Multi-band Metasurface-Driven Surface-Enhanced Infrared Absorption Spectroscopy for Improved Characterization of in-Situ Electrochemical Reactions.

Surface-enhanced spectroscopy techniques are the method-of-choice to characterize adsorbed intermediates occurring during electrochemical reactions, which are crucial in realizing a green and sustainable future. Characterizing species with low coverage or short lifetimes has so far been limited by low signal enhancement. Recently, single-band metasurface-driven surface-enhanced infrared absorption spectroscopy (SEIRAS) has been pioneered as a promising technology to monitor a single vibrational mode during electrochemical CO oxidation. However, electrochemical reactions are complex, and their understanding requires the simultaneous monitoring of multiple adsorbed species in situ, hampering the adoption of nanostructured electrodes in spectro-electrochemistry. Here, we develop a multi-band nanophotonic-electrochemical platform that simultaneously monitors in situ multiple adsorbed species emerging during cyclic voltammetry scans by leveraging the high resolution offered by the reproducible nanostructuring of the working electrode. Specifically, we studied the electrochemical reduction of CO2 on a Pt surface and used two separately tuned metasurface arrays to monitor two adsorption configurations of CO with vibrational bands at ∼2030 and ∼1840 cm-1. Our platform provides a ∼40-fold enhancement in the detection of characteristic absorption signals compared to conventional broadband electrochemically roughened platinum films. A straightforward methodology is outlined starting with baselining our system in a CO-saturated environment and clearly detecting both configurations of adsorption. In contrast, during the electrochemical reduction of CO2 on platinum in K2CO3, CO adsorbed in a bridged configuration could not be detected. We anticipate that our technology will guide researchers in developing similar sensing platforms to simultaneously detect multiple challenging intermediates, with low surface coverage or short lifetimes.

Chaotic chimera attractors in a triangular network of identical oscillators.

A prominent type of collective dynamics in networks of coupled oscillators is the coexistence of coherently and incoherently oscillating domains known as chimera states. Chimera states exhibit various macroscopic dynamics with different motions of the Kuramoto order parameter. Stationary, periodic and quasiperiodic chimeras are known to occur in two-population networks of identical phase oscillators. In a three-population network of identical Kuramoto-Sakaguchi phase oscillators, stationary and periodic symmetric chimeras were previously studied on a reduced manifold in which two populations behaved identically [Phys. Rev. E 82, 016216 (2010)1539-375510.1103/PhysRevE.82.016216]. In this paper, we study the full phase space dynamics of such three-population networks. We demonstrate the existence of macroscopic chaotic chimera attractors that exhibit aperiodic antiphase dynamics of the order parameters. We observe these chaotic chimera states in both finite-sized systems and the thermodynamic limit outside the Ott-Antonsen manifold. The chaotic chimera states coexist with a stable chimera solution on the Ott-Antonsen manifold that displays periodic antiphase oscillation of the two incoherent populations and with a symmetric stationary chimera solution, resulting in tristability of chimera states. Of these three coexisting chimera states, only the symmetric stationary chimera solution exists in the symmetry-reduced manifold.

Heteroclinic switching between chimeras in a ring of six oscillator populations.

In a network of coupled oscillators, a symmetry-broken dynamical state characterized by the coexistence of coherent and incoherent parts can spontaneously form. It is known as a chimera state. We study chimera states in a network consisting of six populations of identical Kuramoto-Sakaguchi phase oscillators. The populations are arranged in a ring, and oscillators belonging to one population are uniformly coupled to all oscillators within the same population and to those in the two neighboring populations. This topology supports the existence of different configurations of coherent and incoherent populations along the ring, but all of them are linearly unstable in most of the parameter space. Yet, chimera dynamics is observed from random initial conditions in a wide parameter range, characterized by one incoherent and five synchronized populations. These observable states are connected to the formation of a heteroclinic cycle between symmetric variants of saddle chimeras, which gives rise to a switching dynamics. We analyze the dynamical and spectral properties of the chimeras in the thermodynamic limit using the Ott-Antonsen ansatz and in finite-sized systems employing Watanabe-Strogatz reduction. For a heterogeneous frequency distribution, a small heterogeneity renders a heteroclinic switching dynamics asymptotically attracting. However, for a large heterogeneity, the heteroclinic orbit does not survive; instead, it is replaced by a variety of attracting chimera states.

Nontrivial twisted states in nonlocally coupled Stuart-Landau oscillators.

A twisted state is an important yet simple form of collective dynamics in an oscillatory medium. Here we describe a nontrivial type of twisted state in a system of nonlocally coupled Stuart-Landau oscillators. The nontrivial twisted state (NTS) is a coherent traveling wave characterized by inhomogeneous profiles of amplitudes and phase gradients, which can be assigned a winding number. To further investigate its properties, several methods are employed. We perform a linear stability analysis in the continuum limit and compare the results with Lyapunov exponents obtained in a finite-size system. The determination of covariant Lyapunov vectors allows us to identify collective modes. Furthermore, we show that the NTS is robust to small heterogeneities in the natural frequencies and present a bifurcation analysis revealing that NTSs are born or annihilated in a saddle-node bifurcation and change their stability in Hopf bifurcations. We observe stable NTSs with winding number 1 and 2. The latter can lose stability in a supercritical Hopf bifurcation, leading to a modulated 2-NTS.

Bifurcations of clusters and collective oscillations in networks of bistable units.

We investigate dynamics and bifurcations in a mathematical model that captures electrochemical experiments on arrays of microelectrodes. In isolation, each individual microelectrode is described by a one-dimensional unit with a bistable current-potential response. When an array of such electrodes is coupled by controlling the total electric current, the common electric potential of all electrodes oscillates in some interval of the current. These coupling-induced collective oscillations of bistable one-dimensional units are captured by the model. Moreover, any equilibrium is contained in a cluster subspace, where the electrodes take at most three distinct states. We systematically analyze the dynamics and bifurcations of the model equations: We consider the dynamics on cluster subspaces of successively increasing dimension and analyze the bifurcations occurring therein. Most importantly, the system exhibits an equivariant transcritical bifurcation of limit cycles. From this bifurcation, several limit cycles branch, one of which is stable for arbitrarily many bistable units.

Attracting Poisson chimeras in two-population networks.

Chimera states, i.e., dynamical states composed of coexisting synchronous and asynchronous oscillations, have been reported to exist in diverse topologies of oscillators in simulations and experiments. Two-population networks with distinct intra- and inter-population coupling have served as simple model systems for chimera states since they possess an invariant synchronized manifold in contrast to networks on a spatial structure. Here, we study dynamical and spectral properties of finite-sized chimeras on two-population networks. First, we elucidate how the Kuramoto order parameter of the finite-sized globally coupled two-population network of phase oscillators is connected to that of the continuum limit. These findings suggest that it is suitable to classify the chimera states according to their order parameter dynamics, and therefore, we define Poisson and non-Poisson chimera states. We then perform a Lyapunov analysis of these two types of chimera states, which yields insight into the full stability properties of the chimera trajectories as well as of collective modes. In particular, our analysis also confirms that Poisson chimeras are neutrally stable. We then introduce two types of "perturbation" that act as small heterogeneities and render Poisson chimeras attracting: A topological variation via the simplest nonlocal intra-population coupling that keeps the network symmetries and the allowance of amplitude variations in the globally coupled two-population network; i.e., we replace the phase oscillators by Stuart-Landau oscillators. The Lyapunov spectral properties of chimera states in the two modified networks are investigated, exploiting an approach based on network symmetry-induced cluster pattern dynamics of the finite-size network.

Birhythmicity, intrinsic entrainment, and minimal chimeras in an electrochemical experiment.

The coexistence of limit cycles in a phase space, so called birhythmicity, is a phenomenon known to exist in many systems in various disciplines. Yet, detailed experimental investigations are rare, as are studies on the interaction between birhythmic components. In this article, we present experimental evidence for the existence of birhythmicity during the anodic electrodissolution of Si in a fluoride-containing electrolyte using weakly illuminated n-type Si electrodes. Moreover, we demonstrate several types of interaction between the coexisting limit cycles, in part resulting in peculiar dynamics. The two limit cycles exhibit vastly different sensitivities with respect to a small perturbation of the electrode potential, rendering the coupling essentially unidirectional. A manifestation of this is an asymmetric 1:2 intrinsic entrainment of the coexisting limit cycles on an individual uniformly oscillating electrode. In this state, the phase-space structure mediates the locking of one of the oscillators to the other one across the separatrix. Furthermore, the transition scenarios from one limit cycle to the other one at the borders of the birhythmicity go along with different types of spatial symmetry breaking. Finally, the master-slave type coupling promotes two (within the experimental limits) identical electrodes initialized on the different limit cycles to adopt states of different complexity: one of the electrodes exhibits irregular, most likely chaotic, motion, while the other one exhibits period-1 oscillations. The coexistence of coherence and incoherence is the characteristic property of a chimera state, the two coupled electrodes constituting an experimental example of a smallest chimera state in a minimal network configuration.

Between synchrony and turbulence: intricate hierarchies of coexistence patterns.

Coupled oscillators, even identical ones, display a wide range of behaviours, among them synchrony and incoherence. The 2002 discovery of so-called chimera states, states of coexisting synchronized and unsynchronized oscillators, provided a possible link between the two and definitely showed that different parts of the same ensemble can sustain qualitatively different forms of motion. Here, we demonstrate that globally coupled identical oscillators can express a range of coexistence patterns more comprehensive than chimeras. A hierarchy of such states evolves from the fully synchronized solution in a series of cluster-splittings. At the far end of this hierarchy, the states further collide with their own mirror-images in phase space - rendering the motion chaotic, destroying some of the clusters and thereby producing even more intricate coexistence patterns. A sequence of such attractor collisions can ultimately lead to full incoherence of only single asynchronous oscillators. Chimera states, with one large synchronized cluster and else only single oscillators, are found to be just one step in this transition from low- to high-dimensional dynamics.

Connecting minimal chimeras and fully asymmetric chaotic attractors through equivariant pitchfork bifurcations.

Highly symmetric networks can exhibit partly symmetry-broken states, including clusters and chimera states, i.e., states of coexisting synchronized and unsynchronized elements. We address the S_{4} permutation symmetry of four globally coupled Stuart-Landau oscillators and uncover an interconnected web of solutions with different symmetries. Among these are chaotic 2-1-1 minimal chimeras that arise from 2-1-1 periodic solutions in a period-doubling cascade, as well as fully asymmetric chaotic states arising similarly from periodic 1-1-1-1 solutions. A backbone of equivariant pitchfork bifurcations mediate between the two cascades, culminating in equivariant pitchforks of chaotic attractors.

Self-Organized Multifrequency Clusters in an Oscillating Electrochemical System with Strong Nonlinear Coupling.

We study the spatiotemporal dynamics of the oscillatory photoelectrodissolution of n-type Si in a fluoride-containing electrolyte under anodic potentials using in situ ellipsometric imaging. When lowering the illumination intensity stepwise, we successively observe uniform oscillations, modulated amplitude clusters, and the coexistence of multifrequency clusters, i.e., regions with different frequencies, with a stationary domain. We argue that the multifrequency clusters emerge due to an adaptive, nonlinear, and nonlocal coupling, similar to those found in the context of neural dynamics.

Lateral silicon oxide/gold interfaces enhance the rate of electrochemical hydrogen evolution reaction in alkaline media.

The production of solar hydrogen with a silicon based water splitting device is a promising future technology, and silicon-based metal-insulator-semiconductor (MIS) electrodes have been proposed as suitable architectures for efficient photocathodes based on the electronic properties of the MIS structures and the catalytic properties of the metals. In this paper, we demonstrate that the interfaces between the metal and oxide of laterally patterned MIS electrodes may strongly enhance the catalytic activity of the electrode compared to bulk metal surfaces. The employed electrodes consist of well-defined, large-area arrays of gold structures of various mesoscopic sizes embedded in a silicon oxide support on silicon. We demonstrate that the activity of these electrodes for hydrogen evolution reaction (HER) increases with an increase in gold/silicon oxide boundary length in both acidic and alkaline media, although the enhancement of the HER rate in alkaline electrolytes is considerably larger than in acidic electrolytes. Electrodes with the largest interfacial length of gold/silicon oxide exhibited a 10-times larger HER rate in alkaline electrolytes than those with the smallest interfacial length. The data suggest that at the metal/silicon oxide boundaries, alkaline HER is enhanced through a bifunctional mechanism, which we tentatively relate to the laterally structured electrode geometry and to positive charges present in silicon oxide: Both properties change locally the interfacial electric field at the gold/silicon oxide boundary, which, in turn, facilitates a faster transport of hydroxide ions away from the electrode/electrolyte interface in alkaline solution. This mechanism boosts the alkaline HER activity of p-type silicon based photoelectrodes close to their HER activity in acidic electrolytes.

Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators.

Oscillatory systems with long-range or global coupling offer promising insight into the interplay between high-dimensional (or microscopic) chaotic motion and collective interaction patterns. Within this paper, we use Lyapunov analysis to investigate whether chimera states in globally coupled Stuart-Landau (SL) oscillators exhibit collective degrees of freedom. We compare two types of chimera states, which emerge in SL ensembles with linear and nonlinear global coupling, respectively, the latter introducing a constraint that conserves the oscillation of the mean. Lyapunov spectra reveal that for both chimera states the Lyapunov exponents split into several groups with different convergence properties in the limit of large system size. Furthermore, in both cases the Lyapunov dimension is found to scale extensively and the localization properties of covariant Lypunov vectors manifest the presence of collective Lyapunov modes. Here, however, we find qualitative differences between the two types of chimera states: Whereas the ones in the system under nonlinear global coupling exhibit only slow collective modes corresponding to Lyapunov exponents equal or close to zero, those which experience the linear mean-field coupling exhibit also faster collective modes associated with Lyapunov exponents with large positive or negative values. Furthermore, for the fastest collective mode we showed that it spreads across both synchonous and incoherent oscillators.

Coupled Dynamics of Anode and Cathode in Proton-Exchange Membrane Fuel Cells.

An external reference electrode was used to monitor individually the evolution of the anodic and cathodic potentials during the stationary as well as oscillatory operation of a Direct Formic Acid Fuel Cell (DFAFC) and a Direct Methanol Fuel Cell (DMFC). Besides evidencing the large activation loss in both cells, we were able to observe how the oscillatory operation of such devices affects their cathodes. In fact, cathodic oscillations synchronized with the cell voltage dynamics were observed, hitherto never reported on fuel cells. We have addressed these phenomena taking into account two main coupling processes: through the proton concentration and a global coupling stemming from the control mode (potentiostatic or galvanostatic).

Bichaoticity induced by inherent birhythmicity during the oscillatory electrodissolution of silicon.

The electrodissolution of p-type silicon in a fluoride-containing electrolyte is a prominent electrochemical oscillator with a still unknown oscillation mechanism. In this article, we present a study of its dynamical states occurring in a wide range of the applied voltage-external resistance parameter plane. We provide evidence that the system possesses inherent birhythmicity, and thus at least two distinct feedback loops promoting oscillatory behavior. The two parameter regions in which the different limit cycles exist are separated by a band in which the dynamics exhibit bistability between two branches with different multimode oscillations. Following the states along one path through this bistable region, one observes that each branch undergoes a different transition to chaos, namely, a period doubling cascade and a quasiperiodic route with a torus-breakdown, respectively, making Si electrodissolution one of the few experimental systems exhibiting bichaoticity.

Cluster singularity: The unfolding of clustering behavior in globally coupled Stuart-Landau oscillators.

The ubiquitous occurrence of cluster patterns in nature still lacks a comprehensive understanding. It is known that the dynamics of many such natural systems is captured by ensembles of Stuart-Landau oscillators. Here, we investigate clustering dynamics in a mean-coupled ensemble of such limit-cycle oscillators. In particular, we show how clustering occurs in minimal networks and elaborate how the observed 2-cluster states crowd when increasing the number of oscillators. Using persistence, we discuss how this crowding leads to a continuous transition from balanced cluster states to synchronized solutions via the intermediate unbalanced 2-cluster states. These cascade-like transitions emerge from what we call a cluster singularity. At this codimension-2 point, the bifurcations of all 2-cluster states collapse and the stable balanced cluster state bifurcates into the synchronized solution supercritically. We confirm our results using numerical simulations and discuss how our conclusions apply to spatially extended systems.

The True Fate of Pyridinium in the Reportedly Pyridinium-Catalyzed Carbon Dioxide Electroreduction on Platinum.

Protonated pyridine (PyH+ ) has been reported to act as a peculiar and promising catalyst for the direct electroreduction of CO2 to methanol and/or formate. Because of recent strong incentives to turn CO2 into valuable products, this claim triggered great interest, prompting many experiments and DFT simulations. However, when performing the electrolysis in near-neutral pH electrolyte, the local pH around the platinum electrode can easily increase, leading to Py and HCO3 - being the predominant species next to the Pt electrode instead of PyH+ and CO2 . Using a carefully designed electrolysis setup which overcomes the local pH shift issue, we demonstrate that protonated pyridine undergoes a complete hydrogenation into piperidine upon mild reductive conditions (near 0 V vs. RHE). The reduction of the PyH+ ring occurs with and without the presence of CO2 in the electrolyte, and no sign of CO2 electroreduction products was observed, strongly questioning that PyH+ acts as a catalyst for CO2 electroreduction.

Symmetries of Chimera States.

Symmetry broken states arise naturally in oscillatory networks. In this Letter, we investigate chaotic attractors in an ensemble of four mean-coupled Stuart-Landau oscillators with two oscillators being synchronized. We report that these states with partially broken symmetry, so-called chimera states, have different setwise symmetries in the incoherent oscillators, and in particular, some are and some are not invariant under a permutation symmetry on average. This allows for a classification of different chimera states in small networks. We conclude our report with a discussion of related states in spatially extended systems, which seem to inherit the symmetry properties of their counterparts in small networks.

An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning.

Manifold-learning techniques are routinely used in mining complex spatiotemporal data to extract useful, parsimonious data representations/parametrizations; these are, in turn, useful in nonlinear model identification tasks. We focus here on the case of time series data that can ultimately be modelled as a spatially distributed system (e.g. a partial differential equation, PDE), but where we do not know the space in which this PDE should be formulated. Hence, even the spatial coordinates for the distributed system themselves need to be identified - to "emerge from"-the data mining process. We will first validate this "emergent space" reconstruction for time series sampled without space labels in known PDEs; this brings up the issue of observability of physical space from temporal observation data, and the transition from spatially resolved to lumped (order-parameter-based) representations by tuning the scale of the data mining kernels. We will then present actual emergent space "discovery" illustrations. Our illustrative examples include chimera states (states of coexisting coherent and incoherent dynamics), and chaotic as well as quasiperiodic spatiotemporal dynamics, arising in partial differential equations and/or in heterogeneous networks. We also discuss how data-driven "spatial" coordinates can be extracted in ways invariant to the nature of the measuring instrument. Such gauge-invariant data mining can go beyond the fusion of heterogeneous observations of the same system, to the possible matching of apparently different systems. For an older version of this article, including other examples, see https://arxiv.org/abs/1708.05406.