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Elucidating the Formation Mechanisms of Silver Nanoparticles, from a Comprehensive Simulation Based on First-Principles Calculations Hosna Sultana, and Eunseok Lee J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09991 • Publication Date (Web): 22 Dec 2017 Downloaded from http://pubs.acs.org on December 26, 2017
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The Journal of Physical Chemistry
Elucidating the Formation Mechanisms of Silver Nanoparticles, from a Comprehensive Simulation Based on First-Principles Calculations Hosna Sultana1 and Eunseok Lee2,* 1
Optical Science and Engineering Program, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA 2 Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA Abstract The nucleation and growth of silver nanoparticles are modeled and simulated based on first-principles calculations. The formation energy of single-crystal and multiply-twinned particles are calculated to elucidate the thermodynamic properties of particles and modeled as a function of geometric parameters. Based on the calculated formation energy and the molecular collision theory, Kinetic Monte Carlo simulations are performed to trace the formation process of silver nanoparticles. In particular, the temporal change of size distribution and morphology are obtained and used to elucidate the governing mechanism in each stage of the formation process. It is demonstrated that the formation process is separated into four phases depending on the power-law time dependence of the particle formation and they are characterized by the size-difference between coalescent particles. The temperature-dependence of size distribution and morphology are also studied to elucidate the underlying mechanisms. The findings are compared with classical theories quantitatively and a strategy to control the morphology of silver nanoparticles is discussed. *Corresponding Author:
[email protected] 1
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1. Introduction Noble-metal nanoparticles (NPs) have attracted attention for various potential applications, such as catalysts, optical device, printed electronics, biomedicine, etc., as well as scientific interests for their properties distinguished from bulk materials.1–4 As these properties are dependent mainly on the size and morphology of NP, controlling their size and morphology is crucial to design NPs with desired properties.5–8 Despite many modifications and extensions of classical theories, a thorough understanding of the nucleation and growth mechanisms of NPs has remained challenging due to the complexities from multiply associated reactions and the difficulty in the in-situ monitoring of the formation process of NPs.2,4,9,10 In classical theories, the process of NP formation is described as the nucleation of crystal seeds (Becker-Döring theory11 and its translation to LaMer burst nucleation model12) followed by growth through molecular addition (e.g. monomer addition).1,4,13 Ostwald ripening describes the further growth NPs as the formation of larger NP by consuming the smaller NPs, which can explain the decay of the number of NPs as the growth progresses.1,4 The effects from dynamics and kinetics has been accounted for by the coalescence mechanism, by which two particles merge to form one particle.1,13 Although they are still seen as the basic model, these classical theories are based on the macroscopic thermodynamics and often fails to explain the processes occurring in a short-time period. Recent progress in experimental tools and atomistic computational methods has led to the probe into microscopic behaviors of NPs and several mechanisms have been suggested accordingly.1,4,5,9,10,14–18
However, the deliverable information from those
measurements and calculations has been limited to the local area of NPs system, short-time period only, and/or depended on empirical inputs and, hence, comprehensive description of the underlying mechanisms of NPs formation has been lacking. In this study, we aim to simulate and explain the comprehensive process of silver NPs formation. Kinetic Monte Carlo (KMC) simulation model of particles formation is developed based on the combination of first-principles calculations and the collision theory. More specifically, the formation energy of a silver particle is obtained as a function of the size and crystal structure of particle via the parameterization of the result of first-principles calculations, and the activation energy barrier for the coalescence of two particles into one particle is modeled based on the parameterized formation energy. The developed KMC model is employed to simulate the formation process of silver NPs comprehensively. The temporal evolution of size distribution and morphology during the formation is obtained at different temperatures and analyzed to unravel the underlying mechanisms and their dependence on temperature. 2. Methods 2.1 Density functional theory calculations (DFT) calculations on particles
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The Journal of Physical Chemistry
Figure 1. Illustration for a few single-crystal and icosahedron silver particles among the ones used in this study. The dashed area indicates the interface between twinned crystals in icosahedrons.
Searching for the stable structures of nanoparticles is challenging due to infinitely large number of possible atomic arrangements, especially for those consisting of more than tens of atoms. In general, typical crystal structures of silver nanoparticles are truncated octahedron, octahedron, decahedron, icosahedron, etc.3,10,14,15 Instead of examining all different crystal structures, we simplified the competition for dominant crystal structure as to whether the dominant particles are single-crystal or multiply-twinned, and focused on two well-known crystal structures of silver NPs: truncated octahedron and icosahedron representing singlecrystal and multiply-twinned crystal types, respectively. The electronic energy (E) was calculated by DFT calculations for truncated octahedral (Agto13, Agto38, Agto79, Agto144, Agto225, Agto342, Agto483, and Agto711) and icosahedral (Agi13, Agi55, Agi147, Agi309, and Agi561) particles; a few of them are illustrated in Figure 1. A few octahedral (Ago19, Ago44, and Ago85) and cuboctahedral particles (Agco13, Agco55) were also examined for comparison in small-size range– their electronic energy were substantially higher than truncated octahedral and icosahedral particles except for small-size range. The formation energy (Ef) of a pristine particle was defined as Ef = E - Natoms𝑒"#$% , where Natoms is the number of atoms comprising the particle and ebulk is the cohesive energy per atom of the bulk state silver. The Ef of the particles sized between two pristine particles were approximated by interpolation. DFT calculations were performed as implemented in VASP19–21 at the generalized-gradientapproximation (GGA) level with the Perdew-Burke-Ernzerhof parametrization.22,23 The size of supercell was designed to have sufficient empty space between particle images (at least 30 Å) to exclude the influence from self-images. 520 eV of cutoff energy was used and k-mesh were adjusted to ensure the accuracy of 1 meV per k-point. DFT calculations on the particles bigger than the examined ones were impossible due to heavy computational load – e.g. the number of atoms of the next smallest icosahedral particles was 923. Instead, we modeled the energy of large-size single-crystal particles and icosahedral particles based on the results from DFT calculation as follows.
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2.2 Energy model of large-size single-crystal particles The equilibrium shapes of pristine single-crystal particles were obtained by Wulff construction, which requires the surface energy of each facet. We considered the (ijk) index facets only, where 0 ≤ i,j,k ≤ 3, because the higher index facets barely emerge in face-centered-cubic (fcc) based crystalline noble metal NPs.24–26 For each facet, a slab structure was prepared to expose the corresponding facet and empty space was inserted in the supercell in the normal direction of the exposed surface. Then, DFT calculation was (+)
performed to calculate the electronic energy of the slab structures and, the surface energy, 𝐸'#() , where the (+)
compound index (i) indicates the exposed surface facet, was defined as 𝐸'#() = 𝐸'$." − 𝑒"#$% 𝑁'$." /2𝑆, where 𝐸'$." is the energy of the slab, 𝑒"#$% is the cohesive energy per atom of bulk material, 𝑁'$." is the number of atoms in the slab, and 2S is the surface area of the slab. Once the equilibrium shapes were predicted by Wulff construction, DFT calculation were performed on the same configurations as the predicted equilibrium shapes (AgW13, AgW19, AgW55, AgW79, AgW147, AgW201, AgW459, Agw675). We parameterized the electronic energy of the equilibrium shape single-crystal particle as: 𝐸= (9)
+
(>)
(9)
(+)
𝑆 (+) 𝐸'#() + 𝑁.567 𝑒"#$% +
9
𝑙 (9) 𝐸:;
𝑛(>) 𝐸?6(@:(
(1)
(>)
where 𝐸:;)
length of edge j, and 𝑛(>) is the number of corner k. 𝐸:;
