# On the thermodynamic stability of liquid capillary bridges.

INTRODUCTION

The study of capillary condensation (the formation of a liquid at a solid-vapour interface in tight cracks or pores at pressures below the saturation pressure) is of great practical importance in nanotechnology applications and in oil recovery, and provides useful insight into phenomena that occur in porous materials. In particular, the behaviour of microscopic liquid capillary bridges between two solid bodies has been of special interest, since it provides an elegant framework for the computation of adhesive forces associated with the capillary condensation process. The existence of a liquid bridge between two solid surfaces generates an attractive force between them (Maeda and Israelachvili, 2002; Stroud et al., 2001). The behaviour of liquid capillary bridges has mostly been investigated from a mechanical perspective, with an emphasis on computing the relevant adhesion forces (Aminu et al., 2004; Christenson, 1993; Christenson and Claesson, 2001; Ducker et al., 1994; Willett et al., 2000). Various system geometries have been considered, such as the liquid bridge occurring between, for instance, two solid spheres (Seville et al., 2000; Willett et al., 2000), two parallel planes (Christenson, 1997; Ducker et al., 1994; Padday et al., 1997; Stroud et al., 2001), two crossed cylinders (Aminu et al., 2004; Christenson, 1997; Luengo et al., 1998; Maeda et al., 2003), or a sharp tip and a plane (Jang et al., 2002). Many of the phenomena related to liquid bridges between crossed cylinders are described by the model configuration of a liquid bridge between a solid sphere and a flat plate (Christenson, 1993; Luengo et al., 1998; Maeda and Israelachvili, 2002; Maeda et al., 2003), which will also be the geometry considered in the present study.

In the present paper, a thermodynamic stability analysis of liquid capillary bridges will be carried out. The approach is similar to that used by other authors (McGaughey and Ward, 2003; Ward et al., 1983; Ward and Levart, 1984) for other system geometries. These studies were mainly concerned with the nucleation of droplets or bubbles. The approach consists of computing the free energy of the considered three-phase (solid-liquid-vapour) system as a function of the radius (or, equivalently, the volume) of a nucleus of the new phase. The stability analysis is carried out in the framework of nucleation theory (Debenedetti, 2006; Hunter, 1993). Usually, the computed free energy of the system has a local maximum (corresponding to an unstable equilibrium state) associated with a small radius of the particle of the new phase. This is generally understood as an energy barrier that needs to be overcome, representing the work necessary for the creation of a two-phase interface. In general (but not always), the system also has a stable equilibrium state, corresponding to a larger value of the particle radius. One or more additional equilibrium states (stable and/or unstable) may also occur (Ward et al., 1983). In constant-volume studies (McGaughey and Ward, 2003; Ward et al., 1983), the Helmholtz free energy is the thermodynamic potential of the system; however, in systems for which the pressure of one phase is controlled, such as in the case of bubble nucleation in a constant-pressure liquid, the appropriate free energy that acts as the thermodynamic potential for the multi-phase system must be determined (Ward and Levart, 1984).

In the present paper, the appropriate free energy function will be derived for a solid-liquid-vapour system in which the liquid phase consists of an axisymmetric capillary bridge, and the system temperature and vapour pressure are held constant. A stability analysis will be carried out based on that function. Some researchers performed a thermodynamic stability analysis for a similar geometry (in which the upper solid object was highly curved); however they did so from a statistical physical perspective, using grand canonical ensemble Monte-Carlo simulations (Jang et al., 2002). The present study is carried out in Gibbs's macroscopic thermodynamic paradigm, which was previously shown to be valid in this context. More specifically, Fisher and Israelachvili (1979, 1981) demonstrated that macroscopic thermodynamics is indeed applicable to two-phase interfaces with radii of curvature down to the order of nanometres. By comparison, the radii of curvature dealt with in the present study are of the order of tens of nanometres or larger.

In the current paper, we will investigate the thermodynamic stability of three-phase systems containing an axisymmetric liquid capillary bridge between two solid bodies, and the dependence of the stability behaviour on the curvatures of the solid bodies, on the distance between them, and on the vapour pressure. This study was inspired by the findings of several researchers (Christenson, 1997; Kohonen et al., 1999; Maeda and Israelachvili, 2002; Maeda et al., 2003) who investigated the formation of liquid bridges between two solid surfaces at close distance (less than 0.1 [micro]m). It was noticed that, as the distance between the two solid surfaces is increased, the density of the liquid bridge remains constant up to a critical distance (approximately equal to the mean radius of curvature of the interface at equilibrium), when it drops abruptly to a value between the density of the bulk liquid and that of the vapour (Maeda et al., 2003). This phenomenon was associated with the occurrence, at the critical distance between the two solid bodies, of a gradient in the density of the condensate, in the directions both normal and parallel to the solid surfaces (Maeda et al., 2003), an effect that had previously been experimentally noticed (Maeda and Israelachvili, 2002). In Jang et al. (2002) and Stroud et al. (2001), the effects reported in Maeda and Israelachvili (2002) were reproduced based on statistical mechanical numerical simulations, in the case of two planar solid surfaces (Stroud et al., 2001), and in the case of a high-curvature upper solid object on a planar surface (Jang et al., 2002). In the present paper, a macroscopic thermodynamics argument, based on the existence of fluctuations around equilibrium states of the system, will be used to support the finding that a diffuse interface does indeed occur at some critical distance between the two solid surfaces. Note that, since in Maeda and Israelachvili (2002) and Maeda et al. (2003) it is reported that the contact angles observed in the experiments are small (e.g. less than 6[degrees] for cyclohexane on a mica surface), for simplicity the present theoretical analysis will be carried out under the assumption that the three-phase contact angle is constant and equal to zero for the considered temperature (24[degrees]C, as in Maeda et al., 2003) and vapour pressures. Though computationally more challenging, the analysis can readily be extended to other small contact angle values, but the qualitative behaviour of the system is expected to be the same.

FREE ENERGY OF THE CONSIDERED THREE-PHASE SYSTEM

In the present section, the approach presented in Ward and Levart (1984) and Elliott (2001) is used to derive the expression of the free energy function of the three-phase system represented in Figure 1. The solid (S) phase consists of an upper sphere (of radius a) and a lower flat plate, both made of the same material. The liquid (L) phase consists of an axisymmetric liquid bridge between the two solid components. The vapour (V) phase is assumed to be connected to a temperature (T) and pressure (P) reservoir. Therefore:

[T.sup.j] = [T.sup.R], for j = L, V, SL, SV, LV (1)

and

[P.sup.V] = [P.sup.R] = [fP.sub.[infinity]] (2)

where f = [P.sup.R]/[P.sub.[infinity]] is the ratio between the pressure in the reservoir and the saturation pressure corresponding to the (single) chemical component present in the fluid. The superscripts indicate that the corresponding physical quantity is associated with the reservoir (R), or a bulk phase (L, V) or a surface phase (SL, SV, LV denoting the corresponding two-phase interfaces). The SL and SV interfaces are modelled by theoretical surfaces placed such that there is no adsorption of the solid component (using "Gibbs' dividing surface approximation"--see Gibbs, 1876; Ward and Sasges, 1998). The solid then becomes part of the reservoir and plays no further role in the thermodynamics other than to define the system geometry. It is assumed that the system contains a single chemical component, and [P.sub.[infinity]] denotes the saturation pressure for the considered chemical component. The LV interface is modelled by a theoretical surface placed such that the corresponding surface tension [[gamma].sup.LV] does not depend on the interface curvature (using "Gibbs' surface-of-tension approximation"--see Gibbs, 1876; Ward and Sasges, 1998). All physical quantities associated with a certain (bulk or surface) phase are assumed to be constant throughout that phase. In particular, the effect of gravity will be neglected throughout this study, since the height of the considered liquid bridges does not exceed 0.1 [micro]m.

[FIGURE 1 OMITTED]

In order to analyze the stability of the considered system, the expression for the free energy of the system must be derived. To that end, first consider the difference form of the fundamental relation (Hunter, 1993) for the reservoir:

[DELTA][U.sup.R] = [T.sup.R][[DELTA]S.sup.R] - [P.sup.R] [DELTA][V.sup.R] + [[mu].sup.R] [DELTA][N.sup.R] (3)

where U, S, V, [mu] and N denote the internal energy, entropy, volume, chemical potential, and number of moles of substance, respectively. The following constraints apply:

[DELTA][N.sup.R] = 0, [DELTA][N.sup.tot] = 0 (4)

[DELTA][U.sup.R] + [summation over (j)][SIGMA][U.sup.j] = 0 (5)

[DELTA][V.sup.R] + [DELTA][V.sup.L] + [DELTA][V.sup.V] = 0 (6)

[DELTA][A.sup.SV] + [DELTA][A.sup.SL] = 0 (7)

where [N.sup.tot] = [summation over (j)][N.sup.j] is the total number of moles in the considered system, and [A.sup.SL], [A.sup.SV], [A.sup.LV] denote the areas of the SL, SV, LV interfaces, respectively. When spontaneous changes occur around an equilibrium state, the entropy of the system and reservoir together increases:

[DELTA][S.sup.R] + [summation over (j)] [DELTA][S.sup.j] [greater than or equal to] 0 (8)

Substituting (4) to (6) and (8) in (3) and making use of (1) and (2) yields:

[P.sup.V] ([DELTA][V.sup.L] + [DELTA][V.sup.V]) + [summation over (j)] ([DELTA][U.sup.j]-[T.sup.j] [DELTA][S.sup.j] [less than or equal to] (9)

From (9) and the difference form of the fundamental relation for the bulk phases:

[DELTA][U.sup.j] = [T.sup.j][DELTA][S.sup.j]-[P.sup.j][DELTA][V.sup.j]+ [[mu].sup.j][DELTA][N.sup.j] = L, V (10)

and for the surface phases:

[DELTA][U.sup.j] = [T.sup.j] [DELTA][S.sup.J] + [[gamma].sup.j] [DELTA][A.sup.j] + [[mu].sup.j][DELTA][N.sup.j], j = SL, SV, LV (11)

(Hunter, 1993), it follows that:

([P.sup.V]-[P.sup.L])[DELTA][V.sup.L]+[[gamma].sup.SL][DELTA][A.sup.SV]+ [[gamma].sup.LV][DELTA][A.sup.LV]+[summation over (j)][[mu].sup.j] [DELTA][N.sup.j][less than or equal to]0 (12)

Using (7), we obtain [DELTA]B [less than or equal to] 0, where

B = ([P.sup.V]-[P.sup.L])[V.sup.L]-([[gamma].sup.SV]-[[gamma].sup.SL]) [A.sup.SL]+[[gamma].sup.LV][A.sup.LV]+[summation over (j)][[mu].sup.j] [N.sup.j] (13)

is the free energy of the considered system. Note that from (9) it follows that

B = [P.sup.V][V.sup.L]+[G.sup.V]+[F.sup.L]+[F.sup.SL]+[F.sup.SV]+ [F.sup.LV] (14)

where [F.sup.j] and [G.sup.j] are the Helmholtz and Gibbs free energies, respectively, corresponding to phase j ( j = L, V, SL, SV, LV).

In an equilibrium state of the considered system, in addition to Equations (1)-(2), the following relations hold (Ward and Sasges, 1998):

[P.sup.L]-[P.sup.V] = 2[[gamma].sup.LV][K.sub.eq] (15)

[[gamma].sup.SV]-[[gamma].sup.SL] = [[gamma].sup.LV] cos[theta] (16)

[[mu].sup.j] = [[mu].sub.eq], for j = L, V, SL, SV, LV (17)

namely, the Laplace and Young equations, and the condition of equality of the chemical potentials of all phases. In the above equations, [theta] denotes the three-phase contact angle, and [K.sub.eq] denotes the mean curvature of the LV interface at equilibrium (which may take on either positive or negative values). The mean radius of curvature of the LV interface at equilibrium:

[R.sub.eq] = 1/[K.sub.eq] = 1/2 (1/[R.sub.1,eq] + 1/[R.sub.2,eq]) (18)

will be understood in a general sense, taking on either positive or negative values.

We choose as a reference state (marked by the subscript "0") the state of the system in which no liquid is present, and we assume that it is an equilibrium state. Then:

[B.sub.0] = [[mu].sup.V.sub.0] [N.sup.V.sub.0] + [[mu].sup.SV.sub.0] [N.sup.SV.sub.0] = [[mu].sub.eq]([N.sup.V.sub.0]+[N.sup.SV.sub.0])= [[mu].sub.eq][N.sup.tot] (19)

Subtracting (19) from (13) yields:

B - [B.sub.0] = ([P.sup.V] - [P.sup.L])[V.sup.L]-([[gamma].sup.SV]- [[gamma].sup.SL])[A.sup.SL]+[[gamma].sup.LV] [A.sup.LV]+[[summation].sub.j] ([[mu].sup.j]-[[mu].sub.eq])[N.sup.j] (20)

If the current state of the system is also an equilibrium state, then (15) to (17) apply, and thus:

B - [B.sub.0] = [[gamma].sup.LV] ([A.sup.LV] - [A.sup.SL] cos [theta] - [2V.sup.L] [K.sub.eq]) (21)

The value of [K.sub.eq] can be determined from the imposed vapour phase pressure, based on expressions for the chemical potentials in the liquid and vapour phases. For an incompressible fluid:

[[mu].sup.L] (T, [P.sup.L]) = [[mu].sup.L](T, [P.sub.[infinity]]) + [[upsilon].sup.L.sub.[infinity]([P.sup.L]-[P.sub.[infinity]]] (22)

and for an ideal gas:

[[mu].sup.V](T, [P.sup.V]) = [[mu].sup.V](T, [P.sub.[infinity]]) + [bar.R]T ln ([P.sup.V]/[P.sub.[infinity]]) (23)

where [v.sup.L.sub.[infinity] is the liquid specific molar volume at saturation for the considered chemical component, [bar.R] is the universal gas constant, and [[mu].sup.j] (T, P) (j = L, V ) is the chemical potential in the phase j at temperature T and pressure P. By (1), (2), (17), and (22) to (23), it follows that

[P.sup.L] = [P.sub.[infinity]] + [bar.R][T.sup.R]/ [v.sup.L.sub.[infinity]] ln([P.sup.R]/[P.sup.[infinity]]) (24)

and using (15) we obtain:

[K.sub.eq] = [P.sub.[infinity]]/2[[gamma].sup.LV] {1-[P.sup.R]/P [infinity]+[bar.R][T.sup.R]/[P.sub.[infinity]][v.sup.L.sub.[infinity]]ln ([P.sup.R]/P[infinity])} (25)

NUMERICAL PROCEDURE

The present paper is concerned with the thermodynamic stability of axisymmetric liquid bridges between an upper solid sphere (of radius a) and a lower planar solid plate (see Figure 1). All numerical simulations in the present study are carried out for n-dodecane at 24[degrees]C. For simplicity, the three-phase contact angle [theta] is assumed to be zero. Various sphere radii will be considered, as well as the limiting case of an upper planar solid plate (equivalent to a solid sphere of infinite radius). The dependence of the free energy of the three-phase system on the width of the liquid bridge will be analyzed. Throughout the present study, for given values of f, a, and H, the LV interface is numerically computed, over a range of values of [alpha] (the half opening angle of the spherical cone determined by the three-phase contact line on the upper solid sphere), by integrating the Laplace Equation (15) in each case. More specifically, once the angle [alpha] (see Figure 1) is known, the coordinates of the upper three-phase contact line

c = a sin [alpha], b = H + a (1-cos [alpha]) (26)

can be computed, and the LV interface shape can be numerically constructed by integrating the three-dimensional system of ordinary differential equations:

[d.sub.[xi]]/d[tau] = cos[phi],d[zeta]/d[tau]=sin[phi],d[phi]/ d[tau]=[R.sub.t](-2[K.sub.m] + sin[phi]/c-[R.sub.t][xi] (27)

(where [tau] = s/[R.sub.t] is the scaled arc-length) with initial conditions

[xi](0) = [zeta](0)=0,[phi](0)=[alpha] (28)

using a fourth-order Runge-Kutta algorithm (with step size [DELTA][tau] = [10.sup.-4]), where

[xi] = c-x/[R.sub.t], [zeta] = b-z/[R.sub.t] (29)

and where, for scaling purposes, the value

[R.sub.t] = b/1 + cos [alpha] (30)

is used, which is the radius of the vertical section of the (unique) toroidal surface corresponding to the current value of [alpha]. In (27), [phi] is the angle between the upwards vertical direction and the direction of the first principal radius of curvature, pointing towards the liquid phase (see Figure 1). The mean curvature [K.sub.m] of the LV interface (which is constant throughout the interface) is a search parameter for the computation of generic bridge shapes (including non-equilibrium shapes). More specifically, for a given value of [K.sub.m], the arc-length integration is stopped at the smallest point [tau] = [[tau].sub.max] for which [phi]([[tau].sub.max])[greater than or equal to][pi] (in which case we also have [phi]([[tau].sub.max])[congruent to][pi]), and then the elevation of the lowest point on the LV interface z([[tau].sub.max]) = b - [R.sub.t][zeta]([tau]max) is computed. For physical consistency, we need to have z([[tau].sub.max])[congruent to] 0, or equivalently [zeta]([[tau].sub.max])[congruent to]b/[R.sub.t]. Therefore, we search for the (unique) value of [K.sub.m] that minimizes [absolute value of [zeta]([[tau].sub.max])-(b/[R.sub.t])]. A standard golden section search procedure is used for that purpose (Press et al., 1992).

The numerical computation of the LV interface using the described algorithm is accurate for D ??0.5 Req. It is worth noting that, alternatively, for D [greater than or equal to] 0.5 [R.sub.eq], the LV interface can accurately be approximated by a toroidal surface (that is, considering the vertical section of the interface to be an arc of a circle). More specifically, though the mean curvature of a toroidal surface varies with elevation, in practice, for the simulations carried out in the present study, the standard deviation of the mean curvature (taken over the range of possible elevations 0 [less than or equal to] z [less than or equal to] b) is generally at most 2.5% of the average mean curvature (taken over the range of possible elevations 0 [less than or equal to] z [less than or equal to] b), for D [greater than or equal to] 6.5 [R.sub.eq]. Thus, the physical LV interface can accurately be approximated by a toroidal surface, for which all quantities of interest in (21) can be computed analytically, thus greatly reducing the computational cost of the simulations.

RESULTS OF NUMERICAL SIMULATIONS

System Stability Dependence on the Distance between the Two Solid Surfaces

For a = 2.5 cm and [P.sup.R] = 0.9[P.sub.[infinity]] (values close to the experimental ones in Maeda et al., 2003), the values of B - [B.sub.0] (given by (21)) are computed over a range of values of the scaled liquid bridge half-width [D.sub.sc] = D/[absolute value of [R.sub.eq]], and for different values of the scaled distance between the upper and the lower plate [H.sub.sc] = H/[absolute value of [[R.sub.eq]], and the results are plotted in Figure 2. It is apparent that the system has an unstable as well as a stable equilibrium configuration (corresponding to a maximum in the free energy at a smaller bridge width and a minimum in the free energy at a larger bridge width, respectively). The free energy variation is similar to that obtained for instance by McGaughey and Ward (2003) (in the case of droplet nucleation in a constant volume, with the Helmholtz free energy as the thermodynamic potential of the system). Thus, in general, for 0 < H < [R.sub.eq], the liquid bridge formation is a nucleation phenomenon, in which the energy barrier to be overcome in order for the new phase to be formed is of order of fJ (femtoJoules). The energy barrier decreases as the distance H between the two solid bodies decreases, until at H = 0 there is no more energy barrier to be overcome, and the liquid phase therefore forms spontaneously, settling into the stable equilibrium configuration illustrated in Figure 3 (in the numerical simulations in this case, a toroidal LV interface approximation was used, which for this particular value of H is very accurate even for very small values of [D.sub.sc]). Thus, for H = 0, the phase transition is not a nucleation process.

[FIGURE 2 OMITTED]

Moreover, in general for H > [R.sub.eq] the formation of the liquid phase is not favourable, as the free energy increases steadily with the bridge width (Figure 2). If the liquid phase already exists (in stable thermodynamic equilibrium) between the two solid bodies, and the distance H is steadily increased (as in the experiments reported in Maeda et al., 2003), the width of the liquid bridge will steadily decrease (as the stable equilibrium configuration corresponds to smaller values of D : see Figures 2 and 4), until, at some critical value of H, the bridge will break (as the existence of the liquid phase becomes energetically unfavourable). This process is illustrated in more detail in Figure 4. It is interesting to note that the liquid bridge breaks for H [congruent to] [R.sub.eq], that is, when the distance between the two solid bodies becomes approximately equal to the equilibrium mean radius of curvature (the "Kelvin radius") of the LV interface. This critical distance is the same as that reported (with a different notation) in Maeda and Israelachvili (2002) and Maeda et al. (2003). It should be stressed that in those studies the Kelvin radius is defined using a different notation from that in the present paper. More specifically, the definition in Fisher and Israelachvili (1981) was used by these authors, namely 1/[R.sub.eq] is defined as 1/[R.sub.1,eq] + 1/[R.sub.2,eq] rather than (1/[R.sub.1,eq] + 1/[R.sub.2,eq])/2 as in the present study and as for instance in Derjaguin and Churaev (1976) (so that 1/[R.sub.eq] is the mean curvature of the surface).

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

As H becomes larger, the free energy minimum becomes shallower, and the stable equilibrium width of the liquid bridge becomes smaller (in general, in this study, deeper free energy minima, and higher free energy maxima, are associated with larger equilibrium bridge widths). From Figure 4 it is apparent that, when the distance H becomes close to [R.sub.eq], the free energies corresponding to the stable and the unstable equilibrium states of the system become approximately equal. In that case (and for [P.sup.R] = 0.9[P.sub.[infinity]]), due to natural fluctuations in free energy, the half-width D of the liquid bridge would be predicted to fluctuate between approximately 40 [R.sub.eq] and 70 [R.sub.eq] (spanning approximately 1.3 ?m, in the case of n-dodecane at [P.sup.R] = 0.9[P.sub.[infinity]]). From a physical point of view, this means that the 1.3 [micro]m-thick extremity of the liquid bridge exhibits an apparent density lower than that of the liquid and higher than that of the vapour. Such a "diffuse LV interface" was hypothesized by Maeda and Israelachvili (2002) and used as one of several possible explanations for some of the phenomena reported in Maeda et al. (2003). The findings of the present study provide a thermodynamic understanding of the results of these authors.

The numerically computed interface shapes, corresponding to both the stable and the unstable equilibria in Figure 4, are illustrated in Figure 5. It is apparent that the vertical section of the LV interface is very close to a circle, and therefore the LV interface shape can overall be approximated by a toroidal surface. It should be stressed that the numerical integration of the interface shape is not a trivial problem, due to the difference in scale between the radius a of the upper sphere and the radius R1 of the vertical section of the LV interface. For values of these parameters similar to those in the experiments reported in Maeda et al. (2003), a/[R.sub.1] is typically of order [10.sup.5]. In order to reduce the numerical errors caused by this difference in scales, in the present study the LV interface shape was computed in each case by starting from a known value of the angle [alpha] (for given values of a, H and f), and integrating the system (27) with the initial conditions (28), from top to bottom, i.e., in the direction of decreasing elevation z.

[FIGURE 6 OMITTED]

System Stability Dependence on the Radius of the Upper Solid Sphere

The manner in which the stability behaviour of the system depends on the radius of the upper solid sphere is now investigated. The variation of free energy with the bridge half-width, for different values of H, is illustrated in Figure 6--in the case of a sphere radius ten times smaller than before (i.e., a = 2.5 mm)--and Figure 7--in the case of an upper planar plate rather than a sphere (equivalent to a = [infinity]). It is apparent that, for the same distance between the two solid bodies, the free energy minimum becomes lower (and, correspondingly, the equilibrium width of the bridge becomes larger) as the radius of the upper sphere increases. In the limiting case when the upper sphere becomes a plane, there is no more energy minimum (no more stable equilibrium configuration), as the free energy decreases steadily as D increases. Therefore, in this situation, once the energy barrier has been overcome, the slit between the two planes will completely fill up with liquid.

[FIGURE 7 OMITTED]

Moreover, the larger the radius of the upper solid sphere, the larger the critical distance H at which the bridge will break. The numerical simulations in the present study indicate a breakage distance of approximately H [congruent] 0.98 [R.sub.eq], H [congruent] 0.99 [R.sub.eq] and H [congruent] [R.sub.eq], for a = 2.5 mm, a = 2.5 cm, and a = [infinity] (upper planar plate), respectively, (see Figures 2, 6 and 7). It is apparent that the curvature of the upper solid body strongly impacts the stability of the liquid bridge. While in the case of two planar solid plates the liquid will completely fill up the slit (i.e., there is no stable equilibrium configuration), when one of the solid surfaces is even slightly curved a stable equilibrium state arises, corresponding to a bridge half-width that is more than 500 times smaller than the radius of the upper sphere (in the simulations in the present study). In the latter case, the liquid will only fill a very small region of the slit. In consequence, in particular, the behaviour of the system with even a slightly curved solid-liquid (and solid-vapour) interface cannot be approximated by the behaviour of the corresponding system with two parallel solid plates.

[FIGURE 8 OMITTED]

System Stability Dependence on Vapour Pressure

For a fixed value of H, the free energy dependence on liquid bridge half-width for a = 2.5 cm was computed, for different values of f = [P.sup.R]/[P.sub.[infinity]] (see (2)). Note that in the numerical simulations carried out up to this point the vapour pressure was kept constant, and in consequence the equilibrium mean radius of curvature [R.sub.eq] of the LV interface was constant (by (25)). In this subsection, due to the fact that the vapour pressure is varied, the distance H between the two solid bodies is set to a value independent of [R.sub.eq]. Given that f is varied between 0.898 and 0.902 (see Figure 8), H is chosen to be equal to 0.98 of the magnitude (absolute value) of the equilibrium mean radius of curvature corresponding to f = 0.9 (which is [R.sub.eq] [congruent to] -43.84 nm). The results are shown in Figure 8. It is apparent that, as the vapour pressure increases, the free energy minimum becomes deeper, (and the stable equilibrium width of the liquid bridge becomes larger), while the free energy maximum becomes smaller (and the unstable equilibrium width of the liquid bridge becomes smaller). In other words, the liquid nucleates easier and easier as the vapour pressure increases, until at a certain pressure the system behaviour changes to a non-nucleating phase transition, in which the entire space fills up with liquid spontaneously, since there is no more energy barrier to be overcome.

From the above discussion it follows that, for a certain (large enough) vapour pressure, the energy barrier corresponding to distances H close to [R.sub.eq] (as in Figure 4) will become smaller than the size of natural random fluctuations in the free energy of the system around its stable equilibrium configuration. In that case, the following phenomenon would occur. When the density of the condensate is measured using multiple-beam interferometry (Israelachvili, 1973; Maeda et al., 2003), then, for a certain range of distances H, the free energy fluctuations would cause the system to sample both the vapour state (corresponding to [D.sub.sc] = 0) and the liquid bridge state (the stable equilibrium state, that is) during the time of measurement (see Figure 4). The multiple-beam interferometer, used to determine the density of the condensate based on measurements of its refractive index, will provide an estimate of the time average of densities around the axis of symmetry of the system (see Maeda et al., 2003). In consequence, the measured (or "apparent") density will be smaller than the density of the liquid and larger than that of the vapour. In that case, one would also expect to observe a variation of the measured density with H, for some range of H. Indeed, as the difference between the free energy local minimum and local maximum values decreases (as the free energy minimum becomes shallower, with the increase in H, as in Figure 4), the energy fluctuations will cause the vapour-only state to be sampled more often than the liquid bridge state (since, in order for the vapour-only state to be sampled, smaller and smaller fluctuations are necessary), leading to a decrease in measured density. This variation is quite steep, since the range of H values between the point when the local maximum and local minimum values of B-[B.sub.0] become close (i.e., with their difference smaller than the size of energy fluctuations) and the point when the liquid bridge state becomes energetically unfavourable, is quite narrow (see Figure 4). These findings provide a thermodynamic explanation of the phenomena observed in the experiments in Maeda et al. (2003).

CONCLUSIONS

In the present paper, a thermodynamic approach was used to analyze the stability of three-phase systems containing an axisymmetric liquid capillary bridge between two solid surfaces. The expression of the free energy of the particular considered system was derived, and the dependence of free energy on the half-width of the liquid bridge was computed, for various distances between the two solid surfaces, for various curvatures of the upper solid surface, and for various vapour pressure values. Numerical computations were carried out using the physical properties of n-dodecane and the geometric distances as in the experiments reported in Maeda et al. (2003). Moreover, since in these experiments very small contact angles were observed, in the present study the three-phase contact angle was assumed to be equal to zero in all cases (the analysis can be repeated, though with greater computational cost, in the case of other small contact angle values, but the qualitative behaviour of the system is not expected to change). It was illustrated that even a very small solid surface curvature causes the system to exhibit a stable equilibrium state, while if no curvature is present the space between the two solid bodies fills up with liquid completely once the initial energy barrier has been overcome. It was also shown that capillary condensation between a sphere and a flat plate, in the limit of zero separation distance, is a non-nucleating phase transition, with no energy barrier to be overcome. Such a phase transition may be expected to occur in a different manner than the nucleating phase transitions such as free droplet or bubble nucleation (McGaughey and Ward, 2003) or nucleation of a bubble in a cone (Ward et al., 1983).

A thermodynamic explanation for the existence of a diffuse interface at a specific distance between the two solid bodies, and for the variation of apparent liquid density with separation of the two bodies, was provided, showing how spontaneous fluctuations around the stable equilibrium state cause a wide range variation of the bridge width. The findings in the present paper are consistent with those previously reported in literature, arising from experimental studies. For the particular geometric dimensions studied, in which moderately curved spheres were brought within tens of nanometres of a solid surface, unlike in other nucleation studies, the unstable and stable equilibrium liquid phase sizes are of the same order of magnitude, manifesting effects of fluctuations that are observable over micrometres. As such, the experimental capillary condensation phenomena in these systems become a probe of our understanding of classical thermodynamic phase nucleation theory.

ACKNOWLEDGEMENTS

This work was supported by the Natural Sciences and Engineering Research Council of Canada. J. A. W. Elliott holds a Canada Research Chair in Interfacial Thermodynamics.

Manuscript received April 3, 2007; revised manuscript received June 21, 2007; accepted for publication June 21, 2007.

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Janet A. W. Elliott * and Ovidiu Voitcu

Department of Chemical and Materials Engineering, University of Alberta, 536 Chemical and Materials Engineering Building, Edmonton, AB, Canada T6G 2G6

* Author to whom correspondence may be addressed. E-mail address: janet.elliott@ualberta.ca

The study of capillary condensation (the formation of a liquid at a solid-vapour interface in tight cracks or pores at pressures below the saturation pressure) is of great practical importance in nanotechnology applications and in oil recovery, and provides useful insight into phenomena that occur in porous materials. In particular, the behaviour of microscopic liquid capillary bridges between two solid bodies has been of special interest, since it provides an elegant framework for the computation of adhesive forces associated with the capillary condensation process. The existence of a liquid bridge between two solid surfaces generates an attractive force between them (Maeda and Israelachvili, 2002; Stroud et al., 2001). The behaviour of liquid capillary bridges has mostly been investigated from a mechanical perspective, with an emphasis on computing the relevant adhesion forces (Aminu et al., 2004; Christenson, 1993; Christenson and Claesson, 2001; Ducker et al., 1994; Willett et al., 2000). Various system geometries have been considered, such as the liquid bridge occurring between, for instance, two solid spheres (Seville et al., 2000; Willett et al., 2000), two parallel planes (Christenson, 1997; Ducker et al., 1994; Padday et al., 1997; Stroud et al., 2001), two crossed cylinders (Aminu et al., 2004; Christenson, 1997; Luengo et al., 1998; Maeda et al., 2003), or a sharp tip and a plane (Jang et al., 2002). Many of the phenomena related to liquid bridges between crossed cylinders are described by the model configuration of a liquid bridge between a solid sphere and a flat plate (Christenson, 1993; Luengo et al., 1998; Maeda and Israelachvili, 2002; Maeda et al., 2003), which will also be the geometry considered in the present study.

In the present paper, a thermodynamic stability analysis of liquid capillary bridges will be carried out. The approach is similar to that used by other authors (McGaughey and Ward, 2003; Ward et al., 1983; Ward and Levart, 1984) for other system geometries. These studies were mainly concerned with the nucleation of droplets or bubbles. The approach consists of computing the free energy of the considered three-phase (solid-liquid-vapour) system as a function of the radius (or, equivalently, the volume) of a nucleus of the new phase. The stability analysis is carried out in the framework of nucleation theory (Debenedetti, 2006; Hunter, 1993). Usually, the computed free energy of the system has a local maximum (corresponding to an unstable equilibrium state) associated with a small radius of the particle of the new phase. This is generally understood as an energy barrier that needs to be overcome, representing the work necessary for the creation of a two-phase interface. In general (but not always), the system also has a stable equilibrium state, corresponding to a larger value of the particle radius. One or more additional equilibrium states (stable and/or unstable) may also occur (Ward et al., 1983). In constant-volume studies (McGaughey and Ward, 2003; Ward et al., 1983), the Helmholtz free energy is the thermodynamic potential of the system; however, in systems for which the pressure of one phase is controlled, such as in the case of bubble nucleation in a constant-pressure liquid, the appropriate free energy that acts as the thermodynamic potential for the multi-phase system must be determined (Ward and Levart, 1984).

In the present paper, the appropriate free energy function will be derived for a solid-liquid-vapour system in which the liquid phase consists of an axisymmetric capillary bridge, and the system temperature and vapour pressure are held constant. A stability analysis will be carried out based on that function. Some researchers performed a thermodynamic stability analysis for a similar geometry (in which the upper solid object was highly curved); however they did so from a statistical physical perspective, using grand canonical ensemble Monte-Carlo simulations (Jang et al., 2002). The present study is carried out in Gibbs's macroscopic thermodynamic paradigm, which was previously shown to be valid in this context. More specifically, Fisher and Israelachvili (1979, 1981) demonstrated that macroscopic thermodynamics is indeed applicable to two-phase interfaces with radii of curvature down to the order of nanometres. By comparison, the radii of curvature dealt with in the present study are of the order of tens of nanometres or larger.

In the current paper, we will investigate the thermodynamic stability of three-phase systems containing an axisymmetric liquid capillary bridge between two solid bodies, and the dependence of the stability behaviour on the curvatures of the solid bodies, on the distance between them, and on the vapour pressure. This study was inspired by the findings of several researchers (Christenson, 1997; Kohonen et al., 1999; Maeda and Israelachvili, 2002; Maeda et al., 2003) who investigated the formation of liquid bridges between two solid surfaces at close distance (less than 0.1 [micro]m). It was noticed that, as the distance between the two solid surfaces is increased, the density of the liquid bridge remains constant up to a critical distance (approximately equal to the mean radius of curvature of the interface at equilibrium), when it drops abruptly to a value between the density of the bulk liquid and that of the vapour (Maeda et al., 2003). This phenomenon was associated with the occurrence, at the critical distance between the two solid bodies, of a gradient in the density of the condensate, in the directions both normal and parallel to the solid surfaces (Maeda et al., 2003), an effect that had previously been experimentally noticed (Maeda and Israelachvili, 2002). In Jang et al. (2002) and Stroud et al. (2001), the effects reported in Maeda and Israelachvili (2002) were reproduced based on statistical mechanical numerical simulations, in the case of two planar solid surfaces (Stroud et al., 2001), and in the case of a high-curvature upper solid object on a planar surface (Jang et al., 2002). In the present paper, a macroscopic thermodynamics argument, based on the existence of fluctuations around equilibrium states of the system, will be used to support the finding that a diffuse interface does indeed occur at some critical distance between the two solid surfaces. Note that, since in Maeda and Israelachvili (2002) and Maeda et al. (2003) it is reported that the contact angles observed in the experiments are small (e.g. less than 6[degrees] for cyclohexane on a mica surface), for simplicity the present theoretical analysis will be carried out under the assumption that the three-phase contact angle is constant and equal to zero for the considered temperature (24[degrees]C, as in Maeda et al., 2003) and vapour pressures. Though computationally more challenging, the analysis can readily be extended to other small contact angle values, but the qualitative behaviour of the system is expected to be the same.

FREE ENERGY OF THE CONSIDERED THREE-PHASE SYSTEM

In the present section, the approach presented in Ward and Levart (1984) and Elliott (2001) is used to derive the expression of the free energy function of the three-phase system represented in Figure 1. The solid (S) phase consists of an upper sphere (of radius a) and a lower flat plate, both made of the same material. The liquid (L) phase consists of an axisymmetric liquid bridge between the two solid components. The vapour (V) phase is assumed to be connected to a temperature (T) and pressure (P) reservoir. Therefore:

[T.sup.j] = [T.sup.R], for j = L, V, SL, SV, LV (1)

and

[P.sup.V] = [P.sup.R] = [fP.sub.[infinity]] (2)

where f = [P.sup.R]/[P.sub.[infinity]] is the ratio between the pressure in the reservoir and the saturation pressure corresponding to the (single) chemical component present in the fluid. The superscripts indicate that the corresponding physical quantity is associated with the reservoir (R), or a bulk phase (L, V) or a surface phase (SL, SV, LV denoting the corresponding two-phase interfaces). The SL and SV interfaces are modelled by theoretical surfaces placed such that there is no adsorption of the solid component (using "Gibbs' dividing surface approximation"--see Gibbs, 1876; Ward and Sasges, 1998). The solid then becomes part of the reservoir and plays no further role in the thermodynamics other than to define the system geometry. It is assumed that the system contains a single chemical component, and [P.sub.[infinity]] denotes the saturation pressure for the considered chemical component. The LV interface is modelled by a theoretical surface placed such that the corresponding surface tension [[gamma].sup.LV] does not depend on the interface curvature (using "Gibbs' surface-of-tension approximation"--see Gibbs, 1876; Ward and Sasges, 1998). All physical quantities associated with a certain (bulk or surface) phase are assumed to be constant throughout that phase. In particular, the effect of gravity will be neglected throughout this study, since the height of the considered liquid bridges does not exceed 0.1 [micro]m.

[FIGURE 1 OMITTED]

In order to analyze the stability of the considered system, the expression for the free energy of the system must be derived. To that end, first consider the difference form of the fundamental relation (Hunter, 1993) for the reservoir:

[DELTA][U.sup.R] = [T.sup.R][[DELTA]S.sup.R] - [P.sup.R] [DELTA][V.sup.R] + [[mu].sup.R] [DELTA][N.sup.R] (3)

where U, S, V, [mu] and N denote the internal energy, entropy, volume, chemical potential, and number of moles of substance, respectively. The following constraints apply:

[DELTA][N.sup.R] = 0, [DELTA][N.sup.tot] = 0 (4)

[DELTA][U.sup.R] + [summation over (j)][SIGMA][U.sup.j] = 0 (5)

[DELTA][V.sup.R] + [DELTA][V.sup.L] + [DELTA][V.sup.V] = 0 (6)

[DELTA][A.sup.SV] + [DELTA][A.sup.SL] = 0 (7)

where [N.sup.tot] = [summation over (j)][N.sup.j] is the total number of moles in the considered system, and [A.sup.SL], [A.sup.SV], [A.sup.LV] denote the areas of the SL, SV, LV interfaces, respectively. When spontaneous changes occur around an equilibrium state, the entropy of the system and reservoir together increases:

[DELTA][S.sup.R] + [summation over (j)] [DELTA][S.sup.j] [greater than or equal to] 0 (8)

Substituting (4) to (6) and (8) in (3) and making use of (1) and (2) yields:

[P.sup.V] ([DELTA][V.sup.L] + [DELTA][V.sup.V]) + [summation over (j)] ([DELTA][U.sup.j]-[T.sup.j] [DELTA][S.sup.j] [less than or equal to] (9)

From (9) and the difference form of the fundamental relation for the bulk phases:

[DELTA][U.sup.j] = [T.sup.j][DELTA][S.sup.j]-[P.sup.j][DELTA][V.sup.j]+ [[mu].sup.j][DELTA][N.sup.j] = L, V (10)

and for the surface phases:

[DELTA][U.sup.j] = [T.sup.j] [DELTA][S.sup.J] + [[gamma].sup.j] [DELTA][A.sup.j] + [[mu].sup.j][DELTA][N.sup.j], j = SL, SV, LV (11)

(Hunter, 1993), it follows that:

([P.sup.V]-[P.sup.L])[DELTA][V.sup.L]+[[gamma].sup.SL][DELTA][A.sup.SV]+ [[gamma].sup.LV][DELTA][A.sup.LV]+[summation over (j)][[mu].sup.j] [DELTA][N.sup.j][less than or equal to]0 (12)

Using (7), we obtain [DELTA]B [less than or equal to] 0, where

B = ([P.sup.V]-[P.sup.L])[V.sup.L]-([[gamma].sup.SV]-[[gamma].sup.SL]) [A.sup.SL]+[[gamma].sup.LV][A.sup.LV]+[summation over (j)][[mu].sup.j] [N.sup.j] (13)

is the free energy of the considered system. Note that from (9) it follows that

B = [P.sup.V][V.sup.L]+[G.sup.V]+[F.sup.L]+[F.sup.SL]+[F.sup.SV]+ [F.sup.LV] (14)

where [F.sup.j] and [G.sup.j] are the Helmholtz and Gibbs free energies, respectively, corresponding to phase j ( j = L, V, SL, SV, LV).

In an equilibrium state of the considered system, in addition to Equations (1)-(2), the following relations hold (Ward and Sasges, 1998):

[P.sup.L]-[P.sup.V] = 2[[gamma].sup.LV][K.sub.eq] (15)

[[gamma].sup.SV]-[[gamma].sup.SL] = [[gamma].sup.LV] cos[theta] (16)

[[mu].sup.j] = [[mu].sub.eq], for j = L, V, SL, SV, LV (17)

namely, the Laplace and Young equations, and the condition of equality of the chemical potentials of all phases. In the above equations, [theta] denotes the three-phase contact angle, and [K.sub.eq] denotes the mean curvature of the LV interface at equilibrium (which may take on either positive or negative values). The mean radius of curvature of the LV interface at equilibrium:

[R.sub.eq] = 1/[K.sub.eq] = 1/2 (1/[R.sub.1,eq] + 1/[R.sub.2,eq]) (18)

will be understood in a general sense, taking on either positive or negative values.

We choose as a reference state (marked by the subscript "0") the state of the system in which no liquid is present, and we assume that it is an equilibrium state. Then:

[B.sub.0] = [[mu].sup.V.sub.0] [N.sup.V.sub.0] + [[mu].sup.SV.sub.0] [N.sup.SV.sub.0] = [[mu].sub.eq]([N.sup.V.sub.0]+[N.sup.SV.sub.0])= [[mu].sub.eq][N.sup.tot] (19)

Subtracting (19) from (13) yields:

B - [B.sub.0] = ([P.sup.V] - [P.sup.L])[V.sup.L]-([[gamma].sup.SV]- [[gamma].sup.SL])[A.sup.SL]+[[gamma].sup.LV] [A.sup.LV]+[[summation].sub.j] ([[mu].sup.j]-[[mu].sub.eq])[N.sup.j] (20)

If the current state of the system is also an equilibrium state, then (15) to (17) apply, and thus:

B - [B.sub.0] = [[gamma].sup.LV] ([A.sup.LV] - [A.sup.SL] cos [theta] - [2V.sup.L] [K.sub.eq]) (21)

The value of [K.sub.eq] can be determined from the imposed vapour phase pressure, based on expressions for the chemical potentials in the liquid and vapour phases. For an incompressible fluid:

[[mu].sup.L] (T, [P.sup.L]) = [[mu].sup.L](T, [P.sub.[infinity]]) + [[upsilon].sup.L.sub.[infinity]([P.sup.L]-[P.sub.[infinity]]] (22)

and for an ideal gas:

[[mu].sup.V](T, [P.sup.V]) = [[mu].sup.V](T, [P.sub.[infinity]]) + [bar.R]T ln ([P.sup.V]/[P.sub.[infinity]]) (23)

where [v.sup.L.sub.[infinity] is the liquid specific molar volume at saturation for the considered chemical component, [bar.R] is the universal gas constant, and [[mu].sup.j] (T, P) (j = L, V ) is the chemical potential in the phase j at temperature T and pressure P. By (1), (2), (17), and (22) to (23), it follows that

[P.sup.L] = [P.sub.[infinity]] + [bar.R][T.sup.R]/ [v.sup.L.sub.[infinity]] ln([P.sup.R]/[P.sup.[infinity]]) (24)

and using (15) we obtain:

[K.sub.eq] = [P.sub.[infinity]]/2[[gamma].sup.LV] {1-[P.sup.R]/P [infinity]+[bar.R][T.sup.R]/[P.sub.[infinity]][v.sup.L.sub.[infinity]]ln ([P.sup.R]/P[infinity])} (25)

NUMERICAL PROCEDURE

The present paper is concerned with the thermodynamic stability of axisymmetric liquid bridges between an upper solid sphere (of radius a) and a lower planar solid plate (see Figure 1). All numerical simulations in the present study are carried out for n-dodecane at 24[degrees]C. For simplicity, the three-phase contact angle [theta] is assumed to be zero. Various sphere radii will be considered, as well as the limiting case of an upper planar solid plate (equivalent to a solid sphere of infinite radius). The dependence of the free energy of the three-phase system on the width of the liquid bridge will be analyzed. Throughout the present study, for given values of f, a, and H, the LV interface is numerically computed, over a range of values of [alpha] (the half opening angle of the spherical cone determined by the three-phase contact line on the upper solid sphere), by integrating the Laplace Equation (15) in each case. More specifically, once the angle [alpha] (see Figure 1) is known, the coordinates of the upper three-phase contact line

c = a sin [alpha], b = H + a (1-cos [alpha]) (26)

can be computed, and the LV interface shape can be numerically constructed by integrating the three-dimensional system of ordinary differential equations:

[d.sub.[xi]]/d[tau] = cos[phi],d[zeta]/d[tau]=sin[phi],d[phi]/ d[tau]=[R.sub.t](-2[K.sub.m] + sin[phi]/c-[R.sub.t][xi] (27)

(where [tau] = s/[R.sub.t] is the scaled arc-length) with initial conditions

[xi](0) = [zeta](0)=0,[phi](0)=[alpha] (28)

using a fourth-order Runge-Kutta algorithm (with step size [DELTA][tau] = [10.sup.-4]), where

[xi] = c-x/[R.sub.t], [zeta] = b-z/[R.sub.t] (29)

and where, for scaling purposes, the value

[R.sub.t] = b/1 + cos [alpha] (30)

is used, which is the radius of the vertical section of the (unique) toroidal surface corresponding to the current value of [alpha]. In (27), [phi] is the angle between the upwards vertical direction and the direction of the first principal radius of curvature, pointing towards the liquid phase (see Figure 1). The mean curvature [K.sub.m] of the LV interface (which is constant throughout the interface) is a search parameter for the computation of generic bridge shapes (including non-equilibrium shapes). More specifically, for a given value of [K.sub.m], the arc-length integration is stopped at the smallest point [tau] = [[tau].sub.max] for which [phi]([[tau].sub.max])[greater than or equal to][pi] (in which case we also have [phi]([[tau].sub.max])[congruent to][pi]), and then the elevation of the lowest point on the LV interface z([[tau].sub.max]) = b - [R.sub.t][zeta]([tau]max) is computed. For physical consistency, we need to have z([[tau].sub.max])[congruent to] 0, or equivalently [zeta]([[tau].sub.max])[congruent to]b/[R.sub.t]. Therefore, we search for the (unique) value of [K.sub.m] that minimizes [absolute value of [zeta]([[tau].sub.max])-(b/[R.sub.t])]. A standard golden section search procedure is used for that purpose (Press et al., 1992).

The numerical computation of the LV interface using the described algorithm is accurate for D ??0.5 Req. It is worth noting that, alternatively, for D [greater than or equal to] 0.5 [R.sub.eq], the LV interface can accurately be approximated by a toroidal surface (that is, considering the vertical section of the interface to be an arc of a circle). More specifically, though the mean curvature of a toroidal surface varies with elevation, in practice, for the simulations carried out in the present study, the standard deviation of the mean curvature (taken over the range of possible elevations 0 [less than or equal to] z [less than or equal to] b) is generally at most 2.5% of the average mean curvature (taken over the range of possible elevations 0 [less than or equal to] z [less than or equal to] b), for D [greater than or equal to] 6.5 [R.sub.eq]. Thus, the physical LV interface can accurately be approximated by a toroidal surface, for which all quantities of interest in (21) can be computed analytically, thus greatly reducing the computational cost of the simulations.

RESULTS OF NUMERICAL SIMULATIONS

System Stability Dependence on the Distance between the Two Solid Surfaces

For a = 2.5 cm and [P.sup.R] = 0.9[P.sub.[infinity]] (values close to the experimental ones in Maeda et al., 2003), the values of B - [B.sub.0] (given by (21)) are computed over a range of values of the scaled liquid bridge half-width [D.sub.sc] = D/[absolute value of [R.sub.eq]], and for different values of the scaled distance between the upper and the lower plate [H.sub.sc] = H/[absolute value of [[R.sub.eq]], and the results are plotted in Figure 2. It is apparent that the system has an unstable as well as a stable equilibrium configuration (corresponding to a maximum in the free energy at a smaller bridge width and a minimum in the free energy at a larger bridge width, respectively). The free energy variation is similar to that obtained for instance by McGaughey and Ward (2003) (in the case of droplet nucleation in a constant volume, with the Helmholtz free energy as the thermodynamic potential of the system). Thus, in general, for 0 < H < [R.sub.eq], the liquid bridge formation is a nucleation phenomenon, in which the energy barrier to be overcome in order for the new phase to be formed is of order of fJ (femtoJoules). The energy barrier decreases as the distance H between the two solid bodies decreases, until at H = 0 there is no more energy barrier to be overcome, and the liquid phase therefore forms spontaneously, settling into the stable equilibrium configuration illustrated in Figure 3 (in the numerical simulations in this case, a toroidal LV interface approximation was used, which for this particular value of H is very accurate even for very small values of [D.sub.sc]). Thus, for H = 0, the phase transition is not a nucleation process.

[FIGURE 2 OMITTED]

Moreover, in general for H > [R.sub.eq] the formation of the liquid phase is not favourable, as the free energy increases steadily with the bridge width (Figure 2). If the liquid phase already exists (in stable thermodynamic equilibrium) between the two solid bodies, and the distance H is steadily increased (as in the experiments reported in Maeda et al., 2003), the width of the liquid bridge will steadily decrease (as the stable equilibrium configuration corresponds to smaller values of D : see Figures 2 and 4), until, at some critical value of H, the bridge will break (as the existence of the liquid phase becomes energetically unfavourable). This process is illustrated in more detail in Figure 4. It is interesting to note that the liquid bridge breaks for H [congruent to] [R.sub.eq], that is, when the distance between the two solid bodies becomes approximately equal to the equilibrium mean radius of curvature (the "Kelvin radius") of the LV interface. This critical distance is the same as that reported (with a different notation) in Maeda and Israelachvili (2002) and Maeda et al. (2003). It should be stressed that in those studies the Kelvin radius is defined using a different notation from that in the present paper. More specifically, the definition in Fisher and Israelachvili (1981) was used by these authors, namely 1/[R.sub.eq] is defined as 1/[R.sub.1,eq] + 1/[R.sub.2,eq] rather than (1/[R.sub.1,eq] + 1/[R.sub.2,eq])/2 as in the present study and as for instance in Derjaguin and Churaev (1976) (so that 1/[R.sub.eq] is the mean curvature of the surface).

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

As H becomes larger, the free energy minimum becomes shallower, and the stable equilibrium width of the liquid bridge becomes smaller (in general, in this study, deeper free energy minima, and higher free energy maxima, are associated with larger equilibrium bridge widths). From Figure 4 it is apparent that, when the distance H becomes close to [R.sub.eq], the free energies corresponding to the stable and the unstable equilibrium states of the system become approximately equal. In that case (and for [P.sup.R] = 0.9[P.sub.[infinity]]), due to natural fluctuations in free energy, the half-width D of the liquid bridge would be predicted to fluctuate between approximately 40 [R.sub.eq] and 70 [R.sub.eq] (spanning approximately 1.3 ?m, in the case of n-dodecane at [P.sup.R] = 0.9[P.sub.[infinity]]). From a physical point of view, this means that the 1.3 [micro]m-thick extremity of the liquid bridge exhibits an apparent density lower than that of the liquid and higher than that of the vapour. Such a "diffuse LV interface" was hypothesized by Maeda and Israelachvili (2002) and used as one of several possible explanations for some of the phenomena reported in Maeda et al. (2003). The findings of the present study provide a thermodynamic understanding of the results of these authors.

The numerically computed interface shapes, corresponding to both the stable and the unstable equilibria in Figure 4, are illustrated in Figure 5. It is apparent that the vertical section of the LV interface is very close to a circle, and therefore the LV interface shape can overall be approximated by a toroidal surface. It should be stressed that the numerical integration of the interface shape is not a trivial problem, due to the difference in scale between the radius a of the upper sphere and the radius R1 of the vertical section of the LV interface. For values of these parameters similar to those in the experiments reported in Maeda et al. (2003), a/[R.sub.1] is typically of order [10.sup.5]. In order to reduce the numerical errors caused by this difference in scales, in the present study the LV interface shape was computed in each case by starting from a known value of the angle [alpha] (for given values of a, H and f), and integrating the system (27) with the initial conditions (28), from top to bottom, i.e., in the direction of decreasing elevation z.

[FIGURE 6 OMITTED]

System Stability Dependence on the Radius of the Upper Solid Sphere

The manner in which the stability behaviour of the system depends on the radius of the upper solid sphere is now investigated. The variation of free energy with the bridge half-width, for different values of H, is illustrated in Figure 6--in the case of a sphere radius ten times smaller than before (i.e., a = 2.5 mm)--and Figure 7--in the case of an upper planar plate rather than a sphere (equivalent to a = [infinity]). It is apparent that, for the same distance between the two solid bodies, the free energy minimum becomes lower (and, correspondingly, the equilibrium width of the bridge becomes larger) as the radius of the upper sphere increases. In the limiting case when the upper sphere becomes a plane, there is no more energy minimum (no more stable equilibrium configuration), as the free energy decreases steadily as D increases. Therefore, in this situation, once the energy barrier has been overcome, the slit between the two planes will completely fill up with liquid.

[FIGURE 7 OMITTED]

Moreover, the larger the radius of the upper solid sphere, the larger the critical distance H at which the bridge will break. The numerical simulations in the present study indicate a breakage distance of approximately H [congruent] 0.98 [R.sub.eq], H [congruent] 0.99 [R.sub.eq] and H [congruent] [R.sub.eq], for a = 2.5 mm, a = 2.5 cm, and a = [infinity] (upper planar plate), respectively, (see Figures 2, 6 and 7). It is apparent that the curvature of the upper solid body strongly impacts the stability of the liquid bridge. While in the case of two planar solid plates the liquid will completely fill up the slit (i.e., there is no stable equilibrium configuration), when one of the solid surfaces is even slightly curved a stable equilibrium state arises, corresponding to a bridge half-width that is more than 500 times smaller than the radius of the upper sphere (in the simulations in the present study). In the latter case, the liquid will only fill a very small region of the slit. In consequence, in particular, the behaviour of the system with even a slightly curved solid-liquid (and solid-vapour) interface cannot be approximated by the behaviour of the corresponding system with two parallel solid plates.

[FIGURE 8 OMITTED]

System Stability Dependence on Vapour Pressure

For a fixed value of H, the free energy dependence on liquid bridge half-width for a = 2.5 cm was computed, for different values of f = [P.sup.R]/[P.sub.[infinity]] (see (2)). Note that in the numerical simulations carried out up to this point the vapour pressure was kept constant, and in consequence the equilibrium mean radius of curvature [R.sub.eq] of the LV interface was constant (by (25)). In this subsection, due to the fact that the vapour pressure is varied, the distance H between the two solid bodies is set to a value independent of [R.sub.eq]. Given that f is varied between 0.898 and 0.902 (see Figure 8), H is chosen to be equal to 0.98 of the magnitude (absolute value) of the equilibrium mean radius of curvature corresponding to f = 0.9 (which is [R.sub.eq] [congruent to] -43.84 nm). The results are shown in Figure 8. It is apparent that, as the vapour pressure increases, the free energy minimum becomes deeper, (and the stable equilibrium width of the liquid bridge becomes larger), while the free energy maximum becomes smaller (and the unstable equilibrium width of the liquid bridge becomes smaller). In other words, the liquid nucleates easier and easier as the vapour pressure increases, until at a certain pressure the system behaviour changes to a non-nucleating phase transition, in which the entire space fills up with liquid spontaneously, since there is no more energy barrier to be overcome.

From the above discussion it follows that, for a certain (large enough) vapour pressure, the energy barrier corresponding to distances H close to [R.sub.eq] (as in Figure 4) will become smaller than the size of natural random fluctuations in the free energy of the system around its stable equilibrium configuration. In that case, the following phenomenon would occur. When the density of the condensate is measured using multiple-beam interferometry (Israelachvili, 1973; Maeda et al., 2003), then, for a certain range of distances H, the free energy fluctuations would cause the system to sample both the vapour state (corresponding to [D.sub.sc] = 0) and the liquid bridge state (the stable equilibrium state, that is) during the time of measurement (see Figure 4). The multiple-beam interferometer, used to determine the density of the condensate based on measurements of its refractive index, will provide an estimate of the time average of densities around the axis of symmetry of the system (see Maeda et al., 2003). In consequence, the measured (or "apparent") density will be smaller than the density of the liquid and larger than that of the vapour. In that case, one would also expect to observe a variation of the measured density with H, for some range of H. Indeed, as the difference between the free energy local minimum and local maximum values decreases (as the free energy minimum becomes shallower, with the increase in H, as in Figure 4), the energy fluctuations will cause the vapour-only state to be sampled more often than the liquid bridge state (since, in order for the vapour-only state to be sampled, smaller and smaller fluctuations are necessary), leading to a decrease in measured density. This variation is quite steep, since the range of H values between the point when the local maximum and local minimum values of B-[B.sub.0] become close (i.e., with their difference smaller than the size of energy fluctuations) and the point when the liquid bridge state becomes energetically unfavourable, is quite narrow (see Figure 4). These findings provide a thermodynamic explanation of the phenomena observed in the experiments in Maeda et al. (2003).

CONCLUSIONS

In the present paper, a thermodynamic approach was used to analyze the stability of three-phase systems containing an axisymmetric liquid capillary bridge between two solid surfaces. The expression of the free energy of the particular considered system was derived, and the dependence of free energy on the half-width of the liquid bridge was computed, for various distances between the two solid surfaces, for various curvatures of the upper solid surface, and for various vapour pressure values. Numerical computations were carried out using the physical properties of n-dodecane and the geometric distances as in the experiments reported in Maeda et al. (2003). Moreover, since in these experiments very small contact angles were observed, in the present study the three-phase contact angle was assumed to be equal to zero in all cases (the analysis can be repeated, though with greater computational cost, in the case of other small contact angle values, but the qualitative behaviour of the system is not expected to change). It was illustrated that even a very small solid surface curvature causes the system to exhibit a stable equilibrium state, while if no curvature is present the space between the two solid bodies fills up with liquid completely once the initial energy barrier has been overcome. It was also shown that capillary condensation between a sphere and a flat plate, in the limit of zero separation distance, is a non-nucleating phase transition, with no energy barrier to be overcome. Such a phase transition may be expected to occur in a different manner than the nucleating phase transitions such as free droplet or bubble nucleation (McGaughey and Ward, 2003) or nucleation of a bubble in a cone (Ward et al., 1983).

A thermodynamic explanation for the existence of a diffuse interface at a specific distance between the two solid bodies, and for the variation of apparent liquid density with separation of the two bodies, was provided, showing how spontaneous fluctuations around the stable equilibrium state cause a wide range variation of the bridge width. The findings in the present paper are consistent with those previously reported in literature, arising from experimental studies. For the particular geometric dimensions studied, in which moderately curved spheres were brought within tens of nanometres of a solid surface, unlike in other nucleation studies, the unstable and stable equilibrium liquid phase sizes are of the same order of magnitude, manifesting effects of fluctuations that are observable over micrometres. As such, the experimental capillary condensation phenomena in these systems become a probe of our understanding of classical thermodynamic phase nucleation theory.

ACKNOWLEDGEMENTS

This work was supported by the Natural Sciences and Engineering Research Council of Canada. J. A. W. Elliott holds a Canada Research Chair in Interfacial Thermodynamics.

NOMENCLATURE A surface area ([m.sup.2]) a radius of the upper solid sphere (m) B free energy of the considered system (J) b elevation of the upper three-phase contact line (m) c radius of the upper three-phase contact line (m) D half-width of the liquid bridge (m) f relative vapour pressure [P.sup.V]/[P.sub.[infinity]] F Helmholtz free energy (J) G Gibbs free energy (J) H distance between the two solid bodies (m) [K.sub.eq] mean curvature of the LV interface at equilibrium (1/m) [K.sub.m] generic mean curvature of the LV interface (1/m) N number of kilomoles of substance (kmol) P pressure (Pa) [bar.R] universal gas constant (J/(kmol x K)) [R.sub.eq] mean radius of curvature of the LV interface at equilibrium (m) [R.sub.t] the radius of the vertical section of the (unique) toroidal surface corresponding to the current value of [alpha](m) [R.sub.1], [R.sub.2] the principal radii of curvature of the LV interface (1/m) S entropy (J/K) s arc-length of the vertical section of the LV interface (m) T temperature (K) U internal energy (J) [v.L.sub.[infinity]] liquid specific volume at saturation ([m.sup.3] /kmol) V volume ([m.sup.3]) [chi] abscissa of the vertical section of the LV interface (m) z ordinate of the vertical section of the LV interface (m) Greek Symbols [alpha] half opening angle of the spherical cone determined by the three-phase contact line on the upper solid sphere (rad) [gamma] surface tension (N/m) [mu] chemical potential (J/kmol) [theta] three-phase contact angle (rad) [tau] scaled arc-length of the vertical section of the LV interface [phi] angle between the upwards vertical direction and the direction of the first principal radius of curvature, pointing towards the liquid phase (rad) [xi] scaled abscissa of the vertical section of the LV interface [zeta] scaled ordinate of the vertical section of the LV interface Subscripts 0 the state in which no liquid is present eq any equilibrium state of the system sc refers to a quantity divided (scaled) by [absolute value of [R.sub.eq]] [infinity] saturation state Superscripts j any bulk or surface phase L the liquid phase LV the liquid-vapour interface R the reservoir S the solid phase SL the solid-liquid interface SV the solid-vapour interface tot total number of kilomoles in the system V the vapour phase

Manuscript received April 3, 2007; revised manuscript received June 21, 2007; accepted for publication June 21, 2007.

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Janet A. W. Elliott * and Ovidiu Voitcu

Department of Chemical and Materials Engineering, University of Alberta, 536 Chemical and Materials Engineering Building, Edmonton, AB, Canada T6G 2G6

* Author to whom correspondence may be addressed. E-mail address: janet.elliott@ualberta.ca

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Author: | Elliott, Janet A.W.; Voitcu, Ovidiu |
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Publication: | Canadian Journal of Chemical Engineering |

Date: | Oct 1, 2007 |

Words: | 6780 |

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