The effective shear viscosity of a uniform suspension of spheres

Physica 88A (1977) 88-121 © North-Holland Publishing Co.

THE EFFECTIVE SHEAR VISCOSITY OF A U N I F O R M SUSPENSION OF SPHERES D. BEDEAUX, R. KAPRAL* and P. MAZUR lnstituut Lorentz, Rijksuniversiteit te Leiden, Nieuwsteeg 18, Leiden, The Netherlands

Received 16 November 1976

A general theory is presented to calculate the wave vector and frequency dependent effective viscosity of a suspension of spheres. The resulting formal expression for the effective viscosity is then used to evaluate the coefficients of the linear and the quadratic terms in an expansion in the volume fraction of the spheres using stick boundary conditions. On the linear level the wellknown Einstein result is obtained. On the quadratic level we find that it is not justified to neglect higher order spatial derivatives of the secondary velocity fields as done in the analysis by Peterson and Fixman. As a consequence we find a value for the Huggins coefficient which is 12% higher than their value.

I. Introduction T h e p r o b l e m of c a l c u l a t i n g the v i s c o s i t y of s u s p e n s i o n s has a long history. T h e earliest a n d b e s t k n o w n c a l c u l a t i o n for a dilute s u s p e n s i o n of s p h e r e s w a s that of E i n s t e i n ~) w h o f o u n d , u s i n g stick b o u n d a r y c o n d i t i o n s o n the s u r f a c e of the s p h e r e s , rt = rt0(i + ~2~b).

(1.1)

H e r e rt is the effective s h e a r v i s c o s i t y of the s u s p e n s i o n , rt0 the s h e a r v i s c o s i t y of the fluid a n d ~b the v o l u m e f r a c t i o n of the s u s p e n d e d s p h e r e s . T h e e x t e n s i o n of this r e s u l t to dilute s o l u t i o n s of p o l y m e r s has r e c e i v e d a c o n s i d e r a b l e a m o u n t of a t t e n t i o n o v e r the years2). F o r m o r e c o n c e n t r a t e d s u s p e n s i o n s it is n e c e s s a r y to c o n s i d e r c o r r e c t i o n s to the v i s c o s i t y w h i c h are of higher o r d e r in the v o l u m e f r a c t i o n . T h e coefficient of the ~b2 t e r m has also b e e n c a l c u l a t e d for b o t h s p h e r e s 3-5) a n d p o l y m e r s 2-6) b y a v a r i e t y of m e t h o d s with o f t e n r a t h e r different n u m e r i c a l results. F r o m a f u n d a m e n t a l p o i n t of view the m o s t s a t i s f a c t o r y c a l c u l a t i o n is the o n e g i v e n b y P e t e r s o n a n d FixmanS), w h o n o t e d the a n a l o g y with the t h e o r y of light s c a t t e r i n g a n d of the d i e l e c t r i c c o n s t a n t . I n their d i s c u s s i o n t h e y n e g l e c t e d higher o r d e r g r a d i e n t s of the s e c o n d a r y v e l o c i t y fields a s s u m * On leave of absence from the Department of Chemistry, University of Toronto, Toronto, Canada M5S 1A1. 88

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ing that the corresponding contribution to the effective viscosity is small. This is, h o w e v e r , b y no m e a n s evident and, as we shall see in this paper, unjustified. We shall present a general f o r m a l i s m for the calculation of the effective shear viscosity for a suspension of spheres as a function of the w a v e vector and the f r e q u e n c y and for arbitrary values of the volume fraction. In our analysis we shall a s s u m e for simplicity that the mass density inside the rigid spheres is equal to the mass density of the incompressible fluid. In principle one m a y also consider the m o r e general case of unequal mass densities. This leads, however, to some additional complications due, for example, to inertial effects of the spheres relative to the fluid in time d e p e n d e n t velocity fields. Within the context of the equal mass density case we take all inertial effects into a c c o u n t contrary to earlier discussions. The starting point of our analysis is the c o n s e r v a t i o n law for the total m o m e n t u m density for the fluid and the suspended spheres. In order to calculate properties which do not depend on the precise positions of the spheres one m a y average o v e r regions large enough to contain m a n y spheres but small c o m p a r e d to the size of the system. More conveniently one m a y average o v e r an ensemble of sphere distributions. Such an averaging procedure leads to the usual conservation laws but now for the average m o m e n t u m density field. Our discussion closely parallels a similar discussion by two of us for the dielectric constantT). Although this general f o r m a l i s m provides a c o m p a c t expression for the effective viscosity of the suspension as a function of the w a v e v e c t o r and the f r e q u e n c y for arbitrary values of the volume fraction, one must of course make a p p r o x i m a t i o n s in order to obtain numerical values. The first approximation which is utilised is a multipole expansion of the one sphere r e s p o n s e function to second order in the w a v e - v e c t o r . Although not explicitly stated, earlier discussions, in particular the one by P e t e r s o n and FixmanS), e m p l o y a similar approximation. In fact P e t e r s o n and F i x m a n neglect higher order derivatives of the s e c o n d a r y velocity fields ti.e. the perturbations of the velocity field generated by the spheres). As we shall show, this approximation is equivalent to a multipole expansion to second order in which two out of three of the second order contributions to the one sphere response function are neglected. The second a p p r o x i m a t i o n which is utilised is an expansion of the effective viscosity in the volume fraction up to the second order which is sufficient for the calculation of the Huggins coefficient. As is to be expected, the coefficient of the linear term in this expansion, for zero w a v e v e c t o r and f r e q u e n c y , depends only on the lowest order term in the multipole expansion whereas the Huggins coefficient depends also on the higher order terms in this expansion and is therefore affected by the level of truncation of this series. In section 2, we derive the formal expression for the effective viscosity of the suspension starting f r o m the c o n s e r v a t i o n of total m o m e n t u m . The pressure tensor field is written as a sum of a b a c k g r o u n d term and an induced part which is only unequal to zero inside and on the surface of the spheres. A

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fluctuating contribution to the viscosity is then introduced as a linear o p e r a t o r which relates the induced pressure tensor field to the gradient of the m o m e n tum field. U p o n averaging, a formal expression is then obtained for the effective viscosity in terms of this fluctuating contribution. In order to express this fluctuating contribution to the viscosity in terms of the r e s p o n s e function of one sphere in an arbitrary external m o m e n t u m field a local field analysis is given in section 3. Subsequently in section 4 we calculate the one sphere response function using a multipole expansion for the case of stick b o u n d a r y conditions and an incompressible fluid. Using the results of sections 2 - 4 we then derive in section 5 an alternative expression for the effective viscosity in terms of n u m b e r density correlation function for the spheres. For this purpose we introduce the analog for the present case of the Clausius-Mossotti function for dielectrics. In section 6 the effective viscosity is calculated in the absence of correlations to all orders in the density. The result has a form similar to the Clausius-Mossotti relation in dielectrics. The contribution f r o m correlations to the Huggins coefficient are calculated in section 7. The resulting value is 12% higher than the value obtained by P e t e r s o n and Fixman. This shows that it is not justified to neglect the higher order derivatives of the secondary velocity fields. We also c o m p a r e our result with the result obtained using the lowest order multipole expansion. This c o m p a r i s o n gives some indication as to the c o n v e r g e n c e of the value of the Huggins coefficient if higher order multipoles are taken into account.

2. Formal theory for the viscosity of a uniform suspension of spheres We consider a suspension of rigid spheres with radius a in an incompressible fluid. The mass densities of the spheres and the fluid are a s s u m e d to be equal and are denoted by p. The linearised equation of conservation of m o m e n t u m is given by:

o ~ g ( r , t) = p

v(r, t) = - d i v P(r, t),

(2.1)

where g, v and P are the m o m e n t u m density, velocity and pressure tensor fields respectively. The velocity field is given by:

v(r,t)={ui(t)+g~i(t)×[r-Ri(t)] vt(r, t)

if [ r - R i ( t ) outside the spheres,

I
(2.2)

where ui, Di and Ri are the velocity, angular velocity and the position of the ith particle respectively, i = 1. . . . . N. F u r t h e r m o r e vt is the velocity field of the fluid between the spheres. We note that in view of the uniformity of the

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mass density div v = 0,

(2.3)

which implies that the normal c o m p o n e n t of v is continuous at the surface of the spheres. We will use stick b o u n d a r y conditions, i.e. v continuous at the surface of the spheres, h o w e v e r , the formal theory of sections 2, 3 and 5 is also valid for other b o u n d a r y conditions. The pressure tensor field is given by

P(r, t) = ~Ps(r, t) if Ir -R~(t)l < a t pt(r, t ) - 2 r / 0 ~ outside the spheres

(2.4)

where Ps is the stress tensor field inside the spheres which need not be further specified. The hydrostatic pressure field of the fluid is p~ (we will not explicitly indicate the unit tensor), 710 is its shear viscosity. The n o t a t i o n ' ' indicates the s y m m e t r i c traceless part of a tensor. Since neither the spheres nor the fluid h a v e an intrinsic angular m o m e n t u m density (local spin density) it follows f r o m the law of c o n s e r v a t i o n of angular m o m e n t u m that the pressure tensor field is e v e r y w h e r e symmetric8):

P ~ ( r , t) = P ~ ( r , t),

(2.5)

where a and /3 indicate cartesian c o m p o n e n t s . Subtracting ~q0V2v = 2 r t 0 d i v ( ' - ~ f r o m both sides of eq. (2.1) one obtains:

(p-~t - rloV2) v = - Vp - V" , =- - Vp + Fi,d,

(2.6)

where we have used the following definitions:

p(r, t)

~0

L pt(r, t)

if I r - R~(t)l < a outside the spheres

~r(r, t) =- P(r, t) - p(r, t) + 2 r / 0 ~ .

(2.7) (2.8)

We shall call (r the induced pressure tensor field. In view of eqs. (2.2), (2.4) and (2.7) the following essential p r o p e r t y follows o'(r, t) = 0, Find(r, t) = 0

outside the spheres.

(2.9)

We note that ~r and Find m a y in principle contain singular contributions on the surface of the spheres. Eq. (2.9) has a very useful collorary

f drFi,o(r,t)= f d r F i , d ( r , t ) = - f d S n . o ( r , t ) = O , [r-R,(t)l
(2.10)

[r-Ri(t)l=a+~

H e r e e is an infinitesimally small positive n u m b e r and n the outward normal on the surface of the sphere. It also follows f r o m eqs. (2.5) and (2.8) that (r is

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a symmetric tensor t r ~ ( r , t) = ~r~(r, t).

(2.11)

As usual an e q u a t i o n f o r the p r e s s u r e field p m a y be f o u n d b y taking the d i v e r g e n c e o f eq. (2.5), w h i c h yields V2p

= -- VV

(2.12)

: O'.

All the effects due to b o u n d a r y conditions are c o n t a i n e d in tr. It is t h e r e f o r e not as y e t n e c e s s a r y to s p e c i f y the f o r m o f these conditions until section 4 w h e r e the o n e - s p h e r e r e s p o n s e f u n c t i o n is calculated. T h e f o r m a l solutions of eqs. (2.6) and (2.12) are given by: v

=

vo- ~[V(p -P0)+

V • tr],

(2.13)

P = P o + ~doVV : or,

(2.14)

w h e r e the " e x t e r n a l " fields vo and po are the solutions in the a b s e n c e of the spheres, ~r = 0. If there are external f o r c e s acting on the fluid, w h i c h w e r e not taken into a c c o u n t in eq. (2.1), vo and po also contain the fields due to these f o r c e s . T h e p r o p a g a t o r s ~ and ~3o are defined f o r m a l l y by: ~--= O

-ago V2

and

~qo----(V2) -I.

(2.15)

M o r e explicit f o r m s of ~ and ~d0 are given in a p p e n d i x C. Substituting eq. (2.14) into eq. (2.13) one obtains v = v 0 - ~3(1 + ~d0Vlr)V : ~r = v o - T V

: or,

(2.16)

w h e r e we h a v e used the t e n s o r o p e r a t o r for the t r a n s v e r s e fields: T - = ~ ( 1 + ~oVV).

(2.17)

U s i n g the fact that TV is traceless with r e s p e c t to the last two indices it follows that the trace o f ~r does not c o n t r i b u t e to v. Since f u r t h e r m o r e tr is s y m m e t r i c we m a y write v = vo-TV

: er.

(2.18)

W e shall in particular be interested in the s y m m e t r i c (traceless) gradient of v w h i c h a c c o r d i n g to this last e q u a t i o n is given by: r-~

V~--~= ~-~Vo- G : or.

(2.19)

H e r e G is a t e n s o r o p e r a t o r o f rank f o u r w h i c h is s y m m e t r i c and traceless both with r e s p e c t to the first as well as to the last two indices G ~ 8 -= l(V~T~V~ + V~T~V~ + V~T~V~ + V~T~sV~).

(2.20)

T h e s y m m e t r i s a t i o n o f G with r e s p e c t to the last two indices is a c o n s e q u e n c e o f the s y m m e t r i c nature o f ~r.

S H E A R V I S C O S I T Y OF A S U S P E N S I O N OF S P H E R E S

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In order to find a c o m p l e t e solution of either eq. (2.18) or eq. (2.19) it is n e c e s s a r y to have constitutive relations which relate ~ to the velocity field v. In view of the linearized nature of our problem these solutions will be linear. Using s y m m e t r y relations which are derived in appendix A and p r o p e r t y (2.10) it is shown in this appendix that ~ can be written solely in t e r m s of the s y m m e t r i c gradient of v ~ r r r r ~ = --2 f ~lb(r, tlr', t') : 'V'v(r', t')' dr' dt'

(2.21a)

or in a more formal o p e r a t o r notation (2.21b)

O" = --21qb : (TV.

The o p e r a t o r ~lb can be interpreted as a " s u b m a c r o s c o p i c " contribution to the viscosity and will be used to find general expressions for the " m a c r o s c o p i c " viscosity of the suspension after averaging o v e r the distribution of the spheres. The explicit f o r m of ~lb depends on the nature of the suspended particles and will be constructed in sections 3 and 4. Substituting eq. (2.21) into eq. (2.19) and solving the resulting equation formally one obtains: 'V--v= (1 - 2G : rib) -1 : r~V0.

(2.22)

Inserting this result into eq. (2.21) one finds ~=

--2Xlb : (1

--

2G : ~lb)-I : ~v0.

(2.23)

The m a c r o s c o p i c fields m a y now be obtained by averaging eqs. (2.22) and (2.23): ~r~v) = ((1 - 2G : rib)-') : '~V0,

(2.24)

,{--~)= --2(~b :(1 -- 2G : ~qb)-') : '-~v0.

(2.25)

H e r e (. . . . ) m a y either be interpreted as an average o v e r regions large c o m p a r e d to the spheres but small c o m p a r e d to the variations in lTv0 or, more conveniently, as an ensemble average o v e r the distribution of the spheres. We shall assume that the distribution functions are translationally invariant. Eqs. (2.24) and (2.25) lead to the following relation b e t w e e n the average induced pressure tensor field and average s y m m e t r i c velocity gradient ( ~ ) = - 2 a n : (~-~v),

(2.26)

where AI? = (l~b ." (1 -- 2G : 1~b)-1) I ((1 -- 2G : 1?b)-l) -l.

(2.27)

For a stationary translationally invariant distribution of spheres one m a y conclude f r o m this equation that Aaq is a convolution operator in (r, t) representation and diagonal in w a v e v e c t o r f r e q u e n c y (k, to) representation.

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Defining the F o u r i e r t r a n s f o r m of a field f ( r , t) by f ( k , to) =- f d r dt e-i(~"-~t)f(r, t),

(2.28)

one m a y t h e r e f o r e write A1q(]¢, ¢olk' , 0o') = A1q(k, 6o)(27r)48(k - k')(~(oJ - ~').

(2.29)

U s i n g f u r t h e r m o r e the isotropic nature of the s y s t e m it then follows in view of the o t h e r s y m m e t r y properties of A~q that A n ( k , o~) = Art~(k, oJ)A + Anz(k, ¢o)A' + an3(k, oJ)A",

(2.30)

w h e r e A, A ' and A" are the following s y m m e t r i c traceless tensors of rank f o u r =--

+ 28,,~8~v - 38,~8v~,

(2.31)

d'~

~- ~(8~ - 3/~J~¢)(&~ - 3/~J~a),

(2.32)

a~

=- -2/~ffl~f~ + ~(/~/~,a~ +/~fsat~ +/~f~&8 +/~a/~aa~),

(2.33)

with /~-= k / k the unit vector in the direction of k. Substituting eqs. (2.30)(2.33) into eq. (2.26) and using the transverse nature of (v) one obtains ( ~ )

= - 2iart (k, w ) ~ ) ,

(2.34)

where Art(k, w) = Artffk, co) + Art3(k, w).

(2.35)

W e note f u r t h e r m o r e that for isotropic s y s t e m s A # ( k = 0, co) m a y not d e p e n d on k. In view of this Artz(k = 0, w) = Art3(k = 0, w) = 0 and zaaq(k = O, w) = Art(k = O, o~)A = A n l ( k = O, w)A.

(2.36)

We shall n o w s h o w that one m a y interpret rt(k, oJ)= rt0+zart(k, w) as the f r e q u e n c y and w a v e v e c t o r d e p e n d e n t effective shear viscosity of the suspension. This b e c o m e s a p p a r e n t if one takes the t r a n s v e r s e part of eq. (2.1) in (k, w) r e p r e s e n t a t i o n - kopv(k, ~o) = - i k • ~

-

(l -/~/~)

= - rtok2v(k, oJ) - i k - ~ ( ~ , w) • (1 - ££),

(2.37)

w h e r e we h a v e also used eq. (2.8). A v e r a g i n g this e q u a t i o n and using eq. (2.34) one then finds - k o p < v ( k , w)) = - [n0 + art(k, oJ)]k2(v(k, w))

(2.38)

w h i c h s h o w s that the total effective shear viscosity is given by rt(k, oo) = no+ Art(k, ~).

(2.39)

Eqs. (2.27) and (2.39) t h e r e f o r e give the effective shear viscosity as a v e r a g e s o v e r the r e s p o n s e f u n c t i o n ~qb w h i c h will be related to the one sphere r e s p o n s e f u n c t i o n in the next section.

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES 3. Local

95

field analysis*

In this section we shall derive an expression for ~b in t e r m s of the external one-sphere r e s p o n s e function and the distribution of the spheres. First consider the case that there is only one sphere in the fluid. In that case one m a y e x p r e s s r ~ i n terms of the s y m m e t r i c gradient of the velocity field analogous to eq. (2.21). Four our present purpose, h o w e v e r , it is more convenient to express "-if'in terms of the s y m m e t r i c gradient of the external velocity field v0.

= - 2 f x ( r - R(t), tlr' - R(t'), t') :'V'vo(r', t')' dr' dt',

(3.1)

where X is the external one-sphere r e s p o n s e function which has the same s y m m e t r y properties as ~b, cf. appendix A, and is in addition spherically symmetric. This r e s p o n s e function will be evaluated in the next section. If there are N spheres in the fluid N

tr(r, t) = ~ ~ri(r, t),

(3.2)

where ari is the induced pressure tensor field of the ith sphere. Eq. (3.1) is still valid for crl if one replaces v0 by the effective velocity field which contains in addition to v0 the velocity fields due to all the other spheres. One then has = -2 | x(r - Ri(t),tlr'-Ri(t'),t'):

J

'V' vl,ef~ --r' , t'~dr'dt',

(3.3)

where the s y m m e t r i c gradient of the effective velocity field is given by, cf. eq. (2.19),

'Vvi,oe(r,t)'=~-

~ f

G(r-r',t-t'):~dr'

dt',

(3.4)

where we have used the fact that G is a convolution operator in ( r , t ) representation. We now introduce a " c u t off" function {~

O(r) =

if r < a + • if r > a + •

(3.5)

where • is an infinitesimally small but positive number. In the subsequent analysis we shall assume that IRdt)-Rj(t)l>2(a + • ) which not only expresses the fact that spheres do not overlap but also that they do not touch * The analysis closely parallels a similar analysis by Hafkenscheid and Vlieger9) in the context of the dielectric properties of crystals.

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D. BEDEAUX et al.

each other. In that case eq. (3.4) m a y be written in the following f o r m

~

)

=~ )

-

.

G(r - r , t - t ' ) O ( r ' - R,(t')) : trj(r, t') ~tr' dr'

f G ( r - r', t - t ' ) O ( r ' = R i ( t ' ) ) : ~

I

dr' dt', (3.6)

w h e r e also eq. (3.2) has been used. S u m m i n g eq. (3.3) o v e r i and substituting eq. (3.6) one obtains

~ ( r , ~ = - 2 f dr' dt' ~ x ( r - Rift), t [ r ' - Rs(t'), t') : {~vo(r', t')' t

- f G ( r ' - r", t' - t")O(r" - R i ( t " ) ) : ~

dr" dt"}.

(3.7)

N o w we define two new o p e r a t o r s 3'b(r, t l r ' , t') =-- ~_~ x ( r -- R i ( t ) , t l r ' -

(3.8)

R i ( t ' ) , t'),

i

H~(r, tlr", t") ~ f dr' dt' ~ x ( r - Ri(t), t l r ' - Ri(t'), t')

:

I

G ( r ' - r", t ' - t " ) O ( r " - Ri(t")).

(3.9)

U s i n g these definitions eq. (3.7) can n o w be written f o r m a l l y as o" = - 2"yb : ~-~v0+ 2/47 : ~ .

(3.10)

O n e m a y n o w eliminate r~v0 using eq. (2.19) which results in o r = - - 2 y b : ~-~V--2yb : G : tr + 2H~ :or.

(3.11)

Solving f o r tr one obtains

~' = -

2(1 + 2 K ~ , ) ' : y b : ~ ,

(3.12)

where K~ ------"IZb: G - H~.

(3.13)

C o m p a r i n g eq. (3.12) with eq. (2.21) one m a y then identify rib = (i + 2K~)-' : ~/b

(3.14)

as the " s u b m a c r o s c o p i c " contribution to the viscosity due to the s u s p e n d e d spheres used in section 2. N o t e that f o r large separations, i.e. low sphere densities, rib is essentially equal to yb. F o r higher densities the d e n o m i n a t o r in eq. (3.14) a c c o u n t s f o r the h y d r o d y n a m i c interactions b e t w e e n the s p h e r e s to all orders in the d e n s i t y of the spheres.

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97

4. The one sphere response function

In o r d e r to calculate the external o n e - s p h e r e r e s p o n s e f u n c t i o n X we c o n s i d e r o n e s p h e r e s u s p e n d e d in an infinite fluid. This s y s t e m is in fact similar to the one c o n s i d e r e d in ref. 11. T h e e q u a t i o n s f o r the v e l o c i t y field v and the i n d u c e d p r e s s u r e t e n s o r field cr f o r the sphere are e x a c t l y the s a m e as given in section 2 but n o w restricted to the special case o f one sphere. A n e x a c t e x p r e s s i o n f o r X can in principle be o b t a i n e d b y solving the o n e sphere p r o b l e m f o r an arbitrary external field v0 using the b o u n d a r y conditions on the s u r f a c e of the sphere. S u c h a calculation can be d o n e in terms of spherical v e c t o r h a r m o n i c s and the resulting e x p r e s s i o n f o r X will be an infinite s u m o v e r these functions. A n explicit c o n s t r u c t i o n of the v e l o c i t y field using these t e c h n i q u e s w a s in f a c t given in ref. 10, but X is not explicitly discussed. In this p a p e r we shall use a multipole e x p a n s i o n f o r the one sphere response function x ( r - R ( t ) , t l r ' - R ( t ' ) , t') = ~ ( - 1)m+n(m !n !)-lXmn(tlt') (S) [V~8(r - R ( t ) ) ] [ ( V ' ) m S ( r ' - R(t'))]. (4.1) mn

In the linear regime one m a y neglect the difference b e t w e e n R ( t ) and R ( t ' ) w h i c h w o u l d give rise to c o n t r i b u t i o n s p r o p o r t i o n a l to the s e c o n d and higher p o w e r s of the v e l o c i t y field, cf. also the d i s c u s s i o n in ref. 11. W e m a y t h e r e f o r e take the position of the particle fixed, R ( t ) = R ( t ' ) = R(O). T a k i n g the origin of the c o o r d i n a t e f r a m e at R(0) eq. (4.1) r e d u c e s to x ( r , tlr', t') = x(r, t - Fir') = ~

(-l)~+n(m!n!)-lX""(t

- t')@ [V"8(r)][(v')"8(r')],

(4.2)

m.n=O

w h e r e we h a v e also used the f a c t that X d e p e n d s only on the time difference. As a f u n c t i o n o f k and oJ this e q u a t i o n m a y be written as* x(k, ~olk', ~o') = X(k, ~olk')2~ra(o~ - ~o') = ~'~ (m !n !)-~Xmn(~O) G (-ik)"(ik')m27ra(oJ - ~o').

(4.3)

m,n = 0

In a p p e n d i x A we s h o w that X satisfies the following s y m m e t r y relation X~,8(r, tlr') = Xav~(r', tlr)

(4.4a)

X~

(4.4b)

or

k, ool k') = Xa~,~(- k', oJ l - k)

* The Fourier transform of an operator is defined as: x(k, ~o]k',~o')=--( dr dt dr' dt' exp [-i(k • r - ~ot)+ i(k' • r' - ~o't')]x(r, t[r', t'). d

98

D. B E D E A U X

et al.

or rtln

X,,o,8~,,...~,,,~,

"'"

~,(m)=

""

/~z~Y~

I .....

/~'1 • - •

~,~(o9).

(4.4c)

This implies the very useful fact that if one has calculated X m" then X "m immediately follows. Furthermore we note that in view of eq. (2.10)

x(r, tlr')=O

for r > a + E .

(4.5)

The s y m m e t r y therefore also implies that

x ( r , tlr') = O,

for r' > a + ~.

(4.6)

The discussions of the effective viscosity in the literature can all be understood if one takes only the lower order multipoles into account. In fact in order to obtain the original Einstein ~) formula it is sufficient to use only the lowest order, X °°, term. The results found by Peterson and Fixman 5) are reproduced if only X °° and X °2 are taken unequal to zero. However, in view of the s y m m e t r y relations given above, X 2° is also unequal to zero and since it is of the second p o w e r in the wave vector it should be taken into account as well. Furthermore we shall find that also X n, which is again of the same order, is unequal to zero. H e n c e for a calculation of X consistent to second order in the wave vector all of these terms should be considered. As we shall see later this will also affect the contribution to the effective viscosity for zero wave vector due to correlations. In appendix B we show that up to second power in the wave vector and for stick b o u n d a r y conditions the external one-sphere response function is given by x(k,~o]k') : x(k, ~olk')a = ~/0v~{1 5 7 2a 2 + Na - ~oa2(k 2 + (k') 2 - ] k . k')}A,

(4.7)

where vs-= ~Ira 3 is the volume of the sphere and a -= (-itop/~0) ~ with Re a > 0.

(4.8)

In eq. (4.7) we also neglected terms of higher than second order in eta. In (r, t) representation eq. (4.7) becomes

x ( r , t [ r', t') = x ( r , t - t'[r')A 5

{

7pa___~20_ 1 a 2 ( V 2 + ( V , ) 2 + 7

= ~ ~qoVsa 1 + 2070 Ot t- 1---0

x 6 ( r ) 6 ( r ' ) 6 ( t - t').

~ I7.17'

)} (4.7b)

The calculations in this paper shall be based on X as given in eq. (4.7). 5. An alternative formula for the effective viscosity In this section we shall make use of the fact that only a finite number of multipoles are taken into account in the external one-sphere response func-

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

99

tion in order to derive a simpler formula for the effective viscosity. In the analysis in this section we shall only go to the second order in the multipole expansion. The essential results of the analysis are, however, still valid if any finite number of multiples is retained. We shall now first simplify the two operators ~% and H~ using the explicit form of X given in eq. (4.7). Upon substitution into eq. (3.8) one obtains

(V 'yb(r, tit', t') = 5 rlovsA ~ / [ 1 4- -7pa2 1-2-0-7+o-O 3-t a 21O

2+(v') 2+ 47 I7. I7')]

× [8(r - Ri(t))B(r' - R i ( t ' ) ) 8 ( t - t')] = ~ "Oov,A

1 -~ 20rio at

1--0

4

x In(r, t ) 3 ( r - r')8(t - t')],

(5.1)

where we have used the other two 8-functions to replace 3 ( r ' - R ~ ( t ' ) ) by 8 ( r - r ' ) and where the number density of the spheres is given by N n(r, t) = ~, 8(r - Ri(t)). (5.2) i=l

In (k, to) representation ~'b becomes, cf. also eq. (4.7), ,,,bCk, tolk', to') = ~,~o,,,a [1 + ~o,,2a 2 - ~oa 2(k ~ + (k') ~ - ] a . k ' ) ] n ( / -

k', t o - to')

= x ( L t o l k ' ) n ( k - k', to - to') = X(/~, t o l k ' ) n ( k

- a ' , ,o - t o ' ) a .

(5.3)

Similarly one finds for H~ upon substitution of eq. (4.7b) into eq. (3.9) H~(r, tlr", t") = ~'rlovsA 5 : f dr' dr'

•

2--~0~+la2(V2+(V')z

4

x [8(r - Ri(t))~5(r' - R i ( t ' ) ) 3 ( t - t')]} x tI(r'

-

r", t' - t " ) O ( r " - Ri(t")),

(5.4)

where the derivatives act only on the functions inside the curly brackets. In the linear regime we may again replace R~(t") in 0 by Ri(t), the difference giving rise to contributions to ~r which are quadratic in the velocity field. One then has : Hv(r, tlr", t") = ~tlov~/t 5

x

f

dr' dt'

7pa 2 3 -----+-~i {[eq 20t/o3t

1 a2 (V2+ (V')2+7 V • V')] 10

× [3(r - R i ( t ) ) 8 ( r - r')8(t - t')]~ J × G ( r ' - r", t ' - t " ) O ( r " - r').

(5.5)

100

D. BEDEAUX et al.

In this e q u a t i o n we have r e p l a c e d ai(t) in the a r g u m e n t of 0 by r ' using the g - f u n c t i o n s b e t w e e n the square brackets. This r e p l a c e m e n t is not automatically justified in that it leads also to terms containing derivatives of 0. T h e s e terms h a v e been omitted in eq. (5.5) in view of the fact that t h e y are only unequal to zero if I r " - r ' I = a + e and t h e r e f o r e do not contribute u p o n substitution into eq. (3.10). It is at this point in the analysis that essential use is being made of the fact that only a finite n u m b e r of multipoles are taken into account. If we n o w define the " c u t - o f f " o p e r a t o r H by H(r,

tlr',

t') = H(r

- r', t -

t') ~ 6 ( r

- r', t - t')O(r'-

r).

(5.6)

T h e o p e r a t o r H, r e d u c e s , using eq. (5.1), to the simple f o r m

H,(r, t[r", t") = f dr' dt' yb(r, tlr', t') : H ( r ' - r", t ' - t")

(5.7)

or in formal o p e r a t o r notation

H~ =

~ b *" H.

(5.8)

If one substitutes eq. (5.8) in eq. (3.14) t o g e t h e r with eq. (3.13) one obtains for the s u b m a c r o s c o p i c c o n t r i b u t o n to the v i s c o s i t y 'l~b = (1

+

2"/b : K) I : yb,

(5.9)

where K = G-

H = G(I - 0).

(5.10)

N o t e that in view o f its definition K(r, tlr', t ' ) = 0 if I r - r'] > a + E. A formal e x p r e s s i o n f o r the effective viscosity m a y n o w be obtained u p o n substitution of eq. (5.9) into eq. (2.27). It is m o r e c o n v e n i e n t h o w e v e r to i n t r o d u c e a new operator y by zlr/---(l+2]z:K)-~:'y

or

~'=AIt:(1--2K:ArI)

~.

(5.11)

W e note that y plays the same role in this p r o b l e m as the C l a u s i u s - M o s s o t t i f u n c t i o n in dielectrics (see e.g. ref. 7). Substitution eq. (5.9) into eq. (2.27) and using eq. (5.11) one obtains after s o m e straight f o r w a r d algebra ]g = ('Yb : (1 - 2H : Yb) 1) _- ((1 - 2H : 'Yb) 1)-1.

(5.12)

In view of the f a c t that "rb is k n o w n in terms of the external o n e - s p h e r e r e s p o n s e f u n c t i o n and the n u m b e r d e n s i t y o f the spheres, this f o r m u l a relates ~, and t h e r e f o r e a~/ to the n u m b e r d e n s i t y correlation functions. It will be the basis of our f u r t h e r analysis. In the next section we shall first c o n s i d e r the c o n s e q u e n c e s of eq. (5.12) in the a b s e n c e o f correlations. D u e to translational invariance and stationarity both y and zl-q are diagonal in (k, to) r e p r e s e n t a t i o n as is K. Eq. (5.11) then r e d u c e s to

A~(k, to) = (1 + 2 y ( k , to) : K(k, to))--I : y(k, to)

(5.13a)

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

101

or

y ( k , oJ) = ATl(k, o~) : (1 - 2 K ( k , ~o) :

(5.13b)

AT(k, ¢o))-1.

In a p p e n d i x C, it is s h o w n that K(k=0, w=0)=

1 ~-n0 A.

(5.14)

A c c o r d i n g to eq. (2.3b) one has

A~q(k = 0, w = O) = Arl(k = 0, co = 0)A.

(5.15)

It follows t h e r e f o r e f r o m eq. (5.13b) that 3, has the f o r m

y ( k = O, oJ = O) = y ( k = 0, ~o = 0)A.

(5.16)

T h u s one finally has f o r the effective shear v i s c o s i t y at z e r o w a v e v e c t o r and f r e q u e n c y , cf. eqs. (2.39), (5.13) and (5.15), -1

rt(k=O,o.,=O)=~o+'y(k=O,a,=O)(1--~no3,(k=O,o.,=O))

,

(5.16)

w h e r e also eq. (5.14) has b e e n used.

6. T h e effective viscosity in the a b s e n c e of c o r r e l a t i o n s

In the a b s e n c e o f correlations one m a y replace yb in eq. (5.12) b y (yb) in w h i c h case eq. (5.12) r e d u c e s to y = (yb)-

(6.1)

Using eq. (5.3) one t h e r e f o r e has in (k, w) r e p r e s e n t a t i o n

y ( k , o~[k', o~') = X ( I , oolk')(n(k - k', oJ - w')).

(6.2)

Since the a v e r a g e density, w h i c h is d e n o t e d b y no, is u n i f o r m and s t a t i o n a r y

(n(k, oJ )) = no(2¢r)48(k )8(oo )

(6.3)

one finds

y ( k , w l k ' , oo') = y ( k , o~)(27r)48(k - k')6(oo - w')

(6.4)

3,(k, o,) = n o x ( k , o,[k) = 3,(k, o,)~1 7 2 2 =-~r/0~bA(1 +y6ot a -~0k2a2),

(6.5)

with

w h e r e ~b--= ~,srl0 is the v o l u m e f r a c t i o n of the spheres. In view o f the app r o x i m a t i o n m a d e in o u r calculation this e q u a t i o n is of c o u r s e only c o r r e c t up to the s e c o n d p o w e r in ka and a a .

102

D. B E D E A U X

e! al.

Up to linear o r d e r in the v o l u m e fraction the d e n o m i n a t o r in eq. (5.13a) m a y be neglected so that the effective shear viscosity of the suspension, cf. eqs. (2.30), (2.35) and (2.39), is given by T / ( k , t o ) = "00[1 q ' - 5 ( ~ ( l -7 2 a 2 -40K I , 2 a 2,1 ± ~6c~ )1

(6.6)

in that case. F o r zero w a v e v e c t o r and f r e q u e n c y one obtains rt(k = 0, w = 0) = rt0(l + ~ b )

(6.7)

which is the well k n o w n Einstein formulaJ). Eq. (6.6) r e p r e s e n t s an extension o f this f o r m u l a to finite values of k and o~ up to the s e c o n d o r d e r in ka and ota.

In the case that the v o l u m e fraction b e c o m e s so large that the d e n o m i n a t o r m a y no longer be neglected we shall restrict o u r s e l v e s to the k = to = 0 case. T h e effective shear viscosity then b e c o m e s , cf. eq. (5.17),

n(k = 0 , to = 0 ) = 7/011 + ~ b ( 1 -

~))-1]

= no(1 + ~b + ~ b 2 + .- -)

(6.8)

w h i c h is a result first o b t a i n e d b y Saito3). This e q u a t i o n predicts a value f o r the H u g g i n s coefficient KH w h i c h is defined by n ( k = 0, to = 0) = rt0(1 + ~4~ + (-~)2KH4~2+ " " ")

(6.9)

of K{a = 0.4.

(6.10)

As we shall see in the next section this coefficient will increase substantially if correlations are taken into a c c o u n t .

7. Correlation contributions to the effective viscosity In this section we shall calculate r/(k = 0, w = 0) up to the s e c o n d o r d e r in the d e n s i t y o f the spheres. F o r this p u r p o s e we i n t r o d u c e the following operator FN =-- y ~ : (1 - 2 H

: ,y~')-~.

(7.1)

T h e o p e r a t o r y~' is given b y eq. (5.1) or eq. (5.3) t o g e t h e r with eq. (5.2). T h e d e p e n d e n c e on N o f these e q u a t i o n s has been m a d e explicit. U s i n g F as given in eq. (7.1) the e x p r e s s i o n f o r -y, eq. (5.12), b e c o m e s 3' = (F) : [1 + 2H : ( F ) ] - ' .

(7.2)

In the linear regime the equal time correlation f u n c t i o n s should be u s e d in the c o n t e x t of this model f o r the calculation of y, eq. (5.12), cf. also the discussion in section 5. In that case o n e m a y take the positions of the s p h e r e s

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

103

to be time independent so that n and yb m a y be written in the form, cf. eqs. (5.1)-(5.3), N

nN(r, to) = 2~r6(to) ~ #(r - ai)

=---2wS(to)nN(r)

(7.3a)

i=1 or N

nN(k, to) = 2~rS(to) ~

e -it'Ri ~

27r6(to)nN(k)

(7.3b)

i=l

and

ybN(r, ,air', to') = -~nov,a{[1 +7otZaZ+~aZ(VZ+ (V')Z + 7F • If')] x [nN(r)6(r - r')]}2cr6(to - to') ------~ ( r , to/r')2crS(to - to')

(7.4)

or the analogous equation in (k, to) representation, eq. (5.3). T h e r e f o r e F'N = F N ( R 1 , R 2 . . . . . R N ) is an o p e r a t o r which depends on the positions of the spheres via the nN d e p e n d e n c e of Yb. In the calculation we m a k e use of the following result, which is p r o v e n in appendix D, for the grand canonical a v e r a g e of F

(F)

no f dR~Ft(R1)+ ~no ' zf

=

dR1 dR2g(Rl-a2)

x [F2(R,, RE) -- F I ( R t ) - Ft(R2)] + O(n~)

(7.5)

where g is the radial distribution function for two spheres. For g we shall take the following f o r m (~

forr<2a for r > 2a

g(r) =

(7.6)

which should be a satisfactory a p p r o x i m a t i o n for low sphere densities consistent with the use of eq. (7.5), cf. also ref. 4. Substitution of eq. (7.5) into eq. (7.2) yields for y "y ---- n o T l

+

noZ'y2+ ~(n 3)

(7.7)

with

=f

dR1FI(RO

(7.8)

f

~'2 = ½J d R l d R 2 { g ( R 1 - R z ) [ F 2 ( R I , R 2 ) - 4F1(RI) : H + 4 [ g ( R t - R2) - 1 ] F I ( R 0 : H : FI(R2)}.

: FI(R2)

-

2Fl(R1)]

(7.9)

104

D. BEDEAUX et al.

W e shall n o w first a n a l y s e yl. T h e o p e r a t o r Fj is given by, cf. eq. (7.1), Fl(Rl) =

'}t~(Rl) : (1 - 2/4 : T~(RJ))-' I

i

= ~ y~(R,) : [2/4 : 'yb(Rl)].

(7.10)

/=0

In view of eq. (7.4) y~,(r, oJIr') contains 8 ( r - R 1 ) 8 ( r ' - R 1 ) and derivatives thereof. T h e o p e r a t o r s H in eq. (7.10) are a l w a y s f o u n d b e t w e e n two of these y~'s so that o n l y / 4 ( r = 0, w) and derivatives o f / 4 in r = 0 a p p e a r in eq. (7.10). B e c a u s e o f the definition of /4, eq. (5.6), these are all zero so that F(R1) = yl(R,).

(7.1 1)

Substitution of this e q u a t i o n into eq. (7.8) t o g e t h e r with the explicit f o r m of l 78, eq. (7.4), yields u p o n integration o v e r Rl ~l(r,

~olr')

= 5~T/0v~A(1 ± ~7c t 2a

2_± ~V2)8(r - r').

(7.12)

This s h o w s that to lowest o r d e r in the d e n s i t y ~, as given b y eq. (7.7) r e d u c e s to the f o r m given in section 6 in the a b s e n c e of correlations as should be expected. N e x t we shall a n a l y s e ~'2. To this end we n e e d F2(Rh R2). Since

"yZ(R1, Rz) = ~,~(Rj) + T~(R2)

(7.13)

/:2 m a y be written as

Fz(Rj, R~) = ~ [T~(R,) + T~(R2)I : {2/4 : [T~(R,) + T~(R2)]}'.

(7.14)

i=0

As

T~(R) : / 4 : y l ( R ) = 0 as d i s c u s s e d after eq. (7.10) eq. (7.14) r e d u c e s to J R 2)] : Z 4'[/4 : T~(Rj) : y~(Rz)]' Fz(Rj, RE) = [yb~(R2) + y~(Rj) : 2/4 : n/b( ,=0

+ [y~(Rj) + y~(R2) : 2/4 : ~,b~(R,)] : ~] 4'[/4 : y~(R2) : / 4 : Tb~(Rj)]'.

(7.15)

/=0

U p o n substitution of eqs. (7.11) and (7.15) into eq. (7.9) we obtain y2 = ~ (Tz.2i+ y2,2i+l)+ yz., = ~] Y2.i, i=t

(7.16)

i=l

where

22, = 22'f dRj dR2g(Rj-R2)'y~(Rj) f

yz.2i+l = 2 z/+j J

dR1 dR2g(Ri

-

: [/4 :

T~(R2) : H : y~(Rj)]',

(7.17a)

R2)'yl(Rz) : H : T~(RI) :

I R i i )<[~"~ : I / b ( 2) : / 4 -" ' ~ b ( R I ) ] ,

(7.17b)

SHEAR VISCOSITY OF A SUSPENSION

OF SPHERES

105

and w h e r e

"y2.~= 2 f dR~ dRE[g(R1 - R2) - 1]~(R~) : H : T~(R2).

(7.17c)

W e shall n o w first evaluate ~,2,1. Writing eq. (7.17c) explicitly in (k, to) r e p r e s e n t a t i o n one has 'y2.1(k, to]k") = TE.l(k, to)(2~)38(k - k") -- 2(27r) -3 f

x

f

dRl dR2[g(R1 - R2) - 1]

1 I dk'y~,(k, tolk';R~):H(k',to):yb(k,tolk

rt ,R2)

= 2(27r) -3 f dRl dRE[g(Ri - R2) - l]

× f dk'x(k , to Ik')H(k', to)x(k', to]k") e itk'-k)' R~ e itk'-~'~"a2. (7.18) w h e r e we h a v e used eq. (5.3) t o g e t h e r with eq. (7.3b) f o r N = 1. U p o n substitution of a n e w variable R = R1 - R2 instead of R~ one m a y p e r f o r m the integration o v e r RE. T h e result is 8 ( k - k"). Eq. (7.18) then yields y2.~(k, t o ) = 2

f dR[g(R)-

f dk'x(k, tolk I)

× H(k', to)x(k', tolk) e i(~'-k)"R.

(7.19)

W e will restrict o u r s e l v e s to the e v a l u a t i o n of this f o r m u l a f o r k = 0 and to = 0. O n e then has, cf. also eq. (5.16), y2,1(k -- 0 , to = 0 ) - "y2,1(k = 0, to = 0 ) A

2

--

f dR[g(R) - 11(2~-) -3 f dkx(0, O]k)H(k, O)x(k, 010) e ik

d.(,-lk2a2)2n(k,O)eikR

f 2(~T/0t's) 2 f

D~

dk(g - 1)(k)(l - ~k2a2)2H(k, 0)(2~r) -3,

(7.20)

w h e r e we h a v e used the explicit f o r m of X, cf. eq. (4.7a), and w h e r e (g - 1)(k) is the F o u r i e r t r a n s f o r m o f g ( R ) - 1 . N o w we use the p r o p e r t y that the integral of H(k, to = 0) o v e r the directions of k gives zero, a p r o p e r t y w h i c h is derived in a p p e n d i x C, to c o n c l u d e that

"yE,l(k

-- 0, to = 0 ) -- 0.

(7.21)

106

D. BEDEAUX et al.

N e x t w e shall e v a l u a t e t h e first n o n - z e r o c o n t r i b u t i o n 3"2,2. A l o n g t h e s a m e l i n e s as in t h e d i s c u s s i o n o f y2,J o n e h a s f o r z e r o k a n d to, u s i n g X = X A [cf. eq. (4.7)], 3"2.2(k = 0, to = 0) = yz,2(k = O, to = O)A 4

/-

f

j dRg(R)(2)-Jdg d k ' x ( O I k ) x ( k l k ' ) H ( k ) :

H ( k ' ) e i<*-k'' R,

(7.22)

w h e r e w e h a v e i n t r o d u c e d t h e s h o r t h a n d n o t a t i o n x ( k l k ' ) = x ( k , to = 0Jk') a n d H ( k ) = H(k, to = 0). T h e i n t e g r a t i o n s o v e r k a n d k' m a y n o w b e p e r f o r m e d a n d one obtains

3"2,2(k = O, to = O) = 4 f dRg(R)x(OIV)X(VIV')X(F'[O) H ( r ) : H(r')ir=r,=l¢, (7.23) w h e r e X(VIV') is n o w a d i f f e r e n t i a l o p e r a t o r , cf. a l s o eq. (4.7), 5

1

2r~2

--

X(Vl V') = ,~rt0vs[ 1 + ~ a ~v t (V') 2 + 7 V . V')].

(7.24)

W e n o t e t h a t X(0IIr') o r X(VI0) a r e o b t a i n e d b y f o r m a l l y r e p l a c i n g 17 o r r e s p e c t i v e l y V' b y z e r o in eq. (7.24). H ( r ) is g i v e n b y t h e f o l l o w i n g e x p r e s s i o n , cf. a p p e n d i x C,

H ( r ) = H(r, to = 0) = 3(8Try/or 3) l(2A - B)O(r) = (5rtous)-~H(r),

(7.25)

where A~o~8 = ~(8~ - 3n~nt3)(6v~ - 3nvns),

(7.26)

1

B,~,8 = - 2n~n~n~n8 + ~(n~n~,$~ + n~n88~r + n~n~6~ + n~ns6~v)

(7.27)

w i t h n -~ r/r. T h e f o u r t e n s o r /4 d e f i n e d in eq. (7.25) is d i m e n s i o n l e s s . U s i n g t h e e x p l i c i t f o r m o f H a n d g, cf. eq. (7.6), it is s t r a i g h t f o r w a r d to p e r f o r m t h e d i f f e r e n t i a t i o n s in t h e i n t e g r a n d o f eq. (7.23) a n d s u b s e q u e n t l y to e v a l u a t e t h e i n t e g r a l . T h i s c a l c u l a t i o n is d o n e in a p p e n d i x E. T h e r e s u l t is yz2(k = 0, to = 0) = 7035(2-1")rtou~ -~ 1 72"0oI'2.

(7.28)

A p p l y i n g t h e s a m e t e c h n i q u e s as u s e d in t h e a n a l y s i s o f y2,~ a n d 3"2,2 t h e g e n e r a l t e r m 3"2,i m a y b e r e d u c e d to

3"23(k = 0, to = 0) = y23(k = 0, to = 0)A 2 i f d R g ( R ) x ( O I V,) X ( V , [ V 2 ) . • • X(V~-,I VJ)X(V~IO) × H ( r O : H(r2) : • • • : H(rj)[r,=r2 . . . . . ,~=R.

(7.29)

I n a p p e n d i x E w e h a v e e v a l u a t e d 3'2.3 u s i n g this e x p r e s s i o n . T h e r e s u l t is y2,3(k = O, to = O) = 2

-19

2

2

(150045)'r/ors = 0.29~oVs.

(7.30)

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

107

W e h a v e not b e e n able to e v a l u a t e y2.j f o r general values of j. W e shall n o w first c o m p a r e o u r results with t h o s e o b t a i n e d if o n l y the lowest o r d e r multipole (L.M.) X °° is retained in the e v a l u a t i o n of X, cf. eq. (4.3), and c o n s e q u e n t l y in the e v a l u a t i o n of y2.1. In that case X in eq. (7.29) 5 r e d u c e s to ~r/0v, and one obtains, with eqs. (7.6) and (7.25), L.M..t k = O, to = O) = ~'tlo~,~ 5 f dRg(R)IYI(R) Y2,i

: 171(R) : • • • : 171(R)

oo

= (~)

rlovs

dRR z

dn(2~A+(-1)iB),

(7.31)

2a

w h e r e we h a v e used the following multiplication table A A

A A A

A A A

B B 0

B

B

0

B

(7.32)

T h e angular a v e r a g e s of A and B m a y easily be e v a l u a t e d and are given b y f dnA

= ½

f

dnB

= ~wA.

(7.33)

T h e integrations m a y n o w be p e r f o r m e d and yield, j > 1, L.M.

1 ./

2

y2.j (k = 0, to = 0) = 4(j - 1)-1(-~)~[1 + 2(-~) ]rlovs.

(7.34)

U p o n s u m m a t i o n o v e r j one then finds, cf. eqs. (7.16) and (7.21), 7 2 yLM'(k = 0, tO = 0) = ~5 ln(~)r/0v~

+. 0.23 + . (2.34 . + 0.36 .

)~0v~2 = 3.13r/0v~.

(7.35)

W e note that the e x p a n s i o n of a logarithm c o n v e r g e s rather slowly. In fact the sum of the first t w o terms (2.71) is still a b o u t 15% b e l o w the final result (3.13). N e x t we c o n s i d e r the case c o n s i d e r e d by P e t e r s o n and FixmanS). Their m e t h o d , in w h i c h the higher o r d e r gradients of the s e c o n d a r y v e l o c i t y fields are neglected, leads to results w h i c h m a y be r e p r o d u c e d in the c o n t e x t o f o u r f o r m a l i s m if one replaces X(VlV') in eq. (7.29) b y X(Irl0). O n e then obtains P.F.zt k = O, to = O) = i'Oovs 5 f dRg(R)S(R) "Y2.j

: S(R)

: . . . : S(R),

(7.36)

w h e r e the f o u r t e n s o r S, w h i c h a p p e a r s j times in this equation, is given b y S(r)

=-2X(VIO)H(r)

= (1 + ~0aEVE)/'l(r).

(7.37)

P e r f o r m i n g the differentiation one has, cf. a p p e n d i x E, S(r)

= 5y3[(1 - y2)A - ~(1 - 2y2)B - ~y2A ]0(r),

(7.38)

108

D. B E D E A U X et al.

w h e r e y =-a/r. Substituting this e x p r e s s i o n in eq. (7.36) and using the multiplication table, eq. (7.32), the explicit f o r m of g, eq. (7.6), and the directional averages, eq. (7.33), one finds 3"2PF(k = 0, to = 0) = 4~'rloV~

f dRR2y3j[( - y2)i+ ½(5 - 6y2) i + (4y 2 - ~)i] 2a

with y --

e'F"k 3` ,J t

(7.39)

a/R. This integral m a y be e v a l u a t e d and gives ( _ 1)i2 3 5i = o , to = o ) =

X [l-{-21 1n=o ~ (-

10)i" ( i ) ( 3 " -

( 5j-3 (-2)2"-J+')\3j- 3 +

~]

2n I J"

(7.40)

F o r j = 2 one has y2.2 p.F = 2-7(195)r10v 2 a result w h i c h was first o b t a i n e d by Vand'2). T h e sum m a y be calculated numerically and yields 3'2PEet k = 0 , t o = 0 ) = ( l . 5 2 + 0 . 1 9 + 0 . 0 7 + ' ' ' ) r / 0 u

2 = 1.82r/0u 2

(7.41)

w h i c h is the result o b t a i n e d by P e t e r s o n and FixmanS). In this case the sum of the first t w o terms is a b o u t 6% b e l o w the final result. In our case, in w h i c h a c o n s i s t e n t calculation was p e r f o r m e d to s e c o n d o r d e r in the multipole e x p a n s i o n , we f o u n d 3"2,2= 1.72rt0v 2 and 3"2.3= 0.29rt0u~. T h e relative m a g n i t u d e of these two terms is the same as in the lowest o r d e r multipole (L.M.) a p p r o x i m a t i o n , cf. eq. (7.35). In view o f this and the e x p e c t e d positive nature of the higher o r d e r c o n t r i b u t i o n we arrive at the following estimate f o r 3'2

y2(k = 0, to = 0) = (1.72 + 0.29 + • • .)r/0v 2 = 2.3 ± 0.1 r/0u2.

(7.42)

C o m p a r i n g this result with the value o b t a i n e d in the lowest o r d e r multipole a p p r o x i m a t i o n (L.M.), cf. eq. (7.35), we see that 3'2 has d e c r e a s e d rather drastically (26.5%) by the use of the multipole e x p a n s i o n up to s e c o n d order. T h e result of P e t e r s o n and F i x m a n , w h o neglect higher o r d e r gradients of the s e c o n d a r y v e l o c i t y fields, is e v e n smaller (42% below the L.M. value). Their a p p r o x i m a t i o n is equivalent to neglecting X 11 and X 2° in the multipole exp a n s i o n f o r X which a c c o u n t for contributions due to these higher o r d e r gradients. O u r result f o r 3"2 is 27% larger than P e t e r s o n and F i x m a n ' s and we t h e r e f o r e c o n c l u d e that it is not c o r r e c t to neglect the higher o r d e r gradients of the s e c o n d a r y v e l o c i t y fields. One must, h o w e v e r , keep in mind that if one goes b e y o n d the s e c o n d o r d e r in the multipole e x p a n s i o n f u r t h e r contributions to 3'2 will appear, w h i c h m a y either increase or d e c r e a s e our result. In view of the 26.5% c h a n g e in the value of 3"2 by the inclusion of terms in the multipole e x p a n s i o n to s e c o n d o r d e r it is to be e x p e c t e d that inclusion of e v e n higher o r d e r terms will affect the value o f y2 quite a p p r e c i a b l y and it w o u l d certainly be w o r t h while to c a r r y the analysis one step further. W e believe,

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

109

h o w e v e r , t h a t it is r e a s o n a b l e to c o n j e c t u r e t h a t a d d i t i o n a l t e r m s will n o t c h a n g e o u r r e s u l t b y m o r e t h a n 10%. L e t us finally c o n s i d e r t h e e f f e c t i v e v i s c o s i t y f o r z e r o w a v e v e c t o r a n d f r e q u e n c y to t h e s e c o n d o r d e r in t h e v o l u m e f r a c t i o n o f t h e s p h e r e s . E x p a n d i n g eq. (5.17) to s e c o n d o r d e r in no, u s i n g eq. (7.7), o n e h a s ~7(k = 0, to = 0) = "00+ Tl(k = 0, to = 0)n0 +

3,~(k = 0, to = 0) + 3,2(k = 0, to = 0)

no2.

(7.43)

C o m p a r i s o n w i t h eq. (6.9) g i v e s f o r t h e H u g g i n s c o e f f i c i e n t KH =

(rl0u~) -1

y2(k = 0, to = 0) + 3,2(k = 0, to = 0) .

(7.44)

S u b s t i t u t i n g 3'1 a n d t h e v a r i o u s v a l u e s o f 3'2 o n e finds, cf. e q s . (7.12), (7.35), (7.41) a n d (7.42) K .L M = 0.90,

(7.45)

K PF = 0.69,

(7.46)

w h i l e w e find K . = 0.77.

(7.47)

W e n o t e t h a t all t h e s e v a l u e s a r e s u b s t a n t i a l l y l a r g e r t h a n t h e v a l u e 0.40 f o u n d if c o r r e l a t i o n s a r e n e g l e c t e d . C o m p a r i s o n o f t h e v a l u e f o u n d b y P e t e r s o n a n d F i x m a n w i t h o u r v a l u e s h o w s a g a i n t h a t c o n t r i b u t i o n s d u e to h i g h e r o r d e r d e r i v a t i v e s o f the s e c o n d a r y v e l o c i t y fields a r e n o n - n e g l i g i b l e . F o r t h e effective v i s c o s i t y w e find

rl(k

= 0, to = 0) = r/0(1 + ~qb + 4.80b z)

(7.48)

to t h e s e c o n d o r d e r in t h e v o l u m e f r a c t i o n . F i n a l l y , w e n o t e t h a t this r e s u l t is c l o s e to t h e v a l u e o f 5.2 + 0.3 f o u n d m o r e r e c e n t l y b y B a t c h e l o r a n d G r e e n 4) by a different analysis.

Acknowledgments O n e of us ( R . K . ) w o u l d like to t h a n k t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada for support and the members of the Lorentz Institute for their h o s p i t a l i t y d u r i n g t h e p e r i o d in w h i c h this r e s e a r c h w a s c a r r i e d out.

Appendix A In o r d e r to d i s c u s s t h e s y m m e t r y o f t h e o n e s p h e r e r e s p o n s e f u n c t i o n it is c o n v e n i e n t to first c o n s i d e r t h e e q u a t i o n o f m o t i o n (2.18) in t h e f o r m v = v o + T " Find,

(A.1)

110

D. BEDEAUX et al.

where Find was defined in eq. (2.6). We next define a function A which gives the response to an external field V0,

Find(r, to) = f dr'A(r, tolr') • v0(r', to).

(A.2)

We will first prove the s y m m e t r y relation

A~(r, tolr' ) = A~(r', oJIr )

(A.3)

and then discuss the connection between A and X which is used in the viscosity calculation. If we consider two arbitrary sets of solutions (v0~, v 1, Filod) and (v~, v 2, Fi2,d) an equivalent statement of the symmetry relation is*

f dr dr'v~,~(r)A~o(rlr')v~,~(r') = f dr dr' v2,~(r)A~B(rlr')vl~(r').

(A.4)

The oJ d e p e n d e n c e has not been explicitly indicated in this equation. Since the derivation is somewhat simpler in k space we write eq. (A.4) in the form f dk d k ' v ~ ( - k ) - A(k[k')- v2(k') =

f dk dk'v~(-k). A(k[k'). v~(k')

(A.5)

and the statement of the s y m m e t r y relation is

A~o(klk') = A ~ ( - k'[- k).

(A.6)

Using the Fourier transform of (A.2) in (A.5), and then using (A.1) to eliminate v0(-k) and noting that T is symmetric we obtain instead of eq. (A.4)

f d k v ' ( - k ) " F~nd(k)= f dkv2(-k) • Filnd(k).

(A.7)

From eqs. (2.1), (2.6) and (2.8) we have Fi.d(k) = -io)pv(k ) + ip(k) + 71ok2v(k ). (A.8) Upon insertion of this expression into eq. (A.7) and noting that k • v(k) = 0 it follows that both sides of eq. (A.7) and therefore of eq. (A.4) are indeed equal for arbitrary choices of v I and v02. The symmetry relation, eq. (A.3) or eq. (A.2) then follows. From eq. (2.10) and eq. (A.2) it follows that

f drA(r[r') = 0

(A.9)

and using the symmetry, eq. (A.3),

f dr'A(rlr') = 0

(A.10)

or in wave vector representation

A ~ ( k = OIk') = A ~ ( - k'lk = O) = O. • W e use the s u m m a t i o n c o n v e n t i o n .

(A. l l)

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

Ill

This relation implies that there is no contribution to A(k[k') which is ind e p e n d e n t of either k or k' and justifies the structure of the r e s p o n s e function in eq. (3.1). Once this f o r m of A is established, the relation b e t w e e n A and the s o m e w h a t more convenient response function X is easily written. Since A is at least first order in k and k' we can define X by the relation

At3~(k [k') =--- 2ik~x,,~,8( k lk')ik '~.

(A. 12)

If we introduce this definition into the Fourier t r a n s f o r m of eq. (A.2) and note eq. (2.6) we find Find,t3(k) = ik~(r~(k) = - 2(2~-)-3ik~ (dk'x,#3=,~(klk')ik'svo.v(k') J

(A. 13)

or in r space

¢r.~(r) = - 2

f

i t ). dr iX~ve(r Ir t )Vsvo,~(r

(A.14)

F r o m eqs. (A.6) and (A.12) we can write

k ~ k ' s x ~ ( k l k ' ) = k" ksx~v~8(- k ' l - k) = k~k'~xsv~(- k ' l - k ).

(A. 15)

Hence

X~t3vs(k [k') = Xs+~ ( - k ' l - k ).

(A. 16)

This is the s y m m e t r y relation given in eq. (4.4b). The relations in eqs. (4.4a) and (4.4c) follow directly f r o m this result. In view of these s y m m e t r y relations it follows that the s y m m e t r i c traceless part of tr couples only to the symmetric traceless part of Vv0. It is this relation which is given in eq. (3.1) and which is used as a starting point of the local field analysis in section 3. Finally one obtains eq. (2.21) for ~-~ in terms of r-ff~vusing the s y m m e t r y relations derived in this appendix and the local field analysis.

Appendix B In this appendix we calculate the one sphere response function with respect to the external velocity field, X, by making use of the multipole expansion of the induced stress tensor ~r. The techniques e m p l o y e d here are similar to those used in ref. 11 for the derivation of a generalization of F a x e n ' s theorem. To begin the calculation we consider the multipole expansion for (r for the case of one sphere or(r, to) = ~ ( - l ) " ( n !)-lo'~(co)Q [V~8(r)], n--0

(B. 1)

where

"_

~aBVl

r.~ . . .

vn

= f dro'~tJr~r~.,

r~,.

(B.2)

112

D. BEDEAUX et al.

W e note that ~r" is o b v i o u s l y s y m m e t r i c for i n t e r c h a n g e of a n y two vi's. In writing the a b o v e e q u a t i o n s we h a v e taken the origin of the c o o r d i n a t e s y s t e m at the c e n t e r of the sphere. F r o m the e q u a t i o n of m o t i o n f o r the v e l o c i t y field, v = v 0 - T V : cr

(B.3)

we can calculate the m o m e n t s of ~r and obtain the X m" by making use of the relation ~r"(to) = - 2

ff'~ (m!) I X " " ( t o ) Q [( .v. .~. +1vo~r , , , to)]r'=O',

(B.4)

m=O

w h i c h is obtained f r o m eqs. (A.14), (4.2) and (B.1). W e n o w explicitly calculate the first two m o m e n t s of ~r. An e x p r e s s i o n for 0 ~r can be o b t a i n e d b y multiplying eq. (B.3) with r and averaging o v e r the v o l u m e and s u r f a c e of the sphere. M a k i n g use of the definition of T in eq. (2.17) we find, r~vt~

-

r~vo.t~ ~ = ( d r ' { V ~ [ 6 ~ - a - 2 V ~ V ~ ] r ~ ( r -

r',

to) ' } o ' ~ ( r

,' t o )

(B.5)

d

and r.v~

v

- r~vo.~

v z

f

dr'{V~[6~ - a - 2 V ~ V ' r ] r ~ ( r

t

- r ' , w ) v }cr~(r , to)

+ a 2 f d r ' { V ' ~ V ' ~ V ' ~ r ~ ( r - r ,' 0) ~}o'~,(r ' , to).

(B.6)

T h e s u r f a c e and v o l u m e a v e r a g e s of a f u n c t i o n A of r in these equations are defined by A ( r ) V = (4¢ra3/3) i

drA(r)

(B.7)

r
and /.

(4rra 2) I

A(r)S=

/

(B.8)

dSA(r).

r=a+~

To simplify eqs. (B.5) and (B.6) we use the following results w h i c h are o b t a i n e d b y s t r a i g h t f o r w a r d calculation"). r~(r

_

t

r, w)

s

= r ' q 3 ( a , oa)(1 + oea)(oer')-3 (oer ' c o s h a r ' -

sinh oer')

for r' < a + ~, r ~ d ( r - r ' , O; = ~1 r ,,~ d ( a , O ) r~d(r-

r ' , to)v= 3 r ~ ( a a )

f o r r' < a + e .

(B.9b)

2c-~(a,0 ) - 9(aa)-2(1 + a a + la2a2)

x (1 + a a ) r~d(r-

(B.9a)

lr~(r

r ' , O) = ~r~(1 ' ' -3(r'/a)2)~(a,

- r', o~)

0)

f o r r' < a + e,

for r ' < a + e.

(B.10a) (B. 10b)

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

l l3

Inserting these expressions into eqs. (B.5) and (B.6) we find, using also eq. (2.9), 1 ~a2a2(r~V _ ~v

= ~ ( a , 0)

f

) + 3(1 + a a + ~a2a2)(l + ota)-l(r~va S - r~vo.~S )

dr'(4o'~, 0 - o'~ - &,oo'ry).

(B.11)

The evaluation of r - ~ ~'~ depends on the nature of the boundary conditions; for stick boundary conditions on the velocity field we have v

1 2

(B.12)

s

1 2

(B.13)

r~v~ = ~a e,,t~-,~O,,

and r~vt3 = ~a ~ v O ~ ,

where E~v is the Levi-Civita tensor. Taking the symmetric traceless part of eq. (B.11) and using eqs. (B.12) and (B.13) one finds f

dr"tr~0(r',to)'

=

-20~r,loa{(l+aa)-l(l+aa-~a

__1

2 2xt~-Is_[_!

a ~r~vo,~

2 2 t~-~v

9t~ a r~vo,~t

(B.14) To cast eq. (B.14) into the form of eq. (B.4) we expand v0(r) in a Taylor series about the center of the sphere. The results are • s ® a2/+22(j + 1) r~vo.a = ~ [(V) zs~ 1,:o, (B. 15) /=o (2j+3)! and v= ~,® a2j+26(j + 1) [(V)2i ~ (2j + 5)!

],=o.

(B.16)

The unperturbed field vo is however a solution of the Navier-Stokes equation in the absence of the sphere, Ot 2 v o ( r )

--

V2V0(r)

=

--

'r/O

1

(B.17)

Vpo(r).

Using the fact that •2po = 0, in view of the incompressibility, it follows that vEJVEvo(r) = a 2~V2vo(r).

(B. 18)

Thus we can write eqs. (B.15) and (B.16) in the forms •

.s

a2

r~vo,~ =~-['V~vo.~(r)'],=0 + a 4

(/=~ 2(j + 1)(ota)2j-2"~ 2~]~-~. / [ V V~v0.~(r)],=o (B.19)

and ~ravo.t3 = ~a2 v,,vo,~O') '1~=o+ a ' (/--~1__6(j ~+j 1)(aa)2J-2~ +-~.v j [ V2"v o v . ( , ) ] _' o .

(B.20)

114

D. BEDEAUX

e t al.

Hence

(l+00a+1002a2)F [ V,,vo~(r) ],=o ([ (1 + 00a) °-k] ' 2

=

f

drff'-~

2

= - 5"Oovs

+ a2 [/_~12(./+ l)(ota)2j-2[ 3(l+00a+~002a2) 002a2 [ (1 + 00a)(2j + 3)! + (2j + 5)!}]

x[V~V~vo,~(r)']~=o}.

(B.21)

In order to obtain X "° we write eq. (B.4) for n = 0 containing only the traceless part = - 2 ~] (m !)-lXm°Q [ ( V ) m ~ l , = o .

(B.22)

rn=0

Comparison of eqs. (B.21) and (B.22) shows that only X °° and X 2° are non-zero and that 00

X.o~( ~o) = X°° ( o ) A ~o,~,

(B.23)

and 20

X,~.,,,,,.~(~o) =

20 X

(~o)a.ov,a,.,,.~,

(B.24)

with xoo(to)=5 (1+ 00a +~ 002a2) 002a2] ~/ovs (1 + eta) t- 60 '

(B.25)

1 2 a 2) 00202 2 -. 2F3(1 .~_ 00a +~00 X2°(w) = 5~/ovsa ~ 2(.i + 1)(00a)LJ L ( i + a - ~ ( 2 - ~ . v I-(2~+~-)!j.

(B.26)

and

The calculation in the text utilizes the one sphere response function in the approximation where terms to second order in the wave vectors and second order in 00a were retained. From eqs. (B.25) and (B.26) the coefficients required for this approximation are 7 2 a 2.1, X°°(oJ) = -~'Oov~(1+~600

(B.27)

l~,/oVsa2"

(B.28)

and X 2 0 ( £ O ) = X02(1.O) =

Before proceeding to calculate X" we note that X°z can also be obtained directly by comparison with the explicit solution for the momentum field outside the sphere given by Landau and Lifshitzl3). Since we will use a similar procedure to calculate X 11 we briefly outline how the identification is made.

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

115

We use eq. (2.18) and insert the definition of the response function, (we only consider zero f r e q u e n c y for simplicity). v(r)-vo(r)

=2 f dr'T(r-r')V:

f d r " x ( r ' ] r " ) : V" vo( r"),

(B.29)

or using the multipole expansion ( - 1)m+n(m !n !)-i / d r ' T ( r - r ' ) V ' : X m"

v ( r ) - vo(r) -- 2 ~

J

m,n=O

Q [(V')"8(r')] f d r " [ ( V " ) " 6 ( r " ) ] : V"vo(r") =2

~" ( - l ) n ( m ! n ! ) ' f d r ' T ( r - r ' ) r ' : X

mn

rn,n =O

Q [( V')" 8 (r')][(V") m V"vo(r")lr"=o.

(B.30)

If we take the symmetric traceless gradient of this equation we obtain ~

-

'V~v0,~(r)' = 2 ~

( - 1)n(m !n !)-~[V~,,... V~nG~¢(r)]

m,n =0

× X s, . nw ~ ..... ~, . ~ r V " ~ . . .

V ~" V ~"v o . , ( r " )]r~=O. (B.31)

For specific types of external fields the hydrodynamic equations can be solved explicitly and comparison with eq. (B.31) will yield expressions for X '~n. For example when I"-'1

v o ( r ) = a °. r;

(B.32)

a ° = a °.

The solution is well known ~3"2)and the symmetric gradient of the velocity field o u t s i d e the spheres takes the form 0

0

(B.33)

where $ ( r ) is defined in eq. (7.37). If eq. (B.32) is used in eq. (B.31) we find a ~ = 2 ~'~ ( - 1) (n .) [ ~,, • • • V,~G~va(r)] Xsv~,~l... on 0 ~a~,.

(B.34)

n=0

Comparison of eqs. (B.33) and (B.34) shows that only the X °° and X °2 coefficients are non-zero and the values are the same as given previously in eqs. (B.27) and (B.28) for zero frequency. We will now use a similar procedure to calculate X" which is the remaining contribution to the response function to second order in the wave vectors. Only the zero f r e q u e n c y value is required. Suppose the velocity and pressure fields in the absence of the sphere are given by vo(r)=al:

rr,

po(r)=O,

(B.35)

116

D. BEDEAUX et al.

where the constant three index tensor ~e~ is s y m m e t r i c and traceless on all pairs of indices. It is then clear that lrv0 is s y m m e t r i c traceless and that V2v0(r) = 0. Outside the sphere the velocity field satisfies the equations "r/0V2v(r) = Vp(r)

V . v(r) = 0.

and

(B.36)

The solution of this equation outside the sphere can be written in the form

v(r) = vo(r) + VxVx[(otJ : VV)f(r)],

(B.37)

p(r) = r/oat: VVVV2[(r),

(B.38)

with the condition that f satisfies the Laplace equation. H e n c e

]:(r) = clr + c2]r.

(B.39)

The constants cl and c2 can be found by fitting the b o u n d a r y conditions on the surface of the sphere. F o r stick b o u n d a r y conditions the results are cl = 7a5/24,

c2 = a7/24.

(B.40)

and the velocity field outside the sphere can be written in the simple f o r m

v(r) = vo(r)+ 18 ( a ) 'or' ~ r r [ ( 7 - ~ 15a2\ - - ) + 3 5 ( - ~ y -a2 l)nn].

(B.41)

where essential use has been made of the s y m m e t r y properties of ee 1. We m a y now use eq. (B.35) to write eq. (B.31) in the f o r m ~ )

-'V~v0,t3(r) ' = 4 ~ (n ! ) - ' ( - 1)~[V,, . . . V,,G~t3~s(r)] X

t/=0 In ,~STgVl~lg I

.

. .

I /~n~gVVl.

(B.42)

The coefficients in the multipole expansion of the r e s p o n s e function can once again be obtained by explicitly computing the s y m m e t r i c gradient of v(r) and comparing with eq. (B.42). In particular, f r o m the f o r m of v(r) it is clear that only X H and X ~3 c o m p o n e n t s are non-zero. The coefficient X H can be obtained f r o m the part of lrv(r) proportional to l/r 4 while X ~3 can be obtained f r o m the part proportional t o lit 6. When the gradient of v(r) is calculated we find that X z~ follows f r o m the solution of the equation, 11 = I[Va(Vt~V~

1

- - 8 B ~ V 2) + V / 3 ( V ~ V ~ - -

8~V2)]V~Vvot

~v(cir).

(B.43)

N o w we use that -(6r/0us) - t a 3 V ~ ( V ~ V , - 8~,V2)V~(r) s y m m e t r i s e d in a, /3 and in /x, 1, is equal to G,m,~. One then has 11

1

7

2

1

4[V~,G~8(r)]xt~.~,.~a~w, = ~ousa [V~G~o~v(r)]ot~.

(B.44)

T h e r e f o r e we have II

Xs~,~,~ = X

It A

~,,~,,~,,

(B.45)

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

117

with X 11 = ~7/ o v s a

2

(B.46)

.

Utilizing the a b o v e results we find the following e x p r e s s i o n f o r the o n e s p h e r e r e s p o n s e f u n c t i o n to s e c o n d o r d e r inota and the w a v e v e c t o r s k and k', 5 [l+70t2a2 x(k, oJlk') -- ~ novs

- T a2 ~ ( k 2 + ( k ' ) 2 - ~7 k • k ' ) ]

(B.47)

or in (r, t) r e p r e s e n t a t i o n [ 5 7 a2p 0 I - 0- -2 ( V 2 + ( V ' ) 2 + Z V- V' ) ] x(r, tlr', t') = ~ novs 1 + 20 T0 Ot 10 4

× 8(r)8(r')8(t - t')

(B.48)

w h i c h are the relations used in the main text.

Appendix C T h e p r o p a g a t o r s ~ and ~o, cf. eq. (2.15), are diagonal in (k, to) r e p r e s e n tation. T h e diagonal e l e m e n t s in this r e p r e s e n t a t i o n are ¢g(k, to) = (-ip¢o + 1"/ok2)-1

and

~0(k, W) = k -2.

(C.I)

In (r, o~) r e p r e s e n t a t i o n o n e has C~(r, wlr',

to') =

~ ( r - r', w)2~'8(w - w'),

(C.2)

ego(r, wlr', w') = ~o(r - r')27rS(w - ¢o'),

(C.3)

~d(r, w) = (4~r+/or) -I e x p ( - o~r)

(C.4)

with and

~0(r) = (4~'r) 1.

Similarly the p r o p a g a t o r s T and G are diagonal in (k, w) r e p r e s e n t a t i o n with diagonal e l e m e n t s T(k, w) = ( - i p t o + T/0k2)-l(1 --/~/~),

(C.5)

G(k, w) = - ½ k 2 ( - i p w + ~lok 2) 1A",

(C.6)

w h e r e 1¢ = Mk and w h e r e zt" is given in eq. (2.33). By F o u r i e r t r a n s f o r m a t i o n of eq. (C.6) with r e s p e c t to k and s u b s e q u e n t l y putting to = 0 one o b t a i n s

G(r, to = 0) = 3(87rr/0r3)-l(2A -- B) -- (5r/o)-IAS(r),

(C.7)

w h e r e A and B are given in eqs. (7.26) and (7.27). T o g e t h e r with eq. (5.6) one then finds eq. (7.25) f o r H(r, ~o = 0). U s i n g eq. (7.33) it follows that the

118

D. B E D E A U X

et al.

directional a v e r a g e of H(r, to = 0) is zero

f H(r, to

0) dn = 0.

(C.8)

As a c o n s e q u e n c e of this it also follows that

f l'l(k, t o = O ) d £ = f d£ f d r e - " r t t ( r , to=O)

=

f

sin kr I"

drr 2 ~

J d n H ( r , to = 0) = 0

(C.9)

0

w h i c h w a s used to p r o v e eq. (7.21). Finally one has

K(k = O, to = O) = f

drG(r, to = 0) = -(57/0) 1A,

(C.10)

r
w h e r e we have used eq. (C.7) and the fact that the directional a v e r a g e of 2 / I - B gives zero, cf. eq. (7.33). This p r o v e s eq. (5.14).

Appendix D C o n s i d e r a f u n c t i o n of the positions of the spheres

F(RI . . . . . RN) = F(n(r)) = F ( 3 ( r - RO + 3(r - R2) • • • + ~3(r - RN)). (D.1) T h e a v e r a g e m a y be written as in a g r a n d canonical e n s e m b l e as

(F) = ~ ' ~, ~ .

d R , . . , dRN e -°wN(R'..... RN~F(N,... RN),

(D.2)

N=I

w h e r e z is the activity, fl = l/kT, WN is the effective potential e n e r g y for N s p h e r e s in the fluid and ~ the g r a n d canonical partition function. W e will virial e x p a n d this e q u a t i o n

z = n o - 2b2n~ + ~(n~), ~'

w h e r e b2 = ½f dRt2(e -°w2~R~2) 1),

l+nofdRl+~?(no 2)

(D.3) (D.4)

e x p a n d i n g ( F ) to s e c o n d p o w e r in no then yields (only N = I and 2 contribute)

( F ) = n o f d R , F ( R , ) + ~n~ f dR, dR2 e ,w:~.,~) × (F(R1, R2) - F(R,) - F(R2)) + ~?(n3).

(D.5)

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

119

Eq. (7.5) n o w follows in view of the f a c t that g(R12) -----e -~w:(R12)+ G(n0).

(D.6)

Appendix E In this a p p e n d i x we calculate 'Y2.2 and T2.3 f o r k = to = 0. I n t r o d u c i n g a as the unit of length eq. (7.23) f o r y2.2 r e d u c e s to

2

with

(E. 1) 9~( V I V') -= 1 + 7 ( V + V') 2 + I V 2 + ~o(V'):,

(E.2)

w h e r e R -----R D . N o w we rewrite the integrand using eq. (7.37)

£(olv)~(vlv')~(v'lo)ft(r)

: fl(r')lr=r,=R

= yC(VlV')S(r ) : S(r')I,=r,=R = [S(r) :

S(r) +-~S(r)

: V2S(r)

+7(I7 + V')2S(r) : S(r')],=,,=s.

N o w we use the f a c t that V4/~ = 0 w h i c h implies that V2S = V2/~ = 10(S - / ~ ) so that (of c o u r s e R - 2 is a l w a y s implied in these equations)

x(OI V)Sc(VlV')~(V'lO)fl(r) : / ~ (r')l r = r ' = R = -~[5S : S - S : H + 7 V : ( S : S)].

(E.3)

U s i n g the explicit f o r m of S and /4 one has

S:S

= 5y6[(5 - 12y 2 + 7y4)A + ~(5 - 16y 2 + 12y4)B + ~y4A],

S :Ft --- H

: S = 5y6[(5 - 6yE)A + ~(5 - 8yE)B].

(E.4) (E.5)

N o t e that A, B and A c o m m u t e with e a c h other, cf. eq. (7.32). T o g e t h e r with eq. (7.33) one then finds f o r the angular a v e r a g e s

1 fdl'l(S:S)=~yl

47r

6,15t _ 4 0 y 2 + 2 8 y 4 ) A ,

1 f d / - / ( S : / ' ) ) = ~ y 6 ( 3 - 4y2) A. 4Ir

(E.6)

(E.7)

F u r t h e r m o r e w e use that o n e m a y substitute

f

f d.(S:S)

(E.8)

120

D. BEDEAUX et al.

and 4 d2 V2f(Y) = Y ~y-~y2f(Y)

(E.9)

inside the integrand. U p o n substitution of eq. (E.3) into eq. (E.1) and using eqs. (E.6)-(E.9) the resulting integral o v e r y = 1 / R is trivial and gives eq. (7.28) for y22. Similar to eq. (E.1) we m a y write [ 15

2\ f

dRR z

f dg/,,~(OI vi)~( Vl Iv2))(( v2[ v3)

2

× ;~(¢310)H(r,) :/4(r2) : H(r3)lr,-~=r,=R.

(E.10)

Using again eq. (7.37) the integrand m a y be written as ~(0[ Vl))~(~l[ V2)~(V2[ ~r3)~( V3]0)/4 (r,) :/4 (r2) :/4 (r3)[ rl=r2=r3=R

= ~(( V, [V2)~((V21V3)S(r,) :/'1 (r2) : S(r3)[,, =,2=~=~.

(E. 1 1)

Substituting the explicit form of ~, eq. (E.2) one finds

~(v,l v2)~(v21 v3)S(r,)

:/~ (r2) : S(r3)l.,=.~=.~=. = S :/~ : S + ~ S :/~ : (V2S) + ~oS : S : ( V ~ )

+ ~ S : V2(/~ : S)

+ (80) 2~,~ _"(V2S) "_(V2S) + 2(80)-2S : (V2/~) : (V2S)

+ 14(80)-2(V2S) : V2(/'! : S) + 14(80)-2S : V2(S -" V2~'~) + (7/80)@2S : Vz(b~ : S ) ,

(E.12)

where we have used that all these four tensors c o m m u t e with each other, cf. eq. (7.32). In the last term in this equation the first V 2 acts only on the first $ and t h e / ' l which is indicated by the line connecting these three entities. The R d e p e n d e n c e on the right-hand side has not been explicitly indicated for ease of notation. F u r t h e r m o r e V2 now acts on the R d e p e n d e n c e but only inside the brackets. Using V2S = V2/q = 1 0 ( S - H) one obtains ,1((I;',[ V2))(( V2l V3)S(r,) : H (r2) -- 3- '2s

:S:S+

~s

:

S(r3)lr,=,a-r3-,

:S:I:I-14S:H:H+~I:I:I:I:I:I

+7S'V2(S:H)+~20S:V2(S:S)--320 + (7/80)2~/q

: S).

: V2(S : ~'~)

(E.13)

U p o n substitution of this result into eq. (E.10) the evaluation of all contributions is straightforward with the exception of the last one. In this last term it is m o s t convenient to use partial integration twice, this leads to two surface terms plus the integral o v e r S : V2(/4 : (V2S) = 10S : V2(/4 : ( S - / - I ) ) which is again of the same nature as the other contributions. P e r f o r m i n g this straightforward but tedious analysis yields eq. (7.30) for ~,2.3.

SHEAR VISCOSITY OF A SUSPENSION OF SPHERES

121

References 1) A. Einstein, Investigations on the Theory of Brownian Motion (Dover, New York, 1975): Ann. Phys. 19 (1906) 289, 34 (1911) 591. 2) See e.g.H. Yamakawa, Modern Theory of Polymer Solutions, (Harper & Row, New York, 1971) and references therein. See also H.L. Frisch and R. Simha, in Rheology, F.R. Eirich, ed. (Academic Press, New York, 1956) vol. I, Chap. 14. B.U. Felderhof, Physica 000 (1976) 000. 3) N. Saito, J. Phys. Soc. Japan 5 (1950) 4, 7 (1952) 447. 4) G.K. Batchelor and J.T. Green, Journ. Fluid Mech. 56 (1972) 401. 5) J.M. Peterson and M. Fixman, J. Chem. Phys. 39 (1963) 2516. 6) S.F. Edwards and K.F. Freed, J. Chem. Phys. 61 (1974) 3626, 62 (1975) 4032. 7) D. Bedeaux and P. Mazur, Physica 67 (1973) 23. 8) S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, Chap. VII (North-Holland, Amsterdam, 1962). 9) L.M. Hafkenscheid and J. Vlieger, Physica 75 (1974) 57. 10) B.U. Felderhof, Force Density Induced on a Sphere in Linear Hydrodynamics, to be published. 11) P. Mazur and D. Bedeaux, Physica 76 (1974) 235. 12) V. Vand, J. Phys. Colloid Chem. 52 (1948) 277. 13) L. Landau and E. Lifshitz, Fluid Mechanics (Pergamon, New York, 1959).