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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

dr→0

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(Tr−1)],(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

0

0.5

1

1.5

2

2.5

3

3.5

0.6 0.8 1 1.2 1.4

Reduced pressure

Reduced temperature

cp

αp

D

κT

CL

CP

boundary

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=h−T 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

=−T∂2g

∂T 2p

,(4.2)

for the isobaric thermal expansion

αp=1

v∂v

∂T p

=1

v∂2g

∂p∂T p

,(4.3)

and for the isothermal compressibility

κT=−1

v∂v

∂p T

=−1

v∂2g

∂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 pr≈1.5

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 pr≤6. 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 pr≈3, 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.

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