Isobaric VaporâLiquid Phase Diagrams for Multicomponent Systems

Article pubs.acs.org/JPCB

Cite This: J. Phys. Chem. B XXXX, XXX, XXX−XXX

Isobaric Vapor−Liquid Phase Diagrams for Multicomponent Systems with Nanoscale Radii of Curvature Nadia Shardt and Janet A. W. Elliott* Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada ABSTRACT: At any given temperature, pressure, and composition, a compound or a mixture of compounds will exist either in a single phase, whether solid, liquid, or vapor, or in a combination of these phases coexisting in equilibrium. For multiphase systems, it is known that the geometry of the interface impacts the equilibrium state; this effect has been wellstudied in single component systems with spherical interfaces. However, multicomponent phase diagrams are usually calculated assuming a planar interface between phases. Recent experimental and theoretical work has started to investigate the effect of curved interfaces on multicomponent phase equilibrium, but these analyses have been limited to isothermal conditions or to a portion of the isobaric phase diagram. Herein, we consider complete vapor−liquid phase diagrams (both bubble and dew lines) under isobaric conditions. We use Gibbsian composite-system thermodynamics to derive the equations governing vapor−liquid equilibrium for systems with a spherical interface separating the phases. We validate our approach by comparing the predicted nitrogen/argon dew points with reported literature data. We then predict complete isobaric phase diagrams as a function of radius of curvature for an ideal methanol/ethanol system and for a nonideal ethanol/water system. We also determine how the azeotropic composition of ethanol/water changes. The effect of curvature on isobaric phase diagrams is similar to that seen on isothermal phase diagrams. This work extends the study of curved-interface multicomponent phase equilibrium to isobaric systems, expanding the conditions under which nanoscale systems, such as nanofluidic systems, shale gas reservoirs, and cloud condensation nuclei, can be understood.

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INTRODUCTION Just 31 articles were published on nanofluidics in 2006; in 2016, researchers around the world published 1392 articles in this field (articles with “nanofluidics” in the title, abstract, or keywords, Scopus, December 19, 2017). Such a rapid increase in publications over the decade underscores the growing interest in fluid phenomena that occur at the nanoscale. One common feature of many nanoscale systems is the presence of curved interfaces in the form of nanobubbles, nanodroplets, or nanopores. As a result of the nanoscale curvature, the phase behavior of fluids in these nanoscale systems differs drastically from that of bulk systems without curved interfaces. The phase behavior of nanoscale systems is particularly important to understand because fluid properties are governed by which phase or phases are present, and predicting fluid properties accurately is important for the design and control of nanoscale systems. Interfacial thermodynamics has been previously used to study the phase behavior of single-component systems with interfaces that have nanoscale radii of curvature. There still lies much opportunity for research in extending the framework of interfacial thermodynamics to multicomponent, multiphase systems. In the natural environment, multicomponent systems with highly curved interfaces can be found from 4 km below the Earth’s crust in hydrocarbon reservoirs1 to up to 15 km above the surface in the troposphere.2,3 In the subsurface, porous © XXXX American Chemical Society

structures can have pore throat diameters in the nanometer range.4−7 These nanopores influence phase equilibrium, and if their effect is unaccounted for, predictions for unconventional hydrocarbon recovery operations may be inaccurate.8−12 Phase diagrams are vital tools in recovering oil and natural gas from shale formations, and the effect of nanoscale curvature on phase diagrams has been calculated for specific compositions of hydrocarbons as a function of temperature and pressure.4,8,13−16 In the atmosphere, clouds can form when water condenses on highly curved nuclei with diameters in the nanometer range.3,17 The number, size, and composition of these nuclei determine the number and size of water droplets, which dictate the size and lifetime of clouds.18 Thus, nanoscale processes can have a significant role in macroscopic properties and phenomena. In industry, catalysts may deactivate if liquid condenses in pores with nanoscale diameters, altering the rate and extent of reaction.19−22 The atomic force microscope (AFM) and surface forces apparatus (SFA) can be affected by condensation between the probe tip and the surface, and the radii of curvature of the condensed phase can reach nanometer scales.23,24 The manufacturing of cosmetics, food, and paint Received: January 5, 2018 Revised: January 30, 2018

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DOI: 10.1021/acs.jpcb.8b00167 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

diagram for a Lennard-Jones azeotropic mixture of Ar/CH4, and Li et al.66 studied how the azeotrope shifts for a LennardJones mixture. For nonideal systems with an azeotrope, Defay and Prigogine43 outline the effect of curvature on the location of the azeotrope under isothermal conditions. Experimental work indicates that the azeotrope of nonideal multicomponent mixtures can be altered by introducing curvature between the vapor and liquid phases.67−69 Several binary mixtures have been distilled with porous plates, and in the theoretical analysis, solid−liquid interactions between the solution and pore walls were identified to significantly alter vapor−liquid equilibrium.67−72 Our previous study examined the effect of interfacial curvature on the isothermal phase diagrams of both ideal and nonideal multicomponent systems.73 Through the framework of Gibbsian thermodynamics, we determined that the pressure in the vapor phase increases for a liquid droplet in a bulk vapor phase and decreases for a vapor bubble in a bulk liquid phase, similar to the trends seen for pure components. Significant effects on the phase envelope were only observed at nanoscale radii of curvature. We also calculated the shift in azeotropic composition as a function of radius of curvature for the nonideal ethanol(1)/water(2) system and determined that the azeotrope can vanish for some radii under isothermal conditions. Conventional separation processes have limited applicability for nonideal systems that exhibit an azeotrope, but new processes could shift the azeotrope by exploiting the effects of nanoscale curvature. The past literature on multicomponent systems with curvature has been limited to isothermal phase diagrams or to a portion of the isobaric phase diagram; in this work, we now extend the scope of study to complete vapor−liquid phase diagrams under isobaric conditions. Under isothermal conditions, liquid properties are generally invariant as a function of pressure and are commonly treated as constant parameters. Isobaric phase diagrams are more challenging to compute, because the liquid properties that govern vapor−liquid phase equilibrium are nonlinear functions of temperature. In other words, the bubble and dew temperatures of a system are more difficult to calculate than the bubble and dew pressures.74 It is important to note that for systems with interfacial curvature, the meaning of the term isobaric may be unclear because each phase has a different pressure; in the current study, the term isobaric will refer to systems that have constant pressure in the vapor phase. In the current study, we derive the equations governing vapor−liquid equilibrium of multicomponent, nonideal systems with interfacial curvature using Gibbsian composite-system thermodynamics, and we choose models that accurately describe the relevant parameters of the system components. We predict the dew temperatures of a nitrogen(1)/argon(2) system and compare the results to experimental measurements as a function of composition. We then predict the complete isobaric phase diagrams for an ideal system (methanol(1)/ ethanol(2)) and a nonideal system (ethanol(1)/water(2)). Finally, the azeotropic composition of ethanol/water is calculated as a function of radius of curvature under isobaric conditions.

can also be described by multicomponent, multiphase thermodynamics with interfacial curvature.25,26 In medicine, nanoscale drug delivery systems show potential for targeting tumors.27−30 The effect of nanoscale curvature is also pertinent in fluid flow through nanochannels31,32 and in phase changes occurring in porous media.26,33 Such phase changes can occur in molecular sieves such as MCM-41 and SBA-15, which are commonly used adsorbents that have pores with diameters of less than 10 nm.11,26,34−36 Extensive theoretical analyses have been carried out for single-component systems with interfacial curvature,37−42 and the effect of curvature on single component systems is outlined in detail by Defay and Prigogine for both constant temperature and constant pressure conditions.43 Single-component phase equilibrium has been investigated for water44,45 and carbon dioxide46 with nanoscale curvature between phases. Recent molecular simulations suggest that the Kelvin equation that describes the increase in vapor-phase pressure as a function of radius of curvature is accurate for water droplets down to a radius of 0.6 nm.47 Defay and Prigogine43 and Vehkmäki48 derived a multicomponent Kelvin equation assuming ideal behavior in the vapor and liquid phases. Another multicomponent form of the Kelvin equation was derived by Shapiro and Stenby,49,50 but it is only valid in the region close to the dew point. It was extended to the bubble point region by Sandoval et al.16 The derivation of the proposed multicomponent form of the Kelvin equation uses a linear approximation in both the dew and the bubble point regions. Using the Shapiro−Stenby approach, Chen et al.51 studied the shift in the dew point, and Sandoval et al.16 calculated the shift in the phase envelope for hydrocarbon mixtures. Gelb et al.26 have summarized theoretical and experimental work on phase equilibria in nanoporous material, while Barsotti et al.11 have reviewed research dealing specifically with the phase behavior of hydrocarbon mixtures. Curvature also affects the equilibrium between solid and liquid phases of single and multicomponent systems,26,52,53 and the effect of confinement in nanoscale capillaries on the phase diagram and the eutectic point of glycerol/water has been studied.54 To predict the phase diagrams of multicomponent systems with interfacial curvature, Tan and Piri55 proposed an empirical model that combines an equation of state with a modified capillary pressure expression that accounts for an adsorbed layer at the capillary wall. However, the model can only be implemented when vast experimental data sets are available, because it relies on a fitting parameter for each pure component that is a function of pore radius and temperature. Using this empirical model, dew temperatures for a binary mixture of nitrogen/argon and bubble temperatures for krypton/argon were predicted and compared to experimental data. Predictions of the complete phase diagrams for nitrogen/argon under isothermal conditions were also made.55 The model was then extended to associating fluids under isothermal conditions.56 Molecular simulations have been used to predict the effect of confinement on both single and multicomponent phase diagrams26 through approaches including density functional theory, Monte Carlo,13,57−63 and molecular dynamics26 for cylindrical pores58,64 and slit pores.60−62,64,65 Many of these simulations have been limited to simpler fluids such as nitrogen,59 argon and krypton,58,62,63 methane,13,59,65 and ethane13 because of the computational intensity that would be required for more complex fluids. Considering confinement in a slit pore, Coasne et al.65 simulated the solid−liquid phase

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GOVERNING EQUATIONS The equilibrium states of a composite system can be determined from Gibbsian thermodynamics. Consider a system B


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of the radius of curvature,48,76 and this means that each component is present at the interface. The differential change in the internal energy of the reservoir is

consisting of one phase in contact with a second phase separated by a spherical interface. This two-phase system is contained in a piston−cylinder device, forming a closed system within a surrounding reservoir. Figure 1 illustrates two

m

dU R = T R dS R − P R dV R +

∑ μiR dNiR i=1

(5)

where m is the number of components in the reservoir. Rearranging eqs 2−5 in terms of dS and substituting into eq 1 gives 1 PL L L d U dV − + TL TL k

−

i=1 k

Figure 1. Schematic diagrams of vapor (V)−liquid (L) systems contained in a reservoir that has pressure PR and temperature TR. The piston−cylinder device imposes the reservoir pressure on the vapor phase. Diagram (a) shows a system with a convex liquid surface, and (b) illustrates a system with a concave liquid surface.

−

i=1

σ LV 1 dU LV − LV dALV LV T T

μi LV T

dNiLV + LV

μi R TR

1 PR R R d U dV + TR TR

dNiR = 0

(6)

(7)

dNiR = 0

(8)

and for each component in the vapor−liquid system: dNiV = −dNiL − dNiLV

(9)

There is a moveable piston between the system and the reservoir, as illustrated in Figure 1a, which permits changes in volume of each phase and of the reservoir. The final constraint is that the total volume of the system and reservoir together is constant:

(2)

k

∑ μiL dNiL

dNiV +

1 PV V V d U dV + TV TV

Next, there is no mass transfer between the system and the reservoir, so for each component in the reservoir:

k

dU L = T L dS L − P L dV L +

T

V

T

dNiL + L

dUV = −dU L − dU LV − dU R

where superscripts L and V denote the liquid and vapor phases, respectively, LV denotes the vapor−liquid interface, and R denotes the reservoir surrounding the vapor−liquid system. The entropies are functions of internal energy (U), volume (V), and number of moles (N). The differential changes of these variables in the bulk vapor and liquid phases are related by the following equations, respectively:40,42,75,76

i=1

μi

i=1

μi L

Equation 6 will yield equilibrium conditions for a system once appropriate constraints are defined. Equilibrium Conditions for a Droplet of Liquid within a Vapor Phase. First, applying the law of conservation of energy to the vapor−liquid system and reservoir gives the following mathematical constraint:

(1)

∑ μi V dNiV

∑ i=1

examples of vapor−liquid systems with curved interfaces that will be analyzed in this Article. Figure 1a shows a liquid droplet within a bulk vapor phase (radius of curvature in the liquid phase), and Figure 1b shows a cylindrical capillary tube containing a liquid phase in contact with a bulk vapor phase, but with a different direction of interfacial concavity as compared to Figure 1a (radius of curvature in the vapor phase). In both cases, the piston−cylinder device imposes the pressure of the reservoir on the vapor phase. An equilibrium state occurs when the entropy (S) of the system is at an extremum; mathematically, this means that

dU V = T V dS V − P V dV V +

∑ i=1 m

−

dS L + dSV + dS LV + dS R = 0

∑

V

k

∑

dV V = − dV L − dV R

(3)

where T is the absolute temperature, P is the pressure, μi is the chemical potential of component i, and k is the number of components. The differential change in internal energy for a curved vapor−liquid interface is40,42,76

(10)

For a spherical liquid drop, the following relationships hold for differential area and volume: dALV = 8πr L dr L

(11)

dV L = 4π (r L)2 dr L

(12)

k

dU LV = T LV dS LV + σ LV dALV +

∑ μiLV dNiLV i=1

(4)

where rL is the radius of the spherical liquid droplet. Substituting eqs 7−12 into eq 6 and collecting like terms gives

where σ is the surface tension and A is the area of the vapor− liquid interface. Using Gibbs’ surface of tension approach for a curved interface ensures that the surface tension is independent C


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The Journal of Physical Chemistry B ⎛ 1 ⎛ 1 1 ⎞ 1 ⎞ ⎜ − V ⎟ dU LV + ⎜ L − V ⎟ dU L LV ⎝T ⎝T T ⎠ T ⎠ ⎡ ⎛ PL ⎛ 1 1 ⎞ PV ⎞ + ⎜ R − V ⎟ dU R + ⎢4π (r L)2 ⎜ L − V ⎟ ⎝T ⎝T T ⎠ T ⎠ ⎣ −

⎤ σ LV L ⎥ dr L + 8 r π T LV ⎦

+

∑ ⎜⎜

⎛μV i V

⎝T

⎛μV

∑ ⎜⎜

i V

⎝T

−

The superscript LV on surface tension is dropped for simplicity. The factor 2/r in eq 23 is the curvature of a spherical surface,77 which is twice the mean curvature.42,78 Governing Equations for Vapor−Liquid Phase Diagrams. Curved Interface. From this point forward, the temperature of the system will be denoted by T based on eq 22. The chemical potential for each component i of an ideal-gas vapor phase is39,74

μi L ⎞ L ⎟⎟ dNi TL ⎠

⎛ yPV ⎞ μi V (yi , T , PV ) = μi V (T , Pisat) + RT ln⎜⎜ i sat ⎟⎟ ⎝ Pi ⎠

μi LV ⎞ LV ⎛ P R PV ⎞ R ⎟ d N + − ⎟ dV = 0 ⎜ i ⎟ ⎝ TR T LV ⎠ TV ⎠

−

(25)

(13)

The chemical potential for each component i of a nonideal, incompressible liquid phase is74,79

For eq 13 to be satisfied for all possible displacements from equilibrium, each of the coefficients multiplying the differentials must be equal to zero. When these coefficients are set to zero, the following equalities can be derived: T V = T L = T LV = T R

= μi L (T , Pisat) + viL(P L − Pisat) + RT ln(γixi)

(14)

2σ LV rL

(15)

μi V = μi L = μi LV

(16)

P R = PV

(17)

P L − PV =

μi L (xi , T , P L)

where yi and xi are vapor and liquid mole fractions, respectively, R is the universal gas constant, vLi is the liquid molar volume of pure component i, and γi is the activity coefficient for L sat component i. The terms μVi (T, Psat i ) and μi (T, Pi ) are the chemical potentials at the reference states, taken to be the chemical potential of the vapor phase and the liquid phase, respectively, at the flat-interface saturation vapor pressure of component i (Psat i ) at a temperature T; at equilibrium, these terms are equal. Equating eqs 25 and 26 as indicated by eq 24 gives

Equations 14−16 are the thermodynamic equations defining the equilibrium conditions for a multicomponent system consisting of a liquid-phase spherical droplet in a vapor phase with temperature and vapor-phase pressure imposed by the surrounding reservoir. Equilibrium Conditions for a Liquid in a Capillary. Following a procedure similar to that outlined for a droplet of liquid, the equilibrium conditions for a liquid phase confined in a capillary with a concave hemispherical interface are V

L

T =T =T PV − P L = L

V

LV

=T

R

⎛ yPV ⎞ RT ln⎜⎜ i sat ⎟⎟ = viL(P L − Pisat) + RT ln(γixi) ⎝ Pi ⎠

μi = μi = μi

LV

P R = PV

⎛ v L(P L − P sat) ⎞ i V sat ⎜ i ⎟ = γ x P exp yP i i i i RT ⎝ ⎠

(19)

⎛ v L(PV + 2σ /r − P sat) ⎞ i V sat ⎜ i ⎟ = γ x P exp yP i i i i RT ⎝ ⎠

(21)

where r is the radius of curvature in the vapor phase. Equilibrium Conditions for Two Phases Separated by a Spherical Interface. The thermal and chemical equilibrium conditions are identical for the liquid droplet and for the liquid in a capillary. The only difference between these systems is the sign of the pressure difference (eq 15 vs eq 19). A common convention is to define a spherical liquid phase as having a positive radius of curvature and a spherical (or hemispherical) vapor phase as having a negative radius of curvature. This convention gives the following equations for equilibrium in the liquid and vapor phases for any multicomponent mixture with a spherical or hemispherical interface between phases:

P L − PV =

2σ r

μi L = μi V = μi LV

(28)

Substituting eq 23 into eq 28 gives73

(20)

V

T V = T L = T LV

(27)

Rearranging eq 27 gives a modified version of Raoult’s Law for a nonideal liquid mixture:80

(18)

LV

2σ rV

(26)

(29)

The exponential factor in eqs 28 and 29 is known as the Poynting factor. Equations for the bubble temperature and dew temperature can be determined by combining eq 29 for each component with ∑iyi = 1 or ∑ixi = 1, respectively. For a given overall mole fraction of the mixture (zi), pressure in the vapor phase, and radius of curvature, the bubble temperature (Tb) and dew temperature (Td) can be calculated with the following equations, given that Pisat, viL, and σ are functions of temperature: k

(22)

PV =

∑ γixiPisat(Tb) i=1

⎛ v L(T )(PV + 2σ(T , x , ..., x )/r − P sat(T )) ⎞ b 1 k−1 i b ⎟ × exp⎜ i b RTb ⎝ ⎠

(23) (24)

(30) D


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The Journal of Physical Chemistry B ⎛ ⎜ k V P = ⎜∑ ⎜⎜ i = 1 γP sat(T ) exp d i i ⎝

yi

(

viL(Td)(PV + 2σ (Td , x1, ..., xk − 1) / r − Pisat(Td)) RTd

)

curvature. The resulting equations for the flat-interface bubble temperature (Tbf) and dew temperature (Tdf) are outlined in thermodynamics textbooks and are given by, with the addition of the Poynting correction:74

⎞−1 ⎟ ⎟ ⎟⎟ ⎠ (31)

Equations 30 and 31 describe the bubble and dew temperatures for a multicomponent system with k components. In this study, we consider the phase diagrams of binary mixtures, so the summations in these equations will only contain two terms, one for each component. The surface tension of a binary mixture simplifies to being a function of only temperature and x1. To solve for the bubble temperature of a binary mixture, the liquid-phase composition is set to the overall composition (x1 = z1), and eq 30 is solved for Tb. The bubble line of the vapor− liquid phase envelope can be calculated for a given radius of curvature by solving eq 30 for all z1 between 0 and 1. Equation 31 can be used to solve for the dew temperature by setting the vapor-phase composition to the overall composition (y1 = z1). Because the liquid-phase activity coefficient and vapor−liquid interfacial tension depend on the liquid-phase composition, another equation is needed to determine this composition. Both sides of eq 29 can be divided by the vapor mole fraction, and then this equation can be written for each component of a binary mixture; the two resulting equations can be equated and then rearranged to give the following expression for liquid phase mole fraction:73

⎛ ⎜ k V P = ⎜∑ ⎜ i = 1 γP sat(T ) exp df i i ⎝

(

(34)

γ2P2sat exp(v2L(PV + 2σ /r − P2sat)/(RT ))

)

∑ γixiPisat(Tbf )

(37)

(38)

i=1

and when the activity coefficients are also unity, eq 37 simplifies to ⎛ k ⎞−1 yi ⎟⎟ P = ⎜⎜∑ sat ⎝ i = 1 Pi (Tdf ) ⎠ V

(39)

When eqs 36 and 37 are solved using an iterative method, the solutions to eqs 38 and 39 can be used as the initial estimates of bubble and dew temperatures substituted into the Poynting correction. When solving eq 37, another equation is needed to determine the liquid-phase composition for evaluating the activity coefficient. For a binary mixture, an equation for the liquid mole fraction at the dew temperature can be derived by using eq 28 for both components and PL = PV for a flat interface, and it is given by ⎛ y γ P sat exp(v L(PV − P sat)/(RT )) ⎞−1 1 1 df 2 1 1 ⎜ x1f = ⎜ + 1⎟⎟ sat L V sat ⎝ y1γ2P2 exp(v2 (P − P2 )/(RTdf )) ⎠

(40)

The flat-interface azeotropic composition can be calculated by setting 2σ/r = 0 in eq 35. Temperature-Dependent Properties of Pure Components. One of the challenges of computing isobaric phase diagrams is accurately accounting for the temperature dependence of liquid properties, including flat-interface saturation vapor pressure, liquid molar volume, and surface tension. The saturation vapor pressure of each component with a planar vapor−liquid interface can be calculated as a function of temperature with an extended Antoine equation:82

where A12 and A21 are system-dependent empirical constants at a given pressure. The activity coefficients for an ideal system are unity. Composition of the Azeotrope for Curved Interfaces. At the azeotropic composition, the mole fraction in the vapor phase is equal to the mole fraction in the liquid phase. Using this property, the following equation can be used to determine the azeotropic composition for a binary system with a curved interface:73 γ1P1sat exp(v1L(PV + 2σ /r − P1sat)/(RT ))

viL(Tdf )(PV − Pisat(Tdf )) RTdf

⎞−1 ⎟ ⎟ ⎟ ⎠

k

PV =

Among the possible activity coefficient models for binary mixtures, we select the Margules equations, given by80,81

ln(γ2) = x12(A 21 + 2(A12 − A 21)x 2)

yi

(36)

When the Poynting correction is unity, eq 36 simplifies to

(32)

(33)

∑ γixiPisat(Tbf ) exp⎜ i=1

⎛ y γ P sat exp(v L(PV + 2σ /r − P sat)/(RT )) ⎞−1 1 1 d 2 1 1 ⎜ x1 = ⎜ + 1⎟⎟ sat L V sat ⎝ y1γ2P2 exp(v2 (P + 2σ /r − P2 )/(RTd)) ⎠

ln(γ1) = x 22(A12 + 2(A 21 − A12 )x1)

⎛ v L(T )(PV − P sat(T )) ⎞ i bf i bf ⎟ RT ⎝ ⎠ bf

k

PV =

=1

ln Pisat = Ai +

(35)

An initial estimate of the azeotropic composition can be determined by setting the exponential Poynting correction factor to 1. At the azeotrope, the bubble and dew temperatures are equal, so eq 30 is chosen to easily solve for temperature, and this temperature can be substituted into eq 35 for the next iteration. This procedure of solving eq 35 followed by solving eq 30 can be repeated until the values of composition and temperature converge. Flat Interface. The governing equations for a system with a flat interface can be derived by using the same Gibbsian composite-system approach that was used for the derivation of bubble and dew temperature equations for a system with

Bi + Ci ln(T ) + DiT Ei T

(41)

where the empirical parameters Ai, Bi, Ci, Di, and Ei are component-specific. The liquid molar volume of each component can be described by the Rackett equation,83,84 but because the Rackett equation is not valid for alcohols,84 the following more generalized form can be used instead:85 ni

viL = ai × bi(1 − T / ci)

(42)

where ai, bi, ci, and ni are component-specific empirical parameters. The surface tension of a pure component is also a function of temperature, and for pure components, a multiple linear E


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Table 1. Extended Antoine Equation Parameters (Equation 41) for Flat-Interface Vapor Pressure (Pa) and the Range of Validity for All Pure Components Considered in This Study82 nitrogen argon methanol ethanol water

Ai

Bi

Ci

58.282 42.127 82.718 73.304 73.649

−1084.1 −1093.1 −6904.5 −7122.3 −7258.2

−8.3144 −4.1425 −8.8622 −7.1424 −7.3037

Di 0.044127 5.7254 × 7.4664 × 2.8853 × 4.1653 ×

10−5 10−6 10−6 10−6

Ei

Tmin (K)

Tmax (K)

1 2 2 2 2

63.15 83.78 175.47 159.05 273.16

126.2 150.86 512.50 514.00 647.10

Table 2. Rackett Equation Parameters (Equation 42) for Liquid Molar Volume (mL/mol) and the Range of Validity for All Pure Components Considered in This Study85 nitrogen argon methanol ethanol water

ai

bi

ci

ni

Tmin (K)

Tmax (K)

89.1527 75.2038 117.802 166.917 55.432

0.2861 0.286 0.27206 0.27668 0.27

126.2 150.86 512.58 516.25 647.13

0.2966 0.2984 0.2331 0.2367 0.23

63.15 83.78 175.47 159.05 290.00

126.20 150.86 512.58 516.25 647.13

Table 3. Polynomial Coefficients (Equation 43) for Surface Tension (N/m) and the Range of Validity for Pure Methanol, Ethanol, and Water82,85 methanol82 ethanol82 water85

θ0,i

θ1,i

θ2,i

θ3,i

θ4,i

θ5,i

Tmin (K)

Tmax (K)

0.03513 0.03764 −0.01073

−7.04 × 10−6 −2.157 × 10−5 1.299 × 10−3

−1.216 × 10−7 −1.025 × 10−7 −6.544 × 10−6

0 0 1.504 × 10−8

0 0 −1.793 × 10−11

0 0 8.635 × 10−15

273.10 273.15 273.16

503.15 503.15 647.10

regression can be performed on experimental data with a polynomial equation:

Table 4. Coefficients (Equation 44) for Surface Tension (mN/m) and the Range of Validity for Pure Nitrogen, Argon, and Krypton86

pi

σi =

∑ θj ,iT j j=0

(43)

nitrogen argon

where pi is the degree of the polynomial used for component i, and θj,i are the empirically determined jth order coefficients for component i. Higher order terms can extend the accuracy of the equation to temperatures up to the component’s critical point. Another common equation used to describe the surface tension of pure components is of the following form:86−88 βi ⎛ T ⎞ ⎜ ⎟ σi = αi⎜1 − ⎟ Tc, i ⎠ ⎝

αi

Tc,i

βi

Tmin (K)

Tmax (K)

29.324108 37.898063

126.2 150.66

1.259 1.278

65 84

120 145

combination of mole fraction and pure component surface tension:43,89 σ = σ1x1 + σ2x 2 (45) Of interest in this study, the methanol/ethanol system has been confirmed to behave ideally from experimental measurements,90 and therefore eq 45 can be used to calculate this mixture’s surface tension as a function of composition. Substituting eq 43 into eq 45 for each component gives the following equation that captures the dependence of surface tension on liquid-phase composition and temperature:

(44)

where αi and βi are fitting coefficients and Tc,i is the liquid’s critical temperature. In this study, a nitrogen(1)/argon(2) system is first selected to compare dew-point predictions with previously published experimental work. Then an ideal methanol(1)/ethanol(2) system and a nonideal ethanol(1)/water(2) system then are selected, and their phase behavior is modeled. The parameters for flat-interface saturation vapor pressure, liquid molar volume, and surface tension are summarized in Tables 1−4 for pure nitrogen, argon, methanol, ethanol, and water. These parameters can be used for temperatures between the listed Tmin and Tmax values. Liquid Mixture Properties. For the nonideal liquid mixture of ethanol(1)/water(2), the constants used in the Margules activity coefficient model are A12 = 1.6252 and A21 = 0.8610 for a pressure of 101 325 Pa.81 In eqs 30, 31, 32, and 35, the surface tension is a function of liquid-phase composition and temperature. For an ideal liquid mixture, surface tension can be calculated using a linear

⎞ ⎛ p1 σ = ⎜⎜∑ θj ,1T j⎟⎟x1 + ⎠ ⎝ j=0

⎞ ⎛ p2 ⎜ ∑ θ T j ⎟x j ,2 ⎟ 2 ⎜ ⎠ ⎝ j=0

(46)

The surface tension of a nonideal mixture such as ethanol/ water has been described as a function of liquid-phase composition by numerous models, such as those proposed by Shereshefsky,91 Connors and Wright,92 and Li et al.93 In the current study, an equation is needed that additionally incorporates the influence of temperature on the nonideal mixture’s surface tension. As shown by Shardt and Elliott,94 an extension of the Connors−Wright model92 (the SE−CW model) is best for calculating the temperature dependence of surface tension for aqueous binary mixtures. The original Connors−Wright model is given by92 F


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Figure 2. Surface tension of ethanol(1)/water(2) as a function of composition and temperature, as measured experimentally by Vázquez et al.95 and as calculated by eq 48.

⎛ b(1 − x1) ⎞ σ = σ2 − ⎜1 + ⎟x1(σ2 − σ1) 1 − a(1 − x1) ⎠ ⎝

β2 ⎛ T ⎞ ⎜ ⎟ σ = α 2 ⎜1 − ⎟ Tc,2 ⎠ ⎝

(47)

β2 β1⎞ ⎛ ⎛ ⎛ b(1 − x1) ⎞ ⎜ ⎛ T ⎞ T ⎞ ⎟ ⎟⎟ − α1⎜⎜1 − ⎟⎟ − ⎜1 + ⎟x1 α2⎜⎜1 − 1 − a(1 − x1) ⎠ ⎜ ⎝ Tc,2 ⎠ Tc,1 ⎠ ⎟ ⎝ ⎝ ⎠ ⎝

where a and b are system-specific parameters fit to surface tension data as a function of x1 at a single temperature, σ2 is the surface tension of pure water, and σ1 is the surface tension of the second component. Substituting the temperature-dependent pure component surface tension (eq 43) into eq 47 gives the following expression that describes the dependence of surface tension on composition and temperature (the SE−CW model):94

(49)

At 83.82 K, the parameters and 95% confidence intervals for nitrogen/argon in eq 49 are a = 0.6295 ± 0.0457 and b = 0.3244 ± 0.0161 for the data reported in Sprow and Prausnitz,96 and the fit is shown in Figure 3.

p2

σ=

∑ θj ,2T j j=0

⎛ p ⎛ b(1 − x1) ⎞ ⎜ 2 − ⎜1 + ⎟x1⎜∑ θj ,2T j − 1 − a(1 − x1) ⎠ ⎝ j = 0 ⎝

p1

⎞

j=0

⎠

∑ θj ,1T j⎟⎟ (48)

Figure 2 compares eq 48 to experimental data for the surface tension of ethanol(1)/water(2) as a function of liquid composition and temperature. For ethanol/water at 293 K, the parameters and 95% confidence intervals for eq 48 are a = 0.958 ± 0.002 and b = 0.926 ± 0.009 for the data reported in Vázquez et al.95 when using eq 43 at 293 K with the coefficients for ethanol and water listed in Table 3. When these parameters are used in eq 48 to predict the surface tension at higher temperatures, the average absolute percent error between model values and experimental points is 0.49%, with a maximum percent error of 1.38%. The parameters are not the same as those reported for ethanol/water in Shardt and Elliott,94 because higher-order polynomials are used in the current study to predict surface tension at temperatures greater than previously considered. For nitrogen(1)/argon(2), available surface tension data as a function of concentration are limited to a single temperature. The SE−CW equation can be used to extrapolate to higher temperatures, because it was shown that the parameters are temperature-independent for the majority of systems previously studied.94 For nitrogen(1)/argon(2), eq 44 can be substituted into eq 47 for each pure component to give

Figure 3. Surface tension as a function of composition at 83.82 K for nitrogen(1)/argon(2) measured experimentally by Sprow and Prausnitz96 and the fit of eq 49 to this data set.

Numerical Methods. For a given pressure, overall composition, and radius of interfacial curvature, solving eqs 30 and 31 with eq 32 yields the bubble and dew temperatures. Fixed-point iteration methods are outlined in Figures 4 and 5 for calculating the bubble temperature and dew temperature, respectively, at a given composition. The tolerance ε for relative error between iteration steps is calculated using97,98 ε = 0.5 × 10−n

(50)

where n is the minimum desired number of accurate significant figures. Once the relative error reaches a value less than the tolerance, iteration is stopped. In this study, the tolerance for relative error is chosen for accuracy to five significant figures. MATLAB R2016b (Natick, U.S.) was used for calculations. G


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Figure 4. Flowchart for calculating the bubble temperature of a binary system with a flat interface and with a curved interface for a given composition in the liquid phase.

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RESULTS AND DISCUSSION Comparison with Experimental Data. Alam et al.99 reported experimental results for the temperature at which phase transitions occur during adsorption and desorption for mixtures of nitrogen(1)/argon(2) in Vycor glass with an average cylindrical pore diameter of 4 nm99,100 at a constant vapor-phase pressure of 300 kPa, as shown by the symbols in Figure 6. The symbols indicate the midpoints of the phase transition, and the error bars on each symbol show the start and end of the transition; the values were digitized from the figure published by Alam et al.99 According to the classical model of adsorption and desorption, only desorption has a geometry resembling the hemispherical interface between the liquid and vapor phases seen in Figure 1b for a cylindrical pore. Adsorption (and capillary condensation) is hypothesized to occur first on the walls of the cylindrical pore, creating a thin cylindrical layer of liquid. The curvature of an adsorbed cylinder of liquid is equal to one-half the curvature of a hemisphere in the same pore (curvature is 1/rp for a cylinder and 2/rp for a sphere with radius rp).77,101,102 That is, r in the governing equations derived herein equals −2 nm for desorption and r equals −4 nm for adsorption. The predictions of dew point

using these radii are plotted in Figure 6 as a function of nitrogen mole fraction. In the context of the adsorption and desorption experiments, the dew point corresponds to the first appearance of liquid during the adsorption process, and it corresponds to the last disappearance of liquid during the desorption process. The dew point predictions agree closely with the experimental data points (the average absolute percent error between predictions and data is 0.45%), and, most importantly, the theoretical predictions are completely independent of the experiments. The governing equations only require the bulk properties of the pure liquids and the pore radius of the experimental porous medium. This is in contrast to the model proposed by Tan and Piri55 that relies on additional fitting parameters for the behavior of pure nitrogen and pure argon in pores; their fitting parameters are functions of temperature and pore radius. Jones and Fretwell100 reported experimental phase-transition temperatures as a function of mole fraction for argon(1)/krypton(2), but the reported trends are not consistent with those reported by Alam et al.99 for nitrogen(1)/argon(2) using the same experimental technique; thus the argon(1)/krypton(2) data cannot be interpreted within the thermodynamic framework proposed in this Article. H


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Figure 5. Flowchart for calculating the dew temperature of a binary system with a flat interface and with a curved interface for a given composition in the vapor phase.

That is, the adsorption curve for argon(1)/krypton(2) is concave up and is proposed to correspond to the bubble point,100 whereas the adsorption curve for nitrogen(1)/ argon(2) is concave down and corresponds to the dew point for each vapor-phase composition. Predictions for Methanol/Ethanol and Ethanol/Water. Given the success of Gibbsian composite-system thermodynamics in predicting the phase behavior of nitrogen(1)/ argon(2) in porous media, systems of methanol(1)/ethanol(2) and ethanol(1)/water(2) are now considered, and predictions are made for a pressure of 101 325 Pa in the vapor phase. Isobaric phase diagrams and x−y plots are illustrated in Figures 7 and 8 for these systems, respectively, at various positive and

negative radii of curvature. The phase envelope outlined in black is for a system with a flat interface between the liquid and vapor phases. For both systems, the bubble and dew temperatures decrease for positive radii of curvature, whereas for negative radii of curvature, the bubble and dew temperatures increase. As the absolute value of the radius of curvature decreases, the temperatures move farther away from the flatinterface phase envelope. If the flat-interface phase envelope were to be used for phase predictions of systems with significant curvature, it would yield increasingly inaccurate results. What is the physical meaning behind the shift of the phase envelope in the presence of interfacial curvature? For a given I


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Figure 6. Prediction (not a fit) of adsorption and desorption dew temperatures as functions of nitrogen mole fraction for nitrogen(1)/ argon(2) in a capillary with a radius of 2 nm and experimental measurements of adsorption and desorption at a vapor-phase pressure of 300 kPa from Alam et al.99 The schematic of the classical model of adsorption and desorption is adapted from Donohue and Aranovich.103

Figure 8. Predictions of (a) the phase diagram and (b) the x−y plot for ethanol(1)/water(2) as a function of interfacial radius of curvature for PV = 101 325 Pa.

Considering a negative radius of curvature, a hemispherical vapor phase condenses at a temperature higher than the dew temperature of a planar liquid−vapor interface, because the bubble is at an elevated pressure as compared to the liquid phase in the capillary. For the bubble point of a system with interfacial curvature, a temperature higher than the bubble temperature of a planar interface is needed to evaporate a liquid phase with a hemispherical interface in a capillary, also because of the elevated pressure in the vapor phase. As a result, a bulk liquid can become superheated until a vapor bubble forms.43 The isobaric x−y curves show trends as a function of radius of curvature similar to those seen for the isothermal x−y plots of each system.73 That is, for methanol(1)/ethanol(2), the curve moves closer to the 45° line for positive radii and away from the line for negative radii. For ethanol(1)/water(2), the x−y curve moves farther away from the 45° line for positive radii and closer for negative radii. For a given vapor-phase composition in either system, a positive radius of curvature decreases the equilibrium liquid mole fraction of ethanol compared to the flat-interface value, and a negative radius increases the equilibrium liquid mole fraction of ethanol. Although the trends are similar, the magnitude of the x−y curve shift differs between isothermal and isobaric conditions. To compare the magnitude of this shift, the isothermal liquid composition for each vapor mole fraction can be calculated as outlined in Shardt and Elliott73 using the equations for flatinterface saturation pressure, liquid molar volume, and surface tension as described in this Article evaluated at 298 K. Figure 9 compares the difference between curved-interface liquid composition and flat-interface liquid composition at the dew point as a function of vapor phase composition for methanol(1)/ethanol(2) and ethanol(1)/water(2) for a radius of curvature of r = 3 nm at constant temperature and at constant PV. Under isothermal conditions, the maximum

Figure 7. Predictions of (a) the phase diagram and (b) the x−y plot for methanol(1)/ethanol(2) as a function of interfacial radius of curvature for PV = 101 325 Pa.

vapor-phase pressure, it is easier to evaporate a liquid droplet (positive radius of curvature) because of the elevated pressure in the liquid phase; a temperature lower than the bubble temperature of a planar liquid interface is therefore needed to evaporate a spherical liquid droplet. Similarly, a temperature lower than the dew temperature for a planar interface is needed to start the condensation of a vapor phase into a spherical liquid droplet because of the higher liquid-phase pressure. The multicomponent vapor phase is supercooled below its planar dew temperature until a phase change occurs to form a highly curved multicomponent liquid phase, as described by Defay and Prigogine for a single component.43 J


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Figure 10. Predictions of the azeotropic composition as a function of radius of curvature for ethanol(1)/water(2) (a) at a pressure of 101 325 Pa in the vapor phase and (b) at a temperature of 298 K.

Figure 9. Difference between the 3 nm curved-interface liquid-phase composition and the flat-interface liquid-phase composition at the dew point as a function of vapor-phase composition for (a) methanol(1)/ ethanol(2) and (b) ethanol(1)/water(2) at a constant PV of 101 325 Pa (solid line) and at a constant temperature of 298 K (dashed line).

difference in liquid composition relative to the flat-interface liquid composition, |x1c − x1f|, is 0.02658, while the maximum difference is just 0.01532 under isobaric conditions for methanol(1)/ethanol(2) at the dew point. For ethanol(1)/ water(2), the maximum isothermal and isobaric differences are 0.1960 and 0.1886, respectively, for r = 3 nm. Therefore, the presence of a nanocurved interface has a greater impact on vapor−liquid equilibrium at a constant temperature of 298 K than it does at a constant PV of 101 325 Pa. For the ethanol(1)/water(2) system, the isobaric azeotropic composition can be calculated as a function of radius of curvature, as illustrated in Figure 10; azeotropic composition is plotted as a function of the inverse radius in Figure 11. The dashed black line in Figure 10 is the azeotropic composition for a system with a flat interface between the vapor and liquid phases. For positive radii of curvature, the azeotrope shifts toward 100 mol % ethanol, whereas for negative radii of curvature, the azeotrope shifts in the opposite direction. Such a trend was also seen under isothermal conditions,73 but, as observed for the x−y curves, the magnitude of the shift differs under constant vapor-phase pressure conditions. The isobaric (PV = 101 325 Pa) azeotrope vanishes for positive radii of curvature less than 17 nm, while the isothermal (298 K) azeotrope vanishes for positive radii of curvature less than 31 nm. It takes a smaller radius of curvature to remove the azeotrope at a constant vapor-phase pressure of 101 325 Pa than it does at a constant temperature of 298 K.

Figure 11. Predictions of the azeotropic composition as a function of inverse radius of curvature for ethanol(1)/water(2) at a pressure of 101 325 Pa in the vapor phase (solid line) and at a temperature of 298 K (dashed line).

curvature between the vapor and liquid phases were derived. On the basis of these conditions, the partial phase diagram of nitrogen/argon and the complete phase diagrams of methanol/ ethanol and ethanol/water systems were calculated. Predictions of dew point for nitrogen/argon agreed with experimental data of adsorption and desorption reported in the literature with an average percent difference of 0.45%. Over all systems, the phase envelopes shifted to lower bubble and dew temperatures for positive radii of curvature and shifted to higher temperatures for negative radii of curvature, with a greater shift for radii closer to zero. For the nonideal ethanol/water system, the azeotropic composition moved toward 100 mol % ethanol for positive radii, as seen for isothermal conditions, but the magnitude of the change differed.

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CONCLUSIONS From Gibbsian composite-system thermodynamics, the equilibrium conditions for a multicomponent system with interfacial K


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The outlined thermodynamic approach can be applied to spherical phases within bulk phases and to phases confined in capillaries. While any solid−liquid interactions that would be present in capillaries are neglected in the present analysis, by doing so, the effect of curvature alone is isolated. It is indeed valuable to ascertain the magnitude of the curvature effect in comparison to other forces so that the relative importance of each contribution can be determined, and a more complete understanding of nanoscale phase equilibrium can be reached in applications ranging from hydrocarbon recovery and atmospheric physics to catalysis and drug delivery.

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AUTHOR INFORMATION

Corresponding Author

*Phone: (780) 492-7963. E-mail: [email protected]. ORCID

Janet A. W. Elliott: 0000-0002-7883-3243 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. William R. Smith of the University of Guelph for bringing some recent literature to our attention. N.S. acknowledges funding from the Natural Sciences and Engineering Research Council of Canada (NSERC), Alberta Innovates and Alberta Advanced Education, the Government of Alberta, and the University of Alberta. J.A.W.E. holds a Canada Research Chair in Thermodynamics.

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