Frenkel line - Revision history

Carl McBride: Reinstated CO2 references and mention of CH4

2015-03-16T18:19:09Z

Reinstated CO2 references and mention of CH4

← Older revision		Revision as of 20:19, 16 March 2015
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	[[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>,		[[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>,		[[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> ~~[http:~~/~~/pubs.acs.org/doi/abs/10.1021/jz5012127 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)]~~</ref>		[[carbon dioxide]] <ref name="StructuralEvolutionJPCL" ></ref><ref name="kostya3"> </ref>,
			and [[methane]] <ref name="kostya3"> </ref>
	have been reported in the literature.		have been reported in the literature.
	==Related Lines==		==Related Lines==

Carl McBride: Corrected article title

2015-03-16T18:15:40Z

Corrected article title

← Older revision		Revision as of 20:15, 16 March 2015
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	Crossing the Frenkel line leads also to some structural crossovers in fluids		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>[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>[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>.		<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]]		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>		<ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref>
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	[[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>,		[[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>,		[[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> [http://pubs.acs.org/doi/abs/10.1021/jz5012127 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)]</ref>		[[carbon dioxide]] <ref name="StructuralEvolutionJPCL" ></ref><ref> [http://pubs.acs.org/doi/abs/10.1021/jz5012127 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)]</ref>
	have been reported in the literature.		have been reported in the literature.
	==Related Lines==		==Related Lines==

Carl McBride: Corrected journal title

2015-03-16T18:13:22Z

Corrected journal title

← Older revision		Revision as of 20:13, 16 March 2015
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	Crossing the Frenkel line leads also to some structural crossovers in fluids		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>[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> [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 '''5''' pp 2785-2790 (2014)]</ref>.		<ref>[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]]		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>		<ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref>

Carl McBride: Replaced link with DOI

2015-03-16T18:06:24Z

Replaced link with DOI

← Older revision		Revision as of 20:06, 16 March 2015
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	Crossing the Frenkel line leads also to some structural crossovers in fluids		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> [http://~~pubs~~.~~acs~~.org~~/doi/abs~~/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 '''5''' pp 2785-2790 (2014)]</ref>.		<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> [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 '''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]]		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>		<ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref>

Carl McBride: Added DOI and corrected a journal name

2015-03-16T17:59:24Z

Added DOI and corrected a journal name

← Older revision		Revision as of 19:59, 16 March 2015
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	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 <ref> [http://~~www~~.~~nature~~.~~com~~/~~srep/2012/120524~~/srep00421~~/full/srep00421.html~~ D. Bolmatov, V. V. Brazhkin, and K. Trachenko "The phonon theory of liquid thermodynamics", Scientific Reports '''2''' 421 (2012)]</ref>, <ref> [http://~~www~~.~~nature~~.com/~~ncomms~~/~~2013~~/~~130816~~/~~ncomms3331/full~~/ncomms3331~~.html~~ Dima Bolmatov, V. V. Brazhkin, and K. Trachenko "Thermodynamic behaviour of supercritical matter", ~~Scientific Reports~~ '''4''' 2331 (2013)~~]</ref>, <ref> [http://physicsworld.com/cws/article/news/2012/jun/13/phonon-theory-sheds-light-on-liquid-thermodynamics "Phonon theory sheds light on liquid thermodynamics", PhysicsWorld, 2012~~]</ref>.		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>
			<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>
			<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>.

	Crossing the Frenkel line leads also to some structural crossovers in fluids		Crossing the Frenkel line leads also to some structural crossovers in fluids

Davidcohen at 23:36, 15 March 2015

2015-03-15T23:36:06Z

Carl McBride: Slight tidy + added internal links

2015-02-26T16:24:25Z

Slight tidy + added internal links

Show changes

Fomin314 at 16:15, 25 February 2015

2015-02-25T16:15:58Z

← Older revision		Revision as of 18:15, 25 February 2015
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	'''Frenkel line''' is a line of change of microscopic dynamics of		'''Frenkel line''' is a line of change of microscopic dynamics 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".

Fomin314: Created page with "'''Frenkel line''' is a line of change of microscopic dynamics of fluids. Below the Frenkel line the fluid are "rigid" and "solid-like" while above it fluids are "soft" and "g..."

2015-02-25T15:56:33Z

Created page with "'''Frenkel line''' is a line of change of microscopic dynamics of fluids. Below the Frenkel line the fluid are "rigid" and "solid-like" while above it fluids are "soft" and "g..."

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

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
allows to explain many principle features of fluids, in
particular, the liquid-gas phase transition, it fails in
explanation of other important issues, such as, for example,
existence in liquids of transverse collective excitations such as
phonons.

Another approach to fluid properties was proposed by J. Frenkel
<ref>[J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, London, 1947]</ref>. It is based on an assumption that at moderate
temperatures the particles of liquid behave similar to the case of
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 based on postulation of some similarity between crystals
and liquids allows to explain many important properties of the
later: transverse collective excitations, large hear 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 one 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. <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
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
scales become comparable one cannot distinguish the oscillations of the particles and theirs jumps to another position. Therefore
the criterion for Frenkel line is given by <math> \tau_0 \approx \tau^* </math>.

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 velocity autocorrelation function (vacf): below the Frenkel
line vacf demonstrate oscillation behavior while above it vacfs
monotonically decay to zero. The second criterion is based on the
fact that at moderate temperature liquids can sustain transverse
excitations which disappear on heating the liquid up. One more
criterion is based on isochoric heat capacities measurements. The
isochoric heat capacity per particle of a monatomic liquid close
to the melting line is close to <math> 3 k_B </math> ( <math> k_B </math> is Boltzmann
constant). The contribution to the heat capacity of potential part
of transverse excitations is <math> 1 k_B </math>. Therefore at the Frenkel
line where transverse excitations vanish the isochoric heat
capacity per particle should be <math> c_V=2 k_B </math>.

Crossing the Frenkel line leads also to some structural changes in
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.

==Related Lines==
*[[Widom line]]
*[[Fisher-Widom line]]

==References==
<references />
[[Category:statistical mechanics]]

← Older revision		Revision as of 01:36, 16 March 2015
Line 1:		Line 1:
	The '''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 fluids 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".
Line 63:		Line 63:
	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://www.nature.com/srep/2012/120524/srep00421/full/srep00421.html D. Bolmatov, V. V. Brazhkin, and K. Trachenko "The phonon theory of liquid thermodynamics", Scientific Reports '''2''' 421 (2012)]</ref>, <ref> [http://www.nature.com/ncomms/2013/130816/ncomms3331/full/ncomms3331.html Dima Bolmatov, V. V. Brazhkin, and K. Trachenko "Thermodynamic behaviour of supercritical matter", Scientific Reports '''4''' 2331 (2013)]</ref>, <ref> [http://physicsworld.com/cws/article/news/2012/jun/13/phonon-theory-sheds-light-on-liquid-thermodynamics "Phonon theory sheds light on liquid thermodynamics", PhysicsWorld, 2012]</ref>.

	Crossing the Frenkel line leads also to some structural ~~changes~~ in fluids		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> [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> [http://pubs.acs.org/doi/abs/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 '''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]]		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>		<ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref>
	as well as [[realistic models]] such as		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>,		[[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>,		[[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>,		[[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="kostya3"~~> </~~ref>~~,		[[carbon dioxide]] <ref> [http://pubs.acs.org/doi/abs/10.1021/jz5012127 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)]</ref>
	and ~~[[methane]] <ref name=~~"~~kostya3~~"> </ref>
	have been reported in the literature.		have been reported in the literature.
	==Related Lines==		==Related Lines==