Registered Charity Number 207890

www.chemistryworldjobs.com

Upload your CV and profile today!

Discover something NEW

Your NEW recruitment site dedicated to chemistry and the chemical sciences

Create a free account to get the most from chemistryworldjobs.com

Get headhunted

Create a profile and publish your CV so potential employers can discover you

Stay ahead of your competition

Set up your job alerts and receive relevant jobs in your inbox as soon as they appear

Discover your next career move

Detailed searches by role, salary and location

Save time and be the first to apply

Online vacancy application

Be efficient

Bookmark jobs that interest you, so you can come back to them later

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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 6397

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: bianchi@cmt.tuwien.ac.at; 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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

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.

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 6399

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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

6400 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 This journal is c the Owner Societies 2011

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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 6401

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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

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,

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 6403

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

one is the non-percolating region).Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

6404 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 This journal is c the Owner Societies 2011

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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 6405

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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

6406 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 This journal is c the Owner Societies 2011

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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 6407

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

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

6408 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 This journal is c the Owner Societies 2011

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.

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 6409

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).

References

1 G. C. Gray and K. E. Gubbins, Theory of Molecular Fluids,Oxford University Press, London, 1984.

2 J. M. Caillol, J. J. Weis and G. N. Patey, Phys. Rev. A: At., Mol.,Opt. Phys., 1988, 38, 4772.

3 S. Klapp and F. Forstmann, J. Chem. Phys., 1997, 106, 9742.4 C. N. Likos, Phys. Rep., 2001, 348, 267.5 T. Tlusty and S. A. Safran, Science, 2000, 290, 1328.6 K. Van Workum and J. F. Douglas, Phys. Rev. E: Stat.,Nonlinear, Soft Matter Phys., 2006, 73, 031502.

7 T. C. Halsey, Science, 1992, 258, 761.8 R. E. Rosensweig, Science, 1996, 271, 614.9 C. N. Likos, H. Lowen, M. Watzlawek, B. Abbas,O. Jucknischke, J. Allgaier and D. Richter, Phys. Rev. Lett.,1998, 80, 4450.

10 C. N. Likos, Soft Matter, 2006, 2, 478.11 K. Solc and W. H. Stockmayer, J. Chem. Phys., 1971, 54, 2756.12 A. Pelissetto and J.-P. Hansen, J. Chem. Phys., 2005, 122, 134904.13 C. Pierleoni, B. Capone and J.-P. Hansen, J. Chem. Phys., 2007,

127, 171102.14 B. Capone, C. Pierleoni, J.-P. Hansen and V. Krakoviack,

J. Phys. Chem. B, 2009, 113, 3629.15 C. Gross and W. Paul, Soft Matter, 2010, 6, 3273.16 F. Lo Verso, C. N. Likos, C. Mayer and H. Lowen, Phys. Rev.

Lett., 2006, 96, 187802.17 F. Lo Verso and C. N. Likos, Polymer, 2008, 49, 1425.18 F. Lo Verso, C. N. Likos and H. Lowen, J. Phys. Chem. C, 2007,

111, 15803.19 F. Lo Verso, A. Z. Panagiotopoulos and C. N. Likos, Phys. Rev.

E: Stat., Nonlinear, Soft Matter Phys., 2009, 79, 010401R.20 F. Lo Verso, A. Z. Panagiotopoulos and C. N. Likos, Faraday

Discuss., 2010, 144, 143.21 S. Yamamoto and S. Hyodo, J. Chem. Phys., 2003, 118, 7937.22 G. Srinivas and J. W. Pitera, Nano Lett., 2008, 8, 611.23 A. B. Pawar and I. Kretzschmar, Macromol. Rapid Commun.,

2010, 31, 150.24 M. Maldovan and E. Thomas, Nat. Mater., 2004, 3, 593–600.25 C. A. Mirkin, R. L. Letsinger, R. C. Mucic and J. J. Storhoff,

Nature, 1996, 382, 607.

26 A. P. Alivisatos, K. P. Johnsson, X. G. Peng, T. E. Wilson,C. J. Loweth, M. P. Bruchez Jr and P. G. Schultz, Nature, 1996,382, 609.

27 A. L. Hiddessen, S. D. Rotgers, D. A. Weitz and D. A. Hammer,Langmuir, 2000, 16, 9744.

28 V. T. Milam, A. L. Hiddessen, J. C. Crocker, D. J. Graves andD. A. Hammer, Langmuir, 2003, 19, 10317.

29 D. Nykypanchuk, M. M. Maye, D. van der Lelie and O. Gang,Nature, 2008, 451, 549.

30 S. Y. Park, A. K. R. Lytton-Jean, B. Lee, S. Weigand,G. C. Schatz and C. A. Mirkin, Nature, 2008, 451, 553.

31 A. van Blaaderen, Nature, 2006, 439, 545.32 V. N. Manoharan, M. T. Elsesser and D. J. Pine, Science, 2003,

301, 483.33 Y.-S. Cho, G.-R. Yi, S.-H. Kim, D. J. Pine and S.-M. Yang,

Chem. Mater., 2005, 17, 5006.34 Y. S. Cho, G. R. Yi, S. H. Kim, M. T. Elsesser, D. R. Breed and

S. M. Yang, J. Colloid Interface Sci., 2008, 318, 124.35 D. J. Kraft, W. S. Vlug, C. M. van Kats, A. van Blaaderen,

A. Imhof and W. K. Kegel, J. Am. Chem. Soc., 2009, 131, 1182.36 D. J. Kraft, J. Groenewold and W. K. Kegel, Soft Matter, 2009,

5, 3823.37 Y.-S. Cho, G.-R. Yi, J.-M. Lim, S.-H. Kim, V. N. Manoharan,

D. J. Pine and S.-M. Yang, J. Am. Chem. Soc., 2005, 127, 15968.38 G. Zhang, D. Wang and H. Mohwald, Nano Lett., 2005, 5, 143.39 G. Zhang, D. Wang and H. Mohwald, Angew. Chem., Int. Ed.,

2005, 44, 7767.40 G. Zhang, D. Wang and H. Mohwald, Chem. Mater., 2006, 18,

3985.41 A. B. Pawar and I. Kretzschmar, Langmuir, 2009, 25, 9057.42 A. B. Pawar and I. Kretzschmar, Langmuir, 2008, 24, 355.43 T. T. Chastek, S. D. Hudson and V. A. Hackley, Langmuir, 2008,

24, 13897.44 C. E. Snyder, A. M. Yake, J. D. Feick and D. Velegol, Langmuir,

2005, 21, 4813.45 A. M. Yake, C. E. Snyder and D. Velegol, Langmuir, 2007, 23,

9069.46 A. Perro, S. Reculusa, S. Ravaine, E. Bourgeat-Lamic and

E. Duguet, J. Mater. Chem., 2005, 15, 3745.47 J. C. Love, B. D. Gates, D. B. Wolfe, K. E. Paul and

G. M. Whitesides, Nano Lett., 2002, 2, 891.48 L. Hong, S. Jiang and S. Granick, Langmuir, 2006, 22, 9495.49 S. Jiang and S. Granick, Langmuir, 2008, 24, 2438.50 N. M. Dixit and C. F. Zukoski, J. Chem. Phys., 2002, 117, 8540.51 J. K. Cheung, V. K. Shen, J. R. Errington and T. M. Truskett,

Biophys. J., 2007, 92, 4316.52 R. P. Sear, J. Chem. Phys., 1999, 111, 4800.53 A. Shiryayev, X. Li and J. D. Gunton, J. Chem. Phys., 2006, 125,

024902.54 N. Wentzel and J. D. Gunton, J. Phys. Chem. B, 2008, 112, 7803.55 G. Pellicane, G. Smith and L. Sarkisov, Phys. Rev. Lett., 2008,

101, 248102.56 W. Smith and I. Nezbeda, J. Chem. Phys., 1984, 81, 3694.57 I. Nezbeda, J. Kolafa and Y. V. Kalyuzhnyi,Mol. Phys., 1989, 68,

143.58 I. Nezbeda and G. Iglesia-Silva, Mol. Phys., 1990, 69, 767.59 M. H. Ford, S. M. Auerbach and P. A. Monson, J. Chem. Phys.,

2004, 121, 8415.60 M. Wertheim, J. Stat. Phys., 1984, 35, 19, ibid. 35.61 M. Wertheim, J. Stat. Phys., 1986, 42, 459, ibid. 477.62 J. P. Hansen and I. R. McDonald, Theory of Simple Liquids,

Academic Press, New York, 3rd edn, 2006.63 R. P. Sear and J. G. Jackson, J. Chem. Phys., 1996, 105, 1113.64 G. Jackson, W. G. Chapman and K. E. Gubbins, Mol. Phys.,

1988, 65, 1.65 G. Chapman, G. Jackson and K. E. Gubbins, Mol. Phys., 1988,

65, 1057.66 E. Bianchi, P. Tartaglia, E. Zaccarelli and F. Sciortino, J. Chem.

Phys., 2008, 128, 144504.67 A. Gil-Villegas, A. Galindo, P. J. Whitehead, S. J. Mills,

G. Jackson and A. N. Burgess, J. Chem. Phys., 1997, 106, 4168.

68 A. Galindo, A. Gil-Villegas, G. Jackson and A. N. Burgess,J. Phys. Chem., 1999, 103, 10272.

69 C. DeMichele, S. Gabrielli, P. Tartaglia and F. Sciortino, J. Phys.Chem. B, 2006, 110, 8064.

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online

6410 Phys. Chem. Chem. Phys., 2011, 13, 6397–6410 This journal is c the Owner Societies 2011

70 C. Vega and P. A. Monson, J. Chem. Phys., 1998, 109, 9938.71 F. Romano, P. Tartaglia and F. Sciortino, J. Phys.: Condens.

Matter, 2007, 19, 322101.72 E. Bianchi, P. Tartaglia and F. Sciortino, J. Chem. Phys., 2008,

129, 224904.73 F. Sciortino, Eur. Phys. J. B, 2008, 64, 505.74 A. Lomakin, N. Asherie and G. Benedek, Proc. Natl. Acad. Sci.

U. S. A., 1999, 96, 9465.75 N. Kern and D. Frenkel, J. Chem. Phys., 2003, 118, 9882.76 J. Chang, A. Lenhoff and S. Sandler, J. Chem. Phys., 2004, 120,

3003.77 G. M. Whitesides and M. Boncheva, Proc. Natl. Acad. Sci.

U. S. A., 2002, 99, 4769.78 P. Buttler, J. Gen. Virol., 1984, 65, 253.79 H. Fraenkel-Conrat and R. C. Williams, Proc. Natl. Acad. Sci.

U. S. A., 1955, 41, 690.80 A. W. Wilber, J. P. K. Doye, A. A. Louis, E. G. Noya,

M. A. Miller and P. Wong, J. Chem. Phys., 2007, 127, 085106.81 A. W. Wilber, J. P. K. Doye and A. A. Louis, J. Chem. Phys.,

2009, 131, 175101.82 A. W. Wilber, J. P. K. Doye, A. A. Louis and A. C. F. Lewis,

J. Chem. Phys., 2009, 131, 175102.83 D. C. Rapaport, Phys. Rev. Lett., 2008, 101, 186101.84 W. K. den Otter, M. R. Renes andW. J. Briels, J. Phys.: Condens.

Matter, 2010, 22, 104103.85 S. C. Glotzer and M. J. Solomon, Nat. Mater., 2007, 6, 557.86 Z. Zhang and S. C. Glotzer, Nano Lett., 2004, 4, 1407.87 S. C. Glotzer, Science, 2004, 306, 419.88 J. P. K. Doye, A. A. Louis, I.-C. Lin, L. R. Allen, E. G. Noya,

A. W. Wilber, H. C. Kok and R. Lyus, Phys. Chem. Chem. Phys.,2007, 9, 2197.

89 E. G. Noya, C. Vega, J. P. K. Doye and A. A. Louis, J. Chem.Phys., 2010, 132, 234511.

90 Z. Zhang, A. S. Keys, T. Chen and S. C. Glotzer, Langmuir, 2005,21, 11547.

91 P. J. Flory, Principles of Polymer Chemistry, Cornell UniversityPress, Ithaca and London, 1953.

92 J. M. Tavares, P. I. C. Teixeira and M. M. T. da Gama, Phys.Rev. E: Stat., Nonlinear, Soft Matter Phys., 2010, 81, 010501.

93 M. von Smoluchowski, Z. Phys., 1916, 17, 557.94 M. von Smoluchowski, Z. Phys. Chem., 1917, 92, 129.95 P. G. J. van Dongen and M. H. Ernst, J. Stat. Phys., 1984, 37,

301.96 S. Corezzi, C. De Michele, E. Zaccarelli, P. Tartaglia and

F. Sciortino, J. Phys. Chem. B, 2009, 113, 1233.97 F. Sciortino, C. De Michele, S. Corezzi, J. Russo, E. Zaccarelli

and P. Tartaglia, Soft Matter, 2009, 5, 2571.98 P. Tartaglia, J. Phys.: Condens. Matter, 2009, 21, 504109.99 F. Sciortino, E. Bianchi, J. F. Douglas and P. Tartaglia, J. Chem.

Phys., 2007, 126, 194903.100 F. Sciortino, C. De Michele and J. F. Douglas, J. Phys.: Condens.

Matter, 2008, 20, 155101.101 E. Bianchi, P. Tartaglia, E. La Nave and F. Sciortino, J. Phys.

Chem. B, 2007, 111, 11765.102 E. Bianchi, J. Largo, P. Tartaglia, E. Zaccarelli and F. Sciortino,

Phys. Rev. Lett., 2006, 97, 168301.103 E. Zaccarelli, J. Phys.: Condens. Matter, 2007, 19, 323101.104 J. Russo and F. Sciortino, Phys. Rev. Lett., 2010, 104, 195701.105 P. R. Sperry, J. Colloid Interface Sci., 1984, 99, 97.106 P. N. Pusey, A. D. Pirie and W. C. K. Poon, Phys. A: Stat. Mech.

Appl., 1993, 201, 322.107 E. H. A. de Hoog, W. K. Kegel, A. van Blaaderen and

H. N. W. Lekkerkerker, Phys. Rev. E: Stat. Phys., Plasmas,Fluids, Relat. Interdiscip. Top., 2001, 64, 021407.

108 G. Foffi, C. De Michele, F. Sciortino and P. Tartaglia, J. Chem.Phys., 2005, 122, 224903.

109 S. Buzzaccaro, R. Rusconi and R. Piazza, Phys. Rev. Lett., 2007,99, 098301.

110 S. Manley, H. Wyss, K. Miyazaki, J. Conrad, V. Trappe,L. J. Kaufman, D. R. Reichman and D. A. Weitz, Phys. Rev.Lett., 2005, 95, 238302.

111 P. J. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino andD. A. Weitz, Nature, 2008, 453, 499.

112 E. Zaccarelli, S. V. Buldyrev, E. La Nave, A. J. Moreno, I. Saika-Voivod, F. Sciortino and P. Tartaglia, Phys. Rev. Lett., 2005, 94,218301.

113 P. Charbonneau and D. Frenkel, J. Chem. Phys., 2007, 126,196101.

114 P. Tartaglia and F. Sciortino, J. Phys.: Condens. Matter, 2010, 22,104108.

115 J. Russo, P. Tartaglia and F. Sciortino, Soft Matter, 2010, 6,4229.

116 J. Russo, P. Tartaglia and F. Sciortino, J. Chem. Phys., 2009, 131,014504.

117 J. M. Tavares, P. I. C. Teixeira and M. M. T. da Gama, Phys.Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 80, 021506.

118 J. M. Tavares, P. I. C. Teixeira and M. M. T. da Gama,Mol. Phys., 2009, 107, 453.

119 J. M. Tavares, P. I. C. Teixeira, M. M. T. da Gama andF. Sciortino, J. Chem. Phys., 2010, 132, 234502.

120 S. Corezzi, C. DeMichele, E. Zaccarelli, D. Fioretto and F. Sciortino,Soft Matter, 2008, 4, 1173.

121 S. Corezzi, D. Fioretto, C. De Michele, E. Zaccarelli andF. Sciortino, J. Phys. Chem. B, 2010, 114, 3769.

122 G. Foffi and F. Sciortino, J. Phys. Chem. B, 2007, 111,9702.

123 F. Romano, E. Sanz and F. Sciortino, J. Phys. Chem. B, 2009,113, 15133.

124 F. Romano, E. Sanz and F. Sciortino, J. Chem. Phys., 2010, 132,184501.

125 A. Giacometti, F. Lado, J. L. G. Pastore and F. Sciortino,J. Chem. Phys., 2009, 131, 174114.

126 F. Sciortino, A. Giacometti and G. Pastore, Phys. Rev. Lett.,2009, 103, 237801.

127 A. Giacometti, F. Lado, J. Largo, G. Pastore and F. Sciortino,J. Chem. Phys., 2010, 132, 174110.

128 F. Sciortino, A. Giacometti and G. Pastore, Phys. Chem. Chem.Phys., 2010, 12, 11869.

129 H. Liu, S. K. Kumar and F. Sciortino, J. Chem. Phys., 2007, 127,084902.

130 H. Liu, S. K. Kumar, F. Sciortino and G. T. Evans, J. Chem.Phys., 2009, 130, 044902.

131 H. Liu, S. K. Kumar and J. F. Douglas, Phys. Rev. Lett., 2009,103, 018101.

132 R. Fantoni, D. Gazzillo, A. Giacometti, M. A. Miller andG. Pastore, J. Chem. Phys., 2007, 127, 234507.

133 M. Noro and D. Frenkel, J. Chem. Phys., 2000, 113, 2941.134 E. G. Noya, C. Vega, J. P. K. Doye and A. A. Louis, J. Chem.

Phys., 2007, 127, 054501.135 G. Villar, A. W. Wilber, A. J. Williamson, P. Thiara,

J. P. K. Doye, A. A. Louis, M. N. Jochum, A. C. F. Lewis andE. D. Levy, Phys. Rev. Lett., 2009, 102, 118106.

136 G. Doppelbauer, E. Bianchi and G. Kahl, J. Phys.: Condens.Matter, 2010, 22, 104105.

137 B. A. H. Huisman, P. G. Bolhuis and A. Fasolino, Phys. Rev.Lett., 2008, 100, 188301.

138 E. Del Gado, J. Phys.: Condens. Matter, 2010, 22, 104117.139 E. Del Gado and W. Kob, Soft Matter, 2010, 6, 1547.140 E. J. W. Verwey and J. T. G. Overbeek, Theory of the Stability of

Lyophobic Colloids, Elsevier, Amsterdam, 1948.141 C. Gogelein, G. Nagele, R. Tuinier, T. Gibaud, A. Stradner and

P. Schurtenberger, J. Chem. Phys., 2008, 129, 085102.142 W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal

Dispersions, Cambridge University Press, Cambridge, 1989.143 P. Debye and E. Huckel, Phys. Z., 1923, 24, 185.144 S. Alexander, P. M. Chaikin, P. Grant, G. J. Morales, P. Pincus

and D. Hone, J. Chem. Phys., 1984, 80, 5776.145 E. Eggen and R. van Roij, Phys. Rev. E: Stat., Nonlinear, Soft

Matter Phys., 2009, 80, 041402.146 N. Boon, E. C. Gallardo, S. Zheng, E. Eggen, M. Dijkstra and

R. van Roij, J. Phys.: Condens. Matter, 2010, 22, 104104.147 C. N. Likos, R. Blaak and A. Wynveen, J. Phys.: Condens.

Matter, 2008, 20, 494221.148 M. Konieczny and C. N. Likos, Soft Matter, 2007, 3, 1130–1134.149 N. Hoffmann, C. N. Likos and J.-P. Hansen, Mol. Phys., 2004,

102, 857.

Dow

nloa

ded

on 2

4 M

arch

201

1Pu

blis

hed

on 1

8 Fe

brua

ry 2

011

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C0C

P022

96A

View Online