Viscosity equation for concentrated suspensions of charged colloidalparticlesA. Ogawa, H. Yamada, S. Matsuda, K. Okajima, and M. Doi Citation: J. Rheol. 41, 769 (1997); doi: 10.1122/1.550875 View online: http://dx.doi.org/10.1122/1.550875 View Table of Contents: http://www.journalofrheology.org/resource/1/JORHD2/v41/i3 Published by the The Society of Rheology Related ArticlesEffects of dispersion and deformation histories on rheology of semidilute and concentrated suspensions ofmultiwalled carbon nanotubes J. Rheol. 57, 1491 (2013) Rheology of sedimenting particle pastes J. Rheol. 57, 1237 (2013) Microstructure in sheared non-Brownian concentrated suspensions J. Rheol. 57, 273 (2013) Ageing, yielding, and rheology of nanocrystalline cellulose suspensions J. Rheol. 57, 131 (2013) Wall slip and flow of concentrated hard-sphere colloidal suspensions J. Rheol. 56, 1005 (2012) Additional information on J. Rheol.Journal Homepage: http://www.journalofrheology.org/ Journal Information: http://www.journalofrheology.org/about Top downloads: http://www.journalofrheology.org/most_downloaded Information for Authors: http://www.journalofrheology.org/author_information

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Viscosity equation for concentrated suspensionsof charged colloidal particles

A. Ogawa,a) H. Yamada, S. Matsuda, and K. Okajima

Fundamental Research Laboratory of Natural & Synthetic Polymers,Asahi Chemical Industry Company, Ltd., 11-7 Hacchonawate, Takatsuki,

Osaka 569, Japan

M. Doi

Department of Applied Physics, Faculty of Engineering,University of Nagoya, Furo-cho, Chikusa, Nagoya 464, Japan

(Received 19 December 1996; final revision received 21 February 1997)

Synopsis

Concentrated suspensions of charged stabilized colloidal particles exhibit very large viscosity at lowshear rate, a strong shear-thinning behavior at intermediate shear rate, and a constant secondNewtonian viscosity at high shear rate. This type of non-Newtonian behavior is affected by manyfactors such as the particle volume fractionf, the particle diameter, the surface electric potentialc0 , salt concentration, etc. The generalized equation for the viscosityh of this system is proposedby applying Eyring’s transition state theory. The surface electric potentialc0 and the thickness ofthe electric double layerk21 are determined by applying the theory to experimental data.Systematic experiments ofh of the model colloidal dispersion systems are carried out as thefunction off and shear rate and the results are satisfactorily reproduced by the present theory. Theeffects of hydrodynamic diameterdh andc0 of the colloidal particle onh are also quantitativelyexplained. ©1997 The Society of Rheology.@S0148-6055~97!01503-4#

I. INTRODUCTION

Charged stabilized colloidal suspensions show a large non-Newtonian viscosity at lowsalt concentration@Fryling ~1963!; Brodnyan and Kelly~1965!; Buscallet al. ~1982a and1982b!; Chen and Zukoski~1990!; Buscall~1991 and 1994!#. The viscosity of the systemdepends on many parameters, such as shear rate~or shear stress!, particle volume frac-tion, particle diameter, surface electric potential, salt concentration, etc. It is important toknow how the viscosity depends on these parameters, and for engineering purposes, it isalso desirable to have an equation that describes the dependence of the viscosity on theseparameters.

Statistical mechanical theories have been developed to predict the viscosity as a func-tion of these parameters. Such theories usually require a rather elaborate computation~such as solving an integrodifferential equation!, and, so far, the calculation has beendone only for dilute suspensions or to the very weak shear flows~linear regime! @Russelet al. ~1989!#.

a!Corresponding author.

© 1997 by The Society of Rheology, Inc.J. Rheol. 41~3!, May/June 1997 7690148-6055/97/41~3!/769/17/$10.00

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On the other hand, many empirical equations describing the non-Newtonian viscosityhave been proposed, but most of them are purely phenomenological, and there are limitednumber of equations that relate the non-Newtonian viscosity with the physical param-eters, such as the interparticle forces and particle volume fractions, etc.

Buscall~1991 and 1994! proposed a viscosity equation that describes the dependenceon the physical parameters. In his equation, the effects of the shear stress and the inter-particle forces are lumped into a single parameter, namely, the effective volume fractionfeff in the Krieger–Dougherty equation@Krieger ~1972!# for the viscosity of hard-spheresuspensions. Though the equation has been successful in reproducing experimental data,the physics underlying the relation between the effective volume fraction and the physicalparameters is not clear.

In this paper, we propose a slightly different semiempirical equation that describes thenon-Newtonian viscosity of charged colloids as a function of the parameters mentionedabove. The equation is based on the classical activation process model, and may not beentirely new. However, we are not aware of any analysis of experimental data based onsuch an equation. To assess the validity of the equation, we conducted experiments formodel systems of styrene–butadiene latex~SBL! dispersions in water varying the particlediameter and concentration. We shall show that the viscosity of such systems is welldescribed by our viscosity equation.

II. THEORY

Let us consider the shear stresss of colloidal suspensions as a function of the particlevolume fractionf and the shear rateg. In general, the stress of colloidal suspensions iswritten as a sum of the viscous stresssv , which arises from the viscosity of the sus-pending media, and the particle stress~or elastic stress! sp , which arises from theinterparticle potential forces@Doi and Edwards~1986!#:

s 5 sv1sp . ~1!

Bothsv andsp are responsible for the non-Newtonian viscosity, but usually the particlestresssp has the dominant contribution. Therefore, we assume that the viscous stress partis given by the Krieger–Dougherty equation@Krieger ~1972!#

sv 5 hv~f!g, ~2!

with

hv~f! 5 hsS12f

fmaxD2~5/2!fmax

, ~3!

wherehs is the viscosity of solvent, andfmax is the maximum volume fraction at whichthe viscosity diverges. In general,fmaxdepends ong, but here we take, for clarity of theargument,fmax 5 0.71 @Russelet al. ~1989!#, which corresponds to the situation ofg → `.

In order to estimate the particle stresssp , we use the following simple model basedon the theory of the activation process@Glasstone, Laidler, and Eyring~1941!#. Weconsider the case that the salt concentration in the system is so low that the interactionrange is much larger than the particle diameterd. Thus, in the absence of shear flow,each particle is sitting around the local minimum of the potential created by other par-ticles@see Fig. 1~a!#. The particles occasionally exchange their positions with their neigh-bors by an activation process. This takes place with mean frequency,

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f 5 f0 expS2UV~f!

kBTD, ~4!

wheref 0 is the characteristic frequency, which can be expressed by the diffusion constant

FIG. 1. ~a! Schematic representation of the apparent activation potential barrierU of Eyring’s transition statetheory in equilibrium. The potential is given by the particle–particle electrostatic repulsive potentialUV ; and~b! the activation potential barrierU in a sheared state. There is an additional bias potentialUS created by theshear stress.

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of the particle and the particle size, andUV(f) is the mean activation energy. We willdiscuss later how to estimateUV(f) from the interparticle potential.

If the system is sheared, the shear stress gives a bias potentialUS for the activationprocess and makes the transition easier as it is shown in Fig. 1~b!. We assume that theactivation energy in such a state is reduced fromUV(f) to

U~f,sp! 5 UV~f!2n†sp , ~5!

where the constantn† denotes the activation volume. Since the interaction range betweenthe particles is larger than the particle size,n† is of the order of the free volume perparticle, i.e.,n† > 1/np , wherenp is the number of particles per unit volume. Sincenp can be expressed by the particle diameterd and the particle volume fractionf as

np 56f

pd3, ~6!

Eq. ~5! can be written as

U~f,sp! 5 UV~f!2k1d

3

fsp , ~7!

wherek1 is a certain numerical constant.We assume that the particle stresssp is given by the viscosityhp(f,sp) correspond-

ing to this activation energy, i.e.,

sp 5 hp~f,sp!g, ~8!

with

hp~f,sp! 5 k2hsf expSU~f,sp!

kBTD, ~9!

wherek2 is another numerical constant. In the prefactor of Eq.~9!, the factorf is put sothat Eq.~9! vanishes in the limitf → 0, and the factorhs is put to make the equationdimensionally correct.

To summarize, the viscosity equation we propose is

h 5 hv~f!1k2hsf expSUV~f!

kBT2k1d

3sp

fkBTD. ~10!

Now we discuss how to estimate the activation energyUV(f) from the interparticlepotentialV(r ) ~the potential acting between the particles whose center-to-center distanceis r !. Let r † be the distance between the particles in the activated state@i.e., the statecorresponding to the maximum of the potential in Fig. 1~a!#, and let r be the meandistance between the particles. Then the activation energy is given by

UV~f! 5 V~r†!2V~r!. ~11!

Usually,V(r †) is much larger thanV( r ), and we may approximate Eq.~11! as

UV~f! 5 V~r†!. ~12!

To estimate the distancer †, we assume thatr † is proportional to the mean distancebetween the particles.

r† 5 k38 r 5 k3df21/3, ~13!

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where k3 is another numerical constant. Equations~12! and ~13! give the activationenergyUV(f) as a function of the interparticle potential and particle volume fraction

UV~f! 5 V~k3df21/3!. ~14!

For actual calculation, we use the following form of the interparticle potential ofcharged colloids@Verway and Overbeek~1948!; McCartney and Levine~1969!#.

FIG. 2. Schematic representation of the styrene–butadiene latex particle~LP!: dL , the diameter of the hard-core part~hatched part! of LP; dh , the hydrodynamic diameter of LP including the hydration layer~soft shellpart!, which contains the polymer brush of the ionic component; andDdh , width of the hydration layer.

TABLE I. Characteristics of styrene–butadiene latex used in the presentexperiment.

Sample dL /nma dh /nm

b Ddh /nmc

WCOOH/wt %d

Ltx 1 100.3 100.8 0.25 3Ltx 2 140.0 143.6 1.80 3Ltx 3 158.0 162.7 2.35 3Ltx 4 200.3 205.3 2.50 3Ltx 5 141.9 144.2 1.15 2Ltx 6 95.9 101.1 2.60 3

aDiameter of LP measured by SEM.bDiameter of LP measured by DLS.cWidth of the layer of hydration containing carboxylic acid on the surface ofLP.dWeight fraction of carboxylic acid monomer initially loaded at the begin-ning of the polymerization.

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V~r! 54pe~d/2!c0

2~r2d/2!

rlnF11

d/2

r2d/2exp~2kL!G, ~15!

where

L 5 r2d, ~16!

is the surface-to-surface distance,c0 is the surface electric potential,e is the dielectricconstant of the medium, andk is the Debye screening parameter given by

k 5 A2nz2e2

ekBT. ~17!

Here,n is the density of free ions,z ionic charge number, ande the elementary electriccharge ( 5 1.603 10219 C). The interaction range of the potentialV(r ) is aboutk21.

If there is no added salt, which is the case in the system studied in the following,n isthe concentration of the counter ions dissociated from the particle, and it may be writtenas

n 5 n0f, ~18!

FIG. 3. The shear rate dependence of viscosity for the Ltx 6 system for various volume fractionf: s, f5 0.1758;h,f 5 0.2343;n,f 5 0.2929;L,f 5 0.3445; and full line, calculated curve@Eq.~10!#.

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wheren0 is the number of ions dissociated from the particle divided by the particlevolume. Hence, Eq.~17! is written as

k~f! 5 k8Af. ~19!

Therefore, if the parametersd, c0 , andk8 are known, we can predict the viscosity of thesuspensions for any volume fraction and stress using Eq.~10!.

III. EXPERIMENT

A. Samples

The samples used in our experiments are styrene–butadiene latices~SBL! dispersed inwater with no added salt. The SBL particles have the structure shown in Fig. 2. Theparticle consists of a hard-core part made of styrene-butadiene and a soft shell part madeof grafted polyelectrolytes solvated with water. The diameter of the core part is denotedby dL and the diameter of the whole particle~including the soft shell! is denoted bydh . The values ofdL anddh are listed in Table I, together withWCOOH, which is theweight fraction of carboxylic acid monomers initially loaded at the beginning of poly-merization.

FIG. 4. The plot of kBT ln(hp /hsf) againstsp of the Ltx 6 system for various volume fractionf: s, f5 0.1758;h,f 5 0.2343;n,f 5 0.2929;L,f 5 0.3445; andg 5 50, 100, 200, 300, 500, 800, 1000,1200, 1500, and 2000 s21.

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The first five latices in Table I~Ltx 1–Ltx 5! are synthesized and supplied by Nish-imoto ~Ashahi Chemical Industry!, and Ltx 6 is commercially available from AsahiChemical Industry Co. Ltd. The styrene–butadiene latex system usually contains manywater soluble components. In this experiment, however, the water soluble componentsare removed as much as possible. This is done by repeating the ultracentrifugation~30 000 rpm, 1 h! and washing the precipitate three times with ion-exchange water. Wealso used ion-exchange water for sample preparation. Thus, we think that the sampleconsists of the SBL particle and counter ions dissociated from the SBL particle andwater.

B. Sample characterization

The diameter of the hard-core partdL was obtained by measuring the diameter of thedry particle by scanning electron microscopy~SEM! as follows: A droplet of diluteSBL/water dispersion~weight fraction of SBLwL being ca. 0.01 wt %! was placed on thesample stage of the SEM~S-800, Hitachi Co. Ltd.! and was dried at room temperature.The SEM micrographs were taken under the accelerating voltage 5 kV with magnification60 000. The micrographs were then analyzed by an imaging analyzer~IP1000PC, AsahiChemical Industry Co. Ltd.! and the number averaged diameterdL was obtained.

FIG. 5. Relationship between the activation energyUV(f) and the volume fractionf of the Ltx 6 system:s,determined by use of Eq.~7!; and full line, fitting curve of Eq.~15! ~fitting parameters,c0 5 30.5 mV, andk8 5 0.101 nm21!.

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On the other hand, the total diameter of the particle was obtained from the effectivehydrodynamic diameterdh that the particle has in dilute solution. This was done asfollows: Dynamic light scattering~DLS! of the SBL/water system with latex contents ca.0.01 wt % was measured by DLS apparatus~DLS-700, Otsuka Electric Co. Ltd.! attemperature 25 °C and scattering angle 90°, and the hydrodynamic diameterdh is ob-tained from the decay of the time correlation function of the scattered intensity. Thethickness of the soft shellDdh is obtained from the difference betweendL anddh ,

Ddh 5dh2dL

2. ~20!

The values ofDdh are also listed in Table I. In the following, we regarddh as the particlediameter, and calculate the volume fraction of SBLf by the equation

f 5 SdhdLD3

wL . ~21!

Here, the density of the latex particle~LP! is taken to be unity, which is confirmed bypicnometer.

FIG. 6. The volume fraction dependence of the viscosity for the Ltx 6 system atg 5 103 s21: s, experimen-tal; broken line, calculated curve@Krieger–Dougherty equation, Eq.~3!#; and full line, calculated curve@Eq.~10!,c0 5 30.5 mV,k8 5 0.101 nm21, k1 5 0.136,k2 5 1,andk3 5 0.905#.

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C. Viscosity measurement

Viscosityh of the system was measured by the rotation-type viscometer~RheoStressRS100, Haake Co. Ltd.; cone/plate angle 1°, diameter 35 mm! at 25 °C.

IV. RESULTS AND DISCUSSION

A. Determination of the surface electric potential c0 and the thickness ofthe electric double layer k21 from Eq. (15)

Figure 3 shows the shear rateg dependence of the steady-state viscosityh for the Ltx6 system at various volume fractionf indicated in Fig. 3. The viscosityh increasessharply with the increase off, and a strong shear-thinning behavior is observed particu-larly in the samples with large volume fraction.

To fit the data with the theoretical curve, we first obtained the particle contribution tothe viscosityhp 5 h 2 hv using Eq.~3!. We then plottedkBT ln(hp /hsf) againstsp5 hpg. An example of such a plot is shown in Fig. 4 withk2 5 1. One can draw astraight line and obtain the activation energyUV(f) by extrapolating the line tosp5 0. The result is plotted in Fig. 5. The data points are then fitted by the theoreticalcurve Eq.~14! with k3 5 0.905. The optimized fitting parameters werec0 5 30.5 mV

FIG. 7. The shear rate dependence of the viscosity for the Ltx 1–4 systems having differentdh (100.82 200.3 nm) at fixed volume fractionf 5 ca. 0.3:s, Ltx 1 ~dh 5 100.8 nm,f 5 0.2994!; h, Ltx 2 ~dh5 143.6 nm,f 5 0.3237!; n, Ltx 3 ~dh 5 162.7 nm,f 5 0.3278!; andL, Ltx 4 ~dh 5 205.3 nm,f5 0.3230!.

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andk8 5 0.101 nm21 ~i.e.,k21 5 22.1 nm forf 5 0.2 andk21 5 15.7 nm forf5 0.4!. The correlation coefficient in the fitting is very close to unity~0.997!, andindicates the validity of our viscosity equation.

Figure 6 shows the volume fraction dependence of the viscosity for the Ltx 6 system,where the circles are the experimental points, the broken line the calculated value of theKrieger–Dougherty equation, and the full line is the calculated value of our viscosityequation. In this plot, the numerical constantk1 , k2 , andk3 are taken to be 0.136, 1, and0.905, respectively.k1 needs not be larger than the volume of a particle. It is difficult tovisualize the activation volumen†: the value will depend on the actual process of therearrangement, and it involves many particles. Thus, we do not think that thek1 valuebeing so small is a problem. Clearly, the observed viscosity is much larger than that givenby the Krieger–Dougherty equation, indicating the importance of the electric repulsion ofthe charged colloids. A similar comparison between the theory and experiments is donein Fig. 3, where the calculated viscosity at each volume fraction is shown by the solidlines. Although the fitting is not perfect, the viscosity equation reproduces a viscositycurve. Notice that the same set of parametersc0 , k8, andk’s have been used in thefitting for all curves in Fig. 3. Considering the crudeness of the model, we may say thatthe model works well.

FIG. 8. The shear rate dependence of the calculated viscosity@Eq. ~10!# having different particle diameters~100, 140, 160, and 200 nm! at fixed volume fractionf 5 0.3: c0 5 40 mV, k8 5 0.08 nm21 ~k21

5 22.8 nmforf 5 0.3!,k1 5 0.262,k2 5 1,andk3 5 0.905.

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B. Particle size dependence of the viscosity

Figure 7 shows the non-Newtonian viscosity for various latex particles that havealmost the same surface charge, but different diameters ranging from 100 to 200 nm. Thevolume fractionf is not exactly the same, but all of them are near 0.3.

It is seen that the viscosity increases with the decrease of the particle diameter. Thisfeature is nontrivial, but it is well reproduced by our viscosity equation. Using the valuesdh 5 100, 140, 160, 200,f 5 0.3, andk8 5 0.08 nm21, we calculated the viscositycurve, and the result is shown in Fig. 8. Although the absolute magnitude of the viscosityis not in perfect agreement, the tendencies that the viscosity increases with the decreaseof the particle size, and that the shear rate characterizing the shear-thinning increases withthe decrease of the particle size are in good accordance with the experimental resultsshown in Fig. 7. Figure 9 shows the calculated viscosity over wide shear rate range. Theabove tendency is clearly observed in Fig. 9. Figure 9 shows that when the effect of theinteraction is strong, the viscosity curve has two characteristic features.~i! The zero shearrate viscosity is very high

h0 5 limg→ 0

h~g! 5 hv~f!1k2hsf expSUV~f!

kBTD. ~22!

FIG. 9. The shear rate dependence of the calculated viscosity@Eq. ~10!# having different particle diameters~100, 140, 160, and 200 nm! at fixed volume fractionf 5 0.3 over the wide shear rate region:c05 40 mV,k8 5 0.08 nm21~k21 5 22.8 nmforf 5 0.3!,k1 5 0.262,k2 5 1,andk3 5 0.905.

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~ii ! In the intermediate range of the shear rate, the slope of the viscosity curve is close to2 1, indicating that there is an apparent yield stress. The magnitude of the yield stress isgiven by

sy 5fUV~f!

k1d3 . ~23!

We now discuss why the viscosity increases with the decreases of the particle diam-eter. According to our theory, this is caused by the increase of the overlapping area of theelectric double layer around each particle. Notice that the thickness of the electric double

FIG. 10. Schematic representation of particles with the same thickness of the electric double layer at fixedvolume fractionf 5 0.3. The diameter of the particles in~b! is two times larger than that in~a!.

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layer k21 remains almost constant when the particle diameter is varied. On the otherhand, the surface-to-surface distanceL is proportional to the particle diameterd if thevolume fraction is kept constant. Thus, the electric double layer around each particleoverlaps more strongly for small particles than for large particles, as it is shown in Fig.10. Therefore, with the decrease of the particle diameter, the repulsive interaction be-tween the particles increases and so does the viscosity. In Fig. 11, we show the results ofthe calculated interparticle potentialUV(f) as a function off. As it is seen in Fig. 11,the interparticle potential increases with the decrease of the particle diameter.

C. Viscosity behavior of latices with different surface electric charge

Figure 12 shows the non-Newtonian viscosity for Ltx 2 and Ltx 5 systems, which havealmost the same diameter but different carboxylic acid contents atf 5 ca. 0.30. Figure12 indicates that the particle with larger carboxylic acid contents gives larger viscosityespecially at low shear rate and gives larger characteristic shear rate for the transition tosecond Newtonian flow. The similar analysis made for Fig. 8 was also made here usingthe values of surface electric potentialc0 5 30, 40, and 50 mV,f 5 0.30, dh5 140 nm, andk21 5 22.8 nm forf 5 0.30. The results are shown in Fig. 13. Figure13 reproduces the behavior shown in Fig. 12. The largerc0 gives largerUV(f) @ as

FIG. 11. The effect of the particle diameter on the relation between the particle–particle electrostatic repulsivepotential UV against the volume fractionf: dh 5 100, 140, 160, and 200 nm,c0 5 40 mV, k85 0.08 nm21, k1 5 0.262,k2 5 1,andk3 5 0.905.

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predicted by Eq.~15!# and larger viscosity, especially at the small shear rate region. Sincethe diameter is the same for both systems examined here, the increase inUV(f) leads toa stronger structure, which calls for large bias for the activation energy for its destruction.This results in the increase of the characteristic shear rate, which specifies the onset of theshear-thinning region.

V. CONCLUSIONS

In this paper, we derived a generalized viscosity equation for concentrated suspensionof charged colloidal particles@Eq. ~10!# with the assumption thath can be expressed bythe simple summation of the viscosity due to disperse medium–particle interactionhvand the viscosity due to particle–particle interactionhp by applying Eyring’s transitionstate theory. The equation takes into account the activation potential barrierU that themoving particles have to pass over. All parametersc0 , k8, and k’s of the colloidalparticle have physical meanings, and can be determined by curve fitting. By using thesevalues, we can reproduce the shear rate and the volume fraction dependencies of theviscosity fairly well. Further, the effects of the particle diameter and the surface electricpotential on viscosity are quantitatively explained by our theory with the fixed otherparameters.

FIG. 12. The shear rate dependence of the viscosity for Ltx 2 and Ltx 5 systems having almost the sameparticle diameter but different carboxylic acid content atf 5 ca. 0.30:s, Ltx 2 ~dh 5 143.6 nm,f 5 0.3237,WCOOH 5 3 wt %!; andh, Ltx 5 ~dh 5 144.2 nm,f 5 0.3148,WCOOH 5 2 wt %!.

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In the present paper, the parametersc0 , andk8 are obtained by curve fitting. How-ever, these parameters can be calculated theoretically from other physical parameters, ormay be obtained from other experiments such as the measurements of the storage and theloss modulus@van der Vorstet al. ~1995!; Buscallet al. ~1982b!#. Such an approach willbe very useful for predicting the complex viscosity behavior of charged colloidal suspen-sions.

ACKNOWLEDGMENTS

The authors express their sincere gratitude to Dr. Makoto Nishimoto of 2nd TechnicalDept., Speciality Chemicals, Asahi Chemical Industry Co. Ltd. for preparing the SBLsamples.

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FIG. 13. The shear rate dependence of the calculated viscosity@Eq. ~10!# having the same particle diameter~ 5 140, 160, and 200 nm! but different surface electric potentialc0 at fixedf 5 0.3: k8 5 0.08 nm21

~k21 5 22.8 nm forf 5 0.3!, k1 5 0.262,k2 5 1, andk3 5 0.905; full line,c0 5 30 mV; broken line,c0 5 40 mV; and dotted line,c0 5 50 mV.

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