Comparative Surface Thermodynamic Analysis of New Fluid Phase

Article pubs.acs.org/Langmuir

Comparative Surface Thermodynamic Analysis of New Fluid Phase Formation between a Sphere and a Flat Plate Leila Zargarzadeh and Janet A.W. Elliott* Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2V4 ABSTRACT: This paper investigates the behavior of confined fluid in the gap between a sphere and a flat plate by examining the curve of free energy of the system versus size of the new phase. Four possible situations corresponding to new phase formation out of confined liquid or vapor at pressures above or below the saturation pressure are studied. Using surface thermodynamics, the feasible shape of the meniscus (concave/convex), the possibility of phase transition, as well as the number and the nature (unstable/stable) of equilibrium states have been determined for each of these four situations. The effects of equilibrium contact angle, separation distance of confinement surfaces, and sphere size have been studied. We show that the number and nature of equilibrium states, along with the effect of different parameters in these four possible situations, can be well described under two categories of new phase formation with (a) concave or (b) convex meniscus. Our results reveal that in the sphere−plate gap, stable coexistence of the liquid and vapor phases is only possible when the meniscus is concave (which corresponds to either capillary condensation or capillary evaporation), and when the sphere and plate are separated by a distance less than a critical amount (where that critical amount is always less than the Kelvin radius). With convex menisci, no stable coexistence of liquid and vapor phase is possible.

■

INTRODUCTION For unconfined pure fluids with no effects of interface curvature, liquid−vapor equilibrium only happens at the saturation pressure. At any pressure other than the saturation pressure an unconfined fluid is stable only in a single-phase form, that is, the stable bulk phase is vapor when pressure is below the saturation pressure (the liquid phase is unstable) and the stable bulk phase is liquid when pressure is above the saturation pressure (the vapor phase is unstable). In contrast, phase behavior of a fluid may be affected as a result of confinement. At small-scale confinements of specific geometry, vapor and liquid phases may coexist at stable equilibrium even at pressures other than the saturation pressure. Also, in very tight confinements, the predicted phase transition (from liquid to vapor or vice versa) might be prevented by the confinement. Confined fluid phenomena are of great practical importance in the oil and gas, chemical and pharmaceutical industries, many geophysical phenomena, and fabrication and function of miniaturized systems, among many other areas. Understanding the basics of confined fluid behavior results in better design, either to employ or prevent these phenomena. Fluid inside a confinement can be placed in one of four categories based on the initial phase type (liquid/vapor) and the pressure of the confined fluid (above/below the saturation pressure). Therefore, a confined fluid is among one of these situations: ① formation of liquid from a confined vapor phase at pressures below the saturation pressurethis is well-known as capillary condensation,1 ② formation of liquid from a confined vapor phase at pressures above the saturation pressure, ③ formation of vapor from a confined liquid phase at pressures above the © 2013 American Chemical Society

saturation pressurethis is well-known as capillary evaporation,1 or ④ formation of vapor from a confined liquid phase at pressures below the saturation pressure. Several approaches have been used to study confined fluid in one or more of these situations. Experimental studies have been performed mostly for capillary condensation.2−4 Many theoretical models of confined fluid behavior have been developed. Various existing models can be categorized, as macroscopic, molecular, or mixed models.5 Macroscopic models can predict over a wide range of fluid phase size, from above approximately 5 nm to several millimeters (the maximum size of the capillary length) and beyond, at a “much reduced computational cost”.4 Molecular dynamic models and mixed models can make predictions for a maximum fluid phase size of 30 and 100 nm, respectively.5 Most of the macroscopic modeling efforts investigate the phenomena from a mechanical point of view, with the focus of calculating the adhesion force between particles as a result of the capillary bridge (capillary neck) for capillary condensation, or capillary evaporation phenomena. Even in studies from the mechanical point of view with the focus on force calculation, some subtle equilibrium thermodynamic assumptions are made. In some of these studies, the exact shape of the meniscus of the new phase is calculated (through the Kelvin equation) assuming thermodynamic equilibrium between the confined phase and the new phase that forms out of the confined phase.4 Received: November 18, 2012 Revised: January 18, 2013 Published: January 29, 2013 3610

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Figure 1. Schematics of typical systems of new phase formation with (a) concave and (b) convex menisci out of an initial bulk phase between a sphere and a flat plate, interacting with the reservoir.

observed at a certain separation distance3 (called the breakage distance9). Above the breakage distance, Elliott and Voitcu9 showed that liquid bridge formation is thermodynamically impossible. For small separation distances (below the breakage distance), Elliott and Voitcu9 found that the free energy as a function of the size of the bridge has two extrema, where the smaller one is a maximum point, corresponding to a nucleation barrier, and the larger one is a minimum point, corresponding to the stable bridge. They also mentioned the plate−plate case as the extreme of the sphere−plate geometry in which the sphere radius is infinity.9 Capillary bridging with concave meniscus was also studied from the thermodynamic point of view (though with a different free energy than in the Elliott and Voitcu article9) for either a liquid bridge or a vapor bridge with concave meniscus inside the gap in the sphere−plate geometry by Andrienko et al.14 They considered the free energy of the system as the sum of bulk and surface terms,14 and got the same number of extrema (two extrema) in the excess free energy as Elliott and Voitcu,9 and stated that the larger bridge corresponds to the minimum (stable) point. They had also found that bridging (where the bridge had concave meniscus) is impossible for distances greater than a certain amount (2γLV/ Δμ where γLV is the liquid−vapor interfacial tension and Δμ is the difference in the chemical potentials of the phase-separated components between the inside of the bridge and the bulk).14 The above-mentioned works only focus on one or two situations and do not vary the contact angle. In this paper, we present a complete, comparative thermodynamic stability analysis of vapor OR liquid nucleation in wetting OR nonwetting confining solids (all the four possible situations ①, ②, ③, and ④) for different contact angles. Results are described in a unifying framework that generalizes the four possible situations into two groups of new phase formation with (a) concave or (b) convex menisci. The problem is investigated under conditions of constant temperature and constant pressure of the initial confined fluid, and zero gravity (or for negligible gravitational effects). The system is closed to mass transfer. Both the initial confined fluid, and the potential new phase which is formed, are assumed to be pure, consisting of component 1. The solid of the confinement is considered to be made up of a nonvolatile, nondissolving component 2 and is

The results from equilibrium assumptions are in good agreement with surface−force apparatus results, where the contact is typically between 0.1 to 1 s.6 Many others use the toroidal approximation for the shape of the bridge, as it is in good agreement with the exact shape calculations.7 Another approach to model confined fluid behavior is thermodynamic stability analysis. First an appropriate free energy of the system is determined, and then stability of the new phase formation out of an initial fluid is investigated through the trend of the free energy versus size of the new phase. It can be determined whether the phase transition is possible, and if so whether the whole initial phase turns into the new phase, or the initial and new phase can coexist at a stable equilibrium.8,9 This approach has been used to describe a variety of phenomena involving liquid−vapor menisci at solid surfaces,8−11 such as proving droplet nucleation to be easier on fluid (soft) surfaces in comparison to rigid surfaces,11 and describing the nucleation and formation of stable bubbles from liquid−gas solutions in conical pits for two different conditions of constant mass and volume,10 and constant mass and pressure.8 Also the equilibrium shape of the bridging bubble between two colloidal spheres of identical size was found by Attard12 using minimization of the constrained Gibbs free energy and a polynomial expansion describing the shape. Attard showed a microscopic bridging bubble to be stable for hydrophobic spheres at small separation distances. He proposed that the force due to the bridging bubble was responsible for the long-range attraction between hydrophobic surfaces in water. From this thermodynamic analysis, the hysteresis in formation/disappearance of the bridging bubble on the approach and separation of the spheres was also explained.12 The confinement geometry of interest in this paper is the gap between a sphere and a flat plate. This geometry is practically important in modeling, such as for nanoscale particles interacting with a surface in humid ambient conditions4 as in atomic force microscopy (AFM).13,14 Elliott and Voitcu9 used thermodynamic stability analysis to study only the liquid capillary bridge with concave meniscus in the sphere−plate geometry, and only for zero contact angle. That study explained the diffuse liquid−vapor interface that had been previously 3611

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also insoluble to the fluid. Solid surfaces are considered to be ideal,15 that is, smooth, rigid, homogeneous, with no appreciable vapor pressure. The effects of three important parameters have been examined: sphere size (RP), sphere−plate separation distance (H), and contact angle (θ). Throughout this investigation, the contact angle is taken to be the equilibrium contact angle and is assumed to be known. If contact angle is a dependent variable as in the work of Ward and Wu16 and Wu et al.,17 then further equations for contact angle will have to be considered. The equilibrium contact angle depends on the solid material, surface manipulation, and adsorption at the solid−liquid interface.18

The constraints of the system are: (a) The system exchanges energy with the reservoir, and the combination of the system plus reservoir is isolated, hence dU R + dU S + dU L + dU V + dU SL + dU SV + dU LV = 0 (4)

(b) The system can exchange volume with the reservoir. dV R + dV S + dV L + dV V = 0

(c) The solid surface is considered to be rigid (no deformation). The solid surface is also assumed to be incompressible, that is, no volume changes happen in the solid.

■

dV S = 0

THEORETICAL BASIS Thermodynamic Stability Analysis. To perform thermodynamic stability analysis, it is supposed that a new phase is formed out of a confined phase. Figure 1 shows schematics of such systems. Thermodynamic analysis consists of the following steps:8 (I) finding the conditions for equilibrium of the new phase, (II) determining the free energy of the system with respect to some reference condition, and (III) analyzing the curve of the free energy versus size of the new phase, which reveals the possibility of phase transition, and the number and type of equilibrium states when phase transition is possible. (a). Conditions for Equilibrium. The conditions for equilibrium of a liquid−vapor system confined in a solid geometry and surrounded by a reservoir can be obtained by extremizing entropy, S. Therefore

R dNres =0

dN1L + dN1V + dN1SL + dN1SV + dN1LV = 0

dN2S = −dN2SL − dN2SV

∑ i=1

T

dN2S = 0

dS ab =

γ 1 dU ab − ab dAab − ab T T

k

∑ i=1

μiab T

ab

(2)

dNiab

(10)

Substituting equations of the form 2 and 3 for every phase and interface into eq 1 and making use of the constraints in eqs 4−10, conditions for equilibrium are obtained, as presented in Table 1.

where U, V, and Ni are internal energy, volume, and number of moles of component i, respectively (these are extensive properties). T, P, and μ are temperature, pressure, and chemical potential, respectively (these are intensive properties). k denotes the total number of components in the phase. For interfaces, denoted ab, dS is19,21 ab

(9)

When the solid−liquid and the solid−vapor interfaces are flat, component 2 is assumed not to be present at the interface according to the “Gibbs dividing surface” approximation, and non−volatility of this component results in

μia

a a dNi

(8)

Due to non−volatility, component 2 can only transfer between the solid phase and the solid−liquid and solid−vapor interfaces. In the case of curved solid interfaces, where all components including component 2 are present at the interface, hence

where superscripts R, S, L, V, SL, SV, and LV denote the reservoir, the solid, liquid, and vapor phases, and the solid− liquid, solid−vapor, and liquid−vapor interfaces, respectively. From the fundamental equation of thermodynamics,19,20 dS for any bulk phase a is given by k

(7)

Component 1 can transfer between bulk phases and interfaces of the system, except to the solid phase which is composed purely of component 2, hence

(1)

1 Pa dS = a dU a + a dV a − T T

(6)

(d) The system is closed and there is no mass exchange between the system and the reservoir.

dS R + dS S + dS L + dSV + dS SL + dS SV + dS LV = 0

a

(5)

Table 1. Conditions for Equilibriuma Conditions for equilibrium for liquid formation out of a bulk vapor phase

T R = T S = T L = T V = T SL = T SV = T LV R

P =P μ1L μ2S

(3)

where γ represents interfacial tension and A is area of the interface. Curvature affects which components are included in the summation in eq 3. For any curved interface all available components are present at the interface, according to the “Gibbs surface of tension” approximation.19 For a flat interface according to the “Gibbs dividing surface” convention,19 it is assumed that the dividing surface is placed such that one of the available components is not present at the interface. In both cases of liquid formation or vapor formation in the confined geometry, the liquid−vapor interface is, in general, curved. The solid−liquid and solid−vapor interfaces may be either curved or flat depending on the geometry of the solid.

V

=

μ1V

=

μ2SL

L

(11)

(12) = μ1SL = μ1SV = μ1LV =

V

(13)

μ2SV only for curved SL and SV interfaces L

SL

SL

SV

SV

(P − P )dV − γ dA − γ dA

LV

LV

− γ dA

=0

(14) (15)

Conditions for equilibrium for vapor formation out of a bulk liquid phase

T R = T S = T L = T V = T SL = T SV = T LV R

P =P

L

μ1L = μ1V = μ1SL = μ1SV = μ1LV μ2S

=

μ2SL

(16)

(17)

=

(18)

μ2SV only for curved SL and SV interfaces

(PV − P L)dV V − γ SL dASL − γ SV dASV − γ LV dALV = 0

(19) (20)

a

Liquid and vapor phases are made up of component 1 and the solid phase is purely component 2. 3612

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⎛ PV ⎞ υ∞L (P L − P∞) = RT ̅ ln⎜ ⎟ ⎝ P∞ ⎠

In eqs 15 and 20, both changes in the interfacial area (dALV) and changes in the volumes of the comprising bulk phases (dVV and dVL) are dependent on curvature of the involved interface. When all these curvature dependent terms are substituted, they result in two well-known conditions for equilibrium: (i) the Young equation21 γ SV − γ SL = γ LV cos θ

V

Either P or P can be described in terms of the other from eq 27 and combined with the Laplace−Young equation, eq 24, to give a new equation for calculating the mean radius of curvature. In the case of liquid formation, where the pressure of the vapor phase is controlled (vapor is the initial, i.e., mother phase), PL is described in terms of PV using eq 27. Substituting PL into eq 24, and choosing Pa − Pb = PV − PL so that the radius of curvature is positive if vapor is inside the curvature, yields the form of the Kelvin equation given below21,25

(21)

where θ is the equilibrium contact angle; the angle that the liquid−vapor interface makes with the solid surface, being measured from the liquid side, and in this work will be assumed to be known; and (ii) the Laplace−Young equation21 ⎛1 1 ⎞ P a − P b = γ ab⎜ + ⎟ R2 ⎠ ⎝ R1

RC =

(22)

a

b

where P is the pressure of one side of the curvature, and P is the pressure of the other side of the curvature. R1 and R2 are the principal radii of curvature used to describe a curved surface at any point. The mean radius of curvature, Rm, is then defined in terms of the principal radii of curvature as follows 1 1⎛ 1 1 ⎞ = ⎜ + ⎟ Rm 2 ⎝ R1 R2 ⎠

(23)

RC =

ab

2γ P −P = Rm b

2γ LV (PV − P∞) −

RT PV ̅ ln P υ∞L ∞

( )

(28)

RC in eq 28 is merely Rm at equilibrium conditions. In the case of vapor formation out of a bulk liquid phase, PV is described in terms of PL using eq 27 and substituted into eq 24, and choosing Pa − Pb = PL − PV so that the radius of curvature is positive if liquid is inside the curvature, yields the following form of the Kelvin equation as previously presented for the case of bubble nucleation in a liquid−gas system.8

and the Laplace−Young eq 22 can be described in terms of this mean radius. a

(27)

L

2γ LV

(

P L − P∞exp

υ∞L (P L RT ̅

− P∞)

)

(29) 26

The Kelvin equation has been shown by Powles to be valid for microscopic drops above the size of validity of homogeneous thermodynamics. For a clean system with no accumulation of contaminates, the Kelvin equation is obeyed by menisci with mean radius as low as eight times the molecular diameters of the material of interest.13 Also bulk thermodynamics and therefore the Kelvin radius are reported to be valid for mean radius of curvature greater than 5 nm for H2O (equivalent to relative vapor phase pressure of 0.9 when considering liquid drop formation).27 From eqs 28 and 29, it can be seen that the sign of the Kelvin radius depends on the value of the bulk pressure. (a) First consider the case of liquid formation out of a bulk vapor phase. With the pressure difference being arbitrarily defined as PV−PL, the Kelvin radius can be obtained from eq 28. Eliminating extremely high bulk phase pressures that are not of interest,28 the relationship between bulk vapor pressure and the sign of the Kelvin radius is

(24)

The principal radii of curvature can be positive or negative, depending on which side of the interface the center of the circular arcs lies. Similarly to the presentation of Middleman,22 arbitrarily one side of the interface is chosen as the “inner” side assuming a positive sign for the radius of curvature, and the other side is chosen as the “outer” side with a negative sign for the radius of curvature. As a result, Pa − Pb = +Pinner − Pouter. The principal radii of curvature have the same signs if both of the centers of their circular arcs are on the same side of the surface. At equilibrium conditions in the absence of gravitational effects, as mentioned by Hunter,21 Pa − Pb must be constant over all parts of the interface; otherwise a fluid flow would occur. As a result, since γab is also constant, (1/R1) + (1/R2), and hence Rm, have to be constant according to eqs 22 and 24. Therefore the interface would be a surface of constant mean curvature. The Kelvin equation can also be obtained by combining some of the conditions for equilibrium, stated in Table 1. With both liquid and vapor phases being single component, assuming the vapor phase to be an ideal gas, and the liquid phase to be incompressible, yields eqs 2523 and 2624 ⎛P V μ V (T V , PV ) = μ V (T V , P∞) + RT ̅ ln⎜ ⎟ ⎝ P∞ ⎠

(25)

μL (T L , P L) = μL (T L , P∞) + υ∞L (P L − P∞)

(26)

⎧ PV ⎪ R C > 0 if 1 ⎪ R C < 0 if P∞ ⎩

V⎞

(30)

It should be noted that at vapor pressure equal to saturation pressure, there would be no pressure difference along the interface (PV=PL) and the interface would be flat. (b) Next consider the case of vapor formation out of a bulk liquid phase. When the pressure difference is arbitrarily defined as PL−PV, the denominator of eq 29 determines the sign of the Kelvin radius. Eliminating extremely high pressures that are not of interest,28 the relationship between the bulk liquid pressure and the sign of the Kelvin radius is

where R̅ is the universal gas constant, P∞ is the saturation pressure of the fluid in bulk, and υL∞ is the specific volume of the pure liquid at the saturation pressure. Equating eqs 25 and 26 and noting equality of temperature results in eq 27 3613

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Table 2. Forms of Free Energy of the System Liquid formation out of a bulk vapor phase Free energy Equivalent form of free energy for system with curved SL and SV interfaces

equation 35

B = [− P SV S + μ2S N2S] + [− P LV L + μ1L N1L] + [μ1V N1V ] + [γ SLASL + μ1SL N1SL + μ2SL N2SL] + [γ SVASV + μ1SV N1SV + μ2SV N2SV ] + [γ LVALV + μ1LV N1LV ] + PVV L

Equivalent form of free energy for system with flat SL and SV interfaces

(41)

B = [− P SV S + μ2S N2S] + [− P LV L + μ1L N1L] + [μ1V N1V ] + [γ SLASL + μ1SL N1SL] + [γ SVASV + μ1SV N1SV ] + [γ LVALV + μ1LV N1LV ]

+ PVV L Vapor formation out of a bulk liquid phase Free energy Equivalent form of free energy for system with curved SL and SV interfaces

(42)

equation 38

B = [− P SV S + μ2S N2S] + [μ1L N1L] + [− PVV V + μ1V N1V ] + [γ SLASL + μ1SL N1SL + μ2SL N2SL] + [γ SVASV + μ1SV N1SV + μ2SV N2SV ] + [γ LVALV + μ1LV N1LV ] + P LV V

Equivalent form of free energy for system with flat SL and SV interfaces

(43)

B = [− P SV S + μ2S N2S] + [μ1L N1L] + [− PVV V + μ1V N1V ] + [γ SLASL + μ1SL N1SL] + [γ SVASV + μ1SV N1SV ] + [γ LVALV + μ1LV N1LV ] + P LV V

⎧ PL ⎪ R C > 0 if 1 ⎪ R C < 0 if P∞ ⎩

Therefore the free energy of the system in which a liquid phase is being formed out of a vapor phase is9,25 B = F S + F L + GV + F SL + F SV + F LV + PVV L

L

V

0 ≥ ΔU + ΔU + ΔU + ΔU R

S

R

L

L

SL

+ ΔU

V

− T (ΔS + ΔS + ΔS + ΔS

SL

SV

+ ΔU

+ ΔS

SV

0 ≥ [ΔU S − T SΔS S] + [ΔU L − T LΔS L + P LΔV L] + [ΔUV − T V ΔS V ] + [ΔU SL − T SLΔS SL]

LV

+ [ΔU SV − T SV ΔS SV ] + [ΔU LV − T LV ΔS LV ]

LV

+ ΔS )

V

+ P (ΔV + ΔV )

+ P LΔV V

(32)

Δ(F S + GL + F V + F SL + F SV + F LV + P LV V ) ≤ 0 (37)

Therefore, the free energy of a system in which a vapor phase is being formed out of a liquid phase is8 B = F S + GL + F V + F SL + F SV + F LV + P LV V

0 ≥ [ΔU S − T SΔS S] + [ΔU L − T LΔS L] + [ΔU SV − T SV ΔS SV ] + [ΔU LV − T LV ΔS LV ] (33)

According to the definitions of Helmholtz (F) and Gibbs (G) free energies, eq 33 is equivalent to L

V

Δ(F + F + G + F

SL

+F

SV

+F

LV

V

(38)

The free energies for the cases of liquid formation out of a vapor phase and vapor formation out of a liquid phase are summarized in Table 2 respectively. In Table 2, equivalent forms of eqs 35 and 38 are also presented, in which the internal energies have been replaced by their corresponding forms from the Euler relation, which for bulk phases is

+ [ΔUV − T V ΔS V + PV ΔV V ] + [ΔU SL − T SLΔS SL]

S

(36)

which is equivalent to

Now the reservoir properties of TR and PR have to be replaced in eq 32 according to the conditions for equilibrium (Table 1) and depending on whether a vapor phase or a liquid phase is forming. (a) In the case of liquid formation out of a bulk vapor phase, where PR = PV, eq 32 is rearranged to

+ PV ΔV L

(35)

While the free energy of the whole system is B, each constituent subsystem has a specific free energy based on its constraints. For example for the solid phase with constant volume (dVS=0) and temperature, the Helmholtz free energy is appropriate. The Gibbs free energy acts as the free energy of the vapor phase which has imposed temperature and pressure (TV = TR, and PV = PR). For the liquid phase where neither the volume nor the pressure is constant, some extra terms appear and the free energy is not in one of those well−known formats. (b) In the case of vapor formation out of a bulk liquid phase, where PR = PL, eq 32 becomes

(31)

(b). Free Energy of the System. The next step is to find the free energy (or thermodynamic potential) of the system that acts as the motivation in any evolution toward equilibrium. It should be noted that while the bulk pressure of the confined fluid is constant (controlled by the reservoir), the pressure of the new phase being formed out of the confined fluid varies with the size of that new phase. Hence Gibbs free energy cannot be the thermodynamic potential of such a system for which only the pressure of the initial phase is constant. Following methods presented earlier,8,9,25 it can be shown that for spontaneous changes about equilibrium S

(44)

k

U a = T aS a − P aV a +

L

+P V )≤0 (34)

∑ μia Nia i=1

3614

(39)

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Table 3. Equivalent Forms of Free Energy with Respect to the Reference Condition Liquid formation out of a vapor phase9

B − B0 = (PV − P L)V L + (γ SL − γ SV )ASL + γ LVALV

Vapor formation out of a liquid phase (similar to that by Ward and Levart8)

B − B0 = (P L − PV )V V + (γ SV − γ SL)ASV + γ LVALV

(45)

⎛1 1 ⎞ L + B − B0 = γ LV ⎜ ⎟V + (− γ LV cos θ)ASL + γ LVALV R2 ⎠ ⎝ R1

⎛1 1 ⎞ V B − B0 = γ LV ⎜ + ⎟V + (γ LV cos θ)ASV + γ LVALV R2 ⎠ ⎝ R1

(47)

LV

B − B0 =

2γ V L + (− γ LV cos θ)ASL + γ LVALV RC

(46) (48)

LV

B − B0 =

(49)

2γ V V + (γ LV cos θ)ASV + γ LVALV RC

(50)

Figure 2. Schematic of possible (but not necessarily stable) liquid bridges(a) concave and (b) convexbetween a sphere and a flat plate, where θ1 and θ2 are the equilibrium contact angles for the lower and upper surfaces (measured from inside the liquid according to the convention), r is the radius of the circle approximating the vertical section of the liquid−vapor interface, d is the liquid bridge half width, H is the minimum distance between the sphere and the flat plate, α is the half filling angle of the liquid in the bridge, y1 is the three phase contact with the lower particle, y2 is the three phase contact with the upper particle, and RP is the radius of the spherical particle.

versus new phase volume. A maximum in the curve of free energy versus new phase size corresponds to an unstable equilibrium state, while minimum points indicate respectively meta−stable or stable equilibrium states for local or absolute minima of the curve. A monotonically increasing curve indicates that new phase formation is unfavorable at the system conditions. In the case of a monotonically descending curve, the new phase will grow forever until all of the initial phase is changed to the new phase form. Geometry-based Equations. The equations developed so far are applicable to any liquid phase formation out of a bulk vapor phase, or vapor formation out of a bulk liquid phase, regardless of the confinement geometry. The required volume and interfacial areas in the equations of Table 3 have to be determined based on the geometry. The geometry we are interested in is the gap between a sphere and a flat plate. (a). Liquid Formation out of a Bulk Vapor Phase. A schematic diagram of liquid formation out of a bulk vapor phase between a sphere and a flat plate is shown in Figure 2 for cases of concave outward and convex outward menisci. As presented in Figure 2, the interface is either concave or convex depending on the solid material. Herein, we introduce the terminology of the concave-to-convex transition contact angle (equivalently wetting/non−wetting transition contact angle), denoted by θt and defined as the contact angle at which the meniscus changes from being concave to being convex. For identical contact angles at the upper and lower solid surfaces (θ1 = θ2 = θ), the concave-to-convex transition contact angle for the case of liquid formation out of a bulk vapor phase between a sphere and a flat plate is (180°−α/2). The meniscus is hence concave for θ < (180°−α/2) and convex for θ > (180°−α/2). A toroidal interface can accurately approximate the liquid− vapor interface for liquid bridge half widths greater than or

and for interfaces is k

U ab = T abS ab + γ abAab +

∑ μiab Niab i=1

(40)

For the liquid−vapor interface, only component 1 occurs in the sum in eq 40. When the solid−liquid or solid−vapor interfaces are curved, both components 1 and 2 occur in the sum in eq 40, whereas when the solid−liquid or solid−vapor interfaces are flat only component 1 occurs. Free energy can only be evaluated with respect to some reference state. The free energy of the reference state is B0, where the subscript 0 denotes the reference state. It is convenient to set the reference state to be an equilibrium condition in which none of the new phase has been formed.29 The system is assumed to be large enough that the intensive properties in the solid phase, and the initial bulk phase, and at the solid−initial phase interface in the reference state do not change after the formation of a small amount of new phase.29 Since B0 is the free energy of an equilibrium condition, for each component the chemical potentials of the initial phase and solid−initial phase interface are equal. Also according to the constraints of the system, the total numbers of moles of each of components 1 and 2 at any condition including the reference state are constant. Following the same procedure as Ward and Forrest,29 the free energies with respect to the reference state, for cases of liquid formation out of a vapor phase and vapor formation out of a liquid phase are summarized in Table 3. (c). Analyzing the Possibility of Phase Transition. At equilibrium conditions the extensive properties of the system take on values that extremize the entropy of that system subject to constraints. As a result, the equilibrium states of the system can be obtained from the extrema of the curve of free energy 3615

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Table 4. Liquid Volume, Solid−Liquid and Liquid−Vapor Interfacial Areas for Liquid Bridge Formation between a Flat Plate and a Sphere VL = π

∫y

y2

F(y)2 dy − VS

1

(51)

π VS = RP3(1 − cos α)2 (2 + cos α) 3 SL

A

2

2

= π[F(y1) + 2RP (1 − cos α)]

ALV = 2π

y2

∫y

F(y) 1 + F′(y)2 dy

1

(52) (53) (54)

Concave meniscus (θ < (180° − α/2)) 2

F(y) = r + d −

y1 = − r cos θ

r −y

Convex meniscus (θ > (180° − α/2))

F(y) = − r + d +

(55)

y1 = r cos θ

(57)

y2 = r cos(θ + α)

r=

2

r=

(61)

d = RP sin α − r[1 − sin(θ + α)]

(56)

(58)

y2 = − r cos(θ + α)

(59)

RP(1 − cos α) + H cos(θ + α) + cos θ

r 2 − y2

(60)

RP(1 − cos α) + H − cos(θ + α) − cos θ

(62)

d = RP sin α + r[1 − sin(θ + α)]

(63)

(64)

Table 5. Vapor Volume, Solid−Vapor and Liquid−Vapor Interfacial Areas for Vapor Bridge Formation between a Sphere and a Flat Plate VV = π

∫y

y2

F(y)2 dy − VS

1

(65)

π VS = RP3(1 − cos α)2 (2 + cos α) 3 SV

A

LV

A

2

2

= π[F(y1) + 2RP (1 − cos α)] = 2π

∫y

y2

F(y) 1 + F′(y) dy

Concave meniscus (θ > (180° + α/2))

F(y) = r + d −

y1 = r cos θ

r −y

2

(68) Convex meniscus (θ < (180° + α/2))

F(y) = − r + d +

(69)

y1 = − r cos θ

(71)

y2 = − r cos(θ − α)

(67)

2

1

2

(66)

d = RP sin α − r[1 − sin(θ − α)]

(74)

R (1 − cos α) + H r= P cos(θ − α) + cos θ

(75)

(70)

(72)

y2 = r cos(θ − α)

(73)

R (1 − cos α) + H r= P − cos(θ − α) − cos θ

r 2 − y2

(76)

d = RP sin α + r[1 − sin(θ − α)]

(77)

equal to 6.5 × RC.9 Pakarinen et al.4 also calculated the exact shape of a liquid meniscus through the Kelvin radius. By comparing the capillary force calculated once for the exact profile and then for the circular approximation, they showed that the circular profile approximation for the meniscus is justified in the validity range of continuum modeling (macroscopic physics).4 For the liquid−vapor interface of constant curvature being approximated by a toroidal surface, the principal radii of curvature, R1 and R2, have the size of |R1| = r and |R2| = d, as shown in Figure 2. The signs of the principal radii of curvature are determined based on the definition of Pa−Pb in eq 22 and according to the concavity of the meniscus. Here Pa−Pb is defined as PV−PL for the case of liquid formation. For a concave meniscus, R1 is positive (R1 = r) and R2 is negative (R2=−d). For the convex case, both radii are negative (R1 = −r and R2= −d). The volume of revolution of a curve, F(y), around the y−axis of symmetry is equivalent to the summation of a sequence of thin flat washers.30 To find the liquid volume (VL), the volume

(78)

of the part of the solid sphere immersed in the liquid (VS) should then be deducted from the calculated volume of revolution.7 ALV can be computed from the surface of revolution around the y−axis of symmetry.31 The appropriate equations giving VL, ASL and ALV are presented in Table 4. To perform the stability analysis for the liquid bridge, VL, ASL and ALV from Table 4 are substituted into eq 49. For a defined problem in which γLV, RC, and θ are known, eq 49 after substitution of all geometric relations would be in terms of r and d, both of which can be written as a function of the half filling angle, α (eqs 61−64). (b). Vapor Formation out of a Bulk Liquid Phase. Similar to the previous case, for vapor formation out of a bulk liquid phase between a sphere and a flat plate, two possible cases of concave or convex meniscus exist. The liquid−vapor interface is similarly approximated by a toroidal interface. For identical contact angles for the upper and the lower solids, the concaveto-convex transition contact angle is (180° + α/2). The meniscus is concave for θ > (180° + α/2) and is convex for θ < (180° + 3616

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α/2). It should be noted that the contact angle is measured from inside the liquid phase according to the convention. With Pa−Pb in eq 22 being defined as PL−PV, for the concave outward meniscus R1 = r and R2 =−d, and for the convex outward case R1 = −r and R2 = −d. The geometric relations for the case of vapor formation out of a bulk liquid phase are presented in Table 5. These are similar to the equations in Table 4 for the case of liquid formation, with the superscript V switched with L, and some changes in y1 and y2 due to the convention of measuring contact angle from the liquid side. Stability analysis for the vapor bridge is performed by substituting VV, ASV and ALV from Table 5 into eq 50. After these substitutions for a defined problem in which γLV, RC, and θ are known, eq 50 would be a function of r and d, both of which can be written as a function of the half filling angle, α (eqs 75−78).

■

RESULTS AND DISCUSSION In this section the shape of the free energy curve, which identifies the existence and the number of stable and/or unstable equilibria, as well as the effect of various parameters for all of the four possible situations of liquid or vapor formation with concave or convex meniscus are presented. ①. Liquid Phase Being Formed from a Bulk Vapor Phase between a Sphere and a Flat Plate: Concave Meniscus. When the pressure difference in eq 22 is defined to be PV−PL, R1 is positive and R2 is negative. Also for various sphere−plate separation distances and equilibrium contact angles of interest, it has been observed that at the equilibrium conditions r is less than d. Hence the mean radius of curvature from eq 23, which is equivalent to the Kelvin radius, is positive. For RC to be positive, the vapor phase pressure must be below the saturation pressure (PV < P∞) according to eq 30. Free Energy Stability Analysis and the Effect of Equilibrium Contact Angle for Liquid Phase Formation out of a Bulk Vapor Phase between a Sphere and a Flat Plate: Concave Meniscus. Curves of the free energy versus scaled concave liquid bridge half width (d/RC) are shown in Figure 3, for a case with the vapor phase pressure below the saturation pressure. From eq 28, the Kelvin radius for the specifications of Figure 3 is 4.39 × 10−8 meters. The distance between the sphere and the flat plate is well below the Kelvin radius (below the breakage distance, as will be discussed in the following section). In Figure 3, the solid line for contact angle equal to 0° is in good agreement with the calculations presented by Elliott and Voitcu,9 who considered only zero contact angle. As can be seen in Figure 3, there is a maximum point in the free energy curve, indicating an energy barrier to be overcome for the phase transition. The phase transition is therefore a nucleation phenomenon.9 Once this barrier is overcome, the bridge grows until it reaches the stable equilibrium size indicated by the minimum point in the free energy. For this case of liquid formation with concave meniscus, as the contact angle increases (gets closer to the concave-to-convex transition contact angle) the height of the energy barrier increases, and the unstable liquid bridge is formed at larger liquid bridge widths. An increase in the contact angle (getting closer to the concave-to-convex transition contact angle) also results in a shallower stable equilibrium state with a smaller stable liquid bridge width. As the increase of the contact angle continues, at a certain contact angle the curve becomes

Figure 3. (a) Effect of the equilibrium contact angle on the free energy versus scaled concave liquid bridge half width between a sphere and a flat plate for n-dodecane at 24 °C, PV = 0.9P∞, H = 0.97RC, and RP = 2.5 cm. (b) Magnification of the unstable equilibrium point.

monotonically increasing where the formation of the liquid turns out to be unfavorable. From further investigations it has been found that a specific number of degrees change in the contact angle results in larger relative changes in the energy barrier and the energy level of the minimum point for contact angles closer to the concave-toconvex transition contact angle.28 This statement only applies to cases where liquid formation is possible, and not cases where liquid formation is unfavorable (the energy curve is monotonically increasing). Effect of Solid Sphere Size on the Stability of the System for Liquid Phase Formation out of a Bulk Vapor Phase between a Sphere and a Flat Plate: Concave Meniscus. In Figure 4, the free energy curves are drawn as the radius of the solid sphere changes from millimeters (105 × RC) to decimeters (5 × 106 × RC). An increase in the radius of the solid sphere results in more stability (deeper free energy minimum) and larger width of the liquid bridge, as shown in Figure 4. This was also concluded by Elliott and Voitcu9 while investigating spheres of two different sizes. For the extreme case of the sphere with radius equal to infinity (liquid bridge between two flat plates), there is no free energy minimum point for the liquid bridge (it is as if the minimum happens at a liquid bridge length of infinity); therefore, all the vapor will change into liquid once the barrier is overcome.9 With the specified separation distance of Figure 4 (H = 0.97 × RC), phase transition becomes impossible (the free energy curve becomes monotonically increasing) as the sphere radius gets smaller than a certain amount (smaller than approximately 2 × 104 × RC, as found by generating curves not shown in Figure 4). 3617

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Figure 4. (a) Effect of the solid sphere size on the free energy versus scaled concave liquid bridge half width between a sphere and a flat plate for n-dodecane at 24 °C, PV = 0.9P∞, θ = 0°, and H = 0.97RC. (b) Magnification of the unstable equilibrium point.

Figure 5. (a) Effect of the solid separation distance on the free energy versus scaled concave liquid bridge half width between a sphere and a flat plate for n-dodecane at 24 °C, PV = 0.9P∞, θ = 10°, and RP = 2.5 cm. (b) Magnification of the unstable equilibrium point.

As can be seen in Figure 4b, the effect of the solid sphere size on the unstable free energy and the unstable liquid bridge width is minor. For example in Figure 4b, while the sphere radius increases by an order of magnitude from 105 × RC (4.39 mm) to 106 × RC (4.39 cm), the energy level of the maximum point decreases only from 1.48 × 10−15 to 1.46 × 10−15 J and the unstable liquid bridge half width decreases only from 13.00 × RC to 12.63 × RC. Even for the geometry of the confined space between two flat plates, which is equivalent to the upper sphere radius being infinite, the energy level of the barrier is 1.46 × 10−15 J and the unstable liquid bridge half width is 12.59 × RC. The reason behind this is the large size of the sphere in comparison to the separation distance of the sphere and plate. For example, the sphere radius of 105 × RC (4.39 mm) is almost 105 times the separation distance in Figure 4. Effect of the Solid Surface Separation Distance on the Stability of the System for Liquid Phase Formation out of a Bulk Vapor Phase between a Sphere and a Flat Plate: Concave Meniscus. Figure 5 shows the free energy curves of the system for five different separation distances, while the contact angle has a non-zero value. From Figure 5, it can be seen that increasing the separation distance between a sphere and a flat plate results in a higher energy barrier with a larger unstable liquid bridge width, and also a less stable (shallower) minimum with shorter stable liquid bridge width. Further increase in the sphere−plate separation distance eventually makes the free energy curve monotonically increasing, indicating the formation of the liquid phase to be unfavorable. The solid surface separation distance above which the liquid formation becomes unfavorable is called the breakage distance (HBreak).9 Elliott and Voitcu9 conducted a similar study for the case when the contact angle is 0°.

For the breakage distance in the case of liquid formation in the gap between a sphere and a flat plate, Fisher and Israelachvili13 developed an approximation in terms of the Kelvin radius and the contact angle, for the cases where RP ≫ | d| ≫ H, and for small contact angles.13 They explained H in terms of d, RP, θ, and RC, where RP, θ, and RC at the pressure of interest are constant. Fisher and Israelachvili13 then define the breakage distance, HBreak, as the maximum of H. In their work some different definitions than ours are used; for example the mean radius of curvature was defined as (1/Rm) = (1/R1 + 1/ R2), rather than (1/Rm) = (1/2)(1/R1 + 1/R2) as in our work and others.9 Translating their notations into ours, the breakage distance is given by13

HBreak

⎧ ⎪ ⎪ = R C cos θ ⎨1 − ⎪ ⎪ ⎩

3

(

32RP cos θ RC

⎫ ⎪ ⎪ ⎬ 1/3 ⎪ ⎪ ⎭

)

(79)

The value of HBreak is less sensitive to RP, and is approximated13 by RCcosθ, which is a good approximation for large enough RP/RC values. For example for n-dodecane at 24 °C, PV = 0.9P∞, and θ = 0°, while the approximation (RC cosθ) suggests a breakage distance of RC, eq 79 gives the breakage distance as 0.990 × RC for a sphere of 2.5 cm (RP/RC = 5.7 × 105) and 0.978 × RC for a sphere of 2.5 mm (RP/RC = 5.7 × 104). The breakage distance is less than RC for any contact angle and any sphere size according to eq 79 . Thus for any sphere− plate separation distance of above or equal to RC (H ≥ RC), liquid formation is certainly unfavorable, regardless of the contact angle and sphere size. Below the Kelvin radius (H < 3618

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RC), H may or may not be below the breakage distance based on the values of contact angle and sphere size. In experiments by Maeda et al.3 on n-hexadecane, two different behaviors have been observed at sphere−plate separation distance equal to HBreak (for the case where contact angle is zero, and HBreak = RC) based on how this separation distance is approached: If the separation distance is initially above HBreak and then decreases to HBreak, the bridge that forms at HBreak has density between the densities of liquid and vapor, and behaves like a fluid above its critical point. If the separation distance is initially below HBreak and then increases to HBreak, the refractive index of the bridge remains that of a bulk liquid at HBreak.3 This can be described as follows: In the first case where the sphere−plate separation distance is initially above and then decreases to HBreak, the free energy curve changes from monotonically increasing (no liquid existence) for H > HBreak, to a curve with an unstable equilibrium state followed by a stable equilibrium state at H = HBreak. Thus in this approach, at H = HBreak liquid has to be formed after passing a nucleation barrier, while the possibility of liquid nucleation did not exist at previous steps where H > HBreak. A longer time is required, due to the necessity of passing a nucleation barrier, before the system can reach its stable equilibrium. As explained previously,9 considering the approximately identical values of the free energy of the stable and the unstable equilibrium at HBreak, the system can fluctuate out of the stable equilibrium condition, that is, natural fluctuations large enough to overcome the nucleation barrier are also large enough to allow the bridge to disappear. This can potentially explain a nonuniform density profile and a “diffuse liquid− vapor interface”.9 In the second process, a sphere and a flat plate are initially separated by a distance below HBreak, which then increases to HBreak. At H < HBreak, with the free energy curve having an unstable and a fairly more stable state, the liquid bridge has already been formed. As the separation distance increases, the liquid bridge which already exists, should only adjust itself to a new (smaller) stable size. Here since the liquid bridge already exists and no nucleation is necessary, no large fluctuations are needed. Hence at H = HBreak, simply the size of the bridge shrinks and gets stable to its new stable size. It should be mentioned that as the focus of this work is the thermodynamic equilibrium of the system, a description of the nonequilibrium transitions as a result of mechanical instabilities (coalescence as a result of van der Waals force when the separation distance is decreasing, and snapping as a result of Rayleigh instability when the separation distance is increasing rapidly3) are not within the scope of this research. At the other extreme when the distance between the sphere and the flat plate decreases to zero, as demonstrated in Figure 6 for different contact angles, the energy barrier is eliminated. Therefore, the phase transition is a non-nucleating spontaneous process that would evolve to the stable equilibrium liquid bridge width. The liquid bridge at zero separation distance is the most stable and has the greatest width compared with other separation distances. Elliott and Voitcu9 showed this case in their work, only for contact angle equal to zero.9 In fact the contact point of the two solids acts as an agent for the new phase formation with concave meniscus. It should be noted that when a flat plate and a sphere are in contact (H = 0), there is always a stable liquid bridge for any contact angle below the concave-to-convex transition contact angle

Figure 6. (a) Free energy versus scaled concave liquid bridge half width between a sphere and a flat plate in contact (H = 0) for ndodecane at 24 °C, PV = 0.9P∞, and RP = 2.5 cm. (b) Magnification of the curve for small values of d/RC.

or any sphere size. For example, for a sphere and a plate at H = 0.97RC (and the conditions of Figure 3) the liquid formation becomes unfavorable for contact angles above 11.5°, whereas with H = 0 liquid formation is favorable even when the contact angle is 89.5° (with the stable scaled bridge width of d/RC = 93). ②. Liquid Phase Being Formed from a Bulk Vapor Phase between a Sphere and a Flat Plate: Convex Meniscus. For liquid formation having convex meniscus, with pressure difference (Pa−Pb) in eq 22 being defined as PV − PL, both R1 and R2 are negative due to their center being located in the liquid phase. Accordingly the mean radius of the curvature from eq 23, and therefore the Kelvin radius are negative. This is only satisfied at vapor phase pressures above the saturation pressure (PV > P∞) according to eq 30. Free Energy Stability Analysis and the Effect of Equilibrium Contact Angle for Liquid Phase Formation out of a Bulk Vapor Phase between a Sphere and a Flat Plate: Convex Meniscus. Curves of free energy versus scaled convex liquid bridge half width (d/RC) are presented in Figure 7 for a case when the vapor phase pressure is above the saturation pressure, and for various contact angles. For the conditions of Figure 7, the Kelvin radius is −4.85 × 10−8 m from eq 28. The formation of a convex liquid bridge between a sphere and a flat plate is a nucleation phenomenon for which an energy barrier has to be overcome. After passing that maximum point, the curve becomes monotonically descending and all the vapor would change into liquid. For liquid formation with a convex meniscus between a sphere and a flat plate, any increase in contact angle (getting farther from the concave-to-convex transition contact angle) causes the barrier to get larger, and the unstable liquid bridge is 3619

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Figure 7. Effect of the equilibrium contact angle on the free energy versus scaled convex liquid bridge half width between a sphere and a flat plate for n-dodecane at 24 °C, PV = 1.1P∞, H = 0.97|RC|, and RP = 2.5 cm.

formed at larger width. Even for the greatest possible contact angle (180°), the curve does not become monotonically increasing; however for the specifications of Figure 7 with the contact angle equal to 180°, the width of the unstable bridge would be so large (d/|RC|=182.53) with such a high energy barrier of 2.24 × 10−13 J (in comparison to d/|RC|=6.46 with an energy barrier of 6.15 × 10−16 J for the case of contact angle equal to 155°, with other conditions kept the same), that nucleation might be practically prevented due to the impossibility of such a large fluctuation in a reasonable time. Effect of Solid Sphere Size on the Stability of the System for Liquid Phase Formation out of a Bulk Vapor Phase between a Sphere and a Flat Plate: Convex Meniscus. In Figure 8, the free energy curves for the system with a convex liquid bridge are presented for different solid sphere sizes. In contrast to the case of concave liquid formation, in Figure 8 as the upper solid sphere radius increases, the level of the energy barrier and the width of the unstable concave liquid bridge would both increase. The highest energy barrier is for the case of the solid sphere of infinite radius, identical to convex liquid bridge formation between two flat plates. For any sphere size other than infinity, the curve of the free energy never becomes monotonically increasing even at the smallest separation distance or for the farthest contact angle from the concave-to-convex transition contact angle, that is, a contact angle of 180°. However, if the sphere is very large, the nucleation happens at a very large liquid width with a relatively high barrier. For example for a sphere of radius 109 × |RC|= 48 m touching the flat plate (H = 0), the unstable bridge has a width of d/|RC|=4.3 × 104 (d∼ 2 mm) with an energy barrier of 3.26 × 10−7 J. For the case of two flat plates the curve can be monotonically increasing if the flat plates’ separation distance is smaller than a certain amount. In Figure 8 where the separation distance is H = 0.97 |RC| convex liquid formation is always possible. Even when the case changes to the case of two flat plates, convex liquid formation happens after passing an energy barrier of 1.94 × 10−15 J at liquid bridge half width of 13.23 × |RC|. Effect of the Solid Surface Separation Distance on the Stability of the System for Liquid Phase Formation out of a Bulk Vapor Phase between a Sphere and a Flat Plate: Convex Meniscus. The effect of the sphere−plate separation distance on the stability of the system with a convex liquid bridge is illustrated in Figure 9. As the sphere−plate separation distance decreases, a higher energy barrier with a larger liquid bridge width has to be

Figure 8. (a) Effect of the solid sphere size on the free energy versus scaled convex liquid bridge half width between a sphere and a flat plate for n-dodecane at 24 °C, PV = 1.1P∞, θ = 160°, and H = 0.97|RC|. (b) Magnification of the unstable equilibrium point.

Figure 9. (a) Effect of the solid surface separation distance on the free energy versus scaled convex liquid bridge half width between a sphere and a flat plate for n-dodecane at 24 °C, PV = 1.1P∞, θ = 160°, and RP = 2.5 cm. (b) Magnification of the unstable equilibrium point.

3620

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There is a maximum point at a very small vapor bridge width (d is much smaller than 6.5 × RC and the toroidal LV surface assumption is no longer valid). Although the value of the energy barrier cannot be trusted because of the very small vapor bridge width, the graph gives a good qualitative description of the behavior of the system. The vapor formation can only happen after passing an energy barrier, that is, the phase transition is a nucleation phenomena. After passing the barrier, the vapor phase grows until it reaches its stable condition, corresponding to the minimum point in the graph. At the stable equilibrium state, the vapor bridge width is above 6.5 × RC, and the toroidal LV surface assumption is valid. Getting closer to the concave-to-convex transition contact angle (equivalently a decrease in the contact angle) results in an increase in the energy barrier and a larger unstable vapor bridge length. It also causes less stability with shorter vapor bridge width in the stable equilibrium state. Getting closer to the concave-to-convex transition contact angle, there is a contact angle above which the curve becomes monotonically increasing, that is, the formation of the concave vapor phase would become unfavorable. In studying the case more closely, it has been found that a specific number of degrees change in the contact angle results in a larger relative change of the energy barrier and the energy level of the stable state for contact angles closer to the concave-to-convex transition contact angle. Effect of Solid Sphere Size on the Stability of the System for Vapor Phase Formation out of a Bulk Liquid Phase between a Sphere and a Flat Plate: Concave Meniscus. The impact of the size of the spherical solid particle on the energy level of the concave vapor bridge is examined in Figure 11.

overcome. In the sphere−plate geometry, even for a sphere and flat plate at contact (H = 0), the free energy curve does not become monotonically increasing (somewhere far enough from the centerline, the distance becomes large enough to allow the formation of the unstable convex liquid bridge); however, the width of the unstable liquid bridge and the size of the barrier to be overcome are both so large that they would be practically impossible to overcome. For example for the case where H = 0 and θ = 180°, with other conditions the same as for Figure 9, the maximum point happens at d = 1016 × |RC| with an energy level of 1.91 × 10−10 J. ③. Vapor Phase Being Formed from a Bulk Liquid Phase between a Sphere and a Flat Plate: Concave Meniscus. With the pressure difference (Pa−Pb) in eq 22 defined as PL−PV, R1 is positive and R2 is negative. For various sphere−plate separation distances and equilibrium contact angles of interest, it has been observed that at the equilibrium conditions r is less than d. Accordingly, Rm from eq 23, and equivalently RC, would be positive. For RC to be positive the liquid pressure must be above the saturation pressure (PL > P∞), based on eq 31. Free Energy Stability Analysis and the Effect of Equilibrium Contact Angle for Vapor Phase Formation out of a Bulk Liquid Phase between a Sphere and a Flat Plate: Concave Meniscus. Curves of the free energy versus the scaled concave vapor bridge half width (d/RC) are shown in Figure 10 while the liquid pressure is above the saturation pressure. With the specifications of Figure 10, the Kelvin radius from eq 29 is 6.22 × 10−4 meters.

Figure 10. (a) Effect of the equilibrium contact angle on the free energy versus scaled concave vapor bridge half width between a sphere and a flat plate for H2O at 20 °C, PL = 1.1P∞, H = 0.5RC (300 μm), and RP = 103RC (62 cm). (b) Magnification of the unstable equilibrium point.

Figure 11. Effect of the solid sphere size on the free energy versus scaled concave vapor bridge half width between a sphere and a flat plate for H2O at 20 °C, PL = 1.1P∞, θ = 180°, and H = 0.5RC (300 μm). (b) Magnification of the unstable equilibrium point. 3621

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In Figure 11, the radius of the spherical particle is changed from several centimeters (102 × RC) to several meters (5 × 103 × RC). As the spherical particle gets larger, the energy barrier gets smaller and the concave vapor phase gets stable at larger bridge width with deeper energy levels. In the case of an upper sphere of infinite radius (equivalent to the case of two flat plates), the energy barrier is at its lowest level, and all of the liquid phase would change into vapor once the energy barrier is overcome, since there is no minimum point (it is as if the minimum happens at a vapor bridge length of infinity). Also with the separation distance of Figure 11 (H = 0.5 × RC), the free energy curve becomes monotonically increasing and phase transition becomes impossible if the sphere radius is smaller than a certain amount (approximately 3.3 × RC, equivalent to 2 mm, as we found from generating curves, not shown in Figure 11). As illustrated in Figure 11b, the effect of the solid sphere size on the free energy and bridge width of the unstable vapor bridge is minor. This is because the separation distance is much smaller than the spherical particle’s radius. The unstable state happens at such a small bridge width that an unstable concave vapor bridge does not sense the size and changes in the curvature of the spherical solid, especially when the contact angle is far from the concave-to-convex transition contact angle. As the concave vapor width gets larger and d gets farther from the center line, the curvature of the solid sphere would significantly affect the energy level of the system, as it does for the stable equilibrium state. Effect of the Solid Surface Separation Distance on the Stability of the System for Vapor Phase Formation out of a Bulk Liquid Phase between a Sphere and a Flat Plate: Concave Meniscus. Figure 12 shows the effect of the sphere− plate separation distance on the free energy curves of the system with a concave vapor bridge. From Figure 12, as the separation distance between the sphere and the flat plate increases, the stable equilibrium state is formed with less stability and shorter vapor bridge width. A larger separation distance also causes a higher energy barrier with a larger unstable vapor bridge width to be overcome. For separation distances higher than a certain amount, the free energy curve becomes monotonically increasing and formation of the vapor phase would be unfavorable. In the case where the contact angle is 180° (farthest from the concave-to-convex transition contact angle) and the sphere radius is 103 × RC, vapor formation becomes unfavorable at, and above, a sphere−plate separation distance of about 0.92 × RC. At constant sphere radius (103 × RC), as contact angle gets less than 180° and becomes closer to the concave-to-convex transition contact angle, the vapor formation would become unfavorable at separation distances less than 0.92 × RC. At constant contact angle, if the sphere size is larger, the concave vapor phase formation becomes unfavorable at a higher sphere−plate separation distance. However, even in the case where the radius of the spherical particle becomes infinity (the case of two flat plates), concave vapor phase formation becomes unfavorable at and beyond H = RC (for contact angle equal to 180°). Hence it can be concluded that for an arbitrary contact angle and sphere size, vapor formation is always unfavorable for H ≥ RC. For H below RC, further investigation is required for any specific contact angle and/or particle size to judge whether the concave vapor formation is favorable.

Figure 12. Effect of the solid surface separation distance on the free energy versus scaled concave vapor bridge half width between a sphere and a flat plate for H2O at 20 °C, PL = 1.1P∞, θ = 180°, and RP = 103RC (62 cm). (b) Magnification of the unstable equilibrium point.

If any concave vapor phase has already been formed, increasing the separation distance would make the stable vapor bridge smaller until it would break at the critical distance. This critical distance is called the breakage distance, and is always less than RC as discussed in the previous paragraph. Modifying eq 79 (which was proposed by Fisher and Israelachvili13) for this case of concave vapor formation out of a bulk liquid phase, based on the negative sign of cos θ (because θ > (180° + α/2)), yields an equation for the breakage distance

HBreak

⎧ ⎪ ⎪ = −R C cos θ ⎨1 + ⎪ ⎪ ⎩

3

(

32RP cos θ RC

⎫ ⎪ ⎪ ⎬ 1/3 ⎪ ⎪ ⎭

)

(80)

This equation works if RP ≫ |d| ≫ H (analogous to the restriction of RP ≫ |d| ≫ H that was mentioned by Fisher and Israelachvili13 in the case of liquid formation with concave meniscus). For H2O at 20 °C, PL = 1.1P∞, θ = 180°, and Rp = 103 × RC (62 cm), the breakage distance from Figure 13 is 0.92 × RC, while the breakage distance given by eq 80 is 0.906 × RC. The energy levels of the unstable and stable equilibrium points become almost equal as the separation distance between the sphere and the flat plate increases to the breakage distance, as shown in Figure 13. The other extreme occurs when the distance between the sphere and the flat plate reduces to zero, as presented in Figure 14 for various contact angles. Concave vapor formation between a sphere and a flat plate that are touching happens through a non-nucleating phenomenon with no energy barrier to be overcome. Decreasing the separation distance to zero also results in the most stable equilibrium with the greatest vapor 3622

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below the saturation pressure are presented in Figure 15 for various contact angles. The Kelvin radius from eq 29 is −6.22 × 10−4 meters at the conditions of Figure 15.

Figure 13. Energy levels of the maximum and minimum points at separation distances close to the breakage distance for a concave vapor bridge between a sphere and a flat plate for H2O at 20 °C, PL = 1.1P∞, θ = 180°, and Rp = 103RC (62 cm). Figure 15. Effect of the equilibrium contact angle on the free energy versus scaled convex vapor bridge half width between a sphere and a flat plate for H2O at 20 °C, PL = 0.9P∞, H = 0.5|RC| (300 μm), and RP = 103|RC| (62 cm).

As can be seen in Figure 15, there exists an energy barrier to be overcome for convex vapor formation out of a bulk liquid phase between a sphere and a flat plate, i.e. the phase transition is a nucleation phenomena. The curve is monotonically decreasing after passing the energy barrier and all the liquid would change into vapor. Also in Figure 15, as the contact angle gets farther from the concave-to-convex transition contact angle (equivalent to contact angle getting smaller), the energy barrier for the formation of the convex vapor phase becomes greater. Even for very large upper sphere radii other than infinity, and with contact angles far from the concave-to-convex transition contact angle (small contact angle in convex vapor formation), the curve is never monotonically increasing. However, an extremely large vapor bridge must be formed with a huge energy barrier to be overcome for nucleation to be possible. For example for RP = 108 × |RC| (62 km) and θ = 0°, the vapor bridge width of the maximum point is 104 × |RC|, with an energy barrier of 4.4 J. At the extreme where the upper sphere turns into a flat plate (radius of infinity), vapor formation with convex meniscus is unfavorable (the free energy curve is monotonically increasing) for contact angles far from the concave-to-convex transition contact angle (small contact angle in convex vapor formation). Effect of Solid Sphere Size on the Stability of the System for Vapor Phase Formation out of a Bulk Liquid Phase between a Sphere and a Flat Plate: Convex Meniscus. The impact of the spherical particle’s size is illustrated in Figure 16 . As shown in Figure 16, an increase in the upper sphere size results in a higher energy barrier and a larger unstable convex vapor bridge. In the extreme case, for an upper sphere with radius of infinity (equivalent to a flat plate), either the highest energy barrier must be overcome for convex vapor bridge formation (in comparison to various cases of spheres with finite size), or vapor formation would not be possible at all (the free energy curve becomes monotonically increasing), depending on the separation distance and the contact angle. For any sphere size other than infinity, the free energy curve never becomes monotonically increasing even at the smallest separation distance or for the smallest contact angle of 0° (the farthest contact angle from the concave-to-convex transition contact angle). However if the sphere is very large, the nucleation

Figure 14. (a) Free energy versus scaled concave vapor bridge half width between a sphere and a flat plate, at contact (H = 0) for H2O at 20 °C, PL = 1.1P∞, and RP = 103RC (62 cm). (b) Magnification of the curve for small values of d/RC.

bridge width (in comparison to a concave vapor bridge formed between a sphere and a flat plate at other separation distances). ④. Vapor Phase Being Formed from a Bulk Liquid Phase between a Sphere and a Flat Plate: Convex Meniscus. Both R1 and R2 are negative in the case of a convex vapor meniscus with the pressure difference (Pa−Pb) in eq 22 being defined as PL−PV. Therefore the mean radius of curvature from eq 23, and equivalently the Kelvin radius are negative. This is only satisfied at liquid pressures below the saturation pressure (PL < P∞) according to eq 31 . Free Energy Stability Analysis and the Effect of Equilibrium Contact Angle for Vapor Phase Formation out of a Bulk Liquid Phase between a Sphere and a Flat Plate: Convex Meniscus. Curves of free energy versus scaled convex vapor bridge half width (d/RC) formed out of liquid at pressure 3623

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Figure 16. Effect of the solid sphere size on the free energy versus scaled convex vapor bridge half width between a sphere and a flat plate for H2O at 20 °C, PL = 0.9P∞, θ = 0°, and H = 0.5|RC| (300 μm). (b) Magnification of the unstable equilibrium point.

Figure 17. Effect of the solid surface separation distance on the free energy versus scaled convex vapor bridge half width between a sphere and a flat plate for H2O at 20 °C, PL = 0.9P∞, θ = 0°, RP = 103|RC| (6.2 km). (b) Magnification of the unstable equilibrium point.

happens at a very large vapor width with a relatively high energy barrier. For example for a sphere of radius 109 × |RC| (∼620 km) touching a flat plate (H = 0), the unstable bridge has a width of d/|RC|=4.4 × 104 (d∼ 28 m) with an energy level of 173 J. In this case the energy level of the nucleation barrier is so large that it might take forever for the nucleation to happen in practice. Effect of the Solid Surface Separation Distance on the Stability of the System for Vapor Phase Formation out of a Bulk Liquid Phase between a Sphere and a Flat Plate: Convex Meniscus. The effect of the sphere−plate separation distance on the free energy of the convex vapor formation is illustrated in Figure 17. As the separation distance gets smaller in Figure 17, convex vapor formation becomes possible after passing a higher energy barrier. Even when a sphere and a flat plate are touching (H = 0), somewhere far enough from the centerline the separation distance becomes large enough to allow the formation of the unstable convex vapor bridge, albeit after passing a huge energy barrier, which might be practically impossible. For example for the case where H = 0 and θ = 0°, with other conditions the same as in Figure 17, the maximum point happens at d = 45.37 × |RC| with an energy level of 1 × 10−4 J.

convex) for potential coexistence of the new phase and the initial phase (liquid−vapor coexistence) was explained. Also for each of these four situations, the effects of three different parameters, contact angle (θ), separation distance (H), and sphere size (RP), were explored. The four possible situations are Situation ①: For a liquid phase to form out of a confined vapor phase with PV < P∞, the meniscus must be concave (θ < (180° − α/2)), that is, confinement of wettable walls. Liquid with concave meniscus may condense out of and stably coexist with the vapor phase at pressures below the saturation pressure (although the liquid phase is not stable in bulk, that is, without confinement, at this pressure). This is well-known as capillary condensation.1 Liquid formation out of a bulk vapor with PV < P∞ might, or might not, be possible depending on the value of different parameters including contact angle (θ), separation distance (H), and sphere size (RP). Situation ②: For a liquid phase to form out of and coexist in an equilibrium state with a confined vapor phase at PV > P∞, the meniscus must be convex (θ > (180° − α/2)), for which the confinement walls have to be nonwettable. The phase transition is a nucleation phenomenon, that is, an energy barrier has to be overcome. It should be highlighted that between a sphere and a flat plate a liquid bridge with a convex meniscus can never exist in a stable equilibrium condition, and all the bulk vapor turns into liquid once the barrier is passed. Depending on the parameters (contact angle (θ), separation distance (H), and sphere size (RP)), the nucleation energy barrier might be so high that the nucleation never happens in practice; but the free energy curve never turns into monotonically increasing (with RP takes other than infinity). Situation ③: For vapor formation out of confined liquid phase at PL > P∞,

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SUMMARY AND CONCLUSIONS In this paper, the behavior of the fluid confined between a sphere and a flat plate was analyzed, using thermodynamic stability. Temperature, bulk phase pressure, and mass of the system were considered to be constant. Four possible situations were considered based on the initial bulk phase (vapor/liquid) and the bulk pressure (below/above the saturation pressure). In each situation, the possible shape of the meniscus (concave/ 3624

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Table 6. Summary of Thermodynamic Stability Analysis for Confined Fluid Inside the Gap between a Sphere and a Flat Plate Number of equilibrium states

Effect of contact angle, getting farther from the concave-toconvex transition contact angle (θt)a

(a) Concave meniscusb

1 unstable 1 stable

→ Smaller barrier → More stability of stable point

(b) Convex meniscusc

1 unstable

→ Larger barrier

New phase formation with

Effect of separation distance, decrease in H → Smaller barrier → More stability of stable point → Larger barrier

Effect of sphere size, increase in RP → Smaller barrier → More stability of stable point → Larger barrier

a This is equivalent to contact angle decrease in cases of liquid formation with concave meniscus out of vapor at PV < P∞ or vapor formation with convex meniscus out of liquid at PL < P∞, and contact angle increase in cases of liquid formation with convex meniscus out of vapor at PV > P∞ or vapor formation with concave meniscus out of liquid at PL > P∞. bLiquid formation out of vapor at PV < P∞ and confinement of wettable walls (capillary condensation) or vapor formation out of liquid at PL > P∞ and confinement of non-wettable walls (capillary evaporation). cLiquid formation out of vapor at PV > P∞ and confinement of non-wettable walls or vapor formation out of liquid at PL < P∞ and confinement of wettable walls.

the meniscus has to be concave (θ > (180° + α/2)), which is possible only if the liquid is confined by nonwettable walls. Some vapor evaporation (with concave meniscus) may happen, even though the pressure is above the saturation pressure. This phenomenon is called capillary evaporation.1,14 Vapor phase formation at this situation might or might not be possible, depending on the value of different parameters including contact angle (θ), separation distance (H), and sphere size (RP). If vapor formation is favorable, after passing an energy barrier, the new vapor phase then grows to and remains at its stable equilibrium size. Situation ④: For a vapor phase to form out of and coexist in an equilibrium state with a confined liquid phase at PL < P∞, the meniscus must be convex (θ < (180° + α/2)) for which confinement walls have to be wettable. The phase transition in this case is a nucleation phenomenon and all the bulk liquid phase turns into vapor phase once the barrier is passed. It should be highlighted that between a sphere and a flat plate a vapor bridge with a convex meniscus can never exist in a stable equilibrium condition. Depending on the parameters of contact angle (θ), separation distance (H), and sphere size (RP), the nucleation energy barrier might be so high that the nucleation will never happen in practice; but the free energy curve never turns into monotonically increasing (with RP other than infinity). The results are summarized in Table 6. The results of the four possible situations can be generalized in two categories, based on the meniscus shape (concave/convex). Defining the concave-to-convex transition contact angle allows us to describe the effects of contact angle in a consistent manner, removing the complexity that is introduced by the convention of measuring contact angle from inside the liquid phase. For example, both an increase in the contact angle of a liquid bridge with concave meniscus, and a decrease in the contact angle of a vapor bridge with concave meniscus, correspond to getting closer to the concave-to-convex transition contact angle. Regarding the effect of the contact angle (θ): (a) New phase formation between a sphere and a flat plate at separation distance H (H ≠ 0) with concave meniscus becomes ultimately impossible for some contact angles close to the concave-toconvex transition contact angle. (b) For new phase formation with convex meniscus, the free energy never becomes monotonically increasing, even for the farthest contact angle from the concave-to-convex transition contact angle. However the energy level of the nucleation barrier might get so large (also depending on separation distance (H), and sphere size (RP)), that it takes so long for the barrier to be overcome, that nucleation never happens in practice.

Regarding the effect of the separation distance (H): (a) New phase formation with concave meniscus becomes unfavorable above a certain distance, called the breakage distance (HBreak).9 For liquid formation with concave meniscus, eq 79 is found in the literature,13 describing the breakage distance for the condition where RP ≫ |d| ≫ H. The breakage distance sensitively depends on the Kelvin radius, and is also a less sensitive function of the sphere radius and contact angle according to eq 79. Equation 80 is the analogous equation for the case of vapor formation with concave meniscus. New phase formation with concave meniscus is certainly not possible for separation distance above the Kelvin radius (H > |RC|) for any sphere size and with any contact angle. Therefore HBreak is always less than |RC|, regardless of the value of contact angle and size of the sphere. At the breakage distance, the free energy of the stable equilibrium is approximately equal to the free energy of the unstable equilibrium. Fluctuations between the unstable and stable equilibrium states can be considered as a reason for a “diffuse liquid−vapor interface”,9 which is experimentally observed at separation distance equal to the breakage distance, in the process of reducing H from H > HBreak to the breakage distance.3 A sphere and a flat plate at separation distance of zero has the most stable bridge with concave meniscus (in comparison with other separation distances) which is formed through a spontaneous non-nucleating phenomenon (zero energy barrier). Investigations showed that for a sphere and a flat plate in contact (H = 0), new phase formation with concave meniscus is always possible (the free energy curve versus size of the new phase never becomes monotonically increasing) for any contact angle (even for the closest contact angle to the concave-to-convex transition contact angle) and any sphere size. (b) For new phase formation with a convex meniscus, reduction of the separation distance, even to H = 0, will never result in a monotonically increasing curve of the free energy versus size of new phase. However, at small separation distances the energy barrier might be so large that it takes so long before it can be overcome, that the nucleation does not happen in practice. Regarding the effect of the sphere size (RP): (a) For new phase formation with a concave meniscus, it is remarkable that the effect of the solid sphere size on the unstable free energy barrier and the unstable liquid bridge width is minor in comparison to its effect on the free energy and the bridge width of the stable equilibrium state. For the extreme condition of two flat plates (when the upper sphere radius is infinity), all of the bulk phase would change into the new phase once the energy barrier is overcome. (b) New phase formation with a convex meniscus, for the extreme condition of two flat plates, 3625

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(7) Tselishchev, Y. G.; Val’tsifer, V. A. Influence of the type of contact between particles joined by a liquid bridge on the capillary cohesive forces. Kolloidn. Zh. 2003, 65, 385−389. (8) Ward, C.; Levart, E. Conditions for stability of bubble nuclei in solid-surfaces contacting a liquid-gas Solution. J. Appl. Phys. 1984, 56, 491−500. (9) Elliott, J. A. W.; Voitcu, O. On the thermodynamic stability of liquid capillary bridges. Can. J. Chem. Eng. 2007, 85, 692−700. (10) Ward, C. A.; Johnson, W. R.; Venter, R. D.; Ho, S.; Forest, T. W.; Fraser, W. D. Heterogeneous bubble nucleation and conditions for growth in a liquid-gas system of constant mass and volume. J. Appl. Phys. 1983, 54, 1833−1843. (11) 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. (12) Attard, P. Thermodynamic analysis of bridging bubbles and a quantitative comparison with the measured hydrophobic attraction. Langmuir 2000, 16, 4455−4466. (13) 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. (14) 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. (15) 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. (16) 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. (17) Wu, J. Y.; Farouk, T.; Ward, C. A. Pressure dependence of the contact angle. J. Phys. Chem. B 2007, 111, 6189−6197. (18) Ghasemi, H.; Ward, C. A. Sessile-water-droplet contact angle dependence on adsorption at the solid-liquid interface. J. Phys. Chem. C 2010, 114, 5088−5100. (19) Gibbs, J. W. On the equilibrium of heterogeneous substances. Trans. Conn. Acad. 2 1876, 108−248 (1878), 343−524. In The scientific papers of J. Willard Gibbs; Ox Bow: Woodbridge, CT, 1993; Vol. I, pp 55−353. (20) Callen, H. B. In Thermodynamics and an introduction to thermostatistics, 2nd ed; Wiley: New York, 1985; p36. (21) Hunter, R. J. In Introduction to modern colloid science, 1st ed; Oxford University Press: Oxford, 2002; pp 136, 140, 142−146. (22) Middleman, S. An Introduction to Fluid Dynamics: Principles of Analysis and Design; Wiley: New York, 1998, pp 4−5. (23) Elliott, J. R.; Lira, C. T. In Introductory chemical engineering thermodynamics; Prentice Hall PTR: Upper Saddle River, NJ, 1999, pp 290−293. (24) Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. d. In Molecular thermodynamics of fluid-phase equilibria, 3rd ed.; PrenticeHall PTR: Upper Saddle River, N.J., 1999; pp 20, 98. (25) Elliott, J. A. W. On the complete Kelvin equation. Chem. Eng. Edu. 2001, 35, 274−279. (26) Powles, J. G. On the validity of the Kelvin equation. J. Phys. A: Math. Gen. 1985, 18, 1551. (27) 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. (28) Zargarzadeh, L. Comparative surface thermodynamic analysis of new fluid phase formation in various confining geometries; M.Sc. Thesis, University of Alberta, January 2012. (29) Ward, C.; Forest, T. On the relation between platelet adhesion and the roughness of a synthetic biomaterial. Ann. Biomed. Eng. 1976, 4, 184−207. (30) Weisstein, E. W. Solid of Revolution. http://mathworld.wolfram. com/SolidofRevolution.html (accessed May 20, 2011).

may be unfavorable (the curve of free energy versus size of the new phase might be monotonically increasing), depending on the values of θ and H. The new phase formation is still thermodynamically favorable for any sphere of finite radius, although the barrier might be so large that such a long time is required before it can be overcome, that the nucleation does not happen in practice. In general, it should be noted that for any new phase formation with convex meniscus between a sphere and a flat plate, the free energy curve of the system never (for any value of θ, H, and RP) changes to monotonically increasing, although overcoming a very large barrier might be practically impossible. This paper presented a comparative thermodynamic stability analysis of new fluid phase formation in the confined geometry created by the gap between a sphere and a flat plate. All four situations of vapor/liquid formation at bulk phase pressure above/below the saturation pressure were studied. The results demonstrated that the shape of the meniscus (concave/convex) determines the number and type of the equilibrium states. The effect of key parameters (contact angle of the liquid−vapor interface with the solid, solid sphere size, and sphere−plate separation distance) on the free energy curves and the possibility of new phase formation were quantified for all situations and could be described with a unified framework (based on the meniscus shape, rather than the initial phase type).

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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. Elliott holds a Canada Research Chair in Thermodynamics. L. Zargarzadeh held scholarships from the University of Alberta.

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