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Superfluid Two-Phase Sound
Abstract
The properties of the sound wave propagating in a superfluid two-phase system (liquid helium and its vapor) are investigated. At a frequency sufficiently low so that the superfluid helium and its vapor are in quasistatic equilibrium, there can propagate only one sound mode whose wave-guide-like velocity depends strongly on the vapor properties as well as the velocity of second sound. Near the lambda point, the mode probes the critical properties of the specific heat of the superfluid.
... Plusieurs ouvrages de référence [62,69,97,19,36] traitent de la propagation d'ondes sonores dans des fluides ayant des interfaces et nous exploiterons les méthodes que les auteurs ont développées pour ce sujet. Par contre la littérature sur le sujet en particulier qui nous intéresse, est assez succincte et seules quelques articles ont abordé ce problème, notamment sur l'interface liquide-gaz pour un mélange d'hélium 3 He-4 He superfluide [83,104,60]. Ainsi nous nous inspirerons largement de ces quelques publications pour établir notre modèle théorique. ...
... Dans le cas précis de superfluides composés d'un mélange d'héliums 3 He-4 He quelques articles ont repris cette condition aux bords [83,104,60]. Notre problème est tout à fait similaire puisqu'à la place d'un mélange d'héliums superfluides nous avons un mélange de neutrons et de protons superfluides. ...
... Nous pourrons utiliser aussi cette condition aux bords pour notre problème. Mais ces mêmes articles [83,104,60] nous ont mis sur la voie pour trouver d'autre conditions aux bords possibles. ...
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... Two-phase sound in superfluid 4He is a mode which propagates in an acoustic waveguide partially filled with liquid. 1 The sound in the vapor above the liquid helium couples to the second sound in the superfluid, resulting in a single propagating mode characterized by both temperature and pressure oscillations. The velocity of the mode depends on the depth of the liquid in the waveguide, varying smoothly from the vapor velocity c~ when there is no liquid to the second-sound velocity c2 when the waveguide is completely filled. ...
... The initial observation of this mode was reported in Ref. 1, which will hereafter be referred to as I. A theory of the mode was formulated there by taking into account the hydrodynamic boundary conditions at the free surface between the liquid and vapor. ...
The attenuation of superfluid two-phase sound in4He is calculated and compared with experimental results. The dispersion relation of this mode (which consists of coupled vapor sound and second sound) is determined by boundary conditions at the free liquid surface. The primary source of attenuation is found to be the evaporation-condensation process at the free surface. A calculation using kinetic models for the nonequilibrium Onsager coefficients yields an attenuation in general agreement with experimental measurements.
... Two-phase sound in superfluid 4He is a mode which propagates in an acoustic waveguide partially filled with liquid. 1 The sound in the vapor above the liquid helium couples to the second sound in the superfluid, resulting in a single propagating mode characterized by both temperature and pressure oscillations. The velocity of the mode depends on the depth of the liquid in the waveguide, varying smoothly from the vapor velocity c~ when there is no liquid to the second-sound velocity c2 when the waveguide is completely filled. ...
... The initial observation of this mode was reported in Ref. 1, which will hereafter be referred to as I. A theory of the mode was formulated there by taking into account the hydrodynamic boundary conditions at the free surface between the liquid and vapor. ...
The surface and bulk tortuosities of sintered ceramic samples have been measured using third- and fourth-sound propagation in superfluid as a probe. Near the pore-space percolation threshold the index of refraction of both sound modes is found to increase rapidly as the porosity is lowered, to much higher values than previously observed. The bulk tortuosity exceeds the surface tortuosity for porosities below 30%, a reversal of the behavior observed at high porosity.
DOI:https://doi.org/10.1103/PhysRevLett.46.1252
A superfluid clamped so that the normal fluid is at rest can, under
pressure-released boundary conditions, support a new propagating thermal
mode whose phase velocity vanishes both at 0°K and
Tλ. Some problems which complicate the observation of
this mode as well as applications to 3He II research are
discussed. It is expected that this thermal mode can propagate even when
the temperature is so low that the wavelength is shorter than the
phonon-phonon mean free path.
This chapter describes the acoustic measurements in superfluid helium. Helium (He), the second lightest element, exists in two stable forms, 3He and 4He. The common isotope 4He condenses at the extraordinarily low temperature of 42°K. The liquid formed at this temperature behaves as a classical fluid and continues to do so until it is cooled to the λ temperature. It is found that when the Navier–Stokes or Euler equations for a classical fluid are linearized and propagating wave solutions are sought, the result is the acoustic solution that propagates at a characteristic velocity, the velocity of sound. The Landau equations also possess acoustic solutions and because of the two-fluid nature of He II, more than one type of wave solution exists. The special nature of the sounds of helium and the extremely low temperatures involved in superfluid helium research impose certain constraints on the transducers used to generate and detect first, second, and fourth sound. It is observed that at frequencies below about 200 MHz, freestanding transducers can be fabricated and operated in direct contact with liquid helium. The very low impedance of helium implies light loading of the transducer, which leads to a number of interesting consequences.
A new thermal mode, fifth sound, has been observed in a superleak
partially filled with superfluid helium. The velocity of the mode in
4He and 3He-4He mixtures is in good
agreement with the theoretical value
c52=(ρnρ)c22.
The index of refraction is found to be a strong function of the
fractional filling of the superleak.
In two-phase sound measurements with 3He-4He mixtures, the vapor sound and second sound are found to propagate as independent modes. This result can be traced to a boundary condition of conservation of 3He atoms at the liquid-vapor interface.
Pressure released bulk superfluid waves are not Doppler shifted by a persistent current in a superleak partially filling the channel.
In our calculations, we restrict ourselves to the case where the volume of the vapor is very small and therefore the evaporation-condensation at the free surface cannot take place. Taking the effects of an external force into consideration, we have three kinds of modes for thick films or two kinds of modes for thin films. At the end, we comment on the fifth sound.
The hydrodynamic boundary conditions at the normal liquid-superfluid interface in stratified3He-4He mixtures are formulated and applied to the propagation of low frequency two-phase sound. Coupling between the first- and
second-sound modes occurs at low temperature because of the large increase of the Fermi-liquid transport coefficients. Starting
with a normal homogeneous mixture, a coupling in a narrow range of temperature near the phase separation point is also predicted.
The hydrodynamic boundary conditions at the liquid-vapor interface of 3He-4He mixtures are formulated and applied to the propagation of low-frequency two-phase sound. It is found necessary to include the diffusive current of the 3He which results from the evaporation-condensation process at the free surface. The theory is then able to describe the crossover behavior of the two-phase sound from pure 4He to 3He-4He mixtures, in good agreement with experiment.
The existence of a new thermal wave in He II has been experimentally confirmed in a channel formed between two plane-parallel glass plates. The thermal wave is the limiting case of second sound in narrow channels under the condition that the channel width is small compared to the penetration depth of a viscous wave. It is a strongly attenuated temperature or entropy wave, which is coupled to pressure oscillations, and it shows dispersion. The phase velocity of this wave mode was determined in dependence on the frequency, the channel width, and the temperature. At temperatures between 1.1 K and the λ point, the results are in very good agreement with a thermohydrodynamic theory. Below 1.1 K the phase velocity is affected by mean-free-path effects of the elementary excitations, and deviations from this theory occur. They can be described by a simple kinetic theory.
The properties of sound waves propagating in a superfluid two-phase system (liquid-helium mixture and its vapor) are investigated as the temperature is lowered for two 3He mass concentrations c=1% and c=0.1% in the liquid. We see a crossover behavior from two independent sounds when T=1.2 K to one coupled mode when T=0.7 K. The crossover temperature depends on c and is higher for weak solutions. This is found to be related to the existence of exchanges which are initiated at the interface by the sound propagation and to the increase of an effective diffusion length LDeff in the liquid: LDeff is the range of the diffusion in the superfluid liquid of 3He atoms and energy which have been exchanged at the interface. The main interest of 3-4He superfluid mixtures is that LDeff may become much larger than the usual diffusion lengths below 1.2 K, due to the rapid increase of the effective 3He mean free path in weak solutions.
Sound propagation has been studied in He II films adsorbed on alumina powder grains (Al2O3). Sound velocity and adsorption isotherm data provide evidence that surf ace tension forces can exceed the van der Waals forces as the film thickness increases. A model of capillary condensation at the points where the powder grains touch accounts for many of the qualitative features of the data. The surface tension sound velocity decreases with increasing powder grain size.
The sound modes of He II in a waveguide partially packed with superleak are investigated for the case of a free surface within the waveguide. In the limit of zero vapor density, two propagating modes are found: one a gravity wave whose velocity depends on the superfluid fraction, and the other a temperature wave which is analogous to adiabatic U-tube osculations. With finite vapor the U-tube mode in pure 4He mixes strongly with the vapor sound; however, in 3He-4He mixtures it is uncoupled from the vapor and is observable. Experimental results are in qualitative agreement with the theory.
On the basis of a set of boundary conditions describing quite generally mass and energy transport processes across the free surface of helium II, the acoustic coefficients of reflection, transmission, and transformation of first sound, second sound, and the sound wave propagating in the vapor are calculated in the case of perpendicular incidence of sound waves against the liquid-vapor phase boundary. Considering rigorously the influences of the Onsager surface coefficients, the isobaric thermal expansion coefficients, and the thermal conductivities of the liquid and the vapor, we derive sets of equations from which the acoustic coefficients are determined numerically. For estimations, simple explicit formulas of the acoustic coefficients are given. It is shown that the evaporation and energy transport processes occurring at the free surface of helium II due to the incidence of sound waves may be connected with appreciable energy dissipation. The surface absorption coefficients of first, second, and gas sound waves are deduced.
The acoustic coefficients of reflection, transmission, and transformation of first-, second-, and gas sound waves incident normally on the free surface of liquid helium have been determined. The experimental results are compared with the theory developed in Part I of this paper. It is shown that equilibrium theories of boundary conditions disagree with the experiments. Nonequilibrium theories, however, which take into account that mass and energy transport processes occur at the liquid-vapor interface of helium II, lead to a very good description of the experimental data below 2 K. Within experimental accuracy the measurements fully confirm several kinetic models of evaporation and energy transport processes. Near the point, deviations from the theory have been found which might be attributed to influences of critical phenomena. The surface absorption coefficients of first-, second-, and gas sound waves could be deduced from the experimental data on the acoustic coefficients, and it is shown that, in agreement with the theory, appreciable energy dissipation occurs at the free surface of helium II when a second-sound or a gas sound wave is incident.
A Kosterlitz-Thouless-like approach is used to describe the superfluid transition of thin 4He films adsorbed on multiply connected surfaces such as packed powders or other porous materials. The elementary topological defect mediating the transition is a string linking two vortices of opposite circulation. The long-range interaction between strings gives rise to a phase transition which has a critical exponent for the superfluid density of nu=0.64.
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