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'''Frenkel line''' is a line of change of microscopic dynamics of
The '''Frenkel line''' is a line of change of thermodynamics, dynamics and structure of
fluids. Below the Frenkel line the fluid are "rigid" and
fluids. Below the Frenkel line the fluids 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
Two types of approaches to the behavior of liquids present in the
literature. The most common one is due to [[Johannes Diderik van der Waals|van der Waals]]. It treats
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 to explain many principle features of fluids, in
allows one to explain many principle features of fluids, in
particular, the liquid-gas phase transition, it fails in
particular, the [[Gas-liquid phase transitions|liquid-gas phase transition]], it fails to
explanation of other important issues, such as, for example,
explain  other important issues such as, for example,
existence in liquids of transverse collective excitations such as
the existence in liquids of transverse collective excitations such as
phonons.
phonons.


Another approach to fluid properties was proposed by J. Frenkel
Another approach to fluid properties was proposed by Jacov Frenkel
<ref>[J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, London, 1947]</ref>. It is based on an assumption that at moderate
<ref>Jacov Frenkel "Kinetic Theory of Liquids", Oxford University Press (1947)</ref>.  
temperatures the particles of liquid behave similar to the case of
It is based on the assumption that at moderate
crystal, i.e. the particles demonstrate oscillatory motions.
[[temperature]]s the particles of liquid behave in a similar manner as a
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 based on postulation of some similarity between crystals
approach is based on postulation of some similarity between crystals
and liquids allows to explain many important properties of the
and liquids,  providing insight into many important properties of the
later: transverse collective excitations, large hear capacity and
latter: transverse collective excitations, large [[heat 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 heats up a fluid from a
qualitatively different. If one [[heat]]s up a fluid from a
temperature close to the melting one up to some high temperature a
temperature close to the [[Melting curve|melting point]]  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.
Line 33: Line 33:
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. <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
Refs.  
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="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>,
scales become comparable one cannot distinguish the oscillations of the particles and theirs jumps to another position. Therefore
<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>.  
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
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
on the [[Phase diagrams: Pressure-temperature plane|pressure-temperature plane]]
on velocity autocorrelation function (vacf): below the Frenkel
(see Refs.  
line vacf demonstrate oscillation behavior while above it vacfs
<ref name="ufn"> </ref>,  
monotonically decay to zero. The second criterion is based on the
<ref name="frpre"> </ref>,  
fact that at moderate temperature liquids can sustain transverse
<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>).  
excitations which disappear on heating the liquid up. One more
One of these criteria is based
criterion is based on isochoric heat capacities measurements. The
on the velocity [[autocorrelation]] function (vacf): below the Frenkel
isochoric heat capacity per particle of a monatomic liquid close
line the vacf demonstrates oscillatory behaviour, while above it the vacf
to the melting line is close to <math> 3 k_B </math> ( <math> k_B  </math> is Boltzmann
monotonically decays to zero. The second criteria is based on the
constant). The contribution to the heat capacity of potential part
fact that at moderate temperatures liquids can sustain transverse
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>.
capacity per particle should be <math> c_V=2 k_B </math>, a direct prediction from the phonon theory of liquid thermodynamics
 
<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>  
Crossing the Frenkel line leads also to some structural changes in
<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>
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>.
<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>.
 
 
 
 
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]]

Latest revision as of 19:19, 16 March 2015

The Frenkel line is a line of change of thermodynamics, dynamics and structure of fluids. Below the Frenkel line the fluids are "rigid" and "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 van der Waals. It treats the liquids as dense structureless gases. Although this approach allows one to explain many principle 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 Jacov Frenkel [1]. It is based on the assumption that at moderate temperatures the particles of liquid behave in a similar manner as a crystal, i.e. the particles demonstrate oscillatory motions. However, while in crystal they oscillate around theirs nodes, in liquids after several periods the particles change the 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 up a fluid from a temperature close to the melting point up to some high temperature, a crossover from the solid-like to gas-like regime appears. The line of this crossover was named Frenkel line after J. Frenkel.

Several methods to locate the Frenkel line were proposed in the literature. The most detailed reviews of the methods are given in Refs. [2], [3]. The exact criterion of Frenkel line is the one based on comparison of characteristic times in fluids. One can define a 'jump time' via

,

where is the particles size and is the 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, . 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 .

There are several approximate criteria to locate the Frenkel line on the pressure-temperature plane (see Refs. [2], [3], [4]). 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 criteria is based on the fact that at moderate temperatures liquids can sustain transverse excitations, which disappear upon heating. One further criteria is based on isochoric heat capacity measurements. The isochoric heat capacity per particle of a monatomic liquid near to the melting line is close to (where is the Boltzmann constant). The contribution to the heat capacity due to 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 [5] [6] [7].

Crossing the Frenkel line leads also to some structural crossovers in fluids [8] [9]. Currently Frenkel lines of several idealised liquids, such as Lennard-Jones and soft spheres [2], [3], [4] as well as realistic models such as liquid iron [10], hydrogen [11], water [12], carbon dioxide [9][12], and methane [12] have been reported in the literature.

Related Lines[edit]

References[edit]

  1. Jacov Frenkel "Kinetic Theory of Liquids", Oxford University Press (1947)
  2. 2.0 2.1 2.2 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)
  3. 3.0 3.1 3.2 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)
  4. 4.0 4.1 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)
  5. D. Bolmatov, V. V. Brazhkin, and K. Trachenko "The phonon theory of liquid thermodynamics", Scientific Reports 2 421 (2012)
  6. Hamish Johnston "Phonon theory sheds light on liquid thermodynamics", PhysicsWorld.com June 13 (2012)
  7. Dima Bolmatov, V. V. Brazhkin, and K. Trachenko "Thermodynamic behaviour of supercritical matter", Nature Communications 4 2331 (2013)
  8. 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)
  9. 9.0 9.1 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)
  10. 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)
  11. 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)
  12. 12.0 12.1 12.2 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)