Initial Rate of Flocculation of Magnetic Dispersions in an Applied

3186

Ind. Eng. Chem. Res. 1996, 35, 3186-3194

Initial Rate of Flocculation of Magnetic Dispersions in an Applied Magnetic Field William D. Young and Dennis C. Prieve* Colloids, Polymers & Surfaces Program and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Initial flocculation rates for aqueous polydisperse (diameters between 5 and 23 nm with a mean of 12 nm) maghemite (γ-Fe2O3) particles were determined in situ using small-angle light scattering. In dilute dispersions, we found that the measured rate was 36% higher in the presence of high applied magnetic fields than it was in the absence of any applied field. Fuchs’ model of the kinetics of Brownian flocculation, using on an orientation-averaged magnetic interaction between particles, predicted less than a 2% increase in the flocculation rate at high magnetic fields for monodisperse 12 nm particles. Although the percentage increase was larger for larger particle sizes, even the largest size observed in our maghemite sample led to a predicted increase of less than 10%, which was well below the 36% increase observed. Our estimates suggest that the Brownian rotation of the particles was not fast enough, compared to translation, to achieve an equilibrium distribution of orientations during flocculation. When, instead of orientation averaging for a given separation distance, the maximum magnetic attraction for two dipoles aligned end to end was used, Fuchs’ theory predicted a greater effect of magnetic field than observed, thus indicating the observations were within theoretical bounds. Introduction For dilute hydrosols, the rate of Brownian collisions between ideal noninteracting particles can be calculated from Smoluchowski’s theory. Brownian collision rates for real particles are also affected by hydrodynamic drag and by van der Waals attractions, which are always active, and possibly by other forces such as electrostatic repulsion, steric repulsion (due to adsorbed polymers), or magnetic interactions. Measurements of flocculation kinetics on real dispersions are mainly of interest for what they reveal about the nature of the interparticle forces: rates much larger than Smoluchowski’s prediction imply strong attraction while much smaller rates imply strong repulsion. A model is needed to make any quantitative inference about the interparticle forces from the measured kinetics. The most common model for Brownian particles is Fuchs’ theory, which considers migration in a force field as a transport mechanism which acts in parallel with diffusion. In this model, the potential energy of interaction between any two particles is a function of the distance separating the two particles. Careful observations of rapid flocculation (high ionic strengths) of model latexes reveal rates which are typically about half the Smoluchowski rate. This is in reasonable agreement with the predictions of Fuchs’ theory, when hydrodynamic drag and van der Waals attractions are considered (Russel et al., 1989). Similar comparisons between Fuchs’ theory and experiments when electrostatic repulsion is important usually show profound disagreement (Prieve and Ruckenstein, 1980). This disagreement remains as one of the unsolved problems of colloid science, which will be the subject of a forthcoming paper by Ofoli and Prieve (Ofoli, 1995). In this paper, we are concerned with the flocculation of particles having a permanent magnetic dipole moment. Magnetic particles are interesting as colloids * Author to whom correspondence should be addressed. Phone: 412-268-2247. Fax: 412-268-7139. E-mail: dcprieve@ cmu.edu.

S0888-5885(96)00039-5 CCC: $12.00

because the particle interactions depend on the orientations of the magnetic dipoles, as well as the separation distance between particles, and because the magnetic interaction tends to be much stronger and have longer range than electrostatic or van der Waals forces. The orientation dependence severely complicates modeling: complete consideration of orientation involves diffusion in five dimensions (two angles for each particle, plus the separation distance) rather than the one dimension in Fuchs’ theory. Most theoretical studies of flocculation kinetics have been for weakly magnetic particles which do not possess a dipole in the absence of an applied magnetic field. For such paramagnetic materials, the induced dipole moment is proportional to the applied field strength and is aligned with the applied magnetic field. In this case, the magnetic interaction is always between two parallel dipoles. Svoboda (1982) used Fuchs’ model to predict the effect of applied field strength on the kinetics of flocculation of paramagnetic Brownian particles. As an alternative to Fuchs’ model, Parker et al. (1984) predicted the trajectories during collisions between two non-Brownian paramagnetic particles. Scholten and Tjaden (1980) used an orientationaveraged potential energy to model the magnetic interaction between a pair of superparamagnetic particles (thermally fluctuating magnetic dipole) in zero applied magnetic field. They assumed a Boltzmann distribution of dipole orientations at a given separation distance and averaged the magnetic dipole-dipole interaction accordingly. The applied magnetic field generally increases the attraction by tending to align the dipoles of interacting particles. Chan and Henderson (1984) used a similar approach to obtain an expression for the orientation-averaged potential energy between a pair of spherical ferromagnetic particles (permanent dipole) in an applied magnetic field. They calculated the average magnetic interaction energy as a function of separation distance and concluded that the effect of the applied magnetic field on the potential energy profiles becomes significant only at high field strengths. © 1996 American Chemical Society

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In the theoretical portion of this paper, we substitute the orientation-averaged potential energy profile obtained by Chan and Henderson into Fuchs’ model and compute the effect of an applied magnetic field on the flocculation kinetics under the same conditions as our experimental studies. Our experiments reveal a significantly greater effect of the applied magnetic field than predicted. Most experimental studies of flocculation of magnetic dispersions are microscopic observations of the formation of chains and very large agglomerates (Hayes, 1975; Peterson and Krueger, 1977; Krueger, 1979; Hayes and Hwang, 1977). Some investigators have tried to quantify their observations by measuring the sedimentation rate (Peterson and Krueger, 1977) or the polarization relaxation times in a magnetic field (Hayes and Hwang, 1977), in order to characterize the relative extent of agglomeration of the dispersions. Charles (1988) gives a good review of this subject. Janssen et al. (1990) measured the relative flocculation rates of electrostatically stabilized aqueous γ-Fe2O3 dispersions as a function of the applied magnetic induction by monitoring the light transmission through the sample during flocculation. The flocculation rate increased sharply as the magnetic field was applied for small fields, then leveled off at high field strengths. This trend with field strength parallels the magnetization curve for this material. In this work, we use small-angle light scattering to monitor the initial rate of flocculation of γ-Fe2O3 dispersions as a function of the applied magnetic field strength. Provided the flocs are not too big, scattering at small angles tends to be independent of floc shape and increases in a predictable manner with floc size (Young and Prieve, 1991). Thus small-angle light scattering provides a more quantitative measure of the extent of flocculation than other techniques with which we are familiar. Our experiments reveal a significantly greater effect of the applied magnetic field than predicted from Fuchs’ model using the orientation-averaged potential-energy profile. Comparing simple estimates of the time for Brownian rotation with the time for Brownian translation, we conclude that orientation averaging is probably not appropriate for our dispersions. Theory Predicting the Stability Ratio. To predict the initial flocculation rate of magnetic particles in a dilute dispersion, we first calculate the potential energy profile between a pair of particles, assuming that the particle concentration was low enough to neglect many-body interactions. By extending the DLVO theory to include magnetic interactions, we write the total interaction energy V as the sum of the individual interactions in the system:

V ) Vmag + Velec + Vvdw

(1)

where Vmag, Velec, and Vvdw are the magnetic, electrostatic, and van der Waals interactions, respectively. The Magnetic Interaction. Even for a single ferromagnetic material like the maghemite used in this study, several types of magnetic response can be obtained depending on the particle size and the applied magnetic field. Particles large enough to be seen with the unaided eye generally consist of multiple magnetic domains, which are regions in which the magnetic dipole

of individual iron atoms are pointed the same direction. Adjacent domains have dipole moments which point in different directions. Application of a strong external magnetic field to such a large particle can cause alignment of the moments of all the domains, resulting in the familiar “permanent magnet”. Particles small enough to consist of a single magnetic domain will possess a permanent dipole moment whose magnitude is proportional to the number of iron atoms present or to the volume of the particle. The proportionality constant relating the magnitude of the magnetic moment to the volume of the particle is a material property called the “magnetization”. Although all the iron atoms in a single-domain, uniaxial, ferromagnetic particle have the same dipole moment, thermal agitation can cause the direction of that moment to reverse spontaneously. The probability of all the atoms simultaneously reversing direction decays exponentially with the number of atoms or with the volume of the particle. Consequently, particles smaller than a few nanometers exhibit “superparamagnetism” in which the direction of the magnetic moment (but not its magnitude) fluctuates with time, even if the orientation of the particle itself is held fixed. In what follows, we assume our particles are large enough not to exhibit superparamagnetism, but are small enough to consist of a single magnetic domain. The magnetic potential energy of two magnetic point dipoles of fixed orientations in an applied magnetic field is given (in cgs units) by the following [For a introduction to magnetism and magnetic units, see Rosensweig (1985).]:

U(r) ) -

[

]

1 3 (m1‚r)(m2‚r) - m1‚m2 r3 r2 [m1‚H + m2‚H] (2)

where the vectors mi (i ) 1, 2) are the magnetic dipoles, r is the dipole separation, and H is the external magnetic field. The magnitude of the magnetic dipole, m, is the product of the magnetization and the particle volume, which we assume is the same for both particles. For γ-Fe2O3 particles we used a magnetization of 350 emu/cm3 (Nunes and Yu, 1989). The first term in eq 2 is the dipole-dipole interaction, and the second term is the energy of two isolated dipoles in a magnetic field. Chan and Henderson (1984) calculated an orientationaveraged magnetic interaction for two particles in a magnetic field, Vmag, by assuming an equilibrium orientation distribution of the magnetic dipoles, and averaging the Boltzmann factor over all possible orientations of the two dipoles and the external magnetic field, according to the expressions

[

exp -

]

Vtot(r) 1 ) kT (4π)3

∫ e-U(r)/kT dΩ1 dΩ2 dΩH

(3)

where k is the Boltzmann constant, T is the temperature, and dΩi (i ) 1, 2, H) is an element of solid angle. They report the following results:

[

exp -

]

∞

Vtot(r) kT

)

∑ (-1)k(2j + 1)(2l + j,k,l)0

( )

2

j k l [ιj(y)]2[ιk(3x/2)][ιl(x/2)] (4) 1) 0 0 0 where x ) m2/kTr3, y ) mH/kT, the ι’s are modified

3188 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

spherical Bessel functions, and the Wigner 3-j symbol is given by

(

j k l 0 0 0

)

Cν-jCν-kCν-l

2

)

(5)

(2ν + 1)Cν

and

(2n)!

Cn )

(6)

2

(n!)

where ν ) (j + k + l)/2. The Wigner 3-j symbol is zero if ν < j, k, or l, or if j + k + l is odd. For r f ∞ (x f 0), Chan and Henderson show that (4) simplifies to

[

exp -

] (

)

Vtot(∞) sinh y ) kT y

2

(7)

The total magnetic energy of two dipoles in a magnetic field, Vtot(r), includes both the dipole-dipole interaction energy, Vmag(r), and the energy of two isolated, noninteracting dipoles in an external magnetic field, Vtot(∞). Therefore, if Vtot(r) and Vtot(∞) are known, the magnetic particle-particle interaction is given by

Vmag(r) ) Vtot(r) - Vtot(∞)

(8)

With these equations, Vmag(r) can be calculated for any particle size, magnetization, or applied magnetic field strength. The Electrostatic Interaction. For weakly interacting identical spheres dispersed in a 1-1 electrolyte, the double-layer interaction can be estimated using the linear superposition approximation (Bell et al., 1970):

( )

kT Velec ) 16 e0

2

( )

x

8πne02 kT

(10)

[

(

)]

A 2a2 2a2 r2 - 4a2 + + ln 6 r2 - 4a2 r2 r2

(11)

where A is the Hamaker coefficient. Retardation effects, which could be estimated from Lifshitz continuum theory, tend to weaken the attraction. In the calculations below, we will see that even when retardation

D

[

∞ exp ∫D(u)

(9)

where n is the number density of cations or anions in the bulk. Although (9) is quantitative only for weakly interacting double layers (i.e., κ(r - 2a) > 2), we will use it for obtaining qualitative insights concerning the role of electrostatic interactions at all separations. If quantitative results are desired, more accurate estimates are available (Sader et al., 1995). The van der Waals Interaction. The nonretarded van der Waals attraction between two identical spheres can be estimated by Hamaker’s linear superposition theory (Hunter, 1987):

Vvdw ) -

effects are ignored, van der Waals attraction is insignificant compared to magnetic interactions. Chan and co-workers (1985) calculated the Hamaker constant for R-Fe2O3 in water using the Lifshitz expression for nonretarded van der Waals interaction. They report A ) 5.0 × 10-13 erg. The Hamaker coefficient for γ-Fe2O3 (maghemite) is difficult to estimate accurately because the dielectric spectra are not available. Instead, we assumed that the Hamaker constant for γ-Fe2O3 would be similar to that of R-Fe2O3 (hematite) and used that value. The Stability Ratio. Interparticle forces cause the flocculation rate to differ from that predicted using Smoluchowski’s theory for ideal noninteracting particles. Fuchs computed the effect of interactions on the steady state rate by considering migration in the force field to act in parallel with diffusion (Hunter, 1987). Spielman (1970) modified Fuchs’ expression to include hydrodynamic interactions which are manifested as a position-dependent diffusion coefficient. With these modifications, Fuchs’ model for the stability ratio (the factor by which the flocculation rate is reduced compared to Smoluchowski) is given by

W ) 2a

e0ψs a2 -κ(r-2a) e tanh 4kT r 2

where ψs is the electrostatic surface potential of the particles, e0 is the protonic charge, a is the particle radius, r is the center to center separation distance,  is the electric permittivity of the medium, and κ is the Debye parameter:

κ)

Figure 1. Small-angle light-scattering apparatus. See text for description of components and operation.

]

V(r) dr kT r2

(12)

where V(r) is the potential energy profile. The effect of separation distance on the diffusion coefficient can be estimated by (Honig et al., 1971)

6u2 + 13u + 2 D∞ ) D(u) 6u2 + 4u

(13)

where u ) (r - 2a)/a and D∞ is the diffusion coefficient for a single particle in an unbounded fluid. Experiment Small-Angle Light Scattering Apparatus. We determined the flocculation rate by measuring the scattering intensity of the flocculating dispersions in an applied magnetic field, using small-angle light scattering. Figure 1 is a schematic of the apparatus: Light from a 5 mW HeNe laser (a) intersects the scattering cell (b) which contains the flocculating dispersion. The scattering cell is a rectangular quartz cuvette with a 1 cm path length. The light scattered by the dispersion at a small angle of 2° passes through a pair of round plates with thin annular slits cut in them (d); the positions of the two slit plates relative to the scattering cell determines the scattering angle. A planoconvex lens system (e) focuses the light that passes through the slits onto the photomultiplier tube (f), and the measured scattering intensity is plotted continuously on a chart recorder. The slit plates and photomultiplier tube are enclosed in a black box which reduces the

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amount of stray room light reaching the photomultiplier tube. More detail on the theory and operation of the apparatus is given by Young and Prieve (1991). The Helmholtz coils [(c) in Figure 1] surround the scattering cell and produce a nearly uniform magnetic field over a region between the coils about 40% as large as the coil radius (Trout, 1988), which is 4 cm for our coils. As pictured in Figure 1, the coils generate a magnetic field which is perpendicular to the incident light beam. If the coils are rotated 90° about a vertical axis, they produce a magnetic field which is parallel to the incident beam. During flocculation experiments, we typically generate magnetic fields up to about 500 oersteds (Oe). By comparison, the Earth’s magnetic field is about 0.4 Oe. At the start of a flocculation experiment, the power supply for the coils was turned on and the current adjusted to give the desired magnetic field strength for that experiment. Then the particle dispersion was mixed with a salt solution (to destabilize the particles) in the scattering cell in the amounts required to give the desired final concentrations. The final particle volume fraction for most of the experiments was on the order of 2 × 10-7. The scattering cell was then immediately inserted into the cell holder in the optical assembly, and the scattering intensity was measured at 2° and plotted continuously on the chart recorder during the experiment. The rate of increase of scattering intensity was easily determined from the chart recorder tracing. The initial flocculation rate of particles that are small compared to the wavelength of light is related to the change in scattering intensity during flocculation by (Young and Prieve, 1991)

1 dn1 1 dI(θ,t) )I0 dt n0 dt

(14)

where I(θ,t) is the measured scattering intensity at a small scattering angle θ and time t, I0 is the initial scattering intensity (at t ) 0), n0 is the initial particle concentration, n1 is the concentration of primary particles at time t, and (n0 - n1)/2 is the concentration of doublets at small times, before larger flocs have formed. The quantity -dn1/dt is the rate of disappearance of singlets from the system and may be identified as the initial rate of flocculation. Equation 14 indicates that the initial slope of the intensity versus time curve for a flocculating dispersion is proportional to the flocculation rate. Our small-angle light-scattering apparatus is a modification of that developed by Lips and Willis (1973). The shorter path length of the spectrophotometer cuvette allows continuous, in situ measurement of the scattering intensity of a flocculating dispersion. In the original version of the apparatus, particles flocculated in a separate container (a beaker, for instance), and small samples of the flocculating dispersion were removed and diluted at discrete time intervals to measure the scattering intensity. Continuous measurement of the scattering intensity is more convenient, provides better estimates of the flocculation rate, and avoids the uncertainty inherent in sampling and diluting the dispersions. Particle Synthesis and Characterization. For the experimental study, we synthesized the magnetic γ-Fe2O3 particles using an aqueous precipitation technique developed by Massart (1981), in which a 2:1

Figure 2. Measured particle size distribution for the γ-Fe2O3 particles used in the experiments. The mean is 12.1 nm with a standard deviation of 3.7 nm.

mixture of FeCl3 and FeCl2 in 2 mol/dm3 hydrochloric acid was added to a 0.7 mol/dm3 ammonia solution. The precipitate was then stirred with 2 mol/dm3 perchloric acid and peptized by decanting the acid and adding water. Although Massart initially suspected that Fe3O4 (magnetite) was produced, he and others confirmed in later work that γ-Fe2O3 was the synthesis product (Bacri et al., 1986; Jolivet and Tronc, 1988). The stock dispersions are stable for at least 6 months, although excessive dilution with water destabilizes them. From transmission electron micrographs of the particles we measured the particle size distribution. Figure 2 is a histogram of the measured particle sizes obtained from analyzing 246 particles. The mean particle diameter is 12 nm with a standard deviation of 4 nm. The size distribution is log-normal (see insert), which is typical for these particles. From the measured particle size distribution and the known synthesis conditions, we calculated the stock concentration to be about 1.5 × 1014 particles/cm3, which corresponds to a volume fraction of about 2.6 × 10-4. The volume fraction of particles in the light-scattering cell during an experiment was a thousand times less than this, or about 2.6 × 10-7. The typical magnetization of particulate γ-Fe2O3 is about 350 emu/cm3, which is 16% lower than the bulk value of 417 emu/cm3 (Nunes and Yu, 1989). The discrepancy has been attributed to surface effects and/ or to the possibility that small amounts of nonmagnetic particles, such R-Fe2O3 or β-FeOOH, are produced during the γ-Fe2O3 synthesis, which would dilute the overall magnetic response of the dispersion. Results and Discussion Predictions of the Flocculation Kinetics. Figure 3 is a plot of the total potential energy profile for a pair of 100 nm γ-Fe2O3 particles. It compares the magnetic interaction in the Earth’s magnetic field (about 0.4 Oe) with the interaction in an applied magnetic field of 500 Oe, or with the addition of the maximum electrostatic repulsion (estimated using (9) with ψs ) ∞ and n ) 10-5 mol/dm3, which gives κ-1 ) 96 nm). All three profiles show a very strong attraction, no matter how large the electrical surface potential might be. The applied magnetic field has virtually no effect on the profile since, owing to the large magnetic dipole moment on these particles; they are completely aligned even by the relatively weak magnetic field of the Earth. van der Waals attractions are negligible compared to the mag-

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Figure 3. Calculated potential-energy profiles for 100 nm γ-Fe2O3 particles in Earth’s magnetic field (about 0.4 Oe), with 500 Oe applied field, and with the maximum electrostatic repulsion which can be estimated using (9). Owing to their large magnetic dipole, the dipoles are already completely aligned by weak magnetic fields, making their interaction insensitive to increases in magnetic field strength. Dispersions of these particles cannot be stabilized electrostatically.

Figure 5. Comparison of the theoretical stability ratios for rapid flocculation (no electrostatic repulsion) of polystyrene and γ-Fe2O3 particles as a function of particle diameter. The sharp drop in the stability ratio for γ-Fe2O3 reflects the strong increase in magnetic attraction with particle size.

Figure 6. Effect of applied magnetic field on the predicted stability ratio for γ-Fe2O3 particles, for several different particle sizes. The stability ratio for each particle size has been normalized to unity at zero applied field for easy comparison. Figure 4. Calculated potential-energy profiles for 25 nm γ-Fe2O3. Owing to a much smaller magnetic dipole (compared to Figure 3), the attraction of these particles can be significantly increased by an applied magnetic field. Dispersions of these particles can be electrostatically stabilized.

netic attraction. Even at the closest separations distance in Figure 3, the van der Waals attraction energy is less than 30 kT. The highly negative potential energy makes aqueous dispersions of these particles unstable with respect to flocculation. Our experimental observations supported this result: we could not disperse in water the 100 nm magnetic particles that we obtained from manufacturers or that we synthesized ourselves. By contrast, it is wellknown that nonmagnetic 100 nm latex particles can be stabilized electrostatically. The difference is the strong magnetic attraction between magnetic particles that lowers the potential energy. Figure 4 shows the same three potential energy profiles for 25 nm γ-Fe2O3 particles, except that we have used a finite surface potential for the particles (ψs ) 50 mV). The profile for low salt and zero applied field shows that the 25 nm magnetic particles are stabilized by a potential-energy barrier between particles of more than 10 kT. When salt is added to neutralize the double-layer repulsion, the potential energy becomes negative for all separation distances and the particles become unstable. Applying a magnetic field lowers the potential energy further, decreases the stability, and increases the flocculation rate of the particles. This

system appears to be ideal for the study of flocculation, since the particles remain stable at low salt concentration and flocculate at high salt concentration with a rate that depends on the magnetic field strength. To further illustrate the difference between magnetic and nonmagnetic particles, Figure 5 compares the stability ratios calculated from (12) for rapid flocculation (Velec ) 0) of maghemite (γ-Fe2O3) and polystyrene particles as a function of particle diameter, in the Earth’s magnetic field. The stability ratio for polystyrene increases gradually with particle size, while the stability ratio for maghemite particles decreases sharply (i.e., the flocculation rate increases) as the particle diameter increases. This result reflects the strong effect of particle size on the magnetic interactions. For very small particles, van der Waals forces dominate magnetic forces, and the flocculation rate for the γ-Fe2O3 particles remains larger (smaller stability ratio) than for polystyrene particles, because the Hamaker constant for metal oxide particles is greater than for latex particles. For larger magnetic particles, magnetic forces dominate; the strong increase in magnetic attraction with particle size causes the flocculation rate to rise sharply (stability ratio drops). In Figure 6, theoretical stability ratios are plotted as a function of applied magnetic field strength for several particle sizes. In order to easily compare the change in stability ratio with magnetic field strength for different sizes, each curve has been normalized so that the

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stability ratio at zero applied field is unity, according to the equation

Wnorm(H) )

W(H) W(0)

(15)

where Wnorm(H) is the normalized stability ratio which appears in Figure 6, and W(H) is the actual value calculated from (12) for the same particle size and magnetization. The curves show that the stability ratio decreases as the magnetic field increases, indicating that the magnetic field increases the particle-particle attraction. The stability ratio at first decreases sharply with the magnetic field at low field strengths, and then gradually levels off as the field strength increases. The extent of the decrease increases with particle size. The decrease in stability ratio (increase in flocculation rate) accompanying an increase in the magnetic field is caused by an increase in the magnetic attraction as particles tend to align more with the external field. If the vectors m and H represents the dipole moment of a given particle and the external magnetic field, while the scalars m and H represent their magnitudes, then a Boltzmann distribution of orientations of a large ensemble of noninteracting particles corresponds to (Chikazumi and Charap, 1964):

〈m‚H〉 1 ) coth(y) - ≡ L(y) mH y

(16)

where 〈...〉 denotes the average over all the particles of the ensemble, y ) mH/kT, and L(y) is called the Langevin function. In the limit of very weak magnetic fields (i.e., y f 0), the dipoles are distributed uniformly over all orientations, resulting in an ensemble average of zero (L f 0). In the opposite limit of very strong magnetic fields (i.e., y f ∞), all the dipoles are perfectly aligned with H, resulting in m‚H ) mH for every particle in the ensemble and L f 1. As the particles become larger, Figure 6 indicates that the maximum effect of the magnetic field on the flocculation rate increases. For the mean diameter of the particles used in the experiments (12 nm), Figure 6 indicates that the stability ratio decreases by less than 2%. At the upper end of the particle size range used in the experiments (20 nm), Figure 6 indicates that the stability ratio decreases by a maximum of 7.5%, corresponding to an 8.1% increase in the flocculation rate. These results will be compared to the experimental results below. Figure 6 also shows that, as the particle diameter increases, the magnetic field strength at which the stability ratio levels off decreases. This could also be inferred from (16), which predicts that complete alignment of the dipoles occurs for mH . kT. Since m is proportional to the particle volume, H required for complete alignment can be expected to be inversely proportional to volume. For the 40 nm particles, the stability ratio levels off at a field strength of about 50 Oe, compared to about 150 Oe for the 20 nm particles. For large maghemite particles (i.e., larger than 100 nm), the Earth’s magnetic field of roughly 0.4 Oe is sufficient to align the particles, and an applied magnetic field will have no additional effect on the initial flocculation rate. Measurement of the Flocculation Kinetics. A. Evidence of Chain Formation in the Magnetic Field. Figure 7 is a superposition of two scans of actual strip-chart tracings of light-scattering intensities recorded for two different directions of the applied mag-

Figure 7. Evolution in scattering intensity from two flocculation experiments superimposed to show the effect of the orientation of the magnetic field on the scattering intensity increase. The applied magnetic field was 280 Oe in both experiments; only the field orientations were different.

netic field. Before starting the experiment, the stripchart output was set to zero at the scattering intensity obtained with filtered water filling the cuvette. Thus the scattering intensity is caused only by particulate matter in the cuvette. The several large spikes occurring during the first 15 min are probably caused by dust fragments settling through the sample volume. Initially, the light scattering intensity increased linearly with time in both cases. A linear increase is typical of dilute flocculating dispersions, with the initial slope being proportional to the initial flocculation rate of the dispersion (see 14)). The two superimposed tracings of Figure 7 were obtained at the same particle concentration and magnetic field strength, but different field orientations. At long times (more than 20 minutes) the scattering intensity with the field parallel to the incident light increased much more rapidly than did the intensity with the field perpendicular to the incident light; in the former the intensity curved upward, while in the latter the intensity curved downward. When the magnetic field was removed, the intensity suddenly jumped to an intermediate value which was the same for both field orientations. The orientation of the magnetic field should not affect the rate of flocculation, other conditions being equal; therefore we suspected the nonlinearity was caused by shape effects on light scattering. With an optical microscope, we (like many others before us) observed that large magnetic particles (100 nm) form long chains in an applied magnetic field, with the chains oriented parallel to the applied magnetic field. While the individual particles are not distinguishable, the chains are easily seen. When the magnetic field rotates, the chains rotate as rigid rods, rather than each individual particle in the chain rotating about its center, which would cause the chains to break up. We expect a similar behavior for the smaller particles and shorter chains found in our flocculation experiments. Alignment and rotation of the chains might explain the differences in scattering intensity between the two magnetic field directions, as we will now explore. According to the Rayleigh-Debye theory of light scattering by small particles, the Rayleigh intensity is multiplied by a light-scattering form factor which accounts for the shape and orientation of a scattering

3192 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

particle with respect to the incident light beam (Kerker, 1969):

I(θ,β) ) IR(θ) P(θ,β)

(17)

where I(θ,β) is the observed scattering intensity at a scattering angle θ, IR(θ) is the Rayleigh scattering intensity which depends on the volume of the particle but not on its shape, and P(θ,β) is the form factor which accounts for the shape and orientation (given by β) of the particle. The form factor has been evaluated for cylinders of arbitrary aspect ratio and orientation. Assuming that we may approximate a long chain of magnetic particles as a thin rod, the form factor for light scattering from such a chain is given by (van de Hulst, 1981)

P(θ,β) ) [E1/2(u)]2 )

(sinu u)

2

(18)

where u ) klsin (θ/2) cos β, θ is the scattering angle (2° in our case), β is the angle between the bisectrix and the axis of the cylinder, k ) 2π/λ, l is the length of the cylinder, and E1/2 is the spherical Bessel function of the first kind, order 0. [The bisectrix is a line bisecting the angle between the line drawn from the scatterer to the detector and the line drawn from the scatterer to the light source.] To estimate the effect of orientation on the light scattering, we do a sample calculation. Only the angle β depends on the orientation of the chain: when the chain is perpendicular to the incident light, β is 1°, and when the chain is parallel the incident light, β is 89°. Using λ ) 476 nm, θ ) 2°, and l ) 6 µm, we obtain a form factor of 0.50 when the chain is perpendicular to the incident beam, compared with a form factor of unity when the chain is parallel to the incident beam. This is roughly the same intensity ratio as in Figure 7 just before the field is removed. Thus alignment of growing chains with the applied magnetic field could easily produce the different scattering intensities observed in Figure 7. The effect of chain orientation on scattering becomes less 10% for chain lengths less than 2.5 µm. The sudden change in intensity that is observed in Figure 7 when the magnetic field was removed was probably caused by Brownian rotation of the highly oriented chains into a randomly oriented configuration. This random configuration scatters the same light intensity regardless of the original magnetic field direction, which may account for the scattering intensity being the same for both orientations after the field was removed. B. Stability Ratio. Stability ratios are experimentally determined from the initial slope of chart tracings like those in Figure 7. In order to compare theory and experiments, we will normalize the experimental stability ratios [as in (15)] so that unity is obtained at zero field strength. Recall that the flocculation rate is related to the time evolution of the scattering intensity by (14). If the initial scattering intensity I0 and particle concentration n0 are reproducible, they will be the same with or without an applied magnetic field and will cancel out, leaving

Wnorm(H) )

| /|

dI dI dt H)0 dt H

(19)

The slope at applied field H appears in the denominator because it is inversely proportional to the stability ratio.

Figure 8. Comparison of experimental stability ratios and theoretical stability ratios for γ-Fe2O3 particles (M ) 350 emu/ cm3). The mean particle diameter was 12 nm for the experiments. To get the same decrease in stability ratio from Fuchs’ theory using an orientation-averaged magnetic interaction, the particle diameter would have to be 61 nmsmuch larger than the largest size reported in Figure 2.

Experimental stability ratios are summarized in Figure 8 along with predictions for monodisperse sols. Qualitatively, the experimental results are similar to the theoretical results. Like the predictions, the experiments show that the stability ratio decreases sharply at first, and then more gradually as the magnetic field is increased. However, the magnitude of the observed decrease in stability ratio was much greater than predicted. For particles with a mean diameter of 12 nm, we observed a 26.5% decrease in the stability ratio at high magnetic fields, while theory predicted less than a 2% decrease for 12 nm particles. Since the particle size distribution is skewed toward larger particles (recall Figure 2), and since small-angle light scattering favors large particles, it may be more appropriate to compare the experimental results to theoretical predictions for particles larger than the observed mean diameter. To predict the observed 26.5% drop for monodisperse particles, we had to use 61 nm as the diameter, which is 2.5 times larger than the largest particle size observed, indicating that the broad size distribution alone cannot account for the larger increase in flocculation rate observed than predicted. Also, for 61 nm particles, the predictions level off at a field strength of about 20 Oe, while the experiments level off at 200-300 Oe. Thus there exists a serious discrepancy between theory and experiment, regardless of what size the particles are assigned. Since Fuchs’ model predicts initial flocculation rates which generally agree with experiments for nonmagnetic particles (Russel et al., 1989), provided doublelayer repulsion is negligible, the source of this discrepancy is most likely the model for magnetic interactions. The main assumption in (3)-(8) is that particles assume a Boltzmann distribution of orientations at each separation distance as they approach each other during a collision. Since avoiding any assumption about the distribution severely complicates the theory, consider another simple model. To maximize the effect of the applied magnetic field, assume the particles are perfectly aligned at all separations when a magnetic field is applied and uniformly distributed in orientation at all separations in the absence of an applied field. This would produce an upper bound on the change in flocculation rates caused by an applied magnetic field. A somewhat smaller relative change would be calculated if we

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averaging “is not valid for permanently magnetic particles: although their rotary Brownian motion makes then behave as superparamagnets, the times involved in this rotation are comparable to the duration of an encounter”. The characteristic time (tt) required for a particle to move by Brownian motion a distance equal to its radius a is

tt ≈

Figure 9. Importance of the relative position of two dipoles which have been aligned by a strong external magnetic field. When the dipoles are side by side (left), they repel each other, but when they are placed end to end (right), they attract.

assume a Boltzmann distribution of orientations in the Earth’s magnetic field (absence of an applied field), but perfect alignment with an applied field. From (2) the maximum dipole-dipole attraction is -2m2/r3, corresponding to a pair of aligned dipoles. This overestimates the actual magnetic interaction and produces the maximum flocculation rate. By dividing this maximum flocculation rate by the predicted flocculation rate at zero applied field from the orientationaveraged model, we predict that the asymptotic decrease in the stability ratio for 12 nm γ-Fe2O3 particles in high magnetic fields was 35.8%. This prediction is larger than the observed decrease of 26.5%, indicating that the experimental results lie well within theoretical bounds, if the orientation distribution can be changed. Since the magnetic dipoles are also fully aligned with the applied magnetic field at high field strengths for the predictions in Figures 6 and 8, we should explain why we predict much larger flocculation rates using -2m2/ r3 for the magnetic interaction compared to orientation averaging at high field strengths. To obtain -2m2/r3 for the magnetic interaction, not only do the two dipole moments m1 and m2 have to be aligned with the applied magnetic field H, but m1 and m2 must also be aligned with the displacement vector r drawn between the dipoles. Figure 9 illustrates the important effect of relative position of the two aligned dipoles. When the two dipoles are arranged side by side, m‚r ) 0 and (2) predicts repulsion. To obtain the strongest attraction, the two dipoles must be aligned end to end. In effect, (3) averages over all orientations of r and thereby predicts a weaker attraction than -2m2/r3. C. Validity of Orientation Averaging. The particles will achieve a Boltzmann distribution of orientations at each separation distance if they can rotate much faster than they translate through the fluid. Otherwise they will not have sufficient time to establish orientation equilibrium. Scholten and Tjaden (1980) used orientational averaging to estimate the interaction energy for superparamagnetic particles in zero applied field. For their superparamagnetic particles, thermal fluctuations of the magnetic dipole occurs on a time scale of 10-9 s, which is much more rapid than the collision time by Brownian motion (10-6 s). In the case of superparamagnetic particles, the magnetic moment fluctuates within the particle, without the particle itself rotating. In our larger particles, we believe the magnetic moment is fixed within the particle as a permanent dipole. Changes in orientation of the dipole occur because of rotation of the particle as a whole. Scholten and Tjaden suspected that orientation

a2 6πηa3 ) Dt kT

where Dt is the translational diffusion coefficient. To obtain the second equation above, we have substituted the Stokes-Einstein expression for Dt. For comparison, the characteristic time for Brownian rotation is

tr ≈

1 8πηa3 ) Dr kT

where Dr is the rotary diffusion coefficient. Clearly translational and rotational Brownian motion occur on the same time scale. A magnetic torque exerted between the particles might cause the rotation to occur more quickly, but any magnetic torque is expected to be accompanied by a magnetic attraction which would cause a similar reduction in the time for collision. These simple arguments suggest that Brownian rotation of the entire particle is never sufficiently rapid to achieve an equilibrium distribution of orientations at every separation distance during a Brownian collision between two particles. Conclusions We have measured the initial rate of rapid flocculation of γ-Fe2O3 particles in an applied magnetic field, and found that strong magnetic fields increase the flocculation rate by up to 36% over the rate in the zero applied field (corresponding to a 26.5% decrease in the stability ratio); the particles were roughly spherical in shape and had a mean diameter of 12.1 nm. At low magnetic fields the flocculation rate increases with an increase in magnetic field strength as the particles become aligned with each other. At high fields the flocculation rate reaches a plateau as the particles become completely aligned with the applied magnetic field. A theory based on an orientation-averaged magnetic interaction between particles predicts less than a 2% increase in flocculation rate at high magnetic fields, which is a much smaller increase than the 36% observed experimentally. Although the theory predicts a greater effect of magnetic fields for larger particles, a particle size 2.5 times larger than the largest size seen experimentally would be needed for agreement. The discrepancy between experiment and theory probably results from the incorrect assumption that particle rotation is fast enough to maintain an Boltzmann distribution of dipole orientations while the particles flocculate. By comparing the characteristic diffusion times for translational and rotational motion of spherical particles, we have shown that translation and rotation occur on approximately the same time scale, indicating that particle rotation is too slow to establish orientational equilibrium during flocculation. Therefore the orientation average might not accurately represent the interac-

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Received for review January 20, 1996 Revised manuscript received May 3, 1996 Accepted May 7, 1996X IE9600396

X Abstract published in Advance ACS Abstracts, August 15, 1996.