scholarly journals Generalized Swift–Hohenberg and phase-field-crystal equations based on a second-gradient phase-field theory

Meccanica ◽  
2020 ◽  
Vol 55 (10) ◽  
pp. 1853-1868
Author(s):  
Luis Espath ◽  
Victor M. Calo ◽  
Eliot Fried
Keyword(s):  
Field Theory ◽  
Free Energy ◽  
Phase Field ◽  
Constitutive Relations ◽  
Energy Imbalance ◽  
Second Gradient ◽  
Microforce Balance ◽  
Broad Generalization ◽  
Gradient Phase ◽  
Phase Field Crystal

Abstract The principle of virtual power is used derive a microforce balance for a second-gradient phase-field theory. In conjunction with constitutive relations consistent with a free-energy imbalance, this balance yields a broad generalization of the Swift–Hohenberg equation. When the phase field is identified with the volume fraction of a conserved constituent, a suitably augmented version of the free-energy imbalance yields constitutive relations which, in conjunction with the microforce balance and the constituent content balance, delivers a broad generalization of the phase-field-crystal equation. Thermodynamically consistent boundary conditions for situations in which the interface between the system and its environment is structureless and cannot support constituent transport are also developed, as are energy decay relations that ensue naturally from the thermodynamic structure of the theory.

scholarly journals Phase-field gradient theory

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Luis Espath ◽  
Victor Calo
Keyword(s):  
Free Energy ◽  
Field Equation ◽  
Field Gradient ◽  
Phase Field ◽  
Constitutive Relations ◽  
The Body ◽  
Second Grade ◽  
Gradient Theory ◽  
Boundary Edge ◽  
Field Equations

AbstractWe propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a ‘generalized Swift–Hohenberg equation’—a second-grade phase-field equation—and its conserved version, the ‘generalized phase-field crystal equation’—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions.

Efficient numerical algorithms for the phase field crystal equation

2021 ◽  
pp. 2150042
Author(s):  
Qingqu Zhuang ◽  
Shuying Zhai ◽  
Zhifeng Weng
Keyword(s):  
Free Energy ◽  
Lower Bound ◽  
Numerical Simulations ◽  
Lagrange Multiplier ◽  
Phase Field ◽  
Numerical Algorithms ◽  
Energy Potential ◽  
Energy Stable ◽  
2D And 3D ◽  
Phase Field Crystal

In this paper, based on the Lagrange Multiplier approach in time and the Fourier-spectral scheme for space, we propose efficient numerical algorithms to solve the phase field crystal equation. The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential. The unconditional energy stability of the three semi-discrete schemes is proven. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.

scholarly journals Second Gradient Electromagnetostatics: Electric Point Charge, Electrostatic and Magnetostatic Dipoles

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1104 ◽  
Author(s):  
Markus Lazar ◽  
Jakob Leck
Keyword(s):  
Field Theory ◽  
Electromagnetic Field ◽  
Point Charge ◽  
Constitutive Relations ◽  
Higher Order ◽  
Gradient Field ◽  
Theoretical Point ◽  
Second Gradient ◽  
Maxwell Electrodynamics ◽  
Constitutive Parameters

In this paper, we study the theory of second gradient electromagnetostatics as the static version of second gradient electrodynamics. The theory of second gradient electrodynamics is a linear generalization of higher order of classical Maxwell electrodynamics whose Lagrangian is both Lorentz and U ( 1 ) -gauge invariant. Second gradient electromagnetostatics is a gradient field theory with up to second-order derivatives of the electromagnetic field strengths in the Lagrangian. Moreover, it possesses a weak nonlocality in space and gives a regularization based on higher-order partial differential equations. From the group theoretical point of view, in second gradient electromagnetostatics the (isotropic) constitutive relations involve an invariant scalar differential operator of fourth order in addition to scalar constitutive parameters. We investigate the classical static problems of an electric point charge, and electric and magnetic dipoles in the framework of second gradient electromagnetostatics, and we show that all the electromagnetic fields (potential, field strength, interaction energy, interaction force) are singularity-free, unlike the corresponding solutions in the classical Maxwell electromagnetism and in the Bopp–Podolsky theory. The theory of second gradient electromagnetostatics delivers a singularity-free electromagnetic field theory with weak spatial nonlocality.

Interfacial free energy adjustable phase field crystal model for homogeneous nucleation

Soft Matter ◽  
2016 ◽  
Vol 12 (20) ◽  
pp. 4666-4673 ◽  
Author(s):  
Can Guo ◽  
Jincheng Wang ◽  
Zhijun Wang ◽  
Junjie Li ◽  
Yaolin Guo ◽  
...  
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Homogeneous Nucleation ◽  
Interfacial Free Energy ◽  
Crystal Model ◽  
Phase Field Crystal

scholarly journals Free energy of the bcc–liquid interface and the Wulff shape as predicted by the phase-field crystal model

2014 ◽  
Vol 385 ◽  
pp. 148-153 ◽  
Author(s):  
Frigyes Podmaniczky ◽  
Gyula I. Tóth ◽  
Tamás Pusztai ◽  
László Gránásy
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Liquid Interface ◽  
Wulff Shape ◽  
Crystal Model ◽  
Phase Field Crystal

scholarly journals Boundary conditions in topological AdS4/CFT3

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Pietro Benetti Genolini ◽  
Matan Grinberg ◽  
Paul Richmond
Keyword(s):  
Field Theory ◽  
Free Energy ◽  
Boundary Conditions ◽  
Smooth Solution ◽  
Three Dimensional ◽  
Field Theories ◽  
Legendre Transformation ◽  
Generating Functional ◽  
Holographic Duals

Abstract We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional $$ \mathcal{N} $$ N = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.

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