Author’s Accepted Manuscript Rheology of a very dilute magnetic suspension with micro-structures of nanoparticles Francisco Ricardo Cunha, Adriano Possebon Rosa, Nuno Jorge Dias www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(15)30454-6 http://dx.doi.org/10.1016/j.jmmm.2015.08.039 MAGMA60521
To appear in: Journal of Magnetism and Magnetic Materials Received date: 27 March 2015 Revised date: 23 July 2015 Accepted date: 7 August 2015 Cite this article as: Francisco Ricardo Cunha, Adriano Possebon Rosa and Nuno Jorge Dias, Rheology of a very dilute magnetic suspension with micro-structures of nanoparticles, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.08.039 This is a PDF file of an unedited manuscript that has been accepted fo publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Rheology of a Very Dilute Magnetic Suspension with Micro-Structures of Nanoparticles Francisco Ricardo Cunha, Adriano Possebon Rosa, Nuno Jorge Dias University of Brasilia, Department of Mechanical Engineering, Laboratory Microhydrodynamic and Rheology - VORTEX, Brasilia-DF, 70910 900, Brazil
Abstract The main objective of this present article is to measure the apparent viscosity of a magnetic suspension in the presence of particles agglomerates of different sizes for several applied magnetic fields, shear rates and particle volume fractions. A secondary goal is to investigate suspension microstructure transition, when subjected to a magnetic field. We show that an employed like virial expansion of two empirical coefficients based only on the experimental data gives a good quantitative description of the magnetorheological suspension effective viscosity up to particle volume fraction less than 0.01. The observed shear rate dependence viscosity is a direct consequence of the stretching, breaking particle structures of different sizes and shapes formed by the action of magnetic attractive force between the polarized particles as observed previously in the context of dense ferrofluids. We have identified even in the limit of a very small particle volume fraction a strong non-linear behavior of the examined suspension due to formation of suspended blobslike aggregates of different sizes and anisotropic chains of particles. These structures are induced by the presence of an external magnetic field and particle-particle magnetic interactions. A histogram of the structure size distribution is also examined. The results of this article are important to those who are interested on the magnetorheological suspensions. Keywords: Magnetorheology, Microstructure, Magnetic fluid, Shear thinning, Blobs-like Aggregates, Anisotropic chains
Email address:
[email protected] (Francisco Ricardo Cunha)
Preprint submitted to Journal of Magnetism and Magnetic Materials
August 10, 2015
1. Introduction Most studies of the characterization of magnetic fluids are motivated by biomedical and industrial applications (Hristov, 1996; Guimar˜aes, 1998; Rinaldi et al., 2005; Cunha and Sobral, 2004; Sartoratto et al., 2005). Recent years have seen rapidly increasing wealth of experimental data and theoretical studies on ferrofluids (Bossis et. al , 2002; Odenbach, 2009). The rheological behavior of a typical magnetic fluid is essentially Newtonian. However, changes in the fluid rheology could be observed when an applied magnetic field induces the formation of particle clusters and structures like chains and new agglomerated structures(Ghasemi, Mirhabibi and Edrissi, 2008). The formation and destruction of magnetically induced structures and the interactions of nanoparticles and aggregates have been explored just more recently (Hosseini et al., 2010). In this context, this paper is devoted to an experimental investigation on the rheology of a dilute magnetic suspension in the presence of micro-structures of nanoparticles induced by the response of the individual particle dipoles to an external magnetic filed. A study on magnetic suspension covering structure changes of these suspensions and some rheological aspects is available in Kroger, Ilg and Hess (2003). The experimental studies made for different magnetic suspensions samples under shear flow have shown that increasing the magnetic field strength yields an increase of the fluids viscosity, the so-called magneto-viscous effect (Odenbach, 2002, 2009; Ilg and Odenbach, 2009; Felderhof, 2001), while increasing shear rate leads to a decrease of the magnitude of the viscosity that corresponds a shear thinning behavior such as observed for elastic liquids (Bird, Armstrong and Hassager, 1987). Zubarev, Fleischer and Odenbach (2005) in the context of polidisperse dense ferrofluids have found that for small shear rate particles aggregates dominates the rheological of these fluids. However, for high shear rate the aggregates are destroyed and the influence of the individual particles dominates the magnetoviscous behavior of the fluid. Under permanent action of an applied magnetic field, Borin, Zubarev, Chirikov and Odenbach (2014) have investigated theoretically and experimentally the long time stress relaxation process in a colloidal magnetic suspension after an applied shear flow being suddenly interrupted. They found that the macroscopic bulk stress of the investigated magnetic fluid can be characterized from a rheological point of view by the presence of linear chains and bulk dense drop-like aggregates. More recently, Gontijo and Cunha (2015) have carried out dynamic simulations of a magnetic suspensions with particles interacting magnetically in 2
a periodic box in the presence of an applied magnetic field. These simulations demonstrate the gradual formation of particle structures like aggregates and its transition to anisotropic chains of aligned particles induced by magnetic effects. For the scale of non-Brownian magnetic particles suspended in a Newtonian fluid interesting new mechanisms as a consequence of irreversible particle interactions have been investigated in presence and absence of a magnetic field and dipole-dipole interactions (Abade and Cunha, 2007; Cunha and Couto, 2011; Cunha, Gontijo and Sobral, 2013a,b). This mechanism is also responsible for magnetic-hydrodynamic dispersion and the break of particle trajectory symmetry due to particle interactions throughout the suspension (Cunha and Couto, 2008). Understanding the mechanisms of particle distributions and structure formation resulting from particle-particle interactions and particle interaction with an external field is very important to predict and characterize the rheology of such complex suspensions (Odenbach, 2002; Bossis et. al , 2002; Odenbach, 2009). More recently, Horv´ath et al. (2015) have used magnetorheological fluid dampers in order to attenuate rotor oscillations of a stepper motor. The magnetic fluid was modeled by the Bingham-Papanastasiou model of viscoplastic media. This paper presents experimental results obtained for a suspension of magnetic particles in the presence of a heterogeneous distribution of Brownian and non-Brownian aggregates subjected to a shear and an uniform magnetic field. The objectives for this first study from our laboratory of microhydrodynamics and rheology, are simply to measure the apparent viscosity of the magnetic suspension for several applied magnetic field, shear rates and particle volume fractions. A secondary goal is to investigate suspension microstructure transition, when it is subjected to a magnetic field in order to understand the direct connections of the structure and rheology of a magnetite magnetic suspension. We show that even a very dilute suspension composed of nanometric magnetic particle of magnetite with particle volume fraction less than 1% when undergoing shear and an applied magnetic field gives rise to a nonlinear rheology as a consequence of suspension anisotropy and particle aggregate in the presence of magnetic particle interactions and a magnetic applied field. We present results of an experimental investigation of the magnetorheological viscosity η/η0 of an anisotropic and heterogeneous magnetic suspension as a function of the non-dimensional applied magnetic field α, particle volume fraction φ and shear rate γ. ˙ We have been able to demonstrate that an employed like virial expansion O(φ3) of two empirical coefficients based on the 3
experimental data gives a good quantitative description of the ferrofluid effective viscosity up to particle volume fraction less than 1%. Actually, our approach provides a quantitative description on the instability of the examined fluid that occurs as a result of the presence of typical aggregates composed of three particles distributed in the fluid domain. A strong non-linear regime of the fluid dominated by suspended blobs-like aggregates and anisotropic chains of particles induced by the presence of an external magnetic field and particle-particle magnetic interactions is identified. The results of this article can be important to those applications involving magnetorheological fluids. 2. Description of the Experiments Magnetic suspensions composed of magnetite nanoparticles with 8 nm of average diameter and density 5.3g/m` dispersed in a Newtonian mineral oil of density 0.87 g/m` and viscosity of ηo = 147.69 ± 4.51 cP at 25o C as the fluid base were investigated. The value of the pure mineral oil viscosity is used as the reference value in order to make the viscosity measurements non-dimensional. The instrument apparatus used to perform the experiments for investigating the rheology of magnetic suspension in the presence of an applied field was a parallel disc geometry model MCR 301 Anto Paar rheometer of radius R which has been adapted a magnetic cell as shown in the schematic of figure (1). The torque T recorded by the rheometer inside the magnetic cell should be related to the magnetic field H, the particle volume fraction φ and the shear rate γ˙ = !ωr/h, where r is the radial coordinate, !ω the angular velocity and h is the gap between the parallel disks. For a fluid with a small yield stress described by τ = η(φ, H, γ) ˙ γ˙ the torque is calculated by the software Rheoplus of the device as being 1 T = πR3 η(φ, H, γ˙ R) γ˙ R . (1) 2 Each value of viscosity corresponds to a convergent average obtained statistically over five experimental realizations. Figure (2) illustrates a typical result which exemplifies how the viscosity measurements were conducted in our experiments. For each time in the plot of this figure is made an average over the five experimental realizations. The final values of the measured viscosity is taken as being the time average at a sufficiently long time (i.e. t ≥ 200 s) corresponding to the steady state rheology. 4
Air filtration system Pneumatic control system Compressor
Sample
Rotating disk
Magnetic Cell
Power Supply
Computer
Figure 1: Instrument schematic diagram for the experimental data on measurements of fluid viscosity as a function of an uniform applied magnetic field. A rheometer MCR 301 of parallel disc geometry is connected to a magnetic cell.
In order to control the problem of the magnetite particle sedimentation in the gap of the rheometer, a typical time scale of an experimental run, tE , should be much smaller than the time scale for a particle to fall a characteristic length `, tP , that means tE tP = `/Us , where Us is the Stokes velocity, Us = 2a2 Δρg/(9µ). Here, a is the particle radius, Δρ is the density difference between particle and fluid, g is the gravity and µ fluid viscosity. In our experiments tE /tP was typically about 10−2 . The density of magnetic flux B can vary from 0 to 1 Tesla by controlling the electrical current between 0 to 5 A. The temperature of the magnetic cell was kept constant at 25 o C by a system of recirculating water using a thermal bath. All magnetic quantities are controlled by the software Rheoplus of the rheometer-magnetic cell. From Ampere’s law the magnetic intensity vector H can be calculated. The software Rheoplus controlling the rheometer gives all fundamental variable of the experiments either magnetic and hydrodynamic. The magnetization of the fluid is quantified by the well-known relation B = µ0 (M + H). A procedure of demagnetization is always performed before applying a new magnetic field. The Joule effect produced inside the magnetic cell by higher applied electric currents can be neglected if the measurements under an applied field is considered only in the period of 100 s. In this interval we can see that the viscosity variation due to the sample heating inside the cell is less than ±1% for an applied field 196000A/m and a shear 5
180
175
Experimental Data Local Average Final Average
hη (cP)
170
165
160
155
150
145 0
100
200
300
400
500
Time (s)
Figure 2: Typical viscosity measurement as a function of the time. The errobars are calculated based on the statistics over five experimental realizations. Here, φ = 0.005, γ˙ = 100s−1 and α = 0.
rate 100s−1 . The system of magnetic cell adapted to the rheometer can be easily set up in our laboratory and it is capable of measuring viscosity as a function of an uniform magnetic field accurately and the instantaneous data can be used to study the statistics (i.e. average and variance) of the viscosity measurements (Rosa, 2014). For Coutte flow u = u(y)i, in a transverse magnetic field H = Hj, the magnetic rotational viscosity ηr is defined by ηr = M1 H2 /2 Ω (Shliomis, 1972; Odenbach, 2009), where Ω is the vorticity. This represents a ratio between the internal magnetic torque and the angular velocity. The magnetic field is represented in non-dimensional terms by the α parameter defined as follows (Rosensweig, 1985) µo mH α = . (2) kT Here, α denotes the non-dimensional magnetic field and it measures the relative importance between the magnetic and Brownian forces in the examined suspension. In equation (2), µo = 4π × 10−7 H/m is the magnetic permeability of the free space, k = 1.38 × 10−23 Nm/K is the Boltzmann constant, T is the absolute temperature, i.e. 298K in this work. The uniform applied magnetic 6
H by the magnetic cell in our experiments is varied from 0 to 196000A/m. The magnetite solid magnetization is Md = 4.46 × 105 A/m (Rosensweig, 1985). The particle dipole moment m used to calculate α is given by m = Md vp ,
(3)
where vp is the particle volume. For particles having a mean diameter of 8nm gives a m = 1.20 × 10−19 Am2 . So, the α parameter can be always calculated. 2.1. Suspension Microstructure In order to investigate possible microstructure transitions of the magnetic suspensions an optical microscopy BX51 Olympus with magnification of 500 times and an UC30 CCD camera of 3.2. megapixels is used. The suspension sample were prepared, homogenized for about 15 minutes and then carefully collected for the an appropriate millimeter and subjected to a magnetic induction field B = 0.3T by approximately 1 minute. The microscopy technique was capable of directly imaging the structure transitions and of the magnetic examined suspension in the presence of a magnetic field for several particle volume fraction. Figure (3) show images of the magnetic suspension structure examined in the present work with the microscope-image system. The images were taken for at a magnetic field null and H0 ≈ 71kA/m for three different particle volume fractions. The images show clearly a structure transition when a magnetic field is applied to the suspension sample. At lower volume fraction, φ = 0.1%, figure (3)a shows a suspension with a heterogeneous distribution of small micro-blobs like aggregates of magnetic nanoparticles ranging for 10 to 100µm. However, these structures are approximately isotropic when compared with the drastic anisotropic ones seen in figure (3) due to the long chains of particles formed in the presence of a magnetic field H0 ≈ 71kA/m (i.e. 0.3 Tesla) for φ = 1%. This findings can also be illustrated by the histograms shown in figure (4). We can see from this plot that in the absence of a magnetic field on the suspension sample the size distribution is much more homogeneous and isotropic with a mean diameter of the aggregates approximately 120µm. The suspension structure size distribution is well-described by a Gaussian probability density function. In contract, when an external magnetic field is applied the suspension presents a non-homogeneous probability distribution, being wellcharacterized by anisotropic long chains structures of appreciable frequency 7
Figure 3: Experimental photographic images of the microstructure transitions for a suspension of magnetite (F e3 O4 ) taken at in the absence of magnetic field (i.e. H0 = 0) and for a magnetic field strength H0 ≈ 71kA/m (B ≈ 0.3T ) for different particle volume fractions (a) φ = 0.1%; H0 = 0. (b) φ = 0.5%; H0 = 0 (c) φ = 0.5%; H0 = 71kA/m. (d) φ = 1%; H0 = 71kA/m. The indicated scale on the bottom of the right hand side of the image is 200 µm for images a, b, c and 600 µm for image d.
8
B=0
Frequency (%)
10
B = 0.3 T Gaussian Distribution
8
6
4
2
0 100
200
300
400
500
600
700
Size (µm) Figure 4: Histogram of the aggregates size distribution. Comparison of the structure size distribution in a dilute magnetic suspension of φ = 1.0% in the absence and with an applied external magnetic field to the suspension sample. We include the Gaussian distribution as a reference for the experimental data.
of occurrence, with size ranging from 100µm to approximately 700µm. This long chain are formed by the particles dipole aligning with the direction of the applied magnetic field. So, the internal attractive magnetic force keeps the chains structures unless the shear flow on the structure was sufficiently strong to degrade or even breakup it. In the next section, we might discuss that the increase in the magnetic effective viscosity with the field can be explained physically by these chainslike structures formed in the suspension and being aligned with the magnetic field as already seen in the figure (3). We could anticipate that for a sufficiently high shear rate of our experiments (i.e. when the magnetic and hydrodynamic time scales appear at the same order of magnitude) aggregate size distribution can indicate the existence of structure break up by the shear flow and can explain the shear thinning behavior observed on the rheological response even at the regime of the very dilute magnetic suspension studied here. 9
3. Results of the Suspension Rheology In this section, we present results of rheometry for this suspension. The effective viscosity of the magnetic suspension relative to the ambient fluid η/η0 was measured as a function of the non-dimensional magnetic field α, the non-dimensional shear rate γ/ ˙ γ˙ max and the volume fraction φ of particles ranging from 0 to 0.01. All of the data reported here were obtained over statistics of five experimental realizations and a time average for a sufficiently long time to reach the steady state rheology. 3.1. Effective viscosity We define the non-dimensional effective viscosity of the dilute magnetic suspension investigated in this work as the fluid ambient viscosity (i.e. the unit) added to the viscosity contribution due to the pure effect of the particle volume fraction and also the one produced by the magnetic field. We can express this simply as η = 1 + Δη(φ, α, C), (4) ηo where Δη takes into account the standard effects due to particle volume fraction φ, the magnetic field α and suspension structure C on the suspension rheology. Here C depends on the particle magnetic interactions and structure transition due to aggregates and clusters formation. Actually , Δη(φ, α, C) can be expressed as being the result of three contributions Δη(φ, α, C) = Δηe (φ) + Δηs (φ, α) + Δηnl (φ, α, C).
(5)
First, Δηe (φ) represents the pure contribution due to particle volume fraction such as an Einstein viscosity(Einstein, 1956); secondly in the presence of a magnetic field there will be an linear effect associated to the individual rotation of the particle dipole moments. Under a field action, particles cannot rotate freely with the vorticity of the shear flow. Now, there is a balance between the mechanical torque and the magnetic torque. This effect arises even at a very dilute regime of a magnetic suspension when the dipole-dipole interactions are neglected. In the present context, we define this contribution as being a rotational viscosity equivalent to Shliomis viscosity, Δηs (φ, α) (Shliomis, 1972, 1994; Bacri, Perzynski, Shliomis and Burde, 1995). So, Δηs (φ, α) can be interpreted as the linear part of the magnetic contribution to the effective viscosity of a very dilute suspension. Finally, 10
Δηnl depends on the particle configuration and will represent a non-linear magnetic contribution due to induced magnetic aggregates and dipole-dipole magnetic interactions between individual particles, blobs-like aggregates and anisotropic chains of different sizes, depending on the particle volume fraction involved. Now, we can propose a way to quantify the nonlinear contribution Δηnl in terms of the ηηo measurements in our laboratory for the regime of dilute magnetic suspensions. This can be expressed in the simple form: Δηnl =
η − (1 + Δηe + Δηs ) . ηo
(6)
In equation (6) Δηe and Δηs can be estimated by theory of infinitely dilute homogeneous monodisperse suspension by calibrating the coefficient using our experimental data. Under these conditions theoretical expressions for Δηe and Δηs are well-known and given respectively by Δηe = k1 φ
and
Δηs = k2 φ
αL(α) , 2 + αL(α)
(7)
where k1 = 5/2 and k2 = 3/2. Here, L(x) is the Langevin function L(x) = coth(x)−1/x (Rosensweig, 1985). Even the real suspension being dilute, it is not monodisperse, statistically homogeneous and isotropic. So, the constants k1 and k2 must be adjusted in order to fit our experimental measurements. Based on our experimental data, the equivalent Einstein dilute regime of η/ηo with φ ranging from 0 to 0.01 can be observed, as shown in figure (5) only in the absence of magnetic field (i.e. α = 0). It is seen a quite linear variation of the non-dimensional effective viscosity with the particle volume fraction at a shear rate of 100s−1 . The data are simply fitted by a O(φ) relationship given by: η = 1 + 11φ, (8) ηo with k1 = 11 for the best fit of the experimental data. This value of k1 is about four times bigger the 5/2 of the Einstein theory. As mentioned above k1 = 11 appears in the O(phi) coefficient only due to existence of particle agglomerates of different sizes and shapes. Otherwise there must be the Einstein multiplier 5/2. Indeed, in our experiments the dispersion even being very dilute it was not statistically homogeneous of identical rigid spherical particles as considered by Einstein’s suspension theory. In contrast, there is a particle distribution characterized by a multi-structures of different length 11
1.2
1.15
hη / hηo
1.1
1.05
1
0.95
0
0.002
0.004
0.006
0.008
0.01
fφ
Figure 5: The linear regime of the non-dimensional effective viscosity as a function of the particle volume fraction for α = 0 and γ˙ = 100s−1 . The solid line in the plot represents the fitting curve given by the relation (8). The experimental errorbars shown in the plot are based on a statistics over five realizations and on a time average for a sufficiently long time of the experiments.
scales due to aggregates of different sizes, shapes and particle volume fraction that depends on the way of the fluid synthesis. So, k1 = 11 reflects the method of the fluid preparation and consequently can not have an universal character. We now present in figure (6) the Δηnl as a function of φ for three different values of the non-dimensional magnetic field α. As expected at very low particle volume fraction, φ ∼ 0.001 this nonlinear contribution goes to zero, because dipole-dipole interactions, which decay with 1/r 3 for torque-dipole and 1/r 4 for force-dipole, and the presence of a number of aggregate structures with a sufficient size for interacting magnetically produce only a second order effects to the suspension rheology. Therefore, even in the presence of an appreciable magnetic field on the suspension sample like α = 5, the nonlinear effects produced by magnetism can be neglected at very low volume fractions. In another words, this means that the rheological contribution from the particle dipole response to an applied uniform magnetic field will always be associated with particle dipole rotation reflected as a linear effect on Δηe . In addition, figure (6) indicates that at higher volume fraction, say 12
1
0.8
DΔhηnl
0.6
0.4
0.2
0 0
0.002
0.004
0.006
0.008
0.01
fφ
Figure 6: Nonlinear contribution of the non-dimensional effective viscosity as a function of particle volume fraction for a constant shear rate γ˙ = 100s−1 . In this plot: the black squares denotes α = 1.45 , black circles α = 2.61 and black triangles α = 4.51.
0.003, but still small the nonlinear magnetic effects on the suspension rheology is already perceptible even for the small applied field α = 1.45. In this volume fraction there is a number of blobs-like aggregates with sufficient size capable to interact magnetically and produce nonlinear deviations on suspension rheology. For φ = 0.01 and α = 4.51 this contribution reaches a value of 0.8 the viscosity of the ambient fluid. In this case the suspension is quite anisotropic and basically composed of anisotropic long-chains formed due to the alignment of the dipole of the magnetic particle with the field direction. Again, a typical structure transition of a heterogeneous magnetic suspension for a fixed applied field and different particle volume fraction has been already presented in figure (3). 3.2. Rheology dependence on the magnetic field Figure (7) shows a plot of the non-dimensional effective viscosity of the magnetic suspension as a function of the non-dimensional applied magnetic field α for a constant shear 100s−1. The results are presented for three particle volume fraction φ. We can see from this result that increasing the magnetic field strength α yields an increase of the suspension viscosity. The saturated values of the viscosity seen in the plots for each volume fraction φ correspond 13
to an equilibrium configuration of particle orientation and size distribution of structures defined, in average, as the balance between the mechanical torque of the shear and the magnetic torque on the aggregate-like structures at large values of α, when the effect of the Brownian motion is neglected. For instance, at the constant shear rate γ˙ = 100−1 and varying the field from zero to its maximum value of α associated to Hmax = 196A/m, the increase in non-dimensional effective viscosity η/η0 is about 3 for the higher volume fraction, φ = 0.01 investigated. We also can see from this plot that even for the lower dilute regime of the suspension examined here, i.e. φ = 0.001, an increase in the saturated value η/η0 of 10% is still observed for α ≥ 5. In this very dilute regime the long-chains anisotropic structures are not formed. These results are direct connected with the formation of network of particles or micro blobs-like aggregates throughout the suspension as shown in figure (3). This magneto-viscous effect in the presence of an uniform magnetic field on the suspension sample has been also observed by previous experimental works for more concentrated suspension of magnetic nanoparticles and nanoaggregates (Odenbach, 2002; Borin, Zubarev, Chirikov and Odenbach, 2014). As a complementary result for this discussion, in the plot of figure (8) the values of the non-dimensional effective viscosity η/η0 as a function of α is shown for a very dilute regime of the magnetic suspension with φ = 0.001 at a constant shear rate. Actually, this results correspond to the insert shown in figure (7). We present a comparison of the η/η0 with the linear regime of the magnetic contribution Δηs (i.e. Shliomis limit) predicted by the equation (7)(Shliomis, 1972). This plot indicates that at the limit of a very dilute suspension subjected to a magnetic field, for instance α ∼ 1 the linear Shliomis theory Δηs fits pretty well the total viscosity η/η0 . It seen that rotation of the non-interacting particle dipole induced by the presence of a magnetic field is the dominant mechanism on the rheology for moderate α in a very dilute regime. In figure (8) the data points and the theoretical curve of Shliomis (1972) are in good agreement for higher values of the non-dimensional magnetic field, α ≥ 3. The particles just develop strong rotational motion entirely induced by the field i.e. η/η0 ≈ 1 + Δηs . This rotational motion increases internal dissipation inside the fluid and consequently produce an increase in the suspension viscosity. The suspension is free of dipole-dipole interactions and particle aggregates of appreciable sizes.
14
5 4.5
1.11
hη / hηo
1.08
4
1.05 1.02
3.5
hη / hηo
0.99 0
3
1
2
3
aα
4
5
6
2.5 2 1.5 1 0
1
2
3
4
5
6
aα
Figure 7: Non-dimensional effective viscosity of the magnetic suspension as a function of the non-dimensional magnetic field, α for a constant shear rate, γ˙ = 100s−1 . The different curves in the plots are associated with three different particle volume fractions. black squares φ = 0.001 (insert) , black circles φ = 0.005 and black triangles φ = 0.01. The errorbars presented in the plot are estimated by statistics over five experimental realizations and the data points are based on a time average at a sufficiently long time to produce steady state rheology.
15
1.12 1.1
hη / hηo
1.08 1.06 1.04 1.02 1
0
1
2
3
4
5
6
7
aα
Figure 8: Non-dimensional viscosity as a function of the non-dimensional magnetic field α for a very dilute magnetic suspension with φ = 0.001 an d a constant shear rate γ˙ = 100s−1. The solid line represents the magnetic contribution at the very dilute limit Δηs (i.e. Shliomis limit) (Shliomis, 1972) given by equation (7) with the best fitting constant based on our experimental data k2 = 165. The errobars are estimated as described before.
16
3.3. Rheology dependence on the shear rate Figure (9) shows a typical shear rate dependence viscosity behavior of the dilute magnetic suspension with φ = 0.01 of magnetite. Shear rates ranging from γ˙ = 20s−1 to γ˙ max = 2000s−1 were used. The experimental data were fitted by an empirical Sisko correlation used for describing generalized Newtonian fluid given by (Bird, Armstrong and Hassager, 1987) η = 1 + k γ˙ n−1 , (9) ηo where k e n are calibrated constants with experimental data. We show in the plot of figure (9) a comparison between experimental data for applied magnetic fields α = 4.51 and α = 0. In the absence of an applied field the magnetic suspension behaves like a Newtonian fluid with an effective viscosity give by Einstein (Einstein, 1956). The shear thinning occurs when the viscous hydrodynamic force related with the applied shear dominates the net magnetic dipole-dipole interaction forces responsible for keeping the particles in the aggregate. The viscous hydrodynamic forces stretch and break the micro-aggregates and chains structures formed by attractive magnetic forces between the particles inside the blob-like structure. Consequently, the suspension flow more easily decreasing its apparent viscosity. Therefore, the shear rate dependence viscosity is only observed if the shear rate produce a mechanical torque sufficiently strong to dominate the magnetic torque, and consequently to break the aggregate-chain-like structures formed in the magnetic suspension after the application of a magnetic field. In another words, the time scale of the applied shear 1/γ˙ should be of same order of a typical magnetic structure time scale with a characteristic length L0 ; ρL0 /Δη 0 . Here, Δη 0 is a typical value of the viscosity increment by magnetic effects due to particle-particle dipole interactions and formation of particle agglomerates. 3.4. Rheology dependence on particle volume fraction In this section we present experimental results of the effective viscosity of the examined magnetic suspension at low particle volume fractions. A connection between the suspension microstructure shown in section §2.1 with the suspension non-dimensional effective viscosity data is depicted in Figure (10). This figure presents a plot of the non-dimensional effective viscosity as a function of the particle volume fraction for a constant shear rate of 100s−1 and three different values of the non-dimensional magnetic 17
5 4.5 4
hη / hηo
3.5 3 2.5 2 1.5 1 0
0.2
0.4
0.6
.
0.8
1
.
gγ / gγmax
Figure 9: Non-dimensional viscosity of the magnetic suspension as a function of the nondimensional shear rate γ/ ˙ γ˙ max for φ = 0.01. In this plot γ˙ max = 2000s−1 denotes the maximum applied shear rate. The circles denotes α = 0 ; The boxes α = 4.51; The solid line represents the Sisko empirical model proposed (Bird, Armstrong and Hassager, 1987) giving by equation (9), with the fitting constants k = 0.25 e n = 0.35.
18
field, α. Under condition of α = 0, the viscosity of the very dilute suspension with φ < 0.01 follows closely the linear Einstein’s O(φ) for all particle volume fraction examined in the interval 0 ≤ φ ≤ 0.008. We can connect this conclusion looking at the inserts of the suspension structure shown in figure (10). In the absence of magnetic field the suspension is composed of small bounding spherical micro-blobs-like aggregates with a distribution in space approximately statistically homogeneous so that an order O(φ) of the Einstein’s law is valid. In this regime, the suspension is composed of small micro-blobs-like aggregates particles approximately spherical, but with different sizes. The micro-blobs of particles are distributed in space almost statistically homogeneous. This justifies the linear behavior observed. In contrast, when a magnetic field is applied , the suspension behavior deviates significantly from Einstein’s linear regime even over surprisingly very small values of φ, i.e. φ ≈ 0.4%. The suspension viscosity increases rapidly with the particle volume fraction requiring higher order corrections terms such as O(φ2), O(φ3) and so on . In addition, we can see from the inserts in figure (10) that when an external magnetic field is applied chain-like structure forms and they align with the magnetic field producing a strong anisotropy and resistance for the shear stress. These anisotropic structures can be seen even at a very dilute regime of our observations (φ ∼ 0.1%). Indeed, this explains the higher values of suspension viscosity when comparing with the viscosity for α = 0. In the presence of a magnetic field there are critical values of particle volume fractions (φc ) for a structure transition occurs from a linear behavior (i.e. Einstein regime) of the dilute magnetic suspension to a nonlinear behavior, involving formation of large number of aggregates and long-chains of particles by action of the interparticle magnetic interactions. Actually, these values define a function φc = G(α, γ/ ˙ γ˙ max ) that appears to be a decreasing function of α and an increasing function of γ˙ in the same interval of volume fractions investigated here. Now, considering the limit of very small volume fraction and motivated by the theories O(φ) and O(φ2) for an isolated and two interacting particles in simple shear, respectively, we use a Taylor’s series in order to represent the function ˙ γ˙ max )φ F (φ, α, γ/ ˙ γ˙ max ) = C1 (α, γ/ ˙ γ˙ max )eC2 (α,γ/ ,
(10)
at the limit φ !→ 0 . This low volume fraction regime is consistent with our experiments. We will fit non-Newtonian effects on the examined magnetic 19
3
hη / hηo
2.5
2
1.5
1 0
0.002
0.004
0.006
0.008
0.01
fφ Figure 10: Non-dimensional viscosity as a function of particle volume fraction φ for four values of the applied magnetic field and a constant shear rate γ˙ = 100s−1 . The inserts of figure represent different microscopy images of the magnetic suspension structure showing configurations of aggregate with different sizes and also very long anisotropic chains for higher particle concentration and magnetic field black-squares α = 0 , the circles denotes α = 1.45 , right-triangles α = 2.61 and left-triangles (upper points) α = 4.51.
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suspension with higher order terms in φ just using a power series approximation where the kth term is given by (k!)−1 C2k φk . Then, η C22 φ2 + O(φ4) = 1 + C1 φ 1 + C2 φ + (11) ηo 2
In general, as mentioned before, the coefficients C1 and C2 are functions of the non-dimensional applied field α and the non-dimensional shear rate γ/ ˙ γ˙ max . It is seen that the proposed series (11) fits very well the experimental data when the expansion is truncated at termO(φ4 ) and C1 = 87 and C2 = 60 are just constants of calibration for a magnetic field α = 2.61 and a shear rate γ˙ = 100s−1. Figure (11) shows a plot of the non-dimensional viscosity as a function of the particle volume fraction. In this result a best fitting curve O(φ4) is used based on the series given in (11). Actually, the experimental data points and the series correlation with C1 = 87 and C2 = 60 are indistinguishable up to φ = 0.5%. In the insert of the same figure the linear regime of the suspension is shown to be valid up to φ ≤ 0.2%. We have demonstrated that using a virial expansion O(φ4 ) of two empirical constant based only on the experimental data without solving a complex Fokker-Planck equation gives a good quantitative description of the ferrofluid effective viscosity up to particle volume fraction less than 1%. This approach also provides a quantitative description on the instability of the investigated fluid that occurs as a result of the presence of aggregates composed of three or more particles distributed in the fluid domain. It seems that the interactions of three-body it already strong enough to explain the dynamical formation of the observed aggregates. Actually the conditional probability for finding a test particle in a position of the suspension domain given that there are two neighboring particles must be proportional to φ3 . The observed shear thinning is a direct consequence of the stretching and orientation of these structures by the applied shear. A quite non-linear regime of the fluid dominated by suspended micro-blobs-like aggregates and long anisotropic chains of particles induced by the presence of an external magnetic field and particle-particle magnetic interactions has been identified. 4. Final Remarks The results presented in this work have shown that under action of magnetic fields there is a strong connection between the rheological behavior of a magnetic suspension and its microstructure even for regimes of very 21
2.5
hη / hηo
1.6 1.4
hη / hηo
1.2 1
2
0
0.002
fφ
0.004
1.5
1 0
0.002
0.004
0.006
0.008
0.01
fφ
Figure 11: Comparison of the data points with the series expansion O(φ3 ) for the nondimensional viscosity as a function of particle volume fraction φ for γ˙ = 100s−1 and α = 2.61. The insert in this plot shows the linear regime of the suspension viscosity up to 0.2% even in the presence of a magnetic field. The solid line represents the fitting O(φ) of the series expansion; the dashed curve is O(φ2 ) of the series and the doted curve is the O(φ3 ). Here, C1 = 87 and C2 = 60.
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dilute suspension. We have explored this micro-macro connection in term of suspension structure transition and the associated rheology, respectively. Some images of the the suspension structure shown distributions of particle micro-blobs-like aggregates and anisotropic chains of particles induced by the presence of a magnetic field for different volume fraction and magnetic field intensity were presented and discussed their link with the suspension rheology. We have captured the increase of the suspension effective viscosity with the applied field as a consequence of structure transition with particle aggregates of different sizes and shapes and long-chains in the case of a sufficient high magnetic field. A histogram of the size structures was calculated, indicating a quite non-homogeneous size distribution with structures ranging from 20µ to 800µm, when a magnetic field is applied at dilute regime of the suspension. In contrast, in the absence of a magnetic field the size distribution is described with good approximation by a normal distribution. We also observe the effect of structure breaking as increasing the shear rate even particles interacting hydrodynamically and magnetically inside the aggregates. Consequently, the apparent viscosity of the suspension becomes smaller with increasing shear rate, resulting in a shear rate dependence viscosity for the examined heterogeneous and anisotropic magnetic suspension. This shear thinning effect has been previously observed experimentally for a dense polydisperse ferrofuid by Zubarev, Fleischer and Odenbach (2005); Borin, Zubarev, Chirikov and Odenbach (2014). This behavior occurs when a time scale of the magnetic forces between the magnetic particles is compatible with the time scale of the viscous force acting on the suspension structures. So, our results have also suggested a shear rate dependence viscosity of the magnetorheological suspension investigated for moderate and high shear rate, even for regimes of very low particle volume fractions, i.e. φ ≤ 0.01. We have captured the linear dependence of the effective viscosity on the particle volume fraction (i.e. Einstein regime) for a very dilute suspension explored in this work with no applied field. However, one of the most interesting phenomena inferred from this experimental study was that in the presence of a magnetic field on the suspension there still exists at low volume fraction a linear behavior of the suspension viscosity identified as a rotational viscosity or Shliomis limit. This is produced by particle rotation as the response of particle dipole interacting with an external field. In this limit the dipole-dipole interactions are neglected. On the other hand, nonlinear effects arise as the volume fraction and the magnetic field increase. The nonlinear effects are clearly associated with larger polidisperse blob-like 23
aggregates throughout the suspension domain and in a more extreme case with the anisotropic chains of particles as the magnetic field is increased. This leads to a flow anisotropy and changes in the viscometric properties of the magnetic suspension. The increase of the particle volume fraction also makes the dipole-dipole interactions more significant inside the structures which leads to a deviation from the linear pure rotational regime in the presence of a magnetic field. An excellent agreement between experimental data and the series model O(φ3) also is observed. Actually, the experimental data points and the series correlation with C1 = 87 and C2 = 60 are indistinguishable up to φ = 0.5%. It should be important to note however that the coefficient C2 is not universal and depend on the magnetic field and on the suspension structure even at this very dilute regime explored here. The approach here also provides a quantitative description on the instability of the magnetic suspension that occurs as a result of the presence of typical aggregates composed of three or more particles distributed in the fluid domain. The observed shear thinning is a direct consequence of the stretching and orientation of these structures by the applied shear. A strong non-linear regime of the fluid dominated by suspended blobs-like aggregates and anisotropic chains of particles induced by the presence of an external magnetic field and particle-particle magnetic interactions has been characterized. Indeed, investigations on the effect of particle volume fraction on the suspension structure transitions (i.e. magnetic and hydrodynamic interactions) and their consequences to the suspension rheology still offers a number of challenging questions and stimulate future researches.
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Abade, G.C. and Cunha, F.R., 2007, Computer simulation of particle aggregates during sedimentation, Comput. Methods Appl. Mech. Engrg. 196, 4597–4612. Bacri J.-C, Perzynski, R. Shliomis, M.I. and Burde, G.I, 1995. Negative viscosity effect in a magnetic fuid, Phys. Rev. Lett., 75, 2128. Bird, R. B., Armstrong, C., Hassager, O., 1987. Dynamics of polymeric liquids - Volume 1: Fluid mechanics, John Wiley & Sons, New York. Borin, D.Y., Zubarev, A.Y., Chirikov, D.N. and Odenbach, S., 2014. Stress relaxation in a ferrofluid with clustered nanoparticles, J. Phys.: Condens. Matter 26, 406002. Bossis, G., Volkova, O., Lacis, S. and Meunier, A. 2002. Magnetorheology: fluids, structures and rheology, LNP, 594, 202–230. Cunha, F. R., Sobral, Y.D., 2004. Characterization of the Physical Parameters in a Process of Magnetic Separation and Pressure Driven Flows of a Magnetic Fluid in a Cylindrical Tube, Physica A, 343, 36–64. Cunha, F. R., Couto, H.L.G, 2008. Transverse Gradient Diffusion in a Polydisperse Dilute Suspension of Magnetic Spheres During Sedimentation, J. Phys: Condens. Matter., 20, 204129. Cunha, F. R., Couto, H.L.G, 2008. On the influence of the hydrodynamic interactions on the aggregation rate of magnetic spheres in a dilute suspension, Journal of Magnetism and Magnetic Materials, 323/1, Pages 77–82. Cunha, F.R., Gontijo, R. G., Sobral, Y. D., 2013. Symmetry breaking of particle trajectories due to magnetic interactions in a dilute suspension, Journal of Magnetism and Magnetic Materials, 326, 240–250. Cunha, F.R., Y.D. Sobral, Gontijo, R.G., 2013. Stabilization of concentration waves in fluidized beds of magnetic particles, Powder Technology, 24 (1C), 219–229. Einstein, A. 1956. Investigations on the Theory of the Brownian Movement, Dover Publications. Felderhof, B.U. 2001. Flow of a ferrofuid down a tube in an oscillating magnetic field, Phys. Rev. E 64, 21508. 25
Gontijo, R.G. and Cunha, F.R. 2015. Dynamic numerical simulations of magnetically interacting suspensions in creeping flow, Powder Technology 279, 146–165. Guimar˜aes, A.P., 1998. Magnetism and Magnetic Resonance in Solids, John Wiley and Sons, INC. Horv´ath, B., Medvegy, T., Kronome, G., Gugolva, Z., Moln´ar, G., Szalai, I., 2015. Controllable Electro- and Magnetorheological Damper for Oscillation Reduction of a Stepper Motor, International Journal of Mechatronics, Eletrical and Comouter Technology, 5(15), 2068–2076. Hosseini, S.M., Fazlali, A., Ghasemi, E., Moghaddam, H. A., Salehi, M. 2010. Rheological properties of a γ-F e2 O3 paraffin-based ferrofluid , Journal of Magnetism and Magnetic Materials, 322, 3792–2796. Hristov, J., 1996. Fluidization of particles in a magnetic 5ed. Part 1: the effect of field line orientation on bed stability, Powder Technol. 87, 59. Ghasemi, E., Mirhabibi, A., and Edrissi, M. 2008. Synthesis and rheological properties of an iron oxide ferrofluid , Journal of Magnetism and Magnetic Materials, 320, 2635–2639. Ilg, P., and Odenbach, S, 2009. Ferrofluid Structure and Rheology, Lect. Notes Phys., 763, 249–325 (in Colloidal Magnetic Fluids), Springer. Kroger, M., Ilg, P., and Siegfried, H 2003. Magnetoviscous model fluids, J. Phys. Condens. Matter, Vol. 15, S1403-S1243. Odenbach, S., 2002. Magnetoviscous Effects in Ferrofluids, Springer-Verlag, Berlin, Heidelberg. Odenbach, S., 2009. Colloidal Magnetic Fluids: Basic, Development and Applications of Ferrofluids, Lect. Notes Phys., 763, Springer-Verlag. Rinaldi, C., Chaves, A., Elborai, S., He, X., Zhan, M., 2005. Magnetic fluid rheology and flows, Current Opinion in Colloid and Interface Science, 10, 141–157. Rosa, A.P, 2014. Numerical and Experimental Analysis of Rheology and Microstructure of Magnetic Suspension, MSc. dissertation, University of Braslia, 1–148 (in Portuguese). 26
Rosensweig, R.E., 1985. Ferrohydrodynamics, Cambridge University Press. Sartoratto, P.P.C., Neto, A.V.S., Lima, E.C.D., Rodrigues de S, A.L.C, and Morais, P.C., 2005. Preparation and electrical properties of oil-based magnetic fluids, J. Appl. Phys. 97, 10Q917. Shliomis, M.I., 1972. Effective viscosity of magnetic suspensions, Sov. Phys. JETP, 34(6), 1291–1294. Shliomis, M.I. and Morozov, K.I., 1994. Negative viscosity of ferrofuid under alternating magnetic field, Phys. Fluids 6, 2855. Zubarev, A.Yu., Fleischer, J., and Odenbach, S., 2005. Towards a theory of dynamical properties of polydisperse magnetic fluids: Effect of chain-like aggregates, Physica A 358, 475–491. Acknowledgments We would like to thank our colleagues Rafael Gabler Gontijo, Paulo C. Morais and Kalil Skeff Neto for helpful and elucidate discussions on magnetic suspensions. Thanks are also due to Patricia Sartoratto who provided the magnetite powder for our experiments. The authors are grateful to CNPq, CAPES and Eletronorte-DF for their generous support to this work.
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