Rheology of Suspensions - Advances in Chemistry (ACS Publications)

3 Rheology of Suspensions Shijie L i u and Jacob H. Masliyah*

Downloaded by UNIV OF MINNESOTA on October 7, 2013 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/ba-1996-0251.ch003

Department of Chemical Engineering, University of Alberta, Edmonton, Alberta T 6 G 2G6, Canada This chapter is an in-depth review on rheology of suspensions. The area covered includes steady shear viscosity, apparent yield stress, viscoelastic behavior, and compression yield stress. The suspensions have been classified by groups: hard sphere, soft sphere, monodis perse, polydisperse, flocculated, and stable systems. The particle shape effects are also discussed. The steady shear rheological be haviors discussed include low- and high-shear limit viscosity, shear thinning, shear thickening, and discontinuity. The steady shear rheology of ternary systems (i.e., oil-water-solid) is also discussed.

to an applied shear stress or strain (J). In other words, rheology is a science of defor mation (a typical response of solids to an applied strain, elasticity) and flow (a typical response of a fluid to an applied shear, viscosity). Some times, the methods that impose a strain are classified as the plasticity approach, whereas the methods that apply shear rates or shear stresses are termed the rheological approach (2). In this review, we focus on the rheological approach, whereas the plasticity approach is dealt with only briefly. RHEOLOGY IS THE STUDY OF THE RESPONSE OF MATERIALS

Definition of Suspension. T h e rheology of suspensions deals with how suspensions respond to an applied stress or strain. T h e term sus pension refers, in general, to dispersions of solids i n fluids, although the term aerosol is conventionally used to refer to dilute suspensions of fine particles in a gas and the term emulsion is used to identify (concentrated) suspensions of particles i n a gas or l i q u i d i n the field of fluidization. However, emulsion is conventionally defined as the dispersion of a liquid in another (immiscible) liquid. In a broader sense, emulsions are also considered as suspensions. In this chapter, we deal mainly w i t h suspen* Corresponding author.

0065-2393/96/0251-0107$24.50/0 © 1996 American Chemical Society

In Suspensions: Fundamentals and Applications in the Petroleum Industry; Schramm, L.; Advances in Chemistry; American Chemical Society: Washington, DC, 1996.

108

SUSPENSIONS: F U N D A M E N T A L S & APPLICATIONS IN P E T R O L E U M INDUSTRY


sions of solids i n liquids, w h i c h are encountered often in the petroleum industry. F o r a gas dispersed i n a l i q u i d (i.e., foam), one should refer to Schramm and Wassmuth (3) and for a l i q u i d dispersed i n another l i q u i d (i.e., emulsion), to P a l et al. (4). Homogeneity of Suspensions. Rheology is " t h e science of de formation and flow." Rheologists normally want to express the relation between deformation and flow on the one hand and stress or strain on the other hand by equations based on a homogeneous material, or a continuum. Obviously, the definition of a suspension entails that it is heterogeneous, that is, a dispersed system consisting of individual par ticles i n a suspending fluid. U n d e r what conditions can we treat a sus pension as a continuous phase from a rheological view point? T h e rheological behavior can for any mixture only be defined when average values of stress and strain are meaningful. Thus, i f a rheological experiment involves a deformation over a length scale that is much greater than the dispersed particle size, then homogeneity or continuum can be observed. It may be useful to apply the representative elementary volume concept (see Chapter 5) for a suspension. O n e may not be i n terested i n the response of any given individual particle but rather i n the bulk response, that is, (volume) averaged properties.

Basic Rheological Concepts T h e relationship between an applied stress or strain and the response of the material, shear rate, or deformation is the aim of the rheology of suspensions. N o r m a l l y , both the stress and the strain are tensors with each having nine components. In simple shear, which is the most common way of determining the rheological behavior, the shear stress a (some literature also uses the symbol τ to stand for the shear stress) can be related to the shear rate y by xy

*xy = G = M 7

(1)

where μ is the dynamic viscosity or simply called viscosity. H o w e v e r , rheological measurements are also performed with other types of flow or stress fields. I f a uniaxial extensional flow field is applied to a material, the stress distribution can be described by °xx -

Vyy = ME*

(2)

where μ is the uniaxial (extensional) viscosity and e is the extensional strain rate. In general, a material can be characterized based on the two types of rheological behavior, that is, viscous and elastic. A solid body is char acterized b y its elastic behavior when the deformation is fully recovered Ε


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109

Rheology of Suspensions


after removal of the applied stress below the rupture point value. A l i q u i d is characterized by its viscous behavior w h e n it flows under any stress. C o m m o n l y , many materials, like the suspensions, are character i z e d by both an elastic and a viscous response, so-called viscoelastic behavior. T h e type of the response depends on the time scale of the experiment. If a small strain or stress is applied very rapidly to a vis coelastic body, it w i l l respond elastically. If the stress or strain is applied for a long time, the material w i l l flow and hence show a viscous response. It is possible to classify a particular system with the Deborah number as given by (3)

D e ^

where £ is the relaxation time and t is the time scale of the experimental measurement. W h e n De is large, the system behaves like a solid (elastic), and when De is small, it flows (viscous). W h e n De is near unity, the response is viscoelastic. T h e relaxation time based on the E i n s t e i n - S m o luchowski relation for an isolated particle is given by E

R

H - & £

(4)

where d is the particle diameter, μ is the viscosity of the continuous phase, k is Boltzmann's constant, k = 1.3806 X 1 0 ~ J / K , and Τ is the absolute temperature. T h e time scale of the experiment, t , for continuous shear flow is {

23

E

t = ~ 7 E

(5)

and for a forced oscillation experiment, it is given by t* = ~ ω

(6)

where ω is the frequency of the oscillation. F o r 10-μπι particles suspended in water at room temperature, t is of the order of 10 m i n . H e n c e , it is only in dilute systems of small particles that complete relaxation of a suspension structure can take place within the shear rate range of most rotational viscometers on the market. R

T i m e - I n d e p e n d e n t R h e o l o g y . In the simplest case, the shear stress σ is independent of time t and is proportional to the shear rate y (equation 1). F o r this case, the fluid is called Newtonian (line 1 of



110


Figure 1). T h e viscosity μ is the slope of the line 1 i n F i g u r e l a . T h e viscosity of a material is a function of temperature and pressure, but otherwise it is a material constant as shown in F i g u r e l b . Newtonian behavior is frequently met w i t h homogenous fluids, but for suspensions it is restricted to dilute nonaggregating (nonflocculated) systems. Frequently, however, the shear stress and shear rate relation is more complicated, as shown in F i g u r e 1. F i g u r e l a depicts some common shear rate response to a shear stress, whereas Figure l b shows the cor responding apparent viscosity variation versus the shear rate. Except for line 1, where the viscosity is independent of shear rate, the apparent viscosity is a function of shear rate (non-Newtonian behavior). F o r ex ample, curve 2 shows an increasing apparent viscosity w i t h shear rate. This system is called shear thickening or dilatant. C u r v e 3 shows a de crease in the apparent viscosity with shear rate and the system is said to be shear thinning. Curves 1 through 3 are normally generalized by a power law relation, where σ = Ky

(7)

n

H e r e η is the consistency (or power law) index and Κ is the consistency of the material. W h e n η > 1, the system is shear thickening, whereas

γ

a)

γ

b)

Figure 1. Classification of rheological behavior in steady shear: curve I, Newtonian; curve 2, shear thickening; curve 3, shear thinning; curve 4, Bingham plastic; and curve 5, nonlinear plastic.


3.

L i u & MASLIYAH


111

for η < 1, the system is shear thinning. A Newtonian material has an index of unity, η = 1. If the decrease in viscosity is very large at small shear rates, the system is sometimes called pseudoplastic (curves 4 and 5). C o m m o n l y , concentrated suspensions show a plastic behavior, that is, there is no response until a limiting y i e l d stress a has been exceeded. If the flow is linear above σ , the system is called Bingham plastic (curve 4) and can be expressed by the Bingham model (5): y

γ

(8)


σ = σ. + ΜρΎ

where the plastic viscosity μ^ is the slope of the flow curve when σ > σ . T h e y i e l d stress σ is sometimes called the Bingham y i e l d stress. T h e curve above the y i e l d stress can also be nonlinear (curve 5). Such be havior can be described by a generalized power-law or the H e r s c h e l Bulkley model: ν

γ

(9)

σ = a + Ky

n

y

or by the Casson model (6):

ι/2

σ

=

, i/2 y

+

(10)

m^m

Kc

where K is Casson constant. It should be noted that fitting the same rheological data to different models can lead to different y i e l d stresses. T h e accuracy of the evaluated y i e l d stress is dependent on the appli cability of the model used and should be treated strictly as a model parameter and not a material property. F o r a real suspension, the y i e l d stress may not be a reality (7). T h e (apparent) y i e l d stress, however, may be used just as a model for convenience (8). Several studies have been devoted for the determination of the y i e l d stress, for example, D z u y and Boger (9), Buscall et al. (JO), C h e n g (11), Yoshimura et al. (12), Astarita (13), Evans (14), V a n der Aerschot and M e w i s (15), Schurz (16), and D e K e e and C h a n M a n F o n g (17). Often, one finds a linear relation at a very small shear rate, a shear thinning behavior at intermediate shear rate, and a linear relation at high shear rate values (see, e.g., V a n D i e m e n and Stein [18] and H u n t e r [19]). In this respect, the M e t e r model may be of special interest i f a system behaves like Newtonian fluid at both low and high shear rates: c

σ =

7 Moo +

(Π)

1 + (σ/σ )" 0

where m is a power index, a is the critical shear stress, and μ and μ^ are the viscosities in the limits of very small shear rates and very high shear rates, respectively. c

0


112


However, the Meter model may have complications in its application in computational flow simulations. A better curve-fitting model emerges by replacing the shear stress on the right hand side of equation 11 w i t h a shear rate: a — y Moo +

Mo - Μα

(12)

ι + (y/y ) c

n

where 7 is a critical shear rate. Equation 12 may be called Cross equation (20) or V a n W a z e r equation (21) and should have similar quality as com pared with equation 11. Another model of interest is given b y P o w e l l and E y r i n g (22),


C

M = Moo + (Mo - Moo)

ln(* 7 + 1)

(13)

PE

(tmj)

n

where £ is a characteristic time constant and η is an index constant, i n most cases, η « 1. PE

Time-Dependent Rheology. T h e rheological properties of sus pensions are often time-dependent. If the apparent viscosity continu ously decreases w i t h time under shear w i t h a subsequent recovery of the viscosity when the flow is ceased, the system is called thixotropic. The opposite behavior is called antithixopy or rheopexy. F i g u r e 2 shows the time-dependent behaviors of suspensions. C u r v e 1 i n Figure 2 i l lustrates a hysteresis produced by a thixotropic suspension, where c o n -

Figure 2, Hysteresis of shear stress: curve 1, thixotropy with slow variation of shear rate; curve 2, rheopexy of slowly increasing, held steady at y , and decreasing y. 1


3.

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113


stant slow increase and decrease of the shear rates are applied. Some times, the loop formed b y the increase and decrease of shear rates may not be singly connected, that is, the two lines may cross. Systems bearing this type of a behavior are, for example, pastes of clay i n water, drilling m u d , and crude oils. M o d e l i n g of the thixotropic behavior has not been well-established (23). F o r the time being, the Sestak model (24), w h i c h is a simplified version of the more general form of the Cheng's model based power-law concept ( H e r s c h e l - B u l k l e y model), may be used + Ky + ( a

σ = σ

n


γ0

y l

^ = a(l - λ) - by X at m

+ Κι7 )λ η

(14a) (14b)

where λ is a memory parameter that has no physical significance; η is a form of power-law index; a denotes the permanent y i e l d stress; σ is the temporary y i e l d stress component; Κ denotes consistencies; and a, b, and m are constants. A l l of the eight model parameters (constants) need to be determined experimentally for each system. C u r v e 2 of Figure 2 shows a typical rheopexic hysteresis where the shear rate is increased slowly at 7 < 7 1 , h e l d constant for some time at 7 = 7 ! and is finally decreased gradually with a constant speed. This behavior is a reverse phenomenon of the thixotropy and is sometimes called antithixotropy. It occurs much less frequently than thixotropy and appears to be restricted to suspensions of very anistropic particles such as needles. In general, the models used for thixotropic systems can be adopted to simulate the behavior of the rheopexic systems. y0

γϊ

Viscoelasticity. Viscoelastic materials are characterized b y a combination of elastic and viscous properties. Thus, the shear stress is not only dependent on the rate of shearing but on the strain 7 as w e l l . In the simplest case, the viscoelastic behavior is governed by σ = Gy + M7

(15)

where G is the shear modulus of the system. T h e four commonly used techniques to extract information on the viscoelastic behavior of suspensions are creep-compliance measure ments, stress-relaxation measurement, shear-wave velocity measure ments, and sinusoidal oscillatory testing (25-27). In general, transient measurements are aimed at two types of measurements, namely, stress relaxation, w h i c h is to measure the time dependence of the shear stress for a constant small strain, and creep measurement, w h i c h is to measure the time dependence of the strain for a constant stress. The most widely used technique is the sinusoidal oscillatory or forced oscillation measurements. Oscillatory measurements consist of subject-


114


ing the medium under study to a continuously oscillating strain over a range of frequencies (Figure 3) and measuring the peak value of the stress σ and the phase difference between the stress and the strain δ, that is, 0

δ = œt

(16)

d

H e r e ω is the frequency of the applied stress. N o r m a l l y , the amplitude of the applied strain, 7 , is small enough to ensure that the system is i n the linear viscoelastic region (i.e., the stress varies w i t h strain linearly). T h e applied strain or the strain resulting from an applied stress can be expressed i n complex form by


0

7 = 7oe'

(17)

M + i )

T h e shear rate is given by y =

at

= i .IM.i 1 »

,Q[

• »

1er

,t 3

ίο

.

••'

... !.. *„,.i._U^ '

' '

ισ ιο° ίο Pe Figure 13. Steady shear viscosity of polystyrene particles of various sizes suspended in differentfluidmedia with a volume fraction of 0.5. (67). 4

ίο

. 2

1

1

Soft Sphere Systems. Suspensions are normally colloidally sta b i l i z e d by the electric double-layer interaction (electrostatic) and at tached polymer chains (steric). The rheological behavior is then strongly dependent on the separation distance between the particles. In elec trostatically stabilized (electrostatic) systems, the range of the electric double-layer interaction potential, expressed by Debye length scale (44), is strongly dependent on the ionic strength of the suspending medium. In sterically stabilized (steric) systems, the thickness of the stabilizing polymer chain attached to the particles can change with the state of the suspending medium due to the degree of solvation of the polymer. It is natural to replace the particle diameter with an effective particle diameter to include the effect of the polymer chain and the electric double layer to the hard sphere. L e t d = eff

(43)

d + 2A = dll -r^)

where Δ is the thickness of the stabilizing polymer layer or the effective electrostatic repulsive layer thickness. F o r electrostatically stabilized systems, Buscall (93) gave a semiempirical model for the effective soft sphere diameter as

4ff = d + £ j l - exp|

8E(r)

dr

(44)

8kT + MT^/KB.


130


where K is a shear effect constant, w h i c h is dependent on the volume fraction of the particles. K « 0.1 may be used (93). E(r) is the pairwise interaction potential. N o r m a l l y , the surface charge is small, hence the term i n the exponential can be treated as small. Equation 44 may be reduced to B

B

MT^eff

E(d )


eS

3

(45)

- kT

The error introduced by this approximation is insignificant, at least when the Peclet number is large. T h e volume fraction of the particles for a soft system is to be replaced by an effective volume fraction. T h e effective volume fraction is given by (46)

0eff - Φ ~ 7 Γ

Figure 14 shows the variation of the steady shear relative viscosity at the high shear limit w i t h the effective volume fraction as defined b y equation 46 for poly(methyl methacrylate) ( P M M A ) suspensions of dif ferent sizes i n decalin sterically stabilized b y means of grafted poly(12hydroxystearic acid) chains w i t h a degree of polymerization of 5. T h e stabilizing polymer layer thickness is 9 ± 1 nm, i n particular, Δ = 9 n m

10

3

17 ' ' ' Ϊ ' ' ' » • •

10

2

r

Eq 40 with = 0.64 max

Ο d = 84 nm Δ d « 475 nm • d = 1220nm

1

J Ο Ο Ο Ο

10

1

-CPtTiT. 101 0.0 0.1

1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I I 1 1 1I I Λ 0.2 0.3 0.4 0.5 0.6

I i

l

0.7

l L

0.8

Φβ«

Figure 14. High shear limit revive viscosity variation with effective volume fraction for various particle sizes (87).


3.


Liu & MASLIYAH

131

for d = 84 n m , Δ = 8 nm for d = 475 n m , and Δ = 10 nm for d = 1220 nm. H a r d sphere scaling principles and the use of ff apply only when the ratio of the thickness of the stabilizing polymer layer to the particle radius 2A/d is small. W h e n the thickness of the polymer layer is i n creased, particularly at high volume fractions, the relative viscosity de creases drastically from that with the same hard sphere scale suspensions (87, 88, 94, 95). Prestidge and Tadros (96), K i m and L u c k h a m (97), and L i a n g et al. (95) used a different approach to estimate the thickness of the stabilizing polymer layer by fitting the viscosity w i t h φ $ to the K r i e g e r - D o u g h e r t y equation while leaving Δ as a free parameter to be de termined in the process. T h e i r studies showed that Δ decreases w i t h increasing volume fraction, indicating a substantial compression of the attached polymer layer. Krieger and E q u i l u z (98) studied the influence of the suspending fluid ionic strength on the rheology of electrostatically stabilized monodisperse polystyrene latex particles. F i g u r e 15 shows some results of K r i e g e r and E q u i l u z (98) for latex particles suspended i n aqueous so lutions of HC1 at various concentration levels (different ionic strength). T h e theoretical model of Buscall (93) is presented as lines in F i g u r e 15. It can be observed that the relative suspension viscosity decreases drase


Β

10

6

c

Aqueuous solution of HCI at Ο deionized • ....... 1.876x1er M ΙΟ ^1.876x1 σ M Δ 1.876x10" M Ο 9.378x1 σ Μ

Ο D

ο

4

5

3

v

D

i° \ Ο

2

n

2

D

10^

l j

ν ν

ίο* 2

ι

ί

I ο \ \ ο -

\

3

ΙΟ"

Ο

•Ϊ

ιο ^

\

!

rοi

^l^HS? •'

10'

1

· .......ι

10°

~° t\

7 10

1

Pe

Figure 15. Variation of steady shear μ with Pe for polystyrene spheres of d = 220nm suspended in aqueous solutions of HCl at φ = 0.4 (98). Γ



132


tically w i t h increasing ionic strength (increasing concentration of H C l ) . T h e decrease is much more substantial at lower shear (smaller Peclet number) region, owing to the fact that the mobility of the particles is strongly restricted by the electrostatic charge at low concentration of H C l . H e n c e , these suspensions are expected to behave solid-like at rest. W h e n the Peclet number is large, the steady shear behavior is controled by the hydrodynamic interactions and hence the relative viscosity varies less w i t h ionic strength. T h e prediction by the soft sphere scaling model of Buscall (93) shows fairly good agreement with the experimental data. F o r sterically stabilized suspensions, the polymer thickness Δ changes with the medium temperature and shear rate. It can be said that the effective polymer thickness decreases with compression pressure exerted on the polymer chain. In other words, for the same suspension, when the volume fraction is low, the polymer layer thickness is likely to have more effect on the steady shear viscosity than when the volume fraction is high. However, when the ratio of the polymer thickness to the particle diameter is large, the suspending fluid inside the polymer layer can reduce the effect of the attached polymer. Figure 16 shows the steady shear relative viscosity variation w i t h the effective Peclet number, Pe , based on the effective particle d i ameter at each temperature level, and the temperature for a P M M A suspension. T h e particles of 0.8 μιη are sterically stabilized by a thick layer of terminally anchored poly(dimethylsiloxane) and suspended in n-hexadecane at the volume fraction of φ = 0.282. T h e data points are eS

10

3

Δ

° ° ο

Γ

Δ

10

ΑΟ

2

Ο Δ • 10

ο

2

Δ - |δ): Ε + 2

M f

E

(53)

where μς is the suspending fluid viscosity and μ is now assigned as the viscosity of the fiber suspension. Equation 5 3 indicates that the shear stress is a strong function of the rate-of-strain as w e l l as the orientation of the fibers. F o r r > 20 and freely rotating particles, Simha (111) found that the Einstein constant for spheroids and rigid rods is given by e

1


k

,

4

15

r */5

r */15 e

e

ln(2r ) - c e

a

ln(2r ) - c + 1 e

a

'

W

where c is a constant. F o r spheroids, c = 1.5 and for r i g i d rods, c = 1.8. Goldsmith and Mason (112) obtained the Einstein constant based on a time-averaged optimum orientation given by a

a

h

=

3[ln(2r ) - c j ( r e

a

e

+ l)

2

(

5

5

)

Haber and Brenner (113) obtained a relation for ellipsoids. T h e Einstein constant for a fiber suspension is also a function of shear rate. T h e ratio of l o w to high shear limit Einstein constant is rj 1.17 for rods and r / 3 1 for discs (114). T h e l o w shear limit Einstein constant is given b y e

for r < 50

k = a + ar , E

0

l

e

(56)

e

where α is a constant. F o r discs, a = 2.51 and a = 0.1127, and for rods, a = 2.34 and a = 0.1636 (115). Shaqfeh and Fredrickson (116) and Shaqfeh and K o c h (117) gave the following formulae: 0

0

x

x

k

=

*P/(3Vp) _ \η(1/φ) + In l n ( l / 0 ) + Α(φ)

l

m K

1

where I is the longest dimension of the fiber, V is the volume of one particle, Α(φ) is a weak function of φ and is dependent on the shape of the fiber. Kitano et al. (82) found that the relative viscosity of a suspension of cylindrical rods can be estimated by the Quemada equation w i t h the maximum packing fraction given by p

0max

= 0.53 - 0 . 0 1 3 r , e

for 5 < r < 30 e

(58)

B y an asymptotic matching of the dilute limit homogeneous viscosity with the Quemada equation, Phan-Thien and Graham (JIS) obtained the following equation valid for a wide range of solid concentration:


136


Mr

1 - 0.5

= 1 + fc

E

φ/φ

(59)

η

(1 - Φ/Φ™)

where k is the intrinsic viscosity of the suspension and may be estimated by equation 56. Figure 17 shows the experimental results on the low shear relative viscosity for suspensions of cylinders and spheroids of Phan-Thien and Graham (J IS) for a good range of particle volume fractions. These data agree with the experimental results of Ganani and P o w e l l (119), although the latter deal only with dilute suspensions. F r o m F i g u r e 17, we observe the agreement between the predictions based on equation 59 and the experimental data is fairly good. Equation 40 may also be used, but the value of the Huggins constant must be provided i n the concentration range shown in F i g u r e 17. W e find that the following formulae may be used with equation 40 based on the data of Kitano et al. (82) and PhanT h i e n and Graham (IIS):


E

(60) k = 3*Ε/Φ,max In a uniaxial elongational flow, the Trouton, or elongational viscosity of an anisometric particle suspension can be estimated by equation 59 as w e l l with an intrinsic viscosity given by H

2r

e

2

3[ln(2r ) e

2

(61)

1.5]

0.4 Φ

Figure 17. Low shear limit viscosity variation with solid volume fraction and aspect ratio for suspensions of spheroids and cylinders (118).



3.

L i u & MASLIYAH


137

Polydispersity Effect. In the previous sections, we discussed the rheology of monodispersed particles, that is, all of the solids have the same particle size and shape. In this section, we consider the case where the suspension is made up of solids having different sizes. Polydispersity can have a significant influence on the rheological behavior of suspensions. F o r the same solid volume fraction, it has been known that particle size distribution can greatly reduce the hydrodynamic resistance of random packs of spherical particles (120) because of the reduction of channel sizes i n the packs as compared w i t h the unimodal systems. Apart from the channel (gap between particles) size effect, a multimodal system can have significantly lower void ratio as w e l l . T h e Quemada equation suggests that lowering of the random packing void ratio alone w o u l d lead to a significant reduction of the suspension viscosity at a given solid volume fraction. Several studies have shown that the use of a bimodal or a continuous broad particle size distribution can lower the viscosity and increase the maximum at tainable solid volume fraction (36, 53-55,121-136). T h e most complete studies to date for concentrated suspensions are the studies of Rodriguez et al. (130) and C h a n g and P o w e l l (55). T h e latter used an orifice vis cometer instead of the traditional rotational viscometers. F o r bimodal sterically stabilized silica suspensions i n cyclohexane, Figure 18 shows the variation of the Einstein and Huggins constants with the relative volume fraction of the larger particles in the total solids, 10r Ο d / d = 1.45 L

8^

:

eF-' ---:;>-

— Δ -

.

0.4

0.6

Δ

d / d = 1.59

ν

d / d = 2.56

•

d / d = 4.07

L

L

s

s

*

m

0.2

s

0.8

L

s

m *

1.0

Figure 18. Variation of the Einstein and Huggins constants with the ratio of the large to small particle size and volume fraction content of the large particles (73). The filled symbols mean the same as the open symbols.


138


e = φJ φ, for different size ratios of large to small particle, d /d . T h e experimental conditions of the data (73) correspond to the low shear limit. It can be seen that the bidispersity has a relatively small effect on the Einstein constant, whereas a more pronounced effect on the Huggins constant can be observed. W h e n the particle size ratio is small, the Huggins constant k experiences a m i n i m u m . H o w e v e r , when the par ticle size ratio is greater than 1.6, a maximum value i n k can be observed. T h e l o w shear limit viscosity at vanishing solid volume fraction for b i dispersed and polydispersed hard sphere systems is also modeled n u merically b y Wagner and Woutersen (137) and Jones (138). These two theoretical studies revealed that the Einstein constant remains constant, whereas the Huggins constant decreases slightly w i t h increasing degree of polydispersivity. F i g u r e 19 shows the steady shear relative viscosity variation w i t h the relative volume fraction of the large particles, € , for various large to small particle size ratios, d /d , and different total solid volume frac tion, φ, for bimodal concentrated dispersions of submicron resin particles at a shear stress level of 0.0155 P a . T h e experimental data are taken from Hoffman (129). W e can observe that the relative viscosity exhibits a minimum near c = 0.8. T h e minimum viscosity behavior is morepronounced at large particle diameter ratios, dijd . T h e bimodal system viscosity can be several order of magnitudes lower than the correspond ing monodispersed systems when the larger particles composed of about L

L

s

H


H

L

h

s

L

s

ν

d / d » 0.68/0.21 ,φ = 0.55

Ο

d / d = 0.68/0.21 ,φ = 0.60

Ο

d /ds = 0.68/0.21 ,φ = 0.65

Δ

d / d = 0 . 9 5 / 0 . 1 4 ^ = 0.65

•

d / d = 0.68/0.27, φ = 0.65

•

Farris (139), theoretical

L

s

L

s

L

L

L

s

s

based on φ = 0.65

,w

0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 e L

Figure 19. Effect ofcomposition of bimodal suspensions ofcolloidally stable polymer particles on the viscosity measured at a steady shear level of 0.0155 Pa (129).


3.

L i u & MASLIYAH

139


8 0 % of the solids in volume. As would be expected, the relative viscosity increases w i t h increasing total solid volume fraction for a given e value as w e l l . The predictions based on Farris (139) are also shown i n the figure. Farris (139) hypothesized that multimodal systems can be treated as suspending large particles (the large size component) into a homo geneous suspension made up of the small particles. This treatment is believed to h o l d for the case where the particle size ratio is very large, so that the large size particles " s e e " the suspension of the small particles as a continuous phase. It holds only for the case when one of the com ponents is dominant in quantity. K i m and L u c k h a m (140) suggested that the relative dynamic viscosity of a bidispersed suspension may be estimated by using the product of the two component relative viscosities each computed from the K r i e g e r Dougherty equation as i f they were alone i n the suspension. This treat ment has been commonly used since Farris (139). H o w e v e r , it is valid only when concentration of either component is large, that is, c 1 or € 0, and the particle sizes are very different from each other, that is, d «d . W h e n the two concentrations are similar, that is, e « 0.5, the estimation of the apparent viscosity using this approach gives a much lower value than the experimental values. This behavior is same for dynamic viscosity as the steady shear viscosity.


L

L

L

s

L

L

Rodriguez et al. (J30) and C h a n g and P o w e l l (55) showed that the traditional Quemada equation and other models can be used to correlate the steady shear viscosity of bidispersed systems p r o v i d e d that the cor rect value of the maximum packing fraction, 0 , is used. F i g u r e 20 shows the universal scaling for both the low and the high shear limit relative viscosities as a function of the reduced solid volume fraction for monodisperse and bidisperse latex particle suspensions. Figure 21 shows the low shear limit maximum packing fraction, 0 , variation w i t h the relative volume fraction of the large particles, e , for various large to small particle size ratios, dijd . W e can observe that the maximum pack ing fraction $ reaches a maximum between e = 0.6 and 0.8 and the maximum value increases w i t h the particle diameter ratio, d /d . T h e phenomenon of higher maximum packing limit, 0 , for bimodal systems than unimodal systems is a reversed relation of the relative shear viscosity versus the relative volume fraction of the large particles, c , and particle diameter ratio, d /d$. Rodriguez et al. (130) showed that the behavior of steady shear viscosity versus the shear stress for a binary system is no different from that of a monodisperse suspension. A universal equation may be obtained for both the monodisperse and bidisperse suspensions. H e n c e , we may further infer from the literature results available that the steady shear viscosity for a polydispersed (multimodal) suspension can be treated the m a x

m a x

L

s

L

m a x

L

s

m a x

L

h



140


10

0.5

0.6

0.7

0.8

0.9

1.0

Figure 20. Universal low and high shear limit viscosity vs. solid volume fraction curve for monodisperse and bidisperse systems (130).

Π

ft' ι ι

8.0

J • » n ιιIι• • • I• ' ' I , I • ι ι , I • • , t I ι . 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

. ι

I

Figure 21. Maximum packing fraction variation with composition of par ticles and large to small size ratio for binary mixtures (55, 130).


3.

Liu & MASLIYAH


141

same way as one w o u l d for a monodisperse (unimodal) suspension by using 0 to normalize φ. m a x


Shear Thickening and Discontinuity Behaviors.

Although

continuous shear thinning behavior is commonly observed, shear thick ening behavior can also be observed for concentrated suspensions of nonaggregating solid particles when subjected to the appropriate con ditions (141). M e t z n e r and W h i t l o c k (142) were probably the first to observe the shear thickening behavior of suspensions. T h e systematic studies on this subject have been conducted by Hoffman (68,143, 144), Strivens (145), Wagstaff and Chaffey (124), W o o d c o c k (146), L a u n (147), Boersma et al. (148,149), Yilmazer and Kalyon (150), and Laun et al. (151). However, a better understanding of the shear thickening behavior is needed to predict the conditions for w h i c h shear thickening w o u l d occur. Figure 22 shows shear thickening behavior of colloidally stable sus pensions for both monodisperse and bidisperse poly(vinyl chloride) par ticles i n dioctyl phthalate. T h e monodisperse system was studied b y Hoffman (68) and the bidisperse system was taken from Boersma et al. (149). The concentrated suspensions of Figure 22 show either continuous or discontinuous shear thickening behavior depending on the solid v o l ume fraction and the composition of the solids. T h e severity of the shear thickening increases w i t h particle concentration, possibly φ/φ^, whereas the critical Peclet number (based on the average particle d i -

10

5

10

4

=* 10

φ, d 0.57, 0.55, 0.51, 0.49, 0.47, 0.57, 0.53,

°

3

• 10H-

10 ΙΟ

Δ

ο •

μπη, 1.25, 1.25, 1.25, 1.25, 1.25, 1.40, 1.40,

a v

dispersity mono mono mono mono mono poly poly

ο 0

9

Λ

° ° o

οο )

0

ο ο

Οο

π

• ν ν • ν

* ,ν

οοο ο

1

1

10°

10

10

1

2

10

3

10

4

Ρθ

Figure 22. Shear thickening behavior of suspensions of monodisperse and bidisperse polyvinyl chloride particles in dioctyl phthalate (68, 149).



142


ameter) for the onset of shear thickening decreases with increasing par ticle concentration. F o r a more concentrated monodisperse system, a large j u m p in the steady shear viscosity is observed at the critical Peclet number and followed by a shear thinning behavior, whereas the tran sition to the next shear thinning behavior is smooth for less concentrated or bidisperse systems. O w i n g to the increase i n maximum packing limit volume fraction or reduction i n the effective solid concentration for multimodal systems, the shear thickening behavior is delayed for the same particle concentration as compared to monodipersed systems (152). Hoffman (68) attributed the shear thickening behavior to some order-disorder transition of particle microstructure, w h i c h was also observed by Ackerson and Pusey (J53) and Wagner and Russel (71). This model was continued by Boersma et al. (148). T h e y assumed that the shear thickening occurs when the shear forces overcome the interparticle forces. Hence, a dimensionless group, N , the ratio of the viscous to the repulsive forces, may be used to describe the transition: d

2ire e £o 0

r

where e is the permittivity of vacuum, e is the dielectric constant of the suspending fluid medium, and £ is the surface charge (zeta) potential of the particles. This group relates to the case of constant surface po tential of electrostatically stabilized systems. Shear thickening is p r e dicted to occur when N > 1. In addition, the shear thickening can occur in sterically stabilized concentrated systems as well (154, 155). Shear thickening behavior appears almost like a j u m p i n the steady shear viscosity versus shear rate, μ versus y, or μ versus Pe plots, es pecially for more concentrated suspensions. T i m e dependence is thus expected to occur. Boersma et al. (149) demonstrated that strong time dependence can be observed w h e n a steady shear is applied near the critical point for w h i c h the shear thickening occurs. L a u n et al. (151) investigated different suspension systems near the transition to shear thickening. T h e i r studies show that the order-disorder assumption may not hold, although there exist some structural changes near the transition. It is certain, however, that shear thickening behavior is associated w i t h the loss of long-range order of the particles. H o w e v e r , the loss of long-range order does not necessarily produce shear thick ening (85, 156, 157). Ackerson and Clark (158), C h e n and Zukoski (159), Stevens et al. (160), and C h e n et al. (161) observed a different k i n d of discontinuous viscosity response to an order-disorder transition for electrostatically stabilized suspensions at a relatively l o w solid concentrations, where the viscosity displays a discontinuous drop at certain shear rate. Apart 0

r

0

d

τ



3.

Liu & MASLIYAH

143


from the obvious changes i n the viscosity versus shear rate plot, C h e n and co-workers (156, 162) observed a rather peculiar discontinuity in the viscosity-shear stress or shear stress versus shear rate plot for a solid concentration between the shear melting and shear thickening values. As shown in F i g u r e 23, there is no visible change i n the viscosity versus shear rate behavior, although the suspension order changed from a strained-crystal microstructure to a shearing layer microstructure. T h e system consisted of a polystyrene latex particles of 0.146 nm diameter suspended i n an aqueous solution of 0.001 M N a C l . T h e solid fraction is 0.33. In the transition region, the long-range orientational order is lost and the suspension takes a polycrystal-line appearance. Despite the changes in the suspension structure, the steady shear viscosity shows little effect on the μ versus y plane (Figure 23). H o w e v e r , when the shear stress is plotted against the shear rate, as shown in F i g u r e 24, discontinuity is observed. It can be concluded that the shear melting, thickening, and the shear stress discontinuity are the results of the shear-induced suspension structure change. The irregularities occur when several suspension structures coexist in the system in separate zones. In the region where the irregularity of melting shear or shear stress discontinuity occurs, the shear stress may decrease w i t h shear rate or remain constant for a particular region of shear rate (J63). O n the shear stress versus shear rate plane, hysteresis exists. The irregularities (shear melting, thickening, and the shear stress discontinuity) are more severe upon increasing shear 10

5

Δ

10

4

3

t 10

2

d. 10

1

10° ΙΟ"

ν τ

! \ * ^

Γ

10

' """I " ' ' ""Ι ι "Ι increasing shear rate decreasing shear rate shear stress controlled metastable, shear stress controlled : 1

Ο • Δ

Γ

1

PC i Strained-crystal ι ι ϋϋπΐ ι ι ι » nul , . η.ml 1 10° ΙΟ 1er 1er ίογ, s"'

10- L 2

4

2

2

1

Shearing layer · ι

10

1

10

2

: · ......

10

3

Figure 23. Steady shear viscosity vs. shear rate diagram for 0.146 polystyrene latex suspension in 10~ M NaCl (162).

μm

3


144


» r - r ™ | -«-r-rrtmy —r-t ,.τπη

""I

7

ο •

Δ

•

·

1

J

[

increasing shear rate decreasing shear rate shear stress controlled metastable, shear stress controlled

0

~ α

θ

θ

1 :

Q


-

1

Δ

PC

Strained-crystal ....1

1er

4

ίο*

1

,

1
^Δ

p

£

Δ

-

1

1

Β

< S Δ


10

2

r

r

10? 0.0

Ί

/

I

1 t t t

1

0.1

I

I

1

I

I

1

0.2

I

1

I

0.3

1

I ι

0.4

ι ι ι I ι ι > ·

0.5

0.6

Figure 27. Low shear viscosity variation with solid volume fraction for weaklyflocculatedsystems. αφ

3

ν

Mr

=

φ

Φη

(64)

+ 1

where a is a constant depending on the structure of the floes. T h e line shown i n F i g u r e 27 is for a = 2.3 Χ 1 0 . O w i n g to the high viscosity values at low shear limit, a y i e l d stress may also be assumed as a con venience for weakly flocculated suspensions. Especially for a concen trated suspension, a long time is needed for the flow to occur after the application of a low shear rate (stress). F i g u r e 28 shows the Bingham yield stress versus the solid volume fraction diagram for several weakly flocculated systems (J 73). It can be observed that the same scaling for the l o w shear viscosity may also be applied to the Bingham y i e l d stress: v

4

v

αφ Υ

Φη

3

Φ

(65)

+ 0ΒΟ

where a and σ are constants that depend on the structure of the sus pension. The curve shown in F i g u r e 28 is for a = 0.24 P a and σ = 0.05 Pa. F r o m F i g u r e 2 8 , one observes that the maximum packing limit is also a function of the particle size. T h e particle size dependence of Bingham y i e l d stress is understood by the fact that different degree of flocculation is obtained for different particle sizes for the same system. Because of the size of a floe, the wall effects become significant i n the viscosity and other rheological measurements for a flocculated susy

Β0

y

Β 0

max


3.

L i u & MASLIYAH

149


1.6,

ν ν •


Δ

i.O

0.1

0.2

0.4

0.3

0.5

0.6

0.7

Φ Figure 28. Bingham yield stress variation with solid volume fraction for weakly flocculated systems of different particle sizes and different layer of thickness (173). pension. Because the measurement devices are l i m i t e d (normally inten tionally) in size, the assumption of a continuum cannot be justified. Apart from the continuum assumption, the wall slip problems can be more severe than for stable systems (51). M o r e studies are needed to elucidate the wall effects of flocculated and stable systems. Buscall et al. (51) pointed out that one common way of avoiding the wall effects is to roughen the surfaces of the measuring vessel. This technique may be appropriate for stable systems provided that the walls are roughened using the same particles and w i t h the same concentration as that in the testing suspension. In addition, the suspension has to behave as a single phase, that is, no apparent relative particle-fluid movement. However, for flocculated systems, it is rather difficult to simulate a sus pension structure on the wall. H e n c e , measurements w i t h a roughened wall are at best as " g o o d " as those w i t h smooth surfaces (Figure 29). It is interesting to note that Buscall et a l . (51) observed no wall effect contribution from the outer cylinder of the cup-and-bob rheometer. In a manner similar to the wall effects for flow through a fixed b e d (188), strong wall effects are present i n the l o w shear regime.

Solids and Droplets Codispersed Systems: Solids-Oil-Water Ternary Mixtures. Mixtures of o i l and solids dispersed i n water and mixtures of water and solids dispersed i n oil are commonly encountered in the petroleum industry (189). It is of great interest i n studying the rheological behavior of ternary systems, where two distinct materials



150


σ, Pa Figure 29. Comparison of flow curves obtained for a nonaqueous acrylic copolymer suspension in low-aromatic white spirit containing poly(isobutylene) (FIB) by a coaxial cylindrical rheometer (51). act as the dispersed phases. T h e ternary systems were also reviewed earlier by Pal et al. (4). T h e rheological behaviors for ternary systems with low solid content, as are commonly encountered i n the petroleum industry, are m u c h simpler to describe than when both of the dispersed materials have high concentration. F o r fluid-in-fluid (liquid-in-liquid and gas-in-liquid) dispersions, Taylor extended Einstein's treatment with interface effects neglected (190, 191) and obtained

2(

Md

+

Mf

)

where μ is the viscosity of the droplet phase fluid. O l d r o y d (192, obtained the Einstein constant by including the interface effects: ά

5μ + 2μ + μ \ά ά

2(μ

ά

{

χ

193)

.

.

+ /if + Mi/5d)

where μ is the interface viscosity and is expressed as ν

Mi = 2μ + 3μ ί 5ί

(68)

β

with μ representing the shear viscosity and μ the extensional viscosity of the interface. 5ί

β1



3.

L i u & MASLIYAH


151

Equations 66 and 68 indicate that the droplet behaves like a solid particle only when the viscosity ratio of the dispersed phase to the con tinuous phase is large. F o r liquid-in-liquid dispersions, the modified Quemada equation, K r i e g e r - D o u g h e r t y equation, and Mooney equation are still applicable provided that the maximum packing limit and the Einstein constant are left as adjustable parameters for a given system. One of the unique rheological features of emulsions is that the ap parent viscosity of the emulsion can drop below the viscosity of the continuous phase when the concentration of the dispersed phase is low, normally below 0.1 i n volume fraction (194). W h e n solids are added to the emulsion, the apparent viscosity can decrease even further and the volume fraction of the dispersed phase at which m i n i m u m viscosity oc curs increases with increasing solids content. F i g u r e 30 shows the ap parent viscosity of water-and-sand-in-bitumen, M , variation with the solid-free water volume fraction, /3 , for two shear rate values. T h e ex perimental data were provided by Yan (private communication), where the system consists of 52 μπι sand particles treated w i t h hexadecyltrimethylammonium bromide ( H A B ) and water droplets of a Sauter mean diameter of 9 μπι dispersed i n bitumen at 60 °C. T h e sand particle volume fraction on water-free basis is 0 = 0.193. T h e range of the water droplet volume fraction, on a solid-free basis, 0 is between 0 and 0.4. It can be observed that a minimum viscosity is present at a solid-free water droplet volume fraction of about 0.1. F o r a lower solid concen tration, ft = 0.113, the minimum apparent viscosity is found at /3 = 0.05 w s b

W

S

W

W

30 α,γ = 0.77 s"

Q_

γ = 6.64 s

1

% :i.o

0.1

0.2

0.3

0.4

1

0.5

Figure 30. Effect of water addition to a sand-in-bitumen suspension at 60 °C. (Data from Ύ. Yan, private communication, 1990.)



152


(J94). T h e viscosity reduction by water addition is not due to the pres ence of the surfactant ( H A B ) . F o r the sand-in-bitumen suspension, the viscosity variation is shown in F i g u r e 31. It can be observed from F i g u r e 31 that the sand-in-bitumen suspensions are slightly shear thinning type. A low shear limit viscosity is observed, although a y i e l d stress may be assumed i f Casson's model is used (194). T h e shear thinning behavior is more severe when the solid volume fraction is increased. Shear thinning behavior is commonly observed in emulsions. W h e n solids are added to emulsions, both shear thinning and shear thickening behaviors are observed. F i g u r e 32 shows the rheograms of clay-and-oil in water measured by Y a n (private communication), where the clayfree o i l volume fraction, β = 0.2. Part of F i g u r e 3 2 , y > 10 s" , was also shown by Yan et al. (195). T h e systems are shear thinning, w h i c h is similar to the pure suspension of clay in water and oil i n water emulsion (J95). In the low shear regime, the systems display some discontinuities on the shear stress versus shear rate plane, and the discontinuity is more severe when the clay volume fraction, 0 , is about 0.12. F o r pure clay in water suspensions, similar behavior is also observed, whereas the discontinuity is much reduced as the o i l concentration is increased. Figure 33 shows the apparent viscosity variation with shear rate for 44 μπ\ glass beads suspended i n a 5 0 % bitumen-in-water emulsion at 25 ° C . T h e viscosity ratio of the dispersed liquid phase (bitumen) to the 1

0

C

40

Ή

'I

1

'—

1

52 μτη sand in bitumen suspensions

30

Φ.(-Α)

20

0.194 A

CO CO Q_

A

A

A

A

A

0.157

h 10 0.113

9 8 7 6

ο οu

5

Q

0.1

o

°OGu

Q

1

10

t

ν ν ο 0 100

S"

1

Figure 31. Viscosity variation of the sand-in-bitumen suspensions with shear rate at different sand volume fractions at 60 °C (194).


3.

Liu & MASLIYAH

10


153

2

5 μιτι (dry) kaolinite clay added into Oil (Bayol-35) in water emulsion,

10

1


Q_

10°

ΙΟ" ΙΟ 1

1

10°

10

10

1

?. s

10

2

3

10

4

1

Figure 32. Rheograms of kaolinite clay and Bayol-35 oil dispersed in water at 25 °C. (Data from Y. Yan, private communication, 1990.)

0.06 44 μπι glass beads added into Bitumen in water emulsion, β = 0.5

0.05

6

φ = 0.213 0.04F

S. | 003

0.02

ο ο σ °~~ο ο ο ο α—ο—ο—ο—ο 0

0.01

0.00" 10

1

10

10

2

3

10

4

t , s"

Figure 33. Viscosity variation with shear rate for 44 μm glass beads sus pended in 50% bitumen-in-water emulsion at 25 °C (196).


154


continuous phase (water), Md/W * 306,000. T h e bitumen droplets have a Sauter mean diameter of 9.1 μπι. This system displays shear thickening behavior. T h e shear thickening appeared to be more pro nounced for the case where the viscosity ratio of the dispersed phase fluid to the continuous phase fluid is high (196). T h e shear thickening behavior as observed by Yan et al. (196) is monotonous, that is, the viscosity does not decrease with increasing shear rate at high shear rates. They found that when μd/μf is small, for the cases of Md/Mf = 2 . 4 (Bayol oil droplets of diameter 10.6 μπ\ in water) and Ma/Mf = 150 (paraffin oil droplets of diameter 10.2 μχη in water), both the emulsions and the ternary systems (all with 44 μ π ι polystyrene beads added) display shear thinning behavior. W h e n μd/μf is large, as for the case of corena oil droplets of diameter 9.6 μπι in water, μd/μf = 1040, the emulsion is Newtonian and the ternary system (with 44 μπι polystyrene beads added) displays shear thickening behavior. W h e n the solid particle size is large, all the ternary systems are nearly Newtonian, irrespective of the values of the viscosity ratio, μd/μf.


s

L i q u i d droplets cannot be treated the same as solid particles in their codispersed systems. This behavior has been indicated by equation 66 or 68, in w h i c h the Einstein constant increases with increasing viscosity ratio of the dispersed phase to the continuous phase. As is shown by Yan et al. (195, 197, 19S), l i q u i d droplet and solid particle effects are additive only when the solid concentration is low, say φ < 0.05, and when both solid particles and l i q u i d droplets have comparable sizes. H o w e v e r , when the particle-to-droplet size ratio is large, the particles and the droplets become additive (192) for a wider solid concentration range (Figures 34 and 35). T h e apparent viscosity of the system may be " a d d e d " in terms of the two distinct model systems: pure emulsion characterized by solid-free dispersed phase volume fraction and pure suspension characterized by the volume fraction of the solids. T h e ad ditive rule for the ternary systems is similar to the rule for bimodal solid particle suspensions due to Farris (139): 5

Msdf Mf

=

MdKffd)

χ

A*sf( 3 i f the viscosity of the fluid forming the droplet phase is not too far from the continuous phase, where no concentration limit is posed on the dispersed droplet phase. The critical size ratio increases with the vis cosity ratio of the fluid forming the dispersed phase to the continuous S

s

d


3.

Liu & MASLIYAH

155


300

250h

G) behavior to a more elastic (G" < G) behavior with increasing solid volume fraction. Figure 39 shows the shear moduli variation w i t h solid volume frac tion for the electrostatically stabilized suspension of 1.4 μπι polystyrene latices i n aqueous solutions of N a C l . A t the lower N a C l concentration (10~ M ) , the double-layer thickness, Ι/κ, is 100 n m , and therefore the suspension show " s o f t " type interaction due to the extended double œ

5


3.

L i u & MASLIYAH 10


159

4 1

1

'

I

'

'

•

'

; G\ G" ο, · d = 0.156 μιη A • d = 0.612 μπι 10 ν , • d = 1.004 μπι 3

I ·

1

•

V Δ V V

ο

2

ο Downloaded by UNIV OF MINNESOTA on October 7, 2013 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/ba-1996-0251.ch003

' ;

1

Χ Ο

ο ο

t

ο

10

1

(0

•

0

ο

• • •

••

•

•

τ

Δ

Δ

Α V

10° I

•

Δ

V

•

• 10""

Δ •

27

10' 0.4

«

0.5

I

I

0.6

I

.

ί

ί

Figure 38. Shear moduli variation with volume fraction at a frequency of 1 Hz for steric polystyrene latex suspensions of various sizes. (209.) layer. F o r the higher N a C l concentration case, the double layer is only 10 nm and is much smaller than the particle size; thus, the suspension is close to a hard-sphere system. Flocculated Systems. T h e viscoelastic responses of flocculated systems are strongly dependent on the suspension structure. T h e sus pension starts to show an elastic response at a critical solid volume frac tion of 0 = 0.05 - 0.07, at w h i c h the particles form a continuous threedimensional network (211-213). T h e magnitude of the elastic response for flocculated suspensions above 0 depends on several parameters, such as the suspension structure, interparticle attraction forces and par ticle size, and shape and volume fraction. Buscall et al. (10) found that the volume fraction dependence of the storage modulus follows a powerlaw behavior. ct

ct

G -

φ

τη

(71)

with a power of m = 2 - 5 (10, 212-216). Figure 40 shows the experimental results of C h e n and Russel (215), which demonstrate that an increase i n the magnitude of the interparticle


160



ce

Φ Figure 39. Shear moduli variation with volume fraction for electrostatic suspension of polystyrene latex in aqueous solution of NaCl at ω = 1 Hz (206).

Figure 40. G variation with ω at temperatures between 20 and 30 °C for a coated silica suspension with d = 112 nm in hexadecane (215).



3.

L i u & MASLIYAH

161


attraction results i n an increase i n the storage modulus, provided that the suspension structure remains minimally disturbed. L o w e r i n g the temperature below 303.2 Κ causes the suspension to form a particle network w i t h stronger elastic responses. T h e change in G depicts the sharp change in the suspension properties from a liquidlike (viscous) at 303.2 Κ to a solidlike (elastic) response at 302.2 K . F u r t h e r reducing the temperature results in an increase i n the magnitude of the inter particle attraction, w h i c h leads to an increase i n the storage modulus. T h e system becomes more elastic. Similar behavior showing the i m p o r tance of the interparticle forces is also reported by Otsubo (217) w i t h different particle sizes instead of varying the temperature. Kawaguchi and co-workers systematically studied the steady shear, transient shear, and dynamic rheological properties of a weakly floc culated silica suspension i n aqueous solutions of H P M C . T o help us understand the viscoelastic behavior of the suspensions, the results of Kawaguchi and R y o (218) and Nakai et al. (J 76) are summarized here. T h e systems shown i n Figures 4 1 - 4 3 are for steric silica suspensions in a 2.0 w t % aqueous solution of H P M C having a molecular weight of 321,000. F i g u r e 41 shows the shear moduli responses with changing frequency for three systems of silica content: 7.5, 5.0, and 2.5 w t % . W e observe, for example, that the loss modulus crosses over the storage modulus at ω « 3 s" for the 5.0 w t % silica suspension. This indicates that the time scale at w h i c h the 5.0 w t % silica suspension switches from a more elastic to a more viscous behavior is about t « 1/3 s. F r o m the 1

E

Figure 41. Shear moduli dependence on frequency for silica suspensions in 2.0% aqueous HPMC solutions (218).




162

Figure 43. Evolution of transient shear stress for 7.5 wt% silica suspensions in 2.0% aqueous HPMC solution (176).


3.

L i u & MASLIYAH


163

steady shear stress versus shear rate diagram (Figure 42), one can observe that when the shear rate is below 3 s~\ the suspension exhibits a rela tively constant shear stress with an increase of shear stress when the shear rate is above 3 s *. T h e time scale is w e l l described by equations 5 and 6. F o r the most concentrated suspension, 7.5 w t % silica, the critical time scale as given by γ is less than 0.1 s as shown i n Figures 41 and 42. F r o m F i g u r e 4 3 , one can observe that the shear stress is a function of time when the shear rate is below 10 s" . As the shear rate is lowered, the time dependence of the shear stress becomes more pronounced and a longer time is needed for the system to reach a steady state. B o t h rheopexy (i.e., shear stress increases w i t h time at a constant applied shear rate) and stress overshooting (i.e., the shear stress is higher than the steady state value) can be observed in F i g u r e 43 at shear rates below 0.07 s" . - 1


1

1

Compression Rheology Callagham and O t t e w i l l (219) were among the first to study the com pressive and decompressive y i e l d stress of suspensions. T h e i r system consisted of fine aqueous montmorillonite clay suspensions. W h e n the tests were performed at different ionic strengths ( 1 0 " - 1 0 M N a C l ) , they found systematic variations i n the compressive y i e l d stress due to the long-range electrostatic double-layer interaction. T h e suspension was compressed at a low ionic strength and it showed a compressive stress response over a much larger clay concentration range, starting at 10 w t % clay. In addition, a hysteresis was found upon decompression. Cairns et al. (220) studied monodispersed P M M A latex with a particle diameter of about 156 nm suspended in dodecane and stabilized by covalently attached poly( 12-hydroxystearic acid) w i t h a thickness of Δ = 9 n m . They found that the resistance to compression starts to i n crease drastically around φ = 0.55, and at φ = 0.566 the compression resistance was very strong, indicating that the suspension w o u l d not yield to compression when the maximum packing limit is reached. Most colloidal stable suspensions show more or less reversible re sponse to compression and decompression. H o w e v e r , in the case of floc culated suspensions, the compressive properties are irreversible. In concentrated flocculated suspensions, a continuous particle network forms. T h e particle network can support some stress up to a critical value. Once this critical stress, also called the compressive y i e l d stress P , is exceeded, the network consolidates to a higher volume fraction with a higher critical stress. Buscall et al. (212) found that the compressive y i e l d stress could be fitted to a power law form with respect to the solid volume fraction 4

_ 1

y

Py

(72)


164


with m « 4. Equation 72 is in agreement with the percolation theory (213) when the solid volume fraction is w e l l above the percolation threshold, φ . T h e y also studied the particle size effect on P and fitted P to d by a power law form ν

y

y

P -


y

(73)

d

n

w i t h a best fit of η = —2, although they w o u l d have preferred η = - 2 . 3 . This can be understood by considering the fact that the compressive y i e l d stress is inversely proportional to the particle surface area. W h e n the surface area is large, it is easier to make the particles to contact w i t h each other and break up the floe structure. F o r a spherical particle, the surface area is proportional to the square of its diameter. T h e compressive y i e l d properties are strongly dependent on the suspension structure. Mills et al. (186) showed that extensive preshearing of a flocculated suspension can result in a much denser floe morphology (in fractal terms), w h i c h was reflected by a decrease in the compressive y i e l d stress by one order of magnitude. T h e compressive y i e l d stress can also be measured with the c e n t r i fuge method, where the sedimentation height as a function of centrifugal acceleration is measured (221-223). Because the centrifugal acceleration can disturb the suspension structure as w e l l , it is expected that the mea sured compressive y i e l d stress depends on the applied forces especially at a lower solid volume fraction. A t a higher solid volume fraction, the centrifuge method can still measure the compressive y i e l d stress that is more or less independent of the acceleration used. H o w e v e r , a strong acceleration is needed to make the suspension consolidate. F i g u r e 44 shows the compressive y i e l d stress variation with volume fraction of solids under various centrifugal accelerations (based on the study of Auzerais et a l . , i.e., reference 23). The system is a strongly flocculated silica suspension in hexadecane w i t h an initial solid volume fraction of 0.096. T h e lines are plotted based on the usual scaling of the flocculated systems:

F o r the high centrifugal speed cases, the (dashed) curve of P = 160 k P a and η = 6 follows the experimental data fairly closely. F o r lower speeds, the maximum packing limit is much reduced. However, the same scaling is indicated in the figure. T h e solid line is for P = 30 k P a and η = 3. T h e maximum packing limit used for the two lines corresponds to the high shear limit. T h e power of 3 is the same as for the low shear viscosity and Bingham y i e l d stress dependence on the solid volume frac tion. T h e higher power for a fixed speed is due to the flocculation of 0

0


3.

L i u & MASLIYAH

165


70000|

60000h

50000

ce 40000

a.


30000

Centrifugal acceleration • 7000 rpm * 6000 rpm ο 5000 rpm • 4500 rpm ο 4000 rpm ο 3500 rpm Δ 2500 rpm ν 1500 rpm

20000

10000

8'l.O

0.7

0.1 Φ

Figure 44. Compressive yield stress variation for a strongly flocculated suspension in hexadecane (d = 0.08 μm) with solid volume fraction and centrifugal acceleration (223). the suspension. H i g h energy input is necessary to break up the floes, and l o w speeds simply w o u l d not make the suspension consolidate completely. Bergstrom et al. (224) studied the compressive yield stress of alumina suspensions in decalin with different degrees of flocculation b y adsorbing different fatty acids at the alumina-decalin interface. F i g u r e 45 shows their results (symbols) as compared with a set of correlations (curves), all based on equation 74, where a power of 3 is used. T h e properties corresponding to those of the figure are tabulated in Table I. It can be observed that both P and are strongly dependent on the magnitude of the attractive forces. As the attractive force increases, the degree of flocculation increases. F o r highly flocculated systems, the maximum packing fraction of the solids is reduced and the force required to break up the floes is greatly increased. Because the compression experiments do not disturb the suspension structure to a significant extent, the max imum packing limit is, b y and large, retained for the flocculated systems. The compression yield stress is directly comparable with the osmotic pressure for a given suspension system. T h e osmotic pressure has been measured b y Barclay et al. (225), Rohrsetzer et al. (226), G o o d w i n et al. (227), and Bonnet-Gonnet et al. (228). T h e osmotic pressure displays the same trend as the compressive y i e l d stress. 0

max



166


Figure 45. Compressive yield stress variation with solid volume fraction for flocculated systems (224).

Summary This chapter has been devoted to an in-depth review of the rheological behavior of suspensions. A most significant problem still exists in today's studies in that the wall effects are not quite well understood. T h e dif ficulty in isolating wall effects has contributed to the scatter of experi mental data from one rheologist to another. Because the rheological studies are directly related to the hydrodynamic behavior of suspensions, the understanding of the rheological behavior cannot be isolated from the hydrodynamic studies. It is for this reason that the rheological studies to date are very important even though the wall effects are present in all the experimental studies. O n e unique scaling behavior has been uncovered in this chapter for the low shear limit viscosity, equation 64, Bingham yield stress, equation 65, and the compressive yield stress-osmotic pressure, equation 74, for Table I.

Parameters of the Flocculated Systems Shown in Figure 45

Added Fatty Acid Propionic Pentanoic Heptanoic Oleic

-Eo (kT)

Po (kPa)

ΦΐΠίΙΧ

54 35 29 11

210 6.0 5.2 0.40

0.54 0.595 0.615 0.63


Liu

3.

& MASLIYAH


167

flocculated systems. A l t h o u g h they are different i n nature, they all show the same scaling behavior for the solid concentration.

List of Symbols ΜΦ) a a> a a v


y

b

C Ca

De d d« d d e

h

0

d d* s

Ε Ε £(r) Eo e G G* G G" Κ Κ Κ K Κι k Β

c

k

E

kn I L M m

weak function of φ, dimensionless empirical constant, dimensionless intrinsic viscosity, dimensionless constant in viscosity scaling equation, dimensionless constant in y i e l d stress scaling equation, dimensional particle shape factors, dimensionless empirical constant, dimensional constant, dimensionless constant, dimensionless Deborah number, dimensionless diameter of particle or dispersed droplet, dimensional effective soft sphere diameter, dimensional diameter of the large particle, dimensional diameter of the orifice, dimensional diameter of the small particle, dimensional tube diameter, dimensional electrostatic potential, dimensional rate-of-strain tensor, dimensional pairwise interaction potential, dimensional surface charge (zeta) potential, dimensional particle interaction parameter, dimensionless shear modulus, dimensional dynamic modulus, dimensional storage modulus, dimensional loss modulus, dimensional kelvin consistency coefficient, dimensional shear effect constant, dimensionless Casson constant, dimensional consistency, temporary part, dimensional Boltzmann's constant, k = 1.3806 X 1 0 ~ J / K Einstein constant or intrinsic viscosity, dimensionless longitudinal Einstein constant, dimensionless transverse Einstein constant, dimensionless Huggins constant, dimensionless longest dimension of the particle, dimensional tube length, dimensional torque, dimensional empirical constant, dimensionless 23


168


Na Pe p P P ρ ρ R Ri Ro r r Τ t td t tpE £R u V ν

the ratio of the viscous to the repulsive forces, dimensionless Peclet number, dimensionless extrusion pressure, dimensional compressive y i e l d stress, dimensional scaling factor for compressive y i e l d stress, dimensional pressure, dimensional particle orientational vector radius, dimensional inner radius, dimensional outer radius, dimensional radial coordinate, dimensional ratio of the long axis and the short axis, dimensionless absolute temperature, dimensional time, dimensional time shift between the stress and strain, dimensional experimental time scale, dimensional P o w e l l - E y r i n g characteristic time, dimensional relaxation time, dimensional axial velocity, dimensional volume of one particle, dimensional velocity, dimensional

e

y


0

e

E

p

Greek α, a β ft ρ

v

parameters, dimensional = (Vi/Vi + Vf), the ratio of the ith (a given dispersed) phase volume, V to the summation of the volume of the ith dis persed phase, V and the continuous phase, Vf, dimensionless polymer chain length, dimensional phase difference between stress and strain strain, dimensionless amplitude of the oscillating strain, dimensionless shear rate, dimensional critical shear rate, dimensional a given shear rate, F i g u r e 2, or the first transitional shear rate, F i g u r e 4, dimensional second and third transitional shear rates, Figure 4, dimensional b

h

Δ δ y 70 7 7c 71 7 , 73 2

€ € e e λ

L

r

0

extensional strain rate, dimensional volume fraction of the large particles i n the total volume of the dispersed phases, dimensionless dielectric constant of the suspending fluid, dimensionless permittivity of vacuum, e = 8.85 Χ 1 0 " C / V m memory parameter, dimensionless 0

1 2


Liu

3.

λ μ μ* μ' μ" μ

Rheology of Suspensions - Advances in Chemistry (ACS Publications)

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