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Self-diffusion of spherical Brownian particles with hard-core interaction
1llA (1982) 181-199
Physica
North-Holland
Publishing
Co.
SELF-DIFFUSION OF SPHERICAL BROWNIAN WITH HARD-CORE INTERACTION Sarwat
fir Physik,
Fakulttit
HANNA,
Walter
Llniversittit
Received
The
N-particle
particles
interacting
self-diffusion constant. the
Oseen
Smoluchowski
coefficient
is found
The hydrodynamic approximation,
hydrodynamic
equation
through hard-core
interaction
HESS
Konstanz,
and Rudolf
D7750
for a system
of spherical
Brownian
to first order in the volume concentration where
is then taken into account
D, = Do(l -0.091#1) is found, due to Felderhof
Fed. Rep. Germany
9 July 1981
to be given by D, = Do(l - 24).
interaction
KLEIN
Konstanz,
is solved exactly
potentials
PARTICLES
gives D, = Do(
whereas
I-
in a perturbative the
4. The
DO is the free diffusion improved
way. form
Using for
the
1.894).
1. Introduction
In recent years there has been considerable interest in the concentration dependence of the collective (or mutual) diffusion coefficient D, and the self-diffusion coefficient D, for macroparticles suspended in fluids. A number of experiments by means of more conventional methods’) and by photon correlation spectroscopy*) have been performed which have led to considerable theoretical work in this field. From the theoretical point of view there is, for general interactions among the macroparticles, no reason why D, and D, should be equal, even to lowest order in the volume concentration C#Jof macroparticles. The diffusion coefficients D, and D, are defined by the hydrodynamic poles of the autocorrelation functions for the total concentration fluctuations 6c(k, t) and for one-particle fluctuations 6Ci(k, t), respectively. These two functions describe different physical situations and will therefore generally lead to different results for D, and D,. In this paper we will concentrate on the lowest-order concentration dependence of the self-diffusion coefficient D, for a system of particles interacting via a (repulsive) hard-core interaction and in addition through the so-called hydrodynamic interaction. The latter arises because of the velocity field which is produced in the solvent by the motion of the tagged particle and felt by neighboring particles. In the discussion we will come back to a comparison of D, with the collective diffusion coefficient D,. 0378-4371/82/0000-0000/$02.75
@ 1982 North-Holland
S. HANNA
182
Earlier starting
theoretical
from irreversible
equations.
Writing
free macroparticle, from the N-particle a hierarchy truncated a result
approaches
include
thermodynamics
D, = DO(l - CX~) where
et al
a calculation and using
of D, by Batchelor’) two-body
Do is the diffusion
hydrodynamic coefficient
of a
Batchelor obtains cy = 1.83. A different approach starts Smoluchowski equation. From it Felderhof4) has derived
of equations
for
s-particle
distribution
functions
which
for systems of low concentration. Felderhof has derived for the collective diffusion coefficient. This approach
can
be
in this way was later
extended by Jones’) to systems of two different kinds of particles. One can also apply the Mori-Zwanzig projection operator formalism to the N-particle Smoluchowski equation’.‘), obtaining in this way exact memory equations for the collective and the self-diffusion propagator. It can be shown that the memory function in the case of collective diffusion vanishes in the hydrodynamic limit faster than k*, so that memory effects are not important for the collective diffusion constant D,. In contrast, this is not the case for selfdiffusion. Since Felderhof’s method is essentially a mean-field theory without memory effects, his conclusions concerning D, are in agreement with those obtained by the Mori-Zwanzig method, whereas the extension to the selfdiffusion problem not included. The main the memory
leads to results
which disagree,
because
memory
effects
are
problem of the Mori-Zwanzig approach lies in the evaluation of functions. In an earlier paper’), concerning charged spherical
particles, a mode-mode coupling approximation for systems of small volume concentrations proximation’). It should used different approaches of course also dependent interaction.
was used. Another possibility is a weak-coupling ap-
also be noted that different authors have not only and various approximations, but that the results are on the particular form used for the hydrodynamic
In this paper we concentrate on the lowest-order concentration dependence of D,. We first solve the two-body problem with hard-core interactions exactly. Using this result we obtain without approximations the solution of the N-particle Smoluchowski equation with hard-core interactions to linear order in the volume concentration 4. This gives an expression for D, for hard-core systems, which includes the effect of the self-memory function to first order in 4 exactly. The hydrodynamic interaction is then treated by a perturbative method to first order in the strength of this interaction. Two different models, the Oseen tensor and Felderhof’s form for the hydrodynamic interaction”) are treated explicitly and are shown to lead to rather different results. Finally, we discuss our findings for D, in relation to earlier ones.
SELF-DIFFUSION
2. Exact solution
OF BROWNIAN
of the two-particle
183
PARTICLES
Smoluchowski
equation
for a hard-core
system The
Smoluchowski
hydrodynamic
equation
interaction,
for
two
interacting
particles,
but
without
has the form
$p(r,,t-2, t) = hl2p(rl,i-2, t), (2.1) ~,2=Do-$($.+P
I
~)+D,,$(&+p~),
I
where p(rl, r2, t) is the probability to find at time t the two particles at r, and r2, respectively, and U(r,,) denotes the two-particle potential, for which we assume a hard-core form 00 for r12 < d, for r12>d,
u(r’2)=(0
(2.2)
d is the particle diameter and r12 the relative distance t-r2 = r, - r2; fi = (knT)-’ and Da = ksT/c, J’ is the friction coefficient. Transforming (2.1) to relative coordinates
and center
of gravity
coordinates, aukI am
&=$D,&+?D,J&(&+/~-.
In the following
we want
to treat
only
particles and omit the center of gravity time 0 the centres of the two particles Then the initial distribution is
R = i(r, + r2), yields (2.3)
)
the
relative
motion
O(x) denotes 0
“(‘)=(I
the unit step function
forxOY
Because of the singular use the relation exp(-pfJ(rr9) Formally function
=
(2.5) nature
of the interaction
(2.2) we may also
(2.6)
eq. (2.1) may be integrated
to the initial
potential
@(r12).
p(r12, t ( r-0) = exp(fir2t)Wrr2subject
two
that at ro> d.
(2.4)
p(r12,O) = 8(rr2 - r-0)0 (r-12- d), where
of the
term in (2.3). Let us assume were separated by a distance
distribution
and gives for the conditional
ro)@(ru(2.4).
d),
distribution
(2.7)
et aI
S.HANNA
184 In spherical
f1 = (I‘+, q) expanded
coordinates
and
the s-function
f1,, = (19,). po) are
in spherical
the
can be expressed
spherical
angles;
as
6(R ~ {I,,)
can
be
harmonics,
(2.9) In spherical
coordinates
the two-particle
diffusion
operator
(2.3) has the form (2. IO)
where
iZ is the square of the angular
momentum
operator,
(2.11) The spherical
harmonics
i’Y;l(n,
are eigenfunctions
of L’, (2.12)
= I(/ + l)Y;“(O),
and therefore exp(d&)Y;l(O)
= Y;“(O)
(2.13)
exp(I&t),
where (2.14) Inserting
(2.8). (2.9) and (2.13) into (2.7) gives (2. IS)
where
X,(r. t 1r,,) is determined
Xl(r. t I rd = exp(RA) Since
we
penetration defining
know of
the
that two
a new function
by
$j 6(r - ro)O(rothe
hard-core
particles k,(r,
we
d).
(2.16)
potential take
this
allows
at no time
into
account
a by
t 1r,J as
X,(r, t 1ro) = O(r - d)JZ,(r, t I rd. Inserting
(2.2)
explicitly
this ansatz into (2.16) and differentiating
(2.17) with respect to time yields
SELF-DIFFUSION
the differential
OF BROWNIAN
PARTICLES
l?S
equation
+ 2D,$(r
- d) i
$(r,
(2.18)
t 1t-0).
Here we have used eq. (2.6) and 6(r - d) =
$
O(r - d).
The problem
of solving
to the problem
of solving
$ JI,(r, together
t 1 rO) = y
(2.19) the two-particle
diffusion
equation
is now reduced
the equation ($
with the boundary
r2i-
I(1 + I))R,(r,
t 1r&
(2.20)
condition (2.21)
this condition expresses the fact that the probability current has to vanish at r = d. Since the probability to find a particle at infinity is zero we have as a second boundary
condition
A(m, t 1ro) = 0,
(2.22)
and from eqs. (2.16) and (2.17) the initial
condition
%(r, 0 1 rO) = i 6(r - ro). In order the initial dition.zero; %(r, r
to solve condition,
(2.23)
(2.20) we look for a particular and for a complementary
their sum has to satisfy I r0)
=
Udr,
t
I r0)
+
Wr,
the boundary t
solution solution
Ur which
satisfies
W, with initial
con-
conditions.
/ r0h
Udr, 0 I r0) = i W - r0),
(2.24)
W,(r, 0 ) ro) = 0. We make the ansatz
CJdr,t 1r-0)= j dq g,(r, q I ro) em2Doq2r 0
(2.25)
IX6
S. HANNA
and find that g,(r. y 1 r,,) satisfies
the Bessel
(
r’ < + 2r $ + (q’r’~ dr-
The
solutions
kind”),
are
the
I(1 + I)))g,(r.
spherical
et al
equation
q 1r,,) = 0.
Bessel
functions
(2.26) of the
first
and
second
j,(q r) and yr(q r), respectively. 1rdidq r) + &(q
gr(r. 4) = A(4 Expanding
the a-function
1rdydq r).
in spherical
(2.27)
Bessel-functions
of the first kind”)
1
(2.28)
2.
we find
AI(~) =
2y? .h(q rd. 77
(2.29)
H,(q) = 0. The Laplace
of Udr, t 1r,,) is
transform *
Udr. z 1rd =
1 dt
e “U,(r, t 1r,,)
0 (2.30) The complementary Laplace transformation
W(r, t I ro) has
solution yields
also
to
satisfy
eq.
(2.20).
zW,(r.zir,,)-++($r’$-l(l+l))W,(r.zlro) = Wl(r, t = 0 ) ro) = 0 because
(2.3 I)
of (2.24). This is again a Bessel
2 2a -$+;z(
[([+I) rC-+20,
2 ))
equation,
Wr(r, 2 I rd = 0,
(2.32)
the solutions are the modified spherical Bessel functions of the first kind i,(rgz/2&) and of the third kind kr(rV’z/2Do)“). Since the il(r%‘z/2Do) are divergent for r+cfl, they can be omitted because of (2.22). Then
+
CI(Z
I rdkdr~/zDDJ.
(2.33)
SELF-DIFFUSION
The
unknown
condition
Cr(z ) ro)
functions
(2.21), which
OF BROWNIAN
completes
finally
187
PARTICLES
determined
by the
boundary
the solution,
m
Cl(&)=-i]dqq’
’ z + 0
f;(b)
where
stands
With (2.19,
__-
jX4
d)h(q
r0)
(2.34)
2Doq2~z/2D,k;(d%‘/)’
for @,(x)/ax Ix+,.
(2.17)
(2.33) and (2.34) we find finally cc
p(r,zIr0)=O(r-d)f
Y~‘(R)YP(Ro)*~I~~~‘~~‘~~~~~
i
I=0
m=-I
0
(2.35)
3. The self-diffusion coefficient The
Fourier-Laplace
tagged
Brownian
C,(k, z) = (e-” Here
fi,
transform
particle
of the
self-diffusion
propagator
rl[~ - firi,lm e” “I).
is the diffusion
of one
is
operator
(3. I)
of an N-particle
system (3.2)
UN is the N-particle UN = f 5 i,j=
i#j
I
potential,
which we assume
as a sum of pair potentials,
LJ(rij).
(3.3)
The tensors Dii describe the hydrodynamic discuss in the next section. If hydrodynamic
interaction interaction
effects, which we will is neglected then
Dij = D()Gij1. ((. . .)) denotes
Furthermore, ((.
* .)> =
(3.4)
1
d{rl(.
a canonical
. ‘) exp(-puN)//
expectation at-1
value,
eW-PU,h
The self-diffusion constant D, is defined from the following self-diffusion propagator in the hydrodynamic limit C,(k, z) = (z + D,k*)-‘,
for k, z small.
(3.5) form
of the
(3.6)
S. HANNA
188
Inserting
et al
(3.6) leads to (3.7)
z)) ‘(I - =C’,(k. 2)))
D, = lim lim{(k’C’,(k.
: -0 A 4
Using (3.1) we get
Employing
the expression
(3.2) for fl,.
and then taking the limit k +O, D,= where
k. (D,,)-
k-l;lrr((k
performing
some partial
integration5
D, is given by
.f,)[z
-- ri,]
(3.9)
‘(k .f,,).
k is a unit vector in the direction
of k and (3.10)
A
similar
operator
result
for
technique.
D,
was
Since
model of hydrodynamic
obtained
Ackerson
interaction,
by
Ackersonh),
restricted
a projection
his treatment
the Oseen-tensor.
absent in ref. [6], because the tensor divergence
using
to a special
the last term in (3.10) is
of D,j vanishes for the Oseen
tensor. In the present order
paper
concentration
corresponding first-order order
we restrict
correction
expression
assuming
term stems from
clusters
will
ourselves
to D,.
The
a noninteracting
N independent
contribute
to U(C’).
equivalent
it is sufficient
Neglecting
now also hydrodynamic
to the calculation zero-order
term
single particle,
two-particle
Since
clusters.
all particles
to treat one cluster and to multiply interaction,
of the first-
is given
are
by the Do. The
All higherstatistically
the result by N.
eq. (3.4), we obtain from (3.9)
and (3.10) D,=Do-lime
z-0
I
drl?(k ^ - fdrd)
x [z - fi(i,J’Ci *fdrd)
(3.11)
expt-pU(rdi,
where
fdrld
= HOP
&
U(rtd = DOB(T&U(r12))i12.
hii,, is now the same two-particle is the hard-core
potential
diffusion
operator
(2.2) and iI2 = r12/r12.
(3.12) as in (2.1),
U(rrz) = U(r,,)
SELF-DIFFUSION
OF BROWNIAN
We now use the properties of the hard-core exp(-pU(rr,)), eq. (2.6) and (2.19), which yields frArr2) exp(- PUG-r,)) = -Do?u &
189
PARTICLES
radial distribution
function
0 G-12- d)
= - Doi128(r12 - d) = -Doi1220(r12-
d)6(r12-
d).
(3.13)
The last equality should be considered as an operator equality, valid in an infinite integral over r12. The factor 2 arises from O(0) = l/2. This property can be verified by considering the O-function and the a-function as limits of continuous functions, e.g. (3.14a)
0 (x) = f + ljr&an-‘3,
(3.14b) Inserting now a a-function 8(r12- ro) behind the resolvent operator and using (2.7) we can write for the self-diffusion coefficient
.
aW12)
drr2 drok - i12 ~ x
in (3.11)
ah2
p(r12, z = 0 1ro)k - io6(ro- d) . >
(3.15)
Writing k^- i,2 = cos 19,~= ~/(47~/3)Y%fl,~) and analogously for k^- io, using the Laplace-transform of (2.15) and the orthogonality relations of the spherical harmonics, I
dRY ;I(fl)Y ;?‘(a)* = 6,,&,,,,,,,
(3.16)
D, becomes
x I drorfiXl(rlz, z = 0 1r,Jb(ro-
d)).
(3.17)
Finally, with (2.17), (2.6) and (2.19) D=D s
0
=
(~-&CD 3
Do( 1 - F
o I
dwdh
‘2
-
4
cd4D,,J?(d, z = 0 1d)).
j
dror%h,
.z =
0
1 ro)Wo
-
4)
(3.18)
S. HANNA
190
In
order
to determine
et al.
%(d, z = 0 1 d), we use (2.33) and (2.34) and perform
the
limit z + 0”)
id4 rd -
0
(p(r,,- r,z)++
$ idy
4 +
f$ qjdq d)) 1
(rl?-
II
rd+& I2
O(rr
(3.19)
-%_@(d-r")). For the special
case r. = r,? = d we find
%(d, z = 0 1d) = (8dDo)
Here again the property yields
the exact
II,= where
(3.20)
‘.
O(0) = l/2 has been used. Substitution
of (3.20) in (3.18)
result (3.21)
Do(l-24).
C#Jis the volume
cf~= $
concentration
of the Brownian
particles,
(d/2)‘c.
(3.22)
4. The effect of the hydrodynamic Neglecting
d)
0
the hydrodynamic
interaction interaction
on the self-diffusion has enabled
coefficient
us to find an exact
solution for the two-particle diffusion equation and an exact result for the first-order concentration correction to the self-diffusion coefficient. This result is however not of much use for the understanding of experiments because of the complete absence of the hydrodynamic interaction. the effect of this interaction in a perturbative way.
Therefore
we include
To lowest order in the concentration we need only the hydrodynamic interaction of a pair of particles. In its most general form the diffusion tensors can then be written as’,“)
@I = 022= &AI + h,(rdP 012 = &I = &CWrP
+ MM1
+ M-&1
The functions hn(r12) will be specified The tensor P projects an arbitrary
- IV, - P)).
(4. la) (4. lb)
later on by choosing particular models. vector on the relative vector r12. In
SELF-DIFFUSION
spherical
OF BROWNIAN
191
PARTICLES
coordinates (4.2)
PV = 1, and all other elements are zero. Using eqs. (4.1) in the general responding &=
two-particle
diffusion
form
of the
operator
diffusion
separates
operator
the
cor-
as
&+A&.
(4.3)
Here fi$ is the operator without hydrodynamic interaction, identical with (2.1) or (2.3), and Ah:, is the additional contribution from hydrodynamic interaction. Omitting again the terms describing the center of gravity motions of the two-particle
cluster
we get (4.4)
Inserting
(4. I), Afi(i:, becomes
+
2Do&.(Mrd
- h(r,z)Xl - P) -
&
=2D,(&(hdrd - h3(r,3))(~2+ P y)
+2Ddhdrd
+2c2ri2(&2 +P y)
- M-d)
- 2Ddhdr12)- h&d k2i’. We now consider
the action
which
to the solution
is identical
Afiim(n2,t 1rd = 2 I=0
(4.5)
of Afit* on the zeroth-order
i
found
in section
solution
2. With (2.19,
pdrr2, t 1ro), (2.17),
Y;"(fl)Yi"(fl~)
m=-I
x [ WI2 -
d)2Q($2 (h&-d - Wd) &
+ Oh - 42Ddhdr12) - Mrd
ia T
-
r12 arlz
a
62-ah2
- O(r,, - d)2DO(hAr,2) - h4(r,& $z [(I + 1) + 2DO(h,(r,2) - h3(r,2)M(r,2-
d) -$-j%(r,2,
t 1rd.
(4.6)
192
S. HANNA
Because
of eq. (2.21),
dfi(i:z&rl5
the fourth
et al
term vanishes and
t I rd (4.7)
with the operator
The
Laplace-transformed
equation
solution
with the operator
of
the
complete
two-partic!e
as given in (4.3) can be written
in a formal
diffusion way as
with
p”(r12. z ( rd = ([z -
dkl
Eq. (4.9) is the perturbative
‘A~i:#‘pdrI~. solution
interaction. The first-order
correction
is
z ( rd
of the problem
(4. IO) including
hydrodynamic
SELF-DIFFUSION
Inserting
OF BROWNIAN
193
PARTICLES
(2.15) and (2.17) gives
pr(r12, z
1 ro) = @(rt2-
x
4
Y;l(fl)Yi"(fl~) 2 2 I=0 m=mI
dr’r’2_%r(r12r z
1 r')&%(r',
(4.12)
z 1 ro).
I
For the calculation
of the diffusion
in section
eqs. (3.9), (3.10) we find
3. From
D,=Do+c -
d3rr2k^ .
I
we proceed
in the same way as
. i ew-pU(b2))
d3rr2(k^ . fl2(r12))[z -
lim c Z-r0
D’,,
coefficient
h721m
I
x (I .fdr12))
(4.13)
exp(-P~(r~2)),
D;, = /A, - Do1 and
where
(4.14)
The tensor
divergence
is”)
=
Do(&
(hdr12) -
+Ddhdrd =
defining the function This result shows
-
h(rd))i
- h(rd
- hdrd + Wd
+2i I
(4.15)
DoHdrdi,
Hl(rr2). that f12(rr2) is proportional
ffdr12) - H2(r12)P
to i
~
(4.16)
where I-Mrr2) = - (1 + hr(rr2) - Mrr2)). Using
a similar
procedure
as in eq. (3.13) we can write the last factors
(4.17) of the
194
S. HANNA
integrand
et al
in (4.13) as
k^* fdrd
expt-pUtrid)
= exp(-pU(r,z))D,,(H,(r,,) + 2H2(d)ti(rIT-
Inserting yields AD,=
a a-function
D,-
behind
the resolvent
Y:‘(0).
(4.18)
in the last term of (4.13)
(4.19) for AD,. Restricting
ourselves
to
(2.17) and (4.12)
AD:“‘=+;_/ I
AD’,” = 7 x
operator
%
d)).
Using (4.9), this is the perturbation series p = p. + pI, we obtain AD, = AD/“’ + AD’,“.
x
J
D,,
x (Hdrd + 2HddP(ro-
With (2.!S),
d))
drr,r%r(rr:, CD;
I
drI?r:?(H,(rl*)0(r,z-d) + Hd46(r12-
I
z = 0 1 r&Hl(rcJ
+ 2HAdP(ro-
dr~~r%H~(r~~)@ (rt2 - d) + HAd)S(r,,
dr’rr2%,(r12,z = 0 1 r’)
I
drg~~,~,(r’,
d)) d)).
(4.20)
- d))
z = 0 1 ro)
x (Hdrd + 2HddF%r(~- d)).
(4.21)
The function .%,(rn, z = 0 1 r,,) has been given in (3.19), the differential operator 2, is defined in eq. (4.8), the only unknown functions are now the h,(r), describing
the specific
form of the hydrodynamic
interaction.
The rest of our
task is a tedious but elementary calculation of the integrals in (4.20), (4.21). We performed these calculations for two specific models of hydrodynamic interaction: the Oseen-approximation and Felderhof’s reflection series”). The Oseen-approximation is obtained from the solution of the linearized NavierStokes equation to lowest order in the ratio d/r. Therefore the Oseen approximation is valid if two particles are far separated. On the other hand, the hydrodynamic interaction will only be important if two particles are close together, and therefore the Oseen approximation must be considered as poor. Nevertheless, because of the simplicity of its analytical form, it is the most
SELF-DIFFUSION
OF BROWNIAN
195
PARTICLES
often used model of hydrodynamic interaction; for this reason we include it here. There have been many attempts to improve the Oseen approximation, solving the linearized Navier-Stokes equation to higher orders in d/r. Batchelo?) has obtained numerical results for the hydrodynamic interaction for all values of the interparticle separation and has given the analytical asymptotic form up to (d/r)4. In this paper we adapt results derived by Felderhof’@) who has presented the asymptotic expansion up to (d/r)‘. Felderhof’s result seems to be the most reliable analytical form of the hydrodynamic interaction for a system of low concentration (to Q(4), which is consistent with our calculations). In table I we specify the two forms of hydrodynamic interaction, and in table II we present the results for AD’,“, AD:“, J d3r& * D;, * h9(rlz- d), and a, where (Y is defined by D, = DO(l - (~4). The mean-field contribution _fd3r& * DiI * b(r12d) has already been calculated by Felderhof and is taken from his work.
5. Discussion
The N-particle Smoluchowski equation has been reduced to a two-particle in solution. equation for a system at low concentration of macroparticles
TABLE I Expressions for the function hi(x), i = 1,. ,4 entering definition (4.la, b) of the diffusion tensor D, which describe hydrodynamic interaction in Oseen- and Felderhof-approximation, respectively. Also given are the functions H)(x), eq. (4.15) HZ(X), eq. (4.17); x stands for d/r
Oseen
hl
0
hz
0
h, h4
3 -x 4 3 -x 8
HI
0
HZ
-1+:x
Felderhof 11 --x154 +mx6 64 --x6 17 1024 3 1 75 -x-jjx 3 +mx’ 4 3 1 -x+16x) 8
the the and
S. HANNA
196
et al
TAHI.E Results for the self-diffusion column
represents
contributions
(4.20)
coefficient
the mean-field and
(4.21).
tributions
D, to first
contribution. respectively.
II order in the volume
The The
second fourth
by giving the coefficient
and third column
concentration columns
summarizes
IY in D, = Do(
4. The first
are the memory the
three
con-
I ~ UC/J)
Hydrodynamic interaction neglected Oseenapproximation Felderhofapproximation
From
the
2+! x
0
- 1.736,
knowledge
of the
distribution
function
we have
calculated
the
self-diffusion propagator C,(k, t). The hydrodynamic pole of this correlation function defines the self-diffusion coefficient II,. The method to obtain D, which has been used in this paper, is equivalent to the approach which applies the Mori-Zwanzig projection operator formalism to the N-particle Smoluchowski equationh.‘). One obtains an exact memory equation for C,(k, t), which can be written as*)
6 C,(k, t) = - K,,(k)C,(k,
dt’M,(k,
t) +
t - t’)C,(k,
t’).
(5.1)
0 Here,
K,,(k)
is the first cumulant
of C,(k, t), which determines
the initial
slope
of log C,(k, t) versus t and M,(k, t) is the memory function. The quantity given in the first column of table II, which represents the mean-field contribution
to ID,, is closely
related
to the
first
K,r(k)/(D,k*) as k + 0. Taking the Laplace-transform the self-diffusion coefficient is
cumulant;
it is the
limit
of (5. I) and using
of
(3.6),
(5.2) It was first pointed out by Ackerson6) that M,(k, t)/k* does in general not vanish as k -+ 0. Therefore, memory effects enter the determination of D,. Although the approach used in this paper avoids a direct calculation of the memory function M,(k, t) the contributions AD!‘) and 406” to D, are just the memory effects. We note that in writing eqs. (3.7) and (5.2) for D, it is implicitly assumed that M&k, z) has no hydrodynamic pole.
SELF-DIFFUSION
OF BROWNIAN
PARTICLES
197
The memory function also enters the velocity-autocorrelation function one particle. Using the continuity equation it is easy to show14) that
!j(Ui(t)
’
Vi(O)) = 1’ ,‘lom & [2K,,(kM(t)
- K(k, t>l,
of
(5.3)
where Vi(t) is the velocity of particle i at time t. This result together with eq. (5.2) leads to co D,= i \ dt(ui(t) * vi(O)).
(5.4)
0
Since the mean-square displacement relation function one obtains’)
W(t) is related to the velocity-autocor-
W(t) = k ((R(t) - R(O))*) = lim k-O !$@
t-
dt’(t - t’)M,(k,
(5.5)
t’)]
I 0
and D,
= lim *+m
w(t).
(5.6)
t
When analyzing experimental data it might be easier to determine first cumulant. From it one can define a quantity’5*‘6) Ds,& k) =
.K,l(k) k
which is sometimes called an effective diffusion coefficient. and (5.6) one should note that this quantity is just D s,eR= lim wO. f-0 t
only the
(5.7) From eqs. (5.5)
(5.8)
Therefore, it neglects all memory effects and differs from the hydrodynamic definition of a self-diffusion coefficient as in eqs. (5.4) and (5.6). The importance of the memory function lies in taking the fluctuations of the surrounding particles into account. Since self-diffusion is a one-particle property these fluctuations are essential. As mentioned in the introduction, this is different in the case of collective diffusion, which is a macroscopic effect. Therefore we believe that an extension of Felderhof’s method4) to the self-diffusion problem is not justified.
198
S. HANNA
et al.
We now turn to a discussion of our results First, it should be noted that memory effects
for D, summarized in table II. in the hard-core case without
hydrodynamic
changing
24).
This
contribution interest
interaction substantial
are very important, shift
(first column
is the
large
is due in table
difference
Oseen-approximation and Whereas the hydrodynamic
to the
fact
that
D, from there
II) in the hard-core between
Felderhof’s interaction
the
two
system.
results
Do to D,,(l ~
is no mean-field Of particular
obtained
form of hydrodynamic taken in Oseen-approximation
for
the
interaction. reduces
the concentration dependence of D, compared to the hard-core system rather drastically, the more reliable form given by Felderhof brings the result almost back to the hard-core case. This may be understood as follows: The Oseenterm represents the long-range part of hydrodynamic interaction and therefore reduces the effects of the very short-range direct interaction rather dramatically. The velocity field produced by the moving macroparticle removes neighboring particles from the collision trajectory leading to an increase of D, compared to the pure hard-core result. Taking into account the short-range term of the hydrodynamic interaction one also includes the reflected velocity fields from other macroparticles. These have the effect of slowing down the motion of the tagged particle, thereby decreasing D, again. In comparing our results with those Batchelo?) obtained D, = Do(l - 1.834),
of earlier papers we first which should be compared
note that with our
mean-field value in the first column of table II. More recently, Marqusee and Deutch? have calculated D, = Do( 1 - a$) from the memory equation approach by using a weak-coupling approximation in evaluating M,(k, t). For the hard-core system cy = 1.33 is obtained compared to cu = 2 in our treatment. The weak-coupling approximation replaces a two-particle propagator by the product of two one-particle propagators. This is an expansion in terms of the strength of the interaction. Such expansion seems to be questionable for the hard-core case, which would explain the rather large difference of the results. These have also included hydrodynamic interaction in Oseen-approximation cy = 0.07 compared to our (Y= 0.09. As mentioned above the Oseen-term collisions
drastically.
Therefore
it is really not so important,
authors finding reduces
how the hard-core
is
treated if hydrodynamic interactions are included. This seems to be the explanation why their result and ours are so close in the Oseen-approximation. Finally, we want to compare D, with the collective diffusion coefficient D,. As noted earlier the collective memory function M,.k. t) vanishes faster than k’ as k +O. Therefore, memory effects vanish for D, to first order in C#LOnly three-particle correlations would contribute if hydrodynamic interaction is included. For a hard-core system one has the well-known result”) D, = DO(I + 84). Defining the collective friction coefficient fC by writing in the
SELF-DIFFUSION
OF BROWNIAN
PARTICLES
199
solvent fixed frame (5.9)
where n is the osmotic pressure, one has fC = fo, f. being the friction coefficient of a free particle. The self-friction coefficient fS is defined by D, = kaT/f,. Therefore, for a hard-core system we have fS = fo( 1 + 24) Z fC. Including now Felderhof’s form for the hydrodynamic interaction, D, is given by4) D, = DO(1 + 1.564), and from (5.9) one finds fC = fO(1 + 6.444). This is to be compared with our result for D, which is D, = Do( 1 - 1.894) and with fS = f,Jl + 1.894). Therefore in all cases considered here, one always has D, # D, and fC # fS. There is no reason why these different transport coefficients should coincide besides in the trivial limit of infinite dilution.
References 1) See, for instance, K.H. Keller, E.R. Canales and S.I. Yum, J. Phys. Chem. 75 (1971) 379. These authors used a diaphragm diffusion cell. 2) A recent comprehensive review of light scattering methods is given by P.N. Pusey and R.J.A. Tough, in: Dynamics of Light Scattering and Velocimetry: Application of Photon Correlation Spectroscopy, R. Pecora, ed. (Plenum, New York, 1981) to be published. 3) G.K. Batchelor, J. Fluid Mech. 74 (1976) I. 4) B.U. Felderhof, J. Phys. All (1978) 929. 5) R.B. Jones, Physica 97A (1979) 113. 6) B.J. Ackerson, J. Chem. Phys. 64 (1976) 242, 69 (1978) 684. 7) W. Dieterich and I. Peschel, Physica 95A (1979) 208. 8) W. Hess and R. Klein, Physica 105A (1981) 552. 9) J.A. Marqusee and J.M. Deutch, J. Chem. Phys. 73 (1980) 5396. 10) B.U. Felderhof, Physica 89A (1977) 373. Functions (Dover, New York, 11) M. Abramowitz and I.A. Stegun, Handbook of Mathematical 1965). Methods for Physicists (Academic Press, New York, London, l-3 G. Arfken, Mathematical 1970). R.B. Bird, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric 13) See, for instance, Liquids, Vol. 1 (Wiley, New York, 1977). in Liquids and Macromolecular Solutions, V. Degiorgio, M. 14) W. Hess, in: Light Scattering Corti and M. Giglio, eds. (Plenum, New York, 1980). 15) P.R. Wills, J. Chem. Phys. 70 (1979) 5865. 16) H.M. Fijnaut, J. Chem. Phys. 74 (1981) 6857. B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 17) See, for instance, 1976).
Physica
North-Holland
Publishing
Co.
SELF-DIFFUSION OF SPHERICAL BROWNIAN WITH HARD-CORE INTERACTION Sarwat
fir Physik,
Fakulttit
HANNA,
Walter
Llniversittit
Received
The
N-particle
particles
interacting
self-diffusion constant. the
Oseen
Smoluchowski
coefficient
is found
The hydrodynamic approximation,
hydrodynamic
equation
through hard-core
interaction
HESS
Konstanz,
and Rudolf
D7750
for a system
of spherical
Brownian
to first order in the volume concentration where
is then taken into account
D, = Do(l -0.091#1) is found, due to Felderhof
Fed. Rep. Germany
9 July 1981
to be given by D, = Do(l - 24).
interaction
KLEIN
Konstanz,
is solved exactly
potentials
PARTICLES
gives D, = Do(
whereas
I-
in a perturbative the
4. The
DO is the free diffusion improved
way. form
Using for
the
1.894).
1. Introduction
In recent years there has been considerable interest in the concentration dependence of the collective (or mutual) diffusion coefficient D, and the self-diffusion coefficient D, for macroparticles suspended in fluids. A number of experiments by means of more conventional methods’) and by photon correlation spectroscopy*) have been performed which have led to considerable theoretical work in this field. From the theoretical point of view there is, for general interactions among the macroparticles, no reason why D, and D, should be equal, even to lowest order in the volume concentration C#Jof macroparticles. The diffusion coefficients D, and D, are defined by the hydrodynamic poles of the autocorrelation functions for the total concentration fluctuations 6c(k, t) and for one-particle fluctuations 6Ci(k, t), respectively. These two functions describe different physical situations and will therefore generally lead to different results for D, and D,. In this paper we will concentrate on the lowest-order concentration dependence of the self-diffusion coefficient D, for a system of particles interacting via a (repulsive) hard-core interaction and in addition through the so-called hydrodynamic interaction. The latter arises because of the velocity field which is produced in the solvent by the motion of the tagged particle and felt by neighboring particles. In the discussion we will come back to a comparison of D, with the collective diffusion coefficient D,. 0378-4371/82/0000-0000/$02.75
@ 1982 North-Holland
S. HANNA
182
Earlier starting
theoretical
from irreversible
equations.
Writing
free macroparticle, from the N-particle a hierarchy truncated a result
approaches
include
thermodynamics
D, = DO(l - CX~) where
et al
a calculation and using
of D, by Batchelor’) two-body
Do is the diffusion
hydrodynamic coefficient
of a
Batchelor obtains cy = 1.83. A different approach starts Smoluchowski equation. From it Felderhof4) has derived
of equations
for
s-particle
distribution
functions
which
for systems of low concentration. Felderhof has derived for the collective diffusion coefficient. This approach
can
be
in this way was later
extended by Jones’) to systems of two different kinds of particles. One can also apply the Mori-Zwanzig projection operator formalism to the N-particle Smoluchowski equation’.‘), obtaining in this way exact memory equations for the collective and the self-diffusion propagator. It can be shown that the memory function in the case of collective diffusion vanishes in the hydrodynamic limit faster than k*, so that memory effects are not important for the collective diffusion constant D,. In contrast, this is not the case for selfdiffusion. Since Felderhof’s method is essentially a mean-field theory without memory effects, his conclusions concerning D, are in agreement with those obtained by the Mori-Zwanzig method, whereas the extension to the selfdiffusion problem not included. The main the memory
leads to results
which disagree,
because
memory
effects
are
problem of the Mori-Zwanzig approach lies in the evaluation of functions. In an earlier paper’), concerning charged spherical
particles, a mode-mode coupling approximation for systems of small volume concentrations proximation’). It should used different approaches of course also dependent interaction.
was used. Another possibility is a weak-coupling ap-
also be noted that different authors have not only and various approximations, but that the results are on the particular form used for the hydrodynamic
In this paper we concentrate on the lowest-order concentration dependence of D,. We first solve the two-body problem with hard-core interactions exactly. Using this result we obtain without approximations the solution of the N-particle Smoluchowski equation with hard-core interactions to linear order in the volume concentration 4. This gives an expression for D, for hard-core systems, which includes the effect of the self-memory function to first order in 4 exactly. The hydrodynamic interaction is then treated by a perturbative method to first order in the strength of this interaction. Two different models, the Oseen tensor and Felderhof’s form for the hydrodynamic interaction”) are treated explicitly and are shown to lead to rather different results. Finally, we discuss our findings for D, in relation to earlier ones.
SELF-DIFFUSION
2. Exact solution
OF BROWNIAN
of the two-particle
183
PARTICLES
Smoluchowski
equation
for a hard-core
system The
Smoluchowski
hydrodynamic
equation
interaction,
for
two
interacting
particles,
but
without
has the form
$p(r,,t-2, t) = hl2p(rl,i-2, t), (2.1) ~,2=Do-$($.+P
I
~)+D,,$(&+p~),
I
where p(rl, r2, t) is the probability to find at time t the two particles at r, and r2, respectively, and U(r,,) denotes the two-particle potential, for which we assume a hard-core form 00 for r12 < d, for r12>d,
u(r’2)=(0
(2.2)
d is the particle diameter and r12 the relative distance t-r2 = r, - r2; fi = (knT)-’ and Da = ksT/c, J’ is the friction coefficient. Transforming (2.1) to relative coordinates
and center
of gravity
coordinates, aukI am
&=$D,&+?D,J&(&+/~-.
In the following
we want
to treat
only
particles and omit the center of gravity time 0 the centres of the two particles Then the initial distribution is
R = i(r, + r2), yields (2.3)
)
the
relative
motion
O(x) denotes 0
“(‘)=(I
the unit step function
forx
Because of the singular use the relation exp(-pfJ(rr9) Formally function
=
(2.5) nature
of the interaction
(2.2) we may also
(2.6)
eq. (2.1) may be integrated
to the initial
potential
@(r12).
p(r12, t ( r-0) = exp(fir2t)Wrr2subject
two
that at ro> d.
(2.4)
p(r12,O) = 8(rr2 - r-0)0 (r-12- d), where
of the
term in (2.3). Let us assume were separated by a distance
distribution
and gives for the conditional
ro)@(ru(2.4).
d),
distribution
(2.7)
et aI
S.HANNA
184 In spherical
f1 = (I‘+, q) expanded
coordinates
and
the s-function
f1,, = (19,). po) are
in spherical
the
can be expressed
spherical
angles;
as
6(R ~ {I,,)
can
be
harmonics,
(2.9) In spherical
coordinates
the two-particle
diffusion
operator
(2.3) has the form (2. IO)
where
iZ is the square of the angular
momentum
operator,
(2.11) The spherical
harmonics
i’Y;l(n,
are eigenfunctions
of L’, (2.12)
= I(/ + l)Y;“(O),
and therefore exp(d&)Y;l(O)
= Y;“(O)
(2.13)
exp(I&t),
where (2.14) Inserting
(2.8). (2.9) and (2.13) into (2.7) gives (2. IS)
where
X,(r. t 1r,,) is determined
Xl(r. t I rd = exp(RA) Since
we
penetration defining
know of
the
that two
a new function
by
$j 6(r - ro)O(rothe
hard-core
particles k,(r,
we
d).
(2.16)
potential take
this
allows
at no time
into
account
a by
t 1r,J as
X,(r, t 1ro) = O(r - d)JZ,(r, t I rd. Inserting
(2.2)
explicitly
this ansatz into (2.16) and differentiating
(2.17) with respect to time yields
SELF-DIFFUSION
the differential
OF BROWNIAN
PARTICLES
l?S
equation
+ 2D,$(r
- d) i
$(r,
(2.18)
t 1t-0).
Here we have used eq. (2.6) and 6(r - d) =
$
O(r - d).
The problem
of solving
to the problem
of solving
$ JI,(r, together
t 1 rO) = y
(2.19) the two-particle
diffusion
equation
is now reduced
the equation ($
with the boundary
r2i-
I(1 + I))R,(r,
t 1r&
(2.20)
condition (2.21)
this condition expresses the fact that the probability current has to vanish at r = d. Since the probability to find a particle at infinity is zero we have as a second boundary
condition
A(m, t 1ro) = 0,
(2.22)
and from eqs. (2.16) and (2.17) the initial
condition
%(r, 0 1 rO) = i 6(r - ro). In order the initial dition.zero; %(r, r
to solve condition,
(2.23)
(2.20) we look for a particular and for a complementary
their sum has to satisfy I r0)
=
Udr,
t
I r0)
+
Wr,
the boundary t
solution solution
Ur which
satisfies
W, with initial
con-
conditions.
/ r0h
Udr, 0 I r0) = i W - r0),
(2.24)
W,(r, 0 ) ro) = 0. We make the ansatz
CJdr,t 1r-0)= j dq g,(r, q I ro) em2Doq2r 0
(2.25)
IX6
S. HANNA
and find that g,(r. y 1 r,,) satisfies
the Bessel
(
r’ < + 2r $ + (q’r’~ dr-
The
solutions
kind”),
are
the
I(1 + I)))g,(r.
spherical
et al
equation
q 1r,,) = 0.
Bessel
functions
(2.26) of the
first
and
second
j,(q r) and yr(q r), respectively. 1rdidq r) + &(q
gr(r. 4) = A(4 Expanding
the a-function
1rdydq r).
in spherical
(2.27)
Bessel-functions
of the first kind”)
1
(2.28)
2.
we find
AI(~) =
2y? .h(q rd. 77
(2.29)
H,(q) = 0. The Laplace
of Udr, t 1r,,) is
transform *
Udr. z 1rd =
1 dt
e “U,(r, t 1r,,)
0 (2.30) The complementary Laplace transformation
W(r, t I ro) has
solution yields
also
to
satisfy
eq.
(2.20).
zW,(r.zir,,)-++($r’$-l(l+l))W,(r.zlro) = Wl(r, t = 0 ) ro) = 0 because
(2.3 I)
of (2.24). This is again a Bessel
2 2a -$+;z(
[([+I) rC-+20,
2 ))
equation,
Wr(r, 2 I rd = 0,
(2.32)
the solutions are the modified spherical Bessel functions of the first kind i,(rgz/2&) and of the third kind kr(rV’z/2Do)“). Since the il(r%‘z/2Do) are divergent for r+cfl, they can be omitted because of (2.22). Then
+
CI(Z
I rdkdr~/zDDJ.
(2.33)
SELF-DIFFUSION
The
unknown
condition
Cr(z ) ro)
functions
(2.21), which
OF BROWNIAN
completes
finally
187
PARTICLES
determined
by the
boundary
the solution,
m
Cl(&)=-i]dqq’
’ z + 0
f;(b)
where
stands
With (2.19,
__-
jX4
d)h(q
r0)
(2.34)
2Doq2~z/2D,k;(d%‘/)’
for @,(x)/ax Ix+,.
(2.17)
(2.33) and (2.34) we find finally cc
p(r,zIr0)=O(r-d)f
Y~‘(R)YP(Ro)*~I~~~‘~~‘~~~~~
i
I=0
m=-I
0
(2.35)
3. The self-diffusion coefficient The
Fourier-Laplace
tagged
Brownian
C,(k, z) = (e-” Here
fi,
transform
particle
of the
self-diffusion
propagator
rl[~ - firi,lm e” “I).
is the diffusion
of one
is
operator
(3. I)
of an N-particle
system (3.2)
UN is the N-particle UN = f 5 i,j=
i#j
I
potential,
which we assume
as a sum of pair potentials,
LJ(rij).
(3.3)
The tensors Dii describe the hydrodynamic discuss in the next section. If hydrodynamic
interaction interaction
effects, which we will is neglected then
Dij = D()Gij1. ((. . .)) denotes
Furthermore, ((.
* .)> =
(3.4)
1
d{rl(.
a canonical
. ‘) exp(-puN)//
expectation at-1
value,
eW-PU,h
The self-diffusion constant D, is defined from the following self-diffusion propagator in the hydrodynamic limit C,(k, z) = (z + D,k*)-‘,
for k, z small.
(3.5) form
of the
(3.6)
S. HANNA
188
Inserting
et al
(3.6) leads to (3.7)
z)) ‘(I - =C’,(k. 2)))
D, = lim lim{(k’C’,(k.
: -0 A 4
Using (3.1) we get
Employing
the expression
(3.2) for fl,.
and then taking the limit k +O, D,= where
k. (D,,)-
k-l;lrr((k
performing
some partial
integration5
D, is given by
.f,)[z
-- ri,]
(3.9)
‘(k .f,,).
k is a unit vector in the direction
of k and (3.10)
A
similar
operator
result
for
technique.
D,
was
Since
model of hydrodynamic
obtained
Ackerson
interaction,
by
Ackersonh),
restricted
a projection
his treatment
the Oseen-tensor.
absent in ref. [6], because the tensor divergence
using
to a special
the last term in (3.10) is
of D,j vanishes for the Oseen
tensor. In the present order
paper
concentration
corresponding first-order order
we restrict
correction
expression
assuming
term stems from
clusters
will
ourselves
to D,.
The
a noninteracting
N independent
contribute
to U(C’).
equivalent
it is sufficient
Neglecting
now also hydrodynamic
to the calculation zero-order
term
single particle,
two-particle
Since
clusters.
all particles
to treat one cluster and to multiply interaction,
of the first-
is given
are
by the Do. The
All higherstatistically
the result by N.
eq. (3.4), we obtain from (3.9)
and (3.10) D,=Do-lime
z-0
I
drl?(k ^ - fdrd)
x [z - fi(i,J’Ci *fdrd)
(3.11)
expt-pU(rdi,
where
fdrld
= HOP
&
U(rtd = DOB(T&U(r12))i12.
hii,, is now the same two-particle is the hard-core
potential
diffusion
operator
(2.2) and iI2 = r12/r12.
(3.12) as in (2.1),
U(rrz) = U(r,,)
SELF-DIFFUSION
OF BROWNIAN
We now use the properties of the hard-core exp(-pU(rr,)), eq. (2.6) and (2.19), which yields frArr2) exp(- PUG-r,)) = -Do?u &
189
PARTICLES
radial distribution
function
0 G-12- d)
= - Doi128(r12 - d) = -Doi1220(r12-
d)6(r12-
d).
(3.13)
The last equality should be considered as an operator equality, valid in an infinite integral over r12. The factor 2 arises from O(0) = l/2. This property can be verified by considering the O-function and the a-function as limits of continuous functions, e.g. (3.14a)
0 (x) = f + ljr&an-‘3,
(3.14b) Inserting now a a-function 8(r12- ro) behind the resolvent operator and using (2.7) we can write for the self-diffusion coefficient
.
aW12)
drr2 drok - i12 ~ x
in (3.11)
ah2
p(r12, z = 0 1ro)k - io6(ro- d) . >
(3.15)
Writing k^- i,2 = cos 19,~= ~/(47~/3)Y%fl,~) and analogously for k^- io, using the Laplace-transform of (2.15) and the orthogonality relations of the spherical harmonics, I
dRY ;I(fl)Y ;?‘(a)* = 6,,&,,,,,,,
(3.16)
D, becomes
x I drorfiXl(rlz, z = 0 1r,Jb(ro-
d)).
(3.17)
Finally, with (2.17), (2.6) and (2.19) D=D s
0
=
(~-&CD 3
Do( 1 - F
o I
dwdh
‘2
-
4
cd4D,,J?(d, z = 0 1d)).
j
dror%h,
.z =
0
1 ro)Wo
-
4)
(3.18)
S. HANNA
190
In
order
to determine
et al.
%(d, z = 0 1 d), we use (2.33) and (2.34) and perform
the
limit z + 0”)
id4 rd -
0
(p(r,,- r,z)++
$ idy
4 +
f$ qjdq d)) 1
(rl?-
II
rd+& I2
O(rr
(3.19)
-%_@(d-r")). For the special
case r. = r,? = d we find
%(d, z = 0 1d) = (8dDo)
Here again the property yields
the exact
II,= where
(3.20)
‘.
O(0) = l/2 has been used. Substitution
of (3.20) in (3.18)
result (3.21)
Do(l-24).
C#Jis the volume
cf~= $
concentration
of the Brownian
particles,
(d/2)‘c.
(3.22)
4. The effect of the hydrodynamic Neglecting
d)
0
the hydrodynamic
interaction interaction
on the self-diffusion has enabled
coefficient
us to find an exact
solution for the two-particle diffusion equation and an exact result for the first-order concentration correction to the self-diffusion coefficient. This result is however not of much use for the understanding of experiments because of the complete absence of the hydrodynamic interaction. the effect of this interaction in a perturbative way.
Therefore
we include
To lowest order in the concentration we need only the hydrodynamic interaction of a pair of particles. In its most general form the diffusion tensors can then be written as’,“)
@I = 022= &AI + h,(rdP 012 = &I = &CWrP
+ MM1
+ M-&1
The functions hn(r12) will be specified The tensor P projects an arbitrary
- IV, - P)).
(4. la) (4. lb)
later on by choosing particular models. vector on the relative vector r12. In
SELF-DIFFUSION
spherical
OF BROWNIAN
191
PARTICLES
coordinates (4.2)
PV = 1, and all other elements are zero. Using eqs. (4.1) in the general responding &=
two-particle
diffusion
form
of the
operator
diffusion
separates
operator
the
cor-
as
&+A&.
(4.3)
Here fi$ is the operator without hydrodynamic interaction, identical with (2.1) or (2.3), and Ah:, is the additional contribution from hydrodynamic interaction. Omitting again the terms describing the center of gravity motions of the two-particle
cluster
we get (4.4)
Inserting
(4. I), Afi(i:, becomes
+
2Do&.(Mrd
- h(r,z)Xl - P) -
&
=2D,(&(hdrd - h3(r,3))(~2+ P y)
+2Ddhdrd
+2c2ri2(&2 +P y)
- M-d)
- 2Ddhdr12)- h&d k2i’. We now consider
the action
which
to the solution
is identical
Afiim(n2,t 1rd = 2 I=0
(4.5)
of Afit* on the zeroth-order
i
found
in section
solution
2. With (2.19,
pdrr2, t 1ro), (2.17),
Y;"(fl)Yi"(fl~)
m=-I
x [ WI2 -
d)2Q($2 (h&-d - Wd) &
+ Oh - 42Ddhdr12) - Mrd
ia T
-
r12 arlz
a
62-ah2
- O(r,, - d)2DO(hAr,2) - h4(r,& $z [(I + 1) + 2DO(h,(r,2) - h3(r,2)M(r,2-
d) -$-j%(r,2,
t 1rd.
(4.6)
192
S. HANNA
Because
of eq. (2.21),
dfi(i:z&rl5
the fourth
et al
term vanishes and
t I rd (4.7)
with the operator
The
Laplace-transformed
equation
solution
with the operator
of
the
complete
two-partic!e
as given in (4.3) can be written
in a formal
diffusion way as
with
p”(r12. z ( rd = ([z -
dkl
Eq. (4.9) is the perturbative
‘A~i:#‘pdrI~. solution
interaction. The first-order
correction
is
z ( rd
of the problem
(4. IO) including
hydrodynamic
SELF-DIFFUSION
Inserting
OF BROWNIAN
193
PARTICLES
(2.15) and (2.17) gives
pr(r12, z
1 ro) = @(rt2-
x
4
Y;l(fl)Yi"(fl~) 2 2 I=0 m=mI
dr’r’2_%r(r12r z
1 r')&%(r',
(4.12)
z 1 ro).
I
For the calculation
of the diffusion
in section
eqs. (3.9), (3.10) we find
3. From
D,=Do+c -
d3rr2k^ .
I
we proceed
in the same way as
. i ew-pU(b2))
d3rr2(k^ . fl2(r12))[z -
lim c Z-r0
D’,,
coefficient
h721m
I
x (I .fdr12))
(4.13)
exp(-P~(r~2)),
D;, = /A, - Do1 and
where
(4.14)
The tensor
divergence
is”)
=
Do(&
(hdr12) -
+Ddhdrd =
defining the function This result shows
-
h(rd))i
- h(rd
- hdrd + Wd
+2i I
(4.15)
DoHdrdi,
Hl(rr2). that f12(rr2) is proportional
ffdr12) - H2(r12)P
to i
~
(4.16)
where I-Mrr2) = - (1 + hr(rr2) - Mrr2)). Using
a similar
procedure
as in eq. (3.13) we can write the last factors
(4.17) of the
194
S. HANNA
integrand
et al
in (4.13) as
k^* fdrd
expt-pUtrid)
= exp(-pU(r,z))D,,(H,(r,,) + 2H2(d)ti(rIT-
Inserting yields AD,=
a a-function
D,-
behind
the resolvent
Y:‘(0).
(4.18)
in the last term of (4.13)
(4.19) for AD,. Restricting
ourselves
to
(2.17) and (4.12)
AD:“‘=+;_/ I
AD’,” = 7 x
operator
%
d)).
Using (4.9), this is the perturbation series p = p. + pI, we obtain AD, = AD/“’ + AD’,“.
x
J
D,,
x (Hdrd + 2HddP(ro-
With (2.!S),
d))
drr,r%r(rr:, CD;
I
drI?r:?(H,(rl*)0(r,z-d) + Hd46(r12-
I
z = 0 1 r&Hl(rcJ
+ 2HAdP(ro-
dr~~r%H~(r~~)@ (rt2 - d) + HAd)S(r,,
dr’rr2%,(r12,z = 0 1 r’)
I
drg~~,~,(r’,
d)) d)).
(4.20)
- d))
z = 0 1 ro)
x (Hdrd + 2HddF%r(~- d)).
(4.21)
The function .%,(rn, z = 0 1 r,,) has been given in (3.19), the differential operator 2, is defined in eq. (4.8), the only unknown functions are now the h,(r), describing
the specific
form of the hydrodynamic
interaction.
The rest of our
task is a tedious but elementary calculation of the integrals in (4.20), (4.21). We performed these calculations for two specific models of hydrodynamic interaction: the Oseen-approximation and Felderhof’s reflection series”). The Oseen-approximation is obtained from the solution of the linearized NavierStokes equation to lowest order in the ratio d/r. Therefore the Oseen approximation is valid if two particles are far separated. On the other hand, the hydrodynamic interaction will only be important if two particles are close together, and therefore the Oseen approximation must be considered as poor. Nevertheless, because of the simplicity of its analytical form, it is the most
SELF-DIFFUSION
OF BROWNIAN
195
PARTICLES
often used model of hydrodynamic interaction; for this reason we include it here. There have been many attempts to improve the Oseen approximation, solving the linearized Navier-Stokes equation to higher orders in d/r. Batchelo?) has obtained numerical results for the hydrodynamic interaction for all values of the interparticle separation and has given the analytical asymptotic form up to (d/r)4. In this paper we adapt results derived by Felderhof’@) who has presented the asymptotic expansion up to (d/r)‘. Felderhof’s result seems to be the most reliable analytical form of the hydrodynamic interaction for a system of low concentration (to Q(4), which is consistent with our calculations). In table I we specify the two forms of hydrodynamic interaction, and in table II we present the results for AD’,“, AD:“, J d3r& * D;, * h9(rlz- d), and a, where (Y is defined by D, = DO(l - (~4). The mean-field contribution _fd3r& * DiI * b(r12d) has already been calculated by Felderhof and is taken from his work.
5. Discussion
The N-particle Smoluchowski equation has been reduced to a two-particle in solution. equation for a system at low concentration of macroparticles
TABLE I Expressions for the function hi(x), i = 1,. ,4 entering definition (4.la, b) of the diffusion tensor D, which describe hydrodynamic interaction in Oseen- and Felderhof-approximation, respectively. Also given are the functions H)(x), eq. (4.15) HZ(X), eq. (4.17); x stands for d/r
Oseen
hl
0
hz
0
h, h4
3 -x 4 3 -x 8
HI
0
HZ
-1+:x
Felderhof 11 --x154 +mx6 64 --x6 17 1024 3 1 75 -x-jjx 3 +mx’ 4 3 1 -x+16x) 8
the the and
S. HANNA
196
et al
TAHI.E Results for the self-diffusion column
represents
contributions
(4.20)
coefficient
the mean-field and
(4.21).
tributions
D, to first
contribution. respectively.
II order in the volume
The The
second fourth
by giving the coefficient
and third column
concentration columns
summarizes
IY in D, = Do(
4. The first
are the memory the
three
con-
I ~ UC/J)
Hydrodynamic interaction neglected Oseenapproximation Felderhofapproximation
From
the
2+! x
0
- 1.736,
knowledge
of the
distribution
function
we have
calculated
the
self-diffusion propagator C,(k, t). The hydrodynamic pole of this correlation function defines the self-diffusion coefficient II,. The method to obtain D, which has been used in this paper, is equivalent to the approach which applies the Mori-Zwanzig projection operator formalism to the N-particle Smoluchowski equationh.‘). One obtains an exact memory equation for C,(k, t), which can be written as*)
6 C,(k, t) = - K,,(k)C,(k,
dt’M,(k,
t) +
t - t’)C,(k,
t’).
(5.1)
0 Here,
K,,(k)
is the first cumulant
of C,(k, t), which determines
the initial
slope
of log C,(k, t) versus t and M,(k, t) is the memory function. The quantity given in the first column of table II, which represents the mean-field contribution
to ID,, is closely
related
to the
first
K,r(k)/(D,k*) as k + 0. Taking the Laplace-transform the self-diffusion coefficient is
cumulant;
it is the
limit
of (5. I) and using
of
(3.6),
(5.2) It was first pointed out by Ackerson6) that M,(k, t)/k* does in general not vanish as k -+ 0. Therefore, memory effects enter the determination of D,. Although the approach used in this paper avoids a direct calculation of the memory function M,(k, t) the contributions AD!‘) and 406” to D, are just the memory effects. We note that in writing eqs. (3.7) and (5.2) for D, it is implicitly assumed that M&k, z) has no hydrodynamic pole.
SELF-DIFFUSION
OF BROWNIAN
PARTICLES
197
The memory function also enters the velocity-autocorrelation function one particle. Using the continuity equation it is easy to show14) that
!j(Ui(t)
’
Vi(O)) = 1’ ,‘lom & [2K,,(kM(t)
- K(k, t>l,
of
(5.3)
where Vi(t) is the velocity of particle i at time t. This result together with eq. (5.2) leads to co D,= i \ dt(ui(t) * vi(O)).
(5.4)
0
Since the mean-square displacement relation function one obtains’)
W(t) is related to the velocity-autocor-
W(t) = k ((R(t) - R(O))*) = lim k-O !$@
t-
dt’(t - t’)M,(k,
(5.5)
t’)]
I 0
and D,
= lim *+m
w(t).
(5.6)
t
When analyzing experimental data it might be easier to determine first cumulant. From it one can define a quantity’5*‘6) Ds,& k) =
.K,l(k) k
which is sometimes called an effective diffusion coefficient. and (5.6) one should note that this quantity is just D s,eR= lim wO. f-0 t
only the
(5.7) From eqs. (5.5)
(5.8)
Therefore, it neglects all memory effects and differs from the hydrodynamic definition of a self-diffusion coefficient as in eqs. (5.4) and (5.6). The importance of the memory function lies in taking the fluctuations of the surrounding particles into account. Since self-diffusion is a one-particle property these fluctuations are essential. As mentioned in the introduction, this is different in the case of collective diffusion, which is a macroscopic effect. Therefore we believe that an extension of Felderhof’s method4) to the self-diffusion problem is not justified.
198
S. HANNA
et al.
We now turn to a discussion of our results First, it should be noted that memory effects
for D, summarized in table II. in the hard-core case without
hydrodynamic
changing
24).
This
contribution interest
interaction substantial
are very important, shift
(first column
is the
large
is due in table
difference
Oseen-approximation and Whereas the hydrodynamic
to the
fact
that
D, from there
II) in the hard-core between
Felderhof’s interaction
the
two
system.
results
Do to D,,(l ~
is no mean-field Of particular
obtained
form of hydrodynamic taken in Oseen-approximation
for
the
interaction. reduces
the concentration dependence of D, compared to the hard-core system rather drastically, the more reliable form given by Felderhof brings the result almost back to the hard-core case. This may be understood as follows: The Oseenterm represents the long-range part of hydrodynamic interaction and therefore reduces the effects of the very short-range direct interaction rather dramatically. The velocity field produced by the moving macroparticle removes neighboring particles from the collision trajectory leading to an increase of D, compared to the pure hard-core result. Taking into account the short-range term of the hydrodynamic interaction one also includes the reflected velocity fields from other macroparticles. These have the effect of slowing down the motion of the tagged particle, thereby decreasing D, again. In comparing our results with those Batchelo?) obtained D, = Do(l - 1.834),
of earlier papers we first which should be compared
note that with our
mean-field value in the first column of table II. More recently, Marqusee and Deutch? have calculated D, = Do( 1 - a$) from the memory equation approach by using a weak-coupling approximation in evaluating M,(k, t). For the hard-core system cy = 1.33 is obtained compared to cu = 2 in our treatment. The weak-coupling approximation replaces a two-particle propagator by the product of two one-particle propagators. This is an expansion in terms of the strength of the interaction. Such expansion seems to be questionable for the hard-core case, which would explain the rather large difference of the results. These have also included hydrodynamic interaction in Oseen-approximation cy = 0.07 compared to our (Y= 0.09. As mentioned above the Oseen-term collisions
drastically.
Therefore
it is really not so important,
authors finding reduces
how the hard-core
is
treated if hydrodynamic interactions are included. This seems to be the explanation why their result and ours are so close in the Oseen-approximation. Finally, we want to compare D, with the collective diffusion coefficient D,. As noted earlier the collective memory function M,.k. t) vanishes faster than k’ as k +O. Therefore, memory effects vanish for D, to first order in C#LOnly three-particle correlations would contribute if hydrodynamic interaction is included. For a hard-core system one has the well-known result”) D, = DO(I + 84). Defining the collective friction coefficient fC by writing in the
SELF-DIFFUSION
OF BROWNIAN
PARTICLES
199
solvent fixed frame (5.9)
where n is the osmotic pressure, one has fC = fo, f. being the friction coefficient of a free particle. The self-friction coefficient fS is defined by D, = kaT/f,. Therefore, for a hard-core system we have fS = fo( 1 + 24) Z fC. Including now Felderhof’s form for the hydrodynamic interaction, D, is given by4) D, = DO(1 + 1.564), and from (5.9) one finds fC = fO(1 + 6.444). This is to be compared with our result for D, which is D, = Do( 1 - 1.894) and with fS = f,Jl + 1.894). Therefore in all cases considered here, one always has D, # D, and fC # fS. There is no reason why these different transport coefficients should coincide besides in the trivial limit of infinite dilution.
References 1) See, for instance, K.H. Keller, E.R. Canales and S.I. Yum, J. Phys. Chem. 75 (1971) 379. These authors used a diaphragm diffusion cell. 2) A recent comprehensive review of light scattering methods is given by P.N. Pusey and R.J.A. Tough, in: Dynamics of Light Scattering and Velocimetry: Application of Photon Correlation Spectroscopy, R. Pecora, ed. (Plenum, New York, 1981) to be published. 3) G.K. Batchelor, J. Fluid Mech. 74 (1976) I. 4) B.U. Felderhof, J. Phys. All (1978) 929. 5) R.B. Jones, Physica 97A (1979) 113. 6) B.J. Ackerson, J. Chem. Phys. 64 (1976) 242, 69 (1978) 684. 7) W. Dieterich and I. Peschel, Physica 95A (1979) 208. 8) W. Hess and R. Klein, Physica 105A (1981) 552. 9) J.A. Marqusee and J.M. Deutch, J. Chem. Phys. 73 (1980) 5396. 10) B.U. Felderhof, Physica 89A (1977) 373. Functions (Dover, New York, 11) M. Abramowitz and I.A. Stegun, Handbook of Mathematical 1965). Methods for Physicists (Academic Press, New York, London, l-3 G. Arfken, Mathematical 1970). R.B. Bird, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric 13) See, for instance, Liquids, Vol. 1 (Wiley, New York, 1977). in Liquids and Macromolecular Solutions, V. Degiorgio, M. 14) W. Hess, in: Light Scattering Corti and M. Giglio, eds. (Plenum, New York, 1980). 15) P.R. Wills, J. Chem. Phys. 70 (1979) 5865. 16) H.M. Fijnaut, J. Chem. Phys. 74 (1981) 6857. B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 17) See, for instance, 1976).