ABSTRACTS
Marc Baus
(Université Libre de Bruxelles, Belgium)
"How Does Confinement Influence the Pressure Profile Across an Interface?"
We consider a liquid-vapour interface confined on one side by a planar
wall perpendicular to the interface. The non-uniform fluid states are
described within a simple van der Waals free-energy density
functional. The pressure profile across the interface is computed both far
away from and in the region close to the wall. The results give a feeling
for how the surface tension measured on a real wall could differ from its
"intrinsic" value defined on an idealized boundary.
"Phase Separation and Effective Interactions in Colloid - Polyelectrolyte Mixtures"
Mixtures of charged colloidal particles and flexible polyelectrolytes are
investigated using the recent theoretical approach based on the "Polymer
Reference Interaction Site Model" (PRISM). The linear polymer chains are
considered as collections of connected charged spherical monomers (sites).
The correlations between monomers and spherical colloids are solved using
the PRISM Ornstein-Zernike equation and simple liquid integral equations.
Mixtures of like-charged colloids and polymers exhibit a demixing of
depletion origin in the semi-dilute regime. The extension of the two-phase
region in the concentration-concentration phase diagram is very sensitive
to the details of the interactions between the different objects like the
colloidal size and charge, the linear charge density of the chains, the
salinity, etc. Random copolymers of charged and hydrophobic monomers are also
investigated and illustrate the subtle coupling between electrostatic and
Van der Waals effects. The results are interpreted in terms of
polymer-averaged effective interactions between colloids.
Johan Bergenholtz
(Göteborg University, Sweden) and
Matthias Fuchs (Technische Universität München, Germany)
"Gel Transitions in Colloidal Suspensions with Attractive Interactions"
The colloidal gel and glass transitions are investigated using the idealized mode coupling
theory (MCT) for model suspensions characterized by short-range attractive interactions.
Results are presented for hard-core attractive Yukawa systems, showing that MCT predicts
low temperature nonergodic states that extend to the critical and subcritical region.
Several features of the MCT nonergodicity transition in these systems agree qualitatively
with experimental observations on the colloidal gel transition, suggesting that the gel
transition is caused by a low temperature extension of the glass transition. The range of
the attraction is shown to govern the way the glass transition line traverses the phase
diagram relative to the critical point, analogous to findings for the fluid-solid freezing transition.
Peter Bolhuis
(University of Cambridge, UK)
"Transition Path Sampling of Capillary Evaporation Between Hydrophobic Particles"
Wim Briels
(University of Twente, The Netherlands)
"Density Functional Calculations Near Structured Walls"
Statistical mechanical density functionals are routinely being used to
calculate the density profiles of liquids near walls and in porous
materials. Usually walls are modelled as structureless mathematical walls,
like flat surfaces, or flat cylinders. The density profile in the liquid
then depends on just one coordinate, making the calculations feasible
without use of any special optimization techniques.
In many applications one is interested in the liquid structure very close
to the wall, as for example in the study of surface diffusion or crystal
growth, or in small or intermediately sized confinements. Obviously in
these cases walls cannot be approximated by flat walls, but their structure
must be taken into account. As soon as lateral structure is introduced
in the liquid-wall interaction, the calculation becomes three-dimensional,
leading to extremely long calculation times.
Density functional theory (DFT) calculations basically amount to varying the
density field until the
grand potential functional has reached a minimum. Whatever minimization
procedure one uses, it will require many function evaluations and/or
evaluations of gradients. In Tarazona's DFT method the grand potential is
written as a local functional of a coarse grained density field. Since the
latter is written as a convolution, profitable use can be made of Fast
Fourier transformation. In this lecture we will describe a method to
perform three-dimensional DFT calculations, which makes extensive use of
FFT methods and of translational symmetry along the corrugated surface. We
will present
results of Lennard-Jones fluids in front of a Lennard-Jones crystal, and in
cylinders composed of Lennard-Jones particles. The results will be
compared with those of corresponding MD simulations.
Jose A. Cuesta
(Universidad Carlos III de Madrid, Spain)
"Density Functional Approach to Depletion Interaction"
The most direct way to obtain depletion interaction in a
binary mixture is to fix the number of big particles and
the excess fugacity of the small ones and to integrate out, in
the partition function, the degrees of freedom of the small
particles. It can be shown that the resulting partition
function corresponds, up to some trivial factors, to that of a
one-component fluid interacting with an effective new
potential containing pair, triplet, etc., interactions.
This is a useful viewpoint if one is concerned, say, with
simulations of a binary mixture, especially in the limit
of large asymmetry (where only the pair potential is
significant). But
the knowledge of the effective potential is of little use
if one is to formulate a density functional description
of the fluid. In that case, if one has a good functional
for the binary mixture, the way to translate it into the
depletion picture is to Legendre-transform the functional,
fixing the chemical potential of the small particles.
Normally the density function of the latter cannot be eliminated
explicitly. Nevertheless this formalism is enough to obtain
the direct correlation function of the effective fluid, a
basic ingredient of the standard density functional theories.
One can go beyond if the limit of infinite asymmetry is taken.
We have made it for a binary mixture of parallel hard
cubes, and have explicitly obtained a density functional for the
effective fluid. It turns out that depletion introduces adhesiveness
between the large cubes. The phase behaviour of this fluid can
be derived and translated into the binary mixture
language. The picture shown in simulations of hard spheres
of fluid-solid or solid-solid demixing also emerges
from this formalism for the fluid of parallel hard cubes.
Alfred Delville
, R.J.M. Pellenq (CNRS-CRMD, France)
"Influence of Electrostatic Interactions on the Stability of Charged Colloids"
We use (N,V,T) Monte Carlo simulations in order to study the electrostatic
interactions between charged colloidal particles in the framework of the
primittive model. These results illustrate the influence of ionic
correlations on the cohesion of a large class of charged lamellar
materials (clays, cement, etc.) neutralized by divalent counterions. We
further investigate the influence of the finite size and shape (discs) on
the mutual interactions between colloids.
"Effective Interactions in Charge-Stabilized Colloidal Suspensions"
Effective interactions between charged hard-sphere macroions in
charge-stabilized colloidal suspensions are investigated using
response theory methods [1]. Modeling the response of the counterion
density to leading nonlinear order in the macroion charge density,
effective pair and three-body interaction potentials are calculated,
together with associated counterion (volume) free energies.
Implications for the thermodynamic phase behaviour
of highly-deionized (low-salt) suspensions will be discussed.
[1] M. J. Grimson and M. Silbert, Mol. Phys. 74, 397 (1991).
Marjolein Dijkstra
, R. van Roij, and R. Evans (University of Bristol, UK)
"Direct Simulation of the Phase Behaviour of Binary Hard-Sphere Mixtures:
Test of the Depletion Potential Description"
Understanding the stability of colloidal mixtures is relevant for many
industrial applications, but is also interesting from a fundamental
statistical physics point of view. Surprisingly, the phase behavior of
even the simplest model colloid mixture, i.e., large and small hard
spheres, is still not established and remains a topic of much debate.
For instance, it is still unclear whether a (stable) fluid-fluid
demixing transition exists for any additive binary hard-sphere mixture.
By integrating out the degrees of freedom of the small spheres in
a binary mixture of large and small hard spheres, we derive an effective
one-component Hamiltonian for the large spheres. Using this effective
Hamiltonian based on pairwise additive depletion potentials in
simulations, we do find a fluid-fluid spinodal instability for binary
hard-sphere mixtures with a size ratio 0.033<q<0.2, but this demixing
transition is metastable with respect to an extremely broad fluid-solid
transition. More surprisingly, we find a stable isostructural
solid-solid transition for q<0.05. We also studied the phase behaviour
by direct simulation of the true binary mixture and we find remarkably
good agreement with those obtained from the effective one-component
Hamiltonian, even at less asymmetric hard-sphere mixtures and even
at high packing fractions, e.g., in the solid phase. These results provide
the first justification for the effective Hamiltonian approach, based on
pairwise additive depletion potentials, for determining the phase equilibria.
Alice P. Gast and
Dean C. Wang (Dept. of Chemical Engineering, Stanford University, USA)
"Soft Sphere Freezing : A Density Functional Model of Freezing in Complex Fluids"
The physical and chemical processes governing the behavior of complex
fluids result from a balance of molecular forces. In this delicate
balance, repulsive interactions often prevail and ordering occurs via a
Kirkwood-Alder transition. Suspensions of monodisperse, charge-stabilized
colloidal particles are useful model systems to study this ordering. We
have found analogous ordering in nonaqueous suspensions of polymeric
micelles. Some of these systems are amenable to modeling with soft
repulsive potential energies such as the Yukawa and power-law potentials.
We will discuss recent results from a density functional model of
ordering. One such theory, the modified weighted density approximation
(MWDA) of Denton and Ashcroft, can successfully model the crystallization
of particles interacting via short-range, repulsive interactions. We have
made an extension of this model by adding a solid reference state, the
modified weighted density approximation with a solid reference state
(MWDA-SRS), to explain and predict crystallization properties for
longer-ranged systems and at high density. We apply the MWDA-SRS model to
the power-law and Yukawa potentials, as models for charge-stabilized
colloidal systems, and polymeric micelles.
We focus our attention on the intriguing question of the stability of the
body-centered cubic solid for long-range repulsive systems. We also look
at features of the ordering process such as the Lindemann parameter at
melting, the Hansen-Verlet rule for freezing and the miscibility gap
between coexisting fluid and solid densities at the phase transition.
"Effective Interactions Between Confined Charged Colloidal Particles"
S. Leroch, Gerhard Kahl
(Technische Universität Wien, Austria) and F. Lado (North Carolina State University, USA)
"Structure and Thermodynamics of a Polydisperse Liquid Mixture"
Willem K. Kegel
(Universiteit Utrecht, The Netherlands)
"Geometric Character of the Hard Sphere Freezing Transition"
A considerable fraction of the elements, when compressed,
undergo a first order phase transition from a disordered state (a
gas, fluid, or liquid) to a face-centred cubic (fcc) or hexagonal
close packed (hcp) crystal. Computer simulations, as well as
experiments show that this freezing transition is already apparent
in the simplest model of interacting atoms: a collection of hard
spheres subject to thermal agitation, implying that crystallisation
only requires excluded volume interactions. Despite this
persuasive evidence, to date, there is no theory from which this
transition follows ‘naturally’, that is, without inserting properties
of the coexisting phases in advance. We apply a rigorous
formalism, using as input purely geometrical properties of small
systems, to obtain the grand distribution function. The freezing
transition appears as two peaks of this function, at a certain
chemical potential, without any a priori information of the
coexisting phases, and is already apparent in systems containing
a number of spheres as small as 8. It is shown that, by this
symmetry breaking transition, the system avoids configurations
that can best be characterised as ‘defective solids’. The
influence of geometrical constraints on the nature of the hard
sphere freezing transition is investigated. It is found that hard
spheres tend to split into a dense and a dilute phase in all
geometries studied, but the transition is significantly less sharp in
a box with cubic symmetry.
"Structure and Thermodynamics of Star Polymers"
Star polymers are macromolecular aggregates consisting of f linear
polymers, one end of which is attached to a common center. Stars offer,
due to their peculiar architecture, a natural "bridge" between colloidal
and polymer science. Based on (a) results from scaling theory,
(b) explicit simulation calculations on the effective force between
two stars and (c) direct comparison with SANS-data, we propose a new
effective potential between two stars in a good solvent. This interaction
is then the basis for theoretical calculations regarding the structure
in the fluid state and phase transformations from the fluid to the solid
state. New phenomena, e.g., an "anomalous structure factor" for the fluid
as well as an unusual topology of the phase diagram result from this interaction. Moreover, there exists a "critical arm number" f_c = 33
such that, for f <= f_c the system never crystallizes, at any density.
Gerhard Nägele
(Universität Konstanz, Germany)
"Viscoelasticity and Generalized Stokes-Einstein Relation in Colloidal Dispersions"
Using a recently developed scaled mode coupling theory (MCT), we
investigate the linear viscoelastic properties of colloidal model
suspensions and discuss possible relations between the (dynamic) shear
viscosity and various diffusion coefficients. Results are presented for
hard sphere suspensions and for particle dispersions with long-ranged
screened Coulomb interactions. The scaled MCT predicts that the shear
viscosity of hard sphere suspensions obeys nearly quantitative generalized
Stokes-Einstein relations (GSE) both with the long-time self-diffusion
coefficient and the long-time collective diffusion measured at the
principle peak of the static structure factor. In contrast, the MCT
predicts that the same GSE relations are violated in case of suspensions
of highly charged particles. We further investigate another empirical GSE
due to Mason and Weitz, which relates the frequency dependent elastic
storage and viscous loss moduli to the particle mean squared displacement.
According to our MCT results, the Mason and Weitz GSE is fullfilled fairly
well for concentrated hard sphere suspensions, but strong deviations are
found for charge-stabilized dispersions. However, remarkably good
agreement is observed between the frequency dependence of the
Laplace-transformed normalized viscoelastic relaxation function and the
Laplace-transformed normalized time-dependent self-diffusion coefficient
of strongly correlated particles.
Roland Roth,
B. Götzelmann, and S. Dietrich (University of Wuppertal, Germany);
M. Dijkstra and R. Evans (University of Bristol, UK)
"Depletion Potential in Hard Sphere Mixtures"
In several experimentally interesting systems, such as mixtures of colloids or
mixtures of colloids and polymers, the particle interaction is almost hard-core-like.
In these systems entropy plays an important role and gives rise
to so-called depletion forces.
We present a new versatile approach for calculating the depletion
potential in a hard sphere mixture within density functional theory. Our
approach is valid for any number of components and for arbitrary densities
of all components. This approach, when applied to binary mixtures in the
dilute limit, in which the density of one component goes to zero, is in
excellent agreement with simulations. The flexibility of our approach
allows us to make contact to some recent experiments.
Daniel Rudhardt,
C. Bechinger, P. Leiderer
(Universität Konstanz, Germany)
"Direct Measurements of Entropic Forces in Colloid/Polymer Mixtures"
The mutual interaction of colloidal particles is well known to be strongly
influcenced by the presence of smaller uncharged particles, e.g. polymer
coils. This phenomenon, which is usually termed depletion interaction, is
very important for the phase behavior and flocculation of polydisperse
mixtures. We have studied the depletion forces between a single colloidal
sphere immersed in a solution of smaller, uncharged polymer coils close to
a flat glass surface by means of total internal reflection microscopy
(TIRM). This method which was originally invented by Prieve et al. [1]
allows the precise measurement of particle-wall interaction potentials and
has prooven to be a powerful tool for the investigation of depletion
forces. We present experiments where both the polymer concentration and
the ratio h=R/r between the radii of the colloids (R) and the polymer
coils (r) are changed. For small h we obtain entirely attractive depletion
forces which are in agreement with an ideal gas theory [2]. For large h,
however, in addition repulsive contributions are observed. We discuss the
role of van der Waals forces to account for this effect.
1. D.C. Prieve, F. Luo, and F. Lanni, Faraday Discuss. Chem. Soc. 83, 297 (1987).
2. D. Rudhardt, C. Bechinger, and P. Leiderer, Phys. Rev. Lett. 81, 1330 (1998).
Matthias Schmidt
(Universität Düsseldorf, Germany)
"Density-Functional Theory for Penetrable Spheres"
Density-functional theory is a powerful tool for the investigation of
liquid and solid properties. We use it to study a system of penetrable
colloids. Their effective interaction is modeled by a step function that
is zero outside the core and has some finite value once two particles
overlap. For this pair potential an ab-initio density functional is
presented. An overview of the derivation that only requires the
geometrical properties of the particles and the exactly known statistical
behaviour of the system under strong confinement is given. The functional
predicts the bulk fluid structure, in satisfactory agreement with liquid
integral theories and simulation, as well as the freezing transition into
an fcc crystal with multiply occupied lattice sites. We discuss various
limits: For both strong confinement and high temperature the theory
becomes exact. Rosenfeld's hard-sphere functional is recovered for zero
temperature.
An-Chang Shi
(Xerox Research Centre of Canada)
"Nature of Anisotropic Fluctuation Modes in Ordered Systems"
The nature of anisotropic fluctuation modes in an ordered structure
is analyzed using general symmetry arguments. It is shown that the
anisotropic fluctuation modes in a periodic phase can be classified using
a wave vector within the irreducible Brillouin zone and a band index. The
spatial profiles of the fluctuation modes are described by Bloch functions
which are plane waves modulated by periodic functions. These general
statements enable the study of the stability and kinetic pathway of
complex ordered structures. The utility of the theory is illustrated by
the Landau-Brazovskii theory of weak crystallization.
Moises Silbert
(University of East Anglia, UK)
"Sterically Stabilised Colloidal Dispersions: Beyond Hard Spheres"
Sterically stabilised colloidal dispersions, such as PMMA, have
long been regarded as archetypal realisations of hard spheres and
hard sphere mixtures. Indeed this is precisely what is found in
studies of these systems as a function of the volume fraction,
at constant temperature. However, as a function of both volume
fraction and temperature, a much richer phase behaviour is found
which goes beyond to what is expected from just hard spheres. For
instance, sterically stabilised colloids are known to exhibit both
upper (high-T) and lower (low-T) two phase behaviour, such that
the former is temperature sensitive but the latter is not.
We present model calculations, based on a double Yukawa potential,
which exhibits some of the rich phase behaviour found
experimentally in these systems. We specifically show that, if we
assume the exponential coefficient of the repulsive Yukawa to be
temperature dependent, the concave up and concave down phases
referred to above do appear in our phase diagram.
We also present preliminary results for the phase behaviour of binary
mixtures of model sterically stabilised colloids.
Mark Stevens
(Sandia National Laboratories, USA)
"Attraction Between Like-Charged Macroions"
Interactions between charged macroions can be quite complex. One of
the intriguing and controversial issues has been whether like-charged
macroions exhibit attractive interactions and whether the source of
this attraction is due to the Coulomb interactions. Even the basic issue
of determining the degree of counterion screening is a
computationally difficult task. I will present simulation results of
charged macroions with explicit, discrete counterions focused on
these issues. Recent algorithms have significantly reduced the cost
of Coulomb calculations. Using such an algorithm, attraction between
charged rods with divalent counterions are found and studied. Similar
simulations for charged spheres find no such attraction. To understand
better the nature of the macroion interactions, counterion screening,
determined in the simulations, will be compared to various theoretical
predictions.
René van Roij
(University of Bristol, UK)
"Gas-Liquid and Gas-Solid Coexistence in Low-Salt Suspensions of Mutually Repelling Charged Colloids"
Experimental observations of gas-liquid and gas-solid
coexistence in low-salt suspensions of charged colloidal
particles have led to the proposition that these charged
particles (with the same charge sign) must attract each
other at long distances. This proposition contrasts the
standard DLVO theory which, in accordance with recent
experimental measurements of the pair-wise interactions,
predicts screened-Coulomb repulsions. In my contribution
I will describe my attempts to resolve these apparently
contradicting results: a simple (but systematic) density
functional theory shows the existence of (i) pair-wise
screened-Coulomb repusions between the colloidal particles,
and (ii) a so-called volume term that can drive liquid-vapour
coexistence at low salt concentrations. These two findings
not only explain the experimental observations, but also
close an apparent gap between the Debye-Hückel theory
for simple electrolytes and the DLVO theory for charged
colloidal suspensions.
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Last updated June 18, 1999