Structure and Stability of Silica Particle Monolayers at Horizontal and

Monolayers of silica particles at horizontal and vertical octane-water interfaces have ... density at the particle-octane interface, 14.1µC/m2, found...
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Langmuir 2005, 21, 7405-7412

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Structure and Stability of Silica Particle Monolayers at Horizontal and Vertical Octane-Water Interfaces Tommy S. Horozov,* Robert Aveyard, Bernard P. Binks, and John H. Clint Surfactant & Colloid Group, Department of Chemistry, University of Hull, Hull, HU6 7RX, U.K. Received April 7, 2005 Monolayers of silica particles at horizontal and vertical octane-water interfaces have been studied by microscopy. It is found that their structure and stability depend strongly on the particle hydrophobicity. Very hydrophobic silica particles, with a contact angle of 152° measured through the water, give wellordered monolayers at interparticle distances larger than 5 particle diameters which are stable toward aggregation and sedimentation. In contrast, monolayers of less-hydrophobic particles are disordered and unstable. Two-dimensional particle sedimentation has been observed in the case of vertical monolayers. The results have been analyzed with a simple two-particle model considering the sedimentation equilibrium as a balance between the long-range electrostatic repulsion through the oil, the gravity force, and the capillary attraction due to deformation of the fluid interface around particles. The value of the charge density at the particle-octane interface, 14.1 µC/m2, found for the most hydrophobic particles is reasonable. It drastically decreases for particles with lower hydrophobicity, which is consistent with the order-disorder transition in monolayer structure reported by us before. The pair interactions between particles at a horizontal octane-water interface have been analyzed including the capillary attraction due to undulated three-phase contact line caused by nonuniform wetting (the contact angle hysteresis). The results are in agreement with the great stability of very hydrophobic silica particle monolayers detected experimentally, even at low pH at the point of zero charge of the particle-water interface, and with the aggregated structure of hydrophilic particle monolayers.

1. Introduction The fact that small solid particles can strongly attach to fluid interfaces was recognized about one century ago.1,2 The interest in particle monolayers at fluid interfaces, however, strongly arose about two decades ago after the pioneering work of Pieranski,3 who showed that small latex particles can form well-ordered structures when attached to the water-air surface. Many studies in this field have been done since,4-16 aiming to verify the theory for melting of two-dimensional crystals5 or to reveal the mechanisms of two-dimensional particle aggregation at horizontal air-water interfaces.6-12 Although less studied in the past, particle monolayers at oil-water interfaces * Author to whom correspondence should be addressed. E-mail: [email protected]. (1) Ramsden, W. Proc. R. Soc. 1903, 72, 156. (2) Pickering, S. U. J. Chem. Soc. 1907, 91, 2001. (3) Pieranski, P. Phys. Rev. Lett. 1980, 45, 569. (4) Kumaki, J. Macromolecules 1988, 21, 749. (5) Armstrong, A. J.; Mockler, R. C.; O’Sullivan, W. J. J. Phys.: Condens. Matter. 1989, 1, 1707. (6) Hurd, A. J.; Schaefer, D. W. Phys. Rev. Lett. 1985, 54, 1043. (7) Onoda, G. Y. Phys. Rev. Lett. 1985, 55, 226. (8) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. A 1992, 46, 2045. (9) Robinson, D. J.; Earnshaw, J. C. Langmuir 1993, 9, 1436. (10) Stankiewicz, J.; Cabrerizo-Vilchez, M. A.; Hidalgo-A Ä lvarez, R. Phys. Rev. B 1993, 47, 2663. (11) Martı´nez-Lo`pez, F.; Cabrerizo-Vilchez, M. A.; Hidalgo-A Ä lvarez, R. J. Colloid Interface Sci. 2000, 232, 303. (12) Ghezzi, F,; Earnshaw, J. C. J. Phys. Condens. Matter 1997, 9, L517. (13) Ho´rvo¨lgyi, Z.; Nemeth, S.; Fendler, J. H. Langmuir 1996, 12, 997. (14) Tolnai, G.; Csempesz, F.; Kabai-Faix, M.; Kalman, E.; Keresztes, Zs.; Kovacs, A. L.; Ramsden, J. J.; Ho´rvo¨lgyi, Z. Langmuir 2001, 17, 2683. (15) Hansen, P. H. F.; Ro¨dner, S.; Bergstro¨m, L. Langmuir 2001, 17, 4867. (16) Szekeres, M.; Kamalin, O.; Grober, P. G.; Schoonheydt, R. A.; Wostyn, K.; Clays, K.; Persoons, A.; De´ka´ny, I. Colloids Surf., A 2003, 227, 77.

are the subject of increased scientific interest during the past few years,17-22 mainly because small solid particles alone can act as very effective emulsion stabilizers.23 Silica particles have been used in a number of monolayer studies,6,14-16,21 most of them at the water-air6,14,16 or organic liquid-air interfaces.15 Recently, we reported a transition from disordered to well-ordered horizontal silica particle monolayers with an increase of particle hydrophobicity and attributted it to Coulombic repulsion acting through the oil phase as a result of charges at the particleoil interface.21 This was supported by our very recent results for vertical emulsion films with particle monolayers at their surfaces.24 It was shown that slightly hydrophobic silica particles did sediment at the bottom of thick vertical films in contrast to the most hydrophobic ones which did not. This interesting observation, however, was not analyzed quantitatively. In contrast to the long range repulsive particle interactions, which are mainly due to electrostatic repulsion through the nonpolar phase (oil or air), the origin of the long-range capillary attraction between small particles at fluid interfaces is still under debate. The capillary attraction is due to overlapping the deformations of the fluid interface around the particles.25,26 The deformation (17) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969. (18) Aveyard, R.; Clint, J. H.; Nees, D.; Quirke, N. Langmuir 2000, 16, 8820. (19) Schwartz, H.; Harel, Y.; Efrima, S. Langmuir 2001, 17, 3884. (20) Stancik, E. J.; Widenbrant, M. J. O.; Laschitsch, A. T.; Vermant, J.; Fuller, G. G. Langmuir 2002, 18, 4372. (21) Horozov, T. S.; Aveyard, R.; Clint, J. H.; Binks, B. P. Langmuir 2003, 19, 2822. (22) Aveyard, R.; Clint, J. H.; Horozov, T. S. Phys. Chem. Chem. Phys. 2003, 5, 2398. (23) Aveyard, R.; Binks, B. P.; Clint, J. H. Adv. Colloid Interface Sci. 2003, 100-102, 503. (24) Horozov, T. S.; Aveyard, R.; Clint, J. H.; Neumann, B. Langmuir 2005, 21, 2330.

10.1021/la050923d CCC: $30.25 © 2005 American Chemical Society Published on Web 06/28/2005

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caused by gravity is negligible for particles whose radius is smaller than several micrometers.26 It has been shown that the undulations of the three-phase contact line due to nonuniform wetting of the particles (contact angle hysteresis) can be a source of deformation leading to a capillary attraction between particles even at large interparticle distances.27,28 It was suggested recently that deformation of the fluid interface around colloid particles can arise due to forces of electrostatic origin.29 Very recently, Danov et al.30 showed both experimentally and theoretically that a deformation of the fluid interface around charged particles should exist due to the so-called “electro-dipping” force pushing particles toward the water. However, the hypothesis for long-range capillary attraction associated with these deformations has been called into question31,32 and is still under discussion. The aims of the present work are (i) to obtain quantitative information about the particle properties affecting their interactions (e.g., the surface charge density at the particle-oil interface) and (ii) to shed more light on the controversial issue of the origin of long-range capillary attraction between small particles at fluid interfaces. To achieve this, we analyze quantitatively the two-dimensional sedimentation equilibrium of the monolayers at vertical oil-water interfaces. Bearing in mind the complexity of this problem, which has not been analyzed in the literature before, we simplify our analysis by using a model for the mechanical equilibrium between two particles at a vertical oil-water interface. We use the obtained information for analyzing our new results for horizontal particle monolayers in order to achieve the second goal formulated above. 2. Materials and Methods Silanized silica particles with diameter 3.00 ( 0.05 µm, density 2.0 g/cm3, and contact angles at the octane-water interface in the range 65°-152° used here were the same as those in our previous study.24 The interfacial tension of the purified octane against water was 50.6 mN/m measured by the ring method at 24 °C. The aqueous phase with pH equal to 5.6 ( 0.3 was deionized water, obtained from a Milli-Q purification unit (Millipore), or 10 mM NaCl (+99.9% Analar grade, BDH). In some experiments pH was adjusted to 2.7 by adding 0.1 M HCl. The experiments were performed at room temperature (24° ( 1° C).

3. Experiments 3.1. Horizontal Monolayers. Horizontal monolayers were formed in a small Petri dish (diameter 2 cm, height 0.7 cm) placed in the center of a larger one with diameter 6 cm and height 3 cm. The smaller dish was filled with the aqueous phase, the water surface was made flat by sucking part of the water out, and octane was added in the larger Petri dish to cover the water. The thickness of the oil layer on the top of the water phase was ∼3 mm. We have found out that this configuration largely suppresses the lateral drift of the particles due to convection. Then, the monolayer of particles at the octane-water (25) Chan, D. Y. C.; Henry, J. D., Jr.; White, L. R. J. Colloid Interface Sci. 1981, 79, 410. (26) Kralchevsky, P. A.; Nagayama, K. Particles at Fluid Interfaces and Membranes; Elsevier Science: Amsterdam, 2001. (27) Stamou, D.; Duschl, C.; Johannsmann, D. Phys. Rev. E 2000, 62, 5263. (28) Kralchevsky, P. A.; Denkov, N. D.; Danov, K. D. Langmuir 2001, 17, 7694. (29) Nikolaides, M. G.; Bausch, A. R.; Hsu, M. F.; Dinsmore, A. D.; Brenner, M. P.; Gay, C.; Weitz, D. A. Nature 2002, 420, 299. (30) Danov, K. D.; Kralchevsky, P. A.; Boneva, M. P. Langmuir 2004, 20, 6139. (31) Megens, M.; Aizenberg, J. Nature 2003, 424, 1014. (32) Foret, L.; Wu¨rger, A. Phys. Rev. Lett. 2004, 92, 058302.

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interface was formed by spreading ∼6 wt% particle suspension in 70 wt% 2-propanol/water (for details see ref 21). The horizontal monolayers were observed from above with a microscope (Optiphot 2, Nikon) with infinitycorrected optics, supplied with extra-long working distance objectives. The images recorded by VCR were processed as described elsewhere.21 3.2. Vertical Monolayers. The experimental setup has been described previously.24 It was used here for studying particle monolayers at the surface of thick vertical octane films in water. They were formed by crossing a dilute horizontal particle monolayer (about 2 × 105 particles/ cm2) by a PTFE frame. The films studied were thicker than 200 µm; therefore, the interactions between the monolayers at opposite film surfaces were negligible. 4. Results 4.1. Horizontal Dilute Particle Monolayers. The structure of horizontal monolayers strongly depends on the particle hydrophobicity expressed by the particle contact angle. The dilute monolayers of particles with low hydrophobicity (i.e., small contact angle) were disordered just after spreading, and significant aggregation occurred with time. This is illustrated in Figure 1a, where an image of a horizontal monolayer of particles (diameter 3 µm) with a contact angle of 65° 1 h after spreading is shown. Most of the particles are inside two-dimensional aggregates. The live observations have shown that aggregated particles are not in close contact. They vibrate keeping on average 4.5 ( 0.2 µm distance between the centers in particle doublets. Dense aggregates of touching particles were observed in the presence of 10 mM NaCl in the aqueous phase (Figure 1c). The horizontal monolayers of more-hydrophobic particles with contact angles of 85° or 99° (not shown here) had a similar disordered structure. The most hydrophobic particles with a contact angle of 152°, however, gave monolayers of hexagonally ordered particles at large separations. An example is shown in Figure 1b, where the average distance between particle centers is 27.8 ( 1.5 µm, i.e., larger than 9 particle diameters. The decrease of pH from 5.6 to 2.7 by adding HCl in the water had no impact on the ordered structure of the monolayer (Figure 2a). The structure of these monolayers remained unchanged for more than 16 h. A small amount of aggregates was present just after spreading, but no further aggregation occurred with time. This is illustrated in Figure 2b, where the number fraction of aggregated particles (i.e., the number of particles in aggregates divided by the total number of particles within one image) measured at different times is shown. Each point in this figure is an average of the values obtained from at least 10 images taken at different locations within the monolayer. 4.2. Vertical Dilute Particle Monolayers-Effect of Gravity. The effect of gravity is negligible for micrometer-size particles at a horizontal fluid interface.26 In contrast, gravity has a dramatic effect on vertical monolayers of particles with low hydrophobicity. The particles with contact angles in the range 65°-99° sedimented with time. This two-dimensional sedimentation was relatively fast, thus giving a well-ordered array of close-packed particles at the bottom of the frame and bare surface at the top about 1 h after formation of the vertical monolayers (Figure 3a). Surprisingly, the vertical monolayers of the most hydrophobic particles did not sediment significantly. Only a small variation of the distance between particle centers with height from 14.5 ( 0.7 µm at the bottom to 15.6 ( 0.7 µm at the top

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Figure 1. Horizontal silica particle monolayers at the octane-water interface without added electrolyte (a, b) and with 10 mM NaCl (c) in the water 1 h after their formation. Particle contact angles measured through the water are 65° (a, c) and 152° (b). The scale bar is equal to 25 µm.

of vertical monolayers in the latter case are remarkably different from those in the case of less-hydrophobic particles. The observed difference is analyzed in the next section. 5. Discussion

Figure 2. An image of a horizontal monolayer of silica particles with a contact angle of 152° at the octane-water interface at pH of the aqueous phase equal to 2.7 (a) and the fraction of aggregated particles as a function of time (b). A small number of aggregates (mainly doublets and triplets) was formed during preparation of the monolayer by spreading. No further aggregation was detected for more than 16 h after formation of the monolayer. The scale bar is equal to 50 µm.

has been detected 1 h after formation of the monolayer (Figure 3b-d). It is seen that the structure and stability

The present results for the structure and stability of dilute horizontal silica particle monolayers are consistent with our previous findings.21 The monolayers of the mosthydrophobic particles with a contact angle of 152° are well ordered even at very large interparticle distances (about 9 particle diameters in Figure 1b). This can be attributed to the long-range Coulomb repulsion through the oil due to charges at the particle-oil interface. In contrast, monolayers of less-hydrophobic particles with a contact angle of 65° are disordered and aggregated (Figure 1a). The aggregated particles, however, are not in close contact in the absence of electrolyte (Figure 1a) but stay well separated at equilibrium distances beyond the range of the van der Waals attraction. Obviously, some longrange attraction is needed to balance the repulsion between particles. The repulsion in this case is mediated mainly through the aqueous phase since 10 mM NaCl in the water is enough to suppress the repulsion and to bring the particles in contact (Figure 1c). This repulsion is not strong enough to oppose gravity, thus the particles in the vertical monolayer sediment giving a well-ordered twodimensional array of close-packed particles at the bottom of the frame (Figure 3a). The same is true for the monolayers of more-hydrophobic particles with contact angles of 85° and 99° which are not shown here. The situation with the most hydrophobic particles is remarkably different. The repulsion between particles with a contact angle of 152° is so strong that they do not sediment but stay well separated at large distances along the whole height (6.2 mm) of the vertical monolayer (Figure 3b-d). The analysis of the stability of the vertical particle

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Figure 3. Vertical silica particle monolayers at the octane-water interface 1 h after their formation in a circular frame with diameter 6.2 mm. The images are taken at the bottom (a, d), top (b), and middle (c) of the frame. Particle contact angles measured through the water are 65° (a) and 152° (b-d). The scale bar is equal to 50 µm.

monolayers based on the balance between the repulsive and attractive interparticle forces, including the external gravity force, should allow the surface charge density at the particle-oil interface to be determined. The rigorous analysis is rather complex and requires the many-body problem to be considered. Below, we present a simpler analysis using a one-dimensional model for the mechanical equilibrium between two particles. 5.1. Mechanical Equilibrium between Two Particles at a Vertical Oil-Water Interface. Two particles with radius R are attached at a vertical water-oil interface parallel to the yz plane, as shown schematically in Figure 4a. The lower particle is fixed, while the upper one can move only along the z axis, thus changing the distance, l, between particle centers. When the mechanical equilibrium is established at interparticle distance, leq, the force balance in the z direction reads

∑FR ) Fg + ∑FA

(1)

i.e., all repulsive forces (the left-hand side of eq 1) balance the attractive forces and the gravity force, Fg (the righthand side of eq 1). We will neglect short-range repulsive and attractive interactions (e.g., hydration repulsion through the aqueous phase, van der Waals, and hydrophobic attraction) which certainly do not play a significant role at separations of particle surfaces larger than 1 µm (see, e.g., refs 11, 22, and 34). Besides the gravity force, the remaining long-range forces are repulsion due to charges at the particle-water interface, repulsion through the oil due to dipoles or charges at the particle-nonpolar fluid interface, and the capillary attraction due to overlapping of the deformed fluid interface around the particles. The deformations could be due to (i) undulations of the three-phase contact line caused by nonuniform wetting of the particles (contact angle hysteresis)27,28 and (ii) electrical stresses in the fluids caused by the charged interfacial particles (a so-called “electro-dipping” force), as was recently suggested29,30 (Figure 4a). 5.1.1. The Gravity Force. The gravity force, Fg, (the difference between the particle weight and the buoyancy force) depends on the position of the particle between the

Figure 4. Sketch of the model of two particles at a vertical oil-water interface (a) and the equilibrium distance between particles, Leq ) leq/2R, as a function of the charge density at the particle-octane interface, σpo (b) at particle contact angles equal to 152° (1) and 65° (2). The deformation of the fluid interface around the particles is partially due to the electro-dipping force acting normal to the oil-water interface and to the undulations of the three-phase contact line with an amplitude δ. The lines in (b) correspond to δ ) 0 nm, while squares and circles correspond to δ ) 10 nm (for details see the text).

two liquids (i.e., on the angle R). It is given by the expression

Fg ) g(Fp - Fw)Vpw + g(Fp - Fo)Vpo

(2)

where g is the acceleration due to gravity and Vpw and

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Vpo are the particle volumes immersed in water and oil, respectively, Fp, Fo, and Fw are the densities of particle, oil, and water. Bearing in mind that Vpw ) {πR3(1 + cos R)2(2 - cos R)}/3 and Vpo ) (4πR3/3) - Vpw, eq 2 gives

Fg )

[

]

Fw - Fo 4 πR3(Fp - Fw)g 1 + (cos3 R - 3 cos R + 2) 3 4(Fp - Fw) (3)

At small deformation of the oil-water interface around the particles, the slope angle ψ ) θ - R is also small; therefore, the angle R can be replaced by the particle contact angle, θ, in the above equation without introducing a significant error. Bearing this in mind, we have calculated that the gravity force, Fg, varies between 0.1470 (θ ) 65°) and 0.1795 pN (θ ) 152°) in our experiments (R ) 1.5 µm, Fp ) 2000 kg/m3, Fo ) 703 kg/m3, Fw ) 1000 kg/m3, g ) 9.81 m/s2). 5.1.2. Repulsion Due to Charges at the Particle-Water Interface. There are two types of long-range repulsion associated with the charges at the particle-water interface: (i) screened Coulomb repulsion through the water and (ii) dipolar repulsion through the nonpolar fluid (air or oil), as suggested by Pieranski.3 Calculation of the pair potential due to screened Coulomb repulsion in water is complicated by the fact that only a portion of the particle surface is immersed in water. Therefore, it should involve numerical integration as has been done for instance in ref 11 using the Derjaguin approximation and taking into account the particle contact angle. It has also been shown there11 that the screened Coulomb pair potential does not change appreciably for hydrophilic particles with contact angles from 0° to ∼85° but sharply drops down and becomes negligibly small for hydrophobic particles with contact angles greater than ∼95°. Therefore, we will assume that the screened Coulomb pair potential in water is UCw ≈ 0 for hydrophobic particles (θ > 90°) while for hydrophilic ones (θ < 90°) is given by the equation for particles totally immersed in water34,35

UCw )

(

4πw0Rψ2pw 1 -

) {

}

exp[-2κR(L - 1)] 1 ln 1 + 2L 2L - 1

(4)

where w is the dielectric constant of water, 0 is the permittivity of vacuum, L ) l/2R is the dimensionless distance between particle centers, ψpw is the surface potential at the particle-water interface and κ ) exc/(w0kT) is the Debye screening parameter, e is the elementary charge, k is the Boltzmann constant, T is the temperature, and c is the bulk concentration (in m-3) of 1:1 electrolyte. Equation 4 is a good approximation at any interparticle distances if κR is not very small.34,35 The relation between the surface charge density at the particle-water interface, σpw, and the surface potential is36 (33) Handbook of Chemistry and Physics, 51st ed.; Weast, R. C., Ed.; CRC: Cleveland, 1970. (34) Kralchevsky, P. A.; Danov, K. D.; Denkov, N. D. In Handbook of Surface and Colloid Chemistry, 2nd ed.; Birdi, K. S., Ed.; CRC Press: New York, 2002; Chapter 5. (35) McCartney, L. N.; Levine, S. J. Colloid Interface Sci. 1969, 30, 345. (36) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1987; Vol. 1, p 335.

σpw ) -

( )

eψpw 4ec sinh κ 2kT

(5)

It can be shown that at the conditions of our experiments (ionic strength ∼6.3 × 10-7 M at pH 5.9) and a reasonable value for the surface potential ψpw ) -100 mV (σpw ) 320 µC/m2 see, e.g., ref 37) UCw for hydrophilic particles becomes smaller than kT at L > 2.1. Therefore, the force of screened Coulomb repulsion through the water, FCw ) ∂UCw/∂l, can be neglected at large interparticle distances. The asymmetric distribution of the free ions around the interfacial particles in water should lead to an effective dipole moment oriented perpendicular to the fluid interface.3 Therefore, adsorbed particles can interact through the oil (air) by long-range unscreened dipolar interactions due to the absence of mobile charges in the nonpolar phase. According to Hurd,38 the pair potential associated with such dipolar interactions, Udw, is

Udw )

q2pw 16π02wκ2R3L3

(6)

where qpw ) Apwσpw ) 2πR2(1 + cos θ)σpw is the total charge at the particle-water interface with area Apw and surface charge density σpw (see also refs 12, 39). The force of dipolar repulsion due to charges at the particle-water interface, Fdw, obtained after differentiation of eq 6 with respect to l is

Fdw )

3π(1 + cos θ)2σ2pw 802wκ2L4

(7)

5.1.3. Repulsion through the Nonpolar Fluid Due to Dipoles at the Particle-Oil Interface. It has been suggested that the particle-air interface carries dipoles originating from polar surface groups; therefore, the increased stability of particle monolayers at air-water interfaces in comparison to bulk dispersions is due to dipole-dipole repulsion through the air.9,11,39 It has been shown21 that at large distances the dipolar pair interaction energy through the oil, Udo, is given by the following asymptotic expression

Udo )

P2o 32πoR3L3

(8)

where  is the dielectric constant of oil and Po ) πpσR2 sin2 θ is the dipole moment of that part of the particle which is in oil. Here, p is the average dipole moment of one polar group (an elementary dipole) and σ is the surface density of polar groups at the particle-oil interface. The dipolar repulsive force through the oil, Fdo ) ∂Udo/∂l is

Fdo )

3P2o 64π0R4L4

(9)

5.1.4. Coulomb Repulsion through the Oil Due to Charges at the Particle-Oil Interface. The pair Coulomb interaction energy, UCo, and the force of Coulomb repulsion, FCo, (37) Behrens, S. H.; Grier, D. G. J. Chem. Phys. 2001, 115, 6716. (38) Hurd, A. J. J. Phys. A: Math. Gen. 1985, 18, L1055. (39) Moncho-Jorda´, A.; Martı´nez-Lo´pez, F.; Quesada-Pe´rez, M.; Cabrerizo-Vı´lchez, M. A.; Hidalgo-A Ä lvarez, R. In Surface and Colloid Science; Matijevic´, E., Borkovec, M., Eds.; Kluwer Academic/Plenum Publishers: New York, 2004; Vol. 17, Chapter 4.

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through the oil between particles at the fluid interface according to ref 17 are

UCo )

FCo )

[

(Apoσpo)2 1 8πR0 L

(Apoσpo)2 2

16πR 0

2

]

(10)

}

(11)

x(3 + cos θ)2 + 4L2

{

L 1 2 L [(3 + cos θ)2/4 + L2]3/2

where σpo and Apo ) 2πR2(1 - cos θ) are the surface charge density and the area of that part of particles which is exposed to oil, respectively. The influence of the aqueous phase on the ‘pure’ Coulomb repulsion due to so-called ‘image charges’ is accounted for by the second term in the large brackets of the above equations (for details, see ref 17). 5.1.5. Capillary Attraction. The capillary attraction between similar particles at a fluid interface is due to overlapping the deformations of the interface around the particles. The deformation caused by gravity is negligible for particles with a radius smaller than several micrometers floating at a horizontal liquid interface. When particles are attached to a vertical oil-water interface, as shown in Figure 4a, the gravity force acts parallel to the fluid interface; hence, it cannot create any deformation. However, if the three-phase contact line around the particles is not perfectly circular but irregular, this will create deformations of the fluid interface irrespective of its orientation (horizontal or vertical). These irregularities (undulations) of the three-phase contact line shape are due to nonuniform wetting of the particles resulting from a topological roughness or chemical inhomogeneity of the particle surface. The latter reflects on the hysteresis of the macroscopic three-phase contact angle, ∆θ, which measures the difference between the advancing (maximum) and the receding (minimum) contact angles. It has been shown that the deformations of the fluid interface around adjacent particles can correlate and the particles adjust their mutual orientation (by rotating in the plane of the liquid interface) to minimize their potential energy.27,28,40,41 A capillary attraction appears which can be considered as a result of interactions between capillary multipoles.28 The capillary quadrupoles give the major contribution to the pair potential of capillary attraction, UcapU, which at the most favorable mutual orientation of the particles (corresponding to the minimum potential energy) is given by27,28

UcapU ) -

3πγowδ2R4c 4R4L4

(12)

where γow is the interfacial tension of the bare oil-water interface, Rc ) R sin θ is the radius of the three-phase contact line projection on the plane of the liquid interface, and δ is the amplitude of the three-phase contact line undulations (see Figure 4a). The force of capillary attraction, FcapU ) ∂UcapU/∂l, from eq 12 is

FcapU )

3πγowδ2 sin4 θ 2RL5

(13)

(40) Bowden, N.; Terfort, A.; Carbeck, J.; Whitesides, G. M. Science 1997, 276, 233. (41) Brown, A. B. D.; Smith, C. G.; Rennie, A. R. Phys. Rev. E 2000, 62, 951.

The amplitude of undulations, δ, in the case of small spherical particles cannot be measured, but its maximum value, δmax, can be estimated from the contact angle hysteresis by using the formula27 δmax ≈ R∆θ/2. The hysteresis of the contact angle in our case was 6° for the particles with an average contact angle of 65° and 4° for the particles with larger contact angles. Hence, the amplitude of undulations of the three-phase contact line should not exceed 78 nm for the less-hydrophobic particles used or 52 nm for the more-hydrophobic ones. Another source of deformation of the fluid interface could be the electric field around particles residing at the interface between liquids with very different dielectric constants.29-32 It has been suggested that the electrically generated deformation should decay logarithmically with distance; hence, a long-range capillary attraction between particles adsorbed at the liquid interface should be expected.29,30 It has been argued, however, that the electrically generated deformation should decay much faster with distance, implying very short-range capillary interactions (decaying as 1/l6).31,32 However, the experimental data presented in ref 30 show a significant deformation of the fluid interface which cannot be due to the gravity alone and has been attributed to the electric field around the charged particle. The extent of deformation measured (at a distance up to ∼2R from the particle center) suggests the corresponding capillary attraction between particles could be important at L e 2 and could contribute to the overall attraction between the particles in the loose aggregates shown in Figure 1a. We will discuss this again later. Finally, taking into account only the forces which are significant at large interparticle distances, eq 1 reduces to

FCo(Leq) + Fdo(Leq) + Fdw(Leq) ) Fg + FcapU(Leq) (14) where Leq ) leq/2R is the equilibrium distance scaled by the particle diameter. The surface charge density at the particle-oil interface, σpo, can be estimated by using eq 14 in combination with eqs 3, 7, 9, 11, and 13 if the equilibrium distance, Leq, the surface charge density at the particle-water interface, σpw, the surface density of polar groups at the particle-oil interface, σ, their dipole moment, p, and the amplitude of the undulations of the three phase contact line, δ, are known. This is done below. 5.2. Estimation of the Surface Charge Density at the Particle-Octane Interface. We have calculated the equilibrium distance between the model particles, Leq, by means of eq 14 as a function of the surface charge density at the particle-octane interface, σpo, assuming reasonable values for the number density of surface polar groups (σ ) 4.6 polar groups/nm2)21 and their dipole moment p ) 5 × 10-30 C m. The latter is greater than the dipole moment of methoxysilane (3.9 × 10-30 C m)33 and close to that of the hydroxyl radical (5.5 × 10-30 C m).33 The results for very hydrophobic particles (θ ) 152°) and least hydrophobic particles (θ ) 65°) are plotted in Figure 4b assuming for the amplitude of undulations of the threephase contact line δ ) 0 (that is no capillary attraction, filled lines) or δ ) 10 nm (squares and circles). The results for particles with contact angles of 85° and 99° lie between curves 1 and 2 in Figure 4b and are omitted for clarity. As could be expected, the smaller surface charge density (therefore smaller repulsion) leads to smaller interparticle distance at equilibrium. In the case of the mosthydrophobic particles (θ ) 152°) at an equilibrium distance equal to 5 particle diameters (dotted line), which corresponds to the average interparticle distance in the vertical

Structure and Stability of Silica Particle Monolayers

monolayer shown in Figure 3b-d, the estimated value of the surface charge density at the particle-octane interface, σpo, is 14.1 µC/m2 (see curve 1 in Figure 4b). This value is not affected by the capillary attraction due to threephase contact line undulations with a small reasonable amplitude (Figure 4b, squares). The obtained value of σpo is close to that assumed in our previous work21 and within the range of the surface charge densities of silica particles in nonaqueous media (from 1 to 80 µC/m2, see ref 21) obtained by others.30,42,43 In our previous study, we assumed that the surface charge density at the particle-octane interface, σpo, does not depend on the particle hydrophobicity.21 Present results for the stability of vertical monolayers (Figure 3) in combination with the results in Figure 4b suggest that the surface charge density, σpo, of particles with contact angles between 65° and 99° is much smaller than that of the most-hydrophobic particles with a contact angle of 152°. For instance, if two particles with a contact angle of 65° have σpo) 14.1 µC/m2, they should stay at an equilibrium distance about 3.5 particle diameters. Instead, the particles in the vertical monolayer in Figure 3a are in close contact (Leq ) 1). In this case, eq 1 (and eq 14) turns into the inequality ∑FR e Fg + ∑FA and the value of σpo cannot be determined. It can be roughly estimated to be smaller than a few µC/m2 (perhaps less than 5). It is also seen that even at σpo ) 0 in the absence of particle attraction (δ ) 0, Figure 4b, curve 2) hydrophilic particles should stay well separated at a distance of ∼2 particle diameters due to dipolar repulsion through the oil (Fdo and Fdw). However, this is not the case, therefore in addition to the gravity force some long-range attraction is needed to bring the particles in contact as in Figure 3a. The capillary attraction due to three-phase contact line undulations with an amplitude of 10 nm should be sufficient (Figure 4b, circles). The situation with morehydrophobic particles with contact angles of 85° and 99° was similar. They both gave a sediment of close-packed particles; hence, the charge density at their surface in contact with octane hardly exceeds several µC/m2. Our present results throw more light on the reasons for the disorder-order transition in horizontal silica particle monolayers observed previously.21 It turns out that the change of the monolayer structure from disordered, aggregated particles at contact angles smaller than 115° to well-separated, ordered particles at contact angles larger than 129° is due to a drastic increase of the charge density at the particle-octane interface at large contact angles, i.e., at very high hydrophobicity. The charging mechanism of the particle-oil interface and the reasons for such a significant increase of the surface charge density still remain unclear. Several mechanisms of surface charging in nonpolar liquids have been proposed.42,44 These are (i) acid-base reactions involving a proton transfer, (ii) adsorption of charged ions, and (iii) donor-acceptor interactions involving an electron transfer.44 The second mechanism is very unlikely in our case because the polar impurities were removed from the octane (see ref 24). Bearing in mind the large density of surface silanol groups, the first mechanism seems plausible. We should point out, however, that in our experiments the surface charge density at the particle-octane interface increases with a decrease of the number of silanol groups at the surface of (42) Labib, M. E.; Williams, R. J. Colloid Interface Sci. 1987, 115, 330. (43) Philipse, A. P.; Vrij, A. J. Colloid Interface Sci. 1989, 128, 121. (44) Lyklema, J. Adv. Colloid Interface Sci. 1968, 2, 65. (45) Persello, J. In Adsorption on Silica Surfaces; Papirer, E., Ed.; Marcel Dekker: New York, 2000; Chapter 10.

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Figure 5. Total pair interaction energy, Utot, of silica particles with a radius R ) 1.5 µm and contact angles 65° (1, 2) and 152° (3) at a horizontal octane-water interface versus distance between particle centers. The surface charge density at the particle-water interface, σpw (in µC/m2), is 320 (1), 16 000 (2, see ref 45), and 0 (3) without (1, 3) and with 10 mM NaCl (2) in the aqueous phase. The surface charge density at the particle-octane interface, σpo (in µC/m2), is 1 (1, 2) and 14.1 (3). The amplitude of undulations of the three-phase contact line, δ, is 15 nm. The density of polar groups at the particle-oil interface is 4.6 nm-2, and the dipole moment of one polar group is 5 × 10-30 C m.

very hydrophobic particles. This is just the opposite of what we should expect if the first charging mechanism is operative. Charging by an electron transfer between the solid and octane is also possible and cannot be excluded. In any case, water (which is always present in our experiments) might play a very significant role in charging of the particle surface in contact with octane.42,44 More experiments are needed to elucidate the origin of the surface charges at the particle-oil interface. 5.3. Total Pair Potential Energy between Particles at a Horizontal Octane-Water Interface. Taking into account the above results for the surface charge density at the particle-oil interface, we have calculated the total pair potential, Utot ) UCw + UCo + Udo + Udw + UcapU, as a function of the interparticle distance, L, using eqs 4, 6, 8, and 12. The results are shown in Figure 5. In the case of hydrophilic particles (θ ) 65°) in the absence of electrolyte, a deep minimum in the pair potential curve is obtained (Figure 5, curve 1). Assuming small reasonable values for the surface charge density at the particleoctane interface (σpo) 1 µC/m2) and δ ) 15 nm for the amplitude of undulations of the three-phase contact line, the position of the potential energy minimum appears at a distance of 4.5 µm between particle centers (L ) 1.5), which corresponds exactly to the experimentally measured distance in the particle doublets (Figure 1a). The minimum disappears at 10 mM NaCl in the aqueous phase (curve 2), and Utot is negative (corresponding to attraction) down to very small interparticle distances where the short-range van der Waals attraction (not included in our analysis) acts. This is in agreement with the close-packed particle aggregates shown in Figure 1c. In the case of very hydrophobic particles (θ ) 152°, curve 3), Utot calculated with the estimated value of σpo (14.1 µC/m2) at the point of zero charge in water (σpw ) 0) is always positive. Utot in this case is ∼3kT even at interparticle distances of ∼20 particle diameters. Hence, the interactions between very hydrophobic particles are totally dominated by the longrange Coulomb repulsion through the octane, which is consistent with the well-ordered structure and great stability of their monolayers (Figures 1b and 2). It is seen that our experimental results for aggregated hydrophilic particle monolayers can be explained by a capillary attraction due to three-phase contact line undulations with a reasonable amplitude. This does not mean that the capillary attraction due to electrically generated deformations of the liquid interface proposed recently29,30 does not exist. However, experimental iden-

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tification of its contribution to the overall capillary attraction seems to be very difficult. This is because there is always some hysteresis of the particle contact angle, but the undulations of the three-phase contact line associated with it should have a very small amplitude (smaller than several tens of nanometers) which can hardly be experimentally measured. 6. Conclusions The structure and stability of silica particle monolayers at the water-octane interface strongly depend on the particle hydrophobicity. Very hydrophobic particles give well-ordered and stable (both toward aggregation and sedimentation) horizontal and vertical monolayers. This is due to the presence of charges at the particle-octane interface, which results in strong long-range Coulomb repulsion between particles through the octane. The estimated value of the surface charge density at the very hydrophobic particle-octane boundary (14.1 µC/m2) is reasonable. Less-hydrophobic silica particles with contact angles smaller than 99° have significantly smaller surface charge density at the particle-octane interface (smaller than several µC/m2). Hence, the Coulomb repulsion through the octane is smaller and insufficient to prevent the particles from aggregating or sedimenting. This finding correlates very well with the order-disorder transition of the monolayer structure occurring within a narrow range of particle contact angles between 129° and 115° reported in our previous work.21 The experimentally observed two-dimensional particle sedimentation giving large domains of well-ordered closepacked particles (Figure 3a) shows yet another possibility for fabrication of ordered two-dimensional particle arrays.

Horozov et al.

The realistic value of the surface charge density at the particle-octane boundary obtained here with a simplistic model for the sedimentation equilibrium in a vertical particle monolayer suggests that this approach can be used for studying the interactions between particles adsorbed at fluid interfaces. Experimental data for sedimentation equilibrium of vertical particle monolayers coupled with an appropriate (many body) theoretical model could give valuable information on the interactions and forces acting between small solid particles at fluid interfaces in the force domain below 0.1 pN where other experimental methods are inapplicable. The long-range attraction between hydrophilic particles observed by us can be explained by the capillary attraction due to three-phase contact line undulations with a small reasonable amplitude. The attraction observed could be at least partially due to the capillary attraction caused by electrically generated deformations of the fluid interface suggested recently.29,30 To determine its contribution to the overall capillary attraction seems to be very difficult by experiments because the actual amplitude of the threephase contact line undulations is unknown. Acknowledgment. T.S.H. and B.P.B. acknowledge financial support from Wacker-Chemie GmbH, Burghausen, Germany for part of this work. T.S.H., R.A., and J.H.C. gratefully acknowledge the provision by the EPSRC of a ROPA Grant (GR/N02778). The authors thank Professor P. A. Kralchevsky for helpful discussions. LA050923D