Fundamentals of Suspensions & Dispersion's

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Process Engineering Guide:
GBHE SPG PEG 300.2

Fundamentals of Suspensions &
Dispersions
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Process Engineering Guide:

Introduction: Fundamentals of
Suspensions & Dispersions

CONTENTS

0

INTRODUCTION

1

NATURE OF SURFACE FORCES

2

STABILITY AND THE STATE OF DISPERSION OF
SUSPENDED PARTICLES

3

MECHANISMS OF FLOCCULATION

4

STRUCTURE OF FLOCCULATED SUSPENSIONS

4.1
4.2

Dilute Suspensions
Concentrated Suspensions

5

STRUCTURE OF STABLE SUSPENSIONS OF
MONODISPERSE PARTICULATES

6

SUMMARY OF STRUCTURES

7

PARTICLE PACKING


8

RHEOLOGY

8.1
8.2

Basic Rheological Concepts
Colloidally Stable Suspensions
8.2.1
8.2.2
8.2.3
8.2.4
8.2.5

8.3

Rheology of Flocculated / Aggregated Systems
8.3.1
8.3.2
8.3.3
8.3.4
8.3.5

8.4

Spherical Particles of around 1 µm
Effect of Particle Size Distribution
Effect of Particle Shape
Submicron Particles
Very Concentrated Systems

Dilute Flocculated Systems
Concentrated Flocculated Systems
Time and History Effects
Slip and Fracture
Behavior of Flocculated Cakes in Compression

Summary of Rheology
Deflocculated Suspensions
Flocculated Suspensions

9

SEDIMENTATION OF SMALL PARTICLES

9.1
9.2
9.3
9.4

Very Dilute Particles
Concentrated Systems
Polydisperse Systems
Flocculated Systems

10

ELECTROKINETIC BEHAVIOR

11

A NOTE ON MAKING DISPERSIONS AND
SUSPENSIONS


12

References

13

Figures

Fig 1a
Fig 1b
Fig 1c

Potential Energy Diagram for Steric Stabilization
PE Diagram for Electrostatic Stabilization
Combined Stabilization

Fig 2&3

DIFFERENT TYPES OF FLOCCULATION MECHANISM IN
WHICH POLYMERIC SPECIES ARE INVOLVED

Fig 4

Rheological Behavior

Fig 5

Relative Viscosity versus Volume Fraction for Polystyrene
Spheres in Water

Fig 6

Time Dependent Flow Behavior of Very Concentrated Suspensions

Fig 7

Flow curves for Flocculated Dispersions


0

INTRODUCTION

It is common when discussing solid-in-liquid suspensions and dispersions to
distinguish between various particle-size regimes, for example to classify
systems as either coarse suspensions or slurries, or fine suspensions or colloidal
dispersions. No such distinctions will be made here and the term suspension is
used as a cover all term.
This process engineering guide and indeed the manual is, however, strongly
biased towards suspensions containing particles with a characteristic dimension
or dimensions less than about 10 µm. The reason for this is that it is in this
regime that surface forces influence the properties of suspensions, thus where
suspensions have the most complex and varied properties and where they pose
the greatest processing difficulties. This process engineering guide is thus
primarily, but not exclusively, to do with colloidal suspensions.
The most important properties from the point of view of their processing behavior
are rheological and transport properties. These are very dependent upon the
state of dispersion of the particles, which in turn depends upon the forces of
interaction between the particles. This process engineering guide provides a brief
account of surface forces and their role in determining the properties of
dispersions and suspensions.
The scheme used is as follows.
First the fundamentals of interparticle forces are discussed and their role in
determining the stability or state of dispersion is described. Having established
why suspended particles may either be well dispersed or aggregated
(flocculated>, the various mechanisms of flocculation are summarized.
The structure of stable and flocculated dispersions is then considered. This is
followed by a discussion of the rheological and transport properties of
suspensions and finally a few comments are made on methods of making
dispersions and fine particles.


1

NATURE OF SURFACE FORCES

In order to create a solid-in-liquid suspension a large amount of solid-liquid
interface has to be created. This requires work to be done as the interfacial
energy is almost invariably non-zero and positive. The disperse state is thus a
high-energy state relative to bulk matter. On a microscopic scale the excess free
energy is manifest as a force of attraction between the particles, the so-called
van der Vaals or dispersion interaction. This, in the absence of any stabilizing
influence, will cause particles to aggregate. The van der Vaals attraction typically
has an effective range of the order of 10 nm and is strong in contact, the energy
of cohesion being orders of magnitude larger than the thermal energy of the
particles. For spherical particles of radius a the interaction potential and force
are given very approximately by:

VA
FA

=
=

- aA/12 (R – 2a)
aA/12 (R – 2a)2

(1)
(2)

respectively with R the centre-to-centre separation of the particles and A a
material constant (the Hamaker constant) which depends upon the spectroscopic
properties of the particles and the medium in which they are dispersed. A is
typically of the order of 2 kT In units of thermal energy (or – 10-20 J). Given this, it
is evident that the attraction is very large compared to the thermal energy of the
particles when the distance of separation approaches something of the order of a
molecular diameter. The van der Vaals attraction is thus large enough to cause
irreversible aggregation, and, In the case of soft or sparingly soluble particles,
sintering or fusion. For this reason fine suspensions can only be made when
there is some source of particle stabilization available that provides an
interparticle repulsion which offsets the effect of the van der Vaals attraction.
Only two types of repulsion are known having sufficient strength and range to
oppose the van der Vaals attraction. These are an electrostatic repulsion that can
arise when the particle surfaces bear an ionic charge and the so-called steric
repulsion that arises when the particles are coated with a layer of solvated chain
molecules (usually Introduced by adding a non-ionic or polymeric surfactant to
the system, although biological cells may be naturally sterically-stabilized>.
Electrostatic stabilization can arise when there is Ionic charge separation at the
solid-liquid interface. Solid particles of most types will acquire an ionic surface
charge in most polar media. However, electrostatic effects are rarely strong
enough to provide a source of stabilization unless the medium is aqueous. The
origin of the surface charge may be one or more of the following:

1)

Adsorption of potential-determining ions at the surface of crystals, eg M+ or
X- onto a metallic salt MX.

2)

Adsorption of non-potential determining inorganic ions, particularly polyions such as polyphosphates, silicates, etc.

3)

Dissociation or protonatlon or surface-groups, eg -COOH, -NH2, -SO3H etc
on latex particles and biological cells and M-OH on metal-oxides.

4)

Isomorphous substitution into crystal lattices to give a non-stoichiometric
crystal, eg replacement of A13+ by Mg2+ or Fe2+ in aluminosilicates.

5)

Adsorption of ionic surfactants such as sodium dodecyl benzene
sulfonate or cetyltrimethyl ammonium bromide (such molecules
are commonly used as dispersants).

The fixed charge is compensated by an equivalent number of counterlons
that tend to cluster round the particle as a result of the high electric field near the
interface (of order 108 Vm-1). This diffuse layer of counterions is referred to as the
diffuse part of the electrical double layer. Any two or more particles experience a
mutual screened Coulomb repulsion of the approximate form:

where εrε0 is the absolute permittivity of the medium and εr is its relative
permittivity or dielectric constant; φE is the electric potential at the edge of the
fixed-charge layer and is typically of order 150 mV for highly charged surfaces (it
spay be measured by electro-kinetic techniques if necessary) ; the screening
parameter U depends upon the ionic strength of the medium and where the latter
is a solution of a symmetrical electrolyte <eg NaCl, or HgSO4, or for
that matter H+ and OH-).

More generally

is proportional to

Ionic strength.


The range of the electrostatic repulsion thus depends upon the ionic composition
of the medium. For 1:1 electrolytes the range can be judged from the
following table.
Electrolyte Concentration
10E-5 mol dmE-3
10E-3
10E-1

Effective Range in Water
(= ~ 4 / Ҝ)
400 nm
40 nm
4 nm

Steric repulsion arises when the particle surfaces are coated with a layer of
solvated molecules of high molecular weight. Suitable species are flexible
polymers and surfactants with oligometric or polymeric head groups. Natural
systems such as biological cells and particles in natural waters may be
adventitiously sterically stabilized, otherwise particles will not normally be
sterically stabilized unless a suitable species has been deliberately added as a
dispersant. Commonly used steric stabilizers include
1)

SOLSPERSE agents, for example those with representative formula

where X is either an acidic or basic group (in the Lewis sense). These are used
to disperse polar particles in organic solvents with the X group anchoring the
molecule to the surface, and the poly(l2-hydroxystearate) chain (PHS) acting as
the stabilizer.
2)

Ethoxylated alkanols and alkyl phenols, eg
CH 3 - (CH2)13 - (CH2CH2O) n - CH3.

These are used as dispersants for hydrophobic particles in water with the longchain alkyl groups acting as the anchor and the ethoxylated chain acting as the
stabilizer.


3) Partially-hydrolyzed polyvinylacetate (*poiyvinylalcohol”)

Molecules of this type with n/m -10 are used as dispersants in water, The acetate
moieties presumably anchor the polymer leaving loops and tails of
polyvinylalcohol penetrating Into solution.
4) Graft polymers, eg

A is used to prepare dispersion polymers in aliphatic hydrocarbons; B is used as
a dispersant in aqueous media.
Steric repulsion arises because of the reluctance of the solvated chains to
interpenetrate or compress as particles collide. The solvated layers in effect
provide a mechanical barrier that prevents close-approach of the particle
surfaces.
The sterlc force and the electrostatic force can be strong enough to offset the
effect of the van der Waals attraction under certain circumstances. The
conditions for stability (le good dispersion) are discussed in the next section.


2

STABILITY AND THE STATE OF DISPERSION OF
SUSPENED PARTICLES

Whether or not suspended particles are stable or unstable and so remain
dispersed or undergo aggregation, depends upon the form of the net Interaction
potential

VTotal = VA + Vi + Vε

(5)

(Vε being the steric potential>.
In the case of particles coated with a steric barrier VS rises rapidly as the layers
come into contact. The conditions for stability are:
1)

The barrier thickness should be greater than the range of VA.
In practice the adsorbed layer should thus be -10 nm or more.

2)

The adsorbed layer should be strongly bound or adsorbed on the
particles.

3) The medium should be a good solvent for the stabilizing chains.
As a result of condition 3, sterically-stabilized particles can be flocculated if the
solvency of the medium is reduced. This can variously be done by adding
miscible non-solvents; adding competing solutes <eg salts in water; these will
often “salt-out” adsorbed polymers If added in sufficient quantity), or by changing
the temperature. On the other hand if conditions 1 to 3 are obeyed, then VTotal is
repulsive at all distances of separation. Because of this, sterically-stabilized
dispersions and suspensions can be considered to be thermodynamically stable
provided the anchoring of the stabilizer is totally irreversible.
Electrostatic stabilization differs from steric stabilization in that electrostatic
dispersions are at best kinetically-stable. This is because the electrostatic
repulsion is never large enough to dominate the van der Waals attraction at very
close distances of approach. The curve of VTotal (Figure 1) thus possesses a
maximum and it is the height of this barrier which determines the stability or state
of dispersion. In rough terms the energy barrier VM can be regarded as being
analogous to chemical activation energy.


The height of the energy barrier depends upon the Ionic strength and
composition of the medium, and the surface potential or surface charge density
of the particles < the two are roughly proportional for surface potentials of 50 mV
or less, generally speaking the potential at the fixed charge boundary is of this
order). It is generally considered that a surface potential of + 20 mV or more is a
necessary but not sufficient condition for stability, this typically corresponds to a
fixed-charge density of the order of one charge per (10 nrn)2 or more. The height
of the energy barrier can be reduced and suspension destabilized by
1)

neutralizing the surface-charge e.g. by changing pH in the case of metal
oxide particles;

2)

screening the repulsion by adding electrolytes.

In the latter case there may, in general, be two contributing mechanisms:
1)

Addition of electrolytes Increases Ҝ (cf equation (3)) and reduces the
range of the repulsion.

2)

Counterions may reduce the effective surface charge by specific
adsorption at the Interface.

However, the first mechanism usually tends to be controlling. The amount of
electrolyte required to reduce the energy barrier to the point where rapid
aggregation occurs is referred to as the critical coagulation concentration (ccc).
The ccc depends upon the valency of the Ions present and particularly the
valency of the counterions. Typical values for negatively charged particles are
shown in the following table.
Electrolyte
NaCl
KCl

ccc
0.1 to 0.5 molar
0.1 to 0.5 molar

BaCl2
MgCl2

0.002 to 0.015 molar
0.002 to 0.015 molar

La(N03)3 @ pH 3 i.e., La3+)

10(-4) to 5x 10 (-4)


These data clearly demonstrate the effect of counterlon valency. Notice that
electrostatic stabilization fails In the presence of only modest concentrations of
divalent and trivalent ions.
Electrostatically-stabilized suspensions are also sensitive to mechanical action
(agitation and shear) unless the particles are small (< 0.5 µm), this is particularly
the case when the dispersion is concentrated and it is usually difficult to prepare
high solids dispersion relying on electrostatic stabilization alone. A measure of
steric stabilization is normally required. In this context it is appropriate to note
that the maximum in the potential energy curve lies quite close to the surface (of
order 1 to 2 nm). Thus a steric barrier much thinner than that required to impart
stability to an uncharged system can be sufficient to improve the stability of
concentrated dispersions in water (cf Figure 1c). This explains the observation
that particles dispersed with anionic surfactants and bulky polyions such as
polyphosphates, low molecular weight polyacrylic acid salts, sodium Dispersal+,
and the like are more + Sodium Dispersol is the sodium salt of sulphonated
naphthalene-formaldehyde condensate of representative formula

stable than simple charged particles with equivalent Ionic charge densities, the
presence of a bulky charging species imparting some measure of steric
stabilization. With the aid of such agents it is possible to prepare concentrated,
stable dispersions.
Example: Emulsifier-free polystyrene particles with particle diameters between
0.2 and 2 pm can readily be made In water by emulsion polymerization. This is
typically done at 10% v/v solids and the resulting particles are stabilized by SO4and COO- groups at the surface. Attempts to concentrate such lattices, e.g. by
rotary evaporation at reduced pressure, almost invariably result In coagulation.
However if sodium DISPERSOL Is added they can be concentrated without
difficulty.


3

MECHANISMS OF FLOCCULATION

Two mechanisms of flocculation or aggregation have been alluded to above ; the
strong aggregation that occurs in the absence of any stabilizing barrier as a
result of the universal van der Waals force between particles, and the incipient
flocculation that occurs when the suspension medium of a sterically-stabilized
suspension is converted by some means into a non-solvent for the stabilizing
chains.
The broad features of these and other types or mechanisms of flocculation are
summarized in the table below and in Figures 2 and 3. In each case in the table a
judgment has been made as to the strength of the cohesion between the
particles. This is a little arbitrary but by "strong” is meant of comparable strength
to van der Waals flocculation, which, generally speaking, is totally irreversible.
By "weak" Is meant weak by comparison to van der Waals flocculation. In most
instances the origin or mechanism of the flocculation is either self-evident or to
some extent explained by the information given, depletion flocculation, however,
requires further explanation and this is given following the table. Otherwise,
flocculation mechanisms are discussed in more detail in reference 1 and the use
of flocculants in “Filtration”, Section 3.7.


Note that certain of these mechanisms require the presence of a soluble or
adsorbed macromolecular species or an immiscible liquid. Depletion flocculation
is a non-specific form of flocculation that can occur non-adsorbing, soluble
macromolecules or very small particles are added to the suspension medium.
The effect arises because the centre of mass of a polymer coil or small particle of
diameter d is excluded from a layer of thickness d/2 close to the surface of a
suspended particle, as a consequence two or more suspended particles which
happen to be closer together than a distance d (surface-to-surface separation at
point of closest approach) are separated by a zone depleted by macromolecules
or small particles. The osmotic stress exerted by the remainder of the suspension
medium on the depletion zone causes solvent to migrate from between the
particles, causing, In effect, a force of attraction. Depletion flocculation is of
potential interest as a means of controlling rheology and structure in surfacecoatings formulations and the effect is discussed in more detail In “Filtration”, In
this context.

4

STRUCTURE OF FLOCCULATED SUSPENSIONS

4.1

Dilute Suspensions

Sufficiently dilute flocculated suspensions consist of discrete aggregates or flocs
built up in most instances by diffusion-controlled aggregation (DCA). DCA tends
to give a rather open structure, and if the forces binding the particles together are
strong so that there is no tendency for rearrangement of the particles during or
after the Initial aggregation, open floes form.
The mean number of nearest neighbors in contact with any one particle being
typically of order 3.
If the flocculation is weak then more compact objects are formed. The reason for
this Is straightforward. Maximizing the number of contacts between particles
minimizes the energy of a floe (entropic effects being small usually) and so If the
flocculation is at all reversible rearrangement and condensation of the floc
structure will occur spontaneously. Close-packed clusters are thus formed when
the flocculation Is very weak, as for example depletion flocculation can be. Closepacked clusters can also be formed by agitation of strongly-flocculated
suspensions under certain conditions.


4.2

Concentrated Suspensions

In concentrated suspensions there Is no room for discrete flocs and above some
critical concentration a continuous network will form throughout the available
space. The critical concentration, usually called the gel-point, generally increases
with particle size and with decreasing strength of flocculation, it being easier to
form a continuous network from open floes than it is from condensed clusters.
In the case of spherical particles the gel point can be anywhere between about
0.1 and 0.3 typically when expressed as a volume fraction, 0.1 being
characteristic of strongly flocculated submicron particles and 0.3 being more
typical of cluster percolation In weakly flocculated systems. Gel-points as low as
0.01 can be observed when the particles are highly acicular. The rheological
properties of suspensions change markedly at the gel-point. If the flocculation is
strong a yield stress develops and the viscosity increases rapidly.
Stable sediments and coherent filter cakes form above the gel-point and so the
location of the latter is a very important processing consideration.

5

STRUCTURE OF STABLE SUSPENSIONS OF MONODISPERSE
PARTICULATES

Stable suspensions can generally be thought of as having a disordered structure.
However, if the repulsion stabilizing the particles happens to be very long-range,
ordering of the particles can occur. Specifically the repulsion has to be on the
same scale as the mean spacing between particles in the suspension, this being
of order 2a(Ø-1/3 - 1) for spherical particles, where Ø is the volume fraction of
solid and a is the particle radius. For the repulsion to be sufficiently long-range
the particles have to be small (a < 0.5 pm) and the ionic strength of the medium
has to be very low (aqueous, electrostatic suspensions) or the steric barrier very
thick. Spherical particles normally assemble onto a face-centered cubic or bodycentered cubic array.
The colloidal crystallites formed can be quite large (a millimeter or more) but
typically ordered suspensions are polycrystalline. When the particles have a
radius of the order 0.1. to 0.5 pm ordered suspensions show a characteristic
iridescence or opalescence.


6

SUMMARY OF STRUCTURES

Stable suspensions normally have a disordered, pseudorandom structure;
ordering can, however, occur in concentrated systems. Flocculated suspensions
variously consist of suspensions of floes, suspensions of clusters, floe networks
and cluster networks depending upon particle size, particle concentration,
strength of aggregation and history.

7

PARTICLE PACKING

There is at present a widespread Interest in very concentrated suspensions in
the form of ceramic green bodies, highly-filled resins, modified cements, high
solids coatings and so on. The absolute limit of solids content that can be
achieved Is clearly bounded by the point where the particles close-pack. Various
theoretical close-packings for monodisperse spheres have the following
densities.
face-centred cubic
rhombohedral
body-centred cubic
random
primitive cubic

0.74
0.74
0.68
0.636
0.524

Sedimentation experiments suggest that particles with sizes of the order of 1 pm
or more probably pack randomly since densities of the order of 0.6 are achieved.
Smaller particles tend to crystallize and higher densities of order 0.7 can be
achieved. The static packing fraction LSPF) is an extreme upper bound. Another
point of Interest is the point where, by extrapolation, the viscosity appears to
diverge. This however turns out not to be a single point but depends upon the
rate of flow.
At very slow flow-rates (see section on rheology) the point of divergence p
appears to correspond roughly to the static packing fraction being typically of the
order 0.6 for larger particles (a 2 0.5 pm> and 0.68 for smaller particles.
However, very concentrated suspensions tend to be dilatant and
'solidify" under the influence of shear and this means that suspensions become
unhandleable well before p is reached. Polydispersity increases the level of
solids content that can be achieved and volume-fractions of the order of 0.8 can
be achieved with multimodal distributions.


8

RHEOLOGY

The rheology of suspensions is one of the key aspects influencing their
processing behavior. Suspension rheology Is very variable, suspensions can
occur as thin fluids, soft gels, thick pastes and weak solids depending upon
particle size, shape and concentration, the state of dispersion, and so on. In this
section the broad features of suspension rheology will be illustrated using
specimen results and data obtained for well-characterized suspensions. A
distinction will be made between Colloidally-stable and flocculated suspensions
since these tend to show very different rheology.
8.1

Basic Rheological Concepts

The concern here will be primarily with the shear flow behavior of
suspensions. Simple molecular liquids in shear flow obey Newton’s law.
This states that the shearing stress Q required to maintain a uniform
laminar shear-flow characterized by a velocity gradient or shear-rate Ɣ Is
proportional to Ɣp the constant of proportionality being the shear viscosity
ᵑ. Two-phase fluids such as suspensions generally show more complex
behavior and may have a variable Viscosity. The viscosity of suspensions
typically depends upon shear stress or shear-rate (whichever is chosen to
be the independent variable characterizing the strength of the flow) and
may also depend upon time and previous history. Suspensions are often
found to be shear-thinning, that is have a viscosity that decreases with
increasing shear-stress or shear-rate, very concentrated suspensions can
however show the converse behavior, shear thickening or dilatancy.
Various types of shear-rate dependent behavior commonly encountered
are illustrated in Figure 4. Type 1, where the viscosity varies between two
extreme Newtonian limits, is typical of weakly flocculated suspensions and
moderately concentrated stable suspensions (although in the latter case
the difference between ᵑ(0) and rho Is often ᶯ oo small as to be of little
consequence). Type II behavior, where ᵑ(O) IS essentially "Infinite" and
the material, in effect, shows a yield stress, is characteristic of strongly
flocculated suspensions, or more generally concentrated suspensions in
which there are strong interparticle Interactions. Types III, IV and V, VI are
typical of very concentrated suspensions; Types III, IV and V are shown
by concentrated, stable suspensions of ever-increasing concentration. In
the case of flocculated suspensions the tendency to shear-thicken at very
high concentrations is often masked by the shear-thinning that flocculation
tends to introduce and something like type VI can be observed.


All of these various types of behavior reflect a dependence of the structure
of the flowing suspension on shear-stress or shear-rate. Thus, roughly
speaking, flocculated suspension shear-thin because the flocculated
structure breaks down with increasing strength of flow. Concentrated
suspensions shear thicken because the particles which have to move
cooperatively for flow to occur lock together if forced to flow too rapidly.
8.2

Colloidally-Stable Suspensions
8.2.1 Spherical Particles with diameters of the order of 1 µm
or more
Stable suspensions of spherical particles with diameters > 1 µm are
found to be Newtonian provided the volume fraction of the solid
phase ɸ is less than 0.5 or so. In this concentration regime the
viscosity depends only upon volume fraction and the nature of the
particle size distribution, and not on shear-rate or absolute particle
size. Viscosity/volume-fraction curves for monodisperse particles
conform to the Dougherty-Krieger equation:

where ᶯ o. is the viscosity of the suspending medium, [ ᶯ] Is the
intrinsic viscosity of the particles and p is a packing constant. It is
expected from theory that ᶯ should be 2.5 for spheres and it has
likewise been argued that p should have a value of 0.6. A fit of
equation (6) to typical data for monodisperse latex particles is
shown in Figure 5. A best fit is obtained using [ ᶯ] = 2.6 (determined
independently > and p = 0.585. Inspection of other available data
on spherical and roughly-spherical particles show that viscosity
concentration curves can usually be fitted using p values In the
range 0.6 i 0.3 and [ ᶯ] values in the range 2.75 + 0.25. By way of
illustration this means that simple suspensions of monodisperse
spherical particles have viscosities only - 12 to 20 times that of the
suspension medium at a volume fraction of 0.5, and only - 2.5 to 4
times at 0.3. Stable suspensions of particles in thin fluids are thus
fluid until very concentrated.


At volume fractions of the order 0.5 or more monodisperse
suspensions start to show shear thickening (cf Figure 4). At volume
fractions less than - 0.54 this tends to be manifest as a continuous
increase of viscosity with shear-rate (or stress) above some critical
shear-rate (or stress). Below the critical point the viscosity
continues the trend shown at lower concentrations (equation (6)),
and it has been suggested that the critical point is actually a critical
level of stress which is dependent upon particle size but
independent of concentration. If this idea is correct then the critical
shear-rate should be inversely proportional to the low-shear
viscosity of the suspension. At higher volume-fractions (0.54 < ɸ <
p> there tends to be some initial shear-thinning followed by
discontinuous shear thickening. (NB Shear thickening is often
called "dilatancy".) The
Viscosity increase can be very large, to the point where shear
thickening makes the suspension effectively unhandleable. As a
rule of thumb ɸ = 0.55 Is a practical handling limit for monodisperse
particles; pipe-flow can sometimes be achieved at higher
concentrations as a result of wall-slip and plug flow but more
complicated and/or homogeneous deformations and flows become
very difficult. Dilatancy or shear-thickening is known to be time
dependent and history-dependent when the particles are large. This
is illustrated in Figure 6 which is a schematic based on data for
glass spheres. Only after very prolonged gentle shearing does the
suspension show the pattern observed for micron-sized particles.
With less severe pretreatment the suspension shows quasiNewtonian behavior. This is probably due to wall-slip In the
rheometer and may not reflect the "true" behavior of the
suspension, which could well appear impossibly dilatant in a
different flow or In a practical context. As a rule of thumb,
monodisperse suspensions can become both difficult to handle and
difficult to characterize rheologically at volume fractions much in
excess of 0.55.


8.2.2 Effect of Particle Size Distribution
The effect of a distribution of particle size is generally to reduce the
viscosity relative to that of a monodisperse suspension at the same
volume fraction and to push the onset of dilatant or shear
thickening behavior to higher volume fractions. In effect, the
packing parameter p is increased. In the case of broad, continuous
distributions the effect of a distribution of sizes cannot easily be
predicted, the effect is however easily measured provided the
particles are denser than the medium since p corresponds to the
volume-fraction in a packed plug produced by prolonged
centrifugation of a concentrated suspension. (ND Segregation
tends to occur if this Is done too dilute.
As a rule of thumb, suspensions of broadly distributed particles are
commonly handleable at volume fractions up to - 0.65 or so rather
than 0.55. Even higher volume fractions can be achieved if the
distribution has discrete peaks or modes chosen so that smaller
particles .fit into the voids left by the larger particles. A method of
calculating p for multimodal systems has been given by Farris (see
“Filtration”), the theoretical basis of the method Is somewhat
dubious, based as it is on static packing considerations (p) is a
dynamic packing parameter, the precise meaning of which is not
yet entirely clear) but It appears to work reasonably well. Again, In
principle, p can be measured by centrifugation but it Is very difficult
to avoid segregation of sizes with discrete modes in the distribution.

8.2.3 Effect of Particle Shape
Rather less is known about the rheology of suspensions of particles
which depart significantly from spherical or roughly polyhedral habit
since very few well-characterized, Colloidally-stable suspensions
have been prepared and examined rheologically.
Light kaolinite clay is one of rather few well-characterized materials
to have been examined. Kaolinite particles are Irregular to
hexagonal lozenges with lateral dimensions typically of the order of
1 by 1 pm and a definite thickness of - 0.1 pm. Natural suspensions
of kaolinite in water are flocculated. Kaolinite particles can however
be deflocculated by the addition of suitable surfactants. The
viscosity-volume fraction curve for moderately concentrated
kaolinite suspensions (0.2 (ɸ ) 0.5) is found to be rather similar to

that shown for spheres In Figure 5, showing that there is no
particular penalty In rheological terms in having particles of platelike habit. Concentrated kaolin suspensions are also found to show
shear thickening behavior at similar volume-fractions to spherical
particles. The position for rod-like particles is less certain. A
researcher has however recently prepared some Colloidally-stable
polymer particles of rod-like shape with an axial ratio of - 3:l or so,
and a size of the order of 1 µm. Suspensions of these were fluid at
volume-fractions in excess of 0.5. These data and the data for
kaolinite suggest that there may be no particular penalty in having a
moderate degree of anisometry. However there is limited data
available In the literature which suggests that more extreme axial
ratios have a more pronounced effect. Kitano et al 1101 studied the
viscosity of suspensions of fibers dispersed in molten polymers, the
fibbers having axial ratios in the range a = 6 to 27. The data
were correlated with an equation of the form (cf equation (6)):

Thus the effect of axial ratio is to reduce the volume-fraction at
which the viscosity appears to diverge, This correlation looks
potentially very powerful but It needs to be confirmed by further
data. In particular it should not be used outside the range of a
indicated. Experiments on very long fibers have shown very
complicated behavior, thus it has been found that whilst the steady
state shear viscosity can be low because of fibre alignment, such
suspensions can show very high viscosities on start-up and in time
varying flows [11]. Given this and the observation that fibre
suspensions even when aligned can show very high extensional
viscosities, it Is always likely that very anisometric suspensions
will show very variable behavior In practical and time-dependent
flow-fields. Further aspects of the rheology of fibre-containing
suspensions, Including a discussion of the role of fibre compliance,
have been given in a review by Nills [12].


8.2.4 Submicron Particles
In order for suspensfons to be Colloidally-stable there has to be a
force of repulsion between the particles. As mentioned above this
may either be sterlc or electrostatic In origin. If the particles are
small then the thickness of the repulsive barrier, that is the
thickness of the steric barrier (δ say), or the thickness of the
electrical diffuse double-layer (which in effect is ~ 2/Ϟ – see
Section:. 2.21, may be significant compared to the particle size.
Should this be the case then the sphere of influence of the particles
will be significantly larger than their particle size. There is thus the
possibility that suspensions of low actual volume-fraction can
appear rheologically-concentrated as a result of overlap of their
sphere of influence. In order to assess the likelihood of this it is
useful to define an effective volume fraction.


as appropriate. In general a suspension might be said to be
rheologically-concentrated if ɸ eff > 0.5 or so. There are then two
possible regimes of behavior depending upon ɸ eff and ɸ. If Ϟa or
a/δ is large so that ɸ eff = ɸ then the rheological behavior of submicron particles is very similar to that described above for particles
with diameters in excess of 1 µm and the viscosity again is found to
conform to the Dougherty-Krieger equation. If however Ϟa < 20 or
so, or a/6 < 10 or so, so that ɸ < ɸ eff then the rheology can be very
different. As a first approximation to the viscosity ɸ eff, replaces ɸ in
the Dougherty-Krieger equation, this expedient however tends to
over-estimate the viscosity because the interaction between the
enlarged particles Is much “softer” than It Is between particles for
which a/δ or Ϟa is larger. Because of the “softness” of the
interaction the collision diameter of the particles depends upon the
force of collision and thus the strength of the flow. The viscosity
thus tends to depend upon, and decrease rapidly with, shear-rate.
A discussion of the rheology of “soft-sphere” suspension (more
generally “soft-particle suspensions”) is beyond the scope of this
review, a qualitative comparison of “soft” and “hard-sphere”
suspensions is however made In the following table.


It is probably fair to say that most sub-micron suspensions met in industry are
hard-sphere-like rather than soft-sphere-like, since for significant departure from
hard-sphere-like behavior to be seen one or more of the following conditions has
to be observed:
(a)

The particle has to be very small;

and/or (b)

the steric barrier has to be very extended, and so adsorbed chains
have to be of high molecular weight;

and/or (c)

the ionic strength of the medium has to be very low (In the case of
aqueous suspensions).

These conditions are rarely met by accident. “Soft-sphere” rheology is however
exploited In the surface-coatings industry where “microgel” particles are
sometimes used to modify the rheology of paints – these are small, sterlcallystabilized particles having an adsorbed layer of thickness comparable to the
core-particle size.

8.2.5 Very Concentrated Systems
Very concentrated, dilatant suspensions can be made to flow under
certain circumstances even though they may be better described as
weak solids rather than very viscous liquids. Thus very
concentrated suspensions can always be pushed down a pipe (but
not necessarily pumped) because of slip at the pipe-wall. Where
slip is observed in a simple confined flow, fracture is often observed
In free-surface flows and other more complex flows (for example
extrusion). In more graphic terms, very concentrated suspensions
in simple fluids have a rheology akin to that of wet sand typically.
However, suspensions of particles in Non-Newtonian media, for
example, In media modified by the addition of soluble
macromolecules or swelling clays of the type used as extrusion
aids In ceramics, can show much more plastic behavior. The
mechanism of action of processing aids of this type Is not at all
clear but It does appear that the processing behavior of very
concentrated suspensions is very dependent upon the rheological
properties of the suspension medium.

8.3

Rheology of Flocculated / Aggregated System

It is useful to distinguish between dilute suspensions comprising more or less
discrete flocs or aggregates and concentrated suspensions where the particles
join to form a ramif led structure that fills the available space.

8.3.1 Dilute Flocculated Suspensions
To a first approximation dilute suspensions of flocculated particles
have a rheology that is qualitatively little different to that of stable
suspensions. A dilute flocculated suspension may be somewhat
more shear-thinning as a result of floe-breakage under shear, the
viscosity may generally be higher because of the enhanced
hydrodynamic volume of the floes but the differences are minor in
processing terms.


8.3.2 Concentrated Flocculated Suspensions
As in Section 5 it is useful to distinguish between strongly or
irreversibly flocculated suspensions having an open network
structure, and weakly or reversibly flocculated suspensions where
there is a tendency to form compact clusters and loose networks of
clusters. Above the gel-point, that is the point where a ramified
structure of some kind forms, the rheology of flocculated
suspensions is both qualitatively and quantitatively different to that
of stable suspensions. Flocculated suspensions are typically highly
shear thinning and can have viscosities many orders of magnitude
larger than those of stable suspensions. Indeed strongly-flocculated
suspensions of sub-microns can be “solid” pastes at quite modest
volume fractions.
Another difference between stable and flocculated suspensions is
that the properties of the latter are very dependent on absolute
particle size, this influences both the structure and the strength of
the network formed. Typical flow curves for flocculated suspensions
are illustrated schematically In Figure 7. The curves to the right
depict the behavior of strongly flocculated suspensions which
typically show a true yield stress ɸ yp, the left-hand curves Illustrate
the behavior of weakly-flocculated suspensions which do not have
a true yield-stress. At higher shear-rates the shear-stress tends to
become proportional to shear-rate in both cases and in this regime
flow curves can usually be represented by the simple Bingham
equation:

Where the parameter nP is referred to as the plastic viscosity and δB
is the extrapolated yield stress or Bingham yield stress. The
Bingham yield stress has no particular fundamental significance; it
is however a useful parameter as it measures the Non-Newtonian
part of the stress required to cause flow.

Thus, it is stress required to cause an appreciable or easily
perceptible rate of flow being of order δB. The process responsible
for the shear-thinning or disruption of the particle network under
shear and as a consequence δB and ɸ yp develop at the gel-point
and Increase rapidly with concentration thereafter. The location of
the gel-point is thus of some significance. Its location depends upon
particle size, particle shape, strength of flocculation, and can also
depend upon previous history. However, the following
generalizations apply.
1)

The gel-point increases with particle size and with increasing
strength of flocculation.

2)

In the case of spherical particles undergoing strong flocculation the
gel-point is not untypically 5-10% v/v solids for sub-micron particles
rising to perhaps 20% when the flocculation is weak.

3)

f the particles are highly anisometric the gel-point can be much
lower. For example, for Bentonite clay particles (very thin plates)
and Attapulgite clay (rods) particles, it is of the order of 1% and
2.5% v/v respectively.

4)

The gel-point Increases with particle-size so that much above a few
microns it approaches 50% v/v or so. Ron-Newtonian effects are
thus small for particles with sizes well In excess of 1 µm unless the
solids content is very high.

A recent study by Mills [5] on model dispersions has shown that the following
observations can be made:
1)

δB Is typically twice ɸ y.

2)

δB or ɸ y typically increase with volume fraction something like $3
above the gel-point.

3)

δB or ɸ y typically decrease something like (particle size) -2. (NB In
polydisperse systems it appears that the number-average size is
important - fines thus dominate the behavior.)


4)

For larger particles (> 1 µm) the plastic viscosity tends to differ little
from the viscosity of a deflocculated suspension. For smaller
particles It can be larger. Nevertheless the Bingham yield stress
usually makes a major contribution to the stress. This is thus a key
parameter.

8.3.3 Time Effects and History Effects
Suspension rheology can be both time and history dependent. These
effects are however beyond the scope of this introduction except to say:
1)

Thixotropy, that Is, a slow a development of structure with time after
shearing, is not uncommon with flocculated systems.

2)

Shearing can sometimes cause compaction of aggregates and an
irreversible decrease in viscosity and yield stress.

3)

Shearing after dilution can sometimes cause the reverse of (2).

8.3.4 Slip and Fracture
Very concentrated systems often show inhomogeneous flow with all of the
deformation occurring at a slip or fracture surface where the stress Is a
maximum (e.g. at the wall of a pipe). Pastes of this type usually show
Bingham plastic type behavior but the apparent yield stress and viscosity
depends upon the flow geometry.

8.3.5 Behavior of Flocculated Cakes in Compression
Fine solids are often separated from liquids by processes such as
filtration, gravity thickening or centrifugation. If the material is not already
flocculated, flocculation Is usually arranged by some means so as to
achieve reasonable rates of separation. Flocculation however leads to the
formation of structured cakes and sediments which are not easily
compressed.


Gravity thickening, for example, produces a thickened phase with a solids
content roughly equal to the gel point. Pressure filtration and centrifugation
can achieve higher concentrations because of the larger compressive
forces acting on the flocculated network but again the lower the gel-point
the lower the ultimate solids content achieved. The mechanical resistance
of the cake to densification can be described by a compressive yield
stress PV (this parameter is discussed extensively in Chapter 3). Pγ like δ
γ, develops rapidly above the gel point and the following observations can
be made:
1)

Pγ typically increases like ɸ 4 above the gel-point.

2)

Pγ is typically two orders of magnitude larger than wy.

3)

The gel-point decreases with decreasing particle size.

The latter means that structured suspensions that appear quite fluid can
be rather "solid-like" In compression and difficult to dandify by mechanical
means Pγ is a very important parameter in solid-liquid separation theory
(cf Chapter 3).

8.4

Summary of Suspension Rheology

The following broad rules apply:
(a) Deflocculated Suspensions
These are quite fluid at volume fractions below 0.5 or so unless the particles are
so small that their collision diameter is much larger than their actual diameter as
a result of long-range repulsive forces. Given this proviso particle size and shape
have little effect provided the particles are not too asicular (α < 10). More
concentrated systems can show shear thickening (dilatant) behavior and be
ultimately paste-like as close packing is approached.


(b) Flocculated Suspensions
Rheology is very dependent upon particle size and shape. Viscosity and yield
stress increase rapidly with concentration above the gel point which Itself Is size
and shape-dependent, At very high concentrations (> 0.5) concentration Is the
dominant variable as It Is with stable systems and in the limit of close-packing
flocculated and deflocculated show rather similar behaviour as a consequence of
slip and fracture at stressed surfaces.
Flocculated networks are much stronger in compression than they are In shear
and can resist large pressures without densifying.

9

SEDIMENTATION OF SMALL PARTICLES

9.1

Very Dilute Particles
Small spherical particles obey Stokes law provided the Reynolds number

is < 0.1 for small particles with a density of 1, Re = 0.1 corresponds to
about 50 µm radius. Stokes law has

Larger particles move more slowly than is Implied by the equation.
For very high Reynolds numbers (> 500) Newton's law gives

In the intermediate region Bird, Stewart and Llghtfoot give

Expressions are available for other shapes of particles, for example for
ellipsoids with semi axes a, b,c the settling velocity is reduced as
compared to that of a sphere of the same volume by a factor


However, K lies In the range 0.95 < K < 2 for aspect ratios (a/√bc)
between 0.125 and 16 so that the settling velocity of a volume equivalent
sphere Is a good approximation for most purposes.

9.2

Concentrated Systems

There is a lot of experimental evidence to the effect that the initial settling rate of
a concentrated system of volume fraction ɸ goes like

9.3

Polydisperse Systems

It is often stated in the literature that whereas dilute mixtures of differing particle
size tend to move more or less independently, concentrated systems (ɸ > 0.2)
undergo “hindered settling” and move en-masse. There is however no evidence
for this in systems that are deflocculated and particles are observed to move
more or less independently unless the volume fraction is very high (> 0.5).


9.4

Flocculated Systems

In dilute systems flocculation Increases the settling velocity by increasing the size
of the sedimenting unit. In concentrated systems (above the gel-point 1 the
converse Is true. In this regime sedimentation, If It occurs at all, occurs by the
network of particles collapsing under Its own weight. For sediments with a yield
stress


10

ELECTROKINETIC BEHAVIOR

Electrically-charged particles migrate in electric fields. This is exploited in
electrical separation methods (electrodecantation, electrodialysis which are
briefly mentioned in Filtration). The mobility of a particle (the velocity of an
electric field gradient of unity) is given by

for particles with diameters of the order of 1 µm or more, (more complicated
expressions apply for smaller particles, these will not be given here), where ζ is
the electrokinetic or Zeta-potential which for highly charged particles (lattices,
clays, alumina, silica etc) Is typically f 50 mV. Particles then typically have
nobilities of a few microns per second per volt per centimeter. Recent work by
Zukoskl (unpublished) has shown that, remarkably, the mobility is independent
of particle concentration for volume fractions less than about 0.55. He also finds
that particles with different nobilities (in sign or magnitude> move independently.
This implies that different types of particles can be separated very efficiently by
electrokinetic methods.

11

A NOTE ON MAKING DISPERSIONSAND SUSPENSIONS

There are two basic methods of making dispersions and suspensions, by
attrition of a solid, or by precipitation from solution. In both cases a suitable
mechanism of stabilization is required In order to prevent recombination or
aggregation of the particles.
Attrition or comminution (Chapter 8) is almost Invariably carried out in the
presence of a surfactant or dispersant which confers stability on the particles
formed. The best stabilizers are however not always the best dispersants or
comminution aids. Adsorption onto newly created particles has to be extremely
rapid In order to prevent their recombination. For this reason polymeric stabilizers
that diffuse slowly are not always as good dispersants as low molecular weight
surfactants In spite of their superior stabilizing power. A combination of low
molecular weight and polymeric surfactant is often useful. It is usually difficult to
get much below a few microns by comminution unless the solid consists of
aggregates or agglomerates. For this reason gas phase methods (plasma or
furnace) which produce fine primary particles are often used to produce solids
that will ultimately be dispersed in liquid. Thus gas-phase Ti02 can be dispersed
down to 0.2 pm by comminution because the primaries are of this size.


Precipitation/growth routes involve the reaction of soluble species that produce
an Insoluble liquid. Control of the stability of the growing particles is the key to
producing fine particles. The best understood and controllable processes of this
type are those used to produce polymer particles (lattices). This is In part
because reaction rates are relatively slow and so controllable, and In part
because it Is relatively easy to find or make powerful stabilizers that either
associate strongly or react with the growing polymer chains.
Fine particles of most Inorganic solids can be made in water by working at high
dilution. Under such conditions the natural electrical surface-charge is often
sufficient to impart stability to the growing particles, and the kinetics of nucleation
and growth can be controlled by adjusting the concentration of reactants. It is,
However, almost always Impossible to produce inorganic particles and other
non-polymeric particles at any reasonable concentration. This is in part because
reaction rates tend to be rapid and in part because of stabilization problems. The
problems can be illustrated by considering one of the few examples where a
concentrated dispersion can be made without difficulty. This is the production of
small (-20 nm) cobalt particles by thermal decomposition of dicobalt
octacarbonyl. Typically a solution of CO2(CO)8, in toluene is heated under an
atmosphere of CL in the presence of a surfactant. Control of the CO over
pressure and the temperature allows the reaction rate to be controlled, and the
extreme difference in character between the metal surface and the solvent allows
oil-soluble surfactants with polar or Ionic head groups to adsorb strongly (head
down> and provide good stabilization (steric stabilization). The implication of this
experience is that polar/ionic/ metallic particle scan be made in organic media
provided soluble precursors can be found This is however very limiting. The
problem with producing concentrated dispersions from aqueous media is almost
certainly one of stabilization. It Is very difficult to find amphilic stabilizing
molecules that comprise two distinct moieties, one of which has strong affinity for
the polar solid surface but little affinity for the medium, and the other of which has
the reverse properties. Given such molecules it would probably be possible to
produce concentrated dispersions of most Inorganic solids, since reaction rates
can probably be controlled In principle by working under conditions where
one reactant is starved.
Some of the available methods of making dispersions are summarized in
the following table.


Fundamentals of Suspensions & Dispersion's

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