Modeling size dependence of melting temperature of metallic nanoparticles

Shandiz, M. Attarian; Safaei, Ali; Sanjabi, S.; Barber, Z. H.

doi:10.1016/j.jpcs.2007.02.049

Cited by 48 publications

(21 citation statements)

References 19 publications

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“…We reported previously that, for nanoparticles smaller than 10 nm ( D < 10 nm), the dependence of melting temperature on size shows good consistency with experimental data using a value of q = 1/4 in our model . Therefore, we assume that D av is the average of diameter of nanoparticles smaller than 10 nm (0 < D ≤ 10 nm), and for the nanoparticle with this size ( D av = (10 + ε)/2 = 5 nm, ε → 0), the value of q is 1/4, or q ( D av ) = 1/4.…”

Section: Introductionsupporting

confidence: 79%

“…We know that by increasing the size of a nanosolid the melting temperature increases, and this means that T mn / T mb is an increasing function of size. This means mathematically that d( T mn / T mb )/d D > 0 or, by replacing X = 1/ D (d X = −d D / D 2 ) we have Because the value of q is different for large and very small nanosolids, , we assume that q is a function of size. For calculation of q , we assume that it is a differentiable function, and then by using eq 3 we can write From inequality 4 and so If we replace the variable X by 1/ D , then we can write − D (d q /d D ) − (1 − q )( bD /( A + bD )) < 0 and then To obtain a function for q , we should solve inequality 7.…”

Section: Introductionmentioning

confidence: 99%

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Modeling the Melting Temperature of Nanoparticles by an Analytical Approach

et al. 2007

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The dependency of the surface-to-volume coordination number upon particle size has been obtained by an analytical function based upon our previous work, and a model for melting temperature as a function of size is presented. This expression for the melting temperature is a nonlinear relation in terms of the reciprocal of size. Also, the geometrical characteristics of nanoclusters have been applied to our general equation for the melting temperature of nanoparticles. It was found that the results of considering the geometrical characteristics of nanoclusters are in good agreement with the melting temperature achieved by the derived analytical function. Finally, this model has been compared with other theoretical models as well as the available experimental data. The predictions are consistent with the experimental data.

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Section: Introductionsupporting

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Modeling the Melting Temperature of Nanoparticles by an Analytical Approach

et al. 2007

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“…3 , the surface energy is proportional to the relative coordination number. Theoretically more elaborated studies lead to descriptions of the type [ 17 , 19 – 20 ] as…”

Section: Reviewmentioning

confidence: 99%

Surface energy of nanoparticles – influence of particle size and structure

Vollath¹,

Fischer²,

Holec³

2018

Beilstein J. Nanotechnol.

172

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The surface energy, particularly for nanoparticles, is one of the most important quantities in understanding the thermodynamics of particles. Therefore, it is astonishing that there is still great uncertainty about its value. The uncertainty increases if one questions its dependence on particle size. Different approaches, such as classical thermodynamics calculations, molecular dynamics simulations, and ab initio calculations, exist to predict this quantity. Generally, considerations based on classical thermodynamics lead to the prediction of decreasing values of the surface energy with decreasing particle size. This phenomenon is caused by the reduced number of next neighbors of surface atoms with decreasing particle size, a phenomenon that is partly compensated by the reduction of the binding energy between the atoms with decreasing particle size. Furthermore, this compensating effect may be expected by the formation of a disordered or quasi-liquid layer at the surface. The atomistic approach, based either on molecular dynamics simulations or ab initio calculations, generally leads to values with an opposite tendency. However, it is shown that this result is based on an insufficient definition of the particle size. A more realistic definition of the particle size is possible only by a detailed analysis of the electronic structure obtained from initio calculations. Except for minor variations caused by changes in the structure, only a minor dependence of the surface energy on the particle size is found. The main conclusion of this work is that surface energy values for the equivalent bulk materials should be used if detailed data for nanoparticles are not available.

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“…In our previous work, we calculated the size dependent thermodynamic parameters (entropy, S m , enthalpy, H m , free energy, g , and solid−liquid interface energy, γ sl ) of nanoparticles and nanofilms based on our model of the size dependent melting temperature of nanosolids , …”

Section: Modelmentioning

confidence: 99%

Modeling of the Heterogeneous Formation of Ni Catalyst Particles for Carbon Nanotube Growth

Sanjabi¹,

Faramarzi²,

Momen³

et al. 2009

J. Phys. Chem. C

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Size dependent thermodynamic parameters (Gibbs free energy, entropy, enthalpy, and surface energy) have been calculated for the heterogeneous transition of Ni nanofilms to nanoparticles. Such particles are often used as catalysts for carbon nanotube formation, and particle size controls the carbon nanotube diameter. The resultant equations are based upon the size dependent melting temperature of nanosolids derived in our previous works, and the results are compared with our recent calculations on the homogeneous formation of Ni catalyst particles. Using this thermodynamic approach, a relation between the minimum diameter of Ni particles and the thickness of the original film is found, and we predict the optimum Ni film thickness to form particles with diameter 20-30 nm. We illustrate that the calculated critical and stable sizes of Ni heterogeneously formed nanoparticles are a function of the wetting angle. For further comparison, a 5 nm Ni film was deposited on a Si substrate and heat treated at 923 K. Atomic force microscopy showed that Ni nanoparticles with spherical cap diameters of 20-200 nm were formed, and their wetting angles were used to confirm the derived equations. The predictions are in good agreement with the experimental results.

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Modeling size dependence of melting temperature of metallic nanoparticles

Cited by 48 publications

References 19 publications

Modeling the Melting Temperature of Nanoparticles by an Analytical Approach

Modeling the Melting Temperature of Nanoparticles by an Analytical Approach

Surface energy of nanoparticles – influence of particle size and structure

Modeling of the Heterogeneous Formation of Ni Catalyst Particles for Carbon Nanotube Growth

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