An experimental study of a model colloid-polymer
mixture
W. Poon, J. Selfe, M. Robertson, S. Ilett, A. Pirie, P. Pusey
To cite this version:
W. Poon, J. Selfe, M. Robertson, S. Ilett, A. Pirie, et al.. An experimental study of a model
colloid-polymer mixture. Journal de Physique II, EDP Sciences, 1993, 3 (7), pp.1075-1086.
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J.
Phys.
France
II
(1993)
3
1075-1086
1993,
JULY
1075
PAGE
Classification
Physics
Abstracts
82.70
64.75
study
experimental
An
W.
C.
Poon,
K.
Department
of
of
Selfe,
J. S.
Physics,
M.
Robertson,
B.
University
The
colloid-polymer
model
a
S.
Edinburgh,
of
M.
Ilett,
A.
Mayfield Road,
D.
mixture
Pirie
and
Edinburgh,
P.
EH9
N.
Pusey
3JZ,
Great
Britain
(Received
Februa~
lo
accepted
1993,
30
1993)
March
bebaviour
of a
model
hard sphere
We
report an experimental study of the phase
non-adsorbing polymer mixture
sterically stabilised
PMMA
(radius
217 nm) and
12.8 nm ) in
cis-decalin.
found'that
the main effect of adding polymer was
polystyrene (r~
It was
colloidal
which
expand the fluid-crystal
coexistence
region of the
suspension,
to
spans
Abstract.
colloid
+
=
=
0.494
#
~
dilute
dense
colloid
with
of
phases
the
possible
sources
polymer in colloidal
polymer
concentrations
Finally, the effect
559)]
of
amount
enough
obtained.
lowers
temperature
transition
into
the
the
«
gel
showed
amount
»
polymer
concentration.
The
separated
zero
polymer partitioning, giving rise to strong
findings are
phase.
These
shown
to
compare
model
[Lekkerkerker
Europhys. Lent. 20
et al.
at
marked
crystal
colloidal
statistical
recent
a
fraction)
volume
=
colloidal
compression
osmotic
favourably
(1992)
(#
0.545
~
and
mechanical
discrepancies are
glasses (#
0.58 )
crystallization was
of
»
of
of
temperature
polymer
needed
on
discussed.
addition
In
the
of
presence
a
small
crystallization. At high
inhibited
all # and a
gel
state
at
Was
Increasing
the phase
behaviour
is reported.
fluid-crystal phase separation as well as
to
cause
was
shown
to
induce
«
»
state.
Introduction.
1.
experiment (see, for example, [1-6]) that the addition of enough nonsuspension of colloidal particles
phase separation to
to a
causes
occur.
Understanding this phenomenon is of practical importance, as well as of fundamental
interest,
polymer mixtures.
colloid + non-adsorbing
industrial
products are, in essence,
since
many
remain
in
both
uncertainties
of
research,
considerable
Nevertheless,
despite a
amount
features,
the
several
of
includes
experiment and theory, Here we report a study, which
new
which
the
colloid-polymer
mixture
in
model
behaviour
of a
phase
two
components
are
individually well
characterized.
Oosawa [7]. When
of this subject is that of Asakura and
earliest
theoretical
discussion
The
colloidal
particles are separated by a distance less than the size of a polymer
surfaces
of two
the
therefore
the particles. The polymer
between
from a depletion region
coil, polymer is excluded
effect
of
another.
The
the
particles
towards
force
which
pushes
osmotic
net
two
exerts
one
a
It
is
known
adsorbing
JOURNAL
DE
from
polymer
PHYSIQUE ii
T
3. N' 7.
JULY
1993
43
1076
polymer can
interparticle
the polymer
thus
(V~ ). The
II
NO 7
polymer-induced
potential to the bare
attractive
depletion
potential
determined
by
«
»
(V~~~ is
of gyration, r~), while its depth is proportional to
a
of this
as
given by
concentration.
Various
(e.g.
size
polymer
the
by adding
described
be
interaction
PHYSIQUE
DE
JOURNAL
range
its radius
theoretical
have
attempts
been
made
determine
to
form
the
of V~~~. In the simplest approximation [8, 9], all intemal
degrees of freedom of the polymer are
ignored, and the polymer coils are treated as interpenetrable. More realistic modelling of the
polymer gives qualitatively
particles if then given by
ssuli~e~
V~ is
0 for
=
0.545,
for
occurs
>
0.494
When
this
4
~
that
~
ratio
is
is
a
less
+
in
phase
is
some
stable
about
colloidal
two
[4, 13].
behaviour
added
of
radius
volume
at
the
V~(r)= co
unperturbed
4
of
potential
~
fluid
calculations
depends
polymer
a,
In
fraction
Coexistence
polymer
0.3,
between
detail
Effective
added
interaction
particles interacting via
by thermodynamic perturbation theory.
crystal.
colloidal
of
than
total
The
phase
V~~~.
of
is treated
V~~~
hard-sphere
colloids
of
[14, 15].
effect
the
The
studied
fluid
0.545
lo-12].
results
V~
effect
of
phase
stable
=
been
has
colloidal
a
the
predict
above
r~
r
Vj~~
The
case
2 a,
polymer),
added
>
known.
paradgmatic
The
V~(r)
4
be
to
similar
is
the
on
predicted
(no
When
crystalline phases
of the
described
type
P°~~i~~~
particle
to
r~2a.
system
[14, 15].
0.494
and
for
ratio,
size
expand
fluid-
the
a
crystal
coexistence
region.
When
i~~
>
0.3,
however,
a
fluid-fluid
type phase
separation
is
also
has
also
a
possible. The case when V~
been
investigated [16].
takes
the
Yukawa
form,
appropriate
for
charged
colloids,
of polymer-induced
phase separation in colloidal
potential approach.
Predictions
resulting from this
interpretation of experimental
data.
the
In
recent
years,
[16-19] have questioned the validity of treating the effect of
however, a number of authors
effective
added polymer via an
two-body depletion potential, since this method is not able to
of polymer
of the partitioning
between
the various
separated
deal with the likely
occurrence
translational
degrees of
phases. Indeed, one attempt [17] at treating the colloidal and polymeric
perturbation theory for mixtures
freedom
strongly
equal footing using thermodynamic
on
an
Until recently
theoretical
most
used
this
suspensions
have
dominated
approach have also
treatments
effective
that such partitioning will
occur.
Experimentally also the situation is less than satisfactory. Much Work has been
colloid+polyelectrolyte),
particularly charged mixtures
(charged
systems,
suggests
performed
in
which
on
the
individually is far from fully
Little
detailed
characterized.
study
composition
the
separated
has
been
reported.
Since
of
phases
data
structure
influenced
interpretation has been
predominantly by theories based on an effective
potential,
for polymer partitioning [5].
evidence
only one experiment to date has provided
Lekkerkerker
issue
of polymer
theoretical
al.
have
addressed
the
In a
recent
et
paper
partitioning [20]. The colloidal particles are taken to be hard spheres of radius a the polymers
with the particles as hard spheres of radius &, but to be interpenetrable
interact
assumed
to
are
interaction
points
far
their
with each
other is
concemed.
of this model,
A
treatment
so
as
«
»
mechanical
using a physically transparent
statistical
approximation, predicts marked partitioTheoretical
ning of the polymer between
separated phases.
phase diagrams in the experimenrepresentation (polymer
concentration
colloid
volume
tally accessible
density-density
versus
for
first
time
which
partitioning,
presented
the
(Fig.
fraction),
show this
I).
were
colloid-polymer
mixture
in this paper,
report the experimental phase diagram of a model
we
poly-methylmethacrylate (PMMA) + polystyrene (PS) in cis-decahydronaphthacolloidal
have
been
themselves
particles [21]
PMMA
The
(c;s-decalin) at room
ierJe
temperature,
extensively in recent years [15, 22-24] and appear to behave like hard spheres. PS is a
studied
below
cis-decalin
is a little
model polymer, Its theta point in
characterized
known and well
well
behaviour
of
the
of
each
component
and
N°
7
PHASE
OF A
DIAGRAM
COLLOID-POLYMER
lo??
MIXTURE
7
f
w
§
x
~J
~
a
,£
lS
#
fl
3
~$
~
[
?
n-
02
O.1
Fig.
I.
Volume
are
imply
O.6
O.7
(11
mixture in which the polymer to colloid
size
predicted phase diagram for a colloid-polymer
according to the theory of [20]. The phase diagram is plotted in terms of an effective
volume
fraction
(4/3) gra~(N~/V). Tie
#
(4/3) gr&~(N~/V), and the colloid
~
coexistence
region. Note that the oblique tie lines
crystal-fluid (C-F)
shown in the two-phase
partitioning of the polymer among the coexisting phases.
considerable
=
=
temperature,
room
under
direct
with
contact
that
so
approximation
it
reasonable
theoretical
the
significant
confirmation
colloid
have
investigated the
we
properties of a « gel »
and
some
(compare [2, 6]).
Materials
is
experimental
our
provide
2.
0.5
Fraction
The
(&la) is 0.08
polymer fraction,
ratio
lines
0A
O.3
colloid
of
work
of
effect
of
obtained
at
studied
mixture
Lekkerkerker
phase
temperature
interpenetrability
coil
assume
The
predicted
the
state
to
conditions.
on
al. ;
et
the
behaviour.
the
data
At
phase
high polymer
here
as
a
first
therefore
makes
presented
below
concentration
one
boundaries.
concentrations
reported
also
are
of
existence
The
samples.
and
particles used in this study
consisted
of
PMMA
stabilised
sterically by thin, locores,
chemically-grafted
layers
of
poly-12-hydroxystearic
acid
[21].
They
suspended in
nm,
were
cis-decahydronaphthalene
Suspensions of this type (without added polymer)
(cis-decalin).
have been
studied
extensively with emphasis on particle dynamics [22], phase behaviour [15],
crystallization [23], and glass
formation
[24].
These
studies
have
established
that
the
interparticle
interaction
approximated by that of hard spheres.
is steep and repulsive and is well
Samples were
dilution
of a stock
solution.
described
previously
made by
concentration
As
or
[15], the
effective
hard sphere
fractions
(4
of the samples
calculated
using
volume
were
The
l5
values
literature
so
that
to
be
of the
freezing
a
217
=
occurs
±
5
nm
densities,
at
the
from
hard
a
pp~~~
sphere
I.18
gcm~
=
value
measurement
4
by
and
0.494.
=
powder
p~~~~i,~
0.894
=
gcm~ ~, and
particle radius
light crystallography
The
was
of
scaled
determined
the
lattice
1078
JOURNAL
of
parameter
crystal
colloidal
a
melting
the
at
polydispersity in the particle radius was
scattering [25] to be
5 9b.
The
polystyrene (PS) was
obtained
PHYSIQUE
DE
II
N° 7
fraction,
volume
by
determined
4
[23].
0.545
The
=
microscopy
electron
dynamic light
and
-
weight
molecular
decalin
To
is
diffusion
12.8
rJJ
12.5
0.6
±
=
Samples
sample
colloid
0
c~
~
in
0.008
~
The
effect
fraction
3.
quoted
The
l. lo.
~
radius
The
theta
polymer
of the
scattering
in
of PS
temperature
calculated
was
dilute
in
solution
;
cis-
from
its
obtained
we
to
0.015
mixer
set
0.63,
~
while
experiments
of
with
tumbled.
and
PS
a
concentrations
the
described
here
solution.
stock
volume
The
fractions
c~ of
were
Each
of
PMMA
spanned
PS
performed
at
room
2 °C.
±
temperature
=0.2±0.0015,
c~
0.00540
=
using
°C,
4
~
first
suspensions
PMMA
bath
a
each
with a colloid
volume
set of samples,
a
varying polymer
concentrations
ranging
from
observed
in the
gcm~ ~ These samples
temperature
were
range
of liquid
whose
controlled
better
than
temperature
to
was
studied
was
with
with
but
°C.
0.5
±
28
to
range
gcm~ ~. The
of
=
8 °C
the
4
of
0.00385
c~
prepared by mixing
homogenised in a vortex
19
temperature,
by
Company.
Chemical
nm.
then
were
M~
hydrodynamic
dynamic light
The
measured
were
was
with
=
[26].
°C
=
constant,
Pressure
~~
390,000
M~
was
from
Results.
Samples
particles
cloudy ;
occurring
were
inspected visually
and
the
however
they
in
bulk
their
ROOM
3.I
decalin,
were
sufficiently
could
be
temperature
are
Samples
out
in
intervals.
large
is
translucent
The
enough
that
difference
that
in
the
processes,
refractive
samples
such
as
indices
of the
appeared quite
crystallization,
seen.
different
The
TEMPERATURE.
mapped
regular
at
-1.49-1.481,
types
4-c~ plane
the
in
of phase
figure 2a.
behaviour
observed
at
room
low
volume
fractions
colloid
(4 ~ 0.49) and low polymer
concentrations
single phases and appeared homogeneous (circles in Fig. 2a). (After days to weeks
gravitational settling of the particles was
observed
however
this settling
much
some
was
slower
than that of the crystalline or gel states
discussed
below,) The spatial arrangement of the
colloidal
particles in these samples is apparently « fluid-like », and individually particles can
explore the whole sample volume by diffusion. This single phase behaviour is an extension of
colloidal
the
fluid phase at 0
0.494
in the polymer free
4
system.
In samples
with higher polymer
concentrations
(squares in Fig. 2a), colloidal
crystallites,
iridescent
specks
under
white
light
illumination,
began
observed
few
be
hours
after
to
«
a
mixing.
Nucleation
appeared to be homogeneous throughout the sample volumes.
Within a day
remained
with
in
~
~
»
and
colloid
polystyrene. The
experimental phase diagram for a mixture of PMMA
gcm-3
fraction
horizontal
colloid
volume
(#).
axis
is
in
the
concentration
polymer
(c~)
gel »
triangles
coexistence
fluid-crystal
fluid ;
asteCircles
single phase colloidal
squares
«
diamonds
glass. The compositions of the two phases into which sample A
fully crystalline
risks
guides to the eye as to the
lines
added
The
various
by filled
marked
are
as
separates
squares.
are
b) Comparison between experiment and theory. The
of various phase
boundaries.
approximate
locations
determined
by experiment (lower
region due to added polymer as
coexistence
expanded crystal-fluid
[20] using
Lekkerkerker
predicted by the theory of
from (a)) and as
lines,
taken
continuous
et a/.
of the
observed
line
continuous
onset
represents the experimentally
0.08 (dotted lines). The upper
3 la
Fig.
a) The
2.
vertical
axis
is
=
=
=
=
gel
state.
=
W
N°
OF
DIAGRAM
PHASE
7
A
COLLOID-POLYMER
1079
MIXTURE
?
AA
~
Da
7
A
A
~
a
~
o
a
~
fl
~
I
~
A
o
I
d
A
A
o
I
~
(Ge)
8
~
jj
A
(F+Q
~~
1
(
A
~
E*
(~
~
~
O
~
O
~
~
O
~
8
O
a
O
o
(Go
Colwd
V~un~e
Frwtdn
(4)
al
?
6
§
m
c
~
j'I
1
(
l
~
o.
i
olume
b)
1080
JOURNAL
crystallites
PHYSIQUE
DE
gravity,
leaving
N° 7
II
fluid
separated from
samples
in this region
appearance,
resembled
0.545
in the polymer free
those at 0.494 ~ #
system [15].
concentrations
(triangles in Fig. 2a) crystallization was
inhibited.
At still higher polymer
Immediately after mixing, samples in this region were visually
identical
samples in the
to
single-phase region found at lower polymer
concentrations.
Preliminary
studies by dynamic
light scattering showed these samples to be effectively « non-ergodic » [27] on the timescale
(~103s) of the
The
amplitudes of the
normalized
measured
intensity
measurement.
correlation
functions
fluid-like
smaller
than those for
samples, indicating the presence of
were
slowly decaying density
fluctuations
the
effective
and
suppression of long-distance
very
diffusion
of
[27]. This, together with the
absence
iridescence,
amorphous but
suggest
an
rigid
of
colloid
almost
particles interspersed with polymer
molecules.
We
will
arrangement
this as the « gel »
The
closest
analogue to this in the polymer free system is the
label
state.
colloidal
0.58 in the polymer free
glass observed
above
4
[15, 24].
system
After a few hours, the gel state
observed
leaving a clear supematant
devoid of
to settle,
was
colloidal
particles. When this settling was complete after a few days, the volume of sediment
consistent
with a
random
close
packed
0.64) of all the
colloidal
(4
arrangement
was
particles originally present. This
crystallize at the interface with the
sediment
observed
to
was
the
so
or
polycrystalline
the
settled
under
phases by
defined
well
colloidal
supematant
boundaries.
In
~
=
~
after
supematant
months.
some
fractions
Polymer-free samples at high colloid
volume
showed
fluid-crystal
(4 m0.53)
coexistence, fully crystalline (asterisks in Fig. 2a) and glassy (diamonds in Fig. 2a) behaviour
with
increasing 4, as observed
previously [15]. On the addition of increasing
of
amounts
initially fully crystalline (0.545
polymer, samples which
underwent
4 ~ 0.58 )
partwere
melting to give a coexisting colloidal fluid. A small
of polymer in a glassy sample was
amount
higher
found to induce
full crystallization,
At
concentrations
of polymer the sample then
even
into
crystal-fluid
coexistence.
At the
highest added
concentrations,
moved
polymer
two
failed to give any crystals,
suggesting gel formation.
samples at high colloid
concentration
compositions of the two phases in one particular sample (sample A in Fig. 2a,
The
actual
composition 4
0.0048 gcm~~) showing
initial
0.lsl,
fluid-crystal
coexistence
were
c~
determined.
ratio
of
of
fluid
crystalline
4.8
The
the
volumes
and
phases was
The
volume
: 1.
determined
fraction of colloid in the crystalline phase was
by light crystallography. The crystal
of
assumed
be
hard-sphere
colloidal
crystal without polymer, a stacking
that
structure
to
was
a
of hexagonally
arranged layers of particles [23]. The spacing of the hexagonal layers was
their
reflection
determined
from
Bragg
and
calculated
from
this
spacing and the
4~~~~~~j
previously
measured
particle radius, 217 nm. This procedure gave 4~~~~~~j
0.655 ± 0.01.
The
of
fluid
crystalline
phases
the
rule
requires
measured
volume
ratio
and
lever
then
to
~
=
=
=
~flutd
°'°~~'
~
The
were
then carefully
were
using a centrifuge,
scattered
light
intensities
two
phases
spun
down
comparing
giving c~ (fluid )
concentrations,
separated.
and
the
with
0.0058
±
=
particles
Colloidal
polymer
those
PS
of
solutions
gcm~ ~. The
0.0008
the
in
concentration
supematant
determined
was
in
decalin
crystal phase
of
was
an
(See
=
uncertainties
3.2
in
estimating
TEMPERATURE
of
twelve
in
these
the
The
DEPENDENCE.
samples (all at 4
0.2
samples ranged from c~
of
amount
±
=
=
0.0015)
0.00385
polymer
in
temperature
was
gcm~
the
sediment
dependent
investigated.
also
to
c~
0.00540
=
The
behaviour
polymer
gcm~
the
reflect
the
centrifuging.)
after
phase
known
redispersed
suspension
volume
of
decalin
and the
polymer
concentration
in the
resultant
equal
estimated
procedure, giving c~(crystal)
0.0019 ± 0.0003 gcm~~
using the same
appendix for more
details
of these
calculations.
The
rather
large error
bars
on
c~
in
phase
by
~
At
of
a
series
concentrations
room
tempera-
N°
DIAGRAM
PHASE
7
samples
fluid-crystal
Fig. 2) all
and gel.
span (see
coexistence
these
ture,
fluid,
OF
COLLOID-POLYMER
A
three
equilibrate at each new
temperature
behaviour.
the phase
The phase diagram for the T-c~ plane at 4
phases as at room
found
temperature
were
allowed
possible phase behaviour : single phase
remixed
by tumbling before being
were
inspection then
bath.
Visual
temperature
of
types
samples
The
in the
to
1081
MIXTURE
determined
of
temperature
5.4
was
change
to
D
A
the
range
at
is
figure
in
shown
all
3.
The
which
over
studied.
temperatures
fluid-crystal
three
same
major
The
coexistence
types
effect
was
A
6
A
6
6
6
6
6
a
a
a
a
a
a
a
a
a
6
6
6
6
6
6
6
6
A
A
A
A
A
A
A
A
A
A
6
A
A
~
~
~
a
a
6
u
u
a
u
u
u
of
observed.
A
a
q
of c~
0.2
=
E
$
~
-
~
u
_q
I
4.6
S
(
~
d
~
4.4
i
h
k
3.8
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
°
°
°
°
284
u
a
D
a
a
D
D
D
u
D
D
D
o
o
o
o
o
o
u
u
u
u
D
u
°
°
°
°
~
~
°
286
Terrperature (K)
Fig. 3.
The effect of temperature.
All samples have
gives polymer
concentration
Circles
(c~) in gcm-3
crystal
coexistence
triangles
gel ».
«
colloid
=
volume
single phase
fraction
~
0.2
vertical
the
axis
=
colloidal
fluid
fluid-
squares
=
=
4.
Discussion.
The
polymer
of
ratio
(r~
=12.8nm)
particle (a
previous
theoretical
fluid-crystal
coexistence
217nm)
colloidal
and
sizes
in
=
experiments
effect
0.494
of
~
is
the
4
~
0.06.
about
For
added
polymer is
0.545
at
c~
0.
this
~
expand
to
Our
=
prediction. The lines
separate the region of fluid-crystal
crystal/glass region at 4
0.545 ;
line
the fully
crystalline
separates
confirm
situation
this
in
all
the
room
temperature
figure
2a
coexistence
the
upper
region
are
from
line
from
results,
the
indicates
the
shown
guide
fluid region
drawn
glassy
the
to
the
onset
region.
at
our
predict that the
region, which spans
in figure 2a,
clearly
studies
4
of the
eye.
~
The
0.494
gel
state
lower
and
lines
from
; the
the
dotted
JOURNAL
1082
The
in figure 2a
indicate
squares
A separated (see Sect. 3.I
solid
sample
which
phase boundary
sample A. It is
the
lines,
on
that
evident
of
measurement
partitioning
and
straight
a
this is the
measured
and
the
line
tie
which
N°
concentrations.
These
of
the
phases
two
should
lie
into
lower
the
on
7
composition of
is made for experimental
in
errors
results
directly
indicate
significant
includes
allowance
when
II
compositions
appendix). These
the
case,
polymer
the
PHYSIQUE
DE
the
overall
polymer between the separated phases.
increased
polymer
concentration
in the fluid phase results in osmotic
The
compression of the
coexisting crystal. The volume
fraction
of the crystal in sample A, 4~~~~i
0.655, is much
0.58, at which a glass transition of the pure (polymer free) colloid has
larger than that, 4
found [15, 24]. We note
that
compression of colloidal crystals has also been
been
osmotic
mixtures
of
colloidal
particles of two
different
observed
in
PMMA
sizes [28].
Quantitative comparison of the measured phase boundaries
with the predictions of [20] is not
straightforward. As mentioned in section I,
Lekkerkerker
that the polymer
assumed
et al. [20]
interpenetrable to one another, but that the centre of each coil was
excluded
from a
coils
were
colloidal
centred
panicle. The polymer
concentration
quoted
sphere of radius a +
on
any
was
of
=
=
as
coils
~
in
the
gr&3c~
3
polymer
effective
an
=
sample
~M,
volume
where
~
Conversion
V.
is
M
~
fraction
volume
the
mass
one
boundary
data) is
at
colloid
fraction
volume
=18nm.
We
see
that
4
=
the
N~
where
experimental
polymer
immediately obvious what value should be used for
simply
obtained by regarding
figure 2b were
required
theory and experiment. The value of
in
~~,
V
the
to
of
~
ar&
3
=
variable
While
coil.
number
of
c~ is via
the
is the
M
known,
is
polymer
relation
it
is
not
phase boundaries
shown
theoretical
fitting parameter » used to connect
experimental fluid-crystal phase
to fit the
0.3 (near the middle of the range of 4 spanned by our
predicted phase
boundaries
then in
qualitative
are
&.
The
as
a
«
with the experimental
results at other
colloid
volume
fractions, but that there are
quantitative
differences.
In
particular,
theoretically
predicted
left-hand
the
phase
boundary is steeper than that found experimentally, and the polymer
concentrations
in the
colloidal
phase (the high
phase boundary) are significantly
underestimated
by the
dense
agreement
clear
theory.
The
radius
fitted
of
value
of
polymer
the
can
be
used
in
compared with various
theoretical
candidates.
The
hydrodynamic
experiments,
measured
by dynamic light scattering at room
our
0.002 gcm~ ~ ), was
12.8 ± 0.6 nm.
polymer
concentration
(c~
r~
Gaussian
coil, the relationship
between
the radius of gyration r~ and the hydrodynamic
For a
Possibly the most realistic quantity
1.51 r~ [29], giving r~
19.3 ± 0.9 nm.
radius r~ is r~
which
thickness
with
should be compared is the
of the depletion layer, f, near a hard wall in a
calculation
field
[30] of this quantity gives f
(21ar '/2) r~
polymer solution. A self consistent
Using
(Gaussian)
infinite
dilution.
r~=19.3nm,
for
unswollen
coils
at
get
we
21.8
It is reassuring that the fitted
value of
18 nm is close to this
value.
±1nm.
f
of r~, via r~, can be compared with the data of Berry [26]. He reported that
Our
measurement
), giving r~(T~) 17.0 nm for M~ 390,000. He
0.0270
for PS in decalin, r~(T~)
of the
Fixman
two-body
interaction
also gave r~ (T)/r~ (T~ as a function
parameter z, a
measure
excluded
volume
in
units
kT.
Berry's
interaction
of
of the
measurements
monomer-monomer
relation
yielded the experimental
temperature
and
low
=
=
=
=
=
=
=
=
=
z
so
that
z
Berry's
0.13
=
results
0.00975
well
,
conditions
experimental
(T
for our
predict that, for r~(To
12.5 °C)
with
the
T~/T]
=
=
compares
)[1
value
=
deduced
292
=
17.0
=
from
the
K, T~
nm,
286
K).
this
At
value
=
r~(T
measured
=
19 °C)
19
=
hydrodynamic
nm
of z,
which
radius,
PHASE
7
N°
r~(T
19 °C
19.3
=
are
0.9
±
=
only slightly
DIAGRAM
swollen
COLLOID-POLYMER
A
Berry's
(above).
nm
at
OF
indicate
also
data
MIXTURE
1083
coils
PS
that
in
cis-decalin
temperature.
room
regarded simply as a parameter to be fitted, there is still clear
disagreement
experiment and theory. We discuss possible origins of this disagreebetween
concentration.
The
starting with the dependence of the properties of the polymer on
ment,
the
Lekkerkerker
al.
[20]
the
is
ideal
in
the
that
theory of
that
polymer
et
sense
assumes
given
ideal
concentration
and that its osmotic
is
by
the
polymer size is independent of
pressure
fraction, see below). In reality, the thickness
form, n
N
kTla V (where a is the free
volume
~
dependent [3 Ii, and the osmotic
concentration
f of the depletion layer (and therefore &) will be
theta
non-ideal.
concentrations
this
will be true
will
be
(At
high
enough
at the
even
pressure
c~,
temperature.) Thus the depletion potential, which scales as lf&2a (see [18] and [20]), will in
general have a
concentration
dependence
complicated than that assumed in the theory.
more
fraction ~' of the volume of solvent occupied by polymer is given in terms of the overall
The
effective
volume
fraction of polymer ~ defined
above by ~
4 ), where as before 4 is
~ / (l
noted
As
above,
even
volume
fraction.
if 3 is
=
'
=
colloid
the
regime (I.e. polymer
dilute
&la
at
In
0.I,
w
described
«
starting point
concentration
of
(see [31]
available
will
~'
at
occur
maximum
value
from
the
Reference
to
crossover
l.
-
~'
of
be
to
dilute
0.3,
around
I
semi-
the
to
figure
shows
which
that
be
can
osmotic
We
therefore
expect that correcting the ideal gas
pre-crossover
».
coefficient
virial
for the polymer in [20] by including the
effect of a
second
used
pressure
would be a good
effect
the
expect
can
we
as
overlap)
coil
variable,
of this
terms
for
accounting
for
fin
on
discussion
a
dependence of lZ
regime is, however,
effect in the fully
of
full theory of the
apparently not yet
semi-dilute
regime).
for the c~
concentration
this
concentration
The
potential
of disagreement
when
between
experiment and theory is revealed
source
it is recognised that the
diameter
of the PS polymer coils,
40 nm,
thicknesses
of the
and the
polymer coatings on the PMMA particles,
m10-15
comparable in magnitude. The
nm,
are
coatings of poly-12-hydroxystearic acid on the particles are thought to be quite tightly packed
[32] but, almost certainly, the « surfaces » of the composite particles are neither
smooth
nor
Limited
penetration of the particles by the polymer is therefore
hard on the scale of a few
nm.
possible.
Finally the theory of [20] is essentially a mean field theory. The free energy of a mixture of
molecules
in total
volume V is written in the form
N~ colloidal particles and N~ polymer
second
A
m
F
where
first
the
polymer
in
a
which
volume
corresponds
term
aV,
volume
accessible
is
the
coordinates
of all
N~
into
the
form
if
above
where
to
to
the
«
makes
the
(.
)
interaction
colloid
a
It
coordinates,
colloidal
fractional
by
free
volume.
self-consistent
should
Polymers
)-
averaging over
colloidal
polymer and colloid is
~~ ~
gra~. This
fraction
4
also
on
an
V
which
Such
calculation
using
pointed out that
unequal footing. The
be
give
would
general
). The
configurations.
solely
to
a
term
included
Ginzburg-Landau
a
the
free
polymer
function
a
only
energy
energy
part,
procedure
dependence
This
in
the
ignores
formulation
be
course,
free
pure
sample
total
of
separates
(2)
contained
rise
is, of
a
total
to
term
of the
«(~)
=
could
second
the
fraction
is the
approximation
(«)
3
fluctuations
a
=
V, and
a
»
In
(r~)
field
denotes
=
the
a
mean
(1)
volume
a
fraction
molecules.
between
volume
in
volume
«((r~)
where
F~(N~, aV)
+
colloid
pure
free
polymer
particles,
the
colloidal
one
F~(N~, V)
=
effect
the
proportional
the theory
type approach.
in
approximation
F~, depends
on
of
the
of
that
the
on
the
a
fluctuations
kT(&4~)
to
for
(I)
means
phase
puts
in
the
behaviour
colloids
colloidal
in
and
volume
JOURNAL
1084
PHYSIQUE
DE
II
N° 7
fraction
there is no polymer
via a (4 ), while
concentration
dependence in the colloid part,
F~. In reality,
however,
that
the
of
added
polymer will perturb the
expect
we
presence
configuration of the colloids and therefore their free energy [17]. In tum, this means that the
averaging indicated in (2) should be performed over the per-tuibed configuration of colloidal
particles, giving rise to a c~ dependence of a. The actual expression used for a (see the
Appendix) does not allow for this possibility.
Work to extend
the theory of [20] to include
of these
complications, and therefore to
some
predicted
determine
their
effect
the
phase
diagram,
is
in
on
progress.
We note
that this
appears to be first experimental study of the effects of adding polymer to
colloidal
crystals and glasses I.e. of the high-i branch of the phase boundary in figure 2a. A
particularly interesting
observation
is the ability of added polymer to induce
crystallization in
colloidal
for this are
the
glass. The
unclear
although it can be speculated
at
present,
reasons
that the
of polymer may
increased
fluctuations
of the free
volume
(see above),
presence
cause
giving rise to local configurations that offer low free energy
barriers
crystallization.
to
effect of
the phase
behaviour
is striking. At the
colloid
concentration
The
temperature
on
studied
in this
work
0.2), the fluid-crystal
coexistence
region is apparently at its
(4
(Fig. 3). Both heating and cooling lead to a
round
about
narrowest
temperature
room
decrease
of polymer
significant expansion of this region. The progressive
in the
amount
fluid
needed
the
single
phase
into
phases
the
is
two
to
to
separate
temperature
cause
as
increased
be
understood
qualitatively as a
combination
of two
effects
individual
coils
can
contribution
of the
second
virial
expand (thus increasing &), and the
the
osmotic
term
to
(14 increases.
Since the depletion potential (in units of k7~ scales as c~ lf& 2 a [18, 20]
pressure
=
that,
expect
we
relevant
A
state.
in
separation
phase
with
agreement
should
when
it
more
detailed
decrease
considering
to
comes
the
with
study
of
observations,
increasing
the
temperature
the
polymer
of
amount
needed
to
cause
Presumably this effect is also
temperature.
of
the
formation
of the gel
temperature
on
effect
effects,
both
experimentally
and
theoretically,
is in
progress.
experiments has not been predicted by theory
little is known
been
of
speculated
that,
under
the
influence
about
state
at
present.
strong enough
a
depletion
particles adopt a
metastable,
fractal-like
attraction,
the
tenuous,
arrangement
interspersed with polymer molecules [2]. The slow settling of this state, leaving a colloid-free
colloidal
molecules
and
particles are still
the
both
the
polymer
that
suggests
supematant,
mobile.
preliminary
light
somewhat
However
dynamic
scattering
measurements,
our
perslow.
We
formed
sample soon after mixing, indicate that these motions
be very
must
on
a
investigate
in
detail
both
of
intriguing
the
and
dynamics
this
intend
structure
state.
to
more
The
gel
state
observed
this
Two
in
our
It has
previous
studies
have
reported
We
have
predictions,
at
shown
moderate
that
the
particularly
polymer
observations
similar
gel
hydroxyethyl
to
ours,
regions of fluid-crystal
added
polymer.
more
of
cellulose
mixtures
polymer and charged
Sperry [2] investigated
aqueous
Particle
inferred
from
magnification
particles of acrylic copolymer.
low
arrangements
were
mixtures
silica spheres, stericallyoptical microscopy. Smits et al. [6] studied
of uncharged
stabilised by alkane
(octadecyl) chains, and polystyrene or poly(dimethyl siloxane) polymer in
various
interactions
between
the
species were quite
cyclohexane. In the first experiment the
complicated ; in the second, the fact that silica particles on their own often fail to crystallize [6]
colloids.
value
hard-sphere
Both
diminishes
somewhat
their
«model»
4 ~ 0.494
at
as
experiments predate [20], so that comparison with that theory was not possible.
conclusion
In
have
shown
that
mixtures
of hard-sphere
PMMA
colloids
and polystyrene
we
constitute
promising model system which we intend to study comprehensively in the future.
a
coexistence
concentrations
and
«
states
»
with
phase diagrams are in reasonable
with
agreement
regard to partitioning of the polymer between
with
recent
separated
theoretical
phases.
N°
We
significant
long-lived
a
suppressed, at high
observed
have
the
as
DIAGRAM
PHASE
7
fornaation
effectively
OF
effect
a
of
of
gel
«
COLLOID-POLYMER
A
temperature
state
in
»,
long-distance
polymer.
added
which
concentrations
of
boundaries,
phase
various
the
on
1085
MIXTURE
well
as
of the
diffusion
colloid
is
P.
B.
Acknowledgements.
of this
Part
work
for
Warren
figure
I.
is
valuable
are
grateful
We
particles,
PMMA
scattering.
by
funded
many
and
and
R.
Unilever
of
use
the
Ottewill
H.
Council.
and
Ms.
for
Research
thank
We
Dr.
generated
providing
the
F.
particles
using
light
these
studentship.
CASE
a
computer
characterizing
(Port Sunlight) for
T.-T.Chui
Mr.
Research
Food
for
and
Professor
to
to
thanks
A.D.P.
Agriculture
the
discussions
that
program
for
Beach
Appendix.
Calculation
of
polymer
the
in
concentrations
separated
the
phases.
mixture
of total
volume V. This sample is spun in a
colloid-polymer
particles.
After
spinning
down, the sediment is observed to
colloidal
to
in the
fraction
f
of
the
volume.
The
polymer
concentration
(now
total
supernatant
occupy
a
this
concentration
be
devoid
particles)
measured
by
light
scattering.
of
be
Let
can
cm3).
is
polymer
concentration
cj (in
The
question
is
what
the
grammes
per
this question, we need to know the
amount
answer
c~ in the original single-phase sample ? To
of polymer in the
sediment.
At the low centrifuging speeds used in our work (w 3 000 rpm ) we
therefore
reasonably
that
do not expect significant sedimenting of the polymer. We
assume
can
sediment
the
concentration
of polymer in the « free
volume » of the
is cj.
The free
volume
of the
sediment
which is
available
for the
volume
is that portion of the
insertion
excluded
closer
than a
of a polymer coil.
Assume
the polymer coil is
from coming
of a colloid
distance
surface
particle, and that the colloidal particles (volume fraction
3 to the
distributed
fraction, a, can be
4) are randomly
hard spheres (radius a). Then the fi.ee volume
estimated
derivable
from the
scaled
particle theory of mixtures [33]
by an expression
Consider
single phase
a
centrifuge
sediment
the
where
y
=
4 ), A
WI (1
=
-By~-Cy~]
4)exp[-Ay
(1-
a
=
3 f
+
f~
3
+
f~,
f~/2
9
B
=
+
3
f~
C
and
3
=
f~ (where
f
~
=
).
a
sediment
The
a
217
The
will
and
nm
=
original
the
random
a
18
m
of
amount
c~ V in
be
3
polymer
sample
c~
packing
close
in
is
4 of
Sect.
cj(a f
=
particles,
of
so
4
that
m
0.64.
In
experiments
our
giving a
0.20.
is then cj afv.
the
sediment
The
total
amount
f) V, giving finally
therefore
cj afv + cj(I
(see
nm
+
text),
main
cjil
f)
I
=
=
f(I
a
of
polymer
)j,
References
[1]
DE
HEK
and
H.
VRIJ
PATHMAMANOHARAN
[2]
[3]
[4]
[5]
SPERRY
P.
R.,
SPERRY
P.
R.,
GAST
A.
VINCENT
PATEL
P.
P.,
B.,
D.
A.,
C.,
J.
DE
HOPFENBERG
J.
RussEL
W.
and
RussEL
H.
H.
B.
Interface
Colloid
EDWARDS
Colloid
HEK
B.
J.,
and
B.,
J.
Sci.
VRIJ
and
and
THOMAS
Sci.
99
HALL
EMMETT
W.
Interface
C.
S,
and
Colloid
A.,
N.
(1984)
K.,
79
(1981)
289 ;
Colloid
Polymer
L.,
Colloid
J.
Sci.
259
Inteifiace
(1981)
Sci.
82
769
(1981)
97.
Inteifiace Sci. 109 (1986) 161.
Surf 31 (1988) 267.
Sci. 131 (1989)
192.
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