Helmholtz Free Energy Research Articles

A multiplicative finite strain deformation for diffusion-induced stress: An incremental approach

Understanding the evolution of lithium concentration and stresses in electrode materials during electrochemical cycling is of paramount importance in studying the fundamental mechanisms contributing to the structural degradation of lithiated electrode materials. In this work, we use the change rate of the volume of a host material in analyzing diffusion-induced volume expansion and obtain the volume expansion as an exponential function of the concentration of solute atoms and the coefficient of linear expansion in contrast to the widely used linear relation between the diffusion-induced volume expansion and the coefficient of linear expansion. Using the principle of multiplicative decomposition in the framework of elastic deformation and thermodynamic free energies, we derive the constitutive relation for diffusion-induced finite deformation of elastic materials. The symmetric Piola-Kirchhoff stress is an exponential function of the coefficient of linear expansion and the concentration of solute atoms. Incorporating the change of strain field with the change in the number of solute atoms in Helmholtz free energy, we derive the chemical potential of the solute atoms in the host material the same as the one given by Li et al. (Zeitschrift für PhysikalischeChemie, 49 (1966) 271–290). Numerical results for the lithiation of a spherical silicon electrode reveal that the spatial distributions of lithium concentration and stresses for small dimensionless time (dilute solution) exhibit similar trends to the corresponding ones calculated from the theory of linear elasticity and nonlinear characteristics for large dimensionless time (concentrated solution).

A Helmholtz Free Energy Equation of State of CO2-CH4-N2 Fluid Mixtures (ZMS EOS) and Its Applications

An equation of state (EOS) of CH4-N2 fluid mixtures in terms of Helmholtz free energy has been developed by using four mixing parameters, which can reproduce the pressure-volume-temperature-composition (PVTx) and vapor-liquid equilibrium (VLE) properties of CH4-N2 fluid mixtures. The average absolute deviation of all the PVTx data available up to 673.15 K and 1380 bar from this EOS is 0.38%. Combining this EOS of CH4-N2 fluid mixtures and the EOS of CH4-CO2 and CO2-N2 fluid mixtures in our previous work, an EOS of CO2-CH4-N2 fluid mixtures has been developed, which is named ZMS EOS. The ZMS EOS can calculate all thermodynamic properties of ternary CO2-CH4-N2 fluid mixtures and the average absolute deviation of the PVTx data from the ZMS EOS is 0.40% for the CO2-CH4-N2 system. The ZMS EOS can be applied to calculate excess enthalpies of CO2-CH4-N2 fluid mixtures, predict the solubility of CO2-CH4-N2 fluid mixtures in brine and water, and quantitatively estimate the impact of the impurities (CH4 and N2) on the CO2 storage capacity in the CO2 capture and storage (CCS) processes. The ZMS EOS can also be applied to calculate the isochores of CO2-CH4-N2 system in the studies of fluid inclusions. All Fortran computer codes and Origin drawing projects in this paper can be obtained freely from the corresponding author.

Three-Phase Fluid Flow Interaction at Pore Scale during Water- and Surfactant-Alternating Gas (WAG/SAG) Injection Using Carbon Dioxide for Geo-Sequestration and Enhanced Oil Recovery

Sequestration of CO2 in geologic formations such as depleted oil reservoirs has emerged as one of the lead solutions to tackle greenhouse gas emissions to reduce pollution and global warming. Supercritical CO2 (sc-CO2) injection in oil reservoirs has proven to be useful as an enhanced oil recovery (EOR) technique along with the benefits of CO2 sequestration. In this study, a tortuous microscopic pore scale model was used to study and investigate the phenomena of water-alternating gas (WAG) and surfactant-alternating gas (SAG) with sc-CO2. The study scrutinizes the dynamics of the pore-level phenomenon in the multiphase WAG and SAG flows at the pore level in detail. Transient computational fluid dynamics (CFD) analysis was used to study the fluid flow characteristics of oil, water, and sc-CO2 at different reservoir pressure and temperature conditions in oil-wet conditions. Governing equations were coupled with EOS (Helmholtz free energy equation) to capture the viscous and intrinsic properties of sc-CO2 due to variations in pressure and temperature conditions. It was found that higher oil recovery does not necessarily indicate higher sc-CO2 sequestration and that temperature harms the displacement mechanism due to unfavorable mobility ratios. Comparing WAG and SAG for the first injection cycle, SAG showed a more diffused interface between displaced and displacing fluid. The additional oil recovery produced in patches was a result of pressure oscillations near the blind pores. Moreover, high vorticity promotes greater intermixing between the displacing and displaced fluid by increasing the rate of interface length. In SAG cases, faster sc-CO2 breakthroughs were observed due to reduced shear stress along the fluid interfaces, which resulted in higher sequestration values in a given time frame. The CO2 sequestration volume in SAG cases was found to be approximately 40% more than in WAG experiments. The study confirms that lower values of oil–water interfacial tension aids in faster and more efficient sequestration of sc-CO2 along with additional oil gain from a given reservoir.

Evolution of gravitational radiation from black hole under the influence of Einstein–Gauss–Bonnet gravity

In our work, we compute the 4-dimensional Einstein–Gauss–Bonnet gravity black hole solution by applying the Newman–Janis approach and also examine the Hawking temperature. The generalized uncertainty principle (GUP) is applied to compute the Lagrangian field equation and by using the semi-classical phenomenon, we analyze the modified Hawking temperature. By taking into account the graphical analysis, we check the stable conditions of the black hole with the influence of rotation parameter, charged parameter and quantum gravity parameter. Furthermore, we compute the logarithmic entropy corrections in the background of corrected temperature and standard entropy for corresponding black hole. We study the behavior of different thermodynamics quantities like Helmholtz free energy, internal energy, Gibbs free energy and heat capacity under these fluctuations. In our analysis, we note that these corrections enhance the stability of this system, so, under the effects of these corrections, the considered geometry becomes more stable.

FeOs: An Open-Source Framework for Equations of State and Classical Density Functional Theory

In this work, we present an open-source software package, referred to as FeOs─Framework for Equations of State and Classical Density Functional Theory. FeOs is a collection of interfaces and data types that can be used (1) to implement thermodynamic equations of state and Helmholtz energy functionals for classical density functional theory, and (2) to compute thermodynamic properties of pure substances and mixtures, phase equilibria, and interfacial properties such as surface tensions and adsorption isotherms. The framework is written in the Rust programming language with a complete Python interface and is designed with a focus on usability and extensibility. It is openly available on GitHub (https://github.com/feos-org/feos). Equations of state can be implemented in Rust, yielding performant code, or as a Python class, which is useful for prototyping and with less emphasis on execution speed. In both cases, the user has to implement a single function: the Helmholtz energy. FeOs then uses generalized (hyper-) dual numbers to evaluate the Helmholtz energy as well as the required exact partial (higher-order) derivatives. Using this type of automatic differentiation delivers performance without the need for implementing any analytical derivatives. The performance is further enhanced by a caching mechanism that avoids duplicate model evaluations. Together with the core interfaces and functionalities for equations of state and classical density functional theory, we provide implementations for multiple models such as the PC-SAFT equation of state (with homo- and heterosegmented group contribution methods) and Helmholtz energy functionals (including segment-based functionals). To showcase a selection of FeOs’ features, an example study of the adsorption of biogas in porous media using the PC-SAFT functional is provided.

Classical density functional theory for interfacial properties of hydrogen, helium, deuterium, neon, and their mixtures.

We present a classical density functional theory (DFT) for fluid mixtures that is based on a third-order thermodynamic perturbation theory of Feynman-Hibbs-corrected Mie potentials. The DFT is developed to study the interfacial properties of hydrogen, helium, neon, deuterium, and their mixtures, i.e., fluids that are strongly influenced by quantum effects at low temperatures. White Bear fundamental measure theory is used for the hard-sphere contribution of the Helmholtz energy functional, and a weighted density approximation is used for the dispersion contribution. For mixtures, a contribution is included to account for non-additivity in the Lorentz-Berthelot combination rule. Predictions of the radial distribution function from DFT are in excellent agreement with results from molecular simulations, both for pure components and mixtures. Above the normal boiling point and 5% below the critical temperature, the DFT yields surface tensions of neon, hydrogen, and deuterium with average deviations from experiments of 7.5%, 4.4%, and 1.8%, respectively. The surface tensions of hydrogen/deuterium, para-hydrogen/helium, deuterium/helium, and hydrogen/neon mixtures are reproduced with a mean absolute error of 5.4%, 8.1%, 1.3%, and 7.5%, respectively. The surface tensions are predicted with an excellent accuracy at temperatures above 20K. The poor accuracy below 20K is due to the inability of Feynman-Hibbs-corrected Mie potentials to represent the real fluid behavior at these conditions, motivating the development of new intermolecular potentials. This DFT can be leveraged in the future to study confined fluids and assess the performance of porous materials for hydrogen storage and transport.

The ion equation of state of plasmas in the warm dense matter regime

The existing “quotidian equation of state (QEOS)” model [More et al., Phys. Fluids 31, 3059 (1988)] has been revised, and an alternative set of formulas is provided for the Helmholtz free energy, internal energy, heat capacity, and pressure. A novel scheme for constructing the ion equation-of-state is proposed based on the additive of solid and fluid components that act throughout the temperature range, in contrast to the QEOS that matches the thermodynamic parameters at selected interfaces. These components are continuous along with their first and second derivatives and obey limiting cases and thermodynamics laws (Lindemann melting, Dulong–Petit law, Grüneisen pressure law, and ideal gas law). Thus, the new scheme eliminates discontinuities in thermodynamics parameters across interfaces and ensures that the thermodynamics parameters are consistent with each other. The Helmholtz free energy, internal energy, heat capacity, and pressure have been approximated with computationally efficient formulas that can be used as parts of other models, e.g., hydro-simulations.

Mechanical modeling of the maturation process for textile reinforced tissue‐engineered heart valves

AbstractBiohybrid tissue‐engineered implants offer promising possibilities to treat cardiovascular diseases. The ability of tissue‐engineered implants to grow and remodel can be utilized to produce implants that can adapt to changes in the physiological conditions within the patient body. To improve the mechanical properties of the implant, a fibre‐reinforced textile scaffold is used. The embedment of a load‐oriented textile scaffold in the implant acts as biomimetic reinforcement and guides the direction of the extracellular matrix (ECM) growth. Our main objective is to design biohybrid heart valves that can withstand tough physiological loading conditions for decades. That makes it necessary to develop accurate and efficient models to support the design process.In this paper, the focus is on the macro‐mechanical modeling of the maturation process for textile reinforced heart valves. A biohybrid heart valve is modeled as a composite structure. The constitutive model is constructed by defining the total Helmholtz free energy as the sum of energies of the valve constituents. The two main constituents are (i) the extracellular matrix (ECM) and (ii) the fiber‐reinforced textile scaffold. The density of protein fibers such as collagen and elastin fibers are treated as internal variables. We introduce a new approach for modeling the densification of protein fibers during the cultivation process which is capable of considering both static and dynamic cultivation processes. The model considers the density change caused by the immunological response as well as the density change due to mechanical stimulation. The textile scaffold anisotropic behavior is introduced using structural tensors. The constitutive models are then embedded into a special finite element technology with reduced integration that enables efficient computations. The finite element simulation results are then validated using the experimental data provided by BioTex (Institute of Applied Medical Engineering, RWTH Aachen University, Germany). In the end, a finite element simulation for an exemplary valve is presented.

Equation of state for the Mie (λr,6) fluid with a repulsive exponent from 11 to 13.

An empirical multi-parameter equation of state in terms of the reduced Helmholtz energy is presented for the Mie (λr-6) fluid with a repulsive exponent λr from 11 to 13. The equation is fitted to an extensive dataset from molecular dynamics simulation as well as the second and third thermal virial coefficients. It is comprehensively compared with the SAFT-VR model and is a more accurate description of the considered fluid class. The equation is valid for reduced temperatures T/Tc from 0.55 to 4.5 and for reduced pressures of up to p/pc = 265. A good extrapolation behavior and the occurrence of a single Maxwell loop down to the vicinity of the triple point temperature are realized.

Noninteger Valence in Ce‐Based Compounds

Herein, a simple thermodynamic approach using the Helmholtz free energy is proposed to estimate the noninteger valence in Ce‐based compounds at the γ→α phase transition. The idea of an electron phase transition within Anderson–Kondo model is used as a microscopic mechanism and applied to Ce–X (X = Y, Al–Sc) within 0–16 at% of X. In the suggested approach, valence of Ce–X compounds depends on volume collapse and the temperatures of start and finish of phase transition as well as external pressure, which can induce γ→α phase transition, and increases with rising the amount of impurity.

Finite-temperature Thomas–Fermi–Dirac-Weizsäcker model in the warm dense matter regime

The Thomas–Fermi–Dirac-Weizsäcker model for finite temperature is presented. Starting with the free energy density, consistent expressions for the Helmholtz free energy, total energy, pressure and virial theorem are derived. The exchange and correlation terms are cast in a form that is suitable for derivation of their contributions to energy and pressure. The Euler–Lagrange equation is written in Schrödinger-like form and solved in spherical symmetry in a Wigner–Seitz cell. The challenges associated with its numerical solution are discussed and a viable approach to obtain a non-negative solution is proposed. The model is benchmarked for two materials, Al and Cu, by comparing the pressure at solid density in the temperature range , as well as the cold pressure curves.

Penentuan Density Matrix Sistem Kuantum Satu Partikel Dimensi Satu dengan Metode Discretized Path Integral (DPI)

A mixed quantum system can be represented by a density matrix. By knowing the density matrix of the system, other thermodynamic properties of the system such as Helmholtz free energy and entropy can be determined. Computation of the density matrix of a one-dimensional single-particle quantum system using the discretized path integral(DPI) method is presented. The results of the DPI method were validated using the finite difference time domain (FDTD) method for a particle in an infinite square well and harmonic oscillator potentials. The results of density matrix, Helmholtz’s free energy and entropy have shown that the DPI method produces correct numerical values compared to the FDTD results. Then, the DPI method is used in the double-well potential by giving variations in barrier potential and temperature of the system. The results show that the particle will be tunneling when the barrier potential energy is more than the kinetic energy of the particle. The particle also can tunnel easily if the system has the higher temperature.

A Reference Equation of State with an Associating Term for the Thermodynamic Properties of Ammonia

Fluid ammonia is highly associated because of strong intermolecular hydrogen bonding. This causes different behavior of thermophysical properties from nonpolar fluids, as reflected, for example, in the heat capacity. In this work, a new equation of state explicit in the Helmholtz energy with independent variables of temperature and density was developed, which includes a new associating term. Its uncertainties in density, vapor pressure, saturated liquid and vapor densities, and caloric properties were estimated by comparisons with experimental data. The new equation of state is valid from the triple-point temperature (195.49 K) to 725 K at pressures up to 1000 MPa and densities up to 53.13 mol dm−3. Physically correct behavior within the region of validity and at extremely high temperatures and pressures, and at temperatures far below the triple-point temperature, was obtained by applying constraints on various properties. The unique physical behavior of ammonia shown in some thermodynamic properties is described in detail, which will provide a preliminary template for developing equations of state for other associating fluids.

Criticality and phase transition of Kerr–anti-de Sitter black hole with quintessence and cloud of strings

In this paper, a thermodynamical study of the Kerr–anti-de Sitter (KAdS) black hole (BH) encircled by quintessence matter and the cloud of strings is presented. We check the effects of the cloud parameter on the thermodynamical variables including Hawking temperature, entropy, Helmholtz free energy (HFE), Gibbs free energy (GFE), and specific heat capacity. These thermodynamic variables hold the Smarr–Gibbs–Dehum (SGD) relation in the appearance of quintessence and the cloud of the string parameter. Then we calculate the critical expressions of the developed variables in two different ways that are “Van-der-Waals (vdW)-type equation of state” (EoS) and the “Maxwell equal-area law” (MEAL). The latter technique is found to be more reasonable to examine the criticality of the complex BHs. Using MEAL, we plot the phase diagram in a temperature–entropy plane and obtain an isobar, which represents the co-existence region of liquid–gas phases. It is concluded that the uncharged BHs exhibit the same phase transition (PT) to vdW fluid as the temperature drops down its critical value. Also, we discuss the influence of thermal fluctuations (TF) on the stability of KAdS BH.

FE{}^textrm{ANN}: an efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining

Herein, we present a new data-driven multiscale framework called FE{}^textrm{ANN} which is based on two main keystones: the usage of physics-constrained artificial neural networks (ANNs) as macroscopic surrogate models and an autonomous data mining process. Our approach allows the efficient simulation of materials with complex underlying microstructures which reveal an overall anisotropic and nonlinear behavior on the macroscale. Thereby, we restrict ourselves to finite strain hyperelasticity problems for now. By using a set of problem specific invariants as the input of the ANN and the Helmholtz free energy density as the output, several physical principles, e. g., objectivity, material symmetry, compatibility with the balance of angular momentum and thermodynamic consistency are fulfilled a priori. The necessary data for the training of the ANN-based surrogate model, i. e., macroscopic deformations and corresponding stresses, are collected via computational homogenization of representative volume elements (RVEs). Thereby, the core feature of the approach is given by a completely autonomous mining of the required data set within an overall loop. In each iteration of the loop, new data are generated by gathering the macroscopic deformation states from the macroscopic finite element simulation and a subsequently sorting by using the anisotropy class of the considered material. Finally, all unknown deformations are prescribed in the RVE simulation to get the corresponding stresses and thus to extend the data set. The proposed framework consequently allows to reduce the number of time-consuming microscale simulations to a minimum. It is exemplarily applied to several descriptive examples, where a fiber reinforced composite with a highly nonlinear Ogden-type behavior of the individual components is considered. Thereby, a rather high accuracy could be proved by a validation of the approach.

A Helmholtz Energy Equation of State for trans-1,1,1,4,4,4-Hexafluoro-2-butene [R-1336mzz(E)] and an Auxiliary Extended Corresponding States Model for the Transport Properties

A Helmholtz Energy Equation of State for trans-1,1,1,4,4,4-Hexafluoro-2-butene [R-1336mzz(E)] and an Auxiliary Extended Corresponding States Model for the Transport Properties

Thermodynamics and evaporation of a modified Schwarzschild black hole in a non–commutative gauge theory

In this work, we study the thermodynamic properties on a non–commutative background via gravitational gauge field potentials. This procedure is accomplished after contracting de Sitter (dS) group, SO(4,1), with the Poincarè group, ISO(3,1). Particularly, we focus on a static spherically symmetric black hole. In this manner, we calculate the modified Hawking temperature and the other deformed thermal state quantities, namely, entropy, heat capacity, Helmholtz free energy and pressure. Finally, we also investigate the black hole evaporation process in such a context.

Probing the Local Environment of Al-Substitution into Ferrihydrite Using DFT + U Calculations

Ferrihydrite (Fh) is a poorly crystalline oxyhydroxide that has attracted attention due to its importance in geochemistry, ion mobility, adsorption, and as a precursor for the formation of more crystalline phases such as goethite and hematite. A chemical model for the isomorphic substitution of the Fe sites by Al is proposed. DFT/plane wave calculations have been performed to provide information about the local environment of Al-substitution into Fh. The structural and electronic properties have been investigated in detail. Al prefers to replace the octahedral Fe sites, which has been evidenced by the comparison of the calculated Al K-edge X-ray absorption near-edge structure spectra with the available experimental data. The effect of the increasing ionicity of the system upon Al-substitution to the quadrupolar coupling constant of 27Al NMR and the Mössbauer quadrupole splitting and isomer shift parameters have been investigated. The results indicate that these parameters are affected by the concentration of Fe-substituted sites. Al-substitution increases the band gap dispersion with the formation of Al electronic states in the conduction band. The standard Helmholtz free energy of Al-substitution has been estimated and discussed in detail.

Equation of State of Periclase Based on Planck–Einstein Functions

The paper presents an extension of the Voronin–Kutsenok method for joint description of both thermochemical and bulk data with combination of Planck–Einstein functions and modified Tait equation. Two approaches based on the Gibbs and Helmholtz energy descriptions were proposed. Magnesium oxide (periclase) was chosen as the test system. The parameters of the equation of state were optimized using published data over a broad range of thermodynamic variables (up to 3000 K and 145 GPa). The predictive power of both approaches was estimated.

Density of States for the Unitary Fermi Gas and the Schwarzschild Black Hole

The density of states of a quantum system can be calculated from its definition, but, in some cases, this approach is quite cumbersome. Alternatively, the density of states can be deduced from the microcanonical entropy or from the canonical partition function. After discussing the relationship among these procedures, we suggest a simple numerical method, which is equivalent in the thermodynamic limit to perform a Legendre transformation, to obtain the density of states from the Helmholtz free energy. We apply this method to determine the many-body density of states of the unitary Fermi gas, a very dilute system of identical fermions interacting with a divergent scattering length. The unitary Fermi gas is highly symmetric due to the absence of any internal scale except for the average distance between two particles and, for this reason, its equation of state is called universal. In the last part of the paper, by using the same thermodynamical techniques, we review some properties of the density of states of a Schwarzschild black hole, which shares the problem of finding the density of states directly from its definition with the unitary Fermi gas.