Tolman Length Research Articles

A diffusion and curvature dependent surface elastic model with application to stress analysis of anode in lithium ion battery

The present paper investigates the influences of surface tension and the stress distribution in the bulk induced by the surface tension, surface curvature and ions’ diffusion on the elastic properties of nanostructures. In the electro-chemical diffusion process, the surface tension, ions diffusion and stress distribution will strongly dependent with each other. Here, considering the curvature effect, a surface elastic model coupled with the ions diffusion is proposed. In the model, the coupling between the stress and ions diffusion is revealed through the chemical potential variation. To reveal the curvature effect on the surface energy, a Tolman length for solid is introduced. As a typical application of the model, we analyze the stress distributions in the silicon anode of the lithium ion battery. In recent years, silicon, due to large theoretical energy density, becomes one of the promising candidate anode materials. However, huge volume changes of silicon in charging and discharging process hinders its application. We depict the stresses provenance and evolution of hollow nanosphere and nanotube induced by lithium ions diffusion. Self-buckling induced by surface stresses of the two nanostructures is also taken into consideration. Critical buckling sizes are analyzed. All results show that hollow nanosphere will be a more suitable structure for electrode. Finally, a new dimensionless number κ related to Young’s modulus, Tolman length and surface energy is proposed to estimate the relative importance of the Tolman length with the intrinsic material length. It is believed that the diffusion and curvature dependent surface elastic model will be a powerful tool to investigate the diffusion behaviors in nanostructured electrode.

An exact formalism to study the thermodynamic properties of hard-sphere systems under spherical confinement

This paper presents a modified grand canonical ensemble which provides a new simple and efficient scheme to study few-body fluid-like inhomogeneous systems under confinement. The new formalism is implemented to investigate the exact thermodynamic properties of a hard sphere (HS) fluid-like system with up to three particles confined in a spherical cavity. In addition, the partition function of this system was used to analyze the surface thermodynamic properties of the many-HS system and to derive the exact curvature dependence of both the surface tension and adsorption in powers of the density. The expressions for the surface tension and the adsorption were also obtained for the many-HS system outside of a fixed hard spherical object. We used these results to derive the dependence of the fluid-substrate Tolman length up to first order in density.

Molecular dynamics determination of the surface tension of silver-gold liquid alloys and the Tolman length of nanoalloys

Using molecular dynamics simulations, an embedded-atom model potential, and the mechanistic route, we have computed the pressure tensor and the surface tension γ of Ag-Au liquid alloys. Although the model generally underestimates γ for pure metals, calculations for a bulk planar slab exhibit nonlinear variations of γ with increasing gold concentration, which agree with experiments and can be accounted for by a perfect solution model. Calculations for various nanoscale droplets containing between 100 and 3200 atoms show a systematic decrease of γ with increasing droplet radius R. The positive Tolman length of the alloy determined from these size variations is estimated to vary slightly with gold concentration. The effects of temperature in the range 1300-1700 K are discussed.

Liquid-drop model for the surface energy of nanoparticles

Based on the liquid-drop model, we have evaluated the Tolman length and surface energy of nanoparticles for different elements and compared with other theoretical models as well as the available simulated data. The predictions of the model show good agreement with the simulated results. Like the cohesive energy and melting temperature, the size-dependency of surface energy is also shape-dependent.

Excess equimolar radius of liquid drops

The curvature dependence of the surface tension is related to the excess equimolar radius of liquid drops, i.e., the deviation of the equimolar radius from the radius defined by the macroscopic capillarity approximation. Based on the Tolman [J. Chem. Phys. 17, 333 (1949)] approach and its interpretation by Nijmeijer et al. [J. Chem. Phys. 96, 565 (1991)], the surface tension of spherical interfaces is analyzed in terms of the pressure difference due to curvature. In the present study, the excess equimolar radius, which can be obtained directly from the density profile, is used instead of the Tolman length. Liquid drops of the truncated and shifted Lennard-Jones fluid are investigated by molecular dynamics simulation in the canonical ensemble, with equimolar radii ranging from 4 to 33 times the Lennard-Jones size parameter σ. In these simulations, the magnitude of the excess equimolar radius is shown to be smaller than σ/2. This suggests that the surface tension of liquid drops at the nanometer length scale is much closer to that of the planar vapor-liquid interface than reported in studies based on the mechanical route.

Numerical approaches to determine the interface tension of curved interfaces from free energy calculations

A recently proposed method to obtain the surface free energy σ(R) of spherical droplets and bubbles of fluids, using a thermodynamic analysis of two-phase coexistence in finite boxes at fixed total density, is reconsidered and extended. Building on a comprehensive review of the basic thermodynamic theory, it is shown that from this analysis one can extract both the equimolar radius R(e) as well as the radius R(s) of the surface of tension. Hence the free energy barrier that needs to be overcome in nucleation events where critical droplets and bubbles are formed can be reliably estimated for the range of radii that is of physical interest. It is found that the conventional theory of nucleation, where the interface tension of planar liquid-vapor interfaces is used to predict nucleation barriers, leads to a significant overestimation, and this failure is particularly large for bubbles. Furthermore, different routes to estimate the effective radius-dependent Tolman length δ(R(s)) from simulations in the canonical ensemble are discussed. Thus we obtain an instructive exemplification of the basic quantities and relations of the thermodynamic theory of metastable droplets/bubbles using simulations. However, the simulation results for δ(R(s)) employing a truncated Lennard-Jones system suffer to some extent from unexplained finite size effects, while no such finite size effects are found in corresponding density functional calculations. The numerical results are compatible with the expectation that δ(R(s) → ∞) is slightly negative and of the order of one tenth of a Lennard-Jones diameter, but much larger systems need to be simulated to allow more precise estimates of δ(R(s) → ∞).

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.

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.

Local pressure components and surface tension of spherical interfaces. Thermodynamic versus mechanical definitions. I. A mesoscale modeling of droplets

We report mesoscale simulations of spherical drops to investigate the surface tension and mechanical properties. The Monte Carlo simulations are performed with the multibody potential commonly used in the many-body dissipative particle dynamics simulations. We establish here the calculation of the local normal and transverse components of the pressure tensor via the perturbation volume within the thermodynamic route. The different profiles of these components are compared to those calculated using the mechanical approach. To complete the mesoscale modeling of drops, we investigate the curvature dependence of the surface tension in order to calculate the Tolman's length, which is found to be negative.

Three hard spheres in a spherical cavity

This work is devoted to furthering the understanding of few- and many-body inhomogeneous systems in the framework of the statistical mechanics of fluids. The three-body system consisting in three hard spheres (HS) confined in a spherical cavity at constant temperature is studied. Its canonical ensemble partition function and thermodynamic properties (such as the free energy, pressures, and fluid-substrate surface tension) are analytically obtained as a function of the cavity radius. This is the first time that a three-body fluid-like system is exactly solved. Symmetry relations between this system and its dual system composed of three HS surrounding a hard spherical object are analyzed. They allow to analytically obtain the canonical partition function of the dual system and its thermodynamic properties. Finally, the behavior of the many-body system of HS in contact with a hard spherical wall in the low density limit, is studied, focusing on the curvature dependence of the fluid-substrate surface tension and finding exact expressions for the Tolman's length and the second order term in curvature.

Atomistic Characterization of Stochastic Cavitation of a Binary Metallic Liquid under Negative Pressure.

We demonstrate the stochastic nature of cavitation in a binary metallic liquid Cu46Zr54 during hydrostatic expansion by employing molecular dynamics (MD) simulations using a quantum mechanics (QM)-derived potential. The activation volume is obtained from MD simulations and transition-state theory. Extrapolation of the pressure dependence of the activation volume from our MD simulations to low tensile pressure agrees remarkably with macroscale cavitation experiments. We find that classical nucleation theory can predict the cavitation rate if we incorporate the Tolman length derived from the MD simulations.

Two interacting particles in a spherical pore

In this work we analytically evaluate, for the first time, the exact canonical partition function for two interacting spherical particles into a spherical pore. The interaction with the spherical substrate and between particles is described by an attractive square-well and a square-shoulder potential. In addition, we obtain exact expressions for both the one particle and an averaged two particle density distribution. We develop a thermodynamic approach to few-body systems by introducing a method based on thermodynamic measures [I. Urrutia, J. Chem. Phys. 134, 104503 (2010)] for nonhard interaction potentials. This analysis enables us to obtain expressions for the pressure, the surface tension, and the equivalent magnitudes for the total and Gaussian curvatures. As a by-product, we solve systems composed of two particles outside a fixed spherical obstacle. We study the low density limit for a many-body system confined to a spherical cavity and a many-body system surrounding a spherical obstacle. From this analysis we derive the exact first order dependence of the surface tension and Tolman length. Our findings show that the Tolman length goes to zero in the case of a purely hard wall spherical substrate, but contains a zero order term in density for square-well and square-shoulder wall-fluid potentials. This suggests that any nonhard wall-fluid potential should produce a non-null zero order term in the Tolman length.

Theoretical Study of Phase Transition, Surface Tension, and Nucleation Rate Predictions for Argon

In this work, a weighted density functional theory has been used to study the equilibrium and metastable processes for argon. In the theoretical approach, the two- and three-body interactions of the fluid molecules are considered simultaneously, and the renormalization group transformation is applied to address the long-range fluctuations inside the critical region. The global phase equilibria, planar and curvature-dependent surface tensions, critical radius, and nucleation rates of argon are investigated systematically. The results are in good agreement with the experimental data. Meanwhile, this work applies a methodology for calculating the curved surface tension in local supersaturated environments, showing that the Tolman length is negligible far from the critical region. Near the critical point, however, the Tolman length becomes positive and appears to diverge.

The Effect of Microdroplet Size on the Surface Tension and Tolman Length

A monomolecular layer model of the surface phase of microdroplets was proposed, and the exact expression for Tolman length was derived in this paper. The Tolman lengths of water, n-pentane, and n-heptane were calculated by the expression, and the values are quite in agreement with the experimental values. By use of the Gibbs-Tolman-Kening-Buff equation, the exact relationship between the microdroplet surface tension and the radius is obtained, and the predicted values agree well with the simulated values. The results show that there is an obvious effect of the size of microdroplets (or nanoparticles) on the surface tension, and the surface tension decreases with decreasing droplet size. For the microdroplets of general liquid, only if their radius approaches or reaches 10(-9) m does the effect become significant.

Effect of Oligosaccharides and their Monosaccharide Mixtures on the Stability of Proteins: A Scaled Particle Study

Experimental results of RNase-A stabilization by sugar osmolytes show that the change in the Gibbs free energy (Δ G D) associated with the equilibrium, N (native) state ↔ D (denatured) state of the protein in the presence of equimolar mixture of monosaccharides is larger than that of the corresponding oligosaccharides at a given temperature and pH. However, at the molar scale, ΔG D obtained in the presence of an oligosaccharide is much higher as compared with ΔG D obtained using individual monosaccharide. We used scaled particle theory (SPT) to explain these experimental observations. The effective length, called Tolman's length that describes the curvature correlations to a surface area or surface tension and in turn contributes to the change in free energy, is discussed. Tolman's length is higher for corresponding monomer mixture than the oligosaccharide molecules. Based on SPT analysis, a geometrical model is proposed for clustering of monosaccharides in the mixture due to high particle density. The cluster is presumed to have weak interaction among them due to larger hydrodynamic radius than that of the bonded molecules of oligosaccharides.

Curvature dependence of surface free energy of liquid drops and bubbles: A simulation study

We study the excess free energy due to phase coexistence of fluids by Monte Carlo simulations using successive umbrella sampling in finite L×L×L boxes with periodic boundary conditions. Both the vapor-liquid phase coexistence of a simple Lennard-Jones fluid and the coexistence between A-rich and B-rich phases of a symmetric binary (AB) Lennard-Jones mixture are studied, varying the density ρ in the simple fluid or the relative concentration x(A) of A in the binary mixture, respectively. The character of phase coexistence changes from a spherical droplet (or bubble) of the minority phase (near the coexistence curve) to a cylindrical droplet (or bubble) and finally (in the center of the miscibility gap) to a slablike configuration of two parallel flat interfaces. Extending the analysis of Schrader et al., [Phys. Rev. E 79, 061104 (2009)], we extract the surface free energy γ(R) of both spherical and cylindrical droplets and bubbles in the vapor-liquid case and present evidence that for R→∞ the leading order (Tolman) correction for droplets has sign opposite to the case of bubbles, consistent with the Tolman length being independent on the sign of curvature. For the symmetric binary mixture, the expected nonexistence of the Tolman length is confirmed. In all cases and for a range of radii R relevant for nucleation theory, γ(R) deviates strongly from γ(∞) which can be accounted for by a term of order γ(∞)/γ(R)-1∝R(-2). Our results for the simple Lennard-Jones fluid are also compared to results from density functional theory, and we find qualitative agreement in the behavior of γ(R) as well as in the sign and magnitude of the Tolman length.

Significance of the Tolman length at a molecular level

It is shown that, by assuming valid a relationship between the work of cavity creation in a liquid and the bulk liquid–vapour surface tension, corrected for curvature effects, and by reproducing calculated ΔGc values to create in water and n-hexane a cavity suitable to host a methane molecule, the obtained estimates of the Tolman length prove to markedly depend on temperature, losing a precise physical meaning.

Response to “Comment on ‘Evaluation of surface tension and Tolman length as a function of droplet radius from experimental nucleation rate and supersaturation ratio: Metal vapor homogeneous nucleation’ ” [J. Chem. Phys. 133, 047101 (2010)

Response to “Comment on ‘Evaluation of surface tension and Tolman length as a function of droplet radius from experimental nucleation rate and supersaturation ratio: Metal vapor homogeneous nucleation’ ” [J. Chem. Phys. <b>133</b>, 047101 (2010)

Comment on “Evaluation of surface tension and Tolman length as a function of droplet radius from experimental nucleation rate and supersaturation ratio: Metal vapor homogeneous nucleation” [J. Chem. Phys. 124, 014506 (2006)

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