Simulation of Metal Clusters and Nanostructures

Abstract

Computational simulations and numerical methods have become an essential tool in the study of metal nanostructures. Given the complexity of the energy landscape, finding the most energetically favorable configuration of even the smallest metal clusters is a task that requires the use of efficient search algorithms and robust interaction potentials, and the study of structural and dynamical properties of metal nanoparticles at finite temperature makes necessary an appropriate scan of the phase space, either by stochastic methods or by solving a large set of equations of motion, performed on a model system sophisticated enough to maintain the main features of the real systems being modeled. This chapter is aimed to present some examples in the use of these models and algorithms, making special emphasis in the use of molecular dynamics simulations to describe the behavior of metal nanoparticles on the melting transition, the sintering of nanoparticles, the proposal of phase diagrams, and the role of substrates and other forms of confinement. Connection with experimental results is also discussed, with several examples on the simulation of scanning transmission electron micrographs and the mechanical manipulation of metal nanowires.

8.1 Introduction

Numerical simulations in the study of materials have been an area of constant growth in the last 50 years or so, and with the development of what we now call nanoscience, their role is nowadays hard to underestimate. Simulations based on ab initio or DFT (density functional theory) calculations give a solid starting point in the search for stable structures in small size clusters and in the prediction of IR and Raman spectra, while atomistic simulations of larger nanostructures, given a suitable interaction potential to describe the forces between atoms, are a great tool to study the structural and dynamical properties of nanomaterials, the testing of thermodynamics-based theories, and the role of the surfaces on which they are deposited. The interplay between real and simulated electron micrographs allows a fair interpretation of the former, with a precision below 1 Angstrom, used in a feedback loop that improves the details of the model as the simulated micrograph is compared with the real one. In this chapter we will review some of the most recent advances in several simulation techniques used in the study of metal nanostructures.

Nanoclusters and nanoparticles may be considered as the elemental blocks of most nanostructures and, hence, the elemental blocks of nanoscience. There is no general agreement on the difference between cluster and nanoparticle, but one possible source of discrimination is to consider a cluster as an entity with a specific number of atoms, while in a particle what is important is to have a narrow size distribution, with no need of making a precise counting of the atoms in it. We will take this approach throughout this chapter. In this sense, we start our discussion reviewing the state of the art on the optimization of small metal clusters, leaving the study of passivated clusters with a specific number of atoms out of this discussion, and only referring the reader to some interesting examples concerning the crystallization of stable thiolated gold clusters and their subsequent structure determination with atomic resolution through X-ray analysis [1], the determination of the structure of the thiolated Au₂₅ cluster using DFT methods [2], the study of the effect of the ligands on the optical and chiroptical properties of the thiolated Au₁₈ cluster by DFT [3], and the prediction of the structure of the Au₁₈(SR)₁₄ cluster [4]. After that, we present several examples of the use of molecular simulations for the study of larger metal nanostructures, with a particular emphasis in the use of molecular dynamics simulations for the investigation of dynamics, structure, and thermodynamic properties.

8.2 Common Potentials for Metallic Systems

Either for the search of global minima or for the calculation of properties of nanostructures via the simulation by stochastic or dynamics methods, it is often needed to define a phenomenological interatomic potential to describe the interactions governing the behavior of the atoms in the nanostructure. Historically, the first potentials developed for this purpose were pairwise, where the force between two atoms depends only on the separation between them, but these simple potentials are known to wrongly describe the relaxation in surfaces in metals and to predict wrongly their elastic moduli. By its own definition, a pairwise potential does not take into consideration angular dependencies, and it fails to describe adequately the interaction where the coordination is low, which explains the failure in the prediction of bulk properties, to the extreme of predicting fcc structures for bcc metals. An example of these first attempts to model the interactions in metals through a pairwise potential is the anharmonic Morse potential [5], which has the form:

$$ \varphi \left({r}_{ij}\right)=D\left[{e}^{-2\alpha \left({r}_{ij}-{r}_0\right)}-2{e}^{-\alpha \left({r}_{ij}-{r}_0\right)}\right] $$

(8.1)

whereD is the dissociation energy and α has units of reciprocal distance. With an appropriate parametrization, the Morse potential is able to produce all the elastic constants as positive and give a general good prediction of properties in perfect crystals, but the predictions are not good for systems with defects (nevertheless, the Morse potential is still used to describe the interaction of metals with other elements). This is of course due to the pairwise potential neglects the nature of metallic bonding, and a solution to overcome this problem is to include in the definition of the potential a many-body term that considers the local electron density surrounding an atom. Several strategies have been designed with this purpose, and here we describe two of the most commonly used: the embedded atom method [6] and the Finnis-Sinclair model [7]. Both strategies use the same general form for the potential:

$$ U=\frac{1}{2}\sum_{i=1}^N\sum_{j\ne i}^N{V}_{ij}\left({r}_{ij}\right)+\sum_{i=1}^NF\left({\rho}_i\right) $$

(8.2)

where the first term corresponds to a pair potential, and the second term is a functional that depends on the local density, such that:

$$ {\rho}_i=\sum_{i=1}^N\sum_{j\ne i}^N{\rho}_{ij}\left({r}_{ij}\right) $$

(8.3)

In the embedded atom method, based on density functional theory, ρ _i is the host electron density at atom i due to the presence of all the other neighboring atoms, F(ρ _i ) is the energy needed to embed the atom i into the electron density, and V _ij is the short-range pairwise potential between atoms i and j due to the repulsion of their cores. There are no explicit expressions for F(ρ _i ), but it has to be defined in a tabular form.

In the Finnis-Sinclair model, F(ρ _i ) takes an explicit form:

$$ F\left({\rho}_i\right)=-c\sqrt{\rho_i} $$

(8.4)

in analogy with the second-moment approximation to the tight-binding model that assumes that the cohesive energy of a solid scale with the square root of its atomic coordination number [8]. The choice of the form of V _ij and ρ _ij gives different Finnis-Sinclair potentials. One of them is the well-known Sutton-Chen potential [9], in which:

$$ {V}_{ij}\left({r}_{ij}\right)=\ \prime{o} {\left(\frac{a}{r_{ij}}\right)}^n $$

(8.5)

$$ {\rho}_{ij}\left({r}_{ij}\right)={\left(\frac{a}{r_{ij}}\right)}^m $$

(8.6)

$$ F\left({\rho}_i\right)=- c\ \prime{o}\sqrt{\rho_i} $$

(8.7)

here, a is a parameter with distance units, ϵ is a parameter with energy units, c is a dimensionless parameter, and n and m are integer numbers, where n must be greater than m. The choice of power law forms for V _ij (r _ij ) and ρ _ij (r _ij ) has the advantage of making the expressions very simple and makes the potential scalable in the same way a Lennard-Jones potential is scalable [10], that is, the results obtained for two different metals with the same set of values of n and m can be converted one into the other by rescaling the units of energy and length [9]. Rafii-Tabar and Sutton [11] expanded the concept to cover alloys of two metals A and B by the definition of the parameters:

$$ {m}^{AB}=\frac{1}{2}\left({m}^{AA}+{m}^{BB}\right) $$

(8.8)

$$ {n}^{AB}=\frac{1}{2}\left({n}^{AA}+{n}^{BB}\right) $$

(8.9)

$$ {a}^{AB}={\left({a}^{AA}{a}^{BB}\right)}^{1/2} $$

(8.10)

$$ {\ \prime{o}}^{AB}={\left({\ \prime{o}}^{AA}{\ \prime{o}}^{BB}\right)}^{1/2} $$

(8.11)

In this way, all the parameters needed to define the Hamiltonian of the alloy can be obtained from the Sutton-Chen parameters of the pure metals, simplifying the implementation of a simulation of an alloy.

Another common many-body Finnis-Sinclair potential is the Gupta potential [12] that has the form:

$$ {V}_{ij}\left({r}_{ij}\right)=A\ \exp \left(-p\frac{r_{ij}-{r}_0}{r_0}\right) $$

(8.12)

$$ {\rho}_{ij}\left({r}_{ij}\right)=\exp \left(-2{q}_{ij}\frac{r_{ij}-{r}_0}{r_0}\right) $$

(8.13)

$$ F\left({\rho}_i\right)=-B\sqrt{\rho_i} $$

(8.14)

This potential, as any Finnis and Sinclair potential, is based in the tight-binding second-moment model, and its form has been chosen such that it has a long-range cutoff that extends up to the fifth-neighbor distance, improving the agreement with experiments. The potential is defined by a small number of parameters (A , r ₀ , p , B , q _ij ), and it is easy to implement into a simulation.

8.3 Global Search of Minima in Metallic Clusters

The global search of the most energetically stable structures of metal clusters is a task of primordial relevance in the study of these structures, since the specific details of the ordering and composition at the surface of the clusters will determine in great measure their catalytic and optical properties. This task is far from simple, specially for nanoalloys, since the energy landscape of a cluster with a specific composition and size may be composed of series of local minima and maxima that make difficult the search of the global minimum. Any searching strategy other than a whole covering of the energy landscape is inherently incomplete, and thus the certainty of finding a global minimum is not 100% in the majority of cases. Because of this, the theoretical search of a global minimum becomes in practice the search of a good local minimum [13]. Several approaches have been used with this purpose. Most of the approaches require of the direct calculation of energy using DFT-based methods or the parametrization of interaction models based on experimental results, DFT results, or a combination of both. In any case, the description of the potential energy surface can become quite complex, as can be noted by the exponential growth in the number of local minima in Lennard-Jones clusters up to 13 atoms as shown in Fig. 8.1 [14] and in the disconnectivity graph shown in Ref. [15] that considers 1467 local minima for a Lennard-Jones cluster of 55 atoms, of which the Mackay icosahedron is the global minimum. The number of local minima is also high for the Lennard-Jones cluster of 31 atoms, and from the disconnectivity graph shown in Fig. 8.2 and extracted from Ref. [16], it can be noted that there are two competing minima, both of them based on icosahedra, but in one of them, the atoms added to a 13-atom icosahedron are arranged as a Mackay structure and the other as an anti-Mackay. There is also a decahedral structure in a relatively low minimum. Actually, the appearance of icosahedral structures in clusters on this range of sizes is expected, since, as was established by Bytheway and Keper [17], clusters described by soft, long-range potentials have a tendency to form icosahedral arrangements. The role of the range of the interactions has been well established by Doye et al. [18], finding that long-range interactions form strained, highly coordinated structures, while potentials with an intermediate range generate icosahedron-like structures, and harder potentials generate decahedra and fcc structures. Nevertheless, the case of bare gold clusters is quite special, since instead of the expected icosahedral symmetry, Garzón et al. [19], using a Gupta potential and DFT calculations, found that clusters formed by 38, 55, and 75 atoms reach their lowest-energy configurations when they take amorphous, disordered conformations. In another pioneering study, Fernández et al. [20], using first-principle generalized gradient approximation density functional calculations, found that while very small clusters of Cu and Ag (sizes up to 6 atoms) have a general tendency to form planar structures, Au planar clusters can be found at larger sizes (up to 12 atoms). This peculiarity in gold clusters is likely to be associated to relativistic effects that are particularly relevant in gold but not in Cu nor Ag.

Fig. 8.1

Exponential growth of the number of local minima on the surface of a Lennard-Jones cluster, as increasing cluster size (Reproduced from Rossi and Ferrando [14]. © IOP Publishing, with permission. All rights reserved)

Fig. 8.2

Disconnectivity graph for a Lennard-Jones cluster of 31 atoms. The two deepest minima found correspond to structures based on icosahedral packing, but with Mackay (the lowest minimum) and anti-Mackay (the second lowest) overlayers. There is also a deep minimum for a decahedral structure (Reprinted with permission from Wales and Bogdan [16]. Copyright (2006) American Chemical Society)

The binding energy in a monometallic cluster will be the result of an interplay between volume and surface contributions [13]. This explains at least partially the tendency of the geometric shape of a particle to be constituted by layers of atoms of a particular geometry (with the exception of the planar and disordered clusters mentioned in the previous paragraphs), where each layer has a specific number of atoms, known as magic numbers. Thus, the smaller icosahedron is made by a central atom surrounded by 12 atoms, for a magic number of 13; the next icosahedron in size is made by 55 atoms, the next by 147 atoms, and so on. A search of a global energy minimum on clusters of these sizes usually considers these geometries as one of their starting points, or at least they are considered as strong candidates for a global minimum. This tendency of the clusters and nanoparticles to arrange themselves in specific geometries has appeared independently using different theoretical strategies [13, 21,22,23,24,25], and it has been corroborated experimentally multiple times through electron microscopy imaging [26,27,28,29].

Chirality in clusters acquires a special interest thanks to the experimental observation by Schaaff and Whetten of a strong optical activity in the metal-based electronic transitions in glutathione-passivated gold clusters [30], as pointed out by Garzón et al. [31], a phenomenon that may be explained by a chiral structure of the metallic core. With this in mind, the group of Garzón and coworkers implemented a method that allowed to show that the metal cores of the passivated clusters are more chiral than the bare clusters and that the degree of chirality decreases with the cluster size. The method is based on the work of Buda and Mislow, and it was built on the framework of the concepts by Hausdorff to measure chirality by taking into consideration the distance between sets [32]. A very relevant result from these calculations is the fact that the passivation in clusters of these sizes plays an essential role in the magnitude of the chirality of the clusters, with the consequent affectation in their electronic and optical properties. More details in the implementation of the Hausdorff chirality measure can be found in Ref. [33].

8.4 Global Search of Minima in Bimetallic Clusters

It is easy to imagine that, if the potential energy surface of a relatively simple structure such as a Lennard-Jones cluster has a high degree of complexity, the problem of finding the most stable structures of nanoalloys must be far from trivial. This is indeed the case, taking into consideration that each cluster of a particular size, composition, and geometry has many different ways in which atoms of the type A and type B are arranged one with respect to the other. These different ways of positioning the atoms of different species are called homotops, a name coined by Jellinek [34]. The number of homotops for a cluster of a particular geometry with N _A atoms of the type A and N _B atoms of the type B increases combinatorially with N = N _A  + N _B , and, even when many of these combinations are geometrically equivalent, the search of a global minimum becomes extraordinarily difficult [35]. This is the main reason why the search of global minima has been restricted to small size bimetallic clusters. The most recent optimization algorithms are robust enough to allow the search of global minima in bimetallic clusters of up to several hundreds of atoms [36]. On the other hand, this restriction is not as unfortunate as it seems at first sight, since for large clusters and particles, it is likely that kinetics plays a role as important as energetics in the atomistic arrangement of a particle, and so a search of a global minimum for a large particle becomes an interesting but to some degree futile exercise. Ferrando, Fortunelli, and Johnston mention, for example, that in a cluster of Pd₄₉Pt₄₉, that is, a total of 98 atoms, the number of homotops is of the order of 10²⁸ for each geometrical structural motif [37]. Several optimization procedures have been used to minimize the energy in bimetallic particles, including genetic algorithms [38], basin-hopping [38], the energy-landscape paving method (ELP) [39], and the parallel excitable walkers (PEW) method [40]. These last strategies were used to minimize small clusters (40 atoms) of AgCu and AuCu, finding for the AgCu particles a tendency to form core-shell structures and reporting that an electronic shell closure favors a fivefold pancake structure; this is one of the first works that demonstrate how the explicit consideration of electronic structure has a dominant role in the kind of geometrical motif that optimizes the energy of a cluster of this size and composition, even if the geometry by itself is apparently nonoptimal (strained bonds in the structure). Paz-Borbón et al. using a genetic algorithm approach [41] combined with a basin-hopping atom-exchange routine [42] investigated the stability of 98-atom Pd-Pt, modeling the interactions using a Gupta many-body empirical potential [38]. They found structures based on defective Marks decahedra but also Leary tetrahedra for clusters in the composition range from Pd₄₆Pt₅₂ to Pd₆₃Pt₃₅.

8.5 Melting and Sintering of Metal Nanoparticles

One of the most interesting questions regarding metal nanoparticles is how temperature affects their local composition and shape. In particular, the problem of predicting the critical temperatures of melting and the possible structural changes previous to the melting transition has been studied by several groups for several systems, both monometallic and bimetallic. The general strategy is to simulate the heating of a particle from a temperature corresponding to the solid state and let the particle to reach a temperature above the melting transition. It is possible to monitor the behavior of the internal energy as a function of the temperature, to build what is called a caloric curve. A jump in the caloric curve corresponding to the latent heat of fusion is a signal of the melting of the particle. Once the particle is molten, the procedure can be reversed and the particle is let to cool down to a temperature below its freezing point. It is common that the melting in freezing points differs in the simulations, and while this hysteresis can be explained using quantum mechanical arguments for the case of small clusters [43], for larger particles, this is likely to be the product of the late appearance of a liquid seed (or a solid seed in the case of freezing) produced by a random fluctuation, from which the melting (or freezing) of the particle starts at a temperature higher (or lower in a freezing process) than the melting temperature. Thus, the temperature at which the jump appears in the caloric curve obtained by simulations may differ to some degree from the true melting temperature. Some simulation techniques are more capable to deal with this issue than others. Conventional molecular dynamics either in the microcanonical or canonical ensemble are exposed to this overheating and undercooling, because of the way the phase space is sampled, following the physical trajectory of the atoms. Stochastic methods such as Monte Carlo may be helpful to avoid overheating and undercooling, depending on the way the random displacements are defined, since the configurations are generated randomly and this allows to pass local energy barriers if they are not too high, but of course all the information on the dynamics of the system is absent. More sophisticated algorithms, such as the replica-exchange molecular dynamics [44], avoid the trapping of the structures in local minima by simulating replicas of the system at several temperatures and exchanging the replicas following a Metropolis criterion, in a similar manner than in Monte Carlo, but without losing the information on the dynamics of the system. In multicanonical Monte Carlo simulations [45], the energy barriers are overcome by the sampling of the states made following the inverse of the density of states of the system, but since this density has to be known a priori, a weight factor (instead of the Boltzmann factor used in conventional canonical Monte Carlo simulations) is built up iteratively, and there are several ways of approaching to this problem. In the next paragraphs, we give several examples using these techniques.

Rodríguez-López et al. [46] studied the melting of AuCu nanoparticles by implementing a set of molecular dynamics simulations, modeling the atomistic interactions by a Sutton and Chen potential. As can be easily understood, the choice of an appropriate potential is essential for the correct description of the properties of the system being simulated; in this case, the choice of a Sutton and Chen (SC) potential is justified by two main reasons: first, it is known that this potential describes appropriately the elastic properties of the system, and second, the SC is a many-body potential that has a simple functional form, dependent on a small number of parameters, and the interaction of atoms of different species can be implemented by simple averaging rules. Rodríguez-López et al. applied this approach to cuboctahedral AuCu particles of 561, 923, and 2057 atoms and 4 stoichiometric concentrations, finding that the particles melt at around 600 K irrespective of the relative concentrations of the two metals. They also found that when one of the metals has a relatively high concentration, the icosahedral phase is stable, while for the other cases, the presence of cuboctahedral motifs is more common. They also reported a premelting transition from cuboctahedral to icosahedral geometry, a transition previously reported for monometallic gold nanoparticles [47] and segregation of the species near the melting point, such that the particles acquire an Au-shell/Cu-core distribution.

Following a similar approach, Mejía-Rosales et al. implemented a set of MD simulations in order to investigate the melting of Au-Pd cuboctahedral nanoparticles of 561 atoms [48]. The geometry and size of the particles were selected based on electron microscopy observations. Several relative concentrations were implemented: Au_100%Pd_0%, Au_75%Pd_25%, Au_50%Pd_50%, Au_25%Pd_75%, and Au_0%Pd_100%. For this set of simulations, the Rafii-Tabar and Sutton version of the SC potential was used to describe the interatomic interactions. The simulations were implemented increasing the temperature of the particles at intervals of 20 K, each temperature simulated in the canonical ensemble. It was found that the external shells of the particles melt first (being a cuboctahedron, the structure of the particle is made of concentric atomic shells); core layers melt simultaneously at a higher temperature than the external shell. A similar premelting behavior was reported by Cleveland et al. for gold nanoparticles [49]. After melting of the outer shell, it was found that gold atoms migrate to the surface to produce a core-shell structure. Unlike the results for AuCu nanoparticles, the results of Mejía-Rosales et al. show a strong dependence of the melting temperature on the relative concentration of the two metals present in the particle: the higher the concentration of Pd, the higher the value of the transition temperature Tm.

Simulating the cooling down of the particles below their melting temperature is a good way to mimic the synthesis conditions of the particles in inert gas condensation reactors. Following a similar strategy of the MD simulations for a heating process, a set of simulations was implemented to freeze Au-Pd melted particles and investigate the structural properties of the resulting frozen particles [50]. Again, the Rafii-Tabar and Sutton version of the SC potential was implemented to describe the interatomic interactions between Au and Pd atoms. In this study, it was found that monometallic particles tend to arrange themselves as icosahedra, with defect-free (111) facets. In comparison, for the bimetallic nanoparticles, the tendency was to form truncated octahedra with rough surfaces, with a high concentration of defects such as kinks, edges, vacancies, and di-vacancies, in agreement with the experimental observations by electron microscopy. In a deeper analysis of the resulting structures, the local order parameter q ₆ was calculated, defined as:

$$ {q}_6(i)={\left(\frac{4\pi }{13}\sum_{m=-6}^6{\left|{q}_{6m}(1)\right|}^2\right)}^{1/2} $$

(8.15)

where

$$ {q}_{6m}(i)=\frac{\sum_{j=1}^{N_n(i)}{Y}_{6m}\left({\overline{r}}_{ij}\right)}{N_n(i)} $$

(8.16)

are the average spherical harmonics of the bonds of the ith atom with their N _n neighbors. This order parameter was used to characterize the ordering of the atomic layers in the particles. It was found in the bimetallic particles a tendency to form fcc domains separated by stacking faults, unlike the pure Au particles, which tended to form icosahedra [51]. More than that, it was also found that in particles with a relatively high concentration of Pd atoms, the surface of the particle was mainly populated by gold atoms with a presence of isolated Pd sites. In particular, three kinds of Pd positions in the surface of the particle were found: the Pd atom one layer below the Au neighbors, the Pd atom in the same layer of the Au neighbors, or the Pd at the top of a hexagonal array. These Pd sites are likely to be important in the high catalytic activity that these nanoalloys are known to have.

In a study involving larger particles, Kart et al. studied the thermodynamical, structural, and dynamical properties of Cu nanoparticles of sizes in the range of 2–10 nm [52]. They used molecular dynamics simulation in the canonical ensemble, and the interactions were modeled through a quantum-corrected version of the Sutton and Chen many-body potential. They heated up the spherical particles from 100 to 1500 K in steps of 100 K, except around the melting transition, where they used increments of 20 K for a better location of the melting points. In order to investigate the properties of the particles, they calculated melting points, heat capacity, radial distribution function, first coordination number, Honeycutt-Andersen index and Lindemann index, mean square displacement, and diffusion coefficient. They obtained the melting temperatures for particles of 2, 4, 6, 8, and 10 nm and compared their results against the prediction of the thermodynamic model in a liquid skin melting that considers the premelting of an external layer at low temperature, until the particle melts as a whole [53]. They found that their results agree in a good degree with the model, as can be noted in Fig. 8.3.

Fig. 8.3

Melting temperature as a function of the nanoparticle size in Cu nanoparticles and its comparison against the prediction of the thermodynamic model in a liquid skin melting (Reprinted from Kart et al. [52]. Copyright 2014, with permission from Elsevier)

Shu et al. investigated the melting of Fe nanoparticles using the simulation method of replica-exchange molecular dynamics that avoids superheating or undercooling by allowing a set of replicas simulated simultaneously at different temperatures to exchange temperature following a Metropolis Monte Carlo criterion [54]. When they used conventional molecular dynamics, they found that the calculated melting temperatures vary depending on the simulation time, what they attributed to superheating and undercooling. When they used the method of replica-exchange molecular dynamics, the exchange of replicas avoids the trapping of the systems in local minima. When the exchange is made, the replicas are being heated and cooled repeatedly. As the gap between temperatures becomes closer, it is possible to find the melting point between the replica with a liquid structure and the replica with a solid structure. The caloric curves do not present the typical hysteresis of the conventional dynamics simulations, and the melting point can be extracted easily from them. Figure 8.4 shows two typical plots of the average total energy per atom as a function of the temperature, where the sudden jumps in the curves locate the melting points unequivocally.

Fig. 8.4

Caloric curves obtained by replica-exchange molecular dynamics for Fe nanoparticles of 523 atoms (top) and 7983 atoms (bottom). The discontinuities in the caloric curves mark the melting transitions (Reproduced from Ref. [54] with permission of The Royal Society of Chemistry)

In another work regarding the melting of nanoparticles, Shen et al. simulated isolated fcc nanoparticles of Fe, covering a wide range of sizes, from 59 to 9577 atoms [55]. They used a Sutton and Chen potential to describe the interatomic interactions, and the simulations were made keeping the temperature constant by using a Berendsen thermostat. They started the set of simulations by heating bcc Fe nanoparticles from 300 to 2000 K with a temperature ramp of 0.25 K/ps. In this way they obtained melted particles. After the heating process, they implemented a cooling procedure to take the particles from 2000 K back to 300 K, using the same temperature ramp as of the heating process. In a third stage, they reheated the nanoparticle with the same temperature ramp. They used a variation of the cluster-type index method to distinguish between bcc, fcc, hcp, and amorphous atoms [56]. With this tool, they found that the structures after cooling had a high frequency of fcc and hcp atoms, and while some relatively small particles presented tetrahedral fcc sections or hcp atoms formed in a fivefold symmetry surrounded by fcc atoms, larger particles had a lamellar structure made of consecutive sections of fcc and hcp atoms. These lamellar structures had been previously reported in results of molecular dynamics simulations of Au-Pd nanoparticles [51]. Since Shen et al. were able to discriminate amorphous from structured atoms, they showed snapshots of the particles before melting and at the melting point, where it is possible to follow the way the structure of the particles is lost. Figure 8.5 shows this evolution. They found that not only surface premelting is present and it has a role in the way the particle melts but also the internal defects play an important role, since these internal defects appear simultaneously with the surface premelting. They also found that fcc and hcp atoms in a fivefold twinning structure and a lamellar structure showed different proportion and melting behavior during the melting process.

Fig. 8.5

Cross-sectional snapshot of melting Fe₉₅₇₇ before melting point (a) and at melting point (b). White and brown balls represent hcp and fcc atoms, while amorphous atoms are colored by transparent gray dots (Reprinted from, Shen et al. [55]. Copyright 2013, with permission from Elsevier)

As it was mentioned before, the use of multicanonical simulations avoids the trapping of the system in local energy minima, since a whole range of the energy spectrum can be sampled by the system, and the melting transition can be investigated. Rapallo et al. [57] compared the results of multicanonical simulations with those of canonical Monte Carlo and molecular dynamics of the melting of Au, Co, and Au/Co nanoparticles. They found a general agreement between the three techniques, but the simulations with the multicanonical Monte Carlo technique located the melting point in a more precise way than the canonical techniques. Nevertheless, the use of multicanonical Monte Carlo at low temperatures may predict structures more energetic than the minima if the number of iterations is low, but this did not affect the location of the melting point.

The surface premelting in metal nanoparticles is a phenomenon with important practical implications. It has been established by means of molecular dynamics simulations that the surface premelting affects the way in which sintering of silver nanoparticles takes place, since the ratio of the size of the neck that originates at the beginning of the sintering process in relation to the particle radius increases linearly in size as the temperature is increased to the surface premelting point [58]. Since sintering of nanoparticles is a mechanism by which it is possible to form films and other nanostructures [59], the understanding of premelting becomes relevant. Alarifi et al. conducted a set of molecular dynamics simulations in order to study the melting and surface premelting in silver nanoparticles with diameters of 4–20 nm [60]. They started defining the nanoparticles as fcc-truncated Marks decahedra, starting with a temperature of 300 K and heating the particles by multiplying the velocities of atoms by a scaling factor and equilibrating the particles for 20 ps at each temperature. For the definition of the interactions, they used the embedded atom model (EAM). For the heating process, they increased the temperature in 50 K intervals initially, but the rate was changed to 10 K intervals in the range of 825–1275 K, to determine the melting points more accurately. One of the results of these series of simulations is exemplified in Fig. 8.6. This is the case of a 18 nm particle. Here, it is shown the behavior of the potential energy per atom as the temperature is increased and the typical atomic arrangements at several temperatures of interest. It can be noted that the particle starts with a solid structure (plots labeled as A and B), corresponding to the extremes of the section of the caloric curve that can be associated with the solid state. As the temperature reaches the point labeled as C, some quasi-liquid ponds appear on the surface. Here the caloric curve changes its behavior and at point D the particle is covered by a quasi-liquid layer that grows into the particle (point E) until it reaches a liquid behavior (point F). The solid center shrinks involved in a quasi-liquid layer (point G) until it loses its crystallinity (point H). After that, the particle melts as a whole (point I), and the caloric curve changes its behavior again. The pronounced changes in the slope of the potential energy versus temperature mark the surface melting and melting critical temperatures.

Fig. 8.6

(a) Potential energy (PE) values during heating of 18 nm Ag particle. (b) Atomic arrangement of the 18 nm Ag particle at different temperatures indicated by letters on the PE curve. Atoms are represented by dots. The lines in the atomic plots of points A and B represent the orientations of the crystallographic planes. Arrows on plot C point toward quasi-liquid ponds. Arrows at plot D point toward solid regions at the surface. Each color in the atomic plots represents a phase (gray, solid; blue, quasi-liquid; red, liquid), and the dashed arcs represent the interfaces between these phases (Reprinted with permission from Alarifi et al. [60]. Copyright (2013) American Chemical Society)

Alarifi et al. compared their results against several theoretical models: the liquid drop model that assumes that the melting temperature of a particle depends on the variation of the cohesive energy and the surface tension [61]; Shi’s model, which is based in Lindemann’s criterion that establishes that a structure melts when the root-mean-square displacement of the atoms reaches a certain fraction of the interatomic distances [62]; Hanszen’s model that assumes that melting and premelting occur at the same temperature and that melting starts at the surface of the nanoparticle forming a liquid layer that expands to the core of the particle when the melting temperature is reached [60]; and Chernyshev’s model, which is based on Shi’s model and defines the premelting temperature as the temperature at which liquid ponds suddenly appear at the surface of the particle [63]. The comparison of the theoretical predictions against the simulation results appear summarized in Fig. 8.7. Here it is worth to note the good agreement between the liquid drop model and Shi’s model with the simulation results for particles of 8, 9, and 10 nm. As the authors point out, this agreement is explained by the fact that for particles of these sizes, they did not find surface premelting, and the models ignored the formation of a liquid shell in the surface. On the other hand, for particles larger than 10 nm, the melting points calculated through the simulations have a good agreement with Hanszen’s model that considers premelting. The premelting temperatures obtained through the simulations are only compared against Chernyshev’s model that is the only model that analyzes the relationship between premelting temperature and the size of the nanoparticle. They found good qualitative agreement with the model.

Fig. 8.7

Melting points (T _m ) and surface premelting points (T _sm ) of different sizes of Ag nanoparticles determined by molecular dynamics simulations and by several theoretical models (Reprinted with permission from Alarifi et al. [60]. Copyright (2013) American Chemical Society)

Buesser and Pratsinis investigated the coalescence of silver nanoparticles by molecular dynamics simulations [64]. They studied the sintering of particles with several morphologies and arrangements with a set of MD simulations in the canonical ensemble and used the parametrization of Foiles et al. [65] of the embedded atom method to describe the interatomic interactions [6]. Sintering was promoted by placing two nanoparticles one next to the other separated by a distance of 3.5 Å. They used the Steinhardt order parameter [66] to define a disorder variable D _i that allows a measurement of the degree of crystalline order in the environment surrounding each atom. In the way they defined D _i , values less than 0.02 indicate an almost perfect fcc environment, and larger values of D _i (ranging from 0.05 up to 0.1) measure deviations from a perfect silver fcc crystal. They followed the dynamics for 100 ns at 800 K, finding that a sintering neck is formed immediately by adhesion, and afterward the concave region at the nanoparticle sintering neck fills up with atoms originally from the surface of the particles, so the sintering takes place by surface diffusion. After this transitory state, the particle coalesces into a proper semispherical shape, with some of the atoms originally at the inside of the particles diffusing into the concave neck region and spreading into the surface of the particle. When they repeated the simulation at a temperature of 900 K, closer to the melting temperature, the sintering process took place in a significant faster pace. They studied the crystallinity dynamics of the nanoparticles while the sintering process takes place, using the disorder variable D _i to analyze deviations of the structure from a perfect fcc crystal. Figure 8.8 shows a series of snapshots of the cross sections of the sintering particles, 3 nm in size, at several times. Here, the coloring of the atoms is according to the value of D _i : blue atoms correspond to an fcc environment, while green and red indicate disordered areas. In this series, the temperature was kept at 800 K. At the beginning of the simulation, it is evident that the interior of the particles is an almost perfect fcc structure, with some disorder near the surface of the particles. At 10 ns of dynamics, it can be noted a disordered boundary (green color), made of atoms that originally were at the surface of the particles. This region remains disordered for most of the simulation, in what is effectively a grain boundary between to fcc regions. In simulations of the sintering of three particles in a line, similar results were obtained.

Fig. 8.8

(a) Snapshots of the cross section of two equally sized sintering particles (original diameter of 3 nm at T = 800 K) colored according to the disorder variable, D _i . Blue (D _i  < 0.02) indicates a perfect fcc-crystal environment and green (D _i  ∼ 0.05) to red (D _i  > 0.1) an increasingly distorted crystal. (b) In the right column, the right-hand-side particle has been rotated by 45° to alter the initial alignment of the crystal planes (red arrow) to demonstrate the role of such lattice defects or mismatches during particle sintering (Reprinted with permission from Buesser and Pratsinis [64]. Copyright (2015) American Chemical Society)

8.6 Phase Diagrams of Metal Nanoparticles

Since there exists a plethora of geometrical motifs possible for metal particles depending on their constituents, size, local distribution of the species, and specific details of the synthesis, there is a need of a comprehensive way of predicting the kind of structures that should be expected for a given set of thermal conditions. One approach to produce a sort of nanoscale phase diagram is through the use of molecular simulations, in particular, using methods of molecular dynamics. One early attempt to use MD simulations for the construction of such a phase diagram was due to Kuo and Clancy, working on gold nanoparticles modeled by a modified embedded atom model (MEAM), by studying the melting and freezing behavior of particles on the range from 2 to 5 nm [67]. Kuo and Clancy reported a transformation from fcc to icosahedral as the system approaches to the melting temperature, followed by a transition to a quasimolten state before the particle melts. A set of simulations of particles supported on silica showed an icosahedral solid-liquid coexistence, what they interpret as proof that the support enhances the structural stability of the particles. With the whole of their results, they constructed a (qualitative) nanophase diagram, shown in Fig. 8.9. Taking a more thermodynamic approach, Barnard et al. [25] used relativistic ab initio calculations to calculate the free energy of formation of gold particles with several structural motifs (Mackay icosahedron, Ino decahedron, Marks decahedron, the symmetrically twinned truncated octahedron, the ideal truncated octahedron, and the ideal cuboctahedron) and made a comparison of the results for the different shapes as a function of size and temperature. The resulting phase diagram is shown in Fig. 8.10. Unlike the results of Kuo and Clancy, the results by Barnard et al. are quantitative, and they cover a large range of particle sizes. It is worthwhile to note that the qualitative diagram of Kuo and Clancy keeps some concordance with the diagram proposed by Barnard: Both diagrams predict icosahedral structures for small particles and low temperatures, a transition to decahedra for larger particles, and a quasi-melting zone before the transition to liquid. While the quantitative diagram by Barnard et al. is a great effort to build an as complete as possible diagram, it must be taken into consideration, as the authors point out that the diagram is built on thermodynamic arguments “and as such is incapable of predicting shapes that result from purely kinetic considerations” [25].

Fig. 8.9

Qualitative phase map for particle structures versus size and temperature according to the melting scenario observed in the results of molecular dynamics simulations (Reprinted with permission from Kuo and Clancy [67]. Copyright (2005) American Chemical Society)

Fig. 8.10

Quantitative phase map of gold nanoparticles, based on relativistic first-principle calculations (Reprinted with permission from Barnard et al. [25]. Copyright (2009) American Chemical Society)

8.7 Phase Diagrams of Bimetallic Nanoparticles

The construction of a sort of phase diagram for bimetallic nanoparticles is a task somewhat different to what is needed to build a phase diagram in monometallic particles. In the former case, the idea is to find how the melting transition depends on the composition of the particle, just as in a regular phase diagram for alloys. Yeo et al. [68] reported a phase diagram for Ag-Au nanoparticles built by using the results of a set of molecular dynamics simulations. Since for such a task they needed to define an adequate interaction between silver and gold atoms, their choice was the Quantum Sutton and Chen (QSC) many-body potential. They modeled AuAg particles of 2 nm in size and several compositions (a range from 90–10% to 10–90% in increments of 10%) and subjected to an annealing process before cooling down the particles at several cooling rates in order to find the freezing transition. They found the liquid to solid transition temperature region by analyzing the caloric curves (configurational energy vs. temperature) and the mean square atomic displacements. With this information, they were able to construct a phase diagram, shown in Fig. 8.11. As can be noted, the transition temperatures are considerably lower than in bulk, and they found that, depending on the relative concentrations, several stable solid structures appear: Ih, amorphous, Ih, and fcc.

Fig. 8.11

Phase diagram of the Ag-Au bimetallic nanoparticles (270 atoms, 2 nm size). The red line (T _L ) represents the surface freezing temperature, and the black line (T _S ) represents the full freezing temperature. Atomistic configurations inside the diagram represent typical structures obtained at the given temperatures. Blue atoms indicate Ag atoms, and yellow ones are Au, while white atoms are the Ag atoms of the Ih-Au package at the surface (Reproduced from Ref. [68] with permission of The Royal Society of Chemistry)

8.8 Supported and Confined Nanoparticles

The role of substrates on the stability of metal nanoparticles at heating processes has been investigated in several manners using molecular dynamics simulations. Fernández-Navarro and Mejía-Rosales used a quantum Sutton-Chen many-body potential to study the dynamics of Pt-Pd nanoparticles, both free and graphite supported, while the particles are subjected to a heating process [69]. To model the metal-carbon interactions, they used a simple Lennard-Jones model. The graphite substrate was modeled as frozen atoms, their presence only justified by their interactions with the nanoparticle, and so the dynamics of the substrate was not studied. As expected, they found that the melting temperatures are considerably lower than those of Pt and Pd at bulk, but, interestingly, they found that the melting temperature increases when the particle is on the graphite support, compared against the results with free particles, with an increase at least 180 K. They also found that Pd atoms tend to remain at the surface and that the Pd atoms wet the graphite surface more than the Pt atoms. Analysis by root-mean-square displacements suggests that surface melting starts from the cluster surface and surface premelting was seen in both free and graphite-supported nanoparticles. Wang et al. using a similar approach but implementing the metal interaction through an EAM potential confined relatively large Au nanoparticles (N = 3990, 8778, 11,970, and 17,157) between two-layer graphene nanosheets, finding the melting temperatures for the particles and reporting that the confined gold particles exhibit a layering ordering even in liquid state [70].

Akbarzadeh and Shamkhali directed their attention to the behavior of the melting of metal (Pd-Pt) nanoparticles confined in single-walled carbon nanotubes [71]. They modeled the metal interactions through a quantum Sutton-Chen potential and used the Lennard-Jones 12–6 potential for the metal-carbon interactions. They found that the core-shell structure of the nanoparticles is not dependent on concentration or kind of confinement but only on the size of the nanoparticle. They also found a large effect of the nanotube chirality on the melting of the particles, such that the particles are more stable in zigzag nanotubes.

In another study on supported nanoparticles, Bochicchio et al. studied Ag-Cu and Ag-Ni nanoalloys of sizes from 100 to 300 atoms adsorbed on MgO(0 0 1) surfaces [36]. They used a global optimization approach to find the low-energy structure of the particles, modeling the metal-metal interactions through a potential based on the second­moment approximation to the tight­binding model, and the interaction with the substrate through a many-body potential fitted on first-principle calculations in the case of nonreactive interfaces, developed to reproduce the DFT energetics prediction of adsorbed atoms. They found that the substrate has a determining role in the stability of the particles and that, unlike the case of isolated particles, there is no marked tendency to produce icosahedral structures. Actually, they found that for Ag-Cu, as the number of Ag atoms is increased, the structure follows the sequence faulted fcc → icosahedral → fcc, while in Ag-Ni particles the increasing of Ag atoms, the structure follows a sequence hcp → faulted fcc–faulted hcp → icosahedral → fcc, which means that a high presence of Ni stabilizes hcp structures. They also found a marked tendency of Ag atoms to migrate to the surfaces of the particles.

An issue of practical relevance in the use of metal nanostructures is how to immobilize the particles in a selective surface. For the design of polymer nanocomposites, for example, one technique consists on filling up the polymer with particles, in such a way that it is possible to tune up the mechanical, optical, and chemical affinity properties of the composite, but this must be done in such a way that agglomeration of particles is avoided, so it must be assured that the particles are strongly attached to the surface. In other example, materials containing metal nanoparticles used in the construction of nanoelectronic devices must have the particles strongly attached to the surface in order to be able to stand mechanical stress, thermal fluctuations, or strong external electric fields, so its mechanical stability is assured. One way of achieving this stability is to cover the metal particles with amines or thiols or to immobilize naked nanoparticles in a porous matrix or in polymer brushes. In a technique proposed by Palmer et al. [72], a highly oriented pyrolytic graphite (HOPG) surface is impacted by small gold clusters produced by inert gas condensation. Each cluster, 20 atoms large and of tetrahedral geometry, lands on the surface with an energy enough to create a nanotunnel on the carbon surface. Once these nanotunnels are created, the surface is exposed to the landing of larger gold nanoparticles (around 1 nm in size). Each of these particles travels in the direction of the surface with an energy corresponding to the soft-landing regime, so when the particle lands, its general shape is preserved, and those particles landing in the proximity of the nanotunnels get trapped by a phenomenon similar to what macroscopically is called capillarity. While it is well established that the particles get trapped by the nanotunnels, the specific details of the mechanism are uncertain due to the limitation on the resolution of the AFM measurements made with this purpose, and it is not known the amount of volume that got trapped into the tunnels nor the final structure of the trapped particles.

In a series of molecular dynamics simulations, de la Rosa-Abad et al. reproduced both the production of nanotunnels and the immobilization of larger particles [73]. These simulations are particularly relevant because for the first time, it is used as parametrization of a Morse potential based on density functional theory (DFT) considering dispersion forces for the description of the metal-carbon interactions. This is a great improvement with respect to the usual choice of using a Lennard-Jones fitting, which is clearly limited since it overestimates the adsorption energy and it doesn’t make any distinction between the carbon atoms at tunnel edges and the atoms at pristine graphene. The gold-gold interactions were described through an embedded atom model potential, and the carbon-carbon interaction was modeled using the adaptive intermolecular reactive empirical bond order (AIREBO), developed by Stuart et al. which considers directionality of the carbon-carbon bonds [74].

As a starting stage, and after thermalizing the graphite layers to 300 K, it simulated the implantation of the Au20 clusters on the surface. Several energies were studied in order to investigate the amount of damage to the surface. It was found that when the metal interactions are governed by the EAM potential, the clusters were able to penetrate until the sixth carbon layer, but when the cluster was rigid, that is, the relative atomic positions were kept fixed, the clusters were able to penetrate until the twelfth carbon layer. It was considered that the experiments should lay somewhere between these two limits, since the pyramidal shape in the EAM cluster was lost at a lower temperature than expected for a cluster of this size, making the model softer than the real cluster, and the rigid cluster was harder than the real one. Once the nanotunnels were created, the first six carbon layers were extracted and subjected to an energy minimization process. Following the experiments of Palmer et al., they used particles of Au₁₄₇, Au₃₀₉, Au₅₆₁, and Au₉₂₃, with icosahedral geometry for the immobilization process. They situated each of the particles close to the nanotunnel and give it a small translation velocity in its direction. As expected, the particles were trapped by the tunnels, and they calculated the degree of insertion into the tunnel by visual inspection and by measuring the height of the particles with respect to the first carbon layer. The results, and their comparison against the experiments, are shown in Fig. 8.12.

Fig. 8.12

Comparison between simulations and experimental measurements of immobilized particle heights, relative to the position of the first graphite layer, as function of the size of the particle (Reproduced from Ref. [73] with permission of The Royal Society of Chemistry)

As it can be noted, a considerable amount of volume of the particle gets inside of the tunnel, with a consequent effect on the height of the particle with respect to the surface. Figure 8.13 shows a top and side views of the immobilized particles, where the insertion of part of the volume of the particles into the nanotunnels is evident. It is remarkable the agreement between the experimental results and the simulations for the case of small particles; nevertheless, the prediction of the simulations for larger particles differs from the experiments, and the source of this disagreement may be related to the underestimation of the affinity between gold and low-coordinated carbon atoms. Another interesting result is that the immobilized particles keep their overall icosahedral shape (with the exception of the smallest one), and this issue must have practical relevance, since this means that the immobilized particles maintain their reactive [111] faces.

Fig. 8.13

Top and side views of four gold nanoparticles immobilized by a tunnel made by a Au₂₀ cluster. Note that with the exception of Au₁₄₇, all the nanoparticles conserve most of their icosahedral structure after immobilization (Reproduced from Ref. [73] with permission of The Royal Society of Chemistry)

8.9 Core-Shell Nanoparticles

The elemental distribution in a bimetallic nanoparticle has a great practical interest, since the potential capacity of fine-tuning the properties of a particle by controlling the distribution of the elements on it may broad their applications in magnetism, catalysis, and optics. Both experiments and theoretical calculations give as possible distributions core-shell particles, particles where the two metals are intermixed, Janus particles where one of the elements lays at one side of the other, multilayered particles, and other possibilities. Thus, the determination of the most stable structures not only must take into consideration the size and geometry but also the local distribution of the chemical species in the particle. In bulk systems, the solubility of two metals will happen if the Hume-Rothery rules are followed: the atomic radii, crystal structures, solubilities, and electronegativities must be similar [75]. In nanoparticles, the situation is more complex, since the system is of finite size and a large part of the atoms are on the surface of the particle. Ferrando, Jellinek, and Johnston [35] mention six particular factors that may influence the appearance of a core-shell structure:

• 1. The relative strengths of the A-A, B-B, and A-B interactions; a relatively strong A-B interaction will produce a tendency to mix the two metals.

• 2. The surface energy of the metals; the metal with the lowest surface energy will have a tendency to migrate to the surface creating a core-shell particle.

• 3. Atomic sizes; the smallest atoms will tend to populate the core of the particle.

• 4. Electronegativity; charge transfer from more to less electronegative metals will favor mixing.

• 5. Effect of surface ligands; the metal that binds more strongly to the ligands will tend to migrate to the surface.

• 6. Electronic and magnetic effects; the arrangement of the metals may be stabilized by forming complete electronic shells or by electron spin interactions.

These factors may be more or less important depending on the size and the relative concentrations of the metals.

In one study by Bochicchio and Ferrando, several weakly miscible systems, including AgCu, Ag-Ni, AgCo, and AuCo, were studied for the most common geometries (fcc, icosahedra, decahedra, and polyicosahedra), using global optimization searches to find the most optimal distribution of the elements on the particles [76]. To describe the interactions, they used an atomistic model based on the second-moment approximation to the tight-binding model (SMATB potential), commonly known as Gupta potential [77]. They used a basin-hopping algorithm for the searches, swapping two atoms of different species at each move (only exchanging chemical species, without varying atomic positions) and relaxing the structure thereafter in order to take it to the nearest local minimum. With this strategy, they found that only the icosahedral structures generated centered cores, until the core is of such a size that a morphological instability develops. They found also that decahedra and fcc generated cores out of the center of the particles; see Fig. 8.14.

Fig. 8.14

Lowest-energy configurations of icosahedral nanoparticles of fixed core size (number of shells k = 4, corresponding to 147 core atoms) for increasing number of shell atoms (and size of the nanoparticle). From top to bottom: N = 561 (number of shells n = 6), N = 923 (n = 7), and N = 1415 (n = 8). Shell atoms are shown as small spheres so that core atoms (bigger spheres) are visible (Reprinted figure with permission from Bochicchio and Ferrando [76]. Copyright (2013) by the American Physical Society)

Yang et al. used molecular dynamics simulations to study the atomic segregation in Fe-Al nanoparticles [78]. They implemented a version of the embedded atom method to define the interactions, and the simulations started with an already formed Fe rhombohedron and an Al icosahedron. At temperatures ranging from 100 K to 300 K, they deposited randomly one by one atoms of the opposite element every 5 ns, until the number of deposited atoms was 500, and the diffusion was investigated. When they deposited Al atoms in the Fe particle, they found that the Al atoms initially occupy sites at the vertices and edges of the particle, but after that diffusion takes place to the other regions of the surface. In the end, the surface of the particle is completely covered by the Al atoms, and a core-shell structure is obtained. When they deposited Fe atoms in the Al icosahedron, the results were different. At 100 K, it was found that all the Fe deposited atoms remained in the surface, forming an Al_core-Fe_shell particle; see Fig. 8.15. Nevertheless, at 200 K and 300 K, temperature is sufficiently high to overcome the energy barrier for the exchange mechanism, and Fe atoms diffuse to the interior of the particle, avoiding the formation of a core-shell structure.

Fig. 8.15

At 100 K, the whole configuration and cross section of the Fe-Al nanoparticle as the deposited Fe atoms is 500. The orange balls show the Fe atoms, and the gray balls show the Al atoms (Reprinted from Yang et al. [78] Copyright (2013), with permission from Elsevier)

8.10 STEM Simulation of Clusters and Particles

Electron microscopy has become the de facto tool for the analysis of the structure of metal nanoparticles. Due to the great advances in resolution in the recent years, it is nowadays very common to find studies that give an atomistic description of the structural properties of metal nanoparticles and nanoalloys, to the degree that it is even possible to make quantitative measurements of internal strains and the precise localization of stacking faults and other kinds of structural defects. Being an inherently two-dimensional technique (the electron microscopy micrograph is the result of the projection of the electron wave function into a plane), the simulation of electron microscopy becomes a very useful tool for the appropriate interpretation of a micrograph, in the sense that through the proposition of a model that describes the features of the real system, it is possible to simulate the interaction of the electron beam with the model, and the resulting simulated micrograph can be compared against the real one with the purpose of identifying the similarities and differences between the proposed model and the real system. In this section we describe how these simulations are performed and give several examples of their use in the case of metal nanoparticles. Special emphasis will be made on the simulation of scanning transmission electron microscopy (STEM), because of the property of this technique of generating micrographs where the intensity signal is strongly dependent on the atomic number of the chemical species present in the sample, so its use in the study of nanoalloys becomes an excellent tool for the analysis of the local composition in a nanoparticle.

The conventional transmission electron microscopy (TEM) technique is based in the interaction of an electron beam transmitted through a thin sample. As the electron passes through the specimen, its wavefunction becomes modified by the interaction of the electron with the sample. The resulting image is magnified and focused onto a screen or detector, and the micrograph is obtained. As any optical device, the resolution of the TEM is largely affected by aberrations produced by the optics of the system; when a simulation of TEM is performed, these aberrations are taken into consideration, together with the energy of the beam, the defocus, and the characteristics of the detector.

A technique complementary to conventional TEM, useful to determine the elemental composition of nanoalloys, is the high-angle annular dark-field scanning transmission electron microscopy or HAADF-STEM. In HAADF-STEM the elastically scattered electrons are collected using an annular (dark-field) detector. The scattered beam carries information about the mass of the electron, such that the intensity signal goes as Zⁿ, where n is a number close to 2. For this reason, this imaging technique is also known as Z-contrast. Theoretical predictions and STEM simulations set the value of n close to 1.4. The fact that the intensity is dependent on Z makes this technique extremely useful for the discrimination of heavy atoms, such as Au, Ag, Pt, and Pd. The beam scans the specimen in order to produce the image.

The simulation technique used to produce HAADF-STEM micrographs is based in what is known as multislice method, which consists in approximating the potential due to the sample by the definition of several slices, where each slice has a thickness of no more than a few interatomic distances. The potential due to the atoms of the specimen laying in the volume corresponding to a particular slice is projected to the central plane of the slice. The wavefunction of the electrons interacting with the slice is calculated, and it is used as input for the interaction of the next slice. The same is made for the next slice, and the process is repeated; thus the whole volume of the specimen is taken into account. Once all the slices were considered, the last wavefunction is used as input for the calculation of the simulated image. Being STEM a scanning technique, the method requires that each point in the micrograph to be calculated at a time, which makes the process computationally expensive and time-consuming. Special care must be taken in the definition of the slices, that should not be too thick nor too thin, and the choice of the slice thickness is strongly dependent on the system to be studied and its orientation relative to the electron beam.

One example of the use of the multislice method together with dynamics simulation is in the work by Khanal et al. [79]. In this work, nanoparticles of Cu-Pt were synthesized by a simple chemical method, and the resulting particles were analyzed by HAADF-STEM. The sample proved to be highly monodisperse in size, and the distribution of copper and platinum was well explained by the results of grand canonical Langevin dynamics simulations that showed a high diffusion of Pt atoms on Cu. This alloying was in accordance with the experimental HAADF-STEM micrographs. More than that, the simulation of HAADF micrographs resulted in simulated micrographs with a high resemblance to the experimental ones, as can be seen in Fig. 8.16.

Fig. 8.16

STEM simulated images of the final configurations of grand canonical Langevin dynamics of Pt deposition on Cu nanoparticles. In (a–d) TO₂₀₁ seed and (e–h) TO₅₈₆ seed. The structures (c, d) and (g, h) were rotated by 30° around the y-axis. Note how the regions enriched in Pt appear brighter (Reprinted from Khanal et al. [79]. doi:https://doi.org/10.3762/bjnano.5.150, under the terms of the Creative Commons Attribution License. http://creativecommons.org/licenses/by/2.0)

Another example is the one by Mayoral et al. [80], where Co-Au nanoparticles were prepared by inert gas condensation, which is a physical technique that has the property of producing non-passivated particles, by creating a supersaturated vapor from the material sputtered from a metal target and condensed in controlled conditions in an inert gas atmosphere. The conditions of the sputtering system were set to create nanoparticles of approximately 5 nm in size. Once the particles were deposed in holey carbon microgrids, the microgrids were subjected to a thermal treatment and let it cool down, and the samples were analyzed by HAADF-STEM. The images confirmed the size of the particles, and the intensity signal in the images showed two kinds of particles, the first of them composed almost entirely of gold (bright signal) and the second of particles mostly composed of cobalt (darker particles). In most of the cases, the particles were fivefold, either decahedra or icosahedra. Since the particles were nonprotected, and due to the facility of cobalt to be oxidized, some presence of cobalt oxide was expected in the zones rich in cobalt. When the density of particles was high enough, some agglomeration occurs, forming either bimetallic particles where one of the sides was rich in Au and the other rich in Co or core-shell structures, with the core of the particle made of gold and the external shell rich in cobalt. Figure 8.17 shows one of these core-shell decahedral particles. Since the difference in intensity due to the two metals is very different, it was appropriate to redraw the HAADF-STEM micrograph in false colors, the red color corresponding to regions rich in Au and the blue regions corresponding to the presence of Co (see Fig. 8.17a). Measurements of interplanar distances gave the clue that the regions rich in cobalt were actually made of cobalt oxide, and so a model reproducing the main features of the particle was prepared considering the oxidation of cobalt; this model is shown in Fig. 8.17b. The regions of interest marked both at the real micrograph and at the model were compared by making a HAADF-STEM simulation on the model; the result is shown in the composition of Fig. 8.17c, where the marked region corresponds to the simulated micrograph, overlapped into the real micrograph, and shows that the model describes appropriately the composition of the particle.

Fig. 8.17

(a) STEM micrograph of gold-cobalt nanoparticle, redrawn in false colors according to intensity. Red color corresponds to gold atoms, blue to cobalt. (b) Atomistic model that describes the Au–CoO interface of the particle shown in (a). Yellow spheres represent gold atoms, blue spheres are cobalt, and red spheres are oxygen atoms. (c) Particle shown in the STEM micrograph of (a). The region marked by the yellow square is the simulated STEM intensity map corresponding to the white square on the model shown in (b) (Reproduced from Ref. [80] with permission from The Royal Society of Chemistry)

8.11 Tensile Strain in Metal Nanowires

The mechanical properties of unidimensional metallic nanostructures have a high relevance from the practical point of view. Metal nanowires can be used as conducting wires for the implementation of nanoelectronic circuits [81] or as AFM tips [82], and their mechanical properties have been measured by bending the nanowire under the influence of the compression from an AFM tip [83]. In this work, it was found that the Young modulus is independent of the nanowire diameter, but the yield strength is until 100 times larger than in bulk for the smallest diameter nanowires. These findings remark the practical relevance that these systems may have, and it is understandable that there exist simulation approaches to the measurement of the strains of the nanowires under tensile deformation.

In a work based on molecular dynamics simulations, Koh et al. modeled the uniaxial tensile strain of platinum nanowires, performing the simulations at several temperatures and strain rates [84]. The wires were segments of solid fcc structures, defined as infinitely long by means of periodic boundary conditions. To model the interactions, they used a Sutton and Chen potential. They defined the axial stress on the nanowire by using the arithmetic mean of the localized axial stress state for each atom. From there, they were able to construct the stress-strain response and Young’s modulus. The simulations showed that at the lowest temperature (50 K), the nanowires presented low ductility, and at 300 K, they found the formation of a helical substructure that enhanced the ductility of the nanowire. At the medium strain rate of 0.4% ps⁻¹, they found that the deformation made the nanowire to change from crystalline to amorphous structure, and at the maximum strain rate of 4.0% ps⁻¹, the deformation was completely amorphous, with the nanowire having a superplastic behavior. They also found a strong lowering of Young’s modulus with respect to the bulk value. An example of the stress-strain relation and some snapshots of the structure of the nanowire at several strains are shown in Fig. 8.18.

Fig. 8.18

Stress-strain response of nanowire at T = 50 K and strain rate = 0.4% ps⁻¹. (a) Stress-strain response with points where snapshots of the nanowire were captured and (b) snapshots of atomic arrangement of platinum nanowire at various strain values (Reprinted figure with permission from Koh et al. [84] Copyright (2005) by the American Physical Society)

In a similar set of simulations, Mejía-Rosales and Fernández-Navarro investigated the stress-strain response of gold nanowires with a structure based on embedded icosahedra [85]. The work was motivated by the experimental synthesis of nanowires with this structure [86] that can be seen in the micrographs of Fig. 8.19. Here, the HAADF-STEM micrographs correspond to helical icosahedral AuAg nanowires, and, as can be seen in high-magnification images, it is notable the icosahedral packing. This unique atomistic arrangement is based on the intercalation of icosahedral motifs forming a Boerdijk-Coxeter spiral. The potential applications of these novel structures will strongly depend on their mechanical response, but since structures of this kind were never reported before, the characterization of their mechanical properties is a work in progress.

Fig. 8.19

(a, b) Aberration-corrected STEM-HAADF images of the Ag-Au nanowires with the helical icosahedral structure in different orientations. The images (c) and (e) are the corresponding high-magnification areas of the icosahedral packing. The image (d) shows the FFT obtained along the axis of the nanowire of (a) showing the fivefold symmetry (Reprinted with permission from Velázquez-Salazar et al. [86]. Copyright (2011) American Chemical Society)

Nevertheless, since the structure of these wires has been well established both by high resolution electron microscopy and by EDS analysis, they were able to build an atomistic model in order to perform a simulation study of tensile strain and compression under several different conditions. A model was built based on the partial overlapping of icosahedra, and the resulting structures keep a high resemblance with the experimental ones. In the model, the wire was totally composed by gold atoms. For the molecular dynamics simulations of the strain, a Sutton and Chen interatomic potential was used. Periodic boundary conditions were applied in the direction of the axis of the nanowire, and the nanowires were subjected to strain at several rates (0.04%, 0.4%, and 4.0% ps⁻¹,) and a temperatures of 300 K, mimicking the conditions used in the study by Koh et al. on the deformation characteristics and mechanical properties of platinum nanowires [84]. For sake of comparison with other already known metal nanowire structures, they also performed the same analysis on wires with an atomistic fcc arrangement and several orientations; see Figs. 8.20 and 8.21. It was found that, in the case of the typical fcc wires, at slow strain rates, the strain-stress curves showed a series of dislocation-relaxation steps, with the crystalline order well-preserved for a large portion of the strain process. In the icosahedral wires, unlike the fcc ones, the regions shared by two adjacent icosahedra suffered local crystalline reordering. In all of the wires, the Young modulus was measured to be a fraction of the bulk modulus. In the compression process, they found that the behavior of the icosahedral wires is very different from the fcc wires, since, while the latter tends to thicken as they are being compressed, the icosahedral wires tend to form coils, as can be seen in Fig. 8.22.

Fig. 8.20

Models for gold nanowires. From left to right: Nanowire along [001] crystallographic direction, nanowire along [011] direction, nanowire along [111] direction, icosahedral nanowire based on the Boerdijk-Coxeter-Bernal (BCB) helix (Figure presented in Ref. [85])

Fig. 8.21

Stress-strain response of the nanowires shown at Fig. 8.20 at T = 300 K and strain rate = 0.04% ps⁻¹. The arrows relate specific points in the stress-strain plot with snapshots of atomic arrangements of the BCB nanowire at various strain values

Fig. 8.22

Molecular dynamics runs of the compression of the BCB nanowire showing that, unlike the fcc wires that under compression tend to increase their cross section by the spontaneous appearance of lattice defects, in the BCB wires, the compression causes the wire the coil into itself, just as it can be noted in the sequence of snapshots above

8.12 Conclusions

We have made a brief review of some examples regarding the use of computational simulations in the study of metal clusters and nanoalloys. Emphasis was wade in the implementation of molecular dynamics for the study of thermal and dynamical behavior of nanoparticles, but it was also discussed the use of optimization techniques in the search of minima in the energy landscape of metal clusters and the importance of the use of electron microscopy simulations in order to have an appropriate way to compare the atomistic models generated by other simulation techniques with the experimental results obtained by real electron micrographs. Although this review is not intended to cover the whole universe of techniques and algorithms, we believe it gives a general idea of the capacities and limits on the use of atomistic modeling in the study of nanostructures and of the relevance of the use of these techniques in the study of metal clusters and nanoalloys.