Diffusive Propagation Research Articles

Kinetics of multicomponent ion-exchange mass transfer in ion exchangers with nonmonotonic behavior of kinetic curves: the displacement effect for diffusion concentration waves of ionic components

A theoretical approach to simulation of the multicomponent mass transfer of ionic components in ion exchangers is reported. A computer model is proposed for use in contemporary theoretical studies of the kinetics of the nonlinear multicomponent mass transfer of ions in the matrix of ion exchangers. Propagating diffusion concentration waves in multicomponent ion-exchange systems for the mass transfer of ions in the ion-exchange matrices of arbitrary forms—namely, planar L membrane, r grain, and ro fiber—are calculated. All the results are presented within the framework of the notion of propagating diffusion concentration waves of ionic components in the matrices of ion exchangers. The idea that is used of concentration waves for the description of the multicomponent kinetics of the diffusion mass transfer yields a clear interpretation of the computer simulation results. The kinetics of the process of the multicomponent mass transfer of components is described theoretically on the basis of computer simulation by solving the set of differential equations in partial derivatives for the mass balance of components. The propagation of concentration waves is illustrated by my simulation graphs and animations showing visually the propagation and interaction-interference of multicomponent concentration waves in the matrices of the various above-mentioned forms. Different ion-exchange systems with simulation of the anomalous behavior of the Fi(T) kinetic curves (non-monotonic and with the presence of the maximum, Fmax) for one of the B+ + C+ ions entering into the R− matrix in the case of the R−A+/(B+ + C+) exchange version, with simultaneous computer simulation of the diffusion propagation of the multicomponent Xi concentration waves within the nonlinear mass transfer kinetics, are examined.

Particle simulations advance insight into diffusion device design, efficiency, and diffuse field development and propagation

After the development of a demonstration to illustrate acoustic diffusion using Ping-Pong balls (DeGrandis, 2011, “Sound Diffusion Explained,” http://acousticsfirst.com/educational-videos-acoustic-sound-diffusion.htm), the idea of virtualizing the concept to gain more insight into diffusion from a particle physics perspective became a goal. Using open-source, Ray tracing modeling, and physics simulation software, the efficiency of different diffusor designs was evaluated. Simulations included individual elements and arrays in varied distribution and varied environments, and their impact on the propagation of sound, as well as the visualization and prediction of the development of an acoustically diffuse field. Integrating a simulated particle emitter to model the excitation, environmental physics to model atmospheric conditions, and 3d models with adjustable surface dampening properties, usable simulations with calculable and predictable properties could be run. New data were collected based on spatial diffusion, isometry, and the speed of which the excitation was broken down from a contiguous wave front into a diffuse particle field. Simulations varied from individual unit efficiency to complex room modeling with multiple emitters and scattering surfaces. This is a collection of the findings from that exercise.

Estimating diffusion propagator and its moments using directional radial basis functions.

The ensemble average diffusion propagator (EAP) obtained from diffusion MRI (dMRI) data captures important structural properties of the underlying tissue. As such, it is imperative to derive an accurate estimate of the EAP from the acquired diffusion data. In this work, we propose a novel method for estimating the EAP by representing the diffusion signal as a linear combination of directional radial basis functions scattered in q-space. In particular, we focus on a special case of anisotropic Gaussian basis functions and derive analytical expressions for the diffusion orientation distribution function (ODF), the return-to-origin probability (RTOP), and mean-squared-displacement (MSD). A significant advantage of the proposed method is that the second and the fourth order moment tensors of the EAP can be computed explicitly. This allows for computing several novel scalar indices (from the moment tensors) such as mean-fourth-order-displacement (MFD) and generalized kurtosis (GK)-which is a generalization of the mean kurtosis measure used in diffusion kurtosis imaging. Additionally, we also propose novel scalar indices computed from the signal in q-space, called the q-space mean-squared-displacement (QMSD) and the q-space mean-fourth-order-displacement (QMFD), which are sensitive to short diffusion time scales. We validate our method extensively on data obtained from a physical phantom with known crossing angle as well as on in-vivo human brain data. Our experiments demonstrate the robustness of our method for different combinations of b-values and number of gradient directions.

A morphological adaptation approach to path planning inspired by slime mould

Path planning is a classic problem in computer science and robotics which has recently been implemented in unconventional computing substrates such as chemical reaction–diffusion computers. These novel computing schemes utilise the parallel spatial propagation of information and often use a two-stage method involving diffusive propagation to discover all paths and a second stage to highlight or visualise the path between two particular points in the arena. The true slime mould Physarum polycephalum is known to construct efficient transport networks between nutrients in its environment. These networks are continuously remodelled as the organism adapts its body plan to changing spatial stimuli. It can be guided towards attractant stimuli (nutrients, warm regions) and it avoids locations containing hazardous stimuli (light irradiation, repellents, or regions occupied by predatory threats). Using a particle model of slime mould we demonstrate scoping experiments which explore how path planning may be performed by morphological adaptation. We initially demonstrate simple path planning by a shrinking blob of virtual plasmodium between two attractant sources within a polygonal arena. We examine the case where multiple paths are required and the subsequent selection of a single path from multiple options. Collision-free paths are implemented via repulsion from the borders of the arena. Finally, obstacle avoidance is implemented by repulsion from obstacles as they are uncovered by the shrinking blob. These examples show proof-of-concept results of path planning by morphological adaptation which complement existing research on path planning in novel computing substrates.

Transient behavior of invisibility cloaks for diffusive light propagation

An ideal invisibility cloak makes any object within itself indistinguishable from its surrounding—for all colors, directions, and polarizations of light. Nearly ideal cloaks have recently been realized for turbid light-scattering media under continuous-wave illumination. Here, we ask whether these cloaks also work under pulsed illumination. Our time-resolved imaging experiments on simple core–shell cloaks show that they do not: they appear bright with respect to their surrounding at early times and dark at later times, leading to vanishing image contrast for time-averaged detection. Furthermore, we show that the same holds true for more complex cloaking architectures designed by spatial coordinate transformations. We discuss implications for diffuse optical tomography and possible applications in terms of high-end security features.

A 4D hyperspherical interpretation of q-space.

3D q-space can be viewed as the surface of a 4D hypersphere. In this paper, we seek to develop a 4D hyperspherical interpretation of q-space by projecting it onto a hypersphere and subsequently modeling the q-space signal via 4D hyperspherical harmonics (HSH). Using this orthonormal basis, we derive several well-established q-space indices and numerically estimate the diffusion orientation distribution function (dODF). We also derive the integral transform describing the relationship between the diffusion signal and propagator on a hypersphere. Most importantly, we will demonstrate that for hybrid diffusion imaging (HYDI) acquisitions low order linear expansion of the HSH basis is sufficient to characterize diffusion in neural tissue. In fact, the HSH basis achieves comparable signal and better dODF reconstructions than other well-established methods, such as Bessel Fourier orientation reconstruction (BFOR), using fewer fitting parameters. All in all, this work provides a new way of looking at q-space.

A phase field model coupling lithium diffusion and stress evolution with crack propagation and application in lithium ion batteries.

Cracking and fracture of electrodes under diffusion during lithiation and delithiation is one of the main factors responsible for short life span of lithium based batteries employing high capacity electrodes. Coupling effects among lithium diffusion, stress evolution and crack propagation have a significant effect on dynamic processes of electrodes during cycling. In this paper, a phase field model coupling lithium diffusion and stress evolution with crack propagation is established. Then the model is applied to a silicon thin film electrode to explore the coupling effects on diffusion and crack propagation paths. During lithiation, simulation results show that lithium accumulates at crack tips and the lithium accumulation further reduces the local hydrostatic stress. Single and multiple crack geometries are considered to elucidate some of the crack patterns in thin film electrodes as a consequence of coupling effects and crack interactions.

Tensorial Spherical Polar Fourier Diffusion MRI with Optimal Dictionary Learning.

High Angular Resolution Diffusion Imaging (HARDI) can characterize complex white matter micro-structure, avoiding the Gaussian diffusion assumption inherent in Diffusion Tensor Imaging (DTI). However, HARDI methods normally require significantly more signal measurements and a longer scan time than DTI, which limits its clinical utility. By considering sparsity of the diffusion signal, Compressed Sensing (CS) allows robust signal reconstruction from relatively fewer samples, reducing the scanning time. A good dictionary that sparsifies the signal is crucial for CS reconstruction. In this paper, we propose a novel method called Tensorial Spherical Polar Fourier Imaging (TSPFI) to recover continuous diffusion signal and diffusion propagator by representing the diffusion signal using an orthonormal TSPF basis. TSPFI is a generalization of the existing model-based method DTI and the model-free method SPFI. We also propose dictionary learning TSPFI (DL-TSPFI) to learn an even sparser dictionary represented as a linear combination of TSPF basis from continuous mixture of Gaussian signals. The learning process is efficiently performed in a small sub-space of SPF coefficients, and the learned dictionary is proved to be sparse for all mixture of Gaussian signals by adaptively setting the tensor in TSPF basis. Then the learned DL-TSPF dictionary is optimally and adaptively applied to different voxels using DTI and a weighted LASSO for CS reconstruction. DL-TSPFI is a generalization of DL-SPFI, by considering general adaptive tensor setting instead of a scale value. The experiments demonstrated that the learned DL-TSPF dictionary has a sparser representation and lower reconstruction Root-Mean-Squared-Error (RMSE) than both the original SPF basis and the DL-SPF dictionary.

A possible origin of gamma rays from the Fermi Bubbles

One of the most exciting discoveries of recent years is a pair of gigantic gamma-ray emission regions, the so-called Fermi bubbles, above and below the Galactic center. The bubbles, discovered by the Fermi space telescope, extend up to ∼50° in Galactic latitude and are ∼40° wide in Galactic longitude. The gamma-ray emission is also found to correlate with radio, microwave and X-rays emission. The origin of the bubbles and the associated non-thermal emissions are still not clearly understood. Possible explanations for the non-thermal emission include cosmic-ray injection from the Galactic center by high speed Galactic winds/jets, acceleration by multiple shocks or plasma turbulence present inside the bubbles, and acceleration by strong shock waves associated with the expansion of the bubbles. In this paper, I will discuss the possibility that the gamma-ray emission is produced by the injection of Galactic cosmic-rays mainly protons during their diffusive propagation through the Galaxy. The protons interact with the bubble plasma producing π°-decay gamma rays, while at the same time, radio and microwave synchrotron emissions are produced by the secondary electrons/positrons resulting from the π± decays.

Lasing threshold of one- and two-photon-pumped dye-doped silica powder

The random laser features around threshold in a ground powder of a silica gel containing Rhodamine 6G-doped silica nanoparticles under one- and two-photon excitation are analyzed. The lasing threshold following two-photon pumping is fifty times higher than after one-photon excitation. Theoretical calculations by using a light diffusive propagation model are in agreement with the experimental results.

Dynamics of Diffusion and Mechanical Waves Interaction under Conditions of Metal Surface Treatment with Particle Fluxes

A coupled isothermal model at the initial stage of a solid body surface treatment with particle flux is presented in this paper. The model takes account of the interaction of two different scale processes: impurity diffusion and mechanical stress wave propagation. The transition to dimensionless variables is briefly described. The examples of the coupled problem solution illustrating the wave interaction under the action of one pulse and two successive pulses are given.

Iterative learning control of inhomogeneous distributed parameter systems—frequency domain design and analysis

This paper aims to construct a design and analysis framework for iterative learning control of linear inhomogeneous distributed parameter systems (LIDPSs), which may be hyperbolic, parabolic, or elliptic, and include many important physical processes such as diffusion, vibration, heat conduction and wave propagation as special cases. Owing to the system model characteristics, LIDPSs are first reformulated into a matrix form in the Laplace transform domain. Then, through the determination of a fundamental matrix, the transfer function of LIDPS is precisely evaluated in a closed form. The derived transfer function provides the direct input–output relationship of the LIDPS, and thus facilitates the consequent ILC design and convergence analysis in the frequency domain. The proposed control design scheme is able to deal with parametric and non-parametric uncertainties and make full use of the process repetition, while avoid any simplification or discretization for the 3D dynamics of LIDPS in the time, space, and iteration domains. In the end, two illustrative processes are addressed to demonstrate the efficacy of the proposed iterative learning control scheme.

Further development of discrete computational techniques for calculation of restricted diffusion propagators in porous media

Magnetic resonance is a well-established tool for structural characterisation of porous media. Features of pore-space morphology can be inferred from NMR diffusion–diffraction plots or the time-dependence of the apparent diffusion coefficient. Diffusion NMR signal attenuation can be computed from the restricted diffusion propagator, which describes the distribution of diffusing particles for a given starting position and diffusion time.We present two techniques for efficient evaluation of restricted diffusion propagators for use in NMR porous-media characterisation. The first is the Lattice Path Count (LPC). Its physical essence is that the restricted diffusion propagator connecting points A and B in time t is proportional to the number of distinct length-t paths from A to B. By using a discrete lattice, the number of such paths can be counted exactly. The second technique is the Markov transition matrix (MTM). The matrix represents the probabilities of jumps between every pair of lattice nodes within a single timestep. The propagator for an arbitrary diffusion time can be calculated as the appropriate matrix power. For periodic geometries, the transition matrix needs to be defined only for a single unit cell. This makes MTM ideally suited for periodic systems.Both LPC and MTM are closely related to existing computational techniques: LPC, to combinatorial techniques; and MTM, to the Fokker–Planck master equation. The relationship between LPC, MTM and other computational techniques is briefly discussed in the paper. Both LPC and MTM perform favourably compared to Monte Carlo sampling, yielding highly accurate and almost noiseless restricted diffusion propagators. Initial tests indicate that their computational performance is comparable to that of finite element methods. Both LPC and MTM can be applied to complicated pore-space geometries with no analytic solution. We discuss the new methods in the context of diffusion propagator calculation in porous materials and model biological tissues.

Second-order approximations for variable order fractional derivatives: Algorithms and applications

Fractional calculus allows variable-order of fractional operators, which can be exploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and/or time. In this paper, we derive two second-order approximation formulas for the variable-order fractional time derivatives involved in anomalous diffusion and wave propagation. We then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous sub-diffusion and super-diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable-order function. Finally, we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super-diffusion, and also simulations of the Burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and/or time. Taken together, our results demonstrate that variable-order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long-standing issues of outflow boundary conditions and monotonicity of integer-order PDEs.

Multi-shell diffusion signal recovery from sparse measurements.

For accurate estimation of the ensemble average diffusion propagator (EAP), traditional multi-shell diffusion imaging (MSDI) approaches require acquisition of diffusion signals for a range of b-values. However, this makes the acquisition time too long for several types of patients, making it difficult to use in a clinical setting. In this work, we propose a new method for the reconstruction of diffusion signals in the entire q-space from highly undersampled sets of MSDI data, thus reducing the scan time significantly. In particular, to sparsely represent the diffusion signal over multiple q-shells, we propose a novel extension to the framework of spherical ridgelets by accurately modeling the monotonically decreasing radial component of the diffusion signal. Further, we enforce the reconstructed signal to have smooth spatial regularity in the brain, by minimizing the total variation (TV) norm. We combine these requirements into a novel cost function and derive an optimal solution using the Alternating Directions Method of Multipliers (ADMM) algorithm. We use a physical phantom data set with known fiber crossing angle of 45° to determine the optimal number of measurements (gradient directions and b-values) needed for accurate signal recovery. We compare our technique with a state-of-the-art sparse reconstruction method (i.e., the SHORE method of Cheng et al. (2010)) in terms of angular error in estimating the crossing angle, incorrect number of peaks detected, normalized mean squared error in signal recovery as well as error in estimating the return-to-origin probability (RTOP). Finally, we also demonstrate the behavior of the proposed technique on human in vivo data sets. Based on these experiments, we conclude that using the proposed algorithm, at least 60 measurements (spread over three b-value shells) are needed for proper recovery of MSDI data in the entire q-space.

GeV-TeV cosmic-ray spectral anomaly as due to reacceleration by weak shocks in the Galaxy

Recent cosmic-ray measurements have found an anomaly in the cosmic-ray energy spectrum at GeV-TeV energies. Although the origin of the anomaly is not clearly understood, suggested explanations include effect of cosmic-ray source spectrum, propagation effects, and the effect of nearby sources. In this paper, we propose that the spectral anomaly might be an effect of re-acceleration of cosmic rays by weak shocks in the Galaxy. After acceleration by strong supernova remnant shock waves, cosmic rays undergo diffusive propagation through the Galaxy. During the propagation, cosmic rays may again encounter expanding supernova remnant shock waves, and get re-accelerated. As the probability of encountering old supernova remnants is expected to be larger than the younger ones due to their bigger sizes, re-acceleration is expected to be mainly due to weaker shocks. Since weaker shocks generate a softer particle spectrum, the resulting re-accelerated component will have a spectrum steeper than the initial cosmic-ray source spectrum produced by strong shocks. For a reasonable set of model parameters, it is shown that such re-accelerated component can dominate the GeV energy region while the non-reaccelerated component dominates at higher energies, explaining the observed GeV-TeV spectral anomaly.

Chemical zonation in olivine-hosted melt inclusions

Significant zonation in major, minor, trace, and volatile elements has been documented in naturally glassy olivine-hosted melt inclusions from the Siqueiros Fracture Zone and the Galapagos Islands. Components with a higher concentration in the host olivine than in the melt (e.g., MgO, FeO, Cr2O3, and MnO) are depleted at the edges of the zoned melt inclusions relative to their centers, whereas except for CaO, H2O, and F, components with a lower concentration in the host olivine than in the melt (e.g., Al2O3, SiO2, Na2O, K2O, TiO2, S, and Cl) are enriched near the melt inclusion edges. This zonation is due to formation of an olivine-depleted boundary layer in the adjacent melt in response to cooling and crystallization of olivine on the walls of the melt inclusions, concurrent with diffusive propagation of the boundary layer toward the inclusion center. Concentration profiles of some components in the melt inclusions exhibit multicomponent diffusion effects such as uphill diffusion (CaO, FeO) or slowing of the diffusion of typically rapidly diffusing components (Na2O, K2O) by coupling to slow diffusing components such as SiO2 and Al2O3. Concentrations of H2O and F decrease toward the edges of some of the Siqueiros melt inclusions, suggesting either that these components have been lost from the inclusions into the host olivine late in their cooling histories and/or that these components are exhibiting multicomponent diffusion effects. A model has been developed of the time-dependent evolution of MgO concentration profiles in melt inclusions due to simultaneous depletion of MgO at the inclusion walls due to olivine growth and diffusion of MgO in the melt inclusions in response to this depletion. Observed concentration profiles were fit to this model to constrain their thermal histories. Cooling rates determined by a single-stage linear cooling model are 150–13,000 °C h−1 from the liquidus down to ~1,000 °C, consistent with previously determined cooling rates for basaltic glasses; compositional trends with melt inclusion size observed in the Siqueiros melt inclusions are described well by this simple single-stage linear cooling model. Despite the overall success of the modeling of MgO concentration profiles using a single-stage cooling history, MgO concentration profiles in some melt inclusions are better fit by a two-stage cooling history with a slower-cooling first stage followed by a faster-cooling second stage; the inferred total duration of cooling from the liquidus down to ~1,000 °C ranges from 40 s to just over 1 h. Based on our observations and models, compositions of zoned melt inclusions (even if measured at the centers of the inclusions) will typically have been diffusively fractionated relative to the initially trapped melt; for such inclusions, the initial composition cannot be simply reconstructed based on olivine-addition calculations, so caution should be exercised in application of such reconstructions to correct for post-entrapment crystallization of olivine on inclusion walls. Off-center analyses of a melt inclusion can also give results significantly fractionated relative to simple olivine crystallization. All melt inclusions from the Siqueiros and Galapagos sample suites exhibit zoning profiles, and this feature may be nearly universal in glassy, olivine-hosted inclusions. If so, zoning profiles in melt inclusions could be widely useful to constrain late-stage syneruptive processes and as natural diffusion experiments.

Raman spectroscopy explores molecular structural signatures of hidden materials in depth: Universal Multiple Angle Raman Spectroscopy.

Non-invasive 3D imaging in materials and medical research involves methodologies such as X-ray imaging, MRI, fluorescence and optical coherence tomography, NIR absorption imaging, etc., providing global morphological/density/absorption changes of the hidden components. However, molecular information of such buried materials has been elusive. In this article we demonstrate observation of molecular structural information of materials hidden/buried in depth using Raman scattering. Typically, Raman spectroscopic observations are made at fixed collection angles, such as, 90°, 135°, and 180°, except in spatially offset Raman scattering (SORS) (only back scattering based collection of photons) and transmission techniques. Such specific collection angles restrict the observations of Raman signals either from or near the surface of the materials. Universal Multiple Angle Raman Spectroscopy (UMARS) presented here employs the principle of (a) penetration depth of photons and then diffuse propagation through non-absorbing media by multiple scattering and (b) detection of signals from all the observable angles.

Non Linear Diffusion and Wave Damped Propagation: Weak Solutions and Statistical Turbulence Behavior

We present mathematical proofs on the existence and uniqueness of weak solutions for a special class of non linear parabolic and hyperbolic equations of mathematical physics subject to colored noise (structured turbulence) as random- initial conditions

A new scheme for the solution of reaction diffusion and wave propagation problems

In this paper, a robust numerical scheme is presented for the reaction diffusion and wave propagation problems. The present method is rather simple and straightforward. The Houbolt method is applied so as to convert both types of partial differential equations into an equivalent system of modified Helmholtz equations. The method of fundamental solutions is then combined with the method of particular solution to solve these new systems of equations. Next, based on the exponential decay of the fundamental solution to the modified Helmholtz equation, the dense matrix is converted into an equivalent sparse matrix. Finally, verification studies on the sensitivity of the method’s accuracy on the degree of sparseness and on the time step magnitude of the Houbolt method are carried out for four benchmark problems.