Self-assembly of monodisperse clusters: Dependence on target geometry
SSelf-assembly of monodisperse clusters: Dependence on target geometry
Alex W. Wilber, Jonathan P. K. Doye, ∗ and Ard A. Louis Physical and Theoretical Chemistry Laboratory, Department of Chemistry,University of Oxford, South Parks Road, Oxford, OX1 3QZ, United Kingdom Rudolf Peierls Centre for Theoretical Physics, University of Oxford,1 Keble Road, Oxford, OX1 3NP, United Kingdom (Dated: November 21, 2018)We apply a simple model system of patchy particles to study monodisperse self-assembly, using thePlatonic solids as target structures. We find marked differences between the assembly behavioursof the different systems. Tetrahedra, octahedra and icosahedra assemble easily, while cubes aremore challenging and dodecahedra do not assemble. We relate these differences to the kineticsand thermodynamics of assembly, with the formation of large disordered aggregates a particularimportant competitor to correct assembly. In particular, the free energy landscapes of those targetsthat are easy to assemble are funnel-like, whereas for the dodecahedral system the landscape isrelatively flat with little driving force to facilitate escape from disordered aggregates.
PACS numbers: 81.16.Dn,47.57.-s,87.15.ak
I. INTRODUCTION
The assembly of nano-structured materials and de-vices presents a great challenge for the future. A verypromising approach is offered by self-assembly processes,in which nanoscopic or colloidal building blocks come to-gether spontaneously to form ordered structures. Whiletop-down approaches to assembly become increasinglychallenging on small length scales, self-assembly offersa bottom-up approach which circumvents many of thesedifficulties, while providing its own unique challenges inthe synthesis and design of the subunits.Current examples of synthetic self-assembling systems,such as micelles and block copolymers, usually lead topolydisperse and imperfectly controlled products. Bio-logical systems, by contrast, display an astonishing vari-ety of ordered and precise self-assembly processes. Forexample, virus replication involves the assembly of hun-dreds of proteins to form highly symmetric and monodis-perse shells (capsids). Such systems provide an inspiringexample of the level of control possible in self-assembly.Future applications in nanotechnology are likely to re-quire this level of sophisticated control in order to formprecisely ordered structures.One attractive aspect of self-assembly is the idea thatthe structure of the final product is entirely determinedby the interactions between the subunits (along with thereaction conditions). Hence, given sufficient understand-ing of the principles of self-assembly, the subunits can bedesigned to produce a given product. In order to imple-ment these designs a great deal of control is required overthe structure of the subunits. One key requirement for so-phisticated self-assembly is the production of anisotropicsubunits, which only attract each other in certain direc-tions. Rational design of this anisotropy, both in theshape and in the interactions, then allows for control overthe resultant self-assembled structures.Recently there has been a great deal of progresstowards synthesising such anisotropic subunits.
Promising avenues included nanoparticles covered withmixed thiol monolayers, two-faced Janus and eventriphasic particles created at interfaces, small colloidalclusters where the gaps between the individual col-loids can be filled with a controlled amount of anothermaterial to create symmetrical ‘patchy’ particles and colloids with patches grown at the sites of contactsin a crystal. The production of functional self-assembling systemswill require not only advanced synthesis techniquesto produce the subunits, but also a strong theoreticalunderstanding of the self-assembly process so that appro-priate designs and conditions can be chosen. Recentlythere has been significant progress in the theory and simulation of monodisperse self-assembly. However, thesestudies have mostly focussed on the self-assembly of virus capsids or other proteincomplexes, andso include protein-like interactions which are morespecific than those which are likely to be available in thefirst generation of synthetic patchy particles.Here we consider a minimal patchy particle model,representing spherical colloids or nanoparticles withanisotropic interactions. Our model is intentionally sim-ple, so that we are easily able to observe and study be-haviours which we expect to be general properties ofmonodisperse self-assembly rather than being specific toour model. This simplicity also allows us to comprehen-sively survey the behaviour, and to connect the kineticswith the form of the underlying free energy landscapes.We have previously used the model to study the self-assembly of 12-particle icosahedra. Here, we build onthis work to consider the effect of the target geometry onthe self-assembly process, using the Platonic solids as ourtargets. We use these examples to address questions suchas whether self-assembly becomes more difficult with in-creasing target size, and whether particular geometricfeatures make some targets particularly difficult to as- a r X i v : . [ c ond - m a t . s o f t ] J u l semble. In doing so, we will pay particular attentionto the main processes which compete with successful as-sembly, and how their impact on the yield can be min-imised. We hope that the design principles we learn fromthis work will offer some guidance to experimental groupsseeking to make practical synthetic self-assembling sys-tems.Although the assembly process studied here is simi-lar to that of virus capsids, in that anisotropic parti-cles come together to form closed, highly symmetric shellstructures, the interaction potential we use has no depen-dence on the torsional angle between interacting parti-cles. While this is likely to be a good choice for modellingsynthetic anisotropic particles, it is not a good represen-tation of the interactions between proteins, and leads tobehaviour not observed for systems of virus capsomers.We consider a model including torsional interactions, andwhich hence more closely mirrors capsid assembly, in theaccompanying paper. II. METHODSA. Model
We make use of a minimal model, designed to con-tain only the essential features required for targeted self-assembly, while allowing for efficient simulation. Themodel consists of spherical particles patterned with at-tractive patches. They are described by a modifiedLennard-Jones potential, in which the repulsive partof the potential is isotropic, but the attractive part isanisotropic and depends on the alignment of patches oninteracting particles. Specifically, the potential is de-scribed by V ij ( r ij , Ω i , Ω j ) = (cid:26) V LJ ( r ij ) r < σ LJ V LJ ( r ij ) V ang ( ˆr ij , Ω i , Ω j ) r ≥ σ LJ , (1)where V LJ , the Lennard-Jones potential, is given by V LJ ( r ) = 4 (cid:15) (cid:20)(cid:16) σ LJ r (cid:17) − (cid:16) σ LJ r (cid:17) (cid:21) . (2) V ang is an angular modulation factor, which depends onthe orientations of the patches on the two interactingparticles, as well as the direction of the vector joiningthem. Specifically, V ang ( ˆr ij , Ω i , Ω j ) = G ij ( ˆr ij , Ω i ) G ji ( ˆr ji , Ω j ) , (3)where G ij ( ˆr ij , Ω i ) = exp (cid:32) − θ k min ij σ (cid:33) , (4) σ gives the width of the Gaussian, θ kij is the angle be-tween patch vector k on particle i and the interparticle (b) (c) (d) (e)(a) FIG. 1: (Colour Online) Single particles and complete clus-ters for the different target structures: (a) tetrahedron, (b)octahedron, (c) cube, (d) icosahedron and (e) dodecahedron. vector r ij , and k min is the patch that minimizes the mag-nitude of this angle. Hence, only the patches on eachparticle that are closest to the interparticle axis inter-act with each other, and V ang is one if the patches pointdirectly at each other. One feature of this potential isthat as σ → ∞ the isotropic Lennard-Jones potential isrecovered. For computational efficiency the potential istruncated and shifted at r = 3 σ LJ , and the crossover dis-tance in Eq. 1 is adjusted so that it still occurs when thepotential is zero.A particular particle is specified by a set of unit vec-tors describing the positions of the attractive patches.For each of our target structures, the patches are placedsuch that they point directly at the neighbouring par-ticles in the target structure. Fig. 1 shows the compo-nent particles and complete clusters for each of our targetstructures, the Platonic solids. Note that for these tar-gets all the particles and patches are equivalent.Somewhat similar patchy particle models have beenused to study gel formation, the crystallizationof proteins and patchy colloids and fibreformation. B. Dynamical simulations
In the simulations of our model we wish to representthe Brownian motion that colloids and nanoparticles un-dergo in solution. As we do not include any solvent par-ticles in our coarse-grained description of the system, asimple and efficient way to represent this dynamics is touse Monte Carlo (MC) where the moves are restricted tobe local, since this ensures that the dynamics are diffu-sive.In particular, we use Metropolis MC in the canonicalensemble, using periodic boundary conditions. The al-lowed move types are small single-particle translationsand rotations. The translational moves are randomlychosen from a cube centred on the selected particle.Rotational moves make use of a quaternion descriptionof the particle’s orientation; the proposed quaternion isgiven by the renormalized sum of the current quaternionand a smaller, randomly generated 4-vector.One potential problem with using single-particle movesis that, although free particles and clusters undergo diffu-sion as required, the relative diffusion rates of clusters ofdifferent sizes can be incorrect with the larger clusters dif-fusing too slowly. However, in practice for systems wherethe main mechanism of cluster growth is by monomeraddition rather than cluster-cluster aggregation, single-particle moves are sufficient. Indeed, preliminary simu-lations using the virtual move MC algorithm, which hasbeen very recently introduced by Whitelam and Geisslerand is designed to overcome this problem by using clus-ter moves, show only very minor differences to thosepresented here. By contrast, we have found that suchan algorithm is crucial for systems designed to assemblehierarchically.
C. Equilibrium simulations
To compute free energy landscapes and the positions ofthermodynamic transitions we make use of umbrella sam-pling. The essential idea is to bias the system such thatfree energy barriers are easier to cross, thus allowing thesystem to explore configuration space much more quickly.Under this scheme, instead of choosing the acceptanceprobabilities using a Boltzmann distribution, we use themodified distribution exp [ − β ( V + w ( Q ))], where Q isan order parameter and w ( Q ) is a weighting function.Canonical averages are simple to obtain from a simula-tion of such a non-Boltzmann (nB) ensemble using theexpression (cid:104) B (cid:105) NV T = (cid:104) B exp [ βw ( Q )] (cid:105) nB , (5)where B is some generic property of the system.Despite considerable effort, it did not prove feasibleto devise an order parameter that facilitated the forma-tion of multiple copies of the target structure. There-fore, we instead restricted our umbrella sampling simu-lations to the formation of a single target structure atthe same density. There are, however, two possible rea-sons why the equilibrium data for this finite-sized systemcan differ from that for bulk. Firstly, it does not allowfor interactions between the target clusters. But sincethe attractions between these clusters are small, this isa good approximation. Secondly, it does not allow forconfigurations where the fraction of the atoms in the tar-get structure is non-integer; in our umbrella samplingsimulations the only possibilities are for all the particlesto be in the complete target structure or none of them.For example, at the midpoint of the transition, the smallsystem will fluctuate in time between the target clus-ter and the monomeric gas state, spending half the timein each state, whereas for a large system one would ex-pect that at any given time the configuration will be amixture of target clusters and monomers with half theparticles in the target clusters. Analytical calculationshave indicated that this restriction has only a relativelysmall effect on the position of the centre of the transi-tion between monomers and the target structure (this iswhat we are most interested in here), but can cause an appreciable narrowing of the transition compared to thethermodynamic limit. A convenient order parameter to study the transitionbetween a monomeric gas and the target structure is thenthe number of particles in the largest cluster in the sim-ulation. The weighting function w ( Q ) was found by it-eratively performing simulations and improving w ( Q ) ateach iteration until the time spent at each value of Q wasapproximately equal.In the case of the dodecahedron we found that theabove order parameter was inadequate because the largerclusters that formed were disordered rather than basedon the dodecahedron. We therefore introduced a second-order parameter, namely the number of pentagons in thelargest cluster, with the aim of aim of promoting the for-mation of dodecahedral clusters. However, this was stillinsufficient to drive the system to assemble successfully.Instead, to finally achieve equilibrium we had tocombine this two-dimensional umbrella-sampling schemewith Hamiltonian exchange. In the latter, additionalexchange moves are introduced that involve swappingconfigurations between simulations with different Hamil-tonians, the purpose being to couple the system of inter-est to one where equilibrium is easier to achieve. In ourcase, we know that successful assembly of dodecahedrais possible for a similar patchy particle model, but wherea torsional component is included in the potential. Specifically, the attractive region of the potential is mod-ulated by an additional factor V tor = exp (cid:18) − φ σ (cid:19) , (6)where φ is a torsional angle between two particles, cho-sen such that φ = 0 in the target structure. In thelimit that σ tor approaches infinity, our original potentialis recovered.In our Hamiltonian exchange scheme we ran an arrayof 20 simulations at different values of σ tor ranging from1.25 radians to ∞ . The intermediate values were chosento maximise the rate of configurational exchange. Theprobability of an exchange move was 0 . σ tor were attempted.The acceptance probability that ensures detailed balanceis p acc (( i, , ( j, → ( i, , ( j, { , exp [ β ( W i, + W j, − W i, − W j, )] } (7)where i and j represent different Hamiltonians, and 1 and2 different configurations. W i, is the combined potentialenergy and umbrella weighting of configuration 1 with theHamiltonian i , i.e. W i, = V i ( r N ) + w i ( Q ( r N )). D. Structural Analysis
Many of our results make use of information on theclusters present in the simulations, and of the ringspresent within those clusters. We define two parti-cles as being bonded if their interaction energy satisfies V ij ≤ − .
4. Particles are part of the same cluster ifthey are connected by an unbroken chain of bonds. Weidentify target clusters by matching the number of parti-cles and bonds in the cluster to the profile for the targetcluster, including some allowance for bonds temporarilyweakened by thermal fluctuation. We define rings usingthe shortest-path algorithm given in Ref. 50; a closedloop is counted as a ring if there is no pair of particleson the ring connected by any chain of bonds shorter (i.e.consisting of fewer bonds) than the route by which theyare connected along the ring.
III. RESULTS
We first wish to identify the conditions under whichsuccessful assembly of the target structure occurs. Be-cause of the relative simplicity of our model, we areable to perform large numbers of simulations, and hencecomprehensively map out the assembly behaviour over arange of parameter space. Fig. 2 shows the dependenceof the final yields of each of the target structures on tem-perature (relative to (cid:15) ) and patch width in simulationsstarting from an initial random geometry after a givennumber of MC cycles. Although the yields in certain re-gions will tend to increase slightly with longer simulationtimes, the form of the diagrams is robust. Also note thatlow (high) temperature is equivalent to strong (weak) in-teractions.To first order the general shapes of the plots are sim-ilar. because the same physical principles apply in eachcase. However, there are also significant differences.Tetrahedra, octahedra and icosahedra assemble readilyover wide ranges of parameter space. By contrast, cubesare more difficult to assemble and are only formed ina smaller region of parameter space, while dodecahedranever form at all. We shall examine the reasons for thesedifferences in detail later.The region of successful assembly is determined by anumber of constraints, both thermodynamic and kinetic.Firstly, we find that at high temperature T the stablestate becomes a gas of monomers and small clusters, andtarget clusters are no longer formed. A second thermo-dynamic constraint arises at high patch width σ , wherethe patches are so wide that the target clusters cease tobe the most stable state of the system. Each patch thenbecomes capable of interacting with more than one neigh-bour, and so the system can both lower its energy andraise its entropy by forming large, unstructured liquid-like droplets.Both of these effects can be seen in Fig. 3, which showsthe average cluster size for each target structure, againas a function of T and σ . At high T the average clustersize approaches unity, signifying a monomer gas, while athigh σ the existence of a liquid phase results in clusterscontaining essentially all the particles in the simulation. FIG. 2: (Colour Online) The percentage yield of target clus-ters formed after 80 000 MC cycles as a function of the patchwidth σ (measured in radians) and the temperature for 1200particles at a number density of 0 . σ − , for particles de-signed to form (a) tetrahedra, (b) octahedra, (c) cubes and(d) icosahedra. No plot is included for particles designed toform dodecahedra, since under no conditions did any dodec-ahedra ever assemble. The white lines show the thermody-namic transition temperature T c for the transition from a gasof clusters to a gas of monomers calculated using umbrellasampling. FIG. 3: (Colour Online) The mean cluster size (averaged overparticles) of systems of particles designed to form (a) tetrahe-dra, (b) octahedra, (c) cubes, (d) icosahedra, and (e) dodec-ahedra, for the same simulations as Fig. 2. The white linesshow T c . FIG. 4: A schematic diagram illustrating the regions of pa-rameter space in which different behaviours are observed. T c is the midpoint of the transition between the cluster gas andthe monomer gas. T lv and T lc are the liquid-vapour phasetransition lines for the monomer and cluster gases, respec-tively. T agg g is the temperature below which aggregates be-come unable to rearrange to form target clusters on the timescale of the simulations, instead becoming trapped in a glass-like state. σ k is the value of σ below which the patches areso narrow that little assembly or aggregation is able to takeplace on the time scale of the simulations. σ min l represents thelowest value of σ at which the liquid state is stable and σ max c the highest value of σ at which the cluster phase is stable. Kinetic constraints become important at low values of T and σ . At low T the system becomes unable to escapefrom incorrect configurations. Since it very likely thatsome incorrect bonds will be formed during the assemblyprocess this results in very low yields, and the systeminstead forms glassy kinetic aggregates. At low values of σ the patches are very narrow. Particles will only rarelybe sufficiently well aligned to feel attractive interactionsfrom their neighbours, and growth is suppressed.Sandwiched in the middle of these regions of inhib-ited assembly is a region of parameter space in which thetarget is both thermodynamically stable and kineticallyaccessible. In this region good yields of the target struc-ture are obtained. These effects are summarised in Fig.4, which shows schematically the different regions of pa-rameter space and the lines which separate them. Wewill now examine these features in more detail, focussingon their dependence on the target geometry. A. The cluster-monomer gas transition
As there are only very weak attractive forces betweenclusters assembled into the target structures, T c , thetemperature above which the stable state is dominatedby monomers rather than target clusters, represents themidpoint of a chemical equilibrium between a monomergas and a cluster gas. The plots in Fig. 2 show that T c H ea t ca p ac it y p e r p a r ti c l e / k Temperature / ε k -1 IcosahedronTetrahedronCube OctahedronDodecahedron
FIG. 5: (Colour Online) The heat capacity C v as a functionof temperature for each of the target structures at σ = 0 . increases with patch width σ for all of the systems. Thisfeature can be explained by considering the free ener-gies of the monomer and cluster gases. At T c , A m and A c , the free energies of the monomer and cluster gasesrespectively, are approximately equal. A m is largely in-dependent of σ , as is the ground state potential energyper particle in the cluster gas, V gsc ≈ m(cid:15)/
2, where m isthe number of patches per particle. However, the entropy S c of the cluster gas increases with σ . At higher σ theclusters are free to undergo larger vibrations, leading toa higher vibrational entropy, and hence a larger T c .While this trend is a generic feature, T c also showsstrong target dependence. Fig. 5 shows the heat capac-ity C v as a function of temperature for each of the targetstructures at σ = 0 .
45. The peaks in the C v curves cor-respond to the cluster-monomer transition, and we de-fine T c as the temperature at which C v is a maximum.Note that the C v plot for dodecahedra exhibits a distinctshoulder to the right of the heat capacity peak, whichis actually indicative that this system is behaving verydifferently from the other four. We shall examine thedodecahedral system in detail in in Section III D.There are two main target-dependent factors affectingthe value of T c . Firstly, the number of patches per par-ticle m is important, since the more patches are presentthe more the cluster gas is energetically stabilised. Thevalue of n , the number of particles per cluster, has theconverse effect. When N particles assemble into N/n clusters there is a reduction in the effective number ofparticles in the system, and hence a corresponding re-duction in the translational entropy. We can obtain acrude estimate for the functional dependence of T c on m and n if we neglect vibrations and ignore the massdependence of the effective ‘particles’: T c ∝ m − n . (8) T c /(cid:15)k − Structure n m Measured PredictedTetrahedron 4 3 0.136 0.126Octahedron 6 4 0.157 0.151Cube 8 3 0.103 0.108Icosahedron 12 5 0.173 0.172Dodecahedron 20 3 0.081 0.099TABLE I: Comparison of T c values at σ = 0 .
45 obtainedfrom umbrella sampling simulations and estimated using Eq.8. The constant of proportionality in Eq. 8 was chosen to givethe best match to the data.
Table I shows estimated values obtained using this equa-tion, along with values obtained from umbrella sam-pling simulations. Despite the simplicity of the model,it matches the data reasonably well, and provides an ex-planation for the ordering of the T c values.The value of T c has an important impact on the abilityof a system to self-assemble, with lower values of T c mak-ing assembly more difficult, because the potential rangeof T over which assembly might occur is decreased. Itis noteworthy that cubes and dodecahedra have the low-est values of T c , and are the hardest of our targets toassemble.As the monomer-cluster transition is a chemical equi-librium, the heat capacity peaks have a finite width. Therelative width is expected to decrease with increasing tar-get size, and this can be seen in Fig. 5, with the excep-tion of the dodecahedral system, again indicating thatthis system is behaving differently. B. The cluster-liquid transition
For all systems, we see the formation of large aggre-gates both at sufficiently large values of σ and at low tem-perature for narrower patches (Fig. 3). The aggregatesformed in these two regions are in fact closely related,with their properties varying smoothly between them, asis especially clear for the dodecahedral case, where thereis no region of cluster formation separating the two re-gions of aggregate formation. Nevertheless, it is naturalto divide these aggregates into two groups. In the first,at high σ , the aggregates are the stable state of the sys-tem. In general these thermodynamic aggregates havesome liquid-like characteristics, being fairly compact andundergoing constant rearrangement. The second groupof aggregates, those found at low temperature and lower σ , are metastable, with the target cluster being the truestable state. These kinetic aggregates also display moregel-like properties, often forming extended networks ofstring-like formations, which may percolate. However,we have not sought to establish whether they form as aresult of kinetically-arrested liquid-vapour phase separa-tion, or gelation of a homogeneous fluid. The aggregates tend to have irregular forms. In thecase of the thermodynamic aggregates, so long as allpatches are pointing inwards, the surface tension is rel-atively low, because the energetic cost of the surface issmall. As a result there is a significantly smaller driv-ing force for forming spherical droplets than for systemswith isotropic interactions. In the case of the kineticaggregates, the reduced ability to rearrange because ofinsufficient thermal energy and narrow patches leads tothe formation of increasingly irregular and ramified ag-gregates at low σ . The energy of the aggregates tendsto become less negative as well, since these increasinglyglassy systems are less able to optimise their configu-rations to achieve lower energies. This effect is clearlyvisible in Fig. 6, which shows the dependence of the finalenergies in the simulations on T and σ for each of thetarget structures.The structures of the aggregates also have a strongdependence on the target. At temperatures and patchwidths close to those where target clusters can success-fully form, the local structure within the aggregates tendsto have a strong similarity to that in the target cluster,so that icosahedron-like groups are visible in the aggre-gates of icosahedron-forming particles, and so on. Thesestructural similarities are visible in Fig. 7 (a)–(d), whichshows snapshots of liquid aggregates near to the regionof successful assembly. The one target which consistentlyproduces aggregates with little resemblance to the tar-get structure is the dodecahedron. The wide splay angle(i.e. the angle between the patches and the symmetryaxis) allows the formation of a wide variety of structures,resulting in highly ramified structures containing manyhexagons and larger rings as well as pentagons (see Fig.7(e)). However as we move to higher values of σ the dif-ferences between the aggregates formed for the differenttargets are lost, so that the dense aggregates formed atvery high σ look essentially the same for all cases.Fig. 7(f)–(j) show typical kinetic aggregate structures.We can see that all of the kinetic aggregates have a ten-dency to form chain-like structures. In larger simula-tions these may sometimes connect together and perco-late across the simulation box. The local structure of thechains is again heavily dependent on the target struc-ture, reflecting the symmetries of the target. We canalso see that the tetrahedra have a greater tendency toform a number of smaller aggregates, which is consistentwith Fig. 3. The narrow splay angle of the tetrahedron-forming particles promotes high curvature and hencesmall clusters. This effect is lost at higher σ , as thepatch positions begin to have less control over aggregatestructure.Interestingly, similar ramified worm-like aggregateshave also been seen for Janus particles that have a hy-drophobic and a charged side. The similarity is probablybecause of the effectively one-sided nature of the attrac-tions for our patchy particles.The extent of the regions in which aggregates are foundis important, as it sets boundaries on the region in whichthe target structure may successfully be formed. In this
FIG. 6: (Colour Online) The normalised final total energy E/ Nm for systems of particles designed to form (a) tetra-hedra, (b) octahedra, (c) cubes, (d) icosahedra, and (e) do-decahedra, for the same simulations as Fig. 2. The energy isnormalised such that a value of − T c . (a) (b) (c) (d) (e)(f) (g) (h) (i) (j) FIG. 7: (Colour Online) Typical aggregate configurations (a)–(e) in the liquid-vapour coexistence region but (a)–(d) close to theregion of successful assembly, and (f)–(j) in the kinetic aggregation regime for particles designed to form (a) and (f) tetrahedra,(b) and (g) octahedra, (c) and (h) cubes, (d) and (i) icosahedra, and (e) and (j) dodecahedra. The values of ( kT /(cid:15), σ ) are(a) (0 . , . . , . . , . . , . . , . . , . . , . . , . . , .
35) and (j) (0 . , . . σ − . section we consider the factors affecting the region inwhich liquid-like aggregates are stable, while in SectionIII C we will return to consider those determining theextent of the kinetic aggregate region.It is useful to define some terms to describe the positionof the transition from the cluster phase to the liquid athigh σ . We define σ max c as the maximum value of σ atwhich the cluster phase is found (which will be at zerotemperature) and σ min l as the minimum value of σ atwhich the liquid phase is found. These two points areconnected by the transition line T lc . All three are shownin Fig. 4. σ max c depends entirely on the energies of the twostates, while the gradient of T lc depends on the entropydifference between the two states.Fig. 6 shows that the dodecahedra- and cube-formingparticles are the most able to satisfy their patchy inter-actions in liquid droplets, and attain the greatest bond-ing energies. The low number of patches per particle m and the wide angles between patches result in fewerconstraints on the liquid structure, making it easier tooptimise patch-patch interactions. We can see a clearcorrelation between the angles between patches and theenergy of the liquid. For the tetrahedra, octahedra andicosahedra, with angles between the patches of 60 ◦ , theparticles would be required to pack very closely togetherin order to satisfy all their bonds, and in fact we see rel-atively small bonding energies per patch. As we increasethe angle, to 90 ◦ for cubes and then 108 ◦ for dodecahe-dra, the energy per patch steadily increases in magnitude.The entropy of the aggregates might be expected tobe correlated with the energy. When there are fewerconstraints it is not only easier to satisfy patch-patchinteractions in a disordered configuration, but there aremore ways of doing so, leading to a higher configurational entropy. We would therefore expect larger aggregate en-tropies for the dodecahedra- and cube-forming particles.We can obtain some information about the relative en-tropies of the aggregate and cluster phases by consider-ing T lc . The shallower the gradient of T lc the greaterthe entropy differences between the aggregate and clus-ter phases, with negative gradients indicating that theaggregates are entropically favoured. Although calculat-ing precise values of T lc would be far from straightfor-ward, we can get an impression of how T lc varies with σ from Fig. 3. From those plots we can see that T lc hasa negative gradient for octahedra, icosahedra and proba-bly cubes, indicating that the liquid has a higher entropythan the cluster gas, and hence that the disorder in theliquid more than compensates for the loss of translationaldegrees of freedom on forming a large aggregate. T lc isclose to vertical for the tetrahedral system indicating thatthe entropies for the liquid and cluster gas are similar—asthe smallest of the targets, the tetrahedral clusters havethe largest translational entropy. The gradient for thedodecahedral systems is unclear from Fig. 3, because ofthe absence of target formation. Later in Sec. III D, wewill see that T lc has a far shallower slope for dodecahe-dra, which is consistent with our expectations based onthe low energy of the aggregates and their large clustersize. C. Mechanisms of assembly and misassembly
The dynamics of the self-assembly simulations dependstrongly on temperature and on the target. Fig. 8 showsthe yields of each of the target structures as a functionof T and time at σ = 0 .
45. These plots show a numberof interesting features.Firstly, in each case the yield approaches equilibriummost rapidly at high temperatures, close to T c . Here thedynamics are relatively fast, and since liquid-like aggre-gates are not stable with respect to the monomer gasat these temperatures, the clusters face no competition.However, close to T c the yield is limited by the finitewidth of the cluster gas-monomer gas transition, withsmaller clusters displaying a broader transition, as is clearin Fig. 8.At slightly lower temperatures the time taken to reachequilibrium becomes longer, both because the time scalefor rearrangement and the breaking of incorrect bonds islonger and because the diffusional time scales increase asthe assembly progresses towards higher yields. As a re-sult the temperature at which optimal yields are obtaineddecreases with time, being determined by the competi-tion between rapid assembly at high temperatures andhigher equilibrium yields at lower temperatures.Moving to still lower temperatures the yield on thetime scale of our simulations falls off, until around T agg g itbecomes severely limited. For target structures of tetra-hedra, cubes and icosahedra, reasonably sharp cut-offsare seen at low T , giving approximate values for T agg g . Inthe case of the tetrahedra the yield does not fall to zerobut rather to a finite value of around 20%, as some yieldof tetrahedra is expected by straightforward chance as-sembly with no rearrangement required. Unlike the othertargets, the plots for octahedra show an approximatelylinear decay in yields with decreasing temperature. Thereasons for this are unclear.Fig. 9(a) shows the yield as a function of time for eachtarget structure (except the dodecahedron) at the opti-mal conditions for the formation of that structure in Fig.2 (i.e. those that give the maximum yield after 80 000cycles). While the final yields obtained in each case ap-proach 100%, the time taken varies by almost two ordersof magnitude. The order of the time taken for tetrahedra,octahedra and icosahedra is consistent with their relativesizes, with larger clusters taking longer to assemble as ex-pected. Some tetrahedra form at very short times via thechance assembly mentioned above. However, the cubestake anomalously long to assemble, which is partly ex-plained by the narrow range of temperature over whichcubes can assemble (Fig. 8(c)) because of the low valueof T c for this system.We will now consider in more detail the dynamics inthe region of parameter space where the target clustersare most stable by considering three sub-regions of thatspace. Firstly, there is the regime where the target clusteris the only structure stable with respect to the monomergas (i.e. T lv < T < T c ). As a result assembly occursby direct nucleation of the target structure by the addi-tion of monomers and other small clusters to a growingcluster, rather than proceeding via an aggregate state.Because aggregates are not stable there are essentiallyno competing states to the target structure.Note that as this region is close to T c , the yield may be P e r ce n t a g e y i e l d o f t e t r a h e d r a Temperature / ε k -1 P e r ce n t a g e y i e l d o f o c t a h e d r a Temperature / ε k -1 P e r ce n t a g e y i e l d o f c ub e s Temperature / ε k -1 P e r ce n t a g e y i e l d o f i c o s a h e d r a Temperature / ε k -1 (b)(a)(c)(d) FIG. 8: (Colour Online) The yields of (a) tetrahedra, (b) oc-tahedra, (c) cubes, and (d) icosahedra after different numbersof simulation steps as a function of temperature T , in simula-tions of systems of 1200 particles at σ = 0 .
45 and at a numberdensity of 0 . σ − . Each data point is an average from fivesimulations. A v e r a g e c l u s t e r s i ze Monte Carlo cycles
TetrahedraOctahedraIcosahedraCubes 0 10 20 30 40 50 60 70 80 90 100 10 100 1000 10000 100000 10 P e r ce n t a g e y i e l d Monte Carlo cycles
TetrahedraOctahedraIcosahedraCubes (a)(b) FIG. 9: (Colour Online) (a) Yields and (b) average clustersizes (weighted over particles) as a function of time at the op-timal conditions for assembly of tetrahedra, octahedra, icosa-hedra and cubes, averaged over 100 simulations. The pa-rameters used were T = 0 . (cid:15)k − , σ = 0 . T = 0 . (cid:15)k − , σ = 0 . T = 0 . (cid:15)k − , σ = 0 . T = 0 . (cid:15)k − , σ = 0 .
45 for cubes. constrained by the finite width of the cluster-monomertransition, i.e. the equilibrium yield of clusters may besignificantly less than 100%. Lower temperatures willgive a higher equilibrium yield. In this region the speedof assembly becomes slower with decreasing σ due to thecomparative rarity of bonding events between particleswith narrow patches.In the second regime both the target clusters and largeliquid aggregates are stable with respect to the monomergas (i.e. T < T c and T < T lv but T > T agg g ), leadingto competition. However, mobility in the liquid dropletsallows them to rearrange to form target clusters. Oncecomplete clusters are formed all their patches point in-wards, such that they experience very little attractive in-teraction with other particles. They are therefore able to“bud off”, separating from the rest of the liquid dropletin which they formed.The optimal conditions for assembly are typicallyfound close to the boundary of these two regimes, at a point in parameter space where both the direct nucle-ation and “budding-off” mechanisms of assembly oper-ate Fig. 9(b) shows the average cluster size against timeunder optimal assembly conditions for each of the targetstructures. For the cubes (and to some extent the octa-hedra) the average size passes through a maximum beforefalling to the target cluster size, indicating clearly thatthe mechanism of aggregation followed by rearrangementplays a significant role. Even though the “budding off”mechanism is not sufficiently prominent to cause a maxi-mum in the cluster size for tetrahedra and icosahedra, itstill contributes to the rapidity of assembly under theseconditions.For T < T agg g assembly is suppressed because the ag-gregates act as kinetic traps. Hence, we define T agg g asthe temperature below which large clusters have insuf-ficient thermal energy to rearrange to form target clus-ters on the time scale of the simulation (and is thereforeweakly dependent on time scale). Below T agg g generallyonly large aggregates are formed, often forming extendedramified networks which may percolate. T agg g is dependent on target geometry for a number ofreasons. The first factor is the degree of order in the ag-gregates. Fig. 7 shows that the aggregates can containa considerable degree of local structure that is similar tothe target, and hence only a relatively small amount ofrearrangement is required for the aggregates to form com-plete clusters. The exception is the dodecahedral case,where the aggregates have very little dodecahedron-likestructure at all, and would require almost total rear-rangement to form the target clusters.A second factor is the stability of the aggregates, be-cause the more stable they are, the smaller is the ther-modynamic driving force for them to rearrange to findthe target structure. This factor is particularly relevantto the cubic and dodecahedral systems, which we sawearlier, had particular low energies and high entropies.A third factor is the size of the aggregates. In generala smaller splay angle in the assembling particles tends tolead to a higher surface curvature and hence to smalleraggregates, which are easier to rearrange. In Fig. 3 wecan see that the low- T aggregates formed with tetrahedraand octahedra as target structures are much smaller thanfor the other shapes.A final factor is the size of the target structures them-selves. Less rearrangement is needed to form a smallertarget. Further, for smaller targets, the growth of aggre-gates is less likely to deviate significantly from the correctassembly pathway. For example, for tetrahedra, some de-gree of successful assembly is observed even at very low T , because a significant proportion tetramers will havecome together by chance to directly form a tetrahedronwithout the need for any rearrangement. However, evenfor octahedra this effect becomes negligible.1 D. The difficulty of assembling dodecahedra
Of the target structures examined in this paper, thedodecahedron is exceptional in the difficulty of its as-sembly. While the other challenging target, the cube,shows a restricted region in which assembly occurs, itis nevertheless easy to obtain high yields by conductinglong simulations with carefully chosen parameters. Bycontrast dodecahedra are never formed with our modelpotential under any set of conditions.In order to more fully understand the difficulty ofassembling dodecahedra we attempted to obtain equi-librium thermodynamic data for a system of 20 parti-cles, sufficient to assemble one dodecahedron. This wasachieved only with considerable difficulty. Indeed, thepathological difficulty of forming a single dodecahedroneven using biasing techniques underlines the extreme un-likeliness of ever forming one under ordinary dynamics,and implies the presence of a large kinetic barrier to as-sembly. Success was eventually achieved by combiningtwo-dimensional umbrella sampling (using the number ofbonds and the number of pentagons in the largest clus-ter as order parameters) with Hamiltonian exchange asdescribed in Section II C.In general, because of the low energy of the targetstructures and the specificity of the interactions, onewould imagine that at moderate σ there must exist atemperature range over which the target is the only sta-ble structure with regard to the monomer gas, and sowhere aggregate formation does not compete with theassembly of the target. One might expect that in thisregion at least, dodecahedra should be able to assem-ble, so long as the time scales are sufficiently long thatthe nucleation barrier can be overcome. However, as wewill see, this is in fact not the case, since over most of therange of σ this region does not exist for the dodecahedralsystem.Fig. 10(a) shows a plot of the heat capacity C v alongwith a plot of the average cluster size for σ = 0 .
45. Attemperatures below the peak in C v the dodecahedron ismost stable. However, we can see that the average clus-ter size remains high beyond this point, indicating thatthe dodecahedron first melts before gradually evaporat-ing at higher temperature. The evaporation of this liquidcluster corresponds to the shoulder in the C v plot.The generality of this picture is confirmed in Fig. 10(b).The higher line in the figure shows the temperature atwhich the average cluster size in the box has a valueof 10 . T m , the temperature at which a dodecahedron melts.The presence of an intermediate liquid state for this 20-particle system, which persists down to σ ≈ .
25, was asurprise to us. For, although we expected the free ener-gies of the monomeric and cluster states to be relativelylittle affected by the small number of particles in theseequilibrium simulations, we expected the liquid state tobe considerably destabilized by the high surface to vol-ume ratio of any 20-particle liquid cluster. That liquid H ea t ca p ac it y C v A v e r a g e c l u s t e r s i ze Temperature T / ε k -1 Average clustersizeC v T e m p e r a t u r e T / ε k - Patch width σ Monomer gasDodecahedronLiquid
T T c (a)(b) FIG. 10: (Colour Online) Equilibrium properties of a sys-tem of 20 dodecahedron-forming particles in the canonicalensemble. (a) The heat capacity C v and average cluster size(weighted by particles) as a function of temperature. (b) T m (the temperature at the peak in C v ) and T . , (the temper-ature at which the average cluster size is 10.5) as a functionof σ . clusters are seen in these simulations is testament to thelow energy and high entropy of the aggregates for thedodecahedral system that we noted earlier. Thus, weexpect the region of stability for the liquid state to belarger for a bulk system than for this 20-particle system,i.e. T . can be considered as a lower bound to T lv , T m as an upper bound to T lc and 0.25 is an upper bound for σ min l These results thus allow us to begin to understand whythe dodecahedra are so hard to assemble. Given that atmoderate σ the stable state of the system at higher tem-peratures is as liquid droplets, the only region in whichdodecahedra might be able to form is at lower tempera-tures, where, although they represent the global free en-ergy minimum, aggregate formation competes with tar-get assembly. In particular, in this region the systemwill aggregate very quickly, and so for a dodecahedron toform these aggregates would then need to be be able torearrange sufficiently, a process which is severely inhib-ited by the slow dynamics at these low temperatures. For2 -6-4-2 0 2 4 6 F r ee e n e r gy o f s t a t e / k T
0 5 10 15 20
Number of triangles in largest cluster N u m b e r o f p a r ti c l e s i n l a r g e s t c l u s t e r -50-40-30-20-10 0 F r ee e n e r gy o f s t a t e / k T
0 2 4 6 8 10 12
Number of pentagons in largest cluster N u m b e r o f p a r ti c l e s i n l a r g e s t c l u s t e r (a)(b) FIG. 11: (Colour Online) Free energy landscapes for sys-tems of (a) 12 icosahedron-forming particles, and (b) 20dodecahedron-forming particles at T = T c and σ = 0 . the dodecahedral system though, such rearrangement ap-pears to be far more difficult than would be expected evenallowing for temperature. This is probably a result ofgeometric factors which are unique, within our set of tar-gets, to the dodecahedron. Because of the widely spacedpatches on the particle surfaces, an enormous range ofincorrect configurations are possible which satisfy thebonding well, and which deviate significantly from thetarget structure even on a local scale. For example, aswell as pentagons, (non-planar) hexagons or larger poly-gons can form without introducing strain. Furthermore,it is possible for bonds to pass through these rings, lead-ing to entangled networks of particles. These structureshave little in common with correctly formed dodecahe-dra, and further, they experience little pressure to rear-range.In order to obtain a clearer picture of the thermody-namic constraints on the dynamics of the system, wemade use of data from the Hamiltonian exchange sim-ulations to plot free energy landscapes in Fig. 11 and12, not only for dodecahedra but also for icosahedra andcubes. The icosahedral system provides a contrasting ex-ample of a successfully assembling system. All are at atemperature corresponding to the maximum in the heatcapacity. -16-14-12-10-8-6-4-2 0 F r ee e n e r gy o f s t a t e / k T
0 0.2 0.4 0.6 0.8 1 1.2
Radial disorder parameter -30-25-20-15-10-5 0 E n e r gy / ε -16-14-12-10-8-6-4-2 0 F r ee e n e r gy o f s t a t e / k T
0 0.2 0.4 0.6 0.8 1
Radial disorder parameter -12-10-8-6-4-2 0 E n e r gy / ε -70-60-50-40-30-20-10 0 F r ee e n e r gy o f s t a t e / k T
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Radial disorder parameter -30-25-20-15-10-5 0 E n e r gy / ε (a)(b)(c) FIG. 12: (Colour Online) Free energy landscapes for systemsof (a) 12 icosahedron-forming particles, (b) 8 cube-formingparticles and (c) 20 dodecahedron-forming particles at T = T c and σ = 0 .
45. The order parameters are the configurationalenergy and a measure of radial disorder, described in the text.
Fig. 11 shows the free energy as a function of the num-ber of particles and the number of correct polygons (tri-angles for the icosahedron and pentagons for the dodec-ahedron) in the largest cluster in the system. In Fig.11(a) we can see clear free energy minima representingthe monomer gas and the complete icosahedron. Betweenthe two minima is a region of higher free energy, with afree energy transition state with a significant amount oftriangular ordering. Once the transition state is passed,the shape of the landscape directs the system towardsforming an icosahedron, facilitating assembly. (Note thatthe high free energy of clusters of 12 particles with 19 tri-angles simply represents the fact that if one bond of an3icosahedron is broken two triangles are lost, and does notrepresent a barrier.)By contrast, the monomeric gas is not a free energyminimum for the dodecahedral system at temperaturesat which the dodecahedron is most stable (Fig. 11(b)).There is no barrier to the formation of a 20-particle clus-ter, so the system will tend to rapidly aggregate, andwill then experience little drive to form a dodecahedron.Indeed, if the system follows the steepest downhill pathit will tend to form a single aggregate containing fewpentagons, so that significant rearrangement would beneeded to form a dodecahedron.Fig. 12 shows the free energy as a function of two differ-ent order parameters, the potential energy and a radialdisorder parameter. The latter is defined as the stan-dard deviation in the distance of particles from the cen-tre of mass, and has a value of zero when all particleslie on a spherical shell (such as in the target structures).Again, the plot for the icosahedron shows two free en-ergy minima, corresponding to a monomer gas and toan icosahedron, separated by a transition state of higherfree energy. Importantly, we see that in order to obtainlower energies the system is forced to steadily reduce itsmaximum radial disorder, such that it is directed into theicosahedral state. The energy landscape is funnel-like. By contrast, the free energy of the dodecahedral sys-tem appears to be almost entirely a function of the con-figurational energy, with very little dependence on theradial disorder. Energies close to that of the global min-imum can be reached by highly disordered states. As aresult the free energy landscape is relatively flat, and thesystem experiences very little driving force to form anordered cluster. We also depict the landscape for cube-forming particles in Fig. 12(b). The intermediate diffi-culty of assembling the cubes is reflected in the broader,less-funneled nature of the lanscape compared to theicosahedron, but there is still some free energy gradienttowards the target structure.Nevertheless, neither of the dodecahedral plots fullyexplain the situation. From these diagrams one mightexpect the formation of dodecahedra to be rare, but notessentially impossible. It appears that there is a largekinetic barrier preventing the formation of dodecahedra,which is not visible in these diagrams due to the choice oforder parameters. Despite considerable efforts, we werenot able to find a combination of order parameters whichwould show this barrier explicitly.
IV. CONCLUSIONS
We have presented a study of the self-assembly of aclass of simple targets, the Platonic solids, using a mini-mal model of assembling patchy particles. We have com-prehensively mapped out the behaviours of systems ofparticles designed to form each of the targets as a func-tion of temperature and patch width. Further, we haveobtained equilibrium data using umbrella sampling and Hamiltonian exchange, allowing us to compare the ther-modynamic properties of the different systems, and toproduce free energy landscapes for the larger targets toelucidate the differences in their behaviour.We find that the behaviour of the systems and thesuccess of self-assembly are strongly dependent on thetarget structure. One key property which varies con-siderably between targets is the stability of disorderedaggregates, which have a complex relationship with theself-assembly process. Over large regions of parameterspace they act as competition and prevent successful as-sembly, but in other regions assembly may proceed byfirst forming aggregates which then rearrange to “budoff” the target structures. As a result the assembly ofcubes, whose constituent particles form particularly sta-ble aggregates, is dominated by this budding mechanism,while dodecahedron-forming particles form aggregates sostable that they effectively block assembly of dodecahe-dra. In general we find that the stability of aggregatesrelative to the target clusters is determined largely bythe spacing between the patches on the particles’ sur-faces. Wider patch spacing leads to fewer constraints onthe structures that the aggregates can adopt while satis-fying all the patchy interactions, which in turn leads toboth lower energies and higher entropies.The stability of the target clusters themselves alsovaries considerably between targets, and depends bothon the number of interactions each particle participatesin, and on the size of the cluster, where greater size tendsin general to decrease stability. High stability is a desir-able trait as it allows assembly at high temperatures (orequivalently for weak interactions), where the breakupand rearrangement of misformed structures is more rapid.We find that the dependence on shape can be such thatcertain shapes never assemble successfully; dodecahedraappear to be essentially impossible to correctly assemblein our model. The dodecahedron is a relatively unstabletarget because of its small number of interactions (eachparticle having only three nearest neighbours) and largesize. The aggregates with which it competes, by contrast,are exceptionally stable. As a result at moderate patchwidths the aggregates are stable to higher temperaturesthan the dodecahedra, and indeed there is no region ofparameter space where the dodecahedron is both ther-modynamically stable and kinetically accessible.These observations collectively suggest a number of de-sign rules for targets that will assemble easily. Firstly, ahigh number of nearest neighbours allows for more in-teraction which stabilise the target structure. Secondly,closely spaced patches serve to inhibit the formation ofstable aggregates which can provide competition. Thesetwo features can be combined by choosing shapes withtriangular faces, and as such we would argue that trian-gular faces are a major advantage in any target structure.Proceeding to larger and more ambitious self-assemblytargets, a difficulty arises. As the target size increases,the competition between the target and disordered ag-gregates increasingly favours aggregates, as the targets4become less entropically favourable. This suggests limitsto the size and complexity of targets of monodisperse self-assembly which can be successfully formed using the sim-ple interactions we have used here. One approach to cir-cumvent this difficulty might be to use multi-componentsystems with different patch types that interact selec-tively, the latter being potentially achievable using DNA-mediated interactions between the particles. An alter-native approach to this difficulty would be to introducefurther constraints to the potential. In particular a po-tential including torsional constraints, i.e. one in whichthe interactions are specific both in direction and in rel-ative orientation, would massively reduce the number of competing configurations available. Protein-protein in-teractions have this kind of specificity, and we expectthat this is crucial in the assembly of large structuressuch as virus capsids. We examine the effect of usingsuch a modified potential in the accompanying paper. Acknowledgments
The authors are grateful for financial support from theEPSRC and the Royal Society. ∗ Author for correspondence D. S. Goodsell,
Bionanotechnology (Wiley-Liss, Hoboken,2004). E. W. Edwards, D. Wang, and M¨ohwald, Macromol. Chem.Phys. , 439 (2007). S. C. Glotzer and M. Solomon, Nature Materials , 557(2007). S. Yang, S.-H. Kim, J.-M. Lim, and G.-R. Yi, J. Mater.Chem. , 2177 (2008). A. M. Jackson, J. W. Myerson, and F. Stellacci, NatureMaterials , 330 (2004). G. A. DeVries, M. Brunnbauer, Y. Hu, A. M. Jackson,B. Long, B. T. Neltner, O. Uzun, B. H. Wunsch, andF. Stellacci, Science , 358 (2007). K.-H. Roh, D. C. Martin, and J. Lahann, Nature Materials , 759 (2005). L. Hong, A. Cacciuto, E. Luijten, and S. Granick, Lang-muir , 621 (2008). K.-H. Roh, D. C. Martin, and J. Lahann, J. Am. Chem.Soc. , 6796 (2006). V. N. Manoharan, M. T. Elsesser, and D. J. Pine, Science , 483 (2003). A. Perro, E. Duguet, O. Lambert, J.-C. Taveau,E. Bourgeat-Lami, and S. Ravaine, Angew. Chem. Int. Ed. (2008). Y.-S. Cho, G.-R. Yi, J.-M. Lim, S.-H. Kim, V. N. Manoha-ran, D. J. Pine, and S.-M. Yang, J. Am. Chem. Soc. ,15968 (2005). Y.-S. Cho, G.-R. Yi, S.-H. Kim, S.-J. Jeon, M. T. Elsesser,H. K. Yu, S.-M. Yang, and D. J. Pine, Chem. Mater. ,3183 (2007). D. J. Kraft, C. M. Vlug, W. S.and van Kats, A. vanBlaaderen, A. Imhof, and W. K. Kegel, J. Am. Chem. Soc. , 1182 (2009). L. Wang, L. Xia, G. Li, S. Ravaine, and X. S. Zhao, Angew.Chem. Int. Ed. , 4725 (2008). D. Endres and A. Zlotnick, Biophys. J. , 1217 (2002). D. Endres, M. Miyahara, P. Moisant, and A. Zlotnick, Pro-tein Sci. , 1518 (2005). A. Zlotnik, J. Mol. Recog. , 479 (2005). R. Zandi, P. van der Schoot, D. Reguera, W. Kegel, andH. Reiss, Biophys. J. , 1939 (2006). M. F. Hagan and D. Chandler, Biophys. J. , 42 (2006). R. L. Jack, M. F. Hagan, and D. Chandler, Phys. Rev. E , 021119 (2007). M. F. Hagan, Phys. Rev. E , 051904 (2008). O. M. Elrad and M. F. Hagan, Nano Lett. , 3850 (2008). H. D. Nguyen, V. S. Reddy, and C. L. Brooks III, NanoLett. , 338 (2007). H. D. Nguyen and C. L. Brooks III, Nano Lett. , 4574(2008). H. D. Nguyen, V. S. Reddy, and C. L. Brooks III, J. Am.Chem. Soc. , 2606 (2009). T. Q. Zhang and R. Schwartz, Biophys. J. , 57 (2006). T. Q. Zhang, W. T. Kim, and R. Schwartz, IEEE T.Nanobio. , 235 (2007). B. Sweeney, T. Q. Zhang, and R. Schwartz, Biophys. J. , 772 (2008). D. C. Rapaport, Phys. Rev. E , 051905 (2004). D. C. Rapaport, Phys. Rev. Lett. , 186101 (2008). G. Villar, A. W. Wilber, A. J. Williamson, P. Thiara,J. P. K. Doye, A. A. Louis, M. N. Jochum, A. C. F. Lewis,and E. D. Levy, Phys. Rev. Lett. , 118106 (2009). Z. Zhang and S. C. Glotzer, Nano Lett. , 1407 (2004). K. Van Workum and J. F. Douglas, Phys. Rev. E ,031502 (2006). A. W. Wilber, J. P. K. Doye, A. A. Louis, E. G. Noya,M. A. Miller, and P. Wong, J. Chem. Phys. , 085106(2007). T. Chen, Z. Zhang, and S. C. Glotzer, Proc. Natl. Acad.Sci. USA , 717 (2007). A. W. Wilber, J. P. K. Doye, A. A. Louis and A. C. F.Lewis, J. Chem. Phys. submitted; arXiv.0907.4811. E. Bianchi, J. Largo, P. Tartaglia, E. Zaccarelli, andF. Sciortino, Phys. Rev. Lett. , 168301 (2006). F. Sciortino, Euro. Phys. J. B , 505 (2008). R. P. Sear, J. Chem. Phys. , 4800 (1999). N. Kern and D. Frenkel, J. Chem. Phys. , 9882 (2003). Z. Zhang, A. S. Keys, T. Chen, and S. C. Glotzer, Lang-muir , 11547 (2006). 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. , 2197 (2007). A. Shiryayev, X. Li, and J. D. Gunton, J. Chem. Phys. , 024902 (2006). B. A. H. Huisman, P. G. Bolhuis, and A. Fasolino, Phys.Rev. Lett. , 188301 (2008). S. Whitelam and P. Geissler, J. Chem. Phys. , 154101(2007). S. Whitelam, E. H. Feng, M. F. Hagan, and P. Geissler, Soft Matter , 1251 (2009). T. F. Ouldridge, A. J. Williamson, J. P. K. Doye and A.A. Louis, J. Phys.: Condens. Matter, to be submitted. Y. Sujita, A. Kitao, and Y. Okamoto, J. Chem. Phys. ,6042 (2000). X. Yuan and A. Cormack, Comp. Mater. Sci. , 343 (2002). J. D. Bryngelson, J. N. Onuchic, N. D. Socci, and P. G.Wolynes, Proteins , 167 (1995). M. M. Maye, D. Nykypanchuk, M. Cuisinier, D. van derLelie, and O. Gang, Nature Materials8