Europhys. Lett., 66 (1), pp. 125–131 (2004)
1 April 2004
Sedimentation and multi-phase equilibria
in mixtures of platelets and ideal polymer
H. H. Wensink and H. N. W. Lekkerkerker(∗)
Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute
Utrecht University - Padualaan 8, 3584 CH Utrecht, The Netherlands
(received 19 November 2003; accepted in final form 26 January 2004)
PACS. 82.70.Dd – Colloids.
PACS. 64.70.Md – Transitions in liquid crystals.
Abstract. – The role of gravity in the phase behaviour of mixtures of hard colloidal plates
without and with non-adsorbing ideal polymer is explored theoretically.
(macroscopic) osmotic equilibrium conditions, we show that sedimentation of the colloidal
platelets is significant on a height range of even a centimeter. Gravity enables the system
to explore a large density range within the height of a test tube which may give rise to the
simultaneous presence of multiple phases.As to plate-polymer mixtures, it is shown that
sedimentation may lead to a four-phase equilibrium involving an isotropic gas and liquid phase,
nematic and columnar phase. The phenomenon has been observed experimentally in systems
of colloidal gibbsite platelets mixed with PDMS polymer.
By analyzing the
It is well known that adding non-adsorbing polymer to a colloidal dispersion induces an
attractive depletion potential of mean force between the colloidal particles [1–3]. For colloidal
spheres, the attractive potential has been shown to give rise to a phase separation in a colloid-
poor “gas” and colloid-rich “liquid” or “solid” phase at sufficiently high concentrations of the
colloid and the polymer [4–8]. Compared to colloidal spheres, the behaviour of dispersions
of rod- and plate-like colloids mixed with polymer is richer due to their possibility to form
liquid-crystal phases, i.e. nematic (N), smectic (Sm) and columnar (C). Recent experiments
on mixtures of colloidal gibbsite platelets and non-adsorbing polymer  have uncovered the
phase behavior of plate-polymer mixtures. A manifestation of the rich phase behaviour of
these mixtures is the observation of a four-phase equilibrium involving both isotropic gas and
liquid phases along with nematic and columnar states. The appearance of multiple phases
seems to conflict with the phase rule of Gibbs which states that the number of coexisting
phases is limited to three for an athermal binary mixture. One of the possible explanations
conjectured by the authors  is that the observation might be due to the polydispersity in
particle size. The presence of many components (i.e. platelets with different diameters and
thicknesses) in principle allows for a coexistence of arbitrarily many phases.
Another possibility to reconcile the experimental results with Gibbs’ phase rule is by
accounting for an external gravitational field. Sedimentation of particles leads to a density
(∗) E-mail: firstname.lastname@example.org
c ? EDP Sciences
H. H. Wensink et al.: Sedimentation in plate-polymer mixtures
with y ≡ φ/(1 − φ) and φ the plate volume fraction. The expression still depends on the
pressure Π(0)of the reference cut sphere system, for which no analytical expression is available
yet. Specific expressions αifor the different liquid-crystal states i can be obtained by inserting
the corresponding EOS˜Π(0)
from the simulation fits. The coefficients are given by
q?1 + 2l − l2?+ q2?2l +?π
q2?1 + 2l − l2?2
2− arcsinl??√1 − l2
?l − l3/3?
C = πq3/6
2?l − l3/3?2
with l = L/D the aspect ratio and q = 2Rg/D the size ratio of the ideal polymer coil and the
platelet. The volume fraction follows from φ = (π/4)ρD3(l − l3/3).
∗ ∗ ∗
We thank G. J. Vroege and M. Oversteegen for a critical reading of the manuscript.
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