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On the Analogy between Lateral Capillary Interactions and Electrostatic Interactions in Colloid Systems Vesselin N. Paunov* Surfactant Science Group, Department of Chemistry, University of Hull, Hull, HU6 7RX, U.K. Received April 8, 1998. In Final Form: June 22, 1998 A formal analogy between the linear theories of the lateral capillary interaction between solid bodies at a liquid-fluid interface and the electrostatic interaction between charged surfaces in electrolyte solution (linear DLVO theory) has been recognized. Thanks to that analogy, the well-developed methods for calculation of electrostatic interactions in DLVO theory can be utilized to explore the less well-studied lateral capillary interactions between particles adsorbed at a liquid-fluid interface. A few examples of the usefulness of such approach to capillary interactions in different geometries are given. A complete electrostatic analogy is found for the lateral capillary interaction between plates or rods, but not in the case of spherical particles. It is demonstrated how the approximations used in the nonlinear theory of electrostatic interaction work for the case of lateral capillary forces in colloid systems. The analogy between electrostatic image forces and capillary image forces is also discussed. The author believes that this study will be helpful for better understanding of the complex phenomena accompanying the capillary interactions between colloid particles attached to a liquid-fluid interface, the formation two-dimensional ordered aggregates of colloid particles in wetting films, etc.
Introduction The lateral capillary interaction between particles attached to a liquid-fluid interface has long been recognized.1 It results from the overlapping of the liquid surface deformations around the particles. Different kinds of capillary interactions may exist because of the different sources of these deformations. For floating particles, the capillary interaction is due to the change of the gravitational potential energy of the particles, when they approach each other.2-4 Such interactions may cause agglomeration of heavy particles attached to a liquid-air interface.5 The situation is quite different when the particles (instead of being freely floating) are captured in a liquid layer.6-9 Recent experimental studies suggested that such “immersion” capillary forces can contribute to the formation of well-ordered arrays of colloid particles on solid10 or liquid11-13 substrates. In this case, the liquid surface deformations around the particles are related to * Permanent address: LTPH, Faculty of Chemistry, University of Sofia, 1126 Sofia, Bulgaria. E-mail for correspondence: vpaunov@ hotmail.com. (1) Gerson, D. F.; Zaijc, J. E.; Ouchi, M. D. In Chemistry for Energy; Tomlinson, M., Ed.; ACS Symposium Series; American Chemical Society: Washington, DC, 1979; Vol.90, p 77. (2) Nicolson, M. M. Proc. Camb. Philos. Soc. 1949, 45, 288. (3) Chan, D. Y. C.; Henry, J. D.; White, L. R. J. Colloid Interface Sci. 1981, 79, 410. (4) Paunov, V. N.; Kralchevsky, P. A.; Denkov, N. D.; Ivanov, I. B.; Nagayama, K. J. Colloid Interface Sci. 1993, 157, 100. (5) Schulze, H. Physicochemical Elementary Processes in Flotation; Elsevier: Amsterdam, 1984. (6) Kralchevsky, P. A.; Paunov, V. N.; Ivanov, I. B.; Nagayama, K. J. Colloid Interface Sci. 1992, 151, 79. (7) Kralchevsky, P. A.; Paunov, V. N.; Denkov, N. D.; Ivanov, I. B.; Nagayama, K. J. Colloid Interface Sci. 1993, 155, 420. (8) Paunov, V. N.; Kralchevsky, P. A.; Denkov, N. D.; Ivanov, I. B.; Nagayama, K. Colloids Surf. 1992, 67, 119. (9) Kralchevsky, P. A.; Nagayama, K. Langmuir 1994, 10, 23. (10) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Nature (London) 1993, 361, 26; Langmuir 1992, 8, 3183. (11) Yoshimura, H.; Matsumoto, M.; Endo, S.; Nagayama, K. Ultramicroscopy 1990, 32, 265. (12) Nagayama, K. Nanobiology 1992, 1, 25. (13) Lazarov, G. S.; Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Nagayama, K. J. Chem. Soc., Faraday Trans, 1994, 90, 2077.
the particle three-phase contact angle, i.e., to intermolecular forces, rather than to gravity. An interesting feature of the lateral “immersion” interaction is that it can be significant even for submicrometer particles.11,12 For large particles, the immersion capillary forces are large enough to allow their direct measurement.14-17 It has been shown that the immersion type of capillary interaction is operative not only between particles in planar liquid layers but also in spherical liquid films18 and lipid bilayers19 (see Figure 1). Recently, it was recognized4,8,20 that there is a close analogy between electrostatic surface forces21,22 and lateral capillary forces. The analogy comes not only from the similar form of the linearized governing equations and boundary conditions for the two-dimensional electrostatic and capillary interaction problem but also from the similar functional form of the forces between the interacting bodies (see below). Although this analogy has been mentioned in the respective articles,8,20 a strict parallel between these quite different kinds of interactions is still missing. The aim of the present study is to show the formal similarities between capillary and electrostatic interactions and to demonstrate how some methods developed for quantifying of electrostatic interactions can be applied to capillary interaction problems. The present paper is organized as follows. The first section considers the capillary forces (14) Camoin, C.; Roussel, J. F.; Faure, R.; Blanc, R. Europhys. Lett. 1987, 3, 449. (15) Velev, O. D.; Denkov, N. D.; Paunov, V. N.; Kralchevsky P. A.; Nagayama, K. Langmuir 1993, 9, 3702. (16) Dushkin, C. D.; Kralchevsky. P. A.; Paunov, V. N.; Yoshimura, H.; Nagayama, K. Langmuir 1996, 12, 641. (17) Dushkin, C. D.; Yoshimura, H.; Nagayama, K. J. Colloid Interface Sci. 1996, 181, 657. (18) Kralchevsky, P. A.; Paunov, V. N.; Nagayama, K. J. Fluid Mech. 1995, 299, 105. (19) Kralchevsky, P. A.; Paunov, V. N.; Denkov, N. D.; Nagayama, K. J. Chem. Soc., Faraday Trans. 1995, 91, 3415. (20) Kralchevsky, P. A.; Paunov, V. N.; Denkov, N. D.; Nagayama, K. J. Colloid Interface Sci. 1994, 167, 47. (21) Derjaguin, B. V. Landau, L. D. Acta Physicochim. USSR 1941, 14, 633. (22) Verwey, E. J. W.; Overbeek, J. Th. G. The Theory of Stability of Liophobic Colloids; Elsevier: Amsterdam, 1948.
S0743-7463(98)00389-8 CCC: $15.00 © 1998 American Chemical Society Published on Web 08/07/1998
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Figure 1. Some examples of lateral capillary interaction in colloidal systems: Lateral “flotation” interaction (a) between two floating particles; (b) between two bubbles, and (c) between a wall and a floating particle, attached to a liquid fluid interface. Lateral “immersion” interaction (d) between two vertical cylinders, partially immersed into a liquid, (e) between particles, captured in a wetting film, (f) between particles trapped into a foam or emulsion film, (g) between particles protruded from a spherical film, and (h) between integral proteins incorporated into a lipid bilayer.
between two plates partially immersed in a liquid and makes an analogy with the case of electrostatic interaction between charged plates in electrolyte solution. Some ideas for the derivation of approximate expressions for the capillary force in the case of large meniscus slope are also proposed. Section 2 deals with the respective interactions between two vertical (parallel) cylinders. In section 3, the similarities between capillary image interactions and electrostatic image interactions between a vertical (parallel) cylinder and a plate are discussed. Finally, the main conclusions are summarized in section 4. 1. Electrostatic and Capillary Interactions between Two Parallel Plates. Let us outline briefly some results of the linear DLVO theory21,22 for the electrostatic interaction between charged walls in electrolyte solution. We will compare these results with the case of lateral capillary interaction between two parallel plates partially immersed in a liquid,
to emphasize the formal analogy between these rather different types of interactions. 1.1. Linear DLVO Theory. For a system containing symmetric Z:Z electrolyte of bulk concentration n0, the electric potential, ψ(x), between two parallel charged walls satisfies the Poisson-Boltzmann equation21-23
φ′′(x) ) κ2 sinh φ(x),
φ(x) ≡ eZψ(x)/kT
(1)
Here κ-1 ) (8πZ2e2n0/kT)-1/2 is the Debye screening length, which determines the decay length of the electrostatic interaction in electrolyte solution. For small potential, φ2(x) , 1, eq 1 can be linearized to read
ψ′′(x) ) κ2ψ(x)
(2)
(23) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Plenum Press: Consultants Bureau, New York, 1987.
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Figure 2. Sketch of the distribution of the electrical potential, ψ(x), between two charged walls at a distance h in solution of electrolyte and the meniscus profile, ζ(x), around two plates at a distance h, partially immersed in a liquid. (a) The case of constant surface charge at the plates, σ1,σ2 ) constant, for the electrostatic problem is similar to the case of constant contact angle, R1,R2 ) constant, for the capillary interaction problem. (b) The case of fixed surface potential at the plates, ψ1,ψ2 ) constant, is analogous to the case of fixed contact line at the plates, H1,H2 ) constant. Note that in this case the inner and the outer electrostatic problems are treated separately with the same boundary conditions, like eq 3.
The solutions of eq 2 along with the boundary conditions for constant surface charge density at the two plates
Figure 3. Comparison of the main cases of monotonic and nonmonotonic interaction in the case of electrostatic surface forces (DLVO theory) between two charged walls in electrolyte solution and the corresponding lateral capillary forces (per unit plate length) between two vertical plates partially immersed in a liquid (see Figure 2): (a) The electrostatic disjoining pressure isotherm at constant surface charge, σ1,σ2 ) constant, and the corresponding case of lateral capillary force at constant contact angle, R1,R2 ) constant. (b) The electrostatic disjoining pressure isotherm at fixed surface potential, ψ1,ψ2 ) constant, and the corresponding case of lateral capillary force at fixed contact line, H1,H2 ) constant. Both cases correspond to linear theories (small potential, φ2 , 1, or small meniscus slope, ζ′2 , 1, respectively).
or constant surface potential
supposed to vanish at x f (∞. As is well-known,23 the osmotic (first) term in eq 7 is repulsive, while the electric (second) term is attractive. For unsymmetrically charged surfaces, the competition between them may lead to a nonmonotonic interaction. The substitution of eqs 5 and 6 into eq 7 gives the well-known expressions23
ψ(-h/2) ) ψ1 ) constant, ψ(h/2) ) ψ2) constant (4)
constant surface charge
|
4πσ1 dψ , )dx x)-h/2
|
4πσ2 dψ ) dx x)h/2
(3)
Πel(h) )
read
constant surface charge 2π cosh κx 2π sinh κx + (σ - σ1) (5) ψ(x) ) (σ1 + σ2) κ κh κ 2 κh sinh cosh 2 2 constant surface potential sinh κx 1 cosh κx 1 + (ψ + ψ2) (6) ψ(x) ) (ψ2 - ψ1) κh 2 1 κh 2 sinh cosh 2 2 see Figure 2. The electrostatic disjoining pressure, Πel(h), between the plates can be calculated via the expression (see, e.g., refs 23-24)
Πel(h) )
dψ 2 2 κ ψ (x) 8π dx
[
2
( )]
(7)
In the outer electrostatic problem the potential, ψ(x) is (24) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992.
2 2 2π 2σ1σ2 cosh κh + σ1 + σ2 (8) sinh2 κh
( )
constant surface potential 2 2 κ2 2ψ1ψ2 cosh κh - ψ1 - ψ2 (9) Πelψ(h) ) 8π sinh2 κh
( )
According to the linear DLVO theory, when the surface potentials have equal sign, ψ1ψ2 > 0, but different magnitude, ψ1 * ψ2, the electrostatic interaction can switch from repulsion at large separations, h, to attraction at small separations. Nonmonotonic interaction can also be realized in the case of constant surface charges of different signs, σ1σ2 < 0, and different magnitude, |σ1| * |σ2|, where the electrostatic attraction at large separations changes to repulsion at small separations. In all other cases, the electrostatic interaction is monotonic (repulsive or attractive); see Figure 3. 1.2. Lateral Capillary Forces between Two Parallel Plates. This is the simplest case of lateral capillary interaction (see Figure 2). Let z ) ζ(x) be the meniscus profile around the plates and h the distance between them.
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In general, ζ(x) satisfies the Laplace equation of capillarity, which for this geometry reads25
ζ′′(x)[1 + ζ′2(x)]-3/2 ) q2ζ(x)
(10)
Here q-1 ) (∆Fg/γ)-1/2 is the capillary length, γ is the meniscus surface tension, and ∆F ) FI - FII is the density difference between the two fluid phases. For small meniscus slope, ζ′2(x) , 1, around the plates, the Laplace equation can be linearized in the form
ζ′′(x) ) q2ζ(x)
(11)
Note that, both the electric potential and the meniscus profile satisfy Helmholtz-type equations (cf. eq 2 and eq 11). The capillary length, q-1 is a counterpart of the Debye screening length, κ-1 in the case of electrostatic interaction. However, for the surface of water in contact with air, the capillary length is q-1 ) 2.7 mm, while the maximal value of the Debye length in aqueous electrolytes is κ-1 ) 0.97 µm; i.e., these two types of interactions have quite different decay lengths. Two cases of boundary conditions at the plates surfaces can be considered: (i) constant contact angles, R1 and R2,
|
|
dζ ) cot R2 dx x)h/2
dζ ) -cot R1, dx x)-h/2
(12)
and (ii) fixed contact lines
ζ(-h/2) ) H1 ) constant,
ζ(h/2) ) H2 ) constant (13)
(or fixed capillary elevations, H1,H2, of the contact lines) due to their attachment to some edge or the boundary between hydrophilic and hydrophobic regions. Note that the condition for small meniscus slope is equivalent to the following conditions to be satisfied 2
2
cot Rk , 1, or (qHk) , 1
acting on the plate surfaces and (ii) the force due to the meniscus surface tension, acting on the contact lines around the plates, i.e.,
(k ) 1, 2) (14)
The meniscus profile between the plates in these two cases reads
constant contact angle (cot R1 + cot R2) cosh qx + ζ(x) ) 2q qh sinh 2 (cot R2 - cot R1)sinh qx (15) qh 2q cosh 2 fixed contact line sinh qx 1 cosh qx 1 + (H + H2) (16) ζ(x) ) (H2 - H1) qh 2 1 qh 2 sinh cosh 2 2 Since the boundary conditions for the Laplace equation, eqs 12-13, and the boundary conditions for the PoissonBoltzmann equation, eqs 3-4 are similar, the solutions for the electric potential and the meniscus shape also have very similar forms (cf. eqs 15-16 and eqs 5-6). The capillary force per unit length (l) of the plates has two components: (i) the force due the hydrostatic pressure (25) Princen, H. M. The Equilibrium Shape of Interfaces, Drops, and Bubbles. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1969 Vol. 2, p 1.
fcap(h) ) Fcap(h)/l )
∫0ζ(x,h)[PI(z) - PII(z)] dz +
γ[1 - cos β(x,h)] (17)
Here PI and PII are the hydrostatic pressures and β(x) is the running slope of the meniscus profile; l is the length of the plates contact lines. We remark that eq 17, which follows directly from Stevin’s principle in hydrostatics, is satisfied for each value of x between the plates, i.e., ∂f/∂x ) 0 (cf. eq 7). Using the identities
PI(z) ) P0 - FIgz,
PII(z) ) P0 - FIIgz
1 cos β(x) ≈ 1 - β(x)2 2
(18) (19)
we arrive at the following expression for the capillary force per unit length of the plates
fcap(h) ≈
dζ γ -q2ζ2(x) + 2 dx
[
2
( )]
(20)
Similarly to the electrostatic problem, ζ(x) is supposed to vanish at x f (∞. We remark that the first term in eq 20, which is the hydrostatic pressure contribution to the force, is always attractive for ∆F > 0, while the surface tension contribution (the second term) is always repulsive. For identical plates, the second term vanishes due to the symmetry, as does the electric term in the disjoining pressure for symmetrically charged plates (see eq 7). The substitution of eqs 15-16 into eq 20 yields
R1,R2 ) constant 2 2 γ 2 cot R1cot R2 cosh qh + cot R1 + cot R2 2 sinh2 qh (21)
()
fcap(h) ) -
(H1,H2 ) constant)
( )
fcap(h) ) -
2 2 γq2 2H1H2 cosh qh - H1 - H2 (22) 2 sinh2 qh
One can see from eqs 21-22 and eqs 8-9, that there is a formal similarity between the capillary force per unit length of the plates and the electrostatic disjoining pressure. The case of constant contact angle at the plates resembles the case of constant surface charge in DLVO theory, while the case of fixed contact line is similar to the case of constant surface potential (see Table 1). However, the sign of the capillary interaction is just opposite to that of the electrostatic interaction. Thus, the case of identically charged plates in DLVO theory always corresponds to electrostatic repulsion, while the case of identical plates always corresponds to attractive capillary forces. Figure 3 summarizes the main cases of nonmonotonic electrostatic and capillary interaction between two plates. Note that the origin of the capillary repulsion both in eqs 21 and 22 when the capillary elevations H1 and H2 have opposite signs comes from the surface tension force. The larger the difference between H1 and H2, the larger the meniscus slope dζ/dx between the plates (see eq 20). The situation is very similar to the famous result of Derjaguin for the electrostatic disjoining pressure between oppositely charged plates, where the same role plays the attractive electric term (cf. eq 7).
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Table 1. Analogous Quantities in the Linear Theories of Lateral Capillary Interaction between Two Vertical Plates, Partially Immersed in a Liquid, and the Electrostatic Interaction between Two Charged Parallel Plates in Electrolyte Solution lateral capillary interaction
electrostatic interaction (DLVO theory)
linear Laplace equation, ζ′′(x) ) q2ζ(x) Meniscus shape, ζ(x) capillary length, q-1 ) (∆Fg/γ)-1/2 constant contact angle, R ) constant fixed contact line, H ) constant capillary force (per unit length), F(h)/l
linear Poisson-Boltzmann equation ψ′′(x) ) κ2ψ(x) electric potential, ψ(x) Debye length, κ-1 ) (8πe2Z2n0/kT)-1/2 constant surface charge, σs ) constant constant surface potential, ψs ) constant disjoining pressure, Π(h)
1.3. Derjaguin Approximation. Although it may seem that there is a full analogy between the electrostatic interaction in the DLVO theory and the lateral capillary interaction theory in their linear variants, it does not follow that all methods of DLVO theory can be automatically applied to capillary interactions. For example, the widely used Derjaguin approximation (in the case of short-ranged interactions) allows one to quantify the interaction force, F, between two different cylinders of radii r1 and r2, (crossed at angle ω) knowing the disjoining pressure, Π(h), between planar surfaces23,26
2πxr1r2 (h) ) sin ω
(cyl)
F
∫h Π(h) dh ∞
(23)
Since the force, F(cyl), in eq 23, is obtained by integration of the surface force over the surface of infinitely long cylinders, it diverges when the cylinders are parallel (ω ) 0) and could not be applied to capillary interaction problems. To obtain a counterpart of the Derjaguin approximation for electrostatic forces and lateral capillary forces between two large vertical cylinders, one should take into account that the problem is essentially two-dimensional. The calculation of the electrostatic force (per unit length of the cylinder) between two parallel charged cylinders in electrolyte solution can be done by the following analysis. We replace the original circular cylinders with equivalent parabolic surfaces, placed at the same distance, h. The x-component of the force, which is acting on the surface ˜ (y)) cos ξ element ds ) 1 dl on the cylinder, is dfx ) Π(h ds, where dl is the elementary length of the cylinder profile and ξ is the slope angle of the cylindrical surface (ξ ) 0at y ) 0). Having in mind that dl ) dy/cos ξ, we can integrate the force acting on the cylindrical surface to obtain the following expression
fx(cyl) ≈ 2
∫0∞Π(h˜ (y)) dy,
h ˜ (y) ≈ h +
(
)
y2 1 1 + 2 r 1 r2
(23a) for the force per unit length of the cylinders. Similar analysis in the case of lateral capillary forces leads to the expression
∫0∞fcap[h˜ (y)] dy,
Fcap(cyl)(h) ≈ 2
h ˜ (y) ≈ h +
(
)
y2 1 1 + (24) 2 r 1 r2
Here, fcap(h) is the lateral capillary force (per unit length) between two parallel plates at separation h (see eqs 21 and 22). Obviously, eq 24 is valid only for the interaction (26) Derjaguin, B. V. Kolloid Zeits. 1934, 69, 155.
between large cylinders, where the contact radii are much larger than the capillary length
(qrk)2 . 1
(25)
It may seem that the capillary interaction between such large bodies is outside the scope of the colloid science, because the capillary length is usually of the order of millimeters (q-1 for the air/water surface). However, when the colloid particles are captured in a thin liquid film, the capillary length can be modified due to the surface forces6,9
q-1 )
Π′ - ) (∆Fg γ γ
-1/2
,
Π′ )
|
dΠ dh ζ)0
(26)
In this case, the capillary length can change from millimeters (thick film) to a few micrometers (thin film) depending on the disjoining pressure derivative, Π′, and eq 25 can be satisfied even for micrometer particles. It should be noted that eq 24 is not limited to the case of small meniscus slope, but it is also valid when the problem is essentially nonlinear, provided that eq 25 is satisfied. 1.4. “Weak Overlap” Approximation. For a large meniscus slope the results outlined in section 1.2 are not valid and obviously a generalization is needed. The situation is similar to the case with the linear DLVO theory for highly charged surfaces (section 1.1). Although both the nonlinear Poisson-Boltzmann equation, eq 1, and the nonlinear Laplace equation, eq 11, can be solved analytically in that geometry,23,25 both the solutions (for ψ(x) and for ζ(x) are expressed in terms of Jacobi’s elliptic functions.27 However, for a weak overlapping of the electric double layers (κh . 1), Verwey and Overbeek21 used a superposition of the nonlinear solutions for the electrical potential of a single plate
(
φ(x) ≈ φˆ 1 x +
h h + φˆ 2 - x 2 2
)
φˆ k(x) ≈ 4 tanh
(
)
( )
φk exp(-κx) 4
k ) 1, 2
(27)
in order to derive an asymptotic (but widely used) expression for the disjoining pressure between two highly charged plates in electrolyte solution
Πel(h) ≈ 64n0kT tanh
( ) ( )
φ1 φ2 tanh exp(-κh) 4 4
κh . 1
(28)
Using the same approach, we derive an asymptotic expression for the capillary force per unit length, when the meniscus slope at the two plates is not small. The (27) Korn, G. A.; Korn, T. M. Mathematical Handbook; McGrawHill: New York, 1968.
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asymptotic of the “nonlinear” meniscus profiles for a single plate reads15
1 ζˆ k ≈ Dk exp(-qx) q qx . 1,
k ) 1, 2
(29)
where
( ) (
Dk ) 4 tan
)
φk φk exp -4 sin2 4 4
φk )
π - Rk 2
k ) 1, 2
(30)
for constant contact angles, and
Dk )
8Yk exp(-2Yk) qHk
Yk ) 1 -
x
1-
q2Hk2 4
k ) 1, 2
(31)
for fixed contact lines at the plates. Following the approach of Verwey and Overbeek21 we approximate the meniscus profile as a superposition of the nonlinear asymptotic of the single plates profiles
(
ζ(x) ) ζˆ 1 x +
h h + ζˆ 2 - x 2 2
)
(
)
(32)
This approximation is reliable only when the menisci of the two plates are weakly overlapping. By substitution of eq 32 into eq 20 we obtain
Figure 4. Sketch of the distribution of the electrical potential, ψ(x,y) around two charged parallel cylinders in electrolyte solution, separated at a distance L from their centers. The same figures represent the meniscus surface, ζ(x,y), around two vertical cylinders, partially immersed in a liquid. (a) The case of constant surface charge, σ1,σ2 ) constant, for the electrostatic problem is similar to the case of constant contact angle, R1,R2 ) constant for the capillary interaction problem. (b) The case of fixed surface potential at the plates, ψ1,ψ2 ) constant, is analogous to the case of fixed contact line at the cylindrical surfaces, H1,H2 ) constant.
operator. The electrostatic boundary conditions at the surfaces of the charged cylinders are
constant surface charge
fcap(h) ≈ -2γD1D2 exp(-qh) qh . 1
(33)
One may expect that eq 33 is valid when the separation, h, between the plates is large compared to the capillary length, q-1. Note that eq 33 is a counterpart of eq 28 for the case of lateral capillary interaction between plates. 2. Electrostatic and Capillary Interactions between Two Parallel Cylinders Here we outline in parallel the similarities between the electrostatic interaction between two charged parallel cylinders (not necessarily of circular cross-section) in electrolyte solution and the lateral capillary interaction between two vertical cylinders partially immersed in a liquid (see Figure 4). We will not compare the numerical values of the electrostatic and the capillary forces of interaction between the cylinders but only emphasize the common approaches. Let us denote by L the distance between the axes of the two cylinders. The respective linearized Poison-Boltzmann equation and Laplace equation have virtually similar forms (see, e.g. refs 7 and 23)
∇II2ψ ) κ2ψ,
ψ ) ψ(x, y)
(34)
∇II2ζ ) q2ζ,
ζ ) ζ(x, y)
(35)
where ∇II ) (∂/∂x, ∂/∂y) is the two-dimensional gradient
4πσk ,
k ) 1, 2 (36)
ψ|Sk ) ψk,
k ) 1, 2 (37)
nk‚∇IIψ|Sk ) constant surface potential
Here n1 and n2 are the unit outer normal vectors to the cylindrical surfaces. Note that the linearized boundary conditions for the capillarity problem
constant contact angle nk‚∇IIζ|Ck ≈ -cos Rk,
k ) 1, 2
(38)
fixed contact line
ζ|Ck ) Hk,
k ) 1, 2
(39)
are again similar to the electrostatic ones (cf. eqs 36 and 37). That is why the solutions of eqs 34-35 must also be similar. For the electrostatic problem, the force per unit length of the cylinders can be calculated by integrating the Maxwellian stress tensor along the intermediate plane between the parallel cylindrical surfaces.28 For small (28) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989.
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Paunov
expression for the capillary force between two cylinders3,15
potentials, the final result reads
fel(L) ≈
8π
∂ψ + dy ∫-∞∞[κ2ψ(0,y)2 - (∂ψ ∂x ) x)0 ( ∂y ) x)0] 2
2
(40)
On the other hand, the following expression has been derived15 for the lateral capillary force between two different cylinders (a sum of hydrostatic pressure and surface tension contributions)
γ Fcap(L) ) 2
∂ζ q ζ(0,y) ∂x
∫-∞[ ∞
2
∂ζ + ∂y x)0
2
( )
2
2
( ) ] dy x)0
(41) Equation 41 has been derived in ref 15 by integration of the forces due to surface tension and hydrostatic pressure acting on an infinite contour, surrounding one of the bodies and passing through the mid plane between them. Its validity is not restricted to large radius of curvature of the solid surfaces, as with the Derjaguin approximation. Besides, it is valid also for the case of a nonlinear problem (large meniscus slope), but only in the limit of a weak overlap (qL . 1). Note that eq 41 and the Derjaguin approximation for capillary forces, eq 24, have different areas of validity, because of the different assumptions used in their derivation. Actually, eq 41 is valid not only for the capillary force between two cylinders but also for the general case of two bodies of an arbitrary shape, provided that the meniscus slope at the mid plane between them is small. Note that the expressions for the electrostatic force per unit length of the cylinders, eq 40, and the respective lateral capillary force, eq 41, also have very similar forms. Since the governing equations, eqs 34 and 35, and the boundary conditions, eqs 36-37 and 38-39, are also quite similar, one could expect similarity in the explicit expressions for the dependence of the respective forces on the distance, L, between the bodies. When the two cylinders are circular (of radii r1 and r2), an approximate solution can be constructed in both cases of linear electrostatic and capillarity boundary value problems by using of a superposition approximation (see, e.g., refs 2, 3, and 15). Thus, for not very small separations between the cylinders, the superposition of the solutions for a single cylinder yields
( x( ) ) ( x( ) )
ψ(x,y) ≈ U1K0 κ
x+
L2 + y2 + 2 U2K0 κ
x-
L2 + y2 (42) 2
( x( ) ) ( x( ) ) x+
L2 + y2 + 2 Q2K0 q
x-
L2 + y2 (43) 2
for the capillarity problem. Here K0(x) is the modified Bessel function of zeroth order and the coefficients are
Uk ) Qk )
4πσk κK1(κrk) cos Rk qK1(qrk)
)
)
ψk K0(κrk) Hk K0(qrk)
(46)
The counterpart of eq 46 for the electrostatic force (per unit length) between two charged cylinders in electrolyte can be obtained from eqs 40 and 42
fel(L) ≈ κU1U2K1(κL) 2
(47)
Using the expansion of the Bessel function, K1(x) ) 1/x + O(x ln x), for small values of x (ref 27), eqs 46 and 47 can be reduced to a remarkably simple form
Fcap(L) ≈ -2πγ fel(L) ≈
Q1Q2 , L
U1U2 , 2 L
qrk , qL , 1 κrk , κL , 1
(48) (49)
For (κrk)2 , 1, eq 44 gives, Uk ≈ 4πσkrk/, and similarly for (qrk)2 , 1, eq 45 yields, Qk ≈ rk cos Rk. According to eq 48, the capillary force between cylinders of constant contact angle does not depend on gravity. Similarly, the electrostatic force (eq 49) does not depend on the electrolyte concentration for cylinders of constant surface charge. In fact, eq 48 represents an analogue of the two-dimensional Coulomb’s law in electrostatics, which justifies the term “capillary charge” introduced in ref 4. Equation 46 has been verified experimentally by capillary force measurements between two vertical cylinders15 of submillimeter radii, as well as between two spherical particles and a spherical particle and vertical cylinder.16,17 Very good agreement between theory and experiment has been found for separations larger than the capillary length for all these systems. Figure 5 summarizes the electrostatic analogies for the lateral capillary interactions between bodies of different geometries. In the case of two spherical particles, however, there is no analogy between electrostatic interaction in electrolyte solution and the respective capillary forces. The obvious reason is that the electrostatic problem in that geometry is three-dimensional, while the capillarity problem remains two-dimensional. In this respect, the lateral capillary interaction between spherical particles is very similar to that between vertical cylinders, which have been used in refs 4, 6-8, 18 to calculate the respective capillary forces. 3. Electrostatic and Capillary Image Forces
for the electrostatic problem, and
ζ(x,y) ≈ Q1K0 q
Fcap(L) ≈ -2πγqQ1Q2K1(qL)
(k ) 1, 2)
(44)
(k ) 1, 2)
(45)
Using eqs 41 and 43, one can derive the following
Recent studies of the capillary interaction between a plate and a vertical cylinder8 and between a plate and floating spherical particle20 showed that there is a close analogy with the electrostatic image forces. As known from electrostatics, a test electric charge, q, placed at a distance L from an interface, feels an electrostatic “image” force, which is equivalent to the interaction with an “image” charge, q′, placed symmetrically with respect to the interface24
Felim )
q′q , (2L)2
- ′ q′ ) q + ′
(50)
(see parts a and b of Figure 6). The ratio of the two dielectric constants (′ and ), which determines the sign of the image force, enters into eq 50 through the electrostatic boundary condition at the interface.
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Langmuir, Vol. 14, No. 18, 1998 5095
Figure 5. Schematic representation of the analogy between lateral capillary forces and electrostatic (DLVO) forces between bodies of different geometries: (a) two vertical plates partially immersed in a liquid, and two charged parallel plates in electrolyte solution; (b) a plate and a vertical cylinder, partially immersed in a liquid, and a charged plate and a charged parallel cylinder in electrolyte solution; (c) two vertical cylinders protruded from a liquid surface, and two charged parallel cylinders in electrolyte solution; (d) two spherical particles captured in a wetting filmsthere is no electrostatic analogy with the respective case of two charged spheres in electrolyte. However, there is a similarity between the lateral capillary forces in (c) and (d).
It was shown in refs 8 and 20 that the capillary interaction between a wall and a particle (at distance L) is equivalent to the interaction of two identical particles located symmetrically with respect to the wall (see Figure 6a). The latter situation is realized when the contact angle at the wall is constant (and exactly equal to 90°). Since the capillary interaction between similar particles is always attractive, such a wall always attracts the particles through capillary image forces
constant contact angle at the wall Fcapim ≈ -2πγqQ22K1(2qL)[1 + O(q2r22)]
(51)
Here Q2 ) r2 sin φ2, is the particle “capillary charge”4 with r2 and φ2 being the contact line radius and the meniscus slope angle, respectively. For a vertical cylinder, Q2 ) r2 cos R2 (cf. eq 45), while for a floating particle3,4 of radius, R2
1 Q2 ≈ q2R23[2 - 4(F2 - FII)/∆F + 3 cos R2 - cos3 R2] 6 (52) It was realized in ref 20 that the capillary image force can also be repulsive, if the contact line at the wall is fixed. Then, for H ) 0 at the wall, the capillary image interaction
Figure 6. A formal analogy exists between electrostatic image forces and capillary image forces. Cases a and b represent the classical electrostatic image forces when an electric charge is placed close to an interface between phases of different dielectric constants, ′ and . The sign of the image charge (and of the image force) depends on the ratio of the dielectric constants (see the text). (c) and (d) represent the analogous capillary image forces, which appear between a wall (plate) and a particle attached to a liquid interface. The capillary image force is repulsive when the contact line at the wall is fixed (c) and it is and attractive when the contact angle at the wall is constant (d).
is equivalent to the interaction between two particles of opposite capillary charges
fixed contact line at the wall Fcapim ≈ 2πγqQ22K1(2qL)[1 + O(q2r2)]
(53)
In both cases, eqs 51 and 53 predict monotonic interactions (repulsion or attraction); see parts c and d of Figure 6. However, when the wall forms its own meniscus, the capillary image force can interfere with the tangential components of the gravity forces acting on the particle. For that situation, the following approximate expression for the lateral capillary force acting on the particle can be derived (see, e.g., refs 16 and 20)
Fcap ≈ -πγ[(-1)λ2qQ22K1(2qL) + 2Q2D1e-qL + q(r2D1e-qL)2][1 + O(q2r22)] (54) The coefficient D1 can be calculated from eq 30 or eq 31. Equation 54 holds for both the case of lateral capillary forces between a spherical particle and a wall and between a vertical cylinder and a wall. It is valid even for a large meniscus slope at the wall, provided that the separation, L, is large compared to the capillary length. The first term in eq 54 is the capillary image force, which is attractive (λ ) 0) for constant contact angle at the wall and repulsive (λ ) 1) for fixed contact line at the wall. The second term is the surface tension force acting on the cylinder (particle) due to the meniscus perturbation by the wall. In the case of a floating particle, this term is equal to the tangential projection of the particle weight along the meniscus of the wall. The third term in eq 54 is the hydrostatic pressure contribution to the lateral capillary force. Note that, due to the competition between different terms in eq 54, a nonmonotonic interaction between the particle and the wall can be realized. For a heavy floating particle and a plate of fixed contact line, D1Q2 > 0, it may happen that the repulsive capillary image force compensates the tangential components of the gravity forces at some particular distance, L*, from the
5096 Langmuir, Vol. 14, No. 18, 1998
Paunov
wall. The equilibrium separation, L*, between a Teflon plate and a submillimeter mercury droplet or copper bead has been measured by Velev et al.,29 and the respective capillary image force has been recalculated from eq 54. More detailed analysis of the different cases of capillary interaction between a floating particle and a wall can be found in ref 20. The capillary image force between a wall and a vertical cylinder has an exact electrostatic analogue, which is the image force between a charged wall and a parallel charged cylinder in electrolyte solution (see Figure 5b). For example, such a configuration can be realized between long rodlike micelles of an ionic surfactant and a plate. An expression, analogous to eq 54 can be derived when the plate is also charged. Thus, for the electrostatic force (per unit length of the cylinder) we obtain the following approximate expression
fel(L) ≈ [(-1)λ2κU22K1(2κL) + 2U2ψ1e-κL + 4 κ(κr2ψ1e-κL)2][1 + O(κ2r22)] (55) (see the Appendix for the details of its derivation). Here the first term is the electrostatic image force which is repulsive for constant surface charge at the wall (λ ) 0) and attractive for constant surface potential at the wall (λ ) 1). The second term in eq 55 comes from the electrostatic interaction of the charged wall and the charged cylinder, and it is repulsive when ψ1U2 > 0 and attractive if ψ1U2 < 0. The third term in eq 55 represents the osmotic contribution to the interaction force, and it is always repulsive. According to eq 55, a nonmonotonic behavior of the electrostatic interaction force between the cylinder and the plate is possible. For an oppositely charged plate and cylinder, and constant surface charge at the plate, the electrostatic image force is repulsive (λ ) 0), the osmotic term is also repulsive, but the second term in eq 55 is attractive. As seen from eq 55, the second term usually dominates the other two at large separations, κL . 1. Thus, at some particular separation, L ) L*, it can be exactly counterbalanced by the osmotic term and the electrostatic image force, which corresponds to a stable equilibrium position between the plate and the cylinder. At this separation, the total interaction force vanishes, fel(L*) ) 0, and the respective electrostatic interaction energy exhibits a minimum (cf. Figure 7). Another situation of nonmonotonic interaction is possible when the surface potential of the plate is fixed, ψ1 ) constant, and the signs of the surface charges at the plate and the cylinder are the same (U2ψ1 > 0). Here, the image force is attractive (λ ) 1), but the next two terms in eq 55 are repulsive and prevail at large separations, which corresponds to unstable equilibrium (energy maximum) at L ) L*, where the interaction force vanishes, fel(L*) ) 0. 4. Conclusions In this paper we emphasize the analogy between the linear theory of the electrostatic interaction of charged bodies in electrolyte solution and the linear theory of the lateral capillary forces in the same geometry. The analogy is due to: (i) the similar form of the linearized Poisson-Boltzmann equation and the linearized Laplace equation (cf. eqs 2 and 11 or eqs 34 and 35), the characteristic decay lengths of these interactions (the Debye length, κ-1 and capillary length, q-1) are quite different; (ii) the similar form of the (29) Velev, O. D.; Denkov, N. D.; Paunov, V. N.; Kralchevsky, P. A.; Nagayama, K. J. Colloid Interface Sci. 1994, 167, 66.
Figure 7. Two cases of similar nonmonotonic interaction between a plate and a vertical cylinder can be observed in both cases of (a) capillary and (b) electrostatic forces. (a) When Q2D1 > 0 and the plate contact line is fixed, the repulsive capillary image force can prevail over the attractive contribution of the surface tension and hydrostatic pressure to the lateral capillary force at some particular distance, L*, from the wall. (b) The repulsive electrostatic image and osmotic forces between a charged plate and a charged cylinder in electrolyte solution can also dominate the attractive electrostatic term (when σ1U2 < 0) and may cause the total force to vanish at some particular separation, L* (see the text).
boundary conditionssconstant surface charge and fixed surface potential in the electrostatic problemsscorresponds to constant contact angle and fixed contact line in capillary interaction problems (cf. eqs 3-4 and 13-14); (iii) the similar functional form of the electrostatic force of interaction (integral of Maxwellian stress tensor over the surface of the body) and the lateral capillary force of interaction (integral of surface tension force along the contact line, and the hydrostatic pressure over the surface of the body). Such a similarity is observed only for twodimensional electrostatic problems (where the Maxwellian stress tensor is two-dimensional). This approach allows one to use electrostatic analogies in many different cases of lateral capillary interactions between (i) two parallel plates; (ii) two vertical cylinders; (iii) a plate and a vertical cylinder, etc. (see Figure 5). In the case of capillary interaction of spherical particles, a direct electrostatic analogy is absent, but there is a close analogy with the case of vertical cylinders. It should be noted that the analogy between capillary forces and electrostatic forces is only formal and the sign of the interaction is just the opposite in the corresponding cases (see Figures 3 and 5) As a result of the analogy with DLVO theory, various approaches can be transferred to the lateral capillary interactions with the appropriate modification. For example, we have obtained a counterpart of the Derjaguin approximation for the case of capillary forces. We also demonstrate how the “weak overlap” approximation of Verwey and Overbeek can be used to derive simple asymptotic expressions for the lateral capillary force, when the meniscus slope is large and the problem is nonlinear. The similarities between electrostatic image forces and capillary image forces are also discussed. An exact electrostatic analogue of the capillary meniscus interaction between a particle and a wall is found and the possibilities for nonmonotonic interactions in both cases are outlined. Finally, the author believes that the successful application of electrostatic analogies to capillary interaction problems will facilitate the description of such complex
Lateral Capillary Interactions
Langmuir, Vol. 14, No. 18, 1998 5097
phenomena like 2D aggregation of colloid particles captured at a liquid interface or into a liquid fim.10-13,30
that eq A.6 resembles its counterpart in the case of lateral capillary forces8
Acknowledgment. The author appreciates the financial support from NATO/Royal Society Postdoctoral grant (ref. FCO/97B/BLL) during the preparation of this paper. The stimulating discussions with Professor R. Aveyard and Dr. J. H. Clint are gratefully acknowledged. The author thanks Professor P. D. I. Fletcher for his comments on the manuscript.
Fcap(L) ≈ γr2
Appendix Electrostatic Interaction between a Charged Cylinder and a Charged Wall in Electrolyte Solution Here we derive an approximate expression for the electrostatic force (per unit length) between a charged cylinder (constant surface charge), and a parallel charged wall in electrolyte solution. The electrostatic force can be obtained by integration of the Maxwellian stress tensor, P ˜ , over the surface of the cylinder
∫0πex‚P˜ ‚er|S
fel(L) ) -2r2
2
dφ
(A.1)
where φ is the azimuthal angle (see Figure 7) and
(ψ ′2 - ψx′2) 8π y
Pyy ) p(ψ) +
(ψ ′2 - ψy′2) 8π x
Pxy ) Pyx ) -
ψ′ψ′ 4π x y
)
cos φ dφ -
∫0πζφ′ sin φ dφ
(A.7)
between a vertical cylinder and a vertical plate partially immersed in a liquid. When the cylinder radius is much smaller than the Debye length, (κr2)2 , 1, and the separation L is not very small, the mean value of the electrical potential, ψ h 2, at the cylinder surface can be well approximated by the expression
ψ h 2(L) ≈ ψ2 + ψ1 exp(-κL) ( U2K0(2κL) r2 , L, (κr2)2 , 1
(A.8)
Here ψ1 and ψ2 are the surface electric potentials at a single wall and a single cylinder, respectively. The term (U2K0(2κL) represents the contribution of the cylinder “image” with respect to the wall (cf. eq 42 with U1 ) (U2). The “+”sign is taken for constant surface charge at the wall, while “-” corresponds to the case of constant surface potential at the wall. Using the approximation
h2 + ψ(φ)|S2 ≈ ψ
dψ h2 r cos φ dL 2
(A.9)
and eq A.6 we derive
fel(L) ≈ -
(A.3)
Here ex, ey, and er are the unit vectors along x-, y-, and r-directions, respectively. For small potentials, we have p(ψ) ≈ p0 + (/8π)κ2ψ2. Further, we will restrict ourselves to the case of constant surface charge at the cylinder surface, i.e.
ψr′|S2 ) -4πσ2/ ) constant
2
2γ sin φ2
P ˜ ) exexPxx + eyeyPyy + (exey + eyex)Pxy (A.2) Pxx ) p(ψ) +
(
∫0π q2ζ2 + r12ζφ′2
[
]
h2 4πσ2r2 dψ (κr2)2ψ h 2(L) + 4 dL
(A.10)
The substitution of eq A.8 into eq A.10 gives eq 55. On the other hand, the same approximations can be used for the analogous capillary interaction problem
h2 + ζ(φ)|C2 ≈ H
dH h2 r cos φ dL 2
(A.11)
1 H h 2 ≈ H2 + D1 exp(-qL) ( Q2K0(2qL) q
(A.4) r2 , L, (qr)2 , 1
(A.12)
Then, by using eqs A.1-A.4 and the identity
er ) ex cos φ + ey sin φ
The substitution of eq A.11 into eq A.7 yields
(A.5) dH h2 dL
h 2(L) + Q2] Fcap(L) ≈ πγ[(qr2)2H
and after introducing polar coordinates we obtain
fel(L) ≈ -
r2 4π
(
)
∫0π κ2ψ2 + r12ψφ′2 2
2
cos φ dφ -
4πσ2
∫0πψφ′ sin φ dφ
(A.6)
Here, the values of the potential, ψ and its derivative, ψφ′ are to be taken at the surface of the cylinder. One sees (30) Velikov, K.; Velev, O. D. Durst, F., Langmuir 1998, 14, 1148.
(A.13)
Then, the combination of eq A.12 and A.13 leads to eq 54. The latter approximation reduces to the result from a superposition of the linear profiles, i.e., D1 f H1 for small meniscus slopes. In other words, it is similar to the nonlinear superposition approximation of Overbeek (weak overlap) (see eq 28). The respective equation for the lateral capillary force obtained by this manner has already been checked experimentally that it predicts the force better than its “linear” version (see ref 16). LA980389S
