Divergence of Tolman's length for a droplet near the critical point

Particle size effect on surface/interfacial tension and Tolman length of nanomaterials: A simple experimental method combining with theoretical

Surface tension and interfacial tension are crucial to the study of nanomaterials. Herein, we report a solubility method using magnesium oxide nanoparticles of different radii (1.8-105.0 nm, MgO NPs) dissolved in pure water as a targeted model; the surface tension and interfacial tension (and their temperature coefficients) were determined by measuring electrical conductivity and combined with the principle of the electrochemical equilibrium method, and the problem of particle size dependence is discussed. Encouragingly, this method can also be used to determine the ionic (atomic or molecular) radius and Tolman length of nanomaterials. This research results disclose that surface/interfacial tension and their temperature coefficients have a significant relationship with particle size. Surface/interfacial tension decreases rapidly with a radius <10 nm (while the temperature coefficients are opposite), while for a radius >10 nm, the effect is minimal. Especially, it is proven that the value of Tolman length is positive, the effect of particle size on Tolman length is consistent with the surface/interfacial tension, and the Tolman length of the bulk does not change much in the temperature range. This work initiates a new era for reliable determination of surface/interfacial tension, their temperature coefficients, ionic radius, and Tolman length of nanomaterials and provides an important theoretical basis for the development and application of various nanomaterials.

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.

Molecular Dynamics Simulation of Tolman Length and Interfacial Tension of Symmetric Binary Lennard–Jones Liquid

The Tolman length and interfacial tension of partially miscible symmetric binary Lennard–Jones (LJ) fluids (A, B) was revealed by performing a large-scale molecular dynamics (MD) simulation with a sufficient interfacial area and cutting distance. A unique phenomenon was observed in symmetric binary LJ fluids, where two surfaces of tension existed on both sides of an equimolar dividing surface. The range of interaction εAB between the different liquids and the temperature in which the two LJ fluids partially mixed was clarified, and the Tolman length exceeded 3 σ when εAB was strong at higher temperatures. The results show that as the temperature or εAB increases, the Tolman length increases and the interfacial tension decreases. This very long Tolman length indicates that one should be very careful when applying the concept of the liquid–liquid interface in the usual continuum approximation to nanoscale droplets and capillary phase separation in nanopores.

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.

Communication: Computing the Tolman length for solid-liquid interfaces

The curvature dependence of interfacial free energy, which is crucial in quantitatively predicting nucleation kinetics and the stability of bubbles and droplets, is quantified by the Tolman length δ. For solid-liquid interfaces, however, δ has never been computed directly due to various theoretical and practical challenges. Here we perform a direct evaluation of the Tolman length from atomistic simulations of a solid-liquid planar interface in out-of-equilibrium conditions, by first computing the surface tension from the amplitude of thermal capillary fluctuations of a localized version of the Gibbs dividing surface and by then calculating how much the surface energy changes when it is defined relative to the equimolar dividing surface. We computed δ for a model potential, and found a good agreement with the values indirectly inferred from nucleation simulations. The agreement not only validates our approach but also suggests that the nucleation free energy of the system can be perfectly described using classical nucleation theory if the Tolman length is taken into account.

Soluble Model Fluids with Complete Scaling and Yang-Yang Features

Yang-Yang (YY) and singular diameter critical anomalies arise in exactly soluble compressible cell gas (CCG) models that obey complete scaling with pressure mixing. Thus, on the critical isochore ρ=ρ(c), C(μ)≔-Td(2)μ/dT(2) diverges as |t|^(-α) when t∝T-T(c)→0^(-) while ρ(d)-ρ(c)∼|t|^(2β) where ρ(d)(T)=1/2[ρ(liq)+ρ(gas)]. When the discrete local CCG cell volumes fluctuate freely, the YY ratio R(μ)=C(μ)/C(V) may take any value -∞<R(μ)<1/2 but "anticorrelated" free volumes are needed for R(μ)>0. More general decorated CCGs, including "hydrogen bonding" water models, illuminate energy-volume coupling as relevant to R(μ).

Finite-size effects on liquid-solid phase coexistence and the estimation of crystal nucleation barriers

A fluid in equilibrium in a finite volume V with particle number N at a density ρ=N/V exceeding the onset density ρ_{f} of freezing may exhibit phase coexistence between a crystalline nucleus and surrounding fluid. Using a method suitable for the estimation of the chemical potential of dense fluids, we obtain the excess free energy due to the surface of the crystalline nucleus. There is neither a need to precisely locate the interface nor to compute the (anisotropic) interfacial tension. As a test case, a soft version of the Asakura-Oosawa model for colloid-polymer mixtures is treated. While our analysis is appropriate for crystal nuclei of arbitrary shape, we find the nucleation barrier to be compatible with a spherical shape and consistent with classical nucleation theory.

Accurate determination of the vapor-liquid-solid contact line tension and the viability of Young equation

In this work, we present a theoretical method to determine the line tension of nanodroplets on homogeneous substrates via decomposing the grand free energy into volume, interface and line contributions. With the obtained line tension, we check the viability of Young equation and find that the chemical potential dependence (or equivalently, droplet curvature dependence) of the interface tensions is crucial for the viability of modified Young equation at the nanometer scale. In particular, the linear relationship between the cosine of contact angle and the curvature of the contact line, which is often used to determine the line tension, is found to be incorrect at the nanometer scale.

Positive Tolman length in a lattice gas with three-body interactions

We present a new method to determine the curvature dependence of the interface tension between coexisting phases in a finite volume from free energies obtained by Monte Carlo simulations. For the example of a lattice gas on a 3D fcc lattice with nearest neighbor three-body interactions, we demonstrate how to calculate the equimolar radius R(e) as well as the radius R(s) of the surface of tension and thus the Tolman length δ(R(s))=R(e)-R(s). Within the physically relevant range of radii, δ(R(s)) shows a pronounced R(s) dependence, such that the simple Tolman parametrization for the interface tension is refutable. For the present model, extrapolation of δ(R(s)) to R(s)→∞ by various methods clearly indicates a positive limiting value.

Universal critical behavior of curvature-dependent interfacial tension

From the analysis of Monte Carlo simulations of a binary Lennard-Jones mixture in the coexistence region, we provide evidence that the curvature dependence of the interfacial tension can be described by a simple theoretical function σ(R)ξ(2)=C(1)/[1+C(2)(ξ/R)(2)], where ξ is the correlation length and R is the droplet radius. The universal constants C(1) and C(2) are estimated. In the model, a Tolman length is strictly absent, but, since its critical behavior is believed to be much weaker than ξ, we argue that it only provides a correction to scaling and does not affect the leading critical behavior, which should be described by the above function for any system in the Ising universality class. The large value of C(2)≃32 implies that conventional nucleation theory becomes inaccurate even for a significantly large droplet radius.

Thermodynamic properties of a symmetrical binary mixture in the coexistence region

A three-dimensional symmetric binary fluid is studied, as a function of temperature, in the two-phase (liquid-liquid) coexistence region via Monte Carlo simulations. Particular focus has been in the understanding of curvature-dependent interfacial tension, which is observed to vary as σ(R) = σ(∞)/[1+2(ℓ/R)(2)], implying that a Tolman length is zero in the limit R → ∞. The length ℓ is found to have a critical divergence the same as the correlation length, but its amplitude is significantly larger (ℓ ~/= 4ξ). Our findings hence imply that the barrier against homogeneous nucleation is significantly reduced (in comparison with the classical nucleation theory) in the critical region. We also report results for the critical behavior of the flat interfacial tension σ(∞) and the concentration susceptibility, as well as the amplitude ratios involving these thermodynamic quantities. Noting that the interatomic potential in our model is described by the Lennard-Jones form that decays faster that 1/r(3), all of our results for critical phenomena are expectedly consistent with the Ising universality class of three spatial dimensions.

Molecular simulation of nitrogen adsorption in nanoporous silica

This article reports on a molecular simulation study of nitrogen adsorption and condensation at 77 K in atomistic silica cylindrical nanopores (MCM-41). Two models are considered for the nitrogen molecule and its interaction with the silica substrate. In the "pea" model, the nitrogen molecule is described as a single Lennard-Jones sphere and only Lennard-Jones interactions between the nitrogen molecule and the oxygens atoms of the silica substrate are taken into account. In the "bean" model (TraPPE force field), the nitrogen molecule is composed of two Lennard-Jones sites and a linear array of three charges on the atomic positions and at the center of the nitrogen-nitrogen bond. In the bean model, the interactions between the sites on the nitrogen molecule and the Si, O, and H atoms of the substrate are the sum of the Coulombic and dispersion interactions with a repulsive short-range contribution. The data obtained with the pea and bean models in silica nanopores conform to the typical behavior observed in the experiments for adsorption/condensation in cylindrical MCM-41 nanopores; the adsorbed amount increases continuously in the multilayer adsorption regime until an irreversible jump occurs because of capillary condensation and evaporation of the fluid within the pore. Our results suggest that the pea model can be used for characterization purposes where one is interested in capturing the global experimental behavior upon adsorption and desorption in silica nanopores. However, the bean model is more suitable to investigating the details of the interaction with the surface because this model, which accounts for the partial charges located on the nitrogen atoms of the molecule (quadrupole), allows a description of the specific interactions between this adsorbate and silica surfaces (silanol groups and siloxane bridges) or grafted silica surfaces. In particular, the bean model provides a more realistic picture of nitrogen adsorption in the vicinity of silica surfaces or confined in silica nanopores, where the isosteric heat of adsorption curves show that the nitrogen molecule in this model is sensitive to the surface heterogeneity.

Complete scaling for inhomogeneous fluids

"Complete scaling," which maps asymmetric fluid criticality onto the symmetric Ising model, is extended to spatially inhomogeneous fluids. This extension enables us to obtain a fluctuation-modified asymmetric interfacial density profile, which incorporates leading effects from the asymmetry of fluid phase coexistence and the asymmetry of the correlation length. The derived asymmetric interfacial profile is used to calculate Tolman's length, the diverging coefficient of the curvature correction to the surface tension. The amplitude of the divergent Tolman length is found to depend on the asymmetry of the correlation length.

Heterogeneous nucleation at a wall near a wetting transition: a Monte Carlo test of the classical theory

While for a slightly supersaturated vapor the free energy barrier ΔF(hom)(*), which needs to be overcome in a homogeneous nucleation event, may be extremely large, nucleation is typically much easier at the walls of the container in which the vapor is located. While no nucleation barrier exists if the walls are wet, for incomplete wetting of the walls, described via a nonzero contact angle Θ, classical theory predicts that nucleation happens through sphere-cap-shaped droplets attracted to the wall, and their formation energy is ΔF(het)(*) = ΔF(hom)(*)f(Θ), with f(Θ) = (1-cosΘ)(2)(2+cosΘ)/4. This prediction is tested through simulations for the simple cubic lattice gas model with nearest-neighbor interactions. The attractive wall is described in terms of a local 'surface field', leading to a critical wetting transition. The variation of the contact angle with the strength of the surface field is determined by using thermodynamic integration methods to obtain the wall free energies which enter Young's equation. Obtaining the chemical potential as a function of the density for a system with periodic boundary conditions (and no walls), the droplet free energy of a spherical droplet in the bulk is obtained for a wide range of droplet radii. Similarly, ΔF(het)(*) is obtained for a system with two parallel walls. We find that the classical theory is fairly accurate if a line tension correction for the contact angle is taken into account.

Blowing bubbles in Lennard-Jonesium along the saturation curve

Extensive molecular simulations of the Lennard-Jones fluid have been performed to determine its liquid-vapor coexistence properties and solvent contact densities with cavities up to ten times the diameter of the solvent from the triple point to the critical point. These simulations are analyzed using a revised scaled-particle theory [H. S. Ashbaugh and L. R. Pratt, Rev. Mod. Phys. 78, 159 (2006)] to evaluate the thermodynamics of cavity solvation and curvature dependent interfacial properties along the saturation curve. While the thermodynamic signatures of cavity solvation are distinct from those in water, exhibiting a chemical potential dominated by a large temperature independent enthalpy, the solvent dewets cavities of increasing size similar with water near coexistence. The interfacial tension for forming a liquid-wall interface is found to be consistently greater than the liquid-vapor surface tension of the Lennard-Jones fluid by up to 10% and potentially reflects the suppression of high amplitude fluctuations at the cavity surface. The first-order curvature correction for the surface tension is negative and appears to diverge to negative infinity at temperatures approaching the critical point. Our results point to the success of the revised scaled-particle theory at bridging molecular and macroscopic descriptions of cavity solvation.

Simulation of vapor-liquid coexistence in finite volumes: a method to compute the surface free energy of droplets

When a fluid at a constant density rho in between the densities of coexisting vapor (rhov) and liquid (rhol) at temperatures below criticality is studied in a (cubic) box of finite linear dimension L , phase separation occurs in this finite volume, provided L is large enough. For a range of densities, one can observe a liquid droplet (at density rhol' slightly exceeding rhol) coexisting in stable thermal equilibrium with surrounding vapor (with density rhov'>rhov, so in the thermodynamic limit such a vapor would be supersaturated). We show, via Monte Carlo simulations of a Lennard-Jones model of a fluid and based on a phenomenological thermodynamic analysis, that via recording the chemical potential micro as function of rho, one can obtain precise estimates of the droplet surface free energy for a wide range of droplet radii. We also show that the deviations of this surface free energy from the prediction based on the "capillarity approximation" of classical nucleation theory (i.e., using the interfacial free energy of a flat liquid-vapor interface for the surface free energy of a droplet irrespective of its radius) are rather small. We also study carefully the limitation of the present method due to the "droplet evaporation/condensation transition" occurring for small volumes and demonstrate that very good equilibrium is achieved in our study, by showing that the radial profile of the local chemical potential from the droplet center to the outside is perfectly flat.

Simple phenomenological model for phase transitions in confined geometry. 2. Capillary condensation/evaporation in cylindrical mesopores

A simple phenomenological model that describes capillary condensation and evaporation of pure fluids confined in cylindrical mesopores is presented. Following the work of Celestini (Celestini, F. Phys. Lett. A 1997, 228, 84), the free energy density of the system is derived using interfacial tensions and a corrective term that accounts for the interaction coupling between the vapor/adsorbed liquid and the adsorbed liquid/adsorbent interfaces. This corrective term is shown to be consistent with the Gibbs adsorption isotherm and assessed by standard adsorption tests. This model reveals that capillary condensation and evaporation are metastable and equilibrium processes, respectively, hence exhibiting the existence of a hysteresis loop inadsorption/desorption isotherm that is well-known in experiment. We extend the phenomenological model of Celestini to give a quantitative description of adsorption on the pore wall and hysteresis width evolution with temperature and confinement. Direct quantitative comparison is made with experimental data for confined argon. Used as a characterizing tool, this integrated model allows in a single fit of an experimental adsorption/desorption isotherm assessing essential characterization data such as the specific surface area, pore volume, and mean pore size.

Diverging curvature correction to the interfacial tension in polymer solutions

Application of polymer scaling to the problem of Tolman's length, a curvature correction coefficient in the interfacial tension, shows that Tolman's length in polymer solutions may become as large as half of the thickness of the interface. Tolman's length depends on the degree of polymerization N and the distance to the critical point of phase separation, Delta T. In the "critical" regime (N 1/2|Delta T|<<1) Tolman's length diverges upon approach to the critical temperature as approximately N 0.348|Delta T|-0.304. In the "polymer" regime (N 1/2|Delta T|>> 1) Tolman's length does not depend on N , but diverges more strongly, as approximately |Delta T|-1, proportional to the thickness of the interface.

Principle of isomorphism and complete scaling for binary-fluid criticality

The extension of the principle of critical-point universality to binary fluid mixtures, known as isomorphism of critical phenomena, has been reformulated in terms of complete scaling, a concept that properly matches asymmetric fluid-phase behavior with the symmetric Ising model. The controversial issue of the proper definition of the order parameter in binary fluid mixtures is clarified. We show that asymmetry of liquid-liquid coexistence in terms of mole fractions originates from two different sources: one is associated with a correlation between concentration and entropy fluctuations, whereas the other source is the correlation between concentration and density fluctuations. By analyzing the coexistence curves of liquid solutions of nitrobenzene in a series of hydrocarbons (from n -pentane to n -hexadecane), we have separated these two sources of asymmetry and found that the leading nonanalytical contribution to the asymmetry correlates linearly with the solute-solvent molecular-volume ratio. Other thermodynamic consequences of complete scaling for binary mixtures, such as an analog of the Yang-Yang anomaly in the behavior of the heat capacity and a curvature correction to the interfacial tension, are also discussed.

Patterned nonadhesive surfaces: superhydrophobicity and wetting regime transitions

Nonadhesive and water-repellent surfaces are required for many tribological applications. We study mechanisms of wetting of patterned superhydrophobic Si surfaces, including the transition between various wetting regimes during microdroplet evaporation in environmental scanning electron microscopy (ESEM) and for contact angle and contact angle hysteresis measurements. Wetting involves interactions at different scale levels: macroscale (water droplet size), microscale (surface texture size), and nanoscale (molecular size). We propose a generalized formulation of the Wenzel and Cassie equations that is consistent with the broad range of experimental data. We show that the contact angle hysteresis involves two different mechanisms and how the transition from the metastable partially wetted (Cassie) state to the homogeneously wetted (Wenzel) state depends upon droplet size and surface pattern parameters.

Conceptual aspects of line tensions

We analyze two representative systems containing a three-phase-contact line: a liquid lens at a fluid-fluid interface and a liquid drop in contact with a gas phase residing on a solid substrate. In addition we study a system containing a planar liquid-gas interface in contact with a solid substrate. We discuss to which extent the decomposition of the grand canonical free energy of such systems into volume, surface, and line contributions is unique in spite of the freedom one has in positioning the Gibbs dividing interfaces. Curvatures of interfaces are taken into account. In the case of a lens it is found that the line tension is independent of arbitrary choices of the Gibbs dividing interfaces. In the case of a drop, however, one arrives at two different possible definitions of the line tension. One of them corresponds seamlessly to that applicable to the lens. The line tension defined this way turns out to be independent of choices of the Gibbs dividing interfaces. In the case of the second definition, however, the line tension does depend on the choice of the Gibbs dividing interfaces. We also provide form invariant equations for the equilibrium contact angles which properly transform under notional shifts of dividing interfaces which change the description of the system but leave the density configurations unchanged. It is shown that in order to accomplish this form invariance, additional stiffness coefficients attributed to the contact line must be introduced. The choice of the dividing interfaces influences the actual values of the stiffness coefficients. We show how these coefficients transform as a function of the relative displacements of the dividing interfaces. Our formulation provides a clearly defined scheme to determine line properties from measured dependences of the contact angles on lens or drop volumes. This scheme implies relations different from the modified Neumann or Young equations, which currently are the basis for extracting line tensions from experimental data. These relations show that the experiments do not render the line tension alone but a combination of the line tension, the Tolman length, and the stiffness coefficients of the line. In contrast to previous approaches our scheme works consistently for any choice of the dividing interfaces. It further allows us to compare results obtained by different experimental or theoretical methods, based on different conventions of choosing the dividing interfaces.

Nature of vapor-liquid asymmetry in fluid criticality

We have investigated the nature and experimental consequences of vapor-liquid asymmetry in near-critical fluids within the framework of "complete scaling" [M. E. Fisher and G. Orkoulas, Phys. Rev. Lett. 85, 696 (2000); Y. C. Kim, Phys. Rev. E 67, 061506 (2003)]. We used the thermodynamic freedom for a choice of the critical-entropy value to simplify "complete scaling" to a form with only two independent parameters, responsible for two different sources of the asymmetry. We then developed a procedure to obtain these two parameters from mean-field equations of state. By combining accurate liquid-vapor coexistence and heat-capacity data, we have unambiguously separated two nonanalytic contributions from the two sources of vapor-liquid asymmetry and proved the validity of "complete scaling." Since the nonanalytic asymmetry effects in the critical region are fully determined by the Ising critical exponents for the symmetric lattice-gas model, there is no need for a special renormalization-group theoretical treatment of "non-Ising" asymmetry in fluid criticality.