Free Energy Function Research Articles

Modeling spider silk supercontraction as a hydration-driven solid-solid phase transition

Spider silks have attracted significant interest due to their exceptional mechanical properties, which include a unique combination of high strength, ultimate strain, and toughness. A notable characteristic of spider silk, still debated from both mechanical and functional viewpoints, is supercontraction–a pronounced contraction of up to half its original length when an unconstrained silk thread is exposed to a wet environment. We propose a predictive model for the hygro-thermo-mechanical behavior of spider silks, conceptualizing this phenomenon as a solid–solid phase transition, similar to the glass transition in rubber, but driven by humidity. As wetting increases, the system undergoes a transition, at the network scale, from a hard, highly crystalline, dry state–where the material behavior is governed by stiff chains elongated along the fiber axis–to a soft, amorphous, wet state, regulated by a rubber-like response. We model these states using a two-well free energy function dependent on molecular stretch, with transition energy modulated by humidity. Based on the methods of Statistical Mechanics, we deduce that supercontraction can be interpreted as a solid–solid phase transition. We elucidate the important role of thermal fluctuations. In particular, the decrease of the critical humidity needed for supercontraction as temperature grows results as an effect of entropic stabilization of the softer rubbery phase. Our model quantitatively predicts the observed experimental behavior, capturing the temperature dependence of humidity-induced supercontraction effects and related cooperative properties.

A dynamic entanglement model for adaptive networks in amorphous polymers with pH-responsive dual-shape memory effect

The pH-responsive shape memory polymers (pH-SMPs) have recently attracted significant attention due to their unique and spontaneous actuation capabilities. However, there are few constitutive models developed to explore the working principles behind these complex shape memory behaviors. In this study, a dynamic entanglement model was developed for describing the pH-responsive shape memory effect (SME) in SMPs, in which the crosslinking points in polymer networks underwent reversible entanglements and disentanglements. Susceptible-Infected-Susceptible (SIS) model was firstly employed to formulate an entanglement probability function, which was used to identify the working principles for entanglements of polymer networks and shape recovery of the pH-SMPs. An entanglement free-energy function was further formulated to characterize the pH-responsive dual-SMEs based on the Flory-Huggins solution theory. Phase transition theory was then used to characterize glass transition behaviors and recovery strains of the pH-SMPs, by combining Gordon-Taylor and Kohlrausch-Williams-Watts (KWW) equations. Finally, the proposed model was verified using experimental results reported in the literature. This study provides a fundamental approach to explore the working principle and constitutive relationship between reversible entanglement and pH-responsive SME in SMPs.

Lattice Fundamental Measure Theory Beyond 0D Cavities: Dimers on Square Lattices

Using classical density functional theory, we study the behavior of dimers, i.e. hard rods of length L=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L=2$$\\end{document}, on a two-dimensional cubic lattice. For deriving a free energy functional, we employ Levy’s prescription which is based on the minimization of a microscopic free energy with respect to the many-body probability under the constraint of a fixed density profile. Using that, we recover the "0D cavity" functional originally found by Lafuente and Cuesta and derive an extension by applying a more general "cluster density functional theory" method introduced by Lafuente and Cuesta as well. Moreover, we introduce a new free energy functional, which is based on approximated configurational probabilities. Both derived free energy functionals are exact on cavities that can hold at most two particles simultaneously. The first functional allows to improve the prediction of the free energy in bulk and both of them improve the prediction in highly confined systems, especially for high packing fractions, in comparison to the "0D cavity" functional.

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.

Free-energy landscape of a polymer in the presence of two nanofluidic entropic traps.

Recently, nanofluidics experiments have been used to characterize the behavior of single DNA molecules confined to narrow slits etched with arrays of nanopits. Analysis of the experimental data relies on analytical estimates of the underlying free-energy landscape. In this study we use computer simulations to explicitly calculate the free energy and test the approximations employed in such analytical models. Specifically, Monte Carlo simulations were used to study a polymer confined to complex geometry consisting of a nanoslit with two square nanopits embedded in one of the surfaces. The two-dimensional weighted histogram analysis method is used to calculate the free energy, F, as a function of the sum (λ_{1}) and the difference (λ_{2}) of the length of the polymer contour contained in the two nanopits. We find the variation of the free-energy function with respect to confinement dimensions to be comparable to the analytical predictions that employ a simplistic theoretical model. However, there are some noteworthy quantitative discrepancies, particularly between the predicted and observed variation of F with respect to λ_{1}. Our study provides a useful lesson on the limitations of using simplistic analytical expressions for polymer free-energy landscapes to interpret results for experiments of DNA confined to a complex geometry and points to the value of carrying out accurate numerical calculations of the free energy instead.

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.