Phase-field System Research Articles

Upper semicontinuous attractor for a hyperbolic phase-field model with memory

We consider a hyperbolic phase-field model with memory which was introduced, justified, and studied in a previous authors'paper, jointly with C. Giorgi. There, we showed in particular that the model defines a dynamical system which possesses a uniformly absorbing set. Here, we deepen the analysis by proving that the system has a uniform attractor of finite fractal dimension, which is upper semicontinuous at zero with respect to a family of attractors associated with a class of parabolic phase-field systems with memory.

Phase-Field Models with Hysteresis

Phase-field systems as mathematical models for phase transitions have drawn increasing attention in recent years. However, while capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past 15 years. In particular, they have proved well-posedness and thermodynamic consistency for hysteretic phase field models which are related to the Caginalp and Penrose–Fife models. In this paper, these results are extended into different directions: we admit temperature-dependent relaxation coefficients and relax the growth conditions for the hysteresis operators considerably; also, a unified approach is used for a general class of systems that includes both the Caginalp and Penrose–Fife analogues.

Hysteresis in phase‐field models with thermal memory

Phase-field systems as mathematical models for phase transitions have drawn increasing attention in recent years. However, while capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming, a new approach has recently been proposed by the last two authors which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. In this paper this approach is extended to cases where the material exhibits an additional thermal memory, i.e., where the heat flux contains a time convolution of the spatial gradient of temperature. It is shown that the corresponding system of field equations admits a unique strong solution that depends continuously on the data of the system. EMAIL:: gilardi@dimat.unipv.it KEYWORDS:: Hysteresis, phase-field models, memory

Hysteresis in phase-field models with thermal memory

Phase-field systems as mathematical models for phase transitions have drawn increasing attention in recent years. However, while capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming, a new approach has recently been proposed by the last two authors which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. In this paper this approach is extended to cases where the material exhibits an additional thermal memory, i.e., where the heat flux contains a time convolution of the spatial gradient of temperature. It is shown that the corresponding system of field equations admits a unique strong solution that depends continuously on the data of the system. EMAIL:: gilardi@dimat.unipv.it KEYWORDS:: Hysteresis, phase-field models, memory

A class of time discrete schemes for a phase–field system of Penrose–Fife type

In this paper, a phase field system of Penrose–Fife type with non–conserved order parameter is considered. A class of time–discrete schemes for an initial–boundary value problem for this phase–field system is presented. In three space dimensions, convergence is proved and an error estimate linear with respect to the time–step size is derived.

A Generalized Phase-Field System

The existence, the estimate, and the uniqueness of a solution to a general phase-field system, motivated by Caginalp's model describing the phase changes, are established. The paper extends the results for the already studied types of nonlinearities related to the model.

Dynamics of a free boundary in a binary medium with variable thermal conductivity

We construct an asymptotic solution of the phase field system with variable thermal conductivity different in domains occupied by different phases. We show that, depending on relations between parameters characterizing the substance, the dynamics of the free interface between the phases is determined by solutions of the classical or modified Stefan problems.

Hugoniot-type conditions and weak solutions to the phase-field system

We consider a new concept of weak solutions to the phase-field equations with a small parameter ε characterizing the length of interaction. For the standard situation of a single free interface, this concept (in contrast with the common one) leads to the well-known Stefan–Gibbs–Thomson problem as ε→0. For the case of a large number M(ε) (M(ε)→∞ as ε→0) of free interfaces, which corresponds to the ‘wave-train’ interpretation of a ‘mushy region’, this concept allows us to obtain the limit problem as ε→0.

Uniqueness and long-time behavior for the conserved phase-field system with memory

This paper is concerned with a conserved phase-field model with memory. Weinclude memory by replacing the standard Fourier heat law with a constitutive assumptionof Gurtin-Pipkin type, and the system is conservative in the sense that the initial mass ofthe order parameter as well as the energy are preserved during the evolution. A Cauchy-Neumann problem is investigated for this model which couples a Volterra integro-differentialequation with fourth order dynamics for the phase field. A sharp uniqueness theorem isproven by demonstrating continuous dependence for a suitably weak formulation. Withregard to the long-time behavior, the limit points of the trajectories are completely characterized.

An Analysis of Phase-Field Alloys and Transition Layers

A phase-field approach to binary alloys is studied. Formal asymptotics of the system of parabolic differential equations leads to new interface relations as part of a macroscopic model which arises in the limit of vanishing interface thickness. Under suitable conditions we prove that the phase-field system has a unique solution which converges to the limiting macroscopic solution. The concentration and phase are monotonic across the interface for a simplified system. Transition layers in concentration are induced due to the change in phase and the change in material diffusion across the interface. Excess impurities may be trapped as a consequence of these layers.

Hysteresis operators in phase-field models of Penrose-fife type

Phase-field systems as mathematical models for phase transitions have drawn a considerable attention in recent years. However, while they are suitable for capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. Well-posedness and thermodynamic consistency were proved for a phase-field system with hysteresis which is closely related to the model advanced by Caginalp in a series of papers. In this note the more difficult case of a phase-field system of Penrose-Fife type with hysteresis is investigated. Under slightly more restrictive assumptions than in the Caginalp case it is shown that the system is well-posed and thermodynamically consistent.

Weak solutions to the phase field system

We consider a new concept of weak solutions to the phase field equations with a small parameter ϵ characterizing the length of interaction. For the standard situation of a single free interface, this concept (in contrast to the common one) leads to the well-known Stefan–Gibbs–Thomson problem as ϵ → 0. For the case of a large number M(ϵ) (M(ϵ) → ∞ as ϵ → 0) of free interfaces, which is related to the “wave-train” interpretation of a “mushy region”, this concept allows us to obtain limiting problems as ϵ → 0.

Asymptotic solution of the conserved phase field system in the fast relaxation case

The conserved phase field system with a small parameter in the n-dimensional case (n[les ]3) is considered. An asymptotic solution, describing the free interface dynamics, is constructed and justified. As the small parameter tends to zero, the limiting solution satisfies the modified Stefan problem with corrected Gibbs–Thomson law.

Weak Solution to Some Penrose–Fife Phase-Field Systems with Temperature-Dependent Memory

In this paper a phase-field model of Penrose–Fife type is considered for a diffusive phase transition in a material in which the heat flux is a superposition of two different contributions: one part is proportional to the spatial gradient of the inverse temperature, while the other is of the form of the Gurtin–Pipkin law introduced in the theory of materials with thermal memory. It is shown that an initial-boundary value problem for the resulting state equations has a unique solution, thereby generalizing a number of recent results.

The Existence of Travelling Wave Solutions of a Generalized Phase-Field Model

This paper establishes the existence and, in certain cases, the uniqueness of travelling wave solutions of both second-order and higher-order phase-field systems. These solutions describe the propagation of planar solidification fronts into a hypercooled liquid. The equations are scaled in the usual way so that the relaxation time is $\alpha\varepsilon^2$, where $\varepsilon$ is a nondimensional measure of the interfacial thickness. The equations for the transition layer separating the two phases form a system identical to that for the travelling-wave problem, in which the temperature is strongly coupled with the order parameter. Thus there is no longer a well-defined temperature at the inteface, as is the case in the more frequently studied situation in which the liquid phase is undercooled but not hypercooled. For phase-field systems of two second-order equations, we prove a general existence theorem based upon topological methods. A second, constructive proof based upon invariant-manifold methods is also given when the parameter $\alpha$ is either sufficiently small or sufficiently large. In either regime, it is also proved that the wave and the wave velocity are globally unique. Analogous results are also obtained for generalized phase-field systems in which the order parameter solves a higher-order differential equation. In this paper, the higher-order tems occur as a singular peturbation of the standard (isotropic) second-order equation. The higher-order terms are useful in modelling anisotropic interfacial motion.

Approximation of the phase-field transition system via fractional steps method

In this paper we prove the convergence of an iterative scheme of fractional steps type for the phase-field transition system. The use of this method simplifies the numerical algorithms due to its decoupling feature. The 2D case is considered and numerical results are reported.

On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type

In this paper we study an initial-boundary value Stefan-type problem with phase relaxation where the heat flux is proportional to the gradient of the inverse absolute temperature. This problem arise naturally as limiting case of the Penrose-Fife model for diffusive phase transitions with nonconserved order parameter if the coefficient of the interfacial energy is taken as zero. It is shown that the relaxed Stefan problem admits a weak solution which is obtained as limit of solutions to the Penrose-Fife phase-field equations. For a special boundary condition involving the heat exchange with the surrounding medium, also uniqueness of the solution is proved.

Inertial manifolds and inertial sets for the phase-field equations

The phase-field system is a mathematical model of phase transition, coupling temperature with a continuous order parameter which describes degree of solidification. The flow induced by this system is shown to be smoothing in H1×L2 and a global attractor is shown to exist. Furthermore, in low-dimensional space, the flow is essentially finite dimensional in the sense that a strongly attracting finite-dimensional manifold (or set) exists.

Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening

We initiate a study of the instability properties of steady solutions of the Cahn-Hilliard (CH) equation and the phase field (PF) system in terms of the coarseness of those solutions. Other spectral results are obtained as well. Steady state solutions for the CH equation with no flux or periodic boundary conditions are also stationary states for the bistable reaction-diffusion equation and the PF system (with constant temperature). Thus, it is meaningful to linearize each of these equations about a common stationary solution and discuss the stability of this state in each dynamical process. Converting the linearized operators to self-adjoint form and expressing eigenvalues in terms of modified Rayleigh quotients enables us to relate the spectra of the three operators. In particular we relate the indices of instability with respect to the three processes. We also discuss the change in the spectrum of the linearized CH operator in one dimension when we change the stationary solution at which it is linearized. We conjecture that the eigenvalues increase with the number of oscillations of the steady state for a given mass, and prove that this is so in certain cases. This lends credence to the idea that motion of these dynamical systems near coarse solutions takes place on longer time scales.

The Dynamics of a Conserved Phase Field System: Stefan-like, Hele-Shaw, and Cahn-Hilliard Models as Asymptotic Limits

The dynamics of a conserved phase field system for a free boundary, that is, u t +1/2lφ t =KΔu, −τφ t =ξ 2 Δ[ξ 2 Δφ+f(φ)/a+2u], are studied asymptotically. Many of the major macroscopic free-boundary problems arise as limits in various scalings