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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 919 records · Page 51
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Data Analytics for Residential PV from Permit to Interconnect (Final Technical Report)

The main objective of this research is to provide novel insights into the effects of permitting, inspection, and interconnection (PII) processes on PV system installations—and in particular, into the relationship between PII processes and adoption timelines. This research can then be used to clarify the potential effect of various process changes on reducing PII timelines, customer cancellation rates, and related costs nationwide. NREL completed this research by assembling a data set of distributed, largely residential rooftop solar systems less than 50 kilowatts in size from participating solar installers. NREL produced five publications describing the effects that PII processes can have on adoption timelines nationwide, in addition to publishing an interactive data viewer with five years of PII cycle time data. This tool can be used by stakeholders to identify potential adoption timelines by local government.

14 SOLAR ENERGY↗
DOE OSTI2023

Reversal of Catalytic Material Substrate Selectivity through Partitioning of Polymers in Hierarchically Ordered Virus-like Particle Frameworks

Control over the selectivity of catalytic materials is a topic of growing interest. Virus-like particle (VLP) based materials such as protein macromolecular frameworks (PMFs) are promising for catalytic applications due to their ease of assembly, modular ability to encapsulate a variety of enzymes, and ease of separation from a reaction mixture. Here we demonstrate the reversal of the initially negative material charge through the titration of a positively charged polymer into the material, causing the reversal of guest molecule uptake and enzymatic activity of PMFs. The charge-inverse material partitions a charged enzyme substrate to concentrate the substrate near an enzyme incorporated within the material, generating up to 5.9-fold increases in enzyme activity toward the partitioned substrate over the excluded substrate. Here we also show that the polymer distributes heterogeneously through the material up to a point of saturation and the effects of guest macromolecules on the lattice parameters of PMFs.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Advanced Polymer Characterization: Modular Operations for Spectral Alignment by Iterative Compression (MOSAIC)

Matrix-assisted laser desorption/ionization (MALDI) mass spectrometry encodes structural information across diverse homo- and copolymer ensembles, yet decrypting these spectra requires a systematic analytical approach. We introduce Modular Operations for Spectral Alignment by Iterative Compression (MOSAIC)─a general cipher algorithm that applies modular arithmetic to filter monomer-derived mass contributions and cluster MALDI peaks by nonconstitutional repeating units (non-CRUs). MOSAIC performs sequential modular operations using monomer mass differences as base units to compress complex spectral data, revealing end-group distributions and comonomer incorporation. As a demonstration, we applied MOSAIC to five copolymers formed by two different polymerization mechanisms. Furthermore, the resulting remainder–mass plots clearly resolve polymer homologs with distinct non-CRUs into visually apparent clusters, enabling intuitive assignment of mass spectral features.

Wang, Hanlin M. [University of Illinois at Urbana−↗
DOE OSTI2026

A Unified Design Theory for Multi-Port Polyphase Transformers Enabling Scalable Power-Multiplexed EV Fleet Charging Systems

This paper presents a unified analytical design theory for multi-port polyphase transformers, targeting scalable and isolated high-power Electric Vehicle (EV) fleet charging systems with power multiplexing capability. As fleet electrification accelerates, conventional one-to-one charger architectures face significant challenges in infrastructure cost, peak power demand, and low utilization of installed power electronics. Power-multiplexed charging architectures, which dynamically distribute power from a shared pool of converter modules across multiple vehicles, have emerged as a promising solution. However, such architectures require scalable, isolated multi-port power interfaces capable of routing energy among multiple inputs and outputs, whose design remains complex and dependent on iterative modeling. To address this gap, the proposed theory provides closed-form expressions for self-inductance, leakage inductance, and mutual coupling terms for arbitrary multi-phase, multi-port transformer structures. The formulation enables direct synthesis of isolated multi-input and multi-output resonant converter systems without reliance on geometry-specific finite-element analysis or extensive parameter extraction. This capability is particularly critical for power-multiplexed systems, where modular converter structures must interface with multiple vehicles while maintaining galvanic isolation and flexible power allocation. The effectiveness of the proposed framework is demonstrated through the design of a 360 kW multi-phase system operating over a 700–900 VDC input and 400–1250 VDC output range. PLECS simulation results confirm accurate prediction of system behavior and validate the applicability of the approach to multi-port, power-multiplexed charging scenarios. The proposed method significantly reduces design complexity while enabling scalable, cost-effective, and fully utilized EV fleet charging infrastructure.

Asa, Erdem [ORNL] (ORCID:0000000190884812)↗
DOE OSTI2023

Sample Size Effects on Petrophysical Characterization and Fluid-to-Pore Accessibility of Natural Rocks

Laboratory-scale analysis of natural rocks provides petrophysical properties such as density, porosity, pore diameter/pore-throat diameter distribution, and fluid accessibility, in addition to the size and shape of framework grains and their contact relationship with the rock matrix. Different types of laboratory approaches for petrophysical characterization involve the use of a range of sample sizes. While the sample sizes selected should aim to be representative of the rock body, there are inherent limitations imposed by the analytical principles and holding capacities of the different experimental apparatuses, with many instruments only able to accept samples at the μm–mm scale. Therefore, a total of nine (three limestones, three shales, two sandstones, and one dolomite) samples were collected from Texas to fill the knowledge gap of the sample size effect on the resultant petrophysical characteristics. The sample sizes ranged from 3 cm cubes to <75 μm particles. Using a combination of petrographic microscopy, helium expansion pycnometry, water immersion porosimetry, mercury intrusion porosimetry, and (ultra-) small-angle X-ray scattering, the impact of sample size on the petrophysical properties of these samples was systematically investigated here. The results suggest that the sample size effect is influenced by both pore structure changes during crushing and sample size-dependent fluid-to-pore connectivity.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Milky Way Satellite Census. I. The Observational Selection Function for Milky Way Satellites in DES Y3 and Pan-STARRS DR1

We report the results of a systematic search for ultra-faint Milky Way satellite galaxies using data from the Dark Energy Survey (DES) and Pan-STARRS1 (PS1). Together, DES and PS1 provide multi-band photometry in optical/near-infrared wavelengths over ~80% of the sky. Our search for satellite galaxies targets ~25,000 deg$^2$ of the high-Galactic-latitude sky reaching a 10$\sigma$ point-source depth of $\gtrsim$ 22.5 mag in the $g$ and $r$ bands. While satellite galaxy searches have been performed independently on DES and PS1 before, this is the first time that a self-consistent search is performed across both data sets. We do not detect any new high-significance satellite galaxy candidates, while recovering the majority of satellites previously detected in surveys of comparable depth. We characterize the sensitivity of our search using a large set of simulated satellites injected into the survey data. We use these simulations to derive both analytic and machine-learning models that accurately predict the detectability of Milky Way satellites as a function of their distance, size, luminosity, and location on the sky. To demonstrate the utility of this observational selection function, we calculate the luminosity function of Milky Way satellite galaxies, assuming that the known population of satellite galaxies is representative of the underlying distribution. We provide access to our observational selection function to facilitate comparisons with cosmological models of galaxy formation and evolution.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Thermal Modeling and Limitations for Power Electronics Embedded in Medium-Voltage Cables

As next-generation energy technologies gain traction and power demand increases, the existing electrical infrastructure faces significant stress, prompting innovative solutions to enhance the grid's capacity and lifespan. This work explores the possibility of embedding medium-voltage (MV) power electronics directly inline with the cable, and the resulting thermal challenges. Since the majority of power distribution cables installed in the U.S. are passively cooled, the work focuses primarily on passive cooling, with an emphasis on the limitations of axial heat spreading within the cable. To date, literature on axial spreading of high incident heat loads on cables and cable environments is limited, typically reporting cases with <10 W of incident heat load. This work will explore the considerations, limits, and tradeoffs of cable-embedded heat loads significantly larger than the cable losses. Both external and internal effects are modeled analytically in nondimensional terms via a Biot number analysis, allowing fundamental limits and tradeoffs to be derived. The work culminates in the design and experimental validation of a cable-embedded thermal system capable of passively dissipating 300 W of heat from a coaxial SiC mosfet switch module over a length of 20 cm, thus validating the possibility of MV cable-embedded power electronics from a thermal standpoint.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2020

An analysis of the spatio-temporal resolution of the immersed boundary method with direct forcing

The immersed boundary method (IBM) with direct forcing is very popular in the simulation of rigid particulate flows. In the IBM, an interaction force is introduced at the interface between fluid and particle in order to approximate the no-slip boundary condition. The interaction force is calculated through dividing the velocity difference (or error) between fluid and particle at the interface by the time step. Here, a dynamic equation for the velocity difference is derived. Additionally, analyses on the dynamic equation provide a few new findings: (i) The interaction force is the solution of a least-squares error problem, with the direct implication that the Lagrangian marker distribution has no effect on the large scale flow structure once the distribution of Lagrangian markers become saturated along the interface (i.e., each marker remains properly correlated with all its neighbors); (ii) The Lagrangian volume-weight is a relaxation factor to control how fast the velocity error decays to the ideal value of zero; (iii) The optimal choice of the Lagrangian volume-weight is the largest value permissible by a stability condition. A comprehensive convergence analysis with regard to the spatial and temporal resolution is presented for the velocity error and also for the shear-stress and surface pressure. In three simple canonical problems, it is analytically and numerically shown that the IBM results converge to the theoretical solutions obtained with precise imposition of no-slip and no-penetration boundary conditions. It is observed that it is not necessary to match the Lagrangian marker volume-weight to that of the local Eulerian cell volume and in fact this matching leads to lower than optimal computational efficiency. However, it is found that extremely high Eulerian grid resolution and small time step have to be used to obtain high precision simulation results. Especially, the time step should be inversely proportional to the particle Reynolds number for low Reynolds number flows. For high frequency oscillation problems, the grid size needs to be reduced by a factor of the square root of the frequency, and the time step to be reduced by a factor of the frequency. The theoretical findings here can be used to alleviate the technical difficulties in simulating non-spherical particles by not requiring the Lagrangian marker distribution to match the Eulerian grids and also in the implementation of IBM on non-uniform Eulerian grids. The present work also provides simple practical guidance on the choice of temporal and spatial resolution so as to control the simulation error a priori.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Understanding solidification of near eutectic alloy using Cellular Automata (CA)

The solidification microstructure of an alloy strongly influences the mechanical properties. Dendritic and eutectic solidification are two important pathways of alloy solidification. The formation of phases, and their size and distribution depend on which pathway to take. In contrast to the Scheil model that has been widely used in describing the solidification pathways of alloys manufactured from conventional casting approach, more sophisticated solidification models are required to describe the state-of-the art manufacturing processes, such as additive manufacturing where distinctly different phase morphology, size and distribution than conventional casting can form, due to the vast difference in cooling conditions. Therefore, this work is to develop a computational Cellular Automata framework which includes the modeling of nucleation and growth of dendritic and eutectic solidification, as well as their competition as a function of alloy composition, undercooling and cooling rate. Further, after individual solidification models were validated against analytical solutions, the models were then combined to predict the competition between dendritic and eutectic solidification in alloys with off-eutectic compositions. This work predicts increased cooling rates suppress the dendritic solidification and promote eutectic solidification. Although quantitative validation is needed, this finding is consistent with qualitative observation in literature.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Scalable Bottom-Up Synthesis of Nanoporous Hexagonal Boron Nitride ( h -BN) for Large-Area Atomically Thin Ceramic Membranes

Nanopores embedded within monolayer hexagonal boron nitride (h-BN) offer possibilities of creating atomically thin ceramic membranes with unique combinations of high permeance (atomic thinness), high selectivity (via molecular sieving), increased thermal stability, and superior chemical resistance. However, fabricating size-selective nanopores in monolayer h-BN via scalable top-down processes remains nontrivial due to its chemical inertness, and characterizing nanopore size distribution over a large area remains extremely challenging. Here, we demonstrate a facile and scalable approach of exploiting the chemical vapor deposition (CVD) process temperature to enable direct incorporation of subnanometer/nanoscale pores into the monolayer h-BN lattice, in combination with manufacturing compatible polymer casting to fabricate centimeter-scale nanoporous atomically thin ceramic membranes. We leverage diffusive transport of analytes including size-selective Ficoll sieving to characterize subnanometer-scale and nanoscale defects that manifest as pores in centimeter-scale h-BN membranes, overcoming previous limitations in large-area characterization of nanoscale defects in h-BN. Our approach opens a new frontier to advance atomically thin membranes to 2D ceramic materials, such as h-BN via facile and direct formation of nanopores, for size-selective separations.

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

Sample preparation with sucrose cryoprotection dramatically alters Zn distribution in the rodent hippocampus, as revealed by elemental mapping

Transition metal ions (Fe, Mn, Cu, Zn) are essential for healthy brain function, but altered concentration, distribution, or chemical form of the metal ions has been implicated in numerous brain pathologies. Currently, it is not possible to image the cellular or sub-cellular distribution of metal ions in vivo and therefore, studying brain-metal homeostasis largely relies on ex vivo in situ elemental mapping. Sample preparation methods that accurately preserve the in vivo elemental distribution are essential if one wishes to translate the knowledge of elemental distributions measured ex vivo toward increased understanding of chemical and physiological pathways of brain disease. Furthermore, the choice of sample preparation is particularly important for metal ions that exist in a labile or mobile form, for which the in vivo distribution could be easily distorted by inappropriate sample preparation. One of the most widely studied brain structures, the hippocampus, contains a large pool of labile and mobile Zn. Herein, we describe how sucrose cryoprotection, the gold standard method of preparing tissues for immuno-histochemistry or immuno-fluorescence, which is also often used as a sample preparation method for elemental mapping studies, drastically alters hippocampal Zn distribution. Based on the results of this study, in combination with a comparison against the strong body of published literature that has used either rapid plunge freezing of brain tissue, or sucrose cryo-protection, we strongly urge investigators in the future to cease using sucrose cryoprotection as a method of sample preparation for elemental mapping, especially if Zn is an analyte of interest.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2026

Intrinsic nonlocality of spin- and polarization-resolved probabilities in strong-field quantum electrodynamics

Spin and polarization are central to precision tests of fundamental physics and for interpreting radiation from astrophysical sources and ultraintense laser-matter experiments. Here, focusing on the fundamental process of nonlinear Compton scattering, we demonstrate that a key assumption underlying current strong-field quantum electrodynamics models, i.e., that emission can be treated as an instantaneous random event sampled from a local differential rate, is inconsistent once emission angles, electron spin, and/or photon polarization are resolved. Namely, even in strictly constant and uniform fields , the resulting fully differential distribution is sign indefinite, yielding negative inferred probabilities. The physical reason is that the photon emission probability builds up over a finite length of the electron trajectory, the formation region, during which the electron direction changes by roughly the same small angle that defines the radiation cone. Therefore, we put forward a new method where we integrate over this formation region analytically to obtain a physically consistent electron spin and photon polarization model. We show that the implementation of our model is compatible with existing Monte Carlo and particle-in-cell workflows. Simulations of a GeV-class electron-laser collision accessible at current petawatt facilities and of emission in a pulsarlike magnetic field are shown to reveal spin and polarization patterns that differ even qualitatively from state-of-the-art local models. In particular, our new model predicts substantial angle-dependent circular photon polarization where the well-known collinear-emission approach yields none, and a pronounced helicity bias in the recoiling electrons absent from current predictions. These findings have direct implications for upcoming strong-field QED experiments and for interpreting polarized radiation from extreme astrophysical environments.

astrophysical electromagnetic fields↗
DOE OSTI2022

Modeling intrinsic galaxy alignment in the MICE simulation

The intrinsic alignment (IA) of galaxies is potentially a major limitation in deriving cosmological constraints from weak lensing surveys. In order to investigate this effect, we assign intrinsic shapes and orientations to galaxies in the light-cone output of the MICE simulation, spanning ~ 5000 deg 2 and reaching redshift Z = 1.4. This assignment is based on a semianalytic IA model that uses photometric properties of galaxies as well as the spin and shape of their host halos. Advancing on previous work, we include more realistic distributions of galaxy shapes and a luminosity-dependent galaxy-halo alignment. The IA model parameters are calibrated against COSMOS and BOSS LOWZ observations. The null detection of IA in observations of blue galaxies is accounted for by setting random orientations for these objects. We compare the two-point alignment statistics measured in the simulation against predictions from the analytical IA models NLA and TATT over a wide range of scales, redshifts, and luminosities for red and blue galaxies separately. We find that both models fit the measurements well at scales above 8h -1 Mpc, while TATT outperforms NLA at smaller scales. The IA parameters derived from our fits are in broad agreement with various observational constraints from red galaxies. Finally, we build a realistic source sample, mimicking DES Year 3 observations and use it to predict the IA contamination to the observed shear statistics. We find this prediction to be within the measurement uncertainty, which might be a consequence of the random alignment of blue galaxies in the simulation.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Integrated System Planning: Emerging Software Requirements in the Power Industry

Power system planning software remains fragmented across organizational boundaries, with specialized tools for capacity expansion, production cost modeling, power flow, and dynamic analysis operating on incompatible data models and assumptions. This article argues that the fragmentation is not merely a technical problem but a predictable consequence of Conway's law: software architectures mirror the departmental structures within which they are developed. Regulatory milestones like Federal Energy Regulatory Commission (FERC) Order 888 formalized these divisions, but the roots trace back to the distinct engineering disciplines-mechanical, chemical, and electrical-that staffed generation and transmission planning departments in vertically integrated utilities. As the industry moves toward integrated system planning (ISP) that coordinates generation, transmission, and distribution investment decisions, the software ecosystem must evolve accordingly. We identify five categories of software requirements to enable this transition: coherent data inputs decoupled from individual applications, unified and extensible data schemas, modular component representations that support multiple abstraction levels, lifecycle management of planning datasets, and well-defined application programming interface (API) contracts that separate data exchange from algorithmic control. We examine how these requirements interact with three common workflow patterns-serial gate clearing, sequential multiapplication, and convergence oriented-and discuss the interface design principles each demands. We then outline a vision for platform-based planning architectures where specialized analytical services compose through standardized interfaces and where artificial intelligence (AI)/machine learning (ML) tools augment decision support within a disciplined software infrastructure. The practices proposed here offer a path from today's siloed tool collections toward collaborative planning ecosystems capable of handling the complexity of modern power system transformation.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

What Is the Smallest Zeolite That Could Be Synthesized?**

Zeolites with a few unit cells are promising as catalyst and adsorbents. The quest to synthesize the smallest zeolites has recently resulted in 4 to 8 nm nanozeolites, about 2 to 4 unit cells. These findings pose the question of what is the smallest zeolite that could be obtained by hydrothermal synthesis. We address this question using molecular simulations and thermodynamic analysis. The simulations predict that amorphous precursors as small as 4 nm can crystallize zeolites, in agreement with the experiments. We find that interfacial forces dominate the structure of smaller particles, resulting in size-dependent compact isomers that have ring and pore distributions different from open framework zeolites. The instability of zeolites smaller than 3±0.5 nm precludes a classical mechanism of nucleation from solution or through assembly of small nanoslabs.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Cobalt‐Doped Bismuth Nanosheet Catalyst for Enhanced Electrochemical CO 2 Reduction to Electrolyte‐Free Formic Acid

Electrochemical carbon dioxide (CO 2 ) reduction reaction (CO 2 RR) to valuable liquid fuels, such as formic acid/formate (HCOOH/HCOO − ) is a promising strategy for carbon neutrality. Enhancing CO 2 RR activity while retaining high selectivity is critical for commercialization. To address this, we developed metal-doped bismuth (Bi) nanosheets via a facile hydrolysis method. These doped nanosheets efficiently generated high-purity HCOOH using a porous solid electrolyte (PSE) layer. Among the evaluated metal-doped Bi catalysts, Co-doped Bi demonstrated improved CO 2 RR performance compared to pristine Bi, achieving ~90 % HCOO − selectivity and boosted activity with a low overpotential of ~1.0 V at a current density of 200 mA cm −2 . In a solid electrolyte reactor, Co-doped Bi maintained HCOOH Faradaic efficiency of ~72 % after a 100-hour operation under a current density of 100 mA cm −2 , generating 0.1 M HCOOH at 3.2 V. Density functional theory (DFT) results revealed that Co-doped Bi required a lower applied potential for HCOOH generation from CO 2 , due to stronger binding energy to the key intermediates OCHO* compared to pure Bi. In conclusion, this study shows that metal doping in Bi nanosheets modifies the chemical composition, element distribution, and morphology, improving CO 2 RR catalytic activity performance by tuning surface adsorption affinity and reactivity.

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

Adsorption of furan, hexanoic acid, and alkanes in a hierarchical zeolite at reaction conditions: Insights from molecular simulations

Hierarchical zeolites containing both micropores and mesopores are valuable catalysts for facilitating reactions of large molecules. Furan acylation by fatty acids is a promising reaction for valorizing biomass, and the self-pillared pentasil (SPP) zeolite was found to perform particularly well for this reaction. To better understand the distribution of molecules in hierarchical zeolites at the elevated temperature (T = 523 K) and the elevated pressure (p >1 bar) associated with typical reaction conditions, unary and binary adsorption were predicted using Monte Carlo simulations in the isothermal–isobaric Gibbs ensemble. Adsorption of six species (furan, hexanoic acid, -hexane, -decane, -tetradecane, and 3,6-diethyloctane) was investigated from vapor, liquid, and supercritical phases, and loadings into the micropores, onto the mesopore surface, and in the mesopore interior of SPP were obtained. As pressure increases, -alkanes fill the micropores before loading the surface and then the interior of the mesopore, while furan and hexanoic acid adsorb strongly to the mesopore surface due to hydrogen bonding interactions with surface silanols. Hydrogen bonding interactions also draw hexanoic acid molecules in the micropore region toward the pore mouths, so their carboxylic acid group forms H-bonds with silanols, while the alkyl tails interact with the micropore walls. Mesopore condensation is observed for molecules below their critical point, and occurs when the Gibbs free energy of transfer into the mesopore interior and onto the mesopore surface converge. When hexanoic acid adsorption occurs in the presence of alkane solvents, then the selectivity and spatial distribution of hexanoic acid in the micropores and on the surface can be tuned by adjusting the fluid pressure and the alkane length and/or branching.

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

Structural Insights into Low and High Recalcitrance Natural Poplar Variants Using Neutron and X-ray Scattering

Here, we used small-angle neutron scattering (SANS) and wide-angle X-ray scattering (WAXS) to study two Populus trichocarpa variants, GW-11012 and BESC-316, with lignin contents of 17.8 and 23.2%, respectively, to understand why the low lignin variant has increased sugar release during liquid hot water (LHW) pretreatment. SANS showed differences in cell wall polymer arrangement in the variants. In GW-11012, the cellulose microfibrils were similar in size to BESC-316 but were not equally spaced and lignin was aggregated rather than homogenously interspersed with other matrix co-polymers. However, after LHW pretreatments, the SANS profiles of the pretreated variants were almost indistinguishable, indicating that the nanoscale structural features could not adequately explain the increased sugar release observed in GW-11012. WAXS analysis revealed higher cellulose crystallinity and larger crystallite size in native and pretreated GW-11012. However, the most significant difference was that the crystallite orientation was approximately 2-fold higher in GW-11012. This suggests that higher crystallite orientation may be favorable for the action of processive cellulases leading to increased sugar release from GW-11012. Our study shows that there is a complex interplay between cell wall polymers during their deposition resulting in changes in lignin distribution and cellulose structure, highlighting the importance of investigating plant cell wall architecture across multiple length scales.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗