Superfluid is a state of matter in which the matter behaves like a fluid with zero viscosity and zero entropy. The substance, which looks like a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and through pores in containers which hold it, subject only to its own inertia.

Known as a major facet in the study of quantum hydrodynamics and macroscopic quantum phenomena, the superfluidity effect was discovered by Pyotr Kapitsa^[1] and John F. Allen, and Don Misener^[2] in 1937. It has since been described through phenomenological and microscopic theories. The formation of the superfluid is known to be related to the formation of a Bose-Einstein condensate. This is made obvious by the fact that superfluidity occurs in liquid helium-4 at far higher temperatures than it does in helium-3. Each molecule of helium-4 is a boson particle, by virtue of its zero spin. Helium-3, however, is a fermion particle, which can form bosons only by pairing with itself at much lower temperatures, in a process similar to the electron pairing in superconductivity.

In the 1950s, Hall and Vinen performed experiments establishing the existence of quantized vortex lines in superfluid helium.^[3] In the 1960s, Rayfield and Reif established the existence of quantized vortex rings.^[4] Packard has observed the intersection of vortex lines with the free surface of the fluid,^[5] and Avenel and Varoquaux have studied the Josephson effect in superfluid helium-4.^[6] In 2006 a group of the University of Maryland has visualized quantized vortices in a most elegant way by using small tracer particles of solid hydrogen.^[7]

Properties[link]

Figure 1 is the phase diagram of ⁴He.^[8] It is a 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 where the difference between gas and liquid disappears. The diagram shows the remarkable property that ⁴He is liquid even at absolute zero. Helium four is only solid at pressures above 25 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.^[9][10] See figure 2, which shows a peak at 2.172 K, the so-called λ-point of ⁴He.

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_n/ρ and ρ_s/ρ, with ρ_n (ρ_s) the density of the normal (superfluid) component, and ρ the total density), depends on temperature and is represented in figure 3.^[11] By lowering the temperature, the fraction of the superfluid density increases from zero at T_λ to one at zero kelvin. Below 1 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_n + ρ_s = constant) which are similar to ordinary sound waves. This effect is called second sound. Due to the temperature dependence of ρ_n (figure 3) these waves in ρ_n are also temperature waves.

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.

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.

Film flow[link]

Many ordinary liquids, like alcohol or petroleum, creep up solid walls, driven by their surface tension. Liquid helium also has this property, but, in the case of He-II, the flow of the liquid in the layer is not restricted by its viscosity but by a critical velocity which is about 20 cm/s. This is a fairly high velocity so superfluid helium can flow relatively easy up the wall of containers, over the top, and down to the same level as the surface of the liquid inside the container, in a siphon effect as illustrated in figure 4. In a container, lifted above the liquid level, it forms visible droplets as seen in figure 5 (see Rollin film).

Superfluid hydrodynamics[link]

The equation of motion for the superfluid component, in a somewhat simplified form,^[12] is given by Newton's law

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec F = M_4\frac{\mathrm{d}\vec v_s}{\mathrm{d}t}.

The mass M₄ is the molar mass of ⁴He and Failed to parse (Missing texvc executable; please see math/README to configure.): \vec v_s

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 4He in the gravitational field the force is given by[13][14]

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec F =- \vec \nabla (\mu + M_4gz).

.

In this expression μ is the molar chemical potential, g the gravitational acceleration, and z the vertical coordinate. Thus we get

Failed to parse (Missing texvc executable; please see math/README to configure.): M_4\frac{\mathrm{d}\vec v_s}{\mathrm{d}t}=- \vec \nabla (\mu + M_4gz).

(1)

Eq.(1) only holds if v_s is below a certain critical value which usually is determined by the diameter of the flow channel.^[15][16]

In classical mechanics the force is often the gradient of a potential energy. Eq.(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.

Fountain pressure[link]

In order to rewrite Eq.(1) in more familiar form we use the general formula

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \mu = V_m\mathrm{d}p - S_m\mathrm{d}T.

(2)

Here S_m is the molar entropy and V_m the molar volume. With Eq.(2) μ(p,T) can be found by a line integration in the p-T plane. First we integrate from the origin (0,0) to (p,0), so at T =0. Next we integrate from (p,0) to (p,T), so with constant pressure (see figure 6). In the first integral dT=0 and in the second dp=0. With Eq.(2) we obtain

We are interested only in cases where p is small so that V_m is practically constant. So

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{0}^{p} V_{m}(p^\prime,0)\mathrm{d}p^\prime = V_{m0}p

(4)

where V_m0 is the molar volume of the liquid at T =0 and p =0. The other term in Eq.(3) is also written as a product of V_m0 and a quantity p_f which has the dimension of pressure

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{0}^T S_{m}(p,T^\prime)\mathrm{d}T^\prime=V_{m0}p_{f}.

(5)

The pressure p_f is called the fountain pressure. It can be calculated from the entropy of ⁴He which, in turn, can be calculated from the heat capacity. For T =T_λ the fountain pressure is equal to 0.692 bar. With a density of liquid helium of 125 kg/m³ and g = 9.8 m/s² 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.(4) and (5) Eq.(3) obtains the form

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(p,T) = \mu_0 + V_{m0}(p-p_{f}).

(6)

Substitution of Eq.(6) in (1) gives

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_0 \frac{\mathrm{d} \vec v_s}{\mathrm{d}t} = - \vec\nabla (p + \rho_0gz-p_{f}).

(7)

with ρ₀ = M₄/V_m0 the density of liquid ⁴He at zero pressure and temperature.

Eq.(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.(5) has only mathematical meaning, but in special experimental arrangements p_f 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 kelvin (T_l=0) and zero pressure (p_l = 0). 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 cm/s). In the steady state v_s=0 so Eq.(7) implies

Failed to parse (Missing texvc executable; please see math/README to configure.): p_{l}+\rho_0gz_{l}-p_{fl}=p_{r}+ \rho_0gz_{r}-p_{fr}

(8)

where the index l (r) applies to the left (right) side of the superleak. In this particular case p_l = 0, z_l = z_r, and p_fl = 0 (since T_l = 0). Consequently

Failed to parse (Missing texvc executable; please see math/README to configure.): 0=p_{r}-p_{fr}.

This means that the pressure in the right vessel is equal to the fountain pressure at T_r.

In an experiment, arranged as in figure 8, a beautiful fountain can be created. The fountain effect is used to drive the circulation of ³He in dilution refrigerators.^[17][18]

Heat transport[link]

Figure 9 depicts a heat-conduction experiment between two temperatures T_H and T_L 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.(7). This pressure drives the normal component from the hot end to the cold end according to

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta p = -\eta_nZ\dot V_n.

(9)

Here η_n is the viscosity of the normal component,^[19] Z some geometrical factor, and Failed to parse (Missing texvc executable; please see math/README to configure.): \dot V_n

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 pipes 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.

Theories[link]

L. D. Landau's phenomenological and semi-microscopic theory of superfluidity of helium-4 earned him the Nobel Prize in physics, in 1962. Assuming that sound waves are the most important excitations in helium-4 at low temperatures, he showed that helium-4 flowing past a wall would not spontaneously create excitations if the flow velocity was less than the sound velocity. In this model, the sound velocity is the "critical velocity" above which superfluidity is destroyed. (Helium-4 actually has a lower flow velocity than the sound velocity, but this model is useful to illustrate the concept.) Landau also showed that the sound wave and other excitations could equilibrate with one another and flow separately from the rest of the helium-4, which is known as the "condensate".

From the momentum and flow velocity of the excitations he could then define a "normal fluid" density, which is zero at zero temperature and increases with temperature. At the so-called Lambda temperature, where the normal fluid density equals the total density, the helium-4 is no longer superfluid.

To explain the early specific heat data on superfluid helium-4, Landau posited the existence of a type of excitation he called a "roton", but as better data became available he considered that the "roton" was the same as a high momentum version of sound.

Bijl in the 1940s,^[20] and Feynman around 1955, ^[21] developed microscopic theories for the roton, which was shortly observed with inelastic neutron experiments by Palevsky.

Landau thought that vorticity entered superfluid helium-4 by vortex sheets, but such sheets have since been shown to be unstable.

Lars Onsager and, later independently, Richard Feynman showed that vorticity enters by quantized vortex lines. They also developed the idea of quantum vortex rings.

Background[link]

Although the phenomenology’s of the superfluid states of helium-4 and helium-3 are very similar, the microscopic details of the transitions are very different. Helium-4 atoms are bosons, and their superfluidity can be understood in terms of the Bose–Einstein statistics that they obey. Specifically, the superfluidity of helium-4 can be regarded as a consequence of Bose-Einstein condensation in an interacting system. On the other hand, helium-3 atoms are fermions, and the superfluid transition in this system is described by a generalization of the BCS theory of superconductivity. In it, Cooper pairing takes place between atoms rather than electrons, and the attractive interaction between them is mediated by spin fluctuations rather than phonons. (See fermion condensate.) A unified description of superconductivity and superfluidity is possible in terms of gauge symmetry breaking.

Superfluids, such as helium-4 below the lambda point, exhibit many unusual properties. (See Helium#Helium II state). A superfluid acts as if it were a mixture of a normal component, with all the properties of a normal fluid, and a superfluid component. The superfluid component has zero viscosity and zero entropy. Application of heat to a spot in superfluid helium results in a flow of the normal component which takes care of the heat transport at relatively high velocity (up to 20 cm/s) which leads to a very high effective thermal conductivity.

Another fundamental property becomes visible if a superfluid is placed in a rotating container. Instead of rotating uniformly with the container, the rotating state consists of quantized vortices. That is, when the container is rotated at speeds below the first critical angular velocity, the liquid remains perfectly stationary. Once the first critical angular velocity is reached, the superfluid will form a vortex. The vortex strength is quantized, that is, a superfluid can only spin at certain "allowed" values. Rotation in a normal fluid, like water, is not quantized. If the rotation speed is increased more and more quantized vortices will be formed which arrange in nice patterns similar to the Abrikosov lattice in a superconductor.

Practical application[link]

Recently in the field of chemistry, superfluid helium-4 has been successfully used in spectroscopic techniques as a quantum solvent. Referred to as Superfluid Helium Droplet Spectroscopy (SHeDS), it is of great interest in studies of gas molecules, as a single molecule solvated in a superfluid medium allows a molecule to have effective rotational freedom, allowing it to behave exactly as it would in the "gas" phase.

Superfluids are also used in high-precision devices such as gyroscopes, which allow the measurement of some theoretically predicted gravitational effects (for an example, see the Gravity Probe B article).

In 1999, one type of superfluid was used to trap light and greatly reduce its speed. In an experiment performed by Lene Hau, light was passed through a Bose-Einstein condensed gas of sodium (analogous to a superfluid) and found to be slowed to 17 m/s (61.2 km/h) from its normal speed of 299,792,458 metres per second in vacuum.^[22] This does not change the absolute value of c, nor is it completely new: any medium other than vacuum, such as water or glass, also slows down the propagation of light to c/n where n is the material's refractive index. The very slow speed of light and high refractive index observed in this particular experiment, moreover, is not a general property of all superfluids.

The Infrared Astronomical Satellite IRAS, launched in January 1983 to gather infrared data was cooled by 73 kilograms of superfluid helium, maintaining a temperature of 1.6 K (−271.4 °C). Furthermore, when used in conjunction with helium-3, temperatures as low as 40 mK are routinely achieved in extreme low temperature experiments. The helium-3, in liquid state at 3.2 K, can be evaporated into the superfluid helium-4, where it acts as a gas due to the latter's properties as a Bose-Einstein condensate. This evaporation pulls energy from the overall system, which can be pumped out in a way completely analogous to normal refrigeration techniques.

Superfluid-helium technology is used to extend the temperature range of cryocoolers to lower temperatures. So far the limit is 1.19 K, but there is a potential to reach 0.7 K.^[23]

21st-century developments[link]

In the early 2000s, physicists created a Fermionic condensate from pairs of ultra-cold fermionic atoms. Under certain conditions, fermion pairs form diatomic molecules and undergo Bose–Einstein condensation. At the other limit, the fermions (most notably superconducting electrons) form Cooper pairs which also exhibit superfluidity. This work with ultra-cold atomic gases has allowed scientists to study the region in between these two extremes, known as the BEC-BCS crossover.

Supersolids may also have been discovered in 2004 by physicists at Penn State University. When helium-4 is cooled below about 200 mK under high pressures, a fraction (~1%) of the solid appears to become superfluid.^[24][25] By quench cooling or lengthening the annealing time, thus increasing or decreasing the defect density respectively, it was shown, via torsional oscillator experiment, that the supersolid fraction could be made to range from 20% to completely non-existent. This suggested that the supersolid nature of helium-4 is not intrinsic to helium-4 but a property of helium-4 and disorder.^[26][27] Some emerging theories posit that the supersolid signal observed in helium-4 was actually an observation of either a superglass state^[28] or intrinsically superfluid grain boundaries in the helium-4 crystal.^[29]

See also[link]

References[link]

Further reading[link]

• London, F. Superfluids (Wiley, New York, 1950)
• D.R. Tilley and J. Tilley, ``Superfluidity and Superconductivity, (IOP Publishing Ltd., Bristol, 1990)
• Hagen Kleinert, Gauge Fields in Condensed Matter, Vol. I, "SUPERFLOW AND VORTEX LINES", pp. 1–742, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online)
• Antony M. Guénault: Basic superfluids. Taylor & Francis, London 2003, ISBN 0-7484-0891-6
• James F. Annett: Superconductivity, superfluids, and condensates. Oxford Univ. Press, Oxford 2005, ISBN 978-0-19-850756-7
• Leggett, A. (1999). "Superfluidity". Reviews of Modern Physics 71 (2): S318–S323. DOI:10.1103/RevModPhys.71.S318.  edit

External links[link]

• Liquid Helium II,Superfluid:demonstrations of Lambda point transition/viscosity paradox /two fluid model/fountain effect/creeping film/ second sound.
• Video including superfluid helium's strange behavior
• Superfluid phases of helium
• Lancaster University, Ultra Low Temperature Physics – Superfluid helium-3 research group.
• http://www.aip.org/png/html/helium3.htm
• http://www.aip.org/pt/vol-54/iss-2/p31.html
• http://web.mit.edu/newsoffice/2005/matter.html
• Superfluid helium as a vacuum
• superfluid hydrodynamics
• The Hindu article on superfluid states

v t e States of matter Solid Liquid Gas / Vapor Plasma Low energy Bose–Einstein condensate · Fermionic condensate · Quantum Hall · Rydberg matter · Strange matter · Superfluid · Supersolid High energy Degenerate matter · QCD matter · Quark–gluon plasma  · Supercritical fluid Other states Colloid · Glass · Liquid crystal · Magnetically ordered (Antiferromagnet, Ferrimagnet, Ferromagnet) · String-net liquid · Superglass Transitions Boiling · Boiling point · Condensation · Critical line · Critical point · Crystallization · Deionization · Deposition · Evaporation · Flash evaporation · Freezing · Ionization · Lambda point · Melting · Melting point · Regelation · Saturated fluid · Sublimation · Supercooling · Triple point · Vaporization Quantities Enthalpy of fusion · Enthalpy of sublimation · Enthalpy of vaporization · Latent heat · Latent internal energy · Trouton's ratio · Volatility Concepts Binodal · Compressed fluid · Cooling curve · Equation of state · Leidenfrost effect · Mpemba effect · Order and disorder (physics) · Spinodal · Superconductivity · Superheated vapor · Superheating · Thermo-dielectric effect

Ben Miller
Miller at the 2008 BAFTA Television Awards
Born	Bennet Evan Miller (1966-02-24) 24 February 1966 (age 46) London, United Kingdom
Education	Natural sciences
Alma mater	St Catharine's College, Cambridge
Occupation	Comedian, director, actor
Spouse	Belinda Stewart-Wilson (Divorced, 2011)
Partner	Jessica Parker
Children	Two sons

Bennet Evan "Ben" Miller (born 24 February 1966^[1]) is an English comedian, actor and director. He is perhaps best known as one half of comedy double act Armstrong and Miller, along with Alexander Armstrong. Together the pair wrote and starred in the Channel 4 sketch show Armstrong and Miller, and the more recent BBC television sketch show The Armstrong and Miller Show. As of 2011, he is starring in the crime drama series Death in Paradise.

Early life and education[link]

Miller was born in London, England, and grew up in Nantwich, Cheshire.^[2] His paternal grandfather was a Lithuanian-born tailor who lived in London's East End;^[3] his father, Michael Miller, was a lecturer in American Literature at the City of Birmingham Polytechnic; and his Welsh mother, Marion, taught English at South Cheshire College.^[3] He has two younger sisters, Leah and Bronwen.

He was educated at Malbank School and Sixth Form College, his local comprehensive school in Nantwich, Cheshire. He then studied natural sciences at St Catharine's College, Cambridge, where he acted with and dated Rachel Weisz.^[4][5] He remained at Cambridge to study for a Ph.D. in quantum physics, entitled "Novel quantum effects in low-temperature quasi-zero dimensional mesoscopic electron systems".

Miller's interest in comedy began when a friend asked him to help ferry around the judges of the National Student Drama Festival, which was being held that year in Cambridge.^[6] Having already finished his undergraduate degree, he then joined the Footlights in 1989 alongside the likes of Andy Parsons, David Wolstencroft and Sue Perkins and went on to direct a revue.^[7]

Career[link]

Miller decided not to continue his doctorate studies and moved to London to pursue a career in comedy.^[8] He was introduced to fellow Cambridge graduate Alexander Armstrong in 1992, at the TBA Sketch Comedy Group, a comedy club which ran at the Gate Theatre Studio, Notting Hill throughout the 1990s. They performed their first full-length show at the Edinburgh Fringe in 1994, and returned in 1996, whereupon they were nominated for the Perrier Comedy Award.^[9]

Their success resulted in the commissioning of the television series Armstrong and Miller, which ran for four series in total from 1997 to 2001 – one on the Paramount Comedy Channel, and three on Channel 4. The duo had their own radio show with the same name on BBC Radio 4 in 1998, which featured many of the sketches and characters from their TV series, and a second show, Children's Hour with Armstrong and Miller, later in the same year. After a six-year break, the show was recommissioned for Hattrick Productions as The Armstrong and Miller Show and is currently in its third series.^[10]

In 2001, Miller starred in Steve Coogan's first feature film, the British comedy The Parole Officer.^[11] In 2003 he played the role of 'Bough', sidekick to Rowan Atkinson's title character in the film Johnny English. In 2004 he co-starred in The Prince and Me. In 2004 and 2005, he starred in two consecutive series of the BBC television series The Worst Week of My Life,^[12] alongside Sarah Alexander,^[2] followed in 2006 by a three-part Christmas special, The Worst Christmas of My Life. Since 2007 he is starring as James Lester in ITV's sci-fi drama Primeval,^[9] and as Mr Jonathan in the Australian film Razzle Dazzle: A Journey Into Dance.^[6]

From 2001 until 2007, Miller provided the voice for the ITV Digital and now PG Tips Monkey, in a popular series of television advertisements featuring Johnny Vegas.^[13]

In 2008, he appeared in three ITV1 series, as television producer Jonathan Pope in Tony Jordan's series Moving Wallpaper. He also starred in Thank God You're Here.

In 2010 he made his directorial debut with the film Huge.

In 2011 (as revealed by Miller on Twitter), he will reprise his role as James Lester in Primeval.

One of his more recent successes, is starring in the hit BBC TV Series Death in Paradise, where he plays DI Richard Poole.

Since November 2011, he has been in The Ladykillers at the Gielgud Theatre playing Louis Harvey.

Production[link]

Miller directed a television pilot, which subsequently became the first episode of Steve Coogan's 2006 British BBC TV Comedy series Saxondale. With Armstrong, he has formed a production company named Toff Media.

Awards[link]

Miller was awarded a Judges' Commendation for his portrayal of Hamlet at the 1990 National Student Drama Festival. He co-wrote MindGym, winner of the first BAFTA Interactive Entertainment Award for comedy in 1998, with Tim Wright and Adam Gee. He and Armstrong won a BCA Award for The Armstrong and Miller Show. In 2010 they also won a BAFTA for The Armstrong and Miller Show.

^[14]

Personal life[link]

Miller was married to Belinda Stewart-Wilson (who guest-starred with him in Series 3 of Primeval), Miller eventually divorcing Stewart-Wilson (2011) after a 2 year separation. The couple lived in Maida Vale. They have a son, Jackson a.k.a Sonny, (born 2006). Miller also has a son, Harrison born late 2011, with his partner, production executive Jessica Parker.^[15][16]

A talented musician, Miller plays the guitar and drums^[2] and likes yoga.^[12] At school Miller played in a band called Aquilae which performed Eagles covers translated into Latin.^{[citation needed]} His fictitious school band, Slide Rule, which featured in sketches from the first A&M TV series, is believed to be an affectionate homage to the legendary Section B.^{[citation needed]} At St Catharine's, he played in a band called the Dear Johns, with Andy Edwards and Eoin Patterson.^{[citation needed]} They released one 7 in single, entitled "Shame".^{[citation needed]}

Miller is often mistaken for Rob Brydon,^[4][12] with whom he appeared on an episode of QI, first broadcast on 20 February 2009 (Series 6. 9). As a joke, they dressed in similar shirts and shared a narcissistic kiss.^[17][18]

Filmography[link]

References[link]

External links[link]

• The Armstrong & Miller Show official website
• Ben Miller at the Internet Movie Database
• Ben Miller hands out his very own BAFTAs