The time complexity of self-assembly

Time efficiency of self-assembly is crucial for many biological processes. Moreover, with the advances of nanotechnology, time efficiency in artificial self-assembly becomes ever more important. While structural determinants and the final assembly yield are increasingly well understood, kinetic aspects concerning the time efficiency, however, remain much more elusive. In computer science, the concept of time complexity is used to characterize the efficiency of an algorithm and describes how the algorithm's runtime depends on the size of the input data. Here we characterize the time complexity of nonequilibrium self-assembly processes by exploring how the time required to realize a certain, substantial yield of a given target structure scales with its size. We identify distinct classes of assembly scenarios, i.e., "algorithms" to accomplish this task, and show that they exhibit drastically different degrees of complexity. Our analysis enables us to identify optimal control strategies for nonequilibrium self-assembly processes. Furthermore, we suggest an efficient irreversible scheme for the artificial self-assembly of nanostructures, which complements the state-of-the-art approach using reversible binding reactions and requires no fine-tuning of binding energies.

Divergent Self-Assembly Pathways to Hierarchically Organized Networks of Isopeptide-Modified Discotics under Kinetic Control

Natural proteins traverse complex free energy landscapes to assemble into hierarchically organized structures, often through stimuli-directed kinetic pathways in response to relevant biological cues. Bioinspired strategies have sought to emulate the complexity, dynamicity, and modularity exhibited in these natural processes with synthetic analogues. However, these efforts are limited by many factors that complicate the rational design and predictable assembly of synthetic constructs, especially in aqueous environments. Herein, a model discotic amphiphile gelator is described that undergoes pathway-dependent structural maturation when exposed to varying application rates of a pH stimulus, investigated by electron microscopy, spectroscopy, and X-ray scattering techniques. Under the direction of a slowly changing pH stimulus, complex hierarchical assemblies result, characterized by mesoscale elongated "superstructure" bundles embedded in a percolated mesh of narrow nanofibers. In contrast, the assembly under a rapidly applied pH stimulus is characterized by homogeneous structures that are reminiscent of the superstructures arising from the more deliberate path, except with significantly reduced scale and concomitantly large increases in bulk rheological properties. This synthetic system bears resemblance to the pathway complexity and hierarchical ordering observed for native structures, such as collagen, and points to fundamental design principles that might be applied toward enhanced control of the properties of supramolecular self-assembly across length scales.

Fabrication of In Situ Nanofiber-Reinforced Molecular Composites by Nonequilibrium Self-Assembly

Although the concept of molecular composites (MCs) is very promising, there are major obstacles arising from the immiscibility of the rigid-rod with the random-coil polymers. Here, we developed a novel method for fabricating an in situ reinforced MC system with nonequilibrium self-assembled nanofibrous structures based on bisphenol A epoxy resin, 4,4'-diaminodiphenylsulfone, bismaleimide, and a polyphenylene ether (PPO) oligomer. A variety of spectroscopic and morphological techniques were used to probe the structural evolution from the emergence of nanofibrils, to growth and aggregation of nanofibers, and then to the formation of in situ reinforced MC with strong interfacial interactions. The in situ nanofibers within the polymer matrix could be formed by the polymerization force extruding the PPO phase through the interspaces within the simultaneous interpenetrating network polymers during the cure process of the thermosetting resin system. Compared to the control sample, the in situ nanofiber-reinforced MC exhibited better thermal properties and flame retardancy. In particular, the obtained MC showed a significant improvement in glass transition temperature and mechanical properties, which were mainly attributed to the restriction of high thermal stability of PPO on the segmental motion of polymer chains, the toughening and reinforcement behaviors of PPO nanofibers on the matrix, and the chemical interaction at the PPO/matrix interface.

Nonequilibrium associative retrieval of multiple stored self-assembly targets

Many biological systems rely on the ability to self-assemble different target structures using the same set of components. Equilibrium self-assembly suffers from a limited capacity in such cases, due to an increasing number of decoy states that grows rapidly with the number of targets encoded. Moreover, improving the kinetic stability of a target at equilibrium carries the price of introducing kinetic traps, leading to slower assembly. Using a toy physical model of interacting particles, we demonstrate that local driving can improve both the assembly time and kinetic stability of multitarget self-assembly, as well as reduce fluctuations around the target configuration. We further show that the local drive can result in a steady-state probability distribution over target structures that deviates from the Boltzmann distribution in a way that depends on the types of interactions that stabilize the targets. Our results illustrate the role that nonequilibrium driving plays in overcoming tradeoffs that are inherent to equilibrium assemblies.

Design principles for nonequilibrium self-assembly

We consider an important class of self-assembly problems, and using the formalism of stochastic thermodynamics, we derive a set of design principles for growing controlled assemblies far from equilibrium. The design principles constrain the set of configurations that can be obtained under nonequilibrium conditions. Our central result provides intuition for how equilibrium self-assembly landscapes are modified under finite nonequilibrium drive.

Dynamic, Directed Self-Assembly of Nanoparticles via Toggled Interactions

Crystals self-assembled from nanoparticles have useful properties such as optical activity and sensing capability. During fabrication, however, gelation and glassification often leave these materials arrested in defective or disordered metastable states. This is a key difficulty preventing adoption of self-assembled nanoparticle materials at scale. Processes which suppress kinetic arrest and defect formation while accelerating growth of ordered materials are essential for bottom-up approaches to creating nanomaterials. Dynamic, directed self-assembly processes in which the interactions between self-assembling components are actuated temporally offer one promising methodology for accelerating and controlling bottom-up growth of nanostructures. In this article, we show through simulation and theory how time-dependent, periodically toggled interparticle attractions can avoid kinetic barriers and yield well-ordered crystalline domains for a dispersion of nanoparticles interacting via a short-ranged, isotropic potential. The growth mechanism and terminal structure of the dispersion are controlled by parameters of the toggling protocol. This control allows for selection of processes that yield rapid self-assembled, low defect crystals. Although self-assembly via periodically toggled attractions is inherently unsteady and out-of-equilibrium, its outcome is predicted by a first-principles theory of nonequilibrium thermodynamics. The theory necessitates equality of the time average of pressure and chemical potential in coexisting phases of the dispersion. These quantities are evaluated using well known equations of state. The phase behavior predicted by this theory agrees well with measurements made in Brownian dynamics simulations of sedimentation equilibrium and homogeneous nucleation. The theory can easily be extended to model dynamic self-assembly directed by other toggled conservative force fields.

Relationship between dynamical entropy and energy dissipation far from thermodynamic equilibrium

Connections between microscopic dynamical observables and macroscopic nonequilibrium (NE) properties have been pursued in statistical physics since Boltzmann, Gibbs, and Maxwell. The simulations we describe here establish a relationship between the Kolmogorov-Sinai entropy and the energy dissipated as heat from a NE system to its environment. First, we show that the Kolmogorov-Sinai or dynamical entropy can be separated into system and bath components and that the entropy of the system characterizes the dynamics of energy dissipation. Second, we find that the average change in the system dynamical entropy is linearly related to the average change in the energy dissipated to the bath. The constant energy and time scales of the bath fix the dynamical relationship between these two quantities. These results provide a link between microscopic dynamical variables and the macroscopic energetics of NE processes.