Non-reciprocal multifarious self-organization

Controlling the Spatiotemporal Self-Organization of Stimuli-Responsive Nanocrystals under Out-of-Equilibrium Conditions

Self-organization under out-of-equilibrium conditions is ubiquitous in natural systems for the generation of hierarchical solid-state patterns of complex structures with intricate properties. Efforts in applying this strategy to synthetic materials that mimic biological function have resulted in remarkable demonstrations of programmable self-healing and adaptive materials. However, the extension of these efforts to multifunctional stimuli-responsive solid-state materials across defined spatial distributions remains an unrealized technological opportunity. This paper describes the use of a nonequilibrium reaction-diffusion process to achieve the synthesis of a multifunctional stimuli-responsive electrically conductive metal-organic framework (cMOF) in a gelled medium with control over particle size and spatial periodicity on a macroscopic scale. Upon integration into chemiresistive devices, the resulting cMOF particles exhibit a size-dependent response toward hydrogen sulfide gas, as determined by their distinct surface-to-volume ratio, porosity, unique synthesis methodology, and unusual microcrystallite morphology compared to their counterparts obtained through bulk solution phase synthesis. Taken altogether, these achievements pave the way toward gaining access to functional nanomaterials with well-defined chemical composition, dimensions, and precisely tailored functions using far-from-equilibrium approaches.

Artificial chemotaxis under electrodiffusiophoresis

HYPOTHESIS

Through a large parameter space, electric fields can tune colloidal interactions and forces leading to diverse static and dynamical structures. So far, however, field-driven interactions have been limited to dipole-dipole and hydrodynamic contributions. Nonetheless, in this work, we propose that under the right conditions, electric fields can also induce interactions based on local chemical fields and diffusiophoretic flows.

EXPERIMENTS

Herein, we present a strategy to generate and measure 3D chemical gradients under electric fields. In this approach, faradaic reactions at electrodes induce global pH gradients that drive long-range transport through electrodiffusiophoresis. Simultaneously, the electric field induces local pH gradients by driving the particle's double layer far from equilibrium.

FINDINGS

As a result, while global pH gradients lead to 2D focusing away from electrodes, local pH gradients induce aggregation in the third dimension. Evidence points to a mechanism of interaction based on diffusiophoresis. Interparticle interactions display a strong dependence on surface chemistry, zeta potential and diameter of particles. Furthermore, pH gradients can be readily tuned by adjusting the voltage and frequency of the electric field. For large Péclet numbers, we observed a collective chemotactic-like collapse of particles. Remarkably, such collapse occurs without reactions at a particle's surface. By mixing particles with different sizes, we also demonstrate, through experiments and Brownian dynamics simulations, the emergence of non-reciprocal interactions, where small particles are more drawn towards large ones.

Non-Reciprocity, Metastability, and Dynamic Reconfiguration in Co-Assembly of Active and Passive Particles

Living organisms often exhibit non-reciprocal interactions where the forces acting on the objects are not equal in magnitude or opposite in direction. The combination of reciprocal and non-reciprocal interactions between synthetic building blocks remains largely unexplored. Here, out-of-equilibrium assemblies of non-motile isotropic passive and metal-patched motile active particles are formed by overlapping bulk interactions with directed self-propulsion. An external alternating current (AC) electric field generates concurrent dipolar and induced-charge electrophoretic forces between the particles which are evaluated using microscopy. The interaction force measurements allow to determine the degree of reciprocity in interactions, which is tunable by designing the active particle and its trajectory. While linearly-propelled active particles evade assembly with passive particles, helically propelled active particles form active-passive clusters with dynamic reconfiguration and long-lived metastability. Large clusters display programmable fluctuations and reconfigurability by controlling the fraction of active particles. The study establishes principles of integrating reciprocal and non-reciprocal interactions in guided colloidal assembly of reconfigurable metastable structures.

Emergent polar order in nonpolar mixtures with nonreciprocal interactions

Phenomenological rules that govern the collective behavior of complex physical systems are powerful tools because they can make concrete predictions about their universality class based on generic considerations, such as symmetries, conservation laws, and dimensionality. While in most cases such considerations are manifestly ingrained in the constituents, novel phenomenology can emerge when composite units associated with emergent symmetries dominate the behavior of the system. We study a generic class of active matter systems with nonreciprocal interactions and demonstrate the existence of true long-range polar order in two dimensions and above, both at the linear level and by including all relevant nonlinearities in the Renormalization Group sense. We achieve this by uncovering a mapping of our scalar active mixture theory to the Toner-Tu theory of dry polar active matter by employing a suitably defined polar order parameter. We then demonstrate that the complete effective field theory-which includes all the soft modes and the relevant nonlinear terms-belongs to the (Burgers-) Kardar-Parisi-Zhang universality class. This classification allows us to prove the stability of the emergent polar long-range order in scalar nonreciprocal mixtures in two dimensions, and hence a conclusive violation of the Mermin-Wagner theorem.

Self-Organized Patterns in Non-Reciprocal Active Droplet Systems

Non-equilibrium patterns are widespread in nature and often arise from the self-organization of constituents through nonreciprocal chemotactic interactions. In this study, we demonstrate how active oil-in-water droplet mixtures with predator-prey interactions can result in a variety of self-organized patterns. By manipulating physical parameters, the droplet diameter ratio and number ratio, we identify distinct classes of patterns within a binary droplet system, rationalize the pattern formation, and quantify motilities. Experimental results are recapitulated in numerical simulations using a minimal computational model that solely incorporates chemotactic interactions and steric repulsion among the constituents. The time evolution of the patterns is investigated and chemically explained. We also investigate how patterns vary with differing interaction strength by altering surfactant composition. Leveraging insights from the binary droplet system, the framework is extended to a ternary droplet mixture composed of multiple chasing droplet pairs to create chemically directed hierarchical organization. Our findings demonstrate how rationalizable, self-organized patterns can be programmed in a chemically minimal system and provide the basis for exploration of emergent organization and higher order complexity in active colloids.

Dissipative Self-Assembly of Patchy Particles under Nonequilibrium Drive: A Computational Study

Inspired by biology and implemented using nanotechnology, the self-assembly of patchy particles has emerged as a pivotal mechanism for constructing complex structures that mimic natural systems with diverse functionalities. Here, we explore the dissipative self-assembly of patchy particles under nonequilibrium conditions, with the aim of overcoming the constraints imposed by equilibrium assembly. Utilizing extensive Monte Carlo (MC) and Molecular Dynamics (MD) simulations, we provide insight into the effects of external forces that mirror natural and chemical processes on the assembly rates and the stability of the resulting assemblies comprising 8, 10, and 13 patchy particles. Implemented by a favorable bond-promoting drive in MC or a pulsed square wave potential in MD, our simulations reveal the role these external drives play in accelerating assembly kinetics and enhancing structural stability, evidenced by a decrease in the time to first assembly and an increase in the duration the system remains in an assembled state. Through the analysis of an order parameter, entropy production, bond dynamics, and interparticle forces, we unravel the underlying mechanisms driving these advancements. We also validated our key findings by simulating a larger system of 100 patchy particles. Our comprehensive results not only shed light on the impact of external stimuli on self-assembly processes but also open a promising pathway for expanding the application by leveraging patchy particles for novel nanostructures.

Driven Self-Assembly of Patchy Particles Overcoming Equilibrium Limitations

Bridging biological complexity and synthetic material design, we investigate dissipative self-assembly in patchy particle systems. Utilizing Monte Carlo and Molecular Dynamics simulations, we demonstrate how external driving forces mitigate equilibrium trade-offs between assembly time and structural stability, traditionally encountered in self-assembly processes. Our findings also extend to biological-mimicking environments, where we explore the dynamics of patchy particles under crowded conditions. This comprehensive analysis offers insights into advanced material design, opening avenues for innovations in nanotechnology applications.

Chemical Programming of Solubilizing, Nonequilibrium Active Droplets

ConspectusThe multifunctionality and resilience of living systems has inspired an explosion of interest in creating materials with life-like properties. Just as life persists out-of-equilibrium, we too should try to design materials that are thermodynamically unstable but can be harnessed to achieve desirable, adaptive behaviors. Studying minimalistic chemical systems that exhibit relatively simple emergent behaviors, such as motility, communication, or self-organization, can provide insight into fundamental principles which may enable the design of more complex and life-like synthetic materials in the future.Emulsions, which are composed of liquid droplets dispersed in another immiscible fluid phase, have emerged as fascinating chemically minimal materials in which to study nonequilibrium, life-like properties. As covered in this Account, our group has focused on studying oil-in-water emulsions, specifically those which destabilize by solubilization, a process wherein oil is released into the continuous phase over time to create gradients of oil-filled micelles. These chemical gradients can create interfacial tension gradients that lead to droplet self-propulsion as well as mediate communication between neighboring oil droplets. As such, oil-in-water emulsions present an interesting platform for studying active matter. However, despite being chemically minimal with sometimes as few as three chemicals (oil, water, and a surfactant), emulsions present surprising complexity across the molecular to macroscale. Fundamental processes governing their active behavior, such as micelle-mediated interfacial transport, are still not well understood. This complexity is compounded by the challenges of studying systems out-of-equilibrium which typically require new analytical methods and may break our intuition derived from equilibrium thermodynamics.In this Account, we highlight our group's efforts toward developing chemical frameworks for understanding active and interactive oil-in-water emulsions. How do the chemical properties and physical spatial organization of the oil, water, and surfactant combine to yield colloidal-scale active properties? Our group tackles this question by employing systematic studies of active behavior working across the chemical space of oils and surfactants to link molecular structure to active behavior. The Account begins with an introduction to the self-propulsion of single, isolated droplets and how by applying biases, such as with a gravitational field or interfacially adsorbed particles, drop speeds can be manipulated. Next, we illustrate that some droplets can be attractive, as well as self-propulsive/repulsive, which does not fall in line with the current understanding of the impact of oil-filled micelle gradients on interfacial tensions. The mechanisms by which oil-filled micelles influence interfacial tensions of nonequilibrium interfaces is poorly understood and requires deeper molecular understanding. Regardless, we extend our knowledge of droplet motility to design emulsions with nonreciprocal predator-prey interactions and describe the dynamic self-organization that arises from the combination of reciprocal and nonreciprocal interactions between droplets. Finally, we highlight our group's progress toward answering key chemical questions surrounding nonequilibrium processes in emulsions that remain to be answered. We hope that our progress in understanding the chemical principles governing the dynamic nonequilibrium properties of oil-in-water droplets can help inform research in tangential research areas such as cell biology and origins of life.

Defect Solutions of the Nonreciprocal Cahn-Hilliard Model: Spirals and Targets

We study the defect solutions of the nonreciprocal Cahn-Hilliard model. We find two kinds of defects, spirals with unit magnitude topological charge, and topologically neutral targets. These defects generate radially outward traveling waves and thus break the parity and time-reversal symmetry. For a given strength of nonreciprocity, spirals and targets with unique asymptotic wave number and amplitude are selected. We use large-scale simulations to show that at low nonreciprocity α, disordered states evolve into quasistationary spiral networks. With increasing α, we observe networks composed primarily of targets. Beyond a critical threshold α_{c}, a disorder-order transition from defect networks to traveling waves emerges. The transition is marked by a sharp rise in the global polar order.

Impact of non-reciprocal interactions on colloidal self-assembly with tunable anisotropy

Non-reciprocal (NR) effective interactions violating Newton's third law occur in many biological systems, but can also be engineered in synthetic, colloidal systems. Recent research has shown that such NR interactions can have tremendous effects on the overall collective behavior and pattern formation, but can also influence aggregation processes on the particle scale. Here, we focus on the impact of non-reciprocity on the self-assembly of a colloidal system (originally passive) with anisotropic interactions whose character is tunable by external fields. In the absence of non-reciprocity, that is, under equilibrium conditions, the colloids form square-like and hexagonal aggregates with extremely long lifetimes yet no large-scale phase separation [Kogler et al., Soft Matter 11, 7356 (2015)], indicating kinetic trapping. Here, we study, based on Brownian dynamics simulations in 2D, an NR version of this model consisting of two species with reciprocal isotropic, but NR anisotropic interactions. We find that NR induces an effective propulsion of particle pairs and small aggregates ("active colloidal molecules") forming at the initial stages of self-assembly, an indication of the NR-induced non-equilibrium. The shape and stability of these initial clusters strongly depend on the degree of anisotropy. At longer times, we find, for weak NR interactions, large (even system-spanning) clusters where single particles can escape and enter at the boundaries, in stark contrast to the small rigid aggregates appearing at the same time in the passive case. In this sense, weak NR shortcuts the aggregation. Increasing the degree of NR (and thus, propulsion), we even observe large-scale phase separation if the interactions are weakly anisotropic. In contrast, systems with strong NR and anisotropy remain essentially disordered. Overall, the NR interactions are shown to destabilize the rigid aggregates interrupting self-assembly and phase separation in the passive case, thereby helping the system to overcome kinetic barriers.

Anomalous Fluctuations in a Droplet of Chemically Active Colloids or Enzymes

Chemically active colloids or enzymes cluster into dense droplets driven by their phoretic response to collectively generated chemical gradients. Employing Brownian dynamics simulation techniques, our study of the dynamics of such a chemically active droplet uncovers a rich variety of structures and dynamical properties, including the full range of fluidlike to solidlike behavior, and non-Gaussian positional fluctuations. Our work sheds light on the complex dynamics of the active constituents of metabolic clusters, which are the main drivers of nonequilibrium activity in living systems.

One-way optomechanical interaction between nanoparticles

Within a closed system, physical interactions are reciprocal. However, the effective interaction between two entities of an open system may not obey reciprocity. Here, we describe a non-reciprocal interaction between nanoparticles which is one-way, almost insensitive to the interparticle distance, and scalable to many particles. The interaction we propose is based on the non-conservative optical forces between two nanoparticles with highly directional scattering patterns. However, we elucidate that scattering patterns can in general be very misleading about the interparticle forces. We introduce zeroth- and first-order non-reciprocity factors to precisely quantify the merits of any optomechanical interaction between nanoparticles. Our proposed one-way interaction could constitute an important step in the realization of mesoscopic heat pumps and refrigerators, the study of non-equilibrium systems, and the simulation of non-Hermitian quantum models.

Escaping Kinetic Traps Using Nonreciprocal Interactions

Kinetic traps are a notorious problem in equilibrium statistical mechanics, where temperature quenches ultimately fail to bring the system to low energy configurations. Using multifarious self-assembly as a model system, we introduce a mechanism to escape kinetic traps by utilizing nonreciprocal interactions between components. Introducing nonequilibrium effects offered by broken action-reaction symmetry in the system pushes the trajectory of the system out of arrested dynamics. The dynamics of the model is studied using tools from the physics of interfaces and defects. Our proposal can find applications in self-assembly, glassy systems, and systems with arrested dynamics to facilitate escape from local minima in rough energy landscapes.

Nanocrystal Assemblies: Current Advances and Open Problems

We explore the potential of nanocrystals (a term used equivalently to nanoparticles) as building blocks for nanomaterials, and the current advances and open challenges for fundamental science developments and applications. Nanocrystal assemblies are inherently multiscale, and the generation of revolutionary material properties requires a precise understanding of the relationship between structure and function, the former being determined by classical effects and the latter often by quantum effects. With an emphasis on theory and computation, we discuss challenges that hamper current assembly strategies and to what extent nanocrystal assemblies represent thermodynamic equilibrium or kinetically trapped metastable states. We also examine dynamic effects and optimization of assembly protocols. Finally, we discuss promising material functions and examples of their realization with nanocrystal assemblies.

Billiards with Spatial Memory

Many classes of active matter develop spatial memory by encoding information in space. We present a framework based on mathematical billiards, wherein particles remember their past trajectories. Despite its deterministic rules, such a system is strongly nonergodic and exhibits intermittent statistics and complex pattern formation. We show how these features emerge from the dynamic change of topology. Our work illustrates how the dynamics of a single-body system can dramatically change with spatial memory, laying the groundwork to further explore systems with complex memory kernels.

Designing 3D multicomponent self-assembling systems with signal-passing building blocks

We introduce an allostery-mimetic building block model for the self-assembly of 3D structures. We represent the building blocks as patchy particles, where each binding site (patch) can be irreversibly activated or deactivated by binding of the particle's other controlling patches to another particle. We show that these allostery-mimetic systems can be designed to increase yields of target structures by disallowing misassembled states and can further decrease the smallest number of distinct species needed to assemble a target structure. Next, we show applications to design a programmable nanoparticle swarm for multifarious assembly: a system of particles that stores multiple possible target structures and a particular structure is recalled by presenting an external trigger signal. Finally, we outline a possible pathway for realization of such structures at nanoscale using DNA nanotechnology devices.

Colloidal Self-Assembly: From Passive to Active Systems

Self-assembly fundamentally implies the organization of small sub-units into large structures or patterns without the intervention of specific local interactions. This process is commonly observed in nature, occurring at various scales ranging from atomic/molecular assembly to the formation of complex biological structures. Colloidal particles may serve as micrometer-scale surrogates for studying assembly, particularly for the poorly understood kinetic and dynamic processes at the atomic scale. Recent advances in colloidal self-assembly have enabled the programmable creation of novel materials with tailored properties. We here provide an overview and comparison of both passive and active colloidal self-assembly, with a discussion on the energy landscape and interactions governing both types. In the realm of passive colloidal assembly, many impressive and important structures have been realized, including colloidal molecules, one-dimensional chains, two-dimensional lattices, and three-dimensional crystals. In contrast, active colloidal self-assembly, driven by optical, electric, chemical, or other fields, involves more intricate dynamic processes, offering more flexibility and potential new applications. A comparative analysis underscores the critical distinctions between passive and active colloidal assemblies, highlighting the unique collective behaviors emerging in active systems. These behaviors encompass collective motion, motility-induced phase segregation, and exotic properties arising from out-of-equilibrium thermodynamics. Through this comparison, we aim to identify the future opportunities in active assembly research, which may suggest new application domains.

Nonreciprocal interactions give rise to fast cilium synchronization in finite systems

Motile cilia beat in an asymmetric fashion in order to propel the surrounding fluid. When many cilia are located on a surface, their beating can synchronize such that their phases form metachronal waves. Here, we computationally study a model where each cilium is represented as a spherical particle, moving along a tilted trajectory with a position-dependent active driving force and a position-dependent internal drag coefficient. The model thus takes into account all the essential broken symmetries of the ciliary beat. We show that taking into account the near-field hydrodynamic interactions, the effective coupling between cilia even over an entire beating cycle can become nonreciprocal: The phase of a cilium is more strongly affected by an adjacent cilium on one side than by a cilium at the same distance in the opposite direction. As a result, synchronization starts from a seed at the edge of a group of cilia and propagates rapidly across the system, leading to a synchronization time that scales proportionally to the linear dimension of the system. We show that a ciliary carpet is characterized by three different velocities: the velocity of fluid transport, the phase velocity of metachronal waves, and the group velocity of order propagation. Unlike in systems with reciprocal coupling, boundary effects are not detrimental for synchronization, but rather enable the formation of the initial seed.

Network Effects Lead to Self-Organization in Metabolic Cycles of Self-Repelling Catalysts

Mixtures of particles that interact through phoretic effects are known to aggregate if they belong to species that exhibit attractive self-interactions. We study self-organization in a model metabolic cycle composed of three species of catalytically active particles that are chemotactic toward the chemicals that define their connectivity network. We find that the self-organization can be controlled by the network properties, as exemplified by a case where a collapse instability is achieved by design for self-repelling species. Our findings highlight a possibility for controlling the intricate functions of metabolic networks by taking advantage of the physics of phoretic active matter.

Self-organization of primitive metabolic cycles due to non-reciprocal interactions

One of the greatest mysteries concerning the origin of life is how it has emerged so quickly after the formation of the earth. In particular, it is not understood how metabolic cycles, which power the non-equilibrium activity of cells, have come into existence in the first instances. While it is generally expected that non-equilibrium conditions would have been necessary for the formation of primitive metabolic structures, the focus has so far been on externally imposed non-equilibrium conditions, such as temperature or proton gradients. Here, we propose an alternative paradigm in which naturally occurring non-reciprocal interactions between catalysts that can partner together in a cyclic reaction lead to their recruitment into self-organized functional structures. We uncover different classes of self-organized cycles that form through exponentially rapid coarsening processes, depending on the parity of the cycle and the nature of the interaction motifs, which are all generic but have readily tuneable features.

Optimization of non-equilibrium self-assembly protocols using Markov state models

The promise of self-assembly to enable the bottom-up formation of materials with prescribed architectures and functions has driven intensive efforts to uncover rational design principles for maximizing the yield of a target structure. Yet, despite many successful examples of self-assembly, ensuring kinetic accessibility of the target structure remains an unsolved problem in many systems. In particular, long-lived kinetic traps can result in assembly times that vastly exceed experimentally accessible timescales. One proposed solution is to design non-equilibrium assembly protocols in which system parameters change over time to avoid such kinetic traps. Here, we develop a framework to combine Markov state model (MSM) analysis with optimal control theory to compute a time-dependent protocol that maximizes the yield of the target structure at a finite time. We present an adjoint-based gradient descent method that, in conjunction with MSMs for a system as a function of its control parameters, enables efficiently optimizing the assembly protocol. We also describe an interpolation approach to significantly reduce the number of simulations required to construct the MSMs. We demonstrate our approach with two examples; a simple semi-analytic model for the folding of a polymer of colloidal particles, and a more complex model for capsid assembly. Our results show that optimizing time-dependent protocols can achieve significant improvements in the yields of selected structures, including equilibrium free energy minima, long-lived metastable structures, and transient states.