Evaluation of finite-size effects in cavitation and droplet formation

Estimating metastable thermodynamic properties by isochoric extrapolation from stable states

The description of metastable fluids, those in local but not global equilibrium, remains an important problem of thermodynamics, and it is crucial for many industrial applications and all first order phase transitions. One way to estimate their properties is by extrapolation from nearby stable states. This is often done isothermally, in terms of a virial expansion for gases or a Taylor expansion in density for liquids. This work presents evidence that an isochoric expansion of pressure at a given temperature is superior to an isothermal density expansion. Two different isochoric extrapolation strategies are evaluated, one best suited for vapors and one for liquids. Both are exact for important model systems, including the van der Waals equation of state. Moreover, we present a simple method to evaluate all the coefficients of the isochoric expansion directly from a simulation in the canonical ensemble. Using only the properties of stable states, the isochoric extrapolation methods reproduce simulation results with Lennard-Jones potentials, mostly within their uncertainties. The isochoric extrapolation methods are able to predict deeply metastable pressures accurately even from temperatures well above the critical. Isochoric extrapolation also predicts a mechanical stability limit, i.e., the thermodynamic spinodal. For water, the liquid spinodal pressure is predicted to be monotonically decreasing with decreasing temperature, in contrast to the re-entrant behavior predicted by the direct extension of the reference equation of state.

Confinement Effects in Droplet Formation on a Solid Particle

Formation of a droplet around a spherical solid particle in supersaturated vapor is considered. The number and stability of equilibrium solutions in a closed small system are studied in the canonical ensemble in comparison to an open system in the grand canonical ensemble. Depending on the system's parameters, two modes exist in the canonical ensemble: the first one with only one solution and the second one with three solutions; the presence of the third solution is due to confinement. The analysis is conducted first on a macroscopic thermodynamic level of description, and then the results are supported by studies within two versions of classical density functional theory: the square-gradient approximation with the Carnahan-Starling equation of state for hard spheres on a completely wettable particle and the random-phase approximation with the fundamental measure theory on a poorly wettable particle. In the latter case, a solution breaking the spherical symmetry is observed at a small total number of molecules.

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.

Surface tension of cavitation bubbles

We have studied homogeneous cavitation in liquid nitrogen and normal liquid helium. We monitor the fluid content in a large number of independent mesopores with an ink-bottle shape, either when the fluid in the pores is quenched to a constant pressure or submitted to a pressure decreasing at a controlled rate. For both fluids, we show that, close enough to their critical point, the cavitation pressure threshold is in good agreement with the Classical Nucleation Theory (CNT). In contrast, at lower temperatures, deviations are observed, consistent with a reduction of the surface tension for bubbles smaller than two nanometers in radius. For nitrogen, we could accurately measure the nucleation rate as a function of the liquid pressure down to the triple point, where the critical bubble radius is about one nanometer. We find that CNT still holds, provided that the curvature dependence of the surface tension is taken into account. Furthermore, we evaluate the first- and second-order corrections in curvature, which are in reasonable agreement with recent calculations for a Lennard-Jones fluid.

Effects of compressibility and wetting on the liquid-vapor transition in a confined fluid

When a fluid is constrained to a fixed, finite volume, the conditions for liquid-vapor equilibrium are different from those for the infinite volume or constant pressure cases. There is even a range of densities for which no bubble can form, and the liquid at a pressure below the bulk saturated vapor pressure remains indefinitely stable. As the fluid density in mineral inclusions is often derived from the temperature of bubble disappearance, a correction for the finite volume effect is required. Previous works have explained these phenomena and proposed a numerical procedure to compute the correction for pure water in a container completely wet by the liquid phase. Here, we revisit these works and provide an analytic formulation valid for any fluid, including the case of partial wetting. We introduce the Berthelot-Laplace length λ = 2γκ/3, which combines the liquid isothermal compressibility κ and its surface tension γ. The quantitative effects are fully captured by a single, nondimensional parameter: the ratio of λ to the container size.

Thermodynamic Stability of Volatile Droplets and Thin Films Governed by Disjoining Pressure in Open and Closed Containers

Distributed thin films of water and their coexistence with droplets are investigated using a capillary description based on a thermodynamic fundamental relation for the film Helmholtz energy, derived from disjoining pressure isotherms and an accurate equation of state. Gas-film and film-solid interfacial tensions are derived, and the latter has a dependence on film thickness. The resulting energy functionals from the capillary description are discretized, and stationary states are identified. The thermodynamic stability of configurations with thin films in systems that are closed (canonical ensemble) or connected to a particle reservoir (grand canonical ensemble) is evaluated by considering the eigenvalues of the corresponding Hessian matrices. The conventional stability criterion from the literature states that thin flat films are stable when the derivative of the disjoining pressure with respect to the film thickness is negative. This criterion is found to apply only in open systems. A closer inspection of the eigenvectors of the negative eigenvalues shows that condensation/evaporation destabilizes the film in an open system. In closed systems, thin films can be stable even though the disjoining pressure derivative is positive, and their stability is governed by mechanical instabilities of a similar kind to those responsible for spinodal dewetting in nonvolatile systems. The films are stabilized when their thickness and disjoining pressure derivative are such that the minimum unstable wavelength is larger than the container diameter. Droplets in coexistence with thin films are found to be unstable for all considered examples in open systems. In closed systems, they are found to be stable under certain conditions. The unstable droplets in both open and closed systems are saddle points in their respective energy landscapes. In the closed system, they represent the activation barrier for the transition between a stable film and a stable droplet. In the open system, the unstable droplets represent the activation barrier for the transition from a film into a bulk liquid phase. Thin films are found to be the equilibrium configuration up to a certain value of the total density in a closed system. Beyond this value, there is a morphological phase transition to stable droplet configurations.

Nucleation and superstabilization in small systems

Phase transitions are known to present peculiarities in small systems that are related to depletion effects of the ambient phase. Mass conservation affects the conditions of thermodynamic equilibrium between a nucleus of the new phase and the matrix as compared with nucleation in infinite systems. This finite-size effect is known to delay the phase transition but can also impede nucleation in very small systems as it stabilizes the initial state, originally metastable in infinite systems. In this work, we investigate this superstabilization effect in the context of classical nucleation theory in multicomponent solutions and we derive an analytical expression for the system size below which nucleation becomes thermodynamically impossible. Comparing with the exact solution, our simple result is shown to accurately predict the superstabilization effect, and can therefore be used, for instance, as a guideline for the design of novel nanomaterials.