Towards ferrofluids with enhanced magnetization

Journal of Magnetism and Magnetic Materials 323 (2011) 1191–1197

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Towards ferrofluids with enhanced magnetization R.E. Rosensweig n Consultant, 34 Gloucester Rd., Summit, NJ 07901, USA

a r t i c l e in f o

abstract

Available online 13 November 2010

It is well known that the saturation magnetization of a sterically stabilized magnetic fluid (ferrofluid) is limited by the presence of a surfactant coating on the surface, and in some cases, by an effectively demagnetized surface layer in the solid magnetic particle. These surface layers take up a disproportionate volume in the colloidal dispersion thereby severely limiting the volume fraction of the core magnetic substance. This work proposes and analyzes Janus particles having the objective of increasing the magnetic loading beyond the present day constraints. Using numerical computation of the virial coefficient it is calculated that the magnetic volume fraction of magnetite ferrofluids might be increased by a factor approaching 2 and that of iron-based ferrofluids by a factor of 3. & 2010 Elsevier B.V. All rights reserved.

Keywords: Ferrofluid Magnetic fluid Magnetic Janus particle Virial coefficient

1. Introduction In a saturating applied magnetic field the magnetic sub-domain particles of a ferrofluid are aligned colinearly with the field. The magnetic attraction between particles is maximum in a head-totail configuration, which can lead to the case of oversize particles and eventually to the formation of chains, and if the chains grow overly long the preparation loses its Newtonian rheology. The concomitant development of a yield stress is undesirable in applications, such as rotary shaft sealing, bearings, dampers, thermomagnetic cooling, and magnetocaloric refrigeration [1]. In addition, oversize agglomerates can settle in magnetic gradient or gravity fields. As it is well known, the chaining, agglomeration, and settling can be prevented using sufficiently small particles with a surface adsorbed layer of surfactant to sterically protect particles against van der Waals agglomeration. Moreover, an effectively dead layer of disoriented magnetic moments is usually present within the surface of the magnetic material [2]. Because the core magnetic particle is so small, typically of the order of 10 nm, the surface layers take up a disproportionate fraction of the particle volume. Thus, the magnetic volume fraction e is given by  3 D2dD e¼ ð1Þ D þ 2dS where dS is thickness of the surfactant layer and dD the thickness of the dead layer. It is assumed that dD ¼0.83 nm equal to the FCC lattice constant for a magnetite particle. Using this value with dS ¼ 2 nm and D ¼10 nm, Eq. (1) yields e ¼0.21. If D could be some way increased the magnetic volume fraction would be enhanced. For example, from Eq. (1) with D ¼20 nm the magnetic volume fraction in a particle increases to about e ¼0.45. n

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This work proposes a method of screening the head-to-tail attraction of magnetic dipolar particles using magnetic Janus particles. As shown in Figs. 1(a–c) these Janus particles consist of an inert component attached to a magnetic component. A variety of shapes can be considered. In one possible mode of deployment a sprinkling of the Janus particles is added to a batch of spherical magnetic particles. Attachment of a Janus particle then prevents the further addition of particles to the end of analogous to a chain termination reaction in the polymerization of molecules. A disadvantage of this method is that two types of Janus particles are required, one with a free north-seeking pole and another with a free south-seeking pole in order to cap both ends of a given chain. Alternatively, using just one type of the Janus particle in this manner may result in an acceptable distribution of chain lengths. A statistical distribution will result in either method of deployment. A further method of deployment considered in this work proposes that the ferrofluid be constituted entirely of Janus particles of one special type. An advantage is that only one kind of Janus particle is needed to prevent chain growth. The ‘monomer’ particles of the type shown in Fig. 1(a) and (b) are not expected to be useful in this context as the inert component of the particle occupies an excessive volume. Instead, a preferred modification, illustrated in Fig. 1(c), uses preformed Janus chains of an acceptable length; more useful are preformed elongated Janus rods, as will be shown. The objective of the analyses that follow is to explore various shapes and sizes of candidate particles of these types and determine the magnetization they confer upon dispersal as a ferrofluid.

2. Methodology Preliminary study using quasi-two-dimensional Monte Carlo simulations yielded clear indication that sprinkling of Janus particles in the collection of spherical particles is effective in

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R.E. Rosensweig / Journal of Magnetism and Magnetic Materials 323 (2011) 1191–1197

Denoting nk the number of chains per unit volume containing k particles it follows that X nk ð5Þ p ¼ kB T k

m

m

The average number of particles N in a chain is given by P kn n N ¼ Pk k ¼ p=kB T k nk

m

ð6Þ

Eliminating p/kBT between Eqs. (6) and (3), and solving for N yields1

m N¼ Fig. 1. Magnetic Janus particles: (a) snowman, (b) rod and (c) chain. Other geometries can be considered. Vector m denotes a magnetic part; nonmagnetic parts are unlabeled. Surfactant coatings and nonmagnetic surface layers are omitted for simplicity.

inhibiting chain formation. However, obtaining a quantitative measure of performance is difficult via that route. The following describes an alternative methodology adopted in its place. Thus, the pioneering study of deGennes and Pincus [3] employed asymptotic virial coefficient analysis to estimate the onset of infinite chaining in a dispersion of bare dipolar particles free of inert surface layers. The present study expands upon that work to include the influence of the surface layers, while using numerical integration to determine the governing range of the transition curves. The governing range is usually intermediate to the asymptotic ranges. A suspension of nanoparticles subjected to thermal fluctuations is assumed to possess behavior that is analogous to that of a gas. Accordingly, the thermodynamic functions of the ferrofluid gas can be expressed in terms of the radial distribution function g. The function g is defined for a one-component gas as the mean ratio of concentration in a volume element dt located at a distance r from the center of a tagged particle to the mean concentration of the mixture. In the limit of low particle number density g is given by   U ð2Þ g ¼ exp  kB T where U is the interaction energy of a pair of particles, i.e., the energy acquired by reversibly transporting a particle from infinite separation to its position r relative to the tagged particle. Eq. (2) is a Boltzmann distribution with kB the Boltzmann constant and T the absolute temperature. Although g strictly applies to dilute suspensions the predicted behaviors will be extrapolated to concentrated suspensions to provide guidance for laboratory experimentation. The equation of state of a sufficiently dilute gas is p ¼nkBT where p is pressure and n is number concentration of particles. For a dispersion of particles in a carrier fluid p represents the osmotic pressure. At higher concentrations the equation of state can be put into the form of an expansion in powers of the concentration. Including the quadratic term the expression is the following in which B(T) is known as the second virial coefficient: p ¼ n þ BðTÞn2 þ    kB T The virial coefficient B is related to g as follows [4]: Z 1 ðg1Þdt B¼ 2 O

ð3Þ

1 1 þnB

If U is specified in Eq. (2), then B can be computed from Eq. (4) and used in Eq. (7) to determine the average number of particles in a chain. For magnetic dipolar particles in an applied magnetic field sufficiently intense, the dipole moments are colinear with the field direction U that is given in modified SI units using B ¼ m0H+M by U¼

m2 ð13cos2 yÞ 4pm0 r 3

where dt is an element of volume, and O is the domain of particle interaction. The pressure p can be expressed alternatively by assuming that chains of different sizes can be treated as a mixture of ideal gases [5].

ð8Þ

where m ¼MDpD3/6 is magnetic moment of a spherical particle, MD is domain magnetization, D is the diameter of a particle, r is the distance from the center of the tagged particle to the center of a field particle and y is the angle between r and the field direction. Thus, g of Eq. (2) may be expressed as    g ¼ exp lx3 3cos2 y1 ð9Þ where x¼ r/D, and l is the coupling coefficient.

l¼

pMD2 D3 =m0 m2 =m0 ¼ 3 144kB T 4pD kB T

ð10Þ

The coupling coefficient of Eq. (10) is the magnetic interaction energy of head-to-tail dipolar particles in contact in ratio to the thermal energy of a particle.

3. Bare particles To illustrate the methodology and calibrate its use consider the simple case of a bare, sub-domain, spherical particle. Fig. 2 illustrates the integration domain. The gray zone indicates a volume of space that is inaccessible to the center of the field particle. From Eq. (4) with the volume element dt ¼r2sin y dcdydr in spherical coordinates, Z 1 B¼ ðg1Þdt 2 O  3  Z 1 Z p Z 2p  1 D ¼ exp l 3 ð3 cos2 y1Þ 1 r 2 sin y dc dy dr 2 D r 0 0 ¼ pD3 IðlÞ ð11Þ where I(l) is the dimensionless iterative integral.   Z 1Z p l IðlÞ ¼ exp 3 ð3 cos2 y1Þ 1 r2 sin y dy dr 1

ð4Þ

ð7Þ

0

r

ð12Þ

in the above r ¼r/D. The number density of particles is given by n¼ f/VP, where f is the volumetric packing fraction of particles and VP ¼ pD3/6 is the spherical volume of the particle. Substitution into 1 Other authors find relationships for N absent the transition to infinite length chains inherent in Eq. (7) when the denominator disappears; see [6,7]. The choice of Eq. (7) permits comparison with results of Ref. [3].

R.E. Rosensweig / Journal of Magnetism and Magnetic Materials 323 (2011) 1191–1197

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and Pincus developed for l b1, namely

H θ

N¼

Ω m m

Fig. 2. Integration domain for bare magnetic spheres. Sphere at center with magnetic moment m is termed the tagged particle; the other sphere is the field particle. Zone within the inner dashed circle is inaccessible to the center of the field particle. The integration domain O extends from the inner to the outer dashed boundary located effectively at infinity.

1

ð15Þ

2

1ð2=3Þððf=l Þe2l Þ

As seen from Fig. 3, there is substantial agreement at the larger values of l which supports our calculation. In the past the asymptotic expression has been used outside the domain of its validity with the peak in the curve at f ¼0.2 interpreted as the maximum concentration that can be achieved without undergoing transition to infinite chaining. However, the numerical curve, which applies at all values, shows there is no such limit on concentration. Of course, maximum packing fraction fT ¼ fM imposes its own limit on the loading. For hexagonal pffiffiffior face-centered close packing of spheres fM is given by f ¼ p=3 2 ¼ 0:7405. The particle diameter at that value of f is determined iteratively in this work from Eq. (14) using trial values of l, then D is computed from the relationship for l given by Eq. (10). For this case it is found that D¼8.72 nm. The corresponding ferrofluid magnetization is MF ¼ fMMD. Using magnetite with domain saturation MD ¼ 560 mT yields MD ¼415 mT. Of course, bare particles are unstable against agglomeration and ferrofluid based on such a system cannot be realized in practice.2 Even the maximally loaded systems are not flowable; we compute their performance as a convenient reference for comparison between different systems. The model prediction for typical ferrofluid particles is considered next.

4. Coated spherical particles This system is illustrated in Fig. 4. The I(l) function differs from that of Eq. (12) only in the lower limit of radius in the integration domain.   Z 1 Z p  l  IðlÞ ¼ exp 3 3 cos2 y1 r2 sin y dy dr ð16Þ 1 þ ðdS =DÞ

0

r

The expression for the coupling coefficient takes a slightly different form as D is defined here and subsequently as the diameter of the solid core including the dead layer and excluding the surfactant layer. Thus, for head-to-tail contacting particles the interaction energy is now expressed as:   p MD2 D3 =m0 dD 6 12 l¼ ð17Þ 144 kB T D

Fig. 3. Predicted transition to infinite chaining for bare spherical dipoles. Curve 1 is the result of numerical integration (this work). Curve 2 is the large l asymptotic result [3]. The curves are in substantial agreement for l 43. The peak in curve 2 is mistakenly equated to the maximum packing fraction fM by some authors.

f¼

Eq. (7) yields N¼

1 16fIðlÞ

ð13Þ

N becomes singular at the transition value of packing fraction given by f ¼ fT.

fT ¼

1 6IðlÞ

Note that D is raised to the third power, while the term in parenthesis carries an exponent of six. This results from incorporating three powers of the diameter in the definition of the dimensionless I(l) function. Replacing Eq. (14) for f in this case is the following, which accounts for the additional volume of the surfactant coating:

ð14Þ

Fig. 3 shows a plot of fT vs. l computed numerically in comparison with a plot of the asymptotic expression of deGennes

ð1 þ 2ðdS =DÞÞ3 6IðlÞ

ð18Þ

A solution is found iteratively assuming a value of D, calculating

l from Eq. (17), then I(l) from Eq. (16) and fT from Eq. (18). This is repeated until fT ¼ fM. The solution for magnetite at fM ¼0.7405 yields D ¼14.24 nm. The reference magnetization is given by MF ¼ fM MD e

ð19Þ

2 The usual 1–2 nm thickness dS of the surfactant layer serves to prevent van der Waals agglomeration over a wide range of particle sizes [8].

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θ=0

δS

θ0 = 0

Ω

Ω1

δD

r1 (θ)

D r

DN m

π θ= 2

Ω2 r2 m

DM

θ2 = Fig. 4. Integration domain for surfactant-coated magnetic spheres having a surface dead layer. A field particle is shown at a distance r from the tagged particle having its center located at the origin. Only the first quadrant cross-section of the inaccessible domain is shown shaded. The outer boundary is effectively infinitely distant.

where the expression for e was given previously as Eq. (1). The calculated value of MF is 137 mT for this case. This value of magnetization will be considered the base case for comparison with the performance of various Janus particles. The value of MF is not inconsistent with the experimental experience, which shows a limiting value of MF 100 mT in quite viscous ferrofluid [8].

The expression for volumetric packing fraction is given by ðDM þ 2dS Þ3 þ ðDN þ 2dS Þ3 6½I1 ðDM ,DN Þ þI2 ðDM ,DN Þ

A model for these particles is shown in Fig. 5. All the particles in the dispersion are assumed to be alike. A non-magnetic sphere of diameter DN is attached to a dipolar magnetic sphere of diameter DM. Both spheres of a given Janus particle bear a surfactant coating, while the magnetic sphere contains a magnetic dead layer in addition. The gray zone in Fig. 5 maps the locations that are inaccessible to the center of the magnetic portion of the Janus field particle. The upper zones break into two spatial domains, O1 and O2. A mirror symmetry of the zones is obtained by reflection across the y ¼ p/2 plane. The zones are spatially bounded as described by the following:   DM þ dS O1 : 0 o y o cos1 DM þ DN þ4dS

 DM þ dS p oyo DM þDN þ 4dS 2 DM þ 2dS o r2 o1

O2 : cos1

ð22Þ

and the ferrofluid magnetization is given by 3

ðDM þ DN þ4ds Þcos y o r1 o 1

2

Fig. 5. Integration domains O1 and O2 for a model of a snowman Janus particle. The shaded zone is inaccessible to the center of a field particle. Dashed circles are centered at the angular limits of the integration domains. The diagram is symmetric by mirror reflection in the plane y2 ¼ p/2.

f¼ 5. Analysis for Janus snowman

π

ð20a; bÞ



ð21a; bÞ

These descriptions of the boundaries furnish the limits on two integrals of the type shown in Eq. (16), with the kernel of the integrals unchanged in form. The expression for l is the same as that given in Eq. (17), and the method of solution the same as described in the previous section. 3 The particles in the figure are shown separated by surfactant layers; the attendant error is small provided dS//D 51.

MF ¼

fM MD ðDM 2dD Þ3 ½ðDM þ 2dS Þ3 þ ðDN þ 2dS Þ3 

ð23Þ

These relationships assuming magnetite as the magnetic substance yield the following results: DN =DM ¼ 1

DM ¼ 22:65 nm

MF ¼ 101 mT

DN =DM ¼ 0:2

DM ¼ 16:98 nm

MF ¼ 155 mT

At DN/DM ¼1 it can be seen that the Janus particle indeed increases in diameter as compared to the 14.14 nm diameter for a coated sphere; however, the magnetization is adversely affected due to the additional non-magnetic volume of the inert portion of the particle. Plotting the dependence of MF on the ratio DN/DM over the range 0–1 yields the plot shown in Fig. 6. The maximum in the curve occurs at a ratio of about 0.2 producing a 14% increase in magnetization relative to that of a coated sphere. Although this is a modest gain it raises interest in exploring the performance of Janus particles with additional magnetic-containing spheres attached to the tail. Nonetheless, we will follow a modified strategy for the following reasons. With two magnetic spheres there are four interactions between the target and the field particles to be taken into account in formulating the energy term U. With three particles the number of interactions is nine, the number of interactions increasing quadratically. In addition, the number of domains of integration increases linearly and difficulties of convergence arise in performing the numerical integrations. These difficulties can be circumvented as described next.

R.E. Rosensweig / Journal of Magnetism and Magnetic Materials 323 (2011) 1191–1197

the pole density of a face is given by MD, hence at the circular end of a cylinder of diameter D the pole strength is given by p¼MD[p(D  dD)2/4]. Thus, the force between poles is given by the product of the magnetic field of the first particle and pole strength of the second particle. The product is inversely proportional to the square of the radius, hence the energy of interaction is proportional to the inverse first power of the radius. The resulting expression for the coupling coefficient that results is given by

20

10 Relative Magnetization, %

1195

0

l¼

-10

p MD2 D3 =m0 64

kB T



12

dD

4 ð24Þ

D

l of Eq. (24) is the interaction energy of the like poles separated by -20 DN DM

-30 0

0.2

0.4

0.6

0.8

1.0

Fig. 6. Magnetization of a ferrofluid based on snowman Janus particles of varying size ratio DN/DM relative to that of a coated spherical particle.

distance D in ratio to the thermal energy of one particle. The kernel of an integral for the virial coefficient B of Eq. (4) now contains contributions from the four interactions between the tagged and field particle poles of which two are equal. In cylindrical coordinates the integral functions take the form    Z 1 Z 1 2 I1 ðD, x, ZÞ ¼ exp  lðDÞ r13 ðr,zÞ x þ Z þ 3ðdS =DÞ þ ðdD =DÞ 0



D z

I2 ðD, x, ZÞ ¼

δD

Z

1



r14 ðr,z, xÞ r23 ðr,z, xÞ x þ Z þ 3ðdS =DÞ þ ðdD =DÞ Z

0

r

Ω1

LM + LN + 3δS + δD

##

1



# 1 r dr dB     exp  lðDÞ

1

1 þ 2ðdS =DÞ

1



1

###

r14 ðr,z, xÞ r23 ðr,z, xÞ

r dr dB

2

r13 ðr,zÞ ð25a; bÞ

where r ¼ r/D, rij ¼ rij/D is normalized distance between poles and Z ¼z/D.

r13 ¼ ðr2 þ Z2 Þ1=2 ¼ r24

LN

"

2

Ω2 LM



r14 ¼ r þ Zx þ 2 " 2



r23 ¼ r þ Z þ x2

dD

2 #1=2

D

dD

2 #1=2

D

ð26a2cÞ

δS z=0 Symmetry plane r=0

D + 2δS

Fig. 7. Construction of integration domains for Janus rod particle.

6. Analysis of cylindrical Janus particles The analysis of cylindrical particles (Janus rods) presents definite advantages. The uniformly magnetized cylinder effectively consists of a uniform distribution of oppositely signed magnetic poles at the ends of the surface. The interaction of a tagged cylinder with a field cylinder is then reduced to the four interactions between these poles regardless of the length of the cylinder. Accordingly, a single analytical framework can capture the behavior for any length of these particles. As an additional benefit, the cylinders are more space-filling than spheres with pffiffiffithe maximum close-packed volume fraction of solids given by f ¼ 3p=6 ¼ 0:9069. The model of this Janus particle is shown in Fig. 7. In formulating the interaction energy between these cylindrical Janus particles the approximation is made such that a pole is concentrated at a point at the center of a face. From an SI pole there emanates one line of induction. Thus, magnetic field H of a pole of strength p at a distance r is given by H¼p/4pm0r2. The magnitude of

x ¼LM/D, where LM is the length of the magnetic solid including the dead layer, and z ¼LN/D where LN is the length of the nonmagnetic part of the Janus cylinder. Expressions for volume fraction f and magnetization MF of the close-packed array are given by f¼

ð1 þ 2ðdS =DÞÞ2 ðx þ z þ2ðdS =DÞÞ 4ðI1 þ I2 Þ

MF ¼ fM MD

ð12ðdS =DÞÞ2 ðx2ðdD =DÞÞ ð1þ 2ðdS =DÞÞ2 ðx þ Z þ 2ðdS =DÞÞ

ð27Þ

ð28Þ

Magnetization for the geometry x ¼ Z ¼1 is inferior to that of the coated sphere base case as might be expected (MF ¼112 mT, D ¼17.57 nm, and fM ¼0.9069). However, magnetization for elongated Janus rods surpasses that of the coated sphere by a factor of 1.88. Further study determined the influence of inert length LN on the magnetization of a Janus Rod with x ¼15. It is found that a maximum of magnetization exists at Z C 0:2, coincidental to what is seen in Fig. 6 for a Janus snowman. The peak value of MF is about 3.4% greater than that of a Janus rod having Z ¼1. Surprisingly, the peak value differs only minutely from that of a rod having no inert tip, i.e., for Z ¼0.

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R.E. Rosensweig / Journal of Magnetism and Magnetic Materials 323 (2011) 1191–1197

1.3

7. Metallic Janus particles

1.2 Relative diameter

Iron will be taken as representative of a highly magnetic metallic species. Iron (MD ¼2100 mT) is much more magnetic than magnetite (MD ¼560 mT). Applying the relationships of Section 4 to a coated sphere of BCC iron with dS ¼ 2 nm and a dead layer having lattice constant 0.287 nm yields a relatively small particle of diameter 6.67 nm with maximum magnetization of only 196 mT. In comparison, using the relationships of Section 6 for the Janus Rod yields a cylindrical diameter D ¼7.09 nm having maximum magnetization of 623 mT. This much larger level of magnetization approaches a range having utility for advanced applications such as in magnetocaloric devices. Cobalt or iron–cobalt particles should yield enhanced performance in the same league. Gadolinium particles and other particles having a near room temperature Curie point could be useful in refrigeration [1]. Table 1 lists computed parameters for Janus rods of iron and magnetite. The corresponding relative particle sizes and magnetizations for Janus rods of various lengths are shown graphically in Figs. 8 and 9, respectively. A further comparison of relative magnetizations of various close-packed samples, including a Janus snowman is displayed in Fig. 10 for magnetite.

Fe3O4 Rods

1.1

Fe Rods 1.0

0.9

Spheres

0

5

10

15 ξ = LM/D

20

25

30

Fig. 8. Transition diameter of Janus rods with LN/D ¼ 1 relative to that of a coated sphere of the same magnetic material. Coated sphere diameters: Fe 6.67 nm; Fe3O4 ¼14.3 nm.

3.5 3.0

Scholten made a ‘best case’ analysis to estimate the maximum magnetization of a ferrofluid based on spherical particles [8]. Using a quite different approach of ‘melting physics’ applied to a particle having domain magnetization MD ¼600 mT, a surfactant coating thickness of 1 nm, absence of any dead layer and packing fraction f ¼0.5 yielded a ferrofluid having magnetization MF ¼ 210 mT and a particle size D¼ 16 nm. Using the same parameters of concentration, domain magnetization, surfactant thickness and absence of a dead layer for a coated sphere the present methodology computes incipient transition at magnetization MF ¼178 mT at f ¼0.5. The associated particle diameter D ¼10.48 nm. Thus, the Scholten estimate yields a larger particle with a correspondingly larger magnetization as a ferrofluid. However, computation of the coupling coefficient for tangent coated spheres yields l ¼4.37 for the Scholten particle vs. 1.04 from our virial computation. It appears that the Scholten particle will chain up freely, while the particle size we compute is virtually free of infinite chaining.

2.5

Table 1 Computed particle size and magnetization of ferrofluids containing close-packed Janus rods. Particle

n ¼LM/D

Iron (Fe)

1 3 5 10 15 30

Magnetite (Fe3O4)

1 3 5 10 15 30

D (nm) 8.77 6.63 6.23 6.23 6.47 7.09 19.1 14.7 14.2 14.6 15.3 16.8

L (nm)

MF (mT)

17.5 26.5 37.4 68.5 104 220

299 391 430 495 542 623

38.2 58.8 85.2 161 245 521

120 167 187 216 232 258

dS ¼2 nm, Z ¼ LN/D¼1, fm ¼ 0.9069, dD(iron)¼0.287 nm and dD(magnetite)¼ 0.83 nm.

Relative Magnetization

8. Comparison to a prior study

Fe Rods

2.0

Fe3O4 Rods

1.5 Spheres

1.0 0.5 0.0

0

5

10

15 ξ = LM/LD

20

25

30

Fig. 9. Magnetization of ferrofluids containing coated Janus rods with LN/D ¼ 1 relative to that of coated spheres of the same magnetic material. Sphere magnetizations: Fe 196 mT; Fe3O4 137 mT.

9. Overview and conclusion Close-packed magnetizations furnish a convenient reference state for comparisons. Dilution of the samples with a carrier fluid is required to attain viscosity low enough to yield flowability as a ferrofluid. Whatever the dilution, the main point is that larger magnetization at a given loading of solids is predicted by a factor approaching two compared to that based on coated spheres and increase by a factor of three for particles based on metallic ferromagnets. Additional small particles added to the interstices of a packing could somewhat enhance the magnetizations. The methodology of this work strictly applies to dilute dispersions of particles in a carrier liquid. Thus, the quantitative results should be considered only indicative of the trends achievable in a concentrated sample where multiparticle interactions predominate. Nonetheless, the work provides directional guidance for the synthesis of ferrofluid possessing an enhanced magnetic moment.

R.E. Rosensweig / Journal of Magnetism and Magnetic Materials 323 (2011) 1191–1197

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been identified yet. Additional geometries of Janus and non-Janus particles remain to be studied.

2.0 1.88

Acknowledgements

Janus Rod, ξ= 30, η= 1

0

Janus Rod, ξ= 15, η= 1

0.5

Janus Snowman DN/DM= 0.2

1.13 1.0

Janus Rod ξ= 5, η = 1

1.36

Coated Sphere

Relative magnetization

1.69

Type of Fe3O4 particle Fig. 10. Relative ferrofluid magnetizations yielded by coated magnetite Janus particles compared to that of coated magnetite spheres.

An example of a laboratory synthesized Janus snowman based on magnetite is found in Ref. [9]; the particle was synthesized for an altogether different biological objective. Notwithstanding the increases in magnetization computed for Janus particles, a surprising result of these studies is that elongated rods with an inert tip absent provide practically the same enhancement. It is possible that the most advantageous geometry has not

The author thanks D. A. Andelman and T. A. Hatton for early discussions and A.O. Cebers, M. Zahn, K. Raj and J. Popplewell for further comments on this work. References [1] R.E. Rosensweig, Refrigeration aspects of magnetic particle suspensions, Int. J. Refrig. 29 (2006) 1250–1258. [2] R. Kaiser, G. Miskolczy, Magnetic properties of stable dispersions of subdomain magnetite particles, J. Appl. Phys. 41 (3)(1970) 1064–1072; A.E. Berkowitz, J.A. Lahut, I.S. Levinson, D.W. Forrester, Spin pinning at ferriteorganic interfaces, Phys. Rev. Lett. 34 (10)(1975) 594–597. [3] P.G. deGennes, P.A. Pincus, Pair correlations in a ferromagnetic colloid, Phys. Condens. Mater. 11 (1970) 189–198. [4] J.E. Mayer, M.G. Mayer, Statistical Mechanics, Wiley, New York, 1940; L.S. Ornstein derived Eq. 4 in his Leiden University Ph.D. Thesis, 1908. [5] I. Perez-Castillo, A. Perez-Madrid, J.M. Rubi, G. Bossis, Chaining in magnetic colloids in the presence of flow, J. Chem. Phys. 113 (15)(2000) 6443–6449. [6] A.O. Tsebers, Association of ferrosols with magnetodipole forces, Magnetohydrodynamics 10 (2)(1974) 135–139 (English translation). [7] K.I. Morozov, M.I. Shliomis, Ferrofluids: flexibility of magnetic particle chains, J. Phys. Condens. Matter 16 (2004) 3807–3818. [8] P.C. Scholten, How magnetic can a magnetic fluid be?, J. Magn. Magn. Mater. 39 (1983) 99–106. [9] H. Gu, J. Yang, C.K. Chang, B. Xu, Heterodimers of nanoparticles, J. Am. Chem. Soc. 127 (2005) 34–35.