concentration. Measurements at one temperature of versus pressure indicate a decreasing dependence of on molar volume asX(³He) increases. Comparison of at T, the minimum of _n in the superfluid phase and the temperature of this minimum is made with previous measurements. Thermal conductivity measurements in the mixtures, carried out simultaneously with those of , revealed no difference in the recorded superfluid transition, contrary to earlier work. In the appendices, we present data from new measurements of the total density for the same mixtures used in viscosity experiments. Furthermore, we discuss the data for _n determined for⁴He and for³He-⁴He mixtures, and which are used in the analysis of the data.     

Thermal transport phenomena near the lambda line of 3He-4He mixtures

Effective thermal conductivities in the absence of mass diffusion have been measured near the lambda line of ³He-⁴He mixtures of up to 33% ³He concentration. The thermal conductivity in the normal phase has a rapidly increasing behavior as the temperature approaches the lambda point, but reaches a finite value there. Thermodiffusion ratios have also been measured, and show weak divergences—if any, weaker than logarithmic. Then, by using previous experimental values of the mass diffusivity given by Ahlers and Pobell, the thermal conductivity and the corresponding thermal diffusivity in the absence of the concentration gradient are deduced. The thermal diffusivity diverges with a critical exponent of approximately 1/3 irrespective of ³He concentration. The diffusivities D₀ and D₂ corresponding respectively to the linewidths of the Rayleigh and the second-sound Brillouin scatterings in the superfluid phase are obtained: D₂ diverges with a critical exponent of approximately 1/3 irrespective of ³He concentration, whereas D ₀ does not diverge at all. As a whole, the mode-coupling theory on the basis of the dynamic scaling hypothesis has been proved to hold in the superfluid transition.Based on a Ph.D. dissertation submitted by Mitsuru Tanaka to the University of Tokyo (1977).     

Universality and the scaling laws in the superfluid phase transition of3He-4He mixtures

From the second-sound velocityU ₂ near the superfluid transition point, the superfluid densities in³He-⁴He mixtures, _s (X) and _s (), were deduced along the paths of constant³He concentrationX and of constant chemical potential difference of³He and⁴He. The following critical exponents of _s are determined: (a) =XX for _s (X) in the(X, T) plane,(b) X for _s (X) in the(, T) plane, and(c) for _s () in the(, T) plane. It is found that and _X change by about 4–6% relative to with increasing³He concentration up toX=0.4 and by 8–10% up toX=0.53. It seems that, belowX=0.53, universality hold for . Values of have been found to be in good agreement with the critical exponent of _s in pure⁴He under constant pressure. The values of and _X forX0.53 are also found to be consistent with the scaling relations in the (,T) plane of³He-⁴He mixture.Work performed in part while at the Electrotechnical Laboratory.     

Transport properties of helium near the liquid-vapor critical point. III. Thermal diffusion ratio,conductivity, and relaxation times of3He-4He mixtures

Measurements of the density and temperature gradient induced by a heat flux in two³He-⁴He mixtures along their critical isochore are reported. From these measurements, the thermal diffusion ratiok _T and the thermal conductivity are derived. In addition, similar measurements were made for³He. It is found that for the mixturesk _T and diverge with respective exponents of =1.23 and 0.6, which differs from the asymptotic predictions for binary mixtures. It therefore appears that the crossover to the expected mixture behavior has not been reached within the investigated range 10^–1>(T-T _c )/T _c >7×10^–4, where the limit close toT _c is set by the very long observed relaxation times. The characteristic times for the establishment of steady state conditions in the density and temperature gradients () and (T) are reported. Within experimental uncertainty they are found to be the same and nearly independent of isotopic composition. They are compared with the predictions from the hydrodynamic diffusion equations that lead to the characteristic times of the diffusion modes. The slowest calculated times are found to be in fair agreement with the observed ones.     

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.     

Thermodynamic derivatives in3He-4He mixtures in the critical region of the superfluid transition

By assuming a phenomenological Ginzburg-Landau-Wilson free energy for³He-⁴He mixtures near the critical surface of the superfluid transition we can express thermodynamic derivatives of the form (?a/?b) _c,d in terms of a correlation function of ¦ψ¦², where Gy is the order parameter. Herea, b, c, and d are any four of the quantities entropys, number densityn, molar concentrationX, temperatureT, pressurep, and chemical potential difference Δ=μ₃?μ₄. Relationships among these thermodynamic derivatives are those of the Pippard-Buckingham-Fairbank approximation in the³He-⁴He case. Some combinations of the coefficients in the free energy are equal to derivatives on the critical surface and remain invariant with respect to renormalization group transformations. A general scheme is also developed to give correlation function expressions for thermodynamic derivatives.     

Thermal transport properties and size effects in very dilute superfluid mixtures of3He in4He

The effective thermal conductivity _eff has been measured for 1.6 K for mixtures with 10^–6–2. Both _eff and the derived impurity mass diffusion coefficient D_iso show an unexpected dependence on the height h of the fluid layer. A scaled representation of D_iso leads to a purely phenomenological model involving an effective length scale, proportional to X^–1 and temperature independent. The relations so obtained are consistent with the observations as well as with the observed transition curve Q_C(X) to non-linear behavior.     

Transport properties of helium near the liquid-vapor critical point. V. The shear viscosity of3He-4He mixtures

A torsional oscillator operating at 158 Hz has been used to measure the shear viscosity for two mixtures,³He_0.65-⁴He_0.35 and³He_0.80–⁴He_0.20. Data for each mixture are reported along near-critical isochores 0.85</ _c <1.18, where _c is the critical density. Just as in a previous paper on pure³He and⁴He, the observed viscosity includes the sum of background and critical contributions and the effect from the earth's gravity nearT _c . The analysis of the data provides (1) the background viscosity versus density and temperatureT, (2) the viscosity ratio / along the various isochores and also along selected isotherms obtained by interpolation, showing the critical contribution, and (3) the shape of the dew-bubble curve for each mixture, as obtained from the maxima in or from the discontinuities in the sloped/dT at the transition from the two-phase to the single-phase regime. A comparison between the data of these mixtures and those for pure fluids is presented. The background viscosity data along isotherms and along isochores are found to vary smoothly from³He to⁴He. The critical viscosity ratio [ / ]( _c ) for mixtures along the critical isochore is comparable with that for the pure fluids, i.e., it diverges weakly asT _c is approached. Hence, just as for previously investigated critical transport properties in³He-⁴He mixtures, there is no significant difference in behavior from that in the pure fluids.     

Anomalous nuclear spin relaxation in liquid 3He-4He mixtures near the lambda transition

The longitudinal nuclear spin relaxation time T ₁ was measured for liquid ³He-⁴He mixtures under saturated vapor pressure in an epoxy sample chamber in which the wall effect is emphasized. The molar concentration of ³He was 0.12, 0.20, 0.35, 0.45, and 1.00. The temperature was regulated to better than 5 K in the range 1.2–2.1 K. A decrease of T ₁ was clearly observed several millikelvins below T (the superfluid transition temperature). T ₁ had a cusp like temperature dependence as a whole where the temperature is far from. T . This anomalous behavior of T ₁ is discussed.Work based on part of a dissertation submitted to the University of Tsukuba in partial fulfillment of the degree of Doctor of Science by one of the authors (T.K.).     

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.     

Nucleation and phase separation in superfluid3He-4He solutions

The kinetics of the phase separation in³He-⁴He superfluid solutions is studied from anomaly in the first sound velocity and the dielectric constant. The histogram of experiments on solution supersaturation is obtained and used to study of phase separation probability. It was shown that above 50 mK the obtained temperature dependence of the supersaturation is qualitatively in accord with the thermal activation mechanism of nucleation. Below 50 mK the data may point towards a transition from classical to quantum mechanism of nucleation.     

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.