Generic, phenomenological, on-the-fly renormalized repulsion model for self-limited organization of terminal supraparticle assemblies

Significance

Engineering nanoassemblies with uniform characteristic dimensions is of great interest given the unique properties of nanoscale building blocks and their collective response in superstructures. We present a generic, particle-based model that predicts the formation of self-limited, or “terminal,” supraparticle assemblies observed in many inorganic, colloidal, and biological systems. The key factor that leads to the self-limiting behavior is shown to be that the repulsion between particles becomes renormalized “on-the-fly” as the particles aggregate. The model explains the monodispersity of terminal assemblies formed from polydisperse nanoparticles, as observed in recent experiments. Our findings not only deepen the understanding of how self-limited, or terminal assemblies form, but also offer versatile approaches to control the dimension and shape of synthetic nanoassemblies.

Abstract

Self-limited, or terminal, supraparticles have long received great interest because of their abundance in biological systems (DNA bundles and virus capsids) and their potential use in a host of applications ranging from photonics and catalysis to encapsulation for drug delivery. Moreover, soft, uniform colloidal aggregates are a promising candidate for quasicrystal and other hierarchical assemblies. In this work, we present a generic coarse-grained model that captures the formation of self-limited assemblies observed in various soft-matter systems including nanoparticles, colloids, and polyelectrolytes. Using molecular dynamics simulations, we demonstrate that the assembly process is self-limited when the repulsion between the particles is renormalized to balance their attraction during aggregation. The uniform finite-sized aggregates are further shown to be thermodynamically stable and tunable with a single dimensionless parameter. We find large aggregates self-organize internally into a core–shell morphology and exhibit anomalous uniformity when the constituent nanoparticles have a polydisperse size distribution.

The spontaneous formation of uniformly sized aggregates observed in inorganic, biological, and colloidal systems (1–14) is of both fundamental and practical interest. Their formation suggests that a generic mechanism is applicable to those chemically distinct systems, whereas their ability to serve as building blocks for engineering nanostructures at larger length scales is of practical interest. Examples of uniform aggregates include filaments, bundles, and toroids with uniform diameters formed by macromolecules (6–9), and regular domains by like-charged macroions on planar and cylindrical surfaces (5, 11). In a recent study, we observed the assembly of positively charged polydisperse CdSe, PbS, and PbSe nanoparticles (NPs) into monodisperse supraparticles (SPs), which form colloidal crystals at sufficiently high density (12). Spherical SPs with uniform size were also obtained from the assembly of similarly charged protein molecules (cytochrome C) and CdTe nanoparticles (13). It was also demonstrated that SPs can serve as a versatile tool to make nanoscale assemblies with unusual spiky shapes and form several constitutive blocks (14). The fact that the characteristic dimensions of these assemblies is highly uniform over a wide range of monomer concentration suggests that their formation from monomer aggregation is thermodynamically self-limited rather than kinetically arrested.

Seminal theoretical studies of uniformly sized aggregates date back to the 1980s, and primarily focused on developing models of aggregation in reactive systems (1–3). An enormous body of theoretical studies that followed has been successful in characterizing the thermodynamic stability of finite-sized aggregates in various systems such as polyelectrolytes (5, 7, 11, 15–19), colloidal suspensions (20–24), and block copolymer and protein solutions (4, 8, 25–30). The optimal size of the assembled clusters is often attributed to the balance of short-ranged attraction and longer-ranged repulsion between the primary building blocks. Nevertheless, uniform clusters are also found in systems where the repulsion is not necessarily long-ranged by nature: it can be energetic (e.g., charge accumulation due to counterions and salt ions on the cluster surfaces as aggregation progresses), entropic (e.g., steric effects due to the nonnegligible size of the counterions and the packing frustration within the aggregates, which prevents counterion penetration), or both (5, 7, 11, 16, 18). Recently, Johnston and coworkers studied the formation of gold colloids in polymeric solutions, showing that the cluster size can be tuned by varying polymer concentration. In this case, the accumulation of the polymeric stabilizers adsorbed on the cluster surfaces provides an effective repulsion between the clusters (29, 30).

Coarse-grained models for the effective pairwise interaction between the primary assembling species (macroions, colloids, and nanoparticles) have been developed to elucidate the formation of finite-sized aggregates (18–21, 23, 25). For instance, Zhang et al. derived effective interactions between the macroions and counterions by integrating over the fast modes of the smaller-sized solvent molecules and salt ions using the random phase approximation, showing that the model reproduced finite-sized clusters at dilute concentrations (18). Kung et al. further coarse-grained over the dynamics of the counterions by incorporating their screening effects into the effective NP–NP potential (19). They demonstrated that it is possible for like-charge NPs in a polar solvent to aggregate due to the steric effects and dipolar interactions of the solvent molecules despite the electrostatic repulsion between the NPs themselves. Sciortino et al. showed that an equilibrium cluster phase is stabilized when the cluster–cluster repulsion increases with the cluster sizes (23), in a similar manner to the Dejarguin–Landau–Verwey–Overbeek model for charged colloids in a dielectric medium (26). An equilibrium cluster phase was also confirmed in protein solutions and colloids (4). However, the equilibrium cluster phases reported in previous computational studies were disordered and only stable at low concentrations (4, 23, 31). It remains unclear whether the ordinary pairwise potentials that have been widely used in the colloidal chemistry/physics and coarse-grained molecular simulation literature can yield ordered cluster phases, i.e., colloidal crystals, as observed in experiments.

Previous simulation studies typically assume that the effective interaction is identical for all pairs of NPs regardless of whether the NPs are forming an aggregate. This assumption neglects the change in the environment surrounding individual NPs and their charge renormalization (32, 33) during aggregation. In previous work (18, 19, 23), the cluster–cluster interaction was derived from the particle–particle interaction after the clusters had been formed. Only recently have Barros and Luijten proposed a numerical technique to resolve the polarization charge distributions in dynamic dielectric geometries arising during assembly (34). They demonstrated that dynamic polarization indeed alters––both qualitatively and quantitatively––predicted self-assembled structures. In the present study, we develop a generic model in which the repulsive interaction between the primary species (macroions, colloids, and NPs) is renormalized as the aggregates grow during the assembly process. The repulsion renormalization in our model is designed to capture the many-body effects associated with the interaction between the primary particles upon aggregation, which are not present in ordinary pairwise potentials. Such many-body effects are highly relevant for systems where, e.g., the primary particles are similarly charged, yet able to aggregate because of short-ranged attraction (12, 13).

The results presented here extend those presented in ref. 12 to consider the effects of repulsion renormalization, to demonstrate SP stability, and to demonstrate that the uniform size of SPs for a given polydispersity of NPs is beyond that expected from statistical considerations. We discuss possible generalization of our model to describe the formation of finite-sized bundles from rod-like polyelectrolytes (7) and of sheets (35) and twisted helical ribbons from colloids (36) and from tetrahedrally shaped NPs (37).

Phenomenological Model

Considering an electroneutral solution of charged NPs, counterions, salt, and solvent molecules, we wish to construct a phenomenological model that implicitly includes the effects of counterions, salt, and solvent molecules into a single-component system in which the NPs interact via an effective pairwise interaction. We focus on the aggregation of NPs due to nonbonded interactions, assuming that the NPs are stable in size and neglecting any further growth during assembly. Note that the terms “aggregates,” “NP clusters,” and “SPs” are often used interchangeably in the present study with the term “supraparticle” referring predominantly to the final product after completion of agglomeration process.

We start with a generic, phenomenological model to capture the interaction between two freely suspended charged NPs in solution:

$$U\left(r_{ij}\right)=U_{\text{LJ}}\left(r_{ij}\right)-U_{\text{LJ}}\left(r_c^{\text{LJ}}\right)+U_{\text{Y}}\left(r_{ij}\right)-U_{\text{Y}}\left(r_c^{\text{Coul}}\right),$$ [1]

where

$$U_{\text{LJ}}\left(r\right)=4\mathit{ϵ}\left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right],$$ [2]

and

$$U_{\text{Y}}\left(r\right)=A_{ij}\frac{\exp \left(-\kappa r/\sigma \right)}{r/\sigma }.$$ [3]

Here $r_{ij}$ is the center-to-center distance between two NPs and κ is the dimensionless inverse screening length of the electrostatic interaction. The Lennard-Jones (LJ) 12–6 potential, $U_{\text{LJ}}\left(r\right)$, represents the combination of the excluded volume and van der Waals interactions. Within the framework of this model, London dispersion forces made the dominant contribution to van der Waals interactions. The parameters ε and σ of the LJ potential are the attraction strength and the NP diameter, respectively. The LJ term is truncated and shifted to zero at $r_c^{\text{LJ}}=2.5\sigma$. When the NP polydispersity is considered, the center-to-center distance between two NPs with different diameters, ${\sigma }_i$ and ${\sigma }_j$, is adjusted by an amount of $\mathrm{\Delta }=\left({\sigma }_i+{\sigma }_j\right)/2-\sigma$.

The short-ranged repulsion between like-charged NPs is modeled by the Yukawa potential, $U_{\text{Y}}\left(r\right)$, with $A_{ij}>0$, to capture the screening effects of the medium including counterions, salt ions, and solvent molecules. The Yukawa potential is truncated and shifted to zero at $r_c^{\text{Coul}}=5.0\sigma$ unless otherwise stated.

In previous attempts to incorporate the collective effects of secondary species (e.g., salt ions and solvent molecules) into the effective potential between primary species, e.g., refs. 18 and 19, it was assumed that the effective interactions between the NPs were invariant over the course of the simulation, regardless of whether the NPs are isolated or part of a cluster. We argue that such an assumption does not reflect the reality of NP interactions. The renormalization of the repulsion between the SPs, i.e., the change in the effective repulsion between NPs residing in the SPs, as the SPs grow in size, takes place in actual dispersions.

Here we adjust the effective pairwise repulsion strength between the NPs, $A_{ij}$, over the course of the simulation, based on their local environment. Specifically, the effective repulsion strength between the NPs from different SPs, $A_{ij}^{inter}$, is different from that between the NPs from the same SP, $A_{ij}^{intra}$, because the dielectric constant inside the SPs is typically smaller than that in the bulk. Moreover, $A_{ij}^{inter}$ is a function of the volume of the interacting SPs, as the effective repulsion between two NPs from different SPs incorporates the many-body effects from constituent NPs.

Consider two particles i and j from two separate SPs, whose volumes are $V_{SP-i}$ and $V_{SP-j}$, respectively. Here the SP volumes, $V_{SP-i}$ and $V_{SP-j}$, are defined as the sum of the volume of their constituent NPs. We choose this definition of the SP volume because it is a geometry-independent measure of the SP size to take into account the fact that the SPs can have arbitrary shapes during aggregation.

We define $V_{SP}=V_{SP-i}+V_{SP-j}$ as the total SP volumes and $V_{SP}^{\left(1\right)}=V_{NP}$ as the volume of a single-particle SP. In our model, the repulsion between particles i and j, $A_{ij}^{inter}$, is renormalized with $V_{SP}$ as

$$A_{ij}^{inter}=A_0+\frac{\partial A_{ij}^{inter}}{\partial V_{SP}}\cdot \left(V_{SP}-V_{SP}^{\left(1\right)}\right),$$ [4]

where $A_0$ is the repulsion strength between freely floating NPs, chosen as the baseline value. $V_{NP}=\pi {\sigma }^3/6$ serves as the unit volume, where σ is the unit length of the system, as dictated by the LJ potential. For polydisperse systems, σ is the mean size of the NPs.

We define a dimensionless quantity s as

$$s=\frac{\partial A_{ij}^{inter}}{\partial V_{SP}}\frac{V_{NP}}{k_{B}T}$$ [5]

to describe how rapidly the inter-SP repulsion per NP increases with the SP volumes. s is a positive number such that $A_{ij}^{inter}$ increases linearly from $A_0$ to $A_{agg}$ until $V_{SP}$ reaches a terminal size $V_{SP}^{max}$:

$$s=\frac{A_{agg}-A_0}{V_{SP}^{max}-V_{SP}^{\left(1\right)}}\frac{V_{NP}}{k_{B}T}=\frac{A_{agg}-A_0}{k_{B}T}\frac{1}{V_{SP}^{max}/V_{NP}-1},$$ [6]

where $A_{agg}$ is the value of $A_{ij}^{inter}$ below which aggregation begins, which is dependent upon temperature $T^{*}=k_{B}T/\mathit{ϵ}$ and $A_0$. $A_{agg}$ is the upper bound on $A_{ij}^{inter}$; as $A_{ij}^{inter}$ approaches $A_{agg}$, NP aggregation halts and $A_{ij}^{inter}$ stops increasing. Thus, Eq. 4 is only applicable for $V_{NP}\le V_{SP}\le V_{SP}^{max}$. For $V_{SP}>V_{SP}^{max}$, $A_{ij}^{inter}=A_{agg}$. In our model, s is the only parameter that dictates the terminal size of the SPs for a given pair of $T^{*}$ and $A_0$. The value of s is system-specific and can be estimated, for example, by the slope of the electrokinetic potential ζ versus SP size curve before the SPs get to their terminal size (12, 13), as will be done below.

During the simulations, the SPs are identified based on the neighbors of NPs within a certain distance of $1.4\sigma -1.5\sigma$ so as to take into account thermal fluctuations and avoid counting close, yet separate, SPs as a single SP. Every NP stores the information of its SP size and radius of gyration, center of mass, and its relative position with respect to the SP center of mass. If the distance from an NP to the SP center of mass is less than the SP radius of gyration by $0.1\sigma$, the NP is considered in the core of the SP; otherwise, it is in the SP shell. Further details on the clustering procedure is given in Model Discussion.

$A_{ij}^{intra}$, the repulsion strength between NPs in the same SP, is determined by their relative positions with respect to the SP center of mass. For NPs in the shell of the SP, i.e., $r_i\sim R_{SP-i}$, $A_{ij}$ scales with the SP volume, and is also bounded by $A_{agg}$ at $V_{SP}^{max}$. Meanwhile, for those in the core of the SP, i.e., $r_i<R_{SP-i}$, $A_{ij}$ is chosen to be $A_0$. In this study, we restrict $V_{SP}^{max}/V_{NP}=13$, the minimum cluster size required to define a core and shell in this way. For intra-SP interactions, we need to specify an SP size threshold, $V_{cs}$, beyond which the repulsion terms between the NPs in the core and in the shell are distinguished. The core–core repulsion is not renormalized with the SP size to prevent the SP from exploding when the SP size exceeds the threshold value by the NPs attaching to the shell. The core–shell repulsion, on the other hand, is renormalized with the SP size to prevent further aggregation of approaching NPs. The threshold is defined relative to $V_{SP}^{max}$ (i.e., $0.8V_{SP}^{max}$ in the present study) instead of an absolute number of NPs, to capture the renormalized core–shell repulsion over a wide range of $V_{SP}^{max}$. The effective NP–NP repulsion strengths used in our simulations are summarized in Table 1.

Table 1.

Parameters for the repulsion between two NPs: $r_i$ is the relative position of an NP to the SP center of mass, $R_{SP-i}$ is the SP radius of gyration

	Pairwise repulsion interaction parameters, $A_{ij}$ and ${\kappa }_{ij}$
Freely floating NPs	$A_{ij}=A_0$
	${\kappa }_{ij}={\kappa }_{medium}=0.2$
NPs from different SPs	$V_{SP-i}+V_{SP-j}<V_{SP}^{max}$:
	$A_{ij}=A_0+s{k}_{B}T\cdot \left(V_{SP-i}+V_{SP-j}-V_{NP}\right)/V_{NP}$
	${\kappa }_{ij}={\kappa }_{ij}^{inter}={\kappa }_{medium}=0.2$
	$V_{SP-i}+V_{SP-j}\ge V_{SP}^{max}$:
	$A_{ij}=A_{agg}$
	${\kappa }_{ij}={\kappa }_{ij}^{inter}={\kappa }_{medium}=0.2$
NPs in the same SP	$r_i<R_{SP-i}\mathrm{\cap }{r}_j<R_{SP-i}$: Both NPs are in the core.
	$A_{ij}=A_{ij}^{intra}=A_0$
	$\kappa ={\kappa }_{ij}^{intra}=0.1$
	$r_i\approx R_{SP-i}\mathrm{\cup }{r}_j\approx R_{SP-i}$: Either NP is in the shell.
	$A_{ij}=A_{agg}{V}_{SP-i}/V_{SP}^{max}$
	${\kappa }_{ij}={\kappa }_{medium}=0.2$

The intra-SP interactions are only considered when the SP size exceeds a threshold, $V_{cs}$, chosen to be $0.8V_{SP}^{max}$ in the present study. See text for explanation.

Although the cutoff of the LJ potential is set at $r_c^{\hspace{0.17em}\text{LJ}}=2.5\sigma$ in the present study, our model should work with smaller values of $r_c$ as long as aggregation can occur (note that changing $r_c^{\hspace{0.17em}\text{LJ}}$ would lead to a change in $A_{agg}$, which in turn changes $V_{SP}^{max}$ for a given value of s). This is because the renormalized repulsion in our model only depends on the SP sizes, not the interaction range. Consequently, we expect that our model should also be applicable to systems with shorter-ranged interactions, e.g., cosolute mediated, depletion interactions.

The required inputs for our simulations, in addition to temperature $T^{*}$, number density ϕ, and repulsion strength $A_0$, include the upper bound $A_{agg}$, the parameter s (or equivalently, the terminal SP size, $V_{SP}^{max}$), and the threshold cluster size used for distinguishing core–shell interactions, $V_{cs}$. Starting from a well-dispersed configuration, the system is equilibrated at constant temperature and volume until the system potential energy plateaus for a sufficiently long period. More details on our model and simulation method can be found in Model Discussion.

Results

In a previous study (12) we used the present model to explain the formation of monodisperse SPs (${\delta }_{SP}\approx 8\text{\%}$) from a suspension of CdSe NPs with substantial size polydispersity (${\delta }_{NP}\approx 25\text{\%}$). It was found that the core–shell morphology of the SP structures was responsible for mitigating the effect of the NP polydispersity. In ref. 12, the increase in the inter-SP repulsion between NPs in our model represents the variation in the electrokinetic potential ζ during aggregation as the SP charge grows with each NP and the requisite counterions accumulate on the SP surface. Given the rate of change of the surface potential with respect to the SP diameter, we were able to estimate $s\sim {10}^{-3}$ from the charge of the NPs $q\approx 3e$ and their average diameter $d_{NP}=5$ nm at room temperature (Simulation Method). Likewise for other systems, an estimate of s can be deduced from the variation of the ζ potential with respect to the diameter of the clusters in the early stages of aggregation.

The parameter s can also be estimated from a conventional model, i.e., without renormalization, using a preliminary analysis in a similar fashion to the work of Sciortino et al. (23), whereby the SP–SP repulsion is obtained by summing the repulsion strength from pairs of NPs from two interacting SPs. The slope of the repulsion strength versus SP size curve gives us an estimate of s. In this case, the relationship may not be linear, and thus s may not be a constant.

In the following, we extend our model to study different values of s over a wide range of density of monodisperse NPs (${\delta }_{NP}=0$) in comparison with the cases where there is no repulsion renormalization. We analyze the stability of the SPs at the terminal size for a given value of s to demonstrate that the SPs are thermodynamically stable. Finally, we compare our model to a simple statistical model to address the unexpected effects of the NP polydispersity (${\delta }_{NP}>0$) on the size polydispersity of the SPs. We assume that the NP diameters are drawn from a normal distribution and NP polydispersity, ${\delta }_{NP}$, is defined as the SD of that distribution, normalized by the average NP size. Meanwhile, the SP size polydispersity, ${\delta }_{SP}$, is defined as the SD of the SP radii of gyration, normalized by the average value, obtained from simulation.

Assembly Process.

Fig. 1 shows finite-sized SPs self-assembled from disordered fluid initial conditions and stable over a wide range of densities for different values of s. In particular, the assembled structures in Fig. 1 A–D are obtained at densities at which the SPs are in close proximity yet do not merge. Self-limiting aggregation occurs when the effective repulsion between NPs from different SPs exceeds their attraction. As s increases, the SP size at which $A_{ij}^{inter}$ reaches $A_{agg}$ decreases, thus reducing the terminal size for the SPs.

Fig. 1.

Representative simulation snapshots for different values of s and number density ϕ. (A) $s=0.42$, (B) $s=0.14$, (C) $s=0.07$, and (D–F) $s=0.09$: (D) $\varphi =0.24$, (E) $\varphi =0.08$, and (F) $\varphi =0.512$. Snapshots are generated by the software VMD.

Fig. 1 D–F shows assembled structures for $s=0.09$ at different densities. It may be argued that the uniform-sized SPs observed in a dilute solution are simply the consequences of limited diffusion or kinetic arrest, because the SPs infrequently interact. However, by increasing ϕ up to 0.5, so that SPs can now interact, we show that the SPs at the terminal size indeed do not coalesce, and that the terminal size does not depend on the system density, as expected for self-limiting assembly.

We emphasize that, without on-the-fly repulsion renormalization, i.e., $s=0$, the NPs instead form disperse SPs with a broad size distribution (Fig. 2A) and a gel-like structure at higher density (Fig. 2B). Even with a long screening length of $1/\kappa =10$, the aggregation is not self-limited as the pairwise short-ranged attraction dominates at the state point under consideration, $T^{*}=0.2$ and $A_0=1.0k_{B}T$. In previous studies (7, 18, 23), the finite-sized clusters were observed in the dilute regime, and the clusters were likely kinetically arrested. By performing these simulations, we show that if the inter-SP repulsion between NPs is not renormalized during the assembly process, the SPs would eventually merge on contact so as to minimize surface tension even with a long screening length. Without repulsion renormalization, the NPs form disordered structures at the given temperature and density for all tested cooling schedules, Coulombic cutoff distances $r_c^{\hspace{0.17em}\text{Coul}}=5.0\sigma -10.0\sigma$, and inverse screening lengths $\kappa =0.1-0.5$.

Fig. 2.

Assembled structures obtained for $s=0$ and $\kappa =0.1$. (A) Aggregates with a broad size distribution at $\varphi =0.01$ with $r_c^{\hspace{0.17em}\text{Coul}}=5.0\sigma$. (B) A gel-like structure at $\varphi =0.3$ with $r_c^{\hspace{0.17em}\text{Coul}}=10.0\sigma$.

Terminal assemblies may be characterized by system potential energy, average number of NPs per SP, and the SP size distribution (Fig. 3). An example of the time evolution of the potential energy for a system consisting of $N={10}^4$ NPs for different values of s is given in Fig. 3A. As aggregation proceeds, the system potential energy decreases to a stable value, which increases with increasing s. This means the contribution of the inter-SP energy to total potential energy increases as the size of the SPs decreases. Additionally, as the number of NPs per SP approaches the terminal value (Fig. 3A, Inset), the SP size, starting from a broad distribution, evolves into a single distinctive peak at the terminal size. These findings demonstrate that the SPs become more and more uniform in size (Fig. 3B), as observed in experiments (12).

Fig. 3.

Assembly process. Time evolution of (A) the potential energy per NP and the average SP size normalized by the terminal size (Inset) for different values of s. (B) Size distributions (in number of NPs) of the assembled SPs for $s=0.09$ at $\varphi =0.37$.

The self-limited aggregation of the NPs is investigated by compressing the system of terminal-sized SPs so that the SPs would come close enough to merge if indeed that was thermodynamically favorable. The compression is performed by periodically rescaling the simulation box dimensions, i.e., every 10 time steps, until the system reaches the specified number density. Our goal in this compression is not to model any experimental compression protocol, but merely to bring the SPs closer together, even in contact, to confirm that they do not merge and are therefore self-limiting.

Fig. 4 shows the dependence of the potential energy per NP on number density, ϕ, after the SPs acquire their terminal size for $s=0.09$. For $0.3\le \varphi \le 0.4$, the monotonic increase in the potential energy with respect to ϕ indicates that the terminal size SPs do not coalesce upon box compression. In contrast, the potential energy is almost unchanged without the repulsion renormalization ($s=0$), as expected for a liquid state in the range of ϕ and $T^{*}$ under consideration. At sufficiently high density, the terminal-sized SPs form a Wigner crystal with face-centered cubic (fcc) ordering, as revealed by the pair correlation function (Fig. 4B) and bond order diagram of the SP centers of mass shown in stereographic projection (Fig. 4B, Inset). This is expected because the SPs are strongly repulsive to each other.

Fig. 4.

Terminal assemblies at higher densities. (A) Potential energy per NP at different number densities ϕ for $s=0.09$ and $s=0$, $\mathit{ϵ}=5k_{B}T$. (B) Radial distribution function (RDF) and bond order diagram (Inset) corresponding to the first peak of the RDF reveals the fcc local ordering of the SPs shown in Fig. 1A for $s=0.42$. The RDFs were calculated for the centers of mass of the SPs (dotted line) and for the NPs (solid line). For the NP–NP pair correlation (solid line), the first peak corresponds to the NP nearest-neighbor separation and the second peak to the largest separation between the particles in a cluster, i.e., an estimate of the SP size.

Stability.

To analyze the stability of the SPs at the terminal size, we compare the potential energy per NP in the SPs of different sizes for a given value of s (Fig. 5). We assume that the NPs form the corresponding energy minimizing cluster (38). As can be seen in Fig. 5, for $s=0$, i.e., without repulsion renormalization, the potential energy decreases as $V_{SP}^{max}/V_{SP}$ increases, and infinite-size SPs are favored. In contrast, for $s>0$ the potential energy decreases until $V_{SP}$ reaches an upper bound value, $V_{SP}^{max}$ at which $A_{ij}^{inter}=A_{agg}$, as mentioned in Model Discussion.

Fig. 5.

Potential energy per NP with different aggregate sizes for different values of s at $T^{*}=0.2$, $\left(A_{agg}-A_0\right)/k_{B}T=5.0$. $r_c^{\hspace{0.17em}\text{Coul}}=5.0\sigma$ and $\kappa =0.1$. The curves for $s=0.09$, $s=0.14$, and $s=0.42$ are shifted vertically for clarity.

The jump in potential energy at the value of $V_{SP}^{max}$ indicates that if we were to force the NPs into an SP whose size is greater than $V_{SP}^{max}$, the SP would be unstable. For instance, when the NPs are initialized in an SP of size $V_{SP}/V_{NP}=13$ and the inter-SP interaction is scaled with $s=0.42$, the NPs in the outer shell of the SP will be strongly repulsive and detach from the SP. We observe that the system evolves until most of the SPs form with $V_{SP}^{max}/V_{NP}=13$ NPs. This analysis again shows that NPs are not energetically favored to attach to the SPs already having the terminal size. In our calculation we neglect entropy because of the strong NP–NP intra-SP interaction and because of the large number of NPs per SP, which leads to the SPs that are mostly spherical in shape. Because SPs of size $V_{SP}^{max}$ after equilibration occur with the highest frequency (compare Fig. 3B), it is reasonable to deduce that the terminal finite size should correspond to a free-energy minimum, as predicted by previous studies (8, 16). The uniform SPs are thermodynamically stable, rather than long-lived, kinetically arrested, clusters. This is consistent with the experimental observation in our previous studies for a variety of materials (12, 13), where the SPs at terminal size do not merge, but pack into colloidal crystals when concentrated to sufficiently high density (12).

Polydispersity.

In this study and in the experiment of ref. 12, a distribution of NPs with relatively large (${\delta }_{NP}\approx 25$%) size polydispersity self-assemble into SPs with very low (ideally $\sim$0% for large SPs) size polydispersity. Here, we estimate how much simple statistical averaging would reduce ${\delta }_{SP}$ relative to ${\delta }_{NP}$ and find that for sufficiently large SPs, ${\delta }_{SP}$ grows with ${\delta }_{NP}$ far more slowly than statistics would predict.

We consider the SP polydispersity to be of the form

$${\delta }_{SP}={\delta }_{SP,0}+\frac{\partial {\delta }_{SP}}{\partial {\delta }_{NP}}{\delta }_{NP},$$ [7]

where ${\delta }_{SP,0}$ is the SP polydispersity at ${\delta }_{NP}=0$; this inherent polydispersity comes from amorphously packing spherical NPs into SPs and varying the total number of NPs for a given value of ${\mu }_{SP}$, the average number of NPs per SP. Our statistical model does not incorporate these effects (${\delta }_{SP,0}=0$) and predicts $\partial {\delta }_{SP}/\partial {\delta }_{NP}$ from statistical averaging.

The statistical model assumes that the SPs are spherical and that the volume of an SP is proportional to the sum of the volumes of its constituent, spherical NPs. For a given ${\mu }_{SP}$, we compute the volume and radius of the SP as

$$V_{SP}=\frac{4\pi }{3}{R}_{SP}^3\propto {\displaystyle \sum _{i=1}^{{\mu }_{SP}}{R}_{NP,i}^3},$$

where $R_{NP}$, the radius of the NPs, is a Gaussian random variable with the normalized SD of ${\delta }_{NP}$, the NP polydispersity. Note that $R_{SP}$ determined this way is equivalent to the SP radius of gyration obtained from our simulation, up to a scaling factor. We estimate the polydispersity of the SPs (${\delta }_{SP}$) by computing the SD of $R_{SP}$ for ${10}^6$ clusters. This statistical model assumes all of the SPs are composed of the same number of particles.

For average numbers of constituent NPs (${\mu }_{SP}$) from 20 to 100, we plot (Fig. 6A) ${\delta }_{SP}$ as a function of NP size polydispersity (${\delta }_{NP}$) predicted by the statistical model and from our simulations. Our model informs us about how we expect ${\delta }_{SP}$ to increase with ${\delta }_{NP}$: We expect that ${\delta }_{SP}$ will increase with ${\delta }_{NP}$, increasing more slowly with larger clusters that allow for more statistical averaging of the polydisperse NPs.

Fig. 6.

Statistical analysis. (A) The polydispersity of the supraparticles (${\delta }_{SP}$) as a function of the polydispersity of the constituent nanoparticles (${\delta }_{NP}$). (B) $\partial {\delta }_{SP}/\partial {\delta }_{NP}$ as a function of average supraparticle size ${\mu }_{SP}$. (C) Cluster asphericity $A_S$ as a function of ${\delta }_{NP}$: Smaller clusters are more aspherical, explaining the departure from our statistical model from small ${\mu }_{SP}$. (D) The radial density profile $n\left(r\right)$ of ${\mu }_{SP}=100$ clusters, which corresponds to $s=0.09$, obtained from three different simulation runs for ${\delta }_{NP}=0.0,0.1$, and 0.2. We find that for higher values of ${\delta }_{NP}$, the inner core’s local density decreases relative to the outer shell, indicating that the SPs formed with a larger ${\mu }_{SP}$ adopt a core–shell structure, which mitigates the effect of ${\delta }_{NP}$ on ${\delta }_{SP}$. Representative snapshots of the SPs formed with (E) ${\delta }_{NP}=0$ and (F) ${\delta }_{NP}=0.2$. The NPs are color-coded based on their diameter. (Right) The plot shows the NP average size profile across the cross-section of the SPs at terminal size for ${\delta }_{NP}=0.2$. The error bars are obtained from averaging over 10–20 SPs of terminal size.

We find deviations in $\partial {\delta }_{SP}/\partial {\delta }_{NP}$ computed from simulations and from our calculations at the smallest and largest SP sizes we study (Fig. 6B). In an attempt to explain this, we calculate the average asphericity parameter, which is defined as (39–41)

$$A_S=\frac{1}{2}\langle \frac{{\displaystyle {\sum }_{i\ne j}{\left({\lambda }_i-{\lambda }_j\right)}^2}}{{\displaystyle {\sum }_i{\lambda }_i^2}}\rangle ,$$ [8]

where ${\lambda }_i$ ($i=x,y,z$) are the eigenvalues of the gyration tensor, for clusters from our simulations and energy minimizing clusters. The $\langle \cdot \rangle$ operator indicates averaging over the clusters and over different simulation configurations for a given value of ${\mu }_{SP}$. We note that the asphericity parameter rapidly decreases between ${\mu }_{SP}=20$ and ${\mu }_{SP}\ge 40$ in both our simulations and the energy minimizing clusters (Fig. 6C). Consequently, the inherent asphericity of clusters of these sizes evidently affects the growth of ${\delta }_{SP}$ with ${\delta }_{NP}$ in this size regime. The deviations in SP size polydispersity relative to the statistical model for small clusters can be explained by the fact that the statistical model assumes spherical averaging of the cluster radius of gyration, i.e., $R_{SP}\propto {\left({\displaystyle {\sum }_{i=1}^{{\mu }_{SP}}{R}_{NP,i}^3}\right)}^{1/3}$. For small clusters, which are more likely aspherical, or even irregular in shape, the effective radius is impacted more greatly. Small clusters also just have more inherent polydispersity because the assumption of perfect volumetric averaging is less accurate.

As the cluster size continues to increase, we find the region in which $\partial {\delta }_{SP}/\partial {\delta }_{NP}$ predicted by our statistical model lies within the error bars computed from our distribution, which assumes spherical clusters of uniform density (Fig. 6B). Clusters in this region are more spherical than those with ${\mu }_{NP}<40$, but not yet large enough to adopt a core–shell morphology; in this region we expect a essentially statistical averaging of NP sizes and find a reasonable agreement with our statistical model.

At ${\mu }_{SP}\ge 80$, we see that $\partial {\delta }_{SP}/\partial {\delta }_{NP}$ of the best fit line is within error bars of 0, in contrast with the monotonically increasing prediction ${\delta }_{SP}$ $\left(\partial {\delta }_{SP}/\partial {\delta }_{NP}\ge 0\right)$ of the statistical model. To explain this discrepancy, we calculate the average radial density profile for ${\mu }_{SP}=100$ clusters in Fig. 6D from ${\delta }_{NP}=0.0,0.1,0.2$. The two peaks in the $\rho \left(r\right)$ indicate the two shells within the cluster. As the clusters grow in size, we expect to recover a homogeneous NP density in the clusters’ interiors, as we observe when ${\delta }_{SP}=0$ (Fig. 6E). However, as ${\delta }_{SP}$ increases, we see the inner core’s density decrease with respect to the outer shell. We believe this core–shell structure of these larger clusters mitigates the polydispersity of the size distribution of the NPs, as observed experimentally (12). Fig. 6F further reveals that larger particles comprise the SP core and that smaller particles comprise the outer shell. The arrangement of the NPs within the large SPs into the core–shell structure is attributed to the difference in the intra-SP and inter-SP repulsions in our model (Model Discussion). That our simulations produce a core–shell morphology like that observed in experiments is another strength of our model.

Discussion

The model we present here is the first, to our knowledge, to capture qualitatively not only the thermodynamic stability of individual SPs at the terminal size, as did previous continuum models (1–3), but also the core–shell morphology and assembly process of the densely packed, ordered cluster phase observed in experiments. The key differences between our model and previous coarse-grained models (18–21, 23, 25) for the formation of finite-sized assemblies are (i) that the inter-SP repulsion between the NPs is renormalized during the simulation, and (ii) that the NP interactions within the SP and from different SPs are treated differently. First, we acknowledge that the first-order approximation of $A_{ij}^{inter}\left(V_{SP}\right)$ with a single parameter s in our model oversimplifies the effective repulsion between the NPs from different SPs, which can be many-body and/or entropic in nature. The linear dependence of $A_{ij}^{inter}$ on $V_{SP}$ also leads to a stronger volume-dependent cluster–cluster repulsion if we were to integrate over all NP–NP pairs, e.g., as done in ref. 23. Although one could further improve the accuracy of $A_{ij}^{inter}\left(V_{SP}\right)$, e.g., by taking into account higher-order terms for three- and four-body effects, we argue that the self-limited assembly behavior persists as long as $A_{ij}^{inter}$ monotonically increases with $V_{SP}$ up to $V_{SP}^{max}$, i.e., s is positive. Therefore, the repulsion renormalization proposed here is applicable not only to systems with charge accumulation, but also to those where the increase in the cluster size leads to energetic/entropic penalties for further attachment of particles to the cluster surfaces. Examples include colloids with short-ranged attraction and long-ranged repulsion, polyelectrolytes with depletion interaction, and micelle-forming polymer-tethered particles. Finally, the difference in inter-SP and intra-SP interactions in our model represents, however crudely, the effects of the dielectric heterogeneity of the media arising over the assembly process. Capturing these dynamical effects accurately requires significantly more rigorous efforts (34, 42–45).

The ability of our model to obtain uniformly sized SPs from self-limited aggregation raises an interesting discussion of whether it would be relevant to other studies such as virus capsid assembly, colloidal crystals, and finite DNA bundles. As revealed by Fig. 1, the NPs within individual SPs arrange into energy-minimizing configurations such as 13-NP clusters in icosahedra for $s=0.42$, 55-NP clusters for $s=0.09$, and 69-NP clusters for $s=0.07$, as reported by ref. 38. Because $A_{ij}^{inter}$ is assumed to be isotropic, the terminal clusters are thus close to a spherical shape. The formation of these terminal energy-minimizing clusters is reminiscent of the mechanism of virus capsid assembly with packing constraints (46) where capsids are formed with a finite number of capsomers and in a regular shape. As the volume of the SPs is restricted to a certain amount, the propensity of assembling energy-minimizing structures is improved as the number of unfavorable configurations is substantially reduced (47). Additionally, the sufficiently strong repulsion between these uniformly sized SPs leads to crystalline structures, which were not observed for $s=0$.

Previous studies (7–9) mostly focused on the most stable cluster size, given the NP charges and Bjerrum length. In this regard, the inter-SP repulsion in our model can be considered as a biasing potential that enhances sampling assembled clusters at a certain size. However, unlike unphysical biasing potentials, e.g., a harmonic potential, our model incorporates physical assembly dynamics in the sense that the parameter s can be estimated from experiments (as already illustrated in Results). As such, our model, to a certain extent, can serve to predict and design SPs at specified sizes. For instance, the size of target assemblies can be adjusted by tuning s, through the degree of “undercooling” $\left(A_{agg}-A_0\right)$.

Finally, we envision that our model can be extended straightforwardly to investigate self-assembly in various soft-matter systems. When the inter-SP repulsion is directional in nature so that the growth of the SPs is limited in specific directions, it is expected that the resulting assemblies will be anisotropic in shape (4, 7, 9, 16, 24, 36, 48). For instance, the finite-sized bundles formed by rod-like polyelectrolytes (7, 9, 16) may result from the lateral repulsion between bundles increasing with their cross-section diameter. Furthermore, when a directional inter-SP repulsion is coupled with building block shape anisotropy, it is possible to rationalize the formation of the sheets formed by tetrahedral-shaped NPs with a uniform thickness (35), and twisted helical ribbons with a uniform width (37). The soft, yet strong, repulsion between the SPs in polydisperse systems could possibly allow for the formation of quasicrystals, as pointed out by Iacovella et al. (49). These interesting topics are all open for future study.

Conclusion

The thermodynamic model proposed reproduces the self-limited assembly observed in recent experiments on NPs, leading to closely packed clusters at high density. Despite the simplicity of our model, we have shown that it captures the balance between short-ranged attraction and accumulated repulsion between NPs in a self-limiting fashion, and that the uniform finite-sized aggregates are thermodynamically stable. The renormalized inter-SP repulsion in our model represents the collective repulsion between charged NPs in the weak screening limit and/or the energetic and entropic effects from the implicitly treated counterions due to aggregation. The model allows us to simulate the formation of colloidal crystal-like structures and to predict the relationship between the polydispersity of the SPs and that of the NPs. We expect further studies extending our model to elucidate the formation of complex soft-matter assemblies such as twisted helical ribbons (37) and quasicrystals (49).

Materials and Methods

Model Discussion.

In our model $A_{ij}^{inter}$, the strength of repulsion between NPs in different SPs, depends on the volume of the interacting SPs. In the weak screening limit, this is relevant because the effective repulsion between two NPs from different SPs incorporates the many-body effects from constituent NPs in each SP. In the strong screening limit and with sufficiently big counterions (16) or with stabilizing polymers (27, 29), the accumulation of the counterion layers (or adsorbed polymeric stabilizers) causes the increase in $A_{ij}^{inter}$ as the SPs grow in size.

Using the argument as above, we propose that the repulsion strength between the NPs in the same SP, $A_{ij}^{intra}$, also depends on the relative positions of the NPs to the SP center of mass. In the present study, we consider a particle in the core if its relative position to the SP center, $r_i$, is less than $\left(R_{SP}-\sigma \right)$, where $R_{SP}$ is the radius of gyration of the SP. If both particles are close to the SP surface, i.e., in the shell of the SP, $A_{ij}^{intra}$ increases with the instantaneous SP volume and reaches $A_{agg}$ when $V_{SP}=V_{SP}^{max}$. In contrast, if either particle is in the SP core, $A_{ij}^{intra}$ is for simplicity assumed to be invariant as the SP grows. In our previous study (12), we tried different values of $A_{ij}^{intra}$ for the core–core interaction to investigate the local packing of the NPs within the SP; in the present study, we set this value to be $A_0$, again for simplicity. Finally, we should note that when the SP size is still less than $0.8V_{SP}^{max}$, we always use $A_{ij}^{intra}=A_0$ regardless of the NPs in the core or in the shell. Although the choices of the SP size threshold (i.e., $0.8V_{SP}^{max}$) and of the core–shell boundary (i.e., $R_{SP}-\sigma$) are relatively arbitrary, we argue that they do not alter the self-limiting behavior of the systems under investigation.

Because van der Waals forces decay much faster than electrostatic forces and are less sensitive to screening effects (27), the total van der Waals attraction increases with the SP radius (26). Using the same renormalization rationale as above, we observe that the rate of change in ${\mathit{ϵ}}_{ij}^{inter}\sim V_{SP}^{-2/3}$, which is substantially slower than that of the electrostatic repulsion. We therefore choose to neglect the variation in the inter-SP LJ attraction between NPs with the SP size in the current study.

Simulation Method.

We use molecular dynamics to simulate a system of N NPs in a 3D cell with periodic boundary conditions. The Langevin thermostat is coupled with the equation of motion of every NP so that the system is equilibrated in the canonical ensemble, i.e., constant volume and temperature. To verify that the simulated assemblies represent thermodynamic equilibrium structures, we consider different numbers of particles in the range of $N={10}^3-{10}^4$ particles from different initial configurations and random seeds, and at different concentrations. The natural units are the mean particle diameter σ, particle mass m, and LJ well depth, ε. The time unit is then defined as $\tau =\sigma \sqrt{m/\mathit{ϵ}}$. As such, the number density $\varphi =N/V$ is expressed in the unit of ${\sigma }^{-3}$ and the screening length κ in the unit of ${\sigma }^{-1}$. The time step used in our simulations is chosen to be $\mathrm{\Delta }t=0.002\tau$. Referring to actual NP systems, these parameters are typically σ = 5 nm, ε = 50–60 kJ/mol, T = 300 K, and $m={10}^{-21}$ kg. Starting from random initial configurations, the systems are equilibrated for $5\times {10}^6-10\times {10}^6$ time steps until the potential energy plateaus, corresponding to a time span of $\approx$2–5 μs. The NP number density is varied from $\varphi =0.01-0.5$ to ascertain that the finite-sized SPs interact yet do not merge. We also anneal stable structures to ascertain that these systems are not kinetically arrested.

The clustering procedure should be performed as frequently as possible. We have tested with different clustering periods at $T^{*}=0.2$, i.e., ${\tau }_c$ = 10, 100, 1,000, and 10,000 time steps, and observed that there are no remarkable differences in the size distribution of the equilibrated structures for ${\tau }_c\le \mathrm{1,000}$. For ${\tau }_c=\mathrm{10,000}$, however, the NPs may form SPs whose size is much bigger than $V_{SP}^{max}$, causing unphysically big repulsive forces, meaning that the aggregation dynamics was not adequately captured. Therefore, we choose ${\tau }_c=\mathrm{1,000}$ time steps to maximize computational efficiency and to avoid unphysical behavior.

In this work, we choose $A_0=1.0k_{B}T$ and $T^{*}=0.2$ for the interaction between two freely floating NPs, where the NPs are energetically favored to aggregate. At that temperature, the solid–liquid repulsion strength $A_{agg}$ is estimated to be $6.0k_{B}T$. According to Eq. 6, the relationship between s and the ratio between $V_{SP}^{max}$ and $V_{NP}$ is $s=5.0/\left(V_{SP}^{max}/V_{NP}-1\right)$. The inverse screening lengths are chosen as ${\kappa }_{ij}^{intra}=0.1$ and ${\kappa }_{ij}^{inter}=0.2$ because the medium outside the SP (e.g., water) often has a larger dielectric constant than inside the SP. For instance, because the counterion concentration inside the SPs is usually lower than in the medium, the inverse screening length inside the SPs is smaller than in the solvent.

We estimated the parameter s from the dependence of the electrokinetic potential on the SP diameter during aggregation (see figure 2e in ref. 12) as follows. From the experimental data points ($d_{SP}/d_{NP}$; ζ) = (9.1;−8.0), (10.1;−11.8), (11.2;−12.0), and (11.8;−17.5), we performed a linear regression for ζ versus $V_{SP}\sim \pi d_{SP}^3/6$ and obtained the rate of change $\partial \zeta /\partial V_{SP}$ = −2.64e + 20 V/${\text{m}}^3$, given the NP diameter $d_{NP}\approx 5.0$ nm. Because the repulsion strength between an SP and an approaching NP is proportional to the electrokinetic potential, the relationship between the repulsion strength between the SP and the approaching NP and $q_{NP}\zeta$ can be approximated by a linear dependence, where $q_{NP}\approx 3\mathrm{e}$ is the charge of the approaching NP. As a result, the variation of the inter-SP repulsion between the NPs, $\partial A_{ij}^{inter}$, with respect to $V_{SP}$ is estimated by $\partial A_{ij}^{inter}/\partial V_{SP}=q_{NP}\partial \zeta /\partial V_{SP}$ = 7.92e + 20 eV/${\text{m}}^3$ = 126.7 J/${\text{m}}^3$. Plugging this value into Eq. 5 and using $V_{NP}=\pi d_{NP}^3/6$ and $T=300$ K gives $s\sim {10}^{-3}$.

The simulations were performed using our in-house code. Our coarse-grained model will be implemented as a plugin for HOOMD-blue (codeblue.umich.edu/hoomd-blue) (50).