Butler Equation Research Articles

The Gibbs and Butler Equations and the Surface Activity of Dilute Aqueous Solutions of Strong and Weak Linear Polyelectrolyte-Surfactant Mixtures: The Roles of Surface Composition and Polydispersity.

In a previous paper, we applied a combination of direct measurements of both surface tension and surface excess in conjunction with the Gibbs equation to explain features of the adsorption and surface tension of mixtures of surfactants and strong linear polyelectrolytes at the air-water interface. This paper extends that model by including (i) the restrictions of the Butler equation for the behavior of the surface tension of mixed systems and (ii) the surface behavior of surfactant and linear weak polyelectrolyte mixtures, for which the inclusion of measurements of the surface excess and composition is shown to be particularly important. In addition, a closer examination of earlier data at higher concentrations provides evidence that the surface layering that is often observed in polyelectrolyte-surfactant systems is also an average equilibrium phenomenon and is driven by particular aggregation patterns that occur in some systems and not in others. Although the successful application of the Gibbs and Butler equations indicates that strong polyelectrolyte-surfactant systems can be described in terms of an average equilibrium over wide ranges of concentration, we have identified two concentration ranges where polydispersity in either polyelectrolyte molecular weight or composition results in significant time dependence of the surface behavior.

Puzzles of Surface Segregation in Binary Pt–Pd Nanoparticles: Molecular Dynamics and Thermodynamic Simulations

Up till now, there have been extremely contradictory opinions and inadequate results concerning surface segregation in binary platinum–palladium (Pt–Pd) nanoparticles, including the problems regarding segregating components, as well as the size and temperature dependences of segregation. Taking into account such a situation, we investigated the surface segregation in Pt–Pd nanoparticles by combining atomistic (molecular dynamics) and thermodynamic simulations. For molecular dynamics experiments, the well-known program LAMMPS and the embedded atom method were employed. In the course of the atomistic simulations, two different sets of parameterizations for the Pt–Pt, Pd–Pd, and Pt–Pd interatomic interaction potentials were used. The thermodynamic simulation was based on solving the Butler equation by employing several successive approximations. The results obtained via atomistic simulation and thermodynamic simulation on the basis of the Butler equation were compared with each other, as well as with predictions that were based on the Langmuir–McLean equation and some experimental data. Both simulation methods (atomistic and thermodynamic) predicted the surface segregation of Pd, which diminishes with the nanoparticle size and with increasing temperature. Our simulation results do not confirm the predictions of some authors on surface segregation inversion, i.e., the reversal from the surface segregation of Pd to the surface segregation of Pt when diminishing the nanoparticle size.

A thermodynamic framework for bulk–surface partitioning in finite-volume mixed organic–inorganic aerosol particles and cloud droplets

Abstract. Atmospheric aerosol particles and their interactions with clouds are among the largest sources of uncertainty in global climate modeling. Aerosol particles in the ultrafine size range with diameters less than 100 nm have very high surface-area-to-volume ratios, with a substantial fraction of molecules occupying the air–droplet interface. The partitioning of surface-active species between the interior bulk of a droplet and the interface with the surrounding air plays a large role in the physicochemical properties of a particle and in the activation of ultrafine particles, especially those of less than 50 nm diameter, into cloud droplets. In this work, a novel and thermodynamically rigorous treatment of bulk–surface equilibrium partitioning is developed through the use of a framework based on the Aerosol Inorganic–Organic Mixtures Functional groups Activity Coefficients (AIOMFAC) model in combination with a finite-depth Guggenheim interface region on spherical, finite-volume droplets. We outline our numerical implementation of the resulting modified Butler equation, including accounting for challenging extreme cases when certain compounds have very limited solubility in either the surface or the bulk phase. This model, which uses a single, physically constrained interface thickness parameter, is capable of predicting the size-dependent surface tension of complex multicomponent solutions containing organic and inorganic species. We explore the impacts of coupled surface tension changes and changes in bulk–surface partitioning coefficients for aerosol particles ranging in diameters from several micrometers to as small as 10 nm and across atmospherically relevant relative humidity ranges. The treatment of bulk–surface equilibrium leads to deviations from classical cloud droplet activation behavior as modeled by simplified treatments of the Köhler equation that do not account for bulk–surface partitioning. The treatments for bulk–surface partitioning laid out in this work, when applied to the Köhler equation, are in agreement with measured critical supersaturations of a range of different systems. However, we also find that challenges remain in accurately modeling the growth behavior of certain systems containing small dicarboxylic acids, especially in a predictive manner. Furthermore, it was determined that the thickness of the interfacial phase is a sensitive parameter in this treatment; however, constraining it to a meaningful range allows for predictive modeling of aerosol particle activation into cloud droplets, including cases with consideration of co-condensation of semivolatile organics.

Thermophysical Assessment of Lead Magnesium Alloy at Different Temperatures

Abstract: The mixing behavior of lead–magnesium liquid alloy is studied using different modeling equations at various temperatures. The quasi-lattice model has been employed too analyze the concentration-dependent thermodynamic and structural properties of lead–magnesium liquid alloy. To validate the model, the obtained theoretical results are compared with experimental results. The viscosity of the alloy has been studied by the Kozlov–Romanov–Petrov equation, the Kaptay equation, and the Budai–Benko–Kaptay model, whereas surface tension has been studied by the Butler equation, tatistical mechanical approach, and the Compound formation model. The primary focus of this study is the interaction parameters among the atoms of the alloy. The alloy shows the moderately interacting and ordering nature within the entire concentration of lead. There is reasonable agreement between the theoretical and experimental data at 973K. The study concludes that the alloy depicts ordering tendency and viscosity and surface tension both decrease with increase in temperature. Keywords: Ordering, Complex formation, Asymmetric, Statistical mechanical approach.

Evaluation of Surface Tension of CaO–SiO2–Al2O3‐Based Mold Flux Containing Ce2O3

The calculation model based on the surface tension of CaO–SiO2–Al2O3‐based mold flux containing Ce2O3 is studied using the modified Butler equation, and the surface tension of mold flux is measured by the sessile drop method. The experimental results show that the calculated surface tension based on the model agrees well with the experimental values, and the absolute value of the error is less than 3.23%. With the content of Ce2O3 in the mold flux increasing from 0% to 20%, the surface tension of the mold flux increases from 0.380 to 0.405 N m−1. The O2− dissociated from Ce2O3 makes the radius of the silicon–oxygen tetrahedral anion decrease, and the ion‐to‐ion interaction force in the melt increase. Therefore, the surface tension of the melt increases with the increase of Ce2O3 content in the mold flux. However, the amount of dissociated O2− is limited, so the effect of Ce2O3 on the surface tension of the mold flux is limited.

Thermophysical properties in the Al-Cu-Ag system: A combined CALPHAD and first-principles study

Thermophysical properties play an essential role in the microstructure simulation during precipitation and solidification processes in materials science and engineering. In this work, a comprehensive investigation on the molar volume and surface tension of the Al-Cu-Ag ternary system was performed by the CALPHAD (CALculation of PHAse Diagram) approach and the first-principles calculation. Based on the critical review of experimental and theoretical data, the molar volume description of pure Ag in the liquid phase, and subsequently the mole fraction dependent thermophysical databases of molar volume and surface tension for the Al-Cu-Ag system including metastable phases were constructed via the CALPHAD approach. Then considering the excess molar volume, the surface tension of solution alloys could be predicted by the Butler equation using a home-made code with the input of thermodynamic and thermophysical parameters. With the increment of temperature, molar volume of the alloys increases while the surface tension decreases. All the calculated results using the present parameters are in satisfactory agreement with the corresponding experimental data, which could be subsequently applied to simulate the microstructure evolution like precipitation kinetics and phase-filed simulation.

Surface tension of liquid Ti, V and their binary alloys measured by electromagnetic levitation

AbstractThe surface tension of the liquid Ti-V system is systematically measured using the oscillating drop technique during electromagnetic levitation. Temperature- and compositional dependence are both investigated. The entire compositional range is covered. A linear decrease with increasing temperature is found for the pure elements as well as for all investigated alloys. The surface tension generally increases with increasing V-content. The obtained data are in good agreement with the Butler model for the ideal solution. Additionally, the Butler model for the regular solution was evaluated in the context of the obtained surface tension data. In contrast to many other Ti-based alloys, the Butler model for the regular solution yields no additional benefit for Ti-V, since there is only a neglectable small deviation between the calculations for the ideal and regular solution. Segregation effects are modeled using the Butler equation for an ideal solution. The findings are discussed considering already existing trends for the mixing behavior of liquid Ti-alloys. The results strongly suggest, that the Ti-V system obeys in general the ideal solution law.

High Temperature Stability of K-Pb Liquid Alloy

The study explores the thermo-physical properties of complex binary liquid potassium-lead alloy at temperature 848 K as a function of concentration by considering complex using different model equations. The Quasi Chemical approximation and the R-K equation are used to investigate features such as free energy, heat of mixing, chemical activity, and concentration fluctuation in the long wave limit at temperature of 848 K. However, at 900 K and 1000 K, these are exclusively examined using the R-K equation. The temperature dependent exponential interaction parameters proposed by Kaptay are taken into account in the RK equation. The study goes on to look at the alloy's viscosity and surface tension using the Budai-Benko-Kaptay model and the Kaptay's improved derivation of Butler equation. The mixing nature of the system is investigated in depth, with a focus on the interaction energy parameters between the alloy's surrounding atoms. The work investigates the fact that the liquid alloy has a moderately interacting as well as ordering character throughout a whole concentration range, and the computed theoretical thermodynamic facts are in reasonable agreement with the corresponding experimental data at 848 K. At greater temperatures, the alloy's tendency goes from ordering to segregating. The alloy's viscosity and surface tension decrease as the temperature rises.

Butler‐based thermodynamic modeling of interfacial energies for in‐vessel corium systems

The modeling of transient phase segregation phenomena in corium multicomponent thermodynamic systems is complicated especially in the liquid miscibility gap which can lead to phase separation (stratification of the liquid phases for corium) and is of tremendous importance. The modeling of the associated phase separation transients in corium stratification is limited by a deficiency in the interfacial energy data, which are difficult to measure experimentally for these high-temperature systems. The limitations of summarizing the interfacial energies as constant coefficients were observed in a recent work on simulation of the stratification kinetics of a corium pool and the necessity to improve the estimation of interfacial energies [1,2]. This work tests the efficacy of a thermodynamic approach to evaluate the interfacial tensions, initially proposed for metallic systems by Kaptay [3]. This method is based on the Butler equation and makes use of existing thermodynamic databases for multicomponent systems constructed by the CALPHAD method. In practice, the application of this method to corium systems was performed using the open-source code openIEC [4]. To do so, this code was modified to enable required features, both functionally and methodically. In particular, the OpenCalphad Gibbs energy minimizer has been interfaced in this framework and benchmarked against previous results. Then, interfacial energies were calculated for two relevant corium systems (U-O and U-O-Zr-Fe) using two different existing databases (TAF-ID and NUCLEA). The first results obtained were analyzed in terms of general trends and compared to existing theoretical aspects. Large differences were observed depending on the thermodynamic database that was used.

Melting and phase diagram of Au–Co alloy at nanoscale by thermodynamic modeling and key experiments

Au–Co nanoparticles (NPs) were prepared by water-phase chemical reduction method with metal precursor. The samples were coated with silica by Stöber method and characterized by ultraviolet and visible spectrophotometer (UV–vis), transmission electron microscopy (TEM) to obtain the absorption spectrum, size, composition distribution of the nanoparticles. The melting behaviors of the prepared Au8Co2 NPs and Au6Co4 NPs were examined by differential scanning calorimetry (DSC). It was observed by DSC measurements that two endothermic effects existed in range of 700.0 K–1350.0 K, corresponding to the melting temperatures of the nanoparticles and the bulk Au–Co alloys, respectively. The surface tension, surface segregation and surface energy of Au–Co NPs of particle size and temperature were calculated by the Butler's equation. On the basis of the obtained experimental results and the Gibbs free energy contributed by surface effects of pure elements and alloys at nanoscale, the nanophase diagrams of Au–Co system with various sizes were optimized. It was indicated that the calculated nanophase diagrams were in good agreement with the experimental results. The calculated eutectic temperatures decreased with the decrease of the size.

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.

Surface tension measurement of Ti–Zr and Ti–Hf alloys at high-temperature by the pendant drop method

High-temperature alloy melts introduced into carbon/carbon (C/C) composites by reactive melt infiltration (RMI) to form ultra-high-temperature ceramics (UHTCs) is currently an important method for improving their ablation resistance. As a fundamental thermo-property, surface tension has an important impact on the infiltration of alloy melts into the C/C composites. However, the surface tension of binary alloy systems, such as Ti–Zr and Ti–Hf, used for RMI has been little studied. In this work, experimental measurements through the pendant drop method and theoretical calculations of the surface tension of Ti–Zr and Ti–Hf alloys with different compositions have been carried out. Meanwhile, inevitably, impurities are introduced during the determination of surface tension at high temperatures, mainly carbon and oxygen. The effect of impurities on the phase composition, distribution, and surface tension of the alloys is investigated. The results demonstrated that the experimental measurements are in good agreement with the theoretical values of Butler's equation. The surface tension of Ti–Zr and Ti–Hf alloys increases with the content of Zr or Hf at the same temperature. Small amounts of oxygen are mainly dissolved in the lattice and have no significant effect on the surface tension. Carbon slightly decreases the surface tension of the alloy.

Application of surface tension in the design of novel Sn-Ag-Cu-based solders

Surface tension is closely related to the wettability of solders, which is a crucial property for the design of Pb-free solders. However, the experimental surface tension data are scarce, especially for the multicomponent systems. In this paper, the surface tension of Sn-Ag-Cu (SAC) based ternary, quaternary and quinary alloys are predicted by various models and verified by experimental data. The calculated results by General Solution model (GSM) combined with binary experimental data are in best agreements with experimental values for most systems. For the systems without enough binary experimental data, GSM model combined with binary Butler equation can also achieve reliable results, which has been applied for predicting the surface tension of SAC305-based quaternary and quinary alloys to guide the design of novel Pb-free solders.

Size- and shape-dependent phase diagram of Ga-Sb nanoparticles

It is well known that the phase diagrams calculated using bulk Gibbs energy alone is insufficient when the dimensions of alloy particles are less than 100 nm. This is due to the surface energy contribution to the Gibbs energy, which increases as surface-to-volume ratio of alloy particles increases. In this work, we have used the CALPHAD approach to calculate the phase diagram of the Ga-Sb system taking into account of the size and shape of alloy particles. To accomplish this, we have added size-dependent surface energy terms and optimized the Gibbs energy expressions of various phases in the Ga-Sb system using surface tension information. The surface tension of the liquid phase is calculated using the Butler equation. The surface energy contribution to the Gibbs energy of the terminal phases is approximated using the surface tension expressions of the elements. The surface energy of the line compound GaSb is computed using the slab model within the framework of Density Functional Theory (DFT). The calculations show that there is lowering of liquidus and eutectic temperatures when size decreases. Among the two shapes considered, the depression is more prominent in the nano-spheres than the nano-rods.

Facile fabrication of ternary metal oxide ZnO/CuO/SnO2 nanocomposite for excellent photocatalytic degradation of fast green dye

ABSTRACT The uncontrolled release of hazardous dyes from textile industries without any pre-treatment has resulted in severe aquatic environment contamination. Owing to their complex aromatic structures, these dyes remain in the water bodies for longer periods. To eradicate toxic dyes from the aquatic ecosystem, there is an immense need to develop advanced scientific approaches. Here, we report the synthesis of a novel ternary metal oxide nanocomposite, ZnO/CuO/SnO2, having high photocatalytic activity. The ZnO/CuO/SnO2 nanocomposite characterisation was done by XRD, FT-IR, UV-Vis, PL, TGA, FE-SEM, TEM, and BET techniques. The composition of the nanocomposite was investigated by XRD analysis, which revealed the presence of three oxide components. The optical properties such as bandgap, were calculated from UV–Vis absorbance data using the Butler equation. The observed bandgap for zinc oxide (ZnO), copper oxide (CuO), tin oxide (SnO2), and ZnO/CuO/SnO2 was noticed as 3.18 eV, 1.3 eV, 3.5 eV, and 2.0 eV, respectively. The ZnO/CuO/SnO2 nanocomposite was applied to degrade the fast green dye under visible light irradiation and exhibited 93.68% degradation efficiency within a short period. Radical trapping experiments were performed to understand the photocatalytic mechanism, superoxide and holes were revealed as the main reacting species. Lastly, the composite catalyst showed excellent photostability with significant retention of photocatalytic efficiency even after four catalytic cycles.

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.

Complex formation of sodium-mercury alloy at molten state

Thermodynamic properties of compound forming binary liquid sodium-mercury alloy at temperature 673 K have been analyzed as a function of concentration by considering complex using quasi chemical approximation. The surface tension of the alloy has been studied by the compound formation model, Butler equation as improved by Kaptay, and Statistical mechanical approach. The mixing behavior of the alloy is studied in detail with more emphasis on the interaction energy parameters between neighboring atoms of the alloy. The study provides the information of moderately interacting as well as ordering nature on the entire range of concentration of the liquid alloy and the computed theoretical thermodynamic data are in good agreement with the corresponding experimental data at 673 K. The surface tension of the alloy computed predicts deviation from the ideal case.

On the problem of stability/instability of bimetallic core-shell nanostructures: Molecular dynamics and thermodynamic simulations

Earlier we put forward a hypothesis about relationship between the spontaneous surface segregation of one of the components of binary A-B nanoparticles and stability/instability of one of the two alternative core-shell nanostructures A@B or B@A. Here, the first symbol (before the @ sign) corresponds to the particle core, and the second (after @) to its shell. In accordance with our hypothesis, stable or at least more stable should be that of the two alternative nanostructures, the shell of which corresponds to the component spontaneously segregating to the surface. Verification of this hypothesis and revealing possible deviations from it have been carried out by using molecular dynamics simulation of Co@Au, Au@Co, Ni@Cu, Cu@Ni, Al@Ni and Ni@Al nanostructures. In addition, the atomistic and thermodynamic simulations were involved to predict the surface segregation in corresponding binary nanoalloys (Co-Au, Ni-Cu and Ni-Al). The thermodynamic simulation of segregation in binary metal nanoparticles was based on numerical solution of the Butler equation for a two-cell particle model. It has been established that the above hypothesis is indeed fulfilled but in some certain temperature ranges. During discussion of the results obtained, the concept was put forward of thermodynamic and kinetic factors of the core-shell nanostructure stability.

Comparison of Thermo-physical Properties of Mg-Ga and Mg-Pb Alloys at 1000 K

Concentration-dependent thermophysical properties of molten Mg-Pb and Mg-Ga alloys at 1000 K was compared using the Redlich Kister equation by optimizing exponential interaction energy parameters based on the R-K polynomials framework. The mixing behavior was investigated by giving more emphasis to the role of temperature-dependent interaction energy parameters. Our study shows that the magnesium gallium alloy is slightly interacting than the magnesium lead alloy. The surface tension and viscosity of both alloys was compared using the Butler equation as improved by Kaptay and KRP (Kozlov-Ronanov-Petrov) approach respectively. The surface tension of Mg-Pb liquid alloy increases but decreases in Mg-Ga alloy with an increase in the concentration of Mg. The viscosity has a nonlinear variation for both alloys with the increase in the concentration of magnesium.

Thermodynamic, surface and viscous properties of ternary Al–Fe–Mn alloy using theoretical modellings

ABSTRACT Thermo-physical properties of ternary Al-Fe-Mn alloys were studied using various theoretical models. The excess free energy of mixing of binary sub-systems of the ternary alloy was computed using an exponentially optimised interaction parameter in the framework of the R-K polynomial. The excess free energy of mixing of the ternary alloy was computed using different theoretical models. The surface properties of the binary and ternary alloys were calculated using Butler equation. The values of surface tensions so obtained were then used to calculate the excess surface tensions of the binary sub-systems. The optimised parameters for the excess surface tension of the binary sub-systems were then used to calculate the surface tension of the ternary alloy in the framework of different modelling equations. The viscosities of the binary sub-systems were computed using the Kaptay, Kozlov-Romanov-Petrov and Moelwyn-Hughes models and the viscosity of ternary alloys was obtained using the Chou, Toop, Kohler and Kaptay models.