Crossing the Widom-line – Supercritical pseudo-boiling

Abstract

Recent publications in the open literature have shown that supercritical fluid states are not homogeneously 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 Clapeyron 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.

Full text

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 ATpb
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
dTsat
=∆s
∆vsat
p,T →pcr,Tcr
−−−−−−−−→
∆s,∆v→0=ds
dvcr
,(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
dTpb
=d ln p
dTcr
=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
dvcr
=1
pcr dp
dTcr
.(6)
The equation for the cpbased Widom line then reads
ppb
pcr
= exp Tcr
pcr ds
dvcr Tpb
Tcr
−1 (7)
or
ppb
pcr
= exp Tcr
pcr dp
dTcr 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
dTcr
.(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
dTcr
= 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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