- Email: [email protected]
Rotational diffusion of a tracer colloid particle
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
Received 23 November
El 4NS,
UK
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 depen037%4371/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
R.B. Jones I Rotationaldiffusion 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 conservation 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].
Here p = 1 lk, T, D(X) is a 6N by 6N configuration
(2.1) dependent
diffusion tensor,
R.B. Jones I Rotational diffusion of a tracer colloid particle II
754
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) equilibrium time correlation function C,,(t) denoting an equilibrium ensemble average)
dX P,(X)A(X)
=
ey’B(X)
then as explained in (I) the can be expressed as ((. . -)
,
(2.4)
where 2 is the formal adjoint of the operator Smoluchowski equation,
9 occurring in the generalized
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
= 1dt e’“‘C,,(t)
= - ( A(X)[iw
+ L.f-‘B(X))
.
(2.6)
II
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
(E,
F)
= j- dX
WW*(X)W)
in terms of which the correlation C,.,(t)
= (E, e*‘F)
,
155
(2.7)
functions can be expressed as
.
(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)
Moreover,
=
(E, e”*F) = (e”‘E, F) = C;.,(t)
= C,,,(t)
.
(2.9)
if we consider the eigenvalue problem for $8,
where (U,, U,,) = 1, we can express the eigenvalues in the form I-L, =
(U,, ,JfU,) = -
I
dX P,(X)(VU,)
(2.11)
- D 0(VU,).
Since the diffusion tensor must be positive definite for a dissipative system we see that all eigenvalues are negative apart from a single zero eigenvalue corresponding to the eigenvector U,,(X) = 1. Expressing the correlation functions in terms of an eigenfunction expansion,
C,*,(t) =
cn (6 e~WJW,,F) = Cn e-‘n’(E,U,)(U,,
we obtain a long time factorization
p$ C,*,(O= (6
wu,,
of the correlation
f9 = (E*) (F) .
F) ,
(2.12)
functions (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 form of Q(X) is then
external magnetic field H. The
(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((& i<,
RJ)} ,
(3.4)
(3.5) where Z’ is the configuration pair interactions and zf =
dR, exp{PMn,
integral associated with the spherically symmetric
sinh E E ’
-H} = 4~ -
(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
tions of Edmonds
[7]. A related set of correlation
where P, is a Legendre polynomial. theorem [7] we have the identity
757
functions is defined by
By the spherical
harmonic
addition
(3.10) We can derive a number of formal properties of the Cl,m,;Imby 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) * If we use the self-adjointness find
of the time translation
G’m’;Im(4 fo = CI*m;,‘m’(f, mf
operator
(3.11)
as in (2.9) we
(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;Immust 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
R. B. Jones I Rotational diffusion of a tracer colloid particle II
7.58
(3.14) Integrate both sides of this equation over the rotation orthonormality [7] of the rotation matrices LB:.?,
group and use the
to get
c CIqiIq(t, 0) Gn’$&~ 0) = !LL!-%E 21+
1
q
= *
qt,
(3.16)
0) .
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)>) from which we get, using (3.17) and Y,,(-nl)
Gm;l,n(~~ H) = (-l)““C,.,;,,(-t, a relation of Onsager-Casimir (3.12) we obtain
)
-ff)
= (-l)‘Y,,,,(n,)
-H
7
(3.18)
,
(3.19)
type [5]. If we combine (3.19) with (3.11) and
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.
R. B. Jones I Rotational diffusion of a tracer colloid particle II
759
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+ 1JE) is a modified Bessel function of the first kind. A long time result follows for C,(t, H) in the form
f;E W,H) = [
ei,(e) * sinh
1
(3.23)
.
4. One and two body contributions In (I) we carried out a cluster expansion of the correlation functions A 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)
z,, = i
-L,
(4.6)
)
{[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)
)
a v;=ij -$+&, -sin1 0, -acp, ’
(4.9)
1
with 6, ,&i unit vectors in the 0,) ‘p, directions. tensors [g-10] have the form
The various two body diffusion
(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
R.B. Jones I Rotational diffusion of a tracer colloid particle II
Z12 = 2;, +
LfF2+ %Y;*
761
(4.11)
)
where
+~(P;;-P::)(RXV).(RXV), CJT,=-
qfqp:;-pyl)(Rxv)*Ll (4.12) + y
(P:‘, - P:‘,)P&X)~(~
Z’i2 = k,T[&;L;
+ (a;;
of;‘L,
-PO&x)
x V) 7
- ~;;)(lt~L,)2]
.
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 _!ZF2vanish when acting on Y,,(O,) so that in a high frequency expansion, -1 =--+ 1 io + &, iw
1 ~~12-~(~12)2+~(%2)3....
(4.13)
(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 to express $ji’(O,)
(~;2+~~2) 12
-l +*a*, iw + 2i2
(4.14)
as’ (4.15)
In 5?;2 itself, (L,)’
and J@- L, commute
with each other but not with the
R. B. Jones / Rotational diffusion of a tracer colloid particle II
762
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,
-1
(2) =
9 Im
iw + Lf;,
ff=o
Y
(4.16)
‘m
Henceforth in the present article we will use the approximate $I:’ defined in (4.16) to evaluate the orientational correlation functions. The perturbative treatment of the external field and of the higher order terms in (4.13) we leave to a later study. When the external field is set to zero, H = 0, the one body operator in (4.6) becomes 2, = D;L;
,
while the operator
(4.17)
L!?i2 becomes
L?i2 = k,T{Pf’,Lf
+ (&;‘I - /3f:)(k*L,)‘}
The one body equation eigenfunctions of L f, LTY,,
= -f(Z + l)Y,,
.
(4.18)
(4.4) can be solved trivially because the Y[,(LZ,) are
(4.19)
.
Thus we can write for $“‘(0,)
the result
Ic,~~‘(fi,) = [-iw + DiI(Z + l)]-‘Y,,(Q)
.
(4.20)
To obtain +~~‘(0,) in the approximation defined by (4.16) requires an analysis which takes account of the fact that i +L, is a component of L, along a direction different from the space fixed axis with respect to which the m label in the Y1,,,is defined. We have to solve the equation -[iw + k,Tp’,‘,(Z?)L: = Yh(fi,).
+ k,T(a~~(R)
- PT’1(R))(R.L,)2]~l~‘(n,)
(4.21)
Since Li and R * L, commute, we can solve (4.21) by expanding Y,,,(&) in terms of the simultaneous eigenfunctions of Lf and & - L, . These eigenfunctions however are just the Y,,(@), the spherical harmonics defined with respect to a rotated frame whose polar axis is along 1. If LY,p, y are the Euler angles of the rotation from the original space fixed frame in which Y,,(n,) is defined to a
R.B. Jones I Rotational diffusion of a tracer colloid particle II
163
frame in which R defines the z axis then we have that (Y= (PR,p = 0, and y is arbitrary, where da, (PRare the polar angles of P in the original space fixed frame. Using the rotation law (3.13) we have
Yh(f47cpl)=
,$_, YqM, 4)qh,
e,, y)
(4.22)
)
and its inverse (4.23) The Y,,(e;, cp;) are eigenfunctions L:Y,,(w
= -v
+ w&u
of both Li and 3
(J? .L,)*,
(~~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 (l) which occurs in the two body contribution *I:) - (Cllm * ;;I - +I;) = c
{[- iw + k,7'1(1 + l)pi’, +
k,Tq*(aI’,
-pan)]-’
4m’
[-iw + DbI(I + l)]-‘}Y~,,9~~,9~~
-
.
(4.26)
(*) - +!J:;)depends upon 0, through From the result above we observe that er,,, 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.$dyl,--* *a
=&jR2dRIdqRdcosB,dy..., *a
(4.27)
764
R. B. Jones I Rotational diffusion of a tracer colloid particle II
where 2a is the closest distance of approach permitted by the pair interactions. Note that we can introduce an integration over all values of the Euler angle y because nothing in the original integrand of (4.3) depends on this angle. By introducing such an integration however we can evaluate the average over the direction of k by use of the orthonormality relations (3.15) for the g$. 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)=
“$‘f~‘m [ dR R2g(R)
i
{[-iw + k,TI(f + l)PIi(R)
q=-/ 2a +
k,Tq2(cq
-
fit’,(R))]-’
-
[-iw + DhZ([ + 1)1-r} .
(4.29)
By using the detailed form of the mobility functions (Y;‘I, pi’, as given elsewhere [8, 91 it can be seen that as R+ x we have k,T@f’,(R)-+ Dfi + 6’((alR)“), while for the difference of mobility functions we find (a;:(R) so that the terms in the summation in (4.29) vanish as P;:(R))+ fWR)“h Re6 at large separation giving an absolutely convergent integration over the separation distance.
5. Long time correlations
at low density
If we approximate the pair distribution function g(R) by its low density we obtain from (3.16), (4.28) and (4.29) the first two value, exp(-pu(R)), terms of a virial expansion of tl(o) in powers of the volume fraction l#J= 4nn,a’/3, Q(W) = Q”(O)
+ q6C’12’(w)+ . . . )
(5.1)
with
ty’(w) =
-1~
.
+
l
(5.2)
D;l(Z + 1) ’ dR
X
,C, {[-iw
-
my’}
.
R2
e~P~(W
+ k,TI(f + l)pl’, +
k,Tq2(a;‘,
- PT:)Im
(5.3)
R. B. Jones I Rotational diffusion of a tracer colloid particle II
We define the orientational
correlation
765
time T! by [6]
m
r, =
I
C,(t) dt = i;(O) .
(5.4)
0
This quantity gives the mean decay time of the correlation function and its reciprocal l/r, can be viewed as a long time quantity to be compared with the short time rotational diffusion coefficient, Di(1 - 0.634). One cannot regard 1 /rr as a long time rotational diffusion coefficient since it is evident from (5.3) that C!(t) is not a single exponential at long times. If we put w = 0 in (5.2) and (5.3) and then use (5.4) we obtain, to first order in r$, the following expression for l/r,: 71-l = D;E(1+ l)[l - +c;(I).
* *] )
(5.5)
where dR R2 e-P~(R)
x $ q=-/
Z(1+ l)(l - &rvA;P;;) f(1+ l)hrqA:/3;‘,
- 87~77A;(cz;; - p;;)q’ + 8n77A;(a;‘,
- p;;)q2
.
(5.6)
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 e p en d 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] . The two body contribution
is dR
X
(5.7)
R*
q$_t Wp[-kJW
e-M0
+ W%‘,+ (aI’, - PI%‘)4 - C!‘)(t)>. (5-B)
R. B. Jones I Rotational diffusion of a tracer colloid particle II
766
It is convenient G(t) ___ c,“‘(t)
to write
= 1 + $7,(t)
+ ...)
thereby introducing the single exponential decay,
function
dR
(5.9) -y,(t) which
measures
the
departure
from
R” e-fiu(R)
x ,E, {exp[-k,V~(~ + l)(Pl’I - Po6) + ((-~frl - P11’,)s’PlI> . (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 ~0by 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 dependent 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)
R. B. Jones I Rotational diffusion of a tracer colloid particle II
767
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.
R. B. Jones I Rotational diffusion of a tracer colloid particle II
768
1 -
m
--& 0 5
-2.0
-
-3.0
-
-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 fraction 4 = 0.05. The dashed line is a plot exponential decay characteristic of an isolated
of the dimensionless variable s = 6DSt for a volume of the function In C:“(t) which shows the single particle without interactions.
Acknowledgement I wish to thank the University which provided the computer above.
of London Central Research Fund for a grant software used for the calculations presented
References [l] (21 [3] [4] [S] [6] [7] [S] [9] [lo] [ll] [12] [13] [14] [15] [16] [17]
R.B. Jones, Physica A 150 (1988) 339, to be referred to as (I). R. Zwanzig, Adv. Chem. Phys. 15 (1969) 325. P.G. Wolynes and J.M. Deutch, J. Chem. Phys. 67 (1977) 733. J.A. Montgomery Jr. and Bruce J. Berne, J. Chem. Phys. 67 (1977) 4589. S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics (North-Holland, Amsterdam. 1962). B.J. Berne and R. Pecora. Dynamic Light Scattering (Wiley, New York, 1976). A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, 1957). R. Schmitz and B.U. Felderhof, Physica A 116 (1982) 163. R.B. Jones and R. Schmitz, Physica A 149 (1988) 373. B. Cichocki, B.U. Felderhof and R. Schmitz, PhysicoChemical Hydrodynamics 10 (1988) 383. S. Kim and R.T. Mifflin, Phys. Fluids 28 (1985) 2033. G.K. Batchelor, J. Fluid Mech. 74 (1976) 1. G.K. Batchelor, J. Fluid Mech. 131 (1983) 155; 137 (1983) 467. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 89 (1988) 1049; 3705. B. Cichocki and R.B. Jones, Z. Phys. B 68 (1987) 513. W. van Megan and S.M. Underwood, J. Chem. Phys. 88 (1988) 7841. W. van Megan and I. Snook, J. Chem. Phys. 88 (1988) 1185.
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
Received 23 November
El 4NS,
UK
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 depen037%4371/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
R.B. Jones I Rotationaldiffusion 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 conservation 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].
Here p = 1 lk, T, D(X) is a 6N by 6N configuration
(2.1) dependent
diffusion tensor,
R.B. Jones I Rotational diffusion of a tracer colloid particle II
754
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) equilibrium time correlation function C,,(t) denoting an equilibrium ensemble average)
dX P,(X)A(X)
=
ey’B(X)
then as explained in (I) the can be expressed as ((. . -)
,
(2.4)
where 2 is the formal adjoint of the operator Smoluchowski equation,
9 occurring in the generalized
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
= 1dt e’“‘C,,(t)
= - ( A(X)[iw
+ L.f-‘B(X))
.
(2.6)
II
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
(E,
F)
= j- dX
WW*(X)W)
in terms of which the correlation C,.,(t)
= (E, e*‘F)
,
155
(2.7)
functions can be expressed as
.
(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)
Moreover,
=
(E, e”*F) = (e”‘E, F) = C;.,(t)
= C,,,(t)
.
(2.9)
if we consider the eigenvalue problem for $8,
where (U,, U,,) = 1, we can express the eigenvalues in the form I-L, =
(U,, ,JfU,) = -
I
dX P,(X)(VU,)
(2.11)
- D 0(VU,).
Since the diffusion tensor must be positive definite for a dissipative system we see that all eigenvalues are negative apart from a single zero eigenvalue corresponding to the eigenvector U,,(X) = 1. Expressing the correlation functions in terms of an eigenfunction expansion,
C,*,(t) =
cn (6 e~WJW,,F) = Cn e-‘n’(E,U,)(U,,
we obtain a long time factorization
p$ C,*,(O= (6
wu,,
of the correlation
f9 = (E*) (F) .
F) ,
(2.12)
functions (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 form of Q(X) is then
external magnetic field H. The
(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((& i<,
RJ)} ,
(3.4)
(3.5) where Z’ is the configuration pair interactions and zf =
dR, exp{PMn,
integral associated with the spherically symmetric
sinh E E ’
-H} = 4~ -
(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
tions of Edmonds
[7]. A related set of correlation
where P, is a Legendre polynomial. theorem [7] we have the identity
757
functions is defined by
By the spherical
harmonic
addition
(3.10) We can derive a number of formal properties of the Cl,m,;Imby 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) * If we use the self-adjointness find
of the time translation
G’m’;Im(4 fo = CI*m;,‘m’(f, mf
operator
(3.11)
as in (2.9) we
(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;Immust 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
R. B. Jones I Rotational diffusion of a tracer colloid particle II
7.58
(3.14) Integrate both sides of this equation over the rotation orthonormality [7] of the rotation matrices LB:.?,
group and use the
to get
c CIqiIq(t, 0) Gn’$&~ 0) = !LL!-%E 21+
1
q
= *
qt,
(3.16)
0) .
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)>) from which we get, using (3.17) and Y,,(-nl)
Gm;l,n(~~ H) = (-l)““C,.,;,,(-t, a relation of Onsager-Casimir (3.12) we obtain
)
-ff)
= (-l)‘Y,,,,(n,)
-H
7
(3.18)
,
(3.19)
type [5]. If we combine (3.19) with (3.11) and
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.
R. B. Jones I Rotational diffusion of a tracer colloid particle II
759
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+ 1JE) is a modified Bessel function of the first kind. A long time result follows for C,(t, H) in the form
f;E W,H) = [
ei,(e) * sinh
1
(3.23)
.
4. One and two body contributions In (I) we carried out a cluster expansion of the correlation functions A 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)
z,, = i
-L,
(4.6)
)
{[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)
)
a v;=ij -$+&, -sin1 0, -acp, ’
(4.9)
1
with 6, ,&i unit vectors in the 0,) ‘p, directions. tensors [g-10] have the form
The various two body diffusion
(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
R.B. Jones I Rotational diffusion of a tracer colloid particle II
Z12 = 2;, +
LfF2+ %Y;*
761
(4.11)
)
where
+~(P;;-P::)(RXV).(RXV), CJT,=-
qfqp:;-pyl)(Rxv)*Ll (4.12) + y
(P:‘, - P:‘,)P&X)~(~
Z’i2 = k,T[&;L;
+ (a;;
of;‘L,
-PO&x)
x V) 7
- ~;;)(lt~L,)2]
.
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 _!ZF2vanish when acting on Y,,(O,) so that in a high frequency expansion, -1 =--+ 1 io + &, iw
1 ~~12-~(~12)2+~(%2)3....
(4.13)
(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 to express $ji’(O,)
(~;2+~~2) 12
-l +*a*, iw + 2i2
(4.14)
as’ (4.15)
In 5?;2 itself, (L,)’
and J@- L, commute
with each other but not with the
R. B. Jones / Rotational diffusion of a tracer colloid particle II
762
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,
-1
(2) =
9 Im
iw + Lf;,
ff=o
Y
(4.16)
‘m
Henceforth in the present article we will use the approximate $I:’ defined in (4.16) to evaluate the orientational correlation functions. The perturbative treatment of the external field and of the higher order terms in (4.13) we leave to a later study. When the external field is set to zero, H = 0, the one body operator in (4.6) becomes 2, = D;L;
,
while the operator
(4.17)
L!?i2 becomes
L?i2 = k,T{Pf’,Lf
+ (&;‘I - /3f:)(k*L,)‘}
The one body equation eigenfunctions of L f, LTY,,
= -f(Z + l)Y,,
.
(4.18)
(4.4) can be solved trivially because the Y[,(LZ,) are
(4.19)
.
Thus we can write for $“‘(0,)
the result
Ic,~~‘(fi,) = [-iw + DiI(Z + l)]-‘Y,,(Q)
.
(4.20)
To obtain +~~‘(0,) in the approximation defined by (4.16) requires an analysis which takes account of the fact that i +L, is a component of L, along a direction different from the space fixed axis with respect to which the m label in the Y1,,,is defined. We have to solve the equation -[iw + k,Tp’,‘,(Z?)L: = Yh(fi,).
+ k,T(a~~(R)
- PT’1(R))(R.L,)2]~l~‘(n,)
(4.21)
Since Li and R * L, commute, we can solve (4.21) by expanding Y,,,(&) in terms of the simultaneous eigenfunctions of Lf and & - L, . These eigenfunctions however are just the Y,,(@), the spherical harmonics defined with respect to a rotated frame whose polar axis is along 1. If LY,p, y are the Euler angles of the rotation from the original space fixed frame in which Y,,(n,) is defined to a
R.B. Jones I Rotational diffusion of a tracer colloid particle II
163
frame in which R defines the z axis then we have that (Y= (PR,p = 0, and y is arbitrary, where da, (PRare the polar angles of P in the original space fixed frame. Using the rotation law (3.13) we have
Yh(f47cpl)=
,$_, YqM, 4)qh,
e,, y)
(4.22)
)
and its inverse (4.23) The Y,,(e;, cp;) are eigenfunctions L:Y,,(w
= -v
+ w&u
of both Li and 3
(J? .L,)*,
(~~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 (l) which occurs in the two body contribution *I:) - (Cllm * ;;I - +I;) = c
{[- iw + k,7'1(1 + l)pi’, +
k,Tq*(aI’,
-pan)]-’
4m’
[-iw + DbI(I + l)]-‘}Y~,,9~~,9~~
-
.
(4.26)
(*) - +!J:;)depends upon 0, through From the result above we observe that er,,, 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.$dyl,--* *a
=&jR2dRIdqRdcosB,dy..., *a
(4.27)
764
R. B. Jones I Rotational diffusion of a tracer colloid particle II
where 2a is the closest distance of approach permitted by the pair interactions. Note that we can introduce an integration over all values of the Euler angle y because nothing in the original integrand of (4.3) depends on this angle. By introducing such an integration however we can evaluate the average over the direction of k by use of the orthonormality relations (3.15) for the g$. 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)=
“$‘f~‘m [ dR R2g(R)
i
{[-iw + k,TI(f + l)PIi(R)
q=-/ 2a +
k,Tq2(cq
-
fit’,(R))]-’
-
[-iw + DhZ([ + 1)1-r} .
(4.29)
By using the detailed form of the mobility functions (Y;‘I, pi’, as given elsewhere [8, 91 it can be seen that as R+ x we have k,T@f’,(R)-+ Dfi + 6’((alR)“), while for the difference of mobility functions we find (a;:(R) so that the terms in the summation in (4.29) vanish as P;:(R))+ fWR)“h Re6 at large separation giving an absolutely convergent integration over the separation distance.
5. Long time correlations
at low density
If we approximate the pair distribution function g(R) by its low density we obtain from (3.16), (4.28) and (4.29) the first two value, exp(-pu(R)), terms of a virial expansion of tl(o) in powers of the volume fraction l#J= 4nn,a’/3, Q(W) = Q”(O)
+ q6C’12’(w)+ . . . )
(5.1)
with
ty’(w) =
-1~
.
+
l
(5.2)
D;l(Z + 1) ’ dR
X
,C, {[-iw
-
my’}
.
R2
e~P~(W
+ k,TI(f + l)pl’, +
k,Tq2(a;‘,
- PT:)Im
(5.3)
R. B. Jones I Rotational diffusion of a tracer colloid particle II
We define the orientational
correlation
765
time T! by [6]
m
r, =
I
C,(t) dt = i;(O) .
(5.4)
0
This quantity gives the mean decay time of the correlation function and its reciprocal l/r, can be viewed as a long time quantity to be compared with the short time rotational diffusion coefficient, Di(1 - 0.634). One cannot regard 1 /rr as a long time rotational diffusion coefficient since it is evident from (5.3) that C!(t) is not a single exponential at long times. If we put w = 0 in (5.2) and (5.3) and then use (5.4) we obtain, to first order in r$, the following expression for l/r,: 71-l = D;E(1+ l)[l - +c;(I).
* *] )
(5.5)
where dR R2 e-P~(R)
x $ q=-/
Z(1+ l)(l - &rvA;P;;) f(1+ l)hrqA:/3;‘,
- 87~77A;(cz;; - p;;)q’ + 8n77A;(a;‘,
- p;;)q2
.
(5.6)
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 e p en d 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] . The two body contribution
is dR
X
(5.7)
R*
q$_t Wp[-kJW
e-M0
+ W%‘,+ (aI’, - PI%‘)4 - C!‘)(t)>. (5-B)
R. B. Jones I Rotational diffusion of a tracer colloid particle II
766
It is convenient G(t) ___ c,“‘(t)
to write
= 1 + $7,(t)
+ ...)
thereby introducing the single exponential decay,
function
dR
(5.9) -y,(t) which
measures
the
departure
from
R” e-fiu(R)
x ,E, {exp[-k,V~(~ + l)(Pl’I - Po6) + ((-~frl - P11’,)s’PlI> . (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 ~0by 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 dependent 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)
R. B. Jones I Rotational diffusion of a tracer colloid particle II
767
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.
R. B. Jones I Rotational diffusion of a tracer colloid particle II
768
1 -
m
--& 0 5
-2.0
-
-3.0
-
-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 fraction 4 = 0.05. The dashed line is a plot exponential decay characteristic of an isolated
of the dimensionless variable s = 6DSt for a volume of the function In C:“(t) which shows the single particle without interactions.
Acknowledgement I wish to thank the University which provided the computer above.
of London Central Research Fund for a grant software used for the calculations presented
References [l] (21 [3] [4] [S] [6] [7] [S] [9] [lo] [ll] [12] [13] [14] [15] [16] [17]
R.B. Jones, Physica A 150 (1988) 339, to be referred to as (I). R. Zwanzig, Adv. Chem. Phys. 15 (1969) 325. P.G. Wolynes and J.M. Deutch, J. Chem. Phys. 67 (1977) 733. J.A. Montgomery Jr. and Bruce J. Berne, J. Chem. Phys. 67 (1977) 4589. S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics (North-Holland, Amsterdam. 1962). B.J. Berne and R. Pecora. Dynamic Light Scattering (Wiley, New York, 1976). A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, 1957). R. Schmitz and B.U. Felderhof, Physica A 116 (1982) 163. R.B. Jones and R. Schmitz, Physica A 149 (1988) 373. B. Cichocki, B.U. Felderhof and R. Schmitz, PhysicoChemical Hydrodynamics 10 (1988) 383. S. Kim and R.T. Mifflin, Phys. Fluids 28 (1985) 2033. G.K. Batchelor, J. Fluid Mech. 74 (1976) 1. G.K. Batchelor, J. Fluid Mech. 131 (1983) 155; 137 (1983) 467. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 89 (1988) 1049; 3705. B. Cichocki and R.B. Jones, Z. Phys. B 68 (1987) 513. W. van Megan and S.M. Underwood, J. Chem. Phys. 88 (1988) 7841. W. van Megan and I. Snook, J. Chem. Phys. 88 (1988) 1185.