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At least 55 records · Page 3

Substituent and Heteroatom Effects on π–π Interactions: Evidence That Parallel-Displaced π-Stacking is Not Driven by Quadrupolar Electrostatics

Stacking interactions are a recurring motif in supramolecular chemistry and biochemistry, where a persistent theme is a preference for parallel-displaced aromatic rings rather than face-to-face π-stacking. This is usually explained in terms of quadrupole–quadrupole interactions between the arene moieties but that interpretation is inconsistent with accurate calculations, which reveal that the quadrupolar picture is qualitatively wrong. At typical π-stacking distances, quadrupolar electrostatics may differ in sign from an exact calculation based on charge densities of the interacting arenes. We apply symmetry-adapted perturbation theory to dimers composed of substituted benzene and various aromatic heterocycles, which display a wide range of electrostatic interactions, and we investigate the interplay of Pauli repulsion, dispersion, and electrostatics as it pertains to parallel-displaced π-stacking. Profiles of energy components along cofacial slip-stacking coordinates support a prominent role for the “van der Waals model” (dispersion in competition with Pauli repulsion), even for polar monomers where electrostatic interactions are significant. While electrostatic interactions are necessary to explain the optimal face-to-face π-stacking distance and to account for the relative orientation of one polar arene with respect to another, we find no evidence to support continued invocation of quadrupolar electrostatics as a basis for π-stacking. Our results suggest that a driving force for offset-stacking exists even in the absence of electrostatic interactions. Consequently, tuning electrostatics via functionalization does not guarantee that slip-stacking can be avoided. This has implications for rational design of soft materials and other supramolecular architectures.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Representation of the QM Subsystem for Long-Range Electrostatic Interaction in Non-Periodic Ab Initio QM/MM Calculations

In QM/MM calculations, it is essential to handle electrostatic interactions between the QM and MM subsystems accurately and efficiently. To achieve maximal efficiency, it is convenient to adopt a hybrid scheme, where the QM electron density is used explicitly in the evaluation of short-range QM/MM electrostatic interactions, while a multipolar representation for the QM electron density is employed to account for the long-range QM/MM electrostatic interactions. In order to avoid energy discontinuity at the cutoffs, which separate the short- and long-range QM/MM electrostatic interactions, a switching function should be utilized to ensure a smooth potential energy surface. In this study, we benchmarked the accuracy of such hybrid embedding schemes for QM/MM electrostatic interactions using different multipolar representations, switching functions and cutoff distances. For test systems (neutral and anionic oxyluciferin in MM (aqueous and enzyme) environments), the best accuracy was acquired with a combination of QM electrostatic potential (ESP) charges and dipoles and two switching functions (long-range electrostatic corrections (LREC) and Switch) in the treatment of long-range QM/MM electrostatics. It allowed us to apply a 10Å distance cutoff and still obtain QM/MM electrostatics/polarization energies within 0.1 kcal/mol and time-dependent density functional theory (TDDFT)/MM vertical excitation energies within 10−3 eV from theoretical reference values.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Electrostatic forces in planetary rings

The average charge on a particle in a particle-plasma cloud, the plasma potential inside the cloud, and the Coulomb force acting on the particle are calculated. The net repulsive electrostatic force on a particle depends on the plasma density, temperature, density of particles, particle size, and the gradient of the particle density. In a uniformly dense ring the electrostatic repulsion is zero. It is also shown that the electrostatic force acts like a pressure force, that even a collisionless ring can be stable against gravitational collapse, and that a finite ring thickness does not necessarily imply a finite velocity dispersion. A simple criterion for the importance of electrostatic forces in planetary rings is derived which involves the calculation of the vertical ring thickness which would result if only electrostatic repulsion were responsible for the finite ring thickness. Electrostatic forces are entirely negligible in the main rings of Saturn and the E and G rings. They may also be negligible in the F ring. However, the Uranian rings and Jupiter's ring seem to be very much influenced by electrostatic repulsion. In fact, electrostatic forces could support a Jovian ring which is an order of magnitude more dense than observed.

Goertz, C. K.↗

A Triboelectric Sensor Array for Electrostatic Studies on the Lunar Surface

The moons electrostatic environment requires careful consideration in the development of future lunar landers. Electrostatically charged dust was well documented during the Apollo missions to cause thermal control, mechanical, and visibility issues. The fine dust particles that make up the surface are electrostatically charged as a result of numerous charging mechanisms. The relatively dry conditions on the moon creates a prime tribocharging environment during surface operations. The photoelectric effect is dominant for lunar day static charging, while plasma electrons are the main contributor for lunar night electrostatic effects. Electrostatic charging is also dependent on solar intensity, Earth-moon relative positions, and cosmic ray flux. This leads to a very complex and dynamic electrostatic environment that must be studied for the success of long term lunar missions.In order to better understand the electrostatic environment of planetary bodies, Kennedy Space Center, in previous collaboration with the Jet Propulsion Laboratory, has developed an electrostatic sensor suite. One of the instruments included in this package is the triboelectric sensor array. It is comprised of strategically selected materials that span the triboelectric series and that also have previous spaceflight history. In this presentation, we discuss detailed testing with the triboelectric sensor array performed at Kennedy Space Center. We will discuss potential benefits and use cases of this low mass, low cost sensor package, both for science and for mission success.

Charging↗

Validation of transfer map calculation for electrostatic deflectors in the code COSY INFINITY

The code COSY INFINITY uses a beamline coordinate system with a Frenet–Serret frame relative to the reference particle, and calculates differential algebra-valued transfer maps by integrating the ODEs of motion in the respective vector space over a differential algebra (DA). In our work, we described and performed computation of the DA transfer map of an electrostatic spherical deflector in a laboratory coordinate system using two conventional methods: (1) by integrating the ODEs of motion using a numerical integrator and (2) by computing analytically and in closed form the properties of the respective elliptical orbits from Kepler theory. We compared the resulting transfer maps with (3) the DA transfer map of COSY INFINITY’s built-in electrostatic spherical deflector element ESP and (4) the transfer map of the electrostatic spherical deflector computed using the program GIOS, which uses analytic formulas from a paper1 by Hermann Wollnik regarding second-order aberrations. In addition to the electrostatic spherical deflector, we studied an electrostatic cylindrical deflector, where the Kepler theory is not applicable. We computed the DA transfer map by the ODE integration method (1), and we compared it with the transfer maps by (3) COSY INFINITY’s built-in electrostatic cylindrical deflector element ECL and (4) GIOS. The transfer maps of electrostatic spherical and cylindrical deflectors obtained using the direct calculation methods (1) and (2) are in excellent agreement with those computed using (3) COSY INFINITY. On the other hand, we found a significant discrepancy with (4) the program GIOS.

43 PARTICLE ACCELERATORS↗

Electrostatic and whistler instabilities excited by an electron beam

The electron beam-plasma system is ubiquitous in the space plasma environment. Here in this paper, using a Darwin particle-in-cell method, the excitation of electrostatic and whistler instabilities by a gyrating electron beam is studied in support of recent laboratory experiments. It is assumed that the total plasma frequency (ω pe ) is larger than the electron cyclotron frequency (Ω e ). The fast-growing electrostatic beam-mode waves saturate in a few plasma oscillations by slowing down and relaxing the electron beam parallel to the background magnetic field. Upon their saturation, the finite amplitude electrostatic beam-mode waves can resonate with the tail of the background thermal electrons and accelerate them to the beam parallel velocity. The slower-growing whistler waves are excited in primarily two resonance modes: (a) through Landau resonance due to the inverted slope of the beam electrons in the parallel velocity and (b) through cyclotron resonance by scattering electrons to both lower pitch angles and smaller energies. It is demonstrated that, for a field-aligned beam, the whistler instability can be suppressed by the electrostatic instability due to a faster energy transfer rate between the beam electrons and electrostatic waves. Such a competition of growth between whistler and electrostatic waves depends on the ratio of ω pe /Ω e . In terms of wave propagation, beam-generated electrostatic waves are confined to the beam region, whereas beam-generated whistler waves transport energy away from the beam.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DFT-based QM/MM with Particle-Mesh Ewald for Direct, Long-Range Electrostatic Embedding

In this work, we present a DFT-based, QM/MM implementation with long-range electrostatic embedding achieved by direct real-space integration of the particle mesh Ewald (PME) computed electrostatic potential. The key transformation is the interpolation of the electrostatic potential from the PME grid to the DFT quadrature grid, from which integrals are easily evaluated utilizing standard DFT machinery. We provide benchmarks of the numerical accuracy with choice of grid size and real-space corrections, and demonstrate that good convergence is achieved while introducing nominal computational overhead. Furthermore, the approach requires only small modification to existing software packages, as is demonstrated with our implementation in the OpenMM and Psi 4 software. After presenting convergence benchmarks, we evaluate the importance of long-range electrostatic embedding in three solute/solvent systems modeled with QM/MM. Water and BMIM/BF 4 ionic liquid were considered as "simple" and "complex" solvents respectively, with water and p-phenylenediamine (PPD) solute molecules treated at QM level of theory. While electrostatic embedding with standard real-space truncation may introduce negligible error for simple systems such as water solute in water solvent, errors become more significant when QM/MM is applied to complex solvents such as ionic liquids. An extreme example is the electrostatic embedding energy for oxidized PPD in BMIM/BF 4 for which real-space truncation produces severe error even at 2-3 nm cutoff distances. This latter example illustrates that utilization of QM/MM to compute redox potentials within concentrated electrolytes/ionic media requires carefully chosen long-range electrostatic embedding algorithms, with our presented algorithm providing a general and robust approach.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Topography and electrostatic interactions of electret fibres in N95 facemasks

Facemasks have been a critical line of defense in the Covid-19 pandemic and have been used for decades to reduce the inhalation of sub-micron aerosols, including bacteria, viruses and nano-materials. These facemasks leverage electrostatic charge properties of non-woven electret fibre mats to remove sub-micron aerosols. However, the topographical and electrostatic properties of these mats have not been well characterized at the nano-scale. In this study, we use atomic force microscopy (AFM) to examine the surface topography of the electret fibres within a commercial respirator fibre mat, use electrostatic force microscopy to map the electrostatic interactions emanating from these fibres, and use synchrotron X-ray scattering to probe polymer morphology. We find that within these mats the fibres exhibit significant heterogeneity in surface topography, electrostatic interactions and polypropylene phase composition. The surface topographies ranged from nano-smooth surfaces to ordered spherulitic structures. Furthermore, the electrostatic interactions varied across fibre length and circumference and are only weakly correlated with surface topographical features. Finally, we use AFM to characterize fibres after filtering a NaCl aerosol and find that the deposited particles are heterogeneous in size and distributed over the fibre surfaces.

36 MATERIALS SCIENCE↗

Chapter Six - Derivation, cross-validation, and comparison of analytic formulas for electrostatic deflector aberrations

Here, we derived first and second order analytic aberration formulas in the horizontal transverse plane for an electrostatic deflector specified by the reference orbit radius, the central angle spanning the deflector, and its inhomogeneity coefficients. The derivation was performed using an iterative order-by-order perturbation method in a Frenet–Serret beamline coordinate system. We produced a C program edabrt for calculation of the first and second order aberrations using the formulas that we derived. The electrostatic deflector aberration formulas from Wollnik (1965) disagree with COSY INFINITY and the aberration formulas that we derived here, and they also deviate from the first and second order symplecticity conditions. The reason for this discrepancy is that the aberration formulas from Wollnik (1965) consider only motion in the main field of the deflector and do not properly account for fringe field effects, in particular the necessarily occurring change of kinetic energy due to the change in potential. The code GIOS uses electrostatic aberration formulas from Wollnik (1965) and exhibits the same discrepancy. We performed a comparison of first and second order electrostatic deflector aberrations for (1) the analytic formulas derived in this work, (2) differential-algebraic (DA) numerical integration of the equations of motion using the code COSY INFINITY, (3) the aberration formulas from Wollnik (1965), (4) aberrations computed using the code GIOS, and (5) the aberration formulas from Wollnik (1965) adjusted to account for the occurring change in potential. An electrostatic spherical deflector and an electrostatic cylindrical deflector were used as test cases for this comparison. There is excellent agreement between methods (1), (2), and (5), and these three methods satisfy the first and second order symplecticity conditions.

43 PARTICLE ACCELERATORS↗

Intramolecular Electrostatic Effects on O 2 , CO 2 , and Acetate Binding to a Cationic Iron Porphyrin

Noncovalent electrostatic interactions are important in many biological and chemical reactions, especially those that involve charged intermediates. There has been a growing interest in using electrostatic ligand designs—placing charges in the second coordination sphere—to improve molecular reactivity, catalysis, and electrocatalysis. For instance, an iron porphyrin bearing four cationic ortho -trimethylanilinium groups, Fe( o -TMA), has been reported to be an exceptional electrocatalyst for both the carbon dioxide reduction reaction (CO 2 RR) and the oxygen reduction reaction (ORR). These reactions involve many different steps, and it is not evident which steps are affected by the four positive charges, or why. By comparing Fe( o -TMA) with the related iron-tetraphenylporphyrin, this work examines how covalently positioned charged groups affect substrate binding and other key pre-equilibria of both the ORR and CO 2 RR, specifically acetate, dioxygen, and carbon dioxide binding. This study is among the first to directly measure the effects of electrostatics on ligand-binding. The results show that adding electrostatic groups to a catalyst design often results in a complex interplay of multiple effects, including changes in pre-equilibria prior to substrate binding, combinations of through-space and inductive contributions, and effects of ionic strength and solution dielectric. The inverse half-order dependence of binding constant on ionic strength is proposed as a clear marker for an electrostatic effect. Overall, the conclusions provide guidance for the increasingly popular electrostatic ligand designs in catalysis and other reactivity.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Stigmatellin Probes the Electrostatic Potential in the QB Site of the Photosynthetic Reaction Center

The electrostatic potential in the secondary quinone (QB) binding site of the reaction center (RC) of the photosynthetic bacterium Rhodobacter sphaeroides determines the rate and free energy change (driving force) of electron transfer to QB. It is controlled by the ionization states of residues in a strongly interacting cluster around the QB site. Reduction of the QB induces change of the ionization states of residues and binding of protons from the bulk. Stigmatellin, an inhibitor of the mitochondrial and photosynthetic respiratory chain, has been proven to be a unique voltage probe of the QB binding pocket. It binds to the QB site with high affinity, and the pK value of its phenolic group monitors the local electrostatic potential with high sensitivity. Investigations with different types of detergent as a model system of isolated RC revealed that the pK of stigmatellin was controlled overwhelmingly by electrostatic and slightly by hydrophobic interactions. Measurements showed a high pK value (>11) of stigmatellin in the QB pocket of the dark-state wild-type RC, indicating substantial negative potential. When the local electrostatics of the QB site was modulated by a single mutation, L213Asp/Ala, or double mutations, L213Asp-L212Glu/Ala-Ala (AA), the pK of stigmatellin dropped to 7.5 and 7.4, respectively, which corresponds to a >210 mV increase in the electrostatic potential relative to the wild-type RC. This significant pK drop (DpK > 3.5) decreased dramatically to (DpK > 0.75) in the RC of the compensatory mutant (AAþM44Asn/AAþM44Asp). Our results indicate that the L213Asp is the most important actor in the control of the electrostatic potential in the QB site of the dark-state wild-type RC, in good accordance with conclusions of former studies using theoretical calculations or light-induced charge recombination assay.

Gerencsér, László↗

A Universal Augmentation Framework for Long-Range Electrostatics in Machine Learning Interatomic Potentials

Most current machine learning interatomic potentials (MLIPs) rely on short-range approximations, without explicit treatment of long-range electrostatics. To address this, we recently developed the Latent Ewald Summation (LES) method, which infers electrostatic interactions, polarization, and Born effective charges (BECs), just by learning from energy and force training data. Here, in this study, we present LES as a standalone library, compatible with any short-range MLIP, and demonstrate its integration with methods such as MACE, NequIP, Allegro, CACE, CHGNet, and UMA. We benchmark LES-enhanced models on distinct systems, including bulk water, polar dipeptides, and gold dimer adsorption on defective substrates, and show that LES not only captures correct electrostatics but also improves accuracy. Additionally, we scale LES to large and chemically diverse data by training MACELES-OFF on the SPICE set containing molecules and clusters, making a universal MLIP with electrostatics for organic systems, including biomolecules. MACELES-OFF is more accurate than its short-range counterpart (MACE-OFF) trained on the same data set, predicts dipoles and BECs reliably, and has better descriptions of bulk liquids. By enabling efficient long-range electrostatics without directly training on electrical properties, LES paves the way for electrostatic foundation MLIPs.

Kim, Dongjin [University of California, Berkeley, ↗

Study of a condition for the mode conversion from purely perpendicular electrostatic waves to electromagnetic waves

One of the mechanisms for generating electromagnetic plasma waves (Z-mode and LO-mode) is mode conversion from electrostatic waves into electromagnetic waves in inhomogeneous plasma. Herein, we study a condition required for mode conversion of electrostatic waves propagating purely perpendicular to the ambient magnetic field, by numerically solving the full dispersion relation. An approximate model is derived describing the coupling between electrostatic waves (hot plasma Bernstein mode) and Z-mode waves at the upper hybrid frequency. The model is used to study conditions required for mode conversion from electrostatic waves (electrostatic electron cyclotron harmonic waves, including Bernstein mode) into electromagnetic plasma waves (LO-mode). It is shown that for mode conversion to occur in inhomogeneous plasma, the angle between the boundary surface and the magnetic field vector should be within a specific range. The range of the angle depends on the norm of the k vector of waves at the site of mode conversion in the inhomogeneous region. The present study reveals that inhomogeneity alone is not a sufficient condition for mode conversion from electrostatic waves to electromagnetic plasma waves and that the angle between the magnetic field and the density gradient plays an important role in the conversion process.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Conversion of electrostatic plasma waves into electromagnetic waves - Numerical calculation of the dispersion relation for all wavelengths.

The dispersion curves have been computed for a wide range of wavelengths from electromagnetic waves to electrostatic waves in a magnetoactive warm plasma with a Maxwellian velocity distribution function. The computation was carried out mainly for the perpendicular propagation mode. The upper hybrid resonance is the connection point of the electrostatic waves and the electromagnetic waves. The electrostatic waves not associated with the upper hybrid resonance are subjected to electron cyclotron damping when the wavelength becomes long. Oblique propagation is allowed for the electrostatic waves in a frequency range from the plasma frequency to the upper hybrid resonance frequency in the long-wavelength region where Landau damping can be neglected and where the electrostatic mode smoothly connects to the electromagnetic X-mode. In a slightly inhomogeneous plasma, the Bernstein-mode electrostatic wave can escape by being converted into the O-mode electromagnetic wave; two reflections take place during this escape process.

Oya, H.↗

Analysis and interpretation of the shocklike electrostatic noise observed during the AMPTE solar wind lithium releases

During the AMPTE (Active Magnetospheric Particle Tracer Explorers) solar wind lithium release on September 11, and again on September 20, 1984, an intense burst of electrostatic noise was observed near the upstream edge of the ion cloud. Comparisons with measurements by the IMP-6 and ISEE-1 spacecraft show that the spectrum and overall features of this noise are very similar to electrostatic noise observed at the Earth's bow shock. A stability analysis using realistic parameters shows that the electrostatic noise can be accounted for by an ion beam-plasma instability caused by the solar wind proton beam streaming through the nearly stationary lithium cloud. The growth rate of this instability is largest when the ion density and solar wind proton density are similar, which explains why the noise only occurs near the outer edge of the ion cloud. The similarity to the noise in the Earth's bow shock suggest that a shock may exist in the solar wind plasma flow upstream of the ion cloud. If the noise is associated with a shock, then it must be an electrostatic shock, since the ion cyclotron radii are too small for the existence of a MHD shock. Since the electrostatic instability occurs at phase velocities near the lithium thermal velocity, the electrostatic turbulence may play a role in heating the lithium ions and transferring momentum from the solar wind to the ion cloud.

Gurnett, D. A.↗

Electrostatic Screen for Transport of Martian and Lunar Regolith

The Martian and Lunar Regolith contain fine particulate including those in the size range from 0.5 to 200 micron [1-2]. Martian dust can be transported and deposited by Aeolian processes, including "Dust Devils". Due to the ultra high vacuum (10e-12 Torr), transport of dust on the Moon is solely a result of collision/ballistic motion. Dust obscuration of solar cells is one of the primary factors limiting the duration of Martian missions, including the Mars Exploration Rovers. Dust contamination in vacuum seals is one of the primarily factors that limited lunar excursions during the Apollo missions. Controlled transportation of dust on Mars and the Moon is important for many reasons, including both contamination mitigation and in situ resource utilization (ISRU). Since both the monopole and dipole electrostatic moments result in non-trivial forces on particles in an electrostatic field, dust particles, whether charged or not, can be transported by electrostatic fields. In the electrostatic screen, alternating waveforms of voltage applied to patterned grids of electrodes will transport dust. The authors will show that the canonical methods for transporting dust via electrostatic screen can be readily applied to transport of Martian and Lunar regolith. Experiments have been performed in ambient, low humidity, Martian, and Lunar conditions. Screen parameters have been examined for application to each regolith, such as grid spacing, trace width, grid voltage, pulse pattern, pulse frequency, and coating type. The authors have also developed an electrostatic screen based on optically transparent conductors that can be placed over solar arrays, windows, visors, lenses, etc.

Immer, C.↗

Active Electrostatic flight for airless bodies

The environment near the surface of asteroids, comets, and the Moon is electrically charged due to the Sun’s photoelectric bombardment and lofting dust, which follows the Sun illumination as the body spins. Charged dust is ever present, in the form of dusty plasma, even at high altitudes, following the solar illumination. If a body with high surface resistivity is exposed to the solar wind and solar radiation, sunexposed areas and shadowed areas become differentially charged. The E-Glider (Electrostatic Glider) is an enabling capability for operation at airless bodies, a solution applicable to many types of in-situ missions, which leverages the natural environment. This platform directly addresses the "All Access Mobility" Challenge, one of the NASA’s Space Technology Grand Challenges. Exploration of comets, asteroids, moons and planetary bodies is limited by mobility on those bodies. The lack of an atmosphere, the low gravity levels, and the unknown surface soil properties pose a very difficult challenge for all forms of know locomotion at airless bodies. This EGlider levitates by extending thin, charged, appendages, which are also articulated to direct the levitation force in the most convenient direction for propulsion and maneuvering. The charging is maintained through continuous charge emission. It lands, wherever it is most convenient, by retracting the appendages or by firing a cold-gas thruster, or by deploying an anchor. Preliminary calculations indicate that a 1 kg mass can be electrostatically levitated in a microgravity field with a 2 m diameter electrostatically inflated ribbon structure at 19kV, hence the need for a “balloon-like” system. The wings could be made of very thin Au-coated Mylar film, which are electrostatically inflated, and would provide the lift due to electrostatic repulsion with the naturally charged asteroid surface. Since the E-glider would follow the Sun’s illumination, the solar panels on the vehicle would constantly charge a battery. Further articulation at the root of the lateral strands or inflated membrane wings, would generate a component of lift depending on the articulation angle, hence a selective maneuvering capability which, to all effects, would lead to electrostatic (rather than aerodynamic) flight.

Schaub, Hanspeter↗

Electrostatic Barriers to Nanoparticle Accessibility of a Porous Matrix

Translocation from one cavity to another through a narrow constriction (i.e., a “hole”) represents the fundamental elementary process underlying hindered mass transport of nanoparticles and macromolecules within many natural and synthetic porous materials, including intracellular environments. This process is complex and may be influenced by long-range (e.g., electrostatic) particle–wall interactions, transient adsorption/desorption, surface diffusion, and hydrodynamic effects. In this study, we used a three-dimensional (3D) tracking method to explicitly visualize the process of nanoparticle diffusion within periodic porous nanostructures, where electrostatic interactions were mediated via ionic strength. The effects of electrostatic interactions on nanoparticle transport were surprisingly large. For example, an increase in the Debye length of only a few nanometers (in a material with a hole diameter of ~100 nm) increased the mean cavity escape time 3-fold. A combination of computational and experimental analyses indicated that this hindered cavity escape was due to an electrostatic energy barrier in the region of the hole, which was quantitatively explained using DLVO theory. These findings explicitly demonstrate that the cavity escape process was barrier-limited and dominated by electrostatic effects.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗