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

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.

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.