On the viscosity of magnetic fluid with low and moderate solid fraction

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Particuology 6 (2008) 191–198

On the viscosity of magnetic fluid with low and moderate solid fraction Zhiqiang Ren a , Yanping Han a , Ruoyu Hong a,b,∗ , Jianmin Ding c , Hongzhong Li b a

Department of Chemical Engineering and Key Laboratory of Organic Synthesis of Jiangsu Province, Soochow University, SIP, Suzhou 215123, China b State Key Laboratory of Multiphase Reaction, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China c IBM, HYDA/050-3 C202, 3605 Highway 52 North, Rochester, MN 55901, USA Received 10 April 2007; accepted 26 January 2008

Abstract The design of a pressurized capillary rheometer operating at prescribed temperature is described to measure the viscosity of magnetic fluids (MFs) containing Fe3 O4 magnetic nanoparticles (MNPs). The equipment constant of the rheometer was obtained using liquids with predetermined viscosities. Experimentally measured viscosities were used to evaluate different equations for suspension viscosities. Deviation of measured suspension viscosities from the Einstein equation was found to be basically due to the influence of spatial distribution and aggregation of Fe3 O4 MNPs. By taking account of the coating layer on MNPs and the aggregation of MNPs in MFs, a modified Einstein equation was proposed to fit the experimental data. Moreover, the influence of external magnetic field on viscosity was also taken into account. Viscosities thus predicted are in good agreement with experimental data. Temperature effect on suspension viscosity was shown experimentally to be due to the shear-thinning behavior of the MFs. © 2008 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved. Keywords: Rheology; Capillary rheometer; Magnetic fluid; Viscosity; Peclet number

1. Introduction Magnetic fluid (MF) is a colloidal suspension of magnetic nanoparticles (MNPs) stabilized by coating surfactants. Recently, investigation on suspension, especially MFs, has attracted much interest (Kimata, Nakagawa, & Hasegawa, 2003; Ramlakhan, Wu, Watano, Dave, & Pfeffer, 2000; Rotariu & Strachan, 2003; Taca & Paunescu, 2000). There are two primary kinds of rheometers to study the rheological properties of materials: the rotating and the capillary rheometers. The rotating rheometers can be further classified into three types, the concentric-cylinder rotating rheometer, the cone-plate rotating rheometer and the parallel-plate rotating rheometer. Although the rotating rheometers have higher efficiency, they can only be applied to low-range shear rates and have low precision of measurement. The capillary rheometers have many advantages over the rotating one, in that it gives wider measuring range, higher accuracy and more applicability to many kinds of fluids/suspensions. In the present study the capillary rheometer is

∗

Corresponding author. Tel.: +86 512 6588 0402; fax: +86 512 6588 0089. E-mail address: [email protected] (R. Hong).

used to measure nanoparticle suspensions. Nanoparticle suspensions (Choi, 1995) have aroused great attention in both academic research and in engineering application (Hu & Dong, 1998; Lee, Choi, & Li, 1999). The efficient utilization, transport, and handling of suspensions require the ability to predict the rheological properties as a function of various physical parameters, including particle content, particles size and size distribution, surface properties of particles, and environmental temperature. The rheology of nanoparticle suspensions has been extensively investigated by Xuan and Li (2000), Li, Li, and Wang (2002) and others (Ramakrishnan & Zukoski, 2000; Tseng & Lin, 2003; Tseng & Wu, 2002; Zaman, Singh, & Moudgil, 2002). In this work, a pressurized capillary rheometer operating at prescribed temperature was constructed, as shown in Fig. 1, for measuring the viscosity of MFs. The scope of this study is to experimentally evaluate previously proposed equations for suspension viscosities. Effects of surfactant (Medina-Valtierra, Frausto-Reyes, Calixto, Bosch, & Lara, 2007) adsorption on MNPs surface, spatial distribution and particle size (MedinaValtierra et al., 2007; Zhang et al., 2008) of MNPs in MFs, are also considered. It is well known that in a dilute suspension the suspended particles are separated from each other and particle–particle

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doi:10.1016/j.partic.2008.01.004

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Nomenclature c C H0 k M Mp n Pe P Pi Q r r¯ ri T

mass content of MF equipment constant in Eq. (14) intensity of applied magnetic field (A/m) the Boltzmann constant (J/K) magnetization of MFs (emu/g) saturation magnetization of MNPs (emu/g) correction coefficient Peclet number total pressure drop (Pa) loss pressure drop (Pa) liquid flow rate (m3 /s) radius of aggregate (m) geometrical mean of the radius of aggregation (m) radius of primary particle (m) absolute temperature (K)

Greek letters γ˙ shear rate (s−1 ) δ thickness of surfactant layer (m) η viscosity of MFs (mPa s) ηs viscosity of solvent (mPa s) η0 viscosity of water (mPa s) μ0 the permeability of vacuum (N/A2 ) ρ density of Fe3 O4 MF (g/cm3 ) density of particles (g/cm3 ) ρ0 ϕ solid volume content

Similar research conducted by Krieger and Dougherty (1959) resulted in the following equation, η = (1 − Kϕ)−2.5/K ,

(3)

in which K is an adjustable parameter. To compare the viscosities for different systems, the Peclet number should be used as a parameter for estimating the onset of shear thickening. Thus, the Peclet number (Yang et al., 2001) based on the suspension viscosity η is: Pe =

˙ i3 8ηγr , kT

(4)

where ri is the radius of aggregate, k the Boltzmann constant, and T is the absolute temperature. The Peclet number for describing the sudden shear-thickening phenomenon has been discussed in many references (Barnes, 1989; Catherall, Melrose, & Ball, 2000; Franks, Zhou, & Duin, 2000; Frith, Haene, Buscall, & Mewis, 1996; Hoffman, 1992; Yang et al., 2001). 2. Experimental procedures

interaction can be neglected. Assuming the particles to be rigid spheres, the viscosity could be estimated by using the Einstein equation, η = ηs (1 + 2.5ϕ),

Roscoe, 1952) proposed other equations for suspension viscosity. Considering the influence of maximum packing effect on viscosity, Mooney (Yang, Li, Gu, & Fang, 2001) proposed the following equation,   2.5ϕ . (2) η = exp 1 − Kϕ

(1)

where η is the suspension viscosity, ηs is the liquid viscosity and ϕ is the volume fraction of particles. The Einstein equation is generally valid for particle volume fraction ϕ less than 2%, although in real applications, particle volume fraction often exceeds this value, calling for modification of Eq. (1). Many investigators (Brinkman, 1952; Goddard & Miller, 1967; Krieger & Dougherty, 1959; Mooney, 1951;

2.1. Rheometer calibration with water The viscosity measurements include absolute and relative viscosity measurements. Generally, relative measurement is easier to be carried out than absolute one, and can give a higher precision. The liquid flow along a tube with radius R and length L is assumed to be fully developed. For Newtonian fluid flow, the liquid flow rate is  Q=

vπr dr =

πR4 P , 8Lη0

(5)

Fig. 1. A pressurized capillary rheometer operating at prescribed temperature. 1: Nitrogen gas cylinder; 2: pressure reducer; 3: filter and surge tank; 4: pressure transducer; 5: thermostatic chamber containing water at prescribed temperature; 6: storage tank of MF; 7: capillary tube; 8: circulating water; 9: digital display; 10: thermometer; 11: magnet.

Z. Ren et al. / Particuology 6 (2008) 191–198

thus, the viscosity, η, is obtained as η0 =

Substitution of Eq. (10) into Eqs. (11) and (12) yields,

πR4 P . 8QL

(6)

In the present investigation, pressurized nitrogen was used to push the liquid through a capillary tube. The primary parameters are the pressure drop (P) between the inlet and outlet of the capillary and the liquid flow rate (Q) through the capillary, which is m Q= . (7) ρw t Actually, there are three parts for the total pressure drops. The first part is due to the acceleration of the testing fluids at the center of the capillary entrance, the second is the additional pressure loss in the inlet and outlet zones, and the last one the kinetic energy loss. For relative viscosity measurement, the first and second parts of pressure loss can be ignored when the capillary tube is long enough (L  R), thus leaving the loss of kinetic energy to be the most important part. The modified Hagen–Poiseuille equation then becomes η0 =

− Pi ) C(P − Pi ) = , 8QL Q

πR4 (P

(8)

where, Pi =

193

nρw Q2 . π 2 R4

(9)

The viscosity of water is sensitive to the variation of temperature and insensitive to pressure of ordinary range. Most capillary rheometers are calibrated at ordinary pressure using water which has known viscosities at various temperatures. In this investigation, two capillary tubes with the same diameter but different lengths were used. When the liquid temperature reached certain fixed value after flowing for some time, the liquid flow rate of the two tubes was maintained the same by adjusting the nitrogen pressure, therefore, Pi1 = Pi2 ,

(10)

And from Eq. (8), P1 − Pi1 =

Qη0 , C1

(11)

P2 − Pi2 =

Qη0 . C2

(12)

P =

Qη0 . C

(13)

where C is given by     1 1 1 , = − C C2 C1

(14)

and C1 or C2 , the equipment constant, is inversely proportional to the length of the capillary tube, L1 or L2 . The experimental data of capillary rheometer using water as reference solution was shown in Table 1. At a prescribed temperature of 15 ◦ C, the equipment constant C of the capillary rheometer with ID of 0.5 mm and length of 208.5 mm is 2.1207 × 10−15 , and another one with ID of 0.5 mm and length of 102 mm is 4.3349 × 10−15 . 2.2. Preparation of MFs MFs with various solid volume fractions were prepared using an ordinary agitator (Hong et al., 2006). The solid volume fraction of MFs was accurately determined by drying the MFs. The dosage of surfactant was fixed at 2%. The MFs were centrifuged, respectively, for 1 h at 6000 rpm to separate large particles. All the MFs were left for 48 h before measurements. All measurements were carried out at 25 ± 0.1 ◦ C except those for studies on temperature effects. The pH of all the MFs remained at 7.0. 2.3. Particle characterization Different methods could result in nanoparticles with different characters (Sarathi, Sindhu, & Chakravarthy, 2007). In the present investigation, the nanoparticles were prepared using the method reported by Hong et al. (2006). The TEM (Hitachi H600-II) image in Fig. 2 shows that the average diameter of primary Fe3 O4 particle is about 7 nm; it also shows aggregation of particles. Fig. 3 depicts the particle size distribution of Fe3 O4 particles (Malvern HPPS5001) in MFs. The mean diameter of the nanoparticles was 13.5 nm, which was larger than that obtained from TEM image. An explanation for this is that some of the primary nanoparticles aggregated in MF. The parameters of the particles used in this study are listed in Table 2.

Table 1 Experimental data of capillary rheometer No.

Length (mm)

Mass flow rate (g)

Measuring time (min)

Flow rate (m3 /s)

Viscosity (cP)

Density (g/cm3 )

Total pressure drop (Pa)

A1 A2 A3 B1 B2 B3

208.5 208.5 208.5 102 102 102

10.77 10.71 10.72 10.62 10.66 10.58

2 2 2 2 2 2

8.98E−08 8.93E−08 8.94E−08 8.86E−08 8.89E−08 8.83E−08

1.1404 1.1404 1.1404 1.1404 1.1404 1.1404

0.999 0.999 0.999 0.999 0.999 0.999

51,267 51,160 51,052 26,689 26,591 26,493

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play the pressure inside the storage tank. The pressure reducer of nitrogen cylinder and the intermediate valve were adjusted to control the pressure inside the storage tank. Finally, the weight of the MF flowing out of the capillary was measured using an electronic balance (precision 0.01 mg), while the elapsed time was also recorded using a stopwatch. The viscosity measurements were completed within 10 min after ultrasonication. 2.5. Generation of magnetic field The magnetic field was imposed through two parallel magnets surrounding the whole capillary. The capillary was at the center of the parallel flats, and the applied magnetic field was oriented in a perpendicular direction to the flowing MFs, so as to insure a uniform and homogeneous magnetic field throughout the region containing MFs. The intensity of the applied magnetic field was controlled by adjusting the distance between the two magnets, and the intensity was measured using a Gaussmeter (Dazhan, Nanjing DZ3326) with the magnetic probe placed between the two magnets. 3. Results and discussion 3.1. Stability tests Fig. 2. TEM of Fe3 O4 nanoparticles (d = 7 nm).

All the experimental samples were centrifuged at 6000 rpm for 1 h. No visible precipitate was found after the centrifugation. Fig. 4 shows the UV–vis absorption spectrum (Hitachi U2810) of samples (ϕ = 7.5%) sedimentated under gravitational field for 50 d, indicating that the absorbance of MF hardly changed after sedimentation for 50 d. Therefore, the MFs exhibited excellent stability under gravitation. 3.2. Viscosity of suspension

Fig. 3. Particle size distribution in MF.

2.4. Viscosity measurement

When the solid volume fraction increased, the viscosity of MF was affected by the hydrodynamic interaction among the nanoparticles (Kruif, Iersel, Vrij, & Russel, 1985). The apparent viscosity (η) of a MF with moderate particle concentrations is

A pressurized capillary rheometer with a stainless steel capillary diameter of 0.5 mm was used to determine the viscosity of the MFs. First, the equipment temperature was adjusted according to experimental requirement. All samples were ultrasonicated for 30 min prior to viscosity measurement. Then, a proper amount of MF was added into the storage tank. Second, nitrogen was introduced from its cylinder to pressurize the MFs storage tank. The pressure transducer was switched on to disTable 2 Parameters of suspension Parameter

Value

Radius of nanoparticles, ri (nm) Radius of aggregates, r (nm) Length of oleate chain, δ (nm) Density of nanoparticles, ρ0 (g/cm3 )

7 13.5 2.7 5.18

Fig. 4. UV–vis absorption of sample (ϕ = 7.5%) under gravity for 50 d.

Z. Ren et al. / Particuology 6 (2008) 191–198

described according to the following proposed equation: 1 η = . ηs 1 + aϕ + bϕ2

3.3. Influence of particle aggregation on viscosity (15)

When ϕ is much smaller than 1, the above equation can be approximated by η ≈ 1 + aϕ. ηs

(16)

For rigid spheres in dilute MFs (ϕ < 0.02), a = −2.5, and Eq. (15) becomes the Einstein equation, Eq. (1). At maximum particle packing, taking ϕr = 0.7405, and the suspensions is at a stationary condition, and we have 1 + aϕr + bϕr2 = 0,

(17)

thus b=−

1 + aϕr . ϕr2

(18)

Using the Einstein equation, i.e. a = −2.5, we obtain b = 1.55 and Eq. (15) becomes (Rosenswieg, 1985): η = ηs (1 − 2.5ϕ + 1.55ϕ2 ).

195

(19)

The particle surface adsorbs surfactants as if the virtual radius is increased. Assuming the primary particles radius is ri and the thickness of surfactant layer is δ, thus the virtual radius of particles is ri + δ, and the virtual volume fraction is (1 + (δ/ri )3 ϕ). Therefore, the MF viscosity η becomes       δ 3 δ 6 2 (20) ϕ + 1.55 1 + ϕ . η = ηs 1 − 2.5 1 + ri ri Fig. 5 shows the comparisons of measured MF viscosities with calculated data from the above equations, showing that the calculated MF viscosities from Eq. (20) involving the thickness of surfactant layer is in better agree with the measured results.

It was found that for some volume fraction range the viscosity of MF increased more rapidly than the prediction of the Einstein equation due to the apparent increase of the hydrodynamic radius of Fe3 O4 MNPs in MF as was already noted in Section 2.3. Aggregation of MNPs in MFs is apparently due to the high contact energy between MNPs surface and the entanglement of polymer chains. The aggregates consisting of nanoparticles and trapped solvent were relatively stable, and the internal forces in the aggregates were relatively in balance. The aggregates formed by the MNPs and the solvent would change their configuration when MF was flowing. Attributed to the presence of aggregation, the virtual volume fraction of MNPs in the MF with aggregates was higher than that of the MFs with monodisperse nanoparticles. But the total mass content is not changed. The Einstein equation may therefore be re-expressed as   cρ η = ηs 1 − 2.5 , (21) ρ0 where c is solid mass fraction, ρ0 is nanoparticle density, and ρ is MF density. Assuming the aggregates in MFs were hard spheres, their hydromechanical radius, r, is a times of the nanoparticle radius (ri ), and their mass is thus a3 times that of the nanoparticles. The volume of an aggregate is V (r) = V =

4 π(ari )3 = a3 V, 3

4 3 πri . 3

(22) (23)

Based on our experimental observations using a laser particlesize analyzer, a model for the structure of aggregates in aqueous media is proposed from which the apparent density of aggregates can be calculated (Hong et al., 2007). The spatial distribution of nanoparticle aggregates can be described by the following logarithm-normal function (Wood & Ashcroft, 1977):  

1 ln(r/¯r ) 2 , (24) n(r) = √ exp − √ r 2π ln σ 2 ln σ where the geometrical mean of r is r¯ , and r¯ can be substituted by ri , σ is an empirical number, normally, σ = 1.5. Thus the average density of an aggregate is defined as   2 m(r)n(r) dr ri rn(r) dr ρ¯ =  . (25) = ρ0  3 V (r)n(r) dr r n(r) dr

Fig. 5. Theoretically predicted and experimentally measured viscosities at different solid volume fractions (shear rate: 25,000 s−1 ).

By combining Eqs. (21) and (25), a modified Einstein equation with consideration of spatial distribution of nanoparticle aggregates is obtained:  3     ρ r n(r) dr  η = ηs 1 − 2.5c = ηs 1 − 2.5ϕ . (26) ρ¯ ri 2 rn(r) dr

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where α is a Langevin parameter, α=

μ0 Mp V (r)H0 , kT

(31)

and Mp is the magnetization of MNPs, L(α) is a Langevin function, L(α) = coth α −

1 , α

(32)

and M = ϕMp L(α).

(33)

So, αL(α) = Fig. 6. Theoretically predicted and experimentally measured viscosities at different solid volume fractions (shear rate: 25,000 s−1 ).

Considering the surface absorption of particles, the following equation is obtained:   (ri + δ)r 3 n(r) dr  η = ηs 1 − 2.5ϕ . (27) ri 3 rn(r) dr

(3/4)πr 3 μ0 H0 M . ϕkT

(34)

When the direction of applied magnetic field (H0 ) is perpendicular to the direction of flow, Eq. (30) can be reduced to, η 1 = 1.5ϕ . ηs (1/ϕ) + (3kT/2πr 3 μ0 H0 M)

(35)

Combination of Eqs. (20) and (27) yields the following equation,  (ri + δ)r 3 n(r) dr η = ηs 1 + 2.5ϕ  3 ri rn(r) dr  2

(ri + δ)r 3 n(r) dr 2  +1.55ϕ . (28) ri 3 rn(r) dr Fig. 6 compares the calculated results from Eq. (28) with experimental data, showing good agreement. Aggregation inevitably existed despite ultrasonication of MFs for 30 min before experimental measurements, though it would be severer without prior ultrasonication. 3.4. Viscosity vs. magnetic field intensity Based on the theory of hydrodynamics and ferrofluids dynamics, viscosity can be influenced by the intensity and direction of magnetic field. The viscosity (ηH ) under magnetic field is proposed to be ηH = η + η.

(29)

When the angle between the direction of applied magnetic field and the swirling vector of magnetic field is β, the ratio between the increment of the viscosity (η) of MFs and the viscosity of carrier liquid is suggested as (Chi, Wang, & Zhao, 1993; Rosenswieg, 1985), η 3 0.5αL(α) = ϕ sin2 β, ηs 2 1 + 0.5αL(α)

(30)

Fig. 7. Predicted and measured viscosities of MF with different solid volume fractions under applied magnetic field: (a) 70 mT and (b) 130 mT.

Z. Ren et al. / Particuology 6 (2008) 191–198

Fig. 8. Viscosity vs. shear rate at different temperatures.

Combining Eqs. (28) and (35) yields the following equation under the magnetic field,  (ri + δ)r 3 n(r) dr η = ηs 1 − 2.5ϕ  3 ri rn(r) dr  2 (ri + δ)r 3 n(r) dr  +1.55ϕ2 ri 3 rn(r) dr

3 1 + ϕ (36) 2 (1/ϕ) + (3kT/2πr 3 μ0 H0 M) where μ0 is the permeability of vacuum, M the magnetization of MFs, H0 the intensity of applied magnetic field, k the Boltzmann constant, and T is the absolute temperature. Fig. 7 compares the calculated results from Eq. (36) with experimental measurements, to verify the validity of the equation for predicting the viscosity of MF at low and moderate volume fractions. 3.5. Viscosity vs. temperature Fig. 8 shows the variation of viscosity over the temperature range of 25–75 ◦ C, showing decreasing viscosity with increasing of temperature. On one hand, increasing temperature enhances particle–particle collisions thus leading to increased suspension viscosity, while on the other hand, the viscosity of the carrier liquid (water) decreases with increasing temperature. However, the second effect is predominant, thus leading to decreasing suspension viscosities with increasing temperature. According to some articles (Lee, So, & Yang, 1999; Yang et al., 2001), certain metal oxide suspensions exhibited anomalous behavior at certain shear rates and temperatures. Fig. 9 shows viscosity as a function of Peclet number, implying that Peclet number could be used to predict the shear-thickening transition point, that is, the MF undergoes shear-thickening behavior for Peclet number between 2 and 10 around 55 ◦ C, after which, viscosity begins to decrease again.

197

Fig. 9. Viscosity vs. Peclet number at various temperatures.

4. Conclusions In the present investigation, viscosity measurements were conducted at a prescribed temperature. A capillary rheometer was set up and the equipment constant was obtained. The viscosities of Fe3 O4 MFs were measured. Based on the Einstein equation, two modified equations were proposed by considering the influence of spatial distribution of nanoparticles, the thickness of surfactant layer, the size distribution of aggregates, etc. By comparing the theoretical predictions with the experimental results, it was found that the proposed Eqs. (28) and (36) were capable of predicting the viscosity of MFs when the solid fraction was low and moderate. It was also found in experimental measurements that the viscosity decreased with increasing temperature first, but when the temperature was around 55 ◦ C a complex behavior of shear thickening was observed. The abnormal behavior of viscosity could be analyzed using the Peclet number. Moreover, the Peclet number could also be used to describe the conditions when no such abnormal behavior occurred. Acknowledgments The project was supported by the National Natural Science Foundation of China (NNSFC, Nos. 20476065 and 20736004), the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry, the State Key Laboratory of Multiphase Reaction of the Chinese Academy of Science (No. 2003-5), the Key Laboratory of Organic Synthesis of Jiangsu Province, the Chemical Experiment Center of Soochow University and R&D Foundation of Nanjing Medical University (NY0586). HRY would also like to thank Prof. A. Zubarev in the Department of Mathematical Physics at the Ural State University References Barnes, H. A. (1989). Shear-thickening (“Dilatancy”) in suspensions of nonaggregating solid particles dispersed in Newtonian liquids. Journal of Rheology, 33, 329–366.

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