On the size dependence of the surface tension

The general form of a differential equation that deduces a size dependence of the surface tension is derived. The well-known Gibbs-Tolman-Koenig-Buff equation for the spherical surface is a particular case of the newly derived one. Analytical solutions to this equation for the spherical, cylindrical, parabolic, and conical surfaces are found.     

Size dependence of the surface energy and surface tension of metal nanoparticles

The specific surface energy of small nanoclusters of transition fcc metals is calculated using the many-particle tight-binding potential and its earlier analog, the Gupta potential. The calculations are performed using both a theoretical model and computer simulations based on the Monte Carlo method. The equimolecular surface is considered as the dividing surface. It is found that at short radii of the nanoclusters, the surface energy and surface tension grow linearly as the particle radius increases. The values of the coefficient of proportionality in the dependence of the specific surface energy on the radius of a nanocluster are compared to the available literature data, experimental and otherwise.     

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.     

Size effect on surface tension of liquid binary alloy droplets

A simple model for size-dependent surface tension of liquid binary alloy droplets has been established based on Bulter’s equation and our model for size-dependent surface tension of pure liquid component. As an example, the surface tension of liquid Bi–Sn alloy droplets are calculated and discussed. The results show that as the size of the liquid alloy droplets decreases, the corresponding surface tension decreases. The component with lower surface tension is enriched in the surface layer at all times while relatively more another component with higher surface tension appear in the surface region when the size decreases. The effect of decreasing size on liquid alloy surface tension is like that of increasing temperature. When size is larger than about 12 nm, the size effect is small and negligible.     

The dependence of surface tension of solid nanoscale films on their thickness

This paper presents a theoretical method to calculate the surface energy dependence on thickness of nanofilms based on the Landau theory. For the first time, it is shown that the surface energy of thin films having free surfaces is greater than the surface energy of macroscopic objects. For nano-objects having free surfaces, it is stated that their interior order parameter is always less than that of macroscopic solids of the same composition. It is obtained that the surface energy of thin films increases with decrease in their thickness passing its maximum meaning. A further decrease in the solid film thickness leads to a monotonic decrease in the surface energy.     

Orientation dependence of the solid/liquid interfacial free energy in binary systems

A first-order theory of compositional segregation at solid/liquid interfaces, based on a pair-bonded, lattice-liquid interfacial model, has been applied to predict the effect of segregation on the orientation dependence of the interfacial free energy in binary metallic systems. The results show that the sharpness of the cusps in the gamma plot is reduced due to preferential segregation at layer edges as compared to layer faces, and cusps may be eliminated under certain conditions. The reduction in cusp sharpness is the greatest when the composition difference of the solid and liquid phases is large and the solutions are appreciably non-ideal. The relative reduction of sharpness due to segregation is less pronounced for cusps which are sharper in the unsegregated condition, so segregation tends to smooth the form of the gamma plot. Graphical results are presented for calculation of segregational anisotropy effects in general systems.     

Temperature dependence of surface magnetization in local-moment systems

We present a theory to study the temperature-dependent behavior of surface magnetization in a ferromagnetic semi-infinite crystal. Our approach is based on the single-site approximation for the s-f model. The effect of the semi-infinite nature of the crystal is taken into account by a localized perturbation method. Using the mean-field theory for the layer-dependent magnetization, the local density of states and the electron-spin polarization are investigated at different temperatures for ordinary and surface transition cases. The results show that the surface magnetic properties may differ strongly from those in the bulk and the coupling constant of atoms plays a decisive role in the degree of spin polarization. In particular, for the case in which the exchange coupling constant on the surface and between atoms in the first and second layer is higher than the corresponding in the bulk, an enhancement of surface Curie temperature and hence the spin polarization can be obtained.     

Size dependence of the temperature coefficient of surface tension for a solid nanoparticle at the interface with vapor

New relationships have been obtained for the temperature coefficient of surface tension for a solid spherical nanoparticle at the interface with vapor dσ/dT as a function of the radius r of the surface of tension for two different cases of two-phase equilibrium (at a constant radius of the curvature r = const and at a constant pressure in the vapor phase P ^β = const), as well as for the case of three-phase (solid nanoparticlevapor-liquid) equilibrium at arbitrary values of r and P ^β. The dependences of dσ/dT, dT/dr, and σ on r have been self-consistently calculated for many metals.     

On the behavior of the surface tension for spin systems in a correlated porous medium

We investigate the behavior of various spin-systems that are subject to the highly correlated and extremely diluted quenched disorder as provided by the fractal aerogel model. For these systems, it is (easily) established that, at all temperatures, the free energy is identical to that of the corresponding uniform system. The surface tension, however, behaves quite differently. Foremost, at any fixed temperature corresponding to the low temperature phase in the uniform system, there is a non-trivial curve in the aerogel phase plane dividing high-temperature behavior (zero surface tension) from low-temperature behavior (positive surface tension). The fractal aerogel has two distinctive phases in its own right: gel and sol. In the gel phase, the spin system has zero surface tension at all temperatures. In one region of the sol phase, the surface tension is shown to be equal to its value in the uniform system. Since part of this region borders on the gel phase, a certain portion of the sol/gel phase boundary is also the dividing line between high- and low-temperature behavior. Evidently, in this case, the surface tension is discontinuous at the phase boundary. on the other hand, there are well-defined length scales that diverge as the phase boundary is approached. This demonstrates an absence of scaling in these systems.     

Surface tension of melts of binary alkali element systems: The sodium-cesium system

Experimental data are presented for the surface tension of melts of the sodium-cesium system in the liquid state, obtained by the large drop method on samples prepared from high-purity components. It is shown that cesium in melts exhibits very high surface activity, attaining in extreme cases ～3000 mN/(m at %). Our results verify data of the activity ratio of cesium in melts with sodium and correlates with the basic criteria for surface activity in binary metal melts.     

Temperature dependence of surface tension and capillary waves at liquid metal surfaces

The temperature dependence of the surface tension γ(T) is treated theoretically and experimentally. The theoretical model based on the Gibbs thermodynamics of a one-component fluid relates ∂γ/∂T to the surface excess entropy density −ΔS. All specific surface effects, namely ordering, capillary waves, and double layer influence the surface entropy, which in turn governs the sign and the magnitude of ∂γ/∂T. Experimental data collected at a free Hg surface in the temperature range from 0°C to 30°C show that ∂γ/∂T is negative. Zh. éksp. Teor. Fiz. 114, 2034–2042 (December 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Finite-size scaling and surface tension from effective one dimensional systems

We develop a method for precise asymptotic analysis of partition functions near first-order phase transitions. Working in a (+1)-dimensional cylinder of volumeL×...×L×t, we show that leading exponentials int can be determined from a simple matrix calculation providedtv logL. Through a careful surface analysis we relate the off-diagonal matrix elements of this matrix to the surface tension andL, while the diagonal matrix elements of this matrix are related to the metastable free energies of the model. For the off-diagonal matrix elements, which are related to the crossover length from hypercubic (L=t) to cylindrical (t=) scaling, this includes a determination of the pre-exponential power ofL as a function of dimension. The results are applied to supersymmetric field theory and, in a forthcoming paper, to the finite-size scaling of the magnetization and inner energy at field and temperature driven first-order transitions in the crossover region from hypercubic to cylindrical scaling.Research partially supported by the A. P. Sloan Foundation and by the NSF under DMS-8858073Research partially supported by the NSF under DMS-8858073 and DMS-9008827     

Influence of the size dependence of surface tension on the dynamics of a bubble in a liquid

The Gibbs dividing surface method is used to derive the differential equation defining the dependence of the surface tension of a bubble in a nonpolar single-component liquid on its radius. Exact and asymptotic solutions of this equation have been obtained. It follows from the calculations that the bubble surface tension increases with decreasing radius. The Rayleigh-Plesset equation describing the bubble collapse dynamics is solved numerically by taking into account the size dependence of surface tension. The size dependence of surface tension is shown to affect significantly the final bubble collapse stage and, on the whole, accelerates this process.     

On the size dependence of surface tension in the temperature range from melting point to critical point

The problem of size dependence of surface tension was investigated in view of a more general problem of the applicability of Gibbs’ thermodynamics to nanosized objects. For the first time, the effective surface tension (coinciding with the specific excess free energy for an equimolecular dividing surface) was calculated within a wide temperature range, from the melting temperature to the critical point, using the thermodynamic perturbation theory. Calculations were carried out for Lennard-Jones and metallic nanosized droplets. It was found that the effective surface tension decreases both, with temperature and particle size.     

Phase field investigation on the initial planarinstability with surface tension anisotropy during directional solidification of binary alloys

Phase field investigation reveals that the stability of the planar interface is related to the anisotropic intensity of surface tension and the misorientation of preferred crystallographic orientation with respect to the heat flow direction. The large anisotropic intensity may compete to determine the stability of the planar interface. The destabilizing effect or the stabilizing effect depends on the misorientation. Moreover, the interface morphology of initial instability is also affected by the surface tension anisotropy.     

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.     

Renormalization-group calculation of the dependence on gravity of the surface tension and bending rigidity of a fluid interface

The surface tension and the bending rigidity of a planar liquid-vapor interface in the presence of vanishing gravity are analyzed using the renormalization group. Based on the density functional theory of inhomogeneous fluids, we show that a term, quartic in the density fluctuations, can be added to the classical capillary-wave model so that a renormalization-group calculation can be performed. By comparing the outcome of such a calculation with rigorous results relating the direct correlation function with surface tension and bending rigidity, we find the scaling dependence of the latter on gravity. The results agree with the expected fact that the interface should become unstable as gravity vanishes.     

Understanding surface tension

This article discusses the presentation of the facts of capillarity at the level of the upper forms of schools or the first year of a university course. The usual formulae are derived consistently from energy considerations, and care is taken to justify such geometrical approximations as are unavoidable. A plea is made that the subject be taught from the beginning in terms of surface energy, so keeping in sight of physical reality.     

Concentration dependence of Fe segregation near the surface in V-Fe alloys after V+ ion irradiation

The effect of Fe segregation near the free surface of V model alloys containing 2, 5, or 7 at % Fe is investigated by x-ray photoelectron spectroscopy. Segregation is induced by 50-keV V⁺ ion irradiation at a temperature of 30–40°C with fluences ranging from 10¹⁹ to 10²¹. Young’s moduli in these alloys are measured by the torsion pendulum method. The degree of Fe segregation is estimated, and its dependence on the irradiation dose and iron concentration in these alloys is analyzed. Correlation is found between the behavior of Young’s moduli and the degree of iron surface segregation as functions of the Fe concentration in the alloys.