Diffusion Matrix Research Articles

Гьольдерова неперервнiсть залежностi коефiцiєнтiв вiд резольвенти крайової задачi Дiрiхле з анiзотропним p-лапласiаном

We discuss the H¨older continuity property for the inverse mapping that identifies the diffusivity matrix A(x) in the main part of anisotropic p-Laplace equation as a function of resolvent operator. In particular, we prove that, within a chosen class of non-smooth admissible matrices the resolvent determines the anisotropic diffusivity in a unique manner and the correspondent inverse mapping is H¨older continuous in suitable topologies.

A cross-diffusion system derived from a Fokker\u2013Planck equation with partial averaging

A cross-diffusion system for two components with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker–Planck equation for the probability density associated with a multi-dimensional Itō process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.

Analyzing Technological Knowledge Diffusion Among Technological Fields Using Patent Data: The Example of Microfluidics

Study of technological knowledge diffusion can provide a basis for R&D planners to invest new R&D projects with a proper direction. In this paper, we introduce a modified method to analyze the diffusion degree of technological knowledge among different technological fields using patent data at the technology patent class level. This method first generates a patent citation network related to a specific research area and then establishes a technological field diffusion matrix according to the concordance between the patent document and the corresponding patent class. After that indicators are set up to measure the diffusion degree of technological field from two aspects of diffusion depth and breadth and then we discern the diffusion type of typical fields. Finally, we conduct microfluidic technology as a case study to prove the feasibility of the method.

Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach

We perform the asymptotic analysis of parabolic equations with stiff transport terms. This kind of problem occurs, for example, in collisional gyrokinetic theory for tokamak plasmas, where the velocity diffusion of the collision mechanism is dominated by the velocity advection along the Laplace force corresponding to a strong magnetic field. This work appeal to the filtering techniques. Removing the fast oscillations associated to the singular transport operator, leads to a stable family of profiles. The limit profile comes by averaging with respect to the fast time variable, and still satisfies a parabolic model, whose diffusion matrix is completely characterized in terms of the original diffusion matrix and the stiff transport operator. Introducing first order correctors allows us to obtain strong convergence results, for general initial conditions (not necessarily well prepared).

A low-Mach Roe-type solver for the Euler equations allowing for gravity source terms

In order to perform simulations of low Mach number flow in presence of gravity the technique from [23] is found insufficient as it is unable to cope with the presence of a hydrostatic equilibrium. Instead, a new modification of the diffusion matrix in the context of Roe-type schemes is suggested. We show that without gravity it is able to resolve the incompressible limit, and does not violate the conditions of hydrostatic equilibrium when gravity is present. These properties are verified by performing a formal asymptotic analysis of the scheme. Furthermore, we study its von Neumann stability when subject to explicit time integration and demonstrate its abilities on numerical examples.

Assessing Electrolyte Transport Properties with Molecular Dynamics

In this work we use estimates of ionic transport properties obtained from molecular dynamics to rank lithium electrolytes of different compositions. We develop linear response methods to obtain the Onsager diffusivity matrix for all chemical species, its Fickian counterpart, and the mobilities of the ionic species. We apply these methods to the well-studied propylene carbonate/ethylene carbonate solvent with dissolved LiBF4 and O2. The results show that, over a range of lithium concentrations and carbonate mixtures, trends in the transport coefficients can be identified and optimal electrolytes can be selected for experimental focus; however, refinement of these estimation techniques is necessary for a reliable ranking of a large set of electrolytes.

Low Design-Cost Fuzzy Controllers for Robust Stabilization of Nonlinear Partial Differential Systems

In general, the design cost of a fuzzy controller for nonlinear partial differential systems (PDSs) is very expensive. In this paper, a new approach for robust fuzzy $H_{\infty }$ stabilization design is developed for a class of $N$ -dimensional nonlinear parabolic PDSs. Further, two low design-cost robust fuzzy controllers, called the robust fuzzy area-controller and point-controller, as well as a normal design-cost robust fuzzy full-controller are proposed for this problem. The difference between the three control designs lies in their controller placement in the spatial domain. First, we present the $N$ -dimensional parabolic Takagi–Sugeno (T–S) fuzzy PDS based on the knowledge-based fuzzy system technique. Next, these three robust fuzzy controllers are constructed via solving diffusion matrix inequality (DMI) problems. With the proposed simple but general method using the Poincare inequality, the linear matrix inequality problems are provided to replace DMI problems for the robust fuzzy $H_{\infty }$ stabilization designs for computational simplicity. Further, the comparison of these three robust fuzzy controllers are demonstrated to enable a designer to select a low-cost option. Finally, a simulation example is provided to illustrate the design procedure and verify the performance of the proposed designs.

Diffusion in Multicomponent Liquids: From Microscopic to Macroscopic Scales.

In spite of considerable research on the nature of aqueous alcohol mixtures that are characterized by microscopic inhomogeneity or incomplete mixing at the molecular level, transport properties have received little attention. We report the results of a study on diffusion in the ternary mixture of water with two alcohols, that is, water + methanol + ethanol, which is investigated on microscopic and macroscopic scales by means of molecular simulation and Taylor dispersion experiments. A novel protocol is developed for the comparison of mutual diffusion coefficients sampled by two fundamentally different approaches, which allows for their critical analysis. Because of complex intermolecular interactions, given by the presence of hydrogen bonding, the analysis of transport processes in this mixture is challenging for not only on the microscopic scale for simulation techniques but also on the macroscopic scale due to unfavorable optical properties. Binary limits of the Fick diffusion matrix are used for validation of the experimental ternary mixture results together with the verification of the validity of the phenomenological Onsager reciprocal relations. The Maxwell-Stefan diffusion coefficients and the thermodynamic factor are sampled by molecular simulation consistently on the basis of given force field models. The protocol for the comparison of the results from both approaches is also challenging because Fick diffusion coefficients of ternary mixtures depend on the frame of reference. Accordingly, the measured coefficients are transformed from the volume-averaged to the molar-averaged frame of reference, and it is demonstrated that both approaches provide not only similar qualitative behavior along two concentration paths but also strong quantitative agreement. This coordinated work using different approaches to study diffusion in multicomponent mixtures is expected to be a significant step forward for the accurate assessment of cross-diffusion.

Parameter estimation for multivariate diffusion processes with the time inhomogeneously positive semidefinite diffusion matrix

ABSTRACTStatistical inference for the diffusion coefficients of multivariate diffusion processes has been well established in recent years; however, it is not the case for the drift coefficients. Furthermore, most existing estimation methods for the drift coefficients are proposed under the assumption that the diffusion matrix is positive definite and time homogeneous. In this article, we put forward two estimation approaches for estimating the drift coefficients of the multivariate diffusion models with the time inhomogeneously positive semidefinite diffusion matrix. They are maximum likelihood estimation methods based on both the martingale representation theorem and conditional characteristic functions and the generalized method of moments based on conditional characteristic functions, respectively. Consistency and asymptotic normality of the generalized method of moments estimation are also proved in this article. Simulation results demonstrate that these methods work well.

Theoretical and numerical investigation of diffusive instabilities in multi-component alloys

Diffusive instabilities of the Mullins-Sekerka type are one of the principal mechanisms through which microstructures form during solidification. In this study, we perform a linear stability analysis for the perturbation of a planar interface, where we derive analytical expressions to characterize the dispersion behavior in multi-component alloys under directional and isothermal solidification conditions. Subsequently, we confirm our calculations using phase-field simulations for different choices of the inter-diffusivity matrices. Thereafter, we highlight the characteristics of the dispersion curves upon change of the diffusivity matrix and the velocity. Finally, we also depict conditions for absolute stability of a planar interface under directional solidification conditions.

Diffusivities of Ternary Mixtures of n-Alkanes with Dissolved Gases by Dynamic Light Scattering

Theoretical approaches suggest that dynamic light scattering (DLS) signals from low-molecular-weight ternary mixtures are governed by fluctuations in temperature as well as two individual contributions from fluctuations in concentration that are related to the eigenvalues of the Fick diffusion matrix. Until now, this could not be proven experimentally in a conclusive way. In the present study, a detailed analysis of DLS signals in ternary mixtures consisting of n-dodecane (n-C12H26) and n-octacosane (n-C28H58) with dissolved hydrogen (H2), carbon monoxide (CO), or water (H2O) as well as of n-C12H26 or n-C28H58 with dissolved H2 and CO is given for temperatures up to 523 K and pressures up to 4.1 MPa. Thermal diffusivities of pure n-C12H26 and n-C28H58 as well as thermal and mutual diffusivities of their binary mixtures being the basis for the ternary mixtures with dissolved gas were studied for comparison purposes. For the investigated ternary mixtures, three individual signals could be distinguished in the time-resolved analysis of scattered light intensity by using photon correlation spectroscopy (PCS). For the first time, it could be evidenced that these signals are clearly associated with hydrodynamic modes. In most cases, the fastest mode observable for ternary mixtures is associated with the thermal diffusivity. The two further modes obviously related to the molecular mass transport are observable on different time scales and comparable to the modes associated with the concentration fluctuations in the respective binary mixtures. Comparison of the experimental data with results from molecular dynamics simulations revealed very good agreement.

Detailed study on self- and multicomponent diffusion of CO2-CH4 gas mixture in coal by molecular simulation

Gas diffusion plays a key role in CO2-enhanced recovery of coal bed methane (ECBM), where more than one types of gases coexist and multicomponent gas diffusion occurs. Such process is now usually described by non-coupled two-component gas diffusion equations which exclude the interactions between gases. Self-diffusion and mutual diffusion of CO2–CH4 mixture are investigated through molecular simulation for the first time. The self-diffusion coefficients of CO2andCH4 decrease with gas concentration but increase with temperature. The mutual diffusion coefficients of binary gas mixture of CO2–CH4incoal are computed through Maxwell–Stefan diffusion theory. A 2D diffusivity matrix |D| (with diagonal element Di and non-diagonal element Dij) is obtained to depict the mutual diffusion of the gas mixture. It is found that CO2 (CH4) diffusion is coupled with CH4 (CO2). The diffusion coupling strength of CO2andCH4 decreases with increasing gas concentration. Temperature positively affects Dij but minimally influences Di, resulting in large ratios of Dij/Di at high temperatures. It means that CO2–CH4 diffusion correlation interactions, which are not present in non-coupled pure gas diffusion equations, are necessary to analyze gas mixture diffusion in coal. In this case, non-coupled pure gas diffusion equations are inadequate for description of CO2–CH4 mixture diffusion. The coupling between the gases can be ignored only at very high temperatures (T>400K), which means a large depth of coal bed in ECBM engineering.

Multicomponent diffusion in silicate melts: SiO2–TiO2–Al2O3–MgO–CaO–Na2O–K2O System

Nine successful diffusion couple experiments were carried out in a 7-component haplobasaltic silicate melt SiO2–TiO2–Al2O3–MgO–CaO–Na2O–K2O system to study multicomponent diffusion at ∼1500°C and 1GPa, typically with compositional gradients in only two components in each experiment. At least two concentration traverses were measured for each experiment. Effective binary diffusion coefficients (EBDC) for monotonic profiles were obtained by an error function fit, and the EBDC of a given component is dependent on its counter diffusing component, especially for SiO2. The EBDC’s of SiO2 vary from 15.7μm2/s when diffusing against Al2O3, to 102.9μm2/s when diffusing against K2O. Furthermore, the multicomponent diffusion matrix was obtained by simultaneously fitting profiles of all oxides in all experiments. Most features in the diffusion profiles, for example uphill diffusion, are captured well by this 6×6 diffusion matrix. The slowest diffusing eigenvector is largely due to the exchange between Si and Al, and the fastest diffusing eigenvector is the exchange of Na with all other components. An anorthite dissolution experiment was also conducted to test whether the diffusion matrix can be applied to mineral dissolution experiments. The calculated diffusion profiles in the melt during anorthite dissolution roughly match the measured profiles, demonstrating the validity and utility of the diffusion matrix in this FeO-free aluminosilicate melt system.

Converting Multi-Shell and Diffusion Spectrum Imaging to High Angular Resolution Diffusion Imaging

Multi-shell and diffusion spectrum imaging (DSI) are becoming increasingly popular methods of acquiring diffusion MRI data in a research context. However, single-shell acquisitions, such as diffusion tensor imaging (DTI) and high angular resolution diffusion imaging (HARDI), still remain the most common acquisition schemes in practice. Here we tested whether multi-shell and DSI data have conversion flexibility to be interpolated into corresponding HARDI data. We acquired multi-shell and DSI data on both a phantom and in vivo human tissue and converted them to HARDI. The correlation and difference between their diffusion signals, anisotropy values, diffusivity measurements, fiber orientations, connectivity matrices, and network measures were examined. Our analysis result showed that the diffusion signals, anisotropy, diffusivity, and connectivity matrix of the HARDI converted from multi-shell and DSI were highly correlated with those of the HARDI acquired on the MR scanner, with correlation coefficients around 0.8~0.9. The average angular error between converted and original HARDI was 20.7° at voxels with signal-to-noise ratios greater than 5. The network topology measures had less than 2% difference, whereas the average nodal measures had a percentage difference around 4~7%. In general, multi-shell and DSI acquisitions can be converted to their corresponding single-shell HARDI with high fidelity. This supports multi-shell and DSI acquisitions over HARDI acquisition as the scheme of choice for diffusion acquisitions.

Robust finite-time stability and stabilization of uncertain Markovian jump systems with time-varying delay

This paper deals with the robust finite-time stability and stabilization problems of uncertain stochastic delayed jump systems, where the uncertainty is in the form of additive perturbations and exists in the drift and diffusion sections simultaneously. Though perturbation, time-varying delay and Brownian motion existing at the same time, two conditions checking its robust finite-time stability are proposed by a mode-dependent parameter approach, which are different from some existing methods. Based on the proposed results, sufficient conditions for the existence of the state-feedback controller are provided with LMIs, which could be solved directly. It is seen that all the features of the underlying system such as time-varying delay, perturbation, diffusion, mode-dependent parameters and uncertain transition rate matrix play important roles in the system analysis and synthesis of finite-time stability. Finally, numerical examples are used to demonstrate the effectiveness and superiority of the proposed methods.

Analysis of Non-Equilibrium Fluctuations In A Ternary Liquid Mixture

From the benchmark values of the diffusion and thermodiffusion coefficients of the tetralin, isobutylbenzene and n-dodecane ternary mixture, and the published optical contrast factors, we have evaluated the theoretical amplitudes of the two composition modes of the refractive-index fluctuations. Shadowgraph experiments have been performed on ground, where the current theory is expected to be correct only for large wave vectors. Two decay times have been observed experimentally. The fastest one being related to the thermal diffusivity of the mixture, while the slower one to mass diffusion. Hence, it has not been possible to distinguish the two eigenvalues of the mass diffusion matrix, a problem also encountered in traditional light-scattering with ternary mixtures of similar-size molecules. Thus, to compare the measured Intermediate Scattering Function with theory, we fix the amplitudes and decay rates to the benchmark values, use the wave number as a fitting parameter, and compare it to the experimental wave number. The good agreement between theory and experiments for the larger wave numbers validates the theory developed for the microgravity conditions.

A meeting point of entropy and bifurcations in cross-diffusion herding

A cross-diffusion system modelling the information herding of individuals is analysed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

Multiscale modelling of transport phenomena for materials with n-layered embedded fibres. Part II: Investigation of fibre packing effects

The case of molecular diffusion is investigated and a method is proposed to predict the effective diffusion properties of transversely isotropic composite materials. This paper is the second one of a set of two consecutive papers published in this volume. The general modelling strategy was devised in the first paper (Hervé-Luanco and Joannès, 2016) and applied to the biphasic case in this one. Closed-form analytical expressions are given to investigate fibre packing effects on the effective transverse diffusivity of any unidirectional composite. The specific case of insulated fibres embedded in a diffusive matrix is studied and a clear procedure is then described to determine the model parameters. It is worth noting that only a very simple cross-section image analysis of the unidirectional composite medium is required to provide the two morphological parameters needed to predict the overall diffusivity tensor. Finally, an application on a real microstructure is provided to demonstrate the efficiency of the suggested turnkey method. Very simple analytical calculations make possible parametric studies less expensive than numerical or experimental characterizations.

Atomic mobility in calcium and sodium aluminosilicate melts at 1200 °C

Multicomponent chemical diffusion in liquids of the quaternary system CaO–Na2O–Al2O3–SiO2 has been studied. Diffusion-couple experiments were performed at 1200°C and for different durations around a central composition of 64.5wt%SiO2, 13.3wt%Na2O, 10.8wt%CaO, 11.4wt%Al2O3, leading to an overconstrained system of equations that was used to determine the diffusion matrix of the system. The dominant eigenvector of the diffusion matrix was found to correspond to the exchange between sodium and calcium, consistent with the results of the ternary soda–lime silica system. On the other hand, neither of the other two eigenvectors of the diffusion matrix of the quaternary system involve sodium. Given a factor of 50 between the dominant and second eigenvalue, diffusion couples involving the exchange of sodium oxide and a network-forming oxide result in strong uphill diffusion of calcium. The second eigenvector, corresponding to the exchange of calcium with silicon and aluminum, is close to the dominant eigenvector found in previous studies of ternary alkaline-earth aluminosilicate systems. Our results therefore suggest that simple systems may be used to understand diffusive mechanisms in more complex systems.

Diffusion in FCC Co-rich Co-Al-W Alloys at 900 and 1000 °C

Diffusion couple experiments between various Co-rich face centered cubic (FCC) alloys in the Co-W-Al ternary system have been conducted at 900 and 1000 °C. Diffusion coefficients have been extracted for the Co-W binary and for ternary alloys at compositions where the diffusion paths cross. In addition, a least squares method has been utilized to optimize diffusion mobility parameters using DICTRA simulations to best fit the experimental concentration versus distance curves. Predictions of the diffusion matrix using the refined mobility database are in good agreement with the values obtained at the diffusion path crossing points.