Size-dependent Surface Energy Research Articles

Size- and shape-dependent phase diagram of Ga-Sb nanoparticles

It is well known that the phase diagrams calculated using bulk Gibbs energy alone is insufficient when the dimensions of alloy particles are less than 100 nm. This is due to the surface energy contribution to the Gibbs energy, which increases as surface-to-volume ratio of alloy particles increases. In this work, we have used the CALPHAD approach to calculate the phase diagram of the Ga-Sb system taking into account of the size and shape of alloy particles. To accomplish this, we have added size-dependent surface energy terms and optimized the Gibbs energy expressions of various phases in the Ga-Sb system using surface tension information. The surface tension of the liquid phase is calculated using the Butler equation. The surface energy contribution to the Gibbs energy of the terminal phases is approximated using the surface tension expressions of the elements. The surface energy of the line compound GaSb is computed using the slab model within the framework of Density Functional Theory (DFT). The calculations show that there is lowering of liquidus and eutectic temperatures when size decreases. Among the two shapes considered, the depression is more prominent in the nano-spheres than the nano-rods.

Modification of Butler equation for nanoparticles

Compared to the conventional use of the Butler equation applied for the calculation of surface energy of a binary alloy, we considered limited size nanoparticles with a large surface to volume ratio and applied additional mass balances to calculate the nanoparticles surface and core composition. Due to limited source of the component with a lower surface energy, its segregation on the surface is less pronounced compared to the unlimited bulk source. Thus, the core is depleted from the surface-preferred component and enriched by the higher surface energy one. The different behaviour of bulk and nanoparticles was demonstrated on the Al-Ti liquid alloy using the Redlich-Kister equation for excess Gibbs energy for the melt. As the depletion effect is size-dependent our approach thus predicts size-dependent surface energy of binary alloy nanoparticles.

Size effects on melting point based on surface energy and coordination number

Size effect on melting point is closely related with the surface energy of nanoparticles, since the increased surface/volume ratio. As the nanoparticle’s size decreases, there have some different surface shape existing, which makes the coordination number are different from each other. Both the surface energy and coordination number of surface atoms are the necessary consideration in understanding the thermal stability of nanoparticles. So, a new model free of any adjustable parameters, based on size-dependent surface energy and calculation of the surface atomic coordination number of nanoparticles, is developed to predict the size-dependent melting point of nanoparticles. The model can be utilized to predict the thermal stability of low-dimensional materials within a large size range. The theoretical predictions in terms of the model are consistent with experimental evidences.

Temperature and size dependent surface energy of metallic nano-materials

In this study, we report a theoretical model for the temperature and size dependent surface energy of metallic nanomaterials. The model is verified by making a comparison with the available simulation and experimental data. Reasonable agreement has been observed between these results. This study reveals that the decrease of surface energy at high temperatures is caused by cohesive energy weakening and bond expansion. With the same nanomaterial size, the sequence of size effects on the surface energy from weak to strong is thin films, nanowires, and nanoparticles. In particular, this work can provide a theoretical basis for the prediction of size dependent surface energy of metallic nanomaterials at different temperatures, which can help in the understanding of the mechanical and thermodynamic properties of metal surfaces.

Effect of electrophoresis time deposition of colloidal gold nanoparticles on inducing the crystallization of amorphous Si thin films

This work investigates the metal induced crystallization of hydrogenated amorphous Silicon (a-Si:H) thin films, processed by plasma enhanced chemical vapor deposition, using Au nanoparticles (AuNPs) as crystallizing mediator metal. Electrophoresis deposition (EPD) of colloidal AuNPs was used to coat the Si substrates before the a-Si:H processing. Then, the crystallization was conducted by isothermal annealing for temperatures above the eutectic (500 and 600 °C). Raman spectroscopy analysis showed that the crystalline volume fraction doubled as the EPD time increases. Scanning electronic microscopy and energy-dispersive X-ray microanalysis revealed that AuNPs induced crystallization at 600 °C was achieved through eutectic reaction with mixing layers. Structural characterizations showed that the low amounts of AuNPs exhibited a metal mediating effect more pronounced than that of continuous Au thin film deposited by dry physical methods. The enhancing effect of AuNPs was assigned to their size dependent high surface energy. Herein, we discuss as well, the effect of the EPD time on the development of Si crystallites on the basis of thermodynamic and kinetic considerations.

Size dependent stability and surface energy of amorphous FePt nanoalloy

The effects of particle size (2–6 nm) and temperature (300–2700 K) on the stability and local structural evolutions of amorphous equiatomic FePt bulk/nanoalloys have been investigated by combining Embedded Atom Model (EAM) with classical molecular dynamics (MD) simulation method. The three dimensional (3D) atomic configuration of amorphous FePt NPs by means of Voronoi analysis reveals that, deformed bcc (d-bcc) and icosahedron (d-ico) type structures are most probable local atomic configurations for 2–6 nm sized Fe50Pt50 NPs both in liquid (1700 K) and glassy (300 K) states. It was shown that nano-scale phase separation takes place around 300 K for 2 nm sized FePt NPs that leads to formation of spherical core-shell segregated structure having Pt-Fe-rich core and Fe-rich surface. Compared to core region, more ordered and high dense configuration of atoms at the surface gives rise to decrease in surface entropy of NPs and hence bringing about surface energy anomaly with positive temperature coefficient above the glass transition temperatures. Below glass transition temperatures of the FePt nanoalloy particles, the thermodynamically stable amorphous phase (TSA) would appear. Glass transition temperature of amorphous Fe50Pt50 NPs increases with increasing particle size and eventually approaches to the glass temperature of bulk system for NPs diameters ≥ ∼16 nm. Results of current predictions are in good qualitative and semi-quantitative agreement with other theoretical and experimental findings reported in the literature.

Size-dependent surface energy of Ar and Si nanovoids

A simple model for the size dependence of liquid–vapor surface energy of a liquid drop, free of any adjustable parameter, has been extended for the size dependence of the surface energy of nanovoids in this work. Considering nanovoids, there is a negative curvature on the void surface, which is completely different from nanoparticles or liquid drops, which have positive curvature surfaces. Thus curvature has an important effect on material properties. Moreover, by introducing negative curvature into the model for deduction of liquid–vapor surface energy, a successful prediction for the behaviour of argon (Ar) and silicon (Si) surface energy on the surface of nanovoids are obtained, which further confirms the role of curvature. Furthermore, it is shown that the surface energy of nanovoids increases as void size is reduced, suggesting a greater difference in energy between surface atoms and interior atoms, especially when void size is smaller than 4 nm. This is owing to the fact that the surface energy gradually approaches that of the bulk when the void size is larger than 4 nm. Thus, the effect of size on the surface energy of voids is significant when the voids are of small dimensions.

Size dependent surface energy of nanoplates: Molecular dynamics and nanoscale continuum theory correlations

A nanoscale gradient continuum theory along with molecular dynamics simulations are employed to investigate the size-dependent surface energy of nanoplates. Molecular dynamics simulations reveal that upon nanoplate thickness reduction, the redistribution of surface energy density along thickness direction causes the decrease of the surface energy of nanoplate free surfaces. Via introducing a calibration benchmark, the length scale model parameter of the gradient continuum theory is methodically determined. The calibrated continuum theory is shown to well predict the size-dependent surface energy and the associated redistribution of surface energy density within nanoplates.

Geometry-dependent band shift and dielectric modification of nanoporous Si nanowires

In order to obtain a detailed understanding of the modulation of electronic properties in nanoporous Si (np-Si) nanowires with containing ordered, nanometer-sized cylindrical pores, we propose a theoretical method to clarify the band shift and associated with the dielectric modification determined by the geometrical parameters, including nanowire diameter, pore size, pore spacing and porosity, in terms of size-dependent surface energy and atomic-bond-relaxation correlation mechanism. Our results reveal that the self-equilibrium strain induced by the atoms located at inner and outer surfaces with high ratio of under-coordinated atoms as well as elastic interaction among pores in np-Si nanowires play the dominant role in the bandgap shift and dielectric depression. The tunable electronic properties of np-Si nanowires with negative curvature make them attractive for nanoelectronic and optoelectronic devices.

Size, shape and temperature dependent surface energy of binary alloy nanoparticles

Surface energy has an important role in determining the properties of nanoparticles. Even though, extensive research has been done on the surface energy of pure nanoparticles, the surface energy of alloy nanoparticles has not been considered enough. In this work, based on the liquid drop model for surface energy of pure nanoparticles and Butler's equation, a model for size dependent surface energy of alloy nanoparticles has been developed. In addition to size and concentration, the model can describe the effects of shape and temperature on surface energy of alloy nanoparticles. Cu − Ag and Cu − Au systems have been studied as two examples and the results have been compared with other theoretical models and available simulated data. Reasonable agreements between the results were observed. The results show that the decreasing particle size decreases surface energy of alloy nanoparticles but decreasing temperature and shape factor increases the value of surface energy.

Pseudo-one-dimensional nucleation in dilute polymer solutions.

Pathogenic protein fibrils have been shown in vitro to have nucleation-dependent kinetics despite the fact that one-dimensional structures do not have the size-dependent surface energy responsible for the lag time in classical theory. We present a theory showing that the conformational entropy of the peptide chains creates a free-energy barrier that is analogous to the translational entropy barrier in higher dimensions. We find that the dynamics of polymer rearrangement make it very unlikely for nucleation to succeed along the lowest free-energy trajectory, meaning that most of the nucleation flux avoids the free-energy saddle point. We use these results to construct a three-dimensional model for amyloid nucleation that accounts for conformational entropy, backbone H bonds, and side-chain interactions to compute nucleation rates as a function of concentration.

Molecular dynamic investigation of size-dependent surface energy of icosahedral copper nanoparticles at different temperature

Molecular dynamic investigation of size-dependent surface energy of icosahedral copper nanoparticles at different temperature

A continuum state variable theory to model the size-dependent surface energy of nanostructures

We propose a continuum-based state variable theory to quantify the excess surface free energy density throughout a nanostructure. The size-dependent effect exhibited by nanoplates and spherical nanoparticles i.e. the reduction of surface energy with reducing nanostructure size is well-captured by our continuum state variable theory. Our constitutive theory is also able to predict the reducing energetic difference between the surface and interior (bulk) portions of a nanostructure with decreasing nanostructure size.

Size-dependent surface energy and Tolman length of TiO2 and SnO2 nanoparticles

In the present paper, we report the calculated results of size dependent surface energy and Tolman length of two metal oxides namely the titanium dioxide (TiO2) and tin dioxide (SnO2) nanoparticles. The methodology used here is based on simple thermodynamic model using parameters for corresponding bulk systems obtained from computer simulations using density functional theory (DFT) based data. We found that the surface energy depends on the size of particles and decreases with decreasing particle size. We also report the surface energy for different planes/facets of TiO2 and SnO2 nano-crystals and observed that the (110) plane/facet is the most stable facet as the value of the surface energy for this facet is minimum among all considered facets. Size dependence of the Tolman length shows an opposite trend than surface energy for both compounds. Tolman length is found to be decreasing with increasing size of SnO2 nanoparticles. Our calculation is found to be in agreement with the experimental results in case of TiO2 nanoparticles.

Size-dependent surface energy density of typically fcc metallic nanomaterials

The surface energy density of nano-sized structural elements exhibits an obvious size-dependent feature. To study this interesting phenomenon, atomistic calculations are carried out in the present paper for different face-centered-cubic (fcc) metallic nano-slabs. Lagrangian and Eulerian descriptions are adopted, respectively, in order to find the varying trends of surface energy densities in an initially un-deformed configuration and a current one. It is found that the Lagrangian surface energy density increases monotonically with an increase of the nano-slab’s thickness in the former no matter what the surface orientation is; while the variation of the Eulerian one is indefinite. The surface relaxation parameters are further simulated for differently oriented surfaces, which gives a very good explanation for the differences between the Lagrangian and Eulerian surface energy densities. The results in this paper should be a useful supplement to theoretical studies on the surface/interface effect of nanomaterials.

Size-dependent interface adhesive energy and interface strength of nanostructured systems

An analytical model about the size-dependent interface adhesive energy and the interface binding strength was developed based on the size-dependent surface energy and the interface cohesive zone model. The model predicts the enhanced interface binding strength between the nanostructured ceramic coatings and the bulk metallic substrates compared to that between the conventional ceramic coatings with micron-scale grains and the same substrates. The interface binding strength between the ceramic coatings and the metallic substrates was measured by the tensile adhesion test method. The result shows that the interface binding strength of the nanostructured coatings with grain diameter of average 70nm increases by 86%. The experimental result is in agreement with the theoretical prediction.

Size of a crystal nucleus in the isothermal crystallization of supercooled liquid

We present an alternative to classical nucleation theory (CNT). We introduce a size-dependent surface energy into the total Gibbs free-energy of formation of a crystal (ΔG). We consider the free-energy in the core part of the total volume of crystal and the free-energy in the surface-layer part of it, separately, for the evaluation of ΔG. As a result, we present an explicit model to evaluate a characteristic size of an initial nucleus that differs from the critical nucleus of CNT, but whose temperature dependence agrees well with that reported for the temperature dependency initial fold length of isotactic polystyrene and polyethylene in the literature. Our model has fitted the experimental data in the literature with only one adjustable parameter that is defined as nucleation constant. The nucleation constant is the Gibbs free-energy difference between the crystal and supercooled liquid phases for the volume of initial nucleus. We also present an expression to approximate the evolution of free-energy in the surface-layer part of crystal during the crystal growth.

Gibbs free energy, surface stress and melting point of nanoparticle

Two approaches to calculating Gibbs free energy of nanoparticle are compared. It is found that the contribution from the vibrational entropy of surface atoms of nanoparticle to its Gibbs free energy can be ignored, and Jiang et al.'s formula [J. Phys. Chem. B 105 (2001) 6275] [27] for calculating surface stress is only valid around room temperature. Furthermore, an approximate relationship between surface stress and surface free energy of nanoparticles is revealed. Finally, the reason why effect of size dependent surface energy on melting point of nanoparticle was neglected is clarified.

Steric-Hindrance-Driven Shape Transition in PbS Quantum Dots: Understanding Size-Dependent Stability

Ambient stability of colloidal nanocrystal quantum dots (QDs) is imperative for low-cost, high-efficiency QD photovoltaics. We synthesized air-stable, ultrasmall PbS QDs with diameter (D) down to 1.5 nm, and found an abrupt transition at D ≈ 4 nm in the air stability as the QD size was varied from 1.5 to 7.5 nm. X-ray photoemission spectroscopy measurements and density functional theory calculations reveal that the stability transition is closely associated with the shape transition of oleate-capped QDs from octahedron to cuboctahedron, driven by steric hindrance and thus size-dependent surface energy of oleate-passivated Pb-rich QD facets. This microscopic understanding of the surface chemistry on ultrasmall QDs, up to a few nanometers, should be very useful for precisely and accurately controlling physicochemical properties of colloidal QDs such as doping polarity, carrier mobility, air stability, and hot-carrier dynamics for solar cell applications.

Size-Dependent Structural Stability and Tuning Mechanism: A Case of Zinc Sulfide

Synchrotron X-ray diffraction and thermodynamic calculation combined with particle size dependent defect ratio, stress distribution, and surface energy have been used to tackle the structural stability and phase transformation of bulk and nano zincblende (ZnS). X-ray diffractions demonstrate that bulk zincblende-type ZnS transforms to a rocksalt phase at ∼15.4 GPa associated with an in situ fracture. The transition pressure remains constant upon decreasing particle size to 15 nm. Below 15 nm, this phase transformation occurs at an elevated pressure; the particle does not break down and instead transforms to a smaller rocksalt grain with 18% volumetric shrinkage relative to zincblende. The two defined critical sizes of 3.4 and 15 nm represent the largest size for a particle that starts a defect extinction and homogeneous stress distribution, respectively, thus enabling interpretation of a combined effect of pressure and particle size. When the particle is larger than 15 nm, the hosted defect (or dislocatio...