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On the characterization of transcritical fluid states

Authors:
Center for Turbulence Research
Annual Research Briefs 2017
1
On the characterization of transcritical fluid states
By D.T. Banuti, M. Raju AND M. Ihme
1. Motivation and objectives
Utilization of supercritical fluids has emerged as a key technology for energy efficiency,
addressing the needs imposed by climate change: a higher operating pressure increases
the combustion efficiency in gas turbines, Diesel engines, and rocket engines (Oschwald
et al. 2006; Oefelein et al. 2012); CO2sequestration, i.e., deposition of supercritical
CO2in subterranean reservoirs, is pursued as a way to effectively reduce the amount of
atmospheric CO2(Benson & Cole 2008). Furthermore, supercritical fluids may play a
fascinating role in the origin of life itself in the neighborhood of deep-sea hydrothermal
vents (Martin et al. 2008).
The classical thermodynamic presentation splits the pure fluid p-Tstate diagram into
four quadrants (Candel et al. 2006), centered around the critical point. The state-space
is thus divided into liquid, vapor, gaseous, transcritical, and supercritical states, with
transitions implied at the critical pressure pcr and temperature Tcr . Recently, another
structuring line, the Widom line as an extension to the coexistence line (Sciortino et al.
1997), has been popularized. This line divides liquid-like and gas-like supercritical states
(Gorelli et al. 2006; Simeoni et al. 2010), and exhibits some properties of a phase transi-
tion (Banuti 2015). The Widom line has since been shown to differ between species as a
function of the acentric factor (Banuti et al. 2016b, 2017a), and even to occur multiply
in certain mixtures (Raju et al. 2017).
However, open questions remain: Specifically, it is unclear whether a physical differ-
ence between liquid, transcritical, and liquid-like states exists. Furthermore, a number of
competing supercritical transition lines have been proposed (Nishikawa & Tanaka 1995;
Brazhkin et al. 2012; Gorelli et al. 2006; Banuti 2015), which differ in their definitions
and yield contradictory results.
The objective of this article is thus to identify the specific differences and similari-
ties between states and transition lines using macroscopic (continuum) and microscopic
(molecular) methods, with the goal of providing a unified view of the fluid state space
based on physical characteristics.
2. Methods
This paper combines theoretical and numerical data, applying continuum and molec-
ular perspectives in the study of supercritical fluid behavior. We selected argon as the
reference fluid of interest: Following the extended corresponding states principle (Reid
et al. 1987), the state plane topology of supercritical fluids can be assumed to be general
(Banuti et al. 2017a), and thus this study can be expected to be of relevance for other,
perhaps more complex, fluids. At the same time, molecular dynamics (MD) modeling in-
fluences are minimized due to the monatomic structure (Tegeler et al. 2016). The critical
temperature and pressure of argon are Tcr = 150.7 K and pcr = 4.863 MPa, respec-
tively. In order to allow for interspecies comparison, the reduced values of temperature
Tr=T/Tcr and pressure pr=p/pcr are used in the following discussion.
2Banuti et al.
The macroscopic continuum behavior was analyzed using reference fluid data from
the NIST database (Linstrom & Mallard 2016). The data are based on a fundamental
Helmholtz equation of state developed specifically for argon by Tegeler et al. (2016). The
equation is fit to experimental and numerical data from an extensive literature review.
Partial derivatives were obtained from numerical differentiation of these data.
The microscopic view is obtained from MD simulations. We used the LAMMPS package
(Plimpton 1995) to run a system with 25,600 Ar atoms in the canonical N-p-T(constant
number of atoms N, constant pressure p, and constant temperature T) ensemble at
different temperatures and pressures. The Ar force field was developed by training the
van der Waals parameters in the ReaxFF reactive force field (van Duin et al. 2001)
against experimental Ar dimer potential energy curves (Ogilvie & Wang 1992, 1993). To
validate the developed ReaxFF force field (Raju et al. 2017), we compared the enthalpy
obtained from isobaric MD-simulations with experimental enthalpy curves obtained from
NIST (Linstrom & Mallard 2016). The MD simulations were performed with a time step
of 0.25 fs using the Nose-Hoover thermostat with a coupling time constant of 10 fs and
Nose-Hoover barostat with a coupling time constant of 100 fs to control the temperature
and pressure of the system, respectively. For each simulation, the system was first energy-
minimized with convergence criterion of 0.1 kcal/˚
A. The system was then equilibrated
over 62.5 ps and the system energy and other properties were averaged for the following
62.5 ps of the production run. To quantitatively investigate the structural characteristics,
we computed the radial distribution function (RDF) (Levine et al. 2011)
g(r) = lim
dr0
p(r)
4π(Npairs/V )r2dr,(2.1)
with the distance between a pair of atoms r, the average number of atom pairs p(r) at
a distance between rand r+ dr, the total volume of the system V, and the number of
pairs of atoms Npairs. The self-diffusion coefficient was obtained from the mean-square
displacement ∆(t)∝ h[r(t)r(0)]2ithrough Einsteins relation (Frenkel & Smit 2001).
3. Thermodynamic states
3.1. Distinguishing state properties
Supercritical fluids have been studied for a long time. Figure 1(a) shows the projected
p-Tstate space of a pure fluid, which is classically divided into four quadrants, cen-
tered around the critical point. Baron Cagnard de la Tour (1822) discovered that fluids
no longer exhibit a liquid-vapor interface when subjected to sufficiently high pressures;
instead, they transform to a uniform medium. This transformation was later explained
with the existence of a critical point (CP), representing an endpoint of the vapor-pressure
curve (Andrews 1869). No unified terminology of the supercritical state space is estab-
lished. Fluids in quadrant I of Figure 1(a) are commonly referred to as liquids (IL) and
vapors (IV). Higher temperatures identify gases in II, that cannot be compressed to a liq-
uid state (Atkins & de Paula 2010). When instead the pressure is raised, the fluid state
in quadrant IV has been referred to as compressed liquid (Oefelein et al. 2012), com-
pressible liquid (Bolmatov et al. 2014), transcritical fluid (Oschwald et al. 2006; Candel
et al. 2006), liquid (Younglove 1982), or a supercritical fluid (Bellan 2000). When both
pressure and temperature exceed the fluid critical values, fluids are commonly considered
supercritical. Banuti et al. (2016a,b, 2017b) pointed out that a supercritical fluid behaves
like an ideal gas for Tr&2 and pr.3.
Transcritical fluid states 3
Tcr
pcr
IV
II
IIIGL
IV
IIILL
IL
Tcr
pcr
IV
II
III
IV
IL
(a)
(b)
Figure 1: Projected p-Tstate plane and supercritical states structure, with coexistence
line and critical point. Subscripts L, V, LL, GL denote liquid, vapor, liquid-like, and gas-
like, respectively. (a) Classical division into four quadrants with critical isotherm and
isobar. (b) Four quadrants and supercritical transition line.
States can be distinguished not only quantitatively (e.g., the liquid density is higher
than the gaseous density), but also by qualitative criteria: In a solid, molecules are
bound in a rigid, orderly structure. Neighboring molecules can be found in a periodic
pattern from other molecules and do not change their place. Movement of molecules is
oscillatory (Atkins & de Paula 2010). Liquids exhibit a similar structure. Movement is
still primarily oscillatory (Brazhkin et al. 2012), but molecules may switch their position
(Bolmatov et al. 2013). As in solids, the molecules are densely packed and can hardly be
compressed further when pressure is applied. Brazhkin et al. (2012) describe liquids as
an intermediate state, sharing properties from both solids and gases. Gases exhibit no
inherent structure. Movement is ballistic-collisional (Brazhkin et al. 2012; Bolmatov et al.
2015), and compressibility is comparably high. Supercritical fluids form the intermediary
between liquids and gases due to the large density inhomogeneities (Nishikawa & Tanaka
1995; Tucker 1999).
3.2. Atomistic analysis
We analyzed the properties of the fluid states in Figure 1(b) with the goal of assessing
whether a physical differentiation of supercritical fluids from liquids and gases can be
justified. The criterion is based on interatomic interaction, where an ideal gas is con-
sidered as the limiting case of vanishing interaction. The corresponding ideal gas RDF
introduced in Eq. (2.1) is then a translated Heavyside function with g= 0 within an
atomic diameter, and g= 1 outside, indicative of no spatial preference of the atoms.
A single peak of the RDF with g > 1 signifies limited interatomic interaction within a
real gas, a higher number of peaks corresponds to a long- range order, characteristic of
a liquid (Fisher & Widom 1969). In order to study the transition from liquid to gas for
various pressures, we carried out MD-simulations of argon, scanning a temperature range
in increments of 5 K at reduced pressures of pr={0.7,1.4,3.0,9.4}. The profiles of the
specific enthalpy computed from the scans are shown in the left column of Figure 2.
The results of the MD-calculations are in good agreement with NIST-data, although
the transition temperatures are slightly overpredicted. The subcritical temperature scan
at pr= 0.7 shows the familiar phase-transition discontinuity in the enthalpy. It is replaced
by a continuous, yet pronounced, crossover at a supercritical pressure of 1.4 pcr, occurring
over a finite temperature interval 0.9< Tr<1.13. We refer to this henceforth as the
4Banuti et al.
Figure 2: Enthalpy (left column) and visualization of the molecular structure of liquid,
transitional, and vapor states (from left to right) from MD computations for argon.
The temperature increases from left to right, the pressure from bottom to top. Reduced
temperature for the liquid and vapor columns of the molecular structure, respectively,
are Tr= 0.5, and Tr= 1.56. The transitional reduced temperatures are 0.95, 1.03, 1.16,
and 1.26, from bottom to top.
‘transitional region’. The transition is weakly discernible at pr= 3 and has completely
vanished at pr= 9.4.
Twelve representative conditions were chosen for liquid, transitional, and vapor states.
Transcritical fluid states 5
RDF
0.5
1.5
2.5
3.5
0
1
2
3
0 5 10 15 20
Radius in Å
pr=9.4
pr=3.0
pr=1.4
pr=0.7
0 5 10 15 20
Radius in Å
pr=9.4
pr=3.0
pr=1.4
pr=0.7
0 5 10 15 20
Radius in Å
pr=9.4
pr=3.0
pr=1.4
pr=0.7
Liquid Transitional Vapor
Figure 3: RDF corresponding to the columns liquid, transitional, and vapor in Figure 2.
The instantaneous atomic distributions in a slice through the computational domain
indicate qualitatively that no appreciable difference can be seen between the low tem-
perature states in the left column. Regardless of pressure, the molecules form a densely
packed liquid. The right column depicts the vapor states at Tr= 1.56. At low pressures,
they exhibit a diluted gaseous character, with little interaction between the molecules.
With increasing pressure, the vapor states approach liquid-like conditions. The transi-
tional states are shown in the center column. We observe a heterogeneous molecular
distribution that homogenizes as the pressure is increased.
A quantitative analysis of the molecular structure using the respective RDF, shown in
Figure 3, supports this assessment. At Tr= 0.5, the RDF is practically indistinguishable
between the four pressures and shows the characteristic multi-peak structure of a long-
range ordered liquid. The transitional fluid exhibits three distinct peaks, indicating a
correlated interaction across three shells surrounding an atom. We see that the peaks
become more pronounced at higher pressures, signifying a higher degree of order. At Tr=
1.56, the fluid resembles a gas. As the compression continues, a second peak is observed
at pr= 3, and a third at a pressure of 9.4 pr, indicative of a liquid molecular structure
(Fisher & Widom 1969). The behavior of the specific enthalpy and RDF suggests that
up to pressures of 3 pcr, the system exhibits a transitional region. At pressures greater
than 3 pcr, the system continuously transforms from a liquid-like to a vapor-like region
with no physically observable difference to distinguish these phases. This suggests that
a homogeneous supercritical fluid phase extends from pressures greater than 3 pcr .
We can conclude that there is no physical difference between liquids and transcriti-
cal fluids. Supercritical fluids may behave like liquids or gases, depending on the exact
conditions. Furthermore, we see that even gases at supercritical temperatures can be
compressed to liquids.
4. Supercritical state transition lines
A number of supercritical transition lines have been proposed in the literature. The
first was by Fisher & Widom (1969), showing that a transition must exist between the
oscillatory decay of the pair correlation function linked to the predominantly repulsive po-
tential in liquids, and a monotonous decay indicative of the attractive potential in gases.
Fisher & Widom emphasized that this transition does not involve thermodynamic singu-
larities and thus does not constitute a phase transition. The first experimental evidence
for a supercritical transition was found by Nishikawa & Tanaka (1995), who identified
6Banuti et al.
a distinct maximum in the correlation length when crossing a supercritical extrapola-
tion of the coexistence line, which they dubbed ‘extension curve’. Stanley’s ‘Widom line’
(Sciortino et al. 1997) was also originally introduced as the locus of maximum correlation
lengths. However, to facilitate evaluation using thermodynamic properties, it is often ap-
proximated as the locus of the thermodynamic response functions (Liu et al. 2005; Xu
et al. 2005), such as isobaric specific heat capacity cp(Xu et al. 2005; Santoro & Gorelli
2008; Ruppeiner et al. 2012; Banuti 2015); isothermal compressibility κT(Sciortino et al.
1997; Abascal & Vega 2010; Nishikawa & Tanaka 1995; Nishikawa & Morita 1997); or
the thermal expansion αp(Okamoto et al. 2003). More recently, the Frenkel-line was
introduced by Brazhkin et al. (2012) and extended by Bolmatov et al. (2013), dividing
rigid from nonrigid liquids, corresponding to a change in molecular motion from pri-
marily oscillatory to primarily ballistic. The crossover takes place where the fluid is no
longer capable of propagating high-frequency tangential shear sound modes. It extends
to arbitrarily high fluid pressures and does not constitute a phase transition (Brazhkin
et al. 2012).
4.1. Dynamic and thermodynamic transitions
The apparent contradiction between the different definitions of the transition lines can be
resolved when we realize that they can be grouped into two different physical phenomena.
On the one hand, we have a dynamic transition from liquid-like to gas-like fluid states.
This transition is reflected by changes in molecular motion from primarily oscillatory
to ballistic, from repulsion to attraction-dominated intermolecular interaction, from os-
cillatory to monotonous decay of the radial distribution function. Across this gradual
transition line, dispersion and propagation of transversal sound modes vanish. This tran-
sition is distinctly not thermodynamic; it extends to arbitrarily high fluid pressures. This
description fits results by Fisher & Widom (1969), Brazhkin et al. (2012), Gorelli et al.
(2006), and Simeoni et al. (2010).
On the other hand, we have a thermodynamic transition reflected by maxima of the re-
sponse functions, and macroscopic changes in fluid properties, such as a drop in density.
The transition resembles the subcritical boiling process and occurs across a continua-
tion of the coexistence line. The transition weakens with growing pressure. For reduced
pressures exceeding three, the effect has become so weak that it is negligible. This de-
scription fits the results of Hendricks et al. (1970), Nishikawa & Tanaka (1995), Oschwald
& Schik (1999), and Banuti (2015). Banuti (2015) proposed an equation that describes
this supercritical line,
pr= exp[As(Tr1)],(4.1)
where As= 5.280 for argon (Banuti et al. 2017a).
4.2. A closer look into the thermodynamic transition
Figure 4 compares MD transition lines for cp,αp,κT, and diffusivity Das extensions
of the coexistence line beyond the critical point. The width of the transitional region is
indicated as determined from the enthalpy (Figure 2). Figure 4 shows that the diffusion-
based Frenkel line lies within the Widom lines based on different response functions.
Consistent with the literature (Fomin et al. 2015; Luo et al. 2014), Figure 4 shows that
the definitions based on different response functions are not equivalent; furthermore the
lines diverge. There is no obvious justification to prefer one response function over any
other as the ‘right’ marker of the thermodynamic crossover.
Transcritical fluid states 7
Figure 4: Comparison of transition lines based on cp,αp,κT,D(symbols); shaded
transitional region based on enthalpy offset, see Figure 2. Data obtained from MD com-
putations.
4.3. The Gibbs free enthalpy and the Widom line
The subcritical phase transition from liquid to vapor across the coexistence line is a first
order phase transition. Thermodynamically, this implies a discontinuous change in the
slope of the Gibbs free energy g=hT s. At supercritical pressures, this discontinuity
has vanished. However, it is interesting to see how the Widom line definitions relate to
the Gibbs energy. Using the Maxwell relations of classical thermodynamics, we obtain
for the isobaric specific heat capacity
cp=∂h
∂T p
=T2g
∂T 2p
,(4.2)
for the isobaric thermal expansion
αp=1
v∂v
∂T p
=1
v2g
∂p∂T p
,(4.3)
and for the isothermal compressibility
κT=1
v∂v
∂p T
=1
v2g
∂p2T
.(4.4)
Thus, the response functions are related to second derivatives of the Gibbs energy, and
hence also are closely interrelated.
The presence of the second derivatives shows that the response functions reflect the
rate of change of the slope of the Gibbs energy. This suggests that while the discontinuity
of (∂g /∂T )pat the phase transition cannot exist anymore under supercritical conditions,
we may find a shadow of this transition in the form of a maximum curvature of the Gibbs
energy. The curvature κof a function yis κ=y00/(1 + y02)3/2. For the condition that
y00 y0, this expression can be approximated as ˜κ=y00. For an isobaric heating process,
8Banuti et al.
Reduced temperature
Reduced pressure
1 1.05 1.1 1.15 1.2 1.25 1.3
2
4
6
8
10
Eqn
Figure 5: Comparison of Widom lines based on κg,cp/T and cp(symbols). Data from
NIST (Linstrom & Mallard 2016). The solid line ‘Eqn’ is Eq. (4.1).
y0and y00 become (∂ g/∂T )pand (2g/∂ T 2)p, respectively, with
∂g
∂T p
=s, (4.5a)
2g
∂T 2p
=∂s
∂T p
=cp
T.(4.5b)
We introduce κgas the magnitude of the curvature of the Gibbs energy,
κg=
cp/Tr
(1 + (s)2)3/2
.(4.6)
Upon approaching the critical point on the Widom line, cpdiverges and the approxima-
tion
˜κg=
cp
T
=cp
T=2g
∂T 2p
(4.7)
is valid.
Figure 5 compares Widom lines based on the the maxima of κg,cp/T and cpwith
Eq. (4.1). We see that the transition temperature does not rise monotonously with pres-
sure, but curves back to lower temperatures at sufficiently high pressures. Up to pr1.5
the lines coincide. At higher pressures, first κg, then cp/T deviate from the cp-curve,
both have an end-point at supercritical conditions.
While the discontinuity in (∂ g/∂T )pacross the coexistence line vanishes at the critical
point, its influence is projected into the supercritical state space as the locus of points at
which the curvature of gexhibits an extremum. We see that the supercritical projection
of the coexistence discontinuity κgcoincides with the maxima of cpfor pr<2.
4.4. An upper limit of the Widom line
Understanding the Widom line as the supercritical projection of the coexistence discon-
tinuity suggests that this effect will weaken with growing distance from the critical point
as the subcritical discontinuity smoothes out.
Figure 6(a) uses NIST (Linstrom & Mallard 2016) data to show how the heat capacity
peak widens, flattens, and moves to higher temperatures as pressures reach pr= 3. At
pr= 6, Figure 6(b) shows that the excess heat capacity of the peak has become so thinly
Transcritical fluid states 9
Reduced temperature
Reduced specific heat capacity
11.5 2
1.6
1.8
2
2.2
2.4
pr =
8
10
6
Reduced temperature
Reduced specific heat capacity
11.5 2
5
10
15
20
25
pr =
2
3
0.8
2
1.2
(a)
(b)
Figure 6: Specific isobaric heat capacity for sub- and supercritical pressures. (a) Low
pressures. (b) High pressures. The peaks widen, flatten, and move to higher temperatures
as the pressure is increased for pr6. At pr= 8, a maximum is hardly discernible and
has moved to lower temperatures; at pr= 10 the peak has vanished altogether. Note
the different temperature scale at higher pressures. Data are from the NIST database
(Linstrom & Mallard 2016).
spread out that a distinct crossover temperature is no longer discernible. As shown in
Figure 5, the peaks move to lower temperature beyond pr= 6, before vanishing altogether
at pr= 10.
Figure 5 shows that the loci of maxima deviate from Eq. (4.1) when they start to curve
back. For κg,cp/T , and cpthis occurs at reduced pressures of 2, 3, and 4, respectively. A
look at the behavior of κgand cpat the corresponding pressures provides insight into the
relevance of this deviation. Figure 7 compares κg,cp/T ,αp, and κTaround the transition
temperature at two pressures. Close to the critical point at pr= 1.2, Figure 7(a) shows
that their respective values at the peak by far exceeds the liquid-like and gas-like values
towards lower and higher temperatures. The transition region is narrow and the peaks
occur at roughly the same temperature, consistent with the validity of Eq. (4.7). The
character of the curves changes when the pressure increases to 2pr(see Figure 7(b)).
The differences between peaks and tails are reduced significantly, and the differences
in the temperature of the respective peaks are more pronounced and move to higher
temperatures in all cases. Thus, Figure 7(b) shows that the peak of κgat pr= 2 has
almost vanished. Similarly, Figure 6 illustrates how the cppeak has flattened and widened
at reduced pressures exceeding 3. We can conclude that the Widom lines diverge when
the respective peak has weakened to the point of losing its physical significance. Similarly,
the peak in the thermodynamic Gibbs curvature κg, Figure 5, vanishes at approximately
2.6 pcr.
We conclude that there is strong evidence for an upper limit to the thermodynamic
Widom line. There is no precise limiting pressure, but instead we observe a universal
weakening in the thermodynamic character of the crossover. Steep property gradients
and response function extrema present at near-critical pressures have weakened to the
point of being negligible at pr3, and vanish for pr>10, consistent with the changing
character of the transitions found for pr= 3.0 and 9.4 in Figure 2.
The respective peaks of the response functions are not isolated local events. Instead,
they represent different stages of the same thermodynamic transition spanning a finite,
broadening transitional region. Thus, each maximum of a response function marks a
10 Banuti et al.
Reduced temperature
Normalized response function
0.8 11.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
Reduced temperature
Normalized response function
0.8 11.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
(a)
(b)
Figure 7: Comparison of response functions and Gibbs curvature at two pressures.
Graphs are normalized to the respective peak values. (a) pr= 1.2. (b) pr= 2.0. Data are
calculated from the NIST database (Linstrom & Mallard 2016).
1Tr
pr
1
2
3
liquid
vapor
transitional
ideal gas
solid
10
non-rigid liquid
Frenkel
Widom
Figure 8: Revised phase state pr-Trstructure.
certain point during the same thermodynamic transition. We found that the cp-based
Widom line is strongly related to the curvature of the Gibbs energy, which projects
the influence of the phase transition into the supercritical domain. A spreading of the
transitional region then naturally causes the different loci to diverge, without stripping
them of physical meaning.
4.5. A revised state diagram
We have shown that the projected pr-Trstructure of the fluid state space is only insuffi-
ciently characterized by the states illustrated in Figure 1(b). Instead, we have presented
evidence for the state regime structure depicted in Figure 8. No difference could be
found between a liquid at subcritical and a compressed liquid at supercritical pressure.
We further showed that a supercritical fluid may behave like a gas at sufficiently high
temperatures. In Figure 8, we use the term ‘vapor’ to denote a real gas with significant
intermolecular interaction, and ‘ideal gas’ to denote a gas with negligible intermolecular
interaction. Only for very low pressures (p < 0.1pcr ) do we find a transition from a liquid
to an ideal gas across the coexistence line; for higher subcritical pressures, the liquid tran-
sitions to a real-gas vapor state instead. For pcr <p<3pcr, the thermodynamic crossover
Transcritical fluid states 11
occurs through a transitional domain, surrounding the heat capacity based Widom line.
This process is spread over a finite temperature interval characterized by strong changes
in fluid properties. When heated to T > 2Tcr , this supercritical fluid behaves like an ideal
gas. At pressures exceeding 3pcr, the phase change-like character vanishes and is replaced
by an almost linear change in enthalpy with temperature. At pressures exceeding 10pcr,
the heat capacity peak has vanished altogether. While the thermodynamic transition
ceases to play a role at p > 3pcr, a structural transition across the Frenkel line may exist
in the fluid for arbitrary pressures.
5. Conclusions
This paper discusses fluid states by studying similarities and differences between liq-
uids, gases, and supercritical fluids, and the transitions between them.
There is no such thing as a homogeneous supercritical state; instead, the state space is
more complex than previously anticipated. We identified four different supercritical fluid
states: classical liquids, a non-rigid liquid unable to sustain transversal sound propaga-
tion, a vapor gas with significant intermolecular interaction, and an ideal gas with no
intermolecular interaction.
By emphasizing the differentiation into the thermodynamic Widom line and the dy-
namic Frenkel line, we are able to categorize the different lines that are currently discussed
in the literature. The Widom line is the supercritical extension to the coexistence line,
associated with a thermodynamic crossover from the classical rigid liquid to the vapor
state. Beyond p > 3pcr, the transition loses its thermodynamic character but retains
the dynamic transition, corresponding to the Frenkel line. The Frenkel line extends to
arbitrarily high fluid pressures, and marks the transition from liquids to non-rigid liq-
uids which no longer exhibit a liquid-like dispersion behavior. Pseudoboiling is no longer
relevant at these pressures.
We introduce the supercritical maximum curvature of the Gibbs energy κgas a ther-
modynamically meaningful interpretation for this transition, showing that the discontin-
uous character of the coexistence line is projected into the supercritical state space. This
projection coincides with the cp-based Widom line. The influence weakens with growing
distance to the critical point and vanishes at 2.5< pr<3. The response functions cp,
κT, and αpcan be expressed in terms of the second derivatives of the Gibbs free energy.
Correspondingly, all functions exhibit maxima in the vicinity of the transition. For tem-
peratures lower than 3pcr, peaks of cpare a valid and simple approximation for extrema
of κg.
Due to the continuous nature of the supercritical liquid-to-vapor crossover, the re-
spective loci of maxima cannot be regarded as isolated events, but should instead be
understood as different stages within the same transition. We demonstrate that the tran-
sition region has a finite width and widens towards higher pressures, encompassing the
transition lines based on other definitions. The widening leads to a divergence of the
different lines without stripping them of physical meaning.
Acknowledgments
Financial support through the Army Research Laboratory with award number W911NF-
16-2-0170 is gratefully acknowledged.
12 Banuti et al.
REFERENCES
Abascal, J. & Vega, C. 2010 Widom line and the liquid-liquid critical point for the
TIP4P/2005 water model. J. Chem. Phys. 133, 234502.
Andrews, T. 1869 The Bakerian lecture: On the continuity of the gaseous and liquid
states of matter. Philos. Trans. R. Soc. London 159, 575–590.
Atkins, P. & de Paula, J. 2010 Physical Chemistry Oxford University Press.
Banuti, D. T. 2015 Crossing the Widom-line – supercritical pseudo-boiling. J. Supercrit.
Fluids 98, 12–16.
Banuti, D. T., Hannemann, V., Hannemann, K. & Weigand, B. 2016aAn effi-
cient multi-fluid-mixing model for real gas reacting flows in liquid propellant rocket
engines. Combust. Flame 168, 98–112.
Banuti, D. T., Raju, M. & Ihme, M. 2016bSupercritical pseudoboiling for general
fluids and its application to injection. Annual Research Briefs, Center for Turbulence
Research, Stanford University, pp. 211–221.
Banuti, D. T., Raju, M. & Ihme, M. 2017aSimilarity law for Widom lines and
coexistence lines. Phys. Rev. E 95, 052120.
Banuti, D. T., Raju, M., Ma, P. C., Ihme, M. & Hickey, J.-P. 2017bSeven
questions about supercritical fluids towards a new fluid state diagram. AIAA 2017-
9144.
Bellan, J. 2000 Supercritical (and subcritical) fluid behavior and modeling: drops,
streams and mixing layers, jets and sprays. Prog. Energy Combust. Sci. 26, 329–366.
Benson, S. & Cole, D. 2008 Co2sequestration in deep sedimentary formations. Ele-
ments 4, 335–331.
Bolmatov, D., Brazhkin, V. & Trachenko, K. 2013 Thermodynamic behavior of
supercritical matter. Nat. Commun. 4, 2331.
Bolmatov, D., Zav’yalov, D., Gao, M. & Zhernenkov, M. 2014 Structural evolu-
tion of supercritical CO2across the Frenkel line. J. Phys. Chem. Lett. 5, 2785–2790.
Bolmatov, D., Zhernenkov, M., & S.N. Tkachev, D. Z., Cunsolo, A. & Cai, Y.
2015 The Frenkel Line: a direct experimental evidence for the new thermodynamic
boundary. Sci. Rep. 5, 15850.
Brazhkin, V. V., Fomin, Y. D., Lyapin, A. G., Ryzhov, V. N. & Trachenko, K.
2012 Two liquid states of matter: A dynamic line on a phase diagram. Phys. Rev. E
85, 031203.
Candel, S., Juniper, M., Singla, G., Scouflaire, P. & Rolon, C. 2006 Structures
and dynamics of cryogenic flames at supercritical pressure. Combust. Sci. Technol.
178, 161–192.
van Duin, A. C. T., Dasgupta, S., Lorant, F. & Goddard, W. A. 2001 ReaxFF:
A reactive force field for hydrocarbons. J. Phys. Chem. A 105, 9396–9409.
Fisher, M. & Widom, B. 1969 Decay of correlations in linear systems. J. Chem. Phys.
50, 3756–3772.
Fomin, Y. D., Ryzhov, V. N., Tsiok, E. N. & Brazhkin, V. V. 2015 Thermody-
namic properties of supercritical carbon dioxide: Widom and Frenkel lines. Phys.
Rev. E 91, 022111.
Frenkel, D. & Smit, B. 2001 Understanding Molecular Simulation. Academic Press,
2nd edition.
Transcritical fluid states 13
Gorelli, F., Santoro, M., Scopigno, T., Krisch, M. & Ruocco, G. 2006
Liquidlike behavior of supercritical fluids. Phys. Rev. Lett. 97, 245702.
Hendricks, R. C., Simoneau, R. J. & Smith, R. V. 1970 Survey of heat transfer to
near-critical fluids. Tech. Rep. TN D-5886. NASA.
Levine, B., Stone, J. & Kohlmeyer, A. 2011 Fast analysis of molecular dynamics
trajectories with graphics processing units–radial distribution function histogram-
ming. J. Comput. Phys. 230, 3556 – 3569.
Linstrom, P. J. & Mallard, W. G., 2016 NIST Chemistry WebBook, NIST Standard
Reference Database Number 69, http://webbook.nist.gov. Gaithersburg MD, 20899:
National Institute of Standards and Technology.
Liu, L., Chen, S.-H., Faraone, A., Yen, C.-W. & Mou, C.-Y. 2005 Pressure de-
pendence of fragile-to-strong transition and a possible second critical point in super-
cooled confined water. Phys. Rev. Lett. 95, 117802.
Luo, J., Xu, L., Lascaris, E., Stanley, H. & Buldyrev, S. 2014 Behavior of the
Widom line in critical phenomena. Phys. Rev. Lett. 112, 135701.
Martin, W., Baross, J., Kelley, D. & Russell, M. 2008 Hydrothermal vents and
the origin of life. Nat. Rev. Microbiol. 6, 805–814.
Morita, T., Nishikawa, K., Takematsu, M., Iida, H. & Furutaka, S. 1997 Struc-
ture study of supercritical CO2near higher-order phase transition line by X-ray
diffraction. J. Phys. Chem. B 101, 7158–7162.
Nishikawa, K. & Morita, T. 1997 Small-angle X-ray-scattering study of supercritical
trifluoromethane. J. Phys. Chem. B 101, 1413–1418.
Nishikawa, K. & Tanaka, I. 1995 Correlation lengths and density fluctuations in
supercritical states of carbon dioxide. Chem. Phys. Lett. 244, 149–152.
Oefelein, J. C., Dahms, R. N., Lacaze, G., Manin, J. L. & Pickett, L. M.
2012 Effects of pressure on fundamental physics of fuel injection in diesel engines. In
Proceedings of the 12th International Conference on Liquid Atomization and Spray
Systems (ICLASS). Heidelberg, Germany.
Ogilvie, J. F. & Wang, F. Y. H. 1992 Potential-energy functions of diatomic molecules
of the noble gases i. like nuclear species. J. Mol. Struct. 273, 277–290.
Ogilvie, J. F. & Wang, F. Y. H. 1993 Potential-energy functions of diatomic molecules
of the noble gases i. like nuclear species. J. Mol. Struct. 291, 313–322.
Okamoto, K., Ota, J., Sakurai, K. & Madarame, H. 2003 Transient velocity
distributions for the supercritical carbon dioxide forced convection heat transfer. J.
Nucl. Sci. 40, 763–767.
Oschwald, M. & Schik, A. 1999 Supercritical nitrogen free jet investigated by spon-
taneous Raman scattering. Exp. Fluids 27, 497–506.
Oschwald, M., Smith, J. J., Branam, R., Hussong, J., Schik, A., Chehroudi,
B. & Talley, D. 2006 Injection of fluids into supercritical environments. Combust.
Sci. Technol. 178, 49–100.
Plimpton, S. 1995 Fast parallel algorithms for short-range molecular dynamics. J. Com-
put. Phys. 117, 1–19.
Raju, M., Banuti, D. T., Ma, P. C. & Ihme, M. 2017 Widom lines and supercritical
phase transitions of binary mixtures. Sci. Rep. 7, 3027.
Reid, R. C., Prausnitz, J. M. & Poling, B. E. 1987 The properties of gases and
liquids, 4th edn. McGraw Hill.
14 Banuti et al.
Ruppeiner, G., Sahay, A., Sarkar, T. & Sengupta, G. 2012 Thermodynamic
geometry, phase transitions, and the Widom line. Phys. Rev. E 86, 052103.
Santoro, M. & Gorelli, F. 2008 Structural changes in supercritical fluids at high
pressures. Phys. Rev. B 77, 212103.
Sciortino, F., Poole, P., Essmann, U. & Stanley, H. 1997 Line of compressibility
maxima in the phase diagram of supercooled water. Phys. Rev. E 55, 727–737.
Simeoni, G., Bryk, T., Gorelli, F., Krisch, M., Ruocco, G., Santoro, M. &
Scopigno, T. 2010 The Widom line as the crossover between liquid-like and gas-like
behaviour in supercritical fluids. Nat. Phys. 6, 503–507.
Tegeler, C., Span, R. & Wagner, W. 2016 A new equation of state for argon covering
the fluid region for temperatures from the melting line to 700 K at Pressures up
to 1000 MPa. In NIST Chemistry WebBook, NIST Standard Reference Database
Number 69 (ed. P. J. Linstrom & W. G. Mallard). National Institute of Standards
and Technology, Gaithersburg MD, 20899, http://webbook.nist.gov, 2016.
de la Tour, C. 1822 Expos´e de quelques r´esultats obtenu par l’action combin´ee de
la chaleur et de la compression sur certains liquides, tels que l’eau, l’alcool, l’´ether
sulfurique et l’essence de p´etrole rectifi´ee. Ann. Chim. Phys. 21, 127–132.
Tucker, S. 1999 Solvent density inhomogeneities in supercritical fluids. Chem. Rev. 99,
391–418.
Xu, L., Kumar, P., Buldyrev, S., Chen, S.-H., Poole, P., Sciortino, F. &
Stanley, H. 2005 Relation between the Widom line and the dynamic crossover in
systems with a liquid–liquid phase transition. PNAS 102, 16558–16562.
Younglove, B. A. 1982 Thermophysical properties of fluids. I. argon, ethylene, parahy-
drogen, nitrogen, nitrogen trifluoride, and oxygen. J. Phys. Chem. Ref. Data 11,
Supplement 1.
... In 1972, Widom demonstrated that the representation of the critical anomalies in the pressure-temperature space are lines emanating from the critical point 18 . Below a reduced pressure (P r = P/P CP ) of 1.5 these lines of maxima in thermodynamic response functions converge on a single line as the liquid-gas critical point is approached 19 . When isobaric processes are analyzed, the most referred Widom line is the one indicating the locus of maxima isobaric heat capacity 20 , as shown in Fig. 1a. ...
... We associated the middle of the plateau to the pseudo-boiling temperature determined from our experiments (T pb(exp) ). Table 1 presents the values of T pb(exp) for each isobar in comparison with the theoretical values determined from NIST data 13 and based on Banuti's theory of pseudo-boiling 19,25,26,33 . The pseudo-boiling temperature increases as the pressure increases and this is consistent with our experimental observations. ...
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... Additionally, there is another line that divides the supercritical state space into sub-regions with liquid-like and gas-like properties. This line is Widom line that it is considered by large changes in density or enthalpy, demonstrating as maxima in the thermodynamic response functions [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. Although both lines are the boundaries between gas-like and liquid-like sub-regions in supercritical area, they are completely separate in nature. ...
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