Semiempirical van der Waals/Cahn–Hilliard theory: Size dependence of the Tolman length

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.

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.

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.

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.

Pathways to self-organization: Crystallization via nucleation and growth

Crystallization, a prototypical self-organization process during which a disordered state spontaneously transforms into a crystal characterized by a regular arrangement of its building blocks, usually proceeds by nucleation and growth. In the initial stages of the transformation, a localized nucleus of the new phase forms in the old one due to a random fluctuation. Most of these nuclei disappear after a short time, but rarely a crystalline embryo may reach a critical size after which further growth becomes thermodynamically favorable and the entire system is converted into the new phase. In this article, we will discuss several theoretical concepts and computational methods to study crystallization. More specifically, we will address the rare event problem arising in the simulation of nucleation processes and explain how to calculate nucleation rates accurately. Particular attention is directed towards discussing statistical tools to analyze crystallization trajectories and identify the transition mechanism.

A perspective on the interfacial properties of nanoscopic liquid drops

The structural and interfacial properties of nanoscopic liquid drops are assessed by means of mechanical, thermodynamical, and statistical mechanical approaches that are discussed in detail, including original developments at both the macroscopic level and the microscopic level of density functional theory (DFT). With a novel analysis we show that a purely macroscopic (static) mechanical treatment can lead to a qualitatively reasonable description of the surface tension and the Tolman length of a liquid drop; the latter parameter, which characterizes the curvature dependence of the tension, is found to be negative and has a magnitude of about a half of the molecular dimension. A mechanical slant cannot, however, be considered satisfactory for small finite-size systems where fluctuation effects are significant. From the opposite perspective, a curvature expansion of the macroscopic thermodynamic properties (density and chemical potential) is then used to demonstrate that a purely thermodynamic approach of this type cannot in itself correctly account for the curvature correction of the surface tension of liquid drops. We emphasize that any approach, e.g., classical nucleation theory, which is based on a purely macroscopic viewpoint, does not lead to a reliable representation when the radius of the drop becomes microscopic. The description of the enhanced inhomogeneity exhibited by small drops (particularly in the dense interior) necessitates a treatment at the molecular level to account for finite-size and surface effects correctly. The so-called mechanical route, which corresponds to a molecular-level extension of the macroscopic theory of elasticity and is particularly popular in molecular dynamics simulation, also appears to be unreliable due to the inherent ambiguity in the definition of the microscopic pressure tensor, an observation which has been known for decades but is frequently ignored. The union of the theory of capillarity (developed in the nineteenth century by Gibbs and then promoted by Tolman) with a microscopic DFT treatment allows for a direct and unambiguous description of the interfacial properties of drops of arbitrary size; DFT provides all of the bulk and surface characteristics of the system that are required to uniquely define its thermodynamic properties. In this vein, we propose a non-local mean-field DFT for Lennard-Jones (LJ) fluids to examine drops of varying size. A comparison of the predictions of our DFT with recent simulation data based on a second-order fluctuation analysis (Sampayo et al 2010 J. Chem. Phys. 132 141101) reveals the consistency of the two treatments. This observation highlights the significance of fluctuation effects in small drops, which give rise to additional entropic (thermal non-mechanical) contributions, in contrast to what one observes in the case of planar interfaces which are governed by the laws of mechanical equilibrium. A small negative Tolman length (which is found to be about a tenth of the molecular diameter) and a non-monotonic behaviour of the surface tension with the drop radius are predicted for the LJ fluid. Finally, the limits of the validity of the Tolman approach, the effect of the range of the intermolecular potential, and the behaviour of bubbles are briefly discussed.

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.

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.

Phase field theory of interfaces and crystal nucleation in a eutectic system of fcc structure: I. Transitions in the one-phase liquid region

The phase field theory (PFT) has been applied to predict equilibrium interfacial properties and nucleation barrier in the binary eutectic system Ag-Cu using double well and interpolation functions deduced from a Ginzburg-Landau expansion that considers fcc (face centered cubic) crystal symmetries. The temperature and composition dependent free energies of the liquid and solid phases are taken from CALculation of PHAse Diagrams-type calculations. The model parameters of PFT are fixed so as to recover an interface thickness of approximately 1 nm from molecular dynamics simulations and the interfacial free energies from the experimental dihedral angles available for the pure components. A nontrivial temperature and composition dependence for the equilibrium interfacial free energy is observed. Mapping the possible nucleation pathways, we find that the Ag and Cu rich critical fluctuations compete against each other in the neighborhood of the eutectic composition. The Tolman length is positive and shows a maximum as a function of undercooling. The PFT predictions for the critical undercooling are found to be consistent with experimental results. These results support the view that heterogeneous nucleation took place in the undercooling experiments available at present. We also present calculations using the classical droplet model [classical nucleation theory (CNT)] and a phenomenological diffuse interface theory (DIT). While the predictions of the CNT with a purely entropic interfacial free energy underestimate the critical undercooling, the DIT results appear to be in a reasonable agreement with the PFT predictions.

The surface tension of a solid at the solid–vacuum interface, an evaluation from adsorption and wall potential calculations

A method for the evaluation of quantities that are experimentally inaccessible such as the surface tension at the solid-vacuum interface and the superficial tension of the fluid in contact with the solid is presented. The approach is based on consideration of an equilibrium of a fluid in solid capillary wherein a balance between surface and capillary forces has been replaced by conceptual alternative interfacial and centrifugal forces. This approach involves the simultaneous numerical solution one the special forms of the Gibbs equation for solid-fluid interface and a generalized Kelvin equation derived earlier. The latter equation takes into account interactions between the solid thick cylindrical wall and confined fluid, this body-body interaction potential has been primarily calculated using the Lennard-Jones (6-12) expression for the atom-atom pair potentials and expressed by hypergeometrical functions having good convergences. All numerical calculations shown here have been performed for the model graphite-argon system at 90 K. Finally, an analysis of the accuracy of the proposed method is considered.

Thermodynamic expressions for the Tolman length

The Tolman length delta [TolmanJ. Chem. Phys. 17, 333 (1949)] measures the extent by which the surface tension of a small liquid drop deviates from its planar value. Despite increasing theoretical attention, debate continues on even the sign of Tolman's length for simple liquids. Recent thermodynamic treatments have proposed a relation between the Tolman length and the isothermal compressibility of the liquid at two-phase coexistence, delta approximately -kappa([script-l])sigma. Here, we review the derivation of this relation and show how it is related to earlier thermodynamic expressions. Its applicability is discussed in the context of the squared-gradient model for the liquid-vapor interface. It is found that the relation is semiquantitatively correct for this model unless one is too close to the critical point.

Phenomenological Predictions of Cohesive Energy and Structural Transition of Nanoparticles

In this paper, it is shown that a liquid-drop model (LDM) can predict the size-dependent cohesive energy (SDCE) of large nanoparticles and clusters (particles with few atoms) quantitatively. The cohesive energy decreases linearly with the inverse of the particle size both for large nanoparticles and clusters though the slopes are different. This indicates that there are three different regions (I-III) of SDCE in the complete size range. Regions I and II represent the SDCE of large nanoparticles and clusters, respectively, while region II represents the intermediate region where the cohesive energy is almost size-independent. Different regions of SDCE correspond to different structures of nanoparticles, and structural transition associated with the particle size can easily be predicted from the SDCE. Analyzing the cohesive energy data on the basis of LDM, it is shown that the surface tension decreases with decreasing size for very small nanoparticles. The Tolman equation can account for the variation of surface tension by predicting the size dependency of the Tolman length.

The Tolman Length: Is It Positive or Negative?

By means of a large-scale molecular dynamics simulation, we show that the Tolman length, although positive, is much smaller in magnitude than previously reported. We found that the range of interparticle interaction can significantly affect the magnitude of the Tolman length. When the range of interaction is longer than five molecular diameters, the Tolman length is on the order of a few hundredths of the molecular diameter, rather than a few tenths known previously.

Multicomponent nucleation: Thermodynamically consistent description of the nucleation work

A thermodynamically consistent formula is derived for the nucleation work in multicomponent homogeneous nucleation. The derivation relies on the conservative dividing surface which defines the nucleus as having specific surface energy equal to the specific surface energy sigma0 of the interface between the macroscopically large new and old phases at coexistence. Expressions are given for the radius of the nucleus defined by the conservative dividing surface and by the surface of tension. As a side result, the curvature dependence of the surface tension sigmaT of the nucleus defined by the surface of tension is also determined. The analysis is valid for nuclei of any size, i.e., for nucleation in the whole range of conditions between the binodal and the spinodal of the metastable old phase provided the inequality sigmaT < or = sigma0 is satisfied. It is found that under the conditions of validity of the analysis the nucleation rate is higher than the nucleation rate given by the classical nucleation theory. The general results are applied to nucleation of unary liquids or solids in binary gaseous, liquid or solid mixtures.