Tolman lengths and rigidity constants from free-energy functionals—General expressions and comparison of theories

Effect of molecular branching and surface wettability on solid-liquid surface tension and line-tension of liquid alkane surface nanodroplets

HYPOTHESIS

Surface nanodroplets have important technological applications. Previous experiments and simulations have shown that their contact angle deviates from Young's equation. A modified version of Young's equation considering the three-phase line tension (τ) has been widely used in literature, and a wide range of values for τ are reported. We have recently shown that molecular branching affects the liquid-vapour surface tension γlv of liquid alkanes. Therefore, the wetting behaviour of surface nanodroplets should be affected by molecular branching. This study conducted molecular dynamics (MD) simulations to gain insight into the wetting behaviour of linear and branched alkane nanodroplets on oleophilic and oleophobic surfaces. We aim to examine the Young equation's validity and branching's effect on fundamental properties, including solid-liquid surface tension γsl and line tension τ.

SIMULATIONS

The simulations were performed on a linear alkane, triacontane (C30H62), as well as four of its branched isomers: 2,6,13,17-tetrapropyloctadecane,2,6,9,10,13,17-hexaethyloctadecane, 2,5,7,8,11,12,15-heptaethylhexadecane and 2,3,6,7,10,11-hexapropyldodecane. Nanodroplets with a diameter of approximately 15 nm were released onto the surfaces, and their contact angles were measured. Additionally, using a novel approach, the solid-liquid surface tension (γsl), the validity of Young's equation and line tension for all alkane and surface combinations are determined.

FINDINGS

It was discovered that the calculated γsl, deviated from the theoretical γsl,Young predicted from Young's equation for all alkanes on oleophilic surfaces. However, this deviation was minimal for branched alkanes on the oleophobic surfaces but more significant for the linear alkane. The findings indicated that γsl < 0 for oleophilic surfaces and γsl > 0 for oleophobic surfaces. Moreover, it was observed that |γsl| was lower for branched molecules and decreased as branching increased. Line tension values were then determined through a novel method, showing τ was positive for oleophilic surfaces ranging from 1.30 × 10-10 to 6.27 × 10-11N. On an oleophobic surface, linear alkane shows a negative line tension of -1.15 × 10-10N and branched alkanes up to two orders of magnitude lower values ranging from -2.09 × 10-12 to 2.43 × 10-11N. Line tension values between -1.15 × 10-10 and + 1.1 × 10-10N are calculated for various linear alkane and surface combinations. These findings show the dependence of line tension on the contact angle and branching, demonstrating that for linear alkanes, τ is significant, whereas, for branched alkanes, line tension is smaller or negligible for large contact angles.

Thermodynamic Theory of Curvature-Dependent Surface Tension: Tolman's Theory Revisited

An exact equation for determining the Tolman length (TL) as a function of radius is obtained within the framework of classical thermodynamics and a computational procedure for solving it is proposed. As a result of implementing this procedure, the dependences of the TL and surface tension on radius are obtained for the drop and bubble cases and various equations of state. As one of the results of the thermodynamic study, a new equation for the dependence of surface tension on radius (curvature effect) alternative to the corresponding Tolman equation and associated with the spinodal point is obtained. The fundamental impossibility to determine the curvature effect analytically from the binodal point, i.e., using the Tolman equation, is established; it is calculated only from the spinodal point and is determined by the characteristics of the system at this point. The sign of the TL asymptotic value debated in the literature in recent decades is uniquely determined in the theory: it is negative for drops and positive for bubbles.

Hierarchical approximations to the nucleation work in the entire range of metastability

The work W to form a nucleus (also known as the critical nucleus) is a key quantity in the description of nucleation phenomena because of its exponentially strong effect on the nucleation rate. The present study provides a general approximate expression for W, which comprises a hierarchy of approximations to the dependence of W on the experimentally controlled overpressure Δp of a nucleating multicomponent phase. This general expression is used to derive explicit formulas for the lowest-order members of the W(Δp) hierarchy as well as for the respective lowest-order approximations to the Δp dependences of the nucleus surface tension, the nucleus radius, the Gibbs-Tolman length, and the stationary nucleation rate. The second-order and the third-order approximations to the W(Δp) dependence are confronted with available W(Δp) data, and the latter is found to agree very well with the data. The results obtained are applicable to homogeneous single-component or multicomponent nucleation from the binodal to the spinodal of the old phase, i.e., in the entire range of the old-phase metastability.

Free energy of critical droplets-from the binodal to the spinodal

Arguably, the main challenge of nucleation theory is to accurately evaluate the work of formation of a critical embryo in the new phase, which governs the nucleation rate. In Classical Nucleation Theory (CNT), this work of formation is estimated using the capillarity approximation, which relies on the value of the planar surface tension. This approximation has been blamed for the large discrepancies between predictions from CNT and experiments. In this work, we present a study of the free energy of formation of critical clusters of the Lennard-Jones fluid truncated and shifted at 2.5σ using Monte Carlo simulations, density gradient theory, and density functional theory. We find that density gradient theory and density functional theory accurately reproduce molecular simulation results for critical droplet sizes and their free energies. The capillarity approximation grossly overestimates the free energy of small droplets. The incorporation of curvature corrections up to the second order with the Helfrich expansion greatly remedies this and performs very well for most of the experimentally accessible regions. However, it is imprecise for the smallest droplets and largest metastabilities since it does not account for a vanishing nucleation barrier at the spinodal. To remedy this, we propose a scaling function that uses all relevant ingredients without adding fitting parameters. The scaling function reproduces accurately the free energy of the formation of critical droplets for the entire metastability range and all temperatures examined and deviates from density gradient theory by less than one k_BT.

Classical density functional theory for interfacial properties of hydrogen, helium, deuterium, neon, and their mixtures

We present a classical density functional theory (DFT) for fluid mixtures that is based on a third-order thermodynamic perturbation theory of Feynman-Hibbs-corrected Mie potentials. The DFT is developed to study the interfacial properties of hydrogen, helium, neon, deuterium, and their mixtures, i.e., fluids that are strongly influenced by quantum effects at low temperatures. White Bear fundamental measure theory is used for the hard-sphere contribution of the Helmholtz energy functional, and a weighted density approximation is used for the dispersion contribution. For mixtures, a contribution is included to account for non-additivity in the Lorentz-Berthelot combination rule. Predictions of the radial distribution function from DFT are in excellent agreement with results from molecular simulations, both for pure components and mixtures. Above the normal boiling point and 5% below the critical temperature, the DFT yields surface tensions of neon, hydrogen, and deuterium with average deviations from experiments of 7.5%, 4.4%, and 1.8%, respectively. The surface tensions of hydrogen/deuterium, para-hydrogen/helium, deuterium/helium, and hydrogen/neon mixtures are reproduced with a mean absolute error of 5.4%, 8.1%, 1.3%, and 7.5%, respectively. The surface tensions are predicted with an excellent accuracy at temperatures above 20 K. The poor accuracy below 20 K is due to the inability of Feynman-Hibbs-corrected Mie potentials to represent the real fluid behavior at these conditions, motivating the development of new intermolecular potentials. This DFT can be leveraged in the future to study confined fluids and assess the performance of porous materials for hydrogen storage and transport.

Mesoscale perspective on the Tolman length

We demonstrate that the multiphase Shan-Chen lattice Boltzmann method (LBM) yields a curvature dependent surface tension σ as computed from three-dimensional hydrostatic droplets and bubbles simulations. Such curvature dependence is routinely characterized, at first order, by the so-called Tolman length δ. LBM allows one to precisely compute σ at the surface of tension R_{s} and determine the Tolman length from the coefficient of the first order correction. The corresponding values of δ display universality for different equations of state, following a power-law scaling near the critical temperature. The Tolman length has been studied so far mainly via computationally demanding Molecular Dynamics simulations or by means of Density Functional Theory approaches playing a pivotal role in extending Classical Nucleation Theory. The present results open a hydrodynamic-compliant mesoscale arena, in which the fundamental role of the Tolman length, alongside real-world applications to cavitation phenomena, can be effectively tackled. All the results can be independently reproduced through the "idea.deploy" framework.

Application of Generalized (Hyper-) Dual Numbers in Equation of State Modeling

The calculation of derivatives is ubiquitous in science and engineering. In thermodynamics, in particular, state properties can be expressed as derivatives of thermodynamic potentials. The manual differentiation of complex models can be tedious and error-prone. In this work, we revisit dual and hyper-dual numbers for the calculation of exact derivatives and show generalizations to higher order derivatives and derivatives with respect to vector quantities. The methods described in this paper are accompanied by an open source Rust implementation with Python bindings. Applications of the generalized (hyper-) dual numbers are given in the context of equation of state modeling and the calculation of critical points.

Direct measurement of curvature-dependent surface tension of an alcohol nanomeniscus

Surface tension is a key parameter for understanding nucleation in the very initial stage of phase transformation. Although surface tension has been predicted to vary with the curvature of the liquid-vapor interface, particularly at the large curvature of, e.g., the subnanometric critical nucleus, experimental study still remains challenging due to inaccessibility to such a small cluster. Here, by directly measuring the critical size of a single capillary-condensed nanomeniscus using atomic force microscopy, we address the curvature dependence of surface tension of alcohols and observe that the surface tension is doubled for ethanol and n-propanol with a radius-of-curvature of ∼-0.46 nm. We also find that the interface of larger negative (positive) curvature exhibits larger (smaller) surface tension, which evidently governs nucleation at the ∼1 nm scale and below, indicating more facilitated nucleation than normally expected. Such well characterized curvature effects contribute to better understanding and accurate analysis of nucleation occurring in various fields including materials science and atmospheric science.

Homogeneous nucleation of carbon dioxide in supersonic nozzles II: molecular dynamics simulations and properties of nucleating clusters

Large scale molecular dynamics simulations of the homogeneous nucleation of carbon dioxide in an argon atmosphere were carried out at temperatures between 75 and 105 K. Extensive analyses of the nucleating clusters' structural and energetic properties were performed to quantify these details for the supersonic nozzle experiments described in the first part of this series [Dingilian et al., Phys. Chem. Chem. Phys., 2020, 22, 19282-19298]. We studied ten different combinations of temperature and vapour pressure, leading to nucleation rates of 10²³-10²⁵ cm^-3 s^-1. Nucleating clusters possess significant excess energy from monomer capture, and the observed cluster temperatures during nucleation - on both sides of the critical cluster size - are higher than that of the carrier gas. Despite strong undercooling with respect to the triple point, most clusters are clearly liquid-like during the nucleation stage. Only at the lowest simulation temperatures and vapour densities, clusters containing over 100 molecules are able to undergo a second phase transition to a crystalline solid. The formation free energies retrieved from the molecular dynamics simulations were used to improve the classical nucleation theory by introducing a Tolman-like term into the classical liquid-drop model expression for the formation free energy. This simulation-based theory predicts the simulated nucleation rates perfectly, and improves the prediction of the experimental rates compared to self-consistent classical nucleation theory.

Nucleation work, surface tension, and Gibbs-Tolman length for nucleus of any size

In the framework of the Gibbs approach to nucleation thermodynamics, expressions are derived for the nucleation work, nucleus size, surface tension, and Gibbs-Tolman length in homogeneous single-component nucleation at a fixed temperature. These expressions are in terms of the experimentally controlled overpressure of the nucleating phase and are valid for the entire overpressure range, i.e., for nucleus of any size. Analysis of available data for bubble and droplet nucleation in Lennard-Jones fluid shows that the theory describes well the data by means of a single free parameter, the Gibbs-Tolman length of the planar liquid/vapor interface. It is found that this length is about one-tenth of the Lennard-Jones molecular-diameter parameter and that it is positive for the bubble nucleus and negative for the droplet nucleus. In a sufficiently narrow temperature range, the nucleation work, nucleus radius, scaled surface tension, and Gibbs-Tolman length are apparently universal functions of scaled overpressure.

Curvature Corrections Remove the Inconsistencies of Binary Classical Nucleation Theory

The study of nucleation in fluid mixtures exposes challenges beyond those of pure systems. A striking example is homogeneous condensation in highly surface-active water-alcohol mixtures, where classical nucleation theory yields an unphysical, negative number of water molecules in the critical embryo. This flaw has rendered multicomponent nucleation theory useless for many industrial and scientific applications. Here, we show that this inconsistency is removed by properly incorporating the curvature dependence of the surface tension of the mixture into classical nucleation theory for multicomponent systems. The Gibbs adsorption equation is used to explain the origin of the inconsistency by linking the molecules adsorbed at the interface to the curvature corrections of the surface tension. The Tolman length and rigidity constant are determined for several water-alcohol mixtures and used to show that the corrected theory is free of physical inconsistencies and provides accurate predictions of the nucleation rates. In particular, for the ethanol-water and propanol-water mixtures, the average error in the predicted nucleation rates is reduced from 11-15 orders of magnitude to below 1.5. The curvature-corrected nucleation theory opens the door to reliable predictions of nucleation rates in multicomponent systems, which are crucial for applications ranging from atmospheric science to research on volcanos.