Anomalous increase in the magnetorheological properties of magnetic fluid induced by silica nanoparticles

Aqueous magnetic fluid is prepared by stabilizing magnetite nanoparticles using lauric acid [14]. The x-ray analysis of uncoated particles confirms single phase spinel ferrite. The parameters determined are: crystal structure: FCC, space-group: Fd3m, lattice parameter (a): 0.8376 ± 0.0003 nm, and crystallite size: 8.9 ± 0.2 nm. The magnetic volume fraction $({{\phi }_{m}})$ and the saturation magnetization of the fluid are 0.0713 and 4.77 kA m⁻¹ (60 G), respectively. The colloidal silica (TM-40) (Sigma-Aldrich) consists of 20 nanometer sized spheres stabilized by sodium ions. The volume fraction of silica nanoparticles $({{\phi }_{{\rm Si}}})$ is 0.182. Three samples of magnetite–silica composite magnetic fluids FT28, FT55, and FT82 are prepared by mixing magnetic fluid and the colloidal silica dispersion (v/v). The description of the code is as follows: FT28 is prepared by mixing 20% magnetic fluids (MF) and 80% colloidal silica (TM-40), FT55 consist of 50% MF and 50% TM-40, and FT82 contains 80% MF and 20% TM-40. The magnetic fluids samples for 20%, 50% and 80% concentrations are also prepared and coded as F20, F50 and F80. The total volume in all the samples is kept constant. All the samples are stabilized at least for 24 h before proceeding for the measurements.

The steady shear magnetorheological behaviour is investigated using a MCR 301 magnetometer (Anton Paar, GmBH). Plate–plate geometry is used for the present study, which has inner plate diameter of 20 mm and the gap within plates 0.1 mm. The magnetic field applied is perpendicular to the direction of shear. Temperature during the measurement is maintained at 20 °C. In the present work, all the data are recorded in constant shear stress mode. In this mode, the strain is measured and apparent viscosity is calculated at the rim of plates using the ratio of shear stress to shear rate. The flow curve is recorded in the 0 to 177 kA m⁻¹ magnetic field range. The general experimental protocol followed is summarized as follows: (i) pre-shearing at a constant shear rate (5 s⁻¹) for 30 s, (ii) the system is left to equilibrate for 60 s, (iii) desired magnetic field is applied for 120 s, and (iv) shear stress logarithmically increases from 0.05 to 50 Pa at the rate of 10 pts/decade. On completion of each measurement, magnetic field is set to zero and magnet is demagnetized using program interfacing. In each measurement a fresh sample is used. Finally, in all cases, experiments are repeated at least three times with a new sample to confirm the reproducibility.

An optical microscope (MAGNUIS MLXi) with a 40x, numerical aperture (NA) = 0.65 air objective is used to record dynamics of chain formation in the samples. Optical images are recorded using a charged couple device (CCD) attached to the microscope. The permanent magnets (0.2T) are placed parallel to the plane of microscope slide to record the structural changes.

The aqueous magnetic fluid and silica suspension (TM-40) exhibits Newtonian viscosity of 1.3 ${\rm mPa}\cdot {\rm s}$ and 5.8 ${\rm mPa}\cdot {\rm s}$ respectively, at 20 °C (not shown here for brevity). Viscosities of the magnetic fluid with different magnetic volume fraction vary linearly, which confirms the stability of magnetic fluid over dilution. Since parent fluids are Newtonian, first of all, it is necessary to identify the nature of all three composite magnetic fluids, i.e. FT28, FT55 and FT82, in an absence of the external magnetic field. Figures 1(a)–(c) shows the dependency of shear stress on shear rate, viscosity on shear rate and viscosity on shear stress for all three fluids in the absence of external magnetic field. On applying shearing force, pre-existing structure breaks, which is a typical characteristic of Newtonian fluid. This can be seen on the log–log plot of shear stress versus shear rate and viscosity versus shear rate, where shear stress increases linearly and viscosity decreases linearly. This typical behaviour is observed in FT28 fluid. Additionally, it is also confirmed from the absence of the low-shear plateau in figures 1(b) and (c). In the case of FT55 and FT82, the plots can be divided into two regions, i.e. low shear plateau and high shear plateau, which is evident from a difference in slopes in both these regions. In FT82 fluid, steady state at low shear rate indicates that pre-existing structure does not rupture completely; this is possible where magnetic fluid possesses columnar or columnar-like structure. On sharing, columns break into chains (if they are not rigid), then at high shear they exhibit Newtonian nature. Depending on the degree of rupture of the structure, the steady state shear stress region obtained at a low shear plateau is known as a static yield stress. In the case of FT55, viscosity observed at low shear rate/shear stress is very high. It is possible that low shear plateau might have shifted to even lower shear rate, which is out of the measurement limit in the present experimental set-up. Nevertheless, in the applied shear rate, structure of the FT55 fluid is already deformed, and hence at moderately high shear rate it becomes Newtonian fluid.

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In the present magnetite–silica composite magnetic fluids, if interaction between diamagnetic SiO₂ and superparamagnetic Fe₃O₄ exists, then it should modify the magnetorheological properties. To corroborate this hypothesis, first of all, magnetic field induced viscosity data of all the fluids are analyzed. Figure 2 shows typical flow (viscosity versus shear rate) curves of FT28, FT55 and FT82 fluids in the absence of and on application of 178 kA m⁻¹ magnetic fields. The experimental data of intermediate field strength is not shown here for brevity; however, theoretical analyses of all the data are incorporated into table 1.

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Table 1.  Best fit parameters to the viscosity curves determined using equation (2).

	c (Pa-s2)
H (kA m−1)	FT28	FT55	FT82
0	0.065	0.038	0.34
16.21	0.125	0.13	2
29.28	0.07	0.34	3.2
41.90	0.08	0.45	5.2
54.07	0.08	0.6	5.5
65.80	0.09	0.65	6.3
93.14	0.08	0.8	6.8
177.04	0.085	0.95	8
n	−0.78	−0.67	−0.9
${{{\boldsymbol{ \eta }} }_{\infty }}$($Pa\cdot s$)	0.011	0.01	0.01

As discussed above, at zero magnetic field, all fluids exhibit Newtonian nature at higher shear rate, however, they behave differently at a low shear region. Such a difference is more pronounced in the presence of magnetic field. The FT28 fluid possesses Newtonian nature even in the presence of the magnetic field. The behaviour of FT55 and FT82 is to a certain extent similar in the absence and presence of magnetic field, except viscosity increases by orders of magnitude. A low shear plateau is not observed in FT55 even at a high magnetic field of 178 kA m⁻¹. This rise in viscosities is dedicated to the magnetic field induced chain or column formation. In order to understand the effect of external magnetic fields on the viscosities of pure and silica added magnetic fluids, the change in the field induced viscosities—so- called magnetoviscous effect (MVE), i.e. R—is calculated using equation (1).

$$\begin{eqnarray}&&R=\;\frac{{{\eta }_{H}}-{{\eta }_{H=0}}}{{{\eta }_{H=0}}}\end{eqnarray} \tag{ 1 }$$

where, the ${{\eta }_{H=0}}$ and ${{\eta }_{H}}$ are the viscosities in the absence and presence of the magnetic field at a constant shear stress. Figure 3 shows the MVE as a function of shear stress at 178 kA m⁻¹ field strength. The MVE data can be divided into two parts, i.e. below the peak and above the peak. The former indicates structure build-up (or shear thickening), and the latter structure break-up (or shear thinning). It is evident from figure 3 that, the magnitude of R is high in F100 compared to F50. Also, a small change in the peak position is also observed. It is known that high concentration of magnetic particles gives rise to the interparticle interaction, and thus field induced chain formations. It is also reflected in the R values. It is known that a fraction of larger particles form chains in the presence of magnetic fields, thus only these particles contribute to the magnetoviscous effect.

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It is derived from the MVE values of F50 and FT55 that silica nanoparticles give rise to the magnetic interaction. The observed high R value and shift in the peak position indicates interaction between silica and magnetite nanoparticles. Since total number of magnetic particles in the fluids remains constant (in F50 and FT55), rise in R is attributed to the increase in the interparticle interaction. The magnetoviscous effect observed in the FT82 fluid is quite high compared to even pure magnetic fluid. This is only possible if strong interaction between silica and magnetite nanoparticles exists. The negligible value of R (∼0.1 to 0.3) observed in FT28 (not shown here for brevity) is attributed to the diamagnetic contribution of silica.

It is summarized based on the collective analysis of FT55 and FT82 that, (i) magnetic field induced viscosity values of silica–magnetic fluids are high compared to pure magnetic fluids, (ii) the rise in the magnetoviscous effect is owing to the presence of larger sized magnetic particles, (iii) hence, on addition of silica nanoparticles the size of the magnetic particles increases, (iv) it results in augmentation of chain and/or column formation. However, detailed analysis is required to confirm these results.

The shear rate dependency of magnetic field induced viscosity (figure 2) follows the power law [8], which is described by equation (2).

$$\begin{eqnarray}&&\eta =c{{\dot{\gamma }}^{n}}+{{\eta }_{\infty }}\end{eqnarray} \tag{ 2 }$$

where, $\dot{\gamma }$ is the shear rate and c, n and ${{\eta }_{\infty }}$ are the fitting constants. Here ${{\eta }_{\infty }}$ is the field independent high shear viscosity. Best fit parameters for all the samples are reported in table 1. It is to be noted that value of an exponent 'n' remains around 0.67 in FT55. This value is close to the ideal value 0.66 (i.e. 2/3), which corresponds to long-range coupling between dipolar chains [19], while, in the case of FT28 and FT82 it is deduced to be 0.78 and 0.9, respectively. Similar values are observed in the inverse ferrofluid (de Gans et al [15], n = 0.8–0.9, Volkava et al [16] n = 0.74–0.87, Remos et al [8], n = 0.65–0.73, Felt et al [17] n = 0.74–0.83, etc). The high value of 'n' suggests the formation of spherical and/or cylindrical chains, which are attributed to the interparticle interaction and explained using a micromechanical model. Here, more pronounced magnetoviscous effect is observed in FT82. This can be seen from the high values of a coefficient of shear rate 'c', which is almost one order of magnitude higher than the other two fluids. This higher value of 'c' indicates that the structurization processes depend not only on the ratio ${{\phi }_{{\rm Si}}}/{{\phi }_{m}}$, but also on shear rate and the strength of applied magnetic field. Therefore, it is important to understand magnetic behaviour of composite magnetic fluids in static condition, which is governed by magnetostatic energy.

Since in the flow curve of composite magnetic fluids, magnetostatic and hydrodynamic forces are expected to dominate, an attempt is made to obtain a master curve using scaling law $\dot{\gamma }/{{H}^{2}}$ (not shown here for brevity). However, they do not overlap in all cases. Therefore, the detailed analyses of flow curves are analysed based on mason number, Mn, which represents dimensionless shear rate and defined as a ratio between hydrodynamic and magnetostatic forces [18],

$$\begin{eqnarray}&&Mn=\frac{8{{\eta }_{c}}\dot{\gamma }}{{{\mu }_{0}}{{\mu }_{cr}}{{\beta }^{2}}{{H}^{2}}}\end{eqnarray} \tag{ 3 }$$

where, ${{\eta }_{c}}$ is the viscosity of magnetic fluid, continuous phase, $\dot{\gamma }$ the shear rate, ${{\mu }_{0}}$ the permeability of vacuum, coupling parameter or constant factor, $\beta =\;({{\mu }_{{\rm pr}}}-{{\mu }_{{\rm f}}})/({{\mu }_{{\rm pr}}}-2{{\mu }_{{\rm f}}})$ with ${{\mu }_{{\rm pr}}}$ the relative permeability of the silica nanoparticles, ${{\mu }_{{\rm f}}}$ the relative permeability of magnetic fluid, and H the magnetic field strength. By using the Mason number (equation (3)) and the shear viscosity definition, $\eta =\tau /\dot{\gamma }$, the modified Mason number can be written as,

$$\begin{eqnarray}&&\frac{\eta }{{{\eta }_{\infty }}}=1+M{{n}^{*}}M{{n}^{\Delta }}\end{eqnarray} \tag{ 4 }$$

Based on the above relation, the values of Δ and Mn* determined for FT28, FT55 and FT82 are: Δ = −0.76, −0.80 and −0.82, and Mn* = 0.141, 0.202, and 0.257, respectively. However, we could not get a good fit at low shear regime. Recently, it has been reported that the low shear rate plateau can be fitted with the structural viscosity model [19], given by,

$$\begin{eqnarray}&&\frac{\eta }{{{\eta }_{\infty }}}={{\left[ \frac{1+{{(Mn/M{{n}^{*}})}^{1/2}}}{{{({{\eta }_{\infty }}/{{\eta }_{0}})}^{1/2}}+{{(Mn/M{{n}^{*}})}^{1/2}}} \right]}^{2}}\end{eqnarray} \tag{ 5 }$$

Where, ${{\eta }_{0}}$ is the low shear viscosity, which was not considered in the equation (4). Hence equation (5) can predict the existence of a low shear viscosity plateau value with smooth transition from the low to high (Newtonian) shear plateau.

Figure 4 shows dimensionless viscosity $(\eta /{{\eta }_{\infty }})$ as a function of normalized Mason number $(Mn/M{{n}^{*}})$ fitted using structural viscosity model (equation (5)) at various magnetic field strengths. This model describes the shear viscosity from the balance between build up (magnetic field-induced clustering) and breakdown (shear induced breakup) of particle aggregates. The values of $M{{n}^{*}}$ determined from the fit are 0.0035, 0.09 and 0.65, respectively for FT28, FT55 and FT82 fluids. This enhancement in $M{{n}^{*}}$ (by order(s) of magnitudes) is attributed to an increment in the field induced clustering. It is evident from figure 4 that the structural viscosity model enables one to fit low shear plateau region in FT28 and FT82 fluids, however, in FT55 theory deviates at low shear region. High values of normalized viscosity predict the existence of certain degree of aggregates or strong structure in FT55. Further findings are needed to understand this phenomenon. Finally, it is envisaged from the magnetic field induced viscosity data and its analysis that the pattern of magnetic field induced structure formation in all three fluids is different. More information and understanding about these structures can be obtained from the rheograms.

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Figure 5(a) shows rheograms (shear stress verses shear rate plots) of three fluids at the typical magnetic field strength. Figure 5(b) shows plots of viscosity (η) verses shear stress (τ), which is important to understand the progressive structure formation on increasing magnetic field and its flow behaviour. Since measurements are carried out in the steady shear stress mode, a vast difference in the shear rate decades is evident in all the fluids. This indicates a difference in the magnetic field induced structure. In the absence of magnetic field, shear stress (τ) linearly varies with the shear rate $\left( {\dot{\gamma }} \right)$. However, on switching on the magnetic field, totally different behaviours are observed in all three fluids.

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Depending on the degree of rupture of the structure, yield stress is defined by the static and dynamic yield stress. The static yield stress is obtained by extrapolating the shear stress at the intermediate pseudoplateau, i.e., $\dot{\gamma }\approx 0.1\;{{s}^{-1}}$ in the log–log rheograms; it is related to the breakage of the structures at their weakest point. Usually static yield stress is a characteristic of the magnetorheological fluids. The dynamic yield stress relates to the complete breakage of the structure, and is usually estimated by fitting the high-shear rate part of the rheogram in linear-scale $(\dot{\gamma }\gt 200\;{{s}^{-1}})$ to the Bingham equation, $\sigma =\;{{\sigma }_{y}}+\eta \dot{\gamma }$, where η is the plastic viscosity. Figure 5 shows the dynamic yield stress values determined at different magnetic fields for all three fluids. It also shows the static yield stress obtained at a constant shear rate 0.5 s⁻¹ at different magnetic fields in FT82 fluid.

In FT28, magnetic field induced shear stress superimposes suggesting a Newtonian behaviour of the fluid. This is expected behaviour in the magnetic fluid having low magnetic volume fraction. The dynamic yield stresses obtained at the different field are almost field independent and remain around 0.12–0.18 Pa. The same is also evident from figure 5(b). Hence, fluid illustrates the balancing between field induced structure formation and shear induced deformation.

In FT55, figures 5(a) and (b) signify low and high plateaus, however, low shear plateau is not very clear. In the presence of magnetic field, rheological behaviour changes to a plastic behaviour. The plastic behaviour is characterized by the appearance of a yield stress, ${{\sigma }_{y}}$, which is necessary stress to induce the flow and manifested by the intermediate pseudoplateau. The observed low shear plateau probably suggests that pseudoplateau might be obtained by increasing magnetic field. The dynamic yield stress values obtained for this fluid that magnetic field induced structure formed in this fluid is stronger than FT28. The ${{\sigma }_{y}}$ increases linearly around 60 kA m⁻¹ and then it saturates. This is the yield stress value to starts the flow, meaning structural deformation process completes around 60 kA m⁻¹.

The FT82 fluid clearly demonstrates low and high shear rate plateau that correspond to static and dynamic yield stresses. At 16 kA m⁻¹ field, noticeable change in the data of FT82 fluids in obtained compared to FT55, suggesting enhancement in the magnetic field induced structure formations. It is noticed from figures 5(a) and (b) and 6(a) and (b) that: (i) static yield stress (σ) linearly increases up to ∼60 kA m⁻¹ and then it saturates, (ii) dynamic yield stress, which is almost double (σ), linearly increases up to 60 kA m⁻¹ and it goes towards saturation. The values of static yield stress confirm that the process of magnetic field induced structure formation completes around 60 kA m⁻¹, whereas non-saturating behaviour of ${{\sigma }_{y}}$ indicates that structure deformation process does not complete in the given field range. The static and dynamic yield stresses are the typical characteristics of magnetorheological fluid (MRF) and also inverse magnetic fluid (inverse ferrofluid). The high value of σ is indicative of formation of layer aggregates. However, it can only be confirmed using various available models.

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As mentioned above, contrast factor or coupling parameter, β, depicts that magnetic interaction exists in the system. Additionally, high value of fitting constant 'n' and variation in $M{{n}^{*}}$ indicates that the field induced structures formed in all three fluids are not the same. It is possible to predict the type of structure formed in each fluid by knowing the dynamic yield stress values and the contrast factor (β). The fact that yield stress strongly depends on the interparticle/interchain interaction can be described using different models, as discussed below.

Chain like model

In this model the effect of interparticle interaction is considered. A simplified form of this model from Klingenberg and Zukoski [10] is expressed as, ${{\tau }_{y,n,KZ}}=1.026\phi {{\beta }^{2}}$. It is assumed here that non-interacting chains of particles are moving parallel to the applied magnetic field and rigidly attached to the wall. Under the assumption that the motion of the chains is not affine with the shear and chain opens between first and second particles, then normalized yield stress is given by, ${{\tau }_{y,n,dG1}}=1.11\phi {{\beta }^{2}}$. This is known as bead-rod model. Contrary to both the above chain models, normalized yield stress is defined by considering the particle size and gap thickness, which is given by,

$$\begin{eqnarray}&&{{\tau }_{y,n,dG1}}=C\phi {{\beta }^{2}}\end{eqnarray} \tag{ 6 }$$

where, C is chain constant and it directly depends on the particle size.

Figure 7 shows chain-like model aggress in the case of FT28 and FT55 fluids, with the chain constant 'C' equal to 0.085 and 0.4, respectively. But it did not agreed with FT82 fluids.

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Micromechanical models

These models are based on the macroscopic description of structure by considering the shape anisotropy of the strained aggregates under small deformation. Internal structure within the aggregates is ignored and only the particle volume fraction is required as an input in this model. In the case of randomly closed spherical aggregates, it is assumed that ${{\varphi }_{a}}=0.64$, but a small change in this value is found. A generalized yield stress model in the case of lattice of magnetic cylinders and stripes is described by Bosis et al [9] as,

$$\begin{eqnarray}&&{{\tau }_{y,n,C/L}}=0.325{{\phi }_{S}}{{\left( {{\mu }_{a}}/{{\mu }_{C}}-1 \right)}^{2}}\frac{1-{{\phi }_{S}}}{C+({{\mu }_{a}}/{{\mu }_{C}}-1)(1-{{\phi }_{S}})}\end{eqnarray} \tag{ 7 }$$

where, relative permeability difference between the aggregates $({{\mu }_{a}})$ and the composite magnetic fluid $\left( \mu \right)$ is quantified as, ${{\tilde{\mu }}_{a}}=({{\mu }_{a}}/\mu )-1$. As a first approximation, a mean field theory is used to described the magnetic permeability of the aggregates in composite magnetic fluid as,$\;{{\mu }_{a}}=\mu ={{\mu }_{C}}\left( 1+2\beta \phi \right)/\left( 1-\beta \phi \right)$, where $\phi$ is the non-magnetic particle volume fraction and ${{\phi }_{s}}\equiv \phi /{{\phi }_{a}}$. In equation (7), C = 2 suggests cylinder-like aggregates, whereas C = 1 implies layered aggregates. Normalized yield stress determined for FT82 is fitted to equation (7) with C = 1. This confirms that high values of 'n', 'Δ' and Mn* obtained are due to formation of layered aggregates in FT82 fluid.

Hence, we summarize from the analysis of magnetorheological measurements that normalized yield stress data of FT28 and FT55 fits chain-like models, whereas FT82 fits to macroscopic bead-like model. Chain constant in FT28 is very small, indicating either weak chain formation or number of chains per unit area are less. In FT55, the chain constant is an order of magnitude higher than the FT28, meaning strong chain formation. FT82 fits to the macroscopic model, with constant C = 1, which indicates layer structure formation. This is further confirmed using microscopic measurements in the presence of external magnetic field.