Soret coefficient of nanoparticles in ferrofluids in the presence of a magnetic field

Soret coefficient of nanoparticles in ferrofluids in the presence of amagnetic fieldElmars Blums, Stefan Odenbach, Ansis Mezulis, and Michail Maiorov Citation: Phys. Fluids 10, 2155 (1998); doi: 10.1063/1.869737 View online: http://dx.doi.org/10.1063/1.869737 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v10/i9 Published by the AIP Publishing LLC. Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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PHYSICS OF FLUIDS VOLUME 10, NUMBER 9 SEPTEMBER 1998

Soret coefficient of nanoparticles in ferrofluids in the presenceof a magnetic field

Elmars BlumsInstitute of Physics, University of Latvia, Salaspils-1, LV-2169, Latvia

Stefan OdenbachZARM, Am Fallturm, University of Bremen, D-28359 Bremen, Germany

Ansis Mezulis and Michail MaiorovInstitute of Physics, University of Latvia, Salaspils-1, LV-2169, Latvia

~Received 12 December 1997; accepted 19 May 1998!

Experiments on a nonstationary separation of nanometer-sized Mn0.5Zn0.5Fe2O particles ofhydrocarbon-based ferrocolloids in a flat vertical thermal diffusion column are performed. By usinga modified separation theory which accounts for a one-dimensional mixed~thermal andconcentration! convection in the column, the Soret coefficient of lyophilized nanoparticles from theseparation curves are calculated. It is shown that in a zero magnetic field particles are transferringtoward decreasing temperatures. The thermal diffusion ratioaT reaches a valueaT'120. Asignificant influence of a uniform magnetic fieldB on particle separation is observed. IfB is orientedalong the temperature gradient“T, a strong decrease in thermal diffusion coefficient takes placewhereas the transversal fieldB'¹T causes an intensification of particle thermophoretic transfer.Both effects qualitatively well agree with theoretical predictions based on a hydrodynamic theory ofparticle thermomagnetophoretic motion. ©1998 American Institute of Physics.@S1070-6631~98!02309-5#

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I. INTRODUCTION

Transport properties of magnetic ultrafine particles pa significant role in the problem of long-term stabilitymagnetic fluids. Usually, in the investigation of colloidal stbility besides the agglomeration problems only gravitatsedimentation and magnetophoretic transfer of the ferroticles are considered. Nevertheless, it may happen thasome magnetofluid devices, e.g., in high speed rotary seain loudspeaker cooling systems in which high temperatgradients are present, the long-term stability of the magnfluid is effected also by thermophoretic nanoparticle transSome experiments, e.g., those of magnetic nanoparticleing in optical interference bands, the so-called forced Rleigh scattering experiment,1 indirectly indicate a high ther-mophoretic mobility of particles in colloidal dispersions2

Recently, a strong Soret effect in ferrofluids was detecmore directly by particle separation measurements in a tmal diffusion column.3,4 Some theoretical considerations alow us to conclude that thermal diffusion of magnetic nanparticles in ferrocolloids may be significantly affectedinternal magnetic field gradients.5 From the analysis ofStokes’s problem for spherical particles, taking into accothe nonpotentiality of thermomagnetic forces, it follows than external uniform magnetic fieldB oriented along the temperature gradient“T may induce a particle motion towarincreasing temperatures whereas in the presence ofB'“Tan opposite direction of particle thermomagnetophoretransfer is awaited.6,7 Both effects may be interpreted aschange of the thermal diffusion coefficient of nanoparticl

The present paper presents results of our research a

2151070-6631/98/10(9)/2155/9/$15.00

Downloaded 11 Sep 2013 to 150.216.68.200. This article is copyrighted as indicated in the abstract

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at investigating the above-mentioned magnetic effectsperimentally. It is well known that direct measurementssmall particle concentration difference in thin ferrofluid laers are extremely difficult. Therefore, we use an indirmethod which is based on particle separation measuremin a thermal diffusion column.

II. THEORETICAL BACKGROUND

The hydrodynamic theory of magnetic particle transin ferrocolloids is based on calculations of the magnephoretic velocity of the particles, when the mass flux is wrten in the form

j52rpD0“w1umrpw.

Hererp is the solid phase density,w is the particle volumefraction in a colloid,D0 is the translation diffusion coeffi-cient ~for diluted colloids containing spherical particlesD0

5kBT/3phd with kB being the Boltzmann constant,T theabsolute temperature,d the particle diameter andh the am-bient liquid viscosity!. The magnetophoretic velocityum in-duced by a magnetic field gradient“H is determined by abalance between the magnetostatic forceFm5m0m“H andthe Stokes hydrodynamic dragFn ~m is the magnetic momenof particle,m0 is the magnetic constant!. For subdomain par-ticles in ferrocolloidsm is considered as being proportionto the magnetizationM p of the particle and its volumeVp ,m5M pVp . Usually, the calculations of particle magnetphoretic mobility are based on a classical solution ofhydrodynamic Stokes problem assuming for the potentiaof the magnetic driving force. Such approaches may be c

5 © 1998 American Institute of Physics

. Reuse of AIP content is subject to the terms at: http://pof.aip.org/about/rights_and_permissions

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2156 Phys. Fluids, Vol. 10, No. 9, September 1998 Blums et al.

sidered as being correct only for homogeneous and isotmal magnetic fluids. In the general case it is necessartake into account for a nonpotentiality of magnetic forcaused by both, the temperature gradient and the concetion nonhomogeneity of a colloid. Such an analysis was pformed by one of us~E.B.! in 1979.5 It was shown that auniform magnetic field directed along the temperature graent induces a magnetic particle motion toward increastemperatures. Further calculations8 led to the conclusion thain the presence of an arbitrarily oriented magnetic fieldparticles are moving not only along“T but also alongB.Recently, a more general analysis of the ferrohydrodynaStokes problem has been performed9 by taking into accountnot only the local distribution of magnetic field near the pticle but also the temperature perturbations caused by aference between the thermal conductivities of particle asurrounding liquid, respectively. From these results it flows that the magnetophoretic velocity of a spherical partmay be represented in the form:6

um5m0d2

18h$3Kmm~H“ !H2H~H“ !m@ 3

5 Km1 380 Km

2

2Kl~ 940 Km1 1

35 Km2 !#1H2

“m@ 65 Km1 21

80 Km2

2Kl~ 340 Km1 2

35 Km2 !#%. ~1!

Here m is the magnetic permeability of the colloid,Km

5(mp2m)/(mp12m) andKl5(lp2l)/(l1lp) are coef-ficients which account for the demagnetization factor (mp isthe magnetic permeability of the particle material! and forthe deformation of the temperature distribution aroundparticle if its thermal conductivitylp is different from that ofthe surrounding liquidl. The first term in~1! represents theclassical Stokes approximation of particle magnetophormotion under the effect of a nonuniform magnetic field~tak-ing into account demagnetization effects!, while the secondand third terms reflect the additional effect of a nonpotenthermomagnetic driving force considered in the linearproximation of both the local magnetic field and the tempeture perturbations around the particle.

If a particle transfer in the presence of a uniform extermagnetic fieldB5const is considered, it follows from~1!,that:~a! whenB is parallel to the magnetic permeability gradie“m:

um52m0d2H

18h“m

12

5Km f 1 ,

~2!

f 15 f 1~Km ,Kl!5123

32Km2

Kl

16 S 124

21KmD ;

~b! whenB is oriented transversally to the“m:

um51m0d2H2

18h“m

6

5Km f 2 ,

~3!

f 25 f 2~Km ,Kl!5117

32Km2

Kl

16 S 1116

21KmD .


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From ~2! and ~3! it is seen that in both cases particles atransferred only along the magnetic permeability gradi“m.

For one-dimensional gradients“5d/dx it follows fromthe Maxwell equation divB50, that

H2dm

dx5

mH

11]M

]H

S ]M

]w

dw

dx1

]M

]T

dT

dxD .

Assuming that the magnetization of the fluid can be dscribed by the Langevin equation

M5wM pL~j!, L~j!5coth j21

j, j5

m0M pVpH

kBT,

~4!

and taking into account relations~2! and~3! one can see thathe mass flux may be written in the classical form~for w!1)

j x52rpS Ddw

dx1

D0aT

T

dT

dxw D , aT5a01am , ~5!

wherea0 is the conventional Soret coefficient andam is anew ‘‘magnetic’’ Soret coefficient which depends on the mtual orientation ofH and“T.

The effective translation diffusion coefficientD also de-pends on strength and direction of the magnetic field. WhB is directed along“w, theory predicts an increase inD inaccordance with

Dp5D0~11 125 KmA f1!. ~6!

The coefficientA depends onw as well as onH:

A5mwjL~j!

11wM p

HjL8~j!

.

By contrast, a magnetic field directed transversally“w causes the opposite effect. In this case the reductiothe translation diffusion coefficient can be written in thform:

Dt5D0~12 65 KmA f2!. ~7!

It is necessary to note that the coefficientKm depends onw aswell as onH. Using the superparamagnetic law~4! we mayrewrite Km in the form

Km5~12w!M pL~j!

3H1~112w!M pL~j!.

The dependenceD5D(H) in the presence of a uniform fieldof various orientation has been observed in a forced Raylescattering experiment in hydrosols containing electricastabilized magnetite particles.10 The relations forDp andDt

~6! and ~7! are in rather good agreement with the resultsthese experiments and with the authors’ theory as welwith other theories11 considering particle interactions.

The magnetic part of the thermal diffusion ratioam alsodepends on the mutual orientation ofB and“T:


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2157Phys. Fluids, Vol. 10, No. 9, September 1998 Blums et al.

~a! B parallel to“T:

amp 5

12

5KmA f1

1

M

]M

]T; ~8!

~b! B transversal to“T:

amt 52

6

5KmA f2

1

M

]M

]T. ~9!

Figure 1 represents the dependenceam5am(H) plottedin accordance to expressions~8! and ~9!. The curves arecalculated using the specific data of the colloid used inthermal diffusion separation experiments, which contasubdomain Mn0.5Zn0.5Fe2O4 particles in tetradecane. It iseen, that in the presence of a longitudinal field thermomnetophoretic transfer toward increasing temperatures ispected whereas a transversal fieldB'“T will cause particlemotion in the opposite direction. The dependence of theefficientsf 1 and f 2 on the parametersKm andKl is relativelyweak—both of them are close to 1 for the colloid considehere—therefore theory predicts a ratio of the thermal difsion ratios of aboutam

p /amt '22.

III. THE UNSTEADY PARTICLE SEPARATION IN ATHERMAL DIFFUSION COLUMN

An attempt to measure the particle thermomagnephoretic mobility in ferrocolloids has already been made1983.12 The investigations were based on nonstationary pticle separation measurements in a vertical thermal diffuscolumn. The obtained results have been interpreted by ua simple theory13 according to which the convection invertical column is caused by a thermogravitation buoyaforce only. In experiments with magnetic colloids such aproximation is not sufficiently correct because even a smvariation of particle concentration during their therm

FIG. 1. Thermomagnetic part of the Soret coefficientam(B) plotted accord-ing to ~8! and~9! for the diluted ferrofluid sample DF39S (Mn0.5Zn0.5Fe2O4

particles in tetradecaneM p5249 kA/m, w052.3231022, ‘‘magnetic’’ di-ameter of the particlesdm511.9 nm): ~1! Bi“T; ~2! B'“T.


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phoretic transfer causes a concentration buoyancy force cparable with the effect of the fluid’s thermal expansion.3

The sketch of the thermal diffusion column used in oexperiments is shown in Fig. 2. The column consists overtical flat channel~a small gapd between two plates odifferent temperaturesT0 andTd) of a lengthL@d and twoseparation chambers of equivalent volumesVc .

Free convection in the channel causes a transfer ofticles in a vertical directionz. The velocity of gravitationsedimentation and Brownian diffusion of the colloidal paticles is considerably less than the velocity of their convtive transport. Therefore, the mean vertical particle fluxj z

may be written in the nondimensional form

j z5rp

d E0

duz~x!wdx,

whereuz(x) is the convection velocity distribution across thchannel andw is the concentration~volume fraction! of par-ticles in a colloid. This vertical flux causes a particle concetration change in the lower (w l) and in the upper (wu) cham-ber of the column:

rp

dw l

dt52

S

Vcj z , rp

dwu

dt51

S

Vcj z ,

whereS denotes the cross-sectional area of the channethe initial stage of separation until the concentration diffenceDw5w l2wu is significantly less than the initial con

FIG. 2. Principal sketch of the thermal diffusion column.L586.5 mm,Lc

559 mm,d50.52 mm, the width of the channeldy@d. The induction coilsof two LC oscillators are mounted inside the lower and the upper separachamber.


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centrationw0 , the particle flux may be calculated assumithat the concentration profile across the channel is timependent but does not depend on vertical coordinatez, w5w(x,t). The most simple analytical results may be otained in the frame of a concentration boundary layerproximation. Employing a mass conservation equation wten in the form of an integral mass balance, we obtainfollowing polynomial function forw5w(x,t) in the concen-tration boundary layer:14,15

w

w02157

km

36km S 12x8

mD 3

, m5A12t.

Here k5aTDT/T is the thermal diffusion parameter (DT5Td2T0 , T is the fluid mean temperature!, t5Dt/d2 is thediffusion Fourier number,m5dc /d is the nondimensionathickness of the concentration boundary layer, whilex85x/d is the distance from the wall. The upper signs corspond to the layer near the wall of a higher temperatureTd ,the lower ones to the second wall of a lower temperatureT0 .Using this concentration profile together with a velocdistribution15 which accounts for an unsteadiness of the ccentration buoyancy force, we obtain the following equatwhich describes the development of the concentration difenceDw in the initial regime of separation:

Dw

w052

kGrTSc

5400

d

Lcm5F S 12

5m

61

10m2

49 D1k

GrTGrw

S 25m2

632

m3

4 D G . ~10!

The thermal and the concentration Grashoff numbersT5gbTDTd3/n2 and Grw5gbww0d3/n2 here include the ki-nematic viscosity of the fluidn and the coefficientsbT

521/r(]r/]T) and bw51/r(]r/]w) which characterizethe dependence of the fluid’s densityr on temperatureT andparticle volume concentrationw; Sc5n/D is the Schmidtnumber.

This dependence is valid fort,1/48 until the thicknessof the concentration boundary layers from both walls reacthe center of the channel atm50.5. From~13! it follows thatin the initial regime of separation a nonlinear dependeDw;t2.5 is awaited. Even at high values of parameterS5kGrw /GrT the influence of the concentration buoyanforce on separation dynamics in this regime is relatively lo

At t.1/48 the concentration profile across the chanstarts to saturate and approximately att'0.25 a steady pro-file w5w(x) is reached. Now, the vertical particle fluxconstant and the concentration difference starts to devlinearly in time. For small values ofk,1 when the steadyconcentration distribution across the channel is linear,obtain

Dw

w051

2

6!

d

Lck~GrT1kGrw!Sct. ~11!

At durable separation, starting after a certain transittime t t @from our previous analytical results12 it follows thatt t'65(GrTScd/Lc)

21] the concentration difference in thchambers starts to saturate and the linear dependence~11!


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may not be observed. The smaller the volume of separachambersVc or the higher the temperature differenceDT,the sooner the linearity of the saturation curves vanishesthe transition regime a time interval exists in which the cocentration differenceDw may be approximated by a squarroot dependence12

Dw

w05g

L

LckAt. ~12!

The coefficientg slightly depends on the ratioL/Lc . Fordevices having approximately equal volumes of the actcolumn V and of the upper and lower containersVc ,g'0.619 is found.4

In long time experiments starting at some timets @forL/Lc'1 ts'53107(GrTScd/L)22] a stationary regime ofparticle separation is reached. According to the classthermal diffusion theories developed forS50, the concen-tration difference in column chambers in the asymptoticgime t.ts is

Dw

w05

504k

GrTScd

L

. ~13!

A theory of the thermal diffusion column accounting fothe complex character of convection in fluids withSÞ0 isnot developed in detail at present. From~11! it follows thatin liquids with S.0, the concentration buoyancy forccauses an increase in the vertical particle flux. In the sarated regime of particle separation the influence of paramS is not so strong: for liquids having positive values ofaT

the increase in the steady particle separation level is expestarting approximately atS.2.16

Usually, in real separation devices, the column widthvaried within an interval of aboutd50.5– 1 mm and thethermal as well as the concentration Grashoff numbers revalues 1–10. Besides, for colloidal particles in ferrocolloithe values of the Schmidt number Sc are very high ('105). The time at which the steady separation regimereached, depends on the convection parameters as well aL and Lc . In our experiments the relaxation timets

5tsd2/D exceeds several tens of hours. To simplify t

measurements as well as to eliminate the difficulties of seration curve analysis due to not very well clarified concetration convection effects, the main attention in our expements is paid to the nonstationary regime of the partiseparation curves.

IV. FERROFLUID SAMPLES AND MEASUREMENTTECHNIQUE

The separation experiments are performed in a vertflat column of widthd50.52 mm and heightL586.5 mm.The heated and cooled walls~copper plates with polishedsurfaces! are connected with two precise thermostats, keing the temperature differenceTd2T0 constant atDT510 °C. In order to lower the temperature inhomogeneitcaused by heat exchange with the environment, the up


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and lower containers of equal volume (Vc'1 cm3, Lc /L50.682) and the outside walls of channel are made fromplastic material with low heat conductivity.

The particle concentration in the upper and lower chaber is determined by measuring the resonance frequencan LC oscillator17 during the separation process. The indutance coils of the LC oscillator are mounted inside the ctainers. The optimum oscillation frequency is chosen takinto account a compromise between the sensitivity of partconcentration measurements and the errors caused by thlaxation of the colloid’s magnetization. The proper adjument of the frequencyf is verified experimentally by measuring the Q factor of the coil. For the ferrofluid samplused in our experiments the oscillation frequency is appromately 70–85 kHz. The high stability LC oscillator has bedescribed earlier in some detail.3 The concentration calibration curve of the coils in the presence of an external mnetic field has been detected experimentally. For colloidsmoderate particle concentration, like we have used inpresent thermal diffusion experiments, the functionf5 f (w) may be regarded being linear. Such a linearity, obously, is expected only for colloids of low magnetic suscetibility x!1. For Langevin-type monodisperse colloids cotaining noninteracting subdomain particles of magnemoment m5M pVp , in a zero external magnetic field thcalibration curvef (w) is proportional to the initial magneticsusceptibility of the colloidx0 which correlates with the particle concentrationn5w/Vp :

x05m0m2

3kBTn.

In the presence of increasing magnetic fields when accorto ~4! the differential magnetic susceptibility decreasessignificant reduction of the resonance frequency dependeon particle concentration takes place.

Additional measurements of the heater and cooler teperature as well as the ohmic resistance of both induccoils are performed to control the temperature in separachambers. By using precise cooling and heating thermosa high stability of the temperature regime during the sepation experiment is achieved. Convection velocity in the chnel is low (umax does not exceed 0.5 mm/s!, therefore, thetemperature in both separation chambers during the expment is practically identical and equal toTa5(T01Td)/2(60.1 °C).

Experiments are performed employing a tetradecabased ferrofluid sample~DF39S! containing chemically co-precipitated Mn0.5Zn0.5Fe2O4 nanoparticles. The colloid isstabilized by using oleic acid as a surfactant. The volufraction of the magnetic phasew05nVp is calculated fromthe colloid’s density. Assuming thatrp of Mn-Zn ferrite isequal to that of bulk magnetite (rp55.243103 kg/m3) wefind the valuew059.2831022 ~undiluted sample!. The satu-ration magnetization of the sampleM s523.1 kA/m is de-tected by an extrapolation of magnetization measuremenH→` assuming the superparamagnetic law~4!. The satura-tion magnetization of Mn-Zn ferrite M p5Ms /w0

5249 kA/m is approximately two times less than that


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bulk magnetite. The asymptotic value of the absolute pymagnetic coefficient2]M /]T of the undiluted colloid instrong fields reaches the value 69.7 A/~m K!. The mean par-ticle magnetic moment measured by a magnetogranulomtechnique18 is m52.22310219 A m2. This value ofm withrespect to the measured value ofM p corresponds to a mea‘‘magnetic’’ diameter of the particles of aboutdm

511.9 nm. The particle Brownian diffusion coefficientcalculated by using the relationD5kBT/(3phdh) assumingthat the ‘‘hydrodynamic’’ diameter of the particlesdh in-cludes not only the magnetic core but also a nonmagnlayer ~being approximately 1 nm! as well as a ‘‘surfacted’’layer ~for oleic acid being approximately 2 nm!. The viscos-ity of the colloid ~at 25 °C) n526.631026 m2/s ~initialsample! andn53.3331026 m2/s ~diluted to 1:4! as well asthe thermal expansion coefficientbT56.7731024 1/K andthe coefficientbw55.19 1/w are detected experimentally.

V. RESULTS AND DISCUSSIONS

Figure 3 shows nonstationary separation curves ofinitial and the diluted~1:4! samples measured in a zero manetic field atDT510 °C. The concentration differenceDw5w l2wu for both samples is positive even in the initial rgion of separation when the effect of concentration buoyaforce is absent. It means that thek value is positive andparticles are therefore transferred toward decreasing tperatures. This direction of thermodiffusive transport of sfacted particles has been measured also in experimentsmagnetite containing organosols.4 The separation curveshow theDw5c1t2.5 ~for t,0.025 as it is predicted theoretically! as well as theDw5c2t0.5 ~also in an agreemen

FIG. 3. Unsteady separation curves of colloids with different particle ccentration,B50. GrT50.842, Grw559.6 ~initial sample,c51) and 14.9~diluted sample,c50.25). The diffusion coefficientD in nondimensionaltime t5(Dt/d2)1/2 is calculated using an approximate ‘‘hydrodynamicdiameter of the particles of about 18 nm.


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with theoretical predictions12! regions of particle separationThe difference in the coefficientsc1 for initial and dilutedsamples reflects, obviously, the effect of the reduced viscity of the diluted sample. According to~10!, with respect toour viscosity measurements, an approximately eightfoldcrease inc1 for the diluted sample is awaited. From thcurves presented in Fig. 1 we can see that the differencthese coefficients is not so high. Obviously, this reflectsfact that not only the viscosity but also the diffusion coefcient in highly concentrated colloids is concentration depdent. Besides, the viscosity of our colloid strongly depenon temperature~in the T0520 °C to Td530 °C range, anapproximately 50% reduction ofh is measured!. It is wellknown13 that such viscosity changes also may effectseparation dynamics in a thermal diffusion column. In ttime interval in which the ‘‘1/2’’ law takes place, the sepration curves in Fig. 3 practically do not depend onw. Inaccordance with~12! a dependence of the coefficientc2 onlyon the Soret parameterk and on the particle diffusion coefficient is expected in this time interval. A small differenceapproximately 15% in the coefficientc2 for both samples,obviously, reflects not only the uncertainty ofD but also thespecifics of rheological properties of highly concentrated cloids. Expressions~10!–~12! are obtained assuming Newtonian behavior of the fluid. Therefore, for further analysis wwill use only the results which correspond to the dilutsample in which viscoelastic effects can be neglected. Us~12! k510.61 ~standard deviation 1.3%! is calculatedwhereas from the initial part of the separation curve withrespect to~10! a lower valuek510.46 ~standard deviation9.5%! is found. Such differences in the results, obviousreflect the uncertainty of the particle diffusion coefficientDwhich is evaluated using a not very well defined particle sand neglecting particle interaction. A 30% reduced valueD ~such correction may be considered as being quite retic! would give practically equivalent values ofk in bothseparation regimes. The value of the Soret coefficienta0

'18 is close to that of magnetite particles reportelsewhere.4

The linear regime~11! of the separation curves is noobserved in the measured data presented in Fig. 3. Due trelatively small volumes of the lower and upper chamber,separation curves start to saturate already before the stconcentration profile across the channel is established~theexperiments correspond tot t50.03 for the diluted sampleand tot t50.2 for the initial one!.

Two series of experiments on particle separation inpresence of a uniform magnetic field have been performTo lower the fluid viscosity, the initial sample has beenluted to 1:4. At DT510 °C the dimensionless parametetake the values GrT50.842, Grw514.9 and Sc52.253105.Figure 4 shows some separation curves measured in theence of a magnetic field oriented horizontally along theated and cooled walls,B'“T. The curves plotted in Fig5 correspond to a second series of experiments when theis oriented transversally to the walls, that is, along the teperature gradient,Bi“T. It is seen, that in the first caseremarkable intensification of particle separation takes plwhereas the field oriented along the temperature grad


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theedy

ed.-

es-

eld-

ent

causes a significant reduction in particle separation intensIn both series a saturation of the magnetic field effect starapproximately atB.0.25 T is reached. All measuremenperformed in the presence of strong magnetic fields indican increase in the dispersion of experimental points. Obously, this peculiarity of measurements reflects the effectreduced sensitivity of the LC oscillator frequency to particconcentration changes in strong fields. The separation cuwhich correspond toB'“T indicate only ‘‘5/2’’ and ‘‘1/2’’regions whereas in the presence ofBi“T in some experi-ments of strongly reduced particle transfer a linear regionalso observed.

In some experiments the particle concentration is ctrolled also by additional magnetization measurements offluid performed before and after the separation. The partconcentration calculated fromMs or from initial magnetic

FIG. 4. Separation curves in the presence of a uniform magnetic fiB'“T. GrT50.842, Grw514.9 (DT510 C, w052.3231022): ~1! zerofield; ~2! 0.03 T; ~3! 0.041 T;~4! 0.17 T; ~5! 0.24 T.

FIG. 5. Separation curves in the presence of a uniform magnetic fiBi“T. GrT50.842, Grw514.9 (DT510 C, w052.3231022): ~1! zerofield; ~2! 0.01 T; ~3! 0.017 T;~4! 0.03 T; ~5! 0.095 T;~6! 0.27 T.


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susceptibility usually is in good agreement with the resuobtained from resonance frequency measurements. As aample, in Fig. 6 magnetization curves for one of the expments~transversal fieldB50.17 T) are presented. From thseparation measurement it follows that at the end of theperiment the relative concentration difference reaches 0.whereas from the magnetization measurements a practiequivalent value 0.494 is obtained. The magnetization invtigations have also been used to perform a magnetogrlometry analysis. Figure 7 represents the distribution curof the magnetic moment of the particles calculated frommagnetization curves shown in Fig. 6 by means of a reguization technique.18 It is seen that there are no noticeabchanges in magnetic moment distribution during the part

FIG. 6. The magnetization curves of a colloid before~1! and after separationprocess in the thermodiffusion column fort510850 s. The samples wertaken from the lower~curve 3! and upper~curve 2! chamber. The magneticfield during the separation processB50.17 T had transversal orientationB'“T.

FIG. 7. The distribution curves of the magnetic moment of the particcalculated from the magnetization curves presented in Fig. 6.


sex-i-

x-90llys-u-ser-

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separation. Similar results are obtained also in experimewith lyophilized magnetite particles.4 Thermal diffusion likethe Brownian diffusion is a collective transport phenomentherefore, the unchanged magnetogranulometry curvesnot evidence of the independence of measured effectsparticle size. Nevertheless, it is necessary to perform ational experiments to examine colloids containing particof different mean sizes. Theory predicts that the thermomnetophoretic mobility of ferroparticles is proportional tod2.

Figure 8 shows the dependence of the thermophormobility of particles on a magnetic field directed in bomutual orientations. The results are calculated from seption curves using expressions~10! and~12!. In order to lowerthe scatter due to experimental error, average values ofresults obtained by using both calculation methods alsopresented. The experimental results agree qualitatively wwith the theoretical estimates described above: A transvemagnetic fieldB'“T causes an intensification of partictransfer toward decreasing temperatures whereas a longinal fieldBi“T reduces the initial positive values ofaT prac-tically to zero. Moreover, from results presented in Fig. 8may see that the reduction ofaT in a longitudinal magneticfield is approximately twice as strong as its increase inpresence of a transversal field, as is also predicted by th@see~8! and ~9!#.

The analysis of the separation curves in thermal difsion columns is an indirect method of thermal diffusion mesurements. When magnetic fluids are considered, it is ne

s

FIG. 8. The effect of a magnetic field on the Soret coefficient of Mn-ferrite nanoparticles stabilized by oleic acid in tetradecane,w052.3231022. The dots represent the results calculated in accordance with forlas ~10! and~12!. The dotted line is plotted taking into account the averavalues of the results which correspond to each experiment.


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sary to take into account the fact that magnetic field minfluence not onlyaT but also other transport coefficients~Dand h at first!. From this point of view the most reliablresults ofaT(H) are these calculated by using the ‘‘1/2’’ la~12! because in this case only the Brownian diffusionparticles may effect the results. The results of translatdiffusion measurements in the presence of a magneticreported by Bacriet al.10 agree very well with our theoryNumerical estimates performed by using expressions~6! and~7! allow us to conclude that in the present experimentschanges inDp andDt are expected to be relatively small, nexceeding 20%. Besides, considering an increase inDp

would lead to a more sufficient reduction ofam in parallelmagnetic field and conversely, a decrease inDt would give asignificantly stronger increase ofam in transversal field.More complicated is the problem of the credibility of resuobtained from relations~10! and ~11! when the calculationsrequire the consideration of changes of the fluid’s viscosThe convection in the channel is very slow and the sheardoes not exceed 10 s21, therefore some effect of a magnetfield on hydrodynamic resistance may be present.19,20 Nev-ertheless, the difference in the results presented in Fig.comparison with the dispersion of experimental points issmall to make any reasonable conclusion. In the presenca longitudinal magnetic fieldBi /“T, another effect—a magnetic Rayleigh–Bernard instability of convective sheflow—may develop. It is well known7 that the internal mag-netic field gradients in nonisothermal ferrofluids alwacause thermomagnetic convection. Critical values ofmagnetic Rayleigh number Rmc at which the fluid layerloses stability usually do not differ significantly from thosof ordinary gravitational convection Rac . For the dilutedsample even in the regime of magnetic saturation Rm dnot exceed the value 760, which is considerably less thancritical value Rac51708 for a horizontal flat fluid layeheated from the bottom. Unfortunately, there is no informtion available at present about the thermomagnetic instaties of free convection in channels. Nevertheless, the chater of particle separation dynamics in our columexperiments allow us to conclude that the shear flow inpresence ofBi“T does not lose stability even in stronfields.

Taking into account the considerations mentioned abowe suppose that reasonable conclusions may be formuonly on a basis of the relative thermal diffusion measuments. Nevertheless, it seems that the magnetic Soret eshown in Fig. 8 is much stronger than the theoretically emated one. Saturated values ofam

t andamp evaluated by us-

ing the measured valuea0518 are almost two orders omagnitude higher than those calculated according to~8! and~9! ~see Fig. 1!. Obviously, such disagreement is evidenfor the necessity to take into account more complicated pcesses to describe the thermal diffusion of nanometer scparticles in colloids.

The phoretic transport of particles in dispersions isfected by the slip velocity at the solid–liquid interface. Exatheories of small particle thermophoresis are developed ofor gaseous suspensions of hard particles or droplets wthe slip characteristics may be calculated by using exis


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gas theories. Some general ideas of a theory of the sliplocity at a solid–liquid interface resulting from a tangenttemperature gradient are reviewed by Anderson.21 Analyzingthe enthalpy flux carried by a forced convection of the fluacross a porous barrier and applying Onsager’s reciprtheory for the slip velocity the following expression for thslip velocity is obtained:22

vs52F 2

hT E0

`

yh~y!dyG¹T,

wherey is the distance from the wall andh(y) is the localexcess specific enthalpy in the interfacial layer. If the sosurface is lyophilic~the liquid phase is attracted on a mlecular level!, thenh(y),0 and the slip velocity is directedtoward higher temperatures. As a result, this theory predthat free particles in colloids stabilized by surfactants womove in the opposite direction, that means in the directiondecreasing temperature. Our experiments agree withconclusion. Obviously, the slip characteristics have totaken into account also when we consider the balancetween magnetic forces and hydrodynamic resistance tolyze the unexpectedly high thermomagnetophoretic mobiof particles in ferrocolloids observed in our experiments. Ufortunately, there is currently no model from which to calclate h~y!.

VI. CONCLUSIONS

The particle separation measurements in a vertical thmal diffusion column indicate a high thermophoretic mobity of magnetic nanoparticles in ferrofluids. Colloidal paticles stabilized in hydrocarbons by use of surfactantsmoving toward lower temperatures. This direction of themophoresis agrees with predictions of the thermodynatheory taking into account the slip velocity of lyophilisolid–liquid boundaries. The Soret coefficient of nanopticles is strongly affected by an external uniform magnefield and its orientation relative to the temperature gradieExperimental results agree qualitatively well with the hydrdynamic theory considering the nonpotentiality of magneforces acting on particles in nonisothermal and nonhomoneous ferrocolloids.

ACKNOWLEDGMENTS

The authors are thankful to G. Kronkalns for providinus with the samples of temperature sensitive Mn-Zn fercontaining ferrofluid. The work has been financially suported by the Latvian Science Council~Grant No. 96.0271!and by the German Ministry of Education and ResearBMBF ~Grant No. LET-001-96!.

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3A. Mezulis, E. Blums, G. Kronkalns, and M. Maiorovs, ‘‘Measurements


ys

np

s

ffu

tic

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Bdeen

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inn-onting

ion,al

of

n

in

u-gn.

v.


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7E. Blums, A. Cebers, and M. M. Maiorov,Magnetic Fluids~de Gruyter,Berlin, 1997!.

8V. A. Miroshnikov, ‘‘Thermophoresis of ferrosuspensions in a magnefield,’’ in Hydrodynamics and Heat Transfer in MHD Flows, edited by E.Blums ~IOP, Salaspils, 1980!, p. 177.

9V. A. Naletova, G. A. Timotin, and I. A. Shkel, ‘‘Force acting on a bodin a nonuniformly heated magnetisable fluid,’’ Fluid Dyn.~USSR! ~En-glish trans.! 24 ~2!, 225 ~1989!.

10J. C. Bacri, A. Cebers, A. Bourdon, G. Demouchy, B. M. Heegard,Kashevsky, and R. Perzinsky, ‘‘Transient grating in a ferrofluid unmagnetic field. Effect of magnetic interactions on the diffusion coefficiof translation,’’ Phys. Rev. E52, 3936~1995!.

11K. I. Morozov, ‘‘The translation and rotational diffusion of colloidal ferroparticles,’’ J. Magn. Magn. Mater.122, 98 ~1993!.

12E. Blums, G. Kronkalns, and R. Ozols, ‘‘The characteristics of mass trafer processes in magnetic fluids,’’ J. Magn. Magn. Mater.39, 142 ~1983!.

13G. D. Rabinowich, R. J. Gurevich, and G. I. Bobrova,Thermodiffusion


.

a-

,’’

-

.rt

s-

Separation in Liquid Dispersions~Nauka I Tehnika, Minsk, 1971! ~inRussian!.

14E. Blums, Yu. A. Mikhailov, and R. Ozols,Heat and Mass Transfer inMHD-Flows ~World Scientific, Singapore, 1987!.

15E. Blums and A. Mezulis, ‘‘Thermal diffusion and particle separationferrocolloids,’’ in Phenomena in Magnetohydrodynamic and Electrocoducting Flows, Proceedings of the Third International ConferenceTransfer Phenomena in Magnetohydrodynamic and ElectroconducFlows ~Aussois, France, 1997!, 2, p. 535.

16O. Encario, J. A. Madariaga, J. Navarro, C. S. Mantamaria, L. A. Carrand J. M. Saviron, ‘‘Non-steady-state density effects in liquid thermdiffusion columns,’’ J. Phys.: Condens. Matter1, 9741~1989!.

17S. Odenbach, ‘‘Forced diffusion in magnetic fluids under the influencea strong magnetic field gradient,’’ J. Phys. B94, 331 ~1994!.

18M. M. Maiorov, ‘‘Magnetization of magnetic fluids and distribution imagnetic moments of ferroparticles,’’ inProceedings of the Tenth RigaMHD Conference~IOP, Riga, 1981! Vol. 1, p. 192~in Russian!.

19O. Ambacher, S. Odenbach, and K. Stierstadt, ‘‘Rotational viscosityferrofluids,’’ Z. Phys. B86, 29 ~1992!.

20S. Odenbach and H. Sto¨rk, ‘‘Shear dependence of field induced contribtions to the viscosity of magnetic fluids at low shear rates,’’ J. MaMagn. Mater.~in press!.

21J. L. Anderson, ‘‘Colloidal transport by interfacial forces,’’ Annu. ReFluid Mech.21, 61 ~1989!.

22B. V. Derjaguin, N. V. Churaev, and V. M. Muller,Surface Forces~Ple-num, New York, 1987!.


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Soret coefficient of nanoparticles in ferrofluids in the presence of a magnetic field