18890 | Phys. Chem. Chem. Phys., 2017, 19, 18890--18901 This journal is© the Owner Societies 2017

Cite this:Phys.Chem.Chem.Phys.,

2017, 19, 18890

Retrograde behavior revisited: implications forconfined fluid phase equilibria in nanopores

Sugata P. Tan * and Mohammad Piri

Many fluid mixtures exhibit retrograde behavior, including those that define natural gases. While the

behavior is well understood for mixtures in bulk, it is not so in nanosize porous space that dominates

shale formations in unconventional reservoirs. The lack of experimental data creates the need for

modeling works to make estimates as good as possible due to immediate needs in gas recovery.

However, such efforts have been straying without firm guidance from systematic studies over what we

have known so far. This article is intended to present the results of such a study that would incite further

investigations in this area of research. Revisiting the retrograde behavior in the bulk is appropriate to

start with, followed by a short review of what we know about fluids confined in nanosize pores. Based

on this information, implications for the behavior of confined mixtures in the retrograde region can be

inferred. The implied features that have been supported by experimental evidence are the locations of

the confined dew point and bubble point at low temperatures, which are both at pressures lower than

their bulk counterparts. Another feature found in this study is completely new, and therefore still open

for further investigation. We reveal that the dew-point and bubble-point curves of confined mixtures

end at moderate pressures on a multiphase curve, beyond which equilibrium occurs among the bulk

and confined phases. The well-known points in the bulk retrograde region, i.e. the critical point and

cricondenbar, are consequently absent in confined mixtures.

1. Introduction

Retrograde condensation in vapor–liquid phase equilibria was firstnoticed by Kuenen in 1892.1 It occurs when a fluid mixturecondenses upon isothermal decompression or isobaric heating,where the opposite, i.e., evaporation, is the more frequently observedprocess. Nevertheless, the retrograde behavior has been discoveredto be very common in fluid mixtures, including oil and natural gases.

In the recovery of natural gases, which starts with fluid at highpressure and high temperature in the reservoir, thegas-like fluid undergoes pressure reduction on its way to theproduction well. On many occasions, the conditions of this processallow the fluid to partly condense into liquid. This retrogradecondensation is the physics responsible for the obstruction of thegas flow to the well, consequently reducing the production.

There have been experimental and theoretical studies as wellas careful reviews on retrograde condensation.2–5 However,comprehensive thermodynamic studies are still required thatcan establish ways to understand more advanced situationssuch as what will happen if the retrograde fluid is entrapped innanosize porous mediums, e.g., unconventional natural gas tobe recovered from shale formations.

It is well known that fluids confined in nanopores havesignificantly different behavior than in bulk.6–9 The ultra-tightconfinement enhances the interactions between the fluids andthe surrounding solid walls, which effectively introduces unusualbehaviors. For confined pure gases, the vapor phase condenses atthe so-called capillary-condensation pressure that is experimentallylower than the bulk saturation pressure, or equivalently condensa-tion temperature higher than the bulk saturation temperature.Moreover, the critical point is lowered both in pressure andtemperature in the nanosize confinement.

Interestingly, despite the long-studied behavior of puregases in nanopores, comparable experimental studies on con-fined fluid mixtures are rare, and studies on their retrogradebehavior are virtually nonexistent. Even for dew points andbubble points, which are important in phase transitions, theexperimental data are scarce for confined fluids. The dataavailable in the literature were measured in the nonretrogradeenvironment using various methods, but none of them wasverified against each other. A differential scanning calorimeter(DSC) and volumetric method were used to measure the bubblepoints of binary mixtures of octane–decane in CPG10 for theformer and methane–octane and methane–decane in SBA11 forthe latter. They both applied the evaporation path, i.e., isobaricheating for the former and isothermal depressurizing for thelatter, but provided opposite conclusions. DSC found that

Department of Petroleum Engineering, University of Wyoming, Laramie, WY 82071,

USA. E-mail: sptan@uwyo.edu

Received 15th April 2017,Accepted 26th June 2017

DOI: 10.1039/c7cp02446k

rsc.li/pccp

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bubble points have lower temperatures in confinement than inbulk, while the volumetric method found the opposite, i.e.,lower pressure than in bulk. The contradicting results may arisefrom factors to be investigated.

Positron annihilation spectroscopy (PAS) has been appliedto measure isobarically, upon cooling, the dew points andbubble points of simple mixtures in nanopores, such as thebinaries argon/nitrogen12 and argon/krypton.13 Both dew pointsand bubble points were measured to be at higher temperaturescompared with their bulk counterparts, which is consistent withthe capillary condensation of pure gases.

The results from separation technology using capillarydistillation14 support the lowering of bubble-point and dew-pointpressures, which in some cases removes the adverse azeotropes.The technology applies porous metal or carbon plates thatevidently show the confinement effects that alter the phasebehavior of fluid mixtures.

For mixtures that exhibit retrograde behavior, no experi-ments have been performed in nanopores in the vicinity of theretrograde region. It is still unknown whether or not the regionshifts to higher temperature and lower pressure in the confine-ment as the nonretrograde fluids do, as described earlier.Setting up the experiments for this purpose may not be trivial,and this motivates researchers to directly resort to modelingand perform calculations on both dew and bubble pointswithout carrying out advanced studies and validation.

So far, there are two different approaches to modeling theretrograde fluids in nanopores, both of which use the cubicequation of state (EOS) coupled with the Young–Laplace equa-tion to account for the confinement effects. The first model stayswith the original critical properties as its EOS parameters, whichmake the mixture’s critical point invariant.15,16 In this model,the dew and bubble pressures are at lower pressures than thebulk in the low temperature range. In contrast, the other modelapplies correlations of experimental data as its EOS parametersto account for the shifts of critical points of the mixturecomponents due to confinement.17 In this model, the dew pointshave pressures higher than that in the bulk, which is contrary tothe results of the first model. Importantly, both models do notreduce to the behavior of a single-component fluid, where thecritical temperature/pressure and saturation pressure shift tolower values in the confinement. For the details of these twomodels, the readers are referred to our recent review article.9

Considering the aforementioned contradictions, both inexperiments and modeling, a systematic approach may be derivedfrom the fundamental thermodynamics in an effort to predict whatmay happen to retrograde fluids in nanosize confinement. In themidst of confusion with the present state of the art, this kind ofeffort is useful at least to provide us with the correct direction totake in the research of nanoconfined fluids.

Such an effort has been made with a mathematical analysisof the shift of the phase envelope due to the capillary pressureintroduced by the confinement. It applies the Young–Laplaceequation coupled with the Soave–Redlich–Kwong (SRK) cubicEOS using the original critical properties as the EOS parametersso that the critical point is the only point that does not shift.16

The analysis suggests that the bubble point and the lowerbranch of the dew point in the retrograde region shift to lowerpressures while the upper branch of the dew point shifts to theopposite direction. However, this analysis did not include theeffects of the interaction between the pore wall and the fluidmolecules, which becomes stronger in smaller pores andcauses the shift of pure-fluid critical properties in nanopores.

This article is intended to improve this type of analysis usingan EOS that is capable of representing the wall–fluid inter-actions. The EOS applies the Perturbed-Chain SAFT coupledwith the Young–Laplace equation with a parameter derivedfrom experimental data to account for the interactions. TheEOS, PC-SAFT/Laplace, has been used to describe various puregases in nanosize pores as well as some binary mixtures ofsimple gases18 and associating molecules with hydrogenbonding.19 Once the parameters of the confined pure compo-nents and the bulk mixture are available, the properties of theconfined mixture can be well predicted. For confined puregases, the EOS is able to reproduce not only the loweringsaturation pressure, but also the shift of the critical point.18

For confined binary mixtures, it also suggests lower pressuresfor dew and bubble points compared with the bulk. Therefore,this EOS may be expected to provide a better description ofconfined fluids in the retrograde region.

Based on the knowledge gathered from previous studies,this article discusses the implied behavior of confined mixturesin the retrograde region. We will start by revisiting the retro-grade behavior in the bulk, and then refresh our currentknowledge of confined fluids. Finally, we will combine thesetwo in trying to describe the confined retrograde fluids, whichis of interest to, e.g., those working on unconventional gasrecovery.

2. Bulk thermodynamics

To describe the retrograde behavior, one may start with theClausius–Clapeyron equation for mixtures, which provides uswith the rate of change of the phase-transition pressure withtemperature. We start with the equation in the bulk:

dPdew

dT¼

PNi¼1

xi �sVi � �sLi� �

PNi¼1

xi �vVi � �vLið Þ(1a)

dPbub

dT¼

PNi¼1

yi �sVi � �sLi� �

PNi¼1

yi �vVi � �vLið Þ(1b)

where s, v, T, P, and N are the molar entropy, molar volume,temperature, pressure, and the number of components, respec-tively. The overbar means the partial molar properties. Thesubscripts ‘‘dew’’ and ‘‘bub’’ refer to the dew point and bubblepoint, while the superscripts V and L refer to vapor and liquid,respectively. The derivation of these equations is presented

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in the Appendix together with the version for the confinedmixture.

The partial molar properties in eqn (1) can be calculatedusing the following basic relationships:

�si ¼ �@mi@T

� �P;x

(2a)

�vi ¼@mi@P

� �T ;x

(2b)

where mi is the chemical potential of the component i in themixture and x is the equilibrium composition, which may be{ yi} for the vapor phase or {xi} for the liquid phase.

The numerator and denominator in eqn (1a) and (1b) are thechange of molar entropy and molar volume when an emergingphase comes to existence at the dew and bubble points,respectively. For example, at the dew point, an emerging liquiddrop with composition {xi} will undergo a change in molarentropy as expressed by the numerator in eqn (1a), while at thebubble point, an emerging bubble with composition {yi} willhave a change in molar volume as expressed in the denominatorof eqn (1b).

On a P–T phase diagram, the pressure may reach stationarypoints if the numerator in eqn (1) is zero, which effectivelyrefers to the maximum dew or bubble pressure that a mixturewith a particular composition may reach. This maximum isknown as the cricondenbar (CB). The same is true if we take thereciprocal of the derivatives in eqn (1), which by the same ideagives the maximum temperature that is known as the cricon-dentherm (CT). At a point known as the critical point (CP) of themixture, both the numerator and the denominator are zero ineqn (1a) and (1b). It borders the dew-point and the bubble-point branches in the phase diagram. Although both thenumerator and the denominator vanish at CP, their quotientis finite, which is the slope of the coexistence curve at CP.

If the changes are represented by Ds and Dv for the molarentropy and molar volume, respectively, then the conditions atCB, CT, and CP are:

CB: Dsdew = 0 or Dsbub = 0 (3a)

CT: Dvdew = 0 or Dvbub = 0 (3b)

CP: Dsdew = Dvdew = Dsbub = Dvbub = 0 (3c)

The subscript ‘‘dew’’ refers to eqn (1a) and ‘‘bub’’ to eqn (1b).Note that CT and CB may either be on the dew-point or thebubble-point branch depending on the mixture chemistry and/orcomposition so that the statement ‘‘or’’ in eqn (3a) and (3b)denotes options whichever applicable. For example, if the mixtureis rich with the more volatile component, CT and CB are both onthe dew-point branch.

By observing the P–T phase diagrams shown in Fig. 1, theslope of the coexistence curve along the phase envelope wasfound to be positive except in the retrograde region, whichmay be defined as being in the range between CB and CT.In the case of CP lying in this region as shown in Fig. 1(a), thedew-point curve is divided by CT into two parts, the upper dewpoints (udp) and the lower dew points (ldp), while CB splitsthe bubble-point curve to the upper bubble points (ubp) and thelower bubble points (lbp). In cases where CP lies outside theretrograde region, CB borders udp1 and udp2 in Fig. 1(b) andCT borders ubp1 and ubp2 in Fig. 1(c). The well-known retro-grade condensation occurs when the pressure is decreasedisothermally crossing udp or udp1 (broken arrows in Fig. 1aand b). If CP is at a lower temperature than CB, as in Fig. 1(b),there is also an isobaric retrograde condensation when thetemperature is increased isobarically crossing udp2. Retrogradeof the second kind3 (dotted arrows) may also occur if thebubble-point curve is encountered upon increasing the pressureor decreasing the temperature.

Fig. 1 Retrograde condensation on P–T phase diagram at constant composition: (a) CB o CP o CT; (b) CP at a lower temperature than CB; (c) CP at alower pressure than CT. The common retrograde condensation (the first kind) occurs as shown by the broken arrows; the second kind occurs as shownby the dotted arrows.

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The consequence of eqn (3a)–(3c) is that the signs of Ds andDv may have to change when one goes across CB, CT, or CP insuch a way that the slope is negative in the retrograde regionand positive elsewhere. The sign of Ds changes when one goesacross CB and CP, while the sign of Dv changes upon crossingCT and CP. These changes of sign on the phase envelopes inFig. 1 are shown in Table 1 for evaporation from high pressureor low temperature; condensation processes have the oppositesigns. In an isothermal retrograde condensation, for example,we may deduce that the first liquid drop emerging in thecondensation undergoes a decrease in entropy but an increasein volume from the vapor to the liquid phase, which is counter-intuitive because the vapor phase usually has a larger molarvolume. In other words, the emerging liquid drop has a smallerdensity than it did in the vapor phase prior to condensation.

The signs in Table 1 are verified using PC-SAFT EOS20 withthe calculated properties for a typical binary mixture in Fig. 2.The example used for Fig. 2 is the binary CO2/n-pentane with79.4 mole% of CO2, of which the P–T diagram belongs to thetype in Fig. 1(b). Note that the slope is negative only along udp1,zero at CB, and infinite at CT. It is finite positive at CP eventhough both the change of molar entropy and the change ofmolar volume are zero, as shown in Fig. 3. The changes arenegative along udp2 for the entropy and along the whole upperdew points for the volume.

3. Confined pure fluids

At capillary condensation, the phase transition takes place froma vapor-like phase a to a liquid-like phase l; both are inequilibrium with the bulk vapor V that is separated from l byan interface. Even though the bulk vapor V does not undergophase transition, its pressure is used to describe the equilibriumfor comparison purposes. For the details of phases involved incapillary condensation, the readers are referred to our reviewarticle (in particular Fig. 1 and its explanation in the text; a and l

were referred to as A and L, respectively, in the article).9

The confined-fluid version of the Clausius–Clapeyron equationanalogous to eqn (1) can be derived with a similar approach tothat for the bulk while recognizing the different pressures acrossthe equilibrium phase boundary:

PV � P‘ ¼ gV‘2

r(4)

In eqn (4), known as the Young–Laplace equation for sphericalinterface between phases V and l, gVl is the surface tensionbetween those phases, while r is the effective pore radius that isusually different from the physically measured value. Completely

Table 1 Signs of the slope dP/dT, Ds, and Dv along the phase envelopeson P–T phase diagrams in Fig. 1 for evaporation from high P or low T

Region dP/dT Ds Dv

Fig. 1(a)ldp + + +udp � + �ubp � � +lbp + + +

Fig. 1(b)ldp + + +udp1 � + �udp2 + � �lbp + + +

Fig. 1(c)ldp + + +ubp2 + � �ubp1 � � +lbp + + +

Fig. 2 (a) P–T phase diagram of binary CO2/n-pentane at 79.4% of CO2.The coexistence curve with CP (filled star), CB (star), and CT (rectangle) iscalculated using PC-SAFT EOS. (b) The corresponding slope.

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wet cylindrical pores have hemispherical shape of phase inter-face at condensation. For other geometries, r is the radius ofequivalent spherical surface. The pressure difference in eqn (4) isalso known as the capillary pressure.

The derivation of the Clausius–Clapeyron equation for con-fined pure gases was published in our previous article.21 Theeffects of confinement to the bulk equation are represented bythe last term in the numerator due to the pressure difference:

dPV

dT

� �pure

¼sV � s‘ � v‘

d

dTgV‘

2

r

� �vV � v‘

(5)

As described in our previous article,21 the numerator in eqn (5)is related to the heat of capillary condensation (Lcc), which isexperimentally and theoretically larger than the heat of con-densation of the bulk:

Lcc

T¼ sV � s‘ � v‘

d

dTgV‘

2

r

� �(6)

Upon experimental observation on the behavior of confinedpure fluids, it is evident that the slope eqn (5) is smaller thanthe corresponding value in the bulk all the way to the porecritical point (TCp), beyond which no capillary condensationcan occur in the confinement. Therefore, the denominator ineqn (5) must be even larger than its bulk counterpart. This ispossible if the molar volume of the vapor phase is larger inequilibria with the confined phase l, which is true as thecondensation pressure is lower than the bulk saturated vaporpressure at the same temperature. The behavior of the numeratorand denominator, as well as the slope dP/dT, is shown in Fig. 4 fora typical confined pure fluid (argon in MCM-41 is taken as anexample here) and compared with the bulk.

Upon comparison with the bulk, there are trends in Fig. 4that are worth mentioning. First, the numerator of eqn (5)would never be zero if it were allowed to go beyond TCp. Second,although the denominator decreases with temperature at ahigher rate than that in the bulk, it is difficult to determinewhether it would reach zero at lower temperature if it wereallowed to go beyond TCp. Third, the slope dP/dT of theconfined fluid would eventually be larger than the bulk if thetemperature were allowed to go beyond TCp. This leads to acrossing point between the condensation curve of the confinedfluid and that of the bulk. All pure fluids in our databasebehave similarly in confinement.

4. Confined fluid mixtures

For confined fluid mixtures, we first need to evaluate theconcept of the dew point and bubble point for phases in theconfinement. As in the bulk, if the measurement is madethrough a condensation path, the onset of condensation isencountered at the dew point and the condensation completesat the bubble point.

The dew point is where the first drop of condensed phase l

appears in the confinement, but note that the drop originatesfrom the confined vapor-like phase a, not from the bulkvapor V. However, as these three phases are in equilibrium atthe dew point, the confined dew-point curve may be defined interms of the composition of the bulk V, rather than theunknown composition of a. This way, the curve is readilycompared in the P–T phase diagram to the correspondingdew points of the bulk.

In contrast, the situation is not that simple for bubble points.We can immediately take the analogy by assigning the confinedbubble point as the condition where the last bubble of phase adisappears in a homogeneous phase l. This means that thepressure and composition on phase diagrams should belongto l, both of which are usually not experimentally measured.

Fig. 3 The change of molar properties along the coexistence curve of thebinary CO2/n-pentane in Fig. 2: (a) entropy, with an inset for zooming in onthe negative region along udp2; (b) volume.

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In experiments that use the evaporation path, the bubble point ismeasured starting from the bulk liquid L with known pressureand composition, analogous to the measurement of the dewpoint that starts from the bulk vapor V with known pressure andcomposition. In this method, it is assumed that the confinedphase l is in equilibrium with L when the experiment starts.However, for most mixtures, the confined bubble point isencountered after the bulk bubble point, which means that thefirst bubble a in the confinement appears in the presence ofboth bulk liquid L and vapor V outside the pores so thataltogether we have four phases in equilibrium.

This complex condition at confined bubble points bringsconsequences to phase-equilibrium calculations. The conventionalmethod of the calculations is to apply the capillary pressure eqn (4)simultaneously with the equality of chemical potentials in twophases, the bulk V and the confined l:

mVi = mli i = 1. . .N (7)

Therefore, there is no way to account for the bulk L. For mostcases, where the bulk V is also present, the bulk pressure is thesame for L and V so that the confined bubble pressure may stillbe presented in terms of this bulk pressure on P–T diagrams,but at a composition of the phase l. We name this conventionalmethod in the presence of bulk V as the Vl method and use themethod in this article. Obviously, it is inapplicable if theconfined bubble point appears in the absence of the bulk V,where eqn (4) is no longer valid as the bulk pressure nowbelongs to L and the surface tension has to be between L and l.

Up to this point, it is also important to note that eventhough the confined vapor-like phase a does not explicitlyappear in eqn (7), it does exist in equilibrium with bothcondensed-phase l and bulk V. In PC-SAFT/Laplace EOS,18

the phase a is modeled to have a density identified as theintermediate root of the van-der-Waals loop in density calculationsthat lies in the metastable region. This way, eqn (7) actuallydescribes three-phase equilibria a, l, and V. However, as the

phase a has little practical use, it is rarely discussed or explicitlycalculated.

With the concept of the dew point and bubble point describedabove, we may now start with simple binary systems, for which theexperimental data are available in the literature. Direct observationof the data provides us with some important features that can laterhelp us describe the systems that exhibit retrograde behavior.

Fig. 5(a) shows the dew-point data of the binary N2/Ar fromisobaric measurements using PAS through condensation upondecreasing temperature.12 It is obvious that the confinedmixture condenses at a higher temperature than the bulk asthe confined pure fluids do. The same behavior is observed inFig. 5(b) for the binary Kr/Ar measured using the samemethod.13 However, upon decreasing temperature, the dewpoint at the onset of the capillary condensation is hardlydetectable,13 perhaps due to the weak interaction between thecomponent gases, which results in high uncertainty for the data(not shown in the figure). On the other hand, the bubble pointsat the completion of condensation were clearly detected to be athigher temperatures than in the bulk. The shift of both dewand bubble points from PAS measurements strongly suggeststhat the phase transition of confined mixtures inherits thebehavior of confined pure fluids, where the transition occursisobarically at a higher temperature or isothermally at a lowerpressure than in the bulk.

The curves in Fig. 5 were calculated using the PC-SAFT/Laplace EOS.18 Note that even if the immiscibility regions of thebulk and confined fluids overlap with each other, as in Fig. 5(b),where the most part of the confined bubble curve is in the bulkimmiscibility region, the Vl method is applicable for both thedew and bubble points. However, it requires the confined phasel always in contact with the bulk V along the bubble-point curvein order to validly use the Vl method; the surface tension has tobe between V and l, not L and l.

Nevertheless, the Vl method gives the composition of confinedphase l, not L. Therefore, the agreement between calculations and

Fig. 4 Behavior of properties with temperature for pure argon in 2.2 nm MCM-41: (a) numerator in eqn (5), which is eqn (6); (b) denominator in eqn (5);(c) the slope of the capillary-condensation pressure, eqn (5). The corresponding bulk properties are also shown for comparison. Other properties,including the heat of capillary condensation, are described in our previous article.21

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the PAS experimental data suggests that the two phases havesimilar compositions. A diffusion of some sort between these twophases was indeed detected in a PAS experiment if they wereallowed to be in full contact with each other.12 A similar situationalso occurred in a series of research on separation using capillarydistillation,14 the experimental data from which can be matched bythe EOS calculations using the Vl method.19

With the above cases in mind, we may now proceed to thethermodynamics. It can be shown that the confinement effect

in eqn (5) for pure gases is simply transferred to mixtures asfollows (see Appendix):

dPVdew

dT¼

PNi¼1

xi �sVi � �s‘i� �

� v‘d

dTgV‘

2

r

� �PNi¼1

xi �vVi � �v‘i� � (8a)

dPVbub

dT¼

PNi¼1

yi �sVi � �s‘i� �

� �v‘id

dTgV‘

2

r

� �� �PNi¼1

yi �vVi � �v‘i� � (8b)

The composition {xi} in eqn (8a) belongs to the confinedcondensed phase l. The molar properties in phases V and l

may be calculated using eqn (2) at their respective pressure andcomposition.

Based on eqn (8), one may expect to find the locations of CBand CT for confined retrograde mixtures using equationssimilar to eqn (3a) and (3b), respectively:

CB:

Dsdew0 � v‘

d

dTgV‘

2

r

� �¼ 0 or

Dsbub0 �Pi

yi�v‘i

d

dTgV‘

2

r

� �¼ 0

(9a)

CT: Dvdew0 = 0 or Dvbub

0 = 0 (9b)

The prime signs are to remind us that the primed propertiesbelong to two equilibrium phases, V and l, that are underdifferent pressures. Therefore, CB and CT for confined fluids ineqn (9) also depend on the behavior of the capillary pressure,which decreases with temperature. We will show later in thearticle that the validity of these equations is challenged.

Sandoval et al.16 derived simple equations that may be usedto predict the direction in which the phase envelope shiftswhen capillary pressure is introduced to the fluid mixture. Inour notation, their equations are rewritten as:

eBP ¼ �vV

Dvbub0 gV‘

2

r

� �; eBT ¼

vV

Dsbub0 gV‘

2

r

� �(10a)

eDP ¼ �v‘

Dvdew0 gV‘

2

r

� �; eDT ¼

v‘

Dsdew0 gV‘

2

r

� �(10b)

where the subscripts BP, DP, BT, and DT represent bubblepressure, dew pressure, bubble temperature, and dew tempera-ture, respectively. The shifts (e) are positive for increases andnegative for decreases relative to the bulk phase envelope. Withwetting fluids in the confinement, the signs of the shift must bedetermined solely by those of the denominators, which may beapproximated using Table 1 as long as the critical point, whichborders the dew curve from the bubble curve, is assumed thesame for confined and bulk fluids. As mentioned before, thisassumption fails to recognize the shift of critical point forconfined pure gases.

Using the signs of property changes in Table 1, Sandoval’sapproach immediately gives us the generic shifts of the phase

Fig. 5 Binary systems in Vycor glass: (a) N2/Ar – dew points; (b) Kr/Ar –bubble points. The experimental data of confined fluids were measuredusing PAS;12,13 the curves are calculated using PC-SAFT20 or PC-SAFT/Laplace.18 The data of bulk N2/Ar are taken from the literature, but at aslightly higher pressure of 3.1 bar.22 Note that Ar has a higher boiling/condensation temperature in (a) but lower in (b) than the secondcomponent.

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envelope as shown in Fig. 6. While CP is invariant by assump-tion, there is no requirement on the invariance of CB and CT.Consequently, CP is the point at which the confined phaseenvelope crosses over that of the bulk while the lbp and ldp ofconfined fluids both shift to lower pressures. For cases given inFig. 1 and 6, CB for confined mixtures is at higher pressurethan the bulk only for case (b), while confined CT is at lowertemperature than the bulk only for case (c).

As critical points have been found to shift for confined puregases, it appears unreasonable for the critical point of the confinedmixture to stay the same as that of the bulk. To account for thepresence of pore critical points, the PC-SAFT/Laplace EOS includesthe fluid–pore wall interaction by introducing an empirical para-meter l to the capillary-pressure formulation eqn (4):18

PV � P‘ ¼ 2gV‘

rp 1� l T ; rp� �� � (11)

where rp is the physical radius of the pores. The parameter l isderived from experimental capillary-condensation data of pure gas.In addition to the fluid species, it also depends on temperature,pore size, and pore chemistry. The following simple-average mixingrule proved effective for confined mixtures:

lmix T ; rp� �

¼X

x‘ili T ; rp� �

(12)

It has been applied to calculate the binary dew and bubble pointsin Vycor glass18 in Fig. 5 and azeotropic binary systems in porousmetal and carbon plates.19

While the EOS works well with mixtures at conditions belowthe pore critical points of the constituting pure fluids, itsapplication under other conditions such as that in the retro-grade region is never tested. In the former conditions, once theparameter li is derived from pure-component capillary con-densation data, the properties of the mixture are predictable.However, in the latter condition, the behavior of li beyond thepore critical point is unknown, so that a prediction in theretrograde region is subject to extrapolation deficiency.

If we may assume that the trends of pure components inFig. 4 are transferrable to mixtures, then some behavior may bededuced. Eqn (9a) would never be satisfied because the entity,which is the numerator in eqn (8) would never be zero, as isshown for confined pure fluids in Fig. 4(a). In other words,confined mixtures do not have CB; the phase envelope is openon the high-pressure portion. This, in turn, leads to theabsence of CP.

5. Discussion

To illustrate the whole situation described above, at least fornow in the absence of experimental data for confined mixtures,we will make our evaluation using the PC-SAFT/Laplace EOS.The binary CO2/n-pentane confined in 2.2 nm MCM-41 nano-pores is taken as an example. The experimental data for thepure components are in our database (CO2 in 2.2 nm pores;23

n-pentane in 2.285 nm pores24). The effect of the different poresizes is assumed to be negligible to the phase behavior of thebinary mixture; this assumption may not be valid quantitativelybut justifiable for qualitative studies in this article. The com-position is chosen to be 79.4% of CO2 for comparison with thebulk in Fig. 2 that has experimental data in the literature.25

A similar situation is observed for other compositions.First, we calculate the confined-phase envelope with l = 0,

which makes the results equivalent to those by Sandoval et al.16

who used the SRK cubic EOS in place of PC-SAFT. As seen inFig. 7, without l the confined envelope follows the genericbehavior in Fig. 6(b) with invariant CP. It is worth noting thatthe experimental data of pure-fluid capillary condensationcannot be reproduced without l.21 On the other hand, if weapply finite values of parameter l, which were correlated frompure-gas experimental data,21 a new situation is observed.

Mimicking the behavior of the bulk, the resulting confinedbubble and dew curves are sandwiched by the capillary-condensation curves of the pure components. Both curves are

Fig. 6 Retrograde condensation of confined mixtures (broken curves) on the P–T phase diagram at constant composition according to Sandoval’sapproach16 compared with that of the bulk (solid curves) for the cases described in Fig. 1.

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at pressures below the corresponding bulk curves and are stilleven lower than those with l = 0. The confined dew curvereaches the maximum temperature at lower pressure andtemperature than that of the bulk CT.

As is further shown in Fig. 7, confined bubble and dewcurves end at some moderate pressures when they cross the

bulk dew-point curve, thus showing no CB, as expected from theprevious lead that eqn (9a) would never be satisfied. This is a newphenomenon that has no analog in the bulk. At the end point ofthe dew-point curve, the confined dew point is the same as its bulkcounterpart, which indicates the emergence of the bulk liquid L.Beyond this point, there is multiple phase equilibrium that involvesthe confined phases and bulk phases. At this time, the appliedconventional Vl method, which does not include the bulk liquid L,cannot account for this multiphase equilibrium.

Definitely, these end points also occur on the correspondingpressure–composition (P–x) or temperature–composition (T–x)phase diagrams; P–x diagrams at three temperatures are shownin Fig. 8. The dew curve and the bubble curve do not meet at theplait point (the maximum pressure in this case) as they do inthe bulk phase, but instead they end when the dew curves crossthose of the bulk, where the first liquid drops emerge in thebulk. This additional phase L reduces the degrees of freedom ofthe phase equilibrium so that it introduces a fixed multiphasepressure at a fixed temperature for binary mixtures shown ashorizontal lines in Fig. 8. The multiphase pressure varies withtemperature and forms a curve connecting the ends of thebubble-point curve and dew-point curve at a fixed compositionon the P–T diagram such as that in Fig. 7; we name this curvethe multiphase curve. As described before, what happens atpressures higher than this curve is beyond the applicability of theconventional Vl method. However, the presence of the multiphasepressure and multiphase curve rules out the existence of CB and CPfor confined mixtures.

As a matter of fact, the confined bubble points calculatedusing the Vl method are located inside the bulk phase envelope,which means that they occur in the presence of the bulk vapor.

Fig. 7 Binary CO2/n-pentane at 79.4% CO2 both in the bulk and nano-pores. Figure (b) is figure (a) with the pressure in logarithmic scale. Theexperimental data: NIST data for bulk pure fluids;27 triangles are the bulkbubble and dew points;25 crosses are the loci of the mixture criticalpoint;26 circles are the confined pure fluids, CO2 in 2.2 nm MCM-4123

and n-pentane in 2.285 nm MCM-41.24 PC-SAFT/Laplace with correlated lwas used to calculate the capillary condensation of confined pure fluids(dashed curves) and to predict the dew and bubble curves of confinedmixture (solid curves); the multiphase curve is shown as the dotted curve.The case with l = 0 is also shown as short-dashed curves.

Fig. 8 P–x diagrams of bulk and confined CO2/n-pentane at threetemperatures. Calculations are shown as dash-dot curves for the bulk(using PC-SAFT) and solid curves for the confined mixtures (using PC-SAFT/Laplace with correlated l). The experimental data for the bulk areincluded: (a) ref. 28; (b) ref. 29; (c) ref. 30.

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This is also a multiphase situation, which may invalidate calcula-tions using the conventional Vl method. For this reason, we willonly discuss the dew points in the rest of this section.

Along the confined dew points, we can calculate the changein molar entropy and molar volume when the first drop ofphase l comes into existence. These changes are shown inFig. 9(a) and (b), respectively. The cases with l = 0 and the bulkare added for comparison. At the same temperature, the con-fined values are higher than in the bulk except when they areclose to the end point of the dew curve.

There is something in Fig. 9(b) and the inset that deservesour attention. It turns out that the change in molar volume inconfinement is not zero at CT regardless of which l is used,which immediately challenges eqn 9(b) and makes it differentfrom the bulk. Further investigation points out that the mixturesurface tension has a slope that changes sign at CT (as shown inFig. 9(c)) from negative infinity to positive infinity with dew-pointpressure, including in the case of bulk surface tension. Thisslope change directly affects the numerator in eqn 8(a), whichsubsequently changes the sign of the dew-point slope at CT.

The numerator of eqn 8(a) is in fact the mixture version ofthe heat of condensation, the pure-fluid version of which iseqn (6). At confined dew points along the ldp, this is the heatreleased upon the formation of the first drop of phase l. Atconfined dew points along the udp, it is the heat absorbed uponthe emergence of the l drop. For the bulk, where it is Dv thatundergoes the change of sign across CT as shown in Table 1,the heat of condensation, which is equal to Ds, stays finite atCT. However, in the case of confined fluids that involve infinity, itwould not make sense to have infinite heat of condensation at CT.

Nevertheless, this situation has nothing to do with the Vlmethod in use here and elsewhere to calculate the confined-phase equilibrium. As shown in Fig. 9(c) for the mixture surfacetension, which is commonly calculated using the well-knownparachor method,16,18 the slope of surface tension changes signat CT even for the bulk. Therefore, the infinite-heat problem isthe reminiscence of this parachor method, which has never

been experimentally validated in the retrograde region. Anycorrection from future research to this parachor method maychange the story described here.

6. Conclusions and remarks

Our study was done in an effort to understand the behavior ofconfined mixtures in high-temperature and high-pressureregions in current situations where experimental data areabsent. Despite the ongoing investigation on the details, somemain features emerged from this study to provide feasibledirections in this area of research.

Applying the PC-SAFT/Laplace EOS and Vl method for thephase equilibrium, we calculated the confined dew and bubblecurves, both of which are located below the corresponding bulkvalues, and are therefore consistent with many of the previousstudies. If the pure-gas values of parameter l are set to zero,which also means that the EOS capability to represent the shiftof critical properties is disabled, the resulting phase envelopebehaves in a similar way to that calculated using other EOS that alsoapply the Vl method. However, if the correlated values of parameterl are used, these confined curves end at moderate pressures on themultiphase curve, i.e., when the confined dew curve crosses thecorresponding bulk curve. Along this multiphase curve, phaseequilibrium occurs simultaneously between the bulk phases andthe confined phases in the nanopores. Due to the presence of thiscurve, CB and CP are absent in confined mixtures.

Some issues have to be carefully noted. As pointed outearlier, comparison with the bulk is straightforward only forthe dew-point curve on P–T diagrams, where the referencepressure is that of the bulk vapor V even though the phasetransition in the confinement is actually from the vapor-like ato the liquid-like l. The composition can also be fixed constantin terms of that of the bulk V. The latter condition is not met,however, for experimental data measured using a static (batch)setup, unless the bulk reservoir is much larger than the volume

Fig. 9 Calculated curves for (a) Ds, (b) Dv, and (c) surface tension along the dew points using PC-SAFT/Laplace EOS with the correlated l and l = 0. Thecorresponding bulk curves is added for comparison. The mixture surface tension was calculated using the well-known parachor method.

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in the porous medium. Flow-through experiments may be abetter choice to keep the bulk-vapor composition constant.31 Acritical problem of experiments in dew-point measurements maybe due to the fact that the amount of condensed phase at dewpoints is too small to detect, including if gravimetric methodsare used (this even happens in measuring the bulk dew points32).

On the other hand, the confined bubble point may not becomparable to its bulk counterpart. First, it is experimentallyimpossible to keep the composition of the phase l constantalong the curve similar to that for the bulk L on a P–T phasediagram. Second, the curve belongs to the confined phase l, thepressure of which is different from that of the bulk and neverdirectly measured. The plot using the bulk pressure on P–Tdiagrams does not reflect this difference. Furthermore, it takesplace at pressures below the bulk bubble points, where the bulkvapor and liquid are in equilibrium in addition to that of theconfined phases, the situation of which may invalidate calcula-tions using the conventional Vl method. In this case, the bulkphase equilibrium has to be accounted for simultaneously witheqn (7). This will be a subject for our subsequent publication.

Another crucial issue that arises is how to calculate themixture surface tension; the well-known parachor methodseems to give an unreasonable infinite heat of condensationat CT. In summary, although the qualitative description in thiswork is thermodynamically consistent with the state-of-the-artabout the confined fluids and the bulk retrograde behavior, it isundoubtedly subject to experimental verification.

Appendix1. Clausius–Clapeyron equation for mixtures in confinement

Along the coexistence curve where the second phase emerges,any change in chemical potential in a phase must be compen-sated for by a corresponding change in the other phase. For acomponent i in the mixture:

dmVi (PV,T,yj) = dmli (Pl,T,xlj ), j = 1. . . N � 1 (A1)

Note that the compositions are subject to consistency equa-tions

Pyj =

Pxj = 1. N is the number of components in the

mixture. Applying the relations in eqn (2):

��sVi dT þ �vVi dPV þ

XN�1j¼1

@mVi@yj

� �T ;PV;yk

dyj

¼ � �s‘idT þ �v‘idP‘ þ

XN�1j¼1

@m‘i@xj

� �T ;P‘;xk

dxj

(A2)

The subscript k runs except for k = j. After implementing eqn (4)for the pressure difference and rearrangement:

�vVi � �v‘i� �

dPV ¼ �sVi � �s‘i� �

dT � �v‘i gV‘2

r

� �

þXN�1j¼1

@m‘i@xj

� �T ;P‘;xk

dxj �@mVi@yj

� �T ;PV ;yk

dyj

" #

(A3)

For dew points where yj is fixed on a P–T diagram, the secondterm in the brackets is zero. Multiplying the equation with xi

and summing up all of N equations:Xx‘i �vVi � �v‘i� �

dPVdew ¼

Xx‘i �sVi � �s‘i� �

dT

�X

x‘i �v‘i gV‘

2

r

� � (A4)

after taking advantage of the Gibbs–Duhem equation at con-stant T and P for the first term in the brackets of eqn (A3):

XNi¼1

x‘idm‘i ¼ 0 (A5)

Rearrangement of eqn (A4) immediately gives eqn (8a). For thebulk, in which the pressure difference vanishes, it giveseqn (1a). Similarly, it is straightforward to derive the equationfor bubble points, eqn (8b), from eqn (A3). With xj fixed, wemultiply it by yi and then sum up all of the N equations to arriveat an equation similar to eqn (A4). The Gibbs–Duhem equationused in this case is:

XNi¼1

yidmVi ¼ 0 (A6)

In the bulk where there is no pressure difference, this willeventually reduce to eqn (1b).

2. EOS parameters

The parameter l as a function of T was correlated over thevalues derived from the experimental capillary-condensa-tion data: CO2 in 2.2 nm MCM-41 and n-pentane in 2.285 nmMCM-41.21

CO2: l = �0.415885 + 0.003905T + 2.46754 � 10�6T2

nC5H12: l = 0.051750 + 0.001014T + 2.140 � 10�6T2

The parachors for the surface-tension calculations are avail-able in our previous paper.21 The binary parameter of themixture surface tension is derived from the experimental dataat 313 K.33 However, because the temperature-dependent dataof the mixture surface tension are not available for the binaryCO2/n-pentane, the behavior of the binary parameter withtemperature is approximated to proportionally mimic that ofCO2/n-butane, the data of which are available.34 This binaryparameter is:

c12 = 0.142218 � 3.393534 � 10�4T

For the bulk-phase calculations, the pure-component para-meters are taken from our previous article,18 while the binaryinteraction parameter is derived from the abundant binaryexperimental data in the literature:

k12 = 0.1767 � 1.502 � 10�4T

In PC-SAFT/Laplace EOS, this binary parameter is also used forthe confined mixture.

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Acknowledgements

We gratefully acknowledge financial support from the HessCorporation, Saudi Aramco, and the College of Engineering &Applied Science and the School of Energy Resources at theUniversity of Wyoming.

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