Faraday Discuss. Chem. SOC., 1987, 83, 33-46 Many-sphere Hydrodynamic Interactions Peter Mazur Instituut-Lorentz, Rijksuniversiteit Leiden, P. 0. Box 9506, 2300 RA Leiden, The Netherlands A scheme for the evaluation of many-sphere hydrodynamic interactions including interactions with container walls is presented. For a system consist- ing of a suspension of N spherical particles in an unbounded incompressible viscous fluid general formulae for the many sphere mobility tensors were obtained with explicit expressions for three- and four-body contributions up to order R-' with R a typical interparticle distance. Explicit expressions were also obtained to order R-"RO"I, n + m G 3, for the three-body problem consisting of two suspended spheres at arbitrary positions within a spherical container of radius R,,.The influence of hydrodynamic interactions on transport properties, in particular their essential non-additivity and the influence of wall effects, is then discussed for self-diffusion and sedimentation. In this paper we present an analysis of many-sphere hydrodynamic interactions which is based on a method of induced forces. These interactions which play an important role in the quantitative understanding of the properties of suspensions, were traditionally studied by the method of reflections, inaugurated by Smoluchowski' for those situations in which the fluid can adequately be described by the quasi-static Stokes equation for incompresssible steady flow. However, owing to the increasing complexity of the prob- lem, essentially only the two-sphere case was analysed by these methods.132 It must be mentioned at this point that Kynch' had already, within the framework of the method of reflections, derived expressions for three- and four-body contributions to the mobility tensors, but his work seems to have remained largely unnoticed.For the discussion of the properties of dilute suspensions only pair interactions need to be taken into a c c o ~ n t . ~ It was usually assumed and hoped that pairwise additivity of hydrodynamic interactions might hold in practice in concentrated suspensions as well [see e.g. ref. ( 5 ) ] . The long-range nature of these interactions, however, was not an argument in favour of such an assumption. With this in mind we developed a systematic scheme, which is reviewed here,6 to treat the full many-body problem (see in this connexion also work by Muthukumar' and Yoshizaki and Yamakawa').We shall first present this scheme in sections 2-6. We then briefly discuss self-diffusion and sedimentation as phenomena for which many-body interactions play an essential role. We consider these properties on the short timescale, i.e. for a time regime such that the relative configuration of suspended particles does not change appreciably. Two points will receive special attention: ( 1 ) the essential non-additivity of hydrodynamic interactions and (2) the influence of very long range hydrodynamic interactions and of wall effects on transport properties. To conclude we mention a number of related problems. 1. Equations of Motion; Formal Solution Consider N macroscopic spheres with radii a j ( i = 1,2.. . N ) , which move with velocities u, and angular velocities oi through an otherwise unbounded incompressible viscous 3334 Many-sphere Hydrodynamic Interactions fluid. The motion of the fluid obeys the quasi-static Stokes equation for all Ir - Ril > a,, i = 1 , 2 . . . N I V P(r) = O v u ( r ) = 0 with Here P is the pressure tensor, p the hydrostatic pressure, u the velocity field and 77 the viscosity of the fluid; Ri denotes the position of the centre of sphere i, while Greek indices label Cartesian components of tensorial quantities. The force Ki and torque T, exerted by the fluid on sphere i are given by where S, is the surface of sphere i (to be precise: the surface of a sphere centred at R, with radius a,+& in the limit c - 0 ) and i, is a unit vector normal to this surface pointing in the outward direction.In order to solve the set of equations (1.1) and (1.2) and subsequently determine the forces and torques from eqn (1.3) and (1.4), boundary conditions at the surfaces of the spheres must be specified. We assume stick conditions u ( r ) = u, + w , A ( r - R, ) for I r - R, I = a,. (1.5) The problem posed by eqn (1.1)-(1.5) may be reformulated by introducing a force density 4 ( r ) induced on the spheres and extending the fluid equations inside the spheres. The fluid equations are then written in the equivalent form for all r (1.6) N V * P = c F , ( r ) v u ( r ) = O j = 1 with F,(r)=O for Ir-R’l>a,. to Inside the spheres the fluid velocity field and pressure field are extended according u ( r ) = u , + w , A ( ~ - R , ) f o r ( r - R , ( s a , (1.7) p (r) = o for 1 r - R, I < a,.(1.8) E ( r ) = a;*.m)~(Ir-R,l- a,). (1.9) The induced force density is then of the form The factor aL2 has been introduced here for convenience. Making use of eqn (1.3), (1.4), (1.6) and (1.9) one can express the force K, and torque T, which the fluid exerts on sphere i in terms of the induced surface forces J;. With Gauss’s theorem one has indeed (1.10) T, = - dn^,n^, A J ; ( ~ ^ ~ ) . ( 1 . 1 1 ) JP. Mazur 35 To solve formally the equation of motion of the fluid we introduce Fourier transforms of e.g. the velocity field: u ( k ) = dr exp ( - i k * r ) u ( r ) . (1.12) I The Fourier-transformed induced force density F ( k ) is defined in a reference frame in which the centre of sphere i is at the origin F j ( k ) = drexp[-ik.( r - R , > J F , ( r ) . (1.13) With these definitions, the equations of motion (1.6) in wavevector representation I become: qk*v( k ) = By applying ;he operator I - LL, N -ikp( k ) + C exp (-ik R i ) 4 ( k ) . (1.14) (1.15) where k^= k / k is the unit vector in the direction of k j = 1 k v ( k ) = 0. and I the unit tensor, to both sides of eqn (1.14) one obtains with eqn (1.15) q k 2 u ( k ) = c exp (-ik R,)(I - ii) ~ ; ( k ) (1.17) which has, assuming that the fluid unperturbed by the motion of the spheres is at rest, the formal solution u ( k ) = c q - 1 ~ - 2 exp (-ik R,)(I - ti) . ~ , ( k ) . (1.18) This equation will serve as the starting point for the calculation of the forces and torques exerted by the fluid on the spheres, and thus of the hydrodynamic interactions which are set up between the spheres by their motion through the fluid.i j 2. Irreducible Tensors ; Induced Force Multipoles and Velocity Surface Moments For the purpose of evaluating hydrodynamic interactions it is convenient to introduce irreducible force multipoles, defined in terms of the surface forces A(&) according to n Here b' is an irreducible tensor of rank I, i.e. the tensor of rank I traceless and symmetric in any pair of its indices, constructed with the vector 6. For I = 1,2,3 one has, see e.g. ref. (9) or (lo), n n b = b, b,bp = b,bp -$6,pb2 - b,bpby = b,bpb, -+(6,pb, + 6,,bp + 6p,b,)b2. According to eqn (l.lO), (1.11) and (2.1) and36 Many-sphere Hydrodynamic Interactions In eqn (2.4) F!*=) is the antisymmetric part of F(2) and E the Levi-Civita tensor, for which one has the identity L : L = -21.The tensors 6f satisfy the orthogonality and completeness conditions6," In the above equations (21+ 1 ) ! 1 = 1 , 3 , 5 . . . (21-1)(2/+ 1). The dot 0 denotes a full 1-fsd contraction between the tensors 7 a n d z with the convention that the last index of 6; is contracted with the first index of etc. A('71) represents an isotropic tensor of rank 21 that projects out the irreducible part of a tensor of rank I : For I = 0, 1 , 2 one has (2.7) A('+') 0 b' = 6' 0 A",') =z With relations (2.5) and (2.6) one shows that the surface-induced force density f;( 6,) has the following expansion in terms of irreducible force multipoles: This expansion, which is written in a coordinate-free way, is equivalent to an expansion in spherical harmonics, to which it can be reduced if polar coordinates are introduced.For the Fourier-transformed induced force density Fi (k), the expansion (2.9) leads to [cf: eqn (1.9) and (1.13)] a? E ( k ) = 1 (21+ l)!!i31(kai)F0 FI'+') / = 0 (2.10) withj,(x) the spherical Bessel function of order I. In deriving the expansion (2.10) from eqn (2.9) use has been made of the identity" n - (--l)Fj/(k). a' sin k ak' k (2.1 1 ) Next to the irreducible induced force multipoles defined above, we also introduce irreducible surface moments of the fluid velocity field. The irreducible surface moment of order rn is defined as (2m + l)!! 4?ra2 I d r F u ( r ) 6( I r - R, I - a , ) n- (2m + l ) ! ! i ~ u ( i ) " = r z l dkj,(k)krnu(k) exp (ik- R,), rn 3 0 .(2.12) I (2rn + l ) ! ! ( 2 d 3 = i" The numerical factor (2m + l)!! is introduced for convenience. The velocity surface moments are the coefficients of an expansions of the fluid velocity field at the surfaces of the spheres in irreducible tensors F. Using the boundary condition (1.5) as well as the orthogonality condition (2.5) one has u( r ) sf = u,, 3iiU( r ) sf = a,L wi ( z m + l ) ! ! ~ u ( u ) "=o, f o r m 3 3 . (2.13)P. Mazur 37 In the next section we shall relate the induced force multipoles to the surface moments of the fluid velocity field through a hierarchy of equations. It is this hierarchy which will then enable us to obtain expressions for the mobility tensors which relate the forces and torques on the spheres to their velocities and angular velocities. 3.Determination of Induced Forces ; Properties of Connectors To determine the induced forces on the surfaces of the spheres we employ eqn (2.12) in the following way: we substitute the formal solution (1.18), together with the expansion (2.10) for F j ( k ) into the last member, and use the results (2.13) for the left-hand side. One then obtains the following set multipoles: where the coefficients, the so-called given by of coupled equations for the irreducible force N c o wjSn2) = C C At,"' Q Fj"' (3.1) j = l m = l connectors which are tensors of rank n+m, are 3(2n - 1)!!(2rn - l)!! y in-" 4T2 A!".") = A A A x dk exp (ik R,)$"-'\/ - kk)k"'lk-2jfl-,( kaj)jm-l( kaj).(3.2) Here R, = Rj - R j ; for i Zj, R , = lR,l> a j + aj. In principle one can determine from the set (3.1) all force multiples in terms of the velocities uj and angular velocities oj and in particular derive expressions for the mobility tensors which relate the forces and the torques exerted by the fluid on the spheres to these quantities. Before establishing these expressions we shall list a number of properties of the connectors A',"."'. ( 1 ) One verifies by inspection that the tensor quantities satisfy a symmetry relation upon simultaneous interchange of the particle indices i and j , the multiple indices n and rn and complete transposition of the tensor indices. (2) One also verifies that the matrix of connectors is positive definite in the following sense: where FI"' is the generalized transpose of the tensor of rank n, F'").One shows that this inequality ensures that the energy dissipation caused by the motion of N spheres is positive, as it should be. (3) Consider now in more detail the self-connectors A$'9m'. Using standard ortho- gonality properties of spherical Bessel functions it follows that A$3m' = -a~lB(n*n)gn,m (3.4) where the tensors B'"."' of rank 2n are independent of the index i. The tensors B(","' have been calculated12 explicitly in terms of tensors A('>'). The first two are where the tensor S of rank 4,38 Many-sp here Hydrodynamic Interact ions (4) Next we discuss the behaviour of the connectors A',"."', i # j , as a function of the interparticle distance R,.Straightforward evaluation of the remaining integral then leads to the result ( 3 . 8 ) A',", m ) ~ G 11'. m ) R ;( n + m - 1 ) + H ',", m R ; ( n + m + 1 ) where the tensors G',"."' and Hf,"'), which only depend on the unit vector fl, = R , / R , , and the radii a, and a,, are given by (3.9) The arrow on d/dR in eqn (3.9) indicates a differentiation to the left. ations, and becomes The expression for H'"."') can easily be further simplified by carrying out the differenti- For the tensor G("3'n' the differentiations can in principle be carried out in a similar formal way. We list here the explicit results for the first few of these tensors G r,' 9' ) = :(I + fg?,, ) (3.12) ( 3 . 1 3 ) (3.14) In eqn ( 3 . 1 3 ) , GY3*') denotes the part of G!,'.') that is (traceless) symmetric in its last two indices.A similar notation is adopted for GfS,*') in eqn (3.14); the tensor D is traceless and symmetric in its first and last two indices and defined by Dapys = 2rffrpryrs -5t rary& + r a r d , , + rpr8Sffy + r p ~ y s f f s ) . ( 3 . 1 5 ) Further explicit expressions for Gif""", with n,m =s 3 and n + m d 5 , may be found in ref. (6). 4. Mobility Tensors In the linear regime considered, the velocities and angular velocities of the spheres are related to the forces and torques exerted on them by the fluid in a way described by the following set of linear coupled equations: In the above equations, p 7 and pRR are translational and rotational mobility tensors, respectively. The tensors p;R and i F couple translational and rotational motion.The mobility tensors account for the hydrodynamic interactions between the spheres through their dependence on their relative positions. The analysis given in the previous sections enables us to express the mobilities in terms of connectors and thereby calculate these quantities as series in powers of inverse distances between the spheres.P. Mazur 39 We shall now, for simplicity's sake, restrict ourselves to the case of free rotation for which the torques T, vanish: T, = a l e : F ~ 2 u ) = 0 (4.3) and evaluate only the mobility tensors p y . By elimination from the set of equations (3.1) of all force multipoles Fin' 2 3, one obtains the following series for this quantity: py= (6~7)-'( aF'6, +AY,')(l + 6,) + C 0 B(2F,2r)-1 0 A t ' , ' ) k # k J Here B ( m , m ) - ' , rn # 2, is the generalized inverse of B(m,m' when actiqg on tensors of rank rn which are irreducible in their first rn - 1 indices.As for B(2r32s)- it follows from eqn (3.6) that this tensor of rank 4 is given by (4.5) B(2.\,2.T)-' - 10 ( 2 , 2 ) --FA . Each term in the series (4.4) has, as a function of a typical interparticle distance R, a given behaviour which is determined by the upper indices of the connectors and their number. Thus according to eqn (3.8) a term in eqn (4.4) with s connectors, s = 1,2,3 . . . , gives contributions proportional to with 1,3 f o r s = 1 P = ( 3s-2+2q f o r s z 2 , q=O,1,2 ,.... (4-6) This implies that py cannot contain terms proportional to R-2 and R-5. Note also that each term in the expression (4.4) containing a sequence of s connectors involves the hydrodynamic interaction between at most s + 1 spheres.Therefore the dominant n- sphere contributions, n 3 2, are of order R-3n+5, where eqn (4.6) has been applied with s = n - 1 and q=O. Explicit expressions for the various terms in the expansion (4.4) can in principle be found, using eqn (3.2) and (3.8)-(3.10) and forming the necessary tensor products. Thus the three-sphere contributions of order R-' to ~7 is given by the product looR-2R - 3 - 2 ( 1 , 2 \ ) . 81 I k k l R,, G , ~ . G ~ ~ , ~ s ) : G ~ ; Y Into this product one then has to insert expressions (3.13) and (3.14) for the correspond- ing G-tensors. In ref. (6) all contributions to the tensors p'f;T, p:R and p;" up to order RP7 are listed explicitly.As a final remark we mention that the series (4.4) can be written in closed form in a matrix notation in which tensors A:,'"' are the n, rn, i , j elements of a generalized matrix.6 5. Wall Effects; The Spherical Container In the preceding sections, we assumed that the suspended spherical particles were moving in an unbounded fluid, and calculated the corresponding mobility tensors which accounted for the hydrodynamic interactions between the spheres. Characteristic of these interactions is their very long range, which is apparent from the explicit expressions for the mobility tensors p:, i Z j . To lowest order in the expansion in connectors these are given by40 Many-sphere Hydrodynamic Interactions As a consequence of this long-range nature, the influence of boundary walls can be of importance even in cases where the vessel containing the suspension is very large.We shall, therefore, discuss in this section an extension of the scheme developed for the evaluation of mobility tensors that includes the effect of a spherical wall bounding the ~uspension.'~ The solution of the problem of N spheres moving in a viscous fluid inside a spherical container may be obtained from the solution to the problem of N + 1 spheres in an unbounded medium studied above, by observing that the analysis given remains valid if one of the spherical boundaries, the container specified by the index i = 0, encloses the other N spheres ( i = 1,2, . . . , N ) and the viscous fluid, provided the induced force Fo on the container is chosen in such a way that F,( r ) = 0 for I r - R,I < a, (5.2) where Ro is the centre of the container and a, its radius, and that the velocity field has, in addition to the extensions (1.7), the extension ~ ( r ) = o for I(r - R,)J z a,.(5.3) The analysis of section 3 then leads to the following set of equations: with connectors A!",") ( i , j = 0 , 1,2 . . . N ) defined again by the integrals (3.2) with the additional conditions R , > a i + a j f o r i , j = l , 2 . . . N, i # j ( 5 . 5 ) Roj<a,-aj f o r j = l , 2 . . . N. (5.6) The particle-particle connectors, therefore, remain unchanged; the particle-container connectors are of a different type but can also be evaluated13 using properties of integrals over Bessel functions. Since the velocity u, and angular velocity oo of the container vanish, one can reduce the set of equations (5.4) for the N + 1 spheres (particles and container) to a reduced set of the form (3.1) for the particles alone, but now in terms of new connectors AEZ!, which incorporate the hydrodynamic interaction with the container, and are given by m A!.",~) 1J;S.C.= + c A$P) . A@"), 1 , j = 1 , 2 . . . N. (5.7) p = l Note that the 'self'-connectors A!,?!:! are not diagonal in their upper indices: different multipoles in the same sphere couple via the container wall. It is in terms of the new connectors that the mobility tensors p,, which are again of the form of eqn (4.19)-(4.22), must now be evaluated. We give the expression for the translational mobility tensors pij (omitting from now on the indices TT) for the case that particle i is concentric with the container, in an expansion to third order in the parameters a / a o and a / R ( a and R are a typical particle radius and an interparticle distance, respectively):P.Mazur 41 ExpIicit expressions for more general cases can be found in ref. (13). 6. The Fluid Velocity Field As we saw (section 5), one needs, for a proper discussion of phenomena such as sedimentation, an expression for the velocity field of the fluid at a point r, caused by the motion of the spheres. Within the linear regime studied and for the case of free rotation, u( r ) may be expressed in terms of the forces exerted by the fluid on the spheres in the following way: N ~ ( r ) = - C Sj(r) K,. j = l The tensors S j ( r ) defined above can be derived from the general expressions for the translational mobilities of N + 1 spheres by putting RN+1= r and taking the limit aN+1+ 0: S j ( r ) ~ lim ) I N + ~ , ~ ( R ~ + , = r ) , j = 1 , 2 .. . N. a’N+,+O This equation expresses the fact that the velocity field can be probed with the aid of an infinitesimally small sphere located at r ; u( r) in eqn (6.1) is the velocity of this test sphere. To lowest order in the expansion of the mobilities in connectors one has for an unbounded suspension as follows from eqn (6.2) and (5.1) for i = N + 1 ( i e . for i denoting the test particle) and j = 1 , 2 . . . N. In eqn (6.3) 6 denotes the unit vector pointing from r to the centre of sphere j . Note that if one puts R, = r, i = 1,2,.. . , N, in eqn (5.1), one has for a given sphere j , to lowest order (and for lRj - rl> uj + ai) 7 af A A )I~( Ri = r ) - Sj( r ) = - ( rjrj - 4). 87r771Rj - rl The left-hand side represents, per unit of force exerted on sphere j , the velocity of a sphere i at position r with respect to the fluid velocity at that point in the absence of sphere i. Note that the R-’ contribution to this relative sphere velocity cancels, but that a long-range R-3 contribution remains. 7. Diffusion To apprehend the influence of hydrodynamic interactions on properties of suspensions, we shall discuss in this survey two transport phenomena. We consider again suspensions of hard spheres of common radius a and mass rn which, except for their short-range hard-sphere interaction and their hydrodynamic coupling, do not exert any direct long-range forces ( e .g . electromagnetic) on each other. The phenomena we shall study here are diffusion, in particular self-diffusion and sedimentation. Another transport phenomenon, viscosity, will not be dealt with, but has been studied using the same methods by Beenakke~,’~ to whose work we refer. As a starting point for our discussion we write down the expression for the short-time wavenumber-dependent diffusion coefficient D ( k ) (cJ: Pusey and Tough”):42 Many-sphere Hydrodynamic Interactions Here kB is Boltzmann’s constant and T the temperature of the system; G ( k ) , the Fourier-transformed density correlation function of the suspended spheres, is given by N G ( k ) = N - ’ C (exp ( i k .( R j - - R j ) ) . ( j = 1 (7.2) The coefficient (7.1) characterizes diffusion on a timescale t such that t,<< t i < t,, with t R = ( 6 7 ~ 7 p - l the characteristic time for the velocity correlation of a Brownian particle, and t, a structural relaxation time in which the configuration of the particles changes appreciably. For typical suspensions t , =: lo-’ s and t , s s. We restrict ourselves to this short time regime because it is both experimentally accessible and theoretically more managable than the long time regime. The (short-time) self-diffusion coefficient is obtained from eqn (7.1) in the limit as k + 00 and is given by D, = lim D( k ) = kTN-’ 1 k*(p ;;) i. kAm I (7.3) The physical meaning of short-time self-diffusion can easily be understood by observing that D, characterizes self-diff usion on a timescale such that the root-mean-square displacement of a particular particle remains much smaller than the average distance between particles.It is so to say the diffusion of a particle in a cage formed by the surrounding particles. The hydrodynamic interactions with other particles play a role, but not yet the direct hard-sphere interaction. We have mentioned that to lowest order in the expansion in connectors the (transla- tional) mobilities pel, for i # j , contain terms of order R-’ and K3 [cf: eqn (5.1)]. Such terms might, when evaluating transport properties, give rise to complications, more specifically to divergent integrals. For self-diffusion where only )rli needs to be considered, the above long-range terms do not contribute.We shall show here that these terms, as known, cause no difficulties in the evaluation of D( k ) , not only in the limit as k -+ 00, i.e. for D,, but even for arbitrary values of k. From eqn (4.4) and (7.2) if follows that to lowest order in the expansion in connectors one has with the Stokes-Einstein diffusion coefficient In eqn (2.1) g ( r ) is the pair correlation function, (7.6) no = N / V the average density of spheres, and A(’,’)( r ) the monopole-monopole con- nector field defined as A(l,I)(,.) (7.7) r / ( R , = 4. A(1.1) Note that since A!.” is only defined for R,, > 2a, the connector field is also only defined in the range r > 2a. However, as g ( r ) = 0 for r < 2a any choice for the continuation of A(’,’)(r) for overlapping spheres leaves the integral in eqn (7.4) unchanged.With this in mind we may define A(”I’(r) for all r as [cf eqn (3.2)] A(’.’)( r ) = ( 2 7 ~ - ~ dk exp (-ik r)A(‘,’)(k) (7.8)P. Mazur with A","(k) = 6 n ( l - &)k-2[jo(ka)]2. Eqn (7.4) may then be written in the form 43 (7.9) d r exp (ik r ) [ g ( r ) - l]A('71)( r ) k" G ( k ) D ( k ) Do + n , L . d r exp (ik. r ) ( 2 7 ~ ) - ~ dk' exp (-ik' r ) A ' ' , ' ) ( k ' ) - i. (7.10) I I Performing the integration over r in the second term yields + n,c 0 aAi'31)( k ) k" d r exp (ik r ) [ g ( r ) - l]A'',')( r) k^ (7.11) (7.12) in view of eqn (7.9). Thus to lowest order in the expansion in connectors only a convergent integral remains [ g ( r ) -+ 1 for r -+ CO, sufficiently fast], which vanishes as k -+ 00.All contributions from higher-order terms in the expansion in connectors, or essentially higher-order terms in an expansion in interparticle distances, yield convergent contributions. This completes the proof that the long-range 1/R and l / R 3 hydrodynamic interaction terms do not give rise to any divergences of D( k ) , in particular of D ( k = 0), the collective diffusion coefficient of the suspension. Let us now consider the self-diffusion coefficient in somewhat more detail. If one inserts into the expression (7.3) for the self-diffusing coefficient the series (4.4), together with the explicit form of the connectors (see section 3), one can, in principle, evaluate D, as a power series in no = N / V (a so-called virial expansion). This has been done up to and including terms of second-order in the density." Up to this order only two- and three-body hydrodynamic interactions need to be considered, since the probability that a given sphere has s neighbours in of order n i .Furthermore, to this order one needs only a knowledge of the hard-sphere pair distribution g ( r ) function to first-order in no and of the three-sphere distribution function g ( R 1 2 , R I 3 , RZ3) to lowest-order. Thus one must insert into the relevant integrals (0 for r < 2a + +[8- 12r/4a + 4 ( r / 4 ~ ) ~ ] for 2a II r ~ 4 a for r > 4a 0 R12<2a or RI3<2a or R2,<2a g ( R12 R13 R23) = I 1 elsewhere. In eqn (7.13) 4 is the volume fraction of suspended spheres (7.13) (7.14) 47T 4 =-a'n,. 3 (7.15)44 Many-sphere Hydrodynamic Interactions Using the above expressions for the distribution functions, it was found that D,/D,= 1 - 1.734 + 0 .8 8 4 ~ + e(4’). (7.16) Only two-body hydrodynamic interaction contribute to the well known term of order # and are, therefore, the only ones to contribute at sufficiently low densities. At higher densities, however, the many-sphere hydrodynamic interactions may not be neglected: two-sphere contributions alone would have led to a value of -0.93#2 for the term of order #*, instead of the value of +0.88#2 in eqn (3.4). This illustrates dramatically the non-additivity of hydrodynamic interactions. We should mention here that in evaluating the coefficients in eqn (7.16), we have in the expansion of the mobility in inverse powers of intersphere distances neglected, both for the two- and three-body case, terms of order R-* and higher.It can be shown however that the terms neglected contribute at most a few percent. It is quite clear from the above results that in a concentrated suspension one fully has to take into account the many-body hydrodynamic interactions between an arbitrary number of spheres. A virial expansion is not appropriate at high densities. However, it is possible to resum algebraically contributions due to hydrodynamic interactions between an arbitrary number of spheres.12 If one then subjects, in combination with this resummation, the expression for D, to a fluctuation expansion, i.e. if one writes D, as a sum of contributions from density correlation functions of higher and higher order, one obtains already from the first two orders numerical values which for volume fractions # < 0, 3 agree reasonably well with experimental results.8. Sedimentation As we have seen, the long-range hydrodynamic interaction terms of order R-’ and K 3 do not contribute to self-diffusion and give rise to convergent integrals in the evaluation of collective diffusion. If one calculates, on the other hand, the velocity of sedimentation in an unbounded medium, this quantity diverges, a fact which is sometimes referred to as the Smoluchowski paradox. Pyun and Fixman16 have shown the way to avoid the difficulty caused by the 1/R divergence by considering sedimentation with respect to the mean volume flow. In this way one indirectly takes into account the backflow caused by container walls.However, even then the R-3 term still gives rise to a conditionally convergent integral and poses, as noted by Burgers,” the problem of a possible depen- dence of the sedimentation velocity on the shape of the vessel containing the suspension. Batchelor18 was able to assign a definite value to the integral in question using an argument based on general considerations of a physical nature (valid for the unbounded system). Ultimately the difficulties mentioned should be resolved by a direct and explicit evaluation of the influence of container walls on the mobilities of sedimenting particles. Such a calculation has been carried out for two geometries, first for the case of a plane wall19 and then also for the case of a spherical container.The latter case has been reviewed in section 5 . For a discussion of sedimentation one also has to consickr the fluid flow caused by the motion of suspended spheres (section 1.6). One is then, using the results found as outlined in the two sections mehtioned, in a position to evaluate the mean particle velocity up and mean fluid velocity uf of a homogeneous distribution of identical spheres, ai = a, = a, sedimenting inside a spherical container.20 These two quantities, calculated at the centre of the container and in the limit that its radius a, tends to infinity, may be written as conditional averagesP. Mazur 45 of= lim C S , ( r = R , ) ( R , , > a forallj O F . (8.2) a0-w ( 1 ) Here (- - - I R,, = 0) denotes an average over those configurations for which RIO = 0, while ( - - - I R,, > a for all j ) denotes an average over configurations for which no suspended sphere overlaps the centre of the container; F is the gravitational force (corrected for buoyancy on each of the particles).To linear order in the volume fraction 4 of suspended spheres calculation of up and of on the basis of eqn (8.1) and ( 8 . 2 ) and eqn ( 5 . 8 ) and (6.2) yields up = [ 1 - 3 . 5 5 4 + e( 4 ' ) ] ( 6 ~ ~ p - ' F of = 1124 + e( +2j](6T77a j-% ( 8 . 3 ) (8.4) We can now also determine the average volume velocity given by o, = &+,+ (1 - 4)uf. From eqn ( 8 . 3 ) and (8.4) it follows that o, = [ 3 4 + e ( 4 ' ) ] ( 6 ~ ~ p - ' F . (8.5) Since, because of incompressibility, the volume flux through any closed surface must vanish, this result, namely that there is a non-vanishing volume velocity at the centre, implies the existence of a vortex of convective flow in the spherical container.Finally, one may evaluate the mean particle velocity with respect to the average volume velocity. The result is V , - V , = [ 1 - 6 . 5 5 4 + e( +')IF. (8.6) This is the result found for this quantity by Batchelor for an unbounded system. It is also what is found for sedimentation perpendicular to and towards a plane wall, in which case o, vanishes, in the limit of an infinitely distant We therefore come to the following conclusions: the average local velocity of a sedimenting particle in an homogeneous suspension depends on the shape of the container, however far the container walls. However, this shape-dependence disappears for the sedimentation velocity with respect to the average volume velocity.These results illustrate once more the essential role played by hydrodynamic many- body interactions. They show that for sedimentation the 'three-body' hydrodynamic interaction of two-particles and the container can in fact, for non-zero values of 4, never be omitted from consideration, even for sufficiently dilute suspensions. The discussion of wall effects and of the non-additivity of hydrodynamic couplings in the previous section thus underscores the relevance and usefulness of the scheme developed and summarized in sections 2-6 for the evaluation of many-sphere hydro- dynamic interactions. The same methods can also be used for the treatment of a number of related problems.Thus they can be applied to the case that the suspended particles are liquid drops instead of solid hard spheres. In this way one extends to many drops the classic Rybczynski- Hadamard theory for a single drop.21 In conclusion, we mention that next to its application to hydrodynamic problems, essentially the same scheme can be used for the analysis of many-body dielectric or diamagnetic interactions in dispersions of dielectric (metallic) or superconducting spheres, respectively. References 1 M. Smoluchowski, Bull. Int. Acad. Polon. Sci. Lett., 1911, l A , 28. 2 See e.g. H. Faxin, Arkiv. Mat. Asiron. Fys., 1925,19A, N 13; J. Happel and H. Brenner, in Low Reynolds Number Hydrodynamics (Noordhoff, Leiden, 1973); J. M. Burgers, Proc. Kon. Ned. Acad. Wet., 1940, 43, 425; 646; 1941, 44, 1045; 1177; B. U. Felderhof, Physica, 1977, 89A, 373.46 Many-sphere Hydrodynamic Interactions 3 G. J. Kynch, J. Fluid Mech., 1959, 5, 193. 4 G. K. Batchelor, J. Fluid Mech., 1976, 74, 1; B. U. Felderhof, J. Phys. A, 1978, 11, 929; R. B. Jones, 5 P. N. Pusey and R. J. A. Tough, in Dynamic Light-scattering, ed. R. Pecora (Plenum Press, New York, 6 P. Mazur and W. van Saarloos, Physica, 1982, 115A, 21; P. Mazur, Helv. Phys. Acta, 1986, 59, 263. 7 M. Muthukumar, J. Chem. Phys., 1982, 77, 959. 8 T. Yoshizaki and H. Yamakawa, J. Chem. Phys., 1980, 73, 578. 9 S. Hess and W. Kohler, in Formeln zur Tensor-Rechnung (Palm und Enke, Erlangen, 1980). Physl’ca, 1979, 97A, 113. 1985). 10 U. Geigenmuller and P. Mazur, Physica, 1986, 136A, 316. 11 P. Mazur and A. J. Weisenborn, Physica, 1984, 123A, 209. 12 C. W. J. Beenakker and P. Mazur, Physica, 1983, 1204 398. 13 C. W. J. Beenakker and P. Mazur, Physica, 1985, 131A, 311. 14 C. W. J. Beenakker, Physica, 1984, 128A, 349 15 P. N. Pusey and R. J. A. Tough, J. Phys. A , 1982, 15, 1291. 16 C. W. Pyun and M. Fixman, J. Chem. Phvs., 1964, 41, 973. 17 J. M. Burgers, Proc. Kon. Ned. Acad. Wet., 1941, 44, 1045; 1177; 1942, 45, 9; 126. 18 G. K. Batchelor, J. Fluid Mech., 1972, 52, 245. 19 C. W. J. Beenakker, W. van Saarloos and P. Mazur, Physica, 1984, 127A, 451. 20 C. W. J. Beenakker and P. Mazur, Phys. Fluids, 1985, 28, 3203; 1985, 28, 767. 21 U. Geigenmuller and P. Mazur, Physica, 1986, 136A, 269. 22 U. Geigenmuller, to be published. Received 8th December, 1986