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.     

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.     

Specific heat of3He-4He mixtures in Vycor glass

The specific heat of a ³ He- ⁴ He mixture filling a narrow pore is calculated in the temperature range 0.5–1 K, including the effect of van der Waals forces due to the walls, and is found to be compatible with the enhancement over the bulk liquid value experimentally observed for mixtures in Vycor glass.     

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).     

Interfacial tension near the tricritical point in3He-4He mixtures

The interfacial tension ₁ of phase-separated³He-⁴He mixtures has been measured between 0.5 K and the tricritical point. The observed decrease of _i on approaching the tricritical point is much stronger than for ordinary critical points. Our data are consistent with a critical exponent =2.     

Ultrasonic propagation in3He-4He mixtures near the tricritical point

We present measurements of the singular sound attenuation _sing in liquid mixtures of³He and⁴He near the superfluid transition and at temperatures above the phase separation curve. The mole fractionX of³He ranged from 0.55 to 0.73 and the frequency /2 varied between 1 and 45 MHz. The temperature range was 0.75–1.4 K, with the greatest emphasis on the tricritical region nearX _t=0.67 andT _t=0.87 K. From the change in slopedU/dT of the sound velocityU, we present a new determination of the phase separation curve, which is compared with previous measurements. The sound attenuation peak along the superfluid transition becomes broader in temperature asT is decreased. In addition, there is also an increase in sound attenuation as the phase separation temperatureT is approached. ForX<X _t these two peaks merge into one asX _t is approached. For a given frequency, the attenuation has a maximum value at the tricritical point. Estimates of the contribution _D of mass diffusion to the attenuation for³He-⁴He mixtures with 0<X<0.55 and comparison with experimental values show that _D becomes relatively more important asX increases, and that atX=0.55 it effectively accounts for all of the observed singular attenuation, at least at megahertz frequencies. Hence we assume that for mixtures withX>0.55 the observed attenuation can be analyzed solely in terms of the diffusive relaxation mechanism. The mass diffusion parameterD is then determined from the data. AtX=0.55,D diverges asT is approached, which is consistent with theoretical expectations and experimental results. NearT _t, there is a crossover to a tricritical regime, and it is found that approximatelyD(T–T _t) ^Z withZ=0.32±0.1. Mode coupling predictions are thatZ=1/2 while recent renormalization group calculations giveZ=1/3. The attenuation curves in the tricritical region at the various frequencies can be represented satisfactorily but not perfectly by a scaling function with a characteristic relaxation time (T–T _t) ^–x withx=1.7±0.15. This time corresponds to order-parameter fluctuations. Its temperature dependence is in excellent agreement with renormalization-group calculations that givex5/3, while expectations from dynamic scaling are thatx=3/2. Our analysis also gives the variation of the amplitudes of andD with the direction of approach toT _t. A comprehensive theory for interpreting all the data, in the normal as well as in the superfluid phase, is lacking at this time.Supported by a grant from the National Science Foundation. A preliminary account of this work was presented inBull. Am. Phys. Soc. 21, 229 (1976).     

Thermal concentration gradients and the thermal diffusion ratio in3He-4He mixtures near the tricritical point

We report a study of transient effects in sound velocity measurements in several³He-⁴He mixtures near the tricritical point, from which by extrapolation we derive the equilibrium thermal concentration gradient and the thermal diffusion ratiok _T . In the superfluid phase, the thermal concentration gradient is found to be proportional to the concentration susceptibility.Research supported by a grant from the National Science Foundation.     

Shear viscosity measurements of liquid 4He and 3He-4He mixtures near the tricritical point

A torsional oscillator cell is described, by means of which simultaneous precision measurements of () and of the molar volume can be made in liquid ⁴He-⁴He mixtures over the temperature range between 0.5 and 3 K. Here is the mass density, the shear viscosity and in the superfluid phase they become the contributions _n and _n of the normal component. The results of for ⁴He near the superfluid transition are compared with the predictions by Schloms, Pankert and Dohm, and by Ferrell. Measurements of () are reported for mixtures with 0.64X0.74, where X is the ³He mole fraction. Those for X = 0.67 and 0.70 are compared with data by Lai and Kitchens. The viscosity experiments show no evidence of a weak singularity at the tricritical point.     

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.).     

Phase diagram and concentration susceptibility of 3He-4He mixtures near the tricritical point

The inverse concentration susceptibility (/\x) _T of ³He-⁴He mixtures has been calculated from high-resolution vapor-pressure measurements and in situ measurements of the dielectric constant very close to the tricritical point. The measurements have been fitted to a scaling-law equation of state. In the normal fluid a logarithmic correction term was necessary to obtain a good fit. In the superfluid a regular correction term, proportional to (x – x _t)², was evaluated. New measurements of the phase-separation curve confirm our earlier measurements on the ³He-rich side and extend the measurements much closer to the tricritical point on the ⁴He-rich side. The tricritical point is at temperature T = 0.8669 ± 0.0005 K and mole fraction x = 0.6716 ± 0.0014.Work performed under the auspices of the U. S. Department of Energy.     

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.     

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.     

Equation of state of a3He-4He mixture near its liquid-vapor critical point

Measurements of the pressure coefficient (P/T)_,x are reported for a ³ He- ⁴ He mixture with a mole fractionX=0.805 of ³ He in the neighborhood of the liquid-vapor critical point. These include data on 16 isochores taken over the density interval–0.50.5 and over the temperature range–0.1 t0.1, where =(– _c )/ _c andt=(T-T _c )/T _c ,with _c andT _c ,respectively, the critical density and temperature of the mixture. From the discontinuity of (P/T)_,x at the boundary between the two-phase and the one-phase regions we determine the dew-bubble curve nearT _c with better precision than was done in recentPVT experiments. From the extrapolation of data not approachingT _c closer than1 mK, (P/T)_,x along the critical isochore appears to be discontinuous atT _c ,while for the isochore / _c 0.92, (P/T)_,x is continuous across the dew curve. It is found that this latter isochore cuts the dew curve at its highest temperature. These observations are discussed in terms of general thermodynamic arguments and theoretical predictions of the asymptotic behavior. We calculate (P/T)_,x from the scaling equation of state proposed by Leung and Griffiths for ³ He- ⁴ He mixtures, using their numerical parameters. In spite of some systematic deviations, especially in the two-phase region, there is in general good agreement with experimental results. In particular, the shape of the measured dew-bubble curve and the apparent discontinuity of (P/T)_,x along the critical isochore show excellent agreement with theory.Work supported by a grant from the National Science Foundation. A report of this work has been presented at the Washington Meeting of the APS [Bull. Am. Phys. Soc. 20, 618 (1975)].     

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.     

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.     

Persistent currents and critical velocities in3He-4He mixtures

Using the method of Doppler-shifted fourth sound, we have investigated the angular velocity _c1 for the onset of vorticity, the saturated critical velocity, and the dependence of _s/ onv _n –v _s for⁴He in three different pore sizes and for three different mixtures of ³ He- ⁴ Hein one of the powders used. These are the first measurements of critical velocities in ³ He- ⁴ He mixtures. We observe only a weak dependence of _c1on the pore size and within the limits of the experiment no dependence on concentrationX up toX=0.11. We find, however, that the maximumv _n –v _s attainable increases as the concentration of ³ He increases forX0.17. The observed dependence of the superfluid density onv _n –v _s is approximately two orders of magnitude stronger than predicted.     

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.     

NMR Studies on 3He-3He and 3He-4He tunneling motions in solid 3He-4He mixtures

Helium-3 nuclear spin relaxation times T ₁, T ₂, and T ₁have been measured for ³He-⁴He solid mixtures at the exchange plateau region (~0.5K). The ³He concentrations X ₃of the samples were 7.2, 2.9, 1.8, 1.4, 0.67, 0.65, and 0.22%, and their molar volumes varied between 19.9 and 20.9cm³/mole in hcp phase. The spectral density function J() for dipolar field fluctuations was determined in the low-frequency branch from T ₁measurements and in the high-frequency branch from conventional T ₁measurements. It was found that J() is given by J() = cJ()|_3–4 + (1–c)J()|_3–3, where J()|_3–4 is the spectral density function due to the ³He-⁴He tunneling motions, and J()|_3–3 is that due to the ³He-³He tunneling motions. Using the Torrey theory, the correlation frequency of the ³He-⁴He tunneling motions was evaluated from T ₁data, and was found to be in good agreement with Landesman 's theory.Supported in part by the Japan Society for the Promotion of Science through a grant to Y.H.     

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.