Session Y35: Nonequilibrium Self-Assembly and Self-Organization I

Design principles for fast and efficient self-assembly processes

Self-assembly is a fundamental concept in biology and of significant interest to nanotechnology. Considerable progress has been made in characterizing and controlling the properties of the resulting structures, both experimentally and theoretically. However, much less is known about kinetic constraints and determinants of dynamical properties like time efficiency, although these constraints can become severe limiting factors of self-assembly processes. I will discuss how the time efficiency and other dynamical properties of reversible self-assembly depend on the morphology (shape) of the building blocks for systems in which the binding energy between the constituents is large. As paradigmatic examples, I will discuss the self-assembly of constituents with triangular, square, and hexagonal morphology into two-dimensional structures of a specified size. A key finding is that the constituents' morphology critically determines the assembly time and how it scales with the size of the target structure. The theoretical analysis reveals three key structural parameters defined by the morphology: The nucleation size and attachment order, which describe the effective order of the chemical reactions by which clusters nucleate and grow, respectively, and the growth exponent, which determines how the growth rate of an emerging structure scales with its size. Using this characterization, it is possible to formulate an effective theory of the self-assembly kinetics, which exhibits an inherent scale invariance. This critical insight leads to general scaling laws that describe the minimal assembly time as a function of the size of the target structure. Finally, I will discuss how these insights on the kinetics of self-assembly processes can be used to design assembly schemes that could significantly increase the time efficiency and robustness of artificial self-assembly processes.   

Asymptotic Designability of Self-Assembled Structures

Self assembly is heavily constrained by kinetic and thermodynamic factors. In some situations, thermodynamic constraints can be avoided with simple recipes, but it is not obvious how to apply these recipes to more complicated assembly scenarios, especially when the number of component species is limited, or when components bind promiscuously. In such cases, it is often unclear whether a target structure (or group of structures) can be made into the thermodynamic ground state, making it hard to predict what structures can be assembled at high yield. Here, we introduce the concept of Asymptotic Designability, which allows us to quickly identify which structures are thermodynamically stable at high yield, and to determine the degree of control over the relative yields within a group of designable structures. We show that checking for Asymptotic Designability can be formulated as a Linear Program, which can be solved efficiently, and that optimizing relative yields is equivalent to solving a system of linear equations. Our asymptotic theory is exact in the limit of infinite binding energies, but it is also predictive for finite energies and concentrations, which allows us to take kinetic constraints into account. These results enable an exact quantification of the fundamental thermodynamic constraints on self-assembly, and they provide a starting point for the description of more advanced self assembly pathways in and out of equilibrium.   

Dynamical phase transition in programmable multicomponent crystals

Complex synthetic materials such as DNA-programmed molecular and colloidal crystals can assemble into multiple distinct polymorphs from a set of shared components. Because molecular self-assembly is an intrinsically nonequilibrium process, the kinetics of the assembly process control the robustness of the assembly of a specific polymorph. We use a lattice model to study the seeded growth of prototypical multicomponent crystals, where different unit cell designs can be encoded through directional interactions. Our simulation results suggest that a dynamical phase transition separates the stable assembly of specific unit cell designs from disordered crystal growth. From our simulations and a dynamical mean-field analysis, we obtain a supersaturation-dependent dynamical phase diagram that reveals the existence of a critical number of encoded unit cells, below which the seeded unit cell design can be reliably grown and beyond which only the disordered phase is stable. Close to the critical number of encoded unit cells, we observe a supersaturation-dependent disordered-phase wetting layer on the growing unit cell, consistent with a first-order dynamical phase transition. Our insights into the nature of this dynamical phase transition suggest design rules for the self-assembly of multicomponent molecular and colloidal crystals, along with strategies for improving the robustness of multi-polymorph crystallization.   

Understanding the mechanisms of crystallization and reconfiguration of anisotropic nanoparticle superlattices

Crystallization is a universal phenomenon. Yet, despite its importance in wide-ranging technologies and natural processes, there is no single theoretical framework that accurately describes diverse crystallization phenomena. Recent breakthroughs in liquid cell transmission electron microscopy (LCTEM) have enabled direct, in situ visualization of nanoparticle systems, including their self-organization into and between distinct superlattices. In this work, we elucidate the complex, solvent-dependent phase behavior and crystallization pathways of polymer-grafted gold nanocubes into ordered superlattices. To explain observations from LCTEM experiments, we develop a computational model that reproduces the experimental phase behavior and show that variable charge screening by the solvent drives the assembly of the nanocubes into different phases. We show that the self-assembly of the different phases follows distinct kinetic pathways; in particular, the assembly of the rhombic phase proceeds via a decoupling of translational and orientational order. Moreover, experiments and simulation reveal a reversible transition between the square-like and rhombic phases and find a hexagonal rotator-like intermediate phase along the transition pathway. These findings open the door for understanding—and hence manipulating—complex microscopic crystallization pathways and phase transition kinetics of anisotropic nanoparticles.   

Elasticity-Mediated Assembly and Transitions of Colloids in an 2D Fluid

With interest in cell trafficking and signaling, or with completely different motivation to engineer pliable contoured coatings with integrated functionality, there has been a drive to identify the principles which govern the assembly of small rigid objects in elastic 2D fluids. Examples of the latter can include the cell membrane or copolymer lamellae. Here, we exploit phospholipid vesicle bilayers containing rigid Brownian plate-shaped domains as platform to explore principles of elasticity-dominated colloidal assembly in a dynamic 2D contour. The plate-shaped colloids, though made from solidified membrane lipids, do not coalesce and instead maintain a fixed shape throughout the experiment, demonstrating a minimal influence of any line tension. Here we demonstrate three classes of domain configurations, controlled by the availability of membrane bending, which is manipulated osmotically. When vesicles are deflated from a spherical shape by as much as 25% volume colloidal plates form a vesicle-encompassing pseudo-hexagonal lattice that maximizes the plate-plate distances. When the vesicles are inflated to within about 2% of their spherical volume, the colloidal plates associate closely but do not touch, assembling into chains. These two classes of assemblies are relatively persistent with long lived structures that suppress Brownian motion. A sharp boundary, near 2% deflation, distinguishes the two classes of assemblies with the domain distances switching sharply at these conditions suggesting cooperative behavior such as phase transition. In the final state, when vesicles are deflated by as much as 5% from a perfect sphere, domains are disordered and dynamic, with the positions changing on the timescale of minutes. This behavior was demonstrated for vesicles whose colloids occupied ~17% of the surface area and were in a size range of 10-40 um and with 4-100 colloids per vesicles, establishing the broad character of this behavior and the extreme utility of bending interactions.   

Multi-objective Optimization for Targeted Self-assembly among Competing Polymorphs

While inverse approaches for designing crystalline materials typically focus on the thermodynamic stability of a target polymorph, the outcome of a self-assembly process is often controlled by kinetic pathways. A prototypical example is the design of an isotropic pair potential to efficiently guide the self-assembly of a two-dimensional honeycomb lattice, which is a challenging problem at low densities due to the existence of competing crystal polymorphs. Here we present a machine-learning guided approach to explore pair potentials that maximizes both the thermodynamic stability and kinetic accessibility of the honeycomb polymorph. We find that the optimal pair potentials exist along a convex Pareto front, indicating a trade-off between these objectives at a low density. Furthermore, we show that the physical origin of this trade-off lies in competition between the honeycomb and triangular polymorphs: specifically, potentials that favor the honeycomb polymorph on short timescales, thus kinetically optimal, instead stabilize the triangular polymorph at long times. We also present a computationally inexpensive optimization method for finding kinetically optimal pair potentials using an ensemble of short trajectories. Our results show that this algorithm can find pair potentials close to the kinetically optimal region of the Pareto front, offering an efficient method for designing fast-assembling metastable honeycomb structures. Our work suggests that in presence of competing polymorphs, optimization for kinetic feasibility may be a better strategy for obtaining Pareto-optimal designs for targeted self-assembly.   

Proofreading mechanism for robust self-assembly

Designing components that can robustly self-assemble into complex structures is a grand challenge. Biological examples show that complex assemblies are possible, surmounting kinetic traps in the free energy landscape, as well as thermal noise. A principled way to improve assembly beyond equilibrium yield is to use proofreading, in which energy is consumed to increase the yield of the desired structures. Here we introduce an explicit proofreading scheme for the assembly of colloidal structures. We consider a patchy particle system and demonstrate that a two staged proofreading mechanism can substantially improve assembly yield and robustness. In the first stage, we learn local rules whereby particles consume energy to increase their binding strengths when they detect a local environment corresponding to the desired structure. The second stage corrects potential errors during the assembly process by adding a reverse pathway to decrease the bond strengths via an intermediate state. The scheme shows significant yield improvements in the regime where structures typically are kinetically trapped. We demonstrate that not only does the scheme increase assembly yield of structures with perfectly designed components, with a much broader temperature range where the desired yield is high, it also exhibits superior robustness against intrinsic quenched disorder. Our findings illuminate a pathway for advancing the programmable design of synthetic living materials, potentially fostering the synthesis of novel biological materials and functional behaviors.   

Theory of Nonequilibrium Symmetry-Breaking Coexistence and Active Crystallization

Crystallization is perhaps the most familiar example of a symmetry-breaking transition. In equilibrium, thermodynamic arguments result in a powerful and convenient set of criteria for determining the coexistence curves associated with these transitions. In recent years, nonequilibrium symmetry-breaking transitions have been routinely observed in a variety of natural and synthetic systems. The breaking of detailed balance, and the resulting absence of Boltzmann statistics, motivates the need for a symmetry-breaking coexistence theory that is independent of the underlying distribution of microstates. Here, we develop such a theory, relying only on mechanics, balance laws, and system symmetries. In doing so, we develop a generalized Gibbs-Duhem relation that results in nonequilibrium coexistence criteria solely in terms of bulk equations of state. We apply our framework to active crystallization, developing a complete description of the phase diagram of active Brownian hard spheres. Our predicted phase diagram quantitatively recapitulates the solid-fluid coexistence curve as well as other key features of active phase behavior, such as the liquid-gas coexistence binodal and solid-liquid-gas triple point. It is our hope that our findings offer a concrete path forward towards the development of a general theory for nonequilibrium coexistence.   

A binary system to investigate photothermally driven self-assembly of colloidal particles and nanocomposites

Photothermal convection and thermophoresis are known to drive the assembly of micro- and nanoscale particles. We demonstrate a simple experimental system using gold nanoparticles and high-intensity LED light to realize rapid, large-scale self-assembly of microscale colloidal particles and surfactant-nanoparticle composites. Photothermal heating of the gold nanoparticle (diameter d = 15 - 50 nm) suspension generates a temperature gradient, causing a convective flow and self-assembly of other particles (D = 0.7 – 10 μm) present in the solution. Interestingly, we find a transition from 2D monolayers to 3D clusters as the assembled particle size gets smaller than 1 μm. We show that the crystalline order of the assembly is determined by the surface charge of the gold nanoparticles that contributes to the screening of the repulsive interactions between the microscale colloidal particles. We demonstrate a patterned light illumination system that can be used to achieve control over the assembly size. Additionally, we use this system to drive the self-assembly of surfactant (CTAB) - gold nanoparticle composite microstructures of various 3D shapes depending on particle concentration and temperature. All these demonstrations show the potential of this binary particle system for studying non-equilibrium multiscale and multi-material assemblies.     

Effect of Calamitic Ligands on Quantum Dot Size and Packing

We investigate the use of calamitic ligands in quantum dot (QD) dispersion and assembly. Six different mesogenic ligands were designed for their ability to connect and stabilize nanoparticles into macroscopic structures. The structures are templated by the liquid crystal (LC) isotropic-to-nematic phase transition and include capsules, and foams. The molecular structure of the host LC phase influences particle distribution, and this distribution can also be tuned depending on the size and surface chemistry of the particle. Ligands play an important role in the self-assembly process. They promote particle dispersion and tune the strength with which the moving phase boundary transports the particles. Particle dispersion can be very challenging because they do not readily incorporate into the host. Our ligands must be able to both promote dispersion and facilitate assembly. We used dynamic light scattering (DLS) to measure effective particle size in dilute solution. Transmission electron microscopy (TEM) analysis using pair autocorrelation functions was used to measure particle separation in dense drop-cast films. XRD was used to measure interparticle separation in 3D assemblies (i.e capsules). Using TEM, we observed that the most calamitic ligands (rod-like) produced a closer packing structure compared to more flexible, shorter ligands. This may be due to stronger short-range attractive interactions. DLS showed that particle radii vary similarly in magnitude. These results help us to understand the role of ligand design for applications that rely on nanoparticle transport in anisotropic fluids.