Langmuir 2002, 18, 8919-8923
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Line Tension on Curved Surfaces: Liquid Drops on Solid Micro- and Nanospheres Abraham Marmur*,† and Boris Krasovitski‡ Departments of Chemical Engineering and Agricultural Engineering, TechnionsIsrael Institute of Technology, 32000 Haifa, Israel Received July 1, 2002. In Final Form: August 29, 2002 Line tension for drops on spherical surfaces is approximately calculated and shown to depend on the contact angle and on the curvature of the spherical surface normalized with respect to the curvature of the drop. It is predicted that (a) the contact angle, at which the transition from positive to negative line tensions occurs, decreases with an increase in the solid surface curvature; (b) line tension is very sensitive to the curvature of the solid surface for low contact angles, but this sensitivity decreases as the contact angle increases; and (c) increased curvature of the solid surface decreases the value of line tension. The significance of line tension to wetting is discussed in terms of a critical contact-line radius, Rc, above which the effect of line tension on the contact angle is experimentally indistinguishable. For acute contact angles, except for relatively low ones, Rc is smaller than a few micrometers and sharply decreases when the contact angle approaches 90°. For obtuse contact angles, Rc is smaller than a few tens of nanometers and also sharply decreases when the contact angle approaches 90°.
Introduction The line tension concept was qualitatively suggested by Gibbs.1 He recognized the fact that three-phase interactions at the contact line between a solid, a liquid, and a vapor phase cannot be simply accounted for by the interfacial tensions of each pair of phases. Later,2-4 the effect of line tension on wetting in a solid-liquid-vapor system was quantitatively expressed by the following equation, which replaced the Young equation as the thermodynamic equilibrium criterion for a drop on a flat, ideal (e.g. smooth, rigid, chemically homogeneous, insoluble, and nonreactive) solid surface:
cos θi ) cos θY -
τ σlR
(1)
In this equation θi is the ideal contact angle (i.e. the contact angle on an ideal solid surface), θY is the contact angle predicted by the Young equation, τ is the line tension, σl is the surface tension of the liquid, and R is the radius of the contact line between the three phases. To estimate the importance of line tension in wetting phenomena, it is obviously necessary to know its order of magnitude. This, however, turned out to be a difficult experimental problem, as witnessed by the fact that reported absolute values for line tension in solid-liquidvapor systems vary by orders of magnitude,5-12 from * Fax: 972-4-829-3088. E-mail:
[email protected]. † Department of Chemical Engineering. ‡ Department of Agricultural Engineering. (1) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs; Dover Publications: New York, 1961; Vol. 1, p 288. (2) Veselovski, V. S.; Pertchov, V. N. Zh. Fiz. Khim. 1936, 8, 245. (3) Pethica, B. A. Rep. Prog. Appl. Chem. 1961, 46, 14. (4) Pethica, B. A. J. Colloid Interface Sci. 1977, 62, 567. (5) Scheludko, A.; Toshev, B. V.; Bojadjiev, D. T. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2815. (6) Good, R. J.; Koo, M. N. J. Colloid Interface Sci. 1979, 71, 283. (7) Drelich, J.; Miller, J. D. J. Colloid Interface Sci. 1994, 164, 252. (8) Duncan, D.; Li, D.; Gaydos, J.; Neumann A. W. J. Colloid Interface Sci. 1995, 169, 256. (9) Drelich, J. Colloids Surf., A 1996, 116, 43. (10) Pompe, T.; Fery, A.; Herminghaus, S. J. Adhes. Sci. Technol. 1999, 13, 1155.
∼10-11 to ∼10-6 N. This large range of uncertainty stems mainly from the difficulty in interpreting contact angle measurements on real (i.e. nonideal) surfaces. Recent measurements10-12 as well as some of the older ones5 tend to agree on values in the range 5 × 10-11 to 10-8 N. These values are also supported by theoretical calculations for solid-liquid-vapor systems.13-20 In the derivation of eq 1 it was a priori assumed that line tension is a constant. However, line tension cannot be considered constant when all possible variations in the system are tested in order to determine the equilibrium state by the minimum energy criterion.17 It must depend on the shape of the drop at each state, since it accounts for the three-phase molecular interactions between the liquid, solid, and vapor, which obviously depend on the system geometry. Therefore, a different line tension value must be associated with each possible state of the drop on the flat solid surface. Taking this into account, eq 1 for a drop on a flat, ideal surface was generalized17 to become
cos θi ) cos θY -
[
]
R(R2 + 3) dτ 1 τσlR (1 + R2) dR
(2)
where R ≡ (1 - cos θi)/sin θi. Another fundamental assumption underlying eq 1 is the ideality of the solid surface. Ideal surfaces hardly exist in reality; therefore, the geometry and heterogeneity of the solid surface need to be considered.21-24 It turns out that when the solid surface is not flat, R, the radius of the (11) Sto¨ckelhuber, K. W.; Radoev, B.; Schulze, H. J. Coloids Surf., A 1999, 156, 323. (12) Wang, J. Y.; Betelu, S.; Law, B. M. Phys. Rev. E 2001, 63, 031601. (13) Szleifer, I.; Widom, B. Mol. Phys. 1992, 75, 925. (14) Dobbs, H. T.; Indekeu, J. O. Phys. A 1993, 201, 457. (15) Indekeu, J. O.; Dobbs, H. T. J. Phys. I France 1994, 4, 77. (16) Widom, B. J. Phys. Chem. 1995, 99, 2803. (17) Marmur, A. J. Colloid Interface Sci. 1997, 186, 462. (18) Getta, T.; Dietrich, S. Phys. Rev. E 1998, 57, 655. (19) Dobbs, H. Physica A 1999, 271, 36. (20) Bauer, C.; Dietrich, S. Phys. Rev. E 2000, 62, 2428. (21) Boruvka, L,; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464. (22) Marmur, A. Colloids Surf., A 1998, 136, 81. (23) Wolansky, G.; Marmur, A. Langmuir 1998, 14, 5292. (24) Swain, P. S.; Lipowsky, R. Langmuir 1998, 14, 6772.
10.1021/la026167i CCC: $22.00 © 2002 American Chemical Society Published on Web 10/03/2002
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Inserting eq 3 into eq 4, one gets
τl ) Uslv + (σsv + σlv - σsl)Ssl
(5)
Since the molecular interactions with the vapor may be neglected, the interfacial tensions, σsv and σlv, may be replaced by the corresponding surface tensions, σs and σl. Then, the expression in parentheses on the right-hand side of eq 5 turns out to be the work of adhesion per unit area, W h ad, between a semi-infinite, solid slab and a similar liquid body: Figure 1. Gedanken experiment used for calculating line tension. The liquid phase is removed to infinite distance from the solid. The effect of the vapor is neglected because of its low density.
contact line, has to be replaced by the geodesic curvature of the contact line. Also, the last term in eq 2, which accounts for the local variations in line tension, can be generalized for rough and chemically heterogeneous surfaces.22,23 To be able to solve the implicit eq 2 or its generalizations22,23 for θi, a relationship between line tension and the geometry of the drop is required. This was achieved for a flat, ideal surface by a simple theoretical approach17 that enables the approximate calculation of line tension from the contact angle. However, the geometry of the drop depends also on the curvature of the solid surface. Therefore, the purpose of the present paper is to extend the previously developed theoretical approach to calculate line tension on spherical, ideal surfaces. This will enable the estimation of the effect of line tension on wetting of spherical particles. In addition, it may serve as a first step toward understanding the very important phenomenon of wetting of particles in general. Theory The energy associated with line tension for a solidliquid-vapor system can be defined as the difference between the actual interfacial energy of the system and the sum of the interfacial energies of the individual interfaces:17
τl ≡ σ/lvSlv + σ/slSsl + σ/svSsv - (σlvSlv + σslSsl + σsvSsv) (3) In this equation l is the length of the three-phase contact line, σ is an interfacial tension in a system that consists of only two phases, σ* is the average interfacial tension (over the whole corresponding interface) under the effect of a nearby third phase, S is an interfacial area, and the subscripts sl, sv, and lv refer to the solid-liquid, solidvapor, and liquid-vapor interfaces. An approximate relationship between the interfacial energies and the molecular interaction between the phases can be developed by following the Gedanken experiment shown in Figure 1. In this “experiment”, the solid and liquid phases, which are initially in contact (Figure 1a) with an interaction energy Uslv, are separated to an infinite distance (Figure 1b) with a zero interaction energy. It is assumed that the molecular interactions with the vapor are negligible because of its low density. The initial interaction energy must equal the difference in interfacial energies before and after separation; therefore
Uslv ) σ/lvSlv + σ/slSsl + σ/svSsv - σlvSlv (σsv + σlv)Ssl - σsvSsv (4)
W h ad ≡ σs + σl - σsl
(6)
Thus, line tension is given by
τlslv ) Uslv + W h adSsl
(7)
The interaction energy Uslv is the sum of all molecular interactions in the system. In the following, only van der Waals interactions will be considered. Again, since the density of the vapor is negligible, it is sufficient to calculate only the interaction between the solid and the liquid. This is expressed by25
Uslv ) -
Asl C
R*+R (r*)-5[R12 ∫RR+δ+C[R22 - (C - R*)2]∫R*-R 2
1
1
1
(R* - r*)2] dr* dR* (8) Here, as also shown in Figure 2, R1 is the radius of the solid sphere, R2 is the radius of the spherical, liquid drop, C is the distance between their centers, Asl is the Hamaker constant, and δ is an average distance between the solid and liquid molecules. δ serves as a cutoff length, to avoid singular values of the interaction energy. R* and r* are the integration variables. Considering the fact that δ is much smaller than the sizes of the spheres, eq 8 leads to the following approximate expression (see Appendix 1 for details):
Uslv = -
[
]
Asl (4R1y0 + y02 - R22) R1(R22 - y02) + 12C δ δ2
(9)
Here, y0 ≡ C - R1 ) -R1 + (R12 - 2R1R2 cos θ + R22)1/2 (note that C > 0 by definition), where θ is the geometric contact angle that the liquid makes with the solid (see Figure 2). The term “geometric contact angle” emphasizes that line tension is calculated on the basis of the shape of the drop at each possible state; θi is the equilibrium value of θ. Throughout the present paper it is assumed that θ * 0. The additional parameters that are needed for calculating τ from eq 7 are given in the following. W h ad is actually the negative of the interaction energy per unit area between the semi-infinite solid and liquid bodies, U h sl, which can be approximated by26
h sl = W h ad ) -U
Asl 12πδ2
(10)
(25) Hamaker, H. C. Physica 1937, 4, 1058. (26) Hiemenz, P. C.; Rajagoplolan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker Inc.: New York, 1997.
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Figure 3. Effect of curvature of the solid surface on line tension. The numbers in the figure indicate the value of r ≡ R1/R2.
In addition, to the same degree of approximation, one can use at equilibrium27
cos θi = -1 + 2
x
σs σl
(15)
Inserting eqs 14 and 15 into eq 13, one gets the following relationship between line tension and the equilibrium contact angle:
τ = 2δσl cos2
(
)
y0 θi cot θi 2 R2 sin θi
(16)
Introducing the following dimensionless variables
τ* ≡
τ ; δσl ξ≡
Figure 2. Geometrical definitions of the system: (a) r ≡ R1/R2 > 1; (b) r < 1.
Ssl ) 2πR1
(
)
R1 - R2 cos θ 1C
(11)
(12)
Substituting eqs 9-12 into eq 7 leads to (note that the coefficient of δ-2 turns out to be identically zero)
(
Asl y0 - cot θ τ=12πδ R2 sin θ
)
(
)
θi ξ cot θi 2 sin θi
(17)
(18)
The main message of the present paper is that the value of the line tension depends not only on the contact angle, as previously elucidated,17 but also on the curvature of the solid surface (relative to the curvature of the liquid drop). As a first step in analyzing the result of the above theoretical development, it is of interest to compare eq 18 with the result of the theory for a flat solid surface. This is easily done by letting R1 f ∞, which, from eq 17, implies that ξ f (-cos θ). Equation 18 then turns into / ) 4 cos2 τflatsurface
(13)
It is more convenient to express line tension in terms of surface tensions rather than Hamaker constants; therefore, the following approximate expression is used17
Asl = 24πδ2xσsσl
2
Results and Discussion
and the length of the contact line is given by
lslv ) 2πR1R2 sin θ/C
( ))
R2 R2 1 - 2 cos θ + R1 R1
eq 16 is expressed as
τ* = 2 cos2
The interfacial solid-liquid area is given by 2
( x
y 0 R1 ) -1 + R2 R2
(14)
θi cot θi 2
(19)
This is exactly the equation that results from the one previously developed for a flat solid surface (eq 21 in ref 17) when eq 15 is introduced into it. The additional effect of the curvature of the solid surface is demonstrated in Figure 3. This figure shows the dimensionless line tension for various values of the (27) Girifalco, L. A.; Good, R. J. J. Phys. Chem. 1957, 61, 904.
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Figure 4. Critical radius of the contact line, above which the effect of line tension is experimentally indistinguishable.
dimensionless radius of the solid sphere (normalized with respect to the radius of the liquid drop), r ≡ R1/R2. Three observations can be made on the basis of these results. First, the contact angle, at which the transition from positive to negative line tensions occurs, decreases with an increase in the solid surface curvature. For flat surfaces this transition is predicted to occur at a contact angle of 90°, while for r ) 0.1, for example, it is expected to occur at ∼35°. Second, line tension is predicted to be very sensitive to the curvature of the solid surface for low contact angles, but this sensitivity decreases as the contact angle increases. It is rather insensitive to the solid surface curvature at contact angles higher than about 120°. Third, increased curvature of the solid surface decreases the value of line tension. In terms of the absolute value of line tension, increased curvature reduces positive line tensions but increases the absolute value of negative line tension. It is of interest at this point to discuss the practical importance of line tension, in terms of its effect on the value of the contact angle. Before discussing the consequences of curvature in this respect, it is worthwhile to discuss the significance of line tension for wetting on flat, ideal surfaces. This is done by a simple, approximate calculation of a critical value, Rc, of the contact-line radius, above which the absolute difference between θi and θY is smaller than 1°. This definition is based on the practical realization that a deviation of about 1° is approximately the lower limit of experimental resolution in contact angle measurements. For simplicity, Rc is calculated from eqs 1 and 19, assuming that the deviation from the Young equation is determined mainly by τ/σlR, while the contribution of the last term in eq 2 or its generalizations is smaller. Under this assumption
Rc 4 cos2(θY/2) cot θY ≈| | δ cos θY - cos(θY - π/180)
meaningless for contact radii above ∼104δ (about a few micrometers). The critical radius sharply decreases when the contact angle approaches 90°: at around 60° the critical radius is ∼102δ (about a few tens of nanometers), and at around 80° it is ∼25δ. For obtuse contact angles, Rc is always smaller than ∼102δ and also sharply decreases when the contact angle approaches 90°: around 100° it is only ∼15δ. Thus, line tension on flat surfaces is expected to be important only for micro- or nanodrops and to strongly depend on the contact angle. The curvature of the solid surface affects the significance of line tension in two independent ways. First, as shown above, it affects the value of line tension. For low contact angles, line tension for a drop on a spherical particle was predicted to be smaller than that on a flat surface; for high contact angles, it is insensitive to the curvature of the solid. Consequently, the critical radius given by eq 20 and shown in Figure 4 can serve as an upper limit for drops on spherical particles as well. The other effect of curvature on the significance of line tension is related to the fact that it is the geodesic curvature of the contact line that counts in eq 1. For a liquid drop on an ideal spherical surface, eq 1 turns into23
τ cos θi ) cos θY - x1 - (R/R1)2 σlR
Again, for simplicity, the additional terms that include derivatives of line tension22,23 are ignored in the present approximate analysis. This equation shows that the effect of line tension on the contact angle for a drop on a spherical particle is actually smaller than that for the case of a flat surface. Thus, the critical radius given by eq 20 and shown in Figure 4 retains its status as a plausible upper limit based on eq 21 too: if the radius of the contact line is above Rc, as given by eq 20, the effect of line tension is predicted to be negligible. Appendix 1 The right-hand side of eq 8 may be written as the following sum:
Uslv ) -
(28) It should be noted that when the contact angle approaches zero, line tension is predicted by eq 19 to diverge. This result, which stems from the lack of consideration of thin film physics in the present model, is, most likely, physically invalid;18,20 therefore, line tension values for very small contact angles should not be calculated using the present model.
Asl [I + I2 + I3 + I4 + I5] 12C 1
(A1.1)
Here
I1 )
[
2R
2R
∫RR+δ+CR22 (R + R1 )3 + (R - R1 )3 + (R +1R )2 2
1
]
[
1
1
1
C + R2 - 2R1 1 dR ) R22 2 (R - R1) (C + R2 - R1)2 δ - R1 3R1 + δ C + R2 + 2R1 + (A1.2) 2 2 (C + R2 + R1) δ (2R1 + δ)2
(20)
In other words, when the radius of the contact line is above Rc, the effect of line tension cannot be experimentally distinguished. Figure 4 shows Rc/δ versus θY and leads to interesting conclusions. For acute contact angles, except for relatively low ones,28 line tension is predicted to be practically
(21)
I2 ) -
2R
∫RR+δ+C(C - R)2(R + R1 )3 dR ) 2
1
[
2R1 ln
]
1
(C + R1)2 2
2(C + R2 + R1)
-
2(C + R1) (C + R2 + R1)
-
]
(C + R1)2 2(C + R1) (C + R2 + R1) + (A1.3) 2 2R1 + δ (2R1 + δ) 2(2R + δ) 1
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2R
∫RR+δ+C(C - R)2(R - R1 )3 dR )
I3 ) -
2
1
[
2R1
1
(C - R1)2
2(C + R2 - R1)
2
-
2(C - R1) (C + R2 - R1)
-
]
(C + R2 - R1) (C - R1)2 2(C - R1) + (A1.4) ln δ δ 2δ2 2
(C - R)
(C + R1)2
∫RR+δ+C(R + R )2 dR ) (C + R
I4 ) -
2
1
1
2
+ R1)
I5 )
- R)2
(C - R1)
∫R +δ (R - R )2 dR ) - (C + R 1
1
2
Uslv ) -
-C-
(C + R2 + R1) (C + R1)2 + R2 - R1 + 2(C + R1) ln 2R1 + δ (2R1 + δ) 2R1 + δ (A1.5) R2+C (C
For all practical purposes δ/R1 f 0, and [δ/(C - R1) or δ/(C + R2 - R1)] f 0. Therefore, I2 and I4 may be ignored compared with I1, I3, and I5. Substituting expressions A1.2, A1.4, and A1.6 into eq A1.1, neglecting all terms that do not diverge as δ f 0, one gets
-
[
+C+
(C + R2 - R1) (C - R1)2 + R2 - R1 - 2(C - R1) ln δ δ δ (A1.6)
(
)
[
Asl (y0 + R2) R1 - δ R22 + 2R1 2y0 - δ ln 12Cδ δ δ
2
- R1)
[ ( ) [ ] ]
(y0 + R2) Asl R1 - δ 2y0 - ln R22 + 2R1 2 12C δ δ δ y02 (y0 + R2) y02 - 2y0 ln + ) 2 δ δ 2δ
2
]
]
y0 (y0 + R2) + y02 - 2y0δ ln (A1.7) 2δ δ
Taking into account that δ ln[(y0 + R2)/δ] f 0 while δ f 0, one gets eq 9. LA026167I
