{"title":"Multicomponent cluster variation method: Application to high entropy alloys","authors":"Vikas Jindal, Shrikant Lele","doi":"10.1016/j.calphad.2025.102825","DOIUrl":"10.1016/j.calphad.2025.102825","url":null,"abstract":"<div><div>Cluster expansion, a method commonly used in computational thermodynamics, expresses the configurational properties of alloys as a sum of products of correlation functions and cluster expansion coefficients (CECs). The CECs are defined using a basis in configurational space that is not unique and changes during the extension of the generalized Ising model from binary alloys to multicomponent alloys. In contrast to the CALPHAD formulation, this change severely restricts the development and use of CEC-based databases. To address this, a new framework is introduced where lower-order CECs remain unchanged in higher-order systems. This is achieved by choosing configurational variables (and corresponding CECs) which are not dependent on a particular choice of basis as correlation functions (CFs). Such a set is provided by cluster variables (CVs), which represent the fraction of cluster configurations of various types. A carefully selected subset of independent CVs is chosen as CFs. Completeness of the basis is demonstrated by deriving expressions for all remaining CVs in terms of these new CFs for disordered bcc, fcc, and hcp structures. A set of CECs for the bcc phase of the Nb-Ti-V-Zr system has been developed using this new formulation. Optimized CECs of all binary and ternary subsystems of the Nb-Ti-V-Zr have been used in its construction. This is a crucial step towards creating a self-consistent computational thermodynamics database. The utility of the Nb-Ti-V-Zr database was demonstrated through calculations of thermodynamic quantities, SRO parameters, and phase diagrams. Additionally, a procedure is outlined to transform existing CALPHAD databases to CEC-based databases under certain assumptions.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102825"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143679058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hongyu Zhang , Zhe Yuan , Fali Liu , Weimin Bai , Yuan Yuan , Maohua Rong , Jiang Wang , Ligang Zhang , Libin Liu
{"title":"Interdiffusion coefficients and atomic mobilities in the bcc phase of the Ti-Fe-Mo system","authors":"Hongyu Zhang , Zhe Yuan , Fali Liu , Weimin Bai , Yuan Yuan , Maohua Rong , Jiang Wang , Ligang Zhang , Libin Liu","doi":"10.1016/j.calphad.2025.102829","DOIUrl":"10.1016/j.calphad.2025.102829","url":null,"abstract":"<div><div>Establishing the atomic mobility database for the Ti-Fe-Mo system is essential for analyzing and simulating diffusion-controlled processes in novel low-cost biomedical titanium alloys. In this study, twelve sets of diffusion couples were prepared at annealing temperatures of 1273 K, 1323 K, and 1373 K, with annealing time of 24 h, 10 h, and 6 h, respectively. The compositional distance profiles at the diffusion couple interfaces were measured using an Electro Probe Micro-Analysis (EPMA). Numerical inverse methods were applied to calculate diffusion coefficients in a high-throughput method, complemented by the Matano-Kirkaldy approach to determine diffusion coefficients at diffusion couple intersections. The results from both methods demonstrated strong agreement. Based on accurate thermodynamic descriptions and diffusion coefficients, the kinetic parameters of the Ti-Fe-Mo sub-binary systems were re-evaluated. Using HitDIC software in conjunction with compositional distance profiles, the atomic mobility database for this system was ultimately established. Additionally, the frequency factor and activation energy, vary with composition, can be derived using the Arrhenius equation.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102829"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xinyi Zhang , Liang Zhang , Taibai Fu , Jiangxing Wang , Bo Wang , Ziqing Xie , Yuling Liu , Yong Du
{"title":"A new algorithm to optimize thermodynamic parameters: Application to the Ag-Pd and La-C systems","authors":"Xinyi Zhang , Liang Zhang , Taibai Fu , Jiangxing Wang , Bo Wang , Ziqing Xie , Yuling Liu , Yong Du","doi":"10.1016/j.calphad.2025.102810","DOIUrl":"10.1016/j.calphad.2025.102810","url":null,"abstract":"<div><div>A new algorithm for optimizing thermodynamic parameters of binary systems is introduced. Unlike traditional algorithms that are commonly single-objective and need extensive manual testing, this algorithm constructs a multi-objective optimization for different types of experimental data. Then, by using weighted sum method, the multi-objective optimization problem is transformed into a single-objective optimization, which is solved by Barzilai-Borwein method. The key advantage of this algorithm is that no restrictions on the selection of initial values are needed. Finally, this algorithm is applied to optimize the thermodynamic parameters in the Ag-Pd and La-C systems. The experimental phase diagrams and thermodynamic properties in these two systems are satisfactorily reproduced by the present calculation.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102810"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143528867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xuemei Ouyang , Quan wang , Fucheng Yin , Yuling Shi , Jingxian Hu
{"title":"Experimental investigation and thermodynamic calculation of the B-Co-Ti ternary system","authors":"Xuemei Ouyang , Quan wang , Fucheng Yin , Yuling Shi , Jingxian Hu","doi":"10.1016/j.calphad.2025.102819","DOIUrl":"10.1016/j.calphad.2025.102819","url":null,"abstract":"<div><div>The phase equilibrium of the B-Co-Ti ternary system is important for the preparation and development of Co-containing TiB<sub>2</sub>-based ceramics. To obtain the liquidus projection, 54 kinds of as-cast alloys were prepared via arc melting. The microstructures and solidification paths of the 54 kinds of alloys were analyzed by scanning electron microscopy, electron probe microanalysis, X-ray diffraction and differential scanning calorimetry. The results show that the proposed liquidus projection of the B-Co-Ti ternary system contains 15 primary precipitation regions: BCC, CoTi<sub>2</sub>, CoTi, Co<sub>2</sub>Ti, Co<sub>3</sub>Ti, αCo, TiB, Ti<sub>3</sub>B<sub>4</sub>, TiB<sub>2</sub>, (B), CoB, Co<sub>2</sub>B, Co<sub>3</sub>B, τ<sub>1</sub> and τ<sub>2</sub>. Six E-type and nine U-type ternary invariant reactions are suggested on the basis of experimental data. The thermodynamic calculation of the B-Co-Ti ternary system was carried out, and a set of self-consistent thermodynamic parameters was obtained. The calculated liquidus projection and 800 °C isothermal sections are in good agreement with the experimental data.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102819"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143632063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sinn-wen Chen , Yi-An Lee , Cheng-Hsi Ho , Jun Zhu , Hsin-Chieh Huang , Te-Wei Lin , Chuan Zhang
{"title":"Phase transformation temperatures of Sn-Ag-Bi-Cu quaternary system","authors":"Sinn-wen Chen , Yi-An Lee , Cheng-Hsi Ho , Jun Zhu , Hsin-Chieh Huang , Te-Wei Lin , Chuan Zhang","doi":"10.1016/j.calphad.2025.102822","DOIUrl":"10.1016/j.calphad.2025.102822","url":null,"abstract":"<div><div>Sn-Ag-Bi-Cu and its constituent systems are important material systems, especially for electronic soldering. Although these alloys have attracted significant interest for industrial applications and have been the subject of various Calphad studies, it is surprising to find that there are significant deviations observed between the calculated and experimentally determined results regarding the phase transformation temperatures. To address this issue, Sn-Ag, Sn-Bi, Sn-Cu, Sn-Ag-Bi, Sn-Ag-Cu, Sn-Bi-Cu and Sn-Ag-Bi-Cu alloys on the Sn-rich side are prepared. Simultaneous thermal analysis and holding-quenching experimental measurements are conducted to determine their liquidus temperatures and invariant reaction temperatures. The experimentally determined results are then compared with the Calphad-calculated results and utilized to improve the thermodynamic parameters in the Calphad models. The calculated results with the revised databases are in good agreement with the experimental measurements.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102822"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143679059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ziyu Cao , Lei Zou , Huimin Zheng , Cuiping Guo , Zhenmin Du
{"title":"Experimental determination and thermodynamic description of the Fe–Nb–Zr system","authors":"Ziyu Cao , Lei Zou , Huimin Zheng , Cuiping Guo , Zhenmin Du","doi":"10.1016/j.calphad.2025.102831","DOIUrl":"10.1016/j.calphad.2025.102831","url":null,"abstract":"<div><div>In the Fe–Nb–Zr system, six primary solidification regions were identified in the liquidus surface projection by analyzing microstructures and phase constituents of 26 as-cast alloys. The equilibrium areas of 4 three phases and 7 two phases were determined by measuring the phase constituents of 24 equilibrated alloys at 1273 K. The solubility of Nb in Fe<sub>23</sub>Zr<sub>6</sub> and λ<sub>2</sub> at 1273 K was ∼0.7 and 9.0 at. %, respectively, and that of Zr in μ and λ<sub>1</sub> at 1273 K was ∼8.1 and 38.4 at. %, respectively. The compound τ with NiTi<sub>2</sub> structure was confirmed to be stable at 1273 K, and the homogeneity range of Nb was ∼10.1–16.8 at. %. Combining the experimental results in literature and those of the current study, the Fe–Nb–Zr system was assessed by CALPHAD method. A set of self-consistent reliable thermodynamic parameters was derived.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102831"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Identifying independent components and internal process order parameters in nonequilibrium multicomponent nonstoichiometric compounds","authors":"Yanzhou Ji , Yueze Tan , Long-Qing Chen","doi":"10.1016/j.calphad.2025.102807","DOIUrl":"10.1016/j.calphad.2025.102807","url":null,"abstract":"<div><div>In CALPHAD-type thermodynamic databases, nonstoichiometric compounds are typically described by sublattice models where the sublattice site fractions represent the occupation probability of different atomic, ionic or defect species on different sublattices. Here, we develop a general procedure and corresponding linear algebra tools for converting the sublattice site fractions to a combination of independent component compositions and internal process order parameters describing the extent of internal atomic exchange, electronic redox and defect generation reactions. We apply them to a number of nonstoichiometric phases in thermodynamic databases and literature. The general procedure can be applied to constructing thermodynamic databases in terms of internal process order parameters for nonstoichiometric phases in multicomponent systems such as high-entropy oxides and alloys, which can be utilized to model their kinetics of nonequilibrium processes and microstructure evolution.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102807"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Veljko Minic , Miljana Popovic , Milena Zecevic , Jelena Miladinovic , Aleksandar Djordjevic
{"title":"Phase diagram evaluation and experimental characterization of the Al-Bi-Ge ternary system","authors":"Veljko Minic , Miljana Popovic , Milena Zecevic , Jelena Miladinovic , Aleksandar Djordjevic","doi":"10.1016/j.calphad.2025.102834","DOIUrl":"10.1016/j.calphad.2025.102834","url":null,"abstract":"<div><div>In this work, the ternary system Al-Bi-Ge has been studied. Phase diagrams were calculated using Pandat software while experimentally tested alloys come from three vertical and two isothermal sections. The three vertical sections Al-BiGe, Bi-AlGe and Ge-AlBi were investigated by differential thermal analysis (DTA) and experimental phase diagrams were compared with calculated ones based on the experimental results. Selected alloys from the isothermal section at 400 °C and as-cast samples were investigated by X-ray diffraction (XRD) and energy dispersive spectrometry (EDS). The microstructure of the investigated samples was characterized by optical microscopy (OM) and the scanning electron microscopy (SEM). The microstructure of all samples annealed at 400 °C, consists of three phases L, (Al) and (Ge), while in the microstructure of as-cast samples phases (Al), (Bi) and (Ge) are detected. Microstructure of as-cast samples were compared with calculated isothermal section at 25 °C. Brinell hardness, electrical conductivity and electrical resistivity were measured for selected ternary alloys. Based on the experimental values and using a suitable mathematical model, these properties were predicted by RSM methodology for all composition ranges.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102834"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143899529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jesus Alejandro Arias-Hernandez , Sun Yong Kwon , Elmira Moosavi-Khoonsari
{"title":"Application of polyhedron model to predict heat capacity of mixed oxides","authors":"Jesus Alejandro Arias-Hernandez , Sun Yong Kwon , Elmira Moosavi-Khoonsari","doi":"10.1016/j.calphad.2025.102836","DOIUrl":"10.1016/j.calphad.2025.102836","url":null,"abstract":"<div><div>The heat capacity of mixed oxides can be estimated using a linear summation of the heat capacities of their structural constituent polyhedra. This approach is particularly useful for hygroscopic and volatile oxides, where experimental data can be difficult to obtain. The present work aims to enhance the polyhedron model (PM) by incorporating contributions from second-order transitions, including magnetic and site order-disorders, into <span><math><mrow><msub><mi>C</mi><mi>p</mi></msub></mrow></math></span> and expanding it to include ZnO and PbO-containing systems in comparison to the previous version of the model. A regression analysis was performed over the new dataset consisting of the properties of 85 compounds in the system Li-Na-K-Ca-Mg-Mn-Fe-Pb-Zn-Al-Ti-Si-O to obtain optimized <span><math><mrow><msub><mi>C</mi><mi>p</mi></msub></mrow></math></span> for 20 constituent polyhedra. We validate the updated PM against experimental data, demonstrating an overall improvement between 7 and 9 % in the estimation of <span><math><mrow><msub><mi>C</mi><mi>p</mi></msub></mrow></math></span> compared to the previous version of the model. We also compare the updated model with well-established models in the literature, such as the Neumann-Kopp Rule, and ab-initio calculations. The PM shows higher precision than NKR and the linear summation nature of PM endows the model with simplicity which contrasts with ab-initio calculations. Additionally, the model has demonstrated an inherent self-correction capability relative to the original input values, as shown for K<sub>2</sub>Si<sub>4</sub>O<sub>9</sub>. The model is also applied to predict the heat capacity of 10 compounds in the Na<sub>2</sub>O-PbO-SiO<sub>2</sub> and Na<sub>2</sub>O-ZnO-SiO<sub>2</sub> systems, where experimental data are lacking.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102836"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144090365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The SGTE collection of binary datasets","authors":"Bengt Hallstedt","doi":"10.1016/j.calphad.2025.102833","DOIUrl":"10.1016/j.calphad.2025.102833","url":null,"abstract":"<div><div>SGTE has collected (and also produced) binary assessments since the 1980's. This was documented with about 450 binary systems in five volumes of the Landolt-Börnstein series. These volumes were accompanied by a database with which the phase diagrams could be calculated, but the database itself was encrypted. It is now time to make the SGTE collection of binary datasets publicly available. The collection now contains 2773 datasets from 1238 different binary systems. By providing this collection in open form it is hoped that Calphad modelling of, in particular, ternary and higher order systems can be further promoted.</div></div>","PeriodicalId":9436,"journal":{"name":"Calphad-computer Coupling of Phase Diagrams and Thermochemistry","volume":"89 ","pages":"Article 102833"},"PeriodicalIF":1.9,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}