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At least 433 records · Page 24

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DOE OSTI2024

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O ( n log n ) instead of the typical O ( n 3 ) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Finally, although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

Seasonal and CO 2 -Induced Shifts of the ITCZ: Testing Energetic Controls in Idealized Simulations with Comprehensive Models

The Tropical Rain Belts with an Annual Cycle and Continent Model Intercomparison Project (TRACMIP) ensemble—a multimodel ensemble of slab-ocean simulations in idealized configurations—provides a test of the relationship between the zonal mean ITCZ and the cross-equatorial atmospheric energy transports (AHT eq ). In a gross sense, the ITCZ position is linearly related to AHT eq , as expected from the energetic framework. Yet, in many aspects, the TRACMIP model simulations do not conform to the framework. Throughout the annual cycle there are large excursions in the ITCZ position unrelated to changes in the AHT eq and, conversely, substantial variations in the magnitude of the AHT eq while the ITCZ is stationary at its northernmost position. Variations both in the net vertical energy input at the ITCZ location and in the vertical profile of ascent play a role in setting the model behavior apart from the conceptual framework. Nevertheless, a linear fit to the ITCZ–AHT eq relationship captures a substantial fraction of the seasonal variations in these quantities as well as the intermodel or across-climate variations in their annual mean values. The slope of the ITCZ–AHT eq linear fit for annual mean changes across simulations with different forcings and configurations varies in magnitude and even sign from model to model and we identify variations in the vertical profile of ascent as a key factor. A simple sea surface temperature–based index avoids the complication of changes in the vertical structure of the atmospheric circulation and provides a more reliable diagnostic for the ITCZ position.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

Integrated hydrogeophysical modelling and data assimilation for geoelectrical leak detection

Time-lapse electrical resistivity tomography (ERT) measurements provide indirect observations of hydrological processes in the Earth's shallow subsurface at high spatial and temporal resolution. ERT has been used in the past decades to detect leaks and monitor the evolution of associated contaminant plumes. Specifically, inverted resistivity images allow visualization of the dynamic changes in the structure of the plume. However, existing methods do not allow the direct estimation of leak parameters (e.g. leak rate, location, etc.) and their uncertainties. We propose an ensemble-based data assimilation framework that evaluates proposed hydrological models against observed time-lapse ERT measurements without directly inverting for the resistivities. Each proposed hydrological model is run through the parallel coupled hydro-geophysical simulation code PFLOTRAN-E4D to obtain simulated ERT measurements. The ensemble of model proposals is then updated using an iterative ensemble smoother. In this paper, we demonstrate the proposed framework on synthetic and field ERT data from controlled tracer injection experiments. Our results show that the approach allows joint identification of contaminant source location, initial release time, and solute loading from the cross-borehole time-lapse ERT data, alongside with an assessment of uncertainties in these estimates. We demonstrate a reduction in site-wide uncertainty by comparing the prior and posterior plume mass discharges at a selected image plane. This framework is particularly attractive to sites that have previously undergone extensive geological investigation (e.g., nuclear sites). It is well suited to complement ERT imaging and we discuss practical issues in its application to field problems.

58 GEOSCIENCES↗

DOE OSTI2025

Uncertainty of 21st Century western U.S. snowfall loss derived from regional climate model large ensemble

Abstract The western United States is dependent on winter snowfall over its major mountain ranges, which gradually melts each year, serving as a natural reservoir for water resources. In a future warmer climate, much of this snowfall could be replaced by rain, making it more challenging to capture and store water. In this study, we utilize an ensemble of dynamically downscaled simulations forced by 14 global climate models (GCMs). These GCMs project wildly different futures, in terms of both temperature and precipitation change, producing significant uncertainty in snowfall projections. Here we exploit the robust statistics of the downscaled ensemble, and diagose the sensitivity of end-of-century snowfall loss across the region to both warming and regional wetting/drying in the driving GCM. The windward slopes of the Sierra Nevada and Cascades are particularly sensitive to warming (losing ~ 15% annual snowfall per degree warming), with little influence of precipitation. By contrast, snowfall loss in the inter-mountain west is less sensitive to warming (~ 5% K −1 ), but is significantly offset/exacerbated by precipitation changes (~ 0.5% snow per 1% precipitation). Combining such sensitivities with the warming and regional precipitation signals in the full CMIP6 ensemble, we can fully quantify likely snowfall loss and its uncertainty at any location, for any emissions scenario. We find that the western U.S. as a whole will lose 34 ± 8% of its total volumetric snowfall by end-of-century under the high-emissions SSP3-7.0 scenario, but 25 ± 6% and 17 ± 6% under the lower-emissions SSP2-4.5 and SSP1-2.6 scenarios.

Norris, Jesse (ORCID:0000000289883326)↗

DOE OSTI2022

Accurate predictions from small boxes: variance suppression via the Zel'dovich approximation

Simulations have become an indispensable tool for accurate modelling of observables measured in galaxy surveys, but can be expensive if very large dynamic range in scale is required. We describe how to combine Lagrangian perturbation theory models with N-body simulations to reduce the effects of finite computational volume in the prediction of ensemble average properties in the simulations within the context of control variates. In particular we use the fact that Zel'dovich displacements, computed during initial condition generation for any simulation, correlate strongly with the final density field. Since all the correlators of biased tracers can be computed with arbitrary precision for these displacements, pairing the Zel'dovich `simulation' with the N-body realization allows hundredfold reductions in sample variance for power spectrum or correlation function estimation. Additionally, Zel'dovich control variates can accurately extend matter or tracer field emulators to larger scales than previously possible, as well as improving measurements of statistics in simulations which are inherently limited to small volumes, such as hydrodynamical simulations of galaxy formation and reionization.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2002

Cloud-Resolving Model Simulations of LBA Convective Systems: Easterly and Westerly Regimes

The 3D Goddard Cumulus Ensemble (GCE) model was used to simulate convection that occurred during the TRMM LBA field experiment in Brazil. Convection in this region can be categorized into two different regimes. Low-level easterly flow results in moderate to high CAPE and a drier environment. Convection is more intense like that seen over continents. Low-level westerly flow results in low CAPE and a moist environment. Convection is weaker and more widespread characteristic of oceanic or monsoon-like systems. The GCE model has been used to study both regimes in order to provide cloud data sets that are representative of both environments in support of TRMM rainfall and heating algorithm development. Two different case are presented: Jan 26,1999, an easterly regime case, and Feb 23,1999, a westerly regime case. The Jan 26 case is an organized squall line and is initialized with a standard cold pool. The sensitivity to mid-level sounding moisture and wind shear will also be shown. The Feb 23 case is less-organized with only transient lines and is initialized with either warm bubbles or prescribed surface fluxes. Heating profiles, rainfall statistics and storm characteristics are compared and validated for the two cases against observations collected during the experiment.

Lang, Stephen E.↗

DOE OSTI2024

Portable, heterogeneous ensemble workflows at scale using libEnsemble

libEnsemble is a Python-based toolkit for running dynamic ensembles, developed as part of the DOE Exascale Computing Project. The toolkit utilizes a unique generator–simulator–allocator paradigm, where generators produce input for simulators, simulators evaluate those inputs, and allocators decide whether and when a simulator or generator should be called. The generator steers the ensemble based on simulation results. Generators may, for example, apply methods for numerical optimization, machine learning, or statistical calibration. libEnsemble communicates between a manager and workers. Flexibility is provided through multiple manager–worker communication substrates each of which has different benefits. These include Python’s multiprocessing, mpi4py, and TCP. Multisite ensembles are supported using Balsam or Globus Compute. We overview the unique characteristics of libEnsemble as well as current and potential interoperability with other packages in the workflow ecosystem. We highlight libEnsemble’s dynamic resource features: libEnsemble can detect system resources, such as available nodes, cores, and GPUs, and assign these in a portable way. These features allow users to specify the number of processors and GPUs required for each simulation; and resources will be automatically assigned on a wide range of systems, including Frontier, Aurora, and Perlmutter. Such ensembles can include multiple simulation types, some using GPUs and others using only CPUs, sharing nodes for maximum efficiency. We also describe the benefits of libEnsemble’s generator–simulator coupling, which easily exposes to the user the ability to cancel, and portably kill, running simulations based on models that are updated with intermediate simulation output. We demonstrate libEnsemble’s capabilities, scalability, and scientific impact via a Gaussian process surrogate training problem for the longitudinal density profile at the exit of a plasma accelerator stage. In conclusion, the study uses gpCAM for the surrogate model and employs either Wake-T or WarpX simulations, highlighting efficient use of resources that can easily extend to exascale.

Dynamic ensembles↗

NASA NTRSDate not supplied

A Framework for Evaluating Climate Model Performance Metrics

The CMIP5 archive contains future climate projections from over 50 models provided by dozens of modeling centers from around the world. Individual model projections, however, are subject to biases created by structural model uncertainties. As a result, ensemble averaging of multiple models is often used to add value to model projections: consensus projections have been shown to consistently outperform individual models. Previous reports for the IPCC establish climate change projections based on an equal-weighted average of all model projections. However, certain models reproduce climate processes better than other models. Should models be weighted based on performance? Unequal ensemble averages have previously been constructed using a variety of mean state metrics. What metrics are most relevant for constraining future climate projections? This project develops a framework for systematically testing metrics in models to identify optimal metrics for unequal weighting multi-model ensembles. A unique aspect of this project is the construction and testing of climate process-based model evaluation metrics. A climate process-based metric is defined as a metric based on the relationship between two physically related climate variables?e.g., outgoing longwave radiation and surface temperature. Metrics are constructed using high-quality Earth radiation budget data from NASA's Clouds and Earth's Radiant Energy System (CERES) instrument and surface temperature data sets. It is found that regional values of tested quantities can vary significantly when comparing weighted and unweighted model ensembles. For example, one tested metric weights the ensemble by how well models reproduce the time-series probability distribution of the cloud forcing component of reflected shortwave radiation. The weighted ensemble for this metric indicates lower simulated precipitation (up to .7 mm/day) in tropical regions than the unweighted ensemble: since CMIP5 models have been shown to overproduce precipitation, this result could indicate that the metric is effective in identifying models which simulate more realistic precipitation. Ultimately, the goal of the framework is to identify performance metrics for advising better methods for ensemble averaging models and create better climate predictions.

Noel C Baker↗

DOE OSTI2025

Combined Meteorological and Hydrologic Uncertainties Shape Projections of Future Soil Moisture in the Eastern United States

Physical hazards pose risks to many critical systems. Designing adaptive measures to mitigate these risks is challenging due to large uncertainties in modeling future hazards and the associated sectoral responses. Here, we help address this challenge in a hydrologic context by examining the combined role of meteorological forcing and hydrologic parameter uncertainties in shaping projections of future soil moisture. By encoding a simple conceptual water balance model in a differentiable programming framework, we facilitate fast runtimes and an efficient calibration, enabling an improved uncertainty analysis. We characterize uncertainty in model parameters by calibrating against different target data sets and by using several loss functions. We then convolve the resulting parameter ensemble with a set of Earth system model projections to produce a large ensemble (2,340 members) of daily soil moisture simulations. Focusing on the eastern United States, we find that most ensemble members project a drying of soils across the region, although some simulate wetter conditions throughout this century. Our ensemble shows an increase in the frequency and intensity of dry extremes while there is less agreement for wet extremes. We conduct sensitivity analyses on several soil moisture signatures to measure the relative influence of meteorological and hydrologic uncertainties across space and time. Both meteorological and hydrologic factors contribute consistently to uncertainty surrounding long-term trends, while changes to both wet and dry soil extremes are typically more sensitive to hydrologic parameter uncertainty. Our results underscore the need to account for varied sources of uncertainty when developing long-term hydrometeorological projections.

Lafferty, David C. [University of Illinois Urbana‐↗

DOE OSTI2020

Future Warming and Intensification of Precipitation Extremes: A “Double Whammy” Leading to Increasing Flood Risk in California

This study focuses on quantifying future anthropogenic changes in surface runoff associated with extreme precipitation in California's Sierra Nevada. The method involves driving a land surface model with output from a high resolution regional atmospheric simulation of the most extreme atmospheric rivers (ARs). AR events were selected from an ensemble of global climate model simulations of historical and late 21st century climate under the “high-emission” RCP8.5 scenario. Average precipitation during the future ARs increases by ~25% but a much lower proportion falls as snow. The resulting future runoff increase is dramatic—nearly 50%, reflecting both the precipitation increase and simultaneous conversion of snow to rain. The “double whammy” impact on runoff is largest in the 2,000–2,500 m elevation band, where the snowfall loss and precipitation increase are both especially large. Overall, this huge increase in runoff during the most extreme AR events could present major flood control challenges for the region.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Monte carlo simulations and experiments of all-silica zeolite LTA assembly combining structure directing agents that match cage sizes

In this work, we investigated the influence of organic structure-directing agents (OSDAs) on the formation rates of all-silica zeolite LTA using both simulations and experiments, to shed light on the crystallization process. We compared syntheses using one OSDA with a diameter close to the size of the large cavity in LTA, and two OSDAs of diameters matching the sizes of both the small and large LTA cavities. Reaction-ensemble Monte Carlo (RxMC) simulations predict a speed up of LTA formation using two OSDAs matching the LTA pore sizes; this qualitative result is confirmed by experimental studies of crystallization kinetics, which find a speedup in all-silica LTA crystallization of a factor of 3. Analyses of simulated rings and their Si–O–Si angular energies during RxMC crystallizations show that all ring sizes in the faster crystallization exhibit lower angular energies, on average, than in the slower crystallization, explaining the origin of the speedup through packing effects.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS2021

Improved Representation of the Global Dust Cycle Using Observational Constraints on Dust Properties and Abundance

Even though desert dust is the most abundant aerosol by mass in Earth’s atmosphere, atmospheric models 25 struggle to accurately represent its spatial and temporal distribution. These model errors are partially caused byfundamental difficulties in simulating dust emission in coarse-resolution models and in accurately representing dust microphysical properties. Here we mitigate these problems by developing a new methodology that yields an improved representation of the global dust cycle. We present an analytical framework that uses inverse modeling to integrate an ensemble of global model simulations with observational constraints on the dust size distribution, 30 extinction efficiency, and regional dust aerosol optical depth. We then compare the inverse model results against independent measurements of dust surface concentration and deposition flux and find that errors are reduced by approximately a factor of two relative to current model simulations of the Northern Hemisphere dust cycle. The inverse model results show smaller improvements in the less dusty Southern Hemisphere, most likely because both the model simulations and the observational constraints used in the inverse model are less accurate. On a global 35 basis, we find that the emission flux of dust with geometric diameter up to 20 μm (PM20) is approximately 5,000 Tg/year, which is greater than most models account for. This larger PM20 dust flux is needed to match observational constraints showing a large atmospheric loading of coarse dust. We obtain gridded data sets of dust emission, vertically integrated loading, dust aerosol optical depth, (surface) concentration, and wet and dry deposition fluxes that are resolved by season and particle size. As our results indicate that this data set is more accurate than current 40 model simulations and the MERRA-2 dust reanalysis product, it can be used to improve quantifications of dust impacts on the Earth system.

aerosols, dust↗

NASA NTRS2021

Improved representation of the global dust cycle using observational constraints on dust properties and abundance

Even though desert dust is the most abundant aerosol by mass in Earth's atmosphere, atmospheric models struggle to accurately represent its spatial and temporal distribution. These model errors are partially caused by fundamental difficulties in simulating dust emission in coarse-resolution models and in accurately representing dust microphysical properties. Here we mitigate these problems by developing a new methodology that yields an improved representation of the global dust cycle. We present an analytical framework that uses inverse modeling to integrate an ensemble of global model simulations with observational constraints on the dust size distribution, extinction efficiency, and regional dust aerosol optical depth. We then compare the inverse model results against independent measurements of dust surface concentration and deposition flux and find that errors are reduced by approximately a factor of 2 relative to current model simulations of the Northern Hemisphere dust cycle. The inverse model results show smaller improvements in the less dusty Southern Hemisphere, most likely because both the model simulations and the observational constraints used in the inverse model are less accurate. On a global basis, we find that the emission flux of dust with a geometric diameter up to 20 µm (PM20) is approximately 5000 Tg yr−1, which is greater than most models account for. This larger PM20 dust flux is needed to match observational constraints showing a large atmospheric loading of coarse dust. We obtain gridded datasets of dust emission, vertically integrated loading, dust aerosol optical depth, (surface) concentration, and wet and dry deposition fluxes that are resolved by season and particle size. As our results indicate that this dataset is more accurate than current model simulations and the MERRA-2 dust reanalysis product, it can be used to improve quantifications of dust impacts on the Earth system.

Dust cycle↗

DOE OSTI2025

Vertical canopy gradients of respiration drive plant carbon budgets and leaf area index

Despite its importance for determining global carbon fluxes, leaf respiration remains poorly constrained in land surface models (LSMs). We tested the sensitivity of the Energy Exascale Earth System Model Land Model – Functionally Assembled Terrestrial Ecosystem Simulator (ELM-FATES) to variation in the canopy gradients of leaf maintenance respiration (R dark ). We ran global and point simulations varying the canopy gradient of R dark to explore the impacts on forest structure, composition, and carbon cycling. In global simulations, steeper canopy gradients of R dark lead to increased understory survival and leaf biomass. Leaf area index (LAI) increased up to 77% in tropical regions compared with the default parameterization, improving alignment with remotely sensed benchmarks. Global vegetation carbon varied from 308 Pg C to 449 Pg C across the ensemble. In tropical forest simulations, steeper gradients of R dark had a large impact on successional dynamics. Results show the importance of canopy gradients in leaf traits and fluxes for determining plant carbon budgets and emergent ecosystem properties such as competitive dynamics, LAI, and vegetation carbon. The high-model sensitivity to canopy gradients in R dark highlights the need for more observations of how leaf traits and fluxes vary along light micro-environments to inform critical dynamics in LSMs.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2024

Reweighting Monte Carlo predictions and automated fragmentation variations in Pythia 8

This work reports on a method for uncertainty estimation in simulated collider-event predictions. The method is based on a Monte Carlo-veto algorithm, and extends previous work on uncertainty estimates in parton showers by including uncertainty estimates for the Lund string-fragmentation model. This method is advantageous from the perspective of simulation costs: a single ensemble of generated events can be reinterpreted as though it was obtained using a different set of input parameters, where each event now is accompanied with a corresponding weight. This allows for a robust exploration of the uncertainties arising from the choice of input model parameters, without the need to rerun full simulation pipelines for each input parameter choice. Such explorations are important when determining the sensitivities of precision physics measurements. Accompanying code is available at https://gitlab.com/uchep/mlhad-weights-validation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

Evaluation of historical precipitation interannual variability in CMIP6 over the United States

Interannual precipitation variability profoundly influences society via its effects on agriculture, water resources, infrastructure, and disaster risks. In this study, we use daily in situ precipitation observations from the global historical climatology network-daily (GHCN-D) to assess the ability of 21 Coupled Model Intercomparison Project Phase 6 (CMIP6) models, including the 50-member fifth-generation Canadian Earth System Model single model initial-condition large ensemble (CanESM5_SMILE), to realistically simulate historical interannual precipitation variability trends within 17 regions of the contiguous United States (CONUS). We assess how accurately the CMIP6 simulations align with observational data across annual, summer, and winter periods, focusing on four key hydrometeorological metrics, including interannual precipitation variability, relative interannual precipitation variability (coefficient of variation), annual mean precipitation, and annual wet day frequency. Our findings reveal that CMIP6 ensemble members generally reproduce the spatial patterns of observed trends in annual mean precipitation. In most regions, models agree well with the signs of observed changes in annual mean precipitation, though discrepancies in trend magnitude are evident. Further, observed trends in winter mean precipitation broadly exhibit a spatial pattern similar to that of the observed annual mean. However, analysis of the CanESM5_SMILE shows that trends in precipitation variability may primarily be the result of model-simulated internal variability, suggesting caution in interpreting multi-model single-realization ensemble results. Challenges in accurately simulating interannual precipitation variability underscore the need for ongoing model refinement and validation to enhance climate projections, especially in regions vulnerable to extreme precipitation events.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Flower‐Type Organized Trade‐Wind Cumulus: A Multi‐Day Lagrangian Large Eddy Simulation Intercomparison Study

Shallow cumulus cloud fields in subtropical marine trade wind environments, particularly over the tropical Atlantic Ocean, show distinct organizational patterns. Among these, Flower‐type clouds are characterized by expansive stratiform cloud patches surrounded by regions of scattered convection. The objectives of this study were (a) to construct a case study of a time period during the EUREC 4 A/ATOMIC field campaign when Flower‐type organization was observed, (b) to evaluate the fidelity of a multi‐model ensemble of large eddy simulations of that case, and (c) to analyze the interaction between cloud and precipitation processes and mesoscale organization in the simulations. The simulations follow a quasi‐Lagrangian trajectory, allowing mesoscale features to develop over time in a domain that follows the boundary‐layer airmass. The results show a broad agreement in simulated thermodynamic properties across different LES codes, with Flower‐type cloud patches appearing within hours of each other. The consensus among models is consistent with observations made during the EUREC 4 A/ATOMIC field campaign on the specific day of interest. The cloud structure reveals three distinct peaks in the joint probability densities of cloud base and cloud top height, with the dominant peak at any given time influenced by the stage of cloud organization. The simulated cloud system evolution reveals consistent occurrence of maxima in liquid water path and rain rate before Flower reaches its maximum length scale. Targeted sensitivity tests reveal a weak relationship between Cloud Droplet Number concentration and the extent/degree/type of organization.

EUREC4A↗

DOE OSTI2024

Reversible Surface Energy Storage in Molecular-Scale Porous Materials

Forcible wetting of hydrophobic pores represents a viable method for energy storage in the form of interfacial energy. The energy used to fill the pores can be recovered as pressure–volume work upon decompression. For efficient recovery, the expulsion pressure should not be significantly lower than the pressure required for infiltration. Hysteresis of the wetting/drying cycle associated with the kinetic barrier to liquid expulsion results in energy dissipation and reduced storage efficiency. In the present work, we use open ensemble (Grand Canonical) Monte Carlo simulations to study the improvement of energy recovery with decreasing diameters of planar pores. Near-complete reversibility is achieved at pore widths barely accommodating a monolayer of the liquid, thus minimizing the area of the liquid/gas interface during the cavitation process. At the same time, these conditions lead to a steep increase in the infiltration pressure required to overcome steric wall/water repulsion in a tight confinement and a considerable reduction in the translational entropy of confined molecules. In principle, similar effects can be expected when increasing the size of the liquid particles without altering the absorbent porosity. While the latter approach is easier to follow in laboratory work, we discuss the advantages of reducing the pore diameter, which reduces the cycling hysteresis while simultaneously improving the stored-energy density in the material.

36 MATERIALS SCIENCE↗