Helmholtz Free Energy Research Articles

Pressure and Isothermal Compressibility of Asymmetric Complex Plasmas in the Poisson–Boltzmann Plus Hole Approximation

ABSTRACTWe consider a two‐component asymmetric complex plasma of finite‐size macroions with charges () and point oppositely charged microions with unit charges. System pressure is calculated within the framework of the Poisson–Boltzmann plus hole approximation by obtaining the Coulomb nonideal parts of interaction energy and Helmholtz free energy. It is shown that both the pressure and plasma isothermal compressibility are positive over the entire range of macroion concentrations. We compared pressure and isothermal compressibility in linearized approximations and in the Poisson–Boltzmann plus hole approximation where the nonlinear screening effect is taken into account and showed a significant difference for some macroions concentrations.

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.

Effects of conical singularity and Wu-Yang magnetic monopole on thermodynamic properties of harmonic oscillator interacting with potential

Abstract This paper investigates the thermodynamic properties of the harmonic oscillator system in a conical singularities background. Moreover, we incorporate a Wu-Yang magnetic monopole (WYMM) and inverse square potential into the quantum system and analyze the resulting effects. Using analytical methods, we derive the energy eigenvalues and study the thermodynamic properties, such as the partition function, Helmholtz free energy, Gibbs's free energy and specific heat capacity at finite temperature $T \neq 0$. Our findings demonstrate that these physical parameters are influenced by the topological parameter, strength of WYMM and potential parameters. These results provide valuable insights into the interplay between thermodynamic properties of a quantum mechanical problem and the gravitational effects.

Equilibrium oxygen isotope fractionation in Mg(OH)2-Ca(OH)2-H2O: Implication from in situ high-temperature and high-pressure vibrational spectra

Abstract Equilibrium oxygen isotope fractionations have been extensively studied between H2O and minerals (like brucite-type hydroxides) in isotope geochemistry. In this study, pure 18O-enriched hydroxides were synthesized through reactions between M3N2 (M = Mg and Ca) powders and H218O water. In situ high-temperature (T) and high-pressure (P) Raman and Fourier transform infrared (FTIR) spectra were systematically collected for these phases. The observed 18O-16O effect on the frequency shifts aligns with the theoretical model, and has little impact on the anharmonic O-H potential well. The calculated intrinsic anharmonic parameters (ai) for both lattice and OH-stretching modes are essentially identical between M(16OH)2 and M(18OH)2, satisfying the classical limit for the Helmholtz free energy at extremely high temperature. Based on the measured vibrational data, the equilibrium oxygen isotope fractionation β(T,P) factors were modeled for both hydroxides. Both the intrinsic and external anharmonic contributions are smaller than the experimental uncertainties, and this model is also validated by ab initio calculation. Next, the 103·lnα(T) profiles were computed between brucite/portlandite and H2O, which are consistent with the reported oxygen isotope exchange measurements above 200 °C. Additionally, the discrepancy between this equilibrium model and the precipitation experiment below 120 °C also provides useful information for investigating the mechanism of kinetic isotope fractionation. The predicted 103·lnαbrucite-portlandite(T) curve also agrees well with points inferred from the 18O-16O exchange measurements, while the pressure effect can be ignored for 18O-16O fractionations. Therefore, vibrational spectroscopic measurements, including both Raman and infrared spectroscopies, have valuable applications in studying equilibrium non-metallic isotope fractionations in minerals.

Corrected thermodynamics and stability of magnetic charged AdS black holes surrounded by quintessence

In this study, we explore the corrected thermodynamics of non-linear magnetic charged anti-de Sitter (AdS) black holes surrounded by quintessence, incorporating thermal fluctuations and deriving the corrected thermodynamic potentials. We analyze the effects of corrections due to thermal fluctuations on various thermodynamic potentials, including enthalpy, Helmholtz free energy, and Gibbs free energy. Our results show significant impacts on smaller black holes, with first-order corrections destabilizing them, while second-order corrections enhance stability with increasing parameter values. The specific heat analysis further elucidates the stability criteria, indicating that the large black holes ensure stability against phase transitions. However, the thermal fluctuations do not affect the physical limitation points as well as the second-order phase transition points of the black hole. Our findings highlight the intricate role of thermal fluctuations in black hole thermodynamics and their influence on stability, providing deeper insights into the behaviour of black holes under corrected thermodynamic conditions.

On the transition between two-phase and single-phase interface dynamics in ammonia sprays

This study explores the application of the density gradient theory to assess the validity of the two-phase spray atomization theory in multi-component ammonia sprays under conditions relevant to practical applications. The research focuses on analyzing the liquid–vapor interface structure at thermodynamic equilibrium. The most recent Helmholtz energy equation of state was employed to compute density profiles for ammonia and the surrounding nitrogen gas. The study investigates species distribution at equilibrium by examining minimal Helmholtz free energy states and evaluates critical parameters, such as interface thickness. The Knudsen number was also calculated as a temperature function to characterize the spray regime. The findings indicate that when the reduced temperature exceeds 0.95, a diffuse interface forms in ammonia-nitrogen systems, supporting using phase-field models for simulating ammonia injection in practical scenarios.

A novel continuum damage evolution model based on the concept of damage driving force for unidirectional composites

A novel damage evolution model for unidirectional (UD) composites is established in this paper in the context of continuum damage mechanics (CDM). It addresses matrix cracking and it is to be applied along with the damage representation established previously. The concept of damage driving force is employed based on the Helmholtz free energy. It is shown that the damage driving force can be partitioned into three parts, resembling closely three conventional modes of fracture, respectively. A damage evolution law is derived accordingly based on the newly obtained expressions of the damage driving force. The fully rationalised Tsai-Wu criterion is employed in the model for predicting the initiation of matrix cracking damage and fibre failure, assisted with the rationalised maximum stress criterion for identifying the damage modes. A mechanism is introduced to describe the unloading behaviour as a part of the proposed model. The predictions were validated against experimental results, showing good agreement with the experiments and demonstrating the capability and effectiveness of the proposed model.

Hierarchical equations of motion for multiple baths (HEOM-MB) and their application to Carnot cycle.

We have developed a computer code for the thermodynamic hierarchical equations of motion derived from a spin subsystem coupled to multiple Drude baths at different temperatures, which are connected to or disconnected from the subsystem as a function of time. The code can simulate the reduced dynamics of the subsystem under isothermal, isentropic, thermostatic, and entropic conditions. The extensive and intensive thermodynamic variables are calculated as physical observables, and Gibbs and Helmholtz energies are evaluated as intensive and extensive work. The energy contribution of the system-bath interaction is evaluated separately from the subsystem using the hierarchical elements of the hierarchical equations of motion. The accuracy of the calculated results for the equilibrium distribution and the two-body correlation functions is assessed by contrasting the results with those obtained from the time-convolution-less Redfield equation. It is shown that the Lindblad master equation is inappropriate for the thermodynamic description of a spin-boson system. Non-Markovian effects in thermostatic processes are investigated by sequentially turning on and off the baths at different temperatures with different switching times and system-bath coupling. In addition, the Carnot cycle is simulated under quasi-static conditions. To analyze the work performed for the subsystem in the cycle, thermodynamic work diagrams are plotted as functions of intensive and extensive variables. The C++ source codes are provided as supplementary material.

A thermodynamically‐based fractional model combined viscoelastic‐viscoplastic‐ductile damage with application to fiber‐reinforced polymer composites

AbstractA thermodynamically‐based fractional viscoelastic‐viscoplastic‐damage constitutive model combined with continuous damage mechanics (CDM) theory was established, in order to describe the rate‐dependent nonlinear behavior of fiber‐reinforced polymer composites (FRPCs). The fractional Helmholtz free energy consists of four contributions: viscoelastic (VE), viscoplastic (VP), hardening and damage, in which the VE and VP parts are constructed by fractional Zener and Scott‐Blair (SB) element forms respectively. The constitutive equation is obtained through Helmholtz free energy for the fractional Zener model, and plastic flow and hardening evolution law are all derived in the process. The ductile damage, coupled to both VE and VP free energy parts, is introduced through fractional damage energy release rates to model the degradation of material properties. The corresponding strain energy release rate and dissipation contributions are also derived. The fractional implicit time integration algorithms of proposed model are presented. The model is applied to validate tests of FRPCs under various loading conditions. The model validation and comparison are presented by simulating experimental data and existing models in the literature. And the corresponding evolution of dissipated energy is discussed to further valid the characterization ability of the model.Highlights A thermodynamical fractional constitutive model was developed for FRPCs. The Helmholtz free‐energy potential for fractional Zener model is adopted. The physical significance of fractional order parameters is explored. Fractional implicit integration algorithm of proposed model is implemented. The validation and comparison of the model are presented under various loads.

Electrically charged black holes in [formula omitted]-Euler–Heisenberg theory

Drawing on the concept of the non-linear dynamics of electromagnetic fields in vacuum in terms of the Euler–Heisenberg system, a static and spherically symmetric black hole solution is derived by coupling the Euler–Heisenberg term and F(R) gravity. In applying the curvature singularity tools, the black hole solution induces a singular behaviour at the centre r=0 while exhibiting a regular shape at high distances. Using the heat capacity (CQ) and the Helmholtz free energy (F), we analyse the thermodynamic stability locally and globally. Furthermore, an investigation is conducted using class tools of Geometrothermodynamics (GTs) features such as Weinhold, Ruppeiner, Hendi–Panahiyan–Eslam–Momennia (HPEM), and Quevedo to identify phase transition points associated with heat capacity, including both physical boundary points (root of the heat capacity) and second critical phase transition points (divergent points of the heat capacity). We have even examined the relevant sparsity of Hawking radiation, showing how the black hole solution behaves as a black body at specific limits. Overall, this study enhances our knowledge of the mergers of a class of non-linear dynamics of electromagnetic fields in a vacuum within F(R) gravity by modelling black hole solutions.

Corrected Thermodynamics of Black Holes in f(R) Gravity with Electrodynamic Field and Cosmological Constant.

The thermodynamics of black holes (BHs) and their corrections have become a hot topic in the study of gravitational physics, with significant progress made in recent decades. In this paper, we study the thermodynamics and corrections of spherically symmetric BHs in models f(R)=R+αR2 and f(R)=R+2γR+8Λ under the f(R) theory, which includes the electrodynamic field and the cosmological constant. Considering thermal fluctuations around equilibrium states, we find that, for both f(R) models, the corrected entropy is meaningful in the case of a negative cosmological constant (anti-de Sitter-RN spacetime) with Λ=-1. It is shown that when the BHs' horizon radius is small, thermal fluctuations have a more significant effect on the corrected entropy. Using the corrected entropy, we derive expressions for the relevant corrected thermodynamic quantities (such as Helmholtz free energy, internal energy, Gibbs free energy, and specific heat) and calculate the effects of the correction terms. The results indicate that the corrections to Helmholtz free energy and Gibbs free energy, caused by thermal fluctuations, are remarkable for small BHs. In addition, we explore the stability of BHs using specific heat. The study reveals that the corrected BH thermodynamics exhibit locally stable for both models, and corrected systems undergo a Hawking-Page phase transition. Considering the requirement on the non-negative volume of BHs, we also investigate the constraint on the EH radius of BHs.

Enlarged Curvature, Torsion and Torque in Helical Conformations and the Stability and Growth of α-Peptide under the Isochoric and Isobaric Conditions: Variatonal Optimization

The torsional deformation behavior of an elastic bar with a circular cross-section was investigated by applying invariant dyadic analysis, where the small finite displacement functions advocated by Saint-Venant (1855) were fully employed. It was found that the previously overlooked circumferential shear force field generated by pure torsion on the side walls of a bar produces an unusual torque term induced by the skew-symmetric part of the deformation tensor and exhibits quadratic length dependence along the z-axis of the bar. The adaptation of this torque term for a helical conformation of α-peptides creates moments acting on the circular cross-sections and is directed along the surface normal of circular cross-sections, which coincides with the tangent vector of the helix. The projection of this torque along the z-axis of the helix varies quadratically with the azimuthal angle. The radial component of the unusual torque, which also lies along the principal normal vector of the helix, starts to perform a precession motion by tracking a spiral orbit around the z-axis, whereas its apex angle decreases asymptotically with the azimuthal angle and finally reaches a finite value depending on the height of the helix along the z-axis. The ordinary torque terms, which are also deduced from the self- and anti-self-conjugate parts of the deformation tensor, have magnitudes half that of the full torque term reported in the literature. The present results were applied to the helical conformation of α-peptides designated by {3.611} to show that the mechanical stability of strained open-ended helical conformations can be successfully achieved by spontaneous readjustments of the surface and bulk Helmholtz free energies under isothermal isochoric conditions. It has been demonstrated that the main contribution to the mechanical stability of α-peptide 3.611 cannot come alone from the electrostatic dipole-dipole interaction potential of the anti-align excess dipole pairs but also from the surface Helmholtz free energy, which is characterized by a binding free energy of -15.5 eV/molecule (-32.56 Kcal/mole) for an alpha-peptide composed of 11 amino acid residues with a critical arc length of approximately 10 nm, assuming that the shear modulus is G = 1GPa and the surface Helmholtz specific free energy density is fs = 800 erg/cm2. This result was in excellent agreement with the experimental observations of the AH-1 conformation of (Glu)n Cys at pH 8. The present theory indicates that only two excess permanent anti-align dipole pairs for one α-Helical peptide molecule is requirement to stabilize the whole secondary structure of the protein that is exposed to heavy torsional deformation during the folding processes which amounts to 7.75 eV/molecule stored electrostatic energy compared to the interfacial Helmholtz free energy of -23.25 eV/molecule, which is exposed to hydrophobic environments.

Thermo-Magnetic properties of non-relativistic particles under the effect of energy-dependent Hellmann potential

This manuscript presents an analysis of the thermal and magnetic properties of particles moving in two dimensions under the influence of an external magnetic field and Aharanov-Bohm flux, considering an energy-dependent Hellmann potential. The energy eigenvalue is obtained using the asymptotic iteration method. Next, we model the system as a canonical ensemble and derive the partition function to acquire analytical expressions for the Helmholtz free energy, entropy, mean energy, and specific heat. We observe that an increase in the external magnetic field and AB flux generally increases the energy eigenvalues, with a stronger effect at lower flux values, while energy eigenvalues decrease with an increase in the screening parameter. Additionally, Helmholtz free energy and entropy functions are sensitive to temperature, particularly in the low-temperature regime, while specific heat exhibits a Schottky anomaly. We also compute the magnetisation and susceptibility functions, finding that both magnetisation and susceptibility increase with the external magnetic field strength and the AB flux.

First-principal analysis and design of TaPO5-based waveguides for photonic circuitry

With the rapid development of telecommunication systems, supercomputers, and large data centers, there is a growing need for advanced photonic materials to overcome the limitations of traditional photonic circuitry. This paper introduces monoclinic TaPO5, an indirect-bandgap semiconductor material, as a compact platform for photonic integrated circuits. TaPO5, with a bandgap of 2.148 eV, is analyzed for its structural, vibrational, thermodynamic, and optical properties using pseudo-potential plane waves within the density functional theory (DFT) framework. Our analysis reveals that TaPO5's thermodynamic properties, such as entropy, internal energy, and heat capacity, increase with temperature, while Helmholtz's free energy decreases. The materials demonstrate favorable optical characteristics, including a refractive index (n) of 2.11, an extinction coefficient (k) of 1.2 × 10⁻⁵, and a dielectric function with a real part of 4.45 and an imaginary part of 5.1 × 10⁻⁵ at a wavelength of 1.55 μm. TaPO5 exhibits promising performance metrics for photonic applications. Specifically, a proposed 1 × 2 multimode interference (MMI) splitter demonstrates 0.03 dB excess loss, a compact footprint of 4 × 18 μm2, and a 50:50 splitting ratio at 1.55 μm. These findings indicate that TaPO5 is a viable material for next-generation photonic integrated circuits, offering theoretical and practical advantages for various applications in photonic technology.

A thermodynamics-based coupled model for multi-phase flowback in deformable dual-porosity shale gas reservoirs

Over the past decades, there has been growing interest in modelling multi-phase flowback in shale gas reservoirs during hydraulic fracturing, primarily due to the significant risk of groundwater pollution. However, the lack of fully coupled simulations and a unified theoretical model has hindered the study of coupled adsorptive hydro-mechanical behaviour and its influence on fluid flowback prediction. In this paper, we have extended the non-equilibrium thermodynamics-based Mixture Coupling Theory to derive the constitutive relationship and governing equation for adsorptive multi-phase fluid flows in deformable dual-porosity media. Helmholtz free energy density has been used to establish the relationship between solids and fluids. The interaction of adsorptive fluids in the pore and fracture space has been fully considered, leading to a new form of constitutive equation, which obtained the molecular adsorptive effect on the rock by introducing adsorption entropy production. The proposed governing equations determine the fully coupled evolution of solid deformation and multi-phase flow, with the consideration of adsorption as well as pore and fracture porosity change. Numerical simulation is then performed to study the flowback of a real shale gas well and the influence of coupled behaviour on model prediction. The simulation shows that the flowback flux is mainly by the fracture (more than 99.9 %) and the overall cumulative volume is 2427.1 m3 within 225 h which accounts for about 5.3 % of the total injected fracturing fluid, and the sensitivity analysis reveals that the manual fracture porosity is the most sensitive factor to the cumulative flowback. This work provides new insights into the flowback process of shale reservoirs in a fully coupled view and the proposed theoretical tool should contribute to the modelling of other adsorptive multi-phase flow problem in deformable dual-porosity media such as carbon storage and coalbed methane production.

Fully polarized Fermi systems at finite temperature.

We propose a simple model of an interacting, fully spin-polarized Fermi gas in dimensions d=2 and d=3, and derive the approximate expression for the energy spectrum and the corresponding formula for the Helmholtz free energy. We analyze the thermodynamics of the system and find the lines of first-order phase transitions between the low- and high-density phases terminating at critical points. The properties of the corresponding phase diagrams are qualitatively different for d=2 and 3, and sensitively depend on the interparticle attraction, which marks a departure from the standard van der Waals theory. The differences originate from the Pauli exclusion principle and are embedded in the fermionic nature of the system under study.

Praseodymium doped ceria: A new sight on coupled thermo-mechanical properties and oxygen vacancy diffusion via statistical moment approach

Applications of praseodymium doped ceria (PDC) in solid oxide fuel cells require a deep insight into the thermal expansion, stresses, and ionic conduction. Nonetheless, the investigation of thermo-mechanical properties and oxygen vacancy diffusion remains incomplete. Based on free Helmholtz energy, we formulate analytical expressions to directly connect thermal expansion coefficient, elastic moduli, oxygen vacancy–dopant binding and oxygen vacancy migration energies by employing statistical moment method (SMM). The SMM calculations capture the full anharmonicity of lattice thermal vibrations. Our analyses report the significant influences of temperature and dopant content on the thermal expansion coefficient, bulk and Young’s moduli, oxygen vacancy–dopant binding and oxygen vacancy migration energies, and ionic conductivity. The role of Pr dopant on the oxygen vacancy diffusion is fully described by combining the symmetric energy profiles related to blocking effect and asymmetric energy profiles generated from trapping effect. By switching van der Waals (vdW) forces on and off in the SMM calculations, we first show that the vdW forces cause a positive effect on the mechanical properties but reduce the thermal expansion and resist the oxygen vacancy diffusion. Our findings are discussed concerning the results from other calculations and experiments in the literature. The insights on the thermo-mechanical properties and oxygen vacancy diffusion for PDC crystal can be generalized and applied to other ceramic materials that are essential for solid state batteries.

Polyconvex inelastic constitutive artificial neural networks

AbstractThe characterization of the material behavior of inelastic materials requires a high degree of expert knowledge to identify and constitutively describe the material response. In addition, specific models are usually pre‐selected in the course of characterization and only the best parameters for these specific models are determined, but therefore not necessarily the best models. Unfortunately, a more general description of these pre‐selected models results in an increased effort during characterization, which is barely practicable by hand. This is where machine learning algorithms may help us. To get the best of both worlds, powerful machine learning and sound thermodynamic considerations, inelastic Constitutive Artificial Neural Networks (iCANNs) discover generic formulations of the Helmholtz free energy and pseudo potential. Constitutive relations guide us towards thermodynamically consistent descriptions of stresses and inelastic strains; a concept that is applicable to a wide range of inelastic phenomena from viscoelasticity, elastoplasticity, phase transformations to growth and remodeling of living tissues. Here, we equip the original iCANN framework to guarantee polyconvexity a priori, which ensures at least one minimizing deformation. We investigate the ability of our iCANN to identify the material behavior of a viscoelastic polymer at different stretch levels and strain rates. We made our source code, data, and example accessible to the public at https://doi.org/10.5281/zenodo.11084354.

Optimización de la energía libre de Helmholtz a través de un programa en Visual Studio 2005 del método Monte Carlo Annealing

In the present work, we present the design of a program in Visual Studio 2005 implementing the Monte Carlo Annealing Method (MCA), to minimize the functional of the Helmholtz energy to study the Isotropic-Nematic phase transition. The program was made in Visual Studio 2005 using the C# programming language, this because it is an object-oriented language, easy interaction with databases, multiple mathematical functions and facilitates the creation of graphs, with this, it was possible to clearly observe the results produced by the program. In addition, by using C# it allows the application to be mounted on a server or Workstation, so that the program can be downloaded from the internet and, in addition, can be updated in real time. In this same context, Microsoft Access was used as the database manager.

Joule-Thomson expansion for charged-AdS black hole with nonlinear electrodynamics and thermal fluctuations by using Barrow entropy

This paper investigates the complex interplay between charged anti-de Sitter black hole thermodynamic behavior and nonlinear electrodynamics. Strong insights are obtained by examining the impact of nonlinear electrodynamics on the Joule-Thomson expansion of charged black holes and assessing the thermal fluctuations with higher-order corrections in the context of Barrow entropy. The research shows the complex interactions influencing thermal properties from Joule-Thomson coefficients are affected by temperature, pressure, and changes in gas composition due to the effects of charges and nonlinear electrodynamics factors on inversion temperature and pressure. Furthermore, analyzing black hole isenthalpic curves provides important insights into the thermodynamic behavior of the system by highlighting discrete heating and cooling zones in the inversion curve. Significant effects of nonlinear dynamics on the thermodynamic parameters of the system are highlighted by the notable variations in the Helmholtz free energy, internal energy, enthalpy, and Gibbs free energy of charged anti-de Sitter black holes. By demonstrating the complex and captivating nature of these interactions, the paper concludes by emphasizing the significant influence of nonlinear dynamics on the thermodynamic parameters of charged black holes within the framework of Barrow entropy.