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.

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.

Bubble Formation in a Finite Cone: More Pieces to the Puzzle.

We investigate the stability of bubble formation, starting with a convex or a concave meniscus, from a liquid solution (of water and a dissolved gas) inside a finite cone at constant temperature and constant liquid pressure (above the saturation pressure of the pure solvent). It is assumed that the dissolved gas (nitrogen) forms a dilute solution at equilibrium, which can be described by Henry's law. The number and nature of equilibrium states are determined with Gibbsian composite-system thermodynamics, both from the intersection of the equilibrium Kelvin radius with the geometry radius and from the extrema in the plot of free energy of the system versus size of the new phase. Bubble stability is studied along the whole growth path, as the bubble grows inside, gets pinned, and grows further outside the finite cone. The changes in the concentration of the liquid bulk phase and the vapor phase during the growth of the bubble are carefully incorporated in the equations. The effects of various parameters, including cone apex angle, cone half mouth radius, contact angle, total number of moles, and initial degree of saturation, on the stability of the bubble are also investigated. Stability of bubble formation from a liquid solution inside a confined geometry such as a finite cone is of interest in areas such as restoring underwater superhydrophobicity and adhesion of particles to the roughness of synthetic biomaterials.

Comparison of the Osmotic Virial Equation with the Margules Activity Model for Solid-Liquid Equilibrium.

The main goal of this paper is to compare the general polynomial forms of the osmotic virial equation and the Margules model for the liquid activity coefficients in binary systems. The coefficients/parameters in each model can be calculated based on best fits to experimental phase equilibrium data. Here, the activity coefficient models were combined with the equation of solid-liquid equilibrium. We obtain the coefficients/parameters for the osmotic virial equation and the one- and two-parameter Margules models and compare the accuracy of each model for fitting the experimental data for five water/solute systems: water/glycerol, water/acetic acid, water/propanoic acid, water/mono-ethylene glycol, and water/sulfolane. In obtaining the osmotic virial coefficients, we present a method to fit the equation to the entire range of data including both the ice-formation region and the solute-precipitation region. In expression of the concentration effect of the solute, we showed that the integration constant that arises from the Gibbs-Duhem equation is dependent on the osmotic virial coefficients. The osmotic virial equation is of great interest for its ability to empirically model a very wide range of aqueous solutions and as such is one of the most widely used solution models in biology.

Thermodynamic Investigation of the Effect of Interface Curvature on the Solid-Liquid Equilibrium and Eutectic Point of Binary Mixtures.

Thermodynamic phase behavior is affected by curved interfaces in micro- and nanoscale systems. For example, capillary freezing point depression is associated with the pressure difference between the solid and liquid phases caused by interface curvature. In this study, the thermal, mechanical, and chemical equilibrium conditions are derived for binary solid-liquid equilibrium with a curved solid-liquid interface due to confinement in a capillary. This derivation shows the equivalence of the most general forms of the Gibbs-Thomson and Ostwald-Freundlich equations. As an example, the effect of curvature on solid-liquid equilibrium is explained quantitatively for the water/glycerol system. Considering the effect of a curved solid-liquid interface, a complete solid-liquid phase diagram is developed over a range of concentrations for the water/glycerol system (including the freezing of pure water or precipitation of pure glycerol depending on the concentration of the solution). This phase diagram is compared with the traditional phase diagram in which the assumption of a flat solid-liquid interface is made. We show the extent to which nanoscale interface curvature can affect the composition-dependent freezing and precipitating processes, as well as the change in the eutectic point temperature and concentration with interface curvature. Understanding the effect of curvature on solid-liquid equilibrium in nanoscale capillaries has applications in the food industry, soil science, cryobiology, nanoporous materials, and various nanoscience fields.

Thermodynamics of Surface Nanobubbles.

In this paper, we examine the thermodynamic stability of surface nanobubbles. The appropriate free energy is defined for the system of nanobubbles on a solid surface submerged in a supersaturated liquid solution at constant pressure and temperature, under conditions where an individual nanobubble is not in diffusive contact with a gas phase outside of the system or with other nanobubbles on the time scale of the experiment. The conditions under which plots of free energy versus the radius of curvature of the nanobubbles show a global minimum, which denotes the stable equilibrium state, are explored. Our investigation shows that supersaturation and an anomalously high contact angle (measured through the liquid) are required to have stable surface nanobubbles. In addition, the anomalously high contact angle of surface nanobubbles is discussed from the standpoint of a framework recently proposed by Koch, Amirfazli, and Elliott that relates advancing and receding contact angles to thermodynamic equilibrium contact angles, combined with the existence of a gas enrichment layer.

Surface thermodynamic analysis of fluid confined in a cone and comparison with the sphere-plate and plate-plate geometries.

The behavior of pure fluid confined in a cone is investigated using thermodynamic stability analysis. Four situations are explained on the basis of the initial confined phase (liquid/vapor) and its pressure (above/below the saturation pressure). Thermodynamic stability analysis (a plot of the free energy of the system versus the size of the new potential phase) reveals whether the phase transition is possible and, if so, the number and type (unstable/metastable/stable) of equilibrium states in each of these situations. Moreover we investigated the effect of the equilibrium contact angle and the cone angle (equivalent to the confinement's surface separation distance) on the free energy (potential equilibrium states). The results are then compared to our previous study of pure fluid confined in the gap between a sphere and a flat plate and the gap between two flat plates.1 Confined fluid behavior of the four possible situations (for these three geometries) can be explained in a unified framework under two categories based on only the meniscus shape (concave/convex). For systems with bulk-phase pressure imposed by a reservoir, the stable coexistence of pure liquid and vapor is possible only when the meniscus is concave.

Comparative surface thermodynamic analysis of new fluid phase formation between a sphere and a flat plate.

This paper investigates the behavior of confined fluid in the gap between a sphere and a flat plate by examining the curve of free energy of the system versus size of the new phase. Four possible situations corresponding to new phase formation out of confined liquid or vapor at pressures above or below the saturation pressure are studied. Using surface thermodynamics, the feasible shape of the meniscus (concave/convex), the possibility of phase transition, as well as the number and the nature (unstable/stable) of equilibrium states have been determined for each of these four situations. The effects of equilibrium contact angle, separation distance of confinement surfaces, and sphere size have been studied. We show that the number and nature of equilibrium states, along with the effect of different parameters in these four possible situations, can be well described under two categories of new phase formation with (a) concave or (b) convex meniscus. Our results reveal that in the sphere-plate gap, stable coexistence of the liquid and vapor phases is only possible when the meniscus is concave (which corresponds to either capillary condensation or capillary evaporation), and when the sphere and plate are separated by a distance less than a critical amount (where that critical amount is always less than the Kelvin radius). With convex menisci, no stable coexistence of liquid and vapor phase is possible.