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# Kinetically guided colloidal structure formation

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

## Significance

The well-studied self-organization of colloidal particles is predicted to result in a variety of fascinating applications. Yet, whereas self-assembly techniques are extensively explored, designing and producing mesoscale-sized objects remains a major challenge, as equilibration times and thus structure formation timescales become prohibitively long. Asymmetric mesoscopic objects, without prior introduction of asymmetric particles with all its complications, are out of reach––due to the underlying principle of thermal equilibration. In the present article, we introduce a strategy to overcome these limitations on the mesoscale. By controlling and stirring the process of diffusion-limited cluster aggregation introducing DNA hybridization-mediated heteroparticle aggregation, we are able to produce finite-sized anisotropic structures on the mesoscale.

## Abstract

The self-organization of colloidal particles is a promising approach to create novel structures and materials, with applications spanning from smart materials to optoelectronics to quantum computation. However, designing and producing mesoscale-sized structures remains a major challenge because at length scales of 10–100 μm equilibration times already become prohibitively long. Here, we extend the principle of rapid diffusion-limited cluster aggregation (DLCA) to a multicomponent system of spherical colloidal particles to enable the rational design and production of finite-sized anisotropic structures on the mesoscale. In stark contrast to equilibrium self-assembly techniques, kinetic traps are not avoided but exploited to control and guide mesoscopic structure formation. To this end the affinities, size, and stoichiometry of up to five different types of DNA-coated microspheres are adjusted to kinetically control a higher-order hierarchical aggregation process in time. We show that the aggregation process can be fully rationalized by considering an extended analytical DLCA model, allowing us to produce mesoscopic structures of up to 26 µm in diameter. This scale-free approach can easily be extended to any multicomponent system that allows for multiple orthogonal interactions, thus yielding a high potential of facilitating novel materials with 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 to reach 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 binary interparticle potentials and their grafting density yields a variety of highly symmetric crystal structures (2⇓⇓–5) and finite-sized structures (6⇓–8). However, crystallization and other commonly used self-assembly processes are equilibrium processes. As all possible configurations have to be sampled in time to ensure equilibrium structure formation, such self-assembly processes rely on the fast diffusion and relaxation times of the building blocks. This requirement is readily met on the nanoscale, but strongly hampers the applicability of known self-assembly techniques on the micrometer to millimeter scale (9). Consequently, nonequilibrium pathways of self-organization also have been studied (10⇓–12), yet the complexity of these systems limits the detailed access to the underlying self-organization pathway, hampering the rational design of mesoscopic structures. Simulations indicate that nonequilibrium systems can offer pathways from compact objects to complex gel structures (13), which have been used to create tetrahedral structures also experimentally (14).

In contrast to crystallization, irreversible diffusion-limited cluster aggregation (DLCA) (15, 16) is a rather rapid process that is very abundant in colloidal self-organization on the micrometer to millimeter scale (16, 17), offering a path for bridging length scales in self-assembly (9, 18, 19). Structure formation is dominated by kinetic arrest rather than energy minimization and therefore exhibits faster kinetics. Using only one spherical particle species in an irreversible homoaggregation process, the resulting structure is an isotropic three-dimensional gel with a well-defined fractal dimension of 1.8 (16, 20), reflecting the fast and uncondensed formation of the aggregates. In stark contrast to equilibrium processes, the structures that are formed during DLCA are heavily influenced by the kinetics of the structure formation process itself (9, 21), yet only uniform self-similar fractal clusters and gels can be formed (16, 17, 22⇓–24). In binary systems it has been shown that this gives rise to an additional degree of freedom, as the final structure of isotropic percolating bigels can be tuned by inducing the aggregation of two colloidal species at different points in time (25). Also the size of aggregates can in principle be limited in binary systems by choosing an asymmetric stoichiometry (26, 27), potentially enabling the assembly of finite-sized structures. However, concepts are still missing to facilitate the production of finite-sized, mesoscopic structures via DLCA and to efficiently use kinetic arrest to assemble higher-order hierarchical structures. Herein, we show that a multicomponent system of spherical colloids interacting in the DLCA regime can be kinetically controlled to assemble complex structures in a hierarchical fashion. We control the specificity of the DLCA processes by using microspheres that are coated with orthogonal DNA strands. Exploiting the high specificity of DNA, the self-organization of multicomponent microspheres can then be guided by the addition of different linker strands at different points in time. We investigate the specific binary and ternary aggregation of microspheres into finite-sized structures and show that these structures can be used as asymmetric building blocks in a hierarchic assembly line. The concept of the presented approach is illustrated in Fig. 1. Multiple spherical colloidal species of different sizes and fluorescent labeling in the micrometer range are coated with different long sticky ssDNA (Fig. 1*A*). To introduce control of size and functionality into the system, these colloids are mixed in a variety of stoichiometries and complexities (Fig. 1*B*). To demonstrate the potential of this approach, we present a time-dependent assembly line of the so-formed functional clusters that results in a rationally designed mesoscopic structure (Fig. 1*C*).

## Binary Aggregation Enables Cluster Size Control

In a binary 1:1 mixture of 1-µm microspheres (coated with the DNA sequences *α* and *β*, respectively), the addition of the complementary linker strand (*A*). This is in excellent agreement with the dimensions expected in a classical DLCA process, where particles diffuse until they meet an aggregation partner to which they stick irreversibly, generating clusters that can then merge to finally form a fractal gel. This clearly shows that the chosen particle grafting densities and linker strand lengths do not allow rearrangements and lead to quick structure formation in the diffusion-limited regime. In classical monocomponent bulk systems, cluster growth cannot be limited, hence structure formation always terminates with complete gelation. Here, in a binary system, binding partners can be screened by adjusting the stoichiometry, which should readily allow the control of the average cluster size, leading to finite-sized structures (Fig. 2*B*). Indeed the average size of the clusters can be tuned from hundreds down to tens of particles by simply varying the stoichiometry *C*). The overproportional binding of one particle type effectively screens the minority particles (β-spheres, shown as green spheres in Fig. 2) from further interaction, yielding aggregation seeds that comprise one minority and several majority spheres (red spheres in Fig. 2). We found that there is a critical ratio at which all possible binding sites of a minority sphere are blocked, before on average two aggregation seeds can merge and thus form larger clusters. This can be readily seen by extending Smoluchowski’s concept of fast aggregation to a binary system (*SI Text*, *Analytical Model for Binary Aggregation at High Stoichiometries*). The fast aggregation rate *W*_{k} of spheres with radius *R* is directly proportional to their concentration *c* and diffusion coefficient *D*, and gives the rate at which two spheres bind: *W*_{k}, the stoichiometry *X*_{α−β,} and the average maximum number of accumulated particles *N*_{max} (*SI Text*, *Analytical Model for Binary Aggregation at High Stoichiometries* and Fig. S1). This model allows for the prediction of a critical stoichiometry *SI Text*, *Analytical Model for Binary Aggregation at High Stoichiometries* and Fig. S2). Remarkably, the critical ratio in this model is universal for all binary DLCA processes and therefore independent of concentration or sphere radius (*SI Text*, *Analytical Model for Binary Aggregation at High Stoichiometries*). Thereby, also the cluster geometry is determined, as the branching probability is inherently length-independent for a scale-free process like DLCA and thus smaller clusters have effectively fewer branches. Whereas at low stoichiometries up to *X*_{growth} branched structures are formed, unbranched and elongated shapes are predominantly obtained above *X*_{growth} (Fig. 2*D*). This tendency toward linearity at high *X*_{α–β} is expressed in the fractal dimension of the formed clusters (Fig. 2*E*). In a classical DLCA process the fractal dimension builds up rapidly with time. However, in the first minutes, where only small clusters are formed, a significantly smaller fractal dimension than observed in the DLCA limit (Fig. 2*B*) emerges. This reflects the tendency of fractal growth to take place at the exposed ends of a growing structure (28). Time-course measurements close to *X*_{growth} show that also at *B*). But, as the branching of the clusters is restricted, *D*_{f} reaches an asymptotic value that lies significantly below the DLCA limit, confirming that the remaining binary clusters in samples with *i*) a regime of fractal growth at low *X*_{α−β}, where large, branched clusters emerge (Fig. 2*E*, gray); (*ii*) a linear regime at intermediate *X*_{α−β}, where the majority of particles is found in small, unbranched structures (Fig. 2*E*, orange); and (*iii*) a compact regime at high *X*_{α−β} of isolated aggregation seeds (Fig. 2*E*, green).

## Ternary Aggregation Leads to Polar Clusters

Due to the inherent nature of fractal growth, the degree of symmetry in binary clusters is still high, inhibiting further assembly. To break this symmetry, another degree of freedom has to be introduced so that the composition of compact clusters can be manipulated. This is achieved by expanding the binary to a ternary aggregation process. To demonstrate this concept, we work in the compact regime *β* (red spheres in Fig. 3) get effectively screened by the majority particles *α*. If a third, equally sized microsphere species *γ* is introduced at the same ratio (*β*-spheres, the β-spheres’ binding sites are isotropically occupied with *α*- and *γ*-spheres (*SI Text*, *Ternary Aggregation of Equally Sized Colloids*). Consistently, lowering *X*_{γ−α} leads to a dominance of *α*-spheres over *γ*-spheres that are found in the seeds. As the average maximum occupancy is *B*), on average only one binding site on a *β*-sphere is occupied by a *γ*-sphere at *SI Text*, *Ternary Aggregation of Equally Sized Colloids*). This polar ratio can be extended drastically to a polar regime by using microspheres of different sizes. By doubling the *γ*-sphere’s radius to 2 µm (=*Γ*-spheres) a geometrical constraint is introduced, which results in an even more effective blocking of potential binding sites on the same *β*-hemisphere for the *α*-spheres, resulting in a large polar regime. At *Γ*-spheres leads to the cross-over to a regime where both *α*- and *Γ*-spheres can bind to a *β*-sphere, yielding true ternary clusters. The majority of the resulting clusters are polar and comprise only one large *Γ*-sphere and multiple small *α*-spheres (Fig. 3, purple markers). Consistently, a further increment of the concentration of *Γ*-spheres leads to the binding of not only one but two *Γ*-spheres to one *β*-sphere. In contrast to equally sized *γ*-spheres, the use of larger *Γ*-spheres preserves polarity also in this regime (Fig. 3, red markers). Only if the concentration of *Γ*-sphere is drastically increased to *Γ*-spheres are bound to one *β*-sphere, polarity is lost again due to an effective screening of the *α*-spheres (Fig. 3, blue markers). The data show that a wide range of concentration of *α*- and *Γ*-spheres can be used to design polar structures with distinct composition (Fig. 3*C*). This variety of compact junction-type structures opens up the possibility to aim for higher-order structures based on a hierarchical DLCA process.

## Multicomponent System Enables Complex Hierarchical Assembly

We demonstrate this concept by a five-particle system, where the microspheres are coated with specific DNA strands *α*, *β*, *Γ*, Δ, and *ε*. *α*-, *β*-, and *ε*-coated spheres are 1-µm microspheres, *Γ*-coated spheres have a diameter of 2 µm, and Δ-coated spheres have a diameter of 6 µm. All particles are present in the solution throughout the complete process; only the linker strands are added subsequently (Fig. 4*A*). First, we trigger ternary aggregation by adding linker *Γ*- and *ε*-spheres can form and bind to the other side of the junction (*B* and *C*). In total, this three-step process results in a tadpole-shaped structure, comprising three different parts. Part one is the 6-µm base sphere Δ, giving the head of the tadpole. Part two is a compact polar junction that we control via ternary aggregation. Part three is a binary cluster in the linear regime that is independently formed by spheres *Γ* and *ε*, constituting the tadpole’s tail. In combination, the five-sphere approach shown here results in an anisotropic and mesoscopic structure of rational design that has been formed by pure DLCA processes of isotropic building blocks. The effective yield of the generated tadpole-shaped structures that reach up to 26 µm in size lies between ∼50% and 70% (Fig. 4*C*, *Inset*).

## Summary

Multicomponent self-assembly is becoming of increasing interest for the field of complex self-assembly (29). Until now, the formation of multiparticle structures has mainly been investigated on the nanoscale, using short pieces of ssDNA as building blocks (30, 31). We show also that colloidal multicomponent systems can be used to create complex mesoscopic structures. We show that kinetic arrest can be used to create finite-sized mesoscopic structures by rational design. Whereas equilibrium-based self-organization techniques rely on the predefinition of the desired structures by precise control of the equilibrium positions, the presented approach is based on the kinetic control of the diffusion dynamics of a multicomponent colloidal system by means such as particle sizes and stoichiometries. This appealingly simple approach can therefore easily be extended to any multicomponent system that allows for multiple orthogonal interactions. As this approach is scale-free it has a high potential of facilitating novel materials with applications spanning from smart materials to optoelectronics to quantum computation (32⇓⇓–35) that require tailored plasmonic excitation bands, scattering, biochemical, or mechanical behavior.

## Materials and Methods

### Preparation of DNA-Coated Microspheres.

Unless otherwise specified, the chemicals used in the current work were purchased from Sigma-Aldrich and used without further purification. Streptavidin-coated polystyrene microspheres *α* and *β* were purchased from Bangs Laboratories, *Γ*- and Δ-microspheres from Polysciences Europe, and Neutravidin-coated polystyrene microspheres *γ* from Life Technologies, and were incubated with biotinylated ssDNA docking strands purchased at Integrated DNA Technologies Europe for at least 12 h (Table S1). The concentration of docking strands was chosen such that ∼

### Sample Preparation.

All samples were prepared in a final buffer of 450 mM sucrose, 150 mM NaCl, 10 mM Tris, and 10 mg/mL BSA. To enable high signal-to-noise ratio (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 a photoactivatable catalyzator for the polyacrylamide (PAM) polymerization [

Purification of junction-type aggregates was performed by exchanging the central *β*-polystyrene microsphere with a ProMag HC 1 microsphere (Bangs Laboratories) that was coated in analogy with the above-described protocol with *β*-docking strands. After 3 h of sample incubation, where the junctions were formed, the Eppendorf tube containing the sample was held close to a neodymium magnet for 30 min. After pellet formation, the supernatant was exchanged three times with final buffer, pipetted into a glass microscopy chamber, and subsequently immobilized by illumination-induced cross-linking of the PAM.

### Confocal Microscopy and Image Analysis.

Imaging was conducted with a Leica SP5 scanning confocal microscope (Leica Microsystems) at a 3D voxel resolution of (120 × 120 × 460) nm^{3} with a 40× water immersion objective. As the particles and clusters are immobilized for imaging by the PAM, imaging was performed at low line rates (<700 Hz) to maximize the SNR. For every sample 4 *z* stacks (123.02 × 123.02 × 80–140) µm^{3}, depending on the microsphere density, were imaged and subsequently analyzed. The 3D particle counting was conducted by a MATLAB script based on particle-tracking algorithms that have been shown to have subpixel resolution (38, 39). Subsequent cluster analysis in all binary aggregation samples was performed with a self-made IGOR PRO 6 connectivity-based clustering algorithm. The algorithm identified clusters by connecting all particles that had a maximum centroid distance of 2 µm (= 2× microsphere diameter). Clusters that were in contact with the image boundaries were excluded from further analysis to reduce artifacts. The mass average of the clusters was calculated as

## SI Text

### Analytical Model for Binary Aggregation at High Stoichiometries.

To gain further understanding of the binary aggregation process that leads to finite-sized structure formation, we developed a simple analytical model. This model is based on Smoluchowski’s concept of fast coagulation (40), but can also be motivated by Langmuir adsorption theory (41, 42). Smoluchowski first introduced the fast coagulation rate *D*, radius *R*, and concentration *c* meet. Using the Stokes–Einstein equation *W*_{k} further condenses to *R*. Note that this is only valid in the limit of spherical particles, as used in this study. Using, for example, anisotropic particles such as prolate ellipsoids leads to a rather complex dependency of *D* on the semiprincipal axes of the ellipsoids (43), preventing further condensation of *W*_{k}. It has been shown that for a real system the hydrodynamic coupling of the particles also has to be taken into account. This reduces the fast coagulation rate by a factor of *A* (minority particle) with radius *R* is surrounded by an infinite number of majority particles *B* with radius *R*, which cannot bind themselves but only the minority particle *A*, we can write the binding probability of *A* to *B* as*N*(*t*) is the number of *B* particles bound to *A* and *A*.

The accumulated number of *B* particles on *A* with time can therefore be written as*B* spheres from infinity to a finite reservoir of *X*_{B-A}, the concentration *c*_{B} is effectively reduced in time by the factor **S2** and subsequently solving for **S4** is valid also for an ensemble of particles, where *N*(*t*) is the average number of accumulated *B* particles on *A* particles and *N*_{max} is the maximum of accumulated particles on *A* particles averaged over all *A* particles. Consequently, *X*_{B–A} is the stoichiometry between *B* and *A* particles (*N*_{max} is best approximated by *A*). As the temperature and the viscosity of the sample can be measured experimentally, there are no remaining free variables in Eq. **S4**. Time-course measurements at **S4** are in good agreement with experimental data (Fig. S1*A*). Only compact clusters with one minority particle per cluster have been evaluated for these data.

Interestingly, Langmuir adsorption theory also leads to the same result. It has been shown elsewhere (41) that an irreversible adsorption of particles from a finite reservoir to a surface can be described by*k*_{on}, but comparison with Eq. **S4** yields

As Eq. **S4** describes the accumulation of majority on minority particles, we can now estimate the stoichiometry at which minority particles are predominantly occupied by majority particles when they meet other minority particles on average for the first time. We do this by evaluating Eq. **S4** at the fast aggregation time τ_{AA} of minority particles*F* on a minority particle on first contact with another minority particle (Fig. S1*C*). If we now ask for the stoichiometry *X*_{growth} at which only 0.5 binding site is available on first contact, we numerically get *X*_{growth} to experimental data, we used the following estimation. As *X*_{growth} marks the point at which approximately 50% of aggregation seeds are able to merge to bigger clusters, it should in turn experimentally reflect the stoichiometry at which the mass average of the clusters is increased by 50% in relation to the compact regime of aggregation seeds. We chose the data point *X*_{growth} (Fig. S2). As Eq. **S8** is not dependent on either the size of the particles or of their concentration, *X*_{growth} can be considered a universal, scale-free constant, applicable to all binary spherical particle systems that are purely diffusion-limited.

### Ternary Aggregation of Equally Sized Colloids.

We investigated a ternary system of three different colloids of the same size (=1 µm), but of different DNA coating *α*, *β*, and *γ*. Upon addition of the linker strands *α*- and *γ*-spheres compete for binding to *β*-spheres. As we perform these experiments in the compact regime (*C*). Obviously, a balanced stoichiometry of *γ*- and *α*-spheres (*β*-sphere (Fig. S3). Decreasing the stoichiometry *X*_{γ–β} results in a depletion of *γ*-spheres, until there is on average only one *γ*-sphere bound to a *β*-sphere. As the maximum number of bound majority spheres to minority spheres is *B*), we measure the stoichiometry at which the coverage equals 1/6.8 and find *γ*-sphere and several *α*-spheres, hence exhibiting polar order (Fig. S3, lowest red dashed line). Increasing the stoichiometry *X*_{γ–β} above 1 symmetrically leads to the same result. At a ratio of *α*-sphere (Fig. S3, highest red dashed line).

## Acknowledgments

The authors thank Alessio Zaccone for stimulating discussions. A.R.B. gratefully acknowledges the hospitality of the Miller Institute for Basic Research in Science at University of California, Berkeley. We gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG) through the Sonderforschungsbereich 1032 and the Nanosystems Initiative Munich (NIM).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: abausch{at}ph.tum.de.

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.

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

Freely available online through the PNAS open access option.

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