Surface Thermodynamic Analysis of Fluid Confined in a Cone and

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Surface Thermodynamic Analysis of Fluid Confined in a Cone and Comparison with the Sphere−Plate and Plate−Plate Geometries Leila Zargarzadeh and Janet A. W. Elliott* Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4 ABSTRACT: The behavior of pure fluid confined in a cone is investigated using thermodynamic stability analysis. Four situations are explained on the basis of the initial confined phase (liquid/ vapor) and its pressure (above/below the saturation pressure). Thermodynamic stability analysis (a plot of the free energy of the system versus the size of the new potential phase) reveals whether the phase transition is possible and, if so, the number and type (unstable/metastable/stable) of equilibrium states in each of these situations. Moreover we investigated the effect of the equilibrium contact angle and the cone angle (equivalent to the confinement’s surface separation distance) on the free energy (potential equilibrium states). The results are then compared to our previous study of pure fluid confined in the gap between a sphere and a flat plate and the gap between two flat plates.1 Confined fluid behavior of the four possible situations (for these three geometries) can be explained in a unified framework under two categories based on only the meniscus shape (concave/convex). For systems with bulk-phase pressure imposed by a reservoir, the stable coexistence of pure liquid and vapor is possible only when the meniscus is concave.

water in spaces enclosed by hydrophobic surfaces5 and is at least partially responsible for the interaction between such hydrophobic surfaces.5,6 Situation ④ describes the heterogeneous nucleation of a stable vapor from an unstable superheated liquid. Many modeling studies address only one of these four situations (mostly capillary condensation) in a specific geometry, such as the gaps between solids in the following geometries: plate−plate,7 sphere−plate,7−10 cone−plate,7−11 (also truncated cone−plate8), cylinder−plate,9,10 sphere− sphere,7,10,12,13 and cone−cone7,10 among many others. There are also some comparative studies comparing one or two of the four situations in a variety of geometries.7,14 Most of these studies, however, consider the phenomena from a mechanical point of view, where the focus is to calculate the adhesion force between particles due to the capillary bridge.7 The approach of this article is thermodynamic stability analysis, where the possibility of phase transition, along with the number and nature (stable, unstable, or metastable) of potential equilibrium states, is investigated on the basis of the extrema of the free energy versus the size of the new phase.1,15−18 Finding an appropriate free energy of the system is critical in this method. This macroscopic modeling of confined fluids from a thermodynamic point of view has been previously used for particular geometries.1,15−17 For example, on the basis of the

1. INTRODUCTION The confinement of a fluid system may cause an unexpected phase transition or prevent an expected one. Confinement phenomena are widely encountered in nature, for example, in reservoirs and many industrial processes, such as in porous catalysts, powdered materials, and microelectromechanical and nanoelectromechanical systems (MEMS and NEMS), among many others. Four possible situations arise on the basis of the initial phase type (liquid/vapor) and its pressure (below/above the saturation pressure). These situations are ① liquid formation from a confined vapor phase (within wettable walls) at pressures below the saturation pressure (where bulk liquid is not normally stable), known as capillary condensation,2 ② liquid formation from a confined vapor phase at pressures above the saturation pressure, ③ vapor formation from a confined liquid phase (within nonwettable walls) at pressures above the saturation pressure (where bulk vapor is not normally stable), known as capillary evaporation,2 and ④ vapor formation from a confined liquid phase at pressures below the saturation pressure. All four of these situations are relevant to real phenomena in various applications. Situation ①, capillary condensation, is responsible for an undesirable reduction in the activity of porous gas-phase catalysts by removing surfaceactive sites from exposure to gas after nanoscale and microscale pores are filled with liquid.3 Capillary condensation can also desirably enhance the action of porous desiccants.4 Situation ② describes the heterogeneous nucleation of a stable liquid from an unstable supercooled vapor. Situation ③, capillary evaporation, can be observed in molecular simulations of © 2013 American Chemical Society

Received: June 18, 2013 Revised: August 29, 2013 Published: September 16, 2013 12950

dx.doi.org/10.1021/la4023135 | Langmuir 2013, 29, 12950−12958

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phase formation out of an initial fluid confined in a cone, as shown in Figure 1. In our previous work,1 we found and

free-energy curves for liquid formed out of vapor in the gap between a sphere and a flat plate, Elliot and Voitcu17 were able to explain the diffuse liquid−vapor interface experimentally observed by Maeda et al.19 in the crossed cylinders geometry. Free energies for capillary bridges between a sphere and a plate or between two spheres were also studied by Attard6 and Andrienko et at.20 Of interest in this work is new phase formation from a confined pure phase inside a cone. Barrer et al.21 calculated the volume of the equilibrium liquid condensate inside a cone, with the equilibrium radius being calculated from the approximate Kelvin equation. However, they considered only the case in which the contact angle is zero. Ward et al.15 showed possible vapor formation and growth from a liquid−gas solution confined in a cone at constant mass and volume. In another study by Ward and Levart,16 from the analysis of a liquid−gas solution inside a cone at constant pressure, temperature, and mass, it was found that stable bubble nuclei with a convex meniscus form (or equivalently that confinement walls should be wettable for stable bubble nuclei to form). Previous thermodynamic stability analysis of fluid confined in a cone15,16,21 considered only one of the four possible situations and did not consider the effect of the equilibrium contact angle. In our previous work,1 we used thermodynamic stability analysis to investigate phase transitions of pure fluids for all of the four possible situations for confinement in the sphere−plate geometry and the plate−plate geometry (as an extreme of the sphere−plate geometry when the radius of the sphere becomes infinite). In each situation, we investigated the effect of the equilibrium contact angle, confinement surface separation distance, and sphere radius. We then summarized the possibility of phase transition, the number and nature of equilibrium states, and the effect of different parameters in the four possible situations under two categories of new phase formation with a concave or convex meniscus. In this article, we consider pure fluid confined in a cone and investigate phase transitions for all four possible situations, along with the effects of the equilibrium contact angle and cone angle. The results are then compared to the previous study of the sphere−plate and plate−plate geometries.1 In all of the situations that we consider, the problem specifications are as follows. The temperature of the system and the pressure of the initial phase are constant. The system is closed to mass transfer. The cone is assumed to be infinitely deep. (For a discussion of the nucleation of a new phase in a finite cone and the edge effects on the growth of the new phase, see the work by Ward et al.15). Both the initial confined phase and the new phase are pure and are made of component 1. The solid is purely component 2 and is nonvolatile, nondissolving, and insoluble in the fluid. Solid surfaces are assumed to be ideal (smooth, rigid, and homogeneous with negligible vapor pressure), as defined by Ward and Neumann.22 The problems were investigated in the absence of gravitational forces (or in situations where their effects are negligible). Also, we assume the contact angle to be a known constant equilibrium contact angle. This is a reasonable assumption when pressure is imposed on one fluid phase by a reservoir and there is only one equilibrium state, as is the case for the cone. Ward et al.23 and Wu et al.24 have described the complexity that arises in other systems when the contact angle is a dependent variable.

Figure 1. Schematic of potential new phase formation out of an initial bulk phase: liquid formation with ① a concave meniscus or ② a convex meniscus out of an initial bulk vapor, and vapor formation with ③ a concave meniscus or ④ a convex meniscus out of an initial bulk liquid, confined in a cone interacting with a constant pressure reservoir.

tabulated equations for the conditions for equilibrium, the resulting Kelvin radius, and the free energy of the system for either liquid formation out of vapor or vapor formation out of liquid. All of those equations are applicable regardless of the confinement geometry. As in our previous work,1 we define the Kelvin radius, RC, in terms of two principal radii of curvature, R1 and R2, at equilibrium 1 1⎛ 1 1 ⎞ = ⎜ + ⎟ RC 2 ⎝ R1 R2 ⎠

(1)

where we arbitrarily choose the sign of the radius of curvature to be positive if the center of the circular arc that defines that radius is in the initial confined phase. Thus the Kelvin equation for the case of liquid formation out of a bulk vapor phase has the following form1,17,25−28 RC =

2γ LV (PV − P∞) −

RT ̅ υ∞L

PV P∞

( )

ln

(2)

where γ is the liquid−vapor interfacial tension, T is the temperature, PV and P∞ are vapor-phase pressure and the saturation pressure, respectively, υL∞ is the specific volume of pure liquid at the saturation pressure, and R̅ is the universal gas constant. The Kelvin radius for the case of vapor formation out of a bulk liquid phase is1,16,28 LV

2. THEORETICAL BASIS 2.1. Thermodynamic Stability Analysis. Here we perform thermodynamic stability analysis of potential new

RC =

12951

2γ LV

(

P L − P∞ exp

υ∞L (P L RT ̅

− P∞)

)

(3)


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where PL is the pressure of the liquid phase. It should be noted that for a system with no contaminants the Kelvin radius can well predict menisci with mean radii as small as 8 times the molecular diameter of the material of interest.29 For water, Fisher et al.30 reported the Kelvin radius validity when the mean radius of curvature is greater than 5 nm.30 Therefore, we assume that using the Kelvin equation (and indeed the composite system thermodynamic framework of Gibbs31) is reasonable in our work because the dimensions of interest are greater than 5 nm. According to eqs 2 and 3, the sign of the Kelvin radius depends on the value of the initial phase pressure (PV in eq 2 and PL in eq 3). Ignoring extremely high initial phase pressures,32 for the case of liquid formation out of a bulk vapor phase1 ⎧ PV ⎪ R C > 0 if 1 ⎪ R C < 0 if P∞ ⎩

eqs 6 and 7 must now be found from the geometry of the system. Figure 2 shows a schematic diagram of liquid formation out of a bulk vapor phase inside a cone for situations of concave outward and convex outward menisci.

Figure 2. Schematic of a potential liquid phase(a) concave meniscus and (b) convex meniscusinside a cone, where θ is the equilibrium contact angle (measured from inside the liquid according to the convention), R1 and R2 are the principal radii of liquid−vapor curvature, which are identical for new phase formation inside a cone, 2β is the cone apex angle, W is the radius of the three-phase contact circle, h is the vertical distance from the apex to the solid−liquid contact line, h′ is the distance between the highest and lowest parts of the liquid−vapor interface, and ⌀ is the half-filling angle of the new phase.

(4)

and for the case of vapor formation out of a bulk liquid phase1 ⎧ PL ⎪ R C > 0 if >1 P∞ ⎪ ⎨ ⎪ PL 90° − β and RC is negative, which is possible only at vapor-phase pressures above the saturation pressure (PV > P∞) according to eq 4. 3.2.1. General Analysis and the Effect of Equilibrium Contact Angle. Free energy versus liquid volume is presented for various contact angles in Figure 5 for a case when the vapor phase pressure is above the saturation pressure. The maximum in the free-energy curves represents an unstable equilibrium state and an energy barrier that has to be overcome (nucleation phenomena) for phase transition to happen. Pure liquid with a convex meniscus cannot coexist in a stable equilibrium with a pure vapor phase. After passing the energy barrier, the whole vapor phase turns into a liquid phase because the free-energy curve is monotonically decreasing after the maximum point. For the specifications of Figure 5, the Kelvin radius is −1.13 × 10−8 m according to eq 2. For liquid formation with a convex meniscus inside a cone, as the equilibrium contact angle decreases (gets closer to the concave-to-convex transition contact angle), the free-energy barrier becomes smaller and the unstable equilibrium is formed with a smaller liquid volume.

(12)

(13)

h′ = R1 − R1 cos ⌀

(14)

where the geometric variables are as defined in Figure 2. Equations 11−14 are then substituted into eqs 8 to 10 to arrive at eqs 15−17 presented in Table 1 for the case of liquid Table 1. New Phase Volume and the Required Surface Areas for the Conical Pit Confinement Volume and Surface Areas for Liquid Formation out of a Bulk Vapor Phase Inside a Cone

⎤ π 3⎡ cos3(θ + β) R1 ⎢ − 2 + 3 sin(θ + β) − sin 3(θ + β)⎥ 3 ⎣ tan β ⎦

VL = ASL

⎡ cos2(θ + β) ⎤ = πR12⎢ ⎥ sin β ⎣ ⎦

LV

A

=

2πR12[1

(15)

(16)

− sin(θ + β)]

(17)

Volume and Surface Areas for Vapor Formation out of a Bulk Liquid Phase Inside a Cone

VV =

⎤ π 3⎡ − cos3(θ − β) R1 ⎢ − 2 + 3 sin(θ − β) − sin 3(θ − β)⎥ 3 ⎣ tan β ⎦

⎡ cos2(θ − β) ⎤ ASL = πR12⎢ ⎥ sin β ⎣ ⎦ ALV = 2πR12[1 − sin(θ − β)]

(18)

(19) (20)

formation with a concave meniscus. For liquid formation with a convex meniscus, following similar steps, equations for volume and surface areas are found to be exactly the same as in eqs 15−17. Similar equations for vapor volume and required surface areas in the case of vapor formation are also presented in eqs 18−20 of Table 1. It should be noted that in the equations of Table 1 the sign of R1 depends on the concavity of the meniscus. In the case of liquid formation, R1 is defined to have a positive sign when the meniscus is concave (θ < 90° − β) and a negative sign when the meniscus is convex (θ > 90° − β). In the case of vapor formation, R1 is positive for the concave meniscus (θ > 90° + β) and negative for the convex meniscus (θ < 90° + β).

3. RESULTS AND DISCUSSION For pure fluid confined in a cone, plots of the free energy versus size of the new phase are presented for each of the four situations. Such plots reveal the number and nature of the equilibrium states. The effects of the equilibrium contact angle and the cone apex angle on the stability are investigated. 3.1. Liquid-Phase Formation from a Bulk Vapor Phase in a Cone: Concave Meniscus. For the meniscus to be concave, θ < 90° − β and RC is positive, which is possible only when the vapor-phase pressure is below the saturation pressure (PV < P∞) according to eq 4. 3.1.1. General Analysis and the Effect of Equilibrium Contact Angle. Free energy versus (a) scaled curvature radius (R1/RC) and versus (b) the liquid volume is shown in Figure 3. There is no energy barrier and only one minimum point in the curves of Figure 3. This means that a stable liquid bridge with concave meniscus can form out of a vapor phase that is confined in a cone at below the saturation pressure, without 12953


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Figure 3. (a) Free energy versus scaled size of the liquid phase formed from H2O vapor in a cone at 20 °C, PV = 0.9P∞, and β = 10° for various contact angles that result in a concave meniscus. (b) Free energy vs volume of liquid under the same conditions for various contact angles (θ) that result in a concave meniscus.

Figure 4. Effect of the cone apex angle (2β) on the free energy vs the volume of the liquid phase formed with a concave meniscus out of H2O vapor in a cone at 20 °C, PV = 0.9P∞, and θ = 10°.

Figure 5. Free energy vs volume of liquid formed out of H2O vapor in a cone at 20 °C, PV = 1.1P∞, and β = 10° for various contact angles (θ) that result in a convex meniscus.

3.2.2. Effect of the Cone Apex Angle. Figure 6 shows the effect of the cone apex angle on the free energy of the liquid phase with a convex meniscus. For a smaller cone apex angle, a higher energy barrier must be overcome and a higher volume of liquid must be formed to overcome the unstable equilibrium, as shown in Figure 6. Thus liquid formation out of a bulk vapor phase within a cone of a solid material that enforces θ > 90° − β becomes less favorable as the apex cone angle decreases. For any cone apex angle, the equilibrium radius of curvature is always the same and equal to the Kelvin radius. 3.3. Vapor-Phase Formation from a Bulk Liquid Phase in a Cone: Concave Meniscus. For vapor formation with a concave meniscus, θ > 90° + β and RC is positive, which is

possible only if the liquid-phase pressure is above the saturation pressure (PL > P∞) based on eq 5. 3.3.1. General Analysis and the Effect of Equilibrium Contact Angle. Figure 7 shows curves of the free energy versus the volume of the vapor being formed out of liquid above its saturation pressure for various equilibrium contact angles. As shown in Figure 7, phase transition happens without passing any energy barrier (maximum point), hence it is not a nucleation phenomenon. The system evolves to a stable equilibrium state at which pure vapor with a concave meniscus coexists with the initial pure liquid. With the specifications of Figure 7, the Kelvin radius from eq 3 is 6.22 × 10−4 m. 12954


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if the liquid-phase pressure is below the saturation pressure (PL < P∞). 3.4.1. General Analysis and the Effect of Equilibrium Contact. Curves of free energy versus the volume of vapor being formed out of liquid above the saturation pressure are presented in Figure 9 for various contact angles. The Kelvin radius from eq 3 is −6.22 × 10−4 m under the conditions of Figure 9.

Figure 6. Effect of the cone apex angle (2β) on the free energy vs the volume of the liquid phase formed with a convex meniscus out of H2O vapor confined in a cone at 20 °C, PV = 1.1P∞, and θ = 170°.

Figure 9. Free energy vs volume of vapor formed out of H2O liquid in a cone at 20 °C, PL = 0.9P∞, and β = 10° for various contact angles (θ), that result in a convex meniscus.

The maximum in the free energy in Figure 9 represents the barrier to be overcome for the nucleation of a vapor phase with a convex meniscus out of a bulk liquid phase below the saturation pressure. Once the energy barrier is overcome, all of the liquid spontaneously turns into vapor because each curve is monotonically decreasing after the maximum point. Figure 9 also shows that as the contact angle gets farther from the concave-to-convex transition contact angle, a greater energy barrier, with a larger volume of vapor should be overcome for the formation of a convex vapor phase. 3.4.2. Effect of the Cone Apex Angle. Figure 10 illustrates the effect of the cone apex angle on the stability of the vapor

Figure 7. Free energy versus volume of vapor formed out of H2O liquid in a cone at 20 °C, PL = 1.1P∞, and β = 10° for various contact angles (θ) that result in a concave meniscus.

As the equilibrium contact angle gets smaller (gets closer to the concave-to-convex transition contact angle) while it is kept above 90° + β, a less-stable equilibrium state would form with less vapor volume. Also a specific number of degrees’ change in the contact angle results in a larger relative change in the energy level of the stable state for contact angles closer to the concaveto-convex transition contact angle (which is 90° + β in the case of vapor formation out of liquid in a cone). 3.3.2. Effect of the Cone Apex Angle. As shown in Figure 8, the cone with the smaller apex angle results in the more stable equilibrium condition, with a higher volume of the vapor phase at equilibrium. 3.4. Vapor-Phase Formation from a Bulk Liquid Phase in a Cone: Convex Meniscus. When vapor is formed with a convex meniscus out of liquid, θ < 90° + β and the Kelvin radius (RC) is negative, which according to eq 5 is possible only

Figure 10. Effect of the cone apex angle (2β) on the free energy versus the volume of the vapor phase formed with a convex meniscus out of the H2O liquid confined in a cone at 20 °C, PL = 0.9P∞, and θ = 10°.

phase with a convex meniscus formed out of a liquid phase at a pressure below the saturation pressure inside a cone. As shown in Figure 10, for a cone with a smaller apex angle, a higher energy barrier with a larger vapor volume must be overcome.

4. CONCLUSIONS The results of situations 1−4 for the cone geometry can be well described by two categories based on the meniscus shape

Figure 8. Effect of the cone apex angle (2β) on the free energy vs the volume of the vapor phase formed with the concave meniscus out of H2O liquid confined in a cone at 20 °C, PL = 1.1P∞, and θ = 170°. 12955


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Table 2. Curves of Free Energy versus Size of the New Phase for New Phase Formation with a Concave Meniscus in Three Different Confining Geometries: Cone, Plate−Plate,1 and Sphere−Plate1

Table 3. Curves of Free Energy versus Size of the New Phase for New Phase Formation with a Convex Meniscus in Three Different Confining Geometries: Cone, Plate−Plate,1 and Sphere−Plate1

(concave/convex), regardless of the initial phase type. Table 2 presents the results for new phase formation with a concave

meniscus, and Table 3 presents the results for new phase formation with a convex meniscus. For comparison, also shown 12956


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Table 4. Complete Picture of Thermodynamic Stability Analysis for a Confined Fluid at a Constant Bulk-Phase Pressure Inside Different Confining Geometries (a) concave meniscusa number of equilibrium states for different confining geometries

cone plate−plate1 sphere−plate1 H≠0

sphere−plate1 H=0 effect of contact angle: getting farther from the concave-to-convex transition contact angle (θt)c effect of separation distance: decrease in H or β

1 1 1 1 1

stable unstable unstable stable stable

(b) convex meniscusb 1 unstable 1 unstable 1 unstable 1 unstable

smaller barrier (maximum point) more stability of the stable equilibrium (minimum point) smaller barrier more stability of the stable equilibrium

larger barrier larger barrier

a Liquid formation out of vapor at PV < P∞ inside the confinement of wettable walls (capillary condensation) or vapor formation out of liquid at PL > P∞ inside the confinement of nonwettable walls (capillary evaporation). bLiquid formation out of vapor at PV > P∞ inside the confinement of nonwettable walls or vapor formation out of liquid at PL < P∞ inside the confinement of wettable walls. cThis is equivalent to a contact angle decrease in cases of liquid formation with a concave meniscus out of vapor at PV < P∞ or vapor formation with a convex meniscus out of liquid at PL < P∞ and a contact angle increase in cases of liquid formation with a convex meniscus out of vapor at PV > P∞ or vapor formation with a concave meniscus out of liquid at PL > P∞.

in Tables 2 and 3 are the results from our previous study1 for the sphere−plate and plate−plate geometries. According to Tables 2 and 3, in a pure system stable liquid− vapor coexistence (indicated by a minimum point in the free energy) can happen only when the new phase is being formed with a concave meniscus (albeit not in the confinement between two flat plates, in which the whole initial phase changes to the new phase). In the case of new phase formation with a convex meniscus, the initial phase turns into the new phase completely once the energy barrier (corresponding to a maximum point in the free energy) is overcome (i.e., the phase transition is a nucleation phenomenon). Therefore, pure liquid and vapor cannot coexist in a stable equilibrium state with a convex meniscus. A stable new phase with a concave meniscus forms spontaneously (without any energy barrier) for the cone geometry, forms spontaneously for the sphere−plate geometry when H = 0, and forms after overcoming an energy barrier for the sphere−plate geometry when H ≠ 0. Although a new phase with a concave meniscus always evolves to a stable equilibrium inside a cone, the phase transition becomes unfavorable (the free-energy curve becomes monotonically increasing) inside the gap between two flat plates or the gap between a sphere and a flat plate with a separation distance above a certain breakage distance1 and/or for contact angles close to the concave-toconvex transition contact angle (only when H ≠ 0). For the case of new phase formation with a convex meniscus, the free-energy curves never become monotonically increasing when the confining geometry is the cone or the gap between a sphere and a flat plate. However, when a sphere and plate are separated by a small separation distance and/or with equilibrium contact angles far from the concave-to-convex transition contact angle, the energy barrier might be so high that nucleation never occurs in practice.1 Inside the gap between two flat plates, the free-energy curve becomes monotonically increasing for small separation distances and/ or equilibrium contact angles far from the concave-to-convex transition contact angle.1 The value of comparing the cone, sphere−plate, and plate− plate geometries (Tables 2 and 3) is the emergence of a pattern that applies across geometries. For example, a wedge may be expected to behave like a cone or like a sphere and plate that touch (H = 0).

For each of the four possible situations and for the three confining geometries, the number and type of equilibrium states, along with the effects of the equilibrium contact angle and the solid separation distance, are summarized in Table 4. The effects of the contact angle can be explained in a consistent manner for both cases of liquid or vapor formation when changes in the contact angle are described by getting closer to, or farther from, the concave-to-convex transition contact angle.1 In the case of new phase formation with a concave meniscus, changing the contact angle by a specific number of degrees results in larger relative changes in the energy level when the contact angle is closer to the concave-to-convex transition contact angle. In general, a tighter confinement facilitates new phase formation with concave meniscus by reducing the energy barrier and/or results in more stability of the stable condition by lowering the energy level of the minimum point. In contrast for new phase formation with a convex meniscus, the tighter the confinement, the higher the energy barrier that has to be overcome. For the cone geometry, the apex angle (2β) determines the tightness of the gap. For the plate−plate and sphere−plate geometries, the separation distance (H) is indicative of the tightness of the confinement. When confinement walls meet at some point (for example, at the apex of a cone or at the point a sphere touches a plate when H = 0), new phase formation with a concave meniscus evolves to its stable size without any energy barrier (therefore it is not a nucleation phenomenon). On the contrary, new phase formation with a convex meniscus inside confinement with some common point requires a very large barrier to be overcome that might be practically impossible. In both cases of a concave and convex meniscus, the free energy is an extremum at the starting point of the curve (i.e., when no new phase has yet formed). It should be noted that a smaller energy barrier corresponds to a smaller volume of the new phase. Also, a more stable equilibrium has a larger volume of the new phase with a lower (more stable) energy level. As well as quantifying the effects of the contact angle and cone apex angle on new phase formation from a constantpressure pure bulk phase inside a cone, this work showed that the thermodynamic stability of the new phase formation inside 12957


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(17) Elliott, J. A. W.; Voitcu, O. On the thermodynamic stability of liquid capillary bridges. Can. J. Chem. Eng. 2007, 85, 692−700. (18) 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. (19) Maeda, N.; Israelachvili, J.; Kohonen, M. Evaporation and instabilities of microscopic capillary bridges RID C-4234-2011. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 803−808. (20) Andrienko, D.; Patrício, P.; Vinogradova, O. I. Capillary bridging and long-range attractive forces in a mean-field approach. J. Chem. Phys. 2004, 121, 4414−4423. (21) Barrer, R. M.; McKenzie, N.; Reay, J. S. S. Capillary condensation in single pores. J. Colloid Sci. 1956, 11, 479−495. (22) Ward, C. A.; Neumann, A. W. On the surface thermodynamics of a two-component liquid-vapor-ideal solid system. J. Colloid Interface Sci. 1974, 49, 286−290. (23) Ward, C. A.; Wu, J. Y. Effect of adsorption on the surface tensions of solid-fluid interfaces. J. Phys. Chem. B 2007, 111, 3685− 3694. (24) Wu, J. Y.; Farouk, T.; Ward, C. A. Pressure dependence of the contact angle. J. Phys. Chem. B 2007, 111, 6189−6197. (25) Hunter, R. J. Introduction to Modern Colloid Science; University Press: Oxford, U.K., 2002; pp 142−146. (26) McGaughey, A. J. H.; Ward, C. A. Droplet stability in a finite system: consideration of the solid−vapor interface. J. Appl. Phys. 2003, 93, 3619. (27) Elliott, J. A. W. On the complete Kelvin equation. Chem. Eng. Educ. 2001, 35, 274−279. (28) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans: London, 1966; pp 219−220. (29) 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. (30) Fisher, L. R.; Israelachvili, J. N. Direct measurement of the effect of meniscus forces on adhesion: a study of the applicability of macroscopic thermodynamics to microscopic liquid interfaces. Colloids Surf. 1981, 3, 303−319. (31) Gibbs, J. W. On the equilibrium of heterogeneous substances. Trans. Conn. Acad. 2 1876, 108−248. Gibbs, J. W. Trans. Conn. Acad. 2 1878, 343−524. Gibbs, J. W. The Scientific Papers of J. Willard Gibbs; Ox Bow Press: Woodbridge, CT, 1993; Vol. I, pp 55−353. (32) Zargarzadeh, L. Comparative Surface Thermodynamic Analysis of New Fluid Phase Formation in Various Confining Geometries. M.Sc. Thesis, University of Alberta, Edmonton, Canada, 2012. (33) Harris, J.; Stöcker, H. Handbook of Mathematics and Computational Science; Springer: New York, 1998.

confined spaces could be understood across geometries based only on the meniscus shape: concave or convex.

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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 Wolfram Mathematica 7.0.1 and 8.0.0 were used for computations and graph production. This research was funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada, including partial support from the NSERC Industry Research Chair in Oil Sands Engineering held by Z. Xu at the University of Alberta. J.A.W.E. holds a Canada Research Chair in Thermodynamics. L.Z. held scholarships from the University of Alberta.

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Surface Thermodynamic Analysis of Fluid Confined in a Cone and

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