Gibbs adsorption equation for planar fluid-fluid interfaces: Invariant formalism

Adsorption of Asphaltenes at Model Oil/Brine Interface: Influence of Solvent Polarity

With the exploitation of heavy oil worldwide, the influence of asphaltene aggregation in the oil phase on the stability of crude oil emulsion has been paid more and more attention. Under this background, the effects of solvent polarity on model oil/brine water interfacial properties and emulsion stability are investigated in this study. It is demonstrated that there is a critical asphaltene concentration for the formation of a stable emulsion. This critical concentration is then found to increase from 80 to 500 ppm with the mixing ratio of methylnaphthalene to n-decane changed from 2:3 to 7:3. The dynamic light scattering experiment shows that the average aggregate size increases abruptly from 132.8 to 261.1 nm at 2:3 mixing ratio of methylnaphthalene to n-decane once the asphaltenes are added to above the critical concentration. Accordingly, the diffusion coefficient of the asphaltenes decreases sharply from 4.36 × 10-12 to 5.68 × 10-13 m2/s. Similar conclusions are also found for the other mixing ratios of 1:1, 3:2, and 7:3. Besides, the aggregation degree of asphaltenes weakens, and the diffusion coefficient enlarges at the same asphaltene concentration with the enhancement of the solvent polarity. Further, the interfacial experiments manifest that the equilibrium interfacial dilation modulus decreases from 38.42 to 23.65 mN/m with the mixing ratio of methylnaphthalene to n-decane increased from 2:3 to 7:3. It can thus be inferred that the structural strength of the interfacial film decreases with the enhancement of the solvent polarity.

Graph-Neural-Network-Based Unsupervised Learning of the Temporal Similarity of Structural Features Observed in Molecular Dynamics Simulations

Classification of molecular structures is a crucial step in molecular dynamics (MD) simulations to detect various structures and phases within systems. Molecular structures, which are commonly identified using order parameters, were recently identified using machine learning (ML), that is, the ML models acquire structural features using labeled crystals or phases via supervised learning. However, these approaches may not identify unlabeled or unknown structures, such as the imperfect crystal structures observed in nonequilibrium systems and interfaces. In this study, we proposed the use of a novel unsupervised learning framework, denoted temporal self-supervised learning (TSSL), to learn structural features and design their parameters. In TSSL, the ML models learn that the structural similarity is learned via contrastive learning based on minor short-term variations caused by perturbations in MD simulations. This learning framework is applied to a sophisticated architecture of graph neural network models that use bond angle and length data of the neighboring atoms. TSSL successfully classifies water and ice crystals based on high local ordering, and furthermore, it detects imperfect structures typical of interfaces such as the water-ice and ice-vapor interfaces.

Molecular modeling of interfacial properties of the hydrogen + water + decane mixture in three-phase equilibrium

The understanding of interfacial phenomena between H2 and geofluids is of great importance for underground H2 storage, but requires further study. We report the first investigation on the three-phase fluid mixture containing H2, H2O, and n-C10H22. Molecular dynamics simulation and PC-SAFT density gradient theory are employed to estimate the interfacial properties under various conditions (temperature ranges from 298 to 373 K and pressure is up to around 100 MPa). Our results demonstrate that interfacial tensions (IFTs) of the H2-H2O interface in the H2 + H2O + C10H22 three-phase mixture are smaller than IFTs in the H2 + H2O two-phase mixture. This decrement of IFT can be attributed to C10H22 adsorption in the interface. Importantly, H2 accumulates in the H2O-C10H22 interface in the three-phase systems, which leads to weaker increments of IFT with increasing pressure compared to IFTs in the water + C10H22 two-phase mixture. In addition, the IFTs of the H2-C10H22 interface are hardly influenced by H2O due to the limited amount of H2O dissolved in nonaqueous phases. Nevertheless, positive surface excesses of H2O are seen in the H2-C10H22 interfacial region. Furthermore, the values of the spreading coefficient are mostly negative revealing the presence of the three-phase contact for the H2 + H2O + C10H22 mixture under studied conditions.

Surfactant Adsorption Layers: Experiments and Modeling

During recent years, great progress has been made in understanding the adsorption of surfactants at liquid interfaces. In addition to tensiometry, new efficient methodologies have been developed, in particular interfacial selective optical methods which allow direct access to the adsorbed amounts and interfacial layer compositions. In addition to these new experimental tools, the thermodynamic description by equations of state now allows one to provide a quantitative picture of surfactant interfacial layers. This is most notable for surfactant layers at water/oil interfaces. Additional knowledge about the structure of interfacial layers was gained through different types of molecular modeling. Improved interrelationships between these three aspects are the challenges for current and future work. Particular attention must be paid to dilational interfacial rheology studies, as these mechanical quantities are much more sensitive to small changes in the interfacial composition and structure.

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.

Neutron reflection and the thermodynamics of the air-water interface

By means of isotopic substitution, measurements of the neutron reflectivity (NR) from a flat water surface generally give model independent measurements of the amount of a chosen solute at the surface irrespective of whether the layer is a mixture or whether there is any aggregation in the bulk solution. Previously, adsorption at air-water interfaces has been determined by applying the Gibbs equation to surface tension (ST) measurements, which requires assumptions about the composition of the surface and about the activity of the solute in the bulk, which, in turn, means that in practice the surface is assumed to consist of the pure solute or of a mixture of pure solutes, and that the activity of the solute in the bulk solution is known. The use of NR in combination with ST-Gibbs measurements makes it possible to (i) avoid these assumptions and hence understand several patterns of ST behaviour previously considered to be anomalous and (ii) to start to analyse quantitatively the behaviour of mixed surfactants both below and above the critical micelle concentration. These two developments in our understanding of the thermodynamics of the air-water interface are described with recent examples.

Intrinsic Dependence of Groundwater Cation Hydraulic and Concentration Features on Negatively Charged Thin Composite Nanofiltration Membrane Rejection and Permeation Behavior

Commercial nanofiltration membranes of different molecular weight cut-offs were tested on a pilot plant for the exploration of permeation nature of Ca, Mg, Mn, Fe, Na and ammonium ions. Correlation of transmembrane pressure and rejection quotient versus volumetric flux efficiency on nanofiltration membrane rejection and permeability behavior toward hydrated divalent and monovalent ions separation from the natural groundwater was observed. Membrane ion rejection affinity (MIRA) dimension was established as normalized TMP with regard to permeate solute moiety representing pressure value necessary for solute rejection change of 1%. Ion rejection coefficient (IRC) was introduced to evaluate the membrane rejection capability, and to indicate the prevailed nanofiltration partitioning mechanism near the membrane surface. Positive values of the IRC indicated satisfactory rejection efficiency of the membrane process and its negative values ensigned very low rejection affinity and high permeability of the membranes for the individual solutes. The TMP quotient and the efficiency of rejection for individual cations showed upward and downward trends along with flux utilization increase. Nanofiltration process was observed as an equilibrium. The higher the Gibbs free energy was, cation rejection was more exothermic and valuably enlarged. Low Gibbs free energy values circumferentially closer to endothermic zone indicated expressed ions permeation.

Prediction of Aqueous Solution Surface Tension of Some Surfactant Mixtures and Composition of Their Monolayers at the Solution—Air Interface

Measurements of the surface tension of the aqueous solution of SDDS mixture with fluorocarbon surfactants (FC) were carried out and considered in light of the surface tension of aqueous solutions of individual surfactants. Similar analyses were made for many other aqueous solutions of binary and ternary mixtures, taking into account the literature data of the surface tension of aqueous solutions of TX100, TX114, TX165, SDDS, SDS, CTAB, CPyB and FC. The possibility of predicting the surface tension of the aqueous solution of many surfactant mixtures from that of the mixture components using both the Szyszkowski, Fainerman and Miller and Joos concepts was analyzed. The surface tension of the aqueous solutions of surfactant mixtures was also considered based on the particular mixture component contribution to the water surface tension reduction. As a result, the composition of the mixed surface layer at the solution–air interface was discussed and compared to that which was determined using the Hua and Rosen concept. As follows from considerations, the surface tension of the aqueous solution of binary and ternary surfactant mixtures can be described and/or predicted.

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.

Surface thermodynamics at the nanoscale

Fluid interfaces with nanoscale radii of curvature are generating great interest, both for their applications and as tools to probe our fundamental understanding. One important question is what is the smallest radius of curvature at which the three main thermodynamic combined equilibrium equations are valid: the Kelvin equation for the effect of curvature on vapor pressure, the Gibbs-Thomson equation for the curvature-induced freezing point depression, and the Ostwald-Freundlich equation for the curvature-induced increase in solubility. The objective of this Perspective is to provide conceptual, molecular modeling, and experimental support for the validity of these thermodynamic combined equilibrium equations down to the smallest interfacial radii of curvature. Important concepts underpinning thermodynamics, including ensemble averaging and Gibbs's treatment of bulk phase heterogeneities in the region of an interface, give reason to believe that these equations might be valid to smaller scales than was previously thought. There is significant molecular modeling and experimental support for all three of the Kelvin equation, the Gibbs-Thomson equation, and the Ostwald-Freundlich equation for interfacial radii of curvature from 1 to 4 nm. There is even evidence of sub-nanometer quantitative accuracy for the Kelvin equation and the Gibbs-Thomson equation.

The Effect of Electrochemical Composite Coatings with LaF3-LaB6 Particles in Nickel-Copper Matrix on the Metallurgical Processes in Arc Welding of Low Alloy Ferrite-Pearlite Steels

Development of welding consumables with fluorides and borides of rare earth metals is a promising area for improving the weldability of low alloy steels. As lanthanum fluoride and boride dissociate, lanthanum and boron dissolve in the weld pool and the welding arc plasma is saturated with fluorine. As a result of FeO, MnO, SiO₂ deoxidation and FeS, MnS desulfurization, refractory lanthanum sulfides and oxides La₂O₃, La₂S₃ are formed in the weld pool, which can be the crystallization nuclei in the weld pool and the origin of acicular ferrite nucleation. The paper proposes a model of metallurgical processes in the arc and weld pool, as well as a model of electrochemical adsorption of Ni²⁺ cations in colloidal electrolytes during electrostatic deposition of nano-dispersed insoluble particles of LaF₃ or LaB₆ on the surface of wire. The paper discusses the constructional design of the welding wire and the technology for forming electrochemical composite coatings with copper and nickel matrix. The composite wires applied in the welding of low alloy steels make it possible to refine the microstructure, increase the tensile strength by 4% and the impact toughness of welds by 20%.

Examining the robustness and concentration dependency of PFAS air-water and NAPL-water interfacial adsorption coefficients

Determining robust values for the air-water or NAPL-water interfacial adsorption coefficient, KIA, is key to characterizing and modeling PFAS transport and fate in several environmental systems. Direct, high-resolution measurements of surfactant adsorption at the fluid-fluid interface were aggregated from the literature. This data set was used to examine the accuracy and applicability of Γ and KIA measurements determined for three PFAS from transport experiments and surface-tension data. The transport-measured Γ and KIA data were observed to be fully consistent with the directly-measured data. Specifically, Γ values for the two methods were entirely coincident in the region of overlapping concentrations, which spanned ~4 orders-of-magnitude. Furthermore, the two data sets adhered to an identical Γ-C profile. These results conclusively demonstrate the accuracy of the transport-measured values. Γ and KIA values determined from the application of the Gibbs adsorption equation to measured surface-tension data were fully consistent with the directly-measured and transport-measured data sets, demonstrating their applicability for representing PFAS transport in environmental systems. The directly-measured data were used to examine the concentration dependency of KIA values, absent the potential confounding effects associated with the use of surface-tension or transport-measured data. The directly-measured data clearly demonstrate that KIA attains a constant, maximum limit at lower concentrations. Two separate analyses of the transport-measured data both produced observations of constant KIA values at lower concentrations, consistent with the directly-measured data. These outcomes are discussed in terms of surface activities, relative surface coverages, and critical concentrations.

Evaluating the Fitness of Combinations of Adsorbents for Quantitative Gas Sensor Arrays

Robust, high-performance gas-sensing technology has applications in industrial process monitoring and control, air quality monitoring, food quality assessment, medical diagnosis, and security threat detection. Nanoporous materials (NPMs) could be utilized as recognition elements in a gas sensor because they selectively adsorb gas. Imitating mammalian olfaction, sensor arrays of NPMs use measurements of the adsorbed mass of gas in a set of distinct NPMs to infer the gas composition. Modular and adjustable NPMs, such as metal-organic frameworks (MOFs), offer a vast material space to sample for combinations to comprise a sensor array that produces a response pattern rich with information about the gas composition. Herein, we frame quantitative gas sensing, using arrays of NPMs, as an inverse problem, which equips us with a method to evaluate the fitness of a proposed combination of NPMs in a sensor array. While the (routine) forward problem is to use an adsorption model to predict the mass of gas adsorbed in each NPM when immersed in a gas mixture of a given composition, the inverse problem is to predict the gas composition from the observed masses of adsorbed gas in the NPMs of the array. The fitness of a given combination of NPMs for gas sensing is then determined by the conditioning of its inverse problem: the prediction of the gas composition provided by a fit (unfit) combination of NPMs is insensitive (sensitive) to inevitable errors in the measurements of the mass of gas adsorbed in the NPMs. For illustration, we use experimentally measured adsorption data to analyze the conditioning of the inverse problem associated with a (IRMOF-1 and HKUST-1) CH₄/CO₂ sensor array.

Negative Pressure within a Liquid-Fluid Interface Determines Its Thickness

The density within the interface between two fluid phases at equilibrium gradually changes from that of one phase to that of the other. The main change in density, according to experimental measurements, practically occurs over a finite distance of O [1 nm]. If we assume that the average stress difference within the interface is on the order of magnitude of ambient pressure, then the Bakker equation implies that for a liquid with surface tensions, say ∼50 mN/m, we get an interface thickness of ∼500 nm. This is certainly too big because it contradicts experimental findings. Alternatively, if the thickness is assumed to be O [10 nm] or less, as is usually believed, the average stress difference must be ∼5 × 10⁶ N/m² or bigger, which is surprisingly high. This paper shows using a few approaches that such a high average stress difference is due to negative stresses in the interface.

Improved Derivation of the Butler Equations for Surface Tension of Solutions

The Butler equation was published in 1932 to describe the equilibrium surface composition and equilibrium surface tension of solutions. Unfortunately, it used the so-called "partial surface tension of a component", which was not properly defined by Butler, leading to a reluctant acceptance of this equation. Although the present author defined the partial surface tension recently in this journal, it is considered an advantage to derive the same key equations of Butler without the need to employ the concept of partial surface tension. This derivation is offered in the present paper, starting from the two fundamental equations of Gibbs. No assumptions are made on the thickness and structure of the surface region, it is only supposed that the surface region has an average composition with a negligible concentration gradient. In this way, the Butler equations are obtained, which have more general validity compared to the original Butler equations derived by supposing a surface monolayer.

Molecular Dynamics Simulations of Membrane Permeability

This Review illustrates the evaluation of permeability of lipid membranes from molecular dynamics (MD) simulation primarily using water and oxygen as examples. Membrane entrance, translocation, and exit of these simple permeants (one hydrophilic and one hydrophobic) can be simulated by conventional MD, and permeabilities can be evaluated directly by Fick's First Law, transition rates, and a global Bayesian analysis of the inhomogeneous solubility-diffusion model. The assorted results, many of which are applicable to simulations of nonbiological membranes, highlight the limitations of the homogeneous solubility diffusion model; support the utility of inhomogeneous solubility diffusion and compartmental models; underscore the need for comparison with experiment for both simple solvent systems (such as water/hexadecane) and well-characterized membranes; and demonstrate the need for microsecond simulations for even simple permeants like water and oxygen. Undulations, subdiffusion, fractional viscosity dependence, periodic boundary conditions, and recent developments in the field are also discussed. Last, while enhanced sampling methods and increasingly sophisticated treatments of diffusion add substantially to the repertoire of simulation-based approaches, they do not address directly the critical need for force fields with polarizability and multipoles, and constant pH methods.

Search for the Source of an Apparent Interfacial Resistance To Mass Transfer of CnEm Surfactants To the Water/Oil Interface

Experiments have shown that relaxation of oil/water interfacial tension by adsorption of alkyl ethoxylate surfactants from water onto an oil droplet is delayed relative to diffusion-controlled adsorption. We examine possible causes of this delay, and we show that several are implausible. We find that redissolution of the surfactant in the oil droplet cannot explain the apparent interfacial resistance at short times because the interface will preferentially fill before any such redissolution occurs. We also perform umbrella sampling with molecular dynamics simulation and do not find any evidence of a free-energy barrier or low-diffusivity zone near the interface. Nor do we find evidence from the simulation that premicellar aggregation slows diffusion enough to cause the observed resistance to interfacial adsorption. We are therefore unable to pinpoint the cause of the resistance, but we suggest that "dead time" associated with the experimental method could be responsible-specifically a local depletion of surfactant by the ejected droplet when creating the fresh interface between the oil and water.

The chemical (not mechanical) paradigm of thermodynamics of colloid and interface science

In the most influential monograph on colloid and interfacial science by Adamson three fundamental equations of "physical chemistry of surfaces" are identified: the Laplace equation, the Kelvin equation and the Gibbs adsorption equation, with a mechanical definition of surface tension by Young as a starting point. Three of them (Young, Laplace and Kelvin) are called here the "mechanical paradigm". In contrary it is shown here that there is only one fundamental equation of the thermodynamics of colloid and interface science and all the above (and other) equations of this field follow as its derivatives. This equation is due to chemical thermodynamics of Gibbs, called here the "chemical paradigm", leading to the definition of surface tension and to 5 rows of equations (see Graphical abstract). The first row is the general equation for interfacial forces, leading to the Young equation, to the Bakker equation and to the Laplace equation, etc. Although the principally wrong extension of the Laplace equation formally leads to the Kelvin equation, using the chemical paradigm it becomes clear that the Kelvin equation is generally incorrect, although it provides right results in special cases. The second row of equations provides equilibrium shapes and positions of phases, including sessile drops of Young, crystals of Wulff, liquids in capillaries, etc. The third row of equations leads to the size-dependent equations of molar Gibbs energies of nano-phases and chemical potentials of their components; from here the corrected versions of the Kelvin equation and its derivatives (the Gibbs-Thomson equation and the Freundlich-Ostwald equation) are derived, including equations for more complex problems. The fourth row of equations is the nucleation theory of Gibbs, also contradicting the Kelvin equation. The fifth row of equations is the adsorption equation of Gibbs, and also the definition of the partial surface tension, leading to the Butler equation and to its derivatives, including the Langmuir equation and the Szyszkowski equation. Positioning the single fundamental equation of Gibbs into the thermodynamic origin of colloid and interface science leads to a coherent set of correct equations of this field. The same provides the chemical (not mechanical) foundation of the chemical (not mechanical) discipline of colloid and interface science.

Surfactant Interactions and Organization at the Gas-Water Interface (CTAB with Added Salt)

We have studied adsorbed layers of cetyltrimethylammonium bromide (CTAB) at air-water interfaces in the presence of added electrolyte. Fast bubble compression/expansion measurements were used to obtain the surface equation of state, i.e., the surface tension vs CTAB surface concentration dependence. We show that while a simple model where the surfactant molecules are assumed to be noninteracting is insufficient to describe the measured response of the surfactant layer, a modified Frumkin equation where the local interactions between the molecular components depend on their surface concentration captures the response. The variation of the effective interactions in the surfactant layer in the model shows that the interactions in the surfactant layer change from effectively repulsive to attractive with increasing surface concentration. Molecular dynamics simulations are performed to probe the origins of the change in the interactions. The simulations indicate that already at low surface concentrations the surfactants aggregate as highly dynamic rafts with surfactant orientation parallel to the interface. Increasing the concentration leads to a change in the assembly morphology at the interface: the surfactant layer thickens and the surfactants sample a range of tilted orientations with respect to the interfacial plane. The change from transient raftlike assemblies to dynamical aggregates at the interface involves a clear increase in the degree of counterion binding: we speculate that the flip of the effective interaction parameter in the model used to interpret the experimental results could result from this. The work here presents basic steps toward a proper understanding of the molecular organization and interactions of surfactants at an air-water interface. This is crucially important in understanding macroscopic properties of surfactant-stabilized systems such as foams, emulsions, and colloidal dispersions.

Adsorption of Methyl Ester Sulfonate at the Air-Water Interface: Can Limitations in the Application of the Gibbs Equation be Overcome by Computer Purification?

We describe a new laboratory synthesis of the α-methyl ester sulfonates based on direct sulfonation of the methyl ester by SO₃ introduced from the vapor phase. This was used to synthesize a chain deuterated sample of αC₁₄MES, which was then used to measure the surface excess of αC₁₄MES directly at the air/water interface over a wide range of concentration using neutron reflection. The adsorption isotherm could be fitted to an empirical equation close to a Langmuir isotherm and gave a limiting surface excess of (3.4 ± 0.1) × 10^-6 mol m^-2 in the absence of added electrolyte. The neutron-measured surface excesses were combined with the integrated Gibbs equation to fit the variation in surface tension with concentration (σ-ln C curve). The fit was exact provided that we used a prefactor consistent with the counterion at the surface being an impurity divalent ion, as has previously been found for sodium diethylhexylsulfosuccinate (aerosol OT or AOT) and various perfluorooctanoates. The critical micelle concentration (CMC) was determined from this fit to be 2.4 ± 0.3 mM in the absence of electrolyte. In the presence of 100 mM NaCl, this contamination was suppressed and the σ-ln C curve could be fitted using the integrated Gibbs equation with the expected prefactor of 1. The new data were used to reinterpret measurements by Danov et al. on an unpurified sample of αC₁₄MES for which computer refinement was used to try to eliminate the effects of the impurities.

Surfactant films in lyotropic lamellar (and related) phases: Fluctuations and interactions

The analogy between soap films thinning under border capillary suction and lamellar stacks of surfactant bilayers dehydrated by osmotic stress is explored, in particular in the highly dehydrated limit where the soap film becomes a Newton black film. The nature of short-range repulsive interactions between surfactant-covered interfaces and acting across water channels in both cases will be discussed.

Interfaces at equilibrium: A guide to fundamentals

The fundamentals of the thermodynamics of interfaces are reviewed and concisely presented. The discussion starts with a short review of the elements of bulk thermodynamics that are also relevant to interfaces. It continues with the interfacial thermodynamics of two-phase systems, including the definition of interfacial tension and adsorption. Finally, the interfacial thermodynamics of three-phase (wetting) systems is discussed, including the topic of non-wettable surfaces. A clear distinction is made between equilibrium conditions, in terms of minimizing energies (internal, Gibbs or Helmholtz), and equilibrium indicators, in terms of measurable, intrinsic properties (temperature, chemical potential, pressure). It is emphasized that the equilibrium indicators are the same whatever energy is minimized, if the boundary conditions are properly chosen. Also, to avoid a common confusion, a distinction is made between systems of constant volume and systems with drops of constant volume.

Size dependence of the surface tension of a free surface of an isotropic fluid

We report on the size dependence of the surface tension of a free surface of an isotropic fluid. The size dependence of the surface tension is evaluated based on the Gibbs-Tolman-Koenig-Buff equation for positive and negative values of curvatures and the Tolman lengths. For all combinations of positive and negative signs of curvature and the Tolman length, we succeed to have a continuous function, avoiding the existing discontinuity at zero curvature (flat interfaces). As an example, a water droplet in the thermodynamical equilibrium with the vapor is analyzed in detail. The size dependence of the surface tension and the Tolman length are evaluated with the use of experimental data of the International Association for the Properties of Water and Steam. The evaluated Tolman length of our approach is in good agreement with molecular dynamics and experimental data.

Relative adsorption excess of ions in binary solvents determined by grazing-incidence X-ray fluorescence

We demonstrate a model-independent method for experimental determination of the relative surface excess of inorganic ions in binary liquid mixtures, based on grazing-incidence X-ray fluorescence. For this purpose, we probe the ion density profiles in a mixture of water and 2,6-dimethylpyridine containing a hydrophilic salt, potassium chloride. Thereby we demonstrate that the proposed method quantifies in a direct manner the difference between cation and anion excess adsorption in binary solvents with a resolution of one excess ion per 200nm² or better.

Surface Tension and Adsorption without a Dividing Surface

The ingenious concept of a dividing surface of zero thickness that was introduced by Gibbs is the basis of the theory of surface tension and adsorption. However, some fundamental questions, mainly those related to the location of the dividing surface and the proper definition of relative adsorption, have remained open over the years. To avoid these questions, the present paper proposes to analyze an interfacial phase by defining a thermodynamic system of constant, but nonzero thickness. The interfacial phase is analyzed as it really is, namely a nonuniform three-dimensional entity. The current analysis redevelops the equation for calculating surface tension, though with different assumptions. However, the main point in the proposed model is that the thermodynamic interfacial system, due to its fixed thickness, conforms to the requirement of first-order homogeneity of the internal energy. This property is the key that allows using the Gibbs adsorption isotherm. It is also characteristic of the Gibbs dividing surface model, but has not always been discussed with regard to subsequent models. The resulting equation leads to a simple, "natural" expression for the relative adsorption. This expression may be compared with simulations and sophisticated surface concentration measurements, and from which the dependence of interfacial tension on the solution composition can be derived. Finally, it is important to point out that in order to calculate the interfacial tension as well as the relative adsorption from data on the properties of the interfacial phase, there is no need to know its exact thickness, as long as it is bigger than the actual thickness but sufficiently small.

Surface dilational rheological properties in the nonlinear domain

The interfacial tension response to dilational deformation of interfacial area exhibits a (more or less) nonlinear behavior, depending on the amplitude of the deformation. Studies of such observable interfacial properties in the nonlinear domain suggest valuable information about the two-dimensional microstructure of the interfacial layer, as well as about the structure time-evolution. In this article, the emphasis is centered on the available mathematical methods for quantitatively analyzing and describing the magnitude and the characteristics of the nonlinear interfacial viscoelastic properties. Specifically, in periodic oscillation experiments the nonlinear behavior can be represented by the combination of a linear part (the surface dilational modulus), with an additional complementary Fourier analysis parameterizing the nonlinearity. Also asymmetric Lissajous plots, of interfacial tension versus deformation, are useful tools for expanding the response nonlinearity into four distinct components relevant to significant points of the cyclic loop. In connection with the mathematical methods, nonequilibrium thermodynamic formulations provide a powerful theoretical framework for investigating the interfacial dynamic properties of multiphase systems. Experimental results for adsorption layers of complex components, available in the literature, show notable nonlinear interfacial viscoelastic behavior. In particular in this review, data are illustrated for solutions of polymers and of polyelectrolyte/surfactant complexes. The observed nonlinear findings reveal formation of complexes, patches, and other different interfacial structures.

Gibbs excess and the calculation of the absolute surface composition of liquid binary mixtures

Adsorption at the liquid-vapor interphase of a liquid binary mixture is traditionally quantified by means of the Gibbs solute excess. Despite several theoretical reviews on the meaning of Gibbs excess defined by the Gibbs dividing surface, it is still misinterpreted as the excess concentration under Guggenheim's finite-depth surface layer approach. In this work, both concepts are clarified in a practical way, aided by a graphical representation without loss of generality. The understanding of both quantities led to the development of a thermodynamic procedure for the calculation of the actual number of solute and solvent molecules at a finite-depth surface layer (not a monolayer), what is called the absolute surface composition. From surface tension and density data, the absolute surface composition of the binary aqueous mixtures of methanol, ethanol, 1-propanol, and 1-butanol was calculated. Results show thermodynamic consistency and agree with experimental reports and with an empirical mixing rule. The increasing alcohol surface concentration throughout the entire concentration range casts doubt on the formation of an alcohol monolayer, as was suggested by other authors. Furthermore, the use of Guggenheim's monolayer model does not reproduce the experimental data, nor does it show thermodynamic consistency.

Film and membrane-model thermodynamics of free thin liquid films

In spite of over 7 decades of effort, the thermodynamics of thin free liquid films (as in emulsions and foams) lacks clarity. Following a brief review of the meaning and measurement of thin-film forces (i.e., conjoining/disjoining pressures), we offer a consistent analysis of thin-film thermodynamics. By carefully defining film reversible work, two distinct thermodynamic formalisms emerge: a film model with two zero-volume membranes each of film tension γ(f) and a membrane model with a single zero-volume membrane of membrane tension 2γ(m). In both models, detailed thermodynamic analysis gives rise to thin-film Gibbs adsorption equations that allow calculation of film and membrane tensions from measurements of disjoining-pressure isotherms. A modified Young-Laplace equation arises in the film model to calculate film-thickness profiles from the film center to the surrounding bulk meniscus. No corresponding relation exists in the membrane model. Illustrative calculations of disjoining-pressure isotherms for water are presented using square-gradient theory. We report considerable deviations from Hamaker theory for films less than about 3 nm in thickness. Such thin films are considerably more attractive than in classical Hamaker theory. Available molecular simulations reinforce this finding.