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Process Engineering Guide: GBHE SPG PEG 300.2

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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 Dilute Suspensions 4.2 Concentrated Suspensions 5 STRUCTURE OF STABLE SUSPENSIONS OF

MONODISPERSE PARTICULATES 6 SUMMARY OF STRUCTURES

7 PARTICLE PACKING



8 RHEOLOGY 8.1 Basic Rheological Concepts 8.2 Colloidally Stable Suspensions

8.2.1 Spherical Particles of around 1 µm 8.2.2 Effect of Particle Size Distribution 8.2.3 Effect of Particle Shape 8.2.4 Submicron Particles 8.2.5 Very Concentrated Systems

8.3 Rheology of Flocculated / Aggregated Systems

8.3.1 Dilute Flocculated Systems 8.3.2 Concentrated Flocculated Systems 8.3.3 Time and History Effects 8.3.4 Slip and Fracture 8.3.5 Behavior of Flocculated Cakes in Compression

8.4 Summary of Rheology Deflocculated Suspensions Flocculated Suspensions

9 SEDIMENTATION OF SMALL PARTICLES 9.1 Very Dilute Particles 9.2 Concentrated Systems 9.3 Polydisperse Systems 9.4 Flocculated Systems 10 ELECTROKINETIC BEHAVIOR 11 A NOTE ON MAKING DISPERSIONS AND SUSPENSIONS



12 References 13 Figures Fig 1a Potential Energy Diagram for Steric Stabilization Fig 1b PE Diagram for Electrostatic Stabilization Fig 1c 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 = - aA/12 (R – 2a) (1) FA = aA/12 (R – 2a)2 (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 poly-

ions 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-).

is proportional to Ionic strength. More generally



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 Effective Range in Water

(= ~ 4 / Ҝ) 10E-5 mol dmE-3 400 nm

10E-3 40 nm 10E-1 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 long-chain 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 ccc NaCl 0.1 to 0.5 molar KCl 0.1 to 0.5 molar

BaCl2 0.002 to 0.015 molar MgCl2 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 SO4-and 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 surface-coatings 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. Close-packed 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 body-centered 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 0.74

rhombohedral 0.74 body-centred cubic 0.68 random 0.636 primitive cubic 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 quasi-Newtonian 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 plate-like 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 sub-micron 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, sterlcally-stabilized 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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