Brownian dynamics in a confined geometry.
Experiments and numerical simulations
Nicolas Garnier, N. Ostrowsky
To cite this version:
Nicolas Garnier, N. Ostrowsky. Brownian dynamics in a confined geometry. Experiments and
numerical simulations. Journal de Physique II, EDP Sciences, 1991, 1 (10), pp.1221-1232.
<10.1051/jp2:1991129>. <jpa-00247585>
HAL Id: jpa-00247585
https://hal.archives-ouvertes.fr/jpa-00247585
Submitted on 1 Jan 1991
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
J.
Phys.
France
II
(J99J)
1
J22J-J232
1991,
OCTOBRE
PAGE
J22J
CJassificafion
Physics
Abstracts
05.40
42.J0
82.70
dynamics
Brownian
numerical
Gamier
N.
in
confined
a
Experiments
geonletry.
and
simulations
and
Laboratoire
Physique de la Matidre
Sophia AnfipoJis,
de
Universitd
Nice
de
(Received
Ostrowsky
N.
April 1991,
18
accepted
17
(*)
Condensde
06034
Cedex,
Nice
France
July 1991)
voisinage immddiat
d'une
au
quasi-dlasfique de la Iumidre en onde
dvanescente.
ddcroissenJent
coeffident
de diffusion, d0 au
ralentissement
On
observe
du
net
un
hydrodynamique des particules trds proches de la paroi. Cet effet est d'autant plus marqud que [es
particules peuvent se rapprocher trds prds de la paroi, c'est-I-dire
que la portde de la rdpulsion
statique
paroilparticule
faible.
possible de tester [es
interactions
statiques
II est
donc
est
Rkswd.-
paroi
dynamique
La
rigide
paroilparticules
analysdes
tion
par
via
une
diffusion
de
de la
paroi,
d'une
lumidre
colloidales
diffusion
of
a
des
diffuseurs
dans
des
par
pidgdes
ou
dynamics
Brownian
The
Altstract.-
particules
de
expdrience de diffusion dynamique de la Iumidre. Los
de dynamique
Brownienne,
particulidrement adaptde
une
voisinage
au
de
technique
la
par
simulation
rdsultats
des
confindes
brownienne
mesurde
est
grinds
«
milieux
»,
poreux
tels
ou
suspension is measured
Quasielasfic Light Scattering
colloidal
que
des
in
donndes
sont
I'interprdtades particules
gels.
I
immediate
the
Technique. A net
hydrodynamic sloving
measured
diffusion
coefficient
due to the
decrease
of the
is observed,
important when the particles
down of the particles very close to the wall. This effect is all the
more
of the static waII/particle repulsive
allowed
the wall, I-e- when
the
closer
to
to get
range
are
for testing the particle/waII static
interactions
interaction
decreases.
It thus provides a
via a
mean
dynamic simulation
The data are analysed by a
Brownian
dynamic fight scattering
measurement.
hindered »
which is proven to be quite valuable to interpret light scattering data from
scatterers,
media or a gel.
such as particles
confined in the neighbourhood of a wall or trapped in a porous
vicinity
of
rigid
a
surface
by
Evanescent
the
Inwoducdon.
1.
The
interactions
suspension
sedimentation
These
The
interactions
(*)
CNRS
between
the
at
are
and
of
adhesion
interactions
static
particles
basis
may
interactions
between
URA190.
the
on
be
suspended in a fiquid and
of important
practical
the
a
charges
confining
wall
the
particle
such
as
the
electrostatic
substrate.
classified
mainly
solid
phenomena
number
a
in
two
include
the
Van
der
carried
by
the
particles
types
:
Waals
attraction
and
the
and
solid
wall.
The
stefic
interactions
example,
problem.
of
DE
JOURNAL
1222
the
bt
II
10
of entropic
origin, present
when
interacting
surfaces
flexible
the
(for
are
undulating vesicles or particles coated with a polymer layer) may be neglected in our
profile in the vicinity
This type of
interactions
non-uniform
concentration
lead to a
wall.
hydrodynamic
particle is perturbed by
original particle.
The
A
PHYSIQUE
number
of
[I]
to
of the
presence
theoretical
of
values
numerical
due
interactions
the
wall
fact
and
numerical
and
interactions
these
the
and
[2]
more
that
the
thus
liquid flow
reacts
studies
back
the
explicited
have
recently
by
created
onto
a
moving
motion
the
form
of the
and
dynamics
of particles in the
molecular
some
their role in the Browniart
motion
[3, 4] have helped understand
vicinity of a rigid wall.
As far as experiments go, the problem is far less
advanced.
Macroscopic experiments have
monitored
the fall of suspended
solid
surface, thus measuring the
friction
balls
onto
a
coefficient
A(z) as a function of the distance z between the particle and the wall [5]. On a
microscopic scale ~particles around lo ~m in diameter) static experiments have studied
more
distribution
the height
of suspended particles above a given transparent plate, from which the
interaction
static
potential
between
and the wall
be
deduced
the particles
[6]. Static
can
Techniques have also been used to
Fluorescence
the
concentration
profile of
measure
fluorescent
particles doped with
probes in the vicinity of a transparent wall [7, 8].
of this
The
is to give
experimental results on the
Browrian
purpose
paper
some
new
dynamics of particles close to a surface, measured by art original method developed in our
quasi elastic light scattering technique using as the
laboratory:
The
incident
light an
which
particles
within
thus
only
probes
than
the penetration
distance less
evanescent
wave
a
depth of the wave (see Sect. 2). To analyse our data, we have performed a
«computer
simulated
light scattering experiment » (see Sect. 3) to generate the correlation
function
of
the
electric
field
scattered
by a
particle
submitted
both
Brownian
the
static
and
to
interactions
hydrodynamic
discussed
above.
simulations
immediate
Q.L.S.
Evanescent
2.
experbnent.
EXPEItIMENTAL
2.I
CONDITIONS.
studied an
latex suspension ~particle diameter
0.09 ~m) whose
aqueous
(cm3 x10~~g/cm~, I-e- mean
particle
distance
between
I ~m) is large
enough to provide a
confortable
signal in the
but
regime,
low enough to
evanescent
wave
allow
neglect
the
different
particle/particle
interactions.
We
salt
have
used
to
us
concentrations
able
be
partially
the
electrostatic
repulsion
between
the
suspended
to
to
so
as
screen
particles and the glass wall, both negatively charged.
Material
We
have
=
concentration
=
Optical set-up (see Fig. I) : The liquid sample is contained in a half cylindrical cell, sealed by
optically flat surface ~polished to A /20) of a larger semi-cylindrical glass prism. This surface
ultrasonically cleaned with distilled
holder is
after each experiment. The sample
water
was
placed on a precision tumtable, so as to easily change the incident angle @~ of the vertically
polarized Argon laser (300 mW at
reflection is
514.5 nm). The critical angle for total
incident
given by the usual relation : sin @~ n~~~~Jn~~. For @;
the
in the
vector
wave
@~,
medium
has a real
2 wn~j~/A parallel to the flat surface of the prism, and an
component k~
imaginary component equal to the inverse of the penetration depth f given by :
an
=
~
=
=
f
This
means
that
we
are
=
(A /2 wn~j~~) [sin~
conducting
a
@I
light scattering
sin~ @~]~
experiment
~~
with
(l)
an
incident
beam
always
bt
BROWNIAN
10
DYNAMICS
IN
A
CONFINED
GEOMETRY
J223
Colloidal
Suspension
Argon
ilter
.
patial
beam
Scope
Micro
PM
EXPERIMENTALLY
2.2
[,
but
at
the
»
and
surface
«
for
recorded
tions
function
linear
of the
homogenous
as
is
illumination
taken
be
the
In
at
(t)
In
whose
heterodyne
a
correlation
of the
scattering
above
experiment,
in its
and
so-called
the
are
recalled.
now
correlation
scattered
func-
angle
critical
=
intensity
the
of the
volume
correlation
expressions
theoretical
function
order
Typical
(f
ml the
in figure 2, illustrating
shown
are
functions
usual
of the
case
r
from
its
particle
and
function
is
a
field, which, with
from, can be written
electric
normalized
bulk
(r,
t
j
time
at
(4
(k~)
~'~
r~i4
exp
=
for
a
Dt
(2)
k~
;
wave
zero.
known
well
k
=
time
the
is the
zero
j~
qrDt
q
scattered
the
at
the
geometry,
starting point
vector
wave
origin at
was
Brownian
particles.
probability density
which
independent
of
P
(k;)
incident
the
scattering
the
where
of the
number
great
a
distance
a
between
position
vector
on
(e"~'~ (~l)
Re
=
difference
the
(t) is the
t
:
first
just
and
J0
:
g
r
correlation
@,
bt
II
FUNCTIONS.
~m)
0.8
=
function
correlation
Bulk
@I
bulk
«
CORRELATION
MEASURED
just below (f
scattering angle
same
PHYSIQUE
DE
JOURNAL
J224
vectors,
and
average
is to
be
time
The
particle
Gaussian
to
at
probability
j
:
(3j
is the
diffusion
coefficient,
related
D=kT/6w~R
bulk
the
particle
radius
to
viscosity
this
density
probability
R and the suspension's
to
compute the average in
~. Using
equation (2) leads to the usual expression for the bulk
correlation
function :
where
g~(t)
exp
=
Surface
t) d~r
e'~'~~~~ P(r,
=
(- Dq
~
(4j
t)
(5)
of a wall, it is useful to decompose the 3D
motions,
motion
parallel and one perpendicular to the
two
one
interactions, the first
wall, along the direction Oz.
to begin vlith, the particles/wall
motion
motion
obeys the usual 2D
Brownian
statistics, and the second
includes
the
one
effect
of
wall.
Accordingly,
the
probability
density
mirror
the
be used in
to
«
»
proper
computing the average in equation (2) is no longer given by equation (3) but now reads :
function
correlation
Brownian
In
:
the
presence
independent
Neglecting,
into
~
( ~ll,
p
where
0
and
In
Z,
Z o,
"
~
"
)-
J~
3/2
(0, zo) and (q, z are the parallel and
t, respectively (see Fig. 5).
the
evanescent
geometry,
wave
one
e
(=
4 Dt
e
perpendicular
must
further
(z
zo12
4 Di
+
coordinates
take
into
zo12
(~)
4 Dt
+ e
of the
account
the
particle
at
fact
that
time
the
unifornfly illuIninated, which requires that the average in equation (2)
field amplitude at the particle's position at time 0
be properly weighted by the
electric
[Eo e ~°'~] and time t [Eo e ~'f
scattering
The
volume
is not
theoretical
forrn
following integral
of
«surface»
the
correlation
function
g~(t)
is
thus
given by the
dz
(7)
:
g~(t)
~°
=
f
~~
e
f
iii
P
(rj,
z,
zo, t
)
e
e'"
~
e'~"~~
~°~
d~q
bt
BROWNIAN
J0
which
been
has
[9]
found
DYNAMICS
equal
be
to
gs(t)
=
to
IN
GEOMETRY
CONFINED
A
:
(- Dq II t) gz(qz, f, D,
eXP
J225
t
(8)
)
of the scattering
parallel and perpendicular to the
components
vector
) is an analytical
function
whose
limited
expansion can easily be
implementd on a micro-computertheoretical
correlation
functions (5) and (8) have been used to draw the solid lines in
The
figure 2, leaving as an adjustable parameter the diffusion
coefficient D. Now, the value found
for the best
adjustment of the surface
correlation
function
always lower than the value
was
found for the bulk
difference
the
being all the greater when the particles were
measurement,
allowed
closer to the wall, I.e. when
the particles/wall
repulsion was
reduced by
to
come
concentration
increasing the salt
of the suspension. This is interpreted as the
combined
effect
of the repulsive
hydrodynamic
and
electrostatic
interactions
between
the particles and the
where
qj and q~ are the
and g~(q~, f, D, t
wall,
wall
as
2.3
we
ROLE
kinds
The
PARTICLES/1VALL
OF
of
interactions
in
results
negatively charged latex particles of
glass
wall, both negatively charged, the
a
DLVO
theory [10] :
~
~~'~
~
~'~
~~'' ~'~~
~
s
following
the
closest
is
c~j~.
a
this
from
the
~~
~~~
~
~
A
Ii
H#1 #21in
WI
particle's
the
are
the
the
"
surface
the
and
[6~~/(2 ~~~~iA
number, e : the
concentration).
salt
surface
potentials
dimensionless
quantity
dielectric
of the
defined
~
H=
To
~
~
derived
be
may
1~°e-
~l1
(91
wall
(I.e.
:
s
z
=
R)
latex
(IO)
Cw~t))~~~
constant
particle
of the
and
the
mediuTn,
glass
wall
the
e
electronic
(typically
the
on
mlo.
of 50
His
Avogadro
the
~j and ~~
order
~~
interaction
:
between
K
(with N~
charge and
two
screening Debye length
is the
x~
notation
distance
electrostatic
(#i+
+
with
introduction,
the
of
studied
i~~~
in
by a position dependent potential
interaction
particles in the vicinity of the wall. As We
radius R suspended in salty Water, in the vicinity
distribution
of
~~ ~~~
mentioned
represented
interactions
non-uniform
a
As
INTERACTIONS.
considered.
be
must
particle/wall
static
U(z),
discuss.
now
repulsive potential
theory [11] :
must
as
:
~~~+
R+s
be
added
~~~
~~~
(ll)
R
the
attractive
Van
der
Waals
potential
derived
from
Hamaker's
with
Am
The
trations,
I kT.
resulting
and
the
interaction
potential
corresponding
U(s) is
BoltzTnan
shown
concentration
in
figure 3a for different
profiles c(s) are plotted
salt
in
concen-
figure
3b.
JOURNAL
1226
~t°t
PHYSIQUE
DE
J0
~~~~
@
total
,oooo~
~OO~O
O
poten~ial
interac~ion
Various
for
(molfl)
concentration
salt
OO
bt
II
°"°oO
~°.
~O
°~
~°COO
'
"OO.
~j
"o«.~
~~
°..o
'
°.OO
°°O.O~
O
O
l~
"
~
_~
""°OOOO.~~
~~~°"OO.Oo
°
°°°Oooooo
*OOOOOOOOO
°
~
°°"'
a
°
...
a)
van
der
potential
Waals
i
c(s)/c~
~
~
S~Un)
o.2
oi
concentration profiles
(mot/l)
concentrations
panicles'
for
various
salt
-2
IO
ao
*
°o~
°a~~~~
a
I
,,......~~~~ll?ff??((11999"i'ioooooooooooooooooooooooooooooooooa.oooooooo«....»ooooooo««
.'
~
~a?..'
~oo......'
a.°°"""
of"
o.
~o°
o.
~o°°
"~
1'
-3
~Q
~~~~oooooo
o°"
o.
°.
~,,o««oO«
d
°
-4
"
o"
o~
]
o
oO
O"
°
o'
O~
o
/
~°
O
o
O
~
O~
."
O
~'
o'
.
a"
IO
b)
0.1
Fig. 3.- Part a)
negatively charged,
shown
The
-5
the
vith
profiles
Total
for
dotted
obtained
Interaction
various
line.
when
salt
potential
(squares)
concentrations.
The
between
attractive
b) Corresponding particles'
ignoring the Van der Waals
Part
~ll~~)
0.2
a
Van
particle
der
BoluJnan
attraction
and
Waals
the
concentration
are
shown
glass wall, both
potential is
profiles (squares).
part of the
with
the
dotted
fines.
potential will only be
attractive
shows that the effect of the Van der Waals
concentration
used (10~~ mol/I) and the
shortest
penetration
highest salt
depth (f
0.2 ~m ), enhancing by lo fb the hydrodynaniic slowing down as will be explained
however
hnportant in all our experiments, as
attractive potential was
in the next section. This
detectable
aggregation, all the more rapid when the salt
it was responsible for a very slow but
Beyond 10~ ~ mol/I, the aggregation was found to be too rapid to
increased.
concentration
was
the flat
surface
of the prism during the
conditions
data
good
insure
evanescent
on
wave
acquisition time (of the order of lo mini.
This
figure clearly
for
detectable
=
the
bt
BROWNIAN
10
effect
The
the
of the
water
the
velocity
V
1)'
~'
(/'
=
z
Using
1227
GEOMETRY
interactions
included
are
vector
CONFINED
IN A
between the particles and the wall,
mediated by
through a position-dependent
friction
A which,
tensor
of the particle, yields the friction
force F experimented by
hydrodynamic
molecules,
multiplied by
the particle :
DYNAMICS
).
(13)
Vz
:
c
Smoluchowsky relation, leads to a position dependent
(s) and D~(s) have been calculated in the literature.
Dj
component
Using Faxen result [12, 4], we computed Dj (s) with the following linfited
diffusion
Einstein
the
tensor
whose
D~(s)
derived
was
series
from
expression [13, 4], keeping the
Brenner's
first
expansion
of the
terrns
seven
:
:
l~b~k
Dz(S)
4
3
vlith
To
a
"
~
~~~
cosh~
n
~
'
for
g~(t)
such
4
function
finfit,
not
relaxation
+
a
(2
(2
a
times,
short
one
can
so
that
enough
small
of the
trivial
a
sinh 2
+ I
n
n
)~ sinh~
+
a
~~ ~~
a
observed
ii
m
Di
that
assume
a
diffusion
its
gilt
Brownian
particles contained in the scattering
c(z) of particles near the wall
concentration
~~~
~~
the
wall
2
j)
ii
in the
short
times
computation of
compared to
the
the
receive
z/f),
leads
Dj (z)
qi
:
Dz(q]+ il12)11
coefficients
function
correlation
exp(-
for
given scattering
The
to
coefficient
except
time.
constant.
particles closer
exponential law
diffusion
matter,
expansion of equation (8) yields
limited
a
dependence
is
gli>(t)
volume
+
n
)
~ ~
position
this
function
correlation
For
(2 n + I
sinh~ (n + 1/2
2 sinh
3
R
account
this
(n + I
) (2
n
(2
=
correlation
In
~~
"
is
(161
Brownian
particle
Djj (z) and D~(z)
of
average
an
can
is
be
equation (16)
confined
to a
considered
as
all
the
over
Taking into account the facts that the
dependent (see Fig. 3b) and that
intensity according to the
higher
and
thus
scatter
a
approximation :
the
to
volume.
is
position
D=(zj (qj
t
+
lif~) ii c(z) exp(-
(-
2
2 zif
dz
~~~~
~~~
)~
c(z)
exp
zif
dz
o
m
Ii
b(f) (q2+ 1/f2)j
(18)
defining the weighted average fi(f) which has been numerically computed for different
10~~ mol/I (see dotted lines in Fig. 4).
penetration depth for [Nacl]
however, that this approach 15 only valid for short times, and the
It must be remembered,
with the very beginning of the
above
result
should
be compared
experimentally
measured
thus
=
1228
JOURNAL
D(I) /
PHYSIQUE
DE
bt
II
10
D~uu~
i
_~_
4
____----------------
n
----
='
&=_==.=.
=-=-=
.=
-
-.=.=.-
(
/R
0
5
20
15
10
coefficient D (f ),
Fig. 4.
normalized
diffusion D~~~, as a function of the
Average diffusion
to the bulk
penetration length f for two salt
concentration
(Q : 10~~mol/1; m
salt
added). The values
no
computed from the « short-time
approximation
shown by the dotted line for [Nacl]
10~ ~ mol/I.
are
10~~ mol/I and [Nacl]
10~~ mol/I are
simulated
for [Nacl]
The
results
indicated
vith
computer
=
=
=
dashed
fines.
correlation
function, I.e. its slope at the origin. Unfortunately, this comparison
precise, as the surface
correlation
function
is far from an
exponential and
origin can only be poorly defined.
for a
We
thus
looked
better
way
experimental data, which led us to the Brownian
dynamics
simulations
we
3.
Brownim
dynandcs
be very
the
cannot
slope at
analyze
its
to
now
our
describe.
sbnviafions.
idea is to simulate a light scattering experiment on a computer
and derive
numerically the
function
correlation
g(t). As we explained in section 2.2, any photon
expected
correlation
experiment
the
normalized
correlation
function
of the electric
fields
by a
scattered
measures
walker at time 0, E(0 ), and by the same
walker an
instant t later, E(t). If the walker at time 0
and t receives the
incident
intensity (constant
illumination
profile), the only difference
same
between
E(0) and E(t) is a phase factor cos (q r (t)) where q is the scattering wave
vector
and r(t) is the
distance
covered by the walker
during the time t, and thus :
The
g(tj
If the
scattering
diffusion
constant
particle
D~, it
is
can
far
be
l~~~)
=
~(°)i
iiE(0j ii
from
any
modelled
=
wall,
by
a
jars (q.r (iii>
thus
random
undergoing
walker
(19)
a
which
Brownian
takes
motion
with
a
time
interval
every
the
simulation
must
(2 D~ r )~'~. The time interval r chosen for
the time decay of the Light Scattering
function, but large
correlation
fluctuations
enough to allow for Brownian
correlation
function
to decay. The light scattering
around
during a time
follows.
walker
interval
is then
obtained
Let a given
move
as
factor
the phase
(q.r(t))
and
r(t)
covering a distance
compute
repeat the
cos
t, thus
experiment a great number of times, the average of the phase factor progressively building the
particle, moving in a
Brownian
function
g(t). In the case of a free
correlation
constant
direction
and in any
compared to
be small
r
a
step
±
bt
BROWNIAN
10
DYNAMICS
IN
CONFINED
A
GEOMETRY
~ll
z
~
w
~~~
~
-~i
(li-$
,."
»'
J'
,
i~
/
z(0)
(
Brownian
of
the
profile,
(I.e.
wall
a
simulations
value
all its
the
walker
submitted
a
L(z)
of
to
a
and
Brownian
walker
a
wall, as we
Langevin
(z)
=
first
coefficient
the
fact
derived
one
is
D.
The
that
by
showing
wall
a
position dependent step
the
and
the
nfirror
the
the
L
The
vicinity of
describe.
now
equation describing the motion of a
Brownian
particle
position-dependent
friction
coefficient
position-dependent
and to a static
MacGamrnon
position-dependent step
[3] derived the expression for the
particle
displacement
accompfishes.
In
problem, the
random
our
along the Oz axis during a time interval r is tile sum of three terms :
from
Errnak
such
wall
the
the
and
Starting
force,
parallel and
perpendicular to the wall
simulations
the well
known
analytical result (see Eq. (5)). If
match
exponential
illuwnation
profile, together with the mirror effect of the
walker
meeting the wall just bounces
back into the suspension see Fig. 5), the
exactly match the solution given in equation (8). The method
however
takes
on
when we
introduce
in addition
the static and hydrodynamic
interactions
between
introduce
now
in the
vector
wave
components
wall.
illumination
we
walker
~z
Scattefing
~)",
''
rigid
5.
~
",, ,-"'
0
effect
."
~
)
.
Fig.
1229
the
not
(2 D~(zi
usual
second
D is
=
random
one
is
constant
differentiating Lo
)"2
r
dD;/dz
Lo
step,
merely
=
±
correction
a
the
whole
respect
to
over
with
+
r
iD=(z)/k11 F=
(2Dr)~'~
dLo
step Lo.
to
This
with
the
a
first
(20)
r
dependent
taking into
z
one,
correction
terra
may
diffusion
account
be
simply
z :
~
dD
~~
~
dL0
+
~
(21)
+
"
2 Dr
which, for dz
Lo, yields the second terra of equation (20). The third
simply the drift of the walker
from
the wall,
due to the
away
F~ computed by taking the gradient of equations (9) and (12).
Now the
random
displacement L'(z) of that same walker parallel
simpler as it includes just the random step, whose magnitude however
=
L'(z)
We
have
thus
r(t) of a walker
procedures as
implemented on
staffing at time
the
walker
moves
a
=
±
MaCII
zero
from
around.
of
term
static
the
to
is
equation (20) is
repulsive force
wall is
position
somewhat
dependent
(2 Dj (z) T)~'~
:
(22)
microcomputer a program computing the position
randomly chosen position r(0), repeating the
a
The
simulated
surface
correlation
function
is
JOURNAL
1230
progressively built up by sumnfing,
weighted phase factor :
PHYSIQUE
DE
for
couple of positions
each
w
ig~(t)j~~~~~~
=
~
~ P~"
°
z
it will
c(z),
accounted
for
in
f
e
[r(0),
r
(t)],
the
10
properly
(
f
e
jq
cos
(r(t)
r(0j)j
(23)
overatipmn
advantage of using a single walker
sample all the distances z from
checked
in figure 6, and the
as
we
The
that
bt
II
following him throughout the
wall according to the
proper
non-uniform
concentration
profile
computation
and
the
distribution
is
law
automatically
is
simulation.
the
profiles obtained by numerical
simulation
using in the walker's step only the
equation 20 (curve I), the first two terms (curve 2) and finally aIJ three terrns (curve 3). The
profile deduced from the BoJtzman
concentration
distribution
using
dashed lines indicate the anaJyticaJ
interaction.
equations (9) plus (10) for the particles/wall
Fig.
Concentration
6.
first
in
terra
for the
particle approaches
only by adding
wall.
It is
profile
is
fitted
well
the importance of all three
in equation (20). Curve I
terms
hydrodynamic repulsion by merely reducing the Brownian step
wall leads to an
of particles in the vicinity of the
accuTnulation
second
(curve
2)
that
the
expected « flat »
concentration
term
in
3)
the
of
the
(n°
static
interaction
potential is
curve
presence
distribution
using equation (9) plus (12) for the particles/wall
illustrates
6
accounting
that
the
as
figure
that
Note
shows
obtained.
The
by
Boltnnan
the
the
the
last
interaction.
Finally
far
too
let
from
away
contribution
z
distorsions
This
on
distance
avoid loosing too much time with the walker sampling a region
intensity is too low to give a significant
(where the illumination
function) we put a
correlation
fictitious
purely reflecting wall at
chosen
good
compromise
between
saving time and avoiding
as a
was
that to
wall
correlation
the
simulated
«
the
the
to
TMs
z~~.
=
mention
us
»
for g~ and the best fit
values off and salt
several
found
4
and
4.
All
salt
fill
to
Results
the
and
last
the
function.
function
correlation
column
yielded
the
concentrations
of
fitted
then
was
simulated
allowed
value
us
to
vlith
the
theoretical
D(f ). Repeating this
plot the dashed lines
expression (8)
procedure for
shown in figure
table1.
discussion.
experimental
concentrations.
results
are
summarized
in
table I for
different
penetration
depths
and
M
10
Table
c;pertinent) periment)
No
=
=
0.
~m
4.35
10-3
f
=
±
0.04
0.85
10-~
f
=
±
0.04
0.81
10-~
f
=
±
0.04
0.79
10-2
f
=
0.08
4.80
±
0.06
~m
3.77
±
0.04
x
0.20
m
3.94
.0
x
=
0.20
m
3.85
.5
x
f
0.20
=
m
3.78
.0
x
f
0.43
=
3.76
measured
the
o~,
~
concentration.
at
extended,
salt, they
so
The
a
that
table.
as
~m
4.05
±
~m
4.09
±
0.79
0.78
0.79
suspensions were prepared in pure
optical set-up was then aligned
coefficient
diffusion
surface
corresponding
the
:
the
The
salt
necessary
Changing
added
solution
was
the
incident
back
and
angle
to
containing the same salt
on
than in
coefficient D~ is larger in the salty solution
extemal
latex
that the dangling chains covering the
less
fully
these
chains
In pure
water,
more
or
are
apart as possible, whereas in the presence of
as far
which
allows the particles to have a larger
structure
suspensions
measured
fact
tips.
groups
compact
more
latex
measured.
the
was
the
0.73
,
was
and
diffusion
their
obtained
data
the
lower
than
from
extracted
note
we
concentration
salt
significantly
±
optics,
OH
expected,
As
the
4.10
[14].
D~(f)/D~~j~
table I and
of
increases
still
the
~m
concentration.
to
at
groups
to
±
follows
f,
the
due
back
4.26
0.73
as
bulk
the
spread
coefficient
ratios
column
of
fold
can
diffusion
to
as
0.04
~m
coefficient
salt
simply
This is
water.
pure
surface
OH~
carry
0.85
coefficient
that
Note
±
0.95
0.I1 0.97
0.85
depth
diffusion
0.85
0.85
different
bulk
±
=
0.74
changing
Without
measured.
@;
floacl]
various
at
0.75
diffusion
penetration
given
I1
0.85
conducted
were
bulk
the
and
water,
for a
0.20
=
experiments
The
D~
±
m
f
4.49
±
0.20
~m
.5
f
123J
for DjD~~j~
results
f
salt
4.35
.43
f
=
simulation
and
GEOMETRY
CONFINED
(experiment)
Depth
f
IN A
Summary of the experimental
and penetration depths f.
I.
concentrations
f
DYNAMICS
BROWNIAN
simulations
are
experimentally
the
that
are
simulations
the
decreased,
is
The
I.
experiments
the
from
but
that
of
pure
the
the
next
to
indicated
in the
last
column
ratio
measured
values
water
summarized
»
obtained
given
at
the
in
last
D~(f)/D~~i~
pure
water
beginning
are
of the
10~~ mol/I.
feel
confident
that the
simulations
in fairly good
As experiments and
agreement,
are
we
interpret light scattering data from
simulations
Brownian
dynamic
will be quite valuable
to
last
column
hindered
that
time
the
made
assuming
such
scatterers,
»
function
correlation
constant
JOURNAL
were
DE
reflecting
PHYSIQUE II
-T
both
I, V
t0,
as
[Nacl]
=
particles trapped
of the
electric
field
in
a
scattered
media or a gel ; it is expected
by such particles Mill decay Mith a
porous
:
OCTOBRE
1991
s3
ii
geometrical
the
constraints
PHYSIQUE
DE
JOURNAL
J232
which
reduce
between
the
the
M
II
particles'
of the
span
diffusive
lo
motion
[15]
down
the
and
iii
physical
the
particles'
summarize,
To
is
interactions
sensitive
a
shown
have
we
the
that
measuring
for
tool
particles
and
walls,
the
which
slow
itself.
motion
Brownian
Evanescent
dynamics
Brownian
Quasielastic Light Scattering Technique
vicinity of a rigid
immediate
in the
surface.
A
down
This
method
the
of the
are
to
interaction
currently
could
very
closer
to
is
observed,
TMs
effect is
coefficient
to
the
wall.
wall, I.e.
the
prove to be very
in the computation
also
stuck
being
get
close
when
the
due
to
more
of
range
hydrodynamic
the
observable
the
when
the
wall/particle
static
decreases.
introducing
remain
diffusion
measured
particles
allowed
surface,
would
of the
decrease
net
slowing
particles
repulsive
done
on
the
wall
to
test
this
sensitive
to
the
residence
a «
again
drifting
possibility.
before
onset
time
in
»
the
of
particles aggregation on
particle
a given
during which
suspension.
Simulations
are
Acknowledgtnents.
The
and
acknowledge
authors
wish
Vanneste
thank
to
and
J. P.
Pierre
Roustan
stimulating
for
Bezot
for
vlith
discussions
his
valuable
help
Ackerson
Bruce
in
the
friendly help in the writing of the
their
and
experimental
Didier
part,
computer
Somette
and
programs.
References
[J]
[2]
[3]
[4]
[5]
[6]
[7J
[8]
[9]
[J0]
[J II
[12]
[J3]
[J4]
[15]
(J976) J.
Inierface Sci. 84 (1981) 497.
MCCAMMON
J. A., J. Chem. Phys. 69 (4) (1978) 1352.
EMAK D. L. and
Chem.
Soc. 83 (J987) J79.
Faraday
Discuss.
G. M.,
WATSON
CLARK A. T., LAL M. and
interface Sci. % (1983).
Colloid.
M. and VAN DE VEN T. G. M., J.
ADAMCzYK
Langmuir 6 (J990) 396-403.
PRIEVE D. C. and FREJ N. A.,
AXELROD D., Biophys. J. 33 (J98J) 435.
BURGHART T. P. and
N. L.,
THomsoN
F., Phys. Rev, Lett. 54 (J985) J948.
RONDELEz
AUSSERE D., HERVET H. and
SORNETTE D., Phys. Rev. Lent. 57 (J986) 17,
N. and
OSTROWSKY
LAN K. H.,
Sci, 33 (1970) 3, 335-359.
L. N., J. Call.
Inter.
MCCARTNEY
BELL G, M., LEVINE S, and
J058.
H. C., Physica 4 (J937)
HAMAKER
Fys. 17 (1923) 27.
Mat.
Astron.
FAXEN H., Arkiv.
H., Chem. Eng. Sci. 16 (J96J) 242.
BRENNER
D. N., J. Chem. Phys. 75 (J98J) J.
PINDER
TROTTER C. M. and
N., to be published.
OSTROWSKY
CELESTINI F., LOBRY L. and
See
for
example
ADAMCzYK
Z.
BATCUELOR
and
VAN
DE
G. K.,
VEN
J.
T. G.
Huid.
M., J.
Mech.
Colloid
74
C.