Maximum Entropy Production Rate Research Articles

Structure and diffuseness model of solid-liquid interface for binary alloys

Using the morphological instability at the solid–liquid interface as a basis by the maximum entropy production rate principle (MEPR), a model is presented on the morphological structure and diffuseness of the interface during directional solidification of binary alloys. It is shown that, the independent diffuseness theory of Cahn and the Jackson roughness criterion can be unified at a limiting condition under this new MEPR solidification model. The model under the principle of MEPR is applied to describe the evolution of atomistically smooth and rough interfaces through the evaluation of the size of the solid–liquid interface and the of number atomic layers. The model is tested with data for binary alloys of aluminium, lead, and tin at varying solute concentrations. The results showed strong agreement with available data from experimental measurements.

Effects of Stochastic Noises on Limit-Cycle Oscillations and Power Losses in Fusion Plasmas and Information Geometry

We investigate the effects of different stochastic noises on the dynamics of the edge-localised modes (ELMs) in magnetically confined fusion plasmas by using a time-dependent PDF method, path-dependent information geometry (information rate, information length), and entropy-related measures (entropy production, mutual information). The oscillation quenching occurs due to either stochastic particle or magnetic perturbations, although particle perturbation is more effective in this amplitude diminishment compared with magnetic perturbations. On the other hand, magnetic perturbations are more effective at altering the oscillation period; the stochastic noise acts to increase the frequency of explosive oscillations (large ELMs) while decreasing the frequency of more regular oscillations (small ELMs). These stochastic noises significantly reduce power and energy losses caused by ELMs and play a key role in reproducing the observed experimental scaling relation of the ELM power loss with the input power. Furthermore, the maximum power loss is closely linked to the maximum entropy production rate, involving irreversible energy dissipation in non-equilibrium. Notably, over one ELM cycle, the information rate appears to keep almost a constant value, indicative of a geodesic. The information rate is also shown to be useful for characterising the statistical properties of ELMs, such as distinguishing between explosive and regular oscillations and the regulation between the pressure gradient and magnetic fluctuations.

Morphological growth criterion for faceted to non-faceted transition in pure materials

A criterion to explain the morphological growth transition from faceted to non-faceted (f-nf) morphology in the case of directional solidification is presented for pure materials. The criterion is based on the number of atomic layers at the solid–liquid interface and allows a foreknowledge of the solidification growth velocity at which f-nf transition becomes significant. The basis for the transition criterion hinges on the Cahn theory of interface structure and the theory of liquid–solid transition by the maximum entropy production rate (MEPR) principle. The criterion is tested with a number of pure materials and compared with experimental measurements for salol. The criterion fits quite well with available experimental data.

Geometry Design of Square Column Heating Devices in Jet Channels

Based on the constructal theory, a heat dissipation optimization model for discrete square column heating devices on thermal conduction bases in 2D jet channels was established. With the total longitudinal section area and the discrete heat source height as constraints, the maximum temperature and entropy production rate of the system were taken as optimization objectives, and the length ratio of each heating device was taken as the optimization variables in the geometry design. The effects of the jet velocity and the heating device spacing on the constructal optimization of the heating device were analyzed. When the jet velocity and the heating device spacing are fixed, there will exist optimal length ratios to minimize the maximum temperature and the entropy production rate of the system, but the optimal length ratios corresponding to different jet velocities and different heating devices spacings are different. The results provide a theoretical guidance for the thermal design of square column heating devices.

Self-Organization, Entropy Generation Rate, and Boundary Defects: A Control Volume Approach.

Self-organization that leads to the discontinuous emergence of optimized new patterns is related to entropy generation and the export of entropy. Compared to the original pattern that the new, self-organized pattern replaces, the new features could involve an abrupt change in the pattern-volume. There is no clear principle of pathway selection for self-organization that is known for triggering a particular new self-organization pattern. The new pattern displays different types of boundary-defects necessary for stabilizing the new order. Boundary-defects can contain high entropy regions of concentrated chemical species. On the other hand, the reorganization (or refinement) of an established pattern is a more kinetically tractable process, where the entropy generation rate varies continuously with the imposed variables that enable and sustain the pattern features. The maximum entropy production rate (MEPR) principle is one possibility that may have predictive capability for self-organization. The scale of shapes that form or evolve during self-organization and reorganization are influenced by the export of specific defects from the control volume of study. The control volume (CV) approach must include the texture patterns to be located inside the CV for the MEPR analysis to be applicable. These hypotheses were examined for patterns that are well-characterized for solidification and wear processes. We tested the governing equations for bifurcations (the onset of new patterns) and for reorganization (the fine tuning of existing patterns) with published experimental data, across the range of solidification morphologies and nonequilibrium phases, for metallic glass and featureless crystalline solids. The self-assembling features of surface-texture patterns for friction and wear conditions were also modeled with the entropy generation (MEPR) principle, including defect production (wear debris). We found that surface texture and entropy generation in the control volume could be predictive for self-organization. The main results of this study provide support to the hypothesis that self-organized patterns are a consequence of the maximum entropy production rate per volume principle. Patterns at any scale optimize a certain outcome and have utility. We discuss some similarities between the self-organization behavior of both inanimate and living systems, with ideas regarding the optimizing features of self-organized pattern features that impact functionality, beauty, and consciousness.

Effects of composite material fin conductivity on natural convection heat transfer and entropy generation inside 3D cavity filled with hybrid nanofluid

The current paper offers a numerical investigation of the combined effects of an inserted composite material fin conductivity ratio and hybrid nanofluid volume fraction on natural convection heat transfer and entropy production inside 3D cubic cavity. In this investigation, a fin is embedded at the left heated wall and is horizontally oriented toward the right cold wall, while the other surfaces are insulated from the surrounding. A broad range of parameters such as composite material conductivity ratio (1 ≤ RC ≤ 103), volumetric fraction of hybrid nanofluid (0 ≤ φ ≤ 0.05) and Rayleigh number (103 ≤ Ra ≤ 105) were considered. Obtained results reveal that the flow structure, the temperature distribution and the entropy generation rate are sensitive to nanoparticles volume fraction and composite material fin conductivity. Moreover, regardless of the latter parameter, the heat transfer is found to be enhanced with adding hybrid nanoparticles. However, the effect of increasing the conductivity on the average Nusselt number and the total generated entropy is significant only for RC > 100. The maximum Nusselt number and entropy production rate are achieved for Ra = 105, φ = 0.05 and RC = 1000.

Mathematical prediction of the spreading rate of COVID-19 using entropy-based thermodynamic model.

In the COVID-19 pandemic era, undoubtedly mathematical modeling helps epidemiological scientists and authorities to take informing decisions about pandemic planning, wise resource allocation, introducing relevant non-pharmaceutical interventions and implementation of social distancing measures. The current coronavirus disease (COVID-19) emerged in the end of 2019, Wuhan, China, spreads quickly in the world. In this study, an entropy-based thermodynamic model has been used for predicting and spreading the rate of COVID-19. In our model, all the epidemic details were considered into a single time-dependent parameter. The parameter was analytically determined using four constraints, including the existence of an inflexion point and a maximum value. Our model has been layout-based the Shannon entropy and the maximum rate of entropy production of postulated complex system. The results show that our proposed model fits well with the number of confirmed COVID-19 cases in daily basis. Also, as a matter of fact that Shannon entropy is an intersection of information, probability theory, (non)linear dynamical systems and statistical physics, the proposed model in this study can be further calibrated to fit much better on COVID-19 observational data, using the above formalisms.

Evolution of cellular morphology in pure materials

The evolution of cellular morphology during interfacial instability for liquid–solid transition for pure unary material systems is studied using the maximum entropy production (generation) rate principle (MEPR) for steady-state directional solidification. This approach is dependent on a quantity called maximum entropy production rate density which inherently contains key solidification parameters that governs cellular evolution for liquid–solid transformation. The maximum entropy production rate density is computationally measured from the solid–liquid interface in diffuse form and considers steady-state solidification at low velocities for both near and far from equilibrium conditions. The results are presented in mathematical expressions for morphological instability that corresponding to the evolution of a cellular morphology which emanates from the solid–liquid interface. The model is formulated to evaluate the solid–liquid interface thickness, the solidification velocity, grain boundary energy, and the size of the cellular morphological at instability. The results are tested with a number of pure single element materials at different temperature gradients which compare well with available experimental data.

Solidification Morphology and Bifurcation Predictions with the Maximum Entropy Production Rate Model

The use of the principle of maximum entropy generation per unit volume is a new approach in materials science that has implications for understanding the morphological evolution during solid–liquid interface growth, including bifurcations with or without diffuseness. A review based on a pre-publication arXiv preprint is first presented. A detailed comparison with experimental observations indicates that the Maximum Entropy Production Rate-density model (MEPR) can correctly predict bifurcations for dilute alloys during solidification. The model predicts a critical diffuseness of the interface at which a plane-front or any other form of diffuse interface will become unstable. A further confidence test for the model is offered in this article by comparing the predicted liquid diffusion coefficients to those obtained experimentally. A comparison of the experimentally determined solute diffusion constant in dilute binary Pb–Sn alloys with those predicted by the various solidification instability models (1953–2011) is additionally discussed. A good predictability is noted for the MEPR model when the interface diffuseness is small. In comparison, the more traditional interface break-down models have low predictiveness.

A flow instability criterion for alloys during hot deformation

Flow instability is an intrinsic behavior of materials, especially during hot deformation, and it will be intensified and propagate internal defects, and finally lead to various damages in the significant perturbation. Some phenomenological models in the literature could present the flow instability behavior, but not available for describing the microstructure changes. To overcome this problem, we introduce a dissipation potential as a function of the plastic strain rate, the dislocation density rate, and the heat transfer rate, D(ε˙p, ρ˙dis, q; ε, ρdis, θ), coupled with strain, dislocation density, and temperature as the boundary condition to record the microstructure evolution and describe the flow instability. In this function, the stored energy rate marks the dislocation density evolution, i.e., the transient microstructure changes, and one parameter ηM, is introduced to evaluate the efficiency of metallurgy. And the flow instability criterion is derived from the principles of maximum dissipation (or maximum entropy production rate) and orthogonality proposed by HANS ZIEGLER. We obtain the necessary conditions of the flow instability are that the dissipation potential is with entirely positive values due to the large plasticity, and the dissipation potential D(ε˙p, ρ˙dis, q; ε, ρdis, θ) is convex, i.e., the associated Hessian matrix is semi-positive. In this work, the function was applied to describe the behavior of Ti6Al4V during hot deformation, and a Kocks-Mecking type model was used to describe the flow stresses as well.

Morphological instability at the solid–liquid interface by the maximum entropy production rate principle

We examine the morphological instability from planar to non-planar cellular morphology with the maximum entropy production rate (MEPR) principle. An expression that quantifies the MEPR density at the solid–liquid interface (SLI) during direction solidification is presented, which leads to an instability criterion for dilute binary alloys. The instability criterion also affords to theoretically calculate the instability solidification growth velocity. The model considers steady state solidification at close-to and far-from equilibrium conditions.

Non-Equilibrium Thermodynamics and Stochastic Dynamics of a Bistable Catalytic Surface Reaction.

Catalytic surface reaction networks exhibit nonlinear dissipative phenomena, such as bistability. Macroscopic rate law descriptions predict that the reaction system resides on one of the two steady-state branches of the bistable region for an indefinite period of time. However, the smaller the catalytic surface, the greater the influence of coverage fluctuations, given that their amplitude normally scales as the square root of the system size. Thus, one can observe fluctuation-induced transitions between the steady-states. In this work, a model for the bistable catalytic CO oxidation on small surfaces is studied. After a brief introduction of the average stochastic modelling framework and its corresponding deterministic limit, we discuss the non-equilibrium conditions necessary for bistability. The entropy production rate, an important thermodynamic quantity measuring dissipation in a system, is compared across the two approaches. We conclude that, in our catalytic model, the most favorable non-equilibrium steady state is not necessary the state with the maximum or minimum entropy production rate.

Calculation of Uniform Corrosion Current Density of Iron in Hydrochloric Acid Solutions based on the Principle of Maximum Entropy Production Rate Applied to Literature Data

In this paper a new procedure for the calculation of uniform corrosion current density is presented. The procedure is based on the following criterion: corrosion current and extent of anodic and cathodic areas are such that the entropy production rate is maximum. Experimental data for the reaction Fe(s) + HCl(aq) = FeCl2(aq) + H2(g), taken from literature, are used to test the new model. A comparison between corrosion current density calculated from weight loss measurements and current density calculated from polarization curves by means of the proposed model is carried out. The proposed model provides a better agreement, between Tafel line extrapolation and gravimetric measurement, in comparison to the mixed potential theory usually adopted.

Prediction of Metabolite Concentrations, Rate Constants and Post-Translational Regulation Using Maximum Entropy-Based Simulations with Application to Central Metabolism of Neurospora crassa.

We report the application of a recently proposed approach for modeling biological systems using a maximum entropy production rate principle in lieu of having in vivo rate constants. The method is applied in four steps: (1) a new ordinary differential equation (ODE) based optimization approach based on Marcelin’s 1910 mass action equation is used to obtain the maximum entropy distribution; (2) the predicted metabolite concentrations are compared to those generally expected from experiments using a loss function from which post-translational regulation of enzymes is inferred; (3) the system is re-optimized with the inferred regulation from which rate constants are determined from the metabolite concentrations and reaction fluxes; and finally (4) a full ODE-based, mass action simulation with rate parameters and allosteric regulation is obtained. From the last step, the power characteristics and resistance of each reaction can be determined. The method is applied to the central metabolism of Neurospora crassa and the flow of material through the three competing pathways of upper glycolysis, the non-oxidative pentose phosphate pathway, and the oxidative pentose phosphate pathway are evaluated as a function of the NADP/NADPH ratio. It is predicted that regulation of phosphofructokinase (PFK) and flow through the pentose phosphate pathway are essential for preventing an extreme level of fructose 1,6-bisphophate accumulation. Such an extreme level of fructose 1,6-bisphophate would otherwise result in a glassy cytoplasm with limited diffusion, dramatically decreasing the entropy and energy production rate and, consequently, biological competitiveness.

Non-steady state mass action dynamics without rate constants: dynamics of coupled reactions using chemical potentials

Comprehensive and predictive simulation of coupled reaction networks has long been a goal of biology and other fields. Currently, metabolic network models that utilize enzyme mass action kinetics have predictive power but are limited in scope and application by the fact that the determination of enzyme rate constants is laborious and low throughput. We present a statistical thermodynamic formulation of the law of mass action for coupled reactions at both steady states and non-stationary states. The formulation uses chemical potentials instead of rate constants. When used to model deterministic systems, the method corresponds to a rescaling of the time dependent reactions in such a way that steady states can be reached on the same time scale but with significantly fewer computational steps. The relationships between reaction affinities, free energy changes and generalized detailed balance are central to the discussion. The significance for applications in systems biology are discussed as is the concept and assumption of maximum entropy production rate as a biological principle that links thermodynamics to natural selection.

The equivalence of minimum entropy production and maximum thermal efficiency in endoreversible heat engines

The objective of this study is to investigate the thermal efficiency and power production of typical models of endoreversible heat engines at the regime of minimum entropy generation rate. The study considers the Curzon-Ahlborn engine, the Novikov’s engine, and the Carnot vapor cycle. The operational regimes at maximum thermal efficiency, maximum power output and minimum entropy production rate are compared for each of these engines. The results reveal that in an endoreversible heat engine, a reduction in entropy production corresponds to an increase in thermal efficiency. The three criteria of minimum entropy production, the maximum thermal efficiency, and the maximum power may become equivalent at the condition of fixed heat input.

Contribution to the study of uniform corrosion by means of the maximum entropy production rate principle

The content of the article is about the principle of maximum entropy production rate applied to the description of uniform corrosion. The corroding metal surface is described as an ensemble of microcells randomly moving on the surface. The theory allows to calculate the ratio of the anodic to cathodic areas, as well as the corrosion current, under the hypothesis that the concentration overvoltages and the ohmic drops are both negligible.

Diffusion of chemically reacting fluids through nonlinear elastic solids: mixture model and stabilized methods

This paper presents a stabilized mixed finite element method for advection-diffusion-reaction phenomena that involve an anisotropic viscous fluid diffusing and chemically reacting with an anisotropic elastic solid. The reactive fluid–solid mixture theory of Hall and Rajagopal (Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body within the context of mixture theory. Math Mech Solid 2012; 17: 131–164) is employed wherein energy and entropy production relations are captured via an equation describing the Lagrange multiplier that results from imposing the constraint of maximum rate of entropy production. The primary partial differential equations are thus reduced to the balance of mass and balance of linear momentum equations for the fluid and the solid, together with an equation for the Lagrange multiplier. Present implementation considers a simplification of the full system of governing equations in the context of isothermal problems, although anisothermal studies are being investigated. The method is applied to problems involving Fickian diffusion, oxidation of PMR-15 polyimide resin, and slurry infiltration, within a one-dimensional finite element context. Results of the oxidation modeling of Tandon et al. (Modeling of oxidative development in PMR-15 resin. Polym Degrad Stab 2006; 91: 1861–1869) are recovered by employing the reaction kinetics model and properties assumed there; the only additional assumed properties are two constants describing coupled chemomechanical and purely chemical dissipation, and standard values for viscosity of air and PMR-15 stiffness properties. The present model provides the individual constituent kinematic and kinetic behaviors, thus adding rich detail to the interpretation of the process in comparison to the original treatment. The last problem considered is slurry infiltration that demonstrates the applicability of the model to account for the imposed mass deposition process and consequent effects on the kinematic and kinetic behaviors of the constituents.

Modeling plastic deformation of post-irradiated copper micro-pillars

We present here an application of a fundamentally new theoretical framework for description of the simultaneous evolution of radiation damage and plasticity that can describe both in situ and ex situ deformation of structural materials [1]. The theory is based on the variational principle of maximum entropy production rate; with constraints on dislocation climb motion that are imposed by point defect fluxes as a result of irradiation. The developed theory is implemented in a new computational code that facilitates the simulation of irradiated and unirradiated materials alike in a consistent fashion [2]. Discrete Dislocation Dynamics (DDD) computer simulations are presented here for irradiated fcc metals that address the phenomenon of dislocation channel formation in post-irradiated copper. The focus of the simulations is on the role of micro-pillar boundaries and the statistics of dislocation pinning by stacking-fault tetrahedra (SFTs) on the onset of dislocation channel and incipient surface crack formation. The simulations show that the spatial heterogeneity in the distribution of SFTs naturally leads to localized plastic deformation and incipient surface fracture of micro-pillars.

On the Clausius-Duhem Inequality and Maximum Entropy Production in a Simple Radiating System

A black planet irradiated by a sun serves as the archetype for a simple radiating two-layer system admitting of a continuum of steady states under steadfast insolation. Steady entropy production rates may be calculated for different opacities of one of the layers, explicitly so for the radiative interactions, and indirectly for all the material irreversibilities involved in maintaining thermal uniformity in each layer. The second law of thermodynamics is laid down in two versions, one of which is the well-known Clausius-Duhem inequality, the other being a modern version known as the entropy inequality. By maximizing the material entropy production rate, a state may be selected that always fulfills the Clausius-Duhem inequality. Some formally possible steady states, while violating the latter, still obey the entropy inequality. In terms of Earth’s climate, global entropy production rates exhibit extrema for any “greenhouse effect”. However, and only insofar as the model be accepted as representative of Earth’s climate, the extrema will not be found to agree with observed (effective) temperatures assignable to both the atmosphere and surface. This notwithstanding, the overall entropy production for the present greenhouse effect on Earth is very close to the maximum entropy production rate of a uniformly warm steady state at the planet’s effective temperature. For an Earth with a weak(er) greenhouse effect the statement is no longer true.