Mean-field Concept Research Articles

Microstructure-Dependent Rate Theory Model of Radiation-Induced Segregation in Binary Alloys

Conventional rate theory often uses the mean field concept to describe the effect of inhomogeneous microstructures on the evolution of radiation induced defect and solute/fission product segregation. However, the spatial and temporal evolution of defects and solutes determines the formation and spatial distribution of radiation-induced second phase such as precipitates and gas bubbles/voids, especially in materials with complicated microstructures and subject to high dose radiation. In this work, a microstructure-dependent model of radiation-induced segregation (RIS) has been developed to investigate the effect of inhomogeneous thermodynamic and kinetics properties of defects on diffusion and accumulations of solute A in AB binary alloys. Four independent concentrations: atom A, interstitial A, interstitial B, and vacancy on [A, B] sublattice are used as field variables to describe temporal and spatial distribution and evolution of defects and solute A. The independent concentrations of interstitial A and interstitial B allow to describe their different generation rates, thermodynamic and kinetic properties, and release the assumptions of interstitial generation and sink strength used in the conventional rate theory. Microstructure and concentration dependent chemical potentials of defects are used to calculate the driving forces of defect diffusions. With the model, the effects of defect chemical potentials and mobilities on the RIS in polycrystalline AB model alloys have been simulated. The results demonstrated the model capability in predicting defect evolution in materials with inhomogeneous thermodynamic and kinetic properties of defects. The model can be extended to materials with complicated microstructures such as a wide range of grain size distribution, coating structure and multiphases as well as radiation-induced precipitation subject to severe radiation damage.

Revisiting the Ostwald Ripening: A Discrete Model Based on Multiple Pairwise Interactions

A discrete multi-particle model of Ostwald ripening based on direct pairwise interactions between precipitates with incoherent interfaces is presented. Although based on the mean field concept, it is a valid alternative to the classical LSW theory. The main differences with respect to the classical approach can be summarized as follows: i) Particles interact the one another; ii) The first Fick’s law is considered to evaluate the fluxes of matter instead of the quasi stationary solution of the concentration field around particles; iii) The rate of matter exchange depends on the average surface-to-surface interparticle distance, a characteristic feature of the system which naturally incorporates the effect of volume fraction of second phase; iv) The multi-particle diffusion is described through the definition of an interaction volume containing all the particles involved in the exchange of solute. The model is in excellent agreement with the experimental data available in the literature. The shape of the quasi-stationary 3D particle size distribution of solid-solid and solid-liquid systems is well predicted from volume fractions of 0.07, 0.30, 0.52 and 0.74. Similarly, a very good prediction of the dependence of the kinetic constant of the coarsening process on the volume fraction of precipitates is obtained with reference to literature data on solid-liquid mixtures in the volume fraction range from 0.20 to about 0.75. For volume fractions below about 0.1 the model predicts broad and right-skewed stationary size distributions resembling a lognormal function. Above this value, a transition to sharper, more symmetrical but still right-skewed shapes occurs.

Fluctuation aspects of isotope theory - III. Isotopic phase similarity in the set of deuterated methanes CHxD4-x

We have proposed the new principle of isotopic phase similarity (IPS) in the previous works I, II, of this series. The fundamental Born-Oppenheimer approximation maintains separately its applicability within each isotopic phase. The fluctuation-thermodynamic formalism is based on the developed earlier predictive model of CVL (congruent vapor-liquid) – diagram. It changes the mean-field concept of the macroscopic VLE (vapor-liquid equilibrium) – diagram to describe adequately the mesoscopic scale of volumes and time-intervals for a fluid state. The well-established by CVL-diagram fluctuation boundaries of the boil- Pb(T)- and dew-Pd(T)-pressures are important ones. They form the essential mesoscopic expansion of the single Pv(T)- vapor-pressure curve postulated by VLE- diagram. This mesoscopic peculiarity of a non-mean-field phase transition was revealed earlier for any one-component fluids. It provides the new insight into the classic theory of isotope effects. In the first instance the conventional interpretation of the vapor-pressure isotope effect (VPIE) and the molar-volume isotope effect needs an independent consideration by the comparison of CVL-diagrams for isotopes. In two previous works I, II we have introduced the undimensional fluctuation calibration complex Zoc=P0M/(RT0ρc) (P0=101.325kPa, T0=1K, M g/mol – molar mass of isotope, ρc - its critical density of mass) which provides the separation of VPIEs for the variety of isotopic families on two main types: (n) normal (Zocn<1) and (i) inverse (Zoci>1) at low temperatures. In this work III we have applied the same IPS-methodology to the set of partially deuterated methanes CHxD4-x of i-type. The set of exclusively n-type pairs {H2O/D2O/T2O; 36Ar/40Ar; 235U/238U} was considered in works I, II. The reasonable predictive capability of IPS-principle determines systematic shifts in the critical parameters of heavier isotopes if only those for the basic lighter isotope (CH4) are known. The most impressive confirmation of IPS-principle is the reasonable concordance of its dew-pressure isotope effect (DPIE)-predictions with the experimental absolute differences between small vapor pressures of isotopes ΔPv=Pv′−Pv. We have proposed and corroborated the concept of two coupled crossover Bancroft'sTbB and Clapeyron'sTlgC-points. They offer the study of the difference between the vaporization enthalpies of isotopes without virial correction for non-ideality of v-phase.

Robust Energy Management of Residential Nanogrids via Decentralized Mean Field Control

Energy management systems (EMS) are viable techniques to reduce the customers billing cost as well as enhancing the grid reliability and efficiency. Uncertainty and information privacy of users are serious concerns in designing EMS of a residential nanogrid with renewable energy resources. This paper addresses a decentralized energy management algorithm for day-ahead scheduling of residential nanogrids using the concept of Mean Field (MF). The aim of each user is to find its optimal demand strategy by minimizing an objective function consisting of energy consumption cost, battery degradation cost, and cost of user discomfort while meeting a suite of constraints. The strategy of a user affects the objective functions of other users through the electricity price; such interaction among the users is hence modeled as a game problem. Each user, without any information exchange with other users, sends its electricity demand to the utility company. The utility company broadcasts the aggregated demand as a common information of the grid (MF term) to all users. The robustness of the algorithm in presence of uncertainties in decision making of the clients and non-residential load prediction error is analytically verified. In particular, it is shown that the algorithm converges to $\epsilon$ -Nash equilibrium point of the game and $\epsilon$ uniformly converges to its minimum value as the population size of the users goes to infinity. Simulation results show the advantages of the proposed method.

Multiscale modelling from quantum level to reactor scale: An example of ethylene epoxidation on silver catalysts

Abstract Ethylene epoxidation is one of the most important selective chemical oxidations in industry. For a controlled transformation of ethylene (ethene) into epoxide, silver is the only commercially suitable catalyst. Although it is usually doped, even with pristine silver activity, selectivity, and stability vary strongly with facets. In this work, we use this reaction on Ag( 111 ) and Ag( 100 ) as a classical formation model to demonstrate the capabilities of physical multiscale modelling, to show why Ag( 100 ) nanocubes offer superior catalysis, and to optimise reactivity. First, we describe the elementary reactions on pristine surfaces with the quantum chemistry calculations, using density functional theory (DFT). The free energies of all intermediates, kinetic rates from the transition state theory and adsorption/desorption equilibria are calculated from first principles. These results are applied to kinetic Monte Carlo (kMC) simulations, where the spatio-temporal evolution of the system on a meso-scale can be followed. The differences in activity, concentration, selectivity, and apparent activation energy are observed, investigated, and analysed. Lastly, mean-field concepts – micro-kinetics and computational fluid dynamics (CFD) – are used to simulate how the synthesis proceeds in a reactor. Mechanism, catalytic coverage and the effects of pressure, temperature, and particle composition, size and shape on the performance are evaluated. We show that multiscale modelling is a powerful instrumental approach for real unit engineering, while the level of detail required is dictated by the purpose of a representation and available resources.

Antisymmetrized, translationally invariant theory of the nucleon optical potential

Earlier work showed how a nucleon optical model wave function could be defined as a projection of a many-nucleon scattering state within a translationally invariant second quantised many-body theory. In this paper an optical potential operator that generates this optical model wave function is defined through a particular off-shell extension of the elastic transition operator. The theory is express explicitly in terms of the many-nucleon Hamiltonian in a mixed representation in which localised target nucleus states feature. No reference to a mean-field concept is involved in the definition. It is shown that the resulting optical model operator satisfies the requirements of rotational and translational invariance and has standard behaviour under the time reversal transformation. The contributions to the optical potential from two different exchange mechanisms are expressed in terms of an effective Hamiltonian involving a nucleon-number conserving one-body interaction. In the weak-binding limit the method reduces to a a version of Feshbach's projection operator formulation of the optical potential with a truncated nucleon-nucleon potential including exchange terms and recoil corrections. Definitions of the nucleon single-particle Green's function and the corresponding Dyson self-energy modified by corrections for translational invariance are presented and different definitions of the optical potential operator are compared.

Reinterpretation of the Mean Field Hypothesis in Analytical Models of Ostwald Ripening and Grain Growth

A long right tail is a common feature of experimental quasi-stationary size distributions of particles and grains that is not explained by the classical theories based on the mean field hypothesis. In this work, it is shown that the “pairwise interaction” approach, here presented in a comprehensive exposition involving both Ostwald ripening and grain growth, is a valid alternative to classical mean field theories since it produces more realistic predictions of the distribution shapes. The new analytical models are based on the mean field concept but rely on a detailed physical description of the elementary interactions responsible for the exchange of matter. They are jointly reviewed and compared with the corresponding classical Lifshitz–Slyozov–Wagner and Hillert models. The interactions are treated as a sum of elementary and specific contributions rather than as a generalized exchange with the mean field. The framework is complemented by the introduction of the “interaction volume” in Ostwald ripening and of the “local grain boundary curvature” in grain growth which are both size-dependent and permit to represent more precisely the local physics of the exchanges. The excellent results obtained in reproducing the experiments without any ad hoc parameters suggest that the mean field hypothesis adopted in the classical theories to describe the environment of a growing particle or grain represents a too drastic approximation. Therefore, it is proposed to replace the classical mean field by a “local mean field,” i.e., the ensemble of actual mean environments interacting with any single element of a given size. This alternative assumption induces a higher growth rate for large particles or grains compared with their respective mean field theories, thus producing right-skewed asymptotic distributions. For particles at small volume fraction the stationary distribution resembles a lognormal function, whereas for grains in normal grain growth regime the Rayleigh distribution is found as solution.

Dissipation Dynamics of Nuclear Fusion Reactions

Based on both the time-dependent Hartree–Fock (TDHF) and the time-dependent density matrix (TDDM) methods, we adopt a macroscopic reduction procedure to investigate the dissipation dynamics of nuclear fusion reactions. The TDDM method is an extension of TDHF, in the sense that it goes beyond the mean-field concept and takes into account two-body correlations explicitly. To investigate the effect of correlations on dissipation, the collective trajectories as well as the friction coefficients for the reaction ¹⁶O + ¹⁶O →³²S are extracted. Our results suggest that two-body correlations play a relevant role in the fusion process.

Fabrication of alumina-PSZ composites via spark plasma sintering and their mechanical properties

Alumina-partially stabilized zirconia (PSZ) composites were fabricated via spark plasma sintering (SPS) technique to produce biocompatible materials with superior mechanical properties. The volume fraction of the composites covered from 100% alumina to 100% PSZ. Their sintering state was examined by optical microscopy, density measurement, and X-ray diffraction, and dense composites without any reaction phases could be fabricated, irrespective of PSZ content. Then, three-point bending tests and hardness tests were conducted. The hardness and elastic modulus agreed with the predictions based on the Voigt model and the Eshelby's equivalent inclusion model combined with the Mori-Tanaka's mean field concept, respectively. While the bending strength of the composites ranged from that of monolithic alumina to that of monolithic PSZ, the fracture toughness of the composites improved as compared with the monoliths of alumina and PSZ. We concluded that the use of alumina and PSZ was effective to fabricate the composites with high mechanical performances.

Re-established congruent vapor-liquid diagram of alkali fluid metals as alternative to crossover VLE - interpretation

The universal predictive methodology termed the congruent vapor-liquid (CVL-) diagram was applied recently to the set of alkali metals for which the high-temperature data of vapor pressure, dew and bubble orthobaric densities cannot be reliably deduced from the direct VLE-experiment. We show in the present work that the unwonted but significant pressure drop in the region of overheated vapor revealed long ago by Ewing et al. for Na, K, Cs and reproduced soon after on for Cs by Novikov becomes explainable due to the concept of a heterogeneous non-Gibbsian interphase introduced by CVL-diagram. Its predictive capability based exclusively on the available data of a condensed phase at atmospheric pressure provides also the fairly well and purely predictive description of two Gibbsian single-phase regions (non-ordered gas-phase and ordered liquid-phase). The crucial rejection from the classical mean-field concept of a unified EOS (common for both coexistent fluid Gibbsian phases) leads, in accordance with GFA (global fluid asymmetry)-principle, to the non-classical exponent β=1/3. It is close to that from the Ising-like (Il) asymptotic criticality. Thus, CVL-diagram provides not only the impressive alternative support and extension of the near-critical Il-universality but it expands GFA-principle on the whole subcritical crossover region of alkali metals.

Power law attenuation of shear waves in fractal media

There are two mechanisms for attenuation of mechanical waves: absorption and multiple scattering. We explore the second mechanism for shear waves in the context of MR elastography. The theory for attenuation was first given in the seismic field (O’Doherty & Anstey, Geophys. Prosp. 1971). Later phase was included, and the mean field concept was applied (Banik et al., Geophys. 1985). Then, a more rigorous theory which also deals with random fractals was developed (Solna, SIAM J ApplMath, 2003; Garnier & Solna, Multiscale Model Simul, 2009). We have developed this theory further by allowing fractality over a limited length scale and also shown that it gives results that are indistinguishable from those of power law absorption models like the fractional Kelvin-Voigt model (Holm & Sinkus, JASA, 2010). This makes it a challenge to understand whether attenuation in a real medium is due to one or the other mechanism or both. Measurement of dispersion has been done for polystyrene microspheres embedded in agarose which is nearly lossless up to 1000 Hz. The phase velocity has a power law variation with exponent up to d = 0.24 (Lambert et al., PRL 2015). According to the theory this is consistent with a fractional dimension which is d less than the Euclidean dimension.

Exact mean field concept to compute defect energetics in random alloys on rigid lattices

In modern materials science modeling, the evolution of the energetics of random alloys with composition are desirable input parameters for several meso-scale and continuum scale models. When using atomistic methods to parameterize the above mentioned concentration dependent function, a mean field theory can significantly reduce the computational burden associated to obtaining the desired statistics in a random alloy. In this work, a mean field concept is developed to obtain the energetics of point-defect clusters in perfect random alloys. It is demonstrated that for a rigid lattice the concept is mathematically exact. In addition to the accuracy of the presented method, it is also computationally efficient as a small box can be used and perfect statistics are obtained in a single run. The method is illustrated by computing the formation and binding energy of solute and vacancy pairs in FeCr and FeW binaries. Also, the dissociation energy of small vacancy clusters was computed in FeCr and FeCr-2%W alloys, which are considered model alloys for Eurofer steels. As a result, it was concluded that the dissociation energy is not expected to vary by more than 0.1eV in the 0–10% Cr and 0–2% W composition range. The present mean field concept can be directly applied to parameterize meso-scale models, such as cluster dynamics and object kinetic Monte Carlo models.

Dipolar interaction and demagnetizing effects in magnetic nanoparticle dispersions: Introducing the mean-field interacting superparamagnet model

A model is developed with the aim of analyzing interacting superparamagnets. Model is built from magnetic dipolar interaction and demagnetizing mean field concepts. A useful expression for effective demagnetizing factors is achieved, which allows for the analysis of non uniform spatial distributions of nanoparticles. This expression is a function of demagnetizing factors associated with specimen and clusters shapes, and of the mean distances between near neighbor nanoparticles and between clusters, relative to the characteristic sizes of each of these two types of objects, respectively. It explains effects of magnetic dipolar interactions such as the observation of apparent nanoparticle magnetic-moments smaller than real ones. It is shown that by performing a minimum set of experimental determinations, model application allows retrieval of intrinsic properties, like magnetic moment and susceptibility in the absence of interactions. It also permits the estimation of mean interparticle and intercluster relative distances, and of demagnetizing factors associated with clusters shape. An expression for magnetic dipolar energy per nanoparticle is also derived. Model experimental test was performed by analysis of results reported in the literature and of original results. They correspond to magnetite particles dispersed in PEGDA-600 polymer, and in PVA ferrogels. Experimental results display different magnetic response when prism shaped specimens are measured along principal directions. Intrinsic properties and structural information were retrieved from the analysis, in excellent agreement with information obtained from FESEM images. In the studied samples nanoparticles were found to be in close contact to each other within almost randomly oriented clusters. Intercluster mean relative-distance was found to vary between 2.2 and 7.5, depending on particles volume fraction.

Self-consistent Green's function embedding for advanced electronic structure methods based on a dynamical mean-field concept

We present an embedding scheme for periodic systems that facilitates the treatment of the physically important part (here the unit cell) with advanced electronic-structure methods, that are computationally too expensive for periodic systems. The rest of the periodic system is treated with computationally less demanding approaches, e.g., Kohn-Sham density-functional theory, in a self- consistent manner. Our scheme is based on the concept of dynamical mean-field theory (DMFT) formulated in terms of Green functions. In contrast to the original DMFT formulation for correlated model Hamiltonians, we here consider the unit cell as local embedded cluster in a first-principles way, that includes all electronic degrees of freedom. Our real-space dynamical mean-field embedding (RDMFE) scheme features two nested Dyson equations, one for the embedded cluster and another for the periodic surrounding. The total energy is computed from the resulting Green functions. The performance of our scheme is demonstrated by treating the embedded region with hybrid functionals and many-body perturbation theory in the GW approach for simple bulk systems. The total energy and the density of states converge rapidly with respect to the computational parameters and approach their bulk limit with increasing cluster (i.e., unit cell) size.

Destruction of large-scale magnetic field in non-linear simulations of the shear dynamo

The Sun's magnetic field exhibits coherence in space and time on much larger scales than the turbulent convection that ultimately powers the dynamo. In the past the α-effect (mean-field) concept has been used to model the solar cycle, but recent work has cast doubt on the validity of the mean-field ansatz under solar conditions. This indicates that one should seek an alternative mechanism for generating large-scale structure. One possibility is the recently proposed ‘shear dynamo’ mechanism where large-scale magnetic fields are generated in the presence of a simple shear. Further investigation of this proposition is required, however, because work has been focused on the linear regime with a uniform shear profile thus far. In this paper we report results of the extension of the original shear dynamo model into the non-linear regime. We find that whilst large-scale structure can initially persist into the saturated regime, in several of our simulations it is destroyed via large increase in kinetic energy. This result casts doubt on the ability of the simple uniform shear dynamo mechanism to act as an alternative to the α-effect in solar conditions.

A micromechanics-based incremental damage theory of bulk metallic glass matrix composites

Based on the Mori–Tanaka’s mean field concept, a micromechanics-based incremental damage theory was developed to investigate the mechanical performance of bulk metallic glass matrix composites under tension. Firstly, the shear band was considered to be a microcrack, and therefore a spherical fictitious inclusion-containing microcracks and the surrounding bulk metallic glass matrix were constructed. Based on the equivalence between the effective elasticity of microcrack-containing media and porous material, the volume fraction of the fictitious inclusion could be determined by the density of microcracks generated during a strain increment. The stress-based Weibull probability distribution function and percolation theory were applied to describe the microcrack evolution that results in the progressive damaging of bulk metallic glass composites. Based on the present model, the impact of shear bands on the tensile ductility was discussed for the composites with various microstructures. The predictions are in fairly good agreement with the experimental data, demonstrating that the developed analytical model is capable of successfully capturing the main features, such as yield strength, strain hardening, and stress softening elongation, of particle-toughened bulk metallic glass. The main conclusions will shed some light on optimizing the microstructures in effectively improving the tensile ductility of bulk metallic glass composites.

A mean-field interatomic potential for a multi-component β-type titanium alloy

By employing the mean-field concept and the embedded-atom method, an interatomic potential was constructed for a β-type titanium alloy, Ti–24Nb–4Zr–7.9Sn (Ti2448). The alloy was assumed to consist of one single special “element”, and its properties were obtained by averaging the interaction between all elements in the alloy. The lattice parameter, shear modulus, bulk modulus, anisotropy ratio and vacancy formation energy from experiments and the cohesive energy from ab initio calculations were used for fitting the potentials. Verification of the present potential was performed by the calculation of the cohesive energies under various lattice structures, the equation of state and point defect energies. Molecular dynamics simulations showed that the present potential is capable of reproducing various manners of plastic deformation, e.g., dislocation nucleation and propagation, deformation twinning, phase transformation, etc. The results show that for such disordered alloys, while retaining the reliability, the mean-field scheme is much simplified compared with traditional methods of potential construction, hence making it possible to study multi-component engineering alloys, such as Ti2448.

Micromechanical modeling of nanocomposites considering debonding of reinforcements

Abstract This paper deals with a constitutive model of particulate-reinforced nanocomposites which can describe the debonding damage, elasto-plastic behavior of matrix and particle size effects on deformation and damage. An incremental damage model of particulate-reinforced composites based on the Mori–Tanaka’s mean field concept, considering the particle size in nanoscale is further developed to consider the debonding of reinforcements in nanocomposites. The applicability of the proposed theory is investigated for nanocomposites consisting of Al2O3 nanoparticles with different size embedded in magnesium alloy (AZ31 and ZK60A) and pure magnesium (Mg) matrix. Based on the present model, analysis of stress–strain response for Al2O3–AZ31 nanocomposite under uniaxial tension is carried out. The effects of particle size and adhesive energy of nanoparticles at interface on stress–strain response of Al2O3–AZ31 nanocomposite is obtained. Moreover, in this paper the effect of debonding of reinforcements on effective Young’s modulus and toughness of the particulate reinforced nanocomposites is demonstrated. When the debonding damage starts to occur, the stress–strain curve for the damaged nanocomposite deviate to lower stress from those for the perfect composite. The influences of adhesive energy at interface and particle size on the stress–strain curve are considerable. Composites with lower adhesive energy at interface and larger particle size have poor toughness due to lower area under the stress–strain curve. In addition, Progressive debonding leads to a loss of stiffness in nanocomposites.

Mechanisms of nanoparticle and bioparticle deposition – Kinetic aspects

Particle deposition phenomena were theoretically analyzed in terms of a hybrid theoretical approach where the bulk transport is described by continuity equations and the surface transport by the random sequential adsorption (RSA) models. RSA modeling furnishes maximum coverages and blocking functions for spherical and anisotropic particles. These parameters are used as boundary conditions for the bulk transport equation. This enables one to derive theoretical results describing kinetics of convection and diffusion-controlled transport of particles to interfaces, which is discussed next. The utility of this theoretical approach is confirmed by experimental data obtained for various colloid systems involving micro- and nanoparticles using optical and fluorescence microscopy AFM, SEM and TEM imaging of particle monolayers. Comparison of theoretical and experimental results shows that the electrostatic interactions play a dominant role affecting the maximum coverage of and the structure of deposited particle monolayers. Particle deposition on heterogeneous surfaces formed by a controlled deposition of nanoparticles and proteins (latex, hematite, fibrinogen) is also considered. These phenomena are properly interpreted in terms of the RSA model considering fluctuations in the particle density. The classical DLVO theory, based on the mean-field (averaged) zeta potential concept proved inadequate. It is also shown that the colloid particle deposition on protein monolayers can be used to efficiently determine the coverage and a real charge distribution under wet, in situ conditions. It is concluded that particle and protein deposition phenomena show deep analogies. This suggests that the results acquired for nanoparticles can be used as convenient reference systems for interpretation of molecular adsorption phenomena inaccessible to direct measurements.

Cluster-virial expansion for nuclear matter within a quasiparticle statistical approach

Correlations in interacting many-particle systems can lead to the formation of clusters, in particular bound states and resonances. Systematic quantum statistical approaches allow to combine the nuclear statistical equilibrium description (law of mass action) with mean-field concepts. A chemical picture, which treats the clusters as distinct entities, serves as an intuitive concept to treat the low-density limit. Within a generalized Beth-Uhlenbeck approach, the quasiparticle virial expansion is extended to include arbitrary clusters, where special attention must be paid to avoid inconsistencies such as double counting. Correlations are suppressed with increasing density due to Pauli blocking. The contribution of the continuum to the virial coefficients can be reduced by considering clusters explicitly and introducing quasiparticle energies. The cluster-virial expansion for nuclear matter joins known benchmarks at low densities with those near saturation density.