The superfluidity of helium-4 is explained until today by a quantum theory: the Bose-Einstein condensation. This theory is rather satisfactory in describ ing the superfluid state of helium-4 because this one is a system made up of bosons (particles of integer spin). However, the discovery of the superfluidity of helium-3 in 1971 called into question the veracity of this quantum theory. In fact, helium-3 being a system composed of fermions (particles of half-integer spin), it cannot be subject to Bose-Einstein condensation. It is to correct this deficiency that we introduce here a classical (non-quantum) theory of superfluids. This new theory makes no difference between the λ transition of bosons and that of fermions. It is based on a fundamental law: “in a superfluid, density is conserved”. In this work, we have shown that this simple law explains not only the zero viscosity of superfluids but also the surprising phenomena observed in the superfluid state, I quote the liquidity of helium at normal pressure down to 0 K, vaporization without boiling, high thermal conductivity, the fountain effect, the ability to go up one side of the wall of a container to come down on the other side and the existence of a critical velocity.
When the temperature of helium-4 ( H 2 4 ) liquid is lowered until it exceeds a certain value Tλcalled the lambda (λ) temperature, under normal atmospheric pressure, a state transition occurs: the liquid passes from the normal state to the superfluid state. This transition is usually called the lambda (λ) transition. As the word “superfluid” indicates, below Tλ, the fluid has zero viscosity. Unable to find an adequate explanation for this phenomenon, physicists directly concluded that it must necessarily have a quantum origin. They then associated the λ transition with the Bose-Einstein condensation [
We recall that according to Clapeyron’s relation:
( d P d T ) L − S = S l − S s v l − v s (1)
where:
S l is the entropy in the liquid state;
S s is the solid state entropy;
v l is the volume in the liquid state;
v s is the solid state volume.
The horizontality of the liquid-solid equilibrium curve at low temperature is given by:
( d P d T ) L − S = 0
⇒ S l ≅ S s (2)
This means that the entropy of the liquid is almost equal to the entropy of the solid at low temperature. This is quite unusual since a solid is generally more spatially ordered than a liquid and therefore has lower entropy. Here since S l ≅ S s , the liquid is as ordered as the solid in the superfluid phase. This implies an alignment of the magnetic spin moments of the atoms in the superfluid as in a ferromagnetic solid. But for this alignment of spin moments to appear, the spins of the atoms of the system must first be non-zero. Superfluid helium-3 atoms exhibit magnetic properties because they have non-zero spins, not because they are fermions. In other words, a non-zero spin-boson system will also exhibit magnetic properties in the superfluid state. It is therefore clear that the Bose-Einstein condensation is no longer sufficient to completely describe the superfluid state at low temperature.
We then introduce here a new classical (non-quantum) theory that makes no difference between the λ transition of bosons and that of fermions. Moreover, it explains not only the zero viscosity of superfluids but also the unusual physical phenomena observed in the superfluid state, I quote [
Who says viscosity, says friction; and who says friction, says collisions between particles. Viscosity is therefore a physical quantity that characterizes the collisions between particles in a fluid. And besides, we can even see it through the unit of measurement used. Indeed, in the international system of units, the kinematic viscosity is measured in m2∙s−1. This simply means that the viscosity is proportional to the cross-section and the frequency of collision of the fluid particles. Zero viscosity therefore means “absence of any collision”. So we have to show here that below Tλ, no more collision is possible in the superfluid. To demonstrate this, we consider a superfluid electron gas, for example in a superconductor. Indeed, it is the superfluidity of the electron gas in the superconducting metal below the critical temperature that is responsible for the infinite conductivity and the Meissner effect of the material [
If E c > E p , then there will be a collision.
If E c < E p , then there will never be a collision.
In the superfluid state, the velocity V e of the electron is so small that we always have E c < E p . Hence the zero viscosity of superfluids. The superfluid being as ordered as a solid, each electron is surrounded by four other electrons: one electron on the left, another on the right, another above and another below. Thus, in the superfluid, the electron is as frozen in a fixed point because it is repelled from all four sides. The distance between two electrons in the superfluid therefore never varies. It is this distance that we have named λ distance. From this, then follows a law that completely defines the superfluid state: in a superfluid, density is conserved. We also call this constant density in the superfluid state the lambda (λ) density. The atom is made up of a cloud of negatively charged electrons that surrounds a positive nucleus, the same reasoning that we have just applied to an electron gas can also be applied to a system composed of neutral atoms. Let us now show that this simple law of constant density in the superfluid state explains all the extraordinary phenomena observed in this state.
1) Critical velocity
For the viscosity to be zero during a flow of the superfluid on a fixed solid plate (i.e., no collision between the atoms of the superfluid and the atoms of the plate), it is necessary that the energy of the Coulomb repulsion E p between the atoms of the superfluid and those of the plate is greater than the kinetic energy of the atoms of the superfluid. In this case, the superfluid moves on the solid plate without touching it by levitating above it. This then imposes the existence of a limit flow velocity called the critical velocity above which E p < E c and the flow becomes normal (non-zero viscosity). When the flow velocity is equal to the critical velocity, we therefore have: E p = E c .
This critical velocity is highlighted when trying to rotate a superfluid. Indeed, if a box containing a superfluid is slowly rotated, the walls of the box rotate but the superfluid remains motionless in a state of rest [
2) Liquidity of helium down to 0 K at normal pressure
Helium is a permanent liquid. This is a unique property of helium; it is the only compound that remains liquid down to absolute zero under normal pressure [
3) Vaporization without boiling
In a liquid in the normal state, boiling occurs because heat travels through it by convection. The fluid near the heat source heats up and its density decreases (increase in volume). Then, following the principle of Archimedes, the less dense fluid rises to make room for the denser fluid. It is this density gradient in the liquid that leads to the formation of bubbles. However, in a superfluid, the λ density is constant and therefore uniform. Hence there is no possible convection in the superfluid and therefore no formation of bubbles: the liquid vaporizes without boiling (evaporation).
4) High thermal conductivity
There are only three means of heat propagation in a system: conduction, convection and radiation. We have just seen that in a superfluid, there cannot be convection. Radiation is also to be excluded because there is no production of any photon in the superfluid. There is, therefore, only one possible means of heat propagation in a superfluid: conduction. However, experiments have shown that the heat conduction in superfluid helium is 1000 times greater than that of copper. Indeed, the conductivity in a superfluid differs completely from the conductivity in a metal. In a solid material, heat is conducted through free electrons (conduction electrons). These electrons transmit kinetic energy from one end of the metal to the other by collision. However, in a superfluid, there can be no collision between particles (zero viscosity). The heat then propagates from
one end of the superfluid to the other in the form of an elastic wave at the speed of sound. In fact, the absence of collision in the superfluid allows us to consider it, in first approximation, as an ideal fluid. The ideal fluid equation is given by:
P V = n R T (3)
where:
P is the pressure;
V is the volume;
T is the temperature;
n is the number of moles;
R is the ideal fluid constant.
As the mass of the superfluid is conserved ( n = C t e ), the volume is also conserved because the λ density of the superfluid is constant.
We then arrive at the equation:
P = C T (with C, a constant)
⇒ Δ P = C Δ T (4)
Which means any variation or gradient in temperature implies a variation or gradient in pressure. It is this pressure gradient that propagates like a wave in the superfluid and which is responsible for the high thermal conduction of superfluids.
5) Fountain effect
When a tube containing superfluid helium is heated, there is a continuous emission of a jet of helium which can reach several tens of centimeters in height at the exit of the tube. This is the fountain effect. The emission ceases when the tube is no longer heated. The height of the jet is variable and increases with the temperature gradient imposed on the system (see
Indeed, Equation (4) tells us that any temperature gradient implies a pressure gradient in the superfluid. If the temperature gradient is high, then the pressure gradient will also be high and can then produce a force of pressure greater than the force of gravity. Hence the fountain effect. As the pressure force increases with the pressure gradient, the jet height will increase with the temperature gradient. As soon as the temperature gradient becomes zero, the pressure gradient also becomes zero and the fountain effect disappears.
6) Ability to go up one side of the wall of a container to go down the other side
When a tube containing superfluid helium is immersed in another container containing superfluid helium, we notice that a thin film located near the wall of the tube rises on one side and descends on the other side. The ascent ceases as soon as equilibrium is regained where the levels inside and outside the tube are identical (See
In fact, when the tube is immersed in the container, a pressure gradient (a pressure difference) is created between the two sides of the tube wall. According to Equation (4), this pressure gradient implies a temperature gradient. The latter then leads to a small-scale fountain effect near the wall of the tube. Hence the
rise of the film is located near the wall. As soon as equilibrium is found, the pressure gradient becomes zero and therefore, the temperature gradient also becomes zero. The fountain effect near the wall disappears and the rise of the film ceases.
In this work, we have introduced a classical theory that better describes the superfluid state than the quantum theory based on Bose-Einstein condensation. This classical theory is based on a fundamental law stated as follows: in a superfluid, density is conserved. We have shown that this law explains not only the zero viscosity of superfluids but also the surprising phenomena that occur in the superfluid state, I quote:
The critical speed V λ corresponds to a fluid flow velocity V for which the kinetic energy E c of the atoms of the superfluid is equal to the Coulomb repulsion energy E p between the atoms of the superfluid and the atoms of the fixed plate. If V < V λ , then E c < E p and the viscosity is zero. If V > V λ , then E c > E p and the viscosity is no longer zero.
The liquidity of helium down to 0 K under normal pressure is due to the fact that superfluid helium retains its density even at 0 K.
Vaporization without boiling is explained by the uniformity and constancy of the density of the superfluid. No convection is therefore possible in the superfluid state.
The high thermal conductivity is due to the fact that in the superfluid the heat propagates as an elastic wave at the speed of sound.
The fountain effect is observed when the superfluid is heated because any temperature gradient implies a pressure gradient in the superfluid state.
The ability to move up one side of the wall of a vessel and down the other side is nothing more than a small-scale fountain effect near the wall. Indeed, any pressure gradient implies a temperature gradient in the superfluid state.
The author declares no conflicts of interest regarding the publication of this paper.
Sorongane, E.W. (2022) Implementation of a Classical Theory for Superfluids. Open Journal of Applied Sciences, 12, 1254-1261. https://doi.org/10.4236/ojapps.2022.127085