Langmuir 2002, 18, 9587-9593
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Drag Forces on a Stationary Particle in Flowing Two-Dimensional Ordered Particle Monolayers: Simulation and Measurement Using Optical Tweezers R. Aveyard, B. P. Binks, J. H. Clint, P. D. I. Fletcher, B. Neumann,† and V. N. Paunov* Surfactant & Colloid Group, Department of Chemistry, University of Hull, Hull, HU6 7RX, United Kingdom
J. Annesley, S. W. Botchway, A. W. Parker, and A. D. Ward CLRC, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon., OX11 0QX, United Kingdom
A. N. Burgess ICI PLC, P.O. Box 90, Wilton Centre, Middlesborough, Cleveland, TS90 8JE, United Kingdom Received June 21, 2002. In Final Form: August 30, 2002 Using a laser tweezer method, we have studied the drag force on a charged polystyrene latex particle adsorbed at the oil-water interface when an ordered 2D monolayer of identical particles is flowed around it. The drag force, measured as a function of the flow velocity and the lattice spacing in the particle monolayer, contains contributions from long-range electrostatic forces between the particles at the liquid interface. The interpretation of the experimental results is based on a model in which the repulsion arises primarily from the presence of a very small net electric charge at the particle-oil interface. This charge corresponds to a fractional ionization of the sulfate groups present at the particle surface exposed to the oil phase which is estimated from an independent experiment (see Aveyard, R.; Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Horozov, T. S.; Neumann, B.; Paunov, V. N.; Annesley, J.; Botchway, S. W.; Nees, D.; Parker, A. W.; Ward, A. D.; Burgess, A. N. Phys. Rev. Lett. 2002, 88, 246102-1) to be approximately 3.3 × 10-4 for the latex sample used. The experimentally determined drag forces are in reasonable agreement with results from computer simulations for a range of values of the velocity and the lattice spacing of the particle monolayer.
1. Introduction The properties of particles adsorbed at liquid surfaces are important in many practical applications including, for example, foam inhibition, the stability and processing of food colloids, paints, and solid-stabilized emulsions. Using micron sized colloidal particles, it is possible to directly visualize the configuration of particle monolayers and to probe the lateral interactions between adsorbed particles.1 It has been found recently that very long-range repulsive forces operate between charged polymer particles, adsorbed at the interface between water and a nonpolar oil.1 This interaction results in the formation of highly ordered monolayers even at a relatively low surface concentration of particles. An image of a monolayer of spherical polystyrene latex particles at the interface between water and a mixture of decane and undecane is shown in Figure 1. The particles, which carry ionizable sulfate groups at their surfaces, have a diameter of 2.7 µm. In this study, we have used a laser optical tweezer method to determine the drag force on an adsorbed particle when an ordered array of identical particles is flowed around it. * To whom correspondence should be addressed. Tel: 44(0) 1482 465660. Fax: 44(0) 1482 466410. E-mail: V.N.Paunov@ hull.ac.uk. † Present address: Colloid Science Group, School of Chemistry, University of Bristol, Bristol, BS8 1TS, U.K. (1) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969.
Figure 1. Optical micrograph of a monolayer of polystyrene spheres, diameter of 2.7 µm, at the interface between water and a mixture of decane and undecane; the average distance between particle centers is 5.8 ( 0.2 µm.
Stillinger2 and later Pieranski3 have shown that repulsion between point charges, located exactly at the interface between water and a medium of low relative permittivity (air or oil), is enhanced over that between (2) Stillinger, F. H. J. Chem. Phys. 1961, 35, 1584. (3) Pieranski, P. Phys. Rev. Lett. 1980, 45, 569.
10.1021/la020578h CCC: $22.00 © 2002 American Chemical Society Published on Web 10/30/2002
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similar charged particles in bulk water. The origin of this effect is due to the asymmetry of the electrical double layer at that part of the particle surface immersed in the aqueous phase, which gives rise to an effective dipole normal to the water-air surface. Theoretical treatments of this phenomenon have subsequently been given by Hurd,4 Earnshaw,5 and Goulding and Hansen.6 The key relevant features of these theoretical treatments are that (i) the long-range interparticle force scales with separation to the power -4 and (ii) the magnitude of the force is inversely proportional to the salt concentration in the aqueous phase. However, for monolayers of charged polystyrene latex particles at an alkane-water interface, it has been observed that the long-range order persists even at high electrolyte concentrations and hence an alternative physical model for the repulsive force has recently been proposed.1,7 In this case, the repulsive force has been attributed to a small amount of residual surface electric charge present at the particle-oil interface resulting from a small fraction of the surface sulfate groups at the particle-oil surfaces being dissociated. Such dissociation may be stabilized by water trapped in microcavities on the rough particle surface.1 In this situation, the interacting charges are effectively shifted from the interface into the oil phase. As shown in the Appendix of ref 1, the pair potential interaction for such particles can be effectively represented as direct charge-charge interaction through the oil phase together with the corresponding imagecharge interaction. In accordance with the experimental findings,1 the lateral repulsion between particles at the oil-water interface resulting from residual particle charges on the particle-oil surfaces is predicted to be insensitive to the presence of even high concentrations of electrolyte in the aqueous phase. Recently, we reported7 the first measurements using optical tweezers of the force between a pair of charged polystyrene spheres at the interface between water and oil (a mixture of alkanes) as a function of the interparticle separation. The experimental results indicate that the force decays with the fourth power of the distance between the particles, similar to the theoretical predictions of Stillinger,2 Pieranski,3 Hurd,4 and ref 1. However, it was also found7 that the interparticle force is insensitive to the electrolyte concentration, which is in contradiction with the predictions of refs 2-4 but in agreement with the model based on the presence of a residual surface charge at the interface between the particle and the oil. Stancik et al.8 have recently investigated the dynamic response and structural distortion of ordered particle monolayers (similar to those discussed here) when subjected to extensional flow. The lattice structure was observed to pass from a hexagonal array (as in Figure 1 here) through a liquid-like state as flow is first applied and finally to a semi-ordered state during steady flow. Following the successful measurement7 of the static force versus separation for a pair of adsorbed particles, in this paper we present experimental results on the drag force on a stationary (trapped) particle when an ordered array of identical particles is flowed around it. The drag force is measured using optical tweezers for different flow (4) Hurd, A. J. J. Phys. A: Math. Gen. 1985, 18, L1055. (5) Earnshaw, J. C. J. Phys. D: Appl. Phys. 1986, 19, 1863. (6) Goulding, D.; Hansen, J.-P. Mol. Phys. 1998, 95, 649. (7) Aveyard, R.; Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Horozov, T. S.; Neumann, B.; Paunov, V. N.; Annesley, J.; Botchway, S. W.; Nees, D.; Parker, A. W.; Ward, A. D.; Burgess, A. N. Phys. Rev. Lett. 2002, 88, 246102-1. (8) Stancik, E. J.; Widenbrant, M. J. O.; Laschitsch, A. T.; Vermant, J.; Fuller, G. G. Langmuir 2002, 18, 4372.
Aveyard et al.
Figure 2. Scheme of the circular trough, external diameter of 25 mm, showing the particle array adsorbed and the laser trap light path. Pinning of the oil-water interface at the steelPTFE junction enabled the formation of a flat interface.
velocities and particle separations in the monolayer, corresponding to different strengths of the interaction between the optically trapped particle and its neighbors in the monolayer. We also perform a computer simulation of the same experiment by using a version of the molecular dynamics method in 2D space and compare the calculated particle drag forces with the experimentally measured values. The paper is organized as follows. In section 2, we describe the experimental setup, the force calibration procedure, and the measurements of the particle drag force. In section 3, we present a computer simulation method for calculation of the particle drag force in particle monolayers by taking into account the interparticle interaction. Section 4 includes the comparison between the experimental and the simulation results and a discussion about the physics of the interparticle interaction. 2. Experimental Section Materials. Monodisperse spherical polystyrene particles (2.7 µm diameter) with sulfate groups located at the surface were obtained from Interfacial Dynamics Corp. (U.S.A.) as a 10 wt % aqueous dispersion. The surface charge density of the particles is quoted by the manufacturer to be σs ) 8.9 µC cm-2. Decane (>99%) and undecane (99%) were purchased from Sigma/Aldrich. These alkanes were columned through chromatographic alumina before use to remove polar components. The spreading solvent, propan-2-ol of analytical grade, was supplied by Sigma/Aldrich and was used as received. Highly purified water of Milli-Q quality was used. Laser Tweezer Apparatus and Particle Monolayer Formation. A detailed description of the laser tweezer apparatus, developed at the Central Laser Facility, CLRC, can be found in ref 9. The cell containing the spread particle monolayer (Figure 2) was mounted on the stage of a Leica DM IRB inverted microscope. A variable power (0-1 W) continuous wave Nd:YAG laser, operating at a wavelength of 1064 nm, entered the cell from below and was focused by the microscope objective (40× magnification; numerical aperture, 0.5; working distance, 2 mm). The optical trap was formed at the waist of the beam focus located in the plane of the oil-water interface. The cell containing the particle monolayer could be moved laterally (i.e. parallel to the plane of the oil-water interface) with a set position/velocity profile using piezoelectric translators under computer control. A CCD camera mounted on the microscope and coupled with fast frame acquisition was used to image the particle monolayers. Particle monolayers were formed by spreading 1 µL of a latex dispersion in propan-2-ol at the interface between ∼290 µL of water and an alkane mixture consisting of 70.5% v/v decane and 29.5% v/v undecane. The dilution of the latex dispersion (typically 300-fold by volume) was adjusted as necessary to obtain different particle separations within the spread monolayers. The volume (9) Nees, D.; Botchway, S. W.; Towrie, M.; Ward, A. D.; Parker, A. W.; Burgess, A.; Central Laser Facility, Rutherford Appleton Laboratory, Annual Report 209 (1999/2000).
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of water used was set in order to form the oil-water interface in the cell at a level where the interface pinned to the poly(tetrafluoroethylene) (PTFE)-metal junction (Figure 2) and produced a flat oil-water interface. The alkane mixture composition was set such that the viscosities of the water and alkane phases were equal. All measurements were conducted in a thermostated room at 21.0 ( 0.3 °C. Drag Force Measurements. A single particle, either isolated or within a 2D ordered array of particles, was trapped at a set laser power. We then applied an oscillatory, lateral movement to the cell for which the position profile was a triangular type waveform with total displacement amplitude of 72 µm (larger than the maximum center-to-center particle separation). The maximum cell velocity vmax, and hence that of the liquid plus particle array surrounding the trapped particle, could be varied in the range from 25 to 400 µm s-1 by adjustment of the frequency of the triangular waveform. At fixed vmax, the trapping laser power was then decreased to determine the critical power P* at the point of focus which was just sufficient to hold the particle within the trap against the forces associated with the fluid plus particle array motion. The viscous drag force, Fdrag, experienced by the trapped particle resulting from the cell movement, was calculated from the Stokes equation
Fdrag ) Bv
B ≈ 6πηR
Figure 3. Laser trap calibration plot of critical laser power P* versus vmax for an isolated particle. Calculation of the drag force corresponding to vmax using eq 2.1 gives the relationship between the trap force and laser power.
(2.1)
where v is the velocity of the cell movement, B is the individual particle drag coefficient, η is the dynamic viscosity of water (equal to that of the oil), and R is the particle radius. The expression for the drag coefficient B in eq 2.1 is only approximate since the flow streamlines are additionally distorted due to the presence of the oil-water interface. From symmetry arguments, if the contact angle, θ, of the particle with the oil-water interface is 90° and the viscosity of the oil phase matches that of the water, eq 2.1 is expected to be accurate. However, the observed contact angle, measured through the water phase for similar particles1 (when the oil phase is octane) is 75 ( 5°. Although the problem for the drag coefficient of a particle in a liquid-liquid interface has not been solved yet, the case of particles at a liquid-gas interface has been considered theoretically.10 On the basis of the numerical results in ref 10, we estimated that for θ ) 75°, eq 2.1 underestimates the drag force with a systematic error of less than 10%, similar to the experimental uncertainties in the measured forces. Similar is the magnitude of the effect of the hydrodynamic interaction11 between the particle and its neighbors for the relatively large lattice spacings used in our experiment. That is why we do not account for the multiple hydrodynamic interactions in our interpretation of the experimental data and use only eq 2.1 to calculate the hydrodynamic drag force. Tweezer Calibration Measurements Using Isolated Particles. Calibration measurements of the variation of the critical laser power P* versus the maximal cell velocity vmax for a single isolated particle in a very dilute monolayer were used to obtain the relationship between force F and laser power P. Figure 3 presents a plot of P* against vmax which is seen to be linear with an intercept of zero. By use of eq 2.1, the maximum trapping force (i.e. the force just sufficient to remove the particle from the trap) was found to be equal to 0.165P* where the force is in units of pN and the laser power is the measured value in mW. A correction for randomly oriented drift velocities of the particles at the interface (due to surface convection and/or nonflat interfacial regions) was not applied; such drift velocities were found to be negligibly small (
