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Collective diffusion of Brownian particles with square well and hydrodynamic interaction.

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Ann. Physik 2 (1993) 201 -229
Annalen
der Physik
0 Johann Ambrosius Barth 1993
Collective diffusion of Brownian particles
with square well and hydrodynamic interaction
Jiirgen Vogel
Institut fiir Theoretische Physik, R.WT..H. Aachen, Templergraben 55. W-5100 Aachen, Germany
Received 27 July 1992, accepted 10 November 1992
Abstract. The Iow wavenumber collective diffusion coefficient of a semi-dilute suspension of
spherical Brownian particles interacting via square well potential and hydrodynamic pair interaction
is considered. The first two nonvanishing terms of an expansion in powers of the wavenumber are
calculated. Analytical expressions are found for extremely narrow wells and in the limit of large well
diameters.
Keywords: Suspension; Brownian motion; Collective diffusion; Square well
Hydrodynamic interaction.
interaction;
1 Introduction
To model a suspension of interacting colloidal particles it is often no sufficient to take
account of the repulsive core of the particle interaction only, but one has to treat the
attractive regime of the interaction as well. Additionally, in general it is not allowed to
neglect the hydrodynamic interaction between the particles. One might think of silica
spheres with attractive coating 111 or silica dispersions with small polymers added [2],
which serve as a means to drive the particles together. Furthermore there are biological
suspensions like proteidwater solutions showing a phase separation at certain critical
densities [3]; thus the effective interaction between the proteins contains an attractive
part besides a repellent core. Unfortunately the critical densities are still far beyond the
densities the theory here is capable to treat.
In a previous article we derived general expressions for the wavenumber and frequency dependent collective diffusion coefficient of a semi-dilute suspension [4]. The results
were used to calculate short- and long-time diffusion coefficients of a suspension of
hard spheres at small wavenumbers [5].
Here the simplest realistic model to include an attractive potential is treated: a semidilute solution of hard spheres surrounded by a finite potential well of given width. We
consider hydrodynamic interaction between pairs of particles and use stick boundary
conditions. The first two nonvanishing terms in an expansion of the long-time diffusion
coefficient in powers of the wavenumber are calculated. Results for the short-time diffusion coefficient are obtained up to equivalent order in wavenumber. Analytical expressions are found in the limit of extremely small and large width of the well.
The next section treats the basic concepts of a statistical description of monodisperse
suspensions. In Section 3 general results previously found for the collective diffusion
202
Ann. Physik 2 (1993)
coefficient of a suspension are reported. The rest of the article presents the calculation
of the short-time and the long-time collective diffusion coefficient for particles interacting with a square well potential.
2 Basic concepts
We give an introduction to the statistical description of a suspension of particles immersed in an incompressible solvent.
As examples one might think of biological suspensions such as blood or milk. Blood
for instance contains different species of large proteins in aqueous solution: there are
the y-globulines with typical diameters of 5 nm and the red blood cells with diameters
around 7.5 pm, to mention just two out of a large variety of protein components [6].
The different species of solutes possess different shapes: the red blood cells are shaped
like a disk while the globulines have a spherical structure. There is yet another factor
which renders suspensions to be such complex systems: the protein diameters cited
above reflect only the average size of the particles. The diameter of red blood cells has
a finite variance.
We will approximate such complex suspensions by a model suspension. Unlike blood
our model suspension contains only rigid hard spheres of radius a. Driven by the impact
of the surrounding fluid molecules the heavier and much bigger spheres undergo Brownian motion in the solvent. Unless the solution is very dilute the heavy particles will interact with each other. There is direct interaction between the particles as well as
hydrodynamic interaction: with direct interaction we mean cg. a simple hard sphere interaction which prevents two particles to interpenetrate each other. In general the
presence of charges on the surface of the particles as well as in the solvent will modify
this simple interaction of two particles. Depending on their center-to-center distance r
the result will be a combination of attractive and repulsive regimes of interaction, see
e.g. [7]. By hydrodynamic interaction we mean interaction mediated by the solvent: a
sphere moving at constant velocity in a quiescent fluid will carry along some fluid. The
velocity field of this fluid is of very long range and exerts additional forces on other
particles. The change in the mobility of particles induced by the presence of other particles is called hydrodynamic interaction.
We focus on the dynamics of the suspended particles. The suspended particles and
the solvent are assumed to have a common temperature kBT = j?-'. The coordinates
of the centers of mass of N suspended spheres are denoted by X = @Z17 . . .,RN).We are
interested in the trajectories of the particles. The particles rotate freely. Let P ( X ,t ) be
the probability to find Nparticles at locations R , , . . .,RN at time t. We repeat a short
derivation of the equation of motion for P(X,t ) . Since the number of particles is conserved, the probability density obeys a continuity equation in the 3 N-dimensional coordinate-space [8]:
a,P(x,t )+
j
cN --aRia [P(X,t ) Vj]
=0
=t
The index i labels the particles. The corresponding probability current
P ( X ,t ) V j ( X t, ) of each particle i consists of two velocity contributions:
J. Vogel, Collective diffusion of Brownian particles
203
The first is a Brownian contribution deriving from an entropic force FB with [9]
The second includes pair forces due to a potential @(X)of pairwise direct interaction
)
two particles
potentials ~ ( tbetween
The N-particle velocities and the N-particle forces are proportional to each other and
=pDQ(X):
related by N 2different 3 x 3 mobility matrices pQ(X)
The mobility matrices pQ(X),
or equivalently the diffusivity matrices Dii(X),depend
on the configuration of all particles. This reflects the presence of hydrodynamic interactions. The diffusivities follow from solutions of the Navier-Stokes equations for the
slow motion of spheres.
With the help of the continuity equation we finally arrive at the generalized
Smoluchowski equation
This is a second order differential equation in space which fixes the time evolution from
an initial value P ( X ,t = to). The equation of motion has an equilibrium solution
p e q (XI
The configurational integral Z @ ) normalizes Peq(X)to 1.
Without hydrodynamic interaction the diffusivity matrices Dijdo not depend on the
configuration X of the particles: Dii = Dodo. The constant Do is the familiar single
particle diffusivity which foIlows from the Stokes-Einstein relation
In general one is not interested in the probability density itself but in averages of
observables. We are interested in the average density of the suspended particles. The
local observable n(r,t ) is a sum of delta functions at the locations of the particles:
n(r)=
c d(r-Ri) .
i
Its time-dependent ensemble average is defined via
(2.9)
204
Ann. Physik 2 (1993)
Instead of treating the probability density P(X,t ) as time-dependent, one might as well
choose the observables to be time-dependent. With the help of the formal solution
P ( X ,t ) = exp (t 9 ) P ( X ,t = 0) of the Smoluchowski equation one can rewrite the average in Eq. (2.9)
(n( r p 1= j d X P (X ,t = 0) n (r,1,X )
.
(2.11)
The local density n (r,t ) = exp ( t a n(r,0) is now time-dependent. The operator Y is the
adjoint operator of 9. It is found to be
(2.12)
In a homogeneous and stationary system the average particle density vanishes. TO learn
about diffusion one has to examine the second moment
5
(n(r‘,0) n (r,t )>= dXP,, (X)
n (r’,0,X)n (r,t,X) .
(2.13)
We omitted the index t on the angular bracket since at t = 0 the system was assumed
to be in equilibrium. From now on the angular brackets denote an equilibrium average.
In standard light scattering experiments the density-density correlation function is measured as a function of wavevector and time, see e.g. [lo].This corresponds to the Fourier
transform of the density-density correlation function:
which is the intermediate scattering function. The index T D denotes the thermodynamic limit: particle number Nand the volume $2 of the suspension tend to infinity but the particle density n = N/Q is fied. With the help of the Laplace transform
of the intermediate scattering function, G(q,a),a generalized frequency- and wavevector-dependent diffusion coefficient Dc(q,w ) can be defined:
(2.15)
Here S ( q ) = F(q, 0) is the static structure factor of the suspension. It is entirely fixed
by the direct interaction between the particles and can be calculated from equilibrium
thermodynamics.
For the simple case of a frequency-independent diffusion constant Dc(q) we can
discuss the temporal behavior of F(q,t ) , the inverse Laplace transform of G(q,0):
it
simply decays exponentially starting at its short-time value S(q):
F ( q , t ) = S ( q ) exP ( - q 2 & ( q ) t )
-
(2.16)
The rate of the exponential decay is fixed by the diffusion constant D,(q). A
frequency- or time-dependent diffusion constant describes more complicated temporal
behavior of the intermediate scattering function which goes beyond simple exponential
J. Vogel, Collective diffusion of Brownian particles
205
decay. In the limit of small wavenumbers q, which corresponds to large distances r-r',
one obtains the usual hydrodynamic diffusion constant Dc which can be determined
from e.g. sedimentation experiments.
In this article we calculate the generalized diffusion coefficient Dc(q,o)up to first
order in the density n. It is then sufficient to consider only the dynamics of independent
pairs of interacting particles. In an earlier article [5] we found explicit expressions for
the low-density collective diffusion coefficient in terms of the Smoluchowski operator,
Eq. (2.12), specialized to two particles. In the next section we will repeat general results
found for Dc(q,o).
3 Collective diffusion coefficient
We examine a semi-dilute suspension of spherical particles of radius a immersed in an
incompressible fluid of viscosity q. The particles occupy a volume fraction
@ = ( 4 n / 3 ) n a 3 ,where n is their number density. The frequency- and wavenumberdependent collective diffusion coefficient Dc(q, o ) can be measured by dynamic light
scattering experiments via the dynamic structure factor of the suspension. It is usual
to split Dc(q, o)in a frequency-independentpart and a frequency-dependent memory
kernel. In the limit of low densities this decomposition results in
Dc(q,0) = DO
+(Lc(q)+ac(q, 0))41+ o(#*).
(3.1)
In the limit of large frequencies o the memory kernel a&, o)vanishes and leaves only the effective diffusion coefficient
Here Dodenotes the diffusion constant of a single particle with stick boundary conditions, namely Do= kBT/(6nqa).In the opposite limit of long times Eq. (3.1) reads
Dc(q>=~otl+(lc(q)+ac(cl))41+0(@~)
. .
(3.3)
In Dc(q,O) and ac(q,O) we dropped the second argument to indicate long-time quantities. Due to the negative definiteness of the memory kernel ac(q) the long-time limit
Dc(q) of the coefficient of collective diffusion is smaller than Defr(q).For a discussion of the meaning of short and long times, respectively, see e.g. [lo].
We will calculate the first nonvanishing expansion coefficients of A&) and a&)
in powers of qa. The coefficient l c ( q ) starts with a constant and contains only even
powers in qa
&(q) = L
g)+ L g)q2a2+higher orders .
(3.4)
The short-time diffusion coefficient contains only even powers of q. The expansion of
a&) starts with q2a2 and includes odd powers as well:
a&) = a(CZ)q2a2+aF)q3a3+higher
orders
.
(3.5)
206
Ann. Physik 2 (1993)
The absence of the constant term stems from an additional conservation law: for a pair
of particles subject to two-particle interactions the total force vanishes [ll]. In the case
of hard spheres with hydrodynamic interaction the expansion coefficients defined above
proved to be finite [5]. We expect this to remain valid for a square well interaction, too.
The form of this direct interaction used here is
(3.6)
with r the center-to-center distance between the two spheres. In addition to a hard
sphere potential an attractive or repulsive shell of width 2a(x, - 1) was added, depending on the sign of the "depth" ulBefore we look at explicit expressions for the diffusion coefficients ac(q) and I&)
we make a few remarks on the dynamics of a semi-dilute suspension. The time-evolution
of the system is modelled with the help of the Smoluchowski operator, Eq. (2.12), which
takes account of hydrodynamic interaction as well as of the direct interaction u(r). For
low densities it is sufficient to look at a pair of particles. As for any two-particle problem the operator .Y(Rl,R2)can be expressed in center-of-mass and relative coordinates:
2 R = R, +R2,r = R2-R1.We then obtain the two-particle Smoluchowski operator
i a
a
.Y(R,r) = ---.[Dll +DlJl*-+2(V-fl(Vv(r))~[D,l -D12]*V.
2 aR
aR
(3.7)
The hydrodynamic two-particle diffusivity tensors Dll( r ) and D12(r)appear in the
center-of-mass diffusion as well as in the diffusion along the relative coordinate r. Since
the system considered is isotropic both tensors can be expressed in terms of standard
hydrodynamic functions A&) and Be@):
& , ( r ) = ~~[1+AII(r)ii+B,,(r)(1-Fi)]
,
D 1 2 ( r=
) Do[AI2(r)ii+Bl2(r)(l-PP)]
.
In the diffusion problem one does not encounter Y ( R , r )but the operator g(q)
which is closely related to Eq. (3.7). Due to the translational invariance of the suspension the center-of-mass coordinate does no longer appear. The function g ( r ) denotes
the two-particle correlation function. The presence of the hydrodynamic diffusivities in
the center-of-mass term Z(q,r ) ,
renders the operator g(q)anisotropic: diffusion along the direction of the wavevector
q is different from diffusion perpendicular to this direction. The tensor ~ ( r ) ,
J. Vogel, Collective diffusion of Brownian particles
207
is an effective relative diffusivity of a pair of particles. For low densities the two-particle
correlation function g ( r ) is
g ( r ) = exp (-Bu(r)) = ~ O ( r - 2 2 ) + ( 1- ~ ) O ( r - 2 2 ~. ~ )
(3.12)
We made use of the Heaviside function O ( r ) and the abbreviation E = exp ( - j u , ) .
This result reflects the general form of the equilibrium solution Peq(X),Eq. (2.7), in
the low-density limit.
In the next sections we will obtain explicit results for the small wavevector coefficients
of collective diffusion in the case of a square well potential.
4 Short-time diffusion coefficient
In this section we present the results for the effective diffusion coefficient Deffin
lowest order of density and for small wavenumbers. According to Eq. (3.2)
In a previous article [5] we found a general expression for the coefficient &(q) in
terms of the two-particle operator g(q)defined in Eq. (3.9):
with the function p,(r)
Usually the short-time diffusion coefficient A c ( q ) is split into five different contributions
In a straightforward calculation starting from Eqs. (4.2) and (4.3) we obtain as the first
term A v which is a pure virial contribution
W
Av(q) = -24
5 & x ~ ( J ( x ) -l ) j 0 ( 2 q a x ) .
(4.5)
0
Here x is used as an abbreviation for r/(2a). The next terms stem from hydrodynamic
interaction: the Oseen correction
and the dipolar term
Ann. Physik 2 (1993)
208
j,(x) denotes a spherical Bessel function of order n. The last two terms LA and As take
account of the higher order corrections in an expansion in l/x of the four
hydrodynamic functions A (r), A 12 (r), Bl ( r ) and B12(r).They read
and
We give the results for the square well potential of Eq. (3.6) for orders qo and q2: The
first six coefficients A(csw)can be calculated exactly:
The remaining three coefficients are integrals over linear combinations of hydrodynamic functions. They have to be done numerically. The general structure in terms of the
hard sphere results A(hs' is
A"w)=&l(h~)+(l-&)L(xl)
,
(4.11)
with the x1 dependent coefficients
(4.12)
and
m
(4.13)
209
J. Vogel, Collective diffusion of Brownian particles
For the numerical integration we used an expansion of the hydrodynamic coefficients
in l/x with up to 840 terms [121. To compensate for the slow convergence like
1 +O(l/ln (xl- 1)) of the integrands close to x1= 1 we employed an extrapolation
scheme introduced in 1131. The result is shown in Fig. 1. Ail three functions AA (xi)and
A$‘)’(’)(xl) decrease monotonically with xl. Table 1 repeats numerical values shown in
Fig. 1 . The respective hard sphere values A(hs) can be read off at x1 = 1. The zeroth and
second order contributions together add up to the result
Ac,h“)(q)= 1.454-1.858q2a2+O(q4a4) ,
(4.14)
obtained previously IS].
1 .8
1.6
1.4
1.2
A
’
0.8
0.6
0.4
0.2
0
1
1.5
2
2.5
3
3.5
4
XI
Fig. 1 Plot of the short time diffusion coefficients -AA(x,), A!$)(x,)and - A f ) ( x , ) as a function of
the square well parameter xl. (According to Eqs. (4.12) and (4.13)).
5 The long-time diffusion memory kernel
In this section we examine the long-time memory kernel a&) in the case of a square
well potential and for small wavenumbers. In the previous article [5] we found general
expressions for the two expansion coefficients a g)(q) and a g ) ( q ) defined in Eq. (3.5).
Since the third order coefficient will follow from the second order solution, we first
start with order q2. The second order coefficient a$) is the sum of a monopole and a
quadrupole part each in form of a simple radial integral:
( I = 0,2) of the perturbed distribution function w1 (4,r )
The radial components h2)(r)
210
Ann. Physik 2 (1993)
are the solutions of the second order differential equation
1 d
2dfj2) 1(1+1)
-Enr ---
r2 dr
r2
dr
Etfj2) = -
1
py
2a2
, I = 0,2
.
(5.3)
Eq. (5.3) is similar to the Poisson equation with a monopolar and a quadrupolar charge
density pj2). The normal and transversal components (with respect to r ) of the diffusivity tensor ~ ( rare
)
The above differential equation can be traced back to the two-particle SmoIuchowski
it is the small wavenumber expansion of the differential equation [5]
operator z(q):
where the small wavenumber expansion of the function pq(r), Eq. (4.3), is provided by
P q @ ) = Do42[Ps)(r)+p~2)(r)P2(COS
(e))l+o(q4a4)-
(5.6)
The two driving terms for I = 0 and I = 2 were [5]:
1
X
pL2’(x) = - - G ( x ) g ’ ( x ) - - g ( x ) [ A I , +2Bll + x W ]
6
3
,
P ~ ’ ( x ) =- -XG ( x ) g ’ ( x ) -2~ g ( ~ ) [ A l , - B l l + ~ W
. ]
(5.7)
3
Here W ( x ) is another hydrodynamic standard function
W ( x )=
G(x)-H(x)
X
+--1dG(X)
2 dx
and x = r/2a.
We like to study square well interaction. The two steps in the square well pair-correlation function g(x), Eq. (3.12), render the driving terms pj2)(x)singular at x = x, = 1
and x = x l . The analogy to the equivalent electrostatics problem with an insulating
sphere of radius x = 1 surrounded by spherical shells of anisotropic dielectric medium
~ ( 7 helps
)
to find the boundary conditions for the “potentials” A2)(x):there are the
conditions of continuous potential and continuous normal components of dielectric
displacement at point xi, with j = 0 and 1:
fj”(x;) =A2’(xi),
J. Vogel, Collective diffusion of Brownian particles
21 1
The two constants are 3c0 = 1 and 3c2 = 2. Additionally, for large distances x = x,,
x 2 - + w ,from the sphere the potential should vanish.
In the language of the diffusion problem the last equation is the conservation of
probability current across the potential step xi. At x = x o two spheres meet and the
normal component of the probability current has to vanish at contact. The general solution of Eq. (5.3) for I = 0,2 in the interval x,<x<xj+l is of the form:
f j 2 ' ( x ) = XQ+1 Q / o ( x )+Yo+i Q / i ( x ) + Ql2(x)
i=
9
.
(5.10)
Qlo(x) and Qlt ( x ) are the solutions of the homogeneous equation, Q,,(x) is a special
solution of the inhomogeneous equation. The boundary conditions Eq. (5.9) relate the
coefficients XQand YQ
Fj
=M/(xj)
Ef:]
(5.1 1)
via a transfer matrix M,(x,)
For this result we made use of the Wronski determinant
Qll( x ) resp. Q12(x)are known from the solution of the hard sphere problem. They all
show a logarithmic singularity for x+l and vanish like x - ' - l resp. proportional to
x - in
~ the limit of large x. Like the hydrodynamic functions G ( x ) and H ( x ) we represent the solutions as a power series expansion in l / x with up to 840 terms [12]. The
other two ascending solutions Qlo(x) of the homogeneous equation can be chosen
regular at contact. Here Qoo(x)is simply a constant
For QzO(x), which behaves like x 2 for large x, a simple power ansatz fails - the
resulting system of algebraic equations for the coefficients becomes singular. A better
ansatz is in terms of the second solution Q Z i ( x ) of the homogeneous equation [14]
3
Q 2 0 ( ~=) Q 2 1 ( ~ [) a l n ( x ) - c ] + x 2 - - x +
16
0)
b,x-"
n
=
323 1
, a = --5 . 2 1 4 .
(5.15)
~
We choose b3 = 1 and the ansatz gives a unique solution from which a and the rest of
the b, follow. The solution Q20(x) becomes regular at x = 1 when we set c = 0.0525,
which cancels the logarithmic singularities up to an accuracy of three digits. We are
now ready to fix the four coefficients X u and Yu: the regularity of the solutionfj2)(x)
at contact and the condition of vanishing potential for large x yield
212
Ann. Physik 2 (1993)
The first two of these four coefficients are already known from the hard sphere solution
QI
which is regular at x = 1. The two remaining coefficients are found from the transfer
matrix MI(xl) to be
(5.18)
The functions C, to E, depend only on the width x1 of the well and follow from the
solutions Q(x):
(5.19)
To rewrite the second order memory coefficient ag) of Eq. (5.1)7 we split off the
singular contributions of pj2)(x)deriving from g’(x) and decompose the range of integration into the two intervals [l,xll and [xl,.x2= 001:
The overbar excludes the delta function parts from pf)(x) of Eq. (5.7).
We represent a!$)as the sum of the hard sphere contribution and a proper square
well contribution. It is not difficult to separate the dependencies on the width and on
the depth of the square well potential:
(5.21)
The depth E enters in a simple way. The three x,-dependent functions
are defined in terms of the radial integrals 1,
J. Vogel, Collective diffusion of Brownian particles
I/(X,F)=
48 *
-1
~ x ’ X ’ 2 F ( X / ) P 1 2 ) ( X I ) / g ( X ) ),
21+11
213
1I X S X 1
,
(5.23)
and the functions JI
24C[ x 3 G ( x ) F ( x ).
J/(X,F ) = -
(5.24)
21+ 1
The function q ( x , ) is always positive. For stick boundary conditions the radial diffusivity G ( x ) vanishes at contact and Jl(1,F) drops out. Setting xl = 1 or E = 1 we
recover the hard sphere result for ag’ IS]:
The numerical values (with an accuracy of three non-zero digits) were
= -0.124
,
= -0.0448
.
(5.26)
At the end of this section we calculate the third order coefficient ag). In our previous paper [ 5 ] we found:
ag) = 2 4 ( ~ & ~
. ))~
(5.27)
The constant yh2) could be read off from the l/r-asymptotics of the second order solution:
A2’(f)= y o(2) 20 ,
f+o3
.
(5.28)
f
A comparison with Eq. (5.10) together with the remarks on the large x behavior of the
functions Q(x) right after Eq. (5.13) lead to the simple result
The hard sphere value
was approximately 0.0880. In the next section we will
discuss the limiting behavior for very wide and very narrow well potentials in more
detail.
6 Limiting values
In this section the limiting values for the collective diffusion coefficient D,(q) for very
large and very small width xi of the potential well will be discussed. Especially for
silica particles with a short range attractive coating [I] it is interesting to look at the
limit of small width xl.
214
Ann. Physik 2 (1993)
6.1 Short-time diffusion coefficient
To examine the short-time diffusion coefficient Deff(q)of Eq. (4.4) for a very narrow
well we have to examine the limit of small tl = x1- 1 for the five different contributions A(5w). Since the first three coefficients, Eq. (4.10),are known analytically, we can
Ap)
restrict our discussion to
and A$””. With the help of the behavior of the
hydrodynamic functions close to contact [151
one obtains
The various contributions add up to
As a check we calculate the value of Ac in the limit of sticky spheres [16]. In this
of the square well tends to zero and the depth E to infinity such
model the width
that 125, E = l / is
~ finite. Here T is a positive (stickiness) parameter:
1
Ap)(q)= 1.454-1.125--q2a2
(6.4)
T
Its zeroth order term is identical to the result obtained in [17].
For large width xithe asymptotics of the hydrodynamic functions AU(xl)and BU(xl)
[ 5 , 181 leads to the first few terms
A,
=
9
8x1 64x: ’
15
--+-
0)
‘!$
5
- -_-75
,
- 2 5 6 ~ ; 256xf
45
9
.
64x1 2 5 6 ~ :
---+-
(6.5)
J. Vogel, Collective diffusion of Brownian particles
215
Beyond xt = 4 the results already agree with the values given in Table 1. The virial contributions of Eq. (4.10) dominate the large x1 behavior of the short-time diffusion:
They do not depend on hydrodynamic interactions at all.
Table 1 Values of the coefficients A,(x,),
A ~ ) ( x , )and L?)(x,) as a function of the width parameter
x1 (according to Eqs. (4.12) and (4.13)).
XI
LA
18’
1.oo
1.01
1.11
1.21
1.31
1.41
1.51
1.61
1.71
1.81
1.91
2.01
2.11
2.21
2.31
2.41
2.51
2.61
2.71
2.81
2.91
3.01
3.11
3.21
3.31
3.41
3.51
3.61
3.71
3.81
3.91
4.01
- 1.8315
-1.81
- 1.62
- 1.48
- 1.37
- 1.28
- 1.20
- 1.13
- 1.07
- 1.01
-0.961
-0.915
- 0.873
- 0.835
-0.800
- 0.768
- 0.738
-0.710
-0.685
-0.661
-0.638
-0.618
- 0.598
- 0.580
- 0.562
- 0.546
-0.531
-0.516
- 0.503
- 0.490
- 0.477
0.465
0.2851
0.270
0.178
0.126
0.0928
0.0698
0.0536
0.0418
0.0330
0.0264
0.0214
0.0175
0.0145
0.0120
0.0101
0.00855
0.00728
0.00623
0.00537
0.00465
0.00405
0.00354
0.0031 1
0.00274
0.00242
0.00215
0.00192
0.00171
0.00154
0.00138
0.00125
0.00113
-
1$2’
- 0.6578
- 0.642
- 0.530
- 0.450
- 0.388
-0.338
- 0.297
0.263
- 0.234
-0.210
-0.189
-0.171
-0.155
-0.142
-0.130
-0.120
-0.110
-0.102
0.0949
0.0883
- 0.0824
0.0771
0.0722
- 0.0678
- 0.0638
- 0.0602
- 0.0568
-0.0537
- 0.0509
0.0482
- 0.0458
- 0.0435
-
-
-
6.2 Memory coefficient
Now we come to the memory coefficient ac(q). Let us start with the easier limit of a
wide potential well with xI% 1.
The behavior of the diffusion coefficient for large x , follows from the solutions
Q ( x ) and the inhomogeneities pI2’(x). The asymptotics of the inhomogeneities pj2’(x)
for large distances x with [5]
216
Ann. Physik 2 (1993)
causes the solutions Q ( x ) of the differential equation Eq. (5.3) to behave like
Qlo(x)= x ' + O ( X ' - ' ) , Q,t(X)=X-'-1+O(x-'-2), Q,(x)= X - ' - ~ + ~ ( X - ' - ~ ) .
(6.8)
This is sufficient to calculate the leading terms in xt for the coefficients Xl0 and Y12:
The last equation when specialized to I = 0 fixes the third order coefficient ag). The
hydrodynamic interaction is long-ranged for large distances x of the two spheres and
tends to vanish for x - + a like [18]
3
G ( x )= l - - + O
4x
($),
H ( x ) = l - - 3+ 0
(6.10)
8x
This, together with Eq. (6.8), allows one to calculate the asymptotics of the J,(x,F).
The radial integrals Z,(xl,F), which appear in a f ) of Eq. (5.22), are well defined.
Eq. (6.7) and Eq. (6.8) confirm this for large arguments. The behavior of the integrand
close to contact will be examined later. We give the numerical values for xt-+ 00 up to an
accuracy of three non-zero valid digits:
Z O ( ~ O , Q O O )-1.45
=
, Z o ( 0 0 9 Q o t ) = -2.48 , l o ( c ~ , Q o ) = -0.124 ,
12(00,QZo)= -1.08
,
Iz(00,Q21>
= -2.06
, 12(c~,QJ= -0.0448
.
(6.1 1)
The two integrals involving Ql are already known from the hard sphere problem: they
= -0.169 (compare Eq. (5.26)). For the other pair of integrals belongadd up to a(2)(hs)
ing to I = 0 one can find the exact relations
by suitably integrating over the differential equation Eq. (5.3) with I = 0.
With the help of the limiting values for large x1
we obtain the memory kernel in the leading order of the width parameter xt
(6.14)
J. Vogel, Collective diffusion of Brownian particles
217
Thus, independent of the details of the square well interaction, the lowest order memory
kernel decreases in the limit of large x1 when the square well is added.
More interesting is the limit of very small well potentials of width = xl- 1. For the
inhomogeneities pj2’(x) we found in this limit [5]
pi2) = -0.385+- 0*0813+0(<1)
In 261
0.613
, pi2) = -o.989--+o((r1)
In 261
.
(6.15)
From the small distance limit of the hydrodynamic functions [15]
G ( x ) = 4 ( ~ - l ) + .. .
, H(x) = 0.401 -
0.532
+...
In (2 (x- 1))
(6.16)
and the structure of the differential equation Eq. (5.3) it follows that the solutions Q(x)
close to x = 1 are either singular like In (x- 1) or approach a constant value with a finite
derivative:
Qoo(x)= 1 ,
Qol(x)= -0.251n(x-l)+...
, Qo(l)=0.103 ,
(6.17)
QZ0(1)=1.07 , Qzl(x)= -1.161n(x-l)+...
, Q2(1)=0.0918 .
Qol can be determined analytically from the Wronski determinant, for the other cases
we extrapolated the numerical values found in the series expansions of the solutions.
The finite values for the derivatives of the regular solutions can be gained directly from
the differential equation Eq. (5.3) by integrating it once:
One verifies that the coefficients X I , and Y,, vanish for <,-O or &+l:
(6.19)
We examine the integrals II(x,F) for arguments x1= 1 + tl.The small t,-expansion of
the integrands follows from Eqs. (6.15) and (6.17):
(6.20)
This proves the existence of the integrals I,(x,F) relevant for collective diffusion. Most
of the integrands behave like 1 +O(l/ln ti), which is the same as for the integrals rele-
218
Ann. Physik 2 (1993)
vant for the short-time diffusion. Only integrands containing QI1diverge like In tl+
0 (1) for small ti.We adapted the extrapolation scheme of [I31 to singularities of this
type.
Together with the small argument limit of the Jl, defined in Eq. (5.24), we arrive at
the final result for the diffusion memory kernel in the limit of a small attractive well
of width Cl a surrounding a hard core of radius 2a :
(6.21)
with the functions
&(tJ= 2.805, ,
SO(<,)= 1.90t:lnt1
,
To(CJ = 0 ,
(6.22)
R2(Cl)= 0.706<, , S 2 ( t 1 )= 0.569<:1n
, T2(<,)= -0.602tlIn(, .
The value of Yoz can be read off from Eq. (6.19). For E not too large, the S, terms can
be neglected. The first corrections to the hard sphere values are then proportional to
C, In t, and linear in tl.
It is peculiar for the collective diffusion that the sticky sphere limit for the memory
kernel is not finite. Clearly, when we let tl approach zero but keep 12t1&= l / r constant, the monopole part shows, unlike the quadrupole term, a logarithmic divergency.
It stems from the hydrodynamic stick boundary conditions we used. The divergency can
for tl+O. In case of e.g.
be traced back to the divergence of the coefficient Xol(tl)
perfect slip the hydrodynamic function G ( x ) near contact would vanish like [19]
G ( x )a
1
(6.23)
In (x- 1)
and consequently Qol(x) would not show any divergency at all, but behave like
(x- 1) In (x- 1) instead, rendering the sticky sphere limit finite.
7 Memory coefficient: numerical results
We present numerical result for the memory kernel ac(q) in lowest orders of the
wavevector. The first few terms were:
ac(q) = a(C2)q2a2+a~)q3a3+higher
orders
.
(7.1)
In Section 4 we found up to this order:
a g ) = 24[YO1+ ( ~ - l ) E ~ ( x , ,) ] ~Yo, = 0.0606 ,
= -0.124 resp.
= -0.0448. The functional
with the hard sphere values
dependence on the depth parameter E = exp ( - B u , ) is shown explicitly.
219
J. Vogel, Collective diffusion of Brownian particles
The four functions RI, Sl, with 1 = 0 resp. 1 = 2, and T2,defined in Eq. (5.22) depend
only on the width x1 of the square well potential. This is also true for Eo defined in
Eq. (5.19). We plot the five xl-dependent functions Ro to T2 in Fig. 2. In Section 5 we
examined their limiting values for very narrow and large square wells. Scaled with their
corresponding behavior for large xl, Eq. (6.13) and Eq, (6.9), they approach unity for
large arguments. In the limit of x1 close to unity they vanish according to Eq. (6.19)
and Eq. (6.22). All functions are monotonic.
1.2
I
I
I
I
I
no hyd.int.
1
0.8
0.6
0.4
0.2
0
1
1.5
2
2.5
3
3.5
4
;el
21
Fig.2 a) to f) The plots in the six figures compare the functions R,(x,)/RT (x,) and S,(x,)/S,"(xl),
I = 0 resp. 2, E,,(x,)/E; (x,) and T2(xI)/T?with their respective values R,(x,)/R,"(x,), . . . without
hydrodynamic interaction.
no hyd.int.
2
1.5
1
2.5
3.5
3
4
Fig. 2 c
21
I
1
6
1
1
no hyd. int.
I
2
1.5
1
2.5
1
1
1
3
3.5
4
Fig. 2d
11
To see the influence of hydrodynamic interaction we cite the solution of the same diffusion problem, namely two spheres interacting via a square well potential, but without
hydrodynamic interaction (201. Again the solution can be written in the form of
Eq. (7.1) and Eq. (7.2), but with different functions
8
R,*(x,)= -(x;-l)
3
32
RZ(x,) = - ( X i - - l )
45
,
S,*(x,) = -
8 (x: - 1)2
3x,
32 (x; - 1)'
, SZ(x,) = --45 x: +2/3
,
1-x:
Y & = Y & + ( & - - l ) -3
, 7-;(x1)=2-
x1-1
3x; + 2
(7.3)
.
22 1
J. Vogel, Collective diffusion of Brownian particles
no hyd.int.
Fig. 2 e
I
I
no hyd.int.
1
Fig. 2 f
1.5
2
2.5
3
3.5
4
x1
The asterisk reminds that hydrodynamic interaction has been neglected in this model.
The value of Y:, is 1/3. Its relation to Y& defines the function E,*, in analogy to
Eq. (5.18). The hard sphere values without hydrodynamic interaction were [ 5 ] :
= -8/3 and a$2)1hS)
= -32/45.
Fig. 2 includes the starred functions Ro,S o . . . without hydrodynamic interaction.
They are scaled with Rr, S r . . ., their behavior for xl+oo. It coincides with the
results of Eq. (6.13). Thus the plots show that the influence of hydrodynamic interaction
is reduced for increasing x,. Around x1= 4 there is still a deviation of up to 30%.
Asymptotic behavior is only reached for larger values of x,. Do the functions with
hydrodynamic interaction always lie below their counter parts without hydrodynamic
222
Ann. Physik 2 (1993)
interaction? Clearly, the intersection of T,*(xJ and T2(x1)disproves this. To decide for
the other five functions the plots are not enough, but we also have to look at the limit
of a small square well parameter x l , which can be read off directly from Eq. (7.3):
These limits must be compared with Eq. (6.22), which takes account of hydrodynamic
interaction. We conclude that both RI and Eo lie below their corresponding values
without hydrodynamic interaction. This is not true for the (absolute values of) SIPsince
the ratios S l / S r logarithmically diverge for small tl.Physically this divergence is not
relevant since experimental values of tl lie beyond
Therefore for the square well
potentials of interest 1 S,/S; I is smaller than unity.
Values of Ro to T2 as a function of x1 are listed in Table 2 and Table 3. The asymptotical values for gl -+ 1 are reached only for very small arguments as can be seen from
Table 3, which gives the values of the six functions Ro to T2 with respect to their small
c,-asymptotics. We note that -Sl(xl) is always smaller than Rl(xl).Since T2 is positive
this means that for repulsive square well interaction ( E c 1) the value of a is always
smaller than the hard sphere value. This statement is independent of the presence of
hydrodynamic interaction.
g’
Table 2 Values of the six coefficients R0(<,),
So(<,), E0(<,),
R2((,),S2(<,) and T2(<,)
for small
argument = xl- 1 (according to Eq. (5.22)). The coefficients are scaled by their limiting values for
<,
<l
-0.
5
10
20
30
40
50
60
70
100
0.97
0.97
0.97
0.97
0.97
0.97
0.97
0.98
0.98
1.5
1.6
1.6
1.7
1.7
1.7
1.7
1.8
1.8
0.96
0.96
0.95
0.95
0.95
0.95
0.94
0.94
0.94
1.2
1.3
1.3
1.3
1.3
1.4
1.4
1.4
1.4
1.5
1.6
1.7
1.7
1.8
1.8
1.8
1.9
2.0
1.2
1.3
1.3
1.3
1.4
1.4
1.4
1.4
1.5
Now we discuss the results for the memory coefficients ag) and a‘).). For this purpose we choose square well parameters found in typical experiments. First we focus on
biological suspensions [3]. As a rough estimate for the parameters relevant for e.g. a certain type of bovine eye lens proteins suspended in water we found x1= 1.5 and E = 2
to 3 1201. This corresponds to a van-der-Wads Iike interaction between the roughly
spherical macromolecules and a value of - ul/kBT around unity. The proteins have a
J. Vogel, Collective diffusion of Brownian particles
223
Table 3 Values of the coefficients Ro(xl), S,(x,) and Eo(xl),R,(x,), S,(x,) and T,(x,) as a function
of the well depth x1(according to Eq.(5.22)). The coefficients are scaled by their limiting values of
xi-+Co-
~
1
1.05
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
0
0.0492
0.0936
0.170
0.233
0.286
0.332
0.372
0.406
0.437
0.465
0.490
0.512
0.533
0.552
0.569
0.585
0.600
0.614
0.626
0.638
0.650
0.660
0.670
0.680
0.688
0.697
0.705
0.712
0.719
0.726
0.733
0
0.01 1 1
0.0347
0.0951
0.158
0.216
0.269
0.316
0.357
0.394
0.427
0.456
0.483
0.507
0.528
0.548
0.566
0.583
0.598
0.613
0.626
0.638
0.650
0.661
0.671
0.680
0.689
0.698
0.706
0.7 13
0.721
0.728
0
0.0753
0.140
0.243
0.324
0.388
0.440
0.484
0.521
0.552
0.580
0.604
0.625
0.644
0.661
0.677
0.691
0.703
0.715
0.726
0.736
0.745
0.754
0.762
0.769
0.776
0.783
0.789
0.795
0.800
0.806
0.811
0
0.0610
0.114
0.201
0.269
0.325
0.372
0.412
0.447
0.477
0.504
0.528
0.550
0.570
0.588
0.604
0.620
0.634
0.647
0.659
0.670
0.681
0.691
0.700
0.709
0.717
0.725
0.732
0.739
0.746
0.752
0.758
0
0.0122
0.0401
0.115
0.192
0.262
0.322
0.373
0.416
0.453
0.485
0.513
0.538
0.560
0.580
0.598
0.614
0.629
0.643
0.656
0.667
0.678
0.689
0.698
0.707
0.716
0.723
0.731
0.738
0.745
0.751
0.757
~~
0
0.193
0.321
0.499
0.615
0.693
0.749
0.789
0.818
0.841
0.859
0.873
0.885
0.894
0.902
0.909
0.915
0.920
0.925
0.929
0.932
0.935
0.938
0.941
0.943
0.945
0.947
0.949
0.951
0.952
0.954
0.955
radius of about 2 to 3 nm. Fig. 3a shows the second order memory kernel ag) for
various values of the depth parameter E. The hard sphere value and the behavior for
large xl proportional to xi were divided out. The three curves start at unity, reach a
minimum below the hard sphere value and saturate at a plateau proportional to c2.
With rising E the minimum is closer to unity. In Fig. 3 b, the third order memory coefficient has roughly the same behavior. Again we divided as) by its large xl-asymptotics
proportional to x f and its hard sphere value. It is interesting that the third order coefficient becomes zero for certain values of x1 and E. To see the roots in the logarithmic
scale of Fig. 3 we added 1 to the ordinate. For growing E the roots shift to smaller xi.
The numerical values can be found from the Eo(xl) column in Table 2 and Table 3 via
Yo,+ (E- l ) E o ( ~ =
~ )0 , Yo, = 0.0606 .
(7.5)
224
Ann. Physik 2 (1993)
1+
XI
Fig.3 Plot of the memory coefficients a") and cfg) as a function of x,. Fig. 3a shows ag)(x,)/
(x:ag)(h))as a function of x1 for various values of E. In Fig. 3 b we plot 1 + aF)(x,)/(xfag)(hS))
as a
function of x1 for various values of E.
Since Yo,is very small one can often use the small behavior of Eo given in Eq. (6.19)
and Table 2. Another system which might be examined consists of coated silica particles
111- Typical values of the thickness of the attractive surface layer lead to a range of
x1= 1.0007 up to xi= 1.005. The corresponding depth parameters were found to be
several k,T which reaches up to E = 200 and beyond. The results for ag) and a g ) in
this range of xiare plotted in Fig. 4. They do not differ qualitatively from Fig. 3. The
J. Vogel, Collective diffusion of Brownian particles
225
100
10
1
1
1.-
1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005
I1
100
1t
I
F
1
I
I
0
,
I
I
I
I
I
I.OOC5 1.001 1,0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005
XI
Fig. 4 Plot of the memory coefficients a $ ) and ag) as a function of xlclose to unity. Fig. 4a shows
a(C2)(xl)/(x:ag)(hS))
as a function of xl for various values of E. In Fig.4b we plot l + a E ) ( x , ) /
(x7ag)(hs))
as a function of xl for various values of E.
effect of hydrodynamic interaction on a g ) is displayed in Fig. 5. We compare the second order coefficient for E = 2.3 and E = 7 with and without hydrodynamic interaction.
Since we divided - a g ) by x: the four curves reach a plateau proportional to & 2 as can
be seen from Eq. (7.1). In the region of smaller x, the effect of hydrodynamic interaction is almost two orders of magnitude, but reduces to about 40% for xl around 1.5.
In almost all cases hydrodynamic interaction reduces the absolute value of ag’. Yet,
226
Ann. Physik 2 (1993)
100
I
I
I
I
I
I
I
I
I
10
0.1
1
0.01
1
1.05
I
1.1
I
1.15
I
1.2
a
1.25
I
I
1.3
1.35
I
I
1.4
1.45
1.5
21
Fig. 5 Plot of the memory kernel a“ as a function of x,. We plot
with and without hydrodynamic interaction.
- a g ) ( x , ) / x : for E = 2.3 and E = 7
beyond E = 33 there are narrow regions of the parameter x1 close to unity (xl5 l.OOl),
where hydrodynamic interaction weakly enlarges ag). In this very range close to contact the hydrodynamic functions vary so strongly that the applicability of the model is
restricted.
Fig.6 makes a similar comparison for the third order memory kernel. We plot
1+ a g ) / x f on a logarithmic scale for E = 1.8, Fig. 6a, and E = 2.3 in Fig. 6b. Hydrodynamic interaction shifts the roots of ag’ to smaller values of x,. Clearly, there are
regions for the parameter x1where hydrodynamic interaction enhances the third order
coefficient as’.
8 Conclusion
We examined the short- and the long-time collective diffusion coefficient of a semidilute suspension of spherical particles interacting via square well interaction.
Hydrodynamic interaction was included. The result for the short-time diffusion coefficient, valid up to order (qa)3,and in first order of the volume fraction @, can be written in the form
with
A c ( q ) = Ag)+A$)q2a2+higherorders
.
(8.2)
J. Vogel, Collective diffusion of Brownian particles
10
I
I
227
I
I
I
I
I
I
I
no hyd.int.
.
I+
with hyd.int.
1
I
I
I
I
I
1.2
1.4
1.6
1.8
2
I
2.2
I
2.4
I
2.6
I
2.8
3
no hyd. int.
t
I
I
1
1.2
I
1.4
I
1.6
I
I
I
I
1.8
2
2.2
2.4
I
2.6
I
2.8
3
XI
Fig. 6 Plot of the memory kernel ag) as a function of the width x l . Fig.6a shows 1 + a $ ) ( x , ) / x f for
E = 1.8, while Fig. 6b sketches 1 + ag)(x,)/xt for E = 2.3. Both figures compare results with and
without hydrodynamic interaction.
contains a memory kernel ac(q) which may be expanded like
a&)
= a(c2)q2a2+a~)q3a3+higher
orders
.
(8.4)
To get an impression of actual values for the diffusion coefficient we chose square well
parameters found in typical experiments. In n b l e 4 we compare the short- and long-
228
Ann. Physik 2 (1993)
time coefficients for the case with and without hydrodynamic interaction. Numerical
values for A&) including hydrodynamic interaction were obtained from Eq. (4.10)
and Table 1, while results for ac(q) were taken from Eq. (7.1) together with Tables 2
and 3. Numerical results without hydrodynamic interaction are from Eq. (4.10) and
Eq. (7.3) together with Eq. (7.1). As can be seen from Table 4 the importance of
hydrodynamic interaction is significant already for hard spheres (xl = 1). Qpical
values of the diffusion constant for potential parameters of biological suspensions [3]
with x1= 1.5 and E = 2.3, or for silica suspensions with added polymers [2], x1= 1.3
and E = 7.5 emphasize this for square well interaction as well. By adding polymers it
is possible to reach rather broad wells with xt = 3 and E = 2.4. Another example
discussed before was a suspension of coated silica particles with a narrow (xl = 1.0014)
but deep attractive square well of E = 202 [l]. The cited tables and equations can serve
to find values of the small wavevector diffusion coefficients for different square well parameters. The results sensitively depend on the magnitude of the square well parameter
x1-
For hard spheres the second order term ag) is small compared to the short-time contribution A($), as can be seen in Table 4. This fact changes when introducing an additional square well interaction - now the sign of the second order contribution can be
Table 4 Values of the short-time diffusion coefficients Ag) and Ag) together with the values of the
memory coefficients ag) and ag)for selected square well parameters x1 and E . Shown are the values
with and without hydrodynamic interaction.
hyd. int.
no hyd. int.
x, = 1
Ip
1.45
A (2)
C
- 1.86
Q
(2)
C
-0.169
Q
(31
C
0.0880
&=l
x, = 1
8
5
&=l
XI
&
= 1.0014
-2.17
&
&
- 0.280
152
--
-8
45
3
-1.1
0.19
- 1.43
0.0638
= 202
x1 = 1.0014
XI
--16
1.24
1.32
= 202
= 1.3
- 35.6
25.7
- 54.2
53.2
- 14.0
13.1
- 16.7
24.2
- 66.3
52.7
= 7.5
= 1.3
X,
&
-111
123
= 7.5
x, = 1.5
-8.93
8.17
E = 2.3
XI =
E
XI
1.5
- 12.8
11.6
= 2.3
=3
- 224
817
- 892
2.09-lo3
& = 2.4
XI
E
=3
= 2.4
- 283
1.08.Id
-1.25.ld
3.34.1d
J. Vogel, Collective diffusion of Brownian particles
229
reversed by the memory kernel. Since the second order terms vanish when one suitably
chooses the square well parameters this might help to decide upon the magnitude of
a possible, small ag) contribution 1101 which will show up when hydrodynamic
threebody interaction becomes relevant.
I am indebted to Prof. V. Dohm for his kind support. 1 thank Prof. B.U. Felderhof for drawing this
problem to my attention and for carefully reading the manuscript.
References
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[2] J. W. Jansen, C. G. de Kruif, A. Vrij, J. Coll. Interf. Sci. 114 (1986) 492
[3] J. A. Thomson, P. Schurtenberger, G .M.Thurston, G.B. Benedek, Proc. Natl. Acad. Sci. USA 84
(1987) 7079; P. Schurtenberger, R. A. Chamberlin, G. M. Thurston, J. A. Thomson, G. B. Benedek,
Phys. Rev. Lett. 63 (1989) 2064; M.L. Broide, C. R. Berland, J. Pande, 0.0.Ogun, G.B. Benedek,
Proc Natl. &ad. Sci. USA 88 (1991) 5660
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