Hierarchy of equations for the surface tension of solids

Of the four main equations in thermodynamics for the surface tension of condensed matter, i.e. the generalized and classical Lippmann equations and the Shuttleworth and Gokhshtein equations, only the classical Lippmann and Gokhshtein equations have been confirmed experimentally. The generalized Lippmann (Couchman–Davidson) equation is considered to be more universal, since three other equations could be derived from it. Although this fact has been widely accepted, it was recently reevaluated in two opposite ways. In the first approach, the experimental verification of the Gokhshtein equation should support the correctness of the generalized Lippmann and Shuttleworth equations. In the second approach, the incompatibility of the Shuttleworth equation with Hermann's mathematical structure of thermodynamics throws doubts upon all its corollaries, including the generalized Lippmann and Gokhshtein equations. However, both of these approaches are here shown to be erroneous, since the Gokhshtein equation cannot be correctly derived from any of the above-mentioned equations, and the opposite is also true: neither the generalized Lippmann nor Shuttleworth equations could be derived from the Gokhshtein equation.     

A kinetic approach to the calculation of surface tension of a spherical drop

In literature, surface tension has been investigated mainly from a Thermodynamics standpoint, more rarely with kinetic methods. In the present work, surface tension of drops is studied in the framework of kinetic theory, starting from the Sutherland approximation to Van Der Waals interaction between molecules. Surface tension is calculated as a function of drop radius: it is found that it approaches swiftly an asymptotic value, for radii of several times the distance of minimum approach D of the Sutherland potential. This theoretical asymptotic value is compared to experimental values of surface tension in plane surfaces of a few liquids, and is found in reasonable agreement.     

A note on surface stress and surface tension and their interrelation via Shuttleworth’s equation and the Lippmann equation

Some aspects of the thermodynamics of solid surfaces, in particular with respect to the surface stress, f, and surface tension, γ, including the case of solid electrodes, are examined in view of their controversial discussion in part of the recent literature. By inspection of the phenomenology that requires a distinction between f and γ, and of a toy model designed to highlight the underlying fundamental science, it is shown that some of the recent publications give misleading conclusions. These include [V.A. Marichev, Surf. Sci. 600 (19) (2006) 4527; E.M. Gutman, J. Phys. Condens. Matter. 7 (48) (1995) L663; D.J. Bottomley, T. Ogino, Phys. Rev. B 63 (2001) 165412]. In spite of claims to the contrary, the validity of the equations of Shuttleworth, Lippmann, and Couchman and Davidson is not impaired by the arguments of the aforementioned articles.     

Vague concept of “reversible cleavage” in the theory of the surface tension of solids

Numerous derivations of the well-known Shuttleworth equation have been based on the unclear concept of “reversible cleavage” leading to the decisive step in any derivation - equalization of the surface free energy and surface stress. This is the key concept in contemporary surface thermodynamics of solids. But “cleavage” is not a surface process and, in this field, it cannot be a reversible operation. Besides, the “reversible cleavage” has no formal definition in the domain of the surface tension of solids that is an abnormal for any exact science. Consequently, this concept and all its corollaries including the Shuttleworth and generalized Lippmann equations have to be recognized as incorrect.     

Investigation of surface properties of liquid transition metals: Surface tension and surface entropy

In the present study, surface properties namely surface tension and surface entropy of liquid transition metals have been reported. The surface entropy of liquid Fe, Co and Ni metals has been investigated using the expression derived by Gosh et al. [R.C. Gosh, A.Z. Ziauddin Ahmed, G.M. Bhuiyan, Eur. Phys. J. B 56 (2007) 177]. To describe interionic interaction the pseudopotential approach has been used and radial distribution functions have been determined from the solution of Ornstein-Zernike integral equation. The calculated values of surface tension and surface entropy agree well with experiment. The present study shows that the expression derived by Gosh et al. leads to a good estimation for the surface entropy.     

Theory of surface tension and its application to simple fluids

We consider a liquid-vapor interface in thermal equilibrium. The tangential component of the pressure tensor is supposed to depend explicitly upon the position and the density profile. Under this hypothesis the mechanical definition of surface tension becomes a finite summation ofN+1 terms related directly to the local compressibility. When the inhomogeneous compressibility equation is considered, the theory provides a microscopic expression of the surface tension coefficient. A calculation for argon near the critical point is done; the agreement with experiment is satisfactory.     

The surface tension of liquid Cu-Fe-Sb alloys

The results of study on the influence of temperature and iron and antimony on the surface tension of liquid ternary Cu-Fe-Sb systems are presented. The measurements were carried out with the sessile drop method, in a broad range of the alloy additions concentration (Fe and Sb). It was demonstrated that the surface tension varies as a linear function of temperature and concentration of iron. It was also demonstrated that antimony, in examined alloys, shows the properties characteristic of a surface-active substance, significantly reducing the surface tension value. The changes of the surface tensions as a function of concentration of antimony were described with the Szyszkowski's equation. Composition of surface layer, enriched with an antimony, was determined basing on the model, which used data regarding properties of binary systems. The surface tension values of Cu-Fe-Sb systems was also computed from model and compared with experimental data. A good agreement was obtained.     

A surface tension study of liquid threads

According to symmetry of liquid threads, definitions of surface tension in axial direction and angular direction are given. The formulas of surface tension in axial direction γ_z and surface tension in angular direction γ_θ are derived. A scheme to calculate Δγ = γ_z − γ_θ is designed. We investigate seven different systems (the numbers of molecules N are 1600, 2240, 2880,3360,4000,4800 and 5280) by molecular dynamics simulations. For liquid threads, Δγ increases with the decreasing radius of dividing surface. It shows that there exists surface tension anisotropy for liquid threads. The results obtained by molecular dynamics simulations support that surface tension is dependent on the dividing surface curvature.     

Model calculation of the surface tension of liquid Ga-Bi alloy

A convenient model, based on some assumptions, for calculating the composition and temperature dependence of the surface tension of binary liquid alloys is reported. The theoretical calculations of the surface tension of gallium-rich-bismuth alloys are presented. The calculated results are compared with the reported experimental data. A relatively good agreement with experimental behavior of the composition dependence of the surface tension was found, but a disagreement was observed with experimental temperature behavior of the surface tension of these alloys. The calculations were conducted in the temperature range from almost 320 K to about 800 K. The surface tension was calculated from eutectic composition (x_Bi = 0.0022) to x_Bi = 0.1, and worked out by linear equations. The model calculation and analysis indicate a first order surface phase transition in this system, which is in accord with experimental findings. For this system, γ decreases linearly with increasing temperature at fixed Bi mole fraction x_Bi, and thus, suggesting a positive surface excess entropy. It is also found that the surface tension isotherms show the linear dependence on the concentration, in the logarithm scale of x_Bi, in the very narrow concentration range.     

Assessing the equation of state and comparing it with other relationships used for determining the surface tension of solids

Some facts regarding the equation of state (EQS) in calculating the surface tension of solids by means of contact angle measurements were manifested. In the present investigation, it was mathematically proved that the surface tension of a solid as estimated by the EQS is in fact equivalent to the Zisman critical surface tension for that same solid. Additionally, the applicability of the EQS's approach in attaining the surface tension of powdered solids by the aid of the capillary rise procedure is also discussed and its limitations are clarified. Furthermore, a methodology was devised so that the surface tension of solids as determined by the EQS could be compared with those calculated by approaches using components of surface tension. This methodology revealed that the applications of approaches based on the geometric mean (i.e. Owens/Wendt and van Oss et al. relationships) are restricted to achieving only high surface tensions of solids.     

The surface tension and density of the Cu-Ag-In alloys

The surface tension and the densities of the Cu-Ag-In alloys have been measured by means of the sessile drop method. The density of these alloys depends linearly on temperature in the case of all the investigated compositions. The surface tension shows a linear dependence on temperature except for the lowest temperatures. For most of the alloys, the surface tension at the lowest temperature is lower than that predicted by the straight line. The experimental values of the surface tension of the Cu-Ag-In alloys are compared with those computed from the model, and quite good agreement is observed.     

A molecular dynamics study of effective parameters on nano-droplet surface tension

The main purpose of this paper is to numerically study the effect of droplet radius, temperature, and surface wettability on droplet surface tension. Moreover, the validity of Young-Laplace equation (Y-L) for nano-droplet is examined. Simulations of droplet surrounded by its vapor and droplet on solid surface are carried out and the results are compared to each other in order to comprehend the role of surface wettability on droplet surface tension. The pair potential for the liquid-liquid and liquid-solid interaction is considered using Lennard-Jones model. Different numbers of atoms and surface wettabilities are employed to generate droplet of different radiuses. In addition, contact angle of droplet on solid surface is computed. Pressure tensor and density profile is locally calculated. Furthermore, liquid pressure is evaluated far from the interface using the virial theorem and gas pressure is obtained using an equation of state. In order to calculate the surface tension, two different approaches are employed; Young-Laplace equation and direct molecular dynamics (MD) simulation. The surface tension increases with increase in droplet radius and it is seen that the surface wettability does not directly influence the surface tension.     

Estimating critical surface tension from droplet spreading area

Critical surface tension (CST) is a measure of solid surface tension and is mainly determined by measuring the contact angle of a droplet on a target solid surface. The concept of CST makes it possible to determine solid surface tension without any unprovable assumptions such as the Fowkes hypothesis. However, it requires somewhat special devices and skills for measuring the contact angle. In this work, we propose a simple method to determine the CST of a solid by measuring the droplet spreading area. This method is developed by combining the conventional CST with a simple analytical droplet model. The difference in estimated CSTs between our method and the conventional one is within 3.0%. Our method enables a quick and simple evaluation of the solid surface tension without special devices for measuring the contact angle.     

Ultrasonically tuned surface tension and nano-film formation of aqueous ZnO nanofluids

Nanofluids’ thermophysical properties and heat transfer performance has been investigated for many years, while research on their surface tension (ST) and wetting behavior is very limited. To assess nanofluids potential as industrial products, a complete picture is required to prove their performance in a specific application. Boiling heat transfer, microfluidics and drug development are among the applications where ST is a variable. ST of water-based ZnO nanofluids were measured in the presence and absence of direct ultrasonication. The experiments covered variation of ST with ZnO concentration (0.05–0.4 vol%), ultrasonication amplitude (40% and 100%) and duration. To the best of the authors’ knowledge, this is the first report of ST– ultrasonication process relation for a nanofluid. Results showed that after direct ultrasonication, nanofluids ST is strongly affected by the temperature raise, and in those cases relative ST may provide a clearer picture. A nano-film over individual and agglomerated nanoparticles spotted via TEM imaging was affected from the ultrasonication. Such a nano-film can play a key role in the anomalous thermal transport and wettability of nanofluids. Statistical analyses revealed that changes in ultrasonication amplitude resulted in a statistically significance difference on nanofluid ST and relative ST. Changes in nanoparticle concentration caused a significant difference on the nanofluid ST while the difference in relative ST was insignificant. Variation of ultrasonication duration caused significant variations on the relative ST while the difference in nanofluid ST was not significant. This work highlights that based on specific applications ST and other related features of any nanofluid can be adjusted employing proper ultrasonication conditions.     

Convexity properties of the surface tension and equilibrium crystals

We study the thermodynamic limit of the orientation-dependent surface tension. Under general conditions, which we show to hold true for a large class of lattice systems, we prove that the limit exists and that it satisfies some convexity properties related to the pyramidal inequality introduced by R. L. Dobrushin and S. B. Shlosman⁽¹⁾. We discuss some consequences of these results for the equilibrium crystal shape.     

Estimation of surface tension for high temperature melts of the binary Ag-X alloys

ABSTRACT

Surface tension is a key property to materials. In this work, the surface tension of the binary alloys Ag-X (Au, Cu, Ce, Bi, Sn, Sb, In, Ni, Y, Pd) is carried out by using Butler Model over enter composition ratio at a certain temperature. According to calculation results, the increasing surface tension of the Ag-X (Cu, Au, Ni, Y, Pd) alloy is accompanied by the composition increases. For Ag-Sn alloy, the surface tension calculated by Butler model is consistent with the experimental result at temperature 1273?K. However, other Ag-X alloys can’t be compared due to the lack of the related experimental data. Although the experimental data about surface tension of the Ag-X alloy are limited, we are possible to make a comparison between the calculated results for the surface tension in this study and the available experimental data. Taken together, the surface tension calculated by Butler model that especially the Ag-Sn alloy are consistent with the experimental results at temperature 1273?K.     

Effect of surface tension on a liquid-jet produced by the collapse of a laser-induced bubble against a rigid boundary

The effect of surface tension on the behavior of a liquid-jet is investigated experimentally by means of a fiber-coupled optical beam deflection (OBD) technique. It is found that a target under water is impacted in turn by a laser-plasma ablation force and by a high-speed liquid-jet impulse induced by bubble collapse in the vicinity of a rigid boundary. The liquid-jet impact is found to be the main damage mechanism in cavitation erosion. Furthermore, the liquid-jet increases monotonously with surface tension, so cavitation erosion rises sharply with increasing surface tension. Surface tension also reduces bubble collapse duration. From the experimental results and the modified Rayleigh theory, the maximum bubble radius is obtained and it is found to reduce with increasing surface tension.     

Thermodynamics and surface properties of liquid Cu–B alloys

The study of the thermodynamic and the surface properties of liquid Cu–B alloys can help better understanding of a complex interfacial chemistry related to liquid Cu–brazes in contact with boride substrates. Despite a simplicity of the Cu–B phase diagram, only a few thermodynamic data are available in the literature, while in the case of the surface properties a complete lack of data is evident. The quasi-chemical approximation (QCA) for the regular solution has been applied to describe the mixing behaviour of liquid Cu–B alloys in terms of their thermodynamic and surface properties as well as the microscopic functions. In view of joining processes related to liquid Cu–brazes/solid boride systems a particular attention is paid to the surface properties of the Cu-rich part of the system and the calculated values are substantiated by the new surface tension experimental data of liquid Cu and Cu–10 at.% B alloy. The tests have been performed by the sessile-drop method under the same experimental conditions. Considering the experimental uncertainties, the effect of oxygen on the surface tension of liquid Cu and Cu–10 at.% B alloy has been analysed by simple model that combines the physical property data included in Butler’s equation with the oxygen solubility data and it gives the same results as Belton’s adsorption equation.     

Thermodynamic approach to competition between surface and volume reactions

Summary Using the monolayer model, a thermodynamic approach is suggested in order to compare the surface reactivity to the volume reactivity under both equilibrium and nonequilibrium conditions (sorption and chemical processes). Capillary parameters, such as surface tension and surface dilation, have a direct influence on the efficiency of chemical reactions and on their stability.

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Riassunto Usando un modello monostrato, si suggerisce un approccio termodinamico per confrontare la reattività di superficie con quella di volume sia in condizioni di equilibrio che di non equilibrio (assorbimento e reazioni chimiche). Parametri capillari, sosì come la tensione di superficie e la dilatazione di superficie, hanno un'influenza diretta sull'efficacia delle reazioni chimiche e sulla loro stabilità.