Jan Svoboda. Magnetic Techniques for the Treatment of Materials

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210 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.37: The eect of the magnetic ﬁeld strength on the degree of ﬂoccula-

tion of strongly magnetic particles (adapted from [K14, K2]).

A condition for magnetic ﬂocculation to take place can be expressed as

+ I

(3.115)

where I

is the electric force as a result of electrical double layer. Since

AA I

for ferromagnetic particles at reasonably strong magnetic ﬁeld, the

probability of ﬂocculation is determined by the interplay between the magnetic

and hydrodynamic forces. As a function of the magnetic ﬁeld strength, the

process of magnetic ﬂocculation proceeds through several stages, as is shown in

Fig. 3.37.

In stage 1, which corresponds to the steady-state phase of the process, in

which I

6 I

, the degree of ﬂocculation #

is proportional to I

and, there-

fore, #

= nK

= At this phase the grade of the ﬂocs is high and ﬂocculation is

selective.

With further increase in the magnetic ﬁeld strength the magnetic interaction

becomes dominant and the ﬂocculation enters phase 2. The process proceeds

non-selectively at a very high rate and the grade of the ﬂocs is reduced. When

the concentration of solids in a suspension is reduced below a certain threshold

value, the mean distance between the particles and their characteristic ﬂoccu-

lation time increase and the process enters an equilibrium phase 3.

3.6.2 Magnetic ﬂocculation of weakly magnetic minerals

In contrast to strongly magnetic materials, the magnetic ﬂocculation of para-

magnetic and diamagnetic materials is a considerably more complex process.

In spite of its signiﬁcant potential, the process has not found any application

on industrial scale as yet. Although several theoretical models that predict

the behaviour of weakly magnetic particles of quasi-colloidal dimensions, in the

3.6. MAGNETIC FLOCCULATION 211

presence of the external magnetic ﬁeld, have been developed over the last twenty

years, their experimental veriﬁcation is still limited and is mainly of qualitative

character.

Stability of colloidal suspensions

The criteria of the stability of colloidal suspensions are governed largely by the

interplay of repulsive and attractive forces between the particles. A detailed

description of this interplay is the basis of the theory of stability of hydrophobic

colloids as proposed by Derjaguin and Landau [D14] and Verwey and Overbeek

[V1] (DLVO theory). The DLVO theory views the stability of colloids as arising

from electrical charges on identical particles and a consequent electrical repul-

sion between them. Opposing this repulsion are the attractive forces of the

universal London-van der Waals type. If the repulsion force can be su!ciently

reduced so that the attractive interaction becomes dominant then the particles

colliding with one another will stick together to form quasi-stable aggregates.

The charge on a particle, together with the equal and opposite charge in

a suspension, comprise the electrical double layer (EDL). Hogg et al. [H17]

showed that the interaction of EDL between two dissimilar particles, under the

condition of low surface potential and small thickness of EDL compared to the

particle size, is given, under the assumption of a constant surface potential, by



+ #

)

4(e

+ e

)

+ #

)

1 + exp([k)

1  exp([k)

+ ln(1  h

2[k

)

(3.116)

where e

and #

are the radii and surface potentials of particles, respec-

tively, 

the dielectric constant of the medium, [ the Debye-Hückel reciprocal

length parameter, and k their distance between the surfaces of the interacting

particles. [ is related to the ionic strength L

of the medium by the expression

[ =

8h

1@2



(3.117)

In the case of identical spherical particles eq. (3.116) reduces to



ln(1 + h

[k

) (3.118)

When the thickness of EDL is not small (i.e. when e[ ? 1), Y

can be expressed

in a simple approximate form:

= 

exp([k)

(2e + k)

(3.119)

Similarly, the London - van der Waals interaction for dissimilar spheres can

be written as

= 

+2(e

+ e

+2(e

+ e

)k +4e

+2(e

+ e

+2(e

+ e

)k +4e

(3.120)

212 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

where D is the Hamaker constant. For identical particles, eq. (3.120) simpliﬁes

= 

+4ek

+4ek +4e

+ln

+4ek

+4ek +4e

(3.121)

When the London forces operate over distances comparable to or larger than

0.1> where  is the wavelength of the intrinsic electronic oscillations of the

atoms, the interaction is reduced by the retardation eect [S43, R15].

Flocculation in a magnetic ﬁeld

When a suspension of colloidal particles is placed in an external magnetic ﬁeld

there is, in addition to the interactions considered so far, a magnetic dipolar

interaction, described by eq. (3.110).

In the strong interaction limit [C14, S34], the mean magnetic interaction

energy can be written as

= 

8



9(k + e

+ e

)

(3.122)

where 

are volume magnetic susceptibilities of particles and K is the magnetic

ﬁeld strength.

In agreement with the assumption of the DVLO theory that double layer and

van der Waals forces are independent and additive, Svoboda [S34] supposed that

the total energy of interaction of colloidal particles in the external magnetic ﬁeld

is, therefore, given by the sum of electrical double-layer energy, van der Waals

interaction energy and magnetic dipolar interaction energy

= Y

+ Y

(3.123)

Generally, the total interaction energy Y

can be either attractive or repulsive

depending on the sign of the individual terms on the right-hand side of eq.

(3.123). Electrical double layer energy is repulsive for interaction of particles

of the same material, while it can be attractive or repulsive for interaction of

unlike materials (e.g. between a valuable mineral and gangue).

The van de Waals interaction is usually attractive, but under certain condi-

tions [V2], determined by relative values of the Hamaker constant of the inter-

acting particle and the medium, the interaction becomes repulsive. Magnetic

interaction can be either attractive or repulsive, depending on the signs of the

magnetic susceptibilities of the interacting particles. It is clear that interac-

tion between diamagnetic and paramagnetic particles is repulsive. On the other

hand, magnetic interaction of identical materials, whether paramagnetic or dia-

magnetic, is always attractive. Magnetic ﬂocculation of diamagnetic particles

is, therefore, possible, in principle.

A curve of the total interaction energy Y

as a function of the interparticle

distance is shown in Fig. 3.38. A potential barrier Y

max

separates two minima: a

closer and deeper primary minimum and a more distant and shallow secondary

3.6. MAGNETIC FLOCCULATION 213

max

min

Figure 3.38: The general form of the total potential energy for a pair of inter-

acting colloidal particles of radius e=

minimum of depth Y

min

. The individual potential energy contributions are

aected by the properties of a suspension in dierent ways. The magnitude and

range of the electric double layer interaction Y

are sensitive to # and [, which

are controlled by the solution conditions (i.e. ionic strength and pH), whereas

the van der Waals and magnetic contributions Y

and Y

are independent of

these parameters.

The magnitudes of the individual potential energy components have a dier-

ent dependence on the particle size: Y

is proportional to the cube of the radius

while Y

and Y

are much less sensitive to particle size as they are proportional

to the ﬁrst power of particle size.

The height of the barrier determines the stability of a suspension. If the

barrier is very low, there is nothing to prevent contact between the particles,

and ﬂocculation will take place at a rate governed by the frequency with which

the particles collide. If there is a secondary minimum in the curve of a potential

energy, particles can ﬂocculate into this potential well. In this instance, the

potential barrier need not be suppressed. Flocs thus formed will have a rather

loose structure as a result of large interparticle distance at which the secondary

minimum is created.

The existence and position of the primary and secondary minima can be

aected by changes in the magnetic ﬁeld, the magnetic susceptibility of the

particles and the pH value of the suspension. The total interaction energy

is shown in Fig. 3.39 as a function of magnetic induction and particle

size. In region I the potential barrier is present and the secondary minimum is

su!ciently close, while in region II the secondary minimum is too distant. In

regions III and IV, in which, owing to high magnetic induction, the potential

214 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.39: Regions in which primary and secondary minima are found in the

(a,B) plane, for hematite (# =30mV,[ =10

1

), d - particle

radius, E - magnetic induction (after [S36, S37]).

barrier is suppressed and the total interaction becomes attractive, and particles

can ﬂocculate into the primary minimum.

It follows from Fig. 3.39 that, for any particle size, a value of the magnetic

induction can be found at which the potential barrier disappears. Wang et

al. [W17] calculated the potential energy of interaction of particles of hematite

and chromite in a magnetic ﬁeld. Typical curves shown in Fig. 3.40 illustrate

that at moderate magnetic ﬁeld strength the potential barrier is eliminated and

ﬂocculation into the primary minimum could take place. A deep secondary

minimum is created at even lower magnetic ﬁeld.

A threshold magnetic ﬁeld, at which the ﬂocculation into the primary min-

imum starts, can be determined provided that the total potential energy of

interaction Y

is negative [S34], or that the stability ratio Z

, which is given by

eq. (3.124), is smaller than, or equal to, unity [S35]. The stability ratio is the

ratio of the initial rate of ﬂocculation to that which would occur if the process

was only diusion controlled (in the absence of an energy barrier) [P11].

exp

(3.124)

3.6. MAGNETIC FLOCCULATION 215

Figure 3.40: The potential energy as a function of interparticle distance for 4

m hematite particles, [ =10

1

(adapted from Wang et al.

[W17]).

Figure 3.41: Threshold magnetic ﬁeld, at which ﬂocculation into a primary min-

imum begins, as a function of particle size (adapted from Svoboda

[S35]).

216 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

The values of the threshold magnetic ﬁeld E

, at which magnetic ﬂocculation

into the primary minimum begins, are shown in Fig. 3.41, as a function of

particle size, for various minerals. It can be seen that as the particle size and

magnetic susceptibility increase, the threshold magnetic ﬁeld decreases.

For ﬁne particles with a volume magnetic susceptibility smaller than 10

3

(SI), the threshold magnetic ﬁeld is too high for magnetic ﬂocculation to be prac-

ticable. This is particularly true for the magnetic ﬂocculation of diamagnetic

materials, which require a magnetic induction of the order of tens of hundreds

of Tesla.

Flocculation in the secondary minimum

As shown in Fig. 3.39, there is a secondary minimum (SM) in addition to

the primary minimum in regions I, II and III. Although in regions I and II the

potential barrier prevents particles from ﬂocculating into the primary minimum,

particles can ﬂocculate into the secondary minimum. The secondary minimum

in region II is too distant from the origin, and consequently, the aggregates will

be packed too loosely. Furthermore, the secondary minimum is too shallow and

the stability of the ﬂocs will be low.

However, in region I, the secondary minimum can be deep enough and close

enough to the origin to render it suitable for practical ﬂocculation.

The mere existence of a secondary minimum does not imply an observable

destabilization of a suspension. The other condition to be satisﬁed is su!cient

depth of SM; the analysis indicates [S36] that, for Y

min

A 1kT(wherekTisthe

unit of thermal energy), the value of the stability ratio Z

for ﬂocculation in

a secondary minimum, as given by eq. (3.125), will be close to unity, and the

ﬂocculation into SM will be rapid.

exp(

)

1  exp(

min

)

(3.125)

where v

is the position of the secondary minimum.

The threshold magnetic ﬁeld E

required to commence the ﬂocculation into

SM can be determined from the condition Z

=1. The dependence on a particle

radius of threshold magnetic ﬁeld E

in SM, which was determined from the

condition of rapid ﬂocculation (Z

=1), is shown in Fig. 3.42.

A comparison of these values with those of the threshold magnetic ﬁeld E

for

ﬂocculation into the primary minimum (Fig. 3.41) indicates that the magnetic

ﬁeld needed to induce rapid ﬂocculation in SM is substantially lower. This

conclusion has been conﬁrmed by detailed calculations by Wang et al. [W17],

an example of which is presented in Fig. 3.40. Experimental conﬁrmation

of the magnetic ﬁeld-induced ﬂocculation is depicted in Fig. 3.43. Figure 3.44

illustrates the experimentally determined onset of rapid ﬂocculation of hematite

and chromite at threshold magnetic ﬁeld of about 0.5 T [W17].

3.6. MAGNETIC FLOCCULATION 217

Figure 3.42: Threshold magnetic ﬁeld for ﬂocculation into a secondary mini-

mum, as a function of particle diameter (adapted from Svoboda

[S35]).

Figure 3.43: The magnetic ﬁeld-induced ﬂocculation of hematite particles at var-

ious values of magnetic induction. pH = 7.8, #  45 mV, g

3.9 m (adapted from Wang et al. [W17]).

218 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Figure 3.44: The onset of the magnetic ﬁeld-induced ﬂocculation of hematite

=3.9m) and chromite (g

=3.6m) particles (adapted

from Wang et al. [W17]).

We have seen that under suitably chosen conditions, the interaction potential

between a pair of colloidal particles can develop a signiﬁcant (> 1 kT) secondary

minimum, while maintaining a large (AA 1 kT) primary maximum. Although

the primary maximum will prevent irreversible coagulation into the primary

minimum, the particles can undergo reversible ﬂocculation into the secondary

minimum.

The above treatment of the ﬂocculation process is applicable only to colloidal

particles. For particles greater than the colloidal range, the thermal motion and

the surface forces play only a limited role, and the process of ﬂocculation will

be governed mainly by the gravitational and hydrodynamic forces.

The kinetics of ﬂocculation

It has been shown that the stability of suspensions depends on the magnitude of

the potential barrier and on the existence and depth of the secondary minimum.

In addition, however, determination of how quickly ﬂocculation will take place

is necessary.

Fuchs [F13] showed that the time taken for the number of particles to be

halved is given by the relation

1@2

3Z

4nWQ

(3.126)

where  is the viscosity of the medium and Q

is the initial concentration of the

particles.

3.6. MAGNETIC FLOCCULATION 219

Figure 3.45: Dependence of the stability ratio Z

(as log Z

) on magnetic in-

duction for assorted sizes of hematite particles (adapted from Svo-

boda [S35]).

It can be seen that the half-time of ﬂocculation is directly proportional to

the stability ratio and indirectly proportional to the initial concentration of the

particles. To achieve a reasonably short ﬂocculation time, low values of Z and

high values of Q

must be used.

In chemical ﬂocculation, the values of Z smaller than 10

1

are reached with

di!culty [V1]. This signiﬁcantly restricts the possibility of accelerated ﬂoccu-

lation. On the other hand, it was shown by Svoboda [S35], that in magnetic

ﬂocculation, the values of the threshold magnetic ﬁeld determined from the con-

dition Z = 1 for rapid ﬂocculation, allow reasonably short ﬂocculation time to

be achieved. As is shown in Fig. 3.45, the values of Z

, of the order of 10

3

can be achieved at reasonably low magnetic ﬁelds so that the ﬂocculation time

into the primary minimum can be realistically short.

Magnetic-hydrophobic ﬂocculation

In order to explain situations in which the original DLVO theory failed, numer-

ous ”non-DLVO” forces have been invented. A variety of forms of long-range

forces such as hydrophobic/hydrophilic interaction have been introduced. For

example, surface-active agents can adsorb at the mineral-water interface and

render the solid hydrophobic. Particle-particle adhesion is favoured by the at-

traction between the hydrophobic coatings on dierent particles and the resul-

tant reduction in the area of hydrophobic mineral-water surface.

A critical review of such extensions of DLVO theory to include ”non-DLVO”

interactions was published by Ninham [N6] who argued that confusion created