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Small-Amplitude Oscillatory Shear Magnetorheology of Inverse Ferrofluids Jose Ramos, Juan de Vicente,* and Roque Hidalgo-Alvarez Biocolloid and Fluid Physics Group, Department of Applied Physics, Faculty of Sciences, University of Granada, C/ Fuentenueva s/n, 18071-Granada, Spain Received January 18, 2010. Revised Manuscript Received March 5, 2010 A comprehensive investigation is performed on highly monodisperse silica-based inverse ferrofluids under smallamplitude oscillatory shear in the presence of external magnetic fields up to 1 T. The effect of particle volume fraction and continuous medium Newtonian viscosity is thoroughly investigated. Experimental results for storage modulus are used to validate existing micromechanical magnetorheological models assuming different particle-level field-induced structures.
1. Introduction Magnetorheological (MR) fluids are smart materials whose mechanical properties can be externally controlled through magnetic fields. They are typically prepared by dispersing micrometersized spherical magnetic particles in nonmagnetic media. Currently, new trends in magnetorheology involve the use of nonspherical particles instead of their spherical counterparts both in rotational1 and oscillatory2 regimes. Even though much effort is being focused now on the use of magnetic particles having different geometries, a profound understanding of the MR properties of even classical sphere-based MR fluids is still missing. The reason for this is basically the lack of monodisperse magnetic particles to validate existing analytical or numerical magnetorheological models. To overcome this problem, inverse ferrofluids (IFFs) are known to be promising workbench candidates to test MR models even though their magnetic driven response is orders of magnitude lower than conventional MR fluids.3 An IFF is formed by dispersing micrometric nonmagnetic particles in a ferrofluid which basically consists of a stable suspension of nanometric magnetite particles. As a consequence, the nonmagnetic particles experience a medium that is magnetic and hydrodynamically continuous. By exposing the IFF to an external magnetic field, dipolar interactions appear between nonmagnetic particles.3 The strength of this interaction can be controlled by varying the strength of the magnetic field and/or the saturation magnetization of the ferrofluid. The interest in using IFFs comes from the fact that nonmagnetic particles are available that are highly monodisperse and susceptible to surface modification. In spite of the interest in using IFFs as models for magnetorheology, little research has been focused on the understanding of their magnetorheological properties if compared to investigations on classical MR fluids. In this sense, the most outstanding works are briefly reviewed next. A comparative study between classical MR fluids and IFFs in steady shear flow was performed by
Volkova and co-workers 10 years ago.4 In their paper, the authors reported the existence of two different yield stresses, one associated to the solid friction with the plates and the other associated to the rupture of the aggregates. Unfortunately, particle size was not well controlled and viscoelastic properties were not discussed. Inverse ferrofluids have been used in the past as model systems for investigating the influence of particle size and particle size distribution. Highly monodisperse IFFs were investigated first by de Gans and co-workers.5,6 Functionalized silica spheres were used having mean diameters in the range from 106 to 380 nm. A very complete characterization was carried out including steady-state and dynamic oscillatory tests. Furthermore, a microstructural rheological model was proposed. For small enough particles, a strong increase of MR properties with particle size was observed. This finding was explained in terms of the average length of the aggregates under the field. Polydisperse IFFs were investigated by Lemaire et al.7 in simple steady shear and Saldivar-Guerrero et al.8 in small-amplitude oscillatory shear. Lemaire et al. did not find any difference in the flow curves between mono- and polydisperse IFFs. In contrast, Saldivar-Guerrero et al. observed an enhanced storage modulus in polydisperse systems compared to monodisperse systems in the linear regime of magnetization. In the case of polydisperse IFFs, there was not a quantitative agreement with the micromechanical model by de Gans et al.5 They found a slow increase of storage modulus with magnetic field strength that was attributed to the existence of thick columnar structures (instead of single-width particle chains), hydrodynamic interactions, and the “poisoning effect”. More recently, Ekwebelam and See9 investigated the effect of particle size distribution on the MR response of IFFs subjected to large amplitude oscillatory shear flow. The ratio of the first to the third harmonic was found to become more pronounced with decreasing particle size as well as with increasing proportion of small particles in bidisperse mixtures.
*To whom correspondence should be addressed. E-mail:
[email protected].
(5) de Gans, B. J.; Blom, C.; Philipse, A. P.; Mellema, J. Phys. Rev. E 1999, 60, 4518–4527. (6) de Gans, B. J.; Duin, N. J.; van den Ende, D.; Mellema, J. J. Chem. Phys. 2000, 113, 2032–2042. (7) Lemaire, E.; Meunier, A.; Bossis, G. J. Rheol. 1995, 39, 1011–1020. (8) Saldivar-Guerrero, R.; Richter, R.; Rehberg, I.; Aksel, N.; Heymann, L.; Rodriguez-Fernandez, O. S. J. Chem. Phys. 2006, 125, 084907–1-7. (9) Ekwebelam, C. C.; See, H. Korea-Aust. Rheol. J. 2007, 19, 35–42.
(1) Bell, R. C.; Karli, J. O.; Vavreck, A. N.; Zimmerman, D. T.; Ngatu, G. T.; Wereley, N. M. Smart Mater. Struct. 2008, 17, 015028–1-6. (2) de Vicente, J.; Segovia-Gutierrez, J. P.; Andablo-Reyes, E.; Vereda, F.; Hidalgo-Alvarez, R. J. Chem. Phys. 2009, 131, 194902–1-10. (3) Skjeltorp, A. T. Phys. Rev. Lett. 1983, 51, 2306–2309. (4) Volkova, O.; Bossis, G.; Guyot, M.; Bashtovoi, V.; Reks, A. J. Rheol. 2000, 44, 91–104.
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To the best of our knowledge, only the micromechanical model proposed by de Gans and co-workers5 has been used to explain the small-amplitude oscillatory shear behavior of IFFs in spite of the fact many other models exist in the MR literature. In this work, the effect of particle volume fraction and medium viscosity are investigated in the small-amplitude oscillatory shear rheology of monodisperse silica-based IFFs for a very wide range of magnetic fields. An extensive literature survey of MR models is presented and experimental results obtained here compared with them.
2. Theoretical Background In this section, we review some of the most relevant theories and models published in the MR literature. We classify existing models in two groups. Those named as “macroscopic” come from magnetic energy minimization principles and assume a bicontinuous structure consisting of spheroidal, cylindrical, or layered particle aggregates. A second group is formed by microscopic models including single-width particle chains sheared under an external field. Among all rheological material functions, those more important in the case of MR fluids are traditionally the yield stress and the storage modulus G0 . The yield stress is associated to the minimum stress value required for the onset of flow and is of special interest in torque transfer applications. Alternatively, the storage modulus provides useful information about the colloidal particle-level structure and is related to the energy storage on the material. On the contrary, the loss modulus G00 is associated to the energy dissipated under flow. Available models revisited below will only focus on storage modulus predictions. In all models considered here for the storage modulus, the ferrofluid is assumed to be a continuous phase on the length scale of the diameter of the dispersed nonmagnetic particles. The shear stress is dominated by the magnetostatic interactions between the nonmagnetic particles, and shear induced deformation is assumed to be affine. Furthermore, field-induced structures are supposed not to interact between them, hence limiting our analysis to low particle concentrations. The storage modulus strongly depends on the interparticle potential of interaction which in turn depends on the interparticle force. As a first approximation, in the magnetic linear regime, this interparticle force is basically a dipolar magnetostatic force (i.e., proportional to the magnetic field squared). As a consequence, a convenient way to describe the viscoelastic properties of MR fluids is by using the normalized storage modulus defined as G0 n = G0 /(μ0μcrH02), where H0 is the external magnetic field strength, μ0 is the magnetic permeability of vacuum, and μcr is the relative magnetic permeability of the ferrofluid. One important difference between MR fluids and IFFs stems from the contrast factor, defined as β =(μp - μc)/(μp þ 2μc), where μp is the magnetic permeability of the particles and μc is the magnetic permeability of the continuous phase. In the case of IFFs, the particles are nonmagnetic and hence the contrast factor reduces to β = (1 - μcr)/(1 þ 2μcr). As a consequence, on the one hand, in the case of IFFs, β can take values between -0.5 (low fields) and 0 (large fields). On the other hand, in the case of classical MR fluids, β can take values between 1 (low fields) and 0 (large fields). In this paper, we will thoroughly investigate the MR behavior for β values in the range from -0.16 to -0.04. Lower values for β are difficult to obtain because very narrow gaps are required in this case to get gap-spanning structures under the applied field (see section 4.3 below). Langmuir 2010, 26(12), 9334–9341
2.1. Macroscopic Models. Macroscopic models were demonstrated in the past to apply well in the prediction of yield stresses of IFFs.10 However, up to now, they have not been tested yet in the prediction of the linear viscoelastic behavior of MR fluids. Basically, these models are based on a mesoscopic description of the structure only taking into account the shape anisotropy of the strained aggregates under small deformation. Internal structure within the aggregates is ignored, and only the particle volume fraction inside is required as an input in the models. For our purpose, this is assumed to be φa = 0.64 (randomly closed spherical aggregates), but slight differences in the results are found when assuming a different value for φa. In general, when a magnetized MR fluid is strained under a simple shear flow, the magnetostatic energy will decrease due to forced nonalignment of the field-induced aggregates with the external field. Then, the restoring stress τ can be written as τ ¼ -
DW Dγ
ð1Þ
being W the magnetostatic energy density and γ the shear strain. Once the magnetic field-induced stress is calculated from eq 1, the storage modulus can be easily obtained by taking the low strain limit of the ratio τ/γ. Three different macroscopic model structures are tested in this work: spheroidal, cylindrical, and layered particles aggregates. Let us first assume a collection of noninteracting spheroidal aggregates (aspect ratio=10)11 with demagnetization factors n1 and n2 in the directions parallel and perpendicular to the magnetic field, respectively. In this case, the normalized storage modulus can be obtained from G0n, S
1 þ 2φβ 1 1 ¼ - φs μ~ 1 - φβ a 1 þ μ~ a n2 1 þ μ~ a n1
! ð2Þ
where φs φ/φa and the relative permeability difference between the aggregate (μa) and the inverse ferrofluid (μ) is quantified by μ~a = (μa/μ) - 1. As a first approximation, a mean field theory12 is used to describe the magnetic permeability of the aggregates μa = μc(1 þ 2βφa)/(1 - βφa) and the inverse ferrofluid μ = μc(1 þ 2βφ)/(1 - βφ), where φ is the nonmagnetic particle volume fraction. Bossis and co-workers10 generalized the model presented by Rosensweig,13 for the shear stress, to the case of a lattice of magnetic cylinders and stripes. From their model, the normalized storage modulus can be inferred as follows: G0n, C=L ¼ φs ðμa =μc - 1Þ2
1 - φs C þ ðμa =μc - 1Þð1 - φs Þ
ð3Þ
where C = 2 for the case of cylinder-like (C) aggregates and C = 1 for the case of layered (L) aggregates. 2.2. Chain Models. The second group of models involves a microscopic description where interparticle forces have to be necessarily considered. The simplest of these models was proposed by Klingenberg and Zukoski.14 In their paper, the force between two isolated magnetizable spheres was calculated (10) (11) (12) (13) (14)
Bossis, G.; Lemaire, E.; Volkova, O.; Clercx, H. J. Rheol. 1997, 41, 687–704. Klingenberg, D. J. AIChE 2001, 47, 247–249. Maxwell-Garnett, J. C. Philos. Trans. R. Soc. London 1904, 203, 385–420. Rosensweig, R. E. J. Rheol. 1995, 39, 179–192. Klingenberg, D. J.; Zukoski, D. F. Langmuir 1990, 6, 15–24.
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Ramos et al. Table 1. Physical Properties of the Ferrofluids Used to Prepare the Inverse Ferrofluidsa
)
including a multipole expansion: 4 a Fm ¼ -12πμ0 μcr β2 a2 H 2 2f cos2 θ - f^ sin2 θ r^ r ð4Þ þ ð fΓ sin 2θÞθ^ )
where a is the particle radius; r is the center-to-center distance; f , f^, and fΓ are the parallel, perpendicular, and gamma coefficients, respectively; and finally, r^ and θˆ represent the radial and tangential unit vector, respectively. The horizontal component of the force required to separate two particles under affine deformation and dipolar approximation is given by 3 ð5Þ FH ¼ πμ0 μcr β2 a2 H 2 4θ - 3θ3 4 and hence, a rough approximation for the storage modulus in the viscoelastic linear region is G0 ¼ lim
θf0
Nc FH =S θ
ð6Þ
where Nc/S = 3φ/2πa2 is the number of chains per unit surface of the plate. As a consequence, a simple equation for the normalized storage modulus results that predicts a quadratic β dependence: G0n, KZ ¼
9 2 φβ 2
ð7Þ
Martin and Anderson15 carried out a refinement of this model by considering multibody interparticle interactions between all particles contained in a free single chain. Considering physical contact between particles and using self-consistent magnetic pair interaction forces, they got the following expression: 9 G0n, MA ¼ ςð3Þ φβ2 k3 4
ð8aÞ
where ς represents the Riemann function and k3 is defined as k k2 k3 ¼ 1 - 4 8
!-1 ,
k ¼ ςð3Þβ ¼ 1:202β
ð8bÞ
Sterically repulsive interactions were later included in the analysis by de Gans et al.5 These interactions prevent particles from interpenetrating. If the repulsive force is short ranged as compared to the size of the particles, the analysis gave the following expression for the modulus: G0n, dG
( ) 3 βςð3Þ -2 βςð3Þ -2 2 ¼ ςð4Þφβ 2 1 þ 1þ 4 2 2
ð9Þ
A different non-Stokesian hydrodynamic description was later used by de Vicente et al.16 According to their model, the following result is obtained in dipolar approximation: G0n, dV1 ¼
9 φ 2 β 8 φa
ð10Þ
(15) Martin, J. E.; Anderson, R. A. J. Chem. Phys. 1996, 104, 4814–4827. (16) de Vicente, J.; Lopez-Lopez, M. T.; Duran, J. D. G.; Bossis, G. J. Colloid Interface Sci. 2005, 282, 193–201.
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ferrofluid
density (kg/m3)
saturation magnetization (kA/m)
F040 1120 25.510 ( 0.014 F200 1050 24.299 ( 0.017 F500 1060 24.468 ( 0.004 a Density values were taken from the supplier’s data sheet.
viscosity (mPa s) 44 198 559
Assuming the expression for the magnetic torque on a thin spheroid given by Halsey and co-workers,17 de Vicente et al. obtained the following outcome:16 G0n, dV2 ¼
1 φ ðμa - 1Þ2 2 φa μ a þ 1
ð11Þ
In this case, the β dependence is embedded in the aggregate permeability μa. As expected, none of these models depend on the viscosity of the ferrofluid. Also, a nearly quadratic dependence with β is predicted in all cases.
3. Materials and Methods 3.1. Materials. Tetraethyl orthosilicate (TEOS) (Acros Organics, 98%), ethanol absolute (Scharlau, reagent grade), ammonia (Scharlau, solution 32%), and ultrapure distilled water (MilliQ Academic, Millipore) were used for the synthesis of silica nanoparticles. Three ferrofluids were purchased from Ferrotec Co. having the same magnetic properties but different Newtonian viscosities. Ferrofluids were coded F040, F200, and F500 (see Table 1). 3.2. Synthesis of Silica Nanoparticles. Spherical monodisperse silica nanoparticles were prepared by condensation polymerization of tetraethyl orthosilicate (TEOS) using the St€ ober method as follows.18 Absolute ethanol, ammonia, and water were mixed in a 500 mL reaction vessel. Then TEOS was added quickly and the reaction mixture was stirred at 350 rpm and room temperature during 1 day. The molar concentration of the mixed solution determined the desired size of silica nanoparticles. In this case, the molar concentrations were water/ammonia = 7.01/3.00. The volume of absolute ethanol was adjusted in each reaction up to 500 mL, and the concentration of TEOS was fixed to 0.2 M. Silica nanoparticles were collected by centrifugation (15 000 rpm, 15 min) and washed by repeating redispersion in absolute ethanol three times. The final product was dried in vacuum oven at 80 C for 24 h. 3.3. Preparation of Inverse Ferrofluids. Silica powders and ferrofluid of the desired viscosity were mixed and ultrasonified for 24 h up to get a homogeneous IFF. Suspensions therefore prepared were found to be stable, and phase separation was not observed during days. The stability mechanism is unknown yet; however, the adsorption of magnetite particles from the ferrofluid on the silica particles could be a possible explanation.5 The volume fractions of silica nanoparticles were chosen to be 12.6, 19.3, and 26.1 vol % assuming a silica density of 2 g/mL. These are the same particle concentrations as those used by de Gans et al.5 to make a straightforward comparative study possible. Nevertheless, it is important to stress here that particles prepared by de Gans and co-workers were 354 nm diameter and nonmagnetic particles used in this work were about double that diameter (see section 4.1 below). 3.4. Magnetorheometry. The small-amplitude oscillatory shear magnetorheological behavior was measured using a plateplate configuration (gap = 300 μm) in a MCR 501 Anton Paar (17) Halsey, T. C.; Martin, J. E.; Adolf, D. Phys. Rev. Lett. 1992, 68, 1519–1522. (18) St€ober, W.; Fink, A. J. Colloid Interface Sci. 1968, 26, 62–69.
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Figure 1. TEM photograph corresponding to silica microparticles.
magnetorheometer. The magnetic field was perpendicular to the plates. The plates have a diameter of 20 mm, and the temperature was maintained at 25 C during the tests. There are several reasons to use parallel plates despite the fact that the shear rate is not constant in this geometry, instead of cone-plate counterparts. First, the field-induced structure in MR fluids strongly depends on the confining geometry;19 by changing the commanded gap distance, it can be easily modified. Second, the average aggregate length is one of the major features determining the rheological response;6 for parallel plates, gap thickness is constant in all the sample volume. External magnetic fields were applied using an electromagnet up to 884 kA/m. This magnetic field strength is enough to fully saturate the fluids. Magnetosweeps were carried out at a strain amplitude of 0.1% (well within the viscoelastic linear region in all cases) and a frequency of 1 Hz according to Laeuger et al.20 The experimental procedure is summarized as follows: (i) precondition at a constant shear rate of 200 s-1 during 30 s, (ii) suspension is left to equilibrate for 1 min, (iii) constant dynamic-mechanical shear conditions are preset (both frequency and strain amplitude are kept constant), and magnetic field is logarithmically increased at a rate of 10 points per decade from 0.175 to 884 kA/m. Then the resulting stress and phase lag between strain and stress signals are measured, and from them both G0 and G00 are calculated.21 To check reproducibility, in all cases experiments were repeated at least three times with fresh new samples. Error bars will not be shown in the figures below because the uncertainty falls within the symbol size.
4. Results and Discussion 4.1. Inverse Ferrofluid Characterization. A transmission electron microscopy (TEM) micrograph corresponding to the silica particles is shown in Figure 1. As observed, particles have a spherical shape. Particle size histograms, shown in Figure 2, reveal a high monodispersity. The most relevant particle size results are summarized in Table 2. The bulk rheology of the three ferrofluids was measured using a cone-plate geometry (50 mm diameter, 1), maintaining the temperature at 25 C with a peltier device. None of these ferrofluids showed viscoelasticity, behaving as purely viscous materials with viscosities 44, 198, and 559 mPa s. Results are (19) Zhu, Y.; McNeary, M.; Breslin, N.; Liu, J. Int. J. Mod. Phys. B 1999, 13, 2044–2051. (20) Laeuger, J.; Wollny, K.; Stettin, H.; Huck, S. A new device for the full rheological characterization of magneto-rheological fluids. In Proceedings of the ERMR2004; World Scientific: Beijing, China, 2005; pp 370-376. (21) Mezger, T. G. The Rheology Handbook, 2nd ed.; Vincentz: Hannover, Germany, Coatings compendia, 2006.
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Figure 2. Particle size distribution (number-average diameters) of silica nanoparticles obtained by TEM. Table 2. Number-, Weight-, and Volume-Average Diameters (dn, dw, dv) and Polydispersity Index (PDI) of Synthesized Silica Nanoparticles dn (nm)
dw (nm)
dv (nm)
PDI
755.6
760.0
757.1
1.006
shown in Table 1. Besides, measured viscosities did not change under the presence of a magnetic field applied, suggesting that nanometric magnetic particles constituting the ferrofluids do not aggregate under the field.22 Ferrofluid magnetization was measured using a Quantum Design (San Diego, CA) MPMS-XL 5.0 T magnetometer. The initial magnetization of the sample was measured from H = 0 to H = 4000 kA/m. The external magnetic field was subsequently swept from þ4000 to -4000 kA/m and then back to þ4000 kA/m. Measurements were carried out at room temperature. To obtain the saturation magnetization, we plotted the magnetization versus the inverse of the applied field (1/H) and carried out a linear fit to the portion of the curve next to the y-axis, which corresponds to small 1/H (large external fields). The point where the linear fit crossed the y-axis was taken as the saturation magnetization (results shown in Table 1). Figure 3 shows bulk magnetization versus external magnetic field (in the range from -2000 to 2000 kA/m) for the three ferrofluids investigated. As observed, a typical sigmoidal Langevin-type curve is obtained. The three ferrofluids have very similar saturation magnetization: 24.76 ( 0.04 kA/m. Permeability curves extracted from magnetization measurements are shown in Figure 3b. In this figure, we show results for the ferrofluid used in the work by de Gans et al.5 As observed, our ferrofluids have a similar magnetic response as those used by de Gans and co-workers. Permeability values shown in Figure 3b will be next used for the calculation of the normalized storage modulus and β contrast factor. 4.2. Small-Amplitude Oscillatory Shear (SAOS) Magnetorheology. SAOS magnetosweep curves are plotted in Figure 4. Here we show typical curves for the storage and loss modulus dependence with the external magnetic field. The storage modulus is associated to the magnetostatic energy stored in the system and hence does depend on the microscopic field-induced structure. At low magnetic fields, particles become magnetized and start to aggregate in the direction of the magnetic field. (22) Odenbach, S. Ferrofluids: magnetically controllable fluids and their applications; Springer: Germany, 2002.
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Figure 3. Magnetic properties of the ferrofluids used (see Table 1). (a) Magnetization curves of the ferrofluids. (b) Permeability curves for the ferrofluids used in this work and that from Figure 3 in de Gans et al.5
Figure 4. Magnetosweep curves for the inverse ferrofluids at different magnetic field strengths, particle volume fractions, and ferrofluid viscosities: (a) ferrofluid viscosity 44 mPa s and (b) particle volume fraction 26.1 vol %.
The average length of these aggregates quickly increases upon increasing the magnetic field. Actually, a simple calculation using the model first proposed by Osipov et al.23 and later adapted by de Gans et al.6 predicts that all aggregates should connect the plates of the rheometer for magnetic fields larger than ∼1 kA/m. In principle, the observed increase in storage modulus upon increasing the field in the range from 10 to 100 kA/m may be associated to the enhanced magnetization of the particles and/or lateral aggregation of initial field-induced chainlike aggregates. At the largest magnetic fields achieved, particles magnetically saturate, reaching their largest magnetic moment. At this stage, the viscoelastic moduli reach a plateau. In fact, the magnetic field associated to the onset of the storage modulus plateau closely corresponds to the saturating magnetic field in Figure 3. A qualitatively similar behavior is found for all IFFs investigated (independently of the silica content and ferrofluid viscosity), suggesting that the physical phenomenon behind their bulk mechanical behavior is alike. Interestingly, the loss modulus increases with the external magnetic field, suggesting that the appearance of chain-chain interactions at high enough volume fractions of the particles contributes more significantly to the energy dissipation in the
system if compared to loose single-width chainlike aggregates existing in the low field region (cf. Figure 4). As expected, in the presence of large enough magnetic fields, the loss modulus is lower than the storage modulus. A full understanding of the loss modulus behavior in magnetorheology is still lacking in the literature. A nonzero G00 value has been frequently ascribed to the existence of loose chains with one or two ends not connected to the surface of the geometry,24 but other mechanisms are present as well at the heart of the problem. 4.2.1. Effect of Silica Concentration. Let us now investigate which is the effect of silica particle volume fraction. As observed in Figure 4a, viscoelastic moduli increase when increasing concentration. Very similar trends in the curves are obtained regardless the ferrofluid viscosity used (results not shown here for the sake of brevity). Such volume fraction dependence is actually expected since more gap-spanning and compacted aggregates would exist if the number of dispersed particles increases. These compacted aggregates would store a larger amount of magnetic energy (larger G0 ) and also dissipate more energy due to more intense hydrodynamic interactions (larger G00 ). Actually, if only single-width chains exist under the field, a linear dependence with concentration is theoretically predicted (see eqs 8-11). It is worth to remark here that a larger ferrofluid viscosity may slow down
(23) Osipov, M. A.; Teixeira, P. I. C.; Telo da Gama, M. M. Phys. Rev. E 1996, 54, 2597–2609.
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(24) McLeish, T. C. B.; Jordan, T.; Shaw, M. T. J. Rheol. 1991, 35, 427–448.
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the aggregation kinetics. However, given that all magnetosweep experiments are run at quasi-equilibrium, ferrofluid viscosity should not qualitatively affect the trends observed for different volume fractions as it happens indeed. 4.2.2. Effect of Ferrofluid Viscosity. As stated before, the storage modulus of a magnetized IFF provides information on the energy storage and arises only from nonhydrodynamic
Figure 5. Loss tangent (tan δ = G00 /G0 ) as a function of magnetic field strength for inverse ferrofluids prepared in different ferrofluid viscosities (44, 198, 559 mPa s). Particle volume fraction = 26.1 vol %.
Article
forces.25 As a consequence, the storage modulus should not depend on the ferrofluid viscosity. In contrast, the effect of ferrofluid viscosity should obviously be embedded in the loss modulus. Actually, the loss modulus captures viscous dissipation phenomena under SAOS-MR having contributions from the magnetic field structuration, interparticle solid friction, and hydrodynamic forces. As a way of example, in Figure 4b, we show results corresponding to 26.1 vol % silica particle concentration and two different ferrofluid viscosities: 44 and 559 mPa s. As expected, the storage modulus does not depend on the ferrofluid viscosity and the loss modulus increases for the more viscous continuous medium. To better understand the effect of ferrofluid viscosity in the MR performance, the loss tangent (tan δ = G00 /G0 ) is calculated as a function of external magnetic field in Figure 5. The loss tangent is a useful material function in this case because both the storage and loss modulus change with the magnetic field strength. In particular, in our case tan δ decreases upon increasing the magnetic field. This means that the magnetic field-induced increase in storage modulus overcomes the increase in loss modulus. Only at very large magnetic fields, in the case of the lowest ferrofluid viscosity employed, tan δ increases possibly due to particle migration. A similar phenomenon was recently observed under the steady-state regime.2 In brief, at large magnetic field strengths, the existence of magnetic field gradients promotes the migration of particles toward the regions of the lowest field. If the ferrofluid viscosity is large enough, it is possible to slow down and eventually hinder the migration. However, particles could certainly move if the continuous medium viscosity is not sufficiently
Figure 6. Comparison between experiments and macroscopic theoretical models for inverse ferrofluids prepared with different ferrofluid viscosities (44, 198, and 559 mPa s) at three different particle volume fractions 12.6, 19.3, and 26.1 vol %: (9) experimental data, (b) experimental data from de Gans et al.,5 (;) spheroidal model (eq 2), (- - -) cylindrical model (eq 3 with C = 2), and ( 3 3 3 ) layered aggregates model (eq 3 with C = 1). Langmuir 2010, 26(12), 9334–9341
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Figure 7. Comparison between experiments and chainlike aggregate theoretical models for inverse ferrofluids prepared with different ferrofluid viscosities (44, 198, and 559 mPa s) at three different particle volume fractions 12.6, 19.3, and 26.1 vol %: (9) experimental data, (b) experimental data from de Gans et al., 5 (;) Klingenberg and Zukoski model (eq 7), (- - -) Martin and Anderson model (eq 8), ( 3 3 3 ) de Gans model (eq 9), (- 3 -) de Vicente model 1 (eq 10), and (- 3 3 -), de Vicente model 2 (eq 11).
large. Of outstanding practical interest is the flow point associated to the magnetic field strength where G0 equals G00 . This point is typically used in SAOS amplitude sweep tests in polymer science to determine the solidlike transition. tan δ values lower than one are associated to a solidlike state, meanwhile tan δ values larger than one are associated to a liquidlike state. Interestingly, as observed from Figure 5, the flow point strongly depends on the ferrofluid viscosity. This finding puts forward that special care must be taken when determining the yield stress using oscillatory tests. 4.3. Comparison with Previous Works and Theoretical Models. In this section, the storage modulus corresponding to the different IFFs investigated in this work are compared to previously obtained results on similar IFFs by de Gans et al.5 and theoretical predictions of macroscopic and chainlike aggregate theoretical models summarized in section 2. In Figures 6 and 7, we show experimental results for the normalized storage modulus G0 n = G0 /(μ0μcrH02) as a function of the contrast factor β (solid squares). Here, the columns of plots have been measured for constant volume fraction, and the rows for constant ferrofluid viscosity. At very low fields (β < -0.16), the experimental normalized storage modulus G0 n presents some scatter associated to the limited torque resolution of our magnetorheometer (results not shown in Figures 6 and 7). Even though we can theoretically reach lower β values (up to -0.5), this is not easy because this would require very narrow gaps between parallel plates for the typical chain length to be of (25) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal dispersions; Cambridge University Press: Cambridge, 1989.
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the order of the gap and hence transfer the torque from one plate to the other. Furthermore, in this low β region, particle aggregates are susceptible to sediment and this may contribute as well to some scatter in the results. This finding is also in agreement with the fact that in the case of using the more viscous ferrofluid this scatter is significantly reduced. As a consequence, only β values larger than -0.16 are shown in the figures. Interestingly, in all cases a nearly quadratic dependence is found in the range -0.16 < β < 0 as expected from macroscopic and microscopic models described in section 2 due to governing dipolar interparticle interactions. At very large β values, all models satisfactorily explain experimental results for the normalized storage modulus. As a reference, results by de Gans et al.5 are also included in Figures 6 and 7 as solid circles. It is worth it to remark that experiments by de Gans et al. were run using a cone-plate geometry. However, negligible differences existed, according to the authors, when the experiments were done using parallel plates. Besides, particles used in this work were around 750 nm diameter (see Table 2), and those used in experiments by de Gans et al. were 354 nm diameter. Nevertheless, this significant difference in size is not expected to be manifested in a different viscoelastic behavior, since gap-spanning chains should exist in both cases.6 By using Osipov theory, we checked that all existing aggregates should connect the plates in the case of IFFs prepared using silica particles having a diameter larger than 200 nm at magnetic fields larger than ∼1 kA/m. As observed from Figures 6 and 7, experimental results obtained here are in very good agreement with those obtained by de Gans et al., especially for the two lowest Langmuir 2010, 26(12), 9334–9341
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concentrations. Slight deviations at large concentrations may be due to a larger magnetic field gradient in the sample volume of the magnetorheological arrangement of de Gans et al. if compared to the one provided by the MCR 501 magnetorheometer used in this work. In fact, a large gradient in the center of the geometry may force nonmagnetic particles to migrate to the center (i.e., the region of lowest field strength), hence reducing the measured torque signal and providing a lower storage modulus. From a careful inspection of Figure 6, we observe that macroscopic models satisfactorily explain experimental observations in the range of concentrations investigated. At low concentration, the spheroidal model captures very well the trend observed. However, the spheroidal model underestimates the experimental results upon increasing the concentration, and now the cylindrical model fits better the experimental values. For the largest concentration, the layered model is the one which better fits the experimental results. As a summary, when particle volume fraction is low enough, noninteracting elongated structures are expected to be formed in the direction of the field and the spheroidal aggregate model satisfactorily explains experimental observations. For larger concentrations, thicker structures are formed that may also mutually interact. However, mutual interactions are not considered in the macroscopic models described here, making them useless at such high concentrations. In conclusion, these results suggest that the spheroidal model satisfactory describes the experiments at low particle content. However, volume fraction dependence is not satisfactorily captured by any of the three macroscopic models considered in this work. In contrast to our findings, de Gans et al.5 and Tang and Conrad26 did claim a proportionality with the particle volume fraction up to approximately 25 vol %. On the other hand, as observed from Figure 7, microscopic models qualitatively capture experimental results as well. The models by de Vicente et al.16 better fit the results for the lowest concentrations. Again, as the concentration increases, the model underestimates experimental results, suggesting that a stronger than linear dependence with the volume fraction is expected in view of the experimental results. This volume fraction dependence is expected to be associated to aggregate-aggregate interactions that are not considered in the simple analytic models described in this work. More work must be done in this direction to elucidate (26) Tang, X.; Conrad, H. J. Rheol. 1996, 40, 1167–1178.
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Article
the effect of particle content in MR properties for medium and large volume fractions. The fact that the de Gans et al. model overestimates experimental results has been argued to be due to the discrete nature of the ferrofluid that provokes a magnetic moment reduction when two silica particles approach each other.5 However, silica particles used in this work are much larger, and as a consequence magnetic moment reduction should be significantly minimized, hence giving a larger elastic response that is not observed.
5. Conclusions Highly monodisperse silica-based (approximately 760 nm diameter) IFFs were prepared, and their magnetic and magnetorheological properties determined under small-amplitude oscillatory shear for a wide range of magnetic fields. Storage modulus values obtained were first compared with previous data from de Gans et al.;5 an excellent agreement was found in spite of the different silica diameters used in the preparation of the IFFs, especially at the lowest silica concentration. Then, results were compared to existing macroscopic and microscopic magnetorheological models; both descriptions satisfactorily captured the storage modulus values at the lowest silica particle volume fraction investigated. Upon increasing the particle volume fraction, deviations from the theory appear, presumably due to aggregate-aggregate interactions and/or the formation of thick columnar structures instead of idealized slender bodies. Apart from the storage modulus, another relevant material function in MR fluids and IFFs is the so-called yield stress. Preferably, steady-state tests have been used in the past. Nonetheless, an estimation of this quantity can also be obtained from oscillatory tests through the flow point (stress value where G0 = G00 ). It is demonstrated here that the flow point strongly depends on the ferrofluid viscosity due to the resulting viscous drag acting on the aggregates under flow. Acknowledgment. This work was supported by MICINN MAT 2006-13646-C03-03 and MEC MAT 2009-14234-C03-03 projects (Spain), by the European Regional Development Fund (ERDF) and by “Junta de Andalucı´ a” P07-FQM-2496, P07FQM-03099 and P07-FQM-02517 projects (Spain). J.R. acknowledges financial support by the “Ministerio de Ciencia e Innovacion: Subprograma Juan de la Cierva” (JCI-2008-2217).
DOI: 10.1021/la100252g
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