THE JOURNAL OF CHEMICAL PHYSICS 134, 184508 (2011)

Magnetophoresis, sedimentation, and diffusion of particles in concentratedmagnetic fluids

Alexander F. Pshenichnikov,1 Ekaterina A. Elfimova,2,a) and Alexey O. Ivanov2

1Institute of Continuous Media Mechanics UB RAS, 1, Korolyov St., 614013, Perm, Russia2Ural State University, 51, Lenin Ave., 620083, Ekaterinburg, Russia

(Received 5 November 2010; accepted 8 April 2011; published online 13 May 2011)

A dynamic mass transfer equation for describing magnetophoresis, sedimentation, and gradient dif-fusion of colloidal particles in concentrated magnetic fluids has been derived. This equation takesinto account steric, magnetodipole, and hydrodynamic interparticle interactions. Steric interactionshave been investigated using the Carnahan-Starling approximation for a hard-sphere system. In or-der to study the effective interparticle attraction, the free energy of the dipolar hard-sphere system isrepresented as a virial expansion with accuracy to the terms quadratic in particle concentration. Thevirial expansion gives an interpolation formula that fits well the results of computer simulation in awide range of particle concentrations and interparticle interaction energies. The diffusion coefficientof colloidal particles is written with regard to steric, magnetodipole and hydrodynamic interactions.We thereby laid the foundation for the formulation of boundary-value problems and for calculationof concentration and magnetic fields in the devices (for example, magnetic fluid seals and accelera-tion sensors), which use a concentrated magnetic fluid as a working fluid. The Monte-Carlo methodsand the analytical approach are employed to study the magnetic fluid stratification generated by thegravitational field in a cylinder of finite height. The coefficient of concentration stratification of themagnetic fluid is calculated in relation to the average concentration of particles and the dipolar cou-pling constant. It is shown that the effective particle attraction causes a many-fold increase in theconcentration inhomogeneity of the fluid if the average volume fraction of particles does not exceed30%. At high volume concentrations steric interactions play a crucial role. © 2011 American Instituteof Physics. [doi:10.1063/1.3586806]

I. INTRODUCTION

Magnetic fluids are stable colloidal suspensions of ferro-and ferrimagnetic nanoparticles in a nonmagnetic liquidcarriers.1 The small size of colloidal ferroparticles (typicallyof the order of 10–20 nm) provides the particle with a perma-nent magnetic moment. It is well known that in the course oftime an initially homogeneous magnetic fluid, filling a cavityof arbitrary shape, becomes spatially inhomogeneous with re-spect to the magnetic phase concentration due to gravitationalsedimentation and magnetophoresis (the motion of particlesunder the action of a nonuniform magnetic field). In the ab-sence of convective motion, the only factor that prevents theconcentration stratification of the fluid is the gradient diffu-sion of particles. The concentration profile in a cavity canbe obtained at some arbitrary time from the solution of theboundary-value problem including Maxwell’s equations forthe magnetic field and the dynamic mass transfer equationwith consideration for the terms attributable to magnetophore-sis and sedimentation of particles.

Up to now, this boundary-value problem has been solvedusing a dilute solution approximation (the volume fraction ofparticles is small compared to unity), which makes it possibleto study the magnetic and diffusion parts of the problem sep-arately and to write the diffusion equation correctly.2–4 How-

a)Author to whom correspondence should be addressed. Electronic mail:Ekaterina.Elfimova@usu.ru.

ever, this approach is not sufficient when we deal with thehigh particle concentrations, i.e., with the range of parametersof special interest from research and application viewpoints.In the case of high particle concentrations, the magnetic anddiffusion problems are strictly interrelated, and the concentra-tion profile depends markedly on steric, magnetodipole, andhydrodynamic interparticle interactions, whose counting is aproblem of great concern.

The influence of interparticle interactions on the diffu-sion processes taking place in magnetic fluids was consideredby a number of researchers, who focused the attention on thecalculation of the diffusion coefficient. In particular, stericand hydrodynamic interactions in the linear (with respectto the particle volume concentration ϕ) approximation weretaken into account using the Batchelor formula for thegradient diffusion coefficient of particles in low concentratedsuspensions:5

D = b0kT (1 + 1.45ϕ) = D0(1 + 1.45ϕ), (1)

where b0 is the mobility of particles in the carrier fluid, kT isthe thermal energy, and D0 is Einstein’s value of the diffusioncoefficient for dilute solutions. The coefficient of gradientdiffusion (1) appears to be the increasing function of particleconcentration, while the particle mobility decreases withconcentration. This effect was explained5, 6 by the influenceof “excluded volume,” stimulating the transition of particlesfrom the high concentration region to the low concentration

0021-9606/2011/134(18)/184508/9/$30.00 © 2011 American Institute of Physics134, 184508-1

184508-2 Pshenichnikov, Elfimova, and Ivanov J. Chem. Phys. 134, 184508 (2011)

region. Also this effect was confirmed by experimental data.7

The region of applicability of the formula (1) is usuallyrestricted by the condition ϕ ≤ 10−2. Biben and Hansenconsidered the non-dilute case.8 They investigated the sedi-mentation of colloidal particles in a monodispersed chargedcolloids. Calculations were based on a simple free energyfunctional with account for excluded volume and Coulombcontributions. This approach was extended by Biesheuveland Lyklema9 for the case of binary mixture of chargedcolloids, and the influence of the gravity, the hard sphere non-penetration and the ion pressure was studied. The formulafor the chemical potential of the magnetic fluid, describingthe excluded volume effect (as in the case of the van derWaals gas) was derived in the work of Cerbers.10 Significantprogress in the problem of taking into account steric interac-tions was achieved in Refs. 11 and 12. The authors derived theformula for the gradient diffusion coefficient of particles inthe Carnagan–Starling approximation for the system of hardspheres13 and, besides, introduced the correction, linear inconcentration, for the effective attraction of spherical dipoles:

D = D0 K (ϕ)

[1 + 2ϕ

4 − ϕ

(1 − ϕ)4− 8

3λ2ϕ

], λ = μ0m2

4πd3kT.

(2)

Here K(ϕ) = b/b0 is the relative mobility of particles in themagnetic fluid, b is the mobility of particles in the magneticfluid, μ0 = 4π10−7 H/m, λ is the coupling constant, m, d arethe magnetic moment and diameter of the particle, respec-tively. In the absence of an external magnetic field the relativemobility K (ϕ) of ferroparticles in a magnetic fluid is a scalarfunction of the particle volume concentration. Mobility K (ϕ)could be expanded in power series over particle volumeconcentration and the linear term was given by Batchelor:5

K (ϕ) = (1 − 6.55ϕ). (3)

The second term in square brackets in Eq. (2) takes intoaccount steric interactions in the full range of particle con-centrations quite accurately and remains unchanged below.

Morozov14, 15 has focused attention on mass transferanisotropy in a system of interacting dipolar particles un-der the presence of an external magnetic field. He has sug-gested the conception of anisotropic diffusion coefficients andhas calculated these renormalized diffusion coefficients tak-ing into account magnetodipole and steric interactions for themagnetic fluid in the infinite plane layer placed in the uniformapplied field H0. The gradient of particle concentration ∇n isdirected across the plane layer so that the magnetic part of theproblem has a trivial solution B = μ0 (H + M) = const orH = const depending on the direction of the applied mag-netic field (M is fluid magnetization). Two field orientations,namely, along the concentration gradient and across it, wereconsidered. In these limiting cases, the diffusion coefficientsin the linear concentration approximation are, respectively,equal to

D|| = b0kT [1 + 1.45ϕ + 2.23χL L2(ξ0)], for H||∇n,

(4)

D⊥ = b0kT [1 + 1.45ϕ − 1.112χL L2(ξ0)], for H ⊥∇n.

(5)

Here χL = μ0m2n/3kT is the initial susceptibility of thefluid, obtained using the Langevin approximation, L(ξ )= coth(ξ ) − 1/ξ is the Langevin function, and ξ0

= μ0m H0/kT is the Langevin parameter.There are two independent reasons of mass transfer

anisotropy in magnetic fluid.14, 15 First, the particle mobilitymight be anisotropic due to field-dipole interaction. The sec-ond one is the anisotropy of thermodynamic forces causedby demagnetizing fields (i.e., cavity shape). The lasts termsin Eqs. (4) and (5) take into account both of these effects.However, it was shown by Morozov15 that the influence ofparticle mobility anisotropy is much weaker (by an order ofmagnitude) than the one of thermodynamic forces anisotropy.This fact has allowed Morozov to neglect the anisotropy ofparticle mobility when studying the diffusion in concentratedmagnetic fluids.15 The obtained results agree well with exper-imental data,16 but the range of its application is limited bythe case of the uniform applied magnetic field and the planelayer of magnetic fluids, when the gradient of particle concen-tration is directed across the layer. Any changes of the cavityshape and/or magnetic field orientation make the expressions(4) and (5) incorrect.

The goal of the present study is to derive a dynamic masstransfer equation for describing magnetophoresis, sedimenta-tion, and gradient diffusion of particles with consideration forsteric, magnetodipole, and hydrodynamic interactions in cav-ities of arbitrary shape and in magnetic field of arbitrary con-figuration. Following Refs. 11 and 15, we do not take into ac-count the weak anisotropy of particle mobility in an externalmagnetic field. This uncrude approximation leads to a greatsimplification of the mass transfer equation. Unlike Refs. 14and 15, we suggest to consider the strong anisotropy of ther-modynamic forces with the help of additional term in themass transfer equation, responsible for magnetophoresis. Inthis way, the anisotropic diffusion coefficients need not to bedefined. The coefficient of gradient diffusion remains a scalarfunction and does not depend on cavity shape and magneticfield configuration.

II. STATIONARY DISTRIBUTION FOR DILUTEMAGNETIC FLUIDS

Let us first consider the problem of stationary distribu-tion of particles in a dilute magnetic fluid, contained in aclosed cavity with fixed isothermal boundaries in the absenceof hydrodynamic flows. The particle concentration is low,the dipolar coupling constant is small, and hence the inter-particle interactions are inessential. This case is of interestbecause the spatial distribution of particles in the cavity canbe found without formulating and solving the boundary-valueproblem and the obtained solution can be further used toanalyze situations that are more complicated. The solutionof the problem is simplified because the force fields arepotential and the cavity walls are impermeable to particles.Under these conditions, the stationary state of the system isat the same time the thermodynamic-equilibrium state (fluxes

184508-3 Magnetophoresis, sedimentation, and diffusion of particles in fluids J. Chem. Phys. 134, 184508 (2011)

of mass, momentum, and energy are absent), and the systemis described by the Boltzmann distribution. The potentialenergy U of a single-domain colloidal particle is

U = −μ0m H (x, y, z) cos θ − �ρVs g(z cos α + x sin α),

(6)

and the distribution function is

f (r, θ ) = exp

(−U (r, θ )

kT

). (7)

Here θ is the angle between the particle magnetic moment mand the field intensity H, �ρ is the density difference betweenthe solid kernel of the particle and the disperse medium, Vs

is the volume of the solid particle kernel, α is the anglebetween the z-axis and the gravity vector g, lying in the xOzplane. The last term in expression (6) describes the influ-ence of the Archimedean force. The number concentrationn(x,y,z) of particles at the arbitrary point of the space can bedetermined by averaging Eq. (7) over the magnetic momentorientations:

n(x, y, z) = Asinh ξ

ξexp[Gγ (z cos α + x sin α)],

Gγ = �ρVs g/kT . (8)

Here the Langevin parameter ξ = μ0m H/kT is the knownfunction of coordinates, and Gγ is the gravitational parameter(the reciprocal height of the barometric distribution). Thenormalization constant A is best determined in terms ofthe average volume concentration of particles in the cavity,therefore it is convenient to change the number concentrationof the particles n for the particle volume concentration ϕ Inthis case, the concentration field in the cavity is described bythe equation

ϕ = 〈ϕ〉sinh ξ

ξexp[Gγ (z cos α + x sin α)]

1

V

∫V

sinh ξ

ξexp[Gγ (z cos α + x sin α)]dv

,

(9)where 〈ϕ〉 denotes the average volume concentration ofparticles.

Since Eq. (9) describes only the equilibrium state ofthe system, it does not include any kinetic coefficients. It isapplicable to the cavities of arbitrary shape (including 3Dproblems) and arbitrary magnetic fields and can be readilyextended to polydisperse suspensions. In this case, we need towrite an Eq. (9) for each fraction and then summarize the left-and right-hand sides of these equations. In the limit of weakfields (ξ� 1), Eq. (9) coincides with the barometric formula.

The main disadvantage of Eq. (9) is that it neglects theinterparticle interactions and, therefore, cannot be used fordescribing concentrated systems, including the case whenthe high concentration of particles appears only in somepart of the system. The equation similar to Eq. (9) was de-rived earlier,4, 17 but it did not take into account particlesedimentation.

III. MAGNETOPHORESIS AND SEDIMENTATIONOF PARTICLES IN CONCENTRATED SYSTEM

The problem of magnetophoresis of ferroparticles in anonuniform magnetic field is solved with account for inter-particle magnetodipole interaction in the framework of themodified model of the effective field (MMEF), whose effi-ciency has been verified in the early works devoted to thestudy of equilibrium magnetization of concentrated magneticfluids.18–20 Following Refs. 19 and 20, the equilibrium mag-netization of magnetic fluids is described by the system ofequations:

M = mnL(ξe), ξe = μ0m He

kT,

He = H + ML (H )

3

[1 + 1

48

d ML (H )

d H

], ML = mnL(ξ ),

(10)

where H is the magnetic field in the fluid, which differs fromthe applied field H0 by the demagnetizing factor, and He isthe effective field. The system of equations (10) agrees wellwith the experimental data18 and the results obtained by theMonte-Carlo methods and molecular dynamics methods.20, 21

A significant difference between the MMEF and the exper-imental magnetization data were observed only in the caseof magnetic fluids with a very high (several tens) magneticsusceptibility.22, 23 Based on the MMEF data, in the knownformula for magnetic force F acting on a particle,24 the fieldintensity H is replaced by its effective value He, He ‖ H,

F = μ0 (m · ∇) He. (11)

Since the characteristic diffusion time of concentration dis-turbances τD ≈ L2/π2D for typical magnetic fluids exceedsthe magnetic moment relaxation time τB ≈ 3ηV/kT (L is thecharacteristic size of the cavity, η is the magnetic fluid viscos-ity) by at least six-seven orders of magnitude, the right-handside of Eq. (11) can be averaged over the time τ satisfying thetwo-sided inequality: τB � τ � τD . Bearing in mind that, inview of the MMEF, the mean value of the magnetic moment isequal to 〈m〉 = mL(ξ e)He/He and assuming that the effectivefield is potential, we obtain from Eq. (11)

F = μ0 mL(ξe) ∇He. (12)

The particle flux density jm is expressed in terms of theirnumber density n and the average drift velocity 〈v〉 = bF

jm = n〈v〉 = μ0bnmL(ξe)∇He.

Assuming that the temperature of the magnetic fluid is ho-mogeneous, we write the magnetophoresis density in theform

jm = nD0 K (ϕ)L(ξe)∇ξe. (13)

The expression for the density js of the sedimentation fluxcaused by the gravitational force is written analogously

js = nD0 K (ϕ) Gγ e. (14)

184508-4 Pshenichnikov, Elfimova, and Ivanov J. Chem. Phys. 134, 184508 (2011)

Here e is the unit vector in the direction of the gravitationalacceleration g, and Gγ is the gravitational parameter obtainedin Eq. (8).

At the end of this section, let us remind once again thatgenerally the mobility of ferroparticles in a magnetic field isanisotropic and depends on the mutual orientation of the mag-netic field intensity and the velocity of particle directed drift.However this anisotropy is weak and we neglect it. The ther-modynamic force anisotropy exceeds mobility anisotropy byan order of magnitude.14, 15 This anisotropy is rather strongand plays a decisive role. Below we take into account the ther-modynamic force anisotropy with the help of Eq. (13) as apart of the total particle volume flux density (20). This term(13) is responsible for magnetophoresis and is proportional tothe effective magnetic field gradient. In this case the coeffi-cient of gradient diffusion remains a scalar function whichleads to great simplification of the specific boundary-valueproblems.

IV. EFFECTIVE ATTRACTION OF PARTICLES

The averaging of the magnetodipole interactions over themagnetic moment orientations gives rise to the effective at-traction between particles and is an additional reason for thedrift of particles in the concentration-inhomogeneous mag-netic fluid. Because the corresponding term in the particle fluxis proportional to the concentration gradient, it can be consid-ered formally as a correction to the diffusion coefficient, like ithas been done in Ref. 12 during the derivation of Eq. (2). Thelast summand of formula (2) takes into account the interpar-ticle magnetodipole interaction in the linear-in-concentrationapproximation. For dense magnetic fluids the diffusion coef-ficient can be obtained using the hard-sphere approximationand the known relationship between the diffusion coefficientand the free energy � of the system. According to Refs. 11and 12, the free energy of hard spherical dipoles can be rep-resented as

� = N0ν0 + Nν0 + NkT ln(ϕ) − kT ln(Qs)

+ kT NϕG(λ, ϕ), (15)

where N0 is the number of molecules of the carrier fluid, N isthe number of ferroparticles in the magnetic fluid, ν0, ν

0 arethe chemical potentials of the carrier fluid and the moleculesof ferroparticles, respectively, Qs is the configuration integralof the hard sphere system,13 G(λ, ϕ) is the function specify-ing the contribution of a magnetodipole interaction to the freeenergy of the dipolar hard sphere system. Calculation of thechemical potential yields the diffusion coefficient

D = D0 K (ϕ)

[1+2ϕ

4−ϕ

(1−ϕ)4−2ϕG − 4ϕ2 ∂G

∂ϕ− ϕ3 ∂2G

∂ϕ2

].

(16)

and the diffusion flux density

jD = −D0 K (ϕ)

[1 + 2ϕ

4 − ϕ

(1 − ϕ)4− ϕ

∂2(ϕ2G)

∂ϕ2

]∇n.

(17)

To obtain G(λ, ϕ), we use the virial expansion in termsof the volume ferroparticle concentration ϕ. Each virial coef-ficient is an effective potential for interactions between the pparticles averaged over the ferroparticle position and the ori-entation of magnetic moments. The order in which the virialcoefficients are taken indicates the number of interacting par-ticles: the coefficient ϕp−2 characterizes the p-particle corre-lations. The virial coefficients are defined by the p-particlecluster integrals using the diagrammatic expansion method.25

In this paper, G(λ, ϕ) is calculated up to ϕ2.

G(λ, ϕ) = 4

3λ2 + 4

75λ4

+[(

2 ln 2 + 1

3

)λ2 − 10

9λ3 − 0.34194λ4

]ϕ

+ 0.96724λ2ϕ2 + . . . . (18)

Using the known computer simulation data,26 we definethe applicability range of Eq. (18): λ ≈ 1, ϕ < 0.3. The slowconvergence of the series and the existence of the alternateterms in the right-hand side of Eq. (18) initiate the physicallyincorrect effects in the range λ > 2. Therefore, expression(18) is modified as

G(λ, ϕ) = 4

3λ2 (1 + 0.04λ2)

(1 + 0.308λ2ϕ)

(1 + 1.28972ϕ + 0.72543ϕ2)

(1 + 0.83333λϕ).

(19)

Approximation (19) coincides with the right-hand side ofEq. (18) with accuracy to the terms of the order of ϕ2, andit provides better agreement with the results of computer sim-ulations (Fig. 1). One can see that formula (19) is in goodagreement with the results corresponding to the parameterrange provided in Ref. 26. It should be mentioned that thepositive value of G(λ, ϕ), in expressions (18) and (19), meansthat the non-central magnetodipole interaction represents it-self in ferroparticle ensemble as the effective interparticleattraction.

FIG. 1. Function of the contribution of magnetodipole interactions to freeenergy as the function of volume particle concentration and dipolar couplingparameter. Dots correspond to computer simulations (Ref. 26) and solid linesare approximation (19).

184508-5 Magnetophoresis, sedimentation, and diffusion of particles in fluids J. Chem. Phys. 134, 184508 (2011)

V. MASS TRANSFER EQUATION

Summarizing the left- and right-hand sides of Eqs. (13)and (14), and (17) and making allowance for the proportion-

ality between the number density n of particles and their vol-ume concentration ϕ, we obtain an expression for the densityof volume flux of particles in the isothermal magnetic fluid

J = D0 K (ϕ)

{ϕL(ξe)∇(ξe) + ϕGγ e −

[1 + 2ϕ(4 − ϕ)

(1 − ϕ)4− ϕ

∂2(ϕ2G)

∂ϕ2

]∇ϕ

}. (20)

The first two terms in braces of Eq. (20) characterize themagnetophoresis and the particle sedimentation in the grav-itational field or in the field of centrifugal forces, the firstand second terms in the square brackets describe the gradi-ent diffusion with the correction for steric interactions, and

the last term in the square brackets is responsible for the ef-fect of magnetodipole attraction. The dynamic mass transferequation in the absence of a convective flow is derived fromEq. (20) in a common way (see, for example, Ref. 27) and canbe written as

∂ϕ

∂t= −div

{D0 K (ϕ)

{ϕL (ξe) ∇ (ξe) + ϕGγ e −

[1 + 2ϕ (4 − ϕ)

(1 − ϕ)4− ϕ

∂2(ϕ2G)

∂ϕ2

]∇ϕ

}}. (21)

This dynamic mass transfer equation (21) for a mag-netic fluid under the action of magnetic field and grav-ity is more general the known ones.2, 11, 14, 15, 28 First, theused gradient diffusion term (16), (19) can be applied tomore concentrated magnetic fluid than the expression (2).Second, in combination with the magnetostatic equationsit can be used for description of the mass transfer pro-cesses in magnetic fluids placed in a cavity of arbitraryshape under the action of a static magnetic field of differentconfiguration.

Let us compare our results with expressions (4) and (5),which have been calculated by Morozov14, 15 in linear approx-imation over concentration for the plane layer of the magneticfluid under the condition when the particle concentration gra-dient is directed across the plane layer and only parallel andperpendicular orientations of a magnetic field are considered.The right-hand part of Eq. (20) could be expanded in powerseries over the particle concentration:

• The expansion of the first two terms in square bracketsis evident.

• Using expression (18) the last summand in squarebrackets gives the linear term over volume concentra-tion 8λ2ϕ/3.

• Expansion of the first term in braces, which is re-sponsible for magnetophoresis, has to be consideredin more details. Assuming the uniformity of an inter-nal magnetic field H, it could be connected with anapplied field H0 and magnetic fluid magnetization Mvia the demagnetizing coefficient κ (H ‖ M ‖ H0):

H = H0 − κ M. (22)

The coefficient κ depends on the geometry of the cavity andthe mutual orientation of the cavity and the applied field. Re-lationship between the corresponding Langevin parameters

follows immediately from Eqs. (22) and (10).

ξ = ξ0 − 24κλϕL(ξe). (23)

Simplifying the magnetization (10) with the linear correctionin concentration only (first-order MMEF), we get the effectiveLangevin parameter

ξe = ξ + 8λϕL(ξ ). (24)

Using combination of Eqs. (23) and (24) and taking into ac-count only linear terms, we obtain

ξe = ξ0 + 8λϕ(1 − 3κ)L(ξ0). (25)

ϕL(ξe)∇ξe = 8λϕ(1 − 3κ)L2(ξ0)∇ϕ

= χL (1 − 3κ)L2(ξ0)∇ϕ.

So, using the Batchelor formula for relative mobility (3), theEq. (20) takes a form

J = −D0

[1 + 1.45ϕ − χL (1 − 3κ)L2(ξ0) − 8

3λ2ϕ

]∇ϕ.

(26)

Hence, the Eq. (20) transforms to the classical Fick’s law, andthe gradient diffusion coefficient of colloidal particles is

D = D0

[1 + 1.45ϕ − χL (1 − 3κ)L2(ξ0) − 8

3λ2ϕ

].

(27)

The anisotropy of the renormalized diffusion coefficient (27)is defined by demagnetizing coefficient κ . In the case ofspherical cavity (κ = 1/3) the anisotropy equals zero. For theplane layer of the magnetic fluid, when the external field isdirected across the plane layer but along the particle concen-tration gradient, the demagnetizing coefficient κ equals one

184508-6 Pshenichnikov, Elfimova, and Ivanov J. Chem. Phys. 134, 184508 (2011)

(κ = 1) and the renormalized diffusion coefficient is a maxi-mum:

D|| = D0

[1 + 1.45ϕ + 2χL L2(ξ0) − 8

3λ2ϕ

]for H || ∇n.

(28)In the other limit case, when the external field is directedalong the plane layer but across the particle concentration gra-dient, the demagnetizing coefficient κ equals zero (κ = 0) andthe renormalized diffusion coefficient approaches the minimalvalue:

D⊥ = D0

[1 + 1.45ϕ − χL L2(ξ0) − 8

3λ2ϕ

]for H ⊥∇n.

(29)The Eqs. (28) and (29) coincide with Eqs. (4) and (5) withaccuracy to small corrections concerned with the anisotropyof mobility and the mutual attraction between particles (lastterm quadratic in λ).

The formulas (28) and (29) hold valid only for the planelayer of the magnetic fluid with known demagnetizing co-efficients, while in Eqs. (20) and (21) there are no restric-tions on the cavity shape and demagnetizing fields. Mass-transfer equation (21) extends the scope of the examinedproblems. Separation of the summands characterizing mag-netophoresis and diffusion provides an adequate formulationof the boundary-value problem on distribution of the magneticphase concentration over an arbitrary cavity with the magneticfluid. The only requirement is that the solution of the diffusionproblem should conform to the solution of the magnetostaticproblem considering the magnetic field intensity inside thefluid.

Terminologically, it seems reasonable to distinguish be-tween the diffusion of particles caused by the Brownian mo-tion and magnetophoresis, i.e., the drift of particles in themagnetic field including demagnetizing field. Magnetophore-sis is responsible for the concentration segregation of themagnetic fluid in the gradient field, and the diffusion of par-ticles equalizes concentration differences. From this stand-point, the question of whether the summand, which takes intoaccount the effective attraction between magnetic particles,should be involved in the diffusion coefficient may be de-bated. However, by analogy with Ref. 11, this summand isinvolved in the diffusion coefficient considered in this study,which allows us to simplify the notation. Hence, by the dif-fusion coefficient of particles in magnetic fluids we mean thequantity

D = D0 K (ϕ)

[1 + 2ϕ(4 − ϕ)

(1 − ϕ)4− ϕ

∂2(ϕ2G)

∂ϕ2

], (30)

which follows from Eq. (16).

VI. INFLUENCE OF INTERPARTICLE INTERACTIONSON MAGNETIC FLUID STRATIFICATION

A. Zero applied field

As can be seen in Eqs. (19) and (30), the existence ofmagnetodipole interparticle interactions leads to a decreasein the diffusion coefficient. The influence of these interac-tions gets stronger as the parameter λ increases, and when

FIG. 2. Diagram of spinodal decomposition of the system in zero appliedfield.

it reaches rather high values, the diffusion coefficient may benegative. In this case, the diffusion flux of particles is directedalong the concentration gradient, and the concentration per-turbation, caused randomly by fluctuations, will increase withtime. The system becomes thermodynamically unstable andstratifies into two phases, weakly and strongly concentrated.The instability gives rise to the phase transition “gas–fluid”.Equating the diffusion coefficient (30) to zero, we obtain thecondition for thermodynamic instability of the system in theabsence of applied field—spinodal decomposition caused bymagnetodipole interactions.

A diagram of decomposition is shown as the plot ϕ =ϕ(λ) in Fig. 2. To the critical point of the diagram there cor-respond ϕ* = 0.034 and λ* = 4.22. At λ < λ* (i.e., at thetemperature higher than the critical one) to the equilibriumstate of the system there corresponds the homogeneous parti-cle distribution, whereas at λ > λ* the system stratifies intoweakly and strongly concentrated phases. Despite the fact thatthe parameter values shown in Fig. 2 lie formally beyond theapplicability of approximation (19), the phase diagram turnsout to be quite realistic. The critical values λ* and ϕ* in thisdiagram fall in the range of values previously obtained byother methods. For example, λ* = 4.08, ϕ* = 0.092 in termsof the effective field model;10 λ* = 2.82, ϕ* = 0.13 in theframework of the thermodynamic theory of perturbation;12, 28

λ* = 4.45, ϕ* = 0.056 in a mean-sphere approximation;29

and λ* = 3.0, ϕ* = 0.034 in the context of Monte-Carlosimulations for a restricted system.30 The above-mentionedanalytical models can be used only within the limited rangeof concentration and dipolar coupling constant, and there-fore a broad scatter in the critical parameters is absolutelynatural.

B. External applied fields

In the external fields (magnetic or/and gravitational),the static distribution of particles in the cavity is ob-tained by equating the full particle flux (20) to zero. Thus,we have

ϕL (ξe) ∇ (ξe) + ϕGγ e

184508-7 Magnetophoresis, sedimentation, and diffusion of particles in fluids J. Chem. Phys. 134, 184508 (2011)

−[

1 + 2ϕ (4 − ϕ)

(1 − ϕ)4− ϕ

∂2(ϕ2G)

∂ϕ2

]∇ϕ = 0. (31)

It is easy to verify that this equation is equivalent to

ln ϕ + 3 − ϕ

(1 − ϕ)3 − ∂(ϕ2G)

∂ϕ

= ln

(sinh ξe

ξe

)− Gγ z + const, (32)

which, being in its implicit form, defines the spatial distribu-tion of particles in the magnetic fluid subjected to the actionof magnetic and gravitational fields. The integration constantin the right-hand side of Eq. (32) can be determined throughthe concentration of particles at some (reference) point insidethe cavity or on its boundary, or through the average volumeconcentration 〈ϕ〉

We used Eq. (32) to perform test calculations of the staticprofile of particle concentration in a vertical cylinder of finiteheight z0 placed in the gravitational field. The magnetic fieldwas absent. The results of calculation were compared withthe data obtained by the Monte-Carlo method. The computersimulation method was similar to the technique described inRef. 31. A colloidal particle is modeled as a hard sphere witha constant value of magnetic moment. The energy of the ithparticle is the sum of dipolar interactions, magnetic and grav-itational potentials:

Ui

kT= Gγ zi − ξ0 cos θi − λ

N∑j=1,

j =i

×[

3(ei · Ri j )(e j · Ri j )

R5i j

− (ei · e j )

R3i j

]. (33)

Here Rij is the distance between the centers of the ith and jthparticles, and θ i is the angle between the applied magneticfield and the magnetic moment of the ith particle, ei is the unitvector in the direction of the magnetic moment of the ith parti-cle. Steric interactions were taken into account by forbiddingthe hard spheres to overlap with each other or with the cylin-der wall. To find the stationary particle distribution profile, thecylinder was divided into 20 horizontal layers of the thicknessequal to the particle diameter. After the establishment of ther-modynamic equilibrium, the local concentration profile wasaveraged over 105 MC-steps. The data from the top and bot-tom layers were not taken into consideration because of theknown boundary effect. The mean concentration of particlesin the cylinder was determined using the rest of 18 layers andappeared to be slightly different from the concentration at theinitial time. Calculations were performed under the assump-tion Gγ z0 = 5 for the system consisting of 103 particles. Theconcentration profiles are given in Fig. 3 for different valuesof λ.

Curve 1 in Fig. 3 corresponds to the non-magnetic parti-cles (λ = 0), and the deflection of this curve from the baro-metric distribution (dashed curve) demonstrates the influenceof steric interactions on the concentration profile. Irregard-less of the relatively low average concentration of particles(slightly higher than 6% by volume), these interactions turned

FIG. 3. Static concentration profile in a vertical cylinder of finite heightplaced in the gravitational field. The magnetic field is absent. Curve 1 cor-responds to λ = 0, 〈ϕ〉 = 0.061; curve 2 is λ = 2, 〈ϕ〉 = 0.062; curve 3 isλ = 3, 〈ϕ〉 = 0.06. Dots are computer simulation data and the solid linescorrespond to formula (32).

out to be significant. Initiation of the magnetodipole interac-tions (curves 2 and 3) markedly strengthens the system strat-ification if λ > 1. In particular, at λ = 3 the stratificationcoefficient P (the ratio between the maximum and minimumvalues of concentration) becomes three times larger than thatobserved for λ = 0. In general, Fig. 3 demonstrates quitegood agreement between Eq. (32) and the results of the MC-simulation for all examined parameters.

Figure 4 presents the plot of the stratification coefficientP versus the average volume concentration, obtained fromEq. (32) under the same conditions as in Fig. 3. The dashedline corresponds to the barometric distribution in dilute so-lutions. The deviation from the barometric distribution indi-cates the influence of interparticle interactions. As the aver-age volume concentration of particles increases from zero tothe maximum possible value ϕm ≈ 0.61, the stratification co-efficient decreases by four (!) orders of magnitude. The ef-fective attraction of the magnetic dipoles plays an importantrole in the stratification of the magnetic fluid in moderatelyconcentrated fluids when the volume particle concentrationis 3%–30%. In this case, the effective attraction is able togenerate a several-fold increase in the inhomogeneity of thefluid. In strongly concentrated magnetic fluid, where the par-ticle volume fraction is higher than 30%, the influence of ef-fective interparticle attraction on the stratification becomesinconsiderable.

VII. INFLUENCE OF APPLIED MAGNETIC FIELD

Equations (21) and (32) describe the spatial distributionof particles in the magnetic fluid. These equations have beenderived without any assumptions concerning the cavity shape.However, the presence of the effective Langevin parameter inEqs. (21) and (32) indicates that for calculation of concentra-tion fields we need to get more information concerning mag-netization and intensity of the field inside the fluid. That is the

184508-8 Pshenichnikov, Elfimova, and Ivanov J. Chem. Phys. 134, 184508 (2011)

FIG. 4. Stratification coefficient P versus the average volume concentration〈ϕ〉, obtained from Eq. (32). Dashed line corresponds to the dilute solutionapproximation, and the numbers of solid curves show the numerical valuesof the dipolar coupling parameter.

reason why Eq. (21) or Eq. (32) should be solved in combina-tion with Maxwell’s equations

rotH = 0, div (H + M) = 0, (34)

and Eq. (10), which show the relation between the magne-tization and the field intensity and particle concentration. Inthe general case (complex geometry of the cavity), the inho-mogeneous distribution of particles in the cavity and the non-linearity of the diffusion equation extremely complicate theproblem even for the numerical solution. The problem is sim-plified for the cavity of simple (ellipsoidal) shape. If the con-centration difference in the cavity is small enough, then themagnetostatic problem with Eq. (34) is solved analytically.24

In this case, we can use formulas (22)–(24). It allowed us toderive the effective value of the Langevin parameter ξ e fromthe Langevin parameter ξ 0 (Eq. (25)) determined through theapplied field H0 and to calculate the equilibrium distributionof the magnetic phase concentration using Eq. (32).

VIII. CONCLUSIONS

In this paper we are studying the mass transfer pro-cesses in a magnetic fluid caused by three different origins:the gradient of the ferroparticle concentration, the gravity orcentrifugal forces, and the gradient of nonuniform magneticfield; the possible convective fluxes are not considered. Thefirst one results in the gradient Brownian diffusion flux jD

of the known Fick’s type (17), and for dense magnetic fluidsthe gradient diffusion coefficient D depends both on the par-ticle concentration and the intensity of interparticle magne-todipole interaction (16). The gravity/centrifugal flux jS iswritten in a usual way (14). In applied magnetic field the masstransfer becomes anisotropic due to the anisotropy of the par-ticle hydrodynamic mobility and the anisotropy of the ther-modynamic forces. The influence of the last reason is muchhigher (by the order of magnitude) than the first one; so we areneglecting the anisotropy of the particle hydrodynamic mobil-

ity and are considering the mobility as the scalar function ofthe particle concentration (3). We suggest to describe the fieldinduced spatial nonuniformity of the mass transfer fluxes bymeans of separate magnetophoreses term jm (13) in the masstransfer equation. Thus, the dynamic mass transfer equation(21) for describing magnetophoresis, sedimentation, and dif-fusion of colloidal particles in concentrated magnetic fluidshas been derived. A general solution (32) has been found forstatic particle distribution. This solution is correct for the cav-ities of arbitrary shape and for the magnetic field of arbitrarygeometry. Steric interactions (excluded volume effects) aredescribed with the help of Carnagan-Starling approximationfor a hard sphere system. This approximation agrees with theresults of computer simulation of static concentration profilein gravitational field (Fig. 3, curve 1).

Magnetodipole interactions demonstrate the double ef-fect. On the one side, they increase the intensity of magne-tophoreses flux due to the additional term in effective mag-netic field (10) acting on each particle magnetic moment. Thisterm, in the right hand part of Eq. (10), is proportional toML (H ) /3 and tends to zero at the limit of strong fields only.On the other hand, the magnetodipole interactions give rise tothe effective interparticle attraction (17) and (19), which in-creases segregation in the system. The magnetic fluid thermo-dynamic instability (spinodal decomposition) can occur forthe case of rather high values of the dipolar coupling constantλ. The result of the instability is the system separation in lowand high concentrated phases (Fig. 2).

The effects of interparticle magnetodipole attraction aredescribed by the last term of Eqs. (20) and (21), which werefound from the free energy virial expansion (18). Obtainedinterpolation formula (19) allows to receive qualitatively cor-rect results for big values of λ; in particular, the calculateddiagram of spinodal decomposition (Fig. 2) predicts quitereasonable values of critical parameters. In zero magneticfields the gravity induced particle stratification is increasinggreatly with the strengthening of effective interparticle attrac-tion (Fig. 3) for low ferroparticle concentration (volume con-centration is less than 20%). At the same time, the interpar-ticle attraction effects are reduced for dense magnetic fluids(ϕ > 30%).

Concerning the hydrodynamic interparticle interactions,they influence only on the dynamics of the mass transfer pro-cess through the relative mobility coefficient K(ϕ). The staticparticle distribution, defined in Eq. (32), does not depend onthe particle mobility.

We suggest to use the obtained dynamic mass transferequation (21) in the boundary-value problems for calculationof concentration distribution and magnetic field geometry inthe devices, based on the concentrated magnetic fluid as aworking fluid (for example, the magnetic fluid seals and theacceleration sensors). According to Eqs. (10) and (20), thedensity of particle flux is determined by the magnetic fieldintensity and the magnetization of fluid inside a cavity. Timedrift of these quantities is correlated with the change in par-ticle concentration. So, the magnetic and diffusion problemsare strictly interrelated and should be solved together. But inthe concentrated magnetic fluids the magnetic moment relax-ation time (10−3–10−4 s) is much less than the characteris-

184508-9 Magnetophoresis, sedimentation, and diffusion of particles in fluids J. Chem. Phys. 134, 184508 (2011)

tic diffusion time τD ≈ L2/π2 D (L is the characteristic sizeof the cavity). The latter exceeds 105 s even for quite smallcavity about 0.1 cm. It means that the magnetic part of theproblem should be solved in quasistatic approximation, forexample, with Maxwell’s equation (34), for known boundaryconditions on the cavity wall.

ACKNOWLEDGMENTS

The work was supported by the Russian Foundation forBasic Research (Grant No. 10-01-96038); the Department ofEnergetic, Machine building, Mechanics and Control Pro-cesses RAS (Project No. 09-T-1-1005); Federal Target Pro-gram “Scientific and Academic – Teaching Staff of Innova-tive Russia” in 2009–2013 and Grant of President of RF MK-1673.2010.2.

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