Application of double diffusivity model to superconductors

Konstantinidis, Dimitris; Eleftheriadis, I.E.; Aifantis, Elias C.

doi:10.1016/s0924-0136(00)00750-0

Cited by 9 publications

(25 citation statements)

References 9 publications

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“…1 and 2 conform qualitatively to predictions based on the deterministic double diffusivity models [11][12][13]; although the amplitudes are higher due to additional energy inputs through stochastic forcing. It is pertinent to remember that the root-mean-squared forms of the respective autocorrelation forms are the dimensional equivalents of the corresponding quantities in the deterministic models in [2,[11][12][13] and hence could be compared on a term-by-term basis. As to the crosscorrelation terms in the spatiotemporal dynamics ( Figs.…”

Section: Discussionsupporting

confidence: 67%

“…We should emphasize here that the simplifying assumption (D 1 = D 2 = D) used in arriving at the above result in no way sullies the implication of this analysis. For all realistic experimental observations [11,12] concerning double diffusivity, including its application in explaining the oxygen diffusivity in barium superconductors [13], the two diffusive constants typically differ by about three orders of magnitude whose exact correlation forms can be analyzed using the representations in the Appendix. Figure 1 uses identical noise strengths γ 1 = γ 2 = 1.…”

Section: A Phase Autocorrelation and Crosscorrelationmentioning

confidence: 99%

“…More specifically, we elaborate on the coupled system of the two partial differential equations proposed in [4], solved in [6], and thereafter extended in [7,8] to explain double porosity in media and through [9,10] to describe heat transport in materials with two temperatures. All of these theoretical enclaves were eventually implemented in [11][12][13] to interpret experimental measurements of diffusion in polycrystals and nanocrystalline aggregates. Some thermodynamic aspects of the deterministic double diffusivity model have been relatively recently discussed in [14], where the approach is probability based.…”

Section: Introductionmentioning

confidence: 99%

“…For microscale polycrystals with even larger grain sizes (∼1 µm), the two families of diffusion paths are represented by the bulk/grain interior and the surrounding/grain boundary space. With the above interpretation, we focus on modeling stochastically enhanced counterpart of the deterministic continuum model proposed and analyzed in [4][5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Double diffusivity model under stochastic forcing

Chattopadhyay

Aifantis

2017

Phys. Rev. E

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The "double diffusivity" model was proposed in the late 1970s, and reworked in the early 1980s, as a continuum counterpart to existing discrete models of diffusion corresponding to high diffusivity paths, such as grain boundaries and dislocation lines. It was later rejuvenated in the 1990s to interpret experimental results on diffusion in polycrystalline and nanocrystalline specimens where grain boundaries and triple grain boundary junctions act as high diffusivity paths. Technically, the model pans out as a system of coupled Fick-type diffusion equations to represent "regular" and "high" diffusivity paths with "source terms" accounting for the mass exchange between the two paths. The model remit was extended by analogy to describe flow in porous media with double porosity, as well as to model heat conduction in media with two nonequilibrium local temperature baths, e.g., ion and electron baths. Uncoupling of the two partial differential equations leads to a higher-ordered diffusion equation, solutions of which could be obtained in terms of classical diffusion equation solutions. Similar equations could also be derived within an "internal length" gradient (ILG) mechanics formulation applied to diffusion problems, i.e., by introducing nonlocal effects, together with inertia and viscosity, in a mechanics based formulation of diffusion theory. While being remarkably successful in studies related to various aspects of transport in inhomogeneous media with deterministic microstructures and nanostructures, its implications in the presence of stochasticity have not yet been considered. This issue becomes particularly important in the case of diffusion in nanopolycrystals whose deterministic ILG-based theoretical calculations predict a relaxation time that is only about one-tenth of the actual experimentally verified time scale. This article provides the "missing link" in this estimation by adding a vital element in the ILG structure, that of stochasticity, that takes into account all boundary layer fluctuations. Our stochastic-ILG diffusion calculation confirms rapprochement between theory and experiment, thereby benchmarking a new generation of gradient-based continuum models that conform closer to real-life fluctuating environments.

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Section: Discussionsupporting

confidence: 67%

Section: A Phase Autocorrelation and Crosscorrelationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Double diffusivity model under stochastic forcing

Chattopadhyay

Aifantis

2017

Phys. Rev. E

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“…Considering boundary layer fluctuations, stochasticity was added in the modeling and first stochastic gradient nanomechanics (SGNM) model was proposed in [8]. Using SGNM model, relaxation time is discussed thoroughly for a specific superconductors [24] in [9]. Also, linear stochastic resonance has been predicted and how stochastic effects start affecting the system is explained in [9].…”

Section: State Of the Artmentioning

confidence: 99%

Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model

Chattopadhyay,

Kundu,

Nath

et al. 2022

Preprint

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Model analogies and exchange of ideas between physics or chemistry with biology or epidemiology have often involved inter-sectoral mapping of techniques. Material mechanics has benefitted hugely from such interpolations from mathematical physics where dislocation patterning of platstically deformed metals [1, 2, 3] and mass transport in nanocomposite materials with high diffusivity paths such as dislocation and grain boundaries, have been traditionally analyzed using the paradigmatic Walgraef-Aifantis (W-A) double-diffusivity (D-D) model [4,5,6,7,8,9]. A long standing challenge in these studies has been the inherent nonlinear correlation between the diffusivity paths, making it extremely difficult to analyze their interdependence. Here, we present a novel method of approximating a closed form solution of the ensemble averaged density profiles and correlation statistics of coupled dynamical systems, drawing from a technique used in mathematical biology to calculate a quantity called the basic reproduction number R 0 , which is the average number of secondary infections generated from every infected. We show that the R 0 formulation can be used to calculate the correlation between diffusivity paths, agreeing closely with the exact numerical solution of the D-D model.

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Variations of heat equation on the half‐line via the Fokas method

Chatziafratis,

Fokas,

Aifantis

2024

Math Methods in App Sciences

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In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

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Application of double diffusivity model to superconductors

Cited by 9 publications

References 9 publications

Double diffusivity model under stochastic forcing

Double diffusivity model under stochastic forcing

Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model

Variations of heat equation on the half‐line via the Fokas method

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