Parallel Diffusion Coefficient Research Articles

Diffusion Behaviors of CaCl2-NaCl Molten Salt under an Electric Field: A Deep Potential Molecular Dynamics Simulation.

CaCl2-NaCl molten salt is one of the widely used electrolytes, and the effect of the electric field on it needs to be considered in the environment of manufacturing metals and alloys as well as in battery applications. To be closer to the state of the CaCl2-NaCl molten salt system in practical applications, this study uses the training-based deep potential, combined with the ionic structure, to analyze the electric field effects on the drift velocity, ionic mobility, ion diffusion, and viscosity of the CaCl2-NaCl system by molecular dynamics simulations. It is shown that the electric field has a stronger effect on the ion diffusion behavior parallel to the direction of the electric field in the CaCl2-NaCl molten salt system. Due to the looser coordinate structure of Na+, the interaction force between Na-Cl is small compared with that between Ca-Cl. Na+ is easily driven and more sensitive to the electric field, whose drift velocity, ionic mobility, and diffusion coefficient are larger than those of the other two ions, with an order of Na+ > Cl- > Ca2+. All ionic drift velocities increase linearly as electric field intensity increases and the ionic mobilities tend to be constant. The diffusion coefficient parallel to the direction of the electric field exponentially increases and the viscosity tends to exponentially decrease with the increase in the electric field intensity. This work is important for accurately predicting the properties and performance of molten salts and their improvement for applications such as solar thermal energy conversion.

Cosmic-ray diffusion in two local filamentary clouds

Context. Hadronic interactions between cosmic rays (CRs) and interstellar gas have been probed in γ rays across the Galaxy. A fairly uniform CR distribution is observed up to a few hundred parsecs from the Sun, except in the Eridu cloud, which shows an unexplained 30–50% deficit in GeV to TeV CR flux. Aims. To explore the origin of this deficit, we studied the Reticulum cloud, which shares notable traits with Eridu: a comparable distance in the low-density region of the Local Valley and a filamentary structure of atomic hydrogen extending along a bundle of ordered magnetic-field lines that are steeply inclined to the Galactic plane. Methods. We measured the γ-ray emissivity per gas nucleon in the Reticulum cloud in the 0.16–63 GeV energy band using 14 years of Fermi-LAT data. We also derived interstellar properties that are important for CR propagation in both the Eridu and Reticulum clouds, at the same parsec scale. Results. The γ-ray emissivity in the Reticulum cloud is fully consistent with the average spectrum measured in the solar neighbourhood, but this emissivity, and therefore the CR flux, is 1.57 ± 0.09 times larger than in Eridu across the whole energy band. The difference cannot be attributed to uncertainties in gas mass. Nevertheless, we find that the two clouds are similar in many respects: both have magnetic-field strengths of a few micro-Gauss in the plane of the sky; both are in approximate equilibrium between magnetic and thermal pressures; they have similar turbulent velocities and sonic Mach numbers; and both show magnetic-field regularity with a dispersion in orientation lower than 10°–15° over large zones. The gas in Reticulum is colder and denser than in Eridu, but we find similar parallel diffusion coefficients around a few times 1028 cm2 s−1 in both clouds if CRs above 1 GV in rigidity diffuse on resonant, self-excited Alfvén waves that are damped by ion-neutral interactions. Conclusions. The loss of CRs in Eridu remains unexplained, but these two clouds provide important test cases to further study how magnetic turbulence, line tangling, and ion-neutral damping regulate CR diffusion in the dominant gas phase of the interstellar medium.

Parallel Diffusion Coefficient of Energetic Charged Particles in the Inner Heliosphere from the Turbulent Magnetic Fields Measured by Parker Solar Probe

Diffusion coefficients of energetic charged particles in turbulent magnetic fields are a fundamental aspect of diffusive transport theory but remain incompletely understood. In this work, we use quasi-linear theory to evaluate the spatial variation of the parallel diffusion coefficient κ ∥ from the measured magnetic turbulence power spectra in the inner heliosphere. We consider the magnetic field and plasma velocity measurements from Parker Solar Probe made during Orbits 5–13. The parallel diffusion coefficient is calculated as a function of radial distance from 0.062 to 0.8 au, and the particle energy from 100 keV to 1 GeV. We find that κ ∥ increases exponentially with both heliocentric distance and energy of particles. The fluctuations in κ ∥ are related to the episodes of large-scale magnetic structures in the solar wind. By fitting the results, we also provide an empirical formula of κ ∥ = (5.16 ± 1.22) × 1018 r 1.17±0.08 E 0.71±0.02 (cm2 s−1) in the inner heliosphere, which can be used as a reference in studying the transport and acceleration of solar energetic particles as well as the modulation of cosmic rays.

The Effect of Solar Wind on Charged Particles’ Diffusion Coefficients

The transport of energetic charged particles through magnetized plasmas is ubiquitous in interplanetary space and astrophysics, and the important physical quantities are the parallel and perpendicular diffusion coefficients of energetic charged particles. In this paper, the influence of solar wind on particle transport is investigated. Using the focusing equation, we obtain parallel and perpendicular diffusion coefficients, accounting for the solar wind effect. For different conditions, the relative importance of the solar wind effect to diffusion is investigated. It is shown that, when energetic charged particles are close to the Sun, for parallel diffusion, the solar wind effect needs to be taken into account. These results are important for studying energetic charged particle transport processes in the vicinity of the Sun.

Effect of temperature and external electric field on proton transport in ordered and amorphous proton exchange membranes: A molecular dynamics study

Proton exchange membrane (PEM) is a key component of proton exchange membrane fuel cell, and its proton conductivity directly affects the performance of fuel cell. Imparting order to Nafion molecules is one of potential effective approaches that can significantly improve the proton transport performance of PEM, but the underlying mechanism of this effect as well as the regulatory mechanism is unknown. In order to investigate the regulatory mechanism of temperature, electric field, and water content on the transport mechanism of the PEM and compare the transport properties between ordered and amorphous membranes, this study examined the proton migration rate of ordered and amorphous membranes under different electric field and water contents at 300 K and 350 K with molecular dynamics methods. The simulation results indicate that the ordered PEM exhibits better proton conductivity compared with the amorphous PEM due to its ordered structure with smoothly connected proton transport channels. The proton conductivity of the ordered PEM with a water content of 16 exhibits a remarkably high proton conductivity of 0.153 S/cm at a temperature of 350 K. Additionally, it was found that increasing electric field intensity, temperature, and hydration level can enhance the diffusion of water molecules and hydrated protons and improve the proton conductivity of the PEM. Especially, at low electric field intensity, ordered PEM demonstrates superior proton conductivity compared to amorphous PEM. However, as the electric field strength increases to 0.7 V/nm, the proton diffusion behaviour of amorphous PEM gradually surpasses that of ordered PEM. Moreover, the electric field can enhance the anisotropic diffusion behaviour of PEM. As the electric field increases from 0 V/nm to 0.7 V/nm, the diffusion coefficient parallel to the direction of the electric field increases by three orders of magnitude, while the diffusion coefficient perpendicular to the direction of the electric field increases by only one order of magnitude.

Diffusion coefficients of electrorheological complex (dusty) plasmas.

Equilibrium molecular dynamics (EMD) simulations have been executed to investigate the parallel (D║) and perpendicular (D┴) diffusion coefficients for three-dimensional (3D) strongly coupled (SC) electrorheological complex (dusty) plasmas (ERCPs). The effects of uniaxial (z-axis) AC electric field (MT) on dust grains have been investigated along with various combinations of plasma parameters (Γ, κ). The new outcomes obtained by mean squared displacement of Einstein relation show diffusion coefficients for low-intermediate to high plasma couplings (Γ) for varying MT. The D║ and D┴ at MT = 0.01 agree well with earlier available data obtained from the Green-Kubo and Einstein relation for 3D SC-Yukawa systems. The simulation data show that D║ increased with an increase in moderate MT strength and that D┴ decreased for the intermediate to large MT strength. Both (D║ and D┴) remained nearly constant for low MT values. The investigations show that the current EMD scheme is more efficient for nonideal gas-like, liquid-like, and solid-like states of SC-ERCPs. It has been demonstrated that present simulation outcomes extended the MT range up to 0.01 ≤ MT ≤ 10 to understand the diffusive and rheological behaviors of dusty plasma systems.

The Ratio of Perpendicular and Parallel Diffusion Coefficients of Low-energy Particles in Turbulent Space Plasmas

Recently an improved nonlinear theory for the transport of energetic particles across a mean magnetic field has been developed. The latter theory is called the field line–particle decorrelation theory and is the first analytical theory that agrees with test-particle simulations without the need of a correction parameter, nor does the theory contain any other free parameter. In the current paper we derive analytical forms for the ratio of perpendicular and parallel spatial diffusion coefficients κ ⊥/κ ∥ of low-energy particles. In the considered limit the latter ratio is constant meaning that it does not depend on particle energy or rigidity. It is shown that the ratio always has the form if a two-dimensional turbulence model is employed. Furthermore, the parameter a 2 depends only on the shape of the turbulence spectrum but not on the magnetic fields. The obtained results can be important for a variety of applications such as studies of solar modulation and diffusive shock acceleration.

Anisotropic cosmic ray diffusion in isotropic Kolmogorov turbulence

ABSTRACT Understanding the time-scales for diffusive processes and their degree of anisotropy is essential for modelling cosmic ray transport in turbulent magnetic fields. We show that the diffusion time-scales are isotropic over a large range of energy and turbulence levels, notwithstanding the high degree of anisotropy exhibited by the components of the diffusion tensor for cases with an ordered magnetic field component. The predictive power of the classical scattering relation as a description for the relation between the parallel and perpendicular diffusion coefficients is discussed and compared to numerical simulations. Very good agreement for a large parameter space is found, transforming classical scattering relation predictions into a computational prescription for the perpendicular component. We discuss and compare these findings, in particular, the time-scales to become diffusive with the time-scales that particles reside in astronomical environments, the so-called escape time-scales. The results show that, especially at high energies, the escape times obtained from diffusion coefficients may exceed the time-scales required for diffusion. In these cases, the escape time cannot be determined by the diffusion coefficients.

Revisiting the Revisited Palmer Consensus: New Insights from Jovian Electron Transport

Abstract Novel insights into the behavior of the diffusion coefficients of charged particles in the inner heliosphere are of great importance to any study of the transport of these particles and are especially relevant with regard to the transport of low-energy electrons. The present study undertakes an exhaustive investigation into the diffusion parameters needed to reproduce low-energy electron intensities as observed at Earth, using a state-of-the-art 3D cosmic ray transport code. To this end, the transport of Jovian electrons is considered, as Jupiter represents the predominant source of these particles in the inner heliosphere, and because a careful comparison of model results with observations taken during periods of good and poor magnetic connectivity between Earth and Jupiter allows for conclusions to be drawn as to both parallel and perpendicular diffusion coefficients. This study then compares these results with the predictions made by various scattering theories. Best-fit parameters for parallel and perpendicular mean free paths at 1 au fall reasonably well within the span of observational values reported by previous studies, but best-fit radial and rigidity dependences vary widely. However, a large number of diffusion parameters lead to reasonable to-good fits to observations, and it is argued that considerable caution must be exercised when comparing theoretical results for diffusion coefficients with diffusion parameters calculated from particle transport studies.

Plasmas, particles and photons—spotlights on multimessenger astronomy

During the past decennia, progress in the area of high-energy astroparticle physics was exceptional, mainly due to the great success of the bridging of particle- and astrophysics both in theory and in the instrumentation of astroparticle physics observatories. Multimessenger data coming from charged cosmic-ray-, gamma-ray- and neutrino-observatories start to shed more and more light on the nature and origin of cosmic rays. At the same time, the development of methods for the investigation of cosmic-ray transport, acceleration and interaction has advanced to the true potential of tying these different pieces of multimessenger data together, this way closing in on the origin of cosmic rays. In recent years, this rapid interplay between modeling and observations has made it clear that it is essential to add the ingredient of plasma physics to the problem. It has been shown that even the interpretation of data of highly relativistic cosmic rays at TeV energies and above is in need of a proper modeling of the plasma physics involved. One of the most important examples is the understanding of wave-particle interactions. In simulations of cosmic-ray transport in the Galaxy, the cosmic-ray diffusion coefficient is typically approximated with a Kolmogorov-type cascade model, resulting in an energy-dependent parallel diffusion coefficient with . Here, we show how the energy dependence of the diffusion coefficient can be investigated systematically as a function of . The complex energy behavior that goes well beyond a simple powerlaw interpretation will be presented together with a formal definition of an energy range that indeed can be approximated as a powerlaw. These results are applied to cosmic-ray transport in the Milky Way. Finally, the transition between the ballistic and diffusive regime will be investigated for astrophysical sources with special focus on relativistic plasmoids of active galaxies.

Study of Electron Anomalous Heat Diffusion Affected by Mode Number of Perturbed Magnetic Island in Tokamak Plasmas

Anomalous electron heat diffusion across high mode magnetic islands and the nonlocal high mode stochastic magnetic field are researched and compared with earlier low mode research work. The work in this paper uses two typical magnetic islands and a high mode stochastic magnetic field that are aroused by five incompact high mode perturbed magnetic islands. It is found that the mode number of the perturbed magnetic island is another key factor of anomalous electron heat diffusion across the minor radius in tokamak plasmas, besides the ratio of the parallel heat diffusion coefficient to the perpendicular coefficient and the width of diffusive layers.

Dynamic measurement of impurity ion transport in runaway electron plateaus in DIII-D

The first dynamic (time-dependent) measurements of impurity ion radial (cross field) and parallel (along-field) diffusion coefficients for post-disruption runaway electron plateaus are presented. Small (∼1 mm diameter) carbon or silicon pellets are fired into the edge of steady-state runaway electron (RE) plateaus, and the resulting radial and toroidal transport of singly charged impurity ions (C+ or Si+) is monitored with spatially distributed visible spectrometers. Consistent with previous steady-state particle balance estimates of Ar+ radial transport, radial (cross field) diffusion coefficients D⊥≈2–5 m2/s are obtained, about 10× larger than expected from neo-classical theory. Parallel diffusion coefficients D∥≈30–80 m2/s are estimated, also much (≈50×) larger than classical. It is speculated at present that these large diffusion coefficients may be due to turbulent transport. Indications of fairly significant (almost 2×) toroidal variation in electron density are seen in the RE plateaus, and this appears to cause some toroidal variation in impurity radial diffusion rates. Indications of slow (≈1 Hz) toroidal rotation in the impurity ions are observed, although the uncertainty in this measurement is large.

Cosmic-ray Transport in Magnetohydrodynamic Turbulence

Abstract This paper studies cosmic-ray (CR) transport in magnetohydrodynamic (MHD) turbulence. CR transport is strongly dependent on the properties of the magnetic turbulence. We perform test particle simulations to study the interactions of CR with both total MHD turbulence and decomposed MHD modes. The spatial diffusion coefficients and the pitch angle scattering diffusion coefficients are calculated from the test particle trajectories in turbulence. Our results confirm that the fast modes dominate the CR propagation, whereas Alfvén and slow modes are much less efficient and have shown similar pitch-angle scattering rates. We investigate the cross field transport on large and small scales. On large/global scales, normal diffusion is observed and the diffusion coefficient is suppressed by M A ζ compared to the parallel diffusion coefficients, with ζ closer to 4 in Alfvén modes than that in total turbulence, as theoretically expected. For the CR transport on scales smaller than the turbulence injection scale, both the local and global magnetic reference frames are adopted. Superdiffusion is observed on such small scales in all the cases. Particularly, CR transport in Alfvén modes show clear Richardson diffusion in the local reference frame. The diffusion transitions smoothly from the Richardson’s one with index 1.5 to normal diffusion as the particle mean free path decreases from λ ∥ ≫ L to λ ∥ ≪ L, where L is the injection/coherence length of turbulence. Our results have broad applications to CRs in various astrophysical environments.

Regimes of cosmic-ray diffusion in Galactic turbulence

Cosmic-ray transport in astrophysical environments is often dominated by the diffusion of particles in a magnetic field composed of both a turbulent and a mean component. This process, which is two-fold turbulent mixing in that the particle motion is stochastic with respect to the field lines, needs to be understood in order to properly model cosmic-ray signatures. One of the most important aspects in the modeling of cosmic-ray diffusion is that fully resonant scattering, the most effective such process, is only possible if the wave spectrum covers the entire range of propagation angles. By taking the wave spectrum boundaries into account, we quantify cosmic-ray diffusion parallel and perpendicular to the guide field direction at turbulence levels above 5% of the total magnetic field. We apply our results of the parallel and perpendicular diffusion coefficient to the Milky Way. We show that simple purely diffusive transport is in conflict with observations of the inner Galaxy, but that just by taking a Galactic wind into account, data can be matched in the central 5 kpc zone. Further comparison shows that the outer Galaxy at >5 kpc, on the other hand, should be dominated by perpendicular diffusion, likely changing to parallel diffusion at the outermost radii of the Milky Way.

Numerical Modeling of Latitudinal Gradients for Galactic Cosmic-Ray Protons during Solar Minima: Comparing with Ulysses Observations

Abstract The latitudinal gradients of galactic cosmic-ray (GCR) protons measured by Ulysses during two successive minima provide a unique opportunity to study the modulation effects in polar regions of the heliosphere. In this work, a GCR modulation model based on numerically solving the Parker transport equation is used to study the latitudinal distribution of GCR protons in the inner heliosphere. Modifications of the standard Parker heliospheric magnetic field, the reduction of particle drifts, the latitudinal-dependent magnetic turbulence characteristics, and the anisotropic perpendicular diffusion coefficient are incorporated in the numerical model to investigate the corresponding modulation effects. It is found that the latitudinal-dependent magnetic turbulence magnitude, which makes the parallel diffusion coefficient decrease with the increasing of latitude, is crucial to obtain the negative latitude gradient in the inner heliosphere during the negative-polarity solar cycle. For the A > 0 period, on the other hand, the latitudinal diffusion coefficient in the inner heliosphere and the reduced drift velocity in the polar region are more important, while the anisotropic perpendicular diffusion coefficient at high latitude might be not essential. Finally, the proton latitudinal gradient and the corresponding differential intensity along the trajectory of Ulysses during its first and third fast latitude scans are computed, and the results show good agreement with the spacecraft observations.

Relationships between spinal cord blood flow measured with flow-sensitive alternating inversion recovery (FAIR) and neurobehavioral outcomes in rat spinal cord injury

In the traumatically injured spinal cord, decreased perfusion is believed to contribute to secondary tissue damage beyond the primary mechanical impact, and restoration of perfusion is believed to be a promising therapeutic target. However, methods to monitor spinal cord perfusion non-invasively are limited. Perfusion magnetic resonance imaging (MRI) techniques established for the brain have not been routinely adopted to the spinal cord. The purpose of this study was to examine the relationship between spinal cord blood flow (SCBF) and injury severity in a rat thoracic spinal cord contusion injury (SCI) model using flow-sensitive alternating inversion recovery (FAIR) with two variants of the label position. SCBF as a marker of severity was compared to T1 mapping and to spinal cord-optimized diffusion weighted imaging (DWI) with filtered parallel apparent diffusion coefficient. Thirty-eight rats underwent a T10 contusion injury with varying severities (8 sham; 10 mild; 10 moderate; 10 severe) with MRI performed at 1 day post injury at the lesion site and follow-up neurological assessments using the Basso, Beattie, Bresnahan (BBB) locomotor scoring up to 28 days post injury. Using whole-cord regions of interest at the lesion epicenter, SCBF was decreased with injury severity and had a significant correlation with BBB scores at 28 days post injury. Importantly, estimates of arterial transit times (ATT) in the injured spinal cord were not altered after injury, which suggests that FAIR protocols optimized to measure SCBF provide more value in the context of acute traumatic injury to the cord. T1-relaxation time constants were strongly related to injury severity and had a larger extent of changes than either SCBF or DWI measures. These findings suggest that perfusion decreases in the spinal cord can be monitored non-invasively after injury, and multi-parametric MRI assessments of perfusion, diffusion, and relaxation capture unique features of the pathophysiology of preclinical injury.

Modeling the solid/liquid interfacial properties of methylimidazole confined in hydrophobic silica nanopores

Many chemical processes proceed in confined environments of silica nanopore, in which the confined reactants or guest adsorbates show drastically different properties from the bulk system. Therefore, the solid/liquid interfacial structure and dynamics of 1-methylimidazole (MeIm) confined in the hydrophobic silica nanopore passivated by methoxysilane groups (-Si(Me)3) were investigated using molecular dynamics (MD) simulations. The mass density of the confined MeIm is layering and pore-size independent, while the negligible layering structure for charge density is observed. The smaller pore size gives rise to a significantly slower reorientation dynamics and translational diffusion of MeIm. In 2.8 nm pore, the parallel diffusion coefficient of MeIm is nearly 10 times larger than the perpendicular one, and their difference becomes smaller with the increase in pore size. Hopefully the insights from this work are helpful to a better understanding of the effects of hydrophobic silica pore on the confined behaviors of liquids.

Turbulence-level dependence of cosmic ray parallel diffusion

ABSTRACT Understanding the transport of energetic cosmic rays belongs to the most challenging topics in astrophysics. Diffusion due to scattering by electromagnetic fluctuations is a key process in cosmic ray transport. The transition from a ballistic to a diffusive-propagation regime is presented in direct numerical calculations of diffusion coefficients for homogeneous magnetic field lines subject to turbulent perturbations. Simulation results are compared with theoretical derivations of the parallel diffusion coefficient’s dependences on the energy and the fluctuation amplitudes in the limit of weak turbulence. The present study shows that the widely used extrapolation of the energy scaling for the parallel diffusion coefficient to high turbulence levels predicted by quasi-linear theory does not provide a universally accurate description in the resonant-scattering regime. It is highlighted here that the numerically calculated diffusion coefficients can be polluted for low energies due to missing resonant interaction possibilities of the particles with the turbulence. Five reduced-rigidity regimes are established, which are separated by analytical boundaries derived in this work. Consequently, a proper description of cosmic ray propagation can only be achieved by using a turbulence-level-dependent diffusion coefficient and can contribute to solving the Galactic cosmic ray gradient problem.

The Invariance of the Diffusion Coefficient with Iterative Operations of the Charged Particle Transport Equation

Abstract The spatial parallel diffusion coefficient (SPDC) is one of the important quantities describing energetic charged particle transport. There are three different definitions for the SPDC: the displacement variance definition , the Fick’s law definition with , and the Taylor–Green–Kubo (TGK) formula definition . For a constant mean magnetic field, the three different definitions of the SPDC give the same result. However, for a focusing field, it is demonstrated that the results of the different definitions are not the same. In this paper, from the Fokker–Planck equation, we find that different methods, e.g., the general Fourier expansion and iteration method, can give different equations of the isotropic distribution function (EIDFs). But it is shown that one EIDF can be transformed into another by some derivative iterative operations (DIOs). If one definition of the SPDC is invariant for the DIOs, it is clear that the definition is also invariant for different EIDFs; therefore, it is an invariant quantity for the different derivation methods of the EIDF. For the focusing field, we suggest that the TGK definition is only an approximate formula, and the Fick’s law definition is not invariant to some DIOs. However, at least for the special condition, in this paper we show that the definition is an invariant quantity to the DIOs. Therefore, for a spatially varying field, the displacement variance definition , rather than the Fick’s law definition and TGK formula definition , is the most appropriate definition of the SPDCs.

Trapping of Cosmic Rays in MHD Turbulence

Abstract Astrophysical plasmas are turbulent and magnetized. The interaction between cosmic rays (CRs) and magnetohydrodynamic (MHD) turbulence is a fundamental astrophysical process. Based on the current understanding of MHD turbulence, we revisit the trapping of CRs by magnetic mirrors in the context of MHD turbulence. In compressible MHD turbulence, isotropic fast modes dominate both trapping and gyroresonant scattering of CRs. The presence of trapping significantly suppresses the pitch-angle scattering and the spatial diffusion of CRs along the magnetic field. The resulting parallel diffusion coefficient has a weaker dependence on CR energy at higher energies. In incompressible MHD turbulence, the trapping by pseudo-Alfvén modes dominates over the gyroresonant scattering by anisotropic Alfvén and pseudo-Alfvén modes at all pitch angles and prevents CRs from diffusion.