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Crossing the Widom-line – supercritical pseudo-boiling

D.T. Banuti∗

German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology, Spacecraft Department,

37073 G¨ottingen, Bunsenstr. 10, Germany; Phone: +49 551 709 2403

Abstract

Recent publications in the open literature have shown that supercritical ﬂuid states are not homo-

geneously distributed but, in fact, can be diﬀerentiated into two distinct regions with gas-like and

liquid-like properties, respectively. These regions are divided by an extension of the coexistence

line, commonly called Widom line. This paper shows that a supercritical analog to subcritical

phase change, pseudoboiling, does exist when crossing this demarcation. The supercritical state

transition does not occur in a phase equilibrium but takes place over a ﬁnite temperature interval.

While subcritical vaporization requires energy to overcome intermolecular attraction, supercritical

state transitions additionally require energy to increase the temperature. It could be shown that

the attractive potential is the dominant energy sink up to a reduced pressure of 1.5 for argon,

nitrogen, oxygen, and water. The eﬀect reduces with growing pressure and becomes negligible for

p/pcr >3. Furthermore, a new equation for this Widom- or pseudoboiling line is given. It exhibits

improved accuracy over previously published equations; performing a limit analysis of the Clapey-

ron equation allows to express its sole parameter purely in terms of thermodynamic variables. This

parameter can then be evaluated from an equation of state or from ﬂuid data - no nonphysical

ﬁtting is required.

Keywords: pseudoboiling, Widom-line, vaporization, Clapeyron, transcritical

1. Introduction

While Andrews [1] introduced the notion of a critical point to terminate the vapor pressure curve

more than 100 years ago, today there is not a common terminology on states exceeding the critical

∗Corresponding author

Email address: daniel.banuti@dlr.de (D.T. Banuti)

Preprint submitted to Elsevier April 6, 2017

temperature or pressure (Bellan [2], Younglove [3]). The common view is that upon exceeding the

critical point, liquids and gases do no longer exist. Instead, a structureless, homogeneous, continuous

supercritical ﬂuid prevails throughout the state space. Furthermore, given the vanishing of latent

heat of vaporization and surface tension, a phase transition no longer occurs.

Fundamentally new insight has been gained in the last two decades, facilitated by advances in

experimental and numerical methods. Inelastic x-ray scattering could be used to study dispersion,

i.e. the dependence of the speed of sound on the frequency, which is a phenomenon found in

liquids but not in gases. Using this technique, Simeoni at al. [4] and Gorelli et al. [5] were able to

attribute this property to supercritical ﬂuids at reduced pressures (pr=p/pcr ) exceeding 1000. They

concluded that the supercritical domain is divided into two regions with distinct respective liquid-

like and gas-like behavior - much like at subcritical conditions. The dividing line is an extension to

the coexistence line. Gorelli suggested to extrapolate the Plank-Riedel equation, Eq. (1), into the

supercritical domain,

ln p

pcr =a+bTcr

T+cln T

Tcr (1)

where a= 4.270, b =−4.271, and c= 1.141 have been ﬁt to neon, oxygen, and nitrogen ﬂuid data.

From a diﬀerent perspective, Sciortino et al. [6] used molecular dynamics computations to

investigate properties of this line emanating from the critical point which was later dubbed ‘Widom’

line. Figure 1 illustrates this new view of supercritical states structure. The work of both groups

is summarized by MacMillan and Stanley [7], Simeoni et al. [4] declare a new chapter to be open,

new theories to be required. All research was concerned with the understanding of thermodynamic

states and the identiﬁcation of the boundary between them.

So far, no quantitative discussion has been given on processes in which this boundary is crossed.

A name, however, exists: ‘pseudoboiling’ has been used by Okamato et al. [8] and Oschwald et

al. [9] to describe a supercritical liquid-like to gas-like transition. The term has originally been

introduced by Kafengauz and Federov [10, 11], who described a phenomenon in cooling pipes which

resembled subcritical boiling - but at supercritical pressures.

The concept is not undisputed: Hall [12] outright rejected it as “irrational and unnecessary

because, clearly, two distinct phases do not exist at supercritical pressures”, whereas Okamato et

al. [8] called it a theoretical problem to be solved. Santoro and Gorelli [13] hypothesize there

might be a “sluggish liquidlike-gaslike transformation mimicking the subcritical liquid-gas phase

transition”. Oschwald et al. [14, 15] suggest that, unlike during subcritical vaporization, heat

2

Tr

log(pr)

1

0

CP

Widom line

liquid-like

gas-like

Figure 1: Fluid state plane and supercritical states structure with a liquid-like super-Widom ﬂuid and a gas-like

sub-Widom ﬂuid. TP=triple point, CP=critical point, Tr=T/Tcr ,pr=p/pcr.

added to the ﬂuid during pseudoboiling will act to both expand and heat it.

It is therefore the purpose of this paper to analyze processes which cross the Widom line from

liquid-like to gas-like ﬂuid states (and vice versa). The main question is: Can the concept of

vaporization/condensation be applied to transitions between supercritical states?

2. Material and Metho d

The approach taken here is mixed analytical and empirical. Thermodynamic relations and

approximations are expressed in analytical form. All ﬂuid data is taken from the NIST webbook

[16]. First, the concept of pseudoboiling is evaluated using nitrogen data by Span et al. [17], with

particular focus on the speciﬁc isobaric heat capacity of states at supercritical pressure. After this

proof of concept, ﬂuid data for argon (Tegeler et al. [18]) and oxygen (Schmidt and Wagner [19])

are investigated in a similar manner because of their expected similarity in behavior, following the

corresponding states principle. Finally, water data (Wagner and Pruss [20]) are shown, explicitly

to point out deviations for more complex ﬂuids.

3

3. Results and Discussion

3.1. Widom Line Equation

The original deﬁnition of the Widom line is the set of supercritical maximal thermodynamic

correlation lengths [4]. However, this property is very hard to come by in a practical manner

(May and Mausbach [21]). As a substitute, response functions - such as the heat capacity, the

isothermal compressibility, and the thermal expansion - are used by several authors (e.g. Liu et al.

[22], Xu et al. [23], Santoro and Gorelli [13], Ruppeiner et al. [24]). Regarding a state transition

as a phenomenon of energy conversion in this paper, the speciﬁc isobaric heat capacity cpis an

appropriate marker. Figure 2(a) illustrates this point using the example of nitrogen.

At the subcritical pressure 3 MPa, cpdiverges at the boiling temperature: here, heat is added

during vaporization without an increase in temperature until all liquid is consumed. A distinct

peak remains visible for supercritical pressures, ﬂattening and moving to higher temperatures as

the pressure increases from 4 MPa to 6 Mpa. Figure 2(b) shows how these peaks line up nicely

when plotted into a log(p)/T diagram. Thus, an equation of the form

ppb

pcr

= exp Tcr

θpb Tpb

Tcr

−1 = exp ATpb

Tcr

−1 (2)

suitably describes the cpbased Widom line, where ‘pb’ and ‘cr’ denote properties along the pseu-

doboiling line and at the critical point, respectively. For dimensional reasons, the integration

constant 1/θpb can be interpreted as the inverse of a characteristic temperature. This parameter

will now be determined from ﬁrst principles.

Along the coexistence line from triple point to critical point, an equilibrium between the liquid

(L) and the vapor (V) phase holds. This can be expressed mathematically by equating the respective

phase diﬀerential Gibbs enthalpies which leads to the classical Clapeyron equation, Eq. (3),

dp

dT=sV−sL

vV−vL

=1

T

hV−hL

vV−vL

.(3)

The last equality holds as Tremains constant during the transition from liquid to vapor. The latent

heat of vaporization is ∆h=hV−hL, the increase in speciﬁc volume and entropy are ∆v=vV−vL

and ∆s=sV−sL, respectively.

Classically, the Clapeyron equation is used up to the critical point. After all, the latent heat of

vaporization vanishes, Eq. (3) loses its signiﬁcance. However, not only does the jump in enthalpy

4

Temperatu re (K )

Cp (J/g*K)

120 130 140 150

5

10

15

20

p / MPa = 3

4

5

6

(a) Nitrogen speciﬁc isobaric heat capacity cp(T) at sub- and

supercritical pressures.

Temperature (K)

Pressure (MPa)

60 80 100 120 140 160

10-2

10-1

100

101

102

Subcritical Vapor Pressure

Supercritical c p, max

Fit cp , max

Tripl e Poin t

Critical Poi nt

(b) Positions of maximum cprelative to coexistence line for

nitrogen.

Figure 2: Construction of heat capacity based Widom line, for the example of nitrogen.

5

(or entropy) vanish - the same is true for the speciﬁc volume. Thus, instead of invalidating the

equation, progressing towards the critical point is mathematically taking the limit of a diﬀerence

fraction, and turning it into a diﬀerential, Eq. (4)

dp

dTsat

=∆s

∆vsat

p,T →pcr,Tcr

−−−−−−−−→

∆s,∆v→0=ds

dvcr

,(4)

where subscript ‘sat’ denotes saturation conditions along the coexistence line. Figure 2(b) shows

that the slope ( d log p/ dT)sat is gradually decreasing in the subcritical region but remains equal

to the slope at the critical point upon entering a supercritical state,

d ln p

dTpb

=d ln p

dTcr

=1

θcr

= const.(5)

Now, combining Eqs. (2), (5), and using the identity ( dln p)/dT= (1/p)( dp/ dT) from calcu-

lus, θpb can be determined as

1

θpb

=1

pcr ds

dvcr

=1

pcr dp

dTcr

.(6)

The equation for the cpbased Widom line then reads

ppb

pcr

= exp Tcr

pcr ds

dvcr Tpb

Tcr

−1 (7)

or

ppb

pcr

= exp Tcr

pcr dp

dTcr Tpb

Tcr

−1.(8)

Note that Eqs. (7) and (8) do not contain any nonphysical ﬁtting parameters and can be evaluated

solely from ﬂuid data. Furthermore, they contain only nondimensional ratios, suggesting a ﬂuid

independence akin to the corresponding states principle.

3.2. Analytical van der Waals Widom Parameter

Using Eq. (8), the coeﬃcient Aintroduced in Eq. (2) can be expressed as

A=Tcr

θpb

=Tcr

pcr dp

dTcr

.(9)

It is now interesting to see whether Amay be evaluated from ﬁrst principles using Eq. (9). The

van der Waals equation of state will serve as an example.

6

As a reference, an AvdW,sat will be determined for Schwabl’s [25] equation for the vapor pressure

curve of a van der Waals ﬂuid

∆pr,sat = 4∆Tr,sat +24

5(−∆Tr,sat)2+O(−∆Tr,sat )5/2.(10)

Parameters ∆φr,sat are deﬁned as ((φ−φcr)/φcr) along the saturation curve. Thus, in the limit

towards the critical point

∆φr,sat

φ→φcr

−−−−→ 0.(11)

Then, higher order terms may be neglected and Eq. (10) reduces to

∆pr,sat,cr = 4∆Tr,sat,cr.(12)

Rearranging and diﬀerentiating, we obtain the reference value of the slope of the vapor pressure

curve at the critical point

dpsat

dTsat vdW,cr

= 4Tcr

pcr

(13)

which corresponds to

AvdW,sat = 4.(14)

Now, AvdW will be determined using Eq. (9). The nondimensional form of the van der Waals

equation can be written

pr=8Tr

3vr−1−3

v2

r

.(15)

The derivative of pressure with respect to temperature is

dp

dT=pcr

Tcr

8

3vr−1.(16)

Then, as vr= 1 at the critical point,

AvdW =Tcr

pcr dp

dTcr

= 4.(17)

Thus, at the critical point, the slope determined from the limit of Clapeyron’s equation introduced

here, Eq. (17), equals the slope of the vapor pressure curve equation, Eq. (14). 1

1Equal fugacities or free Gibbs enthalpies in the respective phases are criteria which determine phase equilibria

in a two phase region. Here, starting with Clapeyron’s equation Eq. (3), a free Gibbs enthalpy equality has been

regarded in the derivation of the vapor pressure curve slope at the critical point. It turned out that the expression

is signiﬁcantly simpliﬁed due to the limiting procedure towards the critical point, Eq. (4). At the critical point only

one phase exists, hence criteria of equal Gibbs enthalpies or fugacities are identically fulﬁlled.

7

3.3. Empirical Widom Parameter

For all practical purposes, van der Waals’ equation of state does not provide suﬃcient accuracy.

Instead, using NIST [16] data for nitrogen, the Widom parameter has been determined to A≈5.6.

Results are shown in Fig. 3. The graph compares Eq. (2) using both the van der Waals coeﬃcient

from Eq. (17) (dashed) and the value from nitrogen data (solid) to the equation suggested by Gorelli

(dash dot), Eq. (1). Overlain are points of maximum heat capacity extracted from the NIST [16]

database for argon, water, nitrogen, and oxygen. Figure 3 shows Eq. (2) in the form of Eq. (8)

to be reasonably accurate in the chosen domain for oxygen and argon in addition to nitrogen.

Water, chosen for its irregular properties, diﬀers stronger. The improved accuracy compared to the

extrapolated Plank-Riedel equation proposed by Gorelli is apparent. Ais a species constant and

has a comparable value for a range of simple ﬂuids.

Figure 3: Comparison of maximal cpdata from NIST with Eq. (2) and Gorelli’s Eq. (1).

3.4. Generalized Isobaric Phase Transitions

Now that an equation tells us where to expect some sort of transition phenomenon, we need

to investigate what to expect when passing through the Widom line. Let T−and T+be the (yet

to be quantiﬁed) start and endpoint of the isobaric transition, such that T−is smaller than the

temperature of maximum cp, and T+is larger. Then, in order to pass isobarically from a liquid-like

8

T

cp

Tpb

T-T+

cp,L

cp,L

cp,pb

cp,iG

Δhst

Δhth

T

cp

Tpb

T-T+

cp,pb

cp,iG

Δhst

Δhth

Figure 4: Finite heat capacity in state transition at supercritical pressures.

to a gas-like state, some amount of energy per unit mass

∆hpb =

T+

Z

T−

cp(T) dT=h(T+)−h(T−) (18)

needs to be added to the ﬂuid. The most important realization is that under supercritical conditions,

added energy is used to both overcome molecular attraction and raise the temperature of the ﬂuid

(Oschwald and Schik [14]). The ﬁrst will be dubbed structural (st) and the latter thermal (th)

contribution. This stands in contrast to subcritical phase changes during which the temperature

remains constant and thus all added energy is used to overcome molecular attraction.

In order to diﬀerentiate the energy needed for structural and thermal changes, both aspects

need to be separated. Figure 4 illustrates the chosen approach. Equation (18) represents the whole

area under the cp(T) curve from T−to T+, the whole amount of energy needed to pass between

both temperatures. The isobaric speciﬁc heat capacity approaches two values: at low temperatures

a liquid heat capacity cp,L, at higher temperatures the ideal gas value cp,iG. While no theoretical

expression is known for cp,L[26], the variation is small and can be neglected compared to cp,max . It

is thus assumed essentially constant for a species over a wide range of pressures. For a calorically

perfect gas, cp,iG =γR/(γ−1), where γis the isentropic exponent and Rthe gas constant. Tpb

is the temperature at which the heat capacity reaches its maximal value cp,pb. Furthermore, let

∆Tpb =T+−T−.

Consider a reference liquid ﬂuid which only raises the temperature upon heat addition. The

energy needed to heat this ﬂuid isobarically from T−to T+is ∆hth =cp,L∆Tpb. The diﬀerence

between the overall change and the thermal contribution is the required excess in energy due to the

structural contribution of pseudoboiling. The structural energy ∆hst is thus ∆hst = ∆hpb −∆hth.

9

Figure 5: Illustration of isobaric supercritical phase change. Determination of interval (T+−T−) for oxygen at

p= 6 MPa using enthalpy asymptotes for the ideal gas hiG and liquid hLreference states.

Now the actual temperature interval has to be determined. We can use the previously used heat

capacities cp,L,cp,iG, and cp,pb to determine characteristic functions in the h/T diagram, Fig. 5.

Then

hiG(T) = γR

γ−1T, (19)

hL(T) = cp,LT+h0,L,(20)

hpb(T) = cp,pb (T−Tpb) + h0,pb .(21)

Per deﬁnition, Eqs. (19) and (20) are independent of pressure; Eq. (21) is valid for the particular

pressure under consideration which determines cp,pb =f(p), h0,pb =f(p), and Tpb =f(p). The

intersections of Eq. (19) and Eq. (21) determine T+, Eq. (20) and Eq. (21) determine T−, valid for

the particular pressure of the regarded process:

T−(p= const) = h0,L−h0,pb +cp,pbTpb

cp,pb −cp,L

,(22)

T+(p= const) = h0,pb −cp,pbTpb

γR

γ−1−cp,pb

.(23)

The required structural enthalpy can now be determined. However, its absolute value is not

necessarily meaningful: Fig. 2(a) shows how the cpdistribution widens and ﬂattens for higher

10

Figure 6: Parameters B1from Eq. (24) as open symbols and B2from Eq. (25) as ﬁlled symbols for Ar, N2, O2, H2O.

pressures. The eﬀect should be notable only if the structural energy is required in a suﬃciently small

temperature interval and if it signiﬁcantly exceeds the thermal energy. A convenient nondimensional

parameter to assess this ratio is thus

B1=∆hst

∆hL

=∆hpb/∆Tpb

cp,L

=∆hpb

∆hL

−1.(24)

The reference factor cp,Lsuggests a simpliﬁed formulation based solely on a ratio of speciﬁc heats.

In order to emphasize the contribution of the state transition over mere heating, the excess speciﬁc

heat cp,pb −cp,Lis used instead of just the maximum speciﬁc heat. The parameter thus reads

B2=cp,pb

cp,L

−1.(25)

Figure 6 shows both parameters for nitrogen, oxygen, water, and argon. The values of the param-

eters are comparable over a range of ﬂuids, especially for the simple ﬂuids nitrogen, oxygen, and

argon. Figure 6 also shows that up to a reduced pressure of 1.5, the structural energy actually

exceeds the thermal energy: i.e. pseudoboiling raises the amount of energy required to heat the

ﬂuid to T+by a factor of two. Even at pr= 3 there is a 10% excess over liquid heating alone.

As expected, the parameter diverges towards the critical point where the heating contribution van-

ishes. Furthermore, the behavior of the approximate B2closely follows the exact B1with an oﬀset,

11

its evaluation however is largely simpliﬁed as transition temperatures need not be known. It thus

appears that B2is better suited for a practical evaluation of the pseudoboiling eﬀect at a given

pressure.

4. Conclusions

Tr

log(pr)

0

1

(2)

(1)

(3)

CP

TP gas-like

liquid-like

pseudoboiling-line

supercritical

subcritical

Figure 7: Structure of the supercritical state space and comparison of subcritical (1) and supercritical isobaric

processes (2),(3).

Pseudoboiling does exist. In this paper, the ﬁrst quantitative analysis has been introduced. This

phase transition between supercritical liquid-like and gas-like states is illustrated in Fig. 7. Pseu-

doboiling in this sense is a continuous, nonlinear, transcritical (i.e. starts at subcritical temperature

and ends at both supercritical temperature and pressure) process resembling subcritical boiling. It

occurs when the Widom- or pseudoboiling-line is crossed. The associated massive reduction in

density and a high speciﬁc heat capacity strongly resemble classical subcritical vaporization, Fig. 7

process (1). At supercritical pressures, the temperature of transition rises with the pressure, i.e.

the transition temperature of Fig. 7 process (3) is higher than in process (2). A simple algebraic

expression with good accuracy has been found for the Widom- or pseudoboiling-line in terms of

reduced pressure and reduced temperature: pr,pb = exp [5.55(Tr−1)]. This equation allows for a

simple determination of the positions of maxima in thermal expansion and heat capacity. Isobaric

pseudoboiling is shown to require energy to both raise temperature (thermal) and overcome molec-

ular attraction (structural). Based on this, two nondimensional parameters are deﬁned which show

that the structural contribution actually exceeds the thermal contribution up to a reduced pressure

12

of 1.5, it becomes negligible above pr≈3. Thus, at supercritical pressures, the critical temperature

loses its signiﬁcance as a transition point and is replaced by the pseudoboiling temperature Tpb .

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