palma_2012

ISSN 0306-0012

0306-0012(2012)41:10;1-D

www.rsc.org/chemsocrev Volume 41 | Number 10 | 21 May 2012 | Pages 3701–4088

Chemical Society Reviews

TUTORIAL REVIEWCarlos-Andres Palma, Marco Cecchini and Paolo SamorìPredicting self-assembly: from empirism to determinism

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View Online / Journal Homepage / Table of Contents for this issue

http://dx.doi.org/10.1039/c2cs15302e

http://pubs.rsc.org/en/journals/journal/CS

http://pubs.rsc.org/en/journals/journal/CS?issueid=CS041010

This journal is c The Royal Society of Chemistry 2012 Chem. Soc. Rev., 2012, 41, 3713–3730 3713

Cite this: Chem. Soc. Rev., 2012, 41, 3713–3730

Predicting self-assembly: from empirism to determinismwzCarlos-Andres Palma,y Marco Cecchini* and Paolo Samorı*

Received 8th November 2011

DOI: 10.1039/c2cs15302e

Self-assembly is one of the most important concepts of the 21st century. Strikingly, despite the

rational design of molecules for biological and pharmaceutical applications is rather well established,

only few are the attempts to formally refine predictions of self-assembly in material science. In the

present tutorial review, we encompass some of the most significant e!orts towards the systematic

study of (thermodynamically stable) self-assembly. We discuss experimental and computer-

simulated self-assembly events in hard-matter, soft-matter and higher symmetry architectures

under the common framework of partition functions. In this framework, we endeavor to correlate

state-of-the-art chemical design, programming and/or engineering of reversible (thermal and

chemical equilibrium) self-assembly with knowledge of the underlying partition function

landscape in a step towards quantitative predictions and ab initio molecular design.

In recent years, the keyword self-assembly has flooded everydomain of natural sciences and found exciting refuge in emergingdisciplines such as material science and nanotechnology.1–3

Within these domains, self-assembly is defined as theprocess by which a system’s components arrange into pre-defined architectures under given boundary conditions.1,2,4–6

These conditions are described here as the action of anoise bath coupled to one or more external fields, withthe noise bath being a random force acting homogeneouslyon the ensemble, i.e. a heat reservoir or a thermostat.Interestingly, without a mediating noise bath, objects do notself-assemble.Molecular self-assembly is conceptually appealing not only

because it is the basis of life,7 but also because it is the path

ISIS & icFRC, Universite de Strasbourg & CNRS,8 allee Gaspard Monge, 67000 Strasbourg, France.E-mail: [email protected], [email protected] Dedicated to Professor Martin Karplus.z Electronic supplementary information (ESI) available. See DOI:10.1039/c2cs15302e

Carlos-Andres Palma

Carlos-Andres Palma completedhis B.Sc. in chemistry at theUniversidad de Costa Rica andPh.D. in physical chemistry atthe Universite de Strasbourgin 2010. After a postdoctoralposition in mass spectroscopyand photochemical reactionsat the Max Planck Institutefor Polymer Research he movedto the Technical University ofMuenchen where he is currentlyworking on reactions andthermodynamics at surfaces.His main interests are inbottom-up (supra)molecular

engineering, from quantitative modeling to (on-surface)synthesisof optoelectronic devices, and natural philosophy.

Marco Cecchini

Marco Cecchini is head of thelaboratory of ‘‘MolecularFunction and Design’’ at theInstitut de Science etd’IngenierieSupramoleculaires(ISIS) of the University ofStrasbourg. He received hisBSc and MSc degrees inChemistry from the Universityof Bologna (Italy) and obtaineda PhD degree in NaturalSciences from the Universityof Zurich (Switzerland).After a post-doctoral trainingwith Martin Karplus betweenthe University of Strasbourg

and Harvard University, he was recently appointed junior groupleader at ISIS (France). Last year, he was awarded a ‘‘Chaired’Excellence’’ CNRS prize and appointed assistant professor atthe University of Strasbourg. His research interests span thedomains of life science and material science. In particular, heaims at the elucidation of the principles of chemical design ofmolecules by means of theoretical and computational approaches.

y Current address: Physik-Department, Technische Universitat Munchen,James-Franck-Straße 1, D-85748 Garching, Germany.

Chem Soc Rev Dynamic Article Links

www.rsc.org/csr TUTORIAL REVIEW

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3714 Chem. Soc. Rev., 2012, 41, 3713–3730 This journal is c The Royal Society of Chemistry 2012

of lowest energy consumption. In addition, the bottom-upfabrication of supramolecular architectures with controlledthree-dimensional arrangements is key to the design of novelfunctional materials.8 In order to be technologically relevant,however, the outcome of self-assembly should be predictedright from the chemical nature of the building blocks, which ofcourse challenges the notion of causality. Causes and e!ects,e.g. as stated by Aristotle,9 have long been our motivation forunderstanding the hidden rules behind natural phenomena.But can one use such knowledge to direct and controlphenomena that are apparently lacking an actuator? Innatural sciences this implies harnessing the complexity ofself-organization of matter and coming up with an ab initio,programmed, or de novo design of molecular building blocksthat are able to arrange spontaneously into predefined archi-tectures. In this tutorial review, we will look into the principlesof chemical design with the ultimate goal of identifying guide-lines towards a deterministic prediction of self-assembly fortechnological applications.

1. Predictability, noise and fields

We first define the system as a collection of building blocks(n particles or molecules) with noise and fields in space andtime. Predicting the possible outcomes of the system is thedomain of statistical mechanics, where the predictability ofone event is subjected to maximization of information.The first important approximation of probability theory istime-independence, which holds in the limit of the ergodichypothesis.10 The second is to assume that events are inde-pendent such that they can be described by statistical weightsor frequencies. Under these assumptions, the predictabilityof a given event xB increases as its frequency f(xB) doesover the others. When normalized over all possible events,

the frequency factor of event B provides a measure of itsprobability.

PB ! f "xB#Pif "xi#

"1#

Also, the predictability of event B increases as its probabilitytends to 1. In the early days of statistical mechanics, LudwigBoltzmann noted that in order to explain the atomistic theoryof gases, the energy distribution of hypothetical particles wasto be maximized.11 From that point on, any event started to beassociated with a collection of microscopic states of the system(or microstates) with energy Ej. The frequency of the event wasthen approximated by

f "xB# !X

j2Be$bEj "2#

which bears the name of Boltzmann’s statistical weight andindicates that the probability of event B depends exponentiallyon the energy of the microstates associated with it, i.e. thelower their energy, the higher the statistical weight of theevent. There are many ways to derive Boltzmann’s factor, allof which must assume at one point the empirical observationthat b = 1/kT.10,12 In order for our first approximation tohold (i.e. the time-independent ansatz) the states variables suchas number of molecules, temperature, and volume (n, T and V,respectively) must be constant. Under this condition, which iscommonly referred to as thermal equilibrium, the probabilityof event B (eqn (1)) can be determined by summing up thestatistical weights of the sub-collection of states that aremicroscopically equivalent to B. Thus,

PB !

Pj2B

e$bEj

Pie$bEi

! ZB

Z"3#

where the normalization factor in the denominator, Z, is calledthe partition function and represents all possible microstates ofthe system under given ensemble conditions. Thermodynamicquantities can then be formulated in terms of the system’spartition function. At a given temperature and volume, theHelmholtz free energy is

F = $kT lnZ|T0,V0(4)

and entropy is defined as

S % $ @F

@T! @"kT lnZ#

@T

!!!!T0;V0

"5#

Incidentally, we note that when the energy levels accessible tothe system, Ej, are similar to one another (Ej - const.), theabsolute entropy provides a measure of the number of system’saccessible states, M, at thermal equilibrium (see ESIz):

S = k lnM|E0,V0(6)

Entropy is indeed a curious quantity because although it is thereason why self-assembly cannot be predicted analytically as aconsequence of the incredibly large number of accessiblestates, without entropy self-assembly does not occur.13

Following eqn (3), the probability of a given self-assembledevent can be predicted from the detailed knowledge of the system’s

Paolo Samorı

Paolo Samorı (Imola, Italy,1971) is full professor anddirector of the Institut deScience et d’Ingenierie Supra-moleculaires of the Universitede Strasbourg, where he is alsohead of the NanochemistryLaboratory. He is a Fellow ofthe Royal Society of Chemistryand junior member of theInstitut Universitaire deFrance. He obtained a Laureain Industrial Chemistry atUniversity of Bologna in1995. In 2000 he received hisPhD in Chemistry from the

Humboldt University, Berlin (Prof. J.P. Rabe). He was apermanent research scientist at Istituto per la Sintesi Organicae la Fotoreattivita of the Consiglio Nazionale delle Ricerche ofBologna from 2001–2008. He has published >150 papers onapplications of scanning probe microscopies beyond imaging,hierarchical self-assembly of hybrid architectures at surfaces,supramolecular electronics, and the fabrication of organic-basednanodevices. He is a member of the advisory boards of Journalof Materials Chemistry and Nanoscale (RSC).

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partition function. In classical statistical mechanics, the latteris approximated by a phase space integral14 in which thedependence of Z on particles velocities (momenta) movingunder the action of a field U(rn) becomes explicit

Z ! h$Dnn!$1

Ze$bPk

p2k=2mk&U"rn#

dpn drn "7#

with D the dimensionality of the system, n the number ofparticles, h the Planck constant, and pk and mk the momentumand the mass of the kth particle, respectively; note that eqn (7)is nothing more than the denominator of eqn (3) where thesummation has been replaced by an integral because of thecontinuum spectrum of energy in the classical framework andthe mechanical energy is expressed as the sum of kinetic energyplus potential energy. Although the phase space integral canbe evaluated for none but the simplest systems, the result ofeqn (7) is important for our theoretical interpretation as itmakes clear that self-assembly emerges from the action of arandom velocity bath (i.e. a heat reservoir) on the system’sdegrees of freedom (Dn). Indeed, temperature plays a criticalrole in self-assembly and, as we shall see, it must be taken intoaccount to achieve a deterministic control over the phenomenon.But, how can one evaluate the statistical weight of a given self-assembled event at thermal equilibrium? Or more precisely,how can one measure the fraction of microstates of the systemcorresponding to the self-assembled event over an astronomicallylarge configurational space? As a first step, we define two eventsonly, an initial event (A) and the sought final event (B) such thatA - B,z and set o! to compute statistics over them. In thisframework, eqn (3) can be e!ectively used to express the relativeprobability of event B over A:

PB

PA! ZB

ZA"8#

or equivalently the di!erence in Helmholtz’s free energybetween the initial and final events

DFB;A ! $kT lnZB

ZA

!!!!T0;V0

"9#

Fig. 1 provides a pictorial representation of such a perspective bysketching the (sub)partition functions ZA and ZB correspondingto the initial and final self-assembly events at thermal equilibrium;the latter should not be confused with the concept of chemicalequilibrium, as we shall see. Because (Helmholtz’s) free energy is astate function, i.e. a path independent property of the system,eqn (9) is indeed a useful result, as it allows us to evaluate DFB,Athrough the most convenient path including unphysical transi-tions15 and non-continuous transformations.16 In addition to that,eqn (9) provides a quantitative way to illustrate Fig. 1 bymeans of projections over ‘‘suitably’’ defined reaction coordinates;see Fig. 2.

The result of our theoretical interpretation (eqn (9)) suggeststhat the sole knowledge of the frequencies of events A and B atthermal equilibrium (i.e. the free energy change involved in thetransformation) would be su"cient for a quantitative predictionof self-assembly. This conclusion, however, relies on two ratherstrong assumptions. First, a pure thermodynamic analysis of

self-assembly provides no information on the kinetic accessibilityof the reaction products, which may eventually not form on thetimescales relevant for technological applications. Second, thethermal equilibrium framework (nVT) does not capture the roleplayed by changes in the number of molecules or equivalentlyconcentrations of the species involved in the self-assembly reac-tion. To tackle the first issue, sensibly more accurate predictionswould be obtained by analyzing the multitude of events aroundthe self-assembled state (B) as well as their interconversion intime. In principle, this can be done by computer simulations,which uniquely allow us to sample large collections of micro-states and thus to evaluate the frequency of the sampled events(A, B, C, . . .) by means of arbitrary order parameters. In thisperspective, a meaningful description of the free energy landscapeunderlying myriads of molecular events clearly compels amultidimensional representation, or hypersurface, whose human-readable projections too often provide an oversimplified pictureof the phenomenon.17 To cope with this, Rao and Caflisch, onthe one hand, and Krivov and Karplus, on the other hand, haveindependently developed tools based on complex networkanalyses in order to describe free-energy landscapes in termsof relative frequency factors.18–20 By clustering microstatessampled by atomisticMolecular Dynamics (MD) at equilibrium

Fig. 1 Pictorial representation of self-assembly at thermal equili-

brium (nVT) as described by eqn (9). For simplicity events A and B

are represented by only a few microstates contributing to the (sub)-

partition functions of the initial ‘‘dis-assembled’’ state (ZA) and the

final ‘‘self-assembled’’ state (ZB). Note that at thermal equilibrium

event B includes two multimeric self-assembled architectures and

monomers. Solvent molecules, if any, are implicit in both events.

Fig. 2 A traditional representation of a two-event partition function

landscape involving the molecular events A and B. The dashed lines

indicate a hypothetical thermal equilibrium path from A to B. The

di!erence in free energy between events A and B is depicted by energy

levels of di!erent height.

z The arrow indicates the direction of the spontaneous transformation.

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based on simple geometric criteria, i.e. RMSD, dihedralangles, strings of secondary-structure elements, etc., the aboveauthors came up with a 2D network representation of the freeenergy hypersurface with nodes and links corresponding tomolecular states and direct transitions between them,18 respec-tively. As shown in Fig. 3, such a representation enables for aprecise identification of the major basins of attraction (events)as well as a quantitative description of the free-energy barriers.20

Moreover, this technology can be used to determine the mostpopulated transition pathway(s) of a folding peptide.21

Strikingly, the network representation in Fig. 3 grasps the(real) predictive nature of the partition function landscape incontrast to the classical yet less intuitive model in Fig. 2.Unfortunately, deterministic analyses of this kind are solimited in time and space22 that performing an actual self-assembly experiment is in most cases more practical. Moreover,only events with relatively large frequencies can be e!ectivelysampled on the simulation timescale.

To address the sampling issue, one must push further oursecond approximation, i.e. the event independent ansatz, so asto factorize the event partition functions,111 e.g. into molecularcontributions. To this aim, we first define each canonical(sub)partition function as the product of n molecular partitionfunctions, one for each independent indistinguishable moleculeas Z = z1z2z3. . ., zn/n! = zn/n!. We note that n! is introducedto correct for the entropy of particle indistinguishability.23

Second we express the probability of self-assembly in terms ofthe free energy per molecule, or chemical potential, as

m % @F

@n! $kT

@

@n"lnZ#jV0;T0

"10#

which in the limit of large n reads23

m ! $kT lnz

n

" #jV0;T0

"11#

where n is the number of (supra)molecules.8 In this special caseof thermal equilibrium, as we shall see, the partition functionformalism uniquely allows for a molecular interpretation of theself-assembly reaction. In a chemical reaction, such as thecomplexation between monomers of species a into species b,reactants and products are related through an algebraic formulabb $ aa = 0, where a and b are called stoichiometric coe"-cients, and the di!erence in chemical potentials is defined asDmb,a % bmb $ ama (see ESIz). By introducing the result ofeqn (11) into the latter definition and rearranging, it gives

Dmb;a ! $kBT lnzbbzaa

& kBT lnnbbnaa

"12#

which expresses the chemical potentials di!erence in termsof molecular partition functions and the number of(supra)molecules (see ESIz for details). When bmb $ ama =0, an important condition occurs. The chemical potentials ofall species involved in the reaction equalize and the system issaid to be at the chemical equilibrium. By arbitrarily choosingthe partition-function term of eqn (12) as a reference (i.e. mob,aor Dmob,a), the chemical equilibrium condition dictates

Dmob;a ! $kT lnnbbnaa

jV0;T0! $kT lnKeqjV0;T0

"13#

where Keq is considered to be one chemical equilibrium con-stant. By comparing eqn (13) with eqn (12), it appears that thepartition-function term on the rhs of the latter is the chemicalequilibrium contribution to the chemical potentials di!erence,while the second term, which depends on the actual number ofmolecules in a and b, may be regarded as an out of chemicalequilibrium correction (see ESIz). Under the assumption ofidealized behavior, i.e. the particles’ indistinguishability ansatz,this result ultimately bridges the concepts of thermal and chemicalequilibrium and allows one to move from the constant-degree offreedom description in Fig. 1 to the chemically more intuitiverepresentation in Fig. 4. Most importantly, the ‘‘partition functionperspective’’ at chemical equilibrium provides a molecular frame-work for the theoretical investigation of self-assembly. In fact, itappears that a detailed knowledge of the molecular partitionfunctions for the initial and final events allows us to predict thedirection of the self-assembly reaction (see eqn (4)), quantify therole of entropy (see eqn (5)), and obtain fundamental insights onthe microscopic origin of self-assembly (see below). Furthermore,eqn (12) captures the out-of-chemical equilibrium character of theself-assembly reaction and predicts how sudden changes inreactants’ concentration will a!ect events populations and fluxes.Finally, as molecular concentrations are observables,

eqn (13) is fundamental for most empirical investigations ofself-assembly. Following the above interpretation of probabilities,it is natural to observe that for a deterministic prediction ofself-assembly ZA { ZB (see eqn (9)), or equivalently zaa { zbb

Fig. 3 The gradient minimum-spanning forest method allows for an

accurate description of the multiple-event partition function landscape of

the (GlySer)2 folding peptide. Notice how the local dynamic states of the

system (circles) can be e!ectively clustered through kinetic pathways into

valleys, thereby creating a tree-like structure showing distinct molecular

events (V2, V3, V4) around the folded native state (V1). Grasping the

probabilistic nature of eqn (8), the method renders the free-energy

representation in Fig. 2 essentially obsolete. Taken from ref. 21

8 With (supra)molecules we refer to all supramolecular architectureswhich could be formed by m building blocks: monomers, dimers,trimers, . . ., m-mers.

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(see eqn (13)) for idealized solutions. From this simple obser-vation, three principles towards achieving thermodynamiccontrol of self-assembly at thermal equilibrium readily emerge:(1) minimize the total number of degrees of freedom (Dn) andsystem’s accessible states in the dis-assembled form (MA) so asto reduce the size of the partition function of the initial eventZA (Low Initial Absolute Entropy). (2) Minimize the energy(of event B) in order to increase the size of the partitionfunction of the final event ZB, which is enthalpically stabilized(Low Final Enthalpy). (3) Maximize the ratio between MB andMA (High Relative Entropy); see Table 1. Interestingly, underthe assumption that rotatory and vibrational motions ofmolecules can be treated as independent, i.e. the harmonic-oscillator rigid-rotor approximation,23 the molecular partitionfunctions of the initial and final states of self-assembly can beseparated into independent contributions

z(V,T) = ztrzrotzvibzelecznucl (14)

each of which is expressed in terms of physico-chemical propertiesof the chemical species involved in the reaction (see ESIz).Importantly, the latter enables for a molecular interpretationof the above thermodynamic principles and translates theminto guidelines for chemical design; see Table 1. In fact, bydeveloping eqn (14) one observes that the absolute entropy ofthe dis-assembled state (a) grows either by increasing the massor the moment of inertia of the molecular building blocks, orby decreasing the frequency of their vibrational modes. Itfollows that in order to minimize the initial absolute entropy asprescribed by the ‘‘Low Initial Entropy’’ principle, optimumbuilding blocks for self-assembly should be light (low-molecularweight), symmetric (isotropic shape), and sti! (rigid). Furthermore,by manipulating eqn (14) it appears that the energy of the finalself-assembled state decreases linearly with the binding strengthper molecule, i.e. the stronger the molecular recognition is, thelower the thermodynamic energy of the assembled architecture. Asthe latter very much depends on the chemical nature of thebuilding blocks (i.e. both the recognition sites and the sca!old),the ‘‘Low Final Enthalpy’’ principle predicts that optimummodules for self-assembly should strongly interact in the finalself-assembled state and maximize their coordination number.Finally, eqn (14) yields the following expression for theentropy change on self-assembly (see ESIz),

DSob;a 'R

3

2lna$"a$ 1#Sa

tr

$ %

tr

&R3

2lnIbIa$"a$ 1#Sa

rot

$ %

rot

&RXk

i

lnvi;avi;b

&"a$ 1#X6

i

lnSi;bvib

!

vib

"15#

where a, I and n are the stoichiometric coe"cient, the moment ofinertia and the normal-mode vibrational frequencies, respectively.Eqn (15) is indeed a useful result, which in the limit of the aboveapproximations is seminal to understand the molecular nature ofself-assembly, as we shall see throughout the tutorial review.Specifically, it shows that the size (loga), the shape (log Ib/Ia),and the softness (log na/nb) of the final architecture are (supra)-molecular properties that could be optimized to predict self-assembly. However, as the entropic cost of association, which

Fig. 4 Pictorial representation of self-assembly under idealized chemical

equilibrium conditions, as described by eqn (13). In contrast to Fig. 1, the

‘‘dis-assembled’’ and the ‘‘self-assembled’’ events are now represented as

collections of microstates involving molecules exclusively populating their

monomeric or their multimeric form, respectively. Noteworthily, micro-

states representing event a or event b include, respectively, na and nbmolecules, the latter being the populations of the dis-assembled and the

assembled state at the chemical equilibrium. As molecules are treated as

independent (see the main text), the canonical partition functions of the

dis-assembled state (event a) and the assembled state (event b) can be

evaluated from the product of na and nb molecular partition functions,

e.g. Za ! znaa =na! Finally, note how molecular events at chemical equili-

brium (a and b) actually represent portions of the thermal equilibrium

event B depicted in Fig. 1. Solvent molecules are not considered.

Table 1 First-step thermodynamic design principles of self-assembly. In the box ma, Ia, and na correspond to the mass, the moment of inertia and thevibrational frequencies of the molecular building blocks in the dis-assembled state a; Eb corresponds to the depth of the potential energy of the groundelectronic state per molecule in the assembled state b; a and b are the stoichiometric coe"cients of the self-assembly reaction

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arises from the loss of three translational and three rotationaldegrees of freedom per binding event, is quite large and growslinearly with a, the softness of the final architecture appears tobe the sole term bringing a sizeable entropy compensation24

and should be carefully considered at the stage of chemicaldesign.25 Optimization of the latter may involve the rationaldesign of the topology of the final architecture or the fine-tuning of interactions between the building blocks (see ESIz).Finally, as the entropic cost of association (Sa

tr and Sarot)

depends on the mass and the moment of inertia of the buildingblocks, eqn (15) predicts that the entropy change on self-assemblycan be sensibly optimized by the use of low-molecular weight,highly symmetric monomers, especially for large extended archi-tectures. In addition, because most of the self-assembly reactionsoccur under fluid conditions, our chemical equilibrium analysisshould be extended to the degrees of freedom of the solvent (s). Inparticular, solvent molecules that are highly structured around thebuilding blocks in the initial monomeric state (s, a), e.g. formingclathrates, may introduce a sizable entropy compensation if theircoordinating ability is lost upon solute–solute association in thefinal solvent state (s, b). This transformation in the solvent, whichmay be seen as the reverse of building blocks self-assembly, canpotentially introduce large relative entropy contributions tosolute association and could be actually engineered as toachieve control over molecular self-assembly.

By and large, the partition-function principles presented inSection 1 (both in their ensemble and molecular formulations)provide a first-step towards the rational design of optimizedbuilding blocks and a suitable environment for self-assemblyunder thermodynamic control. In the following, Sections 2and 3 provide examples to illustrate how these very principlesunderlie most of the empirical realizations of self-assemblyreported so far. Finally, in Section 4 we discuss how the use ofrobust programmable building blocks and computer simula-tions for the design of optimal partition function landscapes arepromising routes to move from design principles to veritabledeterministic predictions.

2. High initial absolute entropy systems

Throw a handful of spherical marbles into a box: what is theprobability that such marbles pack into a perfect crystal?Immediately, the answer that pops up in our mind is ‘‘it depends’’,if not ‘‘slim’’. By restating the problem into a statistical mechanicsframework the question turns into: what is the probability ofcrystallizing hard spheres under a gravity field to a volumefraction of 0.74, which corresponds to the ideal hexagonal closepacking (hcp), through the action of a random force? In any case,our empirical answer based on intuition suggests that the phasediagram of a hard-sphere system is governed by entropy: there areso many accessible states that we cannot access a priori the relativestatistical weights of the self-assembled events. Moreover, theirprobabilities strongly depend on the space (volume) and noise(temperature) of the initial system. This is the realm of granularmaterials and, by analogy, of soft matter in specific cases.

2.1. Hard matter

The organization of hard matter has been extensively studied26

and is receiving increasing attention for micro device fabrication.27

For the purpose of this tutorial review we will consider onlythe self-assembly of non-stochastic (non-Brownian) particles,which constitutes a model system to discuss the role of entropyand noise in molecular self-assembly. In this area of research,Nahmad-Molinari and Ruiz-Suarez have extensively studiedthe assembly of hard macroscopic spheres exposed to agravitational field by kinetic epitaxial growth28 and vibrationalannealing.29 In particular, using a triangular box as a templatethey showed that a defect-free hcp crystal (Fig. 5a) could beformed when metallic beads were dosed into the box undershaking with energy proportional to the size of the crystal. Themicroscopic model proposed by Ruiz-Suarez and co-workersprovides a mechanistic interpretation of self-assembly. Thelatter involves a two-step reaction: first, particles enter thecrystallization box in the ‘‘gas phase’’ and condense intocrystal nuclei by energy dissipation, i.e. they relax into crystalpositions by releasing kinetic energy; second, the collapsedparticles start vibrating in phase and rearrange locally tooptimize the potential energy of the lattice. This interpretationis based on the observation that dosing beads at a rateproportional to the vibrational energy of the shaking systemwas shown to sensibly favor crystallization. Therefore, it wasspeculated that kinetic dosing favors self-assembly because itminimizes energy dissipation. Interestingly, kinetic dosing maybe considered as an ingenious way to decrease the absoluteentropy of the initial system below the bare minimum requiredfor self-assembly.30 Consistently, dosing beads at higher ratesincreases energy dissipation, and thus entropy, and e!ectivelyhinders crystallization. This example nicely illustrates our firstprinciple for predicting self-assembly: ‘‘Reducing the degreesof freedom (N) and the number of accessible states (M) to theabsolute necessary for self-assembly to occur’’.In light of this a new question arises: may the same packing

be achieved in a closed system through the action of aconstant-energy vibrational bath? This is particularly relevantfor our interpretation of self-assembly because the vibrationalbath acts as an energy reservoir and is the mechanical analogousof a temperature bath for molecules. Carvente and Ruiz-Suareztackled the problem by using an optimized vibrational annealingprocedure, which was introduced to overcome granular jammingin confined closed systems,29 i.e. the ball bearing system was

Fig. 5 (a) Single hexagonal-close-packed (hcp) granular crystals

grown by the epitaxial method, in a box of 13 000 steel ball bearings.

(b) Polymorphism in oil-saturated ball bearings showing the body-

centered-tetragonal (bct) and face-centered-cubic (fcc) phases. As the

rectangular box is commensurate with some proportions of the fcc and

bct phases, both may be observed in a self-assembled experiment.

Adapted from ref. 28 and 29.

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annealed at di!erent amplitudes (A) and frequencies (n) whilekeeping the acceleration a = A(2pn)2 constant. By using thesame metallic beads and a rectangular box, the above authorsachieved monodispersed packing into a body-centered-tetragonal(bct) architecture with a density of 0.69. Surprisingly, thearchitecture resulting from vibrational annealing was not theface-cubic-centered (fcc) packing obtained by epitaxialgrowth, which corresponds to the maximal theoretical densityof 0.74. However, the densely packed fcc structure could beobtained by addition of cohesive energy between the beads,e.g. oil. Ruiz-Suarez and co-workers interpreted this phenomenonas an enthalpy/entropy compensation that promotes crystal-lization into a fcc packing by increasing the partition functionof the final self-assembled event (ZB). The introduction ofgranular cohesion, in fact, reduces the ‘‘vibrational’’ energy ofthe beads and minimizes the potential energy of the fcc form.These observations illustrate our second principle for predictingself-assembly: ‘‘Minimize the energy of the final event in orderto increase the size of its partition function, ZB’’. Finally, it wasshown that even in the presence of cohesive forces, i.e. oil, thefcc structure could be transformed into the less-dense bct formby increasing the energy of the vibrational bath. This resultsuggests that for a dry system the free energy landscape under-lying self-assembly shows most probably a single minimum,while it has at least two (bct and fcc) when cohesion isintroduced. Thus, it appears that the subtle interplay betweenenthalpy (gravitational and cohesive contributions) and entropy(translational and vibrational contributions) on self-assemblyactually governs the complexity of granular polymorphism andwill ultimately determine the phase of the crystal (see Fig. 5b).

2.2. Soft matter

A Brownian particle itself can be obtained by molecular self-assembly. In particular, when molecules assemble into archi-tectures featuring limited local crystallinity and a high degreeof conformational freedom, the resulting material is regardedas ‘‘soft’’.31 This class of self-organized systems will be used toexplore our third principle for predicting self-assembly: the‘‘High Relative Entropy’’ principle. The entropy of a softarchitecture is high because the interactions between its buildingblocks are weak and isotropic (mainly of van der Waals type),

yet subject to phase segregation under incompatible solvents.As such, neither local symmetry nor long-range order can befound and the material behaves like a liquid. Indeed, such a‘‘fluid-like’’ character is part of the fascination with soft-matterself-assembly. The resulting architectures are not easily fractured(because defects are ‘‘allowed’’) and extend over long-rangedistances. Not surprisingly, ‘‘floppy’’ bilayers are largely usedin Nature as mesoscopic compartments for cells and organisms,whereas crystalline compartments are mostly found in nanosizedorganisms such as viruses.32 In general, the building blocksinvolved in soft-matter self-assembly are molecules with strongamphiphilic characters (amphiphiles) like detergents andblock-copolymers, which are known to form a wide varietyof self-assembled architectures33 (see Fig. 6).In analogy to hard matter, the self-assembly of objects

featuring short-range isotropic interactions strongly depends onthe initial absolute entropy, which is a function of the volume andthe number of molecules. Therefore, soft-matter self-assemblycritically depends on the concentration of the building blocks,which is sometimes called critical aggregate (or micelle) concen-tration (CAC). Consider the assembly reaction producing avariety of soft-matter architectures (event b) from an idealizedsolution of molecular building blocks (event a). At chemicalequilibrium and for m molecules contributing to the finalarchitectures the reaction’s algebraic equation is ma $ b = 0.

If the latter is rearranged as a$ bm ! 0, the condition of

chemical equilibrium becomes Dmob;a !mbm $ ma ! 0; wherein

the chemical potential di!erence corresponds to the freeenergy change of transferring one molecule from the mono-meric phase to the self-assembly state under standard condi-tions (Dmob,a). The stoichiometric coe"cients of the assemblyreaction being a = 1 and b = 1/m, eqn (13) gives

Dmob;a ! $ kT

mln

nbnma

"16#

where na and nb correspond to the number of particle species(i.e. molecules or supramolecules) at chemical equilibrium.Furthermore, by observing that the actual number of buildingblocks involved in self-assembled architectures, Nb, is relatedto the total number of architectures, nb, by the followingrelation Nb = mnb, eqn (16) can be formulated in terms ofnumber of monomers populating the dis-assembled, Na, or theassembled state, Nb, at equilibrium.

Dmob;a ! $ kT

mln

Nb

mNma

"17#

By rearranging eqn (17) one obtains35

Nb ! mNma e

$mDmo

b;akT "18#

which correlates the probability of self-assembly (Nb) with theconcentration of monomers (Na) and the free-energy changeper monomer upon association. Eqn (18) is indeed a usefulresult. In fact, by simply considering that Nb + Na = Ninit, itis straightforward to see that the actual number of freemonomers in solution, Na, cannot increase beyond

"Na#CAC ! eDmo

b;akT "19#

Fig. 6 A block copolymer and the possible self-assembled materials it

can form. Taken from ref. 34.

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otherwise Nb would exceed the initial number of monomers,Ninit, which violates mass balance (see ESIz). The result ofeqn (19) is called critical aggregate concentration35 and representsa keen strategy to predict the concentration of building blocks(or the number of monomers in a given volume) at which theself-assembled architecture b will start forming. Importantly,eqn (19) can predict the CAC analytically and a priori for anyamphiphile, provided that approximations for Dmob,a are available.Recently, accurate predictions for non-ionic polyoxyethylenesurfactants in water have been reported by Mackie andco-workers based on mean-field theory and the above model.36

Predicting the CAC from first principles (i.e. from the soleknowledge of the chemical structure of the building blocks andthe nature of their interactions) is conceptually an importantstep towards control over the self-organization of matter.

Despite the amazing simplicity by which eqn (19) appears toharness the volumetric (concentration) dependence of soft-matter self-assembly, chemical potentials may only provide aphenomenological description of the assembly reaction withno resolution on the sub-molecular scales. As such, eqn (19) isnot very useful to describe self-assembly events mediated byconformational changes of molecules or chemical events,which are critical for polymers37 and biomolecules.38 In addition,it provides little insights on the microscopic origin of order,which is subtly governed by the chemical nature of the buildingblocks. Indeed, once (CAC) self-assembly has taken place, themorphology and the structural subtleties of the final architectureare subjected to fine molecular design. In this respect, given thelarge variety of possible outcomes of soft-matter self-assembly(see Fig. 6), it would be crucial to know a priori whicharchitecture is thermodynamically most favored for a givenamphiphile, so as to achieve deterministic control. In principle,this can be done through a fine theoretical interpretation ofmolecular partition functions (see Section 1), though predic-tions are expected to be quantitative only for idealized solutionbehavior. Within these approximations, for instance, the shapedependence of self-assembly can be captured by the rotationalcontribution to the relative entropy change, which depends onthe moment of inertia of the final architecture over that of thebuilding blocks (see eqn (15)). Interestingly, the latter provides aquantitative way to access how changes in particles’ symmetry

upon association may change the self-assembly probability. Asan example, eqn (15) has been used to analyze the rotationalentropy change on the assembly of quasi-spherical particles intohighly anisotropic architectures for an increasing number ofmolecules (see ESIz). Fig. 7 shows that despite the sizablechange in the moment of inertia upon association, the shape-dependent entropy compensation is negligible over the rotationalentropy loss per monomer, which is shown to dominate theoverall rotational entropy contribution. Interestingly, the latterappears to be quite sensitive to changes in shape of thebuilding blocks (see Fig. 7c). As such, the rotational entropycost of association can be sensibly reduced by increasing thesymmetry of the self-assembly modules, especially for largeextended architectures. In short, the lighter and more symmetricthe monomers, the smaller the entropic penalty on self-assemblywill be (see Table 1).These results come with no surprise. It is empirically known,

in fact, that the ‘‘floppy’’ shape of the final architecture insoft-matter aggregates is mainly governed by simple geometricalinputs, which spatially maximize the energy of interaction in thefinal self-assembled state (area of head group, length andvolume of hydrocarbon tail).33,35 These foundations have beenestablished for amphiphiles and are now being explored for(di)block-copolymers.31,39 On the road to engineering persistent,structurally defined (low final entropy) supramolecular archi-tectures, this tutorial review will now focus on the chemicaldesign of the building blocks in the broad context of molecularself-assembly. To this end, in the following section we exploresuccessful attempts of empirical and rational design aiming atthe identification of rules and principles towards deterministicstructural predictions.

3. Engineering low final entropy events

Self-assembly into a highly ordered structure can be regardedas a physical transformation associated with a large decreasein entropy. As such, the design of a low-final-entropy eventconceptually corresponds to conceiving ways to allow for theemergence of spontaneous order. Thermodynamically, thiscorresponds to applying the principles highlighted in Table 1with special emphasis on maximizing the enthalpy of the

Fig. 7 Shape dependence of self-assembly. (a) Supramolecular architectures of di!erent shape and di!erent moment of inertia. The architectures

were modeled by placing an increasing number of quasi-spherical particles (C60 fullerene) onto linear (1D), square (2D), and cubic (3D) regular

lattices. The moment of inertia of each building block is Ia = 46 kg3 mol$3 A$6. (b) Rotational entropy loss on self-assembly. Relative rotational

entropies at 300 K are given in kcal mol$1. Despite the strong shape-dependence of the moment of inertia of the final architecture, the shape-

dependent rotational entropy compensation is negligible compared with the rotational entropy loss per monomer. (c) Shape design principle for

self-assembly. The rotational entropy loss is shown to be quite sensitive to changes in shape of the building blocks. As such, larger self-assembled

architectures will be more easily accessible, the more symmetric and lighter the monomers will be; see ESIz for details.

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assembled state, so as to compensate for the entropic penaltyof association. At the molecular level, the latter implies the useof strong and highly anisotropic interactions, such as hydrogen-bonding and metal–organic interactions, which promote self-assembly through directional molecular recognition events.Although crystalline structures are not necessarily the ultimategoal of self-assembly, they provide an ideal playground toelucidate the principles of chemical design of the constitutivecomponents (building blocks). This is the realm of supra-molecular engineering.

3.1. (Supra)molecular engineering

Despite the high degree of final order, engineering architecturesin three dimensions (3D) are known for their unpredictablenature.40 The reason for that is not surprising and relates to thefact that the growth of a multilayered crystal or aggregatefollows prevalently a preferred kinetic pathway. As such, thestatistical probabilities of various events (i.e. partition functions)are strongly dependent on time and space and in practiceBoltzmann’s ergodic and independent assumptions do not hold,i.e. the self-assembly reaction is not under thermodynamic

control. The unpredictability paradigm has been empiricallychallenged by supramolecular and crystal engineering41–44 whichendeavor the rational design of ordered matter. In recent years, asub-realm of the field, known as reticular chemistry,45 has attractedmuch attention because of its ability to produce controlled 3Darchitectures. Indeed, by using building blocks mainly featuringzinc-carboxylate salt/coordination bonds to engineer molecularcrystals, a series of successful attempts have been reported.46

In the following, on the road towards the rational design ofsupramolecular objects with atomic precision we present asystematic investigation of non-covalent self-assembly intodiscrete, one-dimensional, and two-dimensional architectures.These systems often self-organize at thermal equilibrium andas such can be reasonably approximated by Boltzmann’spartition functions.One archetypical example of discrete nano-objects formed under

thermal equilibrium conditions is metal–ligand macrocycles.47

Fujita and co-workers48,49 have carried out extensive studieson the formation of molecular polyhedrons based on squareplanar coordination compounds formed by pyridine moietiesand Pd+2. By varying the angle between the pyridine substituents

Fig. 8 Self-assembly into supramolecular polyhedrons. The crystal structures of the discrete architectures formed by the ligand L and Pd(NO3)2centers (M = Pd2+) are shown along with the chemical structure of the corresponding building blocks. By using pyridine moieties and Pd2+ as

recognition sites, supramolecular objects as di!erent as a cube, a rhombicuboctahedron, and a giant cuboctahedron could be successfully

engineered. Adapted from ref. 48 and 49.

Fig. 9 Softness dependence of self-assembly. (a) Vibrational entropy change on self-assembly for a series of 2D discrete architectures of increasing

size. Increasing the size of the architecture while keeping constant both the inter-particle distance and the interaction strength corresponds to lower

vibrational frequencies, which provide a larger vibrational entropy stabilization of the assembled state; i.e. the softness design principle. (b) Softness

dependence on the strength of the interaction between the building blocks. In the harmonic normal-mode approximation, the softer the spring

constant of the inter-particle potential, the larger the vibrational entropy of the assembled architecture is. Relative vibrational entropies at 300 K

are given in kcal mol$1. Vibrational frequencies were calculated by the program CHARMM50 using MMFF parameters;51 see ESIz for details.

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of the building blocks from 901, to 1271 and 1491, structurallydiverse discrete architectures were obtained under thermo-dynamic control (Fig. 8). The method established a rationalway to engineer supramolecular Platonic solids. In such systems,self-assembly occurs after a long equilibration (B1 day) at 70 1C atconcentration of 0.01 mM in DMSO. Interestingly, many inter-mediate kinetic products were detected during the early stagesof self-assembly, which were then found to undergo ripeninginto the (tentatively) thermodynamic polyhedron structure.

Supramolecular objects made through coordination bondswell exemplify the softness design principle of self-assembly(Table 1). Because coordination bonds are strongly directional,the vibrational frequencies of the assembled architecture can beestimated by normal-mode analysis. To illustrate the softnessdependence principle, the vibrational contribution to theentropy change on self-assembly was determined for a seriesof discrete model architectures of increasing size (Fig. 9).The architectures were modeled by carbon atoms connectedby Morse-potential springs of equal strength and the vibra-tional entropy change (i.e. third term of eqn (15)) determinedby normal-mode analysis (see ESIz for details). The results showthat the larger the supramolecular object is, the larger thevibrational entropy stabilization will be. Also, the latter is shownto depend on the strength of the interactions between thebuilding blocks and appears to increase sensibly when the forceconstant of theMorse potential is decreased from 15.206 md A$1

(i.e. a carbon–carbon double bond) or 4.258 md A$1 (i.e. acarbon–carbon single bond) to 0.206 md A$1 (i.e. an extremelysoft virtual bond); see Fig. 9b. In short, it appears that for agiven architecture the softer the bonds, the larger the thermo-dynamic stabilization; the trends shown in Fig. 9b are expectedto increase even more steeply if anharmonic (configurational)entropies were taken into account. Intriguingly, the softnessdependence principle suggests that if one could arrangeobjects in space such that the resulting architecture were much‘‘softer’’ than the individual monomers (i.e. it would vibrate atmuch lower frequencies), self-assembly could be entropicallydriven.25

In Section 2 of the present tutorial review we discussed howisotropic van der Waals (vdW) interactions may entropicallystabilize an assembled architecture by maximizing the numberof microscopically equivalent states (MB). In supramolecular

chemistry, vdW interactions can be used additionally tostabilize self-assembly enthalpically and promote molecularassociation into highly anisotropic architectures, such as(pseudo) 1D objects. Interestingly, both scopes are achievedby tailoring the molecular design of the building blocks. Largepolyaromatic molecules such as hexabenzocoronenes (HBC),for instance, feature strong anisotropic vdW interactions52 andspontaneously assemble into columnar phases.53 By combiningthe directional stacking capacity of HBC with the amphiphiliccharacter resulting from substituting side chains, Aida andco-workers54–56 succeeded in engineering bilayer nanotubes.Fig. 10 portrays high-resolution Transmission ElectronMicroscopy(TEM) images of the supramolecular architectures formed byshort, chiral and C60-decorated ethylenglycol chains. Althoughthe molecular arrangement has not been documented yet withatomic precision, 2D WAXD investigations revealed a stackingdistance of 0.347 nm between HBC cores, which is typical forstrong van der Waals stacking.52 In addition, the highly orderedarchitecture was shown to be robust to the self-assemblyconditions (e.g. using THF and other solvents at 0.001 Mconcentrations). Although the radius of the nanotubes cannotbe controlled yet, the system represents state-of-the-art supra-molecular wire engineering.Self-assembly at surfaces and interfaces is with no doubt the

most systematically studied field toward the bottom-up fabri-cation of controlled supramolecular architectures. Therein,the assembly reaction is inherently characterized by sizableentropy compensation, as upon confinement at interfaces(i.e. from 3D to 2D) molecules lose at least one translationaland two rotational degrees of freedom. It is therefore notsurprising that supramolecular engineering at surfaces andinterfaces has been demonstrated with exquisite control.41

Following our theoretical interpretation of self-assembly(Table 1), confinement of molecules in 2D is the most practicalstrategy to reduce the initial number of system’s accessiblestates (MA) and thus its absolute entropy. In this field, thedevelopment of chemisorbed self-assembled monolayers repre-sented a milestone.57,58 However, except for a few prototypicalexamples,59 engineering of local order in chemisorbed mono-layers as well as multicomponent patterning have not been yetachieved. Hitherto, the finest examples of molecular controlover the final architecture have been provided by using porous

Fig. 10 Transmission Electron Microscopy (TEM) images of morphologically defined nanotubes (below) from a function-engineered series of

polyaromatic amphiphiles (top). Adapted from ref. 48 and 49.

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physisorbed supramolecular networks as model systems. Inthis field, Barth and co-workers60–62 exploited the ability of thecyanide group to coordinate cobalt atoms on metallic surfaces(Fig. 11a) to create porous metal–organic networks; experi-ments were done under ultra-high-vacuum (UHV) conditionsafter the sublimation of the building blocks at the Ag(111)surface followed by exposure to a beam of cobalt atoms at300 K. We have recently engineered the cavities of a nano-porous bi-component network at the solid–liquid interface.63

At such an interface, the competition between solvation andadsorption (via physisorption) at the surface is critical, andbecause of this reason the use of molecules featuring a gooda"nity for the surface (like those incorporating an all carbonaliphatic or aromatic moiety) is mandatory. To explore self-assembly into a nanoporous bi-component network we havechosen as model systems molecules exposing complementarymelamine and uracil moieties which can undergo molecularrecognition (Fig. 11b); there self-assembly on graphite occurs onthe minute timescale from initial solutions in trichlorobenzene atconcentrations of 10$5 to 10$6 M.Molecular self-assembly at thesolid–liquid interface is also subject to concentration-dependent

polymorphism and phase segregation.63 As for phase segrega-tion, it can be of two types: the first one deals with favoring theadsorption of one component over the other and is knownas competitive adsorption; the second one relates to theformation of extended monocomponent domains at surfaces.Concerning polymorphism, De Feyter and co-workers showedthat it is highly concentration dependent. In fact, by imaging2D patterns formed by dehydrobenzo[12]annulene derivativeson graphite they reported a concentration-dependent equili-brium between the linear densely-packed structure and thehoney-comb network architecture stabilized by side-chainsinterdigitation. In addition, it was shown that depending onthe size of the building blocks the porous architecture wasfavored over the linear polymorph on a well defined concen-tration range.64 In the case of annulene derivatives exposinglong side chains (last two examples in Fig. 11c), for instance, asharp transition from the linear to the porous polymorph wasreported as the system approached monolayer concentrations.By contrast, for the derivatives exposing shorter side chains(first two examples in Fig. 11c), which result in lower interactionenergies as well as reduced binding a"nities for the substrate,

Fig. 11 Scanning Tunneling Microscopy (STM) of engineered nanoporous networks: (a) following ultra-high-vacuum deposition and (b), (c) at

the solid–liquid interface. Adapted from ref. 60–64 and 67.

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uncontrolled polymorphism was observed in most concen-tration ranges. Clearly polymorphism and segregation arisebecause of implicit dependence of the chemical potential onmolecular volumes, such that the same molecule may adsorbexposing di!erent areas and thus with di!erent energies.64–66

In order to avoid polymorphism, a building block must berigid enough to allow for only one possible conformation uponadsorption. Moreover, when monolayers with low densitiesare sought, e.g. porous networks, sub- or near monolayerconcentrations should be used.63

The formation of crystalline patterns involves the associationof hundreds of thousands of molecules. It follows that thetranslational entropy loss associated with crystal growth isexpected to be large and to increase with the size of thesupramolecular adduct. In the limit of the molecular partitionfunction approximations, the first term of eqn (15) allows usto quantify the translational entropy loss on self-assembly.The analysis of the translational entropy change on self-assemblyof model particles into a crystalline architecture (Fig. 12)provides interesting insight: first, the entropic cost of associationis very steep and grows linearly with the size of the aggregate;second, it strongly depends on the molecular weight of thebuilding blocks (ma). Both observations have important con-sequences. The former indicates that self-assembly into largeextended architectures (i.e. nanotubes or 2D crystalline motifs)is possible only through strong enthalpy/entropy or solvent-related entropy/entropy compensations. In the case of amphi-philic nanotubes, the hydrophobic and/or hydrophilic solvente!ects are of crucial importance for large assemblies to occur.

At interfaces, the large enthalpy/entropy compensation isintroduced by the substrate. In fact, it is not surprising thatrigid and planar (poly)-aromatic molecules are excellent candi-dates for 2D crystal engineering, as they strongly adsorb ongraphite52 and gold.68 The second observation related toFig. 12 suggests that the molecular mass of the building blocksis a key parameter that should be optimized at the stage ofchemical design. In particular, it is shown that the lighter themonomers, the larger the self-assembled architecture at equili-brium will be. Along with the shape and softness designprinciples, these results provide novel insight on self-assemblyand will have important consequences on the design of highlyrobust building blocks. For the sake of objectivity, it shouldbe finally noted that theoretical models based on the Sackur–Tetrode equation, such as eqn (15), tend to overestimatethe translational entropy loss, because the molecular volumesof the solute and the solvent are not taken into account and ‘‘freevolume’’ models69 should be used to obtain accurate predictions.The examples presented so far all represent ‘‘secondary-

structure’’ supramolecular engineering, a field in whichmolecules organize into discrete, 1D or 2D objects followingmolecular recognition events. The interest in engineeringhierarchical self-assembly, i.e. the self-organization of secondary-structure objects into higher order architectures, lies in thepossibility of amplifying molecular recognition events to themeso- and the macro-scale or controlling the morphology ofthe final architecture by chemical signals, i.e. binding of small-molecule compounds. A viable way of engineering tertiary-and quaternary-structure self-assembly is to exploit recognition

Fig. 12 Size dependence of self-assembly. (a) The translational entropy cost of self-assembly grows with the size of the architecture (a).(b) Dependence of the translational entropy cost on the mass of the building blocks. Interestingly, the lighter the monomer, the smaller the entropic

cost of association is. Relative translational entropies under standard conditions (i.e. T = 300 K and 1 M concentration) are given in kcal mol$1;

see ESIz for details.

Fig. 13 Cryo-SEM images of (a) an amyloid fiber and (b) the fibrillar architecture formed by an amphiphilic peptide. (c) Molecular dynamics

simulations of the peptide amphiphile sequence given in (b). The violet core is the hydrophobic region while the b-sheets are shown in yellow.

Adapted from ref. 71, 79 and 80.

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events between macromolecules, whose primary structures are onthe kilo Dalton scale. In the following, we will see how macro-molecular self-assembly may mediate the formation of complextertiary and quaternary-structure architectures in close analogyto biological systems, which is the topic of the next section.

3.2. Biomolecular engineering

Interactions in supramolecular chemistry are bio-inspired. Assuch, it is not surprising that the front line of supramolecularengineering is based on non-covalent interactions betweenamino acids. In this regard, proteins and nucleic acids constitutestereotypical models for self-assembly where local structuralorder is provided by secondary-structure elements, suchas a-helices and b-sheets,70 which in turn arrange intotertiary-structure super-elements to form complex three-dimensional objects. In peptide chemistry, one of the moststudied self-assembled architectures is amyloid fibers71

(Fig. 13a) whose formation proceeds by incubation of aqueoussolutions B0.001 M at low pHs for long periods of time.Although their tertiary structure remains atomically unresolved,self-assembly is mediated by the formation of secondary-structureprotofibers or fibrils, which have been recently crystallized fora series of amyloid peptides.72,73 Amyloid fibril formation ismediated by b-sheet assembly and is being a subject ofintensive research to engineer protein architectures.74 Caflisch,Cecchini and co-workers75 performed replica exchangeatomistic simulations of the early steps of aggregation of shortpeptides and measured significantly di!erent self-assemblypropensities for the amyloid-forming heptapeptide GNNQQNYand the soluble nonapeptide SQNGNQQRG from the yeastprion Sup35. Although full atomistic simulations are still veryexpensive, the group has recently investigated the kinetics andpathways of amyloid aggregation by coarse-grained models76

and shown that self-assembly may occur under nucleation(kinetic) control.77 These e!orts represent promising stepstowards the development of deterministic de novo protofiberdesign.

Exploiting b-sheet mediated self-assembly, it appears feasibleto engineer a wide variety of architectures with atomic precision.By mastering the basic principles of soft-matter engineering,78

Stupp and co-workers79,80 focused their endeavors on the designof b-sheet amphiphilic peptides coding for cylindrical micellesformation. Fig. 13b shows typical nanofibers (B8 nm in diameter)incorporating a hydrophobic core. Self-assembly occurs ataqueous concentrations of B10$4 M and pHs around 4.5.The peptide sequences AAAAGGG and SLSLAAA involvedin the b-rich region are typically b-sheet formers, but othersequences with stronger b-propensities may be engineered.81–84

Just recently, molecular dynamics simulations have shown thatthe actual fraction of peptides involved in b-sheet formation isonly 20%,63 thus suggesting that there is still room for optimiza-tion in the b-forming regions of the amphiphilic peptide sequences(Fig. 13c). In addition, other types of anisotropic interactions,such as disulfide-crosslinking,85 could be engineered around thesoft matter to explore the structural and dynamic properties ofresulting architectures. Because of the covalent character of theseinteractions, the so-formed cylindrical micelles would have alladvantages of the soft matter (see Section 2) plus being

exceptionally robust towards changes in the environmentalconditions; e.g. pH, ionic strength, etc.Another area of active research in bio-inspired supra-

molecular engineering involves the study of a-helices. In contrastto b-sheets, a-helices have a marked tendency to assemblethrough intra-molecular recognition events, i.e. interactionsamong chemical groups within the same polymeric chain. Assuch, they cannot be easily used to steer multicomponent self-assembly. One possibility, however, relies on coil-coiling theminto tertiary structures. Woolfson and collaborators86 demon-strated this principle by assembling a 28-residue peptide intoan extended coil-coiled fiber. However, full control over bothshort and long-range structural order was not achieved.Another type of a-prone structurally-defined intermolecularinteractions occurs in collagen’s triple helices.87 Baum, Brodskyand co-workers88 have thoroughly studied these interactions toengineer collagen mimetic peptides.89 Recently, Hartgerink,Fallas and collaborators have exploited the salt-bridginginteraction between lysine (K) and aspartic acid (D) residuesto assemble collagen-like helices frommultiple components.90,91

Fig. 14 compares the single-component trimer with the bis andtris component architectures. Strikingly, the self-assembly ofmultiple components, which takes place in 0.001 M bu!ersolutions at 85 1C for few minutes and incubated for B72 hat ambient temperature, converges to a single register archi-tecture. The method opens up a promising avenue towardsmorphology control over tertiary collagen architectures.

Fig. 14 NMR structures refined by molecular mechanics using the

Amber force field of synthetic (a) monocomponent, (b) bi-component,

and (c) tri-component collagen trimers. The color of the amino acid

sequence corresponds to the space-filling colors of the structures.90,91

Fig. 15 Cryo-TEM images (below) of (a) a tetrahedron, (b) a dodeca-

hedron, (c) a buckyball made through self-assembly between three-

component, three-point-star DNA motifs. Adapted from ref. 94.

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Achieving control over multi-component self-assemblydevelops the potential of knitting higher-order materials. Inthis regard, DNA represents the perfect building block and isde facto the only system known whose self-assembled archi-tecture (B-DNA) is invariant to the primary structure, i.e. thebase pairs sequence. Almost thirty years ago, Seeman demon-strated the principle of ‘‘knitting’’ tertiary structures withDNA.92,93 Recently, Jiang and co-workers94 have engineereda series of DNA polyhedra based on the refinement of theDNA-strand stapling technique (Fig. 15). In this case, self-assembly occurs via the recognition of three components intosecondary structure DNA motifs, which in turn self-organizeinto tertiary structure polyhedra. Importantly, such a hierarchicalassembly occurs in a ‘‘one-pot reaction’’ in aqueous solutionsat concentrations below 10$7 M by slow cooling from 95 1C to22 1C. Unlike metal–organic mediated polyhedron formation,however, hierarchical self-assembly into tertiary architecturesappears to be strongly concentration dependent. In fact,starting from the same secondary structure DNA motif thepolyhedron in Fig. 15b forms at concentration of 5 ( 10$8 M,while that in Fig. 15c assembles at higher concentration,i.e. 5 ( 10$7 M. Conversely, the self-assembly of the three-point-star building-block, which was rationally designed withthe help of a logical algorithm, is relatively concentrationindependent.

By and large, optimizing the rational chemical design ofthe building blocks as well as fine-tuning the experimentalconditions are useful strategies to enhance the predictabilityof self-assembly into ordered architectures. But, how farare we from being able to predict the result of self-assemblyfrom the chemical structure of the building blocks undergiven ensemble conditions? And, how much can we extendthese predictions to improve the chemical design and addnew derivatives to the series? The answer to these questionslies in programming the final architecture and designing itde novo.

4. Programmed and de novo designed architectures

As we have extensively illustrated, the ultimate goal of supra-molecular engineering is to program self-assembly. Becausethe amount of information to process is very large, predictingthe outcome of self-assembly from non-ideal initial conditions(ZA) naturally demands the use of computer programs andalgorithms. While computer assisted-design is widely used in

the life sciences, only recently the field has attracted the attentionof material scientists.95 The level of detail in computer modelingmay range from fully atomistic to simplified (coarse-grained)representation of the building blocks (a) and strongly dependson their propensity (or robustness) to self-organize into a pre-defined architecture (b); we note that a self-assembled event isconsidered to be robust when it does not strongly depend oneither the noise bath (temperature and solvent) or space (volumeand dimensionality) of the system. When the prediction of self-assembly involves the design of new molecular building-blocks,the process is referred to as de novo design96,112 (Fig. 17).Hitherto, the reaction path leading to the final self-assembled

event B from an initially disordered event A has been treated asindependent of the full system’s partition function Z, i.e. thefocus was on thermodynamics. In practice, however, engineeringthe partition functionsZB andZA through Table 1 is not su"cientto formally predict self-assembly. Referring to the famousLevinthal’s paradox,97,98 Fig. 16 illustrates the importance ofdesigning the entire partition function landscape (Z), as wasalready discussed in reference to Fig. 2. In his milestone paperon protein folding98 Cyrus Levinthal estimated that for asystem’s partition function with 10300 degenerate states describingthe event A (ZA) and a single low-energy state describing theevent B (ZB) it would take millions of years for the archi-tecture to self-organize, independently of the underlying

Fig. 16 Illustrative 3D representation of the partition function (free

energy) landscape: (a) Levinthal’s dynamically frustrated ‘‘golf-

course’’, and (b) a rugged funnel-like partition energy landscape.

Adapted from ref. 97.

Fig. 17 Computer-assisted de novo design. Two levels of iterations in

a feedback loop with simulations and experiments are proposed for a

deterministic prediction of self-assembly. First, the principles in

Table 1 are used as guidelines for molecular design. Then, computer

simulations are carried out to probe the underlying partition function

landscape and eventually refine the chemical design. Importantly, the

latter introduces the conceptual link between the chemical (building

blocks) and the physical (noise, space) nature of self-assembly. Finally,

predictions are tested by experiments, which impose new constraints

on the optimization problem, and a new cycle restarts. The refinement

procedure is repeated self-consistently until convergence is reached.

The refinement is naturally 0 when the prediction is 1.

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thermodynamic bias. It follows that for a deterministicprediction of self-assembly one should consider, at least tosome extent, the transition pathway (landscape) from theinitial (A) to the final (B) events. In other words, there is acompelling need to care about kinetics. Empirical andtheoretical solutions to this problem are illustrated in the nextsub-sections.

4.1. Computer algorithms

When the system’s building blocks are robust, straightforwardcomputer algorithms are enough for low-error rates in theprediction of self-assembly. In this case, because the number ofrelevant degrees of freedom is small, predictions and trendsmay be obtained by Monte Carlo simulations99,100 and ad hocenergy optimizations.82,101,113,114 In this context, Seeman andco-workers demonstrated that DNA is perhaps the best repre-sentative as a robust building block. Using DNA crossovers aswell as the ‘‘DNA-staple’’ strategy of Shih and co-workers,102

Rothemund and collaborators103 came up with a simple designalgorithm to program the folding of a single strand of DNAinto complex tertiary structure morphologies (Fig. 18a). In thisexperiment, the self-assembly of a 7176 nucleotide strand fromthe virus M13mp18 occurred by adding a 100 fold excess ofoligoDNA staples and cooling down the mixture from 90 to20 1C in 2 h. Importantly, such a folding strategy is not limitedto planar structures but can be e!ectively used to tailornanometre objects with molecular weights in the mega-Daltonrange and uncommon shapes.104 As an example, the vaseshown in Fig. 18b was formed from a mixture of 10 nM singlestranded M13mp18 DNA (7249 nucleotides) with 10 timesmolar excess of staple strands in a bu!er.105 Despite theamazing control conferred by the above technique over the

resulting architecture, tertiary structure self-assembly has afundamental limit. Nature teaches us that it is hard to designfunnel-like partition functions landscapes for architectures withan exceedingly large number of microstates, so that evolutionselected hierarchical self-assembly as a preferred way to producemesoscopic objects. In this view, quaternary structure self-assembly appears to be not only a better-suited strategy butindeed a requirement. Recently, Winfree and co-workers106

managed for the first time to program the morphology ofquaternary structure self-assembly by using a tertiary foldedDNA seed and tile adapters (Fig. 18c). In particular, theydemonstrated that the use of tile adapters confers full controlover the width of the self-assembled ribbon and most impor-tantly that the position of the tiles in the ribbon could beprecisely engineered. Algorithms of this kind applied to robustbuilding blocks for self-assembly, such as DNA, are nowapproaching quantitative predictions and provide control overthe number of errors and misfolding events. In combinationwith error-minimization protocols these strategies are expectedto provide practical means to control the design of funnel-likepartition functions landscapes for hierarchical self-assembly.

4.2. Atomistic simulations

Atomistic computer simulations provide alternative routes toachieve control over molecular self-assembly. In fact, thesetechniques provide a unique theoretical playground to bridgethe gap between the concepts of chemical design of themolecular components (Table 1) and the resulting archi-tecture. Atomistic simulations are readily accessible throughMolecular Dynamics (MD), which consists of using apotential energy surface inherent to the chemistry of buildingblocks and a simulated noise bath to integrate Newton’s

Fig. 18 Prediction (above) and experiment (below) of programmed DNA self-assembled architectures: (a) tertiary structure self-assembly into a

2D supramolecular object (i.e. a star) designed with the aid of a MATLAB code.103 (b) Tertiary structure self-assembly into a 3D supramolecular

object (i.e. a vase) achieved with no computer help.105 (c) Control over the width of a quaternary structure self-assembled nanoribbon achieved

through the use of (tertiary structure) self-assembled DNA tiles.106

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equations of motion for every degree of freedom.107 However,the timescales that can be e!ectively investigated by classical MDhardly go beyond a few hundreds of ns, such that typical self-assembly phenomena involving hundreds of millions of atomsand phase transformations are currently inaccessible. In this case,approximated descriptions based on a coarse-grained representa-tion of the system, i.e. by reducing the number of degrees offreedom, may e!ectively extend the scope of MD simulations atthe mesoscale on time scales approaching milliseconds.22

Despite the large e!ort devoted to the investigation ofmolecular self-assembly by simulations, only a few attemptshave been reported so far that go beyond the interpretation ofavailable experimental data so as to rationalize the principles ofchemical design.99,108,109 Recently, by using atomistic moleculardynamics simulations and an implicit representation of the substratewe have simulated the self-assembly of the bi-componentsystem depicted in Fig. 11c into a crystalline 2D hexagonalnetwork.108 Besides providing a full atomistic description of thetransition path from the melt to the nanoporous architecture,which was shown to involve a series of polygonal kineticintermediates, the simulations were used to design de novo abi-component system showing improved self-healing abilityin silico. Noteworthily, we demonstrated that the occurrenceof homo coupling (i.e.H-bonding among building blocks of thesame kind) introduces dynamic frustration (i.e. large barriers onthe self-assembly pathway) and e!ectively hinders crystallizationon the simulation (ms) time scales. On the contrary, upon removalof homo recognition either artificially or by chemical design thesystem was found to crystallize rapidly into the experimentallyobserved hexagonal pattern, as shown in Fig. 19b. This exampleillustrates how computer simulations can be e!ectively used tofind ways to carve funnel-like partition function landscapes (Z)13

and translate them into molecular properties and guidelines forchemical design. Analyses of this kind are expected to mediatethe rational design of novel robust self-assembly modules.

5. Summary

Chemistry of complex systems is a burgeoning discipline thatfocuses on the integration of informed molecular componentsinto pre-designed architectures to achieve dedicated tasks orgive rise to emergent properties. Within this context molecular

self-assembly stands out as a unique and elegant approach tothe bottom-up fabrication of advanced functional materials.For this and other reasons, self-assembly in 2D and 3D hasbeen recently referred to as ‘‘a certain growth point of thechemistry of the future’’ by O’Kee!e and co-workers.110

The recent years have witnessed the establishment of threerobust approaches excelling in predictable self-assembly in theframeworks of DNA, Metal–Organic and 2D Supramolecularchemistry. Yet, these fields represent dust grains in the chemicalspace available to the design of building blocks for techno-logical applications. The present tutorial review makes clearthat in order to produce architectures that could respond to thecontinuously increasing needs of material science, one mustbreak with the chemical picture of self-assembly and grasp thepredictive power of partition functions. We have presentedstate-of-art molecular design for self-assembly and interpretedchemically distinct successful attempts reported in the literaturewithin the common framework of statistical mechanics.The ‘‘partition function perspective’’ enables for preliminarytheoretical design of the building blocks, provides fundamentalinsights on the microscopic origin of molecular association bycapturing the role of entropy, and describes how suddenchanges in the reactants concentration may a!ect event popula-tions and fluxes. The emerging theoretical framework as well asthe long series of successful results reported in this reviewprovide unprecedented insights on the first principles of self-organization of matter and will hopefully help the developmentof computer algorithms and methods to predict self-assemblyab initio. Today, fully deterministic predictions of self-assemblyin closed systems are rare, yet we expect them to grow rapidlywith the aid of systematic theory, simulations and experimentalinvestigations. This field is expected to increase boundlessly infuture years, venture beyond the thermal equilibrium boundariesand contemplate open systems where gradients of pressure,volume, temperature and number of molecules determine thebehavior of matter in space and time.

Acknowledgements

We thank Jorge Fallas for providing the original rendering ofFig. 14. This work was financially supported by the ERCproject SUPRAFUNCTION (GA-257305), the ECMarie-Curie

Fig. 19 (a) Atomistic molecular dynamics prediction (above) and scanning tunneling microscopy experiment (below) of 2D bi-component self-

assembly at the solid–liquid interface. Self-assembly into a hexagonal network is the result of the recognition between melamine and uracil-bearing

building blocks on a graphite surface. (b) The self-assembly dynamics of 80 molecules into a single crystal when frustrated homo-recognition events

are artificially removed is shown. Adapted from ref. 108.

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ITN SUPERIOR (PITN-GA-2009-238177), FP7 ONE-P large-scale project no. 212311 and the International Center forFrontier Research in Chemistry (icFRC).

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