Models of Adsorption at a Line of Three-Phase Contact - The Journal

A form of the line analogue of the Gibbs adsorption equation is conjectured, and it is noted that the conjecture is in principle ... Mean-field densit...
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J. Phys. Chem. B 2006, 110, 22125-22132

22125

Models of Adsorption at a Line of Three-Phase Contact† B. Widom Department of Chemistry, Baker Laboratory, Cornell UniVersity, Ithaca, New York 14853-1301 ReceiVed: August 17, 2005

Two model density distributions at a line of three-phase contact for which the adsorptions are readily calculated are analyzed. One of them provides a numerical illustration of a recently found surprising fact about the thermodynamics of adsorption at such contact lines. A form of the line analogue of the Gibbs adsorption equation is conjectured, and it is noted that the conjecture is in principle testable by computer simulation and by experiment.

1. Introduction It was recently found, most surprisingly, that what had long been taken to be the analogue for adsorption at a line of threephase contact of the Gibbs adsorption equation for interfaces1 was in fact incorrect or at least incomplete.2 The purposes of the present paper are to illustrate, with model density distributions, the principles that were derived earlier, and to conjecture what might be the correct adsorption equation for the contact line. The issues are recalled in this Introduction and then illustrated in the subsequent sections. We start by recalling the classical Gibbs adsorption equation for an interface. Figure 1 shows schematically the diffuse interface between two phases R and β, with a Gibbs dividing surface parallel to the interface and at a location that may be chosen at will. The thermodynamic state of a system of c components is fixed by specifying c + 1 field variables, µ1, µ2, ..., µc+1, which may be the c chemical potentials and the temperature, for example. Here only c of these are independently variable because of the constraint of phase equilibrium. Let FRi and Fβi be the densities of the various components in the bulk R and β phases far from the interface. (If µi is the temperature, Fi is the entropy density.) Let ΓRβ i be the adsorption (surface excess per unit area) of i measured with respect to the arbitrarily chosen dividing surface, and in standard Rβ then be the adsorption of i when in particular notation, let Γi(j) the dividing surface is chosen to be that for which ΓRβ j ) 0. While the adsorptions depend on the choice of dividing surface, the interfacial tension, σRβ, does not; it is a uniquely measurable quantity. For simplicity, if c > 1, imagine µ3, ..., µc+1 held fixed, so that only µ1 and µ2 vary, but not independently because of the phase-equilibrium constraint. Let µ1, say, be the independently variable field. Then the Gibbs adsorption equation takes the form

dσRβ Rβ Rβ ) -ΓRβ 1 - Γ2 dµ2/dµ1 ) -Γ1(2) dµ1

(1.1)

Now consider three phases, R, β, and γ, in equilibrium at a line of common contact. As long as the radii of curvature of the contact line and of the three interfaces that meet at that line are greater than microscopic, that line and those interfaces may be taken to be straight and planar, respectively. Figure 2 shows, again schematically, the three-phase equilibrium. The contact line is perpendicular to the plane of the figure, where it appears †

Part of the special issue “Charles M. Knobler Festschrift”.

Figure 1. Diffuse interface (shown hatched) between phases R and β with an arbitrarily located Gibbs dividing surface (solid line).

Figure 2. Two alternative and equally arbitrary choices for the location of the line of mutual contact of three phases R, β, γ, and the associated Gibbs dividing surfaces for the three interfaces. The contact line is taken to be perpendicular to the plane of the figure and so appears as a point, and the dividing surfaces appear as lines meeting at that point.

as a point, while the three interfaces appear as lines meeting at that point. The region of contact is in reality diffuse; the location of the contact line, like that of the dividing surface in Figure 1, may be chosen at will. Figure 2 shows two alternative and equally arbitrary choices for the location of the contact line, and so also of three Gibbs dividing surfaces that meet at that line. One may define line adsorptions Λi in a way that is analogous to the definition of the surface adsorptions Γi. One imagines a volume that contains part of the contact region, asks for the total amount of i in the volume, and subtracts from it the amount that would have been in that volume had the densities Fi and adsorptions Γi been unchanged from their bulk-phase values βγ Rγ FRi , Fβi , and Fγi and distant surface values ΓRβ i , Γi , and Γi throughout the volume. That difference is the linear excess of component i and, per unit length of the contact line, is the line adsorption Λi.1 The calculation of these quantities will be illustrated with model density distributions in sections 2 and 3. Just as Γi depends on the arbitrary location of the dividing surface in Figure 1, so also does Λi depend on the arbitrary location of the contact line in the plane of Figure 2. The ΓRβ i , etc., and Λi, are then functions ΓRβ (r) and Λ (r) of the location i i r of the point in the plane of Figure 2 through which the contact line has been chosen to pass. Λj(r) ) 0 and Λk(r) ) 0 are the equations of two curves in that plane. Λi(j,k) is the adsorption of i in a frame of reference in which the adsorptions of j and k are both 0; i.e., when the contact line is chosen to pass through an intersection of the two curves Λj(r) ) 0 and Λk(r) ) 0.

10.1021/jp058231l CCC: $33.50 © 2006 American Chemical Society Published on Web 01/12/2006

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There is an excess free energy per unit length of the contact line, which is the line tension, τ.1,3-15 Just as the surface tensions σRβ, etc., are invariant to the choice of dividing surfaces, so also is the line tension τ invariant to the choice of location of the contact line. Now suppose, again for simplicity, that if c > 2 the variables µ4, ..., µc+1 are held fixed, so that only µ1, µ2, and µ3 may vary, but not independently because there are now two phaseequilibrium constraints. Take µ2 now to be the only independently variable field. By analogy with the derivation of the Gibbs adsorption equation in the form (1.1), and following a suggestion by Gibbs,3 a line adsorption equation of the form

dτ ) -Λ2(1,3) dµ2

(1.2)

was derived.1 It is now known2 that the curves Λ1(r) ) 0 and Λ3(r) ) 0 are rectangular hyperbolas that may intersect in two or four points and that Λ2(1,3) depends on which of the two or four is chosen to be the location of the contact line. But dτ/dµ2 is a measurable quantity that is independent of any such choice.1,2 Thus, it cannot be equal to -Λ2(1,3); i.e., (1.2), unlike its analogue (1.1), cannot be right. A corrected form of the line adsorption equation will be conjectured in section 4. While Λ2(1,3) depends on which intersection r13 of Λ1(r) ) 0 with Λ3(r) ) 0 is taken for the location of the contact line, it was shown that

Λ2(1,3) +

(

)

dσβγ dσRγ dσRβ e + e + e ‚(r13 - r0) (1.3) dµ2 Rβ dµ2 βγ dµ2 Rγ

does not.2 In the expression (1.3), eRβ, eβγ, and eRγ are unit vectors parallel, respectively, to the Rβ, βγ, and Rγ interfaces in the plane of Figure 2, while r0 is an arbitrary fixed origin from which the several r13 are measured. It should be emphasized that the hyperbola Λi(r) ) 0, which is the locus in the plane of Figure 2 of choices for the location of the contact line for which the linear adsorption of the component i is 0, is in principle determinable by experiment or, in model systems, by computer simulation, just as is the location in an interface of a dividing surface with respect to which a surface adsorption Γi is 0. If the spatial distribution of the several components in the phase equilibrium could be determined, those loci in the real physical space could then be mapped. It would be of great interest to display the rectangular hyperbolas Λi(r) ) 0 in a real or simulated three-phase equilibrium. These will be explicitly obtained in sections 2 and 3 for certain simple, model density distributions. Because of the simplicity of the models, the hyperbolas in those cases coincide with their asymptotes; i.e., each “hyperbola” is then a pair of perpendicular lines. The two line pairs Λ1(r) ) 0 and Λ3(r) ) 0 intersect in four points, P1, P2, Q1, and Q2. In the first of the models treated, in section 2, there is a symmetry that causes the quantity (dσRβ/dµ2)eRβ + ‚‚‚ in (1.3) to vanish, and it is then verified that Λ2(1,3) does indeed have the same value independently of which of the four points, P1 etc., is chosen for the location of the contact line. In the second model treated, in section 3, some of the symmetry is lifted, so that now three distinct values of Λ2(1,3) are found among the four choices P1, P2, Q1, and Q2, while it is verified that the full expression (1.3) still has the same value for all four, as required by the theory. It is the invariance of the expression (1.3) that suggests what is conjectured in section 4 to be the line adsorption equation, correcting (1.2). One implication of the conjectured relation is that there is in the space of the phase equilibrium a line in the

Figure 3. Model density distribution of three components in three phases. In the region R the density F1 ) F while F2 ) F3 ) 0; in β, the density F2 ) F while F1 ) F3 ) 0; in γ, the density F3 ) F while F1 ) F2 ) 0. Within the dashed lines, F1 ) F2 ) F3 ) 0. The model has 3-fold symmetry: the contact angles are all 120° and the interfacial regions have a common thickness, l. The Gibbs dividing surfaces, shown as solid lines, are arbitrarily chosen to be the midlines of the interfacial regions. Their intersection then defines the location of the contact line, shown as a point at the centroid of the equilateral triangle in the middle of the diagram.

plane of Figure 2 that is singled out as playing a special role and that could in principle be identified by experiment or simulation in any real or model system with a three-phase contact line. The results are briefly summarized in the concluding section 5. 2. Fully Symmetric Model Density Distribution In Figure 3 is shown the first of the two model structures we shall consider to illustrate the principles outlined in the Introduction. There are three phases all meeting with 120° contact angles and three components. In the region R outside the dashed lines the density of component 1 has the uniform value F, while the densities of components 2 and 3 are both identically 0. In region β the density of component 2 has the uniform value F while the densities of components 1 and 3 are both 0, and in γ it is the density of component 3 that has the uniform value F while the densities of 1 and 2 are both 0. Everywhere inside the dashed lines the densities of all three components are 0. It is as though each of the three components repels the other two infinitely strongly. This has described the “physical” phase equilibrium. We now choose for the nominal location of the contact line (point, in the plane of the figure) the symmetry point, which we call O, which is the centroid of the equilateral triangle in the middle of the figure. The solid lines meeting at that point, which run through the middle of the respective interfacial regions, are then the Gibbs dividing surfaces of the three interfaces. It is to be emphasized that this choice of location of the contact line, and with it the dividing surfaces, is arbitrary, but is the most convenient given the symmetry of the model. In this simple model each phase remains uniform outside the dashed lines, but there are sharp discontinuities in composition between the regions outside and inside those lines. This is how the model imitates the inhomogeneities in the interfacial and contact regions of real phase equilibria. Our model bulk-phase densities are thus

Adsorption at Three-Phase Contact

J. Phys. Chem. B, Vol. 110, No. 44, 2006 22127

FR1 ) Fβ2 ) Fγ3 ) F Fβ1 ) Fγ1 ) FR2 ) Fγ2 ) FR3 ) Fβ3 ) 0

(2.1)

As in the Introduction, we contemplate there being only three variable fields µ1, µ2, and µ3, of which only one is independently variable. We take them to be the chemical potentials conjugate to the densities F1, F2, and F3, imagining, for simplicity, the temperature to be fixed. In general, the relations that constrain the three µi are two Clapeyron equations, from which follow16

| |

R β R β dµ2 1 F1 - F1 F3 - F3 )β γ β γ dµ1 ∆ F1 - F1 F3 - F3 R β R β dµ3 1 F2 - F2 F1 - F1 )dµ1 ∆ Fβ2 - Fγ2 Fβ1 - Fγ1

with

|

FR - Fβ2 FR3 - Fβ3 ∆ ) 2β F2 - Fγ2 Fβ3 - Fγ3

| |

|

(2.2)

(2.3)

(2.4)

Figure 4. Rectangular region outlined by dashed lines, at the Rγ interface far from the contact line. It extends a distance 1/2l + l1 above the Rγ dividing surface and a distance 1/2l + l2 below it.

For the present model, from (2.1), these give simply

dµ1 ) dµ2 ) dµ3

(2.5)

as the constraints on the three µi, as would have been obvious from the symmetry of the model. Figure 4 shows part of the Rγ interfacial region far from the contact line, and a vertical rectangular region of height l1 + l + l2, also outlined in dashed lines and containing part of the Rγ interface. It represents a region of the equilibrium fluids isolated for the calculation of the adsorptions ΓRγ i . These are calculated with the Rγ dividing surface shown as one of the three solid lines in Figure 3 that meet at the chosen location O of the contact line. The calculated adsorptions are then ΓRγ i (O). With the same fixed spatial distribution of densities but a different choice for the location of the contact line, these adsorptions would have different values. The height l is the common thickness of the three interfacial regions in Figure 3 and, along with the density F, is a model parameter, while the heights l1 and l2 are arbitrary. Then from (2.1) and the definitions of surface adsorptions, we have Rγ Rγ 1 1 ΓRγ 1 (O) ) Fl1 - F(l1 + /2l) ) - /2Fl, Γ2 (O) ) 0, and Γ3 (O) 1 1 ) Fl2 - F(l2 + /2l) ) - /2Fl, all independent of the arbitrary l1 and l2, as they must be. By symmetry we have analogous βγ relations for the ΓRβ i (O) and Γi (O). In summary Rγ Rβ ΓRβ 1 (O) ) Γ1 (O) ) Γ2 (O) )

Γβγ 2 (O)

)

ΓRγ 3 (O)

Figure 5. Equilateral Neumann triangle with sides parallel to and distant l3 from the corresponding sides of the triangle in Figure 3, and with the same centroid. Rβ F and l, the surface adsorptions ΓRγ 1 (O) and Γ1 (O), the bulkphase densities as given in (2.1), and the arbitrary length l3 shown in the figure, one has

(

Λ1(O) ) x3 Fl 23 - F x3 l 23+ ll3 + )

Γβγ 3 (O)

1 ) - Fl (2.6) 2

Rγ Rβ Γβγ 1 (O) ) Γ2 (O) ) Γ3 (O) ) 0

(2.7)

Figure 5 shows a large region around that of the three-phase contact. It is outlined by the large dashed equilateral triangle that contains within it the smaller equilateral triangle of Figure 3. The larger triangle is the Neumann triangle for the present model. Its sides are respectively perpendicular to the interfaces, and for convenience we take its centroid to coincide with that of the smaller triangle within it. With this construction, and from the definition of the line adsorptions as outlined in the Introduction, one may now calculate the three Λi(O) (which are all equal, by symmetry). In terms of the model parameters

1 x3l 2 12

)

1 1 Rβ ΓRγ 1 (O) l3 + x3 l - Γ1 (O) l3 + x3 l (2.8) 6 6

(

)

(

)

Then from (2.6), and the obvious symmetry that makes all three Λi(O) equal

Λ1(O) ) Λ2(O) ) Λ3(O) )

1 x3 Fl2 12

(2.9)

independent of the arbitrary l3, as it must be. We note that in this model and with our particular choice of location of the contact line, the surface adsorptions are all negative or 0 while the line adsorptions are positive. Like the surface adsorptions, the line adsorptions would have had different (and in general unequal) values had the position of the contact line been chosen

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differently, and not in a way that was symmetrically related to the given spatial distributions of the densities. With these data we may now find the locus Λi(r) ) 0; i.e., the rectangular hyperbola that is the locus of points in the plane of Figure 3 with the property that the line adsorption Λi is 0 when the contact line is chosen to pass through any of them. We introduce the x, y coordinate system shown in Figure 6. The origin is at the model’s symmetry point O, so the solid lines in the figure are the same as in Figure 3. The x axis is taken to be along the Rγ dividing surface; it is in the direction of the unit vector eRγ in (1.3). The y axis is perpendicular to it and extends into the bulk γ phase. We seek the hyperbolas Λi(r) ) 0. Their equations are given in ref 2 in terms of the several Γi(O), Λi(O), FRi , etc., and contact angles, for any three-phase equilibrium with a contact line and any choice of origin O. It is shown in Appendix A, from the formulas in ref 2, that for the present model and the present choice of origin and x, y coordinate system, the equation of the “hyperbola” Λ2(r) ) 0 is

(

)(

)

y x 1 y x 1 )0 + + - l l l l x3 x3

(2.10)

Figure 6. x, y coordinate system. The x axis is in the direction of the Rγ interface. The y axis is perpendicular to it and rotated counterclockwise from it. The origin is at the chosen location of the contact line. The solid lines are the Gibbs dividing surfaces meeting at the contact line.

As a consequence of the simplicity of the model density distributions, this “hyperbola”, as is seen, coincides with its asymptotes: it is a pair of perpendicular lines that cross on the negative x axis at x ) -l /x3. It is shown in Figure 7 along with the line pairs Λ1(r) ) 0 and Λ3(r) ) 0 obtained from it by 120° rotations. The equations of the latter two are also in Appendix A. We wish now to calculate Λ2(1,3) and then to evaluate the expression (1.3). Here r13 is any of the intersections of Λ1(r) ) 0 with Λ3(r) ) 0. They intersect in four distinct points, P1, P2, Q1, Q2, with coordinates as obtained from the equations in Appendix A

P1:

x ) (1 - 1/x3)l,

P2:

x ) -(1 + 1/x3)l,

Q1:

x ) y ) l /x3

Q2:

x ) -y ) l /x3

y)0 y)0

(2.11)

We note that the coordinates x1, y1, ..., x4, y4 of the four intersections P1, P2, Q1, Q2 as given in (2.11) satisfy

|

x12 - y12 x22 - y22 x32 - y32 x42 - y42

x1y1 x2y2 x3y3 x4y4

x1 x2 x3 x4

|

y1 y2 )0 y3 y4

(2.12)

as required by the geometry of intersection of two rectangular hyperbolas.2 In Appendix B it is shown from the general formulas in ref 2, with (2.11), that Λ2(1,3) has the common value

1 Λ2(1,3) ) Λ2(P1) ) Λ2(P2) ) Λ2(Q1) ) Λ2(Q2) ) x3 Fl 2 4 (2.13) when r13 is taken to be each of the four points P1, P2, Q1, Q2 in turn. In this model, then, as a consequence of its symmetry, Λ2(1,3) is invariant to the choice of intersection that Λ1(r) ) 0 makes with Λ3(r) ) 0; although in more realistic models, and

Figure 7. Three “hyperbolas” (line pairs in this model) Λ1(r) ) 0, Λ2(r) ) 0, and Λ3(r) ) 0. The solid lines meeting at the contact line are the chosen Gibbs dividing surfaces, with x, y coordinates as in Figure 6.

even in the only slightly less symmetrical model treated in section 3, that is not the case. Since the full expression (1.3) is known to be invariant,2 it must be that in this symmetrical model the quantity in parentheses in (1.3), like Λ2(1,3), must be separately invariant. From the Gibbs adsorption equation for interfaces, for the case in which there are only three variable fields µ1, µ2, and µ3, and the two phase-equilibrium constraints (2.2) and (2.3)

dσRβ dµ1 dµ3 Rβ ) -ΓRβ - ΓRβ 2 - Γ1 3 dµ2 dµ2 dµ2

(2.14)

independently of the location of the dividing surfaces (and analogously for the βγ and Rγ interfaces). Then for the present model, from (2.5) to (2.7)

dσRβ dσβγ dσRγ ) ) ) Fl dµ2 dµ2 dµ2

(2.15)

Adsorption at Three-Phase Contact

J. Phys. Chem. B, Vol. 110, No. 44, 2006 22129

Figure 9. Expanded view of the pentagonal region in the middle of Figure 8, with various distances marked.

Figure 8. Model density distributions of three components in three phases. As in Figure 3 but now the Rβ and βγ interfacial regions are of thickness 2l while that of Rγ is still l.

But also, since all three contact angles are 120°, the sum of unit vectors eRβ + eβγ + eRγ ) 0. Thus, the quantity in parentheses in the expression (1.3) vanishes. That is what allowed Λ2(1,3) by itself to be invariant to which intersection r13 of Λ1(r) ) 0 with Λ3(r) ) 0 was chosen in evaluating it. As will be seen in the following section, and as anticipated, in a less symmetrical model it will no longer be true that Λ2(1,3) is invariant by itself, while the combination (1.3) is still invariant, as required by the general theory.

Λ3(O) ) Λ1(O) ) -F

3. Less Symmetric Model Density Distribution Here we consider a slightly less symmetrical variant of the model treated in section 2. It is shown in Figure 8. The only difference between this and the earlier model depicted in Figures 3 and 5 is that now the thickness of the Rβ and βγ interfacial regions is 2l instead of l, while that of Rγ is still l. We again choose for the contact line the location for which the dividing surfaces that meet at that line (point, in the figure) run through the middle of their respective interfacial regions. We again call this point O. The region of three-phase contact, outlined by dashed lines in the figure, is now pentagonal rather than equilateral-triangular as in Figures 3 and 5. We again take the densities in the various regions as in the earlier model, so that (2.1) again holds for the bulk phases while again F1 ) F2 ) F3 ) 0 in the region within the dashed lines. Then (2.5), which follows from (2.2)-(2.4) with (2.1), remains unchanged. This aspect of the earlier symmetry is thus retained. Rβ Figure 4 still yields the various ΓRγ i (O), but now for the Γi (O) βγ and Γi (O) one must replace the earlier l by 2l, so that instead of (2.6) we now have Rβ βγ βγ ΓRβ 1 (O) ) Γ2 (O) ) Γ2 (O) ) Γ3 (O) ) -Fl

1 Rγ ΓRγ 1 (O) ) Γ3 (O) ) - Fl 2

the bulk outside. Likewise, when the Λi(O) were calculated with the help of Figure 5, it was noted that the result was independent of the arbitrary l3. Thus, it, too, could have been taken to be 0 without affecting the result; i.e., the whole contribution to the Λi(O) came from the inner equilateral triangle that outlined the region of the three-phase contact. These simplifications were all consequences of the densities’ being uniform outside the dashed lines, where they then coincided with their bulk-phase values. The same simplification occurs in the present, less symmetric model, where the net contributions to the Λi(O) then come entirely from within the pentagonal region outlined by the dashed lines in the middle of Figure 8. The geometry of this pentagonal region, as it follows from the construction of Figure 8, is shown separately in Figure 9. By symmetry, Λ3(O) ) Λ1(O) as before, but now Λ2(O) is distinct. From (2.1) and Figure 9, and the definition of the line adsorptions as outlined in the Introduction

(3.1)

while (2.7) holds unchanged. When the ΓRγ i (O) for the earlier model, in section 2, were calculated with the help of Figure 4, it was noted that the result was independent of the arbitrary l1 and l2. Thus, l1 and l2 could have been taken to be 0 without affecting the results; i.e., the whole contribution to the various Γi(O) came from within the dashed lines that outlined the interfacial regions and none from

ΓRβ 1 (O)l

(81 x3l

/x3 -

2

)

+ l 2/x3 -

Γβγ 1 (O)

l /x3 - ΓRγ 1 (O)x3l / 2 (3.2)

Λ2(O) ) -Fl2/x3 - ΓRβ 2 (O) l /x3 Rγ Γβγ 2 (O) l /x3 - Γ2 (O) x3 l / 2 (3.3)

Then from (2.7) and (3.1)

1 Λ3(O) ) Λ1(O) ) x3 Fl 2 8 1 Λ2(O) ) x3 Fl 2 3

(3.4)

We continue to use the x, y coordinate system in Figure 6 and again take the origin to be at the chosen location O of the contact line. The Γi(O) and Λi(O) in (2.7), (3.1), and (3.4) are again the data from which we determine the equations of the hyperbolas Λi(r) ) 0 from the formulas in ref 2. The calculations are in Appendix A. It is again found that the “hyperbolas” Λi(r) ) 0 coincide with their asymptotes, as in the fully symmetrical model in section 2 and again as a result of the model’s especially simple structure. From the results in Appendix A, the line pairs Λ2(r) ) 0 are as in Figure 7 but their crossing on the negative x axis is now at x ) -2l /x3 instead of -l /x3. The two lines of the line pair Λ1(r) ) 0 as obtained in the Appendix are y / l ) (2 + x3)(x / l - 1) and y / l ) -(2 - x3)(x / l + 1). These are qualitatively like that for the earlier model in Figure 7, but now the crossing of the two lines occurs at x ) 1/2x3l, y ) -1/2l, where y ) -x/x3 rather than -x3x as in Figure 7. The line pair Λ3(r) ) 0 is obtained from that of Λ1(r) ) 0 by reflecting

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in the x axis (i.e., by replacing y by -y). These two line pairs again intersect each other in four points P1, P2, Q1, Q2, as in (2.11), but now given by

P1:

x / l ) 1,

y/l)0

P2:

x / l ) -1,

Q1:

x / l ) 2/x3,

y / l ) 1/x3

Q2:

x / l ) 2/x3,

y / l ) -1/x3

y/l)0

(3.5)

If the coordinates of these four intersections are called x1, y1, ..., x4, y4, we again note that (2.12) holds, as required. It is shown in Appendix B, from the general formulas in ref 2, with (3.5), that Λ2(1,3) has the values

(127 x3 + 1)Fl 7 Λ (P2) ) ( x3 - 1)Fl 12

Λ2(P1) )

4. Line Adsorption Equation

2

2

2

5 Λ2(Q1) ) Λ2(Q2) ) x3Fl 2 4

(3.6)

when the point r13 through which the contact line passes is taken to be each of the four points P1, P2, Q1, Q2 in turn. Thus, in this less symmetrical model, unlike in the model of section 2, the adsorption Λ2(1,3) does depend on which of the four intersections of Λ1(r) ) 0 with Λ3(r) ) 0 is taken for the location of the contact line. It now remains to verify that the full expression (1.3) does not. We again have (2.15) for dσRβ/dµ2 and analogous expressions for dσβγ/dµ2 and dσRγ/dµ2. Then from (2.7) and (3.1)

dσRβ dσβγ ) ) 2 Fl dµ2 dµ2 dσRγ ) Fl dµ2

(3.7)

whereas in the more symmetrical model of section 2 all three were Fl (eq 2.16). The common origin r0 in (1.3) from which the various intersections r13 are to be measured had here already been taken to be the origin O of the x, y coordinate system in which the coordinates of P1, P2, Q1, Q2 are expressed in (3.5). Then in terms of the unit vectors eRβ, eβγ, eRγ, with eRβ + eβγ + eRγ ) 0 because the three contact angles are still all 120°, we have for the four points P1, P2, Q1, Q2

P1:

r13 - r0 ) leRγ

P2:

r13 - r0 ) -leRγ

Q1:

1 1 r13 - r0 ) - (2x3 + 1)leRβ - (2x3 - 1)leβγ 3 3

Q2:

1 1 r13 - r0 ) - (2x3 - 1)leRβ - (2x3 + 1)leβγ 3 3 (3.8)

while from (3.7)

dσβγ dσRγ dσRβ eRβ + eβγ + e ) -FleRγ dµ2 dµ2 dµ2 Rγ

that (1.3) does indeed have a common value, (7x3 /12)Fl 2, for each of the four different r13 with which it is calculated. Together with the model in section 2, we now have two concrete, albeit simple, models with which we have illustrated the principles derived in ref 2. The present model, in particular, illustrates the earlier observation2 that the line adsorption Λ2(1,3) is not independent of which of the intersections of Λ1(r) ) 0 with Λ3(r) ) 0 is chosen as the location of the contact line. As discussed in the Introduction, this means that (1.2) cannot be the correct line adsorption equation. But while Λ2(1,3) is not independent of the four possible choices for location of the contact line, the full expression (1.3), as has been illustrated here, is. This leads to a conjecture in the following section of a possible line adsorption equation that would correct (1.2).

(3.9)

Therefore, with the four distinct r13 - r0 in (3.8), and with eRγ‚eRβ ) eRγ‚eβγ ) -1/2, the quantity that is added to Λ2(1,3) in (1.3) is found to have the respective values -Fl 2, Fl 2, -(2/x3)Fl 2, -(2/x3)Fl 2. From these and (3.6), we then find

When all the intersections r13 of the hyperbolas Λ1(r) ) 0 and Λ3(r) ) 0 are measured from a common origin r0, (1.3) takes the same value whichever of those intersections is the point through which the contact line is chosen to pass. It is thus natural to conjecture that the appropriate line adsorption equation, correcting (1.2), may be of the form

(

)

dσβγ dσRγ dσRβ dτ ) -Λ2(1,3) eRβ + eβγ + e ‚(r13 - r0) dµ2 dµ2 dµ2 dµ2 Rγ (4.1) The invariance of the right-hand side to the choice of intersection was illustrated in the models analyzed in sections 2 and 3, particularly in the less symmetrical model in section 3, where the invariance was nontrivial. In that illustration the common origin r0 from which the intersections r13 were measured was taken to be the point at which dividing surfaces running through the middle of the three interfacial regions intersected. That point was also taken as the origin O of the x, y coordinate system in which the calculations were done. By the principle derived in ref 2, had some other origin been chosen from which to measure the various r13 in (1.3), the result would again have been the same for all four of the r13, but that would not in general have been the same common value as with the first choice of origin r0. The separate values of Λ2(1,3) do not change with a change in r0: they depend only on the location in the space of the physical phase equilibrium of the intersections r13 of the two hyperbolas Λ1(r) ) 0 and Λ3(r) ) 0; likewise the quantity in parentheses in (4.1) that multiplies r13 - r0 does not depend on the choice of origin r0; however, r13 - r0, for any one of the intersections r13, does; therefore the right-hand side of (4.1) changes, in general, if, for any one of the intersections r13, one changes the choice of origin r0 from which it is measured. There may, however, be a particular r0 for which (4.1) holds, and that is now our conjecture. This would be the most obvious modification of (1.2) that has the property of being invariant to which intersection r13 is taken on the right-hand side of (4.1). But that would still be true if to that special r0 were added any vector orthogonal to

dσβγ dσRγ dσRβ eRβ + eβγ + e dµ2 dµ2 dµ2 Rγ

(4.2)

The conjecture, then, is that in any plane perpendicular to the contact line, in the physical space of the phase equilibrium, there is a special line in a direction orthogonal to that of the vector (4.2), such that for any point r0 on that line (4.1) holds and does so whichever intersection r13 of the hyperbolas Λ1(r) ) 0

Adsorption at Three-Phase Contact

J. Phys. Chem. B, Vol. 110, No. 44, 2006 22131

and Λ3(r) ) 0 is taken to be the location of the contact line in evaluating Λ2(1,3). The location of that special line, i.e., the locus of the special points r0, would depend, in general, on the thermodynamic state of the phase equilibrium (i.e., here, on µ2). It could in principle be determined by experiment. That would require first measuring the quantities on both sides of (4.1) and then varying r0 until some r0 was found to satisfy the equality with acceptable accuracy. More generally, as noted in the Introduction, in a c-component system there are c + 1 field variables µ1, ..., µc+1, of which, in a three-phase equilibrium, c - 1 are independently variable. The more general form of (1.2) was that for any infinitesimal change dµ1, dµ2, ..., in the state of the three-phase equilibrium, i.e., in the independently variable µi

dτ ) -

∑i Λi dµi

(4.3)

where (it was incorrectly believed at the time1) the right-hand side would be invariant to the choice r of location of the contact line even though the separate Λi(r) depended on it. It is now known2 that the right-hand side of (4.3) is not invariant to the arbitrary choice of r, but that, for any fixed r0

-

∑i Λi(r) dµi - (eRβ dσRβ + eβγ dσβγ + eRγ dσRγ)·(r - r0)

(4.4)

is. Again, the value of this expression for any r depends on r0 (although not on r), but the conjecture is that, for any infinitesimal change dµ1, dµ2, ... in the thermodynamic state of the phase equilibrium, there is in the physical space of the equilibrium phases, in any plane perpendicular to the contact line, a special line in a direction orthogonal to the vector

eRβ dσRβ + eβγ dσβγ + eRγ dσRγ

(4.5)

such that for any r0 on that line the infinitesimal change dτ in the line tension is given by (4.4) independently of the arbitrary choice r for the location of the contact line. This special line that is conjectured to exist is in principle determinable by experiment and, in model three-phase equilibria, by computer simulation. Note that the infinitesimal vector that multiplies r - r0 in (4.4) may equally well be written σRβ deRβ + σβγ deβγ + σRγ deRγ because σRβ eRβ + σβγ eβγ + σRγ eRγ ≡ 0, as the condition of mechanical equilibrium. 5. Summary and Conclusion It had long been thought that (1.2) would be the analogue, for the tension τ of a three-phase contact line, of the Gibbs adsorption equation (1.1) for interfaces,1 but it was recently found that that is not so.2 Here two different model systems were analyzed to illustrate the principles involved. In the model treated in section 2 it was found that Λ2(1,3), the line adsorption of component 2 in a frame of reference in which the line adsorptions of components 1 and 3 both vanish, is indeed independent of which intersection of Λ1 ) 0 with Λ3 ) 0 is taken for the location of the contact line in evaluating Λ1, so for this model there would have been no reason to question the correctness of (1.2). That property of the model, though, proved to be due to its symmetry. That symmetry was partially lifted in the model analyzed in section 3, where Λ2(1,3) was found to depend on which intersection of Λ1 ) 0 with Λ3 ) 0 is taken for the location of the contact line, so (1.2) cannot be correct for that model. While Λ2(1,3) was thus shown not to be invariant in general, it was verified that the extended expression (1.3) is invariant, in agreement with the general theory.2

The invariance of (1.3) led in section 4 to a conjecture of a form of line adsorption equation, (4.1), that corrects the incorrect or incomplete (1.2). An implication of the conjecture is the existence, in any plane perpendicular to the contact line, of a physically defined and experimentally determinable line with the property that (4.1) would hold for every point r0 on that line, although not, in general, for any r0 not on that special line. A more general form of line adsorption equation, when there is more than one independently variable µi, was also conjectured in section 4, and it, too, implies the existence of such a special line in any plane perpendicular to the contact line. It is possible in principle, although it would no doubt be difficult in practice, to determine by experiment17 or by computer simulation18,19 the geometrical structures that are central in the theory, such as the hyperbolas Λi(r) ) 0 and the special lines implied by the conjectured line adsorption equation. It would require measuring with precision the spatial distribution of the several components in the region of three-phase contact. It would be like determining spatial density profiles in an interface but might require even greater ingenuity and the development of new techniques. Acknowledgment. Dedicated to Professor Charles M. Knobler, with gratitude for his inspiring contributions to physical chemistry. This work was supported by a grant from the National Science Foundation and by the Cornell Center for Materials Research. Appendix A It is shown in ref 2 that with an x, y coordinate system as in the present Figure 6, with the surface and line adsorptions Γi and Λi calculated with the contact line at the origin O of this coordinate system, with the bulk-phase densities FRi , etc., and with R, β, γ the contact angles as measured through the phases for which they are named, the equation of the curve Λi(r) ) 0 is the conic

Ai y2 + Bi xy + Ci x2 + Di y + Ei x + Fi ) 0

(A.1)

where

1 1 -Ci ) Ai ) - (FRi - Fβi ) sin 2R + (Fβi - Fγi ) sin 2γ 4 4 1 1 1 Bi ) (Fγi - FRi ) + (FRi - Fβi ) cos 2R + (Fβi - Fγi ) cos 2γ 2 2 2 βγ Di ) -ΓRβ i (O) sin R + Γi (O) sin γ βγ Rγ Ei ) ΓRβ i (O) cos R + Γi (O) cos γ + Γi (O)

Fi ) Λi(O)

(A.2)

It is the relation -Ci ) Ai that identifies this conic as a rectangular hyperbola (perpendicular asymptotes). For the model in section 2 the contact angles are all 120°, βγ Rγ the FRi , Fβi , Fγi are as given in (2.1), the ΓRβ i (O), Γi (O), Γi (O) are in (2.6) and (2.7), and the Λi(O) are in (2.9). From (A.1) and (A.2) we then find for the equation of the “hyperbola” (line pair) Λ2(r) ) 0

(

)(

)

y x 1 y x 1 )0 + + - l l x3 l l x3

(A.3)

as quoted in (2.10). The line pair Λ1(r) ) 0 is likewise obtained

22132 J. Phys. Chem. B, Vol. 110, No. 44, 2006

Widom

as

[

]

1 1 x 1 y (x3 - 1) + (x3 + 1) + × 2 l 2 l x3 1 1 x 1 y (x3 + 1) - (x3 - 1) ) 0 (A.4) 2 l 2 l x3

[

]

while the equation of the line pair Λ3(r) ) 0 is as in (A.4) with y replaced by -y. For the model in section 3 the contact angles are again all 120° and the FRi , Fβi , Fγi are again as given in (2.1); Γβγ 1 (O), Rβ (O), Γ (O) are again all 0, as given in (2.7), while the ΓRγ 2 3 (O), etc., are now in (3.1); and the Λ (O) are in remaining ΓRβ i i (3.4). One then finds from (A.1) and (A.2), for the line pair Λ2(r) ) 0, the equation

(

)(

)

y x 2 y x 2 )0 + + - l l x3 l l x3

(A.5)

while for Λ1(r) ) 0

[yl - (2 + x3)(xl - 1)] [yl + (2 - x3)(xl + 1)] ) 0

(A.6)

The equation of Λ3(r) ) 0 is again obtained from this by replacing y by -y. These are as quoted in the paragraph that includes eq 3.5. Appendix B In ref 2, in addition to the unit vectors eRβ, eβγ, and eRγ in the directions of the three interfaces, there were introduced three additional unit vectors, gRβ, gβγ, and gRγ, respectively perpendicular to and rotated counterclockwise from those in the former set. These two sets of unit vectors appeared together in dyadics that we here call Di

(Fγi - FRi )eRγ gRγ (B.1) Because in the models of both sections 2 and 3 all three of the contact angles are 120°, we have here

erβ + eβγ + erγ ) grβ + gβγ + grγ ) 0 eRβ·eβγ ) eβγ·eRγ ) eRγ·eRβ ) -1/2 2 1 eβγ eRβ x3 x3

gβγ )

1 2 eβγ + eRβ x3 x3

gRγ )

1 1 eβγ eRβ x3 x3

Λ2(P) ) βγ Rγ Λ2(O) + [ΓRβ 2 (O) eRβ + Γ2 (O) eβγ + Γ2 (O) eRγ]‚(P - O) + 1 (P - O)‚D2‚(P - O) (B.3) 2 For the model in section 2, from (2.1), Fβ2 ) F and FR2 ) Fγ2 ) 0; from (2.6) and (2.7), with O now specifically the symmetry point in Figure 3 and the origin of the x, y coordinate βγ Rγ 1 system in Figure 6, ΓRβ 2 (O) ) Γ2 (O) ) - /2Fl and Γ2 (O) ) 1 2 0; from (2.9), Λ2(O) ) /12x3Fl ; and the coordinates of the four intersections P are in (2.11). With these data one calculates from (B.1) to (B.3) that Λ2(1,3), taken in turn to be Λ2(P1), Λ2(P2), Λ2(Q1), Λ2(Q2), has the common value 1/4x3Fl 2, as quoted in (2.13). Similarly, for the model in section 3, Fβ2 ) F and FR2 ) Fγ2 ) Rβ 0 again; ΓRγ 2 (O) ) 0 again but now, from (3.1), Γ2 (O) ) βγ 2 Γ2 (O) ) -Fl; from (3.4), Λ2(O) ) (1/x3)Fl ; and the coordinates of the four intersections P are as in (3.5). With these data one calculates from (B.1) to (B.3) that the values of Λ2(P) with P taken to be P1, P2, Q1, and Q2 in turn, are

(127 x3 + 1)Fl 7 Λ (P2) ) ( x3 - 1)Fl 12 Λ2(P1) )

2

2

2

5 Λ2(Q1) ) Λ2(Q2) ) x3Fl 2 4

(B.4)

as quoted in (3.6). References and Notes

Di ) (FRi - Fβi )eRβ gRβ + (Fβi - Fγi )eβγ gβγ +

gRβ ) -

Let P represent any of the four intersections P1, P2, Q1, and Q2 of the line pair Λ1(r) ) 0 with Λ3(r) ) 0, so that Λ2(P) is any of the four values of Λ2(1,3), and let O be any other point in the plane of Figure 2. Then as follows from eq 2.5 of ref 2

(B.2)

In the coordinate system of Figure 6, eRγ is in the direction of the positive x axis and gRγ is in the direction of the positive y axis.

(1) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982; pp 232-236. (2) Djikaev, Y.; Widom, B. J. Chem. Phys. 2004, 121, 5602. (3) Gibbs, J. W.; The Collected Works of J. Willard Gibbs; Longmans, Green: New York, 1928; Vol. 1, p 288, footnote. (4) de Gennes, P. G.; Brochard-Wyart, F.; Que´re´, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, WaVes; Springer: Berlin, 2004; pp 72-73. (5) Toshev, B. V.; Platikanov, D.; Scheludko, A. Langmuir 1988, 4, 489. (6) Szleifer, I.; Widom, B. Mol. Phys. 1992, 75, 925. (7) Blokhuis, E. M. Physica A 1994, 202, 402. (8) Indekeu, J. O. Int. J. Mod. Phys. B 1994, 8, 309. (9) Robledo, A.; Indekeu, J. O. Europhys. Lett. 1994, 25, 17. (10) Perkovic´ , S.; Blokhuis, E. M.; Han, G. J. Chem. Phys. 1995, 102, 400. (11) Getta, T.; Dietrich, S. Phys. ReV. E 1998, 57, 655. (12) Bauer, C.; Dietrich, S. Eur. Phys. J. B 1999, 10, 767. (13) Rusanov, A. I. Colloids Surf., A 1999, 156, 315. (14) Amirfazli, A.; Neumann, A. W. AdV. Colloid Interface Sci. 2004, 110, 121. (15) Dussaud, A.; Vignes-Adler, M. Langmuir 1997, 13, 581. (16) Widom, B. Colloids Surf., A 2004, 239, 141. (17) Pompe, T.; Herminghaus, S. Phys. ReV. Lett. 2000, 85, 1930. (18) Bresme, F.; Quirke, N. J. Chem. Phys. 2000, 112, 5985. (19) Djikaev, Y. J. Chem. Phys. 2005, 123, 184704.