Size-dependent Melting Temperature Research Articles

Theoretical investigation of size and shape dependent melting temperature of transition metal clusters

Recently, research on material properties has got lots of attention because of their promising technological applications. In this article, we report a theoretical investigation concerning the size and shape effect on the melting temperature of transition metal clusters. Within the thermo dynamical approach, we predict a structural melting temperature for spherical, regular tetrahedral, regular hexahedral and regular octahedral for the sizes around 0.2 nm, 0.5 nm, 1 nm, 1.5 nm and 2 nm respectively. The theoretical investigation of size and shape dependent melting temperature on transition metal clusters (Cr, W &Mo) using thermodynamic model, where the shape and size of clusters are considered by introducing a new parameter, i.e. the shape factor. Based on melting temperature estimations as a function of size and shape, we predict that certain transition metal clusters, such as small size, may be viable for nanoelectronics or nanophotonic devices or not, since the melting temperature is too close to, or in some cases even below, room temperature. The results are well consistent with theoretical and experimental values used in liquid drop model. According to the study, when the size of the cluster decrease the melting temperature of the cluster decrease. On the other hand, when the parameter (shape factor) increase the melting temperature of the cluster decrease or vice versa. In conclusion, both size and shape have big effect on the melting temperature determination of the clusters.

The influence of boundary scattering phase shift on melting behavior of nanoparticles

Based on the Lindemann melting criterion and the lattice dynamical approach, we analytically estimated the melting temperature of nanoparticles with emphasis on the influence of the boundary scattering phase shift parameter α . With the decrease of size, the estimated relation shows that the melting temperature of free standing like nanoparticles will decrease, and the melting temperature of embedded nanoparticles will increase. The increase or decrease of the melting temperature depends on the boundary scattering phase shift parameter. The gradual change of parameter α from − 1 to 1 corresponds to the transition of the surface or environment of nanoparticles from the free boundary condition to the fixed boundary condition. Our analytical estimation qualitatively explains the experimental data, and its predictive capability will help the design of nanoparticles. • Based on the Lindemann criterion, An analytical estimation for size-dependent melting temperature of nanoparticles by lattice dynamical method will be given. • Our theoretical result can fit experimental data of nanoparticles in different environments qualitatively. • A simple boundary scattering phase shift is used to characterize the environment of nanoparticles.

On the influence of the amorphous phase on the stability of crystals in poly(cis‐1,4‐isoprene) networks

AbstractCrosslinked natural rubber and synthetic rubber samples are additivated with up to 9 wt% stearic acid (StA) to better understand the influence of StA on the melting temperature Tm of strain‐induced crystallized poly(cis‐1,4‐isoprene) crystals. To this end, lamellae thicknesses are determined from wide‐angle x‐ray patterns and used to calculate the crystal size dependent melting temperature Tm,calc. Comparing the measured Tm with Tm,calc reveals that Tm deviates downward from Tm,calc and converges Tm,calc with increasing StA concentration until it is identical to Tm,calc, in case of room temperature strain‐induced crystallization. In case of strain‐induced crystallization at 80°C, it was found that Tm is identical with Tm,calc without added StA and deviates upward from Tm,calc with increasing amount of added StA. We suggest that this is due to internal stress onto the polymer crystals exerted by highly strained macromolecules in the surrounding amorphous phase. Whether this stress has a stabilizing or destabilizing effect on the crystals is assumed to depend on its intensity and direction, which can be efficiently altered by the amount and the location of StA crystals in the amorphous phase.

A unified bond energy model for size-dependent melting temperature of freestanding and embedded nanomaterials

The bond energy (BE) model is a simple and practical model that has been developed to predict the melting temperature of nanomaterials. The model contains a material parameter, whose value is assumed to be 0.5 or 0.75. However, the exact value of the material parameter is not known. Available experimental data reveal that 0.5 or 0.75 are not suitable for all materials. In this research, a unified BE model has been developed to calculate the melting temperature of freestanding and embedded nanoparticles. In addition, using the available experimental data, a semi-empirical relationship has been proposed to estimate the value of the material parameter as a function of atomic distances. The results show that for freestanding nanoparticles the value of the material parameter can vary between 0.1 and 1.6. It is also found that the material parameter for embedded nanoparticles correlated with the lattice parameter mismatch between the nanoparticle and the matrix. The results show that if lattice parameter mismatch is less than 11.5%, the material parameter of freestanding and embedded nanoparticles are equal, but increasing lattice parameter mismatch, increases the material parameter of embedded nanoparticles. In addition to the melting temperature of nanoparticles, the proposed model was successfully used to calculate the melting temperature of nanowires.

In situ investigation on melting characteristics of 1D SnCu alloy nanosolder

Nanosoldering can bond various nanomaterials together or connect them with electrodes to form electrical contacts, thus assembling these nanomaterials into functional nanodevices; it is believed to be a promising interconnection technique due to its flexibility, controllability and crucial advantage of avoiding detrimental effects on the nano-objects. In this technique, molten solder as a filler material is introduced between the objects to be joined to form a reliable bond, in which the nanosolder reflow melting is a crucial prerequisite for successful nanosoldering. This work focuses on studying the melting characteristics of one-dimensional 97Sn3Cu nanosolder with low-cost, prominent electrical property and high mechanical reliability, aiming to promote its applications in nanosoldering. The reflow melting of an individual nanosolder has been dynamically observed by in situ heating holder in transmission electron microscopy, where the obtained reflow temperature (530 °C) is much higher than its melting temperature (220.4 °C) because of the external oxide layer confinement. Furthermore, the size-dependent melting temperature of nanosolders with various diameters (20–300 nm) has been investigated by both differential scanning calorimetry and theoretical calculation, revealing that the melting temperature decreases as the diameter goes down, especially for the nanosolders in the sub 80 nm range, where the value decreases significantly. The experimental results are in good agreement with the theoretical predictions. These results pointed out here can be readily extended to other nanosolders.

Modeling of Size Dependent Catalytic Activation Energies of Ag@Pd Spherical Core-Shell Nanoparticles

The core/shell materials have attracted enormous attention as potential applications in diverse areas in biosensors, catalysts, and energy storage systems. Recently, Ag-Pd core-shell nanoparticles (NPs) have been synthesized and shown their superior electro- catalytic activity of small sizes for fuel-cell applications. On this line, the size-dependent catalytic activation energies based on the thermodynamics continuum model have first applied for different sizes of AgcorePdshell spherical NPs as diameters of 4-20nm. We have created AgcorePdshell NPs by following two routes as (i) the constant Ag core size with increasing thicknesses of Pd shells, (ii) the constant Pd shell thicknesses with different Ag core sizes. By expressing the surface to volume ratio for each size of model NPs, the size-dependent melting temperature of spherical Ag@Pd NPs have obtained. It has found that the melting temperatures of Ag@Pd NPs are below than those of the melting points of bulk materials for pure elements. Thus the catalytic activation energies at their melting points have also computed. Both the melting points and catalytic activation energies of those NPs reduce with the mole fraction of Pd atoms decrease. Our results are agreeing with experiment and molecular dynamics simulations.

Effect of aqueous medium on low-frequency dynamics, chemical activity and physical properties of a spherical virus

In this work, we have studied the effect of size and aqueous medium on the low-frequency dynamics, physical properties like melting temperature and glass transition temperature and chemical properties like catalytic activation energy of spherical virus using Lindemann’s criteria and Arrhenius relation under their dynamic limit. The melting temperature and catalytic activation energy decrease with decreasing size of spherical virus. The glass transition temperature which increases with decreasing size of the virus is analyzed through the size dependent melting temperature. The melting temperature and catalytic activation energy of spherical virus of particular size increases when it is embedded in glycerol or water due to mismatch of the physical properties at the interface of virus and surrounding medium. In addition, the glass transition temperature of free and glycerol/water embedded virus using low-frequency vibrational modes has been calculated under the framework of elastic continuum approximation model. The glass transition temperature of spherical virus decreases with size when embedded in glycerol or water. A correlation between and is also drawn for spherical viruses. The study can be useful for spherical virus borne therapy i.e. in detecting and killing of the spherical viruses using a principle based on acoustic phonons (sound waves) resonance.Communicated by Ramaswamy H. Sarma

Size and shape dependent melting temperature of metallic nanomaterials

This study aims to characterize the size and shape dependent melting temperature of nanomaterials. Considering that surface atoms and interior atoms affect the melting of materials in different manners, we thus define an equivalent relationship between the contribution of surface atoms and interior atoms. Based on this definition, a criterion of melting is proposed through introducing a critical energy storage density of melting, the sum of the contribution of surface atoms and the interior atoms. According to the proposed criterion, a new theoretical model without any adjustable parameters is developed to characterize the size effect of melting temperatures of nanomaterials. The model predictions are in good agreement with the available experimental data or molecular dynamics simulations. This model uncovers the quantitative relationship between the melting temperature, size, atomic diameter and shape of nanomaterials. In addition, this model is extended to predict the size dependent glass transition temperatures of polymers. This study can help to better understand and characterize the size dependent melting temperatures of nanomaterials, as well as the size dependent glass transition temperatures of polymers.

Size Effects on Cohesive Energy, Debye Temperature and Lattice Heat Capacity from First-Principles Calculations of Sn Nanoparticles

The size-dependent cohesive energy, melting temperature, Debye temperature and lattice heat capacity are investigated using density functional theory within generalized gradient approximation of Sn nanoparticles. The analyses of the obtained total energies are presented by considering effect of mean bond length and the ratio number of surface atoms to that of its internal with size. The cohesive energy is calculated for Sn nanoparticles and the obtained data are used to determine melting temperature, Debye temperature and lattice heat capacity. The cohesive energy, melting point and Debye temperature drop while the lattice specific heat rise when the size is decreased due to the effects of the elevated bond length stretch. The results obtained are in excellent agreement with the available experimental results for melting point and Debye temperature of Sn nanoparticles. Also, the same trend variations of the lattice heat capacity obtained for Sn nanoparticles to that calculated theoretically in Se and Cu.

Modeling the effects of size and surface broken bond on surface energy of nanocrystals

Based on an analytical formulation for the size-dependent melting temperature, a simple thermodynamic model has been developed to describe the surface energy of nanocrystals γ(D). The results show that, for face-centered cubic, body-centered cubic and hexagonal close packing metals, γ(D) of nanocrystals/nanocavities decreases/increases as D drops. The effect of surface broken-bond number on γ(D) has also been discussed. It is found that reducing the number of surface broken-bonds will slow down the decrease of γ(D). For facets of nanocrystals with different broken-bond number, the difference in size-dependent surface bond contraction determines the variation intensity of surface energy. The model has also been used to describe γ(D) of semiconductor Si nanocavities and bimetallic alloys nanocrystals. The predictions are in good agreement with the reported data and other theory model predictions.

Research into the rationality and the application scopes of different melting models of nanoparticles

A rational melting model is indispensable to address the fundamental issue regarding the melting of nanoparticles. To ascertain the rationality and the application scopes of the three classical thermodynamic models, namely Pawlow, Rie, and Reiss melting models, corresponding accurate equations for size-dependent melting temperature of nanoparticles were derived. Comparison of the melting temperatures of Au, Al, and Sn nanoparticles calculated by the accurate equations with available experimental results demonstrates that both Reiss and Rie melting models are rational and capable of accurately describing the melting behaviors of nanoparticles at different melting stages. The former (surface pre-melting) is applicable to the stage from initial melting to critical thickness of liquid shell, while the latter (solid particles surrounded by a great deal of liquid) from the critical thickness to complete melting. The melting temperatures calculated by the accurate equation based on Reiss melting model are in good agreement with experimental results within the whole size range of calculation compared with those by other theoretical models. In addition, the critical thickness of liquid shell is found to decrease with particle size decreasing and presents a linear variation with particle size. The accurate thermodynamic equations based on Reiss and Rie melting models enable us to quantitatively and conveniently predict and explain the melting behaviors of nanoparticles at all size range in the whole melting process.

Size-dependent cohesive energy, melting temperature, and Debye temperature of spherical metallic nanoparticles

It is necessary to theoretically evaluate the thermodynamic properties of metallic nanoparticles due to the lack of experimental data. Considering the surface effects and crystal structures, a simple theoretical model is developed to study the size dependence of thermodynamic properties of spherical metallic nanoparticles. Based on the model, we have considered Co and Cu nanoparticles for the study of size dependence of cohesive energy, Au and Cu nanoparticles for size dependence of melting temperature, and Cu, Co and Au nanoparticles for size dependence of Debye temperature, respectively. The results show that the size effects on melting temperature, cohesive energy and Debye temperature of the spherical metallic nanoparticles are predominant in the sizes ranging from about 3 nm to 20 nm. The present theoretical predictions are in agreement with available corresponding experimental and computer simulation results for the spherical metallic nanoparticles. The model could be used to determine the thermodynamic properties of other metallic nanoparticles to some extent.

Effect of size and shape on melting and superheating of free standing and embedded nanoparticles

A simple model is proposed to study the size and shape dependence of melting and superheating of nanomaterials. The size dependence of melting temperature of free standing spherical nanosolids viz. Ag, Au, Al and Zn nanowire as well as embedded nanoparticles Pb and Ag is reported. The formulation is used to study the effect of shape on melting temperature during reduction of size. The different shapes of nanomaterials viz. film, icosahedral, wire, spherical, hexahedral, octahedral and tetrahedral are considered. The size and shape dependence of surface atoms, total number of atoms and their ratio is computed. The results obtained are compared with the available experimental data and discussed in the light of recent investigations. A good agreement between model predictions and experimental data supports the validity of the formulation developed. It is concluded that in addition to the size, the shape of nanomaterials also plays an important role during the behaviour of nanomaterials.

Finite size and length effects on bandgap of CdSe nanorods.

As an important property for designing optoelectronic devices, the bandgap Eg of nanorods is tunable with variety of sizes, D, and lengths, L. Unfortunately, the quantitative description of Eg(D, L) is still unavailable. To understand the underlying mechanism of the size and length effects on the bandgap of nanorods, based on size-dependent melting temperature model, a quantitative thermodynamic model without an adjustable parameter was established by considering the variation of the surface-to-volume ratio. Both the size and length have prominent effects on Eg(D, L), and Eg(D, L) increases with decreasing D and L, where the size confinement is stronger than the length confinement for the blue shift of Eg(D, L) due to the different effects of size or length on the A/V variation. The model prediction agrees well with the experimental results of CdSe nanorods. In addition, as the model expansion, for nanocrystals with different shapes, the size effect on the bandgap can be sequenced as follows: Eg(D, L) > Eg(D)nanoparticles > Eg(D)nanowires at L < D, while Eg(D)nanoparticles > Eg(D, L) > Eg(D)nanowires at L > D. The validity of the established model will improve the understanding of the bandgap variations and provide guidelines for designing optoelectronic devices.

Structural and Thermal Properties of Elementary and Binary Tetrahedral Semiconductor Nanoparticles

We report an equation free from fitting parameters as a direct calculation of size-dependent mean bond length for group IV and compounds from the III–V and II–VI binary groups. Size-dependent melting temperature and thermal expansion are also investigated for some materials forming the groups mentioned above. The empirical relation, which is obtained from fitting experimental data of melting enthalpy, is used to recalculate their values as well as entropy. The nanosize dependence of lattice thermal expansion for elements forming group IV is analyzed according to the hard sphere model, while mean ionicity is used for groups III–V and II–VI.

The influence of surface boundary conditions on the phonon contribution to the melting temperature of nanoparticles

The phonon thermal contribution to the melting temperature of nano-particles is inspected. Unlike in periodic boundary condition, under a general boundary condition the integration volume of low energy phonon for a nano-particle is more complex. We estimate the size-dependent melting temperature through the phase shift of the low energy phonon mode acquired by its scattering on boundary surface. A nano-particle can have either a rising or a decreasing melting temperature due to the boundary condition effect, and we found that an upper melting temperature bound exists for a nano-particle in various environments. Moreover, the melting temperature under a fixed boundary condition sets this upper bound.

Quantum concentration for the liquid phase of elements

The novel concept of quantum concentration for the liquid phase of elements was introduced which indicates the number of possible states of atoms per unit volume. A simplified thermodynamic energy model for a solid–liquid phase transition is used to define the quantum concentration for the liquids. Values of quantum concentration for liquid phase of most of elements were calculated and analyzed. It was revealed that the metalloids such as boron, silicon, germanium, antimony, tellurium and bismuth exhibited a highest value of quantum concentrations compared with other elements. Several physical and chemical properties of substances (such as melting temperature and free energy) can be calculated using these data. The size dependence of melting temperatures of Au, Pb and Sn spherically symmetric nanoparticles were calculated and compared well with experimental data and theoretical model based on generalized Gibbs–Thomson predictions.

Logarithmic Size-Dependent Melting Temperature of Nanoparticles

This work describes the logarithmic term in the size-dependent melting temperature of nanoparticles. The Lindemann melting criterion and the thermal phonon contribution are intended primarily as a clear approach for our study of melting temperature. The ratio of the melting temperature (Tmn) of a nanoparticle to the bulk melting temperature (Tmb) is numerically evaluated up to the order of the inverse of the size L of a nanoparticle: Tmn/Tmb = 1 – A/L. We show that the coefficient A is not a constant but has a logarithmic part. The logarithmic term is the dominate term in the factor A(L). Our investigation suggests the behavior of the size-dependent melting is Tmn/Tmb = 1 – (B + C ln L)/(L – 2ls) where B and C are constant factors and ls is the number dimension of the surface layers reconstructed or premelted. The results present a challenge to reach a higher resolution in future thermal experiments in order to distinguish ln L/L behavior from the previously accepted pure 1/L behavior.

Thermodynamic approach to the size dependence of the melting temperatures of films

The size dependence of melting temperature T m of metallic films (tin and copper) on different substrates, including amorphous carbon and another refractory metal (i.e., the dependence of T m on film thickness h) is investigated. It is found that the effect of the interfacial boundary can result in the growth of T m for thin metallic films on carbon substrates with a reduction in film thickness h. For a system with a metallic film on a metallic substrate, the size dependence of T m is less pronounced and T m falls with a reduction in h.

Size-dependent melting temperature of nanoparticles based on cohesive energy

Size-dependent melting temperature of metallic nanoparticles is studied theoretically based on cohesive energy. Three factors are introduced in the present model. The k factor, i.e. efficiency of space filling of crystal lattice is defined as the ratio between the volume of the atoms in a crystal cell and that of the crystal cell. The β factor is defined as the ratio between the cohesive energy of surface atom and interior atom of a crystal. The qs factor represents the packing fraction on a surface crystalline plane. Considering the β, qs and k factors, the relationship between melting temperature and nanoparticle size is discussed. The obtained model is compared with the reported experimental data and the other models.