Theoretical approaches to the rheology of concentrated dispersions

Powder Technology, 51 (1987) 15 - 25

15

Theoretical Approaches to the Rheology of Concentrated Dispersions W. B. RUSSEL

Department of Chemical Engineering, Princeton University, Princeton, NJ (U.S.A.) (Received April 21, 1986; accepted October 1, 1986)

SUMMARY

This review highlights recent work on stable dispersions, specifically hard spheres, and flocculated systems, emphasizing the effects of structure created by many-body interactions. Predictions of weak flow behavior, e.g., low shear viscosity, linear viscoelastic response, and plastic/yield stresses, conform semi-quantitatively with experimental results and provide insight into the mechanisms responsible for shear thinning at higher rates.

INTRODUCTION

Suspensions of sub-micron particles in a liquid respond to flow in a variety of ways depending on the size, concentration, and shape of the particles and the nature and magnitude of the interparticle potentials. The most dramatic phenomena occur when one of the interparticle forces dominates, e.g., strong van der Waals forces for aqueous latices at high ionic strengths [1] or carbon black particles in mineral oil [2], long-range electrostatic repulsions for colloidal crystals [3, 4], and the interactions between adsorbed polymer layers in sterically stabilized suspensions near closest packing [5, 6]. Even with hard sphere interactions though, the theology is significantly shear-thinning at moderate concentrations [7, 8]. In each case, the non-Newtonian phenomena, whether elastic or pseudoplastic, derive from manybody interactions involving both hydrodynamic and interparticle forces. The goal of our work is a quantitative theoretical understanding of the theology of 0032-5910/87/$3.50

colloidal suspensions of industrial relevance. To this end, we focus on well
A NON-EQUILIBRIUM THEORY FOR STABLE DISPERSIONS

Introduction This approach capitalizes on statistical mechanical descriptions of dense fluids at equilibrium to extend exact non-equilibrium treatments of pair interactions to finite concentrations. The former provides predictions for the rest state described by the equlibrium radial distribution function g(r), which accounts for many-body thermodynamic interactions accurately, if not © Elsevier Sequoia/Printed in The Netherlands

16

completely rigorously. The latter requires the incorporation of many-body hydrodynamics and the reduction of an N-body conservation equation to the pair level. Neither are possible exactly, but reasonable approximations are emerging. For charged spheres at low ionic strengths and dilute concentrations, Ohtuski [12] justifiably ignored the hydrodynamic interactions between the spheres. For small perturbations from equilibrium, the superposition approximation for the N-body distribution function led to an integrodifferential equation for the non¢quilibrium pair distribution function. Numerical solutions then determined the structure and the associated rheological parameters for steady shear and small amplitude oscillations. Russel and Gast [13] subsequently addressed concentrated suspensions of hard spheres through a pairwise additive approximation for the hydrodynamics. Neglecting the integral terms representing explicit three-body couplings still left many-body thermodynamic interactions in the form of the potential of meanforce, Vm~ = - - k T In g(r), acting on a pair. The results for the low shear limiting viscosity and the high frequency modulus and dynamic viscosity identified this long-range thermodynamic interaction as the source of the high viscosities, shear thinning, and elasticity which appear at volume fractions ~ > 0.25. Our current efforts strive to improve the hydrodynamics through the pairwise additive force approximation [11] to treat a varietY of concentrated suspensions quantitatively. Below, we review the nature of the theory and the results for hard spheres.

arising from both the applied flow through U, and translation due to the forces Fi = -- ViV -- k TV~ In PN

(3)

generated by gradients in the interparticle potential V and thermodynamic potential k T In PN. For flows weak relative to Brownian motion, Pe = a27/Do ~ 1 for spheres with radius a and diffusivity Do subjected to shear rate 7. This permits the probability density to be expanded about equlibrium and the conservation equation to be integrated over N -- 2 spheres. The pair distribution function then takes the form P2 = n2g(r) 1

a2

2Do

r.E.r

r2

exp(icot)f(r)]

(4)

with n the number density of spheres, r the center-to-center vector, E the rate of strain tensor, and ¢o the frequency of the oscillating strain field. The governing equation, without three-body integrals, reduces to a2¢0 f + -1- -d- r2G -d-f - - G d V m f d f 2D o r 2 dr dr dr k T dr

6H d Vm~ r2 f = --W + r(1 - - A ) d--r k--~

(5)

with the boundary conditions limr(1--A)+G--df =0 } dr

r--~2

(6)

lim f = 0

Formulation

The theory is developed in configuration space in terms of the probability PN d x i .... dXN of finding N spheres at positions xl, ..., XN within a volume V. The conservation equation takes the form

~PN - -

at

N

+ ~

Vi'], = 0

(1)

/=1

with the fluxes

{ y_,N~,j.Fj }

i, = PN U, +

]=1

(2)

Since A -+ 1 and G -+ 0 as r -+ 2, the hard sphere repulsion is not felt directly. In these equations, the potential of mean force plays the same role as the pair potential in a dilute theory. The hydrodynamic functions G, H, and A here and J and W below, which accounted only for pair interactions, are known from solutions to the Stokes equations for two interacting spheres. Solutions to the Percus-Yevick equation, corrected to agree with simulations, determine g(r). For weak flows, the viscous, interparticle, and Brownian forces contribute separately to the bulk stress as

17

5 5 ~2 ~ n = 2~oE exp(icot) 1 + ~ ~b +

152;

+~ 2 ~i=

J(r) g(r)r 2 dr

]

2

2~b2poE exp(icot)

(7)

×f

rail --A(r)]

(r) f(r) dr

2

t

(a)

~B= 9

b:tto E exp(icat)

4

× ; r 2 W(r)g(r) f(r) dr 2

2

with #o the fluid viscosity. For a steady shear flow, ¢o = 0 so f is real and all three stresses contribute to the low shear limiting viscosity 70. But in the high-frequency limit, the timedependent and forcing terms balance in eqn. (5), making f imaginary. Consequently, the stress assumes the form

(

~ H + ~ I + ~ S = 2 7~' -- i

0

-2

-4

t

(b)

E exp(icot) (8)

with the viscous stress determining the dynamic viscosity 7~' and the interparticle and Brownian forces producing G~'. Results For hard spheres at finite volume fractions, the equilibrium pair distribution function g(r) develops an oscillatory structure due to interactions of the pair with other spheres. Under flow, the resulting volume fractiondependent potential of mean force generates a complementary structure in f(r) at both zero and high frequencies {Fig. 1). When f is in phase with dg/dr, the contribution to the stress from the potential of mean force is positive and increases with volume fraction. Consequently, the low shear viscosity {Fig. 2) and the high frequency modulus (Fig. 3) increase rapidly when three-body interactions become important. The difference between 7o and 7~' reflects the contribution of the potential of mean

Fig. 1. Radial dependence of pair distribution functions for hard spheres in weak flows at ~b= 0.50 [13]. (a), Steady shear; (b), small amplitude oscillations.

force to the low shear viscosity. For Pe >>1, viscous stresses would render these thermodynamic contributions insignificant, suggesting that 7(7 = ¢¢) ~ 7~'. Thus, the thermodynamic stresses E I + ZB provide a mechanism for shear thinning even in hard sphere suspensions. The data of Krieger [7] and de Kruif et al. [8] for aqueous polystyrene latices and silica spheres in cyclohexane, respectively, demonstrate the theory to be qualitatively, but not quantitatively, correct (Fig. 2). The divergence of the low shear viscosity at volume fractions of ~b~ 0.63 and the shear thinning for ~ > 0.25 are at least strongly suggested by the predictions for 7o and 770 - - 7 ~ , respectively. The predicted values err in magnitude significantly though, presumably because of the approximate hydrodynamics.

18 2C

ot o

15

~0

o

*0

13° 5

o

o • •

o

O0

0.1

0.2

•

0.3

-'~J I.~

0.4

0.5

0.6

Fig. 2. Comparison of predictions for hard spheres [13] for the low-shear limiting viscosity ( ) and the high-frequency dynamic viscosity (------) with data: Pe = O

aqueous latices [7 ] silica spheres in cyclohexane [8 ]

Pe =oo

o

•

o

•

40

/

10-I

3o o__%,o., o3G~o ZkT

/

,o.3 I0-

0

i

J

J

i

I

I

2O

]0

0

I 0.4

~

I 0.5

L

I 06

Fig. 3. Predictions of the high-frequency modulus for hard spheres [13 ]. APPLICATION OF PERCOLATION THEORIES TO FLOCCULATED NETWORKS Introduction

In contrast to the fluid-like response of stable systems, flocculated suspensions

respond elastically to small deformations with moduli which depend strongly on the volume fraction of particles. As with the stable systems, the elasticity arises solely from the non-hydrodynamic interactions. For aggregation resulting from van der Waals attraction, deformation of the network changes the relative positions of the particles, thereby increasing the total potential energy of the system and producing an interparticle force. If the particles actually fuse, deformation of the network strains the particles themselves, reflecting their internal elasticity. Given the appropriate force law and the exact positions of the particles, one could in principle calculate the elastic deformation of the network. In practice, however, the large number of particles in the network and the absence of detailed information on the structure makes this unappealing. Hence, as with the stable systems, we seek an approach which captures the statistical characteristics of the structure and averages the stresses to determine the effective modulus. The primary difficulty lies in the nonequilibrium nature of the structure which implies a very slow evolution from the initial state toward equlibrium. Recent work on Brownian flocculation of very dilute, unstable dispersions [14, 15] detects flocs containing N particles in a radius R such that N = N o ( R / a ) D with D = 1.75 to 2.0, i.e., less than the dimension of physical space. These correspond to a scale-invariant, or fractal, microstructure with the length scales and mean volume fraction varying radially within the floc as ~fl = ¢~o(r/2a) D - 3 . Other measurements on flocs broken down by a strong shear flow [16], again at infinite dilution, indicate a similar structure but with D = 2.5. In either case, growth at a finite volume fraction ~, in the absence of gravity, will eventually lead to an infinite network with fractal domains of size R / 2 a = (3dPo/Dd~) 1/(3-D). Simulations of colloidal aggregation have successfully explained the fractal dimensions noted above as the consequence of clustercluster aggregation with no subsequent rearrangement (e.g., [17] ). But the extensive computations required appear unattractive for studies of the elasticity of flocculated networks. Instead, we have modelled the randomness of the network through percolation theory and calculated the resulting

19 1.0

moduli through an effective medium theory [18].

The percolation approach generates the structure by placing sites randomly on a lattice with bonds connecting adjacent filled sites. In the absence of the strong correlations caused by the attractions between colloidal particles, an infinite network forms only above a critical fraction Pc of occupied sites. Below this threshold, only finite clusters exist, while just above Pc the probability of a site being part of the infinite cluster increases as ( p - - p c ) m with m = 0.3 to 0.4 and Pc = 0.312 for a simple cubic lattice [18]. The infinite cluster alone has fractal dimension D = 2.49 [ 19]. Presumably, the introduction of correlations between sites would alter both, but no results are available. Transport properties of percolation networks can be calculated exactly by explicit numerical solution of balance laws for each configuration generated in a simulation or approximated analytically by examining a single site embedded in an effective medium. Numerical results for the conductivity K establish t h a t near threshold

K=Ko l 0 a(p --Pc)"

Pc>P P >Pc

(9)

/// ///

0.6 K pP

// 0.4 /

0.2

I I

/

/

I

I~l °°

I

I

[

!

I

""0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 P-FRACTION OF SITES PRESENT

Fig. 4. Results for site percolation on a simple cubic lattice [18]: - - - - - - p P , the fraction of sites in the infinite c l u s t e r ; - K, the conductivity.

formulation of a model for the elastic network thus requires either a lattice with more nearest neighbors or an angular
The effective medium approximation

with Ko the conductivity o f a single bond, since conduction requires a continuous infinite path from site to site. For a simple cubic lattice, n = 1.5 + 0.2 and a = 1.37 [18]. At higher site fractions, the effective medium approximation becomes accurate, determining the conductivity as [ 18] g = g o [ P -- 1.52p(1 - - p ) + O(1 _ p ) 2 ]

0.8

(10)

Figure 4 summarizes these results. The renormalized effective medium approximation of Sahimi et al. [20] provides a similar picture. In general though, the elasticity of a network differs qualitatively from the conductivity. While conductors carry a scalar, current, the bonds in the elastic network transmit force, a vector. For example, for the simple cubic lattice, the infinite network at P =Pc conducts but does n o t support an elastic stress unless the bonds between particles transmit torques. With central forces alone, the bonds rotate freely, allowing the lattice to collapse under simple shear. The

The effective medium approximation focuses on a single site and replaces the remainder of the partially filled network with a homogeneous network in which the bonds have uniform but u n k n o w n force constants. An applied bulk strain e is then transmitted through the effective medium to bonds surrounding the site, which is occupied with probability p and vacant with probability (1 - - p ) . The force constants for the effective medium or, equivalently, the elastic moduli then follow from the self-consistency condition that the average strain in the bonds surrounding the site equals the applied strain,

i.e., e = pe °ccupied + (1 -- p)e e m p t y

(11)

Solution of the boundary value problem for the homogeneous network containing a single inhomogeneous site requires the Green's functions for the deformation of the homogeneous network due to a point force. Proper superposition of these point forces then produces strain fields which satisfy the

20 balance laws t h r o u g h o u t the inhomogeneous network. The calculations are described in detail elsewhere [20 - 23].

Triangular networks with central and bond bending forces To examine the effects of lattice geometry and interaction potential, we [23] applied the effective medium approximation to twodimensional triangular lattices with the force and torque on the ith site proportional to the relative displacements and rotations of the surrounding sites. Two constants, Go and

1.0

0.8

kGo, characterize the magnitudes of the central and bond bending forces, respectively. Figure 5 displays the results for the Poisson ratio v and the bulk modulus K'= 2G'(1 + v)/[3(1 --2v)] as functions of the site fraction p and the ratio k of bond bending to central force constants. For reference, the exact threshold for conductivity is 0.50. Several points are worth noting: (i) The elasticity threshold decreases to that for conductivity when k > 0. (ii) The Poisson ratio decreases to zero at threshold for k > 0. (iii) For k = 1, v = 0, so the bulk modulus and conductivity become equal with K'= K = 2G'/3. (iv) The site fraction at threshold is quite high, i.e., the lattice is over half full. The last fact is responsible for the accuracy of the effective medium for these problems [21].

Application to flocculated dispersions

0.6

K' 0.4

0.2 k.10

0.0 0.5

0.6

0.7

(a)

0.8

0.9

1.0

P 0.35

Data available for the shear modulus and conductivity show several features relevant to these predictions. First, measurements of Mewis and co-workers [2] for carbon blacks in mineral oil indicate G'/K= constant, independent of volume fraction (Fig. 6). Second, those data and two sets obtained with aqueous polystyrene lattices (Fig. 7) exhibit detectable values for G' at ¢ ~ 0.01 to 0.03. Third, the shear moduli depend very strongly on volume fraction. The juxtaposition of these data with the predictions of Fig. 5 leads us to two conclu-

0~

0.3

104

02 04 0.2

103

v

GDC (mho/cm.)

10

2

0.1

1 01

0.0

i 0.5

0.6

!

!

0.7

0.8

I 0.9

1.0

(b) p Fig. 5. Predictions from effective medium theory for site percolation on a two-dimensional triangular lattice as a function of the strength of bond-bending force [23 ]. (a), Bulk modulus; (b), Poisson's ratio.

I

100 10

10

i 1

10

I 2

10

3

10

G' (kPa)

Fig. 6. Conductivity and shear modulus for carbon blacks dispersed in mineral oil [2 ].

21

attached to the infinite cluster. Hence, we strip the lattice of these by defining the volume fraction as

10 3 I I I I I

~m

f

I I I I I I I

G

CPo] I0 z

with Cm = 0.52 for the simple cubic lattice and P(p) the probability t h a t a site resides in the infinite cluster. Kirkpatrick's [18] results for the simple cubic lattice indicate that

I I I I I I

{~0.5(5.32p -- 1.66) m

~ 0.11

,

[

I

I II

0.4 0.6

(12

1.0

(a) i0 7

0.5 < p

V' = Vo(1.45p -- 0.45)"

°i

]

\era /

iO s

8

•

G' = Go

~ (1.45 ~-~ - - 0 . 4 5 )

I0'

(b)

(14)

with 1.3 < n < 1.7. Combining the two results yields the modulus in terms o f the volume fraction as

I0e E

' 0,05

,

, 0.05

,

~ ~ ~ I 0.07 0.10

(13)

with 0.3 < m < 0.4 and Cm = 0.52. In addition, the numerical results for conductivity [18] and the renormalized effective medium approximation [20] determine the conductivity and, hence, the shear modulus as

J

05

p < 0.5

Cmlp

/ ic

(12)

= pP(p)

* 0.20

0.~50

¢

Fig. 7. D a t a f o r t h e s h e a r m o d u l u s f o r a q u e o u s p o l y s t y r e n e lattices. (a), a = 0 . 2 6 p r o , co = 3 0 0 H z [ 2 9 ] ; ( b ) , a = 0 . 0 3 / ~ m , ~0 = 0 [ 3 0 ] .

sions: (i) Despite expectations to the contrary based on interaction potentials between spheres, the bond bending forces appear to be comparable to the central forces. (ii) The randomly filled lattice characteristic of the percolation problem does n o t adequately model a flocculated network with ¢ ~ p. The first indicates t h a t the results cited above for conduction on simple cubic lattices, a much easier problem, apply to the shear modulus. The latter suggests t h a t either correlations must be included explicitly in the percolation model or a more appropriate relationship between the site fraction and the volume fraction developed. We take the latter tack [23]. One failure of the analogy between the flocculated network and the partially filled lattice is the presence of individual sites and clusters not

¢ < 0.5~ (15)

n 0.5¢~ < ¢

Thus, at low volume fractions, i.e., d~ < 0.26, the modulus should vary as Cs with 3.25 < s < 5.67 before turning over to a weaker dependence for ~b> 0.26. The threshold disappears since the elastic backbone has a fractal dimension less than the spatial dimension, D = 2.49 ~ 3, and, therefore, fills space at any volume fraction. The power-law dependence at low volume fractions reflects the ability of additional sites to stiffen the system by securing dangling chains to the elastic backbone. Figure 8 illustrates the predicted dependence of the modulus on volume fraction for n = 1.7 and m = 0.36 such that s = 4.67. The functional form resembles t h a t of the data in Fig. 7 for which the equivalent values of s vary from 2.5 to 5.3. Note though t h a t the range in s from the theory arises from uncertainty in the numerical results, while t h a t in the data must reflect different structures. Thus, the theory captures the general features of the phenomena but cannot describe as y e t the details.

22

i0

0.1 !

0.2 0.3 w

i

0.4 0 5 !

0.6

0.7

t

I04

10-2

o_' % 10"3

ao~

io "5

I

0.1

i

0.2

.......

0.4 0.6

I

1.0

Fig. 8. P r e d i c t e d values o f shear m o d u l u s f r o m [ 15 ] w i t h n = 1.7 and m = 0.36.

A SELF-CONSISTENT FIELD MODEL FOR FLOCCULATED DISPERSIONS

Introduction Application of a steady shear flow to a flocculated suspension necessarily ruptures the network into discrete but strongly interacting flocs whose size decreases with increasing shear rate. The internal structure of each floc should reflect the structure of the original network, with some rearrangement due to the viscous stresses. If the volume fraction of particles varies spatially within the floc, then decreasing the floc size changes the effective volume fraction of the dispersion, thereby altering the stresses and making the theology non-Newtonian. The simplified treatment developed here is based on our recent model for the breakup of isolated flocs subjected to a shear flow [28]. This mean-field theory views the flocs as porous spheres of radius R with radially varying internal volume fraction Cn. In accord with recent results for both Brownian flocculation [14] and the breakup of flocs in shear [16], we assume ~n = ¢o(r/2a) D-3, corresponding to a fractal structure with dimension D varying from 1.7 to 3.0, depending on the history of the floc. The internal flow is governed by Brinkman's equations [24] with local permeability

k(¢n) and local effective viscosity P(¢n) which depend on volume fraction [25] and hence on position. Internal flow transmits forces to the solid skeleton of the floc. The resulting elastic deformation follows from an equilibrium equation which depends on the shear modulus G' and Poisson's ratio v of the floc and includes the viscous drag as a b o d y force. The modulus varies with local volume fraction as discussed in the previous section while the Poisson ratio is held constant. We assume the floc to rupture when the strain energy due to this deformation exceeds a critical value locally, borrowing the yon Mises criterion for yielding of brittle solids [ 26 ]. The model then assumes a monodisperse suspension of flocs of the maximum coherent size. Interactions are approximated by embedding the floc in an effective medium (Fig. 9). The viscosity jumps from the internal value g to that of the pure fluid Po at radius R and then to p* for the effective medium at radius 5R with the Stokes equations governing the fluid motion in both regions. For hard spheres at dilute concentrations, 5 = 2 to reflect the excluded volume which prevents closer approach. At higher concentrations, however, three-body and higher-order interactions increase the number density of spheres near contact, suggesting 5 < 2. With flocs, some interpenetration is inevitable, leaving 5 an u n k n o w n b u t important parameter. Once the full velocity and stress fields are known, setting the average local stress in the suspension equal to that of the effective medium determines p*. The approach has flaws, since mean-field theories are n o t strictly valid for fractals [27]. Also, the present brief treatment avoids solving the field equations by assem-

Fig. 9. U n i t cell f o r s e l f - c o n s i s t e n t field m o d e l .

23 bling existing solutions from the self~onsistent field model for hard spheres [24] and the break-up of porous flocs at infinite dilution [28]. Nevertheless, the t h e o r y predicts features of the theology c o m m o n l y observed with flocculated systems, suggesting the approach has some merit. Mathematical formulation The fluid flow is incompressible throughout but governed by different m o m e n t u m equations in the three regions: for 6 R < r < ~

the form V = VodPnn

(19)

Similarly, S = So~bn"

(20)

implies that yield occurs at a constant strain so t h a t So = Gocmax. The formulation is completed by equating the average stress in the suspension to t h a t in the effective medium, i.e., n

<0>= 2PoE + n i l ; (a -- 2/~oe dr

0 = - - V p + P*V2u

0

= 2/a*E

for R < r < 5R (16)

0 = - - 7 P + laoV2U for0
=

--VP

+ #72u

- - V__ u

k

At the interfaces, the velocities and normal tractions must be continuous while u-+E.r

as

V'o+ V - u = 0 k (17) 2Gv

1 --

2v

tr(e)5

# k

9PoCn 2a 2

= C ~

(22)

A p p r o x i m a t e results for non-draining flocs Results available in the literature provide an indication of the theology expected from this model. Brinkman [24] first constructed the self-consistent field theory for hard spheres with 5 = 1, obtaining in our n o t a t i o n for a simple shear flow o -

go7

(18)

with C < 1 representing a shielding coefficient. The experimental data and the theory cited above suggest a power-law dependence of the shear modulus on volume fraction of

(23)

2.5~beff

1 --

and e the strain. In the absence of solid-solid contact between flocs, a-n = 0 at r = R. The yon Mises criterion assumes the material to yield when o d :% > 2S 2 locally. The constitutive relations for the coefficients in these equations can be assembled from severai sources. For flocs with low internal volume fractions, the permeability is affected only by interactions with nearest neighbors so t h a t [28] p = Po and --

R ¢ = 41rnfl f r2~bfl dr 0

The deformation of the solid matrix of the floc must satisfy

o = 2Ge + ~

Here, o is the viscous stress in the fluid, e the local rate of strain, and nn the number density of flocs. Note t h a t the overall volume fraction of particles in the dispersion is related to nn through

r-*oo.

with

(21)

with 47rR 3

~eff

--

3

nfl

D(Rt3-D =

~)

7o

(24)

In the non-draining limit, our theory for the break-up of isolated flocs [28] related the maximum floc size to the stress through o

14 - -

(25)

Since the theory fails when R ~ a, we add a minimum floc size such that o

-

.[(+:°, ] 1-4

-- 1

(26)

24 Combining these results relates the stress to the shear rate through = 0.375 o

5D2D¢

So

48¢0

0 (1 + m 14~)

So

1/" - P°~/ So

45

(27)

S°lo-

This constitutive equation for o vs. 3, has several interesting features. For example, in the weak flow limit, i.e., # o ~ / S o "* O,

£

[48

So

(28)

indicating creep at a constant stress while the flocs break down. Note that the magnitude of this stress is predicted to correlate with the shear modulus. Alternatively, for So/o -* 0,

1(

o - 1 --B¢/¢o

I~o~ + - - S o 2 1 6 n

;o)

(29)

with B = 5 D 2 D / 4 8 , corresponding to Bingharn plastic behavior. In this limit, the plastic viscosity rh, 1 = po/(1 -- B ¢ / ¢ o ) arises solely from hydrodynamic effects, i.e., is independent of So, characterizing the strength of the interparticle contacts, while the residual or Bingham stress, corresponding to the constant term, varies linearly with S O. The model contains a number of parameters. Some, such as the fluid viscosity and the particle radius, should be known. Others, such as So and n, might be deduced from independent measurements of the shear modulus. The parameter D, motivated by the fractal character of flocs studied at dilute concentrations, as well as ¢o might be determined by static light scattering if the system is transparent. To illustrate the behavior predicted at intermediate shear rates, we set ¢o = 2 D - 3 D / 3 , so that Ce~ = ¢ for R = a. Then, So

2¢

1+ m 45

= -So -

(30)

so that as PoT~So ~ 0 14(5)/1

o

So -

2¢

(5)" -

~ / /

///

O"

I0"~ i 10-7

0.25

, i 10-5

~ "

i i i I 10-3 iO-I /~.oY So

Fig. 10. S t r e s s - r a t e o f strain r e l a t i o n s p r e d i c t e d b y self-consistent field m o d e l w i t h q~o = 2D-3D/3 and n = 4.5.

CONCLUSION For stable suspensions, the non-equilibrium statistical mechanics formalism promises a reasonable description of weak flows for which the structure departs only slightly from equilibrium. Our next step, improving the approximation of the many-body hydrodynamic interactions and evaluating the integral terms neglected here, will be tested against the hard sphere data before moving on to interparticle potentials appropriate for electrostatic, steric, and weak van der Waals forces. For strong flows, there as yet appears no alternative to simulations. Flocculated suspensions still pose considerable difficulties. The percolation theory captures some features of the elasticity of the rest state, but does not reflect appropriately the variation in network structure, i.e., fractal dimension, arising from different histories. Introduction of relevant correlations or the actual physical processes producing the structure may require computer simulations. The self~onsistent field treatment of flowing systems appears to explain the constant-stress region observed with m a n y flocculated systems at low shear rates and the subsequent shear thinning. However, the theory is as yet incomplete and untested.

(31)

and as lZo3,/S o --> oo 0

So

_

1

7¢)

(#o7 + 1 - - 2 . 5 ¢ [So ~nn

(32)

Figure 10 illustrates the smooth transition between these limits for several volume fractions.

ACKNOWLEDGEMENTS

This work was supported by the International Fine Particles Research Institute and the National Science Foundation through Grant CPE-8212327.

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