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Probing DLVO Forces Using Interparticle Magnetic Forces: Transition from Secondary-Minimum to Primary-Minimum Aggregation Ching-Ju Chin,† Sotira Yiacoumi,*,† and Costas Tsouris*,‡ School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0512, and Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6224 Received October 31, 2000. In Final Form: July 13, 2001 The transition from secondary-minimum to primary-minimum aggregation of superparamagnetic colloidal latex particles is investigated in this study. The magnetic induction needed for this transition is calculated from the extended Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, which includes van der Waals, electrostatic, and magnetic-dipole forces as well as non-DLVO hydrophobic attraction. A chain-dipole model is used to estimate the magnetic-dipole potential. The magnetic induction at which the transition from secondary- to primary-minimum aggregation occurs is experimentally determined from visualization of chain formation and breakup. Experimental and theoretical values of transitional magnetic induction show good agreement for the small particles used in this study. The experimental value of the transitional magnetic induction of relatively large particles in a narrow capillary tube is found to deviate significantly from that predicted by theory. However, the transitional magnetic induction obtained from a wide capillary agrees well with the theoretical value. This behavior indicates that, in the narrow capillary, the magnitude of the repulsive force between the particles decreases.
Introduction Because of their importance in both natural and industrial systems, colloidal interactions between particles have drawn significant attention from scientists and engineers for several decades. The Derjaguin-LandauVerwey-Overbeek (DLVO) theory,1,2 which includes van der Waals and electrostatic forces, was developed to explain the interactions between colloidal particles. Particle interactions may cause a colloidal suspension to become unstable (causing primary- or secondary-minimum aggregation) or stable (resulting in no particle aggregation).3 When no energy barrier is present, two particles collide and aggregate in the primary minimum of their potential energy. Aggregates formed by primary-minimum aggregation of single particles are held together by strong van der Waals forces and do not break apart without the application of strong external forces (e.g., shear stress). When both an energy barrier and a secondary attractive sink exist, two interacting particles may aggregate at the secondary minimum of their potential energy and keep a certain distance between their surfaces. This type of aggregation is known as secondary-minimum aggregation. Aggregation of magnetic particles has been studied for several reasons, including the need for the removal of particles from potable water/wastewater and ultrafine mineral streams4 and the investigation of magnetorheological fluids,5,6 that is, magnetic particles immersed in * To whom correspondence should be addressed: syiacoumi@ ce.gatech.edu or
[email protected]. † Georgia Institute of Technology. ‡ Oak Ridge National Laboratory. (1) Derjaguin, B. V.; Landau, L. D. Acta Physicochim. URSS 1941, 14, 733. (2) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (3) Zeichner, G. R.; Schowalter, W. R. AIChE J. 1977, 23, 243. (4) Wang, Y.; Forssberg, E. Miner. Eng. 1992, 5, 895. (5) Fermigier, M.; Gast, A. P. J. Colloid Interface Sci. 1992, 154, 522. (6) Promislow, J. H. E.; Gast, A. P.; Fermigier, M. J. Chem. Phys. 1995, 102, 5492.
a nonmagnetic liquid. Upon the application of a magnetic field, the induced dipoles quickly result in aggregation and chain formation along the direction of the applied field.7,8 The potential energy between colloidal superparamagnetic particles has been studied based on the extended DLVO theory, which incorporates van der Waals, electrostatic, and magnetic-dipole forces.9 It was found that if the electrostatic force is strong enough to prevent aggregation, the presence of a magnetic force, which is of longer range than the electrostatic and van der Waals forces, causes a secondary minimum in the potential energy. Thus, in this case, aggregation occurs at the secondary minimum. It was also shown that secondaryminimum aggregation is reversible after the removal of the magnetic field, while primary-minimum aggregation is irreversible. Thus, whether particle aggregation occurs at the secondary or primary minimum can be determined by visualizing the chain aggregates as the magnetic field is turned off. Mondain-Monval et al.10,11 used magnetic forces to measure the force-distance profile between submicrometer (11) and low-ionicstrength (0.001 M NaCl) solution to prevent particle aggregation.14 The ζ-potential of the suspension was measured by using a Lazer Zee Meter (model 501; Pen Kem, Inc., Bedford Hills, NY) (12) Relle, S.; Grant, S. B.; Tsouris, C. Physica A 1999, 270, 427. (13) Furst, E. M.; Gast, A. G. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 62, 6912. (14) Chin, C. J.; Yiacoumi, S.; Tsouris, C. J. Colloid Interface Sci. 1998, 206, 532.
Figure 1. SEM images and size distributions of particles used in this study: (a) small particles and (b) large particles. and a ZetaPlus ζ-potential analyzer (Brookhaven Instrument Corp., Holtsville, NY). The magnetic colloidal particle suspension was introduced into glass microrectangular capillary tubes of dimensions 50-mm length × 1-mm width × 50-µm thickness (Wale Apparatus, Inc.,
Probing DLVO Forces Using Interparticle Magnetic Forces Hellertown, PA), which are referred to as “narrow tubes” in this study. Glass capillary tubes of dimensions 50-mm length × 2-mm width × 100-µm thickness from the same manufacturer, referred to here as “wide tubes”, were also used for the large particles. The capillary tube was placed between the poles of a watercooled electromagnet (model 2V1; Applied Magnetic Laboratory, Baltimore, MD). Magnetically induced aggregation of the small particles in narrow tubes and the large particles in wide tubes was conducted at a relatively high magnetic field (0.05-0.53 T). Aggregation of the large particles in narrow tubes, on the other hand, was conducted using the remanent magnetism of the poles of the magnet (0-0.005 T). A gaussmeter (model GM1A; Applied Magnetic Laboratory, Baltimore, MD) was employed to measure the local magnetic induction. After the chains were formed, the capillary tube containing the particle suspension sample was moved from the magnetic field to the observation point, where no magnetic field was present, to examine the breakup of the chains. This procedure was repeated by introducing samples of the same solution into new capillary tubes and placing the tubes in different magnitudes of magnetic field. Chain aggregates were observed by using a long-distance microscope lens (QM-100, 250×; Questar Corp., New Hope, PA) connected to a camera (model TC651E; Burl Security Product, Lancaster, PA). The camera was connected to a Panasonic video camera recorder, a monitor (DOT-X, model 15VM939; Dotronix, Inc., New Brighton, MN), and a thermal printer (video copy processor, model P78u; Mitsubishi Electronic America Inc., Somerset, NJ).
Theory Calculations of the interparticle potential in this study were based on the extended DLVO theory, which is the summation of van der Waals, electrostatic, and magneticdipole forces. The non-DLVO hydrophobic attraction due to the surface groups of the particles was also included. van der Waals and Electrostatic Forces. The equation of London-van der Waals attractive potential between spherical particles, Vvdw, was formulated by Hamaker.15 Ho and Higuchi16 included the electromagnetic retardation of van der Waals interaction between unequalsized spheres for s - 2 , 1. Here, s is the dimensionless distance between two spheres: s ) 2rij/(ai + aj); rij is the center-to-center distance between the particles; and ai and aj are the radii of particles i and j, respectively. Two approaches were used to estimate the electrostatic potential, Vel. The Derjaguin approximation (for small separation)17 and the linear superposition approximation (for large separation)18 were used. Detailed descriptions of van der Waals and double-layer interactions are provided in previous publications.14,19 Magnetic-Dipole Potential. Pair-Dipole Model. Because the magnetite particles within the polystyrene are very small (1-20 nm) and randomly distributed, the composite particles are treated as single-domain structures for which the dipole model is applicable.5 Therefore, the volumetric magnetic susceptibility provided by the manufacturer is used to represent the effective volumetric magnetic susceptibility. Magnetic dipoles are attractive along the field direction and repulsive normal to the field direction. The magnetic potential, Vmag, is written as follows:20
Vmag )
4πB2a3i χia3j χj [1 - 3 cos2(R - θ)] 9µ0r3ij
(1)
where B is the magnitude of magnetic induction; χi and (15) Hamaker, H. C. Physica (Amsterdam) 1937, 4, 1058. (16) Ho, N. F. H.; Higuchi, W. I. J. Pharm. Sci. 1968, 436, 57. (17) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Trans. Faraday Soc. 1966, 18, 1638.
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χj are the magnetic susceptibilities of particles i and j, respectively; µ0 is the permeability of free space; R is the angle between the direction of the vertical axis and the direction of the magnetic field; and θ is the angle between the direction of the centerline between particles and the vertical axis. Hence, the interacting colloidal particles form chain aggregates that are aligned along the field direction. The magnetic-dipole interaction is determined for an isolated pair of particles and is referred to in this study as pair-dipole model. Chain-Dipole Model. Particles trapped in a chain also experience magnetic induction from other particles in the same chain. Harpavat21 analyzed the magnetic forces at various joints of a chain of spherical beads by assuming that the field variation within a particle is small as compared to the average field. Paranjpe and Elrod22 considered bead-to-bead interactions in a manner that allowed any bead configuration and derived a simpler model. Zhang and Widom23 examined multipole moments of polarized particles in electrorheological and magnetorheological fluids. Martin and Anderson24 found that it was necessary to include the interactions from other particles in the chain to improve their model of electrorheological fluids. In this study, since the magnetic gradient is small (Table 1), it is assumed that particles in a chain experience the same magnetic induction from the externally applied magnetic field. Therefore, the magnetic induction at particle i (Bi) in a chain of N particles is the summation of the applied induction at particle i (B) and the induction from other particles in the chain (Bij):22 N
Bi ) B +
Bij ∑ j)1
(2a)
j*i
Bij )
[
]
µ0 3rij(mj‚rij) mj - 3 4π r5ij rij
(2b)
4πa3i χi mi ) B 3µ0 i
(2c)
where rij is the center-to-center vector between particles and mi is the moment of particle i in a chain. The magnetic potential energy between any two particles in a chain, Vmag,ik, is referred to here as the chain-dipole model and can be calculated as follows:
Vmag,ik ) mi‚Bk
(3)
Harpavat’s21 analysis showed that the center joint of a chain has the strongest force. Therefore, in this study, we used the total potential between the two center particles in a chain to determine whether the aggregation is in primary or secondary minimum. Hydrophobic Attraction. A hydrophobic surface is inert to water and cannot bind water molecules via (18) Bell, G. M.; Levine, S.; McCartney, L. N. J. J. Colloid Interface Sci. 1970, 33, 335. (19) Yiacoumi, S.; Rountree, D. A.; Tsouris, C. J. Colloid Interface Sci. 1996, 184, 477. (20) Chikazumi, S. Physics of Magnetism; John Wiley and Sons: New York, 1986. (21) Harpavat, G. IEEE Trans. Magn. 1974, 10, 919. (22) Paranjpe, R. S.; Elrod, H. G. J. Appl. Phys. 1986, 60, 418. (23) Zhang, H.; Widom, M. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1995, 51, 2099. (24) Martin, J. E.; Anderson, R. A. J. Chem. Phys. 1996, 102, 4814.
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Figure 2. Magnetic particles and chains in wide capillary tubes after removal of the magnetic field; particle diameter ) 4.5 µm (large particles), ζ-potential ) -50 mV, volumetric magnetic susceptibility ) 0.24, and ionic strength ) 0.001 M NaCl. (a) 0 T; (b) 0.004 T (right after the magnetic field was removed); (c) 0.0021 T (after 2 min); (d) 0.004 T (after 3 min); and (e) 0.005 T (after 2 min).
hydrogen or ionic bonds.25 The water confined in the gap between two hydrophobic surfaces is unable to form clusters and results in an increase of the free energy relative to that of bulk water.26 Therefore, an attraction between hydrophobic surfaces arises as a consequence of water molecules migrating from the gap to the bulk water. The hydrophobic attraction, Vhp, in the range of 0-10 nm is given by25,27
Vhp ) -2πλaiγSLe-(D/λ)
(4a)
γSL ) γSV - γLV cos φ
(4b)
where λ is the characteristic exponential decay wavelength (1-2 nm); D is the shortest surface-to-surface distance; γ is the interfacial tension; subscripts S, L, and V indicate solid, liquid, and vapor, respectively; and φ is the contact angle. Errors and uncertainties in the calculations of this work may come from (i) estimations of various parameters such as the Hamaker constant and (ii) polydispersity in particle size and magnetic susceptibility. An average value of particle size and magnetic susceptibility was used in the calculations, which gives a single value for the transitional magnetic induction, while experimentally the transitional magnetic induction ranged approximately (7% from an average value. Results and Discussion The transition from secondary- to primary-minimum aggregation under a magnetic field is experimentally determined from visualization of chain breakup. Calculations of the potential energy are performed to determine the theoretical transitional magnetic induction and to compare it with the experimental values. Experimental Transitional Magnetic Induction. Upon the application of a magnetic field, particles align and form chains along the direction of the field. If they are not aggregated in the primary minimum of the potential energy, these chains break up after being removed from the magnetic field.9 Figure 2 provides photographs of particles and chains that were formed under different values of magnetic induction applied for a 10-min period. The photographs, taken within 3 min after the field was removed, show only the large particles used in this study. (25) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, 1991. (26) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. Particle Deposition & Aggregation: Measurement, Modelling and Simulation; Butterworth-Heinemann: Oxford, 1995. (27) Shaw, D. J. Introduction to Colloid and Surface Chemistry, 4th ed; Butterworth-Heinemann: Oxford, 1991.
In the absence of a magnetic force, particles were uniformly dispersed and did not aggregate (Figure 2a). Particles remained in chains when the magnetic field was just removed (Figure 2b). Also shown are particles and chains in magnetic fields of 0.0021, 0.004, and 0.005 T. Chains formed at magnetic inductions lower than 0.004 T (e.g., 0.0021 T; Figure 2c) broke into single particles after the field was removed, which provides evidence that particles aggregate into the secondary minimum at magnetic fields lower than 0.004 T. Chains formed at 0.004 T still broke into pieces; however, these chains broke into doublets rather than single particles (Figure 2d). These doublets aggregated in the primary minimum of their potential energy. The end particles broke away from a chain one by one until a doublet was formed from the core particles, implying that the core particles aggregated in the primary minimum while the end particles aggregated in the secondary minimum of the potential energy. This observation also indicates that the magnetic-dipole force is strong at the center and weak at the ends of a chain, which agrees with Harpavat’s simulations.21 When the magnetic induction for chain formation is increased to 0.005 T, the chains remain intact after the field is removed. Some breakup of chains, however, occurs with time at internal nodes (Figure 2e). Once shorter chains are formed from the breakup of longer ones, no further particle breakup is observed, indicating that particles are in the primary minimum of their potential energy. This observation suggests that the breakup of the longer chains is not caused by secondary-minimum aggregation. Instead, the following explanation is offered. Once the magnetic field is removed, the chains deform and move to the bottom (as a result of gravity), which introduces fluid motion and shear stress. This shear stress is believed to be the major reason for the breakup of chains. Chains formed at 0.005 T changed their orientation from vertical to horizontal, and some became entangled after the magnetic force was removed. As Figure 2 indicates, with increasing magnetic induction, the particle suspension that was stable in the absence of a magnetic force can be shifted to secondary-minimum and then to primaryminimum aggregation. The transitional magnetic induction, defined as that at which secondary-minimum aggregation becomes primaryminimum aggregation, was experimentally determined from observations of the breakup of chains formed at different values of magnetic induction as previously described in the Materials and Methods. The series of pictures in Figure 2 represents an example of experimental determination of the transitional magnetic induction. As shown in Table 1, the experimental transitional magnetic induction in narrow tubes was 0.46-0.52 T for the small
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Figure 4. Effect of the number of particles in a chain on the transitional magnetic induction: particle diameter ) 0.23 µm (small particles), Hamaker constant ) 2.23E-20 J, volumetric magnetic susceptibility ) 0.19, ionic strength ) 0.001 M, and ζ-potential ) -49.0 mV.
Figure 3. Dimensionless total potential energy between two particles under different values of magnetic induction: Hamaker constant ) 1.44E-20 J, particle diameter ) 0.23 µm (small particles), ionic strength ) 0.001 M, ζ-potential ) -49.0 mV, and volumetric magnetic susceptibility ) 0.19.
particles and 0.003-0.004 T for the large particles used in our experiments, while that of the large particles in wide capillary tubes was 0.0610-0.0717 T. The wide range reported for the magnetic induction is due to (i) the experimental approach and (ii) the polydispersity in size and magnetic susceptibility of the particles. Theoretical Transitional Magnetic Induction. Figure 3 shows the interparticle potential of small particles used in this work under different values of magnetic induction. The dimensionless total potential, defined as the total potential, VT, divided by kT (k is the Boltzmann constant and T is absolute temperature), is plotted vs the dimensionless separation distance between two particles, which is defined as the surface-to-surface distance over the particle radius (i.e., s - 2). The Hamaker constant for magnetite is reported as 5.0E-20 J28 and 3.2E-21 to 9.0E-21 J for polystyrene.29 In this study, we used the average value of 6.0E-21 J for polystyrene, and the Hamaker constants for both particles were determined as weight averages of magnetite and polystyrene. When the magnetic field is not present (0 T), the interparticle potential is positive, which means that the interaction is repulsive. As the magnetic induction is increased, the location of the secondary minimum shifts to shorter distances, and the magnitude of the energy barrier decreases. Although the potential is negative at all distances, at 0.4 T magnetic induction, an energy barrier still exists that has to be overcome for secondary-minimum aggregation to become primary-minimum aggregation. After the magnetic induction increases to 0.4820 T, the energy barrier disappears. The theoretical transitional (28) Tsouris, C.; Scott, T. C. J. Colloid Interface Sci. 1995, 171, 319. (29) Gingell, D.; Parsegian, V. A. J. Colloid Interface Sci. 1973, 44, 456.
magnetic induction is determined as the magnetic induction at which the energy barrier in the interparticle potential disappears. The van der Waals and electrostatic forces in such solution are also presented in Figure 3. It is shown that the change in the total potential is attributed to the change in the magnetic force since the van der Waals and electrostatic forces remain the same. Also, as the magnetic force is increased, it becomes the dominant component of the total potential. Theoretical calculations of potential energy between magnetic particles have shown that the particles can still form secondary-minimum aggregates when the potential energy is lower than -10kT.4,30 In this work, the potential energy goes from repulsive to attractive forming secondary minimum by addition of the magnetic force. Because the particles are superparamagnetic, that is, the particles are only magnetic when they are in a magnetic field, the potential energy goes back from attractive to repulsive after the particles are removed from the magnetic field. As a consequence, the aggregates break up into single particles.9 The number of particles in a chain needed for potential calculations with the chain-dipole model is usually determined by counting the particles breaking away from the end of a chain. The average numbers of particles in a chain were determined to be 16 and 10, respectively, for the small and large particles used in this study. The gaps between consecutive particles in a chain were also assumed to be the same. For the small particles, the theoretical transitional magnetic induction was found to be 0.4820 T using the pair-dipole model (Figure 3) and 0.4512 T using the chain-dipole model. For the large particles, the theoretical transitional magnetic induction was found to be 0.0726 T using the pair-dipole model and 0.0671 T using the chain-dipole model. Thus, by considering the magnetic induction of other magnetic particles in a chain, the magnetic potential energy between two particles becomes stronger and leads to a lower transitional magnetic induction. Effect of Number of Particles on the Theoretical Transitional Magnetic Induction. Figure 4 shows the effect of the number of particles in a chain on the (30) Ebner, A. D.; Ritter, J. A.; Ploehn, H. J. J. Colloid Interface Sci. 2000, 225, 39.
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Figure 5. Effects of particle size and volumetric magnetic susceptibility on transitional magnetic induction: Hamaker constant ) 2.23E-20 J, ionic strength ) 0.001 M, and ζ-potential ) -49.0 mV.
theoretical transitional magnetic induction. For twoparticle chains, the result is the same for either the chaindipole model or the pair-dipole model. As the number of particles in a chain increases, the center particles experience additional induction from other particles in a chain and have stronger dipole moments. Hence, the transitional magnetic induction required to achieve the same magnetic force to overcome the energy barrier is reduced. The transitional magnetic induction decreases dramatically when the number of particles increases from two to four, indicating that the additional induction from particles next to the center ones is very strong and results in a much stronger interaction between the center particles. When the number of particles in a chain becomes greater than four, the transitional magnetic induction decreases slowly. After the number reaches 10, its effect on the transitional magnetic induction is insignificant. Because the distance between the center and the end of the chain becomes very large, the induction from the end particles to those at the center becomes negligible. This result was found to be consistent for different particle sizes and magnetic susceptibilities. Effect of Particle Size and Volumetric Magnetic Susceptibility on the Transitional Magnetic Induction. Computations also showed that the transitional magnetic induction decreases with increasing particle size and magnetic susceptibility (Figure 5). In addition, as the particle size or magnetic susceptibility becomes very small, the transitional magnetic induction increases dramatically. This result is because the magnetic-dipole force between two particles is proportional to the product of the volumes and the volumetric magnetic susceptibility of the particles (eq 1), while the van der Waals and electrostatic forces are only functions of particle radius. Therefore, when the particles become larger, the energy barrier and the magnetic force required to overcome the barrier increase slightly. Because the product of the particles’ volume increases dramatically with increasing particle size, the magnetic induction required to achieve such magnetic force becomes very small. Comparison of Experimental and Theoretical Values of Transitional Magnetic Induction. The theoretical and experimental values of transitional magnetic induction showed better agreement for the small particles than for the large particles in narrow capillary tubes (see Table 1). Note that the theoretical transitional magnetic induction of the small particles was calculated
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Figure 6. Dimensionless total potential energy with and without including the hydrophobic attraction: particle diameter ) 4.5 µm (large particles), Hamaker constant ) 1.44E-20 J, volumetric magnetic susceptibility ) 2.4, ionic strength ) 0.001 M, magnetic induction ) 0.01 T, ζ-potential ) -50 mV, and interfacial energy between water and polystyrene (with glycidyl ether surface group) ) 45.5 mJ/m2.
using the average size, 0.23 µm, of the polydispersed system. Because 60% of the particles in this system are smaller than the average size, they need stronger magnetic induction to overcome the energy barrier. This is the reason that the lower limit of the experimental transitional magnetic induction is slightly higher than the theoretical one obtained from the chain-dipole model. The hydrophobic attraction was included in the potential energy calculation in an effort to explain the discrepancy observed for the large particles. The large particles have hydrophobic glycidyl ether groups on their surface, and the small particles have carboxylic groups on their surface with a negligible hydration effect (Table 1). The surface tension is 72 mJ/m2 for water and 35 mJ/m2 for polystyrene.27 The contact angle measured by the manufacturer is 110°. On the basis of eq 4b, the interfacial tension between polystyrene with glycidyl ether groups and water is determined to be 45.5 mJ/m2. It is demonstrated in Figure 6 that, by including the hydrophobic attraction, an energy barrier still exists even though the hydrophobic attraction does affect the potential energy significantly. Therefore, the hydrophobic attraction alone could not explain the discrepancy observed for the large particles. The size of the large particles is approximately 1 order of magnitude smaller than the distance between the glass walls of the microrectangular capillary tube in which the suspension is immersed, while that of the small particles is 2 orders of magnitude smaller. Therefore, the effect of the negatively charged glass wall should be stronger for the large particles than for the small ones. To decrease the wall effect, similar experiments with large particles were conducted using microrectangular tubes of 100-µm thickness instead of tubes of 50-µm thickness that have been used before. The experimental transitional magnetic induction for the large particles in wide tubes was 0.06100.0717 T, which agrees with the theoretical result shown in Table 1 (0.0671 T). This result provides the evidence that the repulsive force between similarly charged particles in the confined geometry is decreased. Studies31,32 have shown that the electrostatic force between similarly charged particles changes due to confinement by similarly charged glass plates, pushing the energy barrier closer to (31) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1996, 77, 1897. (32) Sader, J. E.; Chan, D. Y. C. J Colloid Interface Sci. 1999, 218, 423.
Probing DLVO Forces Using Interparticle Magnetic Forces
the origin of the potential energy curve. It has also been reported that the confined walls affect not only the electrostatic interactions between pairwise particles but also the hydrodynamic interactions.33 These phenomena in combination with the presence of the glycidyl ether group on the surface of the large particles, which introduces hydrophobic attraction, can dramatically suppress the energy barrier. Conclusions The purpose of this work was to use the magnetic force to manipulate the total force and experimentally probe the DLVO forces between two particles. Experimental determination of colloidal forces has drawn great attention, and various techniques have been innovated, such as optical tweezers, atomic force microscopy, and internal reflection microscopy, to directly provide such information. Our approach is an indirect method and has some advantages and disadvantages as compared to these techniques. It is simpler and provides an average force measurement over a large number of undisturbed particles. The disadvantage is that it can only work for colloidal magnetic particles and allows measurements only of the force barrier between two particles. By employing novel experimental techniques, however, such as the scattering method of Mondain-Monval et al.,10,11 one could obtain the force-distance profile for colloidal magnetic particles. In summary, we studied the transition from secondaryto primary-minimum aggregation using superparamagnetic particle suspensions of small and large particles under magnetic fields. This transition was determined by visualizing the behavior of chain aggregates after they were removed from the magnetic field. In the absence of a magnetic field, the magnetic particle suspensions were originally stable with respect to aggregation. Values of transitional magnetic induction at which the chains did not break into single particles were determined. The transitional magnetic induction for the small particles was much stronger than that for the large particles used in this work because the magnetic-dipole energy is proportional to the volumes of the particles involved. The (33) Dufresne, E. R.; Squires, T. M.; Brenner, M. P.; Grier, D. G. Phys. Rev. Lett. 2000, 85, 3317.
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potential energy was calculated based on the extended DLVO theory to theoretically determine the transitional magnetic induction. A chain-dipole model was formulated to calculate the magnetic-dipole energy by including the magnetic induction from other particles in the chain. The comparison between experimental and theoretical values of transitional magnetic induction gives a good agreement for the small particles. The comparison between experimental and theoretical values of transitional magnetic induction for the large particles in a 100-µm-thick microretangular tube shows good agreement, while that in a 50-µm tube shows a much smaller experimental transitional magnetic induction. This behavior indicates that the repulsive force between particles decreases in the confined geometry. The effect of the number of particles in a chain on the transitional magnetic induction was also investigated. Calculations showed that the additional induction from neighboring particles in a chain decreases the transitional magnetic induction. A large drop in the transitional magnetic induction occurs when the number of particles in a chain increases from two to four. After the number of particles reaches 10, the distance between the center and the end particles is relatively large; therefore, further addition of particles no longer causes reduction of the transitional magnetic induction. By comparing the experimental and theoretical values of transitional magnetic induction, the DLVO forces were probed. The good agreement observed for small particles and large particles in 100-µm tubes demonstrates that the DLVO theory and the existing formulas for interparticle potential calculations are reliable. Further studies are needed to obtain the force-distance profile and to better understand the behavior of relatively large particles in narrow capillary tubes. Acknowledgment. Support for this research was provided by the National Science Foundation through a Career Award (BES-9702356 to S.Y.) and by the Division of Chemical Sciences, Office of Basic Energy Sciences, U.S. Department of Energy under Contract DE-AC0500OR22725 with UT-Battelle, LLC. The authors are thankful to Dr. Marsha Savage for editing the manuscript. LA0015260
