Observing the Dynamics of an Electrochemically Driven Active Material with Liquid Electron Microscopy.

Electrochemical liquid electron microscopy has revolutionized our understanding of nanomaterial dynamics by allowing for direct observation of their electrochemical production. This technique, primarily applied to inorganic materials, is now being used to explore the self-assembly dynamics of active molecular materials. Our study examines these dynamics across various scales, from the nanoscale behavior of individual fibers to the micrometer-scale hierarchical evolution of fiber clusters. To isolate the influences of the electron beam and electrical potential on material behavior, we conducted thorough beam-sample interaction analyses. Our findings reveal that the dynamics of these active materials at the nanoscale are shaped by their proximity to the electrode and the applied electrical current. By integrating electron microscopy observations with reaction-diffusion simulations, we uncover that local structures and their formation history play a crucial role in determining assembly rates. This suggests that the emergence of nonequilibrium structures can locally accelerate further structural development, offering insights into the behavior of active materials under electrochemical conditions.

CryoEM reveals the complex self-assembly of a chemically driven disulfide hydrogel.

Inspired by the adaptability of biological materials, a variety of synthetic, chemically driven self-assembly processes have been developed that result in the transient formation of supramolecular structures. These structures form through two simultaneous reactions, forward and backward, which generate and consume a molecule that undergoes self-assembly. The dynamics of these assembly processes have been shown to differ from conventional thermodynamically stable molecular assemblies. However, the evolution of nanoscale morphologies in chemically driven self-assembly and how they compare to conventional assemblies has not been resolved. Here, we use a chemically driven redox system to separately carry out the forward and backward reactions. We analyze the forward and backward reactions both sequentially and synchronously with time-resolved cryogenic transmission electron microscopy (cryoEM). Quantitative image analysis shows that the synchronous process is more complex and heterogeneous than the sequential process. Our key finding is that a thermodynamically unstable stacked nanorod phase, briefly observed in the backward reaction, is sustained for ∼6 hours in the synchronous process. Kinetic Monte Carlo modeling show that the synchronous process is driven by multiple cycles of assembly and disassembly. The collective data suggest that chemically driven self-assembly can create sustained morphologies not seen in thermodynamically stable assemblies by kinetically stabilizing transient intermediates. This finding provides plausible design principles to develop and optimize supramolecular materials with novel properties.

Classical Fisher information for differentiable dynamical systems.

Fisher information is a lower bound on the uncertainty in the statistical estimation of classical and quantum mechanical parameters. While some deterministic dynamical systems are not subject to random fluctuations, they do still have a form of uncertainty. Infinitesimal perturbations to the initial conditions can grow exponentially in time, a signature of deterministic chaos. As a measure of this uncertainty, we introduce another classical information, specifically for the deterministic dynamics of isolated, closed, or open classical systems not subject to noise. This classical measure of information is defined with Lyapunov vectors in tangent space, making it less akin to the classical Fisher information and more akin to the quantum Fisher information defined with wavevectors in Hilbert space. Our analysis of the local state space structure and linear stability leads to upper and lower bounds on this information, giving it an interpretation as the net stretching action of the flow. Numerical calculations of this information for illustrative mechanical examples show that it depends directly on the phase space curvature and speed of the flow.

Prevalence of multistability and nonstationarity in driven chemical networks.

External flows of energy, entropy, and matter can cause sudden transitions in the stability of biological and industrial systems, fundamentally altering their dynamical function. How might we control and design these transitions in chemical reaction networks? Here, we analyze transitions giving rise to complex behavior in random reaction networks subject to external driving forces. In the absence of driving, we characterize the uniqueness of the steady state and identify the percolation of a giant connected component in these networks as the number of reactions increases. When subject to chemical driving (influx and outflux of chemical species), the steady state can undergo bifurcations, leading to multistability or oscillatory dynamics. By quantifying the prevalence of these bifurcations, we show how chemical driving and network sparsity tend to promote the emergence of these complex dynamics and increased rates of entropy production. We show that catalysis also plays an important role in the emergence of complexity, strongly correlating with the prevalence of bifurcations. Our results suggest that coupling a minimal number of chemical signatures with external driving can lead to features present in biochemical processes and abiogenesis.

Stochastic paths controlling speed and dissipation.

Natural processes occur in a finite amount of time and dissipate energy, entropy, and matter. Near equilibrium, thermodynamic intuition suggests that fast irreversible processes will dissipate more energy and entropy than slow quasistatic processes connecting the same initial and final states. For small systems, recently discovered thermodynamic speed limits suggest that faster processes will dissipate more than slower processes. Here, we test the hypothesis that this relationship between speed and dissipation holds for stochastic paths far from equilibrium. To analyze stochastic paths on finite timescales, we derive an exact expression for the path probabilities of continuous-time Markov chains from the path summation solution to the master equation. We present a minimal model for a driven system in which relative energies of the initial and target states control the speed, and the nonequilibrium currents of a cycle control the dissipation. Although the hypothesis holds near equilibrium, we find that faster processes can dissipate less under far-from-equilibrium conditions because of strong currents. This model serves as a minimal prototype for designing kinetics to sculpt the nonequilibrium path space so that faster paths produce less dissipation.

Density matrix formulation of dynamical systems.

Physical systems that are dissipating, mixing, and developing turbulence also irreversibly transport statistical density. However, predicting the evolution of density from atomic and molecular scale dynamics is challenging for nonsteady, open, and driven nonequilibrium processes. Here, we establish a theory to address this challenge for classical dynamical systems that is analogous to the density matrix formulation of quantum mechanics. We show that a classical density matrix is similar to the phase-space metric and evolves in time according to generalizations of Liouville's theorem and Liouville's equation for non-Hamiltonian systems. The traditional Liouvillian forms are recovered in the absence of dissipation or driving by imposing trace preservation or by considering Hamiltonian dynamics. Local measures of dynamical instability and chaos are embedded in classical commutators and anticommutators and directly related to Poisson brackets when the dynamics are Hamiltonian. Because the classical density matrix is built from the Lyapunov vectors that underlie classical chaos, it offers an alternative computationally tractable basis for the statistical mechanics of nonequilibrium processes that applies to systems that are driven, transient, dissipative, regular, and chaotic.

Optimizing dynamical functions for speed with stochastic paths.

Living systems are built from microscopic components that function dynamically; they generate work with molecular motors, assemble and disassemble structures such as microtubules, keep time with circadian clocks, and catalyze the replication of DNA. How do we implement these functions in synthetic nanostructured materials to execute them before the onset of dissipative losses? Answering this question requires a quantitative understanding of when we can improve performance and speed while minimizing the dissipative losses associated with operating in a fluctuating environment. Here, we show that there are four modalities for optimizing dynamical functions that can guide the design of nanoscale systems. We analyze Markov models that span the design space: a clock, ratchet, replicator, and self-assembling system. Using stochastic thermodynamics and an exact expression for path probabilities, we classify these models of dynamical functions based on the correlation of speed with dissipation and with the chosen performance metric. We also analyze random networks to identify the model features that affect their classification and the optimization of their functionality. Overall, our results show that the possible nonequilibrium paths can determine our ability to optimize the performance of dynamical functions, despite ever-present dissipation, when there is a need for speed.

Dynamic scaling of stochastic thermodynamic observables for chemical reactions at and away from equilibrium.

Physical kinetic roughening processes are well-known to exhibit universal scaling of observables that fluctuate in space and time. Are there analogous dynamic scaling laws that are unique to the chemical reaction mechanisms available synthetically and occurring naturally? Here, we formulate an approach to the dynamic scaling of stochastic fluctuations in thermodynamic observables at and away from equilibrium. Both analytical expressions and numerical simulations confirm our dynamic scaling ansatz with associated scaling exponents, function, and law. A survey of common chemical mechanisms reveals classes that organize according to the molecularity of the reactions involved, the nature of the reaction vessel and external reservoirs, (non)equilibrium conditions, and the extent of autocatalysis in the reaction network. Varying experimental parameters, such as temperature, can cause coupled reactions capable of chemical feedback to transition between these classes. While path observables, such as the dynamical activity, have scaling exponents that are time-independent, the variance in the entropy production and flow can have time-dependent scaling exponents and self-averaging properties as a result of temporal correlations that emerge during thermodynamically irreversible processes. Altogether, these results establish dynamic universality classes in the nonequilibrium fluctuations of thermodynamic observables for well-mixed chemical reactions.

Benchmarking the Performance of the ReaxFF Reactive Force Field on Hydrogen Combustion Systems.

A thorough understanding of the kinetics and dynamics of combusting mixtures is of considerable interest, especially in regimes beyond the reach of current experimental validation. The ReaxFF reactive force field method has provided a way to simulate large-scale systems of hydrogen combustion via a parametrized potential that can simulate bond breaking. This modeling approach has been applied to hydrogen combustion, as well as myriad other reactive chemical systems. In this work, we benchmark the performance of several common parametrizations of this potential against higher-level quantum mechanical (QM) approaches. We demonstrate instances where these parametrizations of the ReaxFF potential fail both quantitatively and qualitatively to describe reactive events relevant for hydrogen combustion systems.

Critical fluctuations and slowing down of chaos.

Fluids cooled to the liquid-vapor critical point develop system-spanning fluctuations in density that transform their visual appearance. Despite a rich phenomenology, however, there is not currently an explanation of the mechanical instability in the molecular motion at this critical point. Here, we couple techniques from nonlinear dynamics and statistical physics to analyze the emergence of this singular state. Numerical simulations and analytical models show how the ordering mechanisms of critical dynamics are measurable through the hierarchy of spatiotemporal Lyapunov vectors. A subset of unstable vectors soften near the critical point, with a marked suppression in their characteristic exponents that reflects a weakened sensitivity to initial conditions. Finite-time fluctuations in these exponents exhibit sharply peaked dynamical timescales and power law signatures of the critical dynamics. Collectively, these results are symptomatic of a critical slowing down of chaos that sits at the root of our statistical understanding of the liquid-vapor critical point.

Typical Stochastic Paths in the Transient Assembly of Fibrous Materials.

When chemically fueled, molecular self-assembly can sustain dynamic aggregates of polymeric fibers-hydrogels-with tunable properties. If the fuel supply is finite, the hydrogel is transient, as competing reactions switch molecular subunits between active and inactive states, drive fiber growth and collapse, and dissipate energy. Because the process is away from equilibrium, the structure and mechanical properties can reflect the history of preparation. As a result, the formation of these active materials is not readily susceptible to a statistical treatment in which the configuration and properties of the molecular building blocks specify the resulting material structure. Here, we illustrate a stochastic-thermodynamic and information-theoretic framework for this purpose and apply it to these self-annihilating materials. Among the possible paths, the framework variationally identifies those that are typical-loosely, the minimum number with the majority of the probability. We derive these paths from computer simulations of experimentally-informed stochastic chemical kinetics and a physical kinetics model for the growth of an active hydrogel. The model reproduces features observed by confocal microscopy, including the fiber length, lifetime, and abundance as well as the observation of fast fiber growth and stochastic fiber collapse. The typical mesoscopic paths we extract are less than 0.23% of those possible, but they accurately reproduce material properties such as mean fiber length.

Explosion limits of hydrogen-oxygen mixtures from nonequilibrium critical points.

The explosion limits of hydrogen-oxygen mixtures are macroscopic, temperature-pressure boundaries that divide the overall chemistry of hydrogen oxidation into slow-burning and explosive regimes. Here, we demonstrate that it is possible to recover the three chemical explosion limits of H2/O2 mixtures from nonequilibrium stochastic trajectories. This demonstration relies on the finding that, in explosive regimes, these trajectories have the quantitative features of a dynamical phase transition. Through computer simulations for both a generic and a reduced model for hydrogen oxidation, we find only one dominant reactive phase at temperatures below the explosion limits. At temperatures above the limits, however, a second phase transiently emerges from the chemistry. By locating the pseudo-critical temperature where two reactive phases are distinguishable, we construct all three explosion-limit boundaries for model hydrogen-oxygen mixtures of finite size.

Effects of temperature and mass conservation on the typical chemical sequences of hydrogen oxidation.

Macroscopic properties of reacting mixtures are necessary to design synthetic strategies, determine yield, and improve the energy and atom efficiency of many chemical processes. The set of time-ordered sequences of chemical species are one representation of the evolution from reactants to products. However, only a fraction of the possible sequences is typical, having the majority of the joint probability and characterizing the succession of chemical nonequilibrium states. Here, we extend a variational measure of typicality and apply it to atomistic simulations of a model for hydrogen oxidation over a range of temperatures. We demonstrate an information-theoretic methodology to identify typical sequences under the constraints of mass conservation. Including these constraints leads to an improved ability to learn the chemical sequence mechanism from experimentally accessible data. From these typical sequences, we show that two quantities defining the variational typical set of sequences-the joint entropy rate and the topological entropy rate-increase linearly with temperature. These results suggest that, away from explosion limits, data over a narrow range of thermodynamic parameters could be sufficient to extrapolate these typical features of combustion chemistry to other conditions.

Entrance and escape dynamics for the typical set.

According to the asymptotic equipartition property, sufficiently long sequences of random variables converge to a set that is typical. While the size and probability of this set are central to information theory and statistical mechanics, they can often only be estimated accurately in the asymptotic limit due to the exponential growth in possible sequences. Here we derive a time-inhomogeneous dynamics that constructs the properties of the typical set for all finite length sequences of independent and identically distributed random variables. These dynamics link the finite properties of the typical set to asymptotic results and allow the typical set to be applied to small and transient systems. The main result is a geometric mapping-the triangle map-relating sequences of random variables of length n to those of length n+1. We show that the number of points in this map needed to quantify the properties of the typical set grows linearly with sequence length, despite the exponential growth in the number of typical sequences. We illustrate the framework for the Bernoulli process and the Schlögl model for autocatalytic chemical reactions and demonstrate both the convergence to asymptotic limits and the ability to reproduce exact calculations.

Self-Averaging Fluctuations in the Chaoticity of Simple Fluids.

Bulk properties of equilibrium liquids are a manifestation of intermolecular forces. Here, we show how these forces imprint on dynamical fluctuations in the Lyapunov exponents for simple fluids with and without attractive forces. While the bulk of the spectrum is strongly self-averaging, the first Lyapunov exponent self-averages only weakly and at a rate that depends on the length scale of the intermolecular forces; short-range repulsive forces quantitatively dominate longer-range attractive forces, which act as a weak perturbation that slows the convergence to the thermodynamic limit. Regardless of intermolecular forces, the fluctuations in the Kolmogorov-Sinai entropy rate diverge, as one expects for an extensive quantity, and the spontaneous fluctuations of these dynamical observables obey fluctuation-dissipation-like relationships. Together, these results are a representation of the van der Waals picture of fluids and another lens through which we can view the liquid state.

Nonexponential kinetics of ion pair dissociation in electrofreezing water.

Temporally- or spatially-heterogeneous environments can participate in many kinetic processes, from chemical reactions and self-assembly to the forced dissociation of biomolecules. Here, we simulate the molecular dynamics of a model ion pair forced to dissociate in an explicit, aqueous solution. Triggering dissociation with an external electric field causes the surrounding water to electrofreeze and the ion pair population to decay nonexponentially. To further probe the role of the aqueous environment in the kinetics, we also simulate dissociation events under a purely mechanical force on the ion pair. In this case, regardless of whether the surrounding water is a liquid or already electrofrozen, the ion pair population decays exponentially with a well-defined rate constant that is specific to the medium and applied force. These simulation data, and the rate parameters we extract, suggest the disordered kinetics in an electrofreezing medium are a result of the comparable time scales of two concurrent processes, electrofreezing and dissociation.

Nonequilibrium phase coexistence and criticality near the second explosion limit of hydrogen combustion.

While hydrogen is a promising source of clean energy, the safety and optimization of hydrogen technologies rely on controlling ignition through explosion limits: pressure-temperature boundaries separating explosive behavior from comparatively slow burning. Here, we show that the emergent nonequilibrium chemistry of combustible mixtures can exhibit the quantitative features of a phase transition. With stochastic simulations of the chemical kinetics for a model mechanism of hydrogen combustion, we show that the boundaries marking explosive domains of kinetic behavior are nonequilibrium critical points. Near the pressure of the second explosion limit, these critical points terminate the transient coexistence of dynamical phases-one that autoignites and another that progresses slowly. Below the critical point temperature, the chemistry of these phases is indistinguishable. In the large system limit, the pseudo-critical temperature converges to the temperature of the second explosion limit derived from mass-action kinetics.

Extensivity and additivity of the Kolmogorov-Sinai entropy for simple fluids.

According to the van der Waals picture, attractive and repulsive forces play distinct roles in the structure of simple fluids. Here, we examine their roles in dynamics; specifically, in the degree of deterministic chaos using the Kolmogorov-Sinai (KS) entropy rate and the spectra of Lyapunov exponents. With computer simulations of three-dimensional Lennard-Jones and Weeks-Chandler-Andersen fluids, we find repulsive forces dictate these dynamical properties, with attractive forces reducing the KS entropy at a given thermodynamic state. Regardless of interparticle forces, the maximal Lyapunov exponent is intensive for systems ranging from 200 to 2000 particles. Our finite-size scaling analysis also shows that the KS entropy is both extensive (a linear function of system-size) and additive. Both temperature and density control the "dynamical chemical potential," the rate of linear growth of the KS entropy with system size. At fixed system-size, both the KS entropy and the largest exponent exhibit a maximum as a function of density. We attribute the maxima to the competition between two effects: as particles are forced to be in closer proximity, there is an enhancement from the sharp curvature of the repulsive potential and a suppression from the diminishing free volume and particle mobility. The extensivity and additivity of the KS entropy and the intensivity of the largest Lyapunov exponent, however, hold over a range of temperatures and densities across the liquid and liquid-vapor coexistence regimes.

Ignition in an Atomistic Model of Hydrogen Oxidation.

Hydrogen is a potential substitute for fossil fuels that would reduce the combustive emission of carbon dioxide. However, the low ignition energy needed to initiate oxidation imposes constraints on the efficiency and safety of hydrogen-based technologies. Microscopic details of the combustion processes, ephemeral transient species, and complex reaction networks are necessary to control and optimize the use of hydrogen as a commercial fuel. Here, we report estimates of the ignition time of hydrogen-oxygen mixtures over a wide range of equivalence ratios from extensive reactive molecular dynamics simulations. These data show that the shortest ignition time corresponds to a fuel-lean mixture with an equivalence ratio of 0.5, where the number of hydrogen and oxygen molecules in the initial mixture are identical, in good agreement with a recent chemical kinetic model. We find two signatures in the simulation data precede ignition at pressures above 200 MPa. First, there is a peak in hydrogen peroxide that signals ignition is imminent in about 100 ps. Second, we find a strong anticorrelation between the ignition time and the rate of energy dissipation, suggesting the role of thermal feedback in stimulating ignition.

Learning the mechanisms of chemical disequilibria.

When at equilibrium, large-scale systems obey thermodynamics because they have microscopic configurations that are typical. "Typical" states are a fraction of those possible with the majority of the probability. A more precise definition of typical states underlies the transmission, coding, and compression of information. However, this definition does not apply to natural systems that are transiently away from equilibrium. Here, we introduce a variational measure of typicality and apply it to atomistic simulations of a model for hydrogen oxidation. While a gaseous mixture of hydrogen and oxygen combusts, reactant molecules transform through a variety of ephemeral species en route to the product, water. Out of the exponentially growing number of possible sequences of chemical species, we find that greater than 95% of the probability concentrates in less than 1% of the possible sequences. Overall, these results extend the notion of typicality across the nonequilibrium regime and suggest that typical sequences are a route to learning mechanisms from experimental measurements. They also open up the possibility of constructing ensembles for computing the macroscopic observables of systems out of equilibrium.