Hierarchical self-ahith

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scopic twisobjects havsystems thmesostructsurfactantstetrahedralbre-formintriesters [5]

Both thexamine thachiral polysimulationularly twisachiral parend-chains[9] have beently drivedifferent pa

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competing growth mechanisms were identied. Theyrepresent exciting manifestations of theoretical work

*Author for c

One contribution of 18 to a Theme Issue Geometry of interfaces:topological complexity in biology and materials.

Interface Focus (2012) 2, 651657Received 21 OAccepted 6 Mt in systems for which the self-assemblinge no underlying handedness. Experimentalat exhibit such behaviour include helicallyured silica nanobres templated by achiral[1,2], twisted ribbons of self-assembledcadmium telluride nanoparticles [3], certaing dendrimer systems [4] and achiral benzene.eory [6] and simulation [7] have been used toe onset of spontaneous chirality in modelmeric systems. More recently, particle-basedstudies have observed the development of reg-ted superstructures in systems comprisingticles. Specically, rods with both spherical[8] and bolaamphiphilic sphere chain systemsen shown to exhibit such behaviour, appar-n by local packing mismatch between therticle types involved.examine spontaneous chiral symmetry break-context of free self-assembly of brillar

Such brils are found in many biological andstems; they are both important nanocompo-heir own right and, apparently, signicant

conditions such as Alzheimers and Parkinson[10,11]. Hierarchical self-assembly of bres andmicelles is also central to the generation ofmanysilica structures [12]. There is some debate, howthe origins of the chirality observed in varioustructures [2,13]

A number of simulation studies of bre selhave been published in recent years employigrained models of cylindrically symmetricBolhuis [14] considered the behaviour of hadecorated with attractive interaction sitespoles. This patch potential not only promotedmation but also led to weak threadthread inThus, Monte Carlo simulations showed the spmerizing into threads, which subsequentlyinto bundles or bres. In a subsequent study[15] developed a mesoscale simulation modelused to investigate the self-assembly of sofmicelles in dilute solutions. This model wasform exible threads and, for some solvent cthese threads underwent further self-assemblyible and twisted hexagonal bundles. More recmixed experimental/simulation study, the selkinetics of colloidal Janus spheres have been in[16]. Both approaches observed formation omulti-thread bres, some with chiral characteorrespondence (d.j.cleaver@shu.ac.uk).of high-school physics. An issue of more contemporaryinterest, though, is the onset of mesoscopic or macro- selves be pathogenic with respect to neurodegenerativebres from acP. Prybytak1, W. J. Fr

1Materials and Engineering Research InstituShefeld S

2Unilever Discover, Colworth Labor

We investigate, by molecular dynamics simusolvent of Lennard-Jones spheres. When chon these systems, three regimes of self-assemnumerous short threads of discs develop, butQuenching to low temperatures, alternativeextended aggregate which typically comprisespacking defects. For a narrow temperature rachiral bres are found to freely self-assemble.from frustration between the hexagonal packthe pitch being set by the particle shape. Temerging owing to packing alone is pertinentself-assembling systems.

Keywords: self-assembly; structu

1. INTRODUCTION

The mechanism by which molecular handedness leads tooptical chirality on macroscopic length scales is the stuffctober 2011arch 2012 651ssembly of chiraliral particles2 and D. J. Cleaver1,*

Shefeld Hallam University, Howard Street,WB, UKries, Bedfordshire MK44 1LQ, UK

ion, the behaviour of discotic particles in aonic discsphere interactions are imposedy are observed. At moderate temperatures,se threads remain isolated from one another.causes all of the discs to oc into a singleeral distinct sections and contains numerousbetween these regimes, however, defect-freee spontaneous chirality of these bres resultsand interdigitation of neighbouring threads,s demonstration of aggregate-wide chiralitymany biological and synthetic hierarchically

l chirality; computer simulation

intermediate arrangements in certain hierarchical self-assembly processes. For example, it has been hypothesizedthat protobrils, which are intermediate structures

doi:10.1098/rsfs.2011.0104Published online 28 March 2012This journal is q 2012 The Royal Society

sddrij; ui; uj and 1ddrij; ui; uj take the standard solvophobic; this symmetry of amphiphilic behaviourpromotes what is termed chromonic self-assembly inwhich stacking of disc-like molecules is promoted by thedesire to shield their solvophobic faces. Chromonic stack-ing is exhibited by, for example, many dyestuffs, assummarized in an earlier insightful review [23]. Theadditional parameter R tunes the relative inuence ofthe repulsive and attractive components. The interactionpotentials (2.1) and (2.2) are illustrated in gures 1 and 2.

Following the standard convention for molecularsimulations [24], in what follows, we express system par-ameters and observables in terms of the length s0, theenergy 10 and the particle mass m. On adopting thisscheme, reduced temperature is expressed in units of10=kB, where kB is Boltzmanns constant and the unitof reduced time is

ms20=10

p. The MD time step used

here was 0.0015.Initial simulation results for this model, obtained

from small system runs, showed that R provides an effec-tive means by which to vary the quality of the solventand, so, control the formation of brillar structures.With R 1, cooling initially isotropic mixtures led tothe development of numerous linear stacks of discs,

1.0

0.5

0

0 0.5 1.0 1.5 2.0interparticle distance, r(s)

inte

ract

ion

ener

g

Figure 2. The R 1 (sphere-to-edge, red; sphere-to-face, darkblue) and R 5 (sphere-to-edge, light blue; sphere-to-face,black) discsphere interaction potentials for two relative orien-tations of the disc symmetry axis and the inter particle vector.

652 Self-assembly of chiral bres P. Prybytak et al.forms given in the study of Bates & Luckhurst [21]. Hereand in what follows, the sufx f denotes the disc face andthe sufx edenotes its edge. In this study,we considermod-erately discotic particles with aspect ratiosff=see 0:345.The energetic anisotropy of the discdisc interaction is setby the parametrization 1ee=1ff 0:1.

Lennard-Jones solvent particles, of size s0 sff andwell-depth 10, interact with the discs through a tuneableform of the generalized GayBerne interaction [22]. Fora disc i and sphere j, equation (2.1) simplies to

Udsij rij; ui 41dsrij; ui Rsff

rij sdsrij; ui sff

12"

sffrij sdsrij; ui sff

6#; 2:2

sdsrij; ui and 1dsrij; ui adopting the forms given in thestudy of Cleaver et al. [22]. It is straightforward to adjustthe nature of the discsphere interaction by varying theratio 1f=1e. Here, we consider the case 1f=1e 0:2, suchthat the edges of the discs are solvophilic and the facesexplaining the emergence of chirality in systems involvingboth intercalation and hexagonal packing [17,18].

Here, we use molecular dynamics (MD) simulationto investigate bre formation by discotic pseudo-ellipsoidal particles in a solvent of Lennard-Jonesspheres. From these, we show that, in appropriate con-ditions, the types of chiral clusters predicted by the zerotemperature energy landscape calculations of the studyof Chakrabarti et al. [19] can be realized in nite-temperature systems. This is achieved by followingkinetic crossover ideas [20], whereby the yield of stateswhich are thermodynamically stable but kineticallyrestricted is optimized at intermediate temperatures.

In 2, we present details of the model systemand simulation parameters used in this study. Follow-ing this, we present results obtained with oneparametrization of this model in four tempera-ture quenches. In this, we concentrate on structuralaspects of the resultant aggregates. Finally, we charac-terize the structure of the defect-free bre formed atintermediate temperature, discuss its origins and drawsome conclusions.

2. MODEL AND SIMULATION DETAILS

We consider behaviour of oblate GayBerne ellip-soids mixed in a solvent of Lennard-Jones spheres.The discdisc interaction employed is the variantintroduced by Bates & Luckhurst [21]

Uddij rij; ui; uj

41ddrij; ui; uj sffrij sddrij; ui; uj sff

12"

sffrij sddrij; ui; uj sff

6#; 2:1

where sff characterizes the thickness of the discs i and jand the anisotropic shape and well-depth functionsInterface Focus (2012)30

20

10

0

10

0 0.5 1.0 1.5 2.0

inte

ract

ion

ener

gy, U

dd

interparticle distance, r(s)Figure 1. The discdisc interaction potential for three combi-nations of relative particle orientations. Face-to-face, lightblue; edge-to-edge, dark blue; face-to-edge, green.

0.5

y, U

ds

was used (Lx 34.18); this proved signicantly largerthan any of the bre lengths observed in this study.

When these assemblies had relatively few threads(i.e. approx. three to ve), they exhibited considerableexibility, the threads exploring various packing arrange-ments. This exibility reduced, however, as the bundlesgrew. Also, subsequent to this, a small number ofbundlebundle aggregation events took place. One ofthese is apparent from the jump at t 1:4 106 dt inthe moment of intertia time lines shown in gure 4.These time lines show that, even before this jump, the lar-gest bundlewas elongated as two of itsmoments of inertiawere larger than the third. This difference was accentu-ated by the bundlebundle aggregation event, showingthat the two bundles attached end-to-end. Subsequentto this, the three moments of inertia all grew in nearmonotonic fashion, but the smallest grew signicantlymore slowly than the other two. Very little change wasseen over the last two million time steps of the run; bythis stage, virtually all of the discs in the system hadjoined the bre.

A conguration snapshot from the end of this run isshown in gure 5. Here, the discs are colour-coded accord-ing to their orientations, so as to make it easier to

Self-assembly of chiral bres P. Prybytak et al. 653To prepare an equilibrated high-temperature congur-ation, an initial system of 8788 particles (7908 spheresand 880 discs) was melted from a face-centred cubiclattice arrangement and simulated for 7 000 000 MDtime-steps at T 2.3. Once memory of the original lat-tice had been lost, no further transitions were observedin this equilibration run; the system remained isotropicand well mixed throughout. The nal congurationfrom this T 2.3 run was then used as the startingpoint for all other runs; in all cases, this isotropicarrangement was quenched to a lower temperature.

As well as standard thermodynamic observables, anumber of structural characteristics were monitoredin each quench simulation. Threads of discs wereidentied via a standard cluster-identication algor-ithm using a short cut-off distance. Higher orderstructures were then determined via a second-levelcluster-monitoring process which found adjacentthreads. The ndings of this analysis were cross-checked against animations showing system evolution.At regular intervals, the largest aggregate identied ateach given time step was further analysed via calcu-lation of its size, its principal moments of inertia andthe orientational order of its constituent discs. Inaddition to this, post-processing analysis was per-formed on fully formed bres to identify furtherstructural details.

3. RESULTS

An initial quench was made to a temperature of T 2:0.On cooling to this temperature, it was found thatshort thread-like assemblies of two to three discsoccasionally developed. Each of these was found to per-sist for a relatively short time before dissociating intomonomers and dimers.

Performing a deeper quench, to T 1:9, led to morenoteworthy self-assembly processes. Here, signicantlymore and longer threads were observed. These threadswere dynamic objects, with monomers joining andleaving on a regular basis. Once established, thesethreads proved long-lived and several could be tracedover hundreds of thousands of time steps. A typicalconguration from this run is shown in gure 3.consistent with previous simulations of chromonic self-assembly [2527]. These stacks were well solvated anddid not condense into a single aggregate on further cool-inglateral aggregation of the stacks was inhibited bythe strongly associated solvation shells. With R 5,the equilibrium arrangement at temperatures 1.8 was,again, a solution of monomer discs and length-polydis-perse threads of discs. Quenching below T 1:8 withR 5 led, however, to the development of anisotropicmulti-thread aggregates. We employ R 5 in all of thesimulations described below.

In 3, we present results from a series of constantnumber, volume, temperature MD simulations per-formed to investigate the behaviour of the model justdescribed. To reduce the inuence of the periodicboundary conditions, a large simulation box lengthInterface Focus (2012)Disaggregation of these longer threads commonlyinvolved a split into two shorter threads. Anotheroccasional feature seen at this temperature was lateralassociation, leading to second-generation double-thread mini-bundles. At T 1:9, these mini-bundlesproved meta-stable as they always re-divided into separ-ate threads. This initial observation of mini-bundleformation was, though, signicant since it signpos-ted the general propensity of this system to developmulti-thread assemblies.

In an attempt to encourage the formation of moreinvolved assemblies, a lower temperature of T 1:7was used in the next quench simulation. At this temp-erature, after an initial period in which numerousindividual threads were formed (each comprising upto eight monomers), several larger bundles developed.The threads in each of these bundles were approxi-mately parallel, leading to nascent bres whichgradually developed both laterally and longitudinally.

Figure 3. A typical conguration from the equilibrated phaseof the T 1.9 quench. For clarity, only discs in threads oflength 4 or more are shown.

654 Self-assembly of chiral bres P. Prybytak et al.0

5000

10 000

15 000

20 000

1 2 3 4 5 6 7time, dt (106)

mom

ent o

f ine

rtia,

ms

2

Figure 4. Time lines of the principal moments of inertia of thelargest cluster in the T 1.7 quench (largest eigenvalue Il,green; middle eigenvalue Im, blue; smallest eigenvalue Is, red).recognize differing local packing arrangements. Thus, seg-ments of uniform colour are straight, whereas segmentswith gradually changing hue are chiral. As expectedfrom themoment of inertia data, the nal aggregate is cer-tainly bre-like. However, at least three distinct segmentscan be identied: a central straight region and twisted seg-ments (of opposite handedness) at each end. Stackingdefects between these segments can also be identied.Animations of the self-assembly process conrm thatthis segmental structure originated owing to the evol-utionary steps experienced by the bre. Specically,in this quench, three signicant multi-thread bundlesdeveloped independently in the middle stage of the self-assembly. Each of these grew to a sufcient size that itretained its own local packing arrangement throughoutbundlebundle aggregation. This multi-segment-brearrangement was found to be long-lived when the run-time of this simulation was extended, the main segmentsand defect regions persisting indenitely.

On the time scales accessible here, the nal aggregateobtained from this T 1:7 quench proved to be

Figure 5. Final aggregate formed in the T 1.7 quench. Forclarity, no spheres are shown. Discs are colour-coded accord-ing to their orientation; neighbouring threads in chiralsegments therefore have differing colours.

Interface Focus (2012)0

5000

10 000

15 000

20 000

1 2 3 4 5 6 7time, dt (106)

mom

ent o

f ine

rtia,

ms

2

Figure 6. Time lines of the principal moments of inertia of thelargest cluster in the T 1.82 quench (largest eigenvalue Il,green; middle eigenvalue Im, blue; smallest eigenvalue Is, red).kinetically arrested; while the thermal energy in thesystem was sufcient to allow a degree of singleparticle-dissociation type relaxations, the strongerbundlebundle interaction strengths were too great forlarge-scale rearrangements to take place. As a result,the bre had no kinetically accessible route by which toadopt a defect-free structure.

In view of this, subsequent quenching runs wereundertaken at a range of temperatures intermediatebetween T 1:7 and 1.9. Of these, the run atT 1:82 proved optimal in that only one multi-thread bundle was nucleated. As evidenced by thecorresponding moment of inertia time lines (gure 6),this then grew into an elongated bre without exhibit-ing any of the discontinuities seen at T 1:7.Interestingly, the growth time scales apparent fromthese time lines proved relatively insensitive to, forexample, changes to the initial conguration. Thenal conguration from this run, shown in gure 7,was a defect-free, chiral bre which again contained vir-tually all of the discs in the system. As we show in the

Figure 7. Final conguration from the T 1.82 quench. Forclarity, no spheres are shown. Discs are colour-coded accord-ing to their orientation; defect-free helical threads thereforehave gradually changing colours.

4. CHARACTERIZATION ANDDISCUSSION

dista

nce

fr

2

3distance from central axis Il direction (s)

2

2

2

3 3

11

(b)

(c)

Self-assembly of chiral bres P. Prybytak et al. 655To characterize the defect-free chiral bre more fully,each of its threads was tted to the locus of a helixwith centre-line running along the eigenvector corres-following section, the chirality in this bre of achiralparticles resulted from frustration between hexagonalpacking and interdigitation of neighbouring threads.

Table 1. Parametric radius and pitch ts of individualthreads of the T 1.82 bre to ideal helices.

thread radius (see) pitch (see )

1 0.939+ 0.005 32.8+ 0.42 0.960+ 0.004 30.2+ 0.93 0.942+ 0.009 31.6+ 0.74 0.923+ 0.009 33.5+ 0.85 0.976+ 0.005 30.5+ 0.46 1.017+ 0.003 31.7+ 0.47 1.637+ 0.008 32.1+ 0.38 1.734+ 0.003 31.0+ 0.39 1.726+ 0.004 31.8+ 0.210 1.626+ 0.007 33.1+ 0.311 1.640+ 0.006 31.5+ 0.212 1.669+ 0.006 31.8+ 0.313 1.890+ 0.007 31.8+ 0.914 1.907+ 0.007 32.3+ 0.315 1.980+ 0.001 31.5+ 0.216 1.949+ 0.004 32.0+ 0.317 1.974+ 0.005 31.8+ 0.118 1.868+ 0.013 33.3+ 0.119 2.511+ 0.003 32.7+ 0.120 2.484+ 0.006 33.8+ 0.221 2.458+ 0.003 30.4+ 0.2ponding to the bres smallest moment of inertia.Using this approach, parametric ts yielded indepen-dent helical pitch and radius values for 21 of thebres 22 threads (the central thread was found to bestraight and so no t was performed). These ts aregiven in table 1. All of these give a consistent value of 32see for the emergent length scale that is the brepitch. Furthermore, their radial t parameters show atendency to adopt certain preferred values, the rst18 thread radii falling into three distinct groups.

The reason for this grouping is made apparent bygure 8a, which shows all of the disc centres projectedonto a plane perpendicular to the bre axis. This illus-trates that the bre had a straight central thread(layer 0) surrounded by six layer 1 helical threads.Six layer 2 threads then resided in the grooves pro-vided by layer 1 and, similarly, the locations of the sixlayer 3 threads were templated by the grooves inlayer 2. Finally, three layer 4 threads lay in groovesoutside the bre. Snapshots showing the layer 0 andlayer 2 particles in full and (so as to give unobstructedviews of these threads) the other particles as short linesare given in gure 8b,c. Both these reinforce the absenceof defects in this self-assembled structure and, throughtheir orientation-dependent colour coding, show thesmooth helices traced out by the layer 2 threads.

Interface Focus (2012)3(a)

2

1

00 1 2 3123

om c

entra

l axi

s in

I m di

rect

ion

(s)

1This structural characterization also indicates whythe bre formed here was chiral. The chirality wasfounded on the structural stability of a hexagonal7-thread nucleus comprising a straight central threadsheathed in six columns of slightly tilted discs, theouter threads spiralling around so as to maintain theintegrity of the bre. This arrangement was identiedby Chakrabarti et al. [19] as the optimal cluster forn 49 discotic ellipsoids.

The discs in the layer 1 threads were obliged to tiltbecause they had to interdigitate (with each other andwith the central thread) in order to achieve their radial

2

2

2

01

11

1

Figure 8. Views of the T 1.82 chiral bre (a) showing par-ticle centres only, projected onto a plane perpendicular tothe main thread axis and coloured for layer: green, 1; blue,2; red, 3; (b,c) different views showing the particles in layers0 and 2 fully resolved and colour-coded for orientation.

displacements from the central axis (note, from table 1,

formed, the chiral seven-thread core acted as a seed for

surfactant. J. Mater. Chem. 16, 41174122. (doi:10.

656 Self-assembly of chiral bres P. Prybytak et al.the subsequent bre growth.As predicted earlier [17,18], the chirality observed

here develops owing to the interplay of packing entropyand the inter- and intra-thread enthalpic contributions.Provided that the enthalpic contributions are in thecorrect regime, the helical pitch is controlled by pack-ing, i.e. details of the disc-particle shape. We willdescribe the complex kinetics of this hierarchicalgrowth process in a future publication. We also deferour contribution to the intriguing debate regardingthe suggestion that chiral bres may have self-limitingradii [28,29] to this subsequent paper.

It is noteworthy that, for all but the outermost layers,the threads formed in the T 1:82 bre had no defects.In contrast, anisotropic charge transport in discotic liquidcrystals [30,31] via electron hopping has only achieved lim-ited mobilities owing to the marked impact of occasionalin-column defects. This suggests that experimental realiz-ation of defect-free bres of the sort simulated here mayoffer a route to signicantly higher electron mobilitiesthan those achieved using bulk discotics.

In summary, we have shown that a defect-free chiralbre can self-assemble from achiral disc-shaped par-ticles provided that the temperature is set so as tolimit the system to a single nucleation process. Thebre chirality observed here results from the frustrationthat arises when both hexagonal packing and interdigi-tation of neighbouring threads are sought. Thisspontaneous symmetry-breaking through self-assemblyalso has more general implications for nanostructureformation. It demonstrates that it is not always necess-ary for the constituent particles to manifest all of theproperties required of a desired nanostructuregivenappropriately controlled conditions, such propertiesand symmetries can simply develop spontaneously viastructural chirality.

P.P. acknowledges nancial support from Shefeld HallamUniversity and Unilever. This work has beneted fromdiscussions with Chris Care, Tim Spencer, DwaipayanChakrabarti, Gerd Schroder-Turk and Mike Butler.Snapshot images were produced using GBVIEW visualizationsoftware developed by David Michel.

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Self-assembly of chiral bres P. Prybytak et al. 657Interface Focus (2012)

Hierarchical self-assembly of chiral fibres from achiral particlesIntroductionModel and simulation detailsResultsCharacterization and discussionP.P. acknowledges financial support from Sheffield Hallam University and Unilever. This work has benefited from discussions with Chris Care, Tim Spencer, Dwaipayan Chakrabarti, Gerd Schroder-Turk and Mike Butler. Snapshot images were produced using GBview visualization software developed by David Michel.References