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The '''Frenkel line''' is a line of change of thermodynamics, dynamics and structure of
'''Frenkel line''' is a line of change of microscopic dynamics of
fluids. Below the Frenkel line the fluids are "rigid" and
fluids. Below the Frenkel line the fluid are "rigid" and
"solid-like" while above it fluids are "soft" and "gas-like".
"solid-like" while above it fluids are "soft" and "gas-like".


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


Another approach to fluid properties was proposed by Jacov Frenkel
Another approach to fluid properties was proposed by J. Frenkel
<ref>Jacov Frenkel "Kinetic Theory of Liquids", Oxford University Press (1947)</ref>.  
<ref>[J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, London, 1947]</ref>. It is based on an assumption that at moderate
It is based on the assumption that at moderate
temperatures the particles of liquid behave similar to the case of
[[temperature]]s the particles of liquid behave in a similar manner as a
crystal, i.e. the particles demonstrate oscillatory motions.
crystal, ''i.e.'' the particles demonstrate oscillatory motions.
However, while in crystal they oscillate around theirs nodes, in
However, while in crystal they oscillate around theirs nodes, in
liquids after several periods the particles change the nodes. This
liquids after several periods the particles change the nodes. This
approach is based on postulation of some similarity between crystals
approach based on postulation of some similarity between crystals
and liquids,  providing insight into many important properties of the
and liquids allows to explain many important properties of the
latter: transverse collective excitations, large [[heat capacity]] and
later: transverse collective excitations, large hear capacity and
so on.
so on.


From the discussion above one can see that the microscopic
From the discussion above one can see that the microscopic
behavior of particles of moderate and high temperature fluids is
behavior of particles of moderate and high temperature fluids is
qualitatively different. If one [[heat]]s up a fluid from a
qualitatively different. If one heats up a fluid from a
temperature close to the [[Melting curve|melting point]]  up to some high temperature, a
temperature close to the melting one up to some high temperature a
crossover from the solid-like to gas-like regime appears. The line
crossover from the solid-like to gas-like regime appears. The line
of this crossover was named Frenkel line after J. Frenkel.
of this crossover was named Frenkel line after J. Frenkel.
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Several methods to locate the Frenkel line were proposed in the
Several methods to locate the Frenkel line were proposed in the
literature. The most detailed reviews of the methods are given in
literature. The most detailed reviews of the methods are given in
Refs.  
Refs. <ref name="ufn"> [http://iopscience.iop.org/1063-7869/55/11/R01/ V.V. Brazhkin, A.G. Lyapin, V.N. Ryzhov, K. Trachenko, Yu.D. Fomin, E.N. Tsiok, Phys. Usp. 55, 1061 (2012) ]</ref>, <ref name="frpre"> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.85.031203 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, Phys. Rev. E 85, 031203 (2012)]</ref>. The exact criterion of Frenkel line is the one based on comparison of characteristic times in fluids. One can
<ref name="ufn"> [http://dx.doi.org/10.3367/UFNe.0182.201211a.1137 Vadim V. Brazhkin, Aleksandr G Lyapin, Valentin N. Ryzhov, Kostya Trachenko, Yurii D. Fomin and Elena N. Tsiok "Where is the supercritical fluid on the phase diagram?", Physics-Uspekhi '''55''' pp. 1061-1079 (2012)]</ref>,
define a 'jump time' via :<math> \tau_0=\frac{a^2}{6D} </math>, where <math> a </math> is the particles size and :<math> D </math> - diffusion coefficient. This is the time necessary for a particle to move to it's own size. The second characteristic time is the shortest period of transverse oscillations of particles of fluid: <math> \tau^* </math>. When these two time
<ref name="frpre"> [http://dx.doi.org/10.1103/PhysRevE.85.031203 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko "Two liquid states of matter: A dynamic line on a phase diagram", Physical Review E '''85''' 031203 (2012)]</ref>.  
scales become comparable one cannot distinguish the oscillations of the particles and theirs jumps to another position. Therefore
The exact criterion of Frenkel line is the one based on comparison of characteristic times in fluids. One can
define a 'jump time' via  
 
:<math> \tau_0=\frac{a^2}{6D} </math>,  
 
where <math> a </math> is the particles size and <math> D </math> is the [[Diffusion|diffusion coefficient]]. This is the time necessary for a particle to move a distance comparable to it's own size. The second characteristic time corresponds to the shortest period of transverse oscillations of particles within the fluid, <math> \tau^* </math>. When these two time
scales are roughly equal one cannot distinguish between the oscillations of the particles and theirs jumps to another position. Thus
the criterion for Frenkel line is given by <math> \tau_0 \approx \tau^* </math>.
the criterion for Frenkel line is given by <math> \tau_0 \approx \tau^* </math>.


There are several approximate criteria to locate the Frenkel line
There are several approximate criteria to locate the Frenkel line
on the [[Phase diagrams: Pressure-temperature plane|pressure-temperature plane]]
in <math> (P,T) </math> Refs. <ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> [http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.145901 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and Kostya Trachenko, Phys. Rev. Lett. 111, 145901 (2013)]</ref>. One of these criteria is based
(see Refs.  
on velocity autocorrelation function (vacf): below the Frenkel
<ref name="ufn"> </ref>,  
line vacf demonstrate oscillation behavior while above it vacfs
<ref name="frpre"> </ref>,  
monotonically decay to zero. The second criterion is based on the
<ref name="frprl"> [http://dx.doi.org/10.1103/PhysRevLett.111.145901 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and Kostya Trachenko "“Liquid-Gas” Transition in the Supercritical Region: Fundamental Changes in the Particle Dynamics", Physical Review Letters '''111''' 145901 (2013)]</ref>).  
fact that at moderate temperature liquids can sustain transverse
One of these criteria is based
excitations which disappear on heating the liquid up. One more
on the velocity [[autocorrelation]] function (vacf): below the Frenkel
criterion is based on isochoric heat capacities measurements. The
line the vacf demonstrates oscillatory behaviour, while above it the vacf
isochoric heat capacity per particle of a monatomic liquid close
monotonically decays to zero. The second criteria is based on the
to the melting line is close to <math> 3 k_B </math> ( <math> k_B  </math> is Boltzmann
fact that at moderate temperatures liquids can sustain transverse
constant). The contribution to the heat capacity of potential part
excitations, which disappear upon heating. One further
criteria is based on [[Heat capacity#At constant volume|isochoric heat capacity]] measurements.  
The isochoric heat capacity per particle of a monatomic liquid near
to the melting line is close to <math> 3 k_B </math> (where <math> k_B  </math> is the
[[Boltzmann constant]]). The contribution to the heat capacity due to potential part
of transverse excitations is <math> 1 k_B </math>. Therefore at the Frenkel
of transverse excitations is <math> 1 k_B </math>. Therefore at the Frenkel
line, where transverse excitations vanish, the isochoric heat
line where transverse excitations vanish the isochoric heat
capacity per particle should be <math> c_V=2 k_B </math>, a direct prediction from the phonon theory of liquid thermodynamics
capacity per particle should be <math> c_V=2 k_B </math>.
<ref> [http://dx.doi.org/10.1038/srep00421 D. Bolmatov, V. V. Brazhkin, and K. Trachenko "The phonon theory of liquid thermodynamics", Scientific Reports '''2''' 421 (2012)]</ref>  
 
<ref> [http://physicsworld.com/cws/article/news/2012/jun/13/phonon-theory-sheds-light-on-liquid-thermodynamics Hamish Johnston "Phonon theory sheds light on liquid thermodynamics", PhysicsWorld.com  June 13 (2012)]</ref>
Crossing the Frenkel line leads also to some structural changes in
<ref> [http://dx.doi.org/10.1038/ncomms3331 Dima Bolmatov, V. V. Brazhkin, and K. Trachenko "Thermodynamic behaviour of supercritical matter", Nature Communications '''4''' 2331 (2013)]</ref>.
fluids <ref> [http://scitation.aip.org/content/aip/journal/jcp/139/23/10.1063/1.4844135  D. Bolmatov, V. V. Brazhkin, Yu. D. Fomin, V. N. Ryzhov and K. Trachenko, J. Chem. Phys. 139, 234501 (2013)] </ref>.
 
 
 
 
Currently Frenkel lines of several model liquids (Lennard-Jones
and soft spheres <ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref>  and real ones (liquid
iron <ref> [http://www.nature.com/srep/2014/141126/srep07194/fig_tab/srep07194_F1.html Yu. D. Fomin, V. N. Ryzhov, E. N. Tsiok, V. V. Brazhkin and K. Trachenko, Scientific Reports, 4, 7194 (2014)] </ref>, hydrogen <ref> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032126 K. Trachenko, V. V. Brazhkin, and D.Bolmatov, Phys. Rev. E 89, 032126 (2014)] </ref>, water
<ref name="kostya3"> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.91.012112  C. Yang, V. V. Brazhkin, M. T. Dove, and K. Trachenko, Phys. Rev. E, 91, 012112 (2015)] </ref>, <math> CO_2 </math> <ref name="kostya3"> </ref>, <math> CH_4 </math> <ref name="kostya3"> </ref> were reported in the literature.
 
 


Crossing the Frenkel line leads also to some structural crossovers in fluids
<ref>[http://dx.doi.org/10.1063/1.4844135  Dima Bolmatov, V. V. Brazhkin, Yu. D. Fomin, V. N. Ryzhov, and K. Trachenko "Evidence for structural crossover in the supercritical state", Journal of Chemical Physics '''139''' 234501 (2013)]</ref>
<ref name="StructuralEvolutionJPCL" >[http://dx.doi.org/10.1021/jz5012127 Dima Bolmatov, D. Zav’yalov, M. Gao, and Mikhail Zhernenkov "Structural Evolution of Supercritical CO2 across the Frenkel Line", Journal of Physical Chemistry Letters '''5''' pp 2785-2790 (2014)]</ref>.
Currently Frenkel lines of several [[Idealised models|idealised liquids]], such as [[Lennard-Jones model|Lennard-Jones]] and [[Soft sphere potential|soft spheres]]
<ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref> 
as well as [[realistic models]] such as
[[iron|liquid iron]] <ref>[http://dx.doi.org/10.1038/srep07194  Yu. D. Fomin, V. N. Ryzhov, E. N. Tsiok, V. V. Brazhkin, and K. Trachenko "Dynamic transition in supercritical iron", Scientific Reports '''4''' 7194 (2014)]</ref>,
[[hydrogen]] <ref> [http://dx.doi.org/10.1103/PhysRevE.89.032126 K. Trachenko, V. V. Brazhkin, and D. Bolmatov, "Dynamic transition of supercritical hydrogen: Defining the boundary between interior and atmosphere in gas giants", Physical Review E '''89''' 032126 (2014)]</ref>,
[[water]] <ref name="kostya3"> [http://dx.doi.org/10.1103/PhysRevE.91.012112  C. Yang, V. V. Brazhkin, M. T. Dove, and K. Trachenko "Frenkel line and solubility maximum in supercritical fluids", Physical Review E '''91''' 012112 (2015)]</ref>,
[[carbon dioxide]] <ref name="StructuralEvolutionJPCL" ></ref><ref name="kostya3"> </ref>,
and [[methane]] <ref name="kostya3"> </ref>
have been reported in the literature.
==Related Lines==
==Related Lines==
*[[Widom line]]
*[[Widom line]]
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