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Structure and Dynamics of Particle Monolayers at a Liquid-Liquid Interface Subjected to Extensional Flow Edward J. Stancik,† Martin J. O. Widenbrant,† Alex T. Laschitsch,† Jan Vermant,‡ and Gerald G. Fuller*,† Department of Chemical Engineering, Stanford University, Stanford, California 94305, and Department of Chemical Engineering, K.U. Leuven, 3001 Leuven, Belgium Received December 10, 2001. In Final Form: February 11, 2002 Monodisperse spherical polystyrene particles were suspended at the interface between decane and water, and then subjected to extensional flow. Their lattice structure was observed to pass from a hexagonal array at rest, through a liquidlike state as flow was first applied, and finally to a new semi-ordered, anisotropic state during steady flow. This semi-ordered state was shown to be oriented and stretched in the flow direction relative to the original hexagonal structure. The influence of interfacial concentration and extensional rate on particle dynamics is discussed.
Introduction The tendency of colloidal particles to become trapped at interfaces, and their behavior once there, has led to their use in a wide variety of systems including drug delivery, stabilization of foams and emulsions, and development of unique consumer products, to name a few. As their use has become more commonplace, an interest in their study has grown, and various publications1-6describe the structures obtained by the confinement of these particles to two dimensions. The character of these structures has been observed to range from liquid to crystalline1,2 for densely populated interfaces, to having soap or froth structures3 at lower concentrations. The great variety of structures displayed by these particles at an interface has been shown to be dependent on their physical properties (surface charge, wettability, etc.), concentration, and the two phases between which they reside. Studies of particles adsorbed to the air-water interface have found that addition of an electrolyte can induce aggregation.4,5 The attractive forces have been attributed to various sources, ranging from van der Waals to capillary forces,3-6 but are still under debate. A complex dependence of interparticle potential on separation, evidenced by local minima at multiple length scales displayed by particles organized into ring and chain structures, has also been found.7 Another area of study has been that of particles adsorbed to the interface between an organic phase (such as octane) and an aqueous phase which may contain some concentration of electrolyte.8,9The current consensus is that, for particles with ionizable surface groups at the interface between an aqueous solution and a low dielectric material, * To whom correspondence should be addressed. Email: ggf@ stanford.edu. † Department of Chemical Engineering, Stanford University. ‡ Department of Chemical Engineering, K. U. Leuven. (1) Pieranski, P. Phys. Rev. Lett. 1980, 45, 569. (2) Aveyard, R.; Clint, J. H.; Nees, D. Colloid Polym. Sci. 2000, 278, 155. (3) Ruiz-Garcia, J.; Gamez-Corrales, R.; Ivlev, B. I. Phys. Rev. E 1998, 58, 660. (4) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. A 1992, 46, 2045. (5) Williams, D. F.; Berg, J. C. J. Colloid Interf. Sci. 1991, 52, 218. (6) Stamou, D.; Duschl, C.; Johannsmann, D. Phys. Rev. E 2000, 62, 5263. (7) Laschitsch, A.; Vermant, J.; Widenbrant, M. J. O.; Stancik, E. J.; Fuller, G. G. (to be published).
the dominant repulsive force is due to dipole-dipole interactions. The mechanism for creation of the dipole involves dissociation of the surface groups in contact with the aqueous phase such that the bottom portion of the spheres becomes charged, followed by attraction from a solution of counterions to these surface groups to create an electrical double layer. This system, consisting of the negatively charged surface groups and the positively charged double layer, gives rise to a gradient-in-charge perpendicular to the interface, thus creating a dipole. The strength of this dipole depends, in part, on the vertical position of the particles with respect to the interface. While at the air-water interface, polystyrene particles have a contact angle with the aqueous phase on the order of 30 degrees,8 and as such, rest low in the water where their charge can be effectively screened by counterions. Additionally, when the particles are resting at this level, counterions will be present well above the plane of symmetry of the spheres. This situation will serve to lessen the vertical gradient-in-charge, and thus weaken the dipole. One can imagine that if the spheres were lowered further and became entirely submerged, they would be completely surrounded by counterions, thus eliminating any dipole at all. In contrast, at the octane-water interface, the particles experience a contact angle on the order of 75 degrees,8 with the result that they rest higher with a greater portion of their surface excluded from adsorption by counterions. The net effect is that the magnitude of the dipole is increased, and that the dipole is positioned such that it can act, without screening by counterions, through the oil phase.1 Although much has been done to understand the structures that are exhibited by particles at an interface, their dynamics have not been widely studied. Recent reports include studies of interfacial particles and foams subjected to shear7,10 and particles compressed in a Langmuir trough.9 This study focuses on subjecting monodisperse, charged polystyrene particles adsorbed to the decane-water interface to extensional flow. Order transitions induced by the applied flow are reported and the behavior of the particles discussed. Such understand(8) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969. (9) Aveyard, R.; Clint, J. H.; Nees, D.; Quirke, N. Langmuir 2000, 16, 8820. (10) el Kader, A. A.; Earnshaw, J. C. Phys. Rev. Lett. 1999, 82, 2610.
10.1021/la015723q CCC: $22.00 © 2002 American Chemical Society Published on Web 05/02/2002
Particle Monolayers Subjected to Extensional Flow
Figure 1. Extensional flow induced by a four-roll mill showing the (a) extension and (b) compression axes, and (c) the stagnation point.
ing is applicable to practical situations ranging from particles inhaled to deliver medicine through the lungs, to those used to stabilize the interface between shampoo and conditioner in a bottle.
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Figure 2. Particles at interfacial coverages of (a) 5%, (b) 10%, and (c) 30% shown (1) at rest, (2) under steady-state extensional flow, and (3) Fourier transformed while under steady-state extensional flow. Note that the extensional axis runs to the upper left and to the lower right, oriented at 45 degrees, in these images.
Experimental Section Polystyrene spheres (Interfacial Dynamics Corporation) with a mean diameter of 3.0 µm and a surface charge density of 7.5 µC/m2 were acquired in a surfactant-free, deionized water dispersion and subsequently diluted in deionized water (specific resistance 18 MΩ-cm, Millipore) and isopropyl alcohol. The resultant working dispersion contained particles suspended at a concentration of 5.6 × 108 particles/ml in a 20% isopropyl alcohol solution. The isopropyl alcohol aids in the spreading of the particles at the interface. The system was prepared in a clean polystyrene dish by adding an aqueous solution of 10 mM NaCl and subsequently covering its surface with decane (Fisher). A four-roll mill apparatus,11 which essentially consists of four rotating cylinders that can be inserted axially into an interface to create an extensional flow field (Figure 1), was then inserted into the interface. After the working solution was injected at the interface and the particles allowed to come to equilibrium, particle flow behavior was observed using a Nikon Microphot SA microscope centered on the stagnation point of the extensional flow field.
Figure 3. Fourier transforms of images of particles at 5% interfacial coverage (a) at rest, (b) under steady-state extensional flow, (c) during the transition from forward to reverse flow, and (d) under reverse flow.
One can introduce a tensor,
Results and Discussion A series of images is shown in Figure 2 for interfacial particle coverages ranging from 5 to 30% detailing their structure at rest and during steady-state extensional flow at a rate of 0.21 s-1. Although order is present under both conditions, the structure of the particles does change upon the application of flow. One can note that the structure becomes elongated in the direction of extension as applied to the interface by the four-roll mill. The Fourier transforms of the structures under steady flow also make clear that the degree to which the original ordering is perturbed is reduced for systems at higher interfacial concentrations. Generation of the Fourier transforms (Figure 3) of the images of particles under four conditions (at rest, steadystate flow, between forward and reverse flow, and steadystate reversed flow) confirms the above observations and indicates that the structure passes through the original hexagonal lattice during the transition from forward to reverse flow. (11) Higdon, J. J. L. Phys. Fluids A 1993, 5, 274.
d)
[
]
(1)
where x and y indicate the Cartesian coordinates of the primary hexagonal vertexes in the Fourier transform images and the brackets indicate averages, to characterize these images. Note that the origin is located at the center of each. By determining the angle at which the coordinate system must be rotated to transform d into a diagonal matrix, one can solve for the angle,
θ)
(
)
-2 1 tan-1 2
(2)
at which the structure is oriented relative to the unperturbed structure. This is true since diagonalizing d requires that equals zero in the new coordinate system. This requirement is realized by the unperturbed hexagon, in which each vertex is reflected by another across one of the coordinate axes, thus setting it as the reference for the angle. Additionally, solving for the
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Figure 5. An illustration showing how rows of particles shifting in alternate lattice directions give the impression of triangleshaped groups of particles moving in the flow direction in systems with higher interfacial concentrations. The system is shown (a) at rest, and as groups of particles shift in the (b) first, (c) second, and (d) third directions along the lattice. Figure 4. (a) Orientation of particle structure at 5% interfacial coverage by extensional flow. (b) Dimensionless anisotropy induced in particle structure by extensional flow. The times at which flow is (A) started, (B) reversed, and (C) stopped are indicated. The difference in magnitude of the anisotropy developed in systems at 5% and 30% coverage is also shown. Note that the anisotropy has been scaled by the mean square spacing of the points in the Fourier transformed images to make it dimensionless.
eigenvalues of the position tensor d gives
λ( )
x 2 + 42 ( 2 2
(3)
The difference, (λ+ - λ-) is a measure of the anisotropy of the Fourier transform image. Figure 4 shows how these measures of the particle structure develop over time as a system with 5% interfacial coverage is subjected to extension at a rate of 0.42 s-1. Note that the orientation of the Fourier transform approaches the rotation of 45° expected for this flow geometry for both forward and reverse flow. Also note that the anisotropy of the Fourier transform develops after the application of flow, is temporarily reduced during the transition between forward and reverse flow, and finally decays as flow is stopped. This behavior further indicates that the structure has been elongated along the extensional axis as a result of flow, and that it passes through its original hexagonal structure during the transition between forward and reverse flow. A final feature of the Fourier transform is that it allows one to note that the positional order is reduced somewhat during steady-state flow relative to particles at rest, as indicated by a reduced distinctness of individual points in the image. Of interest is the affect of interfacial concentration on particle behavior. At 30% coverage, the structure shows a rotational transition similar to that observed for the less concentrated situations, but is more resistant to induced anisotropy due to the increased magnitude of interparticle repulsions brought about by closer packing. This feature is illustrated in Figure 4b. There are, additionally, some remarkable differences in the particle dynamics observed. For one, relaxation processes appear
much slower at higher interfacial concentrations. In that regime, as a dislocation in the ordered array at rest relaxes, it appears that the particles force their way through an energy barrier, created apparently by the repulsions of the surrounding particles. The manifestation of this phenomenon during steady-state flow is even more dramatic. As flow is applied, dislocations in the lattice form, and create lines along which rows of particles move to relax. These relaxations occur along alternate lattice directions giving the impression of triangle-shaped groups of particles shifting in the flow direction. An illustration depicting this behavior is shown in Figure 5. Essentially, in the concentrated regime, the particles are forced to move by the applied flow before they have time to relax to new lattice positions. As a result, the relaxation that is observed in the system is that of groups of particles shifting together, while the particles within each group maintain their lattice positions relative to each other. This is in contrast to the behavior of particles experiencing flow at low interfacial concentrations, under which they move as individuals in the flow direction, maintaining order by spending more time in lattice positions. These observations suggest that similar behavior would be displayed by particle systems at an equal Peclet number, which is a measure of the relaxation time scale relative to the characteristic flow time scale. A quantitative analysis of the surface structure is obtained by evaluating two correlation functions for the particle system.12 The first of these is a translational correlation function,12 g(r), which describes the average of the local particle density, n(r), relative to average particle density, n, as a function of distance, r, from a reference particle.
g(r) )
n
(4)
The second is an orientational correlation function,12 g6(r), which describes the relationship of a complex order parameter, ψ6j, of a reference particle with that of particles (12) Gray, J. J.; Klein, D. H.; Korgel, B. A.; Bonnecaze, R. T. Langmuir 2001, 17, 2317.
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Figure 6. Dimensionless translational correlation function at 30% interfacial coverage (A) at rest, and at extensional rates of (B) 0.21 s-1 and (C) 0.42 s-1. Functions shifted for clarity.
Figure 8. Semilog plots showing the dimensionless orientational correlation function (a) at 10% and (b) at 30% interfacial coverage showing the system both (A) at rest and (B) under steady-state extensional flow. Figure 7. Semilog plot showing the dimensionless orientational correlation function at 30% interfacial coverage (A) at rest, and at extensional rates of (B) 0.21 s-1 and (C) 0.42 s-1.
at a distance, r. Note that the star refers to the complex conjugate.
g6(r) )
g(r)
(5)
The complex order parameter measures six-fold symmetry:
ψ6(j) )
1 z
∑k exp(6iφjk)
(6)
where φjk is the angle of the line connecting the index particle j and each of the z closest neighbors, k. It achieves a maximum of unity when the index particle has six closest neighbors each separated by 60°. Figure 6 shows g(r) during steady-state flow for the system with an interfacial particle coverage of 30% as the extensional rate is varied from 0 to 0.42 s-1. Note that the distance from the particle, r, has been scaled by particle size, R. One can see that the function exhibits strong, regularly spaced peaks for the system at rest and similar, but dampened oscillations, for the system during steadystate flow. This indicates that the length scale of translational ordering is reduced as flow is applied to the system. Figure 7 shows g6(r) for the same set of conditions. When the particles are at rest, the function remains near unity, indicating a crystalline hexagonal lattice.12,13During steady state flow, however, the magnitude of the function begins to decay with radial position. At lower extensional rates, the function appears to decay linearly on a log plot, indicating power law type behavior. This type of behavior has been attributed to a hexatic structure intermediate between a crystalline lattice and a liquid state.12 At higher extensional rates, a sharp decrease in orientational correlation function can be seen and indicates a more (13) Halperin, B. I.; Nelson, D. R. Phys. Rev. Lett. 1978, 41, 121.
liquidlike state.12,13 One can see that as extensional rate is increased, the magnitudes of the forces provided by the flow overcome the ability of the dipole-dipole forces to maintain the original order in the system. This balance of forces is further demonstrated in Figure 8 which shows that g6(r) is less influenced by a given extensional rate at higher interfacial concentrations. One can note that the function shows a larger drop in magnitude as flow is applied to the less concentrated system. As expected, increasing the magnitude of the forces between the particles reduces the degree to which the original hexagonal lattice is modified by flow. Conclusions It has been demonstrated that extensional flow induces an order transition in particles organized on a hexagonal lattice at the decane-water interface. The resultant structures were oriented and elongated along the extensional axis. An increase in interfacial particle concentration, and thus in the strength of repulsions between the particles, increased the visually observed relaxation time of the system. As such, particles subjected to flow while at higher interfacial concentrations displayed more cooperative movement in their translation through the flow field. The net effect was that a given extensional flow rate became less effective at altering the original hexagonal ordering of the system as the interfacial concentration was increased. Finally, the range of flow strengths over which the dipole-dipole interactions could maintain order in the system was shown to be finite. Acknowledgment. The authors thank R. T. Bonnecaze and Emily Gray for their input on the correlation functions used in this analysis. We also gratefully acknowledge NSF-CTS for the funding of this research. LA015723Q
