An analytical model of tracer diffusivity in colloid suspensions

Physica 133A (1985) 1.52-172

North-Holland, Amsterdam

AN ANALYTICAL MODEL OF TRACER DIFFUSIVITY

IN COLLOID SUSPENSIONS

R.B. JONES and G.S. BURFIELD

Department of Physics, Queen Man C’ollege. M11e End Road, London El 4NS. LJK

Received 21 February 19X5

We present an analytical mode1 of the tracer diffusivity of a colloid particle applicable to a

suspension at a density low enough that only pair interactions of the particles need be considered.

We assume a pair potential consisting of an infinitely repulsive hard core together with an attractive short range tail as well as hydrodynamic pair interactions which are described by the

general mobility series of Schmitz and Felderhof which allow differing hydrodynamic boundary

conditions. Using our earlier low density expansion of the generalized Smoluchowski equation we

obtain an expression for the tracer diffusivity identical to that obtained by Batchelor from a

different derivation. For a class of simple analytic potential functions we show how the tracer

diffusivity may be evaluated by quadratures. We present explicit high order mobility seriec for both

stick and slip hydrodynamic boundary conditions and we use these to evaluate the tracer diffusivity numerically for a range of hard core radii. We distinguish short time and long time components of

the tracer diffusivity and show that the relative magnitude of these components is quite sensitive to the hydrodynamic boundary conditions.

1. Introduction

An interesting set of problems in colloid science concern the ditfusivities

which may be studied in a suspension. In a monodisperse suspension one may

consider the collective diffusivity or the self-diffusivity of a tagged tracer

particle while in a polydisperse suspension one may consider diffusivities for

each of the component species. More generally one can introduce wave vector-

and frequency-dependent diffusivities which can be studied by dynamic light

scattering measurements’-“). All of these diffusivities are of interest because

they depend upon the interactions among the colloid particles and thus give

information about the interparticle forces. In the present paper we study a

model of the tracer particle diffusivity D’ for a suspension of identical

spherically symmetric particles which interact by two-body potentials as well as

by hydrodynamic forces. We assume that the potentials are short ranged and

that the suspension is sufficiently dilute that only pair interactions need be

037%4371/85/$03.30 @ Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

TRACER DIFFUSIVI-IY IN COLLOID SUSPENSIONS 153

considered. Our model is one in which both the potential and hydrodynamic

interactions can be varied and in which an expression for the tracer diffusivity

can be given in closed analytic form so that numerical evaluation is straight-

forward.

Batchelo?‘) has presented accurate numerical results for polydisperse sus-

pensions with variable two body potentials but with a fixed model of the

hydrodynamic interactions. Our model complements his work in that we study

the result of changing the hydrodynamic boundary conditions as well as the

potential interactions and our numerical procedure is different in nature from his.

It is of interest that we arrive at an expression for the self-diffusivity which can be

identified precisely with Batchelor’s result6) although our respective derivations

have quite different starting points.

In section 2 we review different definitions of the tracer diffusivity indicating

how one can distinguish “short time” and “long time” contributions to it. In

section 3 we describe the class of two body potentials considered and we

present accurate high order series expansions of the mobility functions for two

different hydrodynamic models. In section 4 we specialize the results of earlier

work8,9) to give analytic expressions for the “short time” and “long time”

contributions to DT. In sections 5 and 6 we show how one can use our model

potential together with the mobility series to reduce the problem of calculating

DT to simple quadratures. Several key steps in the numerical work depend on

the use of a computer system which can perform algebraic manipulations. In

section 7 we present selected numerical results and indicate the connection

between our derivation and that of Batchelor.

2. Definition of diffusivities

In this paper we discuss the tracer diffusivity of a tagged particle which could

be measured in a diffusion experiment in which the tagged particle moves

through a macroscopic concentration gradient of untagged particles. For sim-

plicity we regard the tagged particle as being identical to the untagged particles

in every respect except for its carrying a label which experimentally might be

an index of refraction different from that of the other particles. The case when

the tagged particle differs in size or interactions from the untagged particles is

easily included in our general method but would greatly complicate the details

of the discussion. In order to compare our formalism with previous work we

first consider different methods of defining a tracer diffusivity.

One simple approach is to obtain the tracer particle diffusivity as the limit of

one of the diffusivities appropriate to a polydisperse suspension of many

species. Thus consider a suspension of spherically symmetric particles of m

1.54 R.B. JONES AND G.S. BURFIELD

different species with the ith species characterized by a number density n’(r, t).

One may then introduce diffusivities D, associated with macroscopic concen-

tration gradients through a generalization of Fick’s law as6,“‘,“)

J,=-2 Dip-h”, p=l

(2.1)

where J, is the flux density of particles of species i. The coefficients D,, depend

upon the equilibrium volume fractions 4’ of the different species, 4’ =

ni4rR:/3, with nh = N’/V, and where N’ is the total number of particles of

species i, V is the volume of the suspension and R, is the radius of a particle of

species i. At low enough densities for suspensions with short range forces the

D,, can be expressed in first order virial expansion as’.“.“)

D,, = D’l4’TiJ (i f i), (2.2)

where Dp = Dfl with 0: the scalar diffusivity of a particle of species i in the

infinite dilution limit and the y,, ri, depend upon the particle interactions. The

scalar tracer diffusivity DT can be obtained from this polydisperse formalism by

considering the tracer particle to constitute a species of zero volume fraction

while all other particles constitute a second species of volume fraction 4 =

n,4rR3/3 where n,, = N/V with N the total particle number for the suspension,

DT = D”(1 + yT~). (2.3)

Light scattering studies of suspensions however require a more general

definition of diffusivities. In these experiments’) one can probe density fluctua-

tions which are space- and time-dependent and characterized by finite wave

number k and frequency w. By linear response methods2.“) one can introduce a

generalized self-diffusivity kernel D,(k, ) w such that DT is the value of D5 in the

hydrodynamic limit

DT = lii ljz D&k, w) . (2.4)

The derivation of the dynamical quantity D,(k, w) shows that it can be split into

two parts,

(2.5)

TRACER DIFFUSIVITY IN COLLOID SUSPENSIONS 155

where the “first cumulant” D:(k) represents an instantaneous response of the

tagged particle to a fixed environment of untagged particles while the “memory

function” M,(k, w) represents the contribution to D,(k, w) arising from the

dynamical evolution of this environment. In dynamic light scattering studies’)

these two components are known as “short time” and “long time” con-

tributions and under appropriate experimental conditions13.14) can be separately

measured. A macroscopic diffusion experiment however is unable to dis-

tinguish these components separately and gives only their combined con-

tribution to DT.

Some earlier attempts to calculate DT for hard spheres”.‘*) inadvertently

omitted the “long time” contribution but more recent workk9) has shown how

to correctly incorporate both contributions. The derivation by Batchelor rests

upon ideas of irreversible thermodynamics while our earlier treatment’) applies

statistical mechanical methods to a microscopic dynamical description of the

suspension by means of the generalized Smoluchowski equation (GSE)“). In

the present paper we will use the general formalism of our earlier work’) where

we computed D:(k) and M,(k, w) to first order in number density in the

hydrodynamical limit (2.4). We will be able to indicate the precise cor-

respondence between our final expression for DT and that given by Batchelor6)

from a different line of argument. For this purpose we will explicitly split yT

into a “short time” component yTS arising from D:(k) and a “long time”

component yTL arising from M,(k, w), so we will write eq. (2.3) as

D= = D”(l + yT4) = D”(l + (yTS + yTL)4).

In subsequent sections we quote analytical expressions for yTS and yTL and we

evaluate them numerically for differing models of the particle interactions.

3. Particle interaction model

For a sufficiently dilute suspension we need consider only pair interactions

which are of two sorts, direct potential interactions and hydrodynamic inter-

actions. The potential interactions are electrodynamic in origin and in principle

involve many body effects but for moderately dense suspensions there is a

widely used effective two-body potential16) which combines a strongly repul-

sive short range core with a longer range tail representing the combined

effects of screened Coulomb repulsion and attractive Van der Waals forces. For

numerical studies in the present article we crudely represent this effective

interaction by a two-body potential with an infinite repulsive hard core at an

interparticle separation r = 2R,, and outside the core a weak short range

156 R.B. JONES AND G.S. BURFIELD

attractive potential. We write it as

b(r) = += , r<2R,,

n

-NE 1

i ) r>2R,, r

(3.1)

where p = l/k,T, with k, Boltzmann’s constant and T the absolute tem-

perature, R is the structural or hydrodynamic radius of the particles (RP > R)

and for simplicity of computation n is an integer such that n 34. Our

numerical method can also deal with a potential which is a general polynomial

in (R/r) but we will use (3.1) for the illustrative numerical studies presented

later.

The hydrodynamic pair interactions”-“) are mediated by the suspending

fluid which is treated as a Newtonian fluid of shear viscosity 7~ described by

creeping motion hydrodynamics’“). The strength of these forces depends on the

boundary interaction of the colloid particle with the surrounding fluid. The

simplest model is to treat the particles as hard spheres (radius R) with stick

fluid boundary conditions at the surface. However for suspensions stabilized by

so-called steric forceslh) there is a coating of the colloid particles consisting of

long molecules which extend into the surrounding fluid so that the simple stick

boundary condition model may not be appropriate. Other fluid interaction

models could employ slip boundary conditions or could treat the particles as

being partly permeable. The strength of the hydrodynamic forces is expressed

through certain mobility functions. At moderate particle separations the

mobility functions may be expanded in series in powers of (R/r) while for very

close approach the series converge slowly and asymptotic expressions must be

used instead. The mobilities are known with high accuracy for all separations

only in the stick boundary condition model’“). However, the series represen-

tation has been derived for any boundary condition in a genera1 form by

Schmitz and Felderhof”.“). Because the hard core repulsion in our model

potential (3.1) prevents extremely close encounters and also because our

numerical technique exploits the nature of the series expansions we have used

the genera1 expressions of Schmitz and Felderhof to evaluate the mobilities in

series through terms of order (R/r)*“. In consequence we are able to study the

effect on yT of altering the boundary condition as well as the potential u(r). To summarize the mobility information’7a’8) we consider two freely rotating

identical spherical particles 1 and 2 acted on by forces F,, F2 respectively which

produce particle velocities Us, u2 related to the forces by

ul= CL:, - F, + p:: * F2 ,

u2 = pi, - F, + py2 - F, . (3.2)


If r denotes the separation of centres of the two particles then the mobility

tensors pi can be expressed in terms of scalar functions a;(r), pi(r) as

(3.3)

Schmitz and Felderhof’7V’s) h ave provided an elegant general expression for

(Y:;, fi:j which enables one to calculate their series expansions to all orders in

(R/r). These general expressions which are given as eqs. (5.6) and (5.7) of their

third paper”) are correct but the evaluation which they give through order

(R/r)” in eqs. (6.1), (6.2) and (6.12) is not correct. We have ascertained this by

use of the symbolic manipulation package MACSYMA to carry out the

algebraic expansion of their eqs. (5.6) and (5.7) through order (R/r)“. By

specializing to stick boundary conditions we were then able to check these

expansions for the coefficients (Y::, (.y:: against the expansion (again performed

using MACSYMA) of the same coefficients derived from the exact solution to

the two sphere problem due to Brenne?).

We may write the form of the series expansions required in the following

way, where we introduce the variable z = Rlr with R the radius of each of the

two particles involved in the interaction:

m

4~Ra~,(z) = 2 ayz*“, n=O

(3.4)

47r~RP;r(z) = c b;.z*“, n=O

cc

47rvRP::(z) = c b;‘z”‘+‘. n=O

We have calculated all terms in these series through zzo for both stick and slip

boundary conditions at the particle surfaces. The relevant expansion

coefficients a:, bz are listed in tables I and II. For the most general models of

the hydrodynamic interaction between a particle and the fluid a set of scatter-

ing coefficients measures the strength of each multipole coupling’8). We have

the expansion of the mobilities through order zzo for general scattering

coefficients stored as a LISP file in the MACSYMA system; if anyone has need

of such expressions they are available on request from one of the authors

(RBJ). The dilute limit diffusion coefficient Do introduced in section 2 is


TABLE I

Mobility expansion coefficients for stick boundary conditions

2 0

3

1 0

5 2

2 11

3 3

4 7

167 5

3 883

6 12

7 102

795 8

8

9 ~ 1141

18683 10 -

8

2

3

0

25

2

-5

131

L

341

3 915 -

1;7

x 8567

4

0

17

24 5

6 23

8 31

I

2 1

i

0

0

0

1X9

64 1439

3 96

689 4201

24 114497

1536 10715

48 77179

80 3313

24 152227

480

96

expressible in terms of one of these scattering coefficients, Ai, as

Do- kBT 45-7A; ’

(3.5)

where AZ is equal to 3R/2 for stick boundary conditions and to R for slip

boundary conditions.

4. Short time and long time contributions

In this section we present analytical expressions for yTS and yTL based on the

general formulae derived by us in an earlier paper’). The short time con-

tribution was given earlier by one of us”) in a form which we can write as


cbrn = n,c* 2

with

TABLE II Mobility expansion coefficients for slip boun-

dary conditions.

n at’ aA b” ” b’* n

4

5

6

1 1

0 0 -1 0

-1 2

-1 6

-5 14

- 22 38

- 78 122

- 262 420

- 883 1486

- 3016

1 1

z 0 0 0 0 1 8 0

1 ; 0

1 3

4 16 2 ? 1

5 139 - 16 40 59

10 96 31 1037

lo 40 11751

c, = - k2Do

g,(r)k . A(r) * k dr ,

704

(4.1)

(4.2)

where

go(r) = exp(-W(r)) (4.3)

is the lowest order virial expansion approximation to the pair correlation

function and where A(r) is a combination of mobility functions,

A(r) = 4wA;D”[ (P;,(r) - &) I+ (a::(r) - L%(r)) F] . (4.4)


Combining these expressions and performing the angular integrations in (4.2)

leads to the expression

Ts- Y - I s e~P”“)[47i&(~~1(z) + 2/?:,(z)) - 3) ,

z (4.5)

where we have changed variable from r to z = R/r and zP = R/2R,.

The long time or memory contribution arises as an additional term in the

projection operator treatment of the GSE8). The contribution to yT requires

the hydrodynamic limit of the memory term as in (2.4) but this is precisely

what we calculated in our earlier paper using a Laplace transform variable

s = -iw instead of the Fourier variable w. The result of that paper can be

quoted in the form

c#dl”yTL = -nod”qS)ls=“, (4.6)

where the species labels (Y, ,f3 refer to the tagged and untagged particles

respectively which are of course assumed identical in their interactions in the

present paper. The quantity d(s) = d”“(s) was expressed as

k2d(s) = j j dr dr'em"""'k * h(r)G(r, r’; .~)k - h(r)). (4.7)

The quantities G and h are defined in our earlier work’) but we will give their

definitions again here in terms of the mobility functions N::, p:; of section 3.

Thus h(r) can be written as

2h(r) = (V@(r))- T(r)- V- T(r), (4.8)

with T(r) a relative mobility tensor:

T(r) = 2[0”7 + A(r) - B(r)] . (4.9)

We have given A(r) already in (4.4) and B(r) is given by

B(r) = 4qA$“[P?z(r)f + (a;;(r)-P::(r))F] . (4.10)

From (4.4) (4.9) and (4.10) we express T(r) as


with

a(r) = 4mA&:‘,(r) - aY2(r)),

b(r) = 4wA%$(r) - &‘2(rN .

(4.11)

(4.12)

The quantity h(r) is written finally as

eSu(‘)r

h(r) = -c(r) - r ’

(4.13)

da(r) + 2(a(r) - b(r)) -

dr

+ a(r) cl t?“(‘) r I dr ’

The kernel G(r, r’; s) is a propagator describing the relative diffusion of a

pair of particles; it is a solution of the Green function equation*)

[S - e’“(‘) v- e-B”“‘T(r) * VjG(r, r’; s) = 6(r - r’), (4.14)

subject to the boundary conditions that the radial component of the relative

particle flux vanishes when the hard cores touch

r * T(r) - VG(r, r’; s) = 0, r = 2R,, (4.15)

and that G itself vanishes as r+ 00. The boundary conditions on G are

explained more fully in the appendix. As shown earlier’) we may expand G in

spherical harmonics and use the form of /z(r) to evaluate the angular integrals

in (4.7) leading to

d(s) = ($)’ j r2 dr / rr2 dr’ c(r)A,(r, r’; s) eP”“‘)c(r’), (4.16)

2R 2R

where A,(r, r’; s) is the only contribution from the harmonic expansion of G to

survive the angular integrations. To obtain y” then we need d(s) at s = 0 and

hence A,(r, r’; 0). The differential equation for A,(r, r’; 0) which follows from

the harmonic expansion of (4.14) is


= - X37D”r2a(r) 6(r - r’) (4.17)

The boundary conditions on A,(r, r’; 0) are that it vanishes as r+ = and, from

(4.19, that at r = 2R,

a(r)+$=O. (4.18)

Finally, we recall a reciprocity relation which G and A, each satisfy’).

empocr)A,(r, r’, s) = empI.““‘A1(r’, r; s), (4.19)

and which is helpful in simplifying the integrations in (4.16).

5. Construction of the kernel A,( r, r’; 0)

In this section we describe a method of solving eq. (4.17) for A,(r, r’; 0) which

depends upon two features of the interaction model described in section 3,

namely that the mobility functions a(r), b(r) are known to high order as series

in the variable z = R/r and that the potential /3u(r) is a polynomial in z. These

features enable us to construct A, explicitly by use of two independent

solutions to the homogeneous form of (4.17) which are obtained by the method

of Frobenius*‘). Throughout this section we express all functions of r as

functions of z = R/r.

The first requirement is to obtain two independent solutions to the homo-

geneous version of eq. (4.17), one of which is regular at z = 0 (r = 3~) the other

singular at this point. In terms of the variable z we write (4.17) as

d24 + f’(z) d4 / Q(z)4 3

dz2 z dz zZ &rD”Ra(z) qz - z’) ,

with P(z), Q(z) given as

P(z)= z 1 da(z) dpv(z)

____-___ a(z) dz I dz *

(5.1)

(5.2)


20) Q(z)= -a(z). (5.3)

For our model potentials /?u(z) both P(z) and Q(z) are regular at z = 0 so we

express them in series form as

P(z) = 2 pnzn+l ) Q(z)= 2 q,,z". (5.4) n=O n=O

The indicial equation for the homogeneous form of (5.1) has two integer

roots, 2 and -1. However, if @(z) has no power of z of degree lower than .z4,

then the coefficients in the series expansions of a(z), b(z) are such (for both

stick and slip boundary conditions) that the second solution corresponding to

the root -1 does not have a logarithmic singularity at z = 0. If v(z) has terms

in z, z2 or z3 then such logarithmic terms are present, but for the present study

we have restricted /?v(z) so that the simpler case holds. We thus can find two

independent solutions fr(z), fz(z),

f,(z) = c ffizn+2 3

n=O

f2(z) = i + i: f’,z”_’ . n=3

(5.5)

The coefficients fi,fi are found by recursion formulae in terms of the

coefficients p,, q, in (5.4)

f+ -

k(k _ 3) - & 5 [@ - l)Pk-l-m + qk-,lf ,i > ka4. (5.6) m 3

The evaluation of the f: begins at k = 1 using as input f:, = 1 while for the f: one begins at k = 4 using as input fz = -1 for stick boundary conditions and

f: = 0 for slip boundary conditions. The detailed mobility series given in (3.4)

and tables I and II enable us to compute the f:, fi explicitly through k = 20.

The two independent solutions f,(z), f*(z) are

Wronskian has the simple form (f’(z) = df(z)/dz)

normalized so that their

(5.7)


For the model interactions of this paper we can now express A,(z, z'; 0)

explicitly in terms of ft(z),fJz). We can do this because the hard core

repulsion prevents any two particles from touching at separation r = 2R. If two

spherical particles do touch at this separation (z = f) then for both stick and

slip boundary conditions the mobility function u(z) vanishes and hence the

vanishing flux boundary condition (4.18) cannot be satisfied by a linear com-

bination of ft(z) and f2(z); rather an asymptotic solution of (5.1) around the

point z = i is necessary such as was carried out by Batchelor and Wen’) for

stick boundary conditions. Unfortunately it is not possible at the moment to

carry out such asymptotic analysis for general boundary conditions. However

if, because of hard core repulsion, the nearest approach is at zP = R/2R, < l,

then a(~,) + 0 and the no-flux condition (4.18) simply requires that dA,/dz

should vanish at zP which can be satisfied by a linear combination off, and fz.

Thus we can express A,(z, z'; 0) in our model as

A,(z, z’; 0) = 8nR$Q!,(z 1 P

) IfXzplfi(--‘) -fX~,lf’~(z’)l e

,‘,(ZI) e-arc:‘)

Ah 2’; 0) = If;(~~)f~(z)-f;(z~)fl(~)I xTRD,)f,(_

1 iP ),

In our numerical calculations we used the MACSYMA system (with infinite

TABLE III

Coefficients p,,, q. for stick boundary conditions

/3v(:‘, , z<z’.

(5.X)

z ;> z’

n _ p*lp, - 2”q, n ~ 2”ilIs, 2”qn

4

5 753 423 16 14828744703

6 16383 945 17 53671276545

7 38913 5919

8 135615 18689

9 1068289 78687

10 2141439 342113

3

9 3

225

783

2

3 9

3

105

II

12 13 14

15

18

19

20

14263041

53410815 188622849 937729023

3217653761

244101316607

878379270145

1128383

5208897 18482239 76194113

1534271931

5

5789611973

5 24683002939

5 91974000069

5 77580683071

1486826970053

5


TABLE IV

Coefficients a, 9” for slip boundary conditions.

n -a - 9” n -Lxl - 9n

0 1 2 11

1 1 1 12

2 1 1 13

3 5 1 14

4 6 3 15

5 13 4 16

29 6 29

4 17

53 7 45 18

4 433

8 109 5 19

833 9

186 20 2o 357

10 397 7

5371 761

40 68249

1496 280

126041 3025 -

280 5926

5796 7

436687 11869 -

280 5007529

22764 - 1680

1858481 46417 -

336 17853043

89928 ~ 1680

33479413 181670 ~

1680 1416267959

36960

precision arithmetic) to generate the input coefficients p,, q,, and then to solve

the recursion formulae (5.6) out to k = 20. The same calculation is possible

with careful use of double precision floating point arithmetic; however to save

computation for anyone who might like to use our model but with a different

potential @(z) we list in tables III and IV the contributions p,,, q, which arise

from the mobility functions alone for IZ = 0, 1, . . . ,20. By Q,, we mean the

partial contribution to (5.2) defined by

6. Reduction to quadratures

(5.9)

By further analysis of expressions (4.16) and (5.8) we can reduce d(0) to a

quantity given by integrations over explicit combinations of the known func-

I66 R.B. JONES AND G.S. BURFIF.LD

tions a(z), j%(z), ft(z), f*(z). The first step follows from the fact that the scalar

function c(r) in (4.13) vanishes inside the hard core separation, r < 2R,, and

possesses a delta function singularity at r = 2R, arising from the derivative of

emSo at that point. We split off this delta function explicitly as

c(r) = D”a(2RJ exp(-/Sv,)6(r - 2RJ + (:(r) , (6.1)

where ,CIv, = lim,_ZR ; ,Bv(r) and

I 0 . 2R<r<2R,.

C(r) = da(r) + 2(a(r) - b(r)) _ N(rj d@(r)

dr r

On insertion of the decomposition (6.1) in the double integral (4.16) four

contributions arise. The reciprocity relation (4.19) implies that the integrand of

(4.16) is symmetric in r, r’ so that the two cross terms in 6(r ~ 2R,) and C(r) can

be combined into one and the double integral can be simplified to give

d(O) = ~~~2i(2R,)zu”n(2R,),’ em(‘(Pp)A1(2Rpr 2R,; 0)

‘[(2R,)‘D”rr(2R,)] em(BL‘n) j dr’ A1(2R,, rl; 0) ,a~q.~~,(,~)

2Kp -, ,

dr r2?(r) dr’ A,(r, r’; 0) eB”““r’2C(r’) (6.3)

ZRp 2Rv

By using the explicit form of A, given in (5.8) together with the Wronskian

(5.7) we get from (4.6) and (6.3) the following result for yTL:

Y TL

1 dz -- 3 Te I

-p”“‘u(z)f2(z) I$ e-a”c*“u(z’)f,(z’)

0 0

i”

+ f;(z,) dz

6f;(zJ Te II 0

(6.4)


Here zP = R/2R, and we use u(z) to denote

da(z) u(z) = -2- WV(z)

dz + 2z[a(z) - b(z)] + z2a(z) 7 .

Having already constructed fi(z), f2(z) explicitly as power series we see that yTL is reduced to quadratures.

7. Numerical results and conclusions

The only other comparable detailed numerical study of yT of which we are aware is due to Batchelor4,6*7) and to Batchelor and Wen’) where the derivation of expressions for yT rested on the ideas of non-equilibrium thermodynamics. It is pleasing to note that the formal expressions given here in section 4 and based on our earlier treatment’) of the GSE can be identified precisely with

Batchelor’s result6’7). To sketch briefly how the connection is made between the two formalisms we note that Batchelor6) expresses DT as

DT = D”(l + K’c$) , (7.1)

with

K, = c$G) + ~41) + y(B) . (7.2)

After specializing Batchelor’s expressions to the case when the tagged particle is identical with the untagged ones and making allowance for the slightly different expression of mobilities one sees that S’(‘) (eq. (3.6) of Batchelor6)) is identical with our yTs in (4.5) above. The terms S’(‘) and SrCB) (eqs. (3.8), (3.9) of

Batchelor6)) are given as integrals over products of functions which involve either the mobilities or the potential pv(r) with a function Q’(r/R) which satisfies a differential equation given in the appendix of that paper6). We observe that in our language Q’(s) (where s = r/R is Batchelor’s variable6)) is simply

m

87TR2 Q’(s) = y-- I ds’ A,(s, s’; 0) es”(s’)s’2c(s’) . (7.3)

Our expression for yTL involves d(0) (see (4.6) and (4.16)) in which the function c(r) occurs quadratically. Upon absorbing one factor of c(r) as in (7.3) to define


Q’ and on writing the other factor of c(r) as the sum of a part involving

mobilities and a part involving the derivative of flu(r) we obtain two con-

tributions to yTL which are respectively the same as SICB’ and S”” so that

Y n = S’PJ’ + S”“,

Batchelor and Wen’) made extensive numerical studies of a class of

potentials flu(r) more realistic than our (3.1) and using a stick boundary

condition model of the hydrodynamic interactions. They studied the regime in

which the hard core repulsion is confined to a very narrow region around the

particles corresponding to 2R < 2R, < 2.2R. In contrast we have restricted our

calculations to the regime 2.2R < 2R, in order that our mobility series give

high precision numerical results. Indeed, with stick conditions our series of

section 3 are accurate to better than one percent down to a separation r = 2.1R

but they have appreciable errors as r approaches 2R. For illustrative purposes

we present numerical results based on a choice of two different attractive

potentials corresponding to the values n = 4 and it = 6 in (3.1); in addition we

study both stick and slip hydrodynamic boundary conditions. In the results

described below we have varied the hard core radius R, in the range 2.2R <

2R, < 3.OR.

In fig. 1 we present graphs of yT, yTS, y’rL for both sets of boundary

conditions but for the case of no potential interaction outside the hard core, i.e.

N = 0 in (3.1). In fig. 2 we present similar graphs but now with an attractive

potential corresponding to n = 6 in (3.1) and with N chosen so that flu(r) = -0.5 at r = 2.2R. For such a potential there is a strong attraction between the

particles when r = 2.2R but not strong enough in relation to the thermal energy

to actually blind the pair as a doublet. As 2R, is varied out to 3R with N fixed

the strength of the potential is greatly reduced. We see that figs. 1 and 2 are

qualitatively similar with the attractive potential tail serving to reduce yT and

yTL by a few percent while increasing -yTS slightly with respect to the case of a

pure hard core without an attractive tail. The most striking feature of both sets

of curves is the contrast between stick and slip boundary conditions. In the

regime 2.2R < 2R, < 2.5R we have /yTSI > lyTLl for stick conditions while the

inequality is reversed for slip conditions. In this regime we find that /yT/ is

smaller for slip conditions than for stick conditions while for 2.8R < 2R, this

relation is reversed. For both sets of boundary conditions yT’, changes much

more rapidly than does yTS as R, is varied. To show that the details of the

potential have only a small influence we give in table V some results for the

II = 6 potential used in fig. 2 together with results for an 12 = 4 potential (again

normalized so that @(r) = -0.5 at r = 2.2R) both with stick boundary con-

ditions. These results show only small changes in yT, yTS, yn-. We cannot

directly compare our numerical results with those of Batchelor and Wen’)

because we are unable to explore reliably the region 2R, < 2.2R where most of


-4.0

-3.0

-2.0

-1.0

2.5 3.0 2.5 3.0

2Rp'R 2Rp/R

Fig. 1. Fig. 2.

Fig. 1. Plots of yTs, yTL and yT (denoted S, L and T) for the case of a pure hard core repulsion with

R, in the range 2.2R <2Rp<3.0R. Solid lines correspond to stick and broken lines to slip

hydrodynamic boundary conditions.

Fig. 2. Plots of yTs, y” and yT (denoted S, L and T) for hard core repulsion together with a short

range attractive tail of the form flu(r) = -IV(R/~)~. The potential is normalized so that pv(2.2R) =

-0.5. As in fig. 1 solid lines denote stick and broken lines slip boundary conditions; the hard core

again lies in the range 2.2R < 2R, < 3.OR.

TABLE V Comparison of two different potentials. The constants N,, N2 are chosen so

that for both potentials @(z) = -0.50 when z = l/2.2.

flu(z) = -N,z4 flu(z) = -Nzz6

2R,lR yTs Yn YT Y TS

Y TL

Y’

2.2 -1.82 -0.40 -2.22 -1.77 -0.40 -2.17

2.3 -1.70 -0.56 -2.26 -1.65 -0.58 -2.23 2.4 -1.61 -0.74 -2.35 -1.56 -0.78 -2.34 2.5 - 1.53 -0.96 -2.49 -1.49 -1.01 -2.50 2.6 -1.46 -1.20 -2.66 -1.42 -1.27 -2.69 2.7 -1.40 -1.47 -2.87 -1.36 -1.55 -2.91 2.8 -1.34 -1.77 -3.11 -1.31 -1.86 -3.17 2.9 -1.29 -2.10 -3.39 -1.26 -2.20 -3.46 3.0 -1.24 -2.47 -3.71 -1.22 -2.58 -3.80


their calculations were performed. However, in the limiting case R, = R with

no attractive tail to the potential they report yT = -2.10. One can see from fig.

1 that this value is consistent with an extrapolation of our results for stick

boundary conditions.

Our two hydrodynamic models represent the extremes of strong and weak

hydrodynamic forces. Our calculations show the relative magnitude of yrs and

YTL to be strongly dependent upon the strength of these forces so that a

separate measurement of each of these components as in light scattering will

give a sensitive probe of the continuum hydrodynamic model used in suspen-

sion studies. A second conclusion is that so long as the attractive short range

tail does not bind the particles it is the hard core repulsive force which gives

the dominant contribution to yTL and moreover for 2.8R < 2R, yT’ is much

larger in magnitude than yTS. In future studies one can readily extend the series

methods of section 5 to include terms in the potential of order (R/r)’ and (R/r)’ to see the effect of longer range attractive potential interactions.

Acknowledgements

One of us (RBJ) wishes to thank Dr. Glenn Sowell for his patient and

effective instruction in the use of the MACSYMA system and Dr. Paul Kyberd

for his help in running the MACSYMA package.

Appendix

In order to see how the boundary conditions given in section 4 for the

propagator G(r, r’; s) arise we recall”) that G is the Laplace transform of a

time-dependent function G,

G(r, r’; s) = em”‘G(r, r’; t) dt . (A.11

The Green function eq. (4.14) becomes in the time domain the differential

equation

aG(r, r’; t)

at - epu(r)V- e--PL.(‘)T(r) - V($(r, r’; t) = 0 (A-2)

subject to an initial condition


(A.3) G(r, r’; t)J,=, = 6(r - r’) .

If we introduce a function

g(r, r’; t) = e -BwG(r, #A; f) ePw)

then (A.2) and (A.3) become

ag(r, r’; t)

at - V - emBu( - V eB”“‘g(r, r’; t) = 0 .

(A.4)

(A.3

g(r, r’; t)l,,o = 6(r - r’) . (‘4.6)

However, (A.5) is the Smoluchowski equation for the relative motion of the particle pair and has the form of an equation of continuity,

ag(r, r’; t)

at + V * i(r, r’; t) = 0 , (A.7)

with a current density

j(r, r’; t) = -e-P”(‘)T(r) - VeP”“)g(r, r’; t)

= -TV Vg- T.(Vpu(r))g. (A.@

The initial condition (A.6) shows that g(r, r’; t) is simply the conditional probability that the particle pair are separated by r at time t given that they were separated by r’ at time t = 0. The current J describes the flow of this conditional probability. Thus both g and 3 must vanish within the hard core

excluded region r < 2R,. We can express j(r, r’; s) in terms of G(r, r’; s) by use of (A.1) (A.4) and (AS)

m

j(r, r’; s) = I

e-“‘i(r, r’; t) dt = -empvcr)T(r) - VG(r, r’; s) e@(“) . (A.9) 0

The vanishing of the radial probability flux r-j at the hard core separation is, in terms of G(r, r’; s), just the boundary condition (4.15). This boundary condition was incorrectly given in the appendix to our earlier paper’).

A more general model of the potential interactions than the one described here would allow finite jump discontinuities in @(r). One of us has studied’) a

model (without hydrodynamic interactions) with a continuous potential for


which A,(r, r’; 0) can be obtained exactly in closed analytic form and for which

as a limiting case we obtain a flu(r) with finite jump discontinuities. The

appropriate conditions on G at such a point are that the combination

G(r, r’; s) eB”(“) is continuous in both r and r’ while the flux j(, r’; s) is

continuous in r implying by (A.9) that VG(r, r’; s) has also a jump discontinuity.

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