J. Phys. Chem. C 2008, 112, 945-950
945
Low-Gradient Magnetophoresis through Field-Induced Reversible Aggregation G. De Las Cuevas,† J. Faraudo,*,‡ and J. Camacho† Departament de Fı´sica, UniVersitat Autonoma de Barcelona and Institut de Ciencia dels Materials de Barcelona ICMAB-CSIC, Campus de la UAB, E-08193 Bellaterra, Spain ReceiVed: July 15, 2007; In Final Form: October 26, 2007
Recent experiments (Yavuz, C. T. et al., Science 2006, 314, 964) show the possibility of low gradient magnetophoretic separation of superparamagnetic nanoparticles in aqueous solution, a process with broad potentially important applications ranging from biomedicine to environmental waste and pollutants removal. Here, we show that the key to low gradient magnetophoresis is the existence of a cooperative mechanism (reversible aggregation) which fuels the magnetophoresis process. The interplay between the different factors determining low gradient magnetophoresis (magnetization of particles, size, ...) is consistently described by a magnetic analogous to the Bjerrum length concept. This concept allows us to formulate a simple criterion predicting the onset of low gradient magnetophoresis separation as a function of the sample properties (e.g., minimum particle radius). These predictions are in agreement with experimental observations. The kinetics of the process depends not only on the properties of the particles but also on concentration. The observed separation times are orders of magnitude shorter than the predictions of present models based on the approximation of noninteracting particles. The separation times of samples with different concentrations and different particles can be described with a unique curve depending on the magnetic Bjerrum length and the concentration.
I. Introduction Magnetic nanoparticles are emerging as key ingredients in the development of new functional materials designed for specific applications.1 The reason for this interest is in the peculiar magnetic behavior known as superparamagnetism exhibited by nanometer sized grains of materials such as iron oxides.2 At zero field, the macroscopic magnetization is zero, and in the presence of an external field, large saturation values of the magnetization can be obtained, hence the term superparamagnetism. The magnetic microspheres employed in typical applications are made of superparamagnetic nanoparticles inserted in a matrix of nonmagnetic material (such as polystyrene or silica) and functionalized by a layer of adsorbed molecules (surfactants, for example). In biomedical applications, a functionalization with appropriate biocompatible molecules (phospholipids, inmunospecfic agents, ...) allows binding to specific targets (red blood cells, lung cancer cells, bacteria, ...) or to deliver drugs in a controlled way.3 In environmental applications, magnetic microspheres are tailored to adsorb target solutes or pollutants.4,5 The essential step in all of these different applications is the removal of the magnetic particles (plus adsorbed biomaterial or pollutants) using inhomogeneous magnetic fields (magnetophoresis).6 Traditionally, this removal is obtained using the socalled high-gradient magnetic separation technique (HGMS), based on the flow of the dispersion containing the particles through a column with high magnetic gradients generated with a packed bed of magnetically susceptible wires (diameter ∼ 50 µm) placed inside an electromagnet.5-8 This technique, initially developed for magnetic clays, can be successfully employed to * Corresponding author. † Universitat Autonoma de Barcelona. ‡ Institut de Ciencia de Materials de Barcelona.
capture functionalized magnetic microspheres with sizes larger than ∼10 nm.5 The physics underlying these magnetophoretic separation processes is relatively well-understood, and it is based on the fact that, in addition to hydrodynamic forces, the particles in the dispersion experience a magnetic force given by:
4 B Fm ) πR3 (M B ‚∇ B )B B0 3
(1)
where M B is the magnetization (per unit volume) of the particle b) is the inhomogeneous magnetic field created by the and B B0(r HGMS column.4,5,7 Theoretical predictions, using typical values for fluid flow and geometry, viscosity, and magnetic properties confirm that very high magnetic gradients are needed in these magnetophoretic processes (typically 104 T/m or even higher).5,6 Surprisingly, recent experiments demonstrate the possibility of magnetic separation of Fe3O4 nanocrystals of diameter 16 nm using low magnetic field gradients (∼20 T/m) in a column separator (see Figure 1 of ref 4). These results cannot be understood in terms of the simple model of noninteracting particles employed in the case of HGMS. A simple estimation of the separation times for noninteracting nanoparticles can be obtained by noting that the magnetic force over a nanosphere b (eq 1) is exactly balanced by the viscous drag B Fd ) 6πRηV (with η ) 10-3 Pa s for water), so we obtain the magnetophoretic velocity:
b V)
2R2 (M B ‚∇ B )B B0 9η
(2)
Assuming that the magnetization has reached the saturation value Msat ≈ 4.7 × 105 A/m,4 we obtain an upper bound for the magnetophoretic velocity of V ∼ 1.5 × 10-7 m/s, several orders of magnitude smaller than the magnetophoretic velocity observed in ref 4. This result suggests that a cooperative
10.1021/jp0755286 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/08/2008
946 J. Phys. Chem. C, Vol. 112, No. 4, 2008 phenomenon such as aggregate or clustering formation (which invalidates the hypothesis of noninteracting particles) is at work. Although clustering induced by a homogeneous magnetic field has been extensively studied (see for example the early works of refs 9-11 or the more recent work in refs 12,13), the case of clustering in inhomogeneous fields was not considered because of the lack of appropriate experimental techniques. Recent experimental studies show that the kinetics of clustering in inhomogeneous weak fields can be studied by using Raman scattering techniques,14,15 but the relation between clustering and magnetophoresis was not analyzed in these works. The evidence discussed so far seems to point toward a close connection between the possibility of LGMS and the formation of field-induced aggregates. If this hypothesis is correct, the kinetics of the magnetophoresis process must be highly dependent on the concentration of the dispersion. The analysis of this dependence for different types of particles may provide essential information on how to predict the kinetics of magnetophoresis processes, a question which is still open. However, at the present time, there are no published studies showing the dependence of the LGMS process on the concentration of the dispersion. Also, there is no theoretical model or equation in the literature allowing a quantitative or even a qualitative prediction of the kinetics of a LGMS process as a function of the properties of the dispersion. In this work, we have conducted a series of new LGMS experiments for a wide range of concentrations (from 0.01 to 10 g/L) and two different particle types. We provide a rationalization of both our new results and the results of other authors in terms of a reversible field-induced aggregation mechanism. We are able to predict the minimum size of particles which can be separated or extracted from a dispersion using LGMS, and we also obtain an estimation of the dependence of the separation time with the particles properties. II. Experimental Section A. Materials. In the experiments reported here, we employ aqueous dispersions of Estapor(R) superparamagnetic microspheres from Merck (references M1-030/40 and M1-020/50, Merck Chimie SAS, France16). They are made of ferrite grains uniformly distributed in a polystyrene matrix (41% and 55% of magnetic content in mass, respectively). The densities of the particles are about 1.1 g/cm3. The diameters of the particles as specified by the fabricant are 0.41 µm (M1-030/40) and 0.2 µm (M1-020/50). These diameters are in agreement with particle size measurements performed with a Malvern Zetasizer Nano ZS. The particles are stabilized by anionic surfactant SDS, with typical ζ potentials between -60 and -70 mV depending on the sample as measured by using a Malvern Zetasizer Nano ZS. All of the particles considered here are superparamagnetic: the magnetization of the particles is completelly reversible; it is zero in the absence of external field, proportional to B0 at low field strengths, and reaches a saturation magnetization Ms for external fields about ≈ 0.1 T. According to the fabricant, the ferrite grains embedded in the microspheres have a saturation magnetization per unit mass about 100 Am2/kg. The differences of magnetic content for each particle type yield different saturation magnetization per unit volume: Ms ≈ 4.5 × 104 A/m for the M1-030/40 particles and Ms ≈ 6 × 104 A/m for the M1-020/50 particles. B. Qualitative Observations of Aggregation in an Inhomogeneous Field. Prior to the quantitative magnetophoresis experiments, we have conducted a series of qualitative observations of magnetophoresis with a microscope, in order to be able
De Las Cuevas et al.
Figure 1. Optical micrograph of a solution of concentration 1 g/L of Estapor(R) M1-030/40 particles under a bar magnet at times (a) 0 s, (b) 120 s, (c) 240 s, and (d) 360 s after placing the magnet. The scale bar in (a) is valid from (a) to (d). Linear aggregates form in the direction of the local field (vertical) and move in the direction of the magnetic field gradient (toward the left), as indicated by white arrows. As time goes, aggregates collide laterally and thicker aggregates are formed (see the movies directly recorded from the microscope17).
to relate the observed macroscopic magnetophoretic behavior of the dispersion with microscopic behaviors of the individual particles. First of all, we have placed a sample with a 1 g/L concentration of M1-030/40 Estapor microspheres under an optical microscope. We have applied a nonuniform magnetic field, generated by a bar magnet. Different observations with different magnet orientations were recorded.17 The main observations can be summarized as follows. After approaching the magnet to the sample, elongated aggregates of particles are formed, which move in the direction of the field gradient (Figure 1). The aggregates have an elongated shape with their long axis in the direction of the local magnetic field, in a way reminiscent of the well-known aggregation of particles in chains in homogeneous magnetic fields.10,12,13 These elongated aggregates move in the direction of the gradient of the magnetic field (see eq 1) which in principle has an arbitrary orientation with the direction of the field itself and hence with the orientation of the major axis of the aggregates. In the case shown in Figure 1, the aggregates move in the direction perpendicular to the long axis of the aggregates. The velocity depends on the size of the aggregates, with the larger aggregates moving faster. Smaller aggregates traveling with slow drift velocities are rapidly captured by larger aggregates, speeding up aggregation. After removal of the magnetic field, all aggregates are dissolved, and we recover the initial dispersed state. C. Magnetophoresis Experiments. The magnetophoresis setup employed in our experiments is the SEPMAG LAB325 2042 apparatus (commercially available from SEPMAG Technologies18). It essentially consists of a cylindrical cavity containing a permanent magnetic field with a uniform gradient of ∼30 T/m pointing toward the walls of the cylindrical vessel. The magnetophoresis experiment is performed by simply placing a bottle of radius 1.5 cm containing 25 mL of aqueous solution inside the SEPMAG cylindrical cavity. In a successful magnetophoresis experiment, the initially brown dispersion becomes transparent in a few minutes, eventually reaching a transparent final state with all particles close to the walls of the bottle. An sketch of the SEPMAG apparatus is shown in Figure 2. In this figure, we also indicate the behavior of the particles during the
Low Gradient Magnetophoresis
Figure 2. Sketch of the behavior of superparamagnetic particles during low gradient magnetic separation in the quadrupole field of a SEPMAG. The particles are assumed to form elongate aggregates with the magnetic dipoles pointing in the direction of the local magnetic field. The magnetic force Fm points in the direction of the magnetic gradient and is balanced by the viscous drag Fd.
Figure 3. Sketch of the method employed to measure the opacity of the dispersion during the magnetophoresis process. The light incides from above and reaches the photoresistance (situated at the bottom of the bottle) after dispersion by the sample. (a) Initially, particles are uniformly distributed in the bottle containing the dispersion, and the opacity of the sample is high. (b) After some time, particles move toward the walls of the container leaving a clean circle behind so that the opacity of the sample decreases.
magnetophoretic experiments. As we have seen in the observations made in section II.B, the particles form elongated aggregates following the local magnetic field. These elongated aggregates move in the direction of the magnetic gradient, which is the radial direction (toward the walls of the container), as shown in Figure 2. A quantitative measurement of the kinetics of the magnetophoresis process is obtained measuring the opacity of the dispersion (i.e., the quantity of light from an external source which is transmitted across the dispersion). A photoresistance (LDR) is placed under the cylindrical cavity containing the solution, which is illuminated from above (see Figure 3). As the intensity of the light detected in the LDR increases, the voltage V measured in the LDR decreases. In this way, changes in the opacity of the sample are measured by changes in V. During the magnetophoresis experiment, we have recorded the evolution of the voltage V(t) from its initial value V(0) until it reaches a minimum, constant value Vmin, corresponding to a transparent sample (all magnetic particles removed from the illuminated zone). Finally, the voltage measured during the magnetophoresis experiment V(t) is converted into a normalized opacity θ by the definition θ(t) ) (V(t) - Vmin)/V(0). The results
J. Phys. Chem. C, Vol. 112, No. 4, 2008 947 are shown in Figure 4 for different concentrations of M1-020/ 50 and M1-030/40 Estapor(R) microspheres. At all measured concentrations (10, 1, 0.1, and 0.01 g/L) of M1-020/50 and M1-030/40 Estapor(R) microspheres, we observe a migration of the magnetic particles toward the walls of the container immediately after introduction of the dispersion in the SEPMAG. As shown in Figure 4, total separation is achieved between less than a minute and about 3 min depending on the concentration. As expected, the magnetophoresis process is reversible: when the bottle is placed outside the SEPMAG, a gentle shake recovers the solution to the initial state. Also, we have checked that subsequent repetitions of the magnetophoretic experiments provide identical results. The results clearly show that the kinetics of the process depends on the concentration of particles. For example, in the case of a 10 g/L solution of M1-030/40 particles, the opacity decays to 10% of its initial value in ts ≈ 32 s. At 0.01 g/L concentration, the magnetophoresis process is slower, and opacity decays to 10% of its initial value in ts ≈176 s. The role of aggregation in the process kinetics can be appreciated by noting that eq 2 predicts, for saturated M1-030/40 particles, a magnetophoretic velocity V ≈ 1.2 × 10-5 m/s, which for a bottle radius of 1.5 cm gives ts ≈ 1250 s. Therefore, experimental results reveal a much faster magnetophoretic process than expected from the noninteracting particles model, with a magnetophoretic velocity increasing with concentration. A more detailed analysis of these experimental results is done in the next section. Here, we would like to remark that the fast separation times obtained in our magnetophoresis experiments and their significant dependence with concentration unambiguously reveal that particle aggregation plays a decisive role. III. Results and Discussion A. Criterion for Low Gradient Magnetophoresis Separation. In the previous section, we have successfully extracted dispersed particles employing a LGMS. However, separation of small particles using magnetic gradients is not always possible.6 Let us remark that the vast majority of the previous studies use high magnetic gradients (HGMS), and only recently, LGMS systems have been developed and studied.4 To the best of our knowledge, there is no theoretical model available predicting under which conditions LGMS can be succesfully obtained. Our objective in this subsection is to rationalize the available observations from the point of view of a field-induced reversible aggregation. We will also propose a testable prediction for the conditions under which LGMS is feasible. As far as we know, the only available LGMS results reported by other authors are the studies of Ibarz et al.19 and Yavuz et al.4 In these studies, the authors employ superparamagnetic particles in order to obtain a high magnetic response from the particles and facilitate the success of LGMS separation. Ibarz et al.19 have attemped to extract 4 nm maghemite nanoparticles (synthesized according to ref 20) using the same SEPMAG LAB325 2042 apparatus that we have employed in section II. However, they report no magnetic separation under time scales of several days. In their pioneering work of LGMS separation, Yavuz et al.4 report separation of pure magnetite particles of 16 nm diameter in water after a few minutes in a 23 T/m batch separator (unfortunately, the exact separation time was not reported). These results and our results from section II are summarized in Table 1. The four experiments summarized in Table 1 encompass a wide range of sizes and saturation magnetizations. In order to formulate the desired criterion for LGMS separation, let us remark that, according to our observations
948 J. Phys. Chem. C, Vol. 112, No. 4, 2008
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Figure 4. Measured opacity as a function of time for water dispersions of particles at different concentrations during magnetophoresis experiments performed in a 30 T/m gradient using a SEPMAG LAB325 2042 apparatus. (a) Estapor(R) M1-030/40 particles and (b) Estapor(R) M1-020/50 particles.
TABLE 1: Summary of LGMS Results Reported by Us and by Other Authors under Different Conditionsa 2R
Ms
gradient
separation times
4 nm 16 nm 200 nm 410 nm
2.7 × 103 A/m 4.7 × 105 A/m 6 × 104 A/m 4.5 × 104 A/m
30 T/m 23 T/m 30 T/m 30 T/m
no separation few minutes 1. In other words, LGMS is possible when the magnetic Bjerrum length is larger than the size of the particles, λB > 2R. In the opposite case, λB < 2R, aggregate formation is not possible since, for two particles in close contact, the thermal energy is larger than the magnetic attraction (the driving force for aggregate formation). The criterion for LGMS based on the magnetic Bjerrum length eq 5 is illustrated in Figure 5 for the different types of particles considered in Table 1. The results (Figure 5) show that our criterion λB > 2R for LGMS separation is in agreement
Low Gradient Magnetophoresis
Figure 5. Log-log plot of the scaled Bjerrum length versus particle diameter. Symbols correspond to a number of particle types: Estapor(R) M1-030/40, Estapor(R) M1-020/50, the 16 nm magnetite particles used by Yavuz et al.4, and the 4 nm maghemite nanoparticles used by Millan et al.20 Dashed lines describe the Bjerrum lengths for particles with the same magnetization per unit volume as the ones given above but with varying sizes. The solid line (λB/2R ) 1) separates the region where aggregates can form so that particles can be rapidly separated by using low magnetic gradients, from the region where aggregates cannot form (shaded region). The cross of a dashed line with the solid line provides the minimum diameter for LGMS to apply for that kind of particle. Arrows indicate the critical diameters for two particles: Estapor(R) M1-030/40 (dmin ≈ 40 nm) and the magnetite particles of ref 4 (dmin ≈ 7 nm).
with the available experimental evidence (Table 1). In the case of the 4 nm nanoparticles synthesized by Millan et al.,20 we obtain λB ∼ 0.04R. We thus predict no low gradient magnetophoretic separation in this case, in agreement with experiments (see Table 1 and Figure 5). In the case of pure magnetite particles as those employed in the work of Yavuz et al.,4 we predict separation under LGMS in the case of particle sizes larger than 7 nm (see arrow in Figure 5), in agreement with their experimental observations. In the case of the Estapor(R) microspheres employed in our magnetophoresis experiments reported in section II, we have λB ≈ 5 µm for M1-030/40 particles and λB ≈ 2 µm for the M1-020/50 particles. In both cases, λB > 2R, so we predict aggregate formation and low gradient magnetic separation. As shown in Figure 5, our experiments correspond to system parameters well inside the region at which low gradient magnetophoresis is possible. B. Magnetophoretic Separation Times. The magnetic Bjerrum concept introduced in the previous section is also useful in the analysis of the magnetophoresis kinetics. In order to characterize the separation of the different dispersions of M1020/50 and M1-030/40 particles, we have computed the time ts corresponding to a decay of the initial opacity of the dispersion to 10% of its initial value. According to the aggregation mechanism proposed in the previous section, we can expect that short separation times ts correspond to situations with strong particle-particle magnetic interaction (the driving force for aggregation). In this model, we can understand the dependence of the separation times ts with concentration by taking into account that particles are separated a typical distance d which depends on the concentration c. Hence, at a given concentration, the relative strength of the particle-particle magnetic interaction can be measured by comparing λB with the typical particleparticle separation d in the suspension. If d/λB , 1, the attractive magnetic energy for particles in the dispersion (typically
J. Phys. Chem. C, Vol. 112, No. 4, 2008 949
Figure 6. Log-log plot of the separation time as a function of the scaled average interparticle distance. Symbols correspond to experimental values for two particle types, M1-030/40 (circles) and M1020/50 particles (squares), at different concentrations. The solid line is the power law fit of the data.
separated a distance d) is greater than the thermal energy, so aggregates are rapidly built-up with a deterministic mechanism. As d/λB increases, aggregation formation is slowed down, and magnetophoretic separation times will increase. We can estimate d as d ≈ (4πR3Fp/3c)1/3 (where Fp ) 1.1 g/cm3 is the density of a microsphere), so we have:
[
]
3kBT Fp d ) λB 2µ0M2 c
1/3
1 R
(6)
The behavior of the separation time ts as a function of d/λB is shown in Figure 6. The experimental results for the different systems considered (M1-020/50 and M1-030/40 particles at different concentrations) show a behavior consistent with a power law of the form:
ts ) t0
[] d λB
R
(7)
where the fit with experimental results gives t0 ) 66 ( 6 s and R ) 0.73 ( 0.10. Of course, eq 7 is valid only when low gradient magnetophoresis is possible; that is, λB > 2R. A very interesting prediction of eqs 6 and 7 is that, for a given material, separation times can be reduced by either increasing the concentration or increasing the particle size. In other words, according to eqs 6 and 7, a reduction in particle size can be compensated by an increase of concentration to obtain the desired separation time. Equation 7 was obtained by a direct fit of our experimental results. However, we can check its validity to predict the separation times for particles made of materials different from those employed in our experiments. For example, if we apply eq 7 to 16 nm diameter magnetite particles (λB ≈ 37 nm) in typical concentrations of the order of 1 g/L (d ≈ 200 nm), we predict separation times of the order of a few minutes. Remarkably, this result is in agreement with the qualitative experiments reported by Yavuz et al.4 despite the fact that their experimental set up is similar but not identical to ours. We regard this agreement as an indication that eq 7 may capture
950 J. Phys. Chem. C, Vol. 112, No. 4, 2008 the essential mechanism underlying low gradient magnetophoresis processes. Finally, let us stress that the separation times empirically obtained and described by eq 7 are orders of magnitude shorter than the ones predicted for noninteracting particles (these are obtained by dividing the radius of the sample container by the velocity, eq 2). As mentioned in section II.C, eq 2 predicts for saturated M1-030/40 particles ts ≈ 1250 s. In the case of the experiments of Yavuz et al.4 the noninteracting model (eq 2) predicts separation times of orders of a day. Our eq 7 predicts for these experiments separation times of the order of few minutes, in agreement with the observations. This difference is clear evidence of the dramatic effect of cooperative effects in the kinetics of magnetic separation processes.
De Las Cuevas et al. Acknowledgment. We thank Lluı´s Martı´nez and Ce´sar Rebollo, from SEPMAG Technologies, for providing us with the SEPMAG LAB325 2042 apparatus, their invaluable help during the experiments, sharing of data, and for endless interesting and useful discussions. We thank Fabrice Sultan from Merck for providing us the samples of Estapor microspheres. We thank Fernando Palacio for sharing unpublished experimental data. We also thank Matgas 2000 A.I.E. for the use of the Nanotechnology Lab (Nanosizer and Zetasizer mesurements). This work was supported by the Spanish CICYT Grant FIS2006-12296-C02-01 and the Catalan Direccio´ General de Recerca Grant 2005-SGR-000-87. The research of J.F. at ICMAB-CSIC is supported by the Spanish Government (Programa Nacional de Potenciacio´n de Recursos Humanos, Plan Nacional de Investigacio´n Cientı´fica 2004-2007).
IV. Conclusions In this paper, we have analyzed the underlying physical mechanisms of a new and fast separation process of magnetic dispersions, the so-called low gradient magnetophoresis process (LGMS). This process is based on the application of low magnetic gradients (smaller than 100 T/m) to a suspension of functionalized superparamagnetic particles. Our experiments show that the underlying mechanism for LGMS separation is the formation of elongated aggregates which move in the direction of the magnetic gradient. We show that the possibility of LGMS separation of dispersions strongly depends on a nontrivial interplay between the properties of the particles, which can be characterized by the magnetic Bjerrum length λB (see eq 5). We have shown that the condition for LGMS separation is λB > 2R, a condition which allows us to predict the minimum size of particles of a given material which can undergo low gradient magnetic separation. Our predictions are in agreement with our experimental results and the results obtained by other authors. Our experiments also show that the kinetics of the LGMS process depends not only on the properties of the particles but also on concentration. We propose that the kinetics of the LGMS process can be characterized by the ratio between the magnetic Bjerrum length and the typical separation between particles, which depends on the concentration of the dispersion (see eq 6). The separation times for different particles and concentrations collapse in a single curve which can be described by a simple power law (see eq 7). In summary, we have provided a characterization of the driving mechanisms behind the low gradient magnetophoresis separation process based in a simple but useful concept (the magnetic Bjerrum length). We hope that the results obtained here will be useful in the technological and biomedical applications of low gradient magnetophoresis.
References and Notes (1) Berkovsky, B. M.; Medvedev, V. F.; Krakov, M. S. Magnetic Fluids, Engineering Applications; Oxford University: Oxford, 1993. (2) Bean, C. P.; Livingstone, J. D. J. Appl. Phys. 1959, 40, 120S. (3) Pankhurst, Q. A.; Connolly, J.; Jones, S. K.; Dobson, J. J. Phys. D: Appl. Phys. 2003, 36, R167. (4) Yavuz, C. T.; Mayo, J. T.; Yu, W. W.; Prakash, A.; Falkner, J. C.; Yean, S.; Cong, L.; Shipley, H. J.; Kan, A.; Tomson, M.; Natelson, D.; Colvin, V. L. Science 2006, 314, 964. (5) Moeser, G. D.; Roach, K. A.; Green, W. H.; Hatton, T. A.; Laibinis, P. E. AIChE J. 2004, 50, 2835. (6) Friedman, G.; Yellen, B. Curr. Opin. Colloid Interface Sci. 2005, 10, 158. (7) Hatch, G. P.; Stelter, R. E. J. Magn. Magn. Mater. 2001, 225, 262. (8) Gerber, R.; Takayasu, M.; Friedlaender, F. J. IEEE Trans. Magn. 1983, 19, 2115. (9) Gast, A. P.; Zukoski, C. F. AdV. Colloid Interface Sci. 1989, 30, 153. (10) Promislow, J. H. E.; Gast, A. P.; Fermigier, M. J. Chem. Phys. 1994, 102, 5492. (11) Liu, J.; Lawrence, E. M.; Wu, A.; Ivey, M. L.; Flores, G. A.; Javier, K.; Bibette, J.; Richard, J. Phys. ReV. Lett. 1995, 74, 2828. (12) Martinez-Pedrero, F.; Tirado-Miranda, M.; Schmitt, A.; CallejasFernandez, J. J. Chem. Phys. 2006, 125, 084706. (13) Martinez-Pedrero, F.; Tirado-Miranda, M.; Schmitt, A.; CallejasFernandez, J. Phys. ReV. E 2007, 76, 011405. (14) Weber, J. E.; Goni, A. R.; Thomsen, C. J. J. Magn. Magn. Mater. 2004, 277, 96. (15) Heinrich, D.; Goni, A. R.; Thomsen, C. J. J. Chem. Phys. 2007, 126, 124701. (16) Estapor Microspheres n 12. A technical newsletter published by Merck Chimie SAS: France. (17) The movies are freely available at our web page http://telemaco.uab.es/magnetophoresis. (18) Sepmag. SEPMAG Technologies, www.sepmag.eu (19) Ibarz, G.; Millan, A.; Palacio, F., unpublished. (20) Millan, A.; Palacio, F.; Falqui, A.; Snoeck, E.; Serin, V.; Bhattacharjee, A.; Ksenofontov, V.; Gutlich, P.; Gilbert, I. Acta Materialia 2007, 55, 2201. (21) Tsouris, C.; Scott, T. C. J. Colloid Interface Sci. 1995, 171, 319. (22) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Dover: Mineola, NY, 1959.
