Helmholtz Free Energy Research Articles

Structural and thermodynamic properties of quasi-2D Mo(1–x)W x (S, Se, Te)2 monolayer alloys: a statistical first principle study

In this work, we report an ab initio study of the structural and thermodynamic properties of two-dimensional transition-metal dichalcogenides (2D-TMDC) alloys, Mo(1–x)W x (S, Se, Te)2, using the cluster expansion framework to compute the Helmholtz free energy of alloys as a function of alloy composition and temperature, in the framework of the generalized quasi-chemical approximation. We consider alloying only on the metal sublayer. Our results indicate a weak dependence of the structural properties (lattice constants, nearest-neighbor bond lengths, and layer width) on the alloy composition (i.e. concentrations of W and Mo atoms), in line with the very similar values of the atomic radii of Mo and W atoms. A stronger dependence on the chalcogen is obtained, a trend that reflects the larger variations in atomic radii among the three chalcogen species. As a function of composition, the structural parameters we examined show similar trends, with negligible bowing (i.e. deviations from a Vegard’s law interpolation between end compounds), for the three alloys. Moreover, already at 300 K the behavior of these structural features as a function of composition is very similar to that of the standard-regular-solution (SRS) high-temperature limit. In contrast, the electronic band gaps of the the three alloys as a function of composition show small but significant bowing, as high as −1% to −2% near the x = 0.5 alloy composition. Similarly to the structural features, the band gaps attain the high-temperature SRS limit already at 300 K. Regarding thermodynamic properties, we obtain negative values of the internal energy of mixing for the three alloys over the full range of compositions. Therefore, the theoretical alloying phase diagram for the three alloys is featureless, with stability of a fully-mixed alloy at all temperatures and compositions, with no miscibility gap (hence no bimodal nor spinodal decomposition lines). The thermodynamic potentials (mixing internal energy, mixing entropy, and mixing free energy) reach the high-temperature limit at ∼1000 K, the temperature range of synthesis of 2D-TMDC alloys. These trends of structural and electronic properties of the 2D-TMDC alloys are due to the very similar atomic radii and the nearly identical coordination chemistry of Mo and W. Our results are in agreement with experimental work on the alloying of Mo and W atoms, for samples of Mo(1–x)W x S2 monolayer alloys, that found that the random mixed alloy is the thermodynamically stable state for this alloy, with no segregation or phase separation.

Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations

We develop a fully data-driven model of anisotropic finite viscoelasticity using neural ordinary differential equations as building blocks. We replace the Helmholtz free energy function and the dissipation potential with data-driven functions that a priori satisfy physics-based constraints such as objectivity and the second law of thermodynamics. Our approach enables modeling viscoelastic behavior of materials under arbitrary loads in three-dimensions even with large deformations and large deviations from the thermodynamic equilibrium. The data-driven nature of the governing potentials endows the model with much needed flexibility in modeling the viscoelastic behavior of a wide class of materials. We train the model using stress–strain data from biological and synthetic materials including human brain tissue, blood clots, natural rubber and human myocardium and show that the data-driven method outperforms traditional, closed-form models of viscoelasticity.

Dehydration requirements for CO2 and impure CO2 for ship transport

CCS is expected to become an important contribution to the mitigation of carbon dioxide emissions. Transportation of CO2 from the capture site to the permanent storage reservoir plays a key role, and transportation by ship is now viewed as a relevant alternative for certain combinations of captured CO2 volume and distance.This paper presents experimental equipment, methods and new results for the water content of carbon dioxide and impure carbon dioxide in equilibrium with hydrates at temperatures between (-55 and -10)°C and pressures between (3 and 120) bar, hence in both vapour and liquid phases. The experimental data are compared with predictions from two thermodynamic models: the Cubic-Plus-Association equation of state and an approach using Multi-Fluid Helmholtz Energy Approximation (MFHEA) equations of state (EoS). In all cases, the hydrate-forming conditions are modelled by the solid solution theory of van der Waals and Platteeuw.

Adsorption in a centrifugal field: A thermodynamic analysis

External fields such as gravitational, centrifugal or electrical fields lead to an uneven spatial distribution of the chemical compounds of a system. In this work, we conceive a scenario where a packing bed is submitted to a centrifugal field, so that centrifugation and adsorption occur simultaneously. Our rotating packed bed is made up by two main regions: one, representing the bulk phase, the other, representing the total pore volume available for adsorption. At constant temperature, total volume, and total amount, the distribution of species of selected binary and ternary mixtures of hydrocarbons, carbon dioxide, and hydrogen sulfide is predicted by minimizing the total Helmholtz energy of the system. Confinement-modified equations of state are employed as thermodynamic model for the adsorbed phase, while the Peng-Robinson equation of state is used in bulk phase calculations. Our purely thermodynamic analysis reveal that coupling both phenomena could be useful to enhance adsorption, or increase the separation obtained when only centrifugation is considered.

Structural and thermal properties of H6Ta2O17

The structural, electronic, mechanical, and thermal properties of H6Ta2O17 are studied using density functional theory. The Helmholtz free energy is computed for various symmetry constraints on the H6Ta2O17 structure during lattice structure optimization and atomic configuration relax- ation. The Helmholtz free energies calculated from Γ-point phonon frequencies demonstrate a pref- erence for slightly lowered symmetry, preferring triclinic symmetry rather than the experimentally determined orthorhombic symmetry. Elastic constants are obtained for the symmetry-enforced (i.e., orthorhombic) structure and are used to calculate the structure’s mechanical and thermal properties including its bulk, shear, and Young’s moduli, Poisson ratio, Pugh ratio, longitudinal and shear sound velocities, and Debye temperature. Using the Born criteria, the symmetry-enforced structure is shown to be mechanically stable. The electronic band structure for H6Ta2O17 is pre- sented and found to exhibit a large indirect band gap.

A revisit of the vapor–liquid interface of the Lennard-Jones fluid

In this work a mean-field crossover generated by the Maxwell construction is defined as the dividing interface for the vapor–liquid interface area. An accurate and physically preferable density-profile equation is used to correlate the local density and to predict various interfacial properties. By using the density gradient theory and a mean-field equation of state for the Lennard-Jones fluid, we are able to extensively explore the interface system in terms of the Gibbs free energy, the Helmholtz free energy and heat capacity. The peak of the intrinsic free energy at the interface reveals the most unstable state and suggests that the interface area is thus divided into vapor- enriched and liquid- enriched sub-regions. The local maxima of the intrinsic isobaric heat capacity echo such a dividing, which is in accordance with the Widom line in the supercritical region. Consequently the mean-field dividing interface is the natural extension of the Widom line into the coexistence region. Hence the entire phase space is coherently divided into liquid-like and gas-like regions in all three (temperature–pressure-volume) planes. Finally, the modeling of adsorption on the interface is briefly discussed.

A thermodynamic framework for non-isothermal phenomenological models of isotropic Mullins effect

The Mullins effect is a common name for a family of intriguing inelastic responses of various solid materials, in particular filled rubbers. Given the importance of the Mullins effect, there have been many attempts to develop mathematical models describing the effect. However, most of the available models focus exclusively on the mechanical response and are restricted to the idealized isothermal setting. We lift the restriction to isothermal processes, and we propose a full thermodynamic framework for a class of phenomenological models of the Mullins effect. In particular, we identify energy storage mechanisms (Helmholtz free energy) and entropy production mechanisms that on the level of stress–strain relation lead to the idealized Mullins effect or to the Mullins effect with permanent strain. The models constructed within the proposed framework can be used in the modelling of fully coupled thermo-mechanical processes, and the models are guaranteed to be consistent with the laws of thermodynamics.

Application of the reference interaction site model self-consistent field method based on the Dirac-Hartree–Fock wave function to a chemical reaction

The reference interaction site model self-consistent field (RISM-SCF) method is a combined method of the electronic structure theory of molecules and the integral equation theory of molecular liquids. The RISM-SCF method based on the Dirac-Hartree-Fock wave function, recently proposed, is applied to a chemical reaction, specifically, a Menshutkin reaction in aqueous solution. The Helmholtz energy profile along the reaction coordinate is calculated and the characteristics of the reaction are discussed based on energy component analysis.

Dependence of predicted bulk properties of hexagonal hydroxyapatite on exchange–correlation functional

Hydroxyapatite (HA) is the main component of human bones and teeth. HA has also been widely applied in various technological fields due to its unique properties. Reliable computational simulations for this critical material often require input parameters obtained from the first-principles density functional theory (DFT) calculations with appropriate exchange–correlation functionals. Previous DFT calculations are insufficient to verify the reliabilities of various functionals, particularly as they fail to assess predictions for multiple properties of the material. In this paper, we first select 18 different functionals to calculate geometric, elastic, electronic, and thermodynamic properties of hexagonal HA bulk crystal. We find that the results from optB86b-vdW and optB88-vdW functionals with dispersion corrections have the overall best agreement with available experimental data. Then, we choose optB88-vdW functional, as well as PBE functional without dispersion corrections as a comparison, to perform extensive first-principles DFT phonon calculations under the quasiharmonic approximation. Various thermodynamic properties (including phonon contributions to internal energy, entropy, and Helmholtz free energy) and thermal parameters (including thermal expansions of volume, thermal expansion coefficients, heat capacities, isothermal bulk moduli, etc.) versus temperature are consequently obtained. By comparing these quantities, we find that the results from optB88-vdW functional have significantly better agreement with available experimental data than those from PBE functional although the latter has been widely used in previous DFT calculations for HA-based material systems.

Structure and physical properties of predicted MAX phase V2SiC under pressure: A first principle calculation

MAX-phase materials are important for technical applications in different fields, where pressure can change the microstructure and thus the properties of the material. In this paper, the structural, electronic, mechanical, thermodynamic, and optical properties of the new ternary layered ceramic MAX phase V2SiC are investigated under different pressures based on DFT simulations. A greater stiffness along the c-axis than along the a-axis direction and a negative enthalpy of formation confirm the thermodynamic stability of the pressurized phase. Although the band structure is altered under pressure, the metallic properties are retained. The coexistence of its covalent, metallic, and ionic bonds is observed in the charge density diagram. The elastic constants satisfy the Born stability criterion, ensuring mechanical stability under pressurized and unpressurized conditions. The increase in υ from 0.239 to 0.279 indicates that the compound changes from brittle to ductile, with the maximum ductility observed at 20 GPa. Phonon calculations show its kinetic stability, and Helmholtz free energy, enthalpy, entropic specific heat capacity, and Debye temperature are also calculated. Optical properties show that V2SiC is suitable for optoelectronic devices in the visible and UV regions and also as a coating material to reduce the heat from sunlight at high pressures up to 10 GPa. This is the first comprehensive theoretical study of the properties of V2SiC and may provide a reference for future research, the results of which still require subsequent experimental verification.

Thermodynamic studies of a rotating polytropic black hole: Outer and interior regions

In this letter, considering the metric of a rotating polytropic black hole in the Boyer-Lindquist coordinates, at first, we derive the thermodynamic parameters such as entropy S, Helmholtz free energy F, internal energy U and Gibbs free energy G and study its dependence on the outer horizon by depicting suitable graphs. Then after reconstruction of the metric of the same in the Eddington-Finkelstein coordinates, we establish the interior volume of the black hole. We further analyze the variations of the interior volume with the small change of the advanced time with respect to the radius. Here we show the existence of a certain value of the radius for which this variation becomes maximum. Moreover, we show the dependence of this maximum value of the radius on the mass of the black hole. We derive the differential form of the interior volume for this limit of the radius and hence the maximal interior volume of the said black hole. Finally, we analyze the same thermodynamic parameters inside the black hole and present a comparative study between the parameters in the outer and interior regions of the black hole.

On the curvature dependence of gradient damage models: Control and opportunities

Curvature dependence of gradient damage models is an often undesired side effect. It originates from a change of the orientation of the non-local gradient term. The resulting curvature dependence then implicitly affects the loading behaviour, the predicted fracture energy and also the damage distribution. Two extended approaches are presented here for curvature control and applied to a micromorphic gradient damage model and to a phase-field model for fracture. Both allow for a better control of the crack path and consider the Hessian of the non-local damage field by means of an additional micromorphic field, since curvature is connected to second derivatives. The first approach is based on an additional curvature-dependent potential within the Helmholtz energy and automatically fulfils the second law of thermodynamics. The second approach modifies the micro forces of the balance law that is responsible for the curvature dependence directly. Unlike the first approach, it allows to eliminate the curvature dependence completely. However, thermodynamic consistency has to be checked explicitly.

Ab initio study of the structural, electronic, magnetic, mechanical, optical, and dynamical properties of the rare-earth dihydrides MH2 (M= Yb, Sc, Eu, Y, Lu and Gd)

Most of the physical properties of rare-earth dihydrides MH2 (M = Yb Sc, Eu, Y, Lu and Gd) have not yet been discovered. The aim of this work is to investigate to the structural, electronic, magnetic, mechanical and optical properties employing density functional theory (DFT) within the VASP code. The computed electronic structures, elastic constants, and lattice parameters of MH2 hydrides are in good agreement with the current experiments and other theoretical studies. The mechanical properties of MH2 (M = Sc, Y, Yb, Eu, Lu and Gd) such as: ductility, elastic moduli, and elastic anisotropy, were evaluated and conferred. The brittle/ductile properties are determined by the bulk/shear ratio B/G and the Cauchy pressure C12–C44. The current work shows that the LuH2, GdH2 and ScH2 hydrides crystallizing in the cubic phase of hydrogen exhibit good mechanical behavior that may be useful for applications in hydrogen storage in the future. In addition, the electronic properties were also investigated in order to present a thorough knowledge of their patterns and verify the metallic behavior trend of these hydrides. The Debye temperature was therefore computed using the single crystal elastic constants and the average sound velocity in accordance with the projected bulk properties of crystalline materials. The optical properties were computed by fitting the dispersion relation of the imagined component. Optical spectra showed a significant absorption coefficient in the in the ultraviolet region of these compounds. The present findings can improve our understanding of rare-earth-based materials towards potential use in the experimental research. Finally, The MH2 (M = Yb Sc, Eu, Y, Lu and Gd) compounds were comprehensively investigated, with a focus on their dynamic characteristics and identification of infrared and Raman active modes. The elastic constant and phonon spectrum calculations revealed the stability of the cubic structure of MH2 under mechanical and dynamic conditions. Furthermore, the study obtained thermal function data, including phonon contributions to internal energy, Helmholtz free energy, entropy, and specific heat at high temperatures.

Crack tip enhanced phase-field model for crack evolution in crystalline Ti6Al from concurrent crystal plasticity FE-molecular dynamics simulations

Dislocations nucleating from a crack-tip contribute to the plasticity evolution in the vicinity of crack, as well as to the crack propagation process. This paper systematically develops a method for enhancing the defect energy, accounting for crack-tip nucleated dislocations in the Helmholtz free energy density functional, associated with the phase-field formulation of crack evolution, in a coupled crystal plasticity finite element phase field (CPFE-PF) model. A concurrent crystal plasticity FE- hyper-dynamics accelerated molecular dynamics (CPFE-MD) model is created. The model that identifies, transfers, and propagates dislocations from the atomistic to continuum domain, is used to generate a dataset of crack tip dislocation density evolution along with different state variables. The paper focuses on crack tip plasticity mechanisms for the crystalline alloy Ti6Al. Using a Bayesian inference approach the critical state variables that affect the evolution of crack tip nucleated dislocation density are inferred. A functional form of the evolution of dislocation density in terms of the state variables is derived by employing a genetic programming based symbolic regression (GPSR) approach. The contribution of nucleated dislocation densities to effective plastic strain evolution at the crack tip is validated using the CPFE-MD model simulations. A comparison of the crack path and other state variables in the vicinity of the crack with and without contributions from the nucleated dislocations shows the importance of this augmentation on crack evolution.

Topological phantom AdS black holes in F(R) gravity

In this paper, we obtain exact phantom (A)dS black hole solutions in the context of F(R) gravity with topological spacetime in four dimensions. Then, we study the effects of different parameters on the event horizon. In the following, we calculate the conserved and thermodynamic quantities of the system and check the first law of thermodynamics for these kinds of black holes. Next, we evaluate the local stability of the topological phantom (A)dS black holes in F(R) gravity by studying the heat capacity and the geometrothemodynamic, where we show that the two approaches agrees. We extend our study and investigate global stability by employing the Gibbs potential and the Helmholtz free energy. In addition, the effects of different parameters on local and global stabilities will be highlighted.

Rayleigh waves in nonlocal generalized thermoelastic media

PurposeThe purpose of this article is to investigate the propagation characteristics (such as particle motion, attenuation and phase velocity) of a Rayleigh wave in a nonlocal generalized thermoelastic media.Design/methodology/approachThe bulk waves are represented with Helmholtz potentials. The stress-free insulated and isothermal plane surfaces are taken into account. Rayleigh wave dispersion relation has been established and is found to be complex. Due to the presence of radicals, the dispersion equation is continuously computed as a complicated irrational expression. The dispersion equation is then converted into a polynomial equation that can be solved numerically for precise complex roots. The extra zeros in this polynomial equation are eliminated to yield the dispersion equation’s roots. These routes are then filtered for inhomogeneous wave propagation that decays with depth. To perform numerical computations, MATLAB software is used.FindingsIn this medium, only one mode of Rayleigh wave exists at both isothermal and insulated boundaries. The thermal factors of nonlocal generalized thermoelastic materials significantly influence the particle motion, attenuation and phase velocity of the Rayleigh wave.Originality/valueNumerical examples are taken to examine how the thermal characteristics of materials affect the existing Rayleigh wave’s propagation characteristics. Graphical analysis is used to evaluate the behavior of particle motion (such as elliptical) both inside and at the isothermal (or insulated) flat surface of the medium under consideration.

P−V criticality of the nonlinear charged black hole solutions in massive gravity’s rainbow

In this paper, we explore the black hole solutions with the rainbow deformed metric in the presence of the exponential form of the nonlinear electrodynamics with asymptotic Reissner–Nordström properties. We calculate the exact solution of metric function and explore the geometrical properties in the background of massive gravity. From the obtained solution, the existence of the singularity is confirmed in proper limits. Using the solutions, we also investigate the thermodynamic properties of the solutions by checking the validity of the first law of thermodynamics. Continuing the thermodynamic study, we investigate the conditions under which the system is thermally stable from the heat capacity and the Gibbs free energy. We also discuss the possible phase transition and the criticality of the system. It was found that the quantum gravitational effects of gravity’s rainbow render the thermodynamic system stable in the vicinity of the singularity. Hence, we obtained a first-order phase transition which is interpreted as the large/small black hole phase transition. From the equation of state, it was found that after diverging at the singularity, the system evolves asymptotically into pressure-less dust as one moves away from the central singularity. We also calculated the quantum work using the change of the Helmholtz free energy.

A micromechanical constitutive model of high-temperature shape memory alloys

In this work, a new micromechanical constitutive model is constructed for high-temperature shape memory alloys. Different from the classical constitutive models, the microstructural patterns, heterogeneities of stress and strain fields at the microscopic scale are addressed. In the proposed model, three different inelastic deformation mechanisms, i.e., reversible martensitic transformation, irreversible transformation-induced plasticity, viscoplasticity, as well as their complex interactions, are considered. In addition to the well-known austenite and martensite phases, a new austenite-martensite interface phase is introduced to address the high local stress induced by the martensitic transformation. The modified Mori-Tanaka's homogenization scheme and Hill's interfacial operator are employed to determine the non-uniform stress fields of the three phases. To satisfy the thermodynamic compatibility of the proposed model, the driving forces and evolution equations of the three inelastic deformation mechanisms are derived based on the constructed Helmholtz free energy and the fundamental laws of irreversible thermodynamics. Inheritances of the plastic deformation induced by moving austenite-martensite (A-M) interface during repeated martensitic transformation and its reverse are incorporated. Temperature evolution caused by internal heat production and heat exchange with the ambient medium is also addressed. To validate the capability of the new constructed model, the predicted results of creep behavior and stress-assisted two-way shape memory effect of NiTiPd high-temperature shape memory alloys are compared with the experimental ones. The proposed model provides a theoretical tool for simultaneously analyzing the deformation behaviors at the macroscopic scale and underlying mechanisms at the microscopic scale.

Thermomechanical response of nonlinear viscoelastic materials

In this article, the thermomechanical properties of the classical and nonclassical viscoelastic materials with particular emphasis on the entropy are studied in the framework of continuum thermodynamics. Specifically, considering the Clausius integral as “Part I” of the second law of thermodynamics in the classical formulation of continuum thermodynamics, the entropy function for various classes of nonclassical thermo-viscoelastic materials is shown to exist. This is achieved by employing the first law of thermodynamics and a scalar potential known as the Helmholtz free energy function. In addition, another function known as the internal rate of production of entropy defined for the classical and nonclassical thermo-viscoelastic materials is adopted to obtain a balance law for the entropy. Finally, upon linearizing the theory, a coupled system of differential equations is obtained for the nonclassical thermo-viscoelastic materials, which under some special circumstances, permits propagation of damped harmonic waves in the viscoelastic medium.

Thermal properties of CaO and SrO and their B1-B2 phase boundaries: Validity of the quasi-harmonic approximation

In this paper, we will further confirm that the validity range of the quasi-harmonic approximation (QHA) can be significantly extended by explicitly calculating the Helmholtz free energy over an appropriate volume range. As in the case of MgO, we performed accurate QHA calculations of the thermal properties of the B1 phase of CaO and SrO over a wide temperature range, as well as their B1-B2 phase boundaries in the pressure–temperature space. This finding contradicts a recent conclusion that the QHA is inadequate for CaO, and it is therefore expected to fail for other alkaline-earth oxides containing more polarizable cations. Moreover, the validity range of the QHA is found to be consistent with the bond ionicity in MgO, CaO and SrO.