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At least 127 records · Page 7

Unified understanding of the impact of semiflexibility, concentration, and molecular weight on macromolecular-scale ring diffusion

Conformationally fluctuating, globally compact macromolecules such as polymeric rings, single-chain nanoparticles, microgels, and many-arm stars display complex dynamic behaviors due to their rich topological structure and intermolecular organization. Synthetic rings are hybrid objects with conformations that display both ideal random walk and compact globular features, which can serve as models of genomic DNA. To date, emphasis has been placed on the effect of ring molecular weight on their unusual behaviors. Here, we combine simulations and a microscopic force-level theory to build a unified understanding for how key aspects of ring dynamics depend on different tunable molecular properties including backbone rigidity, monomer concentration, degree of traditional entanglement, and molecular weight. Our large-scale molecular dynamics simulations of ring melts with very different backbone stiffnesses reveal unanticipated behaviors which agree well with our generalized theory. This includes a universal master curve for center-of-mass diffusion constants as a function of molecular weight scaled by a chemistry and thermodynamic state-dependent critical molecular weight that generalizes the concept of an entanglement cross-over for linear chains. The key physics is how backbone rigidity and monomer concentration induced changes of the entanglement length, interring packing, degree of interpenetration, and liquid compressibility slow down space-time dynamic-force correlations on macromolecular scales. A power law decay of the center-of-mass diffusion constant with inverse molecular weight squared is the first consequence, followed by an ultraslow activated hopping transport regime. Our results set the stage to address slow dynamics and kinetic arrest in different families of compact synthetic and biological polymeric systems.

Science & Technology - Other Topics↗

Conformation and dynamics of ring polymers under symmetric thin film confinement

Understanding the structure and dynamics of polymers under confinement has been of widespread interest, and one class of polymers that have received comparatively little attention under confinement is that of ring polymers. The properties of non-concatenated ring polymers can also be important in biological fields because ring polymers have been proven to be a good model to study DNA organization in the cell nucleus. From our previous study, linear polymers in a cylindrically confined polymer melt were found to segregate from each other as a result of the strong correlation hole effect that is enhanced by the confining surfaces. By comparison, our subsequent study of linear polymers in confined thin films at similar levels of confinements found only the onset of segregation. Here, we use molecular dynamics simulation to investigate the chain conformations and dynamics of ring polymers under planar (1D) confinement as a function of film thickness. Our results show that conformations of ring polymers are similar to the linear polymers under planar confinement, except that ring polymers are less compressed in the direction normal to the walls. While we find that the correlation hole effect is enhanced under confinement, it is not as pronounced as the linear polymers under 2D confinement. Finally, we show that chain dynamics far above Tg are primarily affected by the friction from walls based on the monomeric friction coefficient we get from the Rouse mode analysis.

36 MATERIALS SCIENCE↗

The impact of kidney function on Alzheimer’s disease blood biomarkers: implications for predicting amyloid-β positivity

Impaired kidney function has a potential confounding effect on blood biomarker levels, including biomarkers for Alzheimer’s disease (AD). Given the imminent use of certain blood biomarkers in the routine diagnostic work-up of patients with suspected AD, knowledge on the potential impact of comorbidities on the utility of blood biomarkers is important. We aimed to evaluate the association between kidney function, assessed through estimated glomerular filtration rate (eGFR) calculated from plasma creatinine and AD blood biomarkers, as well as their influence over predicting Aβ-positivity. We included 242 participants from the Translational Biomarkers in Aging and Dementia (TRIAD) cohort, comprising cognitively unimpaired individuals (CU; n = 124), mild cognitive impairment (MCI; n = 58), AD dementia (n = 34), and non-AD dementia (n = 26) patients all characterized by [ 18 F] AZD-4694. Plasma samples were analyzed for Aβ42, Aβ40, glial fibrillary acidic protein (GFAP), neurofilament light chain (NfL), tau phosphorylated at threonine 181 (p-tau181), 217 (p-tau217), 231 (p-tau231) and N-terminal containing tau fragments (NTA-tau) using Simoa technology. Kidney function was assessed by eGFR in mL/min/1.73 m 2 , based on plasma creatinine levels, age, and sex. Participants were also stratified according to their eGFR-indexed stages of chronic kidney disease (CKD). We evaluated the association between eGFR and blood biomarker levels with linear models and assessed whether eGFR provided added predictive value to determine Aβ-positivity with logistic regression models. Biomarker concentrations were highest in individuals with CKD stage 3, followed by stages 2 and 1, but differences were only significant for NfL, Aβ42, and Aβ40 (not Aβ42/Aβ40). All investigated biomarkers showed significant associations with eGFR except plasma NTA-tau, with stronger relationships observed for Aβ40 and NfL. However, after adjusting for either age, sex or Aβ-PET SUVr, the association with eGFR was no longer significant for all biomarkers except Aβ40, Aβ42, NfL, and GFAP. When evaluating whether accounting for kidney function could lead to improved prediction of Aβ-positivity, we observed no improvements in model fit (Akaike Information Criterion, AIC) or in discriminative performance (AUC) by adding eGFR to a base model including each plasma biomarker, age, and sex. While covariates like age and sex improved model fit, eGFR contributed minimally, and there were no significant differences in clinical discrimination based on AUC values. We found that kidney function seems to be associated with AD blood biomarker concentrations. However, these associations did not remain significant after adjusting for age and sex, except for Aβ40, Aβ42, NfL, and GFAP. While covariates such as age and sex improved prediction of Aβ-positivity, including eGFR in the models did not lead to improved prediction for any biomarker. Our findings indicate that renal function, within the normal to mild impairment range, does not seem to have a clinically relevant impact when using highly accurate blood biomarkers, such as p-tau217, in a biomarker-supported diagnosis.

60 APPLIED LIFE SCIENCES↗

Deformed Fredkin model for the ν = 5 / 2 Moore-Read state on thin cylinders

We propose a frustration-free model for the Moore-Read quantum Hall state on sufficiently thin cylinders with circumferences ≲ 7 magnetic lengths. While the Moore-Read Hamiltonian involves complicated long-range interactions between triplets of electrons in a Landau level, our effective model is a simpler one-dimensional chain of qubits with deformed Fredkin gates. We show that the ground state of the Fredkin model has high overlap with the Moore-Read wave function and accurately reproduces the latter's entanglement properties. Moreover, we demonstrate that the model captures the dynamical response of the Moore-Read state to a geometric quench, induced by suddenly changing the anisotropy of the system. We elucidate the underlying mechanism of the quench dynamics and show that it coincides with the linearized bimetric field theory. The minimal model introduced here can be directly implemented as a first step towards quantum simulation of the Moore-Read state, as we demonstrate by deriving an efficient circuit approximation to the ground state and implementing it on an IBM quantum processor. Published by the American Physical Society 2024

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Catalytic Upgrading of Ethanol to C8+ Distillate Range Ethers via Guerbet Coupling and Etherification

Guerbet coupling followed by etherification is a two step process for conversion of ethanol into distillate range molecules. This involves first alcohol oligomerization, followed by acid-catalyzed dehydration. As we will show in this presentation, ethanol can be converted to higher linear and α-branched alcohols by C-C coupling reactions (ethanol oligomerization) as well as to high molecular weight esters by C-O coupling reactions with Cu/MgxAlOy (CuHT) catalysts. Alcohols may be converted to ethers via bimolecular dehydration in a subsequent step. We have identified more than 160 number of species in the products including alcohols, esters, aldehydes, ketonces, and olefins. Alcohols range from C4 to C10. Both alcohols and esters follow a Schulz-Flory chain growth model. We show the relationship between the catalyst properties (BET surface area, acid and base site count and Cu loading and synthesis method) and the performance in the reactions. We also show that physical mixtures of CuHT and HT can have similar product selectivity of low loading CuHT catalysts. The selectivity towards diesel fuel precursor compounds (hereafter ‘DFPC’) increased with conversion until reaching a plateau at high ethanol conversion (~70%). Alcohol selectivity follows a Schultz-Flory distribution at all studied conversions, and that adsorbed ethanol-derived species may undergo surface oligomerization into 1-butanol and higher alcohols before desorbing in a chain-growth mechanism. Zeolite catalysts convert the C4+ alcohols into C8+ ethers in both batch and continuous flow reactors. Selectivities of up to 80% to C8+ ethers at around 70% conversion using a single pass continuous flow system are able to be achieved. The final product feedstocks obtained from etherification have been used in technoeconomic (TEA) and lifecycle analysis (LCA), which indicate a reduction in greenhouse gas (GHG) emissions of 50% relative to conventional diesel. These final feedstocks are undergoing engine testing to elucidate the physiochemical properties and will be compared to diesel #2 ASTM standards as a reference and will serve as a basis to improve the final bio-diesel blendstocks from etherification.

Huber, George↗

Catalytic upgrading of ethanol to C8+ distillate range ethers via Guerbet coupling and etherification

In this presentation we describe the catalytic approaches for conversion of ethanol into diesel fuel ethers. This involves first alcohol oligomerization, followed by acid-catalyzed dehydration. As we will show in this presentation, ethanol can be converted to higher linear and α-branched alcohols by C-C coupling reactions (ethanol oligomerization) as well as to high molecular weight esters by C-O coupling reactions with Cu/MgxAlOy (CuHT) catalysts. Alcohols may be converted to ethers via bimolecular dehydration in a subsequent step. We have identified more than 160 number of species in the products including alcohols, esters, aldehydes, ketones, and olefins. Alcohols range from C4 to C10. Both alcohols and esters follow a Schulz-Flory chain growth model. We show the relationship between the catalyst properties (BET surface area, acid and base site count and Cu loading and synthesis method) and the performance in the reactions. We also show that physical mixtures of CuHT and HT can have similar product selectivity of low loading CuHT catalysts. The selectivity towards diesel fuel precursor compounds (hereafter ‘DFPC’) increased with conversion until reaching a plateau at high ethanol conversion (~70%). Alcohol selectivity follows a Schultz-Flory distribution at all studied conversions, and that adsorbed ethanol-derived species may undergo surface oligomerization into 1-butanol and higher alcohols before desorbing in a chain-growth mechanism. Zeolite catalysts convert the C4+ alcohols into C8+ ethers in both batch and continuous flow reactors. Selectivities of up to 80% to C8+ ethers at around 70% conversion using a single pass continuous flow system are achieved. The final product feedstocks obtained from etherification have been used in technoeconomic (TEA) and lifecycle analysis (LCA), which indicate a reduction in greenhouse gas (GHG) emissions of 50% relative to conventional diesel and diesel fuel prices that are lower than biodiesel. These final feedstocks are undergoing engine testing to elucidate the physiochemical properties and compare well to diesel #2 ASTM standards.

Huber, George↗

Adaptive Variational Quantum Computing Approaches for Green’s Functions and Nonlinear Susceptibilities

Here, we present and benchmark quantum computing approaches for calculating real-time single-particle Green’s functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green’s function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parametrized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.

97 MATHEMATICS AND COMPUTING↗

BOTTLE 2 - Analysis

The use of analysis is foundational to BOTTLE. In particular, we conduct techno-economic analysis (TEA), life cycle assessment (LCA), and supply chain modeling for each new BOTTLE innovation, spanning new deconstruction and upcycling technologies to redesigned polymers. These critical analysis tools are used in parallel with laboratory research to ensure that BOTTLE technologies can ultimately meet our key metrics. The BOTTLE team used analysis early in the current 3-year project cycle to identify research priorities from a scale, energy, and economics perspective (Nicholson et al., Joule 2021), and recently we presented a comprehensive framework for comparing new circularity-focused approaches to linear, incumbent practices (Nicholson et al., Ann. Rev. Chem. Biomolec. Eng. 2022). This presentation will also review case studies in analysis for closed-loop PET recycling across a range of recycling methods, baseline studies that we have conducted for plastics pyrolysis and gasification as another comparator for BOTTLE technologies, and a comprehensive study of existing closed-loop recycling technologies from a full suite of environmental impacts and feedstock quality metrics. We have undertaken studies of circular polymers being developed in the Redesign task, and we will present an exemplary case for a bio-based acrylic polymer that can replace polymethyl methacrylate. Overall, analysis is critical to enable the BOTTLE Consortium to focus on impactful, realistic technologies.

BIOMASS FUELS↗

Effect of Microstructure on Chain Flexibility and Glass Transition Temperature of Polybenzofulvene

Polybenzofulvene (PBF) is a polydiene with a very high glass transition temperature (T g ), which makes it a potential candidate for use as the hard block for high-temperature thermoplastic elastomer applications. The T g s of polydienes are known to be related to chain flexibility. However, no studies have been reported that correlate the chain flexibility to the microstructure of PBF. Herein, we present a study of solution properties of linear PBFs with narrow molecular weight dispersity and having 1,2-addition ranging from 23% to 99%. The materials were prepared by living anionic polymerization under different conditions. Specifically, the chain flexibility as defined by the Flory's characteristic ratio, , and dependence of chain flexibility on microstructure by combined measurements of intrinsic viscosity and molecular weight using triple-detector size exclusion chromatography (SEC) is studied. The persistence lengths (l p ) and chain diameters (d B ) were also estimated using the touched-bead wormlike chain model. To the best of our knowledge, the characteristic ratio of PBF is found to be the highest of all polydienes that have been reported so far.

36 MATERIALS SCIENCE↗

Entanglement entropy of (2 +1)-dimensional SU(2) lattice gauge theory on plaquette chains

We study the entanglement entropy of Hamiltonian SU(2) lattice gauge theory in 2 +1 dimensions on linear plaquette chains and show that the entanglement entropies of both ground and excited states follow Page curves. The transition of the subsystem size dependence of the entanglement entropy from the area law for the ground state to the volume law for highly excited states is found to be described by a universal crossover function. Quantum many-body scars in the middle of the spectrum, which are present in the electric flux truncated Hilbert space, where the gauge theory can be mapped onto an Ising model, disappear when higher electric field representations are included in the Hilbert space basis. This suggests the continuum (2+1)-dimensional SU(2) gauge theory does not have such scarred states.

Astronomy & Astrophysics↗

Sensor selection and tool wear prediction with data‐driven models for precision machining

Abstract Estimation of tool wear in precision machining is vital in the traditional subtractive machining industry to reduce processing cost, improve manufacturing efficiency and product quality. In this vein, fusion of time and frequency‐domain features of commonly sensed signals can provide an early indication of tool wear and improve its prediction accuracy for prognostics and health management. This paper presents a data‐driven methodology and a complete tool chain for the inference of precision machining tool wear from fused machine measurements, such as cutting force, power, audio and vibration signals, and quantify the usefulness of each measurement. Indicators of tool wear are extracted from time‐domain signal statistics, frequency‐domain analysis, and time‐frequency domain analysis. Correlation coefficients between the extracted features (indicators) and the tool wear are used to select the most informative features. Principal Component Analysis and Partial Least‐Squares are used to reduce the dimensionality of the feature space. Regression models, including linear regression, support vector regression, Decision tree regression, neural network regression and Gaussian process regression, are used to predict the tool wear using data from a Haas milling machine performing spiral boss face milling. The performance of the regression models based on subsets of sensors validates the preliminary estimates about the saliency of the sensors. The experimental results show that the proposed methods can predict the machine tool wear precisely, with readily available sensor measurements. Neural network and Gaussian process regression were able to achieve good estimates of tool wear at different machine operating conditions. The most informative signal in predicting tool wear was shown to be the vibration signal. Time‐frequency domain features were the most informative features among the combination of features of three domains. In addition, using partial least squares components extracted from the original features of signals led to higher prediction accuracy.

Han, Seulki↗

Contribution of the Hydroxyl Group on Interfacial Heat Conduction of Monohydric Alcohols: A Molecular Dynamics Study

Monohydric alcohols have been used as promising phase change materials (PCMs) for low-temperature latent heat storage. However, the heat storage/retrieval rates are limited due to the low thermal conductivity of such alcohols. In this work, nonequilibrium molecular dynamics (NEMD) simulations were performed to study the microscopic heat conduction in example monohydric alcohols, i.e., 1-dodecanol (C12H26O), 1-tetradecanol (C14H30O), and 1-hexadecanol (C16H34O). A simplified ideal crystal model was proposed to exploit the potential for improving the thermal conductivity of monohydric alcohols. The effect of ideal crystalline structures, especially the contribution of the hydroxyl group, on the microscopic heat conduction process was analyzed. The thermal conductivity of the ideal crystals of the various monohydric alcohols was predicted to be more than twice as compared to that of their respective solids. The major thermal resistance in the ideal crystals was found around the molecular interfaces, as a result of the excellent heat conduction performance along the linear molecular chains. The calculated vibrational density of states (VDOS) and interfacial heat transfer were then investigated. When the interfaces are surrounded by hydroxyl groups as walls, strong hydrogen bond (HB) interactions were observed. The interfacial heat transfer coefficient of the ideal crystalline structures of 1-tetradecanol was found to reach up to ~735.6 MW/m2 W. It was elucidated that the high interfacial heat transfer rate is clearly related to the stronger intermolecular interactions.

Engineering↗

Heuristic solutions to the single depot electric vehicle scheduling problem with next day operability constraints

This study focuses on the single depot electric vehicle scheduling problem (SDEVSP) within the broader context of the vehicle scheduling problem (VSP). By developing an effective scheduling model using mixed-integer linear programming, we generate bus blocks that accommodate electric vehicles (EVs), ensuring successful completion of each block while considering recharging requirements between blocks and during off-hours. Next day operability constraints are also incorporated, allowing for seamless repetition of blocks on subsequent days. The SDEVSP is known to be computationally complex, deriving optimal solutions unattainable for large-scale problems within reasonable timeframes. To address this, we propose a two-step solution approach: first solving the single depot VSP, and then addressing the block chaining problem (BCP) using the blocks generated in the first step. The BCP focuses on optimizing block combinations to facilitate recharging between consecutive blocks, considering operational constraints. Further, a case study conducted reveals that nearly 100% electrification for Chicago, IL and Austin, TX transit buses is viable yet requires 1.6 EVs at 150-mile range per diesel vehicle.

33 ADVANCED PROPULSION SYSTEMS↗

Generalized Bayesian Framework for Evaluation of Integral Benchmark Experiments

A recently published generalized Bayesian optimization framework has provided a way to retract any or all of the three common assumptions underlying the conventional Generalized Linear Least Squares (GLLS) optimization method based on the concepts introduced in Ref. [2]. These assumptions are: 1. Perfection: The model used for data evaluation and the prior probability distribution function (PDF) of generalized data are perfect. 2. Normality: The prior and posterior PDF are normal. 3. Linearity: The model is linear. In this work we outline how the framework in [1] could be directly adopted for improved evaluation of nuclear criticality integral benchmark experiments (IBEs) by: 1. Removing the first assumption alone by utilizing the concept of imperfections introduced in [1] to enable evaluation in the presence of discrepancies between the data and model or of missing covariance information by a GLLS method that will be seen as a generalization of the conventional GLLS method employed by the TSURFER code, and by 2. Removing the remaining two assumptions by implementing a Markov Chain Monte Carlo method for computation of the posterior PDF in the SAMPLER code, where TSURFER and SAMPLER are the uncertainty quantification (UQ) codes for IBEs in the SCALE code system based on the GLLS and the stochastic method, respectively. The graphic in Figure 1 categorizes the methods discussed in terms of the assumptions that they employ to determine posterior PDFs.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Establishing the Stability of Cross-Linked Poly(ethylene oxide) Blended with Poly(pentyl malonate) against Lithium Metal Anodes

This work focuses on an electrolyte obtained by infusing linear poly(pentyl malonate) (PPM) chains into cross-linked poly(ethylene oxide) (PEO). A lithium salt is added to enable ion transport. Obtaining this electrolyte is only possible because PPM is miscible with PEO irrespective of salt concentration. The properties of this electrolyte are compared against two benchmarks: an electrolyte obtained by infusing linear PEO into cross-linked PEO and cross-linked PEO with no added linear polymer. Measurements of storage moduli and ion transport efficacies are found to be similar across the three electrolytes, suggesting similar stability behavior against lithium metal anodes, based on current theoretical models. Instead, the PPM infused electrolyte outperforms the other two electrolytes. In the cycling tests, the current supported by the PPM infused electrolyte was a factor of 20 higher than that of the PEO infused electrolyte and a factor of 4 higher than that of the cross-linked PEO electrolyte. In the unidirectional plating experiments, the other two electrolytes fail after about 4 h or less, while the PPM containing electrolyte did not fail during the 140 h testing period. Insight into the nature of failure modes was obtained using synchrotron hard X-ray microtomography.

Gido, Lily [University of California, Berkeley]↗

How Cooperatively Folding Are Homopolymer Molecular Knots?

Detailed thermodynamic analysis of complex systems with multiple stable configurational states allows for insight into the cooperativity of each individual transition. In this work, we derive a heat capacity decomposition comprising contributions from each individual configurational state, which together sum to a baseline heat capacity, and contributions from each state-to-state transition. We apply this analysis framework to a series of replica exchange molecular dynamics simulations of linear and 1-1 coarse-grained homo-oligomer models, which fold into stable, configurationally well-defined molecular knots, in order to better understand the parameters leading to stable and cooperative folding of these knots. We find that a stiff harmonic backbone bending angle potential is key to achieving knots with specific 3D structures. Tuning the backbone equilibrium angle in small increments yields a variety of knot topologies. Populations of different knotted states as a function of temperature can also be manipulated by tuning the backbone torsion stiffness or by adding side-chain beads. We find that sharp total heat capacity peaks for the homo-oligomer knots are largely due to a coil-to-globule transition rather than a cooperative knotting step. Furthermore, in some cases, the cooperativity of globule-to-knot and coil-to-globule transitions is comparable, suggesting that highly cooperative folding to knotted structures can be achieved by refining the model parameters or adding sequence specificity.

36 MATERIALS SCIENCE↗

A nonlinear journey from structural phase transitions to quantum annealing

Motivated by an exact mapping between equilibrium properties of a one-dimensional chain of quantum Ising spins in a transverse field (the transverse field Ising (TFI) model) and a two-dimensional classical array of particles in double-well potentials (the “$\phi$ 4 model”) with weak inter-chain coupling, we explore connections between the driven variants of the two systems. Here, we argue that coupling between the fundamental topological solitary waves in the form of kinks between neighboring chains in the classical $\phi$ 4 system is the analog of the competing effect of the transverse field on spin flips in the quantum TFI model. As an example application, we mimic simplified measurement protocols in a closed quantum model system by studying the classical $\phi$ 4 model subjected to periodic perturbations. This reveals memory/loss of memory and coherence/decoherence regimes, whose quantum analogs are essential in annealing phenomena. In particular, we examine regimes where the topological excitations control the thermal equilibration following perturbations. This paves the way for further explorations of the analogy between lower-dimensional linear quantum and higher-dimensional classical nonlinear systems.

74 ATOMIC AND MOLECULAR PHYSICS↗

Chain Flexibility and Structure of a Polyimide Copolymer: Revisiting the Freely Rotating Chain Model

Poly(4–4′-oxydiphenylene-pyromellitimide)-based polyimides─trade name Kapton─have wide-ranging engineering applications owing to their thermal and mechanical stability, but little is known about underlying chain-level characteristics. While theoretical models have conceptualized Kapton as inflexible polycyclic rods separated by freely rotating diphenyl ether hinge groups, the model’s core predictions remain untested and subtleties of the relaxation behavior are missed, which atomistic modeling can resolve. To these ends, we examine all-atom Kapton structures in crystalline and glassy amorphous configurations using a DFT-validated class II force field. Constructing amorphous configurations is challenging, as the fused-ring-containing backbone has slow relaxation dynamics and scaling suggestive of entanglements even in oligomers. In conclusion, we find larger backbone rearrangements of the linear polycyclic segments about ether groups that are consistent with the rod-hinge picture on the monomer scale, whereas a ring rotation analysis suggests partially flexible rod-like segments and involves multiple facile rotational relaxation modes.

Liesen, Nicholas T. [Lawrence Livermore National L↗