J. Phys. Chem. B 2008, 112, 11981–11989
11981
Equilibrium of Multi-Phase Systems in Gravitational Fields Ovidiu Voitcu and Janet A. W. Elliott* Department of Chemical and Materials Engineering, 536 Chemical and Materials Engineering Building, UniVersity of Alberta, Edmonton, Alberta, Canada T6G 2G6 ReceiVed: June 7, 2007; ReVised Manuscript ReceiVed: May 23, 2008
Four necessary conditions for equilibrium of an isolated solid-liquid-vapor system in a gravitational field were derived by Ward and Sasges (1998) in a unified setting, by using an entropy maximization approach, and under the assumption that the liquid-vapor surface tension does not depend on elevation. These are thermal equilibrium, the Laplace and Young equations, and a condition on the chemical potentials of the components present in the system. Gibbs (1876) had obtained the Young equation in a derivation separate from the derivation of the other three conditions and by using an energy minimization approach. However, Gibbs had derived a more general form of the Laplace equation than Ward and Sasges’s. Gibbs’s equation contained a term expressing the contribution of the variation of surface tension with elevation. This equation has since been neglected by most of the scientific community. In the present paper, the same approach as Ward and Sasges’s is used to derive, in an unified setting, the conditions for equilibrium of an isolated solid-liquid-vapor system in a gravitational field but under the assumption that the liquid-vapor surface tension may depend on elevation. The four well-known conditions for equilibrium are obtained, with Gibbs’s generalized Laplace equation instead of the classical Laplace equation. The derivations in this paper were carried out for two different system geometries, namely, for a sessile drop and for a conical capillary tube, and similar conditions for equilibrium were obtained. I. Introduction In his theory of capillarity,1 Gibbs computed the necessary conditions for equilibrium of multiphase fluid systems. For a generic two-phase [for instance, liquid (L)-vapor (V)] fluid system containing an interface and subject to the influence of gravity,2 Gibbs required that the variation of the total energy of the system be zero, subject to constant entropy of the isolated system, and divided this variation into three mutually independent terms. At equilibrium, each of these three terms must vanish. Three equations are obtained, each generating a necessary condition for equilibrium. From the first equation, the thermal equilibrium condition is obtained:
T )T j
for all phases j
(1)
In general, j ) L, V denotes the bulk phases, j ) LV, SL, SV are the corresponding interfaces (S being a solid phase), and Tj is the temperature in phase j. The second equation represents the condition of mechanical equilibrium, from which a Laplacetype equation is derived:
(
PLI - PVI ) γLV
)
1 1 dγLV + + cos φ R1 R2 dz
Figure 1. Sessile drop in a gravitational field.
(2)
where z, PLI , PVI , γLV, R1, and R2 denote the elevation, the liquid and vapor pressure, the surface tension, and the principal radii of curvature, respectively, at a point of the LV interface. The normal vector in a point of the interface makes an angle φ with the vertical direction (see Figure 1). Note that the last term in eq 2 represents the variation of surface tension with elevation, projected along the direction normal to the LV interface. The third equation represents the condition of chemical equilibrium, * Corresponding author. E-mail:
[email protected]. Phone: 1-780492-7963.
from which a condition on the chemical potentials µji of the r chemical components (each denoted by i) present in the system, existing in the different phases j, is found:
µij + Wigz ) λi∗
for all phases j and 1 e i e r
(3)
where Wi is the molar mass of chemical component i and λ*i are constants. In general, j ) L, V, LV, SL, SV. In a separate derivation, when considering a three-phase system in which one of the phases is solid3 (for instance, a solid–liquid–vapor system), Gibbs derived the Young equation as a condition for equilibrium of the three-phase contact line: SL LV γSV b - γb ) γb cos θ
γ jb
(4)
where θ is the contact angle and are the interfacial tensions of the three surface phases j, computed at the three-phase contact line. Even though Gibbs originally derived eq 2 as one of the conditions for equilibrium, he immediately added2 “within moderate differences of level, we may regard [the surface
10.1021/jp074424s CCC: $40.75 2008 American Chemical Society Published on Web 08/27/2008
11982 J. Phys. Chem. B, Vol. 112, No. 38, 2008
Voitcu and Elliott
tension] as constant” (hence, dγLV/dz ) 0). Thus, eq 2 was reduced to what is today referred to as the Laplace equation:
(
PLI - PVI ) γLV
1 1 + R1 R2
)
(5)
Equation 2 seems to have remained unnoticed by all but a few researchers,4 probably because of Gibbs’ simplifying assumption and because of the fact that it appears only once in a paper containing 700 equations.5 In general, the LV surface tension is assumed to be constant with respect to elevation, and the simplified eq 5 is used. The very few researchers who have suggested other possible modifications to the classical Laplace equation (eq 5) have done so in contexts that are not relevant to the current study. There are a number of authors who have looked at this issue from non-Gibbsian viewpoints, either by adding an explicit curvature dependence in the equation of state of the system6–9 or by viewing the surface tension as anisotropic.10–12 In another study, a modified Laplace equation in parametric form was obtained by adding a term describing the interaction of a sessile drop with another colloidal particle.13 These Laplace-type equations are usually translated into systems of ordinary differential equations, which are integrated to compute the shape of various interfaces.13–16 Contact angle and surface tension values for physical systems can then be estimated by fitting such mathematically computed interface shapes to experimental ones.16 In ref 4, eq 2 is obtained in isolation from the other conditions for thermodynamic equilibrium as a consequence of the fact that the value of the work of deformation of a surface element should be independent of the location of the Gibbs dividing surface.1 As mentioned before, Gibbs derived eqs 1–3 as necessary conditions for equilibrium of an isolated two-phase fluid system in a gravitational field,2 and then, in a separate section of ref 1, he derived the Young equation (eq 4) as a necessary condition for equilibrium for a three-phase system in which one of the phases is solid.3 He then simplified eq 2 into the Laplace equation (eq 5) by assuming the LV surface tension to be independent of elevation. By using a standard calculus-of-variations procedure, Ward and Sasges17 derived eqs 1 and 3–5 in a unified setting as necessary conditions for equilibrium for an isolated three-phase (SLV) system in a gravitational field, by assuming a constant LV interfacial tension. The authors considered a liquid and a vapor phase coexisting in a closed straight capillary tube in a gravitational field parallel to the axis of the tube and assumed the LV interface to be axisymmetric. The equations obtained were used to compute the shapes of the two LV interfaces that limit a vapor bubble in the capillary tube. In the present paper, Ward and Sasges’s approach for obtaining the necessary conditions for equilibrium of an isolated three-phase system in a gravitational field is followed. The equilibrium is characterized by an extremum in the total entropy, subject to constraints of constant total energy and constant total number of kilomoles of each chemical component present in the system. Ward and Sasges’s approach appears different from Gibbs’s, which was based on the minimization of the total energy of the system. However, extremization of energy subject to constant entropy and extremization of entropy subject to constant energy are formally equivalent. In the present work, the derivation in ref 17 is repeated with the assumption that the LV surface tension is allowed to vary with elevation. In addition, the most general form of the variations of intensive parameters about the equilibrium state is considered when following the calculusof-variations procedure proposed by Ward and Sasges. Finally,
the differential geometry of the LV interface is computed in a rigorous manner. The procedure for obtaining the necessary conditions for equilibrium is presented in detail for an axisymmetric sessile drop enclosed in a cylinder and situated in a gravitational field. Sessile drops have been extensively studied from a theoretical as well as a practical perspective, because they provide useful insight into various interfacial phenomena. The necessary conditions for equilibrium obtained in this paper are given by eqs 1–4. In other words, equilibrium conditions of the same generality as in Gibbs’s study1 were obtained, but in a unified setting, similar to that in Ward and Sasges’s paper.17 The main contribution of the present paper consists in the unified framework in which all four eqs 1–4 are derived. The same procedure as that for a sessile drop was repeated for a closed conical capillary tube in a gravitational field; only the significant differences compared to the sessile-drop case are pointed out herein. The study of capillary rise in a conical tube has become of interest lately.18,19 As discussed by Asekomhe and Elliott,19 who made improved calculations from the experimental data of ref 18, conical capillary tubes in a gravitational field are important from a practical perspective, providing a simple and inexpensive means of computing the contact angle only on the basis of a measurement of the maximum liquid elevation in the tube. As in ref 17, the computations in ref 19 were also carried out by assuming that the LV surface tension does not vary with height. From a mathematical point of view, a straight capillary tube is a particular case of conical capillary tube, in which the angle β between the tube wall and the horizontal plane is equal to π/2. Thus, the conditions for equilibrium for a straight capillary tube can be obtained from the equilibrium conditions for a conical capillary tube by simply setting β ) π/2. These conditions turn out to be similar to those obtained for a sessile drop. II. Equilibrium of a Sessile Drop in a Gravitational Field Consider the following system in a gravitational field (see Figure 1). Assume that r chemical components existing in liquid as well as vapor phases are enclosed in a cylindrical container that is closed to mass and energy transport. The liquid phase consists of an axisymmetric sessile drop. The gravitational field is parallel to the symmetry axis of the drop, which coincides with the symmetry axis of the cylinder. Also assume that the LV interface is placed such that the surface tension does not depend explicitly on interface curvature (i.e., assume Gibbs surface-of-tension1). For the SL and SV interfaces, Gibbs’s dividing surface approximation is used, and the positions of the dividing surfaces are chosen such that there is no adsorption of the solid component.17 Hence, the entropy per unit area sj of each surface phase is assumed to depend only on the internal energy per unit area uj of that phase and the surface excess number of kilomoles per unit area nji of each chemical component i (1 e i e r):
sj ) sj(uj, nij, for 1 e i e r), j ) LV, SL, SV
(6)
(meaning sj ) for each j ) LV, SL, SV). Similarly, the entropy per unit volume sj of each bulk phase is assumed to depend only on the internal energy per unit volume uj of that phase and the number of kilomoles per unit volume nji of each chemical component i (1 e i e r): sj(uj, nj1,
nj2,...,
nrj )
sj ) sj(uj, nij, for 1 e i e r), j ) L, V
(7)
It is well-known that the intensive properties satisfy the equation:
Equilibrium of Multi-Phase Systems r µij j 1 j dni ds ) j du j T i)1 T
∑
j
J. Phys. Chem. B, Vol. 112, No. 38, 2008 11983
j ) L, V, LV, SL, SV
(8)
In addition, the Euler relations for surface and bulk phases, respectively, can be written as:
conditions for equilibrium (eqs 1 and 3) eventually obtained by Gibbs are accepted as true, by differentiating eqs 3, 9 and 10 with respect to z and using eqs 1 and 8, it follows that
dµij ) -gWi dz
0 e z e H, 1 e i e r, j ) L, V, LV, SL, SV
r
γj ) uj - Tjsj -
∑ µijnij
j ) LV, SL, SV
i)1
r
∑ µijnij
-Pj ) uj - Tjsj -
j ) L, V
(15)
(9)
(10)
r r dµij dγj ) - nij ) g nij(z)Wi dz dz i)1 i)1
∑
i)1
The potential energy, per unit volume for the bulk phase and per unit area for the surface phase, due to the presence of a gravitational field, is given by:17 r
ψj ) zg
∑ Winij
j ) L, V, LV, SL, SV
(11)
i)1
where g is the magnitude of the gravitational acceleration. The bottom of the cylinder is taken as the reference elevation z ) 0. Throughout the following analysis, an orthogonal frame of reference is chosen such that the z-axis coincides with the symmetry axis, pointing upward (that is, opposite to the gravitational field) and the x- and y-axes are included in the horizontal plane determined by the bottom of the cylinder (see Figure 1). Note that if φj is one of the following intensive properties characterizing each phase j, uj, ψj, sj, or nji for some i, 1 e i e r, the total value Φ of that property for the entire system can be computed as follows:
Φ)
∫V
L
φL dV +
∫V
V
φV dV +
∫A
SL
∫A
LV
φLV dA +
φSL dA +
∫A
SV
φSV dA (12)
where Vj, j ) L,V denote the volumes occupied by each bulk phase, and Aj, j ) LV, SL, SV, denote the LV, SL, and SV interfacial areas, respectively. Thus, the total energy E, the total entropy S, and the total number of kilomoles Ni of each chemical component i can be expressed by eq 12 with φj ) uj + ψj, φj ) sj, and φj ) nji, respectively, j ) L, V, LV, SL, SV. As in ref 17, because of the axisymmetry of the system and the fact that the only field acting on the system is parallel to the axis of symmetry, it will be assumed that within a given phase, all intensive properties can at most depend only on the elevation z (assured by combination of equilibrium conditions given by eqs 1 and 3 with the Gibbs phase rule). In other words, it is assumed that within each phase, the intensive properties have constant values throughout each horizontal (that is, x-y) plane. Thus,
uj ) uj(z),
sj ) sj(z),
nij ) nij(z)
0 e z e H,
1 e i e r, j ) L, V, LV, SL, SV (13) and eqs 6 and 7 can be written as follows:
sj ) sj(z) ) sj(uj(z), nij(z), for 1 e i e r), 0 e z e H,
∑
j ) LV, SL, SV (16)
r r dµij dPj nij ) ) -g nij(z)Wi dz i)1 dz i)1
∑
∑
j ) L, V
(17)
Given the similarity of eqs 16 and 17 and the generally accepted fact that pressure does vary with height in a gravitational field, the equilibrium condition for the chemical potentials obtained by Gibbs leads us to the fact that, although the effect may be small in a given circumstance, the LV surface tension must vary with height as well. This assumption will be used throughout the present work and is mathematically more general than the one used in most of the literature, namely, that LV surface tension is independent of elevation. Indeed, (dγj/dz)(zˆ) is equal to zero at some elevation zˆ if and only if ∑i r) 1 Winji(zˆ) ) 0 (which is not in general probable for any curved surface or a flat surface in a multicomponent system). It remains to calculate the magnitude of this omitted effect for a particular circumstance before one can be sure it can be neglected. The dependence of intensive properties on elevation was used as a fundamental assumption in ref 17. However, Ward and Sasges assumed that LV interfacial tension does not vary with elevation. In the present paper, Ward and Sasges’s procedure will be repeated by assuming that γj ) γj(z) for every interfacial phase j. It is assumed that the shape of the LV interface is uniquely determined by the values of the following intensive properties throughout the system: sj(z), uj(z), nji(z), 0 e z e H, 1 e i e r, j ) L, V, LV, SL, SV. Thus, the drop height z0 and half-width x0, as well as the contact radius xb, will be thought of as functionals of sj, uj, nji, 1 e i e r, j ) L, V, LV, SL, SV, viewed as functions of z, 0 e z e H:
z0 ) z0(sj(z), uj(z), nij(z), for 0 e z e H, 1 e i e r, j ) L, V, LV, SL, SV) (18) x0 ) x0(sj(z), uj(z), nij(z), for 0 e z e H, 1 e i e r, j ) L, V, LV, SL, SV) (19) xb ) xb s (z), u (z), nij(z),
(
j
j
for 0 e z e H, 1 e i e r, j ) L, V, LV, SL, SV) (20)
Note that for given values of intensive properties throughout the system, the LV interface can be described by the following equation:
x2 + y2 ) X2I (z)
0 e z e z0, with XI(0) ) xb, XI(z0) ) 0
j ) L, V, LV, SL, SV (14)
(21)
where H is the height of the surrounding solid cylinder. It will also be assumed that the interfacial tension γj corresponding to each surface phase j is a function of the elevation in the field, γj ) γj(z). Gibbs derived the conditions of equilibrium (eqs 1–3) for a generic two-phase system in the presence of gravity, under this general assumption. In the following, it will be shown that this assumption is reasonable. Indeed, if the well-known
where XI is a function of the elevation z. For instance, in the case of a spherical LV interface with radius R and contact angle θ, XI would have the form
XI(z) ) √R2 - (z + R cos θ)2
(22)
Throughout this paper, for every fixed zˆ, 0 e zˆ e H, XI(zˆ) will be thought of as a functional of sj, uj, nji, 1e i e r, j ) L, V,
11984 J. Phys. Chem. B, Vol. 112, No. 38, 2008
Voitcu and Elliott
Dx ) √ExGx - F2x ) XI(z)[1 + ˙ X2I (z)]
LV, SL, SV, viewed as functions of the elevation z, 0 e z e H:
1⁄2
Thus, if φLV is an intensive property characterizing the LV phase,
XI(zˆ) ) XI(zˆ;sj(z), uj(z), nij(z), for 0 e z e H, 1 e i e r, j ) L, V, LV, SL, SV) (23)
∫A
LV
One may set, by definition,
XI(zˆ) ) 0
for z0 e zˆ e H
(24)
By performing simple volume and surface integrations, eq 12 becomes
Φ)π
∫0z X2I (z) φL(z) dz + π∫0z [L2 - X2I (z)]φV(z) dz + H πL2∫z φV(z) dz + ∫A φLV dA + πx2bφSL(0) + 0
0
φLV dA ) 2π
π(L2 - x2b)φSV(0) + 2πL
∫0
H
φSV(z) dz + πL2φSV(H) (25)
where L is the radius of the surrounding solid cylinder. The structure of the integrals in the above equation is similar to that presented in ref 17. For the integral over the LV interface, a parametrization different from Ward and Sasges’s will be used in this paper. To this end, the differential geometry of the LV interface needs to be specified in a rigorous manner.
x ) XI(z)
0 e z e z0, with XI(0) ) xb, XI(z0) ) 0 (26)
If x ) (x, y, z) is the position vector of an arbitrary point on the LV interface, the following parametrization can be used to describe this surface:
x ) (XI(z) cos(R), XI(z) sin(R), z)
0 e z e z0, 0 e R e 2π
˙ (z) sin(R), ˙ x¨zR ) (-X XI(z) cos(R), 0) (33) I Hence, the coefficients of the second fundamental form of the surface are
ex )
〈x¨zz, ˙xz, ˙xR 〉 Dx
x˙R ) (-XI(z) sin(R), XI(z) cos(R), 0) (28) where
I
I
fx )
˙ xzR )
∂2x and so forth. ∂z ∂ R (29)
Hence, the coefficients of the first fundamental form of this surface have the following expressions:
Ex ) 〈˙ xz, ˙ xz 〉 ) 1 + ˙ X2I (z),
〈x¨zR, ˙xz, ˙xR 〉 Dx
) 0 (34)
where 〈 , , 〉 denotes the mixed product. Because of the axisymmetry of the surface, it can be assumed without loss of generality that R ) 0, in other words, that the current point on the surface belongs to the x-z plane. Any vector in the tangent plane to the surface at such a point x ) (XI(z), 0, z) is given by
v ) Vz˙ xz + VR˙ xR ) (Vz˙ XI(z), VRXI(z), Vz)
(35)
The surface can be sectioned by the normal plane containing the vector v, obtaining a planar curve. The radius of curvature of this curve at the point x ) (XI(z), 0, z) has the expression:
|
exV2z + 2fxVzVR + gxV2R 1 ) Rv ExV2z + 2FxVzVR + GxV2R
1 Rv(z)
∂XI (z), ∂z
-X¨I(z) [1 + ˙X2(z)]1 ⁄ 2
|
(36)
The radius of curvature of the LV interface in the present study has the following expression:
˙ xz ) (˙ XI(z) cos(R), ˙ XI(z) sin(R), 1),
˙ XI(z) )
)
XI(z) 〈x¨RR, ˙xz, ˙xR 〉 gx ) ) Dx [1 + ˙X2(z)]1 ⁄ 2
The first-order partial derivatives of x with respect to z and R are given by
∂x , ∂R
(32)
x¨RR ) (-XI(z) cos(R), - XI(z) sin(R), 0)
(27)
˙ xR )
0
x¨zz ) (X¨I(z) cos(R), X¨I(z) sin(R), 0)
III. Differential Geometry of the Liquid-Vapor Interface Note that because of the axisymmetry of the drop, the shape of the LV interface is uniquely determined by the right half (x g 0) of the curve obtained by sectioning the surface through the x-z plane. The drop surface is obtained by completely rotating this curve around the z-axis. The curve can be described by the following equation:
∫0z φLV(z) XI(z)[1 + X2I (z)]1⁄2 dz
The principal radii of curvature at an arbitrary point on the surface will now be computed. To this end, the second-order partial derivatives of x with respect to z and R are needed. These are given by
LV
0
(31)
Gx ) 〈˙ xR, ˙ xR 〉 ) X2I (z), xz, ˙ xR 〉 ) 0 (30) Fx ) 〈˙
where 〈 , 〉 denotes the dot product. The discriminant of the surface will have the expression:
)
-X¨I(z)V2z + XI(z)V2R 1 · [1 + ˙X2I (z)]1 ⁄ 2 [1 + ˙X2I (z)]V2z + X2I (z)V2R
(37)
It can be verified that in the sessile-drop geometry, XI(z) is a concave-down function, hence, X¨I(z) e 0, 0 e z e z0. This assumption was used in the above equation. Denote by R1 the radius of curvature corresponding to a normal section of the surface by a plane including the symmetry axis (the first principal radius of curvature). In the case R ) 0, this means 〈v, j 〉 ) 0
(38)
where i, j, and k are the unit vectors corresponding to the x-, y-, and z-axes, respectively. From eqs 35 and 38, it follows that VR ) 0; therefore, R1 will be given by:
Equilibrium of Multi-Phase Systems
1 R1(z)
)
J. Phys. Chem. B, Vol. 112, No. 38, 2008 11985
-X¨I(z) [1 + ˙X2(z)]3 ⁄ 2
(39)
I
Denote by R2 the radius of curvature corresponding to a normal section of the surface by a plane that is orthogonal to the plane considered above (the second principal radius of curvature). In the case R ) 0, this means 〈v, i〉 ) 0; hence, Vz ) 0, and thus,
1 1 ) 1⁄2 R2(z) X (z)[1 + ˙ X2(z)] I
(40)
I
The curve obtained by sectioning the LV interface with the plane y ) 0 can be parametrized as follows:
x ) XI(z), y ) 0, 0 e z e z0
(41)
The tangent and normal unit vector at a point of this planar curve have the following expressions:
T(z) )
(
˙ XI(z) 1 , 0, 1⁄2 2 ˙ ˙ [1 + X (z)] [1 + X2(z)]1 ⁄ 2 I
N(z) )
(
I
)
Figure 2. Conical capillary tube in a gravitational field.
)
˙ XI(z) -1 , 0, (42) 1 ⁄ 2 2 [1 + ˙X (z)] [1 + ˙X2(z)]1⁄2 I
I
Denote by φ ) φ(z) the angle between the unit vectors N(z) and (-k). It follows that
cos φ(z) ) 〈N(z), -k 〉 )
˙ (z) -X I 2 ˙ [1 + X (z)]1 ⁄ 2
(43)
I
establish what relationships between these intensive properties can be obtained. The above quantities characterize an equilibrium state if the total entropy S of the system is an extremum subject to constant total energy E and total number of kilomoles Ni of each chemical component i, 1 e i e r. Mathematically, S, E, and Ni, 1 e i e r are viewed as functionals of sj, uj, nji, 1 e i e r, j )L, V, LV, SL, SV, which in turn are viewed as functions of z, 0 e z e H. By introducing the Lagrange multipliers λi, 1 e i e r, the following quantity can be defined:
Note that the mapping zfφ(z) is decreasing with
φ(0) ) θ,
r
φ(z0) ) 0,
0 e φ(z) e θ e π
S* ) S - λ0E -
(44)
∑ λiNi
(48)
i)1
where θ is the contact angle. Therefore,
+1 sin φ(z) ) ˙ [1 + X2(z)]1 ⁄ 2
(45)
The expression of S* is given by eq 47 with φj ) τj, j )L, V, LV, SL, SV, where
I
r
It can be shown that because of the axisymmetry of the surface, the center of curvature corresponding to R2 is always on the symmetry axis. In consequence,
τj ) sj - λ0(uj + ψj) -
∑ λinij
(49)
i)1
After having computed the differential geometry of the LV interface, the calculation of the necessary conditions for equilibrium for the considered system can now be carried out.
To obtain the necessary conditions for equilibrium, a standard calculus-of-variations procedure will be used. A method similar to the one employed by Ward and Sasges will be followed, but more general variations about the equilibrium state than those proposed in ref 17 will be considered in this paper. More specifically, the following variations about the equilibrium state of the system will be considered:
IV. Necessary Conditions for Equilibrium
˜ uj(z) ) u j(z) + εj0ηj0(z),
XI(z) sin φ(z) ) R2(z)
(46)
0 e z e H, 1 e i e r, j ) L, V, LV, SL, SV (50)
Substituting eq 32 in eq 25 yields
∫0z X2I (z) φL(z) dz + π∫0z [L2 - X2I (z)]φV(z) dz + z H 1⁄2 X2I (z)] dz + πL2∫z φV(z) dz + 2π∫0 φLV(z) XI(z)[1 + ˙
Φ)π
0
0
0
0
πx2bφSL(0) + π(L2 - x2b)φSV(0) + 2πL
˜ nij(z) ) nij(z) + εijηij(z),
∫0H φSV(z) dz + πL2φSV(H) (47)
The goal of this paper is to find the necessary conditions for equilibrium for the considered physical system. In other words, if jsj(z), ujj(z), njji(z), 0 e z e H, 1 e i e r, j ) L, V, LV, SL, SV describe an equilibrium state of the system, the goal is to
εji
where are small positive real numbers and ηji are functions of z, 0 e z e H. According to the calculus-of-variations procedure, all uj, nji, 1 e i e r, j )L, V, LV, SL, SV will be viewed as independent of each other, that is, one function out of these may be varied at any one time, keeping all other functions at their equilibrium values. In the following discussion, S˜*, z˜0, x˜0, x˜b, X˜I, R˜1, R˜2, and so forth, will denote the quantities associated with a variation in only one of all uj, nji, 1 e i e r, j )L, V, LV, SL, SV about the equilibrium state, all other functions being kept at their equilibrium values. If jsj(z), ujj(z), njji(z), 0 e z e H, 1 e i e r, j )L, V, LV, SL, SV describe an equilibrium state, then
11986 J. Phys. Chem. B, Vol. 112, No. 38, 2008
( ) ˜* ∂S ∂εij
) 0,
Voitcu and Elliott
0 e i e r, j ) L, V, LV, SL, SV (51)
εij)0
πL
∫
∫z˜
˜τ (z) dz + π L
H V
0
˜τ (z) dz + 2π
∫0
z˜0
[L - ˜X2I (z)]˜τV(z) dz + 2
∫
˜z0 LV ˜τ (z) 0
[
]
˙2 ˜ XI (z) XI(z) 1 + ˜
πx˜2b˜τSL(0) + π(L2 - ˜x2b)˜τSV(0) + 2πL
1⁄ 2
˜τ
∫0
1⁄ 2
2 I
(
)
˙2 ∂τ˜LV j˜ ηX 1+˜ XI j i I ∂ni
[ (
(52)
∫0
1⁄ 2
dz +
)
˙2 1+˜ XI
-1⁄ 2
∂2˜ XI ∂εij ∂ z
∫
∂εij ∂ z
(1 + ˙˜X )
[
-1⁄ 2
2 I
(
)
˙ ˙2 ˜τLV˜ XI˜ XI 1 + ˜ XI
∫0
˜z0
-1⁄ 2
( ) ]
dz (53)
˜* ∂S )π ∂εij
∂εij
[
π
˜ ∂X I j i
According to eqs 43 and 44, the first term on the right-hand side of eq 54 is equal to:
(55)
i
Also, note that
[
(
-1⁄ 2
) ] ) ∂τ∂z˜ ˜X ˙˜X (1 + ˙˜X ) + ˙ ˙ ¨ ˙ ˜ X (1 + ˜ X ) + ˜τ ˜ X˜ X (1 + ˜ X) ˙ ˜¨ ˙ ˜τ ˜ X˜ X X (1 + ˜ X)
˙2 ∂ LV˜ ˙ ˜τ XI˜ XI 1 + ˜ XI ∂z
2 I
I I
-1⁄ 2
˜τLV 2I
2 I
-1⁄ 2
LV
I I
LV
∫0
˜z0
[( )
i
]
˜2 ∂X 1 I 1 ∂τ˜LV + + ˜ dz (58) cos φ j ˜ ˜ ∂z ∂εi R1 R 2
˜τLV
V
0
∫z˜
H
0
i
∂τ˜V j ηi dz + 2π ∂nij
∫0
˜z0
(
By using eqs 39, 40, and 43–46, it follows that
2 I
∫0
˜z0
)
˙2 ∂τ˜LV j˜ ηX 1+˜ XI j i I ∂ni
1⁄ 2
dz +
∂τ˜SL ( ) j( ) ˜SV ( ) j( ) 2 ∂τ 2 0 η 0 + π L ˜ x 0 ηi 0 + ( ) i b ∂nij ∂nij SV
SV
i
i
˜ j ∂τ˜ η dz + πL2 j (H)ηij(H) + ∫0H ∂τ∂n j i ∂n
[
(
˜τL - ˜τV + ˜τLV
∂x˜2b ∂εij
)
+
]
˜2 ∂X 1 I 1 ∂τ˜LV + + ˜ dz (59) cos φ j ˜ ˜ ∂z ∂εi R1 R 2
In the above equation, by using eqs 8 and 49, the following equalities hold: j µ ˜j ∂τ˜ j ^ * j and ∂τ˜ ) - i - λ gzW - λ ) 0 if j 0 i i ˜j ∂nij ∂nij T
(60)
^
2 I
2 I I I
dz +
The expression of ∂S˜*/∂εj0 is given by a formula similar to eq 59, where i ) 0 and all ∂τ˜ ˆj/∂nji are replaced by ∂τ˜ ˆj/∂uj. In that case,
-1⁄ 2
LV
1⁄ 2
^
˜2 ∂X ∂x˜2 1 LV( ) I 1 ˜ b ˜τ 0 cos φ ˜(0) j (0) ) ˜τLV(0) cos θ 2 2 ∂ε ∂εj i
)
(
˜] π[˜τSL(0) - ˜τSV(0) + ˜τLV(0) cos θ
) ] ∂ε dz (54)
(
∫0
˙2 ∂τ˜LV j˜ XI ηX 1+˜ j i I ∂ni
i
πL2
-1⁄ 2
dz )
˜z0
L
z)z˜0
˙ ˙2 ∂ LV˜ ˜ ˜τ XIXI 1 + ˜ XI ∂z
1⁄ 2
0
2πL
z)0
1
z˜ ˜ j ∂τ˜ η dz + π∫0 (L2 - ˜ X2I ) j ηij dz + ∫0z˜ ˜X2I ∂τ j i ∂n ∂n
˜ ∂ ∂XI dz ) ∂z ∂εj i ˜ ∂X I
˜τLV˜ XI ˜τLV˜ XI (57) ˜ ˜ R R
By differentiating eq 52 and using eq 58, it follows that
dz )
˜z0 LV˜ ˙ ˜τ XI˜ XI 0
)
˙2 1+˜ XI
1 2
πx˜2b
∂2˜ XI
-1⁄ 2
0
1⁄ 2
∂x˜2 1 LV( ) ˜ b+ ˜τ 0 cos θ 2 ∂εj
where the fact that X˜I(z˜0) ) 0 was used. By assuming that X˜I is a twice continuously differentiable function of (z, εji) (for 0 e z e z˜0 and for small εji > 0) and using integration by parts, it follows that
∫0˜z ˜τLV˜XI˜˙XI(1 + ˜˙X2I )
(
∫
˜z0 LV˜ ˜τ XI 0
)
(
∫
˜z0 LV˜ ˙ ˜τ XI˜ XI 0
∂ ∂εij
1⁄ 2
1⁄ ˜ 2 ∂X ˙ I 2 ˜ 1 + XI dz + j ∂εi
(
∫
˜z0 LV ˜τ 0
)
(
˙2 1+˜ XI
Combining eqs 53–57 yields
) ] dz )
˙2 ∂ LV˜ ˜τ XI 1 + ˜ XI j ∂εi
˜z0
dz )
I
2
In computing (∂S Ward and Sasges’s line of reasoning is followed, carefully keeping track of all derivatives that arise in the calculations. Because of its greater complexity, the partial derivative of the LV integral with respect to εji will first be computed:
(1 + ˜˙X )
LV
2 I
LV
2 I
I I
2 I
∫0H ˜τSV(z) dz + (H)
-1⁄ 2
LV
-3⁄ 2
dz +
πL ˜τ
˜z0 LV˜ ˜τ XI 0
-1⁄ 2
2 I
LV
⁄ ∂εji)εij)0,
˜z0
I
-1⁄ 2
˜*
∫
LV
2 I
I I
2 SV
∂ ∂εij
) ] ) - ∂τ∂z˜ ˜X cos φ˜ + ˙ ˙ ¨ ˙ ˜τ [(1 + ˜ X ) - (1 + ˜ X ) ] + ˜τ ˜ X˜ X (1 + ˜ X) ¨ ˙ ˙ ∂τ˜ ˜ ˜τ ˜ X˜ X [(1 + ˜ X ) - (1 + ˜ X ) ]) ˜+ X cos φ ∂z (
LV
z˜0 ˜ 2 XI (z) 0
2
-1⁄ 2
+1⁄ 2
Note that
˜ S*)π
[
˙2 ∂ LV˜ ˙ ˜τ XI˜ XI 1 + ˜ XI ∂z
∂τ˜ j ∂τ˜j 1 ) 0 if ^j * j and j ) j - λ0 j ˜ ∂u ∂u T
-3⁄ 2
(56)
(61)
By setting εji ) 0 in eq 59, all quantities will take on their equilibrium values. Thus,
Equilibrium of Multi-Phase Systems
0)
( ) ˜* ∂S ∂εij
π
)π εij)0
∫0
z0
X2I
J. Phys. Chem. B, Vol. 112, No. 38, 2008 11987 j
V
H
∂ni
2π πx2b
∫0
z0
0
˙ ∂τLV j ηiXI 1 + X2I ∂nij
(
)
1⁄ 2
SV
SV
i
i
π
∫0
[
V
(
LV
and comparing this equation with eqs 9 and 10 yields
τj )
Thus, eqs 64 and 65 become
γSV(0) - γSL(0) ) γLV(0) cos θ 1 1 PL(z) - PV(z) - γLV(z) + R1(z) R2(z)
( )
]( )
)
(
˜2 ∂X I ∂εij
A similar relation holds for i ) 0, with all ∂τjˆj/∂nji being replaced by ∂τjˆj/∂uj. Because of the arbitrary nature of all variations ηji, 0 e i e r, j ) L, V, LV, SL, SV, from eq 62, it follows that
∂τj ) 0, ∂uj
ˆj
∂τj )0 ∂nij
1 e i e r,
ˆj, j ) L, V, LV, SL, SV (63)
τSV(0) - τSL(0) ) τLV(0) cos θ 1 1 τL(z) - τV(z) + τLV(z) + + ( ) R1 z R2(z)
(
(64)
)
dτLV ( ) z cos φ(z) ) 0, dz
0 e z e z0 (65)
Because in the following discussion all quantities are understood to be at their equilibrium values, the bar notation will be omitted after this point. Then, eq 63 is equivalent to
1 - λ0 ) 0, Tj
-
µij j
T
- λ0gzWi - λi ) 0
Hence
1 )T λ0
j ) L, V, LV, SL, SV
(67)
(i.e., at equilibrium, the temperature is constant throughout the system) and
µij + Wigz ) -Tλi ) -
λi ) λi∗ λ0
V. Equilibrium in a Conical Capillary Tube in a Gravitational Field The above calculations can be repeated for different physical system geometries. For instance, consider a liquid and a vapor phase in an axisymmetric conical capillary tube closed to mass and energy transport (see Figure 2). The envelope of the tube makes an angle β, 0 < β < π with the horizontal plane. In particular, for β ) π/2, a straight capillary tube is obtained. Note that z0 e zb if the LV interface is concave up and zb e z0 if the LV interface is concave down. For simplicity, in the following discussion, only the first case (z0 e zb) will be considered. The reasoning can be repeated similarly for the case zb e z0. In the case of a capillary tube, the LV interface can be parametrized as in eq 21, this time with z0 e z e zb, and the right-hand half of the curve obtained by sectioning the surface with the x-z plane can be described by
z0 e z e zb, with XI(z0) ) 0, XI(zb) ) L(zb) ) Lb (73)
j ) L, V, LV, SL, SV (66)
Tj )
0 e z e z0 (72)
Note that eq 71 is actually the Young equation (eq 4), where γjb ) γj(0), j ) LV, SL, SV are computed at the three-phase contact line, and eq 72 is the generalized Laplace equation (eq 2). Thus, the coupled eqs 1–4 have been obtained as necessary conditions for equilibrium for the considered physical system [eqs 67, 68, 71, and 72].
x ) XI(z)
1 e i e r,
(71)
)
dγLV ( ) z cos φ(z) ) 0 dz
dz εij)0
(62)
ˆj
Pj -γj for j ) L, V and τj ) for j ) LV, SL, SV T T (70)
∂x˜2b + ∂εij εij)0
1 dτLV + + cos φ dz R1 R2 1
∑
j
∑
∫0H ∂τ∂nj ηij dz + πL2 ∂τ∂nj (H)ηij(H) +
τ -τ +τ L
∑
r
r
dz +
π[τSL(0) - τSV(0) + τLV(0) cos θ]
z0
)
r
1 1 µjnj (69) sj - uj + T T i)1 i i
SV ∂τSL ( ) j( ) 2 ∂τ 2 (0)ηij(0) + 0 η 0 + π L x ) ( i b ∂nij ∂nij
2πL
j
∂τV j ηi dz + ∂nij
∫0 (L2 - X2I ) ∂τ j ηij dz+πL2∫z z0
(
µi + Wigz j 1 τ ) s - uj + gz nijWi + ni ) T T i)1 i)1
∂τL j ηi dz + ∂nij
1 e i e r, j ) L, V, LV, SL, SV (68)
which is Gibbs’ condition on the chemical potentials (eq 3). Furthermore, by eqs 49, 66, and 67, it follows that
where L(z) is the radius of the horizontal section of the tube at elevation z, given by
L(z) ) L0 + z cot β
0ezeH
(74)
As in eqs 18–20, z0 and zb are considered to be functionals of sj, uj, nji, 1e i e r, j ) L, V, LV, SL, SV, viewed as functions of z, 0 e z e H, and as in eq 23, for every zˆ, 0 e zˆ e H, XI(zˆ) in eq 73 will again be thought of as a functional of sj, uj, nji, 1 e i e r, j ) L, V, LV, SL, SV, viewed as functions of the elevation z, 0 e z e H. If φj is a per-unit-volume (area) property characterizing each phase j, such as uj, ψj, sj, or nji for some i, 1 e i e r, the total value Φ of that property for the entire system will be given by the formula:
11988 J. Phys. Chem. B, Vol. 112, No. 38, 2008
Φ)π
Voitcu and Elliott
∫0z L2(z) φL(z) dz + π∫zz [L2(z) - X2I (z)]φL(z) dz + 0
b
0
π
∫zz X2I (z) φV(z) dz + π∫zH L2(z) φV(z) dz + b
0
0)
b
2π
( ) ∂S˜* ∂εij
)π εij)0
∫zz φLV(z) XI(z)[1 + ˙X2I (z)]1 ⁄ 2 dz +
π
0
2π csc β
∫z
2π csc β
b
∂τL ( ) j( ) z ηi z dz + ∂nij
L2(z)
(z) ηij(z) dz + ∫zz [L2(z) - X2I (z)] ∂τ ∂nj b
0
i
SV
L(z) φ
0
L
b
H
∫0z
(z)
dz+ π
∫0z L(z) φSL(z) dz+ 0
+ πL2(0) φSL(0) + πL2(H) φSV(H)
∫zz
b
0
2π
(75)
For a straight capillary tube, β ) π/2, and thus, L(z) ) L0 for all z, 0 e z e H. In consequence, eq 75 becomes similar to the expression obtained by Ward and Sasges in ref 17, except for the integral over the LV interface, which in this study is based on a different parametrization of the surface. The procedure previously described in this paper can be repeated similarly for a conical capillary tube, except that now, φ is defined as the angle between N(z) and k; therefore, eq 43 becomes
X2I (z)
V
∂τ ( ) j( ) z ηi z dz+π ∂nij
∫z
zb
0
V
(z) ηij(z) dz + ∫zH L2(z) ∂τ j ∂n b
i
[
]
˙ ∂τLV ( ) j( ) ( ) z ηi z XI z 1 + X2I (z) j ∂ni
1⁄ 2
dz +
SV
2π csc β
∫zH L(z) ∂τ∂nj (z) ηij(z) dz+ b
i
2π csc β πL2(0)
SL
∫0z
0
L(z)
∂τ ( ) j( ) z ηi z dz + ∂nij
∂τSL ( ) j( ) ∂τSV ( ) j( ) 2 ( ) 0 η 0 + πL H H ηi H + i ∂nij ∂nij
( ) ]
2πL(zb) csc β[τSL(zb) - τSV(zb) + τLV(zb) cos θ] ˙ XI(z) cos φ(z) ) 〈N(z), k 〉 ) [1 + ˙X2(z)]1 ⁄ 2
(76) π
I
As z varies between z0 and zb, φ(z) increases from φ(z0) ) 0 to φ(zb) ) φb e π/2. In the case of a conical tube, the contact angle θ can be most generally defined as the angle between the opposite of the tangent unit vector to the LV curve at the contact point and the unit vector (-cos βi - sin βk), parallel to the tube’s envelope, pointing downward. Thus:
cos θ ) 〈-Tb, -cos βi - sin βk 〉 ) 〈Tb, i 〉 cos β +
〈Tb, k 〉 sin β (77) where Tb ) T(zb) is given by eq 42. According to eqs 42, 45, and 76, it follows that
cos θ ) cos φbcos β + sin φb sin β ) cos(β - φb) (78) By defining the variations about the equilibrium state as in eq 50, S˜* is given eq 75 with φj ) τ˜ j, j )L, V, LV, SL, SV, with τj defined as in eq 49. The partial derivative of the LV integral with respect to εji is computed as in eqs 53 and 54, but this time, the limits of integration are z˜0 and z˜b instead of 0 and z˜0. The term in eq 55 now becomes
[
]
˙2 ˙ ˜˜ XI (z˜b) ˜τLV(z˜b)L ˜b) 1 + ˜ bXI(z
-1⁄ 2
˜ ∂X I ∂εij
(z˜b)
(79)
and differentiating X˜I(z˜b) ) L(z˜b) [see eq 73] yields
˜ ∂X I
(z˜b) ) ∂εij
[cot β - ˙˜X (z˜ )] ∂z∂ε˜ I
b
b j i
(80)
The condition for equilibrium in the case of the conical tube then becomes
∫zz
b
0
[
(
τV - τL + τLV
)
∂z˜b ∂εij
+
εij)0
1 1 dτLV + cos φ dz R1 R2
( ) ∂X˜2I ∂εij
dz (81) εij)0
Following a procedure similar to that for the sessile drop, the necessary conditions for equilibrium are eventually found to be eqs 1, 3, and 4, along with the following version of eq 2.
(
PVI - PLI ) γLV
)
1 1 dγLV + cos φ R1 R2 dz
(82)
Note that eq 82 can be viewed as a version of eq2 if φ is viewed as the angle between the direction normal to the interface and the vertical direction, both pointing toward the concavity of the interface, and if, by convention, the two principal radii of curvature are chosen to be positive or negative according to whether the concavity of the interface points toward the liquid or vapor phase, respectively. VI. Conclusions and Discussion In the present paper, the necessary conditions for equilibrium of an isolated SLV system in a gravitational field are computed under the assumption that the LV surface tension is a function of the elevation in the field, in the absence of line tension effects. The procedure proposed by Ward and Sasges17 in the case of a straight capillary tube is refined and used to compute conditions for equilibrium in the case of a sessile drop and of a conical capillary tube. When applying the calculus of variation procedure as in ref 17, a more general form of the variations of perunit-volume (area) properties about the equilibrium state is considered. Moreover, the differential geometry of the LV interface is more rigorously specified. Ward and Sasges’s contribution consisted in computing, in a unified setting and in the context of constant LV surface tension, four equilibrium conditions that Gibbs obtained in two separate derivations. The contribution of the present study consists in recomputing
Equilibrium of Multi-Phase Systems the generalized versions of the same equilibrium conditions, in the same unified paradigm, but including the possibility of LV surface tension dependence on elevation. The conditions for equilibrium obtained in this paper are identical to those computed by Gibbs in the most general setting, before making his simplifying assumption about the LV surface tension. One of the equations derived in the present study is the same generalized version of the Laplace equation originally derived by Gibbs and apparently neglected by most of the scientific community. The equation contains an extra term, which expresses the contribution of the variation of the LV surface tension with elevation. Even though Gibbs neglected this extra term, it is not clear whether it is indeed negligible in all experimental situations. In general, in experiments involving the interaction between gravity and capillarity (such as those carried out under near-zero gravity conditions, reported in ref 20), the negligibility of this extra term should be carefully reevaluated. Acknowledgment. This work was supported by the Canadian Space Agency under Contract 9F007-006050/001/SR and the Natural Sciences and Engineering Research Council of Canada. J.A.W.E. holds a Canada Research Chair in Interfacial Thermodynamics.
J. Phys. Chem. B, Vol. 112, No. 38, 2008 11989 References and Notes (1) Gibbs, J. W. Trans. Conn. Acad. 1876, 2, 108; in The Scientific Papers of Willard Gibbs, J.; Ox Bow: Woodridge, CT, 1993, Vol. I, pp 55-353. (2) Ref 1 pp 276-287. (3) Ref 1 pp 314-331. (4) Rusanov, A. I.; Prokhorov, V. A. Interfacial Tensiometry; Elsevier, 1996. (5) Ref 1 p 281, eq 613. (6) Pasandideh-Fard, M.; Chen, P.; Mostaghimi, J.; Neumann, A. W. AdV. Colloid Interface Sci. 1996, 63, 151. (7) Chen, P.; Susnar, S. S.; Pasandideh-Fard, M.; Mostaghimi, J.; Neumann, A. W. AdV. Colloid Interface Sci. 1996, 63, 179. (8) Gaydos, J. Colloids Surf. A 1996, 114, 1. (9) Abe, S.; Sheridan, J. T. Phys. Lett. A 1999, 253, 317. (10) Rusanov, A. I.; Shchekin, A. K.; Varshavskii, V. B. Colloid J. 2001, 63, 365. translated from Kolloidnyi Zhurnal 2001, 63, 401. (11) Rusanov, A. I.; Shchekin, A. K. Colloids Surf. A 2001, 192, 357. (12) Rusanov, A. I.; Shchekin, A. K. Colloid J. 2002, 64, 186 translated from Kolloidnyi Zhurnal 2002, 64, 209. (13) Miklavcic, J.; Attard, P. J. Phys. A 2001, 34, 7849. (14) Chatterjee, J. J. Colloid Interface Sci. 2003, 259, 139. (15) Berim, G. O.; Ruckenstein, E. J. Phys. Chem. B 2004, 108, 19330. (16) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (17) Ward, C. A.; Sasges, M. R. J. Chem. Phys. 1998, 109, 3651. (18) Jensen, W. C.; Li, D. Colloids Surf. A 1999, 156, 519. (19) Asekomhe, S. O.; Elliott, J. A. W. Colloids Surf. A 2003, 220, 271. (20) Ababneh, A.; Amirfazli, A.; Elliott, J. A.W. Can. J. Chem. Eng. 2006, 84, 39.
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