J. Phys. Chem. B 2009, 113, 13849–13859
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On the Line Tension of Curved Boundary Layers. I. Boundary Thermodynamics† Daniel W. Siderius‡ and David S. Corti* School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907-2100 ReceiVed: February 17, 2009; ReVised Manuscript ReceiVed: March 16, 2009
We present a formally exact thermodynamic treatment of curved boundary layers. Specifically, we extend the boundary layer analysis of Mandell and Reiss [J. Stat. Phys. 1975, 13, 107] for a spherical cavity located within a uniform bulk fluid to the case of a cavity intersecting a hard, structureless wall. We derive various expressions for the line tension of an intersecting cavity, all of which can be evaluated for a hard-sphere fluid using existing versions of scaled particle theory. Since the analysis is similar to the standard approach for describing curved interfacial layers, several boundary analogues to conventional interfacial relations appear. In some instances, we obtain results that apparently have not yet been derived either for boundary layers or for their parallel relations in interfacial thermodynamics. Several results offer interesting insights into the behavior of the line tension of a cavity when the cavity approaches macroscopic sizes or in the specific limit where the cavity no longer intersects the wall. 1. Introduction Scaled particle theory (SPT) provides, through its reliance upon straightforward physical and geometric ideas, several important insights into the behavior of hard-particle fluids.1 One of its main outputs is the reversible work of cavity formation, W (where a cavity is defined to be a spherical region devoid of molecular centers), which can be computed using the following integral
W(λ) ) 4πFkT
∫0λ G(r)r2dr
(1)
where λ is the cavity radius, k is Boltzmann’s constant, T is the absolute temperature, and FG(λ) is defined as the local density of particle centers in contact with the cavity surface, with F equaling the bulk fluid density. Given that the kinetic pressure or stress normal to the surface of the cavity is P1 ) FG(λ)kT, the above relation is obtained through the normal definition of thermodynamic work, where W(0) ) 0 and 4πλ2dλ is the differential change in the volume of the cavity. G(λ) proves to be the central function of SPT; W as well as many other hard-particle properties can be determined directly from G(λ). Using surface thermodynamic arguments for the reversible work of formation of a macroscopic cavity, SPT represents W for λ f ∞ as1
W(λ) )
4π 3 2δ + ··· λ P + 4πλ2γ∞ 1 3 λ
(
)
(2)
The first term is a pressure-volume work contribution, with P denoting the bulk pressure of the fluid. In the second term, γ∞ represents the surface tension (or more correctly, the boundary tension) of a cavity of zero curvature, which is identical to a planar surface, and δ accounts for the dependence of surface tension on curvature (and plays a similar role as the “Tolman length”). Given an approximate relation for G(λ) in the large †
Part of the “H. Ted Davis Special Section”. * To whom correspondence should be addressed. E-mail: dscorti@ ecn.purdue.edu. ‡ Current address: Department of Chemistry and Center for Materials Innovation, Washington University in St. Louis, St. Louis, MO 631304899.
cavity limit, expressions for P, γ∞, and δ can be extracted, all of which are functions of the bulk density of the fluid. The description of W in terms of various surface thermodynamic quantities allows for SPT to be applied to several fluidphase systems, even when the cavity is not in the macroscopic limit. For example, Corti and Reiss2 utilized the work of cavity formation to estimate the depletion force between a hard-sphere colloidal particle, or its equivalent cavity, and a hard, structureless wall. The growth of a cavity that was allowed to overlap the wall was again decomposed into pressure-volume and surface area contributions, with bulk SPT providing the needed pressure and surface tension relations. While the resulting predictions were qualitatively good, problems arose for small colloid diameters and at high bulk densities since the method relied upon SPT expressions that were obtained for a uniform fluid and not for the inhomogeneous fluid that develops near the wall. Errors were also attributed to the neglect of the line tension that develops along the three-phase interface where the cavity, wall, and surrounding fluid meet. This was not, however, an oversight since an expression for the line tension of a cavity intersecting a wall was not known. Recently, Siderius and Corti3–5 developed an extension of SPT to inhomogeneous hard-particle fluids, labeled I-SPT, where the nonuniform fluid density that develops near a wall is explicitly taken into account. In particular, I-SPT begins by noting that the particle centers in contact with the cavity surface are no longer distributed in a spherically symmetric manner and also depend upon the cavity’s distance from the wall. For convenience, the local density is averaged over the cavity surface to yield the following analogue to the bulk SPT relation
W(λ, z) ) W(|z|, z) + 2πFkT
∫|z|λ Gj (r, z)r(r + z)dr
(3)
where z is the fixed location of the cavity center from a given reference plane (and can be either positive or negative), F is the bulk density far away from the wall, and the average stress j (λ,z)kT. The above j 1 ) FG normal to the cavity surface is P equation is only valid when the cavity intersects the wall (coinciding with the chosen z ) 0 reference plane), where W(|z|,z) denotes the work required to insert a cavity that just
10.1021/jp901451t CCC: $40.75 2009 American Chemical Society Published on Web 04/29/2009
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touches the wall, that is, the center of a cavity with radius |z| is placed at a distance z. For z e 0, W(|z|,z) ) 0, while W(|z|,z) > 0 for z > 0. For that portion of the cavity which resides beyond the wall, the differential change in the volume that is exposed to the surrounding fluid is given by 2πλ(λ + z)dλ. When I-SPT is applied to the prediction of depletion forces, knowledge of the line tension is no longer required since its effects are already included in the expression for W(λ,z). On the other hand, the line tension of a cavity that intersects a hard wall can now, in principle, be extracted from I-SPT, so that the behavior of this important quantity can be studied in detail. However, as mentioned earlier, expressions explicitly relating j 1 as well as other surface the line tension of cavities to W and P thermodynamic properties have not yet been derived. To date, Mandell and Reiss6 developed the surface thermodynamic formalism for generating those boundary relations that only correspond to fully spherical cavities contained within a bulk fluid. We therefore extend their boundary thermodynamic analysis to the case in which a cavity, or curved surface, intersects a wall. What follows are the proper thermodynamic relations needed to determine the line tension from known I-SPT, as well as SPT, expressions. (While we explicitly have in mind the analysis of cavities placed within a hard-sphere fluid, nearly all of the relations obtained below are also valid for fluids comprised of particles that interact via soft-core potentials.) In a forthcoming paper, the results obtained here will be used to generate accurate estimates of the line tension of cavities intersecting a hard wall. Additional physical and geometric insights into inhomogeneous fluids should follow from our boundary layer analysis. This improved understanding of inhomogeneous systems is not simply limited to boundary layers but also to the conventional study of curved interfacial layers. The boundary thermodynamic analysis of Mandell and Reiss yielded many analogues with normal interfacial thermodynamics (e.g., droplets or bubbles contained within a host phase). Similarly, our analysis of the line tension of cavities yields analogous relations to those obtained via standard interfacial thermodynamics. We even generate boundary layer results whose corresponding relations in interfacial thermodynamics have apparently not yet been explicitly derived before. Hence, the focus on cavities, the properties of which are well-described by SPT and I-SPT, will also provide important information about the behavior of the line tension of, for example, sessile droplets. Finally, the thermodynamics of curved boundary layers should improve our understanding of the behavior of large hard-sphere colloids as they approach a hard wall exposed to a solvent of much smaller colloidal species. In this so-called Derjaguin limit, the effective force that develops between the large colloid and the wall, arising from the rearrangement of smaller colloids about the large colloid and the wall, can be obtained from a macroscopic work expression similar to eq 2. Using the Derjaguin approximation, which becomes exact in the limit of a macroscopic large colloid, only two terms are needed, the pressure-volume and surface area contributions (the latter accounts for both the increase in surface area of the colloid and the removal from direct contact of the smaller colloids with some portion of the wall). Recently, large deviations between statistical mechanical models, the Derjaguin approximation, and simulation have been noted, particularly in the regime in which the equivalent cavity of the large colloid reaches the separation distance at which it no longer intersects the wall.7 The Derjaguin limit may therefore be approached more slowly than previously thought.7,8 Not normally included in the Derjaguin approxima-
Siderius and Corti
Figure 1. Illustration of the model system for the discussion of spherical boundary thermodynamics. The impenetrable boundary of radius λ, or cavity surface, is indicated by the solid line. Dashed lines represent the angular bounds of the spherical cone with angle ω, while another dashed line of radius a identifies the mathematical dividing surface. The dash-dot line of radius R is the limit of the fluid to be studied. V1 and V2 identify the volume of the regions between λ and a and a and R, respectively.
tion is the line tension of the equivalent intersecting cavity, which appears to be a crucial ingredient in estimating the onset of the Derjaguin limit, particularly for large but not yet macroscopic colloids and for locations in which the cavity just ceases to intersect the wall.8 How the line tension behaves at this point is, however, not precisely known, and further insights into this limiting behavior should be of interest when studying the Derjaguin limit. The paper is organized as follows. In section 2, we revisit the thermodynamics of curved boundary layers for a spherical cavity in a bulk fluid. In section 3, we consider the thermodynamics of a fluid residing near a spherical cavity that intersects a hard wall. From these relations, explicit expressions for the line tension of a cavity are derived in section 4. Conclusions are contained in section 5. 2. Thermodynamics of Curved Boundary Layers: Spherical Cavity in a Bulk Fluid While the thermodynamic formalism for describing a boundary layer created by a spherical cavity within a bulk fluid has already been developed by Mandell and Reiss,6 we nevertheless reproduce most of their results below. For one, the analysis we follow is based upon a related, though slightly more fundamental, approach presented by Hill,9 which provides the more appropriate starting point for our subsequent discussion of cavities that intersect a hard wall. In addition, we derive and later employ various results not explicitly contained in ref 6. Furthermore, some relations given below have clear analogues to several expressions provided in the following sections. We begin by considering a spherical cone of solid angle ω centered at the origin, which also coincides with the center of the cavity of radius λ (see Figure 1). At a distance R from the origin, another (pseudo) boundary is placed. Both the surface of the cavity and the boundary at R are impenetrable to the molecular centers. Fluid particles are introduced between λ and R, with contours of equal density, for example, residing on spherical surfaces centered at the origin. (We also assume that the fluid phase actually persists beyond R and interacts with the fluid within R, so that the outer boundary merely serves to define a given subvolume of interest. While this assumption is not strictly needed, it proves useful in the next section since it automatically eliminates the need to consider various superficial
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quantities at the outer boundary when R f ∞. For example, for R of macroscopic size, the density of the fluid will be strictly uniform and equal to the bulk density at the surface of the outer boundary.) The walls of the cone also have no physical effect. The fluid surrounding the cavity is assumed to be at equilibrium in that all of the thermodynamic properties of the system are completely determined by the following set of variables: R, λ, ω, the temperature T, and the number of particles N. Given this variable set and following the procedure outlined by Hill,9 the full expression for the change in the Helmholtz energy, F, of the system upon a change in one or more of these variables is
dF ) -SdT + µdN + P1ωλ2dλ - P2ωR2dR + ηdω (4) where S is the entropy, µ is the uniform chemical potential of the (equilibrium) inhomogeneous single-component fluid, P1 and P2 are the components of stress normal to the boundaries at λ and R, respectively, and
η≡
∂F ( ∂ω )
(5)
T,N,λ,R
(Without loss of generality, we deal here with a fluid comprised of one component only, though the extension to a multicomponent system is straightforward.) Equation 4 provides an unambiguous starting point for the subsequent analysis since there is no uncertainty about the external work terms associated with dλ, dR, and dω. The term containing dλ is simply the SPT expression for cavity growth in a bulk fluid given by eq 1, while the dR term accounts for the work required to change the radius of the outer boundary. For R of macroscopic size (as we assume here), in which the outer boundary is far enough away for the fluid to have reached a uniform density at R, P2 becomes the bulk pressure of the fluid. Finally, since the system is a firstorder homogeneous function of ω, Euler integration of eq 4 yields6,10
F ) µN + ηω
(6)
So far, only properties that are macroscopic or related to a specific point in the fluid, such as P1, have been introduced. To proceed further, we now introduce a dividing surface of radius a that serves to separate the system into two subvolumes6,9 (see Figure 1)
V1 )
ω 3 (a - λ3) 3
and
V2 )
ω 3 (R - a3) (7) 3
which are separated by a boundary with area A ) ωa2. Solving for dλ, dR, and dω in terms of dV1, dV2, dA, and da and substituting these into eq 4 yields
dF ) -SdT + µdN - P1dV1 - P2dV2 + γdA + [ωa2(P1 - P2) - 2ωaγ]da (8) where
γ)
P1V1 P2V2 [P1V1 + P2V2 - (µN - F)] η + + ) 2 2 2 a ωa ωa ωa2 (9)
or
F ) µN - P1V1 - P2V2 + γA
(10)
in which eq 6 was used. Equation 10 may be regarded as an expression that defines the surface tension of the system.11,12 Since the Helmholtz energy is independent of the choice of the dividing surface, all terms in eq 10 that depend on a must be constant,6 or
(P2 - P1)
a3 + γa2 ) constant 3
(11)
Consequently
1 dγ 1 γ+ a ) a(P1 - P2) 2 da 2
[ ]
(12)
where γ is a function of the dividing surface a but not P1 and P2. The square brackets denote the derivative with respect to the mathematical displacement of the dividing surface, keeping all physical conditions describing the system unaltered, which is to be distinguished from a later derivative of γ with respect to the actual radius of the cavity. Now, the Gibbs surface of tension is conventionally defined as that choice of the dividing surface, as, for which9
[ dγda ]
a)as
)0
(13)
In this case, we find that
(P1 - P2) )
2γs as
(14)
where γs is the surface tension corresponding to as. Equation 14 is formally identical to the well-known Laplace relation, where P1 and P2 would represent the pressures of, for example, the uniform droplet and the surrounding bulk vapor phase, respectively. Given a particular location of the dividing surface, eq 11 requires that
(P2 - P1)
as3 a3 + γa2 ) (P2 - P1) + γsas2 3 3
(15)
which upon substituting for (P2 - P1) using eq 14 leads to6
as2 γ 2a ) + 2 γs 3as 3a
(16)
Again, this relation is formally identical to what appears in the corresponding bubble or droplet interfacial analysis.11,12 At least for a hard-sphere fluid, P1 e P2, so that γs e 0 (highlighting one difference between a boundary tension and an interfacial surface tension). If γs is negative, eq 16 indicates that γ e 0, as well as γ having its maximum value γs (or smallest magnitude) at a ) as. We continue by deriving the boundary layer equivalent of the Gibbs adsorption isotherm. Begin by rewriting eq 8 as
dF ) -SdT + µdN - P1dV1 - P2dV2 + γdA + A ∂γ da (17) ∂a
[ ]
where we have made use of eq 12. Determining dF from eq 10 and equating to the above yields
[ ∂γ∂a ]da
Ndµ ) -SdT + V1dP1 + V2dP2 - Adγ + A
(18) Since the fluid reaches its bulk density at the outer boundary, one defines the adsorption per unit area, Γ, by6
N ) F2(V1 + V2) + ΓA
(19)
which follows from assuming that the total volume V ) V1 + V2 is filled with an homogeneous fluid of bulk density F2 and pressure P2. Substituting this into eq 18 and noting that dP2 ) F2dµ while taking dT ) 0 yields the Gibbs adsorption isotherm
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Γdµ ) -dγ +
∂γ a3 - λ3 d(P1 - P2) + da ∂a 3a2
[ ]
Siderius and Corti
(20)
dF ) 4πP2λ2dλ + 4πd(λ2γλ)
For the surface of tension, we find that
Γsdµ ) -dγs +
as3 - λ3 3as2
( )
d
2γs as
(21)
allowing one to determine how Γs, the adsorption corresponding to a dividing surface at as, varies with the chemical potential or bulk density. As λ f ∞, the above reduces to the planar wall limit of the Gibbs adsorption isotherm, where10
Γ∞dµ ) -dγ∞
(22)
in which γ∞ and Γ∞ are the surface tension and corresponding adsorption for the planar limit, respectively, both being independent of the location of the dividing surface. Next, consider a process for which T and µ are held constant (and therefore the bulk pressure P2 is also fixed) but with a ) λ, that is, the dividing surface is always chosen coincident with the surface of the cavity so that V1 ) 0. From eq 20, we find that
∂γλ ∂γ ) ∂λ ∂a
[ ]
coincide with λ (keeping the same variable set constant as before), we find that
a)λ
∂(λ2γλ) ) λ2(P1 - P2) ∂λ
(24)
Since P1 is a function of λ only (P2 is independent of λ), the above equation is a first-order linear differential equation which can be solved for γλ. Previously, Mandell and Reiss13 only calculated γs as a function of as using results from SPT. No expression for γλ as a function of λ has so far appeared in the literature, at least to our knowledge, though any version of SPT1,13,16–18 with its given dependence of P1 on λ, can be used to generate γλ for a hard-sphere fluid. While the difference between γs and γλ is relatively small, the calculation of reversible works of cavity growth is more conveniently done via a dividing surface that is coincident with λ and not as. To see why, choosing a ) as allows one to rewrite eq 17 for a constant T, N, and R process (total volume is fixed) as
8πγs 2 dF ) 4πP2λ2dλ + λ dλ as
In this form, the determination of ∆F is straightforward. Furthermore, if we are also interested in determining how Γλ, the adsorption defined for a dividing surface coincident with the cavity surface, varies with the chemical potential or bulk density for a fixed value of λ, we see that eq 20 reduces to14
Γλdµ ) -dγλ
(27)
Despite the similarity with eq 22, the above is valid for all values of λ. Finally, we consider the dependence of γλ on the curvature of the cavity as the radius approaches microscopic size. We start from eq 16 to obtain
∂γλ ∂γ ) ∂λ ∂a
[ ]
) a)λ
( )
2γs λ3 - as3 3 asλ3
(28)
which upon substitution of eq 16 into the above yields
∂ ln γλ 2(λ3 - as3) ) ∂ ln λ (2λ3 + as3)
(23)
where γλ is the surface tension corresponding to the dividing surface being located at λ and therefore is a function of T, µ, and λ. In contrast, γ is a function of T, µ, λ, and the dividing surface a. Therefore, ∂γλ/∂λ is the change in γλ with respect to a change in the actual radius λ of the cavity (keeping T and µ fixed), while [∂γ/∂a] denotes how the surface tension varies with only a different choice of the dividing surface (keeping T, µ, and λ fixed). Only for the dividing surface located at λ are these two derivatives equal. Interestingly, this same condition applies for the equimolar dividing surface in the bubble or droplet interfacial analysis.11,12 (The equimolar dividing surface, which is not a relevant descriptor of the fluid surrounding a cavity, can be used to define the size of the bubble or droplet. Hence, it is not unexpected that λ, the actual size of the cavity, arises as the boundary analogue of the equimolar dividing surface.) Applying eq 23 to eq 12, we see that
(26)
(29)
We now define δλ ≡ λ - as, which is the difference between the radius of the cavity and the radius of the surface of tension (this is the opposite of the definition proposed by Mandell and Reiss13). δλ is analogous to the definition put forth by Tolman,15 where λ again corresponds to the equimolar dividing surface in the interfacial analysis. Equation 29 can be rewritten as
d ln γλ ) d ln λ
( )( [ ( ){
( )) ( ) }]
δλ δλ 1 δλ 2 1+ λ λ 3 λ δλ δλ 1 δλ 11+ λ λ 3 λ 2
which is the boundary analogue to the Gibbs-Tolman-KoenigBuff equation12 and is formally exact for all values of λ. In general, δλ is on the order of a few molecular diameters and should become approximately constant and equal to δ∞ as λ f ∞. With γλ f γ∞ as well as δ∞/λ f 0 for λ f ∞, a Taylor series expansion of the righthand side of eq 30 about δλ/λ f 0 yields upon integrating from λ f ∞ to λ the following result
( ) ( )
γλ δ∞ δ∞ )1-2 +2 γ∞ λ λ
2
- ···
(31)
δ∞ plays the same role as the Tolman length, describing the first-order correction to the surface tension as a function of curvature or 1/λ. Although the second-order term in eq 31 is not typically used to modify the surface tension beyond the first-order correction, the above expansion suggests, at least for large radii, that γλ can be well represented by a series in inverse integral powers of λ. Such an expansion is invoked by SPT for the work of cavity formation in the macroscopic limit, where substitution of eq 31 into eq 26 leads to
()
dF ) 4πP2λ2dλ + 8πγ∞λdλ - 8πγ∞δ∞dλ + O
(25)
Although an approximate expression for as as a function of λ can be obtained,13 the above expression is computationally inconvenient. If we instead choose the dividing surface to
(30)
2
1 dλ λ2 (32)
Additional terms in the expansion must begin with 1/λ2 and not 1/λ; otherwise, an unphysical term containing a logarithm would appear upon integration.19,20 Ultimately, the boundary
On the Line Tension of Curved Boundary Layers
Figure 2. Three-dimensional view of the model system used for the boundary thermodynamics of spherical cavities intersecting a hard wall. The fluid is in contact with the wall, and a cavity with radius λ and center coordinate z (not identified here) is introduced. The control volume is a cylindrical wedge of radius R, height Z, and sweep angle ω.
analysis provides formal justification for the use of eq 32 in SPT. We also see that by invoking eq 2, all previous versions of SPT have implicitly utilized the surface tension corresponding to the dividing surface being located at the cavity radius as well as employing a Tolman length that is based on the planar limit, δ∞. While the validity of SPT is of course not in doubt, knowing what forms of the macroscopic work expansion are consistent with surface thermodynamics is an important point that we revisit in the following sections. 3. Thermodynamics of Curved Boundary Layers: Spherical Cavity Intersecting a Hard Wall We now extend the above boundary thermodynamic analysis to the case in which a spherical cavity intersects a hard, structureless wall. For particles without an impenetrable hard core, the center of a given particle can approach the wall. On the other hand, the hard core of a particle may not penetrate the wall, and therefore, the closest approach of a particle center to the wall is σ/2, where σ is the hard-core diameter. For distances less than σ/2, the density of hard-core particle centers is uniformly zero. (In either case, the molecular centers may approach the surface of the cavity.) The plane parallel to and at the distance of σ/2 from the wall therefore acts as an effective hard wall for these hard-core fluids (see Figure 2 of ref 3). In each case, we denote the location of the center of the cavity as z (see Figures 2 and 3), where the z ) 0 plane is either the position of the actual hard wall (for fully penetrable particles) or the position of the effective wall (for impenetrable particles). Throughout the analysis, the location of z is held fixed. Since we only consider cavities that intersect the z ) 0 plane, we are restricted to cavity locations for which -λ e z e λ. As discussed in refs 3 and 5, the local density of particle centers about the surface of the cavity is no longer symmetric, arising from the anisotropic environment induced by the presence of the hard wall, and depends upon the proximity to the wall. Nevertheless, the stress normal to the surface of the cavity can be suitably j 1, as was done in eq 3. averaged to yield an average pressure, P By definition, the reversible work required to change the cavity j 1dVcav, where Vcav is that portion of the radius (for fixed z) is P cavity’s volume extending above z ) 0. As in the bulk spherical j 1 is a function of λ, z, and the bulk density far away case, P from the wall. In our analysis, we can no longer make use of a spherical angle about the center of the cavity since the system is not
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Figure 3. Two-dimensional representation of Figure 2. The center of the cavity is at z (which, for the spherical cap shown, is for z < 0). Volumes V1 and V2 are the regions between the dividing surface, located at a radial distance a from the cavity center, and the cavity of radius λ, and the remainder of the fluid, respectively.
isotropic with respect to an arbitrary change in this coordinate (i.e., we cannot Euler integrate over this variable). If we instead consider a cylindrical coordinate system, we note that the system is symmetric about an angle 0 e ω e 2π that lies completely within a given plane parallel to the wall. We therefore introduce a large cylinder of fixed height Z that encloses the cavity and covers a portion of the wall (see Figures 2 and 3). Furthermore, the radius of the cylinder R and the height Z is chosen to be large enough such that bulk properties are obtained along the top and (nearly all of) the sides of the cylinder, that is, at these distances, the effect of the cavity is no longer felt. We again assume that additional fluid particles exist outside of the cylinder (and still interacts with those particles inside of the cylinder) so that various superficial quantities will not arise along the surface of the cylinder, except where the cylinder intersects the z ) 0 plane. As R is increased, the cylinder will enclose more surface area of the wall (since the wall extends beyond the cylinder). If we assume that the fluid also reaches bulk behavior at the intersection of the z ) 0 plane and the cylinder, the work required to increase R will include an additional (planar) interfacial term in addition to a pressure-volume contribution. (While this interfacial term is not strictly needed to determine the effects of the cavity, it is included here to maintain consistency with previous results for the thermodynamics of a bulk fluid near a planar surface.) The fluid within the cylinder is again assumed to be at equilibrium in that all of its thermodynamic properties are completely determined by the following set of variables: T, N, R, Z, λ, and ω. As before, we start from an expression for the change in the Helmholtz energy that contains no uncertainty in the meanings of the various terms
j 1ω(λ2 + λz)dλ - P2ωZRdR + dF ) -SdT + µdN + P ω γ∞ωRdR - P2 R2dZ + ηdω (33) 2 where
η≡
∂F ( ∂ω )
T,N,λ,R,Z
(34)
The third term on the right-hand side of eq 33 represents the work required to increase the radius of the cavity, where (λ2 + λz)dλ is the differential change in the volume of the cavity above the z ) 0 plane (for fixed z). The fourth and sixth terms denote the pressure-volume work required to increase R and Z, while the fifth term is the free energy-cost of creating additional area of the planar wall, with boundary tension γ∞, that is exposed to the fluid. Note that γ∞ only accounts for the increase in the
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wall area due to a change in R. The area of the wall is also modified when λ changes, but the work of doing so is already accounted for with the third term.3,5 Since the system is a firstorder homogeneous function of ω, Euler integration of eq 33 yields
F ) µN + ηω
(35)
While not immediately apparent, ηω contains pressure-volume as well as all boundary effects within the system: the free energies associated with (1) the cavity surface, (2) the linear interface arising from the intersection of the cavity and the wall, (3) that portion of the wall excluded by the cavity, and (4) the remaining portion of the wall directly exposed to the fluid. As in the analysis of spherical cavities, we would like dF to be expressed in terms of volumes and areas of interfaces (the explicit inclusion of the linear interface along the intersection of the cavity and the z ) 0 plane is delayed until the next section). To do so, we now divide the cylinder into two separate volumes (see Figures 2 and 3) by inserting a spherical dividing surface of radius a g |z| that is centered on z, where we have that
V1 ) ω (a + z)2(2a - z) - ω (λ + z)2(2λ - z) 6 6 ω ω 2 2 V2 ) ZR - (a + z) (2a - z) 2 6 (36) AT ) ω R2 2 A ) ω(a2 + az) in which A is the area of the dividing surface that is concentric with the cavity and AT is the total area of the wall contained within the cylinder and includes that portion of the wall that is covered by the cavity. While the identification of AT with the total of the area of the wall is not necessary, this choice will be justified in subsequent steps. (We note that one can also introduce, along with the above spherical dividing surface, another planar surface that serves to divide the cylinder into three regions. Since this second dividing surface is planar and is introduced above a planar wall whose boundary tension is independent of the location of its dividing surface, the threevolume analysis can be shown to reduce to the currently chosen two-volume analysis. This second planar dividing surface is therefore unnecessary.) Next, we solve for dλ, dR, dZ, and dω in terms of dV1, dV2, dAT, and dA, which when substituted back into eq 33 yields
j 1dV1 - P2dV2 + γ∞dAT + dF ) -SdT + µdN - P j 1V1 P P2V2 γ∞AT ηω j 1 - P2)ω(a2 + dA + (P + + A A A A az) j 1V1 P P2V2 γ∞AT ηω ω(2a + z) da (37) + + A A A A
(
) ( )
(
)
We now define the following new variable
χ≡
j 1V1 P P2V2 γ∞AT ηω + + A A A A j 1V1 P P2V2 γ∞AT (µN - F) (38) ) + A A A A
or upon rearrangement
j 1V1 - P2V2 + γ∞AT + χA F ) µN - P dF can now be written more compactly as
(39)
j 1dV1 - P2dV2 + γ∞dAT + dF ) -SdT + µdN - P j 1 - P2)A - χ ∂A χdA + (P da (40) ∂a ω,z
[
( ) ]
In the absence of the cavity (which no longer requires the introduction of a spherical dividing surface), the above analysis leads to the following expression for the Helmholtz energy21
F0 ) µN - P2VT + γ∞AT
(41)
where VT is the total volume of the cylinder. Equations 39 and 41 indicate that the reversible work of cavity formation is given by
j 1 - P2)V1 + P2Vcav + χA (42) W ) F - F0 ) -(P where VT ) V1 + V2 + Vcav. When the dividing surface is chosen coincident with the cavity, that is, a ) λ so that V1 ) 0, the above reduces to W ) P2Vcav + χA. For any choice of a, the expression for W reveals that χA contains all interfacial, or boundary, contributions, owing to the immediate presence of the cavity in the fluid; these include the free energies of forming (1) the curved interface, (2) that portion of the surface area of the wall deleted by the cavity, and (3) the linear interface residing at the intersection of the cavity and the z ) 0 plane. (Note that AT does not appear in eq 42.) χ is a local quantity in that it only describes the effects of the cavity, all scaled to the area A, and does not include the interfacial free energies assigned to the entire wall contained within the cylinder. The use of a single term to describe all interfacial contributions of interest is not as elucidating as choosing a set of terms that clearly identifies the separate surface and linear contributions, all of which would be scaled accordingly to their appropriate surface area or linear perimeter. We decompose χA into such terms in the next section where the line tension of the cavity is explicitly introduced. The selection of a single term to account for all interfacial contributions proves useful, however, when examining the asymptotic behavior of χ (and therefore the line tension in our subsequent analysis). For example, consider a macroscopic cavity at a fixed value of z. For λ f ∞ and z/λ f 0, that portion of the cavity exposed to the fluid approaches (to leading order) a hemispherical cap with surface area 2πλ2. Since the surface of the cavity is becoming effectively planar, the surface tension of this curved interface approaches γ∞, and the free-energy cost of generating the surface increases as 2πλ2γ∞ (since we are dealing with a macroscopic cavity, the location of the dividing surface is irrelevant and chosen to coincide with the cavity surface for convenience). The portion of the area of the wall erased by the cavity increases as πλ2 (again to leading order) so that the free-energy cost of eliminating this surface area from direct contact with the fluid is given by -πλ2γ∞. Finally, the length of the linear interface along the intersection of the cavity and the z ) 0 plane increases as 2πλ (to leading order). Assuming that the line tension is essentially constant in this macroscopic limit22 and equal to τ∞, the linear free-energy contribution is given by 2πλτ∞. Combining these three terms indicates that χA ) χ2πλ2 ) πλ2γ∞ + 2πλτ∞ or χ∞ ) γ∞/2 as λ f ∞ (which also follows if τ is not constant but does not grow faster than λ). Additional justification for this limiting value of χ is provided at the end of this section, where we derive an asymptotic expansion for χ in terms of inverse powers of λ. Returning to the boundary analysis, we note that since F cannot depend on a (with AT also not depending upon a), eq 39 shows that
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J. Phys. Chem. B, Vol. 113, No. 42, 2009 13855
j 1V1 - P2V2 + χA ) constant -P
(43)
j 1 - P2)A ) χ ∂A (P ∂a
(44)
or
( )
ω,z
[ ∂a∂χ ]
+A
where the square brackets are again used for the derivative of χ with respect to the mathematical displacement of the dividing surface, keeping all physical conditions describing the system unaltered. With the above, dF can be rewritten as
j 1dV1 - P2dV2 + dF ) -SdT + µdN - P
(Note that Γ′ is not the same as Γ, the adsorption along a cavity in a bulk fluid that appears in eq 20.) Substituting this definition of Γ′ into eq 51, again noting that dP2 ) F2dµ while also making use of eq 22, we find for dT ) 0 that
(
Γ′ - Γ∞
)
V1 Aw j 1 - P2) + ∂χ da dµ ) -dχ + d(P A A ∂a (53)
[ ]
j 1 f P2, Γ′ f For the surface of tension and as λ f ∞, where P Γ∞, and Aw/A f 1/2, the above reduces to
Γ∞ dµ ) -dχ∞ 2
∂χ da (45) ∂a
[ ]
γ∞dAT + χdA + A
As for the cavity within a bulk fluid, we can identify a surface of tension at a ) aT such that
[ ∂a∂χ ]
a)aT
)0
corresponding to χ ) χT. Therefore
j 1 - P2 ) P
(
2aT + z aT2 + aTz
(46)
)
which also implies that χ∞ ) γ∞/2. This result is also independent of the location of the dividing surface. Returning to eq 51, let us consider a process for which T and µ, and therefore P2 and γ∞, are constant. If the dividing surface is chosen to coincide with the cavity surface, a ) λ, then
∂χλ ∂χ ) ∂λ ∂a
[ ]
(47)
χT
(54)
(55)
a)λ
Along with eq 44, we find that
which for z ) 0 (hemispherical cap) reduces to
2χ j 1 - P2 ) T P aT
(48)
The hemispherical limit, for geometrical reasons, resembles the j 1 and χT are not, in general, equal bulk relation of eq 14. Since P to P1 and γs, respectively, we do not expect that aT is equal to as. Furthermore, equating eq 43 separately evaluated at a and aT, along with the use of eq 47, yields the following relation
a2(2a + 3z)(2aT + z) + aT2(2aT2 + 4aTz + 3z2) χ ) χT 6aaT(a + z)(aT + z) (49)
∂(Aχλ) j 1 - P2)A ) (P ∂λ
j 1 is known, enables one to determine χλ as a function which, if P of λ. Again, we see that the choice of a ) λ is advantageous since the work of cavity growth at fixed T, N, R, and Z reduces to
dF ) P2dVcav + d(χλA)
(
Γλ′ - Γ∞
χ 2a ) + χT 3aT 3a
(50)
which is formally identical to eq 16. For a hard-sphere fluid, j 1 e P2 for z e 0.3 While not known for sure, it appears likely P j 1 e P2 for z > 0 as well.23 Thus, as follows from eq 47, that P χT e 0. In turn, eq 49 indicates that χ e 0, with χ having its maximum value χT (or smallest magnitude) at a ) aT. The corresponding Gibbs adsorption equation follows from equating eq 45 to the total differential of F obtained from eq 39, which yields
j 1 + V2dP2 - ATdγ∞ - Adχ + A Ndµ ) -SdT + V1dP ∂χ da(51) ∂a
[ ]
In this case, we separate the particles inside of the system into terms related to the volume outside of the cavity as well as the adsorptions occurring along that portion of the wall exposed to the fluid, Γ∞, and the surface of the cavity, Γ′, or
N ) F2(V1 + V2) + Γ∞(AT - Aw) + Γ′A
(52)
where Aw is the area of the wall covered by the cavity, and we have also assumed that the total volume V ) V1 + V2 is filled with an homogeneous fluid of bulk density F2 and pressure P2.
(57)
The above can be easily integrated and would not have been obtained if we had chosen a ) aT. Likewise, for a ) λ (where V1 ) 0) and with the radius of the cavity held fixed, the adsorption isotherm is given by
For z ) 0, the above simplifies to
aT2 2
(56)
)
Aw dµ ) -dχλ A
(58)
a result valid for any cavity size. Finally, we generate an asymptotic expansion for χλ as λ f ∞. As was done for the spherical cavity within the bulk fluid, utilizing eqs 49 and 55 leads to the following result
[(
)(
δχ ∂ ln χλ z z ) 6 -2 +2 1+ ∂ ln λ λ λ λ
) )
)/[(2 + 3 λz )( λz - 2 λ + δχ
2 +
(
1-
δχ λ
)( ( ) ( 2
3
z λ
2
+2 1-
δχ λ
2
+4
(
δχ z 1λ λ
))]]
z λ z 1+ λ 2+
(59)
where δχ ≡ λ - aT is the intersecting cavity equivalent of the Tolman length and is not, in general, equal to δλ introduced in the bulk case. As λ f ∞, the inhomogeneity generated by the cavity is still only felt over a finite distance (as measured from the cavity surface), which is expected to approach a fixed value as the surface area of the cavity becomes that of a hemisphere. Thus, δχ may be considered to reach a constant value, δχ∞, as λ f ∞. Hence, both δχ∞/λ and z/λ should approach zero as λ f ∞. Expanding the above in a Taylor series about 1/λ f 0 and
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Siderius and Corti
then integrating from λ f ∞, where χ approaches a constant value of χ∞, to λ, one finds (after another Taylor expansion of the exponential) that 2 δχ∞ χλ zδχ∞ + 2δχ∞ )1-2 + χ∞ λ λ2 2 δχ∞[z2 + 2zδχ∞ + (8/9)δχ∞ ]
λ3
j 1V1 - P2V2 + γ∞(AT - Aw) + γA + τL F ) µN - P (64)
()
+O
1 (60) λ4
For the hemisphere (z ) 0), the above reduces to
()
()
(61)
which is formally the same as eq 31. Finally, substituting eq 60 into eq 57 shows that
dF ) 2πP2(λ2 + λz)dλ + 2πχ∞(2λ + z)dλ 4πχ∞δχ∞dλ + 2 2 z(zδχ∞ + 2δχ∞ ) - δχ∞[z2 + 2zδχ∞ + (8/9)δχ∞ ]
λ2
dλ +
· · · (62) -1
where again a term proportional to λ does not appear. Future versions of I-SPT should invoke this form of the macroscopic workexpansiontoensureconsistencywithsurfacethermodynamics. In general, δχ∞ should be a function of z (as long as z , λ, the assumption of a constant value of δχ∞ is still valid). For example, consider the increase in the cavity radius at constant T and µ. Substitution of eq 47 into eq 53 shows that
[
j 1 - P2) d (P
(
aT2 + aTz 2aT + z
)]
)
V1 j 1 - P2) d(P A
where L ) ω(a2 - z2)1/2 is the linear interface generated by the intersection of the dividing surface and the z ) 0 plane, Aw ) ω(a2 - z2)/2, and the “line tension”, τ, is defined as follows
τ≡
2 3 δχ∞ χλ δχ∞ 8 δχ∞ 1 +O 4 )1-2 +2 2 χ∞ λ 9 λ3 λ λ
2πχ∞
respectively. With the remaining interfacial contribution being associated with a linear free energy, the Helmholtz energy can be written as
(63)
j 1, V1, and A are functions of λ and z, the above indicates Since P that aT, and therefore δχ ) λ - aT, should also be a function of z and λ. Hence, it is not unreasonable to expect that δχ retains some dependence on z as λ f ∞. Since the cavity approaches a hemisphere for any value of z, we still find that χ∞ is independent of z. Yet, the manner in which the cavity approaches this limit (i.e., the derivatives of the volume and various surface areas) depends upon z, suggesting that all higherorder corrections to χ∞ (including the first) should be functions of z. Since z can be large, one may find that δχ∞ is also large, at least with respect to δ∞, and yet, z , λ, suggesting that δχ∞ still represents a length scale significantly smaller than the cavity radius or the surface of tension aT. 4. Line Tension of a Spherical Cavity Intersecting a Hard Wall Although containing useful information, χ is ultimately a surrogate term that has been introduced to simplify the analysis, particularly when developing an asymptotic expansion of the combined interfacial contributions. The use of a single term to describe all of the surface effects of the cavity hides, however, details about the separate interfacial contributions, specifically the linear interfacial term that is part of χA. We therefore expand our previous analysis and decompose χ into various terms. We begin with eq 39 and then add and subtract γA and γ∞Aw, the free-energy cost of creating the exposed portion of the spherical cavity as if it had been formed far away from the wall (in the bulk phase) and the work of eliminating that portion of the surface area of the wall that is covered by the cavity,
γ∞Aw χA γA + L L L
(65)
Since the cavity is devoid of particles, there is no interfacial contribution between the cavity “phase” and the wall. Also, γ and γ∞ are those values of the surface tension following from the bulk dividing surface analysis. γ is therefore calculated for a fully spherical cavity of radius λ formed in a fluid far away from the wall and at the same state point and with the same radius a of the dividing surface; γ∞ is likewise at the same state point. Equation 65 shows that χA has been separated into three terms, (1) the surface free energy of the cavity surface, γA, (2) the free-energy cost of deleting a portion of the wall exposed to the fluid, -γ∞Aw, and (3) the linear free energy assigned to the intersection of the dividing surface with the z ) 0 plane, τL. In other words, τL describes the excess cost of creating the intersecting cavity based on the removal of some amount of planar surface area and with the spherical interface having been created in the bulk fluid far away from the wall. Substitution of eq 65 into eq 45, along with the evaluation of [∂χ/∂a] using eq 65, allows for dF to be rewritten as
j 1dV1 - P2dV2 + γ∞d(AT dF ) -SdT + µdN - P ∂τ ∂γ +A da (66) Aw) + γdA + τdL + L ∂a ∂a
( [ ] [ ])
where [∂τ/∂a] is again the change in the line tension due to a mathematical displacement of the dividing surface. The term containing γ∞ now accounts for changes in that part of the surface area of the wall beyond the dividing surface (in region V2 only). As before, F must be independent of the location of the dividing surface. By either differentiating eq 64 with respect to a or evaluating ∂F/∂a ) 0 via eq 66, one finds (with T and N held constant)
j 1)(a2 + az) - aγ∞ + γ(2a + z) + τa + x (P2 - P x ∂τ + ∂a ∂γ ) 0 (67) (a2 + az) ∂a
[ ]
[ ]
where x ≡ (a2 - z2)1/2 ) L/ω is the in-plane radius of the linear interface and the expression for A and the derivatives of V1, V2, AT, and Aw were obtained using eq 36. Finally, utilizing eq 12 to replace the last term on the left-hand side yields, upon further rearrangement and division by a, the following result
j 1 - P1)a(1 - cos φ) ) γ cos φ + τ + x ∂τ γ∞ + (P x a ∂a (68)
[ ]
in which cos φ ) -z/a, with φ representing the “contact angle” of the dividing surface at the z ) 0 plane (see Figure 4). Equation 68 resembles the generalized Young equation that describes the relationship between the various interfacial tensions
On the Line Tension of Curved Boundary Layers
J. Phys. Chem. B, Vol. 113, No. 42, 2009 13857
(both surface and linear) of a sessile drop24,25
∂τ τ + x ∂x
[ ]
γSV - γSL ) γLV cos φ +
(69)
where γSV, γSL, and γLV are the solid-vapor, solid-liquid, and liquid-vapor surface tensions. To better compare the two, we rewrite eq 68 as follows
j 1 - P1)(1 - cos φ)√x2 + z2 ) γ cos φ + τ + γ∞ + (P x ∂τ (70) ∂x
[ ]
With the surrounding fluid playing the same role as the vapor phase in eq 69, the correspondence between γ∞ and γSV and between γLV and γ is not surprising. However, since there is no interfacial tension between the “cavity phase” and the wall, a term directly analogous to γSL does not appear. While the contribution proporj 1 - P1) seems to play the same role as γSL, this term tional to (P simply acknowledges that the stress normal to the surface of the cavity is different when the cavity is formed at the wall as compared to its formation far away from the wall. Since γ describes the freej 1, the energy cost based on P1, while the actual cost is related to P j (P1 - P1) term merely accounts for the deviation from the actual free-energy difference. There is no obvious mechanical interpretation of this discrepancy as a force per unit length, or tension, acting at the intersection of the dividing surface and the z ) 0 plane. A similar ambiguity arises for [∂τ/∂x], also contained in eq 69, the mechanical interpretation of which is likewise unclear. (Even the usual mechanical justification of the Young equation, eq 69 without the line tension terms, has been questioned.26,27 When the Young equation is applied to capillary rise, for example, the inherent weaknesses in the standard mechanics-based derivations can be overcome via a proper reinterpretation of the forces involved.28) The Young equation, particularly the generalized version given above, is, strictly speaking, not a statement of mechanical equilibrium but rather a condition on the interfacial tensions, line tension, and contact angle that must be satisfied for a sessile drop in equilibrium, where the free energy of the droplet is at minimum. We may also introduce a particular choice of the dividing surface and identify a so-called “line of tension” where25
∂τ ∂a
[ ]
) a)aL
∂τ ∂x
[ ]
x)xL
)0
(71)
With xL ) - z ) and τL as the value of the line tension at a ) aL, the cavity Young equation, eq 70, becomes (aL2
2 1/2
j 1 - P1)aL(1 - cos φ) ) γa)a cos φ + γ∞ + (P L
τL (72) xL
which for the hemisphere, z ) 0 and cos φ ) 0, reduces to
j 1 - P1)aL ) γ∞ + (P
τL aL
(73)
There is again no reason to expect that aL coincides with either aT or as, so that, for example, γa ) aL may not be the surface tension at which the Laplace equation, eq 14, is satisfied.25 This distinction is often overlooked during the analysis of a sessile drop, though this is most likely unimportant for the large droplets that are typically considered. Furthermore, eq 73 does not impose a constraint on the sign of τL. For a hard-sphere fluid, j 1 - P1 > 0 for hemispherical cavities.3 Therefore, γ∞ < 0, while P depending upon the relative magnitude of these two quantities, τL for a hemispherical cavity could be either positive or negative (or even zero).
Figure 4. Illustration of the contact angle that appears in the cavity Young equation for the two cases, z < 0 and z > 0. The cavity is centered at z, a is the radius of the dividing surface, and x ) (a2 - z2)1/2 is the in-plane radius of the linear interface; φ ) cos-1(-z/a) is the “contact angle” of the dividing surface.
The corresponding adsorption isotherm for the linear interface can be obtained as was done previously. By equating the differential of F obtained via eq 64 with eq 66, we find that
j 1 + V2dP2 - (AT - Aw)dγ∞ Ndµ ) -SdT + V1dP ∂τ ∂γ +A da (74) Adγ - Ldτ + L ∂a ∂a
( [ ] [ ])
We now separate the particles of the system into contributions based on the adsorption at all surfaces, including the adsorption at the linear interface. Λ, the adsorption per unit length along the linear interface, is defined via the following relation
N ≡ F2(V1 + V2) + Γ∞(AT - Aw) + ΓA + ΛL (75) where Γ∞(AT - Aw) represents the adsorption at the planar wall due to V2 and ΓA is the adsorption at the curved interface based on the cavity being created far away from the wall. Substituting the above expression into eq 74, making use of dP2 ) F2dµ as well as the two previously derived adsorption isotherms relating dγ to Γ and dγ∞ to Γ∞ contained in eqs 20 and 22, we find upon setting dT ) 0 that
Λdµ ) -dτ +
(
)
3 3 V1 j 1 - P2) - A a - λ d(P1 d(P L L 3a2 ∂τ P 2) + da (76) ∂a
[ ]
This expression may be considered as the linear analogue of the Gibbs adsorption isotherm, eq 20, and has been previously obtained for the case of three macroscopic phases meeting along a straight line29,30 and for a sessile droplet.25 For a fixed dividing j 1 f P2 and P1 f P2, surface (da ) 0), as λ f ∞, where both P all terms on the right-hand side except -dτ will vanish. This suggests that regardless of the value of z, both τ and Λ approach limiting values related by
Λ∞dµ ) -dτ∞
(77)
both of which are again independent of the location of the dividing surface. Later on, we derive an expression for the limiting value of the line tension. Equation 76 again highlights the convenience of selecting the dividing surface at a ) λ. For a process in which T and µ are held fixed, where V1 ) 0, one finds that
∂τλ ∂τ ) ∂λ ∂a
[ ]
a)λ
(78)
(A similar relation for sessile drops was obtained in ref 25, although the dividing surface was chosen to coincide with the Gibbs surface of tension, as.) In addition, holding T and λ constant and the dividing surface fixed at a ) λ, we find for any cavity size that
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J. Phys. Chem. B, Vol. 113, No. 42, 2009
Λλdµ ) -dτλ
Siderius and Corti
(79)
Finally, substituting eq 78 into eq 68 yields upon rearrangement
∂(τλxλ) j 1 - P1)λ(λ + z) ) λγ∞ + zγλ + (P ∂λ
(80)
which can be used to determine τλ as a function of λ. The above also allows eq 66, upon choosing a ) λ, to be rewritten as
dF ) P2dVcav - γ∞dAw + d(γλA) + d(τλL)
(81)
Finally, again with a ) λ, subtracting eq 41 from eq 64 reveals that W ) F - F0 ) P2Vcav - γ∞Aw + γλA + τλL. This expression for W offers some clues about the behavior of τλ. For example, consider the case in which z e 0. For λ ) |z|, the cavity has not yet been pushed out into the fluid, so that W ) 0. Hence, we note that τλL ) 0. Given that L f 0 at this limit, the value of τλ is not obvious. We know that P2, γ∞, and γλ are not zero, being properties not based upon the intersecting cavity. Nevertheless, their corresponding contributions to W approach zero since these volume and area terms vanish as λ f |z|. Also, these volume and areas vanish faster than L, so that W/L f 0 for λ f |z| (a result that also follows directly from various relations found in ref 3). We therefore conclude that both τλL and τλ vanish as λ f |z|. (Given that the cavity is not exposed to the fluid at this point, one intuitively expects the line tension to be zero.) Does a similar conclusion hold at λ ) z when z > 0? In this case, the center of the cavity resides above the wall (or z ) 0 plane), and the limit of λ f z corresponds to the point at which the cavity ceases to intersect the wall. Here, L and Aw vanish, while W has a positive and finite value. At this limit, γ∞Aw f 0, with both P2Vcav and γλA being nonzero, the sum of which may not add up to W. Consequently, τλL does not in general vanish and may be either positive or negative. Since L f 0, we must conclude that τλ diverges as λ f z. What role this divergence plays in analyses of the onset of the Derjaguin limit for λ f z is unclear.7,8 This interesting behavior of the line tension is also not limited to cavities, as evidence for the divergence of τ at a first-order wetting transition has been noted previously.31,32 We finish this section by deriving an asymptotic expansion for τλ. Beginning with eq 65, we substitute into this relation the asymptotic expansions for γλ (eq 31) and χλ (eq 60) and then Taylor expand L-1 about z/λ f 0. The leading order term, which is proportional to λ, vanishes because χ∞ - γ∞ + γ∞/2 ) 0. What remains is
(
τλ ) γ∞ 2δ∞ - δχ∞ -
(
z + 2
)
)
γ∞ 2 zδχ∞ δ - 2δ∞2 + 2zδ∞ + · · · (82) λ χ∞ 2 Because δ∞, δχ∞, and z are all constants in the analysis, we see that τλ approaches a limiting value, τ∞, defined to be
(
τ∞ ≡ γ∞ 2δ∞ - δχ∞ -
z 2
)
(83)
Although γ∞ < 0 for a hard-sphere fluid, eq 83 apparently does not constrain the sign of τ∞, which may be either positive or negative. The expansion can also be rewritten as
(
τλ ) τ∞ 1 + where
2δL∞ + ··· λ
)
(84)
δL∞
2 2δχ∞ - 4δ∞2 + 4zδ∞ - zδχ∞ ) 8δ∞ - 4δχ∞ - 2z
(85)
acts as a type of “Tolman length” for the line tension. The δL∞, however, is not necessarily the same as λ - aL, indicating that this first-order correction to the line tension is different from either δχ∞ or δ∞. While eq 83 indicates that τλ reaches a limiting value independent of λ, this limit still seems to depend upon the fixed value of z. The explicit appearance of z in eq 83 arises for various reasons. For one, the variations of the relevant volumes and surface areas depend upon z. Furthermore, the line tension is a secondorder effect, in that it is proportional to the linear interface as opposed to the surface area of the cavity. While the intersecting cavity at a given z always approaches a hemisphere as λ f ∞, the “rate” at which this occurs depends upon z. These effects ultimately manifest themselves through the appearance of z in the definition of τ∞. However, as mentioned earlier, we expect that δχ∞ is also a function of z. Hence, it is not immediately apparent how τ∞ should depend upon z. Physical reasoning, as well as some of the above relations, suggests, however, that τ∞ is independent of z and should be equal to the value obtained for z ) 0, the always hemispherical cavity. As the radius of the cavity approaches macroscopic dimensions and as long as λ greatly exceeds z, that portion of the cavity exposed to the fluid should become identical to a hemispherical cap (the z ) 0 limit); the angle of contact between the cavity and the wall will eventually approach a right angle. Hence, the inhomogeneous density profile that develops about the cavity and the part of the wall still in direct contact with the fluid should be the same for any value of z as long as z , λ. The length scales over which these inhomogeneities develop are only felt over microscopic distances from their corresponding surfaces. (As shown in ref 3 j 1(λ,z)0) for λ . |z|; this limit was also j 1(λ,z) f P for z e 0, P found to be true even when λ was not significantly greater than |z|.) Since the effect of the line tension arises from the region focused on the intersection of the cavity and the wall, which is always becoming bounded by a right angle as z/λ f 0, that portion of the inhomogeneity assigned to τ should also become independent of z and equivalent to what develops about the hemisphere, where z is strictly equal to zero. Thus, it is not unreasonable to suspect that τ∞ adopts a common value for any z. Some evidence for this conclusion follows from the study of heterogeneous crystallization in hard-sphere fluids by Auer and Frenkel.22 The line tension of the hard-sphere crystal nucleus was found to be well-described by the following relation (which was presented without formal justification or without specifying a particular choice of the dividing surface)
τ ) τ∞ +
c λ
(86)
In the above, τ∞ and c are constants that are not functions of z. This relation is completely consistent with eq 84 if τ∞ (as well as δL∞) is independent of z. Furthermore, one can argue that the dependence of τ∞ on z may lead to physically inconsistent results. For example, if δχ∞ depends upon z in such a manner that δχ∞ + z/2 still varies as some positive power of z, eq 83 suggests that τ∞ grows without bound as the magnitude of z increases. (Note that z can have any value in our analysis and even be of macroscopic dimensions, as long as z , λ). For a certain range of z, the magnitude of τ∞ will greatly exceed that of γ∞, thereby leading to a linear interface term that contributes more to the free energy than the surface free-energy terms. However, such a result seems to run
On the Line Tension of Curved Boundary Layers
J. Phys. Chem. B, Vol. 113, No. 42, 2009 13859
counter to the underlying physics of the problem. The effects of the linear interface should presumably be on the order of L/A, a small contribution for large λ, and should not be the dominant contribution to the free energy. The fluid within or near the intersection of the cavity and the wall, which comprises only a small fraction of the total number of particles within the fluid, should not control the properties of the entire system. Previously derived surface thermodynamic relations also suggest that τ∞ is independent of z. For example, the cavity Young equation, eq 70, for a ) aL shows that the line tension for z ) 0 is related to the line tension at z as follows
(
)
τ j 1 - P1)aL 1 + z + γa)a z - L ) (P La aL x L L j 1,z)0 - P1)aL,z)0 (P
τL,z)0 (87) aL,z)0
j 1 depend upon z. For λ f ∞, we where, in general, aL and P also expect aL f ∞ for any choice of z (again, given that z/λ f 0), so that the above approaches
j 1 - P1)aL (P
τL τ j 1,z)0 - P1)aL,z)0 - L,z)0 (88) ) (P aL aL,z)0
with xL f aL in the large cavity limit. For z/λ f 0, one again expects that aL,z)0 and aL become identical (to a negligible order of z/λ) as all cavities approach the hemispherical limit. This implies that τL f τL,z)0 for any z. While by no means a rigorous proof, the cavity Young equation is at least consistent with the argument that τ∞ should be independent of z. If τ∞ is to be independent of z, eq 83 requires that δχ∞ be of the following form
δχ∞ ) b -
z 2
(89)
where b is an unknown constant, indicating that
τ∞ ) γ∞(2δ∞ - b)
(90)
Interestingly, τ∞ is related to the properties of the fully spherical cavity, although in the planar limit. This also suggests that in the limit of λ f ∞
aT ) λ + δχ∞ ) λ + b -
z 2
(91)
Of course, the validity of these relations remains to be tested, which may require somewhat intensive molecular simulations. Unfortunately, eq 90 cannot be rigorously verified within the framework of SPT. While SPT of a cavity growing within a bulk fluid can be used to generate estimates of γ∞ and δ∞, any current version of I-SPT, being based on the expansion of χλ provided in eq 60, requires as input an assumed form of the dependence of δχ∞ on z. An expression for δχ∞ leading to eq 90 has already been implicitly invoked within I-SPT.3,5 Although the predictions of the works of cavity growth were in good agreement with simulation results, suggesting that eq 90 is a reasonable assumption, such agreement does not provide independent confirmation of the behavior of τ∞. 5. Conclusions We presented a thermodynamic analysis of the boundary layers formed by a cavity intersecting a hard, structureless wall. We employed the “dividing surface” approach of Gibbs and others, obtaining thermodynamically consistent expressions for the line tension of a cavity that depend on the choice of the dividing surface.
In the end, our analysis yielded some interesting insights into the behavior of the line tension of a cavity. First, various geometric and physical arguments plausibly suggest that the line tension approaches a common value for any center location z as the radius of the cavity approaches macroscopic sizes. I-SPT cannot, unfortunately, independently verify this common limit. Molecular simulations could instead be used, though the required cavity sizes may make such a study computationally prohibitive. Second, we argued that the line tension is expected to diverge as the cavity ceases to intersect the wall. This divergence has apparently been overlooked in the past, and it is presently unclear how this singularity impacts the onset of the Derjaguin limit. Further examination of the boundary thermodynamics of cavities is necessary to fully understand the interplay of the line tension and the Derjaguin limit. In a future paper, we will utilize I-SPT and SPT to generate accurate estimates of the line tension for a range of cavity positions and bulk fluid densities. These studies will determine under what conditions the line tension is positive and when it becomes negative. Information about the linear adsorption assigned to the three-phase contact line will also be generated. We hope that a SPT-based analysis of the line tension of cavities will provide information that may also apply to the behavior of, for example, the line tension of sessile droplets. Acknowledgment. This paper is based upon work supported by the National Science Foundation under Grant No. 0133780. References and Notes (1) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959, 31, 369–380. (2) Corti, D. S.; Reiss, H. Mol. Phys. 1998, 95, 269–280. (3) Siderius, D. W.; Corti, D. S. Phys. ReV. E 2005, 71, 036141. (4) Siderius, D. W.; Corti, D. S. Phys. ReV. E 2005, 71, 036142. (5) Siderius, D. W.; Corti, D. S. Phys. ReV. E 2007, 75, 011108. (6) Mandell, M. J.; Reiss, H. J. Stat. Phys. 1975, 13, 107–112. (7) Herring, A. R.; Henderson, J. R. Phys. ReV. Lett. 2006, 97, 148302. (8) Oettel, M. Phys. ReV. E 2004, 69, 041404. (9) Hill, T. L. J. Phys. Chem. 1952, 56, 526–531. (10) Reiss, H. In Statistical Mechanics and Statistical Methods in Theory and Application; Landman, U. Ed.; Plenum: London, 1977; pp 99-140. (11) Kondo, S. J. Chem. Phys. 1956, 25, 662–669. (12) Ono, S.; Kondo, S. In Encyclopedia of Physics; Flugge, S. Ed.; Springer: Berlin, Germany, 1960; Vol. 10: Structure of Liquids, pp 134280. (13) Mandell, M. J.; Reiss, H. J. Stat. Phys. 1975, 13, 113–128. (14) Mandell, M. H. J. Chem. Phys. 1976, 65, 813–814. (15) Tolman, R. C. J. Chem. Phys. 1949, 17, 333–337. (16) Heying, M.; Corti, D. S. J. Phys. Chem. B 2004, 108, 19756–19768. (17) Siderius, D. W.; Corti, D. S. Ind. Chem. Eng. Res. 2006, 45, 5489– 5500. (18) Siderius, D. W.; Corti, D. S. J. Chem. Phys. 2007, 127, 144502. (19) Tully-Smith, D. M.; Reiss, H. J. Chem. Phys. 1970, 53, 4015– 4025. (20) Stillinger, F. H.; Cotter, M. A. J. Chem. Phys. 1971, 55, 3449– 3458. (21) Heni, M.; Lowen, H. Phys. ReV. E 1999, 60, 7057–7065. (22) Auer, S.; Frenkel, D. Phys. ReV. Lett. 2003, 91, 015703. (23) Siderius, D. W. Ph.D. Thesis, Purdue University, West Lafayette, IN, 2007. (24) Navascue´s, G.; Tarazona, P. Chem. Phys. Lett. 1981, 82, 586–588. (25) Rusanov, A. I.; Shchekin, A. K.; Tatyanenko, D. V. Colloids Surf., A 2004, 250, 263–268. (26) Roura, P.; Fort, J. J. Colloid Interface Sci. 2004, 272, 420–429. (27) Roura, P. Eur. J. Phys. 2007, 28, L27–L32. (28) Pellicer, J.; Manzanares, J. A.; Mafe´, S. Am. J. Phys. 1995, 63, 542–547. (29) Djikaev, Y.; Widom, B. J. Chem. Phys. 2004, 121, 5602–5610. (30) Widom, B. Physica A 2006, 372, 169–172. (31) Varea, C.; Robledo, A. Phys. ReV. A 1992, 45, 2645–2648. (32) Szleifer, I.; Widom, B. Mol. Phys. 1992, 75, 925–943.
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