Rheology Suspensions

8/2/2019 Rheology Suspensions

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Progress in Organic Coatings 40 (2000) 111117

Rheology of sterically stabilized dispersions and latices

Jan Mewis, Jan VermantDepartment of Chemical Engineering, de Croylaan 46, Katholieke Universiteit Leuven, 3001 Leuven, Belgium

Abstract

Steric stabilization is a method that is often used to properly disperse small particles. It can be applied in aqueous as well as non-aqueous

media. The rheologicalproperties of sterically stabilized dispersions are discussed here. The various controlling parameters and the physical

mechanisms involved are reviewed. Brownian hard spheres are used as a reference. Scaling relations are presented that make it possible

to reduce data sets and to predict properties. Viscosity, yield stress, shear thickening and viscoelasticity are included. The rheological

properties are also related to the fundamental colloidal properties of the dispersions under consideration. Quantitative results are available

for monodisperse spherical particles, although the effects of particle size distribution can sometimes be predicted also quite well. In othercases the procedures presented here can be used qualitatively to predict viscosities. 2000 Elsevier Science S.A. All rights reserved.

Keywords: Rheology; Latex; Suspensions; Steric stabilization; Viscosity predictions; Viscosity scaling; Dynamic moduli

1. Introduction

Because of the presence of small particles most liquid

coatings can be considered colloidal suspensions. The col-

loidal stability of the system then determines whether the

particles will remain well dispersed or whether they will

flocculate. When flocculation is not desirable stability can

be induced by electrostatic repulsion between the particles,its application is mainly restricted to aqueous media. Steric

repulsion between layers attached to the particle surface can

provide stability in any suspending medium. Electrostatic

stabilization is very effective but makes the structure and

the rheology sensitive to variations in pH or ionic strength.

Steric, or polymeric, stabilization is not affected by these

parameters but, if the suspending medium is not a good sol-

vent for the stabilizer molecules, the stability might change

with temperature.

Steric stabilization is frequently used as a suitable and ro-

bust way of ensuring proper dispersion of the particles. The

stabilizer layer can be chemically grafted on the particle

surface or, more often, physically adsorbed. The formula-

tion of such materials would obviously be accelerated if the

rheology could be predicted or estimated rather accurately

on the basis of the composition. This is the problem which

is addressed here. Only stabilizer layers of grafted polymers

and adsorbed blockcopolymers or surfactants are consid-

ered, not homopolymers or statistical copolymers as such.

In the latter case the stabilizer layer consists of a complex

Corresponding author.

mixture of tails and loops of the stabilizing polymer,

whereas only tails are present in the other two cases. The

suspending medium can be aqueous or non-aqueous.

2. Brownian hard spheres

Various parameters affect the rheology of stable colloidalsuspensions. In the limiting case of spheres without any

interparticle interactions, only Brownian (thermal) forces

and hydrodynamic forces affect the flow behaviour. The

case of Brownian hard spheres is quite well documented,

experimentally and theoretically, at least for monodisperse

particles (e.g. Ref. [1]). At sufficiently low shear rates Brow-

nian motion will dominate the convective motion caused by

the flow. Under these conditions the equilibrium structure

of the particles that exist at rest is preserved during flow.

As a result the viscosity does not change with shear rate

and the contribution of the Brownian forces to the viscos-

ity is at its maximum. When increasing the shear rate the

Brownian motion will, at a certain stage, become slowerthan the convective motion. From then on the contribution

of the Brownian motion to the viscosity will gradually

decrease with increasing shear rate, whereas the hydrody-

namic contribution remains relatively constant. This causes

the viscosity to drop; a shear thinning region develops. At

still higher shear rates the Brownian contribution levels

off and becomes negligible, but an increase in hydrody-

namic effects can either compensate the decrease, causing a

pseudo-Newtonian high shear plateau, or overcompensate,

producing a shear thickening zone [2].

0300-9440/00/$ see front matter 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 3 0 0 - 9 4 4 0 ( 0 0 ) 0 0 1 4 2 - 9


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112 J. Mewis, J. Vermant / Progress in Organic Coatings 40 (2000) 111117

The viscosity curves for all suspensions with monodis-

perse Brownian hard spheres of a given volume fraction

can be reduced to a single curve [3]. For that purpose the

viscosity has to be divided by the medium viscosity m,

i.e. the relative viscosity r is used, thus scaling for the

hydrodynamic effects of the suspending medium which are

proportional to m. The shear rate has to be substituted

by a dimensionless stress r, which expresses the ratio

between convective and Brownian effects,

r = a3

kT(1)

where a is the particle radius, k the Boltzmanns constant

and Tthe absolute temperature. The resulting representation

scales for the effects of particle size, medium viscosity and

temperature.

The limiting relative Newtonian viscosities at low and

high shear rates depend only on the volume fraction. These

relations are well documented by now [1]. The hydrody-

namic and Brownian contributions can be separated theo-

retically and experimentally [2,4]. A characteristic of the

hydrodynamic forces is that they drop instantaneously to

zero when the flow stops, the other contributions decay over

a finite time scale [5].

At very high volume fractions the situation becomes more

complicated as the details of the surface now become impor-

tant. These effects show up in the viscosities as well as in

the frequency dependence of the storage moduli [4]. Com-

puter simulations are also difficult for such systems because

of the divergence of hydrodynamic forces between particles

when the interparticle distance goes to zero [6]. Experimen-

tally, a sudden shear thickening can be observed in very

concentrated suspensions, at least when the particles are nottoo small. At the moment it is still not quite clear to what

extent the detailed surface conditions of the particles affect

the results.

3. Viscosity curves

The viscosity curves for sterically stabilized suspensions

of monodisperse, spherical particles are similar to those of

suspensions of Brownian hard spheres (see Fig. 1). There is

however a quantitative difference. This is most easily seen

when the limiting Newtonian viscosities are plotted as a

function of volume fraction. When the stabilizer layer is ei-

ther relatively thin with respect to the particle radius or rela-

tively rigid, the viscosity curves still coincide with those of

Brownian hard spheres, provided the stabilizer layer is in-

cluded in the particle volume. The effect of the stabilizer can

then be expressed by means of an effective volume fraction

eff, which can be calculated from the core volume fraction

of the particles themselves, c, the particle radius a and the

thickness of the stabilizer layer

eff = c

1 +

a

3(2)

Fig. 1. Viscosityshear rate curves for sterically stabilized PMMA sus-

pensions (a = 42nm) at different volume fractions, data from [7].

With a known, e.g. from transmission electron microscopy,

the layer thickness can be deduced from intrinsic viscosity

measurements or from dynamic light scattering data. Such

a value is called the hydrodynamic layer thickness. Other

measures for this parameter will be discussed later.

When the stabilizer layer becomes thicker and/or softer,

the previous procedure is not adequate anymore and re-

sults in overestimation of the viscosity, especially at higher

volume fractions [8,9]. This is illustrated in Fig. 2, where

the limiting viscosities at high shear rates are compared for

a number of sterically stabilized PMMA suspensions with

varying particle radius but similar thickness of the stabi-

lizer layer and silica suspensions with different layer thick-

nesses [10]. The lack of superposition could be expected.Indeed, the stabilizer molecules are dissolved by the sus-

pending medium and do not form a rigid layer. This layer

will be compressed whenever particles approach each other

with a certain force. In some cases the effective volume

Fig. 2. Limiting high shear viscosities for various sterically stabilized

PMMA and silica dispersions [10].


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J. Mewis, J. Vermant / Progress in Organic Coatings 40 (2000) 111117 113

fraction becomes even larger than unity, which is clearly

physically meaningless (see Fig. 2). The compression of the

stabilizer layer is determined by the molecular structure of

the layer, including molecular weight, grafting/adsorption

density of the stabilizing polymer and its solubility in the

suspending medium. A discussion on the molecular level is

outside the scope of the present discussion. The softness

of the stabilizer layer can be expressed by a potential for

the particle interaction force. It is important to note that

steric stabilization normally results in rather steep repul-

sion potentials, meaning that the softness of the layer is

limited.

Because of the relatively rigid nature of the stabilizer layer

the viscosity curves still could possibly be treated approxi-

mately on the basis of an hard sphere approach. For such an

approach to be useful it should reduce somehow the viscosi-

ties of different systems to a single curve that reflects hard

sphere behaviour. The viscosity curves of sterically stabi-

lized suspensions, e.g. those in Fig. 2, definitely differ in the

limiting volume fraction of the particles at which the viscos-ity goes to infinity, i.e. in the maximum effective packingeff,max, at low as well as at high shear rates. To compare

the intrinsic shape of the viscosityconcentration curves, the

viscosities can be plotted versus the ratio eff/eff,max. As

illustrated in Fig. 3 this procedure can often superimpose

curves for different dispersions. For very soft particles the

evolution of the viscosity with concentration should be more

gradual than for rigid particles. Clearly, in Fig. 3 the slopes

of the curves at the origin cannot be identical anymore as

they were made to coincide in the representation of Fig. 2.

Yet, the major part of the curves superimposes reasonably

well. It is concluded that some degree of similarity with

hard sphere suspensions still exists, provided a suitable mea-sure for the volume fraction is used. This can be obtained

in several ways. The maximum packing could be used for

this purpose as shown in Fig. 3. Alternatively, a critical con-

centration for phase transition could be used [11]. Below, a

Fig. 3. Scaling the viscosityvolume fraction curves using the maximum

packing, numbers indicate diameters (PMMA data of Fig. 2) [8].

third method will be presented that is based on the particle

interaction potential.

Varying temperature and medium viscosity can be taken

into account for sterically stabilized suspensions by means

of the scaling procedure used for Brownian hard spheres

[12]. When changing the suspending medium its solubility

for the stabilizer molecules could then change as well, thus

affecting the thickness of the layer. As this factor is not cov-

ered by the scaling for Brownian hard spheres, it should be

accounted for separately. When working with media which

are on the edge of solubility for the stabilizer molecules the

viscosity could become very sensitive to temperature be-

cause of weak flocculation setting in. Such a situation is

obviously not desirable in practice.

When the volume fraction reaches the maximum packing

value for zero shear rate the limiting low shear viscosity di-

verges and a yield stress appears. As the viscosity it depends

on volume fraction and the colloidal characteristics of the

system. This will be discussed together with the dynamic

moduli in the next section.

4. Dynamic moduli

Oscillatory measurements, in particular on concentrated

colloidal dispersions, can provide detailed information

about the material under consideration. In such experiments

the viscous and elastic contributions can be separated in,

respectively, the loss (G) and storage (G) moduli. In

Brownian hard spheres the Brownian motion is responsible

for the elastic part, whereas the high frequency limit of

G measures the pure hydrodynamic contribution in the

equilibrium structure. In concentrated, sterically stabilized,systems the stabilizer layers overlap and G measures the

interaction forces between layers on neighbouring particles

during the oscillatory motion. This requires the frequency

to be high enough to avoid particle diffusion, caused by

Brownian motion, during an oscillation. G should become

independent of frequency if this condition is satisfied. The

critical frequency at which this happens, i.e. the relaxation

frequency, depends on particle mobility, exactly as the

critical shear rate does in the viscosity curves. These two

parameters are therefore linked to each other as well as

to the zero shear viscosity which also depends on particle

mobility, of which particle diffusivity is the fundamental

measure. The modulusfrequency curves can also be scaledin a similar fashion as the viscosityshear rate curves.

The overlap of neighbouring layers and the resulting in-

teraction forces will depend on the relative distance between

particles, which will decrease with increasing volume frac-

tion. Hence, the change in moduli with volume fraction can

be used to probe the change of interaction potential with

distance. The ZwanzigMountain equation, derived in the

framework of molecular dynamics, relates the plateau mod-

ulus G to the interaction potential and the distribution

function g(r) of interparticle distances r [13]


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Fig. 4. Interaction potentials as a function of interparticle distance for

two PMMA dispersions and two aqueous latices, calculated from plateau

storage moduli (after Ref. [16]).

G = 2152

0g(r) d

dr

r4 d

dr

dr (3)

with the particle number density. Eq. (3) has been used for

electrostatically as well as sterically stabilized suspensions.

In electrostatically stabilized systems the particles can order

in a crystal-like lattice order in which the distance between

nearest neighbours has a well-defined value [13]. In steri-

cally stabilized systems often a less ordered glassy structure

exists during flow, but using suitable estimates for the av-

erage distance still makes it possible to calculate the inter-

particle potentials [8,14,15]. Some results for non-aqueous

PMMA suspensions and for aqueous latex suspensions with

adsorbed stabilizer are given in Fig. 4 [16]. Three of thefour systems in this figure have stabilizer layers with an hy-

drodynamic thickness of about 10 nm, which is typical for

industrial dispersions.

In the case of adsorbed blockcopolymers or surfactants,

steric stabilization is provided by the moiety of the stabilizer

molecule that is not adsorbed on the particle but is extended

in the suspending medium. Increasing the molecular weight

of this moiety obviously increases the layer thickness of

the stabilizer layer and consequently the repulsion force at

a given interparticle distance (compare the two latices in

Fig. 4). This does not imply that the dispersion becomes

more stable under all conditions, at least for adsorbed layers.

Changing the relative size of the adsorbed and non-adsorbedmoieties affects the strength of the adsorption. The change

that apparently increases the stability will actually reduce the

adsorption strength. At high shear rates this could result in an

irreversible, shear-induced, flocculation. Such a behaviour

has actually been observed in the latex system of Fig. 4

when increasing the length of the hydrophylic part of the

stabilizer molecules.

It is obvious that the relation between plateau storage

modulus and volume fraction is an important characteristic

of a sterically stabilized suspension. It reflects the softness

of the stabilizer layer and provides the means to calculate

the limiting viscosities. Empirically it has also be found

that the ratio between yield stress and plateau modulus is

between 0.02 and 0.04 for many suspensions, irrespective

of stabilizer softness, particles size or volume fraction [8].

Hence these two parameters are affected in the same fashion

by the stabilizer layer.

5. Viscosity predictions based on dynamic moduli

The particle interaction potential, as for instance deter-

mined from the plateau values of the storage moduli, can be

used to estimate the viscosities. Provided the particles are

not too soft, a perturbation theory, based on small devia-

tions from hard sphere behaviour can be applied. As was the

case for the expression of the plateau moduli, it is borrowed

from molecular dynamics theory. In essence it replaces the

real, steep, interaction potential by a hard sphere potential

(i.e. vertical line) at a suitable distance from the particlesurface (see Fig. 5). This distance provides another mea-

sure for the effective thickness of the stabilizer layer, which

is smaller than the hydrodynamic value. The ratio between

the new value, hs, and the hydrodynamic layer thickness

from Eq. (2) depends on the shape of the interaction poten-

tial, i.e. the softness of the stabilizer layer. An approximate

expression for the resulting particle diameter is used

2(a + hs) =

0

1 exp

kT

dr (4)

With Eq. (4) a value for the effective volume can be cal-

culated, hs, which is relevant in the case of non-dilute

suspensions. Using this value superimposes the limitingviscosity curves of various stericially stabilized dispersions,

as demonstrated in Fig. 6 [8,9].

Fig. 6 provides a rational basis for the effective hard

sphere approach in the case of sterically stabilized suspen-

sions, with either grafted or adsorbed stabilizer layers. It

also connects the rheological properties with the colloidal

parameters of the material and allows viscosities to be pre-

dicted over a wide range of conditions, at least for model

Fig. 5. Steric interaction potential (dotted line) and corresponding hard

sphere potential (full line).


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Fig. 6. Superposition of viscosity curves using the corrected volume

fraction hs, PMMA and latex suspensions (numbers indicate diameter)

[16].

systems. In practical formulation problems fundamental

characteristics as interaction potential and particle diffusiv-

ity are often not known in sufficient detail, although some

molecular scaling arguments could still be used. Neverthe-

less, the present approach can always be followed at least in a

semi-quantitative manner. Also, the results suggest that mea-

suring, e.g. the maximum packing is an adequate measure to

characterize a given stabilizer layer and to predict its effect

on the viscosity curves at various particle concentrations.

6. Particle size distribution

In real life systems the particles, even if they are spheri-cal, are seldom monodisperse. The particle size distribution

can have a pronounced effect on the rheology. For large,

non-colloidal particles various mixing rules are available,

as reviewed by Utracki [17]. They can give reasonable

predictions for the viscosity. As could be expected from

the discussion of the previous paragraph, the maximum

packing is still the governing characteristic parameter. A

mixture of different sizes can be packed more densely

than monodisperse particles, which increases the maximum

packing. For large particles the latter can be derived from

geometrical arguments. In colloidal suspensions the situa-

tion becomes more complex. The contribution of Brownian

motion has to be considered, which introduced a differentsize dependence. It reduces the efficiency of packing small

particles between larger ones. For mixtures of colloidal and

non-colloidal particles mixing rules have been proposed

that take the said phenomenon into account [1820]. An

example of such a prediction for the limiting viscosity is

shown in Fig. 7, where measured and predicted limiting

viscosities are compared for mixtures of latex (a = 63nm)

and polystyreen (a = 1545 nm) particles [16]. With each

size a shear rate region for shear thinning can be associated.

As the dimensionless stress contains the particle radius to

Fig. 7. Illustration of predictions for the viscosity in binary mixtures (25%

3m and 75% 127 nm diameter particles) [16].

the power three, Eq. (3), particle size distributions can giverise to a very broad shear thinning region (Fig. 8).

The whole discussion has been limited to spherical par-

ticles. In all other cases the shape has to be considered,

e.g. the aspect ratio (length over diameter) for axisymmetric

particles. For large particles the main effect is a reduction

in maximum packing. For smaller ones the relations given

above cannot be applied quantitatively but qualitatively sim-

ilar strategies can often still be followed.

7. Shear thickening

At sufficiently high volume fractions and shear rates shearthickening will occur in most suspensions. It normally lim-

its the concentration and shear rates that can be used, but

it also turns out to be important in understanding particle

interactions and flow in concentrated suspensions. General

Fig. 8. Broad shear thinning zones in dispersions with two particle sizes

[8].


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discussions on shear thickening are available [21]. This phe-

nomenon is not restricted to colloidal systems, it occurs even

more readily with larger particles. The shear rate or shear

stress at which shear thickening sets in is considered as the

most relevant parameter.

When discussing shear thickening it is important to dis-

tinguish between two different types. At moderate concen-

trations a gradual increase of the viscosity with shear rate

can often be noticed at high shear rates. No abrupt changes

in structure seem to accompany this kind of shear thick-

ening. Experimental as well as simulation results indicate

that gradual shear thickening is normal rather than excep-

tional at the upper limit of the shear thinning region for sus-

pensions with a liquid-like interparticle structure [2,22]. As

mentioned earlier it is caused by a stronger increase with

shear rate of the hydrodynamic contribution to the viscosity

than the simultaneous decrease in Brownian contribution.

The so-called high shear Newtonian plateau is actually a

limiting case of such a gradual shear thickening region.

Under more extreme conditions of volume fraction andshear rate a second type of shear thickening can be en-

countered. In motion-controlled rheometers it is difficult to

study because often the sudden stress jump accompanying

the increase in shear rate is too large to be measured. In

stress-controlled devices it can be seen that, at the onset of

sudden shear thickening, an increase in stress causes in fact

a decrease in shear rate, after which the latter changes errat-

ically (Fig. 9). When continuously increasing and decreas-

ing the shear stress the transition between the two regimes

shows an hysteresis. Both this and the erratic variations

suggest that a statistical process governs the transition.

The mechanisms responsible for the onset of shear thick-

ening are still under debate. Most likely more than onemechanism is possible. Hoffman [23] was the first to show

that at least in some suspensions with monodisperse spheres,

applying high shear rates caused the particles to order in lay-

ers with an hexagonal packing in each layer. This structure

Fig. 9. Sudden shear thickening in a sterically stabilized PMMA suspen-

sion (a = 170nm, eff = 0.636) [8].

would become unstable at a critical shear rate, signalling

the onset of shear thickening. The latter would then corre-

spond to an orderdisorder transition. The instability would

occur because of hydrodynamic effects dominating the sta-

bilizing effect of the interparticle repulsion. This approach

was extended by Boersma and Laven [24] who proposed a

scaling in which the critical shear rate for shear thickening

would be proportional to the product T/ma2. The existence

of such a scaling has been confirmed in systematic exper-

iments in which, however, not always an orderdisorder

transition could be detected at the onset of shear thicken-

ing [25]. Rheo-optical studies on such sterically stabilized

suspensions indicate a flow-induced, hydrodynamic aggre-

gation to be responsible for shear thickening [26]. A similar

result has been obtained in some computer simulations

[22]. As mentioned earlier, hydrodynamic interactions are

difficult to incorporate in such simulations and turn out to

be extremely important [6]. It should be mentioned that in

suspensions of relatively hard spheres the critical shear rate

might scale with a3

rather than with a2

[25].Polydispersity in particle size dramatically postpones

shear thickening. It can still occur although it is unlikely

that it would be preceded by shear layering. In bimodal

dispersions of big and small particles a first approximation

of the critical shear rate can be obtained by plotting this

parameter versus the volume fraction of the big particles

alone [8]. This proofs at least the strong effect of particle

size on shear thickening.

Finally the effect of particle volume fraction should be

considered. At moderate effective volume fractions, often to

about 0.50, shear thickening sets in at a constant shear rate.

At very high volume fractions the critical shear rate tends

to drop drastically and it is rather the critical shear stress cwhich now becomes constant. In that region the product ca

2

provides a scaling factor which takes into account particle

size and volume fraction. The extent of the range in volume

fractions over which the critical shear stress remains constant

seems to be quite variable and to depend on the nature of

the suspending medium.

8. Conclusions

Starting from the reference case of suspensions with

Brownian hard spheres, the rheology of sterically stabilized

suspensions can be quite well understood. The scaling lawsfor Brownian spheres can be used to predict the effect of

temperature and medium viscosity in sterically stabilized

systems as well. The stabilizer layer increases the effective

volume of the particles which results in a viscosity rise.

This could lead to overestimated values for the viscosity,

especially at higher volume fractions, as the stabilizer layer

is partially deformable. This deformability is expressed by

the interparticle potential, which can be used to calculate a

more suitable measure of the effective volume for non-dilute

suspensions. The interparticle potential is normally not


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known, but can be derived from the storage moduli obtained

in oscillatory experiments. On the whole, the rheology of

well characterized suspensions can be explained in terms

of fundamental colloidal parameters such as interparticle

potential and particle diffusivity.

A distribution of particle sizes reduces the viscosities,

especially in more concentrated suspensions. For bimodal

distributions some suitable mixing laws seem to be available.

Scaling relations have also be proposed for the onset of shear

thickening. Medium viscosity, temperature and particle size

can be taken into account. These relations do not describe all

possible conditions but can nevertheless be used to estimate

the effect of changing the composition.

References

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