Free Energy Function Research Articles

Solubility of letrozole in eight pure and five mixed solvents: Measurement, thermodynamic and molecular simulation analysis

In this work, the solubility of letrozole within eight single and five binary solvents was first systematically investigated at 283.15 K to 323.15 K via the gravimetric method. Subsequently, four thermodynamics formulas including modified Apelblat, λh, GSM and Jouyban-Acree were utilized to link the experimental outcomes of letrozole, all models achieved the good fitting performance and the ARD% was less than 2 %. The dissolvability of letrozole in chosen solvents exhibited a positive correlation with temperature, and co-solvency was observed in such mixed solvent systems as ethyl acetate + 1-propanol, and acetonitrile + 1-propanol. Furthermore, the sites of hydrogen bonding donor and acceptor of letrozole and solvents were analyzed through the molecular electrostatic potential surfaces (MEPs). Then solvent effect, solvation free energy and radial distribution function (RDF) analysis gained via molecular simulation were utilized to illuminate experimental phenomena, the outcomes manifested the polarity, cohesive energy density and other properties of solvents along with both solvent–solvent and solute–solvent interactions contributed differently to the dissolution processes. In the end, the apparent thermodynamic identities concerning letrozole within chosen solvent systems were computed under the van’t Hoff and Gibbs formulas, and outcomes showed that dissolution concerning letrozole was a procedure of endothermic and entropy mainly driven.

Assessment of Free Energy Functions for Sand

ABSTRACTThe advantages of constitutive models in energy‐conservation frameworks have been widely addressed in the literature. A key component is choosing an appropriate energy potential to derive the hyperelastic constitutive equations. This article investigates the advantages and limitations of different energy potentials found in the literature based on mathematical conditions to guarantee numerical stability, such as the desired order of homogeneity, positive and non‐singular stiffness within the application range, and equivalent Poisson's ratio from a constitutive modelling standpoint. Potentials meeting the aforementioned criteria are employed to simulate the response envelopes of Karlsruhe fine sand (KFS). Moreover, the performance of the potentials, in conjunction with plasticity theories, is examined. To achieve this, the hyperelastic constitutive equations have been coupled with the bounding surface plasticity model of Dafalias and Manzari to reproduce the soil response in a hyperelastic–plastic frame. Finally, one of the potentials is modified, whereas recommendations for incorporating other appropriate free energy functions into different soil constitutive models are presented. Furthermore, 100 closed elastic strain cycles have been simulated with the bounding surface plasticity model of Dafalias and Manzari considering the original hypoelastic stiffness and hyperelastic–plastic constitutive equations. Using the hypoelastic framework in the simulation led to stress accumulation after 100 closed elastic strain loops, while a reversible response was predicted using the hyperelastic stiffness tensor.

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.

ESCAPE: Energy-based Selective Adaptive Correction for Out-of-distribution 3D Human Pose Estimation

Despite recent advances in human pose estimation (HPE), poor generalization to out-of-distribution (OOD) data remains a difficult problem. While previous works have proposed Test-Time Adaptation (TTA) to bridge the train-test domain gap by refining network parameters at inference, the absence of ground-truth annotations makes it highly challenging and existing methods typically increase inference times by one or more orders of magnitude. We observe that (1) not every test time sample is OOD, and (2) HPE errors are significantly larger on distal keypoints (wrist, ankle). To this end, we propose ESCAPE: a lightweight correction and selective adaptation framework which applies a fast, forward-pass correction on most data while reserving costly TTA for OOD data. The free energy function is introduced to separate OOD samples from incoming data and a correction network is trained to estimate the errors of pretrained backbone HPE predictions on the distal keypoints. For OOD samples, we propose a novel self-consistency adaptation loss to update the correction network by leveraging the constraining relationship between distal keypoints and proximal keypoints (shoulders, hips), via a second “reverse” network. ESCAPE improves the distal MPJPE of five popular HPE models by up to 7% on unseen data, achieves state-of-the-art results on two popular HPE benchmarks, and is significantly faster than existing adaptation methods.

An efficient saddle search method for ordered phase transitions involving translational invariance

In this work, we propose an efficient nullspace-preserving saddle search (NPSS) method for a class of phase transitions involving translational invariance, where the critical states are often degenerate. The NPSS method includes two stages, escaping from the basin and searching for the index-1 generalized saddle point. The NPSS method climbs upward from the generalized local minimum in segments to overcome the challenges of degeneracy. In each segment, an effective ascent direction is ensured by keeping this direction orthogonal to the nullspace of the initial state in this segment. This method can escape the basin quickly and converge to the transition states efficiently. We apply the NPSS method to the phase transitions between crystals, and between crystal and quasicrystal, based on the Landau-Brazovskii and Lifshitz-Petrich free energy functionals. Numerical results show a good performance of the NPSS method.

Multi-scale modeling of damage evolution for particle-filled polymer composites

In this study, a multi-scale modeling framework that spans from molecular chains to macroscopic structure was proposed for a typical particle-filled polymer composite (HTPB propellants). The cohesive zone model (CZM) is utilized in the RVE model of HTPB propellants to capture the debonding phenomena at the AP/HTPB interface. For the HTPB matrix, a free energy function based on the Gaussian chain network is employed. To depict the chain scission behavior, the phase fracture (PF) method along with gradient-damage theory is introduced. Subsequently, microscale fracture behavior under uniaxial tensile load was investigated based on the constructed RVE model, and a phenomenological macroscopic damage model was developed correspondingly. In this developed model, two damage factors related to the debonding of AP/HTPB interface and the growth of voids in matrix are introduced respectively. Thus, it can not only predict the macroscopic stress–strain response, but also can give the microscopic damage evolution information. Overall, this multi-scale modeling framework can offer us a deeper insight into the microstructural changes and the resulting macroscopic mechanical behavior of HTPB propellants.

Phase-field modeling of interdiffusion between dissimilar Fe-Cr-Ni alloys during non-isothermal hot isostatic pressing

Powder metallurgy hot isostatic pressing (PM-HIP) has emerged as a promising alternative to welding for joining dissimilar metals. During HIP, interfacial bonding is mediated by solid state diffusion. The interdiffusion zone across the interface depends on processing conditions, calling for the need for accurate numerical tools capable of simulating interdiffusion and possible phase transformation in order to optimize processing parameters. Here, a phase-field (PF) model based on CALPHAD-based free energy functionals is developed to simulate the interdiffusion and phase evolution between dissimilar Fe–Cr–Ni based steels undergoing HIP and is demonstrated using the interface between 316L and SA508 steels. To overcome the numerical challenges caused by the singular magnetic and entropy terms in the CALPHAD free energy models in the Fe–Cr-Ni system, polynomial functions are fitted with temperature dependent coefficients represented by Fourier series to accurately describe the phase stability of both fcc and bcc phases in the composition and temperature space. This enables simulations of non-isothermal HIP cycles. Diffusivity data from commercial software and literature are taken to parameterize the kinetic parameters. A discrete nucleation model is incorporated for possible phase transformation. The modified thermodynamic models are validated against previous experiments at 923 K and 1273 K. The interdiffusion kinetics are benchmarked against new HIP experiments joining powder and bulk 316L to bulk SA508 with three different HIP cycles. The good agreement between simulations and experiments on both phase stability and interdiffusion indicate that the model is suitable for simulating interdiffusion between Fe–Cr–Ni alloys during HIP cycles. It is also found that using powder and bulk 316L gives similar interdiffusion profiles at elevated temperature when a dense interface forms during HIP.

Neural density functionals: Local learning and pair-correlation matching.

Recently, Dijkman etal. [arXiv:2403.15007] proposed training classical neural density functionals via bulk pair-correlation matching. We show their method to be an efficient regularizer for neural functionals based on local learning of inhomogeneous one-body direct correlations [Sammüller etal., Proc. Natl. Acad. Sci. USA 120, e2312484120 (2023)0027-842410.1073/pnas.2312484120]. While Dijkman etal. demonstrated pair-correlation matching of a global neural free-energy functional, we argue in favor of local one-body learning for flexible neural modeling of the full Mermin-Evans density-functional map. Using spatial localization gives access to accurate neural free-energy functionals, including convolutional neural networks, that transcend the training box.

Thermodynamic Stability at the Two-Particle Level.

We show how the stability conditions for a system of interacting fermions that conventionally involve variations of thermodynamic potentials can be rewritten in terms of one- and two-particle correlators. We illustrate the applicability of this alternative formulation in a multiorbital model of strongly correlated electrons at finite temperatures, inspecting the lowest eigenvalues of the generalized local charge susceptibility in proximity of the phase-separation region. Additionally to the conventional unstable branches, we address unstable solutions possessing a positive, rather than negative, compressibility. Our stability conditions require no derivative of free-energy functions with conceptual and practical advantages for actual calculations and offer a clear-cut criterion for analyzing the thermodynamics of correlated complex systems.

Framework for Laplacian-Level Noninteracting Free-Energy Density Functionals.

A framework for orbital-free Laplacian-level meta-generalized-gradient approximation (meta-GGA) for the noninteracting free-energy-density functionals based upon analysis of the fourth-order gradient expansion is developed. The framework presented here provides a new tool for developing advanced orbital-free thermal functionals at the meta-GGA level of theory. A nonempirical meta-GGA functional, which in the slowly varying density limit correctly reduces to the finite-temperature fourth-order gradient expansion for the noninteracting free energy, is constructed. Application to warm dense helium demonstrates that the developed meta-GGA functional drastically increases the accuracy of orbital-free-density functional theory simulations at temperatures below 40 eV as compared to the lower Thomas-Fermi and GGA rungs.

Extended rubber-elasticity model for non-Gaussian isotropic hyperelasticity of polyelectrolyte hydrogels undergoing curl-to-stretch transition

The nature of extremely high stretchability in hydrogels with a non-Gaussian elasticity has been extensively investigated. However, there are few studies on curl-to-stretch transitions of condensed polymer networks in polyelectrolyte hydrogels and their non-Gaussian hyperelasticity. In this study, an extended rubber-elasticity model was developed to investigate different mechanical behaviors, e.g., S-shaped and J-shaped stress-elongation ratio curves, where the condensed polymer networks in polyelectrolyte hydrogels undergo a curl-to-stretch transition and present a non-Gaussian isotropic hyperelasticity. A free-energy equation was formulated to describe the curl-to-stretch transitions and their thermomechanical conformations of condensed polymer networks, based on the free energy functions of elasticity, electrostatics and hydrophobicity. A constitutive relationship between stress and elongation ratio was then proposed to describe the non-Gaussian isotropic hyperelasticity in polyelectrolyte hydrogels, and effects of segment number, charged segment fraction and salt concentration on the mechanical behaviors were studied. Moreover, a segmental length criterion of condensed polymer networks was proposed to distinguish the S-shaped and J-shaped stress-elongation ratio curves. Finally, experimental data reported in literature were applied to verify the effectiveness of the proposed theoretical models. This study aims to provide a new thermomechanical criterion for non-Gaussian isotropic hyperelasticity in polyelectrolyte hydrogels undergoing the curl-to-stretch transition.

Physicist's view on the unbalanced k-cardinality assignment problem.

The k-cardinality unbalanced assignment problem asks for assigning k "agents" to k "tasks" on a one-to-one basis, while minimizing the total cost associated with the assignment, with the total number of agents N and the total number of tasks M possibly different and larger than k. While many exact algorithms have been proposed to find such an optimal assignment, these methods are computationally prohibitive when the problem is large. We propose an approach to solving the k-cardinality assignment problem using techniques adapted from statistical physics. This paper provides a full description of this formalism, including all the proofs of its main claims. We derive a strongly concave free-energy function that captures the constraints of the k-assignment problem at a finite temperature. We prove that this free energy decreases monotonically as a function of β, the inverse of temperature, to the optimal assignment cost, providing a robust framework for temperature annealing. We also prove that for large enough β values the exact solution to the k-assignment problem can be derived using simple round-off to the nearest integer of the elements of the computed assignment matrix. We show that this framework can be adapted to handle degenerate k-assignment problems. We describe a computer implementation of our framework that is optimized for the GPU parallel architecture, using the library CUDA. This implementation is found to be as efficient as state-of-the-art implementations of parallel Hungarian algorithms on generic assignment problems, and orders of magnitude faster than those algorithms for pathological assignment cases.

Tunable noninteracting free-energy density functionals for high-energy-density physics applications

In this work, we introduce the concept of a tunable noninteracting free-energy density functional and present two examples realized: (i) via a simple one-parameter convex combination of two existing functionals and (ii) via the construction of a generalized gradient approximation (GGA) enhancement factor that contains one free parameter and is designed to satisfy a set of incorporated constraints. Functional (i), constructed as a combination of the local Thomas–Fermi and a pseudopotential-adapted GGA for the noninteracting free-energy, has already demonstrated its practical usability for establishing the high temperature end of the equation of state of deuterium [Phys. Rev. B 104, 144104 (2021)] and CHON resin [Phys. Rev. E 106, 045207 (2022)] for inertial confinement fusion applications. Hugoniot calculations for liquid deuterium are given as another example of how the application of computationally efficient orbital-free density functional theory (OF-DFT) can be utilized with the employment of the developed functionals. Once the functionals have been tuned such that the OF-DFT Hugoniot calculation matches the Kohn–Sham solution at some low-temperature point, agreement with the reference Kohn–Sham results for the rest of the high temperature Hugoniot path is very good with relative errors for compression and pressure on the order of 2% or less.

Elastoplastic damage model and numerical implementation of nano‐silica incorporated concrete

AbstractAn energy‐based isotropic elastoplastic damage model is developed for investigating the elastoplastic damage responses and stress–strain relationships of nano‐silica incorporated concrete. The formulation employs a multiscale micromechanical framework to determine the effective elastic properties of composites at different scales. The stress–strain constitutive relation is derived by splitting the strain tensor into “elastic‐damage” and “plastic‐damage” parts while introducing the homogenized free potential energy function and the undamaged potential energy function. The elastoplastic damage response of the material is further characterized by elastic–plastic‐damage coupling. To construct realistic 3D three‐phase concrete mesostructures in numerical simulations, this paper introduces an encapsulation placement method that avoids particle overlap checking when placing aggregates. This methodology allows adjustments for the aggregate compactness as needed and enhances computational efficiency in concrete mesostructure construction. The numerical results of the modeling show good agreement with the experimental values in the open literature. Further, the influence of nano‐silica addition contents and ITZ (interfacial transition zone) thicknesses on the elastoplastic damage response of nano‐silica incorporated concrete are quantitatively and qualitatively investigated for the optimization of nano‐silica incorporated cementitious composites. The proposed model facilitates simulating and optimizing the mechanical characteristics of nano‐silica incorporated concrete and enhances the computational efficiency of 3D concrete modeling with the introduced encapsulation placement method.

A multiaxial multiscale compressive thermo-mechanical damage model for Fiber Reinforced Cementitious Composites (FRCC)

In the current study, a multiaxial multiscale compressive stress-strain model for Fiber Reinforced Cementitious Composites (FRCC) was innovatively developed. The proposed model was on the basis of the theory of thermodynamics, multiscale theory and internal variable theory. The free energy function and dissipation function were established to characterize the elastic and plastic deformation of FRCC, respectively. The multiscale behaviors of FRCC in meso-scale and macro-scale was taken into account. In meso-scale, the energy dissipation functions of fiber in tension and bond-slip deformation in the interface transition zone (ITZ) were established. The resistance to cracking of cementitious matrix through the fiber bond stress were taken into account. In macro-scale, the thermo-mechanical coupling contributions of plastic damage, thermal damage, yield surface and hardening evolution were creatively considered in the potential function within thermodynamics framework. The influence of types and volume fractions of fibers on the reinforcement mechanism was discussed. When the temperature is higher than the melting point of fiber, the fiber influence coefficients was employed. The sensitivity of yield surface was discussed. The numerical predictions of this developed model were carried out for validation. The nonlinear stress-strain responses of concrete were accurately captured at temperatures ranges from 20 °C to 800 °C compared with the experimental data, which indicates the accuracy and applicability of the proposed model.

A rational multiscale nonlinear constitutive model for freeze–thaw rocks under triaxial compression

The stability and durability of rocks in cold regions are significantly impacted by the degradation of mechanical properties caused by freeze–thaw (F–T) environment. In this work, we shall propose a rational multiscale nonlinear constitutive model based on thermodynamics, micromechanics, and fractional calculus theory to describe the complete deformation and failure process of F–T rocks under triaxial compression. The F–T rocks at the mesoscale are regarded as consisting of porous matrix and cracks, while porous matrix is composed of the micropores and elastic solid grains at the microscale. According to experimental observations, we assume the F–T action mainly causes micropores growth and cracks opening, and mechanical damage is resulted from the initiation and propagation of cracks. In this context, the effects of F–T and mechanical damage on effective elastic properties of rocks can be quantitatively analyzed by using the two-step Mori–Tanaka (M-T) homogenization method. After subtly deriving the total free energy function of F–T rocks under compression, we systematically develop specific criteria for describing open cracks closure deformation, mechanical damage evolution and frictional sliding induced plastic distortion. Note that to correctly capture the plastic deformation characteristics, the non-orthogonal flow rule based on fractional differential calculations is employed. Following that, analytical analyses and numerical implementation of the proposed model are conducted. The performance of the model is evaluated by the simulations with experimental data on two kinds of F–T rocks, and discussions on parameters sensitivity and effects of fractional order are followed.

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

Solutions to the Landau–Lifshitz–Gilbert equation in the frequency space: Discretization schemes for the dynamic-matrix approach

The dynamic matrix method addresses the Landau–Lifshitz–Gilbert (LLG) equation in the frequency domain by transforming it into an eigenproblem. Subsequent numerical solutions are derived from the eigenvalues and eigenvectors of the dynamic matrix. In this work we explore discretization methods needed to obtain a matrix representation of the dynamic operator, a fundamental counterpart of the dynamic matrix. Our approach opens a new set of linear algebra tools for the dynamic matrix method and expose the approximations and limitations intrinsic to it. Moreover, our discretization algorithms can be applied to various discretization schemes, extending beyond micromagnetism problems. We present some application examples, including a technique to obtain the dynamic matrix directly from the magnetic free energy function of an ensemble of macrospins, and an algorithmic method to calculate numerical micromagnetic kernels, including plane wave kernels. We also show how to exploit symmetries and reduce the numerical size of micromagnetic dynamic-matrix problems by a change of basis. This procedure significantly reduces the size of the dynamic matrix by several orders of magnitude while maintaining high numerical precision. Additionally, we calculate analytical approximations for the dispersion relations in magnonic crystals. This work contributes to the understanding of the current magnetization dynamics methods, and could help the development and formulations of novel analytical and numerical methods for solving the LLG equation within the frequency domain.

Dynamic large strain formulation for nematic liquid crystal elastomers

Liquid crystal elastomers (LCEs) are a class of materials which exhibit an anisotropic behavior in their nematic state due to the main orientation of their rod-like molecules called mesogens. The reorientation of mesogens leads to the well-known actuation properties of LCEs, i.e. exceptionally large deformations as a consequence of particular external stimuli, such as temperature increase. Another key feature of nematic LCEs is the capability to undergo deformation by constant stresses while being stretched in a direction perpendicular to the orientation of mesogens. During this plateau stage, the mesogens rotate towards the stretching direction. Such characteristic is defined as semisoft elastic response of nematic LCEs. We aim at modeling the semisoft behavior in a dynamic finite element method based on a variational-based mixed finite element formulation. The reorientation process of the rigid mesogens relative to the continuum rotation is introduced by micropolar drilling degrees of freedom. Responsible for the above-mentioned characteristics is an appropriate free energy function. Starting from an isothermal free energy function based on the small strain theory, we aim to widen it into the framework of large strains by identifying tensor invariants. In this work, we analyze the isothermal influence of the tensor invariants on the mechanical response of the finite element formulation and show that its space-time discretization preserves mechanical balance laws in the discrete setting.

A thermodynamically consistent physics-informed deep learning material model for short fiber/polymer nanocomposites

This work proposes a physics-informed deep learning (PIDL)-based constitutive model for investigating the viscoelastic–viscoplastic behavior of short fiber-reinforced nanoparticle-filled epoxies under various ambient conditions. The deep-learning model is trained to enforce thermodynamic principles, leading to a thermodynamically consistent constitutive model. To accomplish this, a long short-term memory network is combined with a feed-forward neural network to predict internal variables required for characterizing the internal dissipation of the nanocomposite materials. In addition, another feed-forward neural network is used to indicate the free-energy function, which enables defining the thermodynamic state of the entire system. The PIDL model is initially developed for the three-dimensional case by generating synthetic data from a classical constitutive model. The model is then trained by extracting the data directly from cyclic loading–unloading experimental tests. Numerical examples show that the PIDL model can accurately predict the mechanical behavior of epoxy-based nanocomposites for different volume fractions of fibers and nanoparticles under various hygrothermal conditions.