Regular Solution Model Research Articles

Predicting the solid solution structure preference of multi-component alloys

High-entropy alloys (HEAs) provide limitless opportunities to enhance material performance while predicting the thermodynamic properties and the structures rapidly within HEAs remain challenging. Herein, a new method (M2FEn) for rapidly predicting the formation enthalpies is provided. In contrast to Miedema's semi-empirical methodology, which solely calculates the mixing enthalpy, the M2FEn is capable of calculating the formation enthalpies of various solid solution structures, and by ordering these enthalpies, one can determine the preferred solid-solution structure for the single-phase alloy. The predicted structures using M2FEn are in line with first-principles calculations. The effectiveness of different extrapolation models for formation enthalpy has been assessed. The M2FEn with the Ouyang extrapolation model and the regular solution model have achieved a prediction accuracy of 95.8% and 95.4%, respectively for single-phase HEAs that were experimentally prepared. 28 common HEA elements were selected as the chemical space, and the formation enthalpies of all binary, ternary, quaternary, quinary and senary alloys, a total of 499149 various alloys were calculated by M2FEn. It is found that the BCC structure ratio increases with the increase of the principal element of the alloy. The M2FEn should potentially accelerate the development of novel, high-performance HEAs, enabling more efficient and cost-effective materials design processes.

Integrating molecular dynamics simulations and experimental data for azeotrope predictions in binary mixtures.

An azeotrope is a constant boiling point mixture, and its behavior is important for fluid separation processes. Predicting azeotropes from atomistic simulations is difficult due to the complexities and convergence problems of Monte Carlo and free-energy perturbation techniques. Here, we present a methodology for predicting the azeotropes of binary mixtures, which computes the compositional dependence of chemical potentials from molecular dynamics simulations using the S0 method and employs experimental boiling point and vaporization enthalpy data. Using this methodology, we reproduce the azeotropes, or lack thereof, in five case studies, including ethanol/water, ethanol/isooctane, methanol/water, hydrazine/water, and acetone/chloroform mixtures. We find that it is crucial to use the experimental boiling point and vaporization enthalpy for reliable azeotrope predictions, as empirical force fields are not accurate enough for these quantities. Finally, we use regular solution models to rationalize the azeotropes and reveal that they tend to form when the mixture components have similar boiling points and strong interactions.

Solubility of oxazepam in supercritical carbon dioxide: Experimental and modeling

This study thoroughly investigates the solubility of oxazepam in supercritical carbon dioxide (SC-CO2) at temperatures of 308, 318, 328, and 338 K and pressures ranging from 12 to 30 MPa. The solubility measurements revealed a mole fraction of oxazepam ranging from 2.50 × 10−6 to 7.13 × 10−5, with the highest solubility observed at 338 K and 30 MPa. The solubility data were effectively modeled using semi-empirical density-based models (Mendez-Santiago and Teja (MST), Chrastil, Bartle et al., Kumar and Johnston (K-J), and Alwi-Garlapati), two equations of state (Peng-Robinson and modified-Pazuki) and regular solution models. The K-J model emerged as the best fit for the experimental data, boasting the lowest Average Absolute Relative Deviation (AARD) value of 7.73 %. The Peng-Robinson equation of state outperformed the modified-Pazuki equation, with AARD values of 15.71 % and 18.15 %, respectively. The study's key finding is the low solubility of oxazepam in SC-CO2, which suggests that supercritical antisolvent techniques could be effectively employed for synthesizing nanoparticles of this pharmaceutical compound. These findings have practical implications as they provide valuable insights for optimizing drug formulation processes and demonstrate the potential of SC-CO2 in pharmaceutical applications.

Activity Calculation and Vacuum Separation Theoretical Research concerning Ag–Cu, Ag–Sb and Cu–Sb Binary Alloys

The Ag–Cu–Sb system is a key component of lead anode slime and boasts an exceptionally high economic recovery value. In this work, six models, including the Molecular Interaction Volume Model (MIVM), Modified Molecular Interaction Volume Model (M-MIVM), Wilson equation, Miedema model, Regular Solution Model (RSE) and Sub-Regular Solution Model (SRSE), are used to calculate the predicted values of the activity and its deviations with experimental data for binary alloys in the Ag–Cu–Sb system for the first time. The result reveals that the overall means of the average relative deviation and average standard deviation of the M-MIVM are 0.01501 and 3.97278%, respectively, which are about two to six times smaller than those of the other five models, indicating the stability and reliability of the M-MIVM. In the meantime, the predicted data of the Cu–Ag binary alloy at 1423 K, Sb–Ag binary alloy at 1250 K and Sb–Cu binary alloy at 1375 K calculated from the M-MIVM are more reliable and pass the Herington test. Then, the separation coefficient–composition (β–x), temperature–composition (T–x–y) and pressure–composition (P–x–y) of the Cu–Ag, Sb–Ag and Sb–Cu binary alloys are plotted based on the M-MIVM and vacuum theories, showing that the Cu–Ag binary alloy is relatively difficult to separate and that high temperatures or high copper contents are detrimental to obtaining high-purity silver. Meanwhile, theoretical data of the T–x–y diagram are consistent with the available experimental data. These results can guide vacuum separation experiments and industrial production concerning Ag–Cu, Ag–Sb and Cu–Sb binary alloys.

Enthalpy of mixing of solid solution rhomboclase – Indium-substituted rhomboclase (H3O) Fe1−xInx(SO4)2·nH2O

Solid solution of rhomboclase [nominally (H3O) Fe(SO4)2·3H2O] with Fe-In substitution was synthesized by coprecipitation and hydrothermal treatment and investigated by chemical, X-ray diffraction, and thermodynamic analysis in this study. The solid solution is continuous over the whole stoichiometric range despite significant differences in ionic radii between Fe3+ and In3+. Enthalpy of formation of rhomboclase and enthalpies of mixing of solid solution rhomboclase – indium-substituted rhomboclase were determined by acid-solution calorimetry in 5 mol·dm−1 HCl at T = 298.15 K and p = 1 bar. The actual compositions of the studied synthetic end members are (H3O)0.798Fe(SO4)1.899·3.078H2O (rhomboclase with molecular mass of 308.8719 g·mol−1) and (H3O)1.067In(SO4)2.034·3.017H2O (indium-substituted rhomboclase with molecular mass of 384.8377 g·mol−1). The enthalpies of mixing were obtained for a series of four indium-incorporating rhomboclase synthetic samples with molar In/(In + Fe) ratios of 0.115, 0.226, 0.537, and 0.733. The enthalpy of formation from elements in their standard state at T = 298.15 K and p = 1 bar obtained in this study by thermochemical cycle for (H3O)0.798Fe(SO4)1.899·3.078H2O is: ΔfH298.15°=-2815.2±2.8 kJ·mol−1. Mean measured enthalpies of dissolution for (H3O)0.798Fe(SO4)1.899·3.078H2O and (H3O)1.067In(SO4)2.034·3.017H2O are: ΔdissH°=14.47±0.42 kJ·mol−1 and -6.68±0.27 kJ · mol−1, respectively. Enthalpies of mixingΔmixH° have small positive values and can be fitted by a regular solid solution model with a mixing parameter W=4.26±0.32 kJ·mol−1.

Grain boundary segregation models for high-entropy alloys: Theoretical formulation and application to elucidate high-entropy grain boundaries

Grain boundary (GB) segregation models are derived for multi-principal element alloys (MPEAs) and high-entropy alloys (HEAs). Differing from classical models where one component is taken as a solvent and others are considered solutes, these models are referenced to the bulk composition to enable improved treatments of MPEAs and HEAs with no principal components. An ideal solution model is first formulated and solved to obtain analytical expressions that predict GB segregation and GB energy in MPEAs and HEAs. A regular solution model is further derived. The GB composition calculated using the simple analytical expression derived in this study and data from the Materials Project agrees well with a prior atomistic simulation for NbMoTaW. The simplicity of the derived analytical expressions makes them useful for not only conveniently predicting GB segregation trends in HEAs but also analyzing nascent interfacial phenomena in compositionally complex GBs. As an application example, the models are used to further derive a set of equations to elucidate an emergent concept of high-entropy grain boundaries.

Estimation of Activity and Molar Excess Gibbs Energy of Binary Liquid Alloys Pb-Sn, Al-Sn and In-Zn from the Partial Radial Distribution Function Simulated by Ab Initio Molecular Dynamics

For the present, it is difficult to obtain thermodynamic data for binary liquid alloys by experimental measurements. In this study, the molecular dynamics processes of the binary liquid alloys Pb50-Sn50, Al50-Sn50, and In50-Zn50 were simulated by using the ab initio molecular dynamics (AIMD) principle, and their partial radial distribution functions (PRDF) were obtained at different simulation steps. Combined with the relevant binary parameters of the Molecular Interaction Volume Model (MIVM), Regular Solution Model (RSM), Wilson Model, and Non-Random Two-Liquid (NRTL) models. The integral terms containing the PRDF were computed using the graphical integration method to obtain the parameters of these models, thus estimating their activity and molar excess Gibbs energy. The total average relative deviations (ARD) of the activity and molar excess Gibbs energy estimates of the four models for the binary liquid alloys Pb50-Sn50, Al50-Sn50, and In50-Zn50 at full concentration when the PRDF is obtained by the symmetry method are MIVM: 21.59% and 59.35%; RSM: 21.63% and 60.27%; Wilson: 24.27% and 86.7%; NRTL: 23.9% and 83.24%. When the PRDF is obtained by the asymmetric method: MIVM: 22.86% and 68.08%; RSM: 32.84% and 68.66%; Wilson: 25.14% and 82.75%; NRTL: 24.49% and 85.74%. This indicates that the estimation performance of the MIVM model is superior to the other three models, and the symmetric method performs better than the asymmetric method. The present study also derives and verifies the feasibility of Sommer’s equation for estimating the molar excess Gibbs energy and activity of binary liquid alloy systems in the Miedema model by using different equations of enthalpy of mixing versus excess entropy given by Tanaka, Ding, and Sommer. The total ARD of Tanaka, Ding, and Sommer’s relational equations in the Miedema model for estimating the activities and molar excess Gibbs energies of the binary liquid alloys Pb-Sn, Al-Sn, and In-Zn are 3.07% and 8.92%, 6.09% and 17.1%, and 4.1% and 14.77%. The results indicate that the estimation performance of the Miedema model is superior to the other four models.

Experimental and thermodynamic investigation of gemifloxacin solubility in supercritical CO2 for the production of nanoparticles

The effective use of the supercritical technology in the production of new pharmaceutical formulations with greater bioavailability requires the determination of the solubility of pharmaceutical substances in supercritical CO2 (scCO2) at different conditions. In the current work, the solubility of an antibacterial compound, Gemifloxacin, was measured at various temperatures (308 to 338 K), and pressures (120 to 300 bar). This compound has a poor solubility in scCO2 (0.02 × 10−4 to 0.67 × 10−4 mole fraction or 0.0156 to 0.5891 g.kg−1). Furthermore, some popular empirical models and three thermodynamic approaches (SRK and PR equations of state, and the regular solution model) were applied to correlate and predict the solubility of Gemifloxacin in scCO2. The regular solution model with temperature-dependent solubility parameter (mean AARD% = 2.32–4.34) and the SRK-based model (mean AARD% = 7.47), exhibited the highest accuracy in fitting the experimental data.

Estimation of Activity and Molar Excess Gibbs Energy of Binary Liquid Alloys Al-Cu, Al-Ni, and Al-Fe from the Partial Radial Distribution Function Simulated by Ab Initio Molecular Dynamics

To accurately and conveniently obtain the thermodynamic data of binary liquid alloys, a new method is proposed in this study. It combines ab initio molecular dynamics (AIMD) simulation with a thermodynamic model to estimate the activity and molar excess Gibbs energy of binary liquid alloys. Additionally, two methods of grouping the partial radial distribution function (PRDF) of 5000 steps obtained by simulation are proposed for the first time. The PRDF of Al50Cu50, Al50Ni50, and Al50Fe50 is obtained by AIMD simulation. These PRDF are combined with four thermodynamic models to estimate the activity and molar excess Gibbs energy. Furthermore, the estimation results of the four models are compared with those of the Miedema model. The results show that when the first peak of the PRDF is obtained by the symmetric method, the average relative deviation (ARD) of the activity and molar excess Gibbs energy of the four models are, respectively: 28% and 32% for Molecular Interaction Volume Model (MIVM); 162% and 38% for Regular Solution Model (RSM); 508% and 65% for Wilson model; 562% and 67% for Non-Random Two-Liquid (NRTL). When the first peak of PRDF is obtained by non-symmetric method, the average ARD of the activity and molar excess Gibbs energy of the four models are, respectively: 64% and 20% for MIVM; 115% and 26% for RSM; 661% and 70% for Wilson; 727% and 72% for NRTL. In addition, the average ARD of the activity and molar excess Gibbs energy of the Miedema model are 113% and 33%. These data indicate that the estimation performance of the MIVM model is superior to the other four models, and the symmetric method performs better than the non-symmetric method. The grouping treatment of PRDF data effectively improves estimation performance.

Phase and defect diagrams based on spectral grain boundary segregation: A regular solution approach

Recent works have proposed the treatment of alloy defect states with thermodynamics analogous to phase thermodynamics, so they can be analyzed in a manner that is comparable to bulk phase diagrams. We refer to such diagrams containing both bulk phase and defect state information as “phase-and-defect diagrams.” Segregated grain boundaries (GBs) are an example of one such defect state, which enable, for example, the stabilization of nanocrystalline alloys. The recent development of spectral thermodynamic isotherms has demonstrated that a fully spectral description, which captures the complete distribution of local atomic environments in the grain boundary network, is necessary for accurate modeling of segregation behavior. To our knowledge, there is currently no spectral free energy formulation from which a phase-and-defect diagram of the segregated polycrystalline state can be produced. In this work, we present a spectral regular solution model of grain boundary segregation, from which we derive an analytical free energy function for the segregated polycrystalline state. We then use this free energy representation to construct a phase-and-defect diagram for a hypothetical alloy system, providing a framework for future efforts in developing spectral phase-and-defect diagrams.

Estimation of Two Component Activities of Binary Liquid Alloys by the Pair Potential Energy Containing a Polynomial of the Partial Radial Distribution Function

An investigation of partial radial distribution functions and atomic pair potentials within a system has established that the existing potential functions are rooted in the assumption of a static arrangement of atoms, overlooking their distribution and vibration. In this study, Hill’s proposed radial distribution function polynomials are applied for the pure gaseous state to a binary liquid alloy to derive the pair potential energy. The partial radial distribution functions of 36 binary liquid alloy from literatures were used to obtain the binary model parameters of four thermodynamic models for validation. Results show that the regular solution model (RSM) and molecular interaction volume model (MIVM) outperform other models when the asymmetric method calculates the partial radial distribution function. RSM demonstrates an average SD of 0.078 and an ARD of 32.2%. Similarly, MIVM exhibits an average SD of 0.095 and an average ARD of 32.2%. Wilson model yields an average SD of 0.124 and an average ARD of 226%. Nonrandom two-liquid (NRTL) model exhibits an average SD of 0.225 and an average ARD of 911%. On applying the partial radial distribution function symmetry method, MIVM and RSM outperform the other models, with an average SD of 0.143 and an average ARD of 165.9% for MIVM. RSM yields an average SD of 0.117 and an average ARD of 208.3%. Wilson model exhibits average values of 0.133 and 305.6% for SD and ARD, respectively. NRTL model shows an average SD of 0.200 and an average ARD of 771.8%. Based on this result, the influence of the symmetry degree on the thermodynamic model is explored by examining the symmetry degree as defined by the experimental activity curves of the two components.

Solubility measurement of verapamil for the preparation of developed nanomedicines using supercritical fluid

A static method is employed to determine the solubilities of verapamil in supercritical carbon dioxide (SC-CO2) at temperatures between 308 and 338 K and pressures between 12 and 30 MPa. The solubility of verapamil in SC-CO2 expressed as mole fraction are in the range of 3.6 × 10–6 to 7.14 × 10–5. Using four semi-empirical density-based models, the solubility data are correlated: Chrastil, Bartle, Kumar–Johnston (K–J), and Mendez-Santiago and Teja (MST), two equations of state (SRK and PC-SAFT EoS), expanded liquid models (modified Wilson's models), and regular solution model. The obtained results indicated that the regular solution and PC-SAFT models showed the most noteworthy exactness with AARD% of 1.68 and 7.45, respectively. The total heat, vaporization heat, and solvation heat of verapamil are calculated at 39.62, 60.03, and − 20.41 kJ/mol, respectively. Regarding the poor solubility of verapamil in SC-CO2, supercritical anti-solvent methods can be an appropriate choice to produce fine particles of this drug.

Thermodynamics of multi-sublattice battery active materials: from an extended regular solution theory to a phase-field model of LiMnyFe1-yPO4

Phase separation during the lithiation of redox-active materials is a critical factor affecting battery performance, including energy density, charging rates, and cycle life. Accurate physical descriptions of these materials are necessary for understanding underlying lithiation mechanisms, performance limitations, and optimizing energy storage devices. This work presents an extended regular solution model that captures mutual interactions between sublattices of multi-sublattice battery materials, typically synthesized by metal substitution. We apply the model to phospho-olivine materials and demonstrate its quantitative accuracy in predicting the composition-dependent redox shift of the plateaus of LiMnyFe1-yPO4 (LFMP), LiCoyFe1-yPO4 (LFCP), LiCoxMnyFe1-x-yPO4 (LFMCP), as well as their phase separation behavior. Furthermore, we develop a phase-field model of LFMP that consistently matches experimental data and identifies LiMn0.4Fe0.6PO4 as a superior composition that favors a solid solution phase transition, making it ideal for high-power applications.

Critical evaluation and thermodynamic modeling of the Al–Li binary system

Al–Li alloys have extensive applications in aircraft and spacecraft due to their high mechanical performance and lightness in weight. In this work, the thermodynamic property and phase equilibria data of the Al–Li system available in the literature were critically reviewed and evaluated, thermodynamic assessments of the entire Al–Li system were then carried out based on the reliable experimental thermochemistry and phase equilibria data. The liquid phase was described using both the modified quasi-chemical model in pair approximation and the regular solution model. The compound energy formalism and the regular solution model were used to describe the Gibbs energies of the solid phases. The split four sublattice compound energy formalism was used to model the Gibbs free energies of the ordered AlLi (B32) and Al3Li (L12) superlattice structures. The reliable thermodynamic property and phase equilibria data can generally be represented by the currently obtained sets of self-consistent thermodynamic descriptions.

Large Decrease in the Melting Point of Benzoquinones via High-n Eutectic Mixing Predicted by a Regular Solution Model.

Decreasing the melting point (Tm) of a mixture is of interest in cryopreservatives, molten salts, and battery electrolytes. One general strategy to decrease Tm, exemplified by deep eutectic solvents, is to mix components with favorable (negative) enthalpic interactions. We demonstrate a complementary strategy to decrease Tm by mixing many components with neutral or slightly positive enthalpic interactions, using the number of components (n) to increase the entropy of mixing and decrease Tm. In theory, under certain conditions this approach could achieve an arbitrarily low Tm. Furthermore, if the components are small redox-active molecules, such as the benzoquinones studied here, this approach could lead to high energy density flow battery electrolytes. Finding the eutectic composition of a high-n mixture can be challenging due to the large compositional space yet is essential for ensuring the existence of a purely liquid phase. We reformulate and apply fundamental thermodynamic equations to describe high-n eutectic mixtures of small redox-active molecules (benzoquinones and hydroquinones). We illustrate a novel application of this theory by tuning the entropy of melting, rather than the enthalpy, in systems highly relevant to energy storage. We demonstrate with differential scanning calorimetry measurements that 1,4-benzoquinone derivatives exhibit eutectic mixing that decreases their Tm, despite slightly positive enthalpies of mixing (0-5 kJ/mol). By rigorously investigating all 21 binary mixtures of a set of seven 1,4-benzoquinone derivatives with alkyl substituents (Tm's between 44 and 120 °C), we find that the eutectic melting point of a mixture of all seven achieves a large decrease in Tm to -6 °C. We further determine that the regular solution model shows improvement over an ideal solution model in predicting the eutectic properties for this newly investigated type of mixture composed of many small redox-active organic molecules.

Simplified regular solution model for asphaltenes precipitation in crude oil blends

The regular solution model is used with Hankinson-Brobst-Thomson (HBT) molar volume correlation for saturates, aromatics, resins and asphaltenes (SARA) fractions for pure crude oils and blends to predict the precipitation of asphaltenes. The parameters of the HBT correlation were tuned with experimental data for SARA fractions densities. At the same temperature and pressure conditions, the precipitation of asphaltenes is greatly influenced by the presence of other compounds. SARA fractions were obtained according with ASTM D2007 using n-heptane as precipitant solvent. The use of average properties for saturates, aromatics and resins and the assumptions made in the model could reproduce precipitation curves for all study cases. Calculations of asphaltene precipitation prediction for pure oils showed an average absolute deviation between 0.12 and 0.69% for oil of Gulf of Mexico and Cold Lake oil, respectively. For blends it showed values between 0.08 and 0.65% for Cold Lake/Lloydminster-Asphaltene free blend and Cold Lake/Lloydminster-Gulf of Mexico blend respectively. The proposed model maintains its simplicity and can predict the asphaltene precipitation accurately.

Stabilization of Equiatomic Solutions Due to High-Entropy Effect

The stability of solid-solution phases in FCC and BCC lattices was examined in multi-component alloys based on the CALPHAD technique using the compound energy formalism and regular solution model. From the thermodynamic calculations, it was found in ternary systems that the single solid-solution phase became stable around the equiatomic composition where the configurational entropy was the largest value. The transition temperature from the disordered phase to ordered phase(s) or miscibility gap(s) decreased with the increasing number of elements in the system. The order-disorder transition temperature on the FCC lattice was affected by the end member of the ordered phases existing at the equiatomic composition, whereas it was not significant for the order-disorder transition in the BCC lattice. The single solid-solution phase region at equiatomic compositions was affected by variations in the interaction parameters. In multi-component systems, the variations were averaged with increasing the number of elements in the system. This suggests that high-entropy alloys can afford a variety of elements. This study shows that the disordered state can be formed in multicomponent systems around the equiatomic composition and suggests clearly that due to the high-entropy effect, the solution phases are stabilized.

Dilute hydrochloric acid induced switching from antagonism to synergism in the binary mixed systems of cocamidopropyl betaine (CAPB) and sodium dodecyl benzene sulfonate (SDBS)

Abstract A synergism is reported in the binary mixtures of cocamidopropyl betaine (CAPB) and sodium dodecyl benzene sulfonate (SDBS) at mole fractions of 0.25, 0.50, and 0.75, in dilute hydrochloric acid solutions (∼0.05 M) HCl, at 25 °C. Such synergism is confirmed by surface tension values, critical micelle concentration (CMC), and interaction parameter values according to Holland and Rubingh’s model and the satisfaction of Hua and Rosen conditions. Surface excess (Γ) at air/water interface and the minimum area occupied by surfactant molecule (A min) also indicated a reduction in the A min of mixed CAPB/SDBS compared to single surfactants that is due to the strong electrostatic interaction between surfactants heads. The composition of CAPB/SDBS mixed micelles, the interaction parameter (β) and the micellar activity coefficients (f 1 and f 2) were evaluated from the regular solution model for these binary amphoteric/anionic mixed systems. It is clear the CAPB surfactants are converted to the cationic form by charging in the presence of HCl, which enhances the interaction between the oppositely charged CAPB and SDBS heads and overcomes the unfavourable packing between the tails predicted in pure water. Such strong interaction was also reflected on the phase behavior of CAPB/SDBS binary mixed system. Accordingly, a switching from antagonism to synergism is reported for CAPB/SDBS binary mixed system in the presence of dilute hydrochloric acid.

Improvement of compressible regular solution model using Sanchez-Lacombe equation of state for phase behavior prediction of polymer blends

Improvement of compressible regular solution model using Sanchez-Lacombe equation of state for phase behavior prediction of polymer blends

Free Energy of Mixing, Heat of Mixing and Viscosity of Na-K Liquid Alloy at Different Temperatures

The Free energy of mixing, Heat of mixing and Viscosity of Na-K alloy are investigated theoretically at different temperatures using the Regular solution model. The interaction energy is temperature dependent. The theoretical values of interchange energy at different temperatures are obtained by best fit parameter approximation with the help of experimental values at 384K. The properties have been studied with the help of computed theoretical interchange energy at different temperatures using interchange energy and temperature relation. A comparison of theoretical and experimental values at 384K shows that they are in good agreement and using this basis we have computed the values at different temperatures.