Two types of approximations can be employed to the study of clusters in the nanometric regime. On one side nanoparticles are finite-size structures. Therefore, they can be modelled as large molecules, where the total amount of atoms is truly represented (i.e. through the use force fields, or first-principles approaches). However, if we are interested in complex electronic structures, the use of large molecules may not possible. Clusters with large sizes imply the treatment of 1,000 or more atoms which cannot be handled with traditional molecular codes, like Gaussian [46]. Newer algorithms implemented are more suited to address such large structures. For instance, the real space implementation of DFT in GPAW [47] can take into account large metal nanoparticles and oxide clusters and large metal oxide polyanions like -polyoxometallates.
In the description of nanoparticles two different regions can be identified. At small diameters non-scalable regimes appear: this means that the properties of a system with N constituents cannot be directly extrapolated to those of N + 1. From a technological point of view this is a dangerous path as the degree of control in the synthesis and long-term stability cannot ensure that the “active” species can be maintained for sufficiently large time scales. Scalability appears at larger diameters, and thus there is a systematic way to understand the properties of an N + 1 system provided that the N is known. The normal behaviour or activity as a function of the diameter of the particles is shown in Fig. 3.
Non-scalable Regime
Structure
The chemical properties of systems in the non-scalable regimes have been described for several examples. In the limit, the formation of benzene from ethylene on isolated Pd atoms on defects in MgO were described to show that one atom is enough for some interesting chemistry [48]. Larger cluster agglomerates, containing from a few to tenths of atoms, show some of the properties of molecules like (i) the fluxionality, the flexibility of the structure and (ii) a relatively large the number of low-lying configurations where different spin-states, separated by a finite energy, easily surmounted. In many cases, the study of such nanostructures is limited to the electronic and geometric ground state. This simplification can only be employed for applications where the temperatures considered are low and the systems are chemically insulated (such as memory storage) then the individual properties of ground state structures might be enough to represent these systems [49].
In chemical environments or where several relevant configurations might play a role, complex algorithms, usually based on basin hopping and minimization techniques, could serve as a first indication of the number of low-lying states within a small energy difference [50]. The role of these alternative structures implies that the proper description of the chemical phenomena taking place on these scales would require considering at least low-lying structures and weighting the properties by a distribution (i.e. Boltzmann for the equilibrium). Obviously this limits the availability of the atomistic theoretical simulations as they imply a large number of structures and wide sampling. The representativity of low-lying states and their corresponding properties, in particular those related to the chemistry, might be completely irrelevant.
For discrete clusters the energy levels are well separated. This has been exemplified in the comparison of the Density of States for the bulk of gold and two nanoparticles presented in Fig. 4. The HOMO–LUMO gaps are shown so that the convergence with the metal character represented by the bulk is clearly seen and this would account for the scalability at medium to large sizes. For small clusters, there is a group of structures that might behave as noble (or inert) as the parent compounds. These correspond to the appearance of magic numbers, which show the following properties: (i) a large energy difference of the ground state from the lowest to the next configuration, (ii) a close-shell structure with a large gap in the electronic structure and thus a HOMO–LUMO gap. In the case of monoelectronic metals it is very easy to identify which structures will behave as magic clusters. This can be extended to pseudo-monoelectronic metals such as Cu, Ag, and Au. The concept is more complex for metals in which the electronic configuration is less straightforward. Magic numbers can be understood in chemical terms as a kind of aromaticity that has been reported for clusters of carbon (the well-known fullerenes) [51], boron and other compounds.
Magic clusters might also appear for other chemical structures as they are potential energy wells and their appearance depends on (i) the nature of the system (i.e. type of material: metals, oxides, salts); (ii) the redox state of the metals and (iii) the environment (i.e. oxidative, reductive, solvent, surfactants…). Environmental factors might modify not only the surface structure with partial oxides but also generate new stoichiometries for which no models are known, therefore altering the composition and structure of magic clusters. As a consequence, synthetic processes carried out under mild conditions are more prone to show complexity compared to other preparations for which usual “cleaning” procedures, i.e. reduction at high temperatures, simplify the stoichiometry and composition.
Activity
Examples of potential activity in the non-scalable regime have been presented for a number of reactions including the trimerization of ethylene [48], oxidation of CO by Au8 [52] and the selective epoxidation of propyne [53]. For the first case it is clear that the well-anchored structure with a charge-promoted Pd atom sitting on a vacancy site could be enough for the reaction, as it is still active and sufficiently electronic rich to perform the transformation. Moreover, Pd would tend to sit on steps and oxygen vacancies on MgO, if those are present in significant amounts [48]. The concept above is a proof of site isolation presented by several groups in different context. If the isolation of the active centre (and even its promotion) is possible, then the catalyst would have the optimum activity, selectivity and stability. Obviously, the ensemble control would be easier for reactions taking place on isolated atoms than for other requiring more complex configurations. For the second case [52], the CO oxidation on Au8 particles, it was clearly shown that the activity in the reaction has a lateral path, i.e. an O atom remains on the nanocatalyst. This can be detrimental to the overall stability as the O can fill the vacancy healing the anchoring site. So, after one or a few reaction cycles, the active centre is no longer present. The final example, even more interesting from a technological point of view, is the catalyst for epoxidation of ethylene, Ag, cannot be used for propylene due to the high basicity of the oxygen atoms carrying out the reaction [54]. Experiments and computational models have shown that for small clusters, i.e. containing less than ten atoms, the amount of oxygen on the surface might be reduced, leading to a sharp selectivity for the desired compound [53]. Again, the issue for the long-term stability of these silver-based epoxidation nanocatalysts is compromised if enough oxygen is around as agglomeration of the silver catalyst and thus its death is likely.
Scalable Regime
Structure
Scalable regimes corresponds to sizes larger than 1.5 nm diameter, where the properties of a system tend to converge and do not explicitly depend on the number of atoms. The main problem of representing such structures is that at least 103 atoms need to be considered. These are not suited to traditional physic models that exploit plane waves and periodic boundary conditions as they do not benefit for further contractions of the reciprocal space once the direct lattice has a side of about 3 nm. To check for the convergence of cluster properties to the scalable regime, calculations with real space codes have been presented in the literature. For instance, they have shown how the extension and shape of the Au molecular orbitals converge with a diameter of nanoparticles of around 2.7 nm, or 561 atoms [55]. Also the algorithms at the core of the SIESTA package, which employs localized basis sets [56] with a cutoff for the interaction, allows the linear scalability for large enough systems.
Instead of this brute force approach that contains 103 atoms in the calculations of nanoparticles, the traditional view was to separate and study the contribution from different facets of the crystal and then add them up. This approach has been widely employed to understand the activity of nanoparticles and can be summarized as follows. First, the calculations are performed on different facets of the crystal. Thus, the surface energy (i.e. the energy needed to cut a particular facet) for each of the j cuts, γ
j
, is obtained. Finally, the Wulff construction [57] is applied. The Wulff theorem, developed in 1901, states that the lower the surface energy of a facet, the largest contribution it has in the equilibrium structure of a given material. The Gibbs function of the equilibrium nanoparticle, ΔG
i
, thus minimizes the summation for all the surface energies, times the area O
j
of this particular facet:
$$\Updelta G_{i} = \mathop \sum \limits_{j} \gamma_{j} O_{j} .$$
The Wulff construction allows a smart evaluation of the exposed facets of a material with just few cheap calculations, and has been proven exceedingly successful predicting nanoparticle structures. Examples of these structures can be found in Fig. 5 for a prototypical fcc metal and two rutile compounds relevant in industrial processes. Still the Wulff model is oversimplified because it does not consider the energy required to form steps and edges [58]. Obviously, this approximation is less valid when considering small nanoparticles as the number of low-coordinated sites on them is larger. Also, relatively large structures, i.e. >1.5 nm, need to be included for the model to be relevant. Other approximations can be added on top of the simplified Wulff model. One of them relates to the effect of the environment. Clearly when growing under different conditions, e.g. the oxygen pressure, the surface energies change and this might control the facets exposed. This can be transferred to the Wulff’s model and then the surface energy, under oxygen-rich or -lean conditions or with other compounds like CO [58], can be investigated instead of the raw value. The estimation of the surface free energy at a given temperature and pressure can be addressed through first-principles thermodynamics [59], thus adding extra degree of freedoms (and another source of error linked to the ideal gas models employed to account for temperature and pressure effects). This methodology includes the effect of the surroundings through the computation of the corresponding surface Gibbs free energies, by introducing the reaction temperature and the pressures or concentrations of the environment. These constructions are more approximated than the static calculations described before, but provide an insight on the real state of the catalyst that otherwise would be very difficult to determine under experimental conditions; when a sufficiently large pool of configurations is taken into account, a good description of the surface state is obtained. Recent examples on the nature of the self-poisoned Deacon catalyst have pointed out the full coverage of RuO2 surfaces, which might in turn be important for the further development of the reaction. In the Deacon process, coverage effects make surface reoxidation the rate-determining step [60]. This implies that, when growing under different conditions, the nature of the exposed metals is changed. Therefore, instead of the surface energy for the clean surface, γ, that of the environment acting on it N
k
, γ
j
(N
k
), shall be used. The modified equations look then as follows:
$$\Updelta G_{i} = \mathop \sum \limits_{j} \gamma_{j} \left( {N_{k} } \right)O_{j} .$$
Finally, when considering real systems, the role of the interface between different parts (e.g. metal–oxide junctions or oxide–oxide interactions) should also be included within the Wulff construction. As one of the surfaces will be affected by the interaction with the support, symmetric structures will no longer exist, leading to differently wetting types of particles. The wetting equation derived by Young can be written in terms of the new surface energy at the interface between the cluster and the carrier, γ and the clean reference, γ
0. Taking into account the interaction energy per surface, \(\frac{\Updelta E}{O},\) the equation reads as follows [61]:
$$\frac{\gamma }{{\gamma_{0} }} = \frac{{(\gamma_{\text{int}} - \gamma_{0} )}}{{\gamma_{0} }} = 1 + \frac{\Updelta E}{{O\gamma_{0} }}.$$
In turn, the state of the carrier can be affected by the presence of reducing or oxidising environments that modify the quality of the surface (i.e. number of oxygen vacancies) such as for Cu/ZnO [62], or by the presence of water [63]. Furthermore, on some of the carrier surfaces, special active places for nucleation might exist due to the preparation methods [64].
Once the electronic structure is obtained, the different contributions to the activity and selectivity of a given reaction can be calculated in an isolated manner, employing the tools from first principles applied to slabs, and weighting the contributions corresponding to different facets [60]. In principle, the surface amount can be identified in the Wulff construction and then the reaction evaluated and added up. It might result that one of these high energy facets is more active or more selective and thus would be more interesting to show to a larger extent. While this result is important by itself, sometimes this design parameter cannot be employed as the Wulff construction is a thermodynamic sink and the nanoparticle structure will end up being of this kind.
Although the Wulff construction constitutes the simplest model to describe the structure of a nanoparticle, it has several drawbacks related to the lack of information of defects or low-coordinated sites, together with the fact that only nanoparticles that share the same crystal lattice than the bulk can be retrieved. More detailed thermodynamic investigations in the literature have accounted for unusual metal coordinations. Those models can provide a wide description of the morphologies and even phase diagrams, which can then be compared to tomographic experiments [65–68]. Still, they are being developed and do not consider environment effects, which in heterogeneous catalysis turn out to be more important.
In some cases, as a consequence of adopting different preparation methods controlled by kinetics, new crystal structures that would be metastable under other conditions might be the ground state. A beautiful example is Co nanocrystals, for which a different packing configuration, known as the ε crystal, has been shown. Formation of ε-Co is only possible by solution-phase chemistry, namely organometallic route, and generally using a combination of tight binding ligands or surfactants [69]. As this liquid route is not thermodynamically controlled, the surfactants might change the energetics by binding tightly around the growing crystal and the dissolved Co atoms. This is paradigmatic but in the small range confinement might allow magnetization, availability of different spin orderings, or other particular properties. On this issue, calculations are difficult as they need to assess varied structures in order to understand the nucleation and this task is cumbersome [70].
A good example of the complexity that can affect the theoretical study of nanoparticles, is given by Co and Fe containing clusters. Experiments have shown that such Fe- or Co-based nanoparticles form mixed oxides with rather undefined stoichiometries [71, 72]. Even if they keep an important magnetic moment useful during the separation process, the variable oxygen content adds an extra difficulty to the simulations. Actually, DFT-based calculations for Fe oxides have shown the large complexity in assessing properly even only the electronic structure of such strong-correlated systems. Special care shall be taken in correcting the SIE through DFT + U methods. The coupling between spin and orbital moments leads to intricate electronic structures that depend on the U value. Thus, the present theoretical models cannot yet be employed as black-boxes for this type of calculations [73, 74]. In summary, for some nanoparticles in the form of mixed oxides, to address the issue of the nature, stoichiometry, surface termination, dispersion and stability is still a challenge.
Activity
In many cases, when employing nanoparticles, a strong dependence of the activity on the number of atoms is found. The paradigmatic clear example for this corresponds to the activity of gold nanoparticles. While gold is known to be completely inert, when prepared as small particles usually between 2 and 3 nm in diameter, it presents an enhanced activity for oxidation and hydrogenation reactions [75, 76]. In the scalable regime, catalytic properties are closely linked to the large number of low-coordinated sites on these compounds compared to the total number of atoms. Obviously adsorption energies and vacancy formation energies (i.e. either molecule or site activation) do depend on the coordination number and accordingly, the larger the relative number of sites, the better the reactivity.
The case of sponges and membranes is particularly interesting because, under certain conditions, some metals can change their structure by incorporating a large amount of a second compound and obviously these properties might change when nanoparticles are considered. This is the case of Pd for which hydrides are easy to obtain [77]. The contribution of the hydride phase depends on the environment [78]. The hydrogenation capacity acts as a buffer if solvents are present, like in the hydrogenation of alkynes in the presence of alkenes through the Lindlar catalyst; in fact it reduces the amount of hydrogen that is in contact with the metal surface, thus allowing the use of selectivity modifiers as quinoline, that would not be stable otherwise [79]. This kind of cooperative chemistry which relies on multiple elements to achieve a single property is very common in liquid-phase chemistry, but the implications at high pressure conditions or even electrochemical conditions have been less explored. Interestingly enough, some experiments have indicated that there is a difference in the storage ability when reaching the nanosize that enhance the activity in the hydrogenation of large olefins [80].
The Role of Surfactants
Wet synthesis methods usually employ soft-templates to control the shape of nanoparticles. These procedures are highly flexible and can generate a large number of morphologies. Due to their nature, only few examples have been reported in the heterogeneous catalysis literature. In most cases this can be attributed to structural properties. An example was presented by Häkkinen and co-workers [81], showing the structure of a capped nanoparticle as a function of a sulphide-based surfactant. However, the model did not include solvent effects. The final geometry of the nanocluster is thus given by a delicate balance between the metal–metal, surfactant–metal and surfactant–surfactant interactions. In a way, the effect of the surfactant might be seen as the modification of the surface energies, as described in the Wulff model in the previous section [58].
Yet another example on the electronic structure modifications induced by surfactants was presented for the materials that can be employed in quantum dots. Calculations on CdSe nanoparticles show that it is possible to fine tune the HOMO–LUMO gap by adsorbing different types of surfactants without changing the structure (i.e. the local coordination number of the surface atoms). The dipolar moment of the head adsorbed on the surface can slightly modify the position of the states, already different from the bulk values due to the final nature of the structure, resulting in more suitable light adsorption [82].
In the case of wet synthesis, the reactions take place in a liquid phase, where a number of solutes are presented. The system contains at least the metal salt out of which the nanoparticles are generated, the reductive agent, and the surfactant. In many cases morphology modifiers are also added. Such kind of synthesis exhibits a large degree of control for particles with interesting properties in sensing. The enhanced plasmons are then based on the asymmetry that can be induced by controlling the growth. Calculations with charged fragments present some difficulties but in principle a Born cycle can be prepared with different contributions. An example is shown in the cleaning of gold ores which is the inverse of the process [83].
In understanding the activity when surfactants have not been removed, the issue of diffusion to the active site (for reactants) and out to the liquid phase (for products) might be fundamental and compromise the activity of the catalyst. Transport problems of this kind are usually overlooked but they need to be addressed properly if the chemical properties are to be studied [84].