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ISSN 1463-9076
Physical Chemistry Chemical Physics
www.rsc.org/pccp Volume 13 | Number 14 | 14 April 2011 | Pages 6373–6712
COVER ARTICLEBianchi et al.Patchy colloids: state of the art and perspectives
HOT ARTICLEGoerigk and GrimmeA thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions
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Cite this: Phys. Chem. Chem. Phys., 2011, 13, 6397–6410
Patchy colloids: state of the art and perspectives
Emanuela Bianchi,*ab
Ronald Blaakcb
and Christos N. Likoscb
Received 27th October 2010, Accepted 20th December 2010
DOI: 10.1039/c0cp02296a
Recently, an increasing experimental effort has been devoted to the synthesis of complex colloidal
particles with chemically or physically patterned surfaces and possible specific shapes that are far
from spherical. These new colloidal particles with anisotropic interactions are commonly named
patchy particles. In this Perspective article, we focus on patchy systems characterized by spherical
neutral particles with patchy surfaces. We summarize most of the patchy particle models that
have been developed so far and describe how their basic features are connected to the physical
systems they are meant to investigate. Patchy models consider particles as hard or soft spheres
carrying a finite and small number of attractive sites arranged in precise geometries on the
particle’s surface. The anisotropy of the interaction and the limited valence in bonding are the
salient features determining the collective behavior of such systems. By tuning the number, the
interaction parameters and the local arrangements of the patches, it is possible to investigate a
wide range of physical phenomena, from different self-assembly processes of proteins, polymers
and patchy colloids to the dynamical arrest of gel-like structures. We also draw attention to
charged patchy systems: colloidal patchy particles as well as proteins are likely charged, hence the
description of the presence of heterogeneously distributed charges on the particle surface is a
promising perspective for future investigations.
1. Introduction
Theories and simulations of liquids in which the constituent
particles interact by means of spherically symmetric (i.e.,
isotropic) interactions have experienced tremendous progress
in the last few decades. Together with advances in the corres-
ponding experimental techniques, these approaches have led
a Institut fur Theoretische Physik, Technische Universitat Wien,Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria.E-mail: [email protected]; Fax: +43 1 58801-13699;Tel: +43 1 58801-13631
b Institute of Theoretical Physics, Heinrich Heine University ofDusseldorf, Universitatsstraße 1, D-40225 Dusseldorf, Germany
c Faculty of Physics, University of Vienna, Boltzmanngasse 5,A-1090 Vienna, Austria
Emanuela Bianchi
Emanuela Bianchi received herDiploma (2005) and her PhD(2008) degrees in Physics fromthe University of Rome LaSapienza. Subsequently, shewas at the Technical Univer-sity in Vienna with an ErwinSchrodinger Fellowship inMathematics and MathematicalPhysics. Later on, shemoved to the Heinrich HeineUniversity in Dusseldorf asan Alexander von Humboldtpostdoctoral fellow. Currently,she is back at the TechnicalUniversity in Vienna as a post-
doctoral Lise Meitner fellow. Her work has been mainly focusedon the equilibrium properties of patchy colloidal systems, withparticular attention to collective behavior, such as gelation,phase separation, and cluster and crystal formation phenomena.
Ronald Blaak
Ronald Blaak studied Theo-retical Physics at the Univer-sity of Utrecht, where hereceived his Masters degreein 1992 and obtained his PhDin 1997. After a PostDoc atthe University of Amsterdam,he became a Marie-CurieFellow at the UniversityCarlos III in Madrid. From2002 to 2010 he was affiliatedwith the Department of Physicsof the University of Dusseldorfwith an intermediate stay at theDepartment of Chemistry of theUniversity of Cambridge. As of
January 2011, he works as a Scientific Assistant at the Faculty ofPhysics, University of Vienna. His research has been focused on thesimulation and theory of soft-matter systems, like phase behavior ofliquid crystals, Lattice-Boltzmann, nucleation, dipolar fluids,dendrimers, and polyelectrolytes.
PCCP Dynamic Article Links
www.rsc.org/pccp PERSPECTIVE
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6398 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 This journal is c the Owner Societies 2011
to a deeper understanding of the physical mechanisms governing
the structural characteristics, the thermodynamic behavior and
the dynamical properties of classical fluids. These advances,
however, should not obscure the fact that isotropy in the
interactions is an idealization and should be treated either as
an approximation or as a zeroth-order point of departure for the
analysis of more complicated and realistic situations. Indeed,
even on molecular length scales, one readily encounters aniso-
tropy for the simplest of molecules, a fact that has given rise to
the development of the theory of molecular liquids1 and the
correspondingly complex theories of anisotropic fluids.2,3
Moving up in the length scale of the associated particles,
into the mesoscopic domain of soft matter physics, we are
called to deal with effective interactions,4 the microscopic
solvent being usually treated as a continuum. Here, anisotropy
in the interactions naturally arises as a result of an underlying
anisotropy in shape (colloidal ellipsoids, rods, platelets etc.).
Alternatively, one might be dealing with spherical particles
carrying embedded or induced dipoles,5,6 a vast field with
applications in the domain of magneto- and electro-rheological
fluids.7,8 Anisotropy moreover naturally arises in a different
class of soft colloids, which do not have a hard core but rather
consist of polymers with various architectures. Though, for
instance, star polymers with a high number of arms are very-
well approximated as spherical objects9,10 and their fluctuations
around sphericity are weak, polymer chains display instantaneous
shapes that are quite anisotropic, as first shown by Solc and
Stockmayer about 40 years ago.11 In dilute solutions, a
description of polymer chains by means of an angularly-
averaged, effective Gaussian potential is realistic12 and can
be adopted without loss of physical information; however, in
the semidilute regime, a breakdown of the chains into smaller
blobs is necessary if an accurate physical description is
required,13 an approach which re-introduces anisotropy into
the theoretical description.
Anisotropy and the multiblob approach are present from
the outset if one wishes to coarse-grain more complex
macromolecules, such as diblock copolymers with a solvophobic
and a solvophilic group.14,15 When the blocks are organized in
a star architecture with their solvophilic parts grafted on a
common center, telechelic star polymers result.16,17 These soft
colloids with sticky ends are a characteristic example of patchy
particles and the propensity of the patches to associate with
one another gives rise to a number of particular intra-star16,18
and inter-star19,20 association phenomena. Other kinds of soft,
patchy nanoaggregates may also originate from the self
assembly of amphiphiles. The latter self-organize into micelles
and vesicles when dispersed in the specific solvent. Such
aggregates may possibly have patchy surfaces, i.e., the whole
surface of the complex aggregate may show extended regions
where the solvophobic or solvophilic blocks prevail. It has
been shown, for instance, that binary mixtures of two different
amphiphiles in aqueous solutions can give rise to two-components
vesicles21 or to soft patchy micelles.22 The surface patterns
and the specific shapes of such aggregates can be tuned by
changing the composition of the mixture and via the choice of
the hydrophobic–hydrophilic components. A large variety of
systems arises, from spherical aggregates to cylindrical ones,
which have extended patches on the external surface or even
on the internal one.22 Self-assembled patchy complexes of the
above type feature incessant fluctuations in the shape of the
final aggregates.
In this paper, we focus on systems in which there are no
shape fluctuations of the patchy units. More specifically, we
mainly consider spherical units in which the anisotropy of the
interaction comes only from specific patterns on the particle
surface (see Fig. 1). For such systems, we consider most of the
pair potential models that have been developed to date and we
describe how they have been used to study the collective
behavior of patchy systems from a numerical and, possibly,
analytical point of view. Since an extensive, recent review on
the experimental side of fabrication and assembly of patchy
particles is readily available,23 here we make the choice
to focus on the theoretical and numerical achievements.
Fig. 1 Graphical representation of the type of patchy colloid which
we focus on: the spherical particle is decorated on its surface by
extended patches. Patches are usually discrete and limited in number
and the interaction between them, selective or not, gives rise to the
formation of strongly directional bonds between otherwise repulsive
particles. Here we show the specific pattern of four identical patches
arranged on a tetrahedral geometry (the forth patch is hidden behind),
whose importance is related to the possibility of using colloidal
structures with diamond symmetry to create photonic crystals.24
Christos N. Likos
Christos Likos graduated witha Diploma of ElectricalEngineering from the NationalTechnical University of Athensin 1988 and obtained his PhDin Physics from Cornell in1993, working with NeilAshcroft and Chris Henley.After research stays in Munich(Humboldt Fellow), Trieste(Marie Curie Fellow), Julich(Research Fellow), andCambridge (HeinsenbergFellow), he was Professor ofPhysics at the University ofDusseldorf from 2003 to
2010. As of June 2010, he is Professor at the Faculty of Physicsof the University of Vienna. He has held Visiting Professorshipsand Scholarships at Rome, Princeton, Upenn, and at the ErwinSchrodinger Institute in Vienna (2007-2008). He is Coordinatorof the EU-Network ITN-COMPLOIDS, www.itn-comploids.eu.
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Nonetheless, in section 2 experimental patchy systems are
briefly reviewed. In section 3, we describe why, when, and
how patchy models have revealed their importance, either as
simplified descriptions of molecular fluids and globular
proteins or as tunable self assembling systems. In sections 4
and 5, we describe the specific models, the possible theoretical
approaches and the physical phenomena that have been
investigated up to now. We distinguish between patchy models
with spot-like patches on hard particles (section 4), either
spherical or not, or extended patches (section 5), either on
hard or soft repulsive spherical cores. In all these systems,
patchiness is characterized by a correspondence between the
patch number/geometrical arrangement and the bonding
pattern. Such a correspondence does not hold when considering
heterogeneously charged particles. Consideration of particles
with charges non-spherically distributed on their surface is a
promising step forward to understand the behavior of both
colloidal patchy systems and protein solutions. Thus, in
section 6 we report the few patchy models recently developed
for heterogeneously charged systems. Finally, in section 7 we
summarize and draw our conclusions.
2. Experimental patchy colloids
Colloidal particles with chemically or physically patterned
surfaces are generally referred to as patchy colloids. The
presence of discrete patches, usually limited in number,
induces strongly anisotropic, highly directional interactions
between particles. Low valence, directionality, and selectivity
in bonding are the common features of such systems.
Directional, site-specific bonds between colloids can be
induced by functionalizing the particle surface with lock and
key groups, such as DNA oligonucleotides, protein-base
biological cross-linkers, and biotin–avidin or antibody–antigen
binding pairs.25–30 Nonetheless, in this case, positioning the
recognition sites in precise arrangements and numbers on the
particle surfaces is a very challenging task.
The first step towards the realization of patchy colloids has
been the production of non-spherical colloidal clusters with
tunable dimensions, shape and number of constituents.31–33
Such colloidal clusters interact anisotropically because of their
anisotropic shape but they are still not patchy since they
consist of all particles of the same type. The step further has
been the fabrication of colloidal clusters made of different
materials,34–36 such as bidisperse clusters made of particles
with different size ratio (nano- and micro-spheres).34,37
Anisotropic colloidal clusters made of different materials
may be further selectively modified to induce directional
specific bonding. An additional intriguing aspect consists in
the possibility of controlling the angles between the different
cluster components,36 i.e., by changing, the bonding angles
between the patches.
Moreover, spherical patchy colloids with distinguishable
regions on their surface may be realized by decorating uniform
colloidal spheres with metallic patches. For instance, micro-
spheres with well-defined interaction spots have been realized
by patterning the particle surface with gold nanodots.38–40 A
variable number of gold patches have been successfully placed
in highly symmetric arrangements: two to five gold spots
decorate the particle surface in respectively a linear, triangular,
tetrahedral or right-pyramidal geometry.38–40 Recently, colloidal
patchy spheres with multifunctional metallic patches have been
synthesized by creating extended patches of metals with different
chemical properties,41 such as gold and silver. The size and the
relative orientation of such gold/silver patches have been shown
to be tunable to a good extent, while up to now the maximum
number of patches on each sphere is two.41,42
Finally, a chemical layer-by-layer surface modification on
previously masked-off microspheres has been shown to
produce colloids with tunable surface charge patterns made
of absorbed nanoparticles on the microsphere surface.43–45 Up
to now, such partial coating approaches result in the
realization of spherical particles with one patch, whose size
can be effectively controlled.43–45
One-patch colloids whose patch covers an entire hemisphere
are known as Janus particles. The definition of Janus particles
is quite broad as it may refer, for instance, to hydrophobic–
hydrophilic interactions as well as to charged–uncharged,
metallic–polymeric or organic–inorganic functionalizations.
In the last 20 years, the fabrication of Janus particles and
the fine control of the differences between the properties of the
two hemispheres have undergone impressive steps forward,46–49
but few of the developed techniques can be extended to
produce other geometries of patchy colloids.23
Aside of this class of synthetic patchy particles, there are
also some natural, biological macromolecules, for example,
globular proteins, whose anisotropic interaction may be
described by the means of effectively attractive patches on
non- or weakly interacting particles.50–55
3. Once upon a time
The first patchy models were introduced in the 1980s, with the
goal of numerically studying in a simple and general way a
wide class of classical many-body systems, namely the associating
fluids.56 The phenomenon of association (mainly related to
hydrogen bonding) is an effect of orientationally dependent
attractive interactions between molecules. The basic qualitative
features of associating fluids are the presence of strong,
directional bonds in the dense phase and, as a consequence,
coordination numbers much smaller than 12, which is the
average number of nearest neighbors around a molecule in
simple fluids. The primitive models developed for associating
fluids56–59 describe the associating species as repulsive spherical
particles carrying a small number of attractive sites, on which
the association occurs. The total pair potential is then a sum of
the repulsive spherical component plus the attractive aniso-
tropic one. The choice of a hard core repulsive interaction and
of an attractive site–site interaction of a square-well type has
been favored by the parallel development of an analytical
mean field theory able to treat the thermodynamic properties
of such systems. This approach is a thermodynamic perturbation
theory and it was independently developed by Wertheim60,61
to describe fluids with spherically repulsive cores and highly
directional attractive forces.
The Wertheim theory is based on the activity expansion of
the grand partition function, which can be represented by a
sum of topologically distinct diagrams.62 The decoupling
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between attraction and repulsion in the pair interaction
potential allows one to simplify such an expansion, as the Mayer
function for the full pair potential can be decomposed into the
Mayer function for the hard-sphere potential and the Mayer
function for the site–site attraction. The Wertheim approach is
able to further simplify the complex graphical expansion via
the assumption that the conditions of steric incompatibilities
are satisfied: (i) no site can be engaged in more than one bond
(single bonding condition); and (ii) particles do not form ring-
like structures (absence of bonding loops). Thus, only those
graphs involving the formation of tree-like bonding structures
are considered. The resummed cluster expansion of the grand
partition function leads to an expression for the free energy in
terms of the densities of particles in different bonding
states.60,61 In primitive models for associating fluids, the steric
incompatibilities are satisfied due to both the choice of the
potential parameters and to the location of the attractive sites
on/inside the particle. However, it is worth noting that the
theory provides an expression for the free energy of particles
with a number f of attractive sites, independently of the
specific geometric arrangement of the same.60,61 The theory
only assumes that sites are equally reactive and that the
correlation between adjacent sites is missing. Nonetheless, an
analytical theory inspired by Wertheim has been derived to
describe associative fluids in which a correlation between sites
is present.63 In that case, bonding is said to be cooperative, i.e.,
bonds have a strength related to the size of the cluster they
belong to (monomer, dimer, trimer, etc.). The intermolecular
potential is not any more pairwise additive but has both two-
and three-body terms and thus the analytical approach to
describe the system at liquid-like densities has been developed
ad hoc from the associating ideal gas limit.63
The mean field character of the Wertheim theory can be easily
discerned from its reformulation by Chapman and Jackson,64
where the difference in free energy between the associating fluid
and the hard sphere reference fluid is given in terms of the
fraction of non bonded particles. The theory was also extended
to binary mixtures of particles carrying different numbers of
attractive sites. In this case, the additional parameter which is
taken into account is the molar fraction of the two species.65
Recently, a new formulation of the Wertheim theory in terms of
the probability of forming a bond has shown that the free energy
due to bonding can be derived in more than one way, assuming
that the system of associating particles is formally equivalent
to a system of noninteracting clusters in thermodynamic
equilibrium.66 An extension of the Wertheim thermodynamic
perturbation theory has been developed to interpret and/or
predict the behavior of a wide range of substances with potential
industrial applications, as it is possible to treat fluids with strong
hydrogen-bonding associations. The statistical associating fluid
theory (SAFT) has nowadays given rise to a host of branched
developments, which describe for instance chain molecules of
hard-core segments with attractive potentials of variable range,67
or even the phase behavior of electrolyte solutions.68
3.1 Primitive models for water and silica
Primitive models based on capturing the physics of strong
association and low coordination have been developed to
describe diverse classes of network-forming liquids. Such
models adapt the idea of hard core particles decorated with
a fixed number of attractive sites to match the specific features
of the system under investigation, e.g., water or silica.
The first anisotropic patchy potential was indeed introduced
by Nezbeda to describe the structure and the thermodynamic
properties of water.57 In an attempt to mimic the oxygen atom,
the model depicts a particle as a hard sphere with four short-
ranged interaction sites arranged in a tetrahedral geometry.
Sites are of two different types: hydrogen-sites, located on the
hard sphere surface, and ion-pair sites, located under the
particle surface. The hydrogen bonding is modeled by a square
well interaction between hydrogen and lone-pair sites and no
interaction between the like sites occurs. The model has been
proved to reproduce even the anomalies of water58,69 and it
has also been used to describe the solid–fluid equilibrium
behavior as well as the gas–liquid phase separation process.
The phase diagram of such a model shows the presence of a
high-density crystal and a low density solid with tetrahedral
coordination.70 The short range nature of the directional
attraction acts, as in simple fluids, towards the destabilization
of the liquid phase,71 while the low valence in bonding favors
the formation of open networks of bonds which possibly lead
to kinetically arrested states.69
A similar primitive model was introduced to describe
another network forming material: silica.59 The model is a
non-additive hard sphere mixture of particles carrying two or
four sticky sites at a fixed distance from the particle center,
mimicking respectively the silicon and the oxygen atoms. The
only attractive interaction takes place between Si and O sites
and it is modeled as a square well potential, while the inter-
action details are chosen to reproduce the key chemical
features of real silica. The model has been proved to be
powerful in describing dense phases of real silica59 and it has
also been used to investigate the fluid phase close to the
dynamical arrest and the gas–liquid critical point location in
the phase diagram.72 The phase diagram of this model shows
that solid dense phases of the primitive model of silica and
experimental data are in good agreement, and that a liquid
phase is always observed as a metastable phase with respect to
the fluid–solid phase separation, in analogy to the case of the
primitive model for water.
The strong glassy behaviour of such network forming
liquids has been related to the behavior of thermoreversible
colloidal gels.73 Indeed, in the region of intermediate densities
where the low bonding valence stabilizes the liquid phase, the
Arrhenius slowing down of the dynamics is driven by the
attractive interactions rather than packing, as it happens in
colloidal gels.
3.2 Primitive models for proteins
In the late 1990s, a renewed interest in patchy particles has
grown, motivated by the need to tackle the issue of protein
crystallization. The existence of low density crystals and the
tendency to form gel states instead of ordered structures have
been proved to be related to the short range nature and the
high anisotropy of attractions between proteins in solution.
Already from the seminal work on an aeleotopic model for
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globular proteins,74 a simple pair potential approach to get
such features has revealed to be quite powerful.
The minimal model introduced by Sear52 to study the
fluid–fluid and the fluid–solid coexistence in the phase diagram
of globular proteins was inspired by the primitive models of
associating fluids: patchy particles are modeled as hard-
spheres carrying a fixed number of bonding sites in a regular
geometry, so that the pair potential between particles is the
sum of an isotropic hard-core repulsion and a site–site square-
well attraction. In contrast to the majority of primitive models
for associating fluids, all sites are on the particle’s surface and
have the same interaction strength and attractive range, but
they are numbered, so that even sites on a particle interact only
with odd sites on a different particle. Due to an appropriate
choice of the short ranged attraction and to the location of the
bonding sites, the model guarantees the single bonding
condition and minimizes the aggregation of closed bonded
loops. The Kern–Frenkel model75 is another hard sphere
patchy model with square well attractive patches, and it was
introduced to study the gas–liquid critical phenomena when
only a reduced coverage of the particle surface was active in
the bond formation. Indeed, in such a model, patches are
designed to also have an explicit extension on the particle’s
surface. Similar to the Sear model, patches are all identical,
with the same bonding energy, attractive range and, additionally,
the same patch size. In contrast to the Sear model, however,
the patch–patch interaction is not selective. The powerful
feature of the Kern model is that the angular and the radial
characteristics of the patches are independently chosen, while
in the Sear model the interaction range and the patch extension
are defined jointly. On the other hand, while the Kern model
encompasses surface patch coverages both below and above
the threshold that guarantees single bonding between particles,
the Sear model was built to manifestly avoid the formation of
multiple bonds between spot-like patches. The choice of simple
discontinuous pair potentials in both models guarantees that
bonding is properly defined: a bond sets up when two particles
are closer than the patch attraction range and in that case the
pair interaction energy is equal to the depth of the square well.
The study of such models has shown that the fluid–fluid
coexistence moves towards temperatures lower than critical
temperatures observed in isotropic systems and that the
fluid–fluid phase separation is metastable with respect to the
fluid–solid coexistence. Similar results about the metastability
of the fluid phase were also found for a continuous
patchy model where the (weak) isotropic repulsion and the
(strong) anisotropic attraction were both modeled by a 140-35
Lennard-Jones potential.76
In the last five years, there have been several studies showing
how patchiness affects the phase diagram. Inspired by either
the spot-like patch model proposed by Sear (see section 4) or
the extended patch model proposed by Kern (see section 5.1),
a host of minimal models have been designed to explore in a
general way the effect on the phase diagram of the patch
extension, the patch geometrical arrangement, the patch
interaction strength, the patch interaction range or even the
patch specificity. The extensive studies on patchy particles
have not only helped to better investigate collective behaviors
of proteins in solution,50,51,53–55 but are of great importance to
deepen the understanding of a wide a range of different
phenomena as well. Patchy models have indeed played a key
role in studying the relation between chemical and physical
gelation, the dynamics of supercooled liquids and the
formation of kinetically arrested states, the stability of the
liquid phase with respect to crystal formation, the interplay
between locally favored ordered clusters and crystallization or
the assembly of particles into ordered structures.
3.3 Patchy models for self-assembly
Patchy particles are also simple building blocks of mesoscopic
and macroscopic self-assembled equilibrium structures.
Self-assembly is the spontaneous organization of matter from
disordered constituents into well-defined meso- and macro-
scopic structures with given size and symmetry.77
On a molecular scale, traditional examples are the dynamical
assembly of molecules into functional clusters, such as actin
fibers and tubulin, the formation of micelles from block
copolymers or from surfactant molecules exhibiting amphiphilic
interactions (e.g. lipids), and virus capsid proteins assembling
into monodisperse structures, such as icosahedra in HIV
(human immunodeficiency virus) and helical cylinders in
tobacco mosaic virus.6,78,79 Inspired by reversible assembly
of protein units, a large variety of minimal patchy models have
been designed to study the self-assembly of patchy particles
into monodisperse clusters of various symmetry.80–82 Such
models describe the patchy interaction between two particles
via an orientationally dependent Lennard-Jones potential (see
section 5.2). The mechanisms through which aggregation
occurs as well as the interplay between crystallization, kinetic
arrest and clustering have been addressed for different platonic
solids on varying the arrangement of the patches on the
particle surface.80–82 Self-assembly of polyhedral cages has
also been investigated by the means of rigid, non spherical
building blocks carrying anisotropic interaction sites.83,84 In
particular, truncated pyramids with interaction sites on the
lateral faces83 have been specifically designed to self assemble
into viral icosahedral cages, as well as three-legged units with
selective interaction sites on both ends of each leg.84
On a macroscopic scale, self-assembly is a method to build
well defined aggregates ranging from the nanoscale to
extended structures. By virtue of the asymmetry in shapes
and/or the anisotropy in the interactions, patchy particles
significantly extend the possibilities offered by traditional
materials. The relationship between patchiness and equilibrium
target structures has been extensively investigated via coarse-
grained patchy models.85–87 In this case, the anisotropy of the
interparticle interaction is gained via covering the spherical
particles with a number of coarse-grained spherical subunits of
different types arranged in a fixed, predefined geometry on the
particle surface. The local arrangements of patch units, the
total number of surface units and the interactions between like
or complementary units are chosen such that selectivity and
directionality can be tuned and adapted to the particular
system in analysis. Icosahedra, square pyramids, tetrahedra,
rings, chains, twisted and staircase structures, and sheets have
been obtained, on cooling from a disordered state, by a
suitable design of the patches pattern.85–87 Particular attention
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6402 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 This journal is c the Owner Societies 2011
has been paid to the specific case of four identical patches in a
tetrahedral geometry88–90 because of its relevance for photonic
crystals with diamond symmetry.24
4. Spot-like patch models
Various spot-like patch models have been developed, drawing
inspiration from the Sear model for globular proteins.52 Particles
are designed as hard objects carrying on their surface a fixed and
small number of attractive patches (from two up to a maximum
of six), which are spots on the surface and interact with like or
complementary sites on a different particles via a square well pair
potential. These models have been designed to satisfy the single
bonding conditions of each patch and to minimize the formation
of bonded rings. Indeed, because particles cannot overlap, the
site–site square well potential can always be made sufficiently
short-ranged, so that the formation of multiple bonding at a
given site is forbidden. The value of the attraction range for such
models is usually around 10–20 per cent of the particle diameter.
It is worth noting that the single bonding conditions imply a
simple correspondence between the number of sites and the
number of bonds, since a particle with a given number f of
patches can form only up to f bonds and hence f can be
considered the valence or functionality of the particle. Moreover,
the geometrical arrangement of the patches on the particle
surface is often chosen in such a way that double bonding
between the same pair of particles is minimized as well
as the effect of correlation between nearby sites. Given such
general features of the models, the first-order thermodynamic
perturbation Wertheim theory has been widely applied to obtain
the free energy contribution due to bonding in terms of bonding
probability. The bonding probability can be seen as the ratio
between the number of bonds in the system and the maximum
possible number of bonds. Considering the bonding process as a
chemical reaction between two non-bonded sites in equilibrium
into a pair of bonded sites, the bonding probability, pb, results
from the mass action equation as
pb
ð1� pbÞ2¼ rs3e�bDFb ð1Þ
where r is the density of the system, s is the particle diameter,
b = 1/kBT is the inverse of the thermal energy (where kB is the
Boltzmann’s constant) and DFb is the free energy difference
between the bonded and the nonbonded states. The Wertheim
theory provides a simple recipe for DFb in terms of the liquid state
correlation function of the hard-core reference system and the
spherically averaged Mayer function of the site–site interaction.
In particular, given two particles 1 and 2, each of them carrying
f sites, the exponential in the right hand side of eqn (1) reads
e�bDFb ¼ f1
s3
ZgHSð12ÞfSWð12Þdð12Þ ð2Þ
where gHS(12) is the reference hard-sphere fluid pair correlation
function, fSW(12) is the site i-site j Mayer function, and d(12)
represents the integration over all the possible orientations and
interparticle separations of the pair of particles within the range
where bonding occurs.
Since the main assumption of the Wertheim theory is that
particles cluster in open structures without closed bonded
loops and since this hypothesis is also at the heart of the
Flory–Stockmayer theory,91 a combined parameter-free
analytical approach has been developed to describe the
equilibrium properties of these spot-like patch models. The
Flory–Stockmayer theory, developed in the contest of
chemical gelation, provides expressions for the number density
of clusters of a given size as a function of the bonding
probability, which is called the extent of the reaction in the
Flory–Stockmayer language. Combining the cluster size
distribution and the chemical equilibrium equation, it has
been possible to obtain the structural and connectivity properties
of the systems in the temperature-density phase diagram. The
Flory–Stockmayer theory has been recently generalized to
patchy particles with distinct interaction sites.92
An analytical mean-field description is available not only for
the thermodynamic properties of point-shaped patch models
but also for their dynamics. Indeed, the aggregation kinetics of
such systems has been investigated and fully described
by combining the static Wertheim expressions with the
Smoluchowski equation for coagulation.93,94 The Smoluchowski
approach also relies on the assumptions that (i) cyclic structures
do not occur and (ii) at any stage of the aggregation all sites
are equally reactive, regardless of the size of the cluster they
belong to or of the state of the other sites on the same particle.
The Smoluchowski equation was developed to describe the
irreversible growth of clusters by coalescence, but it was lately
extended to reversible aggregation95 by taking into account
the appropriate bonding and breaking rate constants. As a
consequence of conditions (i) and (ii), the bonding and breaking
rate constants only depend on the topology of clusters with a
given size. The second condition is also equivalent to the
assumption that the aggregation process is dominated by
bond-formation and not by diffusion of clusters. If the kinetics
are mainly controlled by bonding, the system is said to be in
its chemical regime. It has been shown95 that the Flory–
Stockmayer cluster size distributions are solutions of the
Smoluchowski equation in the chemical limit. Once the
appropriate bond-breaking processes are taken into account,
the solution of the Smoluchowski equation provides the
evolution in time of the bond probability. For patchy systems
in which reversible and irreversible aggregation take place in
the chemical regime, it has recently been shown96,97 that there
is a direct correspondence between the bonding probability at
a certain time, pb(t), and the equilibrium bonding probability
at a specific finite temperature, pb(T), as it is given from the
Wertheim approach of eqn (1) and (2). The role of the time
in irreversible aggregation has been related to the role of
temperature in reversible aggregation, meaning that the
dynamical evolution of the aggregating systems takes place
via a sequence of equilibrium states.96–98
A self-consistent mean-field approach is thus available to
treat the dynamics and the thermodynamics of spot-like patch
particle models, which can be spherical or not, as described
more in details in sections 4.1 and 4.2.
4.1 Patchy hard spheres
Patchy models consisting of hard-spheres decorated on their
surface by a small number of identical spot-like, short-ranged,
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square-well attractive sites have been used as minimal models
of self-assembly systems. Self-assembly into non-close packed
structures is caused by a very strong inter-particle attraction,
significantly larger than the thermal energy, and by interaction
geometries far from the spherical one. Equilibrium polymer-
ization of linear polymer chains from monomers having
directional bifunctional interactions as well as self-assembly
of branched loop-less clusters of binary mixtures of
bi- and three-functional patchy particles have been deeply
investigated.97,99–101 The presence of even a small fraction of
polyfunctional particles in a chain-forming system leads to the
formation of extended networks and hence it introduces two
phenomena which are missing in chain polymerization: a
percolation transition and a gas–liquid phase separation. A
comprehensive scenario has been developed to describe the
basic equilibrium thermodynamic and dynamic properties of
such systems.97,99–101
Spot-like patch models have also been widely used to
investigate the formation of colloidal gels.66,102–104 A colloidal
gel can be defined as a low density disordered arrested state of
matter in which stable particle networks are formed due to
long-living reversible bonding. Such states are usually
observed at large values of the interaction strength (i.e., values
of the characteristic energy of the attractive potential u0 much
greater than the thermal energy kBT), since such a condition
increases the lifetime of inter-particle bonds and stabilizes the
bonded network, when excluded volume does not favor
caging. On the other hand, large values of u0/kBT also favor
the liquid–gas phase separation process, since the statistical
Boltzmann weight of highly bonded configurations becomes
relevant. It has been shown that for particles interacting with
attractive spherical potentials (besides the hard-core repulsion)
arrest occurs only through interrupted phase separation.105–111
Anisotropically interacting systems in which the bonding
valence is significantly reduced show a reduction of the
gas–liquid phase separation region. Patchy models66,102,104
as well as systems in which coordination numbers are
smaller than 12 have been shown to disfavor dense local
configurations.112,113 The average functionality of the system
is one of the key parameters controlling the stability of the
liquid phase: the reduction of the number of bonded nearest
neighbors broadens the region of stability of the liquid phase
in the temperature-density plane. Fig. 2 shows the shrinking of
the gas–liquid coexistence region in the phase diagram on
decreasing the valence of the patchy system.
Once the gas–liquid separation is strongly disfavored, it is
possible to decouple phase separation from dynamical slowing
down, and hence homogeneous states with u0 c kBT can be
approached in a one-phase system.101,102 The stable (or at least
metastable) fluid phase at low density, either referred as empty
liquid or ideal gel (in the zero temperature limit), can be seen
as an extended network of bonds, whose lifetime is essentially
controlled by the energy scale. On increasing the bond lifetime
(or equivalently on increasing u0/kBT), the occurrence of the
kinetically arrested disordered state of matter at low density
becomes possible. The possibility of forming long-lived
reversible gels has been proved to be robust against changes
in the patches’ geometrical arrangements for systems
composed of patchy particles with fixed and equal number
of patches (lower than six) and for binary mixtures of particles
with a different number of bonding sites such that the average
functionality is low (less than three).66,101,102,104 The same
scenario has been found also in two-dimensional systems,
where the formation of closed bonded loops is highly
entropically favored in comparison to the three-dimensional
counterpart.114,115 A continuous model designed to mimic the
hard-core plus square-well anisotropic potential has been used
to study how the valence influences the gel in a binary mixture
of patchy particles with different functionalities where the
stochiometric ratio of the two species determines the average
valence of the system structure.116 The tendency of patchy
particles to form bonded, low-density networks has also been
exploited to investigate ageing in gel forming systems as
compared to glass forming ones and it has been shown that
violations of the fluctuation–dissipation theorem occur only
for energy-related observables and not for density-related
observables, in contrast to what happens in structural
glasses.104
The vapor–liquid coexistence, the location in the phase
diagram of the critical point and the way it may vanish or
not on tuning the patchiness of the system have been also
explored via a hard sphere model in which the particle surfaces
are patterned with attractive dissimilar spots.117–119 The role
of dissimilar patches has been investigated with particular
attention paid to the richness of the bonded structures and
the possible tailoring of the fluid structural features with
temperature. Specifically, each particle carries three spot-like,
square well patches, two of one type and one of a different
Fig. 2 Suppression of the gas–liquid phase separation on decreasing
the valence of a patchy particle system. The specific patchy model is
the spot-like patch system of ref. 66 and 102: patchy particles are hard
sphere particles of diameter s with a finite number of point-shaped
identical patches, which interact with each others via a square well
potential of range d and attraction depth e. The bottom part of the
Figure shows the temperature–density phase diagram for patchy
particles with 5, 4, and 3 sites arranged on the vertices of respectively
two fused tetrahedra, a tetrahedron, and a triangle, as shown in the
upper part of the Figure. The continuous line delimits the gas–liquid
coexistence region (yellow area) and the dotted line is the percolation
locus, i.e. the threshold at which the formation of a infinite spanning
network occurs (the green area is the percolating region while the blue
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type.117–119 Three different interactions occur between like or
unlike patches and the relative values of the strength of such
interactions tunes the structures of the fluid: from linear non-
interacting chains to non-percolating hyperbranched polymers,
from dimers to chains connected by X-like or Y-like junctions.
The richness of the model comes from the possibility of tuning
the favorite bonding pattern on changing the relative site–site
interaction strengths. Moreover, for fixed ratios of such values,
the temperature influences the effective functionalities of the
particles and hence the prevailing structure. Thermodynamics,
percolation and criticality have been then related to the possible
structures on exploring the range of the energy parameters. The
limit in which the features of the patches favor the formation of
chains connected via Y-junctions has also been shown to have a
strong analogy with the dipolar hard sphere fluid.5
4.2 Patchy hard ellipsoids
A patchy particle model able to combine directional bonding and
non-spherical shapes has been used to study gel formation in
reversible and irreversible aggregation and to connect the two
processes via an analysis of the role of time during chemical
gelation as compared to the role of temperature in physical
gelation.96,98,120,121 The features of the model have been inspired
by the epoxy-amine system120 but they are representative of
colloidal particles functionalized with a small number of patchy
attractive sites. The model is a binary mixture of hard ellipsoids
of revolution carrying on their surface a fixed number of
mutually interactive sites in a predefined geometry. The two
species in the mixture differ from each other in size and number
of attractive sites, while the stoichiometry of the two species is
chosen in such a way that the total number of sites is equal for
both species and the average functionality is less than three. The
interaction potential is the hard ellipsoid potential plus a site–site
square well attractive interaction between sites on different type
of ellipsoids. The model turned out to be well suited for the
application of mean field approaches, since it disfavors the
formation of bonded loops in the clusters and it guarantees
a single bonding condition between sites. The entire poly-
merization process has been described in its structural and
connectivity properties both below and above the percolation
threshold and it has been shown that the irreversible temporal
evolution of the aggregating system can be connected to a
sequence of equilibrium states. It has also been possible to
investigate the interplay of chemically controlled or diffusion
controlled aggregation, finding out that the crossover between
the two regimes is related to the probability of two nearby
clusters to stick as compared to the probability of two distant
clusters to approach each other via diffusion.
5. Extended patch models
Patchy particles with extended patches have been modeled as
either hard core particles interacting via orientationally
discontinuous pair potentials (see section 5.1) or soft particles
(see section 5.2).
The Kern–Frenkel model can be considered the prototype
of the first kind of extended patch models. Such a model and
its extensions have been used to investigate both ordered
and disordered states of matter.122–131 Various theoretical
approaches have been developed and exploited on the different
Kern-inspired patchy models. Once multiple bonding is
appropriately hindered, the theoretical mean-field approach
provided by the Wertheim theory can be applied to predict the
thermodynamic properties of such patchy systems. The
Wertheim theory has been successfully applied to Kern systems
with a hard sphere reference fluid and four tetrahedral patches124
or even to Kern-inspired systems with an isotropic square well
reference fluid and a variable number of patches.129–131 More-
over, standard Kern systems with only one or two patches on the
particle surface have been studied by means of an integral
equation approach. The two cases have been investigated on
changing the areal patch coverage and the integral equation
approach has revealed to be quite powerful to study the fluid in
the regime where multiple bonding between patches is allowed to
occur.125,127 In the case of patches with finite interaction range
and strength, the Ornstein–Zernike equation for the direct
correlation function in terms of the pair correlation function62
has been solved with the reference hypernetted-chain
closure.125,127 The corresponding Baxter limit of the Kern model
with one or two patches has also been investigated. In that case, a
different class of closures132 has been chosen to solve the
Ornstein–Zernike equation.
5.1 Kern and Kern-inspired models
The first patchy model able to independently tune the inter-
action range and the patch extension on the particle surface
has been the Kern–Frenkel model. While in spot-like patchy
models the patch extension is determined by the sphere of
interaction of a spot site, in the Kern model the width and the
range of the patches can be varied independently. The Kern
model was introduced to study the fluid–fluid phase separation
phenomena when only part of the particle surface can participate
in bond formation. To this end, the patch–patch attraction
ranges were chosen to guarantee the stability of the critical
point in the isotropic limit and the coverage percentage of a
small and even number of patches was varied. On reducing the
attractive surface of the particles, the gas–liquid coexistence
region was found to become metastable. As noted before, the
Kern model was conceived to investigate the effect of patchiness
in both single and multiple bonding regimes. Successive
studies have instead focused on the single bond per patch
regime, by appropriately choosing the angular extension and
the range of the patches.122–124 In such a regime, it has been
shown that it is possible to generalize the law of corresponding
states developed for isotropic short-range attractive systems133
to patchy particle systems, at least when they are close to the
gas–liquid critical point.122 Specifically, the critical density and
temperature of Kern patchy systems with three, four and five
patches arranged in a regular geometry have been determined
for different attraction ranges and patch extensions and it has
been shown that, if the number of patches is the same,
the thermodynamic properties scale with the second virial
coefficient, as it happens for isotropic fluids.133
Particular attention has been paid to the Kern model of
patchy particles with four tetrahedrally arranged patches
because of the interest in diamond crystal formation.123,124
Such studies have focused on the effect of the patch range on
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the location of the gas–liquid, the fluid–crystal and the
crystal–crystal coexistence regions in the phase diagram. The
patch–patch attractive range has been varied from 3% to 25%
of the particle diameter, while keeping the patch extension
fixed.123,124 The stability of the liquid phase has been proved
to be metastable when the interaction range becomes less than
15% of the particle diameter, similar to what happens in
isotropically interacting systems, but, in contrast to the
spherical case, the driving force to nucleate a diamond crystal
does not increase upon increasing the interaction range.123
Moreover, the role of the patch–patch attractive range has
been studied with respect to the region of stability of four
crystal phases: diamond cubic, body-centered-cubic (bcc),
face-centered-cubic (fcc) and plastic fcc crystals.124
The effect of the surface patch coverage on the structure of
the fluid has been addressed for Kern patchy systems with one
or two opposite patches.125–128 In both systems, the patch
width values cover the whole range from the full square well
system to the hard sphere one. It has been shown that, for a
given patch width, the liquid phase is favored by a higher
valence127 and that the structure of the system can be tuned
in both its ordered and disordered arrangements.125,127 In
particular, in the two patches case, the gas–liquid phase
coexistence boundaries and the fluid structure can be
determined only when the patch coverage is bigger than
30% of the total particle surface. Below such a value, the
system tends to crystallize. The ordered crystalline structures
are directly controlled by the patchy pattern.127 On decreasing
the patch extension from values such that each patch can form
a maximum of four bonds down to two bonds per patch, the
structure of the system varies from interconnected planes, with
a square or triangular arrangement, to planes interacting
only by excluded volume interaction. Eventually, in the
single-bond-per-patch limit the stable aggregates are polydisperse
chains.127 Crystal formation does not occur in the one patch
case,125 but the morphology of the surface pattern favors the
formation of micelles and vesicles, which can suppress the
phase separation process.126,128
The formation of micelles and vesicles in a one-patch Kern
system has been thoroughly studied in the specific case of a
patch surface coverage equal to half of the total particle
surface.126,128 Patchy particles carrying only one hemispherical
patch are a simple prototype of Janus particles. The phase
diagram of Janus particles shows a gas–liquid phase separation
phenomenon whose critical temperature and density are much
lower than the isotropic counterpart.128 Interestingly, the
fluid–fluid phase separation competes with the aggregation
of particles into micelles or vesicles.126 Indeed, the dense liquid
phase has been found to coexist with a gaslike phase composed
of micelles or vesicles. In the free energy balance, the
entropically favored phase is the liquid, while the gas of
micelles or vesicles is the energetically favored phase, in
contrast to what happens in simple fluids.126 Moreover, unlike
simple liquids, the densities of the two coexisting phases
approach each other on cooling. Investigations on a possible
meeting point of the fluid–fluid coexistence boundaries are
prevented by the formation of a lamellar crystalline phase.126
The stability of micelles, vesicles and lamellae has been
established for a long-range Janus system, while the effect of
interaction ranges closer to the experimental values is still an
open question.128
Kern patchy systems with one or two patches have also been
investigated in their Baxter sticky limit, i.e., when the square well
attraction between the patches becomes infinitely deep and
narrow.132 Both systems have been studied under changes of
the areal patch coverage and the effect of the two geometrical
distribution of the attractive surface has been addressed.132
Similar to the corresponding cases with finite attractive range
and strength between the patches, the liquid phase appears only
over a certain threshold of the surface coverage and it is favored
by a symmetric two patch distribution rather than by one patch
with equivalent attractive surface. The liquid and the percolating
phase of the systems are even more favored when a uniform
adhesive interaction is added to the isotropic hard sphere
repulsion on the top of the patchy interaction.132
A different version of the Kern–Frenkel model complements
the square well patch–patch interaction with an isotropic
square well interaction between particles.129–131 The difference
between this variant and all the previously described models is
that the reference system is not purely repulsive. The isotropic
attraction is chosen to be longer ranged and significantly
weaker than the patch–patch attraction. At the same time,
the patch–patch attractive ranges have been chosen such that,
when varying the number of the patches, the total area-
coverage of the patches does not allow the formation of more
than one bond per patch. The gas–liquid phase separation
phenomena, the critical point temperature and density, and
the gas–liquid coexistence boundaries have been investigated
under changes of the patch number (from three to seven), the
patch geometrical arrangement on the particle surface, and the
total areal coverage of the patches.129,130 Thus, the metastability
of the fluid phase has been addressed on varying the system
parameters from the square well to the hard sphere isotropic
fluid. The generalized law of corresponding states has been
proved to hold only for the patchy hard sphere system and not
for the patchy square well fluid.129,130 The specific case of a
square well patchy system of particles with six uniformly
arranged patches has also been studied with respect to crystal
formation. It has been shown that crystallization in such a
system is induced by the aggregation of clusters with the same
symmetry of the crystal. The self-assembly of long living
clusters takes place in a region of the phase diagram in which
the critical phase separation phenomena do not take place,
and hence it is the self assembly mechanism that leads to the
formation of dense liquid-like regions, thus facilitating the
formation of a critical nucleus, even in absence of critical
fluctuations.131 The same phenomenon was not observed for
five and seven regular patches, meaning that the symmetry of
the aggregates has to be compatible with the crystal structure
for this phenomenon to take place.131
5.2 Patchy soft spheres
Another model in which the patch–patch interaction range
and the patch extension on the particle surface are disconnected
is a continuous orientationally-dependent pair potential built
from an isotropic Lennard-Jones one.80–82,88,89,134,135 Specifically,
two particles interact via an isotropic Lennard-Jones repulsive
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core and the directional patch–patch attraction is given by the
Lennard-Jones attractive part modulated via a Gaussian
angular dependent decay. The cutoff distance of the
Lennard-Jones potential is chosen to be larger than twice the
particle diameter, i.e., patches interact over large interparticle
separations, much larger than the usual square well ranges in
discontinuous patchy models. Moreover, patches are quite
extended on the particles surface, since the Gaussian width
ranges typically from 0.2 rad to 0.5 rad. Nonetheless, a patch
on a particle cannot bond simultaneously to more than one
patch on another particle, since only the smallest of the angles
between the interparticle vector and the patch vectors is taken
into account when calculating the interaction energy of a given
pair of particles. It is worth noting that, in the model, bonds
between more than two particles at a given patch are still
possible and the wider the patches are the more likely these
multiple bonds become. Provided that the patches are
sufficiently narrow, each patch is involved in only one bond
and the single bonding conditions is recovered.
Such a model has been mainly used to tune the self assembly
of clusters with specific shapes and dimensions. The first and
most thoroughly investigated case is the self-assembly of
patchy particles into icosahedral clusters, since most of the
virus capsid proteins often form cages of icosahedral
symmetry.80 On changing the number and the patch location
on the particle surface, it is possible to tune the geometry of
the interparticle interaction and to study self-assembly paths
towards cluster formation as opposed to crystal formation.
Within the described orientational Lennard-Jones model,
interesting insights have been gained on the role of frustration
within the crystallization scenario both in two and three
dimensions.88 The competition between local and global order
can possibly frustrate the kinetics of crystallization, even if the
symmetry of the interactions is compatible with a specific
crystal structure. Cluster assembly into monodisperse target
structures can be tuned via designing patchy particles with a
specific number and geometrical arrangement of
patches.80–82,88 The effect of the target geometry on the
efficiency of the aggregation process has been studied via
comparison of the assembly pathways of different target-shaped
building blocks. Different platonic solids can be identified by
different patchy particles designed to satisfy the bonding
geometry. In particular, cubes, tetrahedra, icosahedra, octahedra
and dodecahedra have been considered.80–82,88 The assembly
of such structures within the described models occurs via both
direct nucleation pathways and a budding mechanism which
leads step-by-step to the desired clusters via subsequent
rearrangements of the bonded particles. The growth of
disordered aggregates and the rearrangement kinetics from
random aggregates to monodisperse clusters are sensitive to
the size and the degree of order of the final and the inter-
mediate structures: the smaller the aggregates are and the
closer to the desired structure the local order is, the easier it
is for them to rearrange into the target. Self-assembly via
budding mechanisms is suppressed when a torsional constraint
in the patch–patch interaction is added.82 Indeed the torsional
constraint strongly reduces the number of possible incorrect
structures and the competition between target and disordered
clusters is reduced as well. The torsional patchy model is hence
less sensitive to the geometry of the target structure, i.e., the
propensity of the patchy building blocks to aggregate does not
show significant differences on assembling cubic, tetrahedral,
icosahedral, octahedral or dodecahedral clusters.82 The
orientational Lennard-Jones patchy model with the torsional
constraint has also been employed to study the self-assembly
pathways of homomeric complexes135 made of patchy
particles with distinguishable patches. The number and the
interaction specificity between the patches have been chosen to
focus on the formation of tetramers and octomers. The
evolution from monomers and dimers to one of the two final
complexes has been investigated with respect to changes of the
relative interaction strengths between different patches.135
Both the torsional and the non-torsional soft patchy models
with identical patches have been used to investigate the self
assembly of the above mentioned platonic clusters.80–82 The
role of the patch width, range and strength on cluster aggregation
has been addressed and some general rules for optimizing the
assembly mechanism can be brought forward.80–82 Narrow
patches guarantee the selection of the desired structure, since
the lowest-energy clusters are expected to saturate the number
of bonds, while non-saturated clusters have a significantly
higher energy. Nonetheless, the narrower the patches are,
the more difficult it is to rearrange an incorrect structure and
the system is likely trapped in glassy kinetic aggregates. On the
other hand, wide patches favor the formation of large
disordered clusters, since the formation of multiple bonds at
a given patch decreases the energy of the aggregates to a value
lower than that of the target cluster. The optimal patch width
is then a compromise between selectivity, kinetic accessibility
and thermodynamics. A similar balance holds for the
patch–patch interaction strength, since strong bonds favor
the thermodynamic stability of the clusters but slow down
the dynamics of rearranging into the correct structure.
Rearrangements of particles have to overcome a higher
energetic barrier also when the attraction range of the patches
is small, so that long ranges increase the yields of the desired
monodisperse clusters. Moreover the assembly process has
been shown to be quite robust against the patch arrangement
on the particle surface, while many and close patches likely
tend to stabilize the target structure as compared to the
disordered aggregates.
Particular attention has been also paid to the cases of six
and four patches arranged on the particle surface in an
octahedral and a tetrahedral geometry respectively.80 It has
been shown that while for particles with six patches the
systems crystalize quite easily in a simple cubic structure, for
particles with four patches the formation of the diamond
lattice is hindered by the aggregation of dodecahedral clusters.
Only for very narrow patches the formation of a diamond
nucleus is energetically favorable with respect to the dodecahedron
aggregation. The complete phase diagrams of such systems
have been lately studied in the case of relatively narrow
patches. Particles with six attractive patches in an octahedral
geometry show four crystalline phases:134 at low pressures the
simple cubic crystal, which is the lowest energy structure; on
increasing the pressure the bcc and the fcc, which are high
density structures both orientationally ordered; and the
disordered (or plastic) fcc for temperatures at which the
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entropic effects are most relevant. The phase diagram of
patchy particles with four patches in a tetrahedral geometry
interestingly shows only three crystal phases, at least for the
long ranged Lennard-Jones attraction between patches: a bcc
structure, an ordered and a plastic fcc crystal. It has been
shown that the bcc structure, which can be viewed as double
tetrahedral interpenetrating structure, is energetically favored
with respect to the formation of the stable diamond structure,
unless the patchy attraction is sufficiently short-ranged.89 The
formation of bonded and crystalline structures has been
investigated in the corresponding two dimensional systems
and it has also extended to systems with different average
valences, patch extensions and pressures.80,136 Some examples
of equilibrium open structures in two-dimensional patchy
systems are shown in Fig. 3.
A similar continuous patchy model built out of a Lennard-
Jones pair potential has been used to describe supramolecular
polymerization of patchy particles with two opposite attractive
patches.137
A different soft patchy model has been developed to study
the structure and the dynamics of gel forming systems.138,139
Such a model is described by a potential made of three terms:
a soft Lennard-Jones-like isotropic pair attraction, a soft
anisotropic repulsion and a three body term which introduce
a local rigidity in the bonded structure. Such a model has been
exploited to investigate the difference between physical gels
obtained via slow cooling or via quenching: the latter is shown
to be more connected and less space-filling than the first.138
Moreover, the arrested glassy dynamics in such a system has
been related to the formation of a stable network of bonded
particles.139
6. Perspectives
Colloids and proteins acquire a charge when dispersed in polar
solvents, due to dissociation of surface groups and/or
preferential adsorbtion of charged species. The electrostatic
interactions between dispersed particles in a liquid medium is a
screened Coulomb interaction. The traditional Derjaguin–
Landau–Verwey–Overbeek (DLVO) description140 is based on
the assumption of symmetric charge distributions. Nonetheless,
an inhomogeneous surface charge can give rise to anisotropic
interactions. Particles with heterogeneously charged surfaces
can thus be regarded as charged patchy particles. The main
feature of such patchy systems is not any more the limited
valence in bonding, but rather a competitive interplay between
attractive and repulsive anisotropic interactions. Moreover,
the presence of charges implies interaction ranges much longer
than the usual bonding distances in patchy systems with
limited valence. The design of simple models for hetero-
geneously charged particles can proceed along two different
paths: (i) the building of minimal models from a close match
with well designed experiments; or (ii) the development of
models from first principles, in order to keep a connection with
the microscopic systems.
A patchy model for lysozyme dispersions has been recently
designed as a minimal model with free parameters, which can
be determined by experimentally accessible information on the
phase behavior of such systems.141 The proposed pair
potential consists of an isotropic DLVO-type repulsion,
complemented by a spherical hard core, and an anisotropic
attraction mediated by two opposite patches on the particle
surface. Using the Kern–Frenkel scheme,75 the patch–patch
attraction can be factorized into a radial and an angular part.
The angular part of the attraction is the Kern–Frenkel
distribution function, while the radial contribution is a
Yukawa-like attraction. The depth and the range of the
attractive Yukawa, as well as the surface coverage of the
patches, are free parameters of the model and they are
determined by using experimental information on the gas–liquid
critical points. The experimentally matched patchy interaction
is found to be stronger and longer ranged than the isotropic
repulsion, while the patch extension is about 70% of the total
particle surface.141 Results for the gas–liquid and fluid–solid
coexistence of such a patchy model are in good agreement with
experiments on lysozyme solutions.141
A first-principles approach to heterogeneously charged
colloids may originate from the statistical mechanical description
of electrolytes which goes by the name of Poisson–Boltzmann
(PB) theory.142 The electrolytic solution is described like a
suspension of impenetrable charged colloidal particles in a
liquid dielectric solvent containing point-like ions and
counterions. By combining the first Maxwell equation of
electrostatic for a linear, homogeneous and isotropic dielectric
with the third Maxwell equation in absence of external fields, a
differential equation of Poisson type is derived to relate the
electric potential and the density distribution of the ionic
charges. Since the charge density follows the Boltzmann
statistics,142 the Poisson–Boltzmann theory is intrinsically a
non-linear problem. The linearized Poisson–Boltzmann (LPB)
theory relies on the assumption of dilute systems, which is
referred as the Debye–Huckel approximation.143 Under the
additional assumption of symmetric charge distributions on
the colloidal surface, the LPB approach leads to the DLVO
description.140
The PB approach has been recently applied to hetero-
geneously charged colloids. Specifically, a Poisson–Boltzmann
cell model for an homogeneously charged sphere144 has been
extended to heterogeneously charged surfaces.145 The linearized
problem has then been solved to study the role of charge
Fig. 3 Examples of equilibrium open structures in two-dimensional
patchy systems where particles interact via a Lennard-Jones pair
potential whose repulsive part is isotropic while the attractive part is
angular dependent, as first introduced by ref. 80. Specifically, the case
of particles decorated with five attractive patches is shown. The
two structures represent the minimum energy configurations at zero
pressure for two different patch widths.136
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heterogeneity on the fluid orientational order.145 Moreover, a
generalization of the charge renormalization approach144 from
homogeneous to heterogeneous systems has been developed in
the non linear screening regime, and it has been applied to
Janus spheres.146 Indeed, Janus particles can be considered as
the simplest example of charged patchy colloids carrying two
oppositely charged patches with a surface extension of an
hemisphere each.
Complex colloids with heterogeneously charged surfaces
can also arise from complexation phenomena. Recent studies147
have shown that the absorption of soft polyelectrolyte stars
(PE-stars) on the surface of oppositely charged colloids leads
to a rich variety of patchy complexes. The equilibrium features
and the self-assembly of such systems can be externally tuned
by varying the ratio between the colloid- and polyelectrolyte
stars- charges and the relative sizes of the two components, as
well as by changing the salinity of the solution and the
functionality of the stars. The specific system of two PE-stars
with low functionality absorbing on a charged colloidal
particle has been investigated in the case of comparable sizes
of the colloidal particle and the PE-stars.147 The relative
charge of the two components and the salt concentration are
hence the key parameters controlling the conformation of the
complex. When the sum of the charges on the two PE-stars is
slightly less than the charge on the colloid and there is no
added salt, the two charged polymers are shown to fully
absorb onto the colloidal surface.147 In this case, the PE-stars
fully absorb on the colloid on opposite poles and assume on its
surface a so-called starfish configuration. The addition of salt
causes a partial detachment of the stars from the particle
surface, driving them from the starfish to the anemone
configuration, in the terminology of ref. 148. Fig. 4 shows a
simulation snapshot of a charged, rigid, spherical colloidal
particle on which two PE-stars of opposite charge have been
irreversibly adsorbed, forming in such a way a self-assembled
patchy colloid. The number of PE-stars that can be adsorbed
can be controlled via the size- and charge ratios.
The resulting heterogeneously charged complex can be seen
as a patchy colloid with positively charged polar patches and a
negatively charged equatorial region, which is free of patches.
The extent and strength of the patches can be controlled either
by changing the salinity of the solution, as mentioned before,
or by using different stars.148 The number of the patches could
be tuned also on changing the charge and the size ratio
between colloids and stars. In a concentrated system, such
patchy colloids interact anisotropically among themselves: the
positively charged patches repel each other, as also do the
negatively charged equatorial parts, while the interaction
between patches and non-patchy parts is attractive. The PB
theory, within the Debye–Huckel approximation, has been
recently extended to describe the screened electrostatic
potential of spherical particles with non spherical surface
charge distributions.149 Such an approach is the starting point
to calculate the effective interaction potential between a pair of
the above-mentioned complexes, which can indeed be cast in a
closed form.
7. Conclusions
Although relatively new in the domain of soft matter/colloidal
science, the research field of patchy colloids has attracted
intensive attention by a large number of workers in the field
and has already reached a high degree of maturity and
richness. The progress achieved and the interdisciplinary
character of the research performed are yet another instance
of the ways in which soft matter systems rekindle interest in
subjects, in this case anisotropically interacting particles,
which once had been introduced and investigated in the
realm of atomic or molecular fluids. Evidently, the task of
investigating the properties of patchy colloids does not
amount to a mere transcription of recipes and techniques
known from atomic systems over to colloidal ones: a host of
novel physical properties and particularities require specific
treatment of their own. Moreover, the possibilities of
manipulating the extent, shape, arrangement and nature of
the patches are much richer in the realm of colloidal science
than in the atomic one (a property that has its counterpart also
for the case of isotropic particles), so that the emerging
phenomena are novel and allow for insights into long-standing
physical questions.
Among the most prominent classes of problems in which
patchy colloids have made possible to open new fields and
address fundamental questions are the ones of self-assembly,
either to local, well-defined structures or to extended ones, and
the issue of dynamical arrest at low concentration, also known
as gelation. Here, limited-valence patchy colloids are a unique
model system with which one can suppress the liquid–gas
separation and observe gel formation without the hindrance
of macroscopic condensation. The game and the investigations
are far from being over. The same holds true for limited-
valence patchy colloids, which are still under investigation,
Fig. 4 Simulation snapshot of a complex between a central, spherical
charged colloid and two, oppositely charged PE-stars that are adsorbed
on its surface at opposing sides of it. The stars have functionality
(number of arms) f = 5 and the star on the back side of the colloid is
only partly visible. The yellow and red beads denote neutral and
charged monomers of the star, respectively, whereas the green and
blue spheres are the counterions of the colloid and the PE-star. The
simulation has been performed under conditions of no added salt.
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regarding their properties on, e.g., formation of ordered
crystals or even quasi crystals, the effect of different kinds
of attractive patches leading to tuning the structure with
temperature, and the inclusion of selective attractions between
different sites. Patchy charged systems are, in our view,
another field that is just emerging, and in which we are dealing
with tunable long range interactions through the addition of
salt, the absence of limited valence and the competition
between attraction and repulsion between the patches,
exemplified in the self-assembled models of inverse patchy
colloids mentioned in the main text. We are confident that
patchy colloids are here to stay and that their investigations
will continue to constitute a very active field of soft matter
research for the years to come.
Acknowledgements
E. B. would like to acknowledge Francesco Sciortino and
Gunther Doppelbauer for fruitful discussions. E. B. additionally
wishes to thank the Alexander von Humboldt Foundation for
financial support through a Research Fellowship as well as the
Austrian Research Fund (FWF) for support through a
Lise-Meitner Fellowship project number M 1170-N16. The
work of R. B. has received support from the Deutsche
Forschungsgemeinschaft (DFG).
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