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Structure and Collapse of Particle Monolayers under Lateral Pressure at the Octane/Aqueous Surfactant Solution Interface† Robert Aveyard,*,‡ John H. Clint,‡ Dieter Nees,§ and Nick Quirke*,| Department of Chemistry, University of Hull, Hull HU6 7RX, United Kingdom, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom, and Department of Chemistry, Imperial College, London SW7 2BY, United Kingdom Received January 18, 2000. In Final Form: March 31, 2000 We report a study of the compression of monolayers of monodisperse spherical polystyrene particles at the interface between aqueous surfactant solutions and octane. The particle size (2.6 µm diameter) was selected so that direct in situ microscopic observation of the monolayer structure could be made during lateral compression and “collapse”. Monolayers have been formed on a miniature Langmuir trough placed on a microscope stage. Our study has focused on (a) the relationship between the monolayer collapse pressure, Πcol, and the interfacial tension, γ*, of the oil/water interface in the absence of a particle monolayer and (b) the mode of monolayer “collapse” at high surface pressure. Interfacial tensions γ* have been adjusted (in the range 50-4 mN m-1) by addition of surfactants over a range of concentration. We find that the monolayer collapses by buckling (folding) when the surface pressure is equal to the surface tension of the oil/water interface. Particle promotion out of the interface is not observed experimentally.
1. Introduction Among others, physicists, physical chemists, and biologists have long had an interest in the behavior of particles at fluid/fluid interfaces. This has stemmed in part from the relevance that solid particles (both in the colloidal size range and larger) have in the behavior of many biological and industrial systems and processes. Some of our own interest arose from the well-known ability of particles to modify the stability of emulsions and foams.1-3 The behavior of particle monolayers at liquid surfaces has received considerable attention, particularly over the last 3 decades. Some 30 years ago Schuller4 studied monolayers of polystyrene (PS) latex particles at the air/ water (A/W) interface and presented surface pressure (Π)surface area (A) isotherms. A similar study was later reported by Sheppard and Tcheurekdjian.5 Measurements of Π-A curves for sterically stabilized spherical polymer particles at the heptane/water interface were used by Doroszkowski and Lambourne6 as a means of estimating the strength of steric barriers in nonaqueous dispersions. More recently there has been considerable interest in the two-dimensional aggregation of monodisperse spherical colloidal particles at liquid surfaces, usually the A/W surface. Aggregate morphology and fractal dimensions have been reported by Horvolgyi and co-workers for † Part of the Special Issue “Colloid Science Matured, Four Colloid Scientists Turn 60 at the Millennium”. * To whom correspondence may be addressed. ‡ University of Hull. § Rutherford Appleton Laboratory. | Imperial College.
(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. Polym. Sci., C 1971, No. 34, 253.
surface-modified glass particles at the A/W surface (see, e.g., ref 7), while Stankiewicz et al.8 and Robinson and Earnshaw9 for example have studied the salt-induced aggregation of PS latex particles at A/W surfaces. Kumaki10 in interesting work on monomolecular PS particle (ca. 50 nm diameter) monolayers has discussed the possible significance of the measured surface pressures. The Π-A isotherms were obtained in three ways, giving rise to quite different curves. The surface pressure can be detected directly as a force on a barrier acting as one of the walls confining the particle monolayer, as in earlier designs of Langmuir troughs. Alternatively, a surface tension probe, e.g., a Wilhelmy plate, can be located through the monolayer. In the case of fluid monolayers of low molar mass molecules, the vertical force acting on the Wilhelmy plate yields the interfacial tension. In the case of particle monolayers, which can become rigid on compression, the significance of the measured quantity is less certain. Indeed, Kumaki found that “surface pressures” obtained with the Wilhelmy plate parallel to and normal to the direction of monolayer compression were quite different, and different from that observed to act on a barrier in the interface. We address this observation in relation to particle size below. In the present work we have been concerned with monolayers of monodisperse spherical PS particles at the interface between aqueous surfactant solutions and octane. The particle size (2.6 µm diameter) was selected so that direct in situ microscopic observation of the monolayer structure could be made during lateral compression. Monolayers have been formed on a miniature Langmuir trough placed on a (reflectance) microscope stage. Our study has focused on (a) the relationship between the monolayer collapse pressure Πcol and the (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, 2055, 2065; Langmuir 1993, 9, 1436. (10) Kumaki, J. Macromolecules 1986, 19, 2258; 1988, 19, 749.
10.1021/la000060i CCC: $19.00 © 2000 American Chemical Society Published on Web 06/17/2000
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interfacial tension γ* of the oil/water interface in the absence of a particle monolayer and (b) the mode of monolayer “collapse” at high surface pressure. The collapse mechanism is very relevant to methods we have proposed previously for the determination of contact angles of particles with liquid interfaces11 and for the estimation of line tensions.12 These methods are based on the assumption that particles are ejected from the interface on collapse. Elsewhere we have considered the behavior of similar particle monolayers below the collapse pressure and obtained an equation of state based on Coulombic repulsion between particles through the oil phase.13 2. Materials and Methods 2.1. Materials. Octane was 99% pure as supplied (Aldrich, England) and was passed through a chromatographic alumina column three times shortly before use. Propan-2-ol (IPA), used as a spreading solvent, was an AnalaR sample from BDH (England) and was used as received. All water was passed through an Elga UHQ II unit fitted with UV irradiation and with cartridges for ion exchange, reverse osmosis, organic adsorption, and ultramicrofiltration. The surface tension of water from the Elga unit was consistently 72.8 ( 0.1 mN m-1 at 293 K. The monodisperse polystyrene (PS) latex particles used (with diameter 2.6 µm) had sulfate groups at the surface and were supplied by Interfacial Dynamics (USA) as a surfactant-free aqueous dispersion. The surface charge density on the particles was 7.7 µC cm-2, which corresponds to an area per sulfate group of 2.08 nm2. The contact angle of the particles with the octane/ aqueous phase interfaces (measured into the aqueous phase) was between 70 and 80°, irrespective of the presence, type, and concentration of surfactant.13 Four surfactants were used to reduce the octane/water interfacial tension in order to probe the effects of changing interfacial tension on the collapse pressure of particle monolayers subjected to lateral pressure on a Langmuir trough (see below). Sodium dodecyl sulfate (SDS) was supplied by Lancaster as a 99% pure sample. Hexadecyltrimethylammonium bromide (CTAB), 99% pure, and decyl β-D-glucopyranoside (decyl β-glucoside DBG), 98% pure, were both from Sigma (U.K.). Cetylpyridinium chloride (CPC, 98% pure) was a Fluka sample. All surfactants were used as received. For the purpose for which they were used, it was unnecessary to purify the samples further. 2.2. The Langmuir Film Balance and Its Use. The Langmuir trough used was a miniature analytical film balance supplied by Nima Technology (England), model 601M. It was modified, in collaboration with the makers, for use with monolayers spread at the oil/water (O/W) interface. This entailed fitting a stainless steel lining to the Teflon trough (as shown in Figure 1) so that the oil/water interface was pinned where the sharp upper edge of the lining meets the Teflon edge. In the absence of the lining the alkane was found to displace the water from contact with the Teflon trough and thus allow leakage of monolayer material. The moving barriers were also constructed of stainless steel, which had the correct wetting properties to prevent monolayer leakage. The trough was charged with the aqueous subphase, and the octane was layered on top, filling the channel around the edge of the trough. The approximate dimensions of the trough are shown in Figure 1. We have found it possible to complete several compression/ expansion cycles without the significant loss of spread particles using this modification of the trough. The barriers run on linear guides and are driven by dc motors through reduction gear boxes, giving smooth dynamic control. Surface pressures (Π) were determined using a pressure sensor, model PS4, with an attached filter paper Wilhelmy plate dipping (11) Clint, J. H.; Taylor, S. E. Colloids Surf. 1992, 65, 61. Clint, J. H.; Quirke, N. Colloids Surf. 1993, 78, 277. (12) Aveyard, R.; Clint, J. H. J. Chem. Soc., Faraday Trans. 1995, 91, 175. (13) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969.
Figure 1. Schematic representation of the Langmuir trough, which could be placed on the microscope stage so that the appearance of particle monolayers could be recorded during compression.
Figure 2. Surface pressure at the octane/water interface produced by adding different volumes of propan-2-ol (IPA) using a microsyringe. into the oil/water (O/W) interface. The plate was aligned either perpendicular to or parallel to the trough barriers (see later). The sensor is designed such that the plate is always suspended at the same position during variation of Π with compression of the monolayer. The maximum interfacial area occupied by a particle monolayer on the trough was 80 cm2 and the minimum area was 20 cm2; the compression rate for all experiments was 2 cm2 min-1, which was the lowest possible on the apparatus. Experiments were carried out at room temperature, which was around 293 K. The operation of the trough is computer controlled using software provided by the makers. The isotherms are displayed on the computer monitor as they are being obtained and can be subsequently printed out. For the microscopic observation of the structure of particle monolayers during compression, the film balance was placed on the stage of a Nikon Optiphot 2 microscope fitted with a JVC TK 1381 digital color CCD camera. A high-
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Figure 3. (a) Defects in a compressed monolayer of 2.6 µm diameter particles at the interface between octane and 1 mM aqueous DBG. The scale bar represents 100 µm. (b) Monolayer structure at the air/water interface formed by allowing a thin layer of octane to evaporate from the octane/water interface containing the monolayer. The particle diameter is 2.6 µm, and the scale bar represents 50 µm. pressure mercury lamp was used for the illumination and the monolayers were observed using 10×, 20×, and 40× reflectance objectives. Images were captured and measured using Lucia, a commercial image processing and analysis system (purchased from Nikon U.K.). For monolayer formation on the trough, aqueous dispersions of particles (as supplied) were mixed with IPA (1:1 or 1:2 by weight). Between 50 and 200 µL of the mixture was then injected into the interface with the trough barriers open. Particles spread readily, even on the surfactant subphases, without significant loss to either bulk phase. The IPA eventually diffuses into the aqueous and oil phases and gives, due to adsorption, a reduction in the oil-water interfacial tension. The surface pressure (interfacial tension lowering) caused by IPA addition (alone) is shown in Figure 2 as a function of the volume added. In a typical experiment, about 85 µL of IPA was added to the system which gives a tension lowering of about 2 mN m-1. After the particles were allowed to spread, systems were left to equilibrate for between 15 min and 1 h before monolayer compression was commenced. In a number of cases the crystallographic defects present in ordered monolayers (see Figure 3a) could be seen to heal during this period of time. It is also interesting to note that if a (thin) layer of octane was allowed to evaporate, the capillary forces generated when the octane/air interface contacted the particle monolayer caused the monolayer to aggregate into very ordered structures as seen in Figure 3b. Kralchevsky et a1.14,15 and Lazarov et a1.16 have discussed the action of capillary forces in such particle monolayers. After each experiment the trough was emptied by sucking off the liquids and then filled and emptied several times with clean water. It was then wiped with ethanol-soaked tissues and filled with ethanol and emptied two or three times. Every few days it was additionally finally cleaned with chloroform. 2.3. Monolayer Pressures Measured Using a Pendant Drop Method. It appears, from Π-A isotherms (see later) that we attain very low interfacial tensions ( 25 mN m-1), the Π-A curves diverge, the “pressure” recorded being higher with the Wilhelmy plate parallel to the barriers (see Figure 4b). As mentioned, a similar
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Figure 5. Surface pressure against surface area for 2.6 µm diameter particles at the octane/water interface. Filled symbols and dashed lines are results obtained using the pendant drop method. Drop areas have been converted to the equivalent Langmuir film balance areas (see text). Results indicated by the full line were obtained using the Langmuir film balance. The inset shows the tension of the particle-covered interface, obtained using the drop profile method, as a function of the equivalent trough area.
Figure 4. (a) A Π-A curve for a monolayer of 2.6 µm diameter particles at the octane/water interface. The surface pressure due to the spreading solvent (2 mN m-1) has been subtracted from the ordinate. (b) Π-A curves for monolayers of 0.21 µm diameter particles at the octane/water interface. Curve i is for the Wilhelmy plate placed parallel to the trough barriers and curve ii for the plate normal to the barriers. Note that the curves are coincident for surface pressures less than about 25 mN m-1.
divergence has been observed for even smaller particles by Kumaki.10 As we will see, we have observed surface pressures at collapse (Πcol) very close to the tensions, γ*, of the octane/ water interfaces in the absence of particles. This means that the tensions of the monolayer-covered interfaces at collapse are close to zero. In an attempt to confirm that such low tensions do in fact exist in particle-covered monolayers, we have determined interfacial tensions quite independently using the drop profile method described in the experimental section. We show results obtained in Figure 5. It is possible with the drop profile method to obtain a low surface density of particles much more conveniently than with the Langmuir trough method. Further, with the drop method there is no need for the use of a spreading solvent; the drop surface spontaneously picks up particles from bulk. It is very likely that at low surface pressures the particles remain trapped in the drop interface. The energy of attachment is of the order of 108 kT per particle. At high surface pressures however (in excess of around 30 mN m-1) particles appear to be progressively lost from the interface. This can be appreciated from the isotherm shown in Figure 5 where results from the Langmuir trough and drop profile techniques are compared. The two isotherms were made coincident for Π ) 20 mN m-1 by
converting the drop area to an equivalent trough area (the abscissa used in Figure 5). Above 30 mN m-1 the Π-A isotherm obtained using the drop method falls to the left (lower area) of the isotherm determined using the Langmuir trough, consistent with particle loss from the drop surface as the drop is retracted. It is also possible that the Laplace equation, used in the profile analysis to obtain tensions, does not apply to interfaces that may be rather rigid. The important point however is that it is clear from the drop profile results that fairly low interfacial tensions (say down to ca. 10 mN m-1ssee inset in Figure 5) are indeed attained just as on the Langmuir trough. 3.2. Effects of Surfactants on Particle Monolayers. Before proceeding to a discussion of the behavior of particle monolayers close to the collapse condition and the significance of the collapse pressure, we first describe the effects that surfactants have on the aggregation of particles in dilute particle monolayers. 3.2.1. Structure of Dilute Particle Monolayers. We show images in Figure 6 of monolayers of 2.6 µm diameter particles resting at octane/aqueous surfactant solution interfaces; for comparison, the highly ordered hexagonal structure obtained in the absence of surfactants is also shown. The cationic surfactant CTAB present at half its critical micelle concentration (cmc) (i.e., at 0.5 mM) reduces the order in the monolayer substantially, but very little particle aggregation is observed. When present at its cmc (1 mM), however, CTAB causes widespread 2-D aggregation. Since the polystyrene particles carry negative charges on their surfaces in water, it is tempting to suppose that the observed effects on monolayer order result in some way from charge neutralization resulting from surfactant adsorption at the particle/aqueous phase interfaces. That charge neutralization is not the cause, however, can be readily appreciated from the observation that the anionic surfactant SDS gives completely parallel behavior to that observed for CTAB. Further, the nonionic surfactant DBG present at half its cmc also reduces the order in the monolayer; we were unable to spread particles at the interface between octane and aqueous DBG at the cmc. We have not yet attempted an analysis of the origin of these effects of surfactants on monolayer structure. As
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Figure 6. Effects of surfactants (as indicated) on the structure of 2.6 µm diameter particle monolayers at the octane/water interface. The cmc values of DBG, CTAB, and SDS are approximately 2, 1, and 9 mM, respectively. All the surfactants reduce the monolayer structure when present at around half the cmc. At the cmc, SDS and CTAB cause aggregation within the monolayers; particles could not be spread on 2 mM DBG. Scale bars represent approximately 50 µm.
mentioned above, our purpose in adding surfactants has been to modulate the o/w interfacial tension γ*. 3.2.2. Collapse Pressures. Surface pressure can be regarded as a two-dimensional osmotic pressure. Particles are strongly anchored to the interface, but solvent molecules (octane and water) can freely enter and leave the monolayer, giving rise to an expanding two-dimensional pressure. Thus there are two opposing effects, one
due to the O/W interfacial tension, γ*, tending to cause contraction of the monolayer and the two-dimensional pressure tending to cause expansion. When the two contributions to the overall (measured) tension, γ, of the monolayer are equal, we can surmise that the onset of monolayer “collapse” occurs. If this is the case, the collapse pressure, Πcol, should be equal to the interfacial tension, γ*.
Structure and Collapse of Particle Monolayers
Figure 7. Surface pressure-area isotherms for 2.6 µm diameter polystyrene latex particles spread at the octane/ aqueous solution interface: (a) no surfactant; (b) 0.1 mM SDS; (c) 0.1 mM decyl β-D-glucopyranoside (DBG); (d) 0.5 mM DBG; (e) 1 mM CTAB.
Figure 8. Relationship between monolayer collapse pressure and the octane/aqueous surfactant solution interfacial tension for systems with varying concentrations of surfactant. The particles have diameter 2.6 µm. Open squares represent systems with DBG, filled squares systems with CTAB, open triangles systems with CPC, and filled triangles systems with SDS. The line is drawn with unit slope and passing through the origin.
To test this proposition we have obtained Π-A isotherms and measured collapse pressures for particle monolayers at the octane/aqueous surfactant solution interface over a range of values of γ*. We have used anionic (SDS), cationic (CTAB, CPC), and nonionic (DBG) surfactants over a concentration range in order to span values of γ* from around 50 down to 5 mN m-1. A series of Π-A curves is shown in Figure 7, where it is seen that collapse pressures, in the presence of various surfactants, cover the range between about 5 and 50 mN m-1. The collapse pressures are plotted against the interfacial tensions in Figure 8; the line is drawn with unit slope, passing through the origin. The remarkable correspondence of the two quantities is evident, and we discuss this later in the section on theory. First, however, it is relevant to consider the mode of monolayer “collapse” at Πcol. 3.3. Mode of Collapse. Clint and Taylor11 have proposed a method for the determination of the contact angle of spherical monodisperse particles with a liquid interface from the collapse pressure Πcol of the monolayer. The method is based on the assumption that particles are
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completely expelled from the monolayer at collapse, and Πcol is equated to the free energy of removal of particles in a unit area from the interface. Others have subsequently used the Langmuir trough approach to obtain contact angles (see, e.g., refs 17 and 18). Aveyard and Clint12 have also proposed a method for the estimation of line tension based on the same assumption. The PS particles used in the present work exhibit some very interesting behavior at liquid surfaces. Monolayers formed at the air/0.01 M NaCl interface indicate that there is interparticle repulsion. Very few particles are aggregated, but the monolayer order is not great. However if octane is layered onto such a monolayer, it becomes highly ordered giving a perfect hexagonal array (as seen in Figure 6), even for a separation of particle surfaces up to three or more particle diameters. Addition of further NaCl (to give 0.1 M and greater) causes widespread aggregation in monolayers at the air/solution interface but has no discernible effect on the structure of monolayers at the octane/water interface. These observations suggest that the interparticle repulsion is mediated through the octane rather than the aqueous phase and is Coulombic in nature. This behavior has been discussed in detail elsewhere.13 Upon compression, particle monolayers at the o/w interface retain their hexagonal order up to surface pressures of around 15-20 mN m-1, after which some distortion of the lattice occurs. In the absence of surfactant, no particle aggregation is observed. We show some monolayer structures in Figure 9. With reference to Figure 4a, at point 1 just below collapse, the monolayer is close packed (Figure 9a) and has a rhombohedral structure. Distortion from the hexagonal structure presumably arises from the anisotropic compression imposed by the Langmuir trough. In any event, the monolayer is clearly planar, within the depth of focus of the microscope. Upon further compression to above collapse (point 2, Figure 4a) it is clear from Figure 9b that the monolayer folds rather than expelling particles. From measurements made on the images, it is concluded that the area per particle does not change significantly within the monolayer; the loss of area on the trough is accommodated by the folding of the monolayer. Yet further compression (point 3, Figure 4a) leads to the formation of large corrugations of the monolayer (parallel to the trough barriers) rather than expulsion of particles. The wavelength of the corrugation depicted in Figure 9c is about 100 µm, and the amplitude about 30 µm, allowance being made for the refractive index (ca.1.40) of the octane layer covering the monolayer. The amplitude was estimated by first focusing on the peaks and then on the troughs of corrugations and determining their vertical separation using the height scale on the focus control. For the polystyrene particles studied in the present work, it appears that the assumption of particle expulsion from the surface at collapse is inappropriate. Clearly then, the Langmuir trough technique should not be used indiscriminately for the determination of contact angles and line tension. It may be of course that the mode of monolayer collapse is dependent on the contact angle and the size of the particles together with the tension of the liquid interface, γ*. However molecular dynamics simulations19 of nanoparticulates display a similar collapse mode to that described here. In any case, in situ observation of (17) Aveyard, R.; Binks, B. P.; Fletcher, P. D. I.; Rutherford, C. E. Colloids Surf. 1994, 83, 89. (18) Mate, M.; Fendler, J. H.; Ramsden, J. J.; Szalma, J.; Horvolgyi, Z. Langmuir 1998, 14, 6501. (19) Fenwick, N.; Bresme, F.; Quirke, N. To be submitted.
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the mode of collapse of monolayers of submicroscopic particles, such as those likely to be used in the determination of line tension, will probably prove difficult. 4. Theoretical Treatment of Monolayer Collapse In this section we focus on the description of the collapse of the particulate arrays described above. In particular we are interested in the situation where the dominance of particle-particle repulsion on compression (the top of region B of Figure 4a) leads to collapse, at a surface pressure equal to the interfacial tension γ*, and to the corrugation of the interface (with the surface pressure remaining constant or only slowly varying beyond the collapse, region C). To understand this behavior for micrometer-sized particles, we apply a theory that has been used successfully to describe similar phenomena observed in molecular dynamics simulation of nanoparticulates at fluid interfaces.19 We seek to reproduce the main features of the experimental pressure-area curves by considering the free energy of a model particle array expressed in terms of particle and fluid surface tensions and a simple particleparticle repulsion. We assume an HCP structure that on compression is able to distort by raising (or lowering) the interface away from the position of the interface at zero surface pressure, while maintaining the particles at their equilibrium contact angles. That is, we do not allow particles to be promoted from the interface. 4.1 The System. We consider N particles in equilibrium at a fluid interface. In the Langmuir trough the particles occupy an area A, having a surface tension γ at a surface pressure Π, in equilibrium with a region, area A*, containing no particles and surface tension γ*. The total interfacial area A + A* is AT. Since the interface containing the particles may buckle, the total surface area is not conserved and we have for a change in the total free energy F of the system
dF ) -S dT - P dV + γ* dA* + γ dA + µ dN (1)
Figure 9. Images of monolayers of 2.6 µm diameter particles at the octane/water interface. (a) Planar monolayer just prior to collapse (point 1 in Figure 4a). (b) Monolayer just above collapse (point 2 in Figure 4a). (c) Corrugated monolayer further past collapse (point 3 in Figure 4a). Scale bars represent 50 µm.
with γ ) (∂F/∂A)N,V,T,A* and where the surface pressure is defined in the usual way as Π ) (γ* - γ). For no buckling A ) A0 and AT is a constant. To obtain Π and γ, we need F(A), the minimum (equilibrium) interfacial free energy. In what follows we hold V, N, and T constant, i.e., dV ) dT ) dN ) 0. To treat a particle array, we must adopt a structure for the particles in the interface. We assume here that the N particles have an HCP arrangement with A/N ) x3r2/2, r being the separation between centers of adjacent particles. Since for our present purposes we are primarily interested in the short-range repulsion in the neighborhood of the knee of the Π-A curve (the top of region B and region C in Figure 4a), we choose to approximate the particle-particle interaction U(r) in this region by a simple quadratic form U(r) ) b(r - c)2, where b is a force constant and c a cutoff length. We take U(r) ) 0 for r > c. Thus at small separations, the particles are held to interact with a Hookean force. This reasonable assumption makes it possible to obtain simple analytic solutions to the equations.20 For an HCP structure, the energy per particle from nearest neighbor particle-particle interactions is 3U(r). The interaction free energy per particle F(r) is approximated as that of the HCP lattice of particles at T ) 0 K, and hence F/N ) 3U(r). The free energy of the (20) We show later that the surface equation of state arising from this assumed interparticle force law reproduces the shape of the experimental Π-A curve satisfactorily in the collapse region (Figure 11).
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F ) Nf(h0) + A(e)γ* + 3bN
(( ) ) 2A(e)
0.5
-c
1/2
3 N
2
(4)
We now seek the elevation e for minimum F at constant A0. The turning points of F are (for f(h0) constant)
∂A(e)/∂e ) 0
(5a)
and
{( )
γ* - 2(31/2)b c
31/2N 2A(e)
0.5
}
-1 )0
(5b)
An interesting physical solution of ∂A(e)/∂e ) 0 is e′ ) 0, where the prime indicates a turning point; hence A(e) ) A0. The second expression in eq 5 is solved by
e′ ) (
[{ (
2(31/2)bc 3N2 4 γ* + 2(31/2)b
) }/{ }] 4
8A0
- A02
0.5
(6)
31/2
The solution e′ ) 0 can be shown19 to be a minimum if
(
)
A0 31/2 2(31/2)bc g N 2 γ* + 2(31/2)b
2
(7)
The equality in eq 7 defines an area AL (which turns out to be that at the onset of collapsessee below) Figure 10. (a) Spherical particle resting in a liquid interface; h0 is the depth of immersion and Ax the area of liquid surface lost due to the presence of the particle (see text). (b) Plan view of a rhombohedral area A of a plane hexagonally packed surface. (c) End view (from point X in (b)) of surface after buckling caused by compression in the direction of the arrows in (b). The projected area in the horizontal plane is A0.
whole system now comprises the surface tension free energy F(A) of the particles in the interface and the interaction free energy F(r), i.e., F ) F(A) + F(r). The surface tension free energy F(A) is given by
F(A) ) Nf(h0) + Aγ*
A(e) ) 2 and
( )( 2A0 1/2
3
1/2
)
3 A0 + e2 8
0.5
(3)
)
2
(8)
then for A0 g AL the minimum free energy Fmin(A0) is at e ) 0, and the surface tension is
γ(A0) )
() ∂F ∂A
)
N,V,T,A*
(
)
∂Fmin(A0) ) ∂A0
{
γ* + 2(31/2)b 1 -
(2)
where f(h0) is the free energy associated with a particle at the equilibrium height h0 in the interface (Figure 10a), corresponding to the local surface tension γ*. Note that the surface tension γ* used to calculate the surface tension contribution to the free energy of the interface (area A) containing the particles is the surface tension γ* of the liquid interface. 4.2. Distortion Mode. We are interested in the surface pressure of the system as a function of the area per particle. We assume that the particles remain in the interface but that the interface itself can expand (by folding) into either phase and increase its surface area. With reference to Figure 10b, let the total (assumed rhombohedral) area containing the particles be A, then for an HCP arrangement, the system has side R ) ((2/30.5)A)0.5. As an approximation to the (perhaps) sinusoidal buckling of the interface, we take two planes starting from opposite edges of the area A and meeting in the middle with an elevation e, as shown in Figure 10. In terms of the in-plane area A0, we have 0.5
(
AL 31/2 2(31/2)bc ) N 2 γ* + 2(31/2)b
}
c (2A0/31/2N)0.5
(9)
from which the surface pressure is given by
{
Π(A0) ) -2(31/2)b 1 -
}
c (2A0/31/2N)0.5
(10)
Defining a cutoff area Ac ) (31/2/2)Nc2 at which Π(A0) ) 0, eq 10 for A0 g AL can be written in the form
{ ()}
Π(A0) ) -2(31/2)b 1 Note that at the area AL
{
γ(AL) ) γ* + 2(31/2)b 1 -
{
Ac A0
0.5
}
c ) (2AL/31/2N)0.5
γ* + 2(31/2)b 1 -
(11)
}
c(γ* + 2(31/2)b) (2(31/2)bc)
) 0 (12)
Hence Π(AL) ) γ*, the surface tension of the system without particles. This corresponds to the knee of the curve in Figure 4a and arises physically from the requirement that the force per unit length due to particle-particle interactions equals the surface tension γ* of the uncovered interface, as discussed above.
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Figure 11. Surface pressure as a function of area for the model presented in the text. Full line is from eq 9a using the parameters b ) 92 mN m-1, Ac ) 44.6 cm2, γ* ) 47.8 mN m-1 for A0 g AL, and Π(A0) ) γ* for A0 < AL. The circles are experimental results from the data set shown in Figure 4a.
For A0 < AL, e ) 0 is no longer a minimum of F, and we consider e * 0. Combining eqs 6 and 8 we obtain
(
e′ ) (
AL2 - A02 8A0/31/2
)
0.5
(13)
which can be shown to correspond to a minimum in F. With this value of e
A(e′) ) 2
( )( ) ( )( 2A0 31/2
0.5
31/2A0 + e′2 8
2A0 31/2
0.5
We have therefore (for e * 0, A0 < AL), A(e′) ) AL. The interfacial area and free energy are both constant with respect to changes in the in-plane surface area A0 for all A0 e AL. As A0 decreases, the interface “bends” (e increases) so that A(e′) ) AL and maintains the balance between the particle-particle forces and the surface tension forces in the corrugated surface. As a result the surface pressure Π for A0 e AL is constant and equal to γ*. It has been mentioned earlier that the assumption of a Hookean force between particles at short separation leads to a surface equation of state (eq 11) that well reproduces the shape of the experimental Π-A curve. This is illustrated in Figure 11 which shows the result of applying the model to the experimental data displayed in Figure 4a for monolayers of 2.6 µm diameter particles at the octane-water interface. As seen, good agreement is obtained by suitable choice of the constants b and c (via Ac) in the collapse region. 5. Conclusions We have demonstrated that particle arrays under lateral pressure at liquid-liquid interfaces collapse by buckling when the particle-particle forces per unit length in the interface are equal to the surface tension (γ*) of the liquid interface, i.e., when Π ) γ*. The collapse pressure Πcol is not a function of the contact angle that the particles make with the fluid interface. The observed behavior of particles in the micrometer size range is similar to that found19 by molecular dynamics simulations for nanoparticulates, and this suggests that the buckling collapse may be a general feature of particle arrays.
0.5
)
31/2A0 AL2 - A02 + 2 2A /31/2 0
)
0.5
) AL (14)
Acknowledgment. The authors gratefully acknowledge the award of EPSRC Grants GR/L50020 (to R.A. and J.H.C.) and GR/L51997 (to N.Q.) for the execution of the work described in this paper. LA000060I
