Expression For Free Energy Research Articles

Two state model for the ML-BOP potential

The coarse-grained machine-learning derived ML-BOP model [Chan et al., Nature Commun. 10, 379 (2019)] provides a monoatomic representation of the water-water interaction potential in which orientational interactions are included as three-body contributions. Despite its simplicity, the model reproduces the phase diagram of water and its anomalies. Here, we show that a two-state Gibbs free energy expression – fitted simultaneously on the temperature and pressure dependence of the density and internal energy – predicts the existence of a liquid–liquid critical point, with critical parameters consistent with previous estimates. We also show that in this model: (i) while the low density liquid is pre-empted by crystal nucleation, the high-density liquid and its spinodal are accessible in numerical studies down to 100 K; (ii) crystallisation requires the presence of a local low density region. Thus, for densities larger than the critical density, spinodal decomposition (or nucleation of the low-density liquid) is a pre-requisite for ice nucleation.

Chemo-elasto-electro free energy of non-uniform system in the diffuse interface context

In the present work, we propose an alternative approach for deriving the free energy formulation of a non-uniform system. Compared with the work of Cahn and Hilliard (1958 J.Chem. Phys.28258-67), our approach provides a more comprehensive explanation for the individual energy contribution in a non-uniform system, including entropy, interaction energy, and internal energy. By employing a fundamental mathematical calculus, we reformulate the local composition within the interface region. Utilizing the reformulated local composition as well as classic thermodynamic principles, we establish formal expressions for entropy, interaction energy, and the internal energy, which are functions of both composition and composition gradients. We obtain a comprehensive free energy expression for a non-uniform system by integrating these energy density formulations. The obtained free energy expression is consistent with the formula type of Cahn and Hilliard and prodives more deeper physical interpretation. Moreover, using the same approach, we derive formulations for elastic energy and electric potential energy in a non-uniform system. However, the proposed approach encounters a limitation in the special case of a non-uniform fluid contacting a solid substrate. Due to the significant difference in the length scales between the solid-fluid and fluid-fluid interfaces, the wall free energy formulation based on the aforementioned concept is unsuitable for this multi-scale system. To address this limitation, we reformulate the wall free energy as a function of the average composition over the solid-fluid interface. Additionally, the previous derivation relies on an artificial treatment of describing the composition variation across the interface by a smooth monotone function, while the true nature of this variation remains unclear. By utilizing the concept of average composition, we circumvent the open question of how the composition varies across the interface region. Our work provides a thorough understanding for the construction of free energy formulations for a non-uniform system in condensed matter physics.

Free energy and extension of stiff polymer chains confined in nanotubes with diverse cross-sectional shapes

The statistical mechanics of stiff polymer chains confined within narrow tubes is a foundational topic in polymer physics, extensively analyzed in prior research. For cylindrical, rectangular, and slit-like confinements, the chains’ free energy and extension adhere to a scaling law consistent with the Odijk theory. While this scaling law may not apply to tubes with different cross-sectional geometries, there is a lack of research examining the behavior of stiff chains in tubes with intricate cross-sectional shapes. In this study, we investigate the partition function of a stiff chain confined within an elliptic tube using the path integral approach, deriving a deflection length in a concise closed form through dimensional analysis. This length scale facilitates straightforward expressions for the chain's free energy and extension. Notably, we discover a shape-independent property of these expressions applicable to tubes with a wide variety of cross-sectional geometries. Extensive numerical simulations are conducted using a biased chain-growth Monte Carlo method, incorporating the Pruned and Enriched Rosenbluth algorithm, to validate the theoretical predictions on the confinement free energy and extension of chains in tubes with differing shapes.

Microscopic derivation of the thin film equation using the Mori-Zwanzig formalism.

The hydrodynamics of thin films is typically described using macroscopic models whose connection to the microscopic particle dynamics is a subject of ongoing research. Existing methods based on density functional theory provide a good description of static thin films but are not sufficient for understanding nonequilibrium dynamics. In this work, we present a microscopic derivation of the thin film equation using the Mori-Zwanzig projection operator formalism. This method allows to directly obtain the correct gradient dynamics structure along with microscopic expressions for mobility and free energy. Our results are verified against molecular dynamics simulations for both simple fluids and polymers.

Multi-phase-field modeling of the dissolution behavior of stoichiometric particles on experimentally relevant length scales

The dissolution of stoichiometric particles within a melt plays a crucial role in various material processes. This study presents a comprehensive phase-field model to analyze the dissolution behavior of these stoichiometric particles under experimental conditions. Our approach addresses the classical phase-field challenges related to modeling stoichiometric compounds and scaling to experimentally relevant lengths in a multi-phase, multi-component context. To overcome the difficulties posed by stoichiometric compounds, we rederive the classical phase-field evolution equations for a multi-phase system, adopting a composition-independent free energy expression for the stoichiometric compound. Additionally, we extend Feyen’s high driving force model [Feyen and Moelans, Acta Materialia, 256 (2023)] to multi-component systems, allowing us to perform quantitative simulations for technologically relevant material systems at experimental length scales within a reasonable computing time. The model’s precision in capturing diffusion-controlled transformations, including dissolution, growth, and the Gibbs–Thomson effect, is validated against analytical solutions for a hypothetical system. The quantitative nature of the model is validated by applying it to the dissolution of Al2O3 particles in CaO–Al2O3–SiO2 slags. We break new ground by conducting three-dimensional simulations for a system size of 875μm×875μm×875μm, directly comparable to confocal scanning laser microscopy experiments, where previous models were limited to two-dimensional simulations and a system size of 2μm×2μm. This validation underscores the model’s proficiency to quantitatively describe the diffusion-controlled dissolution of Al2O3 at the experimentally relevant length scales.

Melting curves of MgSiO3 and CaSiO3 perovskites with cubic structure at extreme conditions

In this work, we derive the analytical expression of Helmholtz free energy, mean nearest neighbor distance between two atoms, lattice constant, and molar volume using statistical moment method (SMM). Then, we present how to determine the melting temperature of cubic perovskite at high pressures by combining SMM and work-heat equivalence principle. The results of the melting theory of cubic perovskite are applied to two perovskites MgSiO3 and CaSiO3 under extreme conditions of the Earth’s lower mantle. Our obtained melting curves of MgSiO3 up to 1400 GPa and CaSiO3 up to 136 GPa were compared with available experiment and other calculations and were in good agreement. The present study provides an effective theoretical tool for finding the melting curves of strongly anharmonic materials under extreme conditions. Our theoretical calculations would be helpful to clarify geological formation process at the Earth’s lower mantle.

Potential energy landscape of a coarse grained model for water: ML-BOP.

We quantify the statistical properties of the potential energy landscape for a recently proposed machine learning coarse grained model for water, machine learning-bond-order potential [Chan et al., Nat. Commun. 10, 379 (2019)]. We find that the landscape can be accurately modeled as a Gaussian landscape at all densities. The resulting landscape-based free-energy expression accurately describes the model properties in a very wide range of temperatures and densities. The density dependence of the Gaussian landscape parameters [total number of inherent structures (ISs), characteristic IS energy scale, and variance of the IS energy distribution] predicts the presence of a liquid-liquid transition located close to P = 1750 ± 100 bars and T = 181.5 ± 1 K.

Criticality Controlling Mechanisms in Nematic Liquid Crystals

We theoretically study the generic mechanisms that could establish critical behavior in nematic liquid crystals (NLCs). The corresponding free energy density terms should exhibit linear coupling with the nematic order parameter and, via this coupling, enhance the nematic order. We consider both temperature- and pressure-driven, order–disorder phase transitions. We derive a scaled effective free energy expression that describes how qualitatively different mechanisms enforce critical behavior. Our main focus is on the impact of nanoparticles (NPs) in homogeneous NP-NLC mixtures. We illustrate that in the case of pressure-driven phase changes, lower concentrations are needed to impose critical point conditions in comparison with pure temperature variations.

Drastic magnetic-field-induced chiral current order and emergent current-bond-field interplay in kagome metals.

In kagome metals, the chiral current order parameter [Formula: see text] with time-reversal-symmetry-breaking is the source of various exotic electronic states, while the method of controlling the current order and its interplay with the star-of-David bond order [Formula: see text] are still unsolved. Here, we reveal that tiny uniform orbital magnetization [Formula: see text] is induced by the chiral current order, and its magnitude is prominently enlarged under the presence of the bond order. Importantly, we derive the magnetic-field ([Formula: see text])-induced Ginzburg-Landau (GL) free energy expression [Formula: see text], which enables us to elucidate the field-induced current-bond phase transitions in kagome metals. The emergent current-bond-[Formula: see text] trilinear coupling term in the free energy, [Formula: see text], naturally explains the characteristic magnetic-field sensitive electronic states in kagome metals, such as the field-induced current order and the strong interplay between the bond and current orders. The GL coefficients of [Formula: see text] derived from the realistic multiorbital model are appropriate to explain various experiments. Furthermore, we discuss the field-induced loop current orders in the square lattice models that have been studied in cuprate superconductors.

Calculation of the entropy for hard-sphere from integral equation method

The Ornstein-Zernike integral equation method has been employed for a single-component hard sphere fluid in terms of the Percus-Yevick (PY) and Martynov-Sarkisov (MS) approximations. The virial equation of state, excess chemical potential, and free energy have been computed in both approximations. The excess entropy has been obtained with both thermodynamic relation and free energy expression for reduced densities of 0.1 to 0.9. It has been shown that the entropy values from thermodynamic relation in the MS approximation are better than those from the PY approximation, especially for high densities, and presents a reasonable comparison with available data in the literature, while the values from free energy expression in both approximations are not only close to each other but also comparable to accurate ones.

Development of an analytical exponential-6 equation of state through Monte Carlo simulations.

The exponential-6 (exp-6) potential is commonly used to model fluids at high densities. In this paper, I propose a new equation of state (EOS) in the form of an analytical expression for the excess Helmholtz free energy of an exp-6 fluid. The EOS is based on extensive Monte Carlo simulations and therefore combines the excellent accuracy of the simulations with the numerical efficiency of a polynomial expression. The mean relative error in compressibility factor and internal energy is 0.14% and 0.25% respectively, which is a significant improvement over statistical mechanical theories. The EOS was implemented into a thermochemical code in order to optimize gas parameters and evaluate its performance on pure gas data, shock compression and detonation properties. Predicted gas densities, heat capacities and speed of sound for pure gases were generally within experimental uncertainties at pressures up to 1GPa and temperatures above 300K. For polar molecules, a simple free energy correction was introduced which greatly improved accuracy at low temperature. Calculated shock Hugoniots showed excellent agreement with experimental values up to 150GPa and 10 000K, and the detonation performance was accurately predicted for a number of different types of explosives.

Non-Local Q-Tensor Approach for Description of Elastic Deformations of Nematic Liquid Crystals at Sub-Micron Scale

Within the framework of phenomenological approach, a generalized concept of a non-local order parameter, that have the form of a traceless correlation tensor function or a tensor integral operator, is introduced. The main relationships, which determine the equilibrium orientational states and the phase transitions of a deformed nematic liquid crystal, are described. The resulting expression for the free energy is quadratic with respect to the gradients of the order parameter tensor. The limiting transition of the non-local order parameter to the local one leads to an expression that complements the classical Oseen–Frank theory of elasticity. The considered model includes eight independent elastic constants. Three of them relate to "bulk" constants analogous to Frank constants, another three ones relate to "surface" constants, and the remaining two constants determine the anisotropic effect of the gradient of the scalar order parameter on the director field. The constants are necessary for the consideration of a liquid crystal state near defects, as well as for the description of orientation effects caused by the inhomogeneity of order parameter. It is shown that in the free energy expression, the "surface" terms contribute to the director orientation in the case of a non-constant scalar order parameter, even in the approximation of rigid boundary conditions.

Analytic Free-Energy Expression for the 2D-Ising Model and Perspectives for Battery Modeling

Although originally developed to describe the magnetic behavior of matter, the Ising model represents one of the most widely used physical models, with applications in almost all scientific areas. Even after 100 years, the model still poses challenges and is the subject of active research. In this work, we address the question of whether it is possible to describe the free energy A of a finite-size 2D-Ising model of arbitrary size, based on a couple of analytically solvable 1D-Ising chains. The presented novel approach is based on rigorous statistical-thermodynamic principles and involves modeling the free energy contribution of an added inter-chain bond ΔAbond(β,N) as function of inverse temperature β and lattice size N. The identified simple analytic expression for ΔAbond is fitted to exact results of a series of finite-size quadratic N×N-systems and enables straightforward and instantaneous calculation of thermodynamic quantities of interest, such as free energy and heat capacity for systems of an arbitrary size. This approach is not only interesting from a fundamental perspective with respect to the possible transfer to a 3D-Ising model, but also from an application-driven viewpoint in the context of (Li-ion) batteries where it could be applied to describe intercalation mechanisms.

Non-Quadratic Strain Gradient Plasticity Theory and Size Effects in Constrained Shear

Abstract A previously proposed strain gradient plasticity theory is extended to incorporate a non-quadratic power law function of the plastic strain gradient in the free energy expression with an exponent of N + 1. The values of N are taken to vary from N = 1 to N = 0. A simple shear problem of a metal layer between rigid boundaries is analyzed. Two stages of plastic deformation are considered. In stage I, the plastic strain is taken to be zero at the boundaries. Stage I ends when a specified magnitude of the plastic strain gradient is attained at the boundaries. In stage II, the magnitude of the plastic strain gradient at the boundaries is fixed at the specified value. With N = 0, a critical plastic strain gradient cannot be specified at the boundaries because the plastic strain gradient is infinite at the boundaries. The theory with N = 0 predicts a constant plateau stress immediately after initial yield, and the dependence of the plateau stress on the layer thickness can fit experimentally observed plateau stress values. However, with N = 0, a stress gap occurs between the initial yield stress and the plateau stress. The theory with 0 < N ≤ 1 and with stage II also can reproduce the experimentally observed dependence of the plateau stress on the layer thickness for any value of N in that range, with an appropriate value of critical plastic strain gradient at the boundaries. The solution for 0 < N ≤ 1 includes that for N = 0 as a limiting case.

Numerical analysis method for intra-stage non-equilibrium two-phase condensing flow in wet steam turbine and its application

To improve the numerical calculation performance of the intra-stage wet steam non-equilibrium condensing flow in a wet steam turbine, this paper proposed a numerically modified wet steam non-equilibrium condensation (NEC) model by considering the interphase heat and mass transfer in the Euler/Euler coordinate system. The model was developed based on nucleation kinetics, and the expressions of the critical droplet formation free energy and the nucleation rate were given. First, combining the experimental data, the non-isothermal effect nucleation model was modified by the surface tension coefficient method. Then, the low-pressure modified droplet growth model, which is suitable for describing any range of Knudsen numbers, was employed to calculate the droplet surface heat and mass transfer process, and the steam/liquid two-phase control equations and the turbulence model were presented. Meanwhile, the parameter transfer and coupling problem between the vapor and liquid phases was solved. Subsequently, the modified model and the original model were adopted to solve the wet steam condensing flow at the 1D Moses-Stein nozzle numerically, and the results showed that the modified model could calculate the distribution characteristics of pressure and droplet radius more accurately and fit the experimental results better. Afterward, the characteristicsof the non-equilibrium condensing flow of wet steam in the steam turbine cascade were investigated. Finally, combined with the experimental result on the pressure distribution in the steam turbine plane static cascade, the validity of the proposed numerically modified model was further verified, which provided a theoretical basis for the high-accuracy numerical analysis of the condensing flow in the nuclear power wet steam turbine cascade.

Substantial enhancement in magnetic and magnetodielectric properties of 0.7(Bi2Fe4O9)-0.3(La0.67Sr0.33MnO3) composite

In pursuit of room-temperature magnetoelectric materials for numerous technological applications various materials are vigorously explored. Such is the motivation behind the synthesis of 0.7(Bi2Fe4O9)-0.3(La0.67Sr0.33MnO3) composite. The phase purity and structural analysis are examined through Rietveld refinement of room temperature X-ray diffraction (XRD) pattern, which shows that the composite has ‘Pbam + Pbnm’ phases. X-ray photoelectron spectroscopy confirmed the presence of mixed valence state of magnetic ions i.e. Fe2+ (41%), Fe3+ (45%), Fe4+ (14%) and Mn2+ (88%), Mn3+ (12%) favouring superexchange and double exchange interactions contributing to enhanced magnetizations. The presence of weak ferromagnetism in the composite is confirmed by large irreversibility in ZFC-FC data, isothermal magnetization curves and Arrott plots throughout the wide temperature range (10–300 K). High value of coercivity (Hc) is observed with an increase in temperature which indicates magnetic anisotropy in the system. Along with that, a plausible magnetodielectric (MD) coupling is evidenced in temperature dependent dielectric (ε′) and tan loss. Later, it is verified through intrinsic MD results carried out at higher frequency. At room temperature, the MD effect is found to be ∼2%. A contrasting behaviour in magnetic field variation of MD% is at 10 K and 300 K. However, a butterfly-shaped MD curve is evidenced where MD% ∼0.21% at 10 K. Lastly, Landau free energy expression confirmed that MD effect emerges from the coupling term ‘γP2M2′ where γ is ∼1.6 × 10−2 (emu/g)−2 and ∼1.1 × 10−2 (emu/g)−2 at 10 K and 300 K respectively. Thus, the above findings highlight the significance of the composite as a potential room temperature magnetoelectric material for device applications.

Unification: Viscosity, permittivity, and hydration forces near an interface

In water-based solutions near interfaces, the hydration forces, dielectric permittivity, and viscosity depend on the short-range order in the arrangement of water molecules. The former two quantities were earlier rationalized by employing the Ginzburg–Landau expression for the solution free energy. Driven by the spirit of unification, we show that the dependence of the viscosity coefficient on the coordinate perpendicular to the interface can be rationalized in this framework as well by using in addition a linear Evans–Polanyi correlation between the activation energy determining viscosity and the solution free energy. In the conventional hydrodynamics, this effect is often formally described by introducing the partial-slip boundary conditions or a stagnant layer near interfaces. We show how the corresponding slip or stagnant-layer length can be explained and quantified.

Electron-Phonon Interaction Contribution to the Total Energy of Group IV Semiconductor Polymorphs: Evaluation and Implications.

In density functional theory (DFT)-based total energy studies, the van der Waals (vdW) and zero-point vibrational energy (ZPVE) correction terms are included to obtain energy differences between polymorphs. We propose and compute a new correction term to the total energy, due to electron-phonon interactions (EPI). We rely on Allen's general formalism, which goes beyond the quasi-harmonic approximation (QHA), to include the free energy contributions due to quasiparticle interactions. We show that, for semiconductors and insulators, the EPI contributions to the free energies of electrons and phonons are the corresponding zero-point energy contributions. Using an approximate version of Allen's formalism in combination with the Allen-Heine theory for EPI corrections, we calculate the zero-point EPI corrections to the total energy for cubic and hexagonal polytypes of carbon, silicon and silicon carbide. The EPI corrections alter the energy differences between polytypes. In SiC polytypes, the EPI correction term is more sensitive to crystal structure than the vdW and ZPVE terms and is thus essential in determining their energy differences. It clearly establishes that the cubic SiC-3C is metastable and hexagonal SiC-4H is the stable polytype. Our results are consistent with the experimental results of Kleykamp. Our study enables the inclusion of EPI corrections as a separate term in the free energy expression. This opens the way to go beyond the QHA by including the contribution of EPI on all thermodynamic properties.

Rigorous formalism for unconventional symmetry breaking in Fermi liquid theory and its application to nematicity in FeSe

Unconventional symmetry breaking due to nonlocal order parameters has attracted considerable attention in many strongly correlated metals. Famous examples are the nematic order in Fe-based superconductors and the star-of-David charge density order in kagome metals. Such exotic symmetry breaking in metals is a central issue of modern condensed matter physics, while its theoretical foundation is still unclear in comparison with the well-established theory of superconductivity. To overcome this difficulty, here we introduce the "form factor" that generalizes the nonlocal order parameter into the Luttinger-Ward (LW) Fermi liquid theory. We then construct a rigorous formalism of the "density-wave equation" that gives the thermodynamically stable form factor, similarly to the superconducting-gap equation. In addition, a rigorous expression of the Ginzburg-Landau free-energy for the unconventional order is presented to calculate various thermodynamic properties. In the next stage, we apply the derived formalism to a typical Fe-based superconductor FeSe, by using the one-loop LW function that represents the free-energy gain due to the interference among paramagnons. The following key experiments are naturally explained: (i) Lifshitz transition (=disappearance of an electron-pocket) due to the bond+orbital order below $T_c$. (ii) Curie-Weiss behavior of the nematic susceptibility at higher T, and the deviation from the Curie-Weiss behavior at lower T near the nematic quantum-critical-point. (iii) Scaling relation of the specific heat jump at $T_c$, $\Delta C/T_c \propto T_c^b$ with $b \sim 3$. (Note that b=0 in the BCS theory.) These results lead to a conclusion that the nematicity in FeSe is the bond+orbital order due to the "paramagnon interference mechanism". The present theory paves the way for solving various unconventional phase transition systems.

Study on density and lattice compression of Al, Cu and AlCu in high temperature and pressure from statistical moment method

We derive analytic expressions of Helmholtz free energy, crystal parameters, nearest neighbor distance and mean nearest neighbor distance between two atoms, volume and equation of state of metal and substitutional alloy with FCC structure on the basis of the statistical moment method. We perform numerical calculations of density and ratio of volume at pressure [Formula: see text] to volume at zero pressure for metals Al, Cu and substitutional alloy AlCu at temperature up to 1500 K and pressure up to 350 GPa when using the Mie–Lennard–Jones [Formula: see text] potential and the coordination sphere method. Our calculation results are compared with the experimental data, other theoretical calculations and molecular dynamics simulations.