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Phase diagram of mixtures of hard colloidal spheres and discs:
A free-volume scaled-particle approach
S. M. Oversteegen and H. N. W. Lekkerkerker
Van’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute, Utrecht University,
Padualaan 8, 3584CH Utrecht, The Netherlands
?Received 28 August 2003; accepted 6 November 2003?
Phase diagrams of mixtures of colloidal hard spheres with hard discs are calculated by means of the
free-volume theory. The free-volume fraction available to the discs is determined from
scaled-particle theory. The calculations show that depletion induced phase separation should occur
at low disc concentrations in systems now experimentally available. The gas–liquid equilibrium of
the spheres becomes stable at comparable size ratios as with bimodal mixtures of spheres or
mixtures of rods and spheres. Introducing finite thickness of the platelets gives rise to a significant
lowering of the fluid branch of the binodal. © 2004 American Institute of Physics.
?DOI: 10.1063/1.1637573?
I. INTRODUCTION
Mixtures of colloidal particles that differ in size and
shape are ubiquitous in industry, food science, and the bio-
logical realm.1,2Knowledge of the phase diagram of those
systems is essential to understand their stability. A range of
colloidal mixtures have been studied extensively by means
of experiments, simulations, and theory. Especially mixtures
of hard spheres with polymers ?see Ref. 3, and references
therein? or hard smaller spheres obtained much attention ?see
Ref. 4, and references therein?. In such systems phase tran-
sitions may take place for purely entropic reasons, i.e., the
species may gain conformational entropy when they are no
longer hindered by the other species which are consequently
expelled. This so-called depletion effect5,6is even more
pronounced in anisometric mixtures of, e.g., rods and
spheres.7–9Here we determine the phase behavior of aniso-
metric mixtures of platelets and spheres by means of the
free-volume theory.10This theory accounts for the volume in
the system that is available to the species in a hypothetical
reservoir and has previously proven to be successful to bi-
modal mixtures of spheres11and to mixtures of rods and
spheres.12,13
II. FREE-VOLUME THEORY
Consider a system of Nshard spheres with diameter ? in
a volume V at temperature T. Suppose this system is in
thermodynamic equilibrium with a reservoir of platelets with
a diameter D and thickness L. The appropriate thermody-
namic quantity to describe this system is the semigrand po-
tential. Applying Widom’s insertion theorem14and elemen-
tary thermodynamic relations, one obtains for dilute platelet
suspensions10–13
??Ns,?p,V,T??A??Ns,V,T??pR?V.
Here A?is the Helmholtz energy of the unperturbed hard
sphere system and ? is the so-called free-volume fraction,
i.e., the relative amount of the volume V that is accessible to
?1?
the platelets. A convenient expression for the pressure pR
exerted by the reservoir of platelets is given by15
pR
RkBT?
?p
1
1??p
R?
3??p
?1??p
R
R?2?
?2?p
R2?3??p
?1??p
R?
R?3
.
?2?
Here ?p
nonsphericity parameter ? follows from the second virial co-
efficient B2and the platelet’s volume vpaccording to
Ris the volume fraction of discs in the reservoir. The
??B2/vp?1
3
.
?3?
Upon comparison with simulations17,18Eq. ?2? is very accu-
rate for both platelets of finite thickness as well as discs. The
mathematics in the following is substantially simplified if we
take discs, i.e., infinitely thin platelets, without losing the
physics. The second virial coefficient then simply reads
B2??2D3/16,
?4?
which converts Eq. ?2? to19
6q3???p
pRvs
kBT?
?
RD3???
?2
16???p
RD3?2??
?4
768???p
RD3?3?, ?5?
where we introduce the aspect ratio q?D/? and the discs’
number density in the reservoir, ?p
Coexistence between phases ? and ? at a given reservoir
pressure is found from
R.
p???s
???p???s
??,
?s
???s
????s
???s
??.
?6?
We obtain the pressure of the system and chemical potential
of the spheres from Eq. ?1? as
p???
T,Ns,?p
??
?V?
?p??pR????s?
??
??s??
?7?
and
JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 51 FEBRUARY 2004
24700021-9606/2004/120(5)/2470/5/$22.00 © 2004 American Institute of Physics
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?s??
??
?Ns?
V,T,?p
??s
??pRvs?
??
??s?.
?8?
Here ?s?Ns(?/6) ?3/V is the volume fraction of the
spheres. The pressure and chemical potential of the unper-
turbed hard-sphere fluid are found from the Carnahan–
Starling equation20
p?vs
kBT?
?s?1??s??s
?1??s?3
2??s
3?
,
?9?
?s
kBT?ln?s?
The equation of state of the hard-sphere crystal is given
by Wood21
?
?s?8?9?s?3?s
?1??s?3
2?
.
?10?
p?vs
kBT?
3?s?cp
?cp??s,
?11?
where ?cp??&/6?0,74 is the closed packed density. The
chemical potential follows from integration of the Maxwell
relation (?p/?Ns)T??(??s/?V)T,
kBT?2.1306?3 ln?
The first term on the right-hand side is an integration con-
stant derived from the absolute free energy of a hard sphere
crystal from Monte Carlo simulations at ?s?0.576.22
?s
?
?s?cp
?cp??s??
3?cp
?cp??s.
?12?
III. SCALED-PARTICLE THEORY
The free-volume fraction ? is determined from the
chemical potential of the discs. Inserting a disc in a sea of
spheres adds, next to the mixing entropy, a work term W to
that chemical potential
?p??p
??kBT ln?p?W.
?13?
On the other hand, it follows from Widom’s insertion
theorem14that
?p??p
??kBT ln?p?kBT ln?.
?14?
The required work to insert a disc in between spheres there-
fore amounts to W??kBT ln? or, alternatively,
??e?W/kBT.
?15?
Looking for the free volume fraction ? is therefore equiva-
lent to finding an expression for W. In order to determine the
insertion work W, the scaled-particle theory considers two
size limits. To that end, we scale the disc diameter D with a
parameter ?. Since the disc is infinitely thin, the thickness
does not need to be scaled.
For the limit ??1 the small discs may be considered to
be points and the overlap volume between the spheres is
therefore negligible. Hence, in that limit the volume avail-
able to the disc is V?Ns?(?D,?), where ?(?D,?) is the
volume of a sphere of diameter ? that is excluded to a disc.
Thus, the free-volume fraction is ??1??s?(?D,?). Using
Eq. ?15? it follows that
W???1???kBT ln?1??s???D,???.
?16?
On the other hand, if ??1, the work required to insert a
large disc between the spheres, will approximately be the
volume work to create a hole with the size of a disc;
W?pvp. However, since the discs are infinitely thin, it ap-
plies that the size of a disc vanishes; vp?0. Hence,
W???1??pvp?0.
?17?
This rather remarkable result also applies to infinitely thin
rods.12,23
The essence of the scaled-particle theory is that we may
add both limits, where we expand W in the limit ??1
around ??0. From Eqs. ?16? and ?17? we obtain
W????W?0???
??0
?W
???
??1
2?
?2W
??2?
??0
?2?pvp
??kBT ln?1??s?0??
2?
kBT?s
1??s?0?0??
?1
kBT?s
1??s?0?0??
kBT?s
?1??s?0?2?0?
2
2??2,
?18?
where ?0, ?0? , and ?0? are the excluded volume, its first and
second derivatives with respect to ?, respectively, evaluated
for ??0.
When we finally scale the disc to the desired size by
putting ??1, we find from Eq. ?18? using Eq. ?15? for the
free-volume fraction
???1??s?0?exp????0??1
2?
2?0???
?s
1??s?0?
?1
2?0?
?s
1??s?0?
2?.
?19?
The problem has now reduced to finding the excluded
volume of a disc around a sphere. Let us place a disc into a
certain orientation and move it in such a way around a
sphere, that its center is as close as possible to the sphere. In
this way we probe the volume that is inaccessible to the disc.
A different orientation of the disc relative to the sphere is
equivalent as redefining the coordinate system. In this way it
is easily seen that all orientations yield the same excluded
volume. For a disc of diameter D around a sphere of diam-
eter ? we find from the body of revolution of a line probing
a circle
??2??
0
?/2?
1
2D??1
4?2?y2?
2
dy
??
4D2???2
8D?2??
6?3.
?20?
This is the same result as found for spheres around a disc.19
Scaling the diameter of the disc with ?, we find for the co-
efficients in Eq. ?19?,
?0??
6?3?vs,
?0???2
8D?2?3
4?qvs,
?0???
2D2??3q2vs.
?21?
2471J. Chem. Phys., Vol. 120, No. 5, 1 February 2004Phase diagram of mixtures of hard colloidal spheres and discs
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Here we introduced the aspect ratio q?D/?. Using ?s
??svs, the free-volume fraction, Eq. ?19?, finally reads
???1??s?exp???A?
?C?
with
2q?
Since we consider dilute disc dispersions, the chemical
potential of the reservoir may be given as ?p
?kBT ln?p
we find the volume fraction of discs in the system as
?p???p
?s
1??s??B?
?s
1??s?
2
?s
1??s?
3??,
?22?
A?3
?
2?q?,
B?9?2
32q2,
C?0.
?23?
R??p
?
R. In equilibrium applies ?p??p
R. Using Eq. ?14?,
R.
IV. RESULTS
At a given number density, ?p
the pressure of the reservoir on the system follows from Eq.
?5?. Together with Eqs. ?22? and ?23? then follows the per-
turbation on the pure hard spheres due to the presence of the
discs in Eqs. ?7? and ?8?.
The set equations of Eq. ?6? can be solved numerically
using Eqs. ?9? and ?10? in Eqs. ?7? and ?8?, respectively, for a
given ?p
‘‘liquid phase’’ rich in spheres ?poor in discs? and a ‘‘gas
phase’’ that is poor in spheres ?relatively rich in discs?. We
Rof discs in the reservoir,
R. This set gives the gas–liquid coexistence of a
calculated points down to the critical point which can be
found numerically from the roots of the first and second
derivatives of the pressure, Eq. ?7?. If we use Eqs. ?9? and
?10? for phase ? and Eqs. ?11? and ?12? for phase ?, the
fluid-crystal equilibrium can be found from the set equations
of Eq. ?6? at given ?p
Since the phase transition takes place at ?p
isotropic–nematictransition
Simulations16,17show that this is first found at a density
?p
equilibrium (G?L) is located inside the F?C region and is
metastable, indicated by the dashed lines in Fig. 1. For large
enough discs (q?D/??0.41) G?L becomes stable, which
leads to the existence of a triple line. This aspect ratio is
comparable to those found for polymer-spheres mixtures
?q?0.4 ?Ref. 24?? and ?infinitely thin? rods ?q?0.3 ?Refs.
12, 13??.
Experimentally the number density of discs in the sys-
tem rather that in the ?hypothetical? reservoir is relevant. The
calculated phase diagrams in the upper row of Fig. 1 are
therefore transformed to experimentally accessible diagrams
in the second row by ?p???p
tie lines to be slanted. As a consequence the triple line opens
up as a triangle.
In an experiment the platelets have a finite thickness. It
seems reasonable as a first approximation to apply the theory
for infinitely thin platelets to the experimental system given
in Fig. 2?a?. The silica spheres have a diameter of ?
?700 nm, whereas the silica coated gibbsite platelets are L
R. Results are given in Fig. 1.
RD3?1, no
expected.ofdiscsis
RD3?4. For relatively small discs ?low q) the gas–liquid
R.10This forces the horizontal
FIG. 1. Calculated phase diagram for mixtures of hard spheres ?volume fraction ?s) with hard discs. The upper row as a function of the reduced reservoir
number density ?pD3, where D is the disc’s diameter, in the second row the corresponding reduced number density in the system. Dashed lines show the
metastable gas–liquid equilibrium.
2472 J. Chem. Phys., Vol. 120, No. 5, 1 February 2004S. M. Oversteegen and H. N. W. Lekkerkerker
Downloaded 28 Jan 2004 to 131.211.152.81. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
Page 4
?30 nm thick and D?200 nm wide. The reduced number
density can be converted to a volume fraction via
?p??pvp??p
?
4D2L??pD3?
4
L
D.
?24?
For the experimental size ratio (q?2/7?0.41) the liquid–
gas phase transition is metastable, so only the fluid-crystal
phase boundary and tie lines are given by the dashed lines in
Fig. 2?b?.
It is straightforward to extend the above theory to plate-
lets of diameter D with a finite thickness L represented by
short cylinders. The second virial coefficient of a reservoir of
cylinders reads25
B2??2D3
16
??D2L
8
?3?????DL2
4
.
?25?
For infinitely thin platelets (L?0) Eq. ?25? reduces to Eq.
?4?. It can be shown that Eq. ?2?, using the second virial
coefficient in Eq. ?3? from Eq. ?25?, gives the same pressure
as from simulations18up to densities ?p
Eq. ?2? using Eq. ?25? suffices for our present purposes.
By applying the scaled particle procedure of Sec. III to a
sphere probing the volume around a cylinder, we find for the
free volume fraction the same functional form as Eq. ?22?
now with the coefficients
2q?
8q2??
RD3?2.0. That is,
A?3
?
2?q?r?1?q?2?,
2?r?
B?9
?
2
?4qr?,
?26?
C?9
2q3r.
Here we defined r?L/D. It is easily seen that for discs
(r?0) Eq. ?26? reduces to Eq. ?23?.
Inserting Eqs. ?25? and ?26? into Eqs. ?2? and ?22?, re-
spectively, and going through the above procedure using the
experimental system of Fig. 2?a?, we find the phase diagram
given by the solid line in Fig. 2?b?. Clearly, the fluid branch
of the binodal goes down by almost a factor 2, whereas from
the tie lines we see that the crystal phase tend to shift to
higher volume fractions of the spheres. Hence, introducing
finite thickness to the platelets leads to a significant lowering
of the binodal. This can also be observed for rods when
infinitely thin needles are replaced by spherocylinders.12,13
If we prepare samples of ?s?0.025 spheres, we may
already expect crystalline ordering of the spheres at ?p
?0.07. This volume fraction is one order of magnitude
larger than experimentally found for dispersions of rods in
spheres (?rod?0.005 for a similar aspect ratio9? but one or-
der of magnitude smaller than predicted for binary hard
spheres mixtures ?we estimate ?small sphere?0.3 for a compa-
rable size ratio26,27?. We predict that fairly low platelet con-
centrations suffice to induce phase separation. We hope to
address this experimentally for systems as shown in Fig. 2.
ACKNOWLEDGMENTS
We are grateful to Dr. J. E. G. J. Wijnhoven for provid-
ing us with the synthesis and TEM-picture of the mixture.
H. H. Wensink is thanked for fruitful discussions. This work
is part of the SoftLink research program of the ‘‘Stichting
voor Fundamenteel Onderzoek der Materie ?FOM?,’’ which
is financially supported by the ‘‘Nederlandse Organisatie
voor Wetenschappelijk Onderzoek ?NWO?.’’
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FIG. 2. ?a? TEM micrograph of a mixture of silica spheres (??700 nm)
with silica coated gibbsite plates (D?200 nm, L?30 nm). ?b? The calcu-
lated phase diagram of the experimental system. The dashed lines give the
phase diagram from discs where the volume fraction of platelets, ?p, is
determined from the reduced number density for discs via Eq. ?24?. Upon
replacing both the second virial coefficient in Eq. ?2? as well as the coeffi-
cients of the free volume fraction in Eq. ?22? by those for thin cylinders, the
solid lines are found.
2473 J. Chem. Phys., Vol. 120, No. 5, 1 February 2004Phase diagram of mixtures of hard colloidal spheres and discs
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2474 J. Chem. Phys., Vol. 120, No. 5, 1 February 2004S. M. Oversteegen and H. N. W. Lekkerkerker
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