Helmholtz Free Energy Research Articles

Chemical potentials in nonhydrostatically stressed anisotropic phases

Summary Chemical potentials are defined as the partial derivatives of the Helmholtz energy with respect to moles of chemical components under conditions of zero domain strain and fixed temperature. Under hydrostatic conditions, chemical potentials are dependent only on state properties. Under nonhydrostatic conditions, they also depend on a “chemical expansivity tensor” - a second-order tensor with unit trace that characterises how the elastic network is compressed to accommodate new material within the local domain element. The five degrees of freedom of this tensor generate a class of chemical potentials. An important group within this class are the “uniaxial chemical potentials”, which quantify the Helmholtz energy change when new material is incorporated via compression along a single axis. Chemical and mechanical equilibrium is achieved when all uniaxial chemical potentials remain constant along their respective axes. The derived expressions apply to both crystalline and amorphous materials. Their utility is demonstrated through solutions to classic phase-equilibrium problems.

Classical density functional theory for alkane adsorption in cationic Faujasites: comparison with grand canonical Monte Carlo simulations

Classical density functional theory for alkane adsorption in cationic Faujasites: comparison with grand canonical Monte Carlo simulations

Automated model discovery for tensional homeostasis: Constitutive machine learning in growth and remodeling.

We present a built-in physics neural network architecture, known as inelastic Constitutive Artificial Neural Network (iCANN), to discover the inelastic phenomenon of tensional homeostasis. In this course, identifying the optimal model and material parameters to accurately capture the macroscopic behavior of inelastic materials can only be accomplished with significant expertise, is often time-consuming, and prone to error, regardless of the specific inelastic phenomenon. To address this challenge, built-in physics machine learning algorithms offer significant potential. We introduce the incorporation of kinematic growth and homeostatic surfaces into the iCANN to discover the Helmholtz free energy and the pseudo potential. The latter describes the state of homeostasis in a smeared sense. To this end, we additionally propose a novel design of the corresponding feed-forward network in terms of principal stresses. We evaluate the ability of the proposed network to learn from experimentally obtained tissue equivalent data at the material point level, assess its predictive accuracy beyond the training regime, and discuss its current limitations when applied at the structural level. Our source code, data, examples, and an implementation of the corresponding material subroutine are made accessible to the public at https://doi.org/10.5281/zenodo.13946282.

Improving the LKP-SJT Equation of State: Application to Alkanes, Carbon Dioxide, and Their Mixtures

The recently introduced modification of the Lee-Kesler-Plöcker equation of state, LKP-SJT, has been further developed, and the results are presented. The new version includes an enhancement of the original approach for calculating the compressibility factor at the critical point. Furthermore, the standard fluid combination used as base points for interpolation has been varied. The results of these calculations are compared with datasets generated from highly accurate equations of state in terms of the Helmholtz energy and with experimental measurements. The investigated fluids include alkanes and carbon dioxide. In comparison to the original version of the LKP-SJT equation of state, improvements were achieved for n-alkanes up to n-dodecane and carbon dioxide. The calculated densities of long-chain alkanes are significantly more accurate, while vapor pressures are less precise. The application of the LKP-SJT to propane - n-alkane and carbon dioxide - n-alkane mixtures up to n-decane confirms its benefits in the density calculation of long-chained alkanes and hence asymmetric mixtures. Calculations of the liquid density for the propane - n-eicosane mixture performed to estimate the extrapolation behavior yield promising results.

A comprehensive study of particle dynamics, thermal fluctuations with Barrow entropy, and greybody factors of quantum-improved charged black holes

This study examines the characteristics and behavior of quantum-improved charged black holes, focusing on the relationships between mass, charge, and the quantum-improved parameter. We explore the consequences of these characteristics on the structure of a black hole’s horizon, particularly investigating the ways that variations in charge and mass modify the horizon radius and size of innermost stable circular orbits. A thorough thermodynamic analysis indicates substantial impacts on corrected energies, including Helmholtz free energy, enthalpy, internal energy, and Gibbs free energy, in the context of Barrow entropy, emphasizing stable and unstable configurations across various quantum correction parameter ranges. Additionally, we analyze wave behavior near the cosmic horizon and its correlation with effective potential, illustrating whether wave frequency influences the gravitational absorption function and the transmission properties of scalar fields near quantum-improved charged black holes. Our findings highlight the complex interaction between black hole dynamics and quantum parameters, facilitating a more profound comprehension of black hole thermodynamics and their significance in curved space-time quantum field theory.

A Nonlocal Cahn–Hilliard–Darcy System with Singular Potential, Degenerate Mobility, and Sources

We consider a Cahn–Hilliard–Darcy system for an incompressible mixture of two fluids we already analyzed in [9]. In this system, the relative concentration difference φ obeys a convective nonlocal Cahn–Hilliard equation with degenerate mobility and singular (e.g., logarithmic) potential, while the volume averaged fluid velocity u is given by a Darcy’s law subject to the Korteweg force μ∇φ, where the chemical potential μ is defined by means of a nonlocal Helmholtz free energy. The kinematic viscosity η depends on φ. With respect to the quoted contribution, here we assume that the Darcy’s law is subject to gravity and to a given additional source. Moreover, we suppose that the Cahn–Hilliard equation and the chemical potential contain source terms. Our main goal is to establish the existence of two notions of weak solutions. The first, called “generalized” weak solution, is based a convenient splitting of μ so that the entropy derivative does not need to be integrable. The second is slightly stronger and allows to reconstruct μ and to prove the validity of a canonical energy identity. For this reason, the latter is called “natural” weak solution. The rigorous relation between the two notions of weak solution is also analyzed. The existence of a global attractor for generalized weak solutions and time independent sources is then demonstrated via the theory of generalized semiflows introduced by J.M. Ball.

Exploring Liquid Crystal Properties through the Two-Phase Thermodynamic Model: Structural, Dynamic, and Thermodynamic Properties.

This work provides a comprehensive analysis of the structural, dynamic, and thermodynamic properties of liquid crystals (LCs) along with their evolution through phase transitions and mesophases. A model of purely repulsive semiflexible spherocylinders is used in a molecular dynamics scheme through simulations involving NPT and NVT combinations. The two-phase thermodynamic model was used to obtain the translational, rotational, and vibrational density of states as well as the absolute values of thermodynamic parameters. We show evidence that during the isotropic-nematic-smectic-solid transitions, the translational diffusion coefficient becomes anisotropic, initially increasing by 15% in the nematic mesophase with a 129% rise along the director vector. Subsequent transitions result in a reduction of the diffusion coefficient by 42% in the smectic phase and 90% in the crystalline phase. Rotational diffusion decreases across all transitions (12, 35, and 26% for nematic, smectic, and solid transitions), although a notable increase in rotation around the principal axis is observed during the last transition. Thermodynamic analysis reveals that the primary contribution to the Gibbs free energy arises from the mechanical term (PV). With regard to the components, rotational motion is the dominant contribution to the Helmholtz free energy in the first transition, while translational motion dominates in the last transition. For the intermediate transition, translational, rotational, and vibrational contributions are comparable. A thorough analysis has been conducted into the Cartesian projections and the principal axes of rotation, in addition to the "solid and gas components" from the two-phase thermodynamic model analysis.

Hydrogen adsorption and properties of goldene: a first-principles study

Goldene, a single-atom Au monolayer with a hexagonal lattice in the P6/mmm space group, exhibits interesting hyrdrogen absorption properties, as revealed using density functional theory calculations. This study focuses on H-adsorbed goldene at different coverage ratios, and provides insights into the energetic and electronic properties of this system, distinguishing it from the well-studied pristine goldene. Hydrogen adsorption on goldene, while energetically comparable to bulk gold, shows a slight reduction in energetic favorability and introduces specific scanning tunneling microscopy images, reported here for the first time. Raman spectra of H-adsorbed goldene at a 1/9 coverage ratio are also first reported here, along with a vibrational mode analysis, highlighting distinct atomic displacement patterns. Finally, for completeness, previously reported results on the dynamical and mechanical stability of pristine goldene are reported, with a special emphasis on the quadratic flexural mode characteristic of 2D materials. New insights into the thermodynamic properties of goldene compared to bulk gold are also discussed. Although bulk gold remains thermodynamically more stable at all temperatures, the vibrational contributions to the Helmholtz free energy favor goldene above 175 K, narrowing the stability gap with temperature. Overall, this study validates goldene's robustness and expands its potential for experimental and theoretical exploration in the context of hydrogen adsorption and functionalized 2D materials more broadly.

Multifluid Mixture Models for the Binary Mixtures of Trifluoroethene (R-1123) with Difluoromethane (R-32), 2,3,3,3-Tetrafluoroprop-1-ene (R-1234yf), and Propane (R-290).

This work presents mixture models for the thermodynamic properties of the mixtures of R-1123 with R-32, R-1234yf, and R-290, which are potential refrigerants for next-generation air conditioning systems. The models are based on the multifluid approximation and employ accurate Helmholtz energy equations of state for the pure fluids. The mixing parameters of the models are determined by fitting experimental data for the (p, ρ, T) behavior and vapor-liquid equilibrium. The range of validity of the models is approximately temperatures from 273 to 400 K and pressures up to 7 MPa. Typical uncertainties (k = 2) in this range are 0.3% for liquid densities, 1.5% for vapor densities, and 1% for bubble-point pressures. The mixture models exhibit qualitatively correct behavior in the critical and extrapolated regions; this is demonstrated by plots of several thermodynamic properties over wide ranges of temperature and pressure.

Finite element analysis of materially uniform dielectric elastomers

In the definition of Noll, a body is uniform if all points are made of the same material. As shown by Noll himself and by Epstein and Maugin, uniformity makes the Helmholtz free energy depend on the material point exclusively through a tensor field, called uniformity tensor or implant tensor or material isomorphism. Uniformity is therefore a particular case of inhomogeneity. In turn, uniformity includes homogeneity as a particular case: indeed, homogeneity is attained when the uniformity tensor happens to be integrable. This work focuses on the non-linear large-deformation behaviour of uniform dielectric elastomers. Building on the foundational works of Toupin, Eringen and others, this work integrates continuum mechanics with electrostatics to develop a finite element framework for analysing uniform dielectric elastomers. This framework allows for considering the inherent inhomogeneity in materials exhibiting non-linear electromechanical coupling such as electro-active polymers. The inhomogeneity is assumed to be self-driven, i.e., not implied by the second law of thermodynamics: rather, it depends on the torsion of the connection (covariant derivative) induced by the uniformity tensor. A MATLAB®-based finite element solver is developed and applied to the simulation of an electromechanical beam-type actuator. The solver is robust and capable of addressing various simulation scenarios. Numerical simulations demonstrate the significant impact of material uniformity on actuator performance. This research provides a tool for future applications in dielectric elastomers, particularly in sensors, actuators and bio-inspired robotics.

A model of elastic softening and displacive phase transitions in anisotropic phases, with application to stishovite and post-stishovite

Summary This paper introduces a comprehensive framework for modelling both instantaneous and time-dependent elastic softening in anisotropic materials at high pressure and temperature. This framework employs Landau Theory, minimizing the Helmholtz energy by varying isochemical parameters (q) that capture structural changes, atomic ordering, and/or electronic spin states. This allows for internally consistent predictions of volume, unit cell parameters, the elastic tensor, and other thermodynamic properties, while allowing large symmetry-breaking strains. The formulation is validated using the stishovite-to-post-stishovite transition. It is demonstrated that, near this transition, both stishovite and post-stishovite exhibit auxetic behaviour in several directions, with post-stishovite also displaying negative linear compressibility along the long axis of its unit cell (either the a or b axis). The new formulation is implemented in the open-source BurnMan software package.

A Nonlinear Breakage Mechanics Model: From Extreme Entire Life Model to Breakage Evolution of Limestone Based on Separation of Helmholtz Free Energy under Cyclic Loading

A Nonlinear Breakage Mechanics Model: From Extreme Entire Life Model to Breakage Evolution of Limestone Based on Separation of Helmholtz Free Energy under Cyclic Loading

Constitutive modelling of non-crystalline solids using Van der Waals Potential

Abstract Non-crystalline solid materials have significant applications in the domains of science and engineering. This study focuses on the use of fundamental concepts of molecular interactions to develop a constitutive equation that can predict the stress-stretch behaviors of these materials. The strain energy density function of the material is derived using Helmholtz free energy of van der Waals potential. It is obtained in terms of excluded volume and number density of the molecules. To make consistent with continuum approximation, the excluded volume and number density are defined in terms of strain invariants of right Cauchy-Green deformation tensor. Finally, the constitutive model is represented in the form of Cauchy stress tensor. The current model is used for predicting finite deformation of non-crystalline solid phase of material. The results derived from the current model are also compared with the experimental results of polyurethane foam and poly vinyl alcohol gel materials. The current constitutive model can also be used for predicting the deformation characteristics of micro/nano components used in engineering systems. This study can provide a basis for the future scope of the constitutive modelling for the non-crystalline solid materials considering their complex molecular structures.

A Helmholtz Energy Equation of State for 3,3,3-Trifluoroprop-1-ene (R-1243zf)

A Helmholtz Energy Equation of State for 3,3,3-Trifluoroprop-1-ene (R-1243zf)

Computational Modeling and Experimental Investigation of CO2-Hydrocarbon System Within Cross-Scale Porous Media.

CO2 flooding plays a crucial role in enhancing oil recovery and achieving carbon reduction targets, particularly in unconventional reservoirs with complex pore structures. The phase behavior of CO2 and hydrocarbons at different scales significantly affects oil recovery efficiency, yet its underlying mechanisms remain insufficiently understood. This study improves existing thermodynamic models by introducing Helmholtz free energy as a convergence criterion and incorporating adsorption effects in micro- and nano-scale pores. This study refines existing thermodynamic models by incorporating Helmholtz free energy as a convergence criterion, offering a more accurate representation of confined phase behavior. Unlike conventional Gibbs free energy-based models, this approach effectively accounts for confinement-induced deviations in phase equilibrium, ensuring improved predictive accuracy for nanoscale reservoirs. Additionally, adsorption effects in micro- and nano-scale pores are explicitly integrated to enhance model reliability. A multi-scale thermodynamic model for CO2-hydrocarbon systems is developed and validated through physical simulations. Key findings indicate that as the scale decreases from bulk to 10 nm, the bubble point pressure shows a deviation of 5% to 23%, while the density of confined fluids increases by approximately 2%. The results also reveal that smaller pores restrict gas expansion, leading to an enhanced CO2 solubility effect and stronger phase mixing behavior. Through phase diagram analysis, density expansion, multi-stage contact, and differential separation simulations, we further clarify how confinement influences CO2 injection efficiency. These findings provide new insights into phase behavior changes in confined porous media, improving the accuracy of CO2 flooding predictions. The proposed model offers a more precise framework for evaluating phase transitions in unconventional reservoirs, aiding in the optimization of CO2-based enhanced oil recovery strategies.

Thermodynamic aspects of higher-dimensional black holes in Einstein–Gauss–Bonnet gravity through exponential entropy

Abstract We assume exponential corrections to the entropy of $5D$ charged AdS
black hole solutions, which are derived within the framework of
Einstein Gauss-Bonnet gravity and nonlinear electrodynamics.
Additionally, we consider two distinct versions of $5D$ charged AdS
black holes by setting the parameters $q\rightarrow0$ and
$k\rightarrow0$ (where $q$ represents charge, $k$ is the non-linear
parameter). We investigate these black holes in the extended phase
space, where the cosmological constant is interpreted as pressure,
and demonstrate the first law of black hole thermodynamics. The
focus extends to understanding the thermal stability or instability,
as well as identifying first and second-order phase transitions.
This exploration is carried out through the analysis of various
thermodynamic quantities, including heat capacity at constant
pressure, Gibbs free energy, Helmholtz free energy, and the trace of
the Hessian matrix. In order to visualize phase transitions,
identify critical points, analyzing stability and provide
comprehensive analysis, we have made the contour plot of the
mentioned thermodynamic quantities and observed that our results are
very much consistent. These investigations are conducted within the
context of exponentially corrected entropies, providing valuable
insights into the intricate thermodynamic behavior of these $5D$
charged AdS black holes under different parameter limits.

A thermodynamically complete multi-phase equation of state for dense and porous metals at wide ranges of temperature and pressure

Abstract A thermodynamically complete multi-phase equation of state (EOS) applicable to both dense and porous metals at wide ranges of temperature and pressure is constructed. A standard three-term decomposition of the Helmholtz free energy as a function of specific volume and temperature is presented, where the cold component models both compression and expansion states, the thermal ion component introduces the Debye approximation and melting entropy, and the thermal electron component employs the Thomas–Fermi–Kirzhnits (TFK) model. The porosity of materials is considered by introducing the dynamic porosity coefficient α and the constitutive P–α relation, connecting the thermodynamic properties between dense and porous systems, allowing for an accurate description of the volume decrease caused by void collapse while maintaining the quasi-static thermodynamic properties of porous systems identical to the dense ones. These models enable the EOS applicable and robust at wide ranges of temperature, pressure and porosity. A systematic evaluation of the new EOS is conducted with aluminum (Al) as an example. 300 K isotherm, shock Hugoniot, as well as melting curves of both dense and porous Al are calculated, which shows great agreements with experimental data and validates the effectiveness of the models and the accuracy of parameterizations. Notably, it is for the first time Hugoniot P–σ curves up to 106 GPa and shock melting behaviors of porous Al are derived from analytical EOS models, which predict much lower compression limit and shock melting temperatures than those of dense Al.

Advanced association theory for monoethylene glycol: Thermodynamic perturbation theory, Monte Carlo simulation, and equation of state parametrization

Thermodynamic perturbation theory (TPT) is a breakthrough in developing an equation of state for systems containing hydrogen bonding. In this work, we derive the association contribution to Helmholtz's free energy...

First-principles molecular-dynamics equationof state of liquid to dense plasma iron.

We computed the equationof state of iron using extended first-principles molecular dynamics simulations, ranging from 7.874g/cm^{3} and 5500K up to 47.2g/cm^{3} and 10^{9}K. We compared the principal Hugoniot curve with semiempirical models, average atom-based model predictions, and shock experiment results. We derived the Helmholtz free energy and the entropy via thermodynamic integration along isochores. We explore the ionization processes at play along the Hugoniot curve by analyzing the evolution of the electronic densities of states in the gigabar regime.

Prediction of lattice constant, volume and density of perovskite MgSiO3 with cubic structure at high pressure up to 135 GPa and high temperature up to 4000 K from statistical moment method

The analytical expression of the Helmholtz free energy, the mean nearest neighbor distance between two atoms, the lattice constant, the volume and the density for perovskite crystals with cubic structure are derived by using statistical moment method (SMM). The article provides numerical results for the temperature and pressure dependence of the lattice constant, the volume and the density of MgSiO3 perovskite in the temperature range from 0 to 4000 K and in the pressure range from 0 to 135 GPa. Our SMM numerical results for MgSiO3 perovskite are compared with that for CaSiO3 perovskite, experimental data and results calculated by other methods. Our other SMM numerical results are new and predictive of experimental results. This is the first application of SMM to study the structural characteristics of cubic MgSiO3 perovskite under extreme conditions of temperature and pressure. On that basis, this research result can be developed to study the thermodynamic, elastic deformation, elastic wave and melting characteristics of cubic MgSiO3 perovskite.