Frenkel line

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In thermodynamics, the Frenkel line is a proposed boundary on the phase diagram of a supercritical fluid, separating regions of qualitatively different behavior. [1] Fluids on opposite sides of the line have been described as "liquidlike" or "gaslike", and exhibit different behaviors in terms of oscillation, excitation modes, and diffusion. [2]

Contents

Other proposed similar boundary lines include for example the Fisher-Widom line and the Widom line.

Overview

Two types of approaches to the behavior of liquids are present in the literature. The most common one is based on a van der Waals model. It treats the liquids as dense structureless gases. Although this approach allows explanation of many principal features of fluids, in particular the liquid-gas phase transition, it fails to explain other important issues such as, for example, the existence in liquids of transverse collective excitations such as phonons.

Another approach to fluid properties was proposed by Yakov Frenkel. [3] It is based on the assumption that at moderate temperatures, the particles of liquid behave in a manner similar to a crystal, i.e. the particles demonstrate oscillatory motions. However, while in crystals they oscillate around their nodes, in liquids, after several periods, the particles change their nodes. This approach is based on postulation of some similarity between crystals and liquids, providing insight into many important properties of the latter: transverse collective excitations, large heat capacity, and so on.

From the discussion above, one can see that the microscopic behavior of particles of moderate and high temperature fluids is qualitatively different. If one heats a fluid from a temperature close to the melting point to some high temperature, a crossover from the solid-like to the gas-like regime occurs. The line of this crossover was named the Frenkel line, after Yakov Frenkel.

Several methods to locate the Frenkel line are proposed in the literature. [4] [5] The exact criterion defining the Frenkel line is the one based on a comparison of characteristic times in fluids. One can define a 'jump time' via

,

where is the size of the particle and is the diffusion coefficient. This is the time necessary for a particle to move a distance comparable to its own size. The second characteristic time corresponds to the shortest period of transverse oscillations of particles within the fluid, . When these two time scales are roughly equal, one cannot distinguish between the oscillations of the particles and their jumps to another position. Thus the criterion for the Frenkel line is given by .

There exist several approximate criteria to locate the Frenkel line on the pressure-temperature plane. [4] [5] [6] One of these criteria is based on the velocity autocorrelation function (vacf): below the Frenkel line, the vacf demonstrates oscillatory behaviour, while above it, the vacf monotonically decays to zero. The second criterion is based on the fact that at moderate temperatures, liquids can sustain transverse excitations, which disappear upon heating. One further criterion is based on isochoric heat capacity measurements. The isochoric heat capacity per particle of a monatomic liquid near the melting line is close to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 3 k_B } (where is the Boltzmann constant). The contribution to the heat capacity due to the potential part of transverse excitations is . Therefore, at the Frenkel line, where transverse excitations vanish, the isochoric heat capacity per particle should be , a direct prediction from the phonon theory of liquid thermodynamics. [7] [8] [9]

Crossing the Frenkel line leads also to some structural crossovers in fluids. [10] [11] Currently Frenkel lines of several idealised liquids, such as Lennard-Jones and soft spheres, [4] [5] [6] as well as realistic models such as liquid iron, [12] hydrogen, [13] water, [14] [15] [16] and carbon dioxide, [17] have been reported in the literature.

See also

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References

  1. ↑ Yoon, Tae Jun; Ha, Min Young; Lee, Won Bo; Lee, Youn-Woo (16 August 2018). ""Two-Phase" Thermodynamics of the Frenkel Line". The Journal of Physical Chemistry Letters. 9 (16): 4550–4554. arXiv: 1806.07608 . doi:10.1021/acs.jpclett.8b01955. PMID   30052454. S2CID   51727309.
  2. ↑ Ghosh, Kanka; Krishnamurthy, C. V. (December 2019). "Frenkel line crossover of confined supercritical fluids". Scientific Reports. 9 (1): 14872. Bibcode:2019NatSR...914872G. doi:10.1038/s41598-019-49574-3. PMC   6795815 . PMID   31619694.
  3. ↑ Frenkel, Jacov (1947). Kinetic Theory of Liquids. Oxford University Press.
  4. 1 2 3 Brazhkin, Vadim V; Lyapin, Aleksandr G; Ryzhov, Valentin N; Trachenko, Kostya; Fomin, Yurii D; Tsiok, Elena N (2012-11-30). "Where is the supercritical fluid on the phase diagram?". Physics-Uspekhi. 55 (11). Uspekhi Fizicheskikh Nauk (UFN) Journal: 1061–1079. Bibcode:2012PhyU...55.1061B. doi:10.3367/ufne.0182.201211a.1137. ISSN   1063-7869. S2CID   119452109.
  5. 1 2 3 Brazhkin, V. V.; Fomin, Yu. D.; Lyapin, A. G.; Ryzhov, V. N.; Trachenko, K. (2012-03-30). "Two liquid states of matter: A dynamic line on a phase diagram". Physical Review E. 85 (3). American Physical Society (APS): 031203. arXiv: 1104.3414 . Bibcode:2012PhRvE..85c1203B. doi:10.1103/physreve.85.031203. ISSN   1539-3755. PMID   22587085. S2CID   544649.
  6. 1 2 Brazhkin, V. V.; Fomin, Yu. D.; Lyapin, A. G.; Ryzhov, V. N.; Tsiok, E. N.; Trachenko, Kostya (2013-10-04). ""Liquid-Gas" Transition in the Supercritical Region: Fundamental Changes in the Particle Dynamics". Physical Review Letters. 111 (14). American Physical Society (APS): 145901. arXiv: 1305.3806 . Bibcode:2013PhRvL.111n5901B. doi:10.1103/physrevlett.111.145901. ISSN   0031-9007. PMID   24138256. S2CID   43100170.
  7. ↑ Bolmatov, D.; Brazhkin, V. V.; Trachenko, K. (2012-05-24). "The phonon theory of liquid thermodynamics". Scientific Reports. 2 (1): 421. arXiv: 1202.0459 . Bibcode:2012NatSR...2E.421B. doi: 10.1038/srep00421 . ISSN   2045-2322. PMC   3359528 . PMID   22639729.
  8. ↑ Bolmatov, Dima; Brazhkin, V. V.; Trachenko, K. (2013-08-16). "Thermodynamic behaviour of supercritical matter". Nature Communications. 4 (1): 2331. arXiv: 1303.3153 . Bibcode:2013NatCo...4.2331B. doi: 10.1038/ncomms3331 . ISSN   2041-1723. PMID   23949085.
  9. ↑ Hamish Johnston (2012-06-13). "Phonon theory sheds light on liquid thermodynamics". PhysicsWorld. Retrieved 2020-03-17.
  10. ↑ Bolmatov, Dima; Brazhkin, V. V.; Fomin, Yu. D.; Ryzhov, V. N.; Trachenko, K. (2013-12-21). "Evidence for structural crossover in the supercritical state". The Journal of Chemical Physics. 139 (23): 234501. arXiv: 1308.1786 . Bibcode:2013JChPh.139w4501B. doi:10.1063/1.4844135. ISSN   0021-9606. PMID   24359374. S2CID   18634979.
  11. ↑ Bolmatov, Dima; Zav’yalov, D.; Gao, M.; Zhernenkov, Mikhail (2014). "Structural Evolution of Supercritical CO2 across the Frenkel Line". The Journal of Physical Chemistry Letters. 5 (16): 2785–2790. arXiv: 1406.1686 . doi:10.1021/jz5012127. ISSN   1948-7185. PMID   26278079. S2CID   119243241.
  12. ↑ Fomin, Yu. D.; Ryzhov, V. N.; Tsiok, E. N.; Brazhkin, V. V.; Trachenko, K. (2014-11-26). "Dynamic transition in supercritical iron". Scientific Reports. 4 (1). Springer Science and Business Media LLC: 7194. arXiv: 1405.6491 . Bibcode:2014NatSR...4E7194F. doi: 10.1038/srep07194 . ISSN   2045-2322. PMC   4244626 . PMID   25424664.
  13. ↑ Trachenko, K.; Brazhkin, V. V.; Bolmatov, D. (2014-03-21). "Dynamic transition of supercritical hydrogen: Defining the boundary between interior and atmosphere in gas giants". Physical Review E. 89 (3): 032126. arXiv: 1309.6500 . Bibcode:2014PhRvE..89c2126T. doi:10.1103/physreve.89.032126. ISSN   1539-3755. PMID   24730809. S2CID   42559818.
  14. ↑ Yang, C.; Brazhkin, V. V.; Dove, M. T.; Trachenko, K. (2015-01-08). "Frenkel line and solubility maximum in supercritical fluids". Physical Review E. 91 (1): 012112. arXiv: 1502.07910 . Bibcode:2015PhRvE..91a2112Y. doi:10.1103/physreve.91.012112. ISSN   1539-3755. PMID   25679575. S2CID   12417884.
  15. ↑ Skarmoutsos, Ioannis; Henao, Andrés; Guardia, Elvira; Samios, Jannis (2021-09-16). "On the Different Faces of the Supercritical Phase of Water at a Near-Critical Temperature: Pressure-Induced Structural Transitions Ranging from a Gaslike Fluid to a Plastic Crystal Polymorph". The Journal of Physical Chemistry B. 125 (36): 10260–10272. doi:10.1021/acs.jpcb.1c05053. hdl: 2117/359088 . ISSN   1520-6106. PMID   34491748. S2CID   237442015.
  16. ↑ Skarmoutsos, Ioannis; Samios, Jannis; Guardia, Elvira (2022). "Fingerprints of the Crossing of the Frenkel and Melting Line on the Properties of High-Pressure Supercritical Water". The Journal of Physical Chemistry Letters. 13 (33): 7636–7644.
  17. ↑ Dima Bolmatov, D. Zav’yalov, M. Gao, and Mikhail Zhernenkov "Evidence for structural crossover in the supercritical state", Journal of Physical Chemistry 5 pp 2785-2790 (2014)

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