Ginzburg-Landau Functionals Research Articles

Phase-field method based simulation of martensitic transformation in porous alloys

Porous materials, characterized by the presence of interconnected pores, exhibit the properties different from their bulk counterparts. One of properties of interest is that the pores can influence the martensitic transformation in shape memory alloys (SMAs), which directly affects the material's shape memory effect and mechanical properties. The martensitic transformation is accompanied by the formation of different martensitic variants, which determine the overall morphology, distribution, and self-accommodation effect of the transformed regions. Previous experimental studies have shown that the presence of pores, particularly at the metal-air interface, can significantly affect the martensitic variant structure, leading to its thinning. This thinning effect has been found to be able to improve the damping performance of the alloy. Experimental observations have indicated that no relief of martensitic variants was found around the metal-air interface, but non-transformed regions were observed. These observations suggest that the metal-air interface in porous materials is not a free surface and plays a crucial role in influencing the martensitic transformation. To further investigate the effect of martensitic variant self-accommodation on different constrained interfaces in porous materials, a three-dimensional phase-field model based on the time dependent Ginzburg-Landau (TDGL) function is proposed in this study. The phase-field model can give a comprehensive understanding of the evolution of martensitic variants and their interaction with the constrained interfaces. Remarkably, the simulation results accord well with the experimental findings, demonstrating the presence of fine martensitic variants near the metal-air interface. The simulations under different interface constraint conditions reveal that increasing the specific surface area of porous materials is an effective strategy to obtain a more refined martensitic variant structure. The system’s total energy is minimized by reducing the strain energy, which leads to the formation of a greater number of fine martensitic variants. This finding suggests that controlling the specific surface area of porous materials can be a promising approach to tailoring the mechanical properties of SMAs for specific applications. In conclusion, the presence of metal-air interface in porous material significantly influences the evolution of the martensitic transformation in SMA. Experimental observations show that the introduction of pore can modify the martensitic variant structure, resulting in improved damping performance. The proposed phase-field model successfully captures the behavior of martensitic variants near constrained interface. The simulation results emphasize the importance of specific surface area in obtaining fine martensitic variant structures. These findings contribute to a more in-depth understanding of the role of porous materials in shaping the properties of SMAs and provide a valuable insight into their design and application in various fields.

Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solutionC±with interior defects, high-index ‘fake defect’ solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds

We use a natural two-parameter min-max construction to produce critical points of the Ginzburg–Landau functionals on a compact Riemannian manifold of dimension $\geq 2$. We investigate the limiting behavior of these critical points as $\varepsilon \to 0$, and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable $(n-2)$-varifold as $\varepsilon \to 0$, suggesting connections to the min-max construction of minimal $(n-2)$-submanifolds.

Magnetic Phase Transitions to an Incommensurate Magnetic Structure in FeGe2 Compound

A symmetry analysis of possible magnetic structures in an incommensurate magnetic phase in FeGe2 compound, resulted from phase transitions from the paramagnetic phase, was performed based on a phenomenological consideration. It is shown that two possible approaches to a such an analysis, the first of which uses the magnetic representation of the space group, and the second one is based on the expansion of the magnetic moment in basis functions of irreducible representations of the space group of the paramagnetic phase, yield the same results. Space group irreducible representations are determined, according to which the transition to an incommensurate structure can occur. The set of these representations appears identical in both approaches. Ginzburg–Landau functionals for analyzing the transitions according to these representations are written. A renormalization group analysis of the second-order phase transitions from the paramagnetic state to the incommensurate magnetic structure is performed. It is shown that a helical magnetic structure can arise in the incommensurate phase as a result of two second-order phase transitions at the transitions temperature.

Магнитные фазовые переходы в несоизмеримую магнитную структуру в соединении FeGe-=SUB=-2-=/SUB=-

AbstractA symmetry analysis of possible magnetic structures in an incommensurate magnetic phase in FeGe_2 compound, resulted from phase transitions from the paramagnetic phase, was performed based on a phenomenological consideration. It is shown that two possible approaches to a such an analysis, the first of which uses the magnetic representation of the space group, and the second one is based on the expansion of the magnetic moment in basis functions of irreducible representations of the space group of the paramagnetic phase, yield the same results. Space group irreducible representations are determined, according to which the transition to an incommensurate structure can occur. The set of these representations appears identical in both approaches. Ginzburg–Landau functionals for analyzing the transitions according to these representations are written. A renormalization group analysis of the second-order phase transitions from the paramagnetic state to the incommensurate magnetic structure is performed. It is shown that a helical magnetic structure can arise in the incommensurate phase as a result of two second-order phase transitions at the transitions temperature.

Asymptotic analysis of the Ginzburg–Landau functional on point clouds

AbstractThe Ginzburg–Landau functional is a phase transition model which is suitable for classification type problems. We study the asymptotics of a sequence of Ginzburg–Landau functionals with anisotropic interaction potentials on point clouds Ψnwherendenotes the number data points. In particular, we show the limiting problem, in the sense of Γ-convergence, is related to the total variation norm restricted to functions taking binary values, which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence.

Nearly Parallel Vortex Filaments in the 3D Ginzburg–Landau Equations

We introduce a framework to study the occurrence of vortex filament concentration in 3D Ginzburg–Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical 3-dimensional domain, in certain scaling limits; it is shown to arise as the $${\Gamma}$$ -limit of a sequence of scaled Ginzburg–Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg–Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically relevant situations where clustering of filaments is expected. In this setting it also constitutes a higher-order asymptotic expansion of the Ginzburg–Landau energy, a refinement over the arclength functional approximation. Our description of the vorticity region significantly improves on previous studies and enables us to rigorously distinguish a collection of multiplicity one vortex filaments from an ensemble of fewer higher multiplicity ones. As an application, we prove the existence of solutions of the Ginzburg–Landau equation that exhibit clusters of vortex filaments whose small-scale structure is governed by the limiting free-energy functional.

A lower bound for the BCS functional with boundary conditions at infinity

We consider a many-body system of fermionic atoms interacting via a local pair potential and subject to an external potential within the framework of Bardeen-Cooper-Schrieffer (BCS) theory. We measure the free energy of the whole sample with respect to the free energy of a reference state which allows us to define a BCS functional with boundary conditions at infinity. Our main result is a lower bound for this energy functional in terms of expressions that typically appear in Ginzburg-Landau functionals.

Theory and application of p-regularized subproblems for p>2

The p-regularized subproblem (p-RS) is the key content of a regularization technique in computing a Newton-like step for unconstrained optimization. The idea is to incorporate a local quadratic approximation of the objective function with a weighted regularization term and then globally minimize it at each iteration. In this paper, we establish a complete theory of the p-RSs for general p>2 that covers previous known results on p=3 or p=4. The theory features necessary and sufficient optimality conditions for the global and also for the local non-global minimizers of (p-RS). It gives a closed-form expression for the global minimum set of (p-RS) and shows that (p-RS), p>2 can have at most one local non-global minimizer. Our theory indicates that (p-RS) have all properties that the trust region subproblems do. In application, (p-RS) can appear in natural formulation for optimization problems. We found two examples. One is to utilize the Tikhonov regularization to stabilize the least square solution for an over-determined linear system; and the other comes from numerical approximations to the generalized Ginzburg–Landau functionals. Moreover, when (p-RS) is appended with m additional linear inequality constraints, denoted by (p-), the problem becomes NP-hard. We show that the partition problem, the k-dispersion-sum problem and the quadratic assignment problem in combinatorial optimization can be equivalently formulated as special types of (p-RS) with p=4. In the end, we develop an algorithm for solving (p-RS).

Dynamical mean-field theory and weakly non-linear analysis for the phase separation of active Brownian particles

Recently, we have derived an effective Cahn-Hilliard equation for the phase separation dynamics of active Brownian particles by performing a weakly non-linear analysis of the effective hydrodynamic equations for density and polarization [Speck et al., Phys. Rev. Lett. 112, 218304 (2014)]. Here, we develop and explore this strategy in more detail and show explicitly how to get to such a large-scale, mean-field description starting from the microscopic dynamics. The effective free energy emerging from this approach has the form of a conventional Ginzburg-Landau function. On the coarsest scale, our results thus agree with the mapping of active phase separation onto that of passive fluids with attractive interactions through a global effective free energy (motility-induced phase transition). Particular attention is paid to the square-gradient term necessary for the phase separation kinetics. We finally discuss results from numerical simulations corroborating the analytical results.

Erratum to: On the Ginzburg–Landau Functional in the Surface Superconductivity Regime

Erratum to: On the Ginzburg–Landau Functional in the Surface Superconductivity Regime

On the Ginzburg–Landau Functional in the Surface Superconductivity Regime

We present new estimates on the two-dimensional Ginzburg-Landau energy of a type-II superconductor in an applied magnetic field varying between the second and third critical fields. In this regime, superconductivity is restricted to a thin layer along the boundary of the sample. We provide new energy lower bounds, proving that the Ginzburg-Landau energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. Estimates relating the density of the Ginzburg-Landau order parameter to that of the 1D problem follow. In the particular case of a disc sample, a refinement of our method leads to a pointwise estimate on the Ginzburg-Landau order parameter, thereby proving a strong form of uniformity of the surface superconductivity layer, related to a conjecture by Xing-Bin Pan.

Numerical approximation of time evolution related to Ginzburg–Landau functionals using weighted Sobolev gradients

Sobolev gradients have been discussed in Sial et al. (2003) as a method for energy minimization related to Ginzburg–Landau functionals. In this article, a weighted Sobolev gradient approach for the time evolution of a Ginzburg–Landau functional is presented for different values of κ. A comparison is given between the weighted and unweighted Sobolev gradients in a finite element setting. It is seen that for small values of κ, the weighted Sobolev gradient method becomes more and more efficient compared to using the unweighted Sobolev gradient. A comparison with Newton’s method is given where the failure of Newton’s method is demonstrated for a test problem.

Convergence of Ginzburg–Landau Functionals in Three-Dimensional Superconductivity

In this paper we consider the asymptotic behavior of the Ginzburg- Landau model for superconductivity in 3-d, in various energy regimes. We rigorously derive, through an analysis via {\Gamma}-convergence, a reduced model for the vortex density, and we deduce a curvature equation for the vortex lines. In a companion paper, we describe further applications to superconductivity and superfluidity, such as general expressions for the first critical magnetic field H_{c1}, and the critical angular velocity of rotating Bose-Einstein condensates.

Nonlocal Free Energy of a Spatially Inhomogeneous Superconductor

The microscopic approach is developed for obtaining of the free energy of a superconductor based on direct calculation of the vacuum amplitude. The free energy functional of the spatially inhomogeneous superconductor in a magnetic field is obtained with help of the developed approach. The obtained functional is generalization of Ginzburg-Landau functionals for any temperature, for arbitrary spatial variations of the order parameter and for the nonlocality of a magnetic response and the order parameter. Moreover, the nonlocality of the magnetic response is the consequence of order parameter's nonlocality. The extremals of this functional are considered in the explicit form in the low- and high-temperature limit at the condition of slowness of spatial variations of the order parameter.

Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

We study the variational convergence of a family of twodimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that if suitably normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.

Phonon-deformational generalized ginzburg-landau functionals for describing martensite transitions between FCC and BCC structures in metals

A method is developed for theoretical description of martensite transitions between fcc and bcc phases of metals based on generalized Ginzburg-Landau functionals (GGLFs), in which the transformation parameters are the “phonon” parameter describing relative slips of close-packed atomic planes and fcc lattice deformations in the region of a martensite inclusion. General expressions are proposed for calculating the deformational contributions to the GGLF. Local strains and local elastic moduli in the region of martensite inclusions are estimated numerically. The possibility of constructing phenomenological expressions for the GGLF using interpolations of available experimental data on the structure and elastic moduli in the fcc and bcc phases is discussed.

Superfluid-insulator transitions in attractive Bose-Hubbard model with three-body constraint

By means of the method of the effective potential, the phase transitions from the Mott insulating state to either the atomic or the dimer superfluid state in the three-body constrained attractive Bose lattice gas are analyzed. Because of the appearance of the Feshbach resonance coupling between the two kinds of order parameters in the derived Ginzburg-Landau function, it is found that the continuous Mott-insulator--superfluid transitions can be preempted by first-order ones. Since the employed approach can provide accurate predictions of phase boundaries in the strong coupling limit, where the dimer superfluid phase can emerge, our work sheds light on the search of this phase in real ultracold Bose gases in optical lattices.

Approximating time evolution related to Ginzburg–Landau functionals via Sobolev gradient methods in a finite-element setting

Sobolev gradients have previously [1] been used to approximate time evolution related to a model A functional in a finite-difference setting in this journal. Here a related approach in a finite-element setting is discussed.

Generalized Ginzburg-Landau functionals for studies of phase transformations between FCC, BCC, and HCP structures in metals and alloys

Generalized Ginzburg-Landau functionals describing transitions between bcc, fcc, and hcp structures are studied using four weakly nonuniform transformation parameters: three tensile strains along the principal crystallographic axes and a “phonon” parameter describing relative slip of close-packed planes. Several versions of transformation paths indicated for each of the three transformations considered, which appear to be the most realistic; explicit expressions for atomic displacements are given for each of these paths. It is shown that the gradient terms in these functionals can be explicitly expressed in terms of the dynamic matrix of the crystal on a transformation path; these dynamic matrices can be determined using interpolations of experimental data on phonon spectra between the initial and final phases. The results are used for estimating the characteristics of interphase boundaries between ferrite and cementite in steels. It is found that the width of such interfaces considerably exceeds the interatomic distance, while the estimated values of the gradient terms are in reasonable agreement with the available experimental data.