Phenomenological Study of Confined Criticality: Insights from theCapillary Condensation of Propane, n‑Butane, and n‑Pentane inNanoporesElizabeth Barsotti,*,† Sugata P. Tan,† Mohammad Piri,† and Jin-Hong Chen‡

†Department of Petroleum Engineering, University of Wyoming, Laramie, Wyoming 82071, United States‡Aramco Services: Aramco Research Center − Houston, Houston, Texas 77084, United States

*S Supporting Information

ABSTRACT: We use the comparison of experimentally measured isotherms forpropane, n-butane, and n-pentane in 2.90, 4.19, and 8.08 nm MCM-41 to show that thecurrent model for the progression of capillary condensation may not hold true for chainmolecules, such as normal alkanes. Until now, the capillary condensation of gases inunconnected, uniformly sized and shaped nanopores has been shown to progress in twodistinct stages before ending in supercriticality of the confined fluid. First, at relativelylow temperatures in isothermal measurements, the phase change is accompanied byhysteresis of adsorption and desorption. Second, as temperature increases, the hysteresiscritical temperature is surpassed, and the phase change occurs reversibly. Althoughpropane followed this progression, we observed a new progression for n-butane and n-pentane, in which hysteresis continues into the supercritical region of the confined fluid.We attribute this behavior to the molecular chain lengths of the adsorbates. Throughfurther comparison of the adsorption, desorption, and critical properties of theadsorbates, we discovered new pressure phenomena of the confined supercritical fluids.

1. INTRODUCTION

Confinement in nanopores is widely known to affect the phasebehavior of fluids.1 For example, confinement has been shown tosuppress the gas-to-liquid phase transition. Studies of thisconfinement-induced phase change, called capillary condensation,have led to subsequent investigations into the criticality of theconfined fluid and the dependency of capillary condensation onpore size. Regarding the former and assuming isothermalmeasurements, two critical temperatures have been attributedto nanoconfined fluids: the hysteresis critical temperature, Tch,and the pore critical temperature, Tcp.

2−4

In relatively well-characterized adsorbents with unconnectedand uniformly sized nanopores, such as MCM-41, isothermsmeasured at low temperatures commonly display hysteresis overthe capillary condensation phase transition (i.e., the phasechange occurs at a lower pressure during desorption thanadsorption). Until now, for simple gases, the width of thehysteresis loop was always found to decrease with increasingtemperature until coincidence of adsorption and desorption atTch.

4 Thus, Tch is the temperature above which no hysteresis isobserved.1,4 Hysteresis has been attributed to different porefilling and emptying mechanisms due to pore wall roughness andphysical processes in metastable states (e.g., cavitation) overdifferent temperature ranges.5

Previous studies have indicated that further increases intemperature above Tch lead to observations of the pore criticaltemperature, Tcp, which is analogous to the bulk criticaltemperature, Tc, although it has been experimentally shown to

occur below Tc.2,6 Whereas two phases, i.e., a vapor-like and a

liquid-like phase, may exist below Tcp, only a supercritical-likephase resides within the nanopores above Tcp, where observedincreases in the density of the confined fluid with pressure aretermed continuous pore f illing.1,3

Experimentally, Thommes and Findenegg derived the porecritical temperatures of sulfur hexafluoride in controlled poreglass from isochores by initially estimating the critical temper-atures from the experimentally measured densities of theconfined fluids and then from coexistence curves of the confinedfluid. The results from both methods were in agreement.6 Morerecently, using experimental isotherms for argon,2,7,8 carbondioxide,2,7,8 nitrogen,7,8 ethylene,7,8 and oxygen2,7,8 in MCM-41,Morishige et al. defined Tcp as the discontinuity in a plot of theslopes of the capillary condensation phase transitions versus thetemperatures at which the isotherms were measured.2,7,8 Both ofthe studies2,6−8 show the pore critical temperature to decreasewith decreasing pore size; however, the rate with which Tcp

decreases with decreases in pore size was observed to depend onthe adsorbate−adsorbent pairings. For example, Morishige et al.experimentally found nitrogen, oxygen, ethylene, carbon dioxide,xenon, and argon to follow a linear trend that passed through theorigin in a plot of (Tc−Tcp)/Tc versus σ/rp, which can be writtenas the proportionality:9

Received: January 14, 2018Revised: February 28, 2018Published: April 3, 2018

Article

pubs.acs.org/LangmuirCite This: Langmuir 2018, 34, 4473−4483

© 2018 American Chemical Society 4473 DOI: 10.1021/acs.langmuir.8b00125Langmuir 2018, 34, 4473−4483

σ− ∼T T T r( )/ /c cp c p (1)

where σ is the Lennard-Jones size parameter and rp is the radiusof the pore.9−11

In the measurements of adsorption isotherms, there are twodegrees of freedom associated with the phase transition ofconfined pure fluid−temperature and pore size−which meansthat if both are fixed, then the capillary condensation pressure iswell determined. At the critical point, the degrees of freedomreduce to one (cf. the bulk critical point has no degree offreedom; it is fixed for an individual substance). Thus, if in theprevious discussion of criticality, temperature, rather than poresize, is held constant, there will also be a critical pore size, dc,below which supercriticality of fluid also occurs.10,12 Therefore,supercriticality in confinement can occur either below the criticalpore size at a fixed temperature or above the critical temperatureat a fixed pore size.In spite of the burgeoning importance of nanopores to fields

including biology, medicine, electrical engineering, fuel storage,and petroleum engineering,13−18 little beyond the informationgathered from the few studies mentioned previously is knownabout the interplay of temperature, pore size, and criticality.1 Inthe course of our research on the confined-phase behavior of n-alkanes, which are themain components of natural gas and oil, wepresent here a study on the confined phase behavior of propane,n-butane, and n-pentane to elucidate the relationships amongcapillary condensation, adsorption, desorption, and the criticalproperties of the confined fluids. First, we measure isotherms forpropane and n-butane at temperatures from 5 to 50 °C in threedifferent pore sizes of MCM-41. Although two decades ago,Ioneva et al. performed a study in which theymeasured isothermsfor propane and n-butane at 283 K in various nanoporousmaterials that had been templated with different surfactants,19

our work provides the first comprehensive data set for bothadsorbates, for Ioneva et al. did not display any isotherms for

propane and only investigated one temperature for eachadsorbate and adsorbent.19 We conclude with measurementsof isotherms for n-pentane in 2.90 nm MCM-41 at 24.8 °C andtwo temperatures (48.6 and 72.1 °C) previously not reported inthe literature.1

2. MATERIALS AND METHODS2.1. MCM-41. Three different samples of MCM-41 were obtained

from Glantreo, Ltd. MCM-41 is a nanoporous silica characterized byunconnected cylindrical pores with an easily tuned, uniform pore sizedistribution.20−22 The simplicity of the MCM-41 pore geometryprecludes pore networking and blocking effects, making it an idealadsorbent for fundamental studies.23 Using a TriStar II 3020 sorptionanalyzer (V1.01, Micromeritics Instrument Corporation) and Micro-Active software (Micromeritics Instrument Corporation), the threesamples were characterized with nitrogen isotherms measured at 77.3 K.Although MCM-41-S and MCM-41-M did not exhibit hysteresis, theH1 hysteresis loop exhibited by MCM-41-L confirmed that the pores inthe sample were of uniform size and unconnected.5 Barrett−Joyner−Halenda (BJH), Dollimore−Heal (D-H), and Non-Local DensityFunctional Theory (NLDFT) analyses of the three isotherms shown inFigure 1 indicated pore sizes of 2.9, 4.2, and 8.1 nm. Full characterizationinformation for the samples can be found in Table 1. Note that, whereasIUPAC recommends NLDFT as the most accurate pore sizecharacterization method, BJH and D-H analyses are both dependenton the Kelvin equation, which has been shown to underestimate poresizes by up to 30%.5 Despite the lower accuracies associated with themethods derived from the Kelvin equation, we include all three methodshere to facilitate easy comparison of this work to both recent and morehistoric studies in the literature, which most commonly report NLDFTand BJH or D-H data, respectively.

Henceforth, we refer to our three MCM-41 samples as MCM-41-S,MCM-41-M, and MCM-41-L, where S, M, and L are used to denote theMCM-41 samples with the small, medium, and large pores, respectively.Note that these are the same adsorbents used in our previous study.24

Following the procedure used by Barsotti et al., approximately 7−9 g ofeach MCM-41 sample was packed into its own titanium core holder.24

Figure 1. Nitrogen master isotherms for the three MCM-41 samples. Adsorption and desorption are indicated by the filled and open symbols,respectively.

Table 1. Properties of the Adsorbents Determined from Nitrogen Adsorption Isotherms at 77 K

sampleBETa surfacearea [m2/g]

BJHb adsorption porediameter [nm]

BJHb desorption porediameter [nm]

D-Hc adsorption porediameter [nm]

D-Hc desorption porediameter [nm]

NLDFTd porediameter [nm]

t-plot microporevolume [cm3/g]

MCM-41-S

1043.01 2.59 2.52 2.78 2.72 2.90 −0.007416

MCM-41-M

832.08 3.51 3.21 3.70 3.44 4.19 0.015355

MCM-41-L

586.18 6.06 5.47 6.32 5.89 8.08 0.021502

aBrunauer−Emmett−Teller25 bBarrett−Joyner−Halenda26 with the Halsey thickness curve and Faas correction. cDollimore−Heal27 dNon-LocalDensity Functional Theory28,29 for nitrogen at 77.3 K in cylindrical pores with oxide surfaces.

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2.2. Adsorbate Dosing. Propane (99%, AirGas, Inc.), n-butane(99%, AirGas, Inc.), and n-pentane (99.8% Alfa Aesar) were used asadsorbates. The adsorbates were introduced to the system using a 6000series dual cylinder Quizix pump (Chandler Engineering). Housing thefluids in the pump prevented contamination of the fluids with air andallowed for pressurization of the fluids for experiments carried out above21 °C. Using the pump, propane and n-butane could be pressurized upto 15 and 2.2 bar, respectively.Isotherms were measured from 5 to 73 °C with temperature control

accuracy of ±0.1 °C using a novel gravimetric apparatus.24,30 Bothadsorption and desorption were measured using the static method(without flow). For adsorption, the amount of fluid in the system wasincreased in stepwise fashion using a series of two-way valves andvariable lengths of eighth inch Hastelloy tubing connecting the Quizixpump to the core holder. For desorption, controlled volumes of fluidwere vacuumed out of the system using a Welch Duo-Seal VacuumPump (Sargent-Welch Scientific Co.). For adsorption and desorption,

the system was closed between pressure increase and decrease steps,respectively. Likewise, all doses were allowed to equilibrate forapproximately 2 h until the system pressure became constant.24

Detailed descriptions of both the experimental apparatus and procedure,including extensive error analysis, can be found in Barsotti et al.24

3. RESULTS AND DISCUSSION

3.1. Determination of Capillary-Condensation Pres-sure. The isotherms for propane and n-butane in all threeadsorbents are presented in Figure 2. The abrupt changes inadsorption at the high-pressure ends of the isotherms, if shown,are the bulk condensation. The changes at intermediate pressurescan be the confined-phase transition (condensation orevaporation) or continuous filling or emptying without phasetransition. Continuous filling and emptying only occur aboveTcp.

Figure 2. Experimentally measured isotherms for (a) propane MCM-41-S, (b) propane in MCM-41-M, (c) propane in MCM-41-L, (d) n-butane inMCM-41-S, (e) n-butane in MCM-41-M, and (f) n-butane in MCM-41-L. Filled symbols are adsorption, and empty symbols are desorption. Nodesorption was measured for n-butane at 13.4 °C in MCM-41-L and MCM-41-S.

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As shown in Figure 2, these changes were not always sharp butoccurred continuously over a range of pressures. In the literature,the absence of an exact phase-change pressure for subcriticalconfined fluid is most commonly attributed to imperfections ofthe adsorbent, such as pore wall roughness and pore-sizedistribution.31 This necessitates the determination of a pressureover the phase transition to represent the capillary-condensation(Pca) or evaporation pressure (Pcd) at temperatures below Tcp.At temperatures above Tcp, we may assign a pressure to

represent the continuous filling, namely the maximum-pore-filling (Pma) or emptying (Pmd), pressure. Of course, prior todetermining Tcp, one cannot tell with confidence whethercapillary condensation, capillary evaporation, continuous porefilling, or continuous pore emptying occurs, particularly atrelatively high temperatures. While Tcp is discussed in the nextsection, we show here that the same method may be used todetermine Pca, Pcd, Pma, and Pmd. Until Section 3.2, we use Pm asan umbrella term to represent these four pressures and includethe subscripts “a” and “d” to represent adsorption anddesorption, respectively.Nearly all available experimental studies identify capillary

condensation and capillary evaporation as the midpoint2,4,9,32,33

of the confined phase transition. However, the midpoint is anonphysical observation, the determination of which requireshighly subjective user-defined limits for the slope of capillarycondensation. Therefore, we instead determine Pm as theinflection point of the abrupt increase in the isotherms. Unlikethe midpoint, the inflection point is a physical phenomenonrepresenting the maximum condensation or evaporation rate. Itwas determined by fitting the Lorentzian function to the firstderivative of the amount adsorbed with respect to pressure (dm/dP) across the abrupt increase. The derivative dm/dP wasdetermined from the experimental data using the centraldifference:

=−−

+ −

+ −

mP

m mP P

dd

i

i

i i

i i

1 1

1 1 (2)

where m and P indicate the data for amount adsorbed (grams)and pressure (bars), respectively.The pressure-dependent first derivative from eq 1 was then

fitted to a Lorentzian function:

ωγγ

Λ =− +

+PP P

C( )( )

2

m2 2

(3)

where ω is the peak height, γ is the half width of the peak at itshalf-maximum height, C is a constant that controls the baseline ofthe function, and Pm is the inflection-point pressure. Note thatthe maximum value of the Lorentzian, max[Λ(P) ], is the sum ofω and C. An example of the Lorentzian fit is given in Figure 3,where the Lorentzian was fit to desorption data for n-butane inMCM-41-S at 49.5 °C and adsorption data for propane inMCM-41-M at 5.3 °C. Pm values for the isotherms are shown in Table 2.To analyze the relationships between Pm and the pore sizes of

the adsorbents, Pm is plotted versus pore size for both adsorbatesin Figure 4. For propane and n-butane, Pm was found to increaseboth with increasing temperature and pore size, as has beencommonly shown in the literature; however, the pressures for n-butane show greater linearity across the range of pores thanpropane. Comparison of the data points at 20.8 °C shows thetrend for propane to be concave up, while that for n-butane isnearly linear.

3.2. Pore Critical Temperature. First, the pore criticaltemperatures of propane were determined. Based on therelationship between the slope of a given isotherm to thesuper- or subcriticality of the adsorbate, we determined Tcp usingthe method introduced by Morishige et al.2 in which thereciprocal of the average slope of the whole pore condensation orcontinuous filling step is plotted against temperature. Tcp wasthen determined as the discontinuity in the slopes of lines fit tothe data points at lower and higher temperatures, as shown inFigure 5. However, instead of using the reciprocal of the averageslope, for better accuracy, we use the reciprocal of the maximum

Figure 3. Typical Lorentzian curve fit: (a) desorption data for n-butanein MCM-41-S at 49.5 °C and (b) adsorption data for propane in MCM-41-M at 5.3 °C.

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of the Lorentzian. Athough it is a different input, max[Λ(P)] isconsistent with the average slope used by Morishige et al.,2 asboth account for the phase transition or continuous filling andthe background adsorption or desorption, which in the case ofmax[Λ(P)] are included in ω and C, respectively. The reciprocalis employed simply to provide better resolution in Figure 5.The discontinuity temperatures, broadly termed Tcx, obtained

for propane are presented in Table 3. These are the pore criticaltemperatures only if the pore size is above its critical value dc. Asshown in Table 3, the discontinuity temperatures increased withpore size, reflecting the well-known behavior of pore criticaltemperatures.7,11 Therefore, all three pore sizes are above dc, and

the temperatures represent Tcp. Note that the propanediscontinuity temperatures were obtained from the adsorptiondata because the hysteresis loop was found to disappear belowTcx, as shown in Figure 2.Subsequently, the reciprocal of the maximum of the

Lorentzian for n-butane in MCM-41-S and MCM-41-M wasplotted in Figure 6. Because closure of the hysteresis loop was notobserved for either adsorbent, both adsorption and desorptiondata are plotted in Figure 6. As shown, MCM-41-S exhibited twodifferent temperatures for adsorption and desorptioin: Tcx,a was25.4 °C and Tcx,d was 28.3 °C. Conversely, Tcx,a was not observedfor MCM-41-M, while Tcx,d was found to be 41.3 °C. Although, aplot is also shown for MCM-41-L, not enough data points wereavailable to accurately determine the discontinuity point forMCM-41-L, due to the maximum pressure of 2.2 bar for theexperiments with n-butane, as discussed in Section 2. Therefore,we exclude Tcx for MCM-41-L from further analyses.Throughout this work, the reliability of this method ofdetermining Tcx from discontinuities in the maximum of theLorentzian was found to depend on the amount of availableexperimental data.

3.3. Hysteresis Critical Temperature. Tch was directlyobserved from Figure 2 to be the temperature above whichhysteresis disappeared. For propane, Tch was approximately 20.8°C in MCM-41-L and 5.3 °C in MCM-41-M and MCM-41-S.Considering the pore critical temperatures determined in Section3.2, this verifies that propane follows the expected progression ofconfined phase behavior from hysteretic capillary condensationto reversible capillary condensation and then to supercriticality.For n-butane, Tch was not observed in any of the pore sizes.

Comparison of the hysteresis of propane to n-butane is shown inFigure 7, in which Pm is plotted versus temperature. As shown inFigure 7a, the adsorption and desorption pressures for propanecoincided before Tcx. Conversely, in Figure 7b, no coincidencewas observed for n-butane even above Tcx. Therefore, we termthe observation of hysteresis above Tcp or below dc supercriticalhysteresis, which has continuous pore emptying along thedesorption branch in addition to continuous pore filling alongthe adsorption branch.Our observation of supercritical hysteresis is, to the best of our

knowledge, the first inMCM-41 and should not be confused withthe prolonged hysteresis that has been documented for naturaladsorbents, such as shale rock,34 and microporous adsorbents,such as aerogels35 and metal organic frameworks,36 which arecharacterized by highly complex, and in many cases irregular,pore networks. For example, in shale rock, supercritical, orotherwise unexpected hysteresis, is attributed to pore blockingeffects.34 In microporous materials, hysteresis is attributed topore blocking,37 diffusion,36 and structural instabilities of theadsorbent, which commonly result in flexibility36 and permanentdeformation38 of the pore network. In our study, the adsorbent isless complex with uniformly sized and shaped nanopores that arenot interconnected, while it is also considerably more rigid thantypical microporous adsorbents, such as aerogels.38 Thus, theobserved phenomenon may not be attributed to pore blockingeffects or to significant structural instabilities.The exact same adsorbent packs were used for all experiments

herein; thus, we infer one major cause of supercritical hysteresisto be the molecular chain length of the adsorbate. The longercarbon chain length of n-butane may be more prone to greaterdeformation in confinement, such as bond angle bending. To testthe presence of supercritical hysteresis as a consequence ofmolecular chain length, we subsequently measured isotherms of

Table 2. Pm Values for All Isotherms Shown in Figure 2a

adsorbate adsorbent temperature [°C] Pma [bar] Pmd [bar]

propane MCM-41-S 5.4 2.11 2.06propane MCM-41-S 10.0 2.54 2.54propane MCM-41-S 20.8 3.44 3.44propane MCM-41-S 28.2 4.29 4.29propane MCM-41-S 35.2 5.29 5.29propane MCM-41-S 42.3 6.39 6.39propane MCM-41-S 49.5 7.52 7.52propane MCM-41-M 5.3 2.55 2.42propane MCM-41-M 8.7 2.76 2.75propane MCM-41-M 11.1 2.97 2.97propane MCM-41-M 13.4 3.15 3.15propane MCM-41-M 16.0 3.55 3.55propane MCM-41-M 20.8 3.76 3.76propane MCM-41-M 28.3 4.71 4.71propane MCM-41-M 35.2 5.77 5.77propane MCM-41-M 42.3 6.79 6.79propane MCM-41-M 49.8 8.25 8.25propane MCM-41-L 5.4 4.29 4.17propane MCM-41-L 10.0 4.98 4.92propane MCM-41-L 16.2 5.81 5.71propane MCM-41-L 20.8 6.54 6.54propane MCM-41-L 28.2 7.89 7.89propane MCM-41-L 35.3 9.30 9.30propane MCM-41-L 42.3 10.95 10.95propane MCM-41-L 49.5 12.96 12.96n-butane MCM-41-S 8.7 0.37 0.32n-butane MCM-41-S 13.4 0.53 -n-butane MCM-41-S 20.8 0.57 0.52n-butane MCM-41-S 28.2 0.75 0.69n-butane MCM-41-S 35.2 0.95 0.85n-butane MCM-41-S 42.3 1.36 1.14n-butane MCM-41-S 49.5 1.66 1.47n-butane MCM-41-M 5.4 0.52 0.40n-butane MCM-41-M 10.0 0.61 0.51n-butane MCM-41-M 16.2 0.74 0.60n-butane MCM-41-M 20.8 0.84 0.75n-butane MCM-41-M 28.2 1.00 0.91n-butane MCM-41-M 35.3 1.28 1.23n-butane MCM-41-M 42.3 1.53 1.45n-butane MCM-41-M 49.5 1.90 1.75n-butane MCM-41-L 8.7 0.99 0.90n-butane MCM-41-L 13.4 1.24 -n-butane MCM-41-L 20.8 1.47 1.35n-butane MCM-41-L 28.2 1.77 1.72

aThe values were determined by fitting the Lorentzian to theexperimental data for each isotherm. The subscripts “a” and “d”designate adsorption and desorption, respectively.

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n-pentane inMCM-41-S at three temperatures. Full adsorption−desorption isotherms for n-pentane in MCM-41-S are shown inFigure 8.The supercriticality of n-pentane is inferred from Figure 9,

where the adsorption branches for n-pentane in MCM-41-S andMCM-41-M at 24.8 °C are plotted. The proximity of the valuesof Pm is due to the similar pore sizes. The smaller slope in thesmaller pore at Pm and the absence of sharp turning near the low-pressure end of the slope is the signature of continuous filling,and thus supercriticality. The corresponding plot for propane inFigure 9 confirms this inference. Therefore, n-pentane issupercritical in MCM-41-S at all temperatues in Figure 8, i.e.,MCM-41-S is below dc, which strongly suggests the existence ofsupercritical hysteresis. Note that a third plot for n-butane is alsoshown in Figure 9, although its interpretation is lessstraightforward due to the ambiguities associated with theapproximation of Tcp from Tcx,a and Tcx,d in Figure 6.

Even at 72.1 °C, which is more than 50 °C above thesupercritical boundary of n-pentane, hysteresis occurred.Furthermore, the width of the hysteresis loop in the supercriticalregion is observed to increase with increasing temperature. Theseresults confirmed the persistent occurrence of continuous fillingin the adsorption branch and continuous emptying in thedesorption branch at high temperatures.

3.4. Comparison of the Phase Diagrams. Comparison ofthe confined and bulk phase behavior of the three adsorbates isshown in Figure 10. Bulk saturation pressures from NIST39 wereused to benchmark the accuracy of our work, while the bulkcritical points were useful for comparison to the values of Tcp

determined from the experimental data herein. NIST reports thecritical pressures (Pc) and temperatures (Tc) for propane, n-butane, and n-pentane to be 96.75 °C and 42.5 bar, 151.85 °Cand 38.0 bar, and 196.65 °C and 33.6 bar, respectively.39 Usingcurves fit to the bulk and confined critical temperatures, thecritical boundaries of the fluids were estimated as shown in

Figure 4. Comparison of the behavior of Pm for (a) propane and (b) n-butane. The circles, triangles, and squares indicate the measurements taken inMCM-41-S, MCM-41-M, and MCM-41-L, respectively. Filled symbols are adsorption, and empty symbols are desorption.

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Figure 10 by fitting the experimental data to a quadraticpolynomial. Because only the critical pore size was known for n-pentane, a full hypothetical critical boundary was not plotted forit.

Despite the need for more data in other pore sizes to achievequantitative accuracy in the estimated critical boundaries, it canstill be seen qualitatively that the subcritical region of the fluid(i.e., the area between the bulk saturation pressure and the criticalboundary) shrinks as molecular chain length increases. Equation1 confirms this trend, when written in terms of Tcp/Tc:

σ− ∼T

T r1 cp

c p (4)

For two different adsorbates, indicated by the subscripts 1 and 2,in the same pore size,

Figure 5. Determination of the discontinuity temperatures as the porecritical temperatures for propane adsorption data in (a) MCM-41-S, (b)MCM-41-M, and (c) MCM-41-L.

Table 3. Discontinuity Temperatures Estimated from theExperimental Data Using the Lorentzian Function

adsorbate adsorbent Tcp [°C]

propane MCM-41-S 30.0propane MCM-41-M 35.3propane MCM-41-L >50

Figure 6.Determination of Tcx for n-butane isotherms: (a) MCM-41-S;(b) MCM-41-M; (c) MCM-41-L, in which no discontinuities wereobserved over the range of experimental temperatures.

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σσ

−−

=T T

T T

1 ( / )

1 ( / )cp,1 c,1

cp,2 c,2

1

2 (5)

If σ1 > σ2,

−−

>T T

T T

1 ( / )

1 ( / )1cp,1 c,1

cp,2 c,2 (6)

which results in

<T T T T/ /cp,1 c,1 cp,2 c,2 (7)

This finding may indicate that hydrocarbons in real nano-porous systems, such as shale reservoirs, may also exist assupercritical fluids. In terms of oil and gas production, this couldinvalidate many of the equations of state currently used forpredicting phase diagrams.Further comparison of the data measured for the fluids in

confinement also indicates that for adsorbates where supercriticalhysteresis is observed, two different phase diagrams may be

Figure 7. Plots to compare the hysteresis of (a) propane to (b) n-butane. Filled symbols are adsorption, and open symbols are desorption.

Figure 8. Isotherms for n-pentane in MCM-41-S. The confined fluid issupercritical over the entire range of pressures24 because the pore size isbelow the critical size at the experimental temperatures.

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necessary to describe the phase behavior of the adsorbate: one foradsorption and one for desorption. For alkanes, specifically, thismay revolutionize the way that shale gas and oil reservoirs areproduced. Because long-chain hydrocarbons constitute aconsiderable fraction of both natural gas and oil, the supercriticalhysteresis observed herein could carry over to the reservoircondition. At reservoir conditions, adsorption−desorptionhysteresis implies differences in fluid injection and productionpressures. This may necessitate the use of different phasediagrams to govern decision making regarding both. The phasediagram of the reservoir fluid is one of the main inputs fordynamic reservoir simulation. Because equations of state canonly capture the equilibrium phase transition, metastability ineither adsorption or desorption may complicate modelingefforts. We intend to focus our future work on studies ofsupercriticality and capillary condensation in both synthetic andnatural nanopores in order to further investigate theseimplications.

4. CONCLUSIONS

We measured isotherms for propane and n-butane in threedifferent pore sizes of MCM-41 and determined the correspond-ing phase-transition pressures. The pressures of both adsorbatesin confinement showed proportionality to the pore size.The pore critical temperatures of propane and n-butane were

estimated by determinations of the discontinuity points of themaximum condensation (or evaporation) rates of the isotherms

across the entire range of measured temperature. However, the

discontinuity points for adsorption and desorption differed for n-

butane in smaller pores and were termed Tcx and indicated new

phenomena that have not been previously reported.In comparing the hysteresis critical temperatures of both

adsorbates as well as their values for Tcx, propane was found to

follow the commonly observed progression of capillary

condensation from hysteresis to reversibility and then to

supercriticality. However, n-butane displayed entirely new

behavior in which no hysteresis critical temperature was

observed, despite the supercriticality of the confined fluid.

Thus, we hypothesize that long-chain n-alkanes may exhibit

hysteresis above their pore critical temperature or below their

critical pore size simply due to the length of these molecules. We

tested this hypothesis through further isotherm measurements

for n-pentane in the smallest pore size where it is known to be

supercritical and found it, too, to exhibit hysteresis. We term the

presence of hysteresis above the pore critical temperature or

below the critical pore size supercritical hysteresis, which consists

of the common continuous pore filling along the adsorption

branch and the new continuous pore emptying along the

desorption branch.

Figure 9.Comparison of adsorption isotherms for n-pentane, n-butane, and propanemeasured in different pore sizes. Note that the data for n-pentane inMCM-41-M is from our previous work.24 Desorption data is excluded to allow for easier visual comparison.

Figure 10. Comparison of the phase diagrams of the three adsorbates both in the bulk39 and in confinement.

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■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.lang-muir.8b00125.

Tables of all the isotherm data presented in Figures 2 and 8(PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail ebarsott@uwyo.edu.

ORCIDElizabeth Barsotti: 0000-0002-4106-5543NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe gratefully acknowledge the financial support of SaudiAramco, Hess Corporation, and the School of Energy Resourcesand the College of Engineering and Applied Science at theUniversity of Wyoming. We also thank Mr. Evan Lowry of thePiri Research Group at the Center of Innovation for FlowThrough Porous Media at the University of Wyoming for hisinsightful discussions.

■ NOMENCLATUREC constant that controls the baseline of the Lorentziandc critical pore sizem amount adsorbedP pressurePc critical pressure of the bulk fluidPca capillary condensation pressurePcd capillary evaporation pressurePm maximum pore filling or emptying pressure equal to the

inflection point of the isothermPma maximum pore filling pressure; it is the capillary

condensation pressure if below the pore critical temper-ature or above the critical pore size

Pmd maximum pore emptying pressure; it is the capillaryevaporation pressure if below the pore critical temper-ature or above the critical pore size

Tc critical temperature of the bulk fluidTch hysteresis critical temperatureTcp pore critical temperatureTcx temperature representing a discontinuity in a plot of the

inverse of the maximum of the Lorentzian versustemperature. It is the pore critical temperature when thepore size is larger than the critical pore size.

γ half width of the Lorentzian peak at its half-maximumheight

Λ(P) Lorentzianσ Lennard-Jones size of a moleculeω peak height of the Lorentzian

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