Thermodynamic Study of the Role of Interface Curvature on Multicomponent Vapor–Liquid Phase Equilibrium

Generalization of Young-Laplace, Kelvin, and Gibbs-Thomson equations for arbitrarily curved surfaces

The Young-Laplace, Kelvin, and Gibbs-Thomson equations form a cornerstone of colloidal and surface sciences and have found successful applications in many subfields of physics, chemistry, and biology. The Gibbs-Thomson effect, for example, predicts that small crystals are in equilibrium with their liquid melt at a lower temperature than large crystals and the positive interfacial energy increases the energy required to form small particles with a high curvature interface. In cases of liquids contained within porous media (confined geometry), the effect indicates decreasing the freezing/melting temperatures and the increment of the temperature is inversely proportional to the pore size. These phenomena can be reformulated for Gaussian maps of macromolecules and can be asked the following question: can one use the equations for predicting the melting temperature and shape of polymer chains in confined geometries? The answer is no, mainly because macromolecules form highly curved surfaces (Gaussian maps), and the equations hold only for simple geometries (sphere, plane, or cylinder). Here, we present general Young-Laplace, Kelvin, and Gibbs-Thomson equations for arbitrarily curved surfaces and apply them to predict temperature distribution on a few protein surfaces. Also, after increased interest toward liquid/liquid phase separation in biology, we derive generic Ostwald ripening and show that for shape-changing condensates, instead of a monotonic growing mechanism, a variety of processes are possible. Due to the generality of equations, we clarify that at appropriate internal/external pressure conditions systems, bounded by surfaces, may adopt any shape and thermal stability is strongly influenced by the geometries of confined spaces.

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.

Thermodynamic Investigation of the Effect of Electric Field on Solid-Liquid Equilibrium

In this study, the thermal, mechanical, and chemical equilibrium conditions are derived for binary solid-liquid equilibrium under the effect of an electric field. As an example, the effect of an electric field on the water/glycerol solid-liquid phase diagram is computed over the complete mole fraction range. We show that the application of an electric field can affect the composition dependent freezing and precipitating processes, changing freezing and precipitating temperatures and changing the eutectic point temperature and mole fraction.

Gibbsian Surface Thermodynamics

Gibbsian composite-system thermodynamics is the framework governing the equilibrium of composite systems, including systems that at equilibrium have more than one value of pressure because of the action of surface tension, semipermeable membranes, or fields, and thus cannot be treated as simple systems. J. W. Gibbs's paper that lays out composite-system thermodynamics, "On the Equilibrium of Heterogeneous Substances", communicated in two parts in 1876 and 1878, is widely regarded as one of the most important pieces of scientific literature of its century. Many scientists adopted and stressed the importance of Gibbsian thermodynamics. In 1960, H. B. Callen wrote a textbook that made Gibbsian composite-system thermodynamics more accessible to thermodynamicists. However, Callen's book left out Gibbs's work on curved fluid interfaces and did not treat the complicated nonideal systems of interest to today's thermodynamicists. In this Feature Article, I have attempted to convey in a comprehensive manner the framework of Gibbsian composite-system thermodynamics including in detail the treatment of systems with interface effects and with nonideal, multicomponent phases. This work lays out the relationships between important equilibrium equations including the following: the Gibbs-Duhem equation, the Gibbs adsorption equation, the Young-Laplace equation, the Young equation, the Cassie-Baxter equation, the Wenzel equation, the Kelvin equation, the Gibbs-Thompson equation, and the Ostwald-Freundlich equation, including nonideal and multicomponent forms. Equations of state that are often useful for Gibbsian composite-system thermodynamics are reviewed including adsorption isotherms and our own work on two semiempirical equations of state: the Elliott et al. form of the osmotic virial equation and the Shardt-Elliott-Connors-Wright equation for the temperature and composition dependence of surface tension. I summarize the work of our group developing Gibbisan composite-system thermodynamics including new equations for such things as the curvature-induced depression of the eutectic temperature or the removal of azeotropes by nanoscale fluid interface curvature. Gibbsian composite-system thermodynamics has broad applications in biotechnology, nanostructured materials, surface textures and coatings, microfluidics, nanoscience, atmospheric and environmental physics, among others.

Gibbsian Thermodynamics of Wenzel Wetting (Was Wenzel Wrong? Revisited)

When a drop is in contact with a rough surface, it can rest on top of the rough features (the Cassie-Baxter state) or it can completely fill the rough structure (the Wenzel state). The contact angle (θ) of a drop in these states is commonly predicted by the Cassie-Baxter or Wenzel equations, respectively, but the accuracy of these equations has been debated. Previously, we used fundamental Gibbsian composite-system thermodynamics to rigorously derive the Cassie-Baxter equation, and we found that the contact line determined the macroscopic contact angle, not the contact area that was originally proposed. Herein, to address the various perspectives on the Wenzel equation, we apply Gibbsian composite-system thermodynamics to derive the complete set of equilibrium conditions (thermal, chemical, and mechanical) for a liquid drop resting on a homogeneous rough solid substrate in the Wenzel mode of wetting. Through this derivation, we show that the roughness must be evaluated at the contact line, not over the whole interfacial area, and we propose a new Wenzel equation for a surface with pillars of equal height. We define a new dimensionless number H = h(1 - λ_solid)/R to quantify when the drop's radius of curvature (R) is large enough compared to the size of the pillars for the new Wenzel equation to be simplified (h is the pillar height; λ_solid is the line fraction of the spherical cap's circumference that is on the pillars). Our new line-roughness Wenzel equation can be simplified to cos θ_W = ρ cos θ_Y when H ≪ 1, where ρ is the line roughness. We also perform a thermodynamic free-energy analysis to determine the stability of the equilibrium states that are predicted by our new Wenzel equation.

Gibbsian Thermodynamics of Cassie-Baxter Wetting (Were Cassie and Baxter Wrong? Revisited)

Over the past decade, there has been a debate over the correct form of the Cassie-Baxter equation, which describes the expected contact angle of a liquid drop on a heterogeneous surface. The original Cassie-Baxter equation uses an area fraction of each solid phase calculated over the entirety of the surface, and its derivation is based on an assumption not all surfaces necessarily satisfy. Herein, we introduce fundamental Gibbsian composite-system thermodynamics as a new approach for deriving the complete set of equilibrium conditions for a liquid drop resting on a heterogeneous multiphase solid substrate. One of the equilibrium conditions is a form of the Cassie-Baxter equation that uses a line fraction determined at the contact line outlining the perimeter of the solid-liquid contact area. We elucidate the practical implications of using the line fraction for common patterns of heterogeneities.

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.

Model for the Surface Tension of Dilute and Concentrated Binary Aqueous Mixtures as a Function of Composition and Temperature

Surface tension dictates fluid behavior, and predicting its magnitude is vital in many applications. Equations have previously been derived to describe how the surface tension of pure liquids changes with temperature, and other models have been derived to describe how the surface tension of mixtures changes with liquid-phase composition. However, the simultaneous dependence of surface tension on temperature and composition for liquid mixtures has been less studied. Past approaches have required extensive experimental data to which models have been fit, yielding a distinct set of fitting parameters at each temperature or composition. Herein, we propose a model that requires only three fitting procedures to predict surface tension as a function of temperature and composition. We achieve this by analyzing and extending the Shereshefsky (J. Colloid Interface Sci. 1967, 24 (3), 317-322), Li et al. (Fluid Phase Equilib. 2000, 175, 185-196), and Connors-Wright (Anal. Chem. 1989, 61 (3), 194-198) models to high temperatures for 15 aqueous systems. The best extensions of the Shereshefsky, Li et al., and Connors-Wright models achieve average relative deviations of 2.11%, 1.20%, and 0.62%, respectively, over all systems. We thus recommend the extended Connors-Wright model for predicting the surface tension of aqueous mixtures at different temperatures with the tabulated coefficients herein. An additional outcome of this study is the previously unreported equivalence of the Li et al. and Connors-Wright models in describing experimental data of surface tension as a function of composition at a single temperature.

A new paradigm on the chemical potentials of components in multi-component nano-phases within multi-phase systems

A new paradigm is offered claiming that the thermodynamic nano-effect in multi-component and multiphase systems is proportional to the increased surface areas of the phases and not to their increased curvatures (as the Kelvin paradigm claims).