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

Abstract

The leading order terms in a curvature expansion of the surface tension, the Tolman length (first order), and rigidities (second order) have been shown to play an important role in the description of nucleation processes. This work presents general and rigorous expressions to compute these quantities for any nonlocal density functional theory (DFT). The expressions hold for pure fluids and mixtures, and reduce to the known expressions from density gradient theory (DGT). The framework is applied to a Helmholtz energy functional based on the perturbed chain polar statistical associating fluid theory (PCP-SAFT) and is used for an extensive investigation of curvature corrections for pure fluids and mixtures. Predictions from the full DFT are compared to two simpler theories: predictive density gradient theory (pDGT), that has a density and temperature dependent influence matrix derived from DFT, and DGT, where the influence parameter reproduces the surface tension as predicted from DFT. All models are based on the same equation of state and predict similar Tolman lengths and spherical rigidities for small molecules, but the deviations between DFT and DGT increase with chain length for the alkanes. For all components except water, we find that DGT underpredicts the value of the Tolman length, but overpredicts the value of the spherical rigidity. An important basis for the calculation is an accurate prediction of the planar surface tension. Therefore, further work is required to accurately extract Tolman lengths and rigidities of alkanols, because DFT with PCP-SAFT does not accurately predict surface tensions of these fluids.

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

“…The difficulty in routinely using this equation is the uncertainty in both the magnitude and sign of the Tolman length. 14,[33][34][35] Classical nucleation theory also assumes perfect thermal equilibrium throughout the system. However, cluster formation involves the release of substantial amounts of latent heat, which requires subsequent collisions with the carrier gas to be dissipated.…”

“…Unlike in the CNT formalism, eqn (34) internally restricts the critical size from being smaller than the monomer radius. It would be imprudent to claim that the Tolman length for pure CO 2 clusters is positive and its value to be r 1 /2 (actually, it is recently estimated to be negative 35 ). The origin of the n 1/3 term is not necessarily only due to the effect of curvature on surface tension, but other physical effects with the same functional form might also be present.…”

Curvature Corrections Remove the Inconsistencies of Binary Classical Nucleation Theory

“…[30], it was shown that square gradient theory (SGT) gave very similar Helfrich coefficients as full density functional theory with a basis in perturbed-chain polar statistical associating fluid theory, even for surface-active mixtures. Since the full density functional theory may give inaccurate predictions for alcohols [30], we have combined SGT with the cubic plus association (CPA) equation of state (EOS) to compute δ and k s for several water-alcohol mixtures [31][32][33]. The methodology is detailed in [29,30], and in the Supplementary Information (SI), which includes Refs.…”

“…Since the full density functional theory may give inaccurate predictions for alcohols [30], we have combined SGT with the cubic plus association (CPA) equation of state (EOS) to compute δ and k s for several water-alcohol mixtures [31][32][33]. The methodology is detailed in [29,30], and in the Supplementary Information (SI), which includes Refs. [34][35][36][37][38][39][40][41].…”

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

“…1(a) to 1(d)), particularly for a very small cluster down to sub-nanometer scale [13,14]. Despite numerous simulation and theoretical studies, the curvature-dependent surface tension is still an issue un-der ongoing debate [15][16][17]. Especially for alcohols, there have been studies that incorporate the curvature correction to the alcohols' surface tension for better analysis of the results in the homogeneous nucleation (HON) rate experiment [18,19].…”