On the Fountain Effect in Superfluid Helium

One of the earliest and perhaps still the most spectacular manifestation of superfluidity is the fountain effect. In this helium vigorously spurts from the open end of a heated tube that is connected by a capillary or microporous frit to a chamber of liquid helium maintained below the condensation temperature.Allen38 ; Balibar17 Closing the heated chamber and measuring the fountain pressure is a common experimental technique for obtaining the entropy.Donnelly98

The commonly accepted physical explanation for the fountain effect as an osmotic pressure is due to Tisza.Balibar17 ; Tisza38 It is based on his two-fluid model, which in turn is based on F. London’s proposal that the $\lambda$-transition in liquid helium is due to Bose-Einstein condensation.FLondon38 The two-fluid model says that helium consists of a mixture of He I and He II, which are excited state and ground state bosons, respectively. The latter comprise the superfluid and their fraction increases with decreasing temperature below the condensation temperature. This gives rise to the notion that osmosis drives He II selectively through the capillary from the low temperature, high concentration chamber to the high temperature, low concentration one. In general in a binary solution the mixing entropy favors concentration equality, and this is what is said to create the osmotic pressure.

H. LondonHLondon39 carried out a thermodynamic analysis of the fountain effect that made a quantitative prediction for the pressure difference for a given temperature difference. His result is historically important as the quantitative experimental verification of this formula was evidence for the picture of superfluid helium being in a state of zero entropy. To the present day his expression remains significant as fountain pressure measurements are used to establish benchmark results against which calorimetric methods for the entropy of helium may be tested.Donnelly98 The results obtained from his expression are also used as calibration standards for instruments.Hammel61

Although this paper does not go beyond the expression of H. London,HLondon39 it does shed significant new light on the physical origin of the fountain effect and on the nature of superfluidity. The major result is to show that the H. London expression is obtained by minimizing the energy of the system with respect to particle transfer between the two chambers, Sec. II.1. Conversely, it shows that the result is inconsistent with maximizing the entropy of the system. This apparent violation of the Second Law of Thermodynamics is peculiar to superfluid flow.

The new derivation and interpretation of the H. London expression given here has a number of consequences for superfluid flow. One is that it rules out the osmotic pressure explanation of the fountain effect due to Tisza.Tisza38 The latter is statistical and based on the mixing entropy, which contradicts the present finding that it is the minimization of energy that underlies the fountain effect. To underscore this point, in Sec. III quantitative calculations are made of the osmotic prediction for the fountain pressure using both calculatedAttard21 ; Attard22a and measuredDonnelly98 data for the condensed boson fraction. It is shown that the predicted fountain pressures do not agree with measured values.Hammel61 Although these calculations are made at the level of ideal solution theory, the quantitative error is so large, and the qualitative difference with energy minimization so stark, that the osmotic explanation for the fountain effect appears nonviable.

This paper is the fourth in a series on superfluidity and superconductivity. Attard22a ; Attard22b ; Attard22c The general formulation of quantum statistical mechanics that underlies them is given in Ref. Attard21, . The author has also given coherent formulations of equilibriumTDSM and non-equilibriumNETDSM thermodynamics and classical statistical mechanics, which results and concepts are freely used below.

In Appendix B the conventional osmotic pressure for a binary solution is obtained using classical ideal solution theory. That result may be compared and contrasted with the present results for a superfluid obtained similarly at the level of an ideal solution.

Consider a system consisting of two chambers at different temperatures below the condensation temperature and containing ⁴He. We give the free energy for a single chamber, and differentiate it to obtain an expression for the chemical potential. This allows us to equate chemical potentials and to obtain the fountain pressure.

The constrained Helmholtz free energy for a single system with $N_{0}$ ground momentum state bosons and $N_{*}$ excited momentum state bosons is given by Attard22a ; Attard21

Here $N=N_{0}+N_{*}$, $\beta\equiv 1/k_{\mathrm{B}}T$, and $\Lambda\equiv\sqrt{2\pi\beta\hbar^{2}/m}$ is the thermal wavelength. The series contains the loop grand potentials, which is a quantum effect that arises from the symmetrization of the wave function. The logarithmic term is the classical or monomer term, and it contains the configuration integral for interacting bosons, $Q(N,V,T)$. Compared to classical statistical mechanics of a binary mixture (see Appendix B), for the condensed boson case there is no $N_{0}!$ in the denominator, and instead of $\Lambda^{3N_{0}}$ there appears $V^{N_{0}}$. These differences are a consequence of the non-locality of the permutations of ground momentum state bosons. Attard22a ; Attard21

We make the incompressible liquid approximation, $V=Nv$. The constrained Gibbs free energy, $G=F+pV$, for chamber $A$ is then given by

Recall that $N_{A}=N_{*A}+N_{0A}$.

Now for an incompressible fluid, since $\mu^{\mathrm{ex}}=p^{\mathrm{ex}}v$ (see Appendix B), the derivative of the configuration integral term vanishes,

This result effectively removes the interactions between the helium-4 atoms, and makes the analysis more or less the same as that for an ideal gas, apart from the way in which the fraction of ground and excited momentum state bosons is calculated or measured. Consequently, inserting the incompressible fluid assumption at this stage of the analysis is a rather serious approximation. It would be better to calculate the chemical potential for interacting atoms, which is quite feasible, and this should be explored in the future.

The derivative of the constrained Gibbs free energy gives the chemical potential, which with the vanishing of the derivative of the configurational integral, is

Here $f_{*A}=N_{*A}/N_{A}$ is the fraction of bosons in excited states in chamber $A$. One can similarly obtain an expression for the excited state chemical potential,

In the equilibrium state, $\mu_{0A}=\mu_{*A}$, in which case one must have

Since the right hand side is a series of positive terms, one must have that $f_{*A}>1/\rho_{A}\Lambda_{A}^{3}$, which places a lower bound on the fraction of excited state bosons (He I) as $T\rightarrow 0$. This result is predicated on the incompressible liquid, ideal solution approximation and also upon the no mixing approximation.

Similar expressions hold for chamber $B$. Hence equating the chemical potential of the two chambers, $\mu_{0A}=\mu_{0B}$, yields the fountain pressure as

Ignoring the loop terms, and taking into account the fact that the excited momentum state fraction increases with increasing temperature, one sees that the higher temperature chamber has the higher pressure. Again neglecting the loop terms, notice the difference between this and the usual osmotic pressure equation (B.4). It is only in the linear regime that they are the same.

The left hand side can be written as a height difference, $[p_{A}-p_{B}]\rho^{-1}\approx mg[h_{A}-h_{B}]$, where $m$ is the molecular mass. (Note that here $g$ is the acceleration due to gravity, not the specific Gibbs free energy, and $h$ is the height, not the specific enthalpy.)

Extensive calculation of the intensive loop Gaussians $g^{(l)}$ up to $l=7$ have been performed for a Lennard-Jones model of ⁴He,Attard21 ; Attard22a and also for ³He.Attard22c The fraction of excited state bosons following condensation has been calculated in the so-called unmixed approximation. Attard22a These have been used in the present expression for the pressure difference. The $\lambda$-transition in Lennard-Jones helium-4 occurs at $T=0.65$. Attard22a

Figure 2 shows the calculated height difference as a function of the low temperature $T_{A}$ for fixed high temperature $T_{B}=0.65$, which is the $\lambda$- temperature in Lennard-Jones ⁴He. It also shows the height difference for fixed low temperature $T_{A}=0.50$, It can be seen that the height difference increases with increasing temperature difference. The magnitude of the fountain effect is surprisingly large considering that the largest temperature difference shown is about 1.5 K. The loop series contribution to the total height difference is on the order of 1–5%. This means that the major part of the height differences in Fig. 2 come from the first line of the right hand side of Eq. (III), which are essentially ideal solution terms.

It is noticeable in the figure that for the same temperature difference, $T_{B}-T_{A}=0.05$, the higher temperature $T_{B}=0.65$ has a height difference $h_{B}-h_{A}=159.2$ m, which is about 50% larger than that at the lower temperature $T_{A}=0.50$, which has a height difference $h_{B}-h_{A}=104.7$ m. This shows that there is a significant decrease in the ratio of height difference to temperature difference with decreasing temperature. This is because the fraction of excited state bosons decreases with decreasing temperature, and to leading order the height difference is proportional to the difference in the excited fraction.

The experimentally measured value at $T_{A}=1.143$ K is 46.8 m for $T_{B}-T_{A}=0.6$ K.Hammel61 The calculated height difference in Fig. 2 is 104.7 m for $T_{B}-T_{A}=0.05\approx 0.51$ K, which is about a factor of 2 too large. The main reason for the disagreement between the calculated and the measured height difference appears to be the use of the incompressible liquid, ideal solution approximation for the calculations.

Figure 3 compares the incompressible fluid osmotic pressure result, the first line of Eq. (III), with the measured fountain pressure. This uses the measuredDonnelly98 rather than calculated excited state fraction. It neglects the loop contributions, which as discussed for the previous figure are expected to be small. From the fact that the measured curves are nearly parallel, one can conclude that the fountain pressure is approximately a function of the temperature difference, $T_{B}-T_{A}$, rather than of $T_{B}$ and $T_{A}$ individually. In any case it can be seen that the incompressible, ideal solution result for the osmotic pressure performs quite badly.

Figure 4 shows the dimensionless parameter $f_{*}\rho\Lambda^{3}$ using measured data for the fraction of He I. Donnelly98 As discussed following Eq. (3.6), equilibrium between ground and excited state bosons implies that this must be greater than unity, which the figure shows is violated for $T\leq 1.5$ K. The tests of the osmotic pressure prediction for the fountain pressure in Fig. 3 are for $T\geq 1.5$ K. Nevertheless that $f_{*}\rho\Lambda^{3}<1$ for $T\leq 1.5$ K is presumably an indication of the failure of the incompressible liquid, ideal solution, and no mixing approximations in the low temperature regime.

The osmotic pressure mechanism offered by TiszaTisza38 as an explanation for the fountain pressure is problematic on physical grounds. In the usual realization of osmotic pressure the solute and the solvent have distinct identities. Hence the only way to equalize the solvent chemical potential is to increase the pressure of the high concentration chamber. In the case of ⁴He, the ground state bosons can become excited state bosons, and vice versa. Hence if the different fractions in each chamber somehow corresponded to a chemical potential difference due to mixing entropy, they could equalize chemical potential simply by changing their state without changing the pressure.

The present calculations show that the osmotic pressure mechanism can be an order of magnitude or more too large for the fountain pressure. Admittedly this is using the incompressible fluid, ideal solution approximation. But since the common understanding of osmotic pressure is that it is due to mixing entropy, and since the latter can be calculated exactly by ideal combinatorics, the failure of the present ideal solution calculations for the fountain pressure argue against osmotic pressure as the physical basis of the fountain effect.

It was found here that H. London’sHLondon39 expression for the temperature derivative of the fountain pressure corresponds to chemical potential equality of the two chambers. This follows from the minimization of energy at constant entropy, which suggests a general principle for superfluid flow and offers an alternative physical explanation for the fountain effect.

Imagine that the high temperature chamber $B$ is initially at saturation. For ⁴He, on the saturation curve the chemical potential from the measuredDonnelly98 enthalpy and entropy decreases with increasing temperature, $\mathrm{d}\mu^{\mathrm{sat}}/\mathrm{d}T<0$. Hence $\mu_{B}^{\mathrm{sat}}<\mu_{A}^{\mathrm{sat}}$. Since $\partial\mu/\partial p=v>0$, the chemical potential of the high temperature chamber can be increased to achieve equality by increasing the pressure beyond its saturation value. This occurs as more bosons arrive in the second chamber because each boson occupies a certain impenetrable volume (i.e. the compressibility is positive). These are the reasons why condensed bosons initially flow down the chemical potential gradient from the low temperature chamber to the high temperature chamber, and why the high temperature chamber subsequently settles at a higher pressure.

One could describe the new principle —that the energy at constant entropy is minimal with respect to particle transfer— as an empirical formula, since the final formula agrees quantitatively with measured results but the axiomatic basis for energy minimization is not well-established. Undoubtedly one could argue that condensed bosons in the superfluid state are in a state of low or zero entropy, and so it would make sense that they respond to mechanical forces rather than to statistical or entropic forces. Also the fact that superfluid flow is flow without viscous dissipation means that the motion of condensed bosons does not produce entropy and so it cannot respond to entropy gradients. And of course condensed bosons in a single quantum state behave as a single particle,Attard21 ; Attard22a which particle responds as a whole to mechanical forces without changing the entropy of the system. Perhaps time and familiarity will raise the status of the present result from empirical to axiomatic.

It is unlikely that the principle of energy minimization at constant entropy is restricted to the fountain effect. Presumably it is a fundamental principle that applies generally for motion without dissipation in Bose-Einstein condensates. As such it provides new insight for the understanding and the analysis of superfluid flow and of superconductor currents.