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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Chromium Tolerant, Highly Active and Stable Electrocatalytic Internal Surface Coating for Cathode of Commercial SOFCs (Final Report)

This project is aimed to develop a chromium (Cr) tolerant, highly active, and stable coating layer on the internal surfaces of the porous composite cathode from commercially available SOFCs. Such coating layer was developed using the additive manufacturing process of Atomic Layer Deposition (ALD) and has been applied on the cathode consisting of either an electronic conductor of LaxSr 1-x MnyO 3-δ (LSM) or mixed ionic and electronic conducting La x Sr 1-x Co y Fe 1-y O 3-δ (LSCF). PI's work has demonstrated that the internal surface of cathode from the commercial cells, can be further tailored using ALD coating to dramatically enhance the cell performance. For instance, ALD layer consisting heterostructured nano composite of nano-Pt and nano-(Mn 0.8 Co 0.2 ) 3 O 4 oxide on the internal surface of porous LSM/YSZ cathode from SOFCs, has resulted in the large reduction of the cell polarizations resistance by up to 53%, and enormous increase of power density over 370%. For the cells with LSCF/Sm 2 O 3 doped CeO 2 (SDC) cathode, the conformal layer of nano-composite consisting of superjacent CoOx and subjacent minimum amount of Pt nano-grains has resulted in the power density enhancement by 126% for the large scale industry tubular cells at 750°C, and both the performance enhancement and nanostructure of the ALD layer are stable over ~ 2000 h continuous operation performed at industry test station. In the meanwhile, those ALD coating layer developed by PI's work is also inherently Cr-tolerant, and could act as physical barrier for preventing Cr diffusion into the cathode backbone, so as to mitigate the Cr poisoning effect on the cathode. In this project, the impact of Cr on the performance of those ALD coated commercial cells has been evaluated. Based on evolution of the cell performance, the ALD coating layer chemistry and ALD coating layer thickness has been optimized to maximize the overall Cr tolerance, cell power density and cell longevity. Different ALD coating has been applied onto the internal surface of LSM/YSZ and LSCF/SDC backbone respectively. The architecture/scaffold structures on the internal surface of different cathode, designed by this project, was catalogued and analyzed using High Resolution Transmission Electron Microscopy (HRTEM), and cell power/durability performance are assured via comprehensive electrochemical performance testing in the industry operation relevant conditions. The impact of the electrochemical operation current density, the water humidity, the cell operation temperature, and cell operation duration on the Cr tolerance of ALD coated cells has been systematically investigated. There is completely different nanostructure degradation mechanisms between LSM and LSCF cells induced by Cr contamination. For the LSCF/SDC baseline cell, With the Cr source, there is no apparent Sr surface segregation phase even for the baseline cell operated for 3000 h at 750 °C. With the Cr source, there is significant amorphous (SrCr)Ox phase accumulated in the original pore region. For the commercial baseline cells, Cr contaminants on the LSM electrode severely impacted the entire cell's electrochemical performance and nanostructure degradation. Those degradations include (1). Peak power density loss of 64 % after 109 h of operation. The dramatic increase in Rp (2). They are cracking at LSM/SSZ interface, LSM grains. SSZ remains intact but with (CrMn)Ox. By contrast, ALD coating (MnCo)Ox/Pt dramatically improves the Cr resistance, as follows (1). ALD-coated cell with a power density is 280-380 % of the baseline cell, depending on the ALD layer thickness. (2). For a cell with a 20 nm thick ALD layer, there is a large performance enhancement (> 200 % power density) induced by ALD coating of Cr-tolerant Mn 0.8 Co 0.2 Ox. (3). For a cell with a 20 nm thick ALD layer, after 168 h at 750 °C power density of the ALD-coated cell is ~ 600% of that baseline cell upon operation with Cr contamination for 109 h. The ALD coating on the internal surface of cathode developed by this project integrated multi-functions. Those multi-functions include (1). Dramatically improving the cell power density for the commercial cells; (2). Dramatically improving contamination resistance of the cathode, for being an excellent protection coating layer sealing off Cr contamination. (3). Dramatically increasing the cell longevity by potentially preventing the microstructure evolution and grain coarsening of the cathode. Overall, this project will provide a simple solution to simultaneously enhance power density and increase the reliability, robustness, and endurance of commercial SOFCs, over the entire operating temperature range of 650-800 °C. For the inherently functional SOFC, the ALD coating of LSM based cathode mitigate the Cr-contamination. Power density of ALD-coated cell is ~ 600% of that baseline cell upon operation with Cr contamination. In addition to SOFCs, the novel on-demand design approach and creation of multifunctional heterogeneous architecture on the electrode surface presented in this work opens further research for their application in other types of fuel cells, batteries, and sensors for which electrochemical reactions on the surface are similarly critical.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Material Properties from ab initio Calculations [Slides]

Summary and Future Direction: Diffusion coefficients - Calculated and compared with experimental viscosity; More work on solid state diffusion coefficients. Heat capacity - Calculated and compared with Ce heat capacity; Need to calculate the solid heat capacity and find more experimental data. Heat of formation - Calculated for one and two Ce atoms; Slab calculations are under way for a surface. Reaction rate - Calculation of a single Ce atom reaction is in progress; Simulations of larger clusters for comparison with ionic reaction experiments. Emissivity and Thermal Conductivity - Single snapshot calculated; Averaging over ionic configurations at different T is in progress.

36 MATERIALS SCIENCE↗
DOE OSTI2025

The Effect of Operational Temperature on the Performance and Durability of Solid Oxide Fuel Cells and Solid Oxide Electrolysis Cells

Solid oxide fuel cells (SOFC) and solid oxide electrolysis cells (SOEC) have received great interest due to their highly effective reversibility as power generation and H2 production system without releasing any greenhouse gases into environment. The LSCF electrode exhibits a higher structural and performance stability under both SOFC and SOEC operation due to its mixed ionic and electronic conductivity, and there is no immediate delamination taking place during the initial several hundred hours operation. However, the LSCF based air electrode still presents significant performance degradation (with the increased resistance) over the prolonged operation, such as over 1000 hours of operation under SOFC and SOEC. The influence factors for the cell’s performance and stability need to be optimized to improve the power generation for SOFC and H2 production for SOEC. The effects of operational temperature on the performance and durability for both SOFC and SOEC are electrochemical operation dependent. The performance and performance durability for the first 1500h were currently studied under optimized operational temperature for reversible SOFC/SOEC.

Fan, Yueying [NETL Site Support Contractor, Nation↗
NASA NTRS1990

Thermal management of advanced fuel cell power systems

It is shown that fuel cell devices are particularly attractive for the high-efficiency, high-reliability space hardware necessary to support upcoming space missions. These low-temperature hydrogen-oxygen systems necessarily operate with two-phase water. In either PEMFCs (proton exchange membrane fuel cells) or AFCs (alkaline fuel cells), engineering design must be critically focused on both stack temperature control and on the relative humidity control necessary to sustain appropriate conductivity within the ionic conductor. Water must also be removed promptly from the hardware. Present designs for AFC space hardware accomplish thermal management through two coupled cooling loops, both driven by a heat transfer fluid, and involve a recirculation fan to remove water and heat from the stack. There appears to be a certain advantage in using product water for these purposes within PEM hardware, because in that case a single fluid can serve both to control stack temperature, operating simultaneously as a heat transfer medium and through evaporation, and to provide the gas-phase moisture levels necessary to set the ionic conductor at appropriate performance levels. Moreover, the humidification cooling process automatically follows current loads. This design may remove the necessity for recirculation gas fans, thus demonstrating the long-term reliability essential for future space power hardware.

Vanderborgh, N. E.↗
NASA NTRS1996

Functional and Multifunctional Polymers: Materials for Smart Structures

The ultimate goal of the research in smart structures and smart materials is the development of a new generation of products/devices which will perform better than products/devices built from passive materials. There are a few examples of multilayer polymer systems which function as smart structures, e.g. a synthetic muscle which is a multilayer assembly of a poly(ethylene) layer, a gold layer, and a poly(pyrrole) layer immersed in a liquid electrolyte. Oxidation and reductions of the active pyrrole layer causes the assembly to reversibly deflect and mimic biological muscles. The drawback of such a setup is slow response times and the use of a liquid electrolyte. We have developed multifunctional polymers which will eliminate the use of a liquid electrolyte, and also because the functionalities of the polymers are within a few hundred angstroms, an improved response time to changes in the external field should be possible. Such multifunctional polymers may be classified as the futuristic 'smart materials.' These materials are composed of a number of different functionalities which work in a synergistic fashion to function as a device. The device performs on the application of an external field and such multifunctional polymers may be scientifically labeled as 'field responsive polymers.' Our group has undertaken a systematic approach to develop functional and multifunctional polymers capable of functioning as field responsive polymers. Our approach utilizes multicomponent polymer systems (block copolymers and graft copolymers), the strategy involves the preparation of block or graft copolymers where the functionalities are limited to different phases in a microphase separated system. Depending on the weight (or volume) fractions of each of the components, different microstructures are possible. And, because of the intimate contact between the functional components, an increase in the synergism between the functionalities may be observed. In this presentation, three examples of multifunctional polymers developed in our labs will be reported. The first class of multifunctional polymers are the microphase separated mixed (ionic and electronic) conducting or MIEC block copolymers. The second class being developed in our labs are the biocompatible conductive materials and the conductive fluids. The final class may be considered microwave active smart polymers.

Arnold, S.↗
NASA NTRS2005

Novel Applications of Magnetic Fields for Fluid Flow Control and for Simulating Variable Gravity Conditions

Static and dynamic magnetic fields have been used to control convection in many materials processing applications. In most of the applications, convection control (damping or enhancement) is achieved through the Lorentz force that can be tailored to counteract/assist dominant system flows. This technique has been successfully applied to liquids that are electrically conducting, such as high temperature melts of semiconductors, metals and alloys, etc. In liquids with low electrical conductivity such as ionic solutions of salts in water, the Lorentz force is weak and hence not very effective and alternate ways of flow control are necessary. If the salt in solution is paramagnetic then the variation of magnetic susceptibility with temperature and/or concentration can be used for flow control. For thermal buoyancy driven flows this can be accomplished in a temperature range below the Curie point of the salt. The magnetic force is proportional to the magnetic susceptibility and the product of the magnetic field and its gradient. By suitably positioning the experiment cell in the magnet, system flows can be assisted or countered, as desired. A similar approach can be extended to diamagnetic substances and fluids but the required magnetic force is considerably larger than that required for paramagnetic substances. The presentation will provide an overview of work to date on a NASA fluid physics sponsored project that aims to test the hypothesis of convective flow control using strong magnetic fields in protein crystal growth. The objective is to understand the nature of the various forces that come into play, delineate causative factors for fluid flow and to quantify them through experiments, analysis, and numerical modeling. The seminar will report specifically on the experimental results using paramagnetic salts and solutions in magnetic fields and compare them to analytical predictions. Applications of the concept to protein crystallization studies will be discussed. The use of strong magnetic fields for terrestrially simulating variable gravity environments and applications supporting the NASA Exploration Initiative will also be briefly discussed.

Ramachandran, N.↗
NASA NTRS2005

Rod-Coil Block Polyimide Copolymers

This invention is a series of rod-coil block polyimide copolymers that are easy to fabricate into mechanically resilient films with acceptable ionic or protonic conductivity at a variety of temperatures. The copolymers consist of short-rigid polyimide rod segments alternating with polyether coil segments. The rods and coil segments can be linear, branched or mixtures of linear and branched segments. The highly incompatible rods and coil segments phase separate, providing nanoscale channels for ion conduction. The polyimide segments provide dimensional and mechanical stability and can be functionalized in a number of ways to provide specialized functions for a given application. These rod-coil black polyimide copolymers are particularly useful in the preparation of ion conductive membranes for use in the manufacture of fuel cells and lithium based polymer batteries.

Mary Ann B Meador↗
NASA NTRS2020

Structural evolution, electrochemical kinetic properties, and stability of A-site doped perovskite Sr(1-x)Yb(x)CoO(3-δ)

Mixed ionic and electronic conducting (MIEC) perovskite SrCoO(3-δ) is a widely studied (electro)catalyst for the oxygen reduction reaction (ORR) and possesses different crystal structures at different temperatures. These temperature dependent phase transitions significantly impact the ordering of oxygen vacancies and electrochemical kinetic properties as well as the reliability of the related devices. Some of the crystal structures formed, e.g. hexagonal phases, turn out to be almost impermeable to oxygen gas. Therefore, it is important to stabilize the crystal structure of SrCoO(3-δ) that favors the ORR over a wide temperature range. Herein, the partial substitution of the A-site Sr with Yb is systematically studied, including synthesis, characterization and analysis of structural evolution, electrochemical kinetic properties, thermal stability, and stability in a CO2-containing atmosphere. The results indicate that Sr(0.90)Yb(0.10)CoO(3-δ) is able to stabilize the tetragonal crystal structure with less ordered oxygen vacancies, leading to polarization resistances of 0.051, 0.115 and 0.272 U sq.cm at 750, 700 and 650 °C, respectively, on symmetrical cells. Sr(0.90)Yb(0.10)CoO(3-δ) demonstrates a very stable surface oxygen vacancy distribution and electronic structure near oxygen vacancies but dissociation of adsorbed oxygen molecules into atomic oxygen is affected by surface Sr segregation, and polarization resistance degradation is mainly induced by surface Sr segregation. Furthermore, Sr(0.90)Yb(0.10)CoO(3-δ) exhibits excellent thermal stability as well as excellent recovery stability and improved polarization performance after a few pure air/CO2-containing air treatment cycles at 700 °C. However, a hysteresis behavior of polarization performance is observed at 650 °C during gas cycling treatment, which may cause long-term degradation of the Sr(0.90)Yb(0.10)CoO(3-δ) electrode. The different polarization behaviors during gas cycling treatment are induced by different sensitivities of the formed surface strontium carbonate and chemisorbed surface oxo-carbonaceous species to different operating temperatures.

electrochemical kinetic↗
DOE OSTI2017

Experimental Determination of Solubilities of Tri-calcium Di-Citrate Tetrahydrate [Ca 3 [C 3 H 5 O(COO) 3 ] 2 •4H 2 O] Earlandite in NaCl and MgCl 2 Solutions to High Ionic Strengths and Its Pitzer Model: Applications to Nuclear Waste Isolation and Other Low Temperature Environments

In this study, solubility measurements on tri-calcium di-citrate tetrahydrate [Ca 3 [C 3 H 5 O(COO) 3 ]2•4H 2 O, abbreviated as Ca 3 [Citrate] 2 •4H 2 O] as a function of ionic strength are conducted in NaCl solutions up to I = 5.0 mol•kg –1 and in MgCl 2 solutions up to I = 7.5 mol•kg –1 , at room temperature (22.5 ± 0.5°C). The solubility constant (log K$0\atop{sp}$) for Ca 3 [Citrate] 2 •4H 2 O and formation constant (logβ$0\atop{1}$) for Ca[C 3 H 5 O(COO) 3 ] – Ca 3 [C 3 H 5 O(COO) 3 ] 2 •4H 2 O (earlandite) = 3Ca 2+ + 2[C 3 H 5 O(COO) 3 ] 3– + 4H 2 O (1) Ca 2+ + [C 3 H 5 O(COO) 3 ] 3– = Ca[C 3 H 5 O(COO) 3 ] – (2) are determined as –18.11 ± 0.05 and 4.97 ± 0.05, respectively, based on the Pitzer model with a set of Pitzer parameters describing the specific interactions in NaCl and M g Cl 2 media.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2020

DOE-BU-26096-4

The infiltration of nanoparticle electrocatalysts into solid oxide fuel cell (SOFC) electrodes has been proven to produce a high density of electrochemically active sites, and reduce charge transfer polarization losses in SOFC electrodes. This is crucial for intermediate temperature operation, as these losses increase greatly at lower temperatures. Nickel-yttria stabilized zirconia (Ni-YSZ) cermets are low-cost, and exhibit excellent stability, but their main disadvantage stems from nickel coarsening and performance loss over their operational lifetimes. Infiltration of electrocatalyst nanoparticles has been shown to mitigate nickel coarsening and the consequent anode degradation. In this work, the effects of these infiltrants have been observed in a standard Ni-YSZ electrode. In addition to nickel, mixed ionic and electronic conducting (MIEC) phases have been infiltrated into Ni-YSZ scaffolds and their performance characterized using electrochemical impedance spectroscopy (EIS). Cross-sectional microscopy of fractured cells has been used to compare electrode microstructure and particle statistics. A model has been proposed to explain the origin of anode performance enhancement from nanoscale electrocatalysts.

Mo, Boshan↗
DOE OSTI2018

H/D Isotope Effect in the Conductivity of CaZr{sub 1 –} {sub x}Sc{sub x}O{sub 3 – α} in Reducing Atmospheres

The ionic (proton and deuteron) conductivity of the system CaZr{sub 1 –} {sub x}Sc{sub x}O{sub 3 – α} (x = 0.03–0.20) is studied experimentally in H{sub 2} + H{sub 2}O + N{sub 2} and D{sub 2} + D{sub 2}O + N{sub 2} reducing atmospheres at pH{sub 2}O = pD{sub 2}O = 3.2 kPa and in the temperature range of 600–900°C. This system exhibits a considerable H/D isotopic effect in conductivity, and its magnitude is estimated. Using theoretical elaborations, we also estimate the magnitude of the thermodynamic isotope effect in the solubility of H{sub 2}O/D{sub 2}O in the considered system under reducing conditions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Molecular Level Differences in Ionic Solvation and Transport Behavior in Ethylene Oxide-Based Homopolymer and Block Copolymer Electrolytes

Block copolymer electrolytes (BCE) such as polystyrene-block-poly(ethylene oxide) (SEO) blended with lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) and composed of mechanically robust insulating and rubbery conducting nanodomains are promising solid-state electrolytes for Li batteries. Here, we compare ionic solvation, association, distribution, and conductivity in SEO-LiTFSI BCEs and their homopolymer PEO-LiTFSI analogs toward a fundamental understanding of the maximum in conductivity and transport mechanisms as a function of salt concentration. Ionic conductivity measurements reveal that SEO-LiTFSI and PEO-LiTFSI exhibit similar behaviors up to a Li/EO ratio of 1/12, where roughly half of the available solvation sites in the system are filled, and conductivity is maximized. As the Li/EO ratios increase to 1/5 the conductivity, of the PEO-LiTFSI drops nearly 3-fold, while the conductivity of SEO-LiTFSI remains constant. FTIR spectroscopy reveals that additional Li cations in the homopolymer electrolyte are complexed by additional EO units when the Li/EO ratio exceeds 1/12, while in the BCE, the proportion of complexed and uncomplexed EO units remains constant; Raman spectroscopy data at the same concentrations show that Li cations in the SEO-LiTFSI samples tend to coordinate more to their counteranions. Atomistic-scale molecular dynamics simulations corroborate these results and further show that associated ions tend to segregate to the SEO-LiTFSI domain interfaces. The opportunity for "excess" salt to be sequestered at BCE interfaces results in the retention of an optimum ratio of uncompleted and complexed PEO solvation sites in the middle of the conductive nanodomains of the BCE and maximized conductivity over a broad range of salt concentrations.

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

Interactive multiscale modeling to bridge atomic properties and electrochemical performance in Li-CO 2 battery design

Li-CO 2 batteries are promising energy storage systems due to their high theoretical energy density and CO 2 fixation capability, relying on reversible Li 2 CO 3 /C formation during discharge/charge cycles. Here, we present a multiscale modeling framework integrating Density Functional Theory (DFT), Ab-Initio Molecular Dynamics (AIMD), classical Molecular Dynamics (MD), and Finite Element Analysis (FEA) to investigate atomic and cell-level properties. The considered Li-CO 2 battery consists of a lithium metal anode, an ionic liquid electrolyte, and a carbon cloth cathode with Sb 0.67 Bi 1.33 Te 3 catalyst. DFT and AIMD determined the electrical conductivities of Sb 0.67 Bi 1.33 Te 3 and Li 2 CO 3 using the Kubo–Greenwood formalism and studied the CO 2 reduction mechanism on the cathode catalyst. MD simulations calculated the CO 2 diffusion coefficient, Li + transference number, ionic conductivity, and Li + solvation structure. The FEA model, parameterized with atomistic simulation data, reproduced the available experimental voltage–capacity profile at 1 mA/cm 2 and revealed spatio-temporal variations in Li 2 CO 3 /C deposition, porosity, and CO 2 concentration dependence on discharge rates in the cathode. Accordingly, Li 2 CO 3 can form large and thin film deposits, leading to dispersed and local porosity changes at 0.1 mA/cm 2 and 1 mA/cm 2 , respectively. The capacity decreases exponentially from 81,570 mAh/g at 0.1 mA/cm 2 to 6200 mAh/g at 1 mA/cm 2 , due to pore clogging from excessive discharge product deposition that limits CO 2 transport to the cathode interior. Therefore, the performance of Li-CO 2 batteries can be improved by enhancing CO 2 transport, regulating Li 2 CO 3 deposition, and optimizing cathode architecture.

Battery performance↗
DOE OSTI2025

Harnessing ionic complexity: A modeling approach for hierarchical ionic circuit design

Since the 1950s, soft ionic devices have evolved from individual components to an expanding library of sensors, actuators, signal transmitters, and processors. However, integrating these components into complex, multifunctional systems remains challenging due to the nonintuitive and nonlinear interactions between ionic elements. In this work, we address these fundamental challenges by developing a lumped element model that enables interrogation of the physics that governs ionic circuits, as well as rapid design and optimization. Our model captures features specific to ionic charge carriers, while preserving the hierarchical design flexibility and computational efficiency of traditional circuit modeling. We demonstrate that our model can not only fit individual device behavior but also accurately predict the behavior of larger circuits formed by combining those devices. Additionally, we show how our tool utilizes the intrinsic nonlinearities of ionic systems to enable extended functionality, revealing how factors such as ion enrichment, ion leakage, and polymer charge density influence performance. Lastly, we present a fully ionic power supply, sensor, control system, and actuator for a soft robot that adapts its motion in response to environmental salt, illustrating the tool’s potential to accelerate advancements in chemical sensing, biointerfacing, biomimetic systems, and adaptive materials.

42 ENGINEERING↗
DOE OSTI2018

The Relative Importance of Saturated Silica Sand Interfacial and Pore Fluid Geochemistry on the Spectral Induced Polarization Response

Adsorption at the solid-pore fluid interface is a key mechanism controlling the mobility of nutrients and contaminants in subsurface soils and sediments. The spectral induced polarization (SIP) method has been shown to be sensitive to the quantity and type of adsorbed ions. Extending previous results, we investigated the relevance of pH, solution conductivity, and ion type on the SIP response of saturated silica sand. We also performed adsorption experiments to evaluate whether adsorption plays a relevant role on the effect of saturating solution conductivity and pH. Given their environmental relevance and different electrochemical characteristics, we focused on exploring the influence of Cu 2+ and Na + adsorption on the SIP signature. The adsorption results confirm the expected and modeled pH influence on the adsorption of both Cu 2+ and Na + . The measured quadrature conductivity spectra indicate that pH and solution conductivity control the electrical double layer electrochemical state and its capacitive behavior. On the contrary, no appreciable SIP signal changes are associated with ion substitution. The adsorption experiments highlight low values of site occupancy for Na and Cu, which suggests that the effects of pH and fluid conductivity are unrelated to their control on the ion adsorption. We interpret the solution conductivity as a proxy for ionic strength. The relative importance of pH and solution conductivity over ion type helps to further constrain the interpretation of SIP results in field geochemical and biogeochemical characterization and monitoring.

58 GEOSCIENCES↗
DOE OSTI2024

Thermally sensitive state change ionic redox transistor

A thermally sensitive ionic redox transistor comprises a channel, a reservoir layer, and an electrolyte layer disposed between the channel and the reservoir layer. A conductance of the channel is varied by changing concentration of ions in the channel layer. The electrolyte layer is configured to undergo a state change at a state transition temperature. Below the state transition temperature, ions in the electrolyte layer are substantially immobile. Above the state transition temperature, ions can move freely between the reservoir layer and the channel across the electrolyte layer in response to a voltage being applied between the channel and the reservoir layer. When the device is cooled below the state transition temperature or temperature range, the ions are trapped in one or more of the layers because the electrolyte layer loses its ionic conductivity. A state of the redox transistor can be read by measuring the conductance of the channel.

Ashby, David Scott↗