Local Free Energy Research Articles

Chemomechanical equilibrium at the interface between a simple elastic solid and its liquid phase.

Applying diffusion coupled deformation theory, we investigate how the elastic properties of a solid body are modified when forced to keep its chemical potential aligned with that of its melt. The theory is implemented at the classical level of continuum mechanics, treating materials as simple continua defined by uniform constitutive relations. A phase boundary is a sharp dividing surface separating two continua in mechanical and chemical equilibrium. We closely follow the continuum theory of the swelling of elastomers (gels) but now applied to a simple two phase one-component system. The liquid is modeled by a local free energy density defining a chemical potential and hydrostatic pressure as usual. The model is extended to a solid by adding a non-linear shear elastic energy term with an effective modulus depending on density. Imposing chemomechanical equilibrium with the liquid reservoir reduces the bulk modulus of the solid to zero. The shear modulus remains finite. The stability of the hyper-compressible solid is investigated in a thought experiment. A mechanical load is applied to a rectangular bar under the constraintof fixed lateral dimensions. The linear elastic modulus for axial loading is evaluated and found to be larger than zero, implying that the bar, despite the zero bulk modulus, can support a weight placed on its upper surface. The weight is stabilized by the induced shear stress. The density dependence of the shear modulus is found to be a second order effect reducing the density of the stressed solid (chemostriction).

The repetitive local sampling and the local distribution theory

AbstractMolecular simulation is a mature and versatile tool set widely utilized in many subjects. However, its methodology development has been struggling with a tradeoff between accuracy and speed, significant improvement of both is necessary to reliably substitute many expensive and laborious experiments in molecular biology and nanotechnology. Previously, the ubiquitous severe wasting of computational resources in molecular simulations due to repetitive local sampling was raised, and the local free energy landscape approach was proposed to address it. The core idea is to first learn local distributions, and followed by dynamic assembly of which to infer global joint distribution of a target molecular system. When compared with conventional explicit solvent molecular dynamics simulations, a simple and approximate implementation of this theory in protein structural refinement harvested acceleration of about six orders of magnitude without loss of accuracy. While this initial test revealed tremendous benefits for addressing repetitive local sampling, there are some implicit assumptions need to be articulated. Here, I present a more thorough discussion of repetitive local sampling; potential options for learning local distributions; a more general formulation with potential extension to simulation of near equilibrium molecular systems; the prospect of developing computation driven molecular science; the connection to mainstream residue pair distance distribution based protein structure studies; and the fundamental difference of the averaging strategy from potential of mean force. This more general development is termed the local distribution theory to release the limitation of strict thermodynamic equilibrium in its potential wide application in general soft condensed molecular systems.This article is categorized under: Molecular and Statistical Mechanics > Free Energy Methods Data Science > Artificial Intelligence/Machine Learning

Unusual stress and strain concentration behaviors at the circular hole of a large monodomain liquid crystal elastomer sheet

Liquid crystal elastomers combine the hyperelasticity of elastomers with the multi-functionality of liquid crystals and have emerged as an important class of soft active materials. Monodomain liquid crystal elastomers under loading exhibit the soft elastic behavior due to the stress induced director rotation and the resulting spontaneous strain. Here, we numerically study their stress and strain concentration behavior by considering the classical example, a large sheet with a small circular hole in the middle under uniaxial loading. The concentration behavior is found to be very different from regular elastomers. Firstly, the concentration factors are much bigger at both small and large strains. Secondly, the locations of the strain concentration may not coincide with that of the stress concentration. Detailed analysis of the director rotation and the resulting spontaneous strain around the free edge of the hole are shown to be the main causes for the unusual concentration behavior. Moreover, under a given strain, the stress level of the LCE sample, and therefore the free energy, is slightly lower than that of the neo-Hookean material, while the local free energy density on the hole edge is much bigger due to the severer concentrations. By considering material parameters obtained from various samples of polysiloxane and polyacrylate side-chain nematic elastomers, we find that their stress and strain concentration behaviors are qualitatively similar but quantitatively quite different. As a result, at a prescribed strain, the samples with a larger maximal spontaneous strain has severer stress and strain concentrations. The stress concentration factor difference scaled by a coupling constant that combines the effect of the two material parameters, r and a, increases as the semi-soft coefficient a decreases. Similar results are found for the scaled maximal spontaneous strains and the scaled strain concentration factor difference at small strains.

The local topological free energy of proteins

Protein folding, the process by which proteins attain a 3-dimensional conformation necessary for their function, remains an important unsolved problem in biology. A major gap in our understanding is how local properties of proteins relate to their global properties. In this manuscript, we use the Writhe and Torsion to introduce a new local topological/geometrical free energy that can be associated to 4 consecutive amino acids along the protein backbone. By analyzing a culled protein dataset from the PDB, our results show that high local topological free energy conformations are independent of sequence and may be involved in the rate limiting step in protein folding. By analyzing a set of 2-state single domain proteins, we find that the total local topological free energy of these proteins correlates with the experimentally observed folding rates reported in Plaxco et al. (2000).

Effect of the Pt concentration on microstructures of Ti-Pt alloys using the first-principles phase field method

The effect of the Pt concentration on microstructures of Ti-Pt alloys is studied by the first-principles phase field (FPPF) method, which enables us to investigate the evolution of microstructures from first principles without using thermodynamic empirical parameters. The method first constructs the local free energy, which is needed in the phase field simulation, by combining density functional theory, cluster expansion theory including atomic vacancies, and potential renormalization theory. Applying this FPPF method to Ti1−xPtx alloys at 800 ∘C, we have confirmed that increasing the Pt concentration as 5, 10, 15 and 20 wt% causes a drastic change in the microstructure of Ti-Pt alloys in accord with the experimental observation by Song et al. (Materials 7 (2014) 3990). In the case of the Pt concentration less than 10 wt%, the microstructure is lamellar- or wavy-shaped everywhere. When the Pt concentration is 15 and 20 wt%, spot-like Ti3Pt precipitates appear in the α-Ti phases with lamellar- or wavy-shaped patterns. These results are confirmed in both 2D and 3D simulations.

Energy Renewal: Isothermal Utilization of Environmental Heat Energy with Asymmetric Structures.

Through the research presented herein, it is quite clear that there are two thermodynamically distinct types (A and B) of energetic processes naturally occurring on Earth. Type A, such as glycolysis and the tricarboxylic acid cycle, apparently follows the second law well; Type B, as exemplified by the thermotrophic function with transmembrane electrostatically localized protons presented here, does not necessarily have to be constrained by the second law, owing to its special asymmetric function. This study now, for the first time, numerically shows that transmembrane electrostatic proton localization (Type-B process) represents a negative entropy event with a local protonic entropy change () in a range from −95 to −110 J/K∙mol. This explains the relationship between both the local protonic entropy change () and the mitochondrial environmental temperature (T) and the local protonic Gibbs free energy () in isothermal environmental heat utilization. The energy efficiency for the utilization of total protonic Gibbs free energy ( including ) in driving the synthesis of ATP is estimated to be about 60%, indicating that a significant fraction of the environmental heat energy associated with the thermal motion kinetic energy (kBT) of transmembrane electrostatically localized protons is locked into the chemical form of energy in ATP molecules. Fundamentally, it is the combination of water as a protonic conductor, and thus the formation of protonic membrane capacitor, with asymmetric structures of mitochondrial membrane and cristae that makes this amazing thermotrophic feature possible. The discovery of energy Type-B processes has inspired an invention (WO 2019/136037 A1) for energy renewal through isothermal environmental heat energy utilization with an asymmetric electron-gated function to generate electricity, which has the potential to power electronic devices forever, including mobile phones and laptops. This invention, as an innovative Type-B mimic, may have many possible industrial applications and is likely to be transformative in energy science and technologies for sustainability on Earth.

"Dividing and Conquering" and "Caching" in Molecular Modeling.

Molecular modeling is widely utilized in subjects including but not limited to physics, chemistry, biology, materials science and engineering. Impressive progress has been made in development of theories, algorithms and software packages. To divide and conquer, and to cache intermediate results have been long standing principles in development of algorithms. Not surprisingly, most important methodological advancements in more than half century of molecular modeling are various implementations of these two fundamental principles. In the mainstream classical computational molecular science, tremendous efforts have been invested on two lines of algorithm development. The first is coarse graining, which is to represent multiple basic particles in higher resolution modeling as a single larger and softer particle in lower resolution counterpart, with resulting force fields of partial transferability at the expense of some information loss. The second is enhanced sampling, which realizes “dividing and conquering” and/or “caching” in configurational space with focus either on reaction coordinates and collective variables as in metadynamics and related algorithms, or on the transition matrix and state discretization as in Markov state models. For this line of algorithms, spatial resolution is maintained but results are not transferable. Deep learning has been utilized to realize more efficient and accurate ways of “dividing and conquering” and “caching” along these two lines of algorithmic research. We proposed and demonstrated the local free energy landscape approach, a new framework for classical computational molecular science. This framework is based on a third class of algorithm that facilitates molecular modeling through partially transferable in resolution “caching” of distributions for local clusters of molecular degrees of freedom. Differences, connections and potential interactions among these three algorithmic directions are discussed, with the hope to stimulate development of more elegant, efficient and reliable formulations and algorithms for “dividing and conquering” and “caching” in complex molecular systems.

Variable-Temperature Electrospray Ionization for Temperature-Dependent Folding/Refolding Reactions of Proteins and Ligand Binding.

Stabilities and structure(s) of proteins are directly coupled to their local environment or Gibbs free energy landscape as defined by solvent, temperature, pressure, and concentration. Solution pH, ionic strength, cofactors, chemical chaperones, and osmolytes perturb the chemical potential and induce further changes in structure, stability, and function. At present, no single analytical technique can monitor these effects in a single measurement. Mass spectrometry and ion mobility-mass spectrometry play increasingly essential roles in studies of proteins, protein complexes, and even membrane protein complexes; however, with few exceptions, the effects of the solution temperature on the stability and structure(s) of analytes have not been thoroughly investigated. Here, we describe a new variable-temperature electrospray ionization (vT-ESI) source that utilizes a thermoelectric chip to cool and heat the solution contained within the static ESI emitter. This design allows for solution temperatures to be varied from ∼5 to 98 °C with short equilibration times (<2 min) between precisely controlled temperature changes. The performance of the apparatus for vT-ESI-mass spectrometry and vT-ESI-ion mobility-mass spectrometry studies of cold- and heat-folding reactions is demonstrated using ubiquitin and frataxin. Instrument performance for studies on temperature-dependent ligand binding is shown using the chaperonin GroEL.

First Steps towards Understanding the Non-Linear Impact of Mg on Calcite Solubility: A Molecular Dynamics Study

Magnesium (Mg2+) is one of the most common impurities in calcite and is known to have a non-linear impact on the solubility of magnesian calcites. Using molecular dynamics (MD), we observed that Mg2+ impacts overall surface energies, local free energy profiles, interfacial water density, structure and dynamics and, at higher concentrations, it also causes crystal surface deformation. Low Mg concentrations did not alter the overall crystal structure, but stabilised Ca2+ locally and tended to increase the etch pit nucleation energy. As a result, Ca-extraction energies over a wide range of 39 kJ/mol were observed. Calcite surfaces with an island were less stable compared to flat surfaces, and the incorporation of Mg2+ destabilised the island surface further, increasing the surface energy and the calcium extraction energies. In general, Ca2+ is less stable in islands of high Mg2+ concentrations. The local variation in free energies depends on the amount and distance to nearest Mg in addition to local disruption of interfacial water and the flexibility of surface carbonate ions to rotate. The result is a complex interplay of these characteristics that cause variability in local dissolution energies. Taken together, these results illustrate molecular scale processes behind the non-linear impact of Mg2+ concentration on the solubility of magnesium-bearing calcites.

Elastic–Plastic Fracture Propagation Modeling in Rock Fracturing via Punch Through Shear Test

Fracture initiation and propagation from a wellbore within a rock formation exhibit nonlinear and inelastic behaviors. When the rock material undergoes plastic deformation prior to failure, the classical Griffith theory is no longer valid. In this study, a variational phase-field approach is applied to model the inelastic behavior of granite rock in a punch through shear test. The rock failure and the fracture initiation and propagation during the loading was simulated and compared to the corresponding experimental investigations. In this numerical approach, the total local free energy is fully coupled with solid deformation and computes the plastic strain rate. The code is scripted in Multiphysics Object Oriented Simulation Environment (MOOSE). The model is shown capable of reproducing the evidenced phenomena from Punch Through Shear (PTS) test encompassing mixed mode fracture pattern Mode I, and Mode II. The numerical results show a good agreement in the stress–displacement curve with experimental data for the critical energy release rate of $${G}_{c}=600\mathrm{N}/\mathrm{m}$$ . Therefore, the granite sample’s fracture toughness for Mode II is calculated to be 4.85 $$\mathrm{MPa}\sqrt{\mathrm{m}}$$ at no confining pressure.

Uncertainty and anharmonicity in thermally activated dynamics

The atomic trajectories of condensed phase systems often reduce to long periods of thermal vibration interspersed by transitions between local free energy minima. The resultant rare event dynamics are exponentially sensitive to the catalog of available transitions, meaning incomplete models can make catastrophically erroneous predictions. This review summarises some recent efforts towards quantifying this uncertainty. I show that Bayesian methods can rigorously measure sampling incompleteness, be propagated to yield a quantified prediction uncertainty and autonomously manage massively parallel simulations. These methods allow uncertainty-controlled investigation of complex atomistic processes with minimal end-user supervision, facilitating high-throughput workflows. For individual transitions rates, I also show how the activation free energy can be evaluated with full treatment of anharmonic thermal vibrations. The developed methods, all freely available, are demonstrated on a wide range of challenging materials science problems.

Study on Ni-Ti alloys around equiatomic composition by the first-principles phase field method

Ni-Ti alloys around equiatomic composition, i.e., those with 45–55 at.% Ni, at 1200 K are studied by using the first-principles phase field (FPPF) method, which does not rely on any thermodynamic empirical parameter. There is an important tolerant aspect of the method, which does not account for the crystal structure at all but accounts for the value of the corresponding local free energy only. Here we show that this tolerance can be used to handle the correct NiTi2 phase in a tetrahedron approximation in cluster expansion theory. With the correct NiTi2 local free energy, the resulting microstructures are in good agreement with the experimental ones. Moreover, since the energy of the most stable structure is generally very close to the less stable structures, even if one assigned an incorrect crystal structure, for example, NiTi3 instead of the correct NiTi2, the method can still reproduce the experimental microstructures, although the shape of the resulting precipitates slightly changes from angular to round. We explicitly demonstrate this characteristic by comparing the results. These features may become a great advantage in predicting unknown alloys by the FPPF method. We also explicitly clarify that the inclusion of interstitial atoms into calculation causes almost no effect on the final microstructures in the present case.

Molecular free energy optimization on a computational graph.

Free energy is arguably the most important property of molecular systems. Despite great progress in both its efficient estimation by scoring functions/potentials and more rigorous computation based on extensive sampling, we remain far from accurately predicting and manipulating biomolecular structures and their interactions. There are fundamental limitations, including accuracy of interaction description and difficulty of sampling in high dimensional space, to be tackled. Computational graph underlies major artificial intelligence platforms and is proven to facilitate training, optimization and learning. Combining autodifferentiation, coordinates transformation and generalized solvation free energy theory, we construct a computational graph infrastructure to realize seamless integration of fully trainable local free energy landscape with end to end differentiable iterative free energy optimization. This new framework drastically improves efficiency by replacing local sampling with differentiation. Its specific implementation in protein structure refinement achieves superb efficiency and competitive accuracy when compared with state of the art all-atom mainstream methods.

2D isotropic-nematic transition in colloidal suspensions of ellipsoids.

Liquid crystals are important condensed matter systems for technological applications, as well as for fundamental studies. An important unresolved issue is the nature of the phase transition in a two-dimensional (2D) liquid crystal system. In contrast to numerous computational studies reported in the last few decades, there have been no convincing experiments to verify these numerical results. Anisotropic colloids provide an excellent experimental model system to study phase transitions, such as crystallization and glass transition in condensed matter physics with single particle resolution. However, using colloids to probe the two-dimensional liquid crystal transition remains a challenge, since the condensed anisotropic colloids usually become stuck in the metastable glassy state rather than approaching their equilibrium liquid crystal phase. Here we report a method of using an external magnetic field to assist a colloidal system of super-paramagnetic anisotropic particles to overcome the local free energy barriers in the metastable states and approach the equilibrium phase. The experiments demonstrate a 2D isotropic-nematic phase transition with increasing packing density. The effects of the anisotropy of the colloidal particles on the 2D isotropic-nematic transition are explored. Our experimental results are compared with those from previous computational work, and quantitative agreements are reached.

Influence of dark matter on phase transitions of XCDM black hole in braneworld based on the framework of GUP

In this paper, under the framework of generalized uncertainty principle (GUP), based on the quantum tunneling radiation, we discuss the influence of dark matter on thermodynamics and phase transition from the X-cold dark matter (XCDM) black hole in Braneworld. It turns out that the existence of the dark matter can give rise to some new relations such as local temperature–mass, heat capacity–mass and local free energy–local temperature, and dark matter can play an important role in phase transitions. It is worth emphasizing that the first-order phase transition, second-order and Hawking–Page-like phase transitions can be observed from the new phase diagrams, and the novel and interesting thermodynamics behavior has been presented under the influence of GUP and dark matter.

A Novel Approach for Understanding Heterogeneous Lithium-Ion Battery Aging Using Phase-Field Model

Aging due to capacity fade is a major cause of concern in the fast growing lithium-ion battery (LIB) industry. In particular, the evolution of Solid Electrolyte Interface (SEI) have been on the focus of battery scientist as one of the primary aging mechanism affecting the cycle life of the battery. According to the recent experimental studies, the formation of SEI possesses a heterogeneous morphology and it is attributed to the (i) anisotropic ionic mobility in the crystal and (ii) surface reactions governing the flux of ions across the electrode/electrolyte interface, depending on the local free energy difference (1, 2). In contrast to most previous study which utilise rough simplification on modelling of SEI growth, we employed a novel aging formulation that captures the heterogeneous multi-layered SEI morphology by including Cahn-Hilliard interstitial diffusion modelling approach coupled with Nernst-Planck and Butler-Volmer equations for mass transport and surface chemistry, respectively. Since the Cahn-Hilliard is able to captures the anisotropic surface intercalation reactions with depending the local free energy difference of active material, our numerical theory shows promising results for better explaining the cause of the heterogeneous SEI growths on active material surfaces than the classical diffusion-limited model. W. Huang, P. M. Attia, H. Wang, S. E. Renfrew, N. Jin, S. Das, Z. Zhang, D. T. Boyle, Y. Li and M. Z. Bazant, Nano letters, 19, 5140 (2019). P. Guan, L. Liu and X. Lin, Journal of The Electrochemical Society, 162, A1798 (2015). Figure 1

Nature of the glass transition in 2D colloidal suspensions of short rods

The glass transition, a many-body system undergoes an apparent dynamic arrest with no appearance of long-range order, is one of the most challenging problems in condensed matter physics. Recently attention has been focused on the puzzling two-step transition observed in colloidal suspensions of ellipsoids. It was observed that micro nematic domains appearing with increasing packing density suggesting that such micro domains may be the structure origin of the two-step glass transition. Here we report an experimental study of monolayers of colloidal rods which exhibits a two-step glass transition with no appearance of pseudo-nematic domains. Instead, it was found that parallel and perpendicularly packed rods form local free energy minima in configurational space, separated by an activation barrier. This barrier increases significantly when rotational glass transition is approached, thereby the rotational motion is frozen while the translational one remains diffusive. We propose that the activation barrier for rotation is the key to the two-step glass transition in suspensions of rods. Such an activation barrier between well-defined local configurations holds the key to understanding the two-step glass transition in general.

Role of substrate softness in stabilizing surface nanobubbles

The contact line pinning and supersaturation theory for the nanobubble stability has attracted extensive concerns from experimental investigators, and some experimenters argue that the contact line pinning is unnecessary. To interpret the experimental observations, we have proposed previously through molecular dynamics simulations that the deformation of soft substrates caused by surface nanobubbles may play an important role in stabilizing surface nanobubbles, while yet no quantitative theory is available for explanation of this mechanism. Here, the detailed mechanism of self-pinning-induced stability of surface nanobubbles is investigated through theoretical analysis. By manipulating substrate softness, we find that the formation of surface nanobubbles may create a deformation ridge nearby their contact lines which leads to the self-pinning effect. Theoretical analysis shows that the formation of nanobubbles on sufficiently rigid substrates or on liquid–liquid interfaces corresponds to a local free energy maximum, while that on the substrates with intermediate softness corresponds to a local minimum. Thus, the substrate softness could regulate the surface nanobubble stability. The critical condition for the self-pinning effect is determined based on contact line depinning, and the effect of gas supersaturation is explored. Finally, the approximate stability range for the surface nanobubbles is also predicted.

Synthesis of Metastable Inorganic Solids with Extended Structures.

The number of known inorganic compounds is dramatically less than predicted due to synthetic challenges, which often constrains products to only the thermodynamically most stable compounds. Consequently, a mechanism-based approach to inorganic solids with designed structures is the holy grail of solid state synthesis. This article discusses a number of synthetic approaches using the concept of an energy landscape, which describes the complex relationship between the energy of different atomic configurations as a function of a variety of parameters such as initial structure, temperature, pressure, and composition. Nucleation limited synthesis approaches with high diffusion rates are contrasted with diffusion limited synthesis approaches. One challenge to the synthesis of new compounds is the inability to accurately predict what structures might be local free energy minima in the free energy landscape. Approaches to this challenge include predicting potentially stable compounds thorough the use of structural homologies and/or theoretical calculations. A second challenge to the synthesis of metastable inorganic solids is developing approaches to move across the energy landscape to a desired local free energy minimum while avoiding deeper free energy minima, such as stable binary compounds, as reaction intermediates. An approach using amorphous intermediates is presented, where local composition can be used to prepare metastable compounds. Designed nanoarchitecture built into a precursor can be preserved at low reaction temperatures and used to direct the reaction to specific structural homologs.

An Active Biomechanical Model of Cell Adhesion Actuated by Intracellular Tensioning-Taxis

Cell adhesion to the extracellular matrix (ECM) is highly active and plays a crucial role in various physiological functions. The active response of cells to physicochemical cues has been universally discovered in multiple microenvironments. However, the mechanisms to rule these active behaviors of cells are still poorly understood. Here, we establish an active model to probe the biomechanical mechanisms governing cell adhesion. The framework of cells is modeled as a tensional integrity that is maintained by cytoskeletons and extracellular matrices. Active movement of the cell model is self-driven by its intrinsic tendency to intracellular tensioning, defined as tensioning-taxis in this study. Tensioning-taxis is quantified as driving potential to actuate cell adhesion, and the traction forces are solved by our proposed numerical method of local free energy adaptation. The modeling results account for the active adhesion of cells with dynamic protruding of leading edge and power-law development of mechanical properties. Furthermore, the morphogenesis of cells evolves actively depending on actin filaments alignments by a predicted mechanism of scaling and directing traction forces. The proposed model provides a quantitative way to investigate the active mechanisms of cell adhesion and holds the potential to guide studies of more complex adhesion and motion of cells coupled with multiple external cues.