Langmuir 1996, 12, 4095-4102
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Magnetorheological Fluid Structure in a Pulsed Magnetic Field Joanne H. E. Promislow† and Alice P. Gast*,‡ Departments of Chemistry and Chemical Engineering, Stanford University, Stanford, California 94305 Received February 2, 1996. In Final Form: May 28, 1996X The pulsed-field structure of an emulsion of monodisperse, magnetizable oil droplets is investigated via optical microscopy. By permitting droplet diffusion in the field-off state, a pulsed field allows minimization of energy through structural rearrangement. For droplets with a magnetic susceptibility of χ ) 2.2 and radius a g 0.32 µm, we find that rearrangement into ellipsoidal aggregates occurs in response to a pulsed magnetic field. The ellipsoid ends are composed of chainlike projections at low pulse frequencies and conical spikes at high pulse frequencies. The conical spikes appear to be energetically favored but cannot form at low pulse frequencies due to the large diffusion distance of the droplets in the field-off state. The eccentricity of the ellipsoids is invariant with field strength in strong fields, but in weak fields we find that the ellipsoids become more elongated as the field strength is lowered. This elongation in weak fields coincides with the formation of more dilute aggregates and gives information about the change in surface structure as field strength decreases.
1. Introduction Magnetorheological (MR) fluids, suspensions of paramagnetic particles in a nonmagnetic fluid, and electrorheological (ER) fluids, suspensions of dielectric particles in a nonpolar liquid, are part of a new class of controllable fluids that has exciting implications for electromechanical devices. The application of a magnetic or electric field to an MR or ER fluid, respectively, causes particle chaining parallel to the field due to the induced dipolar interactions between particles. At high particle volume fraction, the chains crosslink and the suspension effectively solidifies.1,2 Due to this capacity for a dramatic rheological response to an applied field, applications in clutches, brakes and vibration-control systems are envisioned for these controllable fluids.3-5 ER fluids have been the focus of numerous studies in the past decade and are the subject of two review articles.6,7 Studies with MR fluids have become more common in the past several years because MR fluids avoid the charge-related problems associated with ER fluids and thus provide good model systems.8-10 In addition, MR fluids have recently been found to possess greater strength, a wider operational temperature range, and more feasible energy requirements than ER fluids,5 making them better suited for certain applications. The behavior of MR fluids in pulsed magnetic fields (i.e., square wave alternating between field-on and fieldoff) is a new area of research; previous studies of the suspension structure of ER and MR fluids have generally involved a continuous (dc) or alternating field. The response of an MR fluid to a pulsed field is, however, of †
Department of Chemistry. Department of Chemical Engineering. X Abstract published in Advance ACS Abstracts, August 1, 1996. ‡
(1) Rabinow, J. AIEE Trans. 1948, 67, 1308. (2) Winslow, W. M. J. Appl. Phys. 1949, 20, 1137. (3) Shulman, Z. P.; Gorodkin, R. G.; Korobko, E. V.; Gleb, V. K. J. Non-Newtonian Fluid Mech. 1981, 8, 29. (4) Leventon, W. Design News 1993, 185. (5) Carlson, J. D.; Weiss, K. D. Machine Design 1994, Aug. 8, 61. (6) Gast, A. P.; Zukoski, C. F. Adv. Colloid Interface Sci. 1989, 30, 153. (7) Block, H.; Kelly, J. P. J. Phys. D 1988, 21, 1661. (8) Bossis, G.; Mathis, C.; Mimouni, Z.; Paparoditis, C. Europhys. Lett. 1990, 11, 133. (9) Fermigier, M.; Gast, A. P. J. Colloid Interface Sci. 1992, 154, 522. (10) Promislow, J. H. E.; Gast, A. P.; Fermigier, M. J. Chem. Phys. 1995, 102, 5492.
S0743-7463(96)00104-7 CCC: $12.00
great fundamental interest, as well as relevant to MR fluid applications, which require that the fluid respond reproducibly to an external field that is repeatedly being switched on and off. The suspension structure that results from a dc field is determined by kinetics, not energetics, and is rarely the lowest energy formation. As long as the field remains on, the structure is essentially frozen in place. In contrast, a pulsed field permits particle rearrangement so that the suspension can adopt a lower energy structure. While much progress has been made in elucidating the thermodynamically favored structure of ferrofluids,11-13 the equilibrium shape of MR fluid aggregates has remained largely unresolved. This capability for observing the energetically relaxed structure of an MR fluid via a pulsed field is thus very exciting and has significant consequences for understanding the interplay of forces in an MR fluid. The equilibrium shape of an MR fluid aggregate is primarily determined by the demagnetization field within the aggregate and the surface energy that arises from particles on the surface of the aggregate being exposed to a different local field than those in the bulk. Knowledge of the low-energy structure of an MR fluid provides unique information about the structure-determining competition between these dipolar interactions. That the equilibrium shape of an ER or MR fluid was not previously known has been a stumbling block for theories minimizing these dipolar interactions to model ER and MR fluid behavior. Determining the field energy of an aggregate of arbitrary shape is difficult, and thus to date, all theories dealing with equilibrium ER or MR structures have begun by assuming a specific aggregate shape, with debate existing over the assumption of an ellipsoidal versus cylindrical shape.14-17 Being able to accurately determine the low-energy aggregate shape would thus be very beneficial for these models of equilibrium suspension structure. (11) Rosensweig, R. E. Ferrohydrodynamics; Cambridge University Press: Cambridge, 1985. (12) Tsebers, A. O. Magnetohydrodynamics 1982, 137. (13) Bacri. J. C.; Salin, D. J. Phys. Lett. 1982, 43, L649. (14) Halsey, T. C.; Toor, W. J. Statistical Phys. 1990, 61, 1257. (15) 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. (16) Lemaire, E.; Grasselli, Y.; Bossis, G. J. Phys. II 1992, 2. (17) Grasselli, Y.; Bossis, G.; Lemaire, E. J. Phys. II 1994, 4, 253.
© 1996 American Chemical Society
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In this article, we investigate the effects of a pulsed magnetic field on an aqueous suspension of monodisperse, magnetizable oil droplets. We present the first observations of an energetically relaxed MR fluid structure. We find that when droplets larger than 0.6 µm in diameter and with a magnetic susceptibility of χ ) 2.2 are subjected to a strong, pulsed magnetic field, the original cross-linked network that immediately forms gives way to ellipsoidal aggregates. The ellipsoid ends are composed of conical spikes in what appears to be the lowest energy formation, but the details of this end structure depend on pulse frequency. We also find that the eccentricity of the aggregates is independent of field strength for sufficiently strong fields but exhibits a sensitive variation with field strength at low fields. 2. Experimental Section The samples we used are emulsions of ferrofluid-containing oil droplets dispersed in water and stabilized against irreversible aggregation by sodium dodecyl sulfate. We synthesized the emulsions using a fractionation procedure developed by Bibette18 that produces very monodisperse and stable spherical emulsion droplets. For these studies, we used suspensions with droplet radii of 0.18, 0.26, 0.32, 0.38, and 0.46 µm. The ferrofluid, provided by J. Bibette, is in the form of small grains of 100 Å single magnetic domains of the iron oxide Fe2O3 dispersed in octane at 19% by volume. The iron oxide domains are ferromagnetic, but since no long-range order exists between domains, the droplets are superparamagnetic; their magnetization is completely reversible and, at low field strengths, is proportional to the external field through χ, the effective magnetic susceptibility. The suspensions are held on a microscope stand in sealed microrectangular tubes manufactured by Wale Apparatus, 50 µm × 1 mm in cross section and 50 mm in length. A uniform magnetic field is generated in the sample by two coils of copper wire placed one on each side of the sample. After application of the magnetic field, the evolution of suspension structure is recorded with a CCD video camera and digital images, consisting of 510 × 492 pixels with 256 gray levels, are obtained for analysis. When the magnetic field is applied, the emulsion droplets acquire dipole moments m ) 4/3πa3 µ0χH where a is the particle radius, µ0 is the magnetic permeability of a vacuum, and H is the external field. The interaction energy U(r,θ) between two droplets with aligned, identical dipole moments is
U(r,θ) )
m2 1 - 3 cos2 θ 4πµ0 r3
(1)
where r is the distance between sphere centers and θ is the angle between the applied field and the line joining the sphere centers. The dimensionless dipole strength λ provides a ratio of the maximum magnetic attraction between two droplets (i.e., droplets touching and aligned with the external field) to the thermal energy
λ)
-Umax πµ0a3χ2H2 ) kT 9kT
(2)
3. Theory Since this paper is concerned with the low-energy structure of an MR fluid, we will begin by briefly reviewing the dipolar interactions that influence aggregate shape: the demagnetizing field and surface energy. The total energy UT for an ellipsoidal aggregate of magnetizable material in an external magnetic field H is given by19 (18) Bibette, J. J. Colloid Interface Sci. 1991, 147, 474. (19) Landau, L. D.; Lifshitz, E. M.; Pitaevskii, L. P. Electrodynamics of Continuous Media, 2nd ed.; Pergamon Press: New York, 1984.
UT ) -
ma‚H 2
(3)
where ma is the magnetic moment of the aggregate and can be expressed as ma ) mh - ms. The effects of the demagnetization field are accounted for in mh, while ms is related to the surface energy. An ellipsoid with a homogeneous magnetic medium has a magnetic moment mh given by20
[
mh ) µ0Va
χaH
1 + (4πχa)nz
]
(4)
with Va the volume of the aggregate, χa the average magnetic susceptibility of the aggregate, and nz the demagnetizing factor. The demagnetizing factor is determined by the eccentricity of the ellipsoid and decreases as the ellipsoid elongates. Minimizing the demagnetizing field thus favors elongation of the aggregate. The surface energy is a correction to the aggregate dipole moment calculated for a homogeneous medium and arises from the fact that the droplets have a finite size and consequently experience a weaker local field on the surface of the aggregate than in the bulk.17 The actual magnetic moment of the aggregate ma is then reduced from mh by an amount ms ) Ns δm where Ns is the number of droplets on the surface and δm is the difference between the dipole moments of a droplet in the bulk and a droplet on the surface. Minimizing the surface energy thus favors decreasing Ns by adoption of a more spherical aggregate shape. The value of δm is actually very sensitive to the precise structure of the surface and is generally anisotropic as well.21-23 A good estimate of its value can be simply obtained, however, using a mean field approach. For a homogeneous ellipsoid with uniform polarization, Grasselli et al.17 show that ms can also be expressed as
ms ) nσmh
with
nσ )
Ns βφa N 2
(5)
where N is the number of droplets inside the aggregate, φa is the droplet volume fraction in the aggregate, and β) 4 πχp/(4 πχp + 3) with χp the magnetic susceptibility of a particle. Since Ns/N ∝ a, eq 5 shows that the surface tension increases with droplet size and droplet magnetic susceptibility. Results and Discussion Structure Evolution. Figure 1 shows the evolution of suspension structure in a sample (droplet volume fraction φ ) 0.02; a ) 0.32 µm) subjected to a pulsed magnetic field of strength H ) 1480 A/m, corresponding to λ ) 37, and frequency ν ) 2.0 Hz. The immediate response to the applied field is the kinetically-driven formation of the cross-linked network shown in Figure 1a. This network quickly reorganizes into particle-rich and particle-depleted zones as rearrangement occurs during the field-off pulses. This phase-separated state has been predicted to be energetically favorable for the analogous case of dielectric spheres14 and has been observed in MR fluid studies that allow for some equili(20) For consistency within this article, this expression is given in S.I. (rationalized mks) units. Several relevant articles use Gaussian variables, however, in which case the corresponding equation is mh ) Va χaH/[1 + 4πχa nz]. (21) Toor,W. R.; Halsey, T. C. Phys. Rev. A 1992, 45, 8617. (22) Clercx, H. J. H.; Bossis, G. J. Chem. Phys. 1993, 98, 8284. (23) Lobkovsky, A. E.; Halsey, T. C. J. Chem. Phys. 1995, 103, 3737.
MR Fluid Structure in a Pulsed Magnetic Field
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Figure 1. The suspension structure of an MR fluid sample with a droplet volume fraction of φ ) 0.02 and particle radius of a ) 0.32 µm (a) 2 s, (b) 1 min, (c) 3 min, (d) 5 min, (e) 15 min, and (f) 1 h after the application of a pulsed magnetic field of strength H ) 1480 A/m (λ ) 37) and pulse frequency ν ) 2.0 Hz. The field direction is parallel to the long axis of the aggregates.
bration of chains.15-17,24 This is the first observation, however, of the subsequent minimization of energy via the detachment of the concentrated domains from the cell walls and the formation of large ellipsoidal aggregates with spiked ends, as shown in Figure 1f. This ability to observe the low-energy aggregate shape should provide new insight into the structure-determining competition between surface tension and field energy in an MR fluid. In previous studies minimizing the dipolar interactions to model the width of and spacing between aggregates in an equilibrium MR fluid structure, the experimental “equilibrium” structure used for comparison to the theoretical predictions was obtained by increasing an applied dc field at a very slow rate (e80 (A/m)/min or (24) Wirtz, D.; Fermigier, M. Phys. Rev. Lett. 1994, 72, 2294.
0.36 λ/min).15-17 The resulting structure was composed of thin columnar aggregates spanning the cell. The exact shape of the aggregates (i.e., ellipsoidal or cylindrical) could not be discerned. When we apply a dc field increased at a similar rate of 50 A/m (0.043 λ) per min up to a final value corresponding to λ ) 37 to the sample used in Figure 1, we observe the pattern of cross-linked, columnar aggregates shown in Figure 2. This structure is markedly different from the low-energy structure comprising detached ellipsoids that results from exposure to a pulsed field at λ ) 37, indicating that a dc field increased at a very slow rate does not necessarily produce the lowest energy suspension structure. It is also significant for MR fluid technology that a pulsed magnetic field may induce the formation of ellipsoidal
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Figure 2. Suspension structure obtained by applying a continuous field increased at a rate of 50 (A/m)/min up to a final value of H ) 1480 A/m (λ ) 37).
aggregates that detach from the cell walls. Clearly, the suspension shown in Figure 1f cannot support a stress like the original cross-linked network, and hence this relaxation would be disastrous for an MR fluid application. Establishing a weak network, one that is strong enough to prevent particle rearrangement but easily broken by flow, in the field-off state is thus a desirable preventive measure for MR fluid applications. The structure comprising ellipsoidal aggregates in Figure 1f is not a universal equilibrium suspension structure for MR fluids. We found that with samples of smaller droplet size (a e 0.26 µm), applying the same
Promislow and Gast
pulsed magnetic field led to very thin columns that remained connected to the cell walls. Wirtz and Fermigier,24 in the first study involving the application of a pulsed magnetic field to an MR fluid, used particles with a mean radius of 0.75 µm that had a magnetic susceptibility of χ ) 0.9, as compared to χ ) 2.2 for our emulsion droplets, and they also observed the formation of thin, percolated columnar aggregates. This variation in the equilibrium structure can be explained by eq 5. Suspensions of smaller droplets or droplets with a lower magnetic susceptibility have a reduced surface tension. The demagnetizing field can then dominate, and elongated aggregates are favored over the much shorter, thicker ellipsoids we observe with the larger, higher magnetic susceptibility droplets. Dependence of Shape on Pulse Frequency. Figure 3 demonstrates the intriguing variation in the structure of the ellipsoid ends with pulse frequency. All four images are for a sample (a ) 0.32 µm; φ ) 0.02) that has been exposed to a strong magnetic field (λ ) 37), pulsed at a frequency indicated by ν, for 1 h. Above a cutoff frequency (νc ) 25 Hz for a ) 0.32 µm), the relaxation of suspension structure into ellipsoids ceases to occur, as the field-off time becomes too short to allow any significant movement of particles. Parts a and b of Figure 3, corresponding to the two lower frequencies, show aggregate ends consisting of many chainlike projections, while in the two higher frequency images, parts c and d of Figure 3, the ellipsoid ends comprise thick, conical spikes. The size and shape of these spikes appear to be unrelated to the dimensions of the sample cell; we observe very similar patterns using a cell twice as wide and thick (100 µm × 2 mm in cross section). The reason for this frequency-dependent end
Figure 3. Ends of the ellipsoidal aggregates formed in a suspension (φ ) 0.02; a ) 0.32 µm) exposed to a strong (λ ) 37) pulsed magnetic field of pulse frequency (a) 0.033, (b) 0.2, (c) 2, and (d) 10 Hz. Each suspension has been in the pulsed field for 1 h and the structures have achieved their final form.
MR Fluid Structure in a Pulsed Magnetic Field
structure is clear from the experiments. For ν ) 0.033 Hz and ν ) 0.2 Hz, the frequency is low enough that the end structure of the ellipsoids completely dissolves during the field-off pulses and must re-form each time the field is reapplied. In contrast, for ν ) 2 Hz and ν ) 10 Hz, the end structure persists during the field-off pulses and is thus able to slowly rearrange to minimize energy. To examine the transition between frequencies that lead to chainlike projections and frequencies that allow conical spikes, we performed experiments with two sizes of emulsion droplets and compared their average droplet diffusion distances in the field-off state at their respective transition frequencies. The root mean square diffusion distance 〈x2〉1/2 of a droplet during the field-off state can be determined from:25 〈x2〉 ) 2Dt where D ) kT/(6πηa) is the droplet diffusion coefficient and t ) 1/(2ν) is the duration of the field-off pulse. For samples with a ) 0.32 µm, we observe the formation of stable conical spikes for all ν g 1 Hz where 〈x2〉1/2 e 0.82 µm. For samples with a ) 0.46 µm, stable conical spikes exist for ν g 0.7 Hz, corresponding to 〈x2〉1/2 e 0.83 µm. The transition from chainlike projections to conical spikes thus appears to occur at an average droplet diffusion distance during the field-off state of ∼0.8 µm. This dependence on 〈x2〉1/2 is consistent with the mechanism of formation for the end structure. When the field turns off, droplets in a spike begin to diffuse apart. At some critical average droplet separation distance, the droplet concentration at the aggregate ends will be sufficiently dilute that the formation of chains via head-to-tail droplet aggregation will be kinetically favored over the regeneration of smooth conical spikes when the field is reinstated. At the frequency corresponding to this critical separation distance, stable conical spikes can no longer exist. Conical spikes have not been previously observed in an MR fluid. Their presence is indicative of the large energetic penalty for surfaces oriented perpendicular to the field direction.21 In Figures 1 and 3 the field direction is along the long axis of the ellipsoids, and it is clear that all aggregate surfaces are more parallel than perpendicular to the applied field. Conical tips have been seen previously in both dielectric fluid drops in strong electric fields and ferrofluid drops in strong magnetic fields.13,26 A theoretical analysis of the appearance of conical tips in these two systems found that a conical interface is energetically favored when the field is sufficiently high and the dielectric or magnetic susceptibility of the fluid exceeds a critical value of 1.320.27,28 This analysis predicts a cone angle of approximately 30° for a ferrofluid drop with a magnetic susceptibility of χ ) 2.2, while we observe an average cone angle of 13.0 ( 3.0° at ν ) 2 Hz for our MR fluid samples with χ ) 2.2. This discrepancy is not surprising, however, since the theory is for an immiscible dielectric or ferrofluid drop and the surface energy is thus fundamentally different from that in our system. The surface energy inherent in the use of an immiscible fluid drop does not depend on field strength, whereas in our system the surface energy arises solely from the weaker local field at the surface and is proportional to H2. (25) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (26) Taylor, G. I. Proc. R. Soc. London, Ser. A 1964, 280, 383. (27) Li, H.; Halsey, T. C.; Lobkovsky, A. Europhys. Lett. 1994, 27, 575. (28) To be consistent with both the system of units used in this paper (rationalized mks) and those used in ref 27 (Gaussian), we give this critical value in terms of the dielectric or magnetic susceptiblity, since the susceptibility is independent of the system of units employed. The corresponding dielectric constant or magnetic permeability is given by 1 + χ in rationalized mks units and 1 + 4πχ in Gaussian units.
Langmuir, Vol. 12, No. 17, 1996 4099
Figure 4. Final suspension structure for samples (φ ) 0.02; a ) 0.32 µm) after 1 h of exposure to a pulsed (ν ) 2 Hz) magnetic field of strength (a) H ) 1480 A/m (λ ) 37), (b) H ) 874 A/m (λ ) 13), and (c) H ) 638 A/m (λ ) 6.9).
Dependence of Shape on Field Strength. Both the demagnetizing field and surface energy are proportional to H2, suggesting that aggregate shape should be independent of field strength. For all fields satisfying λ > 25, we do find this invariance of aggregate shape with field strength; however, when we apply a relatively low strength pulsed field, the resulting ellipsoids are more elongated than those formed in high fields. This behavior is illustrated in Figure 4. The three images show the final structures resulting when a suspension with a ) 0.32 µm and φ ) 0.02 is exposed to a field with strength H, pulsed
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Figure 5. (a) Final aggregate shape achieved in a sample (φ 0.02; a ) 0.32 µm) after 1 h of exposure to a high-strength (λ ) 38), pulsed (ν ) 2 Hz) field. Parts b-d show the same aggregate after the field has been decreased to (b) λ ) 13, (c) λ ) 6.8 and(d) λ ) 2.7.
at ν ) 2 Hz. Figure 4a shows the high field (H ) 1480 A/m, λ ) 37) structure. This structure forms for all fields above λ) 25. Figure 4b shows that when a relatively low strength field (λ ) 13) is used, the ellipsoids adopt a slightly more elongated shape. In Figure 4c a very low strength field (λ ) 6.9 ) produces ellipsoids so elongated that they remain in contact with both walls. Figure 5 illustrates the evolution in aggregate shape as the applied field strength is decreased. The aggregate in Figure 5a has been exposed to a high-strength (λ ) 38 ) pulsed field of frequency 2 Hz for 1 h and has achieved its final low-energy shape. Stepping down the field to λ ) 13 produces little change as shown in Figure 5b. However, when the field is further reduced to λ ) 6.8, the volume of the ellipsoid increases and the spikes on the end of the ellipsoid begin to extend. Decreasing the field still further to λ ) 2.7 results in another increase in the ellipsoid volume and continued growth of the projections. Figures 6 and 7 show the long-time recovery of multispiked and single-spiked aggregates, respectively, on going from a strong to weak field. Figure 6a shows a sample that has reached its final structure in a high field (λ ) 38) and has many spikes per aggregate end. In Figure 6b the field has been decreased to λ ) 2.7 and the suspension has been allowed to equilibrate for 4 h to adopt its final structure. Although the projections emanating from the spikes have extended to the cell walls, the suspension does not recover the low-field structure that is obtained by applying a low field from the onset (i.e., Figure 4c), because the segments of each aggregate remain attached at their middles. In contrast, Figure 7 shows that recovery of the original low-field structure is possible
if the high-field aggregates have just one spike per end. The path dependence of the final low-field structures shown in Figures 6b and 7b indicates that while a pulsed field does permit structural rearrangement to minimize energy, the absolute lowest energy structure may still not always be accessible. The images in parts c and d of Figure 5 illustrate that the growth of the projections that occurs when the field is lowered coincides with an increase in the volume of the ellipsoid. This volume increase is simply due to the fact that droplet interactions are weaker in low fields and, as in any phase transition, the weaker interactions produce lower density phases and thus the aggregate becomes less compact. We also note that more free droplets are present at the lower field strengths, implying that not only is the aggregate volume growing but the number of particles in the aggregate is decreasing as well. The volume fraction of droplets in an aggregate φa is thus clearly much lower at weaker field strengths. We determined the droplet volume fraction of the aggregates from measurements of their total area, assuming that the high-field volume fraction of droplets in an aggregate is 0.64, the random close-packed value, and that the thickness of the aggregate was invariant due to the cell walls. The sudden decrease in φa as the field is lowered beyond λ ) 13 is illustrated in Figure 8. The formation of elongated aggregates at low field strengths thus appears related to a decrease in φa. Equation 5 indicates that since φa is much lower in weak fields, the surface energy must then also be significantly decreased at low fields since ms ∝ φa. This decrease in surface energy is offset, however, by a similar decrease in
MR Fluid Structure in a Pulsed Magnetic Field
Figure 6. (a) Multispiked ellipsoidal aggregates obtained from the exposure of an MR sample (φ ) 0.02; a ) 0.32 µm) to a strong (H ) 1500 A/m; λ ) 38) pulsed (ν ) 2 Hz) magnetic field. (b) The same aggregates after the strength of the pulsed field have been reduced to H ) 400 A/m (λ ) 2.7) and equilibration has occurred over several hours.
the demagnetizing field. When the droplet volume fraction in an aggregate decreases, the average magnetic susceptibility of the aggregate χa decreases and the effect of the demagnetizing field, represented by the factor 1/(1 + (4πχa)nz) (see eq 4), also diminishes. With a decreased surface energy and demagnetizing field, the driving forces for both elongation and contraction are weakened and thus no simple qualitative argument can be made to explain the aggregate elongation at low field strengths. The shape change nevertheless gives information about the change in surface structure as λ and φa decrease, inspiring more rigorous calculations to elicit this information. In experiments carried out on ferrofluid drops in a magnetic field, the opposite dependence of shape on field strength has been observed; the drop of ferrofluid elongates as the applied magnetic field is increased.13 This behavior does not contradict our observations, however, since the origins of the surface energy in the two systems are very different. Again, in studies of ferrofluid drops in an immiscible fluid, there is a field-independent surface energy due to the use of two immiscible fluids that is not present in our system. The surface energy in our system is a result of the finite size of the droplets and is proportional to H2. This field-dependent contribution to the surface energy is negligible for ferrofluid drops since
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Figure 7. (a) Single-spiked aggregates obtained from exposure of a sample (φ ) 0.02; a ) 0.38 µm) to a high strength (H ) 1500 A/m; λ ) 64) pulsed (ν ) 2 Hz) field. (b) The same aggregates after the field has been reduced to H ) 400 A/m (λ ) 4.5) and equilibration of suspension structure have occurred.
Figure 8. Volume fraction of droplets in an aggregate φa at different values of the interaction parameter λ.
the ferromagnetic grains that make up a ferrofluid are only 100 Å in diameter and the effect would thus be approximately 2 orders of magnitude less than for an MR fluid. 5. Conclusions We have studied the aggregation behavior of monodisperse, paramagnetic emulsion droplets in a pulsed magnetic field. We find that a pulsed field allows droplets
4102 Langmuir, Vol. 12, No. 17, 1996
to explore different arrangements so that a lower energy suspension structure can be adopted. For droplets with radius a g 0.32 µm and magnetic susceptibility χ ) 2.2, we report that the most energetically favorable aggregate shape at high fields is that of an ellipsoid with spiked ends. The end structure of the ellipsoids varies from chainlike projections at low pulse frequencies to conical spikes at high pulse frequencies. The conical spikes appear to be more energetically favorable but are kinetically inaccessible below frequencies corresponding to an average droplet diffusion distance in the field-off state of approximately 0.8 µm. We find that aggregate shape is
Promislow and Gast
independent of field strength at high fields, but when low fields lead to a decrease in the volume fraction of droplets in an aggregate, elongated shapes are favored. Acknowledgment. We thank Be´ne´dicte Deminie`re for her valuable assistance with the emulsion synthesis and J. Bibette for donation of the ferrofluid. We have also benefited from helpful discussions with M. Fermigier. This work was supported in part by the Department of Defense and AT&T Bell Laboratories graduate fellowship programs. We also acknowledge the Stanford Integrated Manufacturing Association for financial support. LA960104G
