Description Of Phase Transitions Research Articles

Error Estimates for a Finite Element Discretization of a Phase Field Model for Mixtures

We derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures. Special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region. The main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system, a suitable regularity of the exact solutions, and a spectrum estimate for the operator of the Allen–Cahn equation. The spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model. This is done by a technique introduced in [X. Feng and A. Prohl, Math. Comp., 73 (2004), pp. 541–567] for phase transitions of pure materials with linear constitutive relations.

Modelling Elasto‐plastic Material Behaviour within the Multi‐Body Frame

AbstractHighly dynamic problems can be well described using multi‐body systems. Nevertheless this method is limited to systems whithout large elasto‐plastic deformations that can e.g. be found during an extrusion process. To model such a system, multi‐body systems can be adequately combined with special particle approaches like the mesoscopic particle method [1], [2]. These mesoscopic particles have internal variables describing hidden degrees of freedom. Interaction forces between particles similar to those on the atomic scale have to be taken into account to describe phase transitions. So materials' properties concerning temperature and pressure can successfully be reproduced. This paper therefore deals with the description of an extrusion process in the multi‐body frame using mesoscopic particles. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Short- and long-term clinical outcomes of diabetics and nondiabetics receiving bare-metal stents versus drug-eluting stents

We describe phase transitions in microfilm assemblies of octanethiol (OT)-capped 2–4 nm-diameter Ag nanoclusters prepared from solutions with OT/Ag+ ratios of ∼50. Using DSC we observe two melting/crystallization-type reversible phase transitions: one at ∼61 °C due to interdigiated unattached octanethiol, and the other at ∼125 °C due to the phase comprised of the assembly of OT-capped nanoclusters. Increased thermal fluctuations weaken the inter-chain hydrophobic interactions between interdigitated OT molecules, leading to both phase transitions. The thiolate bond of OT-molecules bound to Ag nanoclusters are more rigid, thereby requiring a higher temperature to increase the flexibility of the alkyl chain of OT, and to melt the nanocluster assembly. The mobility of the nanoclusters in the melt is limited, and morphological features of the original assemblies are retained during recrystallization. No observable mass loss is detected up to ∼180 °C, above which OT molecules desorb from the Ag nanoclusters.

Phase transitions in large [formula omitted] heavy quark potentials

We describe phase transitions in the heavy quark potential in planar gauge theories having wrapped D5-brane string duals. A new phase transition, previously unnoticed in these models, is driven by the source of a large dimension operator. Another transition, which has already been described in a previous paper, is driven by the presence of light dynamical flavors. Both transitions connect a Coulomb-like phase to a confining linear phase.

The proper treatment of language acquisition and change in a population setting

Language acquisition maps linguistic experience, primary linguistic data (PLD), onto linguistic knowledge, a grammar. Classically, computational models of language acquisition assume a single target grammar and one PLD source, the central question being whether the target grammar can be acquired from the PLD. However, real-world learners confront populations with variation, i.e., multiple target grammars and PLDs. Removing this idealization has inspired a new class of population-based language acquisition models. This paper contrasts 2 such models. In the first, iterated learning (IL), each learner receives PLD from one target grammar but different learners can have different targets. In the second, social learning (SL), each learner receives PLD from possibly multiple targets, e.g., from 2 parents. We demonstrate that these 2 models have radically different evolutionary consequences. The IL model is dynamically deficient in 2 key respects. First, the IL model admits only linear dynamics and so cannot describe phase transitions, attested rapid changes in languages over time. Second, the IL model cannot properly describe the stability of languages over time. In contrast, the SL model leads to nonlinear dynamics, bifurcations, and possibly multiple equilibria and so suffices to model both the case of stable language populations, mixtures of more than 1 language, as well as rapid language change. The 2 models also make distinct, empirically testable predictions about language change. Using historical data, we show that the SL model more faithfully replicates the dynamics of the evolution of Middle English.

Diagrammatic representation of the two-spin correlation function for the generalized heisenberg model

We consider a “generalized Heisenberg model”, with a Hamiltonian given by ℋ(D) = –JΣijS(D)j, where the quantities Si are isotropically-interacting D-dimensional classical spins and the summation is restricted to nearest-neighbor pairs of sites . Thus when D = 1, 2, and 3, this model reduces to the S = 1/2 Ising, plane rotator, and classical Heisenberg models which are useful for describing phase transitions in, respectively, liquid-gas systems, Bose fluids, and magnetic materials. Unfortunately this Hamiltonian is exactly soluble only in the limit D ∞, in which case the solution corresponds to the spherical model of Berlin and Kac. For finite D, we must consider approximation procedures in order to obtain any useful information. In this paper we develop a diagrammatic representation, valid for arbitrary D, for successive terms in the high-temperature expansions of the free energy and the spin-spin correlation function—from which we can obtain series expansions for all the thermodynamic properties of the system.

Maximal attractor and inertial set for Eguchi–Oki–Matsumura equation

We consider the Eguchi–Oki–Matsumura equation describing phase transition in binary alloys. We show that the corresponding semigroup possesses a maximal attractor and prove the existence of an inertial set.

Existence and boundedness of solutions for a singular phase field system

This paper is devoted to the mathematical analysis of a thermo-mechanical model describing phase transitions in terms of the entropy and order structure balance law. We consider a macroscopic description of the phenomenon and make a presentation of the model. Then, the initial and boundary value problem is addressed for the related PDE system, which contains some nonlinear and singular terms with respect to the temperature variable. Existence of the solution is shown along with the boundedness of both phase variable χ and absolute temperature ϑ. Finally, uniqueness is proved in the framework of a source term depending Lipschitz continuously on ϑ.

Can We Describe phase Transition in Insulators within the Landau PT theory Framework?

AbstractBased on studies of simple oxides, this paper demonstrates that the specific energy deposition modes under irradiation induce modifications of materials over different length scales. On the other hand, we show the Landau phase transition theory, widely used to explain the structural stability of materials out of irradiation, can give a general framework to describe the behavior of these oxides under irradiation. The use of X-ray diffraction techniques coupled with the Raman spectroscopy allows defining in a quantitative way the phenomenological parameters leading to predictive results. This paper clearly shows that in two model systems, pure zirconia and spinels, no unexpected new phases are produced in these oxides irradiated at room temperature and with different fluxes. Such a phenomenological approach may be useful to study the radiation tolerance of many crystalline ceramics (e.g. the zirconium based americium ceramics).

Detectability of gravitational waves from phase transitions

Gravitational waves potentially represent our only direct probe of the universe when it was less than one second old. In particular, first-order phase transitions in the early universe can generate a stochastic background of gravitational waves which may be detectable today. We briefly summarize the physical sources of gravitational radiation from phase transitions and present semianalytic expressions for the resulting gravitational wave spectra from three distinct realistic sources: bubble collisions, turbulent plasma motions, and inverse-cascade helical magnetohydrodynamic turbulence. Using phenomenological parameters to describe phase transition properties, we determine the region of parameter space for which gravitational waves can be detected by the proposed Laser Interferometer Space Antenna. The electroweak phase transition is detectable for a wide range of parameters.

Existence of Solutions for a Model Describing the Dynamics of Junctions Between Dislocations

We study a dynamical version of a multiphase field model of Koslowski and Ortiz for planar dislocation networks. We consider a two-dimensional vector field which describes phase transitions between constant phases. Each phase transition corresponds to a dislocation line, and the vectorial field description allows the formation of junctions between dislocations. This vector field is assumed to satisfy a nonlocal vectorial Hamilton–Jacobi equation with nonzero viscosity. For this model, we prove the existence for all time of a weak solution.

On the set of solutions to a variational phase transition problem of continuum mechanics

A variational problem describing phase transitions is considered. It is shown that a multi-vaued function associating with the set of parameters the set of solutions to the problem is continuous and has compact values. Bibliography: 5 titles.

Totally symmetric order parameter in the framework of the phenomenological theory of phase transitions: Ferroelastics

A new approach is proposed for constructing the phenomenological theory of phase transitions. The approach is based on the classical Landau theory with allowance made for the order parameter that corresponds to changes in the charge distribution probability density of a crystal and does not affect the symmetry of the high-symmetry phase. It is demonstrated that this approach makes it possible to describe phase transitions in terms of a nonequilibrium polynomial Landau potential of degree four in the components of the order parameters. The capabilities of the proposed approach are illustrated with three systems that undergo ferroelastic phase transitions.

Landau–Ginzburg theory of phase transitions in fractal systems

We present novel formulation of Ginzburg–Landau theory of phase transitions for some class of fractal systems. Using logarithmic coordinates we derive expression for the free-energy functional, which resembles formula for bulk systems. This allows us to describe phase transitions in the way which is reminiscent to conventional crystalline systems. The presented theory is extended onto antiferroelectric, multisublattice and modulated systems.

Effect of the nonuniformity of the inlet liquid distribution on the trickle-bed reactor output in an exothermic reaction accompanied by evaporation

The effect of the inlet liquid distribution on the operation of a trickle-bed reactor is experimentally investigated, and the observed phenomena are analyzed using a one-dimensional mathematical model with nonequilibrium approaches to describing phase transitions. The modeling shows that, if the fraction of the wetted surface at the reactor inlet is below unity, then, along a sufficiently long bed, there is always a stationary front of complete evaporation and complete conversion. The main conclusion within the framework of the considered one-dimensional model with separate evaporation (on completely wetted granules) and gas-phase hydrogenation (on dry granules) is that safe operation (the absence of a thermal explosion) can be ensured by complete suppression of the reaction in the gas phase.

Gradient flows of non convex functionals in Hilbert spaces and applications

This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space where is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and is (a suitable limiting version of) its subdifferential. We will present some new existence results for the solutions of the equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements and of Young measures. Our analysis is also motivated by some models describing phase transitions phenomena, leading to systems of evolutionary PDEs which have a common underlying gradient flow structure: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.

Thermodynamic properties and phase transitions in the H 2O/CO 2/CH 4 system

Free energy densities as functions of temperature, pressure and the compositions are required for development of a three-component phase field theory able to describe phase transitions involving hydrates. We have evolved the extended adsorption theory due to Kvamme and Tanaka [B. Kvamme, H. Tanaka, J. Phys. Chem. 99 (1995) 7114–7119] by deriving the Gibbs free energy density surface for CO 2 and CH 4 hydrates. The composite Gibbs free energy surface for the liquid phases has been obtained from an SRK equation of state and gas solubilities measured outside the region of hydrate stability. Our comprehensive thermodynamic model has been shown to predict water–hydrate equilibrium properties that are in a good agreement with experimental data. Molecular dynamics simulations of hydrates in contact with water at 200 bar and several temperatures have allowed us to estimate hard-to-establish properties needed as input parameters for the practical applications of proposed theories. The 5–95% interval for the interface width of methane hydrate/liquid water system has been estimated to be 8.5 Å. When the interface Gibbs free energy is known as well, the phase field theory contains no adjustable parameters. This work demonstrates how this theory can be applied to the kinetics of hydrate phase transitions. The process of hydrate growth from aqueous solution was found to be rate-limited by mass transport. As prescribed by the combination of the first and the second laws of thermodynamics, solute concentration near the hydrate approaches the equilibrium between hydrate and aqueous solution rather than the solubility limit.

Long Time Behavior of Solutions to the Caginalp System with Singular Potential

We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature θ and of an order parameter χ. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling χ is characterized by a singular potential W which forces χ to take values in the interval [−1, 1]. We provide reasonable conditions on W which ensure that, from a certain time on, χ stays uniformly away from the pure phases 1 and −1. Combining this separation property with the Lojasiewicz-Simon inequality, we show that any smooth and bounded trajectory uniformly converges to a stationary state and we give an estimate of the decay rate. ∗This work was partially supported by the Italian MIUR PRIN Research Projects Modellizzazione Matematica ed Analisi dei Problemi a Frontiera Libera and Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali, and by the Italian MIUR FIRB Research Project Analisi di Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metodologici, Modellistica, Applicazioni †The work of H.P. was supported by the Grant A1019302 of GA AV CR ‡The work of G.S. was partially supported by the HYKE Research Training Network

Global solution to a singular integro-differential system related to the entropy balance

This paper is devoted to the mathematical analysis of a thermodynamic model describing phase transitions with thermal memory in terms of an entropy equation and a momentum balance for the microforces. The initial and boundary value problem is addressed for the related integro-differential system of partial differential equations (PDEs). Existence and uniqueness, continuous dependence on the data, and regularity results are proved for the global solution, in a finite time interval.

Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity

This paper deals with a thermomechanical model describing phase transitions with thermal memory in terms of balance and equilibrium equations for entropy and microforces, respectively. After a presentation and discussion of the model, the large time behaviour of the solutions to the related integro-differential system of PDE's is investigated.