Generalized Mechanical Equilibrium Condition for Multiphase Contact

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Generalized Mechanical Equilibrium Condition for Multiphase Contact Lines and Multiphase Contact Points Mohammad R. Shadnam and Alidad Amirfazli* Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 2G8, Canada Received August 20, 2002. In Final Form: February 18, 2003 Multiphase contact lines are present in a number of practical cases, and they are also interesting from a theoretical perspective. The equilibrium relation for a multiphase contact line and multiphase contact point is presented using a thermodynamic free-energy formulation. Calculus of variation is applied to the free-energy functional of a multiphase system, and vectorial mechanical equilibrium conditions for contact lines and points are derived. The developed relation for the multiphase contact line could be regarded as the generalization of the classical Neumann triangle relation. It is shown that the mechanical equilibrium condition for a multiphase contact line may graphically be represented by a planar polygon, whose sides consist of the surface and line tensions. The stated mechanical equilibrium condition for a multiphase contact point can be regarded as the zeroth-order analogue of the Laplace equation (second order) and Neumann triangle relation (first order). It is also shown that the mechanical equilibrium condition for a multiphase contact point may graphically be represented by a nonplanar polygon, whose sides consist of the line tensions. To demonstrate the application of the above relations, a model lamina system is discussed in terms of the stability of the multiphase contact lines.

1. Introduction Equilibrium configurations of thin-film laminas (e.g., open-cell and closed-cell foams, as shown in Figure 1) under different boundary conditions are of significant interest in different branches of science and technology. Whereas mathematicians look at these spatial configurations as the solutions of several calculus-of-variation problems1 and minimal surfaces,2 structural engineers look at them as the solutions of complicated optimization problems (e.g., partitioning of the structures for parallel computation).3,4 The equations that describe the equilibrium shape of thin films are also in analogy with the equations governing the temperature field in a circular region generating uniform heat5 and torsion of bars,6 which is of interest to mechanical engineers. Thus, any attempt in the formulation and solution of problems related to equilibrium in such multiphase systems could potentially be important in various fields. Contact lines among more than three bulk phases are not as common as three-phase contact lines but occur in certain practical cases, for example, aluminum-foam and carbonate-beverage production. Thus, a general treatment of the multiphase systems is needed. Such a study would also be interesting from a theoretical point of view. An example that involves several lamina interfaces is foam. In our analysis, the lamina in foam is idealized as a thin interface (negligible thickness) similar to the model * Corresponding author. Telephone: (780) 492-6711. Fax: (780) 492-2200. E-mail: [email protected]. (1) Miliutin, A. A.; Providence, R. I. Calculus of Variations and Optimal Control; American Mathematical Society: New York, 1998. (2) Nitsche, J. C. C. Lectures on Minimal Surfaces; Cambridge University Press: New York, 1989; Vol. 1. (3) Kaveh, A. Comput. Methods Appl. Mech. Eng. 1984, 20, 983998. (4) Kaveh, A. Optimal Structural Analysis; Research Studies Press: London, 1997. (5) Schneider, P. J. Conduction Heat Transfer; Addison-Wesley: Cambridge, MA, 1957. (6) Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. Advanced Mechanics of Materials; John Wiley: New York, 1993.

Figure 1. (a) Open-cell foam (Melamin foam; courtesy of the Johns Manville Corporation, Littleton, U.S.A.). (b) Closed-cell aluminum foam.

proposed by Derjaguin7 for the analysis of thin films. The location of the channels (Plateau’s border) that play an important role in the drainage mechanism of foams8 is represented by linear confluence zones of 0 thickness. In the structure of foam, one could find several contact lines, each shared by a number of interfacial surfaces, as shown in Figure 1b. The equilibrium of the contact lines and contact points, which, for example, are necessary conditions for the stability of foams, is the concern of the present paper. The above mechanical equilibrium conditions in their general form are developed using a thermodynamic approach. An interfacial surface is the geometric intersection of two different bulk phases. A contact (or common) line is (7) Derjaguin, B. V.; Churaev, N. V. Kolloidn. Zh. 1976, 38, 438448. (8) Adamson, A. W. Physical Chemistry of Surfaces; John Wiley: New York, 1990.

10.1021/la0264421 CCC: $25.00 © 2003 American Chemical Society Published on Web 04/23/2003

Generalized Mechanical Equilibrium Condition

Figure 2. Schematic of laminas (assumed to have thickness of 0) that are formed in a wire frame by dipping the wire frame into a surfactant solution. (a) Three-dimensional schematic. (b) Front view of the wire frame and laminas. (c) Front view in which c is vanished. In this case, the contact line (perpendicular to the paper) is created by the intersection of four laminas.

the common intersection of at least three different bulk phases. A contact line could also be thought of as the common intersection of three or more interfacial surfaces, a special case of which is presented in Figure 2c. Topologically, a one-dimensional manifold (i.e., a curve) could be the intersection of an infinite number of twodimensional manifolds (i.e., surfaces). A contact point, in a three-dimensional capillary system, could also be thought of as the common intersection of four or more contact lines (Figure 2a). In the thermodynamic treatment of multiphase systems, Gibbs9,10 assumed the very thin interface region between the two bulk phases in which all transitions occur, to a first approximation, as a two-dimensional mathematical boundary between two bulk phases (dividing surface). Both bulk phases are assumed to extend uniformly right up to the dividing surface.9-11 The descriptive formalism of equilibrium thermodynamics for fluid systems is based on the fundamental equations12 and a minimum principle. For multiphase fluid systems, both the concept of the surface tension and the concept of the line tension arise (9) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs; Dover: New York, 1961; Vol. 1. (10) Donnan, F. G. A Commentary on the Scientific Writings of J. Willard Gibbs; Yale University Press: New Haven, CT, 1936; Vol. 1. (11) Gaydos, J.; Boruvka, L.; Neumann, A. W. Langmuir 1991, 7, 1035-1038. (12) Callen, H. B. Thermodynamics: an introduction to thermo statistics; John Wiley: New York, 1985.

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from a fundamental-equation analysis when applied to systems with multiphase boundary manifolds. The mechanical equilibrium equations were first derived and presented for surfaces (Laplace equation) and three-phase common lines (Young equation) by Gibbs under the assumption of moderate curvatures.9 The graphical representation of the mechanical equilibrium condition for the three fluid common lines was then called the Neumann triangle.13 Buff and Saltsburg generalized the classical theory of capillarity by removing the assumption of moderate curvature.14-16 A comprehensive formulation of the theory of capillarity applicable to different intersections (i.e., surfaces, lines, and points) based on fundamental equations was presented in 1977.17 The presented formulation by Boruvka and Neumann17 is not straightforward, and its application to practical cases is cumbersome.18 In an attempt to facilitate application of the general formulation presented in ref 17, Chen et al.18 recently developed a quadrilateral representation of the mechanical equilibrium condition for three-phase contact lines, which included the line tension effect.19-21 However, in their study the level of generality presented in ref 17 was limited to a three-phase system, and the question of developing a practical mechanical equilibrium formulation to study multiphase systems remained unanswered. The mechanical equilibrium relations presented here for contact lines is not restrictive in the number of phases considered; it views the contact line as the intersecting manifold of an unlimited number of phases (or spaces in mathematical terms). Furthermore, this paper also presents the mechanical equilibrium condition for a multiphase contact point and presents an easy way to understand the vectorial graphical representation. A point to note is that in this paper we have treated the problem using a Gibbsian approach, which precludes the description of a system at the molecular level; that is, similar to any other macroscopic thermodynamic formulation, the role of long-range forces, finite sizes of molecules, and thermal fluctuations at any confluence zone (e.g., interface) cannot be described. The formulism here is a mathematically convenient approach to describe the macroscopic mechanical equilibrium conditions at the dividing surfaces, lines, or points, and it is not intended to describe the force balance for systems where the dimensions approach the molecular size. At the experimental level, a number of arrangements have been considered for the study of equilibrium at the contact lines.22,23 A statistical analysis of the equilibrium configurations of laminas in foam and the validity of the mechanical equilibrium conditions are presented in refs 8 and 24 and the references therein. (13) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longman: London, 1966. (14) Buff, F. P. J. Chem. Phys. 1956, 25, 146. (15) Buff, F. P.; Saltsburg, H. J. Chem. Phys. 1957, 26, 23. (16) Buff, F. P.; Saltsburg, H. J. Chem. Phys. 1957, 26, 1526. (17) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 54645476. (18) Chen, P.; Gaydos, J.; Neumann, A. W. Langmuir 1996, 12, 59565962. (19) Amirfazli, A.; Ha¨nig, S.; Mu¨ller, A.; Neumann, A. W. Langmuir 2000, 16, 2024-2031. (20) Chen, P.; Susnar, S. S.; Amirfazli, A.; Mak, C.; Neumann, A. W. Langmuir 1997, 13, 3035-3042. (21) Amirfazli, A.; Chatain, D.; Neumann, A. W. Colloids Surf., A 1998, 142, 183-188. (22) Scheludko, A.; Toshev, B. V.; Platikanov, D. In The Modern Theory of Capillarity; Goodrich, F. C., Rusanov, A. I., Eds.; AkademieVerlag: Berlin, 1981; p 163. (23) Pompe, T.; Herminghaus, S. Phys. Rev. Lett. 2000, 85, 19301933. (24) Matzke, E. B. Am. J. Bot. 1946, 33, 58-80.

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surfaces. To avoid confusion, in this text all the labels are congruent mod n, that is, when an index assumes a value of n + 1, it means that the index value is 1, and when an index assumes the value of 0, it means that the index value is n.25 The total free energy, Ωt, for the system shown in Figure 3a, which can be assumed enclosed with an arbitrary boundary, is n

Ωt )

∑ i)1

n

ΩVi +

ΩA + ΩL ∑ i)1 i

(1)

where the first set of terms represents the bulk energies, ΩVi, associated with each of the n bulk phases that share a common line; the second set of terms, ΩAi, represents the surface energies; and the last term, ΩL, represents the line energy associated with the common line. When the definition for the canonical free energy is considered and arbitrary variations of the position vector b r inside the system shown in Figure 3a is allowed for, using the divergence theorem,26 the following can be derived:18,27-29 Figure 3. Schematic of a multiphase contact line (the contact line is perpendicular to the plane of the paper) together with the polygon of the forces that represent the mechanical equilibrium at the contact line. The bulk phases are labeled clockwise with bold characters; the interfaces are labeled with regular characters. (a) General case (polygon). (b) Special case of the polygon for a three-phase contact line (quadrilateral). (c) Special case of the three-phase contact line for moderate curvatures (triangle). It should be noted that the schematic illustrates the mathematical model used (i.e., Gibbs’) to describe the system from a macroscopic perspective.

Variational methods are applied to the free-energy functional, and the vectorial mechanical equilibrium equation is derived for the multiphase contact lines. The graphical representation of the mechanical equilibrium condition for multiphase contact lines developed here is a polygon. The polygon sides consist of surface tension vectors and one line tension vector. The developed relation is a generalization of both the classical Neumann triangle relation13 and the recently developed contact line quadrilateral relation.18 The polygonal relation developed can be used for the common line of several bulk phases and includes the effect of the line tension. The vectorial mechanical equilibrium equation is stated for the contact points that are shared among several interfaces and bulk phases. The equilibrium condition can be regarded as the zeroth-order analogue of the Laplace equation (second order) and Neumann triangle relation (first order). It is shown that the mechanical equilibrium condition may be represented graphically by a polygon that is not necessarily planar and whose sides consist of line tension vectors.

∫A Pi(NB i‚δrb) dAi + ∫A

δΩVi ) -

i

Pi(N B i+1‚δr b) dAi+1 (2)

i+1

where Pi is the internal pressure of the ith bulk phase and N B i is the unit outward normal to the ith surface at the common line; the integration is performed on the interfaces. The dAi represents the differential area of the ith surface. The variation of the excess free energy associated with the surfaces after applying the divergence theorem could similarly be written as

∫A γiJi(NB i‚δrb) dA - ∫Lγi(mb i‚δrb) dL

δΩAi ) -

i

(3)

In eq 3, γi is the surface tension of the ith surface, Ji is the mean curvature of ith surface, and m b i is the unit tangent of the ith surface at the common line. The dL represents the differential length of the common line. The sign of Ji depends on the orientation of the unit normal N B i on the surface. The sign convention used by Weatherburn30 was adopted. The variation of the excess free energy associated with the common line where there is no contact point in the system is given by

δΩL ) δ

∫Lσ dL ) -∫LσkB‚δrb dL

(4)

2. Derivation of the Equilibrium Relation for Contact Lines

where σ is the line tension and B k is the curvature vector of the common line. Using eqs 2-4 as expressions of the free-energy variation for the bulk, surface, and line phases and substituting them into the necessary equilibrium condition, δΩt ) 0,

In the idealized structure of foams shown in Figure 1, as discussed in the Introduction, one could find several contact lines that are shared by more than three laminas. The mechanical equilibrium condition for such contact lines is derived in this section. The methodology adopted is the same as the one outlined by Chen et al. in ref 18. In its general form, a multiphase contact line and the constituent phases are shown in Figure 3a. In Figure 3a, the bulk phases are labeled clockwise, and every ith phase is assumed to be located between the ith and (i + 1)th

(25) Adams, W. W.; Goldstein, L. J. Introduction to Number Theory; Prentice Hall: Englewood Cliffs, NJ, 1976. (26) Salas, S. L.; Hille, E.; Garret, J. E. Salas and Hille’s calculus: several variables; John Wiley: New York, 1995. (27) Boruvka, L.; Rotenberg, Y.; Neumann, A. W. Langmuir 1985, 1, 40. (28) Boruvka, L.; Rotenberg, Y.; Neumann, A. W. J. Phys. Chem. 1985, 89, 2714. (29) Boruvka, L.; Rotenberg, Y.; Neumann, A. W. J. Phys. Chem. 1986, 90, 125. (30) Weatherburn, C. E. Differential geometry of three dimensions; Cambridge University Press: Cambridge, U.K., 1930; Vol. 1.


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to the familiar classic Neumann triangle for three-phase systems (cf. Figure 3c).

yields n

δΩt )

∑ i)1

n

δΩVi +

δΩA + δΩL ) ∑ i)1 i

3. Scalar Forms of the Equilibrium Relation for Contact Lines

n

[-∫A Pi(N B i‚δr b) dAi + ∫A ∑ i)1 i

Pi(N B i+1‚δr b) dAi+1 -

i+1

∫A γiJi(NB i‚δrb) dAi - ∫Lγi(mb i‚δrb) dL] ∫LσkB‚δrb dL ) 0 i

(5)

Rearranging the terms of eq 5 results in

X direction:

n

∫ ∑ i)1 {-

Because the measurements could hardly be performed in vectorial forms, developing the scalar forms of the vectorial relation presented in eq 7 is more useful. Taking the direction represented by the line tension vector to be the X direction and the counterclockwise perpendicular to it to be the Y direction, one may write the vector expression as two scalar expressions (the angle next to the ith bulk phase is called θi, compare with Figure 3a).

[γiJi + (Pi - Pi-1)](N B i‚δr b) dAi} A

n

i

σκ +

n

∫L(∑γimb i + σkB)‚δrb dL ) 0

(6)

(8)

Y direction:

i)1

To satisfy the equilibrium condition, that is, δΩt ) 0, each of the integrands in eq 6 should be equal to 0 if δr b is allowed to take any arbitrary value and direction (subject to the phase property constraints). The vanishing of the first summation reveals the mechanical equilibrium condition governing the geometry of the interfaces of any two adjacent bulk phases, which is a generalized form of the Laplace equation for the problem under consideration.17,18 The last integral yields a generalized form of the Neumann triangle relation, which represents the mechanical equilibrium balance among the surface and line tension force vectors at the common line. n

γim b i + σk B)0 ∑ i)1

i-1

γi cos(R + ∑θj) ) 0 ∑ i)1 j)1

(7)

The graphical representation of the mechanical equilibrium condition (eq 7) is a polygon with sides represented by surface tension related vectors, except for one, which is a line tension vector (σκ), as shown in Figure 3a. Constituent vectors of the polygon lie in a plane because the surface tensions act normal to the surface boundary manifold, which is the contact line. The line tension related vector also acts along the line curvature vector. The line curvature vector is always perpendicular to the tangent vector of the line.26 Thus, all the tensions act in directions that are perpendicular to the contact line tangent vector. When eq 7 is applied to the usual threephase contact line, the polygon simplifies to a quadrilateral, as shown in Figure 3b. This quadrilateral representation is the mechanical equilibrium condition that was presented by Chen et al.18 in their analysis. For a radius of curvature less than 2-3 mm, it is shown that the line tension can have a significant effect on the contact angles.19,20,31,32 Given that the line tension term, σ, is always present in association with the line curvature term, κ, it may be seen that if κ is small, that is, the radius of curvature is large (R > 5 mm), the line tension related vector (σk B) becomes negligible. When the line tension term is neglected, the above generalized polygon (eq 7) converts (31) Duncan, D.; Li, D.; Gaydos, J.; Neumann, A. W. J. Colliod Interface Sci. 1995, 169, 256. (32) Gaydos, J.; Neumann, A. W.; In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Marcel Dekker: New York, 1996; Chapter 4.

n

∑ i)1

i-1

γi sin(R +

θj) ) 0 ∑ j)1

(9)

In the previous relations, R, as shown in Figure 3a, represents the angle between the line tension vector, that is, the line curvature vector, and the γ1 vector. Equations 8 and 9 are the generalized forms for both the quadrilateral relation32 and the Neumann triangle.13 By taking the γi values to be 0 for i values greater than 3 and further taking σ to be equal to 0, the scalar forms for the quadrilateral and Neumann triangle relations, respectively, will be recovered. In the case where one of the phases is rigid, the common line is restricted to the rigid surface. If the rigid surface is planar, then δr b as well as B k in eqs 2-6 cannot have any component perpendicular to the rigid surface. Thus, eq 7 is only valid for the projections of the vectors on the rigid surface. In such cases, eq 9 is meaningless and the only mechanical equilibrium equation for the contact line is eq 8. By neglecting the line tension effects and applying eq 8 to the common case of the contact line of three bulk phases (e.g., a sessile drop placed on a solid surface surrounded by the liquid’s vapor), the classical Young’s equation recovers. When the cosine rule was used in the case of the three-phase contact line, several scalar exact and approximate alternative relations could also be obtained, which can be found in refs 18 and 32. The force equilibrium condition for any differential length (dL) of the contact line yields a symmetric scalar form of the mechanical equilibrium condition for the contact line. The directions of the forces lie perpendicular to the contact line, and they are tangent to the two-phase surfaces, that is, the directions given by m b i and the line curvature direction, B k (cf. eq 7). All of the forces are projected in the m b i direction (i ) 1, 2, ..., n) and B k direction. The resultant net force will vanish if the line is in mechanical equilibrium.

m b i direction (i ) 1, 2, ..., n): i-1

∑ j)1

i-1

∑ k)j

γj cos(

n

θk) + γi +

∑

j-1

θk) + ∑ k)i

γj cos(

j)i+1

i-1

σκ cos(R +

∑ θk) ) 0 k)1

(10)

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B k direction: n

frame that is calculated using simple geometrical relations).

i-1

γi cos(R + ∑θj) + σκ ) 0 ∑ i)1 j)1

(11)

The set of n + 1 equations described by eq 11 is a set of dependent homogeneous equations for the surface tensions and σκ because there exist only two independent equations in a plane for the equilibrium of a point. Thus, the necessary condition for the system to have a nontrivial solution is that the determinant of the coefficients vanishes.

|

1 cos θ1 cos(θ1 + θ2) · · ·

cos θ1 1 cos θ2

cos(θ1 + θ2) cos θ2 · · ·

· · ·

1

|

) 0 (12)

Because of the determinant homogeneity, the previous set of equations determines only the ratios of the tensions rather than the tensions themselves. For the orthogonal directions, a similar n-by-n determinant could be derived that contains sine terms instead of cosine ones as another variant. If all the polygon dimensions except one were known (e.g., by the measurement of the interfacial tensions), then it will be possible to use eqs 8 and 9 to measure the line tension of a multiphase contact line. Provided that the proper experimental arrangement is conceived to allow the obtainment of the relevant geometrical information (e.g., by taking a series of photographs), eqs 8 and 9 can be developed for any choice of the coordinate systems; not all mathematically acceptable pairs of equations may be of interest in practice, however. 4. Application of the Generalized Mechanical Equilibrium Conditions to an Idealized Lamina System A contact line that is shared by more than three surfaces is not an imaginary and inaccessible concept. A multiphase contact line of intersecting soap films can be produced by simply dipping a rectangular-cubic or octahedral wire frame into an aqueous surfactant solution. The usual lamina configuration is shown in Figure 2a,b; a less likely configuration is the one shown in Figure 2c. The laminas produced by simply dipping a rectangularcubic wire frame into an aqueous surfactant solution is considered (an ideal schematic is shown in Figure 2). Using an energy approach, the equilibrium and stability of the multiphase contact lines are studied for the idealized system. The results are then compared with the predictions from equations obtained in the previous section and literature. The total energy of the system (U) consists of volume-, surface-, and line-energy terms. When 0 thicknesses for the surfaces are assumed, the total volume is constant; therefore, the volume-energy term can be neglected in a variational analysis. Thus, the total energy for the system will be the summation of the surface and line energies (note that the surface and line energies are proportional to the surface area and line length, respectively). If the length of the frame (l) is much larger than the dimensions a and b, the total energy for the system, per unit length of the frame, can be calculated using eq 13 (the term in square brackets is the surface area per unit length of the

U ) [4xa2 + (b - c)2 + 2c]γ + 2σ

(13)

In eq 13, a, b, and c are the geometrical dimensions given in Figure 2b. Equation 13 can be nondimesionalized by dividing both sides by 2bγ to consider the effect of the frame aspect ratio on the stability of the quadratic contact line (a contact line that is shared by four interfaces, i.e., where the dimension, c, defined in Figure 2b, vanishes). With the normalized total energy denoted by U h , the aspect ratio of the frame a/b by g, and the c/b ratio by d, eq 13 takes the form of

U h ) [2xg2 + (1 - d)2 + d] + σ/bγ

(14a)

Knowing that the surface tension is normally on the order of ∼10-2 mN/m and line tension is on the order of 10-6 mN or less,33 and assuming b is on the order of a centimeter, one finds that the line tension term in eq 14a is at least 2 orders of magnitude less than the surface term and, thus, could safely be neglected.

U h ) 2xg2 + (1 - d)2 + d

(14b)

Figure 4a shows the variation of the free energy for the idealized lamina system with respect to the aspect ratio g and the parameter d (representing the possibility of the formation and stability of a quadratic contact line). Location of the minimum energy, at a constant aspectratio value, corresponds to the equilibrium configuration for the lamina system. For example, Figure 4b depicts two cross sections of the three-dimensional free-energy variation surface shown in Figure 4a at the aspect ratios g of 1.2 and 1.8. As can be seen in Figure 4b, the minimum energy for the associated curve with the aspect ratio of 1.8 occurs at d () c/b) ) 0. This indicates that a quadratic contact line configuration (c ) 0, cf. Figure 2c) represents the equilibrium shape for the lamina system. Similarly, the curve for g ) 1.2 in Figure 4b shows that at the aspect ratio of 1.2, the quadratic contact line is not a stable configuration because the minimum energy is associated with a positive value for d () c/b); in this case, the stable configuration is shown in Figure 2b. To consider the equilibrium configuration(s) at a given aspect ratio, the extremum of the free-energy function for the particular aspect ratio should be found. The local of the extremum for the free-energy function of the system is plotted in Figure 5 (solid line). It can be seen that for the aspect ratios greater than x3, the local of the minimum free energy does not correspond to positive values for d. This means that the absolute minimum for the free energy is at d ) 0. The 0 value of d corresponds to the quadratic contact line (d ) c/b ) 0). Thus, for the aspect ratios greater than x3, the quadratic contact line is an equilibrium configuration, and for g values less than 1/x3, the pattern with two triple lines (Pattern I, cf. Figure 5) is the stable equilibrium configuration. A similar analysis has been performed for the 90°-rotated lamina pattern (Pattern II, cf. Figure 5). The dashed line in Figure 5 shows the local of the minimum energy for Pattern II. It can be seen that for the aspect ratios less than 1/x3 the quadratic contact line is a stable equilibrium configuration, and for g values greater than 1/x3, the pattern with two (33) Amirfazli, A.; Neumann, A. W. Adv. Colloid Interface Sci., in preparation.


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Figure 5. Local of the stable equilibrium configurations for the surfactant-film system in the wire frame for different aspect ratios g. The negative values for d correspond to a stable quadratic contact line. At the aspect ratios between 0 and 1/ x3 and g values greater than x3, the quadratic contact line corresponds to an equilibrium configuration. At the aspect ratios between 1/x3 and x3, three equilibrium configurations are possible: the stable Patterns I and II and the unstable quadratic contact line.

Figure 4. (a) Nondimensionalized potential surface for the surfactant film. (b) Cross sections of the surface depicted in part a at g ) 1.8 and 1.2. The curve associated with the aspect ratio of 1.8 has 0 for its minimum value of the d parameter; therefore, the quadratic contact line corresponds to an equilibrium position. The curves show that the quadratic contact line does not correspond to an equilibrium position for the aspect ratio of 1.2 because the minimum potential occurs for some positive values of the parameter d.

triple lines (Pattern II) is the stable equilibrium. For the g values between 1/x3 and x3, three equilibrium configurations are possible: the stable Patterns I and II and the unstable quadratic contact line. This result is in agreement with the experimental results described in ref 8 (pp 544-546) and the references therein. The obtained final configuration of the laminas in the wire frame is the same as that for the constrained minimal surfaces, obtained using pure algebraic and calculus-of-variation methods.34 Above, we demonstrated the possibility of the existence of a quadratic contact line for the system under consideration. Equations 8 and 9 from the previous section are applied to the case of the quadratic contact line to verify whether these equations can correctly describe the equilibrium configuration for the system. With the fact that the contact line has no curvature noted, eqs 8 and 9 will be simplified to

γ1 + γ2 cos θ1 + γ3 cos(θ1 + θ2) + γ4 cos(θ1 + θ2 + θ3) ) 0 (15) γ2 sin θ1 + γ3 sin(θ1 + θ2) + γ4 sin(θ1 + θ2 + θ3) ) 0 (16)

Figure 6. Equilibrium of the surface tensions at the quadratic contact line in the wire frame shown in Figure 2. The contact line is normal to the plane of the paper.

With the knowledge that all interfaces are in thermodynamic equilibrium and have the same chemical properties, that is, the interfaces all have the same surface tension, eqs 15 and 16 will simplify to

1 + cos θ1 + cos(θ1 + θ2) + cos(θ1 + θ2 + θ3) ) 0 (17) sin θ1 + sin(θ1 + θ2) + sin(θ1 + θ2 + θ3) ) 0

(18)

The correctness of the equilibrium conditions (i.e., eqs 17 and 18) is verified if the described geometrical relation by contact angles θ1-θ4 would conform to the geometry of a quadratic contact line formed in the system under consideration. On the basis of the geometry of the wire frame (Figure 6), one will find

tan(θ1/2) ) tan(θ3/2) ) cot(θ2/2) ) cot(θ4/2) ) b/a (19) Substitution of angles θ1-θ4 from eq 19 into eqs 17 and 18 will satisfy the equilibrium conditions derived in the pervious section. Hence, the applicability of the generalized mechanical equilibrium condition for multiphase contact lines (eqs 8 and 9) is verified for the system described above. (34) Taylor, J. E. Ann. Math. 1976, 103, 489.

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(associated with any arbitrary ordering of the contact lines). The Y direction is counterclockwise perpendicular to the X direction in the plane of the second line tension vector. The Z direction will be the cross product of the unit vectors along the aforementioned two directions. The vector expression, that is, eq 20, can then be stated as three scalar expressions projected in the three directions.

X direction: n

σ1 +

σj cos θj1 ) 0 ∑ j)2

(21)

Y direction: n


(22)

Z direction: Figure 7. (a) Graphical representation of the mechanical equilibrium condition (eq 20) for contact points; the sides represent the line tensions. (b) Polygon of the line tensions for a common point of four confluent contact lines. (c) Spatial representation of the contact-point force balance at the common intersection of four contact lines.

5. Equilibrium Relation for Contact Points Aside from the confluence zones represented by surfaces and contact lines, a system of at least four bulk phases can contain yet another type of confluence zone, that is, those represented by contact points. Contact points could easily be detected in three-dimensional foam structures, as shown in Figure 1. They could also be experimentally produced by simply dipping a wire frame into the aqueous solution of surfactants, for example, soap (Figure 2a). Adopting the same variational algorithm mentioned in the previous sections to minimize the total free energy, one will arrive at the mechanical equilibrium condition for a common point, which is the intersection point of n contact lines (cf. eq 20). Equation 20 was previously derived by Boruvka and Neumann.17 n

σiBt i ) 0 ∑ i)1

(20)

In eq 20, σi is the line tension associated with the ith contact line, and the unit tangent to the ith contact line ending at a common point at the intersection of common lines is denoted by Bti (i ) 1, 2, ..., n). Unlike the surface tension, the line tension may be positive or negative.33,35,36 For ease of understanding the above equilibrium condition, eq 20 may be represented graphically; the shape is a polygon with its sides represented by line tension vectors, as shown in Figure 7a. The polygon is not planar in general. Because the goal of the paper is to present an experimentally relevant general formulation for the mechanical equilibrium conditions, in the next section scalar forms of the above vectorial equilibrium condition (eq 20) are presented. 6. Scalar Forms of the Equilibrium Relation for Contact Points To construct the scalar expressions, one needs to first define a coordinate system. The X direction is taken as the direction represented by the first line tension vector (35) Widom, B. Mol. Phys. 1999, 96, 1019. (36) Van Giessen, A. E.; Bukman, D. J.; Widom, B. Mol. Phys. 1999, 96, 1335.

n


(23)

where cosθj1, cosθj2, and cosθj3 are the first, second, and third cosine directors of the jth line tension vector with respect to the defined coordinate system. The orientation of the coordinate system could be arbitrary, but the experimental factors, for example, observations by photography, may dictate a preference. For simplicity, consider the case of four confluent contact lines (e.g., Figure 7b). The equilibrium condition (eq 20) requires that the line tension vectors form a closed polygon (tetragon in this case). Drawing a diagonal line in Figure 7b will divide the line tension tetrahedral into two triangles. Within each triangle, the cosine rule can be used to derive an expression for the square of the length of the diagonal by using the other two sides of the triangle and the angle between these sides. For example, if one uses the angle θ12, that is, the angle between the unit vectors Bt1 and Bt2 in the plane defined by these two vectors, and the angle θ34, that is, the angle between the unit vectors Bt3 and Bt4 in the plane defined by these two vectors, then a scalar cosine-rule relation can been derived as

σ12 + σ22 + 2σ1σ2 cos θ12 ) σ32 + σ42 + 2σ3σ4 cos θ34 (24) It may be seen from a comparison of eq 24 and Figure 7b that the two angles, θ12 and θ34, occur at opposite corners of the spatial quadrilateral. A similar equation may be written by using the remaining pair of opposite corner angles as another scalar form.

σ22 + σ32 + 2σ3σ2 cos θ32 ) σ12 + σ42 + 2σ1σ4 cos θ14 (25) A more symmetric scalar form of the force-balance expression may be derived by convolving eq 24 with eq 25.

σ22 + σ1σ2 cos θ12 + σ3σ2 cos θ32 ) σ42 + σ3σ4 cos θ34 + σ1σ4 cos θ14 (26) A general determinant form of the equilibrium equations can also be developed for the common point of n confluent contact lines. The force equilibrium condition for the contact point yields a symmetric scalar form of the mechanical equilibrium condition for the contact point.


Langmuir, Vol. 19, No. 11, 2003 4665

The direction of the ith force is tangent to the ith contact line at the common point, that is, the directions given by Bti. All of the forces are projected in Bti direction for i ) 1, 2, ..., n. The resultant net force will vanish if the common point is in equilibrium. n

σj(Bt i‚Bt j) ) 0 ∑ j)1

(27)

In eq 27, the dot product is used to project the spatial line tension vector in Bti direction.26 The set of n equations described by eq 27 is a set of dependent homogeneous equations for the line tensions (n > 3) because there exist only three independent equilibrium equations in a threedimensional space for a point. Thus, the necessary condition for the system to have a nontrivial solution is that the determinant of the coefficients vanishes for n > 3, that is, assuming the Bti vectors represent unit direction vectors.

|

1 Bt 2‚Bt 1 Bt 3‚Bt 1 · · ·

Bt 1‚Bt 2 1 Bt 3‚Bt 2

Bt 1‚Bt 3 Bt 2‚Bt 3 · · ·

· · ·

1

|

)0

(28)

Because of their homogeneity, the above set of equations determines only the ratios of the line tensions rather than the tensions themselves. Such formulation may be useful for the study of idealized foam systems and their stabilities.

7. Conclusions On the basis of a generalization of the Gibbs classical theory of capillarity, both the Neumann triangle relation13 and the newly developed quadrilateral relation18 are extended to account for contact lines that are shared among several bulk phases using variational methods. The mechanical equilibrium relation for multiphase contact lines that could be considered the boundary condition of the Laplace equation was generalized by the addition of several terms, each of which corresponds to one of the tensions in the surfaces that share the contact line. The addition of these terms changes the classic triangle or quadrilateral graphical representation to a polygonal graphical representation. The newly developed contact-point equilibrium relation describes the mechanical equilibrium condition for contact points, and it could be thought of as the boundary condition for the classical Young equation or any version of its generalizations. In the case of the contact point formed by four contact lines, the polygon simplifies to a spatial quadrilateral of the line tensions. Based on the quadrilateral geometry, various scalar forms of the presented mechanical equilibrium conditions at the contact point were derived. The experimental conditions could dictate which forms of the above relations would be appropriate for each application. Acknowledgment. Support from NSERC and Alberta Ingenuity is gratefully acknowledged. M.R.S. also acknowledges the discussions with D. Y. Kowk. LA0264421

Generalized Mechanical Equilibrium Condition for Multiphase Contact

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