Superfluidity

Here is a paper I wrote in March of 2018 about another intriguing physical phenomenon: superfluidity. I hope you find it as cool as you would have to get to observe this phenomenon, which is usually close to absolute zero! Please enjoy, and don’t hesitate to comment in the forum if you have any questions. That way more people, including me, can learn from your question!


“Superfluidity” describes a property of liquid matter, i.e. the property of having zero viscosity, or immeasurably low, to be precise (1). This means that if a superfluid were stirred, it would cycle in endless vortices, conserving  one hundred percent of its kinetic energy.  And if a hole were made in the bottom of the vessel, then the superfluid would flow out very quickly compared to other fluids, e.g. honey. The rate of flow of course depends on the size of the hole, so for comparisons of viscosity assume that the holes are the same size. If one fills a cup with honey and then pokes a hole in it, then the honey will eventually flow out, but very slowly.  A superfluid, on the other hand, would flow out the highest possible rate, only limited by the size of the hole.  (3). To understand this an understanding of viscosity is needed. The fundamental principle behind viscosity is friction: the resistance that one object has to moving over another. In the case of liquids, which lack structure, it is the friction between the molecules or atoms of the liquid itself that causes viscosity. (2). In the case of a superfluid, there is no such internal friction due to special molecular or atomic makeup. That is the fundamental definition of a superfluid, but these materials can exhibit many other strange properties. (1). Along with the property of zero viscosity comes lower density and a significant change in specific heat. But is this property of zero viscosity that produces the strangest effects.

                Superfluidity and superconductivity are normally only exhibited at extremely low temperatures.  This is because the particles that condense must always be indistinguishable. The Debroglie wavelengths of indistinguishable particles must have a high degree of overlap. Debroglie wavelength equals Planck’s constant divided my momentum, mass times velocity. Since large particles, e.g. baseballs have high mass, their Debroglie wavelengths will be infinitesimally small, which is it is so unlikely that large particles will exhibit wave-like behavior. For Debroglie wavelengths to overlap, they need to be very long, achieved by cold temperatures, and it helps to have a high packing density of the particles. (8). Cold temperatures are very costly to generate, so this requirement limits the practical applications of superfluidity and superconductivity. Some materials used to make the superconductors are also very expensive, e.g. niobium. (9).

                But, there exist so called “high temperature superconductors” that exhibit superconductivity at a balmy eighty kelvins. The BCS theory does not account for the existence of these, but they will probably prove very useful in the future, as they do exist. One current use of superconductors is in particle accelerators, such as CERN. (10).

                Superfluids also exhibit extremely high thermal conductivity, according to some sources infinitely high (12). Heat is transmitted so quickly that thermal waves are created. This is possible due to the fact that the particles of a superfluid are in the same quantum state, i.e. if one particle moves, they all are moved. Heat is conducted when excited particles bump into each other. If a particle bumps into superfluid particle it will move at the same time as all of the other molecules in the superfluid, transmitting the heat from one side of the superfluid to the other instantly. (11).

                Having no internal friction allows for some spectacular displays, e.g. if a vessel is filled with liquid helium-4 which is then cooled below 2.17 kelvin, condensing into a superfluid, it will flow up the walls of the vessel and out of the vessel. This effect is caused by minor differences in temperature and atmospheric pressure inside and outside the vessel. These tiny differences are enough to move the superfluid against the force of gravity because friction does not hinder its flow. (3).

                A vessel with tiny, molecule-sized holes in the bottom would hold liquid helium-4, but if it were cooled below 2.17 kelvins, then it would immediately begin flowing through those holes, again as a consequence of its immeasurably low viscosity. (3).

                Just a month after the discovery of superfluidity, in 1932, another odd effect was observed accidentally bay British physicist Jack Allen. He had a long, thin tube sticking above a liquid helium bath and packed with fine emery powder. He shined a flashlight on the apparatus and the emery powder absorbed the light, slightly elevating the temperature of the superfluid. As long as the light shone, a fountain of liquid helium emitted from the end of the tube above the bath. The heat creates a back pressure that forces the superfluid helium up. (13).

                As stated above, a superfluid could be used to create perpetual motion. Of course, however, no one is deceived that this means that a surplus energy could be produced and thereby unlimited energy; a superfluid can only conserve the energy that it is given and no more, it does not produce any of its own. However, it is a special property of helium that it never settles into solid state, not even at absolute zero; it always remains a liquid. This is because the helium atoms are so weakly attracted to one another that the slight jiggling caused by the quantum uncertainty principle is enough to keep them apart, at least at standard pressure. (3). It would be naive to think that infinite energy could be harvested from the quantum uncertainty principle; as physicists learn more about quantum mechanics, they will probably learn how no energy created or destroyed.

                Superfluidity was first demonstrated in two studies published in Nature in 1938 by the Brittish duo John Allen and Don Meisner, and independently by Russian physicist Pyotr Kapitza.  Alan and Meisner measured the flow of liquid helium-4 through long, thin tubes and found that it flowed with zero viscosity at temperatures below 2.17 degrees kelvin, almost absolute zero. Kapitza made comparable observations on the flow of liquid helium-4 between two glass discs. He also hypothesized a connection between superfluidity, the resistanceless flow of atoms or molecules, and superconductivity, the resistanceless flow of electrons, which had been discovered some years earlier in 1911 by Dutchman Heike Kamerlingh Onnes. These two concepts were at the frontier of physics back then, and it wasn’t until 1957 that Bardeen, Cooper, and Schriefer (BCS) devised a complete theory to link the phenomena.  (4).

                However, it was very soon after the discovery of superfluidity that an explanation was offered: Bose-Einstein condensation. This is the process whereby  particles known as bosons condense to form a single, quantum state. (4). A boson is a particle whose spin, or intrinsic angular momentum, is zero or an integer. The only other possible class of particle is the fermion, a particle with half-integer spin.  Spin dictates the energy distribution of a particle. Bosons obey Bose-Einstein statistics and fermions obey Fermi-Dirac statistics. Bose-Einstein statistics allows for an unlimited number of particles to occupy a single energy level, unlike fermi-dirac statistics that follow the Pauli exclusion principle, which dictates that no two associated fermions can occupy the same quantum state. (5). Imagine two fermions, say electrons. Let p equal the probability that electron 1 occupies state a and electron 2 occupies state b. Let p1 equal the probability that electron one occupies state a and let p2 equal the probability that electron 2 is in state b. Electrons are indistiguishable, so it is impossible to tell which electron occupies which state. So  p1 and  p2 must be arranged like this:

p=p1p2-p1p2

The above relationship describes a wave function, and if both electrons occupy the same state, a or b, the wave function will vanish. Physicists draw from this the Pauli exclusion principle. The same equation can be used for bosons, except the minus sign must be changed to a plus sign. It is their ability to condense together in unlimited numbers occupying the same energy state that allows bosons to form bose-einstein condensates. (6). But liquid helium-4 atoms are composed of six fermions (two protons, neutrons, and electrons) and no bosons! Liquid helium-4 atoms can form a bose-einstein condensate only because an even number, e.g. 6, of interacting fermions can form a composite boson. This allows liquid helium-4 atoms to condense into the lowest possible energy state and become a superfluid. (4).

                The BCS theory also offers an explanation for superconductivity. An electron moving through a lattice of superconducting material with attract the lattice toward it causing a ripple in the direction of its motion. An electron moving in the opposite with be attracted to this disturbance and the two electrons are coupled together forming what is called a Cooper pair. These cooper pairs can also act like bosons and condense into a state of zero electrical resistance called superconductivity. (7).

                After WW II, large quantities of the light isotope helium-3 became available  because it was part of the manufacturing process of tritium, to be used in hydrogen bombs. It would seem in the light of the information thus far presented that helium-3, having an odd number of fermions (two, protons, electrons, and a neutron), could not condense into a superfluid. But it might be possible, according to the BCS theory, for the helium-3 atoms themselves to form Cooper pairs and thus become a superfluid. The theoretical properties of this hypothetical superfluid helium-3 were explored in the 1960s, before the actual discovery of this superfluid at temperatures below .003 kelvins in 1972. (4).

                The spin quantum number (S) and the orbital quantum number (L) of Cooper pairs characterize two types of angular momentum. Normal BCS superconductors have S=0 and L=0, but superfluid helium-3 has S=1 and L=1. These non-zero quantum numbers cause helium-3 superfluid to break certain basic symmetries of normal liquid state, namely rotational and time reversal symmetries, entailing a non-trivial topology for the Cooper pairs. A variation of the BCS theory was required to understand this, marking the beginning of unconventional superconductivity, but the strange behaviors of unconventional superconductors were only just beginning to be discovered. (4).

                Superfluid helium-3 has two phases, A and B (in the absence of a magnetic field). The B phase exists over a much wider range of temperatures and pressures. It can also exist in many different excited states due to its lack of rotational symmetry, and these states are classified according to the total angular momentum of the cooper pairs (J), with the possibilities J=0,1, or 2. One remarkable feature of the J=2 state of the B phase is its ability to transmit transverse sound waves, something that was previously only thought possible in rigid solids. (4).

                Since the discovery of superfluid helium-3, numerous other unconventional superconductors have been discovered, e.g. cuprates. But only one other superconducting material has been found to have two superfluid phases, namely UPt3. (4).

                Even today, physicists are finding new superfluid properties that defy understanding. The most recent developments in the study of superfluidity involved helium-3 in ultr-light aerogels that was shown to exhibit never before seen phases of superfluids, that are currently being studied. (4).

                Superfluidity was just one of the amazing and counter-intuitive discoveries about quantum mechanics made in the twentieth century, and scientists continue to learn more about it in the twenty-first century. Scientists are a long way from fully understanding the phenomena harnessing its full potential.


References:

  1. Schmitt, Andreas. (2014). “Introduction to Superfluidity.” Springer. https://arxiv.org/pdf/1404.1284.pdf. Date-accessed: 4/18/2018.
  2. “What is viscosity?” (n.d.) Princeton. https://www.princeton.edu/~gasdyn/Research/T-C_Research_Folder/Viscosity_def.html. Date-accessed: 4/18/2018.
  3. Minkel, J. R. (2009). “Strange but True: Superfluid Helium can Climb Walls.” Scientific American. https://www.scientificamerican.com/article/superfluid-can-climb-walls/. Date-accessed: 4/18/2018.
  4. Halperin, William P. (2018). “Eighty Years of Superfluidity.” Nature. https://www.nature.com/articles/d41586-018-00417-7?error=cookies_not_supported&code=4fe886f1-804a-495e-a2a6-830b84b16621#ref-CR10. Date-accessed: 4/18/2018.
  5. Nave, R. (n.d.). “Spin Classification.” HyperPhysics. http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/spinc.html#c3. Date-accessed: 4/18/2018.
  6. Nave, R. (n.d.). “Pauli Exclusion Principle.” Hyperphysics. http://hyperphysics.phy-astr.gsu.edu/hbase/pauli.html#c2. Date-accessed: 4/18/2018.
  7. Nave, R. (n.d.). “Cooper Pairs.” Hyperphysics. http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/coop.html#c1. Date-accessed: 4/18/2018.
  8. Nave, R. (n.d.). “Wave Nature of Electron.” Hyperphysics. http://hyperphysics.phy-astr.gsu.edu/hbase/debrog.html#c3. Date-accessed: 4/18/2018.
  9. Cooley, Lance. Pong, Ian. (2016). “Cost drivers for very high energy p-p collider magnet conductors.” Fermilab. https://indico.cern.ch/event/438866/contributions/1085142/attachments/1257973/1858756/Cost_drivers_for_VHEPP_magnet_conductors-v2.pdf. Date-accessed: 4/19/2018.
  10. “Superconductivity.” (2018). CERN. https://home.cern/about/engineering/superconductivity. Date-accessed: 4/19/2018.
  11. “Properties of fuperfluid.” (n.d.). http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Rodney_Guritz%20Folder/properties.htm. Date-accessed: 4/19/2018.
  12. “Infinite Thermal Conductivity.” (n.d.). https://superfluidsiiti.weebly.com/index.html. Date-accessed: 4/19/2018.
  13. “Superfluidity II- The Fountain Effect.” (2006). Nature Publishing Group. https://www.nature.com/physics/looking-back/superfluid2/index.html#. Date-accessed: 4/19/2018.

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