A coherent set of model equations for various surface and interface energies in systems with liquid and solid metals and alloys

Analytical Model for Liquid/Liquid Interfacial Energy of Microalloyed Monotectic Alloys

An analytical model is proposed for the calculation of the interfacial energy γL1/L2 between an A-rich matrix liquid and B-rich droplets in an A-B monotectic alloy microalloyed by element I, based on the Gibbs absorption isotherm. The model is applicable to the microalloying element segregated to the liquid/liquid interface or impoverished in the interface. Solidification experimental results validated the model. Calculation results demonstrate that the variation of γL1/L2 with I concentration xI is not always monotonic. It is determined by the dependence of I chemical potential μI on xI, which is closely related to the interaction parameter LAI in the A-rich corner of the liquid A-I alloy. μI increases with xI monotonically when LAI < 2RgTb,A-B (Tb,A-B is the binodal line temperature), and it shows a tendency with xI of first increase and then decrease, followed by an increase when LAI ≥ 2RgTb,A-B. The variation trends of γL1/L2 and μI with xI are opposite for an element I segregated to the liquid/liquid interface and similar for an element I impoverished in the liquid/liquid interface.

Model-free estimation of completeness, uncertainties, and outliers in atomistic machine learning using information theory

An accurate description of information is relevant for a range of problems in atomistic machine learning (ML), such as crafting training sets, performing uncertainty quantification (UQ), or extracting physical insights from large datasets. However, atomistic ML often relies on unsupervised learning or model predictions to analyze information contents from simulation or training data. Here, we introduce a theoretical framework that provides a rigorous, model-free tool to quantify information contents in atomistic simulations. We demonstrate that the information entropy of a distribution of atom-centered environments explains known heuristics in ML potential developments, from training set sizes to dataset optimality. Using this tool, we propose a model-free UQ method that reliably predicts epistemic uncertainty and detects out-of-distribution samples, including rare events in systems such as nucleation. This method provides a general tool for data-driven atomistic modeling and combines efforts in ML, simulations, and physical explainability.

High-Temperature Wetting Behavior and Adhesion Mechanism of Cryolite-Based Molten Salt on SiC Refractory Substrate

The problem of the adhesion of aluminum slag to the inner wall of a vacuum ladle is essential but has not been addressed. Using a high-temperature wettability experimental setup, this paper investigates the mechanism of interfacial wettability, adhesion, and penetration between Na3AlF6-Al2O3-CaF2 cryolite-based molten salt and SiC refractory substrate. The composition of the slag was determined based on the slag obtained in the transfer ladle during the aluminum electrolysis process. We mainly study the effects of different Al2O3 contents in cryolite-based molten salt and temperatures on the contact angle and surface tension. The results indicate that as the Al2O3 content in the slag increases, the contact angle decreases, enhancing the slag's wettability on the SiC substrate. Additionally, a higher Al2O3 content leads to higher slag melting temperatures and surface tension, which improves the slag mobility and enhances the mass transfer and diffusion capabilities of molecules or ions within the slag. The work of adhesion, calculated using the Mills model, also increases with the increasing Al2O3 content. The increased Al2O3 concentration activates the activity of Na3AlF6 in the slag, facilitating the dissolution reactions and improving the wettability between the slag and SiC. Moreover, the wetting behavior of the Na3AlF6-Al2O3-CaF2 slag is primarily influenced by the initial Al2O3 content and its compositional changes. The results show that the slag penetration resistance and mechanical erosion resistance of the ladle lining can be improved by using an SiC-based refractory with an optimized Al2O3 content. This will have important guiding significance for the development, design, and application of inner wall materials for aluminum electrolysis industrial vacuum ladles.

The Generalized Phase Rule, the Extended Definition of the Degree of Freedom, the Component Rule and the Seven Independent Non-Compositional State Variables: To the 150th Anniversary of the Phase Rule of Gibbs

The phase rule of Gibbs is one of the basic equations in phase equilibria. Although it has been with us for 150 years, discussions, interpretations and extensions have been published. Here, the following new content is provided: (i). the choice of independent components is discussed, and the component rule is introduced, (ii). independent state variables are divided into compositional and non-compositional ones, (iii). the generalized phase rule is derived replacing number two in the original phase rule by the number of independent non-compositional state variables introduced above, (iv). the degree of freedom is decreased by the number of compositional constraints in special points (azeotrope and congruent melting) of phase diagrams, (v). a rule is derived connecting the maximum number of coexisting phases with the dimensions of the phase diagram, (vi). examples show how to apply the phase rule to unary, binary and ternary phase diagrams and their sections, (vii). the same is extended with the discussion of calculable and not calculable phase fractions, (viii). it is shown that the current definition of the degree of freedom is not sufficient in the number of cases, (ix). the current definition of the degree of freedom is extended, (x). the application of the generalized phase rule is demonstrated when other non-compositional state variables are applied for nano-phase diagrams, and/or for phase diagrams under the influence of electric potential difference, external magnetic field, mechanical strain or the gravitational field.

On the temperature dependence of surface tension: Historical perspective on the Eötvös equation of capillarity, celebrating his 175th anniversary

The Hungarian baron Roland Eötvös (Eötvös Loránd, 1848-1919) lived in the difficult period between two revolutions in Hungary, but nevertheless he achieved revolutionary results in two fields of science: capillarity (1875-1886) and gravity (after 1886). This paper describes his famous capillary equation published in 1886 in the world-language of the time (German) and in one of the most famous scientific journals of the time (Annalen der Physik und Chemie). In his paper he showed a simple equation for the temperature dependence of surface tension of one-component liquids and more importantly he showed that this quantity approaches zero as temperature tends towards the critical temperature. This result was achieved by measuring the surface tension of 160 (!) different liquids along their boiling lines as function of temperature, in a home-made high-pressure high-temperature equipment, probably the first one of this kind. In this way he extended the meaning of the critical point previously introduced by van der Waals. In this paper, also a modern model of surface tension of one-component liquids is discussed, simplified and compared to the Eötvös equation. It is also shown, how the Avogadro number and the molecular sizes can be determined from the experimental results of Eötvös (note: the Avogadro number was estimated with reasonable accuracy for the first time by Einstein in 1905 from the kinetic theory of liquids). Apparently, it was not that easy to do back in 1886: this becomes obvious from the 1911-paper by Einstein, who gave a wrong estimate for the diameter of Hg atoms (5.19 nm) using the data of Eötvös (the correct value is around 0.3 nm). The Appendix to this paper contains the summary of 1543 handwritten pages on surface tension by Eötvös, including the on-line availability of all pdf files. Note also, that Eötvös used g = 10.0 m/s2 for acceleration due gravity and so he over-estimated his surface tension values and also his Eötvös constant by about 2.0%. The corrected Eötvös constant using his measured values but the correct g-value would be "0.222" vs his published value of "0.227". Probably this uncertainty in the value of g was one of the motives that pushed Eötvös to study gravity after 1886.

The thermal stability and degradation mechanism of Cu/Mo nanomultilayers

The microstructural evolution of Cu/Mo nanomultilayers upon annealing was investigated by X-ray diffraction and transmission electron microscopy. The isothermal annealing process in the temperature ranges of 300-850°C was conducted to understand the thermal behavior of the sample and follow the transformation into a nanocomposite. Annealing at 600°C led to the initiation of grain grooving in the investigated nanomultilayer, and it degraded into a spheroidized nanocomposite structure at 800°C. The sample kept the as-deposited Cu {111}//Mo{110} fiber texture up to 850°C. The residual stress was investigated to explain microstructure changes. The activation energy of degradation kinetics of Cu/Mo nanomultilayers was determined to understand the rate-determining mechanism for the degradation of nanolaminate structures.

Experimental and theoretical aspects of the growth of vertically aligned CNTs by CCVD on AZO substrate

An efficient and reproducible growth of vertically aligned carbon nanotubes by CCVD requires accurate and specific setting of the synthesis parameters and the properties of catalyst thin layers. In this work, the growth of vertically aligned carbon nanotubes onto AZO (= aluminum doped zinc oxide) glass substrate covered by Al2O3 and Fe-Co catalyst layer system is presented. Investigation of the effect of catalyst composition and synthesis temperature on CVD growth revealed the optimum condition of the synthesis. The analysis of as-prepared samples by SEM, TEM and Raman spectroscopy was carried out to prove the structure and quality of carbon deposit. Theoretical considerations have supported speculative ideas about the role of the support layer, the transformation of the catalyst layer in the presence of hydrogen gas and the growth mechanism of carbon nanotubes. The mechanism of CNT growth is modelled and the order of magnitude of experimentally observed vertical linear growth rate of CNT (several nm/s) is reproduced.

A Molecular Dynamics Study of Ag-Ni Nanometric Multilayers: Thermal Behavior and Stability

Nanometric multilayers composed of immiscible Ag and Ni metals were investigated by means of molecular dynamics simulations. The semi-coherent interface between Ag and Ni was examined at low temperatures by analyzing in-plane strain and defect formation. The relaxation of the interface under annealing conditions was also considered. With increasing temperature, a greater number of atomic planes participated in the interface, resulting in enhanced mobility of Ag and Ni atoms, as well as partial dissolution of Ni within the amorphous Ag. To mimic polycrystalline layers with staggered grains, a system with a triple junction between a silver single layer and two grains of nickel was examined. At high temperatures (900 K and 1000 K), the study demonstrated grain boundary grooving. The respective roles of Ni and Ag mobilities in the first steps of grooving dynamics were established. At 1100 K, a temperature close but still below the melting point of Ag, the Ag layer underwent a transition to an amorphous/premelt state, with Ni grains rearranging themselves in contact with the amorphous layer.

Interfacial Energy of Strained Coherent Interfaces and a New Design Rule To Select Phase Combinations for In Situ Coherent Nanocomposites

Nanocomposites show the best performance when their reinforcing phase precipitates in situ from a matrix upon heat treatment and when coherency between the matrix and the reinforcing phase is preserved even upon coarsening the precipitated particles. In this paper, first a new equation is derived for the interfacial energy of strained coherent interfaces. From here, a new design rule is derived in a form of a new dimensionless number to select phase combinations for in situ coherent nanocomposites (ISCNCs). This is calculated from the molar volume mismatch between the two phases, their elastic constants, and the modeled interfacial energy between them. When this dimensionless number is smaller than a critical value, ISCNCs are formed. The critical value of this dimensionless number is found here using experimental data for the Ni-Al/Ni₃Al superalloy. The validity of the new design rule was confirmed on the Al-Li/Al₃Li system. An algorithm is suggested to apply the new design rule. Our new design rule can be simplified to more easily available initial parameters: if the matrix and the precipitate have the same cubic crystal structure the precipitate is expected to form ISCNCs with that matrix if their standard molar volumes differ less than by about 2%.

General Adsorption Model to Describe Sigmoidal Surface Tension Isotherms of Binary Liquid Mixtures

Surface tension (σ) isotherms of liquid mixtures can be divided into Langmuir-type (L-type, including L_I- and L_II-type) and sigmoid-type (S-type, including S_I- and S_II-type). Many models have been developed to describe the σ-isotherms. However, the existing models can well describe the L-type isotherms, but not the S-type ones. In the current work, a thermodynamic model, called the general adsorption model, was developed based on the assumption of surface aggregation occurring in the surface layers, to relate the surface composition with the bulk one. By coupling the general adsorption model with the modified Eberhart model, a two-parameter equation was developed to relate the σ with the bulk composition. Its rationality was examined using the σ data of 10 binary mixtures. The results indicate that the new model can accurately describe the S- and L-type isotherms of binary liquid mixtures, showing a good universality. One advantage of the model is that its two parameters, i.e., the adsorption equilibrium constant (K) and the average aggregation number (n), can be estimated by linear fitting experimental σ data, thereby obtaining unique values. This model suggests that the S- and L_II-type isotherms arise from the surface aggregation (n ≠ 1). In addition, the standard molar Gibbs free energy of surface adsorption (ΔG̃_ad⁰) and the apparent surface layer thickness (τ) were analyzed for 10 binary mixtures. The ΔG̃_ad⁰ data suggest that the order of adsorption tendency is L_I-type ≫ S_I-type ≈ S_II-type > L_II-type, and the strong adsorption usually corresponds to large τ. This work provides a feasible model for describing the S-type isotherms and a better understanding of the surface properties of liquid mixtures.

Wettability of Metals by Water

The wetting behavior of water on metal surfaces is important for a wide range of industries, for example, in the metallurgical industry during the preparation of metallic nanoparticles or electrochemical or electroless coating preparation from aqueous solutions, as well as in the construction industry (e.g., self-cleaning metal surfaces) and in the oil industry, in the case of water–oil separation or corrosion problems. Wettability in water/metal systems has been investigated in the literature; nevertheless, contradictions can be found in the results. Some papers have reported perfect wettability even in water/noble metal systems, while other researchers state that water cannot spread well on the surface of metals, and the contact angle is predicted at around 60°. The purpose of this paper is to resolve this contradiction and find correlations to predict the contact angle for a variety of metals. In our research, the wetting behavior of distilled water on the freshly polished surface of Ag, Au, Cu, Fe, Nb, Ni, Sn, Ti, and W substrates was investigated by the sessile drop method. The contact angle of the water on the metal was determined by KSV software. The contact angle of water is identified as being between 50° and 80°. We found that the contact angle of water on metals decreases linearly with increasing the atomic radius of the substrate. Using our new equation, the contact angle of water was identified on all of the metals in the periodic table. From the measured contact angle values, the adhesion energy of the distilled water/metal substrate interface was also determined and a correlation with the free electron density parameter of substrates was determined.

Extension of the Gibbs-Duhem Equation to the Partial Molar Surface Thermodynamic Properties of Solutions

In this paper, the Gibbs-Duhem equation is extended to the partial molar surface thermodynamic properties of solutions. According to the surface Gibbs-Duhem equations, the sum of the mole fractions of the components in the surface region of a bulk solution multiplied by different partial molar surface quantities should equal zero if summation is taken by all components of the solution. There are four different partial molar surface quantities identified in this paper for which the surface Gibbs-Duhem equation is proven to be valid: (i) the reduced surface chemical potential, (ii) the surface chemical potential, (iii) the partial molar surface area, and (iv) the partial molar excess surface Gibbs energy = the product of partial molar surface area and the partial surface tension. The first one is known since Guggenheim (1940), but the other three are presented here for the first time. It is also demonstrated here how to apply the surface Gibbs-Duhem equations: (i) it is proven that the model equation applied by us recently for the reduced chemical potential [Adv Coll Interf Sci 2020, 283, 102212] obeys one of the surface Gibbs-Duhem equations, (ii) in contrary, it is proven that the model equation suggested by us recently for the partial molar surface area contradicts one of the surface Gibbs-Duhem equations; therefore, a new (and simpler) model equation for the partial molar surface areas of the components is suggested here that obeys the surface Gibbs-Duhem equation. It is also shown that the Butler equation obeys one of the surface Gibbs-Duhem equations. It is also concluded that surface composition in equilibrium should be one that ensures minimum surface tension.

Thermophysical behavior of mercury-lead liquid alloy

Thermophysical properties of compound forming binary liquid mercury-lead alloy at temperature 600 K have been reported as a function of concentration by considering HgPb2 complex using different modelling equations. The thermodynamic properties such as the Gibbs free energy, enthalpy of mixing, chemical activity of each component, and microscopic properties such as concentration fluctuation in long-wavelength limit and Warren-Cowley short range order parameter of the alloy are studied by quasi-chemical approximation. This research paper places additional emphasis on the interaction energy parameters between the atoms of the alloy. The theoretical and experimental data are compared to determine the model’s validity. Compound formation model, statistical mechanical technique, and improved derivation of the Butler equation have all been used to investigate surface tension. The alloy’s viscosity is investigated using the Kozlov-Ronanov-Petrov equation, the Kaptay equation, and the Budai-Benko-Kaptay model. The study depicts a weak interaction of the alloy, and the theoretical thermodynamic data derived at 600 K are in good agreement with the experimental results. The surface tension is slightly different in the compound formation model than in the statistical mechanical approach and the Butler equation at greater bulk concentrations of lead. The estimated viscosities in each of the three models are substantially identical.

Examination of the Butler Equation for the Surface Tension of Liquid Mixtures

The classical Butler equation used to describe surface tension and the surface composition of liquid mixtures is revisited. A straightforward derivation is presented, separating basic chemical thermodynamics and assumptions proper to Butler's model. This model is shown to conceal an approximation not recognized by other researchers. The shortcoming identified consists of not allowing surface standard values to vary with surface tension by virtue of the changing composition. A more rigorous equation is derived and shown to yield the Butler equation in case of incompressible surface phases. It is concluded that the Butler equation slightly overestimates ideal surface tensions. Butler's surface-phase concentrations of the surface-active component are also slightly overestimated in the surface-active component dilute range, being just underestimated at higher concentrations. Despite this, Butler's model stands as a very good standard due to its versatility.