Compression and Structure of Monolayers of Charged Latex Particles

Langmuir 2000, 16, 1969-1979

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Compression and Structure of Monolayers of Charged Latex Particles at Air/Water and Octane/Water Interfaces Robert Aveyard,*,† John H. Clint,† Dieter Nees,‡ and Vesselin N. Paunov§ Department of Chemistry, University of Hull, HU6 7RX, UK, Central Laser Facility, Rutherford Appleton Laboratory, Didcot, OX11 0QX, UK, and Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received July 7, 1999. In Final Form: October 14, 1999 We have studied the compression and structure of compressed monolayers of sulfate polystyrene latex particles on air/water and octane/water interfaces. If compressed sufficiently (on a Langmuir trough) the monolayers at air/water surfaces give rafts of hexagonally packed particles, while those at oil/water interfaces undergo a transition from the originally hexagonal to a rhombohedral structure. We have found that beyond collapse the particle monolayers on both air/water and octane/water interfaces fold and corrugate, and there is no expulsion of individual particles or particle aggregates from the interface. In the case of air/water interfaces, the structuring of particle monolayers (below collapse) was found to be very sensitive to the electrolyte concentration in the aqueous phase. At low electrolyte concentration, a fairly ordered structure resulting from the interparticle repulsion was observed, while at high electrolyte concentration, the particles form 2D clusters. In marked contrast, particle monolayers at octane/water interfaces remain highly ordered as a result of long-range repulsion, even on concentrated electrolyte solution. We attribute the enhanced lateral repulsion between the latex particles at the octane/water interface to the existence of residual surface charges at the particle/octane interface. We propose a simple model, which describes the electrostatic interaction between the adsorbed particles and includes the effect of image forces. From this we have derived an analytic formula for the electrostatic surface pressure vs trough area, which agrees well with the experimental data over a wide range of surface pressure.

1. Introduction There has been a wide interest in the behavior of particle monolayers at liquid surfaces over several decades. Our own relatively recent work stems from the ability of particles in interfaces to modify the stability of foams and emulsions.1-3 Over 3 decades ago Schuller,4 and Sheppard and Tcheurekdjian,5 determined surface pressure (Π)surface area (A) isotherms for monolayers of polystyrene (PS) latex particles at the air/water interface. Measurements of Π-A curves have also been used to estimate the strength of steric barriers between sterically stabilized spherical polymer particles at the oil/water interface.6 More recently studies of the aggregation of particles at the air/water surface, as well as aggregate morphology and fractal dimensions, have been reported.7-12 The forces responsible for the interactions between particles in 3-dimensional dispersions are also operative * Corresponding author. † University of Hull. ‡ Rutherford Appleton Laboratory. § University of Delaware. (1) Aveyard, R.; Clint, J. H. J. Chem. Soc., Faraday Trans 1995, 91, 2681. (2) Aveyard, R.; Binks, B. P.; Fletcher, P. D. I.; Rutherford, C. E. J. Dispersion Sci. Technol. 1994, 15, 251. (3) Aveyard, R.; Cooper, P.; Fletcher, P. D. I.; Rutherford, C. E. Langmuir 1993, 9, 604. (4) Schuller, H. Kolloid-Z. 1967, 216-217, 380. (5) Sheppard, E.; Tcheurekdjian, N. J. Colloid Interface Sci. 1968, 28, 481. (6) Doroszkowski, A.; Lambourne, R. J. Polymer. Sci. C 1971, No. 34, 253. (7) Horvolgyi, Z.; Mate, M.; Zrinyi, M.. Colloids Surf. A 1994, 84, 207. (8) Stankiewicz, J.; Vilchez, M. A. C.; Alvarez, R. H. Phys. Rev. E 1993, 47, 2663. (9) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. A 1992, 46, 2045. (10) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. A 1992, 46, 2055. (11) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. A 1992, 46, 2065. (12) Robinson, D. J.; Earnshaw, J. C. Langmuir 1993, 9, 1436.

in particulate monolayers. However, Pieranski13 noted some time ago that for charged particles at an interface between an aqueous phase and air, or other medium of low dielectric constant, electrostatic repulsion is enhanced over that operating within the bulk aqueous phase, particularly at larger particle separations. At the interface a particle (which is partially immersed in water) has an asymmetric counterion distribution which results in a dipole moment normal to the water interface. Repulsive interactions between the effective dipoles due to neighboring particles occur through the phase of low dielectric constant; repulsion through the aqueous phase is screened by free ions in solution, which leads to an exponentially decaying force law. Interaction between charged particles near a water/air interface have been treated theoretically by Hurd14 and more recently by Goulding and Hansen.15 It has been shown that the corresponding interaction is repulsive and long-ranged. The interaction energy decays inversely with the cube of the separation between the dipoles, and is inversely proportional to the electrolyte concentration. Williams and Berg16 have investigated the adsorption and subsequent aggregation into small clusters of colloidal particles at the air/aqueous electrolyte interface. Interestingly, although in principle dipole repulsion can enhance the stability of particles at interfaces toward aggregation, it was found that surface aggregation commenced at salt concentrations up to 2 orders of magnitude less than those effective in bulk. Even though the 1 µm diameter PS particles used had a ζ potential of -80 mV, it was concluded on the basis of the treatment of Levine et al.17 that the dipole repulsion was negligible; strong dipole repulsion (13) Pieranski, P. Phys. Rev. Lett. 1980, 45, 569. (14) Hurd, A. J. J. Phys. A 1985, 18, L1055. (15) Goulding, D.; Hansen, J.-P. Mol. Phys. 1998, 95, 649. (16) Williams, D. F.; Berg, J. C. J. Colloid Interface Sci. 1992, 152, 218.

10.1021/la990887g CCC: $19.00 © 2000 American Chemical Society Published on Web 12/28/1999

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would have inhibited the observed clustering at the surface. The van der Waals attraction between the PS spheres at the A/W interface occurs in part through air and in part through the aqueous phase; attraction through air is greater than through a condensed phase. Williams and Berg estimated approximately the interparticle attraction from appropriate Hamaker constants for the interaction of polystyrene through air and through water together with a knowledge of the depth of particle immersion, which is simply related to the contact angle of water with polystyrene. Below, we report a study of the forces between PS particles, very similar to the particles used by Williams and Berg, but spread at the octane/ water interface. In connection with Π-A curves for particle monolayers, we note that Kumaki18,19 has discussed the possible significance of measured surface pressures. He has found that Π values measured using a Wilhelmy plate depend on the orientation of the plate with respect to the Langmuir trough barriers. Further, the pressures measured by the plate method were different from those obtained by the direct measurement of the pressure exerted on the trough barrier. We have found that, using the Wilhelmy plate to measure surface pressures, the effect of plate orientation on Π is related to the particle size. Kumaki investigated monomolecular PS particles with diameters around 50 nm. Here we report on a study of monolayers formed from monodisperse spherical PS particles at the interface between octane and aqueous NaCl solutions. Particle diameters range from 0.2 to 2.6 µm, and surface charge densities (in water) from 1.3 to 8.6 µC cm-2; in terms of wettability the particles are described by the supplier as hydrophobic, a point we return to later. We have made direct in situ microscopic observation of the monolayer structure during compression on a miniature Langmuir trough. As will be seen, particles in monolayers at the oil/water interface are highly repulsive in contrast to behavior at the air/water interface, and our main concern here is with the possible origin of the interparticle repulsion. We discuss elsewhere the nature of the monolayer collapse process at high surface pressure.20 2. Materials and Methods 2.1. Materials. Octane (99% pure from Aldrich, England) was passed through chromatographic alumina 3 times to remove polar materials. Water was from an Elga UHQ II unit fitted with UV irradiation and with cartridges for ion exchange, reverse osmosis, organic adsorption, and ultramicrofiltration. At 293 K water from the Elga unit had a surface tension of 72.8 ( 0.1 mN m-1 and the octane/water interfacial tension was 51.7 ( 0.1 mN m-1. Propan-2-ol (IPA), used as a spreading solvent for the particles, was AnalaR grade, and was used as received. The polystyrene latex particles studied had sulfate groups at the surface, and were supplied by Interfacial Dynamics (USA); the relevant properties, given by the maker, are summarized in Table 1. 2.2. The Langmuir Film Balance (Trough) and Microscope. The Langmuir trough was a miniature analytical film balance supplied by Nima Technology (England), Model 601M. It was constructed from Teflon, and it was found that octane displaced the water from contact with the inside of the trough. This wetting problem was overcome by fitting a stainless steel lining inside the trough. In this way the oil/water interface was pinned where the sharp upper edge of the lining meets the Teflon edge. The trough was computer controlled and the Π-A curve (17) Levine, S.; Bowen, B. D.; Partridge, S. J. Colloids Surf. 1989, 38, 325. (18) Kumaki, J. Macromolecules 1986, 19, 2258. (19) Kumaki, J. Macromolecules 1988, 19, 749. (20) Aveyard, R.; Clint, J. H.; Nees, D.; Quirke, N. In preparation.

Aveyard et al. Table 1. Properties of Polystyrene Particles, Supplied as Between 2 and 8 wt % Aqueous Surfactant-Free Dispersions sample

mean diameter/ µma

surface charge density/µC cm-2

area per SO4-/nm2

A B C D E

0.21 ( 0.01 0.40 ( 0.02 0.63 ( 0.01 1.50 ( 0.06 2.60 ( 0.12

1.3 3.4 4.7 8.1 7.7

12.74 4.74 3.44 1.97 2.08

a

( standard deviation of diameter.

could be seen on the monitor during monolayer compression, and subsequently printed out. Surface pressure was measured using filter paper Wilhelmy plates. Further details of construction and operation of the trough are given elsewhere.20 The trough was housed on the stage of a microscope (Nikon Optiphot 2) with an attached color CCD camera (JVC TK 1381 Digital). Monolayers could be observed using 10×, 20×, or 40× reflectance objectives. Monolayer images were captured on a computer and subsequent measurements and processing were made using commercial software (Lucia, from Nikon UK). 2.3. Determination of Contact Angles of Particles Attached to a Liquid Interface. Knowledge of the contact angles, θ, of particles with the oil/water interface is necessary in the theoretical treatment of interparticle forces. The particles are described as being hydrophobic by the makers. Nonetheless the surfaces carry a (negative) charge in water (Table 1) and the particles are supplied as surfactant-free aqueous dispersions. We have used a simple method for the determination of θ, similar to that described by Horvolgyi et al.21 Two vertical glass slides were spaced by particles (slightly larger than those being studied) resting between them. In the case of oil/water interfaces, the oil phase was first added between the plates from the top using a micropipet and then the aqueous phase containing dispersed particles was introduced from below. Some of the particles become attached to the interface. The interfacial particles are then observed from the side by microscope, using transmitted light (Figure 1a). Contact angles are measured directly from the images obtained (see Figure 1b). Contact angles at the oil/water interface are quite high, of the order of 70° to 80°. At the air/water interface however the angles are very low, most of the particle surface area being immersed in the aqueous phase. Indeed, from the images obtained (Figure 1b) it is not entirely clear that the particles are in fact attached to the interface. It is possible however to cause the air/water interface to move by further injection of water. When the interface moves toward the air phase, the particles at the interface move with it, indicating that they are adsorbed and that therefore the contact angle is low, say e30°.

3. Results and Discussion 3.1. Pressure-Area (Π-A) Isotherms. An example of a pressure-area isotherm for particles at an octane/ water interface is shown in Figure 2. In general the observed isotherms for particle monolayers can be broken into three distinct regions, A, B, and C. At high areas the surface pressure rises relatively slowly (region A). As the area is reduced on compression, the surface pressure begins to rise more steeply (region B). Ultimately the rate of pressure increase begins to fall. The locus of the rapid change in slope (C) corresponds to the collapse pressure of the monolayer, Πc. The collapse process is discussed in detail elsewhere.20 The region of rapid rise in pressure prior to collapse results from the repulsion between particles on compression. When the rise starts at high areas and is relatively shallow; this is an indication that the repulsion between particles is very long-range. We note that in the Π-A curves presented the surface pressure due to the spreading solvent (IPA) is subtracted from the ordinate. Usually Π due to IPA was about 2 mN m-1.20 (21) Horvolgyi, Z.; Nemeth, S.; Fendler, J. H. Colloids Surf. A 1993, 71, 207.

Charged Latex Particles at Air/Water and Octane/Water Interfaces

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Figure 1. (a) Cell, formed from spaced glass slides, for the measurement of contact angles of polystyrene spheres with liquid interfaces. The microscope is focused on the center plane of the cell i.e., on the bottom of the meniscus where the particles rest in the interface. (b) Images of 2.6 µm diameter particles at interfaces. The upper image is for particles at the air/0.1 mol dm-3 aqueous NaCl surface and the lower image is for the octane/water interface.

Figure 2. Surface pressure-surface area isotherm for 2.6 µm diameter PS particles at the octane/water interface. The surface pressure (2 mN m-1) due to the spreading solvent has been subtracted from the ordinate. The various letters are referred to in the text.

Like Kumaki18,19 we find that for higher surface pressures, Π-A curves recorded for the smaller particles are different for orientations of the Wilhelmy plate normal and parallel to the trough barriers.20 For particles with diameter 2.6 µm however, the isotherms are indistinguishable for the 2 orientations and are very reproducible. Furthermore, for these particles there is an exact correspondence between monolayer collapse pressures and the tension of the interface in the absence of the particle monolayer over a very wide range of tensions. This leads

us to believe that the surface pressure we measure in the case of the 2.6 µm diameter particles is the reduction in interfacial tension caused by the presence of the particles. 3.2. Monolayers at the Air/Water Interface. Most of our work has been concerned with particle monolayers at the oil/water interface. Before discussing this however we briefly describe some behavior of particle monolayers at the air/water surface. It was not possible to spread the sulfate latex particles efficiently on clean water surfaces in contact with air; it was found necessary to have a low concentration of electrolyte in the aqueous subphase. For NaCl the minimum concentration necessary was about 10 mmol dm-3. In the presence of such low concentrations, dilute particle monolayers exhibit long-range repulsion between particles, which gives rise to a fairly ordered structure, as illustrated in Figure 3a for 2.6 µm diameter particles. The average separation between centers of particles is on the order of 10 µm. Addition of more NaCl screens the interparticle repulsion, and for a subphase of 100 mmol dm-3 NaCl small particle clusters are formed (Figure 3b); on 1 mol dm-3 NaCl large interconnecting networks are formed (Figure 3c). Compression of such monolayers, for all salt concentrations up to 1 mol dm-3, leads to the formation of hexagonally ordered rafts of particles, as illustrated in Figure 4a for particles on 100 mmol dm-3 aqueous NaCl. Here the surface pressure is 12 mN m-1. The gaps between the rafts progressively disappear on further compression up to the collapse pressure. Reexpansion of the collapsed

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Figure 4. Monolayers of 2.6 µm diameter particles at the air/ 100 mmol dm-3 aqueous NaCl interface. (a) Compressed to 12 mN m-1. (b) Reexpanded after collapse. Scale bars represent 50 µm.

Figure 3. Monolayers of 2.6 µm diameter particles at the air/ aqueous NaCl interface. Salt concentrations for (a), (b), and (c) are respectively 10, 100, and 1000 mmol dm-3. Scale bars represent 50 µm.

monolayer results in the formation of areas of close-packed particles with spaces between the rafts as seen in Figure 4b. This kind of behavior is similar to that observed for smaller particles (which have lower surface charge densities) in monolayers at the oil/water interface. 3.3. Monolayers at the Octane/Water Interface. Effects of Particle Size on Behavior during Compression and Expansion. A range of particle sizes has been employed and the surface charge density (of ionized sulfate groups in water) increases with particle size, as seen in Table 1. The smallest particles, 0.21 µm diameter with surface charge density σ ) 1.3 µC cm-2, are not visible individually under the microscope. On compression no changes are seen until the interface starts to fold. On reexpansion the particles adhere and the monolayer apparently consists of rafts of particles with gaps between (Figure 5). Rafts and gaps can be distinguished since the areas of the gaps

Figure 5. Monolayer of 0.21 µm diameter particles at the octane/water interface, after compression and reexpansion. Gaps (darker areas) are plainly seen between islands of adhering particles. Scale bar represents 100 µm.

shrink on recompression. The behavior of the 0.4 µm diameter particles (σ ) 3.4 µC cm-2) is very similar. Particles with diameter 0.63 µm exhibit intermediate behavior between that observed for the smaller particles and that, to be discussed, for the larger particles. On expanding a particle film after compression, only a fraction of the particles remain adhering; some particles redisperse as single entities as is clearly seen in Figure 6. The single particles are mutually repulsive (σ ) 4.7 µC cm-2) and are also obviously repelled from the edges of the adhering rafts. For the larger particles studied (1.5 and 2.6 µm diameters, σ ) 8.1 and 7.7 µC cm-2, respectively) the


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Figure 6. Appearance of a reexpanded monolayer of 0.63 µm diameter particles at the octane/water interface. Single particles are seen in the gaps between adhering particle rafts, and are clearly repelled from the edges of the rafts.

behavior is similar for the two sizes, and we illustrate our observations with reference to results for 2.6 µm particles. The pressure-area isotherm for the particles at the oil/ water interface is depicted in Figure 2 and a sequence of images of the monolayer during compression is reproduced in Figure 7. The image in Figure 7a is of a monolayer with a surface pressure close to zero. The particles are packed more or less hexagonally and separation between centers is ca. 14 µm. On compression or addition of particles to give a coverage of around 0.3, when the surface pressure is 10 mN m-1, the film has perfect hexagonal order over large areas; the separation of centers of the particles in all directions is 4.5 µm (Figure 7b). As the pressure is further increased to 45 mN m-1 the lattice is distorted (Figure 7c). With reference to Figure 8, for a hexagonal lattice, lengths a, b, and c are equal. At surface pressures in excess of about 15 mN m-1 (applied in the direction of the arrows), the packing becomes rhombohedral as illustrated in Figure 8. For the monolayer at Π ) 45 mN m-1 (Figure 7c) the ratio of lengths a () b) to c is approximately 0.87. Values of a, b, and c for monolayers compressed to a range of pressures up to above 40 mN m-1 are shown in Figure 9, from which it is seen that the hexagonal structure becomes increasingly distorted as Π is increased. The collapse pressure for the monolayer is about 48 mN m-1 (Figure 2). The appearance of monolayers above the collapse pressure is shown in Figure 10. Just above the collapse pressure some undulations are apparent, which become larger as Π is increased. Ultimately, large ridges are formed. It is noteworthy that a large degree of order is retained in the monolayer, which remains intact. It appears that neither single particles nor patches of particles are ejected from the monolayer. The wavelength of the corrugations in Figure 10c is about 100 µm, and the amplitude roughly 20 µm. We give a theoretical treatment elsewhere showing why monolayer folding is energetically favored over particle ejection.20 Unlike the situation for small particles, the various structures described for the larger particles are recovered on reexpansion of the monolayer, i.e., the process of

Figure 7. Structure of monolayers of 2.6 µm diameter particles at the octane/water interface. (a) Fractional surface coverage is 0.03 and the surface pressure is almost zero; the distance between centers of particles is ca. 14 µm and the packing is hexagonal. (b) Surface coverage is about 0.3, the surface pressure is 10 mN m-1 and the packing is hexagonal still. The separation between centers of particles is now ca. 4 µm. (c) Monolayer is compressed to 45 mN m-1. The hexagonal structure is lost. Scale bar in (a) represents 100 µm and (b) and (c) 50 µm. See Figures 8 and 9 for the lattice dimensions.

compression and expansion is more or less reversible even when the highest pressures have been attained on compression. It is interesting however that a small fraction of the particles do not separate on expansion, remaining as linear strings of adhering spheres, as shown in Figure 11. Compression presumably forces some of the particles over a repulsive energy barrier. Thomas and McCorkle22 have pointed out that the ends of dendrites of charged particles present less of an electrical potential barrier for particle addition than do the sides. (22) Thomas, I. L.; McCorkle, K. H. J. Colloid Interface Sci. 1971, 36, 110.

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4. Theoretical Description of the Behavior of Particle Monolayers

Figure 8. Packing of particles in monolayers. (a) Hexagonal packing which is observed at low surface pressures. Lengths a, b, and c are equal. (b) Rhombohedral array after anisotropic compression in the direction of the arrows on the Langmuir trough; here a and b are equal and smaller than c.

Figure 9. Two-dimensional lattice dimensions a, b, and c (see Figure 8) as a function of surface pressure for monolayers of 2.6 µm diameter particles at the octane/water interface.

Effects of Electrolyte on Monolayers at Oil/Water Interfaces during Compression and Expansion. Addition of NaCl up to 0.1 mol dm-3 has only little influence on the long range order (repulsion); the symmetry is only slightly reduced. Even for 1 mol dm-3 salt long-range order is still largely retained in dilute monolayers. We show images of particle monolayers on 1 mol dm-3 aqueous NaCl in contact with air (where widespread aggregation occurs) and in contact with octane in Figure 12. That the repulsion in monolayers at the octane/water interface is significantly decreased by salt however, is evidenced by the observation (not shown) that on reexpansion of a compressed monolayer in the presence of 1 mol dm-3 salt most of the particles do not redisperse but remain in islands or chains of adhering particles.

4.1. Physical Interpretation of the Interactions between Interfacial Particles. We attribute the super long-range repulsion between the latex particles at the octane/water interface to the Coloumbic interaction between residual charges at the particle/oil interface. We assume this charge arises as a result of an incomplete binding of H+ or Na+ ions to the sulfate groups at the latex particle surface. It is quite possible that the polystyrene surface, which is not molecularly smooth, traps traces of hydration water localized around the hydrophilic SO4 groups, and this could stabilize a small number of surface charges in contact with the oil phase. Since these charges interact through the oil phase, the interaction is quite insensitive to the electrolyte concentration. The observed behavior of particle monolayers at the air/aqueous solution interface can be accounted for in broad terms as follows. The contact angle of the particles with the aqueous NaCl solutions is low, below 30°, even though the particles are described as hydrophobic (see Experimental Section). This means the particles are deeply immersed in the aqueous phase at the air/water surface. The electrostatic interaction between the charged particle surfaces in water is screened by the electrolyte, except at very low ionic strength. The difference in behavior of particle monolayers at the air/water and octane/water interfaces appears to arise from the different contact angles (between 70° and 80° for the octane/water interface and less than 30° for the air/water surface; see Figure 13b and 13c). One should also bear in mind the possibility of evaporation of water at the air/latex interface, which would lead to better neutralization of the surface charges. In addition, repulsion through the air due to the permanent dipoles on the sulfate groups or any residual free charge (see below) is expected to be small since little of the particle surface is exposed to air. Thus a relatively high concentration of NaCl is able to damp out the repulsion between particles at the air/water surface, leading to spontaneous aggregation (Figure 3b and 3c). Monolayers at the alkane/ water interface are much more repulsive which leads to the formation of the observed highly ordered structures, which persist even in the presence of high concentrations of electrolyte (Figure 12). Below we present the analysis of the monolayer surface pressure before collapse, in terms of the pair interaction potential between the latex particles at the oil/water interface. 4.2. Electrostatic Surface Pressure. Here we analyze the contribution of the electrostatic interactions between latex particles in the monolayer to the lateral force per unit length of the barrier (the effective surface pressure). The particles are assumed to be large enough (micrometersize) to have no effect of Brownian motion on the surface pressure of the monolayer. The electrostatic interaction potential between two charged latex particles adsorbed at the oil/water interface is derived in the Appendix. The interaction force f(L) appears as a sum of the usual Coloumbic force and the image force due to the existence of the oil/water (or particle/water) surface (cf. eq A.23)

f(L) ≈

[

]

q2 1 L 0 L2 (4ζ2 + L2)3/2

(1)

Here q is the total charge of the oil/particle surface (cf. eq A.1), and 0 is the dielectric constant of the oil. In eq 1 the particles are treated as two point charges of magnitude q, placed at a distance ζ from the water interface with the


Langmuir, Vol. 16, No. 4, 2000 1975

Figure 10. Structures of collapsed monolayers of 2.6 µm diameter particles at the octane/water interface. (a) Just above the collapse pressure; incipient folding is seen to be present. (b) On further compression of the monolayer the folding becomes more pronounced. (c) Ultimately large corrugations appear, parallel to the trough barriers, but neither individual particles nor rafts of particles are expelled from the monolayer, which keeps its highly ordered rhombohedral structure. Scale bars all represent 100 µm.

Figure 11. Reexpanded monolayer of 2.6 µm diameter particles at the octane/water interface. Most particles redisperse into an ordered structure but some adhere into linear strings. Scale bar represents 50 µm.

lower side of the particles (see Appendix and eq A.2) and at a distance L from each other. The image charge in the water has opposite sign (since 0 < ) and is placed at a distance 2 ζ from the original charge (see Figure 14). We remark that the interaction force given by eq 1 is the major contribution, i.e., the Coloumbic force between the residual charges on the latex/oil surface. This is true when (i) the particle size is much larger than the Debye length, κR . 1, and (ii) the three-phase contact angle is not very small. For smaller particles however, the Debye screening interaction between the portions of the latex surfaces in contact with water must be taken into account. This can be done in a similar fashion as in the present analysis of the pure Coloumbic part of the interaction, but for the latex particles used (0.5-2.6 µm) this contribution is essential only for pure water, where κ-1 ≈ 1 µm. For ionic strengths of 1 mM and higher, the Debye

screening effect is comparable with the pure Coloumbic interaction only for a particle radius of 50 nm or less (see Figure 17b for particle size effects on the Coulombic interaction energy). With regard to the van der Waals contribution to the interaction force, we believe it has not been reliably estimated for the present configuration of particles on a liquid interface. However, the effect is expected to be somewhere between the van der Waals interaction of two latex particles immersed entirely in water, and those immersed entirely in oil. For the case of water, the Hamaker constant for polystyrene/water/ polystyrene is 0.95 × 10-20 J and for the case of alkane it is even smaller. For this reason we expect that the van der Waals interaction will be important only for particles in close proximity. We now estimate the lateral force acting on the Langmuir trough barrier. The force is transmitted to the barrier by the boundary particles, which interact with the other particles from the monolayer. It is evident from experimental observation that the particles in the monolayer are ordered in a hexagonal lattice even far from the condition of close packing. We denote by D the average center-to-center distance between closest neighbors. The average surface charge density, qΓ of the hexagonal lattice of particles is

qΓ(D) )

q x3D2/2

(2)

The net electrostatic force F(D), acting on the boundary particle can be calculated by direct summation of the x-component of the force between its charge q and the charges of the other particles, assumed for simplicity to be uniformly smeared out around the boundary particle

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Figure 12. Comparison of the structure of monolayers of 2.6 µm diameter particles on 1 mol dm-3 aqueous NaCl in contact with (left image) air and (right image) octane. Scale bars represent 50 µm.

Figure 13. (a) Sketch of the latex particle adsorbed at the oil/water interface. The entrapment of traces of hydration water at the particle/oil interface (due to surface roughness) may stabilize a small part of the surface charges. (b) In the case of the air/water interface, the latex particle is almost entirely in the aqueous phase, the contact angle θ being e30°. (c) In the case of the octane/water interface, the latex particle has a large portion of area in contact with the oil phase; in this case the contact angle is between 70° and 80°.

Figure 14. Schematic representation of the charge-charge interaction near an oil/water interface at small and large distances. The short distance interaction is dominated by a direct charge-charge interaction through oil, while the longdistance interaction is equivalent to the dipole-charge interaction due to the influence of the image charge contribution (see text).

(see Figure 15). Thus

F(D) ≈

fx(Lij) ∑ i*j

By replacing the summation with integration we obtain for F(D)

F(D) ≈ -

[

q2Γ(D) 1 0 L2

∫0πdφ sin φ ∫D/2∞

]

L LdL (3) (4ζ + L2)3/2 2

Taking the integral in eq 3 and recognizing that we have one boundary particle per length D of the barrier we estimate the electrostatic surface pressure Π(D) to be

Π(D) )

[

F(D) 4q2 1 ) 1+ 3 D (1 + 16ζ2/D2)1/2 0D x3

]

1 + (1 + 16ζ2/D2)1/2 ln (4) 2 Finally, by introducing the notations

Figure 15. Sketch of a boundary particle (and its neighbors), which transmits the force acting on the trough barrier. D is the distance between particle centers. The net force is obtained via integration over the x-component of the force of interaction of the boundary particle “0” with the charge of its neighbors “1”, “2”, “3”, etc., which are uniformly smeared out over the lattice around particle “0”.

x≡

A D 2 ) , Ah 2R

( )

β)

ζ2 , R2

(5)

eq 4 can be represented in a form more convenient for comparison with experimental data


Π(x) )

q2 2x30R x

[

3 3/2

1-

1 + (1 + 4β/x)1/2 ln

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]

1 + (1 + 4β/x)1/2 (6) 2

Here x is the ratio between the trough area, A ) Nx3D2/2 and its value, Ah ) N2x3R2 at the close packing point. We remark that eq 6 can also be derived from energy considerations. For highly expanded monolayers (large values of x), the asymptotic behavior of Π(x) is given by

Π(x) )

q2βx3 5β +..., 13 5/2 2 x 02R x

[

]

β,x

(7)

Note that the first term in the large area asymptotic, eq 7, corresponds to charge-dipole interactions, while the first term in eq 6 corresponds to a direct charge-charge interaction through the oil phase. Figure 16 presents the comparison of the experimental Π-A behavior with that obtained from theory (eq 6) for a monolayer composed of sulfate latex particles of diameter 2R ) 2.6 µm and 2.1 nm2 per surface charge in water. The three phase contact angle at the octane/water interface was taken to be 80°. The ionic strength of the aqueous phase is 10 mM. The best fit of the experimental data with eq 6 gives the degree of surface ionization of SO4H groups at the octane/latex interface as R ) 1.03%, i.e., about one percent of the charge on the particle in water. We remark that the interaction between the residual charges completely dominates the electrostatic interaction of the particles through the water phase, which is screened by the electrolyte (except very close to the collapse point, A/Ah ) 1). Note that eq 6 slightly underestimates the surface pressure at A/Ah ≈ 1. The difference is attributed to the electrostatic interaction through the water, which is not taken into account in eq 6. Nevertheless the simple analytical formula for the electrostatic surface pressure describes very well the experimental data. In Figure 17a we plot the interparticle pair potential involving image charges and compare it with the pure Coloumbic interaction (with no images) between the particle residual charges. At small particle separations the two interactions are similar in magnitude, while at large distance they differ by several orders of magnitude since eq A.23 predicts an effective dipole-charge interaction which falls off more rapidly with distance. In Figure 17b we illustrate the effect of particle size on the interparticle interaction energy, all other parameters being held constant. The magnitude of the interaction is seen to decrease rapidly with the decrease of the particle radius.

Figure 16. Surface pressure vs trough area isotherm for 2.6 µm latex particles at the octane/water interface. The surface charge density in water corresponds to 2.1 nm2 per surface charge. The electrolyte concentration is 10 mM. We remark that the surface pressure isotherm is quite insensitive to the electrolyte concentration up to 1M. The circles (O) are experimental data and the solid curve (s) represents the best fit with eq 6, which gives the percentage surface ionization of SO4H groups at octane/latex interface, R ) 1.03% (see the text for more details).

5. Conclusions (1) The structure and the collapse mechanism of compressed monolayers of micrometer-size latex particles at air/water and octane/water interfaces have been revealed by direct microscopic observation. (2) For monolayers at both air/water and oil/water interfaces, collapse occurs by folding rather than by ejection of particles into one or other bulk phase. (3) Strong dependence of the monolayer structure at the air/water interface on the electrolyte concentration was observed, while particle monolayers at the octane/ water interface remained highly ordered even on concentrated electrolyte solution.

Figure 17. (a) Particle interaction energy vs distance for 2.6 µm diameter latex particles at the octane/water interface. All the parameters are the same as in Figure 16. The solid curve (s) represents the result from eq A.23, while the dashed curve gives the result of pure Coloumbic interaction of the particle charges in oil (see the text for details). (b) Particle interaction energy for different radii of the particles. All other parameters are the same as in (a).

(4) It was found that, near collapse, particle monolayers at the octane/water interface change structure from hexagonal to rhombohedral as a result of anisotropic compression.

1978


Aveyard et al.

(5) We attribute the long-range repulsion between the latex particles at the octane/water interface to electrostatic interactions, through the oil phase, of residual charges residing at the particle/octane interface. The difference between the behavior of monolayers at the air/water and octane/water interfaces can be attributed to contact angle differences at the two interfaces. (6) We have developed a simple theoretical model for the interaction between charged particles at a water/fluid interface. An analytic expression for the electrostatic surface pressure of colloidal monolayers of hexagonal structure has been derived. We determine the degree of ionization, R, of SO4H groups at the octane/latex interface by comparison of the theory with the experimental data (R ≈ 1%).

the oil/water interface satisfies the equations

∇2Ψ(1)(r) ) κ2Ψ(1)(r), z e 0, (region 1 - in water), (A.4) ∇2Ψ(2)(r) ) 0, 0 e z e ζ, (region 2 - in oil) ∇2Ψ(3)(r) ) 0, ζ e z, (region 3 - in oil)

Appendix Asymptotic Expression for the Interparticle Interaction Potential. Here we will derive an asymptotic of the electrostatic energy of interaction between two particles residing at an oil/water interface. We consider the case of large (micrometer-size) charged particles with the assumption that a residual surface charge exists at the particle/oil interface. We will neglect the electrostatic interaction between the charged particle/water interfaces since it is effectively screened by the electrolyte at distances larger than a few Debye lengths (much smaller than the particle size). The total residual charge q at the particle/oil interface depends on the particle contact angle θ according to

q)

∫S

op

σsdS )

2πR2σ0s R(1

+ cos θ)

∫S

op

Rσ0s zdS ) πR3Rσ0s sin2 θ

∑k Ψ˜ k(z) exp(ik.s).

(A.7)

In eq A.7 s ) (x,y) and k ) (kx,ky) are the two-dimensional spatial and wave vectors. The electrostatic boundary conditions for eqs A.4-A.6 are

∂Ψ ˜ (2) ∂Ψ ˜ (1) k k (2) | | | ) Ψ ˜ | , ) (A.8) Ψ ˜ (1) k z)0 k z)0 0 ∂z z)0 ∂z z)0 ∂Ψ ˜ (3) ∂Ψ ˜ (2) k k |z)ζ ) | + 4πσk ∂z ∂z z)ζ (A.9)

Ψ ˜ (3) ˜ (2) k |z)ζ ) Ψ k |z)ζ,

where σk ) q/A0 and A0 is the area of the unit cell subdivision of the interface. The proper functional form of the Fourier components, consistent with eqs A.4-A.6 is λz + kz - -kz ˜ (2) ,Ψ ˜ (3) Ψ ˜ (1) k (z) ) Bke , Ψ k (z) ) Ck e + Ck e k (z) )

Dke-kz (A.10) (A.1)

where R is the degree (percentage) of ionization of the surface groups (e.g., -SO4H) at the oil/particle interface and σ0s is the maximal surface charge density. It is expected that the degree of ionization, R, is very small due to the low dielectric constant of the oil. In our treatment, we replace the charged particle with a point charge q located at a position rq ) (0,0,ζ) into the oil phase. We estimate ζ from the following condition for equivalence between the first moments of the real and idealized surface charge distribution

q(ζ - R) )

(A.6)

with κ2 ) 4πe2/kBT∑iZi2ni being the Debye parameter for the aqueous phase. The method of solution of the electrostatic problem we use here is similar to that of Stillinger.23 The system of eqs A.4-A.6 can be solved by Fourier transform24

Ψ(r) ) Acknowledgment. The authors thank the EPSRC and NATO/Royal Society for provision of postdoctoral grants for D.N. and V.P. (ref. FCO/97B/BLL), respectively.

(A.5)

(A.2)

The combination of eqs A.1-A.2 gives

ζ ) R + R(1 + cos θ)/2 ) R(3 + cos θ)/2. (A.3) We remark that for the idealized point charge q, placed in oil, the position of the water/oil interface effectively coincides with the bottom of the particle, since the oil and the particle have very similar dielectric constants. That is why in eqs A.2 and A.3 we have shifted the position of the charge by z ) R. The distribution of the electrostatic potential, Ψ(r), around the point charge q placed at rq ) (0,0,ζ), close to

where λ2 ) κ2 + k2. The integration constants in eq A.10 are determined from the boundary conditions, eqs A.8A.9

Bk )

4πσkk -kζ + 2πσk -kζ e , Ck ) e , Ck ) λ + 0κ 0 2πσk -kζ pe , (A.11) 0 Dk )

2πσk kζ λ - 0k [e - pe-kζ], p ≡ . 0k λ + 0k

(A.12)

As A0 approaches infinity, the k-sum in eq A.7 can be replaced by two-dimensional integration

∑k f

A0

2π ∞ dφ∫0 dkk ∫ 0 2

(A.13)

∫02πexp(ix cos φ)dφ

(A.14)

(2π)

By using the identity

2πJ0(x) )

in combination with eqs A.10-A.13 we obtain (23) Stillinger, F. H. J. Chem. Phys. 1961, 35, 1584. (24) Korn, G. A.; Korn, T. M. Mathematical Handbook; McGrawHill: New York, 1968.


∫0∞dkJ0(kL) ) L1 , ∫0∞dkJ0(kL)e-2kζ )

dkk2J0(ks) exp(λz - kζ), λ ) λ + 0k

∫0∞

Ψ(1)(r) ) 2q

(k2 + κ2)1/2 (A.15) Ψ(2)(r) )

∫0

q 0

∞

dkJ0(ks)(ekz - pe-kz)e-kζ (A.16)

∞

kζ

- pe

-kζ

)e

-kz

(A.17)

Here J0(x) is the Bessel function of zeroth order (see ref 24). The potential energy of electrostatic interaction between two identical charges q located at a distance s ) L from each other and having a vertical position z ) ζ from the interface is

U(L) ) q

1

,

x4ζ2 + L2

dkkJ0(kL) -2kζ exp(-κx4ζ2 + L2) e ) xκ2 + k2 x4ζ2 + L2 (A.21)

∫0∞

we finally obtain

∫0 dkJ0(ks)(e

q Ψ (r) ) 0 (3)

Langmuir, Vol. 16, No. 4, 2000 1979

q2 ∂Ψ(3) | s)L ) ∂z z)ζ 0

∫0∞dkJ0(kL)(1 - pe-2kζ) (A.18)

By expanding p in series for 0/ , 1,

U(L) ≈

1 q2 1 qqim + , qim ≈ 0 L 0 x 2 4ζ + L2 0 -q 1 - 2 exp(-κx4ζ2 + L2) (A.22)

[

The interpretation of the terms in eq A.22 is the following. The first term is the energy of the usual Coloumbic interaction between the two identical charges of magnitude q, separated by a distance L from each other in the oil phase. The second term is the interaction between the second particle of charge q and the image charge qim of the first particle, which is located at a distance (4ζ2 + L2)1/2 from the second particle, symmetrically with respect to the particle/water interface (see Figure 14). Note that the image charge qim has a sign opposite to q and depends on the ionic strength due to the polarization of the electrolyte by the charge q. However, for micrometer-size particles and not extremely low electrolyte concentrations (κR . 1) this dependence is negligible, i.e., qim ≈ -q, which gives

U(L) ≈ 0k λ - 0k ≈1-2 + O(20/2) p) λ + 0k λ

(A.19)

[∫

∞

U(L) ≈

0

dkJ0(kL) -

∫0 dkJ0(kL)e-2kζ + ∞

20

]

∫0∞dk kλ J0(kL)e-2kζ

(A.20)

Using the identities25 (25) Gradshtein, I. S.; Ryzhik, I. M. Table of Integrals Series and Products; Acad. Press: New York, 1980.

[

]

1 q2 1 . 0 L x 2 4ζ + L2

(A.23)

We remark that the interaction potential at large distances between the particles resembles the interaction between an effective dipole and a charge with an asymptotic

and substituting into eq A.18 we obtain

q2 U(L) ≈ 0

]

2q2ζ2 , (ζ/L)2 , 1. 3 0L

(A.24)

The latter equation is equivalent to the energy of interaction between a point dipole µeff ) 2ζq, located at r ) (0,0,0) and directed along the z-axis, and a point charge q, located at r ) (L,0,ζ). The energy of a dipole µeff in the electric field E ) q/(0r2), created by the point charge q is U ) -µeffE cos β, where β is the angle between the direction of the dipole and the direction of the electric field E of the point charge. Since the distance between the charge q and the point dipole µeff is r ) (L2 + ζ2)1/2 ≈ L, and cosβ ) -ζ/r ≈ -ζ/L, E ≈ q/(0L2), by substituting them back into the electrostatic energy U, we obtain eq A.24. LA990887G

Compression and Structure of Monolayers of Charged Latex Particles

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