Suspensions and polymers - Common links in rheology · Key words : rheology, polymer melts, polymer solutions, suspensions 1. Introduction The field of rheology has developed significantly

Korea-Australia Rheology JournalVol. 11, No. 3, September 1999 pp.197-218

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Suspensions and polymers - Common links in rheology

G. Harrison, G. V. Franks, V. Tirtaatmadja, and D.V. Boger*Dept. of Chemical Engineering, The University of Melbourne

Parkville, VIC 3010, Australia

(Received August 6, 1999; final revision received September 21, 1999)

Abstract

Rheological techniques are frequently used to characterize particulate suspensions and polymer systems. Eperimental data frequently show that similar trends and characteristics are found in both systems. Using common examples and illustrations of the rheological behaviour, we attempt to bring together these separate fieland investigate the common links in the different systems. In many cases the similar rheological behaviouobserved in these different systems can be related to the same basic physical principles.

Key words : rheology, polymer melts, polymer solutions, suspensions

1. Introduction

The field of rheology has developed significantly in thelast half century. Theoretical models are constantly beingdeveloped and refined to be better able to predict experi-mental results. The increase in the capability of computershas facilitated the understanding of the deformation of mat-ter using calculations too complex to attempt just ten yearsago. Despite these advances, however, it is still experi-mental methods that often provide the first and best insightinto how a particular material responds to an applied defor-mation. As newer and more user friendly rheometersbecome available, there has been a dramatic increase in theavailable body of experimental literature.

Rheology is a large research field, and investigators frommany different traditional disciplines consider themselvesrheologists. Materials as diverse as polymer melts and so-lutions, gels, emulsions, micelles, particulate and latex sus-pensions and oils are just some of the materials that arecommonly encountered in the modern rheological litera-ture. Given the wide range of interests, and materials to bestudied, it is perhaps not too surprising that there is some-times a lack of communication throughout the entire field.In this paper, it is the intention to try and bring together thebasic concepts in some of the above materials in a way thatallows knowledge and understanding of the rheology topass between subdisciplines.

In order to limit the work, two particular classes of sys-tem are focused upon: particulate suspensions and polymersolutions (and melts). Both these systems may be thoughtof as components (polymers or particles) suspended in a

continuous medium. The reason for this decision is twfold. First, our group has extensive knowledge of the rhology of all of these systems, and has been conducexperiments and publishing papers concerning these mrials for many years. Secondly, there are common linbetween these systems that are manifested in traditiorheological experiments. These links enable rheologistsgain an insight into the similar physical mechanisms bhind the experimental results. Latex suspensions and pmer gels are also included in the discussion. These sysmay be considered as something of an intermediary betwthe other two systems, and this commonality will hefacilitate the comparison between particulate suspensand polymer solutions.

The comparison is limited to experimental results otained in standard rotational rheometers. These teinclude steady shear viscosity, the storage and loss mulus, and the yield stress. These measurements were sen because they are typically used to characterize mdifferent materials in rheological investigations and aacknowledged as material properties. The results are sthat general trends are clearly evident, although specnumerical values will vary, for many materials falling inta particular class of system. Also a large amount of datavailable in the literature on all the systems to be discusOscillatory experiments probe the interactions betwesuspended entities and the microstructure of the samprest without significant deformation occurring. The steashear viscosity probes the hydrodynamic effects as the tem is being deformed, as well as the evolution of tmicrostructure. Material properties such as the extensioviscosity or normal stresses will not be discussed for treasons. First, these measurements are frequently linear and probe primarily hydrodynamic effects. Se

*Corresponding author; [email protected]© 1999 by The Korean Society of Rheology

Korea-Australia Rheology Journal September 1999 Vol. 11, No. 3 197

G. Harrison, G. V. Franks, V. Tirtaatmadja, and D. V. Boger

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ondly, experimental measurements are not available widelyfor all the systems to be discussed. However, many of theconcepts and ideas used to explain the results discussed inthis paper may be used to predict other material properties.

There is a wide range of textbooks and review articlesdiscussing the rheology of suspensions, polymer solutionsand polymer gels. In the suspension and latex area, theinterested reader is referred to Barnes and coworkers(1989), Bergstrom (1994), Russel and coworkers (1991),Tadros (1996), Mewis and Spaull (1976), Buscall andcoworkers (1985). The rheology of polymer solutions andmelts is the focus of books by Larson (1988, 1998), Doiand Edwards (1986), Walters (1975), Barnes and cowork-ers (1989) for example. Lapasin and Pricl (1995) discussthe rheology of polymer gels in some detail.

The remainder of this paper is organized as follows.First, the basic physical principles which apply to all thesystems are introduced. These principles include the inter-actions between the dissolved or suspended material (poly-mers and particles) as well as between the dissolved orsuspended material and the solvent. The specific rheolog-ical behaviour of rigid particle suspensions including latexsuspensions, polymer solutions and polymer gels is thenpresented. Finally, the similarities and differences in thephysical principles governing each of the systems will beexplained in terms of the basic rheological trends.

2. Interactions between Particles and SuspendedMedium

In this section two important factors that influence thebehaviour of the systems under consideration are intro-duced. These factors are: the interactions between the sus-pended entities (polymers and particles) placed in the con-tinuous medium, and the interactions between the solventand the suspended entities. Other factors which also affectthe rheological behaviour, such as the size and shape ofparticles, will not be considered in detail here or in theremainder of the paper.

Intermolecular and surface forces control the interactionswithin and between “particles” (molecules or macroscopicparticles) placed in a continuous fluid medium. In manycases the same type of force produces similar rheologicalbehaviour in different systems. Here, some basic inter-molecular forces are introduced and the mechanisms bywhich they give rise to surface forces between macroscopicbodies are described.

2a. Surface ForcesThe interaction forces between macroscopic particles

arise due to the sum total of the interactions between theindividual molecules that make up the bodies. These inter-action forces are commonly referred to as surface forces. Inmany cases colloid and surface chemists have been quite

successful in understanding these forces and their effecrheological behaviour.

Any particle placed in a liquid medium will interact withother particles in the medium via attractive and/or repulsforces. In addition to Brownian motion and hydrodynamforces these intermolecular and surface forces are duemarily to two effects: 1) interacting charges (electrodnamic and electrostatic) and 2) volume exclusion.

In colloidal particle suspensions, surface forces controand how strongly the particles are either attracted repulsed from each other [Israelachvilli (1992), Ho(1990), Hunter (1987)]. These forces strongly affect trheology of suspensions [Barnes and coworkers (198Bergstrom (1994), Russel and coworkers (1991), Good(1990), Goodwin (1973), Russel (1991), Mewis and Spa(1976), Russel (1980)]. The forces are given by

(1)

where F is the force, V the interparticle pair potential eergy between the two surfaces and D the surface to surseparation distance. Both attractive forces and repulsforces vary in magnitude and extent depending on the pticular mechanism which creates the force.

Attractive forces include van der Waals, electrical doublayer (EDL) (oppositely charged surfaces), hydrophobbridging, and depletion. Repulsive forces include electridouble layer (EDL) (similarly charged surfaces), ste(polymeric), hydration, and structural. The best charactized of these forces both experimentally and theoreticaare the van der Waals attraction and EDL repulsion. Thforces are described by the DLVO (Derjaguin, LandaVerwey, and Overbeek) theory [Derjaguin and Land(1941), Verwey and Overbeek (1948)] where the attractand the repulsive pair potentials are simply summed to fthe total potential energy versus distance relationship the particle-fluid system. This summation technique usally works well also with the other forces not strictly covered by the DLVO theory.

Three different types of interparticle pair potentials wbe considered: strongly attractive (only van der Waals tential), long range repulsive (termed dispersed) aweakly attractive (van der Waals and a short range repsion). A schematic is shown in Figure 1. When there no repulsive forces between the particles, they will drawn together by van der Waals attraction and formstrongly attractive touching network as in Figure 1a. Iflong range repulsion is created between the particles,van der Waals attraction can be completely overwhelmand the pair potential is repulsive as in Figure 1b. shorter range repulsion insufficient in extent to overwhethe van der Waals attraction results in a weakly attracsecondary minimum as in Figure 1c. Repulsive potenti(long or short range) may be created via electrical dou

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Suspensions and polymers - Common links in rheolohy

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layer or steric repulsion and are described in the followingsubsections.

van der Waals AttractionWhenever like molecules (or particles) are placed in a

fluid medium (that has different dielectric properties thanthe molecules or particles), they will be attracted to oneanother by van der Waals attraction. This attraction is dueprimarily to correlated instantaneous dipoles in the atomswithin each molecule (or molecules within each particle).

When the interactions between all the molecules in amacroscopic spherical body of radius (R) are summed up,the van der Waals interaction energy (VVDW) is givenapproximately (when D>>R) by [Israelachvili (1992)]

(2)

where A is the Hamaker constant (which depends on thedielectric properties of the particles and the interveningmedium) and D the surface to surface separation distance.This attractive interaction (if not opposed by a repulsion)leads to pair potentials as shown in Figure 1a and coagu-lation of particles in a suspension. Suspensions of particleswith this type of strongly attractive pair potential are re-ferred to as flocculated or coagulated suspensions.

Electrical Double Layer RepulsionMany polymers and particle surfaces may become

charged in aqueous solutions by the reaction of surfacegroups or sites with acid or base. Examples of such ion-izable groups are carboxylic acid groups (−COOH) onpolyacrylic acid (PAA), amine (−NH2) groups on poly-acrylamide, and surface hydroxyl groups (−OH) on metaloxide surfaces. There is a pH known as the isoelectric point(iep), where the majority of surface sites/groups are neutraland the net charge on the surface/molecule is zero. At a pHbelow or above the isoelectric point, the polymer or par-ticle surfaces can become positively or negatively chargeddue to the addition of either acid (H+) or base (OH-),

respectively [Parks and DeBruyn (1962), James (1987)]cloud of counterions (ions of charge opposite to the sface, eg., Cl-, for HCl additions) shrouds each particle/moecule in order to maintain neutrality of the system. Whthese particles/molecules are pushed together the couion clouds begin to overlap and increase the counterconcentration in the space between the particles/molecuThis gives rise to a repulsive potential due to the osmopressure of the counterions which is known as the Eltrical Double Layer (EDL) repulsion. A measure of ththickness of the counterion cloud (and thus the range ofrepulsion) is the Debye length. When the Debye lengthlarge (small counterion concentration) the particles repulsive at large separation distances so that the vanWaals attraction is overwhelmed as in Figure 1b. Suspsions of particles interacting with long range repulsive ppotentials may be referred to as dispersed or stabilize

The form of the potential energy (VEDL) versus distance(D) relationship for the electrical double layer repulsiobetween macroscopic spherical particles of radius (Rgiven approximately by [Israelachvili (1992)]

(3)

(low constant potential, weak overlap approximatiowhere Ψo is the surface potential (created by the surfacharge), ε the relative permittivity of water, εo the per-mittivity of free space, and κ the inverse Debye length. Theinverse Debye length depends on the square root ofelectrolyte concentration [Israelachvili (1992)]. The Deblength is thus reduced by adding salt, which increasesconcentration of the counterions around the particle. Whsufficient salt is added, the range of the repulsive potenis decreased sufficiently to allow the van der Waals attrtion to dominate at large separation distances. A secondminimum develops and the particle network becomweakly attractive as in Figure 1c.

Steric RepulsionA steric interaction force occurs between particles due

the presence of adsorbed or grafted polymer layers onsurface of the particle. This force consists of two main cotributions: the mixing interaction and the volume exclusiterm. The mixing interaction can be either an attractivea repulsive force, depending on the nature of the adsorpolymer-solvent interaction. When the suspending mediis a good solvent for the adsorbed polymer the mixiinteraction is repulsive while poor solvents create attractmixing interactions. (Solvent quality is described in detin the next section.)

The second component of the steric interaction is the vume exclusion term, which arises from the loss in cofigurational entropy and compression of the adsorbpolymer layer on the approach of a second particle analways repulsive. Steric interaction usually occurs ove

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Figure. 1. Schematic representation of interparticle pair potentials(potential energy (V) versus separation distance (D)).Negative energies are attractive in this representation.a) strongly attractive, b) repulsive, dispersed c) weaklyattractive.



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fairly short range when compared to electrical double layerforces, i.e. only when the particle surface-to-surface dis-tance (D) is slightly larger than twice the adsorbed layerthickness (δ).

The potential energy per unit surface area of flat plateswith terminally anchored polymer molecules in a “mush-room” configuration has been calculated by de Gennes(1987) for a good solvent. By applying Derjaguin's approx-imation [Israelachvili (1992)] and integrating the force, onearrives at the interaction potential between spherical par-ticles of radius (R) in a good solvent:

(4)

where S is the (average) distance between polymerstrands on the particle surface, δ is the adsorbed layerthickness, D is the separation distance between the surfacesof the particles, R is the radius of the particles and kT is thethermal energy. Similar expressions can be derived forpolymer brushes, other geometries and solvents of eithertheta or poor quality [Ploehn and Russel (1990), Napper(1983)].

When δ is small (low molecular weight polymers), thevan der Waals attractive force dominates at large distancesof separation. In a good solvent as the separation distancebecomes less than 2δ, when the polymer layers begin tooverlap, a strong repulsion results as shown in Figure 1c.The potential curve shows a minimum which depends onδ. . . . For sufficiently small δ appreciable attraction prevailssuch that coagulation can occur. If δ is large the van derWaals attraction can be overwhelmed such that a well dis-persed suspension results with pair potential similar to Fig-ure 1b, but without the primary minimum.

2b. Polymer-solvent and Polymer-polymer InteractionsThe interactions between polymer chains and solvent

molecules in polymer solutions are similar in many waysto those of particles coated with polymers. The solubilityof a polymer in a solvent is governed by the free energy ofmixing and thus the relative magnitude of the entropy ofmixing (which is always negative favoring mixing) and theenthalpy of mixing (which is usually positive hinderingmixing). The χ factor (also known as the Flory-Hugginsconstant), is a temperature dependent coefficient that char-acterizes this interaction between the polymer and the sol-vent (the enthalpy of mixing is directly related to χ). Theχ factor can be calculated from the difference in the cohe-sive energy densities of the polymer and solvent [Allcockand Lampe (1981)]. The cohesive energy density of a ma-terial depends upon the interaction pair potentials betweenthe molecules such as electrical and van der Waals inter-actions [Israelachvili (1992)]. A theta solvent is de-scribed as one where the polymer segments have an equalattraction to the solvent and the other segments of the poly-

mer molecule, and the χ factor is equal to 0.5. In a goodsolvent the χ factor of the polymer-solvent system is lesthan 0.5 and in poor solvent conditions χ>0.5.

For dilute polymer solutions, the χ parameter is ameasure of the solubility of the polymer in a particular svent. The configuration of the polymer in solution depenupon the free energy of mixing and the flexibility of thpolymer backbone (generally described by the persistelength of a polymer segment). In an equilibrium stapolymer molecules dissolved in a theta solvent (χ = 0.5)occupy an approximately spherical space when the pmer concentration is sufficiently small that each molecuis essentially independent of the other polymer molecuHowever, a large fraction of this volume is occupied by svent trapped within the polymer molecule. The effectivolume fraction of the polymer molecules in solution wthus be affected by the amount of solvent molecules witthe spherical volume swept out by the polymer molecuLight scattering techniques may be used to measureequilibrium radius of gyration that characterizes the avage size of the molecule and trapped solvent. However,segments of the polymer are continually moving dueBrownian motion, and the polymer configuration is therfore continually changing slightly even at equilibrium.

In a good solvent (χ < 0.5), the polymer segments prefeto interact with the solvent rather than other segmentsthe polymer or other polymer chains due to the naturethe interactions between the polymer and the solvent. Idilute system, this causes the polymer to swell from size associated with the same polymer in a theta solvObviously, as the polymer swells, more solvent is fouwithin the space the molecule occupies and the effecvolume fraction of the molecule increases. A poor solve(χ > 0.5) has the opposite effect: the polymer segmentsmore attracted to other polymer segments than to the vent. This attraction has the effect of compressing the mecule into a smaller configuration than in a theta solvand thus decreasing the effective volume fraction of polymer. In the extreme case, the polymer will not dissoin the solvent, although a more likely occurrence is a nhomogeneous system where the polymer molecules aggate in the solvent.

The specific way in which basic intermolecular forceaffect the interactions between polymers and solventsmore complicated than insoluble macroscopic "hard" pticles or even latex particles. This complexity arises frothe fact that these intermolecular forces control both configuration of each polymer molecule and the interations between the molecule and the solvent and betwpolymer molecules. The configuration of the moleculefurther complicated when the concentration is increassuch that polymer-polymer interactions occur. Entangments are assumed to occur when the effective volufraction of all the polymer molecules in the system, c

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culated using the radius of gyration, exceeds one. At thispoint, the permeable coils are believed to interpenetrate oneanother and entanglements form. Because the "size" of asingle polymer chain characterizes a volume containingboth the polymer molecule and the solvent that is con-tained within the molecule, a relatively small weight per-cent (approximately 1% for a typical 1 million MW poly-mer chain) is necessary for overlapping of the molecules tooccur. Beyond this concentration, interactions betweenpolymer chains in the form of entanglements will occur.Entanglements are different polymer chains continuallycrossing and being intertwined as the molecule undergoesBrownian motion. Although these entanglements are con-tinually being lost and reformed, when viewed as an en-semble of the many polymer chains in a system, averagequantities, such as number of entanglements per chain, willemerge. As the concentration increases further, this con-tinual motion leads to highly entwined polymer networksdue to the length (and, unlike particles, the ability tochange configuration) of the molecules.

Entanglements also act to restrict the motion of the poly-mer chains: if the polymer is held by another polymer dueto an entanglement, its motion and free diffusion is beingrestricted. In response to a large applied deformation, thepolymer will not be able to relax to equilibrium as quicklyas if it were a free chain. Clearly, these entanglements rep-resent a significant difference between the polymer solu-tions and particulate suspensions.

3. Typical Rheological Behaviour

In this section, typical rheological behaviour for the sys-tems under discussion is presented. The rheological behav-iours shown are for particulate suspensions, polymer solu-tions, including melts, and polymer gels. Polymer gels aretreated here in a separate subsection as they can be viewedas an intermediary between polymers in a suspending me-dium and suspensions. The intermolecular interactions ofgels occur due to physical bonds.

These general results are presented here for illustrativepurposes, and are well-known to the suspension or polymercommunities from which they are drawn.

3a. Particulate SuspensionsTypical rheological behaviour of particulate suspensions

is described citing general examples from the literature.The particles discussed are insoluble and essentially non-deformable particles (typically metal oxides or polymer la-tices). Rheological results for non interacting “hard sphere”suspensions are discussed first, followed by suspensions ofrepulsive and attractive systems.

Oxide particles are covalently bonded solid objectswhich are insoluble and non-deformable. These are typ-ically ceramic materials such as silica, alumina, zirconia,

clays and minerals. These materials are usually stabili(metastably prevented from coagulation by a potenbarrier as in Figure 1b) by electrical double layer forccreated by adjusting the pH of the suspending mediumthat there is a charge on the particles surface.

In contrast, sterically stabilized latex particles are pomeric particles with an adsorbed or grafted layer of sfactants or macromolecules on the surface. Without ststabilization, these latex particles are generally insoluand form aggregates or coagulate in the suspending dium. The incorporation (by grafting or adsorption durinsynthesis) of a soluble polymer layer on the surface of latex particles allows the particles to be dispersed duethe repulsive steric interaction previously discussed. Comonly studied systems include the polymethyl metacrylate latex particles with an adsorbed layer of eithpolyhydroxy stearic acid or polydimethyl siloxane supended in organic solvents, and polystyrene or other coymer latices with grafted layers of polyethylene oxidpolystyrene sulphonate or polyvinyl alcohol in aqueosolvent. The polymer layers are generally in the range ofew to tens nm, and are attached to latex particles witdiameter of the order of tens to several hundreds nm, whgives a ratio of the particle radius to the layer thickneranging from less than 10 to approximately 100.

It should be noted that it is also possible to make spensions of oxide particles with steric repulsive forceLikewise latex particles can be dispersed by electridouble layer repulsion.

Hard SpheresNon interacting hard spheres are those in which there

neither attractive nor repulsive forces between the particHard spheres only interact through hydrodynamic aBrownian diffusion forces. Such systems have been stied extensively by many investigators [Jones and cowork(1991), de Kruif and coworkers (1985), van der Werff ade Kruif (1989), van der Werff and coworkers (1989)]. Fiure 2 shows the steady shear viscosity of hard sphere ssuspensions for a range of concentrations [Jones andworkers (1991)]. At low concentrations the suspensiohave viscosities which are independent of shear rate are well predicted by the theoretical expressions of Eins(1906) (for φ < ≈ 0.07) and Batchelor (1977) (for φ < ≈ 0.15).These expressions are discussed in some detail in SeIV. At moderate concentrations the viscosity increasmore rapidly than predicted by Einstein and Batchelor no significant shear thinning is observed. At higher cocentrations a shear thinning region follows the low shviscosity plateau. The hard sphere suspensions appehave a yield stress [Jones and coworkers (1991)] when are tested at volume fractions above that of dense ranpacking (approx. 0.637 volume fraction).

As the volume fraction of the particles in the suspens



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is increased, the rheology gradually changes from a shearrate independent viscosity, to a shear thinning viscosity,and then to a system which possesses a yield stress. In thelow shear plateau of the shear thinning region, the suspen-sion microstructure is not significantly perturbed by theshear because the Brownian forces dominate the hydro-dynamic forces. Thus the equilibrium structure is restoredmore rapidly than it is perturbed. At higher shear rates thehydrodynamic forces begin to dominate and shear thinningoccurs. The shear thinning behaviour is widely believed tobe due to ordering of the particles into favourable flowstructures, such as layers or strings, which reduce the energydissipated under shear [Hoffman (1972), Chen and co-workers (1994), Bilodeau and Bousfield (1998)].

Figure 3 shows the storage modulus of the hard spheresuspensions [Jones and coworkers (1991)]. The elasticbehaviour of the suspension, as measured by G', increasesas the volume fraction approaches the maximum packingdensity. There is a change from liquid-like to solid-like

behaviour as the volume fraction is increased from ab0.60 to 0.70 volume fraction. This is recognized as G' be-comes large and independent of frequency as volume ftion is increased.

Repulsive Particle SystemsRepulsive interactions are generally created between

ticles via either long range electrical double layer or sterepulsion, resulting in interparticle potential energy curvas described in Figure 1b. Figure 4a shows the steady sviscosity of alumina particles dispersed in water via eletrical double layer repulsion [Zhou (1999)]. The behaviois quite similar to that observed for hard spheres (compto Figure 2) in that a change from Newtonian to shear thning viscosities and finally to a material with a yield streis observed with increasing volume fraction. The main dference between repulsive interacting particles and hspheres is that lower volume fractions are required to pduce the deviations from the Einstein-Batchelor relatio

Figure. 2. Steady shear viscosity of 49 nm diameter silica hardsphere suspensions at different volume fractions (φ) asindicated in the figure. Adapted from Jones and co-workers (1991).

Figure. 3. Storage modulus of 49 nm diameter silica hard spheresuspensions at different volume fractions (φ) as indica-ted in the figure. Adapted from Jones and coworkers(1991).

Figure. 4. (a) Steady shear viscosity of 0.55 micron diameter amina suspensions dispersed with electrical double lafor- ces at pH 5.0 in 0.01 M KNO3, at different volumefractions (φ) as indicated in the figure. Adapted fromZhou (1999). (b) Steady shear viscosity of suspensiof copolymer latex particles dispersed with a graftlayer of polyethylene oxide, at different volume frations (φ) as indicated in the figure. Adapted from Jonand coworkers (1992).



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ship, shear thinning and yield stress behaviour with therepulsive particles. These observations can be rationalizedby considering an "effective" volume fraction that includesthe volume of the particle and the additional volume ex-cluded by the repulsive interaction. The viscosity of a sus-pension of repulsive particles is increased above that ofhard spheres even in the limit of very dilute concentrationsbecause of the additional dissipation created by the dis-tortion of the double layer by the flow of solvent aroundthe particle. This effect is known as the primary electro-viscous effect [Hunter (1987,1988)]. At higher concentra-tions a further increase in viscosity is caused by the over-lapping double layers and is known as the secondaryelectroviscous effect [Hunter (1987,1988)].

Figure 4b shows the steady shear viscosity of latex sus-pensions consisting of a copolymer with a grafted poly-ethylene oxide layer, at increasing volume fractions [Jonesand coworkers (1992)]. The effective volume fraction ofthe particles is increased beyond the hard sphere value dueto the presence of the steric repulsive (polymer) layer. Atthe lowest volume fractions the suspensions show shearrate independent behaviour. Because the particle-particleseparation distance is large compared to the range of stericinterparticle forces, only the Brownian diffusion dominates.The viscosity remains constant with shear rate as the par-ticles can diffuse freely in the suspension. As the volumefraction increases the interparticle distance becomes com-parable to the particle radius and the hydrodynamic andsurface interactions dominate. The viscosity increases withincreasing volume fraction and changes from one that isindependent of shear rate to shear thinning. The behaviouris similar to that described for electrical double layer repul-sive particle systems in Figure 4a.

For highly concentrated repulsive suspensions, yieldstress values may also exist as can be seen in Figure 5

[Jones and coworkers (1992)], where the shear stresplotted against shear rate. For volume fractions ab0.688, at low shear rates the stress does not tend to and the yield value may be determined by extrapolatingthe stress axis. Near the maximum volume fraction, whthe interparticle distance becomes less than twice steric stabilizer layer thickness (or twice the Debye lengin electrical double layer stabilized suspensions), larepulsive forces exist between each particle and its mneighbours. These repulsive forces result in a highly veloped structure of particles. In this situation flow wnot occur until sufficient stress is applied to overcome forces that hold the particles in place.

The increase in volume fraction (concentration) of tparticles stabilized with double layer repulsion causeschange from liquid-like to solid-like behaviour. Thichange can be quite dramatic and may occur at fairly volume fractions (especially when the particles are smand the range of the repulsion large). Figure 6a showsstorage moduli for small (10 nm) silica particles stabilizby double layer forces [Persello and coworkers (1994There is a transition to solid-like behaviour above a volufraction of about 0.22, due to the large range of the repsive forces relative to the size of the particles.

Figure 6b shows the storage modulus of suspensionpolybutyl acrylate-styrene latex particles stablized wsurfactants [Raynaud and coworkers (1996)]. The chafrom liquid-like to solid-like behaviour is observed as thvolume fraction is increased above 0.44.

Figure 6c shows the relative magnitude of the storamodulus (G') and the loss modulus (G") for suspensions ofelectrical double layer dispersed silica particles at tdifferent volume fractions [Persello and coworkers (1994At the low volume fraction G" exceeds G' and the behav-iour is liquid-like. At the higher volume fraction the solidlike behaviour is noticed as G' exceeds G" and bothremain independent of frequency.

Figure 6d shows the storage and loss moduli for spensions of polyvinyl acetate stabilized by poly(2-ethhexyl methacrylate) in an organic solvent (Isopar-G), three different weight fractions [Croucher and Milki(1982)]. At the lowest fraction of 0.18, both G' and G"increase almost linearly with frequency, with G">G' at allfrequencies measured, indicating a predominantly viscresponse. As the weight fraction increases to 0.43, probability of particle-particle contact becomes larger, thincreasing the elastic interaction force. This leads to increase in the moduli values and G' is comparable to G".Finally when the weight fraction becomes sufficiently higthe particle-particle distance is comparable to, or evsmaller than, twice the adsorbed layer thickness and results in interpenetration or compression of the layers. TG' is significantly larger than the G", with both showingmuch smaller dependence on frequency and the suspen

Figure. 5. Shear stress versus shear rate for suspensions of copol-ymer latex particles dispersed with a grafted layer ofpolyethylene oxide, at different volume fractions (φ) asindicated in the figure. Adapted from Jones andcoworkers (1992).



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Attractive Particle SystemsStrongly attractive interactions between particles occur

when no special precaution is used to counter the ubiqui-tous van der Waals attraction. The particles come into con-tact with each other in a deep primary minimum energywell as shown in Figure 1a. Weaker attractions as shown inFigure 1c occur when a short range repulsion is present,which negates the van der Waals attraction at small sep-aration distances only. Short range repulsive potentials may

be generated either with the electrical double layer metwith very high salt concentrations or with the steric methwith very short polymer layers. Attraction between the pticles generally produces even higher viscosities and ystresses than non-interacting ("hard sphere") or repulparticles (at identical concentrations) due to the additioattractive surface forces which must be overcome in adtion to the hydrodynamic and Brownian forces. The vcosity of an attractive system is typically shear thinningshown in Figure 7 for titanium dioxide suspensions at vious pH at and around the isoelectric point (iep) [Lidd

Figure. 6. (a) Storage modulus of 10 nm diameter silica particles dispersed with electrical double layer forces at pH 9.0 in 0NO3, at different volume fractions (φ) as indicated in the figure. Adapted from Persello and coworkers (1994). (b) Stmodulus of polybutyl acrylate-styrene latex suspensions stabilized with a mixture of ionic and non-ionic surfactantferent volume fractions (φ) as indicated in the figure. Adapted from Raynaud and coworkers (1996). (c) Storage and loduli of 10 nm diameter silica particles dispersed with electrical double layer forces at pH 9.0 in 0.01M NaNO3, at differentvolume fractions (φ) as indicated in the figure. Adapted from Persello and coworkers (1994). (d) Storage and loss of polyvinyl acetate latex suspensions dispersed with a layer of poly(2-ethyl hexyl methacrylate), at three differenfractions as indicated in the figure. Adapted from Croucher and Milkie (1982).



Figure. 7. Steady shear viscosity of (0.21 microns diameter) tita-nia suspensions at 0.20 volume fraction, at various pHat and around the isoelectric point (iep) as indicated inthe figure. Adapted from Liddell (1996).

Figure. 8. Storage modulus of 56 nm diameter octadecyl-coated sil-ica suspensions in hexadecane at 0.182 volume fraction,at different temperatures as indicated in the figure.Adapted from Chen and Russel (1991).

Figure. 9. Shear modulus versus volume fraction of 1.3 microndiameter alumina suspensions in 1.0 M NH4Cl, at dif-ferent pH as indicated in the figure. Adapted from Chan-nell and Zukoski (1997).

d a

n

t

tee

a-e-ich-theteran-.15ro-veluswsted

and ishisalityas

resis

)]t

Oinalus, thes the

thec-

(1996)]. The attraction between particles is a maximumthe isoelectric point where no charge exists on the surfaof the particles and the highest viscosities are observeall shear rates. As the pH is adjusted away from the iep,surface charge increases creating an electrical double lrepulsion that reduces the magnitude of the attraction. Tresults in lower viscosities throughout the entire shear rrange. The mechanism for the shear thinning behaviousimilar to that described previously for non-interacting arepulsive particles. As the shear rate is increased, attractive particle network (which forms a spanning cluswhen no shear is applied) is broken into smaller asmaller flow units as the shear rate is increased. That ishigher shear rates hydrodynamic forces dominate attractive forces and the particles flow as small clustrather than large aggregates [Bilodeau and Bousfi(1998), Chen and Doi (1999)].

The attractive particle networks produce solid-like behviour at a much lower volume fraction than either the rpulsive or hard sphere systems. Elastic behaviour, in whG' significantly exceeds G" is observed at any volume fraction above that required to form a network that spans container. The volume fraction at which a spanning clusis formed is termed the gel point of the suspension [Chnell and Zukoski (1997)] and may be as low as 0.10 to 0volume fraction. A greater attraction between particles pduces lower gel points. At a given volume fraction (abothe gel point) the (high shear) plateau storage moduwill increase as the attraction is increased. Figure 8 shothe storage modulus of a suspension of octadecyl-coasilica in hexadecane at various temperatures [Chen Russel (1991)]. In this system, the strength of attractioncontrolled by the temperature of the suspension (in tcase changing the temperature changes the solvent quand thus the mixing contribution of the steric interaction

Korea-Australia Rheology Journal September 1999

atces attheyerhisater isd

theernd, athersld

described in the previous section). At higher temperatu(above 30oC) the attraction is small and the behaviour liquid-like. Below 30oC more solid-like behaviour isobserved due to the greater attraction, and G' is inependentof frequency. Figure 9 [Channell and Zukoski (1997shows the storage modulus G' of alumina suspensions adifferenct pH. It shows how G' depends on interparticleattraction when the attraction is controlled by DLVforces. At pH 9 (the iep) there is no charge on the alumparticles and the attraction, as well as the storage moduis at a maximum. As the pH is decreased the charge onparticles increases and thus the attraction decreases. Aattraction decreases so does G' at all volume fractions.Figure 9 also shows that the storage modulus of attractive particle network depends on the volume fra

Vol. 11, No. 3 205


arin-

be- themerer

. Inity,thelowions theandstheses

hemd

oly-lu-edtslso

um.lu-

andosemeearv-ic)-ac-cal to

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tion via a power law relationship in the range of 0.1 < φ< 0.4, (G' = Aφn, where A and n are experimentally deter-mined parameters). According to many researchers [Chan-nell and Zukoski (1997), Yanez and coworkers (1999)] thepower law exponent (n, found to be between 4 and 5 formany systems) is related to the structure and intercon-nectivity of the particle network.

3b. Polymer Solutions Polymer molecules differ significantly from particles in

that they are long chain molecules composed of manyrepeating units. The bonds along the polymer backbone arecontinually rotating, and as a result, despite maintaining anaverage shape and orientation, the molecule itself is con-tinually changing orientation and configuration on a lengthscale much smaller than the equilibrium size.

Dilute solutions are those in which each polymer chain isbelieved (or assumed) to be completely isolated from theother polymer chains, and forms a coil at equilibrium witha (measurable) radius of gyration. When the concentrationis increased such that the effective volume fraction of poly-mer molecules approaches one, polymer molecules beginto interact by becoming entangled. As the polymer molec-ular weight may vary from O(10000 daltons) to O(107 dal-tons), the concentration (known as the critical concen-tration c*) required for a solution to become entangled willdecrease as the molecule becomes longer and occupies alarger equilibrium volume. Polymer molecules may becharacterized by an entanglement molecular weight (tablesexist in The Polymer Handbook (1989), for example),which is the minimum molecular weight below whichentanglements will not occur.

In response to a deformation (such as a flow field foundin a rheological experiment), the polymer molecule itselfcan change both its shape and orientation. In dilute solu-tions, the rheology of the solution is dependent solely onthe dynamics of an individual chain and the number ofchains (i.e. the concentration) in the system. At higher con-centrations in the entangled region, interactions betweenpolymer molecules due to entanglements impact the rhe-ology in a significant way. This is true even for polymersin theta solvent in which there is no significant attraction orrepulsion between polymer molecules. Most researchersbelieve that entanglements between different polymerchains act to form a type of network between the mole-cules. The molecular weight and concentration of the poly-mer will significantly affect the rheological behaviour ofthe polymer solution.

Figure 10 shows the steady shear viscosity for a 411,000molecular weight polystyrene solution (Mw/Mn=1.01)[Graessley and coworkers (1967)] at a range of concen-trations. As expected the viscosity increases witrh increas-ing concentration. The figure also shows that at low shearrates the viscosity reaches a constant plateau (the zero-

shear viscosity, η0) for each concentration. As the sherate is increased, the polymer solutions exhibit shear-thning behaviour. The mechanisms behind the observedhaviour in steady shear may be understood in terms ofshear induced changes in the microstructure and polycharacteristics. At low shear rates the viscosity of polymsolutions remains constant at the zero-shear viscositythis regime, the solution behaves like a constant viscosisotropic fluid, as the stress produced is proportional to applied shear rate. Similar behaviour was observed at shear rates with some of the low concentration suspensdiscussed earlier. Shear thinning occurs because asshear rate is increased the polymer molecules orient align with the flow direction, thus reducing the drag. Athe shear rate is increased further, the alignment with flow becomes more complete, and the viscosity decreafurther.

The relaxation time of a polymer molecule reflects ttime required for the chain to relax back to an equilibriuorientation and configuration following the application ansubsequent removal of stress. It is a property of the pmer molecules primarily rather than a property of the sotion (i.e. system). The molecular relaxation time is affectby the ability of the polymer to relax given the constrainof adjacent molecules hindering free relaxation, and afactors such as the viscosity of the suspending medi

When subject to shear deformations, both polymer sotions and particulate suspensions in the semi-dilute concentrated regime behave as viscoelastic fluids whbehaviour is dependent on the ratio of the relaxation tito the reciprocal of the rate of deformation (i.e. the shrate or frequency). This results in more liquid-like behaiour at low shear rates (or frequency) and solid (or elastlike behaviour at higher deformation rates. If the charteristic relaxation time is small compared to the reciproof the rate of deformation, the system has sufficient time

Figure. 10. Shear viscosity of solutions of 411,000 MW polystrene in n-butyl benzene solvent, at different concentions as indicated in the figure. Adapted from Graeley and coworkers (1967).



x-hater

or acy1)].hetwo

its

neasthehe-

relax to its equilibrium state within the deformation timescale, so that a fluid-like behaviour with constant viscosityis observed and the viscous component of the shear mod-ulus is larger than the elastic component, i.e. G">G'. On theother hand, if the deformation time scale is much smallerthan the relaxation time, e.g. at high shear rates or high fre-quencies of deformation, the system shows a solid-likebehaviour with G' much larger than G".

Oscillatory experiments probe the structure of viscoelas-tic materials, and in particular the spectrum of relaxationtimes in a polymer system. The spectrum emerges as diffe-rent lengths, or segments, of the chain are studied with dif-ferent oscillatory frequencies. This probing of the relaxa-tion times separates oscillatory experiments for particulateand polymer solutions as there is no simple corollary forparticulate suspensions. Figure 11 shows the storage modu-lus of (entangled) polystyrene melts as a function of fre-quency for a range of molecular weights [Onogi andcoworkers (1970)]. The storage modulus is clearly a func-tion of frequency, but as the frequency is increased, thestorage modulus becomes constant for a broad range offrequencies. The region of constant G', known as the pla-teau modulus in polymer rheology, reflects the fact thatthere is little relaxation occurring in this range of fre-quencies. In the melts shown here, the relaxation is inhibi-ted by the entanglements. The value of the plateau modulusis independent of molecular weight, and is a strong func-tion of concentration for entangled polymer ( ~c2-2.25 [deGennes (1976), Graessley (1974), Masuda and coworkers(1972), Nemoto and coworkers (1972)]). However, as themolecular weight of the polymers in the melt increases,the range of frequencies over which G' is independent offreququency increases, reflecting the larger number ofentanglements per chain and the more hindered relaxationprocess. At lower frequencies, the storage modulus is

always a function of frequency. The lower frequency eperimental oscillations probe time scales longer than tof the longest relaxation time of the entangled polymsolution.

Figure 12 shows both the storage and loss modulus f411,000 MW polystyrene solution as a function frequenat several concentrations [Holmes and coworkers (197In this figure, the strong impact of concentration on tstorage and loss modulus is observed. At the lowest concentrations, G" always exceeds G', indicating that thesolution is unentangled (i.e. dilute) and more viscous in

GN0

Figure. 12. Storage and loss moduli for a 411,000 MW polystyrein Aroclor 1232 solvent, at different concentrations indicated in the figure. The open symbols represent loss modulus, G" and the closed symbols represent tstorage modulus, G'. Adapted from Holmes and coworkers (1971).


Figure. 11. Storage modulus of polystyrene melts at 160oC for arange of molecular weights. Adapted from Onagi andcoworkers (1970).

Figure. 13. Shear viscosity of 5 wt% polyacrylamide in aqueous so-lution, at different added salt concentrations. Adaptedfrom Harrison and Boger (1999).


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response to deformation than elastic. With the higher twoconcentrations, at low frequencies the viscous nature of thesolution dominates; however, there is a crossover frequencyabove which the elastic component dominates. This cross-over behaviour is characteristic of an entangled solution. Atfrequencies greater than the crossover frequency, the effectof the entanglements acts to restrict (but not completelyinhibit) relaxation, and G' becomes a weaker function offrequency. This response has in its limit the plateau modu-lus characteristic of highly concentrated polymer solutionsand melts (as was shown in Figure 11).

The solvent quality can have an effect on the rheologicalbehaviour of polymer solutions. Changing the quality ofsolvent changes the interactions between polymer seg-ments both within and between polymer molecules. Dec-reasing the quality of the solvent causes the polymer seg-ments within one molecule to be attracted to each othermore than to the solvent, thus causing the polymer mol-ecule to coil up more tightly into a smaller volume. Thisdecreases the effective volume fraction of the polymersolution as compared to the same concentration (in weightpercent) of polymer in a better solvent. Reducing the effec-tive volume fraction decreases the viscosity. Figure 13shows the shear viscosity of a 5 wt% 3 million molecularweight polyacrylamide solution as a function of added salt(NaCl) [Harrison and Boger (1999)]. The added salt hasthe effect of decreasing the quality of the solvent and thusthe volume occupied by the molecules. As a consequenceof the lower effective volume fraction the shear viscosity ofthe solution decreases.

3c.Polymer GelsGels provide an intermediate system between the rhe-

ology of suspensions (both particulate and latex) and poly-mer solutions. They provide a link to help understand thesimilarities between all the systems under discussion. Poly-mer gels differ from polymer solutions in that there arebonds between different polymer chains. The strength ofthese bonds may vary from very strong to very weak andwill have a significant impact on the rheology. Irreversiblechemical bonds are the strongest, and can be createdthrough the addition of sulfur containing compounds orsimilar crosslinking agents to the solution (or melt), re-sulting in a highly crosslinked elastic solid rubber-like ma-terial. These types of gels are characterized by the fact thatthey behave as viscoelastic solids under small deformationbut rupture rather than flow at a critical strain value. Bondsof lesser strength (such as ionic bonds) that may be revers-ibly formed and broken may produce gels that have vis-coelastic rheological behaviour similar to suspensions inmany cases, due to the breakdown of the network structureinto smaller clusters with increasing shear. Physical bonds,such as van der Waals and hydrogen bonds, are charac-teristic of weaker gels, which show rheological behaviour

with more viscous and less elastic character. Physical bomay be easily broken in response to a deformation, reform when a new equilibrium state is reached; howevthey are stronger than the simple entanglements typicamore concentrated polymer solutions. In addition to tstrength of the bond between the polymer chains, the nber of bonds per unit volume and the structure of the nwork affects the strength of the gelled network. In macases many relatively weak hydrogen bonds work in conto produce helical structures that can be relatively stron

The existence of bonds between different polymer mlecules, rather than the entanglements characteristicconcentrated polymer solutions, gives rise to some sudifferences in the rheology of polymer gels when compared to polymer solutions. As the number of bonincreases the gel becomes stronger and more elastic behaviour. Also, because true bonds are stronger tsimple entanglement effects, gels are generally melastic than polymer solutions at an identical concentratio

The bonds between the molecules in a gel also resthe relaxation processes of the system when comparepolymer solutions. In polymer solutions, molecules ccompletely disentangle from one configuration and diffuinto another. However, the bonds that form the gel netwlimit the ability of the molecules to move, or relax, afreely as in a solution, hence greatly increase the relaxatimes of the system.

Figure 14 shows the shear stress versus shear rate series of scleroglucan aqueous system polymer geldifferent concentrations ranging from 0.13% to 1.13% [Lpasin and coworkers (1990)]. Scleroglucan is a polysaccride that gels by forming helicies held together by intestrand hydrogen bonds. As the concentration increasesshear stress, and hence the viscosity, increases. The calso illustrate that polymer gels are shear thinning as

Figure. 14. Shear stress versus shear rate for a series of scglucan aqueous gels at different concentrations as icated in the figure. Adapted from Lapasin and cworkers (1990).



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shear rate is increased. As observed with the other systemsalready discussed, the gels become more shear thinning asthe concentration is increased.

Figure 14 illustrates that not all gels have the zero-shearrate limit viscosity characteristic of polymer solutions. Asthe concentration is increased, it is apparent that the shearstress appears to level off and reaches a constant value asthe shear rate decreases towards zero. This behaviour ischaracteristic of a yield stress, or a minimum stress belowwhich the gel will not flow. The presence of a yield stresswas also observed in several suspension systems discussedin previous sections. The yield stress exists because thelarge number of hydrogen bonds in the helix structureresults in a stable configuration that shows resistant to flow.Only when a sufficient stress is applied (i.e. the yieldstress) is this structure broken down. Subsequently, ori-entation of the polymer chains occurs, resulting in shearthinning flow behaviour at higher shear rates, in a man-ner similar to that encountered in polymer solutions. How-ever, it is difficult to truly characterize the yield stress of thissystem given that the minimum shear rate explored is just0.2 sec-1. Commercial rheometers of the constant stress typeare capable of shear rates of the order of 10-3 sec-1.

Figure 15 shows the dynamic storage and loss modulus,G' and G" respectively, for a strong gel: amylose in 0.2 MKCl at two different polymer concentrations [Doublier andChoplin (1989)]. Amylose (a constituent of starch) isanother polysaccharide that forms gels via hydrogenbonding and helix formation [Lapasin and Pricl (1995)].The addition of salt produces a strong gel due to cationsbridging and bonding the negatively charged helices intostronger structures. Several trends typical of strong poly-mer gels are evident in this figure. Most importantly, thestorage modulus G' is independent of frequency over abroad range (four orders of magnitude) of frequency. Oneof the definitions of a strong gel is a constant storage mod-ulus. This constant storage modulus indicates that there is

little relaxation occurring over these frequencies and gel is acting like an elastic solid. Second, the loss moduG", is approximately two orders of magnitude smaller ththe storage modulus. This dramatic difference in the mnitude of the two moduli is further evidence that the strogel acts like a solid and the viscous contribution to the rology is dominated by the elastic contribution. As epected, as the polymer concentration is increased, bothstorage and loss moduli increase, indicating that the mpolymer contained in the gel, the greater elastic charait exhibits.

Figure 16 shows the storage modulus of Xanthanwater [Carnali (1992)], at concentrations ranging fro0.3-3% polymer. Xanthan forms a weak gel in water dto the formation of hydrogen bonded helical structures. can be seen in the figure, at lower concentrations, storage modulus is a strong function of frequency, wheras the concentration is increased G' shows less dependencon frequency and the gel is becoming more solid-likClearly, there are significant differences between the strgel shown in Figure 15 and the results for xanthan shohere. Even at the higher concentrations, G' is not inde-pendent of frequency, indicating that there is some relation occurring over the frequency range. Some of thedifferences may be due to the nature of the bonds in xanthan gel. The physical bonds characteristic of this stem are much weaker than those in the strong gel. decrease in G' at lower frequencies is indicative of a morviscoelastic, rather than purely elastic response. Althouthe results are not shown here, as with the strong gels G" islower than G'.

A strong dependence of the storage modulus of strgels on the concentration of polymers is typically observIn many cases the storage modulus is a power law funcof the concentration with a power law exponent grea

Figure. 15. Storage and loss moduli of 1.03 and 1.33 wt% amylosegels in 0.2 M KCl. Adapted from Doublier and Choplin(1989).

Figure. 16. Storage modulus of xanthan solutions at different ccentrations from 0.3 to 3 wt %. Adapted from Carn(1992).



sestherses,msandear

isndearereklyro-

ectre

hei-

onshow

-u-

kee

theicorasede

gels

inning

than 2 [Lapasin and Pricl (1995)]. This behaviour is similarto the storage modulus-volume fraction (or concentration)relationship of attractive particle systems and highlyentangled polymer solutions.

The figures shown above for the polymer gels demon-strate how the rheology of polymer gels may be seen as asort of bridge linking that of particulate suspensions andentangled polymer solutions and melts. In both the simpleshear and the dynamic oscillatory experiments, the resultsindicate the links in the physical underpinnings of all thesystems under discussion. In the following section, thesecommon links are explored further.

4. Discussion-Similarities and Differences

In this section, the rheological trends are used todemonstrate the similarities in the underlying physicalprinciples of all the systems. The experimental results haveshown conclusively that many of the basic rheologicaltrends are very similar for particulate suspensions andpolymer solutions. Despite the substantial differences inthese systems, we believe that the rheology can be ex-plained in general terms by similar ideas.

Table 1 summarizes the trends that occur in the basicshear rheology for all the systems discussed here. Therheological behaviour highlighted in the table only showsgeneral trends that have been observed and may not applyto all situations. The results also vary significantly withconcentration. Consequently, the comparisons are made atthe concentration which highlights the similarities betweenthe systems. The shear viscosity and dynamic propertiesare compared in the concentrated regime where substantialinterparticle interactions between particles in suspensions,or entanglements between polymer molecules in solution,exist. The yield stress row in the table denotes if and atwhat concentration it occurs.

The commonalities observed in the rheological responof the different systems discussed may not extend to oexperiments such as extensional flow and normal stresand there may be significant differences in the systeresponses in these experiments. Both extensional flow normal stresses are characteristic of highly non-linresponses to deformation (the structure of the systemdisturbed by the applied deformation). The materials aexperiments discussed in this work are either in the linviscoelastic regime (i.e. the oscillatory experiments whstructure is probed but not perturbed), or in the weanon-linear regime, simple shear viscosity, where hyddynamic forces are also a factor.

ConcentrationThe first common element in all the systems is the eff

of concentration of the particles or polymer on the natuof rheological behaviour. Generally, one can divide tconcentrations into three distinct regimes: dilute, semdilute and concentrated systems. The polymer solutiand particulate suspensions in these three regimes sdistinctly different rheological behaviour.

In the limit of very dilute concentration of non-interacting particles and polymers, the viscosity of both particlate suspensions [Jones et al. (1991)] and polymer solu-tions [Allcock and Lampe (1981), Schulz and Blasch(1941)] increases linearly with concentration or volumfraction (see Figure 17). Einstein (1906) showed that increase in viscosity is simply due to the hydrodynaminteraction of the particle with the continuous medium. Fslightly higher concentrations, Einstein’s relationship, extended by Batchelor (1977) is appropriate. It is bason the hydrodynamic interactions of the fluid and th“particle” and is as follows:

(5)ηrηηs----- 1 2.5φ 6.2φ2 ...+ + += =

Table 1

insoluble particles soluble polymers

hard spheres repulsive stabilized particles attractive particle networks polymer solutions polymer

Shear viscosity Shear thinning Shear thinning Shear thinning Shear thinning Shear th

Yield stress no** at high φ yes at low φ no yes at low φ

Storage modu-lus

G' = G'(ω)G' = G'(ω) or G' = constant*

G' = constantG' = G'(ω) or G' = constant*

Weak gel G' = G'(ω)Strong gel G' = constant

Loss Modulus

Low ω G" > G'

High ω G" < G'

G" = G"(ω)G" < G' or G" > G"*

G" = G"(ω)G" << G'

Low ω G" > G'High ω G"<G'

G" = G"(ω)G" << G'

* depending on the concentration** except at volume fractions exceeding random dense packing



heeenionifi-

pre-teingly-oilstem.the

rac-

ae

(b)-

-e.

where ηr is the relative viscosity (ηr = η/ηs, where η is theviscosity of the system and ηs the viscosity of the con-tinuous medium) and φ is the volume fraction of "parti-cles". In the case of insoluble particles, φ is simply thevolume fraction. When the particles are soluble polymermolecules, φ is the effective volume fraction of the poly-mer coil and any included solvent. The Einstein-Batchelorrelationship given in Equation 5 holds for both dissolvedpolymers and macroscopic particles, but only in the limitof dilute concentration. In this concentration regime, theparticles or polymer molecules are sufficiently far apartfrom each other that the interactions between “particles”are negligible. For solutions of polymers dissolved in asolvent, the polymer molecules exist as random coils atequilibrium with a (measurable) radius of gyration. Therheology of both the particulate suspensions and polymersolutions is dependent on the dynamics of the particles,with Brownian forces dominating, and the viscosity isindependent of deformation rate.

At somewhat higher concentrations in the semidiluteregion, the viscosity-concentration relationship shows

deviations from the Einstein-Batchelor equation. As tconcentration of "particles" increases, the distance betwparticles becomes comparable to the particle dimensand interactions between "particles" become more signcant. This leads to viscosities much greater than that dicted by Einstein’s simple relationship. In a particulasuspension, the particles interact with many neighbourparticles via hydrodynamic and surface forces. In a pomer solution, as concentration increases the polymer cinterpenetrate one another and form an entangled sysThese interactions play a significant role in determining rheology of the system.

In the case of particulate suspensions, the volume f

Figure. 18. (a) Low shear rate limiting relative viscosity as function of volume fraction for silica hard sphersuspensions and Quemada's equation with φmax=0.63. Adapted from Jones and coworkers (1991). Low shear rate limiting viscosity versus concentration for polystyrene solutions in toluene for different molecular weights as indicated in the figurAdapted from Onogi and coworkers (1966).

Figure. 17. (a) Relative viscosity as a function of volume fractionfor silica hard sphere suspensions. The line in the fig-ure is Einstein’s relationship (1906). Adapted fromJo- nes and coworkers (1991). (b) Relative viscosityas a function of concentration of poly (methyl meth-acrylate) in chloroform. The line is a best fit to thedata. Adapted from Shulz and Blaschke (1941) asreported by Allcock and Lampe (1981).



hly

oftheehendere,er isoname

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tion dependence (at high volume fraction) of the relativeviscosity can be predicted by one of several empiricalrelationships, which usually rely on a maximum volumefraction of packing. The viscosity results for "hard sphere"dispersions at high concentrations are modeled well byQuemada's expression (1984):

(6)

where φmax is the maximum packing volume fraction(usually around 0.64). The above equation is similar to thatof Krieger and Dougherty (1959). This relation can be seenin Figure 18a for a suspension of silica hard spheres [Jonesand coworkers (1991)].

In a similar way to suspensions, the viscosity of polymersolutions in the semi-dilute and concentrated regime is astrong function of concentration. In the limit of dilute con-centration, the (relative) viscosity of the solutions shows alinear relationship with concentration up to the criticalconcentration, c*. Above c*, due to the entanglement in-teractions amongst the molecules, the viscosity of the so-lutions increases drastically with concentration to a powerbetween 4 and 5 [Doi and Edwards (1986)], as shown inFigure 18b for solutions of polystyrene of differentmolecular weights in toluene [Onogi and coworkers (1966)].

For particulate suspensions at high concentrationsapproaching the maximum volume fraction of packing,the particle-particle distance becomes very small comparedto the particle dimension. Strong interparticle interactionsbetween any particle and its many-bodied neighboursdevelop to such an extent that the particles form structurethat resists deformation. The rheology of these highlyconcentrated particulate suspensions is significantlydifferent from the less concentrated suspensions and mayresemble an elastic solid, as shown previously.

In the non-interacting case the influence of one particledispersed in the continuous fluid medium on another is dueonly to hydrodynamic forces. For a suspension of particleswith repulsive double layer interactions, the viscosity isincreased above that of hard spheres because of the addi-tional dissipation created by the distortion of the doublelayer by shear (primary electroviscous effect [Hunter(1987)]). At higher concentration the range of the repul-sive forces act to increase the effective volume fraction ofthe suspension due to the additional excluded volume cre-ated by the repulsive force field, resulting in a soft-sphere system. The additional increase in viscosity due tooverlapping double layers is known as the secondary elec-trovi- scous effect [Hunter (1987)]. Similar overlap ofadsorbed or grafted polymeric steric layers also results inincreased viscosities. The soft inter-particle interactionallows the suspension to flow at a higher effective volumefraction, (although lower actual volume fraction) due to thecompressibility of the soft stabilizer layer. Also in this con-

centration regime all the systems investigated show higshear thinning viscosity behaviour.

Figure 19 shows the relative viscosity as a function volume fraction for suspensions of polystyrene wigrafted polyethylene oxide chains (MW = 2000), at thrdifferent particle sizes [Tadros (1996)]. In this case tvolume fraction φ is based on the particle core radius (anot the effective volume fraction which includes thstabilizer layer thickness). As can be seen from the figuthe ηr-φ curves for the smaller particles shifts to the lowvolume fraction compared to the larger particles. Thisdue to the increasing effects of the “soft” polymer layer the surfaces as the particle size decreases for the sgrafted polymer.

As concentration further increases both particulate fluand polymer solutions exhibit a change from fluid-like more elastic-like behaviour. In the suspension systems,increased concentration brings the particles closer togeThis has two consequences. As the particles come cltogether, the repulsive interparticle forces become stronAlso, as the concentration increases the particles are phcally restricted by other particles in their motion in reponse to an applied deformation. This restriction requithat particles move in conjunction with other surroundiparticles. These two effects tend to increase the restoforces keeping particles in their equilibrium positions, thpromoting elastic behaviour. In polymer systems, the creased concentration results in more entanglementsbonds between the polymer molecules. A consequencthis is that the system appears more elastic and

ηr 1 φφmax---------–

2–

=

Figure. 19. Relative viscosity versus volume fraction of polystyrene dispersed with grafted polyethylene oxide, three different particle sizes as indicated in the figuAdapted from Tadros (1996).



es.t) isingressaksatDoi

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gle-tion

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fluid-like.The influence of concentration on the rheological be-

haviour is very similar in all the systems compared in thiswork. When looked at from the point of view of effectivevolume fraction and the increased interaction between dis-persed phase entities as concentration is increased, thecommonality becomes clear.

Shear ThinningThe results presented in the previous section show that,

at the moderate concentrations (and above), all the systemsinvestigated exhibit shear thinning behaviour as the shearrate is increased. This phenomenon is perhaps the mostcommon rheological similarity between all the systems. Atrest all the systems discussed are in an isotropic, equilib-rium configuration. There is no preferential alignment ofthe particles or molecules. As the deformation is applied,the dispersed phase (particles or polymers) forms a prefe-rential flow structure. In many cases the dispersed phasetends to align with the flow field. This results in an aniso-tropic system that is more efficient for flow, and the vis-cosity subsequently decreases. As the rate of deformationis increased (i.e. at a higher shear rate), the particles ormolecules will align more completely with the flow fieldand the viscosity will decrease further. In all cases theshear thinning behaviour becomes more prominent as theconcentration is increased due to the increased low shearviscosity as dispersed phase entities are crowded closertogether and interact more.

In suspension rheology, the re-orientation of the particlesresults in an alignment of particles that is most favourablefor effective flow. However, in many cases the particles arespherical entities, thus the individual particles themselvescannot aid in alignment. Rather, it is multiple particles thatmust align and configure themselves in concert, rather thanindependently. Suspension rheologists have observed thecoordinated alignment of particles into “sheets” or “super-structures” that result in a decreased viscosity [Chen andcoworkers (1994)].

Polymer solutions shear thin because the long chainpolymer molecules move from an isotropic, equilibriumconfiguration between the entanglements to an oriented,deformed (from equilibrium) configuration, where themajority of molecules are similarly aligned. The inter-molecular effects and interactions, in particular the entan-glements, may actually be reduced due to the anisotropyresulting from the deformation. The net effect of shearthinning results from the similar orientation of many poly-mer segments within the sample.

Attractive particle networks show additional shearthinning due to the need to breakdown the bonds betweenparticles. The additional shear thinning is due primarily tothe increased stress required for flow at low shear rates. Atrest these attractive particles form a space filling elastic

particle network due to the attraction between particlSome finite stress (the yield stress to be discussed nexrequired to make the suspension begin to flow by breakdown the network. At stresses greater than the yield stflow begins. At greater stresses the particle network bredown into smaller and smaller flow units resulting in flow higher shear rates and lower viscosities [Chen and (1999)].

In polymer gels, the segments of polymer chains btween the bonds also align in the flow direction. Raththan maintain the isotropic configuration, the polymbackbone responds to the deformation via alignmeAdditionally, because the imposed deformation results isignificant strain being placed on the sample the bondthe gel may be broken. This behaviour is analogous toloss of entanglements in polymer solutions due to aligment. More bonds may be broken as the shear ratincreased. An equilibrium is reached where the ratebreaking equals the rate of reforming of the bonds.

The steady shear viscosity, in particular shear thinnimay be explained using similar arguments for both pticles and polymers. Major physical differences are thsuspensions rely on a coordinated motion of many pacles. In polymer solutions and gels, the polymer configration changes both due to intra-polymer orientation adue to the interaction between polymers at the entanment or bond level. The stresses caused by the deformadominate these forces and are capable of changingsystem configuration. In all cases, the rest configuratiowhich are defined and controlled by the surface forcbonds, and Brownian motion, are significantly perturbby the deformation to result in more favourable floconditions.

Yield StressThe meaning and existence of a shear yield stress

been a question of debate almost ever since it was cceived [Barnes (1999), Barnes and Walters (1985), Hartand Hu (1989)]. It appears that there exists no real yistress since even at an infinitesimally small stress, material will flow (creep) if the time scale of the experment is sufficiently long [Barnes (1999)]. Nonetheless tengineering reality [Hartnett and Hu (1989)] of a yiestress, which depends upon the time scale of the msurement, is a useful concept [Barnes (1999)]. It will allocomparison of the relative strengths of attractive particulnetworks and polymer gels for instance. Yield stressesparticulate fluids are typically measured by the vatechnique [Nguyen and Boger (1983, 1985)] at a ratestress increase in the order of 5 to 500 Pa/s (the time sof the measurement) [Liddell and Boger (1996)].

Table I indicates that the attractive particle systems alowith the polymer gels, can develop an apparent yield strunder appropriate conditions. For both systems, the y



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stress is only evident at higher volume fractions of particleswhen a space filling percolative network exists. In theattractive particulate systems a minimum volume fractionis required for a yield stress to exist. This volume fractionis known as the gel point (φg) and it decreases as attractiveforces between particles are increased. The gel point of anattractive particulate suspensions is usually of the order of0.05 to 0.15 volume fraction. In polymer gels, a yield stressis found above a critical polymer concentration (unique foreach polymer-solvent system).

In attractive systems the yield stress is a manifestation ofthe attractive forces (bonds) acting between the particles(or polymers). The yield stress is related to the additionalforce required to break the bonds between particles orpolymer chains that hold the network together before flowcan begin. In the attractive suspension, there are attractivesurface forces between the particles. In polymer gels it isthe interpolymer bonds that create the network. Thestrength of the network depends upon the strength ofindividual bonds, the number of bonds per unit volumeand the structure of the network. At low applied shearstress the strength of the network exceeds that of theapplied deformation, and the network deforms elasticallybut does not yield. Only when the applied stress exceeds acritical amount, the yield stress, does the system flow.

In general (although not always) repulsive particle sys-tems do not exhibit yield stresses. Repulsive suspensionscan exhibit yield stress when the range of the repulsion isgreater than the separation distance between dispersedphase entities (i.e. at volume fractions approaching themaximum packing density). The long range repulsiveforces between the particles force the particles to try andmaintain the maximum distance possible from their neigh-bours. This maximum distance is limited by the volumefraction. The forces exerted upon a particle by its neigh-bours holds the particle in a position midway between itsneighbours. Thermal motion is insufficient to move theparticles from this minimum energy position, when themagnitude and range of the forces is sufficient. The yieldstress is the applied stress necessary to overcome theforces resisting deformation of the network in the repul-sive systems.

At volume fractions above that of dense random packing(approx. 0.637 volume fraction) hard sphere suspensionsappear to have a yield stress [Jones and coworkers (1991)].Due to geometric constraints the particles do not have timeto rearrange in response to the applied deformation withinthe time scale of the experiment.

The systems described above that exhibit a yield stressare fundamentally altered when the applied stress exceedsthe yield stress. The surface forces or bonds that form thenetwork must be overcome (or broken) in order for thematerial to flow. However, it should be noted that upon theremoval of an applied stress, the network structure of all

the systems will reform.In general, entangled polymer solutions do not exhibi

yield stress at time scales typically used for rheologimeasurements. The entanglements between chains arfixed bonds in the sense found in polymer gels, or strong surface forces found in the suspensions. Rathermolecules are continually entangling and disentangling, these interactions are capable of sliding along the chairesponse to a deformation. As a consequence, in respto an imposed force, the entangled polymer chains wreorient and deform without requiring a critical appliestress. The interactions between the polymer chains docreate a strong network. Rather than a yield stress, a iting zero-shear viscosity will always be measured in ttimescale achievable in any current experiment.

Viscoelastic BehaviourDynamic oscillatory shear experiments are capable of

rectly probing the structure and viscoelasticity of a sampSo long as the deformation is kept below a limit the rstructure of the sample is not perturbed. Within this regof linear viscoelasticity the interactions between the dpersed phase entities and the rest structure can be intigated without the influence of hydrodynamic effects. Tstorage modulus G'(ω), describes the elastic response ofsample to a small amplitude oscillatory deformation. Lithe other rheological properties already discussed, storage modulus is a function of concentration (or volumfraction). Additionally, for most of the systems, it is alsofunction of frequency ω.

As seen earlier, the storage modulus of particulate spensions is strongly dependent upon the volume fractionparticles and the nature of the interactions, or surfforces, between particles. At very high volume concenttions of non interacting particles, the storage modulus comes independent of frequency (although dependent uconcentration). The constant value of G' is a consequenceof the inability of the particles to significantly rearrangthemselves in response to the oscillatory perturbation to concentration approaching the maximum packing frtion. As the volume fraction of particles is lowered, thstorage modulus is shown to be a function of the frequeof oscillation. With slow oscillations, the particles arcapable of rearranging within the time scale of the pturbation due to the additional unoccupied volume avaable: thus G' is lower. As the frequency is increased, thnetwork structure is unable to re-orient on the time scof the experiment and the network appears more elasG' therefore increases.

There is similar behaviour for the repulsive suspensioalthough solid-like behaviour is observed at lower volumfractions than hard spheres because the overlapping resive forces. The repulsive forces create a potential eneminimum in which the particles reside. The curvature



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this minimum is related to the storage modulus [Buscalland coworkers (1982)]. At low volume fractions, G'increases with increasing frequency. As the volume frac-tion is increased, G' becomes a weaker function of fre-quency and reaches a plateau value at higher frequencies.This is due to the particles both being unable to recon-figure at the higher frequencies and the forces betweenlarge numbers of adjacent particles.

Storage modulus of attractive particles systems isgreater than systems of repulsive or non-interacting par-ticles of the same size at the same volume fraction. Themore solid-like behaviour is due to the deep attractiveenergy minima that tend to keep the particles in their restconfiguration. G' is related to the depth of the potentialwell (through the curvature of the well at its minimum)with deeper wells (stronger attraction) producing greaterstorage moduli [Yanez and coworkers (1996)].

The storage modulus of polymer gels and entangledsolutions is generally described through an understandingof the configuration of a polymer molecule and the inter-actions between different chains. However, these aspects ofthe polymer are inextricably linked to the relaxation timesof the polymer. The concept of polymers having a spec-trum of relaxation times, due to different relaxation mecha-nisms occurring along the chain, is a significant differencebetween the polymer solutions and the suspensions dis-cussed previously. For polymer systems, the frequencymultiplied by a relaxation time is crucial to understandingthe storage modulus. This is because both the storagemodulus and the relaxation time characterize the elasticityof the polymer.

Polymer gels are characterized as either strong or weakgels depending on frequency dependence of the storagemodulus. Strong gels are those gels in which the storagemodulus is constant over a broad range of frequencies. Theconstant value is due to the inhibition of relaxation proc-esses due to the large number of bonds that are acting tomaintain the gel. The small perturbation characteristic ofan oscillatory experiment is not large enough to break thesebonds between the molecules. Furthermore, these bondsare so prevalent in the system that the network responds inthe same elastic manner no matter what frequency (orrelaxation time scale) is probed.

Weak polymer gels demonstrate a frequency dependencein the storage modulus. This is a reflection of the strength(and probably the quantity) of the bonds holding the geltogether. The bonds holding the gel together are lessfrequent along the polymer chains, and as a result somerelaxation can occur in response to a perturbation. There-fore, at low frequencies, the weak gel has a storage mo-dulus that increases with increasing frequencies. Athigher frequencies, the storage modulus becomes a weakerfunction of frequency indicating that the experiment is nowprobing length and time scales where the gel structure is

more established and elasticity dominates.The storage modulus response for entangled polym

solutions is similar to that of weak gels. At low frequencies, which corresponds to time scales longer tthat of the longest relaxation time of the polymer, the stage modulus increases with increasing frequency. Upreaching a certain frequency (that corresponds to ωτ~1,where τ is the longest relaxation (disengagement) timethe polymer), however, the elasticity of the sample remaconstant for a wide range of time scales (and frequencThis constant (as a function of frequency) storage moduis known in polymer rheology as the plateau modulus, athe range of frequencies it encompasses reaches to lofrequencies as the molecular weight of the polymerincreased. The width of the plateau modulus increasemolecular weight increases because the longest relation time of the polymer increases (and therefore the lfrequency limit of the modulus decreases) with increasing molecular weight.

The discussion presented above has demonstrated the storage modulus is best described in terms of the fobetween particles and polymer segments (depending onsystem). For suspensions, the surface forces and crow(volume fraction) effects between adjacent particles demine the storage modulus results. For polymer gels entangled solutions, it is the chain elasticity and spectrof relaxation times associated with that elasticity that is prime determinant of the storage modulus as a functionfrequency.

As shown previously with the shear viscosity and yiestress, the concentration (or volume fraction) of the pticles has a significant impact upon these rheologitrends. Both attractive particle networks and polymer stems show power law dependence of storage modulusthe concentration. The power law exponents are usuallthe range of about 4 to 5 for the particle networks [Chanell and Zukoski (1997), Yanez and coworkers (1999)] abetween 2 and 7 for polymer gels [Lapasin and Pricl (199For entangled polymer solutions, the exponent is lowand has a value of approximately 2 to 2½ [de Genn(1976)]. The very strong dependence of the storage mdulus on the concentration, in all the systems, is relato the similar increasing interconnectivity of the perclating network with increasing concentrations [Stauffand coworkers (1982)].

The loss modulus, G", is a measure of the viscous reponse of a sample to an oscillatory perturbation. As with storage modulus, it is generally a strong function of cocentration. In the hard sphere suspensions, repulsive pensions, and polymer solutions, G" exceeds G' at lowconcentrations, although the concentration dependevaries between the different systems and the different cacteristics of the polymer molecule or particle being invetigated. In this regime, the sample is said to be dominated



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viscous effects. As the concentration is increased for par-ticulate suspensions and sterically stabilized suspensions, G'exceeds G" at all frequencies. This indicates the system isprimarily elastic at all frequencies and the surface forcesacting between the particles restrict viscous dissipation.

For strong polymer gels at high concentrations andattractive particle systems above the gelpoint, the lossmodulus is always one or two orders of magnitude smallerthan the storage modulus. This is a consequence of thebonds maintaining the network structure and limiting theviscous characteristics of the system.

For entangled polymer solutions, the situation is com-plicated by the transient nature of individual entanglementsand the ability of a solution to reconfigure molecularorientation in response to a deformation. At low frequen-cies, G" exceeds G', indicating the dominant viscous natureof the material under these conditions. At higher fre-quencies, the storage modulus exceeds the loss modulus.In this frequency regime, the elasticity of the sample, dueto the entanglements and chain elasticity of the mole-cules, dominates the oscillatory response and in fact leadsto a secondary minimum in the loss modulus.

Polymer Deformability and EntanglementsThe two polymer systems under consideration, polymer

gels and entangled polymer systems, differ from suspen-sions in that the polymer molecules are long chains thatcontinually change configuration (although, on a macro-scopic scale, equilibrium configurations on average rema-in unchanged). In response to a deformation, the confor-mation of the molecule can change dramatically, as thepolymer orients and/or stretches out. These changes im-pact the rheology. Due to the anistropy and departure froman equilibrium configuration, there will be an increase inthe stress in the system. The elastic stress in these systemsrelaxes as the polymer returns to an equilibrium configu-ration. An aspect of the stress in the system that differsfrom that found in particulate and latex suspensions is thatwhen the polymer stretches in length from its equilibriumconfiguration, there may be a substantial increase in stressnot associated with orientation effects. This additionalcontribution to the stress is due to the elasticity of thepolymer chains. As an illustration, consider an elasticband. When it is stretched from its equilibrium shape, itacts to retract to its original condition. The same conceptis found in a polymer solution: the polymer moleculewants to retract to the equilibrium, isotropic configuration.Also unlike the surface forces acting in suspensions thatdecrease as separation distance increases, the elastic stressdue to streching polymer molecules increases signifi-cantly, and nonlinearly, as the molecule is stretched fur-ther from equilibrium. This is because the polymer strandmay be stretched from its initial equilibrium configurationto a configuration approaching its contour length. Given

the differences in entropy, the stress is much greater wthe polymer is highly deformed.

Furthermore, the interactions between “particles” (ipolymer molecules) are highly dependent on the maconfigurations that the molecules can undergo. For polymer gel, the points of interaction between molecuare the chemical or physical bonds that connect the chand form the network. These bonds can be very difficultbreak. For entangled polymer solutions, entanglemebetween adjacent molecules are where chain-chain inaction is centered. In entangled polymer solutions, thare no “bonds” as such. However, due to the concentraof the molecules, and the continual motion of the segmeof the polymer chain due to Brownian forces, the polymcontinually intertwine with one another in a way thresults in, on average, a continually entangled solutiThese entanglements tend to form a weak network betwthe polymer molecules. However, because the entanments continually form and break, and may slide along molecule in response to a deformation, the effect of entanglements on the rheology may be expected to dslightly from the bonds in polymer gels.

Hard particles do not exhibit these characteristics tycal of polymer systems. In many cases, it is the congurational changes in polymer systems that cause typical rheological response. This behaviour contrawith hard particle systems in which it is the forcebetween discrete particles that are responsible for rheological trends.

5. Conclusions

The similarities and differences in the rheology of paticulate suspensions and polymer solutions have been sented and discussed. In many cases the similar rhogical behaviour can be linked to similar physical priciples that are responsible for the characteristic behavof all the systems discussed. The primary similarities athe change from Newtonian to shear thinning behaviowith increasing concentration, the change from fluid-likto solid-like behaviour with increasing concentration athe greater shear thinning and solid-like behaviour attractive networks (both particulate suspensions and gThe primary difference is the ability of polymer moleculeto become entangled and change conformation whinsoluble particles cannot.

It is apparent that due to the similarities in behaviourthe different systems that rheological experiments aloare insufficient to characterize a particular unknown fluIt is necessary to know (by independent means) the phcal and chemical make up of the system. Some questthat need to be answered may include: is the materialganic or inorganic; what is the continuous phase; is solid phase soluble; what is the concentration of the d



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persed phase; what is the size or molecular weight of theparticles; and how do the dispersed phase entities interactwith each other? Only after these and other questions havebeen answered will it be possible to fully understand thesystem and control the conditions to achieve the desiredrheological behaviour.

The ideas in this paper can provide a basis for cross-field discussion. Rheologists, and those who employrheological techniques, have insights that can be valuablein many different research fields. This paper is designed topromote the exchange of ideas and concepts across theboundaries of traditional research groupings, such ascolloids and surface forces and polymer dynamics, andbegin a dialogue between the varied groups of rheologists.

Acknowledgments

Research in particulate fluids in the Advanced MineralProducts Research Centre at the University of Melbourneis funded by an Australian Research Council SpecialResearch Centre Grant. Research in non-Newtonian fluidmechanics at The University of Melbourne is funded by aSpecial Investigator Grant of the Australian ResearchCouncil. The authors would like to acknowledge all theauthors upon whose work this review is based. Thanks toRod Binnington for preparing the figures. Thanks to DaveDunstan for useful discussions.

List of Symbols

A : Hamaker constantc : concentration of polymer solutionc* : concentration at which polymer molecules become

entangledD : surface to surface distance between particlesF : surface forces between particlesG' : storage modulusG" : loss modulus

: plateau storage modulusMN : number average molecular weightMW : weight average molecular weightR : radius of particlesS : distance between polymer strands on particle sur-

facesT : temperature (in Kelvin)V : potential energy between particlesχ :::: Flory-Huggins constantδ : adsorbed polymer layer thicknessε : permeability of waterε0 : permeability of free spaceφ : volume fractionφg : volume fraction at which gel occurs φ : maximum packing volume fractionη : shear viscosity

η0 : zero-shear viscosityηr : relative viscosityηs : viscosity of suspending medium or solventκ : inverse Debye lengthψ0 : surface potential due to surface charges

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Suspensions and polymers - Common links in rheology · Key words : rheology, polymer melts, polymer solutions, suspensions 1. Introduction The field of rheology has developed significantly