Modeling hydrophobic solvation of nonspherical systems: Comparison of use of molecular surface area with accessible surface area

It is shown that the molecular surface and the accessible surface lead to exactly the same results when calculating solvation free energies and transfer free energies, from methods using the surface tension as a parameter if the exact geometric curvature is used with the accessible surface. However, the use of the exact curvature is not necessarily the best approach chemically. Other modifications, including an approximate curvature improves the approach. Such modifications are difficult to include in methods in which the molecular surface rather than the accessible surface is used to calculate solvent effects. A modification of a Gaussian curvature term is necessary if dissociation is to be accounted for properly. The inclusion of a Gaussian curvature term, in addition to the usual mean curvature term, reconciles the difference in magnitude of the microscopic and macroscopic surface tension in the case of the accessible surface area. © 1997 by John Wiley & Sons, Inc.     

First correction to surface tension for the curvature of an interface

Tolman parameter δ_T, which determines the first correction to surface tension for the interface curvature, is calculated in molecular-dynamic experiments performed for Lennard-Jones fluid within the temperature range from the critical to the triple point (and slightly below). It is shown that parameter δ_T is positive and slightly depends on temperature; its absolute magnitude is no larger than 0.1–0.2 molecular diameters and does not coincide with the distance between the equimolecular dividing surface and the surface of tension in a flat interfacial layer, as calculated through the first moment of the pressure tensor. The results of the moleculardynamic experiments are compared with the δ_T values calculated in terms of the extended version of the van der Waals theory of capillarity. It is established that taking into account terms of higher orders than the squared density gradient in the expansion of the free energy of an inhomogeneous system does not reverse the negative sign at δ_T and slightly affects its value.     

The liquid–gas interface of oxygen–nitrogen solutions 2: Description in the Framework of the van der Waals gradient theory

The van der Waals gradient theory (vdW GT) is used to calculate surface tension, density profiles, adsorption, the Tolman length and to determine the position of dividing surfaces in the liquid–gas interface of an oxygen–nitrogen solution. The Helmholtz energy density (HED) is determined via an equation of state (EOS), unified for a liquid and gas, which describes stable, metastable and two-phase states of solutions. The influence parameters are calculated from data on the surface tension of pure components with the use of the mixing rule. At temperatures T > 100 K the vdW GT describes experimental data on the surface tension of oxygen–nitrogen solutions [V.G. Baidakov, A.M. Kaverin, V.N. Andbaeva, The liquid–gas interface of oxygen–nitrogen solutions: 1. Surface tension, Fluid Phase Equilib. 270 (2008) 116–120] within the experimental error. It is shown that the Tolman length, which determines the dependence of surface tension on the curvature of the dividing surface, depends considerably on the solution concentration.     

Multicomponent nucleation: thermodynamically consistent description of the nucleation work

A thermodynamically consistent formula is derived for the nucleation work in multicomponent homogeneous nucleation. The derivation relies on the conservative dividing surface which defines the nucleus as having specific surface energy equal to the specific surface energy sigma0 of the interface between the macroscopically large new and old phases at coexistence. Expressions are given for the radius of the nucleus defined by the conservative dividing surface and by the surface of tension. As a side result, the curvature dependence of the surface tension sigmaT of the nucleus defined by the surface of tension is also determined. The analysis is valid for nuclei of any size, i.e., for nucleation in the whole range of conditions between the binodal and the spinodal of the metastable old phase provided the inequality sigmaT < or = sigma0 is satisfied. It is found that under the conditions of validity of the analysis the nucleation rate is higher than the nucleation rate given by the classical nucleation theory. The general results are applied to nucleation of unary liquids or solids in binary gaseous, liquid or solid mixtures.     

Total Gaussian curvature,drop shapes and the range of applicability of drop shape techniques

Drop shape techniques are used extensively for surface tension measurement. It is well-documented that, as the drop/bubble shape becomes close to spherical, the performance of all drop shape techniques deteriorates. There have been efforts quantifying the range of applicability of drop techniques by studying the deviation of Laplacian drops from the spherical shape. A shape parameter was introduced in the literature and was modified several times to accommodate different drop constellations. However, new problems arise every time a new configuration is considered. Therefore, there is a need for a universal shape parameter applicable to pendant drops, sessile drops, liquid bridges as well as captive bubbles. In this work, the use of the total Gaussian curvature in a unified approach for the shape parameter is introduced for that purpose. The total Gaussian curvature is a dimensionless quantity that is commonly used in differential geometry and surface thermodynamics, and can be easily calculated for different Laplacian drop shapes. The new definition of the shape parameter using the total Gaussian curvature is applied here to both pendant and constrained sessile drops as an illustration. The analysis showed that the new definition is superior and reflects experimental results better than previous definitions, especially at extreme values of the Bond number.     

Prediction of the surface tension of cesium in the whole liquid range

Surface tension is an important quantity, both as physical and technological features. We applied the general form of Lennard-Jones potential model, LJ (m-n), and some thermodynamic arguments in the Kirkwood and Buff equation to calculate the surface tension of liquid cesium. To find out the surface tension in a wide temperature range, only pVT data are required. By this method, we investigated the influence of the potential model on the surface tension. For two selected potential models [LJ (6–3) and LJ (8.5–4)], the values of the calculated surface tension and the predicted surface energy agree with the experimental values.     

On the curvature dependence of the interfacial tension in a symmetric three-component interface

We consider a symmetric interface between two polymers A(N) and B(N) in a common monomeric solvent S using the mean-field Scheutjens-Fleer self-consistent field theory and focus on the curvature dependence of the interfacial tension. In multi-component systems there is not one unique scenario to curve such an interface. We elaborate on this by keeping either the chemical potential of the solvent or the bulk concentration of the solvent fixed, that is we focus on the semi-grand canonical ensemble case. Following Helfrich, we expand the surface tension as a Taylor series in the curvature parameters and find that there is a non-zero linear dependence of the interfacial tension on the mean curvature in both cases. This implies a finite Tolman length. In a thermodynamic analysis we prove that the non-zero Tolman length is related to the adsorption of solvent at the interface. Similar, but not the same, correlations between the solvent adsorption and the Tolman length are found in the two scenarios. This result indicates that one should be careful with symmetry arguments in a Helfrich analysis, in particular for systems that have a finite interfacial tension: one not only should consider the structural symmetry of the interface, but also consider the constraints that are enforced upon imposing the curvature. The volume fraction of solvent, the chain length N as well as the interaction parameter chi(AB) in the system can be used to take the system in the direction of the critical point. The usual critical behavior is found. Both the width of the interface and the Tolman length diverge, whereas the density difference between the two phases, adsorbed amount of solvent at the interface, interfacial tension, spontaneous curvature, mean bending modulus as well as the Gaussian bending modulus vanish upon approach of the critical point.     

Variations of surfactant monolayer surface density in induced steady wave regime

Fluctuations of an insoluble surfactant concentration along the free liquid surface induced by steady surface waves are considered theoretically. The energy of a waved surface is assumed to consist of surface tension, curvature, and van der Waals energy components. Dependencies of the surface tension and the bending stiffness versus the surfactant concentration are assumed to be linear relative to some reference level. The van der Waals energy is taken in the form of interaction term for a thin film. Minimization of the total energy allows the expression for the deviations of concentration to be obtained. The distribution of a surfactant concentration relative to some reference level has been found to be periodic, with a period that is half of the wave period, and the amplitude of oscillations is a function of a wave number that is very similar to the Landau expansion of the free-energy near the critical point in phase transitions.     

Some Physical Properties of Binary Liquid Systems: (2-Butanone + n-Propionic Acid or n-Butyric Acid)

Abstract

Density, viscosity and surface tension of two binary liquid systems: 2-butanone + n-propionic acid, 2-butanone + n-butyric acid have been determined at 20, 30 and 40°C, over the whole compositional range. The excess values of molar volume, viscosity, Gibbs free energy for the activation of flow and surface tension were evaluated. These excess values were fitted to a Redlich-Kister type of equation. The Grunberg-Nissan parameter, d, was also calculated. The binary viscosity data were fitted to the models of McAllister, Heric, Auslander and Teja and Rice. Surface tension data were fitted to the models of Zihao and Jufu, Rice and Teja, and an empirical two-constant model proposed in this study.     

Surface Tension Calculations for Liquid Metals

Abstract

We present a class of models for the surface of a liquid metal, which may be part of an electrochemical interface. The particles of the system, for the purpose of derivation of thermodynamic properties, are the charged ion cores, while the energy of the electrons is evaluated using the electron density functional formalism, previously principally applied to solids. An expression for the surface energy U^s , defined as the energy required to create unit area of surface by separation of a volume of homogeneous metal into two parts, is derived (Eqs. 18–20). The surface tension γ is obtained by differentiating the Helmholtz free energy with respect to the area of the system, keeping volume and particle number constant (Eqs. 27–37). The surface tension is also equal to the difference between the free energy of the system containing a surface and the free energy of a reference system. It thus defines a surface energy through the Gibbs-Helmholtz equation, and this surface energy is shown to be identical to U^s .

The expressions for U^s and γ are made explicit (Eqs. 45–57) by insertion of particular assumptions for the ion-density profile, the electron-density profile, the interionic interaction and pair distribution function, and the electronic energy. Only information about bulk liquid metal is used. The parameter in the electron-density profile is obtained by minimizing the surface energy. The simplest assumption for the interionic interaction, hard-sphere and Coulombic repulsions, requires a choice for the hard-sphere diameter, which is made such that the pressure of bulk metals is given correctly (52–55). For the alkali metals, the surface tension calculated from this model is about half the experimental value in each case, while calculated surface energies are too high (1/5 too high for Cs, but three times too high for Li). For the electrical potential difference between the inside and the outside of a metal, and for the electrochemical potential, agreement with experiment is good. The main reason for the disagreements in the other properties is traced to the simple form used for the ion pair distribution function.     

Surface tension of polystyrene blends: Theory and experiment

Surface tension of linear–linear and star/linear polystyrene blends were measured using a modified Wilhelmy method. Our results show that for both polystyrene blend systems, the surface tension‐composition profile is convex, indicating a strong surface excess of the component with lower surface energy. Star/linear blends display more convex surface tension profiles than their linear–linear counterparts, indicative of stronger surface segregation of the branched‐component relative to linear chains. As a first step toward understanding the physical origin of enhanced‐surface segregation of star polymers, self‐consistent field (SCF) lattice simulations (both incompressible and compressible models) and Cahn‐Hilliard theory were used to predict surface tension‐composition profiles. Results from the lattice simulations indicate that the highly convex surface tension profiles observed in the star/linear blend systems are only possible if an architecture‐dependent, Flory interaction parameter (χ = 0.004) is assumed. This conclusion is inconsistent with results from bulk differential scanning calorimetry (DSC) measurements, which indicate sharp glass transitions in both the star/linear and linear/linear homopolymer blends and a simple linear relationship between the bulk glass transition temperature and blend composition. To implement the Cahn‐Hilliard theory, pressure‐volume‐temperature (PVT) data for each of the pure components in the blends were first measured and the data used as input for the theory. Consistent with the experimental data, Cahn‐Hilliard theory predicts a larger surface excess of star molecules in linear hosts over a wide composition range. Significantly, this result is obtained assuming a nearly neutral interaction parameter between the linear and star components, indicating that the surface enrichment of the stars is not a consequence of complex phase behavior in the bulk. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 1666–1685, 2009     

醋酸离子液体[C2mim][OAc]水溶液的摩尔表面Gibbs自由能

在288.15-318.15 K温度范围内测定了不同浓度离子液体1-乙基-3-甲基咪唑醋酸盐([C₂mim][OAc])水溶液的表面张力和密度;在改进李以圭等人的溶液表面张力模型基础上,提出摩尔表面Gibbs自由能新概念,建立了摩尔表面Gibbs自由能随溶液浓度变化的线性经验方程,利用这个经验方程估算了[C₂mim][OAc]水溶液的摩尔表面Gibbs自由能,并进一步预测了该溶液的表面张力,其预测值与相应的表面张力实验值高度相关并非常相似。由此可见,摩尔表面Gibbs自由能与等张比容极其类似,可能成为预测离子液体及其溶液性质的一种新的半经验方法。在指定溶液浓度下,根据溶液的摩尔表面Gibbs自由能随温度呈线性变化的规律,得到了新的Eötvös方程,与传统的Eötvös方程相比,新Eötvös方程的每一个参数都有明确的物理意义:斜率的负值是摩尔表面熵,截距是摩尔表面焓,在指定浓度的溶液中摩尔表面焓几乎不随温度变化。     

Self-assembly of polymer brushes in the presence of a surfactant: A system of strands

This theoretical study is focused on the formation of a cylindrical microstructure in a planar polymer brush in the presence of a surfactant. It is assumed that the brush may be nonuniform in the direction along the grafting plane and that there are regions with constant concentrations of monomer units and regions occupied only by the surfactant. The surfactant molecule is simulated by a dimer whose parts interact in a different manner with the monomer units of the polymer. At the interface between these regions, dimer molecules are oriented mainly perpendicularly to this interface and the surface tension is reduced. If the surface energy becomes negative, the formation of a structured brush is more favorable in terms of energy than that of a uniform brush. As a result, there may appear a cylindrical microstructure in which grafted macromolecules are united into strands perpendicular to the grafting plane. The stretching of macromolecules and their interaction with the solvent within the strands are described by the Alexander-de Gennes model, whereas the surface energy is calculated with allowance for the surface curvature of strands at a high degree of amphiphilicity of the surfactant molecules. It is shown that the arising strands have radii of the order of the surfactant-molecule length, while the number of macromolecules per strand is proportional to the surface density of their grafting. With an increase in the grafting density, the strand length increases initially, while the volume fraction of the polymer in a strand remains constant. Furthermore, strands start to shorten and their density grows. Structural characteristics are calculated as a function of the parameter characterizing the degree of amphiphilicity of the solvent molecules, their sizes, and their average energy of interaction with monomer units.     

Alternative reference system in the thermodynamics of solid elastic electrodes

Basic quantities in the thermodynamics of the solid elastic electrode are the surface tension tensor g _mn and the work needed for the formation of the surface (interface) γ. It is scarcely mentioned explicitly anywhere that these intensive (specific) quantities are related to the surface of the elastically deformed electrode. On the other hand, in the thermodynamics of the volume elasticity, the free energy density of the deformed solid is related to the volume of the undeformed solid. In this paper, we introduce equivalently the undeformed surface of the solid elastic electrode as reference for both the surface tension tensor and the work of formation of the surface. Generalizing the analysis of two model systems, we deduce the corresponding alternative form of the Shuttleworth equation, where the two quantities appear as generalized force and generalized potential, and discuss consequences for the formulation of the differential of the surface excess of the internal energy.Dedicated to the memory of W. Schwabe     

Applying Tikhonov regularization to process pendant droplet tensiometry data

The problem of obtaining the first and second derivatives of the profile of a pendant droplet is formulated as an integral equation of the first kind. This equation is solved by Tikhonov regularization in which the method of general cross validation is used to guide the selection of the regularization parameter. These derivatives are converted into mean curvature as a function of droplet height. Surface tension is then obtained by regression computation between the mean curvature and two possible algebraic expressions suggested by the Laplace-Young equation. This way of obtaining surface tension is demonstrated by applying it to a number of published droplet profiles. Some of the problems encountered are discussed and solutions suggested.     

Surface Tension of the Vapor–Liquid Interface with Finite Curvature

Based on the division of particles into internal and surface particles, the expression is derived closing the system of equations of classical thermodynamics for curvature-dependent surface tension, equimolar radius, and radius of tension surface. A solution to this system allows one to find the surface tension of new phase nucleus of any size (including minimal) and any sign of surface curvature. The obtained results indicate the weak size dependence of thermodynamic parameters that are the functions of surface tension; it is shown that Tolman's length cannot be determined using experimental determination of these parameters. It is shown that the work of nucleus formation strongly depends on its size and is the function of effective rather than true surface tension. Numerical simulation of clusters by the molecular dynamics method indicates that the pressure inside a fairly small cluster is described by Laplace's formula with the coefficient of surface tension for the plane surface of a liquid that agrees with the proposed theory.     

Study of the surface and interfacial tensions in systems containing a low molar mass liquid crystal

The surface tension of a low molar mass liquid crystal (LMMLC), 4-cyano-4'-n-heptyloxybiphenyl (70CB), was measured as a function of temperature using the pendant drop method, forming drops of different volumes ranging from 5 to 11 mm³. Contact angles formed by drops of 70CB in the nematic and isotropic phases on plates of polystyrene (PS) and of a liquid crystal polymer (LCP), VECTRA A910, were also measured. Only large drops could be used for surface tension analysis. It was shown that in the nematic phase the surface tension of 70CB decreases with increasing temperature, and that in the isotropic phase the surface tension increases with increasing temperature. Using the values of contact angle and of surface tension of 7OCB it was possible to evaluate the interfacial energy between 7OCB and PS and between 7OCB and VECTRA. The interfacial energy between 7OCB and PS, and between 7OCB and VECTRA, decreased with increasing temperature for ranges of temperatures corresponding to both phases of 70CB.     

Surface tensions of smectic liquid crystals

We determine surface tensions σ of smectic liquid crystals from the curvature pressure of smectic films. A new technique is introduced for the comparison of surface tensions of different smectic materials. The method is based on the relation of curvatures of smectic films drawn on communicating vessels. The measurement of the temperature dependence of σ reveals anomalies in the vicinity of phase transitions to low temperature smectic modifications. This anomalous slope dσ/dT can be related to the surface excess entropy of the material in the corresponding temperature range. The surface tension values determined for a number of mesogens fit well into the classification proposed by Mach et al.     

A modified square well model in obtaining the surface tension of pure and binary mixtures of hydrocarbons

A model based on the perturbation theory of fluids was proposed to correlate the experimental data for surface tension of pure hydrocarbons in a wide range of temperature. The results obtained for the pure hydrocarbons were directly used to predict the surface tension for binary hydrocarbon mixtures at various temperatures. In the proposed model, a modified form of the square well potential energy between the molecules of the reference fluid was taken into account while the Lennard–Jones dispersion energy was considered to be dominant amongst the molecules as the perturbed term to the reference part of the model. In general, the proposed model has three adjustable parameters which are chain length, m, size, σ, and energy, ε/κ, parameters, but in some cases the number of parameters was reduced to two, thereby setting the chain length to be unity for pure hydrocarbons. The regressed values of these parameters were obtained using the experimental data for pure hydrocarbons at different temperatures. The results showed that these parameters can be related to the molar mass of hydrocarbons. The model was also extended to predict the surface tension of binary hydrocarbon mixtures using the parameters obtained for the pure compounds. It is worth noting that no additional parameter has been introduced into the model in the extension of the model to the mixtures studied in this work. The results showed that the proposed model can accurately correlate the surface tension of pure hydrocarbons. Also the results showed that the surface tension for binary mixture of hydrocarbons can be accurately predicted using the proposed model over a wide temperature range.     

A general theory for the formation of a new phase. Static cavitation in a nonvolatile liquid

The rate of formation of cavities in a nonvolatile liquid subjected to negative pressure is calculated. The pressure is assumed to be sufficiently moderate, so that the critical bubble (hole) can be treated macroscopically. If this condition is satisfied one may apply the Zel'dovich saddle-point method. However, in distinction from that method, the present paper makes use of a construction of Gibbs grand canonical ensemble, which allows one to avoid the ambiguities that appear in the calculation of the microscopic initial stage of formation of the hole and yields a more accurate value of the kinetic (preexponential) factor. A more accurate value of the work required to form a critical hole, the value which occurs in the exponent, is obtained by taking into account the dependence of the surface tension of the hole on the curvature of the surface. For this purpose we use an expansion of the surface tension in powers of the curvature of the surface of the hole. Since the surface tension occurs in the exponent of the expression for the probability or rate of cavitation, it is necessary to retain the expansion terms which are linear or quadratic in the curvature, whereas the correction factors to the cavitation rate, due to higher-order terms tend to unity as the expansion of the liquid decreases, in contradistinction from the first terms. It is important that for large expansions, and hence for small curvatures of the surface of the critical hole, the series for the surface tension diverges.