Symmetry

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29 pages, 8366 KiB

Open AccessEditor’s ChoiceArticle

Elastic Origin of the Unsymmetrical Thermal Hysteresis in Spin Crossover Materials: Evidence of Symmetry Breaking

by Mamadou Ndiaye, Nour El Islam Belmouri, Jorge Linares and Kamel Boukheddaden

Symmetry 2021, 13(5), 828; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13050828 - 09 May 2021

Cited by 12 | Viewed by 2191

The jungle of experimental behaviors of spin-crossover materials contains a tremendous number of unexpected behaviors, among which, the unsymmetrical hysteresis loops having different shapes on heating and cooling, that we often encounter in literature. Excluding an extra effect of crystallographic phase transitions, we [...] Read more.

The jungle of experimental behaviors of spin-crossover materials contains a tremendous number of unexpected behaviors, among which, the unsymmetrical hysteresis loops having different shapes on heating and cooling, that we often encounter in literature. Excluding an extra effect of crystallographic phase transitions, we study here these phenomena from the point of view of elastic modeling and we demonstrate that a simple model accounting for the bond lengths misfits between the high-spin and low-spin states is sufficient to describe the situation of unsymmetrical hysteresis showing plateaus at the transition only on cooling or on heating branches. The idea behind this effect relates to the existence of a discriminant elastic frustration in the lattice, which expresses only along the high-spin to low-spin transition or in the opposite side. The obtained two-step transitions showed characteristics of self-organization of the spin states under the form of stripes, which we explain as an emergence process of antagonist directional elastic interactions inside the lattice. The analysis of the spin state transformation inside the plateau on cooling in terms of two sublattices demonstrated that the elastic-driven self-organization of the spin states is accompanied with a symmetry breaking. Full article

(This article belongs to the Special Issue Symmetry and Symmetry Breaking: Phase Transitions and Critical Phenomena)

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14 pages, 2881 KiB

Open AccessArticle

A First Order Phase Transition Studied by an Ising-Like Model Solved by Entropic Sampling Monte Carlo Method

by Jorge Linares, Catherine Cazelles, Pierre-Richard Dahoo and Kamel Boukheddaden

Symmetry 2021, 13(4), 587; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13040587 - 02 Apr 2021

Cited by 3 | Viewed by 1981

Abstract

Two-dimensional (2D) square, rectangular and hexagonal lattices and 3D parallelepipedic lattices of spin crossover (SCO) compounds which represent typical examples of first order phase transitions compounds are studied in terms of their size, shape and model through an Ising-like Hamiltonian in which the [...] Read more.

Two-dimensional (2D) square, rectangular and hexagonal lattices and 3D parallelepipedic lattices of spin crossover (SCO) compounds which represent typical examples of first order phase transitions compounds are studied in terms of their size, shape and model through an Ising-like Hamiltonian in which the fictitious spin states are coupled via the respective short and long-range interaction parameters J, and G. Furthermore, an environmental L parameter accounting for surface effects is also introduced. The wealth of SCO transition properties between its bi-stable low spin (LS) and high spin (HS) states are simulated using Monte Carlo Entropic Sampling (MCES) method which favors the scanning of macro states of weak probability occurrences. For given J and G, the focus is on surface effects through parameter L. It is shown that the combined first-order phase transition effects of the parameters of the Hamiltonian can be highlighted through two typical temperatures, T_O.D., the critical order-disorder temperature and T_eq the equilibrium temperature that is fixed at zero effective ligand field. The relative positions of T_O.D. and T_eq control the nature of the transition and mediate the width and position of the thermal hysteresis curves with size and shape. When surface effects are negligible (L = 0), the equilibrium transition temperature, T_eq_. becomes constant, while the thermal hysteresis’ width increases with size. When surface effects are considered, L ≠ 0, T_eq_. increases with size and the first order transition vanishes in favor of a gradual transition until reaching a threshold size, below which a reentrance phenomenon occurs and the thermal hysteresis reappears again, as shown for hexagonal configuration. Full article

(This article belongs to the Special Issue Symmetry and Symmetry Breaking: Phase Transitions and Critical Phenomena)

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9 pages, 2858 KiB

Open AccessArticle

Analogy between Thermodynamic Phase Transitions and Creeping Flows in Rectangular Cavities

by Miron Kaufman and Petru S. Fodor

Symmetry 2020, 12(11), 1859; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111859 - 12 Nov 2020

Cited by 1 | Viewed by 1411

Abstract

An analogy is found between the streamline function corresponding to Stokes flows in rectangular cavities and the thermodynamics of phase transitions and critical points. In a rectangular cavity flow, with no-slip boundary conditions at the walls, the corners are fixed points. The corners [...] Read more.

An analogy is found between the streamline function corresponding to Stokes flows in rectangular cavities and the thermodynamics of phase transitions and critical points. In a rectangular cavity flow, with no-slip boundary conditions at the walls, the corners are fixed points. The corners defined by a stationary and a moving wall, are found to be analogous to a thermodynamic first-order transition point. In contrast, the corners defined by two stationary walls correspond to thermodynamic critical points. Here, flow structures, also known as Moffatt eddies, form and act as stagnation regions where mixing is impeded. A third stationary point occurs in the middle region of the channel and it is analogous to a high temperature thermodynamic fixed point. The numerical results of the fluid flow modeling are correlated with analytical work in the proximity of the fixed points. Full article

(This article belongs to the Special Issue Symmetry and Symmetry Breaking: Phase Transitions and Critical Phenomena)

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13 pages, 448 KiB

Open AccessArticle

Nematic and Smectic Phases: Dynamics and Phase Transition

by Aurélien Bailly-Reyre and Hung T. Diep

Symmetry 2020, 12(9), 1574; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12091574 - 22 Sep 2020

Cited by 3 | Viewed by 4803

Abstract

We study in this paper the dynamics of molecules leading to the formation of nematic and smectic phases using a mobile 6-state Potts spin model with Monte Carlo simulation. Each Potts state represents a molecular orientation. We show that, with the choice of [...] Read more.

We study in this paper the dynamics of molecules leading to the formation of nematic and smectic phases using a mobile 6-state Potts spin model with Monte Carlo simulation. Each Potts state represents a molecular orientation. We show that, with the choice of an appropriate microscopic Hamiltonian describing the interaction between individual molecules modeled by 6-state Potts spins, we obtain the structure of the smectic phase by cooling the molecules from the isotropic phase to low temperatures: molecules are ordered in independent equidistant layers. The isotropic-smectic phase transition is found to have a first-order character. The nematic phase is also obtained with the choice of another microscopic Hamiltonian. The isotropic-nematic phase transition is a second-order one. The real-time dynamics of the molecules leading to the liquid-crystal ordering in each case is shown by a video. Full article

(This article belongs to the Special Issue Symmetry and Symmetry Breaking: Phase Transitions and Critical Phenomena)

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8 pages, 224 KiB

Open AccessArticle

Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking

by Carlo Heissenberg and Franco Strocchi

Symmetry 2020, 12(7), 1120; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12071120 - 06 Jul 2020

Viewed by 1533

Abstract

The matrix elements of operators transforming as irreducible representations of an unbroken symmetry group G are governed by the well-known Wigner–Eckart relations. In the case of infinite-dimensional systems, with G spontaneously broken, we prove that the corrections to such relations are provided by [...] Read more.

The matrix elements of operators transforming as irreducible representations of an unbroken symmetry group G are governed by the well-known Wigner–Eckart relations. In the case of infinite-dimensional systems, with G spontaneously broken, we prove that the corrections to such relations are provided by symmetry breaking Ward identities, and simply reduce to a tadpole term involving Goldstone bosons. The analysis extends to the case in which an explicit symmetry breaking term is present in the Hamiltonian, with the tadpole term now involving pseudo Goldstone bosons. An explicit example is discussed, illustrating the two cases. Full article

(This article belongs to the Special Issue Symmetry and Symmetry Breaking: Phase Transitions and Critical Phenomena)

13 pages, 336 KiB

Open AccessArticle

Function Reconstruction from Reflection Symmetric Radon Data

by Trong Tuong Truong

Symmetry 2020, 12(6), 956; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12060956 - 04 Jun 2020

Cited by 2 | Viewed by 1881

Abstract

In many areas of two-dimensional imaging science, data acquisition arises as an integral process. The inverse process—or image reconstruction—means the solving of a Radon problem mathematically. It may happen that there exists classes of integral data which are mirror symmetric with respect to [...] Read more.

In many areas of two-dimensional imaging science, data acquisition arises as an integral process. The inverse process—or image reconstruction—means the solving of a Radon problem mathematically. It may happen that there exists classes of integral data which are mirror symmetric with respect to a line. Common sense suggests that the occurrence of a symmetry usually provides significant help in the search of the problem solution. Here, we showed an example of the contrary to this popular belief. In fact, to solve such a Radon problem with inherent reflection symmetry, there is a need to split it into two new Radon problems on half-spaces, which do not have solutions at hand. In this paper, a full solution is obtained via geometric inversion mapping of the two original half-spaces Radon problems to the disk interior/exterior Radon problem arising in recent modalities of Compton scattering tomography, which fortunately has explicit worked out inverse formulas. Full article

(This article belongs to the Special Issue Symmetry and Symmetry Breaking: Phase Transitions and Critical Phenomena)

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16 pages, 4708 KiB

Open AccessArticle

Skyrmion Crystals and Phase Transitions in Magneto-Ferroelectric Superlattices: Dzyaloshinskii–Moriya Interaction in a Frustrated J₁ − J₂ Model

by Ildus F. Sharafullin and Hung T. Diep

Symmetry 2020, 12(1), 26; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12010026 - 21 Dec 2019

Cited by 11 | Viewed by 3358

Abstract

The formation of a skyrmion crystal and its phase transition are studied, taking into account the Dzyaloshinskii–Moriya (DM) interaction at the interface between a ferroelectric layer and a magnetic layer in a superlattice. Frustration is introduced in both magnetic and ferroelectric films. The [...] Read more.

The formation of a skyrmion crystal and its phase transition are studied, taking into account the Dzyaloshinskii–Moriya (DM) interaction at the interface between a ferroelectric layer and a magnetic layer in a superlattice. Frustration is introduced in both magnetic and ferroelectric films. The films have a simple cubic lattice structure. The spins inside the magnetic layers are Heisenberg spins interacting with each other via nearest-neighbor (NN) exchange

J^{m}

and next-nearest-neighbor (NNN) exchange

J^{2 m}

. The polarizations in the ferroelectric layers are assumed to be of Ising type with NN and NNN interactions

J^{f}

and

J^{2 f}

. At the magnetoelectric interface, a DM interaction

J^{m f}

between spins and polarizations is supposed. The spin configuration in the ground state is calculated by the steepest descent method. In an applied magnetic field

H

perpendicular to the layers, we show that the formation of skyrmions at the magnetoelectric interface is strongly enhanced by the frustration brought about by the NNN antiferromagnetic interactions

J^{2 m}

and

J^{2 f}

. Various physical quantities at finite temperatures are obtained by Monte Carlo simulations. We show the critical temperature, the order parameters of magnetic and ferroelectric layers as functions of the interface DM coupling, the applied magnetic field, and

J^{2 m}

and

J^{2 f}

. The phase transition to the disordered phase is studied in detail. Full article

(This article belongs to the Special Issue Symmetry and Symmetry Breaking: Phase Transitions and Critical Phenomena)

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22 pages, 1536 KiB

Open AccessArticle

Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling

by Hiroshi Koibuchi, Chrystelle Bernard, Jean-Marc Chenal, Gildas Diguet, Gael Sebald, Jean-Yves Cavaille, Toshiyuki Takagi and Laurent Chazeau

Symmetry 2019, 11(9), 1124; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11091124 - 04 Sep 2019

Cited by 2 | Viewed by 2142

Abstract

Configurations of the polymer state in rubbers, such as so-called isotropic (random) and anisotropic (almost aligned) states, are symmetric/asymmetric under space rotations. In this paper, we present numerical data obtained by Monte Carlo simulations of a model for rubber formulations to compare these [...] Read more.

Configurations of the polymer state in rubbers, such as so-called isotropic (random) and anisotropic (almost aligned) states, are symmetric/asymmetric under space rotations. In this paper, we present numerical data obtained by Monte Carlo simulations of a model for rubber formulations to compare these predictions with the reported experimental stress–strain curves. The model is defined by extending the two-dimensional surface model of Helfrich–Polyakov based on the Finsler geometry description. In the Finsler geometry model, the directional degree of freedom

\vec{σ}

of the polymers and the polymer position

r

are assumed to be the dynamical variables, and these two variables play an important role in the modeling of rubber elasticity. We find that the simulated stresses

τ_{sim}

are in good agreement with the reported experimental stresses

τ_{\exp}

for large strains of up to

1200 %

. It should be emphasized that the stress–strain curves are directly calculated from the Finsler geometry model Hamiltonian and its partition function, and this technique is in sharp contrast to the standard technique in which affine deformation is assumed. It is also shown that the obtained results are qualitatively consistent with the experimental data as influenced by strain-induced crystallization and the presence of fillers, though the real strain-induced crystallization is a time-dependent phenomenon in general. Full article

(This article belongs to the Special Issue Symmetry and Symmetry Breaking: Phase Transitions and Critical Phenomena)

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