Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

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product liquid. After a prolonged period of grinding

in a ball mill (500–1000 h) near-quantitative conver-

sion of the feed solids to colloidal particles of the

order of 9–10 nm results. A representative composi-

tion consists of magnetite particles, oleic acid disper-

sant, and an aliphatic hydrocarbon carrier such as

hexadecane. Oversize material can be removed by

centrifuging or with magnets and the concentration

of particles adjusted by the evaporation or addition

of the carrier solvent.

The domain magnetization of elemental iron or

cobalt exceeds that of magnetite at normal temper-

atures by three- to four-fold giving incentive to the

preparation of more highly magnetizable ferroﬂuids

containing these metals. A problem that arises is the

reactivity of the elemental particles with atmospheric

oxygen. Nakatani et al. (1993) prepared dispersions

of iron-nitride particles, e.g., Fe

N, in oils. Gaseous

ammonia is bubbled through a solution of liquid iron

carbonyl, Fe(CO)

, dissolved in kerosene containing

a polyamine surfactant and produces a well dispersed

colloid with particles of highly uniform size. The iron

nitrides possess a domain magnetization approaching

that of iron and are resistant to oxidation. Ferroﬂuids

having saturation magnetization up to m

M ¼0.233 T

are reported.

3. Colloidal Stability of the Particles

3.1 Stability Against Sedimentation

Ferroﬂuids employ the same principle that keeps the

atmosphere suspended above the earth, preventing it

from collapsing under gravitational pull, to keep the

tiny magnetic particles suspended. In addition, the

ﬁnite size of the particles plays an important role.

Consider a container of ferroﬂuid exposed to a

nonuniform magnetic ﬁeld of sufﬁcient intensity to

magnetically saturate the sample at all points. At an

arbitrary point P the magnetic force acting on a unit

volume of the ﬂuid in a given direction s is given from

the Kelvin force expression for dipolar matter as

mndH/ds where m

is the permeability of free space

(4p 10

7

kgms

2

), m ¼M

is the magnetic

moment of a particle, M

is domain magnetization

(4.46 10

1

for magnetite), V

particle volume

), n the number concentration of particles (m

3

and H magnetic ﬁeld magnitude (Am

1

) where

B ¼m

(H þM). Balance of the forces acting at equi-

librium on a differential slice of the medium centered

about P and oriented perpendicular to s yields the

relationship:

dp ¼ m

mndH ¼ m

efdH ð2Þ

where p is osmotic pressure of the particles (regarding

the particles as a gas occupying the volume of the

ﬂuid space), M

is domain magnetization (4.46 10

for magnetite) e the volume fraction within a coated

particle that is magnetic, and f the local volume

fraction of coated particles in the ferroﬂuid. An

equation of state relates p to the concentration of the

particles.

A semi-van der Waals relationship that accounts

for the volume occupied by the particles is given by:





ð3Þ

where f

is the maximum value of f to which the

colloid can be concentrated as limited by the size

of the coated particles, k the Boltzmann constant

(1.38 10

23

NmK

1

), and T absolute temperature

(K). A representative value of f

is 0.74 correspond-

ing to hexagonal close packed spheres, and a typical

value of e is 0.36 corresponding to 10 nm core par-

ticles coated with a surfactant of 2 nm thickness.

Combining Eqns. (2) and (3) to eliminate p yields a

differential equation governing the distribution of the

particles. Integration yields the plot of Fig. 2 showing

the distribution of the ﬁnite size particles as a func-

tion of the local ﬁeld; concentration never exceeds the

maximum value which is less than one. A curve

showing the distribution for point particles is shown

for comparison. In the application of ferroﬂuids to

zero leakage rotary shaft seals and related devices, a

concentrated ferroﬂuid is generally employed. The

steric hindrance due to the ﬁnite particle size is re-

sponsible for the high retention of ﬂuid homogeneity

required for the successful operation of these devices.

Figure 2

Sedimentation equilibrium of ferroﬂuid particle

concentration in nonuniform magnetic ﬁeld. The steric

hindrance of ﬁnite-size particles limits their maximum

concentration to a physically realistic value.

190

Ferroﬂuids: Introduction

3.2 Stability Against Agglomeration

A typical ferroﬂuid contains on the order of 10

particles per cubic centimeter, and collisions between

particles are frequent. Hence, if the particles adhere

together, agglomeration will be rapid. The particles

are permanently magnetized, so the maximum energy

to overcome the magnetic attraction between a pair

of particles each of radius, R, occurs when the par-

ticles are aligned. The corresponding dipole–dipole

magnetic interaction is given by:

¼

8pm

9ðx þ 2Þ

ð4Þ

where x ¼s/R with s the surface to surface separation

distance.

In addition, van der Waals forces of attraction

arise between neutral particles due to the ﬂuctuating

electric dipole-induced dipole forces that are always

present. It is also necessary to guard against these

forces. London’s model predicts an inverse sixth

power law between point particles which Hamaker

extended to apply to equal spheres with the following

result.

¼

þ 4x

ðx þ 2Þ

þ ln

þ 4x

ðx þ 2Þ

ð5Þ

A is the Hamaker constant, roughly calculable

from the optical dielectric properties of particles and

the medium. For Fe, Fe

,orFe

in hydrocarbon

carrier, A ¼10

19

Nm is representative within a fac-

tor of three.

A technique to prevent particles from approaching

too close to one another is the steric repulsion arising

from the presence of long chain molecules adsorbed

onto the particle surface. A lattice model for the steric

repulsion based on the theory of polymer conﬁgura-

tions in a good solvent (deGennes 1975) yields an

expression for steric repulsion E

¼kT

1  f



 f





2  x



þ 5



ð6Þ

In Eqn. (6) d is coating thickness, a is the length

associated with the lattice spacing, i.e., the correla-

tion length along a chain, and f

is mean volume

fraction of surfactant in the space between the sur-

faces.

The net sum, E

, of the magnetic, van der Waals,

and steric energy terms is decisive for determining

monodispersity of the colloidal ferroﬂuid.

¼ E

þ E

ð7Þ

Figure 3 illustrates calculated behavior for given

size particles with two different thicknesses of coat-

ings. The curve of magnetic energy is the same in

both cases as is the distribution of van der Waals

energy. Magnetic interaction can nearly be neglected

in comparison. Steric repulsion with the thicker coat-

ing (1 nm) yields a net energy curve presenting a bar-

rier energy of nearly 25 kT. Statistically, few particles

cross the barrier and the colloid is stable against ag-

glomeration. The thin coating (0.5 nm) yields a net

energy curve that is attractive at all distances thus

yielding an unstable preparation.

Various sorbent conﬁgurations that confer steric

stability to ferroﬂuids are illustrated schematically in

Fig. 4. A particle coated with a sorbed monolayer of

a long tail molecule having a polar head group, e.g.,

oleic acid, is shown in Fig. 4(a). Figure 4(b) illustrates

a polymeric species having multiple polar groups dis-

tributed along its length. Loops of polymer rather

than tails serve as the protective layer. Because more

than one polar group attaches to the particle surface,

the colloidal stability at elevated temperatures is

enhanced; polybutenylsuccinpolyamines are an ex-

ample. Figure 4(c) illustrates a double layer conﬁg-

uration with inwardly oriented tails of polar

molecules in an outer layer associated with the tails

of the surface-sorbed layer. This conﬁguration serves

to disperse particles in aqueous media. Example

combinations are: sodium oleate/sodium oleate and

sodium oleate/sodium dodecyl benzene sulfonate.

Figure 3

Potential energy versus surface separation distance of

sterically protected 10 nm magnetite particles of a

ferroﬂuid. A net energy barrier greater than about 10 kT

prevents agglomeration of particles.

191

Ferroﬂuids: Introduction

Charge stabilized colloids prepared with the Mass-

art technique use the sorption of small moities to

adjust electrical charge on the particle. Sorption of

tetramethylammonium ion ðNðCH

Þ yields stable

dispersions in alkaline solution (pH49) where hy-

droxyl ion (OH



) furnishes the counterion, whereas

perchlorate ion ðClO



Þ and nitrate ðNO



Þ peptize

particles in acidic media (pHo5) where hydrogen

ions (H

) are the counterions. Magnetic particles

coated with citrate ligands are peptizable at interme-

diate levels of pH and are candidates for medical

application at neutral pH.

4. Phase Transition

Under certain conditions, ferroﬂuids undergo phase

separation in which micronic droplets of a concen-

trated phase exist in equilibrium with a continuous

dilute phase. The appearance of a drop is shown in

Fig. 5(a). The phenomenon can be brought on in

various ways—by application of magnetic ﬁeld in-

tensity in excess of a critical value, addition of a

foreign solvent, change of pH or counter ion con-

centration of an electrically stabilized preparation,

and by reduction of temperature. It is sometimes

possible to take advantage of such a phase separa-

tion, e.g., to sort particles by size, or to induce colors

in thin layers by diffraction of light. In other cases,

where a dilute magnetic ﬂuid is retained by applica-

tion of a ﬁeld gradient, the separation can lead to

undesired weeping of the dilute phase.

Optical methods are useful for detecting the drop-

lets. Direct microscopic observation via light trans-

mitted through a thin layer is possible for droplets

having size of a few micrometers; the layer is trans-

parent red-orange in the homogeneous region. Laser

light scattering being very sensitive to the presence of

small clusters may be more suitable for direct detec-

tion of the very onset of the transition.

Figure 6 shows coexistence curves for a hydro-

carbon-based ferroﬂuid that has particle size d

7.4 nm as determined by electron microscopy. Field

intensity producing transition can be increased using

a sample containing smaller size particles. A sample

with particle sizes d ¼5.0 nm tested homogeneous

at all concentrations in the highest ﬁeld applied

(12.7 Am

1

5. Ferroﬂuid Modiﬁcation

If an excess of a polar solvent such as acetone or

a lower molecular weight alcohol is added to an

Figure 4

Sorbed protective layers of various types yield stably dispersed particles in a ferroﬂuid.

Figure 5

As shown in (a) a stretched droplet of phase-separated

ionic ferroﬂuid exhibits a jump in length as applied ﬁeld

exceeds a critical value; (b): hysteresis results when ﬁeld

is decreased. Viscosity of the drop phase exceeds that of

the dilute phase by a factor of more than 300 as

determined from birefringence (after Bacri and Salin

1983).

192

Ferroﬂuids: Introduction

oleic-acid-stabilized ferroﬂuid in hydrocarbon carrier

the magnetite particles ﬂocculate and settle out in an

external force ﬁeld. Evidently, the surface layer re-

tracts and becomes ineffective in providing steric re-

pulsion. The particles redisperse spontaneously when

fresh solvent is added, showing that the surfactant

remains bound onto the particle surface. The tech-

nique permits the concentration of the colloid to be

adjusted and excess surfactant removed. Removing

the excess generally reduces the viscosity, especially

of high concentration, more magnetic preparations.

Also, the initial solvent, e.g., high volatility heptane,

can be exchanged for one of low volatility, useful in

applications where the ferroﬂuid is exposed to the

atmosphere as in magnetic ﬂuid rotary shaft seals and

acoustic drivers.

Certain surfactants are only weakly adsorbed and

dissolve off into the polar liquid mixture. Contacting

the settled out particles with fresh solvent fails to

produce a stable dispersion. An example is aqueous

ferroﬂuid having particles coated with Aerosol C61,

an amine complex. After washing the precipitated

particles with solvent, a different surfactant may be

added which is capable of strongly adsorbing onto

the particle surface. The new surfactant can be cho-

sen to confer dispersibility of the particles in a totally

different solvent. The technique confers great ﬂexi-

bility in the production of ferroﬂuids in diverse car-

riers such as mineral oils, esters, silicones, etc.

6. Magnetic Properties

6.1 Equilibrium Magnetization

The magnetic torque, t, experienced by a single par-

ticle of volume V and domain magnetization M

whose moment is oriented at angle y to applied in-

duction B is given by t ¼mHsiny where m ¼M

V is

the magnetic moment of the particle. The associated

energy to rotate the particle away from alignment

with the ﬁeld is W ¼mB(1cosy) with the probability

p(y) of ﬁnding a particle oriented between angle y

and y þdy proportional to the Boltzmann factor

exp(W/kT) times the steradians in orientational

space, sinydy, with the component of the moment

along the ﬁeld direction given by mcosy. The mean

value of magnetization for monodisperse particles is

given by the integration:

cosðyÞpðyÞdy

pðyÞdy

¼ cothg 

RLðgÞð8Þ

where L(g) is the Langevin function and g ¼mB/kT.

L(g) ¼g/3 for g51 and L(g) ¼11/g for gb1. Noting

that

m=m ¼ M=M

where M is magnetization of the

ferroﬂuid and M

is the saturation value of M, mag-

netization curves for monodisperse spherical particles

calculated with the domain magnetization of mag-

netite are shown in Fig. 7.

Figure 6

Coexistence curves for a hydrocarbon-based ferroﬂuid

of magnetite particles. Insets show appearance of a thin

layer seen through a microscope: one-phase region

shows homogeneous coloring; cross-sections of

stretched droplets in the two-phase region form a

patterned array due to mutual repulsion; applied ﬁeld is

normal to the layer (after Popplewell and Rosensweig

1992).

Figure 7

Magnetization curves computed from the Langevin

relationship (magnetite, M

¼4.46 10

1

193

Ferroﬂuids: Introduction

For a polydisperse system the shape of the mag-

netization curve depends on the distribution of par-

ticle sizes. Commonly a good ﬁt to the distribution is

given by the log normal distribution function P(d)

where P(d)d(d) is the fraction of total magnetic vol-

ume having diameter between d and d þd(d).

PðdÞ¼

ﬃﬃﬃﬃﬃﬃ

exp 

ðd=d



ð9aÞ

and

PðdÞdðdÞ¼1 ð9bÞ

where ln(d

) is the mean of ln(d), d

is the median

diameter, and s is its standard deviation The most

probable diameter is given by d

¼d

exp(s

), the

mean diameter /dS ¼d

exp(s

/2), and higher mo-

ments /d

S ¼d

exp(n

/2). Taking size distribution

into account the magnetization curve is given by:

MðHÞ¼

Mðd; HÞPðdÞdd ð10Þ

Chantrell et al. (1978) used expansions of the La-

ngevin function together with the log normal distri-

bution to obtain the following expressions for median

diameter, d

, and standard deviation, s, from a mag-

netization curve of a polydisperse system.

s ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð11aÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

18kT



ð11bÞ

where w is the initial susceptility of the ﬂuid and 1/H

is the extrapolation to M ¼0 of the straight line

obtained for high ﬁelds when plotting M ¼f(1/H).

Measured values of magnetization for dilute ferro-

ﬂuids are in good agreement with the superparamag-

netic relationship when the distribution of particle

size is taken into account. Dipole–dipole interaction

between particles elevates the low ﬁeld portion of the

magnetization curve for ferroﬂuids having parameter

V=kTE30 or greater.

Temperature dependence of magnetization is of

interest in problems of thermal convection. The

pyromagnetic coefﬁcient K ¼@M/@T depends on

domain magnetization, ﬂuid expansion coefﬁcient,

particle volume fraction, and thermal disorientation

which can be modeled with the Langevin function.

6.2 Anomalous Susceptibility

The initial susceptibility w ¼M/H of ferroﬂuid con-

taining noninteracting particles can be computed

from the expansion of Eqn. (8) as L(g) ¼M/M

¼g/3

where M

is the saturation magnetization of the

ferroﬂuid and a is the Langevin argument. Thus

w ¼pm

/18kT where M

is the domain mag-

netization of the magnetic particles. For a concen-

trated phase of 12 nm magnetite particles with

¼0.1T the value of w is 3.27. Polydisperse

ferroﬂuid samples posess susceptibilities that are con-

sistent with this estimate. However, as seen previous-

ly, values of w in the droplets of a phase-separated

sample are more than an order of magnitude larger.

An in situ measurement based on elongation behavior

of a droplet as a function of applied magnetic ﬁeld

intensity provides an elegant means for measuring the

susceptibility as discussed in the following.

Using ionic ferroﬂuid the elongated deformation is

obtained in ﬁelds of very low intensity, of the order of

120 Am

1

(1.5 Oe). Above a threshold value of the

magnetic Bond number, B

¼m

,whereR

radius of an initially spherical drop and s

interfacial

tension, the drop experiences an elongational shape

instability characterized by a steep jump of its aspect

ratio. The S-shape curve, see Fig. 5(b) can be ﬁtted to a

theoretical expression assuming an ellipsoidal prolate

shape allowing susceptibilitiy w and s

to be deduced.

The minimum susceptiblity to achieve the hard tran-

sition occurs at about w ¼18. For this data w ¼75710

and s ¼972 10

7

1

with R

¼6.5 mm.

Subjected to a rotating magnetic ﬁeld, the drops

ﬁrst elongate and spin, develop arms reminiscent of

starﬁsh, and undergo other transitions depending on

ﬁeld rotational frequency and intensity.

6.3 Nonequilibrium Magnetization

The vectorial magnetization, M

, in equilibrium with

steady magnetic ﬁeld, H, in motionless ﬂuid is aligned

in the direction of H and possesses magnetization

magnitude M

given, for example, by the Langevin

relationship. However, in the presence of time-vary-

ing magnetic ﬁeld magnitude or orientation the mag-

netization, M, lags behind the ﬁeld and differs from

. Models have been developed (Shliomis 1974,

Martsenyuk et al. 1973) to treat this behavior which

has importance to dynamic ﬂows of ferroﬂuids.

Measurement of the frequency response of mag-

netization of a ferroﬂuid yields information concern-

ing the relaxation mechanism (Fannin et al. 1990).

7. Viscosity of Ferroﬂuid

7.1 Viscosity in Absence of Field

In 1906, Einstein derived the earliest theory for non-

interacting spheres suspended in a liquid based on the

ﬂow ﬁeld of pure rate-of-strain perturbed by the

presence of the sphere. The resulting formula relates

the mixture viscosity, Z, to the carrier ﬂuid viscosity,

194

Ferroﬂuids: Introduction

, and solids fraction, f, in the form Z/Z

¼1 þ5f/2.

This relationship has been extended by Batchelor to

second order in concentration. For higher concen-

trations, a parameterized expression may be assumed

of the form Z/Z

¼1/(1 þaf þbf

) with insistence

that the expression (i) reduce to the Einstein rela-

tionship for small values of f so that a ¼5/2; (ii) the

suspension becomes effectively rigid at a concentra-

tion, f

, indicative of close packing (0.74 for close

packed spheres); and (iii) core particles of volume

fraction f coated with dispersing agent of thickness d

occupies volume fraction f(1 þd/r)

where r is radius

of the core particle. Combining these relationships

results in the expression:

Z  Z

1 þ





=2  1

1 þ



f ð 12Þ

If particles have a tendency to cluster and form a

gel network, the value of f

is less than 0.74. For f

less than 0.4 the coefﬁcient of the last term in Eqn.

(12), and hence the slope of a plot of (Z – Z

)/fZ vs. f

changes sign, a phenomenon that has been observed

for certain ferroﬂuids.

7.2 Viscosity in Magnetic Field

When a magnetic ﬁeld is applied to a ferroﬂuid that is

sheared, the magnetic particles in the ﬂuid tend to

remain aligned in the direction of the orienting ﬁeld

and viscous dissipation in the ﬂuid increases. A the-

oretical treatment (Shliomis 1974) for dilute suspen-

sions of particles with large Ne

el time constant and

accounting for the Brownian rotational motion gives

g  tanhg

g þ tanhg



sin

b ð13Þ

where g is the Langevin parameter, b the angle be-

tween the magnetic ﬁeld, H, and the ﬂuid vorticity,

X ¼X v, where v is the ﬂuid velocity vector. The

vorticity is twice the local average spin rate of the

colloid. Magnetic particles rotate freely when the ro-

tational and magnetic axes are aligned with the local

ﬁeld direction but are hindered otherwise. In laminar

ﬂow through a circular tube (Poiseuille ﬂow) the

vorticity magnitude is constant on concentric circles

in a cross-section normal to the axis of the tube. For

ﬁeld oriented parallel to the axis, sin

b ¼1 while for

perpendicular orientation of the ﬁeld relative to the

tube axis, averaging over b gives /sin

bS ¼1/2.

Hence, for any applied magnetic ﬁeld intensity,

¼2DZ

. As seen in Fig. 8, the theoretical curves

(solid lines) agree well with the experimental data of

McTague (1969) for a dilute suspension of polymer-

stabilized cobalt particles in toluene.

In alternating, linearly polarized magnetic ﬁeld,

the viscosity of a ferroﬂuid can exhibit a substantial

reduction, i.e., a negative viscosity component

(Shliomis and Morozov 1994). The particles of the

ferroﬂuid act as tiny motors rotating faster than the

vorticity spin rate thus helping to propel the ﬂuid.

8. Other Physica l Properties

Mass density and specific heat of ferroﬂuids is esti-

mated as the volumetric average of particle and

matrix liquid values assuming surfactant to possess

properties of the carrier liquid. Pour point and vapor

pressure can be estimated as that of the carrier liquid.

Surface tension at ambient temperatures of sterically-

stabilized aqueous ferroﬂuid is reduced from that of

water (72 mNm

1

) to a typical value of 26 mNm

1

most likely due to free surfactant concentrated at the

air interface. Interfacial tension for ferroﬂuids in var-

ious organic alkanes and esters ranges typically from

20 to 32 mNm

1

depending on the carrier.

The thermal conductivity, l, of a composite con-

sisting of particles in a matrix ﬂuid is described by the

Maxwell formula obtained by comparing the thermal

ﬁeld with the analogous electrical ﬁeld and it is as-

sumed that ferroﬂuids can be described in this man-

ner also.

þ l

þ 2f

ðl

 l

þ l

 f

ðl

 l

ð14Þ

where l

is thermal conductivity of the carrier solution,

the thermal conductivity of the particles, and f

the

particle volumetric concentration. Solids conductivity

may differ significantly from that of the carrier, e.g.,

for g-Fe

and perhaps magnetite l

E6Wm

1

while for an oil l

E0.13 W m

1

. Thus, the solids

volumetric concentration must considerably affect the

Figure 8

Magnetoviscosity of a dilute ferroﬂuid containing

particles of elemental cobalt. Data of McTague (1963)

compared to theory (after Shliomis and Raikher 1980).

195

Ferroﬂuids: Introduction

thermal conductivity of the ferroﬂuid. Data of mag-

netite-in-kerosene ferroﬂuid show a range 1.0ol/

o2.0 with maximum concentration of f ¼0.2. Mag-

netic ﬁeld oriented normal to the temperature gradient

does not affect l.

Ferroﬂuid solutions when sufﬁciently dilute or in a

thin enough layer transmit a light beam, and the

normally isotropic ﬂuid becomes optically aniso-

tropic in an applied magnetic ﬁeld, H. Properties

of interest include birefringence, dichroism, and

Faraday rotation.

Birefringence and dichroism in homogeneous sam-

ples is attributed to the shape anisotropy of particles

and possibly correlations between neighbors. The in-

ﬂuence of an individual particle depends on the ori-

entation of its axis relative to the electric ﬁeld of

the incident light ray. Analysis that averages over

orientations yields a dependence on applied ﬁeld in-

tensity that is the same for both phenomena

(Shliomis and Raikher 1980).

 n

 K

¼ 1 

3LðgÞ

ð15Þ

where n

and K

are components of refractive index

and adsorption coefﬁcient, respectively. Figure 9 il-

lustrates the favorable comparison of data and pre-

dictions for a sterically-stabilized ferroﬂuid. Using

birefringence detection with a sample having volu-

metric particle fraction of less than 10

3

as a sensor,

the viscosity of carrier liquids was measured with an

accuracy of 5% over a viscosity range of seven dec-

ades (Bacri et al. 1985). The viscosity of the solu-

tions was negligibly changed by presence of the

particles, less than 2.5 parts per thousand using the

Einstein relationship. Bibette (1993) produced a

monodisperse emulsion of oil base ferroﬂuid drop-

lets in an aqueous carrier and optically measured

force vs. distance characterizing the electrostatic

double layer surrounding the droplets. Modulation

of the applied magnetic ﬁeld was used to control the

attraction between droplets, hence their separation

distance.

9. Status and Future Directions

Scientific understanding of ferroﬂuids as a material

has progressed greatly. Proceedings of the Interna-

tional Conferences on Magnetic Fluids (Elsevier,

North-Holland, Publishers) are a rich source of in-

formation including bibliographies of patents and the

literature. The inﬂuence of magnetic and thermal

ﬁelds on the hydrodynamics of ferroﬂuids, i.e., ferro-

hydrodynamics, is highly developed. Commercial

applications of ferroﬂuids range from tribology to

instrumentation to densimetric separation of miner-

als, and many other uses are in stages of research.

Materials research challenges for the future include:

development of stable ﬂuids having larger saturation

moments, conjugation of ferroﬂuid particles with

protein species for drug delivery and medical imag-

ing, development of ferroﬂuid inks in various colors,

ferroﬂuids possessing greater monodispersity, stable

ferroﬂuids in liquid metal carrier, and ferroﬂuid com-

posites with tailored properties.

See also: Ferroﬂuids: Applications Ferroﬂuids: Neu-

tron Scattering Studies; Ferroﬂuids: Preparation and

Physical Properties; Ferrohydrodynamics

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Bacri J-C, Perzynski R, Salin D, Cebers A O 1992 Instabilities

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Dependence on magnetic ﬁeld of birefringence and

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Ferroﬂuids: Introduction

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Ferroﬂuids: Magnetic Properties

The magnetic properties of ferroﬂuids are dominated

by the intrinsic magnetic properties of the particles

themselves. These are governed by dynamical effects

due to Ne

el relaxation in the solid and Brownian

motion in the ﬂuid, and inﬂuenced by the interpar-

ticle interaction.

1. Single-domain Particles

The particles have diameters in the range of typically

2–20 nm. The size is generally below the critical limit,

dependent on the specific ferromagnet, where a mag-

netic domain structure can form, and so the particles

can be considered as single domain. Because of the

nanometer-scaled size, ﬁnite-size effects are signiﬁ-

cant. Surface effects inﬂuence the intrinsic properties

(e.g., magnetization, magnetocrystalline anisotropy).

The breaking of symmetry results in site-specific sur-

face anisotropy; the reduced number of magnetic

neighbors and specific electronic and structural fea-

tures lead to magnetic disorder and weakened ex-

change coupling, with possible effects on atomic

moments. All phenomena depend on the surface

microstructure and can vary with the surroundings.

Volume effects inﬂuence the behavior of the ex-

change-coupled spin system against excitation. The

particle volume, V, is small enough for coherent spin

reversal (Stoner–Wohlfarth particle), at least on a

macroscopic viewing scale. Because all the spins ro-

tate in unison, except perhaps near the surface, they

can be dealt with collectively via the particle moment

m. Its magnitude is given by

m ¼ M

V ð1Þ

where M

is the intrinsic magnetization, which is the

magnetic moment per unit volume. It can often be

considered as constantly neglecting its thermal var-

iation. The orientation of m with respect to the lattice

is determined by the total magnetic anisotropy

energy, which results generally from bulk magneto-

crystalline, magnetostatic (shape), and global macro-

scopic surface anisotropies. It can often be considered

uniaxial and is then given by

E ¼ KVsin

y ð2Þ

where K (40) is the effective anisotropy energy den-

sity, generally in the range 10

–10

3

, and y is the

angle with respect to the symmetry (easy) axis.

Since the anisotropy energy decreases with volume,

the energy barriers separating the easy directions can

be comparable to the thermal energy, kT, even below

room temperature; here, k is Boltzmann’s constant

and T the temperature. m is then no longer ﬁxed in

the lattice, unlike in large crystals, but ﬂuctuates

197

Ferroﬂuids: Magnetic Properties

among the easy directions. Ne

el performed the ﬁrst

calculation of the relaxation time and this thermo-

activated phenomenon is called Ne

el or superpara-

magnetic relaxation. The corresponding magnetic

behavior is called superparamagnetism.

A particle responding to a magnetic ﬁeld rotates its

moment. In a solid, the rotation occurs inside the

particle (Ne

el relaxation). In a ferroﬂuid, it can also

occur by bulk rotation (Brownian rotational diffu-

sion). The relaxation rates are generally different and

the two processes can often be treated separately.

2. Ne

el Relaxation

In a uniform magnetic ﬁeld H of strength H, the free

energy of an isolated uniaxial particle with aniso-

tropy barrier E

¼ KV, ﬁxed in space, is given by

E ¼ E

sin

y  m

mHðcos y cos c þ sin y sin c cos jÞð3Þ

where m

is the permeability of free space, c the angle

between the easy axis (unit vector e) and ﬁeld, and j

the longitude of m relative to the plane (e,H). H

m is the anisotropy ﬁeld. The energy has a

bistable form if the ﬁeld is not too large, i.e.,

HoH

ðcÞ with 1XH

X1=2. In Brown’s model,

the equation of motion is given by the Gilbert–Land-

au–Lifshitz equation

1

m  H

þ m  H

ðÞm



2gkT

ðz

1

þ zÞð4Þ

where g is the gyromagnetic ratio (1.7 10

1

and z (E10

1

210

2

) the dimensionless damping

factor of the precession of m in the effective ﬁeld

¼m

1

@E=@m þ hðtÞ, where hðtÞ is a random

white-noise ﬁeld. The probability density of m ori-

entations obeys a Fokker–Planck equation, as is the

case for Brownian motion.

The effective time to surmount the barrier is given

by t

¼ 2t

n¼1

, where l

is the nth eigen-

value, and A

its amplitude. The lowest (a0) eigen-

value, l

, is associated with the overbarrier mode,

and higher ones with intrawell modes. Only the

former is frequently considered and then t

. If the ﬁeld is parallel to the easy axis, the

energy barrier is E

ð17hÞ

with h ¼ H=H

, and t

given by

a ¼

{1; t

¼ t

1 

a þ

875



1

ac1; t

¼ t

1=2

3=2

ð1  h

Þð17hÞ



1

 exp½að17hÞ

;

1

¼ t

1

þ t

1



ð5Þ

1

and t

1



are the probabilities per unit time of m

reversal from the lower to upper minimum and vice

versa, respectively. For H ¼ 0, t

is well approxi-

mated for all barrier heights by

¼ t

f2a½p

1=2

3=2

ða þ 1Þ

1

þ 2

a1

g

1

ðe

 1Þð6Þ

For ac1, t

is often written as t

¼ t

, where the

factor t

is of the order of 10

9

–10

11

3. Brownian Rotational Diffusion

If m is blocked in the lattice, along the easy axis

(y ¼ 0), the free energy is

E ¼m

mHcos c ð7Þ

There is no precessional motion. The equation of

motion for Brownian rotation is given by

m  H

ðÞm½; t

ð8Þ

where V

is the hydrodynamic volume of the particle,

and Z (N s m

2

) the kinematic viscosity of the carrier.

The longitudinal, t

), and transverse, t

, relax-

ation times are given by

x  2LðxÞxL

ðxÞ



LðxÞ

2LðxÞ

x  LðxÞ

ð9Þ

with

x ¼

where LðxÞ¼cothx  1=x is the Langevin function.

For x{1, LðxÞ¼x=3, so t

¼ t

. For xc1,

LðxÞ¼1  1=x and t

¼ t

=2 ¼ t

=x. Equations

(7)–(9) also apply to the solid-state process (with t

replacing t

) if the inﬂuence of the anisotropy is

negligible (HcH

4. Behavior in Zero Field

A moment rotating by both processes has the effec-

tive relaxation time t

eff

given by

eff

ð10Þ

so the process with the shorter time is dominant. In

general, t

, implying a critical volume V

satis-

fying t

¼ t

(e.g., for t

¼ 10

2

and H ¼ 0,

E8 kT K

1

). Smaller particles will relax mostly by

el and larger ones by Brownian process. In a

zero ﬁeld, t

is in the range of approximately 10

10

–

s, and t

of 10

8

–10

4

s, so t

eff

is in the range

198

Ferroﬂuids: Magnetic Properties

10

–10

4

s. A moment will appear blocked or time-

averaged to zero depending on whether t

eff

is longer

or shorter, respectively, than the measuring time, t

Every moment is time-averaged on the scale for d.c.

magnetization measurements, where t

is typically

1–100 s, and so a ferroﬂuid has no spontaneous mag-

netization. Moreover, after the removal of the ﬁeld

the system is demagnetized within a time t{t

,so

ferroﬂuids are intrinsically perfectly reversible in

terms of their d.c. magnetic properties. If particle ro-

tation is frozen out, the relaxation time can reach

a geological scale. Under given experimental (t

; T)

conditions, a particle can appear unmagnetized

) or magnetized (t

) depending on its

size and anisotropy (KV).

5. D.c. Magnetization and Initial Susceptibility

When the measuring time is longer than the relaxa-

tion time, the system is at thermodynamic equilibri-

um and Boltzmann statistics can be applied. If

or t

), the Ne

el relax-

ation is of no consequence and Eqn. (7) leads to the

magnetization given by

M ¼

cos c e

E=kT

sin c dc

E=kT

sin c dc

¼ M

LðxÞð11Þ

If t

(ot

), which can occur only in small ﬁelds,

the system is at equilibrium on the Brownian time

scale (Eqn. (3), MðcÞ); the easy axis orientation ( c)

is changed after a Brownian event and so the mag-

netization at global equilibrium will be M ¼

MðcÞsincdc, which reduces for x{1to

M ¼ M

x=3. For xc1, there is no Ne

el relaxation

(H4H

) and M ¼ M

ð1  1=xÞ as a ﬁrst approxi-

mation. Langevin’s law of paramagnetism describes

the result in each case. By analogy with spins, the

equilibrium behavior is called superparamagnetic.

The initial (low-ﬁeld) susceptibility follows a Curie

law

{1; w

3kT

ð12Þ

If there are n similar particles per unit volume in

the sample, the sample magnetization is M

¼ nM.

In practice, it is necessary to take the size distribution

into account. Experimental devices measure the mag-

netic moment, so M

¼ C

/MS, where C

is the

volume fraction of particles, and M

the average

particle magnetization given by

MðVÞVf ðVÞdV

Z ¼

Vf ðVÞdV ð13Þ

where f ðVÞ is the volume distribution function. If all

particles satisfy the conditions for the low- and high-

ﬁeld approximations of the Langevin function, the

magnetization is given by

low field M

3kTZ

f ðVÞdV ð14Þ

high field Mhi¼M

1 



ð15Þ

where M

is the average intrinsic magnetization. The

data exhibit no remanence and coercivity and show

H/T superposition (Fig. 1). In principle, one can

make use of the magnetization curve to determine the

parameters of the size distribution.

Below the melting temperature of the carrier, the

moments will gradually block with decreasing tem-

perature, according to the distribution of energy

barriers. A blocking temperature T

is deﬁned for a

particle of volume V as the temperature at which

¼ t

. In very small ﬁelds, T

¼ KV=klnðt

Þ for

ac1. If the easy axes are randomly oriented, for

Figure 1

Thermal dependence of reduced magnetization curves of

magnetite ferroﬂuids. Median particle diameter (A)

4.8 nm; (B) 9.6 nm. (Reprinted with permission from

Williams et al. 1993; r Elsevier Science B.V.)

199

Ferroﬂuids: Magnetic Properties