Free Energy Density Research Articles

Elastic properties and constitutive behaviour of graphene at finite temperature and large deformation

In the earlier reported theoretical studies, the bending modulus of the graphene sheet predicted with the inclusion of the dihedral energy term is observed to be almost double as compared to those reported without considering the dihedral energy term. The bending modulus reported considering dihedral energy term is observed to be more accurate when compared with the quantum mechanical calculations. The findings of these studies for bending modulus (with/without considering dihedral energy terms) are reported without considering the effect of finite temperature. In the present work, a multiscale computational framework incorporating the dihedral energy term and finite temperature in the constitutive model is presented for the first time. A temperature-dependent bond length is expressed in terms of strain and curvature tensor through the temperature-related quadratic-type Cauchy-Born rule. The total Helmholtz free energy of the unit cell is expressed as a sum of the potential energy of the unit cell and thermal energy due to the thermal vibrations of atoms arising from the finite temperature. The atomic interactions are modelled through the Tersoff-Brenner potential considering different sets of empirical parameters, including the authors’ recently proposed set of empirical parameters. The stress/moment tensors and the tangent constitutive matrix are obtained from the derivatives of the Helmholtz free energy density with strains and curvatures. The thermal properties, elastic properties, and tangent stiffness coefficients for armchair and zigzag graphene with/without dihedral energy term at finite strain, curvature, and temperature are investigated and discussed in detail. The proposed multiscale model facilitates the accurate prediction of the constitutive behaviour of the graphene in a thermal environment with remarkable improvements for the prediction of bending modulus due to the inclusion of the dihedral energy term at finite temperature.

Anion-Bridged Dual Hydrogen Bond Enabled Concerted Addition of Phenol to Glycal.

A noncovalent organocatalytic concerted addition of phenol to glycal is developed for the stereoselective and regioselective construction of biologically important phenolic 2-deoxyglycosides, featuring wide substrate tolerance. The method relies on an anion-bridged dual hydrogen bond interaction which is experimentally proved by Nuclear Magnetic Resonance (NMR), Ultraviolet and visible (UV-vis), and fluorescence analysis. Experimental evidence including kinetic analysis, Kinetic Isotope Effect (KIE) studies, linear free energy relationship, Hammett plot, and density functional theory (DFT) calculations is provided for a concerted mechanism where a high-energy oxocarbenium ion is not formed. In addition, the potential utility of this method is further demonstrated by the synthesis of biologically active glycosylated flavones. The benchmarking studies demonstrate significant advances in this newly developed method compared to previous approaches.

Localisation of stress-affected chemical reactions in solids described by coupled mechanics-diffusion-reaction models

Chemical reactions in solids can induce chemical expansion of the solid that causes the emergence of the mechanical stresses, which, in turn, can affect the rate of the reaction. A typical example of this is the reaction of Si lithiation, where the stresses can inhibit the reaction up to the reaction locking. The reactions in solids can take place within some volume (bulk reactions) or localise at a chemical reaction front (localised reactions). These cases are typically described by different thermo-chemo-mechanical theories that contain the source/sink terms either in the bulk or at the propagating infinitely-thin interface, respectively. However, there are reactions that can reveal both regimes; hence, there is a need to link the theories describing the bulk and the localised (sharp-interface) reactions. The present paper bridges this gap and shows that when a certain structure of the Helmholtz free energy density is adopted (based on the ideas from the phase-field methods), it is possible to obtain (in the limit) the same driving force for the chemical reaction (hence, the same reaction kinetics) as derived within the theory of the sharp-interface chemical reactions based on the chemical affinity tensor.

On the theory of mechanically induced chemiluminescence in multiple network elastomers

Mechanophores, force-sensitive molecules, can be covalently linked into polymer networks to reveal the damage properties of polymeric materials under deformation. Recent works have introduced bis-adamantyl dioxetane (Ad) into multiple network elastomers. These mechanoluminescent cross-linking molecules can emit luminescence as they break. The experimental results can directly visualize the damage behaviors of multiple network elastomers and reveal the damage mechanism of these materials during deformation. In this work, we develop a constitutive model to describe the complex mechanical behaviors of multiple network elastomers based on observations from mechano-chemiluminescent experiments. The free energy density consists of the contributions from the filler network, the matrix network, and the entangled network. A new type of single-chain model is developed for the filler network, while the classic eight-chain mapping is adopted for the micro–macro transition. The chains in the filler network are progressively damaged under deformation to explain the stress-softening behaviors observed in the small and median strain regions of multiple network elastomers. The entangled network is rooted in individual fragments of the damaged filler network, which can be further damaged in the large deformation regime. The model can comprehensively capture the complex mechanical responses of multiple network elastomers with different values of the prestretching ratio of the filler network, including linear response, necking instability, and re-hardening phenomena in quadruple networks. By assuming that the luminescence intensity of the materials is proportional to the total percentage of damage from both filler and entangled networks, we quantify the relationship between damage and the luminescence intensity from mechanophores, which is further validated in quantifying luminescence intensity against different sets of experimental data on the uniaxial tension of multiple network elastomers. The model is further implemented for finite element analysis, demonstrating the ability to visualize and characterize the damage behaviors around a crack.

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ПОВЕДЕНИЯ СЛОИСТОЙ МУЛЬТИФЕРРОИДНОЙ СТРУКТУРЫ В НЕОДНОРОДНЫХ ЭЛЕКТРИЧЕСКИХ ПОЛЯХ В ОБЛАСТИ СВЕРХВЫСОКИХ ЧАСТОТ

In this work, based on the dispersion characteristics of spin waves and the theory of the magnetoelectric effect, a mathematical model of a multiferroic structure behavior in a non-uniform electric field is constructed. The nature of the exchange interaction is taken into account, and the Rado-Wirtman exchange boundary conditions are also considered based on their representation in terms of free energy density. Based on the constructed mathematical model, the contribution to the dispersion pattern of spin waves of an external electric field of different polarities is estimated. In conclusion, a number of cases of application of the constructed mathematical model in the design of electronically controlled devices for the directional propagation of spin waves, as well as basic devices for neuromorphic computing, are given.

Defect interactions in a two-dimensional sheared lamellar mesophase.

The interaction between two edge dislocations in a sheared lyotropic lamellar liquid-crystalline medium is examined. The model is a mesoscale hydrodynamic model based on a free-energy functional that is minimised for a sinusoidal concentration modulation coupled with concentration and momentum equations. The defect dynamics are analysed as a function of the system size and the Ericksen number, the ratio of the shear stress and the characteristic free-energy density for deformation. Three different regimes are observed as the Ericksen number is increased when the edge dislocations are sheared towards each other, such that there is compression of layers between the defects: (a) defect motion that reduces the cross-stream separation till there is a steady spacing with plug flow between the defects, (b) defect attraction and cancellation resulting in a well-aligned state, and (c) defect creation due to a compressional instability between the defects resulting in an increase in the disorder. When the edge dislocations are sheared away from each other, such that there is extension of the layers between the defects, two distinct regimes are observed as the Ericksen number is increased: (a) bending of layers and a plug flow between the defects at their initial separation, and (b) buckling of the layers leading to creation of more defects and a dynamical steady state between defect creation and cancellation. These regimes are found to be robust for different values of the system size, from 32 to 128 layers, and for different values of the dimensionless groups that characterise the ratio of mass/momentum convection and diffusion.

Exploration of the Character Representation of DNA Chiral Conformations and Deformations via a Curved Surface Discrete Frenet Frame.

While undergoing structural deformation, DNA experiences changes in the interactions between its internal base pairs, presenting challenges to conventional elastic methods. To address this, we propose the Discrete Critical State (DCS) model in this paper. This model combines surface discrete frame theory with gauge theory and Landau phase transition theory to investigate DNA's structural deformation, phase transitions, and chirality. Notably, the DCS model considers both the internal interactions within DNA and formulates an overall equation using unified physical and geometric parameters. By employing the discrete frame, we derive the evolution of physical quantities along the helical axis of DNA, including geodesic curvature, geodesic torsion, and others. Our findings indicate that B-DNA has a significantly lower free energy density compared to Z-DNA, which is in agreement with experimental observations. This research reveals that the direction of base pairs is primarily governed by the geodesic curve within the helical plane, aligning closely with the orientation of the base pairs. Moreover, the geodesic curve has a profound influence on the arrangement of base pairs at the microscopic level and effectively regulates the configuration and geometry of DNA through macroscopic-level free energy considerations.

Holographic thermal observables and M2-branes

We use holography in conjunction with recent results from supersymmetric localization to compute certain thermal observables for 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 holographic SCFTs arising on the worldvolume of N M2-branes. We obtain results for the thermal free energy density on S1 × ℝ2, the Casimir energy on T2 × ℝ, and the three leading coefficients in the large temperature limit of the free energy on S1 × S2 valid to subleading order in the large N limit. As a byproduct of our holographic analysis we also present a conjecture for the structure of the large temperature expansion of the thermal free energy of general 3d CFTs on S1 × S2.

Higher orders for cosmological phase transitions: a global study in a Yukawa model

We perform a state-of-the-art global study of the cosmological thermal histories of a simple Yukawa model, and find higher perturbative orders to be important for determining both the presence and strength of strong first-order phase transitions. Using high-temperature effective field theory, we calculate the free energy density of the model up to O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(y5T4), where y is the Yukawa coupling and T is the temperature. The locations of phase transitions are found using the results of lattice Monte-Carlo simulations, and the strength of first-order transitions are evaluated within perturbation theory, to 3-loop order. This is the first global study of any model at this order. Compared to a vanilla 1-loop analysis, accurate to O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(y2T4), reaching such accuracy enables on average a five-fold reduction in the relative uncertainty in the predicted critical temperature Tc, and an additional ∼ 50% strong first-order transitions with latent heat L/Tc4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ L/{T}_c^4 $$\\end{document}> 0.1 to be identified in our scan.

Mechanism for conversion between twist and writhe of DNA based on chiral gauge potential

It is well known that the linking number is the sum of the writhing number and twisting number based on white’s formula. In this work, we reveal the underlying physics of the transformation between the twist and the writhe of deoxyribonucleic acid (DNA). In the framework of the Landau model, we describe the local interaction between the base pairs by using the chiral gauge potential. With the chiral gauge potential, we develop a geometrical model where the free energy density is given by the geometrical parameters, including the normal curvature, the geodesic curvature, and the geodesic torsion. It is shown that the chiral gauge potential plays a crucial role in the conversion between the twist and the writhe of DNA. By using the invariance of the self-linking number, we find that when the total twisting number is changed by −Δ Q, the writhing number is changed by Δ Q. Our findings not only deepen the understanding of the topological nature of DNA, but also provide a physical model for the research of the conformation of DNA.

Vacuum free energy, quark condensate shifts and magnetization in three-flavor chiral perturbation theory to [formula omitted] in a uniform magnetic field

We study three-flavor QCD in a uniform magnetic field using chiral perturbation theory (χPT). We construct the vacuum free energy density, quark condensate shifts induced by the magnetic field and the renormalized magnetization to O(p6) in the chiral expansion. We find that the calculation of the free energy is greatly simplified by cancellations among two-loop diagrams involving charged mesons. In comparing our results with recent 2+1-flavor lattice QCD data, we find that the light quark condensate shift at O(p6) is in better agreement than the shift at O(p4). We also find that the renormalized magnetization, due to its small-ness, possesses large uncertainties at O(p6) due to the uncertainties in the low-energy constants.

Modeling the thermo-responsive behaviors of polydomain and monodomain nematic liquid crystal elastomers

Liquid crystal elastomers (LCEs) are polymeric materials that combine liquid crystal orientation and rubber elasticity. The mechanical responses of LCEs strongly depend on the nematic order and temperature. Developing a thermo-order-mechanical coupling model is important for the engineering applications of LCEs. In this work, we first synthesize both polydomain and monodomain LCEs. The shape change with temperature under a certain stress level is further characterized. In the theoretical part, the Landau–de Gennes model and neo-classical model are combined to construct the free energy density of LCEs. A linear term accounting for the effect of interior stress on nematic–isotropic transition is incorporated into the Landau–de Gennes free energy density, while a macro-order parameter is defined to describe the polydomain–monodomain transition for polydomain LCEs induced by the external force. Comparison between the model predictions and experimental results shows acceptably consistency for both polydomain and monodomain LCEs. Therefore, this work provides an efficient approach to predict the shape change of LCEs in various scenarios.

A Cahn–Hilliard phase field model coupled to an Allen–Cahn model of viscoelasticity at large strains

We propose a new Cahn–Hilliard phase field model coupled to incompressible viscoelasticity at large strains, obtained from a diffuse interface mixture model and formulated in the Eulerian configuration. A new kind of diffusive regularization, of Allen–Cahn type, is introduced in the transport equation for the deformation gradient, together with a regularizing interface term depending on the gradient of the deformation gradient in the free energy density of the system. The designed regularization preserves the dissipative structure of the equations. We obtain the global existence of a weak solution in three space dimensions and for generic nonlinear elastic energy densities with polynomial growth, comprising the relevant cases of polyconvex Mooney–Rivlin and Ogden elastic energies. Also, our analysis considers elastic free energy densities which depend on the phase field variable and which can possibly degenerate for some values of the phase field variable. We also propose two kinds of unconditionally energy stable finite element approximations of the model, based on convex splitting ideas and on the use of a scalar auxiliary variable respectively, proving the existence and stability of discrete solutions. We finally report numerical results for different test cases with shape memory alloy type free energy, showing the interplay between phase separation and finite elasticity in determining the topology of stationary states with pure phases characterized by different elastic properties.

Nonlinear anisotropic viscoelasticity

In this paper, we revisit the mathematical foundations of nonlinear viscoelasticity. We study the underlying geometry of viscoelastic deformations, and in particular, the intermediate configuration. Starting from the direct multiplicative decomposition of the deformation gradient ▪ , into elastic and viscous distortions ▪ and ▪ , respectively, we point out that ▪ can be either a material tensor ( ▪ is a two-point tensor) or a two-point tensor ( ▪ is a spatial tensor). We show, based on physical grounds, that the second choice is unacceptable. It is assumed that the free energy density is the sum of an equilibrium and a non-equilibrium part. The symmetry transformations and their action on the total, elastic, and viscous deformation gradients are carefully discussed. Following a two-potential approach, the governing equations of nonlinear viscoelasticity are derived using the Lagrange–d’Alembert principle. We discuss the constitutive and kinetic equations for compressible and incompressible isotropic, transversely isotropic, orthotropic, and monoclinic viscoelastic solids. We finally semi-analytically study creep and relaxation in three examples of universal deformations.

Isogeometric smooth crack-band model (isCBM) using spress–sprain relations adapted to microplane theory

Recent optimum fitting of the data from eleven different types of distinctive fracture tests of quasibrittle materials, including the gap tests and the size effect tests, revealed that the crack band model (CBM) with the M7 damage constitutive law outperforms all the other existing computational fracture models for concrete. This is attributed to physically correct boundary conditions, especially those at the crack faces,to the crack front of finite width which allows using a triaxial tensorial damage constitutive law, and to choosing for this law a realistic material behavior description, the M7. Nevertheless the CBM has limitations—the width of the crack band cannot be varied, the statistically smooth damage distribution across the band cannot be resolved, and the propagation direction is biased by the mesh orientation when a regular mesh is used. These limitations have been overcome by the smooth crack band model (sCBM), which uses unconventional homogenization of damage mechanics to yield an additional Helmholtz free energy density, called the “sprain energy”. This energy is a function of the curvature tensor (or a hessian) of the vectorial displacement field, called the “sprain curvature tensor” (or sprain tensor for brevity), which is a sum of the strain gradient tensor and, importantly, the rotation gradient tensor. But, in a previous study, difficulties arose in computational implementation—the use of constant-strain elements required the curvature to be resisted by sprain forces that represented the derivatives of sprain energy with respect to nodal displacements and had to be applied on the nodes of adjacent elements. The need for adjacent nodes is here avoided by a variational derivation of the Isogeometric Smooth Crack-Band Model (isCBM). The microplane damage constitutive model M7 for concrete is adopted, which is made possible by establishing the relations of the sprain stress (or spress, for brevity) and sprain on microplanes of arbitrary orientations. Using Isogeometric Analysis (IGA) based on non-uniform rational B-Splines (NURBS), the excessive displacement curvature and its resistance (spress) can be described by the shape functions with C1-continuity. Computational results show that the orientation bias of regular meshes is eliminated while keeping the constitutive law intact, which is important for compatibility with the existing material laws in general. In addition, comparisons with experimental data reveal that isCBM with M7 can reproduce the correspondence between the size of the fracture process zone and the fracture energy, as indicated by gap tests. Finally, the isCBM does not compromise the strain-path-dependence of material response.

Thermodynamic model to study the structural and electro-optical properties of chiral antiferroelectric smectic C phase exhibited by mesogenic homologous series 3FnHBM6(s) and W-316 mixture

ABSTRACT Using a thermodynamic model developed by us to study the structural and electro-optical properties of Sm C A ∗ phase, we investigate the temperature dependence of helical pitch, tilt angle and spontaneous polarization of a homologous series of fluorinated biphenyl benzoate esters (3FnHBM6, n = 2, 5, 7) and one of the series W-316-3 obtained from the mixture of two chiral liquid crystalline compounds 6F2OBi and 3F6HBM6. The free-energy density of the system exhibiting Sm C A ∗ phase is written as an expansion series in terms of all the relevant degrees of freedom: Q i j , ψ, P, q and the resulting coupling terms between these order parameters. The values of Landau coefficients are determined from the experimental data of antiferroelectric mesogens 3FnHBM6 (n = 2, 5, 7) and W-316-3 in the Sm C A ∗ phase. Using these coefficients, the electro-optical and structural properties have been studied. We have found good agreements between the theoretical and experimental results.

Thermodynamics and Ferroelectric Properties of Pb1-xSrxTiO3 Solid Solutions

Pb1-xSrxTiO3 solid solutions have been utilized to compositionally tune the phase transitions and ferroelectric properties and thus are attractive for potential applications in tunable microwave devices and pyroelectric energy harvesters. However, a reliable thermodynamic description of the Pb1-xSrxTiO3 system is currently lacking. Here, we establish a thermodynamic free energy density function for the Pb1-xSrxTiO3 solid solution system taking into account both the polarization and octahedral tilt, and validated it by reproducing the experimentally measured polarization, lattice parameters, and dielectric constant values. Based on the thermodynamic model, we successfully explain the experimentally observed dielectric anomalies by analyzing the interactions between ferroelectric and octahedral tilt order parameters and constructed a comprehensive temperature-composition Pb1-xSrxTiO3 phase diagram which can be used to guide future experimental synthesis and thin film growth. The thermodynamic model developed in this work can be employed not only to predict the ferroelectric and structural states and their properties for compositions that have not yet been synthesized but also used as input to model the formation of mesoscale domain structures using the phase-field method.

Orbital-free QM/MM simulation combined with a theory of solutions.

In a recent study, we developed a kinetic-energy density functional that can be utilized in orbital-free quantum mechanical/molecular mechanical (OF-QM/MM) simulations. The functional includes the nonlocal term constructed from the response function of the reference system of the QM solute. The present work provides a method to combine the OF-QM/MM with a theory of solutions based on the energy representation to compute the solvation free energy of the QM solute in solution. The method is applied to the calculation of the solvation free energy Δμ of a QM water solute in an MM water solvent. It is demonstrated that Δμ is computed as -7.7 kcal/mol, in good agreement with an experimental value of -6.3 kcal/mol. We also develop a theory to map the free energy δμ due to electron density polarization onto the coordinate space of electrons. The free energy density obtained by the free-energy mapping for the QM water clarifies that each hydrogen atom makes a positive contribution (+34.7 kcal/mol) to δμ, and the oxygen atom gives the negative free energy (-71.7 kcal/mol). It is shown that the small polarization free energy -2.4 kcal/mol is generated as a result of the cancellation of these counteracting energies. These analyses are made possible by the OF-QM/MM approach combined with a statistical theory of solutions.

Wave propagation in generalized thermo-microstretch elastic solid containing voids

A linear theory of generalized thermo-microstretch elastic solid containing voids is formulated. Lord and Shulman [2] theory of thermoelasticity is employed to incorporate thermal effects. Free energy density function is constructed to develop the constitutive relations and field equations for an isotropic homogeneous generalized thermo-microstretch elastic solid containing voids. The possibility of propagation of plane waves is investigated in the medium of infinite extent. It is found that there may exist four sets of coupled longitudinal waves, two sets of coupled transverse waves and an independent longitudinal microrotational wave traveling with distinct speeds. Each set of coupled longitudinal waves is found to be attenuating and dispersive in nature, while an independent longitudinal microrotational wave and the remaining two sets of coupled transverse waves are found to be dispersive but non-attenuating in nature. All the possible waves are influenced by the polar property of the medium; however, all the coupled longitudinal waves are influenced by the stretch, voids and thermal properties of the medium. It is also found that all coupled longitudinal waves exist for all non-negative frequencies, while the independent longitudinal microrotational wave and one of the sets of coupled transverse waves exist only after certain cutoff frequency.

A coupled model of transport-reaction-mechanics with trapping, Part II: Large strain analysis

A coupled finite strain chemo-transport-mechanical formulation with trapping is here proposed to extend a previous work set in the realm of small strain theory in continuum mechanics. The theory is rooted in non-equilibrium rational thermodynamics. The kinematics is based on a multiplicative decomposition of the deformation gradient to account for swelling and shrinking, thermal, elastic and inelastic contributions. Mass balance laws and balance of linear and angular momentum, as well as the laws of thermodynamics for a convecting body, are directly formulated in their material description, after specifications of some standard transformation rules between current and reference configuration. Thermodynamic restrictions are identified based on the functional dependence of the referential Helmholtz free energy density, which is chosen as the thermodynamic potential, and further subjected to a constitutive additive decomposition. Constitutive prescriptions for the chemical potentials, referential heat and mass fluxes, chemical kinetics and the generalized heat equation lead to the establishment of the governing equations. The theoretical framework is complemented by numerical simulations, highlighting the potential of the proposed formulation in multi-physics applications.