Editing Superfluid helium-4 (section)

== Macroscopic theory ==

=== Thermodynamics ===
[[File:Phase diagram of 4He 01.svg|thumb|Fig. 1. Phase diagram of <sup>4</sup>He. In this diagram is also given the λ-line.]]
[[File:He liquid tbotevadobis damokidebuleba TemperaTurastan 350-en.svg|thumb|Fig. 2. Heat capacity of liquid <sup>4</sup>He at saturated vapor pressure as function of the temperature. The peak at T=2.17 K marks a (second-order) phase transition.]]
[[File:Normal and superfluid density 01.jpg|thumb|Fig. 3. Temperature dependence of the relative superfluid and normal components ρ<sub>n</sub>/ρ and ρ<sub>s</sub>/ρ as functions of ''T''.]]

Figure 1 is the [[phase diagram]] of <sup>4</sup>He.<ref>{{cite journal|last1=Swenson|first1=C.|year=1950|title=The Liquid-Solid Transformation in Helium near Absolute Zero|journal=Physical Review|volume=79|issue=4|page=626|bibcode=1950PhRv...79..626S|doi=10.1103/PhysRev.79.626}}</ref> It is a pressure-temperature (p-T) diagram indicating the solid and liquid regions separated by the melting curve (between the liquid and solid state) and the liquid and gas region, separated by the vapor-pressure line. This latter ends in the [[critical point (thermodynamics)|critical point]] where the difference between gas and liquid disappears. The diagram shows the remarkable property that <sup>4</sup>He is liquid even at [[absolute zero]]. <sup>4</sup>He is only solid at pressures above 25 [[bar (unit)|bar]].

Figure 1 also shows the λ-line. This is the line that separates two fluid regions in the phase diagram indicated by He-I and He-II. In the He-I region the helium behaves like a normal fluid; in the He-II region the helium is superfluid.

The name lambda-line comes from the specific heat – temperature plot which has the shape of the Greek letter λ.<ref>{{cite journal|last1=Keesom|first1=W.H.|last2=Keesom|first2=A.P.|year=1935|title=New measurements on the specific heat of liquid helium|journal=Physica|volume=2|issue=1|page=557|bibcode=1935Phy.....2..557K|doi=10.1016/S0031-8914(35)90128-8}}</ref><ref>{{cite book|last1=Buckingham|first1=M.J.|title=The nature of the λ-transition in liquid helium|last2=Fairbank|first2=W.M.|date=1961|isbn=978-0-444-53309-8|series=Progress in Low Temperature Physics|volume=3|page=80|chapter=Chapter III The Nature of the λ-Transition in Liquid Helium|doi=10.1016/S0079-6417(08)60134-1}}</ref> See figure 2, which shows a peak at 2.172 K, the so-called λ-point of <sup>4</sup>He.

Below the lambda line the liquid can be described by the so-called two-fluid model. It behaves as if it consists of two components: a normal component, which behaves like a normal fluid, and a superfluid component with zero viscosity and zero entropy. The ratios of the respective densities ρ<sub>n</sub>/ρ and ρ<sub>s</sub>/ρ, with ρ<sub>n</sub> (ρ<sub>s</sub>) the density of the normal (superfluid) component, and ρ (the total density), depends on temperature and is represented in figure 3.<ref>E.L. Andronikashvili Zh. Éksp. Teor. Fiz, Vol.16 p.780 (1946), Vol.18 p. 424 (1948)</ref> By lowering the temperature, the fraction of the superfluid density increases from zero at ''T''<sub>λ</sub> to one at zero kelvins.  Below 1&nbsp;K the helium is almost completely superfluid.

It is possible to create density waves of the normal component (and hence of the superfluid component since ρ<sub>n</sub> + ρ<sub>s</sub> = constant) which are similar to ordinary sound waves. This effect is called [[second sound]]. Due to the temperature dependence of ρ<sub>n</sub> (figure 3) these waves in ρ<sub>n</sub> are also temperature waves.

[[File:helium-II-creep.svg|thumb|right|Fig. 4. Helium II will "creep" along surfaces in order to find its own level – after a short while, the levels in the two containers will equalize. The [[Rollin film]] also covers the interior of the larger container; if it were not sealed, the helium II would creep out and escape.]]

[[File:Liquid helium Rollin film.jpg|thumb|right|Fig. 5. The liquid helium is in the superfluid phase. As long as it remains superfluid, it creeps up the wall of the cup as a thin film. It comes down on the outside, forming a drop which will fall into the liquid below. Another drop will form – and so on – until the cup is empty.]]

=== Superfluid hydrodynamics  ===
The equation of motion for the superfluid component, in a somewhat simplified form,<ref>S. J. Putterman (1974), ''Superfluid Hydrodynamics'' (Amsterdam: North-Holland) {{ISBN|0-444-10681-2}}.</ref> is given by Newton's law

<math display="block">\vec F = M_4\frac{\mathrm{d}\vec v_s}{\mathrm{d}t}.</math>

The mass <math display="inline">M_4</math> is the molar mass of <sup>4</sup>He, and <math display="inline">\vec v_s</math> is the velocity of the superfluid component. The time derivative is the so-called hydrodynamic derivative, i.e. the rate of increase of the velocity when moving with the fluid. In the case of superfluid <sup>4</sup>He in the gravitational field the force is given by<ref>Landau, L. D. (1941), [http://e-heritage.ru/Book/10088665 "The theory of superfluidity of helium II"], ''Journal of Physics'', Vol. 5, Academy of Sciences of the USSR, p. 71.</ref><ref>Khalatnikov, I. M. (1965), ''An introduction to the theory of superfluidity'' (New York: W. A. Benjamin), {{ISBN|0-7382-0300-9}}.</ref>

<math display="block">\vec F = -\vec \nabla (\mu + M_4 gz).</math>

In this expression <math display="inline">\mu</math> is the molar chemical potential, <math display="inline">g</math> the gravitational acceleration, and <math display="inline">z</math> the vertical coordinate. Thus we get the equation which states that the thermodynamics of a certain constant will be amplified by the force of the natural gravitational acceleration

{{NumBlk2|:|<math display="block">M_4\frac{\mathrm{d}\vec v_s}{\mathrm{d}t}  = -\vec \nabla (\mu + M_4 gz).</math>|1}}

Eq.&nbsp;{{EquationNote|1|(1)}} only holds if <math display="inline">v_s</math> is below a certain critical value, which usually is determined by the diameter of the flow channel.<ref>{{cite journal |doi=10.1016/0031-9163(66)90958-9 |title=The dependence of the critical velocity of the superfluid on channel diameter and film thickness |year=1966 |last1=Van Alphen |first1=W. M. |last2=Van Haasteren |first2=G. J. |last3=De Bruyn Ouboter |first3=R. |last4=Taconis |first4=K. W. |journal=Physics Letters |volume=20 |issue=5 |page=474 |bibcode = 1966PhL....20..474V}}</ref><ref>{{cite book |doi=10.1016/S0079-6417(08)60052-9 |title=Thermodynamics and hydrodynamics of <sup>3</sup>He–<sup>4</sup>He mixtures |chapter=Chapter 3: Thermodynamics and Hydrodynamics of <sup>3</sup>He–<sup>4</sup>He Mixtures |series=Progress in Low Temperature Physics |date=1992 |last1=De Waele |first1=A. Th. A. M. |last2=Kuerten |first2=J. G. M. |isbn=978-0-444-89109-9 |volume=13 |page=167|chapter-url=https://research.utwente.nl/en/publications/thermodynamics-and-hydrodynamics-of-hehe-mixtures(17c67fdc-75ba-4c0f-9e13-c247b78ccf97).html}}</ref>

In classical mechanics the force is often the gradient of a potential energy. Eq.&nbsp;{{EquationNote|1|(1)}} shows that, in the case of the superfluid component, the force contains a term due to the gradient of the [[chemical potential]]. This is the origin of the remarkable properties of He-II such as the fountain effect.

[[File:Integration path in pT diagram 01.jpg|thumb|Fig. 6. Integration path for calculating <math display="inline">\mu</math> at arbitrary <math display="inline">p</math> and <math display="inline">T</math>.]]
[[File:Demo fountain pressure 01.jpg|thumb|Fig. 7. Demonstration of the fountain pressure. The two vessels are connected by a superleak through which only the superfluid component can pass.]]
[[File:Helium fountain 01.jpg|thumb|Fig. 8. Demonstration of the fountain effect. A capillary tube is "closed" at one end by a superleak and is placed into a bath of superfluid helium and then heated. The helium flows up through the tube and squirts like a fountain.]]

=== Fountain pressure ===
In order to rewrite Eq.{{EquationNote|1|(1)}} in more familiar form we use the general formula

{{NumBlk2|:|<math display="block">\mathrm{d} \mu = V_m\mathrm{d}p - S_m\mathrm{d}T.</math>|2}}

Here <math display="inline">S_m</math> is the molar entropy and <math display="inline">V_m</math> the molar volume. With Eq.{{EquationNote|2|(2)}} <math display="inline">\mu(p,T)</math> can be found by a line integration in the <math display="inline">p</math>–<math display="inline">T</math> plane. First we integrate from the origin <math display="inline">(0,0)</math> to <math display="inline">(p,0)</math>, so at <math display="inline">T=0</math>. Next we integrate from <math display="inline">(p,0)</math> to <math display="inline">(p,T)</math>, so with constant pressure (see figure 6). In the first integral <math display="inline">\mathrm{d}T=0</math> and in the second <math display="inline">\mathrm{d}p=0</math>. With Eq.{{EquationNote|2|(2)}} we obtain

{{NumBlk2|:|<math display="block">\mu (p,T)=\mu (0,0)+\int_{0}^{p} V_{m}(p^\prime,0)\mathrm{d}p^\prime
-\int_{0}^T S_{m}(p,T^\prime)\mathrm{d}T^\prime.</math>|3}}

We are interested only in cases where <math display="inline">p</math> is small so that <math display="inline">V_m</math> is practically constant. So

{{NumBlk2|:|<math display="block">\int_{0}^{p} V_{m}(p^\prime,0)\mathrm{d}p^\prime = V_{m0}p</math>|4}}

where <math display="inline">V_{m0}</math> is the molar volume of the liquid at <math display="inline">T=0</math> and <math display="inline">p=0</math>. The other term in Eq.{{EquationNote|3|(3)}} is also written as a product of <math display="inline">V_{m0}</math> and a quantity <math>p_f</math> which has the dimension of pressure

{{NumBlk2|:|<math display="block">\int_{0}^T S_{m}(p,T^\prime)\mathrm{d}T^\prime=V_{m0}p_{f}.</math>|5}}

The pressure <math display="inline">p_f</math> is called the fountain pressure. It can be calculated from the entropy of <sup>4</sup>He which, in turn, can be calculated from the heat capacity. For <math display="inline">T=T_\lambda</math> the fountain pressure is equal to 0.692 bar. With a density of liquid helium of 125&nbsp;kg/m<sup>3</sup> and {{mvar|g}} = 9.8&nbsp;m/s<sup>2</sup> this corresponds with a liquid-helium column of 56 meter height. So, in many experiments, the fountain pressure has a bigger effect on the motion of the superfluid helium than gravity.

With Eqs.{{EquationNote|4|(4)}} and {{EquationNote|5|(5)}}, Eq.{{EquationNote|3|(3)}} obtains the form

{{NumBlk2|:|<math display="block">\mu(p,T) = \mu_0 + V_{m0}(p-p_{f}).</math>|6}}

Substitution of Eq.{{EquationNote|6|(6)}} in {{EquationNote|1|(1)}} gives

{{NumBlk2|:|<math display="block">\rho_0 \frac{\mathrm{d} \vec v_s}{\mathrm{d}t} = - \vec\nabla (p + \rho_0gz-p_{f}).</math>|7}}

with <math display="inline">\rho_0 = M_4/V_{m0}</math> the density of liquid <sup>4</sup>He at zero pressure and temperature.

Eq.{{EquationNote|7|(7)}} shows that the superfluid component is accelerated by gradients in the pressure and in the gravitational field, as usual, but also by a gradient in the fountain pressure.

So far Eq.{{EquationNote|5|(5)}} has only mathematical meaning, but in special experimental arrangements <math display="inline">p_f</math> can show up as a real pressure. Figure 7 shows two vessels both containing He-II. The left vessel is supposed to be at zero kelvins (<math display="inline">T_l=0</math>) and zero pressure (<math display="inline">p_l = 0</math>). The vessels are connected by a so-called superleak. This is a tube, filled with a very fine powder, so the flow of the normal component is blocked. However, the superfluid component can flow through this superleak without any problem (below a critical velocity of about 20&nbsp;cm/s). In the steady state <math display="inline">v_s=0</math> so Eq.{{EquationNote|7|(7)}} implies

{{NumBlk2|:|<math>p_{l}+\rho_0gz_{l}-p_{fl}=p_{r}+ \rho_0gz_{r}-p_{fr}</math>|8}}

where the indexes <math display="inline">l</math> and <math display="inline">r</math> apply to the left and right side of the superleak respectively. In this particular case <math display="inline">p_l = 0</math>, <math display="inline">z_l = z_r</math>, and <math display="inline">p_{fl} = 0</math> (since <math display="inline">T_l = 0</math>). Consequently,

<math display="block">0=p_{r}-p_{fr}.</math>

This means that the pressure in the right vessel is equal to the fountain pressure at <math display="inline">T_r</math>.

In an experiment, arranged as in figure 8, a fountain can be created. The fountain effect is used to drive the circulation of <sup>3</sup>He in dilution refrigerators.<ref>{{cite journal|doi=10.1016/0375-9601(75)90087-0|title=A dilution refrigerator with superfluid injection|year=1975|last1=Staas|first1=F. A.|last2=Severijns|first2=A. P.|last3=Van Der Waerden|first3=H. C.bM.|journal=Physics Letters A|volume=53|issue=4|page=327|bibcode = 1975PhLA...53..327S }}</ref><ref>{{cite journal|title=<sup>3</sup>He flow in dilute <sup>3</sup>He-<sup>4</sup>He mixtures at temperatures between 10 and 150 mK|doi=10.1103/PhysRevB.32.2870|pmid=9937394|year=1985|last1=Castelijns|first1=C.|last2=Kuerten|first2=J.|last3=De Waele|first3=A.|last4=Gijsman|first4=H.|journal=Physical Review B|volume=32|issue=5|pages=2870–2886|bibcode = 1985PhRvB..32.2870C |url=https://research.tue.nl/nl/publications/3he-flow-in-dilute-3he4he-mixtures-at-temperatures-between-10-and-150-mk(d7aefa27-5cff-4379-9b28-e0623ec7de38).html}}</ref>
[[File:Counterflow heat exchange 01.jpg|thumb|Fig. 9. Transport of heat by a counterflow of the normal and superfluid components of He-II]]

=== Heat transport ===
Figure 9 depicts a heat-conduction experiment between two temperatures <math display="inline">T_H</math> and <math display="inline">T_L</math> connected by a tube filled with He-II. When heat is applied to the hot end a pressure builds up at the hot end according to Eq.{{EquationNote|7|(7)}}. This pressure drives the normal component from the hot end to the cold end according to

{{NumBlk2|:|<math display="block">\Delta p = -\eta_nZ\dot V_n.</math>|9}}

Here <math display="inline">\eta_n</math> is the viscosity of the normal component,<ref>Zeegers, J. C. H. ''Critical velocities and mutual friction in <sup>3</sup>He-<sup>4</sup>He mixtures at low temperatures below 100 mK'', thesis, Appendix A, Eindhoven University of Technology, 1991.</ref> <math display="inline">Z</math> some geometrical factor, and <math display="inline">\dot V_n</math> the volume flow. The normal flow is balanced by a flow of the superfluid component from the cold to the hot end. At the end sections a normal to superfluid conversion takes place and vice versa. So heat is transported, not by heat conduction, but by convection. This kind of heat transport is very effective, so the thermal conductivity of He-II is very much better than the best materials. The situation is comparable with [[heat pipe]]s where heat is transported via gas–liquid conversion. The high thermal conductivity of He-II is applied for stabilizing superconducting magnets such as in the [[Large Hadron Collider]] at [[CERN]].