Thermodynamic Study of the Role of Interface Curvature on

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Thermodynamic Study of the Role of Interface Curvature on Multicomponent Vapor−Liquid Phase Equilibrium Nadia Shardt and Janet A. W. Elliott* Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada ABSTRACT: The effect of interface curvature on phase equilibrium has been much more studied for single-component than multicomponent systems. We isolate the effect of curvature on multicomponent vapor− liquid equilibrium (VLE) phase envelopes and phase composition diagrams using the ideal system methanol/ethanol and the nonideal system ethanol/water as illustrative examples. An important finding is how nanoscale interface curvature shifts the azeotrope (equal volatility point) of nonideal systems. Understanding of the effect of curvature on VLE can be exploited in future nanoscale prediction and design.

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INTRODUCTION Vapor−liquid equilibrium (VLE) drives chemical processes that occur under conditions yielding two phases. A phase diagram outlines the temperature or pressure at which a liquid, vapor, or mixture of phases will exist. Curvature of the vapor−liquid interface alters equilibrium pressure in the vapor phase, as described first by the Kelvin equation1 and later corrected by Gibbs.2−4 This equation is valid for a pure component assuming an incompressible liquid and an ideal gas. The Kelvin equation and its liquid−liquid and liquid−solid analogues describe capillary condensation, adsorption hysteresis,5−12 nucleation,13−22 and emulsion stability.23 Single-component systems have been extensively examined theoretically.13−16,24−26 Early experimental work27−29 and more recent work30−33 verified the Kelvin equation, with discrepancies observed for systems containing water.34,35 Using certain capillary plates results in deviations from the Kelvin equation36−43 due to molecular interactions between the liquid and the capillary walls,37 adsorption, or surface contamination.34 Closer experimental agreement was achieved by Shin and Simandl44−48 for multicomponent mixtures. In this work, we apply Gibbsian thermodynamics to multicomponent mixtures to calculate phase envelopes as a function of curvature. We gain an understanding of how an azeotrope shifts with curvature. The thermodynamics of curved interfaces can be applied in atmospheric science,49,50 hydrocarbon recovery,51,52 distillation,42,53,54 and catalyst design.55−57

separated by an interface, the following equation determines equilibrium states: dS L + dS V + dS LV = 0

where S is entropy and the superscripts L, V, and LV denote the liquid phase, the vapor phase, and the liquid−vapor interface, respectively. Substituting the physical constraints of an isolated multicomponent system into eq 1 yields equations for thermal, mechanical, and chemical equilibrium. For an isolated ncomponent liquid−vapor system with a flat liquid−vapor interface, the conditions for equilibrium (in accordance with Gibbs’ dividing surface approach) are found to be

T V = T L = T LV

(2)

PV = P L

(3)

μi L = μi V for i = 1

(4)

μi L = μi V = μi LV for i = 2, 3, ..., n

(5)

where T is absolute temperature, P is pressure, μ is chemical potential, and the subscript i refers to the ith component. For a spherical liquid−vapor interface, the conditions for equilibrium (in accordance with Gibbs’ surface of tension approach) are

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GOVERNING EQUATIONS Gibbsian composite-system thermodynamics2 provides a framework to determine the equilibrium states of a given composite systema system that is made up of simple systems, but that itself is not a simple system, such as a multicomponent multiphase system. A stable equilibrium state occurs when the entropy of the composite system is at a maximum.58 That is, for an isolated system consisting of a vapor and a liquid phase © XXXX American Chemical Society

(1)

T V = T L = T LV

(6)

P L − PV = 2σ /r

(7)

μi L = μi V = μi LV for i = 1, 2, 3, ..., n

(8)

Equation 7 is the Young−Laplace equation for a spherical interface,59 where σ is the surface tension of the liquid mixture and r is the radius of curvature of the interface. As illustrated in Received: October 25, 2015 Revised: January 14, 2016

A

DOI: 10.1021/acs.jpca.5b10450 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A Figure 1, the radius of curvature appearing in eq 7 is defined here to be positive for a convex liquid surface in a capillary or

If the liquid phase is an ideal mixture (γi = 1), eq 13 becomes generalized Raoult’s Law: V yP = xiPisat exp(ViL(P L − Pisat)/(RT )) i

(14)

The exponential term in eq 13 or eq 14 is known as the Poynting correction,61,63,64 and at low pressures and high temperatures, this term becomes unity. Setting the Poynting correction to 1 in eq 14 results in Raoult’s Law, which is commonly used in designing equipment that operates at such moderate conditions. Flat Interface Vapor−Liquid Equilibrium. Vapor−liquid equilibrium calculation procedures for flat interfaces are described in textbooks.61,62 For example, Raoult’s Law given by V yP = xiPisat i

is used under conditions for which the Poynting correction can be neglected, and it is valid when assuming an ideal gas in the vapor phase and an ideal incompressible solution in the liquid phase. The bubble pressure is the pressure at which the first bubble of vapor appears, that is, the equilibrium pressure in the vapor phase for known xi. For a flat interface, the equilibrium pressure in the vapor phase is equal to the pressure in the liquid phase. The bubble pressure can be calculated by combining eq 15 for i components with ∑i yi = 1:

Figure 1. System configurations of positive and negative radii of curvature.

for a droplet of liquid in vapor and to be negative for a concave liquid surface in a capillary or for a bubble of vapor in liquid. The equations for equilibrium can be solved by inserting the equilibrium pressures and temperatures of the phases into the equality of chemical potentials of the liquid and vapor phases (eq 4 and the first equivalence of eqs 5 and 8) and used to determine the composition and quantity of vapor and liquid phases. Equations of state are used to write chemical potentials as functions of temperature, pressure, and composition. For a multicomponent system with assumptions of an ideal gas and an incompressible liquid, the chemical potentials of the vapor and liquid phases for each component are given by60,61 V μi V (T , PV ) = μi V (T , Pisat) + RT ln(yP /Pisat) i

PbV =

(9)

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

PbV =

(11)

∑ xiPisatexp(ViL(PbV − Pisat)/(RT ))

(18)

To obtain an expression for the dew pressure, eq 14 is rearranged to the form

(12)

where A12 and A21 are species-dependent empirical constants at a given temperature. The activity coefficient captures the nonideality of liquid mixtures (e.g., ethanol/water) where the magnitudes of intermolecular interactions are unequal between the components of the mixture. Substituting eqs 9 and 10 into the equality of chemical potentials between the vapor and liquid phases (eqs 4, 5, and 8) results in generalized modified Raoult’s Law:61 V yP = γixiPisat exp(ViL(P L − Pisat)/(RT )) i

(17)

i

and + 2(A12 − A 21)x 2)

⎛ ⎞−1 sat ⎟ ⎜ = ⎜∑ yi /Pi ⎟ ⎝ i ⎠

where the subscript “d” stands for dew. A more general expression of Raoult’s Law should be used in cases of high liquid-phase pressure. For a flat interface between an ideal liquid mixture and an ideal vapor phase, the pressure in the vapor equals the pressure in the liquid (PV = PL), and an expression for bubble pressure can be obtained by combining eq 14 and ∑i yi = 1, as shown in eq 18:

(10)

respectively. Here, yi is the mole fraction of component i in the vapor phase, xi is the mole fraction in the liquid phase, Psat i is the saturation pressure at a given temperature, VLi is the liquid molar volume of pure component i,61 R is the universal gas constant, and γi is the species-dependent activity coefficient. The activity coefficient γi can be modeled, for example, by the Margules equations for a two-component system as follows62

ln(γ2) =

(16)

where the subscript “b” stands for bubble. The dew pressure is the pressure at which the first drop of liquid appears for known yi, that is, the equilibrium pressure in the vapor phase. Rearranging eq 15 in terms of xi for each component and using ∑i xi = 1 gives the following equation to calculate the dew pressure: PdV

x12(A 21

∑ xiPisat i

and

ln(γ1) = x 22(A12 + 2(A 21 − A12 )x1)

(15)

xi =

V yP i

Pisat exp(ViL(PV − Pisat)/(RT ))

(19)

where again the fact that PV = PL for a flat interface is used. Equation 19 is then combined with ∑i xi = 1 to obtain the following dew pressure expression: PdV

(13) B

⎛ ⎞−1 yi ⎟⎟ = ⎜⎜∑ sat L V sat ⎝ i Pi exp(Vi (Pd − Pi )/(RT )) ⎠

(20)


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The Journal of Physical Chemistry A

The vapor-phase pressure at the dew point is determined by using eq 14 with ∑i xi = 1 and the relationship in eq 7 to obtain

Iteration is used to solve eq 18 and eq 20 with an initial guess of PVd calculated from eq 16 and eq 17, respectively, until the value for pressure in the vapor phase converges. For a flat interface between a nonideal liquid mixture and an ideal vapor phase, PV = PL, and using ∑i yi = 1 and eq 13, the bubble pressure is PbV =

∑ γixiPisatexp(ViL(PbV − Pisat)/(RT )) i

PdV

(26)

Equations 25 and 26 are both solved iteratively with an initial guess of 1 for the Poynting correction. For a nonideal liquid mixture, eq 13 is used with ∑i yi = 1 and eq 7 to create an expression for the pressure in the vapor phase of a curved interface at the bubble point:

(21)

By first assuming a Poynting correction value of 1, the bubble pressure can be calculated iteratively using eq 21 together with eqs 11 and 12. Then this calculated value for bubble pressure is used in the exponential term of eq 21 for the next iteration, and this is repeated until the bubble pressure value converges. The dew pressure expression is derived similarly to that of eq 20, where eq 13 is now rearranged and combined with ∑i xi = 1 and PV = PL, to obtain ⎛ ⎞−1 yi V ⎜ ⎟⎟ Pd = ⎜∑ sat L V sat ⎝ i γiPi exp(Vi (Pd − Pi )/(RT )) ⎠

PbV =

(27)

Equations 13 and 7, along with ∑i xi = 1, are combined to obtain the following expression for the pressure in the vapor phase of a curved interface at the dew point:

(22)

PdV

In eq 28, the values of γi and the exponential Poynting correction are functions of liquid mole fraction, which is initially unknown. Therefore, an iterative method is used to calculate the dew pressure and the composition in the liquid and vapor phases. To determine the value of the liquid mole fraction for the dew pressure calculation, eq 13 can be written for each of the two components, each rearranged to form an expression for pressure, and equated. Then, combining this result with eq 7 and simplifying gives

y

− P1sat)/(RT ))

y2

(23)

⎛ γ P sat exp((V L(PV + 2σ /r − P sat))/(RT ))y ⎞−1 1 d 1 1 1 2 ⎜ x1 = ⎜ sat + 1⎟⎟ L V sat ⎝ γ2P2 exp((V2 (Pd + 2σ /r − P2 ))/(RT ))y1 ⎠

Rearranging for x1 gives ⎛ γ P sat exp(V L(PV − P sat)/(RT ))y ⎞−1 1 d 1 1 1 2 ⎜ x1 = ⎜ sat + 1⎟⎟ L V sat ⎝ γ2P2 exp(V2 (Pd − P2 )/(RT ))y1 ⎠

(29) (24)

First, initial guesses of 1 are used for the Poynting correction and Margules activity coefficients in eqs 28 and 29 to estimate the dew pressure and liquid mole fraction values. These estimated dew pressure and liquid mole fraction values are then substituted into eq 28 to update the value for dew pressure, followed by eq 29 to update the liquid mole fraction value. Iteration is repeated until the values of PVd and x1 converge. Each pair of liquid mole fraction and vapor mole fraction values can be plotted to form an x−y composition diagram, a useful tool in the design of chemical processes. It gives a qualitative indication of the ease of separation based on the distance of the x−y curve from a 45° line that indicates equal composition in the two phasesthe smaller the distance, the harder the separation. If the x−y curve intersects the 45° line, an azeotrope exists at this composition and indicates that conventional distillation past this point is not possible. Composition of the Azeotrope. A defining property of an azeotrope is exactly that the mole fraction of each species is the same in the vapor phase as in the liquid phase. That is, xi = yi, so that for a two-component mixture

The values of the Margules activity coefficients and the exponential Poynting correction are updated in the next iteration of evaluating eq 22, followed again by eq 24, until the values of dew pressure and liquid mole fraction converge. Curved Interface Vapor−Liquid Equilibrium. Whereas the equations and calculation procedures for flat interfaces summarized in the preceding section are well-known to chemical engineers, the analogous equations and calculation procedures for curved interfaces have not been developed previously. For a flat interface, the pressures in the liquid and vapor phases are equal, but as the curvature of the interface varies, the pressure changes in the vapor phase according to eq 7, where the surface tension of the liquid mixture, σ, must be evaluated as a function of liquid mole fraction (x1). For the ideal multicomponent VLE of curved interfaces, eq 14 is used with ∑i yi = 1, and eq 7 is substituted for PL to give the expression PbV =

∑ xiPisatexp(ViL(PbV + 2σ /r − Pisat)/(RT )) i

⎛ ⎞−1 yi ⎟⎟ = ⎜⎜∑ sat L V sat ⎝ i γiPi exp((Vi (Pd + 2σ /r − Pi ))/(RT )) ⎠ (28)

γ1x1P1sat exp(V1L(PdV − P1sat)/(RT ))

=

∑ xiγiPisatexp((ViL(PbV + 2σ /r − Pisat))/(RT )) i

Iteration can also be used to calculate the dew pressure from eqs 11, 12, and 22. An initial guess of γi = 1 and a Poynting correction of 1 are used to estimate the dew pressure in eq 22. For a flat interface, an expression for liquid mole fraction is obtained by using PV = PL and rearranging eq 13 for components 1 and 2, followed by equating these expressions to yield

1 γ2x 2P2sat exp(V2L(PdV

⎛ ⎞−1 yi ⎟⎟ = ⎜⎜∑ sat L V sat ⎝ i Pi exp((Vi (Pd + 2σ /r − Pi ))/(RT )) ⎠

(25)

y1 /x1

to calculate the vapor phase pressure at the bubble point of a multicomponent system with curvature.

y2 /x 2 C

=1 (30) DOI: 10.1021/acs.jpca.5b10450 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Table 1 includes the values of flat surface saturation pressures, molar liquid volumes, and surface tensions of pure

Rearranging and substituting generalized modified Raoult’s Law (eq 13) for each component into the numerator and denominator of eq 30 and combining with eq 7 gives γ1P1sat exp((V1L(PV + 2σ /r − P1sat))/(RT )) γ2P2sat exp((V2L(PV + 2σ /r − P2sat))/(RT ))

Table 1. Flat Surface Saturation Pressure in the Vapor Phase, Molar Liquid Volume, and Surface Tension Values for Pure Methanol, Ethanol, and Water72−74

=1 (31)

The pressure in the vapor phase at the azeotropic composition can be calculated from the bubble pressure equation (eq 27). First, the built-in MATLAB (Natick, USA) function fsolve() is used to solve eq 31 with an initial guess of 1 for the exponential Poynting correction factor. Then the calculated x1 is used to determine the value of PV from eq 27. This pressure in the vapor is then substituted back into eq 31, and an updated x1 is calculated. Such an iterative procedure is repeated until the values of azeotropic composition and vapor-phase pressure converge. Surface Tension. The surface tension of a liquid becomes a function of curvature when the radius of curvature approaches the atomic scale,65 with notable deviations for radii less than 10 nm.66 However, in Gibbsian composite system thermodynamics, the convention of placing the dividing surface according to Gibbs’ surface of tension removes this dependence and allows a constant interfacial tension to be used down to the nanometer scale.2 In the current study, for lack of experimental data or theoretical approximations to the contrary for binary systems containing methanol, ethanol, and water, we assume that the surface tension is independent of curvature. Surface tension varies with liquid mole fraction. For the ideal system of methanol(1)/ethanol(2), a linear combination of each component’s surface tension and its liquid mole fraction can be used. The pure liquid surface tension values of methanol and ethanol differ from each other by less than 1 mN/m. For nonaqueous liquid mixtures with such close values of surface tension, a linear composition dependence accurately describes the mixture’s surface tension:67,68 σ = x1σ1 + x 2σ2

methanol ethanol water

flat surface saturation pressure in the vapor, Psat i @ 298 K (kPa)

molar liquid volume, VLi @ 298 K (cm3/mol)

surface tension, σi @ 298 K (mN/m)

16.94 7.87 3.17

40.73 58.69 18.07

22.07 21.97 72.01

methanol, ethanol, and water all at 298 K.72−74 For the ethanol(1)/water(2) system, the two-parameter Margules equation is used to calculate activity coefficients with constants A12 = 1.4473 and A21 = 0.9395 at 298 K.75

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RESULTS AND DISCUSSION In the above theoretical framework, the Poynting correction accounts for the change in chemical potential of a liquid component with pressure assuming incompressibility. Modified Raoult’s Law takes into account nonideality in the liquid phase with activity coefficients. In this study, we use the Margules form of the activity coefficients for the ethanol(1)/water(2) system. Limitations of this study include assuming an ideal gas in the vapor phase and neglecting the effect of any external forces, such as gravity, in our analysis. The hydrogen bonding in water may alter vapor−liquid phase equilibrium at higher pressures in the vapor phase, but at the low pressures considered in the vapor phase, we treat the vapor ideally. The effect of curvature on VLE can be seen for the ideal methanol(1)/ethanol(2) phase envelopes presented in Figure 3, where pressure in the vapor phase is plotted as a function of

(32)

However, for the nonideal system of ethanol(1)/water(2), surface tension is not a linear combination of the two components’ surface tension values.59,69−71 We therefore fit a function to the experimental data from Vázquez et al.72 as seen in Figure 2 to obtain σ(in mN/m) =

2.46 ± 0.14 + (20.1 ± 0.5) x1 + (0.048 ± 0.003)

Figure 3. Phase envelopes of the methanol(1)/ethanol(2) system for a flat interface and for interfaces with radii of curvature of ±3 and ±10 nm.

(33)

methanol liquid mole fraction. The solid and dashed thick black lines show the bubble and dew pressures, respectively, for a methanol(1)/ethanol(2) system with a flat interface. It can be seen that, as expected, the pressures in the vapor phase increase for positive radii of curvature (liquid inside the curvature) and decrease for negative radii of curvature (vapor inside the curvature). Visually, it appears that the phase envelope is simply translated vertically as a function of radius of curvature, but Figure 4 shows that composition is slightly altered by curvature. As a result, it is more difficult to separate the methanol(1)/ ethanol(2) system when the interface has a positive radius of curvature since the x−y curve lies closer to the 45° line, and

Figure 2. Surface tension as a function of ethanol composition in the ethanol/water system, as measured experimentally by Vázquez et al.72 and as given by the model fit to this data. D


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Figure 7. Azeotropic composition as a function of radius of curvature. The dashed line indicates the azeotropic composition for a flat interface between phases.

Figure 4. Vapor mole fraction as a function of liquid mole fraction for methanol(1)/ethanol(2) phase equilibrium for a flat interface and for interfaces with radii of curvature of ±3 nm. The 45° line for the plot is the thin dashed line.

ethanol), shown by the dashed black line. To achieve an azeotropic composition >99 mol% ethanol, a radius of curvature smaller than 66 nm would be desired. Figure 8

separation is easier when the interface has a negative radius of curvature. Figure 5 illustrates the nonideal ethanol(1)/water(2) phase envelopes for flat and curved vapor−liquid interfaces. As seen

Figure 8. Azeotropic composition of an ethanol(1)/water(2) system as a function of the inverse radius of curvature.

Figure 5. Phase envelopes of the ethanol(1)/water(2) system for a flat interface and for interfaces with radii of curvature of ±3 and ±10 nm.

shows azeotropic composition plotted as a function of the inverse radius of curvature. This variable transformation highlights that a flat interface (1/r = 0) has an azeotropic composition of 97.75 mol% ethanol and that the azeotropic composition increases or decreases for positive or negative radii of curvature, respectively. It may be surprising that the azeotropic composition at a curvature of ±100 nm has not yet reached that of a flat interface. Observable departures from flat interface equilibrium values can in fact occur at radii of curvature less than 1000 nm; for a 100 nm pure water droplet in air, the Kelvin equation predicts a vapor pressure 12% higher than the vapor pressure for a flat interface between air and water.76 Also at a radius of curvature of 100 nm, Defay et al. predict a decrease in azeotropic composition from 0.40 to 0.395 for a mixture of nhexane/ethyl iodide, where the value of surface tension and its dependence on composition were arbitrarily chosen.68 In addition to illustrating curvature-dependent phase diagrams, our work lends insight into the predicted change in azeotropic composition with the use of an actual function fit to experimentally measured values of surface tension for ethanol(1)/water(2).

for the methanol(1)/ethanol(2) system, a positive radius of curvature increases the pressure in the vapor phase, and a negative radius of curvature decreases pressure in the vapor phase. Most importantly, for the flat interface at a constant temperature of 298 K, the azeotrope lies at 97.75 mol% ethanol, whereas for positive radii of curvature, the azeotrope shifts toward a liquid composition of ethanol closer to 100 mol%. This is highlighted by Figure 6, where the x-y curve for positive radii of curvature intersects the 45° line at values closer to x1 = 1. In contrast, negative radii of curvature result in azeotropic compositions less than that observed for a flat interface in the ethanol(1)/water(2) system. In Figure 7, as the radius of curvature approaches infinity, the limit of the composition is that of a flat interface (97.75 mol%

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CONCLUSIONS We ultimately investigated the effect of curvature on VLE in methanol(1)/ethanol(2) and ethanol(1)/water(2) mixtures, as well as its effect on the location of the azeotrope for the nonideal mixture. This analysis can be applied to capillaries whose solid−liquid interactions with the binary mixtures are negligible and to droplets or bubbles contained in their respective vapor or liquid phase. For a positive radius of

Figure 6. Vapor mole fraction as a function of liquid mole fraction for ethanol(1)/water(2) phase equilibrium for a flat interface and for interfaces with radii of curvature of ±3 nm. The 45° line for the plot is the thin dashed line. E


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(15) Zargarzadeh, L.; Elliott, J. A. W. Comparative Surface Thermodynamic Analysis of New Fluid Phase Formation between a Sphere and a Flat Plate. Langmuir 2013, 29, 3610−3627. (16) Zargarzadeh, L.; Elliott, J. A. W. Surface Thermodynamic Analysis of Fluid Confined in a Cone and Comparison with the Sphere−Plate and Plate−Plate Geometries. Langmuir 2013, 29, 12950−12958. (17) Vehkamäki, H. Classical Nucleation Theory in Multicomponent Systems; Springer Science & Business Media: Berlin, Germany, 2006. (18) Noppel, M.; Vehkamäki, H.; Winkler, P. M.; Kulmala, M.; Wagner, P. E. Heterogeneous Nucleation in Multi-Component Vapor on a Partially Wettable Charged Conducting Particle. II. The Generalized Laplace, Gibbs−Kelvin, and Young Equations and Application to Nucleation. J. Chem. Phys. 2013, 139, 134108. (19) Eslami, F.; Elliott, J. A. W. Design of Microdrop Concentrating Processes. J. Phys. Chem. B 2013, 117, 2205−2214. (20) Eslami, F.; Elliott, J. A. W. Stability Analysis of Microdrops during Concentrating Processes. J. Phys. Chem. B 2014, 118, 3630− 3641. (21) Eslami, F.; Elliott, J. A. W. Role of Precipitating Solute Curvature on Microdrops and Nanodrops during Concentrating Processes: The Non-Ideal Ostwald-Freundlich Equation. J. Phys. Chem. B 2014, 118, 14675−14686. (22) Elliott, J. A. W.; Voitcu, O. On the Thermodynamic Stability of Liquid Capillary Bridges. Can. J. Chem. Eng. 2007, 85, 692−700. (23) Kabalnov, A. S.; Pertzov, A. V.; Shchukin, E. D. Ostwald Ripening in Two-Component Disperse Phase Systems: Application to Emulsion Stability. Colloids Surf. 1987, 24, 19−32. (24) Lago, M.; Martin, R.; Araujo, M. Liquid-Vapor Isotherm in a Closed Single-Component System with Curved Interfaces. J. Colloid Interface Sci. 2003, 267, 429−444. (25) Melrose, J. C. Thermodynamic Aspects of Capillarity. Ind. Eng. Chem. 1968, 60, 53−70. (26) Buff, F. P. The Spherical Interface. I. Thermodynamics. J. Chem. Phys. 1951, 19, 1591−1594. (27) Gudris, N.; Kulikowa, L. Die Verdampfung kleiner Wassertropfen. Eur. Phys. J. A 1924, 25, 121−132. (28) Thomä, M. Ü ber die an gekrümmten Flüssigkeitsoberflächen auftretenden Dampfdruckänderungen. Eur. Phys. J. A 1930, 64, 224− 236. (29) Skinner, L. M.; Sambles, J. R. The Kelvin Equation - A Review. J. Aerosol Sci. 1972, 3, 199−210. (30) La Mer, V. K.; Gruen, R. A Direct Test of Kelvin’s Equation Connecting Vapour Pressure and Radius of Curvature. Trans. Faraday Soc. 1952, 48, 410−415. (31) Kohonen, M. M.; Christenson, H. K. Capillary Condensation from Vapors of n-Hexane/Perfluoro-n-hexane Mixtures. J. Phys. Chem. B 2002, 106, 6685−6695. (32) Fisher, L. R.; Israelachvili, J. N. Direct Experimental Verification of the Kelvin Equation for Capillary Condensation. Nature 1979, 277, 548−549. (33) Fisher, L. R.; Israelachvili, J. N. Experimental Studies on the Applicability of the Kelvin Equation to Highly Curved Concave Menisci. J. Colloid Interface Sci. 1981, 80, 528−541. (34) Fisher, L. R.; Gamble, R. A.; Middlehurst, J. The Kelvin Equation and the Capillary Condensation of Water. Nature 1981, 290, 575−576. (35) Shereshefsky, J. L. A Study of Vapor Pressures in Small Capillaries. Part I. Water Vapor. (A). Soft Glass Capillaries. J. Am. Chem. Soc. 1928, 50, 2966−2980. (36) Abu Al-Rub, F. A.; Akili, J.; Datta, R. Distillation of Binary Mixtures with Capillary Porous Plates. Sep. Sci. Technol. 1998, 33, 1529−1550. (37) Abu Al-Rub, F. A.; Datta, R. Theoretical Study of Vapor Pressure of Pure Liquids in Porous Media. Fluid Phase Equilib. 1998, 147, 65−83. (38) Abu Al-Rub, F. A.; Datta, R. Theoretical Study of Vapor-Liquid Equilibrium Inside Capillary Porous Plates. Fluid Phase Equilib. 1999, 162, 83−96.

curvature (liquid inside the curvature), the azeotropic composition approaches 100 mol% ethanol, whereas for a negative radius of curvature (vapor inside the curvature), the azeotropic composition shifts in the opposite direction. This paper provides new fundamental understanding and calculation procedures for multicomponent phase equilibrium across curved interfaces, and it introduces curved-interface phase envelope and phase composition diagrams based on Gibbsian thermodynamics. The insight gained can be used to describe a variety of phenomena and provides the foundation needed for accurate prediction and design. For example, the results of this paper indicate that curvature could be used to distill past the azeotrope of nonideal liquid mixtures. Additionally, phase composition in multicomponent systems with curved phase boundaries can be predicted, which is vital in atmospheric physics, material processing, and soft matter nanotechnology.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This research was funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada. J.A.W.E. holds a Canada Research Chair in Thermodynamics.

(1) Thomson, W. On the Equilibrium of Vapour at a Curved Surface of Liquid. Philos. Mag. 1871, 4, 448−452. (2) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs; Ox Bow: Woodbridge, 1993; Vol. 1, pp 55−353. (3) Elliott, J. A. W. On the Complete Kelvin Equation. Chem. Eng. Educ. 2001, 35, 274−278. (4) Kaptay, G. The Gibbs Equation versus the Kelvin and the GibbsThomson Equations to Describe Nucleation and Equilibrium of NanoMaterials. J. Nanosci. Nanotechnol. 2012, 12, 1−9. (5) Butt, H.-J.; Kappl, M. Normal Capillary Forces. Adv. Colloid Interface Sci. 2009, 146, 48−60. (6) Cohan, L. H. Sorption Hysteresis and the Vapor Pressure of Concave Surfaces. J. Am. Chem. Soc. 1938, 60, 433−435. (7) Christenson, H. K. Capillary Condensation in Systems of Immiscible Liquids. J. Colloid Interface Sci. 1985, 104, 234−249. (8) Burgess, C. G. V.; Everett, D. H.; Nuttall, S. Adsorption Hysteresis in Porous Materials. Pure Appl. Chem. 1989, 61, 1845− 1852. (9) Curry, J. E.; Christenson, H. K. Adsorption, Wetting, and Capillary Condensation of Nonpolar Fluids in Mica Slits. Langmuir 1996, 12, 5729−5735. (10) Horikawa, T.; Do, D. D.; Nicholson, D. Capillary Condensation of Adsorbates in Porous Materials. Adv. Colloid Interface Sci. 2011, 169, 40−58. (11) Melrose, J. C. Model Calculations for Capillary Condensation. AIChE J. 1966, 12, 986−994. (12) Mitropoulos, A. C. The Kelvin Equation. J. Colloid Interface Sci. 2008, 317, 643−648. (13) Ward, C. A.; Balakrishnan, A.; Hooper, F. C. On the Thermodynamics of Nucleation in Weak Gas−Liquid Solutions. J. Basic Eng. 1970, 92, 695−701. (14) Eslami, F.; Elliott, J. A. W. Thermodynamic Investigation of the Barrier for Heterogeneous Nucleation on a Fluid Surface in Comparison with a Rigid Surface. J. Phys. Chem. B 2011, 115, 10646−10653. F


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(65) Tolman, R. C. The Effect of Droplet Size on Surface Tension. J. Chem. Phys. 1949, 17, 333−337. (66) Ahn, W. S.; Jhon, M. S.; Pak, H.; Chang, S. Surface Tension of Curved Surfaces. J. Colloid Interface Sci. 1972, 38, 605−608. (67) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2001; Vol. 5, pp 12.12− 12.20, 12.23. (68) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans: London, U.K., 1966; pp 168, 280. (69) Ghahremani, H.; Moradi, A.; Abedini-Torghabeh, J.; Hassani, S. M. Measuring Surface Tension of Binary Mixtures of Water + Alcohols from the Diffraction Pattern of Surface Ripples. Chem. Sin. 2011, 2, 212−221. (70) Khattab, I. S.; Bandarkar, F.; Fakhree, M. A. A.; Jouyban, A. Density, Viscosity, and Surface Tension of Water + Ethanol Mixtures from 293 to 323 K. Korean J. Chem. Eng. 2012, 29, 812−817. (71) Bonnell, W. S.; Byman, L.; Keyes, D. B. Surface Tension of Ethyl Alcohol-Water Mixtures. Ind. Eng. Chem. 1940, 32, 532−534. (72) Vázquez, G.; Alvarez, E.; Navaza, J. M. Surface Tension of Alcohol + Water from 20 to 50 °C. J. Chem. Eng. Data 1995, 40, 611− 614. (73) Lide, D. R.; Kehiaian, H. V. CRC Handbook of Thermophysical and Thermochemical Data; CRC Press: Boca Raton, FL, 1994. (74) Haar, L.; Gallagher, J. S.; Kell, G. S. NBS/NRC Steam Tables: Thermodynamic and Transport Properties and Computer Programs for Vapor and Liquid States of Water in SI Units; Hemisphere: WA, 1984. (75) Gmehling, J.; Onken, U.; Arlt, W. Vapor−Liquid Equilibrium Data Collection: Aqueous−Organic Systems (Supplement 1); DECHEMA: Frankfurt, Germany, 1981. (76) Berg, J. C. An Introduction to Interfaces & Colloids: The Bridge to Nanoscience; World Scientific: NJ, 2010; pp 6, 111.

(39) Abu Al-Rub, F. A.; Datta, R. Isothermal Vapor-Liquid Equilibrium of Ethanol-Water Mixtures + Acetone-Ethanol Mixtures Inside Capillary Porous Plates. Sep. Sci. Technol. 2000, 35, 2203−2225. (40) Coleburn, N. L.; Shereshefsky, J. L. Liquid-Vapor Equilibrium in Microscopic Capillaries. J. Colloid Interface Sci. 1972, 38, 84−90. (41) Wong, N. S. J. The Effects of Capillary Plates on Vapour-Liquid Equilibrium in Aqueous Alcohol Systems. M.Sc. Thesis, McGill University, 1997. (42) Yeh, G. C.; Yeh, B. V.; Ratigan, B. J.; Correnti, S. J.; Yeh, M. S.; Pitakowski, D. W.; Fleming, W.; Ritz, D. B.; Lariviere, J. A. Separation of Liquid Mixtures by Capillary Distillation. Desalination 1991, 81, 129−160. (43) Yeh, G. C.; Yeh, B. V.; Schmidt, S. T.; Yeh, M. S.; McCarthy, A. M.; Celenza, W. J. Vapor−Liquid Equilibrium in Capillary Distillation. Desalination 1991, 81, 161−187. (44) Shin, Y.-O.; Simandl, J. Vapor and Liquid Equilibria in Porous Media. Fluid Phase Equilib. 1999, 166, 79−90. (45) Shapiro, A. A.; Stenby, E. H. Kelvin Equation for a Non-Ideal Multicomponent Mixture. Fluid Phase Equilib. 1997, 134, 87−101. (46) Shapiro, A. A.; Stenby, E. H. Thermodynamics of the Multicomponent Vapor−Liquid Equilibrium under Capillary Pressure Difference. Fluid Phase Equilib. 2001, 178, 17−32. (47) Shapiro, A. A. Comment on “Vapor−Liquid Equilibria in Porous Media. Fluid Phase Equilib. 2000, 173, 159−160. (48) Shin, Y.-O.; Simandl, J. Reply to ‘Comment on “Vapor−Liquid Equilibria in Porous Media”’. Fluid Phase Equilib. 2000, 175, 325. (49) Lewis, E. R. The Effect of Surface Tension (Kelvin Effect) on the Equilibrium Radius of a Hygroscopic Aqueous Aerosol Particle. J. Aerosol Sci. 2006, 37, 1605−1617. (50) Seinfeld, J. H.; Pandis, S. N. Atmospheric Chemistry and Physics From Air Pollution to Climate Change; John Wiley and Sons, Inc: Hoboken, 2006; pp 434−486. (51) Espósito, R. O.; Tavares, F. W.; Castier, M. Phase Equilibrium Calculations for Confined Fluids, Including Surface Tension Prediction Models. Braz. J. Chem. Eng. 2005, 22, 93−104. (52) Firincioglu, T. Bubble Point Suppression in Unconventional Liquids Rich Reservoirs and its Impact on Oil Production. Ph.D. Thesis, Colorado School of Mines, 2013. (53) Abu Al-Rub, F. A. Distillation in Capillary Porous Media for Separation of Biomass Ethanol−Water Mixtures. Ph.D. Thesis, University of Iowa, 1994. (54) Kister, H. Distillation Design; McGraw-Hill Education: New York, 1992. (55) Becker, E. R.; Pereira, C. J. Computer-Aided Design of Catalysis (Chemical Industries); Marcel Dekker, Inc: New York, 1993. (56) Wood, J.; Gladden, L. F. Modelling Diffusion and Reaction Accompanied by Capillary Condensation using Three-Dimensional Pore Networks. Part 1. Fickian Diffusion and Pseudo-First-Order Reaction Kinetics. Chem. Eng. Sci. 2002, 57, 3033−3045. (57) Ostrovskii, N. M.; Wood, J. In Finely Dispersed Particles: Micro-, Nano-, and Atto-Engineering; Hsu, J., Spasic, A. M., Eds.; CRC Press: Boca Raton, FL, 2005. (58) Callen, H. B. Thermodynamics and an Introduction to Thermostatistics; John Wiley & Sons: New York, 1985. (59) Moore, W. J. Physical Chemistry; Prentice-Hall, Inc: Englewood Cliffs, 1972; pp 475−519. (60) Ward, C. A.; Levart, E. Conditions for Stability of Bubble Nuclei in Solid Surfaces Contacting a Liquid-Gas Solution. J. Appl. Phys. 1984, 56, 491−500. (61) Elliott, J. R.; Lira, C. T. Introductory Chemical Engineering Thermodynamics; Prentice-Hall Inc: Upper Saddle River, NJ, 2012; pp 350, 425. (62) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics; McGraw-Hill: New York, 2005; pp 430−482. (63) Poynting, J. H. Change of State: Solid-Liquid. Philos. Mag. 1881, 12, 32−48. (64) Wisniak, J. Historical Development of the Vapor Pressure Equation from Dalton to Antoine. J. Phase Equilib. 2001, 22, 622−630. G


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