Eur. Phys. J. E 15, 271–276 (2004) DOI 10.1140/epje/i2004-10067-3
THE EUROPEAN PHYSICAL JOURNAL E
New problems of particle transfer in ferrocolloids: Magnetic Soret effect and thermoosmosis E. Blumsa Institute of Physics, University of Latvia, Salaspils-1, LV-2169, Latvia Received 27 July 2004 and Received in final form 4 October 2004 / c EDP Sciences / Societ` Published online: 16 November 2004 – ° a Italiana di Fisica / Springer-Verlag 2004 Abstract. The paper deals with critical reviewing of the experiments on thermodiffusion in ferrocolloids. The observed magnetic Soret effect is much stronger than that predicted theoretically. It is shown that the main reason of that is the influence of the magnetic field on mass diffusion. Besides, some measurements are affected by uncontrolled thermal and solutal magnetic convection. In porous media, when macroscopic convection is suppressed, thermodiffusion is accompanied by thermoosmosis as well as by a microconvective mass transfer induced by particle magnetophoresis on filter grains. PACS. 75.50.Mm Magnetic liquids – 82.70.Dd Colloids – 66.10.Cb Diffusion and thermal diffusion
1 Introduction Transport properties of magnetic ultra-fine particles play a significant role in the problem of long-term stability of magnetic fluids. Usually for the colloidal stability research, the agglomeration problems and the magnetophoretic transfer of ferroparticles induced by a nonuniform magnetic field are mainly considered. Under nonisothermal conditions inside the fluid strong gradients of the internal magnetic field appear even if the applied magnetic field is uniform. The corresponding magnetophoretic velocity now is proportional to the temperature gradient, therefore, the field-induced motion of Brownian particles can be considered as an additional mechanism of their thermodiffusive transfer. In Section 2 the latest research results on this magnetic Soret effect are reviewed. Theory predicts a relatively small contribution of the external magnetic field to the resultant Soret coefficient whereas in experiments significantly stronger field effects are detected. The main reason of that is the influence of the magnetic field on mass diffusion. Besides, measurements can be affected by uncontrolled thermal and solutal magnetic convection. One possibility to eliminate the parasitic convection could be performing separation measurements in porous media. However, there arise new problems in interpreting the measurement results. In Section 3 some novel experiments are reported, which reveal that the particle thermophoresis in porous media is accompanied by a remarkable thermoosmotic transfer. In concluding Section 4 a new transfer mechanism (thermomagnetoosmosis) is predicted: in the presence of a magnetic field the a
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particle thermophoresis can be influenced by a microconvective mass transfer induced by particle magnetophoresis near filter grains.
2 Magnetic Soret effect Considering stable colloids without chemical reactions, the mass conservation equation for two-component systems (particles of mass concentration ρi and a carrier liquid) is the following: dρi = − div ji . (1) dt Ultra-fine particles dispersed in fluids obey an intensive Brownian motion. Therefore, the mass transfer in colloids can be assumed similar to that in molecular liquids by involving the concept of gradient diffusion. The mass diffusion coefficient of nanoparticles is determined by the relation D = kT /fv , where fv is the coefficient of hydrodynamic drag force (for spherical particles of radius rfv = 6πηr, with η being the fluid viscosity). Under a non-uniform magnetic field of gradient ∇H the mass flux along with the terms of gradient diffusion and thermodiffusion contains a new term of magnetic sedimentation (for stable nanocolloids we assume ordinary barodiffusion being negligibly small) [1]: ji = −D∇ρi − ρi (1 − ni )DST ∇T ¤ mg £ + µ0 (M i − M 0 )∇H ρi (1 − ni ) . fv
(2)
Here mg is the mass of a particle, µ0 is the magnetic constant, M i and M 0 are the specific magnetizations of particles and carrier liquid, ni is the mass fraction of solid
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phase, and ST is the Soret coefficient. The mass diffusion coefficient of colloidal particles is several orders of magnitude less than that of molecules in liquids. Therefore, the sedimentation processes in ferrofluids play a more significant role than those in molecular liquids. In colloids of non-uniform magnetization M = M (ρi , T ) the magnetophoretic particle transfer can be observed even if the external field is constant. Let us consider a homogeneous magnetic field (flux density) B = const directed along the temperature gradient. From the Maxwell equation div B = div µ0 (H + M) = 0, it follows (we assume M0 = 0): ∇H = −∇M = −
∂M ∂M ∂M ∇ρi − ∇T − ∇H . ∂ρi ∂T ∂H
(3)
The mass flux (2), by taking account of relation (3), can be rewritten as follows (χ = ∂M/∂H is the differential magnetic susceptibility of the ferrocolloid, αT = −M −1 ∂M/∂T and αc = M −1 ∂M/∂ρi are its relative pyromagnetic and solute-magnetic coefficients, respectively): ¶ µ µ 0 α c M 2 mg (1 − ni ) ∇ρi ji = − D + fv (1 + χ) µ ¶ µ 0 α T M 2 mg − ST − ρi D(1 − ni )∇T . (4) fv Dρi (1 + χ) Accordingly, the magnetic stratification of ferroparticles under the influence of the inner gradients of the magnetic field can be regarded as an increase of the mass diffusion coefficient and as a reduction (usually ST > 0 in the surfacted colloids considered here) of the resultant Soret coefficient (the coefficients αT and αc are positive. A more detailed theoretical analysis performed by taking account of the particle magnetic interactions allows a conclusion that the changes in coefficient D are observed not only under a parallel field B k ∇c but also in the case when B is oriented normally to the concentration gradient [2]. The only difference is that now (at B ⊥ ∇c) the field causes a reduction of the mass diffusion coefficient. Thermodiffusion in ferrocolloids in the presence of a uniform field, arbitrary oriented about the temperature gradient, is considered in reference [3]. Analyzing the hydromagnetic Stokes problem for a spherical particle affected by the mean field of a colloid it is shown that in a general case the magnetophoretic velocity of particles um has components directed along the magnetic field H = B/µ0 µ (µ is the fluid magnetic permeability) as well as along the permeability gradient ∇µ. Two special cases, B k ∇µ and B ⊥ ∇µ, when magnetophoresis is directed along ∇µ only, are an exception. If B is oriented parallel to ∇µ, the particles are transferred towards the decreasing permeability [1, 3]: µ ¶ 12 3 2µ0 r2 H 2 p ∇µ · K 1 + K , (5) um = 9η 5 32 whereas at B ⊥ ∇µ they move in a reverse direction: µ ¶ 2µ0 r2 H 2 6 7 n ∇µ · K 1 − K . (6) um = − 9η 5 32
(For simplicity, small terms, which show the influence of the local temperature gradients near the particle due to a difference in thermal conductivities of the particle and the surrounding carrier liquid, are omitted here.) The coefficient K = (µ − µi )/(µi + 2µ) reveals the effect of local perturbations of the magnetic field near the particle on its motion (µi is the magnetic permeability of the particle). Taking into account the dependence of permeability µ = µ(T, ρi ) on the temperature and on the particle concentration, one can calculate the magnetophoretic summands of both the mass diffusion and the thermodiffusion coefficients. The present single-particle approach is valid only for diluted colloids, therefore, the fluid permeability and the parameter K can be calculated under the assumption that the fluid magnetization obeys the Langevin law ¶ µ µ0 mH 1 M = M s ρi coth ξ − , ξ= (7) ξ kT with M s being the specific saturation magnetization of the particle material and m the magnetic moment of subdomain nanoparticles. This hydrodynamic approach to the magnetic component of mass diffusion agrees well with thermodynamic theory [2]. The parallel field B k ∇ρi causes some strengthening of the mass transfer. In contrast, at B ⊥ ∇ρi the diffusion coefficient decreases, the corresponding changes in D n are approximately twice less than those in Dp . Both phenomena are relatively strong; they have been confirmed experimentally, first in the experiments reported in reference [2]. From (5) it follows that the thermophoretic motion of particles under the parallel field is directed towards increasing temperatures (∂M/∂T < 0). In contrast to magnetodiffusivity, the corresponding magnetic component of the Soret coefficient does not predict a very strong field influence: in the regime of magnetic saturation it reaches a value approximately equal to 0.001 K−1 [1]. The influence of a transverse field (motion in an opposite direction) is also approximately twice less (if compare (5) and (6)). The influence of the field on the resultant Soret coefficient depends on the value of the zero-field coefficient ST , which cannot be calculated within the frame of the hydrodynamic model concerned here. The thermodiffusive properties of colloids have not been investigated in detail so far. Approximate theories based on the concept of slip velocity anticipate high positive values of the Soret coefficient for lyophilized nanoparticles in surfacted colloids up to 10−1 1/K [4, 5]. The known experiments [6–9] agree well with that prediction. Surfacted ferrite particles in hydrocarbons are transferred toward decreasing temperatures. The measured Soret coefficients are several orders of magnitude higher than those usually found in molecular liquids, they reach values approximately equal to ST = +0.15 K−1 . Electrically stabilized particles in ionic colloids usually have negative Soret coefficients [9]. Comparing these results with the thermomagnetic mobility of particles calculated from (5) and (6), one can conclude that the magnetic-field effect on the resultant Soret coefficient should be relatively small. The
E. Blums: New problems of particle transfer in ferrocolloids: Magnetic Soret effect and thermoosmosis 3.5
(1) n (2) n
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2 SOR ET C OE FFIC IEN T S T (H )/S T (0 )
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Fig. 1. Experimental results on the magnetic Soret effect in hydrocarbon-based ferrocolloids: (1) magnetite in tetradecane [10]; (2) Mn-Zn ferrite in tetradecane [11]; (3) meghemite in cyclohexane [12]; (4) magnetite in toluene [13]. The index n denotes experiments with B ⊥ ∇T , the index p experiments with B k ∇T . Trend lines correspond to thermodiffusion column experiments [10] and [11].
experiments, employing two different techniques (separation in a thermodiffusion column [10, 11] and optically induced thermal grating in a thin ferrofluid layer [12, 13], have qualitatively well confirmed the theoretical predictions: a parallel field B k ∇T causes a reduction of the resultant Soret coefficient whereas in the presence of a transverse field B ⊥ ∇T the latter increases (see Fig. 1). However, the observed magnetic effects are much stronger than the calculated ones. This result can be interpreted as the influence of the magnetic field on the mass diffusion coefficient (if ∇µ is caused by the temperature gradient, dependencies (5) and (6) represent the thermomagnetophoretic mobility of particles whereas in experiments the Soret coefficients are measured). Grating experiments allow to define both the Soret coefficient ST and the diffusion coefficient D. Analyzing the measurement results presented in reference [13] reveals that the thermodiffusion coefficient DT = ST D in the presence of weak transverse magnetic fields, as anticipated, remains nearly constant. However, under stronger fields the situation changes: now we observe a reduction of DT . Obviously, this is a result of the influence of the parasitic magnetic convection. In grating experiments the fluid heating across the layer is non-uniform and, independently of the field direction, there monotonously develops a non-threshold thermal magnetic convection, which gradually destroys the concentration grating. If B is oriented transverse to the concentration patterns, a strong reduction of DT is observed already in small fields. Obviously, now there develops a solute magnetic convective instability [13, 14]. In
Fig. 2. Steady concentration profiles across the porous layer. The coordinate −0.5 corresponds to the cold lower wall, +0.5 to the upper hot wall. Temperature difference ∆T = 40 K, separation time t = 28.5 h (black triangles) and ∆T = 11 K, t = 24 h (white triangles) [19].
thermodiffusion column experiments the parasitic convection can be driven by demagnetization phenomena near the channel side walls and, if the field is oriented parallel to the temperature gradient, by developing magnetoconvective instabilities (both thermal and solutal), which can destroy the structure of vertical convection [15]. Recently, the development of vertical convective rolls, predicted in theories for convective instability of gravitational convection in an inclined channel [16], has been observed experimentally for a vertical ferrofluid layer in the presence of the magnetic field B k ∇T [17]. Analysis of those experiments have revealed that the critical value of the magnetic Rayleigh number is relatively close to the one in [16] for gravitational convection.
3 New phenomena in porous media One possibility to eliminate the parasitic convection in the thermodiffusion column is to employ a channel filled with porous media. In layers of high porosity, especially if the grains are not very small, the convection velocity is quite enough to achieve a relatively high rate of particle separation even in colloids of low diffusion coefficients. Under Brinkmann’s approximation, the theoretic model of separation is similar to that of a column with a MHD flow [18]. However, there arise new problems in interpreting the measurement results. First of all, we have to reconsider the fluid macroscopic transport coefficients (thermal conductivity, viscosity) needed to calculate the non-dimensional parameters, which determine the convection strength. Moreover, when one analyzes the separation measurements, one should not forget that in capillaryporous layers the thermophoresis in colloids can be accompanied by a particle thermoosmosis. The latter in some
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colloids is so strong that it can even change the direction of particle thermophoretic transfer [19]. As an evidence of the above said, Figure 2 illustrates some profiles of the particle concentration measured in a horizontal non-isothermal porous layer. The layer consists of ten thin round capillary filter sheets (cellulose, “Filtrac” No. 90) of thickness 135 µm and diameter 68 mm, which are cramped between two cooper plates kept at different temperatures. The total thickness of the layer is a = 1.35 mm, its average porosity is 0.33. Before the experiment the filter sheets are carefully saturated with a hydrocarbon-based ferrofluid (magnetite in undecane stabilized by octadecanol, saturation magnetization Ms = 18.1 kA/m, volume fraction of magnetic phase c0 = 0.087, diameter of nanoparticles detected by the magnetogranulometry technique dm = 8.5 nm); the excess of fluid is removed by compression while assembling the cell. Particle concentration profiles in the porous layer are detected from saturation magnetization of each separate filter sheet after the experiment (integral data for each element after drying are measured) [19]. The experiment duration t is at least 24 hours. Since the Brownian diffusion coefficient of nanoparticles is approximately D ≈ 10−11 m2 /s, at t = 24 h the diffusion Fourier number tD = Dt/a2 exceeds the values larger than tD = 0.5, which are sufficient to reach a steady concentration profile µ ¶ x c = exp − S∆T (8) c0 a with S being the porous separation coefficient (∆T is the temperature difference applied to the layer). In the absence of a porous matrix, S represents a conventional Soret coefficient ST . From optical grating measurements it is found that the particles of the examined ferrofluid sample are transferred to a colder region, the Soret coefficient is positive: ST ≈ 0.04 K−1 . Unexpectedly, these particles in the porous layer move in an opposite direction, towards the increasing temperature. The exponential trend lines (best fit) of the experimental results, presented in Figure 2 for both temperature differences, give one and the same value S ≈ −0.1 K−1 . The obtained results obviously testify the influence of the osmotic phenomena on the solute transfer in filter pores. According to the “slip” theory [4], the particle velocity in capillary tubes can be presented as ¶ µ dm (bp − bw )∇T , (9) uT = f dp where bp and bw are the slip-velocity coefficients of the particle and the wall, respectively, and f (dm /dp ) illustrates the hydrodynamic influence of the tube walls on the particle motion (dp is the mean diameter of capillaries). In experiments [19] dm ¿ dp , therefore, this influence is small, f (dm /dp ) = 1. At lyophilized solid/liquid interfaces the slip velocity is directed towards the increasing temperature. Hence, the thermoosmotic fluid motion reduces the resulting particle thermophoretic velocity in the tube. According to existing theories [5, 20], the slip velocity is proportional to the temperature gradient in the surface layer.
The thermal conductivity of ferrites or ferromagnetic metals is much more higher than that of a carrier liquid (hydrocarbons in our samples). Thus, the local temperature gradient in the surface layer of nearly spherical nanoparticles is significantly less than the macroscopic one in the carrier liquid and along the microchannel walls. Therefore, even if the Gibbs absorption in capillary pores is less than that in surfacted nanoparticles, the thermoosmotic velocity in capillary microchannels can play a remarkable role in transport processes. If the osmotic transfer dominates over thermophoresis, the resulting transfer velocity changes even the direction.
4 Magnetic microconvective particle transfer If an external magnetic field is applied to a porous layer, additionally to the ordinary thermoosmotic transfer, one should mind possible changes in transport characteristics caused by a magnetic microconvection inside the pores due to the difference in magnetic susceptibilities of the liquid and the porous matrix. Recently, a strong influence of the magnetic field on the particle transfer through a thin non-isothermal ferrofluid layer with non-magnetic permeable walls (an ultra-fine grid of a period about 10 µm) has been found [21]. The authors interpret those results as a magnetic Soret effect. We propose another transport mechanism —a particle thermomagnetoosmosis [22]. Under non-isothermal conditions, local perturbations of the magnetic field near the grid element bring about both the concentration difference due to magnetophoresis and the thermomagnetic convection directed along the temperature gradient. As a result, there appears a convective particle transfer. Let us consider the membrane as a grid consisting of single spherical grains of magnetic permeability µi different from that of a surrounding liquid µ. An external uniform magnetic field H is oriented parallel to the gradient of permeability ∇µ = −αT ∇T (αT is the fluid pyromagnetic coefficient) and directed toward the azimuthal coordinate ϑ = 0. Under the Stokes approximation (the Reynolds number for micrograins is very small) the convection velocity around a sphere v is governed by the vorticity equation (Ω = rot v) µ0 ∇µ × ∇H 2 (10) ∇ × (∇ × Ω) = 2η with η being the fluid viscosity. For simplicity, we assume that the heat conductivities of the liquid and the sphere are the same, therefore, ∇µ in (10) is equal to the macroscopic one. The mathematical problem is similar to that of particle thermophoresis [1, 23, 24], but in the frame of the linear approximation considered here, the sphere-induced perturbations of the external magnetic field can be calculated by introducing the scalar magnetic potential in a simplified form K ψ = 2 H0 cos ϑ (11) R with no account for smaller terms with ∇µ [23]. Here R is the non-dimensional radial coordinate scaled by the grain
radius R0 and the coefficient K represents the difference in magnetic permeabilities, K = (µ − µi )/(µi + 2µ). Figure 3 illustrates the flow pattern around a non-magnetic sphere immersed in a magnetic fluid [22] (the stream lines are calculated from the analytic solution of convection problems (10) and (11) given in Refs. [23, 24]). The external magnetic field induces six extended convection vortices. Four pared asymmetric vortices cause only a volumetric fluid mixing whereas two other vortices, located in the plane ϑ = ±π/2, induce a macroscopic translation flow (“thermomagnetoosmosis”). The non-magnetic spheres induce a resulting convective flow directed towards the decreasing fluid permeability. Contrary, the magnetic sphere (negative K) causes a convective motion in an opposite direction, towards increasing µ. The local magnetic field gradients induce not only convection but also a magnetophoretic transfer of the particles. According to (2), their velocity in diluted colloids is equal to (here Mp is the particle magnetization)
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MAGNETIZATION GRADIENT R0∇M/Ms
Fig. 4. Mass flux jx induced by one spherical grain (R0 = 10 µm, µI = 1) in a magnetite-based ferrofluid (c0 = 0.1, r = 5 nm) at various B k ∇µ (the x-axis is oriented towards increasing µ). 0
EFFECTIVE SORET COEFFICIENT St (1/K)
Fig. 3. Flow structure near a non-magnetic (K = 0.2) sphere. Angle ϑ0 represents the position of the frontal hydrodynamic attack point.
PARTICLE FLUX DENSITY –jx R0 /c0 D
E. Blums: New problems of particle transfer in ferrocolloids: Magnetic Soret effect and thermoosmosis
R 0=14 µm
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2µ0 r2 u= Mp ∇H . 9η
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(12)
The convection near the sphere corresponds to high values of the Peclet number P e = νR0 /D (the Brownian diffusion coefficient of particles D is very low). The condition P e À 1 allows to analyze the mass transfer problem under a concentration boundary layer approximation. The integral mass conservation equation near the sphere with passive boundaries yields 1+d Zϑ Z ∆c dR = −(1 + d) uR|R=1+d dϑ . Rνϑ c0 1
(13)
ϑ0
Here d = δ/R0 is the non-dimensional boundary layer thickness, uR is the normal component of magnetophoretic velocity (12) at the outer border of the boundary layer (at R = 1 + d), ϑ0 is the angle of frontal hydrodynamic attack (see Fig. 3). If the excess concentration ∆c = c − c0 in the boundary layer is approximated by a polynomial, equation (13) can be solved analytically. Figure 4 illustrates the microconvective mass flux (left part of Eq. (13)) induced in a ferrocolloid by one non-magnetic filter grain. Calculations
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MAGNETIC FIELD B (T)
Fig. 5. Thermomagnetoosmotic separation parameter for a one-layer non-magnetic filter sheet of cross-sectional porosity 0.1 in a magnetite containing a ferrofluid, r = 5 nm, c0 = 0.05.
are performed by introducing the concentration profile in the form of µ ¶3 −m R−1 ∆c = 1− (14) 3+m d that satisfies the conditions for the diffusion boundary layer formed by magnetic sedimentation near an impermeable wall (sedimentation parameter m = uR δ/D). It is interesting to note that the macroscopic mass flux is directed towards the increasing temperature independently of the sign of the difference in magnetic permeabilities for the grid and the surrounding liquid. The resultant “magnetoosmotic” nanoparticle flux in a thin porous filter is proportional to the cross-sectional density n of spheres in the filter sheet. In Figure 5 the microconvective mass transfer through a one-layered sheet is shown in the form of equivalent thermophoretic mobility.
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One can see that negative values of the thermomagnetoosmotic coefficient S can exceed positive zero-field values of the particle Soret coefficient (as said in Sect. 2, they are close to +0.1 in ferrofluids). The experimental results for B k ∇T , reported in reference [21], agree relatively well with the present calculations. At B ⊥ ∇T the resulting flow induced by a membrane grain has an opposite direction, but due to a complicated vortical structure of convection, the integral mass flux should be found numerically. It should be noted that the current analysis is valid only for small magnetophoretic velocities until /m/ ¿ 1. At an intensive particle transfer the excess concentration in the boundary layer is high; it starts to influence the magnetic field and instead of (11) the field distribution around the sphere should be calculated numerically together with the equations of convective mass transfer.
5 Conclusion The main reason of the experimentally detected strong magnetic Soret effect in ferrocolloids is the influence of the magnetic field on the mass diffusion of particles. The thermodiffusion experiments can be disturbed by magnetoconvective instabilities. Thermophoretic transfer of nanoparticles in capillary-porous media is accompanied by thermoosmosis. A new phenomenon, a magnetoosmotic particle transfer in porous filter layers, is theoretically predicted. The author is thankful to the research groups at University Paris-VI (J.C. Bacri, A. Bourdon) and at ZARM, Bremen University (S. Odenbach) for a long-term collaboration. The experiments were performed by A. Mezulis (Riga and Paris) and T. Voelker (Bremen). The work is supported by the Latvian Science Foundation as well as by several international grants including the Alexander von Humboldt Award (FRG).
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