Mobility matrix for arbitrary spherical particles in solution

Physica 149A (1988) 373-394 North-Holland, Amsterdam

MOBILITY MATRIX FOR ARBITRARY SPHERICAL PARTICLES

IN SOLUTION

R.B. JONES

Department of Physics, Queen Mary College, Mile End Road, London El 4NS, England

R. SCHMITZ

Institut fiir Theoretische Physik A, R. W. T. H. Aachen, Templergraben 55, 5100 Aachen Fed. Rep. Germany

Received 26 October 1987

We review the theory of the extended mobility matrix for N arbitrary, spherically symmetric particles immersed in an incompressible fluid. The two-particle mobility functions can be evaluated to any desired order in the inverse interparticle distance by means of an algebraic computer program implementing exact recursion relations. We correct some earlier published expressions and summarize known results for the scattering coefficients which characterize the hydrodynamic properties of the particles. Explicit results are presented for stick and slip hard spheres, for permeable spheres and for fluid droplets.

1. Introduction

In colloidal physics of nondilute suspensions one is often faced with the following computational problem: Given all the forces and torques on the suspended particles and given also a linear fluid velocity at infinity, calculate the translational and rotational velocities of the particles and the dipole moments of the force densities exerted by them on the fluid. Such calculations are essential to determine the concentration dependence of transport quantities in colloidal suspensions, such as effective translational and orientational fric- tion or diffusion coefficients or the effective viscosity. In linear systems (low Reynolds numbers), the answer to the problem can be expressed in terms of a mobility matrix. Its evaluation is very complicated, since the hydrodynamic interactions between all the particles are involved.

What has mainly been studied in the literature is a submatrix of our mobility

0378-4371/ 88 / $03.50 0 Elsevier Science Publishers B S? (North-Holland Physics Publishing Division)

314 R.B. JONES AND R. SCHMITZ

matrix, which expresses the translational and rotational velocities of the particles in terms of the forces and torques acting on them for vanishing flow at infinity. Moreover, attention has usually been restricted to the case that the particles are hard spheres with stick boundary conditions. In this respect we mention here only two recent papers. Mazur and van Saarloos’) have considered an arbitrary number of spheres and evaluated asymptotic results by expanding in a power series in R-I, where R is a typical interparticle distance. They have gone up to order R-‘, for which clusters of up to four spheres contribute. Jeffrey and Onishi*) have carried out extensive and very accurate numerical work on the two-sphere problem, which covers the whole range of R-values.

The results of Jeffrey and Onishi demonstrate that the convergence of the R-‘-expansion, which is good for large distances, becomes very poor when the particles come close. The reason for this behavior is known from lubrication theory, which implies that the mobility matrix exhibits cusp-like singularities (finite values and slopes, but infinite second derivatives) when the particles touch. Nevertheless, whenever integrals over the mobility functions extending over the whole range of R-values are required in calculations for colloidal systems, it is often sufficient to use the R-’ -series representation truncated at about the order Rm*“. On the other hand, when the derivatives of the mobility functions get involved in such integrals, more care is needed, since the effect of the cusp singularities may be badly underestimated by the truncated series. This complication arises, for example, in the problem of the low frequency effective diffusion and viscosity coefficients, whereas the corresponding high frequency coefficients can more easily be evaluated with the aid of the truncated series expansion.

In previous work, one of us (R.S.) and B.U. Felderhof3-‘) have developed on algebraic recursion scheme for an extension of the two-particle mobility matrix by including a linear imposed flow and the force dipole moment terms. Extending the mobility matrix is indispensable for the computation of the effective viscosity in non-dilute colloidal suspensions. The theory is applicable not only for hard spheres with stick boundary conditions, but for any spherically symmetric particles, like hard spheres with partial or perfect slip, penetrable spheres or fluid droplets - to mention only a few examples. In some systems these models might be more realistic in describing the properties of the suspended particles. Independent computations of the extended mobility matrix for two hard spheres with stick boundary conditions have later been published by Kim and Mifflin6).

The method used in refs. 3-5 is based on an expansion of the two-particle flow in terms of complete systems of free flow solutions and the determination of the expansion coefficients by means of a hydrodynamic scattering theory.

MOBILITY MATRIX FOR SPHERICAL PARTICLES 375

Upon iterating the recursion equations one obtains an expansion of the

mobility matrix in powers of R-‘. This approach differs from the commonly used method of reflections7”) or other techniques’) in that it avoids tensor operations which become more and more tedious as the rank of the tensors involved increases. Thus one is able to compute coefficients of relatively high order in R-’ by hand. General results, valid for arbitrary spherically symmetric particles up to the order R-12, have been published before5).

One of us (R.B.J.) has written a computer program, using the SMP package for algebraic manipulations, which performs the iterations up to any desired order in R-‘. By running the program, we have discovered that some of the earlier published expressions’) contain errors. The correction of these errors is one purpose of this paper (section 4). Since for arbitrary spherical particles the complexity of the expressions increases rapidly, we will not publish here any new terms of orders higher than R-12.

Secondly, we will present here tables with explicit results up to order R-2o for several models of two identical particles, namely for hard spheres with mixed slip-stick boundary conditions, for uniformly permeable spheres and for immiscible fluid droplets (section 6). Thirdly, we take the occasion to review the general theory of the N-particle mobility matrix (sections 2 and 3) and to summarize known results for the scattering coefficients, which are involved in the general recursion scheme and which characterize how an arbitrary imposed flow is modified in the presence of one particle (section 5).

We expect that our methods can straightforwardly be generalized to more than two particles.

2. Basic equations

We consider N rigid, non-overlapping, spherically symmetric particles with radii a,, . . . , a,,, immersed in an isothermal and incompressible fluid, and we restrict ourselves to stationary, low-Reynolds number flows. The fluid velocity u(r) follows then from the quasistatic, linear Stokes (or creeping flow) equations’)

$‘2v-Vp=-F, V-v=Q,

where r] is the shear viscosity, p(r) is the pressure and

(2.1)

(2.2)

is the force density exerted by the particles on the fluid. The Fi(r) vanish for


Ir - Rj( > a, and will be specified below. The rigid particle by the velocity field

motion is described

(2.3)

where U,, . . . , U, and a,, . . . , Q, denote the stationary translational and rotational velocities of the particles, respectively. In (2.3) and the following, the positions R, , . . . , R, of the particle centres are assumed to be given. They can be chosen independently from the particle velocities, since our theory is restricted to linear order in the velocities. The step functions in (2.3), defined by 0(x) = 1 for x 2 0 and e(x) = 0 for x < 0, localize u(r) to the region occupied by the particles.

To close the equations, we have still to specify the force densities induced by the particles. This is done by constiturive equations which relate the Fj(r) to the velocity fields. Our theory holds for all constitutive equations which are linear in u(r) - u(r), and which account for the spherical symmetry of the particles. Each particle j may obey a different constitutive equation. We mention just three examples for which we shall present explicit results below.

The most commonly discussed example is a hard sphere with stick boundary conditions. This model can be generalized by allowing for a slip parameter hj with dimension length that can take values between zero (perfect stick) and infinity (perfect slip). The boundary conditions read then”)

n, - (u - U,) = 0

[

h 1 (b-R,1 =a,) > (2.4)

(1 - n,nj) * v-Uj-ajf2jXnj-~a~nj =0 rl

where nj = (r- Rj)l(r- R,( is a unit vector normal to the surface of particle j and u = 277(Vu)” - pl is the stress tensor of the fluid, the superscript s denoting the symmetrized tensor part*. In this case the induced force density can be chosen as lo)

F,(r) = [6(/r - Rjl - a,)1 + aj,$j6’((r - Rj( - a,)(1 - njnj)] *f;.(nj) , (2Sa)

where cj = Ajl(aj + 3A,) is a dimensionless slip parameter andJ(nj) is a surface

* For any second rank tensor T, the components of T” are T& = 4 (Tap + Tpa ).


force density given by

nj $(nJ = -[yzj: vu - p](+ + ) )

(l - njnj)sfi(nj)= - & (1 - njtZj)'U(Uj + )' flj *

(2.5b)

I

The argument (aj + ) on the r.h.s. of (2.5b) indicates that the quantities are to

be evaluated in the limit jr - Rj( + aj f .

Another example is a penetrable particle for which

Fj(r) = -hj(lr - RjJ)[u(r) - u(r)]e(uj - It - Rjl) , (2.6)

where hj(lr - Rjl) is the inverse permeability”). More generally, we could consider

Fj(r) = - 1 Aj(r - R,; r’ - Rj) - [u(r’) - u(r’)] dr’ , (2.7)

where Aj(r, r’) is a symmetric, rotationally invariant kernel that vanishes for r, r’ > ai.

Our last example is a spherical, incompressible fluid droplet that is immiscible to the surrounding fluid. We impose as a constraint that the surface of the droplet moves with rigid body velocity. This behavior corresponds to high surface tension. The immiscibility boundary conditions are 12)

nj * (u - Vi) = 0 (Ir-Rjl=aj+ and (r-Rjl=aj-). W)

In addition, (1 - njnj) * u and (1 - njnj) * u 9 nj are continuous on the particle surface. The normal-normal component of the stress tensor will then have a discontinuity. For this model, the force density is found to be

Fj(r) = V* S,(r) - rjnj6( (r - Rjl - uj) , (2.9a)

where

sj = 2(~~ - ~)(Vu)%(uj - Ir - Rjl) (2.9b)

and


Here, qj is the shear viscosity inside the particle. We could even consider radial

functions 77, = ni(jr - Rjl).

Notice that u(r) in eqs. (2.5) and (2.9) can be replaced by u(r) - U(T),

because a rigid body motion does not contribute there. Hence these constitu-

tive equations can be considered linear in u(r) - u(r), which is one of the

requirements of our theory, as was mentioned before.

3. Grand mobility matrix

Suppose now that an incident flow U”(T) is given by applying suitable forces

or boundary conditions at infinity. u,(r) is then a solution of (2.1) in the

absence of the particles, i.e. for F(r) = 0. Suppose furthermore that a set of

external forces 9 = (9,) . . , SN) and external torques 9 = (.T, , . , &,) is

acting on the particles.

forces and torques are

density. Thus one has

Due to stationarity of the particle motion, the external

balanced by the lowest moments of the induced force

q. = 1 (r - R,) x F,(r) dr

(i=l,...,N). (3.1)

Given 9, 9 and uo(r), it can be shown5) that the fluid velocity u(r) in the

presence of the particles, which tends to U”(T) at infinity, and the set of particle

velocities V = (V,, . , V,) and 0 = (a,, . . , 0,) are uniquely determined.

From these one can find the induced force density F(r) exerted by the particles

on the fluid.

For applications in colloidal systems, where one is mainly interested in

length scales large compared to the particle sizes, it is convenient to use a

multipole representation, in which the linear relationship between the above

quantities is expressed via a grand mobility matrix p(R, . , Rv). Thus one

writes5)

In (3.2), un = (%(R,), . . , u,,(R,,,)), LC)” = (o,(R,), . . , t.O$tN)) and g,, =

(%(R, ), . . . 3 %@iv)) are the incident flow, vorticity and strain, respectively,


evaluated at the particle centres. The latter quantities are defined by

o,=~Vxu,, 90 = (VUOY . (3.3)

Furthermore sC2’) = (%y’), . . . ,91$“‘) is the set of symmetrized dipole mo-

ments of the induced force density, given by

91’“’ = j- [(r - Rj)Fj(r)]” dr (j = 1, . . . , N) . (3.4)

The dots on the r.h.s. of (3.2) denote higher derivatives of the incident flow, while those on the 1.h.s. denote higher multipole moments of the induced force density. These higher orders are usually of less interest and will not be discussed here. The part of the grand mobility matrix, which is explicitly indicated in (3.2), will henceforth be called the mobility matrix. The super- cripts t, r and d characterizing the elements of the mobility matrix refer to translation, rotation and dipole, respectively. It may be noted that these elements have different dimensions.

The mobility matrix has a number of symmetry properties that shall be summarized now. To illustrate our notation for the tensor components of the matrix elements, we give two examples. Thus pi;_2 connects the a-component

of the velocity difference at the centre of particle i, i.e. uo,(Ri) - Uia, with the p-component of the torque acting on particle j, i.e. with Yj.,. Similarly, p:BP connects $zi) with gjW. From the properties of 9@) and go follows*: u

k$@” 7 cL:p,,, symmetric and traceless in (II, V) ,

CL6;pw 9 F&, symmetric and traceless in ((u, p) , (3.5)

G,@” symmetric and traceless in (a, p) and (CL, V) .

Using the Lorentz reciprocal theorem14), one can furthermore show the following symmetry relations:

(3.6)

Next we use the additivity of the induced force densities (cf. eq. (2.2)) to

* Here we use that the trace of 9Fy”’ does not contribute to the flow outside particle j I’).


expand the mobility matrix in terms of contributions arising from clusters of 1,2,. . . , N particles15):

P(R,,..., RN) = i M(R,) + 2 2 M(R,, R,) + . . . . (3.7) &=I k=l I=1

(lfk)

In (3.7), the dots indicate terms involving clusters of three and more particles, which play a role in third and higher order virial coefficients in dense suspensions. Some of the three- and four-particle terms have been evaluated by van Saarloos and Mazur’). We will restrict ourselves here to the one- and two-particle terms. These are of the form

M,(R,) = Mi(Ri)SijGjk (3.8)

and

Mij@kT Rl) = Mij(‘i, Rl)Sik (j=i), M,(R,, R,)6ik6jI (j # i) . (3.9)

To determine the one-particle mobility matrices M,(R,) one has to study the case that particle i is alone in the fluid. To simplify the notation, we take i = 1. From translational and rotational invariance follows that M, is independent of R, and diagonal. Using the Fax&r theorems for the force, the torque and the symmetrized force dipole moment exerted on the fluid by a spherically symmetric particle 16), one obtains immediately

the

MyO, = (4~4,S,,)-‘~,~ ,

MY, = WW~~,,)-~~,~ > (3.10)

m Cap) Mtd ’ CPU)

d&V = $ VA& %,$?v 1

other elements being zero. The symbol --I (a~) in the last equation

denotes a projection onto the symmetric and traceless part in the index pair (@), explicitly

-I(4) I

%, 8,” ‘(pv)= +<s,,s,, + S,,S,,) - &,s,, . (3.11)

Finally, the quantities A:,, , AT,, and As,, appearing in (3.10) are the first few of an infinite set of scattering coefficients of particle 1, which characterize how an arbitrary imposed flow is modified in the presence of the particle (see section

5). To determine the two-particle mobility matrices M,,(R,, Rj) and M,(R,, Rj),

one has to consider the case that particles i and j are alone in the fluid. Taking


i = 1 and i = 1,2 it follows from translational and rotational invariance that the

tensors MIj depend only on R = R, - R,, namely in the following manner:

M;=@(R) = a;(R)k,i, + P;(R)@,,, - k&J,

M;yaJR) = a;;(R)ia I I I I i?,i?” +)+ /3;;(R) (a,,, - i?,R,)k, (“),

M;;JR) = P ;;(R)k &,a,, 9

M;iap(R) = a;;(R)&?, + ,c3ff(R)(Gap - R,k,) ,

I M;;_+,(R) = #(R)i?, q,,&,

-I (‘“),

I 0 I I M;;eB,(R) = cr;;(R) ii,&, (aB)ffr + ,6;;(R) B,(SB,, - ff,i?J (as’,

(3.12)

M;;ae,(R) = P;;(R) , I I?, F,+~ caak, ,

I -l M;,!,v(R) = a;;(R) ri,R,

1 # cap) &I?, (P)

‘(4)

+/3;,!(R) [I

I , I I 6,+8pff, (““)- R,li, (up) R,ff, (pv’

’ 1 + d;(R) [ ‘(@S) 1 Cd)

I , I I I x ‘~&ipv’~~y)- 2 SJQ?, ~rY~S &qi, (@) R,R,‘cCY’ I

)

where R = IRI, 8, = R,IR, and the summation convention is implied. The projected tensors in (3.12) read explicitly

(3.13)


The symmetry relations (3.6) imply that the scalar mobility functions in (3.12) are related by

(3.14)

In view of (3.12) and (3.14), each of the two-particle mobility matrices M,,(R)

and MIZ(R) can be expressed in terms of eleven independent mobility functions, which depend in a complicated way on the interparticle distance R.

4. Two-particle mobility functions

Since the mobility functions are very complicated, it is in general only possible to derive asymptotic expansions, either for widely separated (R-+ x) or nearly touching (R -+ a, + a2) particles. We have evaluated the coefficients in the large distance expansions up to the order R-l’, which provides sufficient accuracy for a number of applications, as was pointed out in the introduction (see also section 6). In the calculations we have applied the SMP algebraic manipulation package to the general recursion scheme presented earlier’-‘) in order to iterate the equations algrebraically. We are therefore in the position to calculate all the mobility functions for any two spherically symmetric particles up to any desired order in R-l in a reliable way.

Since for arbitrary spherical particles the complexity of the expressions increases rapidly, we will not publish any general results here. For expressions up to order R-l2 we refer to eqs. (6.1)-(6.22) in ref. 5 *. For quick reference we shall briefly repeat the notation used there.

The coefficients in the R-’ -expansion of the two-particle mobility functions are expressed in terms of the scattering coefficients A,“,,, B,:,j (n = 0, 1,2, . . .),

Az,j (n = 1,2, . . .) and B,‘,, (n = 2,3, . . .) of the particles j = 1,2. Further-

* The formulae presented there contain also the one-particle contributions, i.e. the terms that

are independent of R.


more, the abbreviations

are used. As has been said before, the scattering coefficients characterize how an arbitrary imposed flow is modified in the presence of a particle. In the next section, we will write down their values for a few exactly solved models.

Unfortunately, by running the program we have discovered that four of the twenty-two expressions given in ref. 5 contain errors. The affected mobility functions are a::(R), /3::(R), a::(R) and P::(R), given by eqs. (6.1), (6.2),

(6.12) and (6.16) in ref. 5. The following corrections should be made:

1) In eq. (6.1) f or a::(R), the three terms proportional to StAz,, (n =

1,2,3) should be multiplied by a factor 2. Thus, in lines 3, 5 and 6 it should read

&SiA”,, , &$A:,, and %S:Ai,, .

2) In eq. (6.2) for P::(R), the three terms proportional to B,“,2 (n = 2,3,4) should be multiplied by a factor 2. Thus, in lines 2, 4 and 6 it should read

3) The last term in the 6th line of eq. (6.12) for a::(R) should read

4) The last term in the 6th line of eq. (6.16) for P;(R) should read

We also remark that the general expression for P;:(R), as quoted in the last formulae in eq. (5.10) of ref. 5, has the wrong sign.

5. Scattering coefficients

The one-particle scattering coefficients AZ, j, B,“,j (n = 0, 1,2, . . .), Ai,j (n=l,2,. . .) and BE,i (n =2,3,. . .) are determined by the constitutive equation for the induced force density Fj(r). To compute them, one decom- poses an arbitrary incident flow into irreducible pieces (denoted by the indices nS, nT and nP) and calculates the scattered flow corresponding to each of


these irreducible flows”). This leads to ordinary differential equations from which the scattering coefficients can be evaluated either numerically, or - in a few cases - exactly. We now summarize the results for three models that can be solved exactly. For brevity, we will omit the particle index j.

First, for hard spheres with mixed slip-stick boundary conditions (cf. eq.

(2.4)) one obtains”)

- BS _ zn + 1 1 35 a2n+3 n 2 1+2nt ’

AT= 1-(n+2)t 2n+l

n l+(n-I)[’ ’

BP’2n-1 l-55 ?A+1

” 2 1 + 2(n - 2)5 a ’

(5.1)

where the slip parameter 5 has been defined below eq. (2.5a). Second, for a uniformly permeable sphere (eq. (2.6) with hj = const) one

finds’) *

A~=2n+3 EL+~(~) 1+ (n + 1)(2n + I)(2n + 3) g,+,(x) -‘azn+l ”

--a 2 I (n +2)x2 I g,-*(x) ’

By = 1 + Vn + Wn + 3) n [ (n + 2)X2 1 a2A,S _ a2n+3 )

(5.2)

2n + 1 B,P=2n_3 ‘+ [

2(2n - 3)(2n - 1) nx* I

a2B~ nm2-a

2n+l ,

where

(5.3)

and g,(x) = ~zn+l,2 (x) is a modified spherical Bessel-function of the first kind18). In the limit A+ 00 these results become identical to the expressions (5.1) for hard spheres with stick boundary conditions ( 5 = 0).

* The expression for At given in eq. (4.3) of ref. 8 contains a misprint: in the first line, the factor

(n + 1) should be replaced by (n + 2).


Finally, in the case of a fluid droplet (eq. (2.8)) one finds after a straightfor-

ward calculation

A; =

B,S=

A; =

B: =

where

2n + 3 + 2v

2(1 f v) a

zn+l

’

zn+l 2n+3

2(1+v)a ’

(n - l)(l - v)

(n - 1) + (n + 2)v a2n+1 ’

2n-l-2v 2n+l

2(1+ v) a ’

(5.4)

v=q/q*. (5.3)

As n1 + CC (v + 0), these expressions converge to the results (5.1) for hard spheres with stick boundary conditions ( 5 = 0), while for n1 + 0 (v + 00) they tend to the complete slip values ( 5 = f ). Our results for As and B f are in agreement with those of Batchelor and Green”).

6. Explicit results for identical particles

In this section we present explicit expansions of the scalar mobility functions

for two identical spherical particles through order Re2’. We tabulate below the numerical coefficients of these series for six different particle models:

(a) hard spheres with stick boundary conditions (6 = 0), (b) permeable spheres with x = 10, (c) permeable spheres with x = 1, (d) immiscible droplets with v = i, (e) immiscible droplets with v = 2, (f) hard spheres with complete slip boundary conditions ( .$’ = f ).

For the last three models ((d)-(f)) the torque on a sphere vanishes identically so that in these cases only the tt, td and dd components of the mobility matrix are given in tables I-XXII.

We denote the expansion coefficients of the mobility functions (in dimen-


sionless form) in powers of a/R by c,. For example, we write

47rnaa;,(R) = 2 c (“)‘I +. . . HZ] n R

(6.1)

The mobility functions depend on a/R through either even powers or odd powers only. Hence at most ten numerical coefficients suffice to specify them to order R-“. The coefficients c, are listed numerically to four significant figures in tables I-XXII. When a number has fewer than four figures it is exact. The symbol (m), m integer, behind some of the numbers indicates multiplica- tion by a factor 10”. To save space in the tabulation only those coefficients are listed which are generically non-vanishing. Any coefficients c, (n < 20) omitted from the tables are identically zero.

The numerical reliability of our truncated expansions of the mobility functions depends firstly on the correctness of the algebraic iteration of the recursion scheme of refs. 3-5 and secondly on the range of particle separations R considered. For the tt sector of the mobility matrix the algebraic computation has been performed independently, using both the MACSYMA and the SMP systems. When specialized to stick had spheres both computations agree and moreover give the same coefficients in (Y:, and cyy* as may be obtained by expanding the bispherical coordinate exact solution for two spheres approach- ing each other as given by Brenner*O). These agreements indicate that the algebraic computation is correct.

The rate of convergence of the series expansions is controlled by the complicated singularities in the mobility functions associated with the lubrication region of close approach (R = 2a). For hard spheres with stick boundary conditions (model (a)) there is accurate independent numerical information about the mobilities at all separations due to Jeffrey and Onishi’) (t and r sectors) and to Kim and Mifflin6) (t, r and d sectors). A comparison of our series expansion with their data shows that for most of the scalar mobility functions the truncated series may be used with good accuracy in the range ~2.1~. The functions (Y::, pi;., By,, (Y;;, ai:, pit, cyty, p:,!, rf; are accurate to 1% or better in this range. The function PI’, is accurate to 2% and the functions (Y::, pi”, are accurate to 3% in the same range. Only the functions

P:, Pi;, Pi”, and PI”, are not well represented by the series at R = 2.1~. For these functions we find that pft and pi: are accurate to 2% for R 2 2.2a, while

Pi’, and pf’, are accurate to 3% for R ~2.3a.

A detailed look at the tables reveals that the expansion coefficients vary markedly from one model to another. Thus the effective transport coefficients of non-dilute suspensions are expected to depend strongly on the character of the suspended particles. For the future we hope to explore this dependence in more detail with the aid of the mobility functions given in the tables.


TABLE I

Model

(4 (b) (4 (4 (4 (f) C4 -2.5 -1.701 -9.462 (-2) -2 -1.5 -1 % 3.667 2.353 1.267 (-1) 1.75 1.429 (-1) -1 CS 7 3.010 4.941 (-2) 4.042 1.272 -1 Cl0 -55.67 -18.03 9.269 (-3) -26.67 -11.03 -5 Cl2 -73.58 -4.835 -2.737 (-2) -62.58 -41.56 -22 Cl4 192 74.67 -8.115 (-2) -30.54 -98.18 -78 Cl6 -99.38 30.23 -1.465 (-1) - 142.6 -277.3 -262 Cl8 -1141 73.68 -2.261 (-1) -718.9 -903.8 -883 %I 2335 1013 -3.267 (-1) -1566 -2983 -3016

Coefficients for the series representation of 4-nr)acryl(R).

TABLE II

(4 (b) (4 C.5 -7.083 (-1) CS -8.333 (-1) Cl0 -2.875 Cl2 - 10.33 Cl4 -28.71 Cl6 -74.54 Cm -223.2 C,, -803.9

-4.070 (-1) -1.738 (-2) -4.674 (-1) -2.212 (-2) -1.282 -2.931 (-2) -3.271 -4.479 (-2) -6.629 -6.692 (-2)

-13.93 -9.541 (-2) -36.90 -1.303 (-1)

-114.3 -1.725 (-1)

-2.292 (-1) -2.183 (-2) 0.125 -1.528 (-1) 1.582 (-1) 0.2 -7.083 (-1) 2.981 (-1) 0.25 -3.3 1.349 (-1) 2.857 (-1) -9.744 -6.557 (-1) 3.125 (-1)

-22.91 -2.762 6.146 (-1) -50.68 -8.229 3.1

-116.7 -21.84 16.69

Coefficients for the series representation of 4nr)a/3yl(R).

TABLE III

Model

C7 - 3.25 - 1.990 - 9.810 (- 2) C9 - 9.5 - 4.412 - 1.158 (- 1) Cl1 - 21.75 - 7.599 -1.362 (-1) Cl3 - 35.75 - 9.897 -1.582 (-1) Cl5 - 81 - 19.72 - 1.817 (- 1) Cl7 - 364.6 - 68.24 -2.073 (-1) Cl9 - 1768 - 234.8 -2.365 (-1)

Coefficients for the series representation of 8Tr)a’py,(R).


TABLE IV

Model

(4 CR -3 Cl0 -6

Cl2 - 10

CL4 - 15

cl6 -21

CIR - 76 c20 - 426

(b) - 1.783

- 2.922

- 4.020

- 5.019

- 5.900

- 16.75

- 74.44

(4 - 8.208 ( - 2)

- 9.283 (- 2)

- 9.932 (- 2)

-1.036 (-1)

-1.067(-l)

-1.100 (-1)

-1.154 (-1)

Coefficients for the series representation of

Sm$a;;(R).

TABLE V

Model

(4 (4 -3.75 -2.551 -1.419 (-1)

-9.75 -5.094 -1.667 (-1)

-18 -7.254 -1.717 (-1) -52.19 -16.46 -1.761 (-1)

-210.8 -46.27 -1.784 (-1)

-800.3 -114.7 -1.798 (-1)

-2646 -236.3 -1.800 (-1)

- 8363 -455.2 -1.796 (-1)


8m&;(R).

TABLE VI

Model

(4 (b) (cl (4 (4 (f)

C5 Cl C9

Cl1 Cl3 Cl5 Cl7 Cl0

12.5 5.786 1.791 (-2) 8 4.5 2 -17.5 -8.236 -2.953 (-2) -6.5 2.143 (-1) 3 -51 -15.60 -1.593 (-2) -24 -5.667 4 233.8 52.35 -6.972 (-3) 83 25.08 13 485 34.06 7.959 (-3) 279.3 129.2 54

-390.4 -212.4 3.497 (-2) 358.3 380.4 204 1143 - 142.9 7.503 (-2) 1103 1165 732 5851 -463.4 1.309 (-1) 3690 3727 2587

Coefficients for the series representation of a-'a::(R)


TABLE VII

Model

(4 0’) (4 (4 (4 w C7 1.667 5.122 (-1) 3.891 (-4) 5.556 (-2) 1.587 (-2) 0 c9 -6 -1.363 2.459 (-3) -4.370 -1.917 0 cl1 -6 1.074 5.978 (-3) -8.611 -5.378 0 Cl3 61.92 16.58 1.524 (-2) 4.356 -8.060 0 cl5 254.1 42.04 3.138 (-2) 58.03 -5.819 0 c17 488.3 64.89 5.579 (-2) 152.6 5.234 0 c,. 261.9 89.50 8.989 (-2) 155.0 13.19 0

Coeffcients for the series representation of a-$::(R).

TABLE VIII

Model

(4 (b) (4

C6 6.25

CS 0

Cl0 -25

Cl2 -56.94

Cl4 67.5

Cl6 688

Cl6 1988

%I 2826

2.893 8.953 (-3) -1.086 -9.979 (-3) -9.648 -1.879 (-2)

-16.75 -2.962 (-2) -2.642 -4.273 (-2) 22.72 -5.823 (-2)

-52.30 -7.638 (-2) -541.7 -9.753 (-2)


P;(R).

TABLE IX

C6 C8 Cl0 Cl2 Cl4 Cl6 Cl8 CT”

Model

(a) (h) (c) (d) (e) (f)

62.5 19.68 3.388 (-3) 32 13.5 4 -56.25 -22.33 -6.006 (-3) -11 7.393 9

-330 -73.26 -4.451 (-3) -122.7 -18 16 753.1 116.5 -3.360 (-3) 160.9 35.32 41

2479 126.7 1.539 (-3) 1005 334.8 144 871.9 -549.6 1.374 (-2) 2226 1298 538

1.359 (4) -283.1 3.628 (-2) 7776 4604 2000 5.095 (4) -1313 7.316 (-2) 2476 (1) 1560 (1) 7359

Coefficients for the series representation of (1/4~~$)a~~(R).


TABLE X

(4 (b) (4 15.63 4.920 8.471 (-4) 12.5 6.131 (-1) -1.016 (-3)

- 105 -23.86 -1.206 (-3) -252.3 -32.00 -1.065 (-3)

501.4 76.47 1.556 (-3) 3931 315.3 8.339 (-3)

1.090 (4) 465.4 2.136 (-2) 1.525 (4) -2.961 4.308 (-2)

(4 8

5.222 -43.41

- 129.6 -23.73 865.6

3407 7504

(4 (0 3.375 1 4.548 3

-6.854 6 -35.38 11 -44.12 24 107.0 72 789.7 263

2820 1002

Coefficients for the series representation of (3/16nqa3)/3ft(R)

TABLE XI

Model

(4 (b) (cl Cd) (4 (0 C8 7.813 2.014 2.465 (-4) 2.222 5.208 (-1) 0 c,,, 7.5 1.444 1.851 (-4) 1.037 -4.167 (-2) 0 ClZ -13.13 -8.718 (-1) 1.599 (-4) - 10.56 -4.253 0 CIJ -13.5 3.635 4.645 (-4) -24.36 -13.22 0 Cl, 204.1 32.48 1.468 (-3) 9.877 -23.08 0 c1R 1249 117.1 3.627 (-3) 238.0 - 17.88 0 c,,, 4670 305.2 7.488 (-3) 1011 48.75 0

Coefficients for the series representation of (3/8nqn’)y~~(R).

TABLE XII

Model

(4 (b) (4 (4 (e) (f)

Cl 1 1 1 1 c? -6.667 (-1) -5.585 (-1) -4.073 (-1) -0.5 c7 12.5 5.786 1.791 (-2) 8 cg -5 -4.622 -2.817 (-2) 3 Cl1 -65.5 -22.57 -2.561 (-2) -22.92 Cl3 113.7 25.02 -1.936 (-2) 32.67 ClS 457.5 32.18 8.253 (-4) 237.9 c17 14.63 -186.1 5.145 (-2) 550.7 Cl9 2142 -153.6 1.527 (-1) 2069

Coefficients for the series representation of 4q~a~~(R).

1 1 -2.857 (- 1) 0

4.5 2 6.429 6 3.561 14

25.67 38 140.1 122 527.6 420

1941 1486


TABLE XIII

Model

(a) (b) (c) (d) (e) (f)

c1 0.5 0.5 0.5 0.5 0.5 0.5 c3 3.333 (-1) 2.793 (-1) 2.036 (-1) 0.25 1.429 (-1) 0 c11 2.953 1.098 2.432 (-3) -8.796 (-2) -1.420 (-1) 1.875 (-1) c13 14.99 4.232 6.091 (-3) 2.940 (-1) -3.129 (-1) 1 c15 52.51 11.48 1.276 (-2) 3.220 3.721 (-1) 3.475 c17 138.0 25.91 2.559 (-2) 8.184 3.736 10 c19 317.1 59.16 5.044 (-2) -5.360 10.62 25.93

Coefficients for the series representation of 4m@~#).

TABLE XIV

Model

(4 (b) (4 C2 -1 -1 -1 c10 -2.5 -7.683 (-1) -5.837 (-4) ClZ -6.219 -2.033 -8.367 (-3) c14 -6.625 -6.038 -2.434 (-2) Cl6 -35.35 -28.89 -5.962 (-2) cl8 -256.8 -111.2 -1.365 (-1) c20 - 1223 -353.5 -3.012 (-1)

Coefficients for the series representation of 87rT&:;(R).

TABLE XV

C3 1 1 1 c13 12 4.240 8.983 (-3) c15 60 17.37 2.540 (-2) cl7 210 50.02 5.415 (-2) c19 630 124.0 1.042 (-1)

Coefficients for the series representation of 8m7a3a;:(R).


TABLE XVI

(a)

Model

(b) (c)

C3 -0.5 -0.5 -0.5

c9 9.375 4.339 1.343 (-2)

c11 30 8.594 9.469 (-4)

c13 48.84 2.360 -3.161 (-2)

Cl5 24.66 -32.15 -1.006 (-1)

c17 23.1 -112.0 -2.386 (- 1)

cl9 799.8 -281.5 -5.070 (-1)


8nr&;(R).

TABLE XVII

Model

(a)

5 2.5

c4 -4

cx 62.5

c10 6.25

c12 -311.3

Cl4 259.1

cl6 1586

cl8 -356.9

c20 1.202 (4)

(b) 1.701

-2.296

19.68

-10.04

-82.62

25.74

19.49

-719.5

-638.1

Cc) Cd) (e) (f) 9.462 (-2) 2 1.5

-9.861 (-2) -2.5 -1.143

3.388 (-3) 32 13.5

-5.749 (-3) 27 26.04

-5.733 (-3) -72.92 28.47

-5.396 (-3) 39.11 89.83

-1.223 (-3) 622.6 402.3

1.279 (-2) 1687 1567

4.616 (-2) 8334 6233

Coefficients for the series representation of i'a:d,(R)

TABLE XVIII

1 0

4

15

42

121

380

1268

4402

Model

(a) (b) (c) (d) (e) (f)

cq 2.667 1.531 6.574 (-2) 1.667 7.619 (-1) 0

c10 -4.167 -8.712 (-1) -3.682 (-5) -1.111 (-1) -2.381 (-2) 0

c12 19.17 5.190 4.007 (-4) 4.179 2.816 (-1) 0

c14 149.3 24.52 1.055 (-3) 31.22 5.295 0

cl6 561.6 60.76 2.182 (-3) 136.7 35.01 0

‘18 1428 105.5 4.342 (-3) 407.3 148.0 0

c20 2705 151.3 8.509 (-3) 809.7 477.4 0

Coefficients for the series representation of amlP:;(R).


TABLE XIX

Model

c3 -2.5 -1.701 -9.462 (-2) cg -15.63 -4.920 -8.471 (-4) cl1 -62.5 -12.20 1.238 (-5) c13 -115 -8.728 7.531 (-4) c15 -57.66 21.39 1.865 (-3) cl7 -56.63 43.08 2.168 (-3) cl9 -1993 -59.03 -1.761 (-3)


Pm.

TABLE XX

Model

(4 W (4 (4 (4 (f)

C3

C5 C-2

Cl1 Cl3

Cl5

Cl7 Cl9

12.5 5.786 -30 -11.85 312.5 66.95 187.5 -14.80

- 1247 -276.9 540 -71.31

4200 -348.1 -1.353 (4) -3410

1.791 (-2) 8 4.5 2 -3.088 (-2) -16 -6 0

6.412 (-4) 128 40.5 8 -1.167 (-3) 168 98.36 36 -1.188 (-3) -133.3 163.1 118 -1.223 (-3) 112.4 423.7 372 -4.437 (-4) 1396 1409 1202

2.672 (-3) 2271 4803 3998

Coefficients for the series representation of (1/4m$)c~~~(R).

TABLE XXI

Model

(4 @I (9 (4 (4 (f) C3 -6.25 -2.893 -8.953 (-3) -4 c5 20 7.900 2.059 (-2) 10.67 c9 -39.06 -8.369 -8.015 (-5) -16 Cl1 -187.5 -24.94 7.994 (-6) -63.56 c13 -150 13.24 2.160 (-4) -89.88 c15 1096 183.9 5.544 (-4) 168.8 Cl7 5184 489.2 9.755 (-4) 1423 cl9 1.125 (4) 759.0 1.308 (-3) 4681

Coefficients for the series representation of (3/16m$)&:(R).

-2.25 -1

4 0

-5.063 -1 -22.64 -6 -57.20 -24 -85.90 -81

12.84 -252 567.9 -759


TABLE XXII

Model

(4 (b) (4 (4 (e) (f) C5 -5 -1.975 -5.147 (-3) -2.667 -1 0

cl? 13.67 1.170 -6.129 (-6) 4.691 1.095 0

Cl5 20.47 -2.851 -3.700 (-5) 19.51 5.938 0

c17 -275 -39.59 -1.311 (-4) 6.975 14.06 0

c19 -2163 - 174.2 -3.576 (-4) -267.9 -3.274 0

Coefficients for the series representation of (3/81rr$)y~~(R)

Acknowledgement

R.B.J. wishes to thank the University of London, Central Research Fund,

for a grant to provide the computer software used in the calculations reported

here.

References

1) P. Mazur and W. van Saarloos, Physica 115A (1982) 21.

2) D.J. Jeffrey and Y. Onishi, J. Fluid Mech. 139 (1984) 261.

3) R. Schmitz and B.U. Felderhof, Physica 113A (1982) 90.



6) S. Kim and R.T. Mifflin, Phys. Fluids 28 (1985) 2033.

7) B.U. Felderhof, Physica 89A (1977) 373.

8) P. Remand, B.U. Felderhof and R.B. Jones, Physica 93A (1978) 465.

9) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Noordhoff, Leiden.

1973) p. 53.

10) B.U. Felderhof, Physica 84A (1976) 569.

11) B.U. Felderhof and J.M. Deutch, J. Chem. Phys. 62 (1975) 2391.

12) U. Geigenmiiller and P. Mazur, Physica 138A (1986) 269.

13) R. Schmitz, Physica 102A (1980) 161.

14) See ref. 9, p. 85 ff.

15) See e.g. B.U. Felderhof. G.W. Ford and E.G.D. Cohen, J. Stat. Phys. 28 (1982) 135.

16) B.U. Felderhof and R.B. Jones, Physica 93A (1978) 457.


18) M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York,

1972), p. 443.

19) G.K. Batchelor and J.T. Green, J. Fluid Mech. 56 (1972) 401, in particular their eqs. (1.4), (1.5).

20) H. Brenner, Chem. Eng. Sci. 16 (1961) 242.

Documents

Mobility matrix for arbitrary spherical particles in solution