Nonlinear Transport Processes Research Articles

Nonlinear equilibria and transport processes in burning plasmas

In this work, we put forward a general phase space transport theory in axisymmetric tokamak plasmas based upon the concept of zonal state (ZS). Within this theoretical framework, the ZS corresponds to a renormalized plasma nonlinear equilibrium consisting of phase space zonal structures (PSZS) and zonal electromagnetic fields (ZFs) which evolve self-consistently with symmetry breaking fluctuations and sources/collisions. More specifically, our approach involves deriving governing equations for the evolution of particle distribution functions (i.e, PSZS), which can be used to compute the corresponding macro-/meso-scale evolving magnetized plasma equilibrium adopting the Chew Goldberger Low description, separating the spatiotemporal microscale structures. The nonlinear physics of ZFs and of geodesic acoustic modes (GAMs)/energetic particle driven GAMs is then analyzed to illustrate the applications of our theory.

Numerical investigation of the VB growth of CaF2 crystal with supercooled crucible wall

Decoupling the nonlinear transport and phase change processes during the growth of calcium fluoride crystals is essential for the preparation of high-quality crystals. In this work, a global model was adopted to predict the melt-crystal interface, convection in the melt, and temperature distribution in the furnace. Our simulation results show that the crucible shoulder experiences severe supercooling, which can cause spontaneous nucleation on the supercooled crucible wall and a periphery-to-center flow along the concave interface. This can result in the formation of grains in the peripheral crystal and impurity accumulation in the middle region, both of which significantly degrade the crystal quality. To address this issue, we proposed using a shorter-shoulder crucible to increase heat absorption and an insulation coat to reduce heat dispersion. Both these approaches are effective in reducing supercooling, but the use of an insulation coat around the supercooled shoulder can effectively improve the shape of the m-c interface while still maintaining the beneficial transitional role of the crucible shoulder. This article presents a technical approach for controlling the melt-crystal interface in the vertical Bridgman growth of calcium fluoride crystals, and is a valuable resource for future experimental research.

Thermolab: A Thermodynamics Laboratory for Nonlinear Transport Processes in Open Systems

AbstractWe developed a numerical thermodynamics laboratory called “Thermolab” to study the effects of the thermodynamic behavior of nonideal solution models on reactive transport processes in open systems. The equations of the state of internally consistent thermodynamic data sets are implemented in MATLAB functions and form the basis for calculating Gibbs energy. A linear algebraic approach is used in Thermolab to compute Gibbs energy of mixing for multicomponent phases to study the impact of the nonideality of solution models on transport processes. The Gibbs energies are benchmarked with experimental data, phase diagrams, and other thermodynamic software. Constrained Gibbs minimization is exemplified with MATLAB codes and iterative refinement of composition of mixtures may be used to increase precision and accuracy. All needed transport variables such as densities, phase compositions, and chemical potentials are obtained from Gibbs energy of the stable phases after the minimization in Thermolab. We demonstrate the use of precomputed local equilibrium data obtained with Thermolab in reactive transport models. In reactive fluid flow the shape and the velocity of the reaction front vary depending on the nonlinearity of the partitioning of a component in fluid and solid. We argue that nonideality of solution models has to be taken into account and further explored in reactive transport models. Thermolab Gibbs energies can be used in Cahn‐Hilliard models for nonlinear diffusion and phase growth. This presents a transient process toward equilibrium and avoids computational problems arising during precomputing of equilibrium data.

A Hybrid Streamline Upwind Finite Volume-Finite Element Method for Semiconductor Continuity Equations

This article presents the construction and analysis of a hybrid numerical method for the discretization of the convection-dominated nonlinear carrier transport process in semiconductor devices. In the construction of the method, the edge streamline upwind (SU) current density model is employed to overcome the nonconvergence brought by the nonlinear property of the current continuity equation, and the mixed finite volume-finite element method (FVFEM) is utilized to effectively extend the edge SU current density model to multidimensional applications. The proposed method is more reliable and flexible than the finite box (FB) method and streamline upwind Petrov–Galerkin (SUPG) method. The performance of several popular SU current density models is compared to find the optimal one. The proposed method is validated by comparing the calculated results with those calculated using commercial software. Based on the streamline upwind-finite volume-finite element method (SU-FVFEM), the finite element method (FEM), and the domain decomposition method (DDM), an in-house parallel computing electrothermal simulator is developed for large-scale semiconductor devices. The performance of the in-house developed parallel simulator is evaluated through simulations of a p-n junction diode and a 3-D bipolar transistor. The results demonstrated that the developed parallel simulator possesses good accuracy, applicability, and scalability.

Explicit physics-informed neural networks for nonlinear closure: The case of transport in tissues

In upscaling methods, closures for nonlinear problems present a well-known challenge. While a number of theoretical methods have been proposed for handling such closures, nonlinearities still remain a significant obstacle for many problems. In this work, we use a combination of formal upscaling and data-driven machine learning for explicitly closing a nonlinear transport and reaction process in multiscale tissues. The classical effectiveness factor model is used to formulate the macroscale reaction kinetics. We train a multilayer perceptron network using training data generated by direct numerical simulations over microscale examples. Once trained, the network is used in an algorithm for numerically solving the upscaled (coarse-grained) differential equation describing mass transport and reaction in two example tissues. The network is described as being explicit in the sense that the network is trained using macroscale concentrations and gradients of concentration as components of the feature space rather than incorporating them as part of a constraint in the optimization process.Network training and solutions to the macroscale transport equations were computed for two different tissues. The two tissue types (brain and liver) exhibit markedly different geometrical complexity and spatial scale (cell size and sample size). The upscaled solutions for the average concentration are compared with numerical solutions derived from the microscale concentration fields by a posteriori averaging. There are three outcomes of this work of particular note.1.Our overall approach results in an upscaled nonlinear PDE. The PDE is closed using a neural network, and our approach results in the definition of the classical effectiveness factor for effecting closure.2.We identify particular source terms for the closure problem that are important for representing the structure of the closure. These source terms involve macroscale concentrations and their gradients. We adopt these source terms to use as explicit features in the learning algorithm. We find the trained networks that include the macroscale source terms generate models that are able to predict the correction factor with increased fidelity over those that do not.3.We find that the trained network exhibits good generalizability, and it is able to predict the effectiveness factor with high fidelity for realistically-structured tissues despite the significantly different scale and geometrical complexity of the two example tissue types.This latter result emphasizes our purposeful connection between conventional averaging methods with the use of machine learning for closure; this contrasts with some machine learning methods for upscaling where the exact form of the macroscale equation remains unknown.

Atomic‐Scale Characterization of Negative Differential Resistance in Ferroelectric Bi2WO6

AbstractNegative differential resistance (NDR), a quantum nonlinear electron transport process, has long attracted research interest owing to its intriguing underlying physics and promising applications in high‐speed electronics. Here, the authors report the NDR behavior in (001)‐oriented ferroelectric Bi2WO6 using scanning tunneling microscopy (STM). The current–voltage characteristics of the diode configuration consisting of an STM tip over the [BiO]+‐[WO4]2−‐[BiO]+ terrace exhibit NDR features. Scanning tunneling spectroscopy combined with density functional theory calculations indicates that the observed NDR results from robust resonant tunneling through confined energy levels within the 2D [WO4]2− layer. The atomically resolved NDR found in such a transition metal oxide offers a unique band structure for novel ferroelectric‐based devices.

Transasymptotics and hydrodynamization of the Fokker-Planck equation for gluons

We investigate the non-linear transport processes and hydrodynamization of a system of gluons undergoing longitudinal boost-invariant expansion. The dynamics is described within the framework of the Boltzmann equation in the small-angle approximation. The kinetic equations for a suitable set of moments of the one-particle distribution function are derived. By investigating the stability and asymptotic resurgent properties of this dynamical system, we demonstrate, that its solutions exhibit a rather different behavior for large (UV) and small (IR) effective Knudsen numbers. Close to the forward attractor in the IR regime the constitutive relations of each moment can be written as a multiparameter transseries. This resummation scheme allows us to extend the definition of a transport coefficient to the non-equilibrium regime naturally. Each transport coefficient is renormalized by the non-perturbative contributions of the non-hydrodynamic modes. The Knudsen number dependence of the transport coefficient is governed by the corresponding renormalization group flow equation. An interesting feature of the Yang-Mills plasma in this regime is that it exhibits transient non-Newtonian behavior while hydrodynamizing. In the UV regime the solution for the moments can be written as a power-law asymptotic series with a finite radius of convergence. We show that radius of convergence of the UV perturbative expansion grows linearly as a function of the shear viscosity to entropy density ratio. Finally, we compare the universal properties in the pullback and forward attracting regions to other kinetic models including the relaxation time approximation and the effective kinetic Arnold-Moore-Yaffe (AMY) theory.

Transport processes in Newtonin and non-Newotonian fluids and fluctuation-dissipation theorems

The subject of this paper is molecular modelling of the transport processes in liquids including nanofluids. The weak and strong nonequilibrium processes are considered. In the first case, the corresponding fluctuation-dissipation theorems are formulated. These theorems determine the relation between the dissipative processes and equilibrium thermal fluctuations of the mass, momentum, and energy in the molecular system. Two different formulations of these theorem are discussed in detail. Various systems are considered: molecular liquids, nanofluids, turbulent liquids, fluids in confined conditions, and microviscosity in biomembranes. In the last part the nonlinear transport processes and rheology of the non-Newtonian fluids are discussed.

Neuronal Oscillations on Evolving Networks: Dynamics, Damage, Degradation, Decline, Dementia, and Death.

Neurodegenerative diseases, such as Alzheimer's or Parkinson's disease, show characteristic degradation of structural brain networks. This degradation eventually leads to changes in the network dynamics and degradation of cognitive functions. Here, we model the progression in terms of coupled physical processes: The accumulation of toxic proteins, given by a nonlinear reaction-diffusion transport process, yields an evolving brain connectome characterized by weighted edges on which a neuronal-mass model evolves. The progression of the brain functions can be tested by simulating the resting-state activity on the evolving brain network. We show that while the evolution of edge weights plays a minor role in the overall progression of the disease, dynamic biomarkers predict a transition over a period of 10 years associated with strong cognitive decline.

Predicting the unexpected in stomatal gas exchange: not just an open-and-shut case.

Plant membrane transport, like transport across all eukaryotic membranes, is highly non-linear and leads to interactions with characteristics so complex that they defy intuitive understanding. The physiological behaviour of stomatal guard cells is a case in point in which, for example, mutations expected to influence stomatal closing have profound effects on stomatal opening and manipulating transport across the vacuolar membrane affects the plasma membrane. Quantitative mathematical modelling is an essential tool in these circumstances, both to integrate the knowledge of each transport process and to understand the consequences of their manipulation in vivo. Here, we outline the OnGuard modelling environment and its use as a guide to predicting the emergent properties arising from the interactions between non-linear transport processes. We summarise some of the recent insights arising from OnGuard, demonstrate its utility in interpreting stomatal behaviour, and suggest ways in which the OnGuard environment may facilitate ‘reverse-engineering’ of stomata to improve water use efficiency and carbon assimilation.

High order numerical schemes for solving fractional powers of elliptic operators

In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic operators. In order to solve these problems numerically it is proposed (Vabishchevich, 2015) to consider equivalent local nonstationary initial value pseudo-parabolic problems. Previously such problems were solved by using the standard implicit backward and symmetrical Euler methods. In this paper we use the one-parameter family of three-level finite difference schemes for solving the initial value problem for the first order nonstationary pseudo-parabolic problem. The fourth-order approximation scheme is developed by selecting the optimal value of the weight parameter. The results of the theoretical analysis are supplemented by results of extensive computational experiments.

Non-linear gas transport inside an ultra-tight Longmaxi shale core under thermal stimulation conditions

Gas adsorption/desorption and non-linear mass transport in ultra-tight shale were very sensitive to temperature and they become even more intricate as temperature changed. In this study, a self-similarity mathematical model was developed to simulate the gas decline process under thermal stimulation conditions. This model not only incorporated slip and free molecular flow but also gas adsorption/desorption. Besides, a canister test was conducted to study the effect of heating temperature on gas production rates of a fresh shale core. The experiment results showed that gas production rate was increased by raising the heating temperature. Free gas is the main source for the gas flow under 55 °C, while adsorbed gas is the main source under 110 °C. Lastly, the declined trend of simulated gas production rate followed a power law of t−0.509+t−0.736 at 55 °C and t−0.169+t−1.828 under at 110 °C, which matched the experimental data well. From the scaling data, it was suggested that free molecular diffusion was the main gas transport form inside the shale both under 55 °C–110 °C. Our experimental results and theoretic model showed that higher heating temperature could promote the gas production rate significantly, and the model can well predict the non-linear gas transport process.

Monsoonal control on a delayed response of sedimentation to the 2008 Wenchuan earthquake.

Infrequent extreme events such as large earthquakes pose hazards and have lasting impacts on landscapes and biogeochemical cycles. Sediments provide valuable records of past events, but unambiguously identifying event deposits is challenging because of nonlinear sediment transport processes and poor age control. Here, we have been able to directly track the propagation of a tectonic signal into stratigraphy using reservoir sediments from before and after the 2008 Wenchuan earthquake. Cycles in magnetic susceptibility allow us to define a precise annual chronology and identify the timing and nature of the earthquake's sedimentary record. The grain size and Rb/Sr ratio of the sediments responded immediately to the earthquake. However, the changes were muted until 2 years after the event, when intense monsoonal runoff drove accumulation of coarser grains and lower Rb/Sr sediments. The delayed response provides insight into how climatic and tectonic agents interact to control sediment transfer and depositional processes.

Experimental and theoretical study of density fluctuations near the stack ends of a thermoacoustic prime mover

This paper proposes the experimental and theoretical study of nonlinear heat transport processes generated by large amplitude acoustic oscillations at the ends of a stack of plates in the presence of a temperature gradient. These processes are notably involved in the operation of thermoacoustic engines. The measurement method, a time-resolved and full-field digital holography interferometry technique, enables to measure the density fluctuations from the optical phase difference between two laser beams. This technique is applied to the analysis of density fluctuations in the vicinity of a stack submitted to a temperature gradient, firstly for the case of (uncontrolled) self-sustained acoustic oscillations generated spontaneously in a standing wave thermoacoustic prime mover, and secondly for the case of an assigned acoustic field whose amplitude is controlled by an external sound source. A theoretical model describing the advective heat transport by sound at the ends of the heated stack is also presented, and numerical simulations are then carried out. The comparison between experimental data and numerical simulations is provided for several stack positions, several sound pressure levels, and several amounts of heat supplied to the stack, and the results show good agreement between the experiments and the model.

An Atlantic streamer in stratospheric ozone observations and SD-WACCM simulation data

Abstract. Observation and simulation of individual ozone streamers are important for the description and understanding of non-linear transport processes in the middle atmosphere. A sudden increase in mid-stratospheric ozone occurred above central Europe on 4 December 2015. The GROund-based Millimeter-wave Ozone Spectrometer (GROMOS) and the Stratospheric Ozone MOnitoring RAdiometer (SOMORA) in Switzerland measured an ozone enhancement of about 30 % at 34 km altitude (8.3 hPa) from 1 to 4 December. A similar ozone increase is simulated by the Specified Dynamics Whole Atmosphere Community Climate (SD-WACCM) model. Further, the global ozone fields at 34 km altitude (8.3 hPa) from SD-WACCM and the satellite experiment Aura/MLS show a remarkable agreement for the location and timing of an ozone streamer (large-scale tongue-like structure) extending from the subtropics in northern America over the Atlantic to central Europe. This agreement indicates that SD-WACCM can inform us about the wind inside the Atlantic ozone streamer. SD-WACCM shows an eastward wind of about 100 m s−1 inside the Atlantic streamer in the mid-stratosphere. SD-WACCM shows that the Atlantic streamer flows along the edge of the polar vortex. The Atlantic streamer turns southward at an erosion region of the polar vortex located above the Caspian Sea. The spatial distribution of stratospheric water vapour indicates a filament outgoing from this erosion region. The Atlantic streamer, the polar vortex erosion region and the water vapour filament belong to the process of planetary wave breaking in the so-called surf zone of the northern midlatitude winter stratosphere.

In Situ Investigation of Mixed Ionic and Electronic Transport across Dense Lithium Peroxide Films

The performance of lithium-oxygen batteries is strongly limited by the insulating properties of the discharge product lithium peroxide Li2O2. Controversial viewpoints have been put forward in the literature regarding the mechanisms of electronic transport across thin Li2O2 films. In this context, the ion transport properties of the films have been largly neglected. We have carried out a detailed study of ion and electron transport across dense Li2O2 films on top of glassy carbon electrodes in an acetonitrile-based electrolyte. To this end, we have combined electrochemical impedance measurements with AFM scratching experiments. The results show that the ionic resistance and the electronic resistance exhibit a very similar dependence on the thickness of the films. This similar thickness dependence can be explained in a straightforward fashion by nonlinear ion and electron hopping transport processes due to large electrical potential and chemical potential gradients in the thin Li2O2 layers. The impedance spectra of our film are at variance with hole tunneling models and with models describing a non-uniform film growth on the electrode.

Dynamics of ion transport and electric double layer in single conical nanopores

Structurally defined nanopores have received great attention in the past decades in part due to the novel mass transport phenomena resolved at such nanoscale solid-liquid interfaces. New capabilities and/or better efficiency are envisioned in molecular sensing, energy storage and conversion, nanofluidics, and membrane transport. This review will focus on the fundamental mass transport processes confined by asymmetric nanoscale structures particularly in conical nanopores. The fabrication of different nanopore devices and the double layer theory are briefly introduced first as background. The nonlinear and dynamic mass transport processes, including ionic current rectification, dynamic concentration polarization, memory effects and capacitive charging/discharging processes are then reviewed. The fundamental understanding and quantification of dynamic transport features offer new physical insights in the nanoscale mass transport processes and has significant implications in related electroanalytical and other applications.

Effect of Porosity Distribution on Intercalation-Induced Stresses and Plating Reaction in Li-Ion Battery

Detailed models that incorporate electrochemical, transport, and thermodynamic processes along with the geometry of the underlying system can be used to predict, monitor and control the internal states of a battery. The porous electrode Pseudo-2 dimensional (P2D) model (1) is one such modeling framework which gives excellent insights into the workings of lithium-ion battery with reasonable computational cost. While various factors (thickness of electrode and current collector, particle radius, filler fractions, porosity etc.) can be varied to get the optimal battery design, here we will focus on the porosity distribution in anode and its effect on battery performance especially on the factors that affect safety and capacity fade (e.g. plating side reaction and intercalation-induced stresses etc.). Porosity and tortuosity are critical parameters for battery performance as they affect reaction rates and transport processes in various ways (e.g. local surface area, overall diffusivity and conductivities of solid phase and electrolyte phase). First design optimization (porosity and thickness) for lithium ion battery can be traced back to the work done by Prof. Newman using reaction zone model (2). Work on optimal porosity distribution considering ohmic drop has been done by Ramadesigan et. al. (3). This presentation will focus on effect of porosity distribution on energy capacity, power rating, intercalation-induced stresses and lithium plating side reactions at various temperatures. The nonlinear dependence reaction rate and transport processes on potential, concentration and temperature makes the analytical treatment of porosity distribution very difficult, hence numerical simulation are performed to assess the performance of lithium ion battery with porosity distribution using P2D model. Anode is vulnerable to various capacity fade mechanics like intercalation-induced stresses (4), lithium plating (5, 6), SEI formation (7) etc., especially at the separator-anode interface. This presentation will also focus on effect of porosity distribution on capacity fade at different location within the anode. Acknowledgements The work presented herein was funded in part by the Advanced Research Projects Agency – Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR0000275.

Nonlinear Transport Processes in Non‐Ideal Media

AbstractThe theory of non‐linear transport is elaborated to determine the Burnett transport properties of non‐ideal multielement plasma and neutral systems. The procedure of the comparison of the phenomenological conservation equations of a continuous dense medium and the microscopic equations for dynamical variables operators in the form of the generalized nonlinear Langevin equations is used for the definition of these properties. In consequence, the microscopic expressions of transport coefficients corresponding to second order thermal disturbances (temperature, mass velocity, etc.) have been found in the long‐wavelength and low frequency limits of correlations functions. Situations are considered when the use of nonlinear transport characteristics (in the Burnett approximation) significantly affects the results of modeling of hydrodynamic processes in dense charged and neutral media. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A design optimization methodology for Li+ batteries

Design optimization for functionally graded battery electrodes is shown to improve the usable energy capacity of Li batteries predicted by computational simulations and numerically optimizing the electrode porosities and particle radii. A multi-scale battery model which accounts for nonlinear transient transport processes, electrochemical reactions, and mechanical deformations is used to predict the usable energy storage capacity of the battery over a range of discharge rates. A multi-objective formulation of the design problem is introduced to maximize the usable capacity over a range of discharge rates while limiting the mechanical stresses. The optimization problem is solved via a gradient based optimization. A LiMn2O4 cathode is simulated with a PEO–LiCF3SO3 electrolyte and both a Li Foil (half cell) and LiC6 anode. Studies were performed on both half and full cell configurations resulting in distinctly different optimal electrode designs. The numerical results show that the highest rate discharge drives the simulations and the optimal designs are dominated by Li+ transport rates. The results also suggest that spatially varying electrode porosities and active particle sizes provides an efficient approach to improve the power-to-energy density of Li+ batteries. For the half cell configuration, the optimal design improves the discharge capacity by 29% while for the full cell the discharge capacity was improved 61% relative to an initial design with a uniform electrode structure. Most of the improvement in capacity was due to the spatially varying porosity, with up to 5% of the gains attributed to the particle radii design variables.