Sedimentation and multi-phase equilibria in mixtures of platelets and ideal polymer
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. By analyzing the (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.
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ABSTRACT: Studies of temperature-dependent phase behaviors of discotic colloids are found infrequently in the literature. We establish here the isotropic-to-nematic (I-N) transition phase diagram of charged platelets in the temperature-versus-volume fraction plane. We discover that the N phase can be melted by increasing temperature, and that coexistent samples are more sensitive to polydispersity at higher temperature and higher concentration.Physical Review E 08/2014; 90(2-1):020504. · 2.33 Impact Factor
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ABSTRACT: Recent work on mineral reactions and microstructures in metamorphic rocks has focused on forward modelling of phase equilibria and on their description through chemical potential relationships which control mass transfer in rocks. The available thermodynamic databases and computer programs for phase equilibria modelling have significantly improved the quantification and understanding of geodynamic processes. Therefore, our current methodological framework seems to be satisfactory. However, the quantification approaches in petrology focus on chemical processes with oversimplified mechanics. A review of the recent literature shows that mechanical effects in rocks may result in the development of pressure variations even on a hand specimen or grain scale. Such variations are critical for interpreting microstructural and mineral composition observations in rocks. Mechanical effects may influence element transport and mineral assemblage in rocks. Considering the interplay of mechanical properties and metamorphic reactions is therefore crucial for a correct interpretation of microstructural observations in metamorphic rocks as well as for quantification of processes. In this contribution, arguments against pressure variations are inspected and disproved. The published quantification procedure for systems with grain-scale pressure variations is reviewed. We demonstrate the equivalence of using Gibbs and Helmholtz energy in an isobaric system and go on to suggest that Gibbs free energy is more convenient for systems with pressure variations. Furthermore, we outline the implications of the new quantification approach for phase equilibria modeling as well as diffusion modeling. The appropriate modification of a macroscopic flux for a system with a pressure variation is derived and a consequence of using mass or molar units in diffusional fluxes is discussed. The impact of ignoring grain-scale pressure variations on geodynamic modeling and our understanding of the processes in the Earth’s interior is assessed. We show that if a pressure variation is overlooked, the error in depth estimates from crustal metamorphic rocks could be as large as the thickness of the crust.Lithos 01/2015; 36. · 3.65 Impact Factor
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ABSTRACT: In this paper we give a general theoretical framework that describes the sedimentation of multicomponent mixtures of particles with sizes ranging from molecules to macroscopic bodies. Both equilibrium sedimentation profiles and the dynamic process of settling, or its converse, creaming, are modeled. Equilibrium profiles are found to be in perfect agreement with experiments. Our model reconciles two apparently contradicting points of view about buoyancy, thereby resolving a long-lived paradox about the correct choice of the buoyant density. On the one hand, the buoyancy force follows necessarily from the suspension density, as it relates to the hydrostatic pressure gradient. On the other hand, sedimentation profiles of colloidal suspensions can be calculated directly using the fluid density as apparent buoyant density in colloidal systems in sedimentation-diffusion equilibrium (SDE) as a result of balancing gravitational and thermodynamic forces. Surprisingly, this balance also holds in multicomponent mixtures. This analysis resolves the ongoing debate of the correct choice of buoyant density (fluid or suspension): both approaches can be used in their own domain. We present calculations of equilibrium sedimentation profiles and dynamic sedimentation that show the consequences of these insights. In bidisperse mixtures of colloids, particles with a lower mass density than the homogeneous suspension will first cream and then settle, whereas particles with a suspension-matched mass density form transient, bimodal particle distributions during sedimentation, which disappear when equilibrium is reached. In all these cases, the centers of the distributions of the particles with the lowest mass density of the two, regardless of their actual mass, will be located in equilibrium above the so-called isopycnic point, a natural consequence of their hard-sphere interactions. We include these interactions using the Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state. Finally, we demonstrate that our model is not limited to hard spheres, by extending it to charged spherical particles, and to dumbbells, trimers and short chains of connected beads.Journal of Physics Condensed Matter 01/2014; 26(7):075101. · 2.22 Impact Factor
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
gradient which facilitates the formation of multiple phases in a vessel of sufficient height. In
this paper we scrutinize the effect of sedimentation in systems of colloidal platelets with and
without added polymer from a simple osmotic compression treatment. We will first consider
a one-component system of colloidal platelets and then study the influence of the polymer-
induced depletion attraction using a mean-field free-volume theory .
Sedimentation equilibrium: one-component system. –
colloidal particles (platelets) in osmotic equilibrium with a dispersing solvent with a chemical
potential µ0subject to a gravitational field along the z-direction of the vessel. We assume
that the concentration profile of the colloids is sufficiently smooth so that the system is locally
in a homogeneous equilibrium state between z and z + dz. This is usually the case if the
particles are not too large and heavy and if the dispersion is not too close to a critical point.
The (macroscopic) condition for sedimentation equilibrium reads
Let us consider a vessel containing
in terms of the osmotic compressibility (∂ρ/∂Π)T,µ0of the dispersion and the buoyant mass
m∗of the colloidal particle (g is the gravitational acceleration). The concentration profile ρ(z)
of the colloids can be obtained from eq. (1) if the osmotic pressure as a function of ρ, i.e. the
equation of state (EOS), is known.
In the present study we will encounter phase-separated samples containing a number of
distinct phases. Since these phases are generally described by different equations of state,
it is convenient to treat each daughter phase i separately and assign a phase height Hi to
each of them. Recasting eq. (1) in dimensionless form by introducing the height parameter
ζ = z/H(i)(with 0 < ζ < 1) and dimensionless plate concentration ci= ρiD3(with D the
plate diameter) corresponding to phase i yields
with β = 1/kBT and˜Hi= Hi/ξ the height of phase i rendered dimensionless by relating it
to the gravitational length ξ = kBT/m∗g which is on the order of 10−3m for the colloidal
dispersions of gibbsite platelets we consider in this paper. The average concentration c0,iin
phase i follows from
where ct,i and cb,i denote the concentrations at the top and the bottom of the phase, re-
spectively. The average concentration c0 of the sample then follows from a simple linear
a given average sample concentration c0. In order to solve eq. (2) for colloidal platelets, we
must know the EOS Πi(ci) for the different liquid-crystal states (viz. isotropic (I), nematic
(N) and columnar (C)) encountered upon densifying these systems. As a quantitative input
we use fits to the EOS obtained from Monte Carlo simulations of hard platelets performed
by Zhang et al. . A polynomial of the K-th order was used as a fitting function so that
the concentration profile from eq. (2) in the different phases.
ic0,i˜Hi/˜H with˜H =?
i˜Hithe dimensionless sample height.
Note that in an experimental situation these concentrations are to be determined from
i, with i = I, N, C. The coefficients an,ipertaining to state i can be
found in ref. . The polynomial form of the EOS allows for a simple analytic solution of
H. H. Wensink et al.: Sedimentation in plate-polymer mixtures
Fig. 1 – a) Phase diagram for colloidal platelets with L/D = 0.05 in a gravitational field. Plotted is
the reduced sample height˜ H = H/ξ vs. the overall plate volume fraction φ0. The three-phase region
opens up at H = 11.15ξ. b) Concentration profile of a sample with overall volume fraction φ0 = 0.157
and vessel height H = 30ξ (corresponding to the open dot in a). Plotted is the relative height z/H
vs. φ. The I-N and N-C phase boundaries are indicated by the horizontal dotted lines.
The effect of gravity on the phase behaviour of the colloidal platelets is presented in fig. 1.
The curves represent so-called cloud curves which indicate the minimum sample height (and
associated overall volume fraction) needed to induce the formation of an infinitesimal amount
of a new phase at the top and/or the bottom of the vessel due to sedimentation of the particles.
On the horizontal axis we find the coexistence densities for the I-N and N-C transitions at zero
gravity, which would correspond to a vessel with potentially zero height. Figure 1a) shows
that a vessel height of about 10 gravitational lengths already leads to significant changes in the
phase diagram. A large three-phase isotropic-nematic-columnar region is encountered which
opens up at the state point indicated by the black dot. At the associated volume fraction
(φ0= 0.291) the system is fully nematic at short sample heights but as soon as the height
exceeds 11.15 gravitational lengths, two additional fractions of an isotropic and columnar
phase are split off simultaneously at the top and bottom of the sample, respectively. To
compare with actual sample heights we use the following expression for the gravitation length,
ξ = kBT/(gvplateρ∗
the colloidal gibbsite platelets dispersed in toluene (plate dimensions D = 180nm, L = 12nm
and buoyant density ρ∗
phase isotropic-nematic-columnar equilibria in fig. 1 may be expected in samples larger than
a centimeter, which is comparable to the typical height of a test tube. Figure 1b) shows an
example of a concentration profile one may encounter experimentally in a sample with overall
plate volume fraction of 15.7% and height of 2.7cm. The scenario is that the system initially
phase-separates into equal portions of an isotropic and nematic phase. At a later stage, a
columnar fraction will be formed at the bottom of the vessel due to slow sedimentation of the
platelets. At sedimentation equilibrium, the I, N and C phases, respectively, occupy 58, 39
and 3% of the system volume.
plate) with vplate=π
4LD2the colloid volume. Using experimental data for
plate= 1.5 103kg/m3) we obtain ξ = 0.9mm. This means that the three-
Plate-polymer mixtures. –
mixed with non-adsorbing ideal polymers (denoted by “p”) in a solvent. The gravitational
length of the polymer is much larger than that of the colloidal particles (ξp? ξ1) due to its
negligible buoyant mass. We may therefore assume that there is no external force acting on
the polymer coils and that the chemical potential of the polymer can be considered constant
throughout the system.The mixture can thus be treated as an effective one-component
We now turn to systems of colloidal platelets (component “1”)
Fig. 2 – Phase diagram of a plate-polymer mixture with L/D = 0.05 and q = 0.355 in the fugacity-
volume fraction plane, reproduced from ref. . On the vertical axis, the region of stable isotropic
gas-liquid (I1-I2) equilibria is confined between a lower critical point at zpD3= 19.233 (dotted line)
and the I1-I2-N triple line at zpD3= 24.454 (lower dashed line). The upper dashed line at zpD3= 104
represents the I1-N-C triple line.
system of colloidal platelets in a gravitational field and the osmotic pressure balance now
reads, analogously to eq. (1),
at constant µp. Similar to eq. (2), we can rewrite this equilibrium condition in dimensionless
form. Substituting the EOS for a colloid-polymer mixture from a free-volume treatment of the
Asakura-Osawa model (see the appendix) yields the following differential equation describing
the colloid density profile c1,i(ζ) in the daughter phase i:
which must be solved along with the auxiliary condition for the overall concentration, eq. (3).
Comparing with eq. (2), we see that the terms between square brackets now represent an ef-
fective (inverse) osmotic compressibility. The first contribution is the inverse compressibility
of the one-component plate system, whereas the second term accounts for the effective deple-
tion attraction between the platelets due to the presence of the polymer. The strength of the
depletion attraction can be varied by changing the fugacity zpof the polymer, related to the
chemical potential via zp= exp[βµp]/V . Note that the result for a one-component system
(eq. (2)) is recovered for zp= 0, as it should. The effective compressibility also depends on the
fraction of free volume αiavailable to the polymer in the liquid-crystal state i. Explicit expres-
sions for αiare given in the appendix. It is easily verified that d2αi/dc2
for all states i = I, N, C, implying that the effective osmotic compressibility is larger than
that of a pure system of plates due to the attractive depletion forces, as we intuitively expect.
In fig. 2 we have depicted a phase diagram for the zero-gravity case reproduced from
ref. . The values for the plate aspect-ratio and the polymer-to-plate size ratio q = 2Rg/D
(with Rgthe polymer radius of gyration) are chosen such as to match the experimental values
for the gibbsite-PDMS mixtures studied by Van der Kooij et al. . The volume fractions in the
1,iis generally positive
H. H. Wensink et al.: Sedimentation in plate-polymer mixtures
Fig. 3 – a) Phase diagram of the same mixture as in fig. 2 in a gravitational field at (constant)
fugacity zpD3= 20. Plotted is the relative sample height˜ H = H/ξ vs. the overall plate volume
fraction. The four-phase region opens up at H = 11.70ξ (black dot). b) Same diagram in a reservoir
fugacity-volume fraction representation at fixed vessel height H = 1.5cm (H/ξ = 16.67).
coexisting phases can be deduced from tie lines given by horizontal lines in this representation.
At low reservoir fugacity the phase behaviour of the mixture differs only marginally from that
of the pure system. At zpD3> 19.233, the isotropic phase becomes unstable with respect to
a demixing into an isotropic gas phase (I1) and a liquid phase (I2). The gas phase is poor in
colloid but rich in polymer, vice versa for the liquid phase. The nematic-columnar transition is
unaffected by the presence of the polymer up to the I1-N-C triple line located at zpD3= 104.
At higher fugacities the depletion attraction is strong enough to induce a transition from an
isotropic gas (I1) to a columnar solid (C) phase, without the intervention of a nematic phase.
In fig. 3a) we have depicted a phase representation, analogous to fig. 1, of the same mixture
in a gravitational field at fixed reservoir fugacity zpD3= 20. Also here, we see that sedimen-
tation leads to remarkably rich phase behaviour; several multi-phase equilibria appear that
are not present in the zero-gravity case in fig. 2. Most notably, a four-phase region opens up
at H/ξ = 11.70 which, recalling that ξ = 0.9mm, is again about a centimeter. An equilibrium
involving isotropic gas, liquid, nematic and columnar phases has been observed in the gibbsite-
PDMS mixtures . We stress that the experimental observation of four distinct phases in a
tube of a few centimeters is related in a fortuitous way to the platelets’ size (and hence their
gravitational length). If the platelets had been much larger, they would rather have formed a
dense, quasi-uniform sediment at the bottom of the tube. If they were much smaller, gravity
may not have been strong enough to enforce a four-phase sedimentation equilibrium.
From an experimental standpoint it is more appropriate to fix the total sample height
rather than the reservoir fugacity. In fig. 3b) we show a representation in terms of the fugacity
vs. the overall volume fraction at fixed sample height H = 1.5cm, which is the typical length
of the test tubes used in experiment . Unlike fig. 2, this representation does not provide
information about the composition of the phases present, it merely indicates which phases can
be expected in a sample with a fixed height and a given overall density and reservoir fugacity.
Comparing with fig. 2, we see that the four-phase region in fig. 3b) must be confined within
the range 19.233 < zpD3< 24.454 since only there both stable I1-I2 and N-C two-phase
equilibria occur at low and high densities, respectively.
We can therefore conclude that gravity enables the colloidal platelets to scan a large density
range within the range of a few centimeters. Since mixtures of plates and ideal polymer display
a number of phase transitions within a relatively small range of concentrations, sedimentation