Dottorato di Ricerca in Fisica
Università degli studi di Roma Sapienza, XXVI Ciclo
PhD Research Project
January 9, 2012
Colloids with anisotropic interactions
Zdenek Preisler
Supervisor: Prof. Francesco Sciortino
In past decades soft matter physics has become an important, well established research
field. In this thesis project, I plan to study the properties and collective behavior of complex
colloids. By the word complexity, I refer to a number of characteristics of the particles
involved, which distinguish them from the far more well suited case of spherical, hard, and
isotropic colloids, the latter being a convenient model system but still only a small subclass
of the large variety of possible colloidal particles. More specifically, I refer to e.g. shape
anisotropy, interaction anisotropy, softness and deformability or multipolar moments, etc.
However, such complexity is not only making these systems diverse and interesting. It is also
giving a rise to a number of an additional variables and parameters making any systematic
evaluation of properties more complex and thus more difficult and time consuming.
Nowadays, we observe significant advances in colloidal synthesis and other fabrication
techniques. These advances allow to generate a new big class of colloid particles having various but well defined shapes and interactions[4]. Interesting is e.g. an anisotropy of interaction
which can be added to the particles using a variety of physical or chemical patterning. Such
complex particles can exhibit, thought anisotropy of interactions and anisotropy of shapes,
a very different collective behavior according to their relative orientations than they would
perform otherwise. To provide an example we can consider a system of Janus colloidal particles, i.e. particles with different chemical properties on the two hemispheres. The disordered
phase diagram of Janus particles has been evaluated recently. A standard liquid-vapor phase
separation has been detected, but the vapor is composed of micelles having lower free energy
then corresponding liquid thus giving an anomalous re-entrant phase transition[6].
The above is only one of many possible examples of the self-assembly, spontaneous organization of matter into more complex arrangement. In general, particles can aggregate in
various structures such as micelles, vesicles, chains, ring and more complex structures. The
design of a set of colloidal molecules on nano or micro scale and then using their self-assembly
properties to generate new complex structures[4] is very promising. The self-assembly of
colloids into colloidal diamond crystal, which can be used in photonic applications, provides
an example of technologically relevant goals. Colloids are thus now being recognized as a
building blocks of new generation self-assembling materials with rich and still unexplored
supra-colloidal structures.
Examples of the self-assembly, bottom-up design, has been already reported. An elegant
experimental example is due by Granick[2]. In his system, patchy colloids – colloidal particles with two attractive spots on their surfaces self-assemble into a two-dimensional Kagome
lattice. Such behavior has also been recently reproduced by simulations[5].
In my thesis I study mainly colloids with anisotropic interactions by means of computer
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Figure 1: Patchy particles represented using Kern-Frenkel potential. Yellow color indicates
a hard sphere colloid and transparent red indicates interaction area. On the left is a Janus
particle and on the right is an example of a tri-block particle.
simulations. In particular, I focus on model patchy colloids, colloidal particles with localized
interaction sites or regions on their surface. I plan to perform an extensive systematic analysis
of their properties such as self assembly, influence of dimensionality (2D or 3D), complexity in
changing parameters, interference between ordered and disordered phases, etc. Our current
research interest are Janus and tri-block Janus colloidal particles. Janus particles are spherical
particle with one patch, while tri-block particles are two patch particles with patches located
on opposite sides of the sphere. They can be fabricated in various ways. An experimental
example of Janus particles can be a spherical silicon particles covered with gold cap from one
half or eventually gold particles covered with silicon from one half.
In order to predict the properties of the system, many different models to describe interactions can be selected as effective potentials. I plan to model patchy colloids by the
Kern-Frenkel potential[6]
u(rij ) = uSW (rij )f (Ωij ),
(1)
where uSW (rij ) is a square-well interaction potential and f (Ωij ) is a function depending on
orientations of two interacting particles.
(


1 if r̂ij · n̂i > cos θ and
f (Ωij ) =
(2)
r̂ji · n̂j > cos θ


0,
where r̂ij is a vector between particles i and j and n̂i,j is an orientation of the particle i, j.
The coverage angle cos θ = 0 corresponds to a half coverage. This model is able to reasonable
reproduce the physical behavior of experimental systems having hydrophobic or hydrophilic
interactions, which are very short range compared to the size of the colloids.
At the moment we are evaluating the phase diagram of Janus and tri–block colloid particles
by mean of computer simulation (see Fig 1).
Our goals are first to find all possible different crystals for one and two patch systems.
Then evaluate their phase diagram as a function of angle and interaction range.
The analysis of crystal structures are particularly interesting because adding anisotropic
interaction can lead to complex crystal structures having application relevance or even better
can allow us to predict how to design colloids which than could form complex crystal structures
having given properties. Our first task is to find all possible different crystals. To achieve that
we implemented a novel floppy box method[3] to perform a crystal search. It is a Monte Carlo
method based on the isotension-isothermal ensemble where we allow the lattice vector of the
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Figure 2: An example of crystal generated using floppy box method, a Janus particle crystal.
simulation box to vary. Only a small number of particles is simulated at constant pressures
and temperatures, with the idea of generating possible unit cells. However, many different
crystal using this method can be found at one state point. Output of these simulations
are thus filtered according to various criteria (e.g. energy, density or symmetry analysis) to
find distinct possible candidates for stable crystal structures (see Fig. 2). For the different
possible candidates the free energy is computed by thermodynamic integration using FrenkelLadd procedure (as a reference system is taken an Einstein crystal) to determine the stable
crystal.
To determine the phase behavior we use Grand Canonical and standard Canonical Monte
Carlo simulations and standard methodologies to evaluate phase coexistence and relative stability of crystal forms. In addition, the possible gas-liquid critical point can be determined
using Gibbs ensemble and coexistence lines between different phases can be computed using
Gibbs-Dunhem integration technique. Free energy calculation for crystals can be again performed using Frenkel-Ladd technique, and for fluids using thermodynamic integration with
ideal gas as a reference system. To determine the above quantities I have written, tested
and optimized my own Monte Carlo code to perform canonical, isobaric-isothermal, Grand
Canonical ensembles and floppy box method as well as free energy calculations for patchy
colloids. In addition I also already implemented various Monte Carlo moves to increase the
efficiency of numerical calculations e.g. aggregation volume bias Monte Carlo moves[1] or
cluster moves and other optimization procedures.
To conclude, I will investigate the phase diagram of colloids with anisotropy interaction,
including gas, liquid and crystal phases as a function of the patch number, patch angle and
interaction range.
References
[1] B. Chen and J. I. Siepmann. Improving the efficiency of the aggregation-volume-bias
monte carlo algorithm. J. Phys. Chem. B, 105:11275–11282, 2001.
[2] Q. Chen, S. C. Bae, and S. Granick. Directed self-assembly of a colloidal kagome lattice.
Nature, 469:381–384, 2011.
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[3] L. Filion, M. Marechal, B. van Oorschot, D. Pelt, F. Smallenburg, and M. Dijkstra.
Efficient method for predicting crystal structures at finite temperature: Variable box
shape simulations. Physical Review Letters, 103(188302), 2009.
[4] S. C. Glotzer and M. J. Solomon. Anisotropy of building blocks and their assembly into
complex structures. Nature Materials, 6:557–562, 2007.
[5] F. Romano and F. Sciortino. Two dimensional assembly of triblock janus particles into
crystal phases in the two bond per patch limit. Soft Matter, 7:5799–5804, 2011.
[6] F. Sciortino, A. Giacometti, and G. Pastore. A numerical study of one-patch colloidal
particles: from square-well to janus. Phys. Chem. Chem. Phys., 12:11869–11877, 2010.
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