Feasibility and limitations of nanolevel high gradient magnetic separation

Separation BPurification Technology Separation and Purification Technology 11 (1997) 199-210

Feasibility and limitations of nanolevel high gradient magnetic separation A.D. Ebner, J.A. Ritter *, H.J. Ploehn Department of Chemical Engineering, Swearingen Engineering Center, University of South Carolina, Columbia, SC 29208, USA Received 4 October 1996; accepted 26 February 1997

Abstract This work proposes a new separation concept denoted as nanolevel high gradient magnetic separation (HGMS) or magnetic adsorption. A magnetic heteroflocculation model describes the magnetic forces between two spherical particles with different sizes and magnetic properties, and reveals the feasibilities and limitations of nanolevel HGMS. The adsorbent particles, composed of antiferromagnetic magnetite, are modeled as large, immobile spheres on the order of 100-500 nm in radius. The adsorbate, paramagnetic colloidal Fe(OH), particles, are treated as freely diffusing small spheres on the order of 20-80 nm in radius. The model assumes that the magnetite particles are dispersed throughout a porous, nonmagnetic, solid matrix and that they are free of convective forces. The model also assumes that magnetic forces alone act on the Fe(OH), particles, opposed only by Brownian motion. When the magnetic force is attractive and overwhelms the randomizing Brownian force, adsorption occurs. The results from this model show the importance of the external field strength, the sizes of the adsorbent and adsorbate particles, and their magnetic properties in developing a practical nanolevel HGMS process. 0 1997 Elsevier Science B.V. Keywords: Feasibility; Limitations;

Nanolevel high gradient magnetic separation

1. Introduction It is well known that magnetism plays an important role as a separating tool in the mineral processing industry [ 1,2]. In recent years, several

different types of magnets with fields ranging from 0.05 to 2 T have been used to separate ferromagnetic and paramagnetic materials from diverse kinds of mineral ores [l-7]. The creation of high magnetic field gradients is of paramount importance in these magnetic separation processes; thus,

* Corresponding author. 1383-5866/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII S1383-5866(97)00021-X

they are collectively referred to as high gradient magnetic separation (HGMS) processes. High gradients can be achieved by magnetically inducing small or thin magnetic elements of high curvature inserted within a separation bed. This principle has motivated researchers to study flocculation mechanisms [S-15], the separation of macromolecules and microorganisms [ 16-181, and the purification of water and other effluents [ 19,201. The objective of this study is to analyze theoretically the feasibility and limitations of the application of magnetic fields in nanoparticle separations (i.e. to develop a basis for nanolevel HGMS). The scenario for the theoretical development is repre-

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sented in Fig. 1, and based on preliminary experimental findings [ 2 1,221. A solution of spherical, colloidal nanoparticles with paramagnetic properties [e.g. Fe(OH), at a relatively high pH] flows through a packed bed inserted inside the bore of an electromagnet. As a first approximation, the colloidal nanoparticles are considered to be in a neutral charge state and free of van der Waals forces. This assumption is not very realistic because van der Waals forces exist even for neutrally charged particles. However, this study is concerned with quantifying and understanding only the magnetic forces between two particles to establish the limitations of a nanolevel HGMS process. Electrostatic and van der Waals effects will be included in a future article. The packing consists of a porous matrix with spherical magnetite particles dispersed throughout the interior surfaces of the porous matrix, the radii of which (0.1-I .6 l.nn) are necessarily larger than those of the nanoparticles ( lo-100 nm). The pores of the matrix are also large enough to permit the free displacement of the colloidal particles, but sufficiently small to avoid the effect of the convective forces associated with the external carrier fluid

(solution). The matrix and the solution are assumed to have similar magnetic properties, which are distinguishable from those of the colloidal and magnetite particles. Furthermore, both materials are considered to behave as a single medium that is essentially unaffected by the magnetic field. When the field of the magnet is turned on, the magnetite particles are magnetically induced, creating a field that contributes to the net field sensed by the colloidal particles. In the absence of convection and when the magnetic force is sufficiently greater than the force associated with Brownian (thermal) motion, the magnetic force created by the field can be attractive and large enough to allow the magnetite to adsorb the colloidal particles. When the field is turned off, the particles are released and dispersed in solution by thermal motion. In the following sections, after reviewing some of the basic concepts of magnetic fields, a theoretical heteroflocculation model is developed to predict the magnetic force exerted on a paramagnetic colloidal nanoparticle by a magnetite particle in a uniform applied magnetic field. This model shows how the magnetic force varies with the size and

Magnetite Particle support \

Fig. 1. Representation of a nanolevel HGMS process, consisting of a packed bed with magnetite particles encased in a porous matrix placed within the bore of an electromagnet.

A.D. Ebner et al. / Separation and Purfication Technology II (1997) 199-210

magnetic properties of both the magnetite and colloidal nanoparticles and also with the strength of the external field. This fundamental information reveals the feasibility and limitations of nanolevel HGMS.

2. Theoretical background and model development 2.1. Magneticjeld, magnetization

Magnetic field vectors can be expressed in terms of either the magnetic field strength H or the magnetic induction B. For diamagnetic or paramagnetic materials, the two field vectors are related by [2X241 (1)

with the magnetic permeability, pL,, as the proportionality constant. In a vacuum, the permeability has the value of ~m=~L,=47c~ 10-7TmA-‘. When a magnetic field passes through a material, the material acquires an induced magnetization M given by M=x,H

(2)

where x,,, is the magnetic susceptibility of the material. In vacuum, xrn= 0. The magnetic induction B can also be expressed as B=,u,(H+M) where M is the induced magnetization material. Eqs. (l)-(3) show that Prn=Pu,(l+Xrn)

2.2. Magnetic dipoles Bar magnets, current-carrying wire loops and molecules are examples of magnetic dipoles. From the definition of B, the force F exerted on a moving positive charge q0 with velocity v is given by F=q,vxB

(5)

and the torque exerted by B on a wire loop enclosing area A and carrying current Z is found to be

magnetic induction and

B = ZlmH

201

(3) of the (4)

which relates the permeability to the susceptibility. The magnetic susceptibility can be used to classify materials according to their magnetic propermaterials have negative ties. Diamagnetic susceptibilities with values on the order of 10e5. while paramagnetic materials have positive susceptibilities two to three orders of magnitude larger. A magnetic field that would attract a paramagnetic molecule would repel a diamagnetic molecule. Other classes of materials, including ferromagnetic, antiferromagnetic and ferrimagnetic, have much larger susceptibilities that are not constant.

z=/ixB

(6)

where /i has magnitude @I=I,4 and vector direction given in terms of the current flow around the loop via the familiar right-hand rule. By analogy with the torque exerted by an electric field on an electric dipole, @ can be identified as the magnetic dipole. Carrying the analogy further, the potential energy of a magnetic dipole with a specified orientation in the field B can be written as U=fi.B

(7)

(7) The motion of electrons around atomic nuclei may cause some atoms and molecules to have permanent magnetic dipole moments ,i&. In the absence of a magnetic field, the magnetic dipoles in a bulk material have random orientations, so the material has zero net magnetic dipole moment. A magnetic field biases the distribution of dipole orientations, thus inducing a net dipole moment. In addition, a magnetic field alters the electron distribution around an atom, thereby producing another contribution to the net dipole moment. The total magnetizability 5i + (&)/( 3kT) [23] reflects the sum of the electronic and orientational contributions to the magnetizability of an atom or molecule of type i. A magnetic field therefore induces a molecular magnetization with an induced dipole given by

(8) The bulk magnetization moments are related by Mi =pLlii

and the induced dipole (9)

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A.D. Ebner et al. 1 Separation and Purification Technology 11 (1997) 199-210

for a pure material, where pi represents the number density of molecules of type i per unit volume. A small particle with volume V, may be treated as a single dipole so that

becomes

Pi = vyp

for the magnetization of the small particle in a medium, This expression for the magnetization of the particle has the character of an excess magnetization: if the particle has the same magnetic properties as the medium, the field produces no magnetization of the particle. Substitution of Eq. (14) into Eq. (12) gives

(10)

An induced dipole has a free energy less than that predicted by Eq. (7) because some energy goes into polarizing the molecule. Beginning with Maxwell’s equations [25] or by analogy with induced electric dipoles [26], the free energy, w of an induced magnetic dipole is given by w= -&

.B= -f~,&

.H

w=

(11)

( ll)taking care not to confuse the permeability of the medium CL, with the dipole pi. For a small particle described by Eq. (10) in a dielectric medium, the free energy of the particle in a magnetic field is given by

W=-$Mp.H

M p,m

(12)

2.3. Magnetic heterojlocculation model This analysis seeks to describe the force exerted on a small particle (p) by a large antiferromagnetic sphere (s) in an applied magnetic field, H,. The force equals the negative of the gradient of the potential defined by Eq. ( 12). To complete the model, the magnetization of the small particle must be established, and the total magnetic field, H, at the particle must be computed. The magnetic field experienced by the particle equals the sum of the applied field and the field created by the magnetization of the large sphere. Consider a small particle with permeability ,L+,; Eqs. (2) and (4) suggest that in a vacuum, the magnetization of the small particle can be written as

=

( 1

PLp-Fo PPO -H PO =

(

PL,

H >

+ip-pm)Hz= _y

By analogy with the treatment of electrostatic interactions in a dielectric medium [25], Eq. (13)

(x, -xmw

( 15)where the second equality employs Eq. (4) to introduce the susceptibilities of the particle and the medium. The question that remains is the computation of the field experienced by the particle in the vicinity of a large (antiferromagnetic) sphere. As mentioned before, the field at the small particle includes contributions from both the applied field and the magnetization field of the sphere. An equation similar to Eq. (14) is written for the sphere as M

= PL,-L H, s,m ~ ( PL, 1

(16)

which gives the magnetization of the sphere as a linear function of the applied field at low field strengths. At high fields the magnitude of M,,, reaches a constant saturation value as expected for antiferromagnetic materials. The Biot-Savart [25,27] law readily provides the magnetic field arising from the magnetized sphere. Fig. 2 depicts the components of this field in spherical coordinates. The Biot-Savart law yields the radial component

K

M,,,

cos 9

27tr3

(13)

(14)

(15)

Hs,,= M

&-Al

~

(17)

and angular component H,,, = ~

K

47cr3

M,,, sin e

(18)

A. D. Ebner et al. J Separation and PuriJcation Technology 11 ( 1997) 199-210

203

particle is given by

1s,

F,, = -VW= 2

pO(xp -x,>V(H’)

and has radial and angular components, tively, given by Eqs. (23) and (24).

(22)

respec-

F sp,r= - ~p&(Xp -xd

[!

2 H,+$s,m

+

cos2 0

a: -H,+3r3A4s,m

(23) 2

F sp,o= v,~o(x, Fig. 2. Field created

by a magnetically

induced

-xm)

sphere.

(24) as the scalar components of the field generated the magnetized sphere with volume by V, = (47raz)/3. Vectorial addition of the applied field to Eqs. (17) and (18) gives H, =

H, + 5

MS,,

!

cos 0

>

sin 8

(19)

(20)

for the radial and angular components of the total magnetic field outside the large sphere, similarly to that reported elsewhere [ 1,271. The square of the magnitude of the field is H2=Hf+H;

(21)

Eq. (15) with Eqs. (19)-(21) give the approximate free energy of the interaction of the small particle with the large sphere. In addition to the approximation of the small particle as a single dipole, the analysis also represents a linear superposition approximation of the complete interaction free energy because it neglects the effect of the field generated by the magnetized small particle on the magnetization of the large sphere. Using the expression for w in Eq. (15), the magnetic force exerted by the sphere on the small

For simplicity, the small particle is treated as a sphere with V, =(4nai)/3. The sphere magnetization MS,, is assumed to be the saturation value. The magnetic force is significant only if its magnitude is substantially greater than the effective Brownian force associated with random thermal motion. The Peclet number (Pe) [ 11,281 represents the ratio of the magnetic force to the Brownian force and is given by Pe=-

aplFspl kT

(25)

where ap is the radius of the small particle. The magnetic force dominates when Pe>> 1. Random thermal motion overwhelms any tendency toward heteroflocculation in the opposite limit. Consequently, Pe can be viewed as the degree of adsorbability of the species under study.

3. Results and discusion The results obtained from the magnetic heteroflocculation model are presented in four sections. First, the critical parameters are discussed along with the selection of the adsorbent and adsorbate

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materials. This is followed by a section on the net forces and their orientation, Then, the effects of the adsorbent (magnetite) and adsorbate [Fe(OH),] sizes are discussed; and finally, the effect of the field strength is analyzed along with the field gradient and its dependence on the radius of the adsorbent. The results are presented in Figs. 3 to 8. 3.1. Criticalparameters, adsorbate selection

and adsorbent and

The Peclet number depends critically on four main variables. The nanoparticle size has an extraordinary effect on Pe due to the fourth-power dependence on its radius. For example, decreasing the nanoparticle radius by two lessens the degree of adsorbability 16-fold. The radius of the sphere

Fig. 3. The modulus of the magnetic a,=40 nm, a,=500 nm and H,=2T.

force

(IFspI) normalized

also has a large influence on Pe because at the magnetic field strengths (H,) considered in this work, the radial component of the magnetic force, F sp,r, is proportional to a5 as shown in Eq. (23). Moreover, magnetically induced spheres of smaller size create larger field gradients in their surrounding as a result of their larger curvature (recall that these field gradients are paramount to designing an effective HGMS system). Lastly, the magnitude of the applied magnetic field, and the magnetic properties of both the sphere and the particle, are also significant. With these in mind, magnetite has ideal characteristics for use as the adsorbent. Magnetite particles are not only spherical in nature; they are also antiferromagnetic, i.e. they exhibit a behavior similar to ferromagnets but with reduced strength due to opposite directions of some of the free spins. Thus, magnetite exhibits

to

the

Brownian

force

as

a function

of

r and

Q for

20.5

A.D. Ebner et al. /Separation and Purijication Technology 11 (1997) 199-210

Fig. 4. Normal component of the ap = 40 nm, a, = 500 nm and H, = 2 T.

force

(F,,,,)

normalized

high magnetization and very low coercive force (hysteresis). This would allow the adsorbate (colloidal particles) to be easily released once the field is turned off. Moreover, when the external field is greater than about 2 T, the induced magnetization of magnetite is at its maximum (saturated) value of about 0.6 Tesla [24,29]. This value is significantly higher than the magnetization of any paramagnetic compound at such an external field. For example, Fe(OH), requires an almost unattainable field of 50 T to obtain a magnetization of the same magnitude. Other materials can be used as the adsorbent, including iron-nickel or ironsilicon compounds, which have even larger saturation magnetization values and less coercive forces than magnetite [29]. However, magnetite is readily available and relatively inexpensive. For simplicity, the magnetization of magnetite is taken to be the

to

the

Brownian

force

as

a

function

of

r/a,

and

0 for

saturation value, since this study is concerned only with magnetic field strengths in the range between 0.5 and 8 T. There are also many paramagnetic particles that are potential adsorbates. In fact, any material that contains the following elements in any of their oxidized forms are candidate adsorbates due to their similar magnetic properties: Ti, V, Cr, Mn, Fe, Co, Ni, the whole series of lanthanides excluding La and Lu, and some actinides such as U. Clearly, this wide range of paramagnetic colloidal (adsorbate) particles makes the idea of a nanolevel HGMS process very attractive from an environmental point of view. In this study Fe(OH), was selected as the adsorbate due to its proven formation of suspended, colloidal precipitate at ranges of pH between 8.5 and 12. Also, both the matrix (e.g. SiOJ and the surrounding liquid (water) are

A.D. Ebner et al. / Separation and Purification Technology I1 (1997) 199-210

206

Fig. 5. Tangen ltial component of the force (F,,.,) normalized nm, a,=500 nm and H,,=2T.

to the Brownian

force as a function

of r/a, and f3 for

a,=40

40

40

t,

1

.

30

30

$

$

20

20

10

10

0

0 0

20

40

60

80

100

ap (W Fig. 6. Pe as a function of ap for different a, with 0=0, r= 3a, and H,=2T.

0

20

40

60

80

100

ap (nm)

Fig. 7. Pe as a function of of aP for different jH,I with 8=0, r = 3u, and a, = 500 mn.

A.D. Ebner et al. 1 Separation and Purification Technology I1 (1997) 199-210

160

80

I \

as = 100

nm

I I

40

!

O0

100

50

150

200

300

250

x(nm)

Fig. 8. Pe as a function surface of the magnetite and H,=2T.

of the distance x (x=r-a,) from the for different a, with O=O, a,=40 nm

assumed to have similar magnetic properties, since their susceptibilities are low compared with those of magnetite and Fe(OH),. Table 1 summarizes the magnetic properties of these materials. 3.2. Net forces and their orientation

207

According to Fig. 4, the radial component of the force points inward (attractive) in the range of 0 between O-60 and 120-180”, and outward (repulsive) in the equatorial zone, i.e. 9 in the range between 60 and 120”. Eq. (23) suggests that the term with a sinus factor is much more important in the equatorial region. In contrast, Fig. 5 indicates that the tangential component, Fe, always points towards the poles. Moreover, Fig. 3 shows that the magnitude of the force and the principal distortion of the field occurs at 0 equal to 40 and 140”, where the forces are mainly attractive (Fig. 4) and several times much more important than those forces in the polar and equatorial zones. The field components also decrease very abruptly in going from the surface to about two a, from the center of the magnetite sphere; thereafter the decrease is much more gradual, but the effect is still prominent at distances greater than 7a,. Also note that the attractive forces are no longer effective at distances greater than 2a, from the surface. Nevertheless, separation may still be effective since a porous matrix containing as little as 2 vol.% magnetite has an average half distance between the magnetite spheres of about 3a, to 4~7,. 3.3. EfSects of adsorbent and adsorbate particle

Figs. 3-5, respectively, show the distribution of the magnetic force and its normal and tangential components as functions of the angle 8 and distance r. These forces have all been normalized to the Brownian motion; thus, their magnitudes are equivalent to Pe. These curves are computed from Eqs. (23)-( 25) with up = 40 nm, a, = 500 nm and H,=2 T. This value for the magnetite particle radius is commonly observed in magnetite powder; it can also be as low as 0.1 urn. Table 1 Properties of the materials tion model

used in the magnetic

Adsorbate (Fe(OH),) susceptibility (I,) Adsorbant (magnetite) saturated magnetization (M,,,) Medium (water and matrix”) susceptibility (x,) “The matrix is assumed

heterofloccula-

1.2 x 1om2 4.8 x lo-’ A mm’ -1.3

x 1o-6

to have the same susceptibility

as water.

sizes

The effects of the adsorbent and adsorbate particle sizes on Pe were studied in terms of their respective radii. The results are plotted in Fig. 6, where Pe is plotted as a function of aP with a, as a parameter. For this case, 19=O, r = 3a, and H,=2 T; thus, FS,,O= 0 and Fsp,r I 0, which implies the magnetic forces are attractive. The size of the colloidal particle has a very dramatic effect on Pe. Under these conditions, colloidal particles with radii less than 40 nm are not adsorbable. This shows very clearly the impracticality of using a nanolevel HGMS process for ionic adsorption. Due to their larger curvature, which generates larger field gradients, better results are obtained with smaller magnetite spheres. However, the effect of increased curvature diminishes quickly with decreasing up due principally to the strong dependence of Pe on the colloidal nanoparticle size. Thus, little improvement is

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A.D. Ebner et al. / Separation and Purijkation

expected by further reducing the size of the magnetite sphere. It is important to note, however, that since a, is changing, so is the distance r (since r/u, is held constant), which implies that the increase in Pe with decreasing a, is caused by increased curvature as well as decreasing distance from the magnetite sphere. These effects are uncoupled in the next section. 3.4. EfSect of$eld strength The effect of the field strength, H,, on Pe was studied as a function of the adsorbate particle radius, aP. The results are plotted in Fig. 7, where Pe is plotted as a function of a, with H, as a parameter. For this case, fJ=O, a,=500 nm, and Y= 3a,, so again the magnetic forces are attractive. It is seen that H, has a marked effect on Pe, but the effect diminishes quickly with decreasing aP. These effects are very similar to the effects of a, on Pe, as seen in Fig. 6. Moreover, under the conditions investigated, and even at high field strengths, colloidal particles with radii less than 20 nm are not adsorbable. Some improvement would be expected, however, for a very high H, and small a,, but not enough to consider adsorbing ions from solution; the thermal (Brownian) effects are simply too overwhelming. It was alluded to earlier that the role of the curvature of the magnetite is important to attaining higher field gradients and that this is achieved with smaller magnetite spheres. However, the range over which these high gradients are effective is very short. Fig. 8 shows this compromise between the field gradient and the effective range of the force. In this figure, Pe is plotted as a function of the distance, x, from the surface of the magnetite particle, with the magnetite particle radius, a, as a parameter. In this case, H,=2 T, up =40 nm, and 8=0, again giving rise to attractive magnetic forces. Very large magnetite particles (10 000 nm) have very small curvature and thus small gradients. As a result, Pe is small at any distance from the surface of the adsorbent, and adsorption never takes place. This clearly illustrates the known limitations of conventional HGMS systems, which typically utilize wire mesh on the order of microns in diameter. In contrast, for very small adsorbent particles, the gradients are larger but so short

Technology II (1997) 199-210

ranged that the adsorbate must closely approach the adsorbent before feeling any appreciable force (a, = 40 nm). However, between these extremes, there are zones in which the curvature is more significant (e.g. for a, = 100 nm) and zones in which the long-range effects are appreciable (e.g. for a, =400 nm). This explains why the curves cross each other in Fig. 8. These results also establish qualitative quidelines on selecting the size of the adsobent particle that would be most effective in a nanolevel HGMS process. 4. Conclusions This work provides a quantitative framework for assessing the utility of applied magnetic fields for creating attractive magnetic forces between adsorbents and adsorbates in a nanolevel HGMS process. The model and results delineate the key roles played by the applied magnetic field strength, the magnetic properties of adsorbent and adsorbate, and the morphology of the adsorbent. The latter factor has not been fully appreciated in previous works. For separation processes based on wire collectors, the large wire diameters and small curvatures lead to relatively low field gradients and low separation efficiency. In contrast, small antiferromagnetic particles with large curvatures produce magnetic fields with high gradients. The model presented here suggests that a separation process employing such an adsorbent might be very efficient for the removal of paramagnetic nanoparticles from solution. Experimental evaluation of this scheme is currently under way. Acknowledgment The authors gratefully acknowledge financial support from the National Science Foundation under grants CTS-9520897 and CTS-9258137, and from Laidlaw Environmental Services, Inc. Appendix Nomenclature aP

2

radius of the colloidal particle (m) radius of the magnetite sphere (m) magnetic vector flux vector (T)

A. D. Ebner et al. J Separation and Purijication Technology II (1997) 199-210

force vector (N) total force vector exerted by the applied field H, and the magnetite sphere over the colloidal particle (N) F sp.r radial component of F,, (N) F tangential component of F,, (N) H magnetic field vector; total magnetic field vector (A m-l) applied magnetic field vector (A m-r) H, radial component (scalar) of the total field H, on the colloidal particle (A m- ‘) tangential component (scalar) of the total HI3 field on the colloidal particle (A m - ‘) radial component (scalar) of the field that HS., arises from a magnetically induced sphere (A m-l) H S.9 tangential component (scalar) of the field that arises from a magnetically induced sphere (A m-l) Boltzmann’s constant= 1.38 x 1O-23 (J K-‘) k induced magnetization vector (A m- ‘) M magnetization vector of an induced dipole Mi of molecule type i (A m- ‘) magnetization vector of a particle comMP posed with molecules type i; magnetization vector of the colloidal particle (A m- ‘) M P.0 magnetization vector of the colloidal particle in free space (A m- ‘) M P.m magnetization vector of the colloidal particle in a medium (A m-l) M s.m magnetization vector of the magnetite sphere in a medium (A m- ‘) Pe Peclet number, defined by Eq. (25) charge (C ) 90 r distance scalar from the center of a magnetically induced particle (m) T temperature (K) U energy scalar of a magnetic dipole with a specified orientation in the field B (J) volume scalar of a particle; volume scalar v, of the colloidal particle (m3) W free energy scalar of an induced magnetic dipole; free energy scalar of the colloidal particle (J)

susceptibility scalar of the colloidal particle vector magnetic moment or dipole (Am-‘) permanent magnetic vector dipole moment of a molecule or atom of type i (A me2) induced magnetic vector moment of molecules of atom of type i; magnetic vector dipole of the colloidal particle (A m-‘) permeability scalar of the medium (T m A-‘) permeability scalar of free space =4n x lop7 (T m A-‘) permeability scalar of a particle p (TmA-‘) angle in the spherical coordinate system as depicted in Fig. 2 number density scalar of molecules of type i (m-3) torque vector exerted on a wire loop when the loop is inserted in a field with induction B (N m-l) electronic magnetabizability scalar of molecules of type i (A mm2 T-l)

F F SP

SP8

References [l] E.G. Kelly, D.J. Spottiswood, Introduction [2] [3] [4] [5] [6]

[7] [8]

[9]

Greek letters Xm

susceptibility scalar of the medium or of a material

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[lo]

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