Isobaric Vapor–Liquid Phase Diagrams for Multicomponent Systems with Nanoscale Radii of Curvature

Oiling-Out: Insights from Gibbsian Surface Thermodynamics by Stability Analysis

The crystallization process is a significant stage in the pharmaceutical industry. During the process of crystallization with cooling, it is possible for a secondary liquid phase to appear before the formation of crystals. This phenomenon is called "oiling out" or liquid-liquid phase separation (LLPS). In this article, we explore the oiling-out phenomenon in a binary system of water and vanillin using stability analysis based on Gibbsian surface thermodynamics. To obtain the full picture of oiling out, we investigated three cases: droplet-solute-lean liquid equilibrium (DLE), crystal-solute-rich liquid equilibrium (CL'E), and crystal-solute-lean liquid equilibrium (CLE). The phase diagram of the system is plotted using the NRTL model for activity coefficients, along with considering the effect of the interfacial curvature on the phase diagram. From the phase boundaries and free-energy diagram of each case, we showed that the occurrence of the oiling-out phenomenon is justified based on the lower energy barrier of the droplet formation compared to that of the crystal formation. However, the energy level of a stable crystal is significantly lower and hence more stable than that of a stable droplet. Finally, we have determined different regions for droplet and crystal formation in the metastable phase diagram based on their supersaturation and provide insight for the oiling-out phenomenon.

Partition Constant of Binary Mixtures for the Equilibrium between a Bulk and a Confined Phase

It is well-known that the thermodynamic, kinetic and structural properties of fluids, and in particular of water and its solutions, can be drastically affected in nanospaces. A possible consequence of nanoscale confinement of a solution is the partial segregation of its components. Thereby, confinement in nanoporous materials (NPM) has been proposed as a means for the separation of mixtures. In fact, separation science can take great advantage of NPM due to the tunability of their properties as a function of nanostructure, morphology, pore size, and surface chemistry. Alcohol-water mixtures are in this context among the most relevant systems. However, a quantitative thermodynamic description allowing for the prediction of the segregation capabilities as a function of the material-solution characteristics is missing. In the present study we attempt to fill this vacancy, by contributing a thermodynamic treatment for the calculation of the partition coefficient in confinement. Combining the multilayer adsorption model for binary mixtures with the Young equation, we conclude that the liquid-vapor surface tension and the contact angle of the pure substances can be used to predict the separation ability of a particular material for a given mixture to a semiquantitative extent. Moreover, we develop a Kelvin-type equation that relates the partition coefficient to the radius of the pore, the contact angle, and the liquid-vapor surface tensions of the constituents. To assess the validity of our thermodynamic formulation, coarse grained molecular dynamics simulations were performed on models of alcohol-water mixtures confined in cylindrical pores. To this end, a coarse-grained amphiphilic molecule was parametrized to be used in conjunction with the mW potential for water. This amphiphilic model reproduces some of the properties of methanol such as enthalpy of vaporization and liquid-vapor surface tension, and the minimum of the excess enthalpy for the aqueous solution. The partition coefficient turns out to be highly dependent on the molar fraction, on the interaction between the components and the confining matrix, and on the radius of the pore. A remarkable agreement between the theory and the simulations is found for pores of radius larger than 15 Å.

Thermodynamic Investigation of Droplet-Droplet and Bubble-Droplet Equilibrium in an Immiscible Medium

In the absence of external fields, interfacial tensions between different phases dictate the equilibrium morphology of a multiphase system. Depending on the relative magnitudes of these interfacial tensions, a composite system made up of immiscible fluids in contact with one another can exhibit contrasting behavior: the formation of lenses in one case and complete encapsulation in another. Relatively simple concepts such as the spreading coefficient (SC) have been extensively used by many researchers to make predictions. However, these qualitative methods are limited to determining the nature of the equilibrium states and do not provide enough information to calculate the exact equilibrium geometries. Moreover, due to the assumptions made, their validity is questionable at smaller scales where pressure forces due to curvature of the interfaces become significant or in systems where a compressible gas phase is present. Here we investigate equilibrium configurations of two fluid drops suspended in another fluid, which can be seen as a simple building block of more complicated systems. We use Gibbsian composite-system thermodynamics to derive equilibrium conditions and the equation acting as the free energy (thermodynamic potential) for this system. These equations are then numerically solved for an example system consisting of a dodecane drop and an air bubble surrounded by water, and the relative stability of distinct equilibrium shapes is investigated based on free-energy comparisons. Quantitative effects of system parameters such as interfacial tensions, volumes, and the scale of the system on geometry and stability are further explored. Multiphase systems similar to the ones analyzed here have broad applications in microfluidics, atmospheric physics, soft photonics, froth flotation, oil recovery, and some biological phenomena.

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.

Bubble Point Pressures of Hydrocarbon Mixtures in Multiscale Volumes from Density Functional Theory

Accurate characterization of the bubble point pressure of hydrocarbon mixtures under nanoconfinement is crucial to the prediction of ultimate oil recovery and well productivity of shale/tight oil reservoirs. Unlike conventional reservoirs, shale has an extensive network of tiny pores in the range of a few nanometers. In nanopores, the properties of hydrocarbon fluids deviate from those in bulk because of significant surface adsorption. Many previous theoretical works use a conventional equation of state model coupled with capillary pressure to study the nanoconfinement effect. Without including the inhomogeneous molecular density distributions in nanoconfinement, these previous approaches predict only slightly reduced bubble points. In this work, we use density functional theory to study the effect of nanoconfinement on the hydrocarbon mixture bubble point pressure by explicitly considering fluid-surface interactions and inhomogeneous density distributions in nanopores. We find that as system pressure decreases, while lighter components are continuously released from the nanopores, heavier components accumulate within. The bubble point pressure of nanoconfined hydrocarbon mixtures is thus significantly suppressed from the bulk bubble point to below the bulk dew point, in line with our previous experiments. When bulk fluids are in a two-phase, the confined hydrocarbon fluids are in a single liquid-like phase. As pore size increases, bubble point pressure of confined fluids increases and hydrocarbon average density in nanopores approaches the liquid-phase density in bulk when bulk is in a two-phase region. For a finite volume bulk bath, we find that because of the competitive adsorption in nanopores, the bulk bubble point pressure increases in line with a previous experimental work. Our work demonstrates how mixture dynamics and nanopore-bulk partitioning influence phase behavior in nanoconfinement and enables the accurate estimation of hydrocarbon mixture bubble point pressure in shale nanopores.

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.