Kinetically guided colloidal structure formationFabian M. Hechta and Andreas R. Bauscha,1

aLehrstuhl fur Zellbiophysik E27, Technische Universität Munchen, 85748 Garching, Germany

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved June 14, 2016 (received for review April 4, 2016)

The self-organization of colloidal particles is a promising approach tocreate novel structures and materials, with applications spanningfrom smart materials to optoelectronics to quantum computation.However, designing and producing mesoscale-sized structures re-mains a major challenge because at length scales of 10–100 μm equil-ibration times already become prohibitively long. Here, we extendthe principle of rapid diffusion-limited cluster aggregation (DLCA) toa multicomponent system of spherical colloidal particles to enablethe rational design and production of finite-sized anisotropic struc-tures on the mesoscale. In stark contrast to equilibrium self-assemblytechniques, kinetic traps are not avoided but exploited to control andguide mesoscopic structure formation. To this end the affinities, size,and stoichiometry of up to five different types of DNA-coated micro-spheres are adjusted to kinetically control a higher-order hierarchicalaggregation process in time. We show that the aggregation processcan be fully rationalized by considering an extended analytical DLCAmodel, allowing us to produce mesoscopic structures of up to 26 μmin diameter. This scale-free approach can easily be extended to anymulticomponent system that allows for multiple orthogonal interac-tions, thus yielding a high potential of facilitating novel materialswith tailored plasmonic excitation bands, scattering, biochemical,or mechanical behavior.

DNA-coated colloids | diffusion-limited cluster aggregation |mesoscopic structure | multicomponent | kinetic arrest

DNA has very successfully been used in colloidal systems toreach precise control over colloidal crystal formation that is

triggered and stabilized by Watson–Crick base pairing (1).Careful design of the used DNA strands that determine the bi-nary interparticle potentials and their grafting density yields avariety of highly symmetric crystal structures (2–5) and finite-sized structures (6–8). However, crystallization and other com-monly used self-assembly processes are equilibrium processes. Asall possible configurations have to be sampled in time to ensureequilibrium structure formation, such self-assembly processes relyon the fast diffusion and relaxation times of the building blocks.This requirement is readily met on the nanoscale, but stronglyhampers the applicability of known self-assembly techniqueson the micrometer to millimeter scale (9). Consequently, non-equilibrium pathways of self-organization also have been studied(10–12), yet the complexity of these systems limits the detailedaccess to the underlying self-organization pathway, hampering therational design of mesoscopic structures. Simulations indicate thatnonequilibrium systems can offer pathways from compact objectsto complex gel structures (13), which have been used to createtetrahedral structures also experimentally (14).In contrast to crystallization, irreversible diffusion-limited

cluster aggregation (DLCA) (15, 16) is a rather rapid processthat is very abundant in colloidal self-organization on the mi-crometer to millimeter scale (16, 17), offering a path for bridginglength scales in self-assembly (9, 18, 19). Structure formation isdominated by kinetic arrest rather than energy minimization andtherefore exhibits faster kinetics. Using only one spherical par-ticle species in an irreversible homoaggregation process, theresulting structure is an isotropic three-dimensional gel with awell-defined fractal dimension of 1.8 (16, 20), reflecting the fastand uncondensed formation of the aggregates. In stark contrastto equilibrium processes, the structures that are formed during

DLCA are heavily influenced by the kinetics of the structureformation process itself (9, 21), yet only uniform self-similarfractal clusters and gels can be formed (16, 17, 22–24). In binarysystems it has been shown that this gives rise to an additionaldegree of freedom, as the final structure of isotropic percolatingbigels can be tuned by inducing the aggregation of two colloidalspecies at different points in time (25). Also the size of aggre-gates can in principle be limited in binary systems by choosingan asymmetric stoichiometry (26, 27), potentially enabling theassembly of finite-sized structures. However, concepts are stillmissing to facilitate the production of finite-sized, mesoscopicstructures via DLCA and to efficiently use kinetic arrest to as-semble higher-order hierarchical structures. Herein, we show thata multicomponent system of spherical colloids interacting in theDLCA regime can be kinetically controlled to assemble complexstructures in a hierarchical fashion. We control the specificity ofthe DLCA processes by using microspheres that are coated withorthogonal DNA strands. Exploiting the high specificity of DNA,the self-organization of multicomponent microspheres can thenbe guided by the addition of different linker strands at differentpoints in time. We investigate the specific binary and ternaryaggregation of microspheres into finite-sized structures and showthat these structures can be used as asymmetric building blocks ina hierarchic assembly line. The concept of the presented ap-proach is illustrated in Fig. 1. Multiple spherical colloidal speciesof different sizes and fluorescent labeling in the micrometer rangeare coated with different long sticky ssDNA (Fig. 1A). To in-troduce control of size and functionality into the system, thesecolloids are mixed in a variety of stoichiometries and complexities(Fig. 1B). To demonstrate the potential of this approach, wepresent a time-dependent assembly line of the so-formed func-tional clusters that results in a rationally designed mesoscopicstructure (Fig. 1C).

Significance

The well-studied self-organization of colloidal particles is pre-dicted to result in a variety of fascinating applications. Yet,whereas self-assembly techniques are extensively explored, de-signing and producing mesoscale-sized objects remains a majorchallenge, as equilibration times and thus structure formationtimescales become prohibitively long. Asymmetric mesoscopicobjects, without prior introduction of asymmetric particles withall its complications, are out of reach––due to the underlyingprinciple of thermal equilibration. In the present article, we in-troduce a strategy to overcome these limitations on the meso-scale. By controlling and stirring the process of diffusion-limitedcluster aggregation introducing DNA hybridization-mediatedheteroparticle aggregation, we are able to produce finite-sizedanisotropic structures on the mesoscale.

Author contributions: F.M.H. and A.R.B. designed research; F.M.H. performed research; F.M.H.and A.R.B. discussed and analyzed data; and F.M.H. and A.R.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: abausch@ph.tum.de.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1605114113/-/DCSupplemental.

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Binary Aggregation Enables Cluster Size ControlIn a binary 1:1 mixture of 1-μm microspheres (coated with theDNA sequences α and β, respectively), the addition of the com-plementary linker strand (αβ) results in fractal cluster growth(Fig. 2). The fractal dimension of the clusters increases contin-uously in time, reaching an asymptotic value of Df = 1.8 within30 min (Fig. 2A). This is in excellent agreement with the dimensionsexpected in a classical DLCA process, where particles diffuseuntil they meet an aggregation partner to which they stick irre-versibly, generating clusters that can then merge to finally form afractal gel. This clearly shows that the chosen particle graftingdensities and linker strand lengths do not allow rearrangementsand lead to quick structure formation in the diffusion-limitedregime. In classical monocomponent bulk systems, cluster growthcannot be limited, hence structure formation always terminateswith complete gelation. Here, in a binary system, binding partnerscan be screened by adjusting the stoichiometry, which shouldreadily allow the control of the average cluster size, leading tofinite-sized structures (Fig. 2B). Indeed the average size of theclusters can be tuned from hundreds down to tens of particles bysimply varying the stoichiometry Xα−β = cα=cβ between 1 and 100(Fig. 2C). The overproportional binding of one particle type ef-fectively screens the minority particles (β-spheres, shown as greenspheres in Fig. 2) from further interaction, yielding aggregationseeds that comprise one minority and several majority spheres(red spheres in Fig. 2). We found that there is a critical ratio atwhich all possible binding sites of a minority sphere are blocked,before on average two aggregation seeds can merge and thusform larger clusters. This can be readily seen by extendingSmoluchowski’s concept of fast aggregation to a binary system(SI Text, Analytical Model for Binary Aggregation at High Stoi-chiometries). The fast aggregation rateWk of spheres with radius R isdirectly proportional to their concentration c and diffusion coefficientD, and gives the rate at which two spheres bind: WK = 8πDRc. In ahomoaggregating system, spheres can bind without restrictions,leading to fast multimerization and fractal growth. However, ina binary system with asymmetric stoichiometries, the aggregation

rate additionally depends on the number of already-boundspheres, as potential binding sites on minority spheres are in-creasingly occupied in time. This leads to an exponential screeningof minority spheres by majority spheres and is determined by Wk,the stoichiometry Xα−β, and the average maximum number ofaccumulated particles Nmax (SI Text, Analytical Model for BinaryAggregation at High Stoichiometries and Fig. S1). This model allowsfor the prediction of a critical stoichiometry Xgrowth ≈ 22.3. At thisstoichiometry the majority of the seeds have on average no bindingsites left for two aggregation seeds to merge when they first meet.Indeed, experimentally we find that at Xgrowth,exp = 21.8± 0.8 theseeds stop to dominate the mass average of the clusters (SI Text,Analytical Model for Binary Aggregation at High Stoichiometries andFig. S2). Remarkably, the critical ratio in this model is universalfor all binary DLCA processes and therefore independent ofconcentration or sphere radius (SI Text, Analytical Model for Bi-nary Aggregation at High Stoichiometries). Thereby, also the clustergeometry is determined, as the branching probability is inherentlylength-independent for a scale-free process like DLCA and thussmaller clusters have effectively fewer branches. Whereas at lowstoichiometries up to Xgrowth branched structures are formed, un-branched and elongated shapes are predominantly obtained aboveXgrowth (Fig. 2D). This tendency toward linearity at high Xα–β isexpressed in the fractal dimension of the formed clusters (Fig. 2E).In a classical DLCA process the fractal dimension builds up rap-idly with time. However, in the first minutes, where only smallclusters are formed, a significantly smaller fractal dimension thanobserved in the DLCA limit (Fig. 2B) emerges. This reflects thetendency of fractal growth to take place at the exposed ends ofa growing structure (28). Time-course measurements close toXgrowth show that also at Xα−β = 18 the fractal dimension growscontinuously in time (Fig. 2B). But, as the branching of the clus-ters is restricted, Df reaches an asymptotic value that lies signifi-cantly below the DLCA limit, confirming that the remainingbinary clusters in samples with Xα−β >   Xgrowth exhibit pronouncedlinearity. Consequently, binary heteroaggregation can be dividedinto three functional regimes: (i) a regime of fractal growth at lowXα−β, where large, branched clusters emerge (Fig. 2E, gray); (ii) alinear regime at intermediate Xα−β, where the majority of particlesis found in small, unbranched structures (Fig. 2E, orange); and(iii) a compact regime at high Xα−β of isolated aggregation seeds(Fig. 2E, green).

Ternary Aggregation Leads to Polar ClustersDue to the inherent nature of fractal growth, the degree ofsymmetry in binary clusters is still high, inhibiting further as-sembly. To break this symmetry, another degree of freedom hasto be introduced so that the composition of compact clusters canbe manipulated. This is achieved by expanding the binary to aternary aggregation process. To demonstrate this concept, wework in the compact regime ðX ≥ 30Þ, where minority particles β(red spheres in Fig. 3) get effectively screened by the majorityparticles α. If a third, equally sized microsphere species γ is in-troduced at the same ratio (Xγ−α = 1) that can, upon the additionof complementary linker strand (βγ), only bind to β-spheres, theβ-spheres’ binding sites are isotropically occupied with α- andγ-spheres (SI Text, Ternary Aggregation of Equally Sized Colloids).Consistently, lowering Xγ−α leads to a dominance of α-spheresover γ-spheres that are found in the seeds. As the averagemaximum occupancy is nmax = 6.8 (Fig. S1B), on average onlyone binding site on a β-sphere is occupied by a γ-sphere atXγ−α ≈ 0.14, thus giving a ratio where polar structures emerge (SIText, Ternary Aggregation of Equally Sized Colloids). This polarratio can be extended drastically to a polar regime by using mi-crospheres of different sizes. By doubling the γ-sphere’s radius to2 μm (=Γ-spheres) a geometrical constraint is introduced, whichresults in an even more effective blocking of potential bindingsites on the same β-hemisphere for the α-spheres, resulting in a

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Fig. 1. Design principle for producing complex mesoscopic structures byDLCA. (A) Colloids with sizes ranging from 1 to 6 μm are coated with longsticky ssDNA and their affinity is tuned by adding a sticky DNA-linker strand.(B) Combining different colloids with varying affinity, stoichiometry, andcomplexity, finite-sized and functional clusters are created. (C) Mesoscopicstructures with a defined complex geometry can be formed by a hierarchicalassembly line.

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large polar regime. At XΓ−α � 1, only isotropic clusters are ob-served again, as the ternary aggregation process is basically re-duced to a binary problem (Fig. 3, green markers). However,increasing the concentration of Γ-spheres leads to the cross-overto a regime where both α- and Γ-spheres can bind to a β-sphere,yielding true ternary clusters. The majority of the resultingclusters are polar and comprise only one large Γ-sphere andmultiple small α-spheres (Fig. 3, purple markers). Consistently, afurther increment of the concentration of Γ-spheres leads to thebinding of not only one but two Γ-spheres to one β-sphere. Incontrast to equally sized γ-spheres, the use of larger Γ-spherespreserves polarity also in this regime (Fig. 3, red markers).Only if the concentration of Γ-sphere is drastically increased toXΓ−α � 1, so that three or four Γ-spheres are bound to oneβ-sphere, polarity is lost again due to an effective screening ofthe α-spheres (Fig. 3, blue markers). The data show that a widerange of concentration of α- and Γ-spheres can be used to designpolar structures with distinct composition (Fig. 3C). This varietyof compact junction-type structures opens up the possibility toaim for higher-order structures based on a hierarchical DLCAprocess.

Multicomponent System Enables Complex HierarchicalAssemblyWe demonstrate this concept by a five-particle system, where themicrospheres are coated with specific DNA strands α, β, Γ, Δ,

and «. α-, β-, and «-coated spheres are 1-μm microspheres,Γ-coated spheres have a diameter of 2 μm, and Δ-coated sphereshave a diameter of 6 μm. All particles are present in the solutionthroughout the complete process; only the linker strands areadded subsequently (Fig. 4A). First, we trigger ternary aggrega-tion by adding linker αβ and Γβ. As we chose XΓ−α = 0.24(Xα−β = 25, XΓ−β = 6), this results in polar-junction-type clusters.In a second incubation step we connect one side of the junctionsto a large type-Δ base sphere by adding linker αΔ ð  XΔ−β = 0.22).Simultaneously, a third step is performed by adding linker Γ«, sothat a binary cluster of Γ- and «-spheres can form and bind to theother side of the junction (X«−β = 10 . . . 100). This third step ofbinary attachment enables us to control the size of the finallyproduced mesoscopic structure (Fig. 4 B and C). In total, thisthree-step process results in a tadpole-shaped structure, com-prising three different parts. Part one is the 6-μm base sphere Δ,giving the head of the tadpole. Part two is a compact polarjunction that we control via ternary aggregation. Part three is abinary cluster in the linear regime that is independently formedby spheres Γ and «, constituting the tadpole’s tail. In combina-tion, the five-sphere approach shown here results in an aniso-tropic and mesoscopic structure of rational design that has beenformed by pure DLCA processes of isotropic building blocks.The effective yield of the generated tadpole-shaped structuresthat reach up to 26 μm in size lies between ∼50% and 70% (Fig.4C, Inset).

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Fig. 2. Binary aggregation enables the formation of finite-sized clusters. (A) In a 1:1 mixture of colloids, binary heteroaggregation leads to the fractalgrowth of a colloidal gel, reaching a well-defined dimension of Df =1.8 within 30 min. (B) Changing the stoichiometry to 1:18 (Xα−β = 18Þ limits the size of theproduced fractal clusters, yielding finite-sized structures of significantly lower fractal dimension. In both A and B, the dashed lines serve as a guide to the eye,showing the final slope at 6 h in each case. (C) Consistently, the average size of these finite-sized structures is tunable over a large range of stoichiometries. Thestoichiometry at which the fractal growth is mainly limited to isolated aggregation seeds is marked by Xgrowth= 22.3, which can be rationalized by a simpleanalytical model based on Smoluchowski’s concept of fast aggregation. Error bars denote the SD of the mass average. (D) Clusters can be classified into threecategories: fractal (branched, gray), linear (unbranched, orange), and compact clusters (green). The three depicted stoichiometries, Xα−β = 4,   9  ,   and  110, illustratethe change in structure distribution from fractal to compact. (E) Both the fractal dimension (black curve) and the percentage of unbranched clusters (blue curve)show a significant dependence on the stoichiometry. The colored areas indicate the regimes of the majority type of clusters.

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SummaryMulticomponent self-assembly is becoming of increasing interest forthe field of complex self-assembly (29). Until now, the formation ofmultiparticle structures has mainly been investigated on the nano-scale, using short pieces of ssDNA as building blocks (30, 31). Weshow also that colloidal multicomponent systems can be used tocreate complex mesoscopic structures. We show that kinetic arrestcan be used to create finite-sized mesoscopic structures by rationaldesign. Whereas equilibrium-based self-organization techniquesrely on the predefinition of the desired structures by precise controlof the equilibrium positions, the presented approach is based onthe kinetic control of the diffusion dynamics of a multicomponentcolloidal system by means such as particle sizes and stoichiome-tries. This appealingly simple approach can therefore easily beextended to any multicomponent system that allows for multipleorthogonal interactions. As this approach is scale-free it has a highpotential of facilitating novel materials with applications spanningfrom smart materials to optoelectronics to quantum computation(32–35) that require tailored plasmonic excitation bands, scatter-ing, biochemical, or mechanical behavior.

Materials and MethodsPreparation of DNA-Coated Microspheres. Unless otherwise specified, thechemicals used in the current work were purchased from Sigma-Aldrich andusedwithout further purification. Streptavidin-coated polystyrenemicrospheresα and β were purchased from Bangs Laboratories, Γ- and Δ-microspheres from

Polysciences Europe, and Neutravidin-coated polystyrene microspheres γ fromLife Technologies, and were incubated with biotinylated ssDNA docking strandspurchased at Integrated DNA Technologies Europe for at least 12 h (Table S1).The concentration of docking strands was chosen such that ∼6 ·104 dockingstrands were present in the incubation solution per 1-μm particle. Conse-quently, 2-μm microspheres were incubated with ∼2.4 ·105 strands per micro-sphere and 6-μm microspheres with ∼2.16 ·106 strands per microsphere topreserve docking strand area density on all microspheres. All docking and linkercombinations used were checked with NUPACK (36) before experiments toexclude cross-talk. After incubation the particle were centrifuged at 1,200relative centrifugal force, the supernatant was removed, and the sample wasresuspended in low-Tris low-salt buffer––150 mMNaCl, 10 mM Tris, pH 8.8. Thiswashing step was performed three times before resuspending the microspheresin a density-matched buffer–450 mM sucrose, 150 mM NaCl, 10 mM Tris, pH8.8––to prevent the microspheres from fast sedimentation during samplepreparation. To gain high monodispersity of the coated mircospheres(polydispersity index≈ 1.1), we vortexed and sonicated the stocks for 30 sbefore storage. In between experiments, the microspheres were stored on arotating device at 4 °C.

Sample Preparation. All samples were prepared in a final buffer of 450 mM su-crose, 150mMNaCl, 10mMTris, and 10mg/mLBSA. To enable high signal-to-noiseratio (SNR) imaging, 4.5% (wt/vol) Acrylamide 4K solution (29:1) (Applichem),0.4% ammonium persulfate, and 140 μM Tris(2,2′-biprydidyl)dichlororuthenium(II)were added to each sample. Tris(2,2′-biprydidyl)dichlororuthenium(II) is aphotoactivatable catalyzator for the polyacrylamide (PAM) polymerization[λmax = 450  nm (37)] and can therefore be used to trigger PAM polymeri-zation at an arbitrary point in time, effectively stopping any diffusion and

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Fig. 3. Ternary aggregation forms compact clusters with defined geometries. (A) The addition of a third, larger colloidal particle at varying stoichiometriesenables the formation of nonisotropic structures with distinct properties. Besides isotropic phases (blue area and blue markers, green area and green markers), aregime of polar ordering (purple area, red and purple markers) is found, where the majority type of formed clusters is of polar order. (B) Representative confocalimage of a purified solution of polar clusters. (C) Change in cluster distribution illustrated by four depicted stoichiometries. Whereas in the blue and greenregimes more than 80% of the clusters are isotropic (I, IV), the majority of clusters remains polar in the purple regime (II, III). Additionally, the exact position in thephase diagram determines the polar substructure (II, III). The shown pictograms of single clusters represent 3D models of confocal data.

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further aggregation of the particles by white-light illumination for ∼3 minwith a Schott KL 1600 light-emitting diode lamp. All binary samples wereprepared at a fixed majority microsphere volume fraction of 2,500 ppm. Allternary and hierarchical aggregation samples were prepared at a fixedminority microsphere concentration of 25 ppm. The specific aggregation inall samples was induced by adding the appropriate linker strands (Table S1)at a final concentration of 58 nM, followed by short pipette mixing. Afterlinker addition the samples were pipetted into a glass microscopy chamberand mounted on a rotating device at 21 °C (∼0.3-Hz rotating frequency).The use of a rotating device is necessary to avoid any sedimentation due toimperfect buoyancy matching during the self-assembly process. This is es-pecially important where larger structures ( ~> 10 microspheres) are assem-bled and therefore small buoyancy mismatches already lead to significantsedimentation. All samples were incubated for at least 3 h, unless statedotherwise, followed by PAM immobilization and 3D confocal imaging.

Purification of junction-type aggregateswas performed by exchanging thecentral β-polystyrene microsphere with a ProMag HC 1 microsphere (BangsLaboratories) that was coated in analogy with the above-described protocolwith β-docking strands. After 3 h of sample incubation, where the junctionswere formed, the Eppendorf tube containing the sample was held close to aneodymium magnet for 30 min. After pellet formation, the supernatant wasexchanged three times with final buffer, pipetted into a glass microscopychamber, and subsequently immobilized by illumination-induced cross-linkingof the PAM.

Confocal Microscopy and Image Analysis. Imaging was conducted with a LeicaSP5 scanning confocal microscope (Leica Microsystems) at a 3D voxel reso-lution of (120 × 120 × 460) nm3 with a 40×water immersion objective. As theparticles and clusters are immobilized for imaging by the PAM, imaging wasperformed at low line rates (<700 Hz) to maximize the SNR. For every sample4 z stacks (123.02 × 123.02 × 80–140) μm3, depending on the microspheredensity, were imaged and subsequently analyzed. The 3D particle countingwas conducted by a MATLAB script based on particle-tracking algorithmsthat have been shown to have subpixel resolution (38, 39). Subsequentcluster analysis in all binary aggregation samples was performed with a self-made IGOR PRO 6 connectivity-based clustering algorithm. The algo-rithm identified clusters by connecting all particles that had a maximumcentroid distance of 2 μm (= 2× microsphere diameter). Clusters that were incontact with the image boundaries were excluded from further analysisto reduce artifacts. The mass average of the clusters was calculated asMw =

PiNi i2=

PiNii. Due to significantly weaker statistics, ternary and hier-

archical aggregation samples were evaluated by manual counting to reducethe significance of counting errors.

ACKNOWLEDGMENTS. The authors thank Alessio Zaccone for stimulatingdiscussions. A.R.B. gratefully acknowledges the hospitality of the MillerInstitute for Basic Research in Science at University of California, Berkeley.We gratefully acknowledge the financial support of the DeutscheForschungsgemeinschaft (DFG) through the Sonderforschungsbereich 1032and the Nanosystems Initiative Munich (NIM).

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Fig. 4. Hierarchical assembly of a mesoscopic structure by kinetic arrest. (A) Schematic assembly line, controlled by the addition of DNA-linker strands atdifferent time points. First, ternary aggregation is induced to create polar clusters by adding two distinct linker strands. As these polar structures representfunctional structures (junctions), they are connected to a 6-μm base particle on one side and to a binary cluster on the other side, which can be controlled inlength by variation of the stoichiometry. (B) Cumulative length distribution of the binary clusters attached to the base particles via junctions for differentstoichiometries X«−β. (Inset) Confocal images show the largest class of binary clusters found in the different samples. (C) The average length of the binaryclusters attachments as a function of the stoichiometry X«−β. The effective yield (number of base particles with attachments/number of all base particles) liesbetween ∼50% and 70% (Inset). Error bars denote SEM of the data.

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