Rotational diffusion of a tracer colloid particle: II. Long time orientational correlations

Physica A 157 (1989) 752-768 North-Holland. Amsterdam

ROTATIONAL DIFFUSION OF A TRACER COLLOID PARTICLE

II. LONG TIME ORIENTATIONAL CORRELATIONS

R.B. JONES Department of Physics, Queen Mary College, Mile End Road, London El 4NS, UK

Received 23 November 1988

We study the orientational correlations of a tracer colloid particle which carries a magnetic moment. We assume the tracer interacts with other particles in the colloid suspension through spherically symmetric pair potentials and hydrodynamic forces. We derive correlation functions identities arising from the microscopic symmetries of rotation, time translation and time reversal. Using a previously derived cluster expansion representation and assuming there is no external magnetic field we obtain an approximate analytic expression for the two-body contribution to the correlation functions valid for all times. In the case of hard spheres with stick hydrodynamic boundary conditions we evaluate the long time correlation functions numerically to show that, within our approximation, the deviation from single exponential decay is small in a low density suspension.

1. Introduction

In a previous article [l] (henceforth called (I)) we studied the rotational correlations of a spherical colloid particle which carries a preferred axis representing either a permanent dipole moment (electric or magnetic) or an axisymmetric polarizability. Describing the dynamics of the interacting colloidal suspension by the generalized Smoluchowski equation [2] including rotations as was done previously by Wolynes and Deutch [3] and Montgomery and Berne [4] we obtained analytic expressions for the short time rotational correlation functions to first order in the suspension density including dependence on an external field. For a hard sphere suspension we used accurate hydrodynamic interaction functions to show that the short time decay of orientational correlations is characterized by a rotational diffusion coefficient Di(l - 0.634) where 4 is the volume fraction of the colloidal particles and 0; is the infinite dilution rotational diffusion coefficient of an isolated sphere.

At intermediate and long times the interactions in the suspension produce a non-exponential decay which cannot be characterized completely by a single diffusion coefficient but rather requires a complete solution to a time depen-

037%4371/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

R.B. Jones I Rotational diffusion of a tracer colloid particle II 153

dent dynamical problem. In the present article we consider the case of a hard sphere suspension with a single tracer carrying a permanent magnetic dipole moment. The rotational diffusion of the dipole axis is coupled to the rest of the suspension by hydrodynamic and potential interactions as well as to any external magnetic field that may be imposed. We will show how approximate orientational correlation functions for the particle can be obtained analytically for all times to first order in volume fraction 4 when the external field is set to zero. In a subsequent article we will treat the field dependence at finite times.

In section 2 we summarize the definitions and time evolution equation introduced in (I) and obtain correlation function identities related to the self-adjointness of the time translation operator. In section 3 we use the microscopic rotational and time reversal symmetries to derive correlation function identities of Onsager-Casimir type [5]. In section 4 we recall the low density dynamical equations derived in (I) and show that in the absence of an external field an approximate analytic expression for the two-body contributions can be obtained for all times. In section 5 we give expressions for the orientational correlation time and for the deviation from single exponential decay. In section 6 we present and discuss numerical results for the long time properties of the correlations.

2. Time evolution equation

In (I) we considered a suspension of N spherical particles of radius a immersed in an incompressible fluid described by the linearized Navier-Stokes equations for creeping flow. In the case when each particle carries a dipole moment the configuration of the ith particle is specified by giving the position of its centre Ri and the direction of its moment which we denote by a unit vector n,. Introducing 3N dimensional vectors X’ = (R, , R,, . . . , RN) and xr=Qz,,n,,..., nN) where superscripts t and r denote translation and rotation respectively, we can describe the configuration of the entire suspension by the 6N dimensional vector X = (X’, X’). The time development of the suspension can be described by a time dependent probability density P(X, t) which obeys a generalized Smoluchowski equation expressing probability conserva- tion in the configuration space.

In the absence of time dependent external forces acting on the system the generalized Smoluchowski equation can be written as

$ = 9P=V-D-[VP+ f3(V@)P]. (2.1)

Here p = 1 lk, T, D(X) is a 6N by 6N configuration dependent diffusion tensor,

754 R.B. Jones I Rotational diffusion of a tracer colloid particle II

V is the 6N dimensional gradient operator in configuration space and Q(X) is the total potential energy of the system including a wall potential and an external potential representing the interactions of a dipole moment with a static external field. The X dependence of the diffusion tensor D(X) expresses the hydrodynamic interactions in the suspension. More detailed expressions for D(X) and V are given in (I).

The normalized equilibrium solution to (2.1) is

(2.2)

where

2 = I dX e-p@(‘) . (2.3)

If we consider two observables A(X), B(X) then as explained in (I) the equilibrium time correlation function C,,(t) can be expressed as ((. . -)

denoting an equilibrium ensemble average)

= dX P,(X)A(X) ey’B(X) , (2.4)

where 2 is the formal adjoint of the operator 9 occurring in the generalized Smoluchowski equation,

c!F=9+=[V-p(V@)].D4’. (2.5)

In trying to calculate C,,(t) it is often easier to work with its Fourier-Laplace transform which is expressible as

x

t&4 = 1 d t e’“‘C,,(t) = - ( A(X)[iw + L.f-‘B(X)) .

II (2.6)

The functions C_,,,(w) are particularly useful in studying long time behaviour of correlations since the long time regime corresponds to the zero frequency limit.

We conclude this section by mentioning two identities for the correlation functions C,,(t) that follow from the form of the time translation operator 2. To give an heuristic derivation of the identities we introduce a Hilbert space structure to the space of observables by defining a scalar product,

R. B. Jones I Rotational diffiuion of a tracer colloid particle II 155

(E, F) = j- dX WW*(X)W) , (2.7)

in terms of which the correlation functions can be expressed as

C,.,(t) = (E, e*‘F) . (2.8)

However, it follows trivially from (2.2) and (2.5) with an integration by parts that the operator 2 is self-adjoint with respect to the scalar product defined above so that

C,.,(t) = (E, e”*F) = (e”‘E, F) = C;.,(t) = C,,,(t) . (2.9)

Moreover, if we consider the eigenvalue problem for $8,

where (U,, U,,) = 1, we can express the eigenvalues in the form

I-L, = (U,, ,JfU,) = - I dX P,(X)(VU,) - D 0 (VU,) . (2.11)

Since the diffusion tensor must be positive definite for a dissipative system we see that all eigenvalues are negative apart from a single zero eigenvalue corresponding to the eigenvector U,,(X) = 1. Expressing the correlation functions in terms of an eigenfunction expansion,

C,*,(t) = c (6 e~WJW,, F) = C e-‘n’(E, U,)(U,, F) , (2.12) n n

we obtain a long time factorization of the correlation functions

p$ C,*,(O = (6 wu,, f9 = (E*) (F) . (2.13)

3. Tracer correlations and symmetries

Just as in article (I) we study the orientational correlation functions of a tracer particle which we choose to be particle 1 with position R, and orientation n,. The direction of q can be specified with respect to a space fixed coordinate system by spherical polar angles (0,) cpl) denoted collectively by 0,. We consider the time evolution of nl(t) for the case that the spherical particles interact through spherically symmetric pair potentials while the tracer alone is

756 R. B. Jones I Rotational diffusion of a tracer colioid particle II

assumed to interact with a static homogeneous external magnetic field H. The form of Q(X) is then

(3.1)

where w is a wall potential and

,y(n,)=x(O,)=-Mn,.H, (3.2)

with A4 the magnitude of the tracer magnetic moment. This choice of potential corresponds to turning off the magnetic moment of all particles except the tracer so there are no long range dipole-dipole magnetic interactions between particles. A further consequence of this assumed potential is that in the thermodynamic limit of a large system when wall effects can be ignored the equilibrium distribution P,(X) becomes translationally invariant and moreover factorizes into translational and rotational parts,

P,(X) = Pk(X*)P;(X’) ) (3.3)

with

C(X’) = -ji exp{ -P 5 u((& - RJ)} , i<,

(3.4)

(3.5)

where Z’ is the configuration integral associated with the spherically symmetric pair interactions and

zf = sinh E

dR, exp{PMn, -H} = 4~ - E ’ (3.6)

E=&%~(HI. (3.7)

The general form of the tracer orientational time correlation functions [6] is

Crm,;,m(f> HI = (Y~,,(n,(O))Y,,(n,(t))), = (Y~,,(a,(O))Y,,(n,(t))), 3

(3.8)

where the Y,,(n,) are the spherical harmonics defined with the phase conven-

R. B. Jones I Rotational diffusion of a tracer colloid particle II 757

tions of Edmonds [7]. A related set of correlation functions is defined by

where P, is a Legendre polynomial. By the spherical harmonic addition theorem [7] we have the identity

(3.10)

We can derive a number of formal properties of the Cl,m,;Im by using the results (2.9) and (2.13) together with the microscopic symmetries of the system. Because we are interested in equilibrium time correlations we can use the time translation invariance or stationarity of the correlation functions. Thus we have

Gd;h(~> HI = (y~,,(n,(O))Y,,(n,(t))),

= (y~,,(a,(-t))Y,,(n,(O))), = G?;h’w~ H) * (3.11)

If we use the self-adjointness of the time translation operator as in (2.9) we find

G’m’;Im(4 fo = CI*m;,‘m’(f, m f (3.12)

Next we can exploit the continuous rotational symmetries as well as the discrete time reversal symmetry which must hold for the microscopic time evolution of the system at the level of the Liouville equation. First consider the case when the external magnetic field vanishes, iY = 0. In such a case the equilibrium ensemble over which we average must be rotationally invariant. This means that the value of Cl,mP;Im must be the same no matter what space fixed frame of reference we use to define the harmonics Y,,(Q). If we choose a space fixed frame that is related to the original one by a rotation with Euler angles CY, /3, y then the harmonics in the original frame are related to those in the rotated frame by [7]

q=-1

where 0; denotes the polar angles defined with respect to the new frame. By rotational invariance however (Y~,,(fl,(O))Y,,(R,(t))) must be the same as (Y&,,(fl;(O))Y,,(R;(t))) so that we have

7.58 R. B. Jones I Rotational diffusion of a tracer colloid particle II

(3.14)

Integrate both sides of this equation over the rotation group and use the orthonormality [7] of the rotation matrices LB:.?,

to get

Gn’$&~ 0) = 21+ 1 q !LL!-%E c CIqiIq(t, 0) = * qt, 0) . (3.16)

Next assume that a non-vanishing external field H acts on the system and use the direction of H to define the polar axis of the coordinates 0, = (0,) cp,). The rotational invariance group is reduced to the subgroup of rotations about the direction of H. This subgroup corresponds to rotations with (Y = 0, p = 0, 0 s y s 21~. If we integrate (3.14) over the rotations about the axis of H we derive the weaker relation

G’m’;lm(~> HI = LmGn;lmk ff) . (3.17)

Finally we can utilise the microscopic time reversal invariance of the system. Under time reversal both the magnetic moment and the external field are reversed (n, + -n,, H-+ -H) so that we have

(Y~,,(n,(O))Y,,(n,(t)))* = (Y~,,(-n,(O))r,,(-n,(-t)>) -H 7 (3.18)

from which we get, using (3.17) and Y,,(-nl) = (-l)‘Y,,,,(n,) ,

Gm;l,n(~~ H) = (-l)““C,.,;,,(-t, -ff) ) (3.19)

a relation of Onsager-Casimir type [5]. If we combine (3.19) with (3.11) and (3.12) we obtain

Cl.m;Im(t, H) = (-l)~+“C~~m;,m(t, -H) . (3.20)

It follows from (3.20) that the simpler correlation functions C[(f, H) defined in (3.9) are even functions of the magnetic field. It is worth noting that if we had an electric dipole moment and external electric field E instead of the magnetic problem treated above, then (3.20) still holds with H replaced by E and both time reversal symmetry and space inversion symmetry used in the derivation.


One further formal result follows from the long time limit relation (2.13)

If we use the expressions (3.3), (3.4) and (3.5) we find

(3.22)

where it(E) = (n/2e)“‘I n+ 1 JE) is a modified Bessel function of the first kind.

A long time result follows for C,(t, H) in the form

ei,(e) * f;E W,H) = sinh .

[ 1 (3.23)

4. One and two body contributions

In (I) we carried A out a cluster expansion of the correlation functions

Cl,,,,,+,(~) to obtain explicitly the separate contributions due to 1,2, . . . , p

body dynamical processes,

The one and two body contributions have the form

Here n, is the number density of the suspension, R = R, - R, is the relative separation vector of particle 2 with respect to the tracer particle 1, g(R) is the spherically symmetric pair distribution function and pzl is the equilibrium distribution function for the tracer orientation defined in (3.5). The functions I,%!; (0,) are solutions to one and two particle dynamical equations,

-b + ~J91s4) = y,,(q) , (4.4)

760 R. B. Jones I Rotational diffusion of a tracer colloid particle II

where

Lq = D;L; - /?Di(L,x) -L, ) (4.6)

z,, = i {[Vi - p(V,u)] - D;; -V, + [Vi - p(V,u)] - D:; - L, r,j=l

+ [Lo - /I(L~x)] - D,: 47, + [L; - P(L;x)] - D,; - L,) . (4.7)

In (4.7) V, is the gradient operator with respect to the position vector R, of particle i and Lj is the dimensionless angular momentum operator associated with the direction of rzi. Since there is no dependence on rz2 in (4.3) the operator L, can be put to zero leaving L, as the only effective operator acting on orientations. It has the form

L, = n, x Vf ) (4.8)

1 a v;=ij -$+&, - - 1 sin 0, acp, ’ (4.9)

with 6, ,&i unit vectors in the 0,) ‘p, directions. The various two body diffusion tensors [g-10] have the form

(4.10) DITpy = k,T{a;(R)ii,&,, + P:f(R)(Gpy - &,liJ} ,

with the aij(R), P,(R) scalar functions of separation and d a unit vector in the direction of R.

In our previous article (I) we were interested in the short time (high frequency) regime in which the operators 2,) 2Zle,, act once only on Y1,(L!,). In that case Zi, greatly simplifies. At longer times however LZ1* acts to all orders on the spherical harmonic and it does not simplify to the same extent because of the dependences which occur in the coefficient functions of 2& itself. Thus eq. (5.9) of (I) is incorrectly printed and should be corrected to eq. (4.5) above. In fact, by expressing Vi, V, in terms of the centre of mass, 4 (R, + R2), and relative coordinate R, and using the tensor structure and symmetries [8,9] of the diffusion tensors in (4.10) we can express Zi, as


Z12 = 2;, + LfF2 + %Y;* ) (4.11)

where

+~(P;;-P::)(RXV).(RXV),

CJT,=- qfqp:;-pyl)(Rxv)*Ll

+ y (P:‘, - P:‘,)P&X)~(~ x V) 7

Z’i2 = k,T[&;L; + (a;; - ~;;)(lt~L,)2]

(4.12)

-PO&x) of; ‘L, .

Note that all reference to the centre of mass variable disappears since it does not occur in either YI,(O,) or in any of the coefficient functions of LZ12. The operator V is the gradient operator with respect to the relative position.

The operators JZi, and _!ZF2 vanish when acting on Y,,(O,) so that in a high

frequency expansion,

-1 1 1

io + &, =--+

~~12-~(~12)2+~(%2)3.... (4.13) iw (io)

one can replace 9’,, by Z’t2 in the linear term and in the quadratic term (by using integration by parts to throw one factor of Zr, onto Yc,,,, in (4.3)), but not in the cubic or higher terms. Since L?i2, Z’y2 and 9~‘;~ do not commute there is no way to solve (4.5) exactly. However, the observation that at short times the dynamics can be expressed in terms of Zi2 alone suggests an approximate perturbative scheme for obtaining $~~)(L?,). We expand

-1 -l +

iw + JZ12 = iw + Z;2 ;,;& (~;2+~~2) -l +*a*, (4.14)

12 iw + 2i2

to express $ji’(O,) as’

(4.15)

In 5?;2 itself, (L,)’ and J@ - L, commute with each other but not with the

762 R. B. Jones / Rotational diffusion of a tracer colloid particle II

external field dependent term L,,y. If we set the external field to zero (H = 0), we obtain an approximation to $,m (‘) which as we shall show, can be expressed ,

in closed analytic form,

9 (2) = -1 Im iw + Lf;, ff=o

Y ‘m

Henceforth in the present article we (4.16) to evaluate the orientational treatment of the external field and of to a later study.

will use the approximate $I:’ defined in correlation functions. The perturbative the higher order terms in (4.13) we leave

When the external field is set to zero, H = 0, the one body operator in (4.6)

becomes

(4.16)

2, = D;L; ,

while the operator L!?i2 becomes

(4.17)

L?i2 = k,T{Pf’,Lf + (&;‘I - /3f:)(k*L,)‘} . (4.18)

The one body equation (4.4) can be solved trivially because the Y[,(LZ,) are eigenfunctions of L f,

LTY,, = -f(Z + l)Y,, . (4.19)

Thus we can write for $“‘(0,) the result

Ic,~~‘(fi,) = [-iw + DiI(Z + l)]-‘Y,,(Q) . (4.20)

To obtain +~~‘(0,) in the approximation defined by (4.16) requires an analysis which takes account of the fact that i + L, is a component of L, along a direction different from the space fixed axis with respect to which the m label in the Y1,,, is defined. We have to solve the equation

-[iw + k,Tp’,‘,(Z?)L: + k,T(a~~(R) - PT’1(R))(R.L,)2]~l~‘(n,)

= Yh(fi,). (4.21)

Since Li and R * L, commute, we can solve (4.21) by expanding Y,,,(&) in terms of the simultaneous eigenfunctions of Lf and & - L, . These eigenfunctions however are just the Y,,(@), the spherical harmonics defined with respect to a rotated frame whose polar axis is along 1. If LY, p, y are the Euler angles of the rotation from the original space fixed frame in which Y,,(n,) is defined to a


frame in which R defines the z axis then we have that (Y = (PR, p = 0, and y is arbitrary, where da, (PR are the polar angles of P in the original space fixed frame. Using the rotation law (3.13) we have

Yh(f47 cpl) = ,$_, YqM, 4)qh, e,, y) ) (4.22)

and its inverse

(4.23)

The Y,,(e;, cp;) are eigenfunctions of both Li and (J? . L,)*,

L:Y,,(w = -v + w&u 3 (~~L,)*y,,M) = -cty,,w;> , (4.24)

so that using (4.22) in (4.21) and then backtransforming with (4.23) gives for

4CW,)

(4.25)

If we again use (4.22) and (4.23) in (4.20) we find for the combination

*I:) - (Cllm (l) which occurs in the two body contribution

* ;;I - +I;) = c {[- iw + k,7'1(1 + l)pi’, + k,Tq*(aI’, -pan)]-’ 4m’

- [-iw + DbI(I + l)]-‘}Y~,,9~~,9~~ . (4.26)

From the result above we observe that er,,, (*) - +!J:;) depends upon 0, through the Y,,,,,(n,), upon the separation R through the scalar mobility functions (Y;;(R), Pi;(R), and upon the direction of R through the rotation matrices 9(‘) . From (4.3) we see that to evaluate the two body contribution to C?lPmC:lm we?must integrate over all directions of R and over all separations R. These integrations we can write as

j-dR...= jR”dR/dcosf.$dyl,--*

*a

=&jR2dRIdqRdcosB,dy...,

*a

(4.27)


where 2a is the closest distance of approach permitted by the pair interactions. Note that we can introduce an integration over all values of the Euler angle y because nothing in the original integrand of (4.3) depends on this angle. By introducing such an integration however we can evaluate the average over the direction of k by use of the orthonormality relations (3.15) for the g$.

Remembering from (3.5) that in zero field P:,(LI,) = 1/47r we finally carry out the averages over 0, to obtain the results

(4.28)

kf,;!Pp,‘(w) = “$‘f~‘m [ dR R2g(R) i {[-iw + k,TI(f + l)PIi(R) 2a

q=-/

+ k,Tq2(cq - fit’,(R))]-’ - [-iw + DhZ([ + 1)1-r} . (4.29)

By using the detailed form of the mobility functions (Y ;‘I, pi’, as given elsewhere [8, 91 it can be seen that as R+ x we have k,T@f’,(R)-+ Dfi + 6’((alR)“), while for the difference of mobility functions we find (a;:(R) -

P;:(R))+ fWR)“h so that the terms in the summation in (4.29) vanish as Re6 at large separation giving an absolutely convergent integration over the separation distance.

5. Long time correlations at low density

If we approximate the pair distribution function g(R) by its low density value, exp(-pu(R)), we obtain from (3.16), (4.28) and (4.29) the first two terms of a virial expansion of tl(o) in powers of the volume fraction l#J = 4nn,a’/3,

Q(W) = Q”(O) + q6C’12’(w) + . . . ) (5.1)

with

ty’(w) = . l -1~ + D;l(Z + 1) ’

dR R2 e~P~(W

X ,C, {[-iw + k,TI(f + l)pl’, + k,Tq2(a;‘, - PT:)Im

- my’} .

(5.2)

(5.3)


We define the orientational correlation time T! by [6]

m

r, = I

C,(t) dt = i;(O) . (5.4) 0

This quantity gives the mean decay time of the correlation function and its reciprocal l/r, can be viewed as a long time quantity to be compared with the short time rotational diffusion coefficient, Di(1 - 0.634). One cannot regard 1 /rr as a long time rotational diffusion coefficient since it is evident from (5.3) that C!(t) is not a single exponential at long times. If we put w = 0 in (5.2) and (5.3) and then use (5.4) we obtain, to first order in r$, the following expression for l/r,:

71 -l = D;E(1+ l)[l - +c;(I). * *] )

where

(5.5)

dR R2 e-P~(R)

x $ Z(1+ l)(l - &rvA;P;;) - 87~77A;(cz;; - p;;)q’

f(1+ l)hrqA:/3;‘, + 8n77A;(a;‘, - p;;)q2 . (5.6)

q=-/

In this last equation 77 is the viscosity of the suspending fluid and AT is a scattering coefficient [S, 91 that measures the drag felt by a single sphere rotating in the fluid. The diffusion coefficient Of, is related to AT by the Einstein relation DT, = k, TISq AT. The coefficient AT and the mobility

functions af’, , PI’, d p d e en on the hydrodynamic boundary conditions at the surface of the spherical colloid particles [8, 91. For stick boundary conditions A; = a3.

From the expressions (5.2) and (5.3) it is easy to carry out the inverse Fourier-Laplace transform to obtain C,(r) explicitly to first order in 4. The one body contribution is

Cl” = exp[- Di1(1+ l)t] . (5.7)

The two body contribution is

dR R* e-M0

X q$_t Wp[-kJW + W%‘, + (aI’, - PI%‘)4 - C!‘)(t)> . (5-B)


It is convenient to write

G(t) ___ = 1 + $7,(t) + . . . ) c,“‘(t) (5.9)

thereby introducing the function -y,(t) which measures the departure from

single exponential decay,

dR R” e-fiu(R)

x ,E, {exp[-k,V~(~ + l)(Pl’I - Po6) + ((-~frl - P11’,)s’Pl- I> . (5.10)

The expressions above, which incorporate the approximation in (4.16),

depend upon the interparticle pair potential u(R) and on the hydrodynamic

boundary conditions at the surface of the particle which affect the coefficient

AT and the functions a;:(R), Pi',(R). In (I) we studied the effect of different

boundary conditions and found that the greatest effect on the rotational

diffusion occurs with stick boundary conditions. In the present case we are

content to evaluate CL(Z) and n(t) for hard spheres (v(R) = 0 for R > 2~) with

stick boundary conditions. As explained in (I) the functions al’;, p’,‘, are

known accurately in the range 2. la d R d ~0 by series expansions [9, lo] in

powers of l/R and in the near touching region (2~ d R i 2.1~) by numerical

values due to Kim and Mifflin [ll].

6. Results and discussion

We have evaluated CL(I) and X(t), as given in the previous section, for hard

spheres with stick boundary conditions at three different Z-values, I = 1,2,3.

The numerical results show that CL(l) and y,(t) are each only weakly depen-

dent upon the 1 value. We find for 1 = 1,2,3

C;(I) = 0.67, (6.1)

with an error of no more than two percent. The dependence on I appears only

in the third significant figure of these calculations. If we compare the short time

rotational diffusion coefficient Dt(l - 0.634) with the long time quantity

[1(1 + l)r,]-’ = D;,(l - 0.674) we see that to first order in volume fraction the

deviations from single exponential decay must be quite small. If we look at the

analogous low density quantities for the translational diffusion of a tracer we

have a short time translational diffusion coefficient [12] 0: = Dh(1 - 1.834)


and a long time diffusion coefficient [13, 141 Dt = Dk(1 - 2.10$). There is a striking similarity between rotational and translational diffusion in the small contribution of non-exponential behaviour at intermediate and long times.

The non-exponential contribution to C,(t) is contained in the function n(t). We have computed this for a range of t values and for I = 1,2,3. Just as with CL(l) the I dependence is weak so we have plotted only one example, X(t), in fig. 1 as a function of the dimensionless time variable s = DiI(Z + 1)t. The deviations from the one particle correlation function C;“(t) are quite small for volume fractions below ten percent. In fig. 2 we plot In C*(t) for 4 = 0.05 and compare with In C:‘(t). It is clear that the deviation from the non-interacting correlation function C:‘) would be difficult to see experimentally at low volume fractions. The small contribution of y,(t) at intermediate times is analogous to the small non-Gaussian contribution to translational tracer diffusion at intermediate times as seen in analytic calculations [15], in experimental light scattering data [16] and in computer simulation studies [17].

The calculations reported above are at zero external field and include the approximation (4.16) to the two body term. If H # 0 then from (3.23) we see that the correlation function C,(t) does not decay to zero as in the instance treated above. This circumstance may make it easier experimentally to observe the correlations at intermediate times. It is important therefore to ask how the external field and the higher order correlations which appear in (4.14) modify the behaviour of C,(t) at intermediate and long times. The relation (3.20)

shows that for low fields the effect on CJt, H) must be second order in H. In a subsequent article we will evaluate numerically these additional contributions by a perturbative calculation starting from the approximate solution presented here.

0.0 1.0 2.0 3.0 A.0 5.0 6.0

S

Fig. 1. Plot of y,(t) as a function of the dimensionless time variable s = 6Dit.


1

- -2.0 - m

--& 0 -3.0 -

5 -L.O -

-5.0 - \

-6.0 L 1 I I I I

0.0 1.0 2.0 3.0 4.0 50 6.0

S

Fig. 2. Plot of In CZ(t) (solid line) as a function of the dimensionless variable s = 6DSt for a volume

fraction 4 = 0.05. The dashed line is a plot of the function In C:“(t) which shows the single

exponential decay characteristic of an isolated particle without interactions.

Acknowledgement

I wish to thank the University of London Central Research Fund for a grant

which provided the computer software used for the calculations presented

above.

References

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Rotational diffusion of a tracer colloid particle: II. Long time orientational correlations