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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Structure evolution of single-site Pt in a metal–organic framework

Heterogeneous single-metal-site catalyst or single-atom catalyst research has grown rapidly due to the accessibility of modern characterization techniques that can provide invaluable information at the atomic-scale. Herein, we study the structural evolution of isolated single Pt sites incorporated in a metal–organic framework containing bipyridine functional groups using in situ diffuse reflectance infrared Fourier transform spectroscopy with CO as the probe molecule. The structure and electronic properties of the isolated Pt sites are further corroborated by x-ray photoelectron spectroscopy and aberration-corrected scanning transmission electron microscopy. Here, we find the prerequisite of high temperature He treatment for Pt activation and CO insertion and inquire into the structural transformation of Pt site process by dynamic nuclear polarization-enhanced solid-state nuclear magnetic resonance spectroscopy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Triplet-pair spin signatures from macroscopically aligned heteroacenes in an oriented single crystal

The photo-driven process of singlet fission generates coupled triplet pairs (TT) with fundamentally intriguing and potentially useful properties. The quintet 5 TT 0 sublevel is particularly interesting for quantum information because it is highly entangled, is addressable with microwave pulses, and could be detected using optical techniques. Previous theoretical work on a model Hamiltonian and nonadiabatic transition theory, called the JDE model, has determined that this sublevel can be selectively populated if certain conditions are met. Among the most challenging, the molecules within the dimer undergoing singlet fission must have their principal magnetic axes parallel to one another and to an applied Zeeman field. Here, we present time-resolved electron paramagnetic resonance (TR-EPR) spectroscopy of a single crystal sample of a tetracenethiophene compound featuring arrays of dimers aligned in this manner, which were mounted so that the orientation of the field relative to the molecular axes could be controlled. The observed spin sublevel populations in the paired TT and unpaired (T+T) triplets are consistent with predictions from the JDE model, including preferential 5 TT 0 formation at z ‖ B 0 , with one caveat—two 5 TT spin sublevels have little to no population. This may be due to crossings between the 5 TT and 3 TT manifolds in the field range investigated by TR-EPR, consistent with the intertriplet exchange energy determined by monitoring photoluminescence at varying magnetic fields.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Single-Molecule Vibrational Characterization of Binding Geometry Effects on Isocyanide–Metal Interactions

Isocyanide−metal binding is governed by σ-donation and π-back-bonding, which affects the isocyanide stretching energy, a characteristic probe for ligand−metal interactions. While extensive correlations exist between structure and spectroscopy in molecular isocyanide−metal systems, interactions of isocyanide with crystalline metallic surfaces, where ligands often bind in various geometries, remain underexplored. Conventional vibrational spectroscopies such as infrared and Raman spectroscopies lack the molecular-scale resolution to distinguish this binding inhomogeneity. In contrast, inelastic electron tunneling spectroscopy with scanning tunneling microscopy (STM-IETS) directly visualizes ligand adsorption geometries and their vibrational signatures. Using STM-IETS, we investigate a metal-adsorbed m -terphenyl isocyanide ligand and find that the adsorption geometry on Cu(100) induces a significant shift in isocyanide stretching frequency, more prominent than replacing Cu(100) with Ag(111). Density functional theory confirms that this shift arises from atomic-scale variations in coordination environments. This study elucidates how precise binding influences the vibrational fingerprints of isocyanide ligands, an often-overlooked factor in understanding the isocyanide−metal interplay.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2023

Dispersion-Engineered Deep Sub-Wavelength Plasmonic Metasurfaces for Broadband Seira Applications

Surface enhanced infrared absorption (SEIRA) spectroscopy is a powerful tool in which plasmonically enhanced electromagnetic fields provide high-sensitivity molecular detection. Most SEIRA platforms operate at a single resonant frequency, which must be tuned to match that of the target molecule, and commonly rely on time-consuming lithographic techniques. This study presents a high-throughput and cost-effective plasmonic metasurface for broadband, tunable, and strong infrared signal enhancement. The platform is built upon the principle of dispersion-engineered plasmonic Fabry–Pérot (FP) nanocavity arrays. It offers 1) tight squeezing of infrared (IR) photons into deep sub-wavelength nano-volumes and 2) spectrally tunable near-field enhancements of up to ≈10 6 , two to three orders of magnitude higher than most optical metasurface systems. By coupling multilayer nano-thin film deposition and nanoskiving fabrication techniques, the dispersive FP metasurfaces can be rapidly and reproducibly constructed in a scalable and lithography-free manner. Using IR spectroscopy, the selective and sensitive label-free detection of a molecular monolayer is achieved at a range of frequencies. An enhancement factor of nearly 10 5 is measured at the carbonyl (C = O) vibrational marker band of the molecule. The confluence of high field enhancement, broadband plasmonic response, and facile fabrication makes this metasurface a promising platform for SEIRA spectroscopy.

36 MATERIALS SCIENCE↗
NASA NTRS1994

Alpha-helical hydrophobic polypeptides form proton-selective channels in lipid bilayers

Proton translocation is important in membrane-mediated processes such as ATP-dependent proton pumps, ATP synthesis, bacteriorhodopsin, and cytochrome oxidase function. The fundamental mechanism, however, is poorly understood. To test the theoretical possibility that bundles of hydrophobic alpha-helices could provide a low energy pathway for ion translocation through the lipid bilayer, polyamino acids were incorporated into extruded liposomes and planar lipid membranes, and proton translocation was measured. Liposomes with incorporated long-chain poly-L-alanine or poly-L-leucine were found to have proton permeability coefficients 5 to 7 times greater than control liposomes, whereas short-chain polyamino acids had relatively little effect. Potassium permeability was not increased markedly by any of the polyamino acids tested. Analytical thin layer chromatography measurements of lipid content and a fluorescamine assay for amino acids showed that there were approximately 135 polyleucine or 65 polyalanine molecules associated with each liposome. Fourier transform infrared spectroscopy indicated that a major fraction of the long-chain hydrophobic peptides existed in an alpha-helical conformation. Single-channel recording in both 0.1 N HCl and 0.1 M KCl was also used to determine whether proton-conducting channels formed in planar lipid membranes (phosphatidylcholine/phosphatidylethanolamine, 1:1). Poly-L-leucine and poly-L-alanine in HCl caused a 10- to 30-fold increase in frequency of conductive events compared to that seen in KCl or by the other polyamino acids in either solution. This finding correlates well with the liposome observations in which these two polyamino acids caused the largest increase in membrane proton permeability but had little effect on potassium permeability. Poly-L-leucine was considerably more conductive than poly-L-alanine due primarily to larger event amplitudes and, to a lesser extent, a higher event frequency. Poly-L-leucine caused two populations of conductive events, one in the 0.1-0.5 pA range, and one in the 1.0-5.0 pA range, whereas nearly all events caused by poly-L-alanine were in the 0.1-0.5 pA range at an applied voltage of +60 mV. The channel-like activity appeared to switch between conductive and nonconductive states, with most open-times in the range of 50-200 ms. We conclude that hydrophobic polyamino acids produce proton-conducting defects in lipid bilayers that may be used to model functional proton channels in biological membranes.

NASA Discipline Exobiology↗
DOE OSTI2021

Linking optical spectra to free charges in donor/acceptor heterojunctions: cross-correlation of transient microwave and optical spectroscopy

The primary photoexcited species in excitonic semiconductors is a bound electron–hole pair, or exciton. An important strategy for producing separated electrons and holes in photoexcited excitonic semiconductors is the use of donor/acceptor heterojunctions, but the degree to which the carriers can escape their mutual Coulomb attraction is still debated for many systems. Here, we employ a combined pump–probe ultrafast transient absorption (TA) spectroscopy and time-resolved microwave conductivity (TRMC) study on a suite of model excitonic heterojunctions consisting of mono-chiral semiconducting single-walled carbon nanotube (s-SWCNT) electron donors and small-molecule electron acceptors. Comparison of the charge-separated state dynamics between TA and TRMC photoconductance reveals a quantitative match over the 0.5 microsecond time scale. Charge separation yields derived from TA allow extraction of s-SWCNT hole mobilities of ca. 1.5 cm 2 V -1 s -1 (at 9 GHz) by TRMC. We find that the correlation between the techniques conclusively demonstrates that photoinduced charge carriers separated across these heterojunctions do not form bound charge transfer states, but instead form free/mobile charge carriers.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Single-Molecule Fluorescence Microscopy Reveals Energy Transfer Active versus Inactive Nanocrystal/Dye Conjugate Pairs

Defect-mediated energy transfer (EnT) is a radiative process that occurs between donor defect states in the forbidden bandgap of semiconductor nanocrystals (NCs) and dye molecules bound to their surfaces. The EnT efficiency depends on the number of dye molecules attached to each NC, the donor–acceptor distance, and the dipole orientation factor between the donor and acceptor, all of which vary across all individual NCs in a sample. While ensemble-level fluorescence spectroscopy measurements have provided average values for donor–acceptor distances, dye-to-NC ratios, and EnT rate constants, questions remain about the impact of donor/acceptor heterogeneity on observed EnT efficiencies. Notably, ensemble-level measurements cannot distinguish between bare NCs and EnT-active versus inactive NC/dye pairs in the same sample batch, limiting the ability to design systems with 100% EnT efficiency. To address this, we studied defect-mediated EnT between AlexaFluor 555 dye acceptors chemically bound to ZnO NC donors at the level of single molecules and single NCs. Interestingly, 20% of bound NC/dye pairs are EnT-inactive, likely contributing to residual defect photoluminescence (PL) observed in ensemble-level measurements and reducing overall EnT efficiency. Single particle-level ZnO defect PL and acceptor fluorescence trajectories exhibited distinct microfluctuations, which are absent in bare ZnO NCs. We hypothesized that our observations can be explained with a competitive dye fluorescence quenching pathway, possibly due to charge transfer between the excited state dye and the ZnO NC. Numerical simulations of single-molecule PL traces for this scenario produced microfluctuations consistent with the experimental results. These findings highlight the impact of sample heterogeneity on EnT processes and provide insights for designing light-harvesting systems with optimized EnT efficiency.

25 ENERGY STORAGE↗
NASA NTRS1993

Hypervelocity impact survivability experiments for carbonaceous impactors

We performed a series of hypervelocity impact experiments using carbon-bearing impactors (diamond, graphite, fullerenes, phthalic acid crystals, and Murchison meteorite) into Al plate at velocities between 4.2 and 6.1 km/s. These tests were made to do the following: (1) determine the survivability of carbon forms and organize molecules in low hypervelocity impact; (2) characterize carbonaceous impactor residues; and (3) determine whether or not fullerenes could form from carbonaceous impactors, under our experimental conditions, or survive as impactors. An analytical protocol of field emission SEM imagery, SEM-EDX, laser Raman spectroscopy, single and 2-stage laser mass spectrometry, and laser induced fluorescence (LIF) found the following: (1) diamonds did not survive impact at 4.8 km/s, but were transformed into various forms of disordered graphite; (2) intact, well-ordered graphite impactors did survive impact at 5.9 km/sec, but were only found in the crater bottom centers; the degree of impact-induced disorder in the graphite increases outward (walls, rims, ejecta); (3) phthalic acid crystals were destroyed on impact (at 4.2 km/s, although a large proportion of phthalic acid molecules did survive impact); (4) fullerenes did not form as products of carbonaceous impactors (5.9 - 6.1 km/s, fullerene impactor molecules mostly survived impact at 5.9 km/s; and (5) two Murchison meteorite samples (launched at 4.8 and 5.9 km/s) show preservation of some higher mass polycyclic aromatic hydrocarbons (PAHs) compared with the non-impacted sample. Each impactor type shows unique impactor residue morphologies produced at a given impact velocity. An expanded methodology is presented to announce relatively new analytical techniques together with innovative modifications to other methods that can be used to characterize small impact residues in LDEF craters, in addition to other acquired extraterrestrial samples.

Bunch, T. E.↗
DOE OSTI2024

Engineering Small HOMO–LUMO Gaps in Polycyclic Aromatic Hydrocarbons with Topologically Protected States

Topological phases in laterally confined low-dimensional nanographenes have emerged as versatile design tools that can imbue otherwise unremarkable materials with exotic band structures ranging from topological semiconductors and quantum dots to intrinsically metallic bands. The periodic boundary conditions that define the topology of a given lattice have thus far prevented the translation of this technology to the quasi-zero-dimensional (0D) domain of small molecular structures. Here, we describe the synthesis of a polycyclic aromatic hydrocarbon (PAH) featuring two localized zero modes (ZMs) formed by the topological junction interface between a trivial and nontrivial phase within a single molecule. First-principles density functional theory calculations predict a strong hybridization between adjacent ZMs that gives rise to an exceptionally small HOMO–LUMO gap. Scanning tunneling microscopy and spectroscopy corroborate the molecular structure of 9/7/9-double quantum dots and reveal an experimental quasiparticle gap of 0.16 eV, corresponding to a carbon-based small molecule long-wavelength infrared (LWIR) absorber.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2020

Structure and Desorption Kinetics of Acetonitrile Thin Films on Pt(111) and on Graphene on Pt(111)

Acetonitrile thin films were prepared on Pt(111) and on graphene on Pt(111) using molecular beam techniques. Temperature programed desorption (TPD) experiments of acetonitrile on Pt(111) displayed first-order kinetics for monolayer desorption and zero-order kinetics for multilayer desorption. We used reflection adsorption infrared spectroscopy (RAIRS) to characterize the orientation of the monolayer on Pt(111). Acetonitrile molecules have been shown to adopt a bridging structure where the C=N bond breaks to form two single bonds to underlying platinum atoms. When Tdeposition = 120 K our RAIRS spectra are consistent with this bridging configuration. However, when Tdeposition was lowered to 60 K molecules retained the C=N bond and adopted a vertical orientation with the nitrogen closest to the platinum. On graphene, the TPD experiments show that both the monolayer and multilayer desorb with zero-order kinetics. Our RAIRS spectra suggest that the monolayer consists of acetonitrile molecules oriented with their molecular axes parallel to the graphene substrate. This orientation did not change when Tdeposition was varied from 60 to 120 K. Our results provide a more complete picture of how acetonitrile molecules interact with Pt(111) and with graphene surfaces.

desorption tin films, molecules, thin films, desor↗
DOE OSTI2021

Atomically Dispersed Copper Sites in a Metal–Organic Framework for Reduction of Nitrogen Dioxide

Metal-organic framework (MOF) materials provide an excellent platform to fabricate single-atom catalysts due to their structural diversity, intrinsic porosity, and designable functionality. However, the unambiguous identification of atomically dispersed metal sites and the elucidation of their role in catalysis are challenging due to limited methods of characterization and lack of direct structural information. Here, we report a comprehensive investigation of the structure and the role of atomically dispersed copper sites in UiO-66 for the catalytic reduction of NO 2 at ambient temperature. The atomic dispersion of copper sites on UiO-66 is confirmed by high-angle annular dark-field scanning transmission electron microscopy, electron paramagnetic resonance spectroscopy, and inelastic neutron scattering, and their location is identified by neutron powder diffraction and solid-state nuclear magnetic resonance spectroscopy. The Cu/UiO-66 catalyst exhibits superior catalytic performance for the reduction of NO 2 at 25 °C without the use of reductants. A selectivity of 88% for the formation of N2 at a 97% conversion of NO 2 with a lifetime of >50 h and an unprecedented turnover frequency of 6.1 h-1 is achieved under nonthermal plasma activation. In situ and operando infrared, solid-state NMR, and EPR spectroscopy reveal the critical role of copper sites in the adsorption and activation of NO 2 molecules, with the formation of {Cu(I)···NO} and {Cu···NO 2 } adducts promoting the conversion of NO 2 to N 2 . This study will inspire the further design and study of new efficient single-atom catalysts for NO 2 abatement via detailed unravelling of their role in catalysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Role of Bimetallic Interactions in the Enhancement of Catalytic CO 2 Reduction by a Macrocyclic Cobalt Catalyst

The use of two metal centers in a CO 2 reduction catalyst that work together synergistically, with similar or complementary functions, can potentially lead to a significant reduction in overpotential, enhance catalytic activity and/or selectivity, and/or enable access to cascade strategies where each metal center catalyzes a different step in the conversion of CO 2 to a fuel. Here, the bimetallic reactivity of two metal centers has been identified as the primary route for the reduction of CO 2 to CO promoted by the macrocycle, [Co(HMD)] 2+ (HMD = 5,7,7,12,14,14-hexamethyl-1,4,8,11-tetraazacyclotetradeca-4,11-diene), based on the experimental characterization of all major steps of the proposed catalytic cycle using pulse radiolysis time-resolved IR (PR-TRIR) spectroscopy, corroborated by density functional theory (DFT) calculations and IR spectroelectrochemistry (IR-SEC). A bimetallic intermediate is formed in situ from two singly reduced [Co(HMD)] + species bridged by a CO 2 molecule, and the presence of a coordinating species, e.g., formate anion, appears to assist in the formation of such an intermediate. In this work, it has been demonstrated that this reactivity enables access to elementary steps with lower energy requirements, resulting in overall catalysis being kinetically more facile compared to the mononuclear pathway. A two-step approach that combines chemical reduction followed by PR-TRIR has been successfully used for probing the structure and reactivity of reactive intermediates involved in the advanced stages of a catalytic cycle, which are rarely interrogated using experimental techniques.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Selectively accessing the hotspots of optical nanoantennas by self-aligned dry laser ablation

Plasmonic nanostructures serve as optical antennas for concentrating the energy of incoming light in localized hotspots close to their surface. By positioning nanoemitters in the antenna hotspots, energy transfer is enabled, leading to novel hybrid antenna-emitter-systems, where the antenna can be used to manipulate the optical properties of the nano-objects. The challenge remains how to precisely position emitters within the hotspots. We report a self-aligned process based on dry laser ablation of a calixarene that enables the attachment of molecules within the electromagnetic hotspots at the tips of gold nanocones. Within the laser focus, the ablation threshold is exceeded in nanoscale volumes, leading to selective access of the hotspot areas. A first indication of the site-selective functionalization process is given by attaching fluorescently labelled proteins to the nanocones. In a second example, Raman-active molecules are selectively attached only to nanocones that were previously exposed in the laser focus, which is verified by surface enhanced Raman spectroscopy. Enabling selective functionalization is an important prerequisite e.g. for preparing single photon sources for quantum optical technologies, or multiplexed Raman sensing platforms.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2023

Mid-infrared spectroscopy with a broadly tunable thin-film lithium niobate optical parametric oscillator

Mid-infrared spectroscopy, an important technique for sensing molecules, has encountered barriers from sources either limited in tuning range or excessively bulky for widespread use. We present a compact, efficient, and broadly tunable optical parametric oscillator surmounting these challenges. Leveraging dispersion-engineered thin-film lithium niobate-on-sapphire photonics and a singly resonant cavity allows broad, controlled tuning over an octave from 1.5–3.3 µm. The device generates >25mW of mid-infrared light at 3.2 µm with 15% conversion efficiency. The ability to precisely control the device’s mid-infrared emission enables spectroscopy of methane and ammonia, demonstrating our approach’s relevance for sensing. Our work signifies an important advance in nonlinear photonics miniaturization, bringing practical field applications of high-speed, broadband mid-infrared spectroscopy closer to reality.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2024

Adsorption and Solvation Modulate Rhodamine B Diffusion in Ethanol/Water-Filled Anodic Alumina Nanopores

Confinement of solvents and solutes within nanoporous materials frequently leads to the emergence of unique mass transport behaviors that, once fully understood, may lead to improved chemical separations. Here, the diffusion of Rhodamine B (RhB) dye within 10 and 20 nm diameter anodic aluminum oxide (AAO) nanopores filled with binary ethanol/water mixtures is investigated. Mixture compositions spanning from pure ethanol to pure water are employed. The results of confocal fluorescence correlation spectroscopy studies reveal that RhB diffusion occurs by a two-component mechanism comprising composition-dependent fast and slow motions, characterized by diffusion coefficients D f and D s . The results are consistent with those of previous studies performed under more limited conditions [J. Phys. Chem. C, 2023, 127, 411-420]. The fast component scales with mixture viscosity and is assigned to hindered bulk-like diffusion in central pore regions. Slow diffusion likely involves adsorption of RhB to the pore surface and may be described by a desorption mediated mechanism. The occurrence of RhB adsorption to the AAO surface is verified at the single-molecule level by wide-field fluorescence imaging of membrane cross-sectional surfaces. Unique composition dependent trends in the autocorrelation amplitude and in D s that mimic bulk RhB solubility are revealed. D s is found to be smallest in pure ethanol and pure water and largest in intermediate mixtures. These results suggest that RhB surface adsorption is strongest in the pure liquids and weakest in mixtures of intermediate composition, where the dye is least soluble, and most soluble, respectively. As a result, molecular dynamics simulations reveal that a water layer appears on the pore surface under most conditions, while RhB is solvated primarily by ethanol. The composition dependence of RhB diffusion is concluded to reflect its solvation dependent interactions with the pore walls.

36 MATERIALS SCIENCE↗
NASA NTRS1988

Diode laser harmonic spectroscopy applied to in situ measurements of atmospheric trace molecules

With the emergence of lead-salt diode laser technology, a broad and important IR spectral region, roughly 3-30 microns, became accessible to tunable laser spectroscopy. More recent advances in fabrication techniques have provided experimenters with relatively high power quasi-single-mode high-temperature lasers that are readily adapted to compact automated instruments for field experiments. An especially attractive capability of diode lasers is the ease of using them for simultaneous multiconstituent sampling in the atmosphere. This paper presents a summary description of field instruments for atmospheric research which employ diode lasers and second-harmonic detection. Representative results obtained with some of these instruments are presented.

Loewenstein, Max↗
DOE OSTI2021

Toward Å-fs-meV Resolution in Electron Microscopy: Systematic Simulation of the Temporal Spread of Single-Electron Packets

Though efforts to improve the temporal resolution of transmission electron microscopes (TEMs) have waxed and waned for decades, with relatively recent advances routinely reaching sub-picosecond scales, fundamental and practical challenges have hindered the advance of combined Å-fs-meV resolutions, particularly for core-loss spectroscopy and real-space imaging. This is due in no small part to the complexity of the approach required to access timescales upon which electrons, atoms, molecules, and materials first begin to respond and transform – attoseconds to picoseconds. Here we present part of a larger effort devoted to systematically mapping the instrument parameter space of a TEM modified to reach ultrafast timescales. With General Particle Tracer, we studied the statistical temporal distributions of single-electron packets as a function of various fs pulsed-laser parameters and electron-gun configurations and fields for the exact architecture and dimensions of a Thermo Fisher Tecnai Femto ultrafast electron microscope. We focused on easily-adjustable parameters, such as laser pulse duration, laser spot size, photon energy, Wehnelt aperture diameter, and photocathode size. In addition to establishing trends and dispersion behaviors, we identify regimes within which packet duration can be 100s of fs and approach the 300-fs laser limit employed here. Overall, the results provide a detailed picture of the temporal behavior of single-electron packets in the Tecnai Femto gun region, forming the initial contribution of a larger effort.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

SERS-based ssDNA composition analysis with inhomogeneous peak broadening and reservoir computing

Surface-enhanced Raman spectroscopy employed in conjunction with post-processing machine learning methods is a promising technique for effective data analysis, allowing one to enhance the molecular and chemical composition analysis of information rich DNA molecules. In this work, we report on a room temperature inhomogeneous broadening as a function of the increased adenine concentration and employ this feature to develop one-dimensional and two dimensional chemical composition classification models of 200 long single stranded DNA sequences. Afterwards, we develop a reservoir computing chemical composition classification scheme of the same molecules and demonstrate enhanced performance that does not rely on manual feature identification.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗