Configurational Entropy Contribution Research Articles

CALPHAD modeling of [formula omitted]- carbide dual ordering in Fe-Al-C ternary alloys

A novel four-sublattice model for the κ phase, denoted as (Fe, Al)3(Fe, Al)1(C, Va)1(C, Va)3 was proposed to improve the thermodynamic prediction, such as equilibrium composition, phase stability of κ-carbide in Fe-Al-C system. The sublattice model explains the transformation from the disordered FCC solid solution to the ordered κ-carbide via concurrent ordering of substitutional and interstitial atoms. The dual ordering model can restrict the irregular contribution of configurational entropy arising at 20 at% C composition, which is an issue with the existing thermodynamic databases. For the CALPHAD assessment, κ-carbide was considered as a single, individual phase that is in equilibrium with the liquid, austenite (γ), ferrite (α) or other intermetallic and carbide phases in the Fe-Al-C system. The formation energy calculated from density functional theory (DFT) showed that Fe3Al–L12 phase is energetically more favorable than the Fe3AlC–E21 phase, and C atoms in sublattice IV are not energetically favorable at all. The assessed parameters provided better accuracy than the existing database in the calculations of isothermal sections, liquidus projection, invariant reactions, and low-temperature phase compositions. The model is highly suitable for the low temperature (<800 °C) phase predictions. Thus, the improved Fe-Al-C model lays the foundation for the thermodynamic and kinetic studies of κ-carbide for designing new Fe-Mn-Al-C alloys and optimizing the heat treatment processes.

Lattice dynamics and free energies of Fe–V alloys with thermal and chemical disorder

Molecular dynamics simulations of Fe-V binary alloys with body-centered cubic as the underlying lattice were performed using a classical potential for chemically ordered and disordered states at finite temperatures for a common set of volumes. The equation of state was fitted to the computational data to obtain temperature- and chemical-order-dependent state functions via the Moruzzi-Janak-Schwarz approximation. Additionally, vibrational entropies that account for both thermal and chemical disorder were calculated for the equiatomic compositions from phonon density-of-states curves computed using effective force constants obtained from fits to the simulations. The latter predicts that the vibrational entropy at room temperature at equiatomicity is higher for the ordered phase than for the solid solution, a peculiar behavior previously observed experimentally. The internal energy of mixing favors ordering at all compositions, with a maximum at equiatomicity that decreases as the solute concentration decreases. The configurational entropy contribution to the free energy of mixing is almost entirely responsible for the stability of the high-temperature disordered phase.

Enhanced stability of triple-halide perovskites CsPbI3-x-yBrxCly (x and y = 0-0.024): understanding the role of Cl doping from ab initio calculations.

Doping of chloride in mixed iodide-bromide perovskites has been shown experimentally to suppress the photo-induced halide-ion segregation and enhance the stability of triple-halide perovskites (THP). However, a fundamental understanding of the effects of Cl doping is yet to be achieved especially when the doping concentration is low. Here we report the results of a state-of-the-art ab initio study of the atomic structure of THP by considering small doping concentrations of Br and Cl in CsPbI3. We find a reduction in the Pb-I bond lengths and tilting of PbI6 octahedra with Cl doping which lead to exothermic heat of mixing and therefore higher stability of THP. Moreover, using quasi-chemical approximation, our results show that there is a very small contribution of configurational entropy to Gibbs free energy at such low doping concentrations and at the operational temperature of 50 °C. This suggests that the favorable heat of mixing value is more important for the stability at low doping concentrations of Cl while a higher concentration of Cl increases the risk of halide segregation. Further calculations on Frenkel defect formation energy of I or Br-interstitial shows that the doping of Cl in I/Br mixed binary-compounds hinders the formation of Frenkel defects. These results support experiments and help to understand the role of chloride in suppressing the halide ion mobility with only a slight increase in the band gap. Accordingly, the THPs manifest a promising pathway for developing single-phase perovskites for solar cells and light-emitting diodes with improved performance and enhanced stability.

A unified approach for calculating free energies of liquid and defective crystals based on thermodynamic integration.

Free energy calculation is fundamentally important in the research of physics, chemistry, and materials. Thermodynamic integration is the most common way to estimate free energies. In the research, we proposed a unified approach using atomic simulations to calculate the free energies of liquid and defective crystals. The new approach is based on thermodynamic integration using two alchemical pathways. Softcore potentials are developed for three-body interatomic potentials to realize the alchemical pathways. Employing the new approach, the free energy of the liquid can be calculated without requiring another reference system. The free energy of the defective crystal can be calculated directly at high temperatures. It avoids the singularity at the integration endpoint caused by the defect diffusion, which is a serious problem in the widely used Einstein crystal method. In addition, the new approach can capture the whole free energy of the defective crystal including the contribution of anharmonic and configurational entropy, which are particularly important at high temperatures. The new method is simple yet effective and can be extended to different materials and more complex liquid and defective crystal systems.

Free‐Energy Parameterization and Thermodynamics in Si–Ge–Sn Alloys

Different models are used to describe and parameterize enthalpy and free energy in GeSnSi alloys over the entire composition range from density‐functional theory calculations and are used to explore if concentrated alloys in the equiatomic concentration range are thermodynamically feasible. It is found that the energetics are very well behaved and follow the expressions of a ternary regular solution rather closely. While a small asymmetry is found in the ideal solution energetics for the three binaries, it is found that the assumption of a symmetric ideal solution as underlying the quasichemical model does not change the general findings. All free energy contributions within this work are parameterized in different approximations, providing a complete parameter set that can be used for further thermodynamic studies of the GeSnSi alloy system. In the obtained results, the enthalpy of mixing of the SiSn system has rather large values, which is maintained more or less unchanged in the ternary system, making a concentrated ternary alloy thermodynamically unfavorable. Calculating the full free energy from adding configurational and vibrational entropy contributions does not change the overall situation.

Ab initio prediction of vacancy energetics in HCP Al-Hf-Sc-Ti-Zr high entropy alloys and the subsystems

We perform ab initio density-functional-theory (DFT) calculations to investigate the vacancy formation and migration energies in HCP Al-Hf-Sc-Ti-Zr high entropy alloys (HEAs) and their subsystems from binaries to quinaries. For the vacancy formation properties, we utilize the special quasi-random structure approach in conjunction with a statistical analysis, from which temperature-dependent formation Gibbs energies as well as averaged atomic environments can be extracted. We show that the temperature-dependent vacancy formation Gibbs energy due to configurational excitations has a negative configurational entropy contribution. For the vacancy migration barriers, obtained from the nudged elastic band method, we apply the local cluster expansion technique to the kinetically resolved activation barriers to explore extended vacancy migration phase spaces. The local chemical environment effect and general trends in terms of the “high entropy” effect are analyzed.

Tuning cellular uptake of nanoparticles via ligand density: Contribution of configurational entropy

The bioactivity of nanoparticles (NPs) crucially depends on their ability to cross biological membranes. A fundamental understanding of cell-NP interaction is hence essential to improve the performance of the NP-based biomedical applications. Although extensive studies of cellular uptake have converged upon the idea that the uptake process is mainly regulated by the elastic deformation of the cell membrane or NP, recent experimental observations indicate the ligand density as another critical factor in modulating NP uptake into cells. In this study, we propose a theoretical model of the wrapping of an elastic vesicle NP by a finite lipid membrane to depict the relevant energetic and morphological evolutions during the wrapping process driven by forming receptor-ligand bonds. In this model, the deformations of the membrane and the vesicle NP are assumed to follow the continuum Canham-Helfrich framework, whereas the change of configurational entropy of receptors is described from statistical thermodynamics. Results show that the ligand density strongly affects the binding energy and configurational entropy of free receptors, thereby altering the morphology of the vesicle-membrane system in the steady wrapping state. For the wrapping process by the finite lipid membrane, we also find that there exists optimal ligand density for the maximum wrapping degree. These predictions are consistent with relevant experimental observations reported in the literature. We have further observed that there are transitions of various wrapping phases (no wrapping, partial wrapping, and full wrapping) in terms of ligand density, membrane tension, and molecular binding energy. In particular, the ligand and receptor shortage regimes for the small and high ligand density are, respectively, identified. These results may provide guidelines for the rational design of nanocarriers for drug delivery.

Modeling the high-temperature phase coexistence region of mixed transition metal oxides fromab initiocalculations

This article describes how to calculate the solubility gap of a complex material containing two types of substitution sites, and including configurational, magnetic and vibrational entropy contributions, from first principles.

Thermodynamics of high entropy oxides

With the hype of “high entropy” alloys and more recently, “high entropy” ceramics and “high entropy” oxides (HEOs), there has been a great push to investigate and characterize systems with 5 or more components. This push has been extremely beneficial for the materials community as it has led to the development of many new systems with targeted applications. However, with our desire to find “new” and “exotic” materials, we have not spent enough time to step back and think deeply about the fundamental thermodynamic constraints that will guide design of future HEOs. Here, we present data-driven discussions with examples that have been collected from the fields of geology and materials science over the past 50 years to highlight critical thermodynamic parameters and principles that can be used for the design of HEOs. The goal of HEOs is to push the limit of the number of components in a single-phase solid solution to achieve unique and tunable properties. True single-phase HEOs are stabilized if the positive entropy of formation more than compensates an unfavorable enthalpy of formation above some critical temperature, making the overall ΔGf negative i.e. the HEO phase is “entropy stabilized”. Under ideal mixing, the number of components in a solid solution does not affect the solubility of an additional component. In real systems, the types of additional components, their structural transformations, and their associated non-ideal interactions influence the solubility limit. Non-ideal interactions can lead to short- or long-range ordering that decreases the overall configurational entropy. Due to the ionic-covalent nature of oxides, this ordering is the norm, not the exception. In the limited cases where mixing is ideal, charge coupled substitutions can work to influence overall configurational entropy contributions due to unique crystallographic sites. Long-range ordering can be minimized by mixing oxide components that have similar charge or are isostructural. Most excitingly, is the realization that surface energies will drastically affect the stability of oxide polymorphs and solubility limits. Thus, nano-materials are an interesting and novel approach that will vastly extend the HEO engineering space. As one can see, there are many avenues for the design and development of “HEOs” that thermodynamics will allow, even though they all may not be driven explicitly by entropy.

Calorimetric Signature of Deuterated Ice II: Turning an Endotherm to an Exotherm.

Calorimetric studies on ice II reveal a surprising H2O/D2O isotope effect. While the ice II to ice Ic transition is endothermic for H2O, it is exothermic for D2O samples. The transition enthalpies are +40 and −140 J/mol, respectively, where such a sign change upon isotope substitution is unprecedented in ice research. To understand the observations we employ force field calculations using two water models known to perform well for H2O ice phases and their vibrational properties. These simulations reveal that the isotope effect can be traced back to zero-point energy. q-TIP4P/F fares better and is able to account for approximately three-fourths of the isotope effect, while MB-pol only catches approximately one-third. Phonon and configurational entropy contributions are necessary to predict reasonable transition enthalpies, but they do not have an impact on the isotope effect. We suggest to use these calorimetric isotope data as a benchmark for water models.

High entropy alloys: Key issues under passionate debate

The present Viewpoint set aims at providing a summary of the recent advancements in the fundamental understanding of high entropy alloys (HEAs) as well as igniting new ideas and activities in this rapidly evolving field of use-inspired basic research. The universality of the core effects in HEAs, ranging from configurational entropy contributions to cocktailing effects are still under a passionate debate and in particular the peer-reviewed articles are meant to provide original perspectives. The various contributions are strongly opinion-based in a variety of areas including diffusion, phase transformations, deformation behavior, corrosion, metastability, structural as well as functional properties. In addition, the impact of the original metallic HEAs onto the field of oxides and ceramics has been illustrated and the role of entropy in high-entropy oxides is critically discussed.

Challenges in design and development of high entropy alloys: A thermodynamic and kinetic perspective

High entropy alloys (HEAs) have brought a renewed interest to thermodynamics and kinetics. Proper understanding and treatment of relevant quantities are important for the effective theoretical design framework. Since the ideal mixing treatment is useful, its validity is evaluated through checking the Lupis and Elliot effect on extensive binary data and temperature dependency effect on limited HEA data. The physical basis of different configurational entropy contributions and mismatch entropy are elucidated. New alloy possibilities such as immiscible HEAs are revealed. Influence of elemental atmospheres and crystalline nature of matrices on diffusion in HEAs are discussed.

Coupling Supervised Molecular Dynamics (SuMD) with Entropy Estimations To Shine Light on the Stability of Multiple Binding Sites.

Exploring at the molecular level, all possible ligand-protein approaching pathways and, consequently, identifying the energetically favorable binding sites is considered crucial to depict a clear picture of the whole scenario of ligand-protein binding. In fact, a ligand can recognize a protein in multiple binding sites, adopting multiple conformations in every single binding site and inducing protein modifications upon binding. In the present work, we would like to present how it is possible to couple a supervised molecular dynamics (SuMD) approach to explore, from an unbound state, the most energetically favorable recognition pathways of the ligand to its protein, with an enthalpic and entropic characterization of the most stable ligand-protein bound states, using the protein kinase CK2α as a prototype study. We identified two accessory binding pockets surrounding the ATP-binding site having a strong enthalpic contribution but a different configurational entropy contribution, suggesting that they play a different role.

Chemical inactivity of GaN(0001) surface – The role of oxygen adsorption – Ab initio picture

Density Functional Theory (DFT) calculations were used to determine adsorption of oxygen at GaN(0001), i.e. Ga-terminated surface. It was shown that at low coverage the oxygen molecule dissociates during adsorption so that the two separate O adatoms are located in H3 sites. Oxygen adatom saturates three Ga broken bonds, modifying their energy by overlap with Op states, so that the three states are degenerate with valence band (VB). The electron counting rule (ECR) indicate on the electron surplus, the excess electrons are donated to other Ga broken bond states, the adsorption energy is equal to 3.74 eV/atom for clean surface. At the first critical coverage θO=516ML, the Fermi level is shifted to conduction band while at the second critical coverage θO=38ML it is shifted down to VBM. The adsorption energy is ΔE12O2(N)=−3.67eV for θO≤724ML, for θO=1548ML and θO=1648ML decreases ΔE12O2(N)=−3.51eV and ΔE12O2(N)=−3.31eV, respectively, for θO=1748 the energy jumps to ΔE12O2(N)=−3.60eV, and for higher coverage θO≥38 the energy rapidly decreases to zero and becomes negative The singular point at θO=1748ML is essential for stability of oxygen coverage of the surface. The equilibrium pressure at low coverage is 10−5 bar for 1500 K and 10−12 bar 1000 K. It is reduced for higher coverage, due to reduction of the energy and configurational entropy contributions. At the coverage θO=1748ML the pressure is reduced by several orders of magnitude, indicating extremely high thermodynamic stability of such coverage, which is responsible for chemical inactivity of GaN(0001) surface observed in experiments, the critical factor for mechano-chemical polishing of the substrates for electronic applications.

Entropic Trapping of a Singly Charged Molecule in Solution.

We demonstrate the ability to confine a single molecule in solution by spatial modulation of its local configurational entropy. Previously we established electrostatic trapping of a charged macromolecule by geometric tailoring of a repulsive electrical interaction potential in a parallel plate system. However, since the lifetime of the trapped state depends exponentially on the electrical charge of the molecule, the electrostatic interaction alone is often insufficient in magnitude to stably confine molecules carrying a net charge of magnitude ≤5 e. Here we show that the configurational entropy of a thermally fluctuating molecule in a geometrically modulated system can be exploited to spatially confine weakly charged molecules in solution. Measurement of the configurational entropy contribution reveals good agreement with theoretical expectations. This additional translational contribution to the total free energy facilitates direct optical imaging and measurement of the effective charge of molecules on the size scale of ∼1 nm and a charge as low as 1 e, physical properties comparable with those of a monovalent ion in solution.

Modeling the Thermodynamic Properties of Solutions Containing Polymer and Electrolyte with New Local Composition Model

A new theory model based on the local composition concept (TNRF-modified NRTL (TNRF-mNRTL) model) was developed to express the short-range contribution of the excess Gibbs energy for the solutions containing polymer and electrolyte. An equation represented the activity coefficient of solvent was derived from the proposed excess Gibbs energy equation. The short-range contribution of interaction along with the long-range contribution of interaction and configurational entropy of mixing were used to correlate the activity coefficient of ternary polymer + electrolyte + water systems and also binary polymer + water and electrolyte + water systems. The long-range interaction and configurational entropy have been given by the Pitzer-Debye-Huckel equation and the Flory-Huggins relation, respectively. The performance of the proposed model in fitting the solvent activity of ternary polymer + electrolyte + water solutions has been compared with that obtained from the ternary NRTL, ternary Wilson, ternary modified NRTL and ternary modified Wilson models. Results comparison was demonstrated the validity of the proposed model for solvent activity of polymer + electrolyte + water solutions.

First-principles study of the phase stability and elastic properties of Ti-X alloys (X = Mo, Nb, Al, Sn, Zr, Fe, Co, and O)

Phase stability and elastic properties of Ti1-x-Xx alloys (X = substitutional Mo, Nb, Al, Sn, Zr, Fe, Co, and interstitial O) in body-centered cubic (bcc) (β phase) and hexagonal close-packed (hcp) (α phase) crystal structures were studied using first-principles calculations. The formation energy was used to determine the dependence of the phase stability on the atomic concentration x (0 ≤ x ≤ 0.5 for substitutional elements and 0 ≤ x < 0.02 for O) of the alloying elements and atomic configurations (ordered, disordered, and other compound structures). The disordered configurations of the atoms in the alloys were considered within the framework of the special quasi-random structures (SQS) method. The composition dependence of elastic constants, isotropic elastic moduli and density of states (DOS) were also evaluated.The predicted formation energies and elastic properties agrees well with previous experimental and theoretical results. Addition of Mo was found to stabilize the β phase significantly while destabilizing the α phase. Nb was found to be a weaker β-stabilizer. In Ti-Al and Ti-Sn systems, the compounds form preferentially at a higher concentration. Zr shows little effect on the phase stability, thus Zr is a neutral element. For Ti-O systems, the octahedral site is the most stable site for interstitial oxygen in both bcc and hcp structures. The comparison of formation energies showed that O acts as an α stabilizer. Considering the configurational entropy contribution, we concluded that temperature effect can be significant to stabilize the phases. The elastic constants calculation revealed that adding Mo, Nb, Fe, or Co increases the mechanical stability of bcc Ti, whereas bcc Ti-Al, Ti-Sn, and Ti-Zr systems are mechanically unstable for all the calculated concentrations. For their hcp counterparts, Ti-Mo and Ti-Nb have lower mechanical stability than the Ti-Al, Ti-O, and Ti-Sn systems. It was found in the electronic structures calculation that increasing addition of Nb, Mo, Fe, and Co weakens the covalent-like bonding of hcp system and strengthens the metallic bonding of bcc system. This variation of the TDOSs of these Ti alloys explains why the bcc phase becomes more stable than the hcp phase at high concentration. Due to the similarity of electronic structure, Zr does not change the DOS of the bcc and hcp phases significantly: The Fermi level in DOS profile of Ti-Zr was found to be identical to that of pure Ti system. Ti-Al and Ti-Sn alloys keep the covalent-like bonding, which explains why hcp Ti-Al and Ti-Sn are stable.

Structural insights into the binding of small ligand molecules to a G-quadruplex DNA located in the HIV-1 promoter

Targeting guanine (G)-rich DNA sequences, folded into non-canonical G-quadruplex (G4) structures, by small ligand molecules is a promising strategy for gene therapy of various diseases. There is experimental proposal that, among eight studied ligands, nitidine chloride – NC and a benzo phenanthridine derivative – BPD have the highest binding affinity for such a sequence (5′-T1G2G3C4C5T6G7G8G9C10G11G12G13A14C15T16G17G18G19−3′) in the HIV-1 promoter, indicating that an anti-HIV-1 prodrug may regulate the expression of the promoter. Herein, this experimental indication is elaborated by using molecular docking simulations and by characterizing the modes of binding of the eight natural molecules to the particular G4. Moreover, the configurational entropy, as an upper bound of the true entropy contribution to the free energy in noncovalent binding, is employed to see into the structural changes experienced by the G4-DNA upon ligand binding. For various parts (complete structure, backbone, system of all bases, bases of G-tetrads) of the G4-DNA structure, a subtle molecular dynamics (MD) is exploited to determine the change of asymptotic (for infinitely long MD simulation) configurational entropy, being the thermodynamic consequence of DNA flexibility change upon complex formation. While NC increases rigidity of G4 (mainly through the system of all nucleobases), BPD increases flexibility of G4 (more than 50% stems from the sugar-phosphate backbone). These insights are further dissected and substantiated by considering the configurational entropy contributions at the level of individual base pairs making the two G-tetrads (G2G7G13G17 and G3G8G12G18) and by exploring the estimates of the total intra-base pair and inter-base pair entropies. This work makes the structural origin of enhanced stability of G4-DNA more certain – useful information when attempting to design optimal G4-DNA binders.

A Thermodynamic Framework Bridging the Composition and Temperature Dependence of Bulk Modulus With Enthalpy of Mixing

The intrinsic thermodynamic links that exist between thermochemical and thermophysical quantities, especially their temperature, pressure, and composition dependence, have seldom been analyzed in sufficient detail in literature. In this connection, an attempt is made to establish a thermodynamic bridge, relating Δo H mix, the standard enthalpy of mixing with Δo B T , the change in isothermal bulk modulus as a result of alloying and its composition and temperature dependence. In essence, by adopting the standard regular and subregular solution approximations to the composition dependence of mixing enthalpy; and furthermore, incorporating separately the configurational (Δo S conf) and vibrational (Δo S Vib) entropy contributions to mixing Gibbs energy change (Δo G mix), simple models have been derived for the composition and temperature variations of excess bulk modulus ΔB T . In particular, a regular or subregular solution analog of the composition variation of ΔB T is shown to be possible if Δo H mix could be described likewise. The vibrational entropy contribution to ΔB T is found to be important only when the change in Gruneisen parameter during alloying turns to be significant. The practical utility of the theoretical framework developed in this study has been demonstrated by applying it to disordered fcc Cu1−x Ni x alloys, wherein it is shown that Δo H mix and Δo B T are linearly correlated, as predicted by the theory.

Separating the configurational and vibrational entropy contributions in metallic glasses

Glassy materials exist in nature and play a critical role in technology, but key differences between the glass, liquid and crystalline phases are not well understood. Over several decades there has been controversy about the specific heat absorbed as a glass transforms to a liquid—does this originate from vibrational entropy or configurational entropy? Here we report direct in situ measurements of the vibrational spectra of strong and fragile metallic glasses in the glass, liquid and crystalline phases. For both types of material, the measured vibrational entropies of the glass and liquid show a tiny excess over the crystal, representing less than 5% of the total excess entropy measured with step calorimetry. These results reveal that the excess entropy of metallic glasses is almost entirely configurational in origin, consistent with the early theories of Gibbs and co-workers describing the glass transition as a purely configurational transition.