Contact Line Quadrilateral Relation. Generalization of the Neumann

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Langmuir 1996, 12, 5956-5962

Contact Line Quadrilateral Relation. Generalization of the Neumann Triangle Relation To Include Line Tension P. Chen,† J. Gaydos,‡ and A. W. Neumann*,† Department of Mechanical Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario M5S-1A4, Canada, Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S-5B6, Canada Received March 26, 1996. In Final Form: August 23, 1996X The contact line quadrilateral relation is a generalization of the classical Neumann triangle relation that includes the effect of line tension. The relation is derived using a minimum free energy principle that may be graphically represented by a quadrilateral whose four sides consist of three surface tension quantities and a line tension term. When the line tension term vanishes, the quadrilateral relation simplifies to the classical Neumann triangle relation. Applications of the quadrilateral relation are illustrated using several axisymmetric capillary geometries.

Introduction to the Concept of Line Tension The concept of line tension and its definition arises as a natural and necessary extension of the requirements of a fundamental equation1 when applied to systems with multiphase boundary curves. Thus, just as surface tension is the two-dimensional analogue of bulk pressure, so is line tension the one-dimensional analogue of surface tension. Line tension, denoted by the symbol σ, is defined in complete analogy to the accepted thermodynamic definitions for pressure and surface tension. If σ is positive, it operates so as to fold-in surface regions along the boundary where several surfaces meet and to constrict the length of the contact line. Even though σ is a welldefined thermodynamic quantity, there are still a large number of problems associated with determining both the magnitude and sign of the line tension, although studies have been conducted since the mid-nineteen thirties. Pethica2 was the first to modify the Young equation of capillarity to include a line tension contribution. He considered the case of a sessile drop on a flat, solid surface and introduced a term σ/Rs where Rs represents the radius of the wetted area below the sessile drop, i.e., the radius of the liquid-solid circle of contact. Lane3 modified Pethica’s analysis and applied his results to the case of a small liquid volume wetting a conical pore (i.e., to model a liquid in contact with an inclined rather than a flat solid surface). Torza and Mason4 obtained force balance expressions for the attachment of two drops, and Pujado and Scriven5 corrected Langmuir’s6 original analysis of the equilibrium conditions of a bubble or drop pressed against another fluid interface in a gravitational field. Thus, a number of distinct experimental arrangements have been considered when attempting to study the effect of line tension.2-42 Regardless of the system considered †

University of Toronto. Carleton University. X Abstract published in Advance ACS Abstracts, November 1, 1996. ‡

(1) Callen, H. B. Thermodynamics; John Wiley and Sons: New York, 1960. (2) Pethica, B. A. Rep. Prog. Appl. Chem. 1961, 46, 14. (3) Lane, J. E. J. Colloid Interface Sci. 1975, 52, 155. (4) Torza, S.; Mason, S. G. Kolloid Z. Z. Polymer. 1971, 246, 593. (5) Pujado, P. R.; Scriven, L. E. J. Colloid Interface Sci. 1972, 40, 82. (6) Langmuir, I. J. Chem. Phys. 1933, 1, 756. (7) Harkins, W. D. J. Chem. Phys. 1937, 5, 135. (8) de Feijter, J. A.; Vrij, A. J. Electroanal. Chem. 1972, 37, 9. (9) Ponter, A. B.; Boyes, A. P. Can. J. Chem. 1972, 50, 2419. (10) Boyes, A. P.; Ponter, A. B. J. Chem. Eng. J. 1974, 7, 314. (11) Toshev, B. V.; Ivanov, I. B. Colloid Polym. Sci. 1975, 253, 558. (12) Ivanov, I. B.; Toshev, B. V. Colloid Polym. Sci. 1975, 253, 593.

S0743-7463(96)00291-0 CCC: $12.00

(e.g., a liquid lens at a fluid-liquid interface or a sessile drop on a solid support) there is one important theoretical point that must be kept in mind: the line tension, in a proper thermodynamic analysis, appears explicitly in both the Neumann triangle relation and the Young equation (13) Ivanov, I. B.; Toshev, B. V.; Radoev, P. On the Thermodynamics of Contact Angles, Line Tension and Wetting Phenomena. In Wetting, Spreading and Adhesion; Padday, J. F., Ed.; Academic: New York, 1978; pp 37-60. (14) de Feijter, J. A.; Rijnbout, J. B.; Vrij, A. J. Colloid Interface Sci. 1978, 64, 258. (15) Good, R. J.; Koo, M. N. J. Colloid Interface Sci. 1979, 71, 283. (16) Platikanov, D.; Nedyalkov, M.; Scheludko, A. J. Colloid Interface Sci. 1980, 75, 612. (17) Starov, V. M.; Churaev, N. V. Colloid J. USSR 1980, 42, 585. (18) Kolarov, T.; Zorin, Z. M. Colloid J. USSR 1980, 42, 899. (19) Scheludko, A.; Toshev, B. V.; Platikanov, D. On the Mechanics and Thermodynamics of Three Phase Contact Line Systems. In The Modern Theory of Capillarity; Goodrich, F. C., Rusanov, A. I.; Eds.; Akademie-Verlag: East Berlin, 1981; pp 163-182. (20) Navascues, G.; Tarazona, P. J. Chem. Phys. 1981, 75, 2441. (21) Navascues, G.; Mederos, L. Surf. Technol. 1982, 17, 79. (22) Churaev, N. V.; Starov, V. M.; Derjaguin, B. V. J. Colloid Interface Sci. 1982, 89, 16. (23) Platikanov, D.; Nedyalkov, M. Contact Angles and Line Tension at Microscopic Three Phase Contacts. In Microscopic Aspects of Adhesion and Lubrication; Georges, J. M., Ed.; Elsevier Science: Amsterdam, 1982; pp 97-106. (24) Marmur, A. J. Colloid Interface Sci. 1983, 93, 18. (25) Marmur, A. Adv. Colloid Interface Sci. 1983, 19, 75. (26) Ponter, A. B.; Yekta-Fard, M. Colloid Polym. Sci. 1985, 263, 1. (27) Churaev, N. V.; Starov, V. M. J. Colloid Interface Sci. 1985, 103, 301. (28) Scheludko, A. D. J. Colloid Interface Sci. 1985, 104, 471. (29) Mederos, L.; Quintana, A.; Navascues, G. J. Appl. Phys. 1985, 57, 559. (30) Ivanov, I. B.; Kralchevsky, P. A.; Nikolov, A. D. J. Colloid Interface Sci. 1986, 112, 97. (31) Kralchevsky, P. A.; Ivanov, I. B.; Nikolov, A. D. J. Colloid Interface Sci. 1986, 112, 108. (32) Kralchevsky, P. A.; Nikolov, A. D.; Ivanov, I. B. J. Colloid Interface Sci. 1986, 112, 132. (33) Nikolov, A. D.; Kralchevsky, P. A.; Ivanov, I. B. J. Colloid Interface Sci. 1986, 112, 122. (34) Nikolov, A. D.; Kralchevsky, P. A.; Ivanov, I. B. A New Method for Measuring Film and Line Tensions. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum: New York, 1986; Vol. 6, pp 1537-1547. (35) Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1987, 120, 76. (36) Wallace, J. A.; Schu¨rch, S. J. Colloid Interface Sci. 1988, 124, 452. (37) Yekta-Fard, M.; Ponter, A. B. J. Colloid Interface Sci. 1988, 126, 134. (38) Li, D.; Neumann, A. W. Colloids Surf. 1990, 43, 195. (39) Wallace, J. A.; Schu¨rch, S. Colloids Surf. 1990, 43, 207. (40) Gaydos, J. Implications of the Generalized Theory of Capillary. Ph.D. Thesis, University of Toronto, 1992. (41) Drelich, J.; Miller, J. D. Part. Sci. Technol. 1992, 10, 1. (42) Duncan, D.; Li, D.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1995, 169, 256.

© 1996 American Chemical Society

Contact Line Quadrilateral Relation

(i.e., the mechanical equilibrium condition for dividing lines) at the lowest level of generality, i.e., the same level in the hierarchy at which the classical Laplace equation is the appropriate equilibrium condition between bulk phases. Further details regarding the derivation of these relations and their generalized forms, including effects from line curvatures and torsion, may be found elsewhere.43-49 However, the interpretation of experimental data, and especially data obtained from liquid lens capillary systems, would be enhanced by a clear picture of the manner in which the Neumann triangle relation is altered by the presence of line tension. In particular, it would be useful to obtain (i) a corresponding graphical representation, i.e. a pictograph, similar to the Neumann triangle picture, for situations where line tension effects are noticeable and (ii) a clear method of conversion from the force balance conditions (which are normally represented by two orthogonal force balance equations) to a type of ‘cosine rule’ Neumann triangle relation in which the cosines of the contact angles are explicitly involved. For a threephase liquid lens system (e.g., oil lens at a liquid-vapor interface) the graphical representation or pictograph, with line tension, corresponds to a quadrilaterial rather than a triangle. Such a representation would be useful in all situations where the Neumann triangle relation has been traditionally employed. One obvious benefit that arises from this representation is a simple, geometric method for manipulating the quadrilaterial pictograph to obtain either a cosine rule expression or a force balance equation projected in a particular direction. Experimental concerns would dictate the choice of variables in the cosine rule expression, and simple considerations, based upon the quadrilateral geometry, would provide the methodology for choosing which quantities would need to be measured. In the next section, we choose a suitable free energy representation for capillary systems and then derive, on the basis of the grand canonical free energy, a modified form of the Neumann triangle relation that includes a line tension contribution. To avoid undue complexity, we shall restrict our investigation to a three-phase lens system with just one line tension term and the usual three surface tensions (cf. Figure 1a). Following Gibbsian thermodynamics,48 the surface phases are modeled as twodimensional dividing surfaces, and the line phase is considered to be a one-dimensional dividing line. Line tension is a mechanical property of the dividing line, and surface tensions are mechanical properties of the dividing surfaces. We shall also assume that the adjacent surfaces forming the contact line are both uniform and isotropic, which, of course, will be satisfied by most pure liquid phases. Finally, we assume that the contact line occurs in a plane of constant external potential, φ(r), so that ∇φ(r) ) 0. This assumption does not limit the analysis, as it is (43) Boruvka, L.; Neumann, A. W. Generalization of the Classical Theory of Capillarity. J. Chem. Phys. 1977, 66, 5464. (44) Gaydos, J.; Budziak, C. J.; Li, D.; Neumann, A. W. Contact Angles on Imperfect Solid Surfaces. In Surface Engineering: Current Trends and Future Prospects; Meguid, S. A., Ed.; Elsevier Applied Science: New York, 1990; pp 100-113. (45) Gaydos, J.; Neumann, A. W. Adv. Colloid Interface Sci. 1992, 38, 143. (46) Gaydos, J.; Neumann, A. W. Adv. Colloid Interface Sci. 1993, 43, 143. (47) Gaydos, J.; Boruvka, L.; Rotenberg, Y.; Chen, P.; Neumann, A. W. The Generalized Theory of Capillarity. In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 1. (48) Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. In The Scientific Papers of J. Willard Gibbs; Dover: New York, 1961; Vol. 1, pp 55-371. (49) Gaydos, J.; Li, D.; Neumann, A. W. Colloid Polym. Sci. 1993, 271, 715.

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Figure 1. Schematic of an axisymmetric liquid lens showing the directions of surface tension forces, γ(j)mj for j ) 1, 2, 3, corresponding binormal directions, mj, and associated outward normals, nj. The direction of the line tension force, σK, is also shown along with the bulk volume phases adjacent to the contact line, denoted by the quantities Vj, where j ) 1, 2, 3.

consistent with virtually all experimental situations considered in the literature.2-42 Once the quadrilateral relation has been developed and a clear physical meaning provided, we shall discuss application to a number of typical geometries that have or could be employed in experimental studies. Derivation of the Neumann Quadrilateral Relation The classical Neumann triangle relation is the appropriate boundary condition when the three surfaces which intersect to form the contact line are all deformable and the contribution from line tension can be ignored. When one of the surfaces is rigid, the Young equation is the appropriate boundary condition. In the general formulation these relations are quite complex because they include a detailed analysis of the higher-order curvature dependence of both the surface and the line boundaries.43,47 In practice, however, the mechanical equilibrium conditions for capillary systems with complex geometry are not of immediate necessity in many experimental studies that deal with either spherical or axisymmetric systems. Furthermore, most estimates of the line tension have been performed using capillary systems which are axisymmetric and where the surfaces forming the contact line are deformable. Thus, we shall restrict our consideration to the equilibrium of three bulk phases which meet in a line of three-phase contact (Figure 1). Macroscopically, the locus of points that forms the contact line is onedimensional and locally linear (i.e., there are no higherorder curvature effects); it is analogous to the macrocopically two-dimensional and locally planar interface between two bulk phases. It is possible to define an excess free energy per unit length, or line tension, that is associated with the contact line and write the total free energy as the sum

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Ωt ) Ω(V1) + Ω(V2) + Ω(V3) + Ω(A1) + Ω(A2) + Ω(A3) + Ω

(L)

δΩ(A) ) (1)

where the first three terms represent bulk phase free energies, the next three terms are surface phase free energies, and the last term is the contribution from the linear phase. The total free energy variation, δΩt, will involve variations of bulk, surface, and linear terms. Consider a single bulk phase inside a fixed volume (where both the volume and the external system boundary are fixed). In this case, the free energy equilibrium principle simply requires that the first variation of Ω(V) vanish. Allowing for arbitrary variations of the position vector r inside the volume V, we have50-52

∫∫∫V - P dV ) ∫∫∫V - δP dV + ∫∫∫V - P δ dV

δΩ(V) ) δ

)

(2)

∫∫∫VP δ dV ) ∫∫∫VP(∇‚δr) dV

∫∫A - P(n‚δr) dA

(4)

(5)

which represents the bulk phase variation of a point along the contact line. Thus, for each bulk phase in our threephase system we have identical expressions; that is,

δΩ(V1) )

∫∫A

1

- P(1)(n1‚δr) dA1 -

∫∫A

2

P(1)(n2‚δr) dA2 (6) where n1 and n2 are unit normals to the surfaces A1 and A2 that bound the volume V1, as sketched in Figure 1b. Similarly, the other bulk phase variations are given by

δΩ(V2) )

∫∫

(2) A2 - P (n2‚δr) dA2 -

3

- P(3)(n3‚δr) dA3 -

∫∫

∫∫A

1

(1)

where the surface tension γ , mean curvature J , and binormal m1 refer to the first surface A1. Similarly, the other surface phase variations are given by

∫∫A γ(2)J(2)(n2‚δr) dA2 - ∫Lγ(2)(m2‚δr) dL 2

(12) and

∫∫A γ(3)J(3)(n3‚δr) dA3 - ∫Lγ(3)(m3‚δr) dL

δΩ(A3) ) -

3

(13) Finally, the variation of the linear energy term in response to the variation of the positions of points along the (continuous) contact line is given by

∫Lσ dL ) ∫L∇1(σ δr) dL ) -∫σK‚δr dL

δΩ(L) ) δ

(14) where K is the curvature vector of the contact line and the scalar product K‚δr represents the extension in length of the contact line per unit length as a result of the variation. Substitution of the bulk, surface, and line variation expressions (using eqs 6-8, 11-13, and 14) into the necessary equilibrium condition δΩt ) 0 yields

∫∫A (γ(1)J(1) + [P(1) - P(3)])n1‚δr dA1 ∫∫A (γ(2)J(2) + [P(2) - P(1)])n2‚δr dA2 ∫∫A (γ(3)J(3) + [P(3) - P(2)])n3‚δr dA1 -

δΩt ) -

1

2

and

∫∫A

∫∫A γ(1)J(1)(n1‚δr) dA1 - ∫Lγ(1)(m1‚δr) dL

δΩ(A1) ) -

A3 -

P(2)(n3‚δr) dA3 (7)

δΩ(V3) )

(10)

where n is the outward unit normal to the surface A and m is a unit vector tangent to the surface A and coincident with the direction of the surface tension force, i.e., the inward directed binormal to the contact line; see Figure 1b. Using this relation we may write the first surface phase variation of a point along the contact line as

(3)

and Gauss’ divergence theorem for the virtual work of internal forces, eq 2, reduces to

δΩ(V) )

∫∫AγJ(n‚δr) dA - ∫Lγ(m‚δr) dL

δΩ(A) ) -

δΩ(A2) ) -

is the virtual work done by internal forces. After employing the relation

∇‚(-P δr) ) -∇P‚δr - P(∇‚δr)

or, upon using the surface analogue of Gauss’ divergence theorem,53

(1)

where the first term on the right-hand side is the virtual work done by the external field, and the second term

)

(9)

(11)

∫∫∫V - ∇P‚δr dV + ∫∫∫V - P(∇‚δr) dV

δW(V) i

∫∫A∇2γ‚δr dA + ∫∫Aγ(∇2‚δr) dA ) ∫∫A∇2‚(γ δr) dA

1

P(3)(n1‚δr) dA1 (8) The corresponding surface phase variations involve both surface and line contributions and may, in analogy with eqs 2 and 5, be written as (50) Boruvka, L.; Rotenberg, Y.; Neumann, A. W. Langmuir 1985, 1, 40. (51) Boruvka, L.; Rotenberg, Y.; Neumann, A. W. J. Phys. Chem. 1985, 89, 2714. (52) Boruvka, L.; Rotenberg, Y.; Neumann, A. W. J. Phys. Chem. 1986, 90, 125.

3

∫

(1) (2) (3) L(γ m1 + γ m2 + γ m3 + σK)‚δr dL ) 0 (15)

With the variation δr unrestricted, each integrand must independently equal zero to satisfy the necessary condition δΩt ) 0. As expected, the Laplace equation of capillarity that is appropriate across each surface comes from the first three surface integrands. The last integrand yields a generalized form of the Neumann triangle relation

γ(1)m1 + γ(2)m2 + γ(3)m3 + σK ) 0

(16)

which represents the mechanical equilibrium balance between the surface tensions and the line tension at the contact line. (53) Weatherburn, C. E. Differential Geometry of Three Dimensions; Cambridge University Press: Cambridge, 1931; Vol. I, p 238.

Contact Line Quadrilateral Relation

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and

y direction: γ(1) sin θκ1 + γ(2) sin(θκ1 + θ12) - γ(3) sin θ3κ ) 0 (18)

Figure 2. Schematic of (a) a side-view of an axisymmetric liquid lens showing the arrangement of surface tensions and contact angles about the contact line and (b) a pictograph of the quadrilateral of forces that result in mechanical equilibrium at the contact line. When the line tension contribution σK is zero, one recovers the classical Neumann triangle pictograph.

Relation 16 may be represented graphically by adding the additional (line tension) vector σK to the triangle of surface tensions that occur in the Neumann triangle relation. The new shape becomes a quadrilateral with the four sides represented by three surface tensions and one line tension; cf. Figure 2. The quadrilateral is planar because we assumed that all surface tensions and the line tension were uniform and isotropic, which means that all tensions act in directions that are normal to the contact line tangent vector t ) n × m. The line curvature K is also normal to t. Given that the line tension σ always occurs in association with the line curvature K it may be easily seen that if K is small (that is, the radius of curvature of the contact line is large), then the linear term will make no significant contribution to the quadrilateral relation and the Neumann triangle relation will be recovered. Variants of the Neumann Quadrilateral Relation To be able to interpret relation 16 in the context of experimental situations, it is desirable to convert the planar vectorial relation that is given in eq 16 into two equivalent, orthogonal scalar relations. This is accomplished by projecting the vector expressions into two orthogonal directions to achieve independent and equivalent scalar force balance equations about the contact line. This may be demonstrated by selecting two directions such that one direction (denoted by the x unit vector) is along the direction in which the line tension force operates, that is, the direction in which the vector K points, and the other direction (denoted by the y unit vector) is perpendicular. With this choice of directions one may write the vector expression (eq 16) as two scalar expressions projected in the two directions

x direction: σK + γ(1) cos θκ1 + γ(2) cos(θκ1 + θ12) + γ(3) cos θ3κ ) 0 (17)

which eliminates the line tension term from the force balance equation in the y direction. Identical expressions, applicable for axisymmetric liquid lens systems, were obtained for the balance of forces in the radial and vertical directions.54 If the surface tensions can be obtained via another independent means, then it would be possible to use these relations (or a similar pair) to obtain the line tension, provided one can perform a measurement of two contact angles. While it may appear from eqs 17 and 18 that one may need to measure three angles, the angles are not completely independent but are related through the relation θ12 + θ23 + θ3κ + θκ1 ) 2π. Theoretically, the choice of angles is arbitrary, but the experimental situation may dictate a preferable pair of angles. This choice may be realized quite easily by manipulating the Neumann quadrilateral pictograph using a main diagonal to divide the quadrilateral into two triangles. Within each triangle, one can use the cosine rule to write down an expression for the square of the length of the diagonal by using the other two sides of the triangle and the contact angle between these sides. From the two triangles, an equation containing two contact angles, three surface tensions, and one line tension follows. For example, if one uses the contact angle θ12 (i.e., the angle between the unit vectors m1 and m2) and the contact angle θ3κ (i.e., the angle between the unit vector m3 and the direction of the line curvature K), then a scalar cosine rule relation can be written as (cf. Figure 2)

[γ(1)]2 + [γ(2)]2 + 2γ(1)γ(2) cos θ12 ) [γ(3)]2 + σ2K‚K + 2σγ(3)xK‚K cos θ3κ (19) It may be seen from a comparison of eq 19 and Figure 2b that the two contact angles occur at opposite corners of the quadrilateral. A similar equation may be written by using the remaining pair of opposite corner contact angles, that is, θ23 and θκ1. If one knows a priori that the line tension term, σK, is much smaller than the surface tension terms, then the square of the line tension term may be dropped in favor of the remaining terms and eq 19 simplifies to

[γ(1)]2 + [γ(2)]2 + 2γ(1)γ(2) cos θ12 ) [γ(3)]2 + 2σγ(3)xK‚K cos θ3κ (20) which may be used to obtain an estimate of the term σK provided the two selected angles can be measured. Wallace and Schu¨rch36,39 used an analogous approach to select a liquid-liquid-fluid system for estimating σ that approximately satisfied θ23 + θ3κ ≈ π so that they only needed to measure the angle θ12. Equation 20 is consistent with the classical form of the Neumann triangle relation, as may be seen by taking the limit σK f 0. Another scalar form of the force balance expression may be easily obtained from the quadrilateral diagram by expressing one of the four quantities (i.e., the three surface tensions and one line tension) in terms of the remaining three. If one were interested in obtaining the line tension (54) Gaydos, J.; Neumann, A. W. Thermodynamics of Axisymmetric Capillarity Systems. In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 2, cf. eqs 128 and 129.

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term as the dependent quantity, then the cosine rule in combination with the quadrilateral diagram (cf. Figure 2b) gives the expression 2

(1) 2

(2) 2

(3) 2

(1) (2)

σ K‚K ) [γ ] + [γ ] + [γ ] + 2γ γ (2) (3)

2γ γ

cos θ12 +

(3) (1)

cos θ23 + 2γ γ

cos θ31 (21)

where

θ31 ) θ3κ + θκ1

(22)

θ12 + θ23 + θ31 ) 2π

(23)

and

Equation 21 may also be obtained by summing the squares of eqs 17 and 18. Three additional equalities may be obtained from eq 21 by permuting the quantities σ2K‚K, [γ(1)]2, [γ(2)]2, and [γ(3)]2 and their corresponding contact angles. These expressions for the cosines of the contact angles are uniquely determined by the ratios of the surface and line tensions. Resolving the net force on any segment of the contact line in directions that lie perpendicular to the contact line and, respectively, in the three interfaces (i.e., directions given by m(i) and the one line curvature direction yields the relations

m(1) direction: γ(1) + γ(2) cos θ12 + γ(3) cos(θ3κ + θκ1) + σxK‚K cos θκ1 ) 0 (24)

γ(1) cos θ12 + γ(2) + γ(3) cos θ23 + σxK‚K cos(θ23 + θ3κ) ) 0 (25) m(3) direction: γ(1) cos(θ3κ + θκ1) + γ(2) cos θ23 + γ(3) + σxK‚K cos θ3κ ) 0 (26) K direction: γ(1) cos θκ1 + γ(2) cos(θκ1 + θ12) + γ(3) cos θ3κ + σxK‚K ) 0 (27) These four equations form a set of dependent homogeneous equations for the surface and line tensions because the determinants of the coefficients vanish; that is,

|

|

cos(θ3κ + θκ1) cos θκ1 cos θ23 cos(θ23 + θ3κ) 1 )0 cos θ23 cos θ3κ 1 cos(θκ1 + θ12) cos θ3κ 1 (28)

cos θ12

Application of the Neumann Quadrilateral Relation to Specific Geometries It is possible to demonstrate the utility of the various equivalent forms of the Neumann quadrilateral relations by considering several common geometries that have or could be used to measure the line tension. In particular, we shall consider (i) a liquid lens, (ii) a pendant drop in contact with a second immiscible liquid, and (iii) a coneshaped capillary. In each case, we shall give the corresponding expression for mechanical equilibrium at the contact line and illustrate how one could obtain an estimate of the line tension from the selected geometry. Arrangement (i): A Liquid Lens. One of the benefits of forming a contact line using only liquids is that the contact line involves only deformable liquid phases with smooth contact lines. Using conventional methods, liquidliquid and liquid-fluid surface tensions can be easily determined to a high degree of accuracy. As mentioned above, in order to apply the quadrilateral relation to a three-phase system, one has to determine at least two contact angles along with three surface tensions to evaluate the line tension. As depicted in Figure 2, for an axisymmetric liquid lens, the angles θκ1 and θ12 are probably the angles which are easiest to measure. The angle θ12 may be obtained by conventional optical methods while θκ1 can be measured through the observation of the slope of the tangent to the liquid lens surface against the horizontal. If these two angles are used, then the cosine quadrilateral relation (eq 21) would be rewritten by rotating the corresponding surface and line tension quantities to yield the expression

[γ(3)]2 ) σ2K‚K + [γ(1)]2 + [γ(2)]2 + 2σγ(1)xK‚K cos θκ1 + 2γ(1)γ(2) cos θ12 + 2σγ(2)xK‚K cos θ2κ (29)

m(2) direction:

1 cos θ12 cos(θ3κ + θκ1) cos θκ1

the ratios of the tensions rather than than the tensions themselves.

which demonstrates that any one of the relations listed in eqs 24-27 follows from the other three. As a consequence of their homogeneity, eqs 24-27 determine only

where the sum of the three contact angles equals 2π. Equation 29 is quadratic in the line tension σ, and its value may be obtained once the magnitude of the curvature |K| of the three-phase contact line is known. The radius of this curvature may be measured by determining the location of the contact line’s periphery so that an estimate of the diameter of the contact circle may be made. This could be accomplished either through a top view of the liquid lens or via proper optical reflection methods. Three potential situations exist depending upon the relative mass density of the three phases involved to form the contact line. If the lower liquid has a much higher density than both the lens liquid and the “bathing” fluid and the density of the lens liquid is only slightly higher than that of the “bathing” fluid, then it is unlikely that the lens deforms the fluid-liquid interface to a significant degree. In this situation, the three-phase contact line system would be similar to the situation of a sessile drop on a solid surface.36,39 The corresponding scalar relations simplify, and one may use eq 17 to write the single expression

σxK‚K + γ(1) cos θκ1 - γ(2) + γ(3) ) 0

(30)

as the only balance relation that applies (i.e., in a direction that is parallel to the horizontal top of the most dense phase). However, there are only a limited number of liquid lens systems that approximately satisfy the conditions necessary to use a sessile drop type approximation. To our knowledge, the first and only attempt using this approach was that of Wallace and Schu¨rch using a substrate-bathing fluid interface modified by a surfactant

Contact Line Quadrilateral Relation

Figure 3. Schematic of a pendant drop in contact with a second liquid. The third surface tension γ(3) represents the interfacial tension between the two liquids involved. The surface tensions γ(1) and γ(2) denote liquid-fluid surface tensions of the pendant drop and the second liquid, respectively. For an axisymmetric arrangement the line tension contribution σK would occur in the same plane as the contact line circle.

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and the ambient fluid) and then best-fit these images (i.e. curves) to the Laplace equation of capillarity. The intersection of the two Laplacian curves would be the position of the three-phase contact line in the cross section of the pendant drop. Once the intersection was found, the slope of the pendant drop profile at the contact line could be determined followed by the two contact angles θ12 and θκ1. The final parameter that needs to be determined is the curvature of the contact line, which may be easily obtained from a measurement of the distance between the two opposite (that is, opposite sides of the axisymmetric axis) intersections of the pendant drop surface with the lower liquid. This measurement would yield the diameter of the contact line. Finally, the line tension would be determined using eq 29. Arrangement (iii): A Cone-Shaped Capillary. A cone-shaped capillary system is another interesting experimental situation that is similar to the sessile drop arrangement for measuring line tension.12-14 The mechanical boundary conditions at the contact line follow eqs 17 and 18

γ(1) cos θ12 + γ(2) - γ(3) - σxK‚K cos θ3κ ) 0 (31) and

γ(1) sin θ12 + σxK‚K sin θ3κ - PN ) 0

Figure 4. Schematic of a cone-shaped capillary illustrating the direction of the line tension contribution σK that lies in the same plane as the contact circle, and the normal stress component PN of the solid surface.

monolayer (where an estimate of the monolayer tension γ(2) was required).36,39 On the other hand, the majority of liquid lens systems cannot be approximated in this fashion, so that one must use the more general Neumann quadrilateral expression, one form of which is given by eq 29. Arrangement (ii): A Pendant Drop in Contact with a Second Immiscible Liquid. An alternative experimental arrangement to measure the line tension involves lowering a pendant drop so that it touches a second liquid; cf. Figure 3. At the contact rim a three-phase contact line is formed between the two liquids and the ambient fluid. One of the key benefits of using a pendant drop is that the extent of contact between the two liquids can be fully controlled by adjusting the drop’s position. Furthermore, the liquid-liquid engulfing problem is avoided. If one knows the three surface tensions, then to obtain an estimate of the line tension, one would need to measure two angles. One strategy would be to measure directly the contact angles θ12 and θκ1 through a physical observation of the slope of the pendant drop at the three-phase contact line. Another strategy for evaluating these two contact angles is to employ the axisymmetric drop shape analysis (ADSA) methodology;55-57 one would obtain images of a part of the pendant drop profile and of the second liquid surface (which is between the second liquid (55) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (56) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1984, 102, 424.

(32)

where θ12 is the contact angle of the liquid against the solid surface, θ3κ is predetermined through the apex angle θA of the cone (θ3κ ) (π - θA)/2), and PN denotes the solid surface stress component in the normal direction. It is necessary to assume that a surface stress exists at the three-phase contact line; otherwise, eq 32 cannot hold, since the left-hand side of the equation is positive definite when PN is absent. The same situation exists in the case of a sessile drop where a normal component of the surface stress has to be assumed to balance the vertical force equation at the contact line. A relevant question relating to a sessile drop on a solid surface was raised by Bikerman.58,59 He objected to the Young equation by pointing out that the conventional proof of this equation was incorrect in that the vertical (to the solid surface) component of liquid surface tension γ(1), namely γ(1) sin θ, was unaccounted for. He further indicated that such a tension component had to be balanced by a force not of a surface origin. These concerns are resolved by eqs 31 and 32. In the experiment of the cone-shaped capillary, one has to measure the contact angle θ12 and the curvature of the contact line κ. θ12 may be obtained through a direct optical observation, and κ can be calculated through the diameter of the contact line which is horizontal. By using eq 31, the line tension can be calculated if the three surface tensions are known. The difficulty in finding the solid surface tensions γ(2) and γ(3) may be resolved by using an equation of state approach under certain conditions.60,61 (57) Lahooti, S.; del Rio, O. I.; Neumann, A. W.; Cheng, P. Axisymmetric Drop Shape Analysis (ADSA). In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 10. (58) Bikerman, J. J. Surface Chemistry, 2nd ed.; Academic Press: New York, 1958; p 340. (59) Bikerman, J. J. Proceedings of the Second International Congress of Surface Activity; Butterworth: London, 1957; Vol. III, p 125. (60) Spelt, J. K.; Li, D. The Equation of State Approach to Interfacial Tensions. In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 5. (61) Li, D.; Neumann, A. W. J. Colloid Interface Sci. 1992, 148, 190.

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Alternatively, the drop size dependence of the contact angle θ12 may be used.62 It is noted that, by changing the circular shape of the cross section to a, say, elliptical shape and observing the shape of the three phase line, one may study the line tension effect on the wetting and adhesion on the solid surface. It is anticipated that the distortion of the threephase contact line is affected by the magnitude of line tension as well as by the local curvature of the cone-shaped capillary. Conclusions The Neumann triangle relation is modified to account for line tension, using the free energy minimum principle. The mechanical equilibrium relation or boundary condition corresponds to the classical Neumann triangle relation but is modified by the addition of the term σK, which converts the triangle of forces to a quadrilateral. From the quadrilateral pictograph it becomes easy to derive (62) Gu, Y.; Li, D.; Cheng, P. J. Colloid Interface Sci. 1996, 180, 212.

Chen et al.

various forms of the mechanical equilibrium conditions at the contact line and to evaluate which form(s) would be appropriate for a particular experimental situation. Several common geometries are discussed and strategies for determining the line tension are proposed. In all cases involving deformable liquid phases (e.g. a liquid lens system), the line tension may be obtained provided two contact angles in addition to three surface tensions are determined. For liquids in contact with ideal solid surfaces, the line tension may be measured after measuring one contact angle and three surface tensions or, alternatively, the drop size dependence of the contact angle and the surface tension of the drop. Acknowledgment. This investigation was supported by the Natural Science and Engineering Research Council of Canada (NSERC) through a Carleton University GR-5 Grant and two NSERC grants, OGP 0155053 (J.G.) and A8278 (A.W.N.). One of the authors (P.C.) is grateful for a University of Toronto Open Fellowship. LA960291I