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Langmuir 2006, 22, 6746-6749
Self-Assembly in Two-Dimensions of Colloidal Particles at Liquid Mixtures Juan Carlos Ferna´ndez-Toledano, Arturo Moncho-Jorda´, Francisco Martı´nez-Lo´pez, and Roque Hidalgo-A Ä lvarez* Biocolloid and Fluid Physics Group, Department of Applied Physics, Faculty of Sciences, UniVersity of Granada, Granada E-18071, Spain ReceiVed April 20, 2006. In Final Form: June 20, 2006 The aim of this work is to simulate the formation of colloidal rings, circular clusters, and voids induced by oily lenses at the air-water interface. The presence of two liquids with different surface tension leads to the formation of a nonhomogeneous interface. In this case, the total interaction potential is assumed to be composed of only two terms; the first one is due to the (repulsive) pairwise dipolar force between partly immersed charged microspheres, whereas the second depends on the position of the particle at the interface and is connected to the interfacial stress caused by the difference of surface tension between both liquids. This simple potential is able to reproduce the experimental rings, circular clusters and voids found by different authors.
The study of the self-assembly of colloidal particles confined at the air-water or oil-water interfaces has received increasing attention in the past few years, due to the rich diversity of structures observed in such systems going from two-dimensional crystals to fractal clusters. Just recently, the spontaneous formation of rings, circular clusters and voids, called mesostructures hereafter, has been reported by several authors.1-13 In fact, the mesostructures are consistent with very different patterns, such as the coexistence of dense and dilute regions, where the typical interparticle distance is of the order of the particle radius. A secondary minimum at such distances has been suggested as a possible explanation. During the past few years, a great deal of effort has been focused on developing a pair interparticle potential able to reproduce these kinds of structures. This new attractive contribution should be of range longer than the van der Waals forces as well as stronger than conventional capillary interactions. Several theories are devoted to show the existence of this minimum. The first theoretical studies are based on extended DLVO (Derjaguin-Landau-Verwey-Overbeek) approaches to colloidal forces,14,15 including capillary, hydrophobic, and dipolar interactions.16 However, this model was not able to explain the formation of mesostructures. More sophisticated theories consider * To whom correspondence should be addressed. E-mail: rhidalgo@ ugr.es. (1) Ruiz-Garcia, J.; Ga´mez-Corrales, R.; Ivlev, B. I. Physica A 1997, 236, 97. (2) Ghezzi, F.; Earnshaw, J. C. J. Phys.: Condens. Matter 1997, 9, L517. (3) Helseth, L. E.; Muruganathan, R. M.; Zhang Y.; Fischer T. M. Langmuir 2005, 21, 7275. (4) Ruiz-Garcia, J.; Ga´mez-Corrales, R.; Ivlev, B. I. Phys. ReV. E 1998, 58 660. (5) Sear, R. P.; Chung S. W.; Markovich, G.; Gelbart, W. M.; Heath, J. R. Phys. ReV. E 1999, 59, R6255. (6) Ghezzi, F.; Earnshaw, J. C.; Finnis M.; McCluney, M. J. Colloid Interface Sci. 2001, 238, 433. (7) Tarimala, S.; Ranabothu, S. R.; Vernetti, J. P.; Dai L. L. Langmuir 2004, 20, 5171. (8) Denkov, N. D. Langmuir 2004, 20, 9463. (9) Chen, W.; Tan, S.; Ng, T.-K.; Ford, W.; Tong P. Phys. ReV. Lett. 2005, 95, 218301. (10) Stancik, E. J.; Kohkan, M.; Fuller, G. G. Langmuir 2004, 20, 90 (11) Stancik, E. J.; Fuller, G. G. Langmuir 2004, 20, 4805. (12) Ferna´ndez-Toledano, J. C.; Moncho-Jorda´, A.; Martı´nez-Lo´pez, F.; Hida´lgo-A Ä lvarez, R. Langmuir 2004, 20, 6977. (13) Wu, Ch-y; Tarimala, S.; Dai, L. L. Langmuir 2006, 22, 2112. (14) Derjaguin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633. (15) Verwey, E. J.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (16) Martı´nez-Lo´pez, F.; Cabrerizo-Vı´lchez, M. A.; Hidalgo-AÄ lvarez, R. J. Colloid Interface Sci. 2000, 232, 303.
lateral capillary interactions as a possible reason for this extra attraction. These capillary forces come from the meniscus deformation around a partly immersed colloid. Different models have been proposed depending on the origin of the interface deformation.17-20 Moreover, in a recent publication, Chen et al.9 proposed an interesting model based on the presence of an inhomogeneous distribution of the charged groups on the particle surface. When the colloidal particles are spread at the air/water interface, this heterogeneous charge distribution provokes an effective electric dipole moment, which can be separated into a perpendicular and a parallel component with respect to the interface plane. The perpendicular component is the typical dipolar interaction characteristic for colloids trapped between a polar and nonpolar medium. However, the parallel component induces a new pairwise anisotropic interaction between the particles. Depending on the relative orientation of the particles confined in the plane, this interaction shows an attractive potential well at a distance of the order of the particle diameter, and as the authors appoint, it can be used to explain the formation of stable bonded clusters, rings, and chains, which are the typical structures observed in the presence of anisotropic dipole-dipole interactions.21 However, the knowledge of the charge distribution in the particle surface is necessary in order to check that model. As concluded from the different theoretical approaches used to account for the formation of mesostructures, the physical origin of this suggested long-range attraction remains nowadays unclear. In fact, the existence of long-range attractive forces between colloidal particles confined at interfaces is one of the greatest controversies of colloid science. All of the theories mentioned above start from the assumption of the existence of a long-range attractive pair potential between colloids confined at the airwater interface. However, the physical reason for the mesostructure formation phenomenon could not rely on any pairwise attractive interaction. Pieranski22 pointed out that the presence of impurities could affect the colloidal monolayer stability. Moreover, the hypothetical presence of some oily films (coming (17) Stamou, D.; Duschl, C. Phys. ReV. E 2000, 62, 5263. (18) Nikolaides, M. G.; Bausch, A. R.; Hsu, M. F.; Dinsmore, A. D.; Brenner, M. P.; Gay, C.; Weitz, D. A. Nature 2002, 420, 299. (19) Megens, M.; Aizenberg, J. Nature 2003, 424, 1014. (20) Oettel, M.; Domı´nguez, A.; Dietrich, S. Phys. ReV. E 2005, 71, 051401. (21) Sinyagin, A. Y.; Belov, A.; Tang, Z.; Kotov, N. A. J. Phys. Chem. B 2006, 110 (14), 7500-7507. (22) Pieranski, P. Phy. ReV. Lett. 1980, 45, 569.
10.1021/la0610755 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/04/2006
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from dissolved styrene oligomers, for instance) as a possible cause for the appearance of mesostructures has been pointed out by Kralchevsky and Denkov.23 The interface is then understood as a two-dimensional emulsion consisting of hydrophobic patches (oil) in a hydrophilic continuous phase (water). Therefore, the confinement of the polymer colloids by finite-size hydrophobic patches would result in an apparent intercolloidal attraction. Theoretical approaches to mesostructure formation based on long-range interparticle attractions are used to assume isotropic interactions and homogeneous interfaces.1,18,20,24-26 However, as can be observed from the pictures of the experimental mesostructures,4,6 even the simplest structures (like rings) cannot be explained using any isotropic force. To show that a heterogeneous interface can induce the appearance of mesostructures, we decided to simulate the behavior of a colloidal monolayer formed by microspheres spread onto a nonhomogeneous air-liquid interface. The only direct interaction between a pair of particles is the dipolar repulsion, Vdip, which is strong enough to avoid particle coagulation. Any other pairwise interaction is assumed to have a negligible effect. Furthermore, there exist two domains with different surface tension values, the air-water (the largest one) and the air-oily phase, which is located on the forming lenses. This implies that the so-called spreading coefficient (S ) γwa - γwo - γoa, where γwa and γoa are the surface tension of the water-air and oil-air interfaces and γwo is the interfacial tension of the water-oil interface) is negative. Consequently, colloidal particles are also affected by an external potential,Vγ. This last interaction comes from the existence of surface tension inhomogeneities at the interface since particle energy depends on whether it is located at the water-air interface, oil-air interface, or water-oil-air triple contact line. For the sake of simplicity, in our model, we assume that flat lenses represent the oily phase. According to this, the total interaction potential on a particle placed at position ri in contact with an oil flat lens of hdrop thickness is given by
Figure 1. (a) Sketch of the interception of a spherical particle with an oil flat lens at the air-nonhomogeneous (oil-water) interface. (b) Representation of the different interfaces in the lens model for the droplet. (c) Picture of a nonhomogeneous air-water interface with droplets of silicone oil (AR200) experimentally obtained. Table 1. Values Assumed for Surface and Interfacial Tensions and Contact Angles of the Different Interfacesa a b c d
γpw
γpo
γpa
γao
γow
θoa
θow
Sspread
hdrop 107
32 32 32 32
14 14 14 31
33 33 33 33
38 38 38 21
38 38 38 53
20 20 20 26
20 20 20 10
-4.62 -4.62 -4.62 -2.97
1.5 3 1.5 1.5
a Surface tensions are expressed in mJ m-2, lengths in meters, and angles in degrees. In all cases, γaw, ddrop, θpw, and θpo were assumed to be 72 mJ m-2, 7.2 × 10-6 m, 89°, and 60°, respectively.
(1)
the oily phase. The areas of the different interfaces depend on this parameter. Here, Spa(r b,hdrop), Spw(r b,hdrop), and Spo(r b,hdrop) represent surface areas corresponding to particle-air, particlewater, and particle-oil interfaces, respectively. Analogously, Sdow(r b,hdrop) and Sdoa(r b,hdrop) are the surface area of the oilwater and oil-air interfaces that would be created if the particles were removed. Thus, the external potential Vγ depends on hdrop and is given by
where N is the total number of particles. According to Horozov et al.,25 the dipolar interaction potential at large distances reduces to
Vγ(b,h r drop) ) Spa(b,h r drop)γpa + Spw(b,h r drop)γpw + r drop)γpo - Sdoa(b,h r drop)γoa - Sdow(b,h r drop)γow Spo(b,h Sdaw(b,h r drop)γaw (3)
N
Vtotal(b r i) ) Vγ(b r i,hdrop) +
Vdip(r) )
Vdip(|b ri - b r j|) ∑ j*i
πfdip2σ2P2R4 sin4 θ 3
40r
(2)
where σ is the charge density on particle surface, fdip is the percentage of charges that form dipoles, P is the dipolar moment, R is the particle radius, θ is the particle contact angle, 0 is the vacuum dielectric permittivity, and is the dielectric constant of air. In calculating Vγ, it is considered that the energy required to create an interface between two media “a” and “b” with a surface tension γab is given by Vb ) Sab‚γab, where Sab is the area of such an interface. The calculation of Vγ involves the study of how the interfaces are modified when a particle moves from position b r1 to b r2. Figure 1, panels a and b, shows the most general case of our problem. Flat lenses of maximum thickness, hdrop, represent (23) Kralchevsky, P. A.; Denkov, N. D. Curr. Opin. Colloid Interface Sci. 2001, 6, 383. (24) Ivlev, B. I. J. Phys.: Condens. Matter 2002, 14, 4829. (25) Foret, L.; Wurger, A. Phys. ReV. Lett. 2004, 92, 058302. (26) Horozov, T. S.; Aveyard, R.; Clint, J. H.; Binks, B. P. Langmuir 2003, 19, 2822-2829.
We consider that the oil lenses are thinner than the particle size (hdrop < 600 nm). According to this model, the interaction between a colloidal particle and an oil droplet has been visualized as the intersection between a sphere and a flat lens (see Figure 1). To calculate the different surface areas it is necessary to integrate over the sphere (to obtain Spw, Spo, and Spa) or over the flat oil lens (to obtain Sdoa and Sdow). It is assumed that the meniscuses of water- and oil-colloidal particle interfaces are flat because of in our experiments the Bond number (B ) gR2∆F/γLV) is small. All symbols have their usual meaning, and ∆F is the density difference between the subphase and the upper phase. Input data values of the external potential Vγ(r b,hdrop) (eq 3) are shown in Table 1. We have assumed that the values ofγwa, γpa, θpw, θpo, θow, and θoa are previously known. The values of γwa and γpa are the typical of the water-air and polystyrene-air interfaces. The contact angles θpw and θpo are taken as fixed values. The others γ values (γoa, γwo, γpo, and γpw) are obtained using the Neumann and Young equations. The spreading coefficient (S) so obtained is negative indicating that the oily phase forms droplets at the air-water interface (see Figure 1c). The values used for interfacial tensions and contact angles of the
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Figure 3. Formation of mesostructures simulated at an air-liquid nonhomogeneous interface. (a and b) Colloidal rings, (c) circular clusters, and (d) voids correspond with the data shown in lines a-d of Table 1.
Figure 2. Vγ(r) versus the reduced distance from the oil-water-air contact line divided by the particle radius. (a and b) Data shown from the lines a and b of Table 1, respectively. (c) Data shown from the line d of Table 1.
oily phase are not related to the values of these properties for a particular oil, but that they are assumed ad hoc. Figure 2 shows the external potential, Vγ(r b,hdrop), as a function of the distance from the particle to the air-water-oil contact line divided by the particle radius. The value of hdrop affects to the external potential as can be seen comparing Figure 2, panels a and b. Also, Figure 2c shows the external potential for values of γao and γow, so as θoa and θow slightly different to those used in the cases in panels a and b. These changes in the values of the interfacial tensions and contact angles of the air-oil and oil-water interfaces dramatically influence the external potential profile, causing an inversion in the potential when the particle is entering into the oily phase. As can be seen in Figure 2b, the energy of a particle inside the oil lens is smaller than the energy when the particle is outside, which means that, in principle, the colloids will tend to emigrate to the oil drops. However, due to the different surface tension values between the air-water and air-oil interfaces, a minimum is found at (r - Rdrop)/Rpart ) -1.5, which prevents particle migration. Removing the particle from the three phase contact line involves a very high-energy cost caused by the creation of a new oil-water interface (Swo). This might explain the appearance of colloidal rings at liquid mixtures.
The simulations were conducted off-lattice using Monte Carlo in a two-dimensional simulation cell of dimensions L × L with periodic boundary conditions. In each simulation, N spheres are randomly placed inside the cell. At any step, the surface areas Spw, Spo, Sdaw, Sdao, Sdwo, and Swo are calculated, for each particle, by numerical integration. Then, the total energy is obtained using eqs 1-3. After that, the particles are displaced (by Brownian movement) to a new position and the new total energy is calculated. Movement is accepted or rejected depending on the differences between the new and the previously calculated energy. The simulation is stopped when a stable situation is reached. Input data for the simulations are as follows: time step (∆t ) 0.0004 s), number of particles (N ) 300), area fraction φ ) 0.10, number of droplets and positions, surface charge density on colloidal spheres σ ) 6 × 1018 e-/m2, fraction of dipoles fdip ) 1%, particle radius R ) 3 × 10-7 m, dipolar moment P ) 4.8 × 10-29 C m assuming a dipole length of 0.3 nm, 0 and ) 1, and the data shown in Table 1. Figure 3, panels a and b, shows the colloidal rings obtained from simulations using the interaction potential given by eq 1. The formation of oil lenses at the air-water interface leads to the existence of a nonhomogeneous interface with hydrophobic (oily lenses) and hydrophilic patches (water surface). When polystyrene microspheres are spread onto this nonhomogeneous interface most of them tend to reach the three phase contact line where they are eventually trapped and form the colloidal rings. Particles at the air-water interface cannot migrate to the oil-air interface (and vice versa), because they have not enough energy to cross the three phase contact line where the energy of the particle has a minimum (Figure 2, panels a and b). This further implies that the average interparticle distance depends on particle accommodation inside hydrophilic and hydrophobic patches at the air-liquid interface. This model explains the appearance of circular rings (see Figure 3, panels a and b, which are common structures observed in colloidal monolayer experiments. It should be noted that a change in hdrop modifies the external potential values (see Figure 2, panels a and b) but not the pattern formation in colloidal rings. The number of colloidal particles that takes
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Figure 4. Pictures a-d display different morphologies of mesostructures at equilibrium obtained with polystyrene particles at the air-water interface contaminated by silicone oil (AR200). (a) Colloidal rings, (b) void, (c) circular clusters, and (d) colloidal rings. More details in ref 12.
part of the ring formation depends on hdrop, and so if hdrop increases, then the width of the potential minimum does and the formation of double or triple rings would also be possible. Other shapes of the mesostructures (circular clusters) could be explained if we consider an initial inhomogeneity in the colloidal distribution due to the (turbulent) spreading of the particles at the interface (Figure 3c shows the structure formed with an initial Gaussian distribution of particles centered on the oil lens). The formation of voids (see Figure 3d) is reached changing the values (see
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Table 1) of γao from 38 to 21 mJ m-2, γow from 38 to 53 mJ m-2, γpo from 14 to 31 mJ m-2, and θow from 20° to 10°. As can be seen in Figure 2c, these changes in the surface and interfacial tensions cause an important modification in the profile of the external potential, Vγ To check the reliability of the proposed simple model, a qualitative comparison with experimental spontaneously formed mesostructures can be made. Figure 4 shows experimental results 30 min after colloidal deposition at an air-water and silicone oil interface. The liquid subphase was ultrapure Milli-Q water, and the colloidal dispersion was composed of polystyrene particles (600 nm diameter) in methanol used as spreading agent and with traces of silicone oil (AR200 from Fluka). The surface particle fraction of the colloidal monolayer ranged from 6% to 8%, and the surface concentration of silicone oil was 3.81 × 10-3 µg/ cm2. It should be noted that, in our experimental findings, these mesostructures appear spontaneously at the interface without introducing external energy in the system, in contrast to the experiments reported by Chen et al.,9 where they supplied kinetic energy by periodically dilating the interface by pipetting a small amount of water in order to overcome the repulsions and to obtain colloidal rings and circular patterns. The simulated colloidal rings, circular clusters, and voids (see Figure 3) are quite similar to those experimentally obtained (see Figure 4). In summary, it seems that the proposed simple interaction model accounting for the dipolar repulsion between particles and heterogeneity in surface tension is able to capture the essential features of mesostructures formation. Acknowledgment. The authors acknowledge the financial support from Spanish “Ministerio de Educacio´n y Ciencia, Plan Nacional de Investigacio´n (I+D+i), MAT 2003-08356-C0401”, by the “European Regional Development Fund” (ERDF) and by the Project No. FQM 392 from Junta de Andalucı´a. Also the authors thank the comments and criticisms made by an anonymous reviewer. LA0610755
